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Quantum Field Theory


The Divergence

This section will examine the origin and the forms of the divergence. The formulation will adopt the Feynman rules in momentum space as shown in Table 03 with a few minor changes (such as 5 ) 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 charge correction (p',p), and the vacuum polarization (k).

Electron Self-energy Figure 03f is the Feynman diagram in momentum space for the electron self-energy (the insert portrays the unperturbed propagator). Using the Feynman rules, the S matrix can be written down for the transition of an electron from state (p, s) to state (p', s') in the form:


where the self-energy correction (p) is a 4X4 matrix given by:


                              -[A + (ip + m)B + ], and m is, at this point, finite but arbitrary.

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



Counting powers of the off-shell 4-momentum |k| = (k2 - E2)1/2 in the numerator and denominator of the integral suggests that it diverges linearly like d|k|. However, there are various methods to trim the divergence, so that it diverges only logarithmically like d|k|/|k| log(), see "footnote".

It can be shown that A = where the fine structure constant = e2/c , and the self-energy becomes infinite as the cut-off (in the upper limit of the k space integration) approaches infinity. This is referred to as ultraviolet divergence since it is caused by the infinite value of |k|. The physical mass of the electron is now the difference between the bare mass m0 and . It is noticed that even when the cutoff is taken to be the Planck energy of 1019 Gev, the correction to the bare mass is /m ~ 0.178. Anyway, the two terms m0 and together render a finite physical mass, i.e., mobs = mR = m = m0 - . The exact cutoff point is not important as long as mR agrees with observation.

Since the modification occurs in the propagator (from x2 to x1), it is necessary to consider its alteration as shown in Figure 03f from the unperturbed state with bare mass m0 (representing the state of no electromagnetic interaction). By using the identity :
1/(X+Y) = 1/X - (1/X)Y(1/X) + with (1/X) = and Y = -i[A + (ip + m)B], it can be shown that the correction to the electron propagator (in momentum space) is given by:


where the term has been moved into the denominator, Z2 = (1 - B) , and is another infinity in the (p) expansion about the point -ip = m. The factor Z2 is absorbed into the electron line to produce the renormalization for the fermion field R = 0/Z21/2, where the subscript 0 denotes bare state and the subscript R the renormalized state. Only the A and B terms are divergent in the Taylor series for (p), and R becomes finite as the result of one infinity from 0 is divided by another infinity from (1-B).

The vertex correction in Figure 03g modifies the first order interaction of the electron with the external potential as shown in the small insert. According to the Feynman rules, the propagator of vertex correction is involved in modification of the charge:
---------- (39b)
This divergent correction to be absorbed by the electric charge is denoted by Z1 , i.e., e e/Z1.

The other divergent correction is labeled as Z3 = (1-C) , where is another infinite number from vacuum polarization, which modifies the photon line as AR = A0/Z31/2 by the divergent integral , where Tr = Trace of a matrix = a11 + a22 + a33 + ... + ann .

Vertex Correction In the S-matrix expansion, each term has a number of the interaction Hamiltonian
HI -e00 0 = -eRR R eR = (Z2Z31/2)e0 (integration over 3-D space is omitted here).

If the renormalized charge further amalgamates the factors Z1, then the effective charge is :
eeff = eR/Z1 = (Z31/2)e0 ,
since the "Ward's Identities" has shown that Z1 = Z2. The infinity is removed by cutoff in the effective charge. It is interpreted as a small reduction of the charge due to the screening by virtual particles.

Thus in terms of the renormalized quantities, all the divergences disappear from the QED formulation, which yield 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 03g Vertex Correction


BTW, as one part of Eq.(39b) gives rise to Z2 , 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.

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 [view large image]

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 .

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