Blueprints of the No-Scale Multiverse at the LHC

0.6 MB·English

by  Tianjun Li

ACT-01-11, MIFPA-11-03
Blueprints of the No-Scale Multiverse at the LHC
Tianjun Li,1,2 James A. Maxin,2 Dimitri V. Nanopoulos,2,3,4 and Joel W. Walker5
1Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, P. R. China
2George P. and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy,
Texas A&M University, College Station, TX 77843, USA
3Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, Woodlands, TX 77381, USA
4Academy of Athens, Division of Natural Sciences,
28 Panepistimiou Avenue, Athens 10679, Greece
5Department of Physics, Sam Houston State University, Huntsville, TX 77341, USA
We present a contemporary perspectiveon theString Landscape and theMultiverse of plausible
string, M- and F-theory vacua. In contrast to traditional statistical classifications and capitulation
2
totheanthropicprinciple,weseekonlytodemonstratetheexistenceofanon-zeroprobabilityfora
1
universematchingourownobservedphysicswithinthesolution ensemble. Wearguefortheimpor-
0
tanceofNo-ScaleSupergravityasanessentialcommonunderpinningforthespontaneousemergence
2
ofacosmologically flatuniversefromthequantum“nothingness”. Concretely,wecontinuetoprobe
n the phenomenology of a specific model which is testable at the LHC and Tevatron. Dubbed No-
a
Scale F-SU(5), it represents the intersection of the Flipped SU(5) Grand Unified Theory (GUT)
J
with extra TeV-Scale vector-like multiplets derived out of F-theory, and the dynamics of No-Scale

6 Supergravity,whichinturnimplyaveryrestrictedsetofhighenergyboundaryconditions. Bysec-
1 ondarilyminimizingtheminimumofthescalarHiggspotential,wedynamicallydeterminetheratio

tanβ ≃15−20 of up-to down-typeHiggs vacuumexpectation values (VEVs),theuniversal gaug-
]
h ino boundary mass M1/2 ≃450 GeV, and consequently also the total magnitude of the GUT-scale
p HiggsVEVs,whileconstrainingthelowenergyStandardModelgaugecouplings. Inparticular,this
- localminimumminimorumlieswithinthepreviouslydescribed“goldenstrip”,satisfyingallcurrent
p experimentalconstraints. Weemphasize,however,thattheoverarchinggoalisnottoestablishwhy
e ourownparticularuniversepossesses anynumberofspecificcharacteristics, butrathertoteaseout
h
what generic principles might govern thesuperset of all possible universes.
[

2 PACSnumbers: 11.10.Kk,11.25.Mj,11.25.-w,12.60.Jv
v
7
9 I. INTRODUCTION Whether the Multiverse be reverie or reality, the con-
1 ceptual superset of our own physics which it embodies
2 Thenumberofconsistent,meta-stablevacuaofstring, must certainly representthe interference of all navigable
1. M- or (predominantly) F-theory flux compactifications universal histories.
0 which exhibit broadly plausible phenomenology, includ- Surely many times afore has mankind’s notion of the
1 ing moduli stabilization and broken supersymmetry [1– heavens expanded - the Earth dispatched from its cen-
1 6], is popularly estimated [7, 8] to be of order 10500. It tral pedestal in our solar system and the Sun rendered
:
v is moreover currently in vogue to suggest that degen- one among some hundred billionstars of the Milky Way,
i eracy of commonfeatures acrossthese many “universes” itself reduced to one among some hundred billion galax-
X
mightstatisticallyisolatethephysicallyrealisticuniverse ies. Finally perhaps, we come to the completion of our
ar fromthe vast“landscape”,muchasthe entropyfunction Odyssey,by realizing that our Universe is one of at least
coaxes the singular order of macroscopic thermodynam- 10500 so possible, thus rendering the anthropic view of
ics from the chaotic duplicity of the entangled quantum ourpositioninthe Universe(environmentalcoincidences
microstate. Weargueherethoughthecounterpointthat explained away by the availability of 1011 ×1011 solar
we are not obliged a priori to live in the likeliest of all systems)functionallyequivalenttothe anthropicviewof
universes, but only in one which is possible. The exis- the origin of the Universe (coincidences in the form and
tence merely of a non-zero probability for our existence contentof physicallaws explained awayby the availabil-
is sufficient. ity, through dynamical phase transitions, of 10500 uni-
Weindulgeforthiseffortthe fancifulimaginationthat verses). Nature’sbountyhasanywayinvariablytrumped
the “Multiverse” of string vacua might exhibit some lit- our wildest anticipations,and though frugal and equani-
eral realization beyond our own physical sphere. A sin- mous in law, she has spared no extravagance or whimsy
gle electron may be said to wander all histories through in its manifestation.
interferingapertures,thoughitsarrivalisultimatelyreg- Our perspective should not be misconstrued,however,
isteredatalocalizedpointonthe target. Thejourneyto as complacent retreatinto the tautology of the weak an-
that destination is steered by the full dynamics of the thropicprinciple. Itisindeedunassailabletruismthatan
theory, although the isolated spontaneous solution re- observed universe must afford and sustain the life of the
flects only faintly the richness of the solution ensemble. observer,including requisiteconstraints,forexample,on
2
the cosmological constant [9] and gauge hierarchy. Our 12,18,19]thatthefluctuationsofadynamicallyevolved
point of view, though, is sharply different; we should be expandinguniversemightspontaneouslyproducetunnel-
abletoresolvethecosmologicalconstantandgaugehier- ingfromafalsevacuumintoanadjacent(likelyalsofalse)
archyproblemsthroughinvestigationofthefundamental meta-stable vacuum of lower energy, driving a local in-
laws of our (or any single) Universe, its accidental and flationary phase, much as a crystal of ice or a bubble
specific properties notwithstanding, without resorting to of steam may nucleate and expand in a super-cooled or
theexistenceofobservers. Inourview,theobserveristhe super-heated fluid during first order transition. In this
output of, not the raison d’ˆetre of, our Universe. Thus, “eternal inflation” scenario, such patches of space will
our attention is advance from this base camp of our own volumetrically dominate by virtue of their exponential
physics, as unlikely an appointment as it may be, to the expansion,recursivelygeneratinganinfinitefractalarray
summit goalof the master theory and symmetries which ofcausallydisconnected“Russiandoll”universes,nesting
governallpossibleuniverses. Insoseeking,ourfirsthalt- each within another, and each featuring its own unique
ing foragemustbe thatofa concretestringmodelwhich physical parameters and physical laws.
can describe Nature locally. Fromjustthespecificlocationonthesolution“target”
where our own Universe landed, it may be impossible to
directly reconstruct the full theory. Fundamentally, it
may be impossible even in principle to specify why our
II. THE ENSEMBLE MULTIVERSE
particular Universe is precisely as it is. However, super-
string theory and its generalizations may yet present to
The greatest mystery of Nature is the origin of the
usaloftierprize-thetheoryofthe ensembleMultiverse.
