Some questions on quantum anomalies

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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

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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
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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
• Vainshteins 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

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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.

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The axial anomaly (AA) basic definitions
AA occurs only at one-loop level
The AVV amplitude
Rosenbergs representation
The anomalous axial-vector Ward identity
Symmetric properties
()
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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
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Dispersion approach to the axial anomaly a
brief review
writing unsubtracted dispersion relations with
respect to we obtain by analogous way
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Vainshteins 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
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Vainsheteins 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 Vainshteins 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.
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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!
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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 Vainshteins
  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
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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

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Fusion of two off-shell gluons
Symmetry of the amplitude
Tensor representation
Formfactors
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Effects of gluon virtualities
Dimensionless parameters
Expansions in the limit
Matrix element
Cross section
on angular distribution
Effects
on matrix element
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Effects of gluon virtualities
with full amplitude
with interference term
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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.