The Instanton Size Modulus and Infrared Limitations

The instanton size $\rho$ is a classical modulus, so an observable requires an integral over all sizes allowed by the geometry and boundary conditions. In an asymptotically free theory, small instantons are weakly coupled, whereas sufficiently large instantons probe $g(1/\rho)\sim1$. A trustworthy calculation must derive the complete power of $\rho$, test both endpoints, and stop before the running coupling leaves the semiclassical regime.

Required background. Instanton measures, zero modes, and determinants supplies the one-loop density and its zero-mode factors. Running couplings and dimensional transmutation supplies $\Lambda$ and the domain of one-loop running.

Helpful background. Dilute instanton ensembles and theta dependence shows where the integrated one-event density enters an observable.

Deriving the size power

For $SU(N)$ with $N_f$ Dirac fermions in the fundamental representation, the charge-one density contains

$$d\rho\, \rho^{-5} (\mu\rho)^{b_0} \left[\frac{8\pi^2}{g^2(\mu)}\right]^{2N} e^{-8\pi^2/g^2(\mu)}, \qquad b_0=\frac{11N}{3}-\frac{2N_f}{3}.$$

Every explicit power has a distinct source:

• $\rho^{-5}d\rho\,d^4x_0$ combines the size and translation measures into a scale-invariant five-dimensional volume element.
• $(\mu\rho)^{b_0}$ comes from regulated nonzero-mode, ghost, and matter determinants and is fixed by the one-loop beta function.
• $g^{-4N}$, written as $[8\pi^2/g^2]^{2N}$, comes from the $4N$ bosonic collective-coordinate Jacobians and carries no additional $\rho$ power at fixed $\mu$.
• A vacuum amplitude with $N_f$ massive fundamental Dirac fermions gains $\prod_f(m_f\rho)$, one dimensionless factor per flavor zero-mode pair.
• A declared observable may add a further $\rho^{p_{\mathcal O}}$ after its instanton profile, external momenta, and spacetime integrations have been evaluated.

Thus the local one-loop power for an observable with $n_m$ mass-saturated fundamental flavors is

$$\int d\rho\,\rho^\alpha, \qquad \alpha=b_0-5+n_m+p_{\mathcal O},$$

apart from logarithms and dimensional constants $\mu^{b_0}\prod m_f$. The value of $p_{\mathcal O}$ must be derived from the actual insertion; it cannot be borrowed from the vacuum measure.

The complete origin of these factors is displayed in the moduli-to-measure chain. The comparison of canonical saddle calculations separates the small coupling that controls local loops from the additional parameter that controls an ensemble.

Renormalization-group form

At one loop,

$$\frac{8\pi^2}{g^2(1/\rho)} = b_0\log\frac{1}{\rho\Lambda}.$$

Using the renormalization-group invariant combination,

$$e^{-8\pi^2/g^2(\mu)}(\mu\rho)^{b_0} = e^{-8\pi^2/g^2(1/\rho)} = (\rho\Lambda)^{b_0},$$

the pure-gauge size distribution becomes, up to a scheme-dependent constant,

$$\frac{dn_I}{d^4x_0\,d\rho} \sim \rho^{-5} (\rho\Lambda)^{b_0} \left[ b_0\log\frac{1}{\rho\Lambda} \right]^{2N}, \qquad \rho\Lambda\ll1.$$

The logarithm is the running version of the bosonic Jacobian factor. It varies slowly compared with the power but must be retained in a precision calculation. The formula itself declares its domain: once $\rho\Lambda$ is order one, neither the one-loop running nor the Gaussian expansion about one instanton is parametrically controlled.

Endpoint tests

For a pure power $\rho^\alpha$,

$$\int_0 d\rho\,\rho^\alpha \quad\text{converges at }0 \quad\Longleftrightarrow\quad \alpha>-1,$$

whereas

$$\int^\infty d\rho\,\rho^\alpha \quad\text{converges at }\infty \quad\Longleftrightarrow\quad \alpha<-1.$$

At $\alpha=-1$, the divergence is logarithmic. These are mathematical endpoint criteria, not guarantees of physical control. In an asymptotically free theory the small-$\rho$ region can be perturbative, so the ultraviolet test is meaningful. The large-$\rho$ criterion is often never reached within the derivation’s domain because $g(1/\rho)$ becomes strong first.

