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PublishedinJHEP StandardModelVacuaforTwo-dimensionalCompactifications Jonathan M. Arnold, Bartosz Fornal and Mark B. Wise California Institute of Technology, Pasadena, CA 91125, USA (Dated:January4,2011) Weexaminethestructureoflower-dimensionalstandardmodelvacuafortwo-dimensionalcompactifications (ona2Dtorusandona2Dsphere). Inthecaseofthetoruswefindanewstandardmodelvacuumforalarge rangeofneutrinomassesconsistentwithexperiment. Quantumeffectsplayacrucialroleintheexistenceof thisvacuum. Forthecompactificationonaspheretheclassicaltermsdominatetheeffectivepotentialforlarge radiiandastablevacuumisachievedonlybyintroducingalargemagneticflux. Wearguethatthereareno two-dimensionalstandardmodelvacuaforcompactificationsonasurfaceofgenusgreaterthanone. I. INTRODUCTION though the ultimate utility of this knowledge remains uncer- tain. 1 While the standard model coupled to gravity has a unique Theresultsofthispaperand[1]showthattherearevacua 1 four-dimensionalvacuum,ithasrecentlybeenshown[1]that of the standard model where either one or two dimensions 0 are compactified. This opens the question: why did we end it may also have a landscape of three-dimensional vacua. 2 up cosmologically in a vacuum with three large spatial di- Theremightevenexistvacuaoflower-dimensionality.Transi- n tionsbetweensuchlower-dimensionalvacuaandthe4Dstan- mensions? Sincethetoroidalandsphericalcompactifications, a whichwefindinthispaper,correspondtoaneffectivetheory dardmodelvacuummayhavemeasurableeffectsonthecos- J withonelargespatialdimensionandanegativecosmological mic microwave background radiation anisotropy ([2–4]; for 1 constant, it is interesting to consider the rate for a tunneling further discussions on transitions between vacua of different dimensionalitysee[5,6]). process that creates a two-dimensional anti de Sitter bubble. h] In this paper we focus on the vacua associated with two- Furthermore,sincethenewvacuaareAdS3×S1,AdS2×T2, -t dimensionalcompactificationsofthestandardmodelcoupled AdS2×S2,theAdS/CFTcorrespondencemayprovideanal- p togravity(ona2Dtorusanda2Dsphere). Theirexactgeom- ternativedescriptionofthestandardmodel[1]. e etryisdeterminedbycompetingcontributionstotheeffective h lower-dimensional potential. Classical contributions come [ fromthefour-dimensionalcosmologicalconstantandthecur- II. COMPACTIFICATIONONA2DTORUS 2 vature term resulting from dimensional reduction, whereas v quantum contributions involve the Casimir energies of stan- Here we demonstrate the existence of lower-dimensional 2 dardmodelparticles. Thebasicformalismusedinthispaper vacuaofthestandardmodelcoupledtogravityinthecaseof 0 3 was developed in [1] and this work extends [1] by explicitly a 2D compactification on a torus. We work in N spacetime 4 findingnewvacuafortoroidalandsphericalcompactifications dimensionsandconsiderthefollowingspacetimeinterval, . ofthestandardmodel.Ourgoalinthispaperistoshowbyex- 0 1 plicit computation that new standard model vacua arise with ds2 =gµν(x)dxµdxν +tij(x)dyidyj, (1) 0 twocompactifieddimensions. 