Strong moduli stabilization and phenomenology

0.4 MB·English

by  Emilian Dudas

EPJ manuscript No.
(will be inserted by the editor)
Strong moduli stabilization and phenomenology
Emilian Dudas1,2,3, Andrei Linde4, Yann Mambrini3, Azar Mustafayev5,6, and Keith A. Olive5
1 Department of Physics, Theory Division, CH-1211, Geneva 23, Switzerland
2 CPhT, Ecole Polytechnique,91128 Palaiseau, France
3 Laboratoire de PhysiqueTh´eorique Universit´eParis-Sud, F-91405 Orsay,France
4 Stanford Instituteof Theoretical Physics and Department of Physics, Stanford University,Stanford, CA 94305, USA
5 William I. Fine Theoretical Physics Institute,
2
1 School of Physics and Astronomy,University of Minnesota, Minneapolis, MN 55455, USA
0 6 Department of Physics and Astronomy,Universityof Hawaii, HI 96822, USA
2
Received: date/ Revised version: date
p
e
S Abstract. We describe the resulting phenomenology of string theory/supergravity models with strong
modulistabilization.TheKLmodelwithF-termuplifting,isonesuchexample.Modelsofthistypepredict
3
universalscalarmassesequaltothegravitinomass.Incontrast,A-termsreceivehighlysuppressedgravity

mediated contributions. Under certain conditions, the same conclusion is valid for gaugino masses, which
]
h likeA-terms,arethendeterminedbyanomalies. Insuchmodels,weareforcedtorelativelylargegravitino
p masses(30-1000 TeV).Wecomputethelowenergyspectrumasafunctionofm3/2.WeseethattheHiggs
- masses naturally takes values between 125-130 GeV. The lower limit is obtained from the requirement of
p
chargino masses greater than 104 GeV, while the upper limit is determined by the relic density of dark
e
h matter (wino-like).
[
CERN-PH-TH/2012-228, CPHT-RR069.0812, UMN–TH–3116/12, FTPI–MINN–12/28, LPT–Orsay-12-92, UH-511-1199-12
1
v
9
Contents willsee,thatwhilestringtheorymodelswithstronglysta-
9
4 bilizedmoduli,providenaturalsolutionstoseveralcosmo-
0 logical problems, they lead to a clear separation in scales
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . 1
. in which the effects of string moduli can be tested in low
9
2 Moduli stabilization and uplifting: a brief review . . 3 energy experiments.
0
2 3 Soft masses for matter fields. . . . . . . . . . . . . . 7 One of the results found in the simplest versions of
1
4 GM Supergravity and Super-GUTphenomenology . 9 the KKLT construction indicates that the mass of the
:
v volume modulus, which describes the “rigidity” of com-
5 Low-energy spectra . . . . . . . . . . . . . . . . . . . 11
i pactification, is of the same order of magnitude as the
X
6 Otherphenomenological aspects . . . . . . . . . . . 13 gravitino mass [2,3]. If one then makes the standard as-
r
sumption that the gravitino mass is in the TeV range or
a 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 16
belowit,KKLTconstructionsbringthescaleofsupersym-
metry breaking in string theory, as well as the masses of
some of the the string theory moduli, down to the LHC
1 Introduction
energyrange.Thisfacthasaninterestingphenomenologi-
cal implication: Supersymmetry breaking in the standard
One of the goals of this paper is to discuss an interesting model may be directly affected by details of the KKLT
interplaybetweenstringtheorymodelswithmodulistabi- construction. Depending on one’s point of view, this may
lization,inflationarycosmology,phenomenologicalmodels be good news, if one tries to study properties of string
of supergravity and the mass of the Higgs boson. Usually theory compactification at LHC, or bad news, if one at-
string theory is associated with an energy scale which is tempts to make predictions independent of the intricacies
manyordersofmagnitude higherthanthe energiesacces- of string theory.
sible at the LHC. This would make it extremely difficult
More importantly, this softness of string theory com-
to test various consequences of string theory. However,
pactificationinthesimplestversionsoftheKKLTscenario
models of moduli stabilization in string theory such as
leads to a specific cosmological problem: vacuum desta-
KKLT[1]allowsonetoinvestigatestringphenomenology,
bilization and decompactification of space if the Hubble
aswellasstringcosmology,fromanewperspective.Aswe
2 Dudaset al.: Strong moduli stabilization and phenomenology
constant during inflation was greater than the gravitino the KL model, as well as any other version of the KKLT
mass [2]. The requirement H O(1) TeV is extremely scenario with strong modulus stabilization.
≤
restrictive;itwouldeliminatemost(thoughnotall)ofthe
Because of the strong modulus stabilization, the KL
presently existing models of inflation. Moreover, a light
scenario leads to some very specific predictions for su-
volume modulus would lead to a novel versionof the cos-
persymmetry breaking and particle phenomenology: It
mological moduli problem, which has plagued supergrav-
describes a certain version of split supersymmetry with
itycosmologyformorethanthreedecades[4].Inaddition,
anomalymediation[3,7]. Moreover,this predictionis sta-
one would still need to solve the cosmological gravitino
blewithrespecttovariousmodificationsoftheKLmodel:
problem, which is another long-standing problem of su-
The same type of supersymmetry breaking and the same
pergravity cosmology [5,6].
pattern of particle masses appears in any version of the
A possible solution to the problem of vacuum desta- KKLT scenario with strong moduli stabilization, which
bilization was proposed back in 2004, in the same paper makes the theory cosmologically consistent [3]. A more
where the existence of this problem was expounded [2]. precise formulation and explanation of this statement is
This solution is realized by adding an extra term to the containedinSection2,wherewegiveabriefreviewofthe
superpotentialoftheKKLTscenario,asinthewell-known KKLT and KL models.
racetrackpotential.Inthisconstruction,thevolumemod-
Moreover, heavy scalars as predicted here are phe-
ulusmasscanbemadearbitrarilylarge,the barrierstabi-
nomenologically interesting for many reasons. Indeed, it
lizingthestringyvacuumcanbemadearbitrarilyhigh,for
is wellknownthat heavy squarkscangreatlyimprovethe
any value of the gravitino mass, and the problem of the
constraintscomingfromSUSYflavorandCPviolatingin-
cosmological vacuum destabilization disappears. To dis-
teractions. In addition, from a theoretical point of view,
tinguishthismodelfromtheoriginalversionoftheKKLT
it is more likely that scalars have heavier masses than
scenario, we will refer to it as the KL model.
fermions, as fermions can be protected from large radia-
It has often been remarkedthat the KL model is very tive corrections whereas scalar particles are generally not
fine-tuned. However,a more detailed investigationof this protected.
