QCD SUM-RULE STUDY OF SCALAR MESONS

.A Thesis Submitted to the College of

Graduate Studies and Research

in Partial Fulfillmenc of the Requirements

for the Degree of Doctor of Philosophy

in the Department of Physics S; Engineering Physics

Cniversity of Saskatchewan

* Saskatoon

BI.

Fang Shi

Spring 1999

@Copyright Fang Shi. 1999. -411 rights reserwd.

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professors who supervised the thesis work or. in their absence. by the Head

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in n-hole or part ~houlc1 be addressecl to:

Head of the Department of Physics and Engineering Physic-5

C-niversitu of Saskatchexan

Saskatoon. Saskatchewan S ï N BE3

ABSTRACT

In this thesis. QCD Laplace mm-riiles for the light quark ( j q twrcnts artn rni-

ployed to study the properties of the non-strange I = O and 1 = 1 light quark

scaiar mesons. This QCD sum-nile analysis allows us to interpret the exper-

imentally observed I = D and 1 = 1 scalar mesons. The Hil(1i.r inctliiality

technique is employer1 CO (let~rrnine rtie rcgioii of v;ilitiity f o r t t i r l (.&'il * r i I l i -

ruIe. and a stability arialysis of t ht. QCD srini-riilta j)rcdic.r ior~ is 1.t ) r i t l i i i . t t v l

throiigh a !donte-Carlo iincertainty sinidation of uncertainties.

The field cheoretical content of the QCD sum rules incorporates piirely-

perturbative QCD contributions to two-loop order. leading contributions frorn

QCD-vacriurn condensates. and the direct single-instanton rontrihirrions in r h r '

instanton-liquici QCD \.iIcuiitn moclel. Singltl-insr i~nt~)tl (wnr r i h r i1m3 ~ L ~ I I t 11 , .

only romponents of the QCD field theon. rhiir distingiiish ht~rwt~t~ri i ~ ) q ) i t i

statcs. and therefore ihey are responsible for h a k i n g t h r niass cltyric~rarv

between the lowest-lying isovector and isoscalar mesons. A novel treatment

of instanton effects in QCD continuum contribution is included in this thesis.

There is also a need to go beyond the narrow resonance approsiniarion for

the scalar channels n-hich arp likely t o rshihit smsirivity t o hroatf rtw)ri;trii.ib

structure. .\ finite-width effect anticipated frorn physical resonanw n-idrhs 1s

incorporated for the hadronic content of the 1 = O and I = 1 QCD suni niies.

In the I = O channel. Our results support interpretation of the J0(980) as

the Iowest-Iykg light quark scaiar meson. indicating that fo(400 - 1200) is

unnaturally decoupled from a light quark non-strange cunent. In the 1 = 1

channel. the restiIts identify ~ ~ ( 2 4 5 0 ) as the Iowest-lying qq resonance. and iiïp

indicative of a non-qq interpretation for ~ ~ ( 9 8 0 ) .

ACKNOWLEDGEMENTS

It was my privilege having been under the guidance of Dr. T. G. Stccle t o

complete my Ph.D. thesis work. I have Iearned and t>twdictt~ci iriiriit~riiirlv frotii

him for the art O € scientific research. the w- of Logira1 ancl critic-al thinkiriq.

and the kintlness of understanding. i tvish co espreçs mu siricwe gratituiltl t u

hirn for his personal interest! guidance. encouragemenc and patience during the

course of the thesis work.

1 would also like to give rny thanks to Dr. V. Elias of Cniversity of \Ytlstf*rn

Ontario for the prodiicriw and pltusant id;iht,raricm r l i r ~ i ~ i ~ ~ i t ~ ~ i r riitn r t i t - ~ i ~

work.

Thr fricntlly atniosphere c i f t h~ S;~.;kacthtwan ; \c-r*~I t~;~ror Liihnr;irory ht*lpcv I

rnake rtiy stay iit Cniwrsity of Sifijhtchemn a lot of fun. 1 tvould like to tharik

the lab's financial supports and al1 the people there.

Special thanks give to rny wîfe. Dr. Weiwei Tan for her understariding.

encouragement. support. help. and abuve dl. her great patienc.~. Ttiatib ;dso

to mu parents. Professors Tianyi Shi and Y u e ~ Fang. for their on-going support

and encouragement in every moment of my Me.

The financial supports frorn Xatural Science and Engineering Research

Council are gratefully acknowledged.

DEDICATION

This thesis is dedicated to the merno- of rny grancimot her Zhici~i Qiii.

Contents

PERMISSION TO USE 1

DEDICATION iv

TABLE OF CONTENTS vii

LIST OF FIGURES x

LIST OF TABLES si

1 Foundations of Quantum Chromodynamics

1 . Histiirical Rrview . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Principles of Perturbative QCD . . . . . . . . . . . . . . . . . .

1.2.1 Gauge Principles . . . . . . . . . . . . . . . . . . . . . .

1 . . Second Quantization . . . . . . . . . . . . . . . . . . .

1.2.3 Regiilarization and Renormalization . . . . . . . . . . . .

1 . 4 Renormalization G roup Eqitat ion . . . . . . . . . . . . .

1 . 2 Scaling and Asymptotic F redom . . . . . . . . . . . . .

1.3 Xon-perturbacive QCD . . . . . . . . . . . . . . . . . . . . . . .

1.3.1 Basis of QCD Sum Rules . . . . . . . . . . . . . . . . . .

1.3.2 iyacuum Expectation Values . . . . . . . . . . . . . . . . 2.3.3 Spontaneous Symmetry Breaking . . . . . . . . . . . . .

1.3.4 Dispersion Relations and Duality . . . . . . . . . . . . .

1.3.5 Borel Transformation nnct S i ~ m Riiles . . . . . .

1.3.6 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3.7 Fermions in Instanton Fields . . . . . . . . . . . . . . . .

1.3.5 QCD Ckcuum S ~ N C ~ U C ~ in Instanton Background . . . .

1.3.9 Single-instantm.Lpprosimation . . . . . . . . . . . . .

2 Motivation 6 1

. . . . . . . . . . . . . . . . . . . 2.1 General Properties of 3[esons 61

2.7 Sciflar SIesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3 Field Theory Calculations and Hadronic Models 68

. . . . . . . . . . . . . . . . . . . . . 3.1 Perturbative Contributions 68

3.1.1 One-Ioop Contributions with Sfass Renormalization Effects ï! - . 3 . 1.2 Tivo-Loop Contributions . . . . . . . . . . . . . . . . . . 1 3

. 3 . L.3 Perturbative Contribution ta Ra Laplace Suni Rules . . . ,9

3.2 QCD Condensate Contributions . . . . . . . . . 4 1

1 .21 Quark Condensate Contrihuticws . . . . . . . <.I -

. . . . . . . . . . . . 3 .2 . Gluon Cundensatc Ciintribi~t ions 57

. . . . . . . . . 3.2.3 Dimension-six Con(Icrisatc Cutitribiitiuris 91

4 QCD Condensate Contribittinm t o Ro B o r t I I . !! I

. . . . . . . . . . . . . . . . . . . . . . 3.3 Inscanton Contributions 96

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Hadronic Models 108

4 QCD Sum-Rule Analysis 118

. . . . 4.1 Ratioride of Our Approach of QCD Stirri-RiiIc~ -4nalvsis 1 lt*

4.1.1 Possible Problrms of Traciicional ;\ppr~iii.htl?i 11"

4.1.2 Traditional Ratio lIethod . . . . . . . . . . . . 1 1 !l

4.1.3 Our Approrich of QCD Strrn-rult . AnaIysis . . . . . . . . 1'11

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Least-k2 Slethods 1'13

4.2.1 Estirnate of QCD Sum-mle Inputs . . . . . . . . . . . . . 123

. . . . . . . . . . . . . . . . . . . . . . 4.2.2 HoIder lnequalit- 12.5 . . . . . . . . . . . . . . . . . . . . 4.2.3 L e a ~ t - ~ ' Approach 131

4 . 4 Search Algorithm . . . . . . . . . . . . . . . 13s

4.3 Best-tir .A nalysis . . . . . . . . . . . . . . . 1 1 1 ..

4.3.1 Best-fit Results fur the 1 = 11 . I Channels . . . . . . . . . l - i3

. . . . . . . . . . . . . . . . . . . . . 4.3.2 Instanton Effects 146

. . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Width Effects 154

4.3.4 Testing of Hadronic SIodels . . . . . . . . . . . . . . . . 156

. . . . . . . . 4 Uncertainty Estimation: Monte-Carlo Simulations 160

5 Conclusions 170

REFERENCES 173

A Conventions and Notations

B Example REDUCE Program

List of Figures

Three- and four-gluon interactions due to the self-coiipling of

gluons in the QCD Lagrangian (1.3). . . . . . . . . . . . . . .

One-loop quark self-energy S ( p ) . . . . . . . . . . . . . . . . .

Contour integral of the spectral density p ( s ) in the comples s

plane wherc o = AI'. p ( s ) eshibits a tliscontiniiity dong rht*

real s a i s . . . . . . . . . . . . . . . . . . . . . . . . . .

One-loop contributions to the scrilar corrciatiun fiirictiari T[(q'i.

Thc injected current carries a nionicntiini of il'. Ttit' r l qiiiirk

and t l qiiark give the sanie cotitrihirtiori. . . . . . . . . . . . .

Two-loop perturbative contributions to the scalar correlation

Function Il($). The injcctcd ciment carries a momentum of q'.

Only cionnecteci ciiagrams contribitte. . . . . . . . . . . . . . .

Quark condensat~ contribiitions in th^ scalrir channels. B y syni-

metry, the two cliagrams give the samp contribiitions. . . . . .

Gluon condensate contribiitions in the scalar channtlls. B! syrii-

tritltry. the first two ciiagrams giw the sanie contribution. . . .

Dimension-sis four-quark condensate contribiitions to the r w -

point correlation function in the scalar channels. . . . . . . . . Single-instanton effects in the scalar correlation function resiilt-

ing in both of the connected and itnnihilated diagrams contribut-

ing to (3.35). . . . . . . . . . . . . . . . . . . . . . . . . . . .

Breit-lyigner shape of resonance widt h. This Brcit-iiïgncr pibiik

is centercd at SI = 1 Ge\- and with ii 200 .\le\- wiclth 14fec.t.

.\ssiiming ii continuum rhreshold occiir at .3,, = '1 Gr\ -' . r hc

shaderl iirea of the Breit-Uïgner tails !.i > s,, and .s < O ) iiril

tmncated in QCD sum rule calculations. . . . . . . . . . . . . . 11 1

An example of the 1 square-pulse approsimation to the Breit-

iVigner resonance shape obtained bu truncating equation(3.143)

to r~ = 4. and by choosing f = 0.701 to ensrrre that the arra

under the Four pulses is equident to the total arca under rhtl

Breit-LVigner ctirve. This particular esample is for a mass JI = . . . . . . . . . . . . . . . . 65051el.' and width ï = 100.\lel-. I l 3

4.1 The shaded area represents the region in the so - T parametu

space consistent with the inequdity for the light quark I = 1

sum-rule (3.150). The input parameter vaiues f,, = 1.5. .\ =

300.1Iel: p = lJ(600.1IeL'). and Lir = 0.1 Ge\'-' are iisctl in

the calculation. The r a~is is in Gr\--' and $0 asis is in Gr I *'. 129

The shaded area represents the region in the s0 - r paranietrr

space consistent with the inequality for the lighc quark I = O

siim-ride (3.150). The input piiranieter values f,, = 1.5. .\ =

300.11el: p = Lj(600MeI.-). and cjr = 0.1 Gel*-' are usecl in the calcuiation. The r a i s is in GeI.'-"nd so avis is in Gel". 130

The ratio of condensate contribution F d ( r . $0) (3.57) over the

total QCD sum-rute !&(r.su) (3.130) in the I = O channel. Thtb

value so = 3.3 Gel '- is chosen. The y-asis is the percentage valiir

of the ratio &"d(r. so)/&(.;. so). and the r axis is in Gr l ' -' 194

The pictiirti shows the relative perturbative contribution (3.44 i

itncertainty with rrspect to the rhangc of continiiiirn thresti-

old s,, in r tic I = channe1. Thp .su = 3.3 Ce1 -2 is ( h i ) -

sen in consistent wich the best-fit tdire. The Y - a i s value is

( r t ( r . 4.0) - r t ! r . 3.0)) /ROI(:. 3.5). and the r - a i s is in

Gel--'. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1:3G

Relative iinc~rtriinty o n the pcrt i~rhatiw ;ind c.ontI~watv c*oti-

trintions to the total QCD sum niIf% (:3.1.50). .\' iisis stiiri(1s for m.,( 7.S,) : the 7 range. 1' = - R i l r . s o i *

in the d i t 1 7 range (ri < r < 72 J ob-

tained from the HiMer inequality. an iipper limit of the relatiw

uncertainty nl~vays esists. The r uis is in . . . . . . . . 137

A simple two-dimension function with a local minimum point pl

and a global minimum point b. where the point s is supposed

as the starting point. . . . . . . . . . . . . . . . . . . . . . . I4U

Contour plot for the 1 = O rhanrid wit h the input parxrnertlr val-

ues r = O.:! Gel: 1,. = 1.5. .\ = 300 31eI-. and p = 1/(600.11t 1-1.142

Contour plot for the I = 1 channei with the input pwanitwr \xi-

ues r = O.2Gel: Jv, = 1.5. .\ = -300 . \ t~ \ i a n d p = 1!(600.\.tc.I'i.143

QCD sum-rule &(Ï. so) from (X1.50) in the I = O channel ( the

solid curve) and the hadronic mode1 with the best-fit parameters

from Table 4.1 (dashed cune) . The r -m is is in Gel--2 and u- axis is in Ge\-4. . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.10 QCD sum-nile Ro(r. s 0 ) froni (3.130) in the 1 = 1 charinel ( thv

solid curve) and the hadronic mode1 with the bat-fit parameters

from Table 4.2 [dashed cune) . The r-mis is in Gel'-' and y-

avis is in GeL*4. . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.11 The osciIlating behaviour of the instanton effect (3.131) with

respect to so in the I = O channel. The T is 6ved at 1

Cep'-'. The y-=is is in GeC'4 and x-auis is in Gel". . . . . .

4.1'1 The oscillating behar-iour of instanton effect (3.131 ) with rt.sptxc.r

to r in the 1 = O channel. TIie .Y,] ~ 1 1 1 1 ~ is fiseil ;1F 3.5 C r I '' consistent xith the best-fit resiilts. The y - i ~ ~ i s is in Gr\ - ' ;incl

s-auis is in Ge I '-'. . . . . . . . . . . . . . . . . . . . . . .

4.13 The iristanton rontrihution &""( 7 . ..;,,) ~o the QCD srirn-nilv

in the I = O channe1 folio~vs equation (4.191 in this tigiirt~.

The input parameter values f,:, = 1.5. -1 = 300 Mel: p =

L/(6OOMeI-). and dr = O. 1 Gelp-' rire rised in the ciil(:iilrition.

The r -mis is in Cf\'-' and su-&.ris is in Gt.1". . . . . . . . . .

-1.14 The wightinq function c i 7 . .<,, I honi I -1.1 1) i n tht* I = 0 i . t i ; ~ r i i i t ~ l

(the riashed curve) m c l the ernpiricd nioriel iihown i is t h scilirl

curve. The y-ixis is in Grl'" and x-auis is i n Gr\'-!. . . . . .

4. I.5 The weighting fttnction F ( 7 . sO) [rom (4.14) in the 1 = 1 i:tianrit.l

(the dashed curve) and the ernpirical rnodel shown as the solid

cume. The y-asis is in Gel'4 and x - a i s is in Ge\*-*. . . . . . .

4.16 Histogram for the fo rnass obtained frorn fits of 300 parameter

sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.11 Histogram for the j,, r.ontintrirrri t tiri>l;holrl ihtiiintvl fri)i~i t i r ' t ,f

-500 parameter sets. The continuum threshold v a h is in Gr-1-'.

4.18 Histogram for the a0 m a s obtained from fits of 500 parametFr

sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.19 Histogram for the a0 continuum threshotd obtained frorti fits of

500 parameter sets. The continuum threshold \-due is in Gel-?.

List of Tables

Light quark properties . . . . . . . . . . . . . . . . . . . . . f i l

Suggested q@ quark-mode1 assigrirnants for nio.;r of r t i ~ kriowi

mesons. Some assignrnents. especiaily for thr 0'- rniiltiplcts

. . . . . . . . . . . . . . . . . . . . . . . . . are controversial. t i::

PDG estirnated m a s and wiclth values of scxlar tr i twris . . . . . t i4

Best fit values for the 1 = O channel (0.3 Ge\--- < r < 1.1 Ge\--') 143

Best fit values for the 1 = 1 channel (0.3GeC--' < T < 1.1 G P I ' - ~ ) 14.7

.Vo-instanton effect results (0.3 Gr\'-' < .; < 1.1 GP\- - : ) . . . . 131

\I'iclth cffwts in the 1 = U rrtiaritiii ivitti itipitt par;initBttbr \-irliios

so = 3.3GeL". JI = l.0Ç;eL:rr = O.OTGr\'". üntl r r;irigtl is

. . . . . . . . . . . . . . . . . . ( 0 . 3 ( ; ~ \ - - ' < r < l .~Gel ' - ' ) 154

LTirtth effects in the I = 1 channel. with input parametcr values

sg = 4.5Gei''. 31 = 1.JGel:u = 0.13Gel-'. and r range 1s

. . . . . . . . . . . . . . . . . . . (0.3Ge\--" 7 < 1.i Ge\*-') 15;

Two finite-wiclth resonance mode1 with same input valiirs . . . 1.57

Tit*o finite-width rpsonnncc rnotlel with differ~nt iripiir vaiut1~ 1?7

One tinicwvidth plus a r i i t r r w rtb>t>[i i \ t l t , tb ~ ~ w l t 4 i v ~ ! t i I U ~ I L [

wicith r = 0.2Gel' . . . . . . . . . . . . . . . . . . . . . . . . 1 :3

Monte Carlo simulation rrsults for tlhe 1 = O c:hariri~l

. . . . . . . . . . . . . . . . . . . (0.3GeCV-' < r < 1.7Gel'-'1 163

4.10 Monte Carlo simulation results for the I = 1 channel

(0.3 GeC'-2 < r < 1 .l Ge\--*) . . . . . . . . . . . . . . . . . . . 168

Chapter 1

Foundations of Quantum Chromodynamics

1.1 Historical Review

Thcri1 are. in principle. four kinds of fundanientai forcc.5 ( intcnct iorn I t l w r r tir:

in nature to our present knowledge: gravitation. electromagnetism I E l i ) . neak

interaction and strong interaction.

Gravitation is described classically by che general theory of relativicy. It

is bclieved chat this interaction proceeds by exchange of Gravitons. ancl is

i.iirrently b~lievetl co play a rityligibltx roltb in ~bwicntarv parric4~ p t i ~ s t r s o i t

t h pti~riortit*ririlor:i~':~l I t~v~ l .

Electromagnetic interactions (e.g. Coiilomb scattering) are carried by tkc

rririsskss spin-one photons. There is only one kind of photon. and the! do not

interact among themselves. Eh1 interaction can be best described by Quantum

cIectrodynamics (QED) which was formulated in about 1990 [l!.

.ifter El1 and gravitation. the remaining strong and weak irittni,tiorih I I I

tiricirre are i i s ~ o ~ i ü t d with nucleons. \ k a k inttmctwtis ( r .q . ~-(IN.;LC.. p - ( l t v ilv 1

describe forces bptween niicleons which involve leptons (1 .e . e . p ) In twak

interactions. IL". \I--(z 80 Ge\-) and Z U ( z 91 Gel-) carry the forces. Thanks

to the pioneering work of Glashow. Weinberg and Salam [a. 31. we now know

t hat the weak and ES[ interactians are connected closely and can be described

bu a single theoq- of electrowak interactions.

The remaining strong interaction is the niain conrcrn in this thcsis. anri

mi l I be descnbed in much more detaii.

The strong interaction describes forces which act among hadrons such as

protons, neutrons, pions, kaons, etc.

In 1933. Yukawa developed a theory that attempted to describe the strong

interaction [4. In his theory. the pion is the carrier of the nurleim forr~s.

.-Utho~igh it is relativelv sucçessfiil. rsperiaI1~ tht. clisrrnxw of n' iri 1947 i, .

the \rtihiv;t t hem- mes several dififrcult ies:

1. At high energies. the proton-neutron forces were not twll cksc.rihcd hy

pion exchange.

2. Due to paritu. the interaction between pions could not be described hy

pion exchiing~.

3. This theor? is rion-rcnornializable and hence h a Liniited predictiv~ pixwr.

A becter theor? is obviously neecIed to describe strong interactions. To date.

Quantum Chromodynamics (QCD) is wide1y accepted as the correct theor!

for the strong interaction. [n the foIIoiving. a brief review of r h ~ hiscoriral

clrv~loprritlnt of QC'D is ziwn.

First I ~ t ' s ;tsk the questinn: nrr rhtb tic3iitrons ; t t d prciri iri:; fiiritiitrtiiwitl + t i -

particles? if the! are nos. what are thcy niade i ~ p of'! Thtl h ~ s r tbvit1tmi.t. - for substnicture h s t cornes from e - p deep inelastic scattering and the piirturl

model. which was introduced at the end of 1960s by Feynman [6]. Bjorken jT]

and others. Deep inelastic scattering happens when a very high mergy dectrori

scatterç with a proton. The. proton b r d s iip inw itn ;~rtiitriirv r i i i r i i l i r~r rd

hadron States. and the invariant mass of the hadrons in thc firial st;itt3 rtiii.;t hi*

a t Ieast CU large as the mass of the proton. EsperÏmentaI eviderice shows rhat

the proton has substructures at distances less than 1 f m.

The iarnous result from deep inelastic scattering is known as Bjorken scaling

[TI. Let us d e h e the knomentum q2 transferred in the sustem. p the mer,?-

Ioss of eIectron. and 1 ~ ' = '% a ~l i rn~ns io~dt~is variahIr. TIir Bjurkthn s i l i i i ? r ' i -

s-s that a t high p and q'< but rnoderate W. the stnwrure func.rions II-, a r r c l

turn out CO he hinctions of u: only, nor of q' and ji separatelu. Ftynrrian

interpreted this phenornenon by the parton mode1 161. which describes rapidly

moving hadron (e-g. the proton) as equitdent to a jet of fiee partides called

partons. So in deep inelastic scattering, electrons scatter. not from the whole

proton. but incoherently from its point-like free constituents (partons). After

accepting the parton structure of hadrons. it is easy to rletcrminr thtl spir! 1 2

of the parton by nieasuring the value of [( %il\, - IL;i/(211'l ).

Even before the discovery of the parton niodel. during che 1950's iind 1960's.

due to the contribution of the discovery of .-resonances". a hadronic population

explosion was observed arnongst strongIy interacting particIes. and hundreds of

hadrons were known. The clüssification of different hadrons w s il big rfidltwzi~

;it t hat t inie. The decisiv~ breakc hroitgh iri t tiis riaprrt n x s niai 1 1 8 iir (';+.Il- l1;iri r i

and bu S ~ * e n i a n in 196 1 [S. . ni. . N.titln plor ring cwtain rhen-k~iciivn hiidruricl 4 ui

a isospin-hypercharge (13 - Z* ) plane. regiIar pattprns iilmays orcitrrd. T ~ P

existence of these patterns suggested that hadrons are composite states of more

fundamental entities known aç "quarks". At chat time. they suggested three

fundamental spin-: - quarks which were denoted as u.d..s qiiarks. Orilv thtx

.1; quark carries the strangeness quantuni niiriilwr. Tlirr cfrsirrit~rl d i t m .

for classifying al1 the then-knort-n hadrons according to thil rcpr~a1nt;itioris

of the flavour SC(3) group. Nowadays. there are believed to be six qitarks

distinguished by u! d. S. c. t. b? the other three flavours being substantialiy more

massive than the (light) u. d, s quarks.

According to the quark modcl. al1 hadrons can be classi6ed into tw grorrps:

baryons and niesons. Baryons are fermionic botincl States of th teck qtlarks (rplq i-

and mesons are bosonic quark-antiquark pair (qq) states. AI1 quarks art? spin

1 3 and cany non-integer electric charges. The quark mode1 was soon accepteci

as the correct t h e o l to describe hadronic states, and physicists identified the

parton as the quark.

In summary. the parton model served as the dynamical evid~nrr for r t i v

existence of quarks.

a A rapidly moving hadron uf niomentiini p is quivalerit to a jet of t T ~ r

particles called partons. Partons travel parallel to the original harlrun.

sharing its momentum.

a Reaction rates for hadrons are obtained from incoherent parcoti r t w r i i i t i

races.

Probability of having a parton of niomentum i p is ,V(i ici(

Ce- soon it was realized that the quark model. if taken seriousl~. posecl a

further puzzle. Let us consider che A'". for example. This partide is made

iip of three u quarks ( u u u ) . The three n quarks are ail in S = 1.1 = il

States. and the synirnerric spariiil w.avct fiirictioii lias trindtt rhv rlirw cpirk> dl

in parallel spin states. which ubvioiisly violatcs Fermi-Dirac st;~tihtii+s. To cicilvtb

this ptizzle. in 1913. Fritzsch and Gell-SIann assigned the quark a new degret1

of freedom [IO]. called colour. LVith the help of the colour degree of freedorn. it

is easy to constmct a totally antisymmetric wave function conectly describing

the A".

How many colour-degrees of freedorn are in the theory'! The ratio of cross

sections for e - P - scattering producing hadrons (inclusive process) tu that u€

c(e"c--*hadrons) producing muons. R = +A.- . indicates there are 3 colours (red. green.

blue) e'asting in nature.

So far. mith the help of three flavour and three colour degrees of freedorn. the

parton-quark mode1 had prowd to be very efficient in qualitatit-el? describing

the spectriim and static properties of niesons and hiiryoris. For ;I c . o r r i p l ( ~ t c ~

strong interaction theo?. the tlynarnical sidt. of rhe rhcory is ritwt~d.

The search for quark dynamics was initiated right after the foiindation of

the parton model. =\II the knom quantum field theories a t that time were

siirveyed as possible candidates for quark dlamics. and were shown not to

solve the following properties of the q iwk interaction:

\Vhen the distances prohetl are very srriall ( 1.r. . tvhim t h t r i i ~ t i i c ~ t i r i i ~ i i

transferred by the probe tvas very high. the situation occiirring iri cltlty)

inelastic scattering), the force between quarks is surprisingly weak and

the quarks move rather like free particles

On the other hand. no free quarks have ever been observecl. so it ih siirtB

chat clver large distances. the forw t ) ~ t i v t m r[~l;trk.s: h . o r t i t ~ s iri(.rtmittdl\.

strong. leading to t l i t b ptitwmienori knotvri ;fi qir;irk cxmfintwivnr

This situation changed after the breakthrough discovecy of asymptotic free-

dom [Il, 12. 131 after the classical work of Yang and Mills [l41. This theory is

a gauge theory similar to quantum electrodynamics though different frorn it in

that the corresponciing gaiige symmetry is not Abrliari ( z .r . the grnerators o f

the synirnetry group are non-commutative). Grr~ss rit i d rsnr~iiritvl tiori-At~c4i;~ri

gauge tield theoties by the use of the renornialization groiip metho11 ( i ~ s r d b t h

discussed later). and found these field theories satisfied the desired pruperty.

which is now called asymptotic freedom: the strong coupling constant is strong

at large distance while weak at short distance. Soon after it was shown that

only non--4belian garige theories eshibir the propPrTy of ii.;\.tiiptotic f t w i l o r i i

. . iimong the known r hvorics in four dimensional spiic.th-tinw . 1 .$ .

Cornhining the quark moclel and the it~~triptotic frtwtoni property of thc.

non-.ibelian gauge t h e o l together, we have reached the celebrated quark dy-

namical theory Quantum Chromodynamics (QCD). The term "chromo" refers

to the coIour symrnetry of the quark systern. There are various reasons for

believing thüt QCD is the best theop- of strong inreraction.

.Jiist like the photon. rvhich is an Abelian p i g e fitbld nifvliiitirig ES1 iritvr-

action betrveen charged particles in QED. thc non-.\hclian gaiigtt field of QC'D

niediates colour interactions between quarks. This non-.\belian gairgc field in

QCD is ca1led the gluon as ic is responsible for binding quarks together. Whilt!

photons have no electric charge* gluons carry colour charges aucl hence the!

can interact ivitIi e x h other in t h t ahsrnue of qii;irks. This priy>cBrtv o f t t i t b

gIuon is an essential ingredient for havin-; asyniptotic freedoni.

In the quark mode1 wich cdoiir iyrnrricwy 5 '1 - (3 ) , . hxlrons appear ;LS

colourless states while quarks c a r y colour quantum numbers. It is assurneci

that onIy colourIess states are physically realized and hence quarks cannot be

observed in isolated states. This also is the reason nhy only rneson (fi) ancl

hayon (qqq) states can be obsen.ed in nature. hpcause thesr r.oniliti;itions trl

qrtr~rks and antiquarks can form colour-singlt~t stiitfx'i.

.-\ccorcIing to rhe property of asymptotir. k~~tioiri of QCD. onc (.iiIi safdv risv

perturbation c heory co calculate short-distance (large energy) react ions. O t i

the other hand. when distances become Iarge in reactions. the effective coupling

constant grows large correspondingiy. and perturbativ~ QCD loses its pntwr in

this region. [n the long-distance (Iow t-nnrq) r~ginn. non-p~rtiirhativi. QC'D

has to be applied to make QCD a complete theory QCD sirm nilrs. insciinroni;.

and Ia~tice-gauge theop are some esamples of non-p~rturbative QCD. In t tiis

thesis. the first two of these non-perturbative QCD rnechods will be presented.