1 For the compactification on a 2D torus the only classi- wheregµν isthemetric(gravitationalconvention)onthenon- v: cal contribution to the effective potential is given by the 4D compactspacetimewithµ,ν = 0,1,...,N −3andtij isthe i cosmological constant term from the 4D action. As such, metric on the compact space with i,j = 1,2. The compact X the Casimir energies of standard model fields contribute sig- coordinatesareyi ∈[0,2π). r nificantly to the shape of the effective potential. We find Weparametrizethe2Dtoruswiththemetric a thatcompetitionsbetweenthesecontributionsproduceanew AdS ×T2 standardmodelvacuumforalargerangeofneu- a2 (cid:18) 1 τ (cid:19) 2 t = 1 , (2) trino masses and for a particular geometry of the toroidal ij τ2 τ1 |τ|2 space. For a discussion of the role of Casimir energies in higher-dimensional toroidal compactifications see for exam- where τ = τ +iτ and a2 are the shape and volume mod- 1 2 ple[7–13]. uli,respectively,allbeingfunctionsofthenoncompactcoor- The effective potential for the compactification on a 2D dinatesxµ. TheN-dimensionalEinstein-Hilbertactionis sphere, on the other hand, contains an extra classical term (cid:90) unique to compactifications on curved spaces. We find sta- S = dN−2xd2y(cid:112)−g t (N−2) bleAdS ×S2vacuawhenthemagneticfluxthroughthe2D 2 (cid:20) (cid:21) sphereislargeenough. × 1MN−2R −(cid:0)ρ +Λ (cid:1) , (3) The work of this paper is not speculative. We determine 2 (N) (N) (N) (N) somevacuaofthestandardmodelwithtwocompactifieddi- mensionsusingonlylongdistancephysics.Understandingthe where g = det(g ), t = det(t ) and M , R , (N−2) µν ij (N) (N) vacuum structure of the standard model is worthwhile even ρ ,Λ aretheN-dimensionalPlanckmass,Ricciscalar, (N) (N) 2 Casimir energy density, and cosmological constant, respec- Casimir energy density for a scalar of mass m in an N- tively. Wenotethat dimensionalspacetimewithN −2dimensionsflatand2di- mensionscompactifiedonatorus, assumingperiodicbound- Λ(N) = Λo(Nbs)−Λq(N.co)rr., (4) aryconditions,is1 wherethesuperscript“obs”indicatestheobservablevalueand ρ (a,τ ,τ ,m)= (N) 1 2 “q.corr.” indicatesthequantumcorrectioninflatspace. AfterperformingareductionoftheactionfromN to(N− 1 1 (cid:88)∞ (cid:90) dN−3k (cid:113) k2+tijn n +m2, (11) 2)dimensionsusingtheexplicitformofthe2Dtorusmetric, (2πa)22 (2π)N−3 i j wearriveat n1,n2=−∞ (cid:90) (cid:20) 1 (cid:18) wheretij istheinverseoft giveninequation(2). Equation S = dN−2x(cid:112)−g (2πa)2 MN−2 R ij (N−2) 2 (N) (N−2) (11)yields (∂ τ )2+(∂ τ )2 (∂ a)2(cid:19) (cid:21) − µ 1 µ 2 +2 µ −V(a,τ) , (5) ρ (a,τ ,τ ,m)= 2τ2 a2 (N) 1 2 twiohneareryVp(oain,τts)w=e(p2eπ2rfao)r2m(cid:0)ρa(Nco)n+foΛrm(Nal)(cid:1)t.raInnsofordrmerattoiofinnodfstthae- −(2π1a)24√Γπ(cid:0)−(4Nπ2−)N22−(cid:1)3 n1,n(cid:88)2∞=−∞(cid:0)tijninj +m2(cid:1)N2−2 .