Super no-scale models in string theory

We consider"super no-scale models"in the framework of the heterotic string, where the N=4,2,1 -->0 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, m_{3/2}<<M_{string}. 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, m_{3/2} ~ M_{string}. The vacuum energy decreases monotonically and converges exponentially to zero, when m_{3/2} varies from M_{string} to 0. 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.


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 nonsupersymmetric models, such as those derived by compactifying the SO(16) × SO(16) tendimensional heterotic string, this vacuum energy density is extremely large [1]. It is generically of order M 4 s , where M s 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 m 3/2 . More precisely, m 3/2 is a flat direction of a classical positive semi-definite potential, V tree ≥ 0. This very fact opens the possibility to generate by quantum effects a vacuum energy of arbitrary magnitude. In N = 1 supergravity language, the no-scale models involve a superpotential w 0 and moduli fields z i , in terms of which the scale of the spontaneous supersymmetry breaking can be expressed as [3], m 2 3/2 = e K |w 0 | 2 = eK|w 0 | 2 Im z 1 Im z 2 Im z 3 , (1.1) where K is the Kälher potential andK is the part of K that is independent of the three moduli z i associated to the breaking of supersymmetry. When w 0 is independent of the z i 's, m 3/2 is undetermined by the minimization condition V tree = 0. In string theory or its associated effective supergravity description at low energy, depending on the choice of supersymmetry breaking mechanism, the z i 's can either be the dilaton-axion field S, or Kähler or complex structure moduli T I , U I 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 S, are stabilized. The magnitude of supersymmetry breaking is determined by |w 0 | 2 = Λ 6 np /M 4 P and the imaginary parts of z i , i ∈ {1, 2, 3}, which can be Kähler or complex structure moduli T I , U I .
In the expression of the superpotential, M P 2.4 · 10 18 GeV is the Planck scale and Λ np = M s exp (−8π 2 /|b|g 2 s ) is the scale of confinement associated to an asymptotically free gauge group, of β-function coefficient b. g s is the string coupling, which relates the string and Planck scales as M s = g s M P . The gaugino condensation breaking mechanism leads naturally to a small gravitino mass, even though the moduli fields Im z i '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 Udualities [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 "coordinatedependent 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 m 3/2 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 M 4 s , 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 m 3/2 , dominate the quantum amplitudes, while the heavier states, whose masses are of order cM s , yield exponentially suppressed contributions, O(e −cMs/m 3/2 ). In practice, cM s 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 supersymmetryà 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 m 3/2 , the effective potential takes the form [10][11][12][13], where n F and n B count the numbers of massless fermionic and bosonic degrees of freedom, while ξ > 0 depends on moduli fields other than m 3/2 . 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].
Notice in Eq. (1.2) the absence of term proportional to Str M 2 Λ 2 co ∝ m 2 3/2 Λ 2 co , where M is the mass operator. Such a term appears in N = 1 and N = 2 supergravities spontaneously broken to N = 0, when the quantum corrections are regularized in the UV by a cut-off scale Λ 2 co = O(M 2 s ). Even if the extremely large term m 2 3/2 Λ 2 co is not present in string theory, the sub-dominant one, proportional to m 4 3/2 , still occurs when n F = n B . This leads a serious difficulty, since it is far too large, compared to the cosmological constant (indirectly) observed by astrophysicists, even when m 3/2 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 n F = n B . These theories generate automatically a 1-loop vacuum energy that is exponentially suppressed, provided m 3/2 is much lower than cM s . 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 nontrivial, as a priori expected [21]. It is important to stress that when these models describe a spontaneous breaking of supersymmetry to N = 0, 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 T 2 × T 2 × T 2 . The moduli T 2 , U 2 and T 3 , U 3 , associated to the 2 nd and 3 rd internal 2-tori, take values such that the right-moving gauge group is enhanced The N = 4 → 0 spontaneous breaking of supersymmetry is realized via a stringy Scherk-Schwarz mechanism [7] that involves the 1 st 2-torus only, and the supersymmetry breaking scale m 3/2 is a function of the associated moduli T 1 , U 1 .
When m 3/2 is of the order of the string scale, a fact that arises when |T 1 | and |U 1 | are O(1), the corrections O(M 4 s e −cMs/m 3/2 ) 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 N = 0 [8,22], 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 m 3/2 is lower than M s . This situation is encountered in two distinct corners of the (T 1 , U 1 )-moduli space, which are T-dual to each other : On the contrary, m 3/2 is greater than M s in the remaining corners of the (T 1 , U 1 )-moduli space, which are also T-dual to one another : When m 3/2 > M s , the model is naturally interpreted as an N = 0 theory realized as an explicit breaking of N = 4 (rather than a no-scale model). It is also interesting to note that when m 3/2 varies from +∞ to 0, V 1-loop decreases monotonically and converges to 0. This behavior imposes the interesting fact that in a cosmological scenario, m 3/2 slides to lower values, thus implying the super no-scale structure to be reached dynamically at a low supersymmetry breaking scale. Actually, slightly deforming the initial background amounts to switching on Higgs scales Y IJ M s smaller than m 3/2 . In this case, some of the n B + n F massless states acquire small masses. In fact, n B and n F are functions of the Y IJ 's, which actually interpolate between distinct integer values. Expanding locally around the initial background, we find whereξ > 0. The structure of this result happens to be valid for any no-scale model that realizes the N = 4 → 0 breaking of supersymmetry. The G α 's are the gauge group factors, and the b α 's are their associated β-function coefficients. The Y IJ 's are their Wilson lines along T 6 . The above result shows that the Wilson lines associated to Cartan generators of an asymptotically free gauge group factor G α (b α < 0), acquire positive squared masses at 1-loop and thus, they are stabilized at the origin, Y IJ = 0. On the contrary, the moduli associated to a non-asymptotically free gauge group factor G α (b α > 0), become tachyonic.
They condense, thus inducing negative contributions to V 1-loop and the Higgsing of G α to subgroups with non-negative β-function coefficients but equal total rank. It is only when b α = 0 that the associated Y IJ 's remain massless.