Universe itself. Modern cosmology is relatively clear re-
gardingtheoccurrenceofahotbigbang,andsubsequent
Planck, grand unification, cosmic inflation, lepto- and
III. THE INVARIANCE OF FLATNESS
baryogenesis,andelectroweakepochs,followedby nucle-
osynthesis, radiation decoupling, and large scale struc-
More important than any differences between various
ture formation. In particular, cosmic inflation can ad-
possible vacua are the properties which might be invari-
dress the flatness and monopole problems, explain ho-
ant, protected by basic symmetries of the underlying
mogeneity, and generate the fractional anisotropy of the
mechanics. We suppose that one such basic property
cosmic background radiation by quantum fluctuation of
must be cosmological flatness, so that the seedling uni-
the inflatonfield[10–14]. Akeyquestionthough,is from
verse may transition dynamically across the boundary
whence the energy of the Universe arose. Interestingly,
of its own creation, maintaining a zero balance of some
thegravitationalfieldinaninflationaryscenariocansup-
suitably defined energy function. In practice, this im-
plytherequiredpositivemass-kineticenergy,sinceitspo-
pliesthatgravitymustbeubiquitous,itsnegativepoten-
tentialenergybecomesnegativewithoutbound,allowing
tialenergyallowingforpositive mass andkinetic energy.
that the total energy could be exactly zero.
Within such a universe, quantum fluctuations may not
Perhaps the most striking revelation of the post-
again cause isolated material objects to spring into ex-
WMAP [15–17] era is the decisive determination that
istence, as their net energy must necessarily be positive.
our Universe is indeed globally flat, i.e. with the net
Fortheexampleofaparticlewithmassmonthesurface
energy contributions from baryonic matter ≃ 5%, dark
of the Earth, the ratio of gravitationalto mass energy is
matter ≃ 23%, and the cosmological constant (dark en-
more than nine orders of magnitude too small
ergy) ≃ 72% finely balanced against the gravitational
potential. Not long ago, it was possible to imagine the G M m
− N E ÷mc2 ≃7×10−10 , (1)
Universe, with all of its physics intact, hosting any ar- (cid:12) R (cid:12)
(cid:12) E (cid:12)
bitrary mass-energy density, such that “k = +1” would (cid:12) (cid:12)
representasuper-criticalcosmologyofpositivecurvature, where GN(cid:12)is the gravit(cid:12)ational constant, c is the speed
and“k =−1”thesub-criticalcaseofnegativecurvature. of light, and ME and RE are the mass and radius of
In hindsight, this may come to seem as na¨ıve as the no- the Earth, respectively. Even in the limiting case of a
tion of an empty infinite Cartesian space. The observed SchwarzschildblackholeofmassMBH,aparticleofmass
energybalanceishighlysuggestiveofafundamentalsym- m at the horizonRS =2GNMBH/c2 has a gravitational
metry which protects the “k = 0” critical solution, such potential which is only half of that required.
that the physical constants of our Universe may not be
G M m 1
divorced from its net content. − N BH = mc2 (2)
(cid:12) R (cid:12) 2
Thisnullenergyconditionlicensesthespeculativecon- (cid:12) S (cid:12)
(cid:12) (cid:12)
nection ex nihilo of our present universe back to the pri- It is important to(cid:12) note that wh(cid:12)ile the energy density for
mordial quantum fluctuation of an external system. In- the gravitational field is surely negative in Newtonian
deed, there is nothing which quantum mechanics abhors mechanics, the global gravitational field energy is not
more than nothingness. This being the case, an extra well defined in general relativity. Unique prescriptions
universehereorthere mightrightlybe consideredno ex- for a stress-energy-momentum pseudotensor can be for-
tratroubleatall! Specifically,ithasbeensuggested[10– mulated though, notably that of Landau and Lifshitz.
3
Any such stress-energy can, however, be made to vanish the brane stacking and intersection multiplicities. The
locally by general coordinate transformation, and it is YukawacouplingsandHiggsstructureareinlikemanners
notevenentirelyclearthatthepseudotensorsoappliedis also specified, leading after radiativesymmetry breaking
an appropriate general relativistic object. Given though of the chiral gauge sector to low energy masses for the
that Newtonian gravity is the classical limit of general chiralfermions and broken gauge generators,each mass-
relativity,itisreasonabletosuspectthattheproperlyde- less in the symmetric limit.
fined field energy density will be likewise also negative, From a top-down view, Supergravity (SUGRA) is an
and that inflation is indeed consistent with a correctly ubiquitous infrared limit of string theory, and forms the
generalized notion of constant, zero total energy. starting point of any two-dimensional world sheet or D-
Auniversewouldthenbeinthissenseclosed,anisland dimensional target space action. The mandatory local-
unto itself, from the moment of its inception from the ization of the Supersymmetry (SUSY) algebra,and thus
quantumfroth;onlyauniverseintotomightsooriginate, the momentum-energy (space-time translation) opera-
emerging as a critically bound structure possessing pro- tors, leads to general coordinate invariance of the action
found density and minute proportion, each as accorded and an Einstein field theory limit. Any available fla-
against intrinsically defined scales (the analogous New- vor of Supergravity will not however suffice. In general,
ton and Planck parameters and the propagation speed extraneous fine tuning is required to avoid a cosmolog-
ofmasslessfields), andexpandingorinflating henceforth ical constant which scales like a dimensionally suitable
and eternally. power of the Planck mass. Neglecting even the question
ofwhethersuchauniversemightbepermittedtoappear
spontaneously,it would then be doomed to curl upon it-
self and collapse within the order of the Planck time, for
IV. THE INVARIANCE OF NO-SCALE SUGRA
comparisonabout10−43secondsinourUniverse. Expan-
sion and inflation appear to uniquely require properties
Inflation, driven by the scalar inflaton field is itself
which arise naturally only in the No-Scale SUGRA for-
inherently a quantum field theoretic subject. However,
mulation [22–26].
