We introduce a new mechanism for producing locally stable de-Sitter or
Minkowski vacua, with spontaneously broken N=1 supersymmetry and no massless
scalars, applicable to superstring and M-theory compactifications with fluxes.
We illustrate the mechanism with a simple N=1 supergravity model that provides
parametric control on the sign and the size of the vacuum energy. The crucial
ingredient is a gauged U(1) that involves both an axionic shift and an
R-symmetry, and severely constrains the F- and D-term contributions to the
potential.Comment: 4 pages, 1 figure, v3: published versio

Fabio Zwirner

Giovanni Villadoro

J. Wess

S. J. Gates

English

arXiv

We introduce a new mechanism for producing locally stable de-Sitter or Minkowski vacua, with spontaneously broken N=1 supersymmetry and no massless scalars, applicable to superstring and M-theory compactifications with fluxes. We illustrate the mechanism with a simple N=1 supergravity model that provides parametric control on the sign and the size of the vacuum energy. The crucial ingredient is a gauged U(1) that involves both an axionic shift and an R-symmetry, and severely constrains the F- and D-term contributions to the potential

VILLADORO, G.

ZWIRNER, FABIO

Archivio istituzionale della ricerca - Università di Padova

PRL 95, 231602 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 DECEMBER 2005de Sitter Vacua via ConsistentD TermsGiovanni Villadoro and Fabio ZwirnerDipartimento di Fisica, Università di Roma ‘‘La Sapienza’’ and INFN, Sezione di Roma, P.le Aldo Moro 2, I-00185 Rome, Italy(Received 26 August 2005; published 2 December 2005)0031-9007=We introduce a new mechanism for producing locally stable de Sitter or Minkowski vacua, withspontaneously broken N  1 supersymmetry and no massless scalars, applicable to superstring and M-theory compactifications with fluxes. We illustrate the mechanism with a simple N  1 supergravitymodel that provides parametric control on the sign and the size of the vacuum energy. The crucialingredient is a gauged U1 that involves both an axionic shift and an R symmetry, and severely constrainsthe F- and D-term contributions to the potential.DOI: 10.1103/PhysRevLett.95.231602 PACS numbers: 11.25.Mj, 04.65.+e, 11.25.WxIntroduction.—Superstring and M-theory vacua withexact or spontaneously broken N  1 supersymmetry de-serve special attention. Their effective D  4 supergrav-ities admit chiral fermions and inherit some strong sym-metry properties from the underlying higher-dimensionaltheory. So far, perturbative compactifications with fluxesand branes could produce, at best, either Minkowski vacuaof the no-scale type, with broken supersymmetry and atleast one complex flat direction, or anti–de Sitter (AdS)vacua with all geometrical moduli stabilized.It is important to go further, exploring the possibleexistence of locally stable de Sitter (dS) or Minkowskivacua, with no residual flat directions. Some interestingattempts along these lines do indeed exist [1–4]. Most ofthem rely on the positive contributions to the potentialassociated with the gauge symmetry of the theory, the so-called D terms. However, the subtle consistency require-ments dictated by the coexistence of the two local symme-tries, supersymmetry and the gauge symmetry, are knownon general grounds [5,6], but were never thoroughly exam-ined in this context.Reference [1] used a superpotential motivated by non-perturbative effects to produce a supersymmetric AdSvacuum, then uplifted the vacuum energy by a positivecontribution to the potential ascribed to D3 branes. So far,however, no consistent supergravity description of such amechanism was found. Reference [2] proposed to replacethe D3-brane contribution with a D-term contribution or-iginated by magnetic fluxes but, as will be clear in thefollowing, such a proposal does not fulfill the above-mentioned consistency requirements. Reference [3] intro-duced nonperturbative superpotentials and D terms thatsatisfy all known consistency conditions, but did not per-form a full minimization of the supergravity potential withrespect to all fields.In the present Letter, we first review the general consis-tency conditions associated with D terms in N  1 super-gravity. We then construct a simple explicit model ful-filling all such conditions. We show that, for a wide rangeof parameters, the model admits locally stable vacua with05=95(23)=231602(4)$23.00 23160spontaneously broken supersymmetry and positive ornegative vacuum energy. The vacuum energy can bezero, or very small and positive, for special values of theparameters. The key features of the model can be present influx compactifications of superstring theories. The crucialone is a gauged U1 symmetry that combines an R sym-metry with an axionic shift, and severely constrains theform of the F- and D-term contributions to the potential.The rigid version of such a symmetry, and some of itsconsequences if unbroken, were previously studied in [7].Supergravity models with gauged R symmetry [8] wereconsidered in [6,9]. The models of [3] have instead agauged axionic symmetry but no gauged R symmetry.We finally comment on the extension of our mechanismto more general models and on its string/M-theory embed-ding, leaving a detailed exploration for future work.D terms in N  1 supergravity.