Note however that the stability of the super no-scale models is always guaranteed when they are considered at finite temperature T , as long as T is greater than m 3/2 . This follows from the fact that in the effective potential at finite temperature -the quantum free energy -, all squared masses are shifted by T 2 , which implies that all moduli deformations are stabilized at Y IJ = 0 [23]. Therefore, in a cosmological scenario where the Universe grows up and the temperature drops, the previously mentioned instabilities (for b α > 0) take place as soon as T 2 reaches m 2 3/2 from above.
In Sect. 4, we consider chains of super no-scale models that realize an N = 2 → 0 or N = 1 → 0 spontaneous breaking of supersymmetry, via Z free 2 or Z free 2 × Z 2 orbifold actions on parent N = 4 → 0 super no-scale models. In the "descendant" theories, Z free 2 is freely acting, which ensures that the sub-breaking of N = 4 → 2 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 [14][15][16].
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 N = 4 → 0 spontaneous supersymmetry breaking, with gauge symmetry that will appear 1-loop effective potential is given as usual in terms of the partition function at genus 1, Z sss , integrated over the fundamental domain F of SL(2, Z), where τ = τ 1 + iτ 2 is the genus-1 Techmüller parameter.
It is also convenient to introduce the O(2N ) characters defined as 8) in terms of which we can write Z N =4 in the following factorized form, where the E 8 character becomesŌ 16 +S 16 .
We then introduce a stringy Scherk-Schwarz mechanism [7] that simultaneously breaks N = 4 → 0 and E 8 × E 8 → SO(16) × SO (16) , spontaneously. This is done by implementing a Z shift 2 orbifold action that shifts the 1 st internal direction, X 1 . The associated lattice shifts h, g ∈ Z 2 are coupled to the spin structure via a non-trivial sign S L , as well as to the SO (16) and SO (16) spinorial characters with another sign S R . In total, this amounts to replacing (2.10) The shift g being coupled by the sign S L S R to the spacetime fermions (a = 1), to the SO (16) spinorial characters (γ = 1) and to the SO(16) spinorial characters (γ = 1), the model will be referred as "spinorial-spinorial-spinorial", or sss-model. Its partition function is 11) which leads to Defining the characters of the shifted (2, 2)-lattice associated to the 1 st 2-torus as the partition function of the sss-model takes the final form (2.14) For comparison, we also display the model where only S L is introduced (S R ≡ 1). The latter realizes the N = 4 → 0 breaking but preserves the full E 8 × E 8 gauge symmetry. Since in that case the shift g is only coupled to the spacetime fermions (a = 1), this model will be referred as "spinorial", or s-model. The associated partition function is with factorized right-moving characters. Z s is similar to the partition function of the initial N = 4 model at finite temperature [8,23]. The latter is obtained by replacing the role of the 1 st internal direction X 1 with that of a compact Euclidean time X 0 of perimeter The spectra of the s-and sss-model can be easily studied by observing that the 1 st 2-torus characters can be written as where the momentum m 1 is redefined as 2k 1 + g, In particular, the scale m 3/2 of N = 4 → 0 spontaneous supersymmetry breaking satisfies In the s-model, the sector O 2,2 O 8Ō16Ō 16 contains tachyonic states when the supersymmetry breaking scale m 3/2 is of order M s . In this case, the integrated partition function i.e. the effective potential is ill-defined and a Hagedorn-like instability actually arises [8,22]. In the N = 4 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 leftmoving character O 8 , is dressed by right-moving characters that start at the massless level, V 16V 16 . Therefore, the level matching condition prevents any physical tachyon to arise for arbitrary T I , U I , {I = 1, 2, 3}. No Hagedorn-like instability occurs and the 1-loop effective potential based on the partition function Z sss is well defined.
However, marginal deformations other than T I , U I 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 N = 4 vector multiplets that realize the U (1) 6+16 Cartan gauge symmetry (the fermionic superpartners are massive). It takes the form and its dimension is 6 × (6 + 16). For small enough deformations away from the sss-model, tachyonic instabilities would not arise. On the contrary, some O(1) Wilson lines deformations can certainly lead to tachyonic modes, when the gravitino mass is of order M s [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 Im T 1 1 and U 1 = O(i), which implies m 3/2 M s [13]. As we will be seen in details in Sect. 3, V 1-loop takes in this regime the following form : 20) where n F and n B are the numbers of fermionic and bosonic massless degrees of freedom 1 , and the functions are shifted complex Eisenstein series of asymmetric weights, where g 1 , g 2 ∈ Z 2 . While n F = 0 for the s-model and V 1-loop scales like m 4 3/2 , we are going to see that the sss-model can be super no-scale.

M s
In order to show that the 1-loop effective potential of the sss-model can be exponentially , when the supersymmetry breaking scale is low, we look for conditions such that the massless fermions and bosons present in the regime Im T 1 1, Given the fact that the states in the sectors O (1) 2,2 1 g , g = 0, 1, have non-trivial winding numbers 2k 1 + 1 along the very large compact direction X 1 , they are super massive. In order to find the massless (or more generally light) states of the sss-model, it is only required to 2,2 V 8Ō16Ō 16 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 (16)×SO (16) , where the factor G (I) arises from the lattice associated to the I th 2-torus. In the regime we consider, I ∈ {2, 3}, may be a higher dimensional group of rank 2. For generic T I , U I , I ∈ {2, 3}, we have G (I) = U (1) 2 , which can be enhanced to SU (2) × U (1), SU (2) 2 or SU (3) at particular points in moduli space. The degeneracy of these massless states is which depends on the moduli T I , U I , I ∈ {2, 3}.
Similarly, the fermionic sector −O 2,2 S 8 (Ō 16S 16 +S 16Ō 16 ) begins at the massless level, with states in the spinorial representations of SO (16) or SO (16) . Their multiplicity is 23) 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 1 st 2-torus. Their momenta along the directions X 1 and X 2 , which are both large, are 2k 1 and m 2 , and their KK masses are of order m 3/2 .