1.2 Principles of Perturbative QCD

1.2.1 Gauge Principles

As discussed in the last section. QCD is a quantized non-Abefian gauge field

theory. We start with an elementary introduction to gauge field ~heories based

on gauge principles and then describe the methods of quantizing gaiigc ticld

t heories.

Gauge field theories are field thcories b a s d on gauge pririciplts. In gcntml.

gauge syrnrnetries can be either abelian or nonabelian. md are realizd ttither

locdly or globally.

Before tfiscrissing the various gauge synmetries. a brief clisc.iission of thtt

Lagrangii~n formdation of I hi1 fit4rl r tworv. wi-liic.h pliivt, il cwir riil rc I I I * I I I r l i t *

contcniporary ilndcrstanding of inrfmctlons and .;vnirtwtrr~*. 13 Ii~~lptiii t r i

hritf t h~ Liigranqiiiri s i r ~stics h w riytiirrrnt*nts:

1. Lorentz invariance, which means thiit Lagritngian L niust tw a sc~;iltir

with appropriate dimensions under Lorentz transformation.

2. Gauge invariance. which mmns that field theori~s sntisk wrriiin giiiirtl

q-mrnetry requiremcnts.

3. Locality. ahich rneans thst L clrpcnrls or1 local prnptbrti~s of firl(1s. r . !

C ciepends on fields air) and a finite number of derivatives. Cwally. cmly

the first deritatix-e of the field û,o(x) is used in Lagrangian formulation.

4. Renormalizability. which is the abiIity to absorb the tfivergrnc~s irito

the arigind Lagrmgian. This is the key riifficdry iri iwri.striic.tiiiq r l i v

t heu-

S. Unitarity. mhich means that the S rnatrk (can be obtained From La-

grangian) satisfis the u n i t q requirement StS = 1.

In the content of classical physics, by "Lagrangian fomuiation" we mean "the

principle of l e s t action". The action S is defined as

The least action principle says fields which actiially occiir in natiire iwrwporiil

to the shortest path in the space-cime. or a stationary value of the action

&S[o] = O. This gives ils a dynarnical principle on tvhich we liase otir for-

mulation of the probl~ni. The l e i ~ ~ t artion principltt Iciids r i ) r t w u i 4 kr1rin.n

Eider-Lagrange qiiation ( 1 . ~ . the clqiiation of motion ti)r r h t * tit?l[l I I .iri,il-

ogous ro Newton's eqtiation of motion for point mtases

Xow tre have the background to clisciiss the gauge syrnrnetp- of the Laqrnngian.

Let's vtart frorn the easiesc one.

1. Global - Abelian gauge invariance ( F.Y. rharqr c.c)nscna-aticin i

Global means the symmetry operations do noc involve space-timc. Abdian

means chat different transformations of the g o u p cornmute with each

other. One esample of a globaLAbelian transformation is the phase t rans-

forniation. q { r ) + o;(s) = r - ' ' ' l % , \ ~ i . ahcri. I ) is a r d riiitribi*r ;iiitl i i i -

dependent of space-cime s. the phase tl is not nieasurablr and chus van h v

chosen arbitrarilv. but once chosen it rnust be the sanie for al1 spacetime.

iA-hen 6 becomes infinitesimal. d'&,(.r) = oijx) - O,(=) = - t B q p J ( z ) .

the global invariance requires 6L: s 0: and Nith the help of the Euler-

Lagrange equation (1.2). one finds that any continuous -met- trans-

formation which leaves the Lagrangian C invariant implies the esistenct*

of a conserved quantity. a statenient of Soether's Thruretri. In oiir i.iL+tb.

the conserved quantity is the consen- cl ciirrpnt .JI' = iq-O whiw tht.

charge is defined by Q = J' d3x JO (2) . The charge defined in t his way is a

constant of motion = O because the surface terms a t infinity become

negligibly srnall J' d3x& JO = J d32@.J, = O. Soether's theorcm is th,>

connection between symrnetries and c~onservritiori i ; iw in tithld t htv)rlt+,.

2. Globa lnon - Abeliangaugeinvariance ( e .y. isospin symmetry )

The peneralization to non--4bclian transformations is k r l y simple in the

global case. The simplest non--1belian invariance is isospiri where the

fields are assumed CO corne in multiplets. The multiplets are cietined i i s

O = ( O * . 0.2. O)\. forrning a t l i f i i ~ for rq)r~wtir~itio115 OC ~ h t x iwbp in qri )I ip

S r ( ? ) involving rotations in isospin spaw. The q;iiiztl triin~ti)rni;itioti

is specified by three piiranietwrs in SK(2) . B = ( a ! . 82.f).1). ch(. transfor- - -

mation is ciefined as o -+ O' = e-'Leo. where is ii niatris citmoting

the generators of SV(?) transformations. The dgebra satisfied by the

generators is [LJ. Lk] = ic!lkLI. where the dlk are called the structure

constants of the groitp and are antisymm~tric under eschange of iiny pair - -

of indices. For an infinitesimal transformation. Jo = o - O' = -1L . Bo .

which makes the Lagrangian invariant under this global gauge transfor-

mation.

3. L o d - Abeiian gauge invariance (QED)

Consider a transformation ol(x) i O;(=) = e-'q~"l')o,ix). wherp H ( . r j

is an arbitra- iunction of x. The local gauge transformation nieans

that one can rneasure and fis the phase 8(r) locally and ciiffctrently

at different points in space-time. The infinitesimal transfnrniiition is

I)o,(x) = -iqid(~)oi(i). The difference occurs when differentiating the

field tenn. û&j(x) f t e - ' q ~ ' ( ~ ) d @ ~ ( x ) . which is an inherent requirement of

the Lagrangian to be invariant under gauge transformation. El I inçerar-

tion is said to be locally invariant because al1 derivatives cnn bc replaccrt

bu "covariant derivatives" D,, (D:! 8, - leq,-4,). By this rrpiaiwriivir.

one can rebuild the reqiiiretl properties D,,o,(r ) 4 F - ' ~ J " ~ ' Djto,(.r 1.

Thus the gauge invariance of the Lagrangian follows as it clirl in tht.

global case.

In QED. .-l,,(x) is the vectar potencial of the photon, the simplest es;irnplta

of a "gaiige field". The vrcrtir potmtial transforiris .LS .-l,,l.r i - .4,,1 r- i = t JUir ) .i,,(r) - ; F. wherr tht. potentid is assaciatrd with tht. nirisskirh ptior un

iincl P is the unit ESI charge. The '-field strength terisor" F,, is cl~fintvl

as F,, = 4-4, -&,.4,. which is itseif invariant iinder local Abelian gauge

transformation. Therefore the photon kinetic energ'; is gaugc invariant if

i~onstructed mith the field çtrength tensor F,,,. In QED. rhr Lngrnngiim

riin br written as C = -iFu,Fuu

4. Localnon - Abeliangaugeinvariance ( QCD)

The first generalization of SI.+(?) to a locally gauge in va ri an^ Lagrarigiaii

is due to Yang and NilIs [11]. In the 't'ang.1Iills case. the SI-(-V) grorip

generators T, obey [Tl. Tk] = tc'JkT, which is the same as thr qlnhitl

non--4belian case. The field niuItrplet 1.s (it4int.d ~3 L. = I r ,. i '. - . r ,

The aim here is to introdt~ce as miIn' wc-tur tieltls -4; (.r j . gaiigt3 tir41 t.

that are anaiogous to the photon field -4, in QED. as is necessart- iri

order to construct a Lagangian inmriant under the local gauge trans-

formation specified by e , ( ~ ) . By analog'- with QED. we seek a .'CO-

txriant derivative" D, instead of the ordinary deritcitiw a#. such tbat

D,u(r) + D;IL'(IJ = e - d N r ~ ~ b ~ ( r j . The following Minirion o f D,.

- - satisfies the above requirement. D, = 3, - igT -4,. where g is the cou-

pling constant. Sirnilar to QED. a generalized field tensor is defined ;I.S

Fi, = 8,Y - W.4: + g~]~[ . -$ .~> and Fi, FIJ"' is gauge invariant. Thus

a gauge invariant Lagrangian can be constructed as folioivs:

It should be noteri that in the above equation. gauge invariance implies

there esists only one interaction pariameter g, which can be deterrnined

by experiments.

After studying the four kinds of gauge syrnmetry c:ases. niiinv siniilaritit*b

beciveen QED (Iocal Abelian case) anci QCD f local non--4brlian cï~.;t.l h i i w h t w i

folinci. incliicling ilse of the covariant deriv;iciv~ r o hiiild gaiqt3 iriv;~ri;ttit t i t l l i f

crançformations. However. QCD and QED stiicly totally clifftwrir iritiwr*tion>.

so it is worthy to highlight the essential differences bet~veen QCD and QED.

The self-coupling of the gauge bosons (gluons) accoiints for the major tiif-

ierence hetween QCD and QED. Due to the qiiadratic rerm of r h ~ field tPrni

in field tensor F,, (Sfobc.4;,-4:). three and four -4; proriiirrs rPrrns a r ~ foiind in

C. and these terms imply the 13 and 4 gluon interactions. is shown iri F i y w

1.1. which do not directly occur from photon interaction5 in QED. It i5 in fact

these interactions of gluons thernselves which are responsible for the property

of kq-mptotic Freedorn of QCD.

1.2.2 Second Quant kat ion

Second quantization is the quantization of gauge fields. especiaIly rhe Y a r i ~

and UiIIs' gauge field. Quantization is not at al1 a unique procedure and a

varie@ of quantization methods ma? esist which Iead to the same ph>-sicai

Figure 1.1: Three- and four-gluon interactions due to the self-coiipling of gluons in the QCD Lagrangian (1.3).

prediction. Basically. there are two ways of cliiantizatiori: thr cïinutiic.;il optw-

tor formalism 1161 and the fiinctional-integrai furnialisni .1 T'. I r i t lit1 r rmlir icmi l

canonical operator formalism. onc regnrcts t i~lds i~ optmt.i)rs ;in(l wts 111) r ; i r i ~ ~ r i -

ical commutation relations for theni. .\Il Grccn's funcrions whirh ch;ir;ii.tt>rizt$

the quantiim thcory of fields may then be calculated as vacuum espectatiori

viiliies of the prodrict of field operators. In thr F(3ynniari fiiiii.tioriii1-i~irt*gr;~l

(path integral) formalism. fields arc c*-niinibtn iind the L;igr;lrigiiiris ;irr* of t tic!

classictil form. Green's fiincrions arc (lcttwniried hy iritegratirig tht. procliic~t O F

fields over al1 possible functional forms with a suitable weight. Coniparin:: tht*

ttvo methods. n-e find three ad\-antages of the path integral over the c.anoriir;tl

operator formalism method:

1. -4s the classical form of the Lagrangian appears in the fiinctional intr-

gral. this formalism is convenient for dealing wirh svsrtms. s11c.h ;LS gaiigrl

symmetries obeyed by the classical Lagrangian.

2. Canonical quantization has difficulties in dealing with non-.I\helian gaugt>

field theorïes, which need the s u p p l e m e n t q contributions of Faddeev-

Popov ghost 6eIds [18, 191.

3. The functional integrai formalism is easier in dealing with the non-perturhariw

regime of QCD.

Because of the above three advantages. in this thesis. only Feynman's fiin(.-

tional integral method will be employed in this thesis as the tool for second

quantization of the non-=\belian gauge field in QCD.

In particle physics. one of the most used results of quantiini nic*chariic.s is

the prediction of cross-section of particular proces5. The rross-st~ccion t l t ~ p t ~ i i t l ~

on the Green's function K. mhich is the probability thiit a pirtic.lt* 11w; i r t v l ,ir

time t , and coordinate r, will be preserit at space-rime ri. t f . h-[.rltf. .r,t, ) =

(r f t f lx , t , ) . \Cè further define the Green's function G(t i . t3) . which is ncwssary

for S matrix elements in quantum mechanics.

where 10) is the ground state. T is the time product and x ( t ) is the Hrisc>nbcrz

position operator. Extension of these ideas to a quantiini field rhcorv leiitls t o

a functional integal representation for the Green's function of field operators

where S is the classical action S = J d4L. and [do] means the integration over

al1 possible field configurations at every space-time point (x. t ) .

After establishing the above relation. one can calculate any Green's function

in the functional integral formalkm nithout reference to the operator languaqe.

Furthemore. the above equation cm be remritten in a more compact forni h~

using the concept of functional derintive with respect CO an external sourrr

first introduced by Schwinger [?O].

Let us consider an external source function J ( x ) and introduce an mificial

source term o ( x ) J ( x ) in the functional integral which gives the generating

function .%'[JI:

-V is the normalization constant which is independent of J ( J ) . The functional

differentiation is defined as,

According to the definition. ive have

The ptiysical interpretation of the generating fiinction is the vacuum to varuritri

transition amplitude in the presence of the source J . 2I.J; (OIO) I J . Thtl

normalization factor guarantees Z[O] = 1.

Sow Feynman's path integral method can be applied to the non-.\belian

gauge field theory (QCD). In order to provide the unitarity in the physical space

of States. ghost-fields ~ ( x ) . which are scalars satis-ing Fermion statistics. are

introduced in the original Lagrangian (1.3). .-\ mathematical tonl r o ct~srriht~

the fermion fields (ghosts and quarks) is known as Grassmann Algebra -Z1\.

In order to obtain the propagator for the gauge fields, one shouId introduce

the gauge-king term. The procedure given by Faddeev and Popov allows one

CO include any gauge-fixing term in the Lagrangian [19]. Because of the in-

troduction of the gauge-fixing term. the QCD Lagrangian is no longer gaiige

invariant. In order to rephrase gairge invariance. Becchi. Route and Str~ra in-

troduced a non-Iinear transformation cailed BRS transformation whicti estends

gauge invariance CO the ghost field and thus leaves the full QCD Lagrangian

invariant [-'LI.

In surnmary. to generate a Green's function for non-.Abelian gauge theories.

the anticornmuting sources JI. Jz fur ghosr fields A. 2.: cal. 4, .Ji. .Ji fur qtiark

fields (1;. (l'/ and comniuting sources ./: for rhr gliion B,I artL iricrdiicwt.

write the final forni of the fiinctional iu:

where ,C is the gauge parameter and

1.2.3 ReguIarization and Renormalization

The key diificulty in QCD is renorrnalizat~on. i\'ith the Lagrangian for QCD

(2.10). the Feynman rules for QCD can be estabIished. Giwn an! quantum

field t h e o l one can constmct the Feynman rules for calcuiating the Greenas

function and S-rnatrk elements in perturbation theory. n-hich in tirrn givcs

the cross section for an arbitrary quark-gluon prowss. It ivill hr srcn 1ati.r

that the lowest-order calculation in a quark-gluon process generaliy r~prodrrrt-s

the parton-model results. Thus the parton picture corresponds to the rrer

ievel ( i - e . no loop diagram) in the penurbative expression based on the QCD

Lagrangian (1.10).

P p k Figure 1.2: One-loop quark self-energy S ( p ) .

In the tree approsimation. however. dynamical rffects of QCD (10 not 4iow

up anci the really important ingretlient of QCD is tii(ldt*ti iii (>CD riuliiirii-4%

corrections to the tree amplitudes. which necessarily iricliide the c-o~irrit)iitioris

of loop diagrams.

in general. the loop contribution to a Green's function generates infinitics

because the momenturn variable in the loop integral ranges over an infinite

range. In other \~orrls. for ;i relativistir theorv. thcrtx is no iiirriii.sic. lit-off

in niorneritum. For csaniple. consider t t i t b oiith-loop c1i;tgrairi for I l l t l q ~ ~ i i r k - ~ t ~ l l

energ? T(p) in QCD. The Feynman diagram shows in Figurta L.2 Itw1.s ro rli th

following self-energy in the Feynman gauge (( = 1).

By simple power countinq in k. iw haw /cl.'k& -- 6. i i t ttio l i rr i i t t i f h - = 1.

thus the divergence cornes from the high-monientum region ikl x. T h t w

divergences will render the calculacion fornially nieaningles. The theory of

renormalization is a method that allows us to consistently isolate and remove

al1 these infinities from the physically measurable quantities. The procdure

which makes divergent integrals tentativelu finite by introdticing a siiititblv

convergence detice is called regularization. Regularization is a pi i r~ly marh-

ematical procedure and has no phisical consequences ( i . e . the regularization

procedure happens in the intermediate stage). AccordingIy it is not a uniqw

procedure. and a variety of schemes exist. The cut-off method and dimensionid

regularization (Dim-Reg) are two widely used schemes.

;\ cut-off is a simple regularization schenies rvh~re the higtl-rriotiicrir~~tii

region. the soilrce of the divergence. is (.lit-off in r hi. cliwrgtw inrtkzriil5. T l i t *

method !vas improved in eariier lireratures in QED ;233. I t . tiotvrvrr. I)rtukh

translation invariance and hence the shifting of momentiirii in the integriil

changes the results. Therefore. it is not suitable for the regularization of gauge

t heories.

Dim-Reg is a metho<l which makes divergent miiltiple integrals convcrqtw

bu reclucing the numher of rriultiple intcgrals. The hasic itica hchirid rhis wtitmi~

is that since the clivergences in Feynnian diagram come frorn the integriitiori

of intemal rnomentum in four-dimensional space. the integrals çan be made

finite by allonring the dimension of space-cime [Il. 2-11 These divergences will

manifest themselves as singularities as the dimension goes to four. Hence. in

Dim-Reg ive keep the space-time dimension D lowr chan 4 and rf@;u.tl riif1

divergent 4 dimension integral bu a D tlinicnsioriiil rntcqral. 5irii.v ir i Diiii-

Reg. nothing has heen violateci espect that space-time is not 4 dinitmiional. iill

physicd requirements are satisfied. Therefore. chis scheme is Lorentz invariant.

gauge invariant. unitary. etc. [?a]. In this sense Dim-Reg is the most suitable

scherne for gauge theow and it is empioyed as the regularization schenie in

this thesis. C'irtuall- al1 modem calculations are done in Dim-Reg.

Still using the above quark self-energ'- esarnple { 1.121. in Dim-Reg schernr.

the incegral form J 3 is calculated in D (continuous) dimensions insted of 4

dimensionai space: the integral t o m is changed to p4-D J' &. The arbitra-

m a s parameter p is introduced to keep the coupling constant go dimensionless

in the Lagrangian (1.10)- In the chirai limit (m, = 0). the quark self-energu

h a the following form [26]:

where c = ?.-,E is the Euler% constant, and we restrict ounelves to the

Feynman gauge ( = 1. As the dimension D approaches -1, the divergence

cornes frorn the terrn.

\Vit h the help of t tie climensionül regiili~riziitioti s i . t i m i t ~ ;III( i r l i t ) i t l j i )v(b c 1ii;it-k

self-energy tnxiirnplf>. ~ve ïan disciiss the ..reriorrnalizatiori". wtiicti riituns. r o -

gether with the redefinition of the field. m u s and coupling constant. che rd-

justment of the norrxialization of the Green's function by suitable multiplicative

factors which r n q eliminate possible infinities in the Green's function.

As an exarnple. using the above quark self-enerq exarnple. the quark prop-

agator in the mitisle~s quark ('itsc c m t ) t l i I t 4 r i r c l ;LS:

where u(p2) is defined as

\Cé can norrnalize the quark propagator s(P) hu a rniiltipliçirtiw fiu.tuï cd Z1

mhich corresponds to the quark field's renormalization constant. The rcnor-

malized propagator SRy is defined as SRv(pj = &'S,, ( J I ) . The quanrity 2'

can be expanded in power of go ( i - e . Z2 = 1 - 22 +- 0 (gi)). The order g,, term

z2 is assiimed to be divergent. Thus.

CVe can choose the divergent term in z2 to just cancel the divergent term in

a ( p 2 ) which renormalizes the whole quark propagator ( i . e . renders it finite in

the c + 0 limit).

Yote here that the w- of elirninating d i v ~ r p n c e s in qiiantiim tliclory is ncir

iiniqiit. hec;iuse there esists an arribiguity in (Icîining thta riivt~rgtwt picc.t* id' r t i t*

Grecn's tiinctiori (tlifferent regiilarization rr~ettioils lead to tliffcrenr. c1iwrgtm.r~

in Green's function). and thiis thcre exists an anibiguity in choosing tht. \\*iiv tci

render the divergent term finite ( i . e . the arbitrariness of choosing the ~2 term in

the propagator). In order to remove chis ambigiiity one has to specify how tci

clef ne the divergent terrn. rvhich ivill bv siihtriictrrl oiit in th^ rc~tiorrriitlizatior~

procrss. The clesc:ription of how to stlhtriirt rlivergtw.t,s iri Grwri f'~tric.rio~is 15

c:iilltvl ;L rt~riorrri;rliz;itiori s(.lirmi!. Two widely ust!d siicfi sctiernes artb niitiirriiil -

subtraction (.LIS) and triodifiecl minimal subtraction (.US).

- 2. .LIS eliminates the [ $ + :,E - In(4;;)j term instead of only cerm in the

Green's fiinction. it is frequentIy riscd in th^ definition of QCD c.oiiplirir;

constant and also in other applications uf gauge ticld t htwics h w i w it

leatls a rather conipact form for the rcnormalizeci C;rtwi's fiinction. hi

the J[S scheme. z2 = -%CF[: ( 4 ~ - -{E - L n ( - i ~ ) ] .

-4s seen from the above description. at each order of the perturbation theory.

the removal of divergences turns out to be a subtraction of divergent p i ~ c e s in

the proper Green-s funceion. The central question of renormalizatiori t h r v

ctepends on n-hether the above siibtraction process iaitn h t b c.cmt;isrt~rirl~ p r -

formed to al1 ordcrs mith a finite nimber of multiplicative factors ( r . r . . r i iv

renormalization constants) and parameters which (such as g and ln) will be

redefined. One can show t hat the renormalizat ion procedures are successfully

çarried out in a restricted class of interactions. LYe use thci p ~ w e r (:orinring

rnethod to determine this particiilar ( . lm. Dctailrd t~r.hniqiitis ;ina oiir lirit'(1 I I I

['IS!.

-Ifter a r ~ r t a i n C is rletrrmined r~norrnaliziible bv p o w r cwintinq. for vrvn

distinct divergent term in the original C. a counter terrn of the sanie striic.riirth

multiplied by zi constant is introdiiced. These counter terms serve <as to cancel

the divergences from the corresponding terms in the original C. and ro giw ;i

finite Grecn's function. It is ~vnrrh norinr: rhiir rht. iiliorr. mttrhid o f 1 ~ 1 i r i i i r i ; i r i r i :

rhr itircrgrntv~~ risinq r.oiintc*r tclrrris is t~sst~nriiillr r hi, .iinit. ;15 r htx r r i i i I r i ~ ~ l i ( ; t r i \ + .

renorrnrilization trirthoii qiveri prrviorislv.

In sunimary. after ;r certain inceraction is determined renormalizable bu rhta

power counting method. counter terms are introduced to cancel the divergences

QCD in the original L. For QCD case. the renormaIized Lagrangian C R 1s:

The Zs are the renormalization constants of the cwr~sponding rrrrrtis of rtw

original Lagrangian. The esplicit forms of the 2s c m be found in [26]. Due

to the gauge invariance requirement, the coupling constants must be universal.

which means that mnny of the Zs are reIated, e.g. g ~ . y , = j0 = = 3. Z1u.w 21

These relations are guaranteed by the Slavnov-Taylor identities i-91 whirih arp

a conscquence of the BRS (Becchi-Rouet-S tord trarisfiirniar ion , ;2'>i. ,

The kcy point r > f the renornialization procws is t l i ; ~ r rio -'ritbir r i m i > " . II- t1

.QC.D introtliired: a11 rhc renurnializctl ternis iri L, , 1.17) hiiw t hi> samb sr.rl1t.r llrt%h

as t he corresporiditig terrns in the original Lagrangian ( 1.10 1.

1.2.4 Renormalization Group Equation

Accortling ro the rrnormii1iz;ition proctlss. ;il1 iiivc~rqrnr.r~s iirv whr r;ic.ti*ci fr( I I I I

tlw Gre~n's ftinct ion systematicall~. orrler bv imltlr in prrrilrbat io r i r fiiv KV.

In the sirbtractiiin proceifure ~ h e r t tlsists ari arbitrariness of huw r c i r l t d i ~ i t ~

a divergent piece in the Green's function. Summing up, the arbitrariness in

subtracting divergences is tivo-fold:

1. arbitrariness of choosing the renormalization condition ( 1.e. scrring iip

the conciition to subtract clivergcnces !.

2. arbitrarin~ss of fising the rrnormalizacion sciik r i I 1 . r . t he nias sc.;iltl ;it

which the subrractions made).

Due to this arbitrariness. there exist many possible expressions For one physical

quantity depending on the choice of the renormalization scheme. Since the!

are obtained from one phpical quanrit! starting from the uriiqrie C. and thw

rlescribe a unique physical phenornenon. they thus have to b~ ~qiiivalcrit. T h

above statement is the idea of Renormalization Group (RG): the phusical con-

tent of the t h e o l shodd be invariant under the transformations that merely

change the normalization conditions. An analytic expression of this property

is @%-en by the RG equation. The renormalization group equation expressps

the effect of a scale change in the cheory or. rnorp au-urnrcly. tlsprtBsscs t h

connection of the renormalizabiIity to scak p transformations. Bv r h t l a t ~ ~ v v

definition. the RG equation m u t tic a difftwntial tqiiation t~sprçwinq tht* nB-

sponse of the Green's function and parameters (e-g. coupling constarit ariti

masses) to the change of renorrnalization scale p [27. 30. 311.

For a .V point Green's function wich \i, glu~ns. .Vc ghosts and quarks.

consicter the relation between the bare rll and the renortrializetl wrws fi i~i( . t iori : ,

rR:

where Zr is the m t d renormillizarion ronstant Zr = ( ~ h - ,, l . '~ ( ii i 21. 1'; .

ri. is r.he c~upling constant. < the gaiige pararnetc?r. p tbt. renornializiirion niasc;

scalp. From ( 1.1s). one fin& that t h ~ r ~ is rio I L cItyt~ti(it~nw OII t fit' riqtir-tii~~i(I

sicle term T,, . As mentianed before. ~ h c IL tfependeriw is ;Ln artifacr of t h *

renormalization process. and can be compensated in the parameters cr and m.

We introduce a dimensionless parameter 1 = 5 . and perform the p$ operation

on both sides of (1.18). the RG group eqiiation c m bc! obtainetl:

where the beta function ?(a): anomalous mass dimension :dm. the d hnction

and 4 are defined as follows:

The rt~nurrnalizatinn grniip eqiriition reprcspntç the idra rhiit i i r w silr 01'

pretlictioris with a renorrnalization scale choic~ p = j t l (ïin hv rcli~ttlrl tct ;mer licbr

choice p = p2 hy the ahcw two rqiiations. Thus thr RG tqiiation i L . 19)

together with the 3. -,, f~inctions guarantee that our tIieory ts tiiuml un a iiriiqiitl

L ancl will give unique physical predictions independent of renormalization

scale p.

1.3.5 Scaling and Asymptot ic F'reedom

\Vhat rw mean bu SC-aling is the responstl of the RG cqiiation ;mtf r tif) river-

nialization parameters such as the coiipling constant anri chr miss . r t r . \ d i ~ t i

scaling al1 momenta by a factor et. The motivation of çtudying the scaling

effect in quantum field theop- was the experimental evidence of Bjorken scal-

ing in deep inelastic scattering in r - p collision ;321. The phvsic-id rriiliiriing I ,f

scaling momentum ( i . e . the change with t ) in s physical prriiws 15 r u r h r i q v

the dependence on the a.:. m to their running qriancities. An? frinction in a

field theoc- depends on a set of four-momenturn (p,). and on the parameters

of the theory (such as the coupling constant and the mas ) , denoted generally

by g,. Replacing the p, b>- rlpi, one can ask how things depend on 7 = e'

as q increases from I to x. The renormülization goiip anaIysis allows t lie 11

dependence to be shifted [rom the niomenturn ~ r r i a b l r s to t4f~ctive piir;iriir1rt3rs

g i ( q ) which x e scale q dependent. These functions are detined as soltition'; to

crrtain ciifferencial equations (RG equations). which are f~rrther deterniinet1 by

the structure of che theory ('161.

Let us scaIe al1 momenta by a factor of q. p, = rlp,. By dimensionai analysis.

WC finri chat the .V point Green's function rR satisfies

where dl- is the mass dimension of the .V point Green's function rR . Together

with (1.19) and set 9 = et. one obtains the fundamental clqiintion of rhc RG

involving the scaIing piirameter t .

The gcneral solution of the above clquation can be obtained via the nitlthoti of

characteristics CO solve Iinear partial (iiffcrential tqtiaciuns. Tht* gvneriil forrri

is as follows:

where the qtiantities fi .,E and r ? ~ itïit detrrmintd . th^ foilon-ing first onlrr

o r d i n q differencial equations:

The beta furictiori J ( a ) (1.24) is of estreme importance becaiise it deter-

mines the running c o u p h g constant and hence further determines the strength

of the interactions of the quantum field theor?. The calctilatiim tif t t i t l htltii

function and the -!,(g) have been perfornitd iip r o foi~r I i i t i p i m i i ~ r :!:3

In two-luop approsiniation the b t u fiiriction 1s:

where a is the running coupling constant related CO g bu a = g'/-1;;. [n SC-(.(3)

cases with three flavour degrees of freedoni ( 1.r. 3 fiavour aritf 3 r r h i i r frtwltmi 1 .

