Parity-violating anomaly in three-dimensional supergravity
E.R. Bezerra de Mello Departamento de Física, Universidade Federal da Paraíba 58000, Joio Pessoa, PE, Brasü
Atotrtrt. The parity-violating part of lhe one-loop order correction of lhe locally supersym- metric effective action for a massless scalar superfield coupled to a background (not quantised) supsrdretbein in three dimensions is presented 1 lit lowest onler appioxtmaiinn to (he gravitational Chetn-Simons term and its lermionic equivalent term are obtained.
1. Introduction
The parity-violating anomalies of ordinary gauge field [I] and gravitational theory [2], as well of a supersymmetric gauge theory [3], all of them defined in a three-dimensional spacetime, were found some time ago. However, an explicit calculation for supergravity in three dimensions has been lacking so far. The purpose of this paper is to present the same kind of anomaly for a locally supersymmetric theory defined by a massless scalar superiield coupled to a background (not quantised) superdreibein in three dimensions. The procedure to obtain this anomaly is the same as for the previous case; we do the calculation of the one-loop order correction of (he (locally supersym- metric) eliective action, and regularise the ultraviolet divergent diagrams by the parity- violating Pauli-Villars subtraction scheme, and at the end of the calculation we take the limit when M% the mass of the massive regulator field, goes to infinity. The regularised diagrams present parity-preserving (generally a non-analytic function of the background field) and parity-violating parts, and so does the etlective action. So, our strategy is very simple, although the calculation to get the parity-violating part of the locally supersymmetric etlective action, f_, is very long.
2. The system and its linear approximation
We sluill start with .i locally supcrsymmeiric action for a massive scalar superfield 4>{x, 6), defined in a three-dimensional spacetime, given below
$•(£)-j dWflE-t-JlV.^'+Wj (2.1a)
where £ is the superdeterminanl of the superdreibein, and
the covariant derivative [4J. (We adopt here the same notation as in 14J, also given in the appendix.) Because the superdreibein, E", contains a large number of degrees of freedom, and some of them can be gauged away, we also present the results of this paper in a non-supersymmetric Wess-Zumino gauge that exhibits the component field content of supergravity most directly [4j:
I 280 with tnd Et»"tmi§y~l-i$'+i—> (2.2c) witht E?-Q. (2.2 d) In this gauge h « h£ is the trace of the ordinary dreibein, and A""*" its remaining part totally symmetric in all indices and traceless. The fields #» - *»M and V*\ are the gravitino Reid components, and a is an auxiliary field. The component E?a of the superdreibein defined by V., = -ÜIV.,V (2.3a) is found to be iN «5 — li[E?{DNEX)+EHDNE?)+i-l)' +"E:E»T?N) (2.36) where the indices M and JV are equal to one when these indices refer to spinorial indices and zero otherwise, and 7Xr = «(W+ KW = W»*lt rest = O. (2.3c) In order to calculate the parity-violating anomaly fcr the supergravity, it suffices to linearise the superdreibein about supersymmetric flat space E? = 8X + kH*! (2.4) and to calculate the one-particle-irreducible (IPI) generating functional given below, ir(£)» S,{E)-S.(E)-S9 (2.56) using an expansion in fe, a dimensionless parameter smaller than I, and keeping only terms of order Oik1). First of all, let us obtain the superdeterminant E in the linear approximation E • J deu EÍ')-1+k u H y -1+*/*::{!