1

Transverse momenta of partons in high-energy scattering

processesPiet Mulders

International School, Orsay June 2012

mulders@few.vu.nl

2

Introduction

• What are we after?– Structure of proton (quarks and gluons)– Use of proton as a tool

• What are our means?– QCD as part of the Standard Model

3

3 colors

Valence structure of hadrons:

global properties of nucleons

duu

proton

• mass• charge• spin• magnetic moment• isospin, strangeness• baryon number

Mp Mn 940 MeV Qp = 1, Qn = 0 s = ½ gp 5.59, gn -3.83 I = ½: (p,n) S = 0 B = 1

Quarks as constituents

4

A real look at the proton

+ N ….

Nucleon excitation spectrumE ~ 1/R ~ 200 MeVR ~ 1 fm

5

A (weak) look at the nucleon

n p + e +

= 900 s Axial charge GA(0) = 1.26

6

A virtual look at the proton

N N + N N_

7

Local – forward and off-forward m.e.

1.

2( ) (' | ) | )( i x GP O tP e t Gx i

2t

1(0) | ( ) |G P O x P

2 (0) | ( ) |G P x O x P

Local operators (coordinate space densities):

P P’

Static properties:

Form factors

Examples:(axial) chargemassspinmagnetic momentangular momentum

8

Nucleon densities from virtual look

proton

neutron

• charge density 0• u more central than d?• role of antiquarks?• n = n0 + p+ … ?

( ) ( )i iG t x

9

Quark and gluon operators

Given the QCD framework, the operators are known quarkic or gluonic currents such as

probed in specific combinationsby photons, Z- or W-bosons

( ) 2 1 13 3 3 ...u d sJ V V V

( ) 21 42 3 sin ...Z u u u

WJ V A V

( ) ' ' ...W ud udJ V A

'5( ) ( ) '( )q qA x q x q x

( ) ( ) ( )qV x q x q x

(axial) vector currents

{ }( ) ~ ( ) ( )qT x q x D q x

( ) ~ ( ) ( )GT x G x G x

energy-momentum currents

probed by gravitons

10

Towards the quarks themselves

• The current provides the densities but only in specific combinations, e.g. quarks minus antiquarks and only flavor weighted

• No information about their correlations, (effectively) pions, or …

• Can we go beyond these global observables (which correspond to local operators)?

• Yes, in high energy (semi-)inclusive measurements we will have access to non-local operators!

• LQCD (quarks, gluons) known!

11

Non-local probing

†2 2 2 2, ...y y y y

x x x xO

2 2 2 2| , | | , |y y y yx xP O P P O P

2. †2 2| | | 0 | ( )y yip ydy e P P P p P f p

Nonlocal forward operators (correlators):

Specifically useful: ‘squares’

Momentum space densities of -ons:

Selectivity at high energies: q = p

12

A hard look at the proton

Hard virtual momenta ( q2 = Q2 ~ many GeV2) can couple to (two) soft momenta

+ N jet jet + jet

13

Experiments!

14

QCD & Standard Model

• QCD framework (including electroweak theory) provides the machinery to calculate cross sections, e.g. *q q, qq *, * qq, qq qq, qg qg, etc.

• E.g. qg qg

• Calculations work for plane waves

__

) .( ( )0 , ( , )s ipi ip s u p s e

15

Soft part: hadronic matrix elements

• For hard scattering process involving electrons and photons the link to external particles is, indeed, the ‘one-particle wave function’

• Confinement, however, implies hadrons as ‘sources’ for quarks

• … and also as ‘source’ for quarks + gluons

• … and also ….

.( )0 , ( , ) ipii p s u p s e

.( ) ii

pX P e

1 1( ). .( ) ( ) i pi

p ipAX P e

PARTON CORRELATORS

16

Thus, the nonperturbative input for calculating hard processes involves [instead of ui(p)uj(p)] forward matrix elements of the form

Soft part: hadronic matrix elements

3

3( , ) | | | | ( )

(2) 0)

)(

2(0j i

Xij X

X

d Pp P P X X P P P p

E

4 .4

1( , ) | |

(2( (

)0) )i p

i ij jp P d e P P

quarkmomentum

INTRODUCTION

_

(0) ( )| |( )j iP PA

17

PDFs and PFFs

Basic idea of PDFs is to get a full factorized description of high energy scattering processes

21 2| ( , ,...) |H p p

1 2 1 1 1 2 2

, ... 1 2 1 1

( , ,...) ... ... ( , ; ) ( , ; )

( , ,...; ) ( , ; )....

a b

ab c c

P P dp p P p P

p p k K

calculable

defined (!) &portable

INTRODUCTION

Give a meaning to integration variables!

