University Quantum Field Theory I · The Lamb Shift Completed at Order α (Z α)⁴ · 15.9
What one loop leaves out, and where the next course begins
Topic 15.08 fell 5.6 MHz short, and this lesson shows the shortfall is a list, not a failure: each missing term has an order in α, Zα, m/M or the proton's size, a sign, and a size you can estimate before looking it up. Then the course ends by naming what the next one builds.
Build the model
Connect the measurement to the mechanism.
Topic 15.08's 1052.19 MHz is the first term of an expansion, and what it misses is the expansion's next terms, each with a size fixed by counting before its coefficient is known. Hydrogen's levels depend on four small numbers: α, from loops of the radiation field and of the electron-positron vacuum; Zα, from exchanges with the nucleus's Coulomb field; m/M, from the nucleus's recoil; and mrₚ, from its size. Every shift is (Zα)⁴m/n³ times powers of these and a coefficient, often carrying a logarithm of Zα.
Counting says where the gap lives. A second loop costs α/π = 0.0023; one more Coulomb exchange inside the same loop costs Zα = 0.0073, and in hydrogen that wins: the largest omission is still one loop, at order α(Zα)⁵, from momentum transfers near m that the small-q expansion of topics 15.02 to 15.04 could not reach. The nucleus enters three separate ways, through its reduced mass, its recoil and its size, and a second loop and a (Zα)² correction to the first finish the ledger within 0.03 MHz of the measurement.
The same entries close the 1S shift. The expansion works only because all four numbers are small, and it stops where Zα is not: in a heavy ion the matching window disappears, the leading term changes sign, and only a calculation exact in Zα survives. That calculation, and the gauge theories beyond QED, belong to the next course.
- Simple definition
- Beyond one loop, hydrogen's Lamb shift is a series in four small numbers, α for loops, Zα for Coulomb exchanges, m/M for recoil and mrₚ for the proton's size, each term being (Zα)⁴m/n³ times powers of them and a computed or quoted coefficient.
- Example
- Hydrogen's 2S₁/₂–2P₁/₂ interval: 1052.19 MHz at order α(Zα)⁴, then +7.14 from α(Zα)⁵, −1.72 from the reduced mass, +0.36 from recoil, +0.14 from the proton's size, +0.10 from two loops and −0.41 from α(Zα)⁶: 1057.80 MHz, against 1057.83 MHz measured.
+7.14 MHz on 2S₁/₂ and nothing on 2P₁/₂: 0.0526 units of C₂ against a gap of 0.0416. The largest omission is still one loop, one Zα further on.
m = 0.511 MeV, an energy; s states only; A₅₀ is 9.291 from the self-energy and 0.327 from vacuum polarisation; ħ = c = 1
|ψ(0)|² and ⟨r⁻³⟩ both scale as mᵣ³: −1.72 MHz on the interval, −13.26 MHz on 1S. It vanishes as M → ∞, Dirac's fixed nucleus.
M = proton mass, m/M = 1/1836.15; ν₄ = 1052.19 MHz for the interval and 8127.44 MHz for 1S, as frequencies ΔE/h
+0.341 MHz on 2S₁/₂ and −0.017 MHz on 2P₁/₂, so +0.36 MHz. There is no α/π: m/M is a quarter of α/π, so recoil rivals the radiative terms.
b(2S) = 6.170, b(2P) = −0.309, b(1S) = 5.445, holding Bethe logarithms and, for s states, ln(1/Zα); mᵣ = mM/(m + M)
+0.138 MHz on 2S₁/₂, 1.02 × 10⁻³ units: the proton's form-factor slope acting as F₁'s did. It is 43 times the 3.2 kHz measurement uncertainty.
rₚ = 0.84 fm, the rms charge radius; mrₚ = 0.511 MeV × 0.84 fm/197.327 MeV fm = 2.18 × 10⁻³; zero for l ≥ 1
+0.127 MHz on 2S₁/₂ and +0.026 MHz on 2P₁/₂: +0.10 MHz on the interval, seventy times less than one more Zα in the first loop.