(12) metrictotheEinsteinframe. Thedesiredtransformationis The sum in the above equation is the extended Chowla- (cid:34)(2πa)2MN−2(cid:35)−N2−4 Selbergzetafunctionandcanbewrittenas(see,appendixA) g → (N) g(E). (6) µν MN−4 µν (N−2) (cid:88)∞ (cid:0)tijn n +m2(cid:1)N2−2 i j Theactionnowtakestheform n1,n2=−∞ (cid:20) S =(cid:90) dN−2x(cid:113)−g(E) (cid:34)1MN−4 (cid:18)R − =mN−2+ aN1−2 2τ˜22−2NζEH(cid:0)2−2N,τ˜2a2m2(cid:1) (N−2) 2 (N−2) (N−2) 2(NN−−42)(∂µa2a)2 − (∂µτ1)22+τ2(∂µτ2)2(cid:19)−Veff(a,τ)(cid:35), (7) +2√π ΓΓ(cid:0)(cid:0)√12−−22NN(cid:1)(cid:1)τ˜2N2ζEH(cid:16)1−2N,a2τ˜m22(cid:17) where 2 +8Γπ(cid:0)2−22N−N(cid:1)τ˜2 (cid:88)∞ p1−2N (cid:16)n2+ a2τ˜m22(cid:17)N4−1 2 n,p=1 (cid:18) (cid:113) (cid:19)(cid:21) V (a,τ)=(cid:34)(2πa)2M(NN−)2(cid:35)−(NN−−42)V(a,τ). (8) ×cos(2πτ˜1np)KN2−1 2πτ˜2p n2+ a2τ˜m22 , (13) eff MN−4 (N−2) whereτ˜ = τ /|τ|2 andζ (s,q)istheEpstein-Hurwitzzeta i i EH Note that the conformal transformation does not exist for functionexpressedby N = 4,andwehavetoworkin(4+(cid:15))dimensions. Wealso assume (cid:15) > 0; otherwise, the stationary point of action (7) √ πΓ(s− 1) wfluocutludant’iotnbseosftatbhleevwoiltuhmreesmpeocdtutloussma2a.llTshpeacceotnimdietiodnespefonrdtehnet ζEH(s,q)=−12q−s+ 2 Γ(s)2 q−s+12 emxiinsitmeniczeinogftahsetaebffleecsttivateiopnoatreyntpiaolinatncdatnakbienegatshileyloimbtiati(cid:15)ne→db0y. +Γ2(πss)q1−42s (cid:88)∞ ns−21Ks−21(2πn√q). (14) Thoseconditionsare n=1 V(a,τ)=0, ∂ V(a,τ)=0, Notethat τ1,2 ∂ V(a,τ)<0, ∂2 V(a,τ)>0. (9) a τ1,2 Λq.corr. =−1(4π)−N2Γ(cid:0)−N(cid:1)mN, (15) (N) 2 2 Theirderivationisgivenin[1]. Theequationsofmotionfor theaction(5)yield so that as N → 4 the potential V(a,τ) can be expressed in 4π2aM(NN−)2R(N−2) =∂aV(a,τ) (10) termsoftheobservedcosmologicalconstantΛo(4b)sandafinite quantumcorrectionρobs, (4) atthestationarypoint.Thismeansthatminimaoftheeffective potentialcorrespondto(two-dimensional)antideSittervacua. In the next step we compute quantum contributions to the effective potential using dimensional regularization. The 1Weneglecttheverysmallcurvatureofthe(N−2)-dimensionalspacetime. 3 Although the neutrino masses have not been determined, we canuseexperimentalmasssplittingsfortheatmosphericand ρobs(a,τ ,τ ,m) (4) 1 2 solarneutrinostogeneratethespectrumgiventhelightestneu- 1 (cid:34)2(am)3/2 (cid:88)∞ 1 (cid:16) (cid:112) (cid:17) trinomassandachoiceofhierarchy. Thisallowsustoinves- =− K 2πpam τ˜ tigatethe potentialfor variousvalues ofthe lightestneutrino (2πa)4 τ˜21/4 p=1p3/2 3/2 2 mass. Experimentally,∆m2atm =(2.43±0.13)×10−3eV2, ∞ ∆m2 =(7.59±0.20)×10−5eV2[16].Weadopttheobser- (cid:88) 1 (cid:16) (cid:17) sol +2τ˜2(am)2 p2K2 2π√pτã2m vationalvalueofΛo(4b)s (cid:39)3.1×10−47GeV4[16]andnumeri- p=1 callysolvethethreeequationsin(9)forawiththeformulafor +4(cid:112)τ˜2 (cid:88)∞ p31/2 (cid:16)n2+ (aτ˜m2)2(cid:17)3/4 Viti(eas,iτn)(9g)ivteonhboylde.qWuaetidoenn(o1t9e)thaendsorleuqtuioirnebthyeat0h.reWeeinneaqruroawl- n,p=1 our analysis to the fundamental region τ1 ∈ (−1/2,1/2], (cid:18) (cid:113) (cid:19)(cid:35) |τ| ≥ 1 and τ2 > 0 [10, 17], as this is the moduli space of ×cos(2πτ˜ pn)K 2πpτ˜ n2+ (am)2 . (16) physicallydistinctvacua. 