We review how vertex constraints inherited from the thermal ground state strongly reduce the integration support of loop fourmomenta 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 2particleirreducible (2PI) loop order, a quantitative analysis enables us to disprove the conjecture voiced in hepth/0609033 that the loop expansion would terminate at a finite order. This reveals the necessity to investigate exact expressions of (at least some) higherloop order diagrams. Explicit calculation shows that although the behaviour of the 2PI threeloop contribution at low temperatures displays hierarchical suppression compared to lower loop orders, its hightemperature expression instead dominates all lower orders. However, an alllooporder resummation of a class of 2PI bubble diagrams is shown to yield an analytic continuation of the lowtemperature hierarchy to all temperatures in the deconfining phase.
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 [
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 twoloop order. These corrections are obtained by a loop expansion of the three effective gauge fields (quasiparticles) obtained after coarsegraining 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 wellcontrolled in the case of the diagrams treated here. A much more detailed and technical presentation of our results can be found in [
This work is structured as follows. In Section
In this section, we explain the origin and structure of sign constraints on massive quasiparticle loop momenta mediated by fourvertices. We state the results of an efficient bookkeeping explained in [
The full set of Feynman rules for the quasiparticles populating the thermal ground state in the deconfining phase is listed in [
Ratio
Loop number  Diagram number 




3  1  0.1667  48  0.00347222 
4  1  0.0463  48  0.00096451 
5  1  0.0139  128  0.00010851 
5  2  0.0123  32  0.00038580 
6  1  0.0044  320  0.00001366 
6  2  0.0036  32  0.00011253 
6  3  0.0033  16  0.00020898 
6  4  0.0033  120  0.00002572 
The only 2PI threeloop diagram (symmetry factor
In agreement with a simple counting argument given in [
The only 2PI fourloop diagram (symmetry factor
The first and second 2PI fiveloop diagram (symmetry factors
The first and most symmetric 2PI sixloop diagram (symmetry factor
The second and third 2PI sixloop diagrams (symmetry factors
The fourth 2PI sixloop diagram (symmetry factor
Despite this drawback, the actual order of magnitude of the higherloop order diagrams is not at all obvious from these sign considerations. Thus it is necessary to consider full expressions of the loop integrals to make definite statements about the convergence properties of the loop expansion. In the next section, we hence discuss the results of explicit calculations up to threeloop order which display hierarchical ordering at low temperatures but a dominating threeloop contribution at high temperatures.
In general, the expansion of the deconfining pressure in SU(2) YangMills thermodynamics reads
Restricting ourselves to the massive sector only, the oneloop pressure reads [
The pressure contribution associated with the twoloop diagram in Figure
The twoloop diagram for the pressure in the massive sector of deconfining SU(2) YangMills thermodynamics (symmetry factor
is defined as the Lorentzinvariant product of the dimensionless (we normalise physical fourmomentum components
In Figure
The twoloop pressure contribution (a) and moduli of the threeloop pressure corrections (b),
The pressure contribution associated with the diagram in Figure
The first sum in (
In the equivalent cases
Summing over these cases, the resulting contribution to
The sum of these cases amounts to
The second sum in (
The polynomial
This complicated expression can be evaluated by Monte Carlo methods for low temperatures (close to the critical temperature
where
The numerical values are obtained using the hightemperature plateau value of the mass and coupling
In Figure
Comparing
Monte Carlo results of
In order to make sense of the hightemperature behaviour of the threeloop diagram, we consider a truncated version of the DysonSchwinger (DS) equation of the fourvertex which reads
When closing legs into two (extra) loops, this becomes the resummation of the class of dihedrally symmetric bubble diagrams:
In the hightemperature limit the Mandelstam variables are constrained like
Hence, for
Solving for
to leading order in
We aimed in this work to provide an insight into how radiative corrections beyond twoloop order to the thermal ground state of SU(2) YangMills theory can be organised. The vertex constraints arising from the thermal ground state have been demonstrated to be insufficient to reduce the loop expansion to a finite number of diagrams. Moreover, explicit calculation of the 2PI threeloop diagram in the massive sector showed that these constraints are also not strong enough to extend the hierarchy in loop orders observed at low temperatures up to high temperatures. Resummation of corresponding classes of diagrams, however, has been demonstrated to be a promising resolution to this problem, yielding wellbounded corrections at all temperatures. The arising small nonthermal (imaginary) corrections to the pressure have been interpreted as a result of inhomogeneities in the thermal ground state constituted of densely packed centers of HarringtonShepard (anti)calorons. At this stage it is not yet clear if further 2PI bubble diagrams in the massive sector are sufficiently constrained prior to resummation, due to lower symmetry and hence likely lower number of possibly equivalent constraints (Section
The subject of how to organise the computation of radiative corrections in deconfining YangMills thermodynamics thus is a broad one. Being of immediate urgency, it would be important to analyse diagrams symmetric under the
The author declares that there are no conflicts of interest regarding the publication of this paper. This includes the funding mentioned in Acknowledgments.
The author thanks his collaborators Thierry Grandou and Ralf Hofmann for their contributions to the related paper [