SUSY GUTs with Yukawa unification: a go/no-go study using FCNC processes
Wolfgang Altmannshofer, Diego Guadagnoli, Stuart Raby, David M. Straub
aa r X i v : . [ h e p - ph ] S e p TUM-HEP-681/08OHSTPY-HEP-T-08-001
SUSY GUTs with Yukawa unification:a go/no-go study using FCNC processes
Wolfgang Altmannshofer a , Diego Guadagnoli a , Stuart Raby b , and David M. Straub a a Physik-Department, Technische Universit¨at M¨unchen,D-85748 Garching, Germany b The Ohio State University, 191 W. Woodruff Ave,Columbus, OH 43210, USA (Dated: October 31, 2018)We address the viability of exact Yukawa unification in the context of general SUSY GUTs withuniversal soft-breaking sfermion and gaugino mass terms at the GUT scale. We find that thispossibility is challenged, unless the squark spectrum is pushed well above the limits allowed bynaturalness. This conclusion is assessed through a global fit using electroweak observables andquark flavour-changing neutral current (FCNC) processes. The problem is mostly the impossibilityof accommodating simultaneously the bottom mass and the BR( B → X s γ ), after the stringent CDFupper bound on the decay B s → µ + µ − is taken into account, and under the basic assumption thatthe b → sγ amplitude have like sign with respect to the Standard Model one, as indicated by the B → X s ℓ + ℓ − data.With the same strategy, we also consider the possibility of relaxing Yukawa unification to b − τ Yukawa unification. We find that with small departures from the condition tan β ≃
50, holdingwhen Yukawa unification is exact, the mentioned tension is substantially relieved. We emphasizethat in the region where fits are successful the lightest part of the SUSY spectrum is basically fixedby the requirements of b − τ unification and the applied FCNC constraints. As such, it is easilyfalsifiable once the LHC turns on. I. INTRODUCTION
The hypothesis of grand unification is able to ad-dress many of the unanswered questions of the Stan-dard Model (SM), like charge quantization or thequantum number assignments of the SM fermions.Moreover, augmenting grand unified theories (GUTs)with supersymmetry (SUSY) not only stabilizes thelarge mass hierarchy between the electroweak (EW)and the GUT scale but also leads to the possibility ofexact gauge coupling unification.Although this unification works remarkably wellin the minimal supersymmetric Standard Model(MSSM), in order to test the idea of grand unifica-tion one needs other, independent observables. A firstcandidate in this respect would be proton decay. How-ever, the absence of a signal at proton decay experi-ments constrains mostly non-supersymmetric GUTs,whereas the strong model-dependence of the protondecay rate contributions from dimension-five opera-tors makes it difficult to draw general conclusions[1, 2, 3, 4, 5, 6, 7, 8] on the viability of SUSY GUTs.A different way to test SUSY GUTs is by theirpredictions for the masses and mixings of the SMfermions. In this respect, SO(10) is especially at-tractive because it unifies all quarks and leptons ofone generation into a representation of the gaugegroup, leading to the opportunity of a unified Yukawacoupling for the fermions of that generation. Thoughthis is not phenomenologically viable for the two lightgenerations, unification of the top, bottom, tau, andtau-neutrino Yukawa couplings might be possible iftan β , the ratio of the vacuum expectation values ofthe two Higgs doublets, is close to 50. It must be taken into account, however, that thesuccess of this Yukawa unification sensitively dependsnot only on tan β , but also on the SUSY spectrumand parameters, because the Yukawa couplings aremuch more sensitive to weak scale threshold correc-tions than are the gauge couplings. In the absenceof a clear signal in favour of supersymmetry or an apriori knowledge of the SUSY spectrum, one wouldneed a set of additional observables to sufficientlyconstrain the allowed ranges for the SUSY spectrumitself, in order to test GUT predictions for fermionmasses. It turns out that flavour-changing neutralcurrent (FCNC) processes – loop-suppressed observ-ables that are highly sensitive to SUSY particle con-tributions – are especially suited for this purpose.In [9], an SO(10) SUSY GUT model proposed byDerm´ıˇsek and Raby (DR) [10], and featuring Yukawaunification, has been reconsidered in a global analysisin light of all the most precise data on FCNCs in thequark sector. While the model successfully describesEW observables as well as quark and lepton massesand mixings [10, 11], in [9] it was found that the si-multaneous description of these observables and allthe FCNC processes considered is impossible unlessthe squark masses are pushed well above the limitsallowed by naturalness and within reach of the LargeHadron Collider (LHC).The aim of this Letter is twofold. First, we showthat the problem pointed out in [9] and mentionedabove is a general feature of SUSY GUT models withYukawa unification and universal sfermion and gaug-ino mass terms at the GUT scale, thus challengingthe viability of these hypotheses, when considered to-gether. Our second aim is then to explore a possibleremedy, namely relaxing the hypothesis of Yukawaunification in favour of the less restrictive t − ν and b − τ Yukawa unifications. The departure from ex-act Yukawa unification can be quantified by the para-metric departure from the condition tan β ≃
50. Asclarified below, this case will be relevant not only forSU(5), but also for SO(10). This study will allowus to address the question whether a range of largetan β .