.jl = -912 and j2 = -S.

\Vith the hrlp of the beta and the -., functions. it is not clificulr to (-al-

d a t e the running coupiing constant and the running mas. i v e choose the

momentum scale to be et = mlp. where q is the typicai momentum under

consideration and p the fked momentum scale ( i t is chosen to satis- f i (0 ) = n).

Here we define a new momentuni scale .\. wherc .\ is renortrialization groilp

invariant i-161.

The momentum scde .\ is often referred as the QCD scaie parameter and is one

of the Free parameters of QCD theop-. which has to be hed bu experiments.

However. it is possible to set some a priori limit which .\ iç the d i i e the ruririirig

coupling becornes too large and pertrirbative theory breaks rlowri. T t i t x r!pii.iil

value for .\ = 300 .Ue1-.

After insening the two-loop beta funcrion (1.27) into (1.24). WC find the

two-loop running coupling constant and the rwo-loop running mass

-i2 B -\ 1261:

ivhere jl and jl- have the ~ a l u e s given before. and r~'(~ ' / . \ r ' ) is defined as.

m is the RG invariant m a s . JI and .& are defined before. -4, = 2 and -.? = 13 II

in SL>(3) case.

In order to demonstrate the asvmptotic freedom property in QCD and

sirnplify our notation. let us just consider the one-Ioop case. In the one-loop

situation. the beta function has the form of Eq.(l.X) with the set to zero.

The important resuIt is that the 3(û) is negative for sufficiently smal1 a as

long as the nurnber of quark flavour .\;f < 16 (for the t h e being. wc havil

esperimental evidence for 6 quarks onlu). Sote. in triis rase. hoth t lie j, a r i ( l

are negative. so the two-loop correction to the beta funcrion ( 1.27) itlso

contributes to its negativeness.

This unique feature of QCD (locally gauge invariant non--4belian vector

gluon t h e o p ) is entirely due to gluon self-interaction contributions. At one-

loop level this is the tripIe-gluon vertex contribution to the two-point flint-tion

T2. From this one-loop beta function. the effectiw riinning coitplirig m n tw

tleriveti:

Bccause of the negativencss of JI . in <_ 16 cases. wr finri whrn t -r s

(ultraviolet Iimit), the running coupling constarit i r l t ) approaciivs rtBro. l . r .

tht! xsymprritir frfwlrirn proprmy: the larger t t i r scaltl paritrnetcr ( t lit. Iitrrthr

chc nionicntiim). or the srrial1er tbt? distiinw bt~twt-t'tm particlt.~. ctie i;niidlw r ~ \ t l

hecotnes and thus the more rcliable the perturbative theov becomes for the

stroiig interactions. This is the enormoiis advantage iind che hpaiity of tht*

QCD. .bymptotically the cheory becomes a free field theory.

In stimrnary. aniong the kriown renornializablc quantum field rhrwrii*s in

four dimensions only QCD (Yang-hIills theor? in gerieral) rnjoys rhtl propcw\.

of asymptotic freedom due to the gluon sclf-interactions. Thr ~ s p ~ r i r r i t ~ n t a l

proof of QCD's asynptotic freedom comes from the deep inelastic scattering

in which quarks behaye like free particies (a + 01 at high energy t t -. x).

Therefore. we are safe to use perturbation t h e o n in the l a r p momentuni rtyjori

of QCD. but in ~ h e low momentiim region. t lit' rtin riing c.oiipling 1.1 mc;r iltir

becorries large. arid espt~cially ~vhen the monltwuni r i tvwi~s~s to th^ imlt'r of

QCD scaling parameter .\. the coiipling constant hecornes so Large that rhtb

perturbation theory is no longer meaningful. In the lom momentum region.

there remain man!- unresolved problenis which wiIl be cliscussed in the next

section.

-4s we know. a lot of phenomena that can be studied in a field theory rely on

perturbation theory. In QCD the miracle of asymptotic freedom. chat is. the

fact chat che running coupling constant (a,) becomes small at short distances.

allon-s ils to use perturbative methods to stirdy man? interesting am1 impi~rtmt

physical phenomena provided that we restrirt oitrsclvrs ro kirlematic- rtyiiin3

where the effective caupling constant becornes small enoiigh. Sonrrht4rss. thth

rast rnajority of ewntç in a typical hadronic reaction fail into a kinematic region

where perturbative QCD tan not be emplo-wd (a, coo large). Exiiniples include

the fundamencal issue of confinement. the determination of hadronic masses

and ciecay widths. vertex f~inccions. etc. Ic is thris of the irtmost iniporra1c.t. ro

Find approaclies to QCD calculations IV hiçh estent1 ri perrurbar ivc ixsp;itti~c)ii ir i

O,. 111ich tias been achievcd in the low-enctrgy non-pertiirhativtb reglori l i t ' r tiv

QCD spectrum. OnIy ttw of the niost artive arrrw of chic i ropit*. QC'D w n i -

ruIes and semi-classical insranton approach of QCD vacuum. will be presenteri.

Other non-perturbative QCD approaches such as Iattice gauge theoris and

effective-field theories are beyond the s c o p ~ of this thess.

1.3.1 Basis of QCD Surn Rules

=\ QCD sum-rule is an anaiytical rnethod designed to provide an approsirnate

calculation scheme for strong coupling QCD and in particular to account for

non-perturbative effects [34]. The basis of the rnethod are certain ideas of the

structure of QCD cacuum and the knowledge of the short distance p r o p ~ r t i ~ s

of QCD. -hocher equivalent definition is: the method of expansion of the rot-

relation function in the vacuum condensaces with the subseqiient rnatching via

the dispersion relation [35]. Ln this approach. we wiil encounter Wilson's op-

erator product expansion (OPE) methods [36.37] which take into account the

QCD wcuum condensate effects. and then in the spirit of duality nith the el-

egant Borelization technology. through the dispersion relation. QCD siirn-rul~s

h i ~ e the poKer of predicting hadronic properties siich as low-lying rcson;inw

masses. dec- constants. etc.

First let us start from hadronic ciments. The original Lagrangian ( 1.3)

is buih frorn the basic rnicroscopic ciegrees of freedorn of QCD. quarks and

gluons. Xeither quarks nor gluons are asymptotic States. Only hadrons that arc

coloiir-sin& bound States are esperirnentally observeci. I r i cirdt>r to stiidy tha.

properties u l tiacirons. it is çonvrmerit to st;irr frorri mipr!. sp;iixb- r hv \ . ; ir . i i i iri i .

inject tliere a qiiark and antiquark pair. iintl tht~n folloiv t htk t ~ v r i l i i r i c m rd r h

valence qiiarks injectecl into the vaciiiini rritvlium.

How to adiieve the injection? T h injection is achieved by twernal tiarlrorii.

currents. The currents are built of quarks and gluons in such a way thal th(.

currents car- interna1 quantum numbers. siich as charge or hypt*rrhargi.. wI1ic.h

coincide ivith the hadrons of interest. For t ~ x ~ i p l t * . t . h I = 1 iïist. ilti) [ I I ~ ~ w ) ~ I I

nncl tire i = O cirse (JO nieson) scalar currrncs an3:

The niost cornmon currents are the vector and asial-vertor rurrmrs s i n c ~

the? actrrdly esist in nature: virtiral photons iind l\* bosons rmplr to vt~rctr

and axial-vector currents. Therefore. the! are experirnentally nrcessihlc in rhia

reaction e*e- i hudrms or hadronic ; decays.

?Ifter constructing the hadronic currents. one needs to calcirlate the corrc-

lation hnction of the current. Basicaiip al1 QCD calculations start €rom the

correlation function which is defined as:

Graphicaily. the correlation function l l ( q 2 ) is the amplitude for ii quark and irs

own antiquark created by a source at point O to meet again at point r. .Usa i t

should be understood that the injected current carries a cotai four rnomenturn

of q-.

The iniaginary part of Il($) for rt > O ( i . r . abovr thtk phys id t hresttold 11f

haciron productioti 1 is callctl the ~p t~ t . t r i i I c l t w i r y {JI 8 1 . i p ( . I = 4:: III/ l li :. - L qZ). Lp m normalkation. it coincides for vwror c:iirrmts !vit ti r titi c.rt,>s w ( . t icm

of e'e- + hadrvns and the axial-vector for the r decq distribution turiction.

so the spectral density p(s) carries full information about the spectrum of

hadrons with giren quantum numbers. Calculating the spectral density esactly

in a systematic way is evc- theoretical physicist's drcam. and hecaiis~ uf t ht'

rtifficuftics in the nori-perturbatire QCD rcyyttitb. at prtbsent im1v ;qqm)siriiitrc8

rncthacls can be iised in practical calciilations.

Having obtiiined the spectral densitu. thc total hadronic clecay ratp of thtl

giren ciment can be ivritten m R = J:' ds p(.s). but since the hadron spectral

functions are sensitive to the non-perturbative effects of QCD. the integrand

cari't be diroctiy calciilated ac present in any s p t ~ m a t i c way. S~vcrrhcless.

the integral itself rat1 be calculattltl ?;ystrni;itic.;il~ 1)- t*xploirin:: r l i t , ;irt;tl\-tic

properties of the correlation fiinctions ii(.sj ;3S!. The correlators arc anidytic

functions of s except along the positive real s a i s where tbeir imagina- parts

have discontinuities. So the integral of p ( s ) can therefore be expressed as a

contour integral mnning frorn s = .Li2 - i c belom the axk to s = M2 + i c above

the a i s . Referring CO Figure 1.3. the integral could also have singularitirs

at Q' = O. By anal?-ticitt-. the integral aro~inti the ~Iriseti vontoiir \ ;mha.

Thus R c m be expresseci as an integral around the contour that runs counter-

clockwise around the circle /.si = .CI2. The advantage of this espression of R

is that it requues the correlator only for complex s of order M' which can be

significantly Iarger than the scaie associateci nrith non-perturbative effects of

QCD.

One thing to mention here is that althuugh our desirrd target is the sprctriil

density p(s ) on the physical CUL ( i . e . the positive miue of q'), a11 theoretical

calculations in QCD are carried out off the physical cut (Q' = -8 > O ) . The

reason is obvious: the QCD Lagrangian (1.3) is formulated in terms of quarks

and gluons. not hadrons. So rvorking in the Euclidean dornain. off the physical

cuts. one can calculate in terms of quarks and gluons. rint1 dispersion rrlatiiiris

iirc user1 to relate the Eucliclean dortiain t i i the ~ > t i ~ i ; i ~ i d rt1t.s.

To calculate the correlator in terms of quarks; and gluons. one niiist iiw r l i t s

\Vilson OPE methods which are forrnulrit~tf in ttie Eiiclirican domain . 1561. . N'il-

son OPE is the ba is of virtually dl analytical calcuht ions of non-pert tirbacive

effects in QCD. and is the theoretical basis of QCD surn rules. The central

itlca is t h etle timc-ordered prodilcr of t t w local (elernentap or I-ompositi~)

operators at stiort distances can tira espi~rirltvl i r i rt1rrris id' c~it i ipl t~t~ w t ot'

reguiar local operators o,(o) [36. 371.

The c-nurnber coefficients C:' are calIed CVilson coefficients.

The operators O,, appcaririq in ! 1.3.5 i ;inb wrisrrairiivl hv r t i c 1 -vrilrnrbrr~

properties of -4(r). B{O) and the unddying quant\ini tieltl rhtwv [ri r t i c t

above equation. al1 short distance singuIarÏties are forced oiit from tfir luc.;il

operator. and built into the c-number coefficients. Since the OPE stuclirs sliort

distance properties (z + O), the coefficients CtB(r) can be determined pertur-

batively. independent of the process involved 1391. Thus in the OPE. wt.e can

separate short-clistance and long-distance effwrs ysternatical1~-. al1 ttip stiorr-

Figure 1.3: Contour integral of the spectral density p( s) in the vornpIt1s * pliine where cr = JI'. p ( s ) eshibits a discontiniiity alorig the r t d .- i~xis.

distance perturbative eKects are built into the singular c-number coefficient C,.

and long-distance. non-perturbatiw eff~cts are emb~dderi in th^ rtypl;ir Io(ïi1

operator O,. The separation of short-<lisçani.tt itrid long-di~lt,tric t. i l f f i v . t > 15 imtb

of the advantages of the OPE method. and enables the OPE to provrtie a iisc~fril

extension of QCD into the non-perturbatiw regime. Usually the O,, rerms in

(1.33) are organized in increasing mass dimension order. generally decreasing

in importance as the dimension increases. In other words. because the total

masç-dimension of ; l ( x ) BIO) is fixed. the c-o~fficients C,(r) ;ire ortlt~tvl .ic.i-oril-

~rig r i i clc<'rc;isirig orr1t)r uf singu1;intp whtln r - O. Si ippos~ iq)~r;ircir~ l i : 1

and B ( L ) respevtively haïe n m s tlimcnsions ri( -4) and i l ( B ) . m i l t ~ p ~ r i i r c i r (),,

hiis mass dimension &O). For sirnplicity. al1 possiblr nnotriiiloris diri~tmiotis

are omitted for now. ive find

Fuiirier-craxisfcirtriaticiri of I 1.35) rrsrilts in thc f o h - i n g esptt3ssiori.

where c, are regilar c-numbers obtained from C, in ( 1.35). By power coirnting.

we fincl

the singularities are represented by the -+ terms in the above equation. W) +

The higher the dimensioc of tocal operators 6,. the less singdar the coef-

ficient C,(x). Hence the dominant operators a t short distance are thme with

the small masi dimensions- Because the OPE is formulaced at short-ciistancw.

the Fourier-transforrning momentum q" is reaçonably large. we can cuncludcb

that the higher the locai operator's mass dimension. the less important its

contribution to the OPE because of the suppression of +-: and after Borel ($1 -P

transformation the higher dimension operator's contribution will he fiirther

siippressecl. Therdore another advanrage of rhe OPE methot! is rtiat i r iisi~;illv

jus involves ii rathrr srnall nurnber of operators. In prirctirc. rh r ~ipt~r;tror.;

which are tised in OPE for paranieterizing thr nori-ptaiirbativtl effwts ;mb h i -

ited : qq :. : GG :. : GGG :. : quGq : and : $q$q :. Detinitions of these

operators will be discussed lacer.

1.3.2 Vacuum Expectation Values

In rnornentrim spacc. applying the OPE co ttie t\\itrpoint c.orrt4ation f i i r i c ~ i c m

ii(x). Ive have

tvhere the Cn(Q2) are the c-nirmber functions as usual. and the

/On)cu,- G (010,lC!j are vacuum ~spectation \dites ( \-E\.I of ()CD optlriirors.

coilectively k n o w as the (.ondensates. The !!!1 1s ttie triits YCD v;u.iiiirii.

The above operator-product espansion (1.39) rnust respect the syrnrnetries of

the quantum field theory being considered. placing restrictions upon the C'EV

appearing in the OPE. For a gauge invariant current in QCD. the \;EV (O,)

must be gauge invariant [39.40\. Applying OPE methods to a gauge invariant.

scalar correkttion function II(Q2) in momwtum spare [41. 4.7- XI!.

where rn, is the Iight quark m a s in the SL'(2)-Bavour limit. which in the OPE

domain is relatively small compared with rnornentiim involvecl (6)'' » r i t i l . TIiv

cpantity IPrrr. coefficients a. b. r . ti and r van ht* cïd(.iilatrd frorii pLrr iirt);iti~.i.

calculations.

The coefficients can be calculated by perturbative niethocis. but the t;zliies

of the VEV condensates cannot be obtained from first principles. There are

two sources for che determination of the C'EV condensates. First, one ni-

stiidy man! different phusical current conelation functions whivh ;il1 inn~lt-il

the sariie basic set of iinknown \.'E\'s. Second. some of rho \'E\'s ;irrb ;ilrwi[\.

tistirriatetl ph~nomenologically froni crrrrent iilgehra stuclics. such xs t h Par-

tiiilIy Consen;ed Axial-vector Current (PCAC).

The lowest dimension (zero) belongs to the trivial unit operator. CVhen one

crilciilates the perturbative contribution to the correlator Il(@). i>nc actuall?.

r.;ilcrilatcs the coefficient of the unit iipimror.

Thi* opcrator of t.hr n t w Ii)wrst ctimtwsiori { t h r w ) is t h clitark 1lr-ii4rv

opt!riit(ir O,, = +I. In thc chiral Iimit t i . c t . ivhm tht. quark m a s tcrm in thti

Lagratigiati (1.3) is set to zero) 0, is the order parameter whose C'EV signals

the spontaneous breaking of the axial SL'(:V]) symmetry and the occurrence of

the corresponding massless pions via the Goldstone mode I-441. In pertttrhatiori

t.heory. the chiral symmetry rernains tinbroken and (O,) variishes idiw i i -d l%. . I r

is tenipting to say thiit (O,,) # O rneasures the clerivations froni r t i v ~itbrtr~rbitriori

~heory. The nl i ie (rnqq) is rietermin~rl frorn the theor- o f vhirnl syrtmirrry

breaking accortling to Gell-1,Iann-Oakes-Renner and using the PC-4C relation

described in [34, 451. .A more detailed discussion will be presented later in

this thesis. PCJIC gives (m, - md)(Üu + 24 = -m:J:. where f7 is t h pion

constant and m, is the pion mifis.

The next highest dimension operator is t hp gluon conct~nsatt* : ( ), ;) . The

gluon-condensate is closely reIated to the semic1;issicai QCD vaciiiini structure

instanton effects. and we will discuss this in more detail later in the t hesis.

The dimension-five operator is the rnived quark-gluon operator

(OpG') = ( @ G q ) . From (I.40), WP know that; this operator's çontribiition ro

rhtt scnlar ehannel's OPE is proportional r i ) 5. si in th^ liglir (li~iirt <.i>sib i v i r t i

r~l i i t iv~ly litrgc momentttm t r i in~f~r . this optwror plavs ;I tii~gligil~lt~ r c h iri r t i c 1

I = 1. O chiinne1 siim riiles. It is V E T ~ importarit. tiowvt~r. in a wiclt* r;irigt. o f

problenis involving bavons [46], and mesons built from one light quark and

one heavy quark [-KI.

At the Itr.el of dimension six. there is one operator hriilt frorti thrw gliiori

field tensors (n, f GGG) and scvtnI foiir-c 1ii;trk o p n t i w of r tir1 r VIN^

Olil = ( V I ïlq2 )(fj.lr2f1,,) ivticre TL.: clrnortl cwr;iin Havoi~r-c~cin.;t~rviti~! i .~~iii t~i-

narioris of c h Lorentz ancf culoiir niiitricw ml ri, is thr Iight qii;irk i i t+ l o f

rlifftwxit kirivciiirs (e.g. u. d..s). The three-gluon operator is espected to hiivtl ii

significant impact in heavy qiiarkoniirm [43]. It is determined from the Instan-

ton vacuum calcirlation i-191 and it does not appear iri the low~st ortltlr 1 = O. 1

nwsoti swn r ~ i l ~ s h ~ r m s ~ th^ r lirw-gluon op~mror's o f f t ~ r 15 ;dsc I 1)rc 1111 1i.r I I I I L L I

ro 5 iL5 stiown in i 1 . - I O ) . v: As fut thri four-qiiiirk upcritturs. ttiiuir \ 'E\ 's ;;irtt [lot k r l c ~ n i l i ( l t y i w 1 i w IV.

The stancIard ~vay CO evaluate them is a vaciiuni saturation ;ipprusiriiatiuri.

which means the four-quark operator can be represented by the product of two

two-quark operators [NI. i. e.

1.3.3 Spantaneous Syrnmetry Breaking

For the sake of brevity. the Lagrangian is definerf as follolvs wichoiir g;iiigtb-

fixing and ghust cerms.

The QCD Lagrangian ivith \;I massless flavoiirs (ml = O ) is knotvn to possrss

a large global synmetry, namely a s y n i r r r y i~nt1t.r SL-1 .Y, ) L @ -51 ' ( .y[ I fi I I I -

clependent rotations of left- and right-hantlrd qiiark fi~lds. This -vriiiri tbrrv i*

ralled chiral symnietry. After setcing m/ = O in the ahow qiiatioti. ttiv QC'D

Lagrangian splics inco two independent quark sectors:

Here (1 tlrtiotes thtt fi;tvoiir w(.tor q = ( I I . d . .< . . . 1 . ;inil y/. . t!,( <r;ltld f i ir l d r ; i l l r i

right hanilril t*otiip<iti~iits of rtw ~ ~ i i i i r k . <il, = (%)il. q~ = ( - ) t I T I i i b

definition of i 5 can he found in Appeiidis 1. Frorii t h . ;~bi,vt! clefiriirioti~. ivtb

find the two quark chiralities live in separate fiavour space which do not talk

to each other (gluon interactions do not change the chiralit?).

Instead of rotating the left- and right-handd quark fields sepiirarclr. ontt

can make equiwlen~ independent vector and asial vector IS(-\-r) rotiitinns tif

the ftill 4-cornponent Dirac spinors. which nimns that che QCD Lagrangiitn 13

invariant tintler these transformation too iri the niassless qriark c:;rstns. In thr

case of mu = rnd = 0. a vector rotation is denoted as u(z) + e-haa(*)'° t.+(x).

and an a i a l rotation is &(x) + e-i"a(z)+'s~(x). under these transformations.

the QCD Lagrangian remains inmriant.

The reason n-hy I L ' ( X ) - J ~ C ' ( X ) is innriant is thar -,, anticomrnrtt~s wirh e v r v

- matrix. thus any bilinear form mith an even number of matrices betwen

ut (x) and w(x) is chiraily symmetric, and ail bilinear tenns nrit h an odd number

are not. Consequently the quark-mass term m,Qq in Eq. (1.13) violates the

symmetries. but since the current-mass of u and d quarks are sniall i rn,, =

-1 SleC*. md = 7 .\le\-). their synirnetry breaking effects slio~il(l t )r wak.

If the a i a l rotation is a chiral symmetry operation. chth c-tiiriil pirrrivr ,)

and u l shoiild have the same mitis. In rcality ttit* split hmvet~ri t h e > chiral

partners p and a l is very large ma, - m, = 1260 - 770 = 300 I f e l ' [.Ni. This

phenomenon implies that in addition to the small expiicit breaking of chiral

symmetry. there must be spontaneous symmetry breaking iSSB). SSB rtit'iiris

a syainletry presem iti the Lagrarigiari is riot prrwntetl in t i l t - ii(.tital pli~~ii . id

;round States. Eqiiivalencly. the vacuum itsclf breaks the syrnrnetry.

IF xuial transfmmations arc ;i syrirrictry operation. it stioiild follow ttiat

Qg = J ~ % ~ ( x ) : ~ * I . = , Ï ~ ~ ~ : ( x ) is an additional conserved quantity. similar to the

total isospin Q" = J~~X$(X)Y~T~~L'(Z). according to Noether's theorem. The

axial current .J: = i (~ )~ , , )~ f ~ ( x ) should also be conserved ( i . e . il,.]: =

O). On the other hand. ive just argued that this symnietry is sporirantwslv

broken for the ground state: thus (i5 10) = i cl) # O. To loncst ordfir. u . ~ tirid

(OIJ~lir,) i; O. The Ici) should be identified ivith sonie field tvhidi coultl br

related to any pseudoscalar. In the absence of chiral symmetry breaking. since

[a. Q5] = O. it follows that ~ 0 1 . u ~ ) = ~ ~ ~ ~ 1 0 ) = O! mhich nieans the particle

must be massless (it is the corresponding, so-called Goldstone boson). The

pion with its estremely small mass is the natural candidate for the Chltlsrone

boson. Imrnediately following the above relation. cornes the famoris PC.-K'

relation:

This equation implies that the violation of axial-vector current conservation

can be described as being due to the coupling to pions. From ttiis starrrne.

point, one can obtain the famous quark condensate relationship:

( m , - m,r)(Olüu +- &[O) = -1;rn:. rvhere ff = 93 . \[pl- ancl rnr = 1-11] .\[FI'.

It has been seen that the non-perturbative effects Iead to the introduction of a

quark condensate. In perturbative QCD, the axial symmetry remains unbroken

and (Qq) = i J d'z Tr[S(x. z)] which vanishes identicallu.

In the SL(3) limits (mu = md = m . = 0). the QCD vacuum brcaks r h v

avial symmetry accordingly (after the symmetry breaking of the remainirig

L,-(3)L @ L'(3)R symmetry. the axial SC(3) and the L(1) symmetry arr not

rnanifest in the particle tlegeneracies). The Goldstone theorem then informs

us that there should be .\y - 1 = 8 appro'iimately massless Goldstone rnesons

in the hadron spectrurn for the breaking of asial SC-(3) symmetry j-15. 51;.

The eigtit Goldstone bosons can be iden~ifieti readily ;L.; ttit> ttirefb pions. ftrrir

kaons ancl one 7 meson. However. since ive nrerl to break rin twra i ~ ~ i i i l Ci 1 \

synimetn.. ive are still one pseudoscalar short. The naturai c-andi(liite is the

remaining pseudoscalar is 4'. but its mass is far too large (mv - 960 .lie\')

[JO1 to be identified as a Goldstone boson. This is the famous C(1) problem.

.homalies in perturbation theory were firçt observecl in 1969 3!. 3;. .\nomaly means the loop cliagram invdving ciitcrnal wctor and asiiil-vtwor

currents could not be regulated in such a way that al1 the r:iirrrrits rtirriiririrrl

conserved. ancl the axial current becomes nonconserwd. From the triangl~

diagrarn involving two gauge fields and the ffavour singlet axial current. one

Ends [Xi]:

This result is not rnodified at higher order corrections in perturbation theop.

The fact that the fiavour singlet curreat bas an anomalous divergence was

welcome in QCD because it can be used to explain the absence of a ninth

Goldstone boson. The right-hand side of the above equation is relateci to th^

famotrs instanton solution which wilI be disçussed latrr. The facc that tht t

PC-AC-type consideration cannot be applied ta the q' irnplies its mistiirta wrh

r h ~ gluonic tvorld throiigh the optmtor FF.

In summary. SSB refers to the breaking of SL-(.Vf)L @ SL'(-Lj). chiral sym-

metry for the QCD of .VI massless quarks into the SU(.Vf)I- symniecry. which

rnust have .Vi almost massless Goldstone bosons according to Goltlsrotie rhtb-

o r m . SSB stlems to soIv~ the hadron nius tltyntwc.v prril)ltwih ,ml r l w I 'I I i

piizzle siicc~ssfiilly. both of t hem connected deeply wit h the non-pcrt iirt~at i\x*

dfects of QCD. The quark c-ondensate is th^ order pararnrccr of rhirid svriinitb-

try and the q' problem is related to FF.

1.3.4 Dispersion Relations and Duality

Lsing several phenornenological numbers ( .\. (mqtij. (CG) ) m l r tir (liiitik iii;~.sc;t*c;

as input. QCD praçtitioners are able to ~rnplo . the OPE methorl tvir.11 t h h l p

of dispersion relations to undersrand an enurniotrs wal th o f the ilata rt+rring

to low-energy hadronic physics. In the ideaI world. one wotild calciilate the

correlator ll(Q2) and the spectral density proportional to I r n i l ( s ) exactlu. In

reality one has to calculate l7(Q2) in the OPE method in the form of a rriin-

cated series. This calculation is done in the EircIidean doniain (posirivr* (1'1

and in terms of quarks and gluons. The complication conies from thtt f w

that al1 physical obsen-ables (such a s cross section. decay width. m a s etc.)

are measured in the blinkowski domain (il = -Q2 > O). The connection be-

tween the Euclidean predictions and the rneasurable quantities is established

via dispersion relations.

k'allen and Lehmann quite long ago shoived that cwo-poirir fiinction rheys ;i

dispersion relation [a-1. Sa]. The dispersion reIation follows from the aniil.vtic.aI

properties of II(Q2) as a function of Q'. the only energ--niorrientuni inwriarit

which appears in a two-point Function. The dispersion relation is:

ivhere the degree of polynomial in the rigtit-hanri si& drptwfs oti t t i ~ t . i )r i iw--

gence properties of Irntt) when t t x. Thr anaiyticity of I I ( I2 - j apürc froni i i

brandi crit almg the negative rea1 axis (6)' < O ) is used to prove this disptlr-

sion rela~ion [56]. -4s we mentioned before. the interest of this representacion

is the spectral function i r m n ( t ) , \ W h J ( x ) a currenc with sperifir. i ~ i i i i r i t i i r i i

nunibers, the spectraI fiincticrn is ttirn dirtbi.tl!- rtbIatvti tci t lie r u r a l i.riis.s-scv.rii,ri

for the protluction ot' the hadronic States nith ttitvr quiintiirli r i i i i~i t~t~:s .

In siininiary. ive have the widcty rised rlisprrsion relation iq r ra tmn

((y = -q l ) :

The key elernent of every calcuIation referring to Siinkowkwn rpantitirs is

quark-hadron duality where a truncated OPE is rtnalytically continried. term

by term. from the Euchdean to the Minkowski domain [si. 581. The smooth

quark curve obtained in this way is supposed to coincide at high energies with

the actud hadronic cross-section. If dualitv is forrnuiated in this way. it is

obviotrs that at Cinite energies deviations from dualit~ must exist because of

the differences betmeen the measured physical crosssection and a srnooth OPE

prediction. The deviation from duality is due to the oniitted components in the

OPE calculation and it will arise when one cru CO predict the spectral ctensity

ImII(s) point-to-point a t large S.

