Onset of magnetic monopole-antimonopole condensation

We determine the critical strength of the effective electric coupling for the onset of Bose condensation of stable magnetic monopoles and antimonopoles in SU(2) Yang-Mills thermodynamics. Two scenarios are considered: infinitely fast and infinitely slow downward approach of the critical temperature. Our results support the claim that the first lepton family and the weak interactions emerge from pure SU(2) gauge dynamics of scale ∼ 0.5MeV. Introduction. A genuine understanding of fundamental physics requires well-controlled conditions. As for four-dimensional Quantum Yang-Mills theory unadulterated thermodynamics seems to be such a setting [1, 2]. Because the term ‘understanding’ not only refers to the ability to derive quantitative results but also to one’s position to actually interpret them in deterministic terms a connection between effective (results) and fundamental (interpretation) fluctuations in the according formulations of the same partition function needs to be made. This connection also serves as the pointer to how a controlled deformation of the original thermodynamical setting can be carried out. In this note we are concerned with the transition between deconfining and preconfining SU(2) Yang-Mills thermodynamics. In approaching this transition from above in a specified way, we ask the question what the critical value for the effective, electric coupling e is at which the thermal ground state of the deconfining phase drastically rearranges such as to attribute mass to a formerly massless gauge mode (condensation of magnetic (anti)monopoles). Technically speaking, the answer to this question is obtained in a surprisingly simple way if the above-mentioned connection between effective and fundamental fluctuations is made. Starting from a useful a priori estimate of the deconfining thermal ground state the computation of thermodynamical quantities is organized into a rapidly converging loop expansion carried by effective gauge-field fluctuations of trivial topology. At high temperatures this expansion indicates the presence of stable and isolated (anti)monopoles of a given number density, and at temperatures not far above the critical temperature Tc the collective dynamics of these defects, essentially exhausted by two-loop corrections, induce, depending on momentum, screening or antiscreening of the tree-level massless, effective gauge mode [3, 4, 5, 6]. Some remarks on deconfining SU(2) Yang-Mills thermodynamics. On the one-loop level the Yang-Mills system is approximated by a gas of noninteracting thermal quasiparticles fluctuating above a thermal ground state. The latter is decribed by an effective inert, adjoint scalar field and a pure-gauge configuration, and the mass spectrum of thermal quasiparticles is made explicit in admissible unitary gauge [1, 2]. On the fundamental level, this situation locally is induced by interacting (anti)calorons of topological charge-modulus unity whose small holonomy [7, 8, 9], temporarily created by the absorption of soft and fundamental propagating gauge fields, is insufficient for the permanent release of their magnetic monopoleantimonopole constituents [10]. This approximation captures the total pressure up to an error smaller than one percent. Effective radiative corrections describe a departure from this situation in so far as the thermal ground state is contributed to by domainized configurations of the adjoint scalar field with the vertices of sufficiently many domain boundaries [11] representing magnetic (anti)monopoles. At high temperature, an average over such configurations is performed implicitly by a particular two-loop diagram for the pressure [3, 4] whose ratio to the one-loop approximation with increasing temperature rapidly approaches a small, negative constant. This constant represents the existence of an average density of highly