Universität Frankfurt

We compute the phase and the modulus of an energy- and pressure-free, composite, adjoint, and inert field $\phi$ in an SU(2) Yang-Mills theory at large temperatures. This field is physically relevant in describing part of the ground-state structure and the quasiparticle masses of excitations. The field $\phi$ possesses nontrivial $S^1$-winding on the group manifold $S^3$. Even at asymptotically high temperatures, where the theory reaches its Stefan-Boltzmann limit, the field $\phi$, though strongly power-suppressed, is conceptually relevant: its presence resolves the infrared problem of thermal perturbation theory.


Introduction
In [1] one of us has put forward an analytical and nonperturbative approach to SU(2)/SU(3) Yang-Mills thermodynamics. Each of theses theories comes in three phases: deconfining (electric phase), preconfining (magnetic phase), and completely confining (center phase). This approach assumes the existence of a composite, adjoint Higgs field φ, describing part of the thermal ground state, that is, the BPS saturated topologically nontrivial sector of the theory. The field φ is generated by a spatial average over noninteracting trivial-holonomy SU(2) calorons [15] which can be embedded in SU (3). The 'condensation' 1 of trivial-holonomy SU(2) calorons into the field φ must take place at an asymptotically high temperature [1], that is, at the limit of applicability of the gauge-field theoretical description. For any physics model formulated in terms of an SU(2)/SU(3) Yang-Mills theory this is to say that caloron 'condensation' takes place at T ∼ M P where M P denotes the Planck mass. Since |φ| ∼ Λ 3 2πT topological defects only very marginally deform the ideal-gas expressions for thermodynamical quantities at T ≫ Λ. Here Λ denotes the Yang-Mills scale. Every contribution to a thermodynamical quantity, which arises from the topologically nontrivial sector, is power suppressed in temperature.
As a consequence, the effective theory is asymptotically free and exhibits the same infrared-ultraviolet decoupling property [1] that is seen in renormalized perturbation theory [2]. Asymptotic freedom is a conceptually very appealing property of SU(N) Yang-Mills theories. It first was discovered in perturbation theory [3].
In the effective thermal theory interactions between trivial-holonomy calorons in the ground state are taken into account by obtaining a pure-gauge solution to the classical equations of motion for the topologically trivial sector in the (nonfluctuating and nonbackreacting) background φ. Thus the partition function of the fundamental theory is evaluated in three steps in the electric phase: (i) integrate over the admissible part of the moduli space for the caloron-anticaloron system and spatially average over the associated two-point correlations to derive the (classical and temperature dependent) dynamics of an adjoint, spatially homogeneous scalar field φ, (ii) establish the quantum mechanical and statistical inertness of φ and use it as a temperature dependent background to find a pure-gauge solution  for the effective gauge coupling e. In the following we will restrict our discussion to the case SU (2). Isolated magnetic charges are generated by dissociating calorons of sufficiently large holonomy [4,5,6,7,8,9,10,11], for a quick explanation of the term holonomy see Fig. 1. Nontrivial holonomy is locally excited by interactions between trivial-holonomy calorons and anticalorons. In [9] it was shown as a result of a heroic calculation that small (large) holonomy leads to an attractive (repulsive) potential between the monopole and the antimonopole constituents of a given caloron.
An attraction between a monopole and an antimonopole leads to annihilation if the distance between their centers is comparable to the sum of their charge radii. Thermodynamically, the probability for exciting a small holonomy is much larger than that for exciting a large holonomy. In the former case this probability roughly is determined by the one-loop quantum weight of a trivial holonomy caloron, while in the latter case both monopole constituents have a combined mass ∼ 4π 2 T ∼ 39 T [5]. Thus an attractive potential between a monopole and its antimonopole is the by far dominating situation. This is the microscopic reason for a negative ground-state pressure P g.s. which, on spatial average, turns out to be P g.s. = −4πΛ 3 T (the equation of ground state is ρ g.s. = −P g.s. ) [1]. In the unlikely case of repulsion (large holonomy) the monopole and the antimonopole separate back-to-back until their interaction is sufficiently screened to facilitate their existence in isolation (as long as the overall pressure of the system is positive). Magnetic monopoles in isolation do not contribute to the pressure of the system 3 . The overall pressure is positive if the gauge-field fluctuations after spatial coarswe-graining are sufficiently light and energetic to over-compensate the negative ground-state contribution, that is, if the temperature is sufficiently large.
