The Complete Two-Loop Integrated Jet Thrust Distribution In Soft-Collinear Effective Theory

TL;DR

The paper resolves the soft part of the two-loop integrated jet thrust distribution in e+e− using a thrust-cone jet algorithm with a veto, addressing the intricate $r$-dependence and non-global logarithms. It develops and deploys sector decomposition, generalized-weight multiple polylogarithms, and an extended coproduct calculus to obtain a compact, exact result that matches SCET resummation and the hemisphere limit. A key finding is that the global, $r$-dependent part reduces to classical polylogs, while the small-$r$ limit reveals a deep link between $\ln r$ terms and non-global logarithms, with the cusp anomalous dimension coefficient $\Gamma_1$ fixed in this structure. The work also demonstrates that knowledge of L-loop hemisphere soft functions can predict large logarithms at L loops outside the standard factorization, offering a path toward systematic multi-loop predictions for exclusive jet observables.

Abstract

In this work, we complete the calculation of the soft part of the two-loop integrated jet thrust distribution in e+e- annihilation. This jet mass observable is based on the thrust cone jet algorithm, which involves a veto scale for out-of-jet radiation. The previously uncomputed part of our result depends in a complicated way on the jet cone size, r, and at intermediate stages of the calculation we actually encounter a new class of multiple polylogarithms. We employ an extension of the coproduct calculus to systematically exploit functional relations and represent our results concisely. In contrast to the individual contributions, the sum of all global terms can be expressed in terms of classical polylogarithms. Our explicit two-loop calculation enables us to clarify the small r picture discussed in earlier work. In particular, we show that the resummation of the logarithms of r that appear in the previously uncomputed part of the two-loop integrated jet thrust distribution is inextricably linked to the resummation of the non-global logarithms. Furthermore, we find that the logarithms of r which cannot be absorbed into the non-global logarithms in the way advocated in earlier work have coefficients fixed by the two-loop cusp anomalous dimension. We also show that, given appropriate L-loop contributions to the integrated hemisphere soft function, one can straightforwardly predict a number of potentially large logarithmic contributions at L loops not controlled by the factorization theorem for jet thrust.

Figures (5)

• Figure 1: The thrust cone jet algorithm used to define the $\tau_\omega$ observable enforces dijet kinematics by imposing a veto on the out-of-jet radiation with total energy greater than $\omega$. The cone size, $r$, is defined to be $\tan^2\left(\alpha\over 2\right)$.
• Figure 2: Final state phase space configurations with a single soft parton. Panel I shows a single soft gluon going into the right jet and panel II shows a single soft gluon going out of all jets.
• Figure 3: Final state phase space configurations containing two soft partons. Panel I shows two soft gluons going into the right jet, panel II shows one soft gluon going into the right jet and another going into the left jet, panel III shows one soft gluon going into the right jet and one soft gluon going out of all jets, and, finally, panel IV shows two soft gluons going out of all jets. In the above, it is in all cases possible to replace the two soft gluons with a soft quark-antiquark pair.
• Figure 4: A hemisphere boundary can be mapped onto a cone boundary by choosing an appropriate Lorentz boost.
• Figure 5: Starting at $\mathcal{O}\left(\alpha_s^3\right)$, there will be contributions to the $\tau_\omega$ soft function which simultaneously probe all three regions of the final state phase space.