issue in [3,7] has demonstrated that the degree of fine-
In this paper, we continue an investigationof the low-
tuningoftheparametersofthismodelisexactlythesame
energy phenomenology of the KL model, as well as all
asinthestandardPolonyimodel,orintheoriginalversion
other versions of the KKLT scenario with strong moduli
of the KKLT scenario: It is determined only by the pos-
stabilization. After a brief review of moduli stabilization
tulated smallness of the gravitinomass.More specifically,
andupliftsinSection2andestablishingthesourcesofsoft
intheKKLTmodel,theconstantterminthesuperpoten-
supersymmetrybreakingmasses inSection 3,we describe
tial, W , must be tuned small. In the KL model, we need
0 the procedure for consistently including radiative elec-
approximatelythesamesmallnumberaddedtoanumber
troweak symmetry breaking in this theory. Because mod-
of O(1) in the superpotential. Recently, a set of super-
els with strong moduli stabilization require heavy scalars
gravityinflationarymodelsincorporatingtheKLscenario
(asinsplitsupersymmetry[12])withrelativelylightgaug-
was proposed, which are very simple and nevertheless are
ino masses (as in models with anomaly mediation [13])
general enough to describe any set of observational pa-
there are difficulties in constructing a UV completion for
rameters n and r to be determined by the Planck satel-
s this anomalously split supersymmetric model with the
lite [7–10]. The KL mechanism of vacuum stabilization
boundaryconditionsimposedbysupergravity.Theseprob-
can be used also in models of chaotic inflation in string
lems and possible solutions will be discussed in Section 4.
theory as proposed in [11].
In Section 5, we describe the sparticle mass spectrum in
Aninterestingfeatureofthisclassofinflationarymod- this theory as a function of gravitino mass. For obvious
els is a controllably small value of the reheating tempera- reasons, we concentrate on the predictions for the Higgs
ture.Thegravitinoproblemmayberesolvedbyasuitably massinsuchmodels.Aswewillsee,togenerateachargino
low reheattemperature or as we will see a large gravitino mass of at least 104 GeV (to be consistent with the LEP
mass which is imposed by the resulting supersymmetric bound [14]) we need a gravitino mass m & 30 TeV.
3/2
sparticle spectrum. As for the cosmological moduli prob- At this value, the Higgs mass is 125 GeV, and rises
≃
lem, supersymmetry breaking is an unavoidable part of slowlyto 130GeVwhenm 1000TeV.InSection6,
3/2
∼ ∼
the KKLTandKLscenario,whichisrelatedto the mech- we consider other phenomenological aspects of the model
anism of uplifting (see next section). As we shall see, if such as the role of 1 TeV gluinos and their detectabilility
this mechanism is realized through F-term uplifting, no at the LHC. We also describe the prospect for dark mat-
light Polonyi fields are required. This addresses the cos- terin these models,aswellasdarkmatter detection.Our
mological moduli problem in the KL scenario, where all conclusions are given in Section 7.
moduli can be superheavy.
These advantages of the KL scenario prompted an in-
vestigation of its consequences for particle phenomenol-
ogy [3,7]. The results of this investigationappeared to be
much more general than initially expected and apply to
Dudaset al.: Strong moduli stabilization and phenomenology 3
2 Moduli stabilization and uplifting: a brief setting D W = 0. It occurs at Imρ = 0, and at a certain
ρ
review value σ of the volume modulus σ =Reρ.
0
After the uplifting to the (nearly Minkowski) dS vac-
2.1 KKLT versus KL uum state, the gravitino mass becomes
aA
TheKKLT(KL)sectorconsistsofasinglechiralfield:the m3/2 ≈ |VAdS|/3≈ 3(2σ )1/2e−aσ0 . (5)
modulusρ.WewilldenoteSMfieldscollectivelyasφ.The 0
p
scalar potential for uncharged chiral superfields in =1 Furthermore, after uplifting
N
supergravity is [15]
3√2
V =eK Ka¯bDaWDbW 3W 2 , (1) DρW = a√σ m3/2 . (6)
− | | 0
(cid:16) (cid:17)
where as usual we defined D W = ∂ W +K W. We de-
a a a The conditions of applicabilityof the KKLT construc-
fineaK¨ahlerpotentialwithano-scale[16]structureinthe
tions are aσ > 1 and σ 1. If one takes aσ 1,
moduli sector and kinetic terms in the matter sector de- 0 0 ≫ 0 ≫
the gravitino mass becomes exponentially small. To have
pending in some unspecified way on the modulus ρ. This
m in the TeV range in the KKLT scenario,one should
can be written as 3/2
take aσ 30.
0
∼
K =−3log(ρ+ρ¯)+hji(ρ,ρ¯)φiφ¯j+K(Si,S¯i)+∆K(φi,φ¯i)+···, The mass of the volume modulus σ in the minimum,
(2) as well as the mass of its imaginary (axionic) component,
where denote terms ofhigher-orderin matter fields φ, is given by m = 2aσ m [3]. For aσ 30, one finds
··· σ 0 3/2 0 ∼
irrelevant for our purposes. In a type IIB string theory m 60m .As aresult,the massofthe volume modu-
σ 3/2
setuporientifoldedbyΩ′ =ΩI6( 1)FL,whereI6 denotes lus i∼s somewhat greater than the gravitino mass, but not
parity in the six internal dimens−ions and ( 1)FL is the by much.This means that the volume stabilizationin the
−
left-handed fermionnumber [17], with D7 and D3 branes, KKLTscenariodescribinglightgravitinosisverysoft;the
the function h(ρ,ρ¯)is aconstantif matter fields originate mass of the volume modulus in this scenario is many or-
from D7-D7 sector, it is given by hj(ρ,ρ¯) = δj/(ρ+ρ¯) ders of magnitude below the string scale or the Planck
i i
for fields in the D3-D3 sector and has specific form for scale. It is this softness of the vacuum stabilization that
fields living at the intersection of various branes. We will leads to the catastrophic decompactification of extra di-
discussthe fieldsS whichareassociatedwithF-termup- mensions during inflation with H &m [2,3].
i 3/2
liftingbelowandwespecify∆K insection4inconnection
The simplest way to avoid this problem is to strongly
with the Giudice-Masieromechanism[18]. The important
stabilize the vacuum by making m greaterthan m by
assumptioninwhatfollowsisthattheupliftfieldsS have σ 3/2
i many orders of magnitude. This was achieved in the KL
a separableK¨ahlerpotential,that canbe justified, for ex-
scenario by using the racetrack superpotential
ample, if the uplift fields arise as D7-D7 states. Indeed,
we will be assuming that the Si are not directly coupled WKL =W0+Ae−aρ Be−bρ . (7)
to matter through either the K¨ahler potential, the super- −
potential, or gauge kinetic function. In each case we will In contrastto the KKLT case, the new degree of freedom
assume the superpotential is separable in the uplift fields offered by Be−bρ allows the new model to have a super-
symmetric Minkowski solution. Indeed, for the particular
W =W(ρ)+W (Si)+g(φi,ρ) , (3)
F choice of W ,
0
where W(ρ) is either the KKLT or KL superpotential,
a b
W is the superpotential associated with uplifting and aA b−a aA b−a
F W = A +B , (8)
0
g(φi,ρ)isthesuperpotentialfortheStandardModel,with − bB bB
(cid:18) (cid:19) (cid:18) (cid:19)
g(0,ρ) = 0 . The possible ρ dependence of the matter
thepotentialofthefieldσhasasupersymmetricminimum
superpotential g is highly restricted by axionic symme-
with W (σ )=0, D W (σ )=0, and V(σ )=0.
