A comparison is only as good as its weaker side, and for the electron that side is now the theory: it needs a value of α that this experiment cannot supply.
🎯 Why this matters
So the most precise test in physics has quietly turned into a measurement. Assume QED and the electron’s anomaly gives α better than anything else can; test QED and α has to come from elsewhere. You cannot have both at once.§5.9a delivered a measurement good to 0.13 parts per trillion. A measurement is worth exactly its comparison, so this page is the other half: what it takes to calculate a number to thirteen digits, and what happens when you do the same exercise for the muon, where the answer does not quite come out right.
The series
The anomaly magnetic anomaly a = (g − 2)/2, the departure of a lepton's gyromagnetic ratio from the Dirac value 2. Schwinger's α/2π ≈ 0.00116 was QED's first prediction; a_e is now known to 0.13 ppt, the most precisely measured quantity in physics. defined in §5.9a — open in glossary is a power series in , Eq. (5.68):
Five QED terms and three that are not QED. The interesting part of the equation is the last three.
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.
The are of order unity, which is the reason the expansion parameter is the only thing setting the convergence rate — and the reason the diagram count is the real cost.
| order | coefficient | diagrams | size of the term | what it buys |
|---|---|---|---|---|
| α¹ | C₂ = 1/2 | 1 | 1.2 × 10⁻³ | Schwinger, 1948 — matched the 1947 measurement on its own |
| α² | C₄ | 7 | 5.4 × 10⁻⁶ | needed by the early 1950s |
| α³ | C₆ | 72 | 1.3 × 10⁻⁸ | still analytic — C₆ is known in closed form |
| α⁴ | C₈ | 891 | 2.9 × 10⁻¹¹ | numerical integration only, from here on |
| α⁵ | C₁₀ | 12 672 | 6.8 × 10⁻¹⁴ | the first term smaller than the 2023 error bar — so it is the last one needed, and it is the one that took decades |
The book prints the counts 72, 891 and 12 672. The sizes are (α/π)ⁿ with the C's of order one; the α² row's count of 7 is the standard result and is given here for continuity. Notice what the last row means: a single number in the fourteenth decimal place of a physical constant cost the evaluation of twelve thousand six hundred and seventy-two integrals.
⚙️ Engineer’s bridge — a series whose term count explodes, and why you still truncate
This is the shape of every perturbative expansion an engineer ever meets, and it has the same two properties.
Truncation is decided by the error bar, not by taste. You stop at the first term smaller than the precision you need — here , because the measurement is good to . Compute one term further and you have wasted a decade; one fewer and the comparison is meaningless. That is the same discipline as choosing a filter order from the required stopband attenuation, or a Taylor expansion’s degree from the ULP of the format you are targeting.
The work per term grows combinatorially, not linearly. 1, 7, 72, 891, 12 672 — roughly a factor of 12 per order. This is not an implementation detail, it is the structure of the problem: the number of ways to wire vertices grows like a factorial. Any engineer who has watched a search space blow up in the branching factor recognises it, and the response is the same one: automate the enumeration and evaluate numerically, which is precisely what happened here from onwards.
The QED series is also, strictly, asymptotic rather than convergent — it eventually diverges, at around order . Nobody cares, because that is a hundred and thirty orders past where anyone works. Asymptotic series that are useless in the limit and superb in practice are the norm in numerical work, not an exception.
Where it breaks: truncating at the first term below your error bar assumes the terms keep shrinking, which is exactly what an asymptotic series eventually stops doing. Here the turnaround is at order 137 and nobody will ever reach it, so the rule is safe — but it is safe by an accident of the coupling’s size, not because the criterion is sound. In a series with a larger expansion parameter the same procedure silently adds error instead of removing it, and there is nothing in the truncation rule itself that warns you.
where each order lands against each era's error bar
import numpy as np
ALPHA = 1 / 137.035999166
x = ALPHA / np.pi
print(f"expansion parameter alpha/pi = {x:.4e}")
print()
print(" order size diagrams smaller than the 1947 bar? the 2023 bar?")
