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^3XS^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, arxiv.org/abs/1205.0290 and arxiv.org/abs/1212.1238, by a competition between monopole-instantons and exotic topological "molecules"---"neutral" or "magnetic" bions. We compute the trace of the Polyakov loop and its two-point 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 theta-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.

Figures (5)

• Figure 1: The conjectured phase diagram of SYM$^*$ in the $m$-$L$ plane. The calculable center-symmetry breaking quantum phase transition, occurring at small-$m,L$---the left-hand corner of the diagram, shown by a thick red line---is conjectured to be continuously connected, upon decoupling the gaugino, to the thermal deconfinement transition in pure YM theory, shown by the thick black line on the right. The dimensionless parameter which is varied, Eq. (\ref{['cmdefined']}), is $c_m \sim {m \over L^2 \Lambda^3}$, with $m \over \Lambda$ and $L \Lambda$ small. For all gauge groups, the calculable quantum phase transition occurs for $c_m$ of order unity. It should be possible to study this phase diagram on the lattice, see Section \ref{['introfive']}.
• Figure 2: An example of the discontinuous change of the Polyakov loop eigenvalues: the ${\mathbb Z}_2$ center-symmetric distribution of the eigenvalues of $\Omega$, in the fundamental representation of $Sp(12)$, for $c_m < c_{\rm cr} \sim 0.614$ (left panel) and the center-broken distribution for $c_m= c_{\rm cr}^+$ (right panel). The eigenvalues on the right panel are plotted for ${g^2 \over 4\pi}=0.4$ to visually enhance the center-breaking effect. The discontinuous change of the eigenvalue distributions across $c_{\rm cr}$ looks similar for all gauge groups.
• Figure 3: The discontinuous change of the string tension (in units of $m_0 R^{-1}$, see Section \ref{['bionpotential0']} for a definition of the scales and Section \ref{['results41']} for more details) for probes in the spinor representation of $Spin(7)$ as a function of $c_m \sim {m\over L^2 \Lambda^3}$ .
• Figure 4: An illustration, for $Spin(6)$ gauge group, of the $\theta$-dependence of the normalized critical transition mass, $c_m$, $c_{\rm \small cr}(\theta) \over c_{\rm \small cr}(\theta=0)$, and the normalized Polyakov loop discontinuity $|{\rm \small Tr}\langle \Delta\Omega(\theta)\rangle|\over |{\rm \small Tr}\langle \Delta\Omega(0)\rangle|$ for $0 < \theta < {10 \over 12}\pi$. The right panel is for the spinor-representation Polyakov loop. The behavior is qualitatively similar for all gauge groups. It has been recently observed in lattice simulations for $SU(N)$D'Elia:2012vvD'Elia:2013eua.
• Figure 5: The continuous line on the bottom shows the most negative eigenvalue of $k_{ij}$ for all $2\le N \le 200$. All eigenvalues $k_i^{diag}$ are shown for $N=10,20,...,190,200$. The $N$ dependence of the most negative eigenvalue is well-fitted by ${\rm min}_i k_i^{diag} \sim - 2.009 \log N$. If the ${\mathbb S}^1$ radius $R$ is kept fixed at large $N$, one finds, from (\ref{['modulispace5']}), a singularity in the moduli space metric at fixed $N g^2({2\over R})$, owing to the onset of strong coupling. On the other hand, when the mass of the lightest $W$-boson ${1 \over N R} = m_W$ is kept fixed, the moduli space metric is smooth, as follows from (\ref{['modulispace6']}) and the $\simeq-2\log N$ asymptotics of the minimal value of $k_i^{diag}$. See discussion in text.