Exponentially suppressed cosmological constant with enhanced gauge symmetry in heterotic interpolating models

A few nine-dimensional interpolating models with two parameters are constructed and the massless spectra are studied by considering compactification of heterotic strings on a twisted circle with Wilson line. It is found that there are some conditions between radius R and Wilson line A under which the gauge symmetry is enhanced. In particular, when the gauge symmetry is enhanced to SO(18) \times SO(14), the cosmological constant is exponentially suppressed. We also construct a non-supersymmetric string model which is tachyon-free in all regions of moduli space and whose gauge symmetry involves E_8.


Introduction
LHC experiments suggest that supersymmetry (SUSY) does not exist at low energy scale.
It is, therefore, natural to consider the possibility that SUSY is broken at the string/Planck scale. For this reason, non-supersymmetric string models [1,2,3], in particular, the SO(16)× SO (16) heterotic string model which is the unique tachyon-free ten-dimensional non-supersymmetric model, are receiving more and more attention. Non-supersymmetric string models, however, always have a problem of stability. Unlike the supersymmetric ones, the cosmological constant is non-vanishing. There are non-vanishing dilaton tadpoles which lead to vacuum instability. Thus, the desired model must both be non-supersymmetric and carry a very small cosmological constant. While several methods to construct such models have been proposed [14,15,16,17,18,19,20,21], in this paper, we try to construct non-supersymmetric heterotic models with a small cosmological constant by focusing on so-called interpolating models [6,8,9,10].
An interpolating model is a (D − d)-dimensional model that continuously relates two D-dimensional models. In this work, we restrict our attention to the case with D = 10 and d = 1 for simplicity. The method of constructing such models is as follows; We start from a ten-dimensional closed string model (called model M 1 ) and compactify this on a circle with a Z 2 twist, which is nothing but the Scherk-Schwarz compactification [4,5]. The resulting nine-dimensional model should have a circle radius R as a parameter, which can be adjusted freely. Because we are considering closed string models, this nine-dimensional model should produce a ten-dimensional model (called model M 2 ) in R → 0 limit as well due to T-duality [28,29]. In particular, if model M 1 is supersymmetric and the Z 2 action contains (−1) F where F is the spacetime fermion number, the compactification causes SUSY breaking and the nine-dimensional interpolating model and model M 2 become non-supersymmetric.
In Ref. [6,7,12,11], it is shown that in the near supersymmetric region of moduli space, the cosmological constant Λ 10 is written as follows: where ξ is a positive constant andã = a −1 = R/ √ α ′ , and N F (N B ) is the number of massless fermionic (bosonic) degrees of freedom. Therefore, the cosmological constant is exponentially suppressed when N F = N B . We would like to have non-supersymmetric models with N F = N B , but the nine-dimensional interpolating models with one parameter R which we will review in section 2 do not have such property no matter how one adjusts the parameter R. In order to generate cases with N F −N B = 0, we need to increase the number of adjustable parameters. One such possibility is to compactify more dimensions. In this work, we instead consider nine-dimensional interpolating models with one more parameter by introducing a constant Wilson line background.

Interpolating models with no Wilson line
In this section, we review the construction of an interpolating model which is originally proposed in Ref. [6], and provide two concrete examples.
In these examples, we provide the interpolations between the ten-dimensional non-supersymmetric SO(16) × SO(16) heterotic string model and one of the ten-dimensional supersymmetric heterotic strings [27] as model The presentation below is based on Ref. [8,9]. 1

