Supersymmetry-Breaking Nonlinear Sigma Models

We consider a novel class of constraints on chiral superfields to obtain supersymmetric nonlinear sigma models in four spacetime dimensions, which strictly combine the internal symmetry breaking with spontaneous supersymmetry breaking. The resultant massless modes can be exclusively Nambu-Goldstone bosons without their complex partners and the goldstino that is charged under the internal symmetry. The massive modes show a peculiar relation among their masses and the scales of symmetry breakings.


Introduction
If supersymmetry (SUSY) is realized in nature, it must be broken at some energy scale.
Models of dynamical SUSY breaking have a possibility to explain smallness of the breaking naturally [1]. These models have some internal (global or gauge) symmetries at least in the UV region. Below the scale of strong dynamics, the SUSY breaking field in an effective theory often appears as a singlet under all the low-energy symmetries except for U (1) R . That is, the effective theory can be described by an O'Raifeartaigh-type model [2]. However, it may be of interest to keep some internal symmetries, under which the SUSY breaking field is charged, even below the dynamical scale. In this case, the massless goldstino due to the SUSY breaking is also charged under these symmetries [3].
Effective theories with spontaneously broken global symmetries may be described by nonlinear sigma models (NLSM) in a general manner. In the non-SUSY case, one of the ways to obtain a NLSM is to impose an algebraic constraint on the fields in a multi-component scalar field theory. In particular, a NLSM describes Nambu-Goldstone (NG) bosons under a symmetric constraint. For example, the O(N ) NLSM in four spacetime dimensions is given by a symmetric Lagrangian with a constraint where ϕ i is a real field with a vectorial index i = 1, · · · , N under the O(N ), ϕ i2 ≡ ϕ i ϕ i , and a denotes a positive constant. Owing to the constraint, the O(N ) symmetry is broken down to O(N − 1) with the corresponding N − 1 NG bosons. A supersymmetric extension of the above construction is naively achieved by straightforward use of the corresponding superfields. Namely, we have O(N )-invariant Kähler potential K and superpotential W with a superfield constraint where X i is a chiral superfield. The above naive extension keeps supersymmetry unbroken in contrast to the internal O(N ) symmetry, which is broken.
In this letter, we consider more general form of constraints such as [( where m and n are positive integers. We show that such a constraint can strictly combine the internal symmetry breaking with spontaneous SUSY breaking [4]. *1 Let us focus on the simplest novel constraint To obtain a SUSY breaking model with this constraint, we only need the field X i with the canonical Kahler potential and the superpotential where µ is a real mass scale. We will analyze the mass spectrum around the vacuum and show that the resultant massless modes are exclusively NG bosons without their complex partners *2 and the charged goldstino due to the SUSY breaking. *3 The rest of the paper is organized as follows. In the next section, we will analyze the constraint to show the SUSY breaking in the model, which suggests a convenient way of changing variables. In section 3, the mass spectrum of the model will be investigated. In section 4, we will present a corresponding linear model with additional massive modes. The final section is devoted to brief discussion.

The model
In this section, we specify our SUSY-breaking nonlinear sigma model in detail based on Eq. (4) and Eq.(5).

The constraint
We consider a chiral superfield *4 which transforms as a fundamental representation under a global O(N ) symmetry: *1 We can utilize the constraint Eq.(3) to obtain SUSY breaking models with separate SUSY-breaking fields. For instance, a simple model is given by the superpotential W = µX i Y i , where Y i is a chiral superfield and µ is a mass scale. The field X i is under the constraint, which gives a part of its lowest component a nonzero expectation value. On the other hand, the F -term of a part of Y i must develop a nonzero expectation value by the equation of motion and thus the SUSY is broken. We note that the corresponding example of UV dynamical model is given by a vector-like model of SUSY breaking [5], where the X i is provided by the mesonic degrees of freedom. *2 In contrast, unbroken SUSY tends to make complex partner scalars also massless [6]. *3 In this paper, the internal symmetry is not gauged. This provides an O'Raifeartaigh-type SUSY-breaking model without singlets of the internal symmetry. *4 We follow the conventions of Ref. [7].
where i = 1, · · · , N and y = x + iθσθ is a four-dimensional coordinate. Let us impose the superfield constraint Eq.(4). This constraint can be solved as follows. First, we define a chiral superfield Then the superfield constraint Eq.(4) is nothing but Z 2 = 0, which is equivalent to the relation among the components We here assume non-vanishing F z , which will be justified retrospectively. Note that this takes the same form as the generic constraint for the goldstino superfield derived in Ref. [8].
In terms of the component fields of X i , this relation is rewritten as

