Volume 159B, number 1 PHYSICS LETTERS 12 September 1985 BOUNDARY EFFECTS ON THE SOLITON MASS IN (1 + 1)-DIMENSIONAL SUPERSYMMETRIC THEORIES Ashoke K. CHATTERJEE and Parthasarathi MAJUMDAR Saha Institute of Nuclear Physics, 92, Acharya Prafulla Chandra Road, Calcutta 700009, lnd[a Received 4 April 1985 A proper incorporation of boundary effects, in terms of a surface term contribution to the one-loop soliton mass in (1 + 1)-dimensional supersymmetric theories, is shown to resolve the problem of renormalizability in the soliton sector, which otherwise arises in a finite space calculation of the soliton mass using a modified version of a set of boundary conditions proposed by Rouhani. The saturation of the Witten-Olive bound for such boundary conditions follows immediately. The evaluation of the lowest-order quantum cor- rections to the soliton mass in (1 + 1)-dimensional supersymmetric theories has been a subject of some study over the past few years, motivated essentially by the issue of the saturation of the Witten-Olive bound [1]. Schonfetd [2], and later, Kaul and Raja- raman [3] have obtained a non.vanishing, primitively divergent one-loop soliton mass correction, arising from the difference in the bosonic and fermionic eigenmode densities, when periodic boundary condi- tions are imposed on the continuum fluctuations in a finite box. The primitive divergence is renormalised away by the vacuum sector scalar mass counterterm, leaving behind a finite soliton mass correction. The saturation of the Witten-Olive bound has also been established in this approach [4]. These results are in agreement with that obtained by Imbimbo and Mukhi [5] employing infinite space methods. It turns out however, that under an alternative set of boundary conditions proposed by Rouhani [6] (al- legedly superior vis-a-vis supersymmetry) or under a modified version of this set to be elaborated in the sequel, the boson and fermion eigenmode densities are identical, leading to an apparently vanishing soli- ton mass correction. Since these new boundary condi- tions do not affect the vacuum sector, the usual in- clusion of the vacuum sector counterterm produces an infinite contribution to the classical soliton mass, which, in the absence of an equally divergent and op- 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) posite in sign quantum correction implies a serious violation of renormalizabflity in the soliton sector. The saturation of the Witten-Olive bound is also in jeopardy for this choice of boundary conditions. This apparent catastrophe is averted by a proper accounting of the boundary effects that arise when the Rouhani (or the modified version) boundary con- ditions are used. Thus, the soliton mass correction has the usual contribution associated with the density difference of bosonic and fermionic continuum eigen- modes, plus a surface term which depends explicitly on the boundary conditions used. Specializing now either to the case of periodic boundary conditions, or to the case of the modified version of Rouhani boun- dary conditions, the appropriate mass correction is seen to emerge, in accord with renormalizability and the saturation of the Witten-Olive bound. Our work has a close kinship with the infinite space local ap- proach of Yamagishi [7], although the relationship of the latter paper to finite space treatments [2,3] is not entirely transparent. The Lagrange density for the (1 + 1)-dimensional supersymmetric scalar-spinor theory is given by [1 ] ./2= ½ [(a/~0) 2 --S2(~0) + i~-~ -- ~'S'(~0)~]. (1) The function S(~o)is such that the theory has classical vacua ~o, and S'(~o±) = -+/~.The theory then admits classical soliton solutions ~Os(X ) with limx._,±=~os(x ) = ~o±.The hamiltonian corresponding to one.loop 37