Caloron-induced tree-level masses for gauge-field modes decay as 1/ √ T when heating up the system. Due to the linear rise of ρ g.s. with T the thermodynamics of the ground state is thus subdominant at large temperatures 4 . The main purpose 3 The reader may convince himself of this fact by computing the energy-momentum tensor on a BPS monopole. 4 Gauge-field excitations are free at large temperatures [12] and contribute to the total pressure and the total energy density like ∼ T 4 . The small residual interactions, which peak close to a 2 ndorder transition to the magnetic phase at T c,E , are likely to explain the large-angle anomalies seen in some CMB power spectra [12,13]. When cooling the system, monopoles and antimonopoles, which were generated by dissociating calorons (large holonomy), start to overlap at a temperature T o slightly higher than T c,E because they are moving towards one another under the influence of an overall negative pressure. The latter is generated by the dominating pressure component generated of the present work is to compute and to discuss the dynamical generation of an adjoint, macroscopic, and composite scalar field φ. This is a first-principle analysis of the ground-state structure in the electric phase of an SU(2) Yang-Mills theory.
The paper is organized as follows: In Sec. 2 we write down and discuss a nonlocal definition, relevant for the determination of φ's phase, in terms of a spatial and scaleparameter average over an adjointly transforming 2-point function. This average needs to be evaluated on trivial-holonomy caloron and anticaloron configurations at a given time τ . In Sec. 3 we perform the average and discuss the occurrence of a global gauge freedom in φ's phase, which has a geometrical interpretation. In Sec. 4 we show how the derived information about a nontrivial S 1 winding of the field by small-holonomy calorons whose monopole constituents attract and eventually annihilate at a given point in space but get re-created elseswhere. Naively seen, negative pressure corresponds to an instability of the system causing it to collapse. We usually imagine a contracting system in terms of a decrease of the mean interparticle distance while tacitly assuming the particles to be pointlike. Despite an overall negative pressure a total collapse in the above sense does not occur in an SU(2) Yang-Mills theory. This can be understood as follows: The mass of an isolated, screened monopole is m ∼ 2π 2 T e , and the effective gauge coupling e is constant if monopoles do not overlap, that is, for T > T o (magnetic charge conservation [1]). At T o the magnetic charge contained in a given spatial volume no longer remains constant in time because of the increasing mobility of strongly screened monopoles. Thus formerly separated monopoles can annihilate and but also get re-created. Therefore the notion of a local collapse ceases to be applicable since the associated particles cease to exist if they are close to one another. If the rate of annihilation equals the rate of re-creation of monopole-antimonopole pairs then we are witnessing an equilibium situation characterized by a temperature despite a negative overall pressure.
φ together with analyticity of the right-hand side of the associated BPS equation and with the assumption of the existence of an externally given scale Λ can be used to uniquely construct a potential determining φ's classical (and temperature dependent) dynamics. In Sec. 5 we summarize and discuss our results and give an outlook on future research.

Definition of φ's phase
In this section we discuss the BPS saturated, topological part of the ground-state physics in the electric phase of an SU(2) Yang-Mills theory. According to the approach in [1] the adjoint scalar φ emerges as an energy-and pressure-free (BPS saturated) field from a spatial average over the classical correlations in a caloronanticaloron system of trivial holonomy in absence of interactions. On spatial average, the latter are taken into account by a pure-gauge configuration solving the classical, trivial-topology gauge-field equations in the spatially homogeneous background φ.
This is consistent since φ's quantum mechanical and statistical inertness can be established. Without assuming the existence of a Yang-Mills scale Λ only φ's phase, that is φ |φ| , can be computed. A computation of φ itself requires the existence of Λ.
