Novel counterexample to the Nelson-Seiberg theorem

We present a new type of counterexample to the Nelson-Seiberg theorem. It is a generic R-symmetric Wess-Zumino model with nine chiral superfields, including one field of R-charge 2 and no R-charge 0 field. As in previous counterexamples, the model gives a set of degenerate supersymmetric vacua with a non-zero expectation value for a pair of oppositely R-charged fields. However, one of these fields appears quadratically in the superpotential, and many other fields with non-zero R-charges gain non-zero expectation values at the vacuum, and so this model escapes the sufficient condition for counterexamples established in previous literature. Thus there are still open problems in the relation of R-symmetries to supersymmetry breaking in generic models.


Introduction
The Nelson-Seiberg theorem relates R-symmetries to superymmetry (SUSY) breaking in generic N = 1 Wess-Zumino models. The original result [1] states that the presence of an R-symmetry is a necessary condition, and a broken R-symmetry is a sufficient condition, for SUSY breaking at the stable vacuum of a generic model. A refinement of this result [2,3] relates the existence of a SUSY vacuum to the numbers of fields with certain R-charges in a model with a polynomial superpotential. However, exceptions [4] to both of these results have been found, in which a model with generic coefficients breaks the R-symmetry at the SUSY vacuum. The source of these exceptions has been identified [5] as pairs of fields with opposite R-charges obtaining vacuum expectation values (VEVs). Features of these exceptions can be summarized into a sufficient condition [6]. To summarize: a sufficient condition for the existence of a SUSY vacuum in a generic R-symmetric Wess-Zumino model is that the number of R-charge 2 fields is less than or equal to the sum of the number of R-charge 0 fields and the number of independent products of oppositely R-charged fields which appear only linearly in cubic terms of a renormalizable superpotential.
In this note, we demonstrate that this sufficient condition is not also necessary, by constructing a generic R-symmetric superpotential which does not satisfy the above condition.
The model nonetheless possesses a set of SUSY vacua where many fields with non-zero Rcharges gain non-zero VEVs. Therefore this model is a counterexample to the Nelson-Seiberg theorem, and escapes the sufficient condition established in previous literature.
The rest of this paper is arranged as follows. Section 2 reviews the sufficient condition for SUSY vacua in R-symmetric Wess-Zumino models which covers all previous counterexamples. Section 3 presents the new counterexample and its vacuum structure, showing that it is a counterexample escaping the previous sufficient condition. Section 4 discusses properties of the SUSY vacuum and implications of the result.

The sufficient condition for SUSY vacua
Here we briefly summarize the results of [2,6]; for details, we refer readers to those papers.
Under a continuous U(1) R-symmetry, where the R-charge for Grassmann numbers θ α is set to 1, the superpotential W (φ i ), built from scalar fields φ i or their corresponding chiral superfields, must have R-charge 2 to make the SUSY action R-invariant. Thus only Rcharge 2 fields may appear as linear terms in the superpotential. Following the convention of [6], we call such fields X i . The terms linear in X i which may appear in a renormalizable superpotential are where a i , b ij , c ijk and d (r)ijk are coefficients, Y j are R-charge 0 fields, and the fields P (r)i and Q (−r)i have opposite R-charges ±r, so that their product is R-neutral. In addition, the assumption is made that the P and Q fields appear only linearly in cubic terms. Thus in addition to W X , other terms which may appear in a renormalizable superpotential are where ξ ijk , ρ ijk , σ (r)ijk , τ (r)ijk , µ ij , ν ijk and λ ijk are coefficients, and A i are fields which have R-charges not equal to 2 or 0 and can not be identified as P or Q fields. The full superpotential contains all possible R-charge 2 terms built from all fields in our classification according to their R-charges. When seeking SUSY vacua, that is, solutions to the F-term equations one can satisfy all the F-term equations coming from derivatives with respect to Y , P , Q and A fields, by assuming that only Y , P and Q fields obtain non-zero VEVs. The number of F-term equations coming from derivatives with respect to X fields is equal to N X , the number of X fields, while the number of independent variables in these equations is equal to the sum of N Y , the number of Y fields, plus N P Q , the number of independent P -Q pair products, which can be expressed as where N P (r) and N Q(−r) are the numbers of P and Q fields with R-charges ±r and the sum is taken only over values of r for which N P (r) and N Q(−r) are non-zero. These equations are always solvable [7] for generic superpotential coefficients if the number of equations is less than or equal to the number of variables, and so a sufficient condition for the existence of SUSY vacua is This condition includes the case N X ≤ N Y , under which the revised Nelson-Seiberg theorem predicts the existence of SUSY vacua [8], and the case N Y < N X ≤ N Y + N P Q which is satisfied by all previous counterexample models [4,5,6]. In the latter case, the facts N X > N Y and that P and Q fields get non-zero VEVs for generic superpotential coefficients indicate that models in this case are counterexamples to both the original Nelson-Seiberg theorem [1] and its revison [2]. In the following section, we shall demonstrate a counterexample which does not satisfy the sufficient condition (6). The model gives a set of SUSY vacua where many fields other than Y , P and Q fields get VEVs. The existence of such a new counterexample means that the sufficient condition presented here is not also a necessary condition for SUSY vacua in R-symmetric Wess-Zumino models.

