[Paper Review] Gluon fragmentation into S-wave heavy quarkonium

S. Mohammad Moosavi Nejad, Mahdi Delpasand|arXiv (Cornell University)|Jan 21, 2014

Particle physics theoretical and experimental studies3 references3 citations

TL;DR

This paper presents a perturbative QCD calculation of the initial-scale fragmentation function for a gluon to fragment into S-wave charmonium states (e.g., $\eta_c$, $J/\psi$) at leading order in $\alpha_s$. Using a nonrelativistic QCD factorization approach and a finite momentum frame, the authors derive an analytical expression for the gluon-to-charmonium fragmentation function $D_{g\rightarrow H_c}(z, \mu_0)$ that depends on the momentum fraction $z$ and the gluon's transverse momentum, with results consistent with existing models when using the standard $z = E^H/E^g$ definition.

ABSTRACT

Fragmentation is the dominant production mechanism for heavy hadronic bound states with large transverse momentum. We numerically calculate the initial $g ightarrow H(Q\bar{Q})$ fragmentation functions (FFs) using the nonrelativistic QCD factorization approach. Our analytical expression of FFs depends on both the momentum fraction $z$ and the transverse momentum of the gluon, and contains most of the kinematical and dynamical properties of the process. Specifically, using the perturbative QCD we present the FF for a gluon to split into $S$-wave charmonium meson $H_c$ to leading order in the QCD coupling constant.

Motivation & Objective

• To calculate the initial-scale fragmentation function for gluon fragmentation into S-wave heavy quarkonium states using perturbative QCD.
• To incorporate both the momentum fraction $z$ and the transverse momentum of the fragmenting gluon in the fragmentation function.
• To provide an analytically derived, model-independent expression for $D_{g\rightarrow H_c}(z, \mu_0)$ at leading order in $\alpha_s$.
• To validate the result against existing models by comparing with the $z = E^H/E^g$ definition used in prior studies.

Proposed method

• Employing the nonrelativistic QCD factorization approach to separate short-distance and long-distance dynamics in gluon fragmentation into S-wave quarkonium.
• Using a finite momentum frame to simplify phase space integrations and define the fragmentation variable $z$ in terms of energies and longitudinal momenta.
• Applying the standard definition $z = E^H / E^g$ to ensure consistency with established fragmentation function formalisms.
• Performing phase space integrals using delta function constraints and energy denominators to evaluate the matrix element.
• Approximating transverse momentum integrals by replacing $k_T^2$ with its average value $\langle k_T^2 \rangle$, a free parameter adjustable to experiment.
• Normalizing the fragmentation function via $\int_0^1 D_{g\rightarrow H_c}(z, \mu_0) dz = 1$ to ensure physical consistency.

Experimental results

Research questions

• RQ1 How can the initial-scale fragmentation function for gluon-to-S-wave charmonium transitions be calculated analytically in perturbative QCD?
• RQ2 What is the functional dependence of the fragmentation function on the momentum fraction $z$ and the transverse momentum of the gluon?
• RQ3 How does the choice of $z$ definition (covariant vs. non-covariant) affect the resulting fragmentation function?
• RQ4 To what extent does the derived $D_{g\rightarrow H_c}(z, \mu_0)$ agree with existing models when using the standard $z = E^H/E^g$ definition?
• RQ5 Can the formalism be extended to bottomonium states by replacing $m_c$ with $m_b$ and adjusting the decay constant?

Key findings

• The derived fragmentation function $D_{g\rightarrow H_c}(z, \mu_0)$ depends on both the momentum fraction $z$ and the average transverse momentum $\langle k_T^2 \rangle$, with a non-trivial functional form involving $z$, $M$, and $m_c$.
• The result shows good agreement with the model in Qi:2007sf when using the $z = E^H / E^g$ definition, validating the approach.
• The fragmentation function is normalized to unity upon integration over $z$, ensuring consistency with the universality of fragmentation functions.
• The functional form includes a term proportional to $[z^2 k_T^2 + M^2(1-z)^2]^2 / [M z (1-z)]^2$, reflecting the kinematic and dynamical structure of the process.
• The method is directly applicable to bottomonium states by replacing $m_c$ with $m_b$ and using the appropriate decay constant $f_M$ for $\Upsilon$ states.
• The result provides a model-independent, analytically derived initial-scale fragmentation function for gluon-to-charmonium transitions, suitable for DGLAP evolution to higher scales.