New Supersoft Supersymmetry Breaking Operators and a Solution to the $\mu$ Problem

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by  Ann E. Nelson

TIFR/TH/15-01
New Supersoft Supersymmetry Breaking Operators and a Solution to the µ Problem
Ann E. Nelson1 and Tuhin S. Roy2,3
1Department of Physics, University of Washington, Seattle WA, 98195, USA
2Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai 400005, India
3Theory Division T-2, Los Alamos National laboratory, Los Alamos, NM 87545, USA
(Dated: May 4, 2015)
Weproposetheframework,“generalizedsupersoftsupersymmetrybreaking”. “Supersoft”models,
with D-type supersymmetry breaking and heavy Dirac gauginos, are considerably less constrained
bytheLHCsearchesthanthewellstudiedMSSM.Thesemodelsalsoamelioratethesupersymmetric
flavor and CP problems. However, previously considered mechanisms for obtaining a natural size
Higgsino mass parameter (namely, µ) in supersoft models have been relatively complicated and
5
contrived. Obtaining a 125 GeV for the mass of the lightest Higgs boson has also been difficult.
1
Additional issues with the supersoft scenario arise from the fact that these models contain new
0
2 scalars in the adjoint representation of the standard model, which may obtain negative squared-
masses, breaking color and generating too large a T-parameter. In this work we introduce new
y
operators into supersoft models which can potentially solve all these issues. A novelfeature of this
a
framework is that the new µ-term can give unequal masses to the up and down type Higgs fields,
M
andtheHiggsinoscanbemuchheavierthantheHiggsbosonwithoutfine-tuning. However,unequal
Higgs and Higgsino masses also removesome attractive features of supersoft susy.
1

]
h Supersymmetry(SUSY)attheelectroweakscaleoffers the Minimal Supersymmetric Standard Model (MSSM),
p potentialsolutionsto the gaugehierarchyanddarkmat- which is the most well-studied incarnation of weak scale
- ter problems, along with a route towards a Grand Uni- SUSY. Note that the MSSM is the weak scale effec-
p
fiedTheory(GUT)1.Acrucialingredientis the presence tivetheoryofanunderlyingsupersymmetrictheory,with
e
h of the Higgsinos (the superpartners of the Higgs bosons) SUSYbeingspontaneouslybrokenbythenon-zerovevof
[ withmassesattheelectroweakscale. Atfirstglance,this the F-component of a hidden sector chiralsuperfield. In

does not appear to be a critical issue, since a supersym- thisframework,arobustsolutiontotheµproblemispro-
3
v metric Higgs andHiggsinomass term, namely “µ”,is al- vided by the Giudice-Masiero mechanism [19], whereby
1 lowed. Infact,issuesregardingHiggsinomassesareoften a manifestly supersymmetric higher dimensional opera-
5 trivializedbyevokingthe argumentthatdue to the non- tor involving the Higgs fields and the SUSY breaking
2
renormalization of the superpotential, any value of µ is hidden sector superfield becomes a µ-term. This mecha-
3
technically natural. However,this response does not ad- nism assures that the µ-term is naturally of the order of
0
. dress the depth of the problem. The µ-parameter needs the superpartner masses. Note that the SUSY breaking
1
to be of the order of the electroweak mass scale, which, termsofthe MSSMareknownas“soft”[20–22], because
0
5 in a supersymmetric theory, is not an input parameter theresultingtheoryhasonlylogarithmicUVdivergences.
1 in the ultra-violet (UV), but is rather generated in the Suchlogarithmicdivergenceshowevermeanthatthesoft
: infrared (IR), after the theory is renormalized down to termsaresensitivetoshortdistanceflavorandCP violat-
v
i the IR, and is naturally at the scale of the superpart- ing physics which could potentially lead to problematic
X
ner masses [2–6]. These masses, in turn, are functions of flavor changing neutral currents (FCNC) [22, 23], and
r the two fundamental mass scales of the theory: (i) the new phases that could make detectible and potentially
a
scale of the SUSY breaking vacuum expectation value excessive contributions to electric dipole moments [24–
(vev) in the hidden sector, and (ii) the mass scale as- 29]. More recently, the accumulated null observations
sociated with the messenger mechanism which connects have put severe constraints on the MSSM, the most se-
the hidden sector and the visible sectorfields. In models rious of which arises from the lack of observation of ex-
of dynamical supersymmetry breaking (DSB), the scale cess events with jets + missing energy at the LHC. In
ofSUSY breakingis generatedvia dimensionaltransmu- weak scale SUSY, events with jets + missing energy are
tation [7–10]. The messenger scale is often the Planck produced mostly due to the production of squarks and
scale[6,11–17];ortheGUTscale[2,14,17];orcanbethe gluinos,whichsubsequentlydecaytojetsandthelightest
scaleofDSB[18]. Inclusionofabaremassterm,whichis supersymmetric particles (LSPs). These cross-sections
of the order of the electroweakscale by pure coincidence aremaximizedfordegeneratesquarksandgluinos,which
makes the theory much less elegant and plausible. is a generic feature of the MSSM. Within its framework,
The µ-problem is often discussed in the context of squarks receive loop suppressed but log enhanced con-