For pure $SU(N)$, $p_{\mathcal O}=n_m=0$, so

$$\alpha_{\rm vac}=\frac{11N}{3}-5.$$

Two examples are

$$\alpha_{SU(2)}=\frac73, \qquad \alpha_{SU(3)}=6.$$

Both integrals converge at $\rho=0$ and grow toward the infrared. If one truncates at $\rho_{\max}=c/\Lambda$ with $c\ll1$, pure $SU(3)$ gives parametrically

$$\int_0^{c/\Lambda}d\rho\,\rho^6\Lambda^{11} \propto c^7\Lambda^4$$

up to logarithms. The strong $c^7$ dependence shows that the result is dominated by the arbitrary edge of the weak-coupling region. Taking $c\sim1$ removes that arbitrariness only by leaving the semiclassical regime. This is the infrared size problem identified in the original instanton calculus; see ‘t Hooft 1976, pp. 3447–3450 and Mariño 2015, § 4.5, pp. 129–146.

Observable insertions can change the diagnosis

Consider a correlator whose instanton profile supplies $p_{\mathcal O}$. A positive $p_{\mathcal O}$ improves small-$\rho$ convergence but worsens formal large-$\rho$ growth; a sufficiently negative power can do the reverse and may introduce ultraviolet contact divergences requiring operator renormalization. Fermion masses add positive powers through $m_f\rho$, but their presence does not restore large-$\rho$ semiclassical control.

External momentum can provide an effective infrared cutoff. Fourier-transformed BPST profiles contain form factors $F(p\rho)$ that decay when $p\rho\gg1$, so a hard observable with $p\gg\Lambda$ can emphasize $\rho\lesssim1/p$. The exact form factor and power must be derived for that observable. One may not replace it by a universal cutoff without changing the calculation.

Compactification can also alter the problem. On $\mathbb R^3\times S^1$, a small circumference and suitable holonomy may replace the unrestricted four-dimensional size family by constituent monopole saddles with different moduli. That is a new controlled regime with explicit boundary data, not a cure applied silently to the $\mathbb R^4$ integral.

What an infrared divergence means

An infrared-growing one-loop density establishes that large configurations receive increasing formal weight. It does not establish an “instanton liquid,” confinement, or any particular cutoff mechanism. Those are additional dynamical claims. The correct conclusion is narrower:

the one-instanton expansion on $\mathbb R^4$ ceases to be predictive for an observable dominated by $\rho\Lambda\sim1$.

A phenomenological size distribution can still be useful if its assumptions and fitted inputs are stated, but it is no longer the controlled one-loop result.

Common pitfalls

Testing only the algebraic endpoint. Formal convergence at infinity does not help if the running coupling becomes strong before the endpoint. Check the validity domain as well as the integral.

Forgetting insertion powers. The vacuum measure, a mass-saturated amplitude, and a hard correlator have different $\rho$ dependence. State every factor before applying an endpoint criterion.

Calling an imposed cutoff a prediction. A cutoff at $\rho\sim\Lambda^{-1}$ is precisely where the weak-coupling derivation fails. Its numerical consequences are model dependent.

Exercises

1. For $SU(3)$ with $N_f=3$ massive fundamental Dirac fermions, find the explicit $\rho$ power in the vacuum amplitude before logarithms.

Solution

The beta-function coefficient is

$$b_0=11-\frac{2}{3}\cdot3=9.$$

The bosonic density gives $\rho^{b_0-5}=\rho^4$, and the three mass factors add $\rho^3$. Thus

$$d\rho\,\rho^7\,m_1m_2m_3$$

times powers of $\Lambda$ and logarithms. It is ultraviolet convergent and strongly infrared weighted; masses do not make the BPST integral controlled at $\rho\Lambda\sim1$.

2. An observable produces a small-$\rho$ integrand $d\rho\,\rho^{-1}$. Classify the endpoint and state the next question.

Solution

The ultraviolet divergence is logarithmic. One must determine whether it is a contact divergence absorbed by renormalization of the inserted operator, whether operator mixing is required, and whether the instanton contribution has been matched in a consistent scheme. It cannot be assigned a finite value from the semiclassical density alone.

References

• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge: Cambridge University Press, 2015. DOI.
• ‘t Hooft, Gerard. “Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle.” Physical Review D 14 (1976): 3432–3450; erratum 18 (1978): 2199. DOI.