As we shall see, the information about the S 1 winding of φ's phase together with the analyticity of the right-hand side of φ's BPS equation uniquely determines φ's modulus in terms of Λ and T .
Let us first set up some prerequisites. We consider BPS saturated solutions to the Yang-Mills equation which are periodic in Euclidean time τ (0 ≤ τ ≤ 1 T ) and of trivial holonomy. The only relevant configurations are calorons of topological charge 5 ±1. They are [15] A where the 't Hooft symbols η aµν andη aµν are defined as The solutions in Eq. (2) (the superscript (A)C refers to (anti)caloron) are generated by a temporal mirror sum of the 'pre'potential Π of a single (anti)instanton in singular gauge [14]. They have the same color orientation as the 'seed' instanton or 'seed' antiinstanton. In Eq. (2) λ a , (a = 1, 2, 3), denote the Pauli matrices.
The 'nonperturbative' definition of the gauge field is used were the gauge coupling constant g is absorbed into the field. 5 Configurations with higher topological charge and trivial holonomy have been constructed, see for example [16,17]. A priori they should contribute to the ground-state thermodynamics of the theory in terms of additional adjoint scalar fields. The nonexistence of these Higgs fields is assured by their larger number of dimensionful moduli -for the charge-two case we have two instanton radii and the core separation -which does not allow for the nonlocal definition of a macroscopic, dimensionless phase, see (9) and the discussion following it.
The scalar function Π(τ, x) is given as [15] Π(τ, where r ≡ |x|, β ≡ 1/T , and ρ denotes the scale parameter whose variation leaves the classical action S = 8π 2 g 2 invariant. At a given ρ the solutions in Eq. From the BPS saturation it follows that the (Euclidean) energy-momentum tensor θ µν , evaluated on A This property translates to the macroscopic field φ with energy-momentum tensor θ µν in an effective theory since φ is obtained by a spatial average over caloronanticaloron correlations neglecting their interactions 7 The field φ is spatially homogeneous since it emerges from a spatial average. If the action density governing φ's dynamics in the absence of caloron interactions 6 Because of periodicity, τ z needs to be restricted as 0 ≤ τ z ≤ β. 7 A spatial average over zero energy-momentum yields zero. contains a kinetic term quadratic in the τ -derivatives and a potential V then Eq. (7) is equivalent to φ solving the first-order equation where parametrizes the potential V and thus also the classical solution to Eq. (8). In the absence of trivial-topology fluctuations it is, however, invisible, see Eq. (7). Only after the macroscopic equation of motion for the trivial-topology sector is solved for a pure-gauge configuration in the background φ does the existence of a Yang-Mills scale become manifest by a nonvanishing ground-state pressure and a nonvanishing ground-state energy density [1]. Hence the trace anomalyθ µµ = 0 for the total energy-momentum tensorθ µν ≡θ g.s. µν + θ fluc µν in the effective theory which includes the effects of trivial-topology fluctuations: Sinceθ g.s.
Without imposing constraints other than nonlocality 9 the τ dependence of φ's phase (the ratio of the two averages φ and |φ| over admissible moduli defomrmations of 9 A local definition of φ's phase always yields zero due to the (anti)selfduality of the (anti)caloron configuration.
A ( C, A)) would naively be characterized as The dots in (9) stand for the contributions of higher n-point functions and for reducible, that is factorizable, contributions with respect to the spatial integrations.
In (9) the following definitions apply: The integral in the Wilson lines in Eqs. (10) is along a straight line 10 connecting the points (τ, 0) and (τ, x), and P denotes the path-ordering symbol.
As a consequence of Eq. (11) the right-hand side of (9) transforms as where the SO(3) matrix R ab (τ ) is defined as 10 Curved integration contours introduce scales which have no physical counterpart on the classical level. Furthermore, shifting the spatial part of the argument (τ, 0) → (τ, z) in (9) introduces a parameter |z| of dimension inverse mass. There is no physical reason for a finite value of |z| to exist on the classical level. Thus we conclude that z = 0.