(α/π)² = 5.40 × 10⁻⁶, so the n = 2 unit is 0.236 MHz; B₄₀ holds the second term of g − 2, −0.328(α/π)²
(Zα)²(−96.8 + 58.4 − 31.2) × 101.7 MHz = −0.38 MHz on 2S₁/₂; polarisation and 2P₁/₂ bring it to −0.41 MHz. The L², 96.8, is why it beats two loops.
L = 9.84 for hydrogen; for 2S, A₆₁ = 5.930 and G = −31.19 (self-energy); all pure numbers, with m an energy
The gap is the next order in Zα, not a second loop
Natural units ħ = c = 1 and the metric (+, −, −, −) hold throughout, and shifts are quoted as ΔE/h. Topic 15.08 left 1052.19 MHz against the measured 1057.83 MHz: a gap of 5.64 MHz, or 0.0416 units of C₂ = 135.64 MHz. The α(Zα)⁴ result used the loop functions at |q| ≪ m (topics 15.02 to 15.04); momentum transfers near m, where an s state's short-distance shape matters, add one more power of Zα. For s states the coefficient is A₅₀ = 9.618, quoted (Karplus, Klein and Schwinger; Baranger, Bethe and Feynman): (α/π)(Zα)⁵mA₅₀/8 = 0.7424 MHz × 9.618 = +7.14 MHz on 2S₁/₂. Its vacuum-polarisation share is derived in a line. Uehling's potential acts within about 193 fm, half the electron's reduced Compton wavelength 1/m = 386 fm, where |ψ|² ≈ |ψ(0)|²(1 − 2Zαmr), the cusp of any ns state; topic 13.08 kept only the 1. Writing Uehling's potential as a sum of Yukawa terms e(−2mur)/r, the −2Zαmr part integrates to (α/π)(Zα)⁵m(4/3)I/n³ with I = ∫ (1 + 1/(2u²))√(u² − 1) u⁻⁵ du from u = 1 to ∞, which is 5π/64: +0.24 MHz. The interval rises to 1059.33 MHz, 1.50 MHz past the measurement.
The nucleus moves: reduced mass and Salpeter's recoil
Every loop so far held the proton fixed. A proton of finite mass M changes two things. The wavefunction carries the reduced mass mᵣ = m/(1 + m/M), and every α(Zα)⁴ entry rides on |ψ(0)|² or ⟨r⁻³⟩, both proportional to mᵣ³. Multiplying 1052.19 MHz by (1 + 1/1836.15)⁻³ = 0.998368 lowers the interval by 1.72 MHz, computed; as M → ∞ the correction vanishes, as it must. The proton also recoils while it exchanges photons with the electron, a two-body effect no external field can imitate. Salpeter found it in 1952 from the relativistic two-body equation that carries his and Bethe's names: (m/M)(Zα)⁵m b/(πn³), with b = 6.170 for 2S and −0.309 for 2P, quoted, raising 2S₁/₂ by 0.341 MHz and lowering 2P₁/₂ by 0.017 MHz, +0.36 MHz in all. This term carries no α: it is pure relativistic recoil, and it competes with the radiative terms of the same power of Zα because m/M = 5.4 × 10⁻⁴ is a quarter of α/π = 2.3 × 10⁻³. Running total: 1057.97 MHz.
The nucleus has a size: a second form-factor slope
The electron feels the proton's charge spread over rₚ = 0.84 fm through the proton's own form factor, GE = 1 − |q|²rₚ²/6 + …, just as topic 15.03 used F₁'s slope. The potential −4πZαGE/|q|² gains (2π/3)Zαrₚ², independent of q: a contact term (2π/3)Zαrₚ²δ³(r). First-order perturbation theory with |ψ(0)|² = (Zαmᵣ)³/(πn³) gives (2/(3n³))(Zα)⁴mᵣ³rₚ² on s states and nothing for l ≥ 1, derived. Dimensions: mass cubed times length squared is a mass when ħ = c = 1; as rₚ → 0 the point nucleus returns. With mrₚ = 2.18 × 10⁻³ it is (π/2)(mrₚ)²/α = 1.02 × 10⁻³ units, +0.14 MHz. Read the joined electron self-energy the same way, through Welton's picture: its 7.49 units are a charge smeared over ⟨r²⟩ = 7.49 × 2α/(πm²) = 5.19 × 10³ fm², about (72 fm)², eighty-six times the proton's radius. Binding has given the electron's cloud the edge that topic 15.02's photon mass could not.
