The Fermi Theory · Elementary Particle Physics

Fermi’s theory predicts the muon’s lifetime from a single measured number and, in the same breath, tells you it cannot be the final answer. Both facts come from that one constant.

🎯 Why this matters

A theory can be quantitatively right and structurally doomed at the same time, and that is the ordinary case rather than a curiosity. The useful question is never whether it is true but where it stops.

Two interactions down. Electromagnetism and QCD share a shape: a massless mediator, a dimensionless coupling, parity conserved, flavour untouched. The weak interaction has none of those properties, and this chapter is a sequence of things it fails to respect.

It starts, in 1933, with the Fermi theory — a theory that is wrong in a spectacularly useful way.

Beta decay, and a desperate hypothesis

Every process in this section is a charged current — one that changes the charge of the fermion passing through it.

A nucleus $(A, Z)$ turns into $(A, Z+1)$ and emits an electron. If that were the whole story it would be a two-body decay and the electron would be monoenergetic. It is not — the spectrum is continuous, and in 1930 Pauli proposed an invisible neutral particle to carry off the balance, calling it a “desperate hypothesis”. At the level the book now works at:

n \to p + e^- + \bar\nu_e \qquad\text{and underneath it}\qquad d \to u + e^- + \bar\nu_e

The mirror process, $\beta^+$ , was found within weeks of the positron’s discovery in 1933:

p \to n + e^+ + \nu_e \qquad u \to d + e^+ + \nu_e

which cannot happen for a free proton — the proton is lighter than the neutron — but proceeds happily inside a nucleus, where the surrounding binding energy pays the difference.

Fermi’s move was to treat the electron–neutrino pair the way a photon is treated in atomic decay: not present beforehand, but created at the moment of the transition. That required a quantized field for the electron and one for the neutrino, at a time when it was not obvious nucleons even obeyed the Dirac equation. From the bilinear covariants of §2.9 he chose the simplest — the vector, by analogy with the electromagnetic current — and wrote the matrix element as a product of two currents.

\mathcal{M} = \frac{\htmlClass{t-G}{G_F}}{\sqrt2}\; \underbrace{\left(\htmlClass{t-h}{\bar\psi_p\gamma^\mu\psi_n}\right)}_{\text{hadronic}} \underbrace{\left(\htmlClass{t-l}{\bar\psi_e\gamma_\mu\psi_\nu}\right)}_{\text{leptonic}}

(7.4)

Bettini p. 275. Two currents, meeting at a single point — which is the assumption that will turn out to be an approximation.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

Fig. 7.1 — neutron beta decay, as Fermi drew it

Click a vertex or an internal line.

Compare every diagram in Chapters 5 and 6: each had an internal line carrying a propagator. This one has none. A four-fermion contact term is what you get when the mediator is too heavy to resolve — and the price is a coupling with units.

Three kinds of weak process

class	examples	what it is used for
leptonic — no hadrons at all	μ⁻ → e⁻ ν̄ₑ ν_μ · ν_μ e⁻ → μ⁻ νₑ	Measuring G_F itself. With no hadrons there is no strong-interaction physics to factor out, so the calculation is exact — which is why the muon lifetime is the definition.
semileptonic — hadrons and leptons	n → p e⁻ ν̄ₑ · π⁻ → μ⁻ ν̄_μ · ν_μ n → μ⁻ p	Almost everything else in this chapter. The hadronic side needs a form factor or a decay constant, which is where the lattice of §6.10 comes in.
non-leptonic — hadrons only	Λ⁰ → p π⁻, i.e. s → u ū d	The hardest to calculate and the easiest to recognise: you spot them by their long lifetimes and by flavour not being conserved.

The recognition rule is worth stating plainly, because it is how every weak process in Chapters 2–4 was identified before any of this theory existed: <strong>a long lifetime, or a change of flavour, means weak</strong>. A strong decay takes 10⁻²³ s and conserves every flavour quantum number; an electromagnetic one takes 10⁻¹⁶–10⁻¹⁹ s and also conserves them. Nothing else changes strangeness.

The constant that has units

⚙️ Engineer’s bridge — a dimensionful coupling is a theory telling you where it breaks

α = 1/137 is a pure number and says nothing about scale. α_s is a pure number too (its value runs, but the quantity is dimensionless). $G_F$ is the Fermi constant , $1.17\times10^{-5}$ GeV⁻², and a coupling with dimensions cannot be the whole story — because you can always ask “what sets that unit?”

That is what makes it an effective theory . Take the question seriously and it answers itself:

\frac{1}{\sqrt{\htmlClass{t-g}{G_F}}} = \htmlClass{t-e}{293\ \text{GeV}}

Supplied — the book gives G_F and its units but never takes the square root. Fermi's 1933 constant contains the electroweak scale, and nothing but dimensional analysis is needed to read it out.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

In 1933, Fermi’s own constant contained the electroweak scale. The W turned up at 80 GeV and the Higgs vacuum expectation value is 246 GeV. The number was sitting in the theory for fifty years before anyone had a machine that could reach it.