Thus we have defined an adjointly transforming scalar in (9). Moreover, only the time-dependent part of a microscopic gauge transformation survives after spatial averaging (macroscopic level).
In (9) the ∼ sign indicates that both left-and right-hand sides satisfy the same homogeneous evolution equation in τ Here D is a differential operator such that Eq. (14) represents a homogeneous differential equation. As it will turn out, Eq. (14) is a linear second-order equation which, up to global gauge rotations, determines the first-order or BPS equation whose solution φ's phase is. Each term in the series in (9) is understood as a sum over the two solutions in Eq. (2), that is, A α = A C α or A α = A A α . As we shall show in Sec. 3, the dimensionless quantity defined on the right-hand side of (9) is ambiguous 11 , the operator D, however, is not.
The quantities appearing in the numerator and denominator of the left-hand side of (9) are understood as functional and spatial averages over the appropriate multilocal operators, being built of the field strength and the gauge field. The functional average is restricted to the moduli spaces of A α = A C α and A α = A A α excluding global color rotations and time translations.
, is always possible and not fixed by a physical boundary condition on the classical level. As we shall see below, the same holds true for the normalization of the right-hand side of (9). Therefore, for each color direction these two ambiguities parametrize the solution space of the second-order linear operator D.
Let us explain this in more detail. For the gauge variant density in (9)  What about the contribution of calorons with a topological charge modulus larger than unity? Let us consider the charge-two case. Here we have three moduli of dimension length which should enter the average defining the differential operator D: two scale parameters and a core separation. The reader may easily convince himself that by the absence of an explicit temperature dependence it is impossible to 12 The 'naked' gauge charge in (9) is needed for a coupling of the trivial topology sector to the ground-state after spatial coarse-graining generating (i) quasiparticle masses and (ii) finite values of the ground-state energy density and the ground-state pressure [1].
define the associated dimensionless quantity in terms of spatial and moduli averages over n-point functions involving these configurations. The situation is even worse for calorons of topological charge larger than two. We conclude that only calorons of topological charge ±1 contribute to the definition of the operator D in Eq. (14) by means of Eq. (9).
Thus the path-ordering symbol can, indeed, be omitted in Eq. (16). The field strength F C aµν on the caloron solution in Eq. (2) is where Π is defined in Eq. (4). For the anticaloron one replacesη by η in Eq. (18).
Using Eqs. (9), (16), and (18), we obtain the following expression for the contribution φ a |φ| C arising from calorons: and The dependences on ρ and β are suppressed in the integrands of (19) and Eq. (20). It is worth mentioning that the integrand in Eq. (20) is proportional to δ(s) for r ≫ β.
A useful set of identities is This, however, would only be the case if no ambiguity in evaluating the integral in both cases existed. But such ambiguities do occur! First, the τ dependence of the anticaloron's contribution may be shifted as compared to that of the caloron. Second, the color orientation of caloron and anticaloron contributions may be different.
Third, the normalization of the two contributions may be different. To see this, we need to investigate the convergence properties of the radial integration in (19). It is easily checked that all terms give rise to a converging r integration except for the following one: Namely, for r > R ≫ β (24) goes over in 4 t a πρ 2 sin(2g(τ, r)) βr 3 .
Thus the r-integral of the term in (24) is logarithmically divergent in the infrared 13 : Recall that g(τ, r) behaves like a constant in r for r > R. The angular integration, on the other hand, would yield zero if the radial integration was regular. Thus a logarithmic divergence can be cancelled by the vanishing angular integral to yield some finite and real but undetermined normalization of the emerging τ dependence.
To investigate this, both angular and radial integration need to regularized.
We may regularize the r integral in (26) by prescribing with ǫ > 0. We have Away from the pole at ǫ = 0 this is regular. For ǫ < 0 Eq. (28) can be regarded as a legitimate analytical continuation. An ambiguity inherent in Eq. (28) relates to how one circumvents the pole in the smeared expression x 1 x 2 α η η C Figure 3: The axis for the integration over θ.
