Complete NLO BFKL impact factors for quarkonium hadroproduction in NRQCD: the case of ${}^1S_0^{[1]}$, ${}^1S_0^{[8]}$, and ${}^3S_1^{[8]}$ states

TL;DR

This work delivers the first complete next-to-leading-order BFKL impact factors for forward/backward hadroproduction of S-wave quarkonia within NRQCD, including the gluon- and quark-initiated channels for both colour-singlet and colour-octet states. It completes the previously obtained one-loop virtual corrections with the full real-emission contributions, demonstrating the cancellation of soft divergences and the factorization of collinear singularities up to one loop. The authors provide explicit analytic and subtraction-based results that establish a consistent NLO framework and enable the first next-to-leading-logarithmic (NLL) studies of forward-backward quarkonium production at hadron colliders. These results advance the application of high-energy factorization to heavy-quarkonium phenomenology and pave the way for matching to NRQCD and other formalisms, including future P-wave state extensions.

Abstract

We present the first complete next-to-leading order calculation of the impact factors for hadroproduction of $S$-wave quarkonium states within the BFKL formalism. We present the computation of the real-emission contributions which completes the recent one of one-loop virtual corrections by one of us for the impact factors for the ${}^1S_0^{[1]}$, ${}^1S_0^{[8]}$, and ${}^3S_1^{[8]}$ NRQCD states. We prove the cancellation of soft divergences between real and virtual contributions, and that the surviving collinear singularities are compatible with factorisation up to one loop for a novel class of processes where BFKL resummation can be applied. Our work indeed represents the first complete NLO quarkonium impact factor in the BFKL framework and paves the way to first next-to-leading-logarithmic-precision studies for hadroproduction of forward-backward quarkonium associated production at hadron colliders.

Figures (6)

• Figure 1: Inclusive production of a quarkonium pair with a large rapidity separation in hadron-hadron collision in the BFKL framework, at cross-section level. The figure shows the production mechanism in the case of two incoming on-shell gluons of momentum $x_1 P_1^+$ and $x_2 P_2^-$, out of the two scattered hadrons of momentum $P_1$ and $P_2$, whose distributions are described by two PDFs. The cross section involves the convolution of the two IFs $V_g^{({\cal Q}_1)}$ and $V_g^{({\cal Q}_2)}$, drawn as dashed ellipses, with a BFKL Green's function, drawn as rectangular box, which takes into account the multiple gluonic emissions treated inclusively, through the exchange of reggeons (drawn as dashed lines). In each IFs $V_{a_1}^{({\cal Q}_1)}$ and $V_{a_2}^{({\cal Q}_2)}$ (here with $a_1=a_2=g$), a $Q \bar{Q}$ pair, of quantum numbers $m_1$ and $m_2$, are produced perturbatively, as described by the impact factors $V_{a_1}^{(m_1)}$ and $V_{a_2}^{(m_1)}$, each of them being depicted as the two upper grey blobs and the two lower blobs respectively. The hadronisation of each state $Q \bar{Q}[m_1]$ or $Q \bar{Q}[m_2]$ into each quarkonium ${\cal Q}_1$ and ${\cal Q}_2$ is encoded in the corresponding LDME, drawn as a rectangular box. This hadronisation may involve an arbitrary number of soft gluonic emissions, whose minimal number depends on the $Q \bar{Q}[m_1]$ state, as illustrated by the two drawn gluons in the LDME boxes through dark blobs. As illustrated in the upper part of the diagram, in the case of NLO real emissions, a hard gluon with transverse momentum ${\boldsymbol{k}}_{1 T}$ and momentum fraction $1-z_1$ may be involved in the short-distance production of the $Q \bar{Q}[m_1]$ state.
• Figure 2: Feynman diagrams for the process $Rg\to Q\bar{Q}$ in the EFT Lipatov95 contributing to the LO IFs (\ref{['eq:LO-IF']}) -- (\ref{['eq:LO-IF-3S18']}). Dashed line denotes the reggeised gluon ($R$) and the lines labelled by $c$ denote a heavy quark $Q$.
• Figure 3: Diagrammatic representation for the factorisation of the amplitude (\ref{['eq:proc-MRK']}) in the MRK (\ref{['eq:A-MRK']}) and QMRK (\ref{['eq:AQMRK^2']}). The dashed lines denote reggeised gluon propagators.
• Figure 4: Feynman diagrams with $Rg$-transition vertex for the subprocess (\ref{['eq:real-proc']}). Heavy quark ($Q$) lines are labelled by $c$. Dashed lines denote reggeised gluons.
• Figure 5: Feynman diagrams with $Rgg$ and $Rggg$ induced vertices for the subprocess (\ref{['eq:real-proc']}). Heavy quark ($Q$) lines are labelled by $c$. Dashed lines denote reggeised gluons.