Table Of Contentt−b−τ Yukawaunificationforµ < 0withasub-TeVsparticlespectrum
Ilia Gogoladzea, Rizwan Khalidb, Shabbar Razac, and Qaisar Shafi
Bartol Research Institute, Department of Physics and Astronomy
University of Delaware, Newark, Delaware 19716
Weshowcompatibilitywithallknownexperimentalconstraintsoft−b−τ Yukawacouplingunification
in supersymmetric SU(4) ×SU(2) ×SU(2) which has non-universal gaugino masses and the MSSM
c L R
parameterµ < 0. Inparticular,therelicneutralinoabundancesatisfiestheWMAPboundsand∆(g−2) is
µ
ingoodagreementwiththeobservations. Weidentifybenchmarkpointsforthesparticlespectrawhichcanbe
testedattheLHC,includingthoseassociatedwithgluinoandstaucoannihilationchannels,mixedbino-Higgsino
stateandtheA-funnelregion.WealsobrieflydiscussprospectsfortestingYukawaunificationwiththeongoing
andplanneddirectdetectionexperiments.
I. INTRODUCTION
1
1
0 SupersymmetricSO(10),incontrasttoitsnon-supersymmetricversion,yieldsthirdfamily(t−b−τ)Yukawaunificationvia
2 theuniquerenormalizableYukawacoupling16·16·10,wherethe10-pletisassumedtocontainthetwominimalsupersymmteric
n standard model (MSSM) Higgs doublets Hu and Hd. The 16-plet contains the 15 chiral fermions per family of the standard
a model(SM)aswellasrighthandedneutrino. Theimplicationsofthisunificationhavebeenextensivelyexploredovertheyears
J
[1, 2]. More recently, it has been argued in [3, 4] that SO(10) Yukawa unification predicts relatively light (≤ TeV) gluinos,
9 which can be readily tested [5] at the Large Hadron Collider (LHC). The squarks and sleptons turn out to have masses in the
1
multi-TeVrange. Moreover,itisarguedin[3,4]thatthelightestneutralinoisnotaviablecolddarkmattercandidate,atleastin
thesimplestmodelsofSO(10)Yukawaunification.
]
h Spurredbythesedevelopmentswehaveinvestigatedt−b−τ Yukawaunification[4,6]intheframeworkofsupersymmetric
p SU(4) ×SU(2) ×SU(2) [7] (4-2-2, for short), with a positive supersymmetric bilinear Higgs parameter (µ > 0). The
c L R
- 4-2-2structureallowsustoconsidernon-universalgauginomasseswhileretainingYukawaunification. Inparticular,assuming
p
e left-rightsymmetry[7,8],ormorepreciselyC-parity[9],wehaveonlyoneadditionalfreeparameterinthesoftsupersymmetry
h breaking (SSB) sector compared to the SO(10) model. In particular, this allow us to distinguish M from M , where M
2 3 2
[ (M )denotestheasymptoticSU(2) (SU(3) )gauginomass. Animportantconclusionreachedin[4,6]isthatwithgaugino
3 L c
non-universality, Yukawaunificationin4-2-2iscompatiblewithneutralinodarkmatter, withgluinoco-annihilation[4,6,10]
3
v playinganimportantrole.
5 Themainpurposeofthispapertoextendthe4-2-2discussiontothecaseofµ<0,whereµdenotesthesupersymmetricHiggs
6 mass parameter. Most authors normally do not consider µ < 0 because it can create serious disagreement with the measured
7 valueofthemuonanomalousmagneticmoment(g−2) [11]. Thenewcontributionto(g−2) fromsupersymmetricparticles
µ µ
2 isproportionaltoµM tanβ/m˜4 [12],wherem˜ istheheaviestsparticlemassintheloop. InSO(10)Yukawaunification,the
. 2
8 scalarmassesareveryheavy(multi-TeV),sothatthesupersymmetry(SUSY)contributionto(g−2) effectivelydecouplesand
µ
0 oneisleftwiththestandardmodelresult[3,4]. Onthecontrary,in4-2-2,withbothµ<0andM <0,theSUSYcontributions
2
0 to(g−2) turnouttobeimportant,aswewillsee,andtherebyyieldsignificantlyimprovedagreementwithexperimentaldata.
µ
1
The outline for the rest of the paper is as follows. In Section II we briefly describe the model and the boundary conditions
:
v forSSBparameterswhichweemployforourscan. InSectionIIIwesummarizethescanningprocedureandtheexperimental
i constraints that we have employed. In Section IV we discuss how µ < 0 leads to better Yukawa unification than µ > 0.