thereistensionbetweenquantummechanicsandgeneral
SUSY is in this case broken while the vacuum energy
relativity. Currently,superstringtheoryisthebestcandi-
densityvanishesautomaticallyattreelevelduetoasuit-
dateforquantumgravity. Thefiveconsistenttendimen-
able choice of the K¨ahler potential, the function which
sionalsuperstringtheories,namelyheteroticE ×E ,het-
8 8 specifies the metric on superspace. At the minimum of
erotic SO(32), Type I, Type IIA, Type IIB, can be uni-
the null scalar potential, there are flat directions which
fied by various duality transformations under an eleven-
leave the compactification moduli VEVs undetermined
dimensional M-theory [20], and the twelve-dimensional
by the classical equations of motion. We thus receive
F-theory can be considered as the strongly coupled for-
without additional effort an answer to the deep ques-
mulation of the Type IIB string theory with a varying
tion of how these moduli are stabilized; they have been
axion-dilatonfield [21]. Self consistency of the string (or
transformedintodynamicalvariableswhicharetobede-
M-, F-) algebra implies a ten (or eleven, twelve) dimen-
termined by minimizing corrections to the scalar poten-
sional master spacetime, some elements of which – six
tial at loop order. In particular, the high energy grav-
(or seven, eight) to match our observed four large di-
itino mass M , and also the proportionally equivalent
mensions–maybecompactifiedonamanifold(typically 3/2
universal gaugino mass M , will be established in this
Calabi-Yau manifolds or G manifolds) which conserves 1/2
2 way. Subsequently, all gauge mediated SUSY breaking
a requisite portion of supersymmetric charges.
soft-terms will be dynamically evolved down from this
Thestructureofthecurvaturewithintheextradimen-
boundary under the renormalization group [27], estab-
sions dictates in no small measure the particular phe-
lishing in large measure the low energy phenomenology,
nomenology of the unfolded dimensions, secreting away
and solving also the Flavour Changing Neutral Current
the“closetspace”toencodethesymmetriesofallgauged
(FCNC) problem. Since the moduli are fixed at a false
interactions. The physical volume of the internal spatial
local minimum, phase transitions by quantum tunneling
manifold is directly related to the effective Planck scale
will naturally occur between discrete vacua.
andbasicgaugecouplingstrengthsintheexternalspace.
Weconjecture,forthereasonsgivenprior,thattheNo-
Thecompactificationisinturndescribedbyfundamental
ScaleSUGRAconstructioncouldpervadealluniversesin
modulifieldswhichmustbestabilized,i.e.givensuitable
the String Landscape with reasonable flux vacua. This
vacuum expectation values (VEVs). The famous exam-
beingthecase,intelligentcreatureselsewhereintheMul-
ple of Kaluza and Klein prototypes the manner in which
tiverse, though separated from us by a bridge too far,
general covariance in five dimensions is transformed to
might reasonablyso concur after parallelexaminationof
gravity plus Maxwell theory in four dimensions when
theirownphysics. Moreover,theymightleverageviathis
the transverse fifth dimension is cycled around a cir-
insightadeeperknowledgeofthe underlyingMultiverse-
cle. The connection of geometry to particle physics is
invariant master theory, of which our known string, M-,
perhaps nowhere more intuitively clear than in the con-
and F-theories may compose some coherently overlap-
text of model building with D6-branes, where the gauge
ping patch of the garment edge. Perhaps we yet share
structure and family replication are related directly to
4
appreciation, across the cords which bind our 13.7 bil- emerges in turn as a suitable candidate scale only when
lion years to their correspondingblink of history, for the substantiallydecoupledfromtheprimaryGUTscaleuni-
common timeless principles under which we are but two fication of SU(3) ×SU(2) via the modification to the
C L
isolatedcondensationsupontwoparticularvacuumsolu- renormalization group equations (RGEs) from the extra
tions among the physical ensemble. F-theory vector multiplets.
In particular, we have systematically established the
hyper-surfacewithinthetanβ,topquarkmassm ,gaug-
t
ino mass M , and vector-like particle mass M pa-
V. AN ARCHETYPE MODEL UNIVERSE 1/2 V
rameter volume which is compatible with the applica-
tion of the simplest No-Scale SUGRA boundary condi-
Though we engage in this work lofty and speculative
tions [22–26], particularly the vanishing of the Higgs bi-
questions of natural philosophy, we balance abstraction
linear soft term B at the ultimate F-SU(5) unification
againstthe measuredmaterialunderpinningsofconcrete µ
scale [45, 46]. We have demonstrated that simultane-
phenomenological models with direct and specific con-
ous adherence to all current experimental constraints,
nection to tested and testable particle physics. If the
most importantly contributions to the muon anomalous
suggestion is correct that eternal inflation and No-Scale
magneticmoment(g−2) [49], the branchingratiolimit
SUGRA models with string origins together describe µ
on (b → sγ) [50, 51], and the 7-year WMAP relic den-
whatis infactourMultiverse,thenwemustasaprereq-
sity measurement [15–17], dramatically reduces the al-
uisitesettletheissueofwhetherourownphenomenology
lowed solutions to a highly non-trivial “golden strip”
can be produced out of such a construction.
with tanβ ≃ 15, m = 173.0 − 174.4 GeV, M =
In the context of Type II intersecting D-brane mod- t 1/2
455−481 GeV, and M = 691−1020 GeV, effectively
els,we haveindeed foundone realisticPati-Salammodel V
eliminating all extraneously tunable model parameters,
which might describe Nature as we observe it [28–30]. If
wheretheconsonanceofthetheoreticallyviablem range
only the F-terms of three complex structure moduli are t
with the experimentally established value [52] is an in-
non-zero, we also automatically have vanishing vacuum
dependently correlated “postdiction”. Finally, taking a
energy, and obtain a generalized No-Scale SUGRA. It
fixedZ-bosonmass,wehavedynamicallydeterminedthe
seems to us that the string derivedGrand Unified Theo-
universalgauginomassM andfixedtanβ viathe“Su-
ries(GUTs),andparticularlytheFlippedSU(5)×U(1) 1/2
X per No-Scale” mechanism [47], that being the secondary
models [31–33], are also candidate realistic string mod-
minimization,orminimum minimorum, ofthe minimum
els with promising predictions that can be tested at the
V oftheHiggspotentialfortheelectroweaksymmetry
min
Large Hadron Collider (LHC), the Tevatron, and other
breaking (EWSB) vacuum.
future experiments.
