Robustness of the O($N$) universality class

TL;DR

This paper investigates the critical exponents of Lorentz-violating O(N) scalar field theories, demonstrating that Lorentz violation does not alter the universal critical exponents, which remain identical to the Lorentz-invariant case.

Contribution

It shows that Lorentz-violating quantum corrections do not affect the universal critical exponents in O(N) scalar theories, confirmed through two independent renormalization methods.

Findings

Critical exponents are unchanged by Lorentz violation.

Renormalization constants and beta functions acquire Lorentz-violating corrections.

Results hold at all loop levels, supported by symmetry arguments.

Abstract

We calculate the critical exponents for Lorentz-violating O($N$) $\lambda\phi^{4}$ scalar field theories by using two independent methods. In the first situation we renormalize a massless theory by utilizing normalization conditions. An identical task is fulfilled in the second case in a massive version of the same theory, previously renormalized in the BPHZ method in four dimensions. We show that although the renormalization constants, the $\beta$ and anomalous dimensions acquire Lorentz-violating quantum corrections, the outcome for the critical exponents in both methods are identical and furthermore they are equal to their Lorentz-invariant counterparts. Finally we generalize the last two results for all loop levels and we provide symmetry arguments for justifying the latter.