1 3/2 2 τ˜2 Itturnsoutthatevenbeforeperformingthenumericalanal- ysiswecanpreciselydeterminethevaluesofτ forwhichthe Afterassuming|τ| = 1equation(16)coincideswiththecor- potential (19) has its extrema. Note that the Casimir energy responding formula given in [12]. In the massless case (16) density(11)isinvariantunderSL(2,Z)modulartransforma- reducesto tions. In particular, the two generators of the modular group (cid:34) 1 ζ(3) π2τ˜2 are T : τ → τ + 1, corresponding to a change of indices ρobs(a,τ ,τ ,0)=− + 2 + (4) 1 2 (2πa)4 2πτ˜2 90 (n1,n2) → (n1,n2−n1),andS : τ → −1/τ,beingequiv- alent to (n ,n ) → (−n ,n ). The same symmetries are (cid:112) (cid:88)∞ (cid:18)n(cid:19)3/2 (cid:35) exhibited b1y th2e potential2(191) since it is a linear combina- 4 τ˜ cos(2πnpτ˜ )K (2πnpτ˜ ) . (17) 2 p 1 3/2 2 tion of the Casimir energies of the particles. It has been ar- n,p=1 guedthatthe√fixedpointsτ =i(forthetransformationS)and Explicitly,thepotentialV (a,τ,m)forasinglescalarcanbe τ = 1/2+i 3/2(forTS)correspondtoextremaofthepo- 1 writtenas tential[10,13,17]. Anumericalanalysisshowsthatonlythe secondonecorrespondstoaminimumofthepotential. Thus, (cid:104) (cid:105) V1(a,τ,m)=(2πa)2 ρo(4b)s(a,τ1,τ2,m)+Λo(4b)s . (18) allnewtwo-dimensional√standardmodelvacuaarecharacter- izedby (τ ,τ )=(1/2, 3/2). 1 2 Forthestandardmodelwesimplymultiplyρobs(a,τ ,τ ,m) (4) 1 2 by the appropriate factor reflecting the number of degrees of (a) For a normal hierarchy the neutrino masses are freedominfourdimensionsandaddaminussignforfermions (m2 ,m2 ,m2 ) = (m2 ,m2 + ∆m2 ,m2 + (i.e.,2forthephoton,2forthegraviton,−4foraDiracneu- ν1 ν2 ν3 ν1 ν1 sol ν1 ∆m2 + ∆m2 ). There are no two-dimensional trino, −2 for a Majorana neutrino) [1, 13–15] and sum over atm sol vacua for m (cid:46) 4.42×10−12 GeV. For all masses degreesoffreedom. ν1 m (cid:38) 4.42 × 10−12 GeV we find precisely one Note that for m (cid:29) 1/a we obtain from equation (16) the ν1 stable vacuum. Table I shows the values of an.h. relation ρobs ∼ exp(−2πam) (up to a factor of τ˜±1/2 in 0 the expon(e4n)t). We restrict our attention to the leng2thscale fGoerVs−ev1e(cid:0)ra1l01m0aGsseeVs−m1ν(cid:39)1.2Tµmhe(cid:1).raTdhiieparloeteoxfpthreesspeodtenin- √ a(cid:29)1/m sothattheCasimirenergiesoftheelectronandall e tialV(a,τ)for τ = 1/2+i 3/2 andseveralmasses heavier standard model particles are negligible compared to fromtableIisgiveninfigure1(a). thecontributionsofthephoton,graviton,andtheneutrinos.2 (b) For an inverted hierarchy the neutrino masses are A. Diracneutrinos (m2 ,m2 ,m2 ) = (m2 +∆m2 −∆m2 ,m2 + ν1 ν2 ν3 ν3 atm sol ν3 ∆m2 ,m2 ). Thistimetherearenotwo-dimensional atm ν3 ForthestandardmodelwithDiracneutrinosthepotentialis vacua for m (cid:46) 1.12×10−12 GeV. For all masses ν3 m (cid:38) 1.12×10−12 GeVweagainfindpreciselyone (cid:20) ν3 V(a,τ)=(2πa)2 4ρo(4b)s(a,τ1,τ2,0) stable vacuum. Values of ai0.h. for a few masses mν3 can be found in table I. The plot of V(a) for some of 3 (cid:21) themassesfromtableIisgiveninfigure1(b). (cid:88) −4 ρobs(a,τ ,τ ,m )+Λobs . (19) (4) 1 2 νi (4) i=1 2Itwillbecomeclearfromfigures1and2thatourresultsholdforthefull rangeofawherethestandardmodelisvalid. 