50 exists, where a successful prediction forthe bottom mass and full compatibility with quarkFCNCs are possible at the same time.
II. YUKAWA UNIFICATION AND FCNCS
It is well-known [12, 13] that, under the assump-tions of a universal sfermion mass m and a univer-sal gaugino mass m / at the GUT scale, and with apositive µ parameter, Yukawa unification prefers theregion in MSSM parameter space characterized by therelations − A ≈ m , µ, m / ≪ m , (1)because they ensure a cancellation of potentially largetan β -enhanced SUSY threshold corrections to thebottom quark mass [14], which could otherwise spoilthe Yukawa unification. Through renormalizationgroup effects, these relations lead to an inverted scalarmass hierarchy (ISMH) [15], i.e., light third genera-tion and heavy first and second generation sfermions.Relations (1), together with the large value oftan β ≈
50 required for Yukawa unification, have animportant impact on the SUSY spectrum and on thepredictions for FCNCs. In particular, the branchingratio of the decay B s → µ + µ − receives large tan β -enhanced contributions from Higgs-mediated neu-tral currents that are proportional to A t (tan β ) /M A [16, 17]. With large tan β and a large trilinear cou-pling A t following from relations (1), the stringentmost recent experimental bound [18]BR( B s → µ + µ − ) exp < . × − (95% C.L.) (2)can only be met with quite heavy A , H , and H + Higgs bosons.Another important process in this respect is thetree-level decay B + → τ + ν . Using the SM fit valuefor the CKM matrix element V ub [19, 20] one obtainsa SM prediction for the branching ratio BR( B + → τ + ν ) SM = (0 . ± . × − (3)that is quite low compared to the experimental value[23, 24, 25]BR( B + → τ + ν ) exp = (1 . ± . × − . (4) This prediction is obtained by normalizing the branching ra-tio to ∆ M d [21]. The value in eq. (3) agrees well with thosereported in [9, 19, 22]. In the MSSM with large tan β , the dominant addi-tional contribution to this process comes from chargedHiggs bosons and is found to interfere always destruc-tively [26] with the SM contribution, thus furtherreducing the theory prediction. Hence, similarly to B s → µ + µ − , the B + → τ + ν decay requires a heavyHiggs spectrum to be in agreement with the experi-mental data. However, given the large experimentaluncertainty in (4), the B s → µ + µ − constraint turnsout usually to be more stringent.Finally, a very important constraint is the inclu-sive decay B → X s γ , which receives the dominantSUSY contributions from a chargino–stop loop anda top–charged Higgs loop. The chargino contribu-tion is tan β -enhanced and, with the large negativetrilinear parameters implied by relations (1), adds de-structively to the SM branching ratio. The chargedHiggs contribution, on the other hand, adds construc-tively to the branching ratio, but is suppressed bythe heavy Higgs masses required to be consistent with B s → µ + µ − . Thus a near cancellation between thetwo contributions, which is necessary in view of thegood agreement between the experimental determina-tion [27]BR( B → X s γ ) exp = (3 . ± . +0 . − . ± . × − (5)and the SM prediction [28]BR( B → X s γ ) SM = (3 . ± . × − , (6)is difficult to achieve.Note that the solution with the chargino contribu-tion so large that the sign of the b → sγ amplitudeis reversed [29] is challenged in our framework by theexperimental data on BR( B → X s ℓ + ℓ − ) [30, 31, 32].In fact, it would lead to a 3 σ discrepancy between theprediction and the