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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| = (k^{2} - E^{2})^{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 =

Since the modification occurs in the propagator (from x

1/(X+Y) = 1/X - (1/X)Y(1/X) + with (1/X) =

where the

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) |

The other divergent correction is labeled as Z

In the S-matrix expansion, each term has a number of the interaction Hamiltonian H _{I} -e_{0}_{0} _{0}_{} = -e_{R}_{R} _{R}_{} e_{R} = (Z_{2}Z_{3}^{1/2})e_{0} (integration over 3-D space is omitted here).If the renormalized charge further amalgamates the factors Z _{1}, then the effective charge is : e _{eff} = e_{R}/Z_{1} = (Z_{3}^{1/2})e_{0} ,since the "Ward's Identities" has shown that Z _{1} = Z_{2}. 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 Z

where

^{...} |

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 [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 . |

- 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 03hb). 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.**

| |

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

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