4 mlightν [GeV] an0.h.[GeV−1] ai0.h.[GeV−1] (τ1,τ2) mlightν [GeV] an0.h.[GeV−1] ai0.h.[GeV−1] (τ1,τ2) √ √ 1.12×10−12 – 2.54×1010 (1/2, 3/2) 0 1.06×1010 5.17×109 (1/2, 3/2) √ √ 4.42×10−12 4.76×1010 1.55×1010 (1/2, 3/2) 5.0×10−12 9.29×109 5.04×109 (1/2, 3/2) √ √ 1.0×10−11 2.23×1010 1.21×1010 (1/2, 3/2) 1.0×10−11 7.69×109 4.79×109 (1/2, 3/2) √ √ 2.0×10−11 1.29×1010 9.28×109 (1/2, 3/2) 2.0×10−11 5.64×109 4.19×109 (1/2, 3/2) √ √ 3.0×10−11 9.46×109 7.66×109 (1/2, 3/2) 1.0×10−10 1.67×109 1.62×109 (1/2, 3/2) √ √ 1.0×10−10 3.39×109 3.28×109 (1/2, 3/2) 1.0×10−9 1.74×108 1.74×108 (1/2, 3/2) √ 1.0×10−9 3.52×108 3.52×108 (1/2, 3/2) TABLE II: Values of an0.h.,ai0.h.,τ1,τ2 for two-dimensional standard TvaAcuBaLfEorIs:eVvaelruaelsliogfhaten0s.th.D,iari0a.ch.n,eτu1t,riτn2ofomratswseos-dmimliegnhstioνna(mlstνa1ndfoarrdnmoromdeall mfoordneolrmvaacluhaiefroarrcsheyv,emralν3ligfhotreisntvMertaejdorhainearanrcehuytr)i.nomassesmlightν (mν1 hierarchy,mν3forinvertedhierarchy).Notethat1010GeV−1(cid:39)2µm. FIG. 1: (a) Plots of V(a) for Dirac neutrinos with normal hierarchy for FIG.2: (a)PlotsofV(a)forMajorananeutrinoswithnormalhierarchyfor m2.a0s×ses10m−ν111G=eV4.(4g2re×en)1,0a−nd123.G0e×V10(p−u1r1plGe)e,V1.0(bl×ue1);0(−b)11PloGtseVofV(re(da)), manνd12.=0×0(1p0u−rp1l1e),G5e.V0×(b1lu0e−);12(bG) PelVots(reodf)V,1(.a0)×fo1r0M−a1j1orGaneaVne(gurtreienno)s, f(oprurDpilrea)c,4n.e4u2tr×ino1s0w−i1th2iGnveeVrte(drehdi)e,r1ar.c0h×yf1o0r−m1ν13G=eV1.(1g2re×en1),0a−n1d22G.0e×V 1w.i0th×in1v0e−rte1d1GhieerVar(cghryeefno)r,amndν32.=0×01(0p−ur1p1leG),e5V.0(b×lue1)0.−12 GeV (red), 10−11GeV(blue). B. Majorananeutrinos Inordertofindnewstandardmodelvacuaweperformanalo- InthecaseofthestandardmodelwithMajorananeutrinos gousstepsasinthecaseofDiracneutrinos. Wefindthatfor thepotentialtakestheform Majorananeutrinostherealwaysexistspreciselyonestandard (cid:20) modelvacuum(thistimeforanyvalueofm )character- V(a,τ)=(2πa)2 4ρobs(a,τ ,τ ,0) √ lightν (4) 1 2 ized also by (τ ,τ ) = (1/2, 3/2). Table II presents the 1 2 valuesofa forafewlightestneutrinomassesforthenormal 3 (cid:21) 0 −2 (cid:88)ρobs(a,τ ,τ ,m )+Λobs . (20) andinvertedhierarchies. Figure2showstheplotofV(a)for (4) 1 2 νi (4) someofthosemasses. i=1 5 III. COMPACTIFICATIONONA2DSPHERE terms) for R (cid:29) 1/(Λobs)1/4 (cid:39) 4 × 1011GeV−1. In the (4) regionofRwhereclassicaleffectsdominate,thepotentialis Now we attempt to find the lower-dimensional standard (cid:104) (cid:105) model vacua for a 2D compactification on a sphere. The V(R)=4π −M2 +R2Λobs . (28) (4) (4) spacetimeintervalweconsideris