A sufficient condition for counterexamples to the Nelson-Seiberg theorem

Several counterexample models to the Nelson-Seiberg theorem have been discovered in previous literature, with generic superpotentials respecting the R-symmetry and non-generic R-charge assignments for chiral fields. This work present a sufficient condition for such counterexample models: The number of R-charge 2 fields, which is greater than the number of R-charge 0 fields, must be less than or equal to the number of R-charge 0 fields plus the number of independent field pairs with opposite R-charges and satisfying some extra requirements. We give a correct count of such field pairs when there are multiple field pairs with degenerated R-charges. These models give supersymmetric vacua with spontaneous R-symmetry breaking, thus are counterexamples to both the Nelson-Seiberg theorem and its extensions.

The proof of both the Nelson-Seiberg theorem and its extensions require generic superpotentials. The term genericness usually refers to generic values of the superpotential parameters. Non-generic models of this type require fine-tuning of parameters, and are not interesting for phenomenology studies. Another type of non-genericness comes from the specific form of the superpotential restricted by the R-symmetry with some special R-charge arrangement. 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. The product of such field pairs act effectively like R-charge 0 fields for the field counts in the revised Nelson-Seiberg theorem [2], so SUSY vacua exist in these models even with more R-charge 2 fields than R-charge 0 fields. Following this idea, a sufficient condition for counterexamples is presented in [24], and new counterexamples are accordingly constructed using multiple field pairs with opposite R-charges.
In this work, we are to present a more general sufficient condition for counterexamples. In addition to the one in [24], we consider the case with multiple R-charge 2 and R-charge 0 fields. We also give a correct count of independent field pairs with opposite R-charges when there are multiple field pairs with degenerated R-charges. The sufficient condition is stated in terms of these field counts. The statement is then supported with examples which give SUSY vacua with spontaneous R-symmetry breaking, as predicted by field counts satisfying the sufficient condition.
The rest part of this paper is arranged as following. Section 2 reviews the Nelson-Seiberg theorem, its extensions and previously found counterexamples. Section 3 presents the sufficient condition with a proof, and discusses properties of vacua from counterexamples. Section 4 gives examples supporting our statement. Section 5 makes the conclusion and final remarks.

The Nelson-Seiberg theorem and counterexamples
In a Wess-Zumino model, SUSY vacua are solutions to the F-term equations where W = W (φ i ), named the superpotential, is a holomorphic function of chiral superfields φ i for i = 1, . . . , n. Such a solution also corresponds to a minimum of the scalar potential with the vacuum expectation value (VEV) V = 0. Einstein summation is used in the expression of V , and the Kähler metric K¯i j is calculated from a Kähler potential K(φ * i , φ j ) which is a real and positive-definite function of φ i 's and their conjugates φ * i 's. The vacuum is determined just by the scalar components z i 's of φ i 's, so W and K are also viewed as functions of z i 's. If (1) has no solution, SUSY must be broken at any vacuum of the model, although we need to assume the existence of a local minimum of V , or build a minimum by introducing various corrections to a runaway direction [25,26,27,28]. Taking the nonexistence of a solution to (1) as the criteria for SUSY breaking, the Nelson-Seiberg theorem and its extensions [1,2,3] give the conditions for SUSY breaking, expressed in terms of R-symmetries under which W has R-charge 2. These theorems are as follows: Theorem 1 (The Nelson-Seiberg theorem) In a Wess-Zumino model with a generic superpotential, assuming the existence of a vacuum at the global minimum of the scalar potential, an R-symmetry is a necessary condition, and a spontaneously broken R-symmetry is a sufficient condition for SUSY breaking at the vacuum.
Theorem 2 (The Nelson-Seiberg theorem revised and generalized) In a Wess-Zumino model with a generic superpotential, assuming the existence of a vacuum at the global minimum of the scalar potential, SUSY is spontaneously broken at the vacuum if and only if the superpotential has an R-symmetry, and one of the following conditions is satisfied: • The superpotential is smooth at the origin of the field space, and the number of R-charge 2 fields is greater than the number of R-charge 0 fields for any possible consistent Rcharge assignment.
• The superpotential is singular at the origin of the field space.
For a generic R-symmetric polynomial W , the criteria for SUSY breaking is just the comparison between R-charge 2 and R-charge 0 field counts, which can be done much more easily and quickly than solving the F-term equations (1). Thus it is possible to do a fast scan of a large number of models based on the field counting criteria. 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 and the superpotential which includes all renormalizable terms of R-charge 2. The R-charge assignment is also uniquely fixed by requiring all terms of W to have R-charge 2. The model has one R-charge 2 field z 1 and no R-charge 0 field, so the field counting criteria predicts SUSY breaking. But the F-term equations have a SUSY solution which also breaks the R-symmetry for generic non-zero values of parameters a and d. Thus the model is a counterexample to both the original and the revised Nelson-Seiberg theorems. The counterexample does not falsify either theorem, since the particular R-charge assignment restrict W to particular form which violates the genericness assumption of the theorems. It is significative to find out the pattern of R-charges in such models rather than simply identify them as counterexamples with non-generic R-charges. One feature of the R-charge assignment (3) is the existence of the oppositely R-charged field pair z 3 and z 4 . They appear as a product z 3 z 4 in the superpotential (4), although z 3 also appears linearly in another cubic term. The work of [24] explores this feature and presents a sufficient condition for counterexamples. New counterexamples are constructed using one R-charge 2 field, no R-charge 0 field, and more than one field pairs with opposite R-charges. At least one pair of oppositely R-charged fields satisfy the condition that they both appear only linearly in the superpotential, and are not involved in any quadratic term. The product of such a field pair gets a non-zero VEV, and acts effectively like an R-charge 0 field to help solving the F-term equations. It is natural to generalize the sufficient condition and the counterexample construction to the case with multiple R-charge 2 and R-charge 0 fields. Such generalization is to be done in the following section.

