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Published by Institute of Physics Publishing for SISSA Received: September 2, 2007 Accepted: October 19, 2007 Published: November 15, 2007

R-symmetry, supersymmetry breaking and metastable vacua in global and local supersymmetric theories

Hiroyuki Abe

Tatsuo Kobayashi Department of Physics, Kyoto University, Kyoto 606-8502, Japan E-mail: [email protected]

Yuji Omura Department of Physics, Kyoto University, Kyoto 606-8501, Japan E-mail: [email protected]

Abstract: We study N = 1 global and local supersymmetric theories with a continuous global U(1)R symmetry as models of dynamical supersymmetry (SUSY) breaking. We introduce explicit R-symmetry breaking terms into such models, in particular a generalized O’Raifeartaigh model. Such explicit R-symmetry breaking terms can lead to a SUSY preserving minimum. We classify explicit R-symmetry breaking terms by the structure of newly appeared SUSY stationary points as a consequence of the R-breaking effect, which could make the SUSY breaking vacuum metastable. We show that the R-breaking terms are basically divided into two categories. One of them does not generate a SUSY solution, or yields SUSY solutions that disappear in the case of supergravity when we tune a parameter so that the original SUSY breaking minimum becomes a Minkowski vacuum. We also show that the general argument by Nelson and Seiberg for a dynamical SUSY breaking still holds with a local SUSY except for a certain nontrivial case, and present concrete examples of the exception. Keywords: Supersymmetry Breaking, Supersymmetric Effective Theories, Supergravity Models.

c SISSA 2007 °

http://jhep.sissa.it/archive/papers/jhep112007044 /jhep112007044 .pdf

JHEP11(2007)044

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan E-mail: [email protected]

Contents 1. Introduction

1

2. Global supersymmetric theory 2.1 R-symmetric model 2.2 Explicit R-symmetry breaking and metastable vacua

3 3 6 9 10 13

4. Explicit R-symmetry breaking in supergravity 4.1 Generalized O’Raifeartaigh model 4.2 Classification of R-breaking terms in supergravity

14 14 16

5. Type-A breaking: Polonyi-like models 5.1 Decoupled case: g1 (φi ) = 0 5.2 Generic case: g1 (φi ) 6= 0

17 18 20

6. Adding type-B breaking: Metastable universe 6.1 Decoupled case: g1 (φi ) = 0 6.2 Generic case: g1 (φi ) 6= 0

21 22 24

7. Conclusion

25

A. Supersymmetric masses involving R-axion

26

1. Introduction Supersymmetric extensions of the standard model are promising candidate for the physics around TeV scale. Supersymmetry (SUSY) can stabilize the huge hierarchy between the weak scale and the Planck scale, and supersymmetric models with R-parity have the lightest superparticle as a good candidate for the dark matter. In addition, the minimal SUSY standard model realizes the unification of three gauge couplings at a scale MGUT ∼ 2 × 1016 GeV, which may suggest some underlying unified structure in the nature. In our real world, the SUSY must be broken with certain amount of the gaugino and scalar masses. The dynamical SUSY breaking has a big predictability of the structure of such SUSY particles. It was shown by Nelson and Seiberg (NS) [1] that a global U(1)R symmetry is necessary for a spontaneous F-term SUSY breaking at the ground state of generic

–1–

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3. R-symmetry in supergravity 3.1 Nelson-Seiberg argument 3.2 Generalized O’Raifeartaigh model in supergravity

1

See for recent works on R-symmetry breaking, e.g. refs. [2 – 9] and references therein.

–2–

JHEP11(2007)044

models with a global SUSY. This predicts an appearance of massless Goldstone mode, Raxion, in dynamically SUSY breaking models with nonvanishing Majorana gaugino masses which breaks U(1)R symmetry.1 Recently, it has been argued by Intriligator, Seiberg and Shih (ISS) [4] that the SUSY breaking vacuum we are living can be metastable for avoiding the light R-axion and also obtaining gaugino masses, and that such situation can be realized by a tiny size of explicit U(1)R breaking effects, whose representative magnitude is denoted by ǫ. Such explicit Rsymmetry breaking terms can lead to a SUSY minimum, but such newly appeared SUSY minimum could be far away from the SUSY breaking minimum, which is found in the R-symmetric model without explicit R-symmetry breaking terms. Furthermore, such Rsymmetry breaking terms would not have significant effects on the original SUSY breaking minimum, because R-symmetry breaking terms are tiny. The distance between the original SUSY breaking minimum and the newly appeared SUSY preserving minima may be estimated as O(1/ǫ) in the field space. Thus, if R-symmetry breaking terms, i.e., the size of ǫ, are sufficiently small, the original SUSY breaking minimum would be a long-lived metastable vacuum. On the other hand, an introduction of gravity into SUSY theories requires that the SUSY must be a local symmetry, i.e., supergravity. In supergravity, the structure of the scalar potential receives a gravitational correction, and also the background geometry of our spacetime is determined by the equation of motion depending upon the vacuum energy. In the above global SUSY model with metastable SUSY breaking vacuum, some fields have large vacuum values at the SUSY preserving vacuum. In such a case, supergravity effects might be sizable. Another important motivation to consider supergravity is to realize the almost vanishing vacuum energy. The global SUSY model always has positive vacuum energy at the SUSY breaking minimum. Supergravity effects could realize almost vanishing vacuum energy. F-flat conditions have supergravity corrections. Thus, the supergravity model with global U(1)R symmetry would have different aspects from the global SUSY model. Furthermore, adding R-symmetry breaking terms would have different effects between global and local SUSY theories. Here we study in detail generic aspects of global and local SUSY theories with R-symmetry and generic behaviors caused by adding explicit R-symmetry breaking terms. That is, we reconsider the above argument for the dynamical SUSY breaking and its metastability by NS and ISS comparing global and local SUSY theories. The important keypoint is to realize the almost vanishing vacuum energy. That is impossible in the SUSY breaking vacuum of global SUSY models, and that is a challenging issue in supergravity models. The vacuum energy may be tuned to vanish, e.g., by the constant superpotential term, which is a sizable R-symmetry breaking term. That would affect all of vacuum structure such as metastability of SUSY breaking vacua and presence of SUSY preserving vacua. Here we study this vacuum structure by using several concrete models, where we start R-symmetric models and add certain classes of R-symmetry breaking terms such that the vanishing vacuum energy is realized.

2. Global supersymmetric theory 2.1 R-symmetric model First, we review briefly the argument by Nelson and Seiberg [1] in R-symmetric global SUSY models. Let us consider the global SUSY model with n chiral superfields zI (I = 1, . . . , n) and their superpotential W (zI ). In the case of global SUSY, F-flat conditions are determined by WzI = 0,

(2.1)

where WzI = ∂zI W . Hereafter we use a similar notation for derivatives of functions H(X) by fields X as HX . The conditions (2.1) are n complex equations for n complex variables, and these can have a solution for generic superpotential. Now, we consider global SUSY models with a continuous global U(1)R symmetry and a nonvanishing superpotential. Since the superpotential has the R-charge 2, there exists at least one field with a nonvanishing R-charge. Suppose that the n-th component zn is such a field with the nonvanishing R-charge, qzn 6= 0. Then, in the following field basis χi =

zi

qz /qzn

zn i Y = zn ,

,

(qχi = 0),

(qY = qzn 6= 0),

(2.2)

where i = 1, 2, . . . , n − 1, the superpotential can be written as WNS = Y 2/qY ζ(χi ).

–3–

(2.3)

JHEP11(2007)044

The following sections are organized as follows. In section 2, we study a structure of dynamical SUSY breaking in R-symmetric models with global SUSY. We consider the generalized O’Raifeartaigh (OR) model [10] following [4]. We introduce explicit R-breaking terms into the model and analyze in detail the newly appeared SUSY vacua as a consequence of the R-symmetry breaking effects. We also examine the stability of the original SUSY breaking vacuum under such R-breaking terms. In section 3, we consider supergravity models with R-symmetry. We extend the argument by NS to the local SUSY theories and study the supergravity OR model. In this section, we also show a special SUSY stationary point, which does not obey the NS condition, and the associated SUSY breaking vacuum in a certain class of R-symmetric supergravity models. We introduce explicit R-breaking terms into the supergravity OR model in section 4 and classify them by the consequent SUSY solutions. In section 5, we study the case with R-symmetry breaking terms (A-type) which might not cause a metastability of SUSY breaking minimum, because corresponding SUSY vacua disappear when we set the vacuum energy at the SUSY breaking minimum vanishing. On the other hand, in section 6, we show that another class of R-symmetry breaking terms (B-type) makes SUSY breaking minimum metastable. Section 7 is devoted to conclusions. In appendix A, we show some general features of R-axion masses, and find that the special SUSY solution exhibited in section 3 is at best a saddle point solution.

