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

1. Problem Statement

The production of heavy quarkonia (specifically $J/\psi$ and $\chi_{cJ}$) at high transverse momentum ($p_T$) in hadronic collisions is dominated by the fragmentation mechanism. Within the Non-Relativistic QCD (NRQCD) factorization framework, this process is described by the convolution of perturbative Short-Distance Coefficients (SDCs) and non-perturbative Long-Distance Matrix Elements (LDMEs).

While Next-to-Leading Order (NLO) QCD corrections for gluon fragmentation into various quarkonium states have been calculated, a critical gap remained: the relativistic corrections (expansion in the heavy quark velocity squared, $v^2$) for gluon fragmentation into P-wave states ($3P_J^{[1,8]}$).

• Previous studies established that for S-wave states, the $O(v^2)$ SDCs are simply proportional to the Leading Order (LO) SDCs by a constant factor.
• It was unknown whether this proportionality holds for P-wave states.
• Furthermore, calculating P-wave fragmentation at $O(v^2)$ introduces complex infrared (IR) divergences in full QCD that require a specific treatment involving S-D mixing to be absorbed into NRQCD LDMEs.

2. Methodology

The authors employed the NRQCD factorization formalism combined with dimensional regularization to compute the fragmentation functions (FFs) up to $O(v^2)$ sub-leading order.

• Matching Procedure: They calculated the fragmentation function $D_{g \to c\bar{c}(n)}$ in two ways:
  1. Full QCD: Calculated directly using Feynman diagrams for $g \to c\bar{c}(3P_J^{[1,8]}) + g$. The amplitude was expanded in powers of the relative momentum $q$ of the $c\bar{c}$ pair.
  2. NRQCD: Expressed as a sum of SDCs multiplied by NRQCD operators (LDMEs).
• Operator Definitions:
  • Defined operators for Color-Singlet (CS) $3P_J^{[1]}$ and Color-Octet (CO) $3P_J^{[8]}$ states.
  • Crucially, introduced S-D mixing operators (specifically $P(3S_1^{[8]}, 3D_1^{[8]})$) to handle the IR divergences that arise at $O(v^2)$.
• Infrared Divergence Cancellation:
  • In full QCD, IR divergences appear at $z \to 1$ (soft gluon emission).
  • These divergences were matched against the renormalization of NRQCD LDMEs (specifically the $O(\alpha_s v^2)$ corrections to the $3S_1^{[8]}$ LDMEs).
  • The inclusion of the S-D mixing term was found to be essential to absorb all IR divergences, a feature not present in the S-wave case.
• Numerical Analysis:
  • Convolved the resulting finite SDCs with the gluon production cross-section at QCD LO.
  • Simulated conditions for the CMS experiment at $\sqrt{s} = 7$ TeV with $|y| < 1.2$.
  • Evaluated the impact using a K-factor defined as the ratio of the $v^2$-corrected prediction to the LO prediction.

3. Key Contributions

1. First Calculation of $O(v^2)$ Relativistic Corrections for P-wave Gluon Fragmentation: The paper provides the analytical results for the SDCs of gluon fragmentation into both CS and CO $3P_J$ states up to $O(v^2)$.
2. Discovery of Non-Proportionality: Unlike the S-wave case (where $d^{(2)} \propto d^{(0)}$), the authors found that for P-wave states, the $O(v^2)$ SDCs are not proportional to the LO SDCs. They depend on the momentum fraction $z$ in a complex, non-trivial way.
3. Necessity of S-D Mixing: The study demonstrates that to obtain IR-finite results for P-wave fragmentation at $O(v^2)$, one must include the S-D mixing contribution in the NRQCD operator basis. This resolves the IR divergences that cannot be absorbed by S-wave LDMEs alone.
4. Analytical Results: The paper provides explicit analytical formulas for the finite parts of the SDCs for $J=0, 1, 2$ in both color-singlet and color-octet channels.

4. Key Results

• SDC Behavior:
  • The finite parts of the LO SDCs ($d^{(0)}_{fin}$) are positive for all channels.
  • The finite parts of the $O(v^2)$ SDCs ($d^{(2)}_{fin}$) are negative and substantial in magnitude, often exceeding the LO terms.
  • For the $J=1$ channel, the ratio $d^{(2)}_{fin}/d^{(0)}_{fin}$ is a constant ($-2.1$), but for $J=0$ and $J=2$, the ratio varies significantly with $z$.
• Phenomenological Impact (K-factors):
  • When convoluted with the gluon cross-section, the ratio of the relativistic correction term to the LO term ($R^{(n)}$) is found to be roughly constant across the $p_T$ spectrum (around $-1.0$ to $-1.4$).
  • Using a typical velocity scaling $\langle v^2 \rangle \approx 0.25$, the relativistic corrections lead to a reduction of the LO cross-section by approximately 30% (specifically 36% for $J=0$, 26% for $J=1$, and 30% for $J=2$).
  • The effect is negative and substantial, meaning LO predictions significantly overestimate the production rate if relativistic effects are ignored.

5. Significance

• Completing the NRQCD Picture: This work fills the last missing piece for the $v^2$ sub-leading order analysis of prompt $J/\psi$ hadroproduction via the fragmentation mechanism.
• Resolving LDME Tensions: The large negative corrections suggest that previous extractions of NRQCD LDMEs (which often relied on LO or NLO QCD only) may be biased. Incorporating these relativistic corrections is indispensable for a consistent fit of LDMEs to experimental data (e.g., from LHC).
• Polarization Predictions: Since the corrections are state-dependent ($J=0,1,2$ behave differently), they will significantly alter theoretical predictions for the polarization of $J/\psi$ and $\chi_{cJ}$ at high $p_T$, potentially helping to resolve the long-standing "polarization puzzle."
• Theoretical Insight: The finding that P-wave SDCs are not proportional to LO terms breaks the pattern observed in S-waves, indicating a more complex interplay between relativistic dynamics and the fragmentation process for higher orbital angular momentum states.

In conclusion, the authors establish that relativistic corrections are not a minor perturbation but a dominant ingredient in the theoretical description of heavy quarkonium production at the LHC, necessitating their inclusion in all future precision studies.