—The gauge-invarianttwo-derivative action for N  1, D  4 supergravity withchiral multiplets i  zi;  i and vector multiplets Va a; Aa is completely fixed by three ingredients [5]. Thefirst is the real and gauge-invariant Kähler function G,which can be written in terms of a real Kähler potentialK and a holomorphic superpotential W asG  K  logjWj2: (1)The second is the holomorphic gauge kinetic function fab,which transforms as a symmetric product of adjoint repre-sentations, plus a possible imaginary shift associated withanomaly cancellation. Generalized Chern-Simons termsmay also be needed [10], but they will not play any rolein the simple case discussed in this Letter. The third are theholomorphic Killing vectors Xa  Xiaz@=@zi, whichgenerate the analytic isometries of the Kähler manifoldfor the scalar fields that are gauged by the vector fields. Inthe following it will suffice to think of G, fab, and Xa asfunctions of the complex scalars zi rather than the super-fields i.The gauge transformation laws and covariant derivativesfor the scalars in the chiral multiplets read2-1 © 2005 The American Physical SocietyPRL 95, 231602 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 DECEMBER 2005zi  Xiaa; Dzi  @zi  AaXia; (2)where a are real parameters. The scalar potential isV  VF  VD  eGGiGi  3 12DaDa; (3)whereGi  @G=@zi, scalar field indices are raised with theinverse Kähler metric Gi k, gauge indices are raised withRef1ab, and Da are the Killing potentials, real solu-tions of the complex Killing equations:Xia  iGik @Da@z k: (4)The general solution to the Killing equation for Da, com-patible with gauge invariance, is thenDa  iGiXia  iKiXia  iWiWXia: (5)Gauge invariance of G requires that K and W be invari-ant up to a Kähler transformationK0  K H  H; W0  WeH; (6)where H is a holomorphic function, thus it will not berestrictive to assume that K is gauge invariant. If W is alsogauge invariant, Eq. (5) reduces to the standard formDa  iKiXia: (7)Otherwise, it must beiWiWXia  a; a 2 R; (8)so that the gauge noninvariance of W can be at most anoverall phase with real parameter a, for the Abelianfactors U1a. The constants a correspond to gaugingsof the R symmetry, and give rise to the supergravityexpression for the D terms [5,6]:Da  iKiXia  a: (9)The a are then the genuine Fayet-Iliopoulos (FI) terms ofsupergravity. For a linearly realized gauge symmetry,iKiXia  KiTaikzk, and we recover the standard expres-sion of [11] for the D terms. For an axionic realization,Xia  iqia, where qia is a real constant, and we obtain the so-called field-dependent FI terms.A few comments are now in order. Equation (5) showsthat D terms are actually proportional to F terms, Fi eG=2Gi. Two facts, frequently forgotten in the recent lit-erature, then become obvious. First, and in contrast withthe rigid case, there cannot be pure D breaking of super-gravity, unless the gravitino mass vanishes and the D-termcontribution to the vacuum energy is uncanceled, as inthe limit of global supersymmetry. Second, if VF admitsa supersymmetric AdS vacuum configuration, hGii  0(8i) and heGi  0, such configuration automatically mini-mizes VD at zero. For this kind of vacua, as already stressed23160in [12], D terms cannot be used to raise the vacuum energyfrom negative to positive or zero. Moreover, for the theoryto be consistent, W must be gauge invariant, up to anoverall phase for U1 factors. This severely restricts thepossibility of constructing superstring-inspired supergrav-ity models with both nonperturbative superpotentials andD terms.D terms associated with a gauged U1 symmetrycannot coexist with ‘‘racetrack’’ (sums of exponentials) orother (e.g., constant plus exponential) superpotentials,when the latter break such a symmetry. This is in agree-ment with some recent results in superstring compactifica-tions, where it was shown that the gauged isometries areprotected from being broken, both by instanton-induced[13] and by flux-induced [14] superpotentials. On the otherhand, a rigid axionic and/or R symmetry, possibly theremnant of a gauged symmetry spontaneously broken atthe string scale, can be explicitly broken to a discretesubgroup by nonperturbative effects.A model with stable de Sitter vacua.