X
In Section V we present the results from our scan and highlight some of the predictions of the 4-2-2 model. The correlation
r
a betweenthespin-independentandspin-dependentdirectdetectionofdarkmatterandYukawaunificationconditionispresented
inSectionVIwherewealsodisplaysomebenchmarkpoints. OurconclusionsaresummarizedinSectionVII.
II. THE4-2-2MODEL
In4-2-2the16-pletofSO(10)matterfieldsconsistsofψ(4,2,1)andψ (¯4,1,2). ThethirdfamilyYukawacouplingψ ψH,
c c
whereH(1,2,2)denotesthebi-doublet(1,2,2),yieldsthefollowingrelationvalidatM ,
GUT
Y =Y =Y =Y . (1)
t b τ ντ
aEmail:[email protected]:AndronikashviliInstituteofPhysics,GAS,Tbilisi,Georgia.
bEmail: [email protected]. Onstudyleavefrom: CentreforAdvancedMathematics&PhysicsoftheNationalUniversityofSciences&Technology,
H-12,Islamabad,Pakistan.
cEmail:[email protected]:DepartmentofPhysics,FUUAST,Islamabad,Pakistan.
2
Supplementing 4-2-2 with a discrete left-right (LR) symmetry [7, 8] (more precisely C-parity) [9] reduces the number of
independent gauge couplings in 4-2-2 from three to two. This is because C-parity imposes the gauge coupling unification
condition (g = g ) at M . We will assume that due to C-parity the SSB mass terms, induced at M through gravity
L R GUT GUT
mediated supersymmetry breaking [13] are equal in magnitude for the squarks and sleptons of the three families. The tree
levelasymptoticMSSMgauginoSSBmasses,ontheotherhand,canbenon-universalfromthefollowingconsideration. From
C-parity, wecanexpectthatthegauginomassesatM associatedwithSU(2) andSU(2) arethesame(M ≡ MR =
GUT L R 2 2
ML). However,theasymptoticSU(4) andconsequentlySU(3) gauginoSSBmassescanbedifferent. Withthehypercharge
2 c c
(cid:112) (cid:112)
generatorin4-2-2givenbyY = 2/5(B−L)+ 3/5I ,whereB−LandI arethediagonalgeneratorsofSU(4) and
3R 3R c
SU(2) ,wehavethefollowingasymptoticrelationbetweenthethreeMSSMgauginoSSBmasses:
R
3 2
M = M + M . (2)
1 5 2 5 3
Thesupersymmetric4-2-2modelwithC-paritythushastwoindependentparameters(M andM )inthegauginosector. In
2 3
order to implement Yukawa unification it turns out that the SSB Higgs mass terms must be non-universal at M . Namely,
GUT
m2 < m2 at M , where m (m ) is theup(down) typeSSB Higgsmass term. The fundamentalparameters ofthe
Hu Hd GUT Hu Hd
4-2-2modelthatweconsiderareasfollows:
m ,m ,m ,M ,M ,A ,tanβ,sign(µ). (3)
0 Hu Hd 2 3 0
Here m is the universal SSB mass for MSSM sfermions, A is the universal SSB trilinear scalar interaction (with the corre-
0 0
spondingYukawacouplingfactoredout),tanβ istheratioofthevacuumexpectationvalues(VEVs)ofthetwoMSSMHiggs
doublets,andthemagnitudeofµ,butnotitssign,isdeterminedbytheradiativeelectroweakbreaking(REWSB)condition. In
thispaperwemainlyfocusonµ < 0aswellasM < 0,asexplainedearlier. Althoughnotrequired,wewillassumethatthe
2
gauge coupling unification condition g = g = g holds at M in 4-2-2. Such a scenario can arise, for example, from a
3 1 2 GUT
higherdimensionalSO(10)[14]orSU(8)[15]modelaftersuitablecompactification.
III. PHENOMENOLOGICALCONSTRAINTSANDSCANNINGPROCEDURE
WeemploytheISAJET7.80package[16]toperformrandomscansovertheparameterspacelistedinEq.(3). Inthispackage,
the weak scale values of gauge and third generation Yukawa couplings are evolved to M via the MSSM renormalization
GUT
groupequations(RGEs)intheDR regularizationscheme. Wedonotstrictlyenforcetheunificationconditiong = g = g
3 1 2
at M , since a few percent deviation from unification can be assigned to unknown GUT-scale threshold corrections [17].
