Minimal Seesaw as an Ultraviolet Insensitive Cure for the Problems of Anomaly Mediation
aa r X i v : . [ h e p - ph ] J u l Minimal Seesaw as an UltravioletInsensitive Cure for the Problems ofAnomaly Mediation
R. N. Mohapatra, N. Setzer, and S. Spinner
Maryland Center for Fundamental Physics and Department of Physics,University of Maryland, College Park, MD 20742, USA
November 5, 2018
Abstract
We show that an intermediate scale supersymmetric left-right see-saw scenario with automatic R -parity conservation can cure the prob-lem of tachyonic slepton masses that arises when supersymmetry isbroken by anomaly mediation, while preserving ultraviolet insensitiv-ity. The reason for this is the existence of light B − L = 2 higgseswith yukawa couplings to the charged leptons. We find these theo-ries to have distinct predictions compared to the usual mSUGRA andgauge mediated models as well as the minimal AMSB models. Suchpredictions include a condensed gaugino mass spectrum and possiblya correspondingly condensed sfermion spectrum. The driving motivations for physics beyond the standard model are: (i) sta-bilizing the Higgs mass against radiative corrections, thereby providing afirmer understanding of the origin of mass; (ii) understanding the neutrinomasses and mixings, especially the extreme smallness of its mass comparedto those of charged fermions; (iii) finding a particle physics candidate for thedark matter of the universe and (iv) solving the strong CP problem. Twoprevalent ideas for resolving them are: supersymmetry (SUSY)—curing (i),1nd the seesaw mechanism[1, 2, 3, 4, 5]—curing (ii)—making SUSY see-saw very enticing. R -parity is assured as an automatic symmetry of thelow energy lagrangian[6, 7, 8] given B − L is a gauged symmetry broken by B − L = 2 higgs fields. Conservation of R -parity would guarantee a stablelightest SUSY particle (LSP),providing a good dark matter candidate[9] aswell as preventing catastrophic proton decay (caused by R -parity breakingterms of the minimal SUSY standard model (MSSM)). Finally, gauged B − L models embedded into the SUSY left-right framework, provide a cure to thestrong CP problem without the need for an axion[10, 11]. This leads us tofocus on the minimal SUSY left-right model and look at further constraintswhen the method of SUSY breaking is considered.The nature and origin of SUSY breaking has been a focus of a greatdeal of attention. The detailed nature of new physics that breaks SUSYis unknown—although there are several interesting suggestions[9]. Here wefocus on SUSY breaking is via anomaly mediation which is related to theradiative breaking of conformal SUSY[12, 13]. Anomaly mediated SUSYbreaking (AMSB) predicts all the soft SUSY breaking parameters in termsof one mass parameter (the mass of the gravitino) and the β and γ functionsof the low energy theory. As such, it is a highly predictive scenario whichavoids the SUSY flavor problem (no new flavor physics is introduced) andsolved the gravitino mass problem.There is, however, a serious problem that afflicts any AMSB model whoselow energy theory is the MSSM: the sleptons have negative mass squaredthereby leading to a ground state that breaks electric charge. Finding a cureto this problem is a difficult task given the predictability of AMSB and thefact that it generally decouples higher scale physics. This forces solutionsto include new couplings in the low energy theory or deflecting the AMSBtrajectories. While proposed solutions along these lines [14, 15, 16, 17, 18,19, 20, 21, 22, 23] are very illuminating they lack an independent motive.In this paper we propose a new way to resolve this problem of AMSBusing the minimal R -parity conserving SUSYLR seesaw model mentionedabove. We present an instance of this class of bottom up seesaw modelsthat has an intermediate seesaw scale (of order 10 GeV or so) and showthat the slepton mass square problem of AMSB is cured. Furthermore, ul-traviolet (UV) insensitivity is preserved; a featured that is shared with onlya