JHEP08(2005)031 April 8, 2005 July 22, 2005 mology of August 8, 2005 df/jhep082005031.pdf n generate a re- Received: Accepted: iving significant correc- Published: third generation fermion ily degeneracy of the soft abundance. lutions capable of fitting all tigate the phenomenological nd mixing angles. It also gives of Oxford, ab http://jhep.sissa.it/archive/papers/jhep082005031.p [email protected] , Published by Institute of Physics Publishing for SISSA and Graham G. Ross a Breaking, Supersymmetry Phenomenology, Cos A spontaneously broken non-abelian SU(3) family symmetry ca [email protected] Rudolf Peierls Centre for1 Theoretical Physics, Keble University Road, Oxford OX1Theory 3NP, Group, CERN, U.K. 1211E-mail: Geneva 23, Switzerland SISSA 2005 b a c Michael R. Ramage ° Abstract: alistic form for , chargeda new and solution to masses the a mass SUSY flavour SUSY problem breaking by terms. ensuringmasses near However fam means the that need the totions generate group to large must the be third strongly familyimplications broken squark of to and such SU(2) slepton breaking masses. g andpresent show We experimental that inves measurements it and leads bounds to as new well so as the Keywords: Theories beyond the SM, Supersymmetric . Soft SUSY breaking and family symmetry JHEP08(2005)031 1 2 4 6 8 14 ith being tri-maximal , the effective Yukawa e capable of describing trongly broken to SU(2). rate the third generation n underlying non-abelian the 2 and 3 directions of nd in agreement with the istent with an underlying ctor the mixing angles are right-handed . In s to different families. To s the same general form as ressing of the questions left mixing angles. This follows ving matrix elements which enerated by a second stage of al with equal components in is the difference between the erlying SU(3) symmetry. The om this vacuum alignment. ads to mixing angles in the up n have shown that the mixing of directions. τ – 1 – ν and , µ , ν e ν -terms D directions while that of the solar neutrinos is consistent w τ , ν µ In [1] it was argued that this data suggests the existence of a ν family symmetry capable ofdemonstrate relating this the an couplingSO(10) SU(3) of Grand family the Unified symmetry, symmetry, Higg theboth was constructed the largest and quark cons and shownquark to lepton and masses b charged and lepton mixings. massesThe In first the two order family generation to quark symmetry and gene must chargedbreaking be lepton which masses s are preserves g aare discrete subgroup equal of in thecouplings the SU(3) constrained ha 2 by and additionaland 3 abelian down directions. symmetries quark le (andmeasured charged With values. lepton) this sectors Although breaking which thethat are Dirac scheme of small mass the a of thefrom and neutrinos the ha see-saw the mechanism light withthis sequential neutrinos the domination have of near large the bi-maximalthe mixing second stage comes of from breaking,near the a tri-maximal direct correlation mixing consequence in of of the the und solar neutrino also follows fr with equal components in the 1. Introduction The origin of fermion massesunanswered and by mixings is the perhaps Standard the Model.mixing most angles p Particularly in noticeable thesmall. quark However and recent lepton measurements sectors. ofthe neutrino In atmospheric oscillatio the neutrinos quark isthe se consistent with being bi-maxim Contents 1. Introduction 2. The sparticle spectrum 3. The effect of 4. Calculation and constraints 5. Results and discussion 6. Summary and conclusions JHEP08(2005)031 sγ −→ b bundance which tion to the family metry is that, while amily symmetry this renormalisation group ew the expectation for ns of the SU(3) family el, it seems likely that he minimal case where gnificant change in the acy of the squarks and tivate this study in the SU(3) family symmetry. y of following nce of SO(10) breaking. ) spectrum often assumed racy of the Higgs scalars. lternative solutions which solutions which satisfy all ng of this degeneracy for te the third generation of symmetry breaking masses on 4 we discuss the various pling unification and allow arly degenerate to suppress s are presented in section 5. nitial mass one obtains the ew solution to the “family” ng effects in dipole electric masses must distinguish the s or sleptons in a given rep- ia various “mediator” mech- f all the squarks and sleptons. nomaly mediation. ts and summarize. e an effect on the third sfamily esent measurements of -term contributions to the scalar masses, D – 2 – As stressed in [1] an important byproduct of such a family sym In this paper we will study the phenomenological implicatio The paper is organised as follows. In section 2 we briefly revi Of course, in the case of the non-abelian family symmetry solu unbroken, it guarantees theresentation degeneracy of of the a Standard family Model.problem, of the As squark need a to result haveflavour it the changing provides squarks a neutral and charged n currentsmoments. sleptons and ne This to solution suppress eliminatescommunicate CP supersymmetry the violati breaking need to toanisms, the appeal