Quantum Electrodynamics and Renormalization

Derives Theorem 100.5. Under Equation (100.7) the derivative acquires an inhomogeneous term,

so that \(\Lag_{0}\mapsto\Lag_{0} +q c\,\bar{\psi}\gamma^{\mu}\psi\,\pp_{\mu}\Lambda =\Lag_{0}+j^{\mu}\pp_{\mu}\Lambda\), which does not vanish for arbitrary \(\Lambda\). The obstruction is that \(\pp_{\mu}\psi\) compares the field at neighbouring points, whose phase conventions are now unrelated. It is removed by supplying a rule of comparison — a connection, in the language of Section 14.6. Postulate a field \(A_{\mu}\) with the transformation law Equation (100.8) and form \(D_{\mu}\) as in Equation (100.9). Then

so \(\bar{\psi}\gamma^{\mu}D_{\mu}\psi\) is invariant, the two phases cancelling. Conversely, any modification \(\pp_{\mu}\to\pp_{\mu}+X_{\mu}\) restoring invariance must satisfy \(X_{\mu}\mapsto X_{\mu}+\ii q\,\pp_{\mu}\Lambda/\hbar\) by the same computation read backwards, which is Equation (100.8) for \(A_{\mu}=\hbar X_{\mu}/\ii q\). Finally

Derives Proposition 100.6. Immediate from Theorem 100.5: the covariant derivative Equation (100.9) depends on the matter field only through the charge appearing in its transformation law Equation (100.7), and no other parameter of the matter sector enters \(D_{\mu}\). That this survives quantization — that radiative corrections do not make the observed charge mass-dependent — is not automatic and is the content of the Ward–Takahashi identity of Section 100.5.2.

Derivation. Derives Phenomenon 100.7. Add the mass term to Equation (100.12); under Equation (100.8) it changes by \(\tfrac{1}{\mu_{0}}(m_{\gamma}c/\hbar)^{2} \left(A^{\mu}\pp_{\mu}\Lambda +\tfrac{1}{2}\pp_{\mu}\Lambda\,\pp^{\mu}\Lambda\right)\), which is not a total derivative and does not vanish. Hence gauge invariance requires \(m_{\gamma}=0\) exactly.

To convert this into something measurable, keep the mass term and drop gauge invariance: the field equation becomes the Proca equation \(\pp_{\mu}F^{\mu\nu}+(m_{\gamma}c/\hbar)^{2}A^{\nu}=\mu_{0}j^{\nu}\), whose static point-charge solution is the Yukawa potential

The verification is the same contour computation as for any Yukawa kernel: the Fourier transform of \((\vect{k}^{2}+\lambda^{-2})^{-1}\) is \(\ee^{-r/\lambda}/4\pi r\), obtained by closing the \(\abs{\vect{k}}\) contour in the upper half plane and picking up the pole at \(\abs{\vect{k}}=\ii/\lambda\) (Theorem 8.24). A Cavendish-type null experiment with concentric conductors bounds any departure from the \(1/r^{2}\) force law, and hence bounds \(\lambda^{-1}\); the laboratory bound \(\lambda>3\times 10^{7}\,\mathrm{m}\) [Williams:1971] translates through \(m_{\gamma}=\hbar/\lambda c\) into

of the order quoted. The bound is a bound and not a measurement: what is established is that if the photon has a mass it is smaller than \(10^{-50}\) of a kilogram, which is \(10^{-20}\) of the electron mass.

Derives Proposition 100.8. Write \(P^{\mathrm{T}}_{\mu\nu}=\eta_{\mu\nu}-p_{\mu}p_{\nu}/p^{2}\) and \(P^{\mathrm{L}}_{\mu\nu}=p_{\mu}p_{\nu}/p^{2}\); these are orthogonal projectors summing to \(\eta_{\mu\nu}\). The gauge-fixed kernel is

whose inverse on each eigenspace is immediate: \(\widetilde{M}^{-1}=-\mu_{0}\hbar^{2} \left(P^{\mathrm{T}}+\xi P^{\mathrm{L}}\right)/p^{2}\). A Gaussian functional integral with phase \(\exp(\ii S/\hbar)\) and quadratic form \(S=\tfrac{1}{2}\int A\,\mathcal{M}A\) has two-point function \(\ii\hbar\mathcal{M}^{-1}\); here \(\mathcal{M}=\widetilde{M}/c\) by Equation (100.2), so the propagator is \(\ii\hbar c\,\widetilde{M}^{-1}\), which is Equation (100.17) after restoring \(P^{\mathrm{T}}+\xi P^{\mathrm{L}} =\eta_{\mu\nu}-(1-\xi)p_{\mu}p_{\nu}/p^{2}\). The \(\ii\epsilon\) prescription is the one that propagates positive frequencies forward in time, fixed by the contour argument of Canonical Quantization of Fields. Dimensionally, \(\mu_{0}\hbar^{3}c/p^{2}\) carries \(\mathrm{kg}^{2}\,\mathrm{m}^{6}\,\mathrm{s}^{-4}\,\mathrm{A}^{-2}\), which is \([A]^{2}\mathrm{m}^{4}\) as the Fourier transform of a product of two potentials must be.

Derives Theorem 100.10. Let \(n\) be the number of photons attached to the loop. The contribution of the loop is, up to a factor common to both orientations,

with \(S(p)=\ii\hbar(\gamma\cdot p+m_{e}c)/(p^{2}-m_{e}^{2}c^{2})\) the fermion propagator and the overall minus sign the one attached to every closed fermion loop by the anticommuting Wick contraction of Canonical Quantization of Fields. Every diagram comes in two versions, differing in the direction in which the charge circulates. Take the transpose of the trace, which leaves it unchanged, and insert \(\mathcal{C}_{\mathrm{D}}^{-1}\mathcal{C}_{\mathrm{D}}\) between every pair of factors. By Equation (100.20), \(\mathcal{C}_{\mathrm{D}}(\gamma^{\mu})\transpose \mathcal{C}_{\mathrm{D}}^{-1}=-\gamma^{\mu}\) and hence \(\mathcal{C}_{\mathrm{D}}S(p)\transpose \mathcal{C}_{\mathrm{D}}^{-1}=S(-p)\), because the transpose reverses the sign of the \(\gamma\cdot p\) term and leaves \(m_{e}c\) alone. The transposed trace is therefore the trace for the reversed orientation, with an extra factor \((-1)^{n}\) from the \(n\) vertices. Summing the two orientations gives \(\left(1+(-1)^{n}\right)T^{\mu_{1}\cdots\mu_{n}}\), which vanishes for odd \(n\).

Derives Proposition 100.13. Two deformations of \(\sigma\) at spacelike separated points \(x\) and \(y\) can be performed in either order and must give the same final surface. Applying Equation (100.21) twice and demanding that the mixed second functional derivative be symmetric gives \(\Ham_{\mathrm{int}}(x)\Ham_{\mathrm{int}}(y) =\Ham_{\mathrm{int}}(y)\Ham_{\mathrm{int}}(x)\), which is Equation (100.22). The condition holds in QED because \(\Ham_{\mathrm{int}}\) is built from local field operators whose commutators vanish outside the light cone (Canonical Quantization of Fields), and it fails for any nonlocal interaction — which is the formal statement of why locality is not optional here.

