This illustrates how the divergences disappear from the expressions for physical observ-ables when they are written in terms of the renormalized quantities. However, one may still have a nagging doubt about the underlying divergences. In fact, the modern view is that the “divergent” momentum integrals are actually cut off physically at the scale at which the theory is replaced by a more complete one, and that Λ should be associated with that scale and not taken to infinity. One can then interpret the renormalizations as finite quantities describing, e.g., the difference between a coupling as measured at a scale much smaller than Λ and the value it would have at Λ. A logarithmically “divergent” term in a weak coupling theory such as QED is then actually a small effect, e.g., 12πe202lnmΛ22 ∼ 0.08 fore20∼e2 and Λ∼MP ∼1019GeV (the Planck scale). Most of the divergences in renormalizable theories are of this logarithmic nature,27 and results are therefore insensitive to the details of the new physics above Λ. Non-renormalizable theories, on the other hand, typically encounter new divergences of order Λ2n at n-loop level, and are very sensitive.
The function Π(q2) in (2.362) is Π(q2) = 1
2π2 Z 1
0
dz z(1−z) ln
m2−q2z(1−z) m2
, (2.363)
which vanishes as Π(q2) → −q2/60π2m2 for q2 → 0. It is well behaved for q2 ≤ 0 (i.e., t-channel exchange, as in Figure 2.18), but has a branch point at q2 = 4m2 associated with thee+e− threshold in s-channel processes. In the limitQ2≡ −q2m2 (positive for t-channel exchange),
Π(q2)→ 1
12π2ln Q2
m2. (2.364)
The replacement in (2.362) is universal for all photon exchanges. It is useful to introduce a runningor effective fine structure constantαef f Q2
∼e2(Q2)/4π=α/(1−4παΠ(−Q2)).
Such considerations are most useful for large Q2, so we will actually define the running quantity as
α Q2
= α(Q20) 1−α(Q3π20)lnQQ22
0
, (2.365)
27The quadratically divergent (but renormalizable) corrections to scalar self-energies, such as the Higgs mass-square in the standard model, are a critical exception. This will be discussed inChapter 10.
where Q20 is an arbitrary reference scale, such asm2. This coincides with αef f for Q2 Q20 ∼m2, and approaches the bare couplingα0=e20/4π forQ2→Λ2 (up to higher-order corrections). Equation (2.365) is equivalent to
1
α(Q2) = 1
α(Q20)−b0lnQ2
Q20, (2.366)
whereb0≡4πb= 1/3π. Thus,α(Q2)−1 runs linearly with lnQ2 at largeQ2. (Higher-order corrections to the vacuum polarization bubble lead to small nonlinear effects). The running α(Q2) therefore increases logarithmically from its value∼1/137 for smallQ2, so the QED interaction strength should be larger for high energy processes. This was motivated here for spacelike momentum transfersq2<0, but continues to hold even in the timelike region, where higher-order corrections are minimized if one usesα(|q2|). Equivalently, the effective Coulomb interaction scales as α(Q ∼ 1/r)/r, and therefore the effective α increases at smaller separations. There is a simple interpretation of this: an electron charge in a dielectric medium is screened for larger separations, leading to an interaction strength falling faster than 1/r. At small separations, however, the screening is less effective and a test charge feels the full electric charge. The same mechanism applies here, except the dielectric is really due to the quantum fluctuations of the vacuum, as represented by the virtual e+e− loop. We will see inChapter 5that the gluon self-interactions in quantum chromodynamics have the opposite effect of antiscreening (for which there is no simple classical analog), leading to a decreasein the strong coupling at large momenta or short distance (asymptotic freedom).
