One-loop divergences of the nonlinear chiral theory

Abstract

An application of dimensional renormalization to chiral-invariant theories is presented. The naive Feynman rules may be used in this framework without introducing any noninvariant corrections to the on-mass-shell S matrix in the one-loop approximation. Moreover, the soft-pion theorem is fulfilled in all orders of perturbation theory.Recently there have been several discussions on perturbation theory in the nonlinear chiral theoIn spite of the nonrenormalizability of this theory it i s expected that chiral invariance should constrain the counterterms in such a way that one may extract some useful results from the one-loop calculations, for instance. In doing any perturbative calculations in this theory we a r e faced with the following questions:(1) I s the Adler condition (the soft-pion theorem) satisfied order by order in perturbation theory?(2) I s the on-mass-shell S matrix computed in perturbation theory invariant under redefinitions of the pion fields?If one worked without any caution (i.e., using naive perturbation theory) the answer to these questions would be negative. However, it was pointed out2 that if one does the perturbation calculations more carefully, by adding an extra term multiplied by 64(0) to the Lagrangian, the Adler condition is automatically satisfied. The second question seems to be more complicated, and in order to give a proper answer one must use an invariant renormalization with respect to a kind of gaugeUsing the background-field technique Ecker and Honerkamp5 succeeded in calculating the one-loop counterterms of the nonlinear chiral-invariant pion Lagrangian. These countert e r m s a r e manifestly chiral-invariant and gaugeinvariant, but they do not have the same structure a s the initial Lagrangian, i.e., the theory remains invariant in the one-loop order but it i s not renormalizable. Unfortunately, all these calculations contain a lot of meaningless things, and one has to introduce a regularization method to deal with them properly.The purpose of this paper i s to show that dimensional regularization6 is very suitable for doing all calculations in this particular theory. We shall show that in this regularization method one-loop diagrams automatically satisfy the Adler condition, and the on-mass-shell S matrix i s invariant under redefinitions of the pion field even though we use a naive perturbation theory.The Lagrangian of our theory can be written as3.' gi, (77) being the metric in a curved isospace with constant curvature f ,-' .All one-loop diagrams can be obtained in a way explained by Coleman and Weinberg7 if one s t a r t s with a prototype diagram, which is a simple circle in this case. Now it is easy to check that in the soft-pion limit (i.e., when all external momenta are zero) all these diagrams contain the integral where n is the space-time dimension. But this term which i s so meaningless in a usual regularization can be treated properly by a convenient definition of the n-dimensional integration.' After this redefinition it can be shown that this t e r m vanishes ...

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