2,2 V 8 (Ō 16S 16 +S 16Ō 16 ) contains light towers of KK modes arising from the 1 st 2-torus. Their momenta along X 1 and X 2 are 2k 1 +1 and m 2 , the oddness of the former implying they cannot be massless. Their degeneracy is 2,2 S 8Ō16Ō 16 contains light KK states, with non-vanishing masses, their momenta being again 2k + 1 and m 2 . Their counting goes as follows : The fact that the number of KK towers with odd momenta equals that of those with even momenta is not a coincidence. In the initial N = 4 theory, among the characters with even γ + γ , those corresponding to spacetime fermions are given a KK mass in the sss-model, while those associated to spacetime bosons are not modified. This feature is common to the s-model, On the contrary, when γ + γ is odd, the sign S R effectively reverses the roles of bosons and fermions. Among the characters with odd γ + γ , those corresponding to spacetime bosons are given a KK mass in the sss-model, while those associated to spacetime fermions are not modified. These facts are opposite to those encountered in the the s-model. The sss case thus leads The condition for the sss-model to be super no-scale is that the numbers of massless fermions and bosons be equal, This imposes [12] d(G (2) ) + d(G (3) ) = 12, which leads for rk(G (2) (4)  Again, the model is naturally super no-scale; the trajectories of the time-dependent moduli associated to the 2 nd and 3 rd 2-tori being attracted to these points.

The T-dual regimes
We have seen that for T 1 → i∞, U 1 = O(i), the sss-model is characterized by a low supersymmetry breaking scale m 3/2 and a super no-scale structure. In the present subsection, our goal is to study the remaining corners of the moduli space where either T 1 or U 1 (but not both) is of order i. We thus define 4 regimes, where the first one is super no-scale with m 3/2 < M s , while the others can be respectively analyzed by defining T-dual moduli, In terms of these new variables, Regime (II) is reached by The relevance of the above definitions of T-dual moduli follows from the fact The third equality is telling us that the sss-model (as well as the s-model) is self-dual under Thus, the corners (I) and (IV) of the 1 st 2-torus moduli space share a common behavior : The sss-model is super no-scale in both limits, and the supersymmetry breaking scale satisfies which is a T-duality invariant expression. On the contrary, the 1 st equality in Eq. (2.32) allows us to rewrite the partition function as which shows that the sss-model is not self-dual under the T-duality transformation (T 1 , U 1 ) → (T 1 ,Û 1 ). Note that in the s-model, this transformation amounts to inter-exchanging the spinorial characters S 8 ↔ C 8 i.e. reversing spacetime chirality. The latter being a matter of convention, Regimes (II) and (III) describe isomorphic particle contents in the s-model.
Finally, the 2 nd equality in Eq. (2.32) guaranties the sss-model (as well as the s-model) is In other words, the identity (2.33) can be rewritten asẐ The above expression guaranties that the corners (II) and (III) of the 1 st 2-torus moduli space yield a common behavior. In the following, we describe the light spectrum and effective potential in these regimes.
The winding numbers along the directions of the T-dual 2-torus whose Kähler and complex structure areT 1 andÛ 1 are 2k 1 + g and m 2 , which implies that in Regime (II), where , the states with non-vanishing 2k 1 + 1 or m 2 are super massive.
Therefore, the pure T-dual KK modes lead exponentially dominant contributions, as follows from the expression of the T-dual 2-torus characters in Regime (II), which for g = 0 are (1) is positive and the second line is obtained by Poisson summation over n 1 and n 2 . For g = 1, the winding numbers cannot vanish, so that The light spectrum arising in Region (II) turns out to be : This sector being self-dual, its massless spectrum is that derived in Sector O , which amounts to n B bosonic and n F fermionic degrees of freedom, In Regime (II), these degrees of freedom are accompanied by light towers of pure T-dual KK modes (pure winding modes for the original 1 st 2-torus), whose momenta are 2n 1 and n 2 , as can be read in the 1 st line of Eq. (2.37). Their masses are of order M s / ImT 1 /2.
2,2 C 8V16V 16 contains light towers of pure T-dual KK modes with momenta 2n 1 + 1 and m 2 . The former being nonzero, these states cannot be massless but their masses are light, of order M s / ImT 1 /2. Their degeneracy is which equals n F .

Note that no light bosonic state arises in Sector
, as can be seen from the right-moving charactersV 16C 16 +C 16V 16 , which start at the massive level, in units of M s . This shows that contrary to the large Im T 1 limit with U 1 = O(i), N = 4 supersymmetry is not recovered in the large ImT 1 limit whenÛ 1 = O(i). If the sss-model realizes a spontaneous breaking of supersymmetry implemented via stringy Scherk-Schwarz compactification on the initial 1 st 2-torus, from the T-dual picture, it realizes a compactification on the T-dual 2-torus of an initially non-supersymmetric model in 6 dimensions. In fact, the dual KK mass scale M s / ImT 1 /2 is not a scale of supersymmetry breaking (spontaneous or not). The no-scale modulus i.e. the spontaneous supersymmetry breaking scale is always m 3/2 , which satisfies in Regimes (II) and (III) . (2.41) In the limit m 2 3/2 → +∞, degrees of freedom decouple, leaving us with an sss-breaking of supersymmetry in six dimensions that is explicit.
