Complex supersymmetry with parity and fermion-number conservation: classification, renormalization, and spontaneous breaking of the Salam–Strathdee framework (1975)

In 1975 Abdus Salam and John Strathdee constructed renormalizable supersymmetric Lagrangians in which a conserved fermion number $\mathbb{F}$ coexists with parity by pairing spinors with scalar 'difermion' partners carrying $\mathbb{F}=2$, the resulting gauge-principle interactions realizing a complex (complexified-charge) extension of supersymmetry — in modern language $N=2$ supersymmetry with the fermion number realized as the $U(1)$ symmetry rotating the two supercharges. The paper, presented at tree level, explicitly left open: (a) the classification of all renormalizable parity- and $\mathbb{F}$-conserving complex-supersymmetric models and of the extended supersymmetry algebras compatible with renormalizability, including a conjectured uniqueness of the renormalizable self-interaction of a single 16-dimensional complex-supersymmetry multiplet; (b) the fate of parity, $\mathbb{F}$ conservation, and the complex-supersymmetry realization under radiative corrections; and (c) whether complex supersymmetry can be spontaneously broken in this class, and whether the Goldstone sector then contains a massless Dirac spinor (a four-component goldstino carrying fermion number) that could split into chiral neutrinos upon further symmetry breaking.

Background

Supersymmetry (SUSY) is a spacetime symmetry whose spinorial charges transform bosons into fermions; ordinary ('$N=1$') supersymmetry has one Majorana (self-conjugate) spinor charge $Q$ with anticommutator $\{Q,\bar Q\}\sim P_\mu$ ($P_\mu$ the momentum). Extended ('$N=2$') supersymmetry has two such charges. 'Complex supersymmetry', the version introduced by Abdus Salam and John Strathdee in the mid-1970s, replaces the real supercharge by a complex spinorial charge, so that its phase rotation defines a conserved unitary symmetry that they called fermion number $\mathbb{F}$; in modern language this is $N=2$ supersymmetry with its $U(1)$ R-symmetry — the internal symmetry rotating the two supercharges — identified with fermion number. The framework can be identified through its usage in later literature: Kallosh's cosmological work, for instance, calls a standard $N=2$ model of a gauge vector plus a charged hypermultiplet (the $N=2$ matter multiplet, containing two complex scalars and two two-component fermions) 'the SSF (Salam–Strathdee–Fayet) model'.

The physical motivation was a structural obstacle. The simplest supersymmetric multiplets contain Majorana (self-conjugate) or single-chirality Weyl (two-component) fermions, and a Majorana mass term violates any conserved additive quantum number such as baryon or lepton number by two units, so fermion number conservation is not automatic in renormalizable supersymmetric theories — renormalizable throughout in the power-counting sense: every interaction operator has mass dimension at most four, so only finitely many couplings are needed. Salam and Strathdee's companion paper 'Supersymmetry and fermion-number conservation' (Nucl. Phys. B 87 (1975) 85) showed that conserving $\mathbb{F}$ forces one to tolerate bosons carrying two units of fermion number — 'difermions' — and the 1975 paper derived the full rules: the fermion-number content of a supermultiplet is governed by its chiral type, with right-handed spinor fermions accompanied by ordinary scalar bosons, left-handed spinor fermions by difermionic bosons, and vector bosons in gauge supermultiplets accompanied by left-handed spinor fermions. Because space reflection (parity) interchanges left- and right-handed chiralities, defining a conserved parity in such theories is nontrivial: the authors showed that naive parity doubling provides no solution unless either supersymmetry or $\mathbb{F}$ is explicitly violated, and that the consistent solution instead makes essential use of local gauge symmetries. For the resulting class of renormalizable Lagrangians, the interactions are governed by a gauge principle and realize the complex extension of the supersymmetry algebra.

The paper was formulated at the classical (tree) level, and its discussion explicitly left several issues open. First, quantum corrections: whether the exhibited classical symmetries — parity, $\mathbb{F}$ conservation, and the realization of the extended (complex) supersymmetry — survive renormalization was not analyzed. Second, uniqueness and classification: the conjectured uniqueness (or near-uniqueness) of the renormalizable self-interaction producing a single 16-dimensional complex-supersymmetry multiplet (the generic massive multiplet of the algebra, with 8 bosonic and 8 fermionic degrees of freedom) was not proved, and a full classification of renormalizable parity- and $\mathbb{F}$-conserving models, and of all extended supersymmetry algebras compatible with renormalizability, was left open. Third, spontaneous breaking: whether complexified supersymmetry can be spontaneously broken in these models, and if so whether the expected massless Goldstone modes (goldstini — the massless fermions associated with spontaneously broken supersymmetry, analogous to Goldstone bosons of broken internal symmetries) include a massless Dirac spinor, i.e. a four-component fermion carrying $\mathbb{F}$ that could split into the observed chiral neutrinos upon further symmetry breaking, was stated as an open dynamical problem. Fourth, as motivation, an appendix sketched a speculative embedding of strong, weak and electromagnetic interactions using several $S_+$ and $S_-$ multiplets (the source's notation for the multiplets of the two conjugate complex supercharges), leaving open the dynamical mechanisms (radiative mass generation, suppression of the parity-violating sector, hierarchy scales set by dimensional couplings) such a unified construction would require.