GUT
The deviation between g = g and g at M is no worse than 3.5%, and it is also possible to get perfect gauge coupling
1 2 3 GUT
unification. Perfectgaugecouplingunificationistypicallynotpossiblewithgauginouniversality. Withnonuniversalityinthe
gaugino sector, one can adjust M /M appropriately to get exact gauge coupling unification. If neutrinos acquire mass via
3 2
TypeIseesaw,theimpactoftheneutrinoDiracYukawacouplingintheRGEsoftheSSBterms,gaugecouplingsandthethird
generationYukawacouplingsissignificantonlyforrelativelylargevalues(∼2orso). Inthe4-2-2modelweexpectthelargest
DiracYukawacouplingtobecomparabletothetopYukawacoupling(∼0.6atM ). Therefore,wedonotincludetheDirac
GUT
neutrinoYukawacouplingintheRGEs.
The various boundary conditions are imposed at M and all the SSB parameters, along with the gauge and Yukawa
GUT
couplings,areevolvedbacktotheweakscaleM . IntheevaluationofYukawacouplingstheSUSYthresholdcorrections[19]
Z √
are taken into account at the common scale M = m m . The entire parameter set is iteratively run between M
SUSY t˜L t˜R Z
and M using the full 2-loop RGEs until a stable solution is obtained. To better account for leading-log corrections, one-
GUT
loopstep-betafunctionsareadoptedforgaugeandYukawacouplings,andtheSSBparametersm areextractedfromRGEsat
i
multiplescalesm =m (m ). TheRGE-improved1-loopeffectivepotentialisminimizedatanoptimizedscaleM ,which
i i i SUSY
effectivelyaccountsfortheleading2-loopcorrections.Full1-loopradiativecorrectionsareincorporatedforallsparticlemasses.
TherequirementofREWSB[20]putsanimportanttheoreticalconstraintontheparameterspace.Anotherimportantconstraint
comesfromlimitsonthecosmologicalabundanceofstablechargedparticles[21]. Thisexcludesregionsintheparameterspace
where charged SUSY particles, such as τ˜ or t˜, become the lightest supersymmetric particle (LSP). We accept only those
1 1
solutionsforwhichoneoftheneutralinosistheLSPandsaturatestheWMAP(WilkinsonMicrowaveAnisotropyProbe)dark
matterrelicabundancebound.
Wehaveperformedrandomscansforthefollowingparameterrange:
3
0≤m ,m ,m ≤20TeV
0 Hu Hd
−2TeV≤M ≤0
2
0≤M ≤2TeV
3
45≤tanβ ≤55
−3≤A /m ≤3
0 0
µ<0, (4)
with m = 173.1GeV [22]. The results are not too sensitive to one or two sigma variation in the value of m . We use
t t
m (m ) = 2.83GeVwhichishard-codedintoISAJET.Thischoiceofparameterswasinfluencedbyourpreviousexperience
b Z
withthe4-2-2wherewesetµ,M >0.
2
Inscanningtheparameterspace,weemploytheMetropolis-Hastingsalgorithmasdescribedin[23]. Allofthecollecteddata
pointssatisfytherequirementofREWSB,withtheneutralinoineachcasebeingtheLSP.Furthermore,allofthesepointssatisfy
the constraint Ω h2 ≤ 10. This is done so as to collect more points with a WMAP compatible value of cold dark matter
CDM
(CDM) relic abundance. For the Metropolis-Hastings algorithm, we only use the value of Ω h2 to bias our search. Our
CDM
purpose in using the Metropolis-Hastings algorithm is to be able to search around regions of acceptable Ω h2 more fully.
CDM
Aftercollectingthedata,weimposethemassboundsonalltheparticles[24]andusetheIsaToolspackage[25]toimplement
thefollowingphenomenologicalconstraints:
m (lightestHiggsmass) ≥ 114.4GeV [26]
h
BR(B →µ+µ−) < 5.8×10−8 [27]
s
2.85×10−4 ≤BR(b→sγ) ≤ 4.24×10−4 (2σ) [28]
0.15≤ BR(Bu→τντ)MSSM ≤ 2.41(3σ) [28]
BR(Bu→τντ)SM
Ω h2 = 0.111+0.028 (5σ) [29]
CDM −0.037
3.4×10−10 ≤∆(g−2) /2 ≤ 55.6×10−10 (3σ) [11]
µ
WeapplytheexperimentalconstraintssuccessivelyonthedatathatweacquirefromISAJET.