—Consider a modelwith a single chiral multiplet S  S;  and Kähler poten-tialK  p logS S  K0; 0<p 2 R; (10)with K0 real and S independent. This form of K is familiarfrom superstring compactifications. Decomposing thecomplex scalar as S  s i, s can stand here for thestring dilaton, a geometrical (Kähler or complex structure)modulus of the compactification manifold, or a combina-tion thereof;  could instead carry the degree of freedom ofsome component of the NS-NS or R-R potentials, or evenof the internal metric. The U1 isometry acting as a shifton the ‘‘axion’’  can be gauged by a vector multiplet. Thecorresponding holomorphic Killing vector is just an imagi-nary constant,XS  iq; q 2 R: (11)The most general form of the superpotential compatiblewith the gauged U1 symmetry is thenW  W0ekS; k 2 R; (12)where W0 is S independent. Equation (12) has the typicalform of the nonperturbative superpotentials induced byinstantons or gaugino condensation [15]. Notice that thegaugedU1 is a combination of the axionicU1, acting asa shift on  and leaving all the other fields invariant, andthe U1 R symmetry, with charge =2 for , =2for the gravitino   and the gaugino , and zero for all thebosonic fields.   kq is the constant FI term. For thegauge kinetic function, we takef  S; (13)as typical of superstring compactifications, but the modelwould still work for f  aS b, the most general formcompatible with the gauged symmetry. Notice, finally, thatthe gauge kinetic function f of the possible gauge group2-2PRL 95, 231602 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 DECEMBER 2005factor associated with instantons or gaugino condensationdoes not need to coincide with the U1 gauge kineticfunction of Eq. (13): this may be of help in obtaining vacuaat weak coupling.The scalar potential of Eq. (3) then readsVF eG0e2ks2sp2s2pkp2s2 3; (14)VD q22skp2s2; (15)where eG0 	 jW0j2eK0 . As required by gauge invariance, Vdoes not depend on: the axion is absorbed by the massiveU1 vector boson via the Higgs effect [16]. However, bothVF and VD depend nontrivially on s. For k < 0, there isalways a stable supersymmetric AdS vacuum at hsi p=2k, but there can be also metastable dS vacua forsuitable values of the parameters. For k > 0 and p  3, Vis positive definite and monotonically decreasing. For k >0 and p < 3, VF is unbounded from below for s! 0, butVD is positive definite and diverges for s! 0. As a result,for a wide range of parameters there is a locally stable dS(or stable AdS) minimum of V at a finite value hsi, withspontaneously broken supersymmetry. At this level, havingapproximate Minkowski vacua requires a tuning of theparameters so that hVDi ’ hVFi  hVi. In principle,this may find an explanation in the correlations of theunderlying string theory. Notice that hsi can be continu-ously varied by rescaling the parameters as:k! 	1k; eG0 ! 	peG0 ; q! 	3=2q; (16)(0<	 2 R), which leads to hsi ! 	hsi. A representativeexample is shown in Fig. 1.Discussion.—The simple model discussed above can beeasily generalized. The inclusion of additional gauge mul-tiplets is straightforward (apart from anomaly cancellation,see below), thus we consider the inclusion of additionalchiral multiplets i, transforming linearly under the axi-onic U1, with charges qi. Since the R charge is fixed tobe vanishing for the zi and =2 for the  i, the corre-FIG. 1. V (solid line), VF (dashed line), and VD (dotted line)for p  1, q  0:3, eG0  1=64, and k  0:1.23160sponding charges under the gauged U1 will be qi andqi  =2, respectively.Consider first the simple case where the full K can bewritten as in (10), with K0 real gauge-invariant function ofthe zi. Assume also a factorized W as in (12), with W0 theanalytic gauge-invariant function of the zi, and f as in (13).We may think of the zi as other moduli of string/M-theorycompactifications, orthogonal to S, or the scalar fields ofthe minimal supersymmetric standard model (MSSM). It iseasy to see that, under mild and plausible assumptions, theminimization of the potential V with respect to the zi and Scan be fully decoupled. If there is a field configuration hziisuch that hGii  0 and qihzii  0 8i, then hsi, as deter-mined in the previous section, and hzii, extremize the fullpotential V with no mass mixing between the zi and s. Alocally stable minimum can be obtained for a wide range ofthe parameters in G0, with the same hVi as before. If,instead, there is a minimum of V such that hVDi  0,then we can decouple a massive N  1 vector multipletand discuss a simpler, but less interesting effective theory.The previous model can be further generalized by in-cluding additional chiral multiplets C	  C	;  	, whichcould stand for some or all the MSSM squarks and leptons,in the approximation of small field fluctuations abouthC	i  0, but relaxing the factorization of K and W. Forexample, we could add to K a K P	jC	j2S Sn	(n	 2 Z), and to W a W polynomial in