18

Example: DIS

19

• Instead of partons use correlators• Expand parton momenta (for SIDIS take e.g. n=

Ph/Ph.P)

Principle for DIS

q

P

2 2P M

( , ) ( , )

( , ) ~ ( ) ( )su p s u p s

p P p m f p

Tx pp P n 2 2. ~p P xM M

. ~ 1x p p n

~ Q ~ M ~ M2/Q

p

20

(calculation of) cross section in DIS

Full calculation

+ …

+ +

+LEADING (in 1/Q)

x = xB = q2/P.q

(x)

Lightcone dominance in DIS

22

Result for DIS

q

P

212

12

2 ( , ) . [ ( , ) ] ( )

[ ( ) ]

T T B

T B

MW P q g dxdp P d p Tr p P x x

g Tr x

p

23

Parametrization of lightcone correlator

Jaffe & Ji NP B 375 (1992) 527Jaffe & Ji PRL 71 (1993) 2547

leading part

• M/P+ parts appear as M/Q terms in cross section• T-reversal applies to (x) no T-odd functions

24

Basis of

partons

‘Good part’ of Dirac space is 2-dimensional

Interpretation of DF’s

unpolarized quarkdistribution

helicity or chiralitydistribution

transverse spin distr.or transversity

25

Off-diagonal elements (RL or LR) are chiral-odd functions Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY

Matrix representation

for M = [(x)+]T

Quark production matrix, directly related to thehelicity formalism

Anselmino et al.

Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712

26

Next example: SIDIS

27

(calculation of) cross section in SIDIS

Full calculation

+

+ …

+

+LEADING (in 1/Q)

Lightfront dominance in SIDIS

Three external momentaP Ph q

transverse directions relevant

qT = q + xB P – Ph/zh or

qT = -Ph/zh

29

Result for SIDIS

2 2

2

2 212

2

2 ( , , )

[ ( , ) ( , ) ] ( )

[ ( , ) ] [ ( , ) ] ( )

h T T

B T h T T T T

T T T

B T h T T T T

MW P P q d p d k

Tr x p z k p q k

g d p d k

Tr x p Tr z k p q k

q

P

Ph

pk

hT B

h

Pq q x P

z

30

Parametrization of (x,pT)

• Also T-odd functions are allowed

• Functions h1 (BM)

and f1T (Sivers)

nonzero!• Similar functions (of course) exist as fragmentation functions (no T-constraints) H1

(Collins) and D1T

Interpretation

unpolarized quarkdistribution

helicity or chiralitydistribution

transverse spin distr.or transversity

need pT

need pT

need pT

need pT

need pT

T-odd

T-odd

32

pT-dependent functions

T-odd: g1T g1T – i f1T and h1L

h1L + i h1

(imaginary parts)

Matrix representationfor M = [±]

(x,pT)+]T

Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712

33

Rather than considering general correlator (p,P,…), one thus integrates over p.P = p (~MR

2, which is of order M2)

and/or pT (which is of order 1)

The integration over p = p.P makes time-ordering automatic. This works for (x) and (x,pT)

This allows the interpretation of soft (squared) matrix elements as forward antiquark-target amplitudes (untruncated!), which satisfy particular analyticity and support properties, etc.

Integrated quark correlators: collinear

and TMD

2.

3 . 0(0) ( )

( . )( , ; )

(2 )q i pT

iij njT

d P dx p n e P P

.

. 0

( . )( (0); )

(2 )( )ij

T

q i pj i n

d Px n e P P

TMD

collinear

(NON-)COLLINEARITY

Jaffe (1984), Diehl & Gousset (1998), …

lightfront

lightcone

34

Summarizing oppertunities of TMDs

TMD quark correlators (leading part, unpolarized) including T-odd part

Interpretation: quark momentum distribution f1q(x,pT) and

its transverse spin polarization h1q(x,pT) both in an

unpolarized hadronThe function h1

q(x,pT) is T-odd (momentum-spin correlations!)

