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Quantum Field Theory

QED Divergences (Vacuum Energy, ..., 2018 Addition)

                                               ( Electron Self-Energy,   Vertex Correction,   Vacuum Polarization,   Vacuum Energy )

This section will examine the origin and the forms of the divergence. The formulation will use the Feynman diagrams in momentum space as shown in Table 03 with a few minor changes to converse the system from nucleon-pion to electron-photon. Since the divergence arises from the virtual particles in the internal line, the main focus will be directed to such divergent integrals in the S matrix. It is customary to designate the infinite integrals for the self-energy (p), the vertex correction (p',p), the vacuum polarization (k), and the vacuum energy . See "Feynman Diagrams and Rules" for more info.

(1) Electron Self-Energy

Electron Self-energy The problem with electron self-energy has been a nuisance in classical electromagnetism for a long time. According to the theory, the self-energy Ee = e2/re (in cgs units) goes to infinity as re approaches zero. The solution is to cut-off the energy scale at the rest mass energy mec2 so that the electron has a finite radius re = e2/mec2 = 3x10-13 cm (Figure 03f,a). In quantum theory, the problem arises from the virtual photon interacting with the electron itself (Figure 03f,b). It can be resolved by the similar technique of cut-off, but a even better method is to cancel the infinity by another infinity. The following is a much reduced mathematical manipulation to show the quantum version of the electron self-energy and its ramification. The formulation starts with the construction of the S matrix Sfi via the Feynman rules.

Figure 03f Electron Self-Energy [view large image]

See "Feynman Diagrams and Rules" for mathematic and graphic symbols with nucleon (N line) replaced by fermion, and meson ( line) by photon).

See "conversion between cgs and natural units".

See "footnote" for regularization schemes, also check out "Richard Feynman at 100".

(2) Vertex Correction

The vertex correction in Figure 03g modifies the first order interaction (bare) of the electron with the external potential as shown in the small insert. The propagator of vertex correction is involved in modification of the electric charge e :
Vertex Correction
---------- (39b)
This divergent correction to be absorbed by the electric charge is denoted by Z1 =
(1 + B) , i.e., e0 (1+B) = e0Z1 e.

Figure 03g Vertex Correction

BTW, as one part of Eq.(39b) gives rise to Z1 , which is divergent at |k| , the other part diverges logarithmically at low value of |k| as |k| 0. This is an example of what is known as the "infrared divergence". A possible way to dispose of this difficulty is to temporarily assign a very small but finite mass to the photon; this can be accomplished by modifying the photon propagator in Eq.(39b) according to the following prescription:

where min is the "mass" of the photon. Under such scheme, virtual photons of very low energies (min) do not get emitted or absorbed. The net result is that, for small values of q = p' - p, that part of Eq.(39b) can be written as follows:
where the second term containing the generalized Pauli matrices can be identified as the second order correction to the electron magnetic moment. It can be shown that the infrared divergence is cancelled by the similar divergence arisen from the bremsstrahlung process to all orders in perturbation theory. Thus, QED is free of divergence seemingly out of a mathematical miracles.

(3) Vacuum Polarization

The process of vacuum polarization as shown in Figure 03h,a produces one more divergent integral :
, where Tr = Trace of a matrix = a11 + a22 + a33 + ... + ann emerges from summing over the initial spin states for unpolarized electron beam..
This correction is labeled as Z3 = (1 - C) , where is another infinity, which modifies the photon line
AR = A0/Z31/2.

In the S-matrix expansion, each term has a number of the interaction Hamiltonian density
Vacuum Polarization I -e00 0 = -eRR R eR = (Z2Z31/2)e0.

If the renormalized charge further amalgamates the factors Z1 (see vertex correction), then the effective charge is :
eeff = eRZ1 = (Z31/2)e0 ,
since Z2Z1 = (1 - B)(1 + B) = 1 - B2 ~ 1 to order . The infinity is removed by cutoff in the effective charge. It is interpreted as a small reduction of the charge solely due to the screening by virtual particles (Figure 03h,b). The electric charge is independent of the type of particle, e.g., whether it's electron or muon.

Thus in terms of the renormalized quantities, all the divergences disappear from the QED formulation, which yields amazing predication with an accuracy up to one part in trillions (as in "Lamb Shift", and "Gyromagnetic Ratio") even though the methodology had been criticized by many prominent physicists.

Figure 03h Vacuum Polarization

BTW, it can be shown that for large momentum transfer Q = |p' - p|, the effective coupling constant in QED including vacuum polarization is modified to:

---------- (39c)

Effective Charge which increases as Q increases (or, equivalently, as the probing distance becomes shorter, see Figure 03ha).

Similar vacuum polarization effects occur in QCD with similar Feynman graph but for gluon exchange between quarks and antiquarks. There is an additional contribution from the gluon self-coupling as shown in Figure 03ha. The effective coupling constant for strong interaction for large Q is now in the form:

where nf = 6 is the effective number of quark flavours for large Q, and -1 is called the

Figure 03ha Effective Coupling Constant

confinement length since s becomes infinite when Q = . The phenomenon of asymptotic freedom in QCD is associated with the fact that s 0 as Q .

(4) Vacuum Energy

Vacuum Energy S-Matrix The Feynman diagram of the S-Matrix for vacuum energy (Figure 03hb) looks rather similar to the vacuum polarization (see Figure 03h) except that the external photon lines becomes an internal propagator. Thus, its occurrence is not connected to any external soruce. It just emerges out of the vacuum according to the rule of Uncertainty Principle t E > . It is one of the four divergence diagrams in QED but customarily ignored since it doesn't contribute to any verifiable process; only until recently when the vacuum energy density becomes the most acceptable choice as the dark energy. Following the Feynman's rules, the S-Matrix in momentum space for the vacuum energy is (where the -i prescription is implicit, and it is a tensor of rank 2) :

Figure 03hb Vacuum Energy S-Matrix [view large image]

See "Lecture 11: Feynman Rules and Vacuum Polarization" for the use of Pauli-Villars Regularization to evaluate the p integration.

The leading term in (k2) is just the photon line renormalization "C" in vacuum polarization; but the remaining terms are a power series of k2 that cannot be cast away such as the case for low energy approximation, since the k integration now runs to . It is not known how to absorb this infinity into anything as it is disconnected to all things. So the best way is to have it ignored if it has nothing to do with the dark energy.

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