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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

The problem with electron self-energy has been a nuisance in classical electromagnetism for a long time. According to the theory, the self-energy E_{e} = e^{2}/r_{e} (in cgs units) goes to infinity as r_{e} approaches zero. The solution is to cut-off the energy scale at the rest mass energy m_{e}c^{2} so that the electron has a finite radius r_{e} = e^{2}/m_{e}c^{2} = 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 S_{fi} via the Feynman rules.
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## 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). |

This divergent correction to be absorbed by the electric charge is denoted by Z
_{1} _{} = (1 + B) _{} , i.e., e_{0} (1+B) = e_{0}Z_{1} e. | |||

## Figure 03g Vertex Correction |

where

^{...} |

This correction is labeled as Z

In the S-matrix expansion, each term has a number of the interaction Hamiltonian density

_{I} -e_{0}_{0} _{0}_{} = -e_{R}_{R} _{R}_{} e_{R} = (Z_{2}Z_{3}^{1/2})e_{0}.If the renormalized charge further amalgamates the factors Z _{1} (see vertex correction), then the effective charge is : e _{eff} = e_{R}Z_{1} = (Z_{3}^{1/2})e_{0} ,since Z _{2}Z_{1} = (1 - B)(1 + B) = 1 - B^{2} ~ 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) |

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 n _{f} = 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 . |

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] |

The leading term in (k

- Dimensional Regularization - This method decreases the dimension of the 4-momentum and thus reduces its share in the numerator. It is the only scheme in perturbative calculations that preserves Gauge Invariance.

**Pauli-Villars Regularization - The divergence of the integral is reduced by assuming the existence of a heavy particle with mass M.**

**Lattice Regularization - This is the most widely used scheme for non-perturbative calculations. Initially, it assumes space-time to be a set of discrete points arranged in lattice (Figure 03hc). The lattice spacing "a" serves as the cutoff. The summation is reverted back to continuous integration by taking a 0 after performing the finite sum. Lattice Regularization preserves Gauge Invariance, and has been very successful in predicting many results of low-energy QCD, including confinement, hardon masses etc.**

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## Figure 03hc Lattice Regularization [view large image] |

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