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Similarly, in the C 1 contribution in .NET
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r~~'):o=
31X2 k   2 op.k 2   In 2
8 p. + ... ) 11 dx' I I I
x {[f3(1 u) :0= _ + (1  f3)(1  v)] [{3u
+ (1  f3)v]  x'f3(1  f3)(u  V)2} (8126) n 2 24 .k2 (~ _ In k
+ ... ) On the contrary, the contributions from Al and Bl possess internal divergences at f3 = 0 or 1. For instance, the terms proportional to kpk. read df3 [f3(1  f3)]2d I2{[f3(1  u) + (1  f3)(1  v)] x [f3u x {X
+ (1 f3)v]  x'f3(1 f3)(u  V)2}d4 f32u(1  u) + (1  Wv(1  v)  x' [ uv f3(1  f3) + op ..
+ (1  ] u)(1  v) 1 2+  + x' But the required expansion at d at 0 and 1, X'2(U  V)2 } 4 is easily obtained if we observe that, for any function F(f3) regular
df3 [f3(1  f3W 1+'/2 F(f3) df3 [f3 1+'/2 + (1  f3)1 +'/2] F(f3) + 0(8) Ultimately, we obtain
~ Op.p [~+ ~ (~ l' 3n 2 82 8 8
2 In k ) +~ In k +In k (1' ~) +... J 2 8
2 p.2 p.2
(8127) The total contribution of Fig. 817b is therefore
(8128) RENORMALIZATION
We should still compute the contribution of the vertex counterterms, as depicted in Fig. 817b'1, bz. But owing to the Ward identity, it is easy to see that they cancel exactly the contribution of the selfenergy counterterms (db az). Indeed, the latter are proportional to (Z2'l  1)[1 1 = (Z2  1)[11, while the former are proportional to (Zl  1)[1]. This is, of course, particular to quantum electrodynamics. Adding the two contributions (8113) and (8128), we finally get the result
r~~l = (kpk a  (j pa k2 )oPl(P) 2 2 oPl(k 2) = _ rx 2 (_ ~ + ~ In k + n 4e 4 J1.2 We observe all the desired features: (8129) ... ) 1. The divergent terms of the form (l/e) In (k 2/J1. 2) which could not be eliminated
by a local counterterm have cancelled. 2. The vacuum polarization tensor is transverse, as expected from the Ward identity. This holds for both the divergent and the finite parts. The former may therefore be renormalized by a transverse counterterm of order h 2 For large euclidean k 2, and up to order rx 2, the renormalized function wR(k 2) is then given by (8130) The foolhardy reader may check that the finite terms that we have neglected are also transverse, or (even better) that the complete expression of w(k 2 ) is independent of the gauge parameter A.. 3. Fortunately, the result (8130) coincides with the one obtained by other authors! We conclude that the dependence on the normalization condition of the selfenergy has dropped out in the sum rea) + reb). This is obvious since the two counterterms actually cancel. 4. Unexpectedly, we find that the divergent terms in 1/e2 [which in a conventional regularization would read In 2 (A 2/J1.2)] and, accordingly, the In 2 (k 2/J1. 2) have disappeared. This is a general property of the vacuum polarization, valid to all orders when we restrict ourselves to diagrams with a single fermion loop. Indeed, it is possible to fix order by order the gauge parameter A. so as to make Z 1 = Z2 equal to one. Consequently, the selected subset of diagrams does not exhibit any internal divergence and its contribution to the gaugeinvariant quantity w(k 2) behaves as a single power of In (A 2 /k 2). This has led to interesting speculations by Johnson, Willey, and Baker and by Adler. Could it possibly be that the coefficient f(rx) of the logarithmic behavior vanishes for a nonzero value of rx Since the same function multiplies In A2 (or l/e), this would seem to indicate that a reordering of the perturbation series might eliminate ultraviolet divergences for some appropriate value of the bare coupling. QUANTUM FIELD THEORY
NOTES
A complete treatment of renormalization for electrodynamics was first given by F. J. Dyson, Phys. Rev., vol. 75, p. 1736, 1949. One can follow the early evolution of the subject in Schwinger's book, "Quantum Electrodynamics," Dover, New York, 1958. The important contributions made by N. N. Bogoliubov and O. S. Parasiuk, Acta Math., vol. 97, p. 227, 1957, appear also in the textbook of the first author and D. V. Shirkov, "Introduction to the Theory of Quantized Fields," Interscience, New York, 1959, and the work of K. Hepp, Comm. Math. Phys., vol. 2, p. 301, 1966, on the convergence of renormalized Feynman integrals is developed in detail in his "Theorie de la Renormalisation," SpringerVerlag, Berlin, 1969. In the early 1970s, W. Zimmermann gave a comprehensive account of the renormalization, including his method of subtracting the integrands, in "Lectures on Elementary Particles and Quantum Field Theory," Brandeis Summer Institute 1970, edited by S. Deser, M. Grisaru, and H. Pendleton, MIT Press, 1970, and in Ann. of Phys. (N. Y.), vol. 77, p. 536, 1973. The elegant but somehow abstract work of H. Epstein and V. Glaser, Annales de l'Institut Poincare, vol. XIX, p. 211, 1973, definitely settles the matter in showing that the procedure preserves all properties of local field theory. Each of the numerous regularization procedures has its own merits. The PauliVillars scheme has been quoted above (Chap. 7). Dimensional regularization is discussed by G. 't Hooft and M. Veltman, Nucl. Phys., ser. B, vol. 44, p. 189, 1972. See also E. R. Speer in "Renormalization Theory," Erice Summer School 1975, edited by G. Velo and A. S. Wightman, D. Reidel Publishing Company, Dordrecht, Holland, and Boston, Mass., 1976. The convergence ofrenormalized integrals was studied by S. Weinberg, Phys. Rev., vol. 118, p. 838, 1960, and by K. Hepp, Comm. Math. Phys., vol. 2, p. 301, 1966. For some aspects of this question in parametric space, see T. Appelquist, Ann. of Phys. (N. Y.), vol. 54, p. 27, 1969, M. Bergere and J. B. Zuber, Comm. Math. Phys., vol. 35, p. 113, 1974, and M. Bergere and Y. M. P. Lam, Comm. Math. Phys., vol. 39, p. 1, 1974. Massless theories are considered in K. Symanzik, Comm. Math. Phys., vol. 34, p. 7, 1973. For infrared cancellations see the work of T. Kinoshita, J. Math. Phys., vol. 3, p. 650, 1962, T. D. Lee and M. Nauenberg, Phys. Rev., ser. B, vol. 133, p. 1549, 1964, and T. Kinoshita and A. Ukawa, Phys. Rev., ser. D, vol. 13, p. 1573, 1976. Our heuristic presentation has been borrowed from E. C. Poggio and H. R. Quinn, Phys. Rev., ser. D, vol. 14, p. 578, 1976. The computation of vacuum polarization to order a 2 was carried out by R. Jost and J. M. Luttinger, Helv. Phys. Acta, vol. 23. p. 201, 1950. See also the textbook of J. D. Bjorken and S. D. Drell, "Relativistic Quantum Fields," McGrawHill, New York, 1965. The possibility of a finite quantum electrodynamics has been studied by K. Johnson, R. Willey, and M. Baker, Phys. Rev., vol. 163, p. 1699, 1967, and S. L. Adler, Phys. Rev., ser. D, vol. 5, p. 3021, 1972.

