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 $\sim 0.5 $MeV.


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 1-3 . Because the term "understanding" not only refers to the ability to derive quantitative results but also refers 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 2 ISRN High Energy Physics 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. For temperatures not far above the critical temperature T c the collective dynamics of these defects are exhaustedly described by twoloop corrections to the pressure 4, 5 . They induce, depending on momentum, screening, or antiscreening of the tree-level massless, effective gauge mode 6, 7 .

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 described 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 1-3 . On the fundamental level, this situation locally is induced by interacting anti calorons of topological charge-modulus unity whose small holonomy 8-12 , temporarily created by the absorption of soft and fundamental propagating gauge fields, is insufficient for the permanent release of their magnetic monopole-antimonopole constituents 13 . 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 14 representing magnetic anti monopoles. At high temperature, an average over such configurations is performed implicitly by a particular two-loop diagram for the pressure 4, 5 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 nonrelativistic and screened magnetic anti monopoles which are released by the rare and irreversible dissociation of anti calorons 7, 13 . The irreversibility of anti caloron dissociation together with the fact that overall magnetic charge is nil due to pairwise monopole-antimonopole creation implies that the chemical potential associated with monopole-antimonopole pairs vanishes in the infinite-volume limit.
As temperature decreases, stable anti monopoles behave like comoving raisins, immersed in an expanding, infinitely extended dough, with their average distance set by the inverse temperature. In the hypothetic limit of isolation, see 7 , the liability of a caloron or an anticaloron to dissociate into a pair of an isolated monopole of mass m M and its antimonopole of mass m A is determined solely by its holonomy 10-13 . For the realistic case of densely packed anti calorons the description of a single caloron by semiclassical methods fails 7 . An average over monopole-antimonopole creation processes, however, leads to a number density n M A of pairs which is determined by the following holonomy-independent sum of masses 10-12 : where T and λ ≡ 2πT/Λ are the dimensionful and the dimensionless versions of the temperature, respectively, and Λ denotes the Yang-Mills scale. The λ dependence of e is a consequence of the renormalization-group invariance of the a priori estimate of the Yang-Mills partition function under the spatial coarse-graining applied to derive the effective ISRN High Energy Physics 3 theory 1-3 . This running of e with temperature describes the screening effects due to instable magnetic dipoles arising from anti calorons whose holonomy is only mildly deformed away from trivial. Notice that e approaches a plateau e ≡ √ 8π very rapidly with increasing λ > λ c 13

Condensation of Monopole-Antimonopole Pairs
Based on the discussion presented by Huang 16 of thermalized, noninteracting Bose particles with mass m a relation was formulated in 17 between the total number density n and the density n 0 of particles residing in the condensate. For statistical weight unity only one species of monopole-antimonopole pairs occurs in an SU 2 Yang-Mills theory one has where μ ≡ m/T , and K 2 x is the modified Bessel function of the second kind. At the onset of Bose condensation, where n 0 is yet zero, the total number density n is given by the number density n fr of freely fluctuating particles So in the fully thermalized system, which takes place if T is slowly lowered towards T c , we have at the onset of Bose condensation In deconfining Yang-Mills thermodynamics, however, a pair of an isolated and screened magnetic monopole and its antimonopole owes its very existence to the presence of a heat bath of given temperature. The according relation between mass and temperature, see 2.1 , together with μ c determines the critical temperature for condensation to be the solution to the following equation: 3.4 Notice that the limit μ c → 0 of 3.3 yields the identity 2ζ 3 2ζ 3 , where ζ z is Riemann's zeta function. Since the left-hand side of 3.3 is monotonic decreasing and the right-hand side is monotonic increasing in μ c , it follows that μ c 0 is the only solution. Thus In deconfining SU 2 Yang-Mills thermodynamics adiabatically slow approach of T c from above , only massless monopoles and antimonopoles condense into a new ground state at the critical temperature T c corresponding to the logarithmic pole in e described by 2.2 .
Alternatively, one may ask the question of what happens in the limit, where T c is rapidly approached from above. As we will see, such an adiabatic sudden approximation fully takes into account the static screening effects imposed by the system at high temperatures but neglects the influence of propagating dual-gauge modes not too far above T c . Since the pole of the coupling e λ , see 2.2 , is logarithmic reflecting the fact that the Yang-Mills scale Λ nonperturbatively interferes with the dynamics of fundamental propagating gauge modes only shortly above λ c 18 , we may consider the limit of large temperatures for the dependence on temperature of the density n M A,as of interacting but statically equilibrated monopole-antimonopole pairs. This situation is relevant for particle collisions at sufficiently high center-of-mass energy where locally a hot spot of deconfining phase is generated whose temperature quickly drops due to cooling and shrinking by evaporation 19 .
Recall that n M A,as is extracted from a particular two-loop correction to the quasiparticle pressure as calculated in the effective theory 7 . One has n M A,as 21.691 −3 T 3 ∼ 9.8 × 10 −5 T 3 .

