§9.1–9.3The Electroweak Interaction, and What Unification Means

Chapter 7 measured the weak interaction and Chapter 8 found the phase hidden in it. Neither asked what the weak interaction is. The answer, and it is the most consequential sentence in the book, is that it is not a separate interaction.

This section builds the claim in three steps: name the symmetry (§9.1), look at what the neutral current actually does (§9.2), and then watch a single rotation angle turn four fields into the photon, the $Z$ and the two $W$s while collapsing 21 independent coupling constants into two (§9.3).

§9.1 Four fields, and who carries which charge

Write $\mathbf{W} = (W_1, W_2, W_3)$ for the SU(2) triplet and $B$ for the U(1) singlet. None of these four is a particle you can detect. The physical mediators are combinations:

The $\mathbf{W}$ triplet has $I_W = 1$ and so carries the charge it couples to — exactly like the gluon in §6.3, and exactly unlike the photon. That single fact will produce the $W\!W\!Z$ vertex of §9.10 and the running of $\sin^2\theta_W$ in §9.8. The $B$ field, by contrast, is a singlet: uncharged under its own group, like the photon.

Hypercharge is defined so that the electric charge comes out right:

The full assignment, for one family, is:

Table 9.1, made live

Everything above collapses into one table of Z-charge factors Z-charge factor c_Z = I_Wz − Q sin²θ_W, the coefficient with which a given fermion chirality couples to the Z⁰ boson. The electroweak analogue of ch06's colour factors, and the reason the 21 neutral-current couplings of three families collapse to two constants (ch09 §9.3). defined in §9.1-9.3 — open in glossary , and it is the reference the rest of the chapter runs on. It is already built — Appendix 3 prints the same data — so here it is with the angle on a slider:

Three things to look for, none of which the printed table can show you.

Q = I_Wz + Y_W/2 holds in all 28 rows. Not approximately — exactly, by construction, because that is what (9.1) says. It is worth checking two or three by hand; the pattern is what makes the assignment non-arbitrary.

Set the slider to zero. Every right-handed $c_Z$ goes to zero and the left-handed ones become $\pm 1/2$. That limit is a theory with no mixing: the neutral current would be pure weak isospin, V−A like the charged current, and blind to right-chirality fermions. The real world is not there, and the distance from it is $\sin^2\theta_W$.

The neutrino’s $c_Z = 1/2$ never moves. It has $Q = 0$, so the $-Q\sin^2\theta_W$ term vanishes identically. Neutrinos are the one place the neutral current is clean, which is why §9.4 measures $\theta_W$ with a neutrino beam.

§9.2 What the neutral current does, and what it refuses to do

Two properties, and the book states both flatly because the experiments of §7.13 established them.

It is flavour-diagonal. The $Z$ couples a fermion to itself — $\bar e e$, never $\bar e\mu$. For quarks the same colour, too: $\bar u_B u_B$, never $\bar u_R u_B$, because the $Z$ carries no colour. This is §7.10’s GIM statement arriving as a property of the gauge group rather than a cancellation between diagrams.

It is not V−A. This is the sharp break with everything Chapter 7 established. The charged current is $\bar\psi_L\gamma^\mu\psi_L$ and nothing else. The neutral current has both chiralities, with different strengths — so for each family there are seven currents (six charged fermions L and R, plus the left neutrino), and with three families, 21 independent weak charges.

That number is the problem the next section solves. A theory needing 21 free constants to describe one interaction is not a theory; it is a table.

The seven currents of the first family, drawn:

The book also states four rules for what the $Z$ talks to, and they follow directly from $c_Z = I_{Wz} - Q\sin^2\theta_W$ being the whole story:

• it couples to both chiralities — because $Q$ is chirality-blind;
• it couples to the $W^\pm$ — because they carry weak isospin;
• it couples to neutral particles with $I_{Wz}\neq 0$, i.e. neutrinos;
• and it couples to nothing with $Q = 0$ and $I_{Wz} = 0$ — which includes the photon and the $Z$ itself.

§9.3 Unification, in one rotation

The history is short and the book tells it well. Glashow (1961) proposed SU(2)⊗U(1) and wrote the photon as a superposition of the two neutral fields, with the coefficients as a sine and a cosine because their squares must sum to 1. Salam and Ward reached the same place in 1964, and the angle they introduced is the weak mixing angle weak mixing angle θ_W, the rotation taking the unmixed fields (W₃, B) to the physical (Z, γ), defined by tanθ_W = g′/g. sin²θ_W ≈ 0.232. Also called the Weinberg angle. One number fixes every neutral-current coupling in the Standard Model, which is why measuring it in a dozen unrelated ways is a real test. defined in §9.1-9.3 — open in glossary .

But a fatal problem remained: the photon is massless and the weak mediators cannot be, and no gauge theory of the time could have both. The missing piece — spontaneous breaking of a gauge symmetry — appeared the same year, and Weinberg (1967) and Salam (1968) put the two together. ‘t Hooft and Veltman proved the result renormalizable in 1971, which is when people started believing it.