These models are moreover quite interesting from a
In the latter case, the Flipped SU(5)×U(1) gauge
X phenomenological point of view [43, 44]. The predicted
symmetry can be broken down to the SM gauge sym-
vector-likeparticlescanbeobservedattheLargeHadron
metry by giving VEVs to one pair of the Higgs fields H
Collider, and the partial lifetime for proton decay in
and H with quantum numbers (10,1) and (10,−1), re- the leading (e|µ)+π0 channels falls around 5 × 1034
spectively. The doublet-triplet splitting problem can be
years,testable at the future Hyper-Kamiokande[53]and
solvednaturally via the missing partnermechanism[33].
Deep Underground Science and Engineering Laboratory
Historically, Flipped SU(5)×U(1) models have been
X (DUSEL)[54]experiments[48,55]. ThelightestCP-even
constructed systematically in the free fermionic string
Higgs boson mass can be increased [56], hybrid inflation
constructions at Kac-Moody level one [33–37]. To ad-
canbenaturallyrealized,andthecorrectcosmicprimor-
dress the little hierarchy problem between the unifica-
dial density fluctuations can be generated [57].
tion scale and the string scale, the Testable Flipped
SU(5)×U(1) model class was proposed, which intro-
X
duces extra TeV-scale vector-like particles [38]. Mod-
els of this type have been constructed locally as exam- VI. NO-SCALE FOUNDATIONS OF F-SU(5)
ples of F-theory model building [39–44], and dubbed F-
SU(5) [43, 44] within that context. Inthetraditionalframework,supersymmetryisbroken
Most recently, we have studied No-Scale extensions of in the hidden sector, and then its breaking effects are
the prior in detail[45–47], emphasizing the essentialrole mediated to the observable sector via gravity or gauge
ofthetripodalfoundationformedbytheF-lippedSU(5) interactions. In GUTs with gravity mediated supersym-
GUT [31–33], two pairs of TeV scale vector-like mul- metry breaking, also known as the minimal Supergrav-
tiplets with origins in F-theory [38, 43, 44, 48] model ity (mSUGRA) model, the supersymmetry breakingsoft
building, and the boundary conditions of No-Scale Su- terms can be parameterized by four universal parame-
pergravity[22–26]. ItappearsthattheNo-Scalescenario, ters: the gaugino mass M , scalar mass M , trilinear
1/2 0
particularlyvanishingoftheHiggsbilinearsofttermB , soft term A, and the ratio of Higgs VEVs tanβ at low
µ
comesintoitsownonlywhenappliedatanelevatedscale, energy, plus the sign of the Higgs bilinear mass term µ.
approaching the Planck mass. M ≃ 7 × 1017 GeV, The µ term andits bilinear soft termB are determined
F µ
the point ofthe secondstageSU(5)×U(1) unification, bytheZ-bosonmassM andtanβ aftertheelectroweak
X Z
5
(EW) symmetry breaking. order. Likewise, the subsequently introduced KKLT [3]
To solve the cosmological constant problem, No-Scale constructions also manifest a No-Scale SUGRA struc-
Supergravity was proposed [22–26]. No-scale Supergrav- ture at the classical level. Indeed, the No-Scale features
ity isdefinedasthe subsetofSupergravitymodelswhich are generically obtained at the tree-level in string the-
satisfythefollowingthreeconstraints[22–26]: (i)thevac- ory compactifications due to the presence of three com-
uum energy vanishes automatically due to the suitable plex extra dimensions. However, this classical level re-
Ka¨hler potential; (ii) at the minimum of the scalar po- sult is rather precariously balanced, and may be spoiled
tential,thereareflatdirectionswhichleavethegravitino by quantum corrections to the superpotential including
mass M undetermined; (iii) the super-trace quantity flux contributions, instanton effects, gaugino condensa-
3/2
StrM2 is zero at the minimum. Without this, the large tion, and the next order α′ corrections. In this sense, we
one-loopcorrectionswouldforceM tobeeitherzeroor considertheKKLTtypeSUGRAmodelsasageneraliza-
3/2
ofPlanckscale. AsimpleK¨ahlerpotentialwhichsatisfies tion or extension of the elemental No-Scale form.
the first two conditions is The No-Scale F-SU(5) model under discussion has
been constructed locally in F-theory [43, 44], although
K = −3ln(T +T − Φ Φ ) , (3) themassoftheadditionalvector-likemultiplets,andeven
i i
Xi the fact of their existence, is not mandated by the F-
theory, wherein it is also possible to realize models with
whereT is amodulus fieldandΦ arematterfields. The
i only the traditional Flipped (or Standard) SU(5) field
third condition is model dependent and can always be
content. We claim only an inherent consistency of their
satisfied in principle [58].
conceptualoriginoutoftheF-theoreticconstruction,and
ThescalarfieldsofEq.(3)parameterizethecosetspace
take the manifest phenomenological benefits which ac-
SU(N +1,1)/(SU(N +1)×U(1)), where N is the
C C C company the natural elevation of the secondary GUT
number of matter fields. Analogousstructures appear in unification phase to M ≃7×1017 GeV as justification
F
theN ≥5extendedSupergravitytheories[59],forexam-
for the greater esteem which we hold for this particu-
ple,N =4forN =5,whichcanberealizedinthecom-
C lar model above other alternatives. There are, though,
pactificationsofstring theory[60,61]. The non-compact
alsodelicatequestionsofcompatibilitybetweenthe local
structure of the symmetry implies that the potential is
F-theoretic model building origins and the purely field-
not only constant but actually identical to zero. For the
theoretic RGE running which we employ up to the pre-
simple example K¨ahler potential given above, one can
sumed high scale. As one approaches the Planck mass
readily check that the scalar potential is automatically
M , consideration must be given to the role which will
Pl
positive semi-definite, and has a flat direction along the
beplayedbyKaluzaKlein(KK)andstringmodeexcita-
T field. Likewise, it may be verified that the simplest tions, and also to corrections of order α′ from stabiliza-
No-ScaleboundaryconditionsM =A=B =0emerge
0 µ tion of the global volume of the six-dimensional internal
dynamically, while M may be non-zeroat the unifica-
1/2 space in association with the establishment of the string
tion scale, allowing for low energy SUSY breaking. scale M ∝(M /R3 ).
