GUT scale unification in heterotic strings

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

GUT

scale

unification

in

heterotic

strings

Carlo Angelantonj

a

,

∗

,

Ioannis Florakis

b

a_{DepartmentofPhysics,UniversityofTorino,INFNSezionediTorinoandArnold-ReggeCentre,ViaPietroGiuria1,10125Torino,Italy} b_{SectionofTheoreticalPhysics,DepartmentofPhysics,UniversityofIoannina,45110Ioannina,Greece}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

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c

t

Articlehistory:

Received21December2018 Accepted23December2018 Availableonline3January2019 Editor: N.Lambert

We present aclassofheterotic compactifications where it ispossible tolower the string unification scale down tothe GUT scale,while preservingthe validityof theperturbative analysis.We illustrate this approach with an explicit example of a four-dimensional chiral heterotic vacuum with _{N =}1 supersymmetry.

©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

The last few decades have been marked by striking progress bothinformaldevelopmentsofStringTheory,aswellasbyefforts toclassifystringvacuathatcouldserveasabasisforsuitable phe-nomenologicalextensionsoftheStandardModel.Inparticular,the stringeffective actionwith

N =

1 supersymmetryhas been suc-cessfully reconstructed attree level. Nevertheless,even the most promising string construction eventually requires a proper incor-poration of quantum corrections before any quantitative contact withlowenergyphenomenologycanbeachieved.

The F2 coupling inthe effectiveaction ofthe heterotic string is perhaps the most extensively studied such case in the litera-ture, dueto its direct relevance for the studyof the running of gaugecouplingsintheframeworkofstringunification, whichhas beenoneofthemostelusive,yetdesired,propertiesofString The-ory. For the one-loop correction in the string coupling constant, and provided the string vacuum admits a description in terms of an exactly solvable worldsheet CFT, the calculation essentially amountstocomputingthetwo-pointfunctionofgaugebosonson the Riemann surface of the worldsheet torus andseparating out the logarithmic contribution of the massless states captured by field theory, from the heavy string states that are encoded into thethresholdcorrection



a[1],

16

π

2 ga2

(

μ

)

=

ka 16

π

2 g2s

+

balog



α

4

π

2 M2_s

μ

2



+ 

a

.

(1)

*

Correspondingauthor.

E-mailaddresses:carlo.angelantonj@unito.it(C. Angelantonj),iflorakis@uoi.gr (I. Florakis).

The running couplings ga

(

μ

)

at scale

μ

associated to a gauge groupfactorGa,realisedasalevel-kaKac–Moodyalgebra,arethen relatedtotheuniversal stringcoupling gs attreelevel,whilethe massless statescontribute via thefamiliar logarithmic term con-trolledbythecorrespondingbetafunctioncoefficientba.Working in the DR scheme results in the specific value for the constant

α

=

8

π

e1−γ

/

3

√

3, with

γ

being the Euler–Mascheroni constant. Giventhe factthat thePlanck massdoesnotrenormalise in het-erotic theories atanyloop order[2], even inthe caseof sponta-neously brokensupersymmetry[3],the stringscale Ms is conve-nientlyrelatedtothePlanckscalebyMs

=

gsMP

/

√

32

π

.

The simplestpropertiesofgauge thresholdsare bestvisiblein vacuathatpreserve

N =

2 supersymmetry.Thestandardparadigm here isthe compactificationon K3

×

T2 spaces, ortheir singular limitsas T4

/

ZN

×

T2 orbifolds,where N

=

2

,

3

,

4

,

6.Withonly a few assumptions requiring the factorisation of the T2 _{and Kac–} Moodylattices,essentiallyimplyingvanishingVEVsfortheWilson lines, the gauge thresholds become functions of the Kähler and complexstructuremoduli T

,

U ofthecompactification2-torus, re-spectively,andmaybedecomposedintotheform[4]



_aN=2

= −

kaY

ˆ

+

ba

ˆ .

