2026-10-01 · chapter Yang–Mills and the mass gapPart 4 Quantisationcounterterms, the renormalisation group

Divergences and renormalisation — rewriting infinities in terms of measured quantities

When a perturbative diagram (4-03) has a loop, an integral over a momentum $k$ not fixed from outside remains. Large $|k|$ are short-distance fluctuations, and in four dimensions there are too many of them for convergence. We cut at $|k|\lt\Lambda$ ($\Lambda$ is the cutoff) and look at how it grows.

$T$ is the diagram closing one propagator, and $I$ the diagram closed with two (a bubble) ($d^4k=2\pi^2k^3dk$). Multiplying the volume $k^3dk$ by $k^{-2}$ for each propagator, $T$ is $k\,dk$ (quadratic divergence) and $I$ is $k^{-1}dk$ (logarithmic divergence). On the position side this corresponds to $G(x)^2\sim|x|^{-4}$ not being integrable at the origin (1-11).

Example. With $m=1$ and $\Lambda=10,10^3,10^6$, $16\pi^2T$ is 95.4, $1.0\times10^6$, $1.0\times10^{12}$, and $16\pi^2I$ is 3.625, 12.816, 26.631. Each time $\Lambda$ is multiplied by 10, $T$ grows about 100-fold and $16\pi^2I$ grows by $2\ln10=4.605$ Computed.

Putting in an artificial scale to make things finite is called regularisation. Regularisation is scaffolding, and the goal is that as $\Lambda\to\infty$ (on a lattice, $a\to0$) the answer does not depend on the scaffolding. Each of the four in Table 1 loses something.

Dimensional regularisation (Bollini–Giambiagi 1972, 't Hooft–Veltman 1972) is the standard tool of perturbation theory, Yang–Mills included. On the other hand, in Table 1 only the lattice row keeps reflection positivity. Reflection positivity is the property needed to assemble the Hilbert space of quantum mechanics (the space of states) from Euclidean correlation functions (4-10). The problem statement writes that there are few ways to recover reflection positivity once lost by regularisation, giving dimensional regularisation and Pauli–Villars as examples (§6.5; quoted in 0-03 §03). Being able to keep from the start a property that is hard to recover once lost is one of the reasons the lattice is chosen on the side of construction.

With $\hbar=c=1$, the dimension $[\,\cdot\,]$ of a quantity is a power of mass (length is $-1$). The action is dimensionless, $d^dx$ is $-d$, and derivatives are $+1$, so the dimensions of fields and couplings are fixed.

In four dimensions both $u$ and $g^2$ are dimensionless. This is what "four dimensions is marginal" means (0-03, 1-12; how to derive the counting of dimensions is (10) of 2-04 §08). The power of momentum of a diagram with $L$ loops and $I$ internal lines is $\omega=dL-2I$ (adding the number of vertices that carry derivatives), and if $\omega\ge0$ it superficially diverges. For $n$ vertices of $\phi^4$ and $E$ external lines, $4n=2I+E$, and from $L=I-n+1$,

($V_3,V_4$ is the number of three-point and four-point vertices of Yang–Mills, and the external lines are gluons). In four dimensions $\omega_{\mathrm{YM}}=4-E$. The dimension of the coupling appears in the coefficient of the order, so there are three cases Known. If positive, $\omega$ falls with the order, and only finitely many diagrams diverge (super-renormalisable). If 0, the types of divergence (values of $E$) are finitely many but appear at all orders (renormalisable). If negative, the types increase without bound (non-renormalisable).

The two-dimensional type has the same form as $T$ and can be removed by Wick ordering (6-03; for three-dimensional Yang–Mills, 6-06). We say "superficial" because even with $\omega\lt0$ a diagram diverges if it has a $\omega\ge0$ subdiagram inside. The theorem that states convergence by a power condition including subdiagrams is Weinberg's 1960 theorem, and the method of subtracting subdiagrams in turn is treated in 4-05.

We follow the procedure at one loop for $\phi^4$ in four dimensions. The coefficients $m_0,u_0$ of the action are called the bare mass and coupling (coefficients under the cutoff, not directly measurable). The four-point strength $\Gamma^4$ (not the diagram symbol $\Gamma$ of 4-03; the four-point function with the external propagators removed, summing only diagrams that do not split in two when one line is cut; $u_0$ at tree level) shifts by the amount of one inserted bubble.

The three terms are the three ways of dividing the external lines into pairs, and $\tfrac12$ is the symmetry factor (counted the same way as 6-08). The bubble with momentum $P$ flowing in reduces to the $I$ of (1) by $\frac{1}{AB}=\int_0^1\frac{dx}{[xA+(1-x)B]^2}$ (a Feynman parameter) and a shift of variable.

The difference between the value cut in the shifted variable and the value cut in the original $k$ is 0 as $\Lambda\to\infty$ ($P=m$, $\Lambda=10^3m$, multiplied by $16\pi^2$, $3\times10^{-7}$) Computed. So we redefine the coupling by a measured quantity. The $\Gamma^4$ measured in the configuration where all three $|P_j|$ are $\mu$ is taken as the renormalised coupling $u_R(\mu)$ at the scale $\mu$.

