Large volume supersymmetry breaking without decompactification problem

0.09 MB·

by  Herve Partouche

Large volume supersymmetry breaking without
decompactification problem
Herve´Partouche
6
1
0 AbstractWeconsiderheteroticstringbackgroundsinfour-dimensionalMinkowski
2 space,whereN =1supersymmetryisspontaneouslybrokenatalowscalem by
3/2
n astringyScherk-Schwarzmechanism.Wereviewhowtheeffectivegaugecouplings
a at1-loopmayevadethe“decompactificationproblem”,namelytheproportionality
J ofthegaugethresholdcorrections,withthelargevolumeofthecompactspacein-

8 volvedinthesupersymmetrybreaking.
1

]
h
1 Introduction
t
-
p
e
Asensiblephysicaltheorymustatleastmeettworequirements:Berealisticandan-
h
[ alyticallyundercontrol.Thefirstpointcanbesatisfiedbyconsideringstringtheory,
whichhastheadvantagetobe,atpresenttime,theonlysetupin whichbothgrav-
1
itationalandgaugeinteractionscanbedescribedconsistentlyatthequantumlevel.
v
Inthisreview,wedonotconsidercosmologicalissuesandthusanalyzemodelsde-
4
6 finedclassicallyinfour-dimensionalMinkowskispace.The“no-scalemodels”are
5 particularlyinteresting since, by definition, they describe in supergravityor string
4 theoryclassical backgrounds,inwhichsupersymmetryis spontaneouslybrokenat
0
an arbitraryscale m in flat space [1]. In other words, even if supersymmetryis
. 3/2
1 notexplicit,theclassicalvacuumenergyvanishes.
0
The most conservative way to preserve analytical control is to ensure the va-
6
lidity of perturbationtheory.In string theory,quantumloopscan be evaluatedex-
1
: plicitly,whentheunderlyingtwo-dimensionalconformalfieldtheoryisitselfunder
v
control.Clearly,thisisthecase, whenoneconsidersfreefieldonthe worldsheet,
i
X forinstanceintoroidalorbifoldmodels[2]orfermionicconstructions[3].Inthese
r frameworks,theN =1 N =0spontaneousbreakingofsupersymmetrycanbe
a implementedat tree leve→l via a stringy version [4] of the Scherk-Schwarzmecha-
nism[5].1 Inthiscase,thesupersymmetrybreakingscaleisoforderoftheinverse
Herve´Partouche
CentredePhysiqueThe´orique,EcolePolytechnique,CNRS,Universite´Paris-Saclay
F–91128Palaiseaucedex,France,e-mail:[email protected]
1Notethatnon-perturbativemechanismsbasedongauginocondensationcouldalsobeconsidered,
butonlyatthelevelofthelowenergyeffectivesupergravity,thusatthepriceofloosingpartofthe
stringpredictability.
1
2 Herve´Partouche
volume of the internal directions involved in the breaking. For a single circle of
radiusR,onehas
M
s
m = , (1)
3/2 R
where M is the string scale, so that havinga low m =O(10TeV) imposesthe
s 3/2
circle to be extremely large, R=O(1017) [6]. Such large directions yield towers
of light Kaluza-Klein states and a problem arises from those chargedunder some
gauge group factor Gi. In general, their contributions to the quantum corrections
totheinversesquaredgaugecouplingisproportionaltotheverylargevolumeand
invalidatestheuseofperturbationtheory.
Tobespecific,letusconsiderinheteroticstringthe1-looplow energyrunning
gaugecouplingg(m ),whichsatisfies[7]
i
16p 2 16p 2 M2
=ki +biln s +D i. (2)
g2(m ) g2 m 2
i s
In this expression, g is the string coupling and ki is the Kac-Moody level of Gi.
s
Thelogarithmiccontribution,whichdependsontheenergyscalem ,arisesfromthe
masslessstatesandisproportionaltotheb -functioncoefficientbi,whilethemassive
modesyieldthethresholdcorrectionsD i.Themaincontributionstothelatterarise
fromthelightKaluza-Kleinstates,whichforasinglelargeradiusyield
1
D i=CiR bilnR2+O , (3)
− R
(cid:18) (cid:19)
whereCi=Cbi Cki,forsomenon-negativeCandC thatdependonothermoduli.
