The Nelson-Seiberg theorem generalized with non-polynomial superpotentials

The Nelson-Seiberg theorem relates R-symmetries to F-term supersymmetry breaking, and provides a guiding rule for new physics model building beyond the Standard Model. A revision of the theorem gives a necessary and sufficient condition to supersymmetry breaking in models with polynomial superpotentials. This work revisits the theorem to include models with non-polynomial superpotentials. With a generic R-symmetric superpotential, singularity at the origin of the field space implies both R-symmetry breaking and supersymmetry breaking. We give a generalized necessary and sufficient condition for supersymmetry breaking which applies to both perturbative and non-perturbative models.


Introduction
Supersymmetry (SUSY) [1,2,3,4,5,6] provides a natural solution to several unsolved problems in the Standard Model (SM), through its extension to the supersymmetric Standard Model (SSM). In this framework, bosons and fermions appear in pairs related by SUSY. So every particle in SM has a SUSY partner called a sparticle, which has similar properties to its corresponding SM particle. The mass spectrum of sparticles will be the same as SM particles if SUSY is a good symmetry at low energy. Since sparticles have not been discovered yet, SUSY must be broken to give them heavy masses escaping the current experimental limit [7]. To avoid the problem of light sparticles in model building, SUSY must be broken in a hidden sector [8] which introduces new fields beyond SM, and then the SUSY breaking effects are mediated to the observable SSM sector by a messenger sector, giving sparticle mass spectrum and coupling constants which may be examined in future experiments. There are two types of SUSY breaking models called F-term and D-term SUSY breaking. This work focuses on F-term SUSY breaking because of its advantages in phenomenology as well as quantum gravity completion.
F-term SUSY breaking models, also called Wess-Zumino models [9,10] or O'Raifeartaigh models [11], involve superpotentials which are holomorphic functions of chiral superfields. In SUSY breaking model buiding, R-symmetries are often utilized because of their generic relation to SUSY breaking vacua discovered by Nelson and Seiberg [12]. Metastable SUSY breaking models [13] also benefits from approximate R-symmetries through an approximate version of the Nelson-Seiberg theorem [14,15]. A revised version of the Nelson-Seiberg theorem gives a combined necessary and sufficient condition for SUSY breaking with the assumption of generic polynomial superpotentials [16], while the original theorem applies to any generic superpotentials but gives separate necessary and sufficient conditions. Although counterexamples are found [17,18], they have nongeneric R-charge assignments so that do not violate both the original and the revised theorems. This work extends the previous analysis to cover models with non-polynomial superpotentials which are often found in dynamical SUSY breaking. We give a generalized theorem on a necessary and sufficient condition for both models with polynomial and non-polynomial superpotentials. It provides a guiding rule for low energy effective SUSY model building to study new physics beyond SM.
The rest part of this paper is arranged as following. Section 2 reviews the original Nelson-Seiberg theorem. Section 3 reviews a revision of the Nelson-Seiberg theorem which gives a necessary and sufficient condition for SUSY breaking with the assumption of polynomial superpotentials. Section 4 gives a proof for our generalized theorem covering both models with polynomial and non-polynomial superpotentials. Section 5 makes the conclusion and final remarks.

The Nelson-Seiberg theorem
This section reviews the original Nelson-Seiberg theorem and its proof [12]. The setup is on a Wess-Zumino model [9,10] which involves a superpotential W (φ i ) as a holomorphic function of chiral superfields φ i , i = 1, . . . , d, and a Kähler potential K(φ * i , φ j ) as a real and positive-definite function of φ i 's and their conjugates. We use Einstein notation to sum up terms with repeated indices throughout this work. Although a minimal Kähler potential K(φ * i , φ j ) = φ * i φ i is often assumed, most of our analysis in this work is valid for generic Kähler potentials. Since the vacuum is determined by scalar components z i 's of φ i 's once the auxiliary F components are solved, W and K are also viewed as functions of z i 's. A vacuum corresponds to a minimum of the scalar potential, which is defined as where Whether SUSY breaking happens or not can be checked by solving the F-term equations A solution to ∂ i W = 0 gives a global minimum of V which preserves SUSY, and non-existence of such a solution means SUSY breaking, although the existence of a SUSY breaking vacua needs to be confirmed by minimizing the scalar potential V . Following the work of Nelson and Seiberg, we are to discuss the possibility of solving ∂ i W = 0 equations, given a superpotential with generic terms and coefficients respecting symmetries in each of the following cases.
• When there is no R-symmetry, a SUSY solution to ∂ i W = 0 generically exists, because there are equal numbers of equations and variables. Introducing a non-R symmetry does not change the situation, because it reduces both equations and variables by a same number.
• When there is an R-symmetry, W must have R-charge 2 in order to make the Lagrangian R-invariant. So there is at least one field with a non-zero R-charge. One can choose such a field z d . With a field redefinition, W is written as where r i 's are R-charges of z i 's. The redefinition makes x has R-charge 2 and y i 's has R-charge 0. Consider the following two types of vacua: There are d − 1 variables to solve d equations.
A generic function f does not allow such a solution to exist. So if such a vacuum with x = 0 does exist, it generically breaks SUSY.
-For a vacuum with x = 0, equations ∂ i W = 0 become The single equation f = 0 can always be solved for a generic function f , and other equations are all satisfied at x = 0. But the field redefinition (5) is usually singular at x = 0 except for some special choices of R-charge assignments. So the existence of a vacuum with x = 0 is unclear.
• Notice that a vacuum with x = 0 spontaneously breaks the R-symmetry, while a vacuum with spontaneous R-symmetry breaking means that there is at least one field z d = 0 with r d = 0, which can be used to make the redefinition (5) with x = 0.
In summary, we have proved the original Nelson-Seiberg theorem: Theorem 1 (The Nelson-Seiberg theorem) In a Wess-Zumino model with a generic superpotential, an R-symmetry is a necessary condition, and a spontaneously broken R-symmetry is a sufficient condition for SUSY breaking at the vacuum of a global minimum.

