IPPP / 07 / 01 Metastable SUSY breaking within the Standard Model

arXiv:hep-ph/0701069v3 26 Jan 2007 Preprint typeset in JHEP style - PAPER VERSION hep-ph/0701069 IPPP/07/01 Metastable SUSY breaking within the Standard Model Steven A. Abel and Valentin V. Khoze Institute for Particle Physics Phenomenology and Centre for Particle Theory, University of Durham, Durham, DH1 3LE, UK [email protected], [email protected] Abstract: We construct a supersymmetric version of the Standard Model which contains a longlived metastable vacuum. In this vacuum supersymmetry is broken and the electroweak symmetry is Higgsed, and we identify it with the physical ground state of the Standard Model. In our approach the metastable supersymmetry breaking (MSB) occurs directly in the SU (2)L × U (1)Y sector of the Standard Model; it does not require a separate MSB sector and in this way it departs from the usual lore. There is a direct link between the electroweak symmetry breaking and the supersymmetry breaking in our model, both effects are induced by the same Higgs fields ϕi , ϕ̃i . In order to generate sufficiently large gluino masses we have to have strong coupling in the Higgs sector, h ≫ 1. Our model results in an extremely compact low-energy effective theory at the electroweak scale with Higgs fields being very heavy, MHiggs ≫ MW and frozen at their vacuum expectation values. 1. Introduction Supersymmetry breaking in a long-lived metastable vacuum (MSB) discovered by Intriligator, Seiberg and Shih [1] is an exciting possibility for model building. MSB scenarios are based on models which contain supersymmetry-breaking metastable vacua in addition to the supersymmetry-preserving (stable) ground states. The existence of supersymmetric vacua in MSB models makes them much less constrained [1] than the more traditionally considered scenarios of dynamical supersymmetry breaking (DSB) [2]) which contain only non-supersymmetric ground states. Furthermore, MSB models are at least as natural as the DSB framework. In particular it has recently been shown that, in Intriligator-Seiberg-Shih (ISS) MSB models, the early Universe would generically have been driven to the metastable vacua by thermal effects [3, 4], and also that once trapped there the lifetime for decay to the true (supersymmetric) vacua is much longer than the age of the Universe for any reasonable choice of parameters [1,3–6], thus realising in an elegant way an early idea of Ellis et al [7]. MSB therefore completes the canon of supersymmetry breaking available to supersymmetric field theory, so from a fundamental viewpoint it is certainly an interesting development. On the other hand, the benefits of the MSB over DSB scenarios for phenomenological applications are less immediately obvious. If the MSB sector forms a hidden sector then its phenomenological consequences are largely determined by the method of mediation to the visible sector. Could one tell in practice if a hidden sector were MSB or DSB? Intuitively it is clear that the more direct the mediation of SUSY breaking to the Standard Model is, the easier this would be. In a series of recent papers [8–12] the ISS-type models with metastable SUSY breaking were used to construct new, simple and calculable models of direct gauge mediation. In this set-up the ISS model forms a hidden sector which breaks supersymmetry; this effect is then communicated to the Standard Model via gauge interactions. In this paper, we will pursue a different and more extreme approach which is to eliminate the need for a hidden sector at all: we will use the ISS model as a basis for visible sector SUSY breaking. There is a more specific reason why ISS models warrant a return to visible sector SUSY breaking, namely the fact that their gauge groups are automatically Higgsed in the metastable minimum and moreover the scale of Higgsing is of the same order as the scale of SUSY breaking. Like O’Raifeartaigh models of old, a natural link between gauge symmetry breaking and SUSY breaking comes for free in MSB. We propose incorporating the ISS-type metastability directly into the electroweak SU (2)L ×U (1)Y sector of the MSSM in order to give a visible sector SUSY breaking that is naturally linked to, and driven by, electroweak symmetry breaking. In this proposal we are thinking of the MSSM-like theory as a magnetic dual theory which is valid in the IR. Indeed just like the ISS models, the model we put forward has a Landau pole in the SU (2)L gauge coupling at some scale ΛL . Above this scale we will assume that there is an (unknown) electric theory which takes over1 ; however the form of the microscopic electric theory has no bearing on phenomenology. The model we propose needs very little 1 We will see that the scale where our theory will require a UV completion is comfortably very high, ΛL > MP l . The –1– extension beyond the conventional MSSM, merely an ISS-like O’Raifeartaigh potential for the Higgs sector and some extra generations of Higgs fields to cause SU (2)L to become strongly coupled at ΛL . At first sight our proposal seems bound to fail because of two familiar “no-go” theorems. The first is the theorem by Nelson and Seiberg [14] that SUSY breaking in a generic theory requires an R-symmetry (where generic means that all operators that are allowed by symmetries appear in the superpotential). This appears to exclude the possibility of non-zero gaugino Majorana masses since they are inconsistent with an unbroken R-symmetry. A spontaneously broken R-symmetry on the other hand implies a massless R−axion which is disallowed on cosmological grounds [15]. The second is a no-go theorem [16] coming from the well-known sum-rule ST r(M 2 ) = 0. This relation holds at tree-level even when SUSY is spontaneously broken, and can be applied to differently charged fields independently, so that for example it predicts m2d˜ + m2s̃ + m2b̃ ∼ (5GeV)2 , obviously completely at odds with experiment [17]. To avoid this tree-level mass relation one has to generate SUSY breaking terms of order ∼TeV at one-loop or higher. This implies that the F -term vev responsible for SUSY breaking must be at least 100 TeV2 as is the case in gauge mediated SUSY breaking for