Probing Transverse-Momentum Dependent Evolution With Groomed Jets

TL;DR

This work introduces a groomed-jet observable that measures the transverse momentum $p_{h\perp}$ and energy fraction $z_h$ of a hadron inside a jet groomed by modified mass drop, establishing a SCET factorization with a Transverse Momentum Dependent Fragmenting Jet Function (TMDFJF) matched to standard fragmentation functions. By performing resummation in both the renormalization scale $\mu$ and rapidity scale $\nu$, the authors derive all-orders and NLL expressions that reveal the rapidity anomalous dimension controls the $z_{cut}$-dependence of the spectrum, while grooming suppresses non-global logarithms and enhances sensitivity to non-perturbative TMD evolution. Numerical results in $e^+e^-$ to dijets with a pion show that different non-perturbative models for the rapidity evolution yield distinguishable low-$p_{h\perp}$ shapes, suggesting the observable can discriminate among universal TMD-evolution scenarios and illuminate the three-dimensional structure of hadrons. The framework provides a pathway to extract non-perturbative inputs from data, with potential extensions to hadron colliders and higher-order resummations.

Abstract

We propose an observable which involves measuring the properties (transverse momentum $p_{h\perp}$ and energy fraction $z_h$) of an identified hadron inside a groomed jet. The jet is identified with an anti-kT/CA algorithm and is groomed by implementing the modified mass drop procedure with an energy cut-off parameter $z_{cut}$. The transverse momentum of the hadron inside the jet is measured with respect to the groomed jet axis. We obtain a factorization theorem in the framework of Soft Collinear Effective Theory (SCET), to define a Transverse Momentum Dependent Fragmenting Jet Function (TMDFJF). The TMDFJF is factorized into collinear and collinear soft modes by matching onto SCET$_+$. We resum large logarithms in $E_J/p_{h\perp}$, where $E_J$ is the ungroomed jet energy, to NLL accuracy and apply this formalism for computing the shape of the $p_{h\perp}$ distribution of a pion produced in an $e^+ +e^-$ collision. We observe that the introduction of grooming makes this observable insensitive to non-global logarithms and particularly sensitive to non-perturbative physics of the transverse momentum dependent evolution at low values of $p_{h\perp}$, which can be probed in the variation of the cut-off parameter $z_{cut}$ of the groomer. We discuss how this observable can be used to distinguish between non-perturbative models that describe universal TMD evolution and provide a window into the three dimensional structure of hadrons.

Figures (4)

• Figure 1: The geometric configuration of the jet and hadron axis relative to beam. Here the jet axis is defined as the momentum of all the particles clustered by the jet algorithm. The vectors $\vec{k}_{\perp}$ and $\vec{p}_{h \perp}$ are two dimensional arrays with components as measured from the corresponding axis.
• Figure 2: The geometric configuration involved in matching the TMDFF to the standard FF. $\theta_{h}$ is the angle that both the hadron (blue line) with momentum fraction $z_h$ of the jet and its initiating parton (red dotted line) make to the jet axis, and is set by perturbative splittings up to power corrections. The initiating parton has momentum fraction $x$ of the jet, so that the fragmented hadron has momentum fraction $\frac{z_h}{x}$ of the initiating parton.
• Figure 3: The up-quark to charged pions TMDFJF at NLL for $z_h=0.4$ (left) and $z_h=0.8$ (right). Other kinematic variables are described in the plots. The dashed (blue) curves corresponds to the calculation in momentum space using the Pavia fits for the CSS model. The parameters for this model are given in Table \ref{['tb:models']}. The solid curves correspond to the momentum space calculation for which we used a profile function to avoid the divergence of the coupling constant at small values of the factorization scale. The HKNS fragmentation functions $D_{u/\pi^+}(x;\mu)$ are taken from Ref. Hirai:2007cx
• Figure 4: The logarithmic derivative of TMDFJF for the three models. All the models agree in the perturbative regime but show significant differences in the non-perturbative region.