Table Of ContentPublishedinJHEP
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√pτ˜a2m 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
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