′ ′
−
WhenCi=O(1),requiringinEq.(2)theloopcorrectiontobesmallcomparedto
thetreelevelcontributionimposesg2R<1.Inotherwords,forperturbationtheory
s
tobevalid,thestringcouplingmustbeextremelyweak,g <O(10 6.5).IfCi>0,
s −
which implies Gi is not asymptotically free, Eq. (2) imposes the running gauge
coupling to be essentially free, g(m )=O(g ), and Gi describes a hidden gauge
i s
group.However,ifCi <0, whichis the case if Gi is asymptoticallyfree,the very
largetreelevelcontributionproportionalto1/g2 mustcancelCiR,uptoveryhigh
s
accuracy, for the running gauge coupling to be of order 1 and have a chance to
describe realistic gauge interactions. This unnaturalfine-tuning is a manifestation
of the so-called “decompactification problem”, which actually arises generically,
whenasubmanifoldoftheinternalspaceislarge,comparedtothestringscale,i.e.
whentheinternalconformalfieldtheoryallowsageometricalinterpretationinterms
ofacompactifiedspace.
To avoid the above described behavior, Ci can be required to vanish. This is
triviallythecaseintheN =4supersymmetrictheories,whereactuallybi=0and
D i=0.TheconditionCi=0remainsvalidinthetheoriesrealizingtheN =4
N =2 spontaneousbreaking, providedN =4 is recoveredwhen the volume→is
senttoinfinity[8].Inthiscase,thethresholdcorrectionsscalelogarithmicallywith
the volume and no fine-tuning is required for perturbation theory to be valid. In
Largevolumesupersymmetrybreakingwithoutdecompactificationproblem 3
Sect. 2, we review the construction of models that realize an N =1 N =0
→
spontaneous breaking at a low scale m , while avoiding the decompactification
3/2
problem.ThecorrespondingthresholdcorrectionsarecomputedinSect.3[9,10].
2 The non-supersymmetric Z Z models
2 2
×
Inthepresentwork,wefocusonheteroticstringbackgroundsinfour-dimensional
Minkowskispaceandanalyzethegaugecouplingthresholdcorrections.At1-loop,
theirformalexpressionis[7,11,12]
D i = d2t 1(cid:229) Q a (2v) P2(2w¯) ki t Z a (2v,2w¯) bi
ZF t2 2a,b b (cid:18) i −4pt 2(cid:19) 2 b − !(cid:12)(cid:12)v=w¯=0
+bilog2e1−g , (cid:2) (cid:3) (cid:2) (cid:3) (cid:12)(cid:12)(cid:12) (4)
p √27
whereF isthe fundamentaldomainof SL(2,Z)andZ a (2v,2w¯) isa refinedpar-
b
titionfunctionforgivenspinstructure(a,b) Z Z .P(2w¯)actsontheright-
2 2 i
∈ × (cid:2) (cid:3)
movingsector as the squared chargeoperator of the gauge groupfactor Gi, while
Q a (2v)actsontheleft-movingsectorasthehelicityoperator,2
b
(cid:2) (cid:3)
1 ¶ 2(q a (2v)) i i q a (2v)
Q ab (2v)= 16p 2 vq ab(cid:2)b((cid:3)2v) −p ¶ t logh ≡ p ¶ t log (cid:2)bh(cid:3) !. (5)
(cid:2) (cid:3)
Fromnow on, we conside(cid:2)r(cid:3)Z Z orbifoldmodels[2] or fermionicconstruc-
2 2
×
tions[3]inwhichthemarginaldeformationsparameterizedbytheKa¨hlerandcom-
plexstructuresT,U ,I=1,2,3,associatedtothethreeinternal2-toriareswitched
I I
on[9,14].Inbothcases, orbifoldsor“moduli-deformedfermionicconstructions”,
N =1supersymmetryisspontaneouslybrokenbyastringyScherk-Schwarzmech-
anism[4].Theassociatedgenus-1refinedpartitionfunctionis
1
Z(2v,2w¯)= (6)
t2(h h¯)2 ×
1(cid:229) 1 (cid:229) 1 (cid:229) ( 1)a+b+abq ab (2v) q ab++HG11 q ab++HG22 q ab++HG33
2 2 2 − h h h h ×
a,b H1,G1 H2,G2 (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
1 (cid:229) S a,hiI,HI Z hi1 H1 Z hi2 H2 Z hi3 H3 Z hiI,HI (2w¯),
2N hiI,giI Lhb,giI,GIi 2,2hgi1(cid:12)G1i 2,2hgi2(cid:12)G2i 2,2hgi3(cid:12)G3i 0,16hgiI,GIi
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
whereournotationsareasfollows:
2 Our conventions for the Jacobi functions q ab (n |t ) (or q a (n |t ), a =1,...,4) and Dedekind
functioncanbefoundin[13].
