Gravitational Gauge Mediation

It is often the case that the naive introduction of the messenger sector to supersymmetry breaking models causes restoration of supersymmetry. We discuss a possibility of stabilizing the supersymmetry breaking vacuum by gravitational interaction.


Introduction
In scenarios where the electroweak scale is stabilized by supersymmetry, it is expected that future collider experiments give us hints of how the standard model sector feels the supersymmetry breaking. It is now important to discuss possible mechanisms for supersymmetry breaking and its mediation.
In field theory, spontaneous breaking of supersymmetry is quite easily realized. An obvious example is the Polonyi model in which a singlet chiral superfield S has a linear superpotential W = µ 2 S. This model also serves as the low energy effective theory of the O'Raifeartaigh model by introducing a non-minimal Kähler potential term K ∋ −(S † S) 2 / Λ 2 with Λ the mass scale of the particles which are integrated out. The same effective theory can also be realized in models with strongly coupled gauge theories such as in the IYIT model in Ref. [1,2]. Recently it has been noticed that a wide class of supersymmetric QCD leads to the above effective theory around the meta-stable vacuum [3].
Even though spontaneous supersymmetry breaking may be a common feature of quantum field theory, mediation of the supersymmetry breaking to the standard model sector is not so simple. Of course, gravity mediation is the simplest way to communicate with the hidden sector. However, we study the possibility of gauge mediation partly because it provides us with a solution to the flavor problem [4] (See [5] for earlier works.).
A naive way of realizing gauge mediation is to introduce vector-like messenger particles q andq which carry standard model quantum numbers and assume a coupling W ∋ −λSqq in the superpotential. Assuming non-vanishing vacuum expectation values in the lowest and the F -component of S, the gaugino masses are obtained by the formula m 1/2 = (α/4π)(F S /S) at one-loop level. However, this model has a supersymmetric and hence global minimum where S = 0 and q =q = µ 2 /λ. Therefore the question is whether it is possible to have a meta-stable vacuum with non-vanishing value of S [6]. For example, in the O'Raifeartaigh model, the sign of the coefficient of the Kähler term (S † S) 2 / Λ 2 is negative, which stabilizes the field S at the origin. Therefore, there is no meta-stable vacuum away from the origin.
Mechanisms of realizing a (local) minimum away from the origin have been discussed in the literature. In Ref. [7], it is shown that the inverted hierarchy mechanism can produce a local minimum at a very large value of S if S is non-singlet under some gauge interaction. The possibility of having a (local) minimum via a non-perturbative effect has also been discussed in Ref. [8] (See also [2,9,10,11] for discussions of the vacuum structure of the IYIT model.). The introduction of a bare mass term for the messenger field also makes the vacuum meta-stable as discussed in Ref. [8,12]. In the context of supersymmetry breaking in chiral gauge theories, it has been discussed that a runaway direction which is lifted only by non-renormalizable operators has a minimum at a very large field value [13].
In this note, we argue that the situation of supersymmetry restoration by the messenger particles can be cured once we include the supergravity effect even if the origin is the unique minimum in the limit of the Planck scale M Pl to infinity. Although the gravity effect is always suppressed by 1/M Pl , it can be large enough to stabilize S away from the origin (S ∼ Λ 2 /M Pl ) when Λ 10 13 GeV.