TMD gluon correlators (leading part, unpolarized)

Interpretation: gluon momentum distribution f1g(x,pT) and

its linear polarization h1g(x,pT) in an unpolarized hadron

(both are T-even)

122 2

1 12

1( , ) ( , )( , )

2

vT T T

g Tg g

T TT

p p gx f x p h x pp g

x M

2 21

]1

[ ( ,( , ) ( , ))2

q q TqT T T

p Px p i

Mf x p h x p

(NON-)COLLINEARITY

35

Twist expansion of (non-local) correlators

Dimensional analysis to determine importance of matrix elements(just as for local operators)maximize contractions with n to get leading contributions

‘Good’ fermion fields and ‘transverse’ gauge fieldsand in addition any number of n.A() = An(x) fields (dimension zero!)but in color gauge invariant combinations

Transverse momentum involves ‘twist 3’.

dim[ (0) ( )] 2n dim[ (0) ( )] 2n nF F

n n n ni iD i gA

T T T Ti iD i gA dim 0:

dim 1:

OPERATOR STRUCTURE

36

Matrix elements containing (gluon) fields produce gauge link

… essential for color gauge invariant definition

Soft parts: gauge invariant definitions

4[ ] . [ ]

[0, ]4( ; )

(2( )

)(0)C i p C

ij j i

dp P e P U P

[ ][0, ]

0

expCig ds AU

P

OPERATOR STRUCTURE

++ …

Any path yields a (different) definition

37

Gauge link results from leading gluons

Expand gluon fields and reshuffle a bit:

1 1 1 1 1 11

1( ) . ( ) ( ) ... ( ) ( ) ...

. .n n

T T

PA p n A p iA p A p p iG p

n P p n

OPERATOR STRUCTURE

Coupling only to final state partons, the collinear gluons add up to a U+ gauge

link, (with transverse connection from AT

Gn reshuffling)

A.V. Belitsky, X.Ji, F. Yuan, NPB 656 (2003) 165D. Boer, PJM, F. Pijlman, NPB 667 (2003) 201

38

Even simplest links for TMD correlators non-trivial:

Gauge-invariant definition of TMDs: which gauge links?

2[ ] . [ ]

[0, ]3 . 0

( . )( , (0) ( ); )

(2 ) jq C i p CTij T i n

d P dx p n e P U P

. [ ][0, ] . 0

( . )( ; )

(2 )(0) ( )ij

T

q i p n

nj i

d Px n e P U P

[] []T

TMD

collinear

OPERATOR STRUCTURE

These merge into a ‘simple’ Wilson line in collinear (pT-integrated) case

39

TMD correlators: gluons2

[ , '] . [ ] [ '][ ,0] [0, ]3 . 0

( . )( , ; )

(2 )(0) ( )C C i p C CT

g Tn

n

nd P dx p n e P U U F PF

The most general TMD gluon correlator contains two links, which in general can have different paths.Note that standard field displacement involves C = C ’

Basic (simplest) gauge links for gluon TMD correlators:

[ ] [ ][ , ] [ , ]( ) ( )C CF U F U

g[] g

[]

g[] g

[]

C Bomhof, PJM, F Pijlman; EPJ C 47 (2006) 147F Dominguez, B-W Xiao, F Yuan, PRL 106 (2011) 022301

OPERATOR STRUCTURE

40

Gauge invariance for SIDIS

COLOR ENTANGLEMENT

[0 , ] [ , ] [ , ] [ ,0 ]

[ ] [ ]† [ ] [ ]†[0, ] [0, ] 1 1[ ] [ ]n n n n

U U U U

W W W p W k

2 1[ ] [ ]† *1 1 1 2 2 2

[ ] [ †]1 1 1 1

[ [ ] ( , )] [ ( , ) [ ]]

ˆ( , ) ( , )

p pSIDIS c q T c q T

q T q T q q

d Tr W p x p Tr x p W p

x p k k

Strategy:transverse moments

Problems with double T-odd functions in DY

M.G.A. Buffing, PJM, 1105.4804

41

Complications (example: qq qq)

[ ](11) [ ](1) [ ](1)1( , ')... ... ( )... ( ') ( ), ( ') ... ... ( )... ( ')

2k k kU p p p p U p U p p p

T.C. Rogers, PJM, PR D81 (2010) 094006

U+[n] [p1,p2,k1]

modifies color flow, spoiling universality(and factorization)

COLOR ENTANGLEMENT

42

Color entanglement

[ ](21) [ ](2) [ ](1)

[ ](1) [ ](1) [ ](1)

[ ](1) [ ](2)

1( , ') ( ) ( ')

41

( ) ( ') ( )41

( ') ( )4

k k k

k k k

k k

U p p U p U p

U p U p U p

U p U p

COLOR ENTANGLEMENT

43

Featuring: phases in gauge theories

'

( ) ( ')x

xig ds A

i iP Pe Px x

.

