1 Introduction and summary

LPTENS–16/04, CPHT–RR033.062016,  July 2016

{centering} Super no-scale models in string theory

Costas Kounnas and Hervé Partouche

Laboratoire de Physique Théorique, Ecole Normale Supérieure,
24 rue Lhomond, F–75231 Paris cedex 05, France
Costas.Kounnas@lpt.ens.fr

Centre de Physique Théorique, Ecole Polytechnique, CNRS, Université Paris-Saclay
F–91128 Palaiseau cedex, France
herve.partouche@polytechnique.edu

Abstract

We consider “super no-scale models” in the framework of the heterotic string, where the spontaneous breaking of supersymmetry is induced by geometrical fluxes realizing a stringy Scherk-Schwarz perturbative mechanism. Classically, these backgrounds are characterized by a boson/fermion degeneracy at the massless level, even if supersymmetry is broken. At the 1-loop level, the vacuum energy is exponentially suppressed, provided the supersymmetry breaking scale is small, . We show that the “super no-scale string models” under consideration are free of Hagedorn-like tachyonic singularities, even when the supersymmetry breaking scale is large, . The vacuum energy decreases monotonically and converges exponentially to zero, when varies from to . We also show that all Wilson lines associated to asymptotically free gauge symmetries are dynamically stabilized by the 1-loop effective potential, while those corresponding to non-asymtotically free gauge groups lead to instabilities and condense. The Wilson lines of the conformal gauge symmetries remain massless. When stable, the stringy super no-scale models admit low energy effective actions, where decoupling gravity yields theories in flat spacetime, with softly broken supersymmetry.

Unité mixte du CNRS et de l’Ecole Normale Supérieure associée à l’Université Pierre et Marie Curie (Paris 6), UMR 8549.

## 1 Introduction and summary

String theory unifies gravitational and gauge interactions at the quantum level. To describe particle physics, one can naturally consider classical models defined in four-dimensional Minkowski spacetime, where string perturbation theory can be implemented to derive the quantum dynamics. However, from a gravitational point of view, the question of the cosmological constant which can be regenerated at 1-loop, must be addressed. In non-supersymmetric models, such as those derived by compactifying the ten-dimensional heterotic string, this vacuum energy density is extremely large [1]. It is generically of order , where is the string scale, and has no chance to be naturally cancelled by any mechanism involving physics at lower energy.

Alternatively, one can consider no-scale models [2], which by definition describe at tree level theories in Minkowski space, where supersymmetry is spontaneously broken at an arbitrary scale . More precisely, is a flat direction of a classical positive semi-definite potential, . This very fact opens the possibility to generate by quantum effects a vacuum energy of arbitrary magnitude. In supergravity language, the no-scale models involve a superpotential and moduli fields , in terms of which the scale of the spontaneous supersymmetry breaking can be expressed as[3],

 m23/2=eK|w0|2=e~K|w0|2Imz1Imz2Imz3, (1.1)

where is the Kälher potential and is the part of that is independent of the three moduli associated to the breaking of supersymmetry. When is independent of the ’s, is undetermined by the minimization condition . In string theory or its associated effective supergravity description at low energy, depending on the choice of supersymmetry breaking mechanism, the ’s can either be the dilaton-axion field , or Kähler or complex structure moduli associated to the six-dimensional internal space. For instance :

- Some initially supersymmetric models can develop non-perturbative effects, such as gaugino condensation[4]. In this case, some of the fields, including , are stabilized. The magnitude of supersymmetry breaking is determined by and the imaginary parts of , , which can be Kähler or complex structure moduli . In the expression of the superpotential, GeV is the Planck scale and is the scale of confinement associated to an asymptotically free gauge group, of -function coefficient . is the string coupling, which relates the string and Planck scales as . The gaugino condensation breaking mechanism leads naturally to a small gravitino mass, even though the moduli fields ’s are of order 1. However, this non-perturbative scenario can only be studied qualitatively at the effective supergravity level, since no fully quantitative derivation from string computations is available yet.

- Alternatively, perturbative or non-perturbative fluxes [5] along the internal space can induce non-trivial superpotentials that break supersymmetry. In some cases, S-,T- or U-dualities [6] can be used to derive semi-quantitative results. In general, there is not yet available full derivation from string computations and so, one must restrict to semi-quantitative descriptions at the effective supergravity level. Some exception however exists, on which we now turn on.

