QUANTUM ELECTRODYNAMICS (QED)

Quantum electrodynamics represents the merger of three great ideas of modern physics:

classical electrodynamics as synthesized by Maxwell, quantum mechanics, and special rela-tivity. The basic formulation of QED was completed by 1930; it combined the Dirac theory of the electron (with its correct predictions of the lowest order electron magnetic dipole moment and the existence of positrons) with the quantization of the electromagnetic field into individual photons. A workable prescription for handling the divergent integrals by renormalization of the electron mass, charge, and wave function was completed by the early 1950s by Bethe, Feynman, Tomonaga, Schwinger, Dyson and others (Schwinger, 1958).

QED therefore became a mathematically consistent and well-defined theory, which was subsequently tested to incredible precision.

In order to consider QED seriously, one must include higher-order (loop) effects in per-turbation theory, both because of the precision of many of the tests and because some effects (such as light by light scattering via the interaction of photons with virtual charged particle loops) only occur at loop level. The calculation of higher-order effects is greatly compli-cated by divergences and the need to renormalize (while maintaining gauge invariance). A systematic study is beyond the scope of this book, and only brief comments are made. The subject is studied in detail in standard texts on field theory. Another complication is that strong interaction effects are very important and cannot be ignored in studying the electro-magnetic interactions of strongly interacting particles (hadrons), such as protons, neutrons, and pions. They even enter at higher orders (through loops involving virtual hadrons) in the electrodynamics of electrons and muons. Fortunately, a great deal can be said about these strong interaction corrections using symmetry principles.

2.12.1 Higher-Order Effects

Analogous to the Hermitian scalar case inSection 2.3.6, QED has logarithmically divergent diagrams such as those shown in Figure 2.17. To take this into account, let us modify the notation in the QED Lagrangian density in (2.218) to

L= ¯ψ(x) (i6∂+e06A−m0)ψ(x)−1

4F_µνF^µν

= ¯ψ(x) (i6∂−m0)ψ(x)−e0J_Q^µA_µ−1

4F_µνF^µν,

(2.348)

where J_Q^µ = −ψγ¯ ^µψ is the electromagnetic current operator defined in (2.294) and ψ is the electron field.e0 andm0 are the bare positron charge and mass, respectively, i.e., the

q

1 2

Figure 2.17

Electron-photon vertex (top left), and one-loop corrections, correspond-ing to the vertex correction (top middle), the vacuum polarization (or photon self-energy) correction (top right), and electron self-energy corrections (bottom).

parameters that appear in L. We redefine e and m as the physical parameters, i.e., the ones actually measured. The renormalizability of QED implies that the divergences enter only in the relations between the bare and physical parameters and in the unobservable wave function renormalization of the fields,²⁶ and that observable quantities are finite to all orders in perturbation theory when expressed in terms ofeandm.

The one and higher-loop vertex corrections imply that the lowest order electron-photon vertex ie₀γ^µ is replaced by a function ie₀Γ^µ(p₂, p₁)/Z₁ that can depend on the external momenta.Z₁is the (divergent) vertex renormalization constant, defined by the requirement that Γ^µ =γ^µ at the on-shell pointp²₂ =p²₁ =m² and q² = (p₂−p₁)² = 0. Similarly, the electron self-energy and vacuum polarization diagrams modify the electron and photon propagators so that on shell they take the same form as the free ones except they are multiplied by divergent wave function renormalization factorsZ_2,3,

SF(p)→ Z₂

6p−m+i, ig^µνDF(q)→ −ig^µν Z₃

q²+i, (2.349) where the position of the pole in SF(p) defines the physical electron mass. It is convenient to associate a factor ofZ₂^1/2orZ₃^1/2from each electron or photon line with the vertex. The otherZ^1/2is associated with the vertex or external state at the other end of the line. The overall vertex factor therefore becomes

ie0γ^µ →ie0

Z2Z₃^1/2 Z1

Γ^µ(p2, p1). (2.350)

26In addition to the ultravioletdivergences (associated with large momenta in the integrals) discussed here, there are also infrared divergences associated with low momentum virtual photons. These can be regulated by introducing a fictitious photon mass; they cancel against similar terms associated with the emission of soft real photons at energies below the threshold of the detector.

The renormalization factorsZ_1,2,3can be combined withe₀to define the physical chargee.

TheWard-Takahashi identityensures thatZ1=Z2 to all orders, so we can identify e=e0

Z2Z₃^1/2 Z1

=e0Z₃^1/2. (2.351)

It can then be shown that Γ^µ(p2, p1) is finite to all orders when expressed in terms of e and m, and in particular, Γ^µ(p2, p1) = γ^µ +O(α) where α = e²/4π ∼ 1/137 is the fine structure constant. The charge renormalization depends only on the photon vacuum polarization diagrams and is therefore the same for all particles, e.g., for electrons with charge−e₀→ −e, and for u-quarks with charge +2e₀/3→2e/3. Similarly, after removing the Z_2,3 factors, the electron and photon propagators behave like the free-field ones near the physicalp²=m²orq²= 0 poles, but can have momentum dependent corrections away from the physical masses. These can also be shown to be finite to all orders when expressed in terms ofeandm. The vacuum polarization corrections will be discussed inSection 2.12.2 in connection with running couplings.