The above remarks suggest that the vacuum energy may be large in Regime (II). To show this is true, we use Eqs (2.37) and (2.38) to write the effective potential in terms of dual moduli aŝ ( 2.42) Contrary to the expression found in Regime (I) for large Im T 1 , the argument of the exponential in the 1 st line, which is proportional to |ñ 1 + 2ñ 2Û1 | 2 , can vanish. Actually, the contribution of the effective potential arising forñ 1 =ñ 2 = 0 grows linearly with the dual volume (2π) 2 ImT 1 /(2M 2 s ). This behavior is drastically different to that encountered in Regime (I), where the potential is exponentially suppressed in Im T 1 (or scales like (n F − n B )m 4 3/2 if n F = n B ) and vanishes in the limit where N = 4 supersymmetry is restored. The remaining terms, with (ñ 1 ,ñ 2 ) = (0, 0), can be treated exactly as is done in Regime (I) and mentioned in the introduction, in the paragraph above Eq. (1.2). They yield light T-dual KK modes of masses O M s / ImT 1 /2 , whose contributions dominate over those arising from the remaining, super heavy string modes. Moreover, as follows from the 2 nd line in Eq. (2.37), these towers of T-dual KK modes regularize the UV, in the sense that up to exponentially suppressed terms, the integral over the fundamental domain F can be extended to the upper half strip, − 1 2 < τ 1 < 1 2 , τ 2 > 0, without introducing divergences. In total, one findŝ (1) 2,2 0 1 (T 1 ,Û 1 ) , while the quantities C and C depend on the 2 nd and 3 rd 2-tori moduli only, (2.44) The final expression of the effective potential in Regime (II) can be simplified tô (2.45) Note that since C + C is nonzero, one obtains in the T-dual 2-torus decompactification limit whereV N 6 =0 1-loop is the effective potential of the obtained non-supersymmetric six-dimensional theory,V which involves the associated partition function For instance, C + C can be evaluated numerically at It is however important to stress that the behavior of the sss-model derived in Regimes (II) and (III) is actually formal. This is due to the fact that in these cases, the 1-loop correction to the classically vanishing vacuum energy density of the universe is very large, O(M 6 s ), as can be seen from the r.h.s. of Eq. (2.46). This fact may cast doubts on the validity of perturbation theory. Moreover, it is expected that in the large T-dual 2-torus limit, the decompactification problem does arise. This should be the case since no N 6 = 2 supersymmetry is recovered in six dimensions (N = 4 in four dimensions) and the towers of T-dual KK modes of masses O M s / ImT 1 /2 should yield large quantum corrections to the gauge thresholds, proportional to the volume (2π) 2 ImT 1 /(2M 2 s ) [13,14]. Finally, taking n F n B , which is satisfied for arbitrary T I , U I , I ∈ {2, 3}, one can extremize the potential (2.45) with respect toÛ 1 , which yields a solutionÛ 1 (1 + i)/2 modulo T-duality.
However, the latter is a saddle point that destabilizes ImÛ 1 to larger and larger or lower and lower values, which brings the theory out of Regime (II).

The intermediate regime
We (1) R | 2 − 1 = 0, thus increasing n B . For instance, taking k 1 = n 1 = 0, these conditions are satisfied for m 2 = −n 2 = ±1 when we sit on the codimension one submanifold of the moduli space that satisfies T 1 = −1/U 1 . These states are 2 gauge bosons and their Wilson lines along the internal space, which enhance the gauge group factor associated to the 1 st 2-torus to G (1) = U (1) × SU (2).
On the contrary, n F does not vary with T 1 , U 1 .
Other extra massless bosons arise in Sector O R | 2 = 0. For instance, taking m 2 = n 2 = 0, these conditions are satisfied for 2k 1 + 1 = 2n 1 + 1 = ±1 when T 1 /2 = −Ū 1 . These modes are two scalars in the bi-fundamental representation of SO (16) × SO (16) , thus with multiplicity  where m 3/2 1. This behavior implies that the term e 4φ V 1-loop , which appears in the effective action in Einstein frame, creates a tadpole for the dilaton φ and imposes the latter to slide at early cosmological times to the weak coupling regime.
Our choice of Re T 1 and U 1 is such that the curve passes through the lines T 1 = −1/U 1 and T 1 /2 = −Ū 1 , when Im T 1 = 1 and 2, respectively. However, no extremum occurs at these points. In Ref. [1], it is shown in general that in non-supersymmetric classical models, the integrated partition function at arbitrary genus-g admits extrema at all "points of maximal enhanced symmetry". The latter are the loci in moduli space where the gauge group is enhanced, with no U (1) factor left. In our case, since G (1) = U (1) × SU (2) and G (1) = U (1) 2 at Im T 1 = 1 and 2, there is no contradiction in not having extrema at these points. Fig. 1

T 6 -moduli and Wilson lines deformations
Once we have found a classical model that yields an exponentially suppressed effective potential at 1-loop, the question of the quantum stability of this background must be addressed.
Actually, the worldsheet CFT admits marginal deformations, which from the spacetime point of view correspond to classical moduli. Since the 1-loop effective potential depends on these scalar deformations, the initial vacuum may be destabilized. In this section, we will study the

Deformation of Background (a)
The  1) and the partition function is given in Eq. (2.14), with Im T 1 1, U 1 = O(i). Denoting G IJ and B IJ the initial internal metric and antisymmetric tensor, 6 × 6 real Y 's are introduced to define their deformed counterparts as while the 6 × 16 remaining ones, are the Wilson lines of SO(16) 2 along T 6 . Our goal is to determine which of the above 6 × 22 deformations acquire at 1-loop positive squared masses or remain massless, while the leftover ones induce tachyonic instabilities.
We first derive a general expression for the 1-loop effective potential, in the regime Let us consider the contribution to the 1-loop partition function arising from a single state s, where F is its fermion number. The left-and right-moving squared masses take the following form, where the "primes" mean that the expressions refer to the deformed background [26], (3.8) where N L , N R denote the oscillator numbers and we have defined In the above expressions, P I andP I are generalized left-and right-moving momenta that depend on the T 6 momenta and winding numbers m I and n I , while Q I , I ∈ {7, . . . , 22}, denote the components of a weight in a representation of the gauge group realized by the extra right-movingφ I 's [27]. Physically, this weight is the charge vector under SO(16) 2 of the state s and its squared length is an even integer. An immediate consequence of the r.h.s. of Eq. (3.7) is that invariance under the modular translation τ → τ + 1 implies that M 2 L − M 2 R = 4L s M 2 s , for some integer L s . Therefore, M 2 L − M 2 R must be invariant under the 6 × 22 continuous deformations, a fact that is easily verified using Eqs (3.9), which yield Next, we note that for small Y -deformations, the term (3.7) integrated over the fundamental domain F leads a contribution of order e −Im T 1 to the effective potential if s has non-trivial winding numbers along either of the two 1 st internal directions, which are large.
