Large hierarchies from approximate R symmetries
Rolf Kappl, Hans Peter Nilles, Saul Ramos-Sanchez, Michael Ratz, Kai Schmidt-Hoberg, Patrick K.S. Vaudrevange
aa r X i v : . [ h e p - t h ] D ec TUM-HEP-705/08; DESY 08-189; LMU-ASC 60/08
Large hierarchies from approximate R symmetries Rolf Kappl , Hans Peter Nilles , Sa´ul Ramos-S´anchez , Michael Ratz ,Kai Schmidt-Hoberg , Patrick K. S. Vaudrevange Physik Department T30, Technische Universit¨at M¨unchen,James-Franck-Strasse, 85748 Garching, Germany Bethe Center for Theoretical Physics and Physikalisches Institut der Universit¨at Bonn,Nussallee 12, 53115 Bonn, Germany Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22603 Hamburg, Germany Arnold Sommerfeld Center for Theoretical Physics,Ludwig-Maximilians-Universit¨at M¨unchen, 80333 M¨unchen, Germany
We show that hierarchically small vacuum expectation values of the superpotential in supersym-metric theories can be a consequence of an approximate R symmetry. We briefly discuss the roleof such small constants in moduli stabilization and understanding the huge hierarchy between thePlanck and electroweak scales. I. INTRODUCTION
One of the major puzzles in contemporary physics isthe existence of large hierarchies in nature, such as the ra-tio between the Planck and electroweak scales M P /m W ∼ . Some of the most promising explanations of suchhierarchies rely on dimensional transmutation. Here thedynamical scale Λ = M P e − a/g (with g and a denot-ing the gauge coupling and a constant, respectively) canbe naturally much smaller than the fundamental scale.However, if one is to embed this mechanism in a morefundamental framework, one often encounters the prob-lem that there has to be a hierarchically small quantityright from the start. Concretely, if one is to make useof the dynamical scale in string theory, one has first tofix the modulus that determines the coupling strength.This in turn often requires the introduction of a smallconstant. One faces then the well-known “chicken-or-eggproblem”.Motivated by results obtained in the framework ofstring theory model building, we present here a poten-tial resolution of the problem. We shall show that, ifthe superpotential in a supersymmetric theory exhibitsan approximate U(1) R symmetry, it generically acquiresa suppressed vacuum expectation value (VEV). Such ac-cidental U(1) R symmetries which get broken at higherorders are naturally present in string compactifications.They arise as remnants from exact, discrete R symme-tries. Such symmetries allow us to control the VEV ofthe (perturbative) superpotential and, in particular, toavoid deep anti-de Sitter vacua. We will discuss the roleof the resulting hierarchically small superpotential VEVsin the context of moduli stabilization in string theory, forgiving a plausible explanation of the huge hierarchy be-tween M P and m W , and for providing, in the context ofa class of promising string models [1], a solution to the µ problem of the minimal supersymmetric standard model(MSSM). II. SUPERSYMMETRIC MINKOWSKI VACUAAS A CONSEQUENCE OF A U(1) R SYMMETRY
Consider a superpotential of the form W = X c n ··· n M φ n · · · φ n M M . (1)Here and in the following we work in Planck units, i.e.we set M P = 1 unless stated differently. Assume that W has an exact R symmetry, under which W has R charge2, W → e α W , (2)and the fields transform as φ j → φ ′ j = e i r j α φ j (3)such that each monomial in (1) has total R charge 2.Let h φ i i denote a field configuration which solves the F -term equations, F i = ∂ W ∂φ i = 0 at φ j = h φ j i ∀ i, j . (4)Consider now an infinitesimal U(1) R transformation, W ( φ i ) → W ( φ ′ i ) = W ( φ i ) + X j ∂ W ∂φ j ∆ φ j . (5)At φ j = h φ j i the superpotential goes into itself, whichcan only be consistent with (2) if W = 0 at φ j = h φ j i .This proves that, if the F equations are satisfied, W van-ishes.A few comments are in order. First, this statementholds regardless of whether the configuration h φ i i pre-serves U(1) R or breaks it spontaneously. Second, in thecontext of supergravity, the statements above imply thatthe D i W vanish for φ i = h φ i i , i.e. also the supergrav-ity F terms vanish and one obtains a supersymmetricMinkowski vacuum. Third, our findings are related to anobservation by Nelson and Seiberg made in [2], where it isstated that, in order to have a theory without supersym-metric ground state, the superpotential has to exhibit acontinuous R symmetry. The statements do, however,not tell us whether or not a theory with a superpotentialexhibiting a continuous R symmetry has a supersymmet-ric ground state or not. Our findings and [2] imply that, ifthere is a continuous R symmetry, there are two options:1. there is a supersymmetric ground state with W = 0(with U(1) R spontaneously broken or unbroken);2. there is no supersymmetric ground state, and in theground state U(1) R is spontaneously broken [2].In this letter we focus on case 1. If the U(1) that actson the scalar components of the superfields gets spon-taneously broken at φ i = h φ i i (which is the case if, forinstance, all h φ i i are non-trivial), it follows then fromGoldstone’s theorem that there is a massless mode, theso-called R axion. III. SMALL CONSTANTS FROMAPPROXIMATE U(1) R SYMMETRIES
Let us now study what happens if the R symmetry is‘slightly’ broken, i.e. by higher order terms. We can writethe superpotential as W ( φ i ) = W ( φ i ) + X j W j ( φ i ) , (6)where W ( φ i ) consists of monomials up to order N − R symmetry while the W j ( φ i ) aremonomials of order ≥ N which break the R symme-try. This means that the superpotential transforms underU(1) R as W ( φ i ) → e α W ( φ i ) + X j e i α R j W j ( φ i ) ≃ W ( φ i ) + i α W ( φ i ) + X j R j W j ( φ i ) (7)with R j = 2, and W ( φ i ) → W ( e i α r i φ i ) ≃ W ( φ i ) + i α X j ∂ W ∂φ j r j φ j . (8)Combining these two expressions and assuming that the F -terms vanish in our vacuum, ∂ W ∂φ i = 0, we see that h W ( φ i ) i = − X j ( R j − h W j ( φ i ) i . (9) This means that in the case of an approximate U(1) R symmetry one obtains suppressed superpotential VEVs,written symbolically as h W i ∼ h φ i ≥ N . (10)In many situations there is a mild hierarchy betweenthe fundamental scale and a typical VEV, h φ i /M P < M P [3]. Accord-ing to the above discussion, the suppression of h W i getsthen enhanced by the N th power of this mild hierarchy,similarly to the Froggatt-Nielsen picture [4].Further, we have seen that there might be a Goldstonemode η . With explicit U(1) R breaking, it will genericallyreceive a mass, m η ∼ h φ i ≥ N − . (The “ −
2” comes fromthe second derivative.) In supergravity theories, h W i setsthe gravitino mass, m / ≃ h W i . (11)This leads then to the expectation that there is a modewhose (supersymmetric) mass scales like m / , m η ∼ m / h φ i . (12)Let us comment that, if one is to include supergravityeffects, W = 0 does not necessarily imply anti-de Sittersolutions (see e.g. the discussion in [5, section 4]). IV. EXPLICIT STRING THEORYREALIZATION
One of the central themes of string theory is the is-sue of moduli stabilization, which is closely connectedto the question of supersymmetry breaking. In the tra-ditional approach, supersymmetry is broken by dimen-sional transmutation [6], e.g. by gaugino condensation [7].However, for this elegant mechanism to work, one needsfirst to fix the gauge coupling, whose strength is given bythe VEV of the dilaton S or another modulus in stringtheory. This can be achieved in various ways: for in-stance, in the race-track scheme [8] one has two com-peting non-perturbative superpotentials which provide anon-trivial minimum of the dilaton potential. The draw-back of this mechanism is that it only works if one hastwo rather large ‘hidden’ gauge groups with rather spe-cial matter contents. A somewhat more economic schemeis that of K¨ahler stabilization [9, 10] where one needsonly one hidden sector. However, in the relevant regimewhere dilaton stabilization may be achieved the theoryis not calculable. More recently, an alternative has beenstudied (with the most prominent example being that ofKKLT [11]) where the superpotential is of the form W = c + A e − a S . (13)The first term c is a constant and the second term reflectshidden