tries and by the origin of matter fields (it is typically an KL 0 ρ KL 0 0
exponential or a modular form of various weight). Pro- Onemayaddanadditionalconstant∆(eitherpositive
vided that the vev’s of matter fields are very small com- or negative) to the superpotential (7). This will shift the
paredtothe Planckscale,the resultsofthe presentpaper minimumofthepotentialdowntotheAdSminimumwith
are largely insensitive to the explicit form of the function V = 3m2 = 3∆2 [3,7], after which one may use
h(ρ,ρ¯) and the ρ dependence of g(φi,ρ). AdS − 3/2 −8σ03
uplifting(asinKKLT)tomakethecosmologicalconstant
The superpotential of the KKLT model is as small as 10−120. Thus one has m2 = ∆2 1,
∼ 3/2 8σ03 ≪
W =W +Ae−aρ . (4) which is the only weak-scale fine-tuning required in the
KKLT 0
KL model. Interestingly, exactly the same level of fine-
where W and a > 0 are constants. In this theory, there tuning of the parameter W is required in the simplest
0 0
is a supersymmetry preserving AdS minimum found by version of the KKLT scenario. This is the standard price
4 Dudaset al.: Strong moduli stabilization and phenomenology
for the desire to protect the Higgs mass by the smallness values for gaugino masses and tri-linear supersymmetry
of supersymmetry breaking. breaking A-terms which are proportional to D W. In [3],
ρ
thecouplingofmattertotheupliftingtermwasneglected
Finally, we turn to uplifting in the theory. Before we
anditwasassumedthatsoftscalarmassesremainedequal
makethe minuscule addition∆to the KLsuperpotential,
to the gravitino mass. However, as can be seen from the
supersymmetry is unbroken, the gravitino mass vanishes,
analysis in Ref. [19], there is a cancelation which leaves
butthevolumemodulusmassisarbitrarilylarge,depend-
onlyatinyscalarmassalsooforderm2 /M .Theresult-
ing on the choice of the parameters A, a, B and b. This 3/2 P
ingspectrumwouldthenbedominatedpurelybyanomaly
mass is virtually unchangedafter adding ∆ anduplifting.
mediation,andsufferfromknownphenomenologicalprob-
Thus,oneachievesthedesiredstrongvacuumstabilization
lems [20,21].
and removes the cosmological constraint H . m . But
3/2
this strong vacuum stabilization has an interesting impli- However, it is in fact relatively simple to recover the
cation for the resulting low-energy phenomenology: Just result given in [3], by using F-term uplifting [22–24] in-
as in the KKLT scenario, DρW(ρ) = 0 in the supersym- steadofantibranes.Infactthispossibilityisquitegeneric
metric AdS minimum prior to uplifting in the KL model. and can be done in various ways, as we now describe in
However,inthesimplestversionoftheKKLTscenariothe two explicit examples. The main idea is to use a SUSY
valueofthefieldρdoesshiftslightlyduringuplifting,and breaking sector for uplifting, preferably with a dynamical
DρW(ρ) becomes (approximately) as large as W, as seen scale leading to a small mass parameter M << 1, which
in Eq. (6). In contrast, strong vacuum stabilization keeps breaks SUSY in the rigid limit M . All masses in
P
moduli practically unchanged during the uplifting. As a this sector will be determined, at th→e t∞ree and one-loop
result, after the uplifting in the KL scenario, and in any level, by the dynamical scale and are much larger than
other version of the KKLT scenario with strong vacuum the gravitino mass. Whereas in the rigid limit the uplift
stabilization, one has W =∆and DρW(ρ) ∆, m3/2. sectorisdecoupledfromtheKLsector,supergravityinter-
| |≪| |
actionscouple the twosectors.However,instrongmoduli
In particular, in the KL model with an uplifting term
σ−2, whichappears in the models with uplifting due to stabilization models like KL, provided the KL modulus
∼ mass and masses of uplifted fields are much larger than
anti-D3 branes in warped space, one has [3]
gravitino mass, supergravity interactions only change the
D W =6√2σ m3/2 m . (9) original KL and uplift sector minima in a very tiny way.
ρ 0 m 3/2 Asaresult,thevacuumstructureisessentiallyunchanged
σ
andthe modulus sector contributionto SUSY breakingis
On the other hand, one can show that in the F-term up-
completely negligible, as will be seen in what follows.
lifting models to be studied below, the uplifting term is
proportional to σ−3, and the result slightly changes, This implies that in the models with strong stabiliza-
tion, there is a certain decoupling of string theory mod-
m
D W =9√2σ 3/2 m . (10) uli from the standard model phenomenology. While this
ρ 0 m 3/2 could seem almost obvious on general grounds, it is not
σ
the case in the simplest versions of the KKLT scenario.
What is most important for us is that in both cases one
Optimistically, this means that investigation of the low
hasD W(ρ) W,D W(ρ) m ,underthecondition
ρ ρ 3/2 energy phenomenology may provide us with a possibil-
≪ ≪
ofstrongvacuumstabilizationm m ,whichensures
σ 3/2 ity to test various mechanisms of moduli stabilization in
≫
vacuumstabilityduringhighenergyinflation.Thus,inde-
string theory.
pendentlyoftheparticularchoiceofthestabilizingsuper-
potential W(ρ), one can simply take D W(ρ) = 0 at the Furthermore, in what follows we will be considering
ρ
minimum of the potential, before and after the uplifting. modelsofstrongstabilizationforallmoduli,includingthe
Andthismeans,asonecaneasilycheck,thatinallmodels F-term uplifting fields Si. Just as strong stabilization of
of such type one has, at the minimum of the potential, the volume modulus ρ was advantageous for cosmology
and inflation, strongly stabilized uplifting fields, provides
W = ∆ =(2σ )3/2m (11) a simple mechanismto avoidcosmologicalproblems asso-
0 3/2
| | | | ciated with these moduli. While we do not enter into the
and details of the cosmology of the these moduli, we take the
3∆
W = . (12) premise that all moduli are strongly stabilized.