IV. SIGNOFµANDYUKAWAUNIFICATION
TofirstappreciatetheimpactofthesignofµonYukawacouplingunificationandtoseewhyµ<0ispreferredoverµ>0,
inFig.1weshowtheevolutionofthetop,bottomandtauYukawacouplingsforarepresentativeYukawacouplingunification
solution. WeobservethatYukawaunificationrequiresrelativelylargethresholdcorrectionstoy . Toquantifythemagnitudeof
b
thethresholdcorrections,wescantheparameterspacegiveninEq.(4)andcalculatethefiniteandlogarithmiccorrectionstoδy ,
i
wheretheindexireferstotop,bottomandtauYukawacouplings.
FIG.1.Evolutionoftop(red),bottom(blue)andtau(green)Yukawacouplingsforµ<0.
Following[3],wedefinethequantityRas,
4
(a) µ>0 (b) µ<0
FIG.2.δy /y (blue),δy /y (green)andδy /y (red)versusR(measureofYukawacouplingunification).
b b τ τ t t
max(y ,y ,y )
R= t b τ (5)
min(y ,y ,y )
t b τ
Thus, R is a useful indicator for Yukawa unification with R ≤ 1.1, for instance, corresponding to Yukawa unification within
10%.
Fig.2showsaplotofδy /y versusRforbothµ>0andµ<0. Pointsinblue,greenandredrepresent,respectively,δy /y ,
i i b b
δy /y andδy /y . Notethatwechoosethesignofδy fromtheperspectiveofevolvingy fromM toM . Fig.2confirms
τ τ t t i i GUT Z
thetrendseeninFig.1thaty andy receivesmallthresholdcorrectionscomparedtoy . Therefore,itisreasonabletofocuson
t τ b
thethresholdcorrectionstoy whilestudyingYukawaunification.
b
The scale at which Yukawa coupling unification is to occur is set by gauge coupling unification and is M . Let us first
GUT
considerthecaseofy (M ) ≈ y (M ). Becausethethresholdcorrectionstoy areverysmall,itisconvenienttothink
t GUT τ GUT t
ofy . TheSUSYcorrectiontothetauleptonmassδm isgivenbyδm =vcosβδy . Inordertogetthecorrectτ mass(m ),
τ τ τ τ τ
onehastogetanappropriateδy . Becauseoftherangeofvaluesofcosβforlargetanβ,thereisfreedomtochoosethevalueof
τ
δy . Itmaybepossibletotradethisfreedominfavoroftop-tauYukawaunificationy (M ) ≈ y (M ). Onethenneeds
τ t GUT τ GUT
thecorrectSUSYcontributiontoδy inordertoachieveYukawacouplingunificationy (M )≈y (M )≈y (M ).
b t GUT b GUT τ GUT
InordertounderstandwhysignofµisverycrucialforYukawaunificationconditionletsfirstanalyzetheanalyticalexpression
ofthresholdcorrectionsforthebottomYukawacoupling. Thedominantcontributiontoδy comesfromthegluinoandchargino
b
(a) µ>0 (b) µ<0
FIG.3.δy /y asafunctionofm .Pointsinblackcorrespondto10%orbetterYukawaunification(R<=1.1).
b b 0
5
loops,andinoursignconvention,isgivenby [19]
g2 µm tanβ y2 µA tanβ
δyfinite = 3 g˜ + t t , (6)
b 12π2 m2 32π2 m2
˜b t˜
whereg isthestronggaugecouplingconstant,m isthegluinomass,m isthesbottommass,m isthestopmass,andA is
3 g˜ ˜b t˜ t
thetoptrilinearcoupling. OnecanseefromFig.2thatinordertoachieveYukawacouplingunificationR ∼ 1, thethreshold
correctionstoy havetobenegative(inoursignconventionforδy )andinasomewhatnarrowinterval(−0.5(cid:46)δy /y (cid:46)−1.5)
b i b b
consideringthefullrangeofpossiblevaluesofδy . Thelogarithmiccorrectionstoy areinfactpositive. Thisleavesthefinite
b b
correctionstoprovideforthecorrectδy tocompensateforthe‘wrong’signofthelogarithmiccorrections. Ifµ>0,thegluino
b
contributionispositive,andsothecontributionfromthecharginoloopmustcancelthecontributionfromthegluinoloopandthe
logarithmiccorrection,aswellasprovidethecorrect(negative)contributiontoδy . Thiscanbeachievedonlyforalargem ,
b 0
asforlargem andforlargeA ,thegluinocontributionscalesasM /m2whilethecharginocontributionscalesasA /m2. It
0 t 1/2 0 t 0
alsoshouldbenotedthatthenumericalfactorforthegluinocontributionislargerthanthanthecorrespondingfactorforchargino
contribution. Therefore,asufficientlylargevalueofA andm isneeded. ThislargerequiredvalueofA isthereasonbehind
t 0 t
therequirementofA /m ∼−2.6forµ>0.
