Universality of Gauge Thresholds in Non-Supersymmetric Heterotic Vacua

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Universality

of

gauge

thresholds

in

non-supersymmetric

heterotic

vacua

Carlo Angelantonj

a

,

Ioannis Florakis

b

,

∗

,

Mirian Tsulaia

c a_{DipartimentodiFisica,UniversitàdiTorino,andINFNSezionediTorino,ViaP.Giuria1,10125Torino,Italy} b_{Max-Planck-InstitutfürPhysik,Werner-Heisenberg-Institut,80805München,Germany}

c_{FacultyofEducationScienceTechnologyandMathematics,UniversityofCanberra,BruceACT2617,Australia}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received31July2014 Accepted1August2014 Availableonline7August2014 Editor:L.Alvarez-Gaumé

Wecomputeone-loopthresholdcorrectionstonon-abeliangaugecouplingsinfour-dimensionalheterotic vacua with spontaneously broken N =2→N =0 supersymmetry, obtained as Scherk–Schwarz reductions of six-dimensional K3 compactifications. As expected, the gauge thresholds are no-longer BPS protected,and receive contributionsalsofrom theexcitations oftheRNS sector. Remarkably,the difference of thresholds for non-abelian gauge couplings is BPS saturated and exhibits a universal behaviourindependentlyoftheorbifoldrealisationofK3.Moreover,thethresholdsandtheirdifference developinfra-redlogarithmicsingularitieswheneverchargedBPS-likestates,originatingfromthetwisted RNSsector,becomemasslessatspeciallociintheclassicalmodulispace.

©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/3.0/).FundedbySCOAP3_.

1. Introduction

Inthelast decadeswe havewitnessed atremendous progress inunderstandingthestructure ofsupersymmetricvacuainString TheoryandM/F-theory. Severalsemi-realisticvacua that incorpo-ratethesalientfeatures oftheMSSM havebeenconstructed and analysedtoaremarkableextent.Theirlow-energyeffectiveaction with

N =

1 supersymmetry hasbeenfullyreconstructedat tree-level,andtheincorporationofquantumand

α

correctionsisstill asubjectofintensestudy.Despitethesesuccessfulendeavours, su-persymmetrybreakinginStringTheoryremainsacompellingopen problemthatstringphenomenologyaspirestoaddress.

Afully-fledgedapproachtospontaneoussupersymmetry break-ing inStringTheory, that admitsan exactly solvableworld-sheet description,isthestringy realisation[1–4]oftheScherk–Schwarz mechanism[5,6],viaspecialfreely-actingorbifolds.Inthisclassof vacua,thesupersymmetrybreakingscaleistiedtothesizeof com-pactdimensions,whiletheexponentialgrowthofstringstatesmay destabilise the classical vacuum dueto the emergence of tachy-onic excitations. This is closelyrelated to the Hagedorn problem ofStringThermodynamics[7]andcanbecircumventedinspecial constructions[8–10].Moreover, it has been recently argued that

*

Correspondingauthor.

E-mailaddresses:carlo.angelantonj@unito.it(C. Angelantonj),

florakis@mppmu.mpg.de(I. Florakis),mirian.tsulaia@canberra.edu.au(M. Tsulaia).

closed string tachyonsemerging fromtwisted orbifold sectors of a class of heterotic vacua with explicitly brokensupersymmetry can actuallyacquire amass byblowing-upthe orbifold singulari-ties[11].

Inallthosecaseswheresupersymmetryis(spontaneously) bro-kenbutthevacuumisclassicallystable, itismeaningfuland im-portanttostudyone-loopradiativecorrectionstocouplingsinthe low-energy effective action. The emergence of one-loop tadpoles for massless statesdoes not impinge on the validity ofthe one-loopanalysis, althoughitmakes theincorporationofhigherloops problematic, unless the back-reaction on the classical vacuum is properlytakenintoaccount[12,13].

Forthisreason,we addressinthislettertheproblemof com-puting one-loop threshold corrections to gauge couplings in a classoffour-dimensionalheteroticvacua withspontaneously bro-ken supersymmetry, that can be built as K3 reductions of the SO

(

16

)

×

SO

(

16

)

constructionof[14]intermsoffreely-acting orbi-folds.Incontrasttoheteroticvacuawithunbrokensupersymmetry, where the moduli dependence of the one-loop corrected gauge couplingsarisesfromtheBPSsector,inthecaseofspontaneously brokensupersymmetrytheamplitude receives contributionsfrom thefulltowerofchargedstringstates,andisno-longertopological. Nevertheless,wefindthatthedifferencebetweengaugethresholds exhibitsaremarkableuniversal structureakintothe

N =

2 super-symmetriccase,duetohighly non-trivialcancellationsinducedby anMSDSspectralflow[15–17]inthebosonicright-movingsector oftheheteroticstring.

http://dx.doi.org/10.1016/j.physletb.2014.08.001

0370-2693/©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/3.0/).Fundedby SCOAP3_.

