§5.8The Evolution of α

The number 1/137 has been on every page of this chapter, and it is wrong. Not mismeasured — wrong in kind. There is no such number. What there is instead is a function, and 1/137 is its value in one particular limit: the value you get when you probe an electron from very far away.

Look closer and you get a different answer. At the mass of the Z the same constant is 1/128.9, and that is not a correction to 1/137 — it is a different point on the same curve. This section is about why the curve exists, what fixes its shape, and how you measure it.

Where the infinities went

Chapter 5 has been quietly dishonest. Every loop diagram in §5.5 — the vertex correction, the self-energy, the photon bubble — integrates over the momentum of the virtual particle running around the loop, and that momentum is unbounded. The integrals diverge. Taken at face value, the second term of the perturbation series is infinite, which is not a small problem for a theory whose entire method is a perturbation series.

Renormalization renormalization absorbing the divergent self-interaction into the definitions of the mass and the charge, on the grounds that the bare quantities are not observable. Bethe's insight of 1947, and what makes QED calculable at all. defined in §5.2-5.3 — open in glossary is the resolution, and its logic is worth stating carefully because it is usually presented as a trick.

Screening, and the one way the analogy breaks

Put a negative charge in a dielectric (Fig. 5.29). The molecules are dipoles; they rotate so their positive ends face the charge. A probe brought in from outside sees the central charge plus the polarized shell, and the shell partly cancels it. That is what a dielectric constant is: the factor by which the medium hides a charge from the outside world.

Now delete the medium (Fig. 5.30). Nothing changes. The vacuum is not empty — it is full of virtual $e^+e^-$ pairs appearing, living for $\Delta t \sim \hbar/\Delta E$, and recombining. A real negative charge orients them exactly as it orients molecules: positrons a little closer, electrons a little farther. This is vacuum polarization vacuum polarization a photon momentarily becoming an e⁺e⁻ pair that re-annihilates. Around a real charge the pairs orient and screen it, so the charge measured from far away is smaller than the bare one — the mechanism behind the running of α. defined in §5.8 — open in glossary , and it screens.

The formula, and the census that sets its slope

Suppose for a moment that the electron were the only charged fermion in the world. Then only $e^+e^-$ pairs can fluctuate, and the result of summing the bubble chain is Eq. (5.49):

Two features to notice before anything else. First, there is a $\mu$ in there — a reference scale — and it is unavoidable. Renormalization theory fixes the shape of the function completely and fixes its overall scale not at all. One number must come from an experiment; after that, every other value is predicted. Second, the dependence is on $|Q|^2$, the absolute value. The sign of $Q^2$ tells you whether you are in the s channel or the t channel; it does not enter the running running coupling the dependence of a coupling on the momentum transfer at which it is measured, α(Q²) = α/[1 − (z_f/3π)α ln(Q²/μ²)]. Not a correction to a constant: the constant does not exist, and which value you get depends on how closely you look. defined in §5.8 — open in glossary .

The world does have more than one charged fermion, and each pair that is light enough to be excited contributes in proportion to the square of its charge. That gives Eq. (5.50), which is the same formula with a counter in it:

The rule for who counts is in the text just after (5.50): in practice, the fermions with $m < Q/2$. It is a sharp-cutoff cartoon of a smooth turn-on, and it is good enough everywhere except at the very bottom of the range — an important exception, and one this page comes back to.

import numpy as np

# Every charged fermion, with its contribution to z_f: the square of its charge,
# times 3 for a quark because each colour is a separate thing to make.
FERMIONS = [
    ("e",   0.000511, 1.0),          ("u", 0.00216, 3 * (2/3)**2),
    ("d",   0.00467,  3 * (1/3)**2), ("s", 0.0934,  3 * (1/3)**2),
    ("mu",  0.10566,  1.0),          ("c", 1.27,    3 * (2/3)**2),
    ("tau", 1.77686,  1.0),          ("b", 4.18,    3 * (1/3)**2),
    ("t",   172.69,   3 * (2/3)**2),
]

def z_f(Q):                       # a pair is resolvable once Q > 2m
    return sum(z for _, m, z in FERMIONS if Q > 2 * m)

print(" Q (GeV)   z_f     what is light enough to fluctuate")
for Q in (0.05, 0.5, 3.0, 50.0, 400.0):
    names = " ".join(n for n, m, _ in FERMIONS if Q > 2 * m)
    print(f"{Q:8.2f}  {z_f(Q):5.3f}   {names}")

print()
print("the two values the book prints:")
print(f"  10 < Q < 100 GeV : z_f = {z_f(50.0):.4f}   book: 6.67")
print(f"  above the top    : z_f = {z_f(400.0):.4f}   book: 8")
print()
print("  read it as 3 leptons + 3 colours x (2 x 4/9 + 3 x 1/9)")
print(f"           = 3 + 3 x {2*4/9 + 3*1/9:.4f} = {3 + 3*(2*4/9 + 3*1/9):.4f}")