(cid:2) (cid:3)
4 Herve´Partouche
TheZ conformalblocksarisefromthethreeinternal2-tori.Thegenus-1sur-
2,2
• facehavingtwonon-trivialcycles,(hi,gi) Z Z ,i=1,2,I=1,2,3denote
associatedshiftsofthesixcoordinatesI.SIim∈ilar2ly×,(H2,G ) Z Z refertothe
I I 2 2
∈ ×
twists, where we have defined for convenience (H ,G ) ( H H , G
3 3 1 2 1
≡ − − − −
G ).Explicitly,wehave
2
G h1I,h2I (T,U )
2,2 g1,g2 I I
I I , when(H ,G )=(0,0)mod2,
 h(h h¯i)2 I I
wZ2h,2ehregh1I1IG,,gh2I2I(cid:12)(cid:12)(cid:12)HGiIIsia=shiftqe(cid:2)d11−l−aHGt4tIIih(cid:3)cqeh¯¯(cid:2)t11h−−aHGtIId(cid:3)edp(cid:12)(cid:12)(cid:12)ehgn1I1IdHGsII(cid:12)(cid:12)(cid:12)o,0nmtohde2Kd (cid:12)(cid:12)(cid:12)a¨hg2I2IhlHGeIIr(cid:12)(cid:12)(cid:12),a0nmdodc2ompleoxthsetrrwucitsu(e7r,e)
2,2
moduli T,U of the Ith 2-torus. The arguments of the Kronecker symbols are
I I
determinants.
When defining each model, linear constraints on the shifts (hi,gi) and twists
• I I
(H ,G )maybeimposed,leavingeffectivelyN independentshifts.
I I
Z denotesthecontributionofthe32extraright-movingworldsheetfermions.
0,16
•
ItsdependanceontheshiftsandtwistsmaygeneratediscreteWilsonlines,which
breakpartiallyE E orSO(32).
8 8
×
Thefirstlinecontainsthecontributionofthespacetimelight-conebosons,while
•
thesecondisthatoftheleft-movingfermions.
S is a conformal block-dependent sign that implements the stringy Scherk-
L
•
Schwarz mechanism. A choice of S that correlates the spin structure (a,b) to
L
someshift(hi,gi)implementstheN =1 N =0spontaneousbreaking.
I I →
The Z Z models contain three N =2 sectors. For the decompactification
2 2
×
problem not to arise, we impose one of them to be realized as a spontaneously
brokenphaseofN =4.ThiscanbedonebydemandingtheZ actioncharacterized
2
by(H ,G )tobefree.Theassociatedgeneratortwiststhe2ndand3rd2-tori(i.e.the
2 2
directionsX6,X7,X8,X9inbosoniclanguage)andshiftssomedirection(s)ofthe1st
2-torus,sayX5 only.Tosimplifyourdiscussion,wetakethegeneratoroftheother
Z , whoseactionischaracterizedby(H ,G ), to notbefree:Ittwists the1st and
2 1 1
3rd 2-tori,andfixesthe 2nd one.Similarly,we supposethattheproductofthetwo
generators,whoseactionischaracterizedby(H ,G ),twiststhe1st and2nd 2-tori,
3 3
andfixesthe3rd one.TheserestrictionsimposethemoduliT ,U andT ,U notto
2 2 3 3
be far from1, in orderto avoid the decompactificationproblemto occurfromthe
remainingtwoN =2sectors.However,ourcareinchoosingtheorbifoldactionis
allowingustotakethevolumeofthe1st2-torustobelarge.