counts = {1: "1", 2: "7", 3: "72", 4: "891", 5: "12 672"}
for n in range(1, 6):
s = x**n
print(f" a^{n} {s:.2e} {counts[n]:>8}"
f"{' yes' if s < 5e-5 else ' no':>10}"
f"{' yes' if s < 1.3e-13 else ' no'}")
print()
print("so 1947 needed one term and 2023 needs five. The measurement improved by")
print(f"a factor {5e-5/1.3e-13:.1e}, and each order of the series is worth {1/x:.0f}x,")
print(f"which takes {np.log(5e-5/1.3e-13)/np.log(1/x):.1f} further orders beyond the first,")
print("so five terms in total -- which is exactly where the calculation stops.")expansion parameter alpha/pi = 2.3228e-03 order size diagrams smaller than the 1947 bar? the 2023 bar? a^1 2.32e-03 1 no no a^2 5.40e-06 7 yes no a^3 1.25e-08 72 yes no a^4 2.91e-11 891 yes no a^5 6.76e-14 12 672 yes yes so 1947 needed one term and 2023 needs five. The measurement improved by a factor 3.8e+08, and each order of the series is worth 431x, which takes 3.3 further orders beyond the first, so five terms in total -- which is exactly where the calculation stops.
The three terms that are not QED
Fig. 5.40 — four ways to correct a vertex, only one of which is calculable to any precision you like
Click a vertex or an internal line.
The book's Fig. 5.40 shows four separate diagrams — (a) an eighth-order self-energy, (b) a virtual muon loop, (c) a virtual Z, (d) hadrons. They differ only in what is put inside the bubble, so the site draws one diagram and names the four fillings on the loop. Click it.
⚠️ One of these terms is not like the others
Look at the three non-QED contributions as fractional uncertainties rather than absolute ones:
- — known to 5 parts in . Its only input is the measured lepton mass ratios, and those are excellent.
- — known to 8 parts in , and small enough that this hardly matters.
- hadronic vacuum polarization the contribution to a lepton's magnetic anomaly from a quark pair inside the loop. It cannot be computed perturbatively, because the loop lives at energies where α_s is large; it must come either from a dispersion integral over measured e⁺e⁻ → hadrons data or from lattice QCD. For the muon it is 95% of the theoretical error bar, and the two methods of computing it disagree. defined in §5.9b — open in glossary — known to 7 parts in , and not small. Its uncertainty alone, , is ninety times the entire experimental error bar on .
The reason is the one §5.8 already met: the hadronic loop involves QCD at low energy, where the coupling is large and perturbation theory does not work. There is no series to truncate. The value has to come from measured data or from a lattice computation.
Hold on to the ratio, because for the muon it gets four orders of magnitude worse, and it is the entire story of the second half of this page.
The electron: theory meets measurement
Summing everything, Aoyama et al. (2018) give Eq. (5.69):
The theoretical electron anomaly, Aoyama et al. (2018), with its three uncertainties kept separate rather than combined — because which one dominates is the point.
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.
Three uncertainties: QED, hadronic, and — much the largest — the uncertainty on itself, which has to come from a different experiment. That third number, 0.720, is 55 times the other two combined, and it is why the section ends by running the comparison backwards: assume the theory, and the electron anomaly becomes the best measurement of there is. That is where §5.8’s came from.