(2)

Thefirstterminthisdecompositionisauniversalcontributiondue to the presence of the gravitational sector that depends only on the level ka, but isotherwise independent ofthe charges of the states. Thesecond termisinstead proportional to thebeta func-tioncoefficientofthe

N =

2 theoryandencodesthecontribution ofthechargedheavystatesrunningintheloop.Usingarguments of modularity, holomorphy and the gravitational anomaly in six

https://doi.org/10.1016/j.physletb.2018.12.054

0370-2693/©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

dimensions,thesecontributionsarecompletelydetermined1 _tobe [4]

ˆ

Y

(

T

,

U

)

=

1 12



F d2

τ

τ

2 2



2,2

(

T

,

U

)

 ˆ¯

E2E

¯

4E

¯

6

− ¯

E34

¯

η

24

+

1008



=

1 2 log

|

j

(

T

)

−

j

(

U

)

|

4

₊

4

π

3T2 E

(

2

;

U

)

+

O

(

e−2πT2

_{) ,}

(3) and[5]

ˆ(

T

,

U

)

=



F d2

τ

τ

2 2





2,2

(

T

,

U

)

−

τ

2



= −

log

α

T2U2

|

η

(

T

)

η

(

U

)

|

4

.

(4)

The modular integrals are performed over the fundamental do-main

F = {

τ

∈ C

+

_,

_|

_τ

_|

_≥

_{1 and}

_|

_τ

₁

_{| ≤}

₁

_/

₂

_}

_{of the moduli space}

ofcomplex structures

τ

=

τ

1

+

i

τ

2 of the worldsheet torus. The modularinvariantNarainlattice



2,2

(T

,

U

)

ofLorentziansignature

(

2

,

2

)

encodes the contributionof Kaluza–Klein and winding ex-citations arising from the T2 compactification, while

η

(

τ

)

is the Dedekind

η

-function and E

ˆ

2

,

E4

,

E6 are the (quasi) holomorphic Eisensteinseriesof modularweight 2,4 and6,respectively. Fur-thermore j(

τ

)

is the Klein invariant function and E(s

;

z) is the real-analytic Eisensteinseries. The subtractionof the linearterm

τ

2 fromtheNarainlatticein(4) simplyamountsto removingthe contributionofmasslessstateswhichhavealreadybeenincluded intothelogarithmictermof(1).

It is remarkable that regardless of the specific details of the construction of the string vacuum, the gauge thresholds are de-composed according to the universal2

N =

2 form (2). This de-composition has very drastic physical consequences. On the one hand,itleadsnaturallytotheunificationofallgaugecouplingsat thescale

MU

=

√

α

2

π

MPgsexp

( ˆ

/

2

) ,

(5)

andgscanbefurtherexpressedintermsofthecommonvalue gU ofthegaugecouplingsatunification,

gs

=

gU



1

+

g 2 U 16

π

2Y

ˆ



−1/2

.

(6)

Ontheother hand,ifone adoptsthe desert doctrineandtriesto setthe string unification scale andcouplingto their GUT values,

MGUT

∼

2

×

1016GeV andgGUT2

=

4

π

/

25,thepreciseexpressionof thegauge thresholds Y and

ˆ

ˆ

reveals that no point inthe T

,

U

modulispaceiscompatiblewiththeunificationdata.3 _Indeed,the

stringunificationscaleinunitsoftheGUTscalecanbewrittenas MU MGUT

=

√

α

4

(

2

π

)

3/2 MP MGUT gGUT



1

+

g2GUT 16π2Y

ˆ

exp

( ˆ

/

2

) .

(7)

1 _{Theexpansioninthesecondlineoftheeq.(3) isvalidonlyinthelargeT} 2 regime.ForT=U thewholeexpansionisoutsidetheregimeofvalidityandone shouldcomputetheintegralusingthemethodsdevelopedin[12] ornumerically. Ineithercase,thefinalexpressionisfinite.

2 _{Actually,thereisanumberofassumptionsrequiredforthissimpleuniversal} formtohold.Ingeneral,thisuniversalitymanifestsitselfinamoregeneralform [6–8].

3 _{See[9] andreferences thereinforareviewofpreviousworksonGUTscale} stringunification.

Since MP

/M

GUT

∼

6

.