The OPE-based predictions require additionally a different type of ctualiry.

One needs to assume that a particular cut of interest in the hadrotiic. ;iniplitir~lc~

is in one-to-one correspondence with the given quark-gluon (:ut. [ri 0tht.r worcls.

it is assiirned that different channels (in terrns of hatlronic pro(-css and iii tclrtris

if quark-gliion process) do not contaminate rach other :59!. This is thtl ~ t * ( ï i l l t d

.-global clualitf [60]. In the OPE caIculations. the cut of the perturbativr

coefficient functions carries has identic- and the abovc global cliiality is tvisil!-

implemented. In this thesis. the global cliialitv is iissiirrierl alrviivs valid. ; ~ t i t l

the cluality in shis context sirnply riieatis [tir 1oc;il rliiiilirv.

IL ~ ; L S been shutt-n that chi* rlifftwriw twtwr~n thtl tlsxr rtviilr ;imI t t i t ~

serics truricatetl at optimal orrler in the OPE is csponential jG1. 621. Instan-

tons, treated in an appropriate way ni11 represent the omitted terms in the

truncated series [63!. The instanton contribution to the correlation function

with large momentum ~ransfer can be graphically interpreted as a rnt.chariisrti

in which the large esternal momentum ici transmitted throiigh a soti cohtwnt

fielcl configiiriition. i . e . the large externa1 mornentiirn is sharetl h'; a vtlry I;tr=;t*

niiniber of quanta so that each quantum is scill relatively soft. It is char that

this mechanism is not represented in the OPE. and thus gives an idea of how

strong deriations from duality might be. The esponential terms nor. sem in

the OPE appear both in the Enchiean. and Minkowski quantities. The rarc of

fa11 off is much faster in the Euciidean domain than in the l[inkowski clorriain.

Conceptually. the exponentia1 t e m in the blinkomki domain determine the

deviations from dualit . Because the exponential deviation from duality can be

related to the existence of instantons, rve d l Ieave this topic for the instanton

part of this thesis.

1.3.5 Borel Transformation and Sum Rdes

In practical applications of the surn-rule rnethod t h e r ~ is a teïtiriic.;il iritlt I i c i c t

needed in addition to dispersion relations in order to stitdy the low-enrrgy

aspects of &CD. The rnethod referred to is Borelization (cornes frorn the Borel

transformation) ['16. 341. and it consists in applying the following operator

to the fiinction iinder consideration:

8 = lirn

wich .II' = $ = fixed. Applying the Borel transformarion to the surn ride in

(1.47). ive have:

where the follo~sing relations h ü v ~ b e ~ n ernployed :'761.

1 (n - l ) !

The admntages in dcaling ni th che Borel-transformed s i m rriies arp obvioiis:

1. First. one improves factorially the convergence of the power series ( 1.50) .

This irnproved accuracy makes the prediction of lowest-lying resonance's

properties more reliable.

2. On the phenomenological side. proceeding to the Borel-transformed (lis-

persion relations n e ai~tomaticallv kiil al1 possibb sutmartiori c-tinstaritil.

Emn more important. che cxponential weight function iti ( 1.19 1, rriiikth,

the integral over the imagina5 pan twll conwrgent. This w i g h fiinï-

tion enhances the lowest-lying resonance's contribution in th^ integrai

mhile contributions of the higher-order resonances are exponentially s i i p

p r esseci.

It is wx th mentioning that there are a wide variety of QCD sum riiles ttiat

can be utilized 16-15. The? include the Laplace sum niles. the Finitr Encrq-

surn niles . 1641. . the Gaussian Transform stim ruleç 1561 . . and sa on. Laplace suni

rules have been most successful in applications.

Laplace sum rules ernpIoy the Borel transformation of the standard disper-

sion rtlIation (si~cti as Eqiiation 1.49). The inreresting point about chi.; ryptt of

sum rules is the presence of the exponential factor in the inctyrariri whid i qiwii

a predorriinrtnr wight tii the Iow-energ?- componcnt of thr hl ro i i i r spiv.rr;ii

function. t 1111s the Laplace Sutri-Rule is pitt~ic.iil;ir ~ ~ ~ i t i ~ b l t ' for t hr, clrirvr111irl;t-

tion of the properties of the lowest-lying resonances of niesons or baryons.

The çimpl~sc spectral rnodel is a dclta function plus continuum.

where the delta function represents the Iowest-lying rpsoniinw <.antribiir ion m t l

the cheta function represents the continuum contribution. This 1s t lit) famuus

Sliifman-\.-ainshtein-Zakharov (SI'Z) QCD sum-rule spectral mode1 (341.

There arp some kinds of modified Laplace SR often ilseci in the litt.ri~tiir~

which are reprcsented as Rk i6-l;. Thr ryiiantit~ Rk is d~fint*d iL\.

where ill' is the Borel transformation parameter iit; ive derineci before and si,

is the starting point of the continuum threshold. It is known chat differrnt

ranks of Rk have different suppression of the non-perturbatire contribution

and the lowest-1-ing resonance contribution. The higher rank RI, enhance the

continuum contribution and suppress the lowest-lying resonance's contribution

co the surn rules. The mosc wed surn rules are Ro. RL :Gai.

In suniniary. QCD surri-riiles hegiri !vit ti sr ii(lv i i f ;i turr iht i o r i fiiric,r io [ ih .

graphically understood aç an csternal ciment injrcted inro rhe \ x * i i i i r i i iv i t t i

very large momentum Q2. in the deep Euclidean domain. The points of in-

jection and annihihion of the quark pair in the vacuum are separatecl by a

small space cime intemal. the injected quark hence has no cime CO interact

with the vacuiim medium. The? propagate as free objects. and thris tvtL ecbt

the pertiirbatiw parc of QCD. For chc nori-pt~rtirrt)ittive part id (.)CD. t t i t k ils-

cernai current's total energy is not large enough. the quark or ;inriqiiark. twitil:

injected. starts evolving according to the clynamical l m s of QCD insrciid o f

acting as free particres. At first. the quarks do not fee1 the impact of the

vacuum medium. -4s the separation between them grows. the effects of the

niediuni became more and niore iriiportant. eventiiiiliy prtlvrwtirig r1ii;irks frorii

appearing in the clctcccljrs (rio f r w qiiarksr. Tttv i r ! i i b t . r i * c t cpirk.. gt.1 1 1 r . t w ~ t l

and rnaterinlize themseives in the form of hadrons [Si.

The general s e p s for a QCD sum-rule calcuIation are:

1. Identifj- an appropriate interpolating field for the hadron of interest and

construct a correlator function.

'7. IdentiFy the tensor structure and invariant fiin(-tions of thr rwrrthtor.

3. Wrïte dispersion reIations for each inkxriant function rt'ith a spectral

model.

4. Construct the OPE for each invariant function,

3. Convert to Borel neighting,

1.3.6 Instantons

QCD surn rules are very successful in predicting some low-energy harlronic

properties tvith just a few parameters a s input, especially in the vector and

mial-vector channels. But still there remain severril ciontraclirtions whic.h QCD

Sl l i i i rtlirts cuiild Liiit i ir iSWXï.

QCD sum rules g i w w- arruratc pretlirtion on vrrror m i l ; ~ s i ; i l - \ w r o r

channels. but provide incornplece results on spin-zero channrlç. hoth for c m -

rerit made of quark and gluon fields such as a. rJ.a and scalar glueballs. This

probleni is first emphasized bu Novikov et al in 1952 [661. For esarnple. using

QCD sum d e s . me find to any order. the a,-, and jll meson's corr~lation ftinr-

tions are identicid. which impiies u,, and fo ~ihiiiild hiivr thi* snrw rIiiwxls. Frorli

the PDG data book ive know there are severaI isospin-partner randidatrs for tro

and fo with different masses [5O]. This phenornenon indicates that sornet hing is

rnissing and better control over high order corrections by OPE methods would

not solve this puzzle.

The second problern is the famous C(1) puzzle [SI. 521. &%y is the 11' meson

estremel- heavy:' It was in principle solved whcn i t au realizetl thilt the C( 1 i

chiral symmetry was esplicitly broken bu the a..ial anomaly (see Section 1 . 3 . 3 ) .

However. the quantitative description of the q' rnass is stiIl rnissing. It is noted

chat this second problem is deeply related to the problems of the spin-zero

mesons.

QCD sum rules use the values of condensates as a first principie (the d u e s

are determined from the experirnental data) rat her t han cteriving r tierii. .-\

field theory method such as QCD which cuuld not explain the migin o f t l i t l

condensates is a incomplete t h e o . The third question is this: iVhere do the

condensates corne from. or whac is the quantitative picture of the QCD vacuum

st rtict ure'?

The concepts of instantons are a natriral sdiition to the abovtl rtlrtv pt1zz1t.s

O - . , 1 6.31. \\+v ~i- i l1 show liow instanrons ~ ) t v t ~ r h t w t h r t ~ p i i ~ ~ l t * ; srq, !)y ' r t b ~ ) Firkt . .

a pedagogicnl ticscripcion of the instanton solution is givcri.

In addition to the gIobal ttiinimuni of the QCD riccion .4", 0 tiist~ti by

perturbation theory. there are man? other local minima called instantons which

have to be taken into account. Instantons are certain configurations of the

Iang-liills potentials .4t(x) satisfying the equation of motion D:b.-l:L = O in

Euclidean space [6iT. The soIution h a hem fmnd hy Bt4avin. Poli.iiktiv t t f i l

in 1915 [6Y. 691: the name .*instanton.' cornes from che suggcsrion of ' r Hoofr

in 1976 . CO\ . . who marie a major contribution to the investigation of insranton

properties.

Physically. one can think of instanton in tn*o wys. on one hand it is a

tiinneling process occiirring in imaginaru rime i this interpretiitiim ht~lotigs ro

1'. Gribov) ' T l 1 ) . . . on the orher hand. it 15 i i locxlizd pc;t~iitlol);trri(~It~ i n r h t l

Euclidean spricc i691.

Before we sttrdy the tunneling phenornena in làng-lIills theory. let lis clar-

ify the definition of the vacuum of the theory. In the Hamilton formulation.

it is convenient to use the so-cailed Weyl gauge -4, = O (here the notation

-4, = =I:Xa/2. n-here the Xa is the norrnalized SI-[';) generator). In rhc \\'tt-1

gauge case. the conjugate momentum r u the field variabl~s .41\.r.i ih litsr r tic1

electric field E, = 80.-l,. The Hamiltonian is given by

where E: is the kinetic and B;' the potential mer= terms. Thc B: is clclinid

as:

1 abc L cl B,D(Et) = rjri ,k[i3j .- i;-8k.- i;+c .4J.4kj -

In the non--4ht.lian giiiigc throry. the chsslcal v;u-iiiini iwrrt*sp{uict:, ro t tio q i i i i y

tirlds which hiive zcro potential enttrgy Ont. fini13 thar i f t htl tit+i hiippcw~ r o l i t '

a piirp gauge -4, = LL-O, IV' (whero the L+ = l ' ( Y ) is the giiiige r ransforrtiat loris 1.

thtx potential rricrgy at sttch points is natrirally zcro. Thus ive defint1 ttic

rln5sical vacria as -4, = 11-4 L i t .

111 o r ( h ro rniirneratr thts chssir;d i-acuii. wr have ta crlassifv ail possible

garige transformations C-ir ). This ri1ean.i t hiir wr liavcl CO st i l i l \ . r q ~ ~ ~ ~ ; i I i ~ t i i ~ v

c~l;issrs of rriapping frotn 3-spric~ R.' into the giiiige groilp .s'['i .\- i. In pr;ic.rirc3.

iw can rrstrict orirselves to matrices satisfyirig C*(S) -r L as r - s i7-j. Sucfi

mapping can be classificd using an integer called the Pontvagin (or iC'inding)

number. mhich counts how many times the group manifold is covered.

[lé conclride that there is an infinitc set of classical vactia rritrniernrctt bv an

integer nrr . Since they are topologicalIy different. one cannot go [rom onc1

mcurim to another by means of a continuous gauge transformation. Therefore

there is no path from one vacuum to another such that the energy remains

zero al1 the -y. The connection of different classical vacua has to be throiigh

a tunneling effect. €rom the above quantum-meclianical r s a m p l ~ i 1.331. wr

know chat we have to look for classical solutions of the Euclidean equation of

motion (the instanton). The best tunneling path is the solu~ion mith minimal

Euclidean action connecting vacua with different minding number (the winding

number in terms of corresponding gauge field A, instead of gauge transforma-

tion L - ( 1 ) is caIled the Chcrn-Sinions numhcr n , . .~ ).

In orticr to mininiizc t h iicririn. IW t~spioir r hr t h r r r v -

and the Iast term is a1wa;s positive. it is dear chat the action is mininial i f t h v

field is iantil self-dual

The above equation also can be used as the definition of the instanton because

one can show directly that the selfduality condition impIies the equation of

motion D, F, = O [631. In addition to that. one can show t har the enPr,q-

momentiim tensor tanishes for the seif-dual fiel&. [n particiilar. th^ srlf-tliiiil

fields have zero (Stinkowki) energ- density. Insen the topolo,PicaI charge UT

(1.57) into the action (1.56). FVe End:

Sr' sr- S 2 > - -

!l - !12

For finite-action field configuration. CJT has to be ;ln inttlger. This (.>in h t b sc t~ i

from the fact that the integrand is a total clerivative.

1 fi,, = - 1fjT2 r J, (.i;ei;i: +

Iriserting the piim gaiige -4, = rL'd,I*' iiito tlir ii1)c)vr iytiatioriil. u-r* hrici rti . ir

(JT = ' 1 . Furthermore. if the griuge putetitial falls off rapidlv i i t tlit* spatial

infinity.

which show rhiit fii4d configurations wirti QT t 0 i.i~riritlct ilifftwnt topol~tz1(~;~1

vriciiii.

Iti perturbation theory. one deals with zero-point quantum-mechanical fliic-

tuations of the Y41 fields near one of the minima. say a t -', = O . The non-

linearity of the 131 theory is taken into account as a perturbation and resiilts

in a series in y' where g is the gauge coupling. In this approach. m e is iip-

parently rnissing a possibility for the svstern to tunnel t o ;inothtlr niininiii. ~ i i y

';, = 1. The tiinneling is a typical non-perttirbative effect. and the instantun

has a direct relation to tunneling effects.

By analogy to the double-ive11 potential problem. the instanton solution can

be written using ~ h e -t Hooft symbol

and &,, is ciefinert by chiinging the 5ign OF thcl Iast rrvo t3q~i ; i t i rm. For t'iirr h * r

properties of thta * t Hooft synihol. sre [TOI. !\*v look for a di t r ion id t l i t ~ -,inIl-tlii;il

eqtiation in the follotving form analogous to thtl clauhI~-~v~II potctirial protdtm:

where (P bas to satisfy the boiindary condition (P O as r - x. hserting

the above qiiation in (1.38). LW End the Beliivin-Polyakov-Schivartz-Tyltpkiti

instanton solution [Mi . . .

rvhere p is an arbitran; parameter characterizhg

(1.65)

the size of the insranton.

The above ~qtiation describes the field of the instanton in the singular Loretitt

gauge. the singularity of -4, i i c r' = O is a salige artikct (~ior phusic.;ii\. iirirl

the field strength and topoIogicaI charge { i ~ n s i r s artB srnooth iit origin. Thr

anti-instanton solution (with topological charge Q = -1) can be obtained by

replacing iji, with q;,

Inserting the above instanton solution into (1.56). we find S = (Sn2iQ7l )/#.

mhich is scaie invariant and independent of instanton size p. The above ac-

tion implies that in the singular gauge the tiinneling probabiiity is Pt,,,,,I,,, =

- 8 ~ ' jg?

For the instanton solution f1.65), one can obviously shift the position of

the instanton to an arbitrary spacetime without changing the action. \Cé

can also rocate the instanton field in colour space by const~mt unitary mat r i r~s

r. For SC(';c) the rotation is characrerimi bu rhe tiital .\;'. - 1 rtirinitwl

of generators minus the (.L*, - 2)' (the nimber of qcrieriirurs rvhit-11 i 1 0 t i i j r

effrct the left upper 2 x 2 corner where the standard SC (2) instanrori i 1.65 l is

residing). chat is 4,Yc - 5 . These degrees of freedom are called the Instanton

orientation in colour space. In total there are 1731.

so-calIed coilective coordinates tlescribing r hc field of instancoris.

1.3.7 Fermions in Instanton Fields

in the prpsence of light fermions. one can determine the fermion propagator in

terrns of the eigenhnctions of the Dirar r)pPr;tlt)r ( 1. D c . , = A r : , I .

The cruciaI property of the instanton. originally discovered by - c Hooft [ÏO]. is

that the Dirac operator has a zero mode i P w o ( x ) = 0 in the instanton field.

For an instanton in the singular gauge. the zero-mode mave f~inctiori is

mhere O"" = cam j& is a constant spinor in which the SC(?) coIoirr iridex cr

is coupIed to the spin indev m = 1,2.

The important properties of instantons are that each instanton (anti-instanton 1

has only one zero-mode. Every instant on contribirt es ontx m i t c o t hi) topo1oq~-

cal charge QT = 1 and has a left-handed zero-mode. n-hile anti-instantons hav\-r

QT = -1 and contribute a right-hand zero mode.

Sow Let us show how the tunneling between topologically different config-

urations (described semiclassically by instantons) explains the a?cid anomal!

( i . e . the L i ( l ) piizzle). The k a 1 anomaly can be described by (1.43)

where is the kiavotir clegree of freetlorn. The rinoniaI\. nléaris that loop (lia-

grarns inmlving externa1 wctor and axial-vector ciment can not be regulated

in siich a way chat ail the currents remained conserveci. The changr in u i a l

chargv is

In ternis of the fermion propagator. AQ5 is giwn bu:

For civeru non-zero A. :;ri is an c~igenv~ctor w t h rigtm-altit. - A . Biit rhis

means that UA and - ~ L : x are orthogonal. so only zero modes can contribute to

(1.71):

Integrating the anorndy equation (1.45). we End the LQ, is related to the

topoiogicai charge, and the unconserveci axial charge is proportional to the

instanton number (nL) mtnus the mti-instanton nurnber (nR).

Thus with the help of instantons. Ive can understand the large r( rnass

quantitativel-

1.3.8 QCD Vacuum St ruc tu re in Instanton Background

The nexc natural questioo is the QCD vacuum structure in the instanton t~iick-

ground. In order to assess the importance of instantons in the QCD vacuum.

two crucial values have to be determined: the instanton density and the in-

stanton size.

The tunneling rate (the instanton tlençity. which tells how ofttw tilt, t i ~ r i ~ i t +

irig Pvents happensl is deterrnined hy the standard gliion c-on<i~nsirr~ f'roni rhtb

QCD surn niles. The determination foIlows che idea t h the non-perturbatiw

fields contributing to ~ h e gIuon condensaces are dominated bu the (weakly in-

teracting) instantons. the condensate is simply proportional tn t hc insranrori

cimsity as seen in (1.56). hecause ewry singlp iiistiintoti c.cmrtihiirt~s i i hnitt-

am»unt .

If we assume chat the average separations of iristantons are largw than

their average size (ta be justified belon-). tve can estimate the total action of

the ensemble as the suni of indhjdual action

Inserting the gluon condensate's estimated d u e . we get the instanton clmsity

n and the average separation distance R.

n 2 L Jm?: R 2 1

= l frn 2OOMeC'

Sext to the instanton density. the typical instanton size is the niost irri-

portant parameter characterizing the instanton rnsrmhlti I t tie CJCD v;i(-iiii~ii

strrictirrs). If instantons are too large (instiiritoris artb iieavilv ovi~rliippcvt 1 . i r

ciocsn't niake any sense to speak of indivic1ii;ri tiinneling rvetits. and stmir*litssr-

cal ~heo- is inapplicable. If instantons are too srnall. then seniichssical theorv

is gooil biit the tiirineling rate is strongly suppressed. First bu Shifrnan (1979)

[34] and then by Shuryak ( 1951) [67]. based on the above cstimatcd iristatitoti

cfensity d u e (il = 1 f m ) . the awragt1 sin. t i f iiisr;iritoris is givtm ln-

with an iipper limit of 1/(500 .\le\-).

M'ith the above two values (n = 1 fm--l. pc = 1/3 fm). ive conclutIe that

the QCD vacuum is a dilute liquici instanton ensemble model with tht> friIlowinr

r:hariirterist ics [ 6 i ] :

1. Since the instanton size is significrintly smali chan the typical separation

R between instantons. p/R - 1/3. the Lxcuurn is fairlv dilute.

2. The fields inside the instanton are very strong. G,, » .\'. this means

that the semiclassical approsimation is valid. and the tvpicd iii~tion ih

large So = Sr2 /g ' (p ) - 10 - 15 » 1 [63!.

3. [ustantons retain their indiciduality and are not destroyed by interartions

[63i.

The dilute liquid instanton mode1 is strongly supported by lattice simulations

[T4, 751 and the agreement with the esperirnental resiilts is quite irnpressiv~.

The average instanton size determines the structure of chiral syninititr.v

breaking. in particular the value of the quark condensate and the pion nirbs.

Yom 1.e show how instantons solve the third puzzle we poseti hefore.

In the presence of Iight quarks (quark condensate problem) one has to deal

with the rntich more complicated problem of quark-induced interactions and

has to be a colIective effect involving infinitely manv instantons. This effect is

most eaçily understood in the contest of the mean-field methori [63. 67. 7131.

The light quark propagator in the mean-field approsimation can he niost

easily cierivetl from the effective partition function [67]. The quark gains a

effective mass . I l (p ) which depends on the momentüm p. At zero momentum.

the effective quark mass (the constituent mass) is given by .\[(O1 2 3.30 M P L *

!-IO. 13\. If IVP replare the current mass by the effective nmss in the quark

propagator . the contribution of a single iristanton is givcn bu l iIi(0). For ii

finitc riensity of instantons. ive cspect:

Buth numbers 51(0) and (iiq) appear to be close to their phenonienologiral

values [63].

In the QCD ground state. chiral synmetry is broken. The order parameter

is the above quark condensate. The presence of a quark condençate irnplies

chat quarks can propagate over long distance. Let us eq la in this in a Iittle bit

more detail. ?\%en instantons interact through fermion exchanges. zero nioti~s

can become delocaiized. forming a collective quark condensate. .A crticl~ pict tlrr

of quark motion in the vacuum can then be formulateci as follo~vs: instantons

act as a potential well. in which Light quarks can form ground States (zero

modes). If instantons forrn an interacting liquid, quarks c m travel over Large

distances by hopping frorn one instanton to another. sirnilar to electrons in a

conductor. Just as the conductivity is determined bu the densitv of s t a t ~ s near

the Fermi surface. the quark condensate is given by che density of eigensriitt~s t i f

the Dirac operacor near zero virtuality. If the distribution of instantoris in tlit*

QCD vacuum is siifficientiy random. there is a nonzero clensity of rigenvaliit+i

near zero. and chiral symrnet- is broken.

The quantum numbcrs of the zero modes produce ver? sperific. ~.orrrlaticitls

between quarks. First. since there is esactly one wro rriork prxr Havoiir. i1ii;irh

with rlifferent fiavours can [rave1 togethtir. biit rliiarks ivirh r t i v s;iriicn tl;iviiiii.

cannot. Firrthermore. si~ice zrro rnodcs haw a tliffrrrrit rhiralitv I I d t - i i ; r i i r l t v I

for instancons. right-hand for ami-instantons). cpiarks Hip tJirir r:tiiraIitt iis th*>.

pass through an instanton. This is very important phenomenological ly because

it distinguishes instanton effects froni pertiirbativ~ interactions. in ~vhich thcl

chiraIity of a massless quark does not change. It also inipli~s that quarks (-an

on[- be eschanged between instantons of opposite charge . 1631. .

It is amusing that the physics of the spontarieous brcaking of diiriil svrii-

met- resembles the so called Mott-Anderson condiictivity in tlisortlererl soli(1-

state systems. Imagine random impurities (atoms) spread over a sample with

[inite density. such that each atom has a locrilized bound srate for an rI~rtron.

Due ta the overlap of these localized clectron states bbelonging to individii;tl

atoms. the levels are sp1it into a band. and the electrons berorne dcli~ixlizetl.

Ieading CO conductivity of the sample. In our case the Iocalized quark zero

modes of individuaI instantons randomly spread over the volume get delocal-

ized due to their overlap, which means chiral symmetry breaking 1731.

In summary Ive find that instantons influence the correlation function as si-

multameous scattering of quarks and anti-quarks on the sanie instiintou. leiiding

to certain effective quark interactions. These interactions are strnrigly rlrpc~ii-

dent on the quark-iintiquark quantum nt~mbers: the! are strong and i~rtriirtiw

in the scalar and eçpecially in the pseudoscalar and axial scalar channels. iirirf

rather meak in the vector and tensor channels [76. ïÏ].

1.3.9 Single-instanton Approximation

Another point of importance is the single-instanton approsiniatiun : i6 ' . Thcl

main idea is that if the distanc~ - y is small cornparrd with the cypi<.;il

instanton separation R. we expect that the contribution froni the instaritori

I = f. closest to the points x and y wi11 dominates over al1 others. One can

rlistinguish this method frorn the diluw liqiiid ripproxirnation. [ i l r hi. c l i l i i r t l -

liquill approsixriatiriri. wtB sysrrrri,tticëiIlv f y ~ i r i t l r h t b cwrt~lirrioii f i i i i i . r i ~ m i r i

terms of the one. two. three. etc. instantons c.ontrihiitiotis. Iii r ho p i ' r w r i c v I 11'

Iight fermions [for 'if > 1). howver. tiiis tntbthod is iiseless b ( u i i s t ~ r t i t w is rio

zero-mode contribution to chirality-violating operators frorn an! finitr ntrrnbi*r

of instantons [63I.

For the propagator in the zerernode zone. this irnpiies.

where the m* is the mecan-field estimate effective quark m a s .

-4s a remit. the propagator in the single-instanton approximation looks like

the zero-mode propagator of single instanton. but for a particle wïth the effec-

tive m a s mg. By explicit calculation. we find the scaiar and pseirdoscaIar [ ri.(!.

the r and q') correlators receive zero-mode contributions. and the contribution

is comparable to the OPE contributions [;TI. This is one of the muons ~ I i y

instantons effects are inchded in chis thesis.

In s u m m q . one Ends the folIowing important one-instanton effects:

1. The interaction is present for scaIar and pseudoscalar correlators. hiit is

absent in the vector and asiril charinels.

2. The sign of the corrections k opposite for scalar and pseirdosca1;tr chan-

nels. but the magnitude of the corrections is same.

3. One-instanton corrections have opposite signs for isospin-one ( fo case)

and isospin-zero (a,, case) correlators.

.\Il thtlse statemencs agree ivith phenornenologic;d obscn-acicm~~. Thcw iti-

stantuns help us su lv~ the first prizzle ive pus4 More. Bcrxiisc rlit* opposirtb-

sign contributions of singleinstanton to rht- a and q' rhannels. IVP ran iintier-

stand the huge m a s spIitting betiveen the tivo Goldstone bosons which have

the idemica1 OPE sum rules expression to al1 order. Due to the opposite con-

tributions of single-instanton CO the isoçpin-one and zero correlation fiinctions.

ne can especc a significant mass sptit between Io and an mtr -sons.

In surnrnary. instancons. one of the rnosc non-trivial properties of non-

-4beiian gauge theories such as QCD. are the topologically non-trivial solu-

tions of Euclidean field equations. describing the tunneling processes between

difFerent classicai EeId configurations. The most important propertv of instan-

tons is the esistence of onIy one zero-mode solution of the eqiratiun of motiim

for each instanton. Roughly speaking. the phenomenological values of strorig

interactions strongl'; suggest chat the QCD tacuum resernbles the -.instanton

liquid" model. the reason is based on two crucial \ahes of instantons: the

tvpical instanton size (p, = & Meb-) is about 1/3 of the average instan-

ton separation ( R = & MeV). The instanton effects enter the correlation

functions in two ways: first is the light quark propagators get dressed in the

iristanton background. and throiigh mean-field rhrory. ontL c m ( ~ l ~ i i l i i r t ~ r t i c&

effective quark rnaçs. Secondly. in the single-instanton approsiniatiori. the iri-

teraction tietween light fermion and instantons strongiy affects tht* currt4ators

in the scalar and pseudoscalar channels while instantons effects are strotigly

siippressed in the vector and asial-vector chünnels. Instanton effects help to

wlve t h IV( l ) puzzle mtf quantitatively espliiin the spontantmsly hrt~;iking of

ibhiral syninletry. The sitiglc-iristatirt,ti ii~~~)rosir~iittit)ii providt.~ rhv orilv kr i~ i tvr i

distinct ion between isospin-one and isospin-zero corrclirt iori h i c t iws. 1vhic.11

heips to understand the rnaçs splitting between the resonaiices in thesc? trvo

channels.

Chapter 2

Motivation

2.1 General Properties of Mesons

QCD is ciirrently accepted as the correct theop- for strong interactions and the

quark mode1 is one of the foiindations of QCD. -4ccording to the quark rnodel.

there are a totd of sis quarks ( i . e . si^ quark fiavounf. namely d. IL . S. c. h. t .

