Heavy quark form factors in the large $β_0$ limit

TL;DR

The paper analyzes heavy-quark form factors in the large-$\beta_0$ limit, where $\beta_0 \alpha_s \sim 1$, by expanding in $1/\beta_0$ and focusing on the leading terms. It provides a detailed framework for matching QCD currents to HQET, including renormalization and RG evolution, and introduces inversion relations that map on-shell massive diagrams to off-shell HQET integrals. In the large-$\beta_0$ regime, it derives explicit expressions for the bare form factors, their renormalization structure, and the associated renormalon ambiguities, with a comprehensive expansion of hypergeometric functions into Nielsen polylogarithms. The results reproduce known higher-order terms for vector form factors and illuminate the analytic structure of heavy-quark currents, including their divergent and finite parts, and the interplay between UV/IR effects inHQET and QCD.

Abstract

Heavy quark form factors are calculated at $β_0 α_s \sim 1$ to all orders in $α_s$ at the first order in $1/β_0$. The $n_f^2 α_s^3$ terms in the recent results [arXiv:1611.07535] for the vector form factors are confirmed, and $n_f^{L-1} α_s^L$ terms for higher $L$ are predicted.

Figures (5)

• Figure 1: Diagrams producing the highest degree of $n_f$ in each order of perturbation theory.
• Figure 2: On-shell massive self-energy integrals and off-shell HQET ones.
• Figure 3: Examples of on-shell massive diagrams which cannot be transformed to off-shell HQET ones by inversion relations.
• Figure 4: One-loop vertex integrals.
• Figure 5: Box diagrams.