In the present work, we focus on geometrical fluxes that realize generalized “coordinate-dependent compactifications” [7, 8]. The latter are similar to that proposed by Scherk and Schwarz in supergravity [9], but upgraded to string theory and furthermore to its gauge sector. In some cases, the mechanism can be implemented at the level of the worldsheet 2-dimensional conformal field theory, thus allowing explicit quantitative string computations, order by order in perturbation. The scale of spontaneous supersymmetry breaking is given by the inverse volume of the internal directions involved in the generalized stringy Scherk-Schwarz mechanism. For the quantum vacuum energy density to not be of order , this volume should be large, and the associated towers of Kaluza-Klein (KK) states should be light, with many consequences :

When their contributions do not cancel each another (a situation that will be central to the present work), the KK states, whose masses are of order , dominate the quantum amplitudes, while the heavier states, whose masses are of order , yield exponentially suppressed contributions, . In practice, can be the string scale, the GUT scale or a large Higgs scale.

These dominant contributions are the full expressions obtained in loop computations done in a pure KK field theory that realizes a spontaneous breaking of supersymmetry à la Scherk-Schwarz. No UV divergence occurs, a fact that is similar to that observed in field theory at finite temperature when the KK modes are Matsubara excitations along the Euclidean time circle and the spectrum at zero temperature is supersymmetric.

At 1-loop, if the model does not contain any scale below , the effective potential takes the form [10, 11, 12, 13],

 V\scriptsize 1-loop=ξ(nF−nB)m43/2+O(M4se−cMs/m3/2), (1.2)

where and count the numbers of massless fermionic and bosonic degrees of freedom, while depends on moduli fields other than . The above result makes sense in the theories that are free of “decompactification problems” [14], which would invalidate the string perturbative approach, due to large threshold corrections to gauge couplings [15, 16]. For instance, models realizing either the or or patterns of spontaneous supersymmetry breaking are consistent at the perturbative level [13].

Notice in Eq. (1.2) the absence of term proportional to , where is the mass operator. Such a term appears in and supergravities spontaneously broken to , when the quantum corrections are regularized in the UV by a cut-off scale . Even if the extremely large term is not present in string theory, the sub-dominant one, proportional to , still occurs when . This leads a serious difficulty, since it is far too large, compared to the cosmological constant (indirectly) observed by astrophysicists, even when is about 10 TeV, which is the order of magnitude of the lowest bound of supersymmetry breaking scale allowed by current observations at the LHC.

This remark invites us to consider “super no-scale models” in string theory [11, 12], which are the subclass of no-scale models satisfying the condition . These theories generate automatically a 1-loop vacuum energy that is exponentially suppressed, provided is much lower than . The “super no-scale models” extend the notion of no-scale structure valid at tree level to the 1-loop level. Note that non-supersymmetric classical models satisfying the even stronger property of boson-fermion degeneracy at each mass level are already know in type II string [17, 18] and orientifold descendants [19, 20]. They are based on asymmetric orbifolds and yield an exactly vanishing vacuum energy at 1-loop. However, contrary to what was initially believed, the 2-loop contribution seems to be non-trivial, as a priori expected [21]. It is important to stress that when these models describe a spontaneous breaking of supersymmetry to , they are super no-scale models in a strong sense and that, when perturbative heterotic dual descriptions are found, the latter appear to be super no-scale models in the weaker sense we have defined i.e. with boson-fermion classical degeneracy at the massless level only [18, 20].

In Sect. 2, we display one of the simplest super no-scale models. It is realized in heterotic string compactified on . The moduli and , associated to the and internal 2-tori, take values such that the right-moving gauge group is enhanced to either or . The spontaneous breaking of supersymmetry is realized via a stringy Scherk-Schwarz mechanism [7] that involves the 2-torus only, and the supersymmetry breaking scale is a function of the associated moduli .

When is of the order of the string scale, a fact that arises when and are , the corrections to the effective potential are not suppressed anymore. Even if these precise terms are those responsible for Hagedorn-like transitions in models where supersymmetry is spontaneously broken to [22, 8], we show that such instabilities are not present in our model. In other words, the theory does not develop classical tachyonic modes. Moreover, the super no-scale structure shows up as soon as is lower than . This situation is encountered in two distinct corners of the -moduli space, which are T-dual to each other : with , and with . On the contrary, is greater than in the remaining corners of the -moduli space, which are also T-dual to one another : with , and with . When , the model is naturally interpreted as an theory realized as an explicit breaking of (rather than a no-scale model). It is also interesting to note that when varies from to 0, decreases monotonically and converges to 0. This behavior imposes the interesting fact that in a cosmological scenario, slides to lower values, thus implying the super no-scale structure to be reached dynamically at a low supersymmetry breaking scale.