Then the F-flat conditions (2.1) are split into two pieces, (2/qY )Y 2/qY −1 ζ(χi ) = 0,

(2.4)

Y 2/qY ∂χj ζ(χi ) = 0.

(2.5)

When we look for an R-symmetry breaking vacuum, hY i = 6 0, these conditions are equivalent to ζ(χi ) = 0,

∂χj ζ(χi ) = 0,

(2.6)

a

where a = 1, 2, . . . , r and i = 1, 2, . . . , s, and the numbers of fields are constrained as r > s. Their R-charges are assigned as qXa = 2 and qφi = 0, and ga (φi ) is a function of φi . In this model, F -flat conditions for Xa are just given by ∂Xa W = ga (φi ) = 0.

(2.8)

These are r complex equations for s complex valuables, that is, these are over-constrained conditions for r > s. Therefore, there is no SUSY solution satisfying (2.8) for generic functions ga (φi ) with r > s. The superpotential of the generalized OR model (2.7) is a specific form of the NS superpotential (2.3). In the generalized OR model, SUSY is always spontaneously broken independently of whether R-symmetry is spontaneously broken or not, or the fields Xa develop nonvanishing vacuum expectation values or not. The simplest OR model is the model with r = 1 and s = 0, and has the superpotential W(OR)1 = f X1 , where f is a constant. Obviously, SUSY is spontaneously broken in this model, because WX1 = f . The basic O’Raifeartaigh model corresponds to the model with r = 2 and s = 1, and g1 (φ) = f + 12 hφ2 and g2 (φ) = mφ, and has the following superpotential, µ ¶ 1 2 W(OR)basic = f + hφ X1 + mφX2 . (2.9) 2

–4–

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which are n complex equations for n − 1 complex variables, that is, these are overconstrained conditions. These cannot be satisfied at the same time for a generic function ζ(χi ), and the SUSY can be broken. This is an observation by Nelson and Seiberg [1] that the existence of an R-symmetry is the necessary condition for a dynamical SUSY breaking, and is also the sufficient condition if the R-symmetry is spontaneously broken, hY i = 6 0. However, the scalar potential, which is obtained from the superpotential (2.3) and the K¨ ahler potential K(|Y |, χi , χ ¯i ), is found to have the global minimum at Y = 0, unless the K¨ ahler potential K(|Y |, χi , χ ¯i ) is non-trivial. Thus, SUSY is not broken dynamically with the NS superpotential (2.3). The O’Raifeartaigh model [10] is a good example of R-symmetric SUSY models, where SUSY is spontaneously broken. Its generalization is shown in ref. [4] as the generalized OR model, which has the following superpotential, X ga (φi ) Xa , (2.7) WOR =

The model has only a SUSY breaking pseudo-moduli space, φ = X2 = 0,

X1 : undetermined,

(2.10)

with WX1 = f as a global minimum of the potential. When integrating out heavy modes X2 and φ, we obtain W(OR)1 as an effective superpotential. However, the flat direction along X1 is lifted at the one-loop level by integrating out φ, and the SUSY breaking vacuum in the quantum corrected OR model is given by φ = X2 = X1 = 0.

(2.11)

a

˜1 + = f˜X

a

X

˜a, g˜a (φi )X

(˜ ga (0) = 0),

(2.12)

a

˜ a = Uab Xb , g˜a (φi ) = gˇa U † and Uab is a constant where fa = ga (0), gˇa (φi ) = ga (φi ) − fa , X ab † unitary matrix defined by fa Uab = f˜b = (f˜, 0, . . . , 0). In the following, we will frequently use this basis of fields and omit the tildes to simplify the notation. In this basis, the F-flat conditions for Xa , eq. (2.8), are written by WXa = ga (φi ) − δa1 f = 0.

(2.13)

P Together with Wφi = a Xa ∂φi ga (φi ) = 0, we find that, if f = 0, there is a solution Xa = φi = 0 and SUSY is not broken. Then it is obvious in the field basis (2.12) that a nonvanishing f is the source of dynamical SUSY breaking in the generalized OR model. In the generalized OR model with the above field basis, the field X1 plays a special role, while each of Xa (a 6= 1) has the qualitatively same character as others Xb (b 6= 1). Thus, the simple model with r = 2 and s = 1, and the superpotential, W(OR)2 = (f + g1 (φ))X1 + g2 (φ)X2 , shows qualitatively generic aspects of the generalized OR model. Its scalar potential is written as V = |f + g1 (φ)|2 + |g2 (φ)|2 + |Wφ |2 , and stationary conditions are obtained as VX1 = Wφ g1′ (φ) = 0, VX2 = Wφ g2′ (φ) = 0, Vφ = Wφ Wφφ + (f + g1 (φ))g1′ (φ) + g2 (φ)g1′ (φ) = 0,

–5–

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These simple models suggest that the tadpole term of Xa is important for SUSY breaking. Indeed, we can show by simple discussion that non-vanishing terms of ga (φi ) at φi = 0 are sources of SUSY breaking. We assume that ga (φi ) are non-singular functions. Then, we can always rewrite the superpotential (2.7) as X X WOR = fa Xa + gˇa (φi )Xa

P where ga′ (φ) = dga (φ)/dφ and Wφ = a Xa ga′ (φ). Unless Wφ does not vanish, we would have over-constrained conditions, i.e., g1′ (φ) = g2′ (φ) = 0 for generic functions. Thus, in general, the solution of the above stationary conditions corresponds to Wφ = X1 g1′ (φ) + X2 g2′ (φ) = 0, (f + g1 (φ))g1′ (φ) + g2 (φ)g1′ (φ) = 0.

(2.14)

The latter is the condition to fix φ. For a fixed value of φ, a ratio between X1 and X2 is fixed by the former condition, but the linear combination X1 g2′ (φ) − X2 g1′ (φ),

(2.15)

2.2 Explicit R-symmetry breaking and metastable vacua In order to have Majorana gaugino masses in addition to soft scalar masses, the R-symmetry must be broken spontaneously or explicitly at the SUSY breaking minimum we are living. On the other hand, as shown in the previous section, the NS argument requires an exact Rsymmetry for the dynamical SUSY breaking. Then, an appearance of an unwanted massless Goldstone mode, an R-axion, is inevitable in such R-symmetry breaking minimum. Does this mean the dynamical SUSY breaking is phenomenologically disfavored ? Recently, it has been argued by Intriligator, Seiberg and Shih [4] that our world must reside in a metastable state, in order to avoid the above conflict between gaugino masses and the R-axion. The arguments are as follows. Consider a theory with an approximate R-symmetry which has a small R-symmetry breaking parameter ǫ. In the limit ǫ → 0, the R-symmetry becomes exact, and the theory possesses a SUSY breaking ground state due to the NS argument. For a nonzero but tiny parameter ǫ, this SUSY breaking minimum still remains as a local minimum of the potential, although there appear SUSY ground states somewhere in the field space due to explicit R-symmetry breaking effects. As long as the parameter ǫ is small enough, the separation between the SUSY breaking minimum and the supersymmetric vacua is large, and the former can be a long-lived metastable vacuum. These facts were exhibited by ISS based on the O’Raifeartaigh model as a simple example of dynamical SUSY breaking model with R-symmetry. Indeed, such O’Raifeartaigh-type model can be realized in some region of the moduli space of SUSY Yang-Mills theories [11]. Here following the discussion by ISS we study generic aspects of explicit R-symmetry breaking terms, and SUSY preserving vacua. We also classify explicit R-symmetry breaking terms in global SUSY models. In addition, we discuss metastability. The simplest R-symmetry breaking term is the constant term WR / = c, but the constant term does not play any role in global SUSY theory. Thus, we do not discuss about adding constant term in this section. It is obvious that when we add any R-symmetry breaking term WR / (Y, χ) to the NS superpotential (2.3), that can relax over-constrained conditions and F-flat conditions can have SUSY solutions.

