Higher-order relativistic corrections to gluon fragmentation into spin-triplet S-wave quarkonium

TL;DR

This work computes the NRQCD short-distance coefficients for gluon fragmentation into a 3S1 quarkonium state through relative order v^4, addressing both single and double soft divergences via analytic subtractions and a two-loop NRQCD operator renormalization. The authors provide a complete matching framework, derive the RG evolution of relevant LDMEs, and present new results for the sum d4,1+d4,2 that governs the color-singlet channel at this order, along with corroborating results for color-octet and color-singlet channels. Numerical analyses reveal a substantial v^4 enhancement in the color-singlet channel near z≈1, driven by soft-divergence remnants, but the effect is not yet large enough to change current phenomenology. The study also shows that gluon fragmentation through 3S1[8] and 3P[J] channels remains phenomenologically significant at high pT, underscoring the importance of higher-order and channel-mouling contributions in quarkonium hadroproduction.

Abstract

We compute the relative-order-v^4 contribution to gluon fragmentation into quarkonium in the 3S1 color-singlet channel, using the nonrelativistic QCD (NRQCD) factorization approach. The QCD fragmentation process contains infrared divergences that produce single and double poles in epsilon in 4-2epsilon dimensions. We devise subtractions that isolate the pole contributions, which ultimately are absorbed into long-distance NRQCD matrix elements in the NRQCD matching procedure. The matching procedure involves two-loop renormalizations of the NRQCD operators. The subtractions are integrated over the phase space analytically in 4-2epsilon dimensions, and the remainder is integrated over the phase-space numerically. We find that the order-v^4 contribution is enhanced relative to the order-v^0 contribution. However, the order-v^4 contribution is not important numerically at the current level of precision of quarkonium-hadroproduction phenomenology. We also estimate the contribution to hadroproduction from gluon fragmentation into quarkonium in the 3PJ color-octet channel and find that it is significant in comparison to the complete next-to-leading-order-in-alpha_s contribution in that channel.

Figures (8)

• Figure 1: Feynman diagram for the vertex that creates a gluon and an eikonal line. The circle represents the operator $G_{a}^{+\nu}$, which creates a gluon with momentum $Q$ and polarization and color indices $\alpha$ and $a$, respectively. $b$ is the color index for the eikonal line. The operator momentum $k$ is the sum of $Q$ and the momentum of the eikonal line.
• Figure 2: Feynman diagram for the fragmentation process $g\to Q\bar{Q}({}^3S_1^{[8]})$. The dashed line represents the final-state cut. The momenta for $Q$ and $\bar{Q}$ on the left side of the cut are $p=\tfrac{1}{2}P+q$ and $\bar{p}=\tfrac{1}{2}P-q$, respectively. The momenta on the right side of the cut are $\tfrac{1}{2}P+q'$ and $\tfrac{1}{2}P-q'$, respectively. Here, $|\bm{q}'|=|\bm{q}|$ in the rest frame of the $Q\bar{Q}$ pair, but we distinguish the directions of $\bm{q}$ and $\bm{q}'$ in order to be able to project out orbital-angular-momentum states in the amplitude and its complex conjugate.
• Figure 3: One of the four Feynman diagrams for the fragmentation process $g\to Q\bar{Q}({}^3P^{[8]})$. Three additional diagrams can be obtained by permuting the gluon-fermion vertices on the left and right sides of the cut.
• Figure 4: One of the 36 Feynman diagrams for the fragmentation process $g\to Q\bar{Q}({}^3S_1^{[1]})$. Thirty-five additional diagrams can be obtained by permuting the gluon-fermion vertices on the left and right sides of the cut.
• Figure 5: One of the four Feynman diagrams for the computation of the LDME $\langle 0|\mathcal{O}_0^{Q\bar{Q}({}^3S_1^{[1]})} [Q\bar{Q}({}^3P^{[8]})]|0\rangle^{(1)}$. The solid circles represent the $Q\bar{Q}$ operators in the LDME. As in the full-QCD calculation, we take the free $Q$ and $\bar{Q}$ states to have momenta $\tfrac{1}{2}P+q$ and $\tfrac{1}{2}P-q$ on the left side of the cut and $\tfrac{1}{2}P+q'$ and $\tfrac{1}{2}P-q'$ on the right side of the cut, where, $|\bm{q}'|=|\bm{q}|$ in the rest frame of the $Q\bar{Q}$ pair, but we distinguish the directions of $\bm{q}$ and $\bm{q}'$ in order to be able to project out orbital-angular-momentum states in the amplitude and its complex conjugate. Three additional diagrams can be obtained by permuting the gluon-fermion and operator vertices on the left and right sides of the cut.