$\delta u\approx\frac{3u_R^2}{32\pi^2}\ln\frac{\Lambda^2}{\mu^2}$ diverges. Putting this back into (4), the divergence is shut up in the form of a difference.

No $\Lambda$ remains on the right-hand side. The content of renormalisation is that predictions written in terms of the measured coupling are finite, and the infinity is pushed into the unmeasurable $u_0$. The divergence of the mass ($T$) is pushed into $m_0$, and from two loops on also into the field normalisation $Z$; for $\phi^4$ in four dimensions these three suffice at all orders. This is the practical meaning of "renormalisable" Known.

ComputedThe numbers in this figure were computed on this machine ((5)(7) integrated numerically in python3; the figure integrates the same formulas on the spot by Simpson's rule, agreeing to 4 digits). Static values ($16\pi^2$ times, $\mu=10m$): the difference at $P=1000m$ is −5.477 at $\Lambda=10^2m$, −9.096 at $10^4m$, −9.099 at $10^8m$ (limit −9.099). At the same $\Lambda$ the bare $B(100m)$ keeps growing: 1.289, 10.208, 28.629.

The reference scale $\mu$ can be chosen freely. How $u_R$ changes when $u_0,\Lambda$ is held fixed and $\mu$ is moved is the β function.

$\Lambda$ vanishes under differentiation. Differentiating (5) numerically, the coefficient approaches $3/16\pi^2=0.018998$: 0.01728 at $\mu=10m$, 0.018997 at $10^3m$ Computed. Integrating at one loop,

and the coupling becomes stronger at shorter distances. For $u_R(\mu_0)=1$, it is 1.78 at $\mu/\mu_0=10^{10}$, and the denominator is 0 at $10^{22.86}$ (the Landau pole; 6-08) ComputedPhysics. Yang–Mills has the opposite sign (4-06).

Example. At one loop in quantum electrodynamics, each Dirac particle of charge 1 (charged lepton) gives $d(1/\alpha)/d\ln\mu=-2/3\pi$. Extending from $1/\alpha=137.036$ to $M_Z=91.188$ GeV with the electron, μ and τ gives 132.20 (only the leading logarithm at one loop) Computed. The PDG 2024 value is $\hat\alpha^{(5)}(M_Z)^{-1}=127.930$. $\overline{\mathrm{MS}}$ is the scheme that, instead of a cutoff, subtracts only the poles of dimensional regularisation (4-05). The superscript (5) indicates the value with the five kinds of quark other than top included. The difference from 132.20 is due to the quarks and the constants of the scheme (3-07).

It can also be said for whole functions. The $n$-point function written in bare quantities knows nothing of $\mu$, and the renormalised $\Gamma^{(n)}_R$ differs by the field normalisation $Z$, so

($\gamma$ is the anomalous dimension; for $\phi^4$ from two loops on). Where masses do not matter, multiplying momenta by $s$ while moving the coupling along with them brings the left-hand side of (10) into a single ordinary derivative in $s$ (the method of characteristics). Ignoring $\gamma$, $\Gamma^4(sP;u,\mu)=\Gamma^4(P;\bar u(s),\mu)$ ($s\,d\bar u/ds=\beta(\bar u)$, $\bar u(1)=u$). Looking at momenta multiplied by $s$ is replacing the coupling by its value at the scale $s\mu$ Known. Gell-Mann–Low 1954 defined the effective charge of the electron $e(\mu)$ as a function of scale (lecture notes: D. Tong, Statistical Field Theory §3.5.3). Stueckelberg–Petermann 1953 are said to have noticed that the transformations of renormalisation form a group. The form of the equation that moves masses is due to Callan 1970 and Symanzik 1970 (the Callan–Symanzik equation), and in textbooks the form (10) is also called by this name. All are statements at each order of perturbation theory.

Regarding $\Lambda$ as the genuine smallest scale, integrating only the fluctuations in $\Lambda/b\lt|k|\lt\Lambda$ and remeasuring lengths $b$ times, all the coefficients of the action move a little. Repeating this is Wilson's 1971 renormalisation group, which implements Kadanoff's idea of blocking (2-08) as integration shell by shell in momentum (a rigorous version on the lattice is 6-05). As a metaphor, it is the operation of lowering the resolution of a photograph one step at a time. Precisely, it is a dynamical system in which the sequence of coefficients of the effective action is carried by a map at each step.

The coefficient of a local term of dimension $\Delta_{\mathcal O}$ has dimension $d-\Delta_{\mathcal O}$, and the dimensionless coefficient is multiplied by $b^{\,d-\Delta_{\mathcal O}}$ at each step towards long distances. The lecture notes (§3.2.2) divide them as 'Relevant if Δ_O < d', 'Irrelevant if Δ_O > d', 'Marginal if Δ_O = d' Known. Irrelevant terms are diluted to about $(E/\Lambda)^{\Delta_{\mathcal O}-d}$ at energy $E$, and at long distances only relevant and marginal terms remain. In four dimensions $\mathrm{tr}F^2$ ($\Delta=4$) is marginal and $\mathrm{tr}F^4$ ($\Delta=8$) is irrelevant (the question of 3-01). The expectation that the same continuum limit is reached regardless of the details of the lattice action (universality; 8-04) also comes from here Physics.