This is a habit worth having. In engineering the same reasoning is routine and usually goes unremarked: a model whose coefficient has units is parameterised by something you have not modelled. An empirical drag coefficient with units of length is telling you there is a length scale in the flow. A time constant in a fitted response is telling you there is a pole you have not written down. Dimensionful phenomenology is always a message about missing structure, and the message includes the scale.

The theory says the same thing a second way, from a completely different direction. Its cross-section rises linearly with $s$ forever, and a probability cannot exceed one. Imposing unitarity gives $\sqrt s \approx 734$ GeV. Two arguments, one from dimensions and one from probability, both landing in the few-hundred-GeV range — and both right.

Where it breaks: the scale a dimensionful coupling names is not the mass of anything. $1/\sqrt{G_F}$ = 293 GeV is neither the W (80 GeV) nor the Higgs vev (246 GeV), because $G_F = g^2/(4\sqrt2 M_W^2)$ hides a coupling inside the mass — so the number is off by whatever $g$ is, and it happens to land between the two things it was pointing at.

Read it as “there is structure somewhere below this” and it is reliable; read it as “there is a particle here” and it is not. Newton’s constant makes the same point unkindly: $1/\sqrt{G_N}$ is $10^{19}$ GeV and nobody expects to find a particle there. The message is that the model is effective, and the scale is an upper bound on where you can still trust it — never an address.

what G_F knows that Fermi did not

import numpy as np
GF, MW = 1.1663788e-5, 80.377          # GeV^-2, GeV
mmu, me = 0.1056583755, 0.51099895e-3

print(f"G_F = {GF:.7e} GeV^-2  -- dimensionful, unlike alpha and alpha_s")
print(f"the scale it names:  1/sqrt(G_F) = {1/np.sqrt(GF):.0f} GeV")
print(f"the unitarity bound: sqrt(2 pi/G_F) = {np.sqrt(2*np.pi/GF):.0f} GeV")
print("  the W is at 80 GeV and the Higgs vev at 246. Fermi's constant contained")
print("  the electroweak scale from 1933; nobody could reach it until 1983.")
print()
print("and where the dimensions actually come from, G_F/sqrt2 = g^2/(8 M_W^2):")
g2 = 8*MW**2*GF/np.sqrt(2)
print(f"  g^2 = {g2:.4f},  g = {np.sqrt(g2):.4f}")
print(f"  alpha_W = g^2/4pi = {g2/(4*np.pi):.4f} = 1/{4*np.pi/g2:.1f}")
print(f"  against alpha = 1/137.  The weak coupling is {(g2/(4*np.pi))*137.036:.1f} times")
print("  STRONGER than the electromagnetic one.")
print()
print("  The weak interaction is not weak.  It is heavy: the 1/M_W^2 in the")
print("  propagator is doing all of the suppression, and at energies above M_W")
print("  it stops -- which is why the Z peak of Sec. 9.7 is a peak at all.")
print()
print("Eq. (7.13): tau(mu) = 192 pi^3/(G_F^2 m_mu^5)")
tau = 192*np.pi**3/(GF**2*mmu**5) * 6.582119569e-25
print(f"  = {tau*1e6:.3f} us  against a measured 2.197 us")
print("  the 0.5% gap is the epsilon of Eq. (7.13), the electron-mass and")
print("  radiative corrections, which are calculable exactly.")

prints

G_F = 1.1663788e-05 GeV^-2  -- dimensionful, unlike alpha and alpha_s
the scale it names:  1/sqrt(G_F) = 293 GeV
the unitarity bound: sqrt(2 pi/G_F) = 734 GeV
the W is at 80 GeV and the Higgs vev at 246. Fermi's constant contained
the electroweak scale from 1933; nobody could reach it until 1983.

and where the dimensions actually come from, G_F/sqrt2 = g^2/(8 M_W^2):
g^2 = 0.4263,  g = 0.6529
alpha_W = g^2/4pi = 0.0339 = 1/29.5
against alpha = 1/137.  The weak coupling is 4.6 times
STRONGER than the electromagnetic one.

The weak interaction is not weak.  It is heavy: the 1/M_W^2 in the
propagator is doing all of the suppression, and at energies above M_W
it stops -- which is why the Z peak of Sec. 9.7 is a peak at all.

Eq. (7.13): tau(mu) = 192 pi^3/(G_F^2 m_mu^5)
= 2.187 us  against a measured 2.197 us
the 0.5% gap is the epsilon of Eq. (7.13), the electron-mass and
radiative corrections, which are calculable exactly.

💡 What this really says — why the fifth power of the mass

Eq. (7.13) has the muon’s lifetime going as $1/m_\mu^5$ , and that exponent is not arbitrary — it is forced by dimensions, exactly like the coupling.

A decay rate has units of energy. $G_F^2$ carries $E^{-4}$ . The only other scale in a muon decay is the muon mass itself, so the rate must be $G_F^2 m^5$ and there is nothing else it could be.