Therefore, we concentrate on the dominant contributions, which arise from the pure momentum states (i.e. with 2n 1 + h = n 2 = 0 in Eq. (2.17)). Choosing one of them, s 0 , with vanishing momenta m 1 = m 2 = 0 (m 1 ≡ 2k 1 + g in Eq. (2.17)), let us gather the contributions to V 1-loop of the KK towers associated to X 1 , X 2 and based on this state. In the initial Background (a), one obtains where F 0 , M 0L , M 0R are the fermion number and left-or right-moving masses of s 0 . The insertion (−1) m 1 in the l.h.s. arises from the fermion number F = F 0 + m 1 . It translates the fact that a mass splitting of order 1/Im T 1 exists between bosons and fermions, as follows from the spontaneous breaking of supersymmetry and can be seen in the partition function (2.14).
This phase e iπm 1 yields in the r.h.s., which is obtained by Poisson summation, a 1 2 -shift of the integerm 1 . This shift implies that the integral in Eq. (3.10) can be extended to the full upper half-strip, −1 < τ 1 ≤ 1, τ 2 > 0, without introducing UV divergences, and that the result differs from that obtained by integrating over F by terms of order e −c √ Im T 1 .
When the Y -deformations are switched on, M 0L , M 0R and more importantly the KK mass are slightly modified. The latter is initially the degree 2 polynomial in m 1 , m 2 , which appears in the argument of the exponential function in the l.h.s. of Eq. (3.10), (and becomes the expression in Eq. (3.26)). However, for small enough Y 's, the full expression after poisson summation is still integrable over the upper half-strip (see Eq. (3.27)). It follows that the integration over τ 1 is straightforward, implying that the surviving dominant contributions to V 1-loop arise from KK states s that are level-matched, L s = 0. Moreover, since the states with vanishing winding numbers along X 1 and X 2 satisfy (3.11) which is independent of m 1 , m 2 , the whole towers of KK modes based on the level-matched states s 0 are level matched as well. Writing the associated contribution, (3.12) and changing the dummy variable of integration τ 2 into x = τ 2 /Im T 1 , we see that when the mass M 0L of the state s 0 in the initial Background (a) is not vanishing, the result is exponentially suppressed. Therefore, we obtain the general expression of the 1-loop effective potential (3.13) where the sum extends over the set of massless states present in the initial Background (a), and M L is the mass of the associated KK mode with momenta m 1 , m 2 along X 1 , X 2 , once the moduli deformations are switched on.
To proceed, we resume the states s 0 of the sss-model in Background (a), which satisfy M 2 0L = M 2 0R = 0. The first condition imposes N L = 1 2 , the second yields N R = 0 or 1, and we recall that their quantum numbers along X 1 , X 2 are m 1 = m 2 = n 1 = n 2 = 0. 14) where Γ Adj SU (2) 4 and Γ Adj SO(16) 2 are the root lattices of SU (2) 4 and SO(16) 2 , we find that :
They are neutral with respect to the gauge group G. Note that the generalized momentum which is a root of squared length equal to 2 of Γ Adj SU (2) 4 .
• Similarly, 8 copies of massless states with F 0 = 0 arise at oscillator level N R = 0 from the root lattice Γ Adj SO (16)  (3.17) In the above formula, 0 k means k consecutive null entries [27]. In total, there are 2 × 112 such roots Q I of squared lengths equal to 2.
-In Sector  Q I is actually one of the 2 × 128 weights of squared lengths equal to 2 [27]. As said in Sec. 2.2, we have a total of n F = 8 × (2 × 128) = 8 × 256 fermionic massless states.
We are ready to compute the contribution to the effective potential (3.13) that arises from the KK towers of states s based on each of the n B + n F states s 0 , which are initially massless in Background (a). The momenta, winding numbers and SO(16) 2 charges of each state s are those of s 0 , up to the momenta m 1 , m 2 along the X 1 , X 2 , which are arbitrary. We first consider the non-Cartan states of SU (2) 4 . For given i ∈ {3, 4, 5, 6} and ∈ {−1, 1}, the contribution of s to the potential involves its squared mass M 2 L given in Eq. (3.8), which is expressed in terms of 4, 5, 6} , (3.20) and the inverse of the metric 3.21) which is In the above equation, G −1 αβ is the inverse of the 2 × 2 matrix G αβ = G αβ , α, β ∈ {1, 2}. The contribution of the 8 copies of KK states associated to the SU (2) 4 root i, is 3.23) where ξ α = √ 2 Y αi +Y αI Y iI and we have expanded at second order in Y 's the e −πτ 2 2P α G −1 αj P j and e −πτ 2 P j G −1 jk P k contributions in the integrand. However, the 2 nd line in Eq.  24) we can use the inverse matrix 2 G αβ to define deformed moduli , (3.25) in terms of which we have 27) where ξ = U 1 ξ 1 − ξ 2 . Expanding the phase in ξ orξ and integrating over τ 2 , one obtains the final contribution, where we have redefined complex moduli as 4, 5, 6} , (3.30) and introduced the dressing coefficient In Eq. (3.28), the scalars Y ji and Y i , for j ∈ {3, 4, 5, 6}, are actually the Wilson lines of the i th SU (2) factor along T 6 , weighted by the associated root √ 2.
We proceed with the contribution V Q 1-loop of the effective potential that arises from the KK modes s based on the state s 0 of right-moving charge Q I , which is either a root of Γ Adj SO(16) 2 or a weight of Γ Spin SO(16) 2 whose length squared equals 2. The novelty is that the former have F 0 = 0, while the latter have F 0 = 1. For such a mode s, we have 4, 5, 6} . (3.32) Comparing with Eq. (3.20), we see that at second order in Y 's, the 8 copies of KK modes yield a contribution identical to V i, 1-loop , up to the overall dressing (−1) F 0 and the exchanges Thus, we immediately conclude that where Y jI , j ∈ {3, 4, 5, 6}, and are the Wilson lines of SO(16) 2 along T 6 .
Finally, we consider the 8 × 24 KK towers of states that are neutral with respect to the gauge group. In this case, F 0 = 0 and P α , P j are like those of Eq. (3.32), with Q I = 0.