Meta-stable vacua in supergravity
The model we will analyze is the following: The chiral superfield S is a singlet field, q andq are the messenger fields which carry standard For λ = 0, this model breaks supersymmetry and S is stabilized at the origin, S = 0, by the supersymmetry breaking effect. However, by turning on the λ coupling, q andq acquire tachyonic mass terms (B-term) which make the vacuum unstable. The true minimum is at S = 0 and q =q = µ 2 /λ where supersymmetry is unbroken. Therefore, there is no supersymmetry breaking vacuum in this model.
We assumed here that the sign of the second term in the Kähler potential is negative as it is the case in the O'Raifeartaigh model. In general, the sign can be positive, and sometimes it is even uncalculable if this term is originated from some strong dynamics. When it is positive, the S field may be stabilized away from the origin and the q andq directions are also stabilized at q =q = 0 by the supersymmetric mass terms. However, we consider the case with a negative sign where the origin is a stable minimum. * As we will see later, the supersymmetry breaking minimum reappears when we turn on the gravity effect even in that case. * In Ref. [11], the presence of a minimum at the origin is shown in the IYIT model.
First, we need to estimate the perturbative quantum correction to the Kähler potential for S coming from the interaction term λSqq, which may be more important than the gravity effect.
We can explicitly calculate the term at one-loop level: where N q is the number of components in q andq. For example, N q = 5 if q andq transforms as 5 and5 under SU(5) GUT . Higher order perturbative contributions, including the dependence on the artificial scale Λ, will be minimized by taking the running coupling constant λ to be the value near the scale S. This term also tends to make S = 0 stable. Now we include the gravity effect in this model. The scalar potential of the supergravity Lagrangian is given by where G ≡ K + log |W | 2 and we set M Pl = 1. G X is the derivative of G with respect to the field X, and G SS † is the inverse of the Kähler metric. D 2 /2 represents the D-term contributions. We can easily find the supersymmetric minimum, that is a solution of the equations: The solution is The gravity effect is a slight shift of the values of order c/(λM 2 Pl ) which is O(µ 2 /(λM Pl )) if we assume the cancellation of the cosmological constant at the meta-stable supersymmetry breaking vacuum below.
Another minimum can be found with the assumption of q =q = 0. The potential is simplified The equation V S = 0 with the phenomenological requirement V = 0, cancellation of the cosmological constant, leads to In the λ → 0 limit, there is no supersymmetric vacuum as in the case without gravity, but the minimum is not at the origin (S ∼ Λ 2 /M Pl ), even though the sign of the (S † S) 2 term in Käher potential is negative. In the supergravity Lagrangian, the origin is no longer a symmetry enhanced point because the R-symmetry is explicitly broken by the c-term. By turning on the λ coupling, the supersymmetric vacuum appears near the origin, but it is separated from the supersymmetry breaking minimum. The disappearance of the supersymmetry breaking minimum by the finite λ coupling seen before was an artifact of the approximation M Pl → ∞.
The stability of the vacuum can be checked by looking at the mass matrices of the S, q, and q fields. The matrices are given by . (10) Therefore, there is a stable minimum when κ ( Λ/M Pl ) 2 and Λ 4 µ 2 M 2 Pl /λ.
There is another phenomenological requirement that the gauge mediation effect, the gaugino masses, is of order 100 GeV. This fixes the relation between the parameters µ 2 and Λ as follows: where M W is the electroweak scale. With this relation, we show in Fig. 1 the parameter region where the supersymmetry breaking vacuum is meta-stable. A gravitational stabilization of the vacuum is possible for Λ 10 13 GeV. For large λ, the one-loop correction to the S potential destabilizes the vacuum, and too small λ leads to unstable q andq direction because of the small supersymmetric mass terms. The tunneling rate into the supersymmetric vacuum is small enough for Λ 10 11 GeV, which is satisfied in whole the viable region [14]. For Λ 10 17 GeV, the possible gravity mediation effect on the gaugino masses from the operator SW α W α /M Pl is larger than the gauge mediation. Note that this term is forbidden if we impose the approximate R-symmetry discussed before.
The mass of the S field depends on Λ. With the relation in Eq. (11), we obtain m S ≃ 100 GeV Λ 10 16 GeV .
The gravitino mass m 3/2 is

