Completing Weinberg's ordinary-energy supersymmetry program: natural Fayet-Iliopoulos terms, anomaly-free U(1) spectra, vacuum stability, and dimension-six proton decay

Weinberg's 1982 paper 'Supersymmetry at ordinary energies. Masses and conservation laws' proposed to build a realistic globally supersymmetric extension of the Standard Model with gauge group $SU(3)\times SU(2)\times U(1)\times \widetilde{U}(1)$, in which an extra abelian factor $\widetilde{U}(1)$ forbids all dimension-4 and dimension-5 baryon- and lepton-number-violating operators, and Fayet-Iliopoulos $D$-terms with electroweak-scale coefficients $\xi$ spontaneously break supersymmetry and the electroweak group at ordinary energies, leaving dimension-6 four-fermion operators as the leading proton-decay mechanism. The paper itself acknowledged that the program was unfinished: the dynamical origin and natural smallness of $\xi$; the choice of extra chiral superfields $X_L$ needed to cancel the $\widetilde{U}(1)$ anomalies without reintroducing renormalizable baryon or lepton violation; the phenomenology of the extra neutral gauge boson $\widetilde{Z}^0$ and of light or massless gauginos; the quantitative size and detectability of dimension-6 proton decay; the avoidance of deeper supersymmetry-preserving charge-breaking minima (the Claudson-Wise obstruction and the tension between adding enough $\widetilde{Y}=+4$ fields and keeping the $\widetilde{Y}^3$ anomaly small); and the treatment of $R$ symmetries.

Background

By 1981 supersymmetry - the proposed symmetry that pairs each boson with a fermionic superpartner and vice versa - was regarded as a candidate solution to the hierarchy problem (the unexplained smallness of the electroweak scale relative to the Planck or grand-unification scale), but no realistic low-energy supersymmetric model existed. In a supersymmetric field theory, particles organize into chiral superfields (each containing a complex scalar and a two-component fermion) and vector superfields (a gauge boson plus its fermionic partner, the gaugino). Interactions are encoded in a holomorphic superpotential, whose scalar components generate the $F$-terms, and in $D$-terms from the gauge sector. For abelian gauge factors, gauge invariance allows a constant term $\xi$ to be added to the $D$-term; this is the Fayet-Iliopoulos term, introduced by Fayet and Iliopoulos in 1974. A Fayet-Iliopoulos term can spontaneously break supersymmetry (and gauge symmetry) at a scale of order $g\xi^{1/2}$.

Weinberg's paper 'Supersymmetry at ordinary energies. Masses and conservation laws' (Phys. Rev. D 26, 287 (1982)) opened by identifying three problems in the minimal $SU(3)\times SU(2)\times U(1)$ supersymmetric model built only from the quark and lepton chiral superfields: the up-type quarks get no masses; baryon number $B$ and lepton number $L$ are violated by renormalizable and hence unsuppressed superpotential terms; and the scalar partners of the quarks and leptons are too light, a difficulty first exposed in Fayet's early models. The model of Dimopoulos and Georgi addresses these by adding Higgs superfields, a discrete symmetry that forbids the renormalizable $B$- and $L$-violating terms, and explicit soft supersymmetry-breaking terms of dimension 2 and 3; but Weinberg found that it still allows dimension-5 $B$- and $L$-violating effective interactions - suppressed by only one power of a superheavy mass - which mediate proton decay at a rate some two to three orders of magnitude beyond the experimental limits of the time.

Weinberg's alternative was to require that the gauge group acting at ordinary energies contain an extra abelian factor $\widetilde{U}(1)$, with charge denoted $\widetilde{Y}$. Gauge invariance under $\widetilde{U}(1)$ then forbids all dimension-4 and dimension-5 $B$- and $L$-violating operators, whatever the superheavy physics above; the leading proton-decay mechanisms are dimension-6 four-fermion operators, suppressed by two powers of the superheavy mass, with a characteristic chiral structure: supersymmetry allows the $LLRR$ combinations of left- and right-handed fields (as expected from heavy vector-boson exchange) but forbids $LLLL$ and $RRRR$. Supersymmetry breaking and electroweak breaking are both driven at ordinary energies by Fayet-Iliopoulos terms of the two $U(1)$ factors; the paper shows that if the scalar potential restricted to the non-quark/lepton fields has a minimum at which a certain mass-squared matrix is positive definite, then quarks and leptons are naturally light while their scalar partners are heavy, resolving the light-scalar problem.