1

×

102, itis clearthat theonlywayfor the string unification scaleto matchthe GUTscale wouldbe forthe string thresholds Y and

ˆ

ˆ

to take suitable values that exactly compensatethisdiscrepancy.However,such achoiceturnsoutto be impossible. In fact, the Narain lattice develops extrema [10] at points fixed under the T-duality symmetry group O

(

2

,

2

;

Z)

=

SL

(

2

;

Z)T

×

SL

(

2

;

Z)U

 Z

2,andsincethemodularintegrals(3) and (4) canbeshowntobeuniformlyconvergentoverthefullmoduli space[11–13],thesepoints arealsoextrema forthethresholds Y

ˆ

and

ˆ

.The minimumofther.h.s. of(7) is achievedattheSU(3) point T

=

U

=

e2πi/3_{, at which Y}

ˆ

_∼

₂₇

_.

_{6 and}

ˆ ∼

₀

_.

_068, imply-ing thatthestringunification scalestill overshootsthe GUTscale roughlybyafactorof20.

Adifferentissueariseswhenconsideringthecaseoflargeextra dimensions.When T2

=

Im

(T

)

becomessufficientlylargeinstring units, the associated Kaluza–Klein scale MKK

∼

1

/

√

T2 becomes muchlower thanthe stringscaleorpossiblyeven theGUTscale, whilethestringunificationscaleispushedabovethePlanckscale exponentially fast.Thereasonis thatin thislimit,thethresholds asymptoticallygrowlinearlywiththevolumeofthe compactifica-tiontorus

ˆ ∼

π

3T2

,

Y

ˆ

∼

4

π

T2

.

(8)

Dependingon thesignof thecorresponding betafunction coeffi-cientba,thegaugecouplingeitherdecouples(ba

>

0)orisdriven to thenon-perturbativeregime (ba

<

0).Thisis theso-called de-compactificationproblem[14] whichreflects thefactthatphysics becomes effectively six dimensional above the KK scale and re-quires a linear dependence of the thresholds on the volume T2. Technically,thislineargrowthinT2 arisesfromthepresenceof

η

(

τ

)

=

q1/24 ∞



n=1

(

1

−

qn

) ,

j

(

τ

)

=

1 q

+

196884 q

+

O

(

q 2

_{) ,}

₍₉₎

intheexpressions for

ˆ

and Y ,

ˆ

withq

=

e2πiτ .Indeed,the com-binations T2

|

η

(T

)

|

4 and j(T

)

are automorphic functions of the modular group SL

(

2

;

Z)T

, reflecting theinvariance of the thresh-oldsunderT-duality.

Anobviouswayto avoidthe decompactificationproblemisto choose the moduli to take values sufficiently close to the string scale,althoughthemismatchbetweenthestringunificationscale andthe GUTscale still persists. Such a choice isalways possible insupersymmetricvacua,where T andU aremoduliandmaybe indeedtreatedasfreeparameters.However,whensupersymmetry isspontaneouslybroken,asforinstanceinthestringrealisationof theScherk–Schwarzmechanism,theT andU scalarsareno-longer moduli, since radiative corrections do generate a non-trivial po-tential.Undercertainconditions[15],thispotentialinduces spon-taneous decompactificationofinternal torii anddrives thetheory intothelarge volumeregimewherethe decompactification prob-lembecomesrelevant.

Atfirstsight,themismatchbetweenthestringunificationscale and the GUTscale on the one hand, andthe decompactification problemat large volume onthe other,appear to be entirely un-correlated problems.Theformer relatesto extremaof theNarain lattice alwaysoccurringatvaluesoftheorderofthestring scale, while the latter is encountered as one approaches the boundary of moduli space. In thiswork, we propose that these two prob-lemsactuallyshareacommonorigin,whichmaybetracedbackto theunbrokenSL

(

2

;

Z)T

×

SL

(

2

;

Z)U

Z

2T-dualitysymmetryofthe Narain lattice. Indeed,asymptotically atlarge volume,



2,2

(T

,

U

)

becomesindependentoftheintegrationvariable

τ

andgrows lin-earlywithT2,henceleadingtothedecompactificationproblem (8).