the C	 andtransforming with the same phase as ekS, e.g., W d	C	CC with q	  q  q  kq. Also in thiscase, for suitable values of the new parameters, there is alocal minimum of the full potential V with hC	i  hG	i 08	, positive squared masses for all the new scalar fieldsC	, and all the remaining features as in the previous model.For a consistent effective theory, all gauge and gravita-tional anomalies associated with our gauged U1 mustvanish: in particular, the cubic (AU13 ), the gravitational(AU1) and the mixed-gauge anomaly (AU1G2 ) if thefull gauge group is U1  G. The fermionic contributionsto the cubic and gravitational anomalies are:TrQ3  323Xi;	qi;	 23; (17)TrQ  212Xi;	qi;	 2; (18)where the contributions from  and  cancel each otherand have been omitted. The remaining ones come from  (see [17]) and possible  i;	, respectively. These contribu-tions must cancel the Green-Schwarz (GS) contributions[18] coming from the variation of  and proportional to q.All the resulting conditions are model dependent, in par-ticular: all of them depend on the matter content; the GScontribution to AU1 depends on higher derivative terms(R2); AU1G2 depends also on the details of G. However,2-3PRL 95, 231602 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 DECEMBER 2005there are, in principle, strong combined constraints on thepossible matter content and on the parameters k and q.The mechanism discussed in this Letter should be rele-vant for the study of superstring and M-theory vacua, withthe anomaly constraints automatically satisfied and thepossibility of determining k and q. N  1 supergravitiesobtained from compactifications with fluxes genericallyallow for some shift symmetry to be gauged.In the heterotic theory, the shift symmetry of the uni-versal axion [19], dual to B, is gauged via the GSmechanism, and fluxes can be used to generate a super-potential W0 [20,21]ZX6H  idJ ^; (19)which can stabilize all geometrical moduli with vanishingF terms and positive masses. A modification of W (or,equivalently, of K) as in (12) would then stabilize also thedilaton S on a dS vacuum, breaking the local symmetriesonly spontaneously.Also in type-IIA compactifications with fluxes, thesuperpotential [21,22]ZX6GeiJ  iH  idJ ^; (20)can produce the stabilization of all bulk moduli withvanishing F terms, with the exception of at least onemassless axion [14,22], associated with a shift symme-try that can eventually be gauged. In this case the roleof S is played by a linear combination of the dilaton andthe complex structure moduli , and its dependencecannot be factorized from the other moduli anymore.Whether in this case an analogous modification of thesuperpotential would allow the lifting of the vacuum en-ergy and the stabilization of all moduli remains an openproblem.Finally, it would be interesting to understand better howthe needed superpotential modifications actually originatefrom string/M-theory, and what the corresponding con-straints on the various parameters are.We thank S. Ferrara for discussions and M. Porrati forcomments. This work was supported in part by theEuropean Programme ‘‘The Quest For Unification,’’Contract No. MRTN-CT-2004-503369.[1] S. Kachru, R. Kallosh, A. Linde, and S. P. Trivedi, Phys.Rev. D 68, 046005 (2003).23160[2] C. P. Burgess, R. Kallosh, and F. Quevedo, J. High EnergyPhys. 10 (2003) 056.[3] E. Dudas and S. K. Vempati, Nucl. Phys. B 727, 139(2005).[4] F. Saueressig, U. Theis, and S. Vandoren, hep-th/0506181.[5] J. Wess and J. Bagger, Supersymmetry and Supergravity(Princeton University, Princeton, NJ, 1992), 2nd ed.; S. J.Gates, M. T. Grisaru, M. Rocek, and W. Siegel,Superspace: Or One Thousand and One Lessons inSupersymmetry, Frontiers in Physics Vol. 58 (Benjamin/Cummings, New York, 1983).[6] P. Binetruy, G. Dvali, R. Kallosh, and A. Van Proeyen,Classical Quantum Gravity 21, 3137 (2004).[7] C. Kounnas and M. Porrati, Phys. Lett. B 191, 91(1987).[8] D. Z. Freedman, Phys. Rev. D 15, 1173 (1977).[9] E. Cremmer, S. Ferrara, L. Girardello, C. Kounnas, andA. Masiero, Phys. Lett. B 137, 62 (1984); A. H.Chamseddine and H. K. Dreiner, Nucl. Phys. B458, 65(1996); D. J. Castano, D. Z. Freedman, and C. Manuel,Nucl. Phys. B461, 50 (1996).[10] L. Andrianopoli, S. Ferrara, and M. A. Lledo, J. HighEnergy Phys. 04 (2004) 005.[11] E. Cremmer, S. Ferrara, L. Girardello, and A. VanProeyen, Nucl. Phys. B212, 413 (1983).[12] K. Choi, A. Falkowski, H. P. Nilles, and M. 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De-Sitter vacua via consistent D terms

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de Sitter Vacua via Consistent
                    
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de-Sitter vacua via consistent D-terms

de-Sitter vacua via consistent D-terms

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