Applications of the Large Mass Expansion

TL;DR

The paper evaluates two main semi-analytic strategies for two-loop self-energy and vertex diagrams with limited kinematic complexity: Taylor expansion (TE) and Large Mass Expansion (LME). It demonstrates how TE, aided by conformal mapping and Padé acceleration, yields analytic coefficients in favorable configurations, while LME provides a systematic asymptotic framework in the presence of a heavy mass, enabling analytic bubbles and convolution with subgraph expansions. Through detailed case studies—notably Z -> bb decays and the muon g-2 context—the authors compare TE and LME and show that LME, especially with Padé resummation, achieves high precision up to and beyond the electroweak scale, with results extending to Bremsstrahlung and diagrams with a single non-zero mass via basis-expansion methods using harmonic sums. The work also discusses automation of diagram generation (TLAMM, DIANA) and introduces a novel basis-based approach for single-mass diagrams, offering practical, semi-analytic tools for accurate SM predictions at two loops.

Abstract

The method of the large mass expansion (LME) is investigated for selfenergy and vertex functions in two-loop order. It has the technical advantage that in many cases the expansion coefficients can be expressed analytically. As long as only one non-zero external momentum squared, $q^2$, is involved also the Taylor expansion (TE) w.r.t. small $q^2$ yields high precision results in a domain sufficient for most applications. In the case of only one non-zero mass $M$ and only one external momentum squared, the expansion w.r.t. $q^2/M^2$ is identical for the TE and the LME. In this case the combined techniques yield analytic expressions for many diagrams, which are quite easy to handle numerically.

Figures (7)

• Figure 1: Kinematics for the decay $Z \to b\bar{b}$.
• Figure 2: Kinematics for anomalous moment ${(g-2)}_{\mu}$.
• Figure 3: Conformal mapping of the complex $y$-plane into the $\omega$-plane.
• Figure 4: Two-loop diagrams with two different masses in internal lines arising in the process $Z \to b \overline{b}$. The notation for the diagrams is chosen according to ft.
• Figure 5: The structure of the LME, see explanations in the text.