0 0
The scenario with µ < 0 is interesting because the gluino contribution to δy has the correct sign to obtain the required
b
b-quarkmass. Thus,weshouldexpectthatwithµ < 0,wecanrealizeYukawaunificationforawiderrangeA values. With
0
thethresholdcontributiontoy proportionaltoM tanβ,andwithM asafreeparameterinthe4-2-2model,wealsoshould
τ 2 2
expectYukawaunificationtooccuroverabroaderrangeoftanβ values.
To proceedfurther, inFig. 3 we present δy /y versus m for µ > 0 and µ < 0, byperforming ascan over theparameter
b b 0
spacegiveninEq.(4). Hereδy includesfulloneloopfiniteandlogarithmiccorrections,andtheblackpointscorrespondto10%
b
orbetterYukawaunification(R≤1.1). ItisclearfromFig. 3thatYukawacouplingunificationwitharelativelylightm ∼400
0
GeVcanberealizedforµ<0. Theµ>0scenariotypicallyrequiresaverylargem (cid:38)8TeV.
0
V. YUKAWAUNIFICATIONANDSPARTICLESPECTROSCOPY
WenowpresenttheresultsofthescanovertheparameterspacelistedinEq.(4). InFig.4weshowtheresultsintheR-m ,
0
R-tanβ,R-A /m andM -M planes. ThegraypointsareconsistentwithREWSBandχ˜0LSP.Thebluepointssatisfythe
0 0 3 2 1
WMAPboundsonχ˜0 darkmatterabundance, sparticlemassbounds, constraintsfromBR(B → µ+µ−)andBR(b → sγ).
1 s
Thegreenpointsbelongtothesubsetofbluepointsthatsatisfyallconstraintsincluding(g−2) . IntheM -M plane,points
µ 3 2
inredrepresentthesubsetofgreenpointsthatsatisfiesYukawacouplingunificationtowithin10%.
IntheR -m planeofFig.4weseethatwithbothµ < 0andM < 0, wecanrealizeYukawaunificationconsistentwith
0 2
allconstraintsmentionedinSectionIIIincludingtheonefrom(g−2) . Thisispossible,aspreviouslynoted,becausewecan
µ
nowimplementYukawaunificationforrelativelysmallm (∼ 400GeV)valuesbecauseµ < 0,and,inturn,(g−2) obtains
0 µ
thedesiredSUSYcontributionwhichisproportionaltoµM . Thisismorethananorderofmagnitudeimprovementonthem
2 0
valuerequiredforYukawaunificationwithµ > 0. WealsoseefromtheR -A /m planethat, asexplainedearlier, Yukawa
0 0
unification for an essentially arbitrary value of A is obtained for µ < 0. Our observation about relaxing the possible range
0
oftanβ thataccommodatesYukawaunifiedmodelsisexplicitlyshownintheR-tanβ plane. Thisalsosuggeststhatweare
perhaps somewhat conservative in limiting the range of tanβ. While it is beyond the scope of this paper, it would be nice to
systematicallysearchforalowerboundontanβ thatisconsistentwiththirdfamilyYukawaunification.
The M - M plane of Fig. 4 has some interesting features. The large gray regions appear because the relic density of
3 2
neutralinos is too high. This may seem peculiar given that we have non-universal Higgs boundary conditions. However, it is
readilyexplainedbythefactthatM <0. Thebulkofthegrayregionhasaneutralinothatistoolight(m (cid:46)30GeV). While
2 χ˜0
itappearspossibletohavearelicdensityofarelativelylightneutralino((cid:46)30GeV)consistentwithWMAP1,BR(B →µ+µ−)
s
andBR(b→sγ),sothatsomeofthisgrayregionmayturnblue,therewillalwaysberegionswheresomeoftheseconstraints
arenotsatisfied.