A sufficient condition for counterexamples
Following the convention in [11] which is also used in [2,3], we make the following field classification in an R-symmetric Wess-Zumino model: Definition 1 (The field classification) Under an R-symmetry, fields are classified according to their R-charges into the following types: • P (r) and Q (−r) fields for a value of r: P (r)i for i = 1, . . . , N P (r) and Q (−r)j for j = 1, . . . , N Q(−r) with R(P (r)i ) = −R(Q (−r)j ) = r, and both P 's and Q's only appear linearly in the superpotential and not in any quadratic terms; • A fields: A i for i = 1, . . . , N A with R(A i ) = r i = 2 or 0, and A's do not satisfy the condition for being classified as P (r) and Q (−r) fields.
For a renormalizable superpotential, both P 's and Q's only appear linearly in cubic terms.
Other oppositely R-charged field pairs which can not be classified as P -Q pairs are identified as A's. This field classification leads to the generic, R-symmetric and renormalizable superpotential All terms of W 1 are at least quadratic in X's and A's. This statement is also true for non-renormalizable superpotentials. So if we look for a vacuum satisfying the first derivatives of W 0 respecting to Y 's, P 's, Q's and A's, as well as all first derivatives of W 1 vanish at such a vacuum. The F-term equations are then reduced to or more generally which also covers the non-renormalizable case. Suppose that the total number of independent quadratic products P (r)j Q (−r)k , or the number of independent P -Q pairs with opposite Rcharges, is N P Q . These quadratic products can be effectively viewed as N P Q independent variables. So there are totally N Y + N P Q variables to solve N X equations. SUSY solutions exist with N X ≤ N Y + N P Q and generic parameters. But the revised theorem predicts SUSY breaking when N X > N Y . Notice also that a solution to (8) or (9) generically gives nonzero VEV's for P 's and Q's, which spontaneously break the R-symmetry. Thus the generic superpotential (6) under the condition gives a counterexample to both the original Nelson-Seiberg theorem and the revised one. The total number of P -Q pairs is But the corresponding quadratic products may not be independent when there are multiple fields with degenerated R-charges. The set contains independent elements, since every quadratic product in this set except P (r)1 Q (−r)1 has either P (r)j or Q (−r)k not appearing in other set elements. Quadratic products not in (12), if existing, can be expressed in terms of the elements of (12): Thus the order of the set (12) gives the number of independent P -Q pairs Notice that N P Q < N P Q if there is at least one r with N P (r) > 1 and N Q(−r) > 1.
In summary, we have obtained the sufficient condition: Theorem 3 (A sufficient condition for counterexamples to the Nelson-Seiberg theorem) In a Wess-Zumino model with a generic R-symmetric superpotential, using the notation in Definition 1, the condition N Y < N X ≤ N Y + N P Q is a sufficient condition for the model to be a counterexample to both the original Nelson-Seiberg theorem and the revised one, where N P Q = r N P (r) + N Q(−r) − 1 is the number of independent P -Q pairs.
The total number of P 's and Q's is The reduced F-term equations (8) or (9) can also be viewed as using N Y + N P +Q variables to solve N X equations. Generically each solution has N Y + N P +Q − N X undetermined variables, or degeneracy of complex dimension N Y + N P +Q − N X . Since N P +Q is greater than N P Q for models with at least one P -Q pairs, SUSY solutions in counterexamples have degeneracy dimension of at least one. The VEV's of P 's and Q's are generically non-zero, and the R-symmetry is spontaneously broken everywhere on the degenerated vacuum. Such a SUSY solution also makes the superpotential (6) to be zero, thus the bound on the superpotential is satisfied [29,30]. These properties of vacua from counterexample models are summarized in the following theorem: Theorem 4 (Properties of vacua from counterexamples) A counterexample model according to Theorem 3 gives one or several SUSY vacua with W = 0 and degeneracy of complex is the total number of P 's and Q's. The R-symmetry is spontaneously broken by the non-zero VEV's of P -Q pairs everywhere on the degenerated vacua.

