The Cosmological Constant and the Electroweak Scale

String theory has no parameter except the string scale, so a dynamically compactified solution to 4 dimensional spacetime should determine both the Planck scale and the cosmological constant $\Lambda$. In the racetrack K\"ahler uplift flux compactification model in Type IIB theory, where the string theory landscape is generated by scanning over discrete values of all the flux parameters, a statistical preference for an exponentially small $\Lambda$ is found to be natural (arXiv:1305.0753). Within this framework and matching the median $\Lambda$ value to the observed $\Lambda$, a mass scale ${\bf m}\simeq 100$ GeV naturally appears. We explain how the electroweak scale can be identified with this mass scale.


Introduction
Cosmological data strongly indicates that our universe has a vanishingly small positive cosmological constant Λ (vacuum energy density) as the dark energy, Λ ∼ 10 −122 M 4 P [2], where the Planck mass M P = G −1/2 N ≃ 10 19 GeV. The smallness of Λ is a major puzzle in physics. In general relativity, Λ is a free arbitrary parameter one can introduce, so its smallness can be accommodated but not explained within the field theory framework. 1 On the other hand, string theory has only a single parameter, namely the string scale M S = 1/ √ 2πα ′ , so everything else should be calculable for each string theory solution. String theory has 9 spatial dimensions, 6 of them must be dynamically compactified to describe our universe. Since both M P and Λ are calculable, Λ can be determined in terms of M P dynamically in each classically stable vacuum solution. So we may find an explanation for a very small positive Λ. This happens if a good fraction of the meta-stable deSitter (dS) vacua in the relevant regions of the string landscape tend to have a very small Λ, as is the case in the racetrack Kähler uplift scenario in flux compactification in string theory [1]. Note that both the Kähler uplift model [3][4][5][6][7] and the racetrack model [8][9][10] are scenarios well explored in string phenomenology.
To simplify the discussion, let us focus on flux compactification of Type IIB theory to 4 dimensional spacetime. Start with the four-dimensional low energy supergravity effective potential V (F i , φ j ), where F i are the field strengths and φ j are the moduli (and dilaton) describing the size and shape of the compactified manifold as well as the coupling. It is known that the field strengths F i in flux compactification in string theory take only quantized values [11], and all parameters like masses and couplings are now functions of discrete flux parameters F i . The string landscape is generated as we scan over all discrete values of F i . That is, V (F i , φ j ) has no free parameter, though it does contain (in principle) calculable quantities like α ′ corrections and geometric quantities like Euler index χ etc.. With no parameters to adjust, the radiative instability problem is absent [12], as ranges of flux parameters scanned over have already included the values to be fixed both before and after radiative corrections.
For a given set of flux parameters F i , we can solve V (F i , φ j ) for its meta-stable vacuum solutions via finding the values φ j,min (F i ) at each solution and determine its vacuum energy density Λ = Λ(F i , φ j,min (F i )) = Λ(F i ). Collecting all such solutions and feeding in a properly normalized probability distribution P i (F i ) for each F i , we can determine the properly normalized probability distribution P (Λ) of Λ of these meta-stable solutions as we sweep through all discrete values of the flux parameters F i . Assuming a"dense discretuum" for each F i , we may treat each P i (F i ) as a continuous function. For smooth P i (F i ), simple probability properties show that P (Λ) easily diverges at Λ = 0 [13], implying that a small Λ is statistically preferred. Ref. [1] finds that the resulting P (Λ) in the racetrack Kähler uplift scenario diverges (i.e., peaks) sharply at Λ = 0. That is, an overwhelmingly large number of meta-stable vacua have an exponentially small positive Λ, so statistically, we should end up in one of them. In short, a dS vacuum with a very small Λ is statistically natural.
In this paper, taking the median value Λ 50 in this racetrack Kähler uplift model to match the observed value, a natural scale emerges, We argue that, in the absence of fine-tuning, statistically, the electroweak scale µ ∼ m. That is, the scalar masses both before and after spontaneous symmetry breaking as well as the resulting vacuum expectation value (vev) ∼ µ of an open string mode φ are of the scale m, within some orders of magnitude. This suggests that string phenomenology should focus in regions of the landscape where Λ is naturally very small.
Following the standard supergravity formalism, the F-term effective potential V (and its minimum) is quadratic in the superpotential W and/or its derivative. So it should not be a surprise that dimensional arguments alone suggest m ∼ |W | 1/3 ∼ |ΛM 2 P | 1/6 . Here we give an explicit string theory scenario to realize this property in a concrete statistical way, when we match the median value Λ 50 in the string landscape to the observed value. However, due to the statistical nature of this result, a reliable estimate of the uncertainty in µ is lacking. It will be interesting to study in greater details this (and other) string theory model(s) the robustness of this intriguing order-of-magnitude property.
After an extensive review of the racetrack Käher uplift model to reach the key point of this paper, we explore the role of a Higgs-like field in this string theory framework. The electroweak scale µ ∼ m emerges naturally in the absence of fine-tuning.

A Racetrack Kähler Uplift Model of Flux Compactification
To be specific, let us review the racetrack Kähler uplift model studied in Ref [1], with the addition of a Higgs-like field. We consider a 6-dimensional Calabi-Yau (CY) manifold M with a single (h 1,1 = 1) Kähler modulus T and two (or three, i.e., h 2,1 = 2 or 3) complex structure moduli U i , so the manifold M has Euler number χ(M) = 2(h 1,1 − h 2,1 ) < 