few of the proposed AMSB solutions. The key to this is the existenceof light doubly-charged higgses[24] and light left-handed triplets and theiryukawa couplings to the lepton superfields. The effects of these doubly-2harged fields can be discovered in low energy experiments as they lead tocharacteristic mass predictions which are different from those of other SUSYbreaking scenarios. We will demostrate these differences between our model,minimal supergravity (mSUGRA), minimal gauge mediated SUSY breaking(mGMSB) and AMSB with a universal scalar mass addition m (mAMSB).Apart from experimental testability, a novel feature of our suggestion is thatthe cure is motivated from independent considerations. These defining phe-nomenological conditions:(i) SUSY(ii) local B − L symmetry as part of the gauge group SU (2) L × SU (2) R × U (1) B − L so that one can implement the seesaw mechanism(iii) B − L symmetry breaking is such that it leaves R -parity unbroken andassuring that there is a naturally stable dark matter candidate(iv) SUSY is broken radiatively by conformal anomalies, hence keeping thesoft terms (renormalization group equation) RGE invariant down tothe TeV scale (UV insensitivity).We will show in Section 3 how these consideration produce slepton pos-itive mass-squares, as well as introduce the model and gives its sparticlespectrum. Section 2 will give a brief overview of AMSB and introduce itsnotation—and terminology—a task to which we now turn. AMSB has many attractive features: a large number of predictions, few pa-rameters, an insensitivity to the UV and a mathematical framework thatelegantly describes its affects. The latter property allows one to express theSUSY breaking effects by analytically continuing parameters into superspace.AMSB then gives a method or set of rules on how to “promote” these param-eters to superfields. To establish these rules, as well as get the basic conceptsof AMSB we start with a generic SUSY theory given by the lagrangian: L = 12 Z d θ K ( D α , Q, W α ) + Z d θ W ( Q, W α ) + h.c. (1)3 W Rθ − +1¯ θ − − dθ + − d ¯ θ + +1Table 1: Weyl weight and R charges of superspace coordinates d W R K +2 0 W +3 +2Table 2: Derived weyl weight and R charge assignments for the K¨ahler andSuper Potentialswhere Q collectively represents the matter content and W α is the gaugecontent—the dependence of K on ¯ D ˙ α , Q † , etc. has been suppressed.AMSB then requires that K and W superconformal. To do this it is neces-sary to introduce the superconformal compensator φ which is an unphysical(in that its scalar and fermionic components may be gauged away) chiralmultiplet with a weyl weight d W ( φ ) = +1 and an R charge of +2 /
3. Thesuperconformal invariance then dictates the φ couplings so that the resultingtheory is invariant under weyl scale transformations and U (1) R .To see the required form for the φ coupling, we first note that the su-perspace coordinate charge assignments (See Table 1) force the K¨ahler po-tential and Superpotential to have the charges shown in Table 2. If we take d W ( ˜ Q ) = d W ( ˜ W α ) = R ( ˜ Q ) = R ( ˜ W α ) = 0 (with ˜ Q being the matter fieldsand ˜ W α the gauge fields, but not in the canonically normalized form), thenwe may write W = f W X W K = e K X K (2)where the “tilded” potentials are functions of only the “tilded” fields. Sincethe “tilded” fields have no charges, the resulting potentials don’t either; hence4ll the transformational weights belong to the X n : d W ( X K ) = +2 d W ( X W ) = +3 R ( X K ) = 0 R ( X W ) = +2Now because the X n carry charges, they can only depend on the conformalcompensator φ (we’ve already removed any other fields’ dependence into thepotentials). Therefore invariance necessitates X K = φ † φ X W = φ (3)We can now write the most general superconformal invariant lagrangian.It is given by L = 12 Z d θ φ † φ e K (cid:16) ˜ D α , ˜ Q, ˜ W α (cid:17) + Z d θ φ f W (cid:16) ˜ Q, ˜ W α (cid:17) + h.c. (4)This picture explicitly demonstrates the φ couplings as required by