Concerning the regularization of the angular integration we may introduce defect (or surplus) angles 2η ′ in the azimuthal integration as In Eq. (30) α C is a constant angle with 0 ≤ α C ≤ 2π and 0 < η ′ ≪ 1. Obviously, this regularization singles out the x 1 x 2 plane. As we shall show below, the choice of regularization plane translates into a global gauge choice for the τ dependence of φ's phase and thus is physically irrelevant: The apparent breaking of rotational symmetry by the angular regularization translates into a gauge choice.
The value of α C is determined by a (physically irrelevant) initial condition. We To see what is going on we may fix, for the time being, the ratio η ′ η for the normalization of the caloron contribution to a finite and positive but otherwise arbitrary constant Ξ when sending η and η ′ to zero in the end of the calculation: Combining Eqs. (26),(29),(31), (21), expression (19) reads: where A is a dimensionless function of its dimensionless argument. The sign ambiguity in (33) arises from the ambiguity associated with the way how one circumvents the pole in Eq. (29) and whether one introduces a surplus or a defect angle in (30).
For the anticaloron contribution we may, for the time being, fix the ratio η ′ η to another finite and positive constant Ξ ′ . In analogy to the caloron case, there is the ambiguity related to a shift τ → τ +τ A (0 ≤ τ A ≤ β) in the anticaloron contribution.
Moreover, we may without restriction of generality (global gauge choice) use an axis for the angular regularization which also lies in the x 1 x 2 plane, but with a different angle α A . Then we have where the choices of signs in either contribution are independent. Eq. (34) is the basis for fixing the operator D in Eq. (14).
To evaluate the function A 2πτ β in Eq. (33) numerically, we introduce the same cutoff for the ρ integration in the caloron and anticaloron case as follows: This introduces an additional dependence of A on ζ. In Fig. 4 the τ dependence of A for various values of ζ is depicted. It can be seen that Therefore we have The numbers ζ 3 Ξ, ζ 3 Ξ ′ , τ C β and τ A β in (37) are undetermined. For each color orientation (corresponding to a given angular regularization) there are two independent  parameters, a normalization and a phase-shift. The principal impossibility to fix the normalizations reflects the fact that on the classical level the theory is invariant under spatial dilatations. To give a meaning to these number, a mass scale needs to be generated dynamically. This, however, can only happen due to dimensional transmutation, which is known to be an effect induced by trivial-topology fluctuations [3]. The result in (37) is highly nontrivial since it is obtained only after an integration over the entire admissible part of the moduli spaces of (anti)calorons is performed.
Let us now discuss the physical content of (37). For fixed values of the parameters ζ 3 Ξ, ζ 3 Ξ ′ , τ C β and τ A β the right-hand side of Eq (37) resembles a fixed elliptic polarization in the x 1 x 2 plane of adjoint color space. For a given polarization plane the two independent numbers (normalization and phase-shift) of each oscillation axis parametrize the solution space (in total four undetermined parameters) of a second-order linear differential equation From (37) we observe that the operator D is Since for a given polarization plane there is a one-to-one map from the solution space of Eq. (38) to the parameter space associated with the ambiguities in the definition (9) we conclude that the operator D is uniquely determined by (9).
What we need to assure the validity of Eq. (7) is a BPS saturation 14 of the solution to Eq. (38). Thus we need to find first-order equations whose solutions solve the second-order equation (38). The relevant two first-order equations are where we have defined φ = |φ|φ(τ ). Obviously, the right-hand sides of Eqs. (40) are subject to a global gauge ambiguity associated with the choice of plane for angular regularization, any normalized generator other than λ 3 could have appeared, see Traceless, hermitian solutions to Eqs. (40) are given aŝ where C and τ 0 denote real integration constants which both are undetermined.
Notice that the requirement of BPS saturation has reduced the number of undetermined parameters from four to two: an elliptic polarization in the x 1 x 2 plane is cast into a circular polarization. Thus the field φ winds along an S 1 on the group manifold S 3 of SU(2). Both winding senses appear but can not be distinguished physically [1].

