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ISRN High Energy Physics
Volume 2012 (2012), Article ID 903106, 31 pages
http://dx.doi.org/10.5402/2012/903106
Research Article

Supersymmetry Breaking in a Minimal Anomalous Extension of the MSSM

1Dipartimento di Fisica dell'Università di Roma “Tor Vergata”, INFN Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy
2National Institute of Chemical Physics and Biophysics, Ravala 10, 10143 Tallinn, Estonia

Received 20 April 2012; Accepted 4 June 2012

Academic Editors: J. R. Espinosa and A. Koshelev

Copyright © 2012 A. Lionetto and A. Racioppi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a supersymmetry breaking mechanism in the context of a minimal anomalous extension of the MSSM. The anomaly cancellation mechanism is achieved through suitable counterterms in the effective action, that is, the Green-Schwarz terms. We assume that the standard MSSM superpotential is perturbatively realized; that is, all terms allowed by gauge symmetries except for the μ-term which has a nonperturbative origin. The presence of this term is expected in many intersecting D-brane models which can be considered as the ultraviolet completion of our model. We show how soft supersymmetry breaking terms arise in this framework, and we study the effect of some phenomenological constraints on this scenario.

1. Introduction

The LHC era has begun and the high-energy physics community is analyzing and discussing the first results. One of the key goals of LHC, beside shedding light on the electroweak (EW) symmetry breaking sector of the standard model (SM), is to find some signature of physics beyond the SM. Supersymmetric particles and extra neutral gauge bosons 𝑍 are widely studied examples of such signatures. A large class of phenomenological and string models aiming to describe the low-energy physics accessible to LHC predict the existence of additional abelian 𝑈(1) gauge groups as well as 𝑁=1 supersymmetry softly broken roughly at the TeV scale. In particular in string theory the presence of extra anomalous 𝑈(1)’s seems ubiquitous. D-brane models in orientifold vacua contain several abelian factors, and they are typically anomalous [157]. In [58] we studied a string-inspired extension of the (minimal supersymmetric SM) MSSM with an additional anomalous 𝑈(1) (see [5967] for other anomalous 𝑈(1) extensions of the SM and see [6871] for extensions of the MSSM). The term anomalous refers to the peculiar mechanism of gauge anomaly cancellation [7274] which does not rely on the fermion charges but rather on the presence of suitable counterterms in the effective action. These terms are usually dubbed as Green-Schwarz (GS) [5967, 7577] and generalized Chern-Simons (GCS) [7885]. They can be considered as the low-energy remnants of the higher-dimensional anomaly cancellation mechanism in string theory. In our model we assumed the usual MSSM superpotential and soft supersymmetry breaking terms allowed by the symmetries (the well-known result [86]). In this paper we address the question of the origin of the latter in the context of a global supersymmetry breaking mechanism. This means that we do not rely on a supergravity origin of the soft terms but rather on a local setup based, for example, on intersecting D-brane constructions in superstring theory in which gravity is essentially decoupled (see, for instance, [87] for, a recent attempt in this direction). Moreover in [58] we made the assumption that all the MSSM superpotential terms were perturbatively realized, that is, allowed by the extra abelian 𝑈(1) symmetries. In the following we assume instead that the 𝜇-term is perturbatively forbidden. The origin of this term is rather non-perturbative and can be associated to an exotic instanton contribution which naturally arises from euclidean D-brane in the framework of a type IIA intersecting brane model (see [88] and references therein).

The paper is organized as follows: in Section 2 we describe the basic setup of the model and we discuss the perturbative and non-perturbative origin of the superpotential terms. We argue how the latter can naturally come from an intersecting D-brane model considered as the ultraviolet (UV) completion of our model. In Section 3 we describe the (global) supersymmetry breaking mechanism that gives mass to all the soft terms. In Section 5 we compute the gauge vector boson masses while in Section 4 we study the scalar potential of the theory in the neutral sector. In Section 6 we describe the neutralino sector while in Section 7 we describe the sfermion mass matrices. In Section 8 we study the phenomenology of our model and the bounds that can be put by some experimental constraints. Finally in Section 9 we draw our conclusions.

2. Model Setup

The model is an extension of the MSSM with two extra abelian gauge groups, 𝑈(1)𝐴 and 𝑈(1)𝐵. The first one is anomalous while the second one is anomaly-free. This assumption is quite generic since in models with several anomalous 𝑈(1) symmetries there exists a unique linear combination which is anomalous while the other combinations are anomaly-free. The charge assignment for the chiral superfields is shown in Table 1. The vector and matter chiral multiplets undergo the usual gauge transformations 𝑉𝑉+𝑖ΛΛ,Φ𝑒𝑖𝑞ΛΦ.(2.1) The anomaly cancellation of the 𝑈(1)𝐴 gauge group is achieved by the four-dimensional analogue of the higher-dimensional GS mechanism which involves the Stückelberg superfield 𝑆=𝑠+2𝜃𝜓𝑆+𝜃2𝐹𝑆 transforming as a shift 𝑆𝑆2𝑖𝑀𝑉𝐴Λ,(2.2) where 𝑀𝑉𝐴 is a mass parameter related to the anomalous 𝑈(1)𝐴 gauge boson mass. It turns out that not all the anomalies can be cancelled in this way. In particular the so-called mixed anomalies between anomalous and non-anomalous 𝑈(1)’s require the presence of trilinear GCS counterterms. For further details about the anomaly cancellation mechanism, see Appendix A (see also, for instance, [58] and [8082]). The effective superpotential of our model at the scale 𝐸=𝑀𝑉𝐴 is given by 𝑊=𝑊MSSM+𝜆𝑒𝑘𝑆𝐻𝑢𝐻𝑑+𝑚Φ+Φ,(2.3) where 𝑊MSSM is given by 𝑊MSSM=𝑦𝑢𝑖𝑗𝑄𝑖𝑈𝑐𝑗𝐻𝑢𝑦𝑑𝑖𝑗𝑄𝑖𝐷𝑐𝑗𝐻𝑑𝑦𝑒𝑖𝑗𝐿𝑖𝐸𝑐𝑗𝐻𝑑,(2.4) which is the usual MSSM superpotential without the 𝜇-term which is forbidden for a generic choice of the charges 𝑞𝐻𝑢 and 𝑞𝐻𝑑. The second term in (2.3) is the only gauge invariant coupling allowed between the Stückelberg superfield and the two Higgs fields. This is the only allowed coupling with matter fields for a field transforming as (2.2). We will argue later about how nonperturbative effects can generate such a term. The last term in (2.3) is a mass term for Φ± which are charged under both 𝑈(1)𝐴 and 𝑈(1)𝐵. These fields have been considered as supersymmetry breaking mediators in the context of anomalous models by Dvali and Pomarol [89]. They play a key role in generating gaugino masses. In the effective lagrangian, beside the usual kinetic terms (they are charged under both 𝑈(1)𝐴 and 𝑈(1)𝐵), the two 𝑈(1)𝐵 fields Φ± couple to the gauge field strength 𝑊𝛼𝑎 through the dimension six effective operator 𝑔=𝑐𝑎Φ+ΦΛ2𝑊𝛼𝑎𝑊𝛼𝑎,(2.5) where 𝑎=𝐴,𝐵,𝑌,2,3, Λ is the cut-off scale of the theory while 𝑐𝑎 are constants that have to be computed in the UV completion of the theory.

tab1
Table 1: Charge assignment.

The non perturbative term in (2.3) is expected to be generated in the effective action of intersecting D-brane models which can be considered as the UV completion of our model. This is the leading order term when the coupling 𝐻𝑢𝐻𝑑 is not allowed by gauge invariance. In string theory there are many axions related to the GS mechanism of anomaly cancellation which are charged under some Ramond-Ramond (RR) form. For example, in type IIA orientifold model with D6-branes, axion fields are associated to the 𝐶3 RR-form (see for a recent review [90]). Instantons charged under this RR-form, such as Euclidean E2-branes wrapping some 𝛾3 3-cycle in the Calabi-Yau (CY) compactification manifold, give a contribution to the holomorphic couplings in the 𝑁=1 superpotential. Our analysis does not rely on any concrete intersecting brane model but rather on the generic appearance of such instanton induced terms. The exponential suppression factor of the classical instanton action is 𝑒Vol𝐸2/𝑔𝑠,(2.6) where Vol𝐸2 is the volume of the 3-cycle in the CY wrapped by a 𝐸2-brane measured in string units while 𝑔𝑠 is the string coupling. Such exponential factor is independent of the 𝑑=4 gauge coupling, and thus this instanton is usually termed as stringy or exotic instanton (see [88, 91] and references therein). Moreover the instanton contribution can be sizable even in the case 𝑔𝑠=1 if Vol𝐸2=1 measured in string units.

