Z'-mediated Supersymmetry Breaking
aa r X i v : . [ h e p - ph ] F e b Z ′ -mediated Supersymmetry Breaking Paul Langacker ∗ , Gil Paz ∗ , Lian-Tao Wang † , Itay Yavin † ∗ School of Natural Sciences, Institute for Advanced Study, Einstein Drive Princeton, NJ 08540 † Physics Department, Princeton University, Princeton NJ 08544 (Dated: October 29, 2018)We consider a class of models in which supersymmetry breaking is communicated dominantly viaa U (1) ′ gauge interaction, which also helps solve the µ problem. Such models can emerge naturallyin top-down constructions and are a version of split supersymmetry. The spectrum contains heavysfermions, Higgsinos, exotics, and Z ′ ∼ −
100 TeV; light gauginos ∼ − ∼
140 GeV; and a light singlino. A specific set of U (1) ′ charges and exotics is analyzed, andwe present five benchmark models. Implications for the gluino lifetime, cold dark matter, and thegravitino and neutrino masses are discussed. PACS numbers: 12.60.Jv, 12.60.Cn, 12.60.Fr
I. INTRODUCTION AND MOTIVATION
To a large extent, the mediation mechanism of super-symmetry (SUSY) breaking determines the low energyphenomenology. A well-studied scenario is gravity me-diation [1]. During the last couple of decades, in orderto satisfy the increasingly stringent constraints from fla-vor changing neutral current measurements, many othermediation mechanisms, such as anomaly mediation [2],gauge mediation [3], and gaugino mediation [4], havebeen proposed (for a review, see [5]). In this letter, wepresent a alternative mechanism in which SUSY break-ing is mediated by exotic gauge interactions, such as anadditional U (1) ′ . Concrete superstring constructions fre-quently lead to additional, non-anomalous, U (1) ′ factorsin the low-energy theory (see, e.g., [6]) with properties al-lowing a U (1) ′ -mediated SUSY breaking. Scenarios withan extra U (1) ′ involved in supersymmetry breaking me-diation have been studied in various contexts [7]. Here,we study a new scenario where Z ′ -mediation is the dom-inant source for both scalar and gaugino masses.Another ingredient we would like to consider is the µ -problem of the Minimal Supersymmetric Standard Model(MSSM). One class of solutions invokes a spontaneouslybroken Peccei-Quinn symmetry (see, e.g., [8]). Fromthe point of view of top-down constructions it is com-mon that such a symmetry is promoted to a U (1) ′ gaugesymmetry [9]. Identifying this U (1) ′ with the mediatorof SUSY breaking sets µ (as well as µB ) to the scale ofthe other soft SUSY breaking parameters, which are ofthe right size whether or not the electroweak symmetrybreaking is finely tuned.In the setup we propose, schematically shown in Fig. 1,visible and hidden sector fields do not have direct renor-malizable coupling with each other. At the same time,they are both charged under U (1) ′ . A supersymme-try breaking Z ′ -ino mass term, M ˜ Z ′ , is generated dueto the U (1) ′ coupling to the hidden sector. The ob-servable sector fields feel the supersymmetry breakingthrough their couplings to U (1) ′ . The sfermion massesare of order m f ∼ M Z ′ / π . The SU (3) C × SU (2) L × MSSM + S DSB+ Exotics Z’ Hidden SectorVisible Sector
FIG. 1: Z ′ -mediated supersymmetry breaking. U (1) Y gaugino masses are generated at higher loop order, M , , ∼ M ˜ Z ′ / (16 π ) , which is 2-3 order of magnitudeslighter than the sfermions. LEP direct searches suggestelectroweak-ino masses >
