If the integrals that we write remain meaningful we find in Visual Studio .NET

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p(k'2) dk'2 k'2(k2 _ k'2)
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+ k2
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1 - J12
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foo dk'2 p(k'2)]
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(7-15)
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According to (7-14) taken for k = 0 and assuming w(O) finite, the last term in brackets in the above
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formula vanishes. Consequently, Eq. (7-13) reads
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'G (k) =
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foo dk'2
(k'2) gpv - k pk v/k'2 P k 2 _k'2
+ kpk v
/1 2
k2-/12/A
(7-16)
This is indeed comparable to Eq. (5-80).
The resummation implied by Eqs. (7-12) and (7-13) has generated a new phenomenon. Indeed, the isolated pole at k 2 = J1.2 of the free theory has at best been moved, and corresponds now to the zero of the expressions k 2 [1 + w(k 2 )J 2 J1. As far as the photon propagator is concerned this is not too drastic. Indeed, J1.2 has only been introduced for convenience, in order to cut off the infrared divergences in intermediate calculations. We shall always have to compute combinations such as jf(k)G pv (k)j2( -k) where hand h are conserved currents [k oj(k) = OJ in order to extract physical information. This combination eliminates the terms in kpk v and enables us to consider the limit J1.2 -+ O. If instead we would have set J1.2 = 0 then G would have taken the form
. gpv zGpv(k) = k 2 [1 + w(k 2 )J
kpk v 1 + w(k 2 ) - A k4 A[1 + w(k 2 )]
J1.2 =
(7-17)
Unless w(O) = 00, the quantity k 2 [1 + w(k 2 )] still vanishes at k 2 = O. In a loose way we may characterize the property k 2 [1 + w(k 2 )] /k2=O = 0 by saying that there is no genuine mass renormalization for photons. But wave function renormalization is there, since the residue of the pole at k 2 = 0 is now equal to [1 + w(O)r 1, instead of being one. One-loop calculations reveal a remarkable circumstance. Namely, under regularization we find that the only potentially divergent and therefore unknown term in w is a constant which may be taken as its value at k2 = O. If only j . G . j is measurable we may use two widely separated identical sources to define what we mean by the charge squared. We define it to be the coefficient of the 1/4nr Coulomb static potential at large distances. This appears here as e2 = e5/[1 + w(O)J = Z3e5 where we have introduced an index zero to characterize the bare coupling constant used up to now in the perturbative series. The above definition of Z 3 agrees with the one given in Chap. 5 in the limit J1.2 = O. We conclude therefore that
1 1 + w(O)
(7-18)
1 - 3n In m2
+ .. ,
To be consistent we have to express all quantities as series in a parameter counting the number of loops L in the bare Feynman diagrams. As we have seen in Sec. 6-2-1, it is natural to multiply such diagrams with a factor h L . For this reason if we concentrate on terms of order hI we may substitute IX for lXo in the finite remainder w[I](k 2 ) - W[I](O), since this is already of order h as is IX - lXo. This also justifies the use of IX in the right-hand side of (7-18).
326 QUANTUM FIELD THEORY
According to this definition of charge the renormalized propagator reads
iG~(k) = k2[1 + wf:~) _ w(O)] + terms in kpk v
(7-19)
It will have residue 1 at the photon pole k 2 = 0 and will be expressed in terms
of the physical charge. We see the renormalization program emerging: undetermined and potentially divergent quantities disappear when Green functions are expressed in terms of physical renormalized quantities. When we set L~ CS In = - In (A2/m 2) the sign was chosen in agreement with condition (7-7). The intuitive content of (7-18) is that the bare charge has been screened at large distances by a factor Z3 = 1 - (()(/3n) log (A2/m 2) smaller than one and positive as long as A2/m 2 is not overwhelmingly large. Vacuum polarization corresponding to the creation of virtual electron-positron pairs has reduced the charge of a test particle as seen by a distant one. We may take another point of view in deriving equations such as (7-19). It amounts to saying that the original lagrangian is only a bookkeeping device enabling us to define the perturbative amplitudes. It was postulated from some intuitive correspondence principle between classical and quantum mechanics. As such it may be amended by quantum corrections. Let us therefore assume that to lowest order .!t is expressed in terms of physical parameters. It wiII then be necessary to construct perturbative corrections of the form (j.!t , called counterterms (the coefficients of which will become infinite as the cutoff goes to infinity), in order to maintain the proper definition of these physical parameters:
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