theory against experiment, and what the agreement excludes
import numpy as np
# Eq. (5.69), theory, and Eq. (5.67), experiment -- both as a_e in units of 1e-12
th, th_err = 1159652182.032, 0.720 # the 0.720 is the uncertainty on alpha
exp = 1.00115965218059 - 1 # g/2 - 1
exp *= 1e12
print(f"theory a_e = {th:.3f} +- {th_err:.3f} (x 1e-12)")
print(f"experiment a_e = {exp:.3f} +- 0.00013")
d = th - exp
print(f"difference = {d:.3f} -> {d/th_err:.1f} sigma")
print()
# What a discrepancy of that size excludes: if the electron had a size R, its
# anomaly would shift by roughly (R / lambdabar_C)^2, so
hbar_c = 197.3269804e-9 # eV m
m_e = 0.51099895000e6 # eV
lam = hbar_c / m_e # the reduced Compton wavelength
R = lam * np.sqrt(th_err * 1e-12)
print(f"reduced Compton wavelength hbar/m_e c = {lam:.4e} m")
print(f"electron radius limit R < lambdabar sqrt(delta a_e) = {R:.2e} m")
print(f" the book prints 3.2e-19 m")
print(f" for scale, a proton is 8.4e-16 m across -- {8.4e-16/R:.0f} times bigger")
print()
# Question 5.2: epsilon_0 from alpha
e_, hbar, c = 1.602176634e-19, 1.054571817e-34, 299792458 # all exact by definition
inv_alpha, inv_alpha_err = 137.035999166, 0.000000015
eps0 = e_**2 / (4 * np.pi * hbar * c) * inv_alpha
print(f"Question 5.2: eps_0 = e^2 / (4 pi alpha hbar c) = {eps0:.10e} F/m")
print(f" every input but alpha is exact, so the relative error is alpha's:")
print(f" {inv_alpha_err/inv_alpha:.1e} -> eps_0 = {eps0:.9e} +- {eps0*inv_alpha_err/inv_alpha:.1e} F/m")theory a_e = 1159652182.032 +- 0.720 (x 1e-12) experiment a_e = 1159652180.590 +- 0.00013 difference = 1.442 -> 2.0 sigma reduced Compton wavelength hbar/m_e c = 3.8616e-13 m electron radius limit R < lambdabar sqrt(delta a_e) = 3.28e-19 m the book prints 3.2e-19 m for scale, a proton is 8.4e-16 m across -- 2564 times bigger Question 5.2: eps_0 = e^2 / (4 pi alpha hbar c) = 8.8541878235e-12 F/m every input but alpha is exact, so the relative error is alpha's: 1.1e-10 -> eps_0 = 8.854187824e-12 +- 9.7e-22 F/m
Erratum — “the difference between experiment and theory on g/2 is 0.7 × 10⁻¹²”
That number is the uncertainty, not the difference. The book’s own two equations give
against a quoted uncertainty of — a difference of 2.0 standard deviations, not 0.7 × 10⁻¹². The printed 0.7 × 10⁻¹² is precisely the third uncertainty in (5.69).
This is not a rounding quibble, because the next sentence — “the agreement allows us to establish deviations from the SM” — and the radius limit that follows both rest on the size of the gap. As it happens the radius bound is unaffected: it uses the uncertainty, and m, matching the printed 3.2 × 10⁻¹⁹ m. So the arithmetic downstream is consistent with reading 0.7 × 10⁻¹² as the error bar, which is what it is.
The 2σ itself is real and much discussed: it tracks a genuine disagreement between the two best independent measurements of (caesium and rubidium recoil), and which one you adopt moves the electron anomaly between roughly agreement and 2.5σ. It is a live question, not a settled one.
The muon: the same measurement, 43 000 times more sensitive
Everything about the muon is the same as the electron except two numbers, and both of them matter enormously.
💡 What this really says — why anyone bothers with the muon at all
The electron anomaly is measured about three million times more precisely than the muon’s — 1.3 × 10⁻¹⁶ against 4.1 × 10⁻¹⁰ in absolute terms, which is 0.13 ppt against 0.35 ppm. And yet the muon is where the search for new physics happens. Why?
Two reasons, and the first is the one everybody quotes.
Because a heavy virtual particle of mass contributes to a lepton’s anomaly in proportion to . The lepton mass is in the numerator. So the muon’s sensitivity to anything heavy is larger than the electron’s by
Why the search for new physics uses the muon even though the electron is measured three million times better. The lepton mass sits in the numerator of the sensitivity, and it is squared.
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.
And the second reason, which the book does not give. A sensitivity ratio of 43 000 would still lose to a three-million-fold precision advantage — if that advantage could be used. It cannot. Testing against theory requires a value of from a different experiment, and that input carries (Eq. 5.69) — five million times the experimental error. The muon’s comparison is limited at .
The two anomalies are limited at the same place, to within 1 %. So the 43 000 is not divided by anything: the muon really is about forty thousand times the better probe of a heavy state, and the electron’s spectacular measurement is spent calibrating instead. Fig. 5.41 makes the point graphically: put a hypothetical unknown particle X in the loop and the muon notices it first.
The catch is that the same factor applies to the hadronic contribution, which is the one nobody can calculate. For the electron the hadronic term is a 7-parts-in-1000 uncertainty on a negligible number. For the muon it is about 40 000 times larger, and it becomes the thing standing between a measurement and a discovery.
