- phenomenological analyses · e d3 d3d k f (x,k ) d (z,p ) sin(3 ) sin(3 -3 ) dq d e d3 d3 d k û...

- Phenomenological Analyses –Parametrizations of transversity, Collins,

Sivers and Boer-Mulders functions

M. Boglione

M. Boglione 2

ℓ

ℓ’

g*

P

k

k’

fq(x)

dDh/q (z)

distribution function

XIncoming proton

elementary cross- section

fragmentation function

h

final detected hadron

)(ˆ)( / zDxf qhqSIDIS

Int. School Niccolo' Cabeo

M. Boglione 3

Unpolarized distribution functions

Δfq↑/p↑(x)Δfsz/+(x)

-

fq/p(x)

-

Helicity distribution functions

qqq

ggg

qqq

ggg

Transversity distribution functions

qqqT


M. Boglione 4

q Δq ΔTq

D.E. Soper, Phys. Rev. D 15 (1977) 1141; Phys. Rev. Lett. 43 (1979) 1847; J.C. Collins and D.E. Soper, Nucl. Phys. B194 (1982) 445; R.L. Jaffe, Nucl. Phys. B 229 (1983) 205.


M. Boglione 5

Very good knowledge of unpolarized distribution functions, q(x,Q2) and g(x,Q2)

Fairly good knowledge of longitudinally polarized, partonicdistributions, Δq(x,Q2); poor knowledge of longitudinallypolarized gluons Δg(x,Q2)

NO direct information on transversely polarized partonicdistributions, ΔTq(x,Q2), from DIS


M. Boglione 6Int. School Niccolo' Cabeo

M. Boglione 7

There is no gluon transversity distribution function

Transversity cannot be studied in deep inelastic scattering because it is chirally odd

Transversity can only appear in a cross-section convoluted to another chirally odd function

+

+–

–

+ –+

–

- ++–

+ –

+–


M. Boglione 8

P

xP

k

Plenty of theoretical and experimental evidence for transverse motion of partons within nucleons,

and of hadrons within fragmentation jets


M. Boglione 9

p p

Q2 = M2

qT

qL

l+

l–*g

qT distribution of lepton pairsin D-Y processes

pT distribution ofhadrons in SIDIS


M. Boglione 10

Distribution and fragmentation functions now depend

on the lightcone momentum fraction (x for the distributions and z for the fragmentations)

on Q2 (→pQCD evolution),

on the intrinsic transverse momentum of the partons,(k for the distributions and p for the fragmentations)

OPEN QUESTIONS:How do TMD’s depend on the intrinsic transverse momentum ?

Gaussian behaviour in the central region …Power law decrease at large transverse momentum…

Does the partonic intrinsic transverse momentum k (p) depend on x (z) ?


M. Boglione 12

Courtesy of Aram Kotzinian


M. Boglione 13

U

L

T

U

L

T

f1(x,k)Unpolarized

g1(x,k)Helicity

h1T(x,k) h1T (x,k)

Transversityg1T(x,k)f1T (x,k)

Sivers

h1 (x,k)

Boer-Mulders

h1L(x,k)

QUARK POLARIZATIONN

UC

LEO

N P

OLA

RIZ

ATIO

N

•Functions in bold face survive k integration•Functions in shaded cells are naïve T-odd •Functions in red box are chirally odd

Courtesy of A. Bacchetta


M. Boglione 14

M. Anselmino, M. Boglione, U. D-Alesio, E. Leader, S. Melis, F. Murgia, PR D71, 014002 (2005), PR D73, 014020 (2006)

X

Aa

M. Anselmino, M. Boglione, U. D-Alesio, S. Melis, F. Murgia, A. Prokudin (2010) in preparation

The distribution function fa/A is the numberdensity of partons of type a inside hadron A


M. Boglione 17

Δfsz/p↑(x,k)

Δfsz/p+(x,k)

Helicity fn.

Δfsx/p+(x,k)Δfsy/p↑(x,k)

Transversity

function

Δfsx/p↑(x,k)

Transversity

function

Δfsy/p(x,k)

Boer-Mulders

function

ΔNfq/p↑(x,k)

Sivers

function

fq/p(x,k)


M. Boglione 18

U

L

T

U

L

T

fq/p(x,k)Unpolarized

Δfsz/p+(x,k)Helicity

Δfsx/p↑(x,k)Δfsy/p

↑(x,k)

TransversityΔfsz/p

↑(x,k)ΔNfq/p↑(x,k)

Sivers

Δfsy/p(x,k)

Boer-Mulders

Δfsx/p+(x,k)

QUARK POLARIZATIONN

UC

LEO

N P

OLA

RIZ

ATIO

N

•Functions in bold face survive k integration•Functions in shaded cells are naïve T-odd •Functions in red box are chirally odd


M. Boglione 19

UnpolarizedSIDIS

Cross Section


M. Boglione 20

TMD-PDF hard scattering TMD-FF

All three fundamental blocks contain phases. The most general expression for the cross-section is obtained when all of them are kept into account.


