Towards massless and massive event shapes in the large-$\beta_0$ limit

We present results for SCET and bHQET matching coefficients and jet functions in the large-$\beta_0$ limit. Our computations exactly predict all terms of the form $\alpha_s^{n+1} n_f^n$ for any $n\geq 0$, and we find full agreement with the coefficients computed in the full theory up to $\mathcal{O}(\alpha_s^4)$. We obtain all-order closed expressions for the cusp and non-cusp anomalous dimensions (which turn out to be unambiguous) as well as matrix elements (with ambiguities) in this limit, which can be easily expanded to arbitrarily high powers of $\alpha_s$ using recursive algorithms to obtain the corresponding fixed-order coefficients. Examining the poles laying on the positive real axis of the Borel-transform variable $u$ we quantify the perturbative convergence of a series and estimate the size of non-perturbative corrections. We find a so far unknown $u=1/2$ renormalon in the bHQET hard factor $H_m$ that affects the normalization of the peak differential cross section for boosted top quark pair production. For ambiguous series the so-called Borel sum is defined with the principal value prescription. Furthermore, one can assign an ambiguity based on the arbitrariness of avoiding the poles by contour deformation into the positive or negative imaginary half-plane. Finally, we compute the relation between the pole mass and four low-scale short distance masses in the large-$\beta_0$ approximation (MSR, RS and two versions of the jet mass), work out their $\mu$- and $R$-evolution in this limit, and study how their implementation improves the convergence of the position-space bHQET jet function, whose three-loop coefficient in full QCD is numerically estimated.