The new counterexample
Consider a Wess-Zumino model with nine fields: X, B, C, Ξ 1 , Ξ 2 , Ξ 3 , A 1 , A 2 and A 3 . The superpotential is given as where a, b, α i , β i , γ i are coefficients. This superpotential possesses a U(1) R-symmetry, under which the fields have the R-charge assignment: This assignment is unique, or equivalently [9], there is no other continuous symmetry of the model. The superpotential above contains all renormalizable terms permitted by this R-symmetry, so it is the form of a generic superpotential given the fields and their R-charges.
For generic values of the coefficients, we have a set of SUSY vacua at with a one complex dimensional degeneracy parameterized by the non-zero VEV of B. Like any SUSY vacuum in generic R-symmetric models, the vacua have the property that the superpotential vanishes term-by-term [10] and satisfies the bound found in [11]. The Rsymmetry is spontaneously broken everywhere on the degeneracy by all the non-zero VEVs of B, C and A i . Thus this model is a counterexample to the Nelson-Seiberg theorem. The model has N X = 1, N Y = 0. Although B and C have opposite R-charges, they can not be identified as P and Q fields because B appears quadratically in β 1 Ξ 1 B 2 and in the quadratic term α 3 Ξ 3 B. Therefore we have N P Q = 0 and the model escapes the previous sufficient condition (6).
We may see the full vacuum structure of the model from the scalar potential where a minimal Kähler potential is assumed. Like any R-symmetric polynomial superpotential which does not contain at least one field of R-charge 2 and at least one field of R-charge 0, the scalar potential has a stationary point at the origin of the field space 1 . In this case, this point is a saddle. Numerical searches also indicate that there are several metastable local minima with |V | > 0, thus SUSY-breaking. Finally, we note that other than ∂ X W , which is uncharged, all the F-terms ∂ i W have positive R-charges. This means that [12,13,14,15], under a complexified R-symmetry all the non-X F-terms will tend to zero as t → +∞. We thus might have a runaway direction as C → ∞ and B → 0. However, as the complexified R-symmetry also takes all other fields to zero in this limit, it coincides with the large-C limit of the SUSY solution (9).

Discussions
As we have shown, the model presented in this work has a field count satisfying N X > N Y + N P Q , which is outside of the previous classes of both the R-symmetric SUSY vacua [8] and the R-symmetry breaking SUSY vacua covered by the sufficient condition [6]. That the SUSY vacua are R-symmetry breaking also indicates that the model is a counterexample to the original Nelson-Seiberg theorem. The existence of such a new counterexample suggests that there are still some unexplored corners in the classification of R-symmetric Wess-Zumino models.
Just like any SUSY vacuum in R-symmetric models, the SUSY vacua in the new counterexample give W = 0 at the SUSY vacuum [10,11,16], and the supergravity version of the model also gives SUSY vacua with zero vacuum energy. One may hope to use the supergravity model as a low energy effective description for flux compactification of type IIB string theory [17,18,19,20], and such string constructions of W = 0 SUSY vacua [21,22,23,24,25,26,27] serve as the first step toward vacua with small superpotentials [28]. But the R-symmetry breaking feature of the vacua means that some complex structure moduli obtain nonzero VEVs, which send the Calabi-Yau manifold away from the R-symmetric point in its moduli space. It is then unnatural to turn on only R-symmetric fluxes and obtain an R-symmetric effective superpotential from the start. Thus silimarly to previous counterexample models, the new counterexample here does not contribute to the string landscape of W = 0 SUSY vacua if we only consider R-symmetric SUSY vacua [23], or string vacua with enhanced symmetries [21,22]. It is still an open question whether these counterexamples could be low energy effective models for other string constructions.