University Quantum Field Theory I · Vacuum Polarisation, Charge Renormalisation and the Running of alpha · 13.9

Muonic hydrogen: where vacuum polarisation dominates

In hydrogen the electron loop is a 27 MHz footnote to the self-energy. Put a muon in the orbit and the same loop becomes the largest correction in the atom, too spread out for the contact shortcut, and the number that replaces it is the one the proton-radius measurement had to subtract first.

Open the lab ↓Practice & worked examples ↓

Build the model

Connect the measurement to the mechanism.

Muonic hydrogen is a proton orbited by a muon, so with the reduced mass mᵣ = 94.96 MeV its Bohr radius shrinks to a = 1/(α mᵣ) = 285 fm. The vacuum-polarisation correction is still the electron loop: in position space it is the Uehling potential of topic 13.07, a superposition of Yukawa potentials whose longest range is 1/(2mₑ) = 193 fm. One ratio decides how to use it: κ = 2mₑa, the orbit in units of that range.

Hydrogen has κ = 274, so the potential acts where the wavefunction is flat and topic 13.08's contact term is good to 0.9%. Muonic hydrogen has κ = 1.47: orbit and cloud are the same size, the 2s density has fallen to 22% of its central value at r = 1/(2mₑ), and the contact term overstates the 2s shift 3.3 times while giving 2p nothing. The replacement is still first-order perturbation theory, but with the whole Uehling potential averaged over non-relativistic 2s and 2p wavefunctions.

The radial integrals close in elementary functions, one integral over the pair spectrum remains, and it puts 2s 205.0 meV below 2p, computed. That is the leading term of the muonic Lamb shift, of order α(Zα)²mᵣ rather than α(Zα)⁴mᵣ³/mₑ², and several hundred times the muon's own self-energy. The model stops at the next layer, Dirac wavefunctions, two-loop and iterated polarisation, recoil and the proton's size, which together move the answer by a few meV and are quoted, not derived.

Simple definition: Muonic hydrogen is a muon bound to a proton; its orbit is small enough to sit inside the electron's vacuum-polarisation cloud, which makes the electron loop the largest radiative correction to its energy levels.
Example: With mᵣ = 94.96 MeV the Bohr radius is 285 fm against a loop range 1/(2mₑ) = 193 fm, so κ = 1.47; the full Uehling potential puts 2s 205.0 meV below 2p, while the contact formula claims −720 meV for 2s alone.

The orbit measured in units of the loop's rangeκ = 2mₑa = 2mₑ/(Zα mᵣ)

κ = 274 for hydrogen, 1.47 for muonic hydrogen, 0.68 for μ⁴He⁺. It decides, before any integral is done, whether the contact form can be trusted.

a = 1/(Zα mᵣ) in fm, with mᵣ = m_μM/(m_μ + M) and mₑ in MeV; 1/(2mₑ) = ħc/(2mₑc²) = 193.08 fm. κ is a pure number.

Contact limit of the 2s shiftΔE₂ₛ(contact) = −α(Zα)⁴ mᵣ³/(30π mₑ²)

−27.08 MHz for hydrogen, −720 meV for muonic hydrogen. The mᵣ³ growth is real; the formula's validity, which needs κ ≫ 1, does not travel with it.

From |ψ₂ₛ(0)|² = 1/(8πa³) in fm⁻³ and ∫d³r V_U = −4Zα²/(15mₑ²), topic 13.08. Energy in meV; equal to −4Zα²mₑ/(15πκ³).

The full shift as an average over the loopΔEₙ/ΔE₂ₛ(contact) = ∫₀^∞ dr G(r) |ψₙ(r)|²/|ψ₂ₛ(0)|²

The contact form sets |ψ(r)|² = |ψ(0)|² under the weight. For muonic hydrogen the 2s average is 0.305, hence the factor 3.3; for 2p it is 0.020, not zero.

G(r) = 4πr²V_U(r)/∫d³r V_U, a normalised weight in fm⁻¹ peaking near r = 0.4/(2mₑ); ψₙ non-relativistic, in fm^(−3/2).