Each qiark crarries spin . baryon ntimber 1 !3. ilnt1 ot her ;ttlclitiorial iI i i i t t i t r i t i i

tliln~btw. C~nwrition;dly. t w h qiiark is ;issigtitd pmit ivv pi irit~ ;iti(l i d i ~ i i t t -

quark negative parity. -4rnong the sis (parlis. the first thrtae I 1 1 . d . .3) ;in. of

tiltimiittl imprirtiinc~. h ~ c n t i s ~ thpu haw r~ la r iwly sniall rnitswl; ancl thtts t h ev

are the cotriponents of almost al1 hadrons fotind so far. Table 2.1 lists sortie

important properties of the three light quarks:

i electric charge 2/3 1 -1/:3 ; -1;3 , I current (bare) mass

isospin (2-component)lz 1 112 1 - i O i strangeness ( 0 1 0 : -1 !

The citrrent niilss means the bitre rtiass tvtiich apprars in rhv (.)C'il 1.a-

gangian ( 1.17). and its relativ~ly small m a s compared rvith hadronir nwss

scales is a good approximation of QCD chiral s~mrnetc-. In contrasr to the

nearIy masstess current mas, the constituent m a s or the s w d e d effective

m a s results from spontaneous chiral synmetry breaking. The dunamica1 m a s

depends on the momentum involved. the above Iisted d u e s are obtained at

zero momentum ivhich can be roughly estirnard as ORP hidf of r t i t ~ p rrltw)r~

m a s or one third of the nucleon mas . Instantons are not thr mly appro;ir.fi

to generate the dynamitai quark mus. The constituent m a s c m also be gm-

erated from the OPE [-!O].

The three light quarks can be identified with three states in the fundamental

representation of Sl'(3)F bu ,qorip theory (assuming m,, = m,, = rri, 1 . H;i(lroiis

ilria th tdor t* ( .onstr~trt~d ;LS Havoiir SC I 3 I c ; t ; i r c ~ I ' t i t w i . i i t i r i t ~ . r l o t i > I ) t ~ l w t w i

group theory iitirl the Liitdronic sptsc.t,riim wrP f i p t c i t w r v t d 1)'. C;t4l-\l;i1iri .iS

who noticed chat when hadrons (of the same spin and parity) iwre plotted

according to two nearlj- conserved quantum numbers certain distinct patterns

appear. These two quantum numbers are the 2-cornponent of isospin IZ and

hypercharge 2 - = B - S. rrhere B is baryon number (1/3 for earh light quark!

:tnd 5' is tht* citrangcncss. One of the higgtlst tririniptis of t i l t * r11i;irk r r i o t l i 4 i .

that altiiost al1 hüdruns car) he i4assititd acrortlingl! into nit~sori m i l Ii;irvoii

families i30].

SIesons are bound states of a quark and an ami-quark ( q ~ ' ) (the Havour

of q and 4' may be different because the strong interaction Hamiitonians are

Havour independent). A11 the established banons are apparently 3-q1iiiik ( y q q i

states. Hadrons are abtained by forniirig borincl s t i i r~s of t t iv f i i t i ( l i i t i i t ~ ~ i t ; i l

representations of SC(3). 2.e. quarks and antiquarks. .\Il hadronic rriultiplrts

are identical to singret. octet 3 and decuplet IO irreducible representations of

SU(3). Oniy 8 and 10 are observed for baryons because of restrictions imposed

by spin (3 @ 3 @ 3 = 1 @ 8 @ 8 @ 10). and on@ 1 and 8 representations are

observed for mesons (3 @ 3 = S $1). For the purpose of this thesis. nte are

mainly concemed with mesons.

If the orbital angular momentum of the qq' state is L. then the parity P

is (- 1) A state qq of quark and its own antiquark is also an eigenstaee

of charge conjugationt with C = ( - L ) ~ " , where the spin S 3 O or 1. Let J

denote the total angular momentum. The L = O states are the pseudoscalars

(JP = O-) and vectors (JP = 1-). the scalar mesons are L = 1 stiittls wirh

( . J P = O-). States with the same I. J P iud ;iclditivtb qiiantiirri riii~titwrs ( . i i I i

niis. tf the? are eigenstates of charge conjugation. the! niust tiavr thr si~rtic'

vaIue of C. Thus the I = O member of ground-state pseudoscalar octet mises

mith the corresponding pseudoscaIar singlet to produce the q and q'? which

appear as members of a nonet (the famous Goldstone nonet). Table 2.2 is a

short summary of the pseudoscaliir. vector and scalar channels relcvarit to t his

r htlsis:

Tihle 2.2: Suggrstrtl ipj qiiark-niodrl iwiignnients for must o f t t i t b ktiowii

mesons. Some assignments. especially for the 0" multiplets are controver- siid

2.2 Scalar Mesons

For the three meson families listed in Table 2.2. the pseudoscalar and vertor

mesons ( the O-' and 1-- c-hannelsl dre WH rsrabIish~ti witk rBsrensivrb ciatii.

Careful (tata analysis and theoretiral calc.tiIarii~n indic-iirr rhctr thr proptwitls

of these mesons. such as m a s and Jecay rvidth. can be well predictrd hv

QCD and they can be cIassified as the bound states of a quark and an anti-

quark. In contrast. the nature of the scalar mesons is a long-standing puzzle.

both from the expenmental identification and fiom the t heoretical calculation.

In principle. QCD should predict the hadron spectrum. But the nurnher of

scaIar resonances found in the energ'. regime helow 2 Ge\' esceecls the n i i n i -

ber of states that conventional quark mode1 ( t .e. the qtj construction OF s d a r

mesons) with the çame quantum numbers can accornmodate. According to the

Particle Data Group [ZO]. there elcist four w e l established mesoris in i = O

channel: f o ( l O O - 1200) a ve- broad structure mith width of 600-1000 1IeC'.

TiibItt '1.3: PDG ~s t i r i in td rnus ami widr h valitru: of scaIar tricsoiis

In addition to the conventional light quark qq' interpretation of scalar me-

son. there are some other candidates for the scalar meson's structure: gluebaIIs.

hybrid mesons and muhiquark-states (qqq<l or qq - y@).

The esistence of gliion self-inreraccinns in QCD sirggelits t h a t idttitiotial

houncl statrs of gIiiuris (ctie strcalltd gliictidl GG.C;C;'Gi ?;lioiilri 111 pritii.iplt.

esist. Theoretical calculations based on lattiw sauge rheory iuid QCD sirni

rules agree that Lightest ghebaii should be a scdar resonance ( JPC = O--)

with a mas of 1600 k 130;11eL' jî9, 80. SL].

Hybrid mesons are qrj states combined with a ghonic escitation [qpG). and

hence esotic (non-qq) quantum numbers are ailorner1 in tiiis itiri*rpri:rarion. Th.

lightest bbr id mesons are expected in the 1.500-2000 >le\- n i i w rarigit in H r r s

tube models [82].

llulti-quark states might exist as a color-singlet configuration of four or

more quarks. -4 four-quark state can be either baglike (qq@) [53]. or a mesonic

molecule originating from a meson-meson bound sate (qq - q$ [S4. The

rnoIeciile mode1 is more important phenomenologicallv. and is acwpred t)v ;i

large population of the high-energy physics hmily. Belotv rhr 2 Gv\- t~ia';:,

regime. there are several bound-state thresholtls chat might be interprtwd ;ih

the scalar meson resonances with molecule structure. The I<T threshold is

about 990 MeV. therefore the f0(980) and ~ ~ ( 9 8 0 ) rnight be interpreted as

Kr molecules. The UIL: and p p thresholds are around 1500 SIeC'. thiis the

f0(1300) ancl a(](l430) might be interpretecl as the> wmbinntion of p f ~ and ;,

niol tdes .

1. fo(9S0) and ~ ~ ( 9 8 0 ) are qq quark mode1 admixtures with K r . r17 ancl r i n

cmtintiiim states [8.5!. The ntwness of the f,](SSO) anri rr , , i 980) t o a 1ïK

threshold tias leacl to a widt4y ir i rcqxrt ; i t i r i r i i)f fir-rtioltwilv -rrtii.riirin

184. 361. as opposetl to light rpj-rcsorian~t~s (liritw r.otribitii~riori~ ~ , t ' v t r

and rid States) [JI!. Howewr. the i~surription rhat thtw Stiittxs iirP Iïlï-

molecules have been subject to recent scrutiny. In particuiar. LIorgari

and Pennington [5ï1 have disputed the K E interpretation of f0(980).

An analysis of TT scattering [SS] is compatible nith h*f;; interpretation

of f0(9SO). but sees ~ ~ ( 9 8 0 ) as ;a dt-namical thmsholtl pffe(-t. i ~ s opp~ci tv f

to a crue resonance state. An e w n more rewnt iinalvsis of ( )P.-\L I h r , i

[89! supports the consistmcy of a rpj int~rpretation of j0(9SO).

2. The conventionai interpretation of ao(L450) and fo(1500) is the quark

mode1 (qQ) tvith instanton induced interaction [901. Others interpreta-

tions of these states are the lowest scalar glueball [Tg. SI] or the bound

states of PP'JI~J [ZOj. On the other hand. Lee and Ckingarten [91! in-

terpret the fo(1500) as a mainly sS state which mixes strongly with the

close-by çcahr gluebal1 which show up as a resonance at 1710 .\[eV.

3. The more controversial state is the a meson (fo(400 - 1200)). because

of its large width (600-1000 MeV). Recent activity 192. 93. 94. 931 in re-

analyzing oid ;r;; and n,V scattering data has lead to the reinstatcnitmt tif

the lowest-lying I = O scalar restiniirictb ttiiit is disriricc from I i i t k f,)i!)SO 1 .

conserv~xiw.oly labeltld by the 1996 and 19% Parriclr Dittii Groiq, -)O'

j'(](-L00 - 1200).

Moreover. the fo(400- 1200) has been wideiy interpreted CO be the a par-

ticle signature of the chiral symmetry breaking anticipateci From ?;ambit

.Jona-Lrisinio (?;.IL) dynamics [96.97]. iind the linear sigma-niciclel i Lo..\/)

spectriim i98. 991. OC equal importance. a clarification of th+) properritks.

or cveri the rsistence of a light a-resonançc is reqiiirerl t r i rlisringiristi

hetween La31 and .\;Ln11 (riorilinear sigma rrioriell ;iltimativrs for tif-

fective theories OF low-energ?; hadron physics [100].

Iri this thesis. we employ QCD Laplace surn-rules as a technique particii-

lady n-el1 suited to relate the field theorrrical rmtent of QCD r o Iinwsr-lviric

resonance properties. Single-instanton effects arr ais0 incliitlrvf ik; titv.t3ssi~rv to

distingiiish the 1 = 0 and I = 1 channt4s' tot;il QCD prrciirhatiw ;incl nori-

perturbative effects. .4nother unique feature of this research is the stucty uf

finite width effects (beyond the traditionai SV2 delta function resonance) in

the hadconic mode1 which is necessa- in dealing with the possibIy large width

of the a meson (the finite nidth effects will be cliscussed Iateri. Kith a11 thest.

ideas. we try CO answer the following questions:

a For the I = 1 channel. which of the two resonances uo(980) and ao(l450)

cari be identified by our QCD sum-rule method as the lowest-lying reso-

nances with the Light qq interpretation? In particular, can we rule out al1

but exotic interpretations of ao(950), and does there esist suni-riile s u p

port for the recently confirmeti rr , , ( l-l.50) hrtiiig the lowvsr-lrirq r p j c h j oc . r

in t his channel?

For the 1 = O channel. which of the four caridiclat~s Tf,l(JOO - 1200).

fo(980). fo(13'70) and f0(1500)] can be incerpreted as the lowest-lying

resonance coupling to the light quark current (1.33)? If the existence of

a a particle ( fo(lOO-1200)) is consistent with out QCD sum-riile analusis.

is such a o meson a broad object. or a relativrly narrower strori!: (.oiipling

dilaton [101]+?

Ti, conclutle. the iiriderstanding of the 0'- scdar rnwons is miportant sinw

the determination of the lowest-lying qq interpretation states in I = O and I = 1

channels is of genuine value as a test of our present understanding of QCD.

particuiar its non-pertiirbativ~ rontmt. Such latvrsr-lying s r m s . n h v n fir\r

conipared with QCD via stim riiles methotls w r e necrssiirily foiirirl ro h i *

degencrate. as purely perturbative and QCD-vacuum condensate contribiitions

to scaiar-ciirrent correlation functions cannot distitigiiish betwen 1 = 0 and

I = 1 channels. However. the instanton component of the QCD vacuum is

known to distinguish between 1 = O and I = 1 scaiar states as discussed earlier

(also see [102]). Such an instanton effect is quite evident in the pseiitloscalar

charinel's large ;r - rl m a s spIit. Similady. the existence of instanton soliitinns

in QCD necessarily imposes the theoretical expectation that a similnr split

occurs betn-een 1 = O and 1 = 1 qq scalar resonance states. with the I = O

state substantiaiiy iighter than its I = 1 isopartner. In this regard. scalar

meson spectroscopy is a genuine test of QCD.

Chapter 3

Field Theory Calculations and Hadronic Models

3.1 Perturbative Contributions

In this chapter. we ca1culrite the prrciirbatiw rspansroti of r h ~ two-pmir w ; . k i r

correlation function to two-loop level. ICé follorv the F~ynmitn's patti integrid

technique as ciiscussed in Chapter L. First ww huilci the hi1 Lagrangian clerisitv

L for QCD with the notation Cnt for the interaction Lagrangian density. then

with the help of generating functional 2 (1.6), we can expand the generating

functional to first order in CZn1 and second order in L,,, respectivelv. rorre-

spotiding to ttie one-loup and two-luop perturba~ivr rontrihiitions. -4 rriricxl

fcature of this calculation is the presence of the riontrivial f rat iw of thcl rtlrior-

rnalizrition constant Z.ir which occiirs in the renorrnalization of the rornposite

operator in the scaiar currents.

FolloMng the previous definition. the Lagrangian density for QCD is:

mhere D, = d, - lg$-4,, - y is the strong wiipling cwristant. and the

field strength is defined bu Fi, = du--î$ - i),-4; 7- Y ~ ~ ~ . - $ - ~ ~ -

Vie can rewrite the Lagrangian density as a sum of the free Lagrangian

(Lo), and the interaction part of the Lagrangian (&) which represents the

deviations from free fields. I'l-iiting the total Lagrangian in rilrms of barc fi4rl.

before reoormalization [26\,

The interaction Lagrangian Ltnt is defined by.

This interaction Lagrangian contains a quark and an antiquark fields (C. L.)

and a gluon field .-$(r). -411 interactions between quark and gluoris ran b t k

clerived froni t tiis Lagrangian.

Based on this observation. the interaction Lagrangian will be employd in

our perturbative calculation. The generating fiinctiond. frorii ivhic.li dl G r t v r i ' -

func+t~ans cari ht) g~nrratrrl . rakes t.tw follnwinq forrri.

Consider the perturbative expansion of the tm-point correlation function T[(q')

wit h the external currents:

which have the same quantum numbers ( JPC = O+) as the scalar mesons. the

current represents the isoscalar channel and JI=l the isovector chatmel.

En the SU(2) Iirnit! we set m, = md = m. The above currents correspond to

renormalization group invariant composite operators j26!.

where O. R stand for bare and renormalized qt~antities. The correlator of these

currents is deiîned as.

In the above espression. the fermion fields 11 ancl r i in rht . c+iirrmts i 1 .TI\ ;iri(l

in the interaction Lagrangian Ln, are bare fields. under renormalization of th^

i~bove correliition fiinction. a mas renormalization constant Z,,, oc.r.rir tm rtir

Ioop order me are working (mo = mZ,,).

Siniilar to che ttlcbnique utilized iri the exparision d thth ~rtwrat in: tiinr-

tional Z. me can espand the e' J'Ltnf"'' in the polver of strong rciiipling viirisriint

y. Froni (3.3) WP know L,,, - y. thertlfore w u n dso rspiintl the tlxpontmitil

part in terms of correspondingly.

The Iowest ordcr term in g is 1 (unit) corresponding to the onc loop pcrturba-

tive contribution and the '2-& term the two-loop perturbative contrihiition.

3.1.1 One-loop Contributions with Mass Renormaiization Effects

Let us start from the one-loop caiculation.

To ohtaineri a fiilly renormaiized correlation function. ive nwd ta r a b into a(--

coirnt the rnass renormalization constant to ensure a final RG invitriitnr rrsiilt

Because only up to two-loop level is nceded in oiir calïularion. n-ti irh i.orrrb-

spontls to orcfer 3 2 5. t herefore. for r htl m a s renormalizatiori i.on3tant.

only ( 3) orcter shoulci be kept. which corresponds to the orle-loop rclnornia1iz;t- -

tion mas constant Z,, [Z6]. In the .US scheme where $ = - In(-lr) - -.E

( i E = 0.37721Z . . . is the Euler-Gamma constant ). the r~ntmn;~l i~; i r ion ni;w

constant is.

here &(R) = 4/3 For SL(3).

Equation (3.9) is easy to reduce to products of free field propagators through

the IVick theorem 1%. 4Sj. In conventional condensate-free perturbation the-

on-, normal-ordered terms corne from the iiïck expansion annihilating the

vacuum. Keep in mind chat onIy quarks of the same Bavor can forrxi norrnai-

ordered terms. we have

Yote both I = 0.1 channels give the itientiçal c~xpression in the lowvr ~wl(&i.

perturbative calctilation. Recall the iree quark propagacor.

where A. B are flavour indices and a, 3 colour indices.

In the above LVick espansion, only conriected diagrams contribtite to the

perturbativc correiator. so therc is no combination such as (0IT;d.r l I L [ 11: ;O!

or ( O I T [ ~ ( 0 ) u ( 0 ) ] 1 0 ) which ,nive the annihilation diagranis.

Sote idso that \\'içk combination is iil\rays h m quiirk point to alitttlltiiïk.

thus the iibove Il'ick espansion in (3.1 1) takes the following forrn.

Perturbative ccalulation implies that the rnornentwn involv~d is large. su i r

is safe to assume p' >> mf, ancl ire can omit the quark m a s in rqiiatiiin 13. L:3 1

to leading order in the chiral limit. Summing over three color indices. and

remembering both u quark and d quark give the same contributiori jS"(p) =

S d ( p ) ] . we have.

Figure :3.1: One-loop contributions to the scalar correlation function l l ( r 1 2 ) . The injected ciment carries a momentuni of The IL quark and d quark giw the same contribution.

The above eqiiation corresponds to Figure 3.1.

Employing ~ h e trace identity Tr!-.P-@"! = Df". wt.e have.

From here. the tlimension-regulation scheme as discussed in the previous chap-

ters should he employed. CCé change our inteqation dimension from 4 to

D = 4 - Zr. .At the end of our calculation. wc sct the hnrk r o wro ihiic-k r o

four-dimension reality). Iri order to kctbp rtie (.oi~pIir~g c.rmsr;irit id t l i t * cx~rrr*l;~-

tion function climensionless. an artificial rnass scale u is iritrodiiiwl:

Csing the identity p - q = ( ( p + q)? - - q2)/ .2. the above ecpatiori Iw(-onim.

The resiilting integrals are tabulated in [16!. ive have:

where ï is the gamma fuaction. The (&)' terni can be rewritten S.

- The ï fiinctions c m be expanded in terrns of I/< and i in -11.5 schrrrie [%]. hi

the final form (3.20). only the constant and i cerms are kept (the reason nill

be tlisciissccl later in this chapter).

Inserting this result back to (3.18). we obtain the one-loop perturbative

result .

Yow taking the mass renorrnalization conscant Z, into accoilnt. which rneans

rn; + m2Z:$, we obtain the one-hop plus the renormalization effect result.

Yotice here we just keep the terms proportional to ln(-q'/u2), because they

are the oniy relevant t e m mhich Nil1 survive after the Borel-transformation.

Other possible t e m in the above equation such as terms proporrion to 0'

~vill disappear after Borel-transformation. t h~ twms propim ionid r t ) \\.il1 1 , t t

ilbsorbed in the procedure of regulation ( ?jrhPnitll.

3.1.2 Two-Ioop Contributions

For the tao-Ioop effects. ive keep the tno Ln, terrns in the expansion of e t ' s C*n''

in (3.5). rvorking in the SC(?) limit (mu = rnd = ml.

Again ive find rrp to two-loop level. both the I = O and I = I channels give

identical results in perturbative calciilation. Applying Wick's t heorern and keep

in niind that al1 quark and antiquark cnnibinatinn is always pointcd finrn quark

to antiquark. In the SC(?) limit. u quark and d quark givr equal contributioris

Figure 3.2: Twdoop perturbative contributions to the scalar correlation func- tion II($). The injected current carries a rnomentum of q". Only connected diagrams contribute.

Sote. only the combinations which give connected diagranis contribute to the

perturbative contribution. The above expression corresponds to the Feynman

diaganis shown in Figure 3.2.

In Figure 3.2 . Diagram ( 1) is eqrihxIfm ro Diagram i '2) n-hirh EIVV i c t f w r i i . ; i i

results. The gluon propagator is.

wtiere < is the gauge parameter. The final result is independent uf gauge

parameter. which is an arbitra- number chosen to make oitr calcuiations easier.

[n our case. Feynman Gauge [< = 1) cari simplify the crilciilations and is

t~rtiploytd in chis ttwiis [anocli~r wtiir~i~mlv 11m1 q.i~i,qv ~ - h o i ( ~ I > r l i t b L , t t ~ , i ~ ~ ~ i

Gaugtl i v t i tw < = 1) 1 .

By direct calculation.

where 1 is a 3 x 3 color-space rinit riiiitris- ;lti(l 1itmw TriAr',\'" = Ifi 2S .

Progranis Iiaw b t m tvrittcn iti REDI'CE ro (Io r ti~stl brrr tiv rwi-loop ;il-

ciiiations (see -4ppentlis for details). Kecp in mind only the ttxrnis proportional

to l n ( - q 2 / u 2 ) are kept since the? have a non-zero Borel transformation resiilt.

The first tno diagrams give the total contribution ll(i')l,2.

The third diagram giws the contribution l l ( ~ ' ) ~

(3.31 l

Combiriing the above tno espressions. tve ohttrin the two-loiil) c.orirrit)iit lori. [ri

the folIorving expression. ive set Q' = - q 2 .

3.1.3 Perturbative Contribution to R,, Laplace Sum Rules

The one-loop contribution witti rrriornialization effects plirs rhr rrv<At)iip I . I ~ I I -

tribution gives the total perturbacive contribixtion to the correlation frinction

to two-loop order.

The ~:urrelation function is noir used in t h Liip1ac.e wm-riilc rsprt*smti t u

calculate &.

The Laplaw siirti-rulc R,, is t~rripIoyci tIiit3 to its proprrtv of ~ti1i;inctwtw t tu,

Iowsr-lying resonance's contribution and esponential suppress~on of rht. pocl-

sible higher e n e r a resonance States beIom continuum threshold. r . e . it is least

sensitive tto continuum contributions. -4s me discussed before. the n(s) is an-

alytic function of s escept dong the positive real s avis where its imiiginan

parts are discontinuoiis. In the complex Q' plane. jiist ahuw and brlon- tlir - ais. Q' takw the following form.

Therefore. above the real s avis ln(Q2) = h ( s ) t ia, below the s avis

ln(Q2) = ln(s ) - 2;;. Thw the discontinuity cornes frorn.

The imaginaru part of il(.s) is defincd consisrrrit wi th clispersion-rt11iitim c . o r i -

vrnt ions:

Insert (3.36). (3.37) and (3.33) into (3.38). we have

inserr this espression into 13.3-1). we liant

The abovt' QCD srirn-riile devehped frmn the renorrnaliziition groilp invitri-

ant currents (1.33) satisfies a simple renorrnaIizaticin group cqtiation [1031.

Cpon renormalization group improvernent. the mass rn and the coupling con-

stant a are nom become r dependent. That is theu becorning rrlnning mass

m = m(r) and ninning coupling constant ri = nlr). The ahove RG rqitiiciori

u t . o r 1 - requires a two-loop corrections to the running mass and riinning roiipliri,

stant as given in 11.29) and (1.31 1. 1-pon RGimprovcmenr. the nariiriil c.hoic.tl

for the scale variable y' in the sum-mle is u' = 1/; j103].

Csing 11-THEhLATIC.4. the fol lo ing integrals can be calculated:

IYhere the -,E is the Euler-Gammaconstant and El is the first order Esponcnriitl-

Integral fiinîtioti. Ctilizinq rlw abt)vil rtlsidri.. n.tl li;ivt~ r t i t t t i ~ i ; t l fi)rrri I ~ t ' ! ~ t + r r i i r -

t~ative c o n t r i h i u n s to Ro.

3.2 $CD Condensate Contributions

In thi section we calculate the non-perturbative condensate effects. The OPE

correlator tükes the following generai form.

wticre the C,\Q2'j are the c-nuinber fuiictirins which can bc. ciilcillartvl ici

perturbacive methods, the (O,),,, are condensates which contain c he non-

pertiirbatiw information. Both of the C,, and 0, st!ries are crtinr.;ittd t o ttiakt.

;in? riili.iiliition possible. For the C',, \VP jrilr kwp r tw lotwst p~*~~iirhiirii-c~ cmtcii-.

For (0, j,.,,., iis discussetl before. only hve of t l i t m ti;ivtl sigdiiïiiit t%-r .+ t III ( . t r-

reI;itiim firnc.rions. they are {qp}. \rvGG'I. <rkGGG!. ,@Gy) i d &,'IrI qrjf i(/ j . I I I

t lie liglit-quark cases (onlu IL and d quarks are invalwd]. ;&G'G) and !(jnCq:

have negligible effects on twcqoint correlation functions since the miscd con-

tiensate (qaGq) and the three-gluon condensate (oGGG) only critrr at cirdcr

rnir12 :42. 431. l n the scalar channels. rhv twicr i l i t i r ionc; i r f r t i t w rin r x , t i i i i * r i -

siires ;ire givcn ;LS 126. 4.). 131.

In the light quark cases m < IOMrl'. terms proportional tu are negligible

compared nith other terms because the typicai energt. scaie involved in QCD

is I Therefore. the coatributions of these two condensates are

safely negligibk compared with other uncertainties involveà in QCD sum-rule

caIcu1ations.

Therefore. in 1 = 0.1 rases. onlv rhree crindensattls (iriy). ~ I I C ; ( ; \ . ;mi

($'iq$'2q)) neetl to be taken inco acmiint in t h e c.alcirlarion.

3.2.1 Quark Condensate Contributions

Let us start from the lowest non-trivial dimension (three) quark condensatrl

(@). tt is the order parameter ofspontaneoris &riil syrnrnetry br~iikiiig. Srart

again from the two-point correlation function ancl tvork in S L ( 2 ) iiniir i r n ! , =

md = rn).

ivhere fl is the true vacuum state d t r r rhc! spont i tneot i~l~ svninitltrt. hrwking.

ancl the crirrents are ciefinetl as in ( L.:3.3). Si~w \.\.l'ic*k's t htwrvni i3 impl i ivc~ 1

with a cliiark and nritiqiiark pair rmcontracterl.

Xote the quark condensate has the same contributions in the I = O and

I = 1 channels. The above equation can be representeci graphicaliy bv Fiqiiri.

3.3.

The tmo diagrams obviousb give the same contributions. Take the frcie

Figure .3.3: Quark condensate contribritions in tht. srdar ihnntl ls . Bv .\-ni-

met-. the two diagrams give the same contribririons.

quark propagator S(p) defined in (3.12) into account and note chat bath [i

quark and d quark contribute identically. 1.e have.

Lcc us consider the quark condensace {Qlcj~t(r)qj;(~)!R) wher~ 1. B :trp t l i ~

fiavorir indices. the cl . 3 are color indices and i. j are spin indices. \VIX espitrid

the quark condensate expression in powers of r p with the understanding thiit

on\? scalars can give a non-zero vacuum expectation value.

For the tirst ttlrni iri (3.32). WP can i t c f in~ ;I srrilar rimstant E siic-h r t w .

Summing over color (a. 3) and spin (1.1) indircs.

For the nest term in the right-hand side of (3.52). however. its cht. der in t iw

is an ordinary one. the vacuum expeçtation value is not gairge inurianr ~vhilv

vie want to relate the terms in (3.32) to the gauge invariant çon~it~risncrs. [r

is cont-enient to work out all the non-perturbacive contribirtions in t hti fisecf

point gauge or the coordinate gauge [26, 1041,

it has been s h o w chat both the fised-point giiuge arid i.ov;iriarit gi i i iy

(ciiscussed in Chapter II yield the sanie gauge invariant gluon wrirlerisate rtb-

sults j1031. In other words. the gauge dependence (whether tixed-poirit gaiigt.

or cowriant gauge is eniployed in the calcuiation) at an interniediace stage

of a crihl i i t i~r i does not iiffect a genuiriely gaiigo inrariant qiianricv 3iit.h A,

the gliiori condensate. This argument Icads us to c-hocai. t ht. rriusr nirilrimit3tir

gauge. fisetl-point gaiige. in thcl gluon corirttinsiitr c-;iicwlatiori.

Thc tised-point gauge allows the %auge invariant contribi~ticins of t tir ttbrni.s

in ( 3 . 2 ) to be estracteci by repIacing o rd ina l derivacives with covariant ones.

in this gauge =IL(x) can be exprased clirectly in terms of the field strength

crnsor F,, . namelu.

It is easy to show that i i series of orriiiiary dorivatiws [:an he repl;w~ri t)lr

symmetrized, covariant derivatives in the fised-point gauge emiuated at s = O

[106. 1071:

Sow we can replace the normal derivative 4, by covariant derivative D, in

the ked-point gauge.

using the equation of motion.