InFig.5weshowtherelicdensitychannelsconsistentwithYukawaunificationinthem -m ,m -m ,m -m and
χ˜±1 χ˜01 g˜ χ˜01 τ˜ χ˜01
m -m planes. AllofthepointsshowninthisfiguresatisfytherequirementsofREWSB,χ˜0 LSP,particlemassboundsand
A χ˜0 1
1
constraintsfromBR(B → µ+µ−)andBR(b → sγ). Thelightbluepointssatisfy,inadditiontheconstraintfrom(g−2) .
s µ
ThegreenpointsformasubsetoflightbluepointsthatsatisfiesYukawaunificationtowithin10%. Theorangepointssatisfyall
theconstraintsmentionedinSectionIII,whiletheredpointsformasubsetoforangepointsthathaveR ≤ 1.1. Thischoiceof
colorcodingisinfluencedfromdisplayingthesparticlespectrumwithandwithoutneutralinodarkmatter,whilestillfocussing
onalltheotherexperimentalconstraints. TheideaistoshowthemyriadofsolutionsthatimplementYukawaunificationandare
consistentwithallknownexperimentalboundsexceptfortheboundonrelicdarkmatterdensityfromWMAP.Theappearance
ofavarietyofYukawaunifiedsolutionswithaveryrichsparticlespectrumisacharacteristicfeatureofµ<0.
6
FIG.4. PlotsintheR-m ,R-tanβ,R-A /m andM -M planes. GraypointsareconsistentwithREWSBandχ˜0LSP.Bluepoints
0 0 0 3 2 1
satisfytheWMAPboundsonχ˜0 darkmatterabundanceparticlemassbounds,constraintsfromBR(B → µ+µ−),BR(B → τν )and
1 s u τ
BR(b→sγ). Greenpointsbelongtothesubsetofbluepointsthatsatisfiesallconstraintsincluding(g−2) . IntheM -M plane,points
µ 3 2
inredrepresentthesubsetofgreenpointsthatsatisfiesYukawacouplingunificationtowithin10%.
WecanseeinFig.5thatavarietyofcoannihilationandannihilationscenariosarecompatiblewithYukawaunificationand
neutralino dark matter. Included in the m - m plane is the line m = 2m which indicates that the A funnel region is
A χ˜0 A χ˜0
1 1
compatiblewithYukawaunification. IntheremainingplanesinFig.5,wedrawtheunitslopelinewhichindicatesthepresence
of gluino, stau and wino coannihilation scenarios. From the m - m plane, it is easy to see the light Higgs (h) and Z
resonance channels. Our results are focussed on relatively lighχ˜t±1neutraχ˜li01nos (m (cid:46) 225GeV). We expect, based on the
χ˜0
1
discussion presented, that other coannihilation channels like the stop coannihilation scenario are also consistent with Yukawa
unificationbutwedidnotfindthembecauseoflackofstatistics.
VI. YUKAWAUNIFICATIONANDDARKMATTERDETECTION
InlightoftherecentresultsbytheCDMS-II[30]andXenon100[31]experiments,itisimportanttoseeifYukawaunification,
withintheframeworkpresentedinthispaper,istestablefromtheperspectiveofdirectandindirectdetectionexperiments. The
questionofinterestiswhetherµ ∼ M isconsistentwithYukawaunification,asthisistherequirementtogetabino-higgsino
1
admixtureforthelightestneutralinowhich,inturn,enhancesboththespindependentandspinindependentneutralino-nucleon
scatteringcrosssections.InFig.6weshowthespinindependentandspindependentcrosssectionsasafunctionoftheneutralino
mass. In the case of spin independent cross section, we also show the current bounds and expected reach of the CDMS and
Xenonexperiments. ThecolorcodingisthesameasinFig.5. AsmallregionoftheparameterspaceconsistentwithYukawa
unificationandtheexperimentalconstraintsdiscussedinSectionIII(redpointsinthefigure)isexcludedaswecansee,bythe
currentCDMSandXENONbounds. Thisshowsthattheongoingandplanneddirectdetectionexperimentswillplayavitalrole
intestingYukawaunifiedmodels.
7
FIG.5. Plotsinthem -m ,m -m ,m -m andm -m planes.AllpointssatisfytherequirementsofREWSB,χ˜0LSP,particle
χ˜±1 χ˜01 g˜ χ˜01 τ˜ χ˜01 A χ˜01 1
massboundsandconstraintsfromBR(B → µ+µ−),BR(B → τν )andBR(b → sγ). Lightbluepointsfurthersatisfytheconstraint
s u τ
on(g−2) . GreenpointsformasubsetoflightbluepointsthatsatisfiesYukawaunificationtowithin10%. Orangepointssatisfyallthe
µ
constraintsmentionedinSectionIIIwhileredpointsformasubsetoforangepointsthathaveR≤1.1.