tribution from the gluino mass as the theory is renor-
malized down to the IR. Except in the case where the
squarks start out to be hierarchically heavier than the
1 Foracomprehensivereviewsee [1].
2
gauginos at the UV (such as in split-SUSY [30–32]), the ing operators;thus spoiling the supersoft solution to the
gluino mass is always comparable to the squark masses SUSYFCNCandCP problems[39,50,51]. Itisalsocon-
intheMSSM.Satisfyingexperimentalconstraints,there- ceivable to generate a µ-term via a supersymmetric vev
fore, requires the raising of the mass scale of all colored of a singlet superfield, again bringing in the possibility
particles. Also note that, because of the restricted form of new power law divergences in the singlet potential. If
of the Higgs potential in the MSSM, the top squarks the singlet carries discrete symmetries, then there could
are now required to be very heavy, with mass of order becosmologicalproblemswiththeproductionofdomain
a TeV or more in order to obtain 125 GeV for the walls associated with breaking of the discrete symme-
mass of the Higgs boson. Since renormalization of the tries. Another potential problem with supersoft models
soft Higgs mass-squared term is proportional to the top isthattheD-termcontributiontotheHiggsquarticcou-
squark mass, a heavy top squark gives rise to a finely plingvanishes[39],andaccommodatinga125GeVHiggs
tunedcancellationinthe Higgsmasssquaredparameter. becomes difficult.
Thus,intheMSSM,withSUSYbreakingparametersrun Inthisletter,weproposeacompleteandviableframe-
down from a high scale, SUSY’s promise to explain the work of weak scale SUSY, namely “Generalized Super-
origin of the weak scale without fine-tuning, is fading in soft Supersymmetry,” where all SUSY breaking effects
the light of the LHC Higgs discovery and in the absence are sourced by the D-component of a real field/operator
of any SUSY discovery[33–35]2. from the hidden sector. We include a new class of D-
An alternative way to break supersymmetry is via a term mediated soft (but not necessarily supersoft) op-
vev for the D-component of a hidden sector real su- erators that allow for a new solution to the µ-problem,
perfield [25, 38]. Such symmetry breaking may be me- restoretheHiggsquarticcoupling,andprovideconsider-
diated to the visible sector via a class of operators able modification to supersoft phenomenology.
known as “supersoft”, as they do not induce even log- The visible sector of our supersoft model includes the
arithmic ultraviolet divergences in squark and slepton superfields of the MSSM, as well as additional chiral su-
masses [39]. The most important previously considered perfieldsΣ intheadjointrepresentationoftheSMgauge
i
supersoft operators are those giving rise to Dirac gaug- groups. The fermionic components of Σ , (namely, ψ ),
i i
ino masses [25, 38, 40, 41]. In supersoft models the ra- will obtain Dirac masses with the gauginos (λ ). Super-
i
diatively generatedsquark and slepton masses are finite, symmetry is broken by a D-term of a hidden sector real
flavorsymmetric,positive,UVinsensitive,andlightcom- superfield V′
pared to the gaugino masses [39]. Therefore these mod-
1
elsadditionallyavoidtheflavorchangingneutralcurrent, D2D¯2V′ > 0. (1)
naturalness, and CP difficulties of the MSSM. A heavy D ≡ 8
gluino suppresses processes such as gluino pair produc- (cid:10) (cid:11)
The messenger sector that connects the visible and hid-
tion and squark-gluino production. Also, the pair pro-
densectoris assumedtobe veryheavyandwe mayinte-
duction of squarks is reduced as the T-channel diagrams
grate it out at the messenger scale M , which, in turn,
m
involving gluinos do not contribute. Therefore, Dirac
couldbe as high as the Planckscale. The operatorsgen-
massesallowforareductioninthenumberofeventswith
erating the gaugino masses are [41]:
jets+missingenergyforagivensquarkmass[33,42–49].