Thestrikingsignatureofspontaneoussupersymmetrybreaking is the emergence of logarithmic singularities atspecial points of theclassicalmodulispace. TheseareascribedtochargedBPS-like statesthatbecomemasslessatpointsofgaugesymmetry enhance-ment,andsurviveinthedifferenceofgaugethresholds.

Thepaperisstructured asfollows:inSection 2we definethe freely-actingorbifoldresponsibleforthe spontaneousbreaking of supersymmetryandpresentthecorresponding one-loop partition function. Section 3is devoted to theevaluation ofgauge thresh-old correctionsfor thenon-abelian gauge couplings andcontains themainresultsofourinvestigation.Finally,inSection 4we dis-cusstherelevantdecompactificationlimitsandcommentontheir physicalinterpretation.

2. Heteroticvacuumwithspontaneoussupersymmetrybreaking

Theclassofnon-supersymmetricvacua that weshallfocuson isobtainedasa Scherk–Schwarz reductionof six-dimensional K3 compactificationsoftheE8

×

E8 heteroticstring.Theycanalsobe

viewedasK3reductionsoftheItoyama–Taylor vacuum[14],that corresponds to a lower-dimensional freely-actingimplementation ofthe non-supersymmetric,non-tachyonic, SO

(

16

)

×

SO

(

16

)

con-struction[18,19].

Forconcreteness,we shallconsidertheT6

/

Z

N

× Z

₂

compacti-fication oftheten-dimensionalE8

×

E8 heterotic string,with

fac-torised T6

₌

_T4

_×

_T2_.The

_Z

N,withN

=

2

,

3

,

4

,

6 rotates

chrystal-lographycallythecomplexified T4 coordinatesas

v

:

z1

→

e2iπ/Nz1

,

z2

→

e−2iπ/Nz2

,

(2.1)

andrealisesthesingular limitofthe K3surface, preserving 8 su-percharges.The

Z

₂isinsteadfreelyactingandisgeneratedby

v

= (−

1

)

Fst+F1+F2

_δ.

_(2.2) Here, Fst is the space–time fermion number, responsible forthe

breakingofsupersymmetry, F1 andF2arethe“fermionnumbers”

ofthetwooriginalE8’s,whereas

δ

actsasanorder-twoshiftalong

theremainingT2_{.Thecombinedactionof}

_δ

_and

₍

₋

₁

₎

Fst _is respon-siblefor thespontaneous breaking ofthe

N =

2 supersymmetry downto

N =

0, whilethe presenceof

(

−

1

)

F1+F2 _{guaranteesthe} classicalstabilityofthevacuum.1

Theone-looppartitionfunctionreads

Z

=

1 2 1



H,G=0 1 N N−1



h,g=0

×



1 2 1



a,b=0

(

−)

a+b

ϑ



a/2 b/2



2

ϑ



a/2+h/N b/2+g/N



ϑ



a/2−h/N b/2−g/N



×



1 2 1



k,=0

¯ϑ



k/2 /2



6

¯ϑ



k/2+h/N /2+g/N



¯ϑ



k/2−h/N /2−g/N



×



1 2 1



r,s=0

¯ϑ



r/2 s/2



8



×

1

η

12

_η

_¯

24

(

−)

H(b++s)+G(a+k+r)+H G

_Γ

2,2



H G



Λ

K3



h g



.

(2.3) Here,

η

is theDedekind function and

ϑ



_α β



are the standard Ja-cobithetaconstantswithcharacteristics.Thesumoverofthespin

1 _{This is no-longertrue when Wilson linesare turnedon, whereby all}

non-supersymmetricheteroticvacuacanbecontinuouslyconnected[20,21].Inthisnote weshallalwaysassumeatrivialWilson-linebackground.

structures a, b, k,



, r and s yield the ten-dimensional E8

×

E8

heterotic-string spectrum, while

(

h

,

g

)

and

(

H

,

G

)

correspond to the

Z

N and

Z

2 orbifolds.Thetwo-dimensionalNarainlatticewith

characteristicsisdefinedas

Γ

2,2



H G



=

τ

2



 m,n eiπG(λ1· m+λ2·n)

_Γ

 m+H₂λ2,n+H₂λ1

(

T

,

U

),

(2.4) with

Γ

m_,n

(

T

,

U

)

=

q 1 4T2U2|m2−Um1+ ¯T(n1+Un2)|2_q

_¯

_4T2U21 |m2−Um1+T(n1+Un2)|2

_,

_(2.5) anddependsontheKählerandcomplexstructuremoduliT and U . As usual, momenta and windingsare labeled by m and



n,



while theintegralvectors

λ

1 and

λ

2encodethefreely-actingshiftof

Z

₂.

Without lossofgenerality, we shallfocus onthe case

λ

1

= (

1

,

0

)

and

λ

2

= (

0

,

0

)

correspondingtoamomentumshiftalongthefirst T2 direction.Allother casescan berelatedtotheformerby suit-ableredefinitionsoftheT andU moduli.