Two loops, one more (Zα)², and the ledger closed
The second loop, quoted: (α/π)²(Zα)⁴mB₄₀/n³, with B₄₀ = 0.539 for s states and 0.109 for 2P₁/₂, gathering the two-loop slope of F₁, two-loop vacuum polarisation and the second term of g − 2. Its n = 2 unit is only 0.236 MHz: +0.127 on 2S₁/₂, +0.026 on 2P₁/₂, +0.10 MHz on the interval. As α → 0 it vanishes faster than every one-loop entry, as a two-loop term should. The first loop at relative order (Zα)² is larger, because its leading piece is −(Zα)²L² with L = ln(1/(Zα)²) = 9.84. With A₆₁ = 5.930 and G = −31.19 for 2S, quoted, it is (Zα)²(−96.8 + 58.4 − 31.2) = −3.71 × 10⁻³ of 101.7 MHz, −0.38 MHz; polarisation adds −0.01 and 2P₁/₂ about +0.02, so −0.41 MHz. Ledger: 1052.19 + 7.14 − 1.72 + 0.36 + 0.14 + 0.10 − 0.41 = 1057.80 MHz, against 1057.8298 ± 0.0032 MHz measured in 2019, quoted. The 0.03 MHz left is the size of what is still omitted, quoted: α²(Zα)⁵, about −0.04 MHz, and finer reduced-mass factors, in the logarithm, on α(Zα)⁵ and on 2P₁/₂'s magnetic term, about +0.07 MHz.
1S closes too, and heavy ions break the series
For 1S the one-loop shift is 8127.44 MHz (topic 15.08). The α(Zα)⁵, size and two-loop entries are contact terms with coefficients common to every s state, so they scale exactly as n⁻³: +57.12, +1.10 and +1.02 MHz. The reduced mass gives −13.26 MHz, computed. Salpeter's +2.41 MHz and the (Zα)² terms' −3.28 MHz are quoted for 1S, since their coefficients depend on n. The total, 8172.55 MHz, sits 0.25 MHz below the 8172.8 MHz inferred from hydrogen's optical frequencies, eight times the n = 2 residue, as omitted contact terms should be. All of it rests on Zα ≪ 1. In hydrogen-like uranium, Zα = 0.671, the binding scale (Zα)²m = 230 keV sits beside m = 511 keV, so topic 15.05's window (Zα)²m ≪ K ≪ m does not exist; the 1S bracket, ln(1/(Zα)²) − ln k₀ + 19/30 = −1.55 units, has turned negative, and the α(Zα)⁵ term, 4.84 units, outweighs it. The measured shift, about 460 eV, quoted, is reached only by calculations exact in Zα (UAQM-13.08).
Where the next course begins
Four subjects this lesson leaned on without building are named, not begun. Bound-state QED: the Furry picture, which puts the exact Coulomb field inside the electron propagator so that heavy ions can be computed to all orders in Zα; the Bethe-Salpeter equation, which Salpeter's recoil needed; and non-relativistic effective field theory, which turns topic 15.07's matching into a systematic expansion in α, Zα and m/M. Path integrals: the same Feynman rules obtained from an integral over field histories, the form in which gauge fixing is done once the gauge group is not abelian. Non-abelian gauge theories: quantum chromodynamics, whose coupling falls at high energy where QED's rises, and whose bound state, the proton, supplied the inputs here that QED cannot compute, its mass and its radius rₚ = 0.84 fm. Spontaneous symmetry breaking: the Higgs mechanism, which gives the W and Z bosons their masses; Z exchange also moves the 2S–2P interval by far less than a hertz. The Wilsonian renormalisation group, reading Unit 6's running as the flow of an effective theory, comes with them.