ρ
2σ
0
2.2.1 F-term uplifting with a non-minimal Polonyi
2.2 F-term uplifting examples
field
In the discussionabove,we assumed uplifting as an effect
arisingsolelyfromstringtheory,forexample,throughthe We begin with a very simple example based on a non-
energyofanti-D3branesplacedinahighlywarpedthroat. minimalversionofthe Polonyimodel,knownas O’KKLT
In the KL model when coupled to matter, the suppressed [24,25]. The O’KKLT model for F-term uplifting is real-
F-term given in equation (9) leads to extremely small ized with the following definitions of K(S,S¯) and W (S)
F
Dudaset al.: Strong moduli stabilization and phenomenology 5
used in eqs. (2) and (3) for a single Polonyi-like field S. ThefieldS attheminimumofitspotentialisreal,and
We take its value is given by
(SS¯)2
K(S,S¯)=SS¯ , (13) √3Λ2
− Λ2 S = . (16)
h i 6
whereweassumethatΛ 1(inPlanckunits).Aswewill
≪ The mass squared of the field S in both directions (real
see, the second term in (13) provides strong stabilization
and imaginary) is given by
forthefieldS.Forthesuperpotential,wecantakesimply,
3∆2 12m2
W (S)=M2S , (14) m2 = = 3/2 m2 , (17)
F S 2σ3Λ2 Λ2 ≫ 3/2
0
as in the Polonyi model, but without an additional con- so it too is strongly stabilized. This is quite important.
stant which is necessary for the fine-tuning of the vanish- Indeed,thecosmologicalmoduliproblemappearsbecause
ingly small value of the cosmological constant. This con- intheminimalPolonyifieldmodel,themassofthePolonyi
stant is already provided by the KKLT/KL superpoten- field S is of the same order as the gravitino mass, which
tial. wassupposedtobeinthe rangeof1TeVorbelow.Inour
In the original O’KKLT model, it was assumed that model,m2S ≫m23/2 andthefieldisconstrainedtolieclose
theterm (SS¯)2 appearsafterintegratingoutsomeheavy to its minimum near S = 0 (for small Λ). Moreover, as
degrees o−f frΛee2dom in the O’Raifeartaigh model. A con- we will soon see, in the models of this class one typically
sistency of this assumption required careful investigation has m3/2 1 TeV. Therefore for sufficiently large m3/2
≫
[24]. However, assuming that this interpretation of the andsufficientlysmallΛ 1,thecosmologicalmoduliand
≪
term (SS¯)2 isavailable,onecansimplyconsiderthisterm gravitino problems will disappear.
− Λ2
as a part of a modified Polonyi model (13), (14) without Strong stabilization of the field S is important in an-
further discussion of its origin [24,25]. other respect as well. Since the field S is strongly sta-
bilized, we can repeat the same procedure that we used
The simplest way to understand the main idea of this
before, and calculate the soft breaking terms in the stan-
scenario is to consider it in the context of the KL model,
dardmodel.Theonlyadditionalparametersthatweneed
oranyotherstronglystabilizedmodelofthattype.Inthis
for these calculations are the values of W and W′ at
case,the position of the AdS minimum of the potential is F F
the minimum of the potentialfor the field S,ignoringthe
strongly fixed. Therefore it is not affected by adding the
standard model fields:
Polonyi field to the theory. In order to find the value of
the Polonyi field and its superpotential, it is sufficient to ∆Λ2
calculatethe values ofthe superpotential W(ρ) ofthe KL WF = | | (18)
2
modelanditsderivativeW (ρ)attheminimumoftheKL
ρ
potential ignoring the Polonyi fields. The results of these and
calculations are given in (11) and (12). These results are ∂ W =√3 ∆ . (19)
S F
thenusedinthecalculationoftheF-termpotentialofthe | |
field S. As a result,
Alternatively, one may wish to abandon any connec- ∆Λ2
tion to string theory and simply consider the supersym- W =W(ρ)+WF =∆+ ∆ ,
2 ≃
metry breaking sector of the Polonyi field S with strong
D W =∂ W +K (W(ρ)+W )
stabilization provided by the K¨ahler potential and super- S S F S F
potentialgivenbyEqs.(13)and(superpol).Forthesuper- √3 1
=√3∆ + Λ2(1+ Λ2)∆ √3∆. (20)
potential, however, we must add back the constant term. | | 6 2 | |≃ | |
For small Λ, we have strong stabilization and the mass
of S can be large, as discussed below and its expectation
value close to 0. So long as we continue to assume that 2.2.2 F-term uplift with a dynamical ISS sector
the gauge kinetic function does not linearly depend on
S, the phenomenological results discussed below will be
unchanged. Asasecondexample,wedisplayhereanotherF-termup-
lifting [23,26] via the ISS mechanism [27], which leads to
Thesecalculations,forstronglystabilizedtheories,show
a qualitatively similar result to the one in the previous
that the field S uplifts the AdS minimum to the nearly
section. The difference is that in the present example the
Minkowski minimum for
corresponding mass scale M has a dynamical origin, that
naturally explain its smallness. The model is defined by
M4 =3∆2 =24σ3m2 , (15)
0 3/2
W = W (ρ) + W (χi) ,
KL F
which determines the choice of the parameter M in (14). K = 3 ln(ρ+ρ¯) + q 2 + q˜2 + S 2 . (21)
− | | | | | |
6 Dudaset al.: Strong moduli stabilization and phenomenology
In (21), χi denotes collectively the fields qa, q˜¯j, Si of the
i a ¯j
ISSmodel[27],wherei,¯j =1 N areflavorindicesand
a,b=1···N are color indices·.·M· ofreover,in (21) V = (ρ+1ρ¯)3 VISS(χi,χ¯¯i) + VKL(ρ,ρ¯) + χ¯M¯iχ2i V1(ρ,ρ¯)
P
WF(χi) = h Tr q˜S q − h M2 TrS , (22) + M13 WISS(χi) V2(ρ,ρ¯)+χi∂iWISS V3(ρ,ρ¯) + h.c.