In type IIA orientifold models with intersecting branes, the complexified moduli, whose imaginary part are the generalized axion fields (depending on the cycle 𝛾𝑖3), can be written as 𝑈𝑖=𝑒𝜑𝛾𝑖3Ω3+𝑖𝛾𝑖3𝐶3,(2.7) where 𝜑 is the dilaton, Ω3 is the CY volume 3-form (which is a complex form), and 𝐶3 is the RR-form. The integral of this form is dual to the axion whose shift symmetry is gauged in the GS mechanism. The generic contribution of an 𝐸2 instanton is formally given by 𝑊𝑛𝑖=1Φ𝑎𝑖,𝑏𝑖𝑒𝑆𝐸2,(2.8) where Φ𝑎𝑖,𝑏𝑖 are chiral superfields localized at the intersection of two D6-branes described by open strings while 𝑆𝐸2 denotes the instanton classical action: 𝑒𝑆𝐸2=exp2𝜋𝑙3𝑠1𝑔𝑠𝛾ΩRe3𝑖𝛾𝐶3.(2.9) This result can be immediately extended to the supersymmetric case which involves the complete Stückelberg multiplet. The appearance of the exponential suppression factor is dictated by the fact that the superpotential is a holomorphic quantity. Thus the only allowed functional dependence on the string coupling 𝑔𝑠=𝑒𝜑 and the axionic field is an exponential. Any other dependence can be excluded due to the shift transformation (2.2).

3. Supersymmetry Breaking

The D-term contribution of the 𝑈(1)𝐴 vector multiplet 𝑉𝐴 relevant to supersymmetry breaking is given, in the limit of vanishing kinetic mixing 𝛿𝑌𝐴,𝛿𝐴𝐵=0, by the following lagrangian: 1=2𝐷𝐴𝐷𝐴+𝑖𝑔𝐴𝑞𝑖𝜙𝑖𝐷𝐴𝜙𝑖+𝜉𝐷𝐴,(3.1) where the sum is extended to all the scalars charged under the 𝑈(1)𝐴. There is no D-term contribution related to the 𝑈(1)𝐵 except that of 𝜙± since all the MSSM chiral fields are uncharged under 𝑈(1)𝐵 (see Table 1). The last term in (3.1) is a tree-level field-dependent Fayet-Iliopoulos (FI) term which comes from the supersymmetrized Stückelberg lagrangian axion=14𝑆+𝑆+2𝑀𝑉𝐴𝑉𝐴2|||𝜃2𝜃2+=𝑀𝑉𝐴𝑆+𝑆𝑉𝐴||𝜃2𝜃2+=𝑀𝑉𝐴𝛼𝐷𝐴+,(3.2) where in the last line 𝛼 denotes the real part of the lowest component of the Stückelberg chiral multiplet 𝑠=𝛼+𝑖𝜑. The fields 𝛼 and 𝜑 are called the saxion and the axion, respectively (with a slight abuse of notation with respect to the previous section where we denoted the dilaton with 𝜑). We assume that the real part 𝛼 gets an expectation value. This gives a contribution to the gauge coupling constants which can be absorbed in the following redefinition: 116𝑔2𝑎𝜏𝑎=116̃𝑔2𝑎𝜏𝑎12𝑏𝑎𝑎𝛼,(3.3) where the gauge factors 𝜏𝑎 take the values 1,1,1,1/2,1/2, and the 𝑏𝑎𝑎 constants are given in (A.2). The tree-level FI term is then given by 𝜉=𝑀𝑉𝐴𝛼.(3.4) Moreover in the following we assume that 1-loop FI terms are absent (see the discussion in [92]). The FI term induces a mass term for the scalars. This can be seen by solving the equations of motion for 𝐷𝐴: 𝐷𝐴+𝑖𝑔𝐴𝑞𝑖𝜙𝑖𝜙𝑖+𝜉=0,(3.5) where the index 𝑖 runs over all chiral superfields. The D-term contribution to the scalar potential is given by 𝑉𝜙𝑖,𝜙𝑖=12𝜉+𝑔𝐴𝑖𝑞𝑖||𝜙𝑖||22.(3.6) The quadratic part gives the scalar mass term 𝑖𝜉𝑔𝐴𝑞𝑖||𝜙𝑖||2=𝑖𝑚2𝑖||𝜙𝑖||2,(3.7) where we have defined 𝑚2𝑖=𝜉𝑔𝐴𝑞𝑖=𝛼𝑔𝐴𝑀𝑉𝐴𝑞𝑖=𝑞𝑖𝑚2𝜉,(3.8) with 𝑚2𝜉=𝛼𝑔𝐴𝑀𝑉𝐴=𝑔𝐴𝜉.(3.9) The typical scale for the mass 𝑚𝜉 is of the order of few hundreds of GeV if 𝑀𝑉𝐴𝛼1 TeV and 𝑔𝐴0.1. It is interesting to note that in this scenario a low sub-TeV supersymmetry breaking scale 𝑚𝜉 is due to the Stückelberg mechanism which gives mass to 𝑉𝐴. This is the most important difference with the scenario proposed in [89], where the scale 𝑚𝜉 is dynamically generated by some dynamics in a strong coupling regime.

Mass terms for the gauginos, that is, 𝜆𝑎𝜆𝑎, are generated by the dimension six effective operator (2.5) in the broken phase where 𝜙± get vacuum expectation value (vev). The contribution coming from this mechanism is 𝑀𝑎=𝑐𝑎𝐹+𝜙+𝐹𝜙+Λ2=𝑐𝑎𝑚𝑣2++𝑣22Λ2,(3.10) where 𝑣±/2=𝜙± and where in the right-hand side we have used the F-term equations of motion for 𝐹±𝐹±=𝜕𝑊𝜕𝜙±=𝑚𝜙,(3.11) having assumed 𝑚 real without any loss in generality. We assume 𝑐𝑎=𝑐 for each 𝑎. This is an assumption of universality as a boundary condition at the cut-off scale Λ which does not affect in a crucial way our analysis. In Section 4 we study the scalar potential of our model and we derive the conditions for having a vev for 𝜙± different from zero. Since we are breaking supersymmetry in the global limit in which the Planck mass 𝑀𝑃, the F-term induced contribution to the scalar masses 𝑚2𝑖𝐹±𝑀2𝑃(3.12) vanishes leaving (3.8) as the leading contribution.

The requirement of gauge invariance of the superpotential implies the following constraints on the 𝑈(1)𝐴 charges 𝑞𝑈𝑐=𝑞𝑄𝑞𝐻𝑢,𝑞𝐷𝑐=𝑞𝑄𝑞𝐻𝑑,𝑞𝐸𝑐=𝑞𝐿𝑞𝐻𝑑,𝑞(3.13)𝑘=𝐻𝑢+𝑞𝐻𝑑2𝑀𝑉𝐴.(3.14) As we said at the beginning of this section, we assume that the net kinetic mixing between 𝑈(1)𝑌 and 𝑈(1)𝐴 vanishes (we postpone the discussion about the kinetic mixing between 𝑈(1)𝐴 and 𝑈(1)𝐵 to the next section). There are two contributions for the 𝑈(1)𝑌𝑈(1)𝐴 kinetic mixing: the 1-loop mixing 𝛿𝑌𝐴 and 𝑏𝑌𝐴 coming from the GS coupling 𝑆𝑊𝑌𝑊𝐴 (see (A.1)). The following conditions imply a bound on the charges 𝛿𝑌𝐴=0𝑓𝑞𝑓𝑌𝑓𝑏=0,𝑌𝐴=0𝑓𝑞2𝑓𝑌𝑓=0,(3.15) where the sum is extended over all the chiral fermions in the theory. Constraints (3.15), can be solved in terms of 𝑞𝑄 and 𝑞𝐿. By using conditions (3.13), we get 𝑞𝐿=143𝑞𝐻𝑢4𝑞𝐻𝑑,𝑞𝑄1=125𝑞𝐻𝑢2𝑞𝐻𝑑.(3.16) The positive squared mass condition for the sfermions 𝑚2𝑓=𝑔𝐴𝑞𝑓𝑀𝑉𝐴𝛼>0,(3.17) implies 𝑞𝑓>0 for all the sfermions having assumed without loss of generality 𝛼>0. Using the constraints (3.13) and (3.16), we get the allowed parameter space 𝑞𝐻𝑢5<0,2𝑞𝐻𝑢<𝑞𝐻𝑑<34𝑞𝐻𝑢.(3.18)