The following relation is obtained by summing of color and spin indices in the

four dimensional world.

trisert E ancl F's expressions back to(3.32).

Insert chis cspression hack to (3.31) and eriiplqy the ;~c:tiri~qrit~ p r t w i r t v l 111

['26T.

where m = rn, = rn,~ is the quark The free quark propagator is S(p) = +=. mas . In the light quark cases where the energy involved in the perturbative

procediire is much larger than the quark-s m a s (5 *< 1). the quark rnass in

the cienorninacor jiist contributes trr I ? ( r r i ) c-orrwtions.

Summing over color indices, we have,

The term Tr(S(p)] gives O(m) corrections. Note that the t.race in the

second term Tr[S(P) : f ] still requires a color summation. and we write

m(4d as (md.

Csing the idericitu i-61

3.2.2 Gluon Condensate Contributions

The nest lowest-dimension conrlensate is the dimension four gluori c.on&matr

(n,GG). ~vtiere the ri, is tiic strong cotipling wnstarit. Still w r k i ~ i g 111 r i i t ,

fked-poinc gaiige (coordinate gaiige) rw.4, = O. tvr cspand tilt' c.r~rrvl;itroii

function to the nest leading order as n-e did for the two-loop pertirrbative

calcuiation.

n-here the interaction Lagangian cakes the following form.

Etriplig-ing \\*ick's t h e o r ~ m iind working in ttitl SV( '21 liriiirs i in,, = iiift =

rn) . I~aving jiist one pair of .4,,.4, iwn-cwntrac.ttvl. \VI) 0 1 ) r ; ~ i r i r l i t , t ; A t nvri~r:

expression.

Taking into accoiint the quark propagator (3.12) and noting the coloirr aigebra

givcs IWO.

Xoce again the gluon-condensate concributions have the same results for botli

I = O and I = l channels. The above expression corresponds tu the diagrams

in Figure 3.4.

where the strong coupling constant a, is equal g'/4ïr.

In order to check the consistency of out calculations. ive ernplov ;inochrr

technique. the plane wave rnethod. to calculatr clir gl11on-<.iiritlt~nsi1tt~ rorirri-

butions again. The plane-mave rnethod begins with i\'ilson's optmcor idrwirv

1361. . after forming a vacuum especcation value of the operator reIation. appro- .

priately chosen d a t e s can be sandwiched CO single out the contribution of a.

given operator.

By esplicit cdculation of the gluon condensate contribution with thp h ~ l p

of the progam REDVCE. ive End the rwo nicrhorls giw identrc;il rtwilt.s. tvti1r.h

is what tvtn tlspt~ct.

3.2.3 Dimension-six Condensate Contributions

The dimension six condensate (Os) (the four-quark condensate) can be ob-

caincci by expanding the two-point conelation function (.3.26). Ieiiving trvo

quark-antiquark pairs uncontracted. Ibeping in minci only cmrin~c.iril cli;itr;r;trris

mtist he tnken into itccoiint. ive finci.

where -4. BI C. D are flavour indices. eu = tl. cd = -1 and zero for other

Its diagrammatic representations are giveri iti Figiire (3.; 1 . Ttic (liagrmis i n

Figure 3.5 where quarks and antiquarks are exchanged are the same but the

sense of the fermionic lines are reversed.

Fignre 3.5: Dirnension-six four-quark condensate contribi~tions ro t h r t~n+pi i i r i r

rorrelation function in the scalar channels.

In order to deal rvith the four quark condensace ive are going t o assime tilt'

vacuum saturation hypothesis [31]. 1.e.

where the j,., is the rxiiirm saturation factor. C-tilizing thr sarrirt rt~.h~iicliicbs

irsed before. the four-qiiark condensate wntribiition to rhtx n(r12\ is givt~n i ~ , 111

[?6!.

3.2.4 QCD Condensate Contributions to &-Borelization

Acldirig ~qiiation (i3.66. 3.16. 3.W) together, iw ohtain r h t ~ QC'D rnnclf~nsart~

contributions to II(Q2) (62" -q").

-4s ive ciis~iisseti beforç.. the QCD Liipl;i(.fb siim-riik R,,(r. .s,,j ih iistd in oiir

analpis insteatl of H(Q2). Hence a cievice is neetled to transter the ll(Q2)

espression to Ro (r. so), the device we refer to is the Borelization and it consists

in applying the FoIlowing operator B to II(Q2).

(-Q2)" B.,p = lim .;.- (n - l)!

Q--r (&)

mhere Q2/n JI2. Two important Borelization identities are.

From the above equation, we 6nd chat the advancages of Borelization are its

propenies of exponentid suppression of large s in the form of l/(s + Q2) . and

the factoria1 suppression of higher order (l/Q? terms.

The QCD condensates part of n(Q2) and RU(r..so) are relattd by r h v clis-

persion relation,

By definition (3.34): we fiud the follotring rehtion.

where L/.II' = r is used. Inserting (13.81) into the above equation. IVP obtain

the QCD condemate contributions to RO.

3.3 Instant on Contributions

CCé have discussed the instanton effects in Chapter 1 and found that it corre-

sponds to the effects of tunneIing from one local vacuum minimum to another

minimum with different topologicai numberç. These non-pertiirbative efftws

are totally ignored in the normal pertiirbatiw rrirchanisni of QCD. I n r t i t b in-

s a n t o n liquirl rnociel. the Light quark propagators get ilressptl in the insrn~iron

h d q r o i i n d and ohtain r he t l ~ n a m i c d tl rfftftlrt ii-P) mass m. . 3. . IV(. iilso eni-

phasizeri chat sirigle-instanton effects are non-negligible in cornparison with the

OPE contributions [:II. Single-instanton corrections to the tw-point corre-

lation ftinctions have opposite signs for isospin onp (1 = 1) ;ind iscispin ;rtaro

( 1 = 0) scalar channels. Sow WC perform an i~splicit t~all.irlaticiii ftir i = il. 1

chhantiels and illi~stratc chi? non-negligiblti rrintribiition to thc cwrr~sponditig

crm~lator . and hcncr illiistratc ttiv itripcirtiiriw of incliiriing insrmrun t4-f~c.r~ iti

t his research.

SOW let 11s start from the two-point correlation function using the ciirrents

defineci in ( 1.33).

Sow employing \S'ick's theorem. and noting thar rliie to insranton rtfrrrs. nor

only connected diagrams contribute. but the annihilation diagrams also con-

tribute ji6. ;]. intuitively. we can think the single-instanton contributes to

the tm-point function as shom in Figure 3.6..

Figure 3.6: Singleinstanton effects in the scaiar correlation function resulting in both of the connectecl and annihilated diagrams rontributing to fS.88l.

LVick's theorem takes the following form for the four-fermion ( r ) interaction

in the instanton background:

where SF is the quark propagator defined in (3.12).

Keep in mind that only quark and antiquark with the same flavoiir can

be contracted. ive tinri the following form of the b ï ck contraction for rt i t* iiri-

Havoiired ciirrent coniponents:

For the flavoured ciirrent components.

Put these results back to (3.89). and work in the SU(2) limits (m, = md =

m. S,=Sd=S\ .

Please note the 7 sign in the above equation, this is the first time we find

a distinction between contributions to the I = 0.1 scalar current correlation

functions (1.33). T h i rnechanism d l be respansible for the breaking of m a s

degeneracy between the I = 0.1 scalar channeis.

As discussed before. for the cases involving light-quarks in the instanton

background. the .t Hooft zero-mode solution is aliwys rrnployed. Ttrr ztw-

niode light quark gains an effective mass m. and the zeremode fermion p r o p

agator is [log!.

which retiiins only the zero-mode. In the singtilar gaiige. the zero-mode ferniion

solution takes the following form j ï O j '

where \i= is the coIour-spin niatris. I denotes the instanton anri ami-insranmi

c:ontributions ITO. 1091.

The quantity o(r) is defined as.

where L- E %.-(-Yc) is the colour orientation rnatris of the inçtanton. Iritegrit-

tion over the instanron orientation as discussed in (1.66). ive have the following

gauge averaged expression for a single instanton and ant i-instanton [IO?. Log].

where CIL. a-. .& are Dirac indices. L'sing the zero-trio& solution of the

ferniions in the iristiinron backgroiincl. rhe light qitark propagiirnr (.an hrl w i r -

t txn ris.

Now ive cari investigate the properties of the correiation functions of I = 0. I

channels as ciefined in (r3.93). Stan from the first term in thc right-tiand siclr

of (3.93).

where the trace of the four colour-spin matrices follows from tqi~ation 13.100 1 .

Employicg the fotIon-ing identities.

In ctie ahove calçulation. a siinimatiuri of r h iihitiï;il r x i n r r i t ~ i ~ r ~ c m o f wiglt~

instanton and anti-instanton. rvhich gives a factor of two. bas hbeen i~i ip~uvtvl .

For the second term in (3.93).

where the trace of colour-spin matrices folIows from [ 3.99) and the srininiat ion

of instanton and anti-instanton contributions is included. The relations

Tr[& = 4u. b arid Tri-.; 4 fl = O are employed.

From the above t w equations ( 3.106 I and ( :3 .LO-j j . 1-c (ïiri criiiclitcli~ r Liiit.

Inserting this resdt back to (3.93): ive have

To this end. w have proved that the instanton contributes oppositely tn isospin

one and isospin zero channels with the same magnitude.

Yote a n important property of the above equation is that effectively only the

annihilation parts of the two-point function contribute to the single instanton

rorrclator.

It is interesting to illustrate the tlifferent roles the instantun a~irl pr1rtiirt);i-

tive QCD p l q e d in the two-point function nlx. ! J I .

In pertiirbatiw QCD iipproach. only the tirst tcrm in the right-hnrid sitle of

above equation contributes. while the other terms tanish identically due to

their annihilation properties. In the single instanton backgroiind approarh. i ~ s

stvn frrorn (3.107). the tirst t i m i in rtitl right-harid sidc i.4 c*;incdt*(I i)v o r i t l t ~ t '

the reniaining ternis. Therefore. only one of the reniainmg iuiri~liiliitir~~i ttBrrri:,

gives the instancon contribution but with an opposice sign for spin ontt iirirl

spin zero channels. In this way. the instanton effects are resporrsible for thv

rriass splitting between the lowest-lying resonances in I = 0.1 channels.

Sow let us calculate the instanton effects in 1 = O chanriels as ;in t~s;irtiplt~.

where the result from (3.105) is used.

and xo is the instanton's position. Additionallu. as discussed in Chapter 1. an

integral over the instanton density n J p ) = n,d(p - p, 1. p, = 1/(600 .\1f \ - t rriii.sr

be p~rfornietl. The value of r r , is given ;L\ 71, = II 3 x 11) - ' G r I ' \ t r w i t i c . l i i c l i ,

(3.109) into che above equittion.

Taking (3.9G) into account. ive ha1.e

,sing the D-dimension identity [26j.

Utilizing the foiiowing results 1261,

and rising the following identity [L 101.

where .IL is the first order Bessel function and Ir'-, is the modified Bessel

function.

Inserting these results back to (3.112). ive obtain the single instanton contri-

bution to the spin one scalar channel's correlation function.

Now we want to calculate the instanton contribution to the Laplace surn rule

Rd+

-4s ive discussed previousl--, for al1 practical QCD calculations. work is d x + s

done in the Euclidean domain (Q' = -q') away from positive real q2 axis. In

the Euclidean domain. Q2 = tezfx above and below the real q2 avis respective-.

C'sing the folloiving ideutities [lll!.

whew the H;" and If:'! are tirsr ordcr Hankel fiincticins.

whcre 1; is the first orcler CVeber function (Bessel function). h i c e chnt f i ., = 7

-,I fi,. iind = \/ t p - : p,. 50 [ha[ nit 1 ;iho<i. r t i ~ r ~ i i l q' iui? r i i l i i - r l i t t f i ~ t ~ ~ .

Belom the real q' asis. n(t) takes the form as

Siibtracting the above two eqiiations ancl takitig chta iniagi~iiiry piirr.

Inserting the expression for Imi l ( t ) back into fi!,,. we hitw thr instiintoii t w -

tribution to the Laplace sum mie &(T, sa) [U21

in conjunction Nith the relation (qq) = -2 between the insranton drnsity

and the quark condensa~e [34. 1081. iw ohtain the following rclaticin.

tnserting this relation back to (3.128). we obtain the final form of the I = O

channel instanton contribution to the Laplace sum-rule &.

where the light quark m a s m(r) denotes the running mass ffunction of 7 ) .

The I = 1 chamel's single-instanton contribution bas the same magnitude but

with an opposite sign.

f t is worth noting that they are the unique instanton correlation results we

obtained [112, 1131.

Prior to our research, the single instanton e f f ~ t s empioyed in the literature

took the following form (in the I = O channel as an example) iï6. 103. Il-L?.

.> 9 3p- r n - ( r 1 ,.,- R f ; J t ( ~ t sO) =

1 6n2T"

where Ka and KI are the zero order and the first order ~Iotlificcl Bcssel fiin(.-

tions. Ttiis forrn of inst;inton c f f~c t s (3.133) on rhc cwrelaiion fiiric.tii,ri r . ;~r i

be iinrlerstoorl iis iin inteqation. rrsiilting ir i rtie following cyii . tbi~iori i t i . 1li.r.

il-il.

tvhere the PLJt(r) has the same espression as in (3.1'31). LCi. can vtd,, c t i ( ~

conclusion by iising the identity [LLO!.

Taking the asymptotiç limits in Eq. (3.131) (so -+ x). ive tind that orir

resiilt 13.131) is equivalent to th^ wir i~ly i i s ~ d ont1 (3.133) T h onlv rliffrriwi*

betwen th es^ t w espressions is t h iriregr;il rii~iges iisr~rf: t h c h 1 1 1 r v ~ l ï d rarigv

is froni 0 to x in (3.151) and the range is frorn O co SI) in (3.1331. 11è will

discuss these two results in Chapter 4 and conclude that our result is the correct

understanding of single-instanton effects in the continuum contributions.

3.4 Hadronic Models

In this thesis. ive employ the Laplace surn rule Ro to investigare the Iorwst-

lying resonance's properties such as the c:orrcsptiriiliiig i.oiitiriiiurr1 oriser .s , , . t t i t h

m a s .U and de--midth ï in the 1 = 0.1 scalar channels.

M e r the field-theory content of IrnIi(tj has been inserted in the abow equa-

tion. this espression can be related to some hatfronic rnodel. t . e . r l i t a hiicirotiic

spectral function. In this rvay our theoretical resuits can bc. r ~ l a t t d t o cvqwri-

rtienta1 [laca. whkh in c i m ~ i w s our QCD suni rtilcs prpdirtion powr.

The narrorr resonancc approximation has been widely used in the lirt~raci~rc~

following the seminal work of SVZ !34!. In that paper. the spectral ansatz is

clescribed as it delta fiinrtion plus a continiium theta function .

wtiere rr are sornc constants. the delta funcrion denotes ttie contribution of the

lowest-lying resonance and the theta function the contribution from above the

continuum omet so. By implementing this ansatz. the narronr width appros-

i rna td subcontinuum resonances contnbute to the Iight-quark (linpar ronihi-

nation of t r and d quarks) nw-point correiation frinrticin iL. a sum o f ( f i - I r a

funct ions.

The coupling coefficient gr is proportional to mf. Hoivever. the c:onst;int of

proportionaiity is expected to be much larger for qq resonances. 1.e. r~sonant-rs

that coupling directly to the field-theoretical operators in thc scalar ciirrenr

(1.33). than for esotic resonances [Ils]. It is for precisely this reason chat

sum-rule searches for non-qp scalar resonance States. such as KI\' molecules

!SA! or glueballs ;Tg]. iitilize correlation funrtions haseri on iippropriiirv I;Iï o r

gluonic crirrcnts chat coupling direcrlv ro siich hadronic cw)rir;i.

Inserting thtl i i b o ~ e spectral ansatz back inro (3.136). the riiirrcw iipprosi-

niittion tiiitlronic. rnoclt4 content of Laplace siini rules c m htl ohriiirit~d.

ivhere Ro(r ) r-ontains the theoretical content of rhe siini-rriltl. sort^ in r t i t b iit)ovt~

narrow width approximation approach. we iniplicitly iissiinit* cht. i t r i t icy)at t~(l

loral cliiiility herween QCD and hacironic phvsics i i b ~ > v ~ sorrits ;ippropriartdv

chosen continuum thrcshold s > so [ST. 351. As discussed in previous chapters.

the condensates and instanton contributions to I'I(q2) are non-pertiirbative cf-

fects which implies the momentum involved is m a l 1 in the region (1ottiin;itcd

by these eîfeferts. L\ïth properly chosen ..,, ( in iilniosr ;il1 ()CD mrii riiIc~+ ;il)-

plications .i0 is sufficient higher thnn 1 Gr \-'). iill field-t hwr!- cwntrwr3 ;il ~n . is

..,, are neariy purely perturbacive. Ttw ~'o~itiniliini wntriburion ;ilw rtBprfwxtits

the effective summation mer excitations which are too weak to be direccly

observed.

Despite its relative success. especially in the calculation of low-resonanw

properties of vector and axial-I-ector channels. the narrow-r~soniinc~ approsi-

mation is insufficient in dealing n-ith broad lowest-lying resonances becaiise it

totally ignores the width effects of the resonances. For ex am pl^. the a nieson

as quoted from [XI] has a width from 400 to 1200 MeV. Witbout finite Nidth

effects. QCD sum rules provide incornpiete results for the 1 = 0.1 channels.

Thiis the narrow-resonance approsimat ion r ~ q u ires signi firant modi fic;ir ion r c

fit the recliiirements of this research.

For Laplace siim rules. a more quantitative est imrite of resonant-e-\vit l t h

effects could bti a replacement of the rlelta hnction in (3.1:39). which slioiild h~

iinderstood as the narron-width limit of the Breit-Wigner shape. by the Breit-

Wigner peak [SOI. and chen substituting into the Laplace sum-rule definition

(3.34).

~ q , . d ( . ~ - r n ; ) = lim l m i - g r / ( . ~ - rnf - ,rn,rr j j r.-lJ .

Howevcr. the Breit-1C'igner shape has an infinite taiI. and significant portions

of that tail may estend above the continuum threshold so or below the s = O

boundary into Euclidean rnomenta as shown in Figure 3.7.

Figure 3.1: Breit-LVigner shape of resonance width. This Breit-\Vigner peak is centered at J I = IGeC' and with a '20031eC- nidth effect. Jlssirming a continuum threshold occur at so = 2GeLW'. the shaded area of thtl B r ~ i t - Wigner tails (.s > su and s < 0) are truncated in QCD sum ru le rnlciilatic~n~.

Such contributions from the Breit-Wigner mil. whether included or trun-

cated amy. can be substantial for resonances with widths in escrss of 100

Iki' 11131. and can be a source of theoretical ~incertainty in Laplace siini-ritles

anrtlysis of broad stibcontinuum resonances.

Tbis uncertainty may be understood as a limitation on the Laplace sum-

rule methodolog-. itself. particiilarly for channels in which the lowvest-resonance

is broad or niore chan one resonance lies belom the continuum thresholcl. 'ion-

lowest-lying resonances are expected to be less stable. and conscquently. to

be subscantially broader than lowest-lying resonances in niost chirnneIs. For

mample. the 1 = 1 pseudoscalar channel haç a narrow lowest-lying resonance

ii and a substantial broader excitation state resonance il(1300) with a nidth ris

large as 600 S[eV [SOI. For this channel. a modified Breit-CVigner wvidth effect

beyond narrow approximation has been employecl with relatively c.onvinriau,

resuIts '1 13. 1161.

Probably an Pvm more important reason co prevent 11s friiiii t~tiiplo~~iti!:

Breit-lfigner form of ividth-effects. is a conipiitationiil otistacle. If rtitx Breit

-wigner shnpe is used direct15 the following integration occiirs in tIir R, sirm-

ruie calculation.

This integration doesn't have a simple form solution. and we have to ernplo~

this integration's result as an intermediate stage in the l e a s X 2 simulation (to

be discussed later). Therefore. the complicated form of the Breit-CVigner shape

integral solution makes our tasks difficult within an! reasonable coniputation

time given available computing power.

In summary. to overcome the computational di6ciilties of Breit-LYigner

shape in the simulation program and the possibly large portion of the infinite

Breit-Wigner tail above the Laplace sum-mle energy release region. a simplified

modification of Breit-LVigner shape is needed.

Instead of the delta function. the Breit-LVigner shape on the right-hand

side of above equation (3.140) can be espressed as a Riemann sum of i ir i i t iiïcii

pulses Pm, centered at s = rnp [llÏ].

= n-cc l i m ' e / n n - i - f ~ + l r [ s . / " ~ - ' I ] (4.113) J = I j - f j - f

tvhere f is an? arbitrarily chosen constant between O and 1. If one approsimates

the resonance shape via (3.143) bu truncating n to sarne finite riiirriler of pulses.

then the approsimation (unlike the n + x limit) becomes sensitivr to thtb

choice o f f . The value for f may be chosen to ensure that the area iinder the

truncated sum is equai to the area under the true resonance shape.

In this thesis. n = -4 is picked. Putting n = 4 back to the above qiiatiori.

ive obtain an area of n by choosing f = 0.10. An example of the four-pulse

approximation is chosen in Figure 3.8.

The accuracy of the finite-wïdth approximation scheme depends on the

number of the square-puises employed. On the one hand, if the approximation

has too few pulses. for example only one or tmo pulses. the Breit-\Vie;ner shaptx

rvoold not be represented accurately, and the final simulation results \vil1 o\w-

estimate the true width effects. On the other hand. if the approsimation tias

too man! pulses (for esample 6 square-pulses approsimation). and when the

resonance midth is large than 200 MeV. the midest pulses may estend belom

s = O or above the continuum threshold so boundary in physical domain. which

is whac WP trt' to avoid. Therefort.. in t hr scalar nieson c.asrs I wtiert. w i r ti

effccts coiild tle l a rg~r than 200 I\lei'). thc four square-pulse npprosirri;it~m i:,

a siiitable choice.

Compared with the narrow-resonance approsimacion (3.139). WC End t hc

non-zero width r, for the corresponding subcontinuum resonance modifies the

hadronic content of &.

In the four pulse approsimation. the width-efftvt function 11-(M. r. 7 ) r t d s

as,

The 1 function can be derit-ed as.

Figure 3.5: An esample of the 4 square-pulse approsimation to the Brtbit- Mïgner resonance shape obtained bv triincating qtiation( 3.143) ro 11 = 4. ;in11 by choosing f = 0.701 to ensure chat the area iincler the tinir piilstv is r*qrriwlr~ir to the total area under the Breit-LVigner curve. This particular rsarriplv is for a m a s -11 = 650 31el ' and width r = 100 .Clel-.

We c m check explicitly that when r approaches to zero the width effecc becausp

LI'(J1, r. r ) + 1 as expected.

Now equaling the theoretical contents with the wide-effect hadronic ansatz.

rve obtain the final equation we mil1 employed in the I = 0. 1 niesons stiidy.

where gr is the strong coupiing coefficient.

Sow let lis investigate the right-hand side of the above equation. r .e. the

liaclronic content of Laplace sum rules. Laplace siim rides are iiserl here si tic:^

theu enhancc the lowst-lving resonancp's c.ontribiition xnrl siipprcmss r t i t x i.ori-

tinuiim contrihtition. If rhe higher-mass rrsoniinr,t.'s riii~ss 1s i t r r i i i ~ i r 1 r l i t , c.c,ri-

tinitttrri cinset so (31; - s o ) , it will be absorbecl in the ciontiriiiirrri part. I F irs

mius is siibstantial belon- so. the highcr-mass siihcontinuiim rpsonant.tb ivill tw

esponentially suppressed due to the e-"'!' terrn in the intrigracion. Thus. it

is safe to conclude that Lapiace sum rules are most sensitive CO the low-Iying

rPsonanws 134. -! 11.

In chis thcsis. LW look at the possibility o f ni or^ r hari one rt3sori;mw h;idroriii

niodcls uid t i r d ouly one subcontinuum tinite-width resonance mode! is rea-

sonable in the 1 = 0.1 scalar channels (Detailed analysis wtll be provided in

the ne-xt chapter). For now, the spectral mode1 is still written as a sum of r

subcontinuurn resonances.

To summarize. the field-theoreticai contribution €rom QCD to the ftmda-

merital Laplace sum-mle Ro(r. su) is related to the phenonienological rriurlt~I

through the following equation.

The field-theor- content of the QCD sum riiles is.

Chapter 4

$CD Sum-Rule Analysis

4.1 Rationale of Our Approach of $CD Sum-Rule Anal-

ysis

4.1.1 Possible Problerns of Traditional Approaches

Haring given the relation between QCD field theoretical contents and the sper-

tral mode1 via dispersion relations. the next task is the QCD sum-rule analÿsis

to estract the properties of certain hadrons in the channe1 tvith corresponding

qiianttirn numhers.

The iinclerlying principle of field theoretical approaches tu hadronic. phth-

nomenolog'. is the duality açsumption. That is. it is possible to simiiltantl-

ously describe a hadron as quarks propagating in the QCD vacuum. and as a

phenomenological fieId with the appropriate quantum numbers. The dualitÿ

matching must be done consistently and systemically in order to make the re-

sults obtainetl Frorri QCD sum-rulrs acceptable under scrutiny. C'nfortiini~rely.

typical QCD siim-riile analyses of hadron p r o p ~ r t i ~ s in the litoratiirr fidl .;horr

in the follotving m y s [Ils!.

1. Selecting a single d u e for the Borel parameter ( r ) which give "nice"

results. -1 Borel regirne ( i - e . Ï range) shouid be selected to meet the

stability and reliability requirements in QCD sum-rute analysis.

2. Selection of the Borel regime without careful regard to the balance o f

the OPE contributions and the continuum contributions (the trontribii-

tions From above the so th re~hold)~ which can be approximated by pure!?

perturbative caiculation.

3. Fixing of parameters (such as the continuum threshold so) to preferred

vaLues. It becomes apparent that this introduces a strong bias to thil

remaining fit parameters. whirh m q not r~flect the proptw IPS [if QCD

correct Iy.

4. Claiming an accuracy for QCD sum-rule predictions without supporting

calculations. OccasionaIly a %tability analÿsis" [119. 1201 is considered.

in which fit parameters are monitored as single indepenclent variablos

whose value is varied once ac a time. However. such analyses P S P ~ O ~ P i l

rcliitively striall çortier ri€ the piiramccrr spi-iw.

In this research. we ty to design it niore rigoruus approiich CO estriwr chta

hadronic parameters frorn the field theop- content of the Laplace sum-rule.

4.1.2 Traditional Ratio Method

One of the cornmon mechocis is the tltlrivativr srirn-ri il^^ m1 rario rtivrhi~(L ortri-

nated irom S\'Z ;34!. The basic iden is that it is pcissibk to isolnrt~ t h ttitwm's

mas of interest as a function of the Borel s d e r. by taking a ratio of t ht. first

derivat ive surn-rule RI (7. so) wit h the fundamentai surn-rule & (7. sol. -4s ive

discussed before. the first derivative mm-rde is defined as,

This approach has some aesthetic iippeal. and hiu widely becorne rhp 111~thoiL

of çhotce for analyzing QCD sum-des. The continuum threshold is selecteci

to niake the ratio of the two sum d e s RL(~ . so ) /&(roso ) as flat a function of

r as possible. Finau. the meson's m a s is selected from the point at which the

ratio is most Bat or stable. Lnfortrinatel-. this method has some stiortrornings:

1. Inherently, this method carries an ambiguous feature. The s0 is rleter-

rnined by --as Rat as possible" and the niasis by ..as most HM". Thcsc c-ri-

teria imply there is no quantitative. mathematical detemination scheme.

The extraction of information from the sum-rule is judged by humaa feel-

ings but not a rigorous method independent of human interference.

'2. The ratio rnethoci does not check the validity of each inclivi<Iuiil sririi

riilc. It is possible to have individual siim riile that is niit valirl tvhiltb

their ratio is a fiâ~ fiinction of r i1181. In acldi~ion. the ratio rilccIio(1

can not ;iccount for the fact that surn rults (10 not w r k rqiiiilly wll.

The Borel regime mhere a sum rule is valid can \*a- from one surn rule

to another. Ic,loreover, large uncertainties in one surn rule can spoiI the

predictions of another siim riile. For example. in reference Il'> . li . it ( I t u i l c r l

cornparison of fiindamental ( R,) and cierivativr i R I ) m i i riilr:, is i ï~ r r i t~ l

out for the 1) nieson ( 1.e. the vcctor chanriel). and found t h ralitl BorvI

regime for the derivative sum rule shrink nearly to non-existmt hiit rhil

ratio stil1 gives a ffat bottom.

In practice. the preclictions based on the fiindamenta1 sum r u l ~ iiïr mcw

rrliable than chose frorn the (lerivativr wrri rrilt~s R,,. .4a n i n c v i ~ w c ; in r t i v

derivative surn riiles &. the pertrirbatiw cmntribiition of R,, h t ~ i ~ i m i t ~ ~ rriow

rlepentlent on the continrrurn threshold s,, i rrtritmbrr Liiphctl simi r d t - 5 iw

clesigned to be relatively insensitive to so)? and the condensate concributio~l

of R,, becomes more dependent on the higher dimensional condensates' con-

tributions. which are not as well estitnated as lower dimensional condensat~s'

contributions. Therefore. we concliide that R,, hecom~s less and less reliabli~

as n increases [116. Ils]. l\;Ioreover. there is no guarantee the .sO is 11niwrs;d

for al1 n.

