Deconfinement and continuity between thermal and (super) Yang-Mills theory for all gauge groups

Abstract

We study the phase structure of N =1 supersymmetric Yang-Mills theory on R 3 × S 1 , with massive gauginos, periodic around the S 1 , with Sp(2N ) (N ≥2), Spin(N ) (N ≥5), G 2 , F 4 , E 6 , E 7 , E 8 gauge groups. As the gaugino mass m is increased, with S 1 size and strong coupling scale fixed, we find a first-order phase transition both for theories with and without a center. This semiclassically calculable transition is driven, as in SU (N ) and G 2 [1,2], by a competition between monopole-instantons and exotic topological "molecules"-"neutral" or "magnetic" bions. We compute the trace of the Polyakov loop and its twopoint correlator near the transition. We find a behavior similar to the one observed near the thermal deconfinement transition in the corresponding pure Yang-Mills (YM) theory in lattice studies (whenever available). Our results lend further support to the conjectured continuity, as a function of m, between the quantum phase transition studied here and the thermal deconfinement transition in YM theory. We also study the θ-angle dependence of the transition, elaborate on the importance of the quantum-corrected moduli-space metric at large N , and offer comments for the future.

QCD on a small circle

“…The photon mixing matrix has been calculated in N = 1 SYM theory to first order in λ [29]. This photon mixing is diagonalized by the same Z N Fourier transform mentioned below, and does not affect the following discussion.…”

“…With fundamental representation fermions (n f N ) with a common mass m q Λ, at large L there are multiple characteristic scales for the masses of particles: the pseudo-Nambu-Goldstone (pNGB) mass scale m pNGB ∼ m q Λ, the glueball and meson mass scale m M ∼ Λ, and the baryon mass scale m B ∼ N Λ. 29 29 If n f N and m q /Λ 1/N 1, then there is an additional scale ΛN −1/2 associated with the mass of the η meson [69].…”

On the global structure of deformed Yang-Mills theory and QCD(adj) on ℝ 3 × S 1 $$ {\mathrm{\mathbb{R}}}^3\times {\mathbb{S}}^1 $$

“…Up to theory-dependent constants and a φ φ φ-independent contribution renormalizing the gauge coupling, which can be found in [33,35] for the various cases, both one loop functions g (1,2) ab from (3.17) are of the form…”

“…The one-loop correction to the metric was calculated for SYM in [33], via the R 3 × S 1 -index theorem [34] in monopole-instanton backgrounds, and in ref. [35] via Feynman diagrams in QCD(adj) and dYM.…”

“…The three vacua are distinguished by the expectation value of the (wrapped on S 1 L ) W H operator, 35 while W H in R 2 has area law due to confinement of its electric part. 33 This is because, while the combinatorics of identification of the minima (3.27) in the case of QCD(adj) is manageable and can potentially be automated, as opposed to the dYM case, we have not found an efficient way to treat all Nc and k. 34 The weights of the fundamental and adjoint (i.e. the nonzero roots α α α) representations are given by (in this simplest case it is easier to revert to an r-component basis)…”