Gauge invariant Lagrangian for non-Abelian tensor gauge fields of fourth rank

TL;DR

This paper constructs a unique gauge-invariant Lagrangian for non-Abelian tensor gauge fields up to rank-4, extending previous work on lower ranks and demonstrating the Lagrangian's uniqueness.

Contribution

It explicitly derives all Lorentz invariant quadratic forms for rank-4 non-Abelian tensor gauge fields and identifies the two unique gauge-invariant combinations, confirming the Lagrangian's uniqueness.

Findings

Only two gauge-invariant quadratic forms exist for rank-4 fields.

The constructed Lagrangian is unique up to rank-4 tensor fields.

Compact expression for Lagrangian coefficients is provided.

Abstract

Using generalized field strength tensors for non-Abelian tensor gauge fields one can explicitly construct all possible Lorentz invariant quadratic forms for rank-4 non-Abelian tensor gauge fields and demonstrate that there exist only two linear combinations of them which form a gauge invariant Lagrangian. Together with the previous construction of independent gauge invariant forms for rank-2 and rank-3 tensor gauge fields this construction proves the uniqueness of early proposed general Lagrangian up to rank-4 tensor fields. Expression for the coefficients of the general Lagrangian is presented in a compact form.