A scale from a dimensionless coupling

Prerequisites: dimensional transmutation, logarithms and a first-order differential equation. Learning outcomes: recover the integrated one-loop running law, distinguish $g$ , $\alpha$ and the ‘t Hooft coupling, and locate the approximation’s failure without interpreting it as a physical pole.

At sufficiently high energy, the coupling becomes weak. Lower the scale and the same formula indicates where its own approximation becomes unreliable.

DIMENSIONAL TRANSMUTATION

A scale from a coupling

Choose SU(N) and a scale μ above Λ. The one-loop formula becomes small at high energy; its growth toward Λ marks the loss of perturbative control.

Gauge group SU(N)N is the fundamental representation dimension. Pure gauge theory, no matter fields.

Scale ratio μ / Λ100

Logarithmic control · 1.12 to 10,000

α(μ) = g²(μ) / 4π0.124

’t Hooft coupling λ(μ) = Ng²(μ)4.68

One-loop expansion indicator0.030Ng² / (16π²)

One-loop α as a function of μ / Λ The curve is clipped at α = 1.5; the formula is never evaluated at or below μ = Λ.

Shading: α ≥ 0.5, an illustrative warning region rather than a rigorous boundary of validity.

High-energy running is under better perturbative control.

One-loop model. μ dg/dμ = −b₀g³, b₀ = 11N/(48π²), so α(μ) = 6π/[11N log(μ/Λ)]. Λ denotes the one-loop integration scale in a specified coupling convention. The divergence of this truncated formula is not a physical pole or a calculation of a mass gap.

What this calculation establishes

Solving the one-loop renormalization-group equation introduces a dimensionful integration constant. This is dimensional transmutation. The numerical value of that scale depends on the renormalization convention.

The low-energy singularity of the one-loop expression marks a failure of the approximation. It is not a physical particle pole or a calculation of the lightest glueball mass. Determining a finite positive ratio between a physical mass and the chosen renormalization scale requires nonperturbative information.

Exercise: At fixed scale ratio, vary N and compare the ordinary coupling with the associated ’t Hooft coupling. Which combination has a particularly simple one-loop large-N scaling?

For the derivation, conventions, and original asymptotic-freedom references, read dimensional transmutation. Then compare with the continuum-limit laboratory and the ultraviolet obligation.

Check the normalization

For pure SU( $N$ ) theory without matter, the convention used here is

\mu\frac{dg}{d\mu}=-b_0g^3,\qquad b_0=\frac{11N}{48\pi^2},\qquad \alpha=\frac{g^2}{4\pi},\qquad \lambda=Ng^2.

Differentiate $g^{-2}$ to find $d(g^{-2})/d\log\mu=2b_0$ . Integrating from a reference $(\mu_0,g_0)$ gives

\frac1{g^2(\mu)}=\frac1{g_0^2}+2b_0\log\frac\mu{\mu_0},\qquad \Lambda=\mu_0\exp\!\left[-\frac1{2b_0g_0^2}\right].

Thus the displayed curve is

\alpha(\mu)=\frac{6\pi}{11N\log(\mu/\Lambda)},\qquad \mu>\Lambda.

Both couplings are dimensionless; $\mu$ and $\Lambda$ have the same energy units. The scale is an integration constant in the selected convention, not a measured glueball mass. The control uses $\log_{10}(\mu/\Lambda)$ ; the equation uses the natural logarithm. The graph never evaluates the denominator at or below its zero.

Worked value and exercise solution

For SU(3) and $\mu/\Lambda=100$ , the formula gives $\alpha\approx0.1240$ , $g^2\approx1.559$ and $\lambda\approx4.676$ . At a larger ratio, the logarithm grows and all three decrease. Recovering $g(\mu_0)=g_0$ by substitution is a useful normalization check.

Solution to the exercise above: at fixed ratio, $\alpha$ and $g^2$ scale as $1/N$ , while

\lambda=\frac{24\pi^2}{11\log(\mu/\Lambda)}

is independent of $N$ in this one-loop pure-gauge model. This does not justify a nonperturbative large- $N$ conclusion. The plot’s shaded $\alpha\geq0.5$ region is an illustrative display convention; it is not an $N$ -independent rigorous error bound. The separate indicator $Ng^2/(16\pi^2)$ makes the group dependence explicit.

Exercise: Can the divergence at $\mu=\Lambda$ be used to assign a particle mass equal to $\Lambda$ ?

Solution: No. It signals the breakdown of this truncated weak-coupling expression. A particle mass concerns the physical spectrum, requiring information not supplied by a running-coupling pole.

The founding primary sources are Gross and Wilczek, Ultraviolet Behavior of Non-Abelian Gauge Theories, Phys. Rev. Lett. 30 (1973), 1343–1346, and Politzer, Reliable Perturbative Results for Strong Interactions?, Phys. Rev. Lett. 30 (1973), 1346–1349. The algebra here checks the stated normalization; it is not a reproduction of their loop computation.