 Q (GeV)   z_f     what is light enough to fluctuate
  0.05  2.667   e u d
  0.50  4.000   e u d s mu
  3.00  5.333   e u d s mu c
 50.00  6.667   e u d s mu c tau b
400.00  8.000   e u d s mu c tau b t

the two values the book prints:
10 < Q < 100 GeV : z_f = 6.6667   book: 6.67
above the top    : z_f = 8.0000   book: 8

read it as 3 leptons + 3 colours x (2 x 4/9 + 3 x 1/9)
         = 3 + 3 x 1.2222 = 6.6667

That reproduces both numbers the book prints, and it also shows what $z_f$ z_f (active-fermion sum) the sum of squared fermion charges light enough to be excited at the scale considered, in units of the elementary charge and counting three colours per quark: 6.67 between 10 and 100 GeV, 8 above the top. Every threshold crossed puts a kink in the running. defined in §5.8 — open in glossary really is: not a property of the electromagnetic interaction, but a headcount of the universe at a given resolution. The 3 in “3 colours” is doing real work here — this same factor of 3 is what §6.4 will measure directly.

The one line worth memorizing

Invert (5.50) and the mess collapses. Equation (5.52):

$1/\alpha$ is a straight line in $\ln Q^2$, with slope $-z_f/3\pi$. That is the whole content of the running, and it is why every plot of a coupling constant in this book and every other has a logarithmic horizontal axis: on that axis the prediction is a ruler-straight line, and the kinks in it are the thresholds.

import numpy as np
INV0, MZ, MEAS, ERR = 137.035999166, 91.1876, 128.936, 0.046

def run_to_MZ(fermions):
    """integrate d(1/a)/dlnQ^2 = -z_f/3pi from Q -> 0 up to the Z"""
    x = np.linspace(np.log(1e-12), np.log(MZ**2), 200001)
    z = np.array([sum(zz for m, zz in fermions if np.exp(u/2) > 2*m) for u in x])
    return INV0 - np.trapezoid(z, x) / (3 * np.pi)

LEP = [(0.000511, 1.0), (0.10566, 1.0), (1.77686, 1.0)]
QRK = [(0.00216, 4/3), (0.00467, 1/3), (0.0934, 1/3), (1.27, 4/3), (4.18, 1/3)]

naive = run_to_MZ(LEP + QRK)
print(f"quark masses as printed in the PDG : 1/alpha(M_Z) = {naive:7.3f}")
print(f"measured (5.54)                    : 1/alpha(M_Z) = {MEAS:7.3f} +- {ERR}")
print(f"                                     off by {naive-MEAS:+.2f}, which is "
      f"{abs(naive-MEAS)/ERR:.0f} sigma")
print()
print("There is no Q at which 'a u-quark pair' becomes producible.  The lightest")
print("thing the vacuum can actually make out of u and d is a PAIR OF PIONS, so")
print("switch u, d, s on at m_pi instead of at their current masses:")
mpi = 0.1396
fixed = run_to_MZ(LEP + [(mpi, 4/3), (mpi, 1/3), (mpi, 1/3), (1.27, 4/3), (4.18, 1/3)])
print(f"                                     1/alpha(M_Z) = {fixed:7.3f} "
      f"({fixed-MEAS:+.2f})")
print()
print("Done honestly, the low-Q part is not calculated at all -- it is measured.")
lep_exact = sum(np.log(MZ**2 / m**2) - 5/3 for m, _ in LEP) / (3 * np.pi) / INV0
had, top = 0.02766, -0.00007          # from the R(s) data of Sec. 5.7, and the top
print(f"  leptons, exact one loop      d(alpha) = {lep_exact:.6f}")
print(f"  hadrons, from measured R(s)  d(alpha) = {had:.6f}")
print(f"  top quark                    d(alpha) = {top:.6f}")
tot = lep_exact + had + top
print(f"  total                        d(alpha) = {tot:.6f}")
print(f"  1/alpha(M_Z) = (1 - d) / alpha(0) = {INV0 * (1 - tot):.3f}   vs {MEAS} measured")
print("  the residual 0.013 is the two-loop leptonic term, left out above")

quark masses as printed in the PDG : 1/alpha(M_Z) = 127.554
measured (5.54)                    : 1/alpha(M_Z) = 128.936 +- 0.046
                                   off by -1.38, which is 30 sigma