example. Say the vev of the Higgs fields breaking electroweak symmetry is µ ∼ g2−1 MW . Then since we want to induce SUSY breaking and electroweak symmetry breaking with the same field this implies F = hµ2 where h is some coupling constant which clearly has to be much greater than one. How can such large couplings - and this is the essence of the problem - be consistent in a calculable theory? We will show that both of these theorems are evaded by the special properties of MSB models. The first crucial point is that, as pointed out by ISS, metastable models do not have to adhere to the Nelson-Seiberg theorem because they have supersymmetric vacua, and indeed in ISS-type models they violate it in an interesting way; the theory at the metastable minimum resembles a standard O’Raifeartaigh model, SUSY is broken and there is a global R-symmetry. However the global supersymmetric minima are recovered by a nonperturbative dynamical term that is generated by the SU (2)L gauge symmetry. The R-symmetry is anomalous under this SU (2)L group and therefore the dynamical term does not respect it. This strongly suggests that other sectors of the theory may dynamically produce R-symmetry violating operators as well whilst leaving supersymmetry intact (as for example the magnetic theory does in the supersymmetric minima). Depending on how the breaking is mediated to the magnetic theory, one does not expect all possible operators to be generated at leading order. The resulting effective superpotential of the IR theory can be only approximately nongeneric, and metastability can still be preserved. (Note that we emphasize “at leading order”; if those operators that destabilize the metastable minimum are small enough, then the decay time of the false vacuum is still sufficiently long to avoid the possibility of decay within the lifetime of the Universe.) The nett effect can be the lifting of the R-axion masses, and the radiative generation of large gaugino masses. In ISS-type models, the breaking of both gauge symmetry and SUSY at the metastable minimum, can be traced back to the gauge singlet field Φji (i, j are ISS-sector flavour indices) of R-charge 2. As already stated, we will require that R-symmetry is broken in the full theory, but that this must electric theory above ΛL can be a string or a field theory which is related to our magnetic theory below ΛL in some way, possibly involving a generalisation of Seiberg duality [13]. –2– be communicated to the low-energy effective theory in a controlled manner so as to maintain the metastability – as a working example we will consider additional SU (3) coloured fields with mass terms. The resulting effective IR theory is non-generic, but this is naturally understood as resulting from the dynamical breaking of R-symmetry. At one-loop and higher this leads to R-symmetry breaking operators of the form T r(Φ) α W Wα , (1.1) WR ≈ const mR where W α is the gluon field-strength superfield, mR represents the scale of R-symmetry breaking, and const ∼ 1/16π 2 takes into account the loop suppression (if this operator is generated perturbatively). In order to motivate WR , in the Appendix we will show how (1.1) can be generated at one loop. It is important that the R-symmetry violating terms (1.1) are holomorphic. Because of this they have a relatively gentle effect on the vacuum structure of our model; with or without them, SUSY is still broken in the metastable minima, and is completely restored in the same SUSY preserving minima. What these operators do is transmit the SUSY breaking from Φ to the gauginos. Indeed, in the metastable vacuum, SUSY is broken by FΦi = hµ2 (for some i) where h and µ are i parameters of our choosing (in the low-energy magnetic theory), and the scale of SU (2)L breaking is MW ≈ g2 µ (note that for more general, i.e. flavour dependent, choices of µ this scale is an upper bound on the scale of supersymmetry breaking). The gaugino masses generated by these terms are then h M2 (1.2) Mλ ≈ const 2 W , g2 mR and the SUSY breaking induced in the gauginos λ is in turn transmitted to the squarks and sleptons through one-loop diagrams. This evades the first ”no-go” theorem, provided that gaugino mass terms are large enough, but how can one accommodate large h in order to evade the second ”no-go” theorem and overcome the loop suppression?2 Naively it seems impossible that the SUSY breaking in the visible sector could ever be large enough: as in gauge-mediation scenarios we require FΦ & 100 TeV2 in order to overcome the one loop suppression factor 1/16π 2 , and the only way to reconcile this with the fact that MW ≈ gµ is to choose a coupling h ≫ 1. One might worry that this would render the theory completely incalculable. We will argue that this is not the case, and that the theory is calculable, at least at energies around the electroweak scale. Ultimately the effective theory is only slightly more complicated than the MSSM itself, and yet the soft-SUSY breaking terms are essentially unsuppressed and completely predicted. The spectrum and phenomenology is expected to be broadly similar to gauge mediation [18] (see [17,19] for a review). One striking difference though is that the Higgs fields become very heavy and essentially decoupled from the theory. Note that there is a connection here with Ref. [8, 20], which provided a general framework for considering MSB models with R-symmetry broken by small parameters. The present work can be thought of as a concrete and minimal realisation of this idea, and indeed at very low scales (i.e. scales far below ΛL ) our model is in effect an example of a “retrofitted” O’Raifeartaigh model. The novelty 2 Note that we are not entitled to take mR ≪ 1 TeV because the approximation in calculating WR breaks down. –3– here is that SUSY breaking and electroweak breaking emerge from the same sector and that there are no hidden sectors. We begin in the following section by reviewing ISS models and introducing the proposed MSSMlike extension of it. As we have already indicated, it is the SU (2)L factor of the gauge group which we are suggesting plays the role of the magnetic dual, and which becomes strong at the scale ΛL & MP l . However, introducing the other multiplets and gauge groups necessary to make the model MSSM-like complicates the vacuum structure. The dynamical restoration of SUSY which leads to the supersymmetric minima involves various parameter-dependent combinations of F and D-flat directions; we will show explicitly how the minima are generated and where they are located. We will also make the following observation: the distance in field space from the metastable origin to the global SUSY restoring minima can be less than, or greater than the Landau pole ΛL and depends on the couplings. Thus, by adjusting couplings, the SUSY restoring minima can be banished beyond the Landau pole. Following this we discuss the generation of R-symmetry breaking terms, and then discuss the resulting phenomenology, and in particular its similarity to gauge mediation. 