SUSY Invariance Research Articles

Supersymmetry shielding the scaling symmetry of conformal quantum mechanics

Renormalization of the inverse square potential usually breaks its classical conformal invariance. In a strongly attractive potential, the scaling symmetry is broken to a discrete subgroup while, in a strongly repulsive potential, it is preserved at quantum level. In the intermediate, weak-medium range of the coupling, an anomalous length scale appears due to a flow of the renormalization group away from a critical point. We show that potentials with couplings in the strongly-repulsive and in the weak-medium ranges can be related by a dynamical supersymmetry. Imposing SUSY invariance unifies these two ranges, and fixes the anomalous scale to zero, thus restoring the continuous scaling symmetry.

Magnetic charge quantization from SYM considerations

An intersecting D3-D3′ system contains magnetic monopole solutions due to D-strings stretched between two branes. These magnetic charges satisfy the usual Dirac quantization relation. We show that this quantization condition can also be obtained directly by SUSY and gauge invariance arguments of the theory and conclude that the independence of physics from a shift of holonomy is exactly equivalent to regarding a Fayet-Iliopoulos (FI) gauge for our set-up. So we are led to conjecture that there is a correspondence between the topological point of view of magnetic charges and SYM considerations of their theories. This picture implies that one can attribute a definite quantity to the integration of the vector multiplet over the singular region such that we can identify it with magnetic flux. It also indicates that the FI parameter is proportional to the magnetic charge so it is a quantized number.

Novel symmetries in an interacting 𝒩 = 2 supersymmetric quantum mechanical model

In this paper, we demonstrate the existence of a set of novel discrete symmetry transformations in the case of an interacting [Formula: see text] supersymmetric quantum mechanical model of a system of an electron moving on a sphere in the background of a magnetic monopole and establish its interpretation in the language of differential geometry. These discrete symmetries are, over and above, the usual three continuous symmetries of the theory which together provide the physical realizations of the de Rham cohomological operators of differential geometry. We derive the nilpotent [Formula: see text] SUSY transformations by exploiting our idea of supervariable approach and provide geometrical meaning to these transformations in the language of Grassmannian translational generators on a [Formula: see text]-dimensional supermanifold on which our [Formula: see text] SUSY quantum mechanical model is generalized. We express the conserved supercharges and the invariance of the Lagrangian in terms of the supervariables (obtained after the imposition of the SUSY invariant restrictions) and provide the geometrical meaning to (i) the nilpotency property of the [Formula: see text] supercharges, and (ii) the SUSY invariance of the Lagrangian of our [Formula: see text] SUSY theory.

Formula omitted] SUSY symmetries for a moving charged particle under influence of a magnetic field: Supervariable approach

We exploit the supersymmetric invariant restrictions (SUSYIRs) on the supervariables to derive the nilpotent N=2 SUSY transformations for the supersymmetric quantum mechanical model of the motion of a charged particle in the X–Y plane (where the magnetic field (Bz) is applied along the Z-direction). The supervariables are defined on a (1, 2)-dimensional supermanifold parametrized by a bosonic “time” variable t and a pair of Grassmannian variables θ and θ̄ (with θ2=θ̄2=0,θθ̄+θ̄θ=0). We take the (anti-)chiral supervariables for our purpose so that the nilpotency property of the N=2 SUSY symmetry transformations could be captured within the framework of supervariable approach. We express the Lagrangian as well as supercharges in terms of the supervariables (that are obtained after the application of the appropriate SUSYIRs) and provide geometrical basis, within the framework of our supervariable approach, for (i) the nilpotency property of the above SUSY transformations (and the corresponding supercharges), and (ii) the SUSY invariance of the Lagrangian.

Supervariable approach to nilpotent symmetries of a couple of N = 2 supersymmetric quantum mechanical models

We derive the on-shell as well as off-shell nilpotent supersymmetric (SUSY) symmetry transformations for the [Formula: see text] = 2 SUSY quantum mechanical model of a (0 + 1)-dimensional (1D) free SUSY particle by exploiting the SUSY-invariant restrictions on the (anti-)chiral supervariables of the SUSY theory that is defined on a (1, 2)-dimensional supermanifold (parametrized by a bosonic variable t and a pair of Grassmannian variables θ and [Formula: see text] with [Formula: see text]). Within the framework of our novel approach, we express the Lagrangian and conserved SUSY charges in terms of the (anti-)chiral supervariables to demonstrate the SUSY invariance of the Lagrangian as well as the nilpotency of the SUSY conserved charges in a simple manner. Our approach has the potential to be generalized to the description of other [Formula: see text] = 2 SUSY quantum mechanical systems with physically interesting potential functions. To corroborate the preceding assertion, we apply our method to derive the [Formula: see text] = 2 continuous and nilpotent SUSY transformations for one of the simplest interacting SUSY systems of a 1D harmonic oscillator.