experimental figure for this branch-ing ratio. Leaving aside, for the moment, the possibility of asign flip in the b → sγ amplitude (we will return tothis issue in section III), the above discussion impliesthat a tension between the prediction for B → X s γ and the bound on B s → µ + µ − should generally beexpected in models with Yukawa unification, as a di-rect consequence of relations (1) and the large valueof tan β . By the nature of the argument, this tensionshould be completely independent of the mechanism(flavour symmetries or other) embedded in the SUSYGUT to explain the light quark masses and mixings.In section IV we will come back to this issue, show-ing that indeed this tension occurs generally in SUSY In the actual numerical analysis we include all the relevantcontributions, in particular gluino-down squark loops as well.The latter are found to play a negligible role. This statement holds, barring non-negligible new physicscontributions to the Wilson coefficients ˜ C eff9 , (see Ref. [30]),which is impossible in our case. GUT models with Yukawa unification and quantifyingthe tension numerically with a χ -procedure.Technically, the most immediate potential remedyto the above mentioned problem seems to be to lowertan β . This in fact alleviates the pressure from B s → µ + µ − , permitting in turn larger Higgs and smallerchargino contributions to B → X s γ and thereby mak-ing possible that those two contributions indeed can-cel to a large extent.Lowering tan β means breaking the unification ofthe top and bottom Yukawa couplings, so that thefull Yukawa unification is relaxed to b − τ unification,occurring, e.g., in SU(5). Such breaking of t − b unifi-cation is actually also a general possibility in SO(10)SUSY GUTs once all the representations needed for arealistic GUT-breaking sector are taken into account.For example, the “minimal breaking scheme” intro-duced by Barr and Raby [33] requires the presenceof a ′ H spinor. In this framework, the MSSM Higgsdoublet H d can naturally be a mixture between a dou-blet contained in the same H representation as thedoublet H u and one doublet contained in this ′ H spinor, since the two have the same quantum num-bers. One then has [3, 4, 34] H u = H ( H ) ,H d = H ( H ) cos γ + H ( ′ H ) sin γ . (7)Consequently, the Yukawa unification relation λ t = λ b is effectively broken to λ b λ t = cos γ . (8)At the EW scale, this relation leads to a value oftan β .
50 parametrically smaller than in the exactunification case, depending on the amount of mixingin the second of eqs. (7).We would like to emphasize that the two casesof SU(5) and SO(10) with minimal breaking schemementioned above are just intended as examples. Ouranalysis will be completely general in SUSY GUTswith b − τ unification.It should be stressed as well that, even without t − b unification, SUSY GUT models with b − τ unificationmaintain in fact most of their predictivity, since therelation between the b and τ Yukawa couplings re-quires the ISMH relations, eq. (1), to be satisfiedin order to obtain a correct prediction for m b . Inaddition, a crucial observation is that b − τ unifi-cation requires tan β either close to unity (which ishowever excluded by the Higgs mass bound [35]) orO(50), because otherwise the predicted bottom quarkmass is in general too large [36, 37]. Although thecase tan β = O(50) can be significantly modified bythe tan β -enhanced threshold corrections to m b men-tioned above, b − τ unification is difficult to achieve fortan β .