–6–

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remains undetermined. That is the pseudo-flat direction, and would be lifted by loop effects. Similarly we can discuss models with several fields Xa and φi (r > s).

The generalized OR model has richer structure in explicit R-symmetry breaking terms. To see such structure, we consider the generalized OR model with three types of typical Rsymmetry breaking terms, i) a function including only φi fields WR / = w(φ), ii) a function including only Xa (a 6= 1), WR / = w(Xa ), and iii) a function including only X1 , WR / = w(X1 ). The first type of R-symmetry breaking terms WR = w(φ) do not change F-flat / conditions for Xa , i.e., ∂Xa W = f δa1 + ga (φi ) = 0. Hence, there is no SUSY solution. For the second type of R-symmetry breaking terms WR / = w(Xa ) (a 6= 1), F-flat conditions are obtained as WX1 = f + g1 (φi ) = 0, for a 6= 1,

a

Thus, if wXa (Wa ) 6= 0 for all of Xa , over-constrained conditions can be relaxed and a SUSY solution can be found. If all of φi vanish, we have g1 (φi ) = 0 and the condition WX1 = 0 can not be satisfied. Hence, the SUSY minimum, which appears by adding WR / = w(Xa ) (a 6= 1), corresponds to the point, where some of φi develop nonvanishing vacuum expectation values. For the third type of R-symmetry breaking terms WR / = w(X1 ), F-flat conditions are obtained as WX1 = f + g1 (φi ) + ∂X1 w(X1 ) = 0, WXa = ga (φi ) = 0 for a 6= 1, X Xa ∂φi ga (φi ) = 0. W φi = a

If r = s + 1, the over-constrained conditions can be relaxed. In this case, the point φi = 0 for all of i can be a solution for WXa = 0 for a 6= 1. Furthermore, the conditions, X f + ∂X1 w(X1 ) = 0, Xa ∂φi ga (φi ) = 0, a

should be satisfied. When R-symmetry breaking terms include X1 and Xa (a 6= 1), over-constrained conditions can be relaxed and a solution for F-flat conditions would correspond to φi 6= 0 for some of φi . The SUSY breaking minimum is found in the generalized OR model without explicit R-symmetry breaking terms, as discussed in the previous subsection. As discussed above, SUSY vacua can appear, when we add the definite form of explicit R-symmetry breaking terms to the generalized OR model. Thus, the previous SUSY breaking minimum is a metastable vacuum, if such R-symmetry breaking effects are small around the SUSY breaking minimum and the SUSY breaking vacuum itself is not destabilized by such R-symmetry breaking terms. As an illustrating example, we consider the basic OR model (2.9) with explicit Rsymmetry breaking terms. ISS introduced an explicit R-symmetry breaking term in the

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WXa = ga (φi ) + wXa (Wa ) = 0 X W φi = Xa ∂φi ga (φi ) = 0.

superpotential,2 W = W(OR)basic + WR / , where WR / =

1 ǫmX22 . 2

In this case, there appears a SUSY minimum, r 2f 1 φ= − , X2 = − φ, h ǫ

(2.16)

X1 =

m , ǫh

which is far away from the (local) SUSY breaking minimum (2.11) for a sufficiently small ǫ ≪ 1. In addition, the SUSY breaking minimum is not destabilized by the above Rsymmetry breaking term (2.16). Then the original SUSY breaking vacuum (2.11) becomes metastable which can be parametrically long-lived for ǫ ≪ 1. Instead, if we consider the following R-breaking term [13] 1 ǫmX12 , 2

(2.17)

the newly appeared SUSY point is found as φ = X2 = 0,

X1 = −

f . ǫm

In this case, the pseudo-moduli space (2.10) disappears at the tree level. However, the SUSY breaking point (2.11) remains as a local minimum due to the one-loop mass for X1 , but becomes metastable. Then the situation is similar to the above example. We easily find that any R-breaking terms which consist of only φ do not restore SUSY. Now, let us study whether the SUSY breaking minimum, which is found without R-symmetry breaking terms, is destabilized by adding R-symmetry breaking terms. We consider the generalized OR model with (r = 2, s = 1), i.e., W(OR)2 , whose stationary conditions (2.14) are studied in the previous subsection. Their solutions are denoted by (0) Xa = Xa and φ = φ(0) . First, we add a small R-symmetry breaking term, WR / = ǫw(X2 ), which depends only on X2 . Then, the scalar potential is written as V = |f + g1′ (φ)|2 + |g2 (φ) + ǫw′ (X2 )|2 + |Wφ |2 , where Wφ = X1 g1′ (φ) + X2 g2′ (φ). In addition, we assume that the stationary conditions of (0) V are satisfied by Xa = Xa + δXa and φ = φ(0) + δφ, and that all of δXa and δφ are of O(ǫ). For example, the stationary condition along φ, Vφ = 0, gives the following condition, ! Ã X X ′′ (0) ′ (0) 2 (0) |ga (φ )| + (f δa1 + ga (φ ))ga (φ )) δφ + ǫg2′ (φ(0) ) w′ (X20 ) = 0, a

a

(0)

where we have used the stationary conditions (2.14) at Xa = Xa and φ = φ(0) . This is the equation to determine δφ. The stationary condition along X1 , VX1 = 0, reduces to g1′ (φ(0) ) δWφ = 0, 2

See also ref. [12].

–8–

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WR / =

where δWφ =

X

ga′ (φ(0) )δXa +

a

X

Xa(0) ga′′ (φ)δφ.

a

Thus, this shows a relation among δXa and δφ unless g1′ (φ(0) ) = 0. On the other hand, the stationary condition along X2 , VX2 = 0, leads to the following equation, (0)

ǫw′′ (X2 ) g2 (φ(0) ) = 0.

V = |f + g1′ (φ) + ǫw′ (X1 )|2 + |g2 (φ)|2 + |Wφ |2 . The stationary conditions along X2 and φ give an equation to determine δφ and a relation among δXa and δφ. However, the stationary condition along X1 , VX1 = 0, leads to ´ ³ (0) w′′ (X1 ) f + g1 (φ(0) ) = 0.

If this condition is not satisfied, the stationary condition at the SUSY breaking vacuum is destabilized. Indeed, the basic O’Raifeartaigh model has f + g1 (φ) = f at φ = 0. Thus, when we add the mass term of X1 , w(X1 ) = 12 mX12 , i.e., w′′ 6= 0, the SUSY breaking minimum become destabilized at the tree level as shown above. Note that this kind of destabilization would be related to the existence of the flat direction (2.15) in the OR model with global SUSY.3 The above discussion shows that adding generic R-symmetry breaking terms can destabilize the SUSY breaking minimum, which is found in the model without such explicit Rsymmetry breaking terms. In order to realize metastability of the original SUSY breaking minimum, we need a certain type of R-symmetry breaking terms. Alternatively, loop-effects would be helpful not to destabilize the original SUSY breaking minimum by R-symmetry breaking terms.

3. R-symmetry in supergravity In the previous section, based on the argument by ISS, we have shown that some sort of explicit R-symmetry breaking terms can restore SUSY, and the original SUSY breaking vacuum can become metastable when a certain (but not generic) class of explicit R-symmetry breaking terms are added and/or loop effects stabilize the original SUSY breaking minimum. The metastable minimum can be parametrically long-lived if the coefficient of the 3

Such flat direction would be lifted by supergravity effects.

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This is not an equation among δXa and δφ, but implies that the stationary condition is (0) destabilized unless w′′ (X2 ) g2 (φ(0) ) = 0. In the above basic O’Raifeartaigh model, we have g2 (φ(0) ) = 0. Thus, the SUSY breaking minimum is not destabilized by adding the mass term of X2 , w(X2 ) = 12 mX2 , i.e., w′′ (X2 ) 6= 0 at X2 = 0. Now, let us add an R-symmetry breaking term, WR / = ǫw(X1 ), which depends only on X1 . Similarly, we can examine stationary conditions of the scalar potential,

3.1 Nelson-Seiberg argument In this subsection, we study the NS argument within the framework of supergravity theory. In the case of supergravity, F-flat conditions (2.1) are modified as DI W ≡ WI + KI W = 0, where K denotes the K¨ ahler potential, K(|Y |, χi , χ¯i ). In the field basis (2.2) with the superpotential (2.3), these are written as Dχi W = Y 2/qY (ζi + Ki ζ) = 0, DY W = (2/qY + Y KY )X 2/qY −1 ζ = 0. Then, we find the following two candidates of R-breaking SUSY solutions in supergravity, ζi = 0,

ζ = 0,

(3.1)

and Dχi ζ = ζi + Ki ζ = 0,

2/qY + Y KY = 0.