Theaboveremarkshaveanimportantconsequence,sincethefinalstringyScherk-
Schwarz mechanism responsible of the N =1 N =0 spontaneous breaking
→
must involvethe moduliT ,U only, for the gravitino mass to be light. Thus, this
1 1
breakingmustbeimplementedviaashiftalongthe1st 2-torus,sayX4,andanon-
trivial choice of S . Therefore, the sector (H ,G )=(0,0) realizes the pattern of
L 1 1
Largevolumesupersymmetrybreakingwithoutdecompactificationproblem 5
spontaneousbreaking N =4 N =2 N =0, while the other two N =2
→ →
sectors,whichhave2nd and3rd 2-torirespectivelyfixed,areindependentofT and
1
U andthusremainsupersymmetric.Asaresult,wehaveinthetwofollowinginde-
1
pendentmodularorbits:
SL=( 1)ag11+bh11+h11g11, when (H1,G1)=(0,0),
−
S =1, when (H ,G )=(0,0). (8)
L 1 1
6
Given the fact that we have imposed (h2,g2) (H ,G ), the 1st 2-toruslattice
1 1 ≡ 2 2
takestheexplicitform
G 2,2hhg1111,,HG22i(T1,U1)=m(cid:229)i,ni(−1)m1g11+m2G2e2ip t¯[m1(n1+12h11)+m2(n2+12H2)]×
e−ImTpt1I2mU1|T1(n1+21h11)+T1U1(n2+12H2)+U1m1−m2|2. (9)
This expression can be used to find the squared scales of spontaneousN =4
N =2andN =2 N =0breaking.ForRe(U ) ( 1,1],theyare →
→ 1 ∈ −2 2
M2 U 2M2
s , m2 = | 1| s , (10)
ImT ImU 3/2 ImT ImU
1 1 1 1
wherethelatterisnothingbutthegravitinomasssquaredofthefullN =0theory.
For these scales to be small compared to M , we consider the regime ImT 1,
s 1
≫
U =O(i).
1
3 Threshold corrections
The threshold corrections can be evaluated in each conformal block [9]. Starting
with those where (H ,G )=(0,0), the discussion is facilitated by summing over
1 1
thespinstructures.Focussingontherelevantpartsoftherefinedpartitionfonction
Z,wehave
21(cid:229)a,b(−1)a+b+ab(−1)ag11+bh11+h11g11q ab (2v)q ab q ab++HG22 q ab−−HG22 =
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
( 1)h11g11+G2(1+h11+H2)q 1−h11 2(v)q 1−h11+H2 2(v), (11)
− 1−g11 1−g11+G2
h i h i
whichshowshowmanyoddq (v) q [1](v) functions(orequivalentlyhowmany
fermionic zero modes in the p1ath i≡ntegr1al) arise for given shift (h1,g1) and twist
1 1
(H ,G ).
2 2
6 Herve´Partouche
ConformalblockA :(h1,g1)=(0,0),(H ,G )=(0,0)
1 1 2 2
Thisblockisproportionaltoq 1 4(v)=O(v4).Uptoanoverallfactor1/23,itisthe
1
contributionoftheN =4spectrumoftheparenttheory,whenneithertheZ Z
2 2
(cid:2) (cid:3) ×
action nor the stringy Scherk-Schwarz mechanism are implemented. Therefore, it
doesnotcontributetothe1-loopgaugecouplings.
ConformalblocksB :(h1,g1)=(0,0),(H ,G )=(0,0)
1 1 2 2
6
They are proportional to q 1−h11 4(v)=O(1). The parity of the winding number
1 g1
− 1
along the compact directiohn X4ibeing h1, the blocks with h1 =1 involve states,
1 1
whicharesupermassivecomparedtothe pureKaluza-Kleinmodes.Theseblocks
arethereforeexponentiallysuppressed,comparedtotheblock(h1,g1)=(0,1).Up
1 1
toanoverallfactor1/22,thelatterarisesfromthespectrumconsideredinthecon-
formal block A, but in the N =4 N = 0 spontaneously broken phase, and
→
contributestothegaugecouplings.
ConformalblocksC :(h1,g1)=(0,0),(H ,G )=(0,0)
1 1 2 2
6
tTohtehyeaarcetipornoopforthtieonhaellitcoitqyo11p(evr)a2tqor.11R−−HGea22so2(nvi)ng=aOsi(nv2t)heanpdredvoiocuosnctarisbeu,ttehetopDarii,tyduoef
(cid:2) (cid:3) (cid:2) (cid:3)
thewindingnumberalongthecompactdirectionX5isH ,whichimpliestheblocks
2
with H =1 yield exponentially suppressed contributions, compared to that asso-
2
ciatedtotheblock(H ,G )=(0,1).Uptoanoverallfactor1/22,thelatterarises
2 2
fromaspectrumrealizingthespontaneousN =4 N =2breaking,whichcon-
C
→
tributestothecouplings.
