Factorization analysis for the fragmentation functions of hadrons containing a heavy quark

TL;DR

This paper develops an effective-field-theory framework to factorize and resum heavy-quark fragmentation functions in the large-x endpoint. By a two-step matching—decoupling heavy-quark pairs to a single fragmentation function in partially quenched QCD, followed by HQET matching in the x→1 region—it separates short-distance dynamics from long-distance bound-state effects and yields analytic RG solutions that resum large logarithms. It provides explicit one- and two-loop results for the endpoint matching coefficient $C_D$, connects perturbative fragmentation to the HQET shape-function formalism, and introduces a phenomenological model for the nonperturbative input $S_{Q/H}$. The framework enables heavy-quark symmetry relations between $B$ and $D$ fragmentation, offers a path to data-driven extraction, and sets the stage for threshold resummation in momentum space, improving predictive power for heavy-hadron production across processes.

Abstract

Using methods of effective field theory, a systematic analysis of the fragmentation functions D_{a/H}(x,m_Q) of a hadron H containing a heavy quark Q is performed (with a=Q,Q_bar,q,q_bar,g). By integrating out pair production of virtual and real heavy quarks, the fragmentation functions are matched onto a single nonperturbative function describing the fragmentation of the heavy quark Q into the hadron H in "partially quenched" QCD. All calculable, short-distance dependence on x is extracted in this step. For x->1, the remaining fragmentation function can be matched further onto a universal function defined in heavy-quark effective theory in order to factor off its residual dependence on the heavy-quark mass. By solving the evolution equation in the effective theory analytically, large logarithms of the ratio mu/m_Q are resummed to all orders in perturbation theory. Connections with existing approaches to heavy-quark fragmentation are discussed. In particular, it is shown that previous attempts to extract log^n(1-x) terms from the fragmentation function D_{Q/H}(x,m_Q) are incompatible with a proper separation of short- and long-distance effects.

Figures (5)

• Figure 1: Examples of fragmentation processes converting a parton $a$ into a heavy hadron $H$. Heavy quarks are denoted by thick lines, while the heavy hadron is drawn as a double line. Only the first process is allowed in "partially quenched" QCD.
• Figure 2: Examples of one-loop diagrams contributing to the perturbative fragmentation functions. The grey dots represent the Wilson lines in expressions (\ref{['Ddef']}) and (\ref{['Sdef']}).
• Figure 3: Scale dependence of the heavy-quark fragmentation function $D_{b/B}(x,m_b,\mu)$ obtained from (\ref{['Dnice']}) using the model functions specified in the text. The overall normalization $N_B$ is left free. In each figure, the three sets of curves refer to $\mu=m_b=4.7$ GeV (blue), $\mu=3$ GeV (red), and $\mu=\mu_0=1.5$ GeV (green). Within each set the curves show the residual dependence on the hard matching scale $\mu_h$. The dotted line shows the phenomenological fit at $\mu=m_b$ obtained in Kniehl:2007yu.
• Figure 4: Heavy-quark symmetry prediction for the charm-quark fragmentation function $D_{c/D}(x,\mu=m_c)$ (solid blue line) obtained from the inverse of relation (\ref{['wow']}) and the phenomenological fit result for the $b$-quark fragmentation function $D_{b/B}(x,\mu=m_b)$ determined in Kniehl:2007yu and overlaid as a dotted line. For comparison we show as red dashed lines the fit results for the charm-quark fragmentation functions of $D^+$ and $D^0$ mesons obtained in Kniehl:2006mw.
• Figure 5: (a,b) Diagrams involving heavy-quark pair production, which yield the contribution $\Delta F_Q(x)$ in eq. (\ref{['cQfull']}). (c) A diagram with identical color structure, which does not involve pair production and hence does not contribute to $c_Q$.