(3.2)

The first conditions (3.1) contain n complex equations for n − 1 complex variables, and the situation is the same as the case of global SUSY (2.6), that is, the solution does not exist for a generic function ζ. On the other hand, the second conditions (3.2) are n complex equations for n complex variables which can have a solution. This corresponds to a SUSY stationary point specialized to R-symmetric supergravity. In this subsection, we analyze the special SUSY stationary solution (3.2) which appears due to purely the supergravity effect and does not obey the NS condition. Then, in the following we assume that there is a solution for 2/qY + Y KY = 0.

– 10 –

(3.3)

JHEP11(2007)044

R-breaking term is sufficiently small with which the SUSY ground state is far from the metastable state in the field space. This argument has been performed in a decoupling limit of gravity. As we find in the above discussion, however, we have to treat a large distance between some separated minima in the field space. This may imply that large vacuum values of some fields might be involved in the analysis, where supergravity effects could become sizable. Moreover, in global SUSY, the SUSY breaking minima always have a positive vacuum energy with the magnitude of the SUSY breaking scale, which never satisfies the observation that the vacuum energy almost vanishes. In such a sense, we would be forced to consider supergravity. Note that, even in supergravity, it is often a hard task to tune the vacuum energy at the stationary points of the scalar potential to be almost vanishing. This might require a large R-symmetry breaking effect specialized to supergravity, i.e., a constant term in the superpotential [14]. The existence of such a special R-symmetry breaking term could also affect the ISS argument of metastability. Loop effects have contributions to the vacuum energy. Here we assume that such loop effects are subdominant, and we tune our parameters such that we realize V ≈ 0 at the tree level. Hereafter we use the unit with MPl = 1, where MPl denotes the reduced Planck scale.

For instance, if the K¨ ahler potential is given by X ˆ i, χ K = cnY |Y |2nY + K(χ ¯i ),

(3.4)

nY =1

the condition (3.3) becomes 2/qY +

X

nY =1

nY cnY |Y |2nY = 0.

Dχi f =0

Again, for the minimal K¨ ahler potential (3.4) with c1 = 1 and cnY >1 = 0, the stationary condition ˆ

2

∂Y v(Y ) = eK(hχi i,hχ¯i i) e|Y | |Y |2/qY −2 (2/qY + |Y |2 ) ´ ³ × |Y |4 + 2(2/qY − 1)|Y |2 + (2/qY )2 − 2/qY = 0,

leads to solutions

|Y |2 = −2/qY ,

(3.5)

and |Y |2 = 1 − 2/qY ±

p

1 − 2/qY .

(3.6)

The first solution (3.5) corresponds to the SUSY saddle point and the second solutions (3.6) are SUSY breaking minima. We can find this kind of SUSY breaking minima in a similar way for more generic K¨ ahler potential. We can study the same system in a different view point. We redefine the field Y as T =−

2 ln Y, aqY

(3.7)

where a is a real constant. In this basis, the K¨ ahler potential and the superpotential (2.3) is written as K = K(T + T¯, χi , χ ¯i ), W = e−aT ζ(χi ).

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(3.8)

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Then, we need at least one negative value of {cnY , qY } to have a solution. In the simplest minimal case with cnY >1 = 0 (and then KY Y¯ = c1 > 0), a negative charge, qY < 0, is required. A nontrivial point of this solution is that this SUSY stationary point is always tachyonic as we can see from the arguments in appendix A. In addition, we can find a SUSY breaking minima along the direction Dχi ζ = 0 (the first condition in eq. (3.2)), if we assume that χi receives a heavy SUSY mass m2χi ≫ m23/2 by the condition Dχi W = 0. This is a reasonable assumption because χi has a vanishing R-charge and ζ(χi ) in W is assumed to be a generic function. The scalar potential along Dχi ζ = 0 is found to be ¯ ´ ³ ¯ = eK KY−1Y¯ |2/qY + KY Y |2 − 3|Y |2 |Y |2(2/qY −1) |ζ|2 . v(Y ) = V ¯

This type of K¨ ahler and superpotential appear in the four-dimensional effective theory derived from superstring theory, where T may be a modulus field associated to some compactified dimensions. In such a case, the K¨ ahler potential is typically given by ˆ i, χ K = −nT ln(T + T¯) + K(χ ¯i ), where nT is a fractional number, and the T -dependence of the superpotential (3.8) may originate from nonperturbative effects such as string/D-brane instanton effects and gaugino condensation effects, where the corresponding gauge coupling is determined by the vacuum value of T . In this case, the scalar potential along Dχi ζ = 0 is given by ¯ ´ ³ ¯ 2 (K − a) − 3 |e−aT ζ|2 , = eK KT−1 v(T ) = V ¯ T ¯ T Dχi f =0

ˆ

∂t v(t) = −eK(hχi i,hχ¯i i) e−at t−nT −1 ³ ´ ×(at + nT ) (a2 /nT )t2 + 2a(1 − 1/nT )t + nT − 3 = 0,

results in a SUSY saddle point t = −n/a and SUSY breaking minima q 2 t = −(nT /a)(1 − 1/nT ) ± (nT |a|/a ) 5/nT + 1/n2T ,

where t = T + T¯. In the literature, there are examples of the models which have this kind of vacuum structure of the potential. Typical superstring models have several moduli TI with the Q K¨ ahler potential K = ln I (TI + T¯I )−nTI . The superpotential induced by some nonperturbative effects is given by P I X W = An e I an TI , n

where An and aIn are constants. If the number of the moduli is the same as or larger than the number of the nonperturbative terms appearing in the superpotential [15], we can define an R-symmetry. A particular linear combination of TI ’s corresponds to T in eq. (3.8) which is determined by the condition that all the remaining combinations corresponding χi ’s receive a heavy mass by the SUSY condition Dχi W = 0. This is possible for certain values of aIn . For the two moduli with double nonperturbative terms, i.e., racetrack models, a detailed analysis was carried out in ref. [16]. We stress that the analysis of the SUSY breaking minimum as well as the SUSY saddle point in this subsection is based on the assumption that all the other fields χi than Y or T are stabilized by Dχi W = 0, that is, by the SUSY masses [17]. We comment that these stationary solutions have a nonvanishing and negative vacuum energy. We need to uplift the SUSY breaking minimum to a Minkowski vacuum in order to identify this minimum as the one we are living. For such purpose, we need another sector which provides the uplifting energy and is well sequestered in order not to spoil the original structure of dynamical

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and then the stationary condition

SUSY breaking. Such sector can be realized by a dynamically generated F-term [18, 19] for which the discussions in the following sections would be important. In summary, there is a possibility of special SUSY stationary solution in R-symmetric supergravity with a generic superpotential. However, it is always a saddle point at best and we find SUSY breaking minima with lower vacuum energy. This may imply that the NS argument for a dynamical SUSY breaking is qualitatively correct also in this case, although there is a SUSY solution. Furthermore, the NS argument still holds in supergravity as long as the K¨ ahler potential satisfies 2/qY + Y KY 6= 0 for any value of Y in the field basis (2.2). For instance, in typical models with qY > 0 and K = |Y |2 , we always find 2/qY +Y KY > 0. 3.2 Generalized O’Raifeartaigh model in supergravity

DXa W = ∂Xa W + (∂Xa K) W X = Mab (Xc , φi ) (gb (φi ) + δb1 f ) = 0,

(3.9)

b

where Mab (Xc , φi ) = δab + KXa Xb . We define its determinant as ∆ ≡ det Mab = 1 +

X

KXa Xa .