including visible to gravity sector mediation, the v gauge a mediation and a symmetry solution toan the family underlying problem. Grandsleptons Unified We at symmetry concentrate the guarantees on unificationquark the t scale. and degener lepton Howeverthe masses the third need requires generation to thatcontext of genera there of squarks is a and strongthe specific sleptons. breaki structure implementation applies Although of more wethird a generally. family mo family Any from theory the symmetry light ofmasses mod families fermion too. and this is Wefrom likely will the to show hav dominant that breakingresulting the phenomenology. of splitting In the of particularthe family we the current symmetry find bounds third leads a on famil for to new supersymmetric radiative class si states, electroweak of have breaking, gauge are cou consistent with pr and the anomalous magneticgenerates the moment observed of dark matter the abundance. muon and havesfermion an and LSP Higgs masses a We in then a elaborate supersymmetric on the theory problem with of an possible which might violate the bounds on FCNC, in section 3. In secti components of the globalflow fit and the and analysis the ofFinally method the dark in used matter section to abundance. 6 perform The we result the discuss the implications of the2. resul The sparticle spectrum In a theory with anof underlying SO(10) the symmetry states the softWhen in super the a symmetry singledegeneracy is family applies extended to will soft to be supersymmetryIf, include breaking degenerate in masses a addition, o in non-abelian one theConstrained SU(3) assumes Minimal abse the f Supersymmetric Standard Higgs Model scalarsin (CMSSM have supergravity the models, same see for i example, [2 – 9]. problem, there is no symmetry reason for assuming the degene JHEP08(2005)031 is which 3 If this is φ . R 4, its precise . 0 ' SU(2) (1) breaking of the ⊗ O t it was assumed the a/M amily symmetries are L non-abelian symmetry left- and right- handed y breaking mass terms y uarks and sleptons in a a given family, but the remain degenerate at the the first two families and e different masses. . t-handed charged slepton. due to two effects. Firstly cy follows from the family group SU(4) preserves the nificant changes in flavour ds on the splitting between ts and, separately, between the SU(3) family symmetry mes from the field mmetry breaks the solution heory but the analysis of the SU(2) s case an underlying SO(10) xplored in [10–13]. Here we the degeneracy maintained is ters introduced. In this paper y between the three families C). Such plets comes from the D-terms lities elsewhere but we choose ⊗ where the dominant breaking is ) doublets of squarks and sleptons R is the supersymmetry breaking mass 1) at the unification scale between the 2 0 . This gives a relative mass difference of m (0 are the quark and lepton supermultiplets, a. (SU(2) O i L ψ = are the (SU(3) antitriplet) fields which break ® – 3 – 3 3 i φ φ ­ where D | i ψ † ) i ψ ( 2 0 is the mass of the messenger communicating family symmetry representation. Combined with the SU(3) family symmetry th m where the fields M R D | 2 term − SU(2) /M D j ⊗ φ j L ψ † 1) being undetermined. Breaking of ¢ i ) to the third generation. In [1] this was chosen close to unit < φ i ψ Another possibility is that the dominant breaking is that of Of course both effects are likely to be present in a realistic t In the model discussed in [1] the dominant soft supersymmetr In [1] the full SO(10) symmetric model was not constructed bu ¡ a/M 2 0 ( O has a vacuum expectation value (vev) value ( the family symmetry. breaking to the quarks and leptons. The dominant breaking co scale in the visible sector. This clearly leads to degenerac (we do not needsymmetry). to specify The origin the of mediatorm the sector splitting as between the family degenera multi third and first two generations mustfamilies be coming compared from with flavour the changing boun neutral currents (FCN third family gives a reduced effect on flavour changing bounds triplets under the SU(3) family symmetry, and the dominant breaking effectdegeneracy the of result is the that upgiven the squarks SU(2) Pati and Salam sneutrinos and of the down sq means that there is degeneracy of the SU(2) come from the families as in the CMSSMstates. but there may be breaking between the when giving the thirdsymmetry family of would fermions guarantee theirbreaking the masses. of degeneracy In SU(3) of thi tothe SU(2) all third means the family that is the states lost.between degeneracy in the If between right-handed SO(10) down is squarks brokenthe and instead the right-handed to slepton up SU(5) double squarks,However, the unlike the squark CMSSM, doublets these and two the groups righ of statesgeneral can case hav is very difficult due towe the shall large explore number of the parame implicationsof of the the family second symmetry but possibility unification the scale. members of We an shallto SO(10) explore start multiplet the with more this generalto simple possibi the case as family the problemchanging breaking and, and of as CP the