ConformalblocksD :(h1,g1)=(H ,G )=(0,0)
1 1 2 2
6
tThhaetyoafrteheprcoopnofrotiromnaallbtoloqck11s−−CHG22, e2x(cve)qpt11th(avt)t2h=e gOe(nve2r)a.toTrhoefsitthueatZionfriseeidaecnttiiocnalrteo-
2
sponsible of the partial sp(cid:2)ontane(cid:3)ousb(cid:2)rea(cid:3)kingof N =4 twists X6,X7,X8,X9 and
shifts X4,X5. The dominant contribution to the threshold corrections arises again
fromtheblock(H ,G )=(0,1),whichdescribesaspectrumrealizingthesponta-
2 2
neousN =4 N =2breaking.
D
→
h1 H
ConformalblocksE : 1 2 =0
g11 G2 6
The remaining conformal(cid:12)(cid:12)(cid:12)blocks(cid:12)(cid:12)(cid:12)have non-trivial determinant hg1111HG22 , which im-
plies q 1−h11 2(v)q 1−h11+H2 2(v)=O(1). However, this condit(cid:12)(cid:12)ion is(cid:12)(cid:12)also saying
1−g11 1−g11+G2 (cid:12) (cid:12)
that(h1h,H )i=(0,0h),whichmieansthemodesintheseblockshavenon-trivialwind-
1 2 6
ingnumber(s)alongX4,X5orboth.Therefore,theircontributionstothegaugecou-
plingsarenon-trivialbutexponentiallysuppressed.
Having analyzedall conformalblockssatisfying (H ,G )=(0,0), we proceed
1 1
with the study of the modularorbit(H ,G )=(0,0), where the sign S is trivial.
1 1 L
6
Largevolumesupersymmetrybreakingwithoutdecompactificationproblem 7
Sincethe1st2-torusistwisted,theseblocksareindependentofthemoduliT ,U and
1 1
thusm .TheycanbeanalyzedasinthecaseofZ Z ,N =1supersymmetric
3/2 2 2
×
models.Actually,summingoverthespinstructures,therelevanttermsintherefined
partitionfunctionZbecome
12(cid:229)a,b(−1)a+b+abq ab (2v)q ab++HG11 q ab++HG22 q ab−−HG11−−HG22 =
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
(−1)(G1+G2)(1+H1+H2)q 11 (v)q 11−−HG11 (v)q 11−−HG22 (v)q 11++HG11++HG22 (v), (12)
whichinvitesustosplitthediscuss(cid:2)io(cid:3)nint(cid:2)hreep(cid:3)arts.(cid:2) (cid:3) (cid:2) (cid:3)
N =2conformalblocks,with fixed 2nd 2-torus:(H ,G )=(0,0)
2 2
Tfixheedybaryetphreonpoonrt-iforneaelatcotiqon11o2f(tvh)eq Z11−−HGc11ha2r(avc)te=rizOe(dv2b)y.(THhe,2Gnd).inAtedrdnianlg2t-htoerucosnis-
(cid:2) (cid:3) (cid:2) 2 (cid:3) 1 1
formalblock A, we obtain an N =2 sector of the theory, up to an overallfactor
1/2 associated to the second Z . This spectrum leads to non-trivialcorrectionsto
2
thegaugecouplings.
N =2conformalblocks,withfixed3rd 2-torus:(H ,G )=(H ,G )
1 1 2 2
Twhhyicahrempearonpsotrhtaiotnthaelt3ordq 211-to2r(uvs)qis11fi−−xHGe11d]2b(vy)t=heOco(vm2b).inAecdtuaacltliyo,n(oHf3t,hGe3g)e=ne(r0a,to0r)s,
(cid:2) (cid:3) (cid:2)
ofthetwoZ ’s.AddingtheconformalblockA,oneobtainsthelastN =2sector
2
ofthetheory,uptoanoverallfactor1/2.Again,thisspectrumyieldsanon-trivial
contributiontothegaugecouplings.