The revised theorem with polynomial superpotentials
This section reviews the revised version of the Nelson-Seiberg theorem and its proof [16]. To avoid singularities in the field space and other complications from the field redefinition (5), we consider the original W without doing any field redefinition in the following proof.
• When there is no R-symmetry, SUSY is generically unbroken according to the original Nelson-Seiberg theorem.
• When there is an R-symmetry, fields can be classified to three types according to their Rcharges: Assuming the superpotential has a polynomial form, we can write down the generic form of W by including all monomial combinations of fields with R-charge 2: Note that not all A i 's can appear in every terms of W 1 . Only those field combinations with R-charge 2 contribute to W 1 with non-zero coefficients. Each term of W 1 contains at least two X i 's or A i 's. This feature is also possessed by non-renormalizable terms of W 1 .
-In the case of N X ≤ N Y , setting X i = A i = 0 makes all first derivatives of W 1 equal zero, then solving f i (Y j ) = 0 gives a SUSY vacuum. Such a solution generically exists because the number of equations, which equals N X , is less than or equal to N Y , the number of variables.
-In the case of N X > N Y , we consider the following two types of vacua: * For a vacuum with X i = A k = 0, all first derivatives of W 1 are set to zero. But generically there is no solution to f i (Y j ) = 0 because the number of equations is greater than the number of variables. SUSY is generically broken if such a vacuum does exist. * For a vacuum with some X i = 0 or A i = 0, which carries a non-zero R-charge, the R-symmetry is spontaneously broken by this field. Then SUSY is generically broken according to the original Nelson-Seiberg theorem.
-If there are more than one consistent R-charge assignments, one should explore all possibilities of R-charge assignments to see whether N X ≤ N Y can be satisfied with one assignment. SUSY is broken only if N X > N Y is satisfied for all possible consistent R-charge assignments.
These exhaust all cases with and without R-symmetries. In summary, we have proved a necessary and sufficient condition for SUSY breaking: Theorem 2 (The Nelson-Seiberg theorem revised) In a Wess-Zumino model with a generic polynomial superpotential, SUSY is spontaneously broken at the global minimum if and only if the superpotential has an R-symmetry and the number of R-charge 2 fields is greater than the number of R-charge 0 fields for any possible consistent R-charge assignment.
The extra freedom to assign different R-charges can be viewed as non-R U (1) symmetries in addition to R-symmetries [19]. A k 's, the fields with R-charges other than 2 and 0, do not appear in the SUSY breaking condition of the revised theorem, but are needed for spontaneous R-symmetry breaking to generate gaugino masses [20,21,22,23,24]. In addition, according to the above proof procedure, A SUSY vacuum in the case with an R-symmetry and N X ≤ N Y also preserves the R-symmetry and gives a zero expectation value to W [25]. Such vacua play important roles in string phenomenology [26,27,28].