The consequence is severe and worth carrying: weak lifetimes are extraordinarily sensitive to the mass of the decaying particle. The τ is 17 times heavier than the muon, so $17^5 \approx 1.4$ million — and the τ lifetime is indeed about a million times shorter. It is also why a free neutron lives 15 minutes while a muon lives 2 μs: the neutron’s decay releases only 0.78 MeV, and it is that released energy, not the neutron mass, that plays the role of $m$ here.

Where the theory announces its own failure

Fermi theory: σ = G_F²s/π, rising for ever
with the W propagator, 1/(M_W² − t)
unitarity limit ≈ 4π/s — a probability cannot exceed 1

The Fermi cross-section climbs until it crosses the unitarity bound at √s ≈ 734 GeV, which is impossible — so the theory must be wrong before then. It is: the contact vertex is really two vertices with a W between them, and once √s reaches M_W the propagator stops the growth. Note that the two curves are indistinguishable below about 10 GeV, which is why the point-like theory survived for fifty years.

⚠️ Two symbols in this chapter that mean something else elsewhere

g now has a fifth meaning. Here it is the weak charge at a W vertex, a dimensionless coupling analogous to $\sqrt\alpha$ and $\sqrt{\alpha_s}$ . Already in this book g has been the metric tensor (§5.1), the gyromagnetic ratio (§5.9a), the strong charge $g_s$ (§6.3) and the gluon parton distribution (§6.2). The rule set in §5.9a stands: g means whatever the current page says it means, and the page must say.

V is about to collide with itself. In §7.3 V is the vector bilinear covariant, as in “V−A”. From §7.11 onwards V is the CKM matrix. Both appear in this chapter and the book does not flag it. On this site the covariants are always written V and A in the phrase “V−A”, and the matrix always carries indices, $V_{ud}$ , $V_{us}$ , or is called “the CKM matrix” in words.

🔢 Worked example — the muon lifetime, from one constant

Everything in Eq. (7.13) is known except the answer, so the muon’s lifetime is a prediction rather than a fit. With $G_F = 1.1663788\times10^{-5}$ GeV⁻² and $m_\mu = 0.1056584$ GeV:

\begin{aligned} \tau_\mu &= \frac{192\pi^3}{\htmlClass{t-g}{G_F^2}\, \htmlClass{t-m}{m_\mu^5}} = \frac{192\times31.006}{(1.1664\times10^{-5})^2\times(0.10566)^5}\;\htmlClass{t-u}{\text{GeV}^{-1}}\\[2pt] &= 3.323\times10^{18}\;\htmlClass{t-u}{\text{GeV}^{-1}} \times \htmlClass{t-u}{6.582\times10^{-25}\;\text{GeV s}} = \htmlClass{t-r}{\mathbf{2.187\;\mu s}} \end{aligned}

(7.13)

The muon lifetime, predicted rather than fitted: everything on the right was measured in other experiments. The 0.5 % gap against 2.197 μs is the calculable ε of Eq. (7.13) — electron-mass and radiative corrections.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

against a measured 2.197 μs. The 0.5 % gap is the $\varepsilon$ of Eq. (7.13) — the electron-mass and radiative corrections — and it is calculable exactly. Note the unit conversion: the formula gives an inverse energy, and multiplying by $\hbar = 6.582\times10^{-25}$ GeV s turns it into a time. Skip that step and the answer is off by twenty-four orders of magnitude.

🔑 If you remember only three things

Three kinds of weak process, one vertex. Leptonic, semileptonic and hadronic differ in what sits at the ends, not in what happens in the middle.
A coupling with units is a coupling with a scale attached. Neither QED nor QCD carries one, which is why neither ever announces a limit of its own.
One number does both jobs. It fixes a lifetime to three digits and marks the energy at which the prediction has to fail.

Where this goes next

§7.2–7.3 replaces $\gamma^\mu$ with $\gamma^\mu(1-\gamma^5)$ and nothing else — and that one change is parity violation.
§7.8 turns the contact vertex back into a W exchange and shows that $G_F/\sqrt2 = g^2/8M_W^2$ , which is where the dimensions came from.
§9.3 explains why $M_W$ has the value it does, and the 293 GeV above turns out to be within a factor of two of the Higgs vev.
§5.7 is the dimensional argument that worked, and worth rereading: there σ ∝ 1/s because the only scale was √s. Here there are two scales, and the argument gives the opposite answer.

✅ Check yourself — the Fermi theory

0/5 answered · 0 correct

1.G_F carries dimensions of GeV⁻², where α and α_s are pure numbers. What does that tell you before any calculation?
2.Is the weak interaction weak?
3.The muon lifetime goes as 1/m_μ⁵. Where does the fifth power come from?
4.The Fermi cross-section σ = G_F²s/π rises linearly with s for ever. Why is that fatal, and what fixes it?
5.Why is G_F measured from the muon lifetime rather than from neutron beta decay, which is far easier to observe?