Derivation of the vertex factor. Derives Proposition 100.14. The interaction part of Equation (100.2) is, by Theorem 100.5,

the factor \(c\) cancelling between Equation (100.2) and Equation (100.6). Expanding \(\exp(\ii S/\hbar)\) to first order produces \((\ii/\hbar)\times(-q\gamma^{\mu})\) at each vertex, which is Equation (100.24). The propagators are the inverses of the quadratic forms, computed in Proposition 100.8 for the photon; for the fermion the quadratic form is \(\gamma\cdot p-m_{e}c\) by Equation (100.14), and \(\ii\hbar(\gamma\cdot p-m_{e}c)^{-1} =\ii\hbar(\gamma\cdot p+m_{e}c)/(p^{2}-m_{e}^{2}c^{2})\) using \((\gamma\cdot p)^{2}=p^{2}\).

Derives Theorem 100.17. Let \(L\) be the number of loops, \(V\) the number of vertices, and \(P_{e}\), \(P_{\gamma}\) the numbers of internal fermion and photon lines. Each loop supplies \(\dd^{4}\ell\), each internal fermion line falls as \(\ell^{-1}\) by Equation (100.23) and each internal photon line as \(\ell^{-2}\) by Equation (100.17), so

The number of independent loops is the number of internal lines less the number of momentum conservation constraints, one per vertex, with one overall constraint left over: \(L=P_{e}+P_{\gamma}-V+1\). Every vertex has exactly two fermion ends and one photon end, and every internal line uses two ends while every external line uses one, so

Eliminating \(P_{e}=V-E_{e}/2\) and \(P_{\gamma}=(V-E_{\gamma})/2\),

which is Equation (100.26): \(V\) cancels, so \(D\) does not grow with the order. Enumerating the non-negative cases, with \(E_{e}\) even by fermion-number conservation:

Every other amplitude has \(D<0\) and is superficially convergent.

Derivation. Derives Phenomenon 100.20. Treat the radiation as quanta of energy \(\hbar\omega\) and momentum \(\hbar\vect{k}\) with \(\abs{\vect{k}}=\omega/c\), and the target as a free electron at rest. Conservation of energy and of momentum give

Squaring the second, \(p^{2}=\hbar^{2}(k^{2}+k'^{2}-2kk'\cos\theta)\), and substituting it together with \(E=\hbar(\omega-\omega')+m_{e}c^{2}\) into the mass-shell relation \(E^{2}=p^{2}c^{2}+m_{e}^{2}c^{4}\) gives

The terms in \(k^{2}\) and \(k'^{2}\) cancel between the two sides, leaving \(m_{e}c\left(k-k'\right)=\hbar kk'\left(1-\cos\theta\right)\), that is \(1/k'-1/k=(\hbar/m_{e}c)(1-\cos\theta)\). Multiplying by \(2\pi\) and using \(\lambda=2\pi/k\) gives Equation (100.27). The shift is purely kinematic: it follows from treating light as a particle of energy \(\hbar\omega\) and momentum \(\hbar\omega/c\), and it involves no property of the electron except its mass. The assumption that fails for tightly bound electrons is that the target is free, which is why an unshifted line survives beside the shifted one.

Derives Proposition 100.21. By Equation (100.27), \(\omega'/\omega =\left[1+(\hbar\omega/m_{e}c^{2})(1-\cos\theta)\right]^{-1}\to1\) when \(\hbar\omega\ll m_{e}c^{2}\). Then \(\omega/\omega'+\omega'/\omega\to2\) and the bracket in Equation (100.30) becomes \(2-\sin^{2}\theta=1+\cos^{2}\theta\). Integrating, \(\int\left(1+\cos^{2}\theta\right)\dd\Omega =4\pi+\tfrac{4}{3}\pi=\tfrac{16}{3}\pi\), so \(\sigma=\tfrac{1}{2}r_{e}^{2}\cdot\tfrac{16}{3}\pi =\tfrac{8}{3}\pi r_{e}^{2}\). Numerically \(r_{e}^{2}=7.9408\times 10^{-30}\,\mathrm{m}^{2}\) and \(\tfrac{8}{3}\pi=8.3776\), giving \(6.652\times 10^{-29}\,\mathrm{m}^{2}\), the accepted Thomson cross-section. The limit is the quantitative statement that classical electrodynamics is recovered when the photon carries away no appreciable recoil, and the departure from it, first seen as the falling cross-section for hard X-rays, is direct evidence that scattering is a quantum process.

Derives Lemma 100.22. Both spinor sums are projectors onto the positive- and negative-frequency solutions of the Dirac equation, and are verified by evaluating both sides in the rest frame, where \(\gamma^{\mu}p_{\mu}=m_{e}c\gamma^{0}\) and the two-dimensional spin sums give \(\tfrac{1}{2}(1\pm\gamma^{0})\cdot2m_{e}c\); the general case follows by boosting, since both sides transform as bispinor matrices. For the photon, the physical sum runs over the two transverse polarizations only and is not covariant. Writing the amplitude as \(\mathcal{M}=\varepsilon^{*}_{\mu}\mathcal{M}^{\mu}\) with \(k_{\mu}\mathcal{M}^{\mu}=0\) — gauge invariance, established to all orders in Section 100.5.2 — the two transverse terms may be completed to the full \(-\eta^{\mu\nu}\) because the extra longitudinal and scalar pieces are proportional to \(k^{\mu}\) or \(k^{\nu}\) and annihilate \(\mathcal{M}^{\mu}\). The replacement is therefore legitimate inside a gauge-invariant amplitude and nowhere else.

Derives Lemma 100.23. \(\tr(\gamma^{\mu}\gamma^{\nu}) =\tfrac{1}{2}\tr\acomm{\gamma^{\mu}}{\gamma^{\nu}} =\eta^{\mu\nu}\tr\identity=4\eta^{\mu\nu}\). For an odd number, insert \((\gamma^{5})^{2}=\identity\) and use \(\acomm{\gamma^{5}}{\gamma^{\mu}}=0\) to move one \(\gamma^{5}\) through \(n\) matrices, picking up \((-1)^{n}\); cyclicity of the trace then gives \(\tr X=(-1)^{n}\tr X\), which vanishes for odd \(n\). The four-index case follows by anticommuting \(\gamma^{\mu}\) to the right through the other three, generating three two-index traces with alternating signs. For the contractions, \(\gamma^{\nu}\gamma^{\alpha}\gamma_{\nu} =\gamma^{\nu}\left(2\eta^{\alpha}_{\ \nu} -\gamma_{\nu}\gamma^{\alpha}\right) =2\gamma^{\alpha}-4\gamma^{\alpha}=-2\gamma^{\alpha}\), and the higher ones follow by the same move applied repeatedly. That these identities carry a dimension-dependent number — the \(4\) in \(\gamma^{\nu}\gamma_{\nu}=4\) — is not innocent: it is exactly what must be tracked when the dimension is continued in Section 100.4.4.

Derives Proposition 100.24. The transition probability per unit time and volume is \(\abs{\mathcal{M}}^{2}\) divided by the normalization of the four external states, and integrated over the available final momenta with the Lorentz-invariant measure \(c\,\dd^{3}p/(2\pi\hbar)^{3}2E\), which is \(\dd^{4}p\,\delta(p^{2}-m^{2}c^{2})\theta(p^{0})\) in disguise and therefore invariant. Dividing by the incident flux \(4\sqrt{(p_{1}\cdot p_{2})^{2}-m_{1}^{2}m_{2}^{2}c^{4}}\), which is the invariant form of \(\abs{\vect{v}_{1}-\vect{v}_{2}}\) times the two densities, gives Equation (100.38). For two final particles the delta function removes three of the six momentum integrals and fixes \(\abs{\vect{p}_{f}}\), leaving

and in the centre-of-mass frame the flux factor is \(4\sqrt{s}\abs{\vect{p}_{i}}\), whence Equation (100.40). The dimensional check is the useful one: \(\dd\Phi_{2}\) carries \(\hbar^{-2}\), \(\overline{\abs{\mathcal{M}}^{2}}\) carries \(\hbar^{4}\) by Proposition 100.14, and the flux carries a squared momentum, so \(\dd\sigma\) carries \(\hbar^{2}/p^{2}\), an area.

