overview on transverse momentum dependent distribution and fragmentation functions m. boglione

Overview on Transverse Momentum Dependent

Distribution and Fragmentation Functions

M. Boglione

M. Boglione 2Glasgow HERMES Symposium

The nucleon has an internal structure x is the fraction of proton momentum carried by the parton The cross section is the incoherent sum of all partonic contributions convoluted with the parton distribution function, which only depends on x at LO

ˆ)( xfqDIS

q

(E,k) (E,k’)q

xP

P

Deep Inelastic Scattering


QCD corrections induce Q2 dependence

f(x) → f(x,Q2)

DGLAP evolution equations exactly predict this Q2 dependence

Q2 Evolution

NOTE : As we will see later, the evolution of some TMD distribution and fragmentation functions is yet unknown


Unpolarized distribution functions

Δfq↑/p↑(x)Δfsz/+(x)

-

fq/p(x)

-

Helicity distribution functions

qqq

ggg

qqq

ggg

Transversity distribution functions

qqqT

Parton distribution functions


Correlator

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.


Parton distribution functions Very good knowledge of unpolarized distribution functions,

q(x,Q2) and g(x,Q2)

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

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


Parton distribution functions


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

++

––

+ –

SIDIS

+–

- ++–

+ –

+–

Drell -Yan

Transversity


ℓℓ’

*g

P

k

k’

fq(x)

dDh/q (z)

distribution function

XIncoming proton

elementary cross- section

fragmentation function

hfinal detected hadron

)(ˆ)( / zDxf qhqSIDIS

Semi Inclusive Deep Inelastic Scattering


Intrinsic TransverseMomentum


P

xP

k

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

and of hadrons within fragmentation jets

Intrinsic Transverse Momentum


Intrinsic Transverse Momentum

p p

Q2 = M2

qT

qL

l+

l–*

qT distribution of lepton pairs in D-Y processes

pT distribution of hadrons in SIDIS


Intrinsic Transverse Momentum 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) ?


Leading twist TMD CorrelatorMulders and Tangermann, NP B461 (1996) 197, Boer and Mulders, PR D57 (1998) 5780


Transverse Mometum Dependent Distribution Functions

Courtesy of Aram Kotzinian

Correlations between spin

and transverse momentum


Transverse Momentum Dependent Distribution Functions

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

UCL

EON

PO

LARI

ZATI

ON

•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

17M. Boglione

General Formalism with Helicity AmplitudesM. 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 number density of partons of type a inside hadron A


General Formalism with Helicity Amplitudes


Δfsz/p↑(x,k)

Δfsz/p+(x,k)Helicity fn.

Δfsx/p+(x,k)Δfsy/p↑(x,k)

Transversityfunction

Δfsx/p↑(x,k)Transversity

function

Δfsy/p(x,k) Boer-Mulders

function

ΔNfq/p↑(x,k)Sivers function

fq/p(x,k)


Correlations between spin

and transverse momentum



U

L

T

U

L

T

fq/p(x,k)Unpolarized

Δfsz/p+(x,k)Helicity

Δfsx/p↑(x,k) Δfsy/p

↑(x,k)Transversity

Δfsz/p↑(x,k)ΔNfq/p

↑(x,k)Sivers

Δfsy/p(x,k) Boer-Mulders

Δfsx/p+(x,k)

QUARK POLARIZATIONN

UCL

EON

PO

LARI

ZATI

ON

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


General SIDISCross Section


SIDIS FACTORIZATION

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.


TMD’s in SIDISTrento

conventions SIDIS in parton model with intrinsic k┴

Factorization holds at large Q2,

and

(Ji, Ma, Yuan)QCDT kP

);,();,(ˆd);,(d 222 QzDQyQxf hq

lqlq

q qlhXlp

pkk

Unpolarized Cross Section


TMD in unpolarized SIDIS → Cahn Effect

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

Azimuthal dependence induced by quark intrinsic motion

CLAS data, arXiv:0809.1153 [hep-ex] F. Giordano and R. Lamb, AIP Conf.Proc.1149:423-426,2009.

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



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

At further dependence on is generated

A cosφ dependence is also generated by Boer-Mulders Collins term, ⊗via a kinematical effect in dΔσ , not included in this fit.

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

Determine the free parameters by fitting experimental data

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


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

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



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]


Polarized SIDIS cross section

30

Polarized SIDIS cross section

Unpolarizedproton and lepton

Longitudinally polarizedproton,polarized lepton

Transverselypolarizedproton,unpolarized lepton

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 : build appropriately “weighted” azimuthal 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. Boglione 31Glasgow HERMES Symposium(Avakian, Efremov, Schweitzer, Metz, Teckentrup, arXiv:0902.0689)

chiral-even TMDs

chiral-odd TMDs

Cahn kinematical effects

TMD’s in polarized SIDIS


TMD’s in SIDISone by one …


The Sivers Distribution Function

)ˆˆ( ),(),(

)ˆˆ( ),(21

),(),(

1/

//,/

kpS

kpSkS

kxfMk

kxf

kxfkxfxf

qTpq

pq

Npqpq

The Sivers function is related to the probability of finding an unpolarized quark inside a transversely polarized proton

X

pS

k

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

The Sivers function is

T-odd



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

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



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

Compass finds Sivers = 0.

