
VOLUME 94  ISSUE 11 
PAGE 867

The analytical singlet α_{s}^{4} QCD contributions into the e^{+}e^{}annihilation Adler function and the generalized Crewther relations

A. L. Kataev
Institute for Nuclear Research RAS, 117312 Moscow, Russia

Abstract
The generalized Crewther relations in the channels of the
nonsinglet and vector quark currents are considered.
These relations follow from the double application of the operator product
expansion approach to the same
axial vectorvectorvector triangle amplitude in two regions,
adjoining to the angle sides
(x,y) (or p^{2},q^{2}).
We assume that the generalized Crewther relations in
these two kinematic regimes result in the
existence of the same perturbation expression for two
products of the coefficient functions
of annihilation and deepinelastic scattering
processes in the nonsinglet and vector channels.
This feature explains
the conformal symmetry motivated cancellations
between the singlet α_{s}^{3} corrections
to the GrossLlewellyn Smith sum rule S_{ GLS} of ν N
deep inelastic scattering and the singlet α_{s}^{3} correction
to the e^{+}e^{}annihilation
Adler function D_{A}^{V} in the product
of the corresponding perturbative series. Taking into account
the BaikovChetyrkinKuhn
4th order result for S_{ GLS}
and the perturbative effects of
the violation of the conformal symmetry in the generalized Crewther relation,
we obtain the analytical
contribution to the singlet α_{s}^{4} correction to the
D_{A}^{V}function.
Its aposteriori comparison with the recent result of
direct diagrambydiagram evaluation of the singlet
4th order corrections
to D_{A}^{V} function demonstrates the coincidence of the predicted
and obtained ζ_{3}^{2}contributions to the singlet term. They
can be obtained
in the conformal invariant limit from the original Crewther relation.
Therefore, on the contrary to previous belief, the appearance
of ζ_{3}terms in the perturbative series in
quantum field theory gauge models
does not contradict to the property of the the conformal symmetry
and can be considered as regular feature.
The BanksZaks motivated
relation between our predicted and the obtained directly
4th order
corrections is mentioned. It
confirms the expectation, previously made by BaikovChetykinKuhn,
that at the 5loop level the generalized Crewther relation in the channel
of vector currents may receive additional singlet contribution, which in
this order of perturbation theory is proportional
to the first coefficient of the QCD βfunction.


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