Examples of counterexamples
The sufficient condition is demonstrated by the following examples of counterexample models. The simplest counterexample (3) and (4) can be rewritten using the notation in in Definition 1, as the R-charge assignment (R(X), R(P ), R(Q), R(A)) = (2, 6, −6, −2) and the superpotential W = X(a + dP Q) It has N X = N P (6) = N Q(−6) = 1, N Y = 0, and thus N P Q = 1. The quadratic product P Q, viewed effectively as an R-charge 0 variable, solves the reduced F-term equation a+dP Q = 0. The SUSY solution is given as and the R-symmetry is broken by the non-zero VEV's of P and Q.
As an example of models with both Y 's and P -Q pairs, the R-charge assignment (R(X 1 ), R(X 2 ), R(Y ), R(P ), R(Q), R(A)) = (2, 2, 0, 6, −6, −2) (19) leads to the superpotential This model has N X = 2, N Y = N P (6) = N Q(−6) = 1, and thus N P Q = 1. The two independent effective variables P Q and Y solve the two reduced F-term equations. The SUSY solution is given as , and the R-symmetry is broken by the non-zero VEV's of P and Q.
As an example of models with some oppositely R-charged field pairs which can not be classified as P -Q pairs, the R-charge assignment leads to the superpotential The A 1 -A 2 pair, although with opposite R-charges, can not be classified as P 's and Q's, because both A 1 and A 2 appear quadraticly in W , and A 1 appears in a quadratic term. This model has N X = N P (4) = N Q(−4) = 1, N Y = 0, and thus N P Q = 1. The SUSY solution is given as and the R-symmetry is broken by the non-zero VEV's of P and Q.
As an example of models which contain multiple P -Q pairs with degenerated R-charges, the R-charge assignment (R(X 1 ), R(X 2 ), R(X 3 ), R(P 1 ), R(P 2 ), R(Q 1 ), R(Q 2 ), R(A)) = (2, 2, 2, 6, 6, −6, −6, −2) (25) leads to the superpotential   This model has N X = 3, N Y = 0, N P (6) = N Q(−6) = 2, and thus SUSY solutions correspond to and the solutions to The three independent effective variables P 1 Q 1 , P 1 Q 2 and P 2 Q 1 solve the three reduced Fterm equations. Then P 1 , P 2 , Q 1 and Q 2 can be expressed in terms of these three variables and a free parameter representing the degeneracy. The analytical solution is a radical expression of several hundred terms, which is too complicated to be presented here. Numerical solutions with some typical choices of coefficient values are listed in Table 1. These solutions, together with (28), give SUSY vacua with R-symmetry breaking by the non-zero VEV's of P 1 , P 2 , Q 1 and Q 2 . All models in this section satisfy N Y < N X ≤ N Y +N P Q , thus are counterexamples to the Nelson-Seiberg theorem according to Theorem 3. The analytically or numerically obtained SUSY solutions have W = 0, degeneracy of N Y + N P +Q − N X , and R-symmetry breaking everywhere on the degenerated vacua. So both Theorem 3 and Theorem 4 are verified by these examples.

Outlook
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 (14). It is still feasible to do a fast survey of a large number of models using the field counting method, even taking into account the counterexamples in this work. Thus the pattern of field counts in counterexamples enables a refined classification of R-symmetric Wess-Zumino models.
Counterexample models built from our sufficient condition have certain properties, which may inspire applications in both phenomenology and formal studies. The R-symmetry breaking SUSY vacua are applicable to model building of tree-level R-symmetry breaking [31,32,33,34], where the R-symmetry breaking sector can be build separately from the SUSY breaking sector. The SUSY vacua with W = 0 become Minkowski SUSY vacua in supergravity, which make up the low-energy SUSY branch of the string landscape [35,36,37]. The degeneracy direction of vacua can be used for cosmological model building with nonperturbative potentials introduced, and the difficulty from the de Sitter swampland conjecture may be avoided [38,39,40,41]. It is still challenging to find ultraviolet completion of these models in strongly coupled SUSY gauge theories or compactification of string theory.