References - Bogoliubov Laboratory of Theoretical Physics - JINR

where
Rμ = ∂μ
�
G a ρν ˜ G ρν,a
�
− 4(DαG να ) a G˜ a
μν . (3)
The contribution [3] of heavy (say, charm) quarks to the nucleon forward matrix
element is
〈N(p, λ)|j (c)
αs
5μ (0)|N(p, λ)〉 =
48πm2 〈N(p, λ)|Rμ(0)|N(p, λ)〉 (4)
c
Note that the first term in Rμ does not contribute to the forward matrix element
because of its gradient form, while the contribution of the second one is rewritten, by
making use of the equation of motion, as matrix element of the operator
〈N(p, λ)|j (c)
αs
5μ (0)|N(p, λ)〉 = 12πm2 〈N(p, λ)|g
c
�
f=u,d,s ¯ ψfγν ˜ G ν
μ ψf|N(p, λ)〉
The parameter f (2)
S
≡ αs
12πm2 2m
c
3 (2)
Nsμf S , (5)
appears in calculations of the power corrections to the first moment
of the singlet part of g1 part of which is given by exactly the quark-gluon-quark matrix
element we got. The non-perturbative calculations are resulting in the estimate ¯ Gc A (0) =
− αs
12π
f (2)
S
( mN
mc )2 ≈−5·10 −4 . The seemingly naive application of this approach to the case of
strange quarks was presented already ten years from now [3] giving for their contribution
to the first moment of g1 roughly −5 · 10 −2 , which is compatible with the experimental
data which is preserved also now despite the problem of matching DIS and SIDIS analysis.
At that time the reason for such a success which was mentioned in [3] (and emerged
due to discussion with Sergey Mikhailov who is participating in this meeting) was the
possible applicability of a heavy quark expansions for strange quarks [4, 5] in the case of
the vacuum condensates of heavy quarks. That analysis was also related to the anomaly
equation for heavy quarks, however, for the trace anomaly, rather than the axial one.
The current understanding may also include what I would call ”multiscale” picture
of nucleon with (squared) strange quark mass (and Λ) being much smaller than that of
nucleon but much larger than (genuine)higher twist parameter. Whether the (”semiclassical”
as Maxim Polyakov calls it) smallness of higher twists holds for consecutive terms
in the series of higher twists may be checked by use of very accurate JLAB data for g1.
The simplest case of course is the non-singlet combination related to Bjorken Sum Rule.
As higher twists are more pronounced at low Q 2 , one should take care on the Landau
singularities which may be achieved by use of Analytic Perturbation Theory (the main
experts in which are here) or Simonov’s soft freezing. The result [6] looks like a first terms
of converging series of higher twists compatible with semiclassical picture.
It is instructive to compare the physical interpretation of gluonic anomaly for massless
and massive quarks. While in the former, most popular, case it corresponds to the
circular polarization of on-shell gluons (recall, that it is rather small, according to various
experimental data ), in the case of massive quarks one deals with very small (because
of small higher twist strength) correlation of nucleon polarization and a sort of polarization
of off-shell gluons. As soon as strange quark mass is not very large, it partially
compensated the smallness of higher twist and this gluon polarization is transmitted to
the non-negligible strange quark polarization. The role of gluonic anomaly is therefore to
produce the ”anomaly-mediated” strangeness polarization.
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