Remark on the Consistent Gauge Anomaly in Supersymmetric Theories

We present a direct field theoretical calculation of the consistent gauge anomaly in the superfield formalism, on the basis of a definition of the effective action through the covariant gauge current. The scheme is conceptually and technically simple and the gauge covariance in intermediate steps reduces calculational labors considerably. The resultant superfield anomaly, being proportional to the anomaly dabc = trT a{T b, T c}, is minimal without supplementing any counterterms. Our anomaly coincides with the anomaly obtained by Marinković as the solution of the Wess-Zumino consistency condition. ⋆ E-mail address: [email protected] † E-mail address: [email protected] ‡ E-mail address: [email protected] § E-mail address: [email protected] The consistent gauge anomaly [1] might be conceptually more natural than the covariant gauge anomaly [2], as it is defined as gauge non-invariance of the effective action of the chiral fermion [3,4]. The consistent anomaly is important because it provides information on the Wess-Zumino Lagrangian [3]. To find an explicit form of the consistent anomaly, one may appeal to the algebraic-geometrical technique [5-8] or directly perform a field theoretical calculation with, say, the Pauli-Villars regularization. As is well-known, however, both approaches can be cumbersome for a theory in higher dimensions, or for a theory with a complicated gauge transformation. In particular, in the field theoretical calculation, gauge non-invariant normal terms (fake anomalies) generally appear. Then, to extract the intrinsic anomaly, one has to find suitable local counterterms to eliminate these normal terms. The covariant gauge anomaly, on the other hand, has the quite restricted possible form due to the gauge covariance. The necessary calculational labors are consequently considerably less. Therefore, a practically useful calculational scheme might be formulated by relating the consistent anomaly with the covariant anomaly (or with a certain gauge covariant expression). In Ref. [9], Banerjee, Banerjee and Mitra gave a field theoretical prescription which provides this kind of calculational scheme. This prescription leads to basically equivalent consequences as the result due to Bardeen and Zumino [4], and that due to Leutwyler [10]. However, the prescription of Ref. [9] is more straightforward and flexible. ¶ In this letter, we give a direct field theoretical calculation of the consistent gauge anomaly in supersymmetric theories, on the basis of the prescription of Ref. [9]. Generally, the treatment of the consistent anomaly with the superfield formalism [12–24] is quite complicated, because the gauge transformation is highly nonlinear and because the gauge superfield has no mass dimension (i.e., an arbitrary function of the gauge superfield is a candidate of the counterterm). The advantage of our treatment in this letter is that the minimal superfield anomaly, being propor¶ It has the application even in chiral gauge theories on the lattice [11]. 2 tional to the anomaly dabc = trT a{T b, T c}, is directly obtained. This minimal-ness is guaranteed by the basic property of the prescription of Ref. [9]. Naturally, the resultant anomaly coincides with that due to Marinković [24], who applied the technique of Ref. [4] to this problem. Also our expressions below have some similarities with that of the work by McArthur and Osborn [23], in which the formulation of Ref. [10] was generalized to supersymmetric theories. Nevertheless, it seems worthwhile to report on our field theoretical calculation, because of simplicity of the basic idea and the treatment. We consider the massless chiral superfield Φ coupled to the external gauge superfield V = V aT a (T a is the representation of the gauge group to which Φ belongs). The classical action is given by ∗