Finally,

Λ

K3



h g



=

⎧

⎪

⎨

⎪

⎩

Γ

4,4 for

(

h

,

g

)

= (

0

,

0

),

kh g  |η|12 |ϑ₁1_//2₂₊+h_g/_/N_Nϑ1/2−h/N 1/2−g/N  |2 for

(

h

,

g

)

= (

0

,

0

),

(2.6)

with

Γ

4,4 being the conventional Narain lattice associated to

the T4_{, k}



0 g



=

16sin4

(

π

g

/

N

)

counting the number of twisted sectors ofthe

Z

N orbifold, andthe remaining k



_h

g



’s withh

=

0 beingdeterminedbymodularinvariance.

As a consequenceofthe Scherk–Schwarzmechanism, thetwo gravitini acquire a mass m3/2

= |

U

|/

√

T2U2, and supersymmetry

isspontaneouslybrokenatagenericpoint intheclassicalmoduli space. The

Z

₂ alsobreaks the E8

×

E7 gauge groupofthe

N =

2

theory down to SO

(

16

)

×

SO

(

12

)

, up to abelian factors. The full spectrumcanbederivedfrom(2.3)usingstandardtechniques.

Notice that,asinthe parentten-dimensionalSO

(

16

)

×

SO

(

16

)

non-supersymmetrictheory[18,19],thespectrumisfreeof tachy-onicexcitationsatagenericpointofthe

(

T

,

U

)

modulispace.This can be verified by looking at the H

=

0, a

=

0 contributions to

(2.3).

3. One-loopthresholdsfornon-abeliangaugecouplings

Although the vacuumconfiguration presented inthe previous sectionisnotsupersymmetric,theabsenceofphysicaltachyonsin theperturbativespectrumimpliesthatitisclassicallystable.Asa result,itisfullyjustifiedandimportanttostudyone-loopradiative correctionstocouplingsinthelow-energyeffectiveaction,in con-trast to higher-loop diagrams that diverge due to theemergence ofone-loop tadpolesback-reactingonthevacuum[12,13].Thisis still anopen probleminStringTheory,andhasrecentlytriggered agrowinginterest[22–24].

Tothisend,weshalladdressherethequestionofquantum cor-rectionstothecouplingsofthenon-abelianSO

(

16

)

×

SO

(

12

)

gauge factors, extending the analysis of [25] to non-supersymmetric vacua.

Thresholdcorrections

Δ

_G associatedtothegroup factor

G

ap-pearintherelationbetweentherunninggaugecouplingg2_G

(

μ

)

of thelow-energytheoryandthestringcouplinggs

16

π

2 g2_G

(

μ

)

=

16

π

2 g2s

+ βG

logM 2 s

μ

2

+ ΔG

,

(3.1)

where,inthecaseathand,the Kac–Moodyalgebrais realisedat levelone,andMssets thestringscale.Theyencodethe

contribu-tionoftheinfinitetowerofmassivestringstatestotheone-loop diagram,andcanbeorganisedas

Δ

_G

≡

i 4

π

NR.N.



F d

μ

1



H,G=0 N

_

−1 h,g=0

Δ

_G



H,h G,g



=

i 4

π

NR.N.



F d

μ

1



H,G=0 N

_

−1 h,g=0

(

−

1

)

H G

×

L



H,h G,g



η

2

Φ

_G



H,h G,g



¯

η

18

Λ

K3



h g



η

4

_η

_¯

4

Γ

2,2



H G



η

2

_η

_¯

2

.

(3.2)

In this expression, d

μ

denotes the SL

(

2

;

R)

invariant measure, whileR.N. standsforthemodular-invariantprescriptionof[28,29]

forregularisingtheinfra-reddivergencesoftheintegral.

Thequantity

L



_GH_,,h_g



encodesthespin-structuresumoverthe integratedworld-sheetcorrelatorsforthefour-dimensionalspace– time fields, whereas

Φ

_G



_H,h

G,g



encodes the contribution of the gauge sector with the relevant trace insertion. They are defined as

L



H,h G,g



≡

1 2



(a,b)=(1,1)

(

−)

a(1+G)+b(1+H)

∂

τ

×

ϑ



_a /2 b/2



η

ϑ



a/2 b/2



ϑ



a/2+h/N b/2+g/N



ϑ



a/2−h/N b/2−g/N



η

3

,

(3.3) and

Φ

_G



H,h G,g



≡

1 4



1

(

2

π

i

)

2

∂

2 z_G

−

1 4

π τ

2

×



₁



k,=0

(

−)

kG+H

¯ϑ



k/2 /2



6

¯ϑ



k/2+h/N /2+g/N



¯ϑ



k/2−h/N /2−g/N



×

1



r,s=0

(

−)

rG+sH

¯ϑ



r/2 s/2



8



(

z_G

)

|

z_G=0

.

(3.4)

Inthelatterequation,itisimpliedthattheVEVz_G isonlyinserted along the particular theta function corresponding to the Cartan chargewhosegrouptraceweareconsidering.

Itisconvenienttoarrangethe4N2 sectorsoftheorbifoldsoas todistinguishtheoriginofthevariouscontributionstothe thresh-olds.The

(

h

,

g

)

= (

0

,

0

)

sectorcorresponds to theItoyama–Taylor construction[14] reducedtofour-dimensions, andisproportional tothe T4 _lattice

_Λ

K3



0

0



,depending onthe invariant T4 _moduli.