4. Scalar Potential

The key ingredient in our model is the instanton-induced term in (2.3) which couples the Stückelberg field to the Higgs fields. The 𝜃2 component of this superpotential term gives the following contribution to the Lagrangian: 𝑊inst||𝜃2=𝜆𝑒𝑘𝑆𝐻𝑢𝐻𝑑||𝜃2=𝜆𝑒𝑘𝑠𝑢𝐹𝑑+𝜆𝑒𝑘𝑠𝐹𝑢𝑑𝜆𝑘𝑒𝑘𝑠𝐹𝑆𝑢𝑑+2𝜆𝑒𝑘𝑠𝑘𝑢𝜓𝑆𝑑+𝑑𝜓𝑆𝑢𝜆𝑒𝑘𝑠𝑢𝑑𝜓𝑆𝜓𝑆,(4.1) where 𝐹𝑢,𝑑 are the F-terms of 𝐻𝑢,𝑑. Solving the F-terms equations for 𝐻𝑢 and 𝐻𝑑, we get the following contributions for the instanton-induced term in the scalar potential: 𝑉inst=2𝜆2𝑒2𝑘𝛼𝑢𝑢+2𝜆2𝑒2𝑘𝛼𝑑𝑑+𝜆𝑘𝑒𝑘𝛼𝑒𝑖𝑘𝜑𝐹𝑆𝑢𝑑.+h.c.(4.2) In the following we assume that 𝛼 gets a vev different from zero and that the mass of this field is much higher than Λ so that its dynamics is not described by the low-energy effective action. From the point of view of the UV completion (e.g., a type IIA intersecting brane model), this amounts to saying that the closed string modulus related to 𝛼 is stabilized. Moreover we made the assumption that the same dynamics that stabilizes 𝛼 also fixes 𝐹𝑆. By supersymmetry the saxion field 𝛼, being part of the Sẗuckelberg multiplet, has a tree-level mass 𝑀𝑉𝐴. Thus if we want to consider a frozen dynamics for 𝛼 at the TeV scale we have to assume a mass parameter for the anomalous 𝑈(1)𝐴 just slightly above the TeV scale; that is, 𝑀𝑉𝐴>1 TeV. In this way the effective instanton-induced potential at a scale 𝐸1 TeV is thus given by 𝑉inst=2𝜆2𝑒2𝑘𝛼𝑢𝑢+2𝜆2𝑒2𝑘𝛼𝑑𝑑+𝜆𝑘𝑒𝑘𝛼𝐹𝑆𝑒𝑖𝑘𝜑𝑢𝑑.+h.c.(4.3) The first two terms are 𝜇-terms while the third one is a b-term. The complete effective scalar potential is given by ||𝜇||𝑉=2+𝑚2𝑢||0𝑢||2+||+𝑢||2+||𝜇||2+𝑚2𝑑||0𝑑||2+||𝑑||2+|𝑚|2+𝑚2𝜙+||𝜙+||2+|𝑚|2+𝑚2𝜙||𝜙||2+𝑏𝑒𝑖𝑘𝜑+𝑢𝑑0𝑢0𝑑+1+h.c.8𝑔22+𝑔2𝑌|0𝑢|2+|+𝑢|2|0𝑑|2|𝑑|22+12𝑔22||+𝑢𝑑0+0𝑢𝑑||2+12𝑔2𝐴𝑞𝐻𝑢||0𝑢||2+||+𝑢||2+𝑞𝐻𝑑||0𝑑||2+||𝑑||2+||𝜙+||2||𝜙||22+12𝑔2𝐵||𝜙+||2||𝜙||22,(4.4) where 𝜇=2𝜆𝑒𝑘𝛼,(4.5)𝑏=𝜆𝑘𝑒𝑘𝛼𝐹𝑆.(4.6) These relations give a solution of the well-known 𝜇-problem since both terms have a common origin (see the analysis in Section 8.2). The soft squared masses are generated by the FI 𝑈(1)𝐴 term: 𝑚2𝑢=𝑞𝐻𝑢𝑚2𝜉,𝑚(4.7)2𝑑=𝑞𝐻𝑑𝑚2𝜉,𝑚(4.8)2𝜙+=𝑚2𝜉,𝑚(4.9)2𝜙=𝑚2𝜉,(4.10) with 𝑚2𝜉 given by (3.8). The scalar potential depends on the following new parameters: 𝛼, 𝐹𝑆, 𝜆, 𝑚, 𝑔𝐴,𝐵, 𝑞𝐻𝑢,𝑑, 𝑀𝑉𝐴.

In order to have a vacuum preserving the electromagnetism, the charged field vevs must vanish. Thus we are left with the problem of finding a minimum for the neutral scalar potential 𝑉0=||𝜇||2+𝑚2𝑢||0𝑢||2+||𝜇||2+𝑚2𝑑||0𝑑||2𝑏𝑒𝑖𝑘𝜑0𝑢0𝑑++h.c.|𝑚|2+𝑚2𝜙+||𝜙+||2+|𝑚|2+𝑚2𝜙||𝜙||2+18𝑔22+𝑔2𝑌||0𝑢||2||0𝑑||22+12𝑔2𝐴𝑞𝐻𝑢||0𝑢||2+𝑞𝐻𝑑||0𝑑||2+||𝜙+||2||𝜙||22+12𝑔2𝐵||𝜙+||2||𝜙||22.(4.11) Since there are no D-flat directions along which the quartic part vanishes, the potential is always bounded from below. To find the minimum we solve 𝜕𝑉0/𝜕𝑧𝑖=0 where the scalar field 𝑧𝑖 runs over {𝜑,0𝑢,0𝑑,𝜙+,𝜙}. The conditions for having a nontrivial minimum boil down to the same condition of the MSSM 𝑏2>||𝜇||2+𝑚2𝑢||𝜇||2+𝑚2𝑑.(4.12) Moreover in order to generate a mass term for the gauginos (see (3.10)), the condition 𝑣0 must hold since 𝑣+=0 due to the positive sign of the coefficient of the 𝜙+ quadratic term in (4.9). This implies the following condition for the coefficient of the 𝜙 quadratic term: |𝑚|2+𝑚2𝜙<0,(4.13) The minimum is attained at 𝜑=𝜙+=0. Actually since the potential for the axion 𝜑 is periodic, the minimum condition holds for 𝜑=2𝑛𝜋/𝑘 with 𝑛. All these minima are physically equivalent, and thus we arbitrarily choose 𝑛=0. The remaining three conditions imply the following constraints on the parameters: 𝑚2𝑑+𝜇2𝑏𝑡𝛽+18𝑔2𝑌+𝑔22𝑣2𝑐2𝛽+12𝑔2𝐴𝑞𝐻𝑑𝑣2𝑞𝐻𝑑𝑐2𝛽+𝑞𝐻𝑢𝑠2𝛽𝑣2𝑚=0,2𝑢+𝜇2𝑏𝑡𝛽118𝑔2𝑌+𝑔22𝑣2𝑐2𝛽+12𝑔2𝐴𝑞𝐻𝑢𝑣2𝑞𝐻𝑑𝑐2𝛽+𝑞𝐻𝑢𝑠2𝛽𝑣2𝑔=0,2𝐴+𝑔2𝐵𝑣2𝑔2𝐴𝑣2𝑞𝐻𝑑𝑐2𝛽+𝑞𝐻𝑢𝑠2𝛽+2|𝑚|2+𝑚2𝜙=0,(4.14) where we have defined in order to keep a compact notation 𝑐𝛽=cos𝛽,𝑠𝛽=sin𝛽,𝑡𝛽=tan𝛽,𝑐2𝛽=cos(2𝛽),𝑠2𝛽=sin(2𝛽),(4.15) and as usual as tan𝛽=𝑣𝑢/𝑣𝑑.