'ie ds A

Pe COLOR ENTANGLEMENT

44

Conclusions

• TMDs enter in processes with more than one hadron involved (e.g. SIDIS and DY)

• Rich phenomenology (Alessandro Bacchetta)• Relevance for JLab, Compass, RHIC, JParc, GSI,

LHC, EIC and LHeC • Role for models using light-cone wf (Barbara

Pasquini) and lattice gauge theories (Philipp Haegler)

• Link of TMD (non-collinear) and GPDs (off-forward)• Easy cases are collinear and 1-parton un-

integrated (1PU) processes, with in the latter case for the TMD a (complex) gauge link, depending on the color flow in the tree-level hard process

• Finding gauge links is only first step, (dis)proving QCD factorization is next (recent work of Ted Rogers and Mert Aybat)

SUMMARY

45

Rather than considering general correlator (p,P,…), one integrates over p.P = p (~MR

2, which is of order M2)

and/or pT (which is of order 1)

The integration over p = p.P makes time-ordering automatic. This works for (x) and (x,pT)

This allows the interpretation of soft (squared) matrix elements as forward antiquark-target amplitudes (untruncated!), which satisfy particular analyticity and support properties, etc.

Thus we need integrated correlators

2.

3 . 0(0) ( )

( . )( , ; )

(2 )q i pT

iij njT

d P dx p n e P P

.

. 0

( . )( (0); )

(2 )( )ij

T

q i pj i n

d Px n e P P

TMD

collinear

(NON-)COLLINEARITY

Jaffe (1984), Diehl & Gousset (1998), …

lightfront

lightcone

46

Large values of momenta

• Calculable!2 2

2 01T p

p

p x Mp

x

2 2( ). 0

2 (1 )p T

p

x p Mp P

x x

2 22 ( ) (1 )

0(1 )

p T pR

p

x x p x x MM

x x

0 0 ( 1)T pp

xp P p x x

x

T Tp 2( , ) ...s

T

p Pp

etc.

Bacchetta, Boer, Diehl, M JHEP 0808:023, 2008 (arXiv:0803.0227)

PARTON CORRELATORS

p0T ~ M

M << pT < Q

hard

47

T-odd single spin asymmetry

• with time reversal constraint only even-spin asymmetries• the time reversal constraint cannot be applied in DY or in 1-

particle inclusive DIS or ee

• In those cases single spin asymmetries can be used to measure T-odd quantities (such as T-odd distribution or fragmentation functions)

*

*

W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)

W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)

W(q;P,S;Ph,Sh) = W(q;P, S;Ph, Sh)

W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)

_

___

_ ____

__ _time

reversal

symmetrystructure

parity

hermiticity

*

*

48

Relevance of transverse momenta in hadron-hadron

scatteringAt high energies fractional parton momenta fixed by kinematics (external momenta)

DY

Also possible for transverse momenta of

partons

DY

2-particle inclusive hadron-hadron scattering

K2

K1

pp-scattering

1 11 1Tp Px p

2 22 2Tp Px p 1 2 21 1

1 2 1 2

. ..

. .

p P q Px p n

P P P P

1 1 2 2 1 2T T Tq q x P x P p p

1 11 1 2 2 1 1 2 2

1 2 1 2

T

T T T T

q z K z K x P x P

p p k k

NON-COLLINEARITY

Care is needed: we need more than one hadron and knowledge of hard process(es)!

Boer & Vogelsang

Second scale!