Derives Theorem 100.25. The single \(t\)-channel diagram gives, by Proposition 100.14 in Feynman gauge,

that is \(\mathcal{M}=q_{1}q_{2}\mu_{0}\hbar c\,t^{-1} \left[\bar{u}_{3}\gamma^{\mu}u_{1}\right] \left[\bar{u}_{4}\gamma_{\mu}u_{2}\right]\). In the non-relativistic limit the spatial components of \(\bar{u}\gamma^{\mu}u\) are suppressed by \(\abs{\vect{v}}/c\) while the time component is \(u^{\dagger}u=2E/c\to2mc\), so the current product becomes \(4m_{1}m_{2}c^{2}\), and the momentum transfer is spacelike, \(t=-\vect{q}^{2}\) with \(\vect{q}\) the three-momentum transfer. Hence

On the other side, an ordinary potential \(V\) treated in first Born approximation and re-expressed with the relativistic normalization of Proposition 100.14 contributes \(\mathcal{M}=-(2m_{1}c)(2m_{2}c)\widetilde{V}(\vect{q})/\hbar c\), the factor \((\hbar c)^{-1}\) arising from \(\exp(\ii S/\hbar)\) together with \(\int\dd t=c^{-1}\int\dd x^{0}\). Equating,

using \(\varepsilon_{0}\mu_{0}c^{2}=1\). Inverting the transform Equation (100.14) in three dimensions,

the angular integral giving \(4\pi\hbar\sin(qr/\hbar)/qr\) and the remaining radial integral \(\int_{0}^{\infty}\sin(x)\,\dd x/x=\pi/2\) after the substitution \(x=qr/\hbar\). That integral is half the imaginary part of the principal-value integral of \(\ee^{\ii z}/z\) along the real axis, which is evaluated by closing in the upper half plane and indenting the contour at the origin, where the half-residue contributes \(\ii\pi\); it is the residue theorem Theorem 8.24 in the indented form carried out in Theorem 17.78, and not the unindented archetype worked in Section 8.7 itself. The sign is the physical one: \(q_{1}q_{2}>0\) gives a positive, repulsive potential. What produced it is the \(\eta_{\mu\nu}\) of the photon propagator contracted with two timelike currents; had the exchanged quantum been a scalar the same computation would have given attraction between like charges, which is the standard argument that the photon must be a vector.

Derives Lemma 100.26. For \(n=2\), substitute directly:

For general \(n\), differentiate Equation (100.42) for \(n-1\) with respect to one of the \(A_{i}\), which multiplies the left-hand side by an extra factor and raises the power on the right; equivalently, use the integral representation \(1/A^{a}=\Gamma(a)^{-1}\int_{0}^{\infty}\dd t\,t^{a-1}\ee^{-tA}\), multiply \(n\) of them together, insert \(1=\int_{0}^{\infty}\dd\lambda\,\delta(\lambda-\sum t_{i})\) and rescale \(t_{i}=\lambda x_{i}\); the \(\lambda\) integral is then a gamma function and the \(x_{i}\) integral is Equation (100.42). The purpose of the identity is geometric: it replaces \(n\) propagator poles at different places by a single quadratic form, which after a shift of the loop momentum depends on the external momenta only through one scalar \(\Delta\).

Derives Lemma 100.27. The denominator \(\ell^{2}-\Delta+\ii\epsilon\) vanishes at \(\ell^{0}=\pm\left(\sqrt{\vect{\ell}^{2}+\Delta}-\ii\epsilon\right)\): one pole just below the positive real axis and one just above the negative real axis. The quadrant between the positive real axis and the positive imaginary axis, and the quadrant between the negative real axis and the negative imaginary axis, are therefore free of singularities, and the arcs at infinity contribute nothing by the decay hypothesis. Cauchy's theorem (Theorem 8.12) applied to the closed contour made of the real segment, the arc and the imaginary segment gives Equation (100.43), the factor \(\ii\) coming from \(\dd\ell^{0}=\ii\,\dd\ell^{0}_{\mathrm{E}}\) and the sign of \(\ell^{2}=(\ell^{0})^{2}-\vect{\ell}^{2} =-\ell_{\mathrm{E}}^{2}\). The \(\ii\epsilon\) prescription is thus not a convenience: it is what makes the rotation legal, and a different prescription would rotate the contour into a pole and give a different — and acausal — answer.

Derives Lemma 100.28. Write the Euclidean integral in polar coordinates. The surface area of the unit sphere \(S^{d-1}\) follows from the Gaussian integral of Fourier Analysis and Integral Transforms — take Equation (17.42) at zero frequency, giving \(\int_{-\infty}^{\infty}\ee^{-t^{2}}\dd t=\sqrt{\pi}\) — evaluated in two ways:

the last step by \(u=r^{2}\) and the integral representation of the gamma function, Equation (17.109). Hence \(\Omega_{d}=2\pi^{d/2}/\Gamma(d/2)\), an expression that is defined for complex \(d\) even though no sphere is — which is the whole content of dimensional continuation. The radial integral is a beta function: with \(u=r^{2}\),

convergent at the origin for \(\Re d>0\) and at infinity for \(\Re d<2n\). Assembling with the measure \((2\pi\hbar)^{-d}\) and cancelling \(\Gamma(d/2)\) gives Equation (100.44). Equation (100.45) follows from Lemma 100.27, which supplies the factor \(\ii\) and turns \((\ell^{2}-\Delta)^{n}\) into \((-1)^{n}(\ell_{\mathrm{E}}^{2}+\Delta)^{n}\); Equation (100.46) follows from the same rotation with \(\ell^{2}=-\ell_{\mathrm{E}}^{2}\) in the numerator and the radial integral run with \(d\to d+2\). The replacement \(\ell^{\mu}\ell^{\nu}\to\eta^{\mu\nu}\ell^{2}/d\) holds because the integral of \(\ell^{\mu}\ell^{\nu}\) against a function of \(\ell^{2}\) is an invariant tensor, hence proportional to \(\eta^{\mu\nu}\), and the constant is fixed by tracing.

Derives Lemma 100.29. From \(\Gamma(1+\varepsilon)=\varepsilon\Gamma(\varepsilon)\) and the Weierstrass product, \(\ln\Gamma(1+\varepsilon) =-\gamma_{\mathrm{E}}\varepsilon+O(\varepsilon^{2})\), so \(\Gamma(\varepsilon)=\varepsilon^{-1}\Gamma(1+\varepsilon) =\varepsilon^{-1}\left(1-\gamma_{\mathrm{E}}\varepsilon+\dots\right)\), which is Equation (100.47). The poles and their residues are recorded, from the Mellin point of view, at Example 17.95. This simple pole is where every ultraviolet divergence of this chapter will appear: an integral that diverges logarithmically in four dimensions converges for \(\Re d<4\) and its analytic continuation has a first-order pole at \(d=4\).

