Quark fragmentation within an identified jet

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 $\sum _h\int dz{\mathcal{G}}_i^h(s,z)$ 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, $\overline{B}\to Xh\ell \overline{\nu }$, with invariant mass $m_{Xh}^2\gg {\Lambda }_{\mathrm{QCD}}^2$. 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 $\overline{B}\to XK\gamma$ which is the fragmentation to a Kaon in $b\to s\gamma$, and for $e^{+}{e}^{-}\to (\mathrm{\text{dijets}})+h$ with measured hemisphere dijet invariant masses.

Figure 1

Kinematic configuration for a pion fragmenting from a jet $X_u\to X\pi$.

Figure 2

Sketch of the hadronic fragmentation process for $\overline{B}\to X\pi \ell \overline{\nu }$.