Erratum — the caption of Fig. 5.41
Fig. 5.41 is captioned “A lowest-order photon–electron vertex diagram including a hypothetical unknown X particle”. Every line in the diagram is labelled μ, and the paragraph it illustrates is about the muon anomaly being 43 000 times more sensitive to a heavy X than the electron’s. It is a photon–muon vertex, and the whole point of the figure depends on that.
The apparatus
Click a numbered marker for what that piece does.
Erratum — three on p. 220, one of them the same slip as Eq. (5.63)
The paragraph recounting the muon anomaly’s history carries three errors, and they are worth separating because only one of them is substantive.
The 1960 value is labelled as the wrong quantity. It is printed as . But is about , not 1.00113 — the quantity that equals 1.00113 is , and it has to be, for the very next clause (“is equal to that of the electron”, whose is 1.00116) to hold. This is exactly the slip in Eq. (5.63) two pages earlier: the anomaly and one-plus-the-anomaly swapped.
Two names. “In 1960, Gawrin et al., using a muon beam produced with the Navis cyclotron” — Garwin et al., at Columbia’s Nevis cyclotron. (The same Garwin who, three years earlier, had co-discovered parity violation in precisely this decay chain, which is what makes the polarization this experiment depends on.)
The Fermilab experiment is misnamed, here and in the caption of Fig. 5.42: it is E989. E821 was the Brookhaven experiment, and the same sentence says so — “the action th[e]n moved to Brookhaven National Laboratory (BNL) from 1989 to 2001 … and finally at FNAL with the E821 experiment” is self-contradictory in one line.
The magic gamma
Muons are unstable, so the experiment is a race: measure as many oscillation periods as possible before the sample is gone. Relativity supplies the extension — at the 2.2 μs lifetime becomes 64 μs in the lab.
But relativity also creates the problem. The electrostatic quadrupoles that keep the beam in place look, in the muon’s own frame, partly like a magnetic field, and is supposed to measure the magnetic field. Equation (5.73):
The muon's anomalous precession in the ring, as printed on p. 220. The first term is the measurement; the second is the contamination that the whole experiment is arranged to switch off.
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.
(that is the book’s printed form — the in the second numerator is a misprint, and the erratum below shows why the book’s own numbers prove it).
Look at the bracket in the second term. It vanishes at one specific beam energy and only there, and that energy defines the magic gamma magic gamma the Lorentz factor γ = 29.3 at which the electric-field term in the muon spin-precession formula cancels to first order, letting a storage ring use electric quadrupole focusing without spoiling the ω_a measurement. defined in §5.9b — open in glossary :
The magic gamma. There is exactly one beam energy at which the electric-field term of Eq. (5.73) has zero coefficient, and the experiment is built to run there and nowhere else.
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.
⚙️ Engineer’s bridge — operate at the null
You cannot remove the electric field: without it the beam falls out of the ring vertically. You cannot measure it well enough either, because the quadrupole field is by construction not uniform, and each muon samples a different bit of it.
So you do neither. You choose the one operating point where the coefficient of the term you cannot control is exactly zero, and the contamination cancels to first order whatever the field’s shape happens to be:
The whole 14-metre ring, the beam energy, the injection line — all of it is built around a number that exists only because a minus sign appeared in Eq. (5.73).
Engineers do this constantly and rarely name it. An oven-controlled crystal oscillator is held at the quartz cut’s turnover temperature, where , so oven ripple stops mattering to first order. A bridge is nulled rather than read. A differential pair is biased where the even-order distortion cancels. In every case the move is the same: when an error term cannot be eliminated or measured, find the operating point where its coefficient vanishes, and build the instrument there.
It is also the most fragile kind of design, and worth noticing why: the null is only first-order, it is only at one energy, and — circularly — it is placed using , the very quantity being measured. That last point is harmless because is already known to six digits and the null only needs to be approximately right; but it is the sort of thing worth checking rather than assuming.
Where it breaks: operating at a null requires the null to stay put. The magic γ is fixed by , which is the quantity being measured — a mild circularity that is harmless only because was already known to six digits beforehand. It also nulls one term at one energy: muons with a momentum spread sit slightly off the null, so the correction returns at second order and has to be computed rather than cancelled. A null is a place, and a real beam occupies a neighbourhood.