Therefore, the effective potential contribution V e 1-loop , for e ∈ {1, . . . , 24}, which arises from the 8 copies of such states, is In order to combine all contributions to the effective potential we have computed, we note that where C(R G )δ ab = tr(T a T b ) and T a , a ∈ {1, . . . , dim G}, are the generators in the representation R G of a gauge group G. Given that, summing over the 4 × 2 roots i, of SU (2) 4 , the 2 × (112 + 128) charges Q of SO(16) 2 and the 24 sets of neutral KK towers, one obtains the final result, In this expression, b SU (2) and b SO(16) are the β-function coefficients of each SU (2) and SO (16) factors, which are obtained using the following contributions of massless degrees of freedom in the to depend on the whole metric of T 6 . 3 Clearly, the stability of an initial no-scale model background requires the term m 4 3/2 to be absent, which is nothing but the super no-scale condition n F = n B . If this is satisfied, we are left with the 2 nd and 3 rd lines in Eq.  (16) , (3.42) which are proportional to m 2 3/2 , as expected for moduli not involved in the supersymmetry breaking [3]. Since |ρ(U 1 )| < 1, Eq. (3.42) leads to the conclusion that any simple gauge group factor that is neither asymptotically free nor conformal, i.e. with b G > 0, yields to local instabilities.
In the sss-super no-scale model we consider here, the SU (2) 4 Wilson lines Y ji and Y i , j, i ∈ {3, 4, 5, 6}, are attracted dynamically to the origin Y ji = Y i = 0, while the SO (16) 2 ones Y jI and Y I , j ∈ {3, 4, 5, 6}, I ∈ {7, . . . , 22}, condense. Due to the periodicity properties of the Wilson lines, this instability is only local and some of the Y jI 's and/or Y I 's are expected to develop large but finite expectation values. Note that since we started with a vanishing effective potential in the super no-scale Background (a), these instabilities imply that V 1-loop becomes negative. We should reach another no-scale model, with new numbers of massless fermions and bosons satisfying n F < n B , and without non-asymptotically free gauge group factors. At this stage, the model would still be in the regime m 3/2 M s , which guaranties no tachyonic instability may arise. However, the scaling of the effective potential now being like −m 4 3/2 , the gravitino mass would be dynamically attracted to larger values. Once it reaches the order of magnitude of the string scale, several scenarios may occur : • A tachyon may arise at tree level, thus inducing a severe Hagedorn-like instability.
• m 3/2 may be stabilized at a (local) minimum, thus yielding an anti-de Sitter vacuum, where a restoration of supersymmetry may or may not occur.
• m 3/2 may continue increasing, with runaway behavior. The model would lead (after T-duality) to an anti-de Sitter theory in higher dimensions, explicitly non-supersymmetric. 3 The gravitino mass m 3/2 involves T 1 , U 1 i.e. B 21 and G αβ only, but the latter is the inverse 2 × 2 matrix

Lifting the instabilities
In the previous sub-section we have shown the existence of two different types of instabilities.
The first ones, arise in the no-scale models having n F = n B , which are due to the nonvanishing of V 1-loop . Actually, the vanishing of the effective potential is required by the dilaton and no-scale modulus stationary condition; namely the absence of dilaton and noscale modulus tadpoles. The second ones are tachyonic instabilities that arise in all no-scale models having positive β-function coefficients. Therefore, it would be relevant to look for super no-scale models without non-asymptotically free gauge group factors. Possibly, one could consider no-scale models with n F > n B , and switch on discrete Wilson lines of order 1 in order to break the non-asymptotically free gauge group factors to products of asymptotically free and/or conformal subgroups.
Another approach is to consider the super no-scale models at finite temperature T . Note that this point of view can be relevant when the models are used in cosmological scenarios. At finite T , the effective potential is nothing but the quantum free energy and all squared masses are shifted by T 2 [23]. Thus, as long as T 2 is greater than m 2 3/2 , the tachyonic instabilities arising from positive β-function coefficients are lifted. For instance, Background (a) of the sss-model is stable during early stages of the cosmological evolution, when T is high. As the Universe grows and the temperature drops, the breaking of SO(16)×SO(16) occurs when T 2 crosses m 2 3/2 and becomes lower. It would be interesting to investigate this phase transition in a dynamical cosmological framework where all moduli fields, including the dilaton and the no-scale modulus, evolve with the temperature.
Another way to bypass the tachyonic instabilities occurring at 1-loop in super no-scale models may be to impose correlations among deformations, in order to preserve those which respect at the quantum level the flatness condition V 1-loop = 0. In the case of Background (a), (16) , ideally the constraint may be implemented, where Y is the total "attractive" Wilson line deformation associated to SU (2) 4 , while H is the total "repulsive" one, associated to SO(16) 2 , It is then legitimate to look for less symmetric super no-scale theories, realizing either an N = 2 → 0 or N = 1 → 0 spontaneous breaking. For this purpose, one may consider no-scale parent theories describing an N = 4 → 0 breaking, and implement Z 2 or Z 2 × Z 2 orbifold actions that yield descendent models satisfying the super no-scale property. However, as was shown in Ref. [13], if no precautions are taken in the choice of orbifold actions, the N = 2 sectors of these models lead generically to gauge coupling threshold corrections [15,16] proportional to the large internal volume [14]. In this case, a fine tuning of the string coupling g s is required to cancel the 1-loop threshold corrections of the gauge couplings of the asymptotically free gauge group factors. In the following, we present a simple strategy that yields N = 2 → 0 or N = 1 → 0 super no-scale models, while evading the above mentioned "decompactification problem".

Chains of N = 4, 2, 1 → 0 super no-scale models
Our goal is to derive a class of N = 2 → 0 and N = 1 → 0 super no-scale models from parent ones that realize the N = 4 → 0 breaking. The next subsection will describe the gauge threshold corrections arising in this case. To begin, we consider any N = 4 heterotic no-scale vacuum obtained by "moduli-deformed fermionic construction" [13,28]. Let us implement a Z 2 or Z 2 × Z 2 orbifold action where at least one of the Z 2 's is freely acting and thus realizes a spontaneous N = 4 → 2 breaking. The resulting vacuum is N = 2 or N = 1 supersymmetric, which is further spontaneously broken to N = 0 by a stringy Scherk-Schwarz mechanism [7] realized along the 1 st internal 2-torus. The latter is chosen to be large, for the supersymmetry breaking scale to be small, m 2 3/2 ∝ M 2 s /Im T 1 . In total, the model describes the N = 4 → 2 → 0 or N = 2 → 1 → 0 pattern of supersymmetry breaking. To be more specific, we request the following [13] : • The generator of the free Z 2 action, denoted as Z free 2 , twists the coordinates of the 2 nd and 3 rd 2-tori, and shifts at least one of the coordinates of the 1 st 2-torus, e.g. (4.1) • In the Z free 2 × Z 2 case, there is no restriction on the second Z 2 . However, in most cases, its generator as well as the product of the latter with the generator of Z free have fixed points.