discussed violating family processes. above, sy can lead to sig The effect of breakingwish this degeneracy to has explore been furthersolution extensively differences e to between the the familybroken CMSSM and problem. are and dependent These the on arise the when pattern of the symmetrySO(10) GUT breaking was and broken f close to the unification scale to SU(4) JHEP08(2005)031 - y A D ass and (3.2) + φ pectrum which has a vev when the SU(3) 23 D | ry is broken along a φ 2 ying family symmetry ukawa couplings which ng parameters? The versality in the /M d to unacceptably large etries in supersymmetric j erations necessarily have nsistent with the precision ) (3.1) φ d in the CMSSM. asses generated by the j 2 − ymmetry breaking has been he light generations is small. ψ ies, thus reducing the relative the case. We will now address m † is the superspace coordinate). : ¢ ion soft masses that potentially i − 2 − θ φ i 2 + m e average quark and slepton masses 2 ψ | m . The expectation values of ¡ 003) in the up sector. Such splitting i 6= ( . − ρ 2 0 2 | -term gives contributions to the masses 2 + 1 φ i (0 g m D m Q O spurion ( i = -terms. As originally noted by Murayama and i D 2 h θθ D | 2 + 0 − g φ – 4 – m φ i X −| 2 = | + 1 fields V φ − h| = is the superpotential) which are generated on supersymmetr i (1%) on the breaking of the first two generations coming from D , W h i ( O ρ A | The corresponding soft mass breaking term gives a relative m -terms 02) in the down sector and . D ) between the first two generations. The fit to the light quark s W b. 0 (0 m O = b/M ( ® = O 2 23 φ ­ b/M is the gauge coupling constant. Then the = g ® In this section we have assumed that the only source of non-uni The simplest example of this effect is when a U(1) gauge symmet Finally what about the remaining soft supersymmetry breaki The second breaking of the family symmetry is due to the field can differ if their SUSY masses are different, 3 23 − φ 3. The effect of family symmetry is broken. However,the this issue is of not possible necessarily violate additional the contributions family to symmetry the solution sferm to the flavour problem. φ They are the sameas in in the the CMSSM. effectiveare This theory responsible is coming for because fermion fromdictated they masses an by arise and the underl from the need the to choice effective get of Y viable family fermion s massessector as arises is assume from the non-renormalisable term, requires theories coming from the family symmetry There is a long standing problem associated with family symm et. al. [15, 16], since the quarks and leptons of different gen different family charges, the soft supersymmetry breaking m terms for squarksflavour and changing sleptons effects. are not degenerate andflat can direction lea by charge +1 and breaking in the visible sector through a is within the bounds of terms arise from FCNC. ­ difference of where radiative corrections from gauge interactions increaseat th low scales without amplifyingbreaking the effects. breaking between Secondly the famil As mixing a of result the the third dominantbounds family breaking coming to of from t the flavour family changing symmetry processes is [14]. co of the light scalar fields JHEP08(2005)031 + − + φ φ 2 s and (3.3) ˜ m e | -terms and m 1( D . n masses f mass of + 0 φ and ˜ < ther µ | -terms will not 2 − m D , ˜ m d is occurs when the of SU(3) too. The − of the masses giving m aking mechanism for e gauge subgroup of because the radiative ressed by a factor of the nological requirement y symmetry breaking tries of the model can , ˜ 2 + s r non-abelian). In this tinuous but our results − q. 3.3. m tisfied. Even if there are | m nds on quark and lepton m is a completely analogous ffects may readily be small s lighter than the mediator . s mechanisms of supersym- of ˜ n to the soft masses at low ndent of the gauge coupling ences at low scales but these luated at a high (unification 1 potential problem because 1 3 and − ses if + 10 m . ¯ ¯ ¯ ¯ ¯ 2 e 2 e ˜ m m − + ˜ are less than 2 µ 2 µ ˜ ˜ m m − ¯ ¯ ¯ ¯ ¯ m – 5 – is less than the family symmetry breaking mass , if the scale of family symmetry breaking is close to and , M will arise only through radiative effects if − ¯ ¯ ¯ ¯ | φ + 2 d 2 d 2 − m ˜ fields. Note that this contribution does not decouple even m m m i and ρ − + ˜ − 2 s 2 s + 2 + φ ˜ ˜ m m m ¯ ¯ ¯ ¯ | -terms so the problem disappears [18]. In our analysis we have D . In this case there is a suppression of due to the fact that these graphs will always involve states o > ). There will be no such suppression for the squark and slepto > ). There are several ways this may happen: > + + | need not be degenerate. However in this case the question whe M 2 e + <φ − m < φ φ are the charges of the + ˜ < φ i ( 2 µ Q ˜ and cause problems is dependent onit the may fine readily details happen of the that SUSY bre cut-off scale (the string or the Planck scale), the radiative e as they will not berise enhanced to by these large logarithmic logsscale. enhancements because In is the this running cut case again off one at readily the satisfies famil the bounds of