49

Lepto-production of pions

H1 is T-odd

and chiral-odd

50

Gauge link in DIS

• In limit of large Q2 the resultof ‘handbag diagram’ survives

• … + contributions from A+ gluonsensuring color gauge invariance

A+ gluons gauge link

Ellis, Furmanski, PetronzioEfremov, Radyushkin

A+

Distribution

From AT() m.e.

including the gauge link (in SIDIS)

A+

One needs also AT

G+ = +AT

AT()= AT

(∞) + d G+

Belitsky, Ji, Yuan, hep-ph/0208038Boer, M, Pijlman, hep-ph/0303034

52

Generic hard processes

C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277 [hep-ph/0406099]; EPJ C 47 (2006) 147 [hep-ph/0601171]

Link structure for fields in correlator 1

• E.g. qq-scattering as hard subprocess

• Matrix elements involving parton 1 and additional gluon(s) A+ = A.n appear at same (leading) order in ‘twist’ expansion

• insertions of gluons collinear with parton 1 are possible at many places

• this leads for correlator involving parton 1 to gauge links to lightcone infinity

53

SIDIS

SIDIS [U+] = [+]

DY [U-] = []

54

A 2 2 hard processes: qq qq

• E.g. qq-scattering as hard subprocess

• The correlator (x,pT) enters for each contributing term in squared amplitude with specific link

[Tr(U□)U+](x,pT)

U□ = U+U†

[U□U+](x,pT)

55

Gluons

2. [ ] [ ']

[ ,0] [0, ]3 . 0

( . )( , ; , ')

((0

)) ( )

2i p C CT n

gn

T n

d P dx p C C e P U PF FU

• Using 3x3 matrix representation for U, one finds in gluon correlator appearance of two links, possibly with different paths.

• Note that standard field displacement involves C = C ’

[ ] [ ][ , ] [ , ]( ) ( )C CF U F U

56

Integrating [±](x,pT) [±]

(x)

[ ] . †[0, ] . 0

( . )( ) (0) ( )

(2 ) T

i p n

n

d Px e P U P

collinear correlator

2[ ] . †

[0, ] [0 , ] [ , ]3 . 0

( . )( , ) (0) ( )

(2 ) T T

i p n T nTT n

d P dx p e P U U U P

57

2[ ] 2 . †

[0, ] [0 , ] [ , ]3 . 0

( . )( ) (0) ( )

(2 ) T T

i p n T nTT T n

d P dx d p e P U i U U P

Integrating [±](x,pT) [±]

(x)

[ ] . †[0, ]

( . )( ) (0) ( )

(2 )[i p n

T

d Px e P U iD P

[0, ] [ , ] [ , ](0) ( . ) ( ) ( ) ]n n n nLCP U d P U gG U P

[ ]

1 11

( ) ( ) ( , )D G

i

x ix x dx x x x

[ ] 2 [ ]( ) ( , )T T Tx d p p x p transve

rse moment

G(p,pp1)

T-even T-odd

58

Gluonic poles

• Thus: [U](x) = (x)

[U](x) =

(x) + CG[U] G

(x,x)

• Universal gluonic pole m.e. (T-odd for distributions)

• G(x) contains the weighted T-odd functions h1(1)(x)

[Boer-Mulders] and (for transversely polarized hadrons) the function f1T

(1)(x) [Sivers]

• (x) contains the T-even functions h1L(1)(x) and g1T

(1)

(x)

• For SIDIS/DY links: CG[U±] = ±1

• In other hard processes one encounters different factors:

CG[U□ U+] = 3, CG

[Tr(U□)U+] = Nc

Efremov and Teryaev 1982; Qiu and Sterman 1991Boer, Mulders, Pijlman, NPB 667 (2003) 201

~

~

59

A 2 2 hard processes: qq qq

• E.g. qq-scattering as hard subprocess

• The correlator (x,pT) enters for each contributing term in squared amplitude with specific link

[Tr(U□)U+](x,pT)

U□ = U+U†

[U□U+](x,pT)

60

examples: qqqq in pp

2 2[( ) ] [ ]

2 2 2

1 52

1 1 1

c cq G

c c c

N N

N N N

2 2 2[( ) ] [ ]

2 2 2

2 1 3

1 1 1

c c cq G

c c c

N N N

N N N

CG [D1]

D1

= CG [D2]

= CG [D4]CG

[D3]

D2

D3

D4

( )

c

Tr U

NU

U U

Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030; hep-ph/0505268

61

examples: qqqq in pp

†2 2

[( ) ] [ ]2 2 2

2 31

1 1 1

c cq G

c c c

N N

N N N

†2 2

[( ) ] [ ]2 2 2

11

1 1 1

c cq G

c c c

N N

N N N

D1

For Nc:

CG [D1] 1

(color flow as DY)

Bacchetta, Bomhof, D’Alesio,Bomhof, Mulders, Murgia, hep-ph/0703153

62

Gluonic pole cross sections

• In order to absorb the factors CG[U], one can define

specific hard cross sections for gluonic poles (which will appear with the functions in transverse moments)

• for pp:

etc.