2. MSSM: M is for metastable Can ISS metastability be successfully embedded in the visible sector of the MSSM? In this section we demonstrate that with minor modification, the MSSM can be transformed into a theory that has all the properties of the macroscopic ISS theory, namely 1. O’Raifeartaigh SUSY breaking at the origin 2. A commensurately Higgsed electroweak sector 3. The gauge group which is Higgsed is IR free and has a Landau pole at scale ΛL 4. SUSY preserving global minima that are generated by strong dynamics This last point is enough to guarantee that the ISS-style metastability has long decay lifetimes since the SUSY preserving minima are generated radiatively and hence the potential is much flatter than it is broad [1]. In addition it guarantees that the metastable minima are preferred in a Universe whose temperature is greater than the SUSY breaking scale [3, 4]. Note that there do exist chiral models with Seiberg duals, and even chiral Seiberg duals of nonchiral theories, but as yet there are no microscopic theories whose magnetic duals are MSSM-like; therefore we will in what follows be working entirely in the IR macroscopic theory and merely be assuming that a UV microscopic theory exists above the Landau pole. This theory may or may not be a field theory which may or may not be related to the macroscopic theory by Seiberg duality, however this is irrelevant to our discussion. In particular we only need show explicitly that the macroscopic theory –4– inherits the same O’Raifeartaigh-like metastable minima as in the explicit ISS theory, and that issue is independent of the microscopic theory. Therefore let us first recapitulate the metastable SUSY breaking minima of ISS. 2.1 The ISS model Intriligator, Seiberg and Shih [1] examined the IR free magnetic dual of an asymptotically free SU (Nc ) theory, in which the magnetic theory has a gauged SU (Nf − Nc ) symmetry and global SU (Nf ) × U (1)B × U (1)R symmetry for degenerate quark mass terms in the microscopic theory. The superpotential of the macroscopic theory is of the form Wcl = h[T r(ϕΦϕ̃) − µ2 T r(Φ)] (2.1) where Φij are the flavour mesons of the IR free theory and ϕai and φ̃ja the fundamental and antifundamentals of quarks under SU (Nc − Nf ). The crucial observation is that for Nf > Nc the F-flatness equation is no longer satisfied due to the so-called rank condition; that is FΦi = h (ϕ̃j .ϕi − µ2 δij ) = 0 j (2.2) can only be satisfied for a rank-(Nf − Nc ) submatrix of the FΦ . The quark mass of the electric theory corresponds to µ2 which is a free parameter. It could be generated in a variety of ways [1, 8, 10, 12], but here we take it as simply the control parameter for both SUSY and gauge breaking. The model is of the standard O’Raifeartaigh type, with supersymmetry being broken at scale µ. The height of the potential at the metastable minimum is given by VT =0 (0) = Nc |h2 µ4 |, (2.3) i.e. there is an equal contribution from each of the non-zero FΦ -terms. The supersymmetric minima are located by allowing Φ to develop a vev. The ϕ and ϕ̃ fields acquire masses of hhΦi and can be integrated out, upon which one recovers a pure SU (Nf −Nc ) Yang-Mills theory with a nonperturbative contribution to the superpotential of the form Wdyn = N hNf detNf Φ ΛNf −3N !1 N . (2.4) This leads to Nc nonperturbatively generated SUSY preserving minima at hhΦji i = µǫ−( 3Nc −2Nf Nc ) j δi (2.5) in accord with the Witten Index theorem, where ǫ = µ/Λ. The minima can be made far from the origin if ǫ is small and 3Nc > 2Nf , the latter being the condition for the magnetic theory to be IR-free. However since we must also have Nf ≥ Nc + 2 the positions of the minima are bounded by the Landau pole and they are always in the region of validity of the macroscopic theory. We shall now adapt this structure to mimic the MSSM. –5– 2.2 A metastable MSSM The main content of the model will be a direct extension of the supersymmetric Standard Model incorporating the above mechanism. Our goal is to embed the metastable ISS model into the Standard Model. Since we are after a minimal such embedding we do not want to treat the ISS model as a hidden sector. Instead we identify the SU (N ) group of the (magnetic) ISS theory with the SU (2)L weak-interaction gauge group of the Standard Model. The Nf pairs of fields ϕ and ϕ̃ are then the Higgs doublets of the Standard Model. (Of course now Nc has no meaning other than as the combination Nf −N = Nf −2, nevertheless we will retain it as a useful parameter.) The Higgs sector superpotential of our model is taken to be WHiggs = h T r[ϕΦϕ̃ − µ2 Φ], (2.6) so that electroweak symmmetry will be automatically Higgsed at the same time as SUSY is broken. The parameter (µ2 )ji = µ2i δij can be generated dynamically, and we will without loss of generality take a flavour basis in which it is a diagonal but non-degenerate matrix in flavour space, so that the U (Nf ) flavour symmetry is explicitly broken to U (1)Nf . We now observe that in order to have SUSY broken by the rank condition we must have Nf ≥ 3. For simplicity we will concentrate here on the minimal case, Nf = 3, and note that our construction can be trivially extended to higher values of Nf . So we have to extend the MSSM to a multi-Higgs model with three generations of Higgs pairs. (It will turn out that only one pair of Higgs doublets will be coupled to