NLO Corrections to Heavy Flavour Distributions in Polarized Deep-Inelastic Scattering
Felix Hekhorn Marco Stratmann


In this talk, we present for the first time two important extensions to our previous
work on inclusive heavy quark (HQ) polarized electroproduction 
at next-to-leading order accuracy <cit.>.
Firstly, we discuss various exclusive distributions related to the HQ DIS structure function
g_1^Q differential in the transverse momentum or rapidity of the HQ or in
the invariant mass and the correlation in azimuthal angle of the HQ pair. These results
have been obtained with a newly developed parton-level Monte Carlo generator.
Secondly, we provide the electroweak contributions to the neutral current (NC) sector in DIS.

In our first paper <cit.>, we followed the concept of the phase-space 
slicing method <cit.>, dealing with all phase-space integrals 
in a largely analytic way. This restricts one to the computation of single-inclusive distributions
either in the rapidity y or in the transverse momentum p_T of the HQ.
To set up a general parton-level Monte Carlo generator for HQ production in polarized DIS, 
that evaluates all phase-space integrations numerically, we adopt the subtraction method 
based on generalized plus distributions <cit.> to isolate all 
singular regions in phase-space.
With the new program at hand, we get access to basically any HQ distribution of interest
including correlated distributions in the invariant mass or the azimuthal angle
of the HQ pair.
We leave the discussion of the relevant technical details 
to forthcoming publications <cit.> and focus here 
on the most important phenomenological results.



    < g r a p h i c s >

Polarized gluon distribution Δ(x,μ_F^2) 
of the DSSV group <cit.> at the scale μ_F^2 = 19
relevant for charm electroproduction,
including the best fit (solid line) and two extreme variations 
("max Δ g" and "min Δ g") allowed by present uncertainties (shaded band).



The main asset of HQ production is its dominance of gluon-induced processes, in case
of DIS photon-gluon fusion, already at the lowest order (LO) approximation of QCD.
To estimate how the measurement of HQ production in polarized DIS at a future
Electron-Ion Collider (EIC) <cit.>
can help to further our knowledge of the so far 
poorly constrained gluon helicity density Δ g, we show in Fig. <ref>
the current uncertainties which are particularly pronounced for small momentum
fractions x, say below x≃ 0.01. Apart from the best fit result given by
the DSSV group <cit.>, we will also utilize in what follows 
their uncertainty estimates, in particular, the two extreme sets labeled
as "max Δ g" and "min Δ g". For our calculations, we use
the MSTW NLO set of unpolarized PDFs <cit.> which was also
adopted in the DSSV analysis to ensure constraints from positivity.



As a first example, we investigate the single-inclusive
p_T-distribution dg_1^/dp_T, for which we define also 
a corresponding p_T-dependent double-spin asymmetry


    A_1,p_T^(x,Q^2,p_T)    = dg_1^/dp_T/dF_1^/dp_T .


Both are shown in Fig. <ref> for fixed momentum fraction x=10^-3 and
photon virtuality Q^2=10 GeV^2 in a range of p_T which should be
accessible at a future EIC. 
As we consider HQs with mass m, we obtain finite results even for p_T=0.
The asymmetry A_1,p_T^ reaches measurable, percent-level values and exhibits a
strong sensitivity to Δ g, clearly dependent on its size and sign as can be gathered
from the results obtained for the minimal and maximal Δ.



    < g r a p h i c s >

Scale dependence of 2x  dg_1^/dp_T shown in
Fig. <ref> at LO (top) and NLO (middle) accuracy
for simultaneous variations of μ_R^2 and μ_F^2
by a factor of 10 around the default choice μ_0^2 = Q^2+4(m^2+p_T^2).
The bottom panel shows the variation of the K-factor.





In Fig. <ref> we illustrate the theoretical uncertainties associated with 
simultaneous variations of the renormalization (μ_R) and factorization (μ_F) 
scales for the p_T-distribution 2x  dg_1^/dp_T computed with the best fit of the DSSV set.
As is expected, the scale dependence is significantly reduced at NLO accuracy and,
more importantly, its size is much smaller than the spread in the results due to
current uncertainty in Δ g as shown in Fig. <ref>.
We note that including a p_T-dependence in the choice of scale μ is important to
obtain sensible results in the high-p_T region.
 
As a first example that can be only obtained with our newly developed Monte-Carlo code, 
we study in Fig. <ref> the invariant mass distribution of the HQ pair, i.e.,
the differential polarized DIS structure function 2x dg_1^/dM 
and its associated double-spin asymmetry A_1,M^. 
The latter is defined in an analogous way to Eq. (<ref>).
Again, we choose x=10^-3 and Q^2=10 GeV^2 to fix the DIS kinematics in the
EIC range. Here, we set the default scale to μ^2 = Q^2 + 4m^2 + p_T,^2 as in
Ref. <cit.> with p_T, referring to the 
transverse momentum of the HQ pair. As for the single-inclusive p_T distributions
shown in Fig. <ref>, the double-spin asymmetry exhibits a nice sensitivity the
the sign and size of the helicity gluon density adopted in the calculation. Again,
the expected values for A_1,M^ are in the percent level range which should
be accessible with the envisioned high luminosity of a future EIC.

Next, we turn to our recent computation of the electroweak contributions to the neutral current (NC) 
DIS structure functions at NLO accuracy <cit.> 
which are mediated by the exchange of a  boson.
Here, we will consider bottom quark production to optimize the kinematical conditions
and use the PDF sets of the NNPDF group <cit.> throughout.
In order to establish a common notation we denote the coupling of a boson b to a 
fermion f as -i e Γ^μ_b,f with e the universal electric charge and

    Γ^μ_b,f   = g^V_b,fΓ^μ_V + g^A_b,fΓ^μ_A = g^V_b,fγ^μ + g^A_b,fγ^μγ^5,    b∈{,},f∈{,,} .

To deal with the subtleties of γ_5 in higher order calculations
we adopt the MVV scheme <cit.>.



We define the relative contributions of the γ Z and Z boson exchange 
to the full NC, parity-conserving HQ DIS structure functions H∈{F_2^,2xg_1^} by


    R^_H(x,Q^2) = 
    (g_,^V-g_,^A)   η_   H^(x,Q^2)/H^NC(x,Q^2)   and   
    R^_H(x,Q^2) = 
    (g_,^V-g_,^A)^2 η_^2 H^(x,Q^2)/H^NC(x,Q^2) ,


respectively. The factor η_ accounts for the difference between the photon and 
the -propagator. The relevance of the electroweak contributions is 
illustrated in Fig. <ref>. One can deduce two important results. 
Firstly, the -interference term and the 
-contribution do not exhibit the same Q^2-dependence, which can be traced back to fact
that we deal here with massive quarks. Secondly, as expected, the electroweak contributions
become increasingly relevant the higher the Q^2 and they should not be neglected in analyses
of DIS data once Q^2 ≳ 10^3 GeV^2.
 


Finally, with the inclusion of the electroweak exchanges, we also get access to the 
parity-violating DIS structure functions <cit.> 
xF_3^, g_4^, and g_L^ 
which are given in Fig. <ref>. 
As they do not contribute in LO to the fully inclusive DIS cross section and, in addition, 
vanish in the case of p_T-differential distributions <cit.>, 
we shown in Fig. <ref> their rapidity dependence for Q^2=10^3 GeV^2
and x=0.1. The LO results were already discussed long ago in Ref. <cit.> but
– to the best of our knowledge – the NLO results are presented here for the first time.


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