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Register a new userThe Yang-Mills deconfinement transition from a high temperature expansion
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The high temperature expansion is an analytical tool to study critical phenomena in statistical mechanics. We apply this method to 3d effective theories of Polyakov loops, which have been derived from 4d lattice Yang-Mills by means of resummed strong coupling expansions. In particular, the Polyakov loop susceptibility is computed as a power series in the effective couplings. A Pade analysis then provides the location of the phase transition in the effective theory, which can be mapped back to the parameters of 4d Yang-Mills. Our purely analytical results for the critical couplings $beta_c(N_tau)$ agree to better than $10%$ with those from Monte Carlo simulations. For the case of $SU(2)$, also the critical exponent $gamma$ is predicted accurately, while a first-order nature as for $SU(3)$ cannot be identified by a Pade analysis. The method can be generalized to include fermions and finite density.
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Euclidean strong coupling expansion of the partition function is applied to lattice Yang-Mills theory at finite temperature, i.e. for lattices with a compactified temporal direction. The expansions have a finite radius of convergence and thus are valid only for $beta<beta_c$, where $beta_c$ denotes the nearest singularity of the free energy on the real axis. The accessible temperature range is thus the confined regime up to the deconfinement transition. We have calculated the first few orders of these expansions of the free energy density as well as the screening masses for the gauge groups SU(2) and SU(3). The resulting free energy series can be summed up and corresponds to a glueball gas of the lowest mass glueballs up to the calculated order. Our result can be used to fix the lower integration constant for Monte Carlo calculations of the thermodynamic pressure via the integral method, and shows from first principles that in the confined phase this constant is indeed exponentially small. Similarly, our results also explain the weak temperature dependence of glueball screening masses below $T_c$, as observed in Monte Carlo simulations. Possibilities and difficulties in extracting $beta_c$ from the series are discussed.
We study the machine learning techniques applied to the lattice gauge theorys critical behavior, particularly to the confinement/deconfinement phase transition in the SU(2) and SU(3) gauge theories. We find that the neural network, trained on lattice configurations of gauge fields at an unphysical value of the lattice parameters as an input, builds up a gauge-invariant function, and finds correlations with the target observable that is valid in the physical region of the parameter space. In particular, if the algorithm aimed to predict the Polyakov loop as the deconfining order parameter, it builds a trace of the gauge group matrices along a closed loop in the time direction. As a result, the neural network, trained at one unphysical value of the lattice coupling $beta$ predicts the order parameter in the whole region of the $beta$ values with good precision. We thus demonstrate that the machine learning techniques may be used as a numerical analog of the analytical continuation from easily accessible but physically uninteresting regions of the coupling space to the interesting but potentially not accessible regions.
The correlations between the modulus of the Polyakov loop, its phase $theta$ and the Landau gauge gluon propagator at finite temperature are investigated in connection with the center symmetry for pure Yang-Mills SU(3) theory. In the deconfined phase, where the center symmetry is spontaneously broken, the phase of the Polyakov loop per configuration is close to $theta = 0$, $pm , 2 pi /3$. We find that the gluon propagator form factors associated with $theta approx 0$ differs quantitatively and qualitatively from those associated to $theta approx pm , 2 pi /3$. This difference between the form factors is a property of the deconfined phase and a sign of the spontaneous breaking of the center symmetry. Furthermore, given that this difference vanishes in the confined phase, it can be used as an order parameter associated to the deconfinement transition. For simulations near the critical temperature $T_c$, the difference between the propagators associated to $theta approx 0$ and $theta approx pm , 2 pi /3$ allows to classify the configurations as belonging to the confined or deconfined phase. This establishes a selection procedure which has a measurable impact in the gluon form factors. Our results also show that the absence of the selection procedure can be erroneously taken as lattice artifacts.
Fermion boundary conditions play a relevant role in revealing the confinement mechanism of N=1 supersymmetric Yang-Mills theory with one compactified space-time dimension. A deconfinement phase transition occurs for a sufficiently small compactification radius, equivalent to a high temperature in the thermal theory where antiperiodic fermion boundary conditions are applied. Periodic fermion boundary conditions, on the other hand, are related to the Witten index and confinement is expected to persist independently of the length of the compactified dimension. We study this aspect with lattice Monte Carlo simulations for different values of the fermion mass parameter that breaks supersymmetry softly. We find a deconfined region that shrinks when the fermion mass is lowered. Deconfinement takes place between two confined regions at large and small compactification radii, that would correspond to low and high temperatures in the thermal theory. At the smallest fermion masses we find no indication of a deconfinement transition. These results are a first signal for the predicted continuity in the compactification of supersymmetric Yang-Mills theory.
In this paper, we provide a thorough study on the expansion of single trace Einstein-Yang-Mills amplitudes into linear combination of color-ordered Yang-Mills amplitudes, from various different perspectives. Using the gauge invariance principle, we propose a recursive construction, where EYM amplitude with any number of gravitons could be expanded into EYM amplitudes with less number of gravitons. Through this construction, we can write down the complete expansion of EYM amplitude in the basis of color-ordered Yang-Mills amplitudes. As a byproduct, we are able to write down the polynomial form of BCJ numerator, i.e., numerators satisfying the color-kinematic duality, for Yang-Mills amplitude. After the discussion of gauge invariance, we move to the BCFW on-shell recursion relation and discuss how the expansion can be understood from the on-shell picture. Finally, we show how to interpret the expansion from the aspect of KLT relation and the way of evaluating the expansion coefficients efficiently.
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