Furthermore,since h

=

g

=

0, one is effectivelydealing withthe SO

(

16

)

×

SO

(

16

)

lattice.Hence, the grouptracesare independent of the choice of gauge group

G

, implying that the difference of thresholdsisindependentoftheT4 moduli.

Anexplicitcalculationyields

Δ

Λ

= −

1 4N

×

122

Λ

K3



0 0



η

12

_η

_¯

24

×



Γ

2,2



0 1



ϑ

₃8

− ϑ

₄8

 ¯

ϑ

₃4

¯ϑ

₄4

 ˆ¯

E2

− ¯ϑ

34

 ¯

ϑ

34

¯ϑ

44

+

8

η

¯

12



− Γ

2,2



1 0



ϑ

₃8

− ϑ

₂8

 ¯

ϑ

₃4

¯ϑ

₂4

 ˆ¯

E2

+ ¯ϑ

34

 ¯

ϑ

24

¯ϑ

34

−

8

η

¯

12



+ Γ

2,2



1 1



ϑ

₄8

− ϑ

₂8

 ¯

ϑ

₄4

¯ϑ

₂4

 ˆ¯

E2

+ ¯ϑ

44

 ¯

ϑ

24

¯ϑ

44

−

8

η

¯

12



,

(3.5)

where E

ˆ

2 istheweight-twoquasi-holomorphic Eisensteinseries.2

Notice that the second andthird lines can be obtainedfromthe first one upon actingwith the SL

(

2

;

Z)

generators S and T S, as demandedbymodularity.

Theremainingcontributionscanbeorganisedas

1



H,G=0 N−1



h,g=0 (h,g)=(0,0)

Δ

_G



H,h G,g



=

N

_

−1 h,g=0 (h,g)=(0,0)



Δ

_G



0,h 0,g



+ ΔG



0,h 1,g



+ ΔG



1,h 0,g



+ ΔG



1,h 1,g



≡ Δ

(u+) G

+ Δ

(Gu−)

+ Δ

G(t+)

+ Δ

(Gt−)

,

(3.6) accordingtothesectorsofthefreely-actingorbifold.Thefirst con-tribution

Δ

(_Gu+), corresponding to

(

H

,

G

)

= (

0

,

0

)

, computes the gaugethresholdsofthe

N =

2 heteroticstringontheorbifoldlimit ofK3. It is thusexpected tobe BPS saturated andthedifference

Δ

(_Gu+)

− Δ

_G(u+) to beuniversal andtodepend onlyon themoduli oftheT2torus3[25].Theremainingterms,connectedamongeach other by S and T S modular transformations, are inherently non-BPS since the freely-acting orbifold acts non-triviallyand breaks supersymmetry.Thisisreflectedbythefactthatthemodular inte-gralnowinvolvesgenuinely non-holomorphiccontributions.

Forconcreteness,weshallpresentexplicitlythevarious contri-butionsinthecaseN

=

2,whereonefinds

Δ

(_SOu+₍₁₆)₎

= −

1 48

Γ

2,2



0 0

 ˆ¯

E2E

¯

4E

¯

6

− ¯

E2₆

¯

η

24

,

(3.7)

Δ

(_SOu−₍₁₆)₎

= −

1 96

Γ

2,2



0 1

 ¯ϑ

4 3

¯ϑ

44

( ¯

ϑ

34

+ ¯ϑ

44

)

[( ˆ¯

E2

− ¯ϑ

34

) ¯

ϑ

34

¯ϑ

44

+

8

η

¯

12

]

¯

η

24

−

1 144

Γ

2,2



0 1

ϑ

4 2

(ϑ

38

− ϑ

48

)

η

12

( ˆ¯

E2

− ¯ϑ

34

) ¯

ϑ

34

¯ϑ

44

+

8

η

¯

12

¯

η

12

,

(3.8) and

Δ

(_SOu+₍₁₂)₎

= −

1 48

Γ

2,2



0 0

 ˆ¯

E2E

¯

4E

¯

6

− ¯

E3₄

¯

η

24

,

(3.9)

Δ

(_SOu−₍₁₂)₎

= −

1 96

Γ

2,2



0 1

 ¯ϑ

8 3

¯ϑ

48

[ ˆ¯

E2

( ¯

ϑ

34

+ ¯ϑ

44

)

+ ¯ϑ

28

−

2

¯ϑ

34

¯ϑ

44

]

¯

η

24

−

1 144

Γ

2,2



0 1

ϑ

4 2

(ϑ

38

− ϑ

48

)

η

12

ˆ¯

E2

¯ϑ

₃4

¯ϑ

₄4

¯

η

12

+

ϑ

24

ϑ

44

|ϑ

24

− ϑ

44

|

2

− ϑ

24

ϑ

34

|ϑ

24

+ ϑ

34

|

2

η

12

_η

_¯

12

¯ϑ

4 3

¯ϑ

44

.