The direction of a marginal coupling is decided by quantum corrections. The lecture notes write 'most couplings which naively appear marginal will end up becoming either marginally relevant or marginally irrelevant due to such corrections', and give 'the Kondo effect and non-Abelian gauge theories' as examples that turn relevant (§3.4). $\phi^4_4$ is on the side that weakens at long distances, Yang–Mills on the side that strengthens. With $t=\ln(\ell/\ell_0)$ towards long distances and the dimensionless $\hat u=u\,\ell^{\,4-d}$, extending $d$ to real values,

(the formula of the lecture notes §3.5 converted to the normalisation of $u/4!$; opposite in direction and in sign to (8)). For $d\lt4$ there is a fixed point at $\hat u_*=16\pi^2(4-d)/3$, and every positive initial value flows into it. This is the Wilson–Fisher 1972 fixed point, the smallest model of universality. For $d=4$ it goes to 0 slowly, for $d\gt4$ fast (6-08) Physics. A figure placing the $[g^2]$ of (2) and the fixed point of (11) next to other quantities that switch at four is the dimension knob of 1-12 §10.

ComputedPhysicsThe numbers in this figure were computed on this machine (closed-form solution; agreeing with RK4). Static values (length ×$10^8$, $\hat u_0=1$ / 100): at $d=3$, 52.6 / 52.6 ($\hat u_*=52.64$); at $d=4$, 0.741 / 2.78; at $d=5$, $9.8\times10^{-9}$ / $3.5\times10^{-7}$. The formula is a one-loop approximation, reliable when $4-d$ is small.

Detour: non-renormalisable theories are useful too — the Fermi interaction

One would like the counterterms of Yang–Mills, once added, to stay within the Yang–Mills form. The gauge-invariant local terms of dimension at most 4 are only $\mathrm{tr}F^2$ and $\mathrm{tr}F\tilde F$ (the θ term; a total derivative that does not enter perturbation theory; 3-05), and the mass term $\mathrm{tr}A_\mu A_\mu$ cannot be written (3-02). However, perturbation theory is done with the gauge-fixed action. There, instead of gauge invariance, BRST symmetry (the symmetry that remains after gauge fixing) constrains the form of the counterterms (4-08). As a result, the gauge field part is kept in the form $\mathrm{tr}F^2$. The divergences are absorbed into $g$ and the field normalisations (including those of the ghost and gauge-fixing terms), and since there is one parameter, the integration constant of the running becomes the only scale $\Lambda_{\mathrm{YM}}$ (a different quantity from the cutoff $\Lambda$; 4-07). Gauge invariance also reduces divergences. The two-point correction of the gluon is restricted to transverse form by the Ward (in the non-abelian case, Slavnov–Taylor) identities Known.

Transverse means $P_\mu\Pi_{\mu\nu}=0$, that is, there is no component along the momentum. Since the second power of $P$ factors out, of the $\omega=2$ in Table 2 only the logarithmically divergent $\Pi$ remains. $\Lambda^2\delta_{\mu\nu}$, corresponding to a mass term, does not appear in gauge-invariant regularisations. A momentum cutoff breaks this, and the construction of Magnen–Rivasseau–Sénéor 1993 also says it 'breaks gauge invariance and requires gauge restoring counterterms'. The abstract of 't Hooft 1971 says that regulator fields that are not gauge invariant can also be used if gauge invariance can be restored by finitely many counterterms in the limit of large regulator mass, and that massless Yang–Mills fields 'appear to be renormalizable' in the usual sense. Gauge fixing and ghosts are in 4-08. The lattice keeps gauge invariance exactly at the stage of regularisation (5-01).

The key is not to mix up three levels.

An early example of the first way of thinking is Dyson 1949 for quantum electrodynamics (an argument removing the divergences at each order by renormalising mass and charge; the complete proof came later with Weinberg 1960 and BPHZ; convergence of the series was not shown). On the third, the problem statement writes that for $\phi^4$ in two and three dimensions the coefficients are chosen 'to depend on κ in the way that perturbation theory suggests' on the cutoff $\kappa$, and 'One then proves that the expectations converge as κ → ∞, even though the specified constants α, . . . diverge.' (§6.2). This is the constructive version of (7) being finite even though the $\delta u$ of (6) diverges.

For four-dimensional Yang–Mills, it is required that the correlation functions agree at short distances with the textbook predictions of asymptotic freedom and perturbative renormalisation theory (§4; quoted in 0-03 §07). §2 writes 'Asymptotic freedom ensures a decisive regularity in the case when classical Sobolev inequalities are borderline'. In this article's reading, "borderline" corresponds to "the coupling is dimensionless in four dimensions" of §03 (8-04), and the sentence is an outlook, not a theorem. That handling the infinities is not itself the centre of the difficulty is in 9-05.

What this chapter can say

Sources and reproduction

Next chapter to read: 4-05 The algebra of divergences (BPHZ, dimensional regularisation, Connes–Kreimer). The sign of the β function is in 4-06.

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