1. Singluarity and Gauge Link in Light Cone Gauge
Jianhua Gao
Shandong Unversity at Weihai
J.H. Gao, “Singularities, boundary conditions and gauge link in the light cone gauge ” arXiv:
J.H. Gao, “A Derivation of the Gauge Link in Light Cone Gauge ” PRD 83, (2011)
The 9th Circum-Pan-Pacific Symposium on High-Energy Spin Physics
October 28-31, 2013, Shandong University, Jinan, China

2. Outline
Introduction
Singularity and regularization in light cone gauge
Derivation of the gauge link in light cone guage
Summary

3. Collinear quark distribution function
Collinear quark distribution function in covariant gauge:
Gauge link along the light cone direction:
All fields are fixed at
Collinear quark distribution function in light cone gauge:

4. TMD quark distribution function
Transverse momentum dependent quark distribution function (TMD):
All fields are fixed at
Gauge link in TMD in SIDIS:
Sivers function
X. Ji and F. Yuan Phys.Lett.B543,66 (2002); A. Belitsky, X. Ji and F. Yuan Nucl. Phys. B 656,165 (2003)

5. Some works on transverse gauge link
X. Ji and F. Yuan Phys.Lett.B543,66 (2002)
A. Belitsky, X. Ji and F. Yuan Nucl. Phys. B 656,165 (2003)
D. Boer, P. Mulders, F. Pijlman Nucl.Phys.B667,201(2003)
A.Idilbi, A.Majumder, Phys. Rev. D80,054022(2009)
I. Cherednikov and N. Stefanis Int.J.Mod.Phys.Conf.Ser.4,135(2011)
… …

6. Some Definition and Notations
Light cone coordinate system:
Vector in light cone coordinate:
Some useful decomposition: where
where
Gauge field:
In light cone gauge:

7. Boundary conditions and singularity
Maxwell equations in coordinate space:
Boundary constraint:
Maxwell equations in momentum space:
In light cone gauge:
Gluon propagator:
A. Bassetto, I. Lazizzera and R. Soldati Phys.Lett. B107,278 (1981)

8. Regularization of the Singularity
Three different boundary conditions:
Advanced:
Retarded:
Antisymmetric:
Typical calculation procedure we often deal with:

9. Gauge link in light cone gauge in SIDIS
Tree diagram in SIDIS:
One gluon exchange contribution:
Quark propagator decomposition:

10. Gauge link in light cone gauge in SIDIS
Pole contribution:
Integrating by parts:

11. Gauge link in light cone gauge in SIDIS
Retarded boundary condition:
Advanced boundary condition:
Antisymmetry boundary condition:

12. Gauge link in light cone gauge in SIDIS
It follows that (for the retarded boundary condition) :
Gauge field at the infinity:
Only keep the first term and integrate by parts:

13. Gauge link in light cone gauge in SIDIS
General n-gluon exchange contribution:
Integrating from to one by one:

14. Gauge link in light cone gauge in Drell-Yan
Tree diagram in Drell-Yan:
One gluon exchange contribution:

15. Gauge link in SIDIS vs Drell-Yan
Retarded boundary condition:
SIDIS:
Drell-Yan:
Advanced boundary condition:
SIDIS:
Drell-Yan:
Antisymmetry boundary condition:
SIDIS:
Drell-Yan:

16. Gauge link in light cone gauge
General n-gluon exchange contribution:
Retarded in SIDIS:
Advanced in Drell-Yan:
Sum over all the orders ( for Drell-Yan):
For the pure gauge , and the equation has the solution:
The gauge link from pure gauge is independent on the link path!
J.H. Gao, arXiv:

17. Gauge link in light cone gauge
More general gauge link in light cone gauge:
Transverse gauge link in light cone gauge:

18. Thanks ! Summary A proper regularization method is provided to deal
with the light cone singularity in high-twist calculations.
A more general gauge link at light cone infinity can be
obtained naturally from the pinched poles.
The difference of the gauge link between SIDIS and
DY process can be obtained directly in our derivation.
The gauge link at light cone infinity is independent on the
path not only for Abelian but also non-Abelian gauge.
Thanks !