Generalization to include non-polynomial superpotentials
The generic form of the superpotential (13) (14) is an essential step of the previous proof, which comes from the assumption of an R-symmetry and a polynomial W . So a superpotential beyond the polynomial expansion may invalidates the proof for the revised theorem. But the proof for the original Nelson-Seiberg theorem does not rely on the polynomial form of W . Models in the scope of the original Nelson-Seiberg theorem but out of the scope of the revised theorem often appear in dynamical SUSY breaking. To achieve a more general theorem to cover these models, we need to analyze W as an arbitrary generic function of fields.
• When there is no R-symmetry, SUSY is generically unbroken according to the original Nelson-Seiberg theorem.
• When there is an R-symmetry, we suppose fields are properly defined so that the origin of the field space preserves the R-symmetry. Thus every field transforms by a complex phase angle under the R-symmetry and can be assigned an R-charge. Fields can be classified to X i 's, Y i 's or A i 's according to their R-charges, just like what we have done before in the previous proof. There are N Y degrees of freedom to choose the origin because any expectation values of Y i 's are invariant under the R-symmetry. The superpotential is supposedly a generic holomorphic function of fields.
-If W is smooth at the origin, it has a Taylor series expansion with a non-zero radius of convergence. The expansion only needs to be done in variables X i 's and A i 's, and all constant coefficients can be replaced with arbitrary functions of Y i 's. The generic expansion from the origin X i = A k = 0 is Note again that each term of W 1 contains at least two X i 's or A i 's. All the previous proof can be carried on to reach the revised Nelson-Seiberg theorem by considering the following two types of vacua. * The discussion on vacua with X i = A k = 0 in the previous proof proceeds without change. Any non-zero radius of convergence of the polynomial expansion (15) (16) ensures the validity of such a vacuum at the origin. * The discussion on vacua with X i = 0 or A i = 0 in the previous proof only involves the original Nelson-Seiberg theorem, which does not rely on the expansion form (15) (16).
-If W is singular at the origin, the vacuum must be away from the origin to ensure a reliable effective theory calculation. The R-symmetry is broken by some field expectation values, and SUSY is broken according to the original Nelson-Seiberg theorem.
By identifying whether W has singularity at the origin of the field space, all cases with polynomial and non-polynomial W 's are covered in our discussion. In summary, we have proved a generalized condition for SUSY breaking in models with generic superpotentials: Theorem 3 (The Nelson-Seiberg theorem generalized) In a Wess-Zumino model with a generic superpotential, SUSY is spontaneously broken at the global minimum 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 R-charge assignment.
• The superpotential is singular at the origin of the field space.
Non-polynomial superpotentials often appears as low energy effective descriptions of dynamical SUSY breaking models, which come from non-perturbative effects in supersymmetric quantum chromodynamics (SQCD) for various number of colors N c and number of flavors N f [29,30,31,32,33]. A non-polynomial Affleck-Dine-Seiberg superpotential [34,35] is generated from gaugino condensation in the case of N f < N c − 1, or from instantons in the case of N f = N c − 1. As and example, the 3-2 model [36] has a gauge group SU (3) × SU (2), a global U (1) symmetry and an R-symmetry U (1) R , with the following chiral superfields: where the representations of SU (3) and SU (2) are written in parentheses, and the subscripts indicate U (1) and U (1) R charges. Assuming SU (3) interactions are much stronger than SU (2) interactions, the superpotential respecting symmetries is The first term is the ADS superpotential coming from SU(3) instantons, and the latter is a treelevel polynomial term. Singularity of the non-polynomial term pushes field expectation values away from the origin, breaks the R-symmetry and SUSY according to the generalized theorem. On the other hand, as a result of vanishing SU (3) and SU (2) D-terms, we can assume all fields have their vacuum expectation values of the same order v. With an approximately minimal Kähler potential at weak coupling limit, the vacuum is calculated by minimizing the superpotential V = ∂ i W 2 . Neglecting constant coefficients, the vacuum expectation values of fields and V are estimated to be The non-zero V indicates a SUSY breaking vacua, which verifies the prediction of the theorem.

Conclusion
The generalized theorem which we have proved in this work provides a tool to build SUSY models R-symmetries, which give either SUSY breaking or SUSY vacua. For models smooth at the origin of the field space, one can arrange R-charges of fields to satisfy either N X > N Y or N X ≤ N Y and get the needed vacua. For models singular at the origin of the field space, field expectation values are pushed away from the origin, and SUSY is broken with a generic superpotential. The theorem applies to both perturbative and non-perturbative models. It allows one to efficiently survey a large number of different models without solving the F-term equations, and select the models with desired vacua to continue explicit model building. It provides a guiding rule for low energy effective SUSY model building to study phenomenology of new physics beyond SM as well as string phenomenology.
It should be noted that non-perturbative effects do not necessarily lead to non-polynomial superpotentials. The form of the superpotential depends on how it is parameterized. For example, in the Kachru-Kallosh-Linde-Trivedi (KKLT) construction for de Sitter vacua in type IIB flux compactifications [37], The superpotential has a tree level contribution W 0 from fluxes, and a non-perturbative correction W corr from D3 brane instantons or gaugino condensation of the gauge theory on a stack of D7 branes, which stabilizes the volume modulus ρ. The exponential form of W corr is smooth at any value of ρ, and no R-charge can be consistently assigned to ρ. An supersymmetric anti-de Sitter vacuum is found at finite ρ by minimizing the supergravity scalar potential with a non-minimal Kähler potential. This model is base on supergravity, thus lies out of the scope of all theorems discussed in this work.