N =1conformalblocks: H1 H2 =0
G1 G2 6
Theremainingblockshavenon-trivialdeterminant, H1H2 =0,whichimpliesthey
(cid:12) (cid:12) G1G2 6
athreemprwopitohrttihoenhaleltiociqty11o(pve)raqto11r−−,HGth(cid:12)11e(rve)squl(cid:12)t11−−isHGp22r(ovp)oqrti11(cid:12)(cid:12)o++nHGa11l++tHG(cid:12)(cid:12)o22 (v)=O(v).Actingon
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
¶ v2 q 11 (v)q 11−−HG11 (v)q 11−−HG22 (v)q 11++HG11++HG22 (v) v=0(cid:181)
(cid:16) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) ¶(cid:2)2 q (v)(cid:3)q (v(cid:17))(cid:12)(cid:12)q (v)q (v) =0, (13)
v 1 2 (cid:12) 3 4
v=0
(cid:16) (cid:17)(cid:12)
thanks to the oddness of q 1(v) and evenness of q 2,3,4(v). Thus, th(cid:12)(cid:12)ese conformal
blocksdonotcontributetothethresholds.
In the class of models we consider, the effective runninggauge couplingasso-
ciatedtosomegaugegroupefactorGi hasauniversalformat1-loop[9].Itcanbe
elegantlyexpressedintermsofthreemoduli-dependentsquaredmassscalesarising
fromthecorrectionsassociatedtotheconformalblocksB,C,D,
8 Herve´Partouche
M2 M2 M2
M2= s ,M2= s ,M2= s ,
B q (U )4ImT ImU C q (U )4ImT ImU D q (U )4ImT ImU
2 1 1 1 4 1 1 1 3 1 1 1
| | | | | | (14)
whichareoforderm2 ,andtwomorescales
3/2
M2
M2= s , I=2,3, (15)
I 16 h (T)4 h (U )4ImT ImU
I I I I
| |
of order Ms2 that encode t(cid:12)(cid:12)he contr(cid:12)(cid:12)ibutionsof the N =2 sectors associated to the
fixed2ndand3rd internal2-tori.Itisalsousefultointroducea“renormalizedstring
coupling”[11],
16p 2 16p 2 1 1
= Y(T ,U ) Y(T ,U ),
g2 g2 −2 2 2 −2 3 3
renor s
1 d2t 3 E¯ E¯
where Y(T,U)= G (T,U) E¯ 4 6 j¯+1008 , (16)
12ZF t2 2,2 (cid:20)(cid:16) 2−pt 2(cid:17) h¯24 − (cid:21)
in which G =G 0,0 is the unshifted lattice, while for q=e2ipt , E =1+
2,2 2,2 0,0 2,4,6
O(q) are holomorphic Eisenstein series of modular weights 2,4,6 and j =1/q+
(cid:2) (cid:3)
744+O(q)isholomorphicandmodularinvariant.Theinversesquared1-loopgauge
couplingatenergyscaleQ2=m 2p 2 isthen
4
16p 2 16p 2 bi Q2 bi Q2 bi Q2
=ki Bln Cln Dln
g2(Q) g2 − 4 Q2+M2 − 4 Q2+M2 − 4 Q2+M2
i renor (cid:18) B(cid:19) (cid:18) C(cid:19) (cid:18) D(cid:19)
bi2ln Q2 bi3ln Q2 +O m23/2 , (17)
− 2 (cid:18)M22(cid:19)− 2 (cid:18)M32(cid:19) Ms2 !
whichdependsonlyonfivemodel-dependentb -functioncoefficientsandtheKac-
Moody level. In this final result, we have shifted M2 Q2+M2 in order
B,C,D → B,C,D
to implementthe thresholdsat which the sectors B,C or D decouple,i.e. when Q
exceedsM ,M orM .Thus,thisexpressionisvalidaslongasQislowerthanthe
B C D
massoftheheavystateswehaveneglectedtheexponentiallysuppressedcontribu-
tionsi.e.thestringorGUTscale,dependingonthemodel.TakingQlowerthanat
leastoneofthescalesM ,M orM ,ther.h.s.ofEq.(17)scalesaslnImT ,which
B C D 1
isthelogarithmofthelarge1st2-torusvolume,asexpectedforthedecompactifica-
tionproblemnottoarise.