FIG.6.Plotsintheσ -m andσ -m planes.ColorcodingisthesameasinFig.5.Intheσ -m planeweshowthecurrentbounds
SI χ˜0 SD χ˜0 SI χ˜0
1 1 1
(blacklines)andfuturereaches(redlines)oftheCDMS(solidlines)andXenon(dottedlines)experiments.Intheσ -m planeweshow
SD χ˜0
1
thecurrentboundsfromSuperK(blackline)andIceCube(dottedredline)andfuturereachofIceCuceDeepCore(redsolidline).
In the case of spin dependent cross section, we show in Fig. 6 the current bounds from the Super-K [32] and IceCube [33]
8
experimentsandtheprojectedreachofIceCubeDeepCore.ItshouldbenotedthatIceCubecurrentlyissensitiveonlytorelatively
largeneutralinomassesandthereforedoesnotconstraintheparameterspacethatwehaveconsidered. Likewise,whileSuper-K
issensitiveinthisregion,theboundsarenotstringentenoughtoruleoutanything. However,fromFig.6weseethatthefuture
IceCubeDeepCoreexperimentwillbeabletoconstrainasignificantregionoftheparameterspace.
It is interesting to comment on the correlation between spin independent and spin dependent cross sections. While both of
thesecrosssectionsareenhancedbythepresenceofalargerhiggsinocomponentintheneutralino,thespinindependentcross
sectionfallsoffas1/m4. InFig.7weplotthecorrelationofthespinindependentcrosssectionversusthespindependentcross
A
sectionform (cid:38)50GeV. Weimposethislowerboundontheneutralinomassonlybecausefor50GeV <m <400GeV,
χ˜0 χ˜0
1 1
boththespinindependentandspindependentfuturereachandspinindependentcurrentboundsarenearlyflat. Thisprovides
onewithanopportunitytoidentifyregionswithsimultaneouslyhighspinindependentandspindependentcrosssectionwhich
mayhavethebesthopeforbeingtestedinongoingandfutureexperiments.
InFig.7wealsoshowlinescorrespondingtoσ =4×10−8,1.5×10−9(pb)andσ =1.5×10−5(pb). Theselinesserve
SI SD
asaguidetodemonstratethecrosscorrelationofspinindependentandspindependentcrosssectionsandtheplausibilityofeither
discoveringaYukawaunifiedneutralinodarkmattersolutionordefinitivelyrulingoutcertainregionsoftheallowedparameter
space. It is to be noted that since ongoing and future experiments require large spin independent and spin dependent cross
sections, neutralino coannihilation channels such as stau coannihilation, bino-wino coannihilation and gluino coannihilation
maybedifficulttotestwithdirectaswellasindirectdetectionexperimentssuchasIceCubeDeepCore.
Letusremarkonthelowmassneutralinosthatwehavefoundinthismodel. Becauseoftherelativenegativesignbetween
M andM ,itispossibleinprincipletohaveM ∼0. Thismeansthatwithinthisframework,theneutralinomaybeaslightas
2 3 1
wewant. Theneutralinomassnonethelessisboundedfrombelowbecauseoftherelicdensityboundsondarkmatter. The4-2-2
modelaswehavepresentedithasalltheingredientsneededtolowertheneutralinomasstothelowestpossiblevalueallowedby
variousconstraints. Wedonotfocusonfindingthelightestneutralinointhismodel. Ofthedatathatwecollected,thelightest
neutralinofoundthatisconsistentwithYukawaunificationhasmass∼ 43GeV. IfwedonotinsistonYukawaunification,we
can get a neutralino as light as ∼ 32GeV. If we are willing to give up neutralino dark matter, then M ∼ 0 consistent with
1
YukawaunificationispossibleasisevidentinFig.5. This,ofcourse,requiresinvokingsomeotherdarkmattercandidatesuch
asaxino.
FIG.7.Correlationofσ andσ .ColorcodingsameasinFig.5.Alsoshownarelinescorrespondingtoσ =4×10−8,1.5×10−9(pb)
SI SD SI
andσ =1.5×10−5(pb).
SD
Finally in Table I we present some benchmark points for the 4-2-2 Yukawa unified model with µ < 0. All of these points
areconsistentwithneutralinodarkmatterandtheconstraintsmentionedinSectionIII.Point1correspondstoasolutionwith
perfectYukawaunification(R = 1.0). Point2representsasolutionwithminimumneutralinomass(43GeV)consistentwith
Yukawaunification. Point3hasasignificantbino-higgsinoadmixtureand,therefore,hasrelativelylargespinindependentand
spindependentneutralino-protonscatteringcross-sections. Point4depictsasolutionwithastopmassofonly826GeV. Finally
in point 5 we show an example with bino-gluino coannihilation channel with the gluino as light as 259GeV. This should be
relativelyeasytofindattheLHC.