100 GeV. We therefore expectthat the sfermions are heavy, typically about 100 TeV. Inthis sense, this scenario can be viewed as a mini-version ofsplit-supersymmetry [10]. In particular, one fine-tuningis needed to maintain a low electroweak scale. This sce-nario does not have flavor or CP violation problems dueto the decoupling of the sfermions. One important differ-ence from split-supersymmetry is the µ -parameter, whichis set by the scale of U (1) ′ breaking. II. GENERIC FEATURES OF Z ′ -MEDIATEDSUPERSYMMETRY BREAKING The visible sector contains an extension of the MSSM.First, we introduce an extra U (1) ′ gauge symmetry. Sec-ond, the µ parameter is promoted into a dynamical field, µH u H d → λSH u H d . S is a Standard Model singletwhich is charged under the U (1) ′ . Third, we includeexotic matter multiplets with Yukawa couplings to S , P i ∈{ exotics } Y i SX i X ci . They are included to cancel theanomalies associated with the U (1) ′ . Such exotics andcouplings generically exist in string theory constructions. A. Features of the Spectrum
We parameterize the hidden sector supersymmetrybreaking by a spurion field X = M + θ F . At the scaleΛ S , supersymmetry breaking is assumed to generate amass M ˜ Z ′ ∼ g z ′ ( F/M ) / π for the fermionic compo-nent of the ˜ Z ′ vector superfield.We assume that all the chiral superfields in the visiblesector are charged under U (1) ′ , so all the correspondingscalars receive soft mass terms at 1-loop, m f i ∼ g z ′ Q f i π M Z ′ log (cid:18) Λ S M ˜ Z ′ (cid:19) ∼ (100 TeV) (1)where g z ′ is the U (1) ′ gauge coupling and Q f i is the U (1) ′ charge of f i , which we take to be of order unity.The SU (3) C × SU (2) L × U (1) Y gaugino masses canonly be generated at 2-loop level since they do not di-rectly couple to the U (1) ′ gaugino, M a ∼ (2) ∼ g z ′ g a (16 π ) M ˜ Z ′ log (cid:18) Λ S M ˜ Z ′ (cid:19) ∼ − GeVwhere g a is the gauge coupling for the gaugino ˜ λ a . It isstraightforward to verify that this is indeed the leading U (1) ′ contribution to the gaugino mass. In particular, ki-netic mixing induced by loops of visible sector fields doesnot contribute significantly due to chiral symmetries.The gravitino mass m / ∼ F/M P depends strongly onthe scale of supersymmetry breaking. Requiring MSSMgaugino masses ≥
100 GeV and assuming √ F , M and Λ S to be of the same order of magnitude, we find √ F ∼ − GeV. This is very different from gauge mediatedsupersymmetry breaking, where the lower scale ( ∼ − F/M P , whichcould be of the same order as Eq. 2. However, its contri-bution to scalar masses ∼ F /M P is negligible comparedwith the Z ′ -mediation. Therefore, we expect the hierar-chy between scalar and gaugino masses to be generic. B. Symmetry breaking and fine-tuning
The U (1) ′ gauge symmetry must be broken by the sin-glet’s VEV h S i . We assume this is triggered by radia-tive corrections to the soft mass m S , especially throughYukawa couplings to exotics. Therefore, successful ra-diative breaking of U (1) ′ usually requires that those cou-plings are not small. h S i is parametrically only an order of magnitude smaller than M ˜ Z ′ . It is therefore reason-able to first determine h S i ignoring the Higgs doublets,and then to consider the Higgs potential for the doubletsregarding h S i as fixed.To generate the electroweak scale Λ EW we must fine-tune one linear combination of the two Higgs doubletsto be much lighter than its natural scale. The full massmatrix for the two Higgs doublets is, M H = m − A H h S i− A H h S i m m = m H u + g z ′ Q S Q h S i + λ h S i m = m H d + g z ′ Q S Q h S i + λ h S i . (3)Generically, one can tune various elements in M H to ob-tain one small eigenvalue ∼ Λ . The up-type Higgsmass term can be driven small or negative due to thelarge top Yukawa coupling. One typically finds solu-tions by tuning | m | ≪ m ∼ g z ′ M Z ′ / π . The tri-linear term is smaller, A H ∼ λg z ′ M ˜ Z ′ / π ∼ λ × H d will not shift the smallereigenvalue significantly. tan β is well approximated bytan β = m /A H h S i ∼ − ∼