• for SIDIS:

for DY:

Bomhof, Mulders, JHEP 0702 (2007) 029 [hep-ph/0609206]

[ ]

[ ]

ˆ ˆ Dqq qq

D

[ ( )] [ ][ ]

[ ]

ˆ ˆU D Dq q qq G

D

C

[ ]ˆ ˆq q q q

[ ]ˆ ˆq q qq

(gluonic pole cross section)

y

( )ˆ q q qqd

ˆqq qqd

63

examples: qgqin pp

2 2[( ) ] [ ]

2 2 2

11

1 1 1

c cq G

c c c

N N

N N N

D1

D2

D3

D4

Only one factor, but more DY-like than SIDIS

Note: also etc.

Transverse momentum dependent weigted

64

examples: qgqg

D1

D2

D3

D4

D5

2 2[( ) ] [ ]

2 2 2

11

1 1 1

c cq G

c c c

N N

N N N

e.g. relevant in Bomhof, Mulders, Vogelsang, Yuan, PRD 75 (2007) 074019

Transverse momentum dependent weighted

65

examples: qgqg

2 2[( ) ] [ ]

2 2 2

11

1 1 1

c cq G

c c c

N N

N N N

Transverse momentum dependent weighted

66

examples: qgqg

†2 2

[( )( ) ] [( ) ] [ ]2 2 2 2

1 1

2( 1) 2( 1) 1 1

c cq G

c c c c

N N

N N N N

2 2[( ) ] [ ]

2 2 2

11

1 1 1

c cq G

c c c

N N

N N N

†2 2

[( )( ) ] [( ) ] [ ]2 2 2 2

1 1

2( 1) 2( 1) 1 1

c cq G

c c c c

N N

N N N N

†2

[( )( ) ] [ ]2 2

1

1 1

cq G

c c

N

N N

†4 2 2

[( )( ) ] [( ) ] [ ]2 2 2 2 2 2

11

( 1) ( 1) 1 1

c c cq G

c c c c

N N N

N N N N

Transverse momentum dependent weighted

67

examples: qgqg

†2 2

[( )( ) ] [( ) ] [ ]2 2 2 2

1 1

2( 1) 2( 1) 1 1

c cq G

c c c c

N N

N N N N

2 2[( ) ] [ ]

2 2 2

11

1 1 1

c cq G

c c c

N N

N N N

†2 2

[( )( ) ] [( ) ] [ ]2 2 2 2

1 1

2( 1) 2( 1) 1 1

c cq G

c c c c

N N

N N N N

†2

[( )( ) ] [ ]2 2

1

1 1

cq G

c c

N

N N

†4 2 2

[( )( ) ] [( ) ] [ ]2 2 2 2 2 2

11

( 1) ( 1) 1 1

c c cq G

c c c c

N N N

N N N N

†2 2

[( )( ) ] [( ) ] [ ]2 2 2

1

2( 1) 2( 1) 1

c c

c c c

N N

N N N

†2

[( )( ) ] [ ]2 2

1

1 1

cG

c c

N

N N

It is also possible to group the TMD functions in a smart way into two!(nontrivial for nine diagrams/four color-flow possibilities)

But still no factorization!