Erratum — the γ in the second term of Eq. (5.73)
The book prints the electric-field coefficient of Eq. (5.73) as . It should be , and the book’s own next sentence is the proof.
Two lines later it states for . Solve for the magic value both ways:
- gives — the printed value.
- gives .
So the equation as printed does not have its cancellation at the the text goes on to use. The Thomas–BMT algebra agrees: subtracting from leaves exactly .
Note that Eq. (5.72) is right — genuinely carries . The was carried down one equation too far.
the magic gamma, and what it buys
import numpy as np
a_mu = 116592061e-11 # Eq. (5.76)
m_mu, m_e = 105.6583755, 0.51099895000 # MeV
tau0, B = 2.1969811e-6, 1.4513 # s, tesla
g = np.sqrt(1 + 1/a_mu)
beta = np.sqrt(1 - 1/g**2)
print(f"magic gamma = sqrt(1 + 1/a_mu) = {g:.2f} the book prints 29.3")
g_bad = (1 + np.sqrt(1 + 4*a_mu**2)) / (2*a_mu) # root of a = gamma/(gamma^2-1)
print(f" had the coefficient really been gamma/(gamma^2-1), the magic value")
print(f" would be {g_bad:.0f}, not 29.3 -- which is the erratum above, in one line")
print(f" muon momentum gamma beta m = {g*beta*m_mu/1000:.3f} GeV/c")
print(f" lifetime, dilated = {g*tau0*1e6:.1f} us (at rest: {tau0*1e6:.2f} us)")
e_, m_kg = 1.602176634e-19, 1.883531627e-28
nu_a = a_mu * e_ * B / (2*np.pi*m_kg)
print(f" anomaly frequency at {B} T = {nu_a/1e3:.1f} kHz -> period {1e6/nu_a:.2f} us")
print(f" oscillations per dilated life = {g*tau0*nu_a:.1f}")
print()
print("without the time dilation there would be 14.7/29.3 = "
f"{tau0*nu_a:.2f} periods -- i.e. no measurement at all.")
print()
print(f"why the muon and not the electron: (m_mu/m_e)^2 = {(m_mu/m_e)**2:.0f}")
print(f" the electron is measured {0.35e-6/0.13e-12:.0f}x more precisely, but the muon is")
print(f" {(m_mu/m_e)**2:.0f}x more sensitive to anything heavy -- so the muon still wins by")
print(f" a factor {(m_mu/m_e)**2/(0.35e-6/0.13e-12):.0f}")magic gamma = sqrt(1 + 1/a_mu) = 29.30 the book prints 29.3 had the coefficient really been gamma/(gamma^2-1), the magic value would be 858, not 29.3 -- which is the erratum above, in one line muon momentum gamma beta m = 3.094 GeV/c lifetime, dilated = 64.4 us (at rest: 2.20 us) anomaly frequency at 1.4513 T = 229.1 kHz -> period 4.37 us oscillations per dilated life = 14.7 without the time dilation there would be 14.7/29.3 = 0.50 periods -- i.e. no measurement at all. why the muon and not the electron: (m_mu/m_e)^2 = 42753 the electron is measured 2692308x more precisely, but the muon is 42753x more sensitive to anything heavy -- so the muon still wins by a factor 0
What the detectors actually record
Eq. (5.71) — N(t) = N₀ e^(−t/γτ) (1 + A cos ω_a t)
- pure decay, e^(−t/τ)
- what the calorimeters count
The frequency is read from the ripple, not from the decay — so what limits the measurement is periods per lifetime, the third box. Drop the asymmetry towards zero and the ripple vanishes into the envelope while the decay is unchanged: a perfectly good decay curve carrying no information at all about ω. That asymmetry is a gift from parity violation, and without it there would be no experiment.
The 24 calorimeters record an arrival time and an energy for every decay positron. That is the entire dataset. The decay is a nuisance — the answer is in the ripple, whose frequency is ω_a and therefore a_μ. Slide γ down towards 1 and watch the measurement disappear: the lifetime shrinks faster than the period, the periods-per-lifetime box collapses, and there is nothing left to fit. That box is why the beam is relativistic, and the magic gamma is why it is relativistic at exactly 29.3.