the SM matter fields.) This is the first and last modification that we need to make to the MSSM-sector; the remaining fields Q, U , D, L, E have the usual charge assignments and number of generations.3 We now want to discuss the symmetry of our model. When all µi parameters in (2.6) are set to zero, this superpotential has U (3) global flavour symmetry. We now turn on the non-vanishing and non-degenerate values for µ’s and order them so that |µ1 | > |µ2 | > |µ3 | > 0. This breaks the flavour symmetry down to U (1)3 which we denote as U (1)Y × U (1)3 × U (1)P Q . We choose the charges of ϕi under these three U (1)’s as follows: U (1)Y : (− 12 , 12 , 0) , U (1)3 : (0, 0, 1) , U (1)P Q : (1, 1, 1) (2.7) The charges of ϕ̃i fields are opposite, and the Φij field transforms in the bifundamental representation under the first two U (1)’s such that the superpotential (2.6) is invariant. We choose this assignment of charges (rather than two traceless combinations and one trace) for reasons that will become clear immediately. The PQ symmetry is the overall U (1) of the broken U (3) flavour symmetry4 . We keep it as a global U (1) of our model which is spontaneously broken by the vevs of ϕ1 and ϕ2 in the metastable vacuum. The U (1)Y symmetry gives rise to the hypercharge when quark and lepton superfields are included. This symmetry is gauged and participates in the spontaneous electroweak symmetry breaking SU (2)L × U (1)Y → U (1)QED by the vevs of ϕ1 and ϕ2 . 3 As we have already mentioned, we will also require an additional sector which breaks the R-symmetry. It is the coupling of this sector to the MSSM sector which yields gaugino masses. However these fields will be chosen so they are a mild perturbation of the vacuum structure, metastability and spectrum of the MSSM-like sector of the theory, which is the subject of this section. 4 Note that the PQ symmetry was the “baryon number” of the original ISS model – we do not use that name to avoid confusion with the conventional B of the MSSM. –6– In addition, there is also an (anomalous) R-symmetry U (1)R as well as the Baryon and Lepton number symmetries. We list the particle content and charges of our metastable SUSY Standard Model in the Table U (1)Y SU (2)L Φij ϕi ϕ̃i L E Q D U 1  ¯  ¯  1 ¯  1 1 1 2 (δi1 U (1)3 − δi2 + δj2 − δj1 ) − 12 + 12 , + 12 , − 12 − 12 1 1 6 1 3 − 23 ,0 ,0 1 2 (δj3 − δi3 ) 0, 0, 1 0 , 0 , −1 0 0 0 0 0 U (1)R U (1)P Q L B 2 0 0 1 1 1 1 1 0 1 −1 0 0 0 1 -1 0 0 0 0 0 0 0 0 − 21 − 21 − 21 − 21 − 21 + 31 − 31 − 31 The U (1)R symmetry will be broken by the R-symmetry sector which we add later. As mentioned earlier, the hypercharge of the Higgs fields is associated with the traceless U (1)Y factor of the parent flavour symmetry. The hypercharges of the remaining fields are determined by anomaly cancellation under this U (1). These fields do not (and indeed need not) fall into obvious representations of the parent flavour symmetry. The U (1)Y hypercharge factor we shall assume to be gauged, and the factor of U (1)3 is assumed to be global, and it will remain unbroken. The third Abelian factor surviving from the broken flavour symmetry – U (1)P Q – is broken spontaneously by the vevs of ϕ and ϕ̃. This implies that there is a single Goldstone boson – the PQ-axion – present in our model in the metastable vacuum (after the three Goldstone bosons of the SU (2)L × U (1)Y gauge group are eaten by the longitudinally polarized vector bosons). It follows that apart from the PQ axion which we will discuss in a moment, there will be no massless scalars arising from the metastable vacuum of our model. The symmetries allow the masses of the quarks and leptons to be generated by the standard Yukawa couplings of the MSSM, WY uk = λU Qϕ2 U + λD Qϕ1 D + λE Lϕ1 E, (2.8) where the λf carry conventional MSSM generation indices. Note that with our assignment of charges, and in particular the U (1)P Q and U (1)3 symmetry, these are all the Yukawa couplings one can write down. The superfields ϕ1 and ϕ2 are the two Higgs doublets, Hd and Hu of the MSSM, and all the remaining ISS chiral fields (i.e. ϕ3 , ϕ̃1 , ϕ̃2 , ϕ̃3 and Φij ) cannot couple to the quarks or leptons. This avoids flavour changing neutral currents appearing at tree-level. The U (1)R symmetry in our model is broken by anomalies to a discrete symmetry which contains Z2 . This Z2 is the conventional R-parity which protects against baryon and lepton number violating operators. Ultimately the only light state remaining in the Higgs sector will be the axion, whose mass is protected by the anomalous PQ symmetry. As it stands this (visible) axion would be disallowed –7– because the Peccei-Quinn scale would be O(MW ). There are two natural ways to make such an axion acceptable: either the PQ symmetry breaking scale is elevated to fP Q ∼ 1011 Gev by some other hidden sector fields which are also charged under U (1)P Q in the sense of [22], or the PQ symmetry is gauged. For the purposes of discussion in this paper we will assume the latter. In this case we add an additional field η to the microscopic theory, which is charged under the U (1)P Q and gets a very large vev ≫ µ. The PQ-axion is the Goldstone boson which is eaten by the longitudinal mode of the massive gauge boson and effectively acquires the mass ghηi ≫ MW . Thus it disappears from the light spectrum. The anomaly is generically cancelled by the Green-Schwarz mechanism in string theory. This generally (although not always) induces Fayet-Iliopoulos (FI) terms into the D-terms of the Lagrangian. This gives a natural mechanism for generating the vev hηi if this is the field which cancels the FI term. An alternative approach, which we have not explored in this paper, but will study elsewhere, is to allow the eventual R-symmetry breaking to also induce PQ-symmetry breaking mass terms for the Higgs fields of the form WHiggs−mass ∼ mij εab ϕai ϕbj . These terms would be equivalent to the ”µ”-terms of the conventional MSSM and would break both P Q and R-symmetry. We will leave a full discussion of these issues to future work. 2.3 The local metastable minima Let us identify the metastable supersymmetry breaking vacuum of our model. The analysis here follows