Change one variable at a time
Make the relationship visible.
Step from 0 to 6: α(Zα)⁵ alone overshoots the solid line by 1.5 MHz, the reduced mass pulls the total under it, and four entries under 0.5 MHz end it 0.03 MHz short. Set recoil to 0 for Dirac's fixed nucleus, or recoil 0.5 and radius 2.13 fm for deuterium's.
TOTAL AFTER THESE ENTRIES1057.80 MHz
MINUS MEASURED 1057.83-0.03 MHz
RECOIL AND SIZE TOGETHER-1.22 MHz
SIZE TERM138 kHz
Live interpretationTOTAL AFTER THESE ENTRIES: 1057.80 MHz. MINUS MEASURED 1057.83: −0.03 MHz. RECOIL AND SIZE TOGETHER: −1.22 MHz. SIZE TERM: 138 kHz
Catch the common trap
Explain before calculating.
Hydrogen's Lamb shift closes to within 0.03 MHz when the terms beyond α(Zα)⁴ are added order by order. Why does the same budget fail for hydrogen-like uranium, U⁹¹⁺, where Zα = 0.671?
Choose an answer to test the model.
Practice & worked examples
Reason from the model, then test the result.
EasyTopic 15.08 left hydrogen's 2S₁/₂–2P₁/₂ interval at 1052.19 MHz against 1057.83 MHz measured. The first omitted term, on s states only, is (α/π)(Zα)⁵mA₅₀/n³ with A₅₀ = 4π(139/128 − ½ ln 2) + 5π/48 (quoted). Using α⁵mc²/h = 2556.83 MHz, find its self-energy and vacuum-polarisation parts on 2S₁/₂, apply the reduced mass for m/M = 1/1836.15, and say how much of the gap is left.
- Unit at n = 2: (α/π)(Zα)⁵m/n³ = 2556.83 MHz × 0.0023228/8 = 0.74238 MHz. Every factor but m is a pure number, so it is an energy, here quoted as a frequency.
- Self-energy: 4π(1.085938 − 0.346574) = 4π × 0.739364 = 9.2911, giving 6.898 MHz. Vacuum polarisation: 5π/48 = 0.32725, giving 0.243 MHz. Together A₅₀ = 9.6184 and 7.140 MHz on 2S₁/₂; 2P₁/₂ gets nothing at this order.
- New total: 1052.19 + 7.14 = 1059.33 MHz, 1.50 MHz above the measurement. The term is 0.0526 units of C₂ = 135.64 MHz, a quarter more than the 0.0416-unit gap, so the next entries must be negative on balance.
- Reduced mass: (1 + 1/1836.15)⁻³ = 0.998368, so every order-α(Zα)⁴ entry shrinks by 0.16321%, and the interval falls by 0.0016321 × 1052.19 MHz = 1.717 MHz.
- Running total 1059.33 − 1.72 = 1057.61 MHz: 0.22 MHz below 1057.83 MHz, 4% of the original 5.64 MHz gap, left for entries each under 0.5 MHz.
Answer+7.14 MHz (6.90 MHz self-energy, 0.24 MHz vacuum polarisation), then −1.72 MHz from the reduced mass: 1057.61 MHz, leaving 0.22 MHz of the 5.64 MHz gap.
MediumCarry the budget to hydrogen's 1S level, whose order-α(Zα)⁴ shift is 8127.44 MHz. For 2S the α(Zα)⁵, size and two-loop entries are +7.140, +0.138 and +0.127 MHz; for 1S Salpeter's recoil is +2.41 MHz and the one-loop (Zα)² terms −3.28 MHz (quoted). Decide which 2S entries carry over by n⁻³, assemble the 1S shift, and compare it with the 8172.8 MHz inferred from optical spectroscopy (quoted).
