# Emergent Inert Adjoint Scalar Field in SU(2) Yang-Mills

Authors: Ulrich Herbst; Ralf Hofmann
Journal: ISRN High Energy Physics
 (2012)

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## Abstract

We compute the phase and the modulus of an energy- and pressure-free, composite, adjoint, and inert fieldϕ 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 ϕ possesses nontrivial S1-winding on the group manifold S3. Even at asymptotically high temperatures, where the theory reaches its Stefan-Boltzmann limit, the field ϕ, though strongly power suppressed, is conceptually relevant: its presence resolves the infrared problem of thermal perturbation theory.


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## Body

## 1. Introduction

In [1, 2] one of us has put forward an analytical and nonperturbative approach to SU(2)/SU(3) Yang-Mills thermodynamics. Each of these 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 [3] 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, 2], 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~MP where MP denotes the Planck mass. Since |ϕ|~Λ3/2πT topological defects only 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, though in a quantitatively different way, the infrared-ultraviolet decoupling property [1, 2] seen in renormalized perturbation theory [4–7]. Asymptotic freedom is a conceptually very appealing property of SU(N) Yang-Mills theories. It first was discovered in perturbation theory [8–11].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) backgroundϕ. Thus the partition function of the fundamental theory is evaluated in three steps in the deconfining 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 aμbg to the Yang-Mills equations describing the trivial-topology sector. Together, ϕ and aμbg constitute the thermal ground state of the system. The fact that the action for the ground-state configurations ϕ and aμbg is infinite is unproblematic since the corresponding, vanishing weight in the partition function is associated with a nonfluctuating configuration and therefore can be factored out and is cancelled when evaluating expectation values in the effective theory. (iii) Consider the interactions between the macroscopic ground state and trivial-topology fluctuations in terms of quasiparticle masses of the latter which are generated by the adjoint Higgs mechanism2 and impose thermodynamical self-consistency to derive an evolution equation 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 [12–23]; for a quick explanation of the term holonomy, see Figure 1. Nontrivial holonomy is locally excited by interactions between trivial-holonomy calorons and anticalorons mediated by plane-wave configurations. In [18] 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 once 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 ~4π2T~39T [14]. 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 Pg.s. which, on spatial average, turns out to be Pg.s.=-4πΛ3T (the equation of ground state is ρg.s.=-Pg.s.) [1, 2]. 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 system3. The overall pressure is positive if the gauge-field fluctuations after spatial coars graining are sufficiently light and energetic to over compensate the negative ground-state contribution, that is, if the temperature is sufficiently large.Figure 1
Possible values of the Polyakov loop𝒫 at spatial infinity on a given gauge-field configuration. A small holonomy corresponds to values close to the center elements depicted by the dots. Crosses indicate examples for a large holonomy.Caloron-induced tree-level masses for gauge-field modes decay as1/T when heating up the system. Due to the linear rise of ρg.s. with T the thermodynamics of the ground state is thus subdominant at large temperatures4. The main purpose 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 Section 2 we write down and discuss a nonlocal definition, relevant for the determination of ϕ’s phase, in terms of a spatial and scale-parameter 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 Section 3 we perform the average and discuss the occurrence of a global gauge freedom in ϕ’s phase, which has a geometrical interpretation. In Section 4, we show how the derived information about a nontrivial S1 winding of the field 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 Section 5 we summarize and discuss our results and give an outlook on future research.


