Gauge Theory Color-Thermal Spectrum

• Gauge theory color-thermal spectrum is a manifestation of color-sector thermality in non-Abelian Yang–Mills backgrounds, quantifying the thermal distribution of color eigenvalues via worldline instanton techniques.
• Its derivation employs semiclassical methods and topological winding modes, leading to a Bose–Einstein-like distribution with quadratic backreaction corrections.
• The framework maps gauge theory thermality to Hawking radiation through double copy correspondence and is supported by lattice simulations demonstrating symmetry restoration.

The gauge theory color-thermal spectrum is a fundamental manifestation of color-sector thermality in non-Abelian Yang–Mills backgrounds, arising from semiclassical and worldline instanton analyses. This spectrum encodes the statistical emission rate of color-charged probes or pairs, exhibiting a thermal (Planckian or Bose–Einstein) distribution in the color charge quantum number. Nonperturbative treatments reveal the spectrum as a consequence of topological winding modes, with implications for backreaction and deep connections to gravitational Hawking radiation via the double copy correspondence.

1. Worldline Instanton Formalism and Color-Thermal Spectrum

The nonperturbative derivation exploits the Euclidean worldline path integral of a scalar probe in a non-Abelian Yang–Mills root of a Schwarzschild background. In the sector of color-charge eigenvalue $\lambda$ for $SU(N_c)$, the non-Abelian Wilson loop diagonalizes, yielding an effective abelianized background with action

$$\Gamma_E[A] =\int_{0}^{\infty}\!\frac{dT}{T}\, \int_{x(0)=x(T)}\!\mathcal D x(\tau)\, \exp\Biggl[-\int_{0}^{T}\!d\tau\, \Bigl(\tfrac{\dot x^2}{4} -\,i\,g\,Q\,\lambda\;\frac{k\!\cdot\!\dot x}{r} + m^2\Bigr)\Biggr],$$

where $k^\mu=(1,1,0,0)$ is the Kerr–Schild null vector, and $\alpha\equiv gQ\lambda$ is the effective charge (Carrasco et al., 25 Jan 2026).

The dominant contributions at weak coupling or large charge arise from saddle-point “worldline instantons,” closed loops in Euclidean spacetime whose classical action encodes vacuum decay and probe emission. In the massless limit, these instantons exhibit $n$-fold topological winding around $r=0$, and the on-shell action accumulates as $S_E(n) = 4\pi n g Q \lambda$ per winding.

2. Topological Origin and Spectral Resummation

Each topologically distinct instanton sector ($n$ windings) yields a contribution $e^{-S_E(n)}$ to the emission probability. Summing over all $n\geq1$ generates a geometric series directly analogous to the sum over Matsubara windings in thermal field theory: $$\rho(\lambda)\propto\sum_{n=1}^\infty e^{-4\pi n g Q |\lambda|} = \frac{1}{e^{|\lambda|/T_c}-1}$$ with the emergent color-temperature

$T_c = \frac{1}{4\pi g Q}.$

This functional structure is the Bose–Einstein (Planck) factor, but in color charge eigenvalue $\lambda$ rather than energy. At leading order ($n=1$), the spectrum is approximately Boltzmann: $\rho(\lambda)\propto e^{-|\lambda|/T_c}$, with the full Planckian correction realized only after summing over all windings (Carrasco et al., 25 Jan 2026).

3. Color Phase-Space Structure and Large-$N_c$ Limit

The inclusive emission rate in a non-Abelian gauge background is not only weighted by the dynamical Planck factor but also by the density of color eigenstates. For an $SU(N_c)$ probe in the large-$N_c$ limit, the color eigenvalue $\lambda$ (for a fixed color direction $c^aT^a$) exhibits a distribution governed by the Wigner semicircle law: $$\rho(\lambda) = \frac{2}{\pi R^2} \sqrt{R^2-\lambda^2}\quad \mathrm{for}~|\lambda|\leq R,$$ with $\rho(\lambda)=0$ outside this interval. Thus, the differential spectrum becomes

$$\frac{dN}{d\lambda} = \frac{1}{e^{\beta_\text{color}\lambda}-1} \times \frac{2}{\pi R^2} \sqrt{R^2-\lambda^2},$$

where $\beta_\text{color}=2\pi g Q_0/E_p$ encodes the effective color temperature, and $C=g Q_0/E_p$ (Carrasco et al., 3 Nov 2025).