To conclude,we would like to mentiontwo importantremarks.First of all, we
stress that the Z Z models, where a Z is freely acting and a stringy Scherk-
2 2 2
Schwarzmechanis×mresponsibleofthefinalbreakingofN =1 takesplace,have
non-chiral massless spectra. This is due to the fact that in the N =1, Z Z
2 2
×
models, chiral families occur from twisted states localized at fixed points. In the
modelswehaveconsidered,fixedpointslocalizedonthe2ndand3rd2-toricanarise
but are independentof the moduli T ,U i.e. m . Thus, taking the large volume
1 1 3/2
Largevolumesupersymmetrybreakingwithoutdecompactificationproblem 9
limitofthe1st 2-torus,whereN =2supersymmetryisrecovered,oneconcludes
thatthetwistedstatesareactuallyhypermultipletsi.e.couplesoffamiliesandanti-
families.
Second,we pointoutthatin themodelsanalyzedin the presentwork,thecon-
formalblockBisthe onlynon-supersymmetricandnon-negligiblecontributionto
thepartitionfunctionZ,andthustothe1-loopeffectivepotential.InRef.[10,15],it
isshownthatinsomemodels,thelatterispositivesemi-definite.Themotionofthe
moduliT ,U andT ,U isthusattractedtopoints[?],wheretheeffectivepotential
2 2 3 3
vanishes,allowingm tobearbitrary.Inotherwords,thedefiningpropertiesofthe
3/2
no-scalemodels,namelyarbitrarinessofthesupersymmetrybreakingscalem in
3/2
flatspace,whicharevalidattreelevel,areextendedtothe1-looplevel.Thisvery
fact, characteristic of the so-called “super no-scale models”, may have interesting
consequenceson the smallnessof a cosmologicalconstantgeneratedat higheror-
ders.InRef.[17],othermodelshaving1-loopvanishingcosmologicalconstantare
alsoconsidered,whichhoweversufferfromthedecompactificationproblem.
Acknowledgements IamgratefultoA.FaraggiandK.Kounnas,withwhomtheoriginalwork[9]
hasbeendoneincollaboration.IwouldalsoliketothankC.Angelantonj,I.AntoniadisandJ.Rizos
forfruitfuldiscussions,andtheLaboratoiredePhysiqueThe´oriqueofEcoleNormaleSupe´rieure
forhospitality.
References
1. E.Cremmer,S.Ferrara,C.KounnasandD.V.Nanopoulos,“Naturallyvanishingcosmologi-
calconstantinN =1supergravity,”Phys.Lett.B133(1983)61;
J. R.Ellis,C.Kounnas andD.V.Nanopoulos, “Phenomenological SU(1,1) supergravity,”
Nucl.Phys.B241(1984)406.
2. L.J.Dixon,J.A.Harvey,C.VafaandE.Witten,“Stringsonorbifolds,”Nucl.Phys.B261
(1985)678;
L.J.Dixon,J.A.Harvey,C.VafaandE.Witten,“StringsonorbifoldsII,”Nucl.Phys.B274
(1986)285.
3. I.Antoniadis,C.Bachas,C.KounnasandP.Windey,“Supersymmetryamongfreefermions
andsuperstrings,”Phys.Lett.B171(1986)51;
I.Antoniadis,C.P.BachasandC.Kounnas,“Four-dimensionalsuperstrings,”Nucl.Phys.B
289(1987)87;
H. Kawai, D. C. Lewellen and S. H. H. Tye, “Construction of fermionic string models in
four-dimensions,”Nucl.Phys.B288(1987)1.
4. R.Rohm,“Spontaneous supersymmetrybreakinginsupersymmetricstringtheories,”Nucl.
Phys.B237(1984)553;
C.KounnasandM.Porrati,“Spontaneoussupersymmetry breakinginstringtheory,”Nucl.
Phys.B310(1988)355;
S.Ferrara,C.Kounnas,M.PorratiandF.Zwirner,“Superstringswithspontaneouslybroken
supersymmetryandtheireffectivetheories,”Nucl.Phys.B318(1989)75;
S. Ferrara, C. Kounnas and M. Porrati, “Superstring solutions with spontaneously broken
four-dimensionalsupersymmetry,”Nucl.Phys.B304(1988)500;
C.KounnasandB.Rostand,“Coordinatedependentcompactificationsanddiscretesymme-
tries,”Nucl.Phys.B341(1990)641.
10 Herve´Partouche
5. J.ScherkandJ.H.Schwarz,“Spontaneousbreakingofsupersymmetrythroughdimensional
reduction,”Phys.Lett.B82(1979)60.