P
and WKL is given in Eq. (7). As explained in [27], the + , (cid:2) (cid:3)(25)
sector qa, q˜¯j has a perturbative description in the free ···
i a
magnetic range N > 3N. SUSY is broken in (22) by wherebycomparing(25)with(24)wecancheckthatV
f 1
the “rankcondition”,i.e.F-termsofmesonfields(FS)ji = m23/2MP2, and V2,V3 ∼m3/2MP3. The contribution to th∼e
hq˜jqa M2δj cannot be set simultaneously to zero. vacuum energy from the ISS sector,in the globallimit, is
a i − i V = (N N)h2 M4.Sinceweareinterestedin30
ISS f
As is transparent in (21), the KL and the ISS sectors h i − −
1000 TeV scale gravitino masses, it is clear that the first
areonlycoupledthroughgravitationalinteractions.Inthe
twotermsintherhsof(25),V andV ,aretheleading
ISS KL
type II orientifold setup, if the ISS gauge group comes
terms.Consequently,there shouldbe avacuumveryclose
fromD3branes,thedynamicalscaleintheelectrictheory
to an uplift KL vacuum ρ = ρ and the ISS vacuum
0
andthereforealsothe massparameterM inthe magnetic χi = χi. The cosmologichaliconstant at the lowest order
theorysuperpotential(22)dependonthedilatonS,which h i 0
is given by
we assume is already stabilized by NS-NS and RR three-
formfluxes[17].AsintheO’KKLTmodel,thisdecoupling (N N)h2M4
f
between the uplift field(s) and modulus ρ is instrumental Λ = VKL(ρ0,ρ¯0) + (ρ−+ρ¯ )3 , (26)
in getting the uplift of the vacuum energy. 0 0
At the globalsupersymmetrylevelandbefore gauging
which shows that the ISS sector plays indeed the role of
thecolorsymmetry,theISSmodelhasaglobalsymmetry
un uplifting sector of the KL model. In the zeroth order
G = SU(N) SU(N ) SU(N ) U(1) U(1)′
f L f R B approximation and in the large volume limit σ 1, we
× × × × × 0
U(1) ,brokenexplicitlytoSU(N) SU(N ) U(1) ≫
R f V B findthattheconditionofzerocosmologicalconstantΛ=0
× × ×
U(1) bythemassparameterM.Inthesupergravityem-
R implies roughly
bedding (22),the R-symmetryU(1) is explicitlybroken.
R
We considerhereonly the ungaugedtheoryforsimplicity, 3 W 2 h2 (N N) M4 . (27)
f
inwhichtheSU(N)ispartoftheglobalsymmetrygroup. | | ∼ −
For the effects of the gauging, see e.g. [23] in the related If we wantto have a gravitino mass in the 30 1000TeV
contextoftheKKLTuplift.Atthe globalsupersymmetry range, we need small values of M (10−5 −10−6)M .
P
level, the metastable ISS vacuum is Since M in the ISS model has a d∼ynamical−origin, this
is natural. Moreover, the metastable ISS vacuum has a
S = 0 , q = q˜T = MIN , (23) significantlylargelifetime exactlyinthis limit.Therefore,
0 0 0 0 the claimed value of the gravitino mass is natural in our
(cid:18) (cid:19)
model and compatible with the uplift of the cosmological
where I is the N N identity matrix and M Λ , constant.
N m
× ≪
where Λ M denotes the mass scale associated with
the Landmau≤poleP for the gauge coupling in the magnetic In the rigid MP →∞ limit, the ISS fields have masses of
order
theory. The first question to address is the vacuum struc-
ture of the model. In order to answer this question, we
tree level m hM ,
0
start from the supergravity scalar potential (1). By using − ∼| |
h2M
(21)-(22), we find one loop m | | . (28)
1
− ∼ 4π
eχ¯¯iχi (ρ+ρ¯)2 2
V = (ρ+ρ¯)3 3 DρW Of course the goldstone bosons of the broken global sym-
(cid:26) metries are massless for the time being. It is easy to re-
(cid:12) (cid:12)
+ ∂iW + χ¯¯iW(cid:12)2 − 3(cid:12)|W|2) . (24) mmoetvreythfreosme mthaessvleesrsysbtaetgeisnnbiyngbrbeyakhinagvinthgesegvloebraall msyamss-
Xi (cid:12)(cid:12) (cid:12)(cid:12) pitayrcaomrreetcetriso,nMsg2iTverStr→ee-levieMlmi2aSsiis.eNstootitcheethpasetusduop-emrgordauvl-i
SinceM M ,thevev’sintheISSmodelarewellbe- fieldsoftheISSmodel.APsexplainedinmoregeneralterms
P
≪
low the Planck scale.Then an illuminating wayof rewrit- in [27], these corrections are subleading with respect to
ing the scalar potential (24) is to expand it in powers of masses arising from the one-loop Coleman-Weinberg ef-
the fields χi/M , in which case it reads1 fective potential in the global supersymmetric limit. This
P
can be explicitly checked starting from the supergravity
1 In most of the formulae of this letter, M = 1. In some scalar potential (24) and expanding in small fluctuations
P
formulae, however, we keep explicitly M . around the vacuum (23) to the quadratic order.
P
Dudaset al.: Strong moduli stabilization and phenomenology 7
SimilarlytothepreviousO’KKLTexample,thereisno pensates that of S [28]. In our uplift examples, it would
moduli problem in the present setup: both the ρ modulus mean that S couplings in h and the S-dependence of
A
and the ISS fields are much heavier than the gravitino Yukawas are suppressed by the mass parameter M2
mass. e−2aT. For example h = h0(1+β e−2aTS) or y =∼
A A A ijk
y0 (1+c e−2aTS).However,the“anomalous”symmetry
ijk ijk
does not forbid couplings in the K¨ahler potential, which
we have argued against earlier.
3 Soft masses for matter fields
Undertheassumptionsabovewenowshowthatstrong
modulistabilizationwithanyF-termupliftsleadstosmall
While the particular form of the KL superpotential was A-terms which are dominated by anomaly contributions.
instrumental in our analysis, the relation DρW(ρ) W, As we will see, this fact alone forces one to large scalar
≪
which we use in this section for computing soft terms for masses and hence a large gravitino mass. This is accept-
matter fields has a much more general validity. It follows able if the symmetry preventing a linear coupling of S to
directly from our requirementof strongvacuum stabiliza- matter is operative, and hence we are restricted to small
tion, which solves the problem of decompactification dur- gauginomassesalsodominatedbyanomalycontributions.
ing inflation with H & m3/2, as well as the cosmological Couplings in the K¨ahler potential of the type S†Sφ†φ, on
moduli problem. the other hand, are invariant under all symmetries and,
if present, they can change scalar masses in what follows.
Anadditionalassumptionwewillmakeinwhatfollows
We will comment on their possible effects below.