35. Therefore the strategy to lower tan β isnot a trivial one in our context, since b − τ unificationpushes by itself tan β to high values. With the above arguments, departure from thirdgeneration Yukawa unification and restriction to b − τ unification seems to be a promising approach to retainthe predictivity of GUT models, while at the sametime possibly removing tensions in FCNC observables,thanks to tan β <
50. The rest of our Letter is thusan attempt to address the following two questions: • Is the tension between FCNC observables a gen-eral feature of GUT models with third gener-ation Yukawa unification and universal massesfor sfermions and gauginos at the GUT scale; • Is this tension relieved when tan β is (slightly)below 50, i.e., if one moves from exact Yukawaunification but retains b − τ unification.These issues will be studied through a numerical pro-cedure to be described below. III. PROCEDURE
We assume, at scales higher than the GUT scale M G , the existence of a grand unified group entail-ing b − τ unification. Beneath M G the grand unifiedgroup is broken to the SM group G SM ≡ SU (3) c × SU (2) L × U (1) Y and the MSSM running is performeddown to the EW scale. As for the GUT scale bound-ary conditions to this running, we include a unifiedgauge coupling α G , allowing for a GUT scale thresh-old correction ǫ to the strong coupling constant, theYukawa couplings for up- and down-type fermions ofthe third generation λ t and λ b , and a right-handedneutrino mass M R . At M G we also assume a softSUSY-breaking sector, consisting of a universal tri-linear coupling A , a universal sfermion mass m , auniversal gaugino mass m / , as well as non-universalHiggs mass parameters m H u , m H d .We run all the parameters using one-loop RGEs forthe soft sector and two-loop RGEs for the Yukawaand gauge couplings [38]. To take correctly into ac-count the effects of right-handed neutrinos present inSO(10) and required for the see-saw mechanism, weinclude the contribution of a third-generation neu-trino Yukawa coupling (with initial condition λ ν τ = λ t ) in all RGEs between M G and M R [39, 40, 41].In our framework, there are thus no potentially largelogarithmic GUT scale threshold corrections to eitherYukawa unification or to Higgs splitting, which wouldbe present if such contribution were neglected in theRGEs. The remaining GUT scale threshold correc- For our purposes, this can be assumed to happen in one singlestep. On the Yukawa couplings of the lightest two generations wewill comment later on in this section. In contrast with statements made in the literature and inaccord with the results of [10], we find that neutrino Yukawa tions to the Yukawa couplings are expected to be small[13].At the EW scale, we finally have the two additionalfree parameters tan β and µ . The total number of freeparameters is then 13 and they are collected in table I. Sector α G , M G , ǫ SUSY 5 m , m / , A , m H u , m H d Yukawas 2 λ t , λ b neutrino 1 M R SUSY (EW scale) 2 tan β , µ TABLE I: Model parameters. Unless explicitly stated,they are intended at the GUT scale.
After calculating the SUSY and Higgs spectra andthe threshold corrections to third generation fermionmasses [42], we evaluate the flavour-changing observ-ables using the effective Lagrangian approach of [47].Thereafter, in order to have a quantitative test of themodel, we construct a χ function defined as χ [ ~ϑ ] ≡ N obs X i =1 ( f i [ ~ϑ ] − O i ) ( σ i ) exp + ( σ i ) theo , (9)composed of the quantities given in tables II and III. Observable Value( σ exp ) Observable Lower Bound M W . M h . M Z . M ˜ χ + G µ . M ˜ t . /α em . α s ( M Z ) 0 . M t . . m b ( m b ) 4 . M τ . In eq. (9) O i indicates the experimental value of theobservables listed in tables II and III and f i [ ~ϑ ] the cor-responding theoretical prediction, which will be func-tion of the model parameters listed in table I, collec-tively indicated with ~ϑ . The χ function is minimizedupon variation of the model parameters, using theminimization algorithm MIGRAD , which is part of the
CERNlib library [53].Some comments are in order on the determinationof the errors. First, one can note that among the effects are not sufficient to explain the large Higgs splittingrequired for successful EWSB to occur (see [10], footnote 15). We calculate the Higgs VEVs and M A following [42] and use FeynHiggs [43, 44, 45, 46] to obtain the light Higgs mass.