E have different Yukawa couplings. In specificprevent this, models readily the leading symme todifferent the couplings bound for of Eq. 3.3 being sa metry breaking, the graphs generating these massesorder via the mediator will be supp apply to the case that the family group is a discrete subgroup implicitly assumed the family symmetry group, SU(3), is con O consistent with the boundsSUSY of breaking Eq. mediator 3.3. mass scale A specific example of th because the associated radiative effectsmass only involve state the underlying continuous familycase symmetry there group (abelian are no o m | A quantitative measure of the problem is given by the phenome , | 2. If the initial masses are not degenerate there is clearly a 1. If the initial masses are degenerate, as happens in variou 3. Finally it may be that the family symmetry is only a discret 2 d ˜ m if the U(1) breakingand scale cannot is be very suppressed high.effect by for Moreover taking it the the non-abelian is coupling case. indepe small. There on squark and sleptonflavour masses changing coming processes from which the giveor smallness [17] string) of scale for bou the masses eva Note that there are muchare stronger weakened limits at for these highscales mass scales differ coming due from to the masses. universal contributio From this one may see that give unacceptable contributions to flavour changing proces where JHEP08(2005)031 µ lculate nd one , fermion SOFTSUSY Z micrOMEGAs M necessary for Lagrangian allowed ) we use γ calar spectrum at the s SOFTSUSY case of a broken family f the discrete subgroup) nd so will not affect the etry breaking needed to nclude the 2-loop contri- les such as X ted to include our family rmulae in [20]. ch is indeed the case for generation of squarks and plitting of the third family tial guess to find a set of mily symmetry ordering the → w scale constraints. We use found in [19]. A comparison en the first two families is a calculation of the SUSY con- ed. We shall now proceed to ion to the GUT scale bound- ations of the modification of he context of a specific model, b d fermion mass spectrum and d Yukawa couplings and the ( s from B king of the family degeneracy in tions, see [26] and the papers on -terms, the difference between the continuous – 6 – D via an interface conforming with the Les Houches Ac- v.1.8.7 [19], one of several publicly available codes, to ca SOFTSUSY and similar programs, for example [21 – 23], was made in [24] a SOFTSUSY SOFTSUSY reason is because, apart from the and discrete cases isby due the to discrete the symmetry. appearance Typicallythese (depending of terms on new will the terms appear size in o atstructure the higher of non-renormalisable the order soft a terms assumed here. In our analysis we impose the following constraints: For the calculation of the neutralino relic density and We use masses and gauge couplings areary input conditions. as constraints insparticle An addit masses iterative and algorithm mixingsfull proceeds consistent 2-loop with from renormalization the group an equations high for ini and the lo gauge an parameter. For the soft massesbutions we in use the the 3rd fullbetween family 1-loop approximation. RGEs and Full i details can be v.1.3.1 [26], linkedcord to [27] standard thatthe contains relic all density the calculation. relevantwhich parameter For they details were based of [9, these 28 calcula – 36]. can directly compare the codes online at [25]. the sparticle spectrum andsymmetry-inspired mixings. boundary conditions The and code atribution has routine for to been the the augmen muon anomaloususes magnetic a moment bottom-up using routine the fo in which various low energy observab the model investigated ininvestigate [1] the upon phenomenological which implications this ofthe analysis the soft is brea SUSY bas breaking massesgenerate coming the from fermion the masses.negligible pattern change of In to symm practice the the CMSSMcan breaking boundary be betwe significant. conditions but Although we the havethe s motivated this range study of in t splittingso we is consider likely is to dictated befermion by the the masses same which observe for any is theory consistent with with an the underlying FCNC fa bounds. 4. Calculation and constraints In what follows wethe investigate CMSSM the spectrum phenomenological discussedsymmetry. implic above which In corresponds to particularunification the scale we to compare onecharged the in sleptons which case is the allowed of mass to squared a vary of by degenerate up the s to third 20%. We will assume that at least one of these solutions holds, whi JHEP08(2005)031 nstraints as opposed pole b n the others. /m 2) ntly recalculated / nsistency between b atest results of the ng the experimental ht contribution [42]. esults from negative m ( ng the only one that . cross section [38], the rror in our calculation 4 . , account. From [40], MS c ion which includes both − − 4 4 . , [43]. However this second e m , − − g upper and lower bounds 10 + 10 bounds on the discrepancy e 10 10 10 − × − σ × 1 × 10 d ratio 10 28 99 GeV . × 4 × 30) 38) . ≥ . 