TMDs in nuclei

Jian ZhouTemple University

Based on paper: Phys.Rev.D77:125010,2008. e-Print: arXiv:0801.0434 [hep-ph] by Liang, Wang and JZ.

Outline:

Brief review on kT broadening phenomena

Nuclear TMDs and kT broadening Nuclear dependent azimuthal asymmetrySummary

nuclear dependent effectInclusive process (not too small x)

weak dependence on target sizesingle hard scattering ( the nuclear PDF)coherent multiple scattering power suppression 1/Q2 A1/3

In order to explore strong nuclear dependence effect, there are few ways to go.

strong nuclear effect:small x region, multiple scales process, no power suppression energy loss, kT broadening

kT broadening and higher-twist collinear approach

It turns out that kT broadening is proportional to gluon distribution in the medium.Moreover,

Transverse momentum distribution at low pT is ill-defined in fixed order perturbative calculation

Moment of pT-distribution is less sensitive to low pT region:

Momentum broadening: sensitive to the medium properties

Baier, Dokshitzer, Mueller, Peign and Schiff

kT broadening in Drell-Yan

First considered in a QED model. Bodwin, Brodsky and Lepage

kT broadening calculated in the collinear factorization,in the covariant gauge, longitudinal gluon carry small transverse momentum. Guo in the light cone gauge, transverse gluon with collinear momentum. Fries

In the collinear factorization, Double scattering contribute to kt broadening

kT broadening in various processes

1 Di-jet(photon-quark) imbalance Luo, Qiu and Sterman2 Single jet in SIDIS Guo3 heavy quarkonia in d+A kang and Qiu

A lot of models for twist-4 collinear correlations are available,Guo; Qiu and Vitev; Fries; Osborne and Wang

Assume nucleon is weakly bounded, gluon and quark come from the different nucleon,

Conclusion:

Resummation

Such resummation is also achieved in the

Multiple scattering resummed in the collinear factorization: Majumder and Muller

Wilson line approach Kovner and Wiedemann,SCET Idilbi and Majumder; D’Eramo, Liu and RajagopalTMD factorization Liang, Wang and JZ

Nuclear TMDs Our starting point:

where,Ji, Ma, Yuan

In the light cone gauge(A+=0), L|| =1

These gauge links not only make the TMDs gauge invariant but also lead to physical consequences such as single-spin asymmetry and nuclear dependent effect.

Belitsky, Ji and Yuan

kT broadening and nuclear TMD

In the light cone gauge A+=0,

Partialintegration:

kT broadening and nuclear TMD

( )( ( )( )0)D y Dd A y A Ayy

To isolate the leading nuclear effect, we neglect

Weakly bound approximation

where,

Strong nuclear size dependent effect

Integrate over kT

kT distribution and nuclear TMD Infinite multiple scattering effect have been encoded in the gauge link, one should

be able to reach resummation formula by manipulating the gauge link.

Using this relation again,

kT distribution and nuclear TMD

Expand the exponential factor, neglect covariant derivative

weakly bound approximation

( )( ( )( )0)D y Dd A y A Ayy

Odd power of the operator vanish under the parity invariance, we are left only with the even-power terms of the expansion,

Transport operator:

Maximal two-gluon correlation approximation

The combinatorial factor for grouping 2n number of gluon field operators into n pairs,

each gluon pair attaching to different nucleon in nuclei, so that we have themaximum nuclear size enhancement

Courtesy of Gao

Gaussian distribution

Inserting this expression into nuclear TMD, one ends up with,

Replace the delta function with , and integrate over

where

Taking into account intrinsic transverse momentum in a nucleon, nuclear TMD modified as,

Azimuthal asymmetry in SIDIS

Unpolarized cross section,

High pT, gluon radiationGeorgi and Politzer

Low pT, parton intrinsictransverse momentum Cahn

Nuclear dependent effect: jet production in SIDIS

Gao, Liang and Wang

Twist-3 TMD distribution

free partons g=0, using equation of motion,

Reproduce the well known Cahn effect result (due to the finite kT) Cahn

Nuclear dependent effect: jet production in SIDIS

with given twist-2 and twist-3 TMDs ina nucleon, one can then calculate nucleardependence of the azimuthal asymmetry.

To illustrate it qualitatively, using an ansatz of the Gaussian

Conclusion: the azimuthal asymmetry is suppressed by the kT broadening.

Nuclear TMDs:

Gao, Liang and Wang

Nuclear dependent effect: direct photon production in SIDIS

As long as lT<<Q, TMD factorization is valid,

TMD factorization formula reads,

Fragmentation TMDs, : the probability of finding a photon in a quark

where,

In particular, is the counterpart of in the fragmentation section.

Nuclear dependent effect: direct photon production in SIDIS

Fragmentation functions are perturbative calculable in QED.

Transverse momentum conservation:

When,expand structure functions with respect to kT around pT=lT

Finally, structure functions take form,

Nuclear dependent effect: direct photon production in SIDIS

At High lT, twist-4 collinear factorization apply,

lT<<Q, TMD factorization; lT>>ΛQCD, collinear factorization.

One may expect TMD factorization and Collinear factorization yield the same result in the overlap region ΛQCD<<lT<<Q where both apply.

The relevant study is on the way...

Courtesy of Gao

Summary:We demonstrate that the leading nuclear

effect comes from the gauge link in the nuclear TMDs.

Azimuthal asymmetry is suppressed due to the kT broadening.

Outlook:The scale evolution of kT broadening.