2026-10-01 · chapter Yang–Mills and the mass gapPart 3 Yang–Mills theory (classical)the Standard Model

Yang–Mills inside the Standard Model — a product of three groups, with matter and the Higgs added

The conventions are the same as in 3-02 §02 ($\hbar=c=1$, the metric $(-,+,+,+)$, Hermitian generators, $D_\mu=\partial_\mu-igA_\mu$).

The Standard Model is a gauge theory with three groups side by side, and the PDG review defines it as “the renormalizable field theory with gauge group $G_{\mathrm{SM}}=\mathrm{SU}(3)_C\times\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y$” Known ($C$ is colour, $L$ is left-handed, $Y$ is hypercharge). Each factor carries a Yang–Mills field, and the number of fields is $8+3+1=12$: the gluons $G^a_\mu$, $W^i_\mu$ and $B_\mu$.

The field strengths of $G$ and $W$ have commutator terms, in the form of (7) of 3-02. $B_{\mu\nu}=\partial_\mu B_\nu-\partial_\nu B_\mu$, like the electromagnetic field, has no self-interaction. The action on matter is collected into one covariant derivative.

$T^a$ acts only on colour triplets and $\sigma^i/2$ (1-04) only on doublets; on singlets the term is dropped. Since the group is not simple, there is one coupling constant for each factor, three in all (1-03). Example. $g_s\approx1.22$ is obtained from the strength of the strong coupling $\alpha_s=g_s^2/4\pi$ measured at the scale of the Z mass, $\alpha_s(m_Z)=0.1180$ (the $\overline{\mathrm{MS}}$ convention; the note to Table 3 in §06). $g\approx0.65$ and $g'\approx0.35$ are tree-level estimates worked back from the W and Z masses in Table 3 of §06 Computed.

Quarks and leptons are spin-1/2 fields. We do not use what is inside the components (the Dirac matrices), only how they turn under the groups. What is needed is that such a field splits into a left-handed part $\psi_L$ and a right-handed part $\psi_R$ (for zero mass, the way the spin turns relative to the direction of motion), and that SU(2) acts only on the left-handed part Known. Spatial reflection $\mathbf x\mapsto-\mathbf x$ (parity) exchanges left-handed and right-handed, so a force acting only on the left-handed part is not invariant under this reflection. This is called parity violation.

Computed$Q$ was computed as exact fractions. Known(93.1) of the PDG review of grand unification writes the right-handed fields as their conjugates $u^c,d^c,e^c$ (§03; colour becomes $\bar{\mathbf3}$ and the sign of $Y$ flips), and attaches $Y$ as a subscript, normalised to twice its value. For example, the $u_R$ of Table 1 ($(\mathbf3,\mathbf1)$, $Y=2/3$) is the $(\bar{\mathbf3},\mathbf1)_{-4/3}$ of $u^c$. The Higgs is $(\mathbf1,\mathbf2)_1$ in the PDG and corresponds to the $Y=1/2$ of Table 1. Converted, they agree with Table 1.

One generation has 15 components counting colour, and there are three generations (PDG) Known. The gauge principle does not decide which field goes into which representation (3-01), and the PDG too states that the “apparently rather complex group-theoretic data” of Table 1 are not explained.

First we prepare one word. Taking the complex conjugate of each component of a representation $\psi\mapsto\Omega\psi$ gives $\psi^*\mapsto\Omega^*\psi^*$, which is again a representation; we call it the conjugate representation. The conjugate of the triplet is written $\bar{\mathbf3}$ (the antitriplet), and in SU(3) it is not equivalent to $\mathbf3$. In U(1), $e^{iY\alpha}$ becomes $e^{-iY\alpha}$, so the sign of $Y$ flips Known. For the SU(2) doublet $\sigma^2\Omega^*\sigma^2=\Omega$ ($\sigma^2$ is the second Pauli matrix, not a square), so the conjugate is a representation equivalent to the original and is not distinguished Computed.