The above statement is valid provided that there are no tachyonic instabilities, which can be developed at the 1-loop level. In order to study this issue, we consider in Sect. 3 the response of under all possible small moduli deformations of the lattice, namely the -metric and antisymmetric tensor, and Wilson lines. The associated moduli , , cover the full classical moduli space around the initial extended symmetry point based on the gauge group . Actually, slightly deforming the initial background amounts to switching on Higgs scales smaller than . In this case, some of the massless states acquire small masses. In fact, and are functions of the ’s, which actually interpolate between distinct integer values. Expanding locally around the initial background, we find

 V\scriptsize 1-loop=ξ(nF−nB)m43/2−~ξm23/2∑αbαrankGα∑J=16∑I=1(YIJMs)2+⋯+O(M4se−cMs/m3/2), (1.3)

where . The structure of this result happens to be valid for any no-scale model that realizes the breaking of supersymmetry. The ’s are the gauge group factors, and the ’s are their associated -function coefficients. The ’s are their Wilson lines along . The above result shows that the Wilson lines associated to Cartan generators of an asymptotically free gauge group factor (), acquire positive squared masses at 1-loop and thus, they are stabilized at the origin, . On the contrary, the moduli associated to a non-asymptotically free gauge group factor (), become tachyonic. They condense, thus inducing negative contributions to and the Higgsing of to subgroups with non-negative -function coefficients but equal total rank. It is only when that the associated ’s remain massless.

Note however that the stability of the super no-scale models is always guaranteed when they are considered at finite temperature , as long as is greater than . This follows from the fact that in the effective potential at finite temperature – the quantum free energy –, all squared masses are shifted by , which implies that all moduli deformations are stabilized at [23]. Therefore, in a cosmological scenario where the Universe grows up and the temperature drops, the previously mentioned instabilities (for ) take place as soon as reaches from above.

In Sect. 4, we consider chains of super no-scale models that realize an or spontaneous breaking of supersymmetry, via or orbifold actions on parent super no-scale models. In the “descendant” theories, is freely acting, which ensures that the sub-breaking of is spontaneous, so that the models are free of decompactification problems [13]. The drawback of this chain of models is that the final spectrum is non-chiral, as opposed to that of the super no-scale models based on non-freely acting orbifolds and constructed in Ref. [11], which however suffer from decompactification problems [15, 16, 14].

Finally, additional remarks and perspectives can be found in Sect. 5.

## 2 N=4→0 super no-scale model

In this section, we built and analyze in more details one of the simplest super no-scale models, already presented in Ref. [12]. It is constructed in heterotic string and realizes the spontaneous supersymmetry breaking, with gauge symmetry that will appear to be either or . The 1-loop effective potential is given as usual in terms of the partition function at genus 1, , integrated over the fundamental domain of ,

 V\scriptsize 1-loop=−M4s(2π)4∫Fd2τ2τ22Zsss, (2.4)

where is the genus-1 Techmüller parameter.

### 2.1 Partition function

Our starting point is the “parent” , heterotic string compactified on , whose partition function has the following factorized form :

 ZN=4=O(0)2,2O(1)2,2O(2)2,2O(3)2,212∑a,bZ(F)4,0[ab]Z0,8+8, (2.5)

where denotes the contribution of the left-moving 2-dimensional fermions, super-partners of the coordinates in light-cone gauge, and is that of the right-moving compact bosons, which give rise to the affine characters in the adjoint representation,

 Z(F)4,0[ab]=(−1)a+b+abθ[ab]4η4,Z0,8+8=⎛⎝12∑γ,δ¯θ[γδ]8¯η8⎞⎠⎛⎜⎝12∑γ′,δ′¯θ[γ′δ′]8¯η8⎞⎟⎠, (2.6)

where the spin structure and .

denotes the contributions of the spacetime light-cone coordinates, while , , arise from the coordinates of the three internal 2-tori and can be expressed in terms lattices :

 O(0)d−2,d−2=1(√τ2η¯η)d−2,O(I)2,2=Γ2,2(TI,UI)η2¯η2,I∈{1,2,3}. (2.7)

We denote by the unshifted -lattice. More generally, the shifted lattice to be used in a moment is defined as , where we limit ourselves to shifts and ,

 Γ2,2[h1,h2g1,g2](T,U)=∑m1,m2n1,n2eiπ(g1m1+g2m2)q12|pL|2¯q12|pR|2, (2.8)

where and

 pL =1√2ImTImU[Um1−m2+T(n1+12h1)+TU(n2+12h2)], pR =1√2ImTImU[Um1−m2+¯T(n1+12h1)+¯TU(n2+12h2)]. (2.9)

and are given as usual in terms of the internal metric and antisymmetric tensor , ,

 TI=i√G2I−1,2I−1G2I,2I−G22I−1,2I+B2I,2I−1, UI=i√G2I−1,2I−1G2I,2I−G22I−1,2I+G2I,2I−1G2I−1,2I−1,I∈{1,2,3}. (2.10)

In the above expressions, (or , , to be used later) are the Jacobi elliptic functions and is the Dedekind function, following the conventions of Ref. [24].

