In these expressions j and js denote the naive divergences of the currents and in Visual Studio .NET

Generate PDF417 in Visual Studio .NET In these expressions j and js denote the naive divergences of the currents and

In these expressions j and js denote the naive divergences of the currents and
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F~v F~v
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= ap. "Yv - ov"Yp. - i[ "Yp., "YvJ - i[d'p., d'vJ = op.d'v - ovd'p. - i[.sotp., "YvJ - i["Yp., d'vJ
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(11-226)
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Neither the structure nor the coefficient appearing in expressions such as (11-213), (11-221), or (11-225) are modified by higher-order corrections provided a suitable definition is given to the renormalized operators. To see this, let us consider the case of electrodynamics, for instance. The argument relies on the existence of a regularization of the photon propagator, in the form -iFp.v(1 + 02jA4)pv, for instance, such that all diagrams of interest except those with one loop become superficially convergent. Indeed, the new power counting gives for a diagram with EF external fermion lines, EA photon lines or current insertions and L loops: w = 8 - iEF - EA - 4L. Therefore amplitudes with L :::::: 2 will superficially con-
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verge. It is, of course, understood that the standard renormalization of quantum electrodynamics has disposed of all internal divergences. This regularization is gauge invariant and does not modify the structure of the axial current. Consequently, the only anomalies are those arising from one-loop subdiagrams which we have just analyzed. We note that it is possible to define a different axial current with a normal Ward identity but violating gauge in variance. In electrodynamics, for instance, such would be the case for (11-227) which satisfies (11-228)
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These different possibilities can be interpreted in a different language. We could try to construct directly the operator j"s in terms of the fields t/t and I{/ in the presence of an external potential. Since combinations such as t/t(x)l{/(y) are singular in the limit x --+ y we separate the arguments by an infinitesimal space-like interval B, therefore defining
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j\f)~(x, B) = I{/(x
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+ B)Y"Yst/t(x)
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(11-229)
This new operator is not gauge invariant. Following Schwinger'S suggestion, this is corrected as follows. We multiply (11-229) by a phase factor involving the integral of the vector potential along a space-like path from x to x + B along which the various components commute:
j"s(X,B)
I{/(x
+ B)y"Yst/t(x)exp [ -ie
dz P AiZ )]
(11-230)
If we heuristically use the equation of motion, the divergence of this operator can be written as
o"j"s(x, B)
2mil{/(x
+ B)Yst/t(X) exp [ -
dz P Ap(Z)] - iej"s(x,
B)F~,(x)BV[l + O(B)]
(11-231)
The second term on the right-hand side is singular in the limit B --+ O. The vacuum matrix element of j"s in the presence of an external field has a l/B behavior from which we recover in the limit (11-232)
The axial anomaly modifies the PCAC expression in the presence of electromagnetic interactions [Eq. (11-213)]. The added term is a hard one, changing the renormalization properties of the axial current. We may return to nO decay and express the amplitude as
e~e2 Tl'iq2)
= (m;; - q2)<y(kt, er), y(k2' e2) 1n(O) 10)
(11-233)
which, according to (11-213), may also be written
e~e2 Tl'iq2) = m;; --;/q2 [<Y(kt, e1), y(k2' e2) 1 A~(O) 10) 8
m" "
(11-234)
560 QUANTUM FlliLD THEORY
To lowest order in IX the second matrix element is equal to (lX/n)BtBzcllvpakfk'5., while the first one vanishes in the limit q2 = O. We therefore reach the conclusion that (11-235) in agreement with (11-199). This low-energy theorem is in fact valid to all orders in IX due to the nonrenormalization property of the anomaly. On the other hand, the extrapolation to = m 2 will depend on the order of the approximation. We conclude that anomalies are not an artefact but a true product of renormalization, involving a deeper aspect of field theory. Each time a soluble model is available we can check them explicitly as in the two-dimensional Schwinger model of electrodynamics with massless fermions which exhibits a computable anomaly of the form
(11-236) The appearance of anomalous dimensions in the asymptotic behavior (Chap. 13) will offer a new insight into these phenomena.
NOTES
Unitary symmetry is described by its fathers M. Gell-Mann and Y. Ne'eman in ''The Eightfold Way," Benjamin, New York, 1964. The quark model was independently proposed by M. Gell-Mann, Phys. Lett., vol. 8, p. 214, 1963, and by G. Zweig (unpublished CERN report, 1963). The spectrum associated to symmetry breaking was discussed by J. Goldstone, Nuov. Cim., vol. 19, p. 154, 1961; Y. Nambu and G. Jona-Lasinio, Phys. Rev., vol. 122, p. 345, 1961; and J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev., vol. 127, p. 965, 1962. The impossibility of continuous symmetry breaking in dimension two was shown by N. D. Mermin and H. Wagner, Phys. Rev. Lett., vol. 17, p. 1133, 1966, in the context of statistical mechanics, and was extended to field theory by S. Coleman, Comm. Math. Phys., vol. 31, p. 259, 1973. The proof of phase transitions in lattice systems with a discrete symmetry is due to R. E. Peierls, Phys. Rev., vol. 54, p. 918, 1938. The ground-state invariance was analyzed by S. Coleman, J. Math. Phys., vol. 7, p. 787, 1966. A general survey of symmetry problems is given by G. S. Guralnik, C. R. Hagen, and T. W. Kibble in "Advances in Particle Physics," edited by R. L. Cool and R. E. Marshak, Interscience, New York, 1968; and by S. Coleman in "Laws of Hadronic Matter," edited by A. Zichichi, Academic Press, New York, 1975. The abundant work on current algebra may be traced in the books of S. Adler and R. Dashen, "Current Algebras," Benjamin, New York, 1968; V. de Alfaro, S. Fubini, G. Furlan, and C. Rossetti, "Currents in Hadron Physics," North-Holland, Amsterdam, 1973; and in the lectures by S. Weinberg in "Lectures on Elementary Particles and Quantum Field Theory (Brandeis, 1970)," edited
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