## 2. Definition ofϕ’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, 2] the adjoint scalar ϕ emerges as an energy- and pressure-free (BPS saturated) field from a spatial average over the classical correlations in a caloron-anticaloron 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 will see, the information about the S1 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(2.1)DμFμν=0
which are periodic in Euclidean time τ (0≤τ≤1/T) and of trivial holonomy. The only relevant configurations are calorons of topological charge5  ±1. They are [3](2.2)AμC(τ,x)=η¯aμνλa2∂νln⁡Π(τ,x)orAμA(τ,x)=ηaμνλa2∂νln⁡Π(τ,x),
where the ’t Hooft symbols ηaμν and η¯aμν are defined as(2.3)ηaμν=ϵaμν+δaμδν4-δaνδμ4,η¯aμν=ϵaμν-δaμδν4+δaνδμ4.
The solutions in (2.2) (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 [24–26]. They have the same color orientation as the “seed” instanton or “seed” anti-instanton. In (2.2) λa, (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.The scalar functionΠ(τ,x) is given as [3](2.4)Π(τ,x)=Π(τ,r)≡1+πρ2βrsinh⁡(2πr/β)cosh⁡(2πr/β)-cos⁡(2πτ/β),
where r≡|x|, β≡1/T, and ρ denotes the scale parameter whose variation leaves the classical action S=8π2/g2 invariant. At a given ρ the solutions in (2.2) can be generalized by shifting the center from z=0 to z=(τz,z) by the (quasi)translational invariance of the classical action6  S. Another set of moduli is associated with global color rotations of the solutions in (2.2).From the BPS saturation(2.5)Fμν[A(C,A)]=(+,-)F̃μν[A(C,A)]
it follows that the (Euclidean) energy-momentum tensor θμν, evaluated on Aμ(C,A), vanishes identically(2.6)θμν[A(C,A)]≡0.
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 caloron-anticaloron correlations neglecting their interactions7(2.7)θ¯μν[ϕ]≡0.
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 contains a kinetic term quadratic in the τ-derivatives and a potential V then (2.7) is equivalent to ϕ solving the first-order equation(2.8)∂τϕ=V(1/2),
where V(1/2) denotes the “square root” of V8, V≡tr(V(1/2))†V(1/2). In (2.8) the right-hand side will turn out to be determined only up to a global gauge rotation, see Figure 2. Already at this point it is important to remark that the Yang-Mills scale parametrizes the potential V and thus also the classical solution to (2.8). In the absence of trivial-topology fluctuations it is, however, invisible, see (2.7). 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, 2]. Hence the trace anomaly θ̃μμ≠0 for the total energy-momentum tensor θ̃μν≡θ¯μνg.s.+θμνfluc in the effective theory which includes the effects of trivial-topology fluctuations: since θ¯μνg.s.=4πTΛ3δμν and θμνfluc∝T4 for T≫Λ the trace anomaly dies off as Λ3/T3.Figure 2
Possible directions of winding of(ϕ/|ϕ|)(τ) around the group manifold S3 of SU(2). The angles γ,δ are arbitrary but constant. They are determined by the choice of plane in which angular regularization is carried out when computing ϕ’s phase, see below. The angle α(τ) parametrizes the S1 winding of ϕ/|ϕ|.Without imposing constraints other than nonlocality9 the τ dependence of ϕ’s phase (the ratio of the two averages ϕ and |ϕ| over admissible moduli deformations, Aμ(C,A) would naively be characterized as(2.9)ϕa|ϕ|(τ)~tr⁡[β01!∫d3x∫dρλa2Fμν[Aα(ρ,β)]((τ,0)){(τ,0),(τ,x)}[Aα(ρ,β)]×Fμν[Aα(ρ,β)]((τ,x)){(τ,x),(τ,0)}[Aα(ρ,β)]+β-12!∫d3x∫d3y∫dρλa2Fμλ[Aα(ρ,β)]((τ,0)){(τ,0),(τ,x)}[Aα(ρ,β)]×Fλν[Aα(ρ,β)]((τ,x)){(τ,x),(τ,y)}[Aα(ρ,β)]×Fνμ[Aα(ρ,β)]((τ,y)){(τ,y),(τ,0)}+β-23!∫d3x∫d3y∫d3u∫dρλa2Fμλ[Aα(ρ,β)]((τ,0)){(τ,0),(τ,x)}[Aα(ρ,β)]×Fλν[Aα(ρ,β)]((τ,x)){(τ,x),(τ,y)}[Aα(ρ,β)]×Fνκ[Aα(ρ,β)]((τ,y)){(τ,y),(τ,u)}Fκμ[Aα(ρ,β)]((τ,u))×{(τ,u),(τ,0)}+⋯β01!∫d3x∫dρ].