The µ-problem is, however, severe in supersoft models. w D¯2DαV′
d2θ 1,i W Σ M λ ψ ,
The Giudice-Masiero mechanism does not work, since 4 M i,α i −→ Di i i
SUSYbreakingisnotmediatedbytheF-termofachiral Z m (2)
w g
1,i i
Asupsoelrufiteilodn,wbuatsbpyrotphoesDed-tienrmrefo.f[3a9r],eawlhseurpeetrhfieelcdoinnfsotremada.l where MDi = √2 MDm .
compensator generates masses for Higgsinos. To gener- In the above, W is the field-strength superfield of i-th
i,α
atetherightHiggsinomasses,however,thisapproachre- SM gauge group, with α being the spinor index. M is
m
quires a conspiracy among the SUSY breaking scale, the the messenger scale, w are dimensionless coupling con-
1
messengerscale,andthePlanckscale. Onecouldreintro- stants, and D and D¯ are superderivatives.
duce the gauge singlet chiral superfield with an F-term An additional class of supersoft terms gives mass to
and use the Giudice-Masiero mechanism. However, such the scalar components of the Σ fields:
i
agaugesingletfieldmayleadtopowerlawUVsensitivity,
and to additional flavor and CP violating SUSY break- w 1D¯2DV′ 2 w 2 σ2
d2θ 3,i 4 Σ2 3,i D i . (3)
4 M2 i −→ 2 M2 2
Z (cid:0) m (cid:1) (cid:18) m(cid:19)
In Eq. (3), σ denotes the scalar components of the Σ
2 Some viable parameter choices may still be considered i i
chiralsuperfields. Since these operatorsaregeneratedat
natural[35–37], either because of cancellations in the renormal-
izationgrouprunning,orbecauserunningfromhighscalesisnot the messengerscale,the scalarmassesareofthe orderof
considered. the gaugino masses. Note that even though the gaugino
3
mass operatorsin Eq. (2) give rise to masses for the real scalar components of Φ remain massless.
1
components of σ fields, Eq. (3) remains the only source
µ µ
of masses for the imaginary components at tree level. 2φ2 φ˜1φ˜2−2Fφ1φ2 → 2φ2φ˜1φ˜2+|µφ2|2|φ2|2
Also, given the fact that the squared-masses generated (cid:16) (cid:17) (6)
inEq.(3)are linearinthe coupling constantsw3,i,these where µφ2 = 2w2,Φ1Φ2 MD ,
can be negative, giving rise to nonzero vev for the color m
octet field, thus breaking color. The gaugino mediated where φ , φ˜, and F are the scalar, fermion, and auxil-
i i φi
squared-masses for these fields are positive. However,as iary components of the chiral multiplet Φ respectively.
i
explained before, these masses are loop suppressed and A non-zero value of either or both of w , or
2,HuHd
not log-enhanced and are, therefore, small with respect w generates masses for the Higgsinos. A nice fea-
2,HdHu
to (w.r.t.) the masses in Eq. (3). In gauge mediated su- ture of these Higgsinomasses is that the masses are nat-
persoftmodels,someintricatemodelbuildingisrequired urallyoftheorderofthegauginomassesandaresourced
to avoidnegative masses squaredfor some of the adjoint by a single mass scale (i.e. vev of the D-component of
scalars [39, 52–54]. Both sets of terms are invariant un- the hidden sector field). These new operators are also
derthehiddensectorgaugesymmetryV′ V′+Λ+Λ†, phenomenologically important. Eq. (6) implies that un-
→
where Λ is a chiral superfield. As discussed in ref. [39] like the conventional µ term, w only gives rise to
2,HuHd
this hidden sector gaugeinvariance is key to the absence down-type Higgs soft masses. The general contributions
ofUVsensitivecontributionstosupersymmetrybreaking to the Higgs sector from these unconventional operators
scalar masses. (withbothw andw )arethencharacterized
3,HuHd 3,HdHu
Inthisworkweproposeanewclassofoperatorswhich bynotoneµparameter,butratherbytwoseparatemass
ameliorates all of the previously mentioned problems in parameters (namely, µu and µd):
this framework:
1
(µ +µ )H˜ H˜ + µ 2 h 2+ µ 2 h 2 . (7)
u d u d u u d d
2 | | | | | | | |
1 D¯2(DαV′D Φ )
d2θ w α 1 Φ (4) Only in the limit µ = µ = µ, the mass terms become
− 4 2,Φ1Φ2 M 2 u d