M. Boglione 21

SIDIS in parton modelwith intrinsic k┴

Factorization holds at large Q2,

and

(Ji, Ma, Yuan)

QCDT kP

);,();,(ˆd);,(d 222 QzDQyQxf h

q

lqlq

q q

lhXlp

pkk


M. Boglione 22

EMC data, µp and µd, E between 100 and 280 GeV

CLAS data, arXiv:0809.1153 [hep-ex] F. Giordano and R. Lamb, arXiv:0901.2438 [hep-ex]

W. Käfer, COMPASS collaboration, talk at Transversity 2008, Ferrara


M. Boglione 23

Assume a simple, factorized form for the TMD distribution and fragmentationfunctions, with a gaussian dependence on the intrinsic transverse momentum

Normalized Gaussian(one free parameter)

Normalized Gaussian(one free parameter)


x and z dependencefrom world data fits

M. Boglione 24

Determine the value of the two free parameters by fitting experimental data

A cosφ dependence is also generated by Boer-Mulders⊗Collins term, via a kinematical effect in dΔσ ,

not included in this fit.

At further dependence on cos(2φ) is generated

EMC data, µp and µd, E between 100 and 280 GeV

M.Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin, Phys. Rev. D71:074006,2005.

(Gaussian widths)


M. Boglione 25

W. Käfer, COMPASS collaboration, talk at Transversity 2008, Ferrara

Comparison withM. Anselmino, M. Boglione, A. Prokudin, C. TürkEur. Phys. J. A 31, 373-381 (2007) does not include the Boer – Mulders contribution


M. Boglione 26

PT dependence of data in agreement with a Gaussian k⊥ dependence of unpolarized TMDs

Hint of a z-dependence at small z values

No hint of x dependence in the explored region

solid line →Gaussian TMD’s with

CLAS data, arXiv:0809.1153 [hep-ex] CLAS data, arXiv:0809.1153 [hep-ex]


M. Boglione 27

General SIDISCross Section


29

Unpolarizedproton and lepton

Longitudinally polarizedproton,polarized lepton

Transverselypolarizedproton,unpolarizedlepton

Transverselypolarizedproton,polarized lepton

Studying Sivers, Collins and other mechanisms is complicated by the fact that all these effects mix and overlap in the polarized SIDIS cross section and azimuthal asymmetries

Way out : appropriate azimuthal moments of the asymmetries !

The F structure functions contain all the TMD’s

Kotzinian, NP B441 (1995) 234, Mulders and Tangerman, NP B461 (1996) 197; Boer and Mulders, PR D57 (1998) 5780, Bacchetta et al., PL B595 (2004) 309, Bacchetta et al., JHEP 0702 (2007) 093, M. Anselmino, M. Boglione, U.D’Alesio, S. Melis, F. Murgia, A. Prokudin, (in preparation).

M. BoglioneInt. School Niccolo' Cabeo

M. Boglione 30

(Avakian, Efremov, Schweitzer, Metz, Teckentrup, arXiv:0902.0689)

chiral-even TMDs

chiral-odd TMDs

Cahn kinematical effects


M. Boglione 31

TMD’s in SIDISone by one …


M. Boglione 32

)ˆˆ( ),(),(

)ˆˆ( ),(2

1),(),(

1/

//,/

kpS

kpSkS

kxfM

kkxf

kxfkxfxf

q

Tpq

pq

N

pqpq

The Sivers function isrelated to the probabilityof finding an unpolarizedquark inside a transverselypolarized proton

X

p

Sk

The Sivers function inbedsthe correlation between the proton spin and the quark transverse momentum

The Siversfunction is

T-odd


M. Boglione 33

hX'phX'p

hX'phX'p

UT

dσdσ

dσdσA

q

h/q2q/p

2

hS

2

q

q

ShSh/q2q/p

N2

hS

2

q

)φ-sin(φ

UT

)p(z,D dQ

dσ )k(x,f kddφdφe

)φ-sin(φ )φsin( )p(z,D dQ

d )k(x,f Δkd dφdφe

A Sh

Distribution fn. → probability to find quark q carrying a light-cone fraction x of the parent proton momentum, an intrinsic transverse momentum k, at scale Q2.

Fragmentation fn.→ describes the hadronization of the struck quark into the final, detected hadron.