Observable Value( σ exp )( σ theo ) Ref.∆ M s / ∆ M d BR( B → X s γ ) 3.55(26)(46) [27]10 BR( B → X s ℓ + ℓ − ) 1.60(51)(40) [51, 52]10 BR( B + → τ + ν ) 1.41(43)(26) [23]BR( B s → µ + µ − ) < . × − [18]TABLE III: Flavour-changing observables used in the fit.The BR( B → X s ℓ + ℓ − ) is intended in the range q ℓ + ℓ − ∈ [1 ,
6] GeV . observables in table II, some have a negligible exper-imental error. In this case, we took as overall un-certainly 0.5% of the experimental value, which weconsider a realistic estimate of the numerical error as-sociated with the calculations. Concerning the theo-retical errors on the flavour observables (table III), wenote the following: the error on ∆ M s / ∆ M d takes intoaccount the SM contribution, dominated by ξ andthe new physics contributions, dominated by scalaroperators; the error on BR( B + → τ + ν ), after nor-malization by ∆ M d [21], is dominated by the lattice“bag” parameter ˆ B d and the relevant CKM entries;the error on BR( B → X s γ ) is taken as twice thetotal theoretical error associated with the SM calcu-lation [28]; finally the error on BR( B → X s ℓ + ℓ − ) istaken as 25% of the experimental result, and is esti-mated from the spread of the theoretical predictionsafter variations of the scale of matching of the SUSYcontributions.In evaluating the χ function, we also included thebounds reported in tables II and III. These con-straints are in the form of suitably smoothened stepfunctions, which are added to the χ -function of eq.(9). If any of the constraints is violated, the step func-tions add a large positive number to the χ , while forrespected constraints the returned value is zero, sothat the χ is set back to its ‘unbiased’ definition (9).A step function was also included in order to enforcethe desired sign for the b → sγ calculated amplitude,thus permitting to systematically explore both casesof like sign or flipped sign with respect to the SM one.In the case of flipped sign, large SUSY contributionsare necessary such that SUSY is not quite a correc-tion to the SM result, but rather the opposite. As aconsequence, one would need a theoretical control onthe SUSY part at least as good as that on the pureSM calculation. In the absence of this knowledge, theamplitude in the flipped-sign case is generally verysensitive to variations of the matching scale, and theassociated theoretical error hard to control. In or-der to be able to estimate as reliably as possible the This choice is quite conservative, considering that in our casevariations of the calculated BR( B → X s γ ) upon variation ofthe SUSY matching scale in the huge range [0.1, 1] TeV aretypically around 4%. However, we feel it is justified in thecase of large cancellations among new physics contributions. b → sγ amplitude also in the flipped-sign case, wehave taken advantage of the SusyBSG code [54], whichis directly called by the fitting procedure.As already mentioned in section II, the scenariowith flipped b → sγ amplitude leads to χ & σ discrepancy in B → X s ℓ + ℓ − . We calculated the BR( B → X s ℓ + ℓ − )using the results of Ref. [55]. We will address the sce-nario with flipped b → sγ amplitude quantitatively insection IV.An important observation is in order at this point,justifying why our analysis should be valid for any SUSY GUT model with b − τ unification and universalsoft terms as in table I. As already mentioned in sec-tion II, any such model prefers the region of parameterspace leading to ISMH, i.e., third generation sfermionmasses much lighter than first and second generationones, the latter being of O( m ). One should as wellconsider that, assuming hierarchical Yukawa matricesand large tan β , it is sufficient to include only the 33-elements of Yukawa matrices in the RGEs, that is,take GUT-scale boundary conditions for the Yukawacouplings as Y u,d = diag(0 , , λ t,b ) . (10)This will have a negligible effect on the determinationof the first and second generation sfermion masses,given their heaviness. This observation makes it pos-sible to separate the effects of specific Yukawa tex-tures, which depend on the flavour model one embedsinto the SUSY GUT, from those genuinely due to theunification of Yukawa couplings. The adoption of thisstrategy brings us to the following remarks: • Given the approximation we adopt for the initialconditions on the Yukawas, we do not need toassume any particular flavour model. The low-energy input of the CKM matrix and of the fer-mion masses other than third generation ones,necessary for the calculation of many among theobservables included in the fit, is then taken di-rectly from experiment . • Due to ISMH, the lighter stop is always thelightest sfermion and in fact its tree-level masscan be very small. Therefore, we include theone-loop corrections to the light stop mass to en-sure our solutions are consistent with the lowerbound in table II. In practice, due to its light-ness, we calculate the stop pole mass with thesame accuracy as the pole masses of the thirdgeneration fermions, the W , Z , and the Higgsbosons. Specifically, the CKM input is taken from the new physicsindependent CKM fit [19]. • The validity of our approximation, eq. (10), waschecked by performing the full analysis also witha SUSY GUT model with specific flavour tex-tures, namely the DR model [10] (with Yukawaunification relaxed as in eq. (8)). Our resultswere not significantly affected.
