Fermionic and Scalar Corrections for the Abelian Form Factor at Two Loops

TL;DR

The paper addresses two-loop corrections to the Abelian form factor arising from massless fermion and scalar loops in a theory with a massive gauge boson. It employs an evolution-equation framework to organize Sudakov logarithms up to NNLL and computes the exact $n_f$ and $n_s$ contributions for arbitrary $s/M^2$, providing explicit expressions and high-energy expansions. It finds that power-suppressed terms vanish rapidly at high energies, while subleading logarithms can be numerically large and partially cancel leading terms, underscoring the necessity of including them for precise electroweak predictions and informing full four-fermion calculations. Overall, the work clarifies the structure of two-loop corrections in a simplified Abelian setting and highlights the significant role of subleading terms in future collider phenomenology.

Abstract

Two-loop corrections for the form factor in a massive Abelian theory are evaluated, which result from the insertion of massless fermion or scalar loops into the massive gauge boson propagator. The result is valid for arbitrary energies and gauge boson mass. Power-suppressed terms vanish rapidly in the high energy region where the result is well approximated by a polynomial of third order in ln(s/M^2). The relative importance of subleading logarithms is emphasised.

Figures (5)

• Figure 1: Two-loop diagrams contributing to the fermionic and scalar form factor. The circle represents a fermion loop for the $n_f$-part or a scalar loop for the $n_s$-part of the form factor.
• Figure 2: The one-loop contribution $\hat{{\cal F}}^{(1)}$ of the Abelian form factor as defined in Eq. (\ref{['eq:FFnfscaling']}). Plotted are the exact result (\ref{['eq:F1complete']}) and the complete logarithmic approximation (\ref{['eq:F1logs']}).
• Figure 3: The one-loop contribution $\hat{{\cal F}}^{(1)}$ of the Abelian form factor as defined in Eq. (\ref{['eq:FFnfscaling']}). Plotted are the individual contributions of the large logarithms as well as the complete logarithmic approximation.
• Figure 4: The fermionic contribution $\hat{{\cal F}}^{(2)}_{n_f}$ and the scalar contribution $\hat{{\cal F}}^{(2)}_{n_s}$ of the Abelian form factor at two loops as defined in Eq. (\ref{['eq:FFnfscaling']}). Plotted are the exact result and the complete logarithmic approximation.
• Figure 5: The fermionic contribution $\hat{{\cal F}}^{(2)}_{n_f}$ and the scalar contribution $\hat{{\cal F}}^{(2)}_{n_s}$ of the Abelian form factor at two loops as defined in Eq. (\ref{['eq:FFnfscaling']}). Plotted are the individual contributions of the large logarithms as well as the complete logarithmic approximation.