Here,wewish toarguethat bothproblemsmayinfactbe re-solvedsimultaneouslyprovidedtheT-dualitygroupisbrokenina suchawayastorealisethereplacement

SL

(

2

; Z)

T

→ 

1

(

N

)

T

,

(10)

with



1

(N)

T beingthecongruencesubgroup



1

(

N

)

=



a b c d



∈

SL

(

2

; Z)



_

a

,

d

=

1

(

mod N

),

b

=

0

(

mod N

)



,

(11)

actingontheKählermodulus.Sincethemodulidependenceofthe thresholds enters the amplitude only through the KK and wind-ing lattice sum, the breaking (10) implies that K3 and T2 no longerfactorise,buttheK3mustbeellipticallyfiberedoverthe T2_. Thelattercanbe straightforwardlyrealisedasanexactly solvable CFTintermsoffreely-acting

ZN

orbifolds,combiningtwistsalong theK3directionstogether withshifts alonga non-trivialcycleof the T2_{. As a result, the amplitude for the factorised K3}

_×

_T2 _is replacedby



F d2

τ

τ

2 2

⎛

⎝

1 N



h,g∈ZN

A



h g



⎞

_{⎠ }

2,2

→



F d2

_τ

τ

2 2 1 N



h,g∈ZN

A



h g





2,2



h g



,

(12)

whereh labels the orbifold sectorswhile the summationover g

implements the orbifold projection. Following [13], upon partial unfoldingonecanrecastther.h.sas

1 N



F d2

τ

τ

2 2

A



0 0





2,2



0 0



+

1 N



FN d2

τ

τ

2 2

A



0 1





2,2



0 1



,

(13)

where

F

N is the fundamental domain of the Heckecongruence subgroup



0

(N)

.Forthegauge thresholds,thehelicitysupertrace in

A



0₀



vanishes identicallyinaccordancewiththefact thatthe

(h,

g)

= (

0

,

0

)

sector of the orbifold preserves all sixteen super-charges. Therefore the large volume behaviour of the thresholds depends on the way the coordinate shift is implemented in the Narainlattice. Although instring theory one hasthe freedom to shiftacoordinate X orits dual X ,

˜

orevenboth, therequirement that the T-duality group contains



1

_(N)

T automatically selects a geometric action,amountingto amomentum shiftalonga direc-tion

λ

1

+ λ

2U of T2,

λ

i

∈ ZN

. This results in the shifted Narain lattice



2,2



₀ 1



(

T

,

U

)

=

T2



mi_,ni_∈Z e−2πiT det(A)− πT2 τ2 U2  (1 U)Aτ₁_2

,

(14) with A

=



n1 _m1

₊

λ1 N n2 m2

+

λ2 N



.

(15)

Forthisconstruction,the

ˆ

andY contributions

ˆ

tothethresholds takethegenericform

ˆ =



FN d2

τ

τ

2 2



2,2



₀ 1



(

T

,

U

) ,

ˆ

Y

=



FN d2

τ

τ

2 2



2,2



₀ 1



(

T

,

U

)

N

(

τ

) ,

(16)

with

N beingaweakquasi-holomorphicmodularformof



0

(N)

with vanishing constant term. It can be shown [13,16] that the dominantbehaviourofthethresholdsatlarge-T2 is

ˆ ∼ −

log

(

α

fN

(

U

)

T2

)

+

O

(

e−2πT2

) ,

Y

ˆ

∼

O

(

T2−1

) ,

(17) where fN

(U

)

is a function ofthe complex structure modulus U , whose explicitexpression depends onthe order N of the freely-actingorbifold. Contrarytothe casewherethe T-dualitygroup is maximal, thereplacement(10) forcesthethreshold

ˆ

to become unbounded frombelow. Therefore,independently ofthe new ex-tremaof

ˆ

,onemayalwaysfindsuitablevaluesofthevolumeT2 to matchthe string unification andGUTscales. In fact, imposing

MU

=

MGUTin(7) onefinds T2

g2_GUT 128

π

3_f N

(

U

)



MP MGUT



2

∼

50

,

(18)

where,wehaveassumedthat fN

(U

)

=

O(1

)

forstring-scalevalues oftheU modulus,whichistypicallythecasefororbifold construc-tions, wherethediscretesymmetry isonlycompatiblewithfixed valuesofU .