The construction of interpolating models
Let us start from a flat ten-dimensional closed string model M 1 whose partition function is where Z + + represents the contribution from the fermionic and the internal parts of string and Z (n) B from the bosonic parts of string: Let us first consider the circle compactification: The left-and right-moving momenta along the compactified dimension are respectively for n, w ∈ Z. After the circle compactification, the partition function of model M 1 becomes In order to obtain two different ten-dimensional models at R → ∞ and R → 0 limits, we have to consider the compactification on a twisted circle. We choose T Q as the Z 2 twist where T acts on the compactified circle as a half translation: T :X 9 →X 9 + πR.
Here,X 9 is the T-dualized coordinate for the compactified dimension andR = α ′ /R is the T-dualized radius. 2 We denote by Q a Z 2 action that acts on the internal part of the string and that determines the two ten-dimensional models at the limits.
Next, let us check the behaviors of Λ α,β as a → 0 (R → ∞) and as a → ∞ (R → 0). For a → 0 limit, it is the part with zero coefficients of a −2 in the exponential in Eq. (8) that give non-vanishing contributions. So only the lattices containing zero winding number are non-vanishing in the large R limit: where x = a(n + α). Consequently, we see as a → 0 On the other hand, in a → ∞ limit, the non-vanishing contributions come from the lattices with zero momentum in Eq. (8): where y = (w + β)/a. Consequenly, we see as a → ∞ Coming back to Eq. (6), we can rewrite as using Λ α,β . An interpolating model is obtained from Z (9)+ + by orbifolding with the Z 2 action T Q. A half translation T affects the lattices Λ α,β and acts such that only the states with even winding numbers survive: where Z + − is defined as the Q-action of Z + + . The modular invariance requires the twisted sector [34,35]. By using Eq. (10), we see that under S transformation, Z where Z + − (−1/τ ) ≡ Z − + (τ ). Furthermore, when T Q acts on Z (9)− + , we obtain where Z − − is defined as the Q-action of Z − + . As a result, the total partition function which is modular invariant is In accordance with Eq. (14), we see that Z (9) int reproduces model M 1 in a → ∞ limit. Note that the original model is reproduced as R → 0 as we have adopted the convention that a half translation T is introduced with regard to the T-dualized coordinate. If T were introduced with regard to the ordinary coordinate, the interpolating model would reproduce the original model M 1 in R → ∞ limit. On the other hand, in a → 0 limit, Z (9) int produces model M 2 whose partition function is That is, model M 2 is obtained by Q-twisting model M 1 , which means that model M 2 is related to model M 1 by the Z 2 action Q.

Two examples
In this subsection, we review two examples of nine-dimensional interpolating models which are tachyon free for all radii.
As the first example, let us choose the ten-dimensional SO(16) × SO(16) heterotic model as model M 1 and the ten-dimensional supersymmetric SO(32) heterotic model as model M 2 : In this case, in the language of subsection 2.1, The Z 2 action Q which relates the SO(16) × SO(16) model to the supersymmetric SO(32) model isR OC , which is defined as the reflection of the right-moving SO(8) characters: Using this Z 2 action Q and the modular transformation of SO(2n) characters S : we have Thus, from Eq. (19), we obtain the partition function of the interpolating model: We can see that the first and the third lines of Eq. Let us see the massless spectrum of this model from the partition function (27). For a generic radius 0 < R < ∞, massless states can appear only when n = w = 0, so we can find out the massless states by expanding the first line of Eq. (27) in q. We list the expansion of each character in Appendix B.1. Then, for a generic radius, the massless spectrum of the model is • the nine-dimensional gravity multiplet: graviton G µν , anti-symmetric tensor B µν and dilaton φ; • the gauge bosons transforming in the adjoint representation of SO(16) × SO(16) × • a spinor transforming in the (16,16) where U(1) G,B implies the Abelian factors generated by G µ9 and B µ9 . Note that this model has no points at which the gauge symmetry is enhanced in the region 0 < R < ∞. Also, there are no points at which the cosmological constant is exponentially suppressed, that is, In R → ∞ limit, the number of fermions is equal to that of bosons at each mass level including the massless level, which means that SUSY is restored in the limit.
In the second example, let us choose the SO (16) In this case, the Z 2 action Q is R V C which is defined as the reflection of one of the two left-moving SO(16) characters: The partition function of this interpolating model is obtained in a similar way to the first example: For a generic radius 0 < R < ∞, the massless spectrum of this model is • the nine-dimensional gravity multiplet: graviton G µν , anti-symmetric tensor B µν and dilaton φ; • the gauge bosons transforming in the adjoint representation of SO(16) × SO(16) × • a spinor transforming in the (128, 1) ⊕ (1, 128) of SO(16) × SO (16).
In this case, there are no points either where the gauge symmetry is enhanced or the cosmological constant is exponentially suppressed.