The dots in (2.9) stand for the contributions of higher n-point functions and for reducible, that is, factorizable, contributions with respect to the spatial integrations. In (2.9) the following definitions apply:(2.10)|ϕ|≡12tr⁡ϕ2,{(τ,0),(τ,x)}[Aα]≡Pexp⁡[i∫(τ,0)(τ,x)dyβAβ(y,ρ)],{(τ,x),(τ,0)}[Aα]≡Pexp⁡[-i∫(τ,0)(τ,x)dyβAβ(y,ρ)].
The integral in the Wilson lines in (2.10) is along a straight line10 connecting the points (τ,0) and (τ,x), and 𝒫 denotes the path-ordering symbol.Under a microscopic gauge transformationΩ(y) the following relations hold:(2.11){(τ,0),(τ,x)}[Aα]⟶Ω†((τ,0)){(τ,0),(τ,x)}[Aα]Ω((τ,x)),{(τ,x),(τ,0)}[Aα]⟶Ω†((τ,x)){(τ,x),(τ,0)}[Aα]Ω((τ,0)),Fμν[Aα]((τ,x))⟶Ω†((τ,x))Fμν[Aα]((τ,x))Ω((τ,x)),Fμν[Aα]((τ,0))⟶Ω†((τ,0))Fμν[Aα]((τ,0))Ω((τ,0)).
As a consequence of (2.11) the right-hand side of (2.9) transforms as(2.12)ϕa|ϕ|(τ)⟶Rab(τ)ϕb|ϕ|(τ),
where the SO(3) matrix Rab(τ) is defined as(2.13)Rab(τ)λb=Ω((τ,0))λaΩ†((τ,0)).
Thus we have defined an adjointly transforming scalar in (2.9). Moreover, only the time-dependent part of a microscopic gauge transformation survives after spatial averaging (macroscopic level).In (2.9) the ~ sign indicates that both left- and right-hand sides satisfy the same homogeneous evolution equation in τ(2.14)D[ϕ|ϕ|]=0.
Here 𝒟 is a differential operator such that (2.14) represents a homogeneous differential equation. As it will turn out, (2.14) 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 (2.9) is understood as a sum over the two solutions in (2.2), that is, Aα=AαC or Aα=AαA. As we will show in Section 3, the dimensionless quantity defined on the right-hand side of (2.9) is ambiguous11; the operator 𝒟, however, is not.The quantities appearing in the numerator and denominator of the left-hand side of (2.9) 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 Aα=AαC and Aα=AαA excluding global color rotations and time translations.Let us explain this in more detail. For the gauge variant density in (2.9) an average over global color rotations and time shifts τ→τ+τz (0≤τz≤β) would yield zero and thus is forbidden12. The nonflatness of the measure with respect to the separate ρ integration in the numerator and the denominator average in (2.9) transforms into flatness by taking the ratio. Since the integration weight exp(-S) is independent of temperature on the moduli space of a caloron, the right-hand side of (2.9) must not exhibit an explicit temperature dependence. This forbids the contribution of n-point functions with n>2, and we are left with an investigation of the first term in (2.9). In the absence of a fixed mass scale on the classical level an average overspatial translations would have a dimensionful measure d3z making the definition of a dimensionless quantity ~ϕ/|ϕ| impossible. We conclude that the average overspatial translations is already performed in (2.9). Since one of the two available length scales ρ and β parametrizing the caloron or the anticaloron is integrated over in (2.9), the only scale responsible for a nontrivial τ dependence of ϕa/|ϕ| is β.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 operator𝒟: 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 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 (2.14) by means of (2.9).