Thefirstconditionin(27)yieldstheclassicalsolution ds2 =g (x)dxµdxν +R2(x)(cid:0)dθ2+sin2θdφ2(cid:1) , (21) µν M whereθ ∈ [0,π], φ ∈ [0,2π). TheN-dimensionalEinstein- R0 = (cid:113) (4) (cid:39)4.3×1026m, (29) Hilbert action is again given by equation (3) and, after the Λobs (4) reduction to a (N − 2)-dimensional spacetime, it takes the form whichisapproximatelythecurrentHubbleradius,butthisso- lutionisunstablesincethesecondconditionisnotfulfilled. (cid:34) S =(cid:90) dN−2x(cid:112)−g (4πR2)1MN−2 In the region 1/M(4) (cid:28) R (cid:46) 1/(Λo(4b)s)1/4 the squared (N−2) 2 (N) Planck mass still dominates the potential (the fact that quan- (cid:18) (∂ R)2(cid:19) (cid:35) tumcorrectionsbecomelargerthanthecosmologicalconstant × R +2 µ −V(R) , (22) termisirrelevant)andtherearenostablevacuasincetheequa- (N−2) R2 tionV(R)=0hasnosolutions. However, it is possible to find stable vacua in the pres- where ence of a magnetic flux. This introduces an additional term (cid:34) MN−2 (cid:35) in the potential (28) proportional to N2/(eR)2 (where N is V(R)=(4πR2) − (N) +ρ +Λ . (23) thenumberofmagneticfluxquanta,eistheelectroncharge) R2 (N) (N) andyieldsexactlyonestablesolution In the case of the sphere the potential includes not only the  (cid:113) 1/2 M2 − M4 − N2Λobs Casimir energy density and the cosmological constant term, (4) (4) 2e2 (4) R=  , (30) but also the curvature of the sphere (in the case of the torus 2Λobs (4) thisadditionaltermisobviouslyzero). Weagainperformaconformaltransformationofthemetric √ valid for N < 2eM2 /(Λobs)1/2 and the region of R andwritedowntheactioninthe(N−2)-dimensionalEinstein (4) (4) where quantum corrections are negligible compared to both frame. Thedesiredtransformationis classical terms. The existence of a stable vacuum for R (cid:29) (cid:34)(4πR2)MN−2(cid:35)−N2−4 4×1011GeV−1requires1030 (cid:46)N (cid:46)1061,wheretheupper gµν → MN−4(N) gµ(Eν), (24) boundcorrespondstoR(cid:46)M(4)/(2Λo(4b)s)1/2 (cid:39)1042GeV−1. (N−2) Taking into account also the region of R where just the squared Planck mass dominates over the quantum correc- andtheactiontakestheform tions, stable vacua exist for the whole range 1017 (cid:46) N (cid:46) (cid:90) (cid:113) (cid:34)1 (cid:18) 1061, where the lower bound is set by the experimentally S = dN−2x −g(E) MN−4 R tested region of validity of the standard model, that is R (cid:38) (N−2) 2 (N−2) (2+(cid:15)) 1/(100GeV). 2(N −2)(∂ R)2(cid:19) (cid:35) Wehavealsoinvestigated2Dcompactificationsonsurfaces − µ −V (R) , (25) ofgenusg > 1. Thecurvatureofsuchmanifoldsisnegative, N −4 R2 eff therefore the squared Planck mass term enters the potential withapositivesign. Intheregionwherequantumeffectsare where negligibletheequationforthevolumemoduluscorresponding (cid:34)(4πR2)MN−2(cid:35)−(NN−−24) to the first condition in (27) has no solutions. This indicates (N) thatthereareno2Dstandardmodelvacuaforsuchcompacti- V (R)= V(R). (26) eff MN−4 