100 TeV. The Higgs mass is somewhat heav-ier than the typical prediction of the MSSM, due to the U (1) ′ D term and the running of the effective quarticcoupling from M ˜ Z ′ down to the electroweak scale.It is possible to tune with all the parameters, suchas g z ′ and λ , of the same order. In addition, thereis an interesting limit when g z ′ ≪ λ . Generically,we expect h S i ∼ M ˜ Z ′ / π . The singlino mass is ∼ g z ′ Q S h S i /M ˜ Z ′ ∼ g z ′ M ˜ Z ′ / π ≪ M ˜ Z ′ . Moreover,since | m H u | ∝ g z ′ M Z ′ / π , to fine-tune m ∼ Λ we expect the parameters to be chosen so that the sin-glet’s VEV is even smaller h S i ∼ ( g z ′ /λ ) M ˜ Z ′ / π . There-fore, it is possible to have the singlino be very light m ˜ S ∼ (cid:0) − − − (cid:1) M ˜ Z ′ . In certain cases, the Z ′ gauge-boson, M Z ′ ∼ g z ′ Q S h S i , could even be light enough tobe produced at the LHC. III. MODEL PARAMETERS ANDLOW-ENERGY SPECTRUM
The free parameters are g z ′ , λ , the exotic Yukawa cou-plings, the U (1) ′ charges, M ˜ Z ′ , and the supersymmetrybreaking scale Λ S . The charges are chosen to cancel allthe anomalies. A minimal choice, which also leads to alight wino ( M < M , ), involves the introduction of 3families of colored exotics ( D ) and two uncolored SU (2)-singlet families ( E ). Normalizing the down-type Higgscharge to unity, Q = 1, we are left with two independentparameters, which we choose to be the up-type Higgs andthe left-handed quark charges, Q and Q Q respectively.Several additional constraints need to be satisfied by the Q − − − − − Q Q − − − − − g z ′ λ Y D Y E h S i × × × × × tan β
20 29 33 45 60 M M
710 195 180 340 123 M m H
140 140 140 140 140 m ˜ Q × × × × m ˜ L × × m /
890 3600 810 3 0.1 m ˜ S m Z ′ × . × . × M ˜ Z ′ = 10 GeV. The masses of the first two generations of squarks andsfermions are typically larger than that of the third. The in-put parameters λ , g z ′ and Y D,E are defined at Λ S . The spec-tra are calculated using exact Renormalization Group Equa-tions (RGE) (see, e.g., [12]). There is a theoretical uncer-tainty due to multiple RGE thresholds. This mainly affects m H , leading to a several GeV uncertainty. The gravitino massis calculated by m / = Λ S /M P assuming Λ S ∼ √ F . Therecould be deviations from this relation in some SUSY breakingmodels which could lead to a gravitino mass that is differentby up to a couple orders of magnitude (typically lower). Fordetails, see [11]. choices of charges and other parameters. U (1) ′ has to bespontaneously broken by radiative corrections. It mustallow appropriate fine-tuning to break the electroweaksymmetry. Moreover, since U (1) ′ D -terms could con-tribute to scalar masses with either sign, one must checkfor the existence of charge or color breaking minima.We have found several regions in the ( Q Q , Q ) spacewhere a solution satisfies all the constraints. A detailedscan will be presented in a forthcoming publication [11].The results exhibit a variety of patterns for the low en-ergy spectrum. In Table I, we display five representa-tive models. Different ordering of the MSSM gauginoand singlino masses could give rise to very different phe-nomenology. The singlino mass typically has more vari-ation since it is determined by fine-tuning. The appear-ance of a light Z ′ in the spectrum, shown in model 5(with σ × BR( Z ′ → ℓ ¯ ℓ ) &
