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'' 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 G(i)(h) can be computed in terms of perturbatively calculable coefficients, J(ij)(s, z/x), integrated against standard nonperturbative fragmentation functions, D-j(h)(x). We also show that Sigma(h) integral dzG(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, (B) over bar -> Xhl (v) over bar, with invariant mass m(Xh)(2) >> Lambda(2)(QCD). Our analysis yields a simple replacement rule that allows any factorization theorem depending on an inclusive jet function Ji to be converted to a semi-inclusive process with a fragmenting hadron h. We apply this rule to derive factorization theorems for (B) over bar -> XK gamma which is the fragmentation to a Kaon in b -> s gamma, and for e(+)e(-) -> (dijets) + h with measured hemisphere dijet invariant masses.