In the previous discussion we treated the scalar potential in an exact way. In the following we want to introduce some useful approximation in order to compute the mass eigenstates. Let us go back to the minima equations (4.14). Supposing 𝑣𝑣,(4.16) we can neglect all the 𝑔𝐴𝑣 terms. With this approximation, the minima equations read 𝑚2𝑑+𝜇2𝑏𝑡𝛽+18𝑔2𝑌+𝑔22𝑣2𝑐2𝛽=0,(4.17)𝑚2𝑢+𝜇2𝑏𝑡𝛽118𝑔2𝑌+𝑔22𝑣2𝑐2𝛽𝑔=0,(4.18)2𝐴+𝑔2𝐵𝑣2+2|𝑚|2+𝑚2𝜙=0,(4.19) where we have defined 𝑚2𝑑=𝑚2𝑑12𝑔2𝐴𝑞𝐻𝑑𝑣2,𝑚2𝑢=𝑚2𝑢12𝑔2𝐴𝑞𝐻𝑢𝑣2.(4.20) Equations (4.17) and (4.18) have the same functional form as in the MSSM case. Moreover 𝑣 does not depend on any parameter of the visible sector. Within this approximation the dynamics of the fields 𝜙± is decoupled from that of the Higgs sector, and thus the Higgs potential can be studied by fixing 𝜙± at their vevs. We get ||𝜇||𝑉2+𝑚2𝑢||0𝑢||2+||+𝑢||2+|𝜇|2+𝑚2𝑑||0𝑑||2+||𝑑||2+𝑏𝑒𝑖𝑘𝜑+𝑢𝑑0𝑢0𝑑+1+h.c.8𝑔22+𝑔2𝑌||0𝑢||2+||+𝑢||2||0𝑑||2||𝑑||22+12𝑔22||+𝑢𝑑0+0𝑢𝑑||2+12𝑔2𝐴𝑞𝐻𝑢||0𝑢||2+||+𝑢||2+𝑞𝐻𝑑||0𝑑||2+||𝑑||212𝑣22,(4.21) neglecting further constant terms in 𝑣. Close to the minima, the relevant term in the last line of (4.21) is the double product of the Higgs part with the 𝑣2 term. Hence by using (4.16) we finally get 𝑉,𝜑||𝜇||2+𝑚2𝑢||0𝑢||2+||+𝑢||2+||𝜇||2+𝑚2𝑑||0𝑑||2+||𝑑||2+𝑏𝑒𝑖𝑘𝜑+𝑢𝑑0𝑢0𝑑+1+h.c.8𝑔22+𝑔2𝑌||0𝑢||2+||+𝑢||2||0𝑑||2||𝑑||22+12𝑔22||+𝑢𝑑0+0𝑢𝑑||2.(4.22) This potential has the same form (except for the contribution of the exponential term in 𝜑) of the MSSM potential, and the corresponding minima equations are exactly given in (4.17) and (4.18). Thus all the well-known MSSM results apply here [93].

In particular one of the constraints is 𝑡𝛽1.2 [93] which implies (The presence of the extra field 𝜑 does not affect this result since the minima conditions are the same as the MSSM.) 𝑚2𝑢<𝑚2𝑑. By using (4.20) we get 𝑔𝐴𝑞𝐻𝑢𝑀𝑉𝐴1𝛼2𝑔𝐴𝑣2<𝑔𝐴𝑞𝐻𝑑𝑀𝑉𝐴1𝛼2𝑔𝐴𝑣2.(4.23) By assuming 𝑀𝑉𝐴>1 TeV, 𝑣 in the TeV range, 𝑔𝐴𝑂(0.1), the term between brackets is positive and we get the following constraint: 𝑞𝐻𝑢<𝑞𝐻𝑑(4.24) for the 𝑈(1)𝐴 Higgs charges.

4.1. Higgs Mass Matrices

We discuss the mass eigenvalues starting from the exact form of the scalar potential (4.4), switching to the approximated expression (4.22) when needed. In the neutral sector the singlet scalar 𝜙+ does not mix with any other scalar, so it is a mass eigenstate with square mass 𝑀2𝜙+=2|𝑚|2.(4.25) The same holds for the imaginary part of 𝜙 which becomes the longitudinal mode of the gauge vector 𝑍2. The mass matrix for the real scalar fields {𝜑,Im(0𝑢),Im(0𝑑)} is given by 𝑆(Im)=𝑏𝑡𝛽𝑏𝑏𝑡𝛽1𝑏𝑘𝑣𝑠𝛽𝑏𝑘𝑣𝑐𝛽𝑏𝑘2𝑣2𝑐𝛽𝑠𝛽.(4.26) The determinant of this matrix is zero. Two eigenvalues are zero which correspond to the Goldstone modes of 𝑍0 and 𝑍1. The physical massive state is an axi-Higgs state with mass given by 𝑀2𝐴0=2𝑏𝑠2𝛽11𝑞16𝐻𝑢+𝑞𝐻𝑑2𝑣2𝑀2𝑉𝐴𝑠22𝛽,(4.27) where we used the relation (3.14). The mass matrix for the real scalar fields {Re(0𝑢), Re(0𝑑), 𝜙𝑅Re(𝜙)} reads as 𝑆(Re)=14𝑔2𝐸𝑊+𝑔2𝐴𝑞𝐻𝑑2𝑣2𝑐2𝛽+𝑏𝑡𝛽1𝑏4𝑔2𝐸𝑊𝑔2𝐴𝑞𝐻𝑑𝑞𝐻𝑢𝑣2𝑐𝛽𝑠𝛽14𝑔2𝐸𝑊+𝑔2𝐴𝑞𝐻𝑢2𝑣2𝑠2𝛽+𝑏𝑡𝛽1𝑔2𝐴𝑞𝐻𝑑𝑣𝑣𝑐𝛽𝑔2𝐴𝑞𝐻𝑢𝑣𝑣𝑠𝛽𝑔2𝐴+𝑔2𝐵𝑣2,(4.28) where 𝑔2𝐸𝑊=(𝑔2𝑌+𝑔22). The matrix can be diagonalized exactly, but the results are cumbersome and difficult to read. It is much more convenient starting from the approximated potential (4.22) neglecting the mixing between Higgses and 𝜙. In this case we can apply the MSSM equations and get the following mass eigenvalues: 𝑀20,𝐻0122𝑏𝑠2𝛽2𝑏𝑠2𝛽14𝑔2𝑌+𝑔22𝑣22𝑔+2𝑏2𝑌+𝑔22𝑣2𝑠2𝛽,𝑀2𝜙𝑅𝑔2𝐴+𝑔2𝐵𝑣2.(4.29)

The charged sector is unchanged with respect to the MSSM, so 𝑀2𝐻±=2𝑏𝑠2𝛽+𝑀2𝑊.(4.30)

As in the standard MSSM case, the mass of the lightest Higgs 𝑀0 has a theoretical bound. It is a well-known problem in the MSSM that the upper bound [94, 95] is not compatible with the LEP bound [96]. In our case the bound is increased due to the presence of 𝐷𝐴-term corrections 𝑀20<14𝑔2𝑌+𝑔22𝑣2𝑐22𝛽+14𝑔2𝐴𝑣2𝑞𝐻𝑑+𝑞𝐻𝑢+𝑞𝐻𝑑𝑞𝐻𝑢𝑐2𝛽2,(4.31) where the first term is the MSSM bound. In principle, for arbitrary high values of 𝑔𝐴𝑞𝐻𝑑, 𝑔𝐴𝑞𝐻𝑢, we get an increasing upper bound. However, as in the standard MSSM case, 𝑀20 undergoes to relatively drastic quantum corrections [93]. Hence in Section 8 we consider tree-level masses for all the particles except for 0 for which we use the 1-loop corrected expression (see (8.10)).