Radial integrals done, one integral leftΔEₙₗ = −(2α/3π)(Zα/a) ∫₁^∞ du ω(u) Jₙₗ(κu)

Each Yukawa term integrates against e^(−r/a) in closed form. For muonic hydrogen the prefactor is 7.831 eV, giving −219.58 meV for 2s and −14.58 meV for 2p.

ω(u) = (1 + 1/(2u²))√(u² − 1)/u²; J₂ₛ(x) = [(1+x)⁻² − 2(1+x)⁻³ + 1.5(1+x)⁻⁴]/2; J₂ₚ(x) = (1+x)⁻⁴/4. Zα/a in MeV.

The vacuum-polarisation Lamb splittingE₂ₚ − E₂ₛ = (α/3π)(Zα/a) ∫₁^∞ du ω(u) (κu)²/(1 + κu)⁴

Large κ gives (2α/3π)(Zα/a)/(5κ²), the contact value; κ → 0 gives (2α/3π)(Zα/a)/12. At κ = 1.47 the true 205 meV is under a third of either.

Uses J₂ₛ − J₂ₚ = x²/(2(1+x)⁴); energies in meV, κ and u pure numbers. Computed: 205.007 meV for μp, 1665.8 meV for μ⁴He⁺.

Finite proton size, the term being measuredΔE_fs(2s) = (2πZα/3)|ψ₂ₛ(0)|² rₚ² = (Zα)⁴ mᵣ³ rₚ²/12

It raises 2s by 3.70 meV for rₚ = 0.841 fm, 1.8% of the Lamb shift, and can be read off only after the 205 meV Uehling term has been subtracted.

rₚ: proton charge radius in fm; 2p is unshifted at this order. Leading term 5.197 meV fm⁻², computed; the full 5.2275 is quoted.

Two lengths decide whether the contact form survives

Natural units ħ = c = 1 and the metric (+, −, −, −) throughout, with ħc = 197.327 MeV fm for lengths. Topic 13.07 turned the finite Π̂₂ into the Uehling potential, quoted here: V_U(r) = −(Zα/r)(2α/3π)∫₁^∞ du ω(u) e^(−2mₑru), a superposition of Yukawa potentials whose longest range is 1/(2mₑ) = 193.08 fm. The atom supplies the second length. With m_μ = 105.658 MeV and mₚ = 938.272 MeV the reduced mass is mᵣ = 94.964 MeV and the Bohr radius is a = ħc/(α mᵣc²) = 284.75 fm. Their ratio κ = 2mₑa is 1.475; ordinary hydrogen, with a = 5.295 × 10⁴ fm, has κ = 274.2. The contact form of topic 13.08 assumes |ψ(r)|² still equals |ψ(0)|² wherever V_U acts. At r = 193 fm the hydrogen 2s density has changed by 0.7%. In muonic hydrogen the same radius is ρ = r/a = 0.678, where (1 − ρ/2)²e^(−ρ) = 0.222: the density has already fallen to a fifth. No small correction repairs that, so the whole potential has to be used.

Scaling hydrogen's number gives −720 meV, and it is wrong

The contact result is ΔE₂ₛ = −(4Zα²/(15mₑ²))|ψ₂ₛ(0)|² with |ψ₂ₛ(0)|² = 1/(8πa³), that is −α(Zα)⁴mᵣ³/(30πmₑ²). The atom enters only through mᵣ³, so hydrogen's −27.08 MHz, computed with its reduced mass 0.51072 MeV, scales by (94.964/0.51072)³ = 6.43 × 10⁶: −1.120 × 10⁻⁷ eV becomes −0.720 eV. The equivalent form −4α²mₑ/(15πκ³) gives the same −720.1 meV from κ = 1.4748 alone. The number has the right dimension, since α²mₑ is an energy, and the right hydrogen limit, and it is still 3.3 times too large, because the formula is only the first term of an expansion in 1/κ. In hydrogen the full Uehling integral, computed below, gives −26.84 MHz against the contact −27.08 MHz, a 0.9% correction that matches the quoted next term, smaller by the factor 1.23Zα. For the muon 1/κ = 0.68, and an expansion in 0.68 is not an expansion. The contact form is also silent about 2p, whose wavefunction vanishes at the origin, although the 2p orbit lies inside the same cloud.

