TMDs in nuclei
Jian Zhou
Temple 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 asymmetry
 Summary
nuclear dependent effect
Inclusive process (not too small x)
weak dependence on target size
single 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
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
It turns out that kT broadening is proportional to gluon distribution in the medium.
Moreover,
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 Sterman
2 Single jet in SIDIS
Guo
3 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
Multiple scattering resummed in the collinear factorization: Majumder and Muller
Such resummation is also achieved in the
Wilson line approach
SCET
TMD factorization
Kovner and Wiedemann,
Idilbi and Majumder; D’Eramo, Liu and Rajagopal
Liang, Wang and JZ
Nuclear TMDs
Our starting point:
Ji, Ma, Yuan
where,
Belitsky, Ji and 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.
kT broadening and nuclear TMD
Partial
integration:
In the light cone gauge A+=0,
kT broadening and nuclear TMD
To isolate the leading nuclear effect, we neglect
 dy

A  (0)D ( y ) D ( y ) ( y ) A  A
Integrate over kT
Weakly bound
approximation
Strong nuclear size dependent effect
where
,
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
Transport operator:
Expand the exponential factor, neglect covariant derivative

d
y
 A  (0)D ( y) D ( y) ( y) A  A
Odd power of the operator vanish under the parity invariance, we are left
only with the even-power terms of the expansion,
weakly bound approximation
Maximal two-gluon correlation approximation
The combinatorial factor for grouping 2n number of gluon field
operators into n pairs,
Courtesy of Gao
each gluon pair attaching to different nucleon in nuclei, so that we have the
maximum nuclear size enhancement
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 radiation
Georgi and Politzer
Low pT, parton intrinsic
transverse 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
Gao, Liang and Wang
Nuclear TMDs:
with given twist-2 and twist-3 TMDs in
a nucleon, one can then calculate nuclear
dependence 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 dependent effect: direct photon production
in SIDIS
As long as lT<<Q, TMD factorization is valid,
TMD factorization formula reads,
where,
Fragmentation TMDs,
In particular,
: the probability of finding a photon in a quark
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,
Courtesy of Gao
lT<<Q,
lT>>ΛQCD,
TMD factorization;
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...
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.