The asymmetry deserves a sentence of its own, because it is not free. It is the product of the beam polarization and the weak decay asymmetry, and both come from parity violation in the weak interaction (Chapter 7). Pions decaying at rest produce 100% polarized muons; the muons then decay with the highest-energy positrons preferentially along the spin. Without parity violation the muons would be unpolarized, the positrons isotropic, — and Eq. (5.71) would be a featureless exponential carrying no information about whatsoever.
🔬 Experiment card — the muon g−2 ring, Brookhaven and Fermilab
Apparatus
A 14 m storage ring with an exceptionally uniform 1.45 T dipole field, mapped continuously by NMR probes; four electrostatic quadrupoles for vertical focusing; an inflector to get the beam in; 24 segmented calorimeters on the inner wall. The muons are made by firing protons at a target, selecting forward pions, and taking the muons from their decay — which arrive already 100% polarized. Beam momentum 3.094 GeV/c, and not a value chosen for convenience.What is measured
Two things, and their ratio: from the ripple frequency in the positron arrival times, and from the NMR map. Both must reach sub-ppm. Note the structure — it is §5.9a’s principle again: the quantity extracted is the difference frequency , never itself.The result
FNAL’s first result, Eq. (5.75): , 0.46 ppm. Combined with the earlier Brookhaven measurements, Eq. (5.76):What it proved
That the experimental side is not the problem. Two laboratories, three decades, independent systematics, one answer. Whether it disagrees with the Standard Model is now entirely a question about a theoretical number — and specifically about one term in it.The number everything now depends on
The muon’s hadronic vacuum-polarization term is about 40 000 times larger than the electron’s, and it cannot be computed perturbatively — at the energies where it lives, is large and the expansion of Eq. (5.68) has no analogue.
The book’s quoted route is lattice QCD lattice QCD computing QCD by discretizing spacetime onto a finite grid and evaluating the path integral numerically. The only first-principles route to quantities where the coupling is too large for a perturbative series — hadron masses, the LO-HVP term in a_μ, the deconfinement transition. defined in §6.10-6.11 — open in glossary : discretize spacetime, put the fields on the grid, and integrate numerically. The Budapest–Marseille–Wuppertal collaboration reached sub-percent accuracy on the leading term, Eq. (5.77):
which gives the full theoretical prediction, Eq. (5.78):
the comparison, and where the error bars actually come from
import numpy as np
exp, exp_e = 116592061, 41 # Eq. (5.76), x 1e-11
th, th_e = 116591953, 58 # Eq. (5.78), x 1e-11
hvp, hvp_e = 7075, 55 # Eq. (5.77), the lattice LO-HVP
d = exp - th
comb = np.hypot(exp_e, th_e)
print(f"experiment {exp} +- {exp_e} (x 1e-11)")
print(f"theory {th} +- {th_e}")
print(f"difference {d:>9} +- {comb:.0f} -> {d/comb:.1f} sigma"
" the book says 1.5")
print()
print("where the theory error bar comes from:")
print(f" LO-HVP alone contributes +- {hvp_e} ({hvp_e/th_e*100:.0f}% of the total {th_e},")
print(f" and {(hvp_e/th_e)**2*100:.0f}% of its variance). Everything else is noise beside it.")
print()
print("what it would take to call this a discovery:")
for target in (3, 5):
need = d / target
print(f" {target} sigma needs a combined error of {need:.0f}e-11;")
print(f" with the experiment at {exp_e}, theory must reach "
f"{np.sqrt(max(need**2 - exp_e**2, 0)):.0f}e-11"
+ (" -- impossible while the experiment sits at 41" if need < exp_e else ""))
print()
print(f"LO-HVP is {hvp/th*1e6:.0f} ppm of a_mu, and it is known to {hvp_e/hvp*100:.2f}%.")
print("A 0.77% error on a 60 ppm term is a 0.5 ppm error on the answer -- which is")
print("larger than the 0.35 ppm experiment. The lattice is the whole ballgame.")experiment 116592061 +- 41 (x 1e-11) theory 116591953 +- 58 difference 108 +- 71 -> 1.5 sigma the book says 1.5 where the theory error bar comes from: LO-HVP alone contributes +- 55 (95% of the total 58, and 90% of its variance). Everything else is noise beside it. what it would take to call this a discovery: 3 sigma needs a combined error of 36e-11; with the experiment at 41, theory must reach 0e-11 -- impossible while the experiment sits at 41 5 sigma needs a combined error of 22e-11; with the experiment at 41, theory must reach 0e-11 -- impossible while the experiment sits at 41 LO-HVP is 61 ppm of a_mu, and it is known to 0.78%. A 0.77% error on a 60 ppm term is a 0.5 ppm error on the answer -- which is larger than the 0.35 ppm experiment. The lattice is the whole ballgame.