- The α(Zα)⁵, size and two-loop entries are contact terms: |ψ(0)|² = (Zαmᵣ)³/(πn³) times a coefficient the same for every s state. Each grows by 2³ = 8: +57.12, +1.10 and +1.02 MHz.
- Reduced mass, computed: 1 − (1 + 1/1836.15)⁻³ = 0.0016321 of 8127.44 MHz is −13.26 MHz.
- The quoted entries do not scale: eight times 2S's +0.341 MHz of recoil would be +2.73 MHz, not +2.41, and eight times 2S's −0.389 MHz of (Zα)² terms would be −3.11 MHz, not −3.28, because their coefficients hold Bethe logarithms, harmonic sums and ln n, all of which change with n.
- Total: 8127.44 + 57.12 + 1.10 + 1.02 − 13.26 + 2.41 − 3.28 = 8172.55 MHz, an energy of 3.380 × 10⁻⁵ eV.
- Against 8172.8 MHz the ledger is 0.25 MHz, 0.003%, short: eight times the 0.03 MHz left at n = 2, as omitted terms that are themselves contact terms, such as α²(Zα)⁵, should be.
Answer1S rises 8172.55 MHz ≈ 8.17 GHz (3.380 × 10⁻⁵ eV), 0.25 MHz below the inferred 8172.8 MHz; the α(Zα)⁵ term alone moved it 57 MHz.
HardHydrogen's interval was measured in 2019 as 1057.8298 ± 0.0032 MHz (quoted). With ΔE(2S) = (1/12)(Zα)⁴mᵣ³rₚ² and mᵣ = 0.510721 MeV, find dν/drₚ at rₚ = 0.84 fm and the radius uncertainty 3.2 kHz alone allows. Compare the predictions for rₚ = 0.8409 fm and 0.8751 fm. Then find the coefficient for muonic hydrogen, mᵣ = 94.965 MeV, and the share of its 202.37 meV interval (quoted) that rₚ = 0.8409 fm makes.
- Coefficient: (1/12)α⁴mᵣ³/(ħc)² = (1/12)(2.8357 × 10⁻⁹)(0.13321 MeV³)/(38 938 MeV² fm²) = 8.085 × 10⁻¹⁶ MeV fm⁻², which divided by h is 0.19548 MHz fm⁻². Dimensions: MeV³ over MeV² fm², times fm², is MeV.
- Sensitivity: dν/drₚ = 2 × 0.19548 × 0.84 = 0.3284 MHz fm⁻¹, so 0.0032 MHz fixes rₚ to ±0.0097 fm, 1.2% of it, provided every other entry is known to better than 3 kHz.
- Candidates: 0.19548 MHz fm⁻² × (0.8751² − 0.8409²) fm² = 0.19548 × 0.05869 MHz = 11.5 kHz, 3.6 times the measurement's uncertainty; the first is muonic hydrogen's radius, the second the 2014 average of hydrogen spectroscopy and electron scattering. The 2019 radius, 0.833 ± 0.010 fm (quoted), sides with the smaller.
- Muonic hydrogen: (94.965/0.510721)³ = 185.94³ = 6.429 × 10⁶, so the coefficient is 0.19548 MHz fm⁻² × 6.429 × 10⁶ × h = 5.20 meV fm⁻² at leading order: the muon's orbit is 186 times smaller and sits that much closer to the proton.
- Share: 5.20 meV fm⁻² × 0.8409² fm² = 3.68 meV, 1.82% of 202.37 meV, against 0.013% in hydrogen. There Uehling's potential puts 2S below 2P, and the size term narrows the gap. As rₚ → 0 both shares vanish.
Answerdν/drₚ = 0.328 MHz fm⁻¹, so 3.2 kHz fixes rₚ to ±0.010 fm; the two radii differ by 11.5 kHz, 3.6σ; muonic hydrogen's coefficient is 6.43 × 10⁶ times larger, 5.20 meV fm⁻², and rₚ makes 1.8% of its interval.