su-perconformal invariance at a cost of using non-canonically normalized fields.It is possible to return to the usual fields by defining Q = φ ˜ Q D α = φ † φ / ˜ D α W α = φ / ˜ W α (5)with the last equation being a consequence of the second. To illustrate howthese definitions return us to the canonical fields, we must write the potentialsschematically as e K = ˜ Q † e W ˜ Q + . . . (6) f W = L ˜ Q + M ˜ Q + Y ˜ Q + λ Λ ˜ Q + . . . + ˜ W α ˜ W α + . . . (7)It is then clear that the lagrangian of Eq. (4), combined with the field redef-initions Eq. (5), leads to a lagrangian L = 12 Z d θ (cid:2) Q † e W Q + . . . (cid:3) + Z d θ (cid:20) Lφ Q + M φQ + Y Q + λ Λ φ Q + . . . + W α W α + . . . (cid:21) + h.c. (8)5everal comments are in order regarding Eq. (8): first, the presence ofa linear or a mass term leave a φ in the superpotential resulting in thebreaking of superconformal invariance at tree level—something relevant forthe MSSM because of the Higgs’s mass term. Second, the nonrenormalizableterms always break superconformal invariance, and will always have the pairΛ φ to some power—as these terms are usually thought of as the result of athreshold or cutoff, this form will be important when we discuss intermediatethresholds and renormalization below. Finally, if no dimensionful couplingsare included ( L → M →
0, Λ → ∞ ), the lagrangian is superconformalinvariant at tree level; however, this is not true at loop level.When including quantum corrections a mass parameter, µ , will be in-troduced upon which the couplings (and the wave function renormalizationconstant Z ) depend. The mass parameter will also require some type of reg-ulator which can be chosen to be a cutoff Λ. This regulator is convenient touse because we have already established that such a cutoff must be pairedwith φ should it give rise to nonrenormalizable terms of the form in Eq. (8) .Thus, because it is necessary for µ to always appear in the ratio µ/ | Λ | , theeffect of µ is to promote the renormalized parameters to superfields throughthe rule µ → µ p φ † φ (9)The promotion of Z ( µ ) to a superfield Z ( µ ) and 1 /g ( µ ) to the superfield R ( µ ) gives rise to soft SUSY breaking terms. To obtain an expression forthose terms it is convenient to chose a gauge where φ = 1 + F φ θ (10)This leads to the following form for the soft SUSY breaking parameters m Q = 12 | F φ | dd ln µ γ Q (cid:0) Y (ln µ ) , g b (ln µ ) (cid:1) = 12 | F φ | (cid:20) β g a ∂γ Q ∂g a + β Y ∂γ Q ∂Y + c.c. (cid:21) (11) A Q = β Y Q F φ (12) M λ a = β g a g a F φ (13) The result that the UV cutoff gets paired with a φ is independent of whether or notit yields nonrenormalizable terms; however, it is a convenient illustration here SU (2) L × U (1) Y are infrared free, their β functions arenegative and hence the sleptons get negative mass-squares. This is a funda-mental problem because it implies the breakdown of electric charge in theground state. Before AMSB models can be phenomenologically viable thisproblem must be solved, but it is worth pursuing a solution because AMSBprovides decoupling of UV physics in an elegant manner (we discuss this be-low), naturally suppressed FCNC (the SUSY breaking parameters depend onthe Yukawa couplings and are diagonalized with them), and high predictivepower with a minimal number of arbitrary parameters (essentially all softSUSY breaking terms depend on only F φ ). It is therefore of great interestto seek reasonable models where the slepton mass-squares are made positivewithout destroying those good features. We will present such a model in Sec-tion 3, where we demonstrate that extending the MSSM to include neutrinomass—generated by an R -parity conserving seesaw mechanism—will simplyand effectively achieve this goal. Yet for the moment we will continue ourreview of AMSB and address the decoupling of higher scale physics.To illustrate the UV insensitivity of AMSB, consider a threshold Λ ≫ M ≫ F φ —such a scale may be an explicit mass term in the superpotentialor the vev of the scalar