IV. RESULTS
In order to address the questions outlined in theintroduction, we have explored in the m vs. λ t /λ b plane the class of GUT models defined in section III.Concretely, we fixed m to values ≥ λ t /λ b to values ≥ β tovalues ≤
50) and minimized the χ function (9) uponvariation of the remaining model parameters. Theminimum χ value provides then a quantitative testof the performance of the model. The results of oursurvey are reported in the four panels of Fig. 1. Inparticular, panels (a) to (c) report the χ contours assolid lines in the m vs. tan β plane. As reference,also the values of λ t /λ b are reported on a right-handvertical scale. Superimposed to the χ contours are:in panels (a) and (b), the deviations of respectivelyBR( B → X s γ ) and m b from the central values in ta-bles II-III in units of the total error; in panel (c) themass contours of the lightest stop. Finally, panel (d)shows the χ contributions from BR( B → X s γ ), m b and all the rest (as three stacked contributions, rep-resented by solid lines) vs. m in the special caseof λ t /λ b = 1, corresponding to exact Yukawa unifica-tion.Various comments are in order on these plots.1. Panel (d) shows that, for any m . χ contribution from B → X s γ alone is no lessthan roughly 4, corresponding to no less than 2 σ deviation from the result of eq. (5). Therefore,in the case of Yukawa unification, agreementamong FCNCs is only achieved at the price ofdecoupling in the scalar sector. One shouldnote in this respect the quite conservative choiceof the B → X s γ error, already mentioned infootnote 8. The apparent non-monotonic be-haviour of the B → X s γ χ -profile is due tothe fact that, for m . m b (which in fact gets much worse).2. For m . . b → sγ ampli- Of course such test cannot be attached a statistically rigorousmeaning, since, e.g., the χ -entries are not all independentlymeasured observables. For similar findings in the context of Bayesian analyses ofthe CMSSM, see, e.g., Ref. [56].
789 10152025 1 Σ Σ Σ Σ m (cid:144) TeV t a n Β Λ t (cid:144) Λ b - Χ >
25 or no fit H a L Σ Σ Σ Σ Σ Σ Σ Σ m (cid:144) TeV t a n Β Λ t (cid:144) Λ b - Χ >
25 or no fit H b L m (cid:144) TeV t a n Β Λ t (cid:144) Λ b - Χ >
25 or no fit H c L m (cid:144) TeV Χ c on t r i bu ti on B ® X s Γ m b else I Χ L tot H d L FIG. 1: Panels (a)-(c): χ contours (solid lines) in the m vs. tan β plane. Superimposed as dashed lines are the pullsfor BR( B → X s γ ) (panel (a)) and for m b (panel (b)) and the lightest stop mass contours (panel (c)). Panel (d): χ contributions vs. m in the special case of exact Yukawa unification. All the plots assume a SM-like sign for the b → sγ amplitude, except for panel (d), where also the total χ for the flipped-sign case is shown as a dot-dashed line. tude, discussed in section III. However, at thequantitative level, this requirement (implying aquite light stop mass) turns out to be difficultto reconcile with that of successful predictionsfor the bottom mass and/or for EW observ-ables. The solutions we found have χ & m b alone. On top of it one has toadd the χ contributions from FCNCs, being & σ discrepancy inBR( B → X s ℓ + ℓ − ). The total χ for the fitswith flipped-sign b → sγ amplitude is reportedin figure 1 (d) as a dot-dashed line. 3. From panels (a)-(c) one can note a region ofsuccessful fits for m & . tan β .