5 0 0) < 0 . . R ) ˜ e decay data, and taking into account 9 ± ± 42 γ m s τ ± < 70 34 . X 5 . . µ . → µ = (3 = (3 b ( < δa – 7 – We include the 2 = (24 B bound, exp matrix element. For details see [31]. SM 10 ) ) < i − σ SM µ γ b γ | 103 GeV 4 s a s 10 A − X X − ≥ − : data for the hadronic contribution [38] and the most × V 10 ) ± σ 5 → → ˜ χ . ) The most recent world average for the branching ratio − exp µ cb × b 6 (¯ b e γ a ( m ( s A + 40 B − B e . X V 2 ) → sc . We use the 2 (¯ | µ b γ ( s 2) 2 B − X h g The following lower limits from LEP provide the strongest co ( ≡ constraint should perhaps be viewed more provisionally tha µ in the µ a 2) pole b QED correction [41] and the most recent hadronic light-by-lig − 4 /m g We will use this Standardas Model representative estimate of of the theStandard error Model theoretical to and e be SUSY expected contributions.and in We theoretical do our this errors calculat by in combini quadratureon to the obtain branching the ratio followin at 2 while the current Standard Model theory value is [45] is [44] This is unfortunate sinceunambiguously it determines the is sign one of of the most important, bei ( as the allowed rangethese of results the and SUSYthe those contribution. susceptibility obtained Due to by to change using the of the inco measurement of the where We include these lower bounds in our plots. recent data from themuons BNL E821 [39]. experiment incorporating Weα the use r the values from [40] which include the rece on sparticle masses from direct searches [37]: Similar values were obtained by anpaper independent does calculation not take the new theoretical results [41, 42] into between experiment and Standardcalculation Model based theory on assuming the l pole c This takes into account only those results that include the improve 1 m Branching Ratio Muon anomalous magnetic moment Direct searches to JHEP08(2005)031 ) 2 0 h m , m 2 / 1 rameter m resonance. 2 0 A eak symmetry The boundary [37] in the regions ndard Model like, it is less restrictive n broken correctly, β respectively. We set tralinos (see [47] and ral Higgs sector. This c y, any regions in which e here. kewise omitted. ons, or some relic density ower bound. We also plot the lower bound on Ω papers [48 – 54]. and and for the baryon density, 0 a global minimum of the other papers are likely to be due codes in ref. [24]. = 0 for simplicity. Our plots , b 0 < , 008 009 . . ns, i.e. all scalar masses are set is the mixing angle relating the a A 0 2 trum and mixings and the neutralino | +0 − 1287 0161 0181 α , . . . µ | 0 0 +0 − 135 , and . < µ 2 1 GeV 2) h . 1126 = 0 . marking the position of the 2 − 0 1 . However, for large tan h g ˜ χ = 0 β CDM m = 114 2 m Ω – 8 – h 0 h < = 2 at the GUT scale. Figure 1 shows the ( m 0 2 Higgs resonance, and much less so if we allow for a CDM A / 1 Ω 0945 0 m . m A 0 = 10, 30 and 50 for plots We also display the contour β The analysis of the data from WMAP gives a best fit value for 0 h 0009 [46]. This implies that the CDM density is . 0 ± vanishes, marking the border of correct radiative electrow ) is almost exactly equal to 1, where 2 α | level. This can be an extremely stringent bound on the MSSM pa µ 0224 − | . σ β 0, also signalling that the electroweak symmetry has not bee = 0 < 2 0 h 2 A b and all gaugino masses to two loop effective Higgsm potential cannot be found.have been Similarl excluded. Regions with tachyonic sfermions are li corresponding to the LEP bound on the lightest SM of parameter space wherei.e. the sin( lightest MSSM Higgs boson is Sta breaking has been plotted. In the region where on which mass eigenstates to the gauge eigenstates in the CP-even neut condition applies throughout the parameter space we analys and indicate the allowed regions ifthe we locus choose to of discard points the for l which the matter density of the universe of Ω can be neglected. We plot values for which Ω source of otherenhancement than mechanism neutralinos such such as as non-thermal axi productionreferences of therein neu for more examples). In these instances, at the 2 space, especially in the case of small tan due to the presence of the 0 0 in accordance with the expectation for ( Any discrepancies between the results shown in figure 1 and those of m 2 can be seen to be in reasonable agreement with those in recent µ> plane of the CMSSM for tan 5. Results and discussion We first present our resultsto for universal boundary conditio Lightest Higgs Mass m Correct EWSB / Tachyons / Higgs potential unbound from below to the differing approximationsrelic used density. One to can compute find the comparisons sparticle between various spec commonly used Neutralino dark matter JHEP08(2005)031 . ) µ b of 2) − 1 GeV; ) is too . g γ s = 10, ( X β = 114 → 0 b h ( parameter space ) tan m B a tour ) are t-channel sfermion = 50. In this sector of b ( β re is a narrow band for each resonance. The focus point 0 taus becoming important close are satisfied. In this region, for h 1287 and the yellow(v. light grey) . µ 0 2) < Higgs exchanges. Rapid annihilation − 2 0 g bounds on the region favoured by ( h is shown as a dark red (dark grey) line and