The specific K¨ahlerpotentialofEq.(3)hasbeeninde- S Pl global
Themostimportantquestioniswhetherourmodelcan
pendentlyderivedinbothweaklycoupledheteroticstring
infactbeembeddedintoagloballyconsistentframework.
theory [60] and the leading order compactification of M-
Without such, we do not know the concrete K¨ahler po-
theoryonS1/Z [61]. Notethatinbothcases,theYang-
2 tentialoftheSMfermionsandHiggsfields,andcannotby
Mills fields span a ten dimensional space-time. It is not
thismeansexplicitlycalculatethesupersymmetrybreak-
obtained directly out of F-theory, as represented for ex-
ingscalarmassesandtrilinearsoftterms. Thisconstruc-
amplebythestrongcouplingliftfromTypeIIBintersect-
tion remains elusive though, and is beyond the reach of
ingD-branemodelbuildingwithD7-andD3-branes[39–
the current work. Regardless, one may anticipate that
42],wheretheYang-MillsfieldsontheD7-branesoccupy
in such a globally consistent model, a string scale of or-
an eight dimensional space-time. Nevertheless, it is cer- der 1017 GeV would indeed be realized, as in the weakly
tainly possible in principle to calculate a gauge kinetic
coupled heterotic string theory, tying in nicely with our
function,Kahlerpotentialandsuperpotentialinthecon-
na¨ıvely projected value for M . It seems additionally
F
textofTypeIIBinterectingD-branemodelbuilding,and
that a field-theoretic application of the No-Scale bound-
the F-theory could thus admit a more general definition
ary conditions might prove to be validated in this case.
of No-Scale Supergravity, as realized by a K¨ahler poten-
Moreover,we would not necessarily require the presence
tial like
of instanton effects or gaugino condensation for stabi-
lization of the modulus T as in the KKLT mechanism.
K = −ln(S+S)−ln(T +T )
1 1 Thisiscrucial,becausesucheffectscanhavethenegative
−ln(T +T )−ln(T +T ), (4)
2 2 3 3 side effect of destroying the leading No-Scale structure.
In fact, we could have no gaugino condensation at all,
where only three of the moduli fields S andT mayyield
i or the superpotential from gaugino condensation might
non-zero F-terms.
only depend on S, as again exemplified in the Type IIB
In Ref. [2], No-Scale Supergravity was obtained in the
intersecting D-brane models [62].
Type IIB and F-theory compactifications at the leading
6
Such considerations, coupled with the demonstrated uum. Since V depends on M , the gaugino mass
min 1/2
testability and phenomenological success of the first or- M is consequently dynamically determined by the
1/2
deranalysisinthesimplestNo-ScaleSUGRAframework, equation dV /dM =0, aptly referredto as the “Su-
min 1/2
argue for a continuing study of the generalized No-Scale per No-Scale” mechanism [47].
SUGRA picture. It is important to note that there ex- It could easily have been that in consideration of the
ist several such generalizations, including the previously above technique, there were: A) too few undetermined
mentioned Type II intersecting D-brane models [28–30], parameters, with the B = 0 condition forming an in-
µ
mirage mediation of flux compactifications [63, 64], and compatibleover-constraint,andthusdemonstrablyfalse,
theextractionofSUSYbreakingsofttermsfromthelead- or B) so many undetermined parameters that the dy-
ingordercompactificationofM-theoryonS1/Z [65–69]; namic determination possessed many distinct solutions,
2
inthelattercasewehavepreviouslyobtained(inadiffer- orwassofarseparatedfromexperimentthatitcouldnot
ent model context) a generalization employing modulus possiblybe demonstratedto be true. The actualstateof
dominated SUSY breaking [69]. affairs is much more propitious, being specifically as fol-
Inthispaper,however,wemaintaina“firststepsfirst” lows. The three parameters M ,A,B are once again
0 µ
perspective, concentrating on the simplest No-Scale Su- identically zero at the boundary because of the defining
pergravity and reserving any such extensions for the fu- K¨ahler potential, and are thus known at all other scales
ture. The potentialforstringy modificationsduly noted, as well by the RGEs. The minimization of the Higgs
we then essentially aim to study an F-theory inspired scalar potential with respect to the neutral elements of
variety of low energy SUSY phenomenology, remaining bothSUSYHiggsdoubletsgivestwoconditions,thefirst
agnosticastothe detailsofthe K¨ahlerstructure. Never- ofwhichfixesthemagnitudeofµ. Thesecondcondition,
theless, by studying the simplest No-Scale Supergravity, whichwouldtraditionallybeusedtofixB ,insteadhere
µ
we may still expect to encapsulate the correct leading enforces a consistency relationship on the remaining pa-
order behavior. We likewise maintain the simplicity of a rameters, being that B is already constrained.
µ
leading order approximation by neglecting consideration In general, the B = 0 condition gives a hypersurface
µ
of any stringy threshold corrections, the substantive on- of solutions cut out from a very large parameter space.
set of which is anyway expected to be deferred to M , If we lock all but one parameter, it will give the final
F
the true GUT scale of this model. It should be added value. If we take a slice of two dimensional space, as
thatsince the running ofthe gaugecouplingsis logarith- has been described, it will give a relation between two
micallydependentuponthemassscale,thecontributions parameters for all others fixed. In a three-dimensional
to the RGEs from the string and KK mode excitations view with B on the vertical axis, this curve is the “flat
µ
are quite small. direction” line along the bottom of the trench of B =0
µ
solutions. In general,we must vary at least two parame-
ters rather than just one in isolation, in order that their
mutual compensation may transport the solution along
VII. THE SUPER NO-SCALE MECHANISM
thiscurve. The mostnaturalfirstchoiceisinsomesense
the pair of prominent unknown inputs M and tanβ,
The single relevant modulus field in the simplest 1/2
as was demonstrated in Ref. [47].
stringy No-Scale Supergravity is the K¨ahler modulus T,
Havingcometothispoint,itisbynomeansguaranteed
a characteristic of the Calabi-Yau manifold, the dilaton
that the potential will form a stable minimum. It must
coupling being irrelevant. We consider the gauginomass
be emphasized that the B =0 No-Scale boundary con-
M as a useful modulus related to the F-term of T, µ
1/2 ditionisthecentralagentaffordingthisdetermination,as
stipulating, in other words, that the gauge kinetic func-
it is the extraction of the parameterized parabolic curve
tion must depend on T. This is realized, for example,
of solutions in the two compensating variables which al-
in the Type IIB intersecting D-brane models [62] where
lows for a localized, bound nadir point to be isolated by
gauge kinetic functions explicitly depend on both S and
the Super No-Scale condition, dynamically determining
T , as in Eq. (4). Again, since the F-theory may be con-
i bothparameters. The backgroundsurfaceofV for the
min
sideredasastronglycoupledformulationoftheTypeIIB
full parameter space outside the viable B =0 subset is,
string theory, it is natural to believe that the gauge ki- µ
in contrast, a steadily inclined and uninteresting func-
neticfunctionunderthisliftdependsonT aswell. While
tion. In our prior study, the local minimum minimorum
the limitisquite suggestive,lackingstillaconcreteglob-
of V for the choices M =1000 GeV and m =173.1
ally consistent embedding, we cannot definitively prove min V t
GeV dynamically established M ≃ 450 GeV, and
that the superpotential remains unperturbed by T. 1/2
tanβ ≃ 15 − 20. Although we have remarked that
Proceeding tentatively as such, the F-term of T gen-
M andtanβ havenodirectlyestablishedexperimental
erates the gravitino mass M , which is proportionally 1/2
3/2 values, they are severely indirectly constrained by phe-
equivalent to M . Exploiting the simplest No-Scale
1/2 nomenology in the context of this model [45, 46]. It is
boundary condition at M and running from high en-
F highly non-trivial that there should be accord between
ergy to low energy under the RGEs, there can be a sec-
thetop-downandbottom-upperspectives,butthisisin-
ondary minimization, or minimum minimorum, of the
deed precisely what has been observed [47].