There is no Q at which 'a u-quark pair' becomes producible.  The lightest
thing the vacuum can actually make out of u and d is a PAIR OF PIONS, so
switch u, d, s on at m_pi instead of at their current masses:
                                   1/alpha(M_Z) = 129.002 (+0.07)

Done honestly, the low-Q part is not calculated at all -- it is measured.
leptons, exact one loop      d(alpha) = 0.031421
hadrons, from measured R(s)  d(alpha) = 0.027660
top quark                    d(alpha) = -0.000070
total                        d(alpha) = 0.059011
1/alpha(M_Z) = (1 - d) / alpha(0) = 128.949   vs 128.936 measured
the residual 0.013 is the two-loop leptonic term, left out above

Measuring it, twice, from opposite sides of zero

The coupling cannot be observed directly — no experiment returns “α”. What is measured is a cross-section or a lifetime, and α is extracted by comparing it to a QED calculation. The book uses two processes, chosen because they sit on opposite sides of $Q^2 = 0$.

Time-like: $Q^2 = s > 0$

Annihilate and watch what comes out: $e^+e^- \to f^+f^-$. Here $Q^2 = s$ is the collider energy squared, so the machine’s energy is the probe scale, and scanning energy scans the curve directly.

Fig. 5.32 is the result: $1/\alpha$ from PEP, DORIS, PETRA, TOPAZ, TRISTAN and OPAL, plotted against $Q$ up to 200 GeV, with the horizontal line at 137 for comparison. The points sit below it and fall.

Space-like: $Q^2 = t < 0$

To get to negative $Q^2$ you need an exchange, not an annihilation, and the cleanest one available at an $e^+e^-$ collider is elastic scattering off itself: Bhabha scattering bhabha scattering elastic e⁺e⁻ → e⁺e⁻. Because the initial and final states are identical it gets both an s-channel and a t-channel diagram, and the t channel is what gives access to Q² < 0: |Q|² = −t = (s/2)(1 − cosθ), so the scattering angle is the knob that sets the probe scale. Used at LEP to map α(t) over four decades of Q², and used at every e⁺e⁻ collider to measure the luminosity. defined in §5.8 — open in glossary , $e^+e^- \to e^+e^-$.

Because the initial and final states are identical, Bhabha gets both an s-channel and a t-channel diagram (Fig. 5.34) — unlike $e^+e^- \to \mu^+\mu^-$ in §5.7, which had only the s channel. Here the t channel is the one you want, so you work far from the Z peak where the s channel would dominate, and subtract what is left of it.

The useful feature is Eq. (5.56):

The scattering angle is the probe scale. One beam energy, and the detector delivers a whole range of $Q^2$ at once, sorted by where the electron landed.

import numpy as np

# Eq. (5.56): a t-channel process reaches |Q|^2 = -t = (s/2)(1 - cos theta).
# Forward scattering transfers nothing; only backward scattering probes deep.
rs = 198.0
s = rs**2
print(f"Bhabha scattering at sqrt(s) = {rs:.0f} GeV,  s = {s:.0f} GeV^2")
print("  theta   |cos|    |Q|^2 (GeV^2)")
for th in (5, 25.8, 45, 60, 90, 180):
    q2 = s / 2 * (1 - np.cos(np.radians(th)))
    print(f"  {th:5.1f}   {abs(np.cos(np.radians(th))):.3f}   {q2:9.0f}")
print(f"  the kinematic maximum at 90 deg is s/2 = {s/2:.0f} GeV^2")
print(f"  and reaching the quoted 21600 GeV^2 at 90 deg needs "
      f"sqrt(s) = {np.sqrt(2*21600):.0f} GeV")
print()

