Homeotic Supersymmetry

It is commonly believed that unbroken supersymmetry (SUSY) implies that all members of a supermultiplet have the same mass. We demonstrate that this is not true, by exhibiting a simple counterexample. We employ the formalism of homeotic fermions, in a simple model where CPT conjugate fermions have different masses. This model can be supersymmetrized to a hypermultiplet of fields which form a representation of the conventional N=2 SUSY algebra. Nevertheless, CPT conjugate states in this hypermultiplet have different masses. These surprising results do not violate either the CPT theorem nor the Haag-Lopuszanski-Sohnius theorem.


INTRODUCTION
Supersymmetries are the only possible extensions of the four dimensional Poincaré invariance observed ubiquitously in particle interactions [1,2]. Supersymmetry (SUSY) plays a fundamental role in string theory, and there are many strong phenomenological motivations for believing that supersymmetry is realized in nature, in spontaneously broken form. On the other hand, no superpartner particles have yet been observed, and spontaneously broken supersymmetry makes a prediction for the cosmological vacuum energy density which is too large by at least 60 orders of magnitude. Thus it is important to push the boundaries of our fundamental understanding of supersymmetry, and more especially to look for novel ways of expressing supersymmetry in physical systems.
It is widely believed that unbroken supersymmetry implies that all members of a supermultiplet have the same mass. For example Sohnius' authoritative review article states explicitly that "supersymmetry must be broken in nature where elementary particles do not come in massdegenerate multiplets" [3]. This belief is based upon the strong constraint that any conserved supercharge Q must commute with the 4-momentum operator P µ , which in turn implies the O'Raifeartaigh theorem: In spite of these facts, we demonstrate in this letter that exact supersymmetry does not always imply mass-degenerate multiplets. We employ the formalism of homeotic fermions [4], developed previously by us as a toy model for CP T violation in the neutrino sector [5,6,7,8,9]. Our starting point is an extremely simple model where CP T conjugate fermions have different masses. This model can be supersymmetrized to a hypermultiplet of fields which form a representation of the conventional N =2 SUSY algebra. Nevertheless, CP T conjugate states in this hypermultiplet have different masses. Each fermion state is still mass-degenerate with a boson state, however the hypermultiplet is not reducible to a pair of mass-degenerate supermultiplets. These surprising results do not violate either the CP T theorem [10] nor the Haag-Lopuszański-Sohnius theorem [1].

HOMEOTIC FERMIONS
Homeotic fermions, like Dirac fermions, are denoted by 4-component complex spinor fields ψ(x). In the free homeotic theory the Fourier-transformed fermion fields obey the equation of motion: which differs from the Dirac equation by the presence of ǫ(p 0 ), the sign function of delta calculus. Solutions of (2) are solutions of the Klein-Gordon equation; the homeotic theory can be regarded as "the other square root" of the Klein-Gordon equation. The homeotic case is usually neglected because the equation of motion is nonlocal in position space: where P denotes the principal value integral, which we will now assume throughout. Despite this nonlocality, it was demonstrated in [4] that causal interacting homeotic field theories exist, in the sense of having a well-defined perturbative S-matrix. As noted in [4], the combination of a homeotic mass term with a Dirac mass term violates CP T . Consider the simple free theory defined by the lagrangian (4) If we define the CP T operator such that the Dirac mass term is CP T even, then the homeotic mass term is CP T odd, and vice-versa. CP T conjugate spinors in this theory have mass-squared eigenvalues (m d ± m h ) 2 .
We would now like to extend this theory of 4component complex spinors to a supermultiplet of fields which furnish a representation of the standard N =2 SUSY algebra. In the 4-component notation of Sohnius [3] the relevant parts of the N =2 algebra are: where the Q i , i=1, 2, are symplectic Majorana spinor supercharges, and Z is the (antihermitian) central charge operator, which commutes with all of the other generators of the algebra. The supercharges form a doublet under the SU (2) R symmetry of N =2 SUSY; the index i is raised and lowered with the two-dimensional Levi-Civita tensors ǫ ij =ǫ ij . In our derivation we will need a number of identities for bilinears of symplectic Majorana spinors:ζ Let A i (x) denote a doublet of complex scalar fields, and ψ(x) a 4-component complex fermion field. We want these fields to form an N =2 hypermultiplet, i.e., to furnish a representation of the algebra (5). The fundamental relation between the fields is one additional input is required: where F i (x) is a doublet of complex bosonic auxiliary fields; we will specify the precise form of F i (x) later. Expressions (7) and (8), together with the algebra (5), the Jacobi identities, and the identities (6), now imply: These relations define a set of hypermultiplet fields A i (x), ψ(x), and F i (x) which form a representation of N =2 SUSY. These relations determine uniquely the equations of motion for the dynamical fields A i (x) and ψ(x), once we specify the auxiliary fields F i (x). If e.g. we write then (8) combined with (9) implies that the A i (x) satisfy the Klein-Gordon equation, while (7) combined with (9) implies that ψ(x) obeys the Dirac equation.