The $Y$ of Table 1 is strongly constrained. Even when the classical action is gauge invariant, invariance under transformations that turn left and right separately can break at the stage of quantisation; this is called a gauge anomaly. The abstract of Gross–Jackiw (1972) states that it “may destroy this gauge invariance in perturbation theory” Known. We do not treat the derivation, and use only the fact that the condition becomes arithmetic. Taking the conjugate of a spin-1/2 field turns right-handed into left-handed, so aligning the right-handed fields to left-handed by conjugation (the sign of $Y$ flips, and $u^c$ becomes $-2/3$), the condition is that all the following sums are 0. The brackets on the right show which combination of groups each anomaly condition belongs to.

The sums are weighted by the dimension of the representations of the groups not in the combination (in the sum over doublets $Q_L$ is multiplied by the number of colours, 3; in the sum over triplets and antitriplets $Q_L$ is multiplied by 2). $\sum Y^3$ and $\sum Y$ run over the 15 components counting colour and doublet components, and the latter is the condition for the combination with gravity, $\text{gravity}^2\,\mathrm U(1)$. The abstract of Bouchiat, Iliopoulos and Meyer (1972) states that in models with leptons and quarks “the troublesome anomalies cancel” Known. Example. Fixing the charges and changing only the number of colours $N_c$ gives Table 2.

ComputedAdded as exact fractions. For general $N_c$, the $\mathrm{SU}(2)^2\mathrm U(1)$ column is $N_c/6-1/2$ and the $\mathrm U(1)^3$ column is $(3-N_c)/4$. The $\mathrm{SU}(3)^3$ condition has no column in the table; it is $\#\mathbf3-\#\bar{\mathbf3}=2-2=0$ for $N_c=3$ (the two $Q_L$ and the two $u^c,d^c$). KnownSU(2) has another condition too: the abstract of Witten (1982) states that an SU(2) gauge theory with an odd number of left-handed doublets (and no other representations) is mathematically inconsistent. One generation has $3+1=4$ doublets, an even number.

For $\mathrm U(1)^3$, the quarks contribute $-3/4$ and the leptons $+3/4$; they cancel only when a whole generation is present, and only when there are 3 colours Computed. In pure Yang–Mills, with no spin-1/2 fields, this condition does not arise.

Detour: it fits into SU(5)

The SU(3) part is quantum chromodynamics (QCD). The PDG calls QCD “the SU(3) component of the SU(3)×SU(2)×U(1) Standard Model”, and writes that quarks come in three colours in the fundamental representation and gluons are of 8 kinds in the adjoint representation Known.

The sum runs over the 6 kinds of quark ($u,d,s,c,b,t$), $\gamma^\mu$ is a $4\times4$ matrix acting on spin, and $\bar\psi=\psi^\dagger\gamma^0$. It is the form of (9.1) of the PDG with the sign of the covariant derivative matched to 3-02, and the last term is pure Yang–Mills. The $\gamma^\mu$ term is left in the PDG's convention (metric $(+,-,-,-)$), and its content is not used in this chapter. The mass term $m_q\bar\psi\psi$ multiplies left and right; colour turns left and right the same way, so it can be written, but SU(2) turns only the left-handed part, so $\bar u_Lu_R$ is not invariant, and in the Standard Model mass enters through the Higgs field (§07) Known. Colour singlets can be made from two properties of SU(3).

The former is the form of mesons, the latter of baryons (proton, neutron). Example. Under $\Omega=\mathrm{diag}(e^{i\alpha},e^{i\beta},e^{-i(\alpha+\beta)})$ the phase of $\psi^1\psi^2\psi^3$ does not change. The elements of the centre of 1-04, $\mathbb Z_3$, also give 1 when taken three times. Jaffe–Witten state confinement as the physical particle states being “SU(3)-invariant” Known (for the mechanism, 7-02).

A mass term breaks gauge invariance (3-02 §08), yet the W and Z are heavy particles. What bridged this gap was the idea of placing the minimum of the potential of a charged scalar field away from $\phi=0$. We look at it in a U(1) model.

Let $\lambda\gt0$. If $\mu^2\lt0$, the minima line up on the circle $|\phi|=v/\sqrt2$ ($v=\sqrt{-\mu^2/\lambda}$), and the gauge transformation $\phi\mapsto e^{i\alpha(x)}\phi$ moves along this circle. Turning the phase at each point and writing $\phi=(v+h)/\sqrt2$ (real),

follows. The $h=0$ part is exactly the mass term forbidden in (14) of 3-02, and the masses are as follows Known. The action stays invariant, and the solution (the position of the minimum) brings in the mass.

