Fragmentation to a jet in the large $z$ limit

TL;DR

The paper develops a soft-collinear effective theory (SCET) framework to factorize the fragmentation function to a jet (FFJ) in the large-$z$ endpoint with small jet radius $R$. By introducing a collinear-soft mode, the FFJ is expressed as a product of an in-jet integrated jet function and a collinear-soft function, enabling simultaneous resummation of large logarithms in $R$ and $1-z$ through RG evolution to next-to-leading-logarithmic (NLL) accuracy plus NLO matching. It provides explicit NLO results for the jet and soft functions, derives their anomalous dimensions, and demonstrates how the RG flow reproduces the correct endpoint behavior and DGLAP limits. The authors also discuss potential nonglobal logarithms (NGLs) at two loops, offer a phenomenological estimate for their size in the large-$N_c$ limit, and present numerical results showing substantial improvements over naive DGLAP-based resummation, with implications for jet physics and extensions to related observables.

Abstract

We consider the fragmentation of a parton into a jet with small radius $R$ in the large $z$ limit, where $z$ is the ratio of the jet energy to the mother parton energy. In this region of phase space, large logarithms of both $R$ and $1-z$ can appear, requiring resummation in order to have a well defined perturbative expansion. Using soft-collinear effective theory, we study the fragmentation function to a jet (FFJ) in this endpoint region. We derive a factorization theorem for this object, separating collinear and collinear-soft modes. This allows for the resummation using renormalization group evolution of the logarithms $\ln R$ and $\ln(1-z)$ simultaneously. We show results valid to next-to-leading logarithmic order for the global Sudakov logarithms. We also discuss the possibility of non-global logarithms that should appear at two-loops and give an estimate of their size.

Figures (6)

• Figure 1: Phase space for the real gluon emission in the collinear-soft function. In the $(k_+,k_-)$ plane, the region above the border line $k_-=t^2k_+$ gives the out-jet contribution and the region below gives the in-jet contribution. $\Lambda_+$ is the maximum value for the distribution of $\ell_+$ and can be chosen arbitrarily.
• Figure 2: $D_{J_q/q}(z)$ (left panel) and $D_{J_g/g}(z)$ (right panel) with different jet energies. Red, blue, and black curves correspond to jet energy $E_J$ equal to 500, 1000, and 2000 GeV, respectively. The jet radius is chosen to be $R=0.2$ and the factorization scale is $\mu_f=E_J$. Error estimation is described in the text.
• Figure 3: $D_{J_q/q}(z)$ (left panel) and $D_{J_g/g}(z)$ (right panel) with different jet radii. Red, blue, and black curves correspond to the jet radius $R$ equal to $0.1$, $0.2$, and $0.4$ respectively. The jet energy is $E_J=1000$ GeV and the factorization scale is $\mu_f=E_J$.
• Figure 4: Comparison of the result using leading DGLAP evolution and the resummed result at $\mathrm{NLL_G+NLO}$ from the factorization approach. The orange (green) curves are obtained using leading DGLAP evolution with FFJs running from $\mu_c=E_JR~(\mu_{cs}=E_JR(1-z))$ to $\mu_f = E_J$. Blue curves are the resummed result of the FFJs. The jet radius is $R=0.2$ and jet energy is $E_J=1000$ GeV.
• Figure 5: Comparison of the resummed results with (orange) and without (blue) resumming the NGLs. Here $R=0.2$ and $E_J=1000$ GeV.