The paper was explicit that this program was incomplete. (i) The electroweak-scale coefficients $\xi$ are put in by hand: although they do not receive large perturbative corrections (a result the paper credits to Witten, and that Fischler, Nilles, Polchinski, Raby and Susskind established in published form), the paper leaves open why they are present at all and small compared with superheavy scales, expressing the hope that nonperturbative effects might generate terms of order $\exp(-\text{constant}/g^2)$. (ii) The extra $\widetilde{U}(1)$ introduces Adler-Bell-Jackiw anomalies (quantum violations of classical current conservation that render a gauge theory inconsistent unless the sums of charges cancel); the paper works through an attempt embedded in $E_6$ (the exceptional Lie group used in some grand-unified models) that fails (it gives unacceptably large neutrino masses and reintroduces unsuppressed proton decay) and then offers a cancellation scheme - an $SU(3)$ octet with $\widetilde{Y}=-2$, an $SU(2)$ triplet, singlet pairs, and additional neutral chiral superfields $X_L$ - that works for precisely three generations but leaves the neutrino-mass, proton-decay and extra-generation problems open. (iii) The models predict new neutral currents mediated by the extra neutral gauge boson $\widetilde{Z}^0$ (whose mass mixing with the ordinary $Z^0$ is, in the simplest pattern, 'probably inconsistent with experiment') and massless fermionic gluon superpartners. (iv) The dimension-6 proton-decay rates are never evaluated quantitatively. (v) In the Added Notes, Weinberg reports (from M. Claudson and M. Wise) that when leptons are included the illustrative model of his Appendix B acquires a deeper supersymmetry-preserving minimum at which electric charge is broken; the fix - enough chiral superfields with $\widetilde{Y}=+4$ - is 'very difficult' to reconcile with keeping the $\widetilde{Y}^3$ anomaly small. (vi) The analysis of $R$ symmetries (symmetries rotating the superspace Grassmann coordinate, which keep gauginos massless) was deferred to a future paper, which appeared as Farrar and Weinberg's sequel in 1983.

The sequel carried out that deferred analysis and reported severe additional obstructions; the field subsequently turned to supergravity-mediated soft breaking and to $R$-parity, superseding the program's motivation, and the components that remain without closure in the literature are a mechanism for naturally small $\xi$, an anomaly-free and vacuum-stable $\widetilde{U}(1)$ spectrum, and quantitative dimension-6 proton decay.

Problem Statement

The source paper's program can be stated as a single completion question: does there exist a fully specified, internally consistent, renormalizable globally supersymmetric theory with gauge group $SU(3)\times SU(2)\times U(1)\times \widetilde{U}(1)$ acting at ordinary energies, in which supersymmetry and the electroweak group are spontaneously broken at the electroweak scale by Fayet-Iliopoulos $D$-terms, and which simultaneously satisfies all of the conditions the paper identified as unresolved? These conditions are: (1) a dynamical origin for the Fayet-Iliopoulos coefficients - an explanation of why the $\xi$ terms are present at all and are naturally small compared with the superheavy masses; (2) an anomaly-free chiral spectrum - all Adler-Bell-Jackiw anomaly traces including $\mathrm{Tr}(\widetilde{Y}^3)$ vanish - achieved with a definite choice of the extra $SU(3)\times SU(2)\times U(1)$-neutral chiral superfields $X_L$, without reintroducing renormalizable or dimension-5 $B$- or $L$-violating interactions, without unacceptably large neutrino masses, and without an experimentally excluded pattern of $\widetilde{Z}^0$-$Z^0$ mass mixing and neutral currents; (3) vacuum stability - the desired charge-conserving supersymmetry-breaking vacuum is the global minimum, or at least metastable with a tunneling lifetime exceeding the age of the universe, evading the Claudson-Wise type deeper supersymmetry-preserving charge-breaking minima, including the requirement of enough $\widetilde{Y}=+4$ fields to exclude supersymmetric solutions without introducing a large $\widetilde{Y}$ anomaly; (4) an acceptable gaugino and $R$-symmetry sector, confronting the sequel's finding that such theories tend to obey a global $R$ symmetry with unacceptable consequences (an axion-like or ninth-pseudoscalar Goldstone boson, or loss of QCD asymptotic freedom) and a gluino mass bounded near or below 1 GeV; and (5) quantitatively evaluated dimension-6 proton decay - rates implied by the allowed $LLRR$ four-fermion operators, their hadronic matrix elements, and whether the resulting lifetimes are consistent with existing bounds and within the reach of future experiments. These conditions jointly constrain one and the same model, which is what makes them a single scientific objective rather than independent questions. A complete answer establishes all five conditions for one and the same explicit model; because the conditions are a conjunction over one fixed class of theories, a rigorous demonstration that any single one of them cannot be satisfied by any theory in that class already establishes that no theory in the class satisfies all five, and equally closes the program, negatively.

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