Althoughtheideaofusingfreely-actingorbifoldswhichrestore

N =

4 supersymmetryatlarge volumeasa methodto eliminate thedecompactificationproblemhasalreadybeendiscussedin[17,

18],thefactthatthissametechniquesimultaneouslyallowsoneto lowerthestringunificationscalearbitrarilyclosetotheGUTscale hadnotbeenpreviouslyconsidered.

Theparadigmwehaveexposedsofaractuallyenjoysunbroken

N =

2 supersymmetryinfour-dimensions,andisthus incompati-blewiththechiralnatureoftheobservedspectrumofelementary particles. For this reason, any attempt to apply our program to stringvacuawithsensiblelow-energyphenomenologyrequiresan extensionoftheaboveanalysistochiral

N =

1 orbifold compacti-fications.Insuchvacua,thresholdcorrectionsdecomposeas



a

=

da

+



i



β

a,i

ˆ

(i)

−

kaY

ˆ

(i)



,

(19)

whereda aremoduliindependentconstantsoriginatingfrom

N =

1 subsectors, while

ˆ

(i) _{and Y}

ˆ

(i) _{are moduli-dependent}

contri-butions ofthe i-th

N =

2 subsector. The associated

N =

2 beta function coefficients are denoted

β

a,i and maybe related to six-dimensionalanomalycoefficients[19].

Achievinggaugecouplingunification(atanyscale)isnolonger automatic butrequiressomeadditionalconditionsonthecharged spectrum.Namely,thequantity

a

≡

ba ka log



α

4

π

2 M2_s M2_U



+

da ka

+



i

β

a,i ka

ˆ

(i)

_,

₍₂₀₎

mustbethesameforallunifiedgaugegroupfactors,i.e.

a

= b

foralla,b,andreducestotheconditionsdiscussedin[20] inthe special caseda

=

0 and

a

=

0. It should be stressed that these conditionsareespeciallyrelevantformirage-likeunification,where the gauge group in thefull string construction is alreadybroken downtotheStandardModelgroupfactors.Inthecaseof‘true’ uni-fication,instead,thereisonlyonerelevantgaugegroupabovethe unificationscaleand,therefore,thepreviousconditionsbecome re-dundant anditiseasytochoosethemoduli Ti tomatchtheGUT data. Nevertheless, our analysisis quite general and maybe ap-plied in both cases.4 In the mirage scenario with at most three

4 _{Ingeneral,itmaybenecessarytofurthergeneralise(10) toincludealso the} group1(N)T forcertainN=2 subsectorsinordertoallowvaluesTi,2<1,which

gaugegroupfactors,theconditions

a

= b

withMU

=

MGUTcan alwaysbesatisfiedforasuitablechoiceofthemoduliTi.

It is a well-known fact that heterotic vacua with

N =

1 su-persymmetrycanberealisedasorbifoldlimitsT6

_/

_{ofCalabi–Yau} spaces,with



asuitablediscretesymmetrygroupofthe compact-ificationandchargelattices,preservingexactlyfourKillingspinors. In such vacua, thresholds are typically moduli-independent, un-less



contains elements whichpreserve eight supercharges, cor-responding to sectorsenjoying enhanced

N =

2 supersymmetry. Thresholdsthendecomposeaccordingtoeq.(19),andthe decom-pactification problem becomes relevant. Therefore, the breaking pattern(10) oughttobeimplementedforallsuchsectors.Special careisrequiredsothatchiralitybenotspoiledbythisprocedure.

Infact, in the simplest instances of

Z

2

× Z

2 orbifolds,where each

Z

2 factor realisesa freely-acting K3, T-duality isbroken ac-cordingto(10) andthedecompactificationproblemisnotpresent. However,asdiscussedin[18],chiralityistheninevitablylost.5_This

isduetothefactthat,althoughprojectedwithrespecttothefull

Z

2

× Z

2 group, theuntwistedsector ofthisorbifoldisnonchiral, whilethethree twistedsectorsonly involvepartial

Z

2 projectors which yield an enhanced

N =

2 supersymmetry. Indeed, on the one hand,the vertex operators of the untwistedfermions are in representations of the full ten-dimensional Lorentz group, upon whichthe

Z

2’shavearealaction,thusbeingunabletodistinguish between four-dimensional chiralities. On the other hand, the si-multaneousactionoftwistsandshiftsisincompatible witha full projection in the twisted sectors, since the independent twisted modularorbitvanishesidentically.

