2026-10-01 · chapter Yang–Mills and the mass gapPart 4 Quantisationthe β function, 11N/3

Asymptotic freedom — the sign that weakens the coupling at short distances, and what is inside 11N/3

The renormalised coupling $g$ has a value only once the scale $\mu$ at which it is measured is fixed (a momentum; as a distance, roughly $1/\mu$) (4-04). How it changes is the β function, and the right-hand side is an expansion in small $g$.

If $\beta\gt0$, the coupling grows at short distances (large $\mu$); QED and four-dimensional $\phi^4$ are on this side (6-08). If $b_0\gt0$, it weakens at short distances. This is asymptotic freedom. Dropping $b_1$ and higher and integrating, $1/g^2$ becomes a linear function of $\ln\mu$ (the same formula as in 7-01).

Formula (2) means that $1/g^2$ grows in proportion to the logarithm of the factor by which $\mu$ has grown. Whether $\mu$ is multiplied by 10 or by 100, it grows in fixed steps, and the coupling weakens only slowly.

Example. Writing $\alpha$ for the square of the coupling divided by $4\pi$, one-loop QED with electrons only gives $d(1/\alpha)/d\ln\mu=-2/(3\pi)\approx-0.212$ (a rewriting of the formula in §2.4.1 of Tong's lecture notes), and pure SU(3) gives $+1.751$. In the latter, $1/\alpha$ grows by about 4.03 each time $\mu$ grows by a factor of 10 (0-03) Computed.

For a gauge group with $n_f$ Dirac particles (representation $R_f$) and $n_s$ complex scalars (representation $R_s$) added, the one-loop coefficient is as follows (§2.4 and §2.7 of Tong's lecture notes; formula (5.3) of 't Hooft 1999 is the same formula with a different normalisation) Known.

$T(R)$ is the number fixed by $\mathrm{tr}_R(T^aT^b)=T(R)\,\delta^{ab}$: $\tfrac12$ for the fundamental representation and $C_A$ for the adjoint representation ($N$ for SU(N); 1-04). For pure SU(N), $b_0=\tfrac{11N}{48\pi^2}$. The apparent numbers change with the choice of variable, so we list them in Table 1 (the four rows say the same thing).

ComputedEvaluated with python3. $b_0^{(\alpha)}=33/(12\pi)$ for SU(3) agrees with formula (9.3) of the PDG and the formula for $b_0$ right after it.

The last row is in the language of the lattice (5-01). The one-loop prediction is that the same physics is kept if $\beta_{\mathrm L}$ is raised by $\kappa\ln2$ (0.258 for SU(2), 0.579 for SU(3)) each time the lattice spacing $a$ is halved Computed. $\kappa=11/(3\pi^2)$ for SU(2) is the same number as the slope $\kappa$ in the record kept on this machine, and is the known one-loop coefficient written in lattice conventions (10-02) Known.

Why is the sign reversed? The background field method places a slowly varying external field (the background field), integrates out the quantum fluctuations around it at one loop, and looks at how the coefficient $1/g^2$ of the action of the background field changes. Calculated this way (§2.4.2 of Tong's lecture notes), the gluon contribution splits in two. The term that counts the components of the field like scalars gives, after the excess is subtracted with ghosts, $-\tfrac13$ in the screening direction, the same as in electromagnetism. The term that comes from spin 1 (the lecture notes' 'the paramagnetic piece') gives $+4$ in the antiscreening direction. The overall minus sign comes from here ('This is where the overall minus sign comes from') Known.

The photon carries no charge and takes no part in the vacuum polarisation of QED, but the gluon carries colour charge, so it does take part (3-02). Wilczek's Nobel lecture also writes 'it is the effect of virtual gluons that is responsible for antiscreening'.

