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

TL;DR

The paper introduces the semi-inclusive jet function J_i(z, ω_J, μ) within Soft Collinear Effective Theory to describe how a parton forms a jet with radius R and energy fraction z. It shows that J_i evolves via timelike DGLAP equations, enabling ln R resummation for inclusive jet cross sections at small R, and it provides explicit NLO calculations for quark and gluon jets with cone and anti-k_T algorithms. The authors demonstrate that inclusive jet production shares the same short-distance hard functions as hadron production, with D_i^h replaced by J_i, and develop LL_R and NLL_R resummations matched to fixed NLO results, applying the framework to e+e- and LHC pp collisions. Numerical results reveal significant small-R effects and reduced theoretical uncertainties, illustrating the practical impact for precise jet cross-section predictions at the LHC.

Abstract

We introduce a new kind of jet function: the semi-inclusive jet function $J_i(z, ω_J, μ)$, which describes how a parton $i$ is transformed into a jet with a jet radius $R$ and energy fraction $z = ω_J/ω$, with $ω_J$ and $ω$ 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 $J_q(z, ω_J, μ)$ and $J_g(z, ω_J, μ)$ to the next-to-leading order (NLO) for cone and anti-k$_{\rm T}$ algorithms. We demonstrate that the renormalization group (RG) equations for $J_i(z, ω_J, μ)$ 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 $J_i(z, ω_J, μ)$ and those for the so-called unmeasured jet functions $J_i(ω_J, μ)$, 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 $D_i^h(z, μ)$ replaced by $J_i(z, ω_J, μ)$. 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.

Figures (11)

• Figure 1: Feynman diagrams that contribute to the semi-inclusive quark jet function. The quark that initiates the jet has momentum $\ell = (\ell^- = \omega, \ell^+, 0_\perp)$, with $\omega = \omega_J/z$ and $\omega_J$ the jet energy.
• Figure 2: Three situations that contribute to the semi-inclusive quark jet function: (A) both quark and gluon are inside the jet, (B) only quark is inside the jet, (C) only gluon is inside the jet.
• Figure 3: Feynman diagrams that contribute to the semi-inclusive gluon jet function $J_g(z, \omega_J)$. The gluon that initiates the jet has momentum $\ell = (\ell^- = \omega, \ell^+, 0_\perp)$, with $\omega = \omega_J/z$ and $\omega_J$ the jet energy. The dotted loop in (B) is the ghost loop, while the dashed loop in (D) and (E) are collinear quark loops, the mirror diagrams of (F) and (G) are not shown here but are included in the calculations.
• Figure 4: The semi-inclusive jet function with evolution (red) and without evolution (blue) for several values of the jet radius parameter $R=0.99,\, 0.7,\, 0.5,\, 0.3,\, 0.1,\, 0.05$. Using the DGLAP evolution equations, the semi-inclusive jet function is evolved to a final scale of $\mu=250$ GeV. In order to perform the correct matching to NLO, we need to perform the evolution of the LO and NLO jet functions separately for both quarks $J_q^{(0),(1)}$ and for gluons $J_g^{(0),(1)}$ as shown in the four panels. Note that the initial condition for the evolution of the LO jet function is given by a delta function which is illustrated in the left two panels by a blue straight line.
• Figure 5: Tree-level matching onto the operators for single inclusive hadron/jet production in $e^+e^-\to h X$ or $e^+e^-\to {\rm jet} X$. The red vertical line is the final-state cut.