The relation of such renormalized quantities to physical on-shell amplitudes and matrix elements is considered in field theory texts, but it should be plausible from the above discussion that the on-shell matrix element of−J_Q^µ(x) between physical electron states is

hp~₂s₂|ψ¯(x)γ^µψ(x)|~p₁s₁i= ¯u₂Γ^µ(p₂, p₁)u₁e^iq·x, (2.352) whereq=p2−p1and thexdependence follows from translation invariance, Equation (1.15) on page 3. The form of Γ^µ(p2, p1) is strongly restricted by symmetry considerations, which continue to hold in the presence of the strong interactions. In particular, Lorentz invariance implies that the r.h.s of (2.352) must be a four-vector, which can only be constructed from p^µ₁, p^µ₂, and the Dirac matrices. The Gordon identities in Problem 2.10 indicate that it is sufficient to consider p^µ_1,2 in the combinationq^µ only. Furthermore, QED (and the strong interactions) are reflection invariant. Therefore, using (2.276) the most general allowed form is

¯

u₂Γ^µ(p2, p₁)u1= ¯u₂

γ^µF₁(q²) +iσ^µν

2m q_νF₂(q²) +q^µF₃(q²)

u₁, (2.353) where theform factorsF1,2,3(q²) are Lorentz invariant functions ofq². Charge conservation,

∂µJ_Q^µ = 0, which can be derived to all orders from the equations of motion or from the Noether theorem of Section 3.2.2, requires that q²F3(q²) = 0. One does not expect a δ function to develop to any order, soF3(q²) must vanish. Also, the Hermiticity ofJ_Q^µ implies Γ^µ(p2, p1) = Γ^µ(p1, p2), (2.354) where Γ^µ is the Dirac adjoint. FromTable 2.2,F1,2must therefore be real. Finally, we have normalized so thatF1(0) = 1.

We have seen thateF1(0) is just the physical electric charge. To interpretF2(0), consider the interaction of an electron with a static classical gauge potentialA_µ(x) (cf., (2.66) and Problem 2.24). The matrix element of the interaction Hamiltonian

H =−e0

Z

d³~xψ(x)γ¯ ^µψ(x)Aµ(x) (2.355) between one-electron states is

h~p₂s₂|H|p~₁s₁i=−e¯u₂Γ^µu₁ Z

A_µ(~x)e^−i~q·~xd³~x≡ −e¯u₂Γ^µu₁ A˜_µ(~q). (2.356)

Expanding the fermion bilinear in (2.356) to linear order in the non-relativistic limit

|~p_i| m, this reduces (Problem 2.30) to φ^†_s2

−2meA˜0(~q) +e(~p1+~p2)·A(~~˜ q) +e[1 +F2(0)]~σ·B(~~˜ q)

φ_s1, (2.357) where φ_s1,2 are the two-component Pauli spinors and B~˜ is the Fourier transform of the magnetic fieldB~ =∇ ×~ A~. The conventional non-relativistic interaction Hamiltonian for an e⁻ in an external field is Bohr magneton,S~ =~σ/2 is the spin operator, and g is the electron g-factor, which is not predicted in the non-relativistic theory. The three terms correspond to the Coulomb interac-tion, and the orbital and spin magnetic moment interactions, respectively. The momentum space matrix element ofH_I (corrected to agree with our covariant state normalization) is

h~p2s2|H|~p1s1i= 2m Z

d³~x e^−i~q·~xφ^†_s2HIφs₁, (2.359) with the ~p operators in HI replaced by the ~p2,1 eigenvalues. This coincides with (2.357) provided that one identifies theg-factor asg= 2 [1 +F2(0)], where the 2 is the relativistic Dirac contribution (from γ^µF1(0)), and F2(0) = (g−2)/2 is the anomalous QED term, usually denoted bya_e. The leading contribution, from the vertex diagram inFigure 2.17, is F₂(0) =α/2π, as first calculated by Schwinger. The calculation is sketched inAppendix Eas an illustration of the techniques for calculating Feynman loop integrals. The relation of field theory matrix elements to non-relativistic potentials is further discussed in Problem 2.33.

2.12.2 The Running Coupling

We saw in (2.351) that the renormalization of the electric charge is due to the vacuum polarization diagram ofFigure 2.17, with the divergent parts of the electron self-energy and vertex diagrams cancelling by the Ward-Takahashi identity. Now, let us consider the vacuum polarization as a function ofq². The one-loop vacuum polarization diagram inFigure 2.18, can be added to the tree level photon propagator to yield (see, e.g., Peskin and Schroeder, 1995)

where we have also included a factor ofe²₀from the outside vertices. Λ is anultraviolet cutoff of the divergent momentum integral. In practice, the cutoff must be introduced in a way that respects gauge invariance, such as in the Pauli-Villars or dimensional regularization schemes. Π(q²) is a finite function that vanishes atq² = 0, so we can identify the photon wave function renormalization factor as

which diverges for Λ→ ∞. The r.h.s. of (2.360) can be rewritten

−ig_µν

where the corrections to the last form are higher order ine²₀or e². One can show that this form holds and is finite to all orders, provided Π(q²) is expressed in terms of e² and m². Thee²ⁿΠⁿ term in the expansion (1−e²Π)⁻¹= 1 +e²Π +e⁴Π²+· · · corresponds to the diagram withnseparate vacuum polarization bubbles along the line.

e₀ q e₀ e₀ e₀

e₀ e₀