Asymptotic Freedom, Perelman Entropy, and IR Memory
The UV fixed point of QCD as an entropy minimum, the Perelman analogue for Yang–Mills, and confinement as geometric surgery
1. Asymptotic Freedom of QCD
Quantum chromodynamics is based on the gauge group $SU(3)_c$ with coupling constant $g_3$. The one-loop beta function for QCD with $N_f$ active quark flavors is:
$$\beta_3(\mu) = \mu \frac{dg_3}{d\mu} = -\frac{g_3^3}{16\pi^2}\left(\frac{11 N_c - 2 N_f}{3}\right)$$
For $N_c = 3$ colors and $N_f \leq 16$, the coefficient is negative:
$$\beta_3 = -\frac{7 g_3^3}{16\pi^2} < 0 \qquad (N_f = 5)$$
The negativity of the beta function implies asymptotic freedom: the coupling$g_3(\mu) \to 0$ as the energy scale $\mu \to \infty$. The UV fixed point is the free theory. Conversely, as $\mu$ decreases toward $\Lambda_{\rm QCD} \approx 200$ MeV, the coupling grows without bound — this is confinement.
The running coupling in terms of $\alpha_s = g_3^2 / (4\pi)$ takes the explicit form:
$$\alpha_s(\mu) = \frac{\alpha_s(M_Z)}{1 + \frac{b_0\, \alpha_s(M_Z)}{2\pi}\ln\!\left(\frac{\mu}{M_Z}\right)}, \qquad b_0 = \frac{11 N_c - 2 N_f}{3}$$
2. The Perelman Entropy Analogy
Perelman's $\mathcal{W}$-entropy is monotonically non-decreasing along the Ricci flow:
$$\mathcal{W}(g, f, \tau) = \int_M \bigl[\tau(R + |\nabla f|^2) + f - n\bigr](4\pi\tau)^{-n/2} e^{-f}\, d\mu_g$$
Its time derivative satisfies:
$$\frac{d\mathcal{W}}{dt} = 2\tau \int_M \left|R_{ij} + \nabla_i\nabla_j f - \frac{g_{ij}}{2\tau}\right|^2 (4\pi\tau)^{-n/2} e^{-f}\, d\mu_g \;\geq\; 0$$
This drives the geometry toward gradient Ricci soliton fixed points. The analogy with asymptotic freedom is precise: just as the QCD beta function drives the coupling toward the free UV fixed point, Perelman's entropy drives the geometry toward the soliton fixed point. Both are monotone flows on a space of couplings.
3. The Yang–Mills Perelman Functional
We define a Perelman-type functional for Yang–Mills theory by coupling the YM Lagrangian density to a dilaton-like weight function $f$:
$$\mathcal{F}_{\rm YM}(A, f) = \int_M \Bigl(\mathrm{tr}(F_{\mu\nu} F^{\mu\nu}) + |\nabla f|^2\Bigr) e^{-f}\, d^4x$$
The gradient flow of $\mathcal{F}_{\rm YM}$ yields a Yang–Mills flow coupled to a diffusion equation for $f$:
$$\frac{\partial A_\mu}{\partial t} = -D^\nu F_{\nu\mu} + \nabla_\mu(\nabla_\nu A^\nu), \qquad \frac{\partial f}{\partial t} = -\Delta f + |\nabla f|^2 - \mathrm{tr}(F^2)$$
The first equation is the Yang–Mills heat flow (with a DeTurck-type gauge-fixing term), while the second is the backward heat equation for the weight, entirely parallel to Perelman's system for Ricci flow.
