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

2. Introduction Main Parts Summary
Outline
Introduction
Main Parts
Summary

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

4. 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.

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

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

7. Choosing and using we get the vector current curl equation:

8. The Quantized Theory : quantized : classical external field anomaly:

9. < >: 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.

10. 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.)

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

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

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

14. Using and we can write (1)

15. The order contribution to is given by

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

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

18. After multiplying by we get the order contribution to :

19. regularization: two auxiliary masses and We write (3)

20. 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

21. So eq.(3) becomes
(3’)

22. 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.

23. 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.

24. 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.

25. 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.