(3.10) Intheseexpressions E4 (E6)istheweight-four(-six)holomorphic

Eisensteinseries.Again,theremaining termscanbe computedby the action of the generators S and T S of SL

(

2

;

Z)

on the corre-sponding

Δ

(u−)_{contributions.}

As anticipated,Eqs. (3.7) and (3.9) compute the thresholds to the

N =

2 supersymmetricE8 andE7gaugefactors.Eqs.(3.8)and

2 _{Wheneverthecharacteristicsofthethetaconstantsequal0, 1/2 weemploythe}

lightnotationintermsofthe

ϑ

α’s.

3 _{ThedifferenceofgaugethresholdsisindeeduniversalfortheT}4_/Z

Norbifolds, thoughinmoregeneralconstructionsthey mayexhibitanon-universalstructure [26,27].

(3.10)involvecontributionsfromBPSstateswhosemassesarenow deformedbythefreeactionofthe

Z

₂ orbifold.

The BPS contributions to these amplitudes can be integrated overthe SL

(

2

;

Z)

fundamental domain

F

or,afterpartial unfold-ing,overthefundamental domain

F

0

[

2

]

ofthe

Γ

0

(

2

)

congruence

subgroup,followingtheproceduredevelopedin[28–30].The non-BPS contributions can be shown to be exponentially suppressed

[31]inthelarge T2 _{volumelimit,andarethusnegligibleat}

low-energies.Weshallnot indulgehereinthefullcomputation ofthe thresholds,butratherfocusontheirdifference.Onefinds

Δ

_SO(u+₍₁₆) ₎

− Δ

_SO(u+₍₁₂) ₎

= −

36

Γ

2,2



0 0



,

(3.11)

thatreproducestheresultof[25],and

Δ

_SO(u−₍₁₆) ₎

− Δ

_SO(u−₍₁₂) ₎

= −

1 6

Γ

2,2



0 1

ϑ

12 2

η

12

−

8

.

(3.12)

Surprisingly, the non-holomorphic contributions to the thresh-oldscancel when takingtheir difference,and reduce to a purely holomorphic BPS-like term. As we shall show, the difference of gaugethresholds exhibitsaremarkableuniversalbehaviour, inde-pendentlyof the details ofthe T4

/

Z

N orbifold. Indeed,the

non-holomorphiccontributiontothedifferenceofthresholdsreads

−

ϑ

28

|ϑ

34

+ ϑ

44

|

2

¯ϑ

34

¯ϑ

44

η

12

_η

_¯

12

−

ϑ

₂4

ϑ

₄4

|ϑ

₂4

− ϑ

₄4

|

2

¯ϑ

₃4

¯ϑ

₄4

η

12

_η

_¯

12

+

ϑ

24

ϑ

34

|ϑ

24

+ ϑ

34

|

2

¯ϑ

34

¯ϑ

44

η

12

_η

_¯

12

=

12



O2₈V8

+

3V83

 ¯

O28V

¯

8

− ¯

V83



,

(3.13)

whereintheright-handside wehaveintroducedtheSO

(

8

)

char-acters. Although, this term looks completely non-holomorphic it actuallypossessesa BPS-likestructureduetoaremarkableMSDS identity[16,10]

¯

O2₈V

¯

8

− ¯

V83

=

8

,

(3.14)

whichreflects ahiddenMSDS spectralflow inthebosonic sector oftheglobal

N = (

2

,

2

)

superconformalsymmetry onthe world-sheet[17,31].Asaresult,Eq.(3.13)reducestothepurely holomor-phiccontribution(3.12).

Toevaluate theintegrals,we firstnotice thatthe combination

ϑ

₂12

/

η

12_{correspondstoanautomorphicfunctionoftheHecke}

con-gruence subgroup

Γ

0

(

2

)

. Moreover, it is regular at the cusp at

τ

=

i

∞

whileithasasimplepoleatthecusp

τ

=

0.4 Thisis suf-ficienttoidentify[30]

ϑ

₂12

η

12

=

F

0

(

1

,

1

,

0

)

−

16

= ˆj

2

(

τ

)

−

24

,

(3.15)

where

F

0

(

1

,

1

,

0

)

isthemeromorphicweight-zeroNiebur–Poincaré

series attached to the cusp at

τ

=

0 of

Γ

0

(

2

)

, and

ˆj

2

(

τ

)

is the

Fricketransform[30]ofthe

Γ

0

(

2

)

Hauptmodul

j2

(

τ

)

=

η

24

₍

_τ

₎

η

24

₍

₂

_τ

₎

+

24

.

(3.16)

Themodularintegralscanbestraightforwardlycomputedusing theresultsof5 _{[28–30,32]toyield}

4 _{Weremindherethatthecompactificationofthefundamentaldomain}

_F

0[2]of Γ0(2)requiresaddingtwopoints,i.e. thetwocusps,τ=i∞andτ=0.See,for instance,[30].

5 _{Thefirstintegralwasactuallyoriginallycomputedin[25]byunfoldingthe}

fun-damentaldomainagainsttheNarainlattice.

R.N.