The charged fields are the combinations that carry charge:

and the neutral pair is a rotation:

The Lagrangian makes the same point in one line. Grouping the interaction terms around the physical fields:

The two constraints, and what is left free

Requiring the photon term to be the electric charge gives (9.20), and the same demand applied through the other route gives (9.21):

Two equations, three unknowns — so exactly one free parameter survives, and it can be taken to be $\theta_W$. Squaring and adding the reciprocals gives a relation with no angle in it at all:

import numpy as np
alpha, GF, s2 = 1/137.035999, 1.1663788e-5, 0.232
s, c = np.sqrt(s2), np.sqrt(1 - s2)
e = np.sqrt(4*np.pi*alpha); g, gp = e/s, e/c

print(f"inputs: alpha = 1/137.036, G_F = {GF:.5e} GeV^-2, sin^2(theta_W) = {s2}")
print("\nthe angle and the two couplings")
print(f"  theta_W               = {np.degrees(np.arctan2(s, c)):.1f} deg        book: \"not small, about 29 deg\"")
print(f"  e = sqrt(4 pi alpha)  = {e:.4f}")
print(f"  g  = e / sin(theta_W) = {g:.4f}")
print(f"  g' = e / cos(theta_W) = {gp:.4f}")
print(f"  g'/g = {gp/g:.4f} = tan(theta_W)            consistency check on (9.17)")
print(f"\n  ** g > e:  the WEAK coupling is {g/e:.2f}x the electromagnetic one **")

print("\nthe two shares of 1/alpha, Eq. (9.23)")
print(f"  4 pi / g'^2 = cos^2/alpha = {4*np.pi/gp**2:5.1f}       book: 105.2")
print(f"  4 pi / g^2  = sin^2/alpha = {4*np.pi/g**2:5.1f}       book:  31.8")
print(f"  sum                       = {4*np.pi/g**2 + 4*np.pi/gp**2:5.1f}       = 1/alpha, as (9.22) demands")

A = np.sqrt(np.pi*alpha/(np.sqrt(2)*GF))
MW, MZ = A/s, A/s/c
print("\nthe mass predictions, Eqs. (9.27) and (9.28)")
print(f"  M_W = {A:.2f} GeV / sin(theta_W) = {MW:.1f} GeV")
print(f"  M_Z = M_W / cos(theta_W)       = {MZ:.1f} GeV")
print( "  book quotes \"approximately 80 and 91\"")
mw, mz = 80.377, 91.1875
print(f"\n  measured:  M_W = {mw:.3f},  M_Z = {mz:.3f}")
print(f"  the tree-level values are LOW by {(1-MW/mw)*100:.1f}% and {(1-MZ/mz)*100:.1f}% -- not rounding.")
print( "  radiative corrections (Sec. 9.8, 9.11) supply the rest, and the")
print( "  top-quark loop is most of it.")
print("\nrunning the prediction backwards from the measured masses")
print(f"  sin^2(theta_W) = 1 - (M_W/M_Z)^2 = {1-(mw/mz)**2:.4f}")
print( "  vs the 0.232 assumed above: a 4% difference in the ANGLE is what")
print( "  the 4% difference in the masses is telling you.")

inputs: alpha = 1/137.036, G_F = 1.16638e-05 GeV^-2, sin^2(theta_W) = 0.232

the angle and the two couplings
theta_W               = 28.8 deg        book: "not small, about 29 deg"
e = sqrt(4 pi alpha)  = 0.3028
g  = e / sin(theta_W) = 0.6287
g' = e / cos(theta_W) = 0.3455
g'/g = 0.5496 = tan(theta_W)            consistency check on (9.17)

** g > e:  the WEAK coupling is 2.08x the electromagnetic one **

the two shares of 1/alpha, Eq. (9.23)
4 pi / g'^2 = cos^2/alpha = 105.2       book: 105.2
4 pi / g^2  = sin^2/alpha =  31.8       book:  31.8
sum                       = 137.0       = 1/alpha, as (9.22) demands

the mass predictions, Eqs. (9.27) and (9.28)
M_W = 37.28 GeV / sin(theta_W) = 77.4 GeV
M_Z = M_W / cos(theta_W)       = 88.3 GeV
book quotes "approximately 80 and 91"

measured:  M_W = 80.377,  M_Z = 91.188
the tree-level values are LOW by 3.7% and 3.1% -- not rounding.
radiative corrections (Sec. 9.8, 9.11) supply the rest, and the
top-quark loop is most of it.

running the prediction backwards from the measured masses
sin^2(theta_W) = 1 - (M_W/M_Z)^2 = 0.2231
vs the 0.232 assumed above: a 4% difference in the ANGLE is what
the 4% difference in the masses is telling you.

The prediction that could have killed the theory

$G_F$ has been known since the 1930s and $\theta_W$ is measurable at low energy. Eq. (7.55) relates $g$ to both — so the $W$ mass is not a free parameter:

And the ratio of the two masses is the angle itself, with the couplings gone:

What the Standard Model contains, and what it does not

The book closes §9.3 with a list, and it is a fair summary of the rest of the chapter:

And what it does not contain, stated here because it is easy to lose in 84 pages: the theory predicts no masses. Not $M_W$ on its own (it needs $G_F$ and the angle as inputs), not $M_H$, and not a single fermion mass — each of those is a Yukawa coupling put in by hand. What the theory predicts is relations among them. That is a weaker claim than “explains mass”, and it is the reason §9.20 ends with a list of open questions rather than a full stop.

Where this goes next

§9.4 measures $\theta_W$, and does it the hard way: the CHARM2 experiment counts neutrino–electron elastic scatterings, a process four orders of magnitude rarer than neutrino–nucleus scattering, precisely because being purely leptonic it carries no hadronic uncertainty. The ratio of $\nu$ to $\bar\nu$ cross-sections depends on the angle alone.

Then §9.5 turns the machinery on itself: with the masses predicted and $c_Z$ known for every fermion, every partial width of the $W$ and $Z$ follows from one number. §9.7 is 1983, when they were found where the theory said.