Jet Shape Resummation Using Soft-Collinear Effective Theory

TL;DR

This paper develops a soft-collinear effective theory (SCET) framework to resummation of large logs in jet shapes, showing that the integral jet shape reduces, up to power corrections, to a ratio of jet energy functions. The authors perform NLL resummation via renormalization-group evolution, and validate the approach by comparing with CMS measurements and Pythia simulations, finding good agreement in proton-proton collisions. They highlight that soft recoil is suppressed with a recoil-free jet axis, enabling a clean factorization where hard and soft functions cancel in the jet-shape ratio. The work establishes a robust baseline for jet-shape studies and outlines clear paths toward NNLL precision and heavy-ion applications in forthcoming research.

Abstract

The jet shape is a classic jet substructure observable that probes the average transverse energy profile inside a reconstructed jet. The studies of jet shapes in proton-proton collisions have served as precision tests of perturbative Quantum Chromodynamics (QCD). They have also recently become the baseline for studying the in-medium modification of parton showers in ultra-relativistic nucleus-nucleus collisions. The jet shape is a function of two angular parameters $R$ and $r$, which can be at hierarchical scales. Its calculation suffers from large logarithms of the ratio between the two scales, and these phase space logarithms can conveniently be resummed in the framework of soft-collinear effective theory (SCET). We find that, up to power corrections, the integral jet shape can be expressed in a factorized form which involves only the ratio between two jet energy functions. Resummation is performed at next-to-leading logarithmic order using renormalization-group evolution techniques. Comparisons to jet shape measurements at the LHC are presented to verify the dominant role of the collinear parton shower and to identify the kinematic region in which power-suppressed soft modes and non-perturbative effects may play a role.

Figures (7)

• Figure 1: The integral (left) and differential (right) jet shapes of quark and gluon jets of size $R=0.3$ in proton-proton collisions, plotted as an illustration of their differences. Jet shape contains the information about the transverse energy distribution inside a jet. On average, quark jets are more localized whereas gluon jets are more spread out.
• Figure 2: Schematic event topology of $N$-jet production with collinear and soft radiation. Jets are reconstructed using a jet algorithm with a parameter $R$. The energy $E_r$ inside a cone of size $r$ in $J_1$ is measured, as well as its transverse momentum $p_T$ and rapidity $y$. An energy cutoff $\Lambda$ outside the jets is imposed to ensure the $N$-jet configuration.
• Figure 3: The renormalization group evolution for the jet energy functions. The jet energy functions $J^{E_r}_\omega(\mu)$ of quark jets and gluon jets are calculated at ${\cal O}(\alpha_s)$ at the natural scale $\mu_{j_r}$, and they are evolved to a common renormalization scale $\mu$. At the natural scale there are no large logarithms in the jet energy functions. Large logarithms of the form $\log r/R$ in jet shapes are resummed by the renormalization group evolution between the two jet scales $\mu_{j_r}$ and $\mu_{j_R}$.
• Figure 4: The integral (left) and differential (right) jet shapes for quark and gluon jets reconstructed using the anti-$\rm k_T$ algorithm with $R=0.3$, with a fixed jet energy $E_J=100$ GeV plotted as an illustration. The dashed lines are the SCET calculations at leading-order (LO), whereas the solid lines are the ones at next-to-leading logarithmic order (NLL).
• Figure 5: The integral (left) and differential (right) jet shapes in proton-proton collisions with center of mass energy at $\sqrt{s_{\rm NN}}=2.76$ TeV are plotted as a function of $r$, which is the angle from the jet axis. Jets are reconstructed using the anti-$\rm k_T$ algorithm with $R=0.3$. The cuts on the transverse momenta and rapidity of the jets ($p_T^{\rm jet}>100$ GeV and $0.3<|y^{\rm jet}|<2$) are imposed. The dots are the CMS data with negligible experimental uncertainties. The shaded blue boxes are the LO (light) and NLL (dark) results for anti-$\rm k_T$ jets, with the theoretical uncertainties estimated by varying the jet scales between $\frac{1}{2}\mu_{j_R}<\mu<2\mu_{j_R}$. As we can see, the NLL results agree with the data much better than the LO results. The shaded green boxes are the NLL results for cone jets, plotted as an illustration of the algorithm dependence in jet shapes.