Relativistic corrections to gluon fragmentation into the $^3P_{J}^{[1,8]}$ states

TL;DR

The paper computes relativistic $v^2$ corrections to gluon fragmentation into the $^3P_J^{[1,8]}$ quarkonium Fock states within NRQCD. Using Collins-Soper fragmentation and a two-step perturbative program, it shows that $S$-$D$ mixing is necessary to absorb infrared divergences in full QCD and performs a matching to obtain finite short-distance coefficients, which are not simply proportional to their leading-order counterparts. The numerical results reveal negative, sizable $v^2$ corrections with moderate $z$-dependence and a near-constant ratio for certain channels; upon convolution with gluon production at the LHC, these corrections reduce LO predictions by about 30%, indicating that relativistic effects are essential for accurate $J/\psi$ phenomenology and LDME determinations.

Abstract

We compute relativistic corrections to the gluon fragmentation functions to ${}^3P_J^{[1,8]}$ Fock states of heavy quarkonium within non-relativistic QCD factorization framework. We find that, at $\mathcal{O}(v^2)$ sub-leading order, the $S$-$D$ mixing effect must be taken into account to absorb the infrared divergence of spin-triplet $P$-wave production within full QCD into the NRQCD long-distance matrix elements. Unlike the $S$-wave case, we find that the short-distance coefficients of the fragmentation functions at leading and sub-leading order are no longer proportional to each other. However, upon convolution with the gluon production cross section, their ratios are almost constant across the whole $p_T$ region. We find the relativistic corrections to be negative and substantial, which makes them a non-negligible ingredient in the study of $J/ψ$ production at the LHC.

Figures (5)

• Figure 1: The representative Feynman diagrams for gluon fragmenting into $c\bar{c}(^3P_J^{[1,8]})$ at $\alpha_s$ LO.
• Figure 2: The finite part of the SDCs of the fragmentation functions of $g\to c\bar{c} (^3P_J^{[1,8]})$ for $J=0$(left), $J=1$(middle), and $J=2$(right) at QCD LO. The dashed line is for $d_{\mathrm{fin}}^{(0)}(z)\times c^{[1,8]}$, and the solid line is for $d_{\mathrm{fin}}^{(2)}(z)) \times c^{[1,8]}$, where $c^{[1]}=3m_c^5/\alpha_s^2$ for color-singlet and $c^{[8]}=15m_c^5/(16\alpha_s^2)$ for color-octet.
• Figure 3: The ratios between the finite part of the SDCs, $d_{\mathrm{fin}}^{(2)}[g\to c\bar{c}(^3P_J^{[1,8]})]$ and $d_{\mathrm{fin}}^{(0)}[g\to c\bar{c}(^3P_J^{[1,8]})]$, for $J=0,1,2$, in which the $\delta$ and "+" function terms are dropped.
• Figure 4: The ratios $R(n)$ for $n=$$^3P_{J}^{[1,8]}$ as function of $p_T$ under CMS conditions at the $\sqrt{s}=7$ TeV LHC in rapidity range of $|y|<1.2$.
• Figure 5: The K-factors $K(n)$ for $n=$$^3P_{J}^{[1,8]}$ as function of $p_T$ under CMS conditions at the $\sqrt{s}=7$ TeV LHC in rapidity range of $|y|<1.2$. The central solid line is calculated at $\langle v_n^{2}\rangle^{H}=0.25$, and the band is obtained by varying $\langle v_n^{2}\rangle^{H}$ from 0.2 to 0.3.