Calculation of Higher Mass-Dimensional Effective Lagrangians in Quantum Field Theory

TL;DR

This paper develops a systematic method for calculating low-energy one-loop effective Lagrangians in non-Abelian quantum field theories, enabling higher-order corrections up to eighth mass-dimensions with explicit invariants.

Contribution

It introduces a general prescription using Lorentz and gauge-invariant monomials to compute higher-mass dimensional effective Lagrangians, extending previous calculations up to 12 mass-dimensions.

Findings

Reproduced zeroth-order corrections with covariant derivatives.

Extended calculations to higher-order corrections up to 8 mass-dimensions.

Provided explicit invariants for corrections up to 16 mass-dimensions.

Abstract

A prescription for calculating low-energy one-loop higher-mass dimensional effective Lagrangians for non-Abelian field theories is constructed in the spirit of quasilocal background field method. Basis of Lorentz and gauge-invariant monomials of similar mass-dimensions acting as building blocks are matrix-multiplied in a specified order (usually dictated by a permutation of tensorial indices) generating the much needed invariants. The same set of building blocks is used to generate higher-order corrections for a specific mass-dimension. Though the gauge group, the spacetime dimensions, the order of corrections that can be included, and the mass-dimensions that can be formed are all kept arbitrary in the prescription, we constructed basis invariants from 3 up to 12 mass-dimensions to accommodate higher-order corrections up to fourth-order. With these basis, we pursued solving the zeroth-order corrections leading to invariants from 2 up to 16 mass-dimensions, for first-order from 4 up to 8 mass-dimensions, second and third order corrections from 6 up to 8 mass-dimensions. As a result, we have reproduced the zeroth-order corrections showing dependence on the covariant derivative of the background matrix potential. Previous calculation was done up to 12 mass-dimensions but this dependence was not shown in closed form. For higher-order corrections, the case for 4 up to 6 mass-dimensions are also reproduced. Finally, we calculated the case for 8 mass-dimensions which is reduced only by exploiting the antisymmetry of the fieldstrength tensor and the freedom to throw away total derivatives.