It is also convenient to introduce the characters defined as

 O2N V2N S2N =θ[10]N+(−i)Nθ[11]N2ηN, C2N =θ[10]N−(−i)Nθ[11]N2ηN, (2.11)

in terms of which we can write in the following factorized form,

 ZN=4=O(0)2,2O(1)2,2O(2)2,2O(3)2,2(V8−S8)(¯O16+¯S16)(¯O′16+¯S′16), (2.12)

where the character becomes .

We then introduce a stringy Scherk-Schwarz mechanism [7] that simultaneously breaks and , spontaneously. This is done by implementing a orbifold action that shifts the internal direction, . The associated lattice shifts are coupled to the spin structure via a non-trivial sign , as well as to the and spinorial characters with another sign . In total, this amounts to replacing

 O(1)2,2 ⟶12∑h,gSL[a;hb;g]Γ2,2[h,0g,0](T1,U1)η2¯η2 with SL[a;hb;g] =(−1)ga+hb+hg, Z0,16 ⟶12∑γ,δ12∑γ′,δ′SR[γ,γ′;hδ,δ′;g]¯θ[γδ]8¯η8¯θ[γ′δ′]8¯η8 with SR[γ,γ′;hδ,δ′;g] =(−1)g(γ+γ′)+h(δ+δ′). (2.13)

The shift being coupled by the sign to the spacetime fermions (), to the spinorial characters () and to the spinorial characters (), the model will be referred as “spinorial-spinorial-spinorial”, or sss-model. Its partition function is

 Zsss= O(0)2,2O(2)2,2O(3)2,212∑h,gΓ2,2[h,0g,0](T1,U1)η2¯η2× (2.14)

 Zsss=O(0)2,2O(2)2,2O(3)2,212η2¯η2 [Γ2,2[0,00,0](T1,U1)(V8−S8)(¯O16+¯S16)(¯O′16+¯S′16) +Γ2,2[0,01,0](T1,U1)(V8+S8)(¯O16−¯S16)(¯O′16−¯S′16) +Γ2,2[1,00,0](T1,U1)(O8−C8)(¯V16+¯C16)(¯V′16+¯C′16) −Γ2,2[1,01,0](T1,U1)(O8+C8)(¯V16−¯C16)(¯V′16−¯C′16)]. (2.15)

Defining the characters of the shifted -lattice associated to the 2-torus as

 (2.16)

the partition function of the sss-model takes the final form

 Zsss=O(0)2,2O(2)2,2O(3)2,2 [O(1)2,2[00](V8(¯O16¯O′16+¯S16¯S′16)−S8(¯O16¯S′16+¯S16¯O′16)) +O(1)2,2[01](V8(¯O16¯S′16+¯S16¯O′16)−S8(¯O16¯O′16+¯S16¯S′16)) +O(1)2,2[10](O8(¯V16¯C′16+¯C16¯V′16)−C8(¯V16¯V′16+¯C16¯C′16)) +O(1)2,2[11](O8(¯V16¯V′16+¯C16¯C′16)−C8(¯V16¯C′16+¯C16¯V′16))]. (2.17)

For comparison, we also display the model where only is introduced (). The latter realizes the breaking but preserves the full gauge symmetry. Since in that case the shift is only coupled to the spacetime fermions (), this model will be referred as “spinorial”, or s-model. The associated partition function is

 (2.18)

with factorized right-moving characters. is similar to the partition function of the initial model at finite temperature [8, 23]. The latter is obtained by replacing the role of the internal direction with that of a compact Euclidean time of perimeter , where is the temperature.