Up to now, only HERMES data

reveal non-zero Sivers effect.

We need further check s!



Determining the Sivers function

Models →

Fits →

M. Anselmino, M. Boglione, J.C. Collins, U. D’Alesio, A.V. Efremov, K. Goeke, A. Kotzinian, S. Menze, A. Metz, F. Murgia, A. Prokudin, P. Schweitzer, W. Vogelsang, F. Yuan

The first and 1/2-transverse moments of the Sivers quark distribution functions. The fits were constrained mainly (or solely) by the preliminary HERMES data in the indicated x-range.

The curves indicate the 1-σ regions of the various parameterizations.

Bacchetta, Gamberg, Goldstain, Mukherjee, Metz, Amrath, Shaefer, Yang, Brodsky, Schmidt, Hwang, Scopetta, Courtoy, Frattini, Vento, Radici, Pasquini, Yuan …


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

Extraction of Sivers Function from SIDIS experimental data

New data,

old fit !


Sivers Function …… to do list

Update the fit using new analysis from HERMES and COMPASS and, possibly, more modern fragmentation functions

Study evolution equations

Test new parametrizations with non-gaussian intrinsic transverse momentum dependence

Comparison SIDIS vs. Drell-Yan


Spin-orbit correlations


A non-zero Sivers function requires non-zero orbital angular momentum !

Lattice [P. Haegler et al.]

Lu>0

Ld<0

[Matthias Burkardt]


3D view of the proton

Courtesy of Alexei Prokudin


The Boer-Mulders Distribution Function

)ˆˆ( ),(2

1),(

2

1

)ˆˆ( ),(2

1),(

2

1),(

1/

///,

kps

kpsks

qq

pq

qpq

Npqpq

kxhM

kkxf

kxfkxfxfq

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

X

pqs

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



W. Käfer, COMPASS collaboration, talk at Transversity 2008, FerraraF. Giordano and R. Lamb, AIP Conf.Proc.1149:423-426,2009.

L. P. Gamberg and G. R. Goldstein, Phys.Rev.D77:094016,2008.

Diquark spectator model



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

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

There are two contributions to <cos 2φ>:•Boer-Mulders + Collins (no suppression in 1/Q)•Cahn effect at O(k

2/Q2), which turns out to be large



?


)ˆˆ( ),(

),(

)ˆˆ( ),(2

1 ),(),(

1/

//,/

pps

ppsps

qqq

hqh

qqqh

Nqhqh

pzHMz

ppzD

pzDpzDzDq

The Collins function is 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

The Collins Fragmentation Function

M. Boglione 47

The Collins Effect in SIDIS

The Collins effect in SIDIS couples to transversity

COMPASS 2002-2004

S. Levorato for the COMPASS Collaboration, Transversity 2008

COMPASS proton data


• 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.

Simultaneous determination of Transversity and Collins functions

BELLEThrust axis method

BELLEHadronic plane method


Simultaneous determination of Transversity and Collins functions

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

Models for the Collins functions


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

The last three TMD’s …


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

COMPASS data, arXiv:0705.2402

The last three TMD’s …

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

52Glasgow HERMES Symposium

Pretzelosity

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

P. Schweitzer, talk given at INT Program 09-3, Seattle

53Glasgow HERMES Symposium

• The Collins fragmentation function is universal (no initial/final state interactions, no effects induced by requiring color gauge invariance)

J. Collins and A. Metz, Phys. Rev. Lett. 93,252001 (2004), F. Yuan, arXiv:0903.4680

• The Sivers distribution function (naively time reversal odd) is subject to initial/final state interaction – color gauge invariance requirements induce color factors (process dependence).

C. J. Bomhof, P. J. Mulders,F. Pijlman, Eur. Phys. J. C 47, 147 (2006)

Example:

YanDrell

pq

NSIDIS

pq

N ff )()(

//

Universality of Sivers and Collins functions

Mark Schlegel, talk given at INT Program 09-3, Seattle


Drell-Yan Processes

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


Hot topics

Universality Factorization

Evolution

Orbital Angular Momentum

TMD’s


Present and future measurements …see talk by Marco Contalbrigo!


Enjoy the

Symposium !

Enjoy Loch Lomond!

Thank you !

overview on transverse momentum dependent distribution and fragmentation functions m. boglione

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