sector strong dynamics, i.e. S is related to thegauge coupling, Re S ∝ /g , and a is related to the β -function of the hidden gauge group. In the KKLT setup,the constant comes from fluxes. The minimum of thescalar potential for S occurs at a point where | a S A e − a S | ∼ | c | . (14)The VEV of W , i.e. the gravitino mass, is of the sameorder. In order to have MSSM superpartner masses at theTeV scale, the gravitino mass cannot exceed O (100) TeV,hence | c | . − (15)in Planck units. The small scale in this setting is there-fore not explained by dimensional transmutation butoriginates from the smallness of the constant c . KKLTand others argue that, due to the large number of vacua,some of them might have such c by accident.In what follows, we will exploit the observation of sec-tion III that small VEVs of the (perturbative) superpo-tential can be explained by an approximate U(1) R sym-metry. We will use this in order to discuss moduli sta-bilization in the context of the heterotic string. We fo-cus on orbifold compactifications [12] since they possessmany (and well-understood) discrete symmetries, which,as it turns out, imply approximate U(1) R symmetries ofthe superpotentials describing the effective field theoriesderived from these constructions. As we shall see, super-potential VEVs of the order 10 −O (10) can naturally beobtained. Orbifold compactifications allow us to embedthe MSSM into string theory [14,13,1].In our calculations we focus on the models of the ‘het-erotic MiniLandscape’ [15,1]. These models exhibit thestandard model gauge group and the chiral matter con-tent of the MSSM. They are based on the Z -II orbifoldwith three factorizable tori (see [16,13] for details). Thediscrete symmetry of the geometry leads to a large num-ber of discrete symmetries governing the couplings of theeffective field theory [17, 18] (cf. also [16,13,19]). Apartfrom various bosonic discrete symmetries, one has a[ Z × Z × Z ] R (16)symmetry; other orbifolds have similar discrete symme-tries. Further, in almost all of the MiniLandscape modelsthere is, at one-loop, a Fayet-Iliopoulos (FI) D -term, V D ⊃ g X i q i | φ i | + ξ ! , (17)where the q i denote the charges under the so-called‘anomalous U(1)’. It turns out that, in all models withnon-vanishing FI term, ξ is of order 0 . φ i with the followingproperties: • giving VEVs to the φ i allows us to cancel the FIterm; • there is no other field that is singlet under the gaugesymmetries left unbroken by the φ i VEVs.These properties ensure that the h φ i i can be consistentwith a vanishing D -term potential and that the F -termsof all other massless modes vanish, implying that it is suf-ficient to derive the superpotential terms involving onlythe φ i fields. A crucial property of these superpotentialsis that they exhibit accidental U(1) R symmetries that getonly broken at rather high orders N . As discussed, thiscan be regarded as a consequence of high-power discrete R symmetries (equation (16)). N depends on the cho-sen φ i configuration; as a general rule we find that themore φ i fields are considered, the lower N values emerge.For instance, in a model where only seven fields are con-sidered, we obtain N = 26, on the other hand, in themodel 1 of [1] with 24 fields switched on, U(1) R gets bro-ken at order 9.Given non-trivial solutions to the F -term equations, φ i ∂ W ∂φ i = 0 , with φ i = 0 , (18)one can use complexified gauge transformations to en-sure vanishing D -terms as well [20]. Although D -termconstraints do not fix the scale of the h φ i i in general, therequirement to cancel the FI term introduces the scale √ ξ ∼ . V D = V F = 0 in the regime | φ i | <