(3.10)

a

If there is no solution for ∆ = 0, the matrix Mab has an inverse matrix and consequently the F-flat conditions (3.9) are reduced to the same ones as eq. (2.8) in the global SUSY, ga (φi ) + δa1 f = 0, which does not allow a solution for r > s in general. However, in the limit f → 0 in the tilde basis (2.12), these equations are satisfied at φi = 0. Thus, the constant f represents the typical size of SUSY breaking effects and ga (φi ) as the global SUSY case. We comment that the situation changes if there exists a solution of ∆ = 0. Actually, the condition ∆ = 0 is an analogue of the second condition in eq. (3.2). Then, we can carry out a similar analysis as in the previous subsection also for this OR model. That is straightforward and is omitted here. Note that the condition ∆ = 0 is never satisfied for a minimal K¨ ahler potential, X X K = |Xa |2 + |φi |2 . (3.11) a

i

In the following, we just assume that there is no solution for ∆ = 0. We comment that, even in supergravity, the scalar potential is positive, V > 0, in the generalized OR model (2.12) with the minimal K¨ ahler potential (3.11). In this case, the scalar potential is written as £ ¤ ¯a Xb }(gb + δb1 f ) + |Xa Dφ ga |2 . V = eK (¯ ga + δa1 f¯){δab + (|Xc |2 − 1)X i

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Now we consider the generalized OR model (2.7) in supergravity. The F-flat conditions (2.8) for Xa become

For any vacuum values of Xa , we can always rotate their basis as ˆ a = (0, . . . , 0, X ˆc , 0, . . . , 0), Uab Xb = X by a unitary matrix U(Xa ), and in this basis we can write ˆ c |2 − 1/2)2 + 3/4}|ˆ e−K V = {(|X gc |2 +

X a6=c

|ˆ ga |2 +

X i

ˆc Dφ gˆc |2 > 0, |X i

4. Explicit R-symmetry breaking in supergravity Here we study explicit R-symmetry breaking terms in supergravity and examine whether SUSY solutions can be found by adding explicit R-symmetry breaking terms to the NS model and the generalized OR model. In the previous section, we have pointed out that there is a SUSY stationary point when the condition (3.3) or the condition ∆ = 0 is satisfied. In the following sections, we consider the models, where such conditions are not satisfied, and SUSY is broken in the NS and generalized OR models even within the framework of supergravity like global SUSY theory. First we consider the NS model with explicit R-symmetry breaking terms WR / = w(Y, χi ). The total superpotential is written as, W = Y 2/qY ζ(χi ) + w(Y, χi ). In this case, F-flat conditions of supergravity theory, DY W = Dχi W = 0, do not lead to over-constrained conditions for any non-vanishing function w(Y, χi ). It is remarkable that within the framework of supergravity theory the constant term WR / = c breaks R-symmetry and even such term is enough to relax the over-constrained conditions. 4.1 Generalized O’Raifeartaigh model Let us study more explicitly the generalized OR model with explicit R-symmetry breaking terms WR / = w(Xa , φi ). The total superpotential is written as, W = f X1 +

r X

ga (φi )Xa + w(Xa , φi ).

a=1

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where gˆa = (U † )ab (gb + δb1 f ). Note that gˆa are now Xa -dependent functions. As discussed above, the conditions, gˆa (φ) = 0, can not be satisfied at the same time. Thus, the vacuum energy must be positive, V > 0. Since typical magnitudes of gˆa (φ) would be of O(f ), we would estimate V ∼ f 2 . To realize the almost vanishing vacuum energy V ≈ 0 at this SUSY breaking minimum, we need a negative and sizable contribution to the vacuum energy, which can be generated by R-symmetry breaking effects, e.g., the constant term in the superpotential.

First, we consider the case with the constant R-symmetry breaking term, WR / = c. In this case, F-flat conditions are written explicitly as Ã ! r X ga (φi )Xa + c = 0, DXa W = f δa1 + ga (φi ) + KXa f X1 + Dφi W =

X a

Xa ∂i ga (φi ) + Kφi

Ã

a=1 r X

f X1 +

a=1

!

ga (φi )Xa + c

= 0.

a=1

Thus, the situation is quite similar to the case with WR / = c. To have a SUSY solution, it is required that hW i, hw(φi )i and at least (r − s) vacuum values of KXa must be nonvanishing. Sizes of hW i and hw(φi )i are expected to be comparable with f . Finally, we consider the case that R-symmetry breaking terms include Xa fields, WR / = w(Xa , φi ). In this case, F-flat conditions along Xa , DXa W = 0, are written as DXa W = f δa1 + ga (φi ) + ∂Xa w(Xa , φi ) + KXa W = 0. When KXa W is sufficiently small, the above F-flat conditions correspond to F-flat conditions in global SUSY theory. In such a case, we have a SUSY solution when w(Xa , φi ) depend on at least (r − s) Xa ’s. Otherwise, the global SUSY solution can not be found, but a SUSY solution with hw(Xa , φi )i = 6 0 and hW i = 6 0 can be found within the framework of supergravity theory. Such situation is similar to the case with WR / = c. We have discussed that the NS model and generalized OR model with R-symmetry breaking terms have SUSY solutions with hW i = 6 0 in supergravity theory. If the SUSY breaking minimum, which is found without R-symmetry breaking terms, is not destabilized

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The former conditions are not always over-constrained for c 6= 0. Furthermore, the vacuum expectation value of W and at least (r − s) vacuum values of KXa are required to be nonvanishing. Otherwise, the former conditions become over-constrained for generic functions ga (φi ). Furthermore, when KXa for a 6= 1 does not vanish, all vacuum values of φi can not vanish to satisfy DXa W = ga (φi ) + KXa W = 0. Thus, a SUSY solution can be found by adding WR / = c. This solution corresponds to the AdS SUSY minimum, because non-vanishing hW i is required and the scalar potential at this point is evaluated as V = −3eK |W |2 < 0. The values of the constant c and hW i must be sizable, because this AdS SUSY point disappears in the limit that c → 0 or hW i → 0. Magnitudes of c and hW i are expected to be comparable with f when KXa = O(1). Hence, we can find the new type of SUSY solution, which can not be found in global SUSY theory. However, that requires large values of c and hW i, and may have sizable effects on the previous SUSY breaking minimum, which is found in the generalized OR model without R-symmetry breaking terms. Similarly, we can discuss the case that R-symmetry breaking terms include only φi fields, i.e., WR / = w(φi ). In this case, F-flat conditions along Xa , DXa W = 0, are written as ! Ã r X ga (φi )Xa + w(φi ) = 0. DXa W = f δa1 + ga (φi ) + KXa f X1 +

4.2 Classification of R-breaking terms in supergravity In this subsection and the following sections, we consider minutely the previous discussions about the explicit R-symmetry breaking in the supergravity framework by examining concrete examples. We introduce the explicit R-symmetry breaking terms WR / into the above supergravity OR model, WR / = c(φi ) +

1X m ǫab (φi )Xa Xb + · · · , 2

(4.1)

a,b

where c(φi ) and ǫab (φi ) are generic functions of φi including φ-independent constants, and the ellipsis denotes the higher order terms in Xa . Note that, as mentioned before, only the ǫab (φi ) terms are relevant to the recovery of SUSY vacua in the case of global SUSY. Now we have the total superpotential, W = WOR + WR / . The F-flat conditions (2.13) are modified as µ ¶ X X −1 −1 DXa W = Mab gb (φi ) + δa1 f1 + Mbc ǫcd (φi )Xd + ∆ KXb WR = 0. (4.2) / b

c,d

Here we find that all the terms in WR / including c(φi ) are accompanied by Xa in the above F-flat conditions and then have a possibility for restoring SUSY, contrary to the case of global SUSY explained in the previous section. Most notably, just a constant superpotential WR / = c,

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(4.3)

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by the presence of R-symmetry breaking terms, the previous SUSY breaking minimum would correspond to a SUSY breaking metastable vacuum. However, a sizable vacuum value of superpotential is required unless ∂Xa w(Xa , φi ) 6= 0 for at least (r − s) Xa fields. Such large superpotential (even if that is a constant term) would affect the stability of the previous SUSY breaking minimum. Furthermore, we have another reason to have a large size of hW i at the previous SUSY breaking minimum. At the previous SUSY breaking minimum, the vacuum energy is estimated as V ∼ |f |2 > 0 for hW i = 0. To realize the almost vanishing vacuum energy, V ≈ 0, we need a non-vanishing value of hW i, which are comparable with f . In this case, supergravity effects at the previous SUSY breaking minimum are not negligible. This purpose to realize V ≈ 0 has the implication even for the case that R-symmetry breaking terms include more than (r−s) Xa fields. In this case, we can find a (global) SUSY solution even for hW i = 0. However, realization of V ≈ 0 requires a sizable vacuum value of hW i, although values hW i at the SUSY breaking minimum and SUSY preserving minimum are not the same. Hence, it is quite non-trivial whether one can realize a metastable SUSY breaking vacuum with V ≈ 0 in supergravity theory, which has a SUSY minimum. We will study this possibility concretely by using simple classes of the generalized OR models in the following sections. We will concentrate ourselves to the minimal K¨ ahler potential (3.11) in most cases of the following discussions.