The apparent incompatibility between the breaking (10) and chiralitycan be lifted if theorbifold group



hasa complex ac-tionontheuntwistedfermions,sothatdifferentspinorialweights carry different charges with respect to



, generating chiralityin theuntwistedsectoritself.Tothisend,we presenta T6

/

Z

3

× Z

₃ orbifoldconstructionwherethedecompactificationproblemis ab-sent,thespectrumenjoysfour-dimensionalchirality,andthestring unificationscalecanbeloweredtomatchtheGUTscale.

Thefirst

Z

3 factoractsonthethreecomplexcoordinates zi of the factorised T6

₌

_T2

_×

_T2

_×

_T2_{, with fixed complex structure} Ui

=

e2πi/6, as rotations generated by v

= (

1₃

,

1₃

,

2₃

)

, and corre-spondstothecelebratedZ orbifold[22] withstandardembedding and vanishing Wilson lines. It preserves four supercharges, the gauge group is broken to E6

×

SU

(

3

)

×

E8 and the chiral spec-trum comprises 3 copies of chiral multiplets in the representa-tion

(

27

,

3

,

1

)

fromthe untwisted sector aswell as 27copies in

(

27

,

1

,

1

)

and81copiesin

(

1

,

3

¯

,

1

)

.Sincev rotatesallthreezi,the gaugethresholdsarec-numbersindependentofthe compactifica-tionmoduli.

The second

Z

₃ factor, withgenerator w

= (

1₃

+ δ,

−

1₃

+ δ,

δ)

, actsassimultaneous opposite rotations ofthe first two T2_{’s and} order-threeshifts

δ

:

zi

→

zi

+

₃1

(

1

+

Ui

)

onallthreecomplex co-ordinates,thuspreservingeightsupercharges.Noticethattheshift

δ

is a symmetry of the tori with complex structure Ui

=

e2πi/6 sinceitpermutesthethree points z

ˆ

(_ik)

=

k₃

(

1

+

Ui

)

,k

=

0

,

1

,

2,of eachT2,fixedunderthe2

π

/

3 rotation.

Thefullorbifoldgroupisthuscomprisedofthenineelements

(

1

+

v

+

v2

)(

1

+

w

+

w2

)

=

1

+

v

+

v2

+

w

+

w2

+

v w

+

v2w2

+

v2w

+

v w2

,

(21) maybeeasilyobtainedbyreplacingthecorrespondingmomentumshiftsby wind-ingshifts.

5 _{Adifferentwaytoresolvethedecompactificationproblem,whichdoesnotspoil} chiralityandisbasedonacancellationbetweentheuniversalandrunningpartsof thegaugethresholds,wasrecentlyconsideredin[21].

out of which one can recognise four independent

Z

3 subgroups, associated to v, w, vw and v2w, respectively. While v clearly

generatesthe T6

/

Z

3 Calabi–Yau, theothersgeneratethree freely-actingK3’s whichall meet thecondition (10). Again,the discon-nected,twisted,modularorbitsareabsentinthisT6

/

Z

3

×Z

₃ com-pactification because

δ

permutes the fixed points. Nevertheless, four-dimensionalchiralityisnowpreservedbothintheuntwisted andinthe v-twistedsectors.

The spectrum enjoys

N =

1 supersymmetry, with a

non-Abelian gauge group E6

×

E8 and charged matter in the chiral representations 3

× (

27

,

1

)

fromthe untwistedsectorsaswell as inthechiralrepresentations9

× (

27

,

1

)

fromthev-twistedsector. Thetwistedsectorsassociatedtothefreely-actingorbifoldsare in-steadallmassive.

