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Some questions on quantum anomalies. Roman Pasechnik Moscow State University, Moscow & Bogoliubov Lab of Theoretical Physics, JINR, Dubna 46-th Cracow School of Theoretical Physics, May 27 – June 5, 2006. Outline. ?. Classical symmetries. Quantum symmetries.

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Some questions on quantum anomalies

Some questions on quantum anomalies

Roman Pasechnik

Moscow State University, Moscow

&

Bogoliubov Lab of Theoretical Physics, JINR, Dubna

46-th Cracow School of Theoretical Physics, May 27 – June 5, 2006


Outline
Outline

?

Classical symmetries

Quantum symmetries

There is no the general principle allowing us to transfer

classical symmetries on quantum level

Anomaly appears then there is the breaking of some

classical symmetry in quantum theory

See for review, for example:

S. Adler, “Anomalies to all orders” hep-th/0405040 and “Anomalies” hep-th/0411038


My talk includes
My talk includes:

  • One of the applications of axial anomaly: the muon anomalous magnetic moment

  • Useful definitions concerning to the axial anomaly

  • Brief description of the dispersive approach to the axial anomaly

  • Vainshtein’s non-renormalization theorem: dispersive point of view

  • One of the applications of trace anomaly: the Higgs boson production in a fusion of two gluons

  • The calculation of off-shell effects on the amplitude and cross section


Motivation
Motivation

  • There is a class of electro-weak 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 presence of the external magnetic field to first order in this field. This is the motivation for studying anomalous AVV amplitude in detail.


Some questions on quantum anomalies

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
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”

at one loop:


Some questions on quantum anomalies

Dispersion approach to the axial anomaly: a brief review

writing unsubtracted dispersion relations with respect to we obtain by analogous way


Some questions on quantum anomalies

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 :


Some questions on quantum anomalies

Vainshetein’s non-renormalization theorem:

dispersion point of view

We have two dispersion relations for AA. The equaling of l.h.s. of this relations

with and being interchanged gives

(***)

is the same with the 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's mass.


Some questions on quantum anomalies

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!


Some questions on quantum anomalies

R.S.Pasechnik, O.V.Teryaev, PRD73, 034017, ’06

  • The dispersive approach to the axial anomaly is postulated to be valid in the higher orders of perturbation theory

  • 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

  • It is proposed to expand the Vainshtein’s non-renormalization theorem for arbitrary fermion's masses in the triangle loop for above cases. But this work is still in progress now…

! But: Kirill Melnikov, hep-ph/0604205

non-vanishing two loop QCD mass corrections

to the AVV correlator exist that is opposite to our result


Some questions on quantum anomalies

Standard Model Higgs boson production

The dominant production mechanism at hadron colliders is

via gluon-gluon fusion

The amplitude for on-shell gluons is well-known

(effective Lagrangian approach):

  • We posed the following problems:

  • to take into account the non-zeroth gluon virtualities in the amplitude including finite (not infinite) masses of quarks in the loop

  • to calculate the matrix element and inclusive cross-section in the framework of kt-factorization approach


Some questions on quantum anomalies

Fusion of two off-shell gluons

Symmetry of the amplitude

Tensor representation

Formfactors


Some questions on quantum anomalies

Effects of gluon virtualities

Dimensionless parameters

Expansions in the limit

Matrix element

Cross section

on angular distribution

Effects:

on matrix element


Some questions on quantum anomalies

Effects of gluon virtualities

with full amplitude

with interference term


Some questions on quantum anomalies

R.S.Pasechnik, O.V.Teryaev, A.Szczurek, Eur. Phys. J. C, in press

  • We have analyzed the effect of the non-zeroth virtualities of external gluons on the amplitude of a scalar Higgs boson production. We found a new term in the amplitude compared to the recent effective Lagrangian calculation.

  • The relative drop of the averaged square of the matrix element is about 1% or less at relevant physical parameters, so this effect could be verified in the high precision experiments only.

  • The effect of the non-zeroth virtualities on the angular distribution is much more significant due to a quick growth of the second formfactor.