5. Vector Mass Matrix

We now discuss the vector mass matrix. All the neutral scalars could in principle take a vev different from zero; hence we assume 𝜙±=𝑣±2,0𝑢,𝑑=𝑣𝑢,𝑑2.(5.1) The neutral vector square mass matrix in the base (𝑉𝐵,𝑉𝐴,𝑉𝑌,𝑉32) is 𝑉=𝑔2𝐵𝑣2𝜙𝑔𝐴𝑔𝐵𝑣2𝜙𝑔2𝐴𝑐2𝛽𝑞𝐻𝑑2+𝑠2𝛽𝑞𝐻𝑢2𝑣2+𝑣2𝜙+𝑀2𝑉𝐴012𝑔𝐴𝑔𝑌𝑞𝐻(𝛽)𝑣214𝑔2𝑌𝑣2102𝑔𝐴𝑔2𝑞𝐻(𝛽)𝑣214𝑔𝑌𝑔2𝑣214𝑔22𝑣2,(5.2) where 𝑣2𝜙=𝑣2++𝑣2,𝑣2=𝑣2𝑢+𝑣2𝑑,𝑞𝐻𝑠(𝛽)=2𝛽𝑞𝐻𝑢𝑐2𝛽𝑞𝐻𝑑.(5.3) By taking 𝑀𝑉𝐴>1 TeV (see Section 4), 𝑉𝐴 can be considered as decoupled from the low-energy gauge sector (namely, 𝐸1 TeV), and we can ignore with very good approximation any mixing term (The kinetic mixing between 𝑈(1)𝐴 and 𝑈(1)𝐵 deserves some comment, in particular if we relax the 𝑀𝑉𝐴>1 TeV assumption. Actually the presence of this mixing turns out to be irrelevant for the phenomenology of the visible sector. Anyway one has to take into account that for Tr(𝑞𝐴𝑞𝐵)0 such a mixing arises at the 1-loop level. In such a case it can be assumed that the two 𝑈(1)'s are in the kinetic diagonalized basis with Tr(𝑞𝐴𝑞𝐵)=0 thanks to some additional heavy chiral multiplet charged under both 𝑈(1)𝐴 and 𝑈(1)𝐵. These multiplets generate a counterterm in the effective theory that cancels against 𝛿𝐴𝐵 making the net kinetic mixing term equal to zero. This mechanism is analogous to the anomaly cancellation one where the GS mechanism can be generated by an anomaly-free theory with some heavy chiral fermion integrated out of the mass spectrum [8082].) involving 𝑉𝐴. From now on we will apply this approximation.

Since 𝑉𝐵 is a hidden gauge boson, it is decoupled from the SM sector. The charged vector sector is unchanged with respect to the MSSM, so 𝑊±𝜇=𝑉21𝜇𝑖𝑉22𝜇2,𝑀2𝑊=14𝑔22𝑣2.(5.4)

6. Neutralinos

In comparison with the standard MSSM, we now have five new neutral fermionic fields: 𝜓𝑆, 𝜆𝐴, 𝜆𝐵, 𝜙±. However under the assumption 𝑀𝑉𝐴>1 TeV, 𝜓𝑆 and 𝜆𝐴 are not in the low-energy sector because of the 𝑀𝑉𝐴 mass term (we stress that the 𝜓𝑆𝜆𝐴 sector presents a different parameters’ choice with respect to [9799], where we realized a scenario in which the mixing between 𝜓𝑆 and 𝜆𝐴 was suppressed). Thus we have neutralinomass1=2𝜓0𝑇𝑁𝜓0+h.c.,(6.1) where 𝜓0𝑇=𝜆𝐵,𝜙,𝜙+,𝜆𝑌,𝜆02,0𝑑,0𝑢.(6.2) In this basis the neutralino mass matrix 𝑁 is written as 𝑁=𝑀𝐵𝑔𝐵𝑣00𝑚0000𝑀10000𝑀2𝑔0001𝑣𝑑2𝑔2𝑣𝑑2𝑔00001𝑣𝑢2𝑔2𝑣𝑢2𝜇0,(6.3) where 𝜇 is given in (4.5). We recall that gaugino masses arise from the Dvali-Pomarol term (2.5).

𝑁 factorizes in a 4×4 MSSM block in the lower right corner, and in a 3×3 new sector block in the upper left corner. The new sector block is given by the 𝜆𝐵 and 𝜙± contributions. This last block has a MSSM-like structure that can be easily understood just considering the superpotential (2.3) and the gaugino masses (3.10) and by recalling that 𝜙 gets a vev 𝑣 different from zero, while 𝑣+=0.

Finally there are also corrections coming from the anomalous axino couplings: F-term couplings of the type 𝑏𝑎𝑎𝐹𝑆𝜆𝑎𝜆𝑎, D-term couplings of the type 𝑏𝑎𝑎𝜓𝑆𝜆𝑎𝐷𝑎, and corrections coming from the superpotential term 𝑒𝑘𝑆𝐻𝑢𝐻𝑑+h.c. However such corrections are always subdominant, and thus we neglect them with very good approximation.

We assume the lightest supersymmetric particle (LSP) in our model comes from the neutralino sector. In Section 8 we show the parameter regions in which this holds true. In order to ensure that the neutralino is the LSP, we keep fixed the gravitino mass 𝑚3/2𝑂 (TeV) in the limit 𝑀𝑃.

7. Sfermion Masses

The sfermion masses receive several contributions. As we have seen in Section 3, the leading contribution comes from the induced soft masses (3.8). But there are further contributions. We have MSSM-like contributions: F-term corrections proportional to the Yukawa couplings and 𝐷𝑌 and 𝐷2 term correction from the Higgs sector. Moreover there are 𝐷𝐴 term corrections from the Higgs and 𝜙 sector. As an aside, the appearance of such terms in the low-energy action, given our assumption 𝑀𝑉𝐴>1 TeV, can be understood in terms of quantum corrections to, Kähler potential [100]. Considering the first two families, we neglect the corresponding Yukawa couplings (the so-called third family approximation). In this approximation the mass eigenvalues are given by 𝑚2̃𝑢𝐿𝑚2̃𝑐𝐿=𝑚2𝑄+13𝑔2𝑌𝑔22Δ𝑣28+𝑞𝑄𝑚2𝐷𝐴,𝑚2̃𝑢𝑅𝑚2̃𝑐𝑅=𝑚2𝑈𝑐𝑔2𝑌Δ𝑣26+𝑞𝑈𝑐𝑚2𝐷𝐴,𝑚2𝑑𝐿𝑚2̃𝑠𝐿=𝑚2𝑄+13𝑔2𝑌+𝑔22Δ𝑣28+𝑞𝑄𝑚2𝐷𝐴,𝑚2𝑑𝑅𝑚2̃𝑠𝑅=𝑚2𝐷𝑐+𝑔2𝑌Δ𝑣212+𝑞𝐷𝑐𝑚2𝐷𝐴,𝑚2̃𝜈𝑒=𝑚2̃𝜈𝜇=𝑚2𝐿𝑔2𝑌+𝑔22Δ𝑣28+𝑞𝐿𝑚2𝐷𝐴,𝑚2̃𝑒𝐿𝑚2𝜇𝐿=𝑚2𝐿𝑔2𝑌𝑔22Δ𝑣28+𝑞𝐿𝑚2𝐷𝐴,𝑚2̃𝑒𝑅𝑚2𝜇𝑅=𝑚2𝐸𝑐+𝑔2𝑌Δ𝑣24+𝑞𝐸𝑐𝑚2𝐷𝐴.(7.1) The first terms on the right-hand side 𝑚2𝑄,𝑈𝑐,𝐷𝑐,𝐿,𝐸𝑐 are the corresponding soft masses (3.8), the second terms are the 𝐷𝑌,2 contributions with Δ𝑣2=𝑣2𝑢𝑣2𝑑=𝑣2𝑐2𝛽, while the last terms are the 𝐷𝐴 corrections given by 𝑚2𝐷𝐴=12𝑞𝐻𝑢𝑣2𝑢+𝑞𝐻𝑑𝑣2𝑑𝑣2,(7.2) There is an approximated degeneracy between the sfermions with the same charges.

