YM-Thermodynamics 2024, 26: Deconfining Thermal Ground State
Assume two basic princples:
Without external sources, up to admissable gauge transformations, a thermodynamical system in the infinite volume limit guarantees that the spatial isotropy and homogeneity of an effective local field if this field is not associated with the propagation of energy-momentum by fundamental gauge fields. Partial ensemble averages, combined with spatial averages yields a nonzero result iff the emerging effective field ϕ is a rotational scalar independent of space, disregarding gauge transformations. The emerging field ϕ also doesn't depend on time.
In SU(2), SU(3) YM-theory a simple description of thermodynamics emerges by applying coarse-graining to the self-dual nonpropagating fundamental field configuration. ϕ emerges upon spatial coarse-graining over selfdual configurations and inherits their nonpropagating nature, so all interaction in UV-range mediated by trivial topology can be cast into an effective pure-gauge configuration a. This solves the effective inert field ϕ. This method is not applicable to field theories with global symmetry, whereas the coarse-graining method should be.
a lifts the vanishing energy density (assumed a priori) associated with the nonpropagating effective field ϕ to a finite value. UV fluctuations for k = 0 act as energy-density shifters (no deformation, so while retaining the background), ϕ is the nonresolvability at the resolution set by the background. Explicit interactions of coarse-grained k = 0 sectors with ϕ in the effective theory doesn't transfer energy-momentum to it.
The spatial coarse-graining method over interacting calorons begins with an a priori estimate for the thermal ground state, obtained through averaging over trivial |k| = 1 calorons. This generates ϕ, and solves the YM-equations in the background, yielding the pure-gauge configuration a for UV interactions. A properly weighted average over all domainizations is obtained in the effective theory by loop-expansion.
Restricted first to SU(2), only a scalar field with constant modulus can describe the ground-state thermodynamics without breaking isotropy. Spacetime dependce would break spacetime homogeneity. ϕ can only consistently participate in the thermodynamics of the entire YM-system, by coupling to the effective propagating excitations in the k = 0 sector. The k = 0 modes are wave-functions-renormalized versions of the fundamental modes, and the effective modes become adjoint representations of the Lie algebra gauge invariance of the effective action, so ϕ is also in adjoint representation. As a scalar field ϕ transforms homogeneously under gauge transformations.
Over noninteracting calorons, coarse-graining on trivial holonomy begins with defining and evaluating the kernel of a differential operator D by a family of adjointly transforming phases {ϕᵢ}. D is linear, of second order and uniquely determined by {ϕᵢ}. |ϕ| being constant implies D annihilating the entire field ϕ and the consistency of the BPS saturation, while solving Dϕ = 0 requires a first-order equation to model a potential V. This introduces an integration constant with mass dimension one, which will turn out to be the YM-scale.
The options of defining {ϕᵢ} locally are determined by the homogeneous transformation under adjoint representation. A local definition would imply a polynomial form for the field strength, with powers
This makes a local definition nonsensical. Instead, some additional initial principles are required for a nonlocal construction.
The emergence of ϕ is a product of an incomplete application of the partition function. No explicit T-dependence should occur in {ϕᵢ}.
{ϕᵢ} as well as the integration over spatial point separations must come with a flat measure. For the same reason, Wilson lines between spatial points follow straight lines.
Integrations over gauge orbits and over compactified time are forbidden.
The gauge-field configuration a must be stable under quantum fluctuations and have nonvanishing quantum weight in the infinite-volume limit to contribute to the partition function (under trivial holonomy)
From these, in su(2) coordinates, for calorons of trivial holonomy where caloron and anticaloron are spatially cetered at 0
The integrand in the exponent varies along a fixed direction in su(2), so path-ordering can be ignored. The centering at 0 is a consequence of the explicit power term of β⁻³ to make the phases dimensionless. Alternatively any other centering would have to be fixed, which is incompatible with spatial isotropy. The only integrable modulus of the caloron is ρ, which would have to be integrated on flat measure.
Calorons with |k| > 1 and n>2-point functions as integrands, the translational moduli, there are m > 1 parameters of dimension length. A k = 2 caloron of trivial holonomy has 3 dimensionful moduli: 2 scale parameters, and the distance between the two poles of topological charge. The n-point function, integrated n - 1 times over space, along with the flat measure over the caloron parameters, give rise to the mass dimension equation 2n - 3(n - 1) - m = 3 - n - m. This mass dimension needs to vanish, difficult, since for n ≥ 2 or m > 1, then 3 - n - m ≠ 0. For higher topological charge, the n-point functions are excluded in the definition of {ϕᵢ} with unique modulo global gauge rotation.
Derive the potential V from the differential operator D by taking the Wilson Lines
The behavior of the integral is checked by Taylor-expansion to ensure that the integral exists, and that it doesn't diverge in any problematic manner. A (dynamic) boundary can be imposed on the integrand, and by following the calculation to its end, one receives a functional F that adheres to the adjoint construction of the (anti-)caloron model.
where the sine term is the only contributor that gives rise to a radial divergence. Said divergence is logarithmic in nature, and since only spatial derivatives are involved, said term is based in magnetic-magnetic correlations.
Logarithmically divergent radial integrals multiply vanishing angular integrals, so since the regularizations are pairwise independent, {ϕᵢ} may be treated as a simple global gauge rotation, and rotational symmetry is retained. For sufficiently large cutoffs ξ, the result for {ϕᵢ} should become independent of ξ, and it should represent a two-fold copy of the kernel of D, where D is modeled on the space of real, smooth and period-β functions of τ. There are two real parameters for each polarization of the scalar field, each pair of which span completely ker(D) in the space of smooth and real functions in question. This makes D unique.
Linearity of D and the norm of the field are spacetime-independent, so D is expected to annihilate the entire field. The kinetics can be determined as usual, through Euler-Lagrange. Based on the independence of the calorons' action density on the temperature, coarse-graining prohibits any explicit temperature dependence in the field's effective action density and its equation of motion. The adjoint scalar field is also subject to a Euclidean Lagrangian density with explicit dependence on β. We are interested in the potential.
Motion in the field occurs in as three-dimensional vectors in su(2), and is independent of space and time. Using the BPS saturation requirement, get a first order differential equation for the potential, which consists of only the field norm and the Yang-Mills scale
The field represents a spatially homogeneous background for coarse-grained propagating gauge fields, breaking the SU(2) down to U(1). At maximal resolution of |ϕ|, determined by the coarse-graining, the infinite volume can derive a close approximation of D. No other interaction exists between ϕ and the k = 0 sector, and the nonperturbative emergence of the YM-scale is not influenced by it. The YM-scale sets the pole position for the evolution of the fundamental gauge coupling in the k = 0 sector. The value of Λ is determined by the gauge coupling g at a given resolution greater than the YM-scale. Assume the perturbative expansion to be sufficiently close to asymptotic behavior to truncate the beta-function at lower orders, and the regime close to the pole to be smoothly extrapolated from small coupling constants.