Quark fragmentation within an identified jet

Abstract

We derive a factorization theorem that describes an energetic hadron $h$ fragmenting from a jet produced by a parton $i$, where the jet invariant mass is measured. The analysis yields a ``fragmenting jet function'' ${\mathcal{G}}_{i}^{h}(s,z)$ that depends on the jet invariant mass $s$, and on the energy fraction $z$ of the fragmentation hadron. We show that ${\mathcal{G}}_{i}^{h}$ can be computed in terms of perturbatively calculable coefficients, ${\mathcal{J}}_{ij}(s,z/x)$, integrated against standard nonperturbative fragmentation functions, ${D}_{j}^{h}(x)$. We also show that $\ensuremath{\sum}_{h}\ensuremath{\int}dz{\mathcal{G}}_{i}^{h}(s,z)$ is given by the standard inclusive jet function ${J}_{i}(s)$ which is perturbatively calculable in QCD. We use soft collinear effective theory and for simplicity carry out our derivation for a process with a single jet, $\overline{B}\ensuremath{\rightarrow}Xh\ensuremath{\ell}\overline{\ensuremath{\nu}}$, with invariant mass ${m}_{Xh}^{2}\ensuremath{\gg}{\ensuremath{\Lambda}}_{\mathrm{QCD}}^{2}$. Our analysis yields a simple replacement rule that allows any factorization theorem depending on an inclusive jet function ${J}_{i}$ to be converted to a semi-inclusive process with a fragmenting hadron $h$. We apply this rule to derive factorization theorems for $\overline{B}\ensuremath{\rightarrow}XK\ensuremath{\gamma}$ which is the fragmentation to a Kaon in $b\ensuremath{\rightarrow}s\ensuremath{\gamma}$, and for ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}(\mathrm{\text{dijets}})+h$ with measured hemisphere dijet invariant masses.