The mass matrix for the third family sfermions is parametrized as 2𝑓=𝑀𝑓2𝐿𝐿𝑀𝑓2𝐿𝑅𝑀𝑓2𝐿𝑅𝑀𝑓2𝑅𝑅,(7.3) where the off-diagonal terms are generated by F-term corrections proportional to the Yukawa couplings. The stop mass matrix elements are 𝑀̃𝑡2𝐿𝐿=𝑚2𝑡+𝑚2𝑄+13𝑔2𝑌𝑔22Δ𝑣28+𝑞𝑄𝑚2𝐷𝐴,𝑀̃𝑡2𝑅𝑅=𝑚2𝑡+𝑚2𝑈𝑐𝑔2𝑌Δ𝑣26+𝑞𝑈𝑐𝑚2𝐷𝐴,𝑀̃𝑡2𝐿𝑅=𝜇𝑚𝑡𝑡𝛽1.(7.4) The sbottom mass matrix elements are 𝑀̃𝑏2𝐿𝐿=𝑚2𝑏+𝑚2𝑄+13𝑔2𝑌+𝑔22Δ𝑣28+𝑞𝑄𝑚2𝐷𝐴,𝑀̃𝑏2𝑅𝑅=𝑚2𝑏+𝑚2𝐷𝑐+𝑔2𝑌Δ𝑣212+𝑞𝐷𝑐𝑚2𝐷𝐴,𝑀̃𝑏2𝐿𝑅=𝜇𝑚𝑏𝑡𝛽.(7.5) The stau mass matrix elements are 𝑀̃𝜏2𝐿𝐿=𝑚2𝜏+𝑚2𝐿𝑔2𝑌𝑔22Δ𝑣28+𝑞𝐿𝑚2𝐷𝐴,𝑀̃𝜏2𝑅𝑅=𝑚2𝜏+𝑚2𝐸𝑐+𝑔2𝑌Δ𝑣24+𝑞𝐸𝑐𝑚2𝐷𝐴,𝑀̃𝜏2𝐿𝑅=𝜇𝑚𝜏𝑡𝛽.(7.6) The tau sneutrino mass is 𝑚2̃𝜈𝜏=𝑚2𝐿𝑔2𝑌+𝑔22Δ𝑣28+𝑞𝐿𝑚2𝐷𝐴,(7.7) where 𝑚𝑡, 𝑚𝑏, and 𝑚𝜏 are the masses of the corresponding standard fermions (i.e., further F-term contributions proportional to the Yukawa couplings). The structure of the diagonal terms of (7.3) is the same as in (7.1): soft masses, MSSM D-term contribution, and 𝐷𝐴 term correction. Furthermore we stress that there is a mass degeneracy between the three sneutrinos ̃𝜈𝑒,𝜇,𝜏 since the soft masses (3.8) are flavor blind.

8. Phenomenology

In the following we derive the phenomenological consequences of our scenario. Following our assumption of having a mass parameter for the anomalous 𝑈(1)𝐴 just slightly above the TeV scale, we fix 𝑀𝑉𝐴=10 TeV. The mass scale in the gaugino sector Λ is set to be 𝑂(𝑀𝑉𝐴).

8.1. Charge Bounds

The model parameter space can in principle be constrained by precision EW measurements [101]. However, since 𝑀𝑉𝐴=10 TeV, every value of 𝑔𝐴𝑞𝐻𝑢 and 𝑔𝐴𝑞𝐻𝑑 is allowed by EW precision data if |𝑔𝐴𝑞𝐻𝑢|,|𝑔𝐴𝑞𝐻𝑑|0.1. So the only relevant constraints are (3.18) and (4.24) that are plotted, respectively, with a red and a blue region, in Figure 1 in the plane (𝑔𝐴𝑞𝐻𝑢, 𝑔𝐴𝑞𝐻𝑑).

903106.fig.001
Figure 1: Higgs couplings bounds. The yellow spot represents our charge choice.
8.2. Free Parameters

Here we discuss which parameters remain free in our model after all the constraints discussed in the previous sections are imposed. Our choice for the Higgs 𝑈(1)𝐴 charges corresponds to the yellow spot in Figure 1: 𝑔𝐴=0.1,𝑀𝑉𝐴𝑞=10TeV,𝐻𝑑1=3,𝑞𝐻𝑢2=5.(8.1) In order to fix the remaining parameters (𝛼, 𝐹𝑆, 𝜆, 𝑚, 𝑔𝐵) we assume 𝑣246 GeV, and then we choose some benchmark value for 𝑔𝐵 and 𝑣 in the 𝑈(1)𝐵 sector (we recall that 𝑣+=0  (see Section 4)): (A)𝑔𝐵=0.4,𝑣=5TeV,(B)𝑔𝐵=0.1,𝑣=4TeV.(8.2) The next step is to solve the minima conditions (4.14) determining 𝐹𝑆, 𝜆, 𝑚 as function of 𝛼. In the limit in which 𝑣2𝑀𝑉𝐴𝛼,𝑣2, we get 𝜆218𝑒2𝛼𝑔𝐴(𝑞𝐻𝑑+𝑞𝐻𝑢)/𝑀𝑉𝐴𝑔𝐴𝑔𝐴𝑣22𝛼𝑀𝑉𝐴𝑞sec(2𝛽)𝐻𝑑𝑞𝐻𝑢+𝑞𝐻𝑑+𝑞𝐻𝑢,𝐹𝑆𝑒𝛼𝑔𝐴(𝑞𝐻𝑑+𝑞𝐻𝑢)/𝑀𝑉𝐴𝑀𝑉𝐴tan(2𝛽)4𝑞𝐻𝑑+𝑞𝐻𝑢𝜆𝑞𝐻𝑑𝑞𝐻𝑢2𝛼𝑀𝑉𝐴𝑔𝐴𝑣2,|𝑚|2𝑔𝐴𝛼𝑀𝑉𝐴12𝑔2𝐴+𝑔2𝐵𝑣2.(8.3) In Appendix B we report the exact formulae. Thus the only remaining free parameters are 𝑡𝛽 and 𝛼, and we perform the following analysis of the mass spectrum as a function of 𝑡𝛽 and 𝛼. A lower bound on 𝛼 as a function of 𝑣 can be obtained, given the approximation (4.16); from (4.19), 𝛼|𝑚|2𝑔+1/22𝐴+𝑔2𝐵𝑣2𝑔𝐴𝑀𝑉𝐴,(8.4) where we used the relation 𝑚2𝜙=𝑚2𝜉=𝛼𝑔𝐴𝑀𝑉𝐴.(8.5) Thus the lower bound on 𝛼 is obtained simply by setting |𝑚|=0: 𝛼𝑚𝑔1/22𝐴+𝑔2𝐵𝑣2𝑔𝐴𝑀𝑉𝐴.(8.6) The condition 𝛼>𝛼𝑚 must hold since otherwise we would have a massless scalar field in the spectrum (see (4.25)). Another lower bound, 𝛼𝑏, can be obtained from the condition (4.12), by solving the minima conditions (4.14) and by substituting the corresponding 𝐹𝑆, 𝜆 and 𝑚 values (8.3). The resulting lower bound can be expressed as 𝛼>max𝛼𝑚,𝛼𝑏.(8.7) No upper bound can be imposed; hence we decide to perform our analysis by considering 𝛼100 TeV.

The parameters 𝜆 and 𝐹𝑆 are of a particular phenomenological importance since they appear in the 𝜇 and 𝑏 terms (see (4.5) and (4.6)). In the case A, 𝜇 is in the range (900,6000)  GeV and 𝑏 is in the range (50,1200)  GeV while in the case B, 𝜇 is in the range (500,6000)  GeV and 𝑏 is in the range (25,1200)  GeV. These values are in the right range to solve the 𝜇-problem.

8.3. Mass Spectrum

(A)With such choice the gauge vector sector is completely fixed up to a 𝑡𝛽 dependence. Anyway even such a dependence can be safely ignored with a very good approximation in the new gauge sector since the mixing is strongly suppressed. So for each 𝑡𝛽 value, we have 𝑀𝑍1𝑀10TeV,𝑍22TeV,(8.8) where with 𝑍1 we denote the 𝑉𝐴-like vector.(B)As in the previous case, we just give the 𝑍1,2 masses 𝑀𝑍1𝑀10TeV,𝑍2400GeV,(8.9) where as in the previous case 𝑍1 is 𝑉𝐴-like.