## 3. Computation of Two-Point Correlation

Before we perform the actual calculation let us stress some simplifying properties of the solutionsAμ(C,A) in (2.2).The path-ordering prescription for the Wilson lines{(τ,0),(τ,x)} and {(τ,0),(τ,x)}† in (2.10) can actually be omitted. To see this, we first consider the quantity P(C,A)(τ,rst) defined as(3.1)P(C,A)(τ,rst)≡Ai(C,A)(τ,srt)ti,
where 0≤r≤∞, 0≤s≤1, (i=1,2,3). The vector t denotes the unit line tangential along the straight line connecting the points (τ,0) and (τ,x)≡(τ,rt). We have(3.2){(τ,0),(τ,rt)}(C,A)=Pexp⁡[ir∫01dsP(C,A)(τ,srt)],
where(3.3)P(C,A)(τ,srt)=∓12t⋅λ∂4ln⁡Π(τ,sr).
Thus the path-ordering symbol can, indeed, be omitted in (3.2). The field strength FaμνC on the caloron solution in (2.2) is(3.4)FaμνC=η¯aμν(∂κΠ)(∂κΠ)Π2+η¯aμκΠ(∂ν∂κΠ)-2(∂κΠ)(∂νΠ)Π2-η¯aνκΠ(∂μ∂κΠ)-2(∂κΠ)(∂μΠ)Π2,
where Π is defined in (2.4). For the anticaloron one replaces η¯ by η in (3.4). Using (2.9), (3.2), and (3.4), we obtain the following expression for the contribution (ϕa/|ϕ|)|C arising from calorons:(3.5)ϕa|ϕ||C~i∫dρ∫d3xxar[(∂4Π(τ,0))2Π2(τ,0)-23∂42Π(τ,0)Π(τ,0)]×{(∂4Π(τ,r))2-(∂rΠ(τ,r))2Π2(τ,r)4cos⁡(2g(τ,r))[∂r∂4Π(τ,r)Π(τ,r)-2(∂rΠ(τ,r))(∂4Π(τ,r))Π2(τ,r)]+sin⁡(2g(τ,r))[4(∂4Π(τ,r))2-(∂rΠ(τ,r))2Π2(τ,r)+2∂r2Π(τ,r)-∂42Π(τ,r)Π(τ,r)]},
where(3.6)g(τ,r)≡∫01dsr2∂4ln⁡Π(τ,sr),(3.7)(∂4Π(τ,0))2Π2(τ,0)-23∂42Π(τ,0)Π(τ,0)=-163π4ρ2β2π2ρ2+β2(2+cos⁡(2πτ/β))[2π2ρ2+β2(1-cos⁡(2πτ/β))]2.
The dependences on ρ and β are suppressed in the integrands of (3.5) and (3.6). It is worth mentioning that the integrand in (3.6) is proportional to δ(s) for r≫β. A useful set of identities is(3.8)FμνC(τ,x)=FμνA(τ,-x){(τ,0),(τ,x)}C=({(τ,x),(τ,0)}C)†==({(τ,-x),(τ,0}A)†.{(τ,0),(τ,-x)}A
Equation (3.8) states that the integrand for (ϕa/|ϕ|)|A can be obtained by a parity transformation x→-x of the integrand for (ϕa/|ϕ|)|C. Since the latter changes its sign, see (3.5); one naively would conclude that(3.9)ϕa|ϕ|=ϕa|ϕ||C+ϕa|ϕ||A=0.
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 (3.5). It is easily checked that all terms give rise to a converging r integration except for the following one:(3.10)2xarsin⁡(2g(τ,r))∂r2Π(τ,r)Π(τ,r).
Namely, for r>R≫β (3.10) goes over in(3.11)4taπρ2sin⁡(2g(τ,r))βr3.
Thus the r-integral of the term in (3.10) is logarithmically divergent in the infrared13:(3.12)4taπρ2β∫R∞drrsin⁡(2g(τ,r)).
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 ther integral in (3.12) by prescribing(3.13)∫R∞drr⟶βϵ∫R∞drr1+ϵ
with ϵ>0. We have(3.14)βϵ∫R∞drr1+ϵ=βϵ∫0∞dr(r+R)1+ϵ=1ϵ-log⁡(Rβ)+12ϵlog⁡2(Rβ)+⋯.
Away from the pole at ϵ=0 this is regular. For ϵ<0 (3.14) can be regarded as a legitimate analytical continuation. An ambiguity inherent in (3.14) relates to how one circumvents the pole in the smeared expression(3.15)12η∫-ηηdϵ(1ϵ±i0-log⁡(Rβ)+12ϵlog⁡2(Rβ)+⋯)=∓πi2η-log⁡(Rβ)+⋯,(η>0,η≪1).
Concerning the regularization of the angular integration we may introduce defect (or surplus) angles 2η′ in the azimuthal integration as(3.16)∫0πdωsin⁡ω∫02πdθ⟶∫0πdωsin⁡ω∫αC±η′αC+2π∓η′dθ,
see Figure 3. In (3.16) αC is a constant angle with 0≤αC≤2π and 0<η′≪1. Obviously, this regularization singles out the x1x2 plane. As we will 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.Figure 3
The axis for the integration overθ.The value ofαC is determined by an (physically irrelevant) initial condition. We have(3.17)∫0πdωsin⁡ω∫αC±η′αC+2π∓η′dθta~∓πη′(δa1cos⁡αC+δa2sin⁡αC).
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:(3.18)lim⁡η,η′→0η′η=Ξ.