Interpolating models with Wilson line
where G is rank r group which has eight non-zero roots, then we obtain interpolating models in which N F − N B = 0.
However, in this work, we will add one parameter by turning on Wilson line. In other words, we will generalize interpolating models by considering a twisted circle with a constant background. We expect that there are some conditions between parameters under which the gauge symmetry is enhanced as in Ref. [22,23,24,26]. In this section, we construct ninedimensional interpolating models with two parameters by considering the compactification on a twisted circle with Wilson line.
Let us write the uncompactified dimensions as X µ (µ = 0, · · · , 9) and the internal ones as X I L (I = 1, · · · , 16) for a ten-dimensional heterotic string model, and compactify the X 9direction on a twisted circle. Furthermore, we switch on a constant Wilson line background with the components of µ = 9 and I = 1 by adding to the worldsheet action It is only the momentum lattice of the center-of-mass mode that is affected by turning on Wilson line A. The addition of the constant Wilson line background corresponds to the boost on the momentum lattice [22,23,25]: where is the left-moving momentum of the X I=1 L -direction and m ∈ Z for the NS (anti-periodic) boundary condition and m ∈ Z + 1/2 for R (periodic). Here, M ℓ L -p R and R ℓ L -p L represent the boost on the ℓ L -p R plane and the rotation on the ℓ L -p L plane respectively. The boost M ℓ L -p R is written in terms of A as follows: We use A to write R ℓ L -p L as follows: Therefore, after turning on Wilson line, we have The above equations mean that the left-and right-moving momenta of X µ=9 in Eq.
It is convenient to introduce a modular parameterτ in terms of the parameter of the twisted circle and Wilson line asτ Note that |τ | 2 = 1/a 2 , which means that the radial coordinate corresponds to radius R and the angular coordinate to Wilson line A. Usingτ , momenta (36) are rewritten as for m ∈ Z + γ, n ∈ Z + α, w ∈ 2(Z + β). From these momenta (44) Therefore the fundamental region of moduli space is 3 where O 3 If the Z 2 twist T Q acted trivially, then n and w would be both integers. Then, in addition to the shift (45), the momentum lattices would be invariant underτ → −1/τ with the replacement n ↔ w. This transformation would correspond to T-dual transformation, so the two limiting ten-dimensional models would be the same and the fundamental region would become − √ 2/2 ≤τ 1 ≤ √ 2/2 and |τ | ≥ 1.
We will refer to O We obtain As a result of these equations, we find the total partition function of the interpolating model:

The limiting cases
Next, let us see the limiting cases a → 0 and a → ∞ of the interpolating model (52). In the partition function (52), only the momentum lattices (38) depend on a, so we need to see the behavior of Λ (α,β) (γ,δ) in these limiting cases. As in the cases without Wilson line, the non-vanishing contributions come from the parts with zero winding number (momentum) in a → 0 (a → ∞) limit, and Λ (α,1/2) (γ,δ) (Λ (1/2,β) (γ,δ) ) vanishes as a → 0 (a → ∞). As a → 0, we find where x ≡ a(n + α) where the physical radius at the large (small) R region. In fact, from Eq. (36) we see Note that the effect of Wilson line is found only with the physical radii in the limiting cases.
In terms of the boosted characters, Eq. (53) and Eq. (54) respectively imply