Ref. [1] closely, but with the important modification that flavour symmetry is broken by the µi terms, so that there are no (uneaten) Goldstone modes. The rank condition ensures that SUSY is broken at the origin of Φ, and SU (2)L is Higgsed. Without loss of generality take |µ1 | > |µ2 | > |µ3 | > 0. The D-terms in the potential ensure that ϕi = ϕ̃i , and the F -terms for the Higgs sector are of the form VF = |h|2 |ϕi ϕ̃j − µ2 δji |2 + |h|2 |Φji ϕ̃j |2 + |h|2 |ϕi Φji |2 , (2.9) so that, as in the original ISS model, the Higgs vevs are at T ϕ = ϕ̃ = µ1 0 0 0 µ2 0 ! , (2.10) the non-zero F -term is FΦ33 = hµ23 , and the height of the potential at the metastable minimum is given by V (0) = |h2 µ43 |. (2.11) The vevs can be written succinctly using a block notation Φ= Y Z Z̃ X ! ; ϕT = –8– σ ρ ! ; ϕ̃ = σ̃ ρ̃ ! (2.12) where for example Y is a 2 × 2 matrix and X is (for the minimal assumption that Nf = 3) a single field. Defining flavour indices i = 1, 2 and colour indices a = 1, 2 the vevs are ! µ1 0 hσi = hσ̃i = 0 µ2 j hρa i = hρ̃a i = hY ii = hZii = hZ̃ii = 0 hXi = X0 . It is straightforward to identify those fields that gain a mass at tree level from the F -terms. Defining the eigenstates (for simplicity we take µ2i to be real in the remainder of this subsection) 1 1 ρa± = √ (ρ ± ρ̃∗ )a ; (σ± )ai = √ (δσ ± δσ̃)ai 2 2 (2.13) we find the F -term contributions to the mass-squareds at tree level to be m2ρa+ = h2 (µ2a + µ23 ) m2ρa− = h2 (µ2a − µ23 ) m2σ+ = h2 µi µa m2 i = h2 (µ2i + µ2j ) Y j m2Z = m2 i = h2 µ2i Z̃ i 2 2 mσ− = mX = 0 . (2.14) Note that the higher minima (i.e. those involving FΦ11 = hµ21 or FΦ22 = hµ22 instead of FΦ33 6= 0) are unstable to decay into this one since the ρ− and ρ̃− become tachyonic. Also in the case of degenerate µi these same states become additional Goldstone modes reflecting the enhanced flavour symmetry. The traceless part of the states Re(σ− ) are lifted by the D-terms, and the 3 traceless components of Im(σ− ) are eaten to become the longitudinal components of the W ± and Z. This leaves X and T r(σ− ) as pseudo-moduli, the latter being associated with the spontaneously broken but anomalous PQ symmetry. As in [1] one can now evaluate the one-loop contribution to these mass-squared and find that they are O(h4 µ2 /16π 2 ). We conclude that all these Higgs fields are massive and in the limit h ≫ 1 become very heavy. In this large-h limit all Higgs fields will decouple from physics at and around the electroweak scale. The PQ-axion is also removed from the spectrum by choosing the unitary gauge of the gauged PQsymmetry as we have explained above. To complete the discussion of the model around the metastable vacua, let us estimate the position of the Landau pole. This can be evaluated from the usual expression for the Wilsonian gauge coupling with a beta-function which, for the particle content listed, is b0 = −Nf : 2 − g8π(µ) e 2 =  µ ΛL –9–  Nf . (2.15) Taking Nf = 3, αSU (2) = 1/30 and µ = 100 GeV the Landau pole is found to be comfortably much greater than the Planck scale. Since at the Planck scale the physics is supposed to be modified anyway by the inclusion of gravity, knowledge of the electric theory is not even required. If extra SU (2) fields are introduced into the model or into the R-messenger sector, the number of flavours may be increased (as we shall see later). Two or more extra flavours implies that there is a ΛL below MP l . In what follows we will always assume that the UV completion of the theory would be required at ΛL ∼ MP l . 2.4 The global minima: dynamical restoration of SUSY Having established the existence of non-supersymmetric vacua, now we want to show that in this model there are also supersymmetric ground states. In these ‘true’ vacua supersymmetry will be restored dynamically as in the ISS model. To find these vacua we will first need to determine the dynamical superpotential of the model and then solve the resulting F-flatness equations. The behaviour of the SU (2)L group factor will of course be affected by the extra doublets of the Standard Model. The number of extra SU (2)L fundamentals is 12 (i.e. three generations each of L, and of Q × 3 colours); in order to be general we shall call this number n, and also will use SU (N ) for SU (2)L , 3(L + 3 × Q) = 12 := n , SU (2)L := SU (N ). (2.16) The first coefficient of the β-function of the SU (N ) gauge theory is b = 3N − 12 n − Nf . As our first step, we note that dynamical supersymmetry restoration requires vevs along more than just the Φ direction. Indeed giving large vevs to Φ gives masses to 2Nf fundamental fields (ϕi and ϕ̃i ), so the beta function at scales below these masses becomes b = 3N − 12 n. Since we wish to have N = 2 and n = 12 this gives a β-function for SU (2)L which is coincidentally zero and we conclude that there can be no dynamically generated term which is solely dependent on Φ. Clearly dynamical supersymmetry breaking requires that we integrate out more flavours to reverse the sign of the β function. In order to do this we must search along directions that give masses to the other fundamentals of SU (N ), but do not break SU (N ) and that are gauge invariant monomials (i.e. D-flat). As well as Φ itself, there are 42 independent monomials that one could consider U DD ; U U U EE ; U U DE, (2.17) where we have suppressed flavour indices [21]. In general, giving vevs to a combination of these directions will give masses to some of the flavours which can then be integrated out. In our model all fundamental matter fields (n of Q’s and L’s and 2Nf of ϕ’s and ϕ̃’s) will become massive along these directions and can be integrated out. In the IR the theory will become a pure SYM with the gauge group SU (N ). This gauge group confines, and the expression for the nonperturbative superpotential Wdyn of this IR theory is uniquely determined by the gaugino condensation to be Wdyn = N (ΛSU (N ) )3 . (2.18) Here ΛSU (N ) is the dynamical scale of the pure SYM. To derive the dynamical superpotential of the original theory with all matter fields present we use (2.18) and the matching relations for the Λ scales – 10 – of the theory below and above each mass threshold. In order to keep the discussion