Gaugethresholds receivemoduli-dependentcontributions only from the three freely-acting orbifolds and the residual T-duality groupis



3_i=1



1

(

3

)

Ti,actingontheKählermodulispace.

Perform-ingthepartialunfoldingof[13],wemaycastthegaugethresholds as



E8

=



i=1,2,3



ˆ

Y(i)

−

20

ˆ

(i)



+

d8

,



E6

=



i=1,2,3



ˆ

Y(i)

−

8

ˆ

(i)



+

d6

,

(22)

whered8 andd6 arec-numbercontributionsfromthe Z orbifold, while

ˆ

Y(i)

=

1 144



F3 d2

τ

τ

2 2



2,2



₀ 1



(

Ti

,

Ui

)

 ˆ¯

E2E

¯

4

(

3E

¯

4X

¯

3

−

2E

¯

6

)

2

η

¯

24

+

E

¯

4

(

2E

¯

42

−

3X

¯

3E

¯

6

)

2

η

¯

24

+

1152



,

(23) and

ˆ

(i)

₌



F3 d2

τ

τ

2 2



2,2



₀ 1



(

Ti

,

Ui

) ,

(24)

involve the Narain lattice of eq. (14) with the momentum shift

(λ

1

,

λ

2

)

= (

1

,

1

)

.AsidefromtheDedekindfunctionandthe Eisen-stein series of SL

(

2

;

Z)

, the



0

(

3

)

modular integrals involve the holomorphic weight-two



0

(

3

)

modular form X3

=

E2

(

τ

)

−

3E2

(

3

τ

)

.

The integrals in

ˆ

(i)_{’s can be evaluated by standard methods}

andread

ˆ

(i)

₍

_T i

,

Ui

)

= −

log

⎡

⎣

α

33Ti,2Ui,2







η

3

₍

_T i

/

3

)

η

(

Ti

)

η

3

₍

1+Ui 3

)

η

(

Ui

)







2

⎤

⎦ ,

−

log



_α

33Ti,2f3

(

Ui

)



+

O

(

e−2πT2/3

_{) ,}

(25)

andindeedonlyexhibitslogarithmicgrowthasadvocatedin(10). Aconvenientprocedurefortheevaluationoftheuniversalparts

ˆ

Y(i) _{amounts to representing the quasi-holomorphicfunctions in}

the integrands as linear combinations of Niebur–Poincaré series

F

a

(s,

κ

,

0

)

associatedtothetwocusps

a

=

0

,

∞

of



0

(

3

)

[13],

E4

(

2E2₄

−

3X3E6

)

2

η

24

ˆ

E2E4

(

3E4X3

−

2E6

)

2

η

24

=

20

F

_∞(3)

(

1

,

1

,

0

)

−

4

F

_∞(3)

(

2

,

1

,

0

)

−

36

F

₀(3)

(

1

,

2

,

0

)

+

8

F

(3) 0

(

2

,

2

,

0

)

−

36

F

(3) 0

(

1

,

1

,

0

)

+

12

F

(3) 0

(

2

,

1

,

0

)

+

144

.

(26)

The fundamental domain

F

3 may then be unfolded against the Niebur–Poincaréseriesaccordingtothemethodsdevelopedin[13,

16],andtheaboveintegralscan bethen explicitlyevaluated.The resultingexpressionisrathercumbersomeandinvolves combina-tions of real-analytic modular functions ofthe T and U moduli.

Forthesakeofsimplicity,weshallonlydisplay thecontributions totheleading,possiblysingular,behaviouratlargevolumes.

Potentially divergent terms originate either from the Niebur– Poincaréserieswiths

=

1,orfromtheconstanttermsinthelinear decomposition(26)

ˆ

Y_singular(i)

∼

log



|

j

(

T

)

−

744

|

1/3

|

j_∞

(

T

/

3

)

+

3

|





j∞

(

T

/

3

)

+

231 j_∞

(

T

/

3

)

−

12





9



,

(27)

where j_∞istheKleinfunctionof



0

(

3

)

attachedtothecuspat

∞

. Althoughit isclearthat a logarithmic growthin T2 isabsent, as expectedfromthefactthattheuniversalcontributionsarealways IRfinite,theadditionalabsenceoflineargrowthisduetoa non-trivial cancellation between the Klein functions. This is a direct consequence of our approach (10). In conclusion, Y

ˆ

_singular(i) is ac-tuallyfiniteanddecaysexponentiallyforlargevolume.

