The ratio $R={\rm d}σ_L/{\rm d}σ_T$ in heavy-quark pair leptoproduction as a probe of linearly polarized gluons in unpolarized proton

TL;DR

The paper investigates whether the Callan-Gross ratio R = dσ_L/dσ_T in heavy-quark leptoproduction can reveal the presence and magnitude of linearly polarized gluons inside an unpolarized proton. It employs LO TMD factorization to express cross sections in terms of the unpolarized gluon density f1^g and the linearly polarized density h1^{⊥g}, deriving analytic expressions for R with and without gluon polarization. A key result is that R can reach about 2 when gluons are unpolarized and can be enhanced up to Q^2/(4 m^2) when including h1^{⊥g}, with R^h(r) = 2(1−r)/[1+r+4λ(3−r)] linking R to the polarization parameter r. Consequently, measurements of σ_L and σ_T (and their q_T^2 dependence) at future facilities like the EIC and LHeC could determine the gluon’s linear polarization, advancing our understanding of gluon TMDs, though NLO corrections for the polarized case remain to be computed.

Abstract

We study the Callan-Gross ratio $R={\rm d}σ_L/{\rm d}σ_T$ in heavy-quark pair leptoproduction, $lN\rightarrow l^{\prime}Q\bar{Q}X$, as a probe of linearly polarized gluons inside unpolarized proton, where ${\rm d}σ_T$ (${\rm d}σ_L$) is the differential cross section of the $γ^*N\rightarrow Q\bar{Q}X$ process initiated by a transverse (longitudinal) virtual photon. Note first that the maximal value for the quantity $R$ allowed by the photon-gluon fusion with unpolarized gluons is large, about 2. We calculate the contribution of the transverse-momentum dependent gluonic counterpart of the Boer-Mulders function, $h_{1}^{\perp g}$, describing the linear polarization of gluons inside unpolarized proton. Our analysis shows that the maximum value of the ratio $R$ depends strongly on the gluon polarization; it varies from 0 to $\frac{Q^2}{4m^2}$ depending on $h_{1}^{\perp g}$. We conclude that the Callan-Gross ratio in heavy-quark pair leptoproduction is predicted to be large and very sensitive to the contribution of linearly polarized gluons. For this reason, future measurements of the longitudinal and transverse components of the charm and bottom production cross sections at the proposed EIC and LHeC colliders seem to be very promising for determination of the linear polarization of gluons inside unpolarized proton.

Figures (4)

• Figure 1: Definition of the azimuthal angles $\varphi_Q$ and $\varphi_{\bar{Q}}$ in the nucleon rest frame.
• Figure 2: Callan-Gross ratio $R\,(K_{\perp})\equiv R\,(z=1/2,K_{\perp})$ in charm ( left panel) and bottom ( right panel) electroproduction as a function of $K_{\perp}=|\vec{K}_{\perp}|$ at several values of $Q^2$.
• Figure 3: Callan-Gross ratio $R\,(z)\equiv R\,(z,\vec{K}_{\perp}^2\!=m^2+Q^2/4)$ in heavy quark leptoproduction as a function of $z$ at several values of $\lambda$.
• Figure 4: Maximum value of the Callan-Gross ratio with the contribution of linearly polarized gluons, $R^h(r)$, as a function of $r$ at several values of $\lambda$.