- theory (5.78), 116 591 953 ± 58
- experiment (5.76), 116 592 061 ± 41
⚠️ What “agrees within 1.5 standard deviations” is really saying
The book’s closing sentence is accurate and easy to under-read. Three things are worth being explicit about.
The theory bar is longer than the experimental one — 58 against 41. That has been true since about 2006 and it is the reason the field’s effort has moved from building rings to computing hadronic loops. Fermilab has since published further improvements to the 41; none of them change the picture while the 58 stands.
Essentially all of the 58 is one term. The LO-HVP uncertainty of 55 is 95% of it — 90% of the variance. Every other contribution to a quantity known to twelve digits, summed, is a rounding error next to one number that has to be computed by putting QCD on a grid.
And that number is contested. The book quotes the lattice result (5.77). There is a second, older route to the same quantity — the dispersion integral over measured hadrons data, the same machinery §5.8 needed for — and it gives a smaller LO-HVP, which would make the discrepancy roughly 4σ instead of 1.5σ. The two methods disagree with each other by more than either’s stated uncertainty. Whether there is new physics in the muon anomaly is, as of the book’s writing, a question about which of two calculations of a hadronic loop is right.
That is a genuinely uncomfortable place for the most sensitive test of the Standard Model to be, and it is worth carrying into Chapter 6, where the lattice technique is explained.
| electron | muon | |
|---|---|---|
| measured to | 0.13 ppt | 0.35 ppm — 2.7 million times coarser |
| sensitivity to a heavy X | 1 | (m_μ/m_e)² ≈ 43 000 |
| net reach for new physics | ×1 | ×12 better, despite the worse measurement |
| hadronic term | 1.69 × 10⁻¹², a 0.7% uncertainty on a negligible number | 7075 × 10⁻¹¹, a 0.77% uncertainty that is 95% of the total error bar |
| limiting factor | the independent measurement of α | lattice QCD |
| what it is used for | measuring α — run backwards, it gives (5.53) | searching for physics beyond the SM |
| status | 2.0σ from the book's own numbers (see the erratum above) | 1.5σ with the lattice HVP; ~4σ with the data-driven one |
Two measurements of the same quantity for two particles that differ only in mass — and they have ended up as completely different instruments. The electron became a metrology standard; the muon became a search.
🔑 If you remember only three things
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The two halves of the comparison are built by different communities. One number comes out of a trap and the other out of a decade of diagram counting, and neither side can check the other’s.
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Sensitivity and precision are separate currencies. The muon buys the first and pays in the second, which is why both experiments are still running.
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Every improvement now moves the bottleneck instead of removing it. A better trap and α limits you; a better α and the hadronic terms do.
Where this goes next
- §6.10 is lattice QCD: how you compute a hadronic loop by putting spacetime on a grid, and why sub-percent accuracy on it took until the 2020s.
- §5.8 supplies the other route to the same hadronic integral — the dispersion relation over measured hadrons. The tension between the two methods is the open question of this page.
- §7.2 is parity violation, which supplies the polarization and the decay asymmetry that make Eq. (5.71) have a ripple at all.
- §9.20 is the global fit — the same exercise as this page, run over every measurable in the Standard Model at once.
✅ Check yourself — the anomalies aₑ and a_μ
0/5 answered · 0 correct
1.Why does the calculation stop at (α/π)⁵ rather than at some other order?
2.The electron is measured 2.7 million times more precisely than the muon. Why is the muon still the better place to look for new physics?
3.What problem does the 'magic' Lorentz factor γ = 29.3 solve?
4.In Eq. (5.71), N(t) = N₀e^(−t/τ)(1 + A cos ω_a t), what fixes how well ω_a can be measured?
5.The muon result is quoted as agreeing with theory 'within 1.5 standard deviations'. What is the honest reading of that?