Ultraviolet completion
There are several possibilities for the underlying microscopic models which give the effective theory defined in Eqs. (1) and (2). An obvious example is the O'Raifeartaigh model as discussed before. The mass scale of the fields which are integrated out is identified with Λ.
There is another interesting possibility that the scale Λ is identified with the dynamical scale of the strongly coupled gauge theory and the linear term µ 2 S originates from the mass term of the quarks in that theory. This possibility is realized quite simply in the models of Ref. [3]. In the SO(N c ) gauge theory with N c − 4 flavor quarks, there is a branch where the quarks confine and there is no non-perturbatively generated superpotential. If there is a mass term for the quarks T in the N c dimensional representation, W = mT 2 , the low energy effective theory is where M is the meson field M ∼ T 2 . Therefore there is no supersymmetric vacuum in this branch. By assuming the presence of the coupling of the messenger fields q andq to the operator T 2 , the effective superpotential is identical to that in Eq. For other numbers of flavors, it is shown that there are meta-stable supersymmetry breaking vacua, for example, in SU(N c ) with N c + 1 ≤ N f < 3N c /2 [3]. However, in those cases, the relation between the dynamical scale Λ and the parameters in Eqs. (1) and (2), ( Λ, µ 2 ), is Λ ∼ √ mΛ and µ 2 ∼ mΛ, respectively. With this relation, µ 2 ∼ Λ 2 , we cannot satisfy the relation in Eq. (11). † Exceptions are SO(N c ) with N c − 3 and N c − 2 flavor theories where there is no non-perturbatively generated superpotential. In the N c − 3 flavor model, there is a branch where low energy effective theory has superpotential [15]: The chiral superfield a consists of N f gauge singlet fields, and f (t) is an unknown function of t = (det S)(Sa 2 ) with f (0) = 0. We expect that the coupling constant between S and a 2 is O(1) at low energy. In that case, the vacuum with a = 0 may be meta-stable for Λ 10 16 GeV according to Fig. 1. In the N c − 2 flavor model, the low energy effective theory is U(1) gauge theory with a similar superpotential: where a + and a − are the monopoles and t = det S. The same conclusion applies in this case.
For these N f = N c − 3 and N f = N c − 2 models, it may be possible that the fields a or in the scenario where the Higgs fields are composite particles of the strong dynamics which we are discussing [16]. Moreover, the same dynamics can be responsible for the dynamical breaking of the gauge symmetry in grand unified theories as shown in Ref. [16]. In the conventional picture, the electroweak scale appears as a consequence of the supersymmetry breaking, and there was a puzzle that the supersymmetric parameter, the µ-parameter, must be the same size as the supersymmetry breaking parameters. This puzzle was particularly sharp in the scenario of gauge mediation. However, in this scenario, the electroweak scale is the scale which drives the supersymmetry breaking and therefore there is no coincidence problem. There are many possibilities for the origin of the scale O(100 GeV) such as the dynamical scale of another strongly coupled gauge theory. A more attractive possibility of relating it to the cosmological constant term, c-term in Eq. (2), is pointed out in Ref. [16].
The IYIT model also gives the same effective theory in Eqs. (1) and (2). The model is an Sp(N c ) gauge theory with N f = N c + 1 flavors with the superpotential: where S is a singlet field and an anti-symmetric 2N f × 2N f matrix, Q is the quark in the 2N c dimensional representation, and y is the coupling constant. By the quantum modified constraint, the effective superpotential is with Λ being the dynamical scale. Therefore µ 2 = yΛ 2 . Near the origin of S, the correction to the effective Kähler potential is calculated in Ref. [11] to be δK ∼ − y 2 (4π) 2 which stabilize the origin of the potential. The relation Λ ≃ 4πΛ/y is obtained. In order to satisfy the relation in Eq. (11), the coupling constant is determined to be y ≃ 10 −4 . With this small value of y, the field S is stabilized at a large value, S ∼ (4π) 2 Λ 2 /(y 2 M Pl ) which must be smaller than Λ/y so that the effective Kähler potential in Eq. (19) is reliable. This constraint gives an upper limit on Λ to be Λ 10 12 GeV which translates into Λ M Pl /10. This is consistent with the region in Fig. 1.

Summary
We considered a gravitational stabilization mechanism of the supersymmetry breaking vacuum in a simple gauge mediation model. The gravitational interaction splits the supersymmetry breaking and supersymmetric vacua for large enough values of the "cut-off" scale Λ 10 13 GeV.
The low energy effective model we analyzed can arise from many microscopic theories of supersymmetry breaking. Therefore the mechanism we discussed is applicable to a wide class of models. The model possesses R-symmetry. The R-symmetry is unbroken at the origin of the field S. However, the explicit breaking of the R-symmetry in supergravity (by the cosmological constant) shifts the vacuum from the origin. There is no unwanted Goldstone mode associated with the R-symmetry breaking since the symmetry is broken explicitly [17].