Derives Theorem 100.31. By Proposition 100.14, with the closed-fermion-loop factor \(-1\) and the vertex Equation (100.24) evaluated at the electron charge \(-e\), so that each vertex is \(+\ii e\gamma^{\mu}/\hbar\),

with \(\ell'=\ell+q\). The numerical prefactor is \(-(-e^{2}/\hbar^{2})(\ii\hbar)^{2}=-e^{2}\), so

where, by Lemma 100.23,

the odd-trace terms vanishing. Combine the two denominators with Lemma 100.26 and shift \(k=\ell+xq\); the denominator becomes \(\left(k^{2}-\Delta\right)^{2}\) with \(\Delta\) as stated. Dropping the terms odd in \(k\) and using \(k^{\mu}k^{\nu}\to\eta^{\mu\nu}k^{2}/d\) from Lemma 100.28,

Now apply Equation (100.45) and Equation (100.46) with \(n=2\). The first term gives

the leading minus sign being the \((-1)^{n-1}\) of Equation (100.46) at \(n=2\), and using \(\Gamma(1-\tfrac{d}{2})=\Gamma(2-\tfrac{d}{2})/(1-\tfrac{d}{2})\), which converts the leading \(\Gamma(1-d/2)\) — the quadratic divergence, with a pole already at \(d=2\) — into \(\Gamma(2-d/2)\). The third term gives \(+\ii\hbar^{-d}(4\pi)^{-d/2}\Gamma(2-\tfrac{d}{2}) \left(m_{e}^{2}c^{2}+x(1-x)q^{2}\right)\Delta^{\frac{d}{2}-2}\), and since \(m_{e}^{2}c^{2}+x(1-x)q^{2}=\Delta+2x(1-x)q^{2}\) the two \(\eta^{\mu\nu}\) contributions cancel except for \(2x(1-x)q^{2}\eta^{\mu\nu}\). Collecting with the \(q^{\mu}q^{\nu}\) term,

The cancellation is the whole point: it is forced by gauge invariance (equivalently by Equation (100.18)), it removes the quadratic divergence that would otherwise have generated a photon mass term \(\propto\Lambda^{2}\eta^{\mu\nu}\), and it is what Section 100.4.4 means by saying that dimensional regularization respects the symmetry while a momentum cutoff does not. Writing \(d=4-2\varepsilon\) and stripping the tensor structure according to Equation (100.49),

where \(\mu_{0}ce^{2}/\hbar=4\pi\alpha\) by Equation (100.4) and \(8/(4\pi)^{2}=1/2\pi^{2}\), so that the numerical factor is \(4\pi\alpha/2\pi^{2}=2\alpha/\pi\); the expansion of \(\Gamma(\varepsilon)\) is Lemma 100.29. This is Equation (100.50). Note that every \(\hbar\) has cancelled: \(\Pi_{2}\) is a pure number, as it must be, since it corrects a dimensionless coupling.

Derives Corollary 100.32. Each insertion contributes \(\widetilde{D}\left(\ii\Pi\right)\widetilde{D}\); using Equation (100.49) and the transverse projector, the \(n\)th term is \(\Pi_{2}^{n}\) times the free propagator, so the sum is \(\left(1-\Pi_{2}\right)^{-1}\) times it. A mass would require a pole at \(q^{2}\neq0\), that is a term in \(\Pi^{\mu\nu}\) surviving at \(q=0\); the explicit \(q^{2}\eta^{\mu\nu}-q^{\mu}q^{\nu}\) forbids it.

Derives Proposition 100.33. Subtracting Equation (100.50) at \(q^{2}=0\) removes the pole and the scale:

Expanding the logarithm for small \(\abs{q^{2}}\) gives \(x(1-x)q^{2}/m_{e}^{2}c^{2}\), and \(\int_{0}^{1}x^{2}(1-x)^{2}\dd x=1/30\), whence Equation (100.53). The effective coupling seen at momentum transfer \(\vect{q}\) is therefore \(\alpha\left[1+\alpha\vect{q}^{2}/15\pi m_{e}^{2}c^{2}\right]\) for spacelike \(q^{2}=-\vect{q}^{2}\), and Theorem 100.25 then gives

whose second term is independent of \(\vect{q}\) and therefore, on transforming back, proportional to \(\delta^{3}(\vect{r})\):

using \(e^{2}/\varepsilon_{0}=4\pi\alpha\hbar c\). Taking the expectation value in a hydrogenic \(S\) state with \(Z=1\), where \(\abs{\psi_{n00}(0)}^{2}=1/\pi n^{3}a_{0}^{3}\) and \(a_{0}=\hbar/\alpha m_{e}c\) (The Hydrogen Atom),

Numerically, with \(\alpha^{5}=2.0693\times 10^{-11}\) and \(m_{e}c^{2}=8.1871\times 10^{-14}\,\mathrm{J}\), the \(n=2\) value is \(\delta E_{2}=-1.798\times 10^{-26}\,\mathrm{J}\), that is \(\delta E_{2}/h=-27.1\,\mathrm{MHz}\). The sign is the physically important part: the vacuum screens, so the attraction is stronger at short distance and the \(2S\) level, which has support at the origin, is pulled down relative to \(2P\), which has none. This is one of the three one-loop contributions to the Lamb shift of Section 100.8.1, and the only one whose sign is opposite to the total; in muonic atoms, where the Bohr radius is smaller by \(m_{\mu}/m_{e}\approx207\) and the electron loop is no longer short-ranged compared with the orbit, the same term becomes the dominant QED correction, which is why muonic hydrogen [Pohl:2010] [Antognini:2013] measures the proton radius so sharply.

Derives Theorem 100.35. The rules of Proposition 100.14 give

whose numerical prefactor is \((-e^{2}/\hbar^{2})(\ii\hbar)(-\ii\mu_{0}\hbar^{3}c) =-e^{2}\mu_{0}c\hbar^{2}=-4\pi\alpha\hbar^{3}\), so that

with \(\hbar^{4}(2\pi\hbar)^{-d}(\bar{\mu}/\hbar)^{4-d}\) reducing to \((2\pi)^{-4}\) at \(d=4\) — every \(\hbar\) again cancels, and \(\Sigma_{2}\) comes out with the dimension of a momentum, as the mass term it corrects requires. Combine the denominators with Lemma 100.26, weight \(x\) on the photon line — equivalently \(1-x\) on the electron line — and shift \(k=\ell-(1-x)p\); completing the square gives \((k^{2}-\Delta)^{2}\) with \(\Delta\) as stated, and the numerator's \(\gamma\cdot(p-\ell)=\gamma\cdot(xp-k)\) then supplies the \(x\) that multiplies \(\gamma\cdot p\). In the numerator use Equation (100.37) in \(d\) dimensions, \(\gamma^{\mu}\gamma^{\alpha}\gamma_{\mu}=-(2-2\varepsilon) \gamma^{\alpha}\) and \(\gamma^{\mu}\gamma_{\mu}=d\), drop the terms odd in \(k\), and there remains \(-\left(2-2\varepsilon\right)x\,\gamma\cdot p +\left(4-2\varepsilon\right)m_{e}c\), which at \(\varepsilon=0\) is \(-2x\,\gamma\cdot p+4m_{e}c\). Applying Equation (100.45) with \(n=2\),

and \(4\pi/(4\pi)^{2}=1/4\pi\) together with the factor \(2\) pulled out of the bracket gives the \(\alpha/2\pi\) of Equation (100.56). The logarithm rather than a linear divergence is visible in Equation (100.57) itself: the numerator is linear in \(\ell\) and the denominator quartic, so the would-be linearly divergent piece is odd in the shifted momentum and integrates to zero, leaving a logarithm.