We will not give the exact values of the 𝑍0 mass. It is enough for our purposes to know that they are compatible with the bounds of Section 8.1. Both case A and B are compatible with CDF bounds about 𝑍 direct production [102].

Recent LHC data have restricted the most probable range for the Higgs particle mass to be [115.5,131] GeV (ATLAS) [103] and [114.5,127] (CMS) [104]. Moreover, there are hints observed by both CMS and ATLAS of an excess of events that might correspond to decays of a Higgs particle with a mass in a range close to 125  GeV. So, in Figures 2 and 3 we give region plots showing the allowed values of 𝛼 and 𝑡𝛽 for case A (B) and Λ/𝑐=(5)10TeV. The red region is the one in which 𝑀20|1-loop[124,126]GeV where the 0 mass is computed considering 1-loop corrections. Since it turns out that the top squarks have small mixing angle and considering the limit 𝑀𝐴0𝑀𝑍0, we have [93] 𝑀20|||1-loop𝑀20|||tree+34𝜋2𝑠2𝛽𝑦2𝑡𝑚2𝑡𝑚̃𝑡ln1𝑚̃𝑡2𝑚2𝑡𝑀20|||tree+32𝜋2𝑚4𝑡𝑣2𝑚̃𝑡ln1𝑚̃𝑡2𝑚2𝑡,(8.10) where 𝑀0|tree is the tree-level 0 mass and we used 𝑚𝑡=𝑦𝑡𝑣𝑢/2=𝑦𝑡𝑣𝑠𝛽/2. There is an approximated inverse correlation between 𝛼 and 𝑡𝛽 in the 0 mass allowed region because the 1-loop correction in (8.10) increases for increasing values of 𝛼 or 𝑡𝛽. The 0 mass allowed region is almost the same for case A and B because of two reasons: (i)the mixing with 𝜙± is suppressed,(ii)the parameters 𝑚𝑢,𝑚𝑑,𝜇,𝑏 in the scalar potential (4.22) are ruled by the square mass parameters 𝑔𝐴𝛼𝑀𝑉𝐴 and (𝑔𝐴𝑣)2, and the first one turns out to be dominant.

fig2
Figure 2: Allowed 𝛼 and tan𝛽 values for case A (up) and case B (down), Λ𝑐=5 (left) and Λ𝑐=10 (right). The red region is the one in which 𝑀20|1-loop[124,126]GeV, the magenta region is the one in which 𝑀20|1-loop[114.5,131]GeV and the blue region satisfies all the mass bounds on the sparticles (from PDG) and requires a neutralino LSP. The yellow dots are our benchmark points.
fig3
Figure 3: Allowed 𝛼 and tan𝛽 values for case A (up) and case B (down), Λ𝑐=5 (left) and Λ𝑐=10 (right). The red region is the one in which 𝑀20|1-loop[124,126]GeV, the magenta region is the one in which 𝑀20|1-loop[114.5,131]GeV and the blue region satisfies all the mass bounds on the sparticles (from preliminary LHC data) and requires a neutralino LSP. The yellow dots are our benchmark points.

The magenta region satisfies a milder constraint on the light Higgs boson: 𝑀20|1-loop[114.5,131]  GeV. In order to be more conservative, we imposed the joint constraints of ATLAS and CMS.

The blue region satisfies all the mass bounds on the sparticles and requires a neutralino LSP. We considered two possibilities: one more optimistic (Figure 2) using the PDG bounds [105, 106] and one more conservative (Figure 3) using recent LHC data [107, 108]. The combination of the gluino mass bound with a neutralino LSP is a strong constraint that reduces drastically the allowed parameter space. In some cases there is not even a blue region, which means that we cannot satisfy simultaneously all the mass bounds and have a neutralino LSP, so they are completely ruled out. When the gluino mass bound is from PDG, case A is allowed; otherwise it is completely ruled out and only case B for Λ/𝑐=10TeV presents allowed regions. We notice that case A favors low 𝛼 values, while case B favors high 𝛼 values. For every allowed case we choose a benchmark point (yellow spots in Figures 2 and 3):(i)case A, Λ𝑐=5, 𝛼=3 TeV and 𝑡𝛽=50 so that 𝑀0|1-loop121.6 GeV; (ii)case A, Λ𝑐=10, 𝛼=5 TeV and 𝑡𝛽=10 so that 𝑀0|1-loop124.7 GeV; (iii)case B, Λ𝑐=10, 𝛼=4.5 TeV and 𝑡𝛽=50 so that 𝑀0|1-loop125.1 GeV;(iv)case B, Λ𝑐=10, 𝛼=50 TeV and 𝑡𝛽=2.5 so that 𝑀0|1-loop130.1 GeV, and we give the full mass spectrum in Figures 4 and 5.

fig4
Figure 4: Mass spectrum, case A, Λ𝑐=5, 𝛼=0.3 TeV and 𝑡𝛽=50 (a), Λ𝑐=10, 𝛼=0.5 TeV and 𝑡𝛽=10 (b).
fig5
Figure 5: Mass spectrum, case B, Λ𝑐=10, 𝛼=0.45 TeV and 𝑡𝛽=50 (a), 𝛼=8 TeV and 𝑡𝛽=2.5 (b).

All the benchmark points share some common features. (i)The LSP is the lightest neutralino of the new sector: in case A it is a combination of 𝜙± and 𝜆𝐵 while in case B is almost a pure 𝜆𝐵. (ii)An approximated mass degeneracy of 𝐻0, 𝐴0, and 𝐻± holds, and their masses satisfy the bounds of [96, 109]. (iii)The lightest sleptons is a sneutrino, except for 𝑡𝛽=50 when it is ̃𝜏1(iv)The lightest squark is ̃𝑢𝐿, except for 𝑡𝛽=50 when it is ̃𝑏1(v)The first and second family left-handed squarks/sleptons are likely to be lighter than their right-handed counterparts. This is at odds with the usual MSSM cases [93]. (vi)𝐶1(2) is close in mass with 𝑁MSSM1(4). 𝐶2 and 𝑁MSSM4 are heavier than all sfermions. (vii)The gluino is close in mass to 𝐶1 and 𝑁MSSM1 which are gaugino-like. Moreover it is lighter than all the squarks except for point (i). So it turns out to be long lived, specially in case B where the approximated mass degeneracy involves also the LSP. Long-lived gluinos bind with SM quarks and gluons from the vacuum during the hadronisation process and produce R-hadrons. R-hadrons are among the most interesting searches at LHC. Anyway we will come back to this point with a more detailed study in a forthcoming paper. (viii)There is an approximated mass degeneracy between ̃𝑒𝑅 and ̃𝑢𝑅 because using the charge constraints (3.16) and (8.1) we get 𝑞𝐸𝑐=3 and 𝑞𝑈𝑐2.9. (ix)𝑚𝜙𝑅<𝑚𝜙+ except for point (i).

Case B points deserve some more comments.

𝜙+ and 𝑁new2,3 are out of the plot of point (iv) because they are heavier than 6 TeV. 𝑍2 is among the lightest not SM particle, so it can decay only into SM particles, because of energy and R-parity conservation. So 𝑍2 is long lived, because SM particles are coupled to 𝑍2 only through the suppressed 𝑉𝐴,𝐵 mixing or through the Higgs scalars which present a tiny mixing with 𝜙±.

It is not an easy task to compare the resulting spectrum we get for our model with those related to the rich zoology of supersymmetry breaking scenarios. It is worth to stress anyway that the two representative spectrums showed in Figure 5 which encode the key features of our scenarios listed above are not reproduced in any of the benchmark points showed in [110, 111].