component of the superfield X . In either case weassume that below M there are no remnant singlets in the effective theory;this is the same as requiring that as Λ → ∞ , M remains finite. The previouscondition ensures that the effective theory’s lagrangian has the schematicform L eff = L Q + M − n f (cid:0) Q, ψ Q (cid:1) + M = 12 L Q + M + Z d θ (cid:18) Q M φ + Q M φ + . . . (cid:19) + h.c. (14)where n > L Q represents the part of the lagrangian involving only thevarious components of the matter superfields Q . This form of the effectivetheory (which is of the exact same form as the nonrenormalizable termsof Eq. (8)) makes explicit that the additional SUSY breaking effects fromthe threshold M go as F φ /M ≪
1. Thus, the rule µ → µ/ | φ | completelyparameterizes all the SUSY breaking in both the high-scale and low-scaletheories resulting in the maintenance of the AMSB trajectory below M . we use the notation X as the scalar component of the superfield X M → M φ + F φ " c F φ M + c (cid:18) F φ M (cid:19) + · · · θ ≈ M φ (15)in addition to which there is the requirement pairing Λ with φ . The quantumcorrections of the lagrangian of Eq. (14) force M to appear in the effectivetheory as ln | M/ Λ | and ln µ/ | M | (which comes when M is used as a cutoffin loop calculations). Using the replacement rules on these quantities givesln (cid:12)(cid:12)(cid:12)(cid:12) M Λ (cid:12)(cid:12)(cid:12)(cid:12) → ln (cid:12)(cid:12)(cid:12)(cid:12) M φ Λ φ (cid:12)(cid:12)(cid:12)(cid:12) = ln (cid:12)(cid:12)(cid:12)(cid:12) M Λ (cid:12)(cid:12)(cid:12)(cid:12) (16)ln µ | M | → ln µ | M φ | (17)and once again only µ → µ/ | φ | is required to capture all the SUSY breaking.The above argument may disturb the reader since the β functions changewhen crossing the threshold; however, what is actually happening is thatwhen the threshold is crossed, the removal of the heavy fields adds a term∆ β that results in a shift of the higher-scale β functions, β + , to the lower-scale β functions, β − . This property, namely∆ β + β + = β − (18)is the one that keeps the theory in the AMSB form.The UV decoupling of AMSB presents a major obstacle for fixing thenegative mass-squares of the MSSM since any high-scale tinkering will leavelittle to no evidence at the low scale. The new feature of models combining AMSB and SUSYLR is that the ef-fective theory below the v R scale contains Yukawa couplings to both theleft- and right-handed electrons in addition to those of the MSSM; hencethe slepton masses can be made positive. Thus, the marriage of SUSYLRwith AMSB gives positive slepton mass-squares and the resulting theorycombines the prodigious predictive power of AMSB, explains small neutrinomasses (through the seesaw mechanism), and retains a natural dark mattercandidate (the LSP is stable due to R -parity conservation).8 .1 The Model The particle content of a SUSYLR model is shown in Table 3. As the model isleft-right symmetric, it contains both left- and right-handed higgs bosons—inthis case B − L = ± R -parity may be preserved (a task forwhich the B − L = 1 doublets are not suitable). The presence of SU (2) L and SU (2) R triplets means that parity is a good symmetry until SU (2) R breaks.While the seesaw mechanism may be achieved with only SU (2) R higgs fields,demanding parity forces the left-handed triplets to be present these togetherthen yield positive slepton masses.Fields SU (3) c × SU (2) L × SU (2) R × U (1) B − L Q (3 , , , + ) Q c (¯3 , , , − ) L (1 , , , − L c (1 , , , +1)Φ (1 , , , , , , +2)¯∆ (1 , , , − c (1 , , , − c (1 , , , +2)Table 3: Assignment of the fermion and Higgs fields’ representations of theleft-right symmetry group (except for U (1) B − L where the charge under thatgroup is given.)The parity-respecting SUSYLR superpotential is then W SUSYLR = W Y + W H + W NR (19)9ith W Y = i y aQ Q T τ Φ a Q c + i y aL L T τ Φ a L c + i f c L cT τ ∆ c L c + i f L T τ ∆ L (20) W H = ( M ∆ φ − λ S S ) (cid:2) Tr (cid:0) ∆ c ¯∆ c (cid:1) + Tr (cid:0) ∆ ¯∆ (cid:1) (cid:3) + 12 µ S