The full Uehling shift reduces to one integral

First-order perturbation theory still applies, ΔEₙₗ = ⟨nl|V_U|nl⟩, with non-relativistic wavefunctions whose a carries the reduced mass. Do the r integral first, for one Yukawa component e^(−λr)/r with λ = 2mₑu, and write x = λa = κu. With |R₂₀|² = (1 − r/2a)²e^(−r/a)/(2a³) and ∫rⁿe^(−βr)dr = n!/β⁽ⁿ⁺¹⁾, the 2s average is a⟨e^(−λr)/r⟩ = J₂ₛ(x) = [(1+x)⁻² − 2(1+x)⁻³ + 1.5(1+x)⁻⁴]/2; with |R₂₁|² = r²e^(−r/a)/(24a⁵) the 2p average is J₂ₚ(x) = (1+x)⁻⁴/4. Both return 1/4 at x = 0, the Coulomb ⟨a/r⟩ for n = 2, which is the check. What is left, ΔEₙₗ = −(2α/3π)(Zα/a)∫₁^∞ du ω(u)Jₙₗ(κu), is smooth, and eight-point Gauss quadrature in u = 1/sin θ already reaches 10⁻⁵. For muonic hydrogen (2α/3π)α²mᵣ = 7.831 eV and the integrals are 0.028040 and 0.0018614, so ΔE₂ₛ = −219.58 meV and ΔE₂ₚ = −14.58 meV, computed. The difference, 205.007 meV, agrees with the published non-relativistic 205.0074 meV, quoted.

Two limits bracket the answer, and both miss it

Subtract the two J's and a clean function is left, J₂ₛ − J₂ₚ = x²/(2(1+x)⁴), so E₂ₚ − E₂ₛ = (α/3π)(Zα/a)∫₁^∞ du ω(u)(κu)²/(1+κu)⁴. When κ ≫ 1 the factor tends to (κu)⁻², and ∫₁^∞ ω(u)u⁻²du = 2/5 returns the contact value (2α/3π)(Zα/a)/(5κ²), 720 meV at muonic hydrogen's κ. When κ ≪ 1 the whole atom sits inside the cloud, where V_U is the logarithmically running charge of topic 13.05: both levels drop by large, nearly equal amounts, and only the change of the logarithm across the orbit separates them, giving (2α/3π)(Zα/a)/12 = 653 meV. Muonic hydrogen, at κ = 1.47, gets 205.0 meV, under a third of either, because the cloud is neither inside nor outside the orbit. All three are α²mᵣ times a pure number, so the dimensions agree. The parametric order has moved from α(Zα)⁴mᵣ³/mₑ² in hydrogen to α(Zα)²mᵣ here, which is how a one-loop correction reaches 3.2 × 10⁻⁴ of the 632 eV binding energy of n = 2.

Why the electron loop outweighs the muon's own corrections

Hydrogen has the opposite order: the self-energy raises 2s by about 1 GHz while vacuum polarisation lowers it by 27 MHz (UAQM-13.09), a ratio near 40. The self-energy belongs to the orbiting particle and scales like α(Zα)⁴mᵣ³/m² with a logarithm; scaling Bethe's 1040 MHz by that factor, 150, estimates 0.65 meV, and the full muon self-energy with muon-loop polarisation moves the muonic splitting by −0.67 meV, quoted. The electron loop is instead a property of the vacuum with its size fixed by mₑ, and a heavier orbiter samples it at higher density, until the finite range stops the growth at order α(Zα)²mᵣ. Polarisation over self-energy goes from about 1/40 to about 300. Heavier loops lose by the same logic run backwards: the muon loop's range 1/(2m_μ) = 0.93 fm makes it a contact term again, and hadronic polarisation adds about 0.01 meV, quoted. The next electronic terms are the two-loop Källén–Sabry potential, about 1.51 meV, and the Uehling potential iterated in second order, about 0.15 meV, both quoted and each over a hundred times smaller than the leading term.

