On the superfield effective potential in three dimensions
A. F. Ferrari, M. Gomes, A. C. Lehum, J. R. Nascimento, A. Yu. Petrov, E. O. Silva, A. J. da Silva
aa r X i v : . [ h e p - t h ] J u l On the super(cid:28)eld e(cid:27)e tive potential in three dimensionsA. F. Ferrari,1, ∗ M. Gomes,2, † A. C. Lehum,2, 3, ‡ J. R. Nas imento,4, § A. Yu. Petrov,4, ¶ E. O. Silva,4, ∗∗ and A. J. da Silva2, †† ∗ Ele troni address: alysson.ferrariufab .edu.br † Ele troni address: mgomesfma.if.usp.br ‡ Ele troni address: lehumfma.if.usp.br § Ele troni address: jroberto(cid:28)si a.ufpb.br ¶ Ele troni address: petrov(cid:28)si a.ufpb.br ∗∗ Ele troni address: edilberto(cid:28)si a.ufpb.br †† Ele troni address: ajsilvafma.if.usp.brtheories, espe ially non ommutative ones [8℄.Therefore a natural problem is the development of a manifestly super ovariant methodology forthe al ulation of the e(cid:27)e tive potential in a three-dimensional supersymmetri (cid:28)eld theory. Somedi(cid:30) ulties related to this subje t were pointed out in [9, 10℄. One important point is that in writingthe va uum expe tation value of a s alar super(cid:28)eld Φ( x, θ ) = A ( x ) + θ α ψ α − θ F ( x ) , we would havein general < Φ( x, θ ) > = a − θ f , (1)with a and f onstants ( < ψ α > = 0 to preserve Lorentz invarian e). If we allow f = 0 , theba kground-dependent propagator for the quantum (cid:28)eld Φ be omes non-lo al in the θ variable,making umbersome the al ulation of supergraphs. On the other hand, if f = 0 , the ba kgroundsuper(cid:28)eld < Φ > would be independent of the grassmanian oordinate of the superspa e and, as a onsequen e, every superspa e integral of a polinomial of ba kground super(cid:28)eld would identi allyvanish. In four spa etime dimensions, su h di(cid:30) ulties an be surmounted, for example, by using themethodology developed in [3, 4℄ for the evaluation of the super(cid:28)eld e(cid:27)e tive potential. However, thestru ture of three-dimensional supersymmetri models di(cid:27)ers in relevant aspe ts when omparedto the four-dimensional theories. In parti ular, in three-dimensions there are neither hiral noranti- hiral super(cid:28)elds, whi h play a fundamental role in the approa h of [3, 4℄. In this work, we willshow how the above mentioned method must be modi(cid:28)ed for three-dimensional theories. We shallwork on a non ommutative spa etime, but our method also an be applied in the ommutative ase.For simpli ity, we will restri t ourselves to the ase of a theory involving a single s alar super(cid:28)eld.We start with the following three-dimensional super(cid:28)eld theory whi h is des ribed by a generals alar super(cid:28)eld a tion (see f.e. [11℄): S [Φ] = Z d z (cid:20)