Transverse momentum dependent weighted

68

‘Residual’ TMDs

• We find that we can work with basic TMD functions [±](x,pT) + ‘junk’

• The ‘junk’ constitutes process-dependent residual TMDs

• The residuals satisfies (x) = 0 and G(x,x) = 0, i.e. cancelling kT contributions; moreover they most likely disappear for large kT

† †

†[( )( ) ]

[( )( ) ] [ ] [( )( ) ] [ ]

( , )

( , ) ( , ) ( , )

T

T T T

x p

x p x p x p

[ ] [ ] [ ] [ ]( , ) 2 ( , ) ( , ) ( , )T T T Tx p x p x p x p

definite T-behavior

no definiteT-behavior

69

2 21 1( , () , )( , )

2T

T Tq q q

Tf x p hp P

x p iM

x p

5

1 1( , )2

[ , ]

2 2

T

q q TqT

q q q qPnp

T

p Px p i

M

pM i P n

Me h

M

f h

f g

1 (1

( )2

) (2

)q q qf xM

x P e x

1 1

1( ) ( ) ( )

2 2( )q qq qM m

x ef x x f xx Mx

p m

70

2 21 1( , () , )( , )

2T

T Tq q q

Tf x p hp P

x p iM

x p

1( )2

( )q qf xP

x

2.

3 . 0

( . )( , ()

(2 )0) ( )i pT

T n

d P dx p e P P

2 .

. 0

( . )( ) ( , ) (0) ( )

(2 ) T

i pT T n

d Px d p x p e P P

(1)1

22 2

12( ) ( , )

2 2 2( )q qT

T Tq pP P

x d p h x pM

h x

2 .

. 0

( . )( ) ( , ) ( (0) ( ))

(2 ) T

i pT nTT T

d Pix d p p x p e P P

71

2 2 2 21 12 ( , ) ( , ) ( , )qq T

Tq

T T

kz z k iD z z k H x k K

Mz

1

5

12 ( , )

[ , ]

2

T

q TT

h

K

q

nkT

hq

h

q q

h

q

q

kz z k i K

M

k i

D H

E D GK n

MM M

H

1

2 [ , ]( ) )

2(qq

hq qD z E

i K nHz K M

z

1

1( ) .( .

2) .qq D z k mz

72

2 21 1( , ) , ) , )

2( (q Tq

U Tq

T Tf x p hp P

x p iM

x p

1

525

21( , ) ( , ) , )

2(

L

Tqq qT L TLL T LS Sg x p

p Px px p

Mh

73

1( )2

( )qqU f

Pxx

51( )2

( )qqLL g

PxSx

1 5(2

)) (q TqT

Pxx h S

74

75

1-parton unintegrated

• Resummation of all phases spoils universality

• Transverse moments (pT-weighting) feels entanglement

• Special situations for only one transverse momentum, as in single weighted asymmetries

• But: it does produces ‘complex’ gauge links

• Applications of 1PU is looking for gluon h1

g (linear gluon polarization) using jet or heavy quark production in ep scattering (e.g. EIC), D. Boer, S.J. Brodsky, PJM, C. Pisano, PRL 106 (2011) 132001

2 2 2 21 2 1 2

2 2 2 21 1 2 1 2 2

... ... ( )

... ...

T T T T T T T

T T T T T T

d q q d p d p q p p

d p p d p d p d p p

COLOR ENTANGLEMENT

76

Full color disentanglement? NO!

COLOR ENTANGLEMENT

2 1 2 1 1 1 2 2 1 1 1 1

[ ] [ ] [ ]†[0 , ][0 , ] [ , ][ , ] [ , ] [ ,0 ] [0 , ] [0 , ] [0 , ]

[ ] [ ]2 1[ ] [ ]

n n n

n n

U U U U W W W

W p W p

1 1( ) (0 )

1[ ,0 ]

2[ , ] 2[ ,0 ]

1[ , ]

1 2[ , ][ , ]

†

†

[( ) ] * [( ) ]1 1

[( ) ] * [( ) ]2 2

~ [ ( ) ( ) ]

[ ( ) ( ) ]

c b a

b ac

Tr p k

Tr p k

2 2( ) (0 )

Loop 1:

X

a

b*

b*

a

M.G.A. Buffing, PJM; in preparation

1 2[0 , ][0 , ]

77

Result for integrated cross section

1 2 ... 1ˆ~ ( ) ( ) ( )...a b ab c cabc

x x z

Collinear cross section

[ ] 2 [ ]( ) ( , )C CT Tx d p x p

[ ]... ...ˆ ˆ D

ab c ab cD

(partonic cross section)

APPLICATIONS

Gauge link structure becomes irrelevant!

