Download Pasechnik

Dispersive approach to axial
anomaly and hadronic
contribution to g-2
Roman Pasechnik
Bogoliubov Lab of Theoretical Physics, JINR, Dubna
&
Moscow State University, Moscow
SPIN-Praha-2005
My talk includes:

Some motivations and significance of our work

Useful definitions concerning to the axial anomaly

Brief description of the dispersive approach to the axial
anomaly

Short review of Vainshtein’s non-renormalization theorem

Non-renormalization theorem from dispersive point of view

Calculation of two loop axial anomaly and interpretation

Formulation of the Vainshtein’s theorem for massive case

Summary
SPIN-Praha-2005
Motivations and significance


There is a class of electroweak contributions to the muon g-2 containing a
fermion triangle along with a virtual photon and Z boson
For the determination of the muon anomalous magnetic moment (g-2) we are
interested in the
transition between virtual Z and
in the presense
of the external magnetic field to first order in this field
The axial anomaly (AA): basic definitions
AA occurs only at one-loop level
The AVV amplitude
Rosenberg’s representation
The anomalous axial-vector Ward identity
Symmetric properties
(*)
Dispersion approach to the axial anomaly:
a brief review
where
Imaginary parts satisfy non-anomalous Ward identity
With (*) we get
Therefore the occurrence of the axial anomaly is equivalent to a “sum rule”
Dispersion approach to the axial anomaly:
a brief review
writing unsubtracted dispersion relations with respect to
we obtain by analogous way
Vainshtein’s non-renormalization theorem
Let
is a source of a soft photon with polarization vector
then
It is well-known that in the chiral limit at one-loop level
(**)
or
in the chiral limit
There is the symmetry of the amplitude under permutation
As a result the relations (**) get no the perturbative corrections from gluon exchanges
The anomaly is expressed only through
:
Vainshetein’s non-renormalization theorem:
dispersion point of view
We have two dispersion relations for anomaly. The equaling
of l.h.s. of this relations with and being interchanged gives
(***)
is the same with imaginary part of (**) for real external photons in the chiral limit
at the one-loop level.
In difference from Vainshtein’s approach within the dispersion approach we have
two dispersion relations for axial anomaly including both structures
If the relation (***) gets no the perturbative corrections in the higher orders
then it can provide the non-renormalization theorem for transversal part
of the triangle for arbitrary fermion mass.
To check this we have to calculate
in two loop approximation
Calculation of two loop axial anomaly
We have calculated the imaginary part of the third
formfactor
corresponding to the full two loop
amplitude in both kinematics. The result
is zero!
Summary


The Ward identity is proved up to two loop
level in both cases of the external momenta
corresponding to two real photons and one
real and one virtual photons
Vainshtein’s theorem is expanded for arbitrary
fermion masses in the triangle loop for above
cases