Perturbative aspects of deformed Yang-Mills theory

Center-stabilized $SU(N)$ Yang-Mills theories on ${\mathbb{R}}^{3}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ are QCD-like theories that can be engineered to remain weakly-coupled at all energy scales by taking the ${S}^{1}$ circle length $L$ to be sufficiently small. In this regime, these theories admit effective long-distance descriptions as Abelian $U(1{)}^{N\ensuremath{-}1}$ gauge theories on ${\mathbb{R}}^{3}$, and semiclassics can be reliably employed to study nonperturbative phenomena such as color confinement and the generation of mass gaps in an analytical setting. At the perturbative tree level, the long-distance effective theory contains ($N\ensuremath{-}1$) free photons with identical gauge couplings ${g}_{3}^{2}\ensuremath{\equiv}{g}^{2}/L$. Vacuum-polarization effects, from integrating out heavy charged fields, lift this degeneracy to give $\ensuremath{\lfloor}\frac{N}{2}\ensuremath{\rfloor}$ distinct values, ${g}^{2}(\frac{2}{L})\ensuremath{\lesssim}{g}_{3,\ensuremath{\ell}}^{2}L\ensuremath{\lesssim}{g}^{2}(\frac{2\ensuremath{\pi}}{NL})$. In this work, we calculate these corrections to one-loop order in theories where the center-symmetric vacuum is stabilized by $2\ensuremath{\le}{n}_{f}\ensuremath{\le}5$ massive adjoint Weyl fermions with masses of order ${m}_{\ensuremath{\lambda}}\ensuremath{\sim}\frac{2\ensuremath{\pi}}{NL}$, (also known as ``deformed Yang-Mills,'') and show that our results agree with those found in previous studies in the ${m}_{\ensuremath{\lambda}}\ensuremath{\rightarrow}0$ limit. Then, we show that our result has an intuitive interpretation as the running of the coupling in a ``lattice momentum'' in the context of the nonperturbative ``emergent latticized fourth dimension'' in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$, fixed-$NL$ limit.

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