Almost-zero-energy Eigenvalues of Some Broken Supersymmetric Systems

For a quantum mechanical system with broken supersymmetry, we present a simple method of determining the ground state when the corresponding energy eigenvalue is sufficiently small. A concise formula is derived for the approximate ground state energy in an associated, well-separated, asymmetric double-well-type potential. Our discussion is also relevant for the analysis of the fermion bound state in the kink-antikink scalar background. ∗Email address: [email protected] †Email address: [email protected] 1 Supersymmetry(SUSY) and its breaking are fundamental issues in theoretical particle physics. There have also been numerous applications of SUSY to quantum-mechanical 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][5]. This approach can sometimes be extended to parameter ranges corresponding to (spontaneously) broken SUSY [6,7]. 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, a simple perturbative scheme leading to an easy evaluation of Ē can be developed. Our discussion 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) so that the related potentials VR±(x) ≡ σ R ± σ′ R(x) and VL±(x) ≡ σ L ± σ′ L(x) may have the general shapes shown in Figs.1 and 2, respectively. Then the superpotential appropriate to our case is obtained by combining these two types of functions as W (x) = σR(x− l1) + σL(x− l2) + v (5) with L ≡ |l1− l2| taken to be reasonably large (so that W (x) may have a flat basin between the points x = l2 and x = l1). 1 See Fig.3 for the schematic plots of W (x) and the related 1At the later stage of our discussion we will use the fact if l∗ denotes a certain point in the flat basin, the approximation σR(x− l1) + v ≃ 0 for x < l∗ or σL(x− l2) + v ≃ 0 for x > l∗ is valid. 2 potentials V±(x) ≡ W (x) ± W ′(x). Both W (∞) and W (−∞) being positive, this corresponds to the case of broken SUSY [1,2]; but, for the posited superpotential (with L large), the ground state energy Ē is expected to be rather small. [Our superpotential will be an even function of x if σL(x) happens to be the mirror image of σR(x), i.e., σL(x) = σR(−x), and (l1, l2) are equal to ( L 2 ,− 2 )]. For W (x) specified as above, the corresponding Hamiltonians H± involve the potentials which can approximately be described by the sum of two well-separated potentials (aside from a constant term), i.e., V±(x) ∼ VR±(x− l1) + VL±(x− l2)− v . (6) These correspond to asymmetric double wells even whenW (x) is an even function, and hence the well-known approximation schemes used for symmetric double wells (e.g., instanton methods, tight-binding approximations) would not be much useful. [Note that, for the ground state of our Hamiltonian with the potential V+ ∼ VR++VL+−v (or V− ∼ VR−+VL−− v), the tight-binding approximation is plainly not available — while the local Hamiltonian involving VL+ (or VR−) allows a zero-energy bound state, no zero-energy state exists for the local Hamiltonian involving VR+ (or VL−)]. But, for a supersymmetric system, one can always consider a pair of coupled first-order differential equations instead of the secondorder Schrödinger equations. Our perturbative approach for the ground state is based on the analysis of these first-order equations, and as a result we obtain a remarkably simple formula for the lowest eigenvalue Ē. It is simply the square of the product of the two zeroenergy eigenfunctions (allowed with the potentials VL+(x− l2) and VR−(x− l1) separately), evaluated at an arbitrary chosen point l∗ in the flat middle region of the superpotential W (x). See the expression (21) below. Consider a matrix Hamiltonian H = ( 0 A† A 0 ) (7) with A = ∂x +W (x) , A † = −∂x +W (x) . (8) The corresponding eigenvalue equation, HΨ(x) = ωΨ(x) with Ψ(x) = (