Transverse Axial Vector and Vector Anomalies in U(1) Gauge Theories Wei-Min Sun Department of Physics Nanjing University 2004.8.5 Collaborators: Hong-Shi Zong Xiang-Song Chen Fan Wang

Introduction Main Parts Summary Outline Introduction Main Parts Summary

Introduction Longitudinal and Transverse Anomalies The Motivation for Studying Transverse Anomaly transverse anomaly transverse WT identities fermion-gauge-boson vertex function DS equation

The Main Methods for Calculating Transverse Anomaly the point-splitting method the perturbative method the path-integral method The Present Status of the Study of Transverse Anomaly K.-I. Kondo, Int. J. Mod. Phys. 12(1997), 5651. H. X. He, Phys. Lett. B507(2001), 351. W.-M. Sun et al, Phys. Lett. B569(2003), 211.

The Classical Theory The equation of motion of the Dirac field Main Parts The Classical Theory The equation of motion of the Dirac field

Choosing and using we get the axial vector current curl equation:

Choosing and using we get the vector current curl equation:

The Quantized Theory : quantized : classical external field anomaly:

< >: vacuum expectation value in the presence of external field The v.e.v. in the above two equations are functionals of the external field . The anomaly terms, if exist, are also functionals of . Our method: calculate perturbatively (in powers of ) the v.e.v. of both sides of the above two equations and see whether there exists an anomaly term.

The perturbative expansion of the v. e. v The perturbative expansion of the v.e.v. generates a series of one-loop diagrams: (a typical one-loop diagram for the order contribution of the v.e.v.)

The axial vector current: The order contribution to is given by where

Pauli-Villars regularization: and : polynomials in of degree coefficient of for large

Therefore and ensures that be convergent for all . Such conditions may be realized through the introduction of two auxiliary masses:

Using and we can write (1)

The order contribution to is given by

regularization: two auxiliary masses and Using we can write (2)

To calculate the order contribution one needs to calculate the order contribution to . The latter is given by

After multiplying by we get the order contribution to :

regularization: two auxiliary masses and We write (3)

In the second integral in the RHS of eq.(3), we first rename the integration (summation) variables (indices): and then make the shift to obtain

So eq.(3) becomes (3’)

Comparing eqs.(1),(2) ,(3’), we find that the axial vector current curl equation is satisfied (at the level of v.e.v.) and there is no transverse anomaly for the axial current.

By parallel procedure one finds that the vector current curl equation is satisfied (at the level of v.e.v.) and there is no transverse anomaly for the vector current.

Other regularization schemes. dimensional regularization: in dimensions. The whole arguments given above in the language of Pauli-Villars regularization can be presented in the language of dimensional regularization without essential changes, and the conclusions are the same.

Summary Using perturbative methods we show that there are no transverse anomalies for both the axial vector and vector current in U(1) gauge theories. The Pauli-Villars regularization and dimensional regularization give the same results on the transverse anomaly of the axial vector and vector current.