Derives Corollary 100.36. On shell, \(\gamma\cdot p=m_{e}c\) and \(p^{2}=m_{e}^{2}c^{2}\), so \(\Delta\to(1-x)^{2}m_{e}^{2}c^{2}\) when \(m_{\gamma}\to0\), and Equation (100.56) gives \(\Sigma_{2}=\frac{\alpha}{2\pi}m_{e}c\int_{0}^{1}(2-x)\dd x \left[\varepsilon^{-1}+\dots\right] =\frac{3\alpha}{4\pi}m_{e}c\left[\varepsilon^{-1}+\dots\right]\), since \(\int_{0}^{1}(2-x)\dd x=3/2\). The correspondence \(\varepsilon^{-1}\leftrightarrow \ln\left(\Lambda^{2}/m_{e}^{2}c^{2}\right)\) between the two regulators, established in Section 100.4.4, gives the second form.

Derives Proposition 100.37. Set \(m_{e}=0\) in Equation (100.12). The Lagrangian is then invariant under the chiral rotation \(\psi\mapsto\ee^{\ii\theta\gamma^{5}}\psi\), because \(\gamma^{\mu}\) anticommutes with \(\gamma^{5}\) while the mass term does not, and the interaction \(-j^{\mu}A_{\mu}\) is built from \(\gamma^{\mu}\) alone. A symmetry of the classical Lagrangian that survives regularization — and dimensional regularization preserves this one, away from the anomaly of Discrete Symmetries and CPT — forbids the generation of a term that violates it. Hence no mass can be generated when \(m_{e}=0\), so \(\delta m\) must vanish with \(m_{e}\), and by dimensional analysis the only remaining possibility is \(\delta m\propto m_{e}\times\)(logarithm). Numerically, \(3\alpha/2\pi=3.484\times 10^{-3}\) and \(51.5\times3.484\times 10^{-3}=0.179\). The contrast with a scalar field is the whole reason the point is worth making: a scalar mass has no symmetry protecting it, its correction goes as \(\Lambda^{2}\) rather than \(m\ln\Lambda\), and the resulting sensitivity of the Higgs mass to the cutoff is the naturalness problem recorded in Electroweak Unification and the Higgs Boson.

Derives Lemma 100.40. Using \(\gamma^{\nu}\gamma^{\mu}=\eta^{\nu\mu} -\ii\sigma^{\nu\mu}\), which is the split of a product into its symmetric and antisymmetric parts,

by the Dirac equations \(\bar{u}(p')\gamma\cdot p'=m_{e}c\bar{u}(p')\) and \(\gamma\cdot p\,u(p)=m_{e}c\,u(p)\). Expanding, \(\gamma\cdot p'\gamma^{\mu}=p'^{\mu}-\ii p'_{\nu}\sigma^{\nu\mu}\) and \(\gamma^{\mu}\gamma\cdot p=p^{\mu}-\ii p_{\nu}\sigma^{\mu\nu}\), and \(\sigma^{\nu\mu}=-\sigma^{\mu\nu}\), so the antisymmetric terms combine into \(\ii\sigma^{\mu\nu}(p'-p)_{\nu}\).

Derives Theorem 100.41. By Proposition 100.14 the one-loop vertex is

with \(\ell'=\ell+q\). The numerical prefactor is \(\left(-\ii e^{3}/\hbar^{3}\right) \left(-\ii\mu_{0}\hbar^{3}c\right)\left(\ii\hbar\right)^{2} =e^{3}\mu_{0}c\hbar^{2}\), so, dividing by \(\ii e/\hbar\) and using \(\mu_{0}ce^{2}/\hbar=4\pi\alpha\),

every \(\hbar\) having cancelled between \(\hbar^{4}\) in the prefactor and \((2\pi\hbar)^{-4}\) in the measure. The answer is already constrained to its final form: the correction is \(\alpha\) times a dimensionless integral over a dimensionless integrand, so \(F_{2}(0)\) can only be \(\alpha\) times a pure number.

The numerator, by Equation (100.37), is

Combine the three denominators by Lemma 100.26 with parameters \(x\) on \(\ell'^{2}-m_{e}^{2}c^{2}\), \(y\) on \(\ell^{2}-m_{e}^{2}c^{2}\) and \(z\) on \((\ell-p)^{2}\), and \(x+y+z=1\). Using \(p^{2}=p'^{2}=m_{e}^{2}c^{2}\) and hence \(2p\cdot q=-q^{2}\), the combined denominator is \((k^{2}-\Delta)^{3}\) after the shift \(k=\ell+xq-zp\), with

Terms odd in \(k\) integrate to zero. The \(F_{2}\) part of the answer comes entirely from the terms in \(N^{\mu}\) proportional to \((p+p')^{\mu}\), so those are what must be collected. At \(k=0\),

using \(q=p'-p\). Sandwiching between \(\bar{u}(p')\) and \(u(p)\) and using the Dirac equations repeatedly, together with \(\gamma\cdot p\,\gamma^{\mu}\gamma\cdot p=2p^{\mu}\gamma\cdot p -m_{e}^{2}c^{2}\gamma^{\mu}\) and \(2p\cdot p'=2m_{e}^{2}c^{2}-q^{2}\), the four cross terms of \(-2\,\gamma\cdot\ell\,\gamma^{\mu}\gamma\cdot\ell'\) reduce to

while the second piece of \(N^{\mu}\) contributes \(4m_{e}c\left[2z\,p^{\mu}+(y+z-x)q^{\mu}\right]\). Writing \(p^{\mu}=\tfrac{1}{2}\left[(p+p')^{\mu}-q^{\mu}\right]\) and \(p'^{\mu}=\tfrac{1}{2}\left[(p+p')^{\mu}+q^{\mu}\right]\) and collecting the coefficient \(b\) of \((p+p')^{\mu}\),

and with \(x+y=1-z\) this is \(3-3z+2z-(1-z)^{2}-2=z-z^{2}=z(1-z)\), so \(b=2m_{e}c\,z(1-z)\). By the Gordon identity Equation (100.61) read backwards, \((p+p')^{\mu}=2m_{e}c\,\gamma^{\mu}-\ii\sigma^{\mu\nu}q_{\nu}\) between the spinors, so the coefficient of \(\ii\sigma^{\mu\nu}q_{\nu}/2m_{e}c\) in \(N^{\mu}\) is \(-2m_{e}c\,b=-4m_{e}^{2}c^{2}z(1-z)\).

It remains to do the momentum integral. From Lemma 100.26 with \(n=3\) the Feynman combination carries a factor \(2\), and from Equation (100.45) with \(n=3\), \(d=4\),

which is finite: the \(F_{2}\) part needs no regulator, and this is why \(a_{e}\) is a prediction while \(F_{1}\) is not. Assembling,

the prefactor being that of Equation (100.63), in which every \(\hbar\) has already cancelled. At \(q^{2}=0\), Equation (100.64) gives \(\Delta=(1-z)^{2}m_{e}^{2}c^{2}\), so

the \(y\) integral running over the length \(1-z\) of the simplex. Numerically \(\alpha/2\pi=1.161410\times 10^{-3}\).

Derives Proposition 100.43. In the cutoff scheme, \(\int^{\Lambda}\dd^{4}\ell_{\mathrm{E}} \left(\ell_{\mathrm{E}}^{2}+\Delta\right)^{-2} =2\pi^{2}\int_{0}^{\Lambda}\ell^{3}\dd\ell \left(\ell^{2}+\Delta\right)^{-2} =\pi^{2}\left[\ln(\Lambda^{2}/\Delta)-1+O(\Delta/\Lambda^{2})\right]\); in dimensional regularization the same integral is \(\pi^{2}\left[\varepsilon^{-1}-\gamma_{\mathrm{E}}+\ln4\pi -\ln(\Delta/\bar{\mu}^{2})\right]\) by Equation (100.44) and Lemma 100.29. The two differ by a \(\Delta\)-independent constant, which is Equation (100.67); and in a difference at two values of \(\Delta\) the constant cancels together with \(\Lambda\) and \(\bar{\mu}\).

