On the Almost-zero-energy Eigenvalues of Some Broken Supersymmetric Systems

For a quantum mechanical system with broken supersymmetry, it is demonstrated that reformulation of the problem as that of a (1 + 1)-D Dirac equation allows an easy determination of the ground state if the corresponding energy eigenvalue is sufficiently small. A simple expression is derived for the approximate ground state energy in an associated, wellseparated, asymmetric double-well-type potential. Our discussion is also relevant for the analysis of the fermion bound state in the kink-antikink scalar background. 1 E-mail: [email protected] 2 E-mail: [email protected] Supersymmetry(SUSY) and its breaking are fundamental issues in theoretical particle physics. There have also been numerous applications of SUSY to quantummechanical potential problems [1, 2], based on the observation that the spectrum of the Hamiltonian H+ = − d dx2 + V+(x) , V+(x) =W (x) +W ′(x) (1) (W (x) is the superpotential, and we set ~ = 2m = 1) is related through SUSY to that of the partner Hamiltonian H− = − d dx2 + V−(x) , V−(x) = W (x)−W ′(x) . (2) This formalism has provided us with a number of exactly solvable quantum mechanical systems for which energy eigenvalues and eigenfunctions can be found in closed forms. The key properties that made such feat possible are unbroken SUSY, manifested by the vanishing energy for the ground state of H− (or H+), and shape invariance under the change of parameters for the given potentials [3]-[6]. This approach can sometimes be extended to parameter ranges corresponding to (spontaneously) broken SUSY, and authors of Refs. [7, 8] have found several additional exactly solvable systems by such consideration. But, with SUSY broken, the ground state energy is no longer equal to zero and this jeopardizes the possibility of obtaining exact analytic results by the SUSY-based method in a crucial way. In this work we will show that, in some broken SUSY case for which the lowest energy Ē(> 0) for the Hamiltonian H+ or H− is sufficiently small, reformulating the system into that for a Dirac Hamiltonian (defined on a line) makes possible an easy evaluation of Ē. The conspicuous role assumed by the Dirac operator as regards the zero eigenmodes is well-known [9], and we here extend this role to the case of almost-zero-energy eigenmodes in a restricted sense at least. Our method finds useful application in studying the almost-zero-energy fermion modes in the background of a soliton-antisoliton pair. The superpotential relevant for our discussion is given as follows. Let σR(x) be a generic function with the properties σR(x) > 0 , for x > 0 and |x| not very small , σR(x) → −v , for x < 0 and |x| not very small (3) and σL(x) the one with the properties σL(x) > 0 , for x < 0 and |x| not very small , σL(x) → −v , for x > 0 and |x| not very small (4)