TMDs in nuclei

1. 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.

2. Outline: • Brief review on kT broadening phenomena • Nuclear TMDs and kT broadening • Nuclear dependent azimuthal asymmetry • Summary

3. 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

4. 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

5. 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

6. Assume nucleon is weakly bounded, gluon and quark come from the different nucleon, 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 Conclusion:

7. Resummation Multiple scattering resummed in the collinear factorization: Majumder and Muller Such resummation is also achieved in the Wilson line approach Kovner and Wiedemann, SCET Idilbi and Majumder; D’Eramo, Liu and Rajagopal TMD factorization Liang, Wang and JZ

8. Nuclear TMDs Our starting point: Ji, Ma, Yuan where, Belitsky, Ji and Yuan In the light cone gauge(A+=0), L|| =1 These gauge linksnot only make the TMDs gauge invariant but also lead to physical consequencessuch as single-spin asymmetry and nuclear dependent effect.

9. kT broadening and nuclear TMD Partial integration: In the light cone gauge A+=0,

10. kT broadening and nuclear TMD To isolate the leading nuclear effect, we neglect Integrate over kT Weakly bound approximation Strong nuclear size dependent effect where,

11. 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,

12. kT distribution and nuclear TMD Transport operator: Expand the exponential factor, neglect covariant derivative 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

13. Maximal two-gluon correlation approximation The combinatorial factor for grouping2n 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

14. Replace the delta function with , and integrate over Gaussian distribution Inserting this expression into nuclear TMD, one ends up with, where Taking into account intrinsic transverse momentum in a nucleon, nuclear TMD modified as,

15. Azimuthal asymmetry in SIDIS Unpolarized cross section, High pT, gluon radiation Georgi and Politzer Low pT, parton intrinsic transverse momentum Cahn

16. 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

17. 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.

18. In particular, is the counterpart of in the fragmentation section. Nuclear dependent effect:direct photon production in SIDIS As long as lT<<Q, TMD factorization is valid, TMD factorization formula reads, where, Fragmentation TMDs, : the probability of finding a photon in a quark

19. 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,

20. Nuclear dependent effect:direct photon production in SIDIS At High lT, twist-4 collinear factorization apply, Courtesy of Gao 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...

21. 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.