SIMPLE EXAMPLE OF HOMEOTIC SUPERSYMMETRY
We obtain the simplest example of homeotic supersymmetry by specifying the auxiliary fields as follows: Applying the relations (7-9), we determine the equations of motion to be: The action which reproduces (11), (12) is given by where It is easy to check that all four pieces of the action are separately invariant under the standard N =2 SUSY transformations of the fields: We can proceed further to construct the conserved supercurrent in terms of the component fields. However the usual Noether procedure does not yield a conserved current; this is a generic feature of nonlocal field theories, and was noted in our previous work with regard to the fermion number current of the homeotic fermion theory. Let us first review that case, in which we employed a trick of Pauli's construct the conserved fermion number current: In a supersymmetric theory the conserved bosonic current J µ (x) must belong to a supermultiplet of conserved currents. In particular, we can immediately obtain an expression for the supercurrent j µi , by applying the supersymmetry transformations (15) to the component fields in (16). The result is: Using the equations of motion (11), (12), one finds that j µi (x) is conserved on-shell. It thus represents the supercurrent modulo possible "improvement" terms [3].

MASS SPECTRUM
Both the bosonic and fermionic parts of the action (13) violate CP T . Let us focus first on the bosonic sector. We can expand the fields A i (x) in positive and negative frequency plane wave solutions of the equations of motion (12): We quantize the theory by assuming that a pi , b pi satisfy the commutation relations of creation/annihilation operators. It follows that the general commutator of A i (x) with its conjugate Π j (x) is given by Thus the equal-time commutator is canonical, but the general commutator is not. Using (19) we can now verify that the supercharge extracted from (16) satisfies (7).
Another novel feature appears when we construct the bosonic part of the hamiltonian in terms of a pi , b pi . The canonical hamiltonian is not diagonalized in the basis defined by (18); it is instead diagonalized in the basis defined by where In this basis the bosonic hamiltonian is showing that the CP T conjugate single particle states have a mass-squared splitting equal to |4m d m h |. A similar analysis for the fermions diagonalizes the fermionic part of the hamiltonian in terms of the anticommuting Fock operatorsã ps ,b ps , where s is the spin label: Again the CP T conjugate states have a mass-squared splitting equal to |4m d m h |.

COMMENTS
It would be interesting to extend the above construction to produce an interacting theory. A conventional N =2 hypermultiplet can interact with an N =2 vector multiplet, or can have self-interactions describing a nonlinear sigma model [11]. For the homeotic case neither extension appears entirely straightforward.
The homeotic N =2 hypermultiplet has 8 real on-shell degrees of freedom. We have just seen that half of these describe a boson-fermion pair with mass |m d +m h |, while the other half describe a boson-fermion pair with mass |m d − m h |.
Since an ordinary N =2 hypermultiplet can be split into two N =1 chiral multiplets, it is important to ask whether our homeotic N =2 hypermultiplet is reducible into two N =1 multiplets. The answer is no. This can be seen by imposing the Majorana condition on the fermions in (4), and observing that the homeotic mass term then vanishes identically. Alternatively, one notes that the usual decomposition of the hypermultiplet into chiral multiplets can be written in the Fock basis as obviously in the homeotic case this would mix operators with different dispersion relations. The irreducibility of the homeotic N =2 hypermultiplet is in fact very analogous to the irreducibility of the ordinary N =1 chiral multiplet. In this case one finds the 4 on-shell degrees of freedom consist of two CP T conjugate pairs; each pair has one boson and one fermion state, related by supersymmetry. However it is not possible to reduce the multiplet, due to the non-existence of Majorana-Weyl spinors in four dimensions [12]. This is the analog of the non-existence of homeotic Majorana spinors in four dimensions.
As a parting remark, let's inquire how one might attempt to contruct a supermultiplet in which bose-fermi degeneracy is violated. At the level of the on-shell hamiltonian, this does not appear to be particularly difficult. Let H 0 (m) be a mass term with mass parameter m for a free supersymmetric hamiltonian containing two species of fermions, and construct a new hamiltonian defined by where F 1 and F 2 are the fermion number operators for the two species of fermions. Clearly the single particle eigenstates of H 1 have different masses for bosons and fermions: the bosons have mass m+m ′ while the fermions have mass m − m ′ . It is easy to see that, acting on single particle states: from which it follows that, acting on single particle states: and thus supersymmetry is unbroken. The challenge, of course, is to realize such a scheme in field theory. We would like to acknowledge helpful discussions with Bill Bardeen, Piyush Kumar, Chris Quigg, and Erick Weinberg. This research was supported by the U.S. Department of Energy Grants DE-AC02-76CHO3000 and DE-FG02-90ER40560.