Not yet reviewed by Pith; the record is open.

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 a jet 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 the jet and the corresponding parton $i$ that initiates the jet, respectively. Within the framework of Soft Collinear Effective Theory (SCET) we calculate both $J_q(z, \omega_J, \mu)$ and $J_g(z, \omega_J, \mu)$ to the next-to-leading order (NLO) for cone and anti-k$_{\rm T}$ algorithms. We demonstrate that the renormalization group (RG) equations for $J_i(z, \omega_J, \mu)$ follow exactly the usual DGLAP evolution, which can be used to perform the $\ln R$ resummation for {\it inclusive} jet cross sections with a small jet radius $R$. We clarify the difference between our RG equations for $J_i(z, \omega_J, \mu)$ and those for the so-called unmeasured jet functions $J_i(\omega_J, \mu)$, widely used in SCET for {\it exclusive} jet production. Finally, we present applications of the new semi-inclusive jet functions to inclusive jet production in $e^+e^-$ and $pp$ collisions. We demonstrate that single inclusive jet production in these collisions shares the same short-distance hard functions as single inclusive hadron production, with only the fragmentation functions $D_i^h(z, \mu)$ replaced by $J_i(z, \omega_J, \mu)$. This can facilitate more efficient higher-order analytical computations of jet cross sections. We further match our $\ln R$ resummation at both LL$_{R}$ and NLL$_{R}$ to fixed NLO results and present the phenomenological implications for single inclusive jet production at the LHC.