-*//::. Uy (2.3M we have % 3*2 - 3( I + kj)2 = 3+bkj, where ^ = h + flV* - 92a. We can see that in the Wess-Zumino gauge h - I + kh and the gravitino and auxiliary fields are of O(k). So, we have and 281 Now we have E-l+4** (2.6) The S,(E) given by (2J6), (2.1) and (2.4) is < I 1 5,(E)«Sl(H) = Sy'(Hf+S , '(//)+S', '(H) (2.7a) where "(«)- -è f (2.76) J 2 'A«)- f d xd (27c) -2m* J d*x d'«#*d'«#*22. . (2.7 The propagator, in momentum space. Tor the massive scalar superlield is found from the Tree generating functional J W*]exp{i J dJ (2.8) J l J J to be with According to our approximation, the three Feyninau diugr-jius that we have to consider are given in figure 1. Calculating these three diagrams we gel the one-loop order correction to the three-dimensional supergravity action I (//). Une can see that these three diagrams are ultraviolet divergent. So in order to obtain a unite expression to V(H), we use the Pauli-Villars subtraction scheme. explicitly given in the next section. The regularised expression. I "(HI, will present a parity-preserving (generally a non-analytic function or the background field) and a parity-violating part associated with the anomaly. 282 —o >o —o— l«l 1*1 «ri Hptc I. The tottcst-ofder Ff>—IMI diagrwNS. . 3. Parity-violating anomaly As was said in the last section, we shall develop the calculation ol" the diagrams given in figure I. The graphs (a) and (6) appear in the first order ol' the expansion (2.5a) and (r) in the second order. We shall start by a brief comment on the tadpole diagrams (a) and Kb). 3.1. Tadpok diagrams Calculating these two diagrams we can see the external momenta iq) factor out of the loop integral and we get for V%{H) contributions of the kind [/<'«<•' where P[m, A) is a polynomial in the mass m and cut-olf momentum A. Such contributions are not relevant t.o the parity-violating anomaly, they vanish by the Pauli-Villars regularisation scheme which we employ in our calculation. 3.2. Two-point Junctions to l'(H) The only diagram that contributes to the parity-violating anomaly ol the SUC.RA elfective action, V \H), is (c). This diagram is obtained by the Wick ordering of the second-order expression in (2.5a). It is bilinear in the super field H*{.\. 0). where N assumes vectorial and spinorial indices, so we have vector-vector, spinor-spinor and vector- spinor components to r,,,(H) = rlZ'(H). All of these components present an ultra- violating divergence, and we shall employ the parity-violating scheme to regularise them. The expression to Vtlf{H), given by (2.5a), is: irIJI(W)«-J with where H,7»(-H^, H*'\. 3.2.1. Vector-vector component. Using the algebra for the D operator given in the appendix and after a few steps, the 9 integration can be easily done so that the 0 integrals reduce to a single point in 0-space, and the integration in the momenta reduce to only two. So, we have: 283 CUUJS)] (3 3a) where J-2.2. Sywor-yworcowyowfiit. The calculation of this component to IM_2'(H). is similar to the last one. so using the algebra for the D operator we set the following expression: d\t l2O*U)*(a (3.4) 3JJ. Sfmor-vtctor component. For this coasponent we have two contributions from (3J») and one from (3Je). They ait: 4. Conclusion and discussion We have obtained a parity-violaling anomaly to the three-dimensional supergravity, F., calculating the one-loop order corrections of the locally supersymmetric ellective action for a massless scalar superfield coupled with a background superdreibein and renormalising the divergent diagram by the parity-violating Pauli-Villurs subtraction scheme. We present I", in terms of the supcrdreibein and its supersymmetric covariant derivative, and also we present the anomaly by the component field or the superdreibein and compare our result with the previous one obtained for the ordinary dreibein and gravitino. Because our calculation was done in the linear approximation, i.e. linearising the superdreibein about supersymmetric flat space, we can compare our result with the similar one when linearised. Appendix In this appendix we present our convention, the notation (also given in 14J) and some algebra of the XT-operator in momentum space, useful for us in 13. Notation and convention where ^, r• 1,2 and a >0,1,2. 284 Abo we «te lhe followiag coaveatioa for lhe [I] Rt«KfcANIM4ffc|*.JbkD292M» [2] Vtaori» IIM» Mpt. Lm. I7SS 17» VM der •« J J. Pinnfci R D vrf RM S IM» flhw. tm. ITM17 [3] Nhammam BR má HOttnSimsntft. Lm. Dee N IM» flkj* Km D M 9912 [4] Ch2 151 285