Next-to-leading order QCD corrections to inclusive heavy-flavor production in polarized deep-inelastic scattering

We provide a first calculation of the complete next-to-leading order QCD corrections for heavy flavor contributions to the inclusive structure function $g_1$ in longitudinally polarized deep-inelastic scattering. The results are derived with largely analytical methods and retain the full dependence on the heavy quark’s mass. We discuss all relevant technical details of the calculation and present numerical results for the heavy quark scaling functions. We perform important crosschecks to verify our results in the known limit of photoproduction and for the unpolarized electroproduction of heavy quarks. We also compare our calculations to the available, partial results in the polarized case, in particular, in the limit of asymptotically large photon virtualities, and analyze the behavior of the scaling functions near threshold. First steps towards phenomenological applications are taken by providing some estimates for inclusive charm production in polarized deep-inelastic scattering at a future electron-ion collider and studying their sensitivity to the polarized gluon distribution. The residual dependence of heavy quark electroproduction on unphysical factorization and renormalization scales and on the heavy quark mass is investigated.

Figure 1

PGF process ${\gamma }^{*}\mathrm{g}\to Q\overline{Q}$ at LO accuracy. A second Feynman diagram (not shown) is obtained by reversing the heavy quark lines.

Figure 2

Examples for Feynman diagrams contributing to the one-loop virtual corrections to the PGF process. The dotted line in the gluon self-energy correction (e) can represent either a quark, gluon, or ghost loop. Remaining diagrams are obtained by appropriate crossing.

Figure 3

Selected Feynman diagrams contributing to the real gluon emission process in (25). Remaining diagrams are obtained by appropriate crossing.

Figure 4

Feynman diagrams (a), (b) and (c), (d) denote the light quark initiated Bethe-Heitler and Compton process, respectively, that start to contribute to HQ electroproduction at NLO accuracy. Similar contributions arise for incoming light antiquarks.

Figure 5

The top, middle, and lower panel show the Born $c_{k,\mathrm{g}}^{(0)}$, and the NLO $c_{k,\mathrm{g}}^{(1)}$ and ${\overline{c}}_{k,\mathrm{g}}^{(1)}$ contributions, respectively, to the total cross section ${\sigma }_{k,\mathrm{g}}$ at NLO in Eq. (71) as a function of $\eta$ for three different values of $Q^2$ and fixed HQ mass $m=4.75\mathrm{GeV}$. The open and closed symbols denote the projections $k=T$ and $k=P$, respectively.

Figure 6

The top, middle, and lower panel show the light-quark scaling functions $c_{P,q}^{(1)}$, ${\overline{c}}_{P,q}^{(1)}$, and $d_{P,q}^{(1)}$, respectively, at NLO accuracy for three different values of $Q^2$ and fixed $m=4.75\mathrm{GeV}$.

Figure 7

The DIS charm structure functions $2x{F}_1^{\mathrm{c}}$ (left-hand side) and $2x{g}_1^{\mathrm{c}}$ (right-hand side) at LO and NLO accuracy as a function of $x$ for two different values of $Q^2$. The lower panels show the respective $K$-factors, see text; the result for $g_1^{\mathrm{c}}$ is difficult to display because of the zero near $x=7\times {10}^{-3}$. All results were obtained for $m=m_{\mathrm{c}}=1.5\mathrm{GeV}$ and ${\mu }_F^2={\mu }_R^2=4m^2+Q^2$.

Figure 8

The shaded bands illustrate the spread in the predictions for $2x{g}_1^{\mathrm{c}}$ at NLO and LO accuracy (upper row) due to the uncertainties of the DSSV helicity PDFs as a function of $x$ for two different values of $Q^2$. The solid lines refer to the best fit of DSSV. The lower panels show the gluon and light-(anti)quark initiated NLO contributions to $g_1^{\mathrm{c}}$, evaluated from the second and third row of Eq. (85), respectively, and their corresponding spread due to PDF uncertainties.

Figure 9

The spin asymmetry $A_1^{\mathrm{c}}$ for charm quark electroproduction (upper panel) as defined in Eq. (63) at LO and NLO accuracy for $Q^2=10{\mathrm{GeV}}^2$. The shaded bands represent the spread in predictions estimated from the uncertainties of the DSSV set of helicity PDFs. The small inset zooms into the phenomenologically interesting small-$x$ region. The lower panel gives the corresponding $K$-factor for the optimum set helicity PDFs from the DSSV group.

Figure 10

The scale dependence of $F_1^{\mathrm{c}}$, $g_1^{\mathrm{c}}$, and $A_1^{\mathrm{c}}$ at LO and NLO accuracy for two different pairs of $x$ and $Q^2$ values for ${\mu }^2={\mu }_F^2={\mu }_R^2$ in the range ${\mu }_0^2/10\le {\mu }^2\le 10{\mu }_0^2$, where ${\mu }_0^2=4m^2+Q^2$ is our default choice of scale. In each panel, the results are normalized to the ones obtained for ${\mu }^2={\mu }_0^2$.

Figure 11

Similar to Fig. 10 but now for independent variations of ${\mu }_F^2$ (on the “f-axis”) and ${\mu }_R^2$ (“r-axis”) in the computation of the double-spin asymmetry $A_1^{\mathrm{c}}$ at NLO accuracy for $x={10}^{-3}$ and $Q^2=10{\mathrm{GeV}}^2$ (l.h.s.) and $x=0.1$ and $Q^2=100{\mathrm{GeV}}^2$ (r.h.s.). Again, all results are normalized to the one obtained for ${\mu }_F^2={\mu }_R^2={\mu }_0^2=4m^2+Q^2$. The base of each plot shows lines of constant ratio (dashed lines), and the solid line indicates the choice ${\mu }_F^2={\mu }_R^2$ adopted in Fig. 10.

Figure 12

Dependence of the double-spin asymmetry $A_1^{\mathrm{c}}$ for charm electroproduction on the choice of $m^2$ for two pairs of $x$ and $Q^2$ values. All results are normalized to the one obtained for our default choice of the charm quark pole mass $m=m_0=1.5\mathrm{GeV}$ (vertical dotted line).