Some questions on quantum anomalies

1. 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

2. 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

3. 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

4. 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.

5. 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 (*)

6. 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:

7. Dispersion approach to the axial anomaly: a brief review writing unsubtracted dispersion relations with respect to we obtain by analogous way

8. 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 :

9. 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.

10. 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!

11. 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

12. 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

13. Fusion of two off-shell gluons Symmetry of the amplitude Tensor representation Formfactors

14. Effects of gluon virtualities Dimensionless parameters Expansions in the limit Matrix element Cross section on angular distribution Effects: on matrix element

15. Effects of gluon virtualities with full amplitude with interference term

16. 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.