Z. Phys. C - Particles and Fields 30, 79-81 (1986) Particles ftir Physik C and F lds 9 Springer-Verlag 1986 Effective Lagrangian for a Supersymmetric Non-Abelian Gauge Theory Pinaki Roy Electronics Unit, Indian Statistical Institute, Calcutta - 700035, India Received 25 June 1984 Abstract. We evaluate the one-loop effective poten- tial for a supersymmetric SU(2) gauge theory in the presence of a constant external magnetic field and discuss the minima of the resulting effective poten- tial. 1. Introduction In the last few years, there have been a number of studies concerning the effect of an external magnetic field on both pure Yang-Mills theory [1-3], and gauge theories with spontaneously broken non-abe- lian gauge symmetry [4]. This problem is important for various reasons, e.g. colour confinement in QCD [5]. However, so far, no investigation on the effects of a strong external magnetic field on supersym- metric models has been carried out*. In the present paper, we shall consider a supersymmetric, non- abelian gauge theory in the presence of a strong external magnetic field and evaluate the effective Lagrangian to one-loop order. In carrying out the calculations, we shall use the background field tech- nique due to Schwinger [63 and De-Will [7]. The paper is organised as follows: in Sect. 2 we describe the model; in Sect. 3, the evaluation of the one-loop correction to the effective potential is shown; in Sect. 4, we discuss the minimal of the resulting effec- tive potential and Sect. 5 is denoted to a discussion. 2. The Model The model that we consider is described by the following Lagrangian: L: 1 a 2 1 2 1 2 --~(Fr +5(DuA) +~(D.B) +itfiDO + i g klrn.771*'ak "nk m 1 2." klm AI B,,,)2 qJ ta -tD 7s)~b -~g te (2.1) * For a method involving vacuum energy summation method see [14 3 where DuA k=a~A k klm 1 m +ge AuA etc. Here A,B, and A, stands for triplets of scalar, pseudo-scalar, Dirac spinor and gauge mesons respectively. All these fields are in the adjoint representation of the group SU(2) and the Lagrangian (2.1) is invariant under supersymmetric transformations and local S U(2) gauge transformations. It is to be noted that only the charged particles are affected by the magnetic field and as such we write AI+iA 2 a + -- BI +_iB 2 B+-_ ~,+__O~ +_iO 2 (2.2) Now, we shift the neutral fields A 3 and B 3 as fol- lows: A3~A3+q~ B3~Ba+a. (2.3) By this choice we have SU(2)--, U(1) and we identify A~ with the photon. The charged Higgs scalars are then given by o A -+ _ q S B - + H -+ _ (4) 2+ ~r2) ~ (2.4) and the Goldstone particles are given by G + - _( oA• +aB • ((I{) 2 -~ 0-2) 89 " (2.5)