Combining (3.12), (3.15), (3.17), and (3.7), expression (3.5) reads:(3.19)ϕa|ϕ||C~±323π7β3Ξ(δa1cos⁡αC+δa2sin⁡αC)∫dρ[lim⁡r→∞sin⁡(2g(τ,r))]×ρ4π2ρ2+β2(2+cos⁡(2πτ/β))[2π2ρ2+β2(1-cos⁡(2πτ/β))]2≡±Ξ(δa1cos⁡αC+δa2sin⁡αC)A(2πτβ),
where 𝒜 is a dimensionless function of its dimensionless argument. The sign ambiguity in (3.19) arises from the ambiguity associated with the way how one circumvents the pole in (3.15) and whether one introduces a surplus or a defect angle in (3.16). Furthermore, there is an ambiguity associated with a constant shift τ→τ+τC (0≤τC≤β) in (3.19).For the anticaloron contribution we may, for the time being, fix the ratioη′/η to another finite and positive constant Ξ′. In analogy to the caloron case, there is the ambiguity related to a shift τ→τ+τA (0≤τA≤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 x1x2 plane, but with a different angle αA. Then we have(3.20)ϕa|ϕ|=ϕa|ϕ||C+ϕa|ϕ||A=±Ξ(δa1cos⁡αC+δa2sin⁡αC)A(2π(τ+τC)β)±Ξ′(δa1cos⁡αA+δa2sin⁡αA)A(2π(τ+τA)β)≠0,
where the choices of signs in either contribution are independent. Equation (3.20) is the basis for fixing the operator 𝒟 in (2.14).To evaluate the function𝒜(2πτ/β) in (3.19) numerically, we introduce the same cutoff for the ρ integration in the caloron and anticaloron case as follows:(3.21)∫dρ⟶∫0ζβdρ,(ζ>0).
This introduces an additional dependence of 𝒜 on ζ. In Figure 4 the τ dependence of 𝒜 for various values of ζ is depicted. It can be seen that(3.22)A(2πβτ,ζ⟶∞)⟶272ζ3sin⁡(2πβτ).
Therefore we have(3.23)ϕa|ϕ|~272ζ3(±Ξ(δa1cos⁡αC+δa2sin⁡αC)sin⁡(2πβ(τ+τC))±Ξ′(δa1cos⁡αA+δa2sin⁡αA)sin⁡(2πβ(τ+τA)))≡ϕ̂a.
The numbers ζ3Ξ, ζ3Ξ′, τC/β, and τA/β in (3.23) 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 [8–11]. The result in (3.23) 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.𝒜 as a function of (2π/βτ) for ζ=1,2,10. For each case the dashed line is a plot of max𝒜×sin((2π/β)τ). We have fitted the asymptotic dependence on ζ of the amplitude of 𝒜 as 𝒜((2π/β)τ=π/2,ζ)=272ζ3,(ζ>10). The fit is stable under variations of the fitting interval. For the case ζ=10 the difference between the two curves cannot be resolved anymore.
(a)(b)(c)Let us now discuss the physical content of (3.23). For fixed values of the parameters ζ3Ξ, ζ3Ξ′, τC/β, and τA/β the right-hand side of (3.23) resembles a fixed elliptic polarization in the x1x2 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(3.24)Dϕ̂=0.
From (3.23) we observe that the operator 𝒟 is(3.25)D=∂τ2+(2πβ)2.
Since for a given polarization plane there is a one-to-one map from the solution space of (3.24) to the parameter space associated with the ambiguities in the definition (2.9) we conclude that the operator 𝒟 is uniquely determined by (2.9).What we need to assure the validity of (2.7) is a BPS saturation14 of the solution to (3.24). Thus we need to find first-order equations whose solutions solve the second-order equation (3.24). The relevant two first-order equations are(3.26)∂τϕ̂=±2πiβλ3ϕ̂,
where we have defined ϕ=|ϕ|ϕ̂(τ). Obviously, the right-hand sides of (3.26) are subject to a global gauge ambiguity associated with the choice of plane for angular regularization, any normalized generator other than λ3 could have appeared, see Figure 2. Now the solution to either of the two equations (3.26) also solves (3.24),(3.27)∂τ2ϕ̂=±2πiβλ3∂τϕ̂=2πiβλ32πiβλ3ϕ̂=-(2πβ)2ϕ̂.
Traceless, hermitian solutions to (3.26) are given as(3.28)ϕ̂=Cλ1exp⁡(∓2πiβλ3(τ-τ0)),
where C and τ0 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 x1x2 plane is cast into a circular polarization. Thus the field ϕ winds along an S1 on the group manifold S3 of SU(2). Both winding senses appear but cannot be distinguished physically [1, 2].