1, and find that theyexist. We explicitly verify that for such solutions the su-perpotential is hierarchically small, h W i ∼ h φ i N , where h φ i denotes the typical size of a VEV. A very importantproperty of many of these configurations is that all fieldsacquire (supersymmetric) masses. Hereby typically onlyone field has a mass of the order m η (see equation (12))while the others are much heavier. We have also checkedthat these features are robust under supergravity correc-tions.Altogether we find that in the models under consider-ation one obtains isolated supersymmetric field configu-rations with | φ i | < h W i is hierarchically small.Before discussing applications, let us compare ourfindings to other recent results [21]. There, using thestringy selection rules, so-called ‘maximal vacua’ wereconstructed in which the superpotential vanishes termby term (and to all orders). In our approach, each su-perpotential term composed out of φ i fields acquires anon-trivial VEV, but to the order at which the accidentalU(1) R is exact, all terms cancel non-trivially. At higherorders, a non-trivial VEV of W gets induced.Let us now briefly sketch how this can be used in orderto stabilize the dilaton, whose VEV determines the gaugecoupling. After integrating out the φ i fields, one is leftwith a superpotential of the form (13), W eff = c + A e − a S , (19)where c = h W i = 10 −O (10) , and A e − a S describes somenon-perturbative dynamics, such as gaugino condensa-tion [7,22,23,24]. As we have discussed before in equa-tion (14), this superpotential leads to a non-trivial mini-mum for the dilaton. In the MiniLandscape models, real-istic gauge couplings are correlated with favorable sizes ofthe dynamical scale, A e − a S /M ∼ TeV [25]. Hence, fortypical expectation values h W i = 10 −O (10) one obtainsreasonable gauge couplings. The fixing of the T -moduliand other issues such as ‘uplifting’ will be studied else-where.Another application of our findings concerns the µ term of the MSSM. In [26] it has been proposed thatin models in which the field combination h u h d (with h u and h d denoting the up-type and down-type Higgs fields,respectively) is completely neutral w.r.t. all symmetriesthere is an interesting relation between the Higgs masscoefficient µ and h W i , µ ∼ h W i . (20)The heterotic MiniLandscape [15] contains many modelsin which the Higgs pair (and only the Higgs pair) hasthis property. Apart from the above property, such mod-els exhibit ‘gauge-top unification’, i.e. the top Yukawacoupling is of the order of the gauge coupling, as well asmany other desirable properties. In a concrete example,the benchmark model 1A of [1], it was found that solvingthe F -term equations for the superpotential up to order 6always leads to h W i = 0. We have now obtained a betterunderstanding of this fact: there is a U(1) R symmetrythat holds up to order 11, explaining this property. It isamazing to see that these models, constructed in order toreproduce the MSSM spectrum and gauge interactions,exhibit so many appealing properties automatically. V. CONCLUSIONS
We have shown that approximate U(1) R symmetriescan explain the appearance of hierarchically smallconstants. We find that at configurations where the F -term equations are solved, the superpotential goeslike h W i ∼ h φ i N with h φ i denoting a typical expectationvalue and N being the order at which U(1) R gets broken.We have analyzed various heterotic orbifold models andfound that there, due to the presence of high-powerdiscrete R symmetries, approximate U(1) R s are generic.We have explicitly solved the F -term equations in severalmodels, thus obtaining points in field space in which the F - and D -term potentials vanish, and confirmed that,for | φ i | <
1, the superpotential is hierarchically small.We have argued that such suppressed superpotentialexpectation values can be the origin for the appearanceof large hierarchies in nature: they fix the scale ofthe gravitino mass, which in schemes with low-energysupersymmetry sets the weak scale, and can be used tostabilize the string theory moduli at realistic values.
Acknowledgments.
We would like to thankF. Br¨ummer, T. Kobayashi and J. Schmidt foruseful discussions. This research was supported by theDFG cluster of excellence Origin and Structure of theUniverse, the European Union 6th framework programMRTN-CT-2006-035863 ”UniverseNet”, LMUExcellentand the SFB-Transregios 27 ”Neutrinos and Beyond”and 33 ”The Dark Universe” by Deutsche Forschungsge-meinschaft (DFG). [1] O. Lebedev et al., Phys. Rev.
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