is that there is no direct coupling inthe K¨ahlerpotential,
superpotentialandgaugekineticfunctionbetweenmatter Soft terms for matter fields generated in supergravity
fieldsandtheupliftfields.Theabsenceoflinearcouplings in the limit M with fixed gravitino mass m
P 3/2
to the SUSY breaking uplift fields in the gauge kinetic [29] have a nice ge→om∞etrical structure. For F-term SUSY
function and superpotential for matter fields can be ar- breaking, they are given by [30]
gued at various levels:
- At the level of symmetries in the second uplift exam- m2i¯j =m23/2 (Gi¯j −Ri¯jαβ¯GαGβ¯ ) ,
fpolrembausneddeornchIiSrSalmsyomdeml,ettrhieesmSeUso(nNfi)elds SSijUt(hNer)e t,rbanros-- (B µ)ij =m23/2 (2∇iGj +Gα∇i∇jGα) ,
f L f R
ken only by mass terms. It is expected ×that couplings to (A y)ijk =m23/2(3∇i∇jGk+Gα∇i∇j∇kGα) ,
MSSMfieldsrespectchiralsymmetriesoftheupliftsector,
µ =m G ,
ij 3/2 i j
therefore linear couplings to S should be absent. ∇
1
-The upliftPolonyiorISSsectordoesbreakSUSYinthe ma = (Re h )−1m ∂ h Gα , (29)
1/2 2 A 3/2 α A
rigid limit in the absence of additional couplings to mat-
ter and moduli fields. When these additional couplings where G = K +ln W 2, y are Yukawa couplings, h
ijk A
are present, supersymmetry tends to be restored, espe- are the gauge kinet|ic |functions and denotes K¨ahler
i
cially for those couplings which break the R-symmetry covariantderivatives ∇
of the uplift sector. It is possible that the vacuum we
G =∂ G ΓkG , (30)
are discussing will still be a local miminum with a very ∇i j i j − ij k
long lifetime, however the absence of new couplings helps
in avoiding new supersymmetric minima (this argument, where Γikj = Gk¯l∂iGj¯l is the K¨ahler connection. Greek
indices α,β in (29) refer to SUSY breaking fields S and
however,does not pertain to the gauge kinetic function).
- From a string theory viewpoint, the linear terms in su- ρ,latinindices refertomatter fields,whereasRi¯jαβ¯ isthe
perpotentialspresentinbothofourexamplesdonotarise RiemanntensoroftheK¨ahlerspacespannedbychiral(su-
at tree-level in string perturbation theory. They can arise per)fields. In our models with strong moduli stabilization
nonperturbatively by D-brane instanton effects. In this anddecouplingbetweenupliftfieldsandmatterfields,the
case S is actually a field charged under an “anomalous” curvature terms in the scalar masses of matter fields are
U(1) . This U(1) is broken close to the string scale negligible and we find to great accuracy
X X
by field-dependent Fayet-Iliopoulos terms, depending on m2 =m2 , (31)
some modulus field called T in what follows. The axionic 0 3/2
fieldintheT multipletisshiftednonlinearlyunderU(1)X, where the gravitino mass is given by
T T+iδ α,whereαis the gaugetransformationpa-
→ GS 1
trhame epteerrt,uarbnadtiivseelaetveenl, ucopupbylintghseoUf S(1)aXregvaeurgyerefisetlrdi.ctAedt m23/2 =eG = 8σ03|W(ρ)+WF(S)|2, (32)
by the U(1)X symmetry. The gauge kinetic function hA and fixes the universal mass scale for scalars. For the
must clearly be invariant, and therefore S cannot appear O’KKLT model described above, so long as Λ2 1, the
there perturbatively. Instantonic effects are proportional dominant contribution to the gravitino mass com≪es from
to the D-instantonactionSinst =e−2πT, which has a spe- W(ρ) = ∆ at the minimum (see eq. (11)). The trilinear
cific U(1)X charge. Linear terms in S can arise nonper- terms are given by
turbatively in h and W via the gauge invariant combi-
A
nation e−2aTS, where the U(1) charge of e−2aT com- (Ay) =eKKαβ¯D W(K + )W , (33)
X ijk β α α ijk
∇
8 Dudaset al.: Strong moduli stabilization and phenomenology
whereW =∂ ∂ ∂ g,whereg(φi,ρ)isthesuperpotential togauginomassesandA-termsarisefromloopcorrections
ijk i j k
formatterfields(3).Inourcase,moreexplicitlytheyequal and give [19]
b g2 FC
(Ay)ijk =eK Kρρ¯DρW(Kρ+∇ρ)+KSS¯DSWKS Wijk, ma1/2 = 16aπa2 C0 (40)
h i (34) and
where we used, according to the arguments given above, γ +γ +γ FC
i j k
our hypothesis that Yukawas depend very weakly on S. Aijk =− 16π2 C . (41)
For bilinears B-terms, keeping also Giudice-Masiero like 0
terms, we find Here ba = 11,1, 3 for a = 1,2,3 are the one-loop beta
−
function coefficients, γ are the anomalous dimensions of
i
(Bµ)ij =eKKαβ¯DβW(Kα+ α)Wij m3/2eK/2Wij + the matter fields yi and
∇ −
m2 (2+Gα )K m2 Γα(2G +GβG ) . (35)
Not3i/c2e that in∇ouαr caijse−, sin3c/e2DijW aαnd K αβS¯ are very FCC0 =−31eK/2Kαβ¯KαD¯β¯W¯ +m3/2 ≃ m3/2 (42)
ρ S
∼
small, we find negligibly small A-terms. More precisely,
isrelatedtotheconformalcompensatorandequalstovery
we find that the dominant contribution to A is given by
0 high accuracy m in the models we consider.
S¯D¯S¯W¯ sothatatthetree-levelonefindsthattheA-terms 3/2
are given by Because of the loop suppression factor in Eq. (40), we
are forced to relatively large ( (10-1000) TeV) gravitino
1 ∆Λ2 1 masses in order to have acceptOably large gaugino masses3
A | | = m Λ2 (36)
0 ≃−(2σ )3/2 2 2 3/2 Thus, the sparticle spectrum consists of relatively light
0
gauginoswhosemassesaredeterminedfromanomalyme-
and are extremely small if Λ 1. This expression for diationandlargesoft scalarmassesfixedby the gravitino
A0 is valid so long as m3/2/m≪σ Λ2 1. For Λ2 mass yielding a spectrum reminiscent of split supersym-
≪ ≪ ≪
m /m 1theparameterA isproportionaltom2 /m ,metry[12]. The problemoftachyonicscalarsnormallyas-
3/2 σ ≪ 0 3/2 σ
so in both cases A m . Thus we are driven to small sociated with anomaly mediated models is absent here.
0 3/2
≪
values of A as a direct consequence of strong stabiliza-
0 In what follows, we will examine the phenomenolog-
tion. On the other hand, the µ and Bµ parameters in the
ical consequences of the above model. In particular, we
Higgs sector are given by
will see that it is difficult to construct consistent models
if one wants to maintain the possibility of radiative elec-
µ=m G =eK/2W +m K =µ +m K ,
3/2 12 12 3/2 12 0 3/2 12 troweak symmetry breaking. If the input supersymmetry
Bµ=(A m )µ +2m2 K . (37) breakingscaleischosentobetheGUTscale(i.e.thescale
0− 3/2 0 3/2 12
at which gauge coupling unification occurs), one can not
where W12 = ∂H1∂H2W, K12 = ∂H1∂H2K, and µ0 = choose arbitrarily large universal scalar masses and insist
eK/2W12 is the µ-term in the absence of Giudice-Masiero ona well defined electroweaksymmetry breaking vacuum
terms. By combining eqs. (37), we find (i.e., µ2 > 0). This difficulty can be alleviated in at least
two ways:
B =(A m )µ0 + 2m23/2K , (38) - increasing the supersymmetry breaking scale, Min >
0− 3/2 µ µ 12 MGUT. This is the case that we study in detail in the
next section.