What the proton-radius measurement took from this number

The proton's finite size raises s levels by (2πZα/3)|ψ(0)|²rₚ², for 2s (Zα)⁴mᵣ³rₚ²/12: 5.197 meV fm⁻² at leading order, computed, and 5.2275 meV fm⁻² with higher-order pieces, quoted. Because |ψ(0)|² grows as mᵣ³, the size term is 1.8% of the muonic Lamb shift against about 10⁻⁴ in hydrogen, which is why the experiment was worth doing. In 2010 the CREMA collaboration at PSI drove the 2s–2p transition with a 6 μm laser and found rₚ = 0.84184(67) fm against the then-accepted 0.8768(69) fm, a 5σ disagreement, quoted. A radius is what remains after theory is subtracted: of the 206.03 meV of radius-independent theory, quoted, 205.01 meV is the Uehling term computed here. The disputed 0.32 meV is 0.16% of it, and that term is a closed calculation from α, mₑ and mᵣ with no adjustable parameter and a relativistic correction of only about 0.02 meV, so it could not hide the gap. The independent test came later: electron scattering and hydrogen spectroscopy moved towards 0.84 fm, and CODATA 2018 adopted 0.8414(19) fm, quoted.

Change one variable at a time

Make the relationship visible.

Interactive model

Orbiting particle mass1.00 m_μ

Start at one muon mass: the weight overlaps the falling 2s curve and the contact formula is 3.3 times too big. Drag down to 0.1 m_μ and the cloud squeezes towards r = 0 while the dashed level climbs towards 1; drag up to 3 m_μ and the overlap with 2p grows.

ORBIT ÷ CLOUD κ = 2mₑa1.47

CONTACT 2s SHIFT-720.1 meV

FULL UEHLING 2s SHIFT-219.6 meV

VP SPLITTING E(2p) − E(2s)205.0 meV

Live interpretationORBIT ÷ CLOUD κ = 2mₑa: 1.47. CONTACT 2s SHIFT: −720.1 meV. FULL UEHLING 2s SHIFT: −219.6 meV. VP SPLITTING E(2p) − E(2s): 205.0 meV

Catch the common trap

Explain before calculating.

Misconception check

The tempting move is to treat hydrogen's contact formula as a law and scale it: the shift goes as mᵣ³, the muon's reduced mass is 186 times the electron's, so multiply −27.08 MHz by 6.4 × 10⁶. Every step of that arithmetic is right and the answer, −720 meV, is 3.3 times too large, because the formula is not a law but the leading term of an expansion in 1/κ = 1/(2mₑa), the loop's range divided by the orbit. In hydrogen 1/κ = 0.0036 and the next term is a 0.9% correction; in muonic hydrogen 1/κ = 0.68 and the expansion means nothing. The same reasoning predicts no vacuum-polarisation shift for 2p, because ψ₂ₚ(0) = 0, while the full potential lowers 2p by 14.6 meV. Before using any contact potential, compare its range with the size of the state it acts on.

The contact formula gives muonic hydrogen's 2s level a vacuum-polarisation shift of −720 meV, while the full Uehling potential gives −220 meV. What is the main reason for the difference?

Choose an answer to test the model.

Practice & worked examples

Reason from the model, then test the result.

EasyMuonic deuterium is a muon bound to a deuteron of mass 1875.613 MeV. Find its Bohr radius and κ = 2mₑa, predict the 2s vacuum-polarisation shift from the contact formula, and decide whether that prediction can be trusted.

Reduced mass: mᵣ = m_μm_d/(m_μ + m_d) = 105.658 × 1875.613/1981.271 MeV = 100.024 MeV, 5.3% above muonic hydrogen's 94.964 MeV because the heavier deuteron recoils less.
Bohr radius: a = ħc/(α mᵣc²) = 197.327 MeV fm × 137.036/100.024 MeV = 270.3 fm.
Orbit over range: κ = 2mₑ/(α mᵣ) = 2 × 0.51100 × 137.036/100.024 = 1.4002, so the orbit is only 40% wider than the loop's range 1/(2mₑ) = 193.08 fm.
Contact formula: ΔE₂ₛ = −4α²mₑ/(15πκ³) = −2309.8 meV/1.4002³ = −841 meV, the same as muonic hydrogen's −720.1 meV × (100.024/94.964)³.
Verdict: 1/κ = 0.71, so the expansion behind the contact form fails as badly as for muonic hydrogen. The full Uehling integral of this lesson gives −245.3 meV for 2s and a 2p − 2s splitting of 227.6 meV, computed, so the contact value is 3.4 times too large.