VII. CONCLUSIONS
We have shown that Yukawa coupling unification consistent with known experimental constraints is realized in a SUSY
SU(4) ×SU(2) ×SU(2) model. Withµ < 0Yukawacouplingunificationisachievedform (cid:38) 400GeV,asopposedto
c L R 0
m (cid:38) 8TeV forµ > 0, bytamingthefinitecorrectionstotheb-quarkmass. ByconsideringM < 0andM > 0gauginos
0 2 3
9
andµ<0,wecanobtainthecorrectsignforthedesiredcontributionto(g−2) . Thisenablesustosimultaneouslysatisfythe
µ
requirementsoft−b−τ Yukawaunification,neutralinodarkmatterand(g−2) ,aswellasavarietyofotherknownbounds.
µ
Wehavedemonstratedtheexistenceofavarietyofcoannihilationscenariosinvolvinggluino,winoandstau,inadditiontothe
light Higgs, Z and A resonance solutions. The Yukawa unified solutions may also have relativley large spin independent and
spindependentinteractioncrosssectionswithnucleonsinthecaseofmixedbino-Higgsinodarkmatter.Finally,withinthe4-2-2
model, it is possible to obtain relatively low neutralino masses ∼ 43GeV (∼ 30GeV without Yukawa unification) consistent
withneutralinodarkmatter.
Point1 Point2 Point3 Point4 Point5
m 1027 1800 1210 980 1720
0
M -665 -81 -414 -126 -538
1
M -1475 -543 -940 -517 -943
2
M 550 611 374 460 70
3
tanβ 49.1 52.8 50.6 47.0 47.6
A /m 0.26 1.06 -1.15 -1.08 -1.25
0 0
m 743 1919 1231 1090 295
Hu
m 1505 2395 1745 1869 1729
Hd
m 114 115 114 115 115
h
m 847 573 781 1100 1006
H
m 841 569 776 1090 1000
A
m 852 581 787 1100 1010
H±
m 280,341 43,352 168,242 56,337 233,782
χ˜0
1,2
m 352,1236 380,513 246,795 371,476 1210,1216
χ˜0
3,4
m 342,1225 355,509 239,786 338,475 782,1217
χ˜±
1,2
m 1321 1470 955 1110 270
g˜
m 1771,1489 2170,2130 1550,1410 1400,1320 1818,1697
u˜L,R
m 1053,1410 1400,1440 822,1040 826,965 1070,1248
t˜1,2
m 1773,1512 2180,2160 1550,1440 1400,1370 1820,1730
d˜L,R
m 954,1399 1350,1430 774,1020 724,906 992,1245
˜b1,2
m 1391 1810 1340 1000 1807
ν˜1
m 1211 1420 1100 759 1550
ν˜3
m 1393,1096 1820,1820 1340,1250 1010,1040 1809,1763
e˜L,R
m 500,1212 885,1420 641,1110 462,765 1170,1554
τ˜1,2
σ (pb) 4.02×10−8 4.1×10−9 4.1×10−8 9.5×10−10 1.1×10−10
SI
σ (pb) 8.4×10−5 7.5×10−6 1.7×10−4 8.2×10−6 2.9×10−8
SD
Ω h2 0.08 0.11 0.09 0.08 0.11
CDM
R 1.01 1.11 1.09 1.07 1.08
g /g (M ) 0.98 0.98 0.99 0.98 1.00
3 1 GUT
TABLE I. Sparticle and Higgs masses (in GeV), with m = 173.1GeV. All of these benchmark points satisfy the various constraints
t
mentionedinSectionIIIandarecompatiblewithYukawaunification. Point1exhibits‘perfect’Yukawaunification,point2hasthelightest
neutralino,point3shows‘large’spinindependentandspindependentcross-sections,points4and5correspondtothelighteststopandgluino
respectively.Point5alsoprovidesanexampleofbino-gluinocoannihilationchannel.
10
ACKNOWLEDGMENTS
WethankHowieBaerandAzarMustafayevforvaluablediscussions. ThisworkissupportedinpartbytheDOEGrantNo.
DE-FG02-91ER40626(I.G.,R.K.,S.R.andQ.S.)andGNSFGrantNo. 07 462 4-270(I.G.).
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