Derives Theorem 100.48. Consider the Green function \(G^{\mu}(x;y,z)=\avg{T\,j^{\mu}(x)\,\psi(y)\,\bar{\psi}(z)}\) with \(j^{\mu}\) the current Equation (100.6). Differentiating, the divergence picks up two contributions: one from \(\pp_{\mu}j^{\mu}=0\), which vanishes, and one from differentiating the time-ordering symbol, which produces equal-time commutators. Using the canonical relation of Canonical Quantization of Fields, \(\comm{j^{0}(x)}{\psi(y)}\delta(x^{0}-y^{0}) =-qc\,\delta^{4}(x-y)\,\psi(y)\), and similarly for \(\bar{\psi}\) with the opposite sign,

Fourier transforming with Equation (100.14) turns \(\pp_{\mu}\mapsto-\ii q_{\mu}/\hbar\) on the left and the delta functions into unity on the right, so that

the left-hand side being the amputated vertex reattached to two full propagators. Multiplying by \(S^{-1}\) on both sides gives Equation (100.71); the normalization is fixed by comparing the tree-level terms, where \(\Gamma^{\mu}=\gamma^{\mu}\) and \(q_{\mu}\gamma^{\mu}=\gamma\cdot(p+q)-\gamma\cdot p\) identically. Equation (100.72) follows by dividing by \(q\) and letting \(q\to0\). The only inputs are current conservation, which is Noether's theorem applied to the exact symmetry of Theorem 100.5, and the canonical commutator; neither is perturbative, so the identity holds order by order and also non-perturbatively.

Derives Corollary 100.49. By Definition 100.46, \(\delta_{2}\) is fixed by the derivative of \(\Sigma\) at the mass shell and \(\delta_{1}\) by \(\Gamma^{\mu}(p,p)=Z_{1}^{-1}\gamma^{\mu}\) at \(F_{1}(0)=1\). Take Equation (100.72) on shell: the right-hand side is \(\gamma^{\mu}-\pp\Sigma/\pp p_{\mu}\), whose divergent part is \(\gamma^{\mu}(1-\delta_{2}^{\text{div}})\) by the definition of \(\delta_{2}\), while the left-hand side is \(\gamma^{\mu}(1-\delta_{1}^{\text{div}})\) by the definition of \(\delta_{1}\). Hence \(\delta_{1}=\delta_{2}\) to all orders, so \(Z_{1}=Z_{2}\), and Equation (100.68) collapses to Equation (100.73). Proposition 100.6 asserted universality at tree level; this is the statement that it is not destroyed at any order, which is what had to be shown, since \(Z_{1}\) and \(Z_{2}\) are separately mass-dependent and only their ratio is not.

Derives Proposition 100.55. Both schemes are related by a finite redefinition \(\alpha\mapsto\alpha'=\alpha(1+c_{1}\alpha+\dots)\), \(m\mapsto m'=m(1+d_{1}\alpha+\dots)\) with \(c_{i},d_{i}\) finite. An observable computed to order \(\alpha^{n}\) in one scheme, re-expanded in the other's parameters, differs only at order \(\alpha^{n+1}\) — that is, by terms of the order that has not been computed. Once the parameters are eliminated in favour of two measurements, the prediction for a third quantity is scheme independent at every order at which it is complete. This is the criterion that separates convention from physics in this chapter: \(\alpha\), \(m_{e}\), \(\xi\), \(\bar{\mu}\) and the choice of regulator are conventions; the relation between Equation (100.62) and the Lamb shift is not.

Derivation. Derives Phenomenon 100.57. Let \(\mathcal{M}_{0}\) be the amplitude for a process in which an electron of momentum \(p\) is scattered into momentum \(p'\), and attach one outgoing photon of momentum \(k\) and polarization \(\varepsilon_{(\lambda)}\). The photon may be emitted from the outgoing leg or the incoming one. In the first case the electron propagator immediately before emission carries momentum \(p'+k\), so by Equation (100.23) it contributes

using \(p'^{2}=m_{e}^{2}c^{2}\) and \(k^{2}=0\). As \(k\to0\) the denominator vanishes linearly, so this term dominates; in the numerator \(\gamma\cdot k\) may be dropped and \(\left(\gamma\cdot p'+m_{e}c\right)\gamma^{\mu} \varepsilon^{*}_{\mu}u \to 2p'\cdot\varepsilon^{*}\,u\) by the Dirac equation. The emission from the incoming leg gives the same structure with \(p'\to p\) and an opposite sign, because there the propagator carries \(p-k\). Hence the amplitude factorizes,

with \(\mathcal{J}^{\mu}\) the eikonal current, of dimension \(\mathrm{C}\,\mathrm{s}/\mathrm{kg}/\mathrm{m}\); the whole dependence on the hard process has dropped out. Squaring, summing over the two polarizations with Equation (100.34), and integrating over the photon phase space of Equation (100.39), the expected number of photons emitted is

which is dimensionless as it must be, the bracket carrying the inverse square of a momentum and \(\dd^{3}k/\abs{\vect{k}}\) its square. Every term in the bracket scales as \(\abs{\vect{k}}^{-2}\), while \(\dd^{3}k=\abs{\vect{k}}^{2}\dd\abs{\vect{k}}\,\dd\Omega\), so the photon-momentum dependence is exactly

and \(\int_{0}^{\omega_{\mathrm{max}}}\dd\omega/\omega\) diverges logarithmically at the lower limit. The number of photons emitted below any energy is infinite; the energy they carry, \(\int\hbar\omega\,\dd\avg{n}\propto\int\dd\omega\), is finite, which is why nothing physical is wrong. The angular integration for an ultrarelativistic deflection through momentum transfer \(q\) gives

the double logarithm of Bloch and Nordsieck [Bloch:1937] and Weinberg [Weinberg:1965]. Since \(\avg{n}\to\infty\) as \(\omega_{\mathrm{min}}\to0\), the probability of emitting no photon at all, which for the Poisson distribution the soft emissions obey is \(\ee^{-\avg{n}}\), is zero. The elastic peak of any real experiment is the set of events in which no photon above the detector threshold was emitted, and it is therefore a property of the threshold.

Derivation. Derives Phenomenon 100.62. The measured coupling at momentum transfer \(q\) is the one that appears in the dressed propagator Equation (100.52): the product of two charges and one photon exchange carries \(\alpha/\left[1-\Pi_{2}(q^{2})\right]\) where \(\alpha\) is the Thomson-limit value fixed by Definition 100.46. Since the condition there was \(\Pi_{2}(0)=0\) after subtraction, the effective coupling is

By Proposition 100.33 the subtracted polarization is

and for \(-q^{2}\gg m_{e}^{2}c^{2}\) the logarithm becomes \(\ln\left[m_{e}^{2}c^{2}/\left(-x(1-x)q^{2}\right)\right]\). With \(\int_{0}^{1}x(1-x)\dd x=1/6\) and \(\int_{0}^{1}x(1-x)\ln\left[x(1-x)\right]\dd x =2\int_{0}^{1}x(1-x)\ln x\,\dd x=2\left(-\tfrac{1}{4} +\tfrac{1}{9}\right)=-\tfrac{5}{18}\),

so that the effective coupling grows logarithmically with \(-q^{2}\). Numerically, running from \(q^{2}=0\) to \(-q^{2}=m_{Z}^{2}c^{2}\) with only the electron in the loop gives \(\ln\left(m_{Z}/m_{e}\right)^{2}=\ln\left(1.78\times 10^{5}\right)^{2}=24.2\) and \(\alpha\left(24.2-5/3\right)/3\pi=0.0174\), which is the electron's share of the shift. The muon and the tau contribute \(0.0092\) and \(0.0048\) by the same formula with their own masses, and the quarks contribute about \(0.0276\), weighted by squared charge and by three colours and evaluated from data rather than from Equation (100.50) for the reason given below; the total \(\Delta\alpha\approx0.059\) is the observed \(\alpha^{-1}(m_{Z}^{2}c^{2})\approx129\). The physical reading is Remark 100.34: a probe of higher momentum transfer penetrates further into the polarization cloud and sees a larger charge.

