Advances in High Energy Physics

Volume 2017 (2017), Article ID 8593678, 9 pages

https://doi.org/10.1155/2017/8593678

## Massive Fluctuations in Deconfining SU(2) Yang-Mills Thermodynamics

Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

Correspondence should be addressed to Ingolf Bischer; ed.grebledieh-inu.syhpht@rehcsib

Received 3 April 2017; Revised 17 July 2017; Accepted 10 August 2017; Published 27 September 2017

Academic Editor: Sally Seidel

Copyright © 2017 Ingolf Bischer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We review how vertex constraints inherited from the thermal ground state strongly reduce the integration support of loop four-momenta associated with massive quasiparticles in bubble diagrams constituting corrections to the free thermal quasiparticle pressure. In spite of the observed increasingly suppressing effect when increasing 2-particle-irreducible (2PI) loop order, a quantitative analysis enables us to disprove the conjecture voiced in hep-th/0609033 that the loop expansion would terminate at a finite order. This reveals the necessity to investigate exact expressions of (at least some) higher-loop order diagrams. Explicit calculation shows that although the behaviour of the 2PI three-loop contribution at low temperatures displays hierarchical suppression compared to lower loop orders, its high-temperature expression instead dominates all lower orders. However, an all-loop-order resummation of a class of 2PI bubble diagrams is shown to yield an analytic continuation of the low-temperature hierarchy to all temperatures in the deconfining phase.

#### 1. Introduction

There is a variety of topologically nontrivial solutions to classical equations of motion in SU(2) gauge theory on a flat Euclidean spacetime manifold. That the trivial vacuum may not be the relevant one at nonzero temperature becomes apparent in the problems of the standard perturbative approach, in particular in the infrared problem already pointed out by Linde in 1980 [1]. Divergences in the soft-magnetic sector, as encountered in small-coupling expansions at high temperature [2–8], motivated by asymptotic freedom [9–11], invalidate the perturbative expansion starting at some finite order [12] and hint at relevant substructures that are missed. Indeed, lattice-based studies relate topological configurations to fundamental properties of Yang-Mills theory [13–15]. An approach to finding a thermal ground state estimate that includes gauge field configurations of nontrivial topology reveals that Harrington-Shepard (anti)calorons [16] of topological charge are the constituents of this ground state with spatially densely packed centers and overlapping peripheries. Their contribution is manifest in the nontriviality of the spatial and scale-parameter average (spatial coarse-graining) of the two-point field-strength correlator in association with the magnetic field of an (anti)caloron [17]. Lattice gauge theory qualitatively reproduces certain aspects of this correlation in infrared sensitive thermodynamical quantities such as the pressure, provided that the* differential* method is used which appeals to the nonperturbative beta function [18, 19]. However, this function needs to be approximated. On the other hand, the* integral* method [20], which does not rely on the beta function but introduces an integration constant, yields results that are largely disparate, the reason being the choice of integration constant (no negative pressure) and finite-volume artifacts [21].

In this work, we give an overview of recent proceedings in the treatment of radiative corrections to the pressure of this thermal ground state beyond two-loop order. These corrections are obtained by a loop expansion of the three effective gauge fields (quasiparticles) obtained after coarse-graining over the ground state constituent configurations, two of which become massive by an adjoint Higgs mechanism. We find that resummation of infinitely many diagrams is necessary to obtain a finite result which after resummation is well-controlled in the case of the diagrams treated here. A much more detailed and technical presentation of our results can be found in [22].

This work is structured as follows. In Section 2, we present a nonexhaustive way of using constraints in the massive sector to reduce the number of possible loop-momentum configurations in bubble diagrams in a purely combinatorical way. In Section 3, we state the contributions of all bubble diagrams in the massive sector up to three loops and conclude why resummation is necessary. This resummation of a particular family of diagrams is finally demonstrated in Section 4 and followed by a summary and conclusions in Section 5.

#### 2. Sign Constraints in Massive Bubble Diagrams

In this section, we explain the origin and structure of sign constraints on massive quasiparticle loop momenta mediated by four-vertices. We state the results of an efficient book-keeping explained in [22] in terms of the ratio of the number of nonexcluded sign configurations and the number of a priori possible sign configurations. To close the section, an explanation of why nonvanishing diagrams exist at any finite loop order is given.

The full set of Feynman rules for the quasiparticles populating the thermal ground state in the deconfining phase is listed in [17]. Here, we restrict the discussion to 2PI diagrams, by which we mean bubble diagrams that do not become 1PI contributions to a polarisation tensor upon cutting any single line, including only the two massive fields (corresponding to two su(2) algebra directions that are broken by the thermal ground state and obtain a mass by an adjoint Higgs mechanism). This implies that only four-vertices may appear. The first important fact for what follows is that those massive fields propagate strictly on-shellwhere is any four-momentum, is the mass, is the effective gauge coupling, and is the gauge invariant modulus of the inert, adjoint scalar field associated with densely packed (anti)caloron centers in the thermal ground state [17, 23] which sets the scale of maximal resolution. The second important fact is that the scattering channels at four-vertices are restricted not to resolve higher energies than this scale. By this we mean that each four-vertex hosts a superposition of channels corresponding to the three Mandelstam variables , , and constrained by . By virtue of the on-shellness, each constraint on a Mandelstam variable implies a restriction of the energy-signs of the respective loop momenta according to [24]Hence, for all scattering channel combinations in a diagram, one can exclude several sign configurations. We define the ratio of a diagram by the sum (over channel combinations) of the numbers of nonexcluded sign configurations divided by the number of a priori possible sign configurations times the number of channel combinations (, where denotes the number of vertices). In Figure 1 through Figure 5 we give all 2PI diagrams up to six-loop order and their respective values of . All results are listed in Table 1.