10 fb), could result in a spec-tacular signal and help untangle the underlying model.This generically happens in the case where the singlinois very light.A wino as the lightest supersymmetric particle (LSP)and its nearly degenerate charged partner (the degener-acy is lifted at one-loop by about 160 MeV [13] and al-lows the decay ˜ W + → ˜ W + π + , which results in a 4 cmdisplaced vertex) have been studied extensively [14], es-pecially in connection with anomaly mediated models [2]. It can annihilate efficiently into gauge bosons. For purethermal production the dark matter density is too low forthe several hundred GeV mass range we have assumed.However, it can be considerably larger for non-standardcosmological scenarios.Due to small mixings, at most of the order λv/µ tan β ,the decay chain involving the singlino and wino will havea long life-time which could result in a displaced vertex.For example, depending on whether the decay is two orthree-body, the life-time for ˜ S → h ( ∗ ) + ˜ W or ˜ W → h ( ∗ ) + ˜ S is in the range of 10 − − − s. This couldgive an interesting signature in case of M > M ˜ S , or M ˜ S > M if the Z ′ is light enough and has an appreciablebranching ratio for decay into the singlino.There is a wide range of possible gravitino masses, m / ∼ − − GeV. With typical assumptions aboutcosmology, m / is strongly constrained by Big Bang Nu-cleosynthesis (BBN). If the gravitino is not the LSP, wetypically require either it to be heavy ( >
10 TeV) so itdecays before BBN, or that the reheating temperatureis less than about 10 − GeV [15]. In the case thatthe gravitino is the LSP and the next to lightest super-symmetric particle (NLSP) is the wino, we require thegravitino to be lighter than about 100 MeV [16]. It isparticularly problematic when the singlino is the NLSPsince its decay to the gravitino is further suppressed, un-less the singlino density is strongly diluted by some latetime entropy generation. We also note that decaying intoa light gravitino, m / ∼ MeV, is not observable on col-lider time scales since the NLSP is neutral.Since the squarks are heavy the gluino decays off-shell[10]. Its life-time is very sensitive to g z ′ and is given by, τ ˜ g = 4 × − sec (cid:18) m ˜ Q TeV (cid:19) (cid:18) M (cid:19) ∝ g z ′ . (4)Even though the life-time is long enough for the gluino tohadronize it is too short to result in a displaced vertex.Since the scalars are heavy, one-loop contributions tomost flavor observables (such as b → sγ ) are highly sup-pressed. There are also two loop contributions to EDMand muon g −
2. However, those are suppressed as com-pared with the Split SUSY scenario [10] since the Hig-gsinos are heavy and the singlino-wino mixing is small.The exotic matter in this model is very heavy and doesnot enter any collider phenomenology.
IV. DISCUSSION
In this letter, we discussed the generic feature of su-persymmetry breaking dominantly mediated by an extra U (1) ′ . We have used a U (1) ′ which forbids a µ term.Such a requirement gives additional constraints and pre-dicts interesting low energy phenomenology, such as theexistence of a light singlino and Z ′ in various regions ofthe parameter space. However, Z ′ -mediation is possiblein a wider range of U (1) ′ models, such as U (1) B − L . Weexpect the hierarchy between the soft scalar masses andthe gaugino masses to be generic, although the detailedpattern of soft terms could be quite different. Consider-ing Z ′ -mediation in a broader range of models is certainlyworth pursuing.The model presented here does not provide a seesawmechanism for neutrino mass. However, in a simple vari-ant the U (1) ′ symmetry forbids Dirac Yukawa couplings Y ν H u Lν c at the renormalizable level, but allows them tobe generated by a higher-dimensional operator [17], W ν = c ν SM P H u Lν c . (5)This naturally yields small Dirac neutrino masses of order(0 . c ν ) eV for h S i = 100 TeV.There are several scenarios for the decays and lifetimesof the heavy exotic particles [18] and for gauge unifica- tion. These depend on the details of the U (1) ′ chargeassignments, and will be discussed in [11]. Acknowledgments
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