φS + 13 κ S S + λ abN N Tr (cid:0) Φ Ta τ Φ b τ (cid:1) + 13 κ N N (21) W NR = λ A M Pl φ Tr (cid:0) ∆ ¯∆ (cid:1) + λ cA M Pl φ Tr (cid:0) ∆ c ¯∆ c (cid:1) + λ B M Pl φ Tr(∆∆) Tr (cid:0) ¯∆ ¯∆ (cid:1) + λ cB M Pl φ Tr(∆ c ∆ c ) Tr (cid:0) ¯∆ c ¯∆ c (cid:1) + λ C M Pl φ Tr (cid:0) ∆ ¯∆ (cid:1) Tr (cid:0) ∆ c ¯∆ c (cid:1) + λ D M Pl φ Tr(∆∆) Tr(∆ c ∆ c ) + ¯ λ D M Pl φ Tr (cid:0) ¯∆ ¯∆ (cid:1) Tr (cid:0) ¯∆ c ¯∆ c (cid:1) + · · · (22)We have assumed that the singlet couplings absent from Eq. (21) are zero orsmall enough that they can be neglected. This condition is necessary to keepone singlet light ( N ) so that below the right-handed scale v R the theory isthe NMSSM with some additional particles. Although this may seem rather ad hoc , we do it out of convenience rather than necessity: the low scaletheory must be such that it avoids an MSSM higgs bilinear b term that istoo large[25]; the superpotential given above happens to be one. However,it is not the only one and several alternative methods exist[16, 12] to avoidthis problem. As any of these alternatives are equally valid, and because theexact form of the electroweak scale theory is irrelevant to the conclusions, wemerely select to use the superpotential above.The superpotential of Eq. (21) dictates that h S i = M ∆ λ S φ (23) h ∆ c i (cid:10) ¯∆ c (cid:11) = h S i (cid:18) M ∆ κ S λ S + µ S λ S (cid:19) φ (24)Eq. (23) should be evident from the form of the superpotential; Eq. (24)requires Eq. (21) to be recast as W H ⊃ (cid:20) − λ S Tr (cid:0) ∆ c ¯∆ c (cid:1) + 12 µ S φS + 13 κ S S (cid:21) S (25)10he inclusion of the nonrenormalizable terms of Eq. (22) (which are necessaryif R -parity is conserved [26, 24])will shift the vevs of ∆ c , ¯∆ c , and S by ∼ M /M Pl ≪ M ∆ so they may be safely be ignored. It is worth noting thatthe nonrenormalizable terms are only irrelevant because as M Pl → ∞ thevevs all remain finite; that is, they depend at most on 1 /M Pl .Because the nonrenormalizable operators are insignificant, Eqs. (23) and(24) are still valid and the theory respects the AMSB trajectory below v R :the advertised UV insensitivity. Yet even though the particles remain ontheir AMSB trajectory, the negative slepton mass-squares problem is stillsolved. This comes about because of the additional yukawa couplings f and f c which survive to the lower-scale theory.The existence of the f coupling at the lower scale can be seen from thesuperpotential Eq. (21): when S gets the vev of Eq. (23), the mass termfor the SU (2) L triplets vanishes while the SU (2) R triplets also get a vev, sotheir mass term remains. This would leave ∆ and ¯∆ massless below the right-handed breaking scale except that the non-renormalizable terms contributea mass through λ C M Pl φ Tr (cid:0) ∆ ¯∆ (cid:1) Tr (cid:0) ∆ c ¯∆ c (cid:1) → λ C M Pl φ h ∆ c i (cid:10) ¯∆ c (cid:11) Tr (cid:0) ∆ ¯∆ (cid:1) ≃ λ C v R φM Pl Tr (cid:0) ∆ ¯∆ (cid:1) (26)The same mass value of v R /M Pl is also responsible for f c surviving tothe low scale, but this time in the context of light-doubly charged particles.It is well known that the class of SUSYLR models considered here havelight doubly-charged particles[24] with a mass as mentioned above. Thequestion that needs to be addressed here is “how light?” If their mass is large, F φ ≪ m DC ≪ v R , then these particles merely introduce another trajectorypreserving threshold which decouples from the lower scale theory. For theright-handed selectron this would be disastrous as it would have a purelynegative AMSB contribution to its mass. Thus, it makes sense to demandthat the doubly-charged particles have a mass m DC . F φ .The existence of the SU (2) L triplets and the doubly-charged particlesbelow or around m / means that their couplings remain in the low-scalesuperpotential and are therefore important. For the sleptons, the relevantterms are W ⊃ f c ∆ c −− e c e c + i f L T τ ∆ L (27) We denote the scalar component of the superfield X as X f c and f allow the scalar e c and e mass-squares to be positive. Assuming that f , f c are diagonal in flavorspace (an assumption validated by lepton flavor violating experiments[27]),we need only f ≃ f ≃ f c ≃ f c ≃ O (1) to make the sleptons posi-tive. The only constraint here is from muonium-antimuonium oscillations[28]which demands that f c f c / √ m DC ≈ f f / √ m DC < × − G F ; how-ever, with both the doubly-charged fields and SU (2) L triplets having a mass m DC ≃ F φ ∼