general, let us assume that the i’th set of fundamentals is integrated out at the scale Ei , whereupon the β function changes from bi → bi+1 . Specifically we take i = 0 at the highest scale so that b0 = b = 3N − 21 n − Nf , Λ0 = ΛL (2.19) and take E1 ≥ E2 ≥ . . . ≥ Ei ≥ Ei+1 ≥ . . . En to be the masses of each of the n matter fundamentals Q and L. The matching of Wilsonian gauge coupling constants at each mass threshold Ei gives e−8π 2 /g 2 (E i) = (Ei /Λi )−bi = (Ei /Λi−1 )−bi−1 b b −bi−1 i−1 Λbi i = Λi−1 Ei i => (2.20) These relations relate Λ0 = ΛL of the high-energy theory to the scale Λn of the gauge theory with all n fundamentals integrated out. We now need to integrate out the remaining 2Nf of ϕ and ϕ̃ matter fields to descend to the pure SYM with ΛSU (N ) . Masses of ϕ and ϕ̃ fields are set by the vevs of Φij so that the mass to the Nf power is given by (mϕ,ϕ̃ )Nf = hNf detNf hΦi (2.21) We thus obtain the following general expression for the dynamically generated superpotential Wdyn = N 3N −Nf −n/2 Nf h (detNf Φ) ΛL n Y (b −b ) Ei i i−1 i=1 !1 N (2.22) where bi = 3N − Nf − 12 (n − i). Note that the dependence on any particular energy scale is dependent only on the change in β-function from the states that are integrated out there, so that Wdyn behaves correctly if we let any of the thresholds coalesce. As long as the eventual β-function is positive (i.e. b0 is negative) at the high scale we are assured of generating such a dynamical superpotential, Wdyn . We can now solve the F-flatness condition for the superpotential,
∂ Wdyn − h(µi Φii ) = 0 ∂Φ to find hΦkk 1/(Nf −N )  Nf 1 Y 2  µ = 2 µk i=1 i 3N −Nf −n/2 ΛL n Y (b −b ) Ei i i−1 i=1 (2.23) !− 1 Nf −N , k = 1, . . . , Nf (2.24) Substituting back into W we find Wmin 1  Nc Nf Y 2 = Nc  µ i  i=1 3N −Nf −n/2 ΛL n Y (b −b ) Ei i i−1 i=1 !− 1 Nc , Nc := Nf − N (2.25) As in the original ISS model we appear to be running to Nc minima in Φ. Here Nc does not necessarily have an intrinsic meaning beyond its definition Nc = Nf − N , but we will use it in formulae below. – 11 – We now replace the thresholds Ei with holomorphic fields corresponding to the masses induced by non-zero vevs of gauge invariant monomials. We shall consider an R-parity conserving theory in order to avoid proton-decay, in which case the dynamically generated superpotential can only depend on the 27 U U DE monomials. Turning on a vev in this direction, which we shall refer to as X14 , X14 = U U DE (2.26) generates masses for the quark and lepton fields via the standard Yukawa couplings of the MSSM. These masses are given by the |Fϕ |2 and |Fϕ̃ |2 terms in the potential, and are EQ ≡ q 2|λ2U | + |λ2D | X1 (2.27) EL ≡ λE X1 (2.28) With this prescription the potential is a runaway to large values of X1 . However this is because as it stands the potential is nongeneric (in the sense of Nelson and Seiberg [14]). Consider making the λ U U DE where M & ΛL is a high superpotential generic by adding to it the nonrenormalizable term M mass scale, which is allowed by all the symmetries of the theory including R-parity (note that the fact that X1 appears in Wdyn means that this term must have been allowed in the original superpotential). In order to satisfy constraints from proton decay, we shall implicitly assume that M ∼ MP l . To simplify the discussion let us integrate out all n of the SU (N ) fundamentals at the scale λt X1 (in other words, for simplicity of presentation we will not distinguish between the different Yukawa couplings and we will also set all µi = µ) Wmin = Nc µ 2Nf Nc − 1  λ 4 3N −Nf −n/2 Nc (λt X1 )n/2 ΛL X + M 1 (2.29) Minimizing this potential we find the supersymmetric minima at hX1 i =  nM 8λt  Nc n/2+3Nc −2Nf 2Nf −n/2 µ λt ΛL ! 1 4Nc +n/2 . (2.30) To be inside the Landau-pole circle we would like X1 to be less than ΛL ; we may rewrite it as hX1 i = ΛL  n 8λt  Nc −n/2 λt ! 1 4Nc +n/2 2Nf −Nc ǫ 4Nc +n/2 ξ 4Nc +n/2 (2.31) where ǫ = µ/ΛL (2.32) ξ = ΛL /M. (2.33) 2Nf Setting all the Yukawas to be of order unity, the constraint hX1 i < ΛL translates into ξ > ǫ Nc . This is easy to achieve as ǫ is naturally ≪ 1 and ξ can be . 1. The second requirement is that – 12 – µ ≪ hΦi < ΛL where the first inequality is needed to ensure that hΦi is far away from the origin and there is no tunnelling to the metastable vacuum. From (2.24) we have −1 hΦi ∼ h µ 2N Nc − 1  3N −Nf −n/2 Nc n/2 hX1 i ΛL In our model N = 2, Nf = 3, Nc = 1 and n = 12 and we have  0.4  0.6 µ ΛL −1 hΦi ∼ ΛL h ≪ ΛL ΛL M  0.6  0.6 ΛL −1 ΛL ≫µ hΦi ∼ µ h µ M (2.34) (2.35) (2.36) and both inequalities can be easily satisfied for not too large h. At very large values of h the last equation, however, may not be valid and we will discuss this case in the next subsection. For now we conclude that any generic superpotential (i.e. one which includes nonrenormalizable operators allowed by the symmetries), dynamically restores the supersymmetry at a scale below the Landau pole. Of course one may worry that the nonrenormalizable operator is generated by physics at a scale M & ΛL which is the region outside of validity of the macroscopic theory. However this is perfectly consistent; the microscopic theory is expected to generate all operators allowed by the symmetries, and even if they are nonrenormalizable, they can be simply rewritten in terms of the fundamental fields of the macroscopic theory. 