Higher-point Energy Correlators: Factorization in the Back-to-Back Limit & Non-perturbative Effects

Abstract

N-point energy correlators are powerful observables for studying strong interactions, with applications ranging from extractions of the strong coupling $α_s$ to probes of jet modification in heavy-ion collisions and determination of the top-quark mass. Their practical use has, however, been limited by the complicated phase space for large N. Using a recently introduced parametrization that simplifies this structure, we study projected N-point correlators in two regimes: factorization in the back-to-back limit and leading non-perturbative effects in the collinear limit. While results in the back-to-back regime were previously limited to the energy-energy correlator, our approach allows us to derive the factorization theorem for arbitrary N. We compute the new ingredient, a one-loop jet function, needed for the next-to-next-to-leading-logarithmic resummation, which enables future $α_s$ extractions with complementary systematics. We further determine the analytic structure of leading non-perturbative power corrections for arbitrary N, including their dependence on the center-of-mass energy Q, the value of N, and the angular scale $x$. We present the first results for non-integer N<1, finding that the classical scaling in $x$ acquires an N-dependent modification, and that a new non-perturbative matrix element $\tildeΩ^{[N]}$ appears. In a certain approximation, $\tildeΩ^{[N]}$ can be related to the standard parameter $Ω_1$ relevant for N>1. Our analytic predictions are tested against the hadronization model in Pythia, finding good agreement. The results presented in this paper demonstrate the significant advancements enabled through our new parametrization of energy correlators.

Figures (11)

• Figure 1: An illustrative diagram comparing the (a) traditional parametrization for PENCs against the (b) new parametrization for $N=4$. In (a) all $\binom{N}{2}$ pairwise distances are needed to determine the largest separation $\chi_L =\max_{j,k}\{\theta_{i_k,i_j}\}$. By contrast, in (b) the PENCs are parametrized relative to the special particle $i_s$, with the largest separation characterized by $\chi =\max_{j}\{\theta_{i_s,i_j}\}$. In both cases all particles are summed over, including the special particle in (b). The dashed line illustrates the boundary of the jet, which is not needed in $e^+e^-$ collisions.
• Figure 2: The PE3C distribution in $e^{+}\, e^{-} \to {\rm jets}$, obtained from Pythia at $Q=1$ TeV, for the traditional parametrization and our new parametrization logarithmic in (a) $\chi$ and (b) $\pi - \chi$. The relative difference is shown in the bottom panel of the figure. The vertical dashed line indicates the transition between the perturbative and non-perturbative region, and the plot range is chosen to highlight the collinear region in (a) and the back-to-back regime in (b). As visible from the figure, the differences in the perturbative region are relatively small: for (a) a few percent and up to about 10% for (b), while difference in the transition to the non-perturbative region are larger but below $10\%$ in (a) and up to $30 \%$ in (b). The larger differences for the back-to-back regime are explained in the text.
• Figure 3: Schematic of the back-to-back regime, where the special particle is in one of the jets and the measurement is sensitive to details of the collinear radiation in the opposite jet, through their distance to the antipode of the special particle $-i_s$, i.e. whether these particles are included in $z_{\rm disk}$ shown in gray. The soft radiation produces an overall recoil, such that the jets are no longer exactly back-to-back.
• Figure 4: Coefficient $\Delta \mathcal{F}_{N-1}$ of the plus-distribution due to recoil in \ref{['eq:Delta_J']} for $N=3$, (a) averaged over $\phi$ and (b) for specific $\phi$ values. The $N$-dependence is largely an overall factor.
• Figure 5: Log-log plot showing the non-perturbative contributions extracted from Pythia simulations for $N=0.1,0.5,2$ and $5$. The size of these corrections is significantly larger for $N<1$ than for $N>1$ values. Moreover, as can be seen when $N$ approaches $0$, the scaling region does not have a simple power law structure (it's not quite a straight line anymore). The vertical dashed lines represent the perturbative scaling region where leading non-perturbative corrections suffice and can be studied. The light gray band shows the variation of these boundaries by $\pm 20\%$, which will be used in fits.