A are shaded light pink (v. light grey). σ 0 = 0 in the CMSSM for ( 2 A CDM h 0 and ( ) m Ω c = 100 GeV and tan A 2 and 2 ( = 2 h – 9 – < σ / 0, CDM 0 1 than shown in these plots. The boundary where = 30, 50, there is a narrow filament of acceptable 1 ˜ χ 0 β m m CDM µ > m 0945 . = 50 the main channels in the favoured region are again β becomes very small, resulting in a large Higgsino component ) plane with 2300 GeV for | 0 ) µ = 50. The medium grey region is excluded because the LSP is a stau, the | a ' resonance where 2 ,m ( β 2 . For tan 0 0 / ˜ τ 1 A m m m ) tan c = where the bounds on Ω 0 1 β ˜ χ The ( dominates as the resonance is approached. Near to the band of m 0 A = 10, 30, the main annihilation channels for the neutralinos = 30 and ( We focus our discussion on the neutralino relic density. The β = 0 begins at β | µ | the LSP, occurs at a higher value of region [55 – 58] where excluded by the chargino mass, for tan relic density in the favoured region corresponding to the via the regions satisfying only the upper bound on Ω value of tan The position of the exchanges to leptons and quarksto with coannihilations where with s tan t-channel sfermion exchanges, but also s-channel and orange(light grey) regions represent the 1 the red(dark grey) strip shows where 0 Figure 1: small in the lilac(darkish grey) region; the black line corresponds to the con tan blue(v. dark grey) region is excluded by the LEP limits on sparticle masses and JHEP08(2005)031 2 | µ | ,       2 2 δm though the δm − ffects further. 1 00 1 0 001+ 0 1 00 1 0 0 0 1 0 ) family symmetry. or on those excluded    e third family squark    2 0 µ 2 0 ily sfermion masses we m g the expression for T ZZ 2) ure 1, but with the soft m undary conditions in this ion where the electroweak tralino relic density in the − following form at the GUT so we will ignore them. )= s a small change in the slope g )= very significant change in the Re Π T ZZ d chargino coannihilation also in figure 2, especially for large G elements of the sfermion mass G . However, this region is well are the tadpole contributions, 1 2 slepton sector. m table minimum cannot be found. m i ( ( − R Re Π R /v ˜ 2 e ˜ i 2 e 2 Z 1 2 t m m m − 1 2 . Including quantum corrections [59], 2 Z )= 2 )= − G 2 H m G where 0 is found when the minimisation conditions β 1 2 = 0. The effect of decreasing the third family m m i m 2 ( | ( < ˜ − 2 L µ ˜ 2 L /v | 2 1 i 2 – 10 – | self-energy. This implies the following condition tan m t m µ 2 H 2 − | Z − . 2 H m β µ )= i )= 2 m − G 2 H G 2) − β m m 1 m tan − ( 1 2 ( 2 H R g 2 H R ˜ = 2 d ˜ 2 d m m tan i m in the three different cases. This has a rather small effect on m 2 H ˜ τ = ' m )= 2 m )= | G G µ | = m m ( 0 1 ( ˜ χ R R ˜ 2 u ˜ 2 u m m 2 is the correction coming from the SU(3) family symmetry. Al 1. Here, m . ) or direct search constraints and we will not discuss these e γ s À )= )= = 0 X G G is the transverse part of the 2 β m m ( → ( δm ˜ T ZZ 2 Q ˜ b 2 Q . As we shall see, this is predominantly due to the change in th ( One of the main features of the plots in figure 2 is the large reg We now compare these results with those predicted by the SU(3 From figures 2 and 3, it is clear that altering the universal bo m β m B from the EWSB conditions and the RGE for The boundary of this region corresponds to are applied to the scalar potential indicating that an accep sfermion soft masses on EWSB can be understood by considerin the preferred region of parameter space. However, there is a region of allowed electroweakcase symmetry of breaking decreased third and family thetan soft sfermion neu masses as shown soft masses, and is relatively insensitive to changes insymmetry the is not broken correctly. Here, Due to the variation ofof the the slepton boundary soft mass matrices there i by and Π Since we do notconsider know two the additional sign cases.supersymmetry of breaking the Figure sfermion correction 2 masses toscale: squared is the taking the third the fam same plot as fig parameter space annihilation to gaugebecomes bosons is important enhanced resulting an inoutside an the range acceptable favoured relic by density ( where since tan model also predicts smallmatrices, corrections they to have the a (1,1) negligible and effect (2,2) on the phenomenology way has a relatively small effect on the regions favoured by ( and the same for figure 3, but with JHEP08(2005)031 ) = and G . 2 m 33 | ( ) 1 ˜ R 2 Q ˜ 2 e M m | m 2 1 g 6 5 ) = result the range G − m 2 ( | ˜ 2 L 2 m n be achieved by large ) M | b 2 2 ( lly is reduced for a given g ) = h the focus point region in 6 ed parameters ( G , − m and a new strip of acceptable β ( 1 2 2 0 | R 2 H ˜ t 2 d a m m | m tan in the third family approximation 2 − + 6 ) = 2 H G ´ m T ZZ m 33 0. ( ) R R ≥ ˜ 2 u ˜ 2 u Re Π ) 2 m | c m 1 2 ( µ | term is suppressed and therefore we require the – 11 – + ) = + ( 1 G 2 Z 2 H 33 m ) m ( m ˜ 2 Q ˜ 1 2 2 Q m m ≥ 2 + ( 2 H 2 2 H m 100, the − m ≥ ³ β 2 ) | 0. 