singluarity and gauge link in light cone gauge
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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 !
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