3.5
At high temperatures isolated, stable, and screened anti monopoles are nonrelativistic 7 . If temperature is lowered towards T c in a sufficiently rapid way then the generation of almost massless stable monopoles and antimonopoles by strong screening occurs quickly enough to not affect their nonrelativistic nature endowed by high-temperature physics. To catch up in velocity anti monopoles must interact via the exchange of dual-gauge modes that are close to their mass shell and therefore propagate at a speed close to the velocity of light. In contrast to the portion of magnetic screening induced by an increased activity of instable monopole-antimonopole pairs and described by the effective-theory a priori estimate for the thermal ground state this part of the thermalization of anti monopoles-a loop correction in the effective theory-thus requires a finite amount of time. where the expression to the far right is obtained by considering μ c,M A 1 of n c or, equivalently, of 1/2π 2 times the right-hand side of 3.3 . To summarize, 3.6 determines μ c,M A in a situation where highly nonrelativistic, stable, and isolated anti monopoles are adiabatically fast deprived of their mass by cooling enhanced instantaneous screening by instable dipoles so that no time is available for them to start moving.
The solution to 3.6 is μ c,M A 7.04 × 10 −3 . Note the amusing fact that the nonrelativistic nature of monopole-antimonopole pairs is assured by the large-mass limit of n c while the solution to 3.6 actually corresponds to a small mass on the scale of temperature. The resolution of this apparent puzzle is grounded in the fact that the sudden approximation employed does not admit a thermodynamical interpretation: the expression for n M A,as at T T c is analytically continued down to T c . Using

Mass of Charged Vector Bosons in the Standard Model
The ratio of the charged-vector-boson-mass m W to the electron mass m e , as experimentally measured, is given as Obviously, this value for e c lies in the range given by 3.9 .

Summary
In this paper we have considered two scenarios for the onset of magnetic monopoleantimonopole condensation at the deconfining-preconfining transition in SU 2 Yang-Mills thermodynamics: infinitely slow and infinitely fast downward approach of T c . In the former situation, we have shown that pairs of stable monopoles and antimonopoles do only condense when they are massless, that is, at the pole position for the effective electric coupling e. This is a consequence of the fact that due to the irreversibility of the anti monopole creation process dissociation of large-holonomy anti calorons and due to overall charge neutrality pairwise creation in an infinite spatial volume the chemical potential associated with pairs is nil. Concerning the case of an infinitely fast approach of T c , we obtain a lower bound on the value e c of the critical coupling. Our results are compatible with the claim that a pure SU 2 Yang-Mills theory of scale Λ e ∼ m e ∼ 0.5 MeV is responsible for the emergence of the first lepton family and the weak interactions of the Standard Model of Particle Physics. Due to the experimental fact of a universal electric coupling of the photon-likely to emerge from an SU 2 Yang-Mills theory of scale Λ CMB ∼ 10 −4 eV 5, 6, 23 -to all charged leptons it is clear that Nature's SU 2 Yang-Mills theories of the same electric-magnetic parity mix maximally.