9. Conclusions

In this paper we presented a viable mechanism to generate soft supersymmetry breaking terms in the framework of a minimal supersymmetric anomalous extension of the SM. The crucial ingredient is a non perturbative term in the superpotential (2.3) which couples the Stückelberg field 𝑆 to the Higgs sector. This term is related to the generation of a suitable 𝜇 and 𝑏 terms (see (4.5) and (4.6)) in the low-energy effective action when the Stückelberg gets vev. We argued about the origin of this term from an exotic instanton in an intersecting D-brane setup. We computed the spectrum of our model as a function of the saxion vev 𝛼 and for different choices of the remaining free parameters. We checked our results against known phenomenological bounds, namely, current lower bounds on the mass of the scalar and fermionic superpartners. We analyzed a scenario in which the anomalous sector is the source of the soft supersymmetry breaking terms while the corresponding vector and Stückelberg multiplets are not present in the low-energy effective action. For what concerns the non anomalous sector, we took into account two different cases (dubbed case A and case B).

As we stated in Section 8, by applying some phenomenological constraints we were able to derive some bounds on the saxion vev 𝛼, which is the relevant parameter setting the mass scale of the scalars. The strongest constraints on 𝛼 and 𝑡𝛽 come from the combined requirement of 𝑀20|1-loop[124,126]GeV or ([114.5,131]GeV), a neutralino LSP, and that all mass bounds (specially the gluino one) are fulfilled. In Figure 2 (pre-LHC bounds) and 8.3 (preliminary LHC bounds), we summarize the allowed regions for 𝛼. In the first case, by requiring a phenomenological appealing neutralino LSP, we get an allowed 𝛼 of few TeV up to 10 TeV for the A and B scenarios respectively. In the second case (preliminary LHC bounds), we get that only the B scenario is allowed with 𝛼5 TeV. These results can be seen as a bound that a concrete D-brane model has to satisfy. We deserve this analysis for future work.

In Figure 5 we explicitly showed two benchmark mass spectrums for our model with 𝛼 and 𝑡𝛽 which fulfill the above bounds. The cases shared different peculiar features: the LSP is the lightest neutralino of the new sector, there is a near mass degeneracy between 𝐻0, 𝐴0 and 𝐻±, and ̃𝑒𝑅 and ̃𝑢𝑅 the lightest sleptons is a sneutrino except for 𝑡𝛽=50 when it is stau, the lightest squark is a ̃𝑢𝐿 except for 𝑡𝛽=50, when it is a sbottom, the first and second family left-handed squarks/sleptons are typically lighter than their right-handed counterparts. Moreover in case B the gluino is long lived and can produce R-hadrons. It turns out that these features are not reproduced in any of the widely studied benchmark points presented in [110, 111].

Appendices

A. Anomalous Lagrangians

The Lagrangian involved in the anomaly cancellation procedure is 𝑆=14𝑆+𝑆+2𝑀𝑉𝐴𝑉𝐴2|||𝜃2𝜃22𝑎𝑔2𝑎𝑏𝑎𝑎𝑊𝑆Tr𝑎𝑊𝑎+𝑔𝑌𝑔𝐴𝑏𝑌𝐴𝑆𝑊𝑌𝑊𝐴𝜃2,+h.c.(A.1) where the index 𝑎=𝐴,𝐵,𝑌,2,3 runs over the 𝑈(1)𝐴, 𝑈(1)𝐵, 𝑈(1)𝑌, 𝑆𝑈(2), and 𝑆𝑈(3) gauge groups respectively, and the constants 𝑏𝑎𝑏 are fixed by the anomaly cancellation.

Since we have only one anomalous 𝑈(1), we can avoid the use of GCS terms, distributing the anomalies only on the 𝑈(1)𝐴 vertices. So we have 𝑏𝐴𝐴𝑔=𝐴𝒜𝐴𝐴96𝜋2𝑀𝑉𝐴,𝑏𝑌𝑌𝑔=𝐴𝒜𝑌𝑌32𝜋2𝑀𝑉𝐴,𝑏22𝑔=𝐴𝒜2216𝜋2𝑀𝑉𝐴,𝑏33𝑔=𝐴𝒜3316𝜋2𝑀𝑉𝐴,𝑏𝑌𝐴𝑔=𝐴𝒜𝑌𝐴32𝜋2𝑀𝑉𝐴,(A.2) where the 𝒜’s are the corresponding anomalies: 𝒜𝐴𝐴=10𝑞𝐻𝑑39𝑞𝐻𝑑2𝑞𝐿+3𝑞𝑄9𝑞𝐻𝑑𝑞𝐿2+3𝑞𝑄27𝑞𝐻𝑢327𝑞𝐻𝑢2𝑞𝑄27𝑞𝐻𝑢𝑞𝑄2+3𝑞𝐿3,𝒜𝑌𝑌1=27𝑞𝐻𝑑+7𝑞𝐻𝑢+3𝑞𝐿+9𝑞𝑄,𝒜22=12𝑞𝐻𝑑+𝑞𝐻𝑢+3𝑞𝐿+9𝑞𝑄,𝒜333=2𝑞𝐻𝑑+𝑞𝐻𝑢,𝒜𝑌𝐴=5𝑞𝐻𝑑2+6𝑞𝐻𝑑𝑞𝐿+𝑞𝑄𝑞𝐻𝑢5𝑞𝐻𝑢+12𝑞𝑄,(A.3) where we used the constraints (3.13). Imposing the conditions (3.16) we get 𝒜𝐴𝐴=1641168𝑞𝐻𝑑3+1776𝑞𝐻𝑑2𝑞𝐻𝑢996𝑞𝐻𝑑𝑞𝐻𝑢2+53𝑞𝐻𝑢3,𝒜(A.4)𝑌𝑌=114𝑞𝐻𝑑+𝑞𝐻𝑢,𝒜(A.5)221=4𝑞𝐻𝑑+𝑞𝐻𝑢,𝒜(A.6)333=2𝑞𝐻𝑑+𝑞H𝑢,𝒜(A.7)𝑌𝐴=0.(A.8) We recall that (A.8) is not a consequence of (3.16), but rather (3.16) is a consequence of imposing (A.8) in order to cancel the 𝑈(1)𝑌𝑈(1)𝐴 kinetic mixing.

B. Exact Fixed Parameters

In this Appendix section we give the exact values for the 𝐹𝑆, 𝜆, 𝑚 parameters determined in Section 8.2. Solving the minima conditions (4.14), we get 𝜆2=𝑒2𝛼𝑔𝐴(𝑞𝐻𝑑+𝑞𝐻𝑢)/𝑀𝑉𝐴×32𝑔𝐴𝑞sec(2𝛽)𝐻𝑑𝑞𝐻𝑢8𝛼𝑀𝑉𝐴+𝑔𝐴𝑣2𝑞𝐻𝑑+𝑞𝐻𝑢(cos(4𝛽)+3)4𝑔𝐴𝑣28𝛼𝑔𝐴𝑀𝑉𝐴𝑞𝐻𝑑+𝑞𝐻𝑢2𝑣22𝑔2𝐴𝑞𝐻𝑑2+𝑞𝐻𝑢2+𝑔2𝑌+𝑔22+4𝑔2𝐴𝑣2𝑞𝐻𝑑+𝑞𝐻𝑢,𝐹𝑆=𝑒𝛼𝑔𝐴(𝑞𝐻𝑑+𝑞𝐻𝑢)/𝑀𝑉𝐴×𝑀𝑉𝐴tan(2𝛽)8𝑔𝐴𝑞𝐻𝑑+𝑞𝐻𝑢𝜆×𝑔𝐴𝑞𝐻𝑑𝑞𝐻𝑢4𝛼𝑀𝑉𝐴+𝑔𝐴𝑣2𝑞𝐻d+𝑞𝐻𝑢2𝑔𝐴𝑣2+𝑣2𝑔cos(2𝛽)2𝐴𝑞𝐻𝑑𝑞𝐻𝑢2+𝑔2𝑌+𝑔22,|𝑚|2=𝑔𝐴𝛼𝑀𝑉𝐴12𝑔2𝐴+𝑔2𝐵𝑣2+𝑔2𝐴𝑣2𝑞𝐻𝑑𝑐2𝛽+𝑞𝐻𝑢𝑠2𝛽.(B.1)

Acknowledgments

A. Lionetto acknowledges M. Bianchi, E. Kiritsis, and R. Richter for useful discussions and comments. A. Racioppi acknowledges M. Raidal for discussions and the ESF JD164 contract for financial support.

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