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We introduce a new kind of jet function: the semi-inclusive jet function$J_i(z, \omega_J, \mu)$, which describes how a parton $i$ is transformed into ajet with a jet radius $R$ and energy fraction $z = \omega_J/\omega$, with$\omega_J$ and $\omega$ being the large light-cone momentum component of thejet and the corresponding parton $i$ that initiates the jet, respectively.Within the framework of Soft Collinear Effective Theory (SCET) we calculateboth $J_q(z, \omega_J, \mu)$ and $J_g(z, \omega_J, \mu)$ to the next-to-leadingorder (NLO) for cone and anti-k$_{\rm T}$ algorithms. We demonstrate that therenormalization group (RG) equations for $J_i(z, \omega_J, \mu)$ follow exactlythe usual DGLAP evolution, which can be used to perform the $\ln R$ resummationfor {\it inclusive} jet cross sections with a small jet radius $R$. We clarifythe difference between our RG equations for $J_i(z, \omega_J, \mu)$ and thosefor the so-called unmeasured jet functions $J_i(\omega_J, \mu)$, widely used inSCET for {\it exclusive} jet production. Finally, we present applications ofthe new semi-inclusive jet functions to inclusive jet production in $e^+e^-$and $pp$ collisions. We demonstrate that single inclusive jet production inthese collisions shares the same short-distance hard functions as singleinclusive hadron production, with only the fragmentation functions $D_i^h(z,\mu)$ replaced by $J_i(z, \omega_J, \mu)$. This can facilitate more efficienthigher-order analytical computations of jet cross sections. We further matchour $\ln R$ resummation at both LL$_{R}$ and NLL$_{R}$ to fixed NLO results andpresent the phenomenological implications for single inclusive jet productionat the LHC.

The semi-inclusive jet function in SCET and small radius resummation for inclusive jet production