i.e., c(φi ) = c and ǫab (φi ) = 0, can restore SUSY. In this case with the minimal K¨ ahler potential (3.11), we find a solution for eq. (4.2) as ¯ a = −c−1 ∆ ga (φi ), X

(4.4)

P where ∆ = 1 + a |Xa |2 defined in eq. (3.10) is real and positive. From eq. (4.4), Xa can be written in terms of φi , and then ∆ is given by p P |c|2 ± |c| |c|2 − 4 a |ga (φi )| P ∆= , 2 a |ga (φi )|

a

where hφi i are solutions of Dφi W = 0 under the condition (4.4). Because X1 is distinguished in the superpotential (2.12), we divide the generic Rbreaking terms (4.1) into two pieces: (A)

WR / = WR /

(B)

+ WR / ,

where (A)

WR / (Xa6=1 ; φi ) = c(φi ) +

1 X m ǫab (φi )Xa Xb + · · · , 2

(4.6)

a,b6=1

(B)

WR / (X1 ; Xa6=1 , φi ) =

X a6=1

1 m ǫa1 (φi )Xa X1 + m ǫ11 (φi )X12 + · · · . 2

(4.7)

(A)

The ellipses denote the higher order terms in terms of Xa6=1 in WR / , and those of X1 and (B)

Xa6=1 in WR / . Without loss of generality, we can assume that ǫ11 (0) is real and positive among ǫab (0), which is referred as ǫ in section 6.

5. Type-A breaking: Polonyi-like models In this section, we study the effect of R-breaking terms (4.6) which we call the A-type breaking, (A) W = WOR + WR / . Because this type of breaking terms does not contain X1 , we find the Polonyi model [20] W |Xa6=1 =0,φi =0 = WPolonyi ≡ f X1 + c,

(5.1)

in the hypersurface Xa6=1 = 0, φi = 0 of the scalar potential, where c = c(0). This hypersurface would be a stationary plane in the Xa6=1 - and the φi -directions if ∂φi ga6=1 (0) are sufficiently large, which correspond to SUSY masses for Xa6=1 and φi on that plane.

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which should be a real number. Therefore, in order for the SUSY solution (4.4) to be valid, the constant superpotential c must satisfy the condition X 4 |ga (hφi i)|2 ≤ |c|2 , (4.5)

Moreover, if mi1 and/or hij 1 in g1 (φi ) = mi1 φi + hij 1 φi φj + · · · ,

(5.2)

are nonvanishing, the Polonyi model in this hypersurface can be affected/modified by a tree-level SUSY mass and/or a one-loop SUSY breaking mass for X1 . Then, we further classify the A-type breaking models into two cases, g1 (φi ) = 0 and g1 (φi ) 6= 0. 5.1 Decoupled case: g1 (φi ) = 0 In the case with g1 (φi ) = 0, the superpotential of the A-type breaking models is written as W = f X1 +

ga (φi )Xa + c(φi ) +

a6=1

1 X mǫab (φi )Xa Xb + · · · 2 a,b6=1

1 = c + f X1 + µAB ΦA ΦB + · · · , 2

where ΦA = (Xa6=1 , φi ) with the index A = (a 6= 1, i). The SUSY mass matrix µAB is given by the R-breaking components, µa6=1,b6=1 = mǫab (0), µij = ∂φi ∂φj c(0) and the R-symmetric components, µa6=1,i = 2∂φi ga (0). After the unitary rotation which makes µAB diagonal, the above superpotential takes the form of 1 W = c + f X1 + µA Φ2A + · · · , 2

(5.3)

where µA represents the eigenvalues of µAB . Because of the SUSY mass µA , the field ΦA would be integrated out without affecting the low energy dynamics of X1 , because X1 is completely decoupled in the present case.4 Then, the effective action for X ≡ X1 is just determined by the Polonyi superpotential (5.1), where the phase of c and f can be eliminated by the U(1)R rotation and the rephasing of X1 . Assuming the minimal K¨ ahler potential (3.11) for simplicity, the effective ¯ = x satisfying the stationary scalar potential is minimized by a real vacuum value X = X condition VX = eG GX (GXX + G2X − 2) = 0, where G = K + ln |W |2 and GXX + G2X − 2 = f W −1 (x3 + f −1 cx2 − 2f −1 c), GX = f W −1 (x2 + f −1 cx + 1).

(5.4) (5.5)

The F-flat condition for X corresponds to GX = 0, and the SUSY breaking stationary point is determined by the condition GXX + G2X − 2 = 0. As we declared, we persist in obtaining a vanishing vacuum energy at the SUSY breaking minimum. Then in addition to the stationary condition GXX + G2X − 2 = 0, we set 4

We may have to assume that the K¨ ahler mixing is also zero or negligible between X1 and the others.

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X

¯

V = eG (GX X |GX |2 − 3) = 0. In this case, we have to take a definite value of the constant c and find two solutions √ √ (x, f −1 c) = ( 3 − 1, 2 − 3), (5.6) and

x± =

p 1 (−f −1 c ± (f −1 c)2 − 4), 2

(5.8)

if the R-breaking constant c satisfies

|f −1 c| ≥ 2.

(5.9)

Note that this condition (5.9) corresponds to eq. (4.5) in the previous general argument for the generalized OR model. The mass eigenvalues of (ReX, ImX) are computed as W±2 (x2± − 2)(x2± + 1) and W±2 (x2± − 1)(x2± + 2) at this SUSY AdS stationary point where ¯ p 1 ¯¯ ¯ |W± | = |f x± + c| = ¯f (f −1 c ± (f −1 c)2 − 4)¯ > 0, 2 and then we obtain

2

V = −3eG = −3ex± |W± |2 < 0. Remark that, in the vanishing (one of) R-breaking limit, c → 0, the condition (5.9) is not satisfied, and the SUSY solution (5.8) disappears. In the other words, this SUSY solution is a consequence of the R-breaking constant term c in the superpotential. Due to the appearance of this SUSY solution, there is a possibility that the SUSY breaking point determined by GXX + G2X − 2 = 0 becomes a metastable vacuum as in the case of global SUSY explained previously. However, this is not the case. Interestingly, if we tune the R-breaking constant super√ potential c as f −1 c = 2 − 3 so that the solution (5.6) with the vanishing vacuum energy is realized, the condition (5.9) is not satisfied and the SUSY stationary solution (5.8) disappears. In such a sense, the constant R-breaking term c does not lead to a metastability of SUSY breaking Minkowski minimum (5.6).

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√ √ (x, f −1 c) = (− 3 − 1, 2 + 3). (5.7) √ √ The mass eigenvalues of (ReX, ImX) are computed as (2 3f 2 , (4 − 2 3)f 2 ) for the first √ √ solution (5.6), and (−2 3f 2 , (4 + 2 3)f 2 ) for the second one (5.7), at this SUSY breaking Minkowski stationary point where W = f . Then, only the first solution (5.6) can be a minimum of the potential, while the second one (5.7) is a saddle point. We comment that φi and Xa6=1 directions would not possess tachyonic masses at these points for sufficiently large SUSY mass µA compared with the SUSY breaking mass f . Therefore, the candidate for our present universe, where the SUSY is broken with (almost) vanishing vacuum energy, is the first solution (5.6). In addition to a SUSY breaking solution satisfying GXX + G2X − 2 = 0, we have a SUSY solution GX = 0 due to the R-breaking effect c 6= 0, that is,

Next, we consider the SUSY stationary solutions outside the Polonyi slice Xa6=1 = 0, φi = 0. For the superpotential (5.3), the F-flat directions are determined by DΦA W = KA W + µA ΦA + · · · = 0, DX1 W = f + KX1 W = 0,

which can be satisfied by distinguishing a single field ΦB 6= 0 for ∃ B as −1 −1 f ΦB (µB + · · ·) = −KX W = −KB 1

ΦA = KA = 0

(for ∃ B),

(5.10)