The running ofα(Q2) is described by therenormalization group equation(RGE) dα(Q2)
dlnQ2 =β(q2) =b0α2(Q2) +O(α3), (2.367) where b0 = 1/3π is due to the one-loop diagram in Figure 2.18, and the other terms are higher-loop contributions to the vacuum polarization bubbles. Equation (2.366) is the solu-tion in one-loop approximasolu-tion. The running in (2.366) is valid forQ2m2; there is little effect for Q2 .m2. However, for Q2 m2µ, where mµ ∼ 200me is the muon mass and me≡m is thee− mass, one should also include the effect of the muon loop in the photon propagator, i.e.,b0 →2/3π, and
1
α(Q2) = 1
α(m2µ)− 2 3πln Q2
m2µ (2.368)
forQ2> m2µ. Thus,b0changes discontinuously and 1/αhas a kink near the particle thresh-old. (The exact form of the threshold, including constant terms, whether it occurs atmµ or 2mµ, etc, depends on the details of the renormalization scheme.) Quark loops also contribute to the vacuum polarization and the running. If one could ignore the strong interactions, then a quark of flavorr(e.g., u, d,s) would contribute a term 3qr2/3πto b0 forQ2> m2r, where qris the quark electric charge in units ofe,mr is its mass, and the 3 is because there are 3 quark colors. Thus,
b0 = 1 3π
X
mr<Q
Crq2r, (2.369)
where the sum includes both quarks and charged leptons, withCr= 1 (leptons) andCr= 3 (quarks). Unfortunately, this is a poor approximation for the strongly interacting particles, which cannot really be treated as free quarks at low energies. Multiple gluon exchanges between the quarks in the vacuum polarization diagram, or hadronic bound state effects,
invalidate the quark part of (2.369). A more reliable approximation can be obtained by replacing the perturbative loop by a dispersion relation
3qr2 3π lnQ2
Q20 → Z ∞
4m2π
F(Q2, s)R(s)ds (2.370) in the 1/α equation.R(s)≡σe+e−(s)/(4πα2/3s) is the ratio of the total cross section for e+e− → hadrons at CM energy √s divided by the cross section for e+e− → µ+µ− and F(Q2, s) is a known function (see, e.g., Eidelman and Jegerlehner, 1995). The low-energy part ofR(s) can be taken from experiment, while the high-energy part is predicted by QCD.
One therefore finds that 1/α decreases from∼137 forQ∼0 to∼129 at the the Z mass.
The extrapolation can be done quite reliably, but the small (O(0.02%)) uncertainty is still the largest theoretical uncertainty in the precision electroweak program. Closely related hadronic uncertainties are also significant in the interpretation of the measured anomalous magnetic moment of the muon.
The runningα(Q2) effect was sketched here in anon-shell renormalization scheme, i.e., ewas defined in terms of the electron-photon vertex on shell, andmas the location of the pole in the propagator. In the minimal subtraction schemes (’t Hooft, 1973; Bardeen et al., 1978) one defines renormalized couplings and masses at an arbitrary renormalization scale µ, at which the poles in dimensional regularization are subtracted. Both the masses and charges run as a function of µ. Higher-order corrections to physical processes involving a single large scaleQare usually minimized by evaluating the couplings and masses atµ=Q.
2.12.3 Tests of QED
Quantum electrodynamics is the most successful theory in physics when judged in terms of the theoretical and experimental precision of its tests. A detailed review is given in (Ki-noshita, 1990). The classical atomic tests of QED, such as the Lamb shift, atomic hyperfine splittings, muonium (µ+e− bound states), and positronium (e+e− bound states) are re-viewed in (Karshenboim, 2005). More recent results and the experimental values ofαand other physical constants are surveyed in (Mohr et al., 2012). Measurements of α and of possible deviations from QED are listed inTable 2.3.
.
TABLE 2.3 Most precise determinations of the fine structure constantα=e2/4π and other QED quantities.a
Experiment Quantity Value Precision
ae= (ge−2)/2 α−1 137.035 999 157(33) 2.5×10−10 h/m(Rb) α−1 137.035 999 049(90) 6.6×10−10
Solar wind mγ <10−18 eV −
CMB Qγ <10−35 −
e−6→νγ τe− >6.6×1028 yr − Neutrality ofSF6 |Qp+Qe−|, Qn <10−21 −
aQi is the electric charge of particlei in units ofe. Detailed descriptions, caveats, and references are given in (Mohr et al., 2012; Aoyama et al., 2015; Patrignani, 2016).
e e γ
e e
γ
e, µ, τ
e e
γ
Figure 2.19