How to obtain φ's modulus
Here we show how the information about φ's phase in Eq. (42) can be used to infer its modulus. Let us assume that a scale Λ is externally given which characterizes this modulus at a given temperature T . Together, Λ and T determine what the minimal physical volume |φ| −3 is for which the spatial average over the caloron-anticaloron system saturates the infinite-volume average appearing in (9).
We have In order to reproduce the phase in Eq. (42) a linear dependence on φ must appear on the right-hand side of the BPS equation (8). Furthermore, this right-hand side ought not depend on β explicitly and must be analytic in φ 15 . The two following possibilities exist: 15 The former requirement derives from the fact that φ and its potential V are obtained by functionally integrating over a noninteracting caloron-anticaloron system. The associated part of the partition function does not exhibit an explicit β dependence since the action S is β independent on the caloron and anticaloron moduli spaces. Thus a β dependence of V or V (1/2) can only be generated via the periodicity of φ itself. The latter requirement derives from the demand that the thermodynamics at temperature T + δT to any given accuracy must be derivable from the thermodynamics at temperature T for δT sufficiently small provided no phase transition occurs at T . This is accomplished by a Taylor expansion of the right-hand side of the BPS equation (finite radius of convergence) which, in turn, is the starting point for a perturbative treatment with expansion parameter δT T . or has a finite radius of convergence. According to Eq. (42) we may write , and the equation of motion (45) can be derived from the following action: Notice that a shift V → V +const is forbidden in Eq. (50) since the relevant equation of motion is the first-order equation (45).
After the spatial average is performed the action S φ is extended by including topologically trivial configurations a µ in a minimal fashion: D µ φ and an added kinetic term. Here e denotes the effective gauge coupling. Thus the effective Yang-Mills action S is written as where G µν = G a µν λ a 2 and G a µν = ∂ µ a a ν − ∂ ν a a µ − eǫ abc a b µ a c ν .
In Eqs. (44) and (45) which follows from the action in Eq. (51). A pure-gauge solution to Eq. (52), de-scribing the ground state together with φ, is As a consequence of Eq. (53) we have D µ φ ≡ 0, and thus a ground-state pressure P g.s = −4π Λ 3 T and a ground-state energy-density ρ g.s = 4π Λ 3 T are generated in the electric phase: The so-far hidden scale Λ becomes visible by averaged-over caloron-anticaloron interactions encoded in the pure-gauge configuration a bg µ .

Summary and Outlook
Let us summarize our results. We have derived the phase and the modulus of The BPS saturated and classical field φ possesses nontrivial S 1 winding on the group manifold S 3 . The associated trajectory on S 3 becomes circular and thus a pure phase only after the integration over the entire admissible parts of the moduli spaces is carried out. Together with a pure-gauge configuration the adjoint scalar field φ generates a linear temperature dependence of the ground-state pressure and the ground-state energy-density where the pure-gauge configuration solves the Yang-Mills equations in the background φ and, after the spatial average, describes the interactions between trivial-holonomy calorons. The pure-gauge configuration also makes explicit that the electric phase is deconfining [1]. Since trivial-topology fluctuations may acquire quasiparticle masses on tree-level by the adjoint Higgs mechanism [1] the presence of φ resolves the infrared problem inherent in a perturbative loop expansion of thermodynamical quantities [12]. Since there are kinematical constraints for the maximal hardness of topologically trivial quantum fluctuations no renormalization procedure for the treatment of ultraviolet divergences is needed in the loop expansion of thermodynamical quantities [12] performed in the effective theory. These kinematical constraints arise from φ's compositeness emerging at distances ∼ |φ| −1 . The usual assertion that the effects of the topologically nontrivial sector are extremely suppressed at high temperature -they turn out to be power suppressed in T -is shown to be correct by taking this sector into account. The theory, indeed, has a Stefan-Boltzmann limit which is very quickly approached. It turns out to be incorrect, however, to neglect the topologically nontrivial sector from the start: assuming T ≫ Λ to justify the omission of the topologically nontrivial sector before performing a (perturbative) loop expansion of thermodynamical quantities does not capture the thermodynamics of an SU(2) Yang-Mills theory and leads to the known problems in the infrared sector [18].