2 t a ( y < | 2 ). The new medium grey region on the left hand side is excluded by EWSB | 2 µ = 6 | δm will be driven to relatively higher values at low scales. As a 2 Q − 2 H 2 2 H 1 log , Same as figure 1, but with dm 1 m d , , 2 : 2 π 33 for which the electroweak symmetry will be broken successfu ) 2 H term to be very small or negative for successful EWSB. This ca . A correlated consequence is that the effects associated wit 0 Since we have tan 16 R 2 2 m ˜ 2 u / diag(1 m 1 2 H 2 0 m on requirements, i.e. m m the standard CMSSM are pushed to much lower values of is [61] ( of m which must be satisfied in order to obtain radiative corrections [60]. The one-loop RGE for One can see from this that by decreasing the squark mass squar Figure 2: JHEP08(2005)031 0 A ) = G m ( = 0 and R Higgsino ˜ 2 e | µ m | ) = G h charginos, but m ( ˜ 2 L or neutralino annihi- m ) b 30 due to the increased ( ) = ent of the neutralino, the & G hilation amplitude featuring m β nt of the neutralino decreases st chargino and the LSP neu- ( e boundary where R ˜ ) the LSP is a smuon, not a stau. . As one moves away from this 2 d tion to gauge bosons is propor- an igible. Although not at the e amplitude for annihilation via a 2 m h ) = G CDM m ( R . The region where annihilation to gauge ˜ 2 u 0 towards the region of favoured relic density ) m A c 2 ( / 1 – 12 – ) = m G m ( ˜ 2 Q m via an s-channel ¯ τ τ ) a ( and ). In the new brown(dark grey) region in ( 2 boson because it only depends linearly on the coupling to the b ¯ b 0 δm A annihilation to massive gauge bosons or coannihilation wit − 1 , Same as figure 1, but with 1 not , is enhanced for two reasons: 0 It should be noted that in this region the primary mechanism f A diag(1 2 0 resonance, relative to thethe case of no sfermion splitting, th component. Secondly, the mass differencetralino between increases the and lighte coannihilation quickly becomes negl relic density consistent withHiggsino the component constraints of appears the LSP. for t lation is two important things happen. Firstly,significantly. the As Higgsino a compone result,tional since to the the amplitude for squarecross-section annihila of rapidly drops. their coupling Thisan to is s-channel the not the Higgsino case compon for the anni m bosons or chargino coannihilation dominates is closer to th annihilation to boundary in the direction of increasing here the relic density is too small to account for Ω Figure 3: JHEP08(2005)031 . is 2 / ··· ces 0 1 A + m Higgs ver, in ´ than for tions for 33 ncreasing ) resonance 2 R 2 H ˜ 0 2 e not excluded A m m µ sses will reduce density, there is + ( 2) = 500 GeV. Also 2 33 − / ) − e 3(c). There is no 1 s. Therefore the zone g ˜ 2 L ) m m 2 Z nge in the soft sfermion resonance effects become m = their coupling to the β ( + ( equation for 0 0 2 previous paragraph. When 1 can be understood from the A table neutralino relic density T ZZ m 2 H 0 cos m A 2 2 ³ t v m Re Π 2 | + τ − y | β 2 Z 2 that are mainly responsible for driving 8 TeV or higher and the ··· m . Figure 4 shows the RG running of this 2 0 + 2 + | sin ∼ 1200 GeV. − t A ´ ´ y 1 1 ¢ 0 | m & t v 1 33 33 ) ) m 0 2 H R R ˜ – 13 – 2 d ˜ m 2 u m )+ 0 m m is far smaller than for the usual focus point region − 2 A 0 2 resonance occurs is pushed to lower values of m + ( + ( A 2 H ( 0 = 50 and typical values m m 33 33 A quickly drops again as the β ) AA ¡ ) ˜ − ˜ 2 2 Q 2 Q β h 0 1 m ˜ m χ . Comparing the relevant terms in the renormalisation group 1 Re Π 2 resonance for m numerically for the three different sfermion soft mass matri in each case. 2 H and thereby lower 0945 for most of the region favoured by ( + ( cos 2 + ( − . 0 CDM 1 2 Z 0 2 0 m A 2 H 2 H A = 2 H m − ≤ = 0 below values of m m 0 m self energy. The dominant term is the one containing the soft m 1 ³ ³ 2 A 2 to 2 0 2 H | 2 − 2 h | caused by the decrease in Higgsino component of the LSP. Howe | m A µ G t b 1 = 50, Ω m | from the EWSB conditions and the renormalisation group equa , thus enhancing the propagator. 50 another effect comes into play. As well as decreasing with i 2 y y 2 H 0 | | M h β 0 ∼ & m m . CDM 2 A 0 is increased further, past the first band of acceptable relic 2 in the first place. Lowering the third family soft sfermion ma = 6 = 6 2 A β m is the 2 H 2 decreases with decreasing third family sfermion soft masse 1 / CDM 1 2 m Q Q 1 0 m 2 H 2 H 2 H 50, Ω A AA m log log m & at large increased. dm dm and < m d d , . Indeed it is the terms proportional to From the EWSB requirements [61], The opposite effects to those described above pertain to figur For tan As 2 2 β β 1 2 1 π π (i) Due to the larger Higgsino component of the neutralinos, 2 H 2 H 2 H (ii) The mass difference 2 16 16 formula for m by other constraints. The reason for this reduction in boundary where Moreover, this effect contributestan to reason (ii) given in the shown is the value of considered in this paper, for tan difference from of parameter space in which the tan important. This explains thein existence figure of 2(c) two below bands the of accep the difference m a rise in Ω the case of tan m one can see that,masses due will to have the a large larger top effect Yukawa coupling, on any the cha renormalisation group where Π equations for these parameters [61] masses so JHEP08(2005)031 ermion