The same split can be used for matter. We count each state of a massless particle (spin $s$, component $s_z$ along the magnetic field) as contributing to $16\pi^2b_0$ the amount

each. $(2s_z)^2$ is the spin (paramagnetic) part, $-\tfrac13$ the orbital (diamagnetic) part, and $(-1)^{2s}$ the sign of a fermion loop. A gluon ($s_z=\pm1$) gives $\tfrac{11}{3}C_A$, a Dirac particle $-\tfrac43T$, and a complex scalar $-\tfrac13T$, and all of formula (3) comes out (Table 2). In this chapter we set down formula (5) as a way of counting that reproduces the known coefficients. That this rule reproduces all the coefficients of formula (3) was checked on this machine Computed. The derivation of the rule itself is not treated in this chapter. The analogy that views asymptotic freedom as paramagnetism is said to go back to Nielsen 1981 (Tong's note on the literature; Tong also mentions a private attribution to 't Hooft).

ComputedThe numbers in this figure and table were computed on this machine (formula (5) evaluated with python3 rationals, agreeing with the known coefficients; the figure adds up the same formula on the spot). If JavaScript is disabled, this table can be read instead.

Detour: the vacuum is a paramagnet

Matter lowers $b_0$. With $n_f$ Dirac particles in the fundamental representation, formula (3) is $16\pi^2b_0=\tfrac{11N}{3}-\tfrac{2n_f}{3}$, and the theory is asymptotically free only when $n_f\lt11N/2$ (less than 16.5 for SU(3)) Computed. 't Hooft's recollection (1999) has the same number. There are 6 real quarks, and QCD is on the asymptotically free side.

For an abelian group $C_A=0$, so only the matter terms remain in (3), and $b_0\lt0$ (the direction of QED in §01). For $U(1)_Y$ of the Standard Model (normalised by $Q=T_3+Y$; the hypercharges are in Table 1 of 3-07), replacing $T(R)$ by $Y^2$ and counting one left-handed Weyl component as half a Dirac particle, the three generations of fermions ($\sum Y^2=10/3$ per generation) and the Higgs doublet give

and at one loop the coupling grows at short distances Computed. This is a different number from $b_1$ in §05 (the two-loop coefficient).

For theories without Yang–Mills fields, the abstract of Coleman and Gross (1973) states 'No renormalizable field theory without non-Abelian gauge fields can be asymptotically free.' (a claim at the level of calculating β functions) Known. §6.2 of the problem statement also places the fact that four-dimensional $\phi^4$ is not asymptotically free next to triviality (6-08), and calls Yang–Mills 'the natural candidate for four-dimensional CQFT'. This is one reason why Yang–Mills is the protagonist of the problem.

The two-loop coefficient was found by Caswell and by Jones, each in a 1974 paper Known. For pure SU(N) it is positive, and rewriting the PDG formula for $\alpha$ in terms of $g$ gives the following form (checked for SU(3) Computed).

$b_0$ and $b_1$ do not change when the definition of the coupling (the renormalisation scheme) is changed. Changing to another coupling $\tilde g=g+c\,g^3+d\,g^5+\cdots$ and rewriting $\mu\,d\tilde g/d\mu=(1+3cg^2+5dg^4)\beta(g)$ in terms of $\tilde g$ gives

and $c,d$ enter only from $\tilde b_2$ on (checked with series of rationals Computed). The PDG also writes that $b_2$ and beyond are 'renormalization-scheme-dependent', and Capitani et al. 1999 call $b_0,b_1$ 'The universal coefficients' Known. This universality is the footing for §07.

Example. For pure SU(3), starting from the same integration constant $\Lambda$ (4-07), $\alpha$ at $\mu/\Lambda=10$ is 0.248 at one loop and 0.186 at two loops, and even at $10^{10}$ it is 0.0248 and 0.0232, a difference of 7% Computed. The correction is a "log of a log" and is slow to disappear. Adding matter to make $b_1\lt0$, the β function vanishes at $g_*^2=-b_0/b_1$, and for SU(3) with $n_f=16$ an infrared fixed point with $\alpha_*\approx0.042$ appears Computed (Banks–Zaks 1982 Physics). The coefficients have been computed up to five loops (the PDG's list of references; for the meaning of the series as a whole, 4-12).

The negative sign had appeared in calculations before 1973. According to 't Hooft's recollection (1999), at the time it was thought to be universal that the β function is positive, and quantum field theory was thought unable to explain Bjorken scaling (the regularity, in high-energy scattering, that the constituents inside a hadron appear to move almost freely at short distances).