2.5 (Lack of ) tunnelling out of the metastable vacuum when h ≫ 1 One question that we should clarify is metastability at h ≫ 1. As has been argued in the literature, for h ∼ 1 it is rather easy to find lifetimes for the metastable vacua that are longer than the age of the Universe. The criterion is that there should be less than one tunnelling event in the past light cone of the Universe which translates into SE & 400 (2.37) where SE is the Euclidean bounce action. As a rule-of-thumb, the latter is given by SE ∼ 2π 2 hΦi4 , ∆V (2.38) so that potentials that are wider than they are tall have longer lifetimes. In the pure ISS model this gives a not very severe bound on hΦi. In detail one finds that SE = which translates into 2π 2 N 3 hΦi4 , 3h2 Nf2 µ4 √ hΦi & 3 h. µ – 13 – (2.39) (2.40) However as we have seen hΦi scales as µh−1 times by some large dimensionless functions. In pure ISS we find (3N−N ) f 2π 2 −4 (Nf −N) ≫ 400. ǫ 6 3h (2.41) where ǫ = µ/ΛL . In fact the power of h−6 can be found by a simple scaling argument without even evaluating the action; defining Φ̂ = hΦ and x̂ = h2 x in the bounce action one finds SE (h) = h−6 SE (1). Now for certain values of N and Nf this can make the bounds far more restrictive for large h. If for 9 example Nf = 5N/2 then one requires ǫ ≪ 0.04 h− 2 which can be a severe bound on ΛL . We now find a pleasant surprise for our model. As we observed, SUSY restoration involves other flat directions of the MSSM. Although the supersymmetric minima are indeed found at small values of hΦi as h ≫ 1, the vevs along the other flat directions are independent of h, since they depend only on the thresholds induced by the Yukawa couplings of the MSSM-sector. Hence at large h the rule-of-thumb still applies but with hΦi replaced by the vevs of the other flat directions which as we have seen can be as large as ΛL . Thus large h destroys the metastability in the pure ISS model, but in our model it does not. 3. R-symmetry breaking In order to generate Majorana masses for gauginos we need to break the R-symmetry of the model. The easiest way to do this is by adding an appropriate higher-dimensional operator to the superpotential. The unique leading-order operator of this type is of the form WR = 2 gA T r(Φ) α A WA Wα , 2 16π mR (3.1) where WA is the field-strength chiral superfield of the gauge field of type A (where A = 1, 2, 3 distinguishes between the different types of gauge fields, SU (3) colour, U (1)Y or SU (2)L , in the Standard 2 /(16π 2 ) comes from the fact that this operator is generated radiatively, and Model); the factor of gA mR is the mass scale of the R-symmetry violating effects. In writing down (3.1) we have made use of the fact that the T r(Φ) is a gauge singlet field and as such can be coupled to the combination WAα WαA . This operator breaks R-symmetry since Φ has R-charge 2, and each of the W fields has R-charge equal to 1. It can be straightforwardly generated by additional massive fields which couple to the gauge fields, which can either be integrated out or included as new degrees of freedom in the low energy theory. An example of the latter is discussed in the Appendix, where it is shown how the operator (3.1) can be generated at one-loop. In the remainder of the paper we will simply assume the presence of the R-symmetry violating operator WR . – 14 – 3.1 Generation of gaugino masses The superpotential (3.1) generates Majorana masses MλA for the gauginos λA via Z g2 T rhFΦ i g2 T r(Φ) α A WA Wα ∋ A 2 λA λA . d2 θ A 2 16π mR 16π mR (3.2) The vev for the F -term for Φ is non-vanishing in the metastable vacuum and is given by hFΦ33 i = −hµ23 , (3.3) which determines the gaugino masses to be, MλA = h 2 gA µ23 . 16π 2 mR (3.4) It is remarkable that we are able to obtain these SUSY-breaking gaugino masses (3.4) from the manifestly supersymmetric superpotential (3.1), (2.6) directly and without adding any new degrees of freedom to our model. The gaugino masses are generated by TrhFΦ i. The fact that TrhFΦ i = 6 0 is of course the consequence of the rank condition and is the key feature of our metastable model which is responsible for SUSY breaking. To get a rough estimate of the values of gaugino masses we can take mR ∼ µ3 ∼ 100GeV and h ∼ 16π 2 /g 2 ≫ 1. Then Mλ ∼ µ ∼ 100GeV. As we have already anticipated in the Introduction, we see that in order to get sizable gluino masses we have to assume that the Higgs sector of our model is sufficiently strongly coupled, i.e. h ∼ 16π 2 /g2 ≫ 1. In the following subsection we will argue that this requirement of h ≫ 1 does not render the theory incalculable, instead it actually simplifies it at energy scales not much above the electroweak scale (i.e. below the Higgs mass scale MH ∼ hµ ≫ MW ∼ gµ). 3.2 Theory at large h and decoupling of Higgses Clearly the perturbation theory in powers of h breaks down when h ≫ 1. Here we would like to point out that this fact does not necessarily render our model incalculable. Instead it signals the decoupling of all the Higgs fields from physics at scales low compared to their masses, i.e. at the electroweak scale. The primary effect of large h is to ensure that all the Higgs masses are very large. We have already seen that treated in perturbation theory these masses receive their first non-vanishing contributions h2 at the order hµ or 16π 2 µ. There of course can be significant corrections to these masses from higher orders of perturbation theory in h. We will assume that the full masses of all the Higgses stay large, i.e. of the order MHiggs ∼ hµ or higher as h ≫ 1. When estimating quantum corrections from these heavy Higgs fields on internal lines to Feynman diagrams for various processes at energies below MHiggs one would be required to use the full (resummed) masses. It is then easy to see by power counting that the whole Higgs sector decouples from – 15 – the Standard Model physics at these energy scales. The h2 enhancement of the Higgs self-interactions as in (2.9) will be overcome by the factors of h2 from the Higgs masses in the propagators. 4. Discussion We have presented an extremely compact formulation of the visible sector supersymmetry breaking in a supersymmetric version of the Standard Model. Supersymmetry breaking is a consequence of the ISS-type metastable vacuum in our model guaranteed by the superpotential WHiggs = h T r[ϕΦϕ̃ − µ2 Φ]. (4.1) The Yukawa interactions allowed by the symmetries of the model are precisely of the minimal standard form WY uk = λU Qϕ2 U + λD Qϕ1 D + λE Lϕ1 E. (4.2) The SUSY breaking induced by the rank condition of (4.1) is communicated to gauginos via the R-symmetry breaking effective superpotential term WR = 2 gA T r(Φ) α A WA Wα . 