(for A 6= B),

For this value of ΦB , the remaining condition DX1 W = 0 is satisfied by X1 = f /µB . Note that the number of this SUSY points is nX + nφ − 1 because the solution (5.11) is valid for every choice of B = (b 6= 1, j). In order for the solution (5.11) to be valid, the parameter µB must satisfy µ2B + cµB + f 2 ≤ 0. This leads to the same condition (5.9) for the R-breaking constant term c as in the Polonyitype SUSY solution. In summary, the A-type breaking terms (4.6) can restore SUSY in the generalized OR model (2.7) or equivalently (2.12) in general. However, if we tune the R-breaking constant term in the superpotential so that the SUSY breaking minimum has a vanishing vacuum energy, i.e., (5.6), the SUSY solutions (5.8) and (5.11) disappear. Therefore, in this sense, the A-type R-symmetry breaking terms do not lead to a metastability of the SUSY breaking (Minkowski) vacuum aside from a possibility of the existence of more complicated SUSY solutions than (5.11). 5.2 Generic case: g1 (φi ) 6= 0 Now we turn on a nonvanishing g1 (φi ) as in eq. (5.2). With this term, the tree-level (field dependent) mass matrices in the φi = 0 plane contain the following contributions, VX1 X¯1 |φl =0 = |mi1 |2 + · · · , ¯

¯¯

2 ¯ jk ¯ j1 + 4hik Vφi φ¯¯j |φl =0 = mi1 m 1 h1 |X1 | + · · · ,

¯ Vφi φj |φl =0 = hij 1 f + ···, ¯

VX1 φi |φl =0 = 2hij ¯ j1 X1 + · · · , 1 m

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(5.12)

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where the ellipsis represents the higher order terms of ΦB . The first line gives two complex equations for two complex variables X1 and Φ∃ B , which have a solution in general. For example, if the K¨ ahler potential is minimal (3.11), all the parameters in the superpotential are real and there is no higher order terms of ΦB (no ellipses in the above expressions), then the solution for eq. (5.10) is found as µ ¶ c f2 2 + 2 + 1 > 0, |ΦB | = −2 ΦA6=B = 0. (5.11) µB µB

where the ellipses represent the original terms involving Xa6=1 , those coming from c(φi ), and the supergravity corrections. Here the doubled indices are summed up. The K¨ ahler covariant derivatives of the superpotential in the hypersurface φi = 0, Xa6=1 = 0 are given by DX1 W |φl =0 = f + KX1 W,

DXa6=0 W |φl =0 = 0,

Dφi W |φl =0 = mi1 X1 .

¯¯

2 ¯jk Vφi φ¯¯j |φl =0 = 4hik 1 h1 |X1 | + · · · ,

¯ Vφi φj |φl =0 = hij 1 f + ···.

From the second equation, we observe that some linear combinations of Re φi and Im φj become tachyonic in the φi = 0 plane if |hij 1 f | dominate the SUSY mass for φi . The X1 dependence in the first one indicates that a SUSY breaking mass of X1 is generated at the one-loop level, which is proportional to hij 1. Therefore, the effective potential after integrating out φi and Xa6=1 is given by V = V (0) + V (1) ,

¯

V (0) = eG (GX X |GX |2 − 3),

V (1) = m2X |X|2 ,

(5.13)

where X ≡ X1 , G = K + ln |W |2 , and the effective superpotential W = WPolonyi is shown in eq. (5.1). The one-loop mass mX is determined by hij 1 as well as f , which would be considered as an independent parameter in the effective action. The stationary condition VX = 0 results in [19] X ≃ 2f c/m2X , for c ∼ f ∼ mX ≪ 1 in the unit with MP l = 1, and the vanishing vacuum energy at this minimum requires √ c = f / 3 + O(f 3 /m2X ). The SUSY is broken at this Minkowski minimum with DX W = f + O(f 2 ) and W = √ f / 3 + O(f 2 ).

6. Adding type-B breaking: Metastable universe In the previous section, we have analyzed the generalized OR model with the explicit Rsymmetry breaking terms (4.7) which do not involve the source field X1 for the dynamical SUSY breaking.

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From the third equation, we find that φi can not be integrated out prior to X1 by the F-flat condition Dφi W = 0 unlike before. This is because, with the nonvanishing mi1 , the source field X1 for SUSY breaking shares a common SUSY mass with φi as shown in eq. (5.12). In this case, the purely X1 -direction is no longer special in the scalar potential. We have to treat X1 and φi at the same time. The analysis is quite complicated, and then we consider the case with mi1 = 0 in the following, where g1 (φi ) starts from the quadratic term in φi , and the φi can be integrated by their F-flat conditions Dφi W = 0 resulting φi = 0. We will comment about the case with mi1 6= 0 in section 6.2 together with more general R-breaking terms. The components of the mass matrices (5.12) are now reduced to

In this section, we study more general case with the R-breaking terms (4.6) including X1 , i.e., (B) (A) + WR W = WOR + WR / . / In the type-B breaking terms (4.6), the first term with ǫa6=1,1 (0) gives the common SUSY mass for X1 and Xa6=1 in the φi = 0 plane. Then the situation is similar to the case with a nonvanishing mi1 in eq. (5.2), that is, we can not integrate out Xa6=1 prior to X1 , and we will include this case also in section 6.2. By setting ǫa6=1,1 (0) = 0, the superpotential in the hypersurface φi = Xa6=1 = 0 is given by 1 W = f X + mǫX 2 + c + · · · , 2

(6.1)

6.1 Decoupled case: g1 (φi ) = 0 As in the previous section, we first consider the case with g1 (φi ) = 0, where X1 is decoupled from the others in the superpotential. In this case the hypersurface φi = Xa6=1 = 0 would be stable in the φi -, Xa6=1 -direction as in section 5.1. The effective theory in this slice is described by the superpotential (6.1). With the minimal K¨ ahler potential (3.11), real parameters f , c, m and no higher order terms (ellipsis) in the superpotential (6.1) for simplicity, the SUSY breaking and SUSY stationary conditions are respectively given by eqs. (5.4) and (5.5). In the limit ǫ → 0 of eq. (6.1), the SUSY breaking solution is given by eq. (5.6). Then we can find the deviation of X from this point assuming ǫ ≪ 1 and m ∼ c1/3 ∼ f 1/2 . We find a SUSY breaking minimum with a vanishing vacuum energy at √ ǫm + O(ǫ2 ), (6.2) δX = − XSB = X0 + δX, X0 = 3 − 1, 2f where the constant superpotential term c is tuned as √ √ c = (2 − 3)f + (2 3 − 3)ǫm + O(ǫ2 ).

(6.3)

On the other hand, a SUSY solution, XSUSY ≃ −

2f , ǫm

(6.4)

arises as a consequence of the B-type R-breaking term represented by the parameter ǫ, although the vacuum energy is set to be vanishing at the SUSY breaking minimum. This is unlike the case of SUSY solutions (5.8) and (5.11) caused by the introduction of A-type R-breaking terms (4.6). The shift of SUSY breaking minimum δX in eq. (6.2) is rewritten as 1 1 , δX/X0 ≃ √ X 3 − 1 SUSY and we find 1 |XSUSY | > √ ∼ O(1), 3−1

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where X ≡ X1 , ǫ = ǫ11 (0) and the ellipsis stands for the higher order terms in X.