Answera = 270 fm and κ = 1.40; the contact formula predicts −841 meV for 2s, which cannot be trusted: the full Uehling potential gives −245 meV (computed), a factor 3.4 smaller.

MediumThe theory of muonic hydrogen's Lamb shift is quoted as E₂ₚ − E₂ₛ = 206.0336 meV + 0.0332 meV − (5.2275 meV fm⁻²)rₚ², and the measured value is 202.3706 meV. Extract rₚ, find the splitting the older radius 0.8768 fm would have produced, and say how accurately the Uehling term must be known.

Solve for rₚ²: (206.0336 + 0.0332 − 202.3706) meV ÷ 5.2275 meV fm⁻² = 3.6962/5.2275 fm² = 0.70707 fm², so rₚ = 0.84087 fm.
With rₚ = 0.8768 fm the size term is 5.2275 × 0.76878 = 4.0188 meV, so the predicted splitting would be 206.0668 − 4.0188 = 202.0480 meV.
The measurement lies 0.3226 meV above that; dividing by h = 4.1357 × 10⁻¹⁵ eV s gives 78.0 GHz, against a line located to under 1 GHz.
Against the Uehling term, 205.007 meV computed in this lesson, the gap is 0.3226/205.007 = 1.57 × 10⁻³: hiding it would need the one-loop integral to be wrong in its third digit.
The experimental uncertainty, 0.0023 meV, is 1.1 × 10⁻⁵ of the Uehling term. That is the precision the theory must carry, which is why the relativistic correction, about 0.017 meV, and the two-loop term, 1.51 meV, are included, both quoted.

Answerrₚ = 0.8409 fm; 0.8768 fm would give 202.048 meV, 0.32 meV or 78 GHz below the measurement, which is 0.16% of the 205 meV Uehling term, a term the theory carries to better than 10⁻⁵.

HardIn the muonic helium ion μ⁴He⁺ (Z = 2, m_α = 3727.379 MeV) the orbit lies inside the electron loop's range. Compute the vacuum-polarisation shifts of 2s and 2p from the full Uehling potential, compare 2s with the contact formula, and explain why the 2p − 2s splitting is not 16 times muonic hydrogen's 205.0 meV.

mᵣ = 105.658 × 3727.379/3833.037 MeV = 102.746 MeV, a = ħc/(Zα mᵣc²) = 197.327 × 137.036/(2 × 102.746) fm = 131.59 fm, and κ = 131.59/193.08 = 0.6815: the orbit is now smaller than the loop's range.
The radial integrals keep their form, ΔEₙₗ = −(2α/3π)(Zα/a)∫₁^∞ du ω(u)Jₙₗ(κu), with prefactor (2α/3π)Z²α²mᵣ = 1.54855 × 10⁻³ × 21.8854 keV = 33.8906 eV.
Quadrature at κ = 0.6815 gives ∫ω J₂ₛ du = 0.061292 and ∫ω J₂ₚ du = 0.0121405, so ΔE₂ₛ = −33.8906 eV × 0.061292 = −2077.2 meV and ΔE₂ₚ = −33.8906 eV × 0.0121405 = −411.4 meV: 2p lies 1665.8 meV above 2s, computed.
Contact: −4Zα²mₑ/(15πκ³) = −2 × 2309.8 meV/0.68154³ = −14.59 eV, which overstates the 2s shift by 14.59/2.077 = 7.0, twice as badly as in muonic hydrogen.
Scaling: contact behaviour, Z⁴mᵣ³, would multiply muonic hydrogen's splitting by 16 × (102.746/94.964)³ = 20.3; the inside-the-cloud limit (2α/3π)(Zα/a)/12 scales as Z²mᵣ, a factor 4 × 1.082 = 4.33. The computed 1665.8/205.0 = 8.13 lies between, since κ = 0.68 is in neither limit.

AnswerΔE₂ₛ = −2077 meV and ΔE₂ₚ = −411 meV, so 2p lies 1665.8 meV above 2s (computed); the contact −14.6 eV overstates 2s sevenfold, and the splitting is 8.1 times muonic hydrogen's, between the Z²mᵣ and Z⁴mᵣ³ scalings.