F d

μ

Γ

2,2

(

T

,

U

)

= −

log T2U2

η

(

T

)

η

(

U

)



4

,

(3.17) R.N.



F0[2] d

μ

Γ

2,2



0 1



(

T

,

U

)

= −

log T2U2



ϑ

4

(

T

)ϑ

2

(

U

)



4

,

(3.18) and R.N.



F0[2] d

μ

Γ

2,2



0 1



(

T

,

U

)

θ

12 2

(

τ

)

η

12

₍

_τ

₎

= −

2 log

 ˆ

j

2

(

T

/

2

)

− ˆj

2

(

U

)



4



j2

(

U

)

−

24



4



.

(3.19)

Combiningthevariouscontributions,onefinds

Δ

SO(16)

− Δ

SO(12)

=

36 log



T2U2



η

(

T

)

η

(

U

)



4



−

4 3log



T2U2



ϑ

4

(

T

)ϑ

2

(

U

)



4



+

1 3log

 ˆ

j

2

(

T

/

2

)

− ˆj

2

(

U

)



4



j2

(

U

)

−

24



4



.

(3.20)

Again, thevarious termshavea clearphysicalinterpretation. The firstlinegeneralisesthecelebratedresultof[25].Thepresenceof second term is ascribed to the modified Kaluza–Klein masses of BPSstates,thatareindeedaffectedby thefreeactionofthe orbi-fold. In fact, since the

Z

₂ orbifold corresponds to a spontaneous breaking ofsupersymmetry, themodel in(2.3) contains precisely thesameexcitationsastheE8

×

E8heteroticstringonT2

×

T4

/

Z

N,

whose masses are continuously deformedby the scale of super-symmetrybreaking.Asaresult,thedualitygroup isbrokendown tothesubgroup

Γ

0

(

2

)

T

× Γ

0

(

2

)

U ofSL

(

2

;

Z)

T

×

SL

(

2

;

Z)

U.

While the first two contributions are regular at any point in theclassical modulispace, theterminthesecondline, particular tothisvacuumwithbrokensupersymmetry,possesseslogarithmic singularitiesatthelocusT

/

2

=

U andits

Γ

0

(

2

)

images.Theorigin

ofthesesingularitiesisascribedtomassivechargedBPS-likestates that become masslessatspecial pointsin modulispace. To man-ifest their originintheperturbative spectrum,itis convenientto expresstheir contributionto(2.3) intermsofthe SO

(

2n

)

charac-ters 1 2

(

O4O4

× ¯

V12O

¯

4V

¯

16

)



Γ

2,2



1 0



+ Γ

2,2



1 1



.

(3.21)

Thesestatesincludetheleft-movingNSvacuumanditsstringy ex-citations,whiletheright-movingsectorismasslessandbelongsto thebi-fundamentalrepresentation

(

16

,

12

)

oftheSO

(

16

)

×

SO

(

12

)

gaugegroup.Theyalwayscarry non-trivialmomentumand wind-ingquantumnumbers,andthelighteststateshavemass

m2_O4O4

=

|

T

/

2

−

U

|

2 T2U2

.

(3.22)

Indeed,thesestatesbecomemasslessatthepointT

/

2

=

U ,where p2

R

=

0, and are responsible for the logarithmic divergence in (3.20).

Notice that the fact that extra massless states emerge from the

Z

₂ twisted sector is compatiblewith the fact that the term

ϑ

₂12

/

η

12_{,originatingfromtheun-twistedsector, hasapoleatthe}

cusp

τ

=

0 butis regular at

τ

=

i

∞

. In fact, the two cusps are relatedby an S modular transformationthat also relatesthe un-twisted andtwisted sectors.As aresult,thesingularityofthe u₋ sector at

τ

=

0 is to be understood as the map under S of the physicalinfra-redsingularityofthetwistedsector.

One cancompute thegaugethresholds alsointhecaseofthe other singular limits ofK3, namely N

=

3

,

4

,

6. Although the re-sultofthethresholdsdependsontheparticularvalue of N,their

difference exhibits a remarkable universal behaviour. In fact, one finds

Δ

SO(16)

− Δ

SO(12)

=

α

log



T2U2



η

(

T

)

η

(

U

)



4



+ β

log



T2U2



ϑ

4

(

T

)ϑ

2

(

U

)



4



+

γ

log

 ˆ

j

2

(

T

/

2

)

− ˆj

2

(

U

)



4



j2

(

U

)

−

24



4



,

(3.23)

with

(

α

,

β,

γ

)

= (

36

,

−

₃4

,

1₃

)

for the

Z

2 and

Z

3 orbifolds,

(

α

,

β,

γ

)

=

5₈

(

36

,

−

4₃

,

₁₅8

)

for the

Z

4 orbifold and

(

α

,

β,

γ

)

=

35 144

(

36

,

−

4 3

,

1

3

)

forthe

Z

6 orbifold.

Thisuniversalitystructureisadirectconsequenceofthe univer-salbehaviour ofthe

N =

2 thresholds[25,33],whichispreserved bythefreeactionofthe

Z

₂.