4. Monotonicity of the YM Perelman Functional
The key result is that $\mathcal{F}_{\rm YM}$ is monotonically non-decreasing along the coupled flow. Differentiating and using the flow equations:
$$\frac{d\mathcal{F}_{\rm YM}}{dt} = 2\int_M \bigl|D^\nu F_{\nu\mu} + \nabla_\mu \nabla_\nu A^\nu\bigr|^2 e^{-f}\, d\mu \;\geq\; 0$$
Equality holds if and only if the connection satisfies the Yang–Mills equation $D^\nu F_{\nu\mu} = 0$in the gauge $\nabla_\nu A^\nu = 0$. These are the YM solitons — the analogues of Ricci solitons. The monotonicity parallels Perelman's:
$$\text{Perelman: } \frac{d\mathcal{F}}{dt} = 2\int |R_{ij} + \nabla_i\nabla_j f|^2 e^{-f}\, d\mu \geq 0$$
$$\text{YM: } \frac{d\mathcal{F}_{\rm YM}}{dt} = 2\int |D^\nu F_{\nu\mu} + \nabla_\mu\nabla_\nu A^\nu|^2 e^{-f}\, d\mu \geq 0$$
In both cases, the integrand is a squared gradient of the Euler–Lagrange equations weighted by the Perelman measure $e^{-f}\, d\mu$.
5. Confinement as Geometric Surgery
At the confinement scale $\Lambda_{\rm QCD}$, the perturbative description breaks down. In Perelman's Ricci flow, singularities are resolved by surgery: one cuts out the singular neck region and caps off the resulting boundaries.
$$\Lambda_{\rm QCD} = M_Z \exp\!\left(-\frac{2\pi}{b_0\, \alpha_s(M_Z)}\right) \approx 200\;\text{MeV}$$
The analogy is:
- Perelman surgery scale $\delta > 0$: the curvature threshold at which one performs topological surgery
- QCD confinement scale $\Lambda_{\rm QCD}$: the energy scale at which the coupling becomes non-perturbative and color is screened
In both cases, a monotone functional ($\mathcal{W}$ or $\mathcal{F}_{\rm YM}$) controls the flow until a singular regime is reached, where a qualitative topological change (surgery / confinement) must occur.
6. IR Memory and Color Screening
At energies far above $\Lambda_{\rm QCD}$, soft gluon theorems produce a color memory effect. The soft gluon factor for emission of a soft gluon with color index $a$ is:
$$S^{(0)}_a = g_3 \sum_k \frac{p_k \cdot \epsilon}{p_k \cdot q}\, T^a_k$$
where $T^a_k$ are the color generators acting on particle $k$. This leads to a permanent change in the gauge field configuration — the color memory. However, at IR scales below $\Lambda_{\rm QCD}$, confinement screens all color charges:
$$\Delta A^a_\mu\big|_{\rm IR} = 0 \quad \text{(color memory is screened by confinement)}$$
This is the infrared analogue of Perelman's surgery: the surgery procedure changes the topology, eliminating certain geometric memory effects. Confinement similarly eliminates color memory, leaving only color-singlet hadrons.
$$\text{Pre-surgery: } \Delta\Psi \neq 0 \quad \longleftrightarrow \quad \text{UV: } \Delta A^a_\mu \neq 0$$
$$\text{Post-surgery: topology changed} \quad \longleftrightarrow \quad \text{IR: color confined}$$
7. The Perelman Lambda Invariant for YM
Perelman's $\lambda$-invariant is the infimum of $\mathcal{F}$ over all dilaton fields:
$$\lambda(g) = \inf_f \left\{ \mathcal{F}(g, f) \;\Big|\; \int_M e^{-f}\, d\mu = 1 \right\}$$
The YM analogue defines:
$$\lambda_{\rm YM}(A) = \inf_f \left\{ \mathcal{F}_{\rm YM}(A, f) \;\Big|\; \int_M e^{-f}\, d^4x = 1 \right\}$$
The sign of $\lambda_{\rm YM}$ distinguishes phases: $\lambda_{\rm YM} < 0$ corresponds to the confining regime, while $\lambda_{\rm YM} \to 0$ at the UV fixed point where the connection becomes flat. This precisely mirrors how $\lambda(g) < 0$ on hyperbolic manifolds and $\lambda(g) = 0$ on flat space.
Simulation: QCD Running Coupling and Perelman Entropy
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