that will be used in the next section for phenomenology. - Allow for direct couplings between the uplift field(s) S
If K12 = 0, we get µ = µ0 and B = A0 m3/2, which is and the Higgs in the Kahler potential, by terms of the
−
just the familiar mSUGRA relation B0 =A0 m0. type S†SH†H . In this case, Higgs soft scalar masses ac-
− i i
h(ρ)Fδor a, osnueitagbenleercahtoeisceunofivgearsuaglegkaiungeitnicofmunacstsieosns hαβ = qFuir=e aedKd/it2iKonSaS¯lDcorWre.ctTiohnesyparoreponrotiolnoanlgetro e|FquSa|2l,twohtehree
αβ S S
other scalar masses and are not necessarily degenerate
√2σ0 anymore. These boundary conditions for scalar masses
m = D W(ρ) ∂ lnReh , (39)
1/2 6 ρ ρ thenresemble those assumedinnon-universalHiggsmass
models. This problem can be traced directly back to our
where, according to our decoupling hypothesis, we have
assumption of strong moduli stabilization and small A-
assumedthath doesnotexplicitlydependonS 2.Incon-
terms. With large A-terms, there is no difficulty in ob-
trast to the universal scalar masses which are equal to
taining electroweak vacuum solutions with large m and
the gravitino mass, m is proportional to D W and is 0
1/2 ρ µ2 >0.
suppressed by m /m .
3/2 σ
As a result, we obtain models resembling those medi- 3 A posteriori, we know that for very small A0/m0 the re-
atedby anomalies[13],where the dominantcontributions quirement for relatively large Higgs masses would lead us to
the same conclusion regarding large scalar masses, which to
2 Ifweallowacouplingoftheformh =h0(1+β e−2aTS), controltherelicdensitywouldalsorequireanomalymediation
A A A
we would find a suppression m1/2∝m23/2/MP. for gaugino masses.
Dudaset al.: Strong moduli stabilization and phenomenology 9
Another challenge presented in these types of models masses are run down to low energy using standard renor-
stems from the mSUGRA relation between B and A . malization group evolution. In contrast to the CMSSM,
0 0
Unlike CMSSM models [31,32], this relationforces one to the gravitymediatedpartofgauginomassesandA-terms
solve for tanβ for a given choice of m ,m , and A . In in the KL model are extremely small and their dominant
1/2 0 0
the present context, we expect no solutions as there is in contributions are determined by anomalies at any scale
effectonlyasinglefreeparameter,namelym .However, using Eqs. (40) and (41). In the CMSSM, µ and B are
3/2
an interesting extension of minimal supergravity is one solved for in terms of m and tanβ:
Z
where terms proportional H H are added to the K¨ahler
1 2
potential as in the Giudice-Masiero mechanism [18]. By m2 m2tan2β+ 1m2(1 tan2β)+∆(1)
introducing a non-minimal coupling to the K¨ahler poten- µ2 = 1− 2 2 Z − µ ,
tan2β 1+∆(2)
tial, one can effectively fix tanβ. If in addition, we take − µ
1
Min >MGUT, we can in fact formulate a consistent phe- Bµ = (m2+m2+2µ2)sin2β+∆ , (43)
nomenological model. −2 1 2 B
In the next section we briefly review the GM exten- (1,2)
where ∆ and ∆ are loop corrections [33–35], and
B µ
sion to mSUGRA and the consequences of taking M >
in m aretheHiggssoftmasses(hereevaluatedattheweak
1,2
M . As a result, we are forced to consider a specific
GUT scale). In mSUGRA models, however, B can not be de-
GUT and here for simplicity, we take minimal SU(5) as a
termined independently as it must respect its boundary
concreteexample.Insection5,wepresentthemainresults
condition B =A m at M . Instead, one must solve
0 0 0 in
of the paper which include the low energy spectrum as a −
for tanβ (and µ) using the electroweak symmetry break-
functionofthegravitinomass.Inparticular,thisamounts
ing conditions [36,37]. In this sense, the KL models we
tothe gauginoandHiggsmassesasallofthe othersuper-
aredescribingaremorereminiscentofmSUGRAthanthe
symmetric scalars are very heavy. Other phenomenologi-
CMSSM.
calaspectsofthe modelssuchasgluinoproductionatthe
LHC and the direct andindirectdetection ofdarkmatter Thereare,however,twoimmediatepotentialproblems
are discussed in section 6. with the framework as described: 1) There is no guaran-
teethatreasonablesolutionsfortanβexistwhilerequiring
B =A m atM .Indeeditisknown[36,37]thatonly
0 0 0 in
−
alimitedportionofparameterspace(definedbym ,m
0 1/2
4 GM Supergravity and Super-GUT andA )possessessolutionsfortanβ.2)Thereisnoguar-
0
phenomenology antee that solutions with µ2 > 0 exist when m0 is very
large. This of course is a well known issue present in the
CMSSM. For fixed m and A , there is an upper limit
1/2 0
As described above, the KL phenomenological model has to m for which there are solutions to (43) with µ2 > 0
0
onefreeparameter,m ,whichwhenextendedtoinclude known as the focus point or hyperbolic branch [38]. This
3/2
a Giudice-Masiero term, has two free parameters, which upper limit is also present in mSUGRA models as well,
we take to be m and tanβ. This is to be compared particularly when A /m is small (as it is the case un-
3/2 0 0
with mSUGRA models which have 3 free parameters or der consideration). As we now describe, neither problem
CMSSM models with 4 free parameters. In the present is criticaland there are known and relatively simple solu-
context, the gaugino masses, scalar masses, and A-terms tions to both.
are all determined by the gravitino mass. The alternative
To tackle the problem of tanβ, consider a Giudice-
ofcoupling the uplift fields S directly to the Higgssector,
Masiero (GM) -like contribution to K of the form [18],
in order to obtain non-universal Higgs masses which are
differentfromthe gravitinomass,mentionedinthe previ-
∆K =c H H +h.c., (44)
H 1 2
oussection,willnotbepursuedhereforsimplicity.Onthe
otherhand,directcouplingsofupliftfieldstosquarksand where c (equal to K in the previous section) is a con-
H 12
sleptons λ S†Sφiφ† in the Kahler potential wouldgener- stant, and H are the usual MSSM Higgs doublets. The
ij j 1,2
ically lead to flavor dependence and therefore to FCNC presence of ∆K affects the boundary conditions for both
effects. Evenfor 30 50 TeV scalarmasses,which will be µ and the B term at the supersymmetry breaking input
−
our typical values in what follows, FCNC effects require scale, M . The µ term is shifted as seen in Eq. (37) to
in
somedegree ofdegeneracy.This is actually the mainphe-
nomenologicalreasonweareimposing nodirectcouplings µ0+cHm0. (45)
between uplift fields and matter fields in our paper.