Both mechanisms play an important role in determining total cross section and spin asymmetries

Two soft mechanisms at work in SIDIS:


M. Boglione 34

)k(x,f)h(k(x)2)k(x,fΔ q/pqq/p

N

N

11with

ba

)b(ax)(1x (x)

q

b

q

a

q

)b(a

qqba

qqqq

qq

qq

N

N N

2

2

M

kk2e)h(k

eM

2

2

k

k

2q/pq/p ek

1(x)f)k(x,f

1

)k(x,f2

|)k(x,fΔ|

q/p

q/p

N

Positivity Bound automatically satisfied

|Nq(x)| and |h(k)| are smaller than 1 for any x and for any k


M. Boglione 35

M.Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, S. Melis, F. Murgia, A. Prokudin, C. Türk, Eur.Phys.J.A39:89-100,2009.

Valence quark

Sea quarks

11 parameters

fit


M. Boglione 36

+ production at HERMES K+ production at HERMES

K0 fragmentation functions

M.Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, S. Melis, F. Murgia, A. Prokudin, C. Türk, Eur.Phys.J.A39:89-100,2009.


M. Boglione 37

New data

(2009), old fit !

HERMES Collaboration, Phys.Rev.Lett.103:152002,2009.


M. Boglione 38

Courtesy of F. Bradamante

COMPASS vs HERMES, problems?

COMPASS proton data

S. Levorato for the COMPASS Collaboration, Transversity 2008

COMPASS Collaboration, Phys. Lett. B673: 127-135, 2009

COMPASS deuteron data

Until recently, Compass found

Sivers = 0. Only HERMES data revealed Sivers effect. Need further

check s!Int. School Niccolo' Cabeo

M. Boglione 39

new

new

Compared with the predictions from M. Anselmino et al. Eur.Phys.J.A39:89-100,2009.

COMPASS proton data –new re-analysis of Sivers asymmetry

Anna Martin for COMPASS Collaboration - DIS2010


M. Boglione 42

Courtesy of Alexei Prokudin


M. Boglione 43

Courtesy of Alexei Prokudin


M. Boglione 44

)ˆˆ( ),(

),(

)ˆˆ( ),(2

1 ),(),(

1/

//,/

pps

ppsps

qq

q

h

qh

qqqh

N

qhqh

pzHMz

ppzD

pzDpzDzDq

The Collins functionis related to the probability that a transversely polarized struck quark will fragment into a spinless hadron

The Collins function is

chirally odd

X

qqs

p

The Collins function inbeds the correlation between the fragmenting quark spin and the transverse momentum of the produced hadron


M. Boglione 45

FRAGMENTATION PROCESS

Little portrait of φqh

φqh is the azimuthal angle of the detected hadron h around the

direction of the parent fragmenting quark q. Technically it is the azimuthal angle of p in the helicity frame of q.

xy

z

Ph

k’

PT


cos fqh1 = PT/|p| cos (fh-f) – zh k/PT

sin fqh1 = PT/|p| sin (fh-f)

Int. School Niccolo' Cabeo M. Boglione 46

The Collins function is chirally odd, so it must appear in physical processesconvoluted with another chirally odd function.

In SIDIS it appears , for instance, in combination with either transversity or Boer-Mulders distribution functions.

The transversity distribution function contains basic information on the spinstructure of the nucleons.

Being related to the expectation value of a chiral odd operator, transversityappears in physical processes which require a quark helicity flip; therefore it cannot be measured in usual DIS.

Transversity function can be measured in single or double polarized Drell-Yan → planned experiments in polarized pp at PAX

At present, the only chance of gathering information on the transversitydistribution function is SIDIS, where it appears associated to the Collins fragmentation function.

DOUBLE PUZZLE: we cannot determine the transversity distribution function if we do not know the Collins fragmentation function.


OUR STARTEGY :

A global fit of HERMES and COMPASSSIDIS single spin asymmetries combinedwith BELLE data on e+e-→h1h2X will allowus to determine both the transversitydistribution and the Collins fragmentationfunctions of u and d quarks.

48

The Collins effect in SIDIS couples totransversity

COMPASS 2002-2004

S. Levorato for the COMPASS Collaboration, Transversity 2008

COMPASS proton data

M. BoglioneInt. School Niccolo' Cabeo

M. Boglione 49

• We need to determine two convoluted unknown functions– Fix one of the two functions according to some theoretical model and use

SIDIS data to determine the other (see for example Efremov, Goeke,Schweitzer)

– Perform a simultaneous fit of SIDIS and e+e-→h1h2X BELLE data.

