Some questions on quantum Some questions on quantum anomaliesanomalies

Roman Pasechnik

Moscow State University, Moscow&

Bogoliubov Lab of Theoretical Physics, JINR, Dubna

46-th Cracow School of Theoretical Physics, May 27 – June 5, 46-th Cracow School of Theoretical Physics, May 27 – June 5, 20062006

Outline

Classical symmetries Quantum symmetries

?

Anomaly appears then there is the breaking of some classical symmetry in quantum theory

There is no the general principle allowing us to transfer classical symmetries on quantum level

See for review, for example:S. Adler, “Anomalies to all orders” hep-th/0405040 and “Anomalies” hep-th/0411038

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

My talk includes:My talk includes:

MotivationMotivation

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.

The axial anomaly (AA): basic definitionsThe axial anomaly (AA): basic definitions

AA occurs only at one-loop levelAA occurs only at one-loop level The AVV amplitudeThe AVV amplitude

Rosenberg’s representationRosenberg’s representation

Symmetric propertiesSymmetric propertiesThe anomalous axial-vector Ward identityThe anomalous axial-vector Ward identity

(*)

Dispersion approach to the axial anomaly: Dispersion approach to the axial anomaly:

a brief reviewa 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:

Dispersion approach to the axial anomaly: Dispersion approach to the axial anomaly:

a brief reviewa brief review

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

Vainshtein’s non-renormalization theoremVainshtein’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

There is the symmetry of the amplitude under permutation in the chiral limit

(**)

As a result the relations (**) get no the perturbative corrections from gluon exchanges

The anomaly is expressed only through :

Vainshetein’s non-renormalization theorem: Vainshetein’s non-renormalization theorem: dispersion point of viewdispersion point of view

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 If the relation (***) gets no the perturbative corrections in the higher orders then it can provide the non-renormalization theorem for transversal part then it can provide the non-renormalization theorem for transversal part of the triangle for arbitrary fermion's mass.of the triangle for arbitrary fermion's mass.

(***)

We have two dispersion relations for AA. The equaling of l.h.s. of this relations with and being interchanged gives

Calculation of two loop axial anomalyCalculation 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!

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…

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

! But: Kirill Melnikov, hep-ph/0604205

non-vanishing two loop QCD mass corrections to the AVV correlator exist that is opposite to our result

Standard Model Higgs boson productionStandard 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:

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

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

Fusion of two off-shell gluonsFusion of two off-shell gluons

Symmetry of the amplitude Tensor representation

Formfactors

Effects of gluon virtualitiesEffects of gluon virtualities

Dimensionless parameters

Expansions in the limit

Matrix element

Effects: on matrix element on angular distribution

Cross section

Effects of gluon virtualitiesEffects of gluon virtualities

with full amplitude

with interference term

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.