Volume 171, number 2,3 PHYSICS LETTERS B 24 April 1986 SELF-DUAL FACTORIZATION OF THE PROCA EQUATION WITH CHERN-SIMONS TERM IN 4K- 1 DIMENSIONS Samir K. PAUL.and Avinash KHARE Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India Received 6 November 1985; revised manuscript received 28 January 1986 We show that the Proca equation with Chern-Simons term propagates a self-dual field with two distinct masses in 4K - 1 dimensions. In the special case of three-dimensions (K = 1) it is further shown that the corresponding lagrangian is the free gauge part of the abefian Higgs model with Chern-Simons term after the Higgs mechanism has taken place. Some unusual features of the nonabelian Higgs model with Chern-Simons term are also pointed out. Some times back Townsend et al. [1] have shown that it is possible to f'md a "square root" of the Proca equation in (2 + 1) dimensions (our metric conven- tion is + - -) a FUU+m2AV=O, (1) gt which may be written as a "self-dual" field equation A u = (1/2m)euu ° Fv°. (2) They showed that whereas eq. (1) propagates two massive modes, eq. (2) propagates only one massive mode. Later on, in an interesting paper, Deser and Jackiw [2] showed that the corresponding self-dual lagrangian £SD -- -51m2A u AU+~meUVOAuFvo (3) is equivalent to the topological lagrangian of Deser et al. [3] = F~A ° (4) ~'T -~FvFUV + ~meuvo via a Legendre transformation (the equivalence is ob- vious once one notices that the relation (2) is equiva- lent toA u = (1/rn)*Fu; *F u being dual to the field strength Fur ). Deser and Jackiw [2] also showed that both (3) and (4) describe a massive gauge-invariant lagrangian with "spin" either +1 or -1 depending on whether m is >0 or <0. The purpose of this note is to point out a novel feature of the Proca equation with Chern-Simons (C-S) term. The lagrangian which gives rise to such 244 an equation is £TM =-X~F- uv FUV + ~ l aeUVaF ' , uv A~+ ~m2AuA#" (5) This £TM can also be obtained from £T by the addi- tion of the "normal" gauge meson mass term. The point is that £TM without the topological term propa- gates two massive modes and is a parity-conserving theory. The topological term, however, violates parity. Further the lagrangian (5) still propagates only two modes since the Lorentz condition au A u = 0 (6) is still implied by the field equation which follows from £TM ~u FUr + m2AV + ½ #eW't~F~t~ = 0. (7) The question is: in what way are the two massive modes propagated by (5) so as to reflect parity viola- tion in the theory? The other question which we discuss (and as we shall see below is related to the question discussed above) is with regard to the Higgs mechanism for the topological lagrangian (4). Because of the Higgs mech- anism, the topologically massive gauge field should "eat up" one scalar field and propagate two massive modes rather than one which is propagated by £SD. How does that exactly happen in the parity-violating model characterized by £SD ? The purpose of this note is to answer the two ques-