General [formula omitted] supersymmetric quantum mechanical model: Supervariable approach to its off-shell nilpotent symmetries

Using the supersymmetric (SUSY) invariant restrictions on the (anti-)chiral supervariables, we derive the off-shell nilpotent symmetries of the general one (0+1)-dimensional N=2 SUSY quantum mechanical (QM) model which is considered on a (1, 2)-dimensional supermanifold (parametrized by a bosonic variable t and a pair of Grassmannian variables θ and θ̄ with θ2=θ̄2=0,θθ̄+θ̄θ=0). We provide the geometrical meanings to the two SUSY transformations of our present theory which are valid for any arbitrary type of superpotential. We express the conserved charges and Lagrangian of the theory in terms of the supervariables (that are obtained after the application of SUSY invariant restrictions) and provide the geometrical interpretation for the nilpotency property and SUSY invariance of the Lagrangian for the general N=2 SUSY quantum theory. We also comment on the mathematical interpretation of the above symmetry transformations.

Makarov potential in relativistic equation via Laplace transformation approach

Exact solutions and corresponding normalized eigenfunctions of the Dirac equation are studied for the Makarov potential by using the Laplace transform approach under the pseudospin symmetry. By using the ideas of SUSY and shape invariance, we obtain the energy eigenvalues equation. The wave functions of the angle part are obtained by using the Nikiforov–Uvarov method, too. Finally, we also discuss the special cases of this potential and the valence energy states can be produced from our solution for the hole state by taking appropriate transformation of parameters.

The SUSY-QCD β function to three loops

A number of $\overline {\mbox {\textsc{D}R}}$ renormalization constants in softly broken SUSY- QCD are evaluated to three-loop level: the wave function renormalization constants for quarks, squarks, gluons, gluinos, ghosts, and ε-scalars, and the renormalization constants for the quark and gluino mass as well as for all cubic vertices. The latter allow us to derive the corresponding β functions through three loops, all of which we find to be identical to the expression for the gauge β function obtained by Jack et al. (Phys. Lett. B 386:138, 1996, hep-ph/9606323 ) (see also Pickering et al. in Phys. Lett. B 510, 347, 2001, hep-ph/0104247 ). This explicitly demonstrates the consistency of DRED with SUSY and gauge invariance, an important pre-requisite for precision calculations in supersymmetric theories.

(0,2) Gauged linear sigma model on supermanifold

AbstractWe construct (0,2), D = 2 gauged linear sigma model on supermanifold with both an Abelian and non‐Abelian gauge symmetry. For the purpose of checking the exact supersymmetric (SUSY) invariance of the Lagrangian density, it is convenient to introduce a new operator for the Abelian gauge group. The operator provides consistency conditions for satisfying the SUSY invariance. On the other hand, it is not essential to introduce a similar operator in order to check the exact SUSY invariance of the Lagrangian density of non‐Abelian model, contrary to the Abelian one. However, we still need a new operator in order to define the (0,2) chirality conditions for the (0,2) chiral superfields. The operator can be defined from the conditions assuring the (0,2) supersymmetric invariance of the Lagrangian density in superfield formalism for the (0,2) U(N) gauged linear sigma model. We found consistency conditions for the Abelian gauge group which assure (0,2) supersymmetric invariance of Lagrangian density and agree with (0,2) chirality conditions for the superpotential. The supermanifold ℳ︁m|n becomes the super weighted complex projective space WCPm‐1|n in the U(1) case, which is considered as an example of a Calabi‐Yau supermanifold. The superpotential W(ϕ,ξ) for the non‐Abelian gauge group satisfies more complex condition for the SU(N) part, except the U(1) part of U(N), but does not satisfy a quasi‐homogeneous condition. This fact implies the need for taking care of constructing the Calabi‐Yau supermanifold in the SU(N) part. Because more stringent restrictions are imposed on the form of the superpotential than in the U(1) case, the superpotential seems to define a certain kind of new supermanifolds which we cannot identify exactly with one of the mathematically well defined objects.

(0,2) Gauged linear sigma model on supermanifold

Göttinger Dissertation. Reprint of “Über Elementarakte mit zwei Quantensprüngen”, Annalen der Physik 9 [401], 273–294 (1931). Translation by Daniel C. Koepke, Montana State University, Department of Physics, Bozeman, MT 59717-3840, USA; e-mail: dkoepke@physics.montana.edu

Ring structure of SUSY [formula omitted] product and 1/2 SUSY Wess–Zumino model

There are two types of non(anti)commutative deformation of D=4, N=1 supersymmetric field theories and D=2, N=2 theories. One is based on the nonsupersymmetric star product and the other is based on the supersymmetric star product. These deformations cause partial breaking of supersymmetry in general. In case of supersymmetric star product, the chirality is broken by the effect of the supersymmetric star product, then it is not clear that Lagrangian or observables including F-terms preserve part of supersymmetry. In this Letter, we investigate the ring structure whose product is defined by the supersymmetric star product. We find the ring whose elements correspond to 1/2 SUSY F-terms. Using this, the 1/2 SUSY invariance of the Wess–Zumino model is shown easily and directly.