2 16π mR (4.3) In our model both supersymmetry and the electroweak gauge symmetry is broken by the same parameters µi ∼ 100 GeV. More precisely, vector bosons get their masses from µ1 and µ2 while SUSY is broken by µ3 where |µ3 | < |µ1,2 |. This provides us with a non-trivial direct link between SUSY breaking and the electroweak breaking. The price we have to pay for this very low SUSY breaking scale is that we have to have strong coupling in the Higgs sector, h ≫ 1, in order to generate sufficiently large gaugino masses from (4.3), (4.1). However, we have argued that the primary role of large-h is to make all the Higgs fields very heavy compared to the electroweak scale. At such energy scales all the Higgs fields become non-dynamical, frozen at their vevs, and essentially the whole Higgs sector decouples from the theory. The masses for squarks and sleptons in our model are generated from the gaugino masses above at one-loop level in a way which is completely analogous to the gauge mediation scenarios [23], [18,19]. In conclusion, we have constructed an extremely compact model of spontaneous supersymmetry breaking in the visible sector which gives a softly broken supersymmetric Standard Model as the low-energy effective theory. Acknowledgements We thank Sakis Dedes, Ken Intriligator, Joerg Jaeckel and Stefan Forste for useful conversations. – 16 – APPENDIX: R-symmetry breaking with new massive fields In the case that R-symmetry is broken by fields which remain in the low energy theory, it may be necessary to add additional flavours (i.e. make Nf = 4) in order to enhance the rank condition and preserve metastability. As an example, consider the following addition to the superpotential (as in Ref. [10]): κ T r(Φ) f˜ · f + mf f˜ · f , (A-1) where f and f˜ are the new R-neutral fields transforming in the fundamental and anti-fundamental representations of the gauge group of type A, and κ and mf are constants. In (A-1) they are coupled to the gauge singlet T r(Φ) and they have an R-symmetry violating mass-term mf . It is easy to see that at one-loop the f and f˜ fields indeed give contributions of the type (3.1). They remain in the spectrum as massive fields coupled to the gauge supermultiplets of type A, e.g. to gluons and gluinos. Adding f ,f˜ to the theory is the simplest way of generating operators of the form eq.(3.1), but one should worry about the fact that the new R-symmetry breaking fields f and f˜ in (A-1) interfere with the rank condition (2.2), possibly destroying the metastability of the non-supersymmetric vacuum. In the minimal model with Nf = 3, the F -term hFΦ33 i can in principle be set to zero (and invalidate p the rank condition) by turning on large vevs hf i = hf˜i = f0 = h/κ µ3 , corresponding to a new supersymmetric vacuum a distance f0 along the f -direction from the metastable vacuum at the origin. This breakdown of the rank condition is easily avoided by the simple expedient of adding a fourth generation of ϕ, ϕ̃. Note that this does not introduce any new Goldstone modes, since the new global U (1) symmetry is exact and unbroken (just like the U (1)3 before), and neither does it introduce new Yukawa couplings with the SM matter fields since the new ϕ4 , ϕ̃4 fields are neutral under hypercharge. It easily follows in Nf = 4 models, that there are no (perturbative) supersymmetric vacua and ISS-type metastability is completely restored. In fact in this case there are two candidates for the metastable vacua. The first is similar to the one we have already explored, and is characterized by hf.f˜i = 0 hΦij i = 0  µ21  0  hϕi ϕ̃j i =   0 0 0 µ22 0 0 0 0 0 0 0 0 0 0    ,  (A-2) with hFΦ33 i = hµ23 and hFΦ44 i = hµ24 . The vacuum energy of this minimum is V+ = |h|2 (|µ3 |4 + |µ4 |4 ), where we have ordered |µ1 | > |µ2 | > |µ3 | > |µ4 | > 0. We should ensure that there are no tachyonic directions around this vacuum. Along directions orthogonal to f and f˜, the analysis of [1] ensures that all mass-squareds are positive. Along the f and f˜ directions themselves, diagonalizing their mass matrix yields5 mass eigenstates ξ± = √12 (f ± eiθ f˜∗ ) 5 Assuming κh to be real, without loss of generality. – 17 – where θ = Arg[µ23 + µ24 ]. Their mass-squareds are m2ξ± = m2f ∓ κh|µ23 + µ24 |. The dominant contribution to the effective operator (3.1), g2 T r(Φ) α A WA Wα , (A-3) WR = κ A 2 16π mR comes from the loop with the lightest eigenstate ξ− circulating in the loop, and this determines mR in the denominator of (A-3) to be q mR = mξ− = m2f − hκ |µ23 + µ24 | . (A-4) To maximize the contribution of the operator (A-3) to the gluino masses, as in subsection 3.1, we √ take mR ∼ µ ∼ 100 GeV and hκ ≫ 1. This implies that mf ∼ hκ µ. The second candidate vacuum is characterized by h 2 (µ + µ24 ) 2κ 3 mf hΦ33 i + hΦ44 i = −  κ µ21 − 21 (µ23 + µ24 ) 0  1 2 0 µ2 − 2 (µ23 + µ24 )  hϕi ϕ̃j i =   0 0 0 0 hf.f˜i = 0 0 0 0 0 0 0 0      (A-5) There are two vanishing FΦii -terms, hFΦ11 i = hFΦ22 i = 0, and two nonvanishing F -terms, hFΦ33 i = −hFΦ44 i = h (µ24 − µ23 ) , 2 (A-6) where the µ′ s are now ordered so that these are the least possible F -terms. (Note that here we do not 2 necessarily require |µ1 | > |µ2 | > |µ3 | > |µ4 |). The vacuum energy is V+ new = |h|2 |µ23 − µ24 |2 . Generally, this configuration breaks whatever symmetry f and f˜ couple to, so is certainly not a desirable vacuum state if f and f˜ are charged under SU (3)c . Thus for the purposes of this paper we want the system to be in the first vacuum, but the second 2 vacuum (A-5) always has lower energy since V+ = |h|2 (|µ3 |4 + |µ4 |4 ) ≥ Inf( |h|2 |µ2i − µ2j |2 ). Therefore one should ensure that the decay time is sufficiently long. The tunnelling rate from the first to the second vacuum is indeed suppressed due to a large separation between the two vacua in the f - and in the Φ-directions. A simple estimate of the bounce action gives SE ∼ 2π 2 4π 2 hfieldsi4 ≈ 2 ∆V κ (A-7) where hfieldsi denotes the separation in the field space between the two vacua. In our case it is p ∼ h/κ µ for both, the f - and the Φ-directions. SE is easily made & 400 to suppress tunnelling by choosing κ . 0.3. Note that this conclusion would have been similar if we had stayed with Nf = 3, the only difference being that the second vacuum would have been supersymmetric. With Nf = 4 all perturbative vacua are metastable. – 18 – Thus we have demonstrated that the system can be trapped in the metastable vacuum of the first type where the effective operator WR in (A-3), is generated. 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