ComputedThe numbers in this figure and table were computed on this machine (by (9)). In the $\mu^2\ge0$ rows the scalar mass is that of the two real components of $\phi$ (both $\sqrt{\mu^2}$); in the $\mu^2\lt0$ rows it is that of the radial $h$.

Degrees of freedom and the Goldstone boson

Example. We count the degrees of freedom. A massless vector field has 2 polarisations, because of its four components, one is removed by gauge transformations and one by the constraint (2-03 §04). With a mass term the one removable by gauge transformations is gone and only the constraint remains, so $4-1=3$ Known. Then before, there are the massless $A$ (2) and $\phi$ (2 real components); after, the massive $A$ (3) and $h$ (1); both make 4. The phase that disappeared has become the longitudinal oscillation of $A$.

A massless excitation that appears when a continuous symmetry is spontaneously broken is called a Goldstone boson Known. The motion along the valley in Figure 1 corresponds to it, and when there is a gauge field, this direction can be removed by a gauge transformation. Higgs (1964) wrote “the longitudinal degrees of freedom of these particles (which would be absent if their mass were zero) go over into the Goldstone bosons when the coupling tends to zero”, and called it the “relativistic analog” of Anderson's (1963) remark on the plasmon in superconductors. In the same year Englert–Brout and Guralnik–Hagen–Kibble also published papers Known. Higgs also cautioned that “nothing is proved about the quantized theory”.

"Spontaneous symmetry breaking" is a way of speaking in a gauge-fixed description. Elitzur (1975) argued that in lattice gauge theory without gauge fixing, local symmetries are not spontaneously broken Known (3-01 §07).

In the Standard Model the same thing is done with SU(2)×U(1) and the Higgs doublet $\Phi$ of Table 1. What leaves one of the minima of $V=\lambda(\Phi^\dagger\Phi-v^2/2)^2$, $\Phi_0=(0,\,v/\sqrt2)$, unchanged is the U(1) of the charge $Q$, which is $T_3+Y=-\tfrac12+\tfrac12=0$ on the lower component. In Jaffe–Witten's words, the structure group is reduced “to a U(1) subgroup (diagonally embedded in SU(2) × U(1))” Known. Letting (2) act on $\Phi_0$,

follows. $W^3$ and $B$ appear only in the combination $gW^3-g'B$, so we rotate the plane $(W^3,B)$ by the weak mixing angle $\theta_W$ ($\tan\theta_W=g'/g$).

$W^\pm=(W^1\mp iW^2)/\sqrt2$. This is the same as (10.3)(10.5) of the PDG, and the PDG cautions that these are masses “at tree level, i.e., to lowest order in perturbation theory” Known. ComputedBuilding the mass matrix of $4\times4$ from Pauli matrices, the eigenvalues agreed with (12) for three choices of $(g,g',v)$ (relative differences below $10^{-15}$, of the order of rounding error).

ComputedThe numbers in this figure and table were computed on this machine (by (12)). For $g'/g=0$ the photon is $B$ itself. 0.536 is the value from the measured values of Table 3.

Example. Working (12) back from the PDG 2024 values gives Table 3.

ComputedIn the PDG, the on-shell $\sin^2\theta_W=0.22348$ and $1/\alpha$ is 137.036 at low energy, and 127.930 at $m_Z$ ($\overline{\mathrm{MS}}$). On-shell and $\overline{\mathrm{MS}}$ are names for ways of including quantum corrections (renormalisation conventions), and quantities with the same name differ slightly in value (4-04). The $\lambda$ of Table 3 is the value in the $V=\lambda(\Phi^\dagger\Phi-v^2/2)^2$ convention of the text. The PDG's $\lambda$ becomes $m_H/v=0.5085$ with the normalisation $m_H=\lambda v$ ((10.5a)), and the two are related by $\lambda=\lambda_{\mathrm{PDG}}^2/2$. $m_W$ is the average excluding CDF (2022).

