Scheme dependence of two-loop HTLpt-resummed $\text{SYM}_{4,4}$ thermodynamics

TL;DR

This paper compares two regularization schemes, DRG and RDR, in two-loop HTLpt calculations of ${ m SYM}_{4,4}$ thermodynamics, showing minimal differences for small coupling and residual scheme dependence at higher orders.

Contribution

It assesses the scheme dependence of two-loop HTLpt results in ${ m SYM}_{4,4}$, emphasizing the importance of supersymmetry-preserving regularization.

Findings

For $ ext{lambda} \, extless\, 6$, results are numerically similar across schemes.

Scheme-independent coefficients are obtained at order $ ext{lambda}$, $ ext{lambda}^{3/2}$, and $ ext{lambda}^2 \, \log\text{lambda}$.

Residual scheme dependence appears at order $ ext{lambda}^2$.

Abstract

The resummed thermodynamics of ${\cal N}=4$ supersymmetric Yang-Mills theory in four space-time dimensions ($\text{SYM}_{4,4}$) has been calculated previously to two loop order within hard thermal loop perturbation theory (HTLpt) using the canonical dimensional regularization (DRG) scheme. Herein, we revisit this calculation using the regularization by dimensional reduction (RDR) scheme. Since the RDR scheme manifestly preserves supersymmetry it is the preferred scheme, however, it is important to assess if and by how much the resummed perturbative results depend on the regularization scheme used. Comparing predictions for the scaled entropy obtained using the DRG and RDR schemes we find that for $\lambda \lesssim 6$ they are numerically very similar. We then compare the results obtained in both schemes with the strict perturbative result, which is accurate up to order $\lambda^2$, and a generalized Pad\'{e} approximant constructed from the known large-$N_c$ weak- and strong-coupling expansions. Comparing the strict perturbative expansion of the two-loop HTLpt result with the perturbative expansion to order $\lambda^2$, we find that both the DRG and RDR HTLpt calculations result in the same scheme-independent predictions for the coefficients at order $\lambda$, $\lambda^{3/2}$, and $\lambda^2 \log\lambda$, however, at order $\lambda^2$ there is a residual regularization scheme dependence.