If this happens, in order not to induce large threshold corrections to the gauge couplings, we impose the 2 nd and 3 rd 2-tori moduli T 2 , U 2 and T 3 , U 3 not to be far from i. For instance, they can sit at extended symmetry points.
• The stringy Scherk-Schwarz mechanism responsible for the final supersymmetry breaking is realized as a 1 2 -shift along the 1 st 2-torus, say X 1 , coupled to one of the R-symmetry charges, such as the helicity a.
Once the above restrictions are satisfied and m 3/2 M s , the effective potential of the Z free 2 and Z free 2 × Z 2 models turn out to be 1 2 and 1 4 of that of the "parent" N = 4 → 0 theory, up to exponentially suppressed contributions [13], Therefore, considering any N = 4 → 0 super no-scale model, such as the sss one, as a "parent" theory, one obtains automatically a chain of "descendant" models realizing the N = 2 → 0 or N = 1 → 0 breaking, with exponentially suppressed vacuum energy at 1-loop.

Threshold corrections without decompactification problem
As shown in Ref. [13], the gauge coupling threshold corrections of the Z free 2 and Z free 2 × Z 2 descendant theories derived from no-scale models realizing the N = 4 → 0 breaking of supersymmetry turn out to have a universal form, free of decompactification problem. In the following, we present the running gauge coupling associated to a gauge group factor G α , in the Z free 2 × Z 2 case. At low supersymmetry breaking scale m 3/2 , it is expressed in terms of moduli-dependent masses of order m 3/2 that encode the dominant contributions arising from five conformal blocks, which naturally appear in the left-moving piece of the partition function,  All other conformal blocks give either vanishing contributions, like the N = 4 block A, (h, g) = (H 1 , G 1 ) = (H 2 , G 2 ) = (0, 0), or the N = 1 ones, which have H 1 H 2 G 1 G 2 = 0. Or, their contributions are exponentially suppressed, as is the case for the blocks E and F , which have (H 2 , G 2 ) = (0, 0) and h H 1 g G 1 = 0, and realize N C = 2 → 0 and N D = 2 → 0 spontaneously broken phases.
Absorbing in a "renormalized string coupling" the universal contribution to the gauge coupling [16], where E 2,4,6 = 1 + O(q) are the holomorphic Eisenstein series of modular weights 2,4,6 and j = 1/q + 744 + O(q) is holomorphic and modular invariant, the final result for the running gauge coupling g α (Q) at energy scale Q is [13], It only depends on the Kac-Moody level k α of the gauge group factor G α and on 5 modeldependent β-function coefficients b α B,C,D and b α 2,3 . The terms in the 1 st line are associated to the N = 0, N C = 2 and N D = 2 spectra, which arise respectively in the conformal blocks B, C and D, while those in the 2 nd line arise from the N I = 2 spectra, I ∈ {2, 3}.
Note that in the 1 st line of Eq. (4.6), we have shifted M 2 B,C,D → Q 2 + M 2 B,C,D , in order to extend the validity of the result to values of Q above the threshold scales M B,C,D at which the conformal blocks B, C or D decouple. Therefore, Q is allowed to be as large as the lowest mass, which is of order cM s , of the massive states we have neglected the exponentially suppressed contributions. At low energy, i.e. Q lower than the three scales M B , M C , M D , the r.h.s. of Eq. (4.6) behaves as − 1 (1) when Im T 1 is large and U 1 = O(i). No volume term O(Im T 1 ) being present, the models evade the decompactification problem.
As already stated in the previous subsection, up to exponentially suppressed terms, the 1-loop effective potentials in the Z free 2 × Z 2 models we consider here come only from the conformal block B where N = 4 supersymmetry is spontaneously broken to N = 0, In this expression, n F − n B is the number of massless fermions minus the number of massless bosons in the "parent" N = 4 → 0 theory. Actually, 1 4 (n F − n B ) turns out to be the same quantity in the final N = 1 → 0 "descendant" model. This is a consequence of the underlying "non-aligned" N C = 2, N D = 2 and N I = 2, I ∈ {2, 3}, supersymmetries. Thus, when the initial N = 4 → 0 model is super no-scale, we have n F − n B = 0, which guaranties the Z free 2 and Z free 2 × Z 2 descendant orbifold theories to be super no-scale models as well.

T 2 × T 2 × T 2 -moduli and Wilson lines deformations
Starting from an N = 4 → 0 no-scale model, the moduli space that survive Z 2 or Z 2 × Z 2 orbifold actions in the "descendant" models is reduced. This follows from the fact that several deformations are frozen to some discrete values, in order to respect the factorization of the internal 6-torus as T 2 × T 4 or T 2 × T 2 × T 2 . For instance, in the sss-model, the scalars Y i in Eq. (3.30) are fixed to 0. However, new moduli fields arise generically from the massless scalars of the twisted sectors. Therefore, the stability and quantum flatness condition of the N = 2 → 0 and N = 1 → 0 no-scale models must be reconsidered.