The spectra of the s- and sss-model can be easily studied by observing that the 2-torus characters can be written as

 O(1)2,2[hg]=1η2¯η2∑k1,m2n1,n2q12|p(1)L|2¯q12|p(1)R|2, (2.19)

where the momentum is redefined as ,

 p(1)L =1√2ImT1ImU1[U1(2k1+g)−m2+T12(2n1+h)+T1U1n2], p(1)R =1√2ImT1ImU1[U1(2k1+g)−m2+¯T12(2n1+h)+¯T1U1n2]. (2.20)

In particular, the scale of spontaneous supersymmetry breaking satisfies

 m23/2=|U1|2M2sImT1ImU1. (2.21)

In the s-model, the sector contains tachyonic states when the supersymmetry breaking scale is of order . In this case, the integrated partition function i.e. the effective potential is ill-defined and a Hagedorn-like instability actually arises [22, 8]. In the theory at finite temperature, this phenomenon is nothing but the well known Hagedorn instability, which takes place when . On the contrary, the situation happens to be drastically different in the sss-model. The reason is that the sector with reversed GSO projection, which is characterized by the left-moving character , is dressed by right-moving characters that start at the massless level, . Therefore, the level matching condition prevents any physical tachyon to arise for arbitrary , . No Hagedorn-like instability occurs and the 1-loop effective potential based on the partition function is well defined.

However, marginal deformations other than can be switched on. Beside the dilaton, the classical moduli space can be parameterized by the 6 scalars of the bosonic degrees of freedom of the vector multiplets that realize the Cartan gauge symmetry (the fermionic superpartners are massive). It takes the form

 SU(6)×SO(6+16)SO(6)×SO(16) (2.22)

and its dimension is . For small enough deformations away from the sss-model, tachyonic instabilities would not arise. On the contrary, some Wilson lines deformations can certainly lead to tachyonic modes, when the gravitino mass is of order [1]. Note however that theories where all potentially dangerous moduli deformations have been projected out do exist, as shown explicitly in a four-dimensional orientifold model constructed in Ref. [25].

Before concluding this subsection, we give the expression of the 1-loop effective potential of the s- and sss-model, when and , which implies [13]. As we will be seen in details in Sect. 3, takes in this regime the following form :

 V\scriptsize 1-loop=nF−nB16π7M4s(ImT1)2E(1,0)(U1|3,0)+O(M4se−c√ImT1), (2.23)

where and are the numbers of fermionic and bosonic massless degrees of freedom111The factor appearing in the exponentially suppressed terms depends on all moduli but and the dilaton. It is of order , where is the lowest mass above the pure KK mass scale . In the s- and sss-model, it is of order , but can be in other cases a large Higgs scale or GUT scale (See Sect. 3)., and the functions

 E(g1,g2)(U|s,k)=∑~m1,~m2′(ImU)s(~m1+g12+(~m2+g22)U)s+k(~m1+g12+(~m2+g22)¯U)s−k (2.24)

are shifted complex Eisenstein series of asymmetric weights, where . While for the s-model and scales like , we are going to see that the sss-model can be super no-scale.

### 2.2 The super no-scale regime, m3/2≪Ms

In order to show that the 1-loop effective potential of the sss-model can be exponentially suppressed, , when the supersymmetry breaking scale is low, we look for conditions such that the massless fermions and bosons present in the regime , satisfy [12].

Given the fact that the states in the sectors , , have non-trivial winding numbers along the very large compact direction , they are super massive. In order to find the massless (or more generally light) states of the sss-model, it is only required to analyze the sectors , .

Sector

The bosonic sector contains massless degrees of freedom, which are associated to the graviton, antisymmetric tensor, moduli fields (dilaton, Wilson lines, internal metric and antisymmetric tensor) and to a vector boson in the adjoint representation of a gauge group , where the factor arises from the lattice associated to the 2-torus. In the regime we consider, but , , may be a higher dimensional group of rank 2. For generic , , we have , which can be enhanced to , or at particular points in moduli space. The degeneracy of these massless states is

 nB≡d(Bosons[00]) =d(V8)[d(O(0)2,2)+d(O(1)2,2[00])+d(O(2)2,2)+d(O(3)2,2)+d(¯O16)+d(¯O′16)] =8×[2+2+d(G(2))+d(G(3))+8×15+8×15] =8×[244+d(G(2))+d(G(3))]––––––––––––––––––––––––––––––––, (2.25)

which depends on the moduli , .

Similarly, the fermionic sector begins at the massless level, with states in the spinorial representations of or . Their multiplicity is

 nF≡d(Fermions[00])=d(S8)[d(¯S′16)+d(¯S16)]=8×(128+128)=8×256––––––––, (2.26)

which is independent of the point in moduli space we sit at. Moreover, the above bosonic and fermionic degrees of freedom are accompanied by light towers of pure KK states associated to the 2-torus. Their momenta along the directions and , which are both large, are and , and their KK masses are of order .

Sector

The bosonic sector contains light towers of KK modes arising from the 2-torus. Their momenta along and are and , the oddness of the former implying they cannot be massless. Their degeneracy is

 d(Bosons[01])=d(V8)[d(¯S′16)+d(¯S16)]=8×256––––––––, (2.27)

which equals .

Similarly, the fermionic sector