The leading contribution to the universal thresholds, comes instead from integrals involving the Niebur–Poincaré series with

s

=

2 inthedecompositions(26),andarisesfromthezero-mode6

I₃(0)

(

s

; a) =

22s

√

4

π

(

s

−

₂1

)

T₂1−sEa

(

s

,

0

;

U

)

(28)

ofthefunctionI3

(s

;

a)

introducedin[16].Asaresult,theuniversal thresholddecaysinverselywiththevolume,

ˆ

Y(i)

(

Ti

,

Ui

)

∼

c3

(

Ui

)

Ti,2

+

O

(

e−2πTi,2/3

_{) ,}

₍₂₉₎ where the c3

(U

i

)

are order-one c-numbers related to the real-analytic Eisenstein series Ea

(s,

0

;

U

)

associated to thetwo cusps

ofthefundamentaldomainof



0

(

3

)

.

Wewouldliketostress,thatthelargevolumebehaviour

ˆ

(i)

_{∼ −}

_log



α

33Ti,2f3

(

Ui

)



,

Y

ˆ

(i)

∼

c3

(

Ui

)

Ti,2

,

(30)

isnot anaccidental featureofthe specificexamplewe have pre-sentedhere,butitisactually agenericproperty of heterotic com-pactificationswhichsatisfytherequirement(10).

Gravitationalthresholdsforthe R2 termactuallyenjoy a simi-larproperty.Inanyheteroticstringcompactificationtheyaregiven bythe integralofthe E

ˆ

2 dependent termintheintegrandofthe universalthresholds Y ,

ˆ

forinstanceeq.(26) in theexplicitmodel consideredhere.AsimilaranalysisasforY can

ˆ

becarriedout ex-plicitlyanditisstraightforwardtoseethattheycangrowatmost logarithmicallywiththe volume,providedcondition (10) is satis-fied.

6 _{Actually, the function I}

3(s;a) refers to 0(3) modular integrals involving Niebur–Poincaréseriesassociatedtothecuspaandwithgeneriss butwithκ=1. Theresultforintegerκ>1 canbeobtainedbytheactionoftheHeckeoperator H(κ3)actingontheU modulus[16].

The unification of gauge couplings atthe GUTscale isa phe-nomenologically appealingpossibilitywhichhasalreadybeen ad-dressed in the string literature. However, past treatments either requirednon-trivialWilsonlines[23,24] orfacedthe decompacti-ficationproblemwhichdrivesthetheoryintothenon-perturbative regimeveryclosetotheGUTscale.

In thiswork,we haveshownthat gaugethresholdcorrections can be generated that allow fora precise matching ofthe string and GUTscales for both

N =

1 and even

N =

2 vacua, without imposingconditionswhicharetoorestrictiveandunnatural.In or-derforthistooccur,thecompactificationhastobreaktheoriginal T-dualitygroupdowntoproductsof



1

(N)

subgroupssothatboth gauge andgravitationalthresholdscanatmostexperiencea loga-rithmic growthatlarge volume,which guarantees thevalidity of theperturbativeanalysis.Thebreaking(10) oftheT-duality group can be easily achieved via freely acting orbifolds, although it is conceivablethat compactificationsonorbifoldsofnon-factorisable torii might induce a similar effect. We have worked out an ex-plicit examplewith a chiral spectrum in fourdimensions where thisparadigmis,forthefirsttime,atwork.

Acknowledgements

We would like to warmly thank Ignatios Antoniadis, Roberto Contino,EmilianDudas,JohnRizos,Stephan StiebergerandEnrico Trincherini forenlightening discussions. C.A. wouldlike to thank the CERN Theory Divisionand I.F.would liketo thank the CERN TheoryDivisionaswell asthePhysicsDepartmentofthe Univer-sityofTorinofortheirkindhospitalityduringvariousstagesofthis work.

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