Quaternionic Formulation of Supersymmetric Quantum Mechanics

Quaternionic formulation of supersymmetric quantum mechanics has been developed consistently in terms of Hamiltonians, super partner Hamiltonians, and supercharges for free particle and interacting field in one and three dimensions. Supercharges, super partner Hamiltonians and energy eigenvalues are discussed and it has been shown that the results are consistent with the results of quantum mechanics.     

Supersymmetry in quantum mechanics

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable. In this lecture I review the theoretical formulation of supersymmetric quantum mechanics and discuss many of its applications. I show that the well-known exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials and shape invariance. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Further, it is pointed out that the connection between the solutions of the Dirac equation and the Schrödinger equation is exactly same as between the solutions of the MKdV and the KdV equations.     

Supersymmetric preon models with three-fermion generations

A class of supersymmetric preon models is considered in which the hypercolour groupG _HC and the unbroken flavour groupG _f anomalies are zero without needing spectators. It is shown that forG _HC=SU(2) and SU(3) quarks and leptons as composites can be obtained satisfying ’t Hooft’s anomaly matching conditions. For the case ofG _HC=SU(3),G _f can accommodate a horizontal symmetry group to describe just three generations.     

Supersymmetric preon models with gauged colour-flavour symmetry

We have conducted a search for globally supersymmetric preon models with gauged colour-flavour symmetries. Theories with both two- and three-preon composites, and colour-flavour groups from E₆ down to the standard model, are examined under the following conditions: asymptotically-free metacolour, anomaly-free gauged symmetries, and Pauli principle obeyed. It is found that there are no models with three or more supersymmetric families. If supersymmetry is broken, one model with four families emerges. The purely fermionic preon theories can also be considered as the light sector of a chiral supersymmetric theory, with supersymmetry breaking at the preon level.     

Supersymmetric classical mechanics

The purpose of the paper is to construct a supersymmetric Lagrangian within the framework of classical mechanics which would be regarded as a candidate for passage to supersymmetric quantum mechanics. The authors felicitate Prof. D S Kothari on his eightieth birthday and dedicate this paper to him on this occasion.     

Split supersymmetry in brane models

Type-I string theory in the presence of internal magnetic fields provides a concrete realization of split supersymmetry. To lowest order, gauginos are massless while squarks and sleptons are superheavy. For weak magnetic fields, the correct Standard Model spectrum guarantees gauge coupling unification with sin² ϑW=3/8 at the compactification scale of M _GUT ⋍ 2 × 10¹⁶ GeV. I discuss mechanisms for generating gaugino and higgsino masses at the TeV scale, as well as generalizations to models with split extended supersymmetry in the gauge sector.     

Constraints on sparticle spectrum in different supersymmetry breaking models

We derive sum rules for the sparticle masses in different models of supersymmetry breaking. This includes the gravity-mediated models (SUGRA models) as well as models in which supersymmetry breaking terms are induced by super-Weyl anomaly (AMSB models). These sum rules can help in distinguishing between these models. In particular, we obtain an upper bound on the mass of the lightest neutralino as a function of the gluino mass in SUGRA and AMSB models.     

A class of partially solvable two-dimensional quantum models with periodic potentials

The supersymmetric approach is used to analyse a class of two-dimensional quantum systems with periodic potentials. In particular, the method of SUSY-separation of variables allowed us to find a part of the energy spectra and the corresponding wave functions (partial solvability) for several models. These models are not amenable to conventional separation of variables, and they can be considered as two-dimensional generalizations of Lamé, associated Lamé, and trigonometric Razavy potentials. All these models have the symmetry operators of fourth order in momenta, and one of them (the Lamé potential) obeys the property of self-isospectrality.     

A generalization of Hamiltonian mechanics

For a symplectic manifold the Poisson bracket on the space of functions is (uniquely) extended to a graded Lie bracket on the space of differential forms modulo exact forms. A large portion of the Hamiltonian formalism is still working.     

On messengers and metastability in gauge mediation

One notoriously difficult problem in perturbative gauge mediation of supersymmetry breaking via messenger fields is the generic presence of a phenomenologically unacceptable vacuum with messenger vevs, with a lower energy than the desired (“MSSM”) vacuum. We investigate the possibility that quantum corrections promote the latter to the ground state of the theory, and find that this is indeed feasible. For this to happen, the couplings of the messengers to the goldstino superfield must be small, and this implies an additional suppression of the MSSM soft terms with respect to the supersymmetry breaking scale. This in turn sets a lower limit on the masses of the messengers and of the supersymmetry breaking fields, which makes both sectors inaccessible at colliders. Contrary to other scenarios like direct gauge mediation, gaugino masses are unsuppressed with respect to scalar masses.     

