How to include exclusive $J/ψ$ production data in global PDF analyses

TL;DR

This work shows that exclusive $J/\psi$ photoproduction can be incorporated into global PDF analyses at NLO within collinear factorization by using an optimal factorization scale $\mu_F = M_\psi/2$ to absorb double logarithms into the PDFs and a $Q_0$-subtraction to avoid double counting. The Shuvaev transform is employed to relate generalized parton distributions to forward PDFs at small $x$, enabling a direct link between exclusive production and the gluon PDF. Comparisons with HERA data demonstrate compatibility of existing PDFs, while the high-precision LHCb ultraperipheral data have the potential to constrain the gluon PDF down to $x \sim 3\times 10^{-6}$ over a broad kinematic range at a fixed low scale. The approach provides a path to significantly reduce low-$x$ PDF uncertainties and to incorporate exclusive $J/\psi$ measurements into future global fits, with implications for precision phenomenology at the LHC and beyond.

Abstract

We compare the cross section for exclusive $J/ψ$ photoproduction calculated at NLO in the collinear factorization approach with HERA and LHCb data. Using the optimum scale formalism together with the subtraction of the low $k_t<Q_0$ contribution from the NLO coefficient function to avoid double counting we show that the existing global parton distribution functions (PDFs) are consistent with the data within their uncertainties. However, at low $x$ the uncertainties of the present global PDFs are large. On the other hand, the accuracy of the LHCb data are rather good. Therefore, these data provide the possibility to directly measure the gluon PDF over the very large interval of $x$, $10^{-6}<x<10^{-2}$, at a fixed low scale.

Figures (6)

• Figure 1: (a) LO contribution to $\gamma p \to V +p$. (b) NLO quark contribution. For these graphs all permutations of the parton lines and couplings of the gluon lines to the heavy-quark pair are to be understood. Here the momentum $P\equiv (p+p^\prime)/2$ and $l$ is the loop momentum. Note that the momentum fractions of the left and right partons are $x=X+\xi$ and $x'=X-\xi$ respectively; for the upper gluons we have $x' \ll x$ and so $x\simeq 2\xi$.
• Figure 2: LO and LO+NLO contributions to the imaginary part of the $\gamma p \to V +p$ amplitude as a function of the $\gamma p$ centre-of-mass energy, $W$, with $\mu_F = m_c$ before (left panel) and after (right panel) the double counting correction has been implemented, as explained in the text. The dashed, continuous and dot-dashed (red) curves correspond to three choices of the factorization scale $\mu_f$: namely $\mu_f^2=2m_c^2,~m_c^2,~Q_0^2,$ respectively, where $m_c^2=M^2_\psi /4=2.4$ GeV$^2$. Here $Q_0=1.3$ GeV is the starting scale of the input PDFs from CTEQ6.6 CTEQ6.6 which were used. The dotted black curve is the LO contribution.
• Figure 3: The gluon LO+NLO and quark NLO contributions to the imaginary part of the $\gamma p \to J/\psi +p$ amplitude for two different choices of the factorization scale $\mu_f^2=\mu_R^2=m^2_c,~2m_c^2$ shown by the continuous and dashed curves respectively. CT14 global PDFs CT14 are used and the 'optimal' scale $\mu_F=m_c$ is chosen.
• Figure 4: The $\gamma p\to J/\psi+p$ data obtained at HERA HERA and LHCb LHCb compared with the predictions obtained using the PDFs taken from three different sets of global partons NNPDFMMHTCT14 with $\mu_f = m_c$ (solid lines). The dashed line for the CT14 prediction, corresponding to $\mu_f^2 = 2m_c^2$, is added to demonstrate the scale stability of our NLO predictions; but note that our optimal choice $\mu_f^2 = m_c^2$ agrees better with the HERA data.
• Figure 5: The two diagrams describing exclusive $J/\psi$ production at the LHC. The vertical lines represent two-gluon exchange. Diagram (a), the $W_+$ component, is the major contribution to the $pp \to p+J/\psi +p$ cross section for a $J/\psi$ produced at large rapidity $Y$. Thus such data allow a probe of very low $x$ values, $x\sim M_{\psi} {\rm exp}(-Y)/\sqrt{s}\,$; recall that for two-gluon exchange we have $x\gg x'$. The $q_T$ of the photon is very small and so the photon can be considered as a real on-mass-shell particle.