Based on the above argument, we ernploy the fundamentai sum-rule &(so, r )

as the theoretical prediction tool rather than the ratio metbod discussed above.

The utilkation ofRo(r. sa) to eatract information €rom the QCD field-theoreti(ïil

calculation is the content of next section.

4.1.3 Our Approach of QCD Sum-rule -4nalysis

In this thesis. a method of quantitatively determining of phenomenological

quantities with and Nithout an uncertainties estimate will be presented. The

following is the outline of otir approach:

2. Developing a least-\;' approach in a multiple parameter set to estract

the desired information from the QCD sum-nrle.

3. St4er.t itig the optimization algorithm to clrtrrmin~ t ht> Imsr \ ' w t iiin r I I I .

; range clcterniiried from the ;~bovçb step.

4. Estracting the uptirnizacion values of the spectral ansatz's p;ir;ittirtt3rs

by minimizing the y' d u e (Best-fit approüch).

5 . Estimating uncertainties associated with the corresponding channels' QCD

siim-rule.

The first step \vil1 help us to obcain a rigorous quantitative regiori in which

the contitiuum and OPE contribution tri11 be baliinced so as not to (l~stroy

the stability of the QCD sum-rule. From the Hiilder estimation. we hope the

first tm possible problems in the traditional analyses can be avoided. In the

second step. we use the least-x2 method to get rid of the wgue nature of

the traditional QCD ratio methods [3f 1 (wilI be discuswd later). In thc final

step. dl sources of uncertainties associated with scalar chânnels are taken inro

account sirnultaneously to search the stability and accuracy of the anal-is

results.

4.2 Least-x2 Methods

4.2.1 Estimate of QCD Sum-rule Inputs

In the QCD sum-rule equation (3.150). several inputs art1 nercitvl ro c.orl(i~i(.r r htb

sum-nile analysis. Since the RG invariant niass riz is absorbeci in the right-tiarid

side of (3.150). the values of the needed input parameters for this QCD suni-riilt*

are the QCD scale -1. the quark condensate (Qq). the gluon condensate (crGG).

the vacuum saturation factor f,,, the dimension SLY condensate (crqqQq) and

the instanton size p,. In assigning the standard values and the uncertaintics

for these parameters. tve follow two generai principles:

1. me woiild like to select the standard values in according with the generallv

accepted values in the literatiire.

2 . we tvould like to select the uncertainties conservatively enough siich chat

the QCD suni-ririe approach can test the validity of oiir present iinder-

standing of QCD and at the same tirnc. a e noiilti likc to wvcr i f i rriii(41

as possible the clifferent parameters' values argiied and rrnplo!rd in riicl

literactire.

Now ive discuss these standard values and their uncertainties respectively.

The QCD scale -1 (normaiization point) is dependent on the renornial-

ization scheme employed. It is generally foiincl that the QCD siim-rtiiv

analysis is insensitive to a change in .\ 1-12. :34 . Throughoiit this ttitnsis.

- ive choose the three-fiavour -11s scheme calue .\ = 300 S k i ' [33. 50!.

For the quark condensate (mqq). we use the standard value in most of the

QCD sum-rule anaiysis obtained from PC+X Gell-Mann-Okubo relation

[122!. (mqq) = -f;rn?/2 Nith the physicai values of na, = 240 NP\* and

= 93 MeV. If we assume mu + md = 11 Me\-. this implies the sran-

dard value (qq) = -250MeLp3. However. QCD sum-rule considerations

of octet baryon magnetic moments [123] prefer a smaller magnitude at

(Qq) = -225~CleL-~ which is about 10% smaller than the standard value

we used.

0 The gluon condensate (aGG) estimated from charmoniiini siim riiles :J-I:

hi& beconie the standard value for Borel sum riiles ( +C;C;\ = 0.0 1-IGt- I ' ' nith a variation about 30% [11S. 1241.

O Relatively Little is known about the magnitude of the dimension-sis. four-

quark condensate. Its value is obtained by employing the so-called var-

itum saturation hypothesis (qqfq) = J,!s(qq)2>- where the f,!, is c t i t ~ viirriiirti

saturation factor. Early arguments placet1 the vnliie of chic; c~oti(lt.risatt~

trithin 10% of the façtorized values [106!. wherc f,, = 1. H o w ~ v ~ r . ochtlr

analyses (:laimecl significant vioiation of factorimion for these four-quark

operators in both meson and nucleon sum rules [120. 124. 125. 1261. In

references [124. 1271, the authors claimed the violation is as large as a

factor of 2. Therefore. in this work. we choose the average value f,., = 1.5

with a variation range 1 c f,, < 2 .

O The single instanton size ,oc used in this research follo~vs the rlassic.al

paper of Shuryak (6Ïj and SL'Z i34 where fi = 1/(600 .\le\-). This

number has been accepted as a standard value for single-instanton effects

in most of the QCD sum-nile analysis. In SVZ's paper. the authors

concluded that it is completely unreasonable to speak abouc an iristanton

with dimension exceeding 1/300 Me\-. Thus ive give a L.j% variaciun

space for the instanton size.

The fint thing needed for this QCD sum-mie analysis is the determination of

the valid r ranges for both I = 0.1 channels. The determination of a \alid - range in a given channe1 has alwqç been a sotlrce of iincertaintv iti QCD siini

riile analyses even since its innovation 13-17. - . In that paper. SI'Z argiied . '34:: .

1. the siim rule are espected to be t-alid as long as the power corrections ( thc

higher dimension condensate t ems) do not dominate. and r is bounded

from below that point.

2 . The loir. r hoiinclary is cfetermined such chat the integral ovtlr t t i v imss

secrion is dominater1 by a single resoniirico and rhe c.cmtiniiiini iwtitrihii-

tiens are not too large-

Following the above two starenients. S i - Z roiighly estiniated a valid 7 ratigr

for the p rneson application. From then on. rnost of the QCD stim rtiltb iippli-

cations determine the r range trithout sy-;rpmaticrii arirl rigoroiis txii~tho(1s. tri

conclusion. the determination of the r rqime in the litcratiirt. ;il~viiys fdls i i i r o

the following two categorics:

1. The determination rnethod has no quantitative andysis. The r range is

just roughly estimated such as the SV2 case.

2 . The determination method has some quantitative analusis. but the crite-

rion utiIized lack of solid ground. One example is presented in [ I . 181. . Iti

that paper. the .; range is chosen such that the highest dimension opera-

tor(s) contribute no more than 10% to the QCD side n-hile the continuum

contribution is l e s than about 30% of the total phenomenological side.

The former sets a criterion for the convergence of the OPE (1.e. the high

.; bounday) whiIe the Iatter controls the continuum contribution ( 1 . r .

thc low r boundap-). However. the selection of I O 5 iind 3 ' X is kirid o f

arbitrary and lack a solid theorctical argument.

In order to avoid the above controversy in detennining r regime. we cmploy

a new technique. the QCD sum-mIe inequalities method developed in [118].

This rien- technique is based on the Holder integrai inequdity. which provides

fundamental constraincs on QCD stim rules. These ronstrainci; rtiiisr tw i a r -

istierl if the QCD sum rtiles are to cotisistently describe iritrgr;itrtl pliysitxi

cross-sections. which are related to the integration of imagina- part of twr

point correlation functions. An important feature of the inequality nierhod

is that thesc çonstraints do not require any phenomenological input (such r t s

the 10% of higher-dimension OPE contribution and the 50% of the cxmtintiiirii

wntribiition as statcd in Leinweber's ivorki. r h i v f o r ~ it providtxs iinivcmid

qiiarititiitivti cotistraints or1 any QCD jiini riiles.

L-sing rhis techniqiie. non-trivial inforniation conscrnining the corititiui~ai

thrcshuld (do) and sum-rule energv scale r will be obtained. This inforniation

then provides insight into the issues conceming the continuum hypothesis and

the energy range in which the sum-rules are reliable. To our purpose here. our

main concem is to obtain the r range suitable for our Laplace sum rules;. Theri

~ve use the continuum ttireshold so obtained from QCD suni-nile anaiysis to

compare tt-ith inequality's provided so as a consiscency check.

Holder's inequality for integrals defined over a mesure dp is jl29. 1301.

= 1 and p. q >_ 1. L'ben p=q=l) the HiiIder inequality reduces tu the

well-knom Schwarz inequaiity- ï h e key idea in appiyïng Holder's inequaiity to

sum mles is recognizing that for a typicai correlation hnction H ( q " ) . ImIf(q")

is positive because of its relation to physical cross-sections and can thus serves

as the measure dp = ImT[(t) dt in (4.2).

Returning to (4.2) with dp = IrnII( t ) d t . the Laplace srim r i i k R,, is clt4iritrl

as in ( 1.32)- Set

where a - b = 1 and t L = O. t2 = so. we find

(Il2 ( t ) / ~ d p ) l i p = (A'' e-at"lrnlI(t) d l ) = [&(api. t 1

Put (4.4): (4.3) and (4.6) back to (42) . we obtain the ii$ Holder inequality

relation.

In the above equation. if we set ap;- = r,,, and bqr = Ï,,,. 1/p = i anci

l /q = 1 - d. we have

where r,,, <_ r,,. To analyze the inequaiity (4.8). ive use

- ,,, - r,, = 67 = 0.1 GeI/--2 to perform a local anaIysis i13lI. The inequality

is insensitive to the value of 67 provided that 6r is reasonably smail (in QCD

57 - O.1GeV-:! appears sufficient).

The results of chis inequaiity analysis for both 1 = O and 1 = 1 channels

are ihstraterl in Figure (4.1) and (4.2). corresponding tu specific \-alws of

the condensates and the vacuum saturatian factor f,, used in the progrilni.

In each figure the shaded region represents the admissibie (sO. T ) parirameters

space where the Holder inequality is satisfied.

Referring to the literature [ES. 131). chere are dways tmo common features

chat persist in an incclualit'; zinalysis of valici QCD sum n d ~ s :

1. The existence of a Loiver bound on the continuuni thresholtl .+il.

2. The esistence of a upper boiind oti the wrn ritle energy parameter 7 .

This can be viewed as a test of both the validity of the continuum hypothesis

( i.e. the duality hypothesis above continuum threshold so) and of the upper

bound of r (the Iowest energy r = 1/31'. where .II is the Borel paracricter).

The second feature gives a constraint ati the conderisatr. svries. A s swn t'rotri

Figure (4.3). condensate contributions to the QCD sunl rulc R,](T. si, ) twcortir

more and more important in the high r regon. which denotes the truncated

tiigher-dimension condensates and other negiected or unknown effects (such as

the tacuurn saturation factor) weights in the total sum mle become substantial

[34. LIS]. In the Hdder inequaiitv anaIysis. beyond a certain r valiie in the*

corresponding channei's ( r - a) space- theçr contiensate effects which arcB rior

included in the field content calculation become so large that the integratetl

property of ImIl(t) is broken. therefore. the upper bound Ï parameter indicated

a maximai condensate contribution allomed in the QCD sum rule. From Figure

tau

Figure 4.2: The shaded area represents the region in the so - r parameter space consistent -<th the inequality for the Light quark I = O sum-mie (3.130). The input parameter values f,, = 1.5. .\ = 300hfel.: p = 1/(600MeC*). and 87 = 0.1 Ge\--' are used in the calcuiation. The r &.sis is in Gel--' and .?O

a i s is in Gel -'.

(4.1) and Figure (4.2). we obtain an upper bound of r - 1.7Ge\'-' in the I = O

channel. and - 1.1 GeC'-' in 1 = I channe!.

The Iomer bound T parameter is obtained by the start point h m w h r w

a relatively flat bottom (T - .so) boundary can be observed. In horh I = O. 1

channels. Ï,,, = 0.3 Ge\'-' is obtained. The low r boundary givcs a con-

straint on the continuum contribution (which states above a certain point s ~ .

00 LT al1 contribution to JO e IrniI(s)ds can be calculated by purely perturbacive

QCD (1.e. the duality hypothesis). Below the boundary point Ï,,,,,. che (.on-

riniiiim contribution grows so large that a toc-al variation of ; i I ~ T ) will \ -roI; ir t~

the inequaIity constraints on the integratiori property of Irnll(t) . ttiat is t h *

Ro(r sa) is too sensitive to so d u e below the r,,,. This constraint corresponds

to Si'Z's conclusion that the one-resonance should dominate the integraI over

the cross-section and the continuum should not become too strong.

ive conclude.

O For the I = 1 channel. 0.3GeI'-' < 7 < 1.1 Ge\'-'.

4.2.3 Least-x2 Approach

In this thesis. a procedure based on a overall fit of the Borel-paranieter ( 7 )

dependence of a sum nile's field-theoretiral cwntrnt to tht. dt~petirfrric-r iinrir-i-

pated from resonance properties has been usecf. This niethoci has rir;iirri \r.iilt)

attention in the community and has been successfulIy employed in various

channels [llS]. For esample. properties of the first pion-excitation state il'

have been obtained by fitting the QCD Borel-parameter dependence for the

lowest II'-sensiti~e Laplace sum rule to its corresponding hadronic contents

[116. 1171. in this thesis. we obtain ri iL-minimizing wighted ~ P W sqiimB fit [ i f

G'"'(T. .sO) to the ;-dependence ariticipated from the c-orrespunding h;rrIroriir-

models. In other worcls. ouc fit is generated bu obtaining d u e s for lnr. T,. .,,

and the resonance's coupling factor a, that minimize a least-square fit of QCD

field content with properly chosen weighting function,

mIicre the Mt-tiand sicie Ro(r . .sri j has the following form.

3m' 1 rn' YStn2;r A - ; rnfq) - -- .) (fkqGG\ - -j,..((khjq)+

16 7 2 7

In oiir case. tve denote the y L weighting function as f ( r . SrIecring .Y

evenly distributed points in the allowed r parameter space (Borel scale region]

- ,, . 1 = 1 - - .V. the y' is defined as.

The seIection of weighting function is important in this work. Two criteria

guide the seIecting of the weighting function ~ ( r . so):

1. The weighting function shoiild refl ect the iincertaincies açsociaretl t u t 11

Laplace siim riile &(Ï. sO).

2. The uncertairities in &(r . sO) are not iiniforrnly distrihtitctd throughortc

the Borel regime, therefore. the weighting function should manifest this

property as well.

Yow we investigate the total uncertainty of sum d e &{T. sO). Let us start from

the condensate uncertainties of &. The goss effect comes from the uncertainty

of quark condensate. the gIuon condensate and the foilr-quark condensate.

For now. the vacuum saturation hypothesis is açsumed as esactlv in chtl

four-quark condensace caIçulation. that is (Qqqq) = (ijq)' where j,. = 1 (the

variation of jvf,, nr31 be left in the Monte Car10 simulation). The uncertainty

of the dimension six condensate cornes from the accumu1ation of the qiiark

condensate (@j iincertainty.

Figure (4.3) is the ratio of total conclcnsat~ ~ t f c r r s iivcr R,, i n t l w 1 = I I

chrinnel. Ive tind the çoriclens;ite effwtç bwornr laro;~ on Litpl;ic.r ~tirn rtilil

only aE the high erid of r region mainly because th^ four quark condensate

term contribution in (3.150) (gq)'7 gcom as Ï increases.

FVe can conctude that the uncertainties asociated with the total conden-

sace contribution vd( T. so) 0n1y have significant effects a t high r rqime. Thtl

condensate irncertainties (esccpt the f,, tinr~rtainty) cwrric h m rhr clii;irk i 9 0 r i -

clensatel unçertaincy. the gluon condensate iincertaincy. and t h c h iiriwrr,~itiritb~

from the tinknown truncated higher-dimension condensates. In th^ high - regime. the four-quark condensate h s the dominant effect.

Cncertainties in perturbative part of Laplace sum rule corne from the

higher-loop correction to Ra h o p 2 3) and even more important. wriie frorri

rhe relatiwly large .so oncertainty. The Lap1ac.e srim riiltl is rle.signrti r i ) twr,ic t

infocmacion for the loaest-lying resonance in a giwn channel. 30 IC 1s r~ l~ i r iwlv

insensitive to the continuum t hreshoid so value. Therefore. the uncertainty

associated s i t h so is relatively large. and we conclude the perturbative part

of QCD uncertainty in this thesis work main- cornes €rom the so uncertainty

The distribution of this uncertaine with respect to r is s h o w in Figure (4.4).

We find the uncertainty are large at loir r region and negiigible at the high 7

.- /-' /'-

0.00 O .O 0.5 1 .O 1.5 2.0

tau

Figure 4.3: The ratio of condensate contribution Rg"d(r,so) (3.84) over the total QCD sum-mle &(r.so) (3.150) in the I = O channel. The value s,, = 3.3GeLd is chosen. The y-axis is the percencage value of the ratio & m d ( ~ . ~ ~ ) / & ( i . ~ ~ ) . and the 7 uis is in Gel'-'.

end. Conservatively perturbative uncertainties lie within 30% of total pertur-

bative effect R ~ ' ( Ï . sa ) at the small r range 134. 411. Furtherniorc. wr firici

that the higher-loop corrections can be understootl eqiiiçrilently as a c.hange in

the contiuuum threshold sa [IlTl.

The instanton contributions to R,,(T. sci ) am relatively srnaII in thr valid r

and sa region. The maximum contribution is about 10% of the total sum nile

and locatetl a t the low r regime which the pcrtiirbative tincrrtaintv has definite

dominant effect. Thiis for now, the instanton unwrtainty ran hr safrlv tgnoribct

in the twighting function (We have tester1 the variation of iiistiitirori w v ,,, i r i

thp valitl regirnr. m l foiind the changeci of instanton size hiu nilnor c 4 f t ~ t on

t htx h~st-fit vaIiit1s. in bot h çhannels. the change of n i a s is lctsh t hiiri I50 l I ~ . \ * i .

Its uncertainty effects will be included dong with others in the ,Ilonte Carlo

simulation process.

Concliiding froni the aboïe iincwtainty disciission and the tnm srlwtinq

critcria for c(.sU. 7 ) . tve find thnt the wighting functiori f(.sn. r ) rriiisr htl pro-

portional to the total &(r..sO) uncercainty. which is the siirii of perrirrbativc~

iincertainty ancl condensate iincertainty. That is.

From Figure (4.4) and (4.3). we notice.

must have the shape shown in Figure (4.5).

Lsing the valid Ï range obtained from Hlilder inequality analysis (for I = O

channel. 0.3 Gelm-' < r < l .rGe\ '-'). and utilizing thcl iint:tartiiinrie rst i-

c ( - 5 0 ) - ~ R O ( ~ . J O I - mated above. n-e conclude that - - 0.3 is a coriserwciw 1iippfnr

0.00 0.0 0.5 1 .O 1.5 2.0

tau

Figure 4.4: The picture shows the relative perturbative contribution (3.44) uncertainty with respect to the change of continuum threshoId SB in the I = O channel. The so = 3.3 Gel/' is chosen in consistent with the best-fit t*aIue. The 'r--axis value is ( ~ ~ ~ ( r . 4 - O ) - ~ ~ ~ ( 7 . 3 . 0 ) ) /4pt(~.3.5). and the T - ~ S L S is in Ge\--'

Figure 4.3: Relative uncenainty on the pertiirbative and condensate contrin- tions CO the total QCD sum nile (3.150). S mis stands for t t i ~ - range.

- J&tr.so) 1 =- R O I ~ . s o i - In the valici r range (7 , < 7 < 7') o b t a i n ~ d frnrri rhtb Hii lr i fbr

inequalrty. an iipper limit of the relative unncertainty aln-ays esiss . Thth 7 asis is in Gel--'.

limit in the valid r range. Therefore. n e identify the weighting function C ( Ï . .su)

as

This espression can be eaçily verified chat it rnatrhrs r tir riva i i t+c*riori i.rirt>ri;i

rnentioned above. A s will be shown later. t hc coefficient of 0.3 iri ( 4.141 is jiisr

r i owrall constant in the Ierist-L2 calculütion. the change of this coefficient has

no eRect on the fitting procedures.

4.2.4 Search Algorithm

In this investigation. ive d l try t o rnininiize thr \' 6lric~ion.

The parameter set to be fitted in the one resonance modei 1s (su. (1. .U. r). Orir

ohjectiw here is to tind an optimal algorichni idiic*h wiII rimirn ;t nitriïniirrii

value of k' trIien fitting the field-theor~tiral content 111 th15 tr i i i l t i r l1t i i t~ni l~o11i~I

paranieter space.

In this work. we utilize a modified version of DowuhiII Simplex algorithm

[132] and a derivative Levenberg-blarquardt aigorithm [133) to rninimize the

L2 in (4.13). Cornparing these two rnethods. we find the sirnplex rnethod is su-

perior in converging to the globaI minimum point in a multidimensional spaw.

The adramage of this rnetfiod is that it requires only function evahation. h c

no deriutives involved. The Levenberg-Slarquardt method uses derivatives io

End the steepest descent step in rnt~ltidimensiond space. Sot involving the

gradient of function in the Simples algorithm. we hope our algorithm can eaç-

ily converge to the true global minimum point if several local minima exkt in

the very compIicated multi-dimensional space.

A simples is the geometrical figure consisting. in 'i' dimensions. of S - 1

points and al1 their interconnecting Iine segments. polygonal faces. etc. The

downhill simples method now takes a series steps. most of the steps just moving

the point of simples where the function is larges through the opposite face of

the simplex to a lower point. and after these reflections the simplex pulls itself

down in around the loivest point (the glabd minimum point). In our approach.

the convergence is reached if the difference between the largest value point and

the loivest value point in the simplex is ri. thousand times smaller than thr

average point. The abow downhi11 simplex routine is called .Irnoeba. The key

feature of our algorithm utilized in the QCD sitm-rule anaIysis is to restart

another Amoeba right after one Arnoeba ciaims to find a minimum. while the

initial step 1i.e. the volume of the .V + 1 point simplex) of the second Arnoeba

is properly chosen to give the second simples the ability to compare one locot

minima ivith another if they eEst. For ~'rarnple. as s h o w in Figure (4.6).

where the function has two local minima pl and fi. If the aigorithm starts

from point n. the first Amoeba tvill converge co loca1 minima pl which is not a

global minimum point we are searching for. E3y choosing a proper initiai step

(in this case is the distance from pl to f i ) . tbe second -4rnoeba c m initialize a

simples covering hoth points and eventitaily converge to the gIohal tnininiuni

point p?. If derivacive methods such as Levenberg-lhrquardt algorithm artb

used. the gradient of function couId not tell the differences among several

minima after the program leads to any of the Iocal minimal t' points.

in this research. we chose -V = 20 (divided the valid r range into 20 eveniy

distributed points) to obtain the least-y'. Cn practice. .V = 10 is sufficient to

give a stable estimation.

Figure 4.6: -4 simple twn-tlime~ision ftinctiim with a Local niininiuni poirir pl and a global minimum point p-, where the point s is siipposed ; ~ 5 r t i t b sr;lrririg point.

In order to iIlustrate the local minima phenomena and even more impor-

tantly. to gain a quantitative approximation of the distance arnong Iocal mini-

mal points. ive reduce the rniiltidimensionûl spaw to thr~e-dim~nsiona1 spitc~

Meer recitir.ing the titting parameters to two iri the 1' timctiori. cwm)llr plotc,

c m be used to examine the distribution of Iocal minima. The k' f~inrtiori

tiefined in (4.11) is a function of four variables y' = y2(r. M. 5,. a) . NON-

we fix ï value as an input. then the variable number is reduced to three,

y' = So. a). At local minimum. we have ' 0. and after solving this

expression.

Insert the ahove espression back to (4.1:). and set r = 0.2 GeC'. we have the

contour pIots for I = 0.1 channels as shown in Figures 4.7 and 4.8.

In order to make our QCD mm-riile andvsis consistent. + ~ ~ p t ~ i . i i t l l ~ r t w c11i;tl-

ity byputhcsis of the QCD calculation above a thresholcl valire s+ t w rspiic:itly

set iin important criterion in our program. m2 < .$o. This criteriori will rtile

out certain $-dimension points. which wilI contaminate the validity of Laplace

sum-de.

From Figure 4.7 and Figure 4.8. ive End that in both channels there do exist

several local minima. especiaIly in I = 1 channel. More important. froni t t i ~ s ~

plots. ~ v e c m roughly estimate initial steps for the second .\rnoeh;i roiitinv

in Our program. chis d l ensure the program always converge to the global

minimum point.

ive compare the simpiex program with the derivative Levenberg-Marquardt

program esplicitly with different input parameter values. and find our sirnplex

program a lway converges to the globaI minimum Ieast-y* point whiIe the L w -

Figure 4.7: Contour plot for the I = O channel with the input parameter miues i' = 0.2 Gel.: f,, = 1.5. -1 = 300 Mek: and p = 1/(600MeV).

Figure 4.8: Contour plot for the I = 1 clhanne1 with the input pnr;irnrttar ~ ; i l i i t ~ ~

r = O.2Gel: f,, = 1.5. .\ = 300 .lie\: and p = Ii(600.\~ir\7.

enberg program sometimes Cinds onIy local minima.

4.3 Best-fit Analysis

4.3.1 Best-fit Results for the I = 0 , l Channeis

.ifter obtaining the QCD field theory contents, ttic hadronic mod~l . rhr r;tlitl -

ritrige . the 1' ftinction and the siiitable algorithm. tve can obtain thr paranicttbr

vdiies (m. r. so. a) which give a best fit to the Laplace surn rule Ru (r. si,).

We employ the standard set of input values for QCD sum rule as discussed

before: ( m m ) = - f;rnS/:! where m, = 14031el: f, = 93JIeLP. (aCG) =

0.0-~5Cel-~. ~ r ( ÿ ~ ) ~ = 0.0001YGel*6. .\ = 0.3GeL: f,., = 1.3 and institntun s i z ~

pc = 1/(600 J l e l* ) .

Titil~s 4.1 iind 4.2 list the lowest-resonance's parilmeter valii~s ~LS swn frorri

il light-quark interpretation. The input parameter ridues are chosrn to givth

minimum y' values. Our programs are designecl to converge CO the global

minimum points starting from an? point in the multi-dimensional spaw. st,

th^ output r~sttlts are insetisitiv~ co thv input vnliitls. S~v~rzht4f~ss . a quo(l

giicss of the iripiit iïiliies c m always Save siibstantial cwrnpiiti~tioriiil ririiit.

Table 4.1: Best fit values for the I = O c+h;iniiel (0.3 Gr\'-' < r < ~ . T G F I ' - ? )

Table 4 2 : Best fit values for the I = 1 channel (0..3Ge\'-' < r < 1.1 Ge\--')

input

L 1 , 1 1 I

output 1 2.059 x 10-' 1 4.51 1 1.50 1 0.160 1 0.197 1

lem-y'

The best-fit gives a lowest resonance around 966 MeV in the I = O chanrit4

1 output 1 7.758 x 10-" 3.61

input

sO(GeL") 3.5

sO(GeV2) 4.5

Ieast-x'

0.966

hIIGeV) 1.0

M(GeV) 1.5

0.0731 0.240

a(GeV7 0.07

a (GeV4) 0.15

r (GeC') 0.2

F (Gel.') 0.2

and ri resonance around 1500 Slel- in the I = 1 channel, and both rrsonmtm

have a width around 'LOO Me\.'.

In the I = O channel. because the 100 MeV width we obtained is sub-

stantially narrower than the 600-1200 MeV width for the fo(400 - 1300) [50].

we can conclude that fo(980) is properly interpreted as the lowest-lying light-

quark resonance in the I = O chanriel. suggesting ai rion-yq intcrprtutim

f,][4tIO - 1200).

[il the I = 1 channel, clearly the light-quark resonance intrrprtutiuti r u l ~ s

ciut the (to(9SO) ~ i s the lowest-lying resonance. The data obtained from ~ h e QCD

sum-rule analysis agrees with the interpretation of the ao( l4ÙOl as the lowst-

Lying +y resonnnce . $01. . This supports a exotica interpretation for n0i9SO).

Figiircs ( -1.9) itnd (4.10) illustrate rhts tit of tidronic iiio(lt4 nit t i r 1 1 t h h r -ht

parameter Ï ~ I I I P S IO thta QCD siiiti-riilr Ro( 7 . Y r i ) in thtl I = O. 1 ihtiiirls.

4.3.2 Instanton Effects

iié have emphasized the importance of instanton effects piayed in this research.

They are the only terms distinguishing between the two scalar channels' QCD

fielcl-t hcoretiçal concent expressions (3.130). and çonsequently i.;itise rhv r i i i hh

splic between the two channels' lowesr-lying rt>soriarice riiuses.

The QCD surn-mie is based on the qtiark(g1uon)-hadron duality. Several

authors argued that one of the most important rispects vidating the duality

is the existence of oscilIating (esponentiai contribution) terms absent in the

OPE calculation 13.5. 611, and instanton effects are responsible for the miss-

ing ex-gonential terms in the QCD field content espression [61. 761. Fotiorving

these arguments. we expect that instanton effects should explore the oscilla-

tion featrire of the missing esponentiaI terms. Equation (3.127) does show

that IrniI'""'(t) - tk; (fipc) fi^^). and because Bessel functions 1; and JI

- QCD sum-rule +O) - - - - best-fit result

0.000 0.0 0.5 1 .O 1.5 2.0

tau

Figure 4.9: QCD sum-nile Ro(r.so) from (3.150) in the 1 = O channel c t h e solid cun-e) and the hadronic mode1 with the best-fit parameters from Table 4.1 (dashed curve). The Ï-a-is is in Ge\--' and !-ais is in Ge\-".