# The Landau pole: Eq. (5.51)'s denominator has a zero.
INV_Z, MZ, zf = 128.936, 91.1876, 8.0
L = MZ * np.exp(3 * np.pi * INV_Z / (2 * zf))
print(f"Landau pole: set 1 - (z_f/3pi) alpha(M_Z) ln(Q^2/M_Z^2) = 0 with z_f = {zf:.0f}")
print(f"  Lambda_EM = M_Z exp(3 pi / (2 z_f alpha)) = {L:.2e} GeV   book: 1e35")
print(f"  for scale: the Planck mass is 1.22e19 GeV, so this is "
      f"{L/1.22e19:.0e} times higher")
print(f"  and the corresponding distance is {197.327e-18/L:.1e} m "
      "(the proton is 1e-15 m)")

Bhabha scattering at sqrt(s) = 198 GeV,  s = 39204 GeV^2
theta   |cos|    |Q|^2 (GeV^2)
  5.0   0.996          75
 25.8   0.900        1954
 45.0   0.707        5741
 60.0   0.500        9801
 90.0   0.000       19602
180.0   1.000       39204
the kinematic maximum at 90 deg is s/2 = 19602 GeV^2
and reaching the quoted 21600 GeV^2 at 90 deg needs sqrt(s) = 208 GeV

Landau pole: set 1 - (z_f/3pi) alpha(M_Z) ln(Q^2/M_Z^2) = 0 with z_f = 8
Lambda_EM = M_Z exp(3 pi / (2 z_f alpha)) = 8.80e+34 GeV   book: 1e35
for scale: the Planck mass is 1.22e19 GeV, so this is 7e+15 times higher
and the corresponding distance is 2.2e-51 m (the proton is 1e-15 m)

Comparing the measured differential cross-section with the one computed at fixed α gives Eq. (5.57):

The square is because the amplitude carries one factor of α (two vertices, $\sqrt\alpha$ each) and the cross-section is its square. In Fig. 5.35(a) the dotted curve — fixed α — is visibly below the data at large $|\cos\theta|$, which is to say at large $|Q|^2$; the solid curve, with α running, goes through them.

The pitfall at the end

Look again at Eq. (5.50). It is a fraction, and its denominator $1 - (z_f/3\pi)\,\alpha\ln(Q^2/\mu^2)$ decreases as $Q$ grows. Push $Q$ far enough and it reaches zero. There α is infinite.

This is the Landau pole landau pole the energy Λ_EM ≈ 10³⁵ GeV at which the running α diverges, from setting the denominator of the running formula to zero. Far above the Planck scale and of no practical consequence, but it shows QED is not logically complete on its own. defined in §5.8 — open in glossary , found by Landau in 1955. Solving for it with $z_f = 8$ and the LEP anchor gives Eq. (5.58): $\Lambda_{\text{EM}} \approx 10^{35}$ GeV — the snippet above returns $8.8 \times 10^{34}$, agreeing with the book’s one-significant-figure statement.

Three honest remarks about it:

• It has no practical consequence. $10^{35}$ GeV is sixteen orders of magnitude above the Planck scale, at a distance of $10^{-51}$ m. No experiment will ever go there, and long before you did, gravity would have to be part of the theory.
• It is not an artefact of the approximation. The obvious objection is that (5.50) is only the leading term of a series, so of course it misbehaves when extrapolated absurdly far. But including higher orders moves the pole; it does not remove it.
• It is not repaired by unification either. Chapter 9 shows that the electromagnetic and weak interactions merge at around 100 GeV, which changes the evolution of α above that scale. The divergence moves higher. It still exists.

The conclusion is uncomfortable and worth stating plainly: QED, on its own, is not a logically complete theory. It is an extraordinarily accurate description of a range of energies, and it announces its own incompleteness at the top of that range. That is a different situation from a theory being wrong, and it is the normal condition of physics — but it is unusual for a theory to compute the scale of its own failure as precisely as this one does.

Where this goes next

• §5.9 is where $\alpha^{-1}(0) = 137.035999166$ (5.53) comes from: not from a scattering experiment at all, but from the electron’s magnetic moment measured to 0.13 parts per trillion and the SM relation between the two.
• §6.4 runs $\alpha_s$ with the same machinery and the opposite sign. The widget on this page takes a theory="qcd" prop for exactly that.
• §9.1 unifies the electromagnetic and weak couplings, which is a statement about two of these lines crossing.
• §10.5 extrapolates all three couplings towards a grand unification scale — the most consequential thing anyone has ever done with a straight line on a log plot.