Derives Proposition 100.63. From Equation (100.83) at one loop, \(\alpha(\bar{\mu})^{-1}=\alpha^{-1} -\left(3\pi\right)^{-1}\ln\left(\bar{\mu}^{2}/m_{e}^{2}c^{2}\right)\) for a single unit-charge fermion. Differentiating with respect to \(\ln\bar{\mu}\), \(-\alpha^{-2}\,\dd\alpha/\dd\ln\bar{\mu}=-2/3\pi\), which is Equation (100.84). Each additional species contributes an identical loop with \(e^{2}\) replaced by \(q_{f}^{2}\) and multiplied by its number of colours, whence the sum; each decouples below its own mass, which is the decoupling of heavy states [Appelquist:1975], implicit in the mass dependence of \(\Delta\) in Equation (100.50). The sign is fixed by the fermion loop being a screening medium (Remark 100.34), and it is the opposite of the non-abelian case [Politzer:1973] [Gross:1973]. The systematic apparatus — the Callan–Symanzik equation [Callan:1970] [Symanzik:1970], anticipated by Gell-Mann and Low [GellMann:1954] and, earlier still, by Stueckelberg and Petermann [Stueckelberg:1953] — is The Renormalization Group.

Derives Proposition 100.64. \(\alpha(\bar{\mu})^{-1}=\alpha^{-1} -\left(3\pi\right)^{-1}\ln\left(\bar{\mu}^{2}/m_{e}^{2}c^{2}\right)\) vanishes when \(\ln\left(\bar{\mu}^{2}/m_{e}^{2}c^{2}\right)=3\pi/\alpha\), that is at \(\bar{\mu}=m_{e}c\exp\left(3\pi/2\alpha\right)\). With \(\alpha^{-1}=137.036\) the exponent is \(3\pi\times137.036/2=645.8\), so \(\exp(645.8)=10^{280.4}\), and multiplying by \(m_{e}c^{2}=0.511\times 10^{6}\,\mathrm{eV}\) gives Equation (100.85).

Derivation of the dominant term. Derives Phenomenon 100.66. Bethe's calculation [Bethe:1947] is non-relativistic and treats the electron in the Coulomb field of the proton, coupled to the quantized transverse field of Canonical Quantization of Fields. In the Coulomb gauge the coupling of an electron of charge \(-e\) is, to first order in the vector potential,

with \(\omega_{k}=c\abs{\vect{k}}\) and \(V\) the quantization volume; the normalization is the one that makes the field energy \(\sum\hbar\omega_{k}\left(n_{\vect{k}\lambda} +\tfrac{1}{2}\right)\). Second-order perturbation theory for the state \(\ket{n}\) with no photons present, summed over intermediate atomic states \(\ket{m}\) and one-photon states, gives

writing \(E_{nm}=E_{n}-E_{m}\) and \(\vect{p}_{mn} =\bra{m}\hat{\vect{p}}\ket{n}\), and using the dipole approximation \(\ee^{\ii\vect{k}\cdot\vect{x}}\approx1\), legitimate because the relevant wavelengths are long compared with the atom. Converting the mode sum, \(\sum_{\vect{k}}\to V(2\pi)^{-3}\int\dd^{3}k\), and using \(\sum_{\lambda}\int\dd\Omega\, \abs{\vect{\varepsilon}_{(\lambda)}\cdot\vect{p}}^{2} =\tfrac{8}{3}\pi\abs{\vect{p}}^{2}\) together with \(k^{2}\dd k/\omega=\omega\,\dd\omega/c^{3}\),

where \(e^{2}\hbar/2\varepsilon_{0}m_{e}^{2}\cdot \left(3\pi^{2}c^{3}\right)^{-1} =2\alpha\hbar^{2}/3\pi m_{e}^{2}c^{2}\) by \(e^{2}/\varepsilon_{0}=4\pi\alpha\hbar c\). The integral diverges linearly, which is the divergence of Theorem 100.35 seen in non-relativistic disguise. Split it:

The first piece contributes \(-\left(2\alpha\hbar/3\pi m_{e}^{2}c^{2}\right) \omega_{\mathrm{max}}\sum_{m}\abs{\vect{p}_{mn}}^{2} =-\left(2\alpha\hbar\omega_{\mathrm{max}}/3\pi m_{e}c^{2}\right) \avg{\hat{\vect{p}}^{2}/m_{e}}_{n}\) by completeness, that is a term proportional to the expectation of the kinetic energy: it is a change in the electron's mass, present for a free electron as much as a bound one, and it is removed by the mass renormalization of Corollary 100.36. This subtraction is the whole of Bethe's insight, and it is renormalization performed before anyone had a name for it. What remains is

with \(E_{mn}=-E_{nm}\), the logarithm arising from \(\int^{\omega_{\mathrm{max}}}\dd\omega/(E_{nm}-\hbar\omega)\). Bethe cut the integral off at \(\hbar\omega_{\mathrm{max}}=m_{e}c^{2}\), the energy above which the non-relativistic treatment is meaningless; the covariant calculation [Kroll:1949] later confirmed that this is the right scale and supplied the constant beside the logarithm. Defining the Bethe logarithm by \(\ln\avg{\Delta E}\) such that the weighted average may be taken outside the sum, the remaining sum is a commutator:

using \(\comm{H}{\hat{\vect{p}}}=\ii\hbar\nabla V\) and \(\comm{\hat{p}_{j}}{\ii\hbar\pp_{j}V}=\hbar^{2}\nabla^{2}V\). For the Coulomb potential \(V=-e^{2}/4\pi\varepsilon_{0}r\) one has \(\nabla^{2}V=\left(e^{2}/\varepsilon_{0}\right)\delta^{3}(\vect{r})\), so the sum equals \(2\pi\alpha\hbar^{3}c\,\abs{\psi_{n}(0)}^{2}\) and vanishes for every state with \(\ell\neq0\) — which is why the effect separates \(2S\) from \(2P\) at all. With \(\abs{\psi_{n00}(0)}^{2}=\alpha^{3}m_{e}^{3}c^{3}/\pi n^{3}\hbar^{3}\) (The Hydrogen Atom),

Numerically, for \(n=2\) the prefactor is \(4\alpha^{5}m_{e}c^{2}/24\pi=8.99\times 10^{-26}\,\mathrm{J}\), that is \(h\times135.6\,\mathrm{MHz}\); Bethe's estimate \(\avg{\Delta E}\approx17.8\,\mathrm{Ry}=242\,\mathrm{eV}\) gives a logarithm \(\ln\left(5.11\times 10^{5}/242\right)=7.66\) and hence \(1040\,\mathrm{MHz}\), against the \(1000\,\mathrm{MHz}\) Lamb and Retherford had measured. That a divergent theory, subtracted by hand, reproduced a measured microwave frequency to four percent within weeks of the measurement is the event from which the whole of Section 100.5 follows.