ΜB f 0.2 0.1 0

Εmf

-0.1 -0.2 -0.3 -0.5

0

0.5

1

Figure 1: Parameter region (white) of µB and ǫ allowing the SUSY solution (6.5). All the parameters are assumed to be real and the constant term c is fixed by the vanishing vacuum energy condition (6.3) at the SUSY breaking minimum (6.2). In the shaded region, the SUSY solution (6.5) is not allowed and the SUSY breaking solution (6.2) does not become metastable due to the Rbreaking effect parameterized by ǫ. We find no allowed region in the limit ǫ → 0 which corresponds to the solution (5.11).

in order for the shift δX to reside in a perturbative region, |δX/X0 | < 1. This means that the vacuum value of |X| at the newly appeared SUSY vacuum must be larger than the Planck scale MPl = 1, where the supergravity calculation might not valid. It would be possible that the potential is lifted for |X| > 1 by the effect of quantum gravity, the above SUSY vacuum is washed out and the SUSY breaking minimum remains as a global minimum. If the supergravity approximation is valid even for |X| > 1 by any reason, we obtain a constraint on the R-breaking parameter ǫ as √ ǫ < 2( 3 − 1)|f /m|, from the above condition. We also find a SUSY minimum outside the hyperplane φi = Xa6=1 = 0, which is a generalization of eq. (5.11), given by ½ µ ¶¾ f2 2 ǫm 2 µB + c + |ΦB | = − 1+ ≥ 0, µB µB − ǫm 2(µB − ǫm) ΦA6=B = KA6=B = 0, f X = , (6.5) µB − ǫm where we assumed the minimal K¨ ahler potential (3.11), and the absence of the higher order terms of X in the superpotential for concreteness. In the limit ǫ → 0, this solution is

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-1

reduced to (5.11). In contrast to (5.11), the above solution (6.5) does not disappear in all of the parameter region, even after the vacuum energy at the SUSY breaking minimum is set to zero as in eq. (6.2). Such parameter region of µB and ǫ allowing the SUSY solution is shown in figure 1. In the shaded region, the SUSY solution (6.5) is not allowed and the SUSY breaking solution does not become metastable due to the R-breaking effect represented by ǫ. Note that we find no allowed region along the ǫ = 0 axis, which corresponds to the case of the solution (5.11). 6.2 Generic case: g1 (φi ) 6= 0

XSB =

2f c (1 + O(ǫ2 )), m2X

(6.6)

where the R-breaking constant √ c = f / 3 + O(f 3 /m2X ; ǫ2 ), is determined by the vanishing vacuum energy condition. The SUSY ground state in the hyperplane φi = Xa6=1 = 0 which originates from the Rbreaking parameter ǫ is the same as eq. (6.4), and the above breaking minimum becomes metastable. Unlike (6.2), the SUSY breaking minimum (6.6) is not affected by the Rbreaking term at O(ǫ) due to the one-loop mass mX , that is, the SUSY minimum (6.4) is independent of the SUSY breaking minimum (6.6) at this order. There might exist SUSY points analogous to eq. (6.5) outside the hypersurface φi = Xa6=1 = 0 also in this case, but the solution would be more complicated due to the nonvanishing hij 1 in eq. (5.2). i Finally we comment about the case with m1 6= 0 in eq. (5.2). In this case, as mentioned in section 5.2, the field X1 has a SUSY mass with the same magnitude as those of φi ’s as shown in eq. (5.12). Then the field X1 in the field basis (2.12) is no longer special. In this generalized OR model with most general R-breaking terms, the total superpotential would be written as X 1 X mǫab (φi )Xa Xb + · · · W = f X1 + ga (φi )Xa + c(φi ) + 2 a=1

a,b=1

1 = c + f X1 + µIJ ΦI ΦJ + · · · , 2

where ΦI = (Xa , φi ), I = (a, i) and the ellipses denote the higher order terms in ΦI . The SUSY mass matrix µIJ is given by the R-breaking components, µab = mǫab (0), µij = ∂φi ∂φj c(0) and the R-symmetric components, µai = 2∂φi ga (0). Note that µ1i = 2∂φi g1 (0) =

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Finally we introduce nonvanishing g1 (φ). As in section 5.2, we first consider the case with mi1 = 0 in eq. (5.2). In this case we can still integrate φi and Xa6=1 by use of Dφi W = DXa6=1 W = 0 resulting in φi = Xa6=1 = 0. The remnant of these heavy fields would be the one-loop mass mX for X1 = X in eq. (5.13). The effective scalar potential is in the same form as eq. (5.13) but the effective superpotential W in G = K+ln |W |2 is now replaced by eq. (6.1). For ǫ ≪ c ∼ f ∼ mX ≪ 1 in the unit with MPl = 1, we can obtain a SUSY breaking Minkowski minimum

2mi1 . After the unitary rotation which makes µIJ diagonal, the above superpotential takes the form of 1 W = c + f U1I ΦI + µI Φ2I + · · · , 2 where UIJ is the rotation matrix and µI represents the eigenvalues of µIJ . The F-flat conditions, DI W = WI + KI W = 0, allow a solution in general and SUSY would not be broken for mi1 ∼ f .

7. Conclusion

Acknowledgments The authors would like to thank K. Choi and T. Higaki for useful discussions. H. A. and T. K. are supported in part by the Grand-in-Aid for Scientific Research #182496 and #17540251, respectively. T. K. is also supported in part by the Grant-in-Aid for the 21st Century COE “The Center for Diversity and Universality in Physics” from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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We considered N = 1 global and local supersymmetric models with a continuous global U(1)R symmetry, and studied the effect of explicit R-symmetry breaking terms in detail. In global supersymmetric models, based on the argument by ISS, we have shown that some sort of explicit R-symmetry breaking terms can restore SUSY, and the original SUSY breaking vacuum can become metastable when a certain (but not generic) class of explicit R-symmetry breaking terms are added and/or loop effects stabilize the original SUSY breaking minimum. We have executed similar analyses in R-symmetric supergravity models. First we examined the general argument by NS in supergravity and found that it also holds with local SUSY except for the nontrivial case where the K¨ ahler potential allows solution for the second condition in eq. (3.2). We presented concrete examples of this exception. These models lead to AdS SUSY stationary solutions and associated SUSY breaking vacua with lower vacuum energy. We found the general argument that this class of SUSY solutions corresponds to at best a saddle point, referring to appendix A. Then, we studied the generalized OR model in supergravity with explicit R-symmetry breaking terms. We analyzed the structure of newly appeared SUSY stationary points as a consequence of the R-breaking effect and classified them. We have shown that these SUSY solutions disappear for type-A breaking terms (4.6), when we tune the R-breaking constant term in the superpotential such that the original SUSY breaking minimum has a vanishing vacuum energy. In this sense, the introduction of explicit R-breaking terms does not always lead to a metastability of the SUSY breaking vacuum. On the other hand, the introduction of type-B breaking terms (4.7) could cause a metastability of SUSY Minkowski minimum. We examined a parameter region which yields metastable vacuum in some concrete examples.

A. Supersymmetric masses involving R-axion In this appendix, we show some general results for the SUSY masses for the scalar component of an R-axion multiplet. For this analysis, it is convenient to redefine the R-charged superfield Y by R=

2 ln Y, qY

where R can be interpreted as the R-axion superfield. (Note that R = −aT in eq. (3.7).) In this basis, the K¨ ahler potential and the superpotential (2.3) is written as ¯ χi , χ K = K(R + R, ¯i ), W = eR ζ(χi ).

(A.1)

GRR = KRR + W −1 WRR − (W −1 WR )2 = KRR = KRR¯ ,

Gχi R = Kχi R + W −1 Wχi R − (W −1 Wχi )(W −1 WR ) = Kχi R = Kχi R¯ . Substituting these into the general formulae for the second derivatives at the SUSY point, ¯ ¯ ¯ = eG (GM N GM I GN¯ J¯ − 2GI J¯), VI J¯¯ ¯DK W =0 ¯ = −eG GIJ , VIJ ¯ DK W =0

we find

¯ ¯ ¯ ¯ = −KRR¯ m23/2 , = VRR ¯ VRR¯ ¯ DK W =0 DK W =0 ¯ ¯ ¯ ¯ = −Kχi R¯ m23/2 , = Vχi R ¯ Vχi R¯ ¯ DK W =0

DK W =0

(A.2) (A.3)

where m23/2 = eG is the gravitino mass square. From eq. (A.2), the mass squared eigenvalues of (Re R, Im R) can be calculated as 0 and −2m23/2 with the canonical kinetic terms normalized by KRR¯ > 0. The first massless eigenmode corresponds to the R-axion scalar associated to the spontaneously broken global U(1)R symmetry. The second negative-definite eigenvalue indicates that the special SUSY solution (3.2) is at best a saddle point solution. Note that the gravitino mass m3/2 is nonvanishing at this point and the vacuum energy is negative. We also find from eq. (A.3) that the mixing-mass between R and χi is vanishing if the K¨ ahler (kinetic) mixing is vanishing, Kχi R¯ = 0. In this case, the R-axion direction is completely separated from the other fields χi , that is, the above mass eigenvalues of R-axion multiplet become exact in this case. Finally we comment that the second derivatives (A.2) and (A.3) are all vanishing at the SUSY point (3.1) where m3/2 = 0. In this case, both the real and the imaginary scalar component of R-axion multiplet remain massless. Note that eq. (3.2) may also allow a solution even in this case if ζ is not a generic function.

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From eq. (A.1), we find W −1 ∂Ym W = 1 where m = 1, 2, . . . ., and obtain

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