A counterexample to the Nelson-Seiberg theorem

Abstract

We present a counterexample to the Nelson-Seiberg theorem and its extensions. The model has 4 chiral fields, including one R-charge 2 field and no R-charge 0 filed. Giving generic values of coefficients in the renormalizable superpotential, there is a supersymmetric vacuum with one complex dimensional degeneracy. The superpotential equals zero and the R-symmetry is broken everywhere on the degenerated vacuum. The existence of such a vacuum disagrees with both the original Nelson-Seiberg theorem and its extensions, and can be viewed as the consequence of a non-generic R-charge assignment. Such counterexamples may introduce error to the field counting method for surveying the string landscape, and are worth further investigations.

A sufficient condition for counterexamples to the Nelson-Seiberg theorem

“…In this work, we investigate features of counterexamples to the Nelson-Seiberg theorem, and successfully proved a theorem which provides a sufficient condition for counterexamples. The scope of the theorem covers all previously found counterexamples in literature [23,24]. The sufficient condition is expressed as the comparison between field counts of different Rcharges, and the count of independent P -Q pairs has the simple expression (3.9).…”

“…The accuracy of such a scan is affected by the existence of counterexamples with generic parameters and non-generic R-charges. The simplest counterexample model [23] of this type has four fields {z 1 , z 2 , z 3 , z 4 } with R-charges…”

“…Such non-generic models do not suffer from parameter tuning, and can be referred to as counterexamples to the Nelson-Seiberg theorem with non-generic R-charges. The first counterexample model of this type has been discovered recently [23] and thereafter generalized [24]. A common feature of these models is that they have at least one field pairs with opposite R-charges.…”

Nonexistence of supersymmetry breaking counterexamples to the Nelson-Seiberg theorem

“…While these relations have found successful applications in SUSY phenomenology beyond SM and compactification of string theory, counterexamples to the Nelson-Seiberg theorem and its revisions are constructed in literature, with generic superpotential coefficients and non-generic R-charges [20,21]. Their features are summarized into the following sufficient condition [22]:…”

“…These relations are applied in the study of both SUSY phenomenology beyond the Standard Model (SM) [10][11][12][13][14] and compactification of string theory [15][16][17][18][19]. However, the recently discovered counterexample models [20,21] have vacuum solutions contradicting with the claims from the Nelson-Seiberg theorem and its revisions. These counterexamples have generic coefficients respecting symmetries and do not suffer from fine-tuning, thus are not neglectable in phenomenology studies.…”