The massless spectrum
Let us see the massless spectrum of this interpolating model for a generic set of values of a and A. As is done in section 2, we can find out massless states from the parts with zero momentum and zero winding number of the partition function (52). By expanding the characters in q, 4 we find the following massless states for a generic set of values of a and A: • the nine-dimensional gravity multiplet: graviton G µν , anti-symmetric tensor B µν and dilaton φ; • the gauge bosons transforming in the adjoint representation of SO(16) × SO(14) × • a spinor transforming in the (16,14) of SO(16) × SO (14).
Note that, compared to the first example in subsection 2. There are some conditions under which the additional massless states appear: Using a and A, this condition is rewritten as for any integer n 1 . Under this condition, we find that the following additional massless states appear: • two vectors transforming in the (1,14) of SO(16) × SO (14); • two spinors transforming in the (16, 1) of SO(16) × SO (14).
These massless vectors and spinors come fromV 8 O  under which the gauge symmetry is enhanced to SO(18) × SO (14). The green semi-circles correspond to condition (III) and we plot four orbits with w 3 = ±1, ±3.
Using a and A, this condition is rewritten as for any odd integer w 3 . The additional massless states are • two conjugate spinors transforming in the (1, 64) of SO(16) × SO (14).
We plot these conditions in the fundamental region (47) ofτ -plane in Fig. 1. The Table 1 summarizes the conditions under which the additional massless states appear in this model.
The table shows only the conditions with w = 0 because we are interested in the large R region where Eq. (1) is valid. Next, let us include Wilson line in the second example of subsection 2.2. The starting point is the same as at subsection 3.1 but the Q action is R V C in this case. According to the construction in subsection 2.1, we find that the total partition function is Using the limiting behaviors of the boosted characters (56), we can see that this interpolating model (59) reproduces the supersymmetric E 8 × E 8 model and the SO(16) × SO(16) model as a → 0 and a → ∞ respectively, for any value of A.

The massless spectrum
Let us see the massless spectrum of this interpolating model for a generic set of values of a and A. By expanding the partition function (59) in q, we find • the nine-dimensional gravity multiplet: graviton G µν , anti-symmetric tensor B µν and dilaton φ; • the gauge bosons transforming in the adjoint representation of SO(16) × SO(14) × • a spinor transforming in the (128, 1) of SO(16) × SO (14).
We plot these conditions in the fundamental region (47) ofτ -plane in Fig. 2. The Table 2 summarizes the conditions under which the additional massless states appear in this model. Finally, let us mention that in these models considered in this section, it is straightforward to calculate tree and one-loop scattering amplitudes of massless particles to obtain signals of broken supersymmetry [36,37,38,39].

Conclusions
We have constructed nine-dimensional interpolating models with two parameters by considering the compactification on a twisted circle with the constant Wilson line background (31), and have studied the massless spectra of these models. Furthermore, we have found some conditions between circle radius R and Wilson line A under which additional massless states are present. In the nine-dimensional model that interpolates between the tendimensional supersymmetric SO(32) model and the ten-dimensional SO (16) (14). Especially, under the second condition, the massless fermionic and bosonic degrees of freedom become equal, which means that the cosmological constant is exponentially suppressed. Recent references related to this point include [40,41,42]. According to Ref. [13], which is carried out in the type I dual picture [30], the brane configuration with We have however found the conditions under which the gauge symmetry is enhanced to SO(16) × SO (16) or SO(16) × E 8 .
As one of the future works, we have to discuss the stability of Wilson line as in Ref. [13,40,41,42] A Notation for the partition functions We summarize the notation for some functions that appear in the partition functions. The Dedekind eta function is where q = e 2πiτ . The theta function with characteristics is defined by Especially, when α and β are 0 or 1/2 and z = 0, we use the following shorthand notations: These theta functions satisfy the Jacobi's abstruse identity: We write the SO(2n) characters in terms of the theta functions as follows: In terms of the characters, the Jacobi's abstruse identity is

B The expansions of the characters
In string theories, we can see the spectrum of each mass levels by expanding the partition function in q. In this appendix, in order to see the massless states, which are the coefficients of q 0 , we shall expand the SO(8) and SO(16) characters, which appear in the partition function of some heterotic models 5 .

B.1 The case with no Wilson line
For section 2, we expand η −8 (O 2n , V 2n , S 2n , C 2n ) where η 8 is the contribution from X m and the SO(2n) characters are from ψ m or X I L , where m = 2, · · · , 10 and I = 1, · · · , 16: Note that the lowest order terms of (72), (73) and (74) correspond to the degrees of freedom of the identity, the vector and the spinor (the conjugate spinor) respectively, and the second 5 There are five ten-dimensional heterotic models whose partition functions are expressed in terms of the characters SO (8)  The right moving parts of the partition functions are expanded as The left moving parts of the partition functions in some heterotic models might include where the sum is taken over n ∈ Z + α and w ∈ 2(Z + β). As we are interested only in the left-moving parts of the partition function, we expand the following products:    V 16 ) will never be massless.