- QCD sum-rule (I=1 channel) best-fit result

tau

Figure 4.10: QCD sum-mie &(i .so) from (3.150) in the 1 = 1 channel (the solid cun-e) and the hadronic mode1 with the best-fit parameters from Table 4.2 (dashed curve). The 7-asis is in Gel--' and !-&sis is in Gel-".

-0.004 - - - .. - -. .- - . - . 0 .O 2.0 4.0 6.0 8.0

continuum threshold

Figure 4.11: The oscillating behavioiir of the instariton r k t (3.131) with respect to .SO in the I = O channel. The r value is fised at 1 Gelv-'. The y-a..is is in Gel.-.' and 'c-asis is in Gel.".

are oscillating. thereiore the instanton effects are oscillating as 4 1 . Figures

(4.1 1) and (4.12) do illustrate explicitly that instanton effect has an oscillaring

behariour over the continuum threshold so and r range.

In order to illustrate the instanton effect on the mass split between the two

lowest-resonance in scalar channels. the ioIlowing table tabulates the results

tvith no instanton effect in both chaanels. Ln order to make the results corn-

parable. we choose the same r range ( Q - ~ G ~ E " ~ < r < 1.7 GebT-') as the fo

case.

One set of inputs ive used is the same as the I = O channel best fit's

tau

Figure 4.12: The osciilating bebaviour of instanton effect (3.131) with respect to T in the I = O channel. The so value is L.ed a t 3.5 Ge\" consistent with the best-fit results. The y-axis is in Gel.-' and s-axis is in Ge\'-'.

Table 4.3: 30-instanton effect results (0,3 Gel,'-? < .r < 1.1

input and another set used the I = 1 channel best fit's input. 1C'c fintl horh

1 input i output

inputs converge to a mass around 1.15MeC' without instanton effects. \\-hen

instanton effects are included and the valid r ranges are selected according to

the inequality estimation. a mass split as large as 500:lIel' occurs frorn the

instanton effects (see the results in Table 4.1 and 4.2).

By psplicit ralciiiation. we find instanton effects are relatively large at low

r while negligible at the high r . and instanton effeccs have a large contribution

to the total Laplace surn-rule in the I = 1 ctiannel than in rhe I = O channe1

(.At r = 0.3 Gel'-'. the ratio R:yt/ RFta' - LZ% in the i = 1 dianriel while

R r s L / R V ' - 3% in the I = O channel). Cornpared mith the non-instanton

resiilts (Table 4.3) with the instanton-included results (Table 4.1 and 4.3) in

both channels. WC can conclude chat the instanton effects on splitting niasses

between the two channels rnainly from increasing the lowest-lying I = 1 rhrin-

least-Y' 1 so(Ge\--) 1 S1(Gelm) ; 3.5 1.0

nel's resonance m a s .

a (GeL*-') 1 T(Gel'i 0.07 1 O I

, least-t' input /

Now Re need to mention the instanton effect on the continuum part of the

QCD siim-rule. LVe have found chat several authors have ignored the instanton

effects on the continuum part [76. 10Bt 1141. which is the contribution to the

3.32 x IO-'

r (Gel-) 1 0 2 2.10 1

3.35 1 1.1'1 , 0.072

a (Gel'" 0.15

sO(Ge1-') 4.5

0.196 1

0.083

' lI(Gel') 1.5

1 output / 4.44 x 1 0 - ~ 1 3.70 1.17

&(T. sO) from above sot the rt (r. so) can be defined as.

In references [76: 108. 1141, the continuum contributions emt are calciilated

totally bu perturbation QCD without instanton effect. Thus their instanton

contributions to Ro jr. so) are equivalent to.

- - ,.' F-o: (3, K I - (4.H) 1 6 r 2 ~ ' r n L ( ~ ) [ o r 2 r 1

where I\;, and KI are the zero order and the first order SIutlifiect B m e l fiin(.-

tioris. Corripareci with our instanton espression (3.132 ).

ibë find the difference betmeen the two is a difference of

Jc Irnil'wt(t)e-7tdt. The net effects of missing instanton contributions frorn

above the continuum threshold have a rninor influence on the simulation results

such as mass and so. Severtheless. it is conceptuaI1y incorrect to understand

the instanton effects as in (4.18). The Holder inequdity analysis of QCD stini-

rule with Eq. (4.19) f o m OF instanton effects is shotvn in Figure 4.13. which

verifies our argument graphically.

Compared with Figure (4.2): we Iose one of the physical features of valid

QCD sum rule inequality analysis. a global lower boundary on the continuuni

threshold sg in Figure (4.13). This picture indicates that at low i regirne.

duality starts almost from so - O. a result that is clearly nonphysical. Ctobally.

ive clon't have a continuum threshold.

4.3.3 Width Effects

Som we stiidy the finite width effects on the scalar channels. The width rveight-

ing function CC'[r. SI. r] is obtained via the four pulse approximation l t d ing to

equation (3.146). In order to see the effects of different width ï on the simula-

tion results. w~ tixed the midth r as an input and chariged its values diiririg t'i~<.ti

~it1lll~~lt10ti p r o ( ~ 5 S . t lit4 ~ M l i i b k ~ il1 t h \ ' f ~ l ( T 1 0 1 1 t'hii~lgf' t o t hrlv *II. .\!. I

The followitig t,ahies stiow the sirriii1;ition resiilts.

Tiible 4.4: Lç'idtfi effwts in the I = 0 chantirl. with input p;iriiIiitwr v;il i ic~s

.so = :Li Gel". JI = 1.0 Gel: n = 0.0TGel"'. and r range is (0.3 Gel'-' < r < 1.TGel'-')

input r (Gel.') 1 1 k - j .su (Gel'" ) ..1I(GeiW) / a (Gel") j 0.0 / 2.426 x IO-' :3.0:37 ' 0.834 0.062

Tibles 4.4 and 4.5 indicate that the values of the lowest-lying resonance's

mass and the continuum threshold (.su) in both 1 = 0. I ctiantiels inçrea';~ wch

the width r. In the I = O channel. the results clearly rule out a light and broder a

particle contributing in the lowest-Iying scalar resonance. With zero-width

input. the lowest-possible m a s starts from 850 MeV. when the width effect

increases to 500 MeV. which is slightly comparable with the low cncf of rhe

wirlth estimüted value [rom PDG [XI] (600-1200 '\.Ici'). the Inus grow to or.iBr

1600 MeC' which is substantial larger chan che high end of PDG [.?O] estimateri

a particle's mass (400-1200 SIeV). These data indicate the lowest-lying light-

quark (&) resonance in the I = O channel could not be a cr particle. I t also

shows us that the lowest-lying resonance couId not be j0(1300) which has a

niirrow width Iess chan 100 MeV [30]. because the substantid large wirlth

(400-,500 SIe\7 fur the m u s arourd 1 3 l O Ur\' in out iisd iriilrh ~ ; t t n t i l ; ~ r ~ ~ ~ ~ i .

t h e v e r . from the wiritti effect analysis. WC cannot rule ciiit i l cont ribiicioti of

h)(1370) to the Iowest-lying resonance in the 1 = O channel. The Io( 1370)

rcsonance hiis a pritential co overlap with the high-end width data in the I = O

channel (jo(L3ïO) has a ver? large width from 200 to 500 MeV estimated by

PDG). Coriibiriirig the resirlts from Tables 4.4 and 4.1. ive confirm that oiir

TabIe 4.3: SVidth effects in the I = 1 channel. with input parameter r-alues .SU = 4.: Ge\.-'. -11 = 1.3 GeC: a = 0.13 GeV.'. and r range is (0.3 Gel*-' < r < 1.1 Ge C'-')

Similarly for the 1 = I channel. Table (4.3) indicates that even with zero

width (the Iowest possible mass case). the mas 1390 JIe\- is stiii Far iiiv;i\.

from ~ ~ ( 9 8 0 ) . This further confirms our best-fit conciusion that ~ ~ ( 1 4 . 5 0 ) is

the lowest-lying light-quark resonance as seen from Our QCD sum-rule. In this

sense, ~ ~ ( 9 8 0 ) is unnaturally decoupled from the scalar quark current. mhich

supports a possible KK molecule interpretation for this resonance as suggested

by 186. 114. 1341.

4.3.4 Testing of Hadronic Models

As a final application of the best-fit algorithm. we test t a ious hadronic models

and try to End the one mhich fits the QCD field content best, i.e. the one having

the l e a ~ t - ~ ' value. -4s we have discussed in Chapter 3. there are three hadronic

models in the I = 0.1 channels that should be considered:

1. Orie firiite-width resoniince rnodel.

2 . Orie finite-tvidt h resonance plus a finite-widt h excited resonance rnodel.

3. One finite-width resonance plus a narrow-width resonance model.

Among these candidates. 1 and 2 are the most possible ones. due to the pos-

sible ver? large tviclth of the lowest-lying rcsonance in sraliir rhanrirls ilricl th i l

wide nature of escited state resonances (generally speaking. excicd staces arcX

less stable than ground states). However. Ive test al1 of the three candidates

explicitly in the I = O channel.

The one finite width model is nothing but the best fit results from Table

4.1. while the two finite-width resonance mode1 is defined as

Equating the hadronic model to the QCD sum rule &(r. s0) and foilom the

same f o m of X* function, we have 7 parameters ( S O , U L , I' l ,m~>ap, r2:m2) to

fit. Observe the different candidate resonances existing in the 1 = O channel

[50], tve 6nd the possible two subcontinuum resonances m u t be one around

950 hleC' another one around 1.500 .LIeiP, thus tve use these tmo numbers as our

inicial input griess.

Table 4.6: Two finite-width resonance mode1 wit h same input values

f input E t p u t 1

r~sr~tiançe niotirI is tIit1 split of couplirig crieiiïcicnt rr betwen al üncl CL?.

Table 4.7: Ttvo finite-width resonance mode1 with different input values

Table 4.7 use different mas d u e s as inprit. the output results do not

show two cleariy distinguished resonances below che continuum threshold. One

resonance is centered at 530 MeV mith a IÏS Me\- width largely overlapped

tvith another resonance centered at 1120 MeV mith a 239 MeV nidth. Besides

we observe the least-X* value in this model is the same order of magnitude

as chat from one finite-width model s h o m in Table 4.1. Identification of the

$30 .Lie\' and 1120 MeV as two separate resonances is in con~riidiçtion wirh

rtie results of the leading I = 1 pscudoscalar ciment surn riiltb. for whic.11 ;i

fit including ttvo coatributing subcontiniium resonances (7 and a') leatls ro a

k2 an order of magnitude lower than of a single resonance fit [1161. Thus ive

can safely conclude the two hite-width resonance model gives one finite width

subcontinuum resonarice as output of our program.

For the one finite plus a narrom resonance model. tve write the hadronic

iinsatz as

Iinowing the results of one and two finite-width resonance models. we can

conclude the triclth is around '200 MeV. so rve fis this value w an input in tht.

program. Thiis. this model has tive parameters I sr,. c l , . mi. (1 , . r i l r ) t u fit r l i t ~

QCD sum mie Ro(r . so)

Table 4.8: One finite-njdth plus a narrotv resonance model ~ i t h input width r = 0.2GeV

Comparing the coupling coefficients of the two output resonances c~n te r~c i

at 924 lie\‘ and 1340 MeV respective1. ive End that the narrow higher n i a s

resonance couples CO the Iight quark current ( a - ) 20 times smalIer than that of

the finite width resonance at 935 MeV (uL). Thus. the excited-state contribu-

tion is t o t d y negligible. and we c m conclude only one iinite resonance exists

in this model. AS before. the y' is not substantially reduced compared to the

sin& resonance rnodel.

Sumrnarizing the conciusions €rom the abovc thrce h;idronir rnott~ls. r w i ~ i n

state that the Leading I = O scalar-curr~nt siini-ride is rloniinatcti bv il sitiqlt.

finice-widr h resonance. The same conclusion applies to the I = 1 charinel.

4.4 Uncertainty Estimation: Monte-Carlo Simulations

One of the common shortcomings in traditional QCD sum rule analysis is

claiming a accuracy stacement for QCD sum rule prediction rvithout supporting

calculations. Even with a stability anal~sis. rior al1 pa ran i~ tm ' iincrrriiincirs

are i.onsider~d simdtiineoiisl': These are the problcms ive are rrying r o avoid

in r.his srction.

In this section. me mi11 estimate the uncertaincies of the Ieast-y' analysis

method originating from various uncertainty sources in the QCD parameters

employed in the surn rule calculation. LVe nriil seek the cornbined effects of al1

these uncertainties integrated together. not sirnply individual QCD pararnerer

variations. In this tvay. ire r d 1 learn hotr the iinrerrainties in QCD paranicrm

are mapped into uncertainties in the phenomenological fit parameters.

The standard ra-alues and the associated uncertainties of the free parame-

ters used in this analysis (-1. f,,. (Qq). (aGG). (mjqqq). so and p,) have been

discussed earlier. The iincertainties related CO so (the continuum uncertaincy 1.

(qq) . jnGG). !Û@J@~) are combined togethrr into the wi;liting fttnrtion f i r. -,, t in the \' forn~ulation. rvhich is 30% of toral Laplace sirm ride &(r. - , , I . rhta

vaciiiirn saturation factor /., can vary from 1 to 2 and instanton size has a

15% uncertainty. Figures (4.14) and (4.15) show the combinations of power

law and continuum uncertainties. i.e. 47. sa) = 0-3RO(r. sa). are me11 defined

by the following empincai formula. The &(r. so) is enurnerated at the best-fit

~a lues (-1 = 0.3 GeC: f,, = 1.5).

1. For chc I = O case. the input so = .3..1)GfiC-'. €(Y) = oierp(-1.0301:37r0 -'M \

rvhere n l = 0.01i1-1 Gel *".

2. For the 1 = 1 case, the input so = 4.5 GeV2, E ( T ) = a2 T ezp(-3.ï4~~5~0.5267)

where a2 = 2.1547GeV4.

- ernpirical model weighting function

0.000 0.0 0.5 1 .O 1.5 2.0

tau

Figure 4.14: The weighting function ~ ( r . so) from (4.14) in the I = O channel (the dashed cutve) and the empincal model shown as the soiid cuwe. The u-ais is in Gel-" and x-asïs is in GeC'-'.

The simulation of the uncertainties of the continuum part and contlerisare';

contribution is through the random variation of a, and a? in the r tbov~ r~vo

ernpirical espressions. al and a:! are constrained to regions

so that the simulation does not exceed the error estimation of &(T. s0)-

Yow we can conduct the uncertainty estimation by performing a Monte-

Carlo simulation with random variation in the parameter set (p,. a l . al and

- - - - weighting function

- empirical model

tau

Figure 4.1.5: The weighting function ~ ( r . so) from (4.14) in the I = 1 channel (the dashed curve) and the empirical mode1 shown as the solid curve. The --a+ is in Gel-" and x-asis is in Ge\'-'.

-0.01714Gel-~ < al < 0.OlÏl-l Gel-"

1 < f,, < 2 (4.22I

The criterion i JI2 < sO) discussed in the brst-fit sirnrilation ic; still twiplii~ril

to niake the bionte-Carlo siniulatioti consistent ivith the ciuality hypothes~s.

In practice. 500 point simulations are performed in each channel. howevw.

a 100 point siniulation is sufficient to get a stable uncertainty estimation. The

following figures are the histograms plots of the fitted parameter from the 500

simulaticins in both cases.

Froni Figures (4.16). (4.17). (4.13) and (4.19). ive find the miuses arr niwIy

centered around the best-fit vaiue (mr=o = 1000.CfeC: mlz l = IJIOO.\Iel-)

with a relatively narrow region of uncertainty space in both scdar channeIs.

whtle the d u e s for s0 vary across a reiatively large uncertainty space (in the

I = O channel 3.OGel" < so < 4.5GeV2. in the I = 1 channel 4.OGel" <

s o < 6 3 Ge\*?). This should not be suppressing because Laplace surn niles ;ire

designed to be sensitiw to lowsr-Iying resonance's contribution. and thry $vth

a relatively larger uncertaintv prediction on the continuum threshold rdue.

The 90% confidence level intertal 500-point Monte-Carlo simulation resuits

are presented in Tables 1.9 and 4.10.

For the I = O channel. we ünd a m a s with a central t-aiue 1000 Ilel- and

an average width of 193 'ifet-, ruling out the possibility of lo(13TOj (accordiug

to j5Ol. Io(13Ï0) Lias a mass 1200 + 1300 Me\ - with a 200 -. 500 JIEL- widthr

as a dominant contribution to the lowest-lying light-quark resonance in QCD

MASS (GeV)

Figure 4.16: Histogram for the fo m a s obtained from fits of $00 parameter sets.

continuum threshold

Figure -LIT: Histogram for the JO continuum threshold obtained from fitç of 300 parameter sets. The continuum threshold value is in GeC".

MASS (GeV)

Figure 4-18: Histogrnm for the a0 mass obtained from fits of 300 parameter sets.

continuum threshold

Figure 4.19: Histogram for the a0 continuum threshold obtained frorn fits of 500 parameter sets. The continuum threshold value is in GeV2.

Table -1.9: 'Ionte CarIo simdation resdts for the I = O channe1 (0.3 Gel7-% T < l.TGeV-?)

. . . . . ,

f input ) 3.5 1 1 .O 1 0.07 i 0.2 ; I

j j 3.73 I 0.45 1 1.000 k 0.087 1 0 . 0 ~ 9 I 0 . ~ 3 1 0.19 i 0.14 I

Table 4.10: Monte CarIo simulation results for the I = 1 channel (0.3 GeG--' c r < 1.1 Ge\.'-?)

siim rtile. it is also definitely not a a-Iike particle nhich needs a Iight and

broder structure. and the best fit simdation (TabIe 4.1) rules out a dominant

cmmibirtion of JO( 1-500) as the lowest-lying resonance. Therefore. togcther

v-ith the best-fit r ~s t i l t s . w ronfirm again char the l o w s r - l ~ i r i g rtlsoriarir.tb ir i

the I = O channel is the fo(9S0) wic.h a Light-quark interpretation.

For the I = 1 channe!. withouc an? question. the lowst-lying resonance

is ~ ~ ( 1 4 3 0 ) . The ao(9S0) must be unnaturdly decoupled [rom the Iight-quark

current suggescing an esotica structure can be used to interpret the ao(9SO).

The ividth ranges obtained from Monte-CarIo sirnuhtion for the two scaliir

ctiannttis (50 11eG- < TlEo < 3JO .\[ri'. 110 .Uri' < Tr , , < 331 J l r I- i ; inb

also consistent with che above two conclusions (jo(9SOI and a*( L - 4 3 ) are the

lowest-lying resonances). From Table 2.3. the PDG estirnateci varues for the trw

mesons are 40 MeC- < r < 100 MeV for fo(980) and 232 MeC- < r < 278 Me\'

for ~ ~ ( 1 4 5 0 ) are embedded in the Monte-Carlo estimations.

The much higher I = I value for the continuum threshold in Table 4.10 is

comparable to the d u e s for s o obtained via sum n i l ~ analysis of thc I = I

psetidoscalar channel [111. 1311. In both I = 0.1 channels. sum ride rnechod-

ology r~ould siiggest that Borel stability not occur for values of ski' tliat are

less than the masses of contributing subcontinuum resonances. The large t-alue

O € mass evident forn Table 4.10 necessarily requires values of so Iarger that of

the I = O channeI. consistent mit h the values of so actually listed.

-4s nientioned before. the Holtler inequality provides a valuable criterion for

st udying the validity and self-consistency of a QCD siim-ride ralci~lation. So t

al1 QCD siim riiks are consistent with Holder inequality constraints. Inconsis-

tencies between QCD sum-rule calculations and Holder inequalities have becn

found in the literature [131]. which indicates those QCD sum-rule calculations

do not çonsistently clescrihe iritegrated physical cross-sections. t . e. t hose calcu-

laticiris cririltl riot provitir a (.onvinring description of a rcsonanw and p t i ~ ~ i c * i i l

r.onriniiiinis. Thcrrforc. i t is ivorth çheckitig oiir sittiulation rwiilts ivith Hiil(lrr

ineqtirt1ities. Comparing the s'0 ~ ~ I U C S obtainecl from our Nonte-Ciirlo simu-

lations (For the r = 0 channel. 3.%Ge\*' < .$O < 4.1SGel-': for the i = 1

channel 4.23GeL" < .so < 5.6iGeC") with Figure (4.1) and Figure (4.2). we

find the above tivo so Monte-Car10 estimated talues lie Nithin the correspond-

ing valid si, - r rpgion from Molder ineqiiality analyis. Thus KT cmnrlirrlc rhrir

oiir leiist-1' simulation results are consistent n-ith Hiilder inequidity c-imstrriincs

on the corresponding channel's QCD sum-mie.

Chapter 5

Conclusions

The goal of t his t hesis is deterrnining which of several empirically justifiable in-

terpretations of the experimentd observed reçonances in the scalar channels are

compatible with theoretical constraints based on QCD sum-rule methodolop.

Cornpared with traditional derivative sum-ruk ratio methods. wc.e finrl thi~t

the l e ; ~ t - ~ ' estriiccicia of the propertips (if r t w Iowst-lving scxlar rtit*sori~, oti-

tüinetl froni a global fit to the shape of the fun[iarnencal QCD Laplact* suiri ri i l i .

of the light-quark non-strange currents. has several reliability ancl valiciity ad-

vantages. To calculate the least-x2. a modified version of the Downhill simples

algorithm is empioyed. Cornpared esplicitly with the derivative Levenberg-

Siarquardt algorithm. ive Find that the no-derivative-involvecf simplex rnethoti

is superior in converging to the global rniriimrini point providtd sotrit* prdirrii-

nary knotvletige is gained about the tottgh separations among local niiriirriii.

One of the unique features of this thesis is the iitilization of the Holder

inequality technique to determine the valid Borel scale r range for our QCD

sum-mle analysis. The Holder inequality provides fundamental constraints for

the QCD stirn-rulc to consistently describe integrated physical cross-wctions.

CVith no need for an! phenomenologica1 inputs. the Holder inrquality siipplit~s

a rigorous method for deterrnining the valid r range and so space for a given

QCD sum-rule.

The single-instanton contribution is incorporated in the field theoreticd

content of the scalar QCD sum rules. instanton effects are of ultimate impor-

tant in this study since they are the ody distinct components in the I = 0.1

chmuel's QCD sum rules to any loop order. Therefore instantons are respon-

sible for breaking the m a s degeneracy between the loaest-Iying isovwtor m i 1

isoscaIar mesons. A novel treatment of inscanton effets in the QCD continuum

contribution is deveioped in this thesis. The Holder inequality analysis shorvs

that the nidelu accepted formuta for instanton effects is incorrect.

-4 four-pulse approsimation finite width effect is incorporatecl intu the

hadronic contribution of QCD Laplace siim riiles ro civPrcrinw rhr niirroti. res-

onance niorfel's Iiniits in study the lowesr-lying sr-alar resc;uriatlctb proprrtirs.

From the best-fit simulation results. tve find a Lowst-luing resonance's tinitc-

width cffcct elevares a sum-rule determination of that resonance's mas. Several

possible hadronic models are explicitly tested. and ive conciucie that the scalar

riirri rules are dorninated by a single finite-width resonance.

T h ht~st-fit giws ;i Iowst-lying rPson;inctb armnrl 966 \Ir\' iri r I i t 1 I = i l

<kmni?L ;inci i i lightesc resm;uiw ;iroiinil 1500 l i t > \ ' iti t h i = 1 c h i t i d .

both resonancc's witlths are aroiind 200 Me\-. Thtw results indirate IL,^[ 1 4 X i

Iieing the lotvest-lying qq meson in the I = 1 channei. suggesting iin esotic

interpretation of the ~~(980). In the 1 = O channel. the results cIearly rule out

a dominant contribution of the jo(-!OO - 1200) co the light quark current's mm-

rrile. and the f0(950) is identified as the Lo~twst-Iying qq isoscalar rcwnanci..

Thr stability analyses of the QCD siim-ride best-fit predictiuns art1 w r i -

tfiicted through a Monte-Carlo simulation based on the uncertainties of the

perturbative contribution. the instanton contribution. the condensate contri-

bution. and the vacuum saturation hypothesis. The results of the fitting pro-

ceditre at 90%. confidence level are as folIows: For the I = O channel. the

Ionest-lying resonance's rnass is .II = 11.000 i 0.0871 Gel-. the rota1 width is

r = (0.19 0.14) Gel-. the r:orrespunding continuiini ttiresholrl is s,, = ( 3 . 7 3 = 0.45) Gel -' and the resonance's coupling coefficient is n = (0.079r0.023) Gel -'. For the I = I chamel, the results are .LI = (1.55 k 0.1 1) Gel/: = (0.22 f

0.11) GeV. $0 = (4.95 I 0.72) Gel/"- and a = (0.170 3~ 0.035) GeV? These

results are in good agreement nith the best-fit predictions and consistent with

the Holder inequality constraints.

In conclusion. j0(950) is identified as the Llotvest-lying light qirark resoniinw

from our QCD sum-ruie analpis in rhe I = O channel. This concIusion is s u p

ported by [Sa. $7.891. The fo(400 - 1200) is unnaturally decoupled froni a light

quark current. we suggest a gluebail or a mixture of gluebal1 and qriarkonium

structure for this c~nt ro~ers ia l a meson j13JI. An in-depth study of this pos-

sibility is in progress in our research group. In the I = 1 channel. n-r rondiidt~

that uO( 14301 is rht. Ligtitest resonancc. This inrlicaces ;i nori-clcl i ~ t r i i r r i i r r

for aolSSOI. and hence a Kr-rnoiecule structure for rhis resonance is preferred

[86. S i ] .

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Appendix A Convent ions and Notations

In chis thesis. natural units are empioyd.

In these units. the position in a contravariant coordinate is denoted bv a 4- vector P.

The covariant coordinate x, is defined by

.,TV = ylrvxv = ( t . -2) (.4.3\

wherc the matris gu, is defines as

The Slinkowski spacc rnatriv is defined by the following line element.

where the Einstein sumrnation convention of "sum over repeated indices" is followed.

ln the h = c = I units susteni, the Lagrnngian for frre Dirac fwnlion O!'

mass rn is

n-here the -;, are the 4 x 4 Dirac matrices satisfy the algebra

-4 cornmon notation is 4 = d ' ~ ~ . The Dirac representation of the y matrices is defined as.

where t.he I in the above equation is 2 x 2 unir nlacri'c [: 1 . ne 3marr i t5

are the Pauli matrices

The eight trtlct?Iess generators X of S1>(3 LIann matrices.

) group are called the 3 x 3 GeLI-

O 1 0 - 1 O

O O O 0 0 0

L O O 0 0 1 ,\, = (, -1 O ) . ,\, = ( O O O ) ,A.,-!,

O 0 O 1 0 O

O O -1 0 0 0

O 0 I)

O 1. O O O - 2 (.LI:)

The general representations Ta of SL2(3) obey a Lie algebra

where the ami-symmetric fabc are knonm as the Lie group structure constants.

Appendix B Exarnple REDUCE Program

The foUowing is an example REDUCE program used in calculating the gluon condensate contribution to the light-quark scalar current correlation hrnction. This calculation corresponds to the third plot in Figure (3.4) in Chapter 3.

The indented Iines are the actual REDUCE codes and the others are the output.

> off allfac:

> > vecdim 4+2*ep;

> > > for al1 y let gamma(y) =l/y-r+l/2* (r*r+z) *y;

*** gamma declared operator

*** gamma1 declared operator

*** gamma2 declared operator

*** gamma3 declared operator

> > > f o r al1 y let ngammal (y) =l/ (-l+y) *gamma(y) ;

*** ngammal declared operator

> > for a l l y l e t n g m a 2 ( y ) = l / ( - 2 + y ) *ngammal(y) ;

*** ngamma2 declared operator

*** ngamma3 declared operator

> > > > > for al1 p let f p ( p ) = i * ( g ( l , p ) ) / ( p . p ) ;

*** f p declared operator

> > f o r al1 p l e t vcg(p)=- i*g*g( l ,p ) ;

*** vcg declared operator

> > let p l . p l = x , p2.p2=y, p3 .p3=z, (pl-q) . (pl-q)=o;

> > l e t q.q=q2, pi.q=-(w-q2-x)/2;

> > let p2=pl;

> l e t p3=pl-q;

> let y=x;

> let z=w;

> > index v,tau,la

>

, rho

> amps :=(v. tau*la.rho-v.rho*la. tau) *ampl;

2 2 2 2 2 2 2 amps := ( - 4*ep *g *q2 + 4*ep *g *w + 4*ep *g *x - lO*ep*g *q2

> > remind v, tau, la, rho ;

> > > let denp=(x*w) **8;

> > ampl :=amps*denp$ > > for al1 m,n match x**m*w**n=f î (8-m,8-n) ;

> > amps :=ampl;

*** fi deciared operator

> > > let f i(2,3)=1/16/pi/pi/q2^3*gamma(ep) *ngammal (ep) *gamma3(-ep) /2/ngammal(Z*ep) ;

> let f i(2,1)=1/16/pi/pi/q2*gamma(ep) *gammal(epl *gammal(-ep)/gmal (2*ep) ;

> let fi (1,3) =1/16/pi/pi/q2'2*gammal (ep) *ngammal (ep) *gamma2 (-ep) /2/gamma(2*ep) ;

> > let fi (112) =1/16/pi/pi/q2*gammal Cep) * g a m m a ( a a l ( 2 * e p ) ;

> >

bye;