Derivation. Derives Phenomenon 100.67. The leading term is Theorem 100.41, \(a_{e}=\alpha/2\pi=1.161410\times 10^{-3}\), derived in full in Section 100.4.3. It already accounts for the measured value to one part in \(10^{3}\), and the remainder is the rest of the series. Because the vertex correction was shown there to be \(\alpha\) times a dimensionless integral, the whole expansion is a power series in \(\alpha/\pi\) with pure numbers as coefficients,

with, from [Aoyama:2019],

These are the mass-independent coefficients \(A_{1}^{(2n)}\) of that reference; the mass-dependent terms are the separate \(a_{e}^{\mu,\tau}\) of Equation (100.90). \(C_{5}\) is known only numerically, its uncertainty is the error of the numerical integration, and the value quoted supersedes the same group's earlier \(6.675(192)\). \(C_{2}\) and \(C_{3}\) are known in closed form; \(C_{4}\) required the numerical evaluation of \(891\) diagrams and \(C_{5}\) of \(12672\), and an independent recomputation of the fifth-order term [Volkov:2019] does not agree with the published value within its quoted uncertainty — a fact recorded here because a single unverified computation would otherwise be load-bearing. The remaining terms are small: muon and tau loops contribute \(2.75\times 10^{-12}\), hadronic vacuum polarization and hadronic light-by-light about \(1.69\times 10^{-12}\), and the electroweak contribution about \(0.03\times 10^{-12}\) [Aoyama:2019]. Summing at \(\alpha^{-1}=137.035999177\) reproduces \(0.00115965218\), in agreement with [Fan:2023].

The honest statement of the test requires one more step, and it is not the one usually made. The series is a function of \(\alpha\), so the comparison needs \(\alpha\) measured somewhere else. The two best independent determinations, both from photon-recoil interferometry, are

and they disagree by more than five combined standard uncertainties. Propagating the difference through \(a_{e}=\alpha/2\pi+\dots\) gives a spread in the predicted \(a_{e}\) of about \(1.4\times 10^{-12}\), roughly ten times the experimental uncertainty \(1.3\times 10^{-13}\) of [Fan:2023]. What QED is confirmed to is therefore the precision of its input — about one part in \(10^{9}\) of \(a_{e}\) — and not the thirteen figures that the precision of \(a_{e}\) alone would suggest. The arithmetic is carried out in Experiment: The Electron Anomalous Magnetic Moment, and the conclusion is worth stating plainly here because the looser claim is repeated everywhere.

Derivation of the effective Lagrangian and the scale. Derives Phenomenon 100.69. The amplitude is the \(E_{e}=0\), \(E_{\gamma}=4\) entry of Theorem 100.17, with superficial degree of divergence \(D=0\): naively logarithmically divergent. It is in fact finite, and for the same reason Theorem 100.31 was only logarithmically divergent — gauge invariance forces each of the four photon legs to appear through its field strength rather than its potential, so the amplitude carries four explicit powers of momentum and the would-be divergence is cancelled. There is accordingly no counterterm for photon–photon scattering, and the process is a parameter-free prediction.

At photon energies well below \(m_{e}c^{2}\) the electron loop may be integrated out, leaving a local interaction among the field strengths. Lorentz and gauge invariance and the \(C\), \(P\), \(T\) symmetry of Section 100.1.4 permit only two independent quartic invariants, and the coefficients follow from expanding the exact one-loop effective action of a constant field [Heisenberg:1936] [Schwinger:1951a]:

The dimensions check: \(\varepsilon_{0}E^{2}\) is an energy density, so the bracket with its prefactor \(\varepsilon_{0}^{2}\) carries \(\mathrm{J}^{2}/\mathrm{m}^{6}\), and \(\hbar^{3}/m_{e}^{4}c^{5}\) carries \(\mathrm{m}^{3}/\mathrm{J}\), leaving \(\mathrm{J}/\mathrm{m}^{3}\) as an addition to Equation (100.12) must. Compare Equation (100.91) with the Maxwell term \(\tfrac{\varepsilon_{0}}{2}\left(E^{2}-c^{2}B^{2}\right)\) for a pure electric field. Their ratio is \(4\alpha^{2}\varepsilon_{0}\hbar^{3}E^{2}/45m_{e}^{4}c^{5}\), and using \(\varepsilon_{0}\hbar c/e^{2}=1/4\pi\alpha\) it collapses to

with \(E_{\mathrm{cr}}\) the Schwinger field [Schwinger:1951a], at which the work done on an electron over a Compton wavelength equals its rest energy and at which the vacuum becomes unstable against pair creation. Two statements must be kept apart here, because the literature runs them together. \(E_{\mathrm{cr}}\) is the scale that makes the expansion parameter \(E/E_{\mathrm{cr}}\) of order one, and it is the field at which the weak-field expansion Equation (100.91) stops being an expansion at all, since pair creation sets in with a rate carrying \(\exp\left(-\pi E_{\mathrm{cr}}/E\right)\). The two terms of the Lagrangian are not equal there: the loop factor \(\alpha/45\pi=5.16\times 10^{-5}\) makes the ratio that small at \(E=E_{\mathrm{cr}}\), and it would reach unity only at \(\sqrt{45\pi/\alpha}\approx139\) times \(E_{\mathrm{cr}}\), that is about \(1.8\times 10^{20}\,\mathrm{V}/\mathrm{m}\) — a field the theory as written cannot be extrapolated to. Laboratory static fields are some ten orders of magnitude below \(E_{\mathrm{cr}}\), which is why the vacuum looks linear.

The corresponding photon–photon cross-section for unpolarized light of energy \(\hbar\omega\ll m_{e}c^{2}\) in the centre-of-mass frame is

the sixth power of the energy being the direct consequence of the four powers of field strength in Equation (100.91) together with the two-body phase space. The \(\alpha^{4}\) and the sixth power together make the cross-section utterly negligible for visible light: at \(\hbar\omega=1\,\mathrm{eV}\) it is of order \(10^{-70}\,\mathrm{m}^{2}\).

What makes the process observable is a heavy-ion collider. A nucleus of charge \(Ze\) passing at small impact parameter carries a coherent electromagnetic field whose equivalent photon flux scales as \(Z^{2}\), so a photon–photon luminosity in Pb–Pb collisions is enhanced by \(Z^{4}=82^{4}\approx4.5\times 10^{7}\) over the proton case. ATLAS reported evidence at \(4.4\) standard deviations [Aaboud:2017] and then observation at \(8.2\) [Aad:2019], with a fiducial cross-section of about \(7.8\times 10^{-36}\,\mathrm{m}^{2}\) — about \(78\) nanobarns — consistent with the QED prediction.

Derives Proposition 100.71. Route (i) is Equation (100.90) solved for \(\alpha\); route (ii) is kinematic, since a photon of known frequency transfers a known momentum to an atom of known mass and the resulting recoil frequency shift gives \(h/m_{\mathrm{at}}\), from which \(\alpha\) follows through the displayed relation with \(R_{\infty}\) and the mass ratio measured independently; route (iii) uses the theoretical expression for the hyperfine interval, which is proportional to \(\alpha^{2}\) times the magnetic moment ratio. The routes share no systematic error, and two of them do not presuppose the QED series at all. That is what converts \(\alpha\) from a parameter fitted to one measurement into a measured constant of Nature, and it is what makes the anomaly a test rather than a definition — the distinction insisted on in Remark 100.56.