## 4. How to Obtainϕ’s Modulus

Here we show how the information aboutϕ’s phase in (3.28) 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 (2.9).We have(4.1)ϕ=ϕ(β,Λ,τβ).
In order to reproduce the phase in (3.28) a linear dependence on ϕ must appear on the right-hand side of the BPS equation (2.8). Furthermore, this right-hand side ought not depend on β explicitly and must be analytic in ϕ15. The two following possibilities exist:(4.2)∂τϕ=±iΛλ3ϕ
or(4.3)∂τϕ=±iΛ3λ3ϕ-1,
where ϕ-1≡ϕ/|ϕ|2. Recall that(4.4)ϕ-1=ϕ0-1∑n=0∞(-1)nϕ0-n(ϕ-ϕ0)n
has a finite radius of convergence. According to (3.28) we may write(4.5)ϕ=|ϕ|(β,Λ)×λ1exp⁡(∓2πiβλ3(τ-τ0)).
Substituting (4.5) into (4.2) yields(4.6)Λ=2πβ
which is unacceptable since Λ is a constant scale. For the possibility in (4.3) we obtain(4.7)|ϕ|(β,Λ)=βΛ32π=Λ32πT
when substituting (4.5) into (4.3). This is acceptable and indicates that at T≫Λϕ’s modulus is small. The right-hand side of (4.3) defines the “square-root” V(1/2) of a potential V(|ϕ|)≡tr(V(1/2))†V(1/2)=Λ6trϕ-2, and the equation of motion (4.3) can be derived from the following action:(4.8)Sϕ=tr⁡∫0βdτ∫d3x(∂τϕ∂τϕ+Λ6ϕ-2).
Notice that a shift V→V+const is forbidden in (4.8) since the relevant equation of motion is the first-order equation (4.3).After the spatial average is performed the actionSϕ is extended by including topologically trivial configurations aμ in a minimal fashion: ∂τϕ→∂μϕ+ie[ϕ,aμ]≡Dμϕ and an added kinetic term. Here e denotes the effective gauge coupling. Thus the effective Yang-Mills action S is written as(4.9)S=tr⁡∫0βdτ∫d3x(12GμνGμν+DμϕDμϕ+Λ6ϕ-2),
where Gμν=Gμνaλa/2 and Gμνa=∂μaνa-∂νaμa-eϵabcaμbaνc.In (4.2) and (4.3) the existence of the mass scale Λ (the Yang-Mills scale) was assumed. One attributes the generation of a mass scale to the topologically trivial sector which, however, was assumed to be switched off so far. How can a contradiction be avoided? The answer to this question is that the scale Λ remains hidden as long as topologically trivial fluctuations are switched off; see (2.7). Only after switching on interactions between trivial-holonomy calorons within the ground state can Λ be seen [1, 2]. Let us repeat the derivation of this result: in [1, 2] we have shown that the mass squared of ϕ-field fluctuations, ∂|ϕ|2V(|ϕ|), is much larger than the square of the compositeness scale |ϕ|. Moreover ∂|ϕ|2V(|ϕ|) is much larger than T2 for all temperatures T≥Tc,E where Tc,E denotes the critical temperature for the electric-magnetic transition. Thus ϕ is quantum mechanically and statistically inert: it provides a (nonbackreacting and undeformable) source for the following equation of motion:(4.10)DμGμν=2ie[ϕ,Dνϕ]
which follows from the action in (4.9). A pure-gauge solution to (4.10), describing the ground state together with ϕ, is(4.11)aμbg=πeTδμ4λ3.
As a consequence of (4.11) we have 𝒟μϕ≡0, and thus a ground-state pressure Pg.s.=-4πΛ3T and a ground-state energy-density ρg.s.=4πΛ3T 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 aμbg.


## 5. Summary and Outlook

Let us summarize our results. We have derived the phase and the modulus of a statistically and quantum mechanically inert adjoint and spatially homogeneous scalar fieldϕ for an SU(2) Yang-Mills theory being in its electric phase. This field and a pure-gauge configuration together suggest the concept of a thermal ground state since they generate temperature-dependent pressure and energy density with an equation of state corresponding to a cosmological constant. The existence of ϕ originates from the spatial correlations inherent in BPS saturated, trivial-holonomy solutions to the classical Yang-Mills equations at finite temperature: the Harrington-Shepard solutions of topological charge modulus one. To derive ϕ’s phase these field configurations are, in a first step, treated as noninteracting when performing the functional average over the admissible parts of their moduli spaces. We have shown why adjoint scalar fields arising from configurations of higher topological charge do not exist.The BPS saturated and classical fieldϕ possesses nontrivial S1 winding on the group manifold S3. The associated trajectory on S3 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, 2]. Since trivial-topology fluctuations may acquire quasiparticle masses on tree level by the adjoint Higgs mechanism [1, 2], the presence of ϕ resolves the infrared problem inherent in a perturbative loop expansion of thermodynamical quantities [27]. 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 [27] 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 [28].



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*Source: 373121-2012-03-05.xml*