4. Decompactificationlimits

It is instructive to study Eq. (3.23) in the decompactification limits.Forconvenience,weshallassumeasquaredT2 _with T

=

i R1R2 and U

=

i

R2 R1

,

(4.1)

so that the masses of the two gravitini andof the O4O4 states

read m2_3/2

=

1 R2₁ and m 2 O4O4

=

1 4



R1

−

2 R1

2

.

(4.2)

Inthe R1

→ ∞

limit,

N =

2 supersymmetry isrecovered,and

theleadingbehaviourofEq.(3.23)

lim R1→∞

[Δ

SO(16)

− Δ

SO(12)] =

π α

3 R1



R2

+

1 R2

+ . . .

(4.3) grows linearly with the T2 volume. This is expected from scal-ing arguments, since in six dimensions the gauge coupling has length dimension

−

1. The term proportional to

β

in (3.23) only grows logarithmically with R1 asa result of supersymmetry

en-hancementsince,chargedstateslighter thanthe supersymmetry-breaking scale are effectively BPS-like and thus contribute loga-rithmicallytothedifferenceofthresholdcorrections, whereasthe infinitetowers of chargedstatesheavierthanm3/2 havean

effec-tive

N =

4 supersymmetry and, thus, do not contribute. Finally, the term proportional to

γ

is exponentially suppressed because the lightest charged states O4O4 have mass mO4O4

m3/2 and

effectivelydecouple.

IntheR2

→ ∞

limit,theleadingbehaviourof(3.23)is

lim R2→∞

[Δ

SO(16)

− Δ

SO(12)]

=

π α

3 R2



R1

+

1 R1

+

π

β

R2 R1

+

2

π γ

R2



R1

−

2 R1

−





R1

−

2 R1





+ . . . .

(4.4) As expected, the term proportional to

α

is again linearly diver-gentwiththe T2 _{volume.The termproportionalto}

_β

_{now scales}

as R2

/

R1, andconsistently vanishes asm3/2

→

0. The term

pro-portionalto

γ

dependson thescale of supersymmetrybreaking. When R1

>

√

2 it is exponentially suppressed because m3/2

<

mO4O4,whereas when R1

<

√

2 it scales as R2

(

2

/

R1

−

R1

)

.This

isaconsequenceofthefact that,inthe R1

→

0 limit,

supersym-metry isexplicitly broken, andthistermgrows withthe volume R2R

˜

1

∼

R2

/

R1oftheT-dualtorus.

Notice that inthe R1

→

0 limit,the freely-actingorbifold

de-generatesintoanexplicitbreakingofsupersymmetry.Thisimplies that the universal behaviour (3.23) should hold also in the case whentheten-dimensionalO

(

16

)

×

O

(

16

)

theoryof[18,19]is com-pactifiedon T2

×

K3.Asa result,asimilar universalbehaviour of the thresholddifferencesis expectedto arisealso when T2

×

K3 is replaced by a generic Calabi–Yau manifold. It would be inter-esting toinvestigatewhetherEq.(3.23) alsoholdswhenthe ten-dimensional heterotic string, whether supersymmetric or not, is compactifiedonamanifold thatdoesnotpreserveany supersym-metry.

Acknowledgements

WearegratefultoB. Piolineforfruitfuldiscussions.M.T.would like to thank the Departmentof Physics, theUniversity of Auck-land,wherepartofthisworkhasbeenperformed,foritskind hos-pitality.TheworkofC.A.hasbeensupportedinpartby the Euro-peanERC AdvancedGrant No. 226455“Supersymmetry, Quantum GravityandGaugeFields”(SUPERFIELDS)andinpartbythe Com-pagniadiSanPaolocontract“ModernApplicationinStringTheory” (MAST)TO-Call3-2012-0088.TheworkofM.T.hasbeensupported inpartbyanAustralianResearchCouncilgrantDP120101340.M.T. would alsolike to acknowledgegrant 31/89 ofthe Rustaveli Na-tionalScienceFoundation.

References

[1]R.Rohm,Spontaneoussupersymmetrybreakinginsupersymmetricstring the-ories,Nucl.Phys.B237(1984)553.

[2]C.Kounnas,M.Porrati,Spontaneoussupersymmetrybreakinginstringtheory, Nucl.Phys.B310(1988)355.

[3]S.Ferrara,C.Kounnas,M.Porrati,F.Zwirner,Superstringswithspontaneously brokensupersymmetryandtheireffectivetheories,Nucl.Phys.B318(1989) 75.

[4]C.Kounnas,B.Rostand,Coordinatedependentcompactificationsanddiscrete symmetries,Nucl.Phys.B341(1990)641.

[5]J.Scherk,J.H.Schwarz,Spontaneousbreaking ofsupersymmetrythrough di-mensionalreduction,Phys.Lett.B82(1979)60.

[6]J.Scherk,J.H.Schwarz,Howtogetmassesfromextradimensions,Nucl.Phys. B153(1979)61.

[7]J.J. Atick,E.Witten,TheHagedorntransitionandthe numberofdegreesof freedomofstringtheory,Nucl.Phys.B310(1988)291.