6. I.Antoniadis,“ApossiblenewdimensionatafewTeV,”Phys.Lett.B246(1990)377.
7. V. S. Kaplunovsky, “One loop threshold effects in string unification,” Nucl. Phys. B 307
(1988)145[Erratum-ibid.B382(1992)436][hep-th/9205068];
L.J.Dixon,V.KaplunovskyandJ.Louis,“Modulidependenceofstringloopcorrectionsto
gaugecouplingconstants,”Nucl.Phys.B355(1991)649;
I.Antoniadis,K.S.NarainandT.R.Taylor,“Highergenusstringcorrectionstogaugecou-
plings,”Phys.Lett.B267(1991)37;
P.MayrandS.Stieberger,“Thresholdcorrectionstogaugecouplingsinorbifoldcompactifi-
cations,”Nucl.Phys.B407(1993)725[hep-th/9303017];
E.KiritsisandC.Kounnas,“Infrared regularizationofsuperstringtheoryandtheoneloop
calculationofcouplingconstants,”Nucl.Phys.B442(1995)472[hep-th/9501020].
8. E.Kiritsis,C.Kounnas,P.M.PetropoulosandJ.Rizos,“Solvingthedecompactificationprob-
leminstringtheory,”Phys.Lett.B385(1996)87[hep-th/9606087];
E. Kiritsis, C. Kounnas, P. M. Petropoulos and J. Rizos, “String threshold corrections
in models with spontaneously broken supersymmetry,” Nucl. Phys. B 540 (1999) 87
[arXiv:hep-th/9807067].
9. A.E.Faraggi,C.KounnasandH.Partouche,“Largevolumesusybreakingwithasolutionto
thedecompactificationproblem,”Nucl.Phys.B899(2015)328[arXiv:1410.6147[hep-th]].
10. C.KounnasandH.Partouche, “StringyN =1superno-scalemodels,”arXiv:1511.02709
[hep-th].
11. E.Kiritsis,C.Kounnas,P.M.PetropoulosandJ.Rizos,“UniversalitypropertiesofN =2
andN =1heteroticthresholdcorrections,”Nucl.Phys.B483(1997)141[hep-th/9608034];
P. M.Petropoulos andJ. Rizos,“Universal moduli dependent stringthresholds inZ Z
2 2
×
orbifolds,”Phys.Lett.B374(1996)49[hep-th/9601037].
12. C. Angelantonj, I. Florakis and M. Tsulaia, “Universality of gauge thresholds in non-
supersymmetricheteroticvacua,”Phys.Lett.B736(2014)365[arXiv:1407.8023[hep-th]];
C. Angelantonj, I. Florakis and M. Tsulaia, “Generalised universality of gauge thresh-
olds in heterotic vacua with and without supersymmetry,” Nucl. Phys. B 900 (2015) 170
[arXiv:1509.00027[hep-th]].
13. E.Kiritsis,“Stringtheoryinanutshell,”PrincetonUniversityPress,2007.
14. C.KounnasandB.Rostand,“Deformationsofsuperstringsolutionsandspontaneous sym-
metrybreaking,”HellenicSchool1989:0657-668.
15. C.KounnasandH.Partouche,“Superno-scalemodelsinstringtheory,”toappear.
16. F.Bourliot,J.Estes,C.KounnasandH.Partouche,“Cosmologicalphasesofthestringther-
maleffectivepotential,”Nucl.Phys.B830(2010)330[arXiv:0908.1881[hep-th]];
J.Estes,C.KounnasandH.Partouche,“SuperstringcosmologyforN4=1 0superstring
→
vacua,”Fortsch.Phys.59(2011)861[arXiv:1003.0471[hep-th]];
J.Estes,L.LiuandH.Partouche,“MasslessD-stringsandmodulistabilizationintypeIcos-
mology,”JHEP1106(2011)060[arXiv:1102.5001[hep-th]];
L. Liu and H. Partouche, “Moduli stabilization in type II Calabi-Yau compactifications at
finitetemperature,”JHEP1211(2012)079[arXiv:1111.7307[hep-th]].
17. S.Abel,K.R.DienesandE.Mavroudi,“Towardsanonsupersymmetricstringphenomenol-
ogy,”Phys.Rev.D91(2015)12,126014[arXiv:1502.03087[hep-th]].