10 TeV, this is easily satisfied. Furthermore, this constraintlimits the range for v R as m DC ≃ v R /M Pl ≃ F φ implies that v R ≃ –10 GeV.The amazing result is that AMSB and SUSYLR yield a sfermion sectorthat depends on very few parameters: F φ , f c , f c , in addition to the usualtan β and sgn µ (because of parity, f = f c and f = f c ). Interestingly, twoof the new parameters—the f c yukawa couplings—also have implications forneutrino oscillations. We now present the resulting mass spectrum for this model. For this analysiswe start by running the parameters of the Standard Model up to M SUSY ,match at that point to the NQNMSSM (Not-Quite NMSSM: the NMSSMwith doubly-charged particles, left-handed triplets, and two additional Higgsdoublets), and use the appropriately modified RGEs of [29] to get to theright-handed scale. Without loss of generality we assume that only one up-type Higgs and one down-type Higgs get a vev[30]. Additionally, we take thestandard simplifying assumption that only the third generation higgs yukawacouplings are important.Figure 1 shows the mass spectrum of the general SUSYLR model and theMSSM with other popular SUSY breaking scenarios (the figure is truly onlycomprehensible in color—a form available on line at http://arXiv.org ).The comparison was obtained by matching the gluino mass between themodels, and then running the masses down to the scale Q using ISAJET[31].The spectra in Figure 1 contain the generic features, though the figure wasgenerated using the points listed in Table 4. It is also interesting to notethat the heavier sfermion mass eigenstates are mostly right-handed contraryto most mSUGRA and GMSB scenarios.One of the more striking features of the SUSYLR+AMSB spectrum isthat gaugino sector masses are all relatively close to each other. This is12 massKeySUSYLRmAMSBmSUGRAmGMSB˜ B ˜ W ˜ g ˜ u ˜ d ˜ ν e ˜ e ˜ u c ˜ d c ˜ e c ˜ t ˜ b ˜ ν τ ˜ τ ˜ t ˜ b ˜ τ ˜ B ˜ W ˜ g ˜ u ˜ d ˜ ν e ˜ e ˜ u c ˜ d c ˜ e c ˜ t ˜ b ˜ ν τ ˜ τ ˜ t ˜ b ˜ τ ˜ B ˜ W ˜ g ˜ u ˜ d ˜ ν e ˜ e ˜ u c ˜ d c ˜ e c ˜ t ˜ b ˜ ν τ ˜ τ ˜ t ˜ b ˜ τ ˜ B ˜ W ˜ g ˜ u ˜ d ˜ ν e ˜ e ˜ u c ˜ d c ˜ e c ˜ t ˜ b ˜ ν τ ˜ τ ˜ t ˜ b ˜ τ Figure 1: The mass spectrum for the superpartners of the Standard Model forfour different models (in four different colors): SUSYLR+AMSB, mAMSB,mSUGRA, and mGMSB. Note that for the SUSYLR+AMSB, ˜ t and ˜ b aremostly right-handed; in contrast with the usual mSUGRA or mGMSB caseswhere they are typically mostly left-handed.13USYLR+AMSB AMSB+ m mGMSB mSUGRAtan β = 15 tan β = 15 tan β = 15 tan β = 15sgn µ = +1 sgn µ = +1 sgn µ = +1 sgn µ = +1 Q = 550 GeV Q = 558 GeV Q = 899 GeV Q = 537 GeV F φ = 30 TeV F φ = 30 TeV Λ = 90 TeV m = 190 GeV m = 290 GeV M mess = 180 TeV m / = 285 GeV v R = 135 EeV A = 241 GeV f = f c = 0 . f = f c = 0 . m to F φ / π . Q is the scale at which the the masses are reportedby ISAJET. Because it is not widely known, we remind the reader that themetric prefix E in the above table means “exa” and is 10 .unique from the popular scenarios displayed in Figure 1 and is due to thecontributions of the SU (2) L and U (1) Y extended particle content at lowenergy. Such a massive wino consequently relaxes the naturalness argumentsmade in [32, 33]. These arguments proceed along the lines that squark massesand the µ term must be below around 1 TeV to preserve the naturalnessof SUSY. Therefore, a naturalness upper bound can be put on the winomass. Such an upper bound suggests that run II of the tevatron should haveexplored most of the viable wino parameter space, which would not be thecase here.Furthermore, we can