fications,evenafterintroducingmagneticflux. (N−2) Theconditionsforastablestationarypointareobtainedsimi- larlyasinthetoruscaseand,sincethistimethepotentialisa IV. CONCLUSIONS functiononlyofR,theyare Wehaveinvestigatedthevacuumstructureofthestandard V(R)=0, ∂ V(R)<0. (27) model coupled to gravity with two spatial dimensions com- R pactified (on a 2D torus and on a 2D sphere). We have Notethatthe4-dimensionalPlanckmassislarge(M(4) (cid:39) calculated the potential, which, apart from the cosmological 1.22×1019GeV) and the 4-dimensional cosmological con- constantterm,containsalsoCasimirenergiesofthestandard stantistiny(Λobs (cid:39)3.1×10−47 GeV4). Wefindthatquan- modelparticlesand,forthespherecase,includesacurvature (4) tum corrections are negligible (with respect to both classical term. 6 Forthe2Dtoruscompactificationwedidouranalysissep- Afterusing(A1)torewrite(A2)andexpressingitintermsof aratelyforthestandardmodelwithDiracandMajorananeu- thenewvariables w(cid:48) =−iπw and z(cid:48) =z/w wearriveat trinos. Foreachofthosecasesweperformedthecalculation ∞ assuminganormalhierarchy,aswellasaninvertedhierarchy (cid:88) e−(n+z(cid:48))2w(cid:48) oftheneutrinomasses. Wehaveshownthatforthestandard n=−∞ model with Majorana neutrinos there always exists precisely (cid:114) (cid:34) ∞ (cid:35) onenewtwo-dimensionalstandardmodelvacuum.Inthecase = π 1+2(cid:88)e−πw2n(cid:48)2 cos(2πnz(cid:48)) (A3) of Dirac neutrinos we have also found exactly one standard w(cid:48) n=1 model vacuum, but it exists only for a subset of the experi- mentallyallowedneutrinomasses. Theminimumvalueofthe with Re(w(cid:48)) > 0. Notice that the corresponding formula in lightestDiracneutrinomassforwhichthereisavacuumde- [19]ismissingafactorof2. pendsonthetypeofhierarchy. Thetoroidalshapeofthenew Wenowexpressthegeneralformofthedoublesumby vacuum for both Dirac an√d Majorana neutrinos is character- ∞ ized by (τ1,τ2) = (1/2, 3/2); its volume modulus a2 de- (cid:88) (cid:0)Ap2+Bpn+Cn2+Q(cid:1)−s pendsonthemassofthelightestneutrinoandwefindatobe n,p=−∞ ontheorderofmicrons. Inourworkweusedperiodicbound- ary conditions for the fermions. It is worthwhile to consider = 1 (cid:88)∞ (cid:90) ∞dtts−1e−(Ap2+Bpn+Cn2+Q)t. (A4) otherboundaryconditions. Γ(s) n,p=−∞ 0 Thecompactificationona2Dsphereisconsiderablydiffer- ent since the potential contains an additional curvature term We assume A, Q > 0 and 4AC −B2 > 0. Writing the resultingfromdimensionalreduction. Forlargeenoughradii quadraticformas the potential becomes classical. A straightforward classical Ap2+Bpn+Cn2 =A(cid:0)p+ B n(cid:1)2+ ∆n2, (A5) analysis shows that a new stable two-dimensional standard 2A 4A modelvacuumexistsonlyafterintroducingalargemagnetic where∆ = 4AC −B2,andusingrelation(A3)withrespect flux. Finally, we have argued that for 2D compactifications totheindexpforz(cid:48) =Bn/(2A)andw(cid:48) =At,wearriveat onsurfacesofgenusg > 1therearenonewstandardmodel vacua. ∞ (cid:88) (cid:0)Ap2+Bpn+Cn2+Q(cid:1)−s The calculations