The subsequent history matters for placing the question. The 'complex supersymmetry' terminology disappears from the indexed literature after the late 1980s (Saidi's 1988 classification of $N=2$ complex supersymmetric representations is its last direct continuation); the modern descendant is the standard $N=2$ formalism, in particular harmonic superspace — an $N=2$ superspace enlarged by coordinates on the sphere $SU(2)/U(1)$ — which gives unconstrained superfield descriptions of $N=2$ matter, Yang–Mills and supergravity theories and their general hypermultiplet self-couplings. Related modern questions — partial supersymmetry breaking $N=2\to N=1$, goldstino effective actions for spontaneously broken $N=2$, and exact results for $N=2$ gauge theories — are active, but they are posed without the parity and conserved-fermion-number constraints that define the present problem.

Problem Statement

Within the class of four-dimensional, power-counting renormalizable (every interaction operator of mass dimension at most four, so only finitely many couplings) Lagrangian field theories that (a) realize the complex (complexified-charge, $N=2$-type) supersymmetry algebra of Salam and Strathdee with its conserved fermion number $\mathbb{F}$ acting as the phase rotation of the complex supercharge, and (b) conserve parity, determine the viability and the complete structure of the framework by answering the three linked questions the 1975 source left open:

1. Classification and uniqueness: classify all such renormalizable models — their gauge sectors, matter (spinor, scalar, and difermion) content, and allowed self-interactions — and all extended supersymmetry algebras (with or without central charges, i.e. additional conserved scalar quantities that commute with every generator and can appear as constants in the supersymmetry anticommutation relations) compatible with renormalizability under the parity and $\mathbb{F}$-conservation constraints; in particular, prove or refute the source's conjecture that the renormalizable self-interaction producing a single 16-dimensional complex-supersymmetry multiplet is unique (or essentially unique).

2. Quantum consistency: determine whether the parity invariance, the $\mathbb{F}$ conservation, and the realization of the complex supersymmetry algebra exhibited at tree level (i.e. in the classical Lagrangian before quantization) survive radiative (loop) corrections — the corrections generated by closed loops of virtual particles in perturbation theory — or establish which of them are necessarily violated or deformed by quantization.

3. Spontaneous breaking: determine whether complex supersymmetry can be spontaneously broken within this class, and if so whether the resulting Goldstone sector contains a massless Dirac spinor — a four-component goldstino carrying fermion number, equivalently two Majorana goldstini forming a Dirac pair — which, upon further symmetry breaking, could split into the observed chiral neutrinos. The question is whether complex supersymmetry is spontaneously broken in this class with parity and $\mathbb{F}$ intact, leaving a massless Dirac goldstino in the Goldstone sector, or whether instead either no vacuum in the class spontaneously breaks complex supersymmetry while conserving parity and $\mathbb{F}$, or any such breaking necessarily breaks $\mathbb{F}$ (or parity) as well, so that no Dirac goldstino carrying a conserved fermion number arises. The complementary positive statement — that spontaneous fermion-number breaking cannot accompany spontaneous supersymmetry breaking, conjectured by Huq on the basis of tree-level plus one-loop analyses of three models — would instead secure the unbroken fermion number that the Dirac-goldstino scenario requires, and establishing or refuting it in general is part of this same question.

These three questions jointly serve one objective — establishing whether the complex-supersymmetry framework can be realized as a consistent quantum field theory with realistic spontaneous breaking — so an answer to one feeds the others (a classification bounds the models in which breaking and renormalization can be analyzed). The paper's appendix proposes, as motivation, a speculative unification of strong, weak, and electromagnetic interactions in this framework using several $S_+$ and $S_-$ multiplets, requiring dynamical mechanisms (radiative mass generation, suppression of the heavy parity-violating sector, hierarchy scales set by dimensional couplings); this programmatic component is part of the source question and is motivation for questions 1–3 rather than a further question posed here.

References

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