11 1 2 1

[ ( )] [ ]...2

,1

( , ) ( ) ( )...ˆ~ T

C D Da b ab c c

D abcT

x p x zd

d p

(1PU)

78

Result for single weighted cross section

[ ( )] [ ]1 1 2 ... 1

,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

p x x z

Single weighted cross section (azimuthal asymmetry)

[ ] 2 [ ]( ) ( , )C CT T Tx d p p x p

APPLICATIONS

11 1 2 1

[ ( )] [ ]...2

,1

( , ) ( ) ( )...ˆ~ T

C D Da b ab c c

D abcT

x p x zd

d p

(1PU)

79

Result for single weighted cross section

[ ( )] [ ]1 1 2 ... 1

,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

p x x z

[ ( )] [ ]1 1 2 ... 1

,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

p x x z [ ( )] [ ]

1 1 2 ... 1,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

p x x z [ ( )] [ ]

1 1 2 ... 1,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

p x x z

APPLICATIONS

11 1 2 1

[ ( )] [ ]...2

,1

( , ) ( ) ( )...ˆ~ T

C D Da b ab c c

D abcT

x p x zd

d p

(1PU)

80

Result for single weighted cross section

[ ( )] [ ]1 1 2 ... 1

,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

p x x z [ ] [ [ ] [ ])] (( ) ( ) ( , )C C U C C

G Gx x C x x

universal matrix elements

G(x,x) is gluonic pole (x1 = 0) matrix element (color entangled!)

T-even T-odd

(operator structure)

1( , )G x x x APPLICATIONS

Qiu, Sterman; Koike; Brodsky, Hwang, Schmidt, …

G(p,pp1

)

11 1 2 1

[ ( )] [ ]...2

,1

( , ) ( ) ( )...ˆ~ T

C D Da b ab c c

D abcT

x p x zd

d p

(1PU)

81

Result for single weighted cross section

[ ( )] [ ]1 1 2 ... 1

,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

p x x z [ ] [ [ ] [ ])] (( ) ( ) ( , )C C U C C

G Gx x C x x

universal matrix elements

Examples are:

CG[U+] = 1, CG

[U] = -1, CG[W U+] = 3, CG

[Tr(W)U+] = Nc

APPLICATIONS

11 1 2 1

[ ( )] [ ]...2

,1

( , ) ( ) ( )...ˆ~ T

C D Da b ab c c

D abcT

x p x zd

d p

(1PU)

82

Result for single weighted cross section

[ ( )] [ ]1 1 2 ... 1

,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

p x x z [ ] [ [ ] [ ])] (( ) ( ) ( , )C C U C C

G Gx x C x x

1 1 2 ... 1

1 1 2 [ ] ... 1

ˆ~ ( ) ( ) ( )...

ˆ( , ) ( ) ( )...

T a b ab c cabc

G a b a b c cabc

p x x z

x x x z

( ([ ] [ ][ ] ... . .

).

)ˆ ˆU C D Da b c G ab c

D

C (gluonic pole cross section)

APPLICATIONS

11 1 2 1

[ ( )] [ ]...2

,1

( , ) ( ) ( )...ˆ~ T

C D Da b ab c c

D abcT

x p x zd

d p

(1PU)

T-odd part

83

Higher pT moments

• Higher transverse moments

• involve yet more functions

• Important application: there are no complications for fragmentation, since the ‘extra’ functions G, GG, … vanish. using the link to ‘amplitudes’; L. Gamberg, A. Mukherjee, PJM, PRD 83 (2011) 071503 (R)

• In general, by looking at higher transverse moments at tree-level, one concludes that transverse momentum effects from different initial state hadrons cannot simply factorize.

( ), ( , ), ( , , )G GGx x x x x x

SUMMARY

1 1 1[ ] ... 2( ) ( ... ) ( , )NNT T T Tx d p p p traces x p

84

DIS

q

P

2 2q Q

2 2P M2

2 .B

QP q

x 2 2P M

.Bq x P

nP q

85

DIS

q

P

n P

.Bq x P

n nP q

Basis 1

ˆq

zQ

2ˆ Bq x Pt

Q

Basis 2

86

Principle for DY

1 1

1 1 1 1

( , ) ( , )

( , ) ~ ( ) ( )su p s u p s

p P p m f p

• Instead of partons use correlators

• Expand parton momenta (for DY take e.g. n1= P2/P1.P2)

Tx pp P n 2 2. ~p P xM M

. ~ 1x p p n

~ Q ~ M ~ M2/Q

(NON-)COLLINEARITY