48, corresponding to a moderate break-ing of t − b unification, since it corresponds to0 . & ( λ t /λ b − & . B → X s γ and m b , pushing tan β to respectively lower andlarger values. In this region, the lightest stopmass is below order 1 TeV, but not less thanroughly 800 GeV.5. If m is not too large, the interesting region Observable Exp. Fit Pull M W M Z G µ /α em α s ( M Z ) 0.1176 0.1159 0.8 M t m b ( m b ) 4.20 4.28 M τ BR( B → X s γ ) 3.55 2.72 BR( B → X s ℓ + ℓ − ) 1.60 1.62 0.0∆ M s / ∆ M d BR( B + → τ + ν ) 1.41 0.726 BR( B s → µ + µ − ) < . χ : Input parameters Mass predictions m M h µ M H M / M A A − M H + β . m ˜ t /α G . m ˜ t M G / . m ˜ b ǫ / % − . m ˜ b m H u /m ) . m ˜ τ m H d /m ) . m ˜ χ M R / . m ˜ χ λ u . m ˜ χ +1 λ d . M ˜ g b − τ unification. Dimensionful quantities are expressed inpowers of GeV. Higgs, lightest stop and gluino masses are pole masses, while the rest are running masses evaluated at M Z . Observable Exp. Fit Pull M W M Z G µ /α em α s ( M Z ) 0.1176 0.1144 M t m b ( m b ) 4.2 4.41 M τ BR( B → X s γ ) 3.55 3.69 0.310 BR( B → X s ℓ + ℓ − ) 1.60 4.41 ∆ M s / ∆ M d BR( B + → τ + ν ) 1.41 0.561 BR( B s → µ + µ − ) < . χ : Input parameters Mass predictions m M h µ M H M / M A A − M H + β . m ˜ t /α G . m ˜ t M G / . m ˜ b ǫ / % − . m ˜ b m H u /m ) . m ˜ τ m H d /m ) . m ˜ χ M R / . m ˜ χ λ u . m ˜ χ +1 λ d . M ˜ g t − b − τ unification and flipped b → sγ amplitude. Conventionsas in table IV. is clearly distinguished from the correspondingcase with exact Yukawa unification, as far asthe fit quality is concerned. By looking at pan-els (a)-(c), one can in fact recognize that thegradient of χ variation is close to vertical for,say, m = 7 TeV, when increasing tan β fromaround 48. On the other hand, increasing m makes the case of breaking t − b unification moreand more indistinguishable, for the fit perfor-mance, from the decoupling regime of point 1.6. An example of a fit in the interesting region isreported in table IV and shown in panels (a)-(c)as a black square. Note that the prediction forBR( B → X s γ ) still tends to be on the lower sideof the experimental range from eq. (5), with acentral value around 2 . × − . As a compari-son, a representative fit in the region with exactYukawa unification, featuring a flipped b → sγ amplitude, is reported in table V.We note, point 1. above implies that, for anySUSY GUT with Yukawa unification, compatibility with FCNC observables and with the now preciselyknown value of m b selects the “partially decoupledsolution”, advocated in Tobe and Wells [57], as theonly phenomenologically viable. The low- m solu-tion originally found in [12, 13] is disfavoured whencombining all the most recent data. This conclusionhas been here quantitatively assessed with a χ pro-cedure. Points 3. to 6. illustrate instead that compatibilityamong all the considered observables at values of m of order 7 TeV can be recovered at the price of relaxing t − b − τ unification to just b − τ unification and withoutmodifying universalities in the soft terms at the GUTscale. Note that, in our approach, exact Yukawa unification canbe enforced, so that the lower bound on m emerges trans-parently as a tension among observables. Instead, in, e.g.,[58], it is low-energy observables (like m b ) to be fixed and alarge value for m is needed for Yukawa unification to occurwithin a given tolerance. We conclude this section with a few additional re-marks. A first interesting issue is whether the gen-eral tension involving FCNCs and m b studied in thisLetter may be relieved if, instead or in addition tolowering tan β , one allows for a complex phase in thetrilinear coupling A t . The latter in our analysis is real,since we take real A . A complex phase φ t in A t wouldinduce a cos φ t suppression factor in the leading char-gino correction to BR( B → X s γ ). In the presence ofa complex A t , however, the tan β -enhanced charginocontribution to the bottom mass would also becomecomplex, so that a (chiral) redefinition of the down-quark fields would be necessary in order to end upwith real and positive masses. This may impact non-negligibly the overall size of SUSY corrections to m b [13, 14] as well as