The color phase-space density $\rho(\lambda)$ quantifies how many probe channels participate at fixed $\lambda$, fundamentally shaping the observed spectrum. For strong coupling or large shell charge (large $C$), the Planck-like behavior dominates; otherwise, the semicircle spectral weight is apparent.

4. Backreaction, Quadratic Corrections, and Casimir Structure

Incorporating finite source charge depletes $Q$ by the emitted charge $\lambda'$, modifying the emission probability to account for backreaction: $$Q \to Q-\lambda',\qquad X_{\rm full}(\lambda)= 4\pi g |\lambda| \left( Q - \frac{|\lambda|}{2} \right).$$ This results in a universal quadratic correction to the exponent: $$\Delta S = -2\pi g \lambda^2\qquad \to\qquad P_{\rm pair}\sim \exp\!\bigl[-4\pi g |\lambda|\left(Q-\tfrac{|\lambda|}{2}\right)\bigr].$$ This structure mirrors the Parikh–Wilczek correction $\propto \omega^2$ for black-hole backreaction, ensuring conservation of both total color charge in gauge theory and energy in gravity (Carrasco et al., 25 Jan 2026).

Representation-theoretic analysis interprets the quadratic term as a change in the quadratic Casimir: $$\Delta C_2\propto Q^2-(Q-\lambda)^2=2Q\lambda-\lambda^2$$, with the linear component giving the leading thermal weight and the quadratic component encoding backreaction.

5. Double Copy and Gravitational Analogue

The gauge theory color-thermal spectrum admits a direct double copy mapping to gravitational Hawking radiation. The dictionary can be summarized as:

Yang–Mills Quantity	Corresponds To—Gravity
Color $Q$	Mass $M$
Eigenvalue $\lambda$	Energy $\omega$
Casimir $Q^2$	Horizon area $M^2$
$4\pi g\lambda(Q-\lambda/2)$	$8\pi G_N\omega(M-\omega/2)$

The worldline instanton’s winding number $n$ generates the Planck spectrum in either color or energy, with the quadratic correction encoding backreaction (color depletion or mass loss per emission) (Carrasco et al., 25 Jan 2026, Carrasco et al., 3 Nov 2025). The leading linear term reproduces the Hawking temperature $T_H = 1/(8\pi G_N M)$. Thus, black hole energy thermality arises as the double copy of color thermality in the non-Abelian gauge root, and the unitary correlations among successive quanta are precisely mapped.

6. Thermal Meson Spectra in Lattice Gauge Theories

Lattice simulations further elucidate color-sector thermal spectra. For $SU(2)$ gauge theory with two fundamental Dirac fermions, the screening masses of light mesons in various channels (pseudoscalar, scalar, vector, axial-vector) are measured as functions of temperature $T/T_c$. Data demonstrate:

• Below $T_c$: significant mass splittings between parity partners.
• Above $T_c$: rapid restoration of global $SU(4)$ symmetry—vector and axial-vector masses degenerate; delayed restoration of $U(1)_A$, with pseudoscalar and scalar degeneracy only for $T \gtrsim 1.5T_c$.
• Screening mass ratios $R_V=(M_{AV}-M_{V})/(M_{AV}+M_{V})$ and $R_S=(M_{S}-M_{PS})/(M_{S}+M_{PS})$ confirm symmetry restoration patterns (Lee et al., 2017).

These results evidence thermalization and symmetry properties in lattice gauge spectra, paralleling features of the color-thermal spectrum derived semiclassically.

7. Significance, Misconceptions, and Outlook

The gauge theory color-thermal spectrum formalizes the statistical emission of color through a thermal factor in the color quantum number, arising from nonperturbative worldline instantons and encoded in the double copy correspondence. The spectrum does not imply energy thermality in gauge theory; instead, the observable is the eigenvalue $\lambda$ in color space. In gravity, the double copy acts to map color thermality into familiar Hawking thermality in energy. The quadratic backreaction correction ensures conservation and unitarity, paralleling gravitational results.

A common misconception is identifying energy thermality directly in non-Abelian gauge theory emission; in fact, it is color quantum number thermality that is primary (Carrasco et al., 3 Nov 2025). The gravitational Hawking spectrum emerges from this under double copy.

Lattice results, spectral phase-space considerations, and worldline instanton techniques collectively underpin the interpretation and computation of the color-thermal spectrum, with implications for quantum gravity, black hole microphysics, and nonperturbative gauge theory dynamics.