SUSY breaking
We next investigate the SUSY breaking in our model. The superpotential is given by Eq.(5).
In the vacuum, the constraint Eq.(9) leads to while the F -term for the field X i is given by where the corrections due to the constraint is vanishing under ψ i = 0. We see that a part of F i must be nonzero in the vacuum due to the constraint and hence SUSY is spontaneously broken in this model. The claim F z = 0 is also confirmed.

Changing variables
We now provide a convenient change of variables to be adopted in the following analyses.
Let us assume without loss of generality that only the chiral superfield X 1 in X i has nonzero expectation values to satisfy the constraint Eq.(10) and the F -term equation Eq.(11) as The chiral superfield Z defined in Eq. (7) has an expectation value Z = 2aF θ 2 in the vacuum. We expand these fields around the vacuum as X 1 ≡ X 1 +X 1 and Z ≡ Z +Z. Then Eq.(7) results in whereī = 2, · · · , N . By iterative use of this equation, the variable set can be changed from X i toZ and Xī so as to be valid up to arbitrarily higher-order fluctuation terms of the fields.
This serves to analyze the mass spectrum around the SUSY-breaking vacuum in the next section and also to construct a possible linear model, which we will provide in section 4.

Mass spectrum
We now investigate the mass spectrum of the model around the vacuum that breaks both the SUSY and the O(N ) global symmetry spontaneously. The variableX 1 can be replaced withZ and Xī using Eq.(12) repeatedly: where the ellipses denote the higher-order terms that do not contribute to masses of the fields.
In the third equality, we utilized the constraint asZ 2 = −2 Z Z . It is thus straightforward to replace X 1 in our original Lagrangian withZ and Xī: where the ellipsis denotes the higher-order interaction terms. Let us further redefine the superfieldZ asZ → 2aZ to canonically normalize the field. Then, we obtain the Lagrangian in terms of the component fields as We may use the equations of motion for Fī and Fz with the restricted scalar modez eliminated by means of the constraint Eq. (8). That leads to The fermion ψz is massless and none other than the goldstino in this model.  10), which can be written as Substituting this expression into the scalar potential, we obtain We now expand the fields xī with their real and imaginary parts as xī = (ξī + iηī)/ √ 2 where ξī and ηī are real fields. Then the potential is given by We see that there are no quadratic terms of ξī, which correspond to (N − 1) NG bosons, as is expected.

A linear model
We proceed to consider an example of linear models which is with no constraint and effectively realize our constraint after integrating out certain massive modes. The Lagrangian is given by *5 This corresponds to x i ψ i = x 1 ψ 1 in the original variables.
Here, the superfield Z is defined by Eq. (7) and all the fields X i are independent degrees of freedom. That is, the superfield constraint discussed so far is not imposed in this section. To obtain a meta-stable vacuum at z = 0, we assume f (Z, Z † ) to take a higher-order form in Z and Z † such as where b, c, and d are real constants and M denotes a mass scale. As in the previous section, we can change the variables fromX 1 toZ and Xī using Eq.(12) repeatedly: Then the Lagrangian is rewritten as where we have used µ = −F/a and rescaled the field Z as Z → 2aZ to canonically normalize the field. Note that we here adopt Z rather thanZ as a variable.
Hence the scalar potential is given by where K Z,Z † is the derivative of the Kahler potential in Eq.(23) with respect to Z and Z † .
To have a meta-stable vacuum at z = 0, firstly we have to cancel out the linear terms in z.
Thus we are led to set to obtain the masses of two real scalars as (26) *6 It is not so special to have such a parameters choice. What we need is just some non-zero VEV of x 1 , which is actually determined by the higher dimensional operators.