Transcript 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
?
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:
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One of the applications of axial anomaly: the muon anomalous
magnetic moment
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Useful definitions concerning to the axial anomaly
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Brief description of the dispersive approach to the axial anomaly
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Vainshtein’s non-renormalization theorem: dispersive point of view
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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
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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 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”
at one loop:
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 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.
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!
R.S.Pasechnik, O.V.Teryaev, PRD73, 034017, ’06
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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
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:
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 gluons
Symmetry of the amplitude
Formfactors
Tensor representation
Effects of gluon virtualities
Dimensionless parameters
Expansions in the limit
Matrix element
Cross section
Effects:
on matrix element
on angular distribution
Effects of gluon virtualities
with full amplitude
with interference term
R.S.Pasechnik, O.V.Teryaev, A.Szczurek, Eur. Phys. J. C, in press
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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.