minimumofthe Higgspotential V forthe EWSB vac-
min
7
VIII. THE GUT HIGGS MODULUS
An alternatepair ofparametersforwhichone may at-
tempt to isolate a B = 0 curve, which we consider for
µ
the first time in this work,is that of M and the GUT
1/2
scale M , at which the SU(3) and SU(2) couplings
32 C L
initially meet. Fundamentally, the latter corresponds to
the modulus which sets the total magnitude of the GUT
Higgs field’s VEVs. M could of course in some sense
32
be considereda“known”quantity,takingthelowenergy
couplings as input. Indeed, starting from the measured
SM gauge couplings and fermion Yukawa couplings at
the standard 91.187 GeV electroweak scale, we may cal-
culate both M and the final unification scale M , and
32 F
subsequently the unifiedgaugecoupling andSM fermion
YukawacouplingsatM ,viarunningoftheRGEs. How-
F
ever, since the VEVs of the GUT Higgs fields H and H
are considered here as free parameters, the GUT scale FIG.1: Theinterrelatedvariationofsin2(θW),theGUTscale
M32 must notbe fixed either. As a consequence,the low M32 (logarithmic axis),andtheZ-boson massMZ isdemon-
strated for the parameter strips which preserve B = 0 and
energySMgaugecouplings,andinparticulartheSU(2) µ
L
gauge coupling g , will also run freely via this feedback µ = 460 GeV at MF. The variation in MZ is linked dom-
from M . 2 inantly to motion of the EW couplings via sin2(θW). Also
32 shown is the corresponding predicted proton lifetime in the
We consider this conceptual release of a known quan- leading(e|µ)+π0channels,inunitsof1034years,withthecur-
tity, in order to establish the nature of the model’s de- rent lower bound of 1.0×1034 years indicated bythedashed
pendence upon it, to be a valid and valuable technique, horizontal purple line.
and have employed it previously with specific regards to
“postdiction” of the top quark mass value [46]. Indeed,
forcingthe theoreticaloutputofsuchaparameterisonly
The parameter ranges for the variation depicted in
possible in a model with highly constrainedphysics, and Fig. 1 are M = 91.18− 92.64, sin2(θ ) = 0.2262−
Z W
it may be expected to meet success only by intervention 0.2357, and M = 1.5×1015 −1.04×1016 GeV, and
32
of either grand coincidence or grand conspiracy of Na-
likewise also the same for Figs. (2-8), which will fea-
ture. Simultaneous to the recognition of the presence
ture subsequently. The minimum minimorum falls at
of a second dynamic modulus, we lock down the value
the boundary of the prior list, dynamically fixing M ≃
32
of µ, which by contrast is a simple numerical parame- 1.0×1016 GeV and placing M againin the vicinity of
1/2
ter, and ought then to be treated in a manner consis-
450GeV. The low energySM gaugecouplings are simul-
tent with the top quark and vector-like mass parame-
taneously constrained by means of the associated Wein-
ters. For this study, we choose a vector-like particle berg angle, with sin2(θ ) ≃ 0.236, in excellent agree-
W
mass M = 1000 GeV, and use the experimental top
V ment with experiment. The corresponding range of pre-
quark mass input mt = 173.1 GeV. We emphasize that dicted proton lifetimes in the leading (e|µ)+π0 modes is
the choice of MV = 1000 GeV is not an arbitrary one, 2.5×1031−5.7×1034years[48]. IftheGUTscaleM be-
32
sinceaprioranalysis[46]hasshownthata1TeVvector- comes excessively light, below about 7×1015 GeV, then
like mass is in compliance with all current experimental
proton decay would be more rapid than allowed by the
data and the No-Scale Bµ=0 requirement. The constant recently updated lower bound of 1.0×1034 years from
parameter µ is set consistent with its value prior to the
Super-Kamiokande [70].
variation of the GUT modulus.
We are cautious against making a claim in precisely
In actual practice, the variation of M is achieved in
32 the same vein for the dynamic determination of M ≃
Z
the reverse by programmatic variation of the Weinberg
91.2GeV,sinceagainthecrucialelectroweakHiggsVEV
angle, holding the strong and electromagnetic couplings
is not a substantial element of the variation. However,
at their physically measured values. Figure 1 demon-
in conjunction with the radiative electroweak symmetry
strates the scaling between sin2(θ ), M (logarithmic
W 32 breaking [71, 72] numerically implemented within the
axis), and the Z-bosonmass. The variationof M is at-
Z SuSpect 2.34 code base [73], the fixing of the Higgs
tributed primarily to the motion of the electroweak cou-
VEVandthedeterminationoftheelectroweakscalemay
plings, the magnitude of the Higgs VEV being held es-
also plausibly be considered legitimate dynamic output,
sentiallyconstant. We ensurealsothatthe unifiedgauge
if one posits the M scale input to be available a priori.