achieve regions in parameter space where F φ is lowerthan would be possible in other AMSB models without violating these nat-uralness bounds. Specifically we can investigate a point in parameter spacesuch as F φ = 15 TeV, tan β = 15, f c = f = 1 and f c = f = 1 . f sare at the right-handed scale) with a spectrum given in Table 5. Here eventhe sfermion sector has very little hierarchy in it. Such spectra are exoticcompared to typical mSUGRA and mGMSB type models although they arepossible in deflected AMSB[34].From a cosmological point of view, there is a potential problem with theincrease in SU (2) L and U (1) Y gauge coupling strengths at the right-handed14article masses (GeV)˜ t b u d t b u c d c ν τ τ τ ν e e e c B W G F φ = 15 TeV, tan β = 15 and at theright-handed scale f c = f = 1 and f c = f = 1 .
6. Masses are evaluated at Q = 325 GeV.scale: they cause tachyonic squark masses at that scale (remember thesegauge couplings give a negative contribution in Eq. (11)). Theories withtachyonic squark masses have been studied in the GUT framework and werefound to be safe albeit unsavory[35]. Large reheating temperatures will causecharge violating vacua to disappear[36] and tunneling rates to the bad vacuaare too small in most of the parameter space[37, 38] to cause a problem.Continuing along cosmological lines, both mass spectrums shown aboveindicate that the sneutrino is the LSP in this model. Both the tau andelectron sneutrinos are LSP candidates depending on the relative sizes of f and f . Although sneutrino dark matter is highly constrained[39, 40], therecould be other dark matter candidates such as light singlet fields mixed withHiggsinos. It could also be that the sneutrinos generated from late decay ofthe gravitino are dark matter. We are currently investigating these scenarios.15inally, let us consider the sleptons masses—the main purpose of thispaper. As advertised earlier, these are positive and depend on just a fewparameters: F φ , f , f (since we have preserve parity at the high scale inthis paper f c = f and f c = f at the right-handed scale) and to a lesserextent on tan β and the right-handed scale. The relative sizes of the massesare controlled by relative f coupling: the larger the coupling the larger themass, e.g. increasing f would raise the mass of the left-handed slepton. Suchan affect contrasts strongly with other non-AMSB models with light doubly-charged higgses where the right-handed stau mass drops with increase in f c type coupling [41, 42]. We have presented a new way to solve the negative mass-squared sleptonproblem of AMSB using a minimal, bottom-up extension of the MSSM thatincorporates neutrino masses (via the seesaw mechanism), solves the strongCP problem, and resolves the R -parity violation problem of the MSSM. Slep-ton masses are rescued from the red by their couplings to both remnantdoubly-charged fields and left-handed triplets. Constraints from low energyphysics and the non-decoupling of these additional fields require the seesawscale to be around 10 GeV clearly distinguishing our model from GUTseesaw models.The model we presented has soft terms which remain on their AMSBtrajectory down to the SUSY scale. We have shown the sparticle spectrumfor this model and compared it with typical predictions from other SUSYbreaking scenarios finding significant deviations, especially in the gauginosector. Furthermore, in some regions of parameter space it is possible toproduce a spectrum with little hierarchy between sleptons and squarks.
We are indebted to Zackaria Chacko for his discussion and proof-readingof our paper. We would also like to thank Markus Luty and Ken Hsieh forhelpful discussion on AMSB. Furthermore, Michael Ratz has our appreciationfor his discussion on the early universe vacua. Finally we would like toacknowledge the assistance of Craig Group for help with the online tool16UPERSIM. This work was supported by the National Science Foundationgrant no. Phy-0354401.
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