the pattern of CP violation in B -physics observables. In addition, a complex phase in A t is also constrained by the electric dipole momentsof the electron and neutron. Addressing these issuesquantitatively goes beyond the scope of the presentLetter, where we confine ourselves to real GUT-scalesoft terms.As a second remark, we note that, in our analysis ofthe parameter space, we restricted ourselves to posi-tive values of the µ parameter. This is because, fornegative µ , the ISMH solution is lost [9], thus lead-ing to much heavier third generation sfermions for agiven value of m (typically m ˜ t & . a µ , whereas asizeable positive contribution is currently favoured byexperiment.In fact, in the region preferred by the χ function,the contributions to a µ are much smaller than wouldbe needed to explain the E821 result [59, 60]. This is awell-known fact [9, 11] and we decided not to includethis observable in our χ function to obtain resultsunbiased by this issue, which still needs to be settled.We also chose not to include the constraint from the‘standard’ neutralino relic density calculation. Thelatter yields a posteriori a too large relic density in theinteresting region of parameter space for well-knownreasons, i.e., the heaviness of sfermions and that ofthe pseudoscalar Higgs, making the A -funnel regionunattainable. This constraint can however be circum-vented in many ways, such as a late-time entropy in-jection (see, e.g., [61]). For the above reasons, we preferred to restrict theset of observables considered in the analysis to thoselisted in tables II and III, on which consensus is broad. The requirement that SO(10) SUSY GUTs with Yukawa uni-fication do account in full for the standard CDM abundancehas been recently reconsidered in [62].
V. CONCLUSIONS
In this Letter we have studied the viability of thehypothesis of t − b − τ Yukawa unification in SUSYGUTs under the assumption that soft-breaking termsfor sfermions and gauginos be universal at the GUTscale. We found that this hypothesis is challenged,unless the squark spectrum is pushed well above 1TeV. Our conclusion is assessed through a global fitincluding EW observables as well as quark FCNC pro-cesses. The origin of the difficulty is mostly in the spe-cific parameter region chosen by Yukawa unification,which guarantees the correct value for the bottommass and implies ISMH. In this region, it is impossi-ble to accommodate simultaneously the experimentalvalue for BR( B → X s γ ) and the severe upper boundon BR( B s → µ + µ − ). This statement holds under theprior assumption that the sign of the b → sγ ampli-tude be the same as in the SM. For m . . b → sγ amplitude. In this instance it is however difficult toachieve agreement on the b → sγ prediction and, si-multaneously, on those of EW observables and/or ofthe bottom mass. In addition, this possibility entailsin our case a 3 σ discrepancy in B → X s ℓ + ℓ − data.We have shown that our above conclusions holdirrespectively of the flavour model embedded in theSUSY GUT, since, due to ISMH, first and secondgeneration squark masses are much heavier than thirdgeneration ones.We have also addressed the possibility of relaxingYukawa unification to b − τ unification, which in factallows to maintain most of the predictivity of SUSYGUTs. In this case, the tension underlined above is infact largely relieved. The fit still prefers large valuesof 46 . tan β .
48, as a compromise between FCNCsand m b , pushing tan β to respectively lower and largervalues. The range for tan β corresponds to a moderatebreaking of t − b Yukawa unification, in the interval λ t /λ b − ∈ [0 . , . &
800 GeV, a light gluino around 400 GeVand lightest Higgs, neutralino and chargino close tothe lower bounds. This spectrum implies BR( B s → µ + µ − ) in the range 2 to 4 × − and BR( B → X s γ ),on the lower side of the “acceptable” range, ≈ . × − . We stress that the requirements of b − τ unifi-cation and the FCNC constraints are enough to makethe above figures, exemplified in table IV, basicallya firm prediction within the interesting region, henceeasily falsifiable once the LHC turns on. Acknowledgments
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