Singluarity and Gauge Linkin Light Cone Gauge

Jianhua Gao

Shandong Unversity at Weihai

J.H. Gao, “Singularities, boundary conditions and gauge link in the light cone gauge ” arXiv:1309.4970 J.H. Gao, “A Derivation of the Gauge Link in Light Cone Gauge ” PRD 83, 094017(2011)

The 9th Circum-Pan-Pacific Symposium on High-Energy Spin PhysicsOctober 28-31, 2013, Shandong University, Jinan, China

Outline

• Introduction• Singularity and regularization in light cone gauge• Derivation of the gauge link in light cone guage• Summary

Collinear quark distribution function

Collinear quark distribution function in covariant gauge:

Collinear quark distribution function in light cone gauge:

Gauge link along the light cone direction: All fields are fixed at

TMD quark distribution function

Transverse momentum dependent quark distribution function (TMD):

Gauge link in TMD in SIDIS:

X. Ji and F. Yuan Phys.Lett.B543,66 (2002); A. Belitsky, X. Ji and F. Yuan Nucl. Phys. B 656,165 (2003)

All fields are fixed at

Sivers function

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)

• … …

Some Definition and Notations

Light cone coordinate system:

Vector in light cone coordinate:

Some useful decomposition: where

where

Gauge field:

In light cone gauge:

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)

Regularization of the Singularity

Advanced:

Retarded:

Antisymmetric:

Typical calculation procedure we often deal with:

Three different boundary conditions:

Gauge link in light cone gauge in SIDIS

One gluon exchange contribution:

Tree diagram in SIDIS:

Quark propagator decomposition:

Gauge link in light cone gauge in SIDIS

Pole contribution:

Integrating by parts:

Gauge link in light cone gauge in SIDIS

Retarded boundary condition:

Advanced boundary condition:

Antisymmetry boundary condition:

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:

Gauge link in light cone gauge in SIDIS

General n-gluon exchange contribution:

Integrating from to one by one:

Gauge link in light cone gauge in Drell-Yan

One gluon exchange contribution:

Tree diagram in Drell-Yan:

Gauge link in SIDIS vs Drell-Yan

Retarded boundary condition:

Advanced boundary condition:

Antisymmetry boundary condition:

SIDIS:

Drell-Yan:

SIDIS:

Drell-Yan:

SIDIS:

Drell-Yan:

Gauge link in light cone gaugeGeneral n-gluon exchange contribution:

Sum over all the orders ( for Drell-Yan):

Retarded in SIDIS:

Advanced in 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:1309.4970

Gauge link in light cone gauge

More general gauge link in light cone gauge:

Transverse gauge link in light cone gauge:

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 !