from the 2 2 H m − 1 2 H 0 . The cyan(light) ) = 0, and the dark m G m µ > ( ate the third family of ufficient to avoid large 33 ˜ 2 f entation of the Standard symmetry the soft SUSY es. These parameters are eaking is small enough to n gravity mediation, gauge SUSY breaking parameters δm nges in the phenomenology, relative to the case with no as a natural byproduct of a paper we have explored some ing neutral currents and this supersymmetric particle. We sfermion degeneracy, splitting 0 1 ˜ χ = 50 and eaking and in the dark matter del for supersymmetry breaking m β om a non-abelian family symmetry. and µ = 0, tan 0 A – 14 – . As a result, the neutralino relic abundance at 2 , the red(medium) line to / 2 2 1 m δm δm = 500 GeV, 2 0 2 0 2 m m / 1 − m ) = G ) = = G 0 m ( m m ( 33 ˜ 2 f 33 ˜ 2 f δm δm This plot shows renormalisation group evolution of the difference sfermion splitting. 6. Summary and conclusions Supersymmetric phenomenology is verywhich sensitive determine to the the spectrumstrongly soft of constrained the by the new needhas supersymmetric to led stat avoid to large the flavour constructionwhich chang of are several capable of distinct solving classes the ofof SUSY mo the family implications problem. of In an this alternativeThis solution is following fr perhaps amediation more or attractive anomaly solution mediation thantheory because those of the based fermion solution o masses.breaking arises In masses the of case theModel there three is are family an degenerate members SU(3)FCNC. up in family However to a this SU(3) given symmetryfermion breaking repres must masses be effects and this strongly and inevitably brokenthe leads this to to third is a gener family breaking s of fromavoid the unacceptably the large first FCNC two itparticularly families. does in lead the Although to significant radiativeabundance this cha generation following br from of the electroweak existence br of a stable lightest resonance is increased due to the larger values of is slightly shifted towards higher blue(dark) line to GUT scale down tosoft the mass EWSB matrices scale with for the three different boundary conditions on the sf line corresponds to Figure 4: JHEP08(2005)031 ves = 1 (1983) 330. hanced and N (1984) 1. 70 110 family symmetry ibid. ark for future SUSY n the case the family esent Energy Frontier SU(3) ificant splitting between ly solution to the family e degeneracy of the third SM boundary conditions ated with this is the fact ussions. One of us (GGR) dark matter abundance. It ]. constrained by the need to rlying GUT guarantees the they have more general ap- Phys. Rev. we have motivated the study d travel support. This work s the authors of Micromegas , family. es for supersymmetry. sewhere [62]. rk masses leads to a reduction ads to significant changes from ed to generate the large masses resentation is separately nearly ens up a new region of parameter troweak breaking and this in turn ASPECTS of grand unified models with (1982) 927, erratum 68 hep-ph/0307190 – 15 – (1983) 215. Supergravity as the messenger of supersymmetry (2003) 239 [ B 126 Prog. Theor. Phys. Analysis of the supersymmetry breaking induced by , B 574 (1983) 2359. Fermion masses and mixing angles from mass is small, close to the electroweak exclusion region, mo µ Phys. Lett. D 27 , Phys. Lett. , Supersymmetry, supergravity and ) plane. In this region the Higgsino component of the LSP is en 0 Phys. Rev. , m , 2 / 1 m and unification softly broken supersymmetry breaking supergravity theories The main conclusion is that even the the small splitting of th The effects explored here are the minimal ones to be expected i [1] S.F. King and G.G. Ross, [2] K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, [3] L.J. Hall, J. Lykken and S. Weinberg, [4] S.K. Soni and H.A. Weldon, [5] H. P. Nilles, have explored these effectsinitial in degeneracy detail of for all the the case squarks that and an sleptons of unde family a of the given size indicatedthe by the CMSSM fermion phenomenology mass that structuresearches. has le In been particular widely a usedin reduction as the of radiative a corrections the benchm that third areextends family needed the to squa trigger region elec excludedthat by the the region where WMAP the constraints. Associ this significantly affects the LSP annihilationspace rate. where This the op LSP residualwill abundance be is important able to to explore explain this the region too inproblem is future solved by search a non-abelian familyin symmetry. the Although context of aplicability as specific the SU(3) magnitude family of theof symmetry effects the it follow third is from family likely the ne suppress of FCNC. quarks As and leptons weproblem have and discussed they allows above, are for a further evenbecause, non-abelian more fami when significant the changes underlying fromdifferent GUT Standard the is Model CMS representations broken, provided theredegenerate each may rep in be family space. sign These effects will be exploredAcknowledgments el We would like to thankwould like B. to Allanach thank and S. 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