Politzer's title carried a question mark: 'Reliable Perturbative Results for Strong Interactions?' Known. 't Hooft writes of the 1973 papers 'I was surprised, not about the result, which I had known all along, but about the stir they caused', and goes on to say that the real problem is confinement. The problem of this book also lies beyond that (7-01).

On the lattice (5-01) the lattice spacing $a$ is the cutoff, and the bare coupling $g_0$ runs with $a$. By the universality of §05 its β function also begins with the same $b_0,b_1$, and integrating to two loops gives

(the form of formula (2.7) of Capitani et al. 1999 with $\mu=1/a$) Known. We compare the lattice $\Lambda_{\mathrm L}$ with the $\Lambda$ of the continuum $\overline{\mathrm{MS}}$ scheme. $\overline{\mathrm{MS}}$ is the standard definition of the coupling in continuum theory, which in dimensional regularisation subtracts only the poles of the divergences and the constants that come with them (4-05). The two $\Lambda$ differ only by a constant factor fixed at one loop; for the Wilson action $\Lambda_{\overline{\mathrm{MS}}}/\Lambda_{\mathrm L}=28.81$ (SU(3)) and 19.82 (SU(2)). Computed from the coefficient $k_1$ of Lüscher–Weisz 1995, the SU(3) value agreed with 28.80934 in formula (4.1) of the same paper Computed. That formula (8) holds for lattice numbers is called asymptotic scaling (5-05; an early test is Creutz 1980).

At today's precision — $r_0/a$ for pure SU(3)

The length standard $r_0$ is fixed from the force $F$ between a pair of static colour charges by $r_0^2F(r_0)=1.65$ (7-01). The naive estimate of $r_0\Lambda_{\overline{\mathrm{MS}}}$, obtained by multiplying $r_0/a$ of Necco–Sommer 2002 by formula (8) and the ratio of Λs, is a quantity that should be constant if asymptotic scaling holds.

ComputedThe numbers in this figure and table were computed on this machine. $r_0/a$ is from Table 1 of Necco–Sommer 2002 (* converts $r_c/a$ with $r_c/r_0=0.5133$), and the step (data) is from the interpolation formula (2.6) of the same paper (— is out of range). The estimate is $(r_0/a)\times a\Lambda_{\mathrm L}\times28.81$. The values in the figure agree with python3. If JavaScript is disabled, this table can be read instead.

The estimate keeps growing from 0.28 to 0.46 and has not yet reached the non-perturbative value 0.602(48). The expansion in the bare coupling converges slowly (with one loop alone, 0.09 to 0.14). Still, the direction and the order of magnitude agree, and the higher $\beta_{\mathrm L}$, the closer to the prediction. In the method that measures the running coupling non-perturbatively with the size of the box as the scale (Lüscher–Weisz–Wolff 1991), the measured coupling is 'closely matched by the perturbative evolution', and 0.602(48) comes from there (Capitani et al. 1999) Known. All of these, however, are comparisons of finite-lattice numbers with perturbation theory, and show neither the existence nor the uniqueness of the continuum limit (5-09, 9-05).

The problem statement first introduces asymptotic freedom as a physical explanation (§1), and gives it two roles. One is a requirement: the correlation functions must agree at short distances with the predictions of asymptotic freedom and perturbative renormalisation theory (§4; quoted in 0-03 §07). Its role is to check at short distances that the theory built is "Yang–Mills". The other is a tool: 'Asymptotic freedom ensures a decisive regularity in the case when classical Sobolev inequalities are borderline' (§2) Known. This can be read as saying that four is the borderline dimension in which classical estimates fall just short, and asymptotic freedom makes up the shortfall (1-12; for Bałaban's work, 6-07).

Asymptotic freedom says "short distances are easy", and at the same time says "at long distances there is no handle for an expansion". The difficulty of the problem lies on the latter side (6-10, 6-11).

What this chapter can say

Sources and reproduction

Next chapter to read: 4-07 Dimensional transmutation and Λ (the $\Lambda$ that appears when formula (2) is extended to the long-distance side). The continuation on the lattice side is 5-05.

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