An exception however exists, for the models arising from N = 4 → 0 no-scale theories, on which a Z free 2 or Z free 2 × Z 2 orbifold action is implemented, as described in Subsect. 4.1. In this case, modulo the constraint of the Γ 6,6 lattice factorization, the structure of the deformed effective potential is as in Eq. (3.38), up to the multiplicative factor 1 2 or 1 4 , and fully arises from the untwisted sector. Due to the free action of Z free 2 , the 1 st 2-torus is not fixed under any orbifold group element, so that no twisted massless states and thus no new moduli sensitive to the stringy Scherk-Schwarz mechanism is introduced. On the contrary, twisted massless states are allowed in the conformal blocks where the 2 nd or 3 rd 2-tori are fixed. However, being N 2 = 2 or N 3 = 2 supersymmetric at tree level, new moduli deformations exist, but remain exactly flat directions at 1-loop and therefore do not show up in the effective potential at this order. Thus, in the study of the quantum stability of the N = 4 → 2 → 0 or N = 2 → 1 → 0 models obtained by Z free or Z free 2 × Z 2 orbifold actions, only the β-function coefficients of the "parent" N = 4 → 0 theory are relevant. The resolution of an instability in a chain of N = 4, 2, 1 → 0 no-scale models is thus universal, in the sense that it is independent of the specific spectra of the "descendant" theories.

Conclusion
In this work, we focus on no-scale string models [2] where the spontaneous N = 4, 2, 1 → 0 breaking of supersymmetry is implemented at the perturbative level by geometrical fluxes.
This setup realizes a "coordinate-dependent string compactification" [7,8], in the spirit of the Scherk-Schwarz mechanism introduced in supergravity [9]. The gravitino mass scale m 3/2 is related to the inverse volume of the compact space involved in the supersymmetry breaking. Even thought supersymmetry is broken, the classical effective potential is positive semi-definite, V tree ≥ 0 [2], while the supersymmetry breaking scale m 3/2 is undetermined by the flatness condition.
At the quantum level, the 1-loop effective potential receives non-trivial corrections. The latter are however under control, at least in the regime of low supersymmetry breaking scale, m 3/2 < cM s , in which case one has The above formula arises from the contributions of the light KK towers of states associated to the large internal space, and remains valid in the string context we consider even when the no-scale models realize the N = 2 → 0 or N = 1 → 0 breaking. These facts lead us to consider the situation where the numbers of massless fermionic and bosonic degrees of freedom are equal, n F = n B [11,12]. In this case, V 1-loop vanishes modulo exponentially suppressed terms and we refer to these theories as "super no-scale string models". At the 1-loop level, they satisfy the flatness condition, as well as the absence of dilaton and no-scale In both examples, if V 1-loop is exponentially suppressed when m 3/2 < M s , it is not suppressed when m 3/2 = O(M s ). However, no Hagedorn-like instability takes place in this regime [8,22], which means that no state becomes tachyonic at any point of the (T 1 , U 1 )-moduli space.
Moreover, in the regime where m 3/2 > M s , the model is more naturally interpreted as an explicitly non-supersymmetric theory, rather than a no-scale model. Altogether, V 1-loop turns out to be positive and increases monotonically with m 3/2 . Therefore, in a cosmological context, the dynamics drives naturally these models to the super no-scale regime, where the supersymmetry breaking scale is small.
We also examine the local stability of the model with gauge symmetry G = U (1) 2 × SU (2) 4 × SO (16) 2 , under small moduli perturbations of the internal Γ 6,6+16 lattice. The analysis actually applies to all no-scale string models realizing an N = 4 → 0 breaking via stringy Scherk-Schwarz mechanism [7,8] along a large 1 st internal 2-torus, wether they are super no-scale, i.e. with n F = n B , or not. The rank of the gauge group being always 6 + 16, we find the following three possible behaviors of the moduli Y IJ , I ∈ {1, . . . , 6}, J ∈ {1, . . . , 6 + 16} : • For J associated to a Cartan generator of an asymptotically free gauge group factor G α (b α < 0), the Y IJ 's acquire 1-loop masses of order m 3/2 , and are therefore stabilized at the origin, Y IJ = 0.
• For J corresponding to a Cartan generator of a non-asymptotically free gauge group factor (b α > 0), the Y IJ 's aquiere negative squared masses, which leads instabilities. They condense and break G α to subgroups with non-negative β-function coefficients.
• The last Y IJ 's, associated to gauge group factors with b α = 0, remain massless.
Thus, in the examples we considered, the SO(16) × SO (16) Wilson lines yield a destabilization of the initial background. However, we stress that in all super no-scale models, the quantum instabilities are harmless when the theories are considered at finite temperature T , provided that T > m 3/2 . This follows from the fact that finite temperature induces effective mass terms proportional to T 2 (Y IJ ) 2 , which screen all tachyonic contributions −m 2 3/2 (Y IJ ) 2 . Therefore, in the framework of string cosmology at finite temperature [23], a phase transition happens when T approaches m 3/2 from above, which drives the initial model to a new phase without non-asymptotically free gauge group factors.
A particular class of super no-scale models, which realize the spontaneous N = 2 → 0 or N = 1 → 0 breaking of supersymmetry, can be constructed easily. They are built from parent N = 4 → 0 super no-scale models, on which a Z free 2 or Z free 2 × Z 2 orbifold action is implemented. The fact that the Z free 2 group is freely acting ensures that the partial N = 4 → 2 breaking is spontaneous, which yields important consequences [13]. First, the 1-loop effective potential in the descendant models is simply 1 2 or 1 4 of that of the parent theory. Second, the threshold corrections to the gauge couplings are not proportional to the volume of the large internal submanifold involved in the stringy Scherk-Schwarz mechanism.
This fact guaranties the validity of the string perturbative expansion, i.e. solves the socalled "decompactification problem". In the descendent theories, the space of untwisted moduli, which are those appearing in the effective potential, is reduced, as follows from the factorization of the internal space required by the orbifold action.
To conclude, we mention that it would be very interesting to study in super no-scale models the higher order corrections in string coupling to the effective potential. This would allow to see wether insisting on the flatness condition would yield additional restrictions on the models. One can also construct super no-scale theories by implementing Z 2 or Z 2 × Z 2 orbifold actions on N = 4 → 0 no-scale models, where each Z 2 admits fixed points [11].
In this case, our analysis of the moduli deformations must be completed, since the effective potential does depend on twisted moduli sensitive to the final breaking of supersymmetry to N = 0. However, if these models are compatible with the physical requirement of possessing chiral spectra, the decompactification problem has to be readdressed.