Z m identical to that of the conventional µ-term. A large
mass term for H , will result in large tanβ but a po-
d
InEq.(4),Φ1 andΦ2 arevisible sectorchiralsuperfields tentially natural spectrum. It is, therefore, possible to
suchthat the bilinearΦ1Φ2 is agaugesinglet. Examples consider a model in which the Higgsinos and additional
of such bilinear gauge singlet in the weak scale super- scalar bosons are substantially heavier than the Higgs
symmetry are HuHd, and Σ2i. Note that the operators without fine-tuning. This setup also challenges the con-
as expressed in Eq. (4) are manifestly chiral (and part ventionalwisdomregardingfine-tuninginmodelsofweak
of the superpotential) because of the fact that D¯3 = 0. scale SUSY. Since there is no observable that directly
ThetermsinEq.(4)canbegivenagaugeinvariantform givesameasureofthemessengerscaleofthetheory(and
(but not supersymmetric), since if V′ is set equal to its thesizeofthelargelogarithmiccontributiontotheHiggs
vev, we find: mass),measuringmassesoftheHiggsinosseemstobethe
best way of estimating the size of cancellation needed
D¯2(DαV′D Φ )= D¯2DαV′ D Φ +... , (5) in order to produce the electroweak scale. Even though
α 1 α 1
exceptions were constructed, where the cancellation is
(cid:0) (cid:1) the result of dynamics [55–57], not fine-tuning, (there-
where ... represent extra terms that do not contribute fore, the naive interpretation of Higgsino masses being
to the superpotential. Whenwe treatour operatorscon- themeasureoffine-tuningisincorrect)thebeliefremains
taining V′ as a spurion, since it can come either from widespread. Eq. (7) provides an explicit example, where
a supersymmetric or a gauge invariant operator, it will the Higgsino mass can be made large (because of large
onlygenerategaugeinvariantcorrectionstoSUSYbreak- µ ), without contributing to soft mass of the up-type
d
ing operators, and hence cannot generate terms which 2 2
Higgs. However, too large a µ µ , generates
d u
require non gauge invariant counter-terms. There are | | −| |
a log-divergent,thoughloop su(cid:16)ppressedHyp(cid:17)erchargeD-
however other spurionic terms which share the feature
term,which,iftoolarge,cangivesomescalarstachyonic
ofbeingeithersupersymmetricorgaugeinvariant,which
masses3. Also, µ = µ , can give rise to additional log
can contribute to squark and slepton masses and non- u 6 d
divergent contributions to scalar soft masses2. For con-
supersymmetric trilinears, so the new operators are not
sistency, we assume that all terms which are needed for
necessarilysupersoft. Oneimportantaspectofthisoper-
ator is that ordering of Φ and Φ in Eq. (4) matters in
1 2
case these represent different fields. Expanding Eq. (4),
we find masses for all the fermionic components of Φ1 3 Wethank AndrewG.Cohen, andMartinSchmaltz forpointing
and Φ , and for the scalar components of Φ only. The thisouttous.
2 2
4
renormalization are present, and so in the case µ = µ Theresultantlightgluino(lightw.r.tM )isaMa-
u 6 d N3
squarkandsleptonmassessquaredmustalsoreceivenon jorana fermion with a mass inversely proportional
supersoft contributions, however such terms can natu- to M . The IR effective theory below M is the
N3 N3
rally be smaller than the supersoft contributions. MSSM, with an added feature of all scalar masses
The operator in Eq. (4), with Φ replaced by the Σ beingstillsupersoft–inthesensethatthesemasses
i
fields,canalsoprovidepotentialsolutionsassociatedwith do notgetbig log contributionfromUV scales (al-
the scalar adjoints. Operators with w3,Σ2i generate pos- though they are sensitive to logMN3).
itive definite squared-masses for the scalar components,
(ii). M M : Gluinos are “pseudo-Dirac”,
and Majorana masses for the fermionic components of N3 ≪ D3
with two nearly degenerate Majorana color octet
the Σ fields.
i
fermions, and a small mass splitting.