However, since we solve for µ at the weak scale, its UV
From Eq. (31), we expect scalar mass universality at
value is fixed by the low energy boundary condition. In
some renormalization scale, M . In the CMSSM, this
in contrast,the boundary condition on µB shifts from µ B
scaleisusuallyassociatedwiththeGUTscale4.Ifso,these 0 0
to
µ B +2c m2. (46)
4 TheGUTscale,M ,isdefinedasthescalewhereSU(2) 0 0 H 0
GUT
and U(1) gauge couplings unify and is approximately 1.5× WecanaddtheGMtermtobetterconnectthesolutionof
1016 GeV. the minimization conditions to a supergravity boundary
10 Dudaset al.: Strong moduli stabilization and phenomenology
condition at M . Indeed, by allowing c =0, we can fix There are now two µ-parameters, µ and µ , as well as
in H H Σ
tanβ and derive µ and Bµ at the weak sca6 le. By running two new couplings, λ and λ′. Results are mainly sensitive
ourderivedvaluesofB(M )andµ(M )upto the GUT to λ and the ratio of the two couplings. In what follows,
W W
scale, we can write we will fix λ′ =0.1.
Bµ(MGUT)=(A0−m0)µ0(MGUT)+2cHm20, (47) conTdiotiognen,ewrealwizreittehe GM solution for the B0 boundary
whichis preciselyeq.(37)ofthe previoussection.Strictly
speaking, (47) is valid at tree-level in SUGRA and does 1
∆K =c + c TrΣ2+h.c., (49)
not include anomaly contributions. However, the latter H 1 2 Σ
H H 2
are small compared to tree-level values of m , B and µ,
0
so (47) is anexcellent approximation.In what follows,we where are scalar components of the Higgs five-plets
1,2
H
use Eq. (47) to derive the necessary value of cH. andΣ is the scalarcomponent ofthe adjointHiggs.Thus
in principle, we have two extra parameters which can be
Of course, one must still check, whether the solution
adjustedtorelatetheCMSSMandsupergravityboundary
for c is reasonable(i.e., perturbative). In [37], it was in-
H
conditions for M >M . Nevertheless, these parame-
deed shown that over much of the mSUGRA parameter in GUT
tershavevirtuallynoeffectonthesparticlemassspectrum
space c . O(1). For fixed tanβ and A /m , c is rea-
H 0 0 H
otherthanallowingustofixtanβ inaconsistentmanner.
sonablysmallformostchoicesofm andm .Exceptions
1/2 0
lying in the region where m1/2 ≫m0 and the lightest su- For Min > MGUT, scalar mass universality is defined
persymmetric particle (LSP) is the gravitino. When A0 intermsofthescalarcomponentsoftheHiggsesandmat-
is large,these offending regionsare further compressedto ter fields in the 5 and 10 representations. At the GUT
small m0. Thus by allowing non-zero cH, we can always scale, these must be matched to their Standard Model
satisfythemSUGRAboundaryconditionforB0andcheck counterparts. More importantly is the matching of the µ
a posteriori that cH is small. and B-terms from SU(5) to Standard Model parameters.
These have been discussedextensively in Ref. [37,49] and
As noted above, in the CMSSM and mSUGRA, there
we do not repeat that analysis here.
is generally an upper limit to m for fixed m ,A , and
0 1/2 0
tanβ determined by µ2 = 0 in Eq. (43). While it is com- There is one aspect of the matching of soft terms at
mon to assume that the input supersymmetry breaking M that is specific to the present model. Dominant
GUT
scale is equal to the GUT scale, it is quite plausible that contributionstogauginomassesandA-termsareprovided
M may be either below [39] (as in models with mirage bytheconformalanomaly(40,41),withbetafunctionsand
in
mediation[19,26,40]) orabove[37,41–45] the GUT scale. anomalousdimensionscomputedwiththespectrumatthe
Increasing M increases the renormalization of the soft givenenergyscaleE.FortheMSSMforexample,gaugino
in
masses which tends in turn to increase the splittings be- masses at scale E are given by
tweenthephysicalsparticlemasses[43].Asaconsequence,
thefocus-pointsolutionforµ2 =0oftenmovesouttovery b g2(E)FC
ma (E)= a a . (50)
large values of m0. This feature of super-GUT models is 1/2 16π2 C0
essentialforKLmodeldescribedhere.Notethatwhilethe
introduction of Min adds a free parameter to the model, AboveMGUT,ontheotherhand,wehaveaunified(SU(5)
aswewillsee,ourresultsareveryinsensitivetothechoice in our case) theory, with a unified gauge coupling g
GUT
ofMin.Forconsistencywiththe KLparadigm,weshould and a unified beta function bGUT. The unified gaugino
also only consider values of Min <mσ. mass is then given by
To realize M >M , we need to work in the con-
in GUT b g2 (E)FC
text of a specific GUT. Here, we use the particle content mGUT(E) = GUT GUT (51)
and the renormalization-groupequations (RGEs) of min- 1/2 16π2 C0
imal SU(5) [43,46], primarily for simplicity: for a recent
reviewofthissamplemodelanditscompatibilitywithex- andits valuehas to be takeninto accountfor the running
periment, see [47]. As this specific super-GUT extension of soft terms between Min and MGUT. However, there is
of the CMSSM was studied extensively in Refs. [41,48], no matching at MGUT between (50) and (51). The mis-
we refer the reader there for details of the model. matchistobeinterpretedasathresholdeffect,duetothe
decoupling of heavy GUT particles at M . The argu-
GUT
The model is defined by the superpotential
ment is completely similar for the A-terms.
1
W5 = µΣTrΣˆ2+ λ′TrΣˆ3+µH ˆ1 ˆ2+λˆ1Σˆ ˆ2 The additional running between Min and MGUT in
6 H H H H CMSSM-like models is very efficient at raising the up-
+(h10)ijψˆiψˆj ˆ2+(h5)ijψˆiφˆj ˆ1, (48) per limit on m0 [37,41,42,44] provided the Higgs cou-
H H
plingλissufficientlylarge.InmSUGRA-likemodels,how-
where φˆi (ψˆi) correspondto the 5 (10) representationsof ever,we are still faced with the difficulty of satisfying the
superfields, Σˆ(24), ˆ (5) and ˆ (5) represent the Higgs B boundary condition and the GM parameters must be
1 2 0
adjoint and five-pletHs. Here i,jH= 1..3 are generation in- added. As shown in [37], for M & 1017 GeV, and fixed
in
dicesandwesuppresstheSU(5)indexstructureforbrevity. tanβ,A ,m ,andm ,valuesofc areonlysmallwhen
0 0 1/2 H