[8]C.Angelantonj, M.Cardella,N. Irges,Analternativefor modulistabilisation, Phys.Lett.B641(2006)474,arXiv:hep-th/0608022.

[9]C. Angelantonj, C. Kounnas,H. Partouche, N.Toumbas, Resolutionof Hage-dornsingularityinsuperstringswithgravito-magneticfluxes,Nucl.Phys.B809 (2009)291,arXiv:0808.1357[hep-th].

[10]I. Florakis, C. Kounnas, N. Toumbas, Marginal deformations of vacua with massiveboson–fermiondegeneracysymmetry,Nucl.Phys.B834(2010)273, arXiv:1002.2427[hep-th].

[11]M. Blaszczyk, S. GrootNibbelink, O. Loukas,S. Ramos-Sanchez, Non-super-symmetricheteroticmodelbuilding,arXiv:1407.6362[hep-th].

[12]W.Fischler,L.Susskind,Dilatontadpoles,stringcondensatesandscale invari-ance,Phys.Lett.B171(1986)383.

[13]W.Fischler,L.Susskind,Dilatontadpoles,stringcondensatesandscale invari-ance.2,Phys.Lett.B173(1986)262.

[14]H.Itoyama,T.R.Taylor,SupersymmetryrestorationinthecompactifiedO(16)× O(16)heteroticstringtheory,Phys.Lett.B186(1987)129.

[15]C.Kounnas,Massiveboson–fermiondegeneracyandtheearlystructureofthe universe,Fortschr.Phys.56(2008)1143,arXiv:0808.1340[hep-th].

[16]I. Florakis, C. Kounnas, Orbifold symmetry reductions of massive boson– fermiondegeneracy,Nucl.Phys.B820(2009)237,arXiv:0901.3055[hep-th]. [17]A.E.Faraggi,I.Florakis,T.Mohaupt,M.Tsulaia,Conformalaspectsofspinor–

vectorduality,Nucl.Phys.B848(2011)332,arXiv:1101.4194[hep-th]. [18]L.Alvarez-Gaume,P.H.Ginsparg,G.W.Moore,C.Vafa,AnO(16)×O(16)

het-eroticstring,Phys.Lett.B171(1986)155.

[19]L.J.Dixon,J.A.Harvey,Stringtheoriesinten-dimensionswithoutspace–time supersymmetry,Nucl.Phys.B274(1986)93.

[20]P.H.Ginsparg, C.Vafa,Toroidalcompactificationofnon-supersymmetric het-eroticstrings,Nucl.Phys.B289(1987)414.

[21]V.P. Nair, A.D. Shapere, A. Strominger, F. Wilczek, Compactification ofthe twistedheteroticstring,Nucl.Phys.B287(1987)402.

[22]E.Dudas,G.Pradisi,M.Nicolosi,A.Sagnotti,Ontadpolesandvacuum redefini-tionsinstringtheory,Nucl.Phys.B708(2005)3,arXiv:hep-th/0410101. [23]N.Kitazawa,Tadpoleresummationsinstringtheory,Phys.Lett.B660(2008)

415,arXiv:0801.1702[hep-th].

[24]R.Pius,A.Rudra,A.Sen,Stringperturbationtheoryarounddynamicallyshifted vacuum,arXiv:1404.6254[hep-th].

[25]L.J.Dixon,V.Kaplunovsky,J.Louis,Modulidependenceofstringloop correc-tionstogaugecouplingconstants,Nucl.Phys.B355(1991)649.

[26]P.Mayr,S.Stieberger,Thresholdcorrectionstogaugecouplingsinorbifold com-pactifications,Nucl.Phys.B407(1993)725,arXiv:hep-th/9303017.

[27]E.Kiritsis,C.Kounnas,P.M.Petropoulos,J.Rizos,Stringthresholdcorrectionsin modelswithspontaneouslybrokensupersymmetry,Nucl.Phys.B540(1999) 87,arXiv:hep-th/9807067.

[28]C. Angelantonj, I. Florakis, B. Pioline, A new look at one-loop integrals in stringtheory,Commun.NumberTheoryPhys.6(2012)159,arXiv:1110.5318 [hep-th].

[29]C.Angelantonj, I.Florakis, B.Pioline, One-loopBPSamplitudes asBPS-state sums,J.HighEnergyPhys.1206(2012)070,arXiv:1203.0566[hep-th]. [30]C. Angelantonj, I. Florakis, B. Pioline, Rankin–Selberg methods for closed

stringsonorbifolds,J. HighEnergyPhys. 1307(2013)181,arXiv:1304.4271 [hep-th].

[31] C.Angelantonj,I.Florakis,M.Tsulaia,inpreparation. [32] C.Angelantonj,I.Florakis,B.Pioline,inpreparation.

[33]E.Kiritsis, C.Kounnas, P.M.Petropoulos,J. Rizos, Universalityproperties of N=2 andN=1 heteroticthresholdcorrections,Nucl.Phys.B483(1997) 141,arXiv:hep-th/9608034.