F
coupling,SM fermionYukawacouplings,andspecifically
By extracting a constant µ slice of the V hyper-
min
also the Higgs bilinear term µ≃460 GeV, are each held
surface, the secondary minimization condition on tanβ
stable at the scale M to correctly mimic the previously
F is effectively rotated, albeit quite moderately, relative to
described procedure.
the procedure of Ref. ([47]). The present minimization,
8
referencing M , M and tanβ, is again dependent
1/2 32
upon M and m , while the previously described [47]
V t 790
determination of tanβ was, by contrast, M and m in-
V t
variant. Recognizing that a minimization with all three
parameters simultaneously active is required to declare 780
allthreeparameterstohavebeensimultaneouslydynam-
ically determined, we emphasize the mutual consistency 770
of the results. We again stress that the new minimum
Vmin(h)
minimorum is also consistent with the previously adver- 760
tised golden strip, satisfying all presently known experi-
mental constraints to our available resolution. It more- 750
over also addresses the problems of the SUSY breaking
scale and gauge hierarchy [47], insomuch as M1/2 is de- 93.00
460 92.75
termined dynamically. 458 92.2592.50
456 92.00
M1/2 454 452 91.2591.5091.75 MZ
790 450 91.00 Minimum Minimorum
EWSB Vacua FIG. 3: Three-dimensional graph of (M ,M ,∆V (h))
780 Z 1/2 min
space(greencurve). Theprojectionsontothethreemutually
perpendicular planes (red curves) are likewise shown. M ,
Z
770 M , and ∆V (h) are in units of GeV. The dynamically
1/2 min
preferred region, allowing for plausible variation, is circled
760 Vmin(h) and tipped in gold.
750
790
93.00
92.75
92.50
92.25 780
35 92.00
30 25 20 91.5901.75 MZ
tan 15 91.25 770
10 91.00
Minimum Minimorum
Vmin(h)
760
FIG. 2: Three-dimensional graph of (M ,tanβ,∆V (h))
Z min
space(greencurve). Theprojectionsontothethreemutually
perpendicular planes (red curves) are likewise shown. M 750
Z
and∆V (h)areinunitsofGeV.Thedynamicallypreferred
min
region, allowing for plausible variation, is circled and tipped 460
in gold. 35 458
30 456
25
454
20
tan 15 452 M1/2
10 450 Minimum Minimorum
IX. THE MINIMUM MINIMORUM OF THE
FIG. 4: Three-dimensional graph of (M ,tanβ,∆V (h))
ELECTROWEAK HIGGS POTENTIAL 1/2 min
space(greencurve). Theprojectionsontothethreemutually
perpendicular planes (red curves) are likewise shown. M
1/2
In supersymmetric SMs, there is a pair of Higgs and∆V (h)areinunitsofGeV.Thedynamicallypreferred
min
doublets H and H which give mass to the up-type region, allowing for plausible variation, is circled and tipped
u d
quarks and down-type quarks/charged leptons, respec- in gold.
tively. The one-loop effective Higgs potential in the ’t
Hooft-Landau gauge and in the DR scheme is given by
n m2(φ) 3
V = i m4(φ) ln i − , (7)
V = V (H0, H0)+V (H0, H0) , (5) 1 64π2 i (cid:18) Q2 2(cid:19)
eff 0 u d 1 u d Xi
where where m2 and m2 are the supersymmetry breaking
Hu Hd
soft masses, g and g are respectively the gauge cou-
2 Y
V0 = (µ2+m2Hu)(Hu0)2+(µ2+m2Hd)(Hd0)2 plings of SU(2)L and U(1)Y, ni and m2i(φ) are respec-
−2BµµHu0Hd0+ g22+8gY2 (Hu0)2−(Hd0)2 2 ,(6) ttihveelryenthoermdaelgizraeetioonf fsrceaeldeo.mInanodurmnaussmfeorricφail,raensudltQs iins
(cid:2) (cid:3)
9
176.6
790
780 176.5
770
176.4
Vmin(h) v = (v2u + v2d )1/2
760
176.3
750
176.2
460
460 92.75
458 176.6 92.50
458 92.25
456 176.5 456 92.00
454 176.4 454 91.75
M1/2 452 176.3 v = (v2u + v2d )1/2 M1/2 452 91.2591.50 MZ
450 176.2 Minimum Minimorum 450 91.00
FIG. 5: Three-dimensional graph of (v,M ,∆V (h)) FIG.7: Three-dimensionalgraphof(M ,M ,v)space(pur-
1/2 min Z 1/2
space(greencurve). Theprojectionsontothethreemutually ple curve). The projections onto the three mutually perpen-
perpendicular planes (red curves) are likewise shown. M , dicular planes (red curves) are likewise shown. M , M ,
1/2 Z 1/2
v, and ∆V (h) are in units of GeV. The dynamically pre- and v are in units of GeV.
min
ferred region, allowing for plausible variation, is circled and
tipped in gold.
0.742
891xx01110601155 0.740
7x1015
6x1015 0.738
5x1015
4x1015
0.736
3x1015 M32 g = (g22 + g2Y )1/2
0.734
2x1015
0.732
93.00
1015 460 92.5092.75
92.75 458 92.25
92.50 456 92.00
460 458 91.7952.0902.25 M1/2 454 452 91.2591.5091.75 MZ
456 91.50 MZ 450 91.00
454
M1/2 452 91.25
450 91.00 FIG. 8: Three-dimensional graph of (MZ,M1/2,g) space
(royal blue curve). The projections onto the three mutually
FIG. 6: Three-dimensional graph of (MZ,M1/2,M32) space perpendicular planes (red curves) are likewise shown. MZ
(blue curve). The projections onto the three mutually per- and M are in unitsof GeV.
1/2
pendicularplanes(redcurves)arelikewiseshown. M ,M ,
Z 1/2
and M32 are in unitsof GeV.
M = 1000 GeV, m = 173.1 GeV, and µ(M ) ≃ 460
V t F
the figures, we shall designate differences in the fourth- GeV, we present the one-loop effective Higgs potential
rootoftheeffectiveHiggspotentialas∆Vmin(h)≡Ve1ff/4, ∆Vmin(h) in terms of MZ and tanβ in Fig. 2, in terms
measuredin units ofGeV relativeto anarbitraryoverall of MZ and M1/2 in Fig. 3, in terms of M1/2 and tanβ
zero-offset. in Fig. 4, and in terms of v and M1/2 in Fig. 5, where
WehaverevisedtheSuSpect 2.34codebase[73]toin- v = v2 +v2, v = hH0i, and v = hH0i. These fig-
u d u u d d
corporateourspecializedNo-ScaleF-SU(5)withvector- ures pclearly demonstrate the localization of the mini-
like mass algorithm, and accordingly employ two-loop mum minimorum of the Higgs potential, corroborating
RGE running for the SM gauge couplings, and one-loop the dynamical determination of tanβ ≃ 15 − 20 and
RGE running for the SM fermion Yukawa couplings, µ M ≃450 GeV in [47].