The effective Kähler potential,metastable vacua and R-symmetry breaking in O'Raifeartaigh models

Much has been learned about metastable vacua and R-symmetry breaking in O'Raifeartaigh models. Such work has largely been done from the perspective of the superpotential and by including Coleman–Weinberg corrections to the scalar potential. Instead, we consider these ideas from the perspective of the one loop effective Kähler potential. We translate known ideas to this framework and construct convenient formulas for computing individual terms in the expanded effective Kähler potential. We do so for arbitrary R-charge assignments and allow for small R-symmetry violating terms so that both spontaneous and explicit R-symmetry breaking is allowed in our analysis.     

Spontaneous supersymmetry breaking in matrix models from the viewpoints of localization and Nicolai mapping

In the previous work, it was shown that, in supersymmetric (matrix) discretized quantum mechanics, inclusion of an external field twisting the boundary condition of fermions enables us to discuss spontaneous breaking of supersymmetry (SUSY) in the path-integral formalism in a well-defined way. In the present work, we continue investigating the same systems from the points of view of localization and Nicolai mapping. The localization is studied by changing of integration variables in the path integral, which is applicable whether or not SUSY is explicitly broken. We examine in detail how the integrand of the partition function with respect to the integral over the auxiliary field behaves as the auxiliary field vanishes, which clarifies a mechanism of the localization. In SUSY matrix models, we obtain a matrix-model generalization of the localization formula. In terms of eigenvalues of matrix variables, we observe that eigenvalues' dynamics is governed by balance of attractive force from the localization and repulsive force from the Vandermonde determinant. The approach of the Nicolai mapping works even in the presence of the external field. It enables us to compute the partition function of SUSY matrix models for finite N (N is the rank of matrices) with arbitrary superpotential at least in the leading nontrivial order of an expansion with respect to the small external field. We confirm the restoration of SUSY in the large-N limit of a SUSY matrix model with a double-well scalar potential observed in the previous work.     

Generalized spin and pseudo-spin symmetry: Relativistic extension of supersymmetric quantum mechanics

We consider the Dirac equation in 1+1$1+1$ space–time dimension with vector, scalar and pseudo-scalar coupling. In the traditional spin (or pseudo-spin) symmetry, the difference between (or sum of) the scalar and vector potentials is a constant. Here, however, we introduce an alternative symmetry where the scalar or pseudo-scalar potential is proportional to the vector potential. This leads to a model with significant extensions to supersymmetric quantum mechanics. We present a formal solution of the problem but give explicit analytic results for specific examples.     

Statistical mechanics of superfluid models

It is shown that the equilibrium thermodynamic properties of some superfluid models can be calculated by using density matrices defined in suitable irreducible representations. The approach enables one to introduce in a fundamental manner temperature-dependent order parameters and can be used to obtain general formulations of the problems of superfluids.     

Goldstino and sgoldstino in microscopic models and the constrained superfields formalism

We examine the exact relation between the superconformal symmetry breaking chiral superfield (X) and the goldstino superfield in microscopic models of an arbitrary Kahler potential (K) and in the presence of matter fields. We investigate the decoupling of the massive sgoldstino and scalar matter fields and the offshell/onshell-SUSY expressions of their superfields in terms of the fermions composites. For general K of two superfields, we study the properties of the superfield X after integrating out these scalar fields, to show that in the infrared it satisfies (offshell) the condition X³=0 and X²≠0. We then compare our results to those of the well-known method of constrained superfields discussed in the literature, based on the conjecture X²=0. Our results can be used in applications, to couple offshell the (s)goldstino fields to realistic models such as the MSSM.     

Hierarchy generation in compactified supersymmetric models

The problem of generating a large hierarchy in compactified supersymmetric models is re-examined. It is shown how, even for the class of models for which Str M² is non-vanishing, a combination of non-perturbative effects and radiative corrections may lead to an exponentially large hierarchy. A corollary is that the couplings of the effective field theory in the visible sector should be small, i.e., perturbation theory should be applicable.     

Particle mass limits in minimal and nonminimal supersymmetric models

A review of the Higgs and neutralino sector of supersymmetric models is presented. This includes the upper limit on the mass of the lightest Higgs boson in the minimal supersymmetric standard model, as well as models based on the standard model gauge groupSU(2) _L xU(l) _Y with extended Higgs sectors. We then discuss the Higgs sector of left-right supersymmetric models, which conserveR-parity as a consequence of gauge invariance, and present a calculable upper bound on the mass of the lightest Higgs boson in these models. We also discuss the neutralino sector of general supersymmetric models based on the SM gauge group. We show that, as a consequence of gauge coupling unification, an upper bound on the mass of the lightest neutralino as a function of the gluino mass can be obtained.