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It is also mentioned that divergences (infinities) appear with certain loop diagrams in QED. The origin of these infinities can be traced back to the singular nature of the inverse-square law 1 / r

- Pauli-Villars regularization - A fictitious particle of mass M is introduced to modify the propagator such that it behaves now as 1/k
^{4}, which is usually enough to render all graphs finite in the off mass-shell integral. Then the limit of M^{2}is taken at the end of calculation to decouple the unphysical particle from the theory. - Dimensional regularization - The action is generalized to arbitrary dimension d, where there are regions in complex d space in which the Feynman integrals are all finite. The Feynman graphs pick up poles in d space to absorb the divergences of the theory into the physical parameters as d is reduced to four.
- Lattice regularization - This is the most widely used in QCD for non-perturbative calculations. The lattice spacing serves as the cutoff for the space-time integrals.

Mathematically, the divergences come from the integration of the 4-momentum p (or sometimes denoted as k) for the virtual particles (both fermion and boson) to infinity. The integration "4-volume" element d

D = 4 - F - 2B - nV ---------- (39a)

where F and B are respectively the number of fermion and boson propagators (internal lines), V is the number of vertices in the Feynman diagram, which is divergent if D 0. The coupling constants g for any gauge theory is dimensionless, so n = 0. Depending on whether n = 0, n > 0, or n < 0, the corresponding theories are renormalizable, super-renormalizable, or non-renormalizable.

According to the formula for D, only a small variety of graphs in a renormalizable theory are divergent. It is summerized in Table 04 below:

- In Table 04:
- The time axis in the Feynman diagrams runs horizontally from left to right.
- The processes are related to quantum electrodynamics (QED).
- The number of Ferimon and boson internal lines (propagator) F and B can be counted from each diagram.
- The degree of divergence D is computed from Eq.(39a) with n = 0 for gauge theories. D is often referred as "primitive" divergence. For example, the electron-electron scattering diagram has no primitive divergence since D = -2. However, there is a subdiagram in its upper corner that looks like the vertex diagram, which is know to have D = 0 producing logarithmic divergence. In other words, divergences in subdiagrams are not counted as primitive divergences for the whole diagram.
- A gauge theory has a high degree of symmetry, which allows some primitive divergences to be cancelled and reduces the degree of divergence to D'. This is the case for the photon self-energy and the photon-photon scattering diagrams, which are actually primitively convergent.
- It is known that the divergences of a diagram are summarized in D' + 1 divergent parameters. The vacuum diagram and its divergence is usually ignored, on the grounds that it is not related to any external particles and cannot be measured experimentally. If we assume that the bare mass m
_{0}, and the bare charge e_{0}are adjustable infinite quantities, then they can be combined with the divergent parameters to re-formulate the theory into one that does not involve infinities. The mass m_{R}, and charge e_{R}in this new form will be "dressed" and agree with experimental measurements. The bare and renormalized quantities are related by the following formulas:

m_{R}= m_{0}+ m

e_{R}= [(Z_{1}Z_{2}Z_{3}^{1/2})]e_{0}= (Z_{3}^{1/2})e_{0}

It also requires that the fields should be transformed like:

_{R}= Z_{2}^{-1/2}_{0}

A_{R}= Z_{3}^{-1/2}A_{0} - The vacuum diagram in which virtual e
^{-}e^{+}pairs and virtual photons are created and annihilated together, give terms that modify the vacuum energy. Energy shifts in perturbation theory are to be expected, but since there is no unperturbed vacuum with which to compare, such shifts are not measurable. The cosmological constant of general relativity gives a measure of the vacuum energy density that is very small. Thus, the vacuum energy density, whatever its origin, is taken to be zero.

A non-renormalizable theory has n < 0, so D' grows indefinitely with V. Since D' + 1 divergent parameters are required in each case, a non-renormalizable theory contains an infinite number of divergent parameters. Even if we had an infinite number of bare parameters to adjust, there would be an infinite number of parameters in the theory. It would take a prohibited long time to measure all of them. The gravitational constant G = 6.7087x10

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