1/2
term, and SUSY breaking soft terms. For our choice of Additionally, we exhibit the (M ,M ,M ) space
Z 1/2 32
10
in Fig. 6, the (M ,M ,v) space in Fig. 7, and the canbe reconstructedfromimpending LHCeventsbyde-
Z 1/2
(MZ,M1/2,g) space in Fig. 8, where g = g22+gY2. termining the gauginos M1, M2, and M3 at the elec-
Fig. 6 demonstrates that M ≃ 1.0 × 101p6 GeV at troweak scale, which will in turn require knowledge of
32
themassesfortheneutralinos,charginos,andthegluino.
the minimum minimorum, which correlates to M ≃
Z
91.2 GeV, or more directly, sin2(θ )≃0.236. Together, Likewise, tanβ can be ascertained in principle from a
W
distinctiveexperimentalobservable,aswasaccomplished
the alternate perspectives of Figs. 6, 7, and 8 complete
for mSUGRA in [74]. We will not undertake a compre-
the view given in Figs. 2, 3, 4, and 5 to visually tell
hensive analysis here of the reconstruction of M and
the story of the dynamic interrelation between the M , 1/2
Z
tanβ,butwillofferfornowacursoryexaminationoftyp-
M , and M scales, as well as the electroweak gauge
1/2 32
ical events expected at the LHC. We leave the detailed
couplings, and the Higgs VEVs. The curves in each of
compilationoftheexperimentalobservablesnecessaryfor
these figures represent only those points that satisfy the
validation of the No-Scale F-SU(5) at the LHC for the
B = 0 requirement,asdictated by No-Scale Supergrav-
µ
future, and we especially encourage those specializing in
ity, serving as a crucial constraint on the dynamically
such research to investigate the No-Scale F-SU(5).
determined parameter space. Ultimately, it is the signif-
For the benchmark SUSY spectrum presented in Ta-
icance of the B =0 requirementthat separates the No-
µ
ble I, we have adopted the specific values M = 453,
Scale F-SU(5) with vector-like particles from the entire 1/2
tanβ = 15 and M = 91.187. We expect that higher
compilation of prospective string theory derived models. Z
order corrections will shift the precise location of the
By means of the B = 0 vehicle, No-Scale F-SU(5) has
µ
minimum minimorum a little bit, for example, within
surmounted the paramountchallenge of phenomenology,
the encircled gold-tipped regions of the diagrams in the
that of dynamically determining the electroweak scale,
prior section. We have selected a ratio for tanβ at the
the scale of fundamental prominence in particle physics.
lowerend ofthis rangefor consistencywith our previous
We wish to note that recent progress has been made
study [47], and to avoid stau dark matter.
inincorporatingmoreprecisenumericalcalculationsinto
ourbaselinealgorithmforNo-ScaleF-SU(5)withvector-
like particles. Initially, when we commenced the task of
TABLEI:Spectrum(inGeV)forthebenchmarkpoint. Here,
fullydevelopingthephenomenologyofthismodel,theex-
M =453 GeV,M =1000 GeV,m =173.1 GeV,M =
1/2 V t Z
treme complexity of properly numerically implementing
91.187 GeV,µ(MF)=460.3 GeV,∆Vmin(h)=748GeV,Ωχ
No-Scale F-SU(5)with vector-like particles compelled a =0.113,σ =2×10−10pb,andhσvi =1.8×10−28cm3/s.
SI γγ
gradualstrategyforconstructionandpersistentenhance- The central prediction for the p→(e|µ)+π0 proton lifetime is
mentofthealgorithm. Preliminaryfindingsofaprecision around4.9×1034years. Thelightestneutralinois99.8%Bino.
iymeaprroWveMdAalPgorrietlhicmdienndsiictaytecothnasttrcaoinmtspliraenqcueirwesitha tshliegh7t- χe01 94 χe±1 184 eeR 150 et1 486 ueR 947 mh 120.1
upward shift to tanβ ≃ 19 − 20 from the value com- χe02 184 χe±2 822 eeL 504 et2 906 ueL 1032 mA,H 916
puted in Ref. [45], suggesting a potential convergence to χe03 817 νee/µ 498 τe1 104 eb1 855 deR 988 mH± 921
even finer resolution of the dynamical determination of χe04 821 νeτ 491 τe2 499 eb2 963 deL 1035 eg 617
tanβ given by the Super No-Scale mechanism, and the
value demanded by the experimental relic density mea- At the benchmark point, we calculate Ω = 0.113 for
χ
surements. We shall furnish a comprehensive analysis of the cold dark matter relic density. The phenomenology
the precision improved algorithm at a later date. ismoreoverconsistentwiththe LEPlimitonthelightest
CP-even Higgs boson mass, m ≥114 GeV [75, 75], the
h
CDMSII[76]andXenon100[77]upperlimitsonthespin-
X. PROBING THE BLUEPRINTS OF THE independentcrosssectionσ , andthe Fermi-LAT space
SI
NO-SCALE MULTIVERSE AT THE COLLIDERS telescopeconstraints[78]onthephoton-photonannihila-
tion cross section hσvi . The differential cross-sections
γγ
We offer in closing a brief summary of direct collider, andbranchingratioshavebeencalculatedwithPGS4[79]
detector, and telescope level tests which may probe the executing a call to PYTHIA 6.411 [80], using our spe-
blueprints ofthe No-ScaleMultiversewhichwehavelaid cialized No-Scale algorithm integrated into the SuSpect
out. As to the deepquestionofwhether the ensemble be 2.34codeforinitialcomputationofthesparticlemasses.
literal in manifestation, or merely the conceptual super- The benchmark point resides in the region of the
set of unrealized possibilities of a single island Universe, experimentally allowed parameter space that generates
we pretend no definitive answer. However, we have ar- the relic density through stau-neutralino coannihilation.
gued that the emergence ex nihilo of seedling universes Hence, the five lightest sparticles for this benchmark
whichfuelaneternalchaoticinflationscenarioisparticu- point are χ0 < τ± < e < χ0 ∼ χ±. Here, the
1 1 R 2 1
larlyplausible,andevennatural,withinNo-ScaleSuper- gluino is lighter than all the squarks with the exception
gravity, and our goal of probing the specific features of of the lighteest stoep, so aell squareks willepredominantly
ourownUniversewhichmightimplicateitsoriginsinthis decay to a gluino and hadronic jet, with a small per-
construction are immediately realizable and practicable. centage of squarks producing a jet and either a χ± or
1
The unified gaugino M at the unification scale M χ0. The gluinos will decay via virtual (off-shell) squarks
1/2 F 2
e
e