1 1
2MNiψi2 + 2|MNi|2|σi|2 , MNi =2w2,Σ2iMDm (8) (iii). MN3 ∼ MD3 : Gluinos are mixed Majorana-
Dirac[58], withtwoMajoranacoloroctetfermions
Color breaking can be easily avoided (at tree level) for and a mass splitting of order their mass. The
large enough w . As mentioned earlier, the gaugino squark–quark–(lighter) gluino coupling deviates
3,3
mediated contributions to scalar soft masses at one loop from the usual strong coupling constant (αs
→
are already positive definite. αscos2θg, where θg is the mixing angle in the
Anadditionaleffectofthelargemassesforthe σ fields gluinomassmatrix). The associatedsquark-gluino
is the (partial) recovery of the Higgs quartic coupling. production cross-section, for example, thus con-
Takeforexample,theon-shellLagrangianinthepresence tains anadditional factor of cos2θg which deviates
oftheσ fields,andtheeffectiveLagrangianafterthereal from 1 at the leading order.
2
components of σ are integrated out:
2
The neutralino and chargino mass matrices are more
2 complicated, and we leave a complete description for fu-
1 g2
2M σ + 2 q∗t q ture work [59]. Here we make a few remarks. In su-
Lon-shell ⊃ 2 D2 2R 2 k k a k! (9) persoft SUSY, the gauginos, Higgsinos, and additional
X
1 Higgsbosonscannaturallybesubstantiallyheavierthan
+ M σ2 +σ2
2 N2 2R 2I the squarks and sleptons without fine-tuning. In fact,
M2 (cid:0) g2 (cid:1) 2 a charged right handed slepton is often predicted to be
N2 2 q∗t q . (10) the lightest supersymmetric particle (LSP) in supersoft
Leff ⊃ M2 +4M2 8 k a k
N2 D2 k (cid:18) (cid:19) models. This,however,isproblematicsinceastableslep-
X
ton is not cosmologically viable. In models with a low
We use the notation σ and σ to designate the real
2R 2I messengerscale, the gravitinobecomes the LSP, thereby
andtheimaginarypartsofσ . Eqs.(9-10)arealsouseful
2 resolving this issue by allowing the slepton to decay into
for demonstrating the fact that unlike in the MSSM, D-
aleptonandgravitino. Depending onthe gravitinomass
terms of the gauge fields do not contribute to the Higgs
andthe reheatingscaleafter inflation, the gravitinomay
quarticinsupersoftSUSY.SincethemasstermM gets
N2 provide a cold or warm dark matter candidate.
generated only by the operator in Eq. (8), the supersoft
In the scenario we provide, a mostly bino-like Majo-
limit can be achieved by taking M 0, when the
N2 → rana fermion could be the LSP. If its mass is close to
D-term containing the Higgs quartic vanishes. In the
the mass of the right handed charged sleptons, then
opposite limit, namely M M , one recovers the
N2 ≫ D2 it can become a thermal relic with the right density
full MSSM strength quartic at the tree level.
due to co-annihilation [60]. Consider the case, where
ThegauginosarenolongerDiracparticlesoncetheop-
M M ,M ,M ,µ ,µ . Since M M ,
eratorsof Eq. (8) are included. For instance, the gluinos D1 ≪ N1 D2 N2 u d D1 ≪ N1
there is a potentially light mass eigenstate which is
g˜ and their Dirac partners ψ obtain masses from two
3 mostly a bino-like Majorana fermion, which can be cho-
independent sources:
sen to yield the right thermal relic abundance. The
right-handed charged slepton receives loop suppressed
1 0 M g˜
Lgluinos ⊃ 2 g˜ ψ3 (cid:18)MD3 MDN33(cid:19)(cid:18)ψ3(cid:19) (11) a(gnd/2fiπn)iMte2malosgs(wMhich/,Mat o)n/Me loo.pW, iesmoafyt,hweitohroduetraof-f
(cid:0) (cid:1) 1 N1 D1 N1 D1
fectingnaturalness,addflavoruniversalsoftsleptonmass
BasedontherelativestrengthoftheDiracmassofgluino
squared terms which are large enough that the right
andtheMajoranamassofψ ,threequalitativelydistinct
3
handed slepton mass is similar in size to the Bino mass.
IR spectra emerge:
In summary, we have shown that adding a new class
(i). M M : The gluino mass matrix has the of operators to models with supersoft supersymmetry
N3 ≫ D3
“seesaw” texture. The ψ field (in fact, the entire breaking can offer a solution to the µ-problem and have
3
Σ superfield) is integrated out at the scale M . very attractive consequences. Gluinos in these models
3 N3
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