The 132.1 for $1/\alpha$ is 3.6 % smaller than the low-energy 137.036. The $v$ of Table 3 was taken from $G_F$ (a constant fixed by the muon lifetime), so the quantum correction $\Delta r$ linking $G_F$ and $\alpha$ is not included. Squaring (10.22b) of the PDG electroweak review gives $m_W^2\sin^2\theta_W\,(1-\Delta r)=\pi\alpha/(\sqrt2\,G_F)$ ($\sin^2\theta_W$ is on-shell), and the PDG gives $\Delta r=0.03685$ Known. The tree-level $1/\alpha$ corresponds to $(1-\Delta r)/\alpha$, and $137.036\times(1-0.03685)=131.99$ agrees with the 132.1 of Table 3 to 0.09 % Computed (quantum corrections in 4-04). The 127.930 at $\overline{\mathrm{MS}}$ is a quantity with a different convention, listed only for reference. The degrees of freedom are 12 both before and after (from the 8 of $W^i,B$ and the 4 of the doublet, to the 9 of the heavy $W^\pm,Z$, the 2 of the photon and the 1 of H) Computed.

Weinberg (1967) wrote that the masses of the W and Z come “not from a mass term put in at the beginning”, and in 2012 ATLAS and CMS reported the observation of the Higgs particle Known.

What $v$ gives directly is the masses of the W and Z, and the masses of quarks and leptons through the Yukawa couplings Known. A Yukawa coupling is a term $\bar\psi_L\Phi\psi_R$ that is a product of the Higgs field and spin-1/2 fields; for up-type quarks $\tilde\Phi=i\sigma^2\Phi^*$ is used (a doublet of $Y=-\tfrac12$; the conjugate of §03). In every term the sum of $Y$ is 0 (for example $\bar Q_L\Phi d_R$ is $-\tfrac16+\tfrac12-\tfrac13=0$), so it is gauge invariant Computed. Setting $\Phi$ to $\Phi_0$ gives a mass term proportional to $\bar\psi_L\psi_R$. But this hardly explains the mass of the proton. The sum of the masses of the core u, u, d is about 1% of the proton's.

ComputedValues from the PDG 2024 Summary Tables ($m_u,m_d$ is $\overline{\mathrm{MS}}$, 2 GeV). Quark masses depend on the definition, so (13) is a guide to orders of magnitude. The rest is thought to come from the dynamics of QCD, whose scale is the $\Lambda$ produced by quantisation (4-07) Physics.

The pion is light, about 15% of the proton ($m_{\pi^\pm}=139.57$ MeV) Computed. According to the PDG, in the limit where the masses of the three light quarks are set to 0, the flavour symmetry that turns left and right separately (chiral symmetry) breaks, and “eight massless Goldstone bosons, the π’s, K’s, and η” appear Physics. That QCD is thought to have no gap. This is the “chiral symmetry breaking” that Jaffe–Witten list as the third property of QCD Known. Pure Yang–Mills has no quarks, and its lightest states are thought to be glueballs (5-08) Physics. The gap $\Delta$ of the problem statement is a scale that should arise from the Yang–Mills field alone, relying on neither quarks nor the Higgs field (4-11).

The problem statement concerns pure Yang–Mills theory for any compact simple group (0-03). Table 4 shows what to remove from the Standard Model to get there.

There are three differences. The group: it is not simple. The abelian U(1)$_Y$ part is not asymptotically free (the property that the coupling weakens at short distances) (Coleman–Gross, 4-06 §04) Known, and at one loop the coupling grows at short distances Physics. Massless particles: the photon remains, so the Standard Model as a whole has no gap, and a gap can be asked about only for the strong-force part. Where mass comes from: in the Higgs mechanism the scale $\mu^2$ is in the classical action and visible from the solution, but the classical action of pure Yang–Mills has no scale, and the gap is thought to arise from quantisation Physics. Jaffe–Witten too, in §2, give the Higgs mechanism as an example of “classical properties”, and the mass gap and confinement as examples of “quantum properties that do not have classical analogs” Known.

What was left out remains in extension problems. Jaffe–Witten list “other four-dimensional gauge theories (incorporating additional fields that preserve asymptotic freedom)” (0-03), and at the time of writing the problem description they call the two-dimensional abelian Higgs model of Brydges–Fröhlich–Seiler “the only complete example of an interacting gauge theory satisfying the axioms” Known. A special limit of the four-dimensional Higgs model is in 6-10, and a picture of where the groups come from is in 7-07.

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