BELLEThrust axis method

BELLEHadronic plane method



Unpolarized distribution

x and z dependencefrom world data fits

Gaussian widths

Gaussian k andp dependences


x and z dependence from world data fits

Gaussian k andp dependences

TransversityDistributionFunction

Collins FragmentationFunction


NuT (varying between -1 and +1)

NdT (varying between -1 and +1)

NfavC (varying between -1 and +1)

NunfC (varying between -1 and +1)

M2

a

β

γ

δ

9 parameters fit


As the two different sets of Belle data are based on a different analysis of the same experimental events, they are strongly correlated.

Therefore, we treat them separately:

FIT I → HERMES + COMPASS + BELLE_cos(φ1+φ2)

FIT II → HERMES + COMPASS + BELLE_cos(2φ0)

By using 2006-BELLE data, Fit I and Fit II are perfectly compatible within the errors; the asymmetries they generate are hardly distinguishable

2008-BELLE data have fantasticly small error bars, but checking their consistency is more problematic. Here we show the results we obtain fromFIT I (HERMES + COMPASS + BELLE_cos(φ1+φ2))

M. Boglione 54

M.Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, S. Melis, F. Murgia, A. Prokudin, C. Türk

R. Saidl et al., Phys. Rev. D 78 (2008) 032011 R. Saidl et al., Phys. Rev. D 78 (2008) 032011


Predictions

M. Boglione 56

)ˆˆ( ),(2

1),(

2

1

)ˆˆ( ),(2

1),(

2

1),(

1/

///,

kps

kpsks

q

q

pq

qpqpqpq

kxhM

kkxf

kxfkxfxfq

The Boer-Mulders functionis related to the probabilityof finding a polarized quark inside an unpolarized proton

X

p

qs

k

The Boer-Mulders function inbeds the correlation between the quark spin and its transverse momentum

The Boer-Mulders

function is chirally odd and T-odd


M. Boglione 57

The angular distribution in the unpolarized SIDIS can be written as

is the usual azimuthal-independent contribution

subleading - Cahn + Boer-Mulders effect

+...other unknown twist-4 terms?

V. Barone, S. Melis and A. Prokudin

ArXiv:0912.5194 (to be published in Phys. Rev. D)


58

Simple parametrization of the Boer-Mulders functions:

for valence quarks

for sea quarks

2 parameters

fit

Sivers function as determined by Anselmino et al, Eur.Phys.J.A39:89-100,2009.

Int. School Niccolo' Cabeo M. BoglioneA. Prokudin, talk at Workshop on Transverse Spin Physics, Beijing (2008)

M. Boglione 59


DSS FF

Gaussians: <k�>=0.25 (GeV/c)2

<p�>=0.20 (GeV/c)2

(from Cahn-cosφ effect)

2

2

GRV98 PDF

Simulated evolution (unp.)

FIT I (C term)

HERMES proton and deuteron

target (x,z,PT) charged hadrons

COMPASS deuteron target

(x,z,PT) charged hadrons

Experimental data


M. Boglione 60

and both negativeFIT I

Bound saturated


M. Boglione 61

Cahn effect (Twist-4)comparable

to BM effect

Same sign of Cahn contribution

for positive and negative pions

BM contribution opposite in sign

for positive and negative pions

FIT I

FIT I


M. Boglione 62

?


M. Boglione 63

Better description of HERMES but the BM is unchanged

FIT II

FIT II

COMPASS HERMES

<k�>=0.25 (GeV/c)2

<p�>=0.20 (GeV/c)2

2

2

<k�>=0.18 (GeV/c)2

<p�>=0.20 (GeV/c)2

2

2

~ EMC ~ HERMES MC


64V. Barone, S. Melis, A. Prokudin ArXiv: 0912.5194

Int. School Niccolo' Cabeo M. Boglione

M. Boglione 65

what about the last 3 TMDs? any relation with the others?

neglecting twist-3 contributions

similar to the Wandzura-Wilczek relation

supported by experiment

for a recent model of all twist-2 TMDs see Bacchetta et al., arXiv:0807.0323


M. Boglione 66

HERMES data, PRL 84 (2000) 4047; PL B562 (2003) 182

COMPASS data, arXiv:0705.2402

H. Avakian., A.V. Efremov, P. Schweitzer, F. Yuan → Bag Model predictionsarXiv:0805.3355 [hep-ph]


67

Spin-orbit correlationsLight-cone SU(6) quark-diquark model, Ma, Schmidt (1998)

P. Schweitzer, talk given at INT Program 09-3, Seattle M. BoglioneInt. School Niccolo' Cabeo

M. Boglione 69

A. Prokudin, talk at Workshop on Transverse Spin Physics, Beijing (2008)


- phenomenological analyses · e d3 d3d k f (x,k ) d (z,p ) sin(3 ) sin(3 -3 ) dq d e d3 d3 d k û...

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