in this paper are not speculative physics. They contribute to determining the vacuum structure of the n,p=−∞ standardmodelanddependonlyonlongdistancephysics. = 1 (cid:88)∞ (cid:90) ∞dtts−1e−(4∆An2+Q)t Γ(s) n=−∞ 0 Acknowledgment ×(cid:113)Aπt(cid:34)1+2(cid:88)∞ e−πA2pt2 cos(cid:0)πnpBA(cid:1)(cid:35). (A6) p=1 The work of the authors was supported in part by the Then=0contributionto(A6)is U.S. Department of Energy under contract No. DE-FG02- 92ER40701. (cid:88)∞ (cid:16) (cid:17)−s 2A−s p2+ Q +Q−s. (A7) A p=1 Making use of the property of the modified Bessel functions AppendixA:GeneralizedChowla-Selbergformula ofthesecondkind Herewederive,followingthestepsoutlinedin[18,19],the (cid:90) ∞duus−1e−α2u−βu2 =2(cid:18)αβ(cid:19)sKs(2αβ), (A8) formula for the regularized double sum in equation (12), in 0 whichthedivergentpiecesoftheanalyticcontinuationoccur thecontributionofthesumovernon-zerongives as poles in gamma functions for particular values of N. We (cid:12) startbywritingdowntheformulafortheJacobithetafunction (cid:88)∞ (cid:0)Ap2+Bpn+Cn2+Q(cid:1)−s(cid:12)(cid:12) (cid:12) (cid:12) ∞ n,p=−∞ n(cid:54)=0 θ3(z,w)= (cid:88) eiπ(n2w+2nz), (A1) =2(cid:113)π Γ(s−12)(cid:0)∆(cid:1)12−s(cid:88)∞ (cid:16)n2+ 4AQ(cid:17)12−s n=−∞ A Γ(s) 4A ∆ n=1 where z, w are complex numbers and Im(w) > 0. We note + 8πs A−2s−14 (cid:0)∆(cid:1)−2s+14 (cid:88)∞ ps−21 (cid:16)n2+ 4AQ(cid:17)14−2s thatthefollowingJacobiidentityholds[20] Γ(s) 4A ∆ n,p=1 (cid:18) √ (cid:113) (cid:19) θ3(cid:0)wz,−w1(cid:1)=√−iweiπzw2θ3(z,w). (A2) ×cos(cid:0)πnpBA(cid:1)Ks−21 πpA∆ n2+ 4A∆Q . (A9) 7 We point out that the corresponding formula in [19] has an wheretheEpstein-Hurwitzzetafunction errorinthedefinitionofbandiscorrectonlyaftersubstituting b→2b. Wefinallyarriveat ζ (s,q)= (cid:88)∞ (cid:0)n2+q(cid:1)−s (A11) EH (cid:88)∞ (cid:0)Ap2+Bpn+Cn2+Q(cid:1)−s n=1 can also be regularized through the above procedure and is n,p=−∞ (cid:16) (cid:17) given by formula (14). From the form of the metric (2) and = Q−s+2A−sζ s,Q EH A formula(12)weimmediatelyidentify √ 22s πAs−1Γ(s− 1) (cid:16) (cid:17) + 2 ζ s− 1,4AQ |τ|2 2τ 1 ∆s−12 Γ(s) EH 2 ∆ A= a2τ , B =−a2τ1 , C = a2τ , +Γπ(ss)2s+√52∆A14−2s (cid:88)∞ ps−12 (cid:16)n2+ 4A∆Q(cid:17)14−2s Q2 =m2, s=22−2N . 2 (A12) n,p=1 (cid:18) √ (cid:113) (cid:19) Aquickcheckshowsthat∆=4/a4 >0,thusformula(A10) ×cos(cid:0)πnpB(cid:1)K πp ∆ n2+ 4AQ , (A10) applies. Substitutionof(A12)into(A10)yieldsrelation(13). A s−1 A ∆ 2 [1] N.Arkani-Hamed, S.Dubovsky, A.NicolisandG.Villadoro, [10] W.Buchmuller,R.CatenaandK.Schmidt-Hoberg,Enhanced Quantum horizons of the standard model landscape, JHEP symmetriesoforbifoldsfrommodulistabilization,Nucl.Phys. 0706,078(2007)[arXiv:hep-th/0703067]. B821,1(2009)[arXiv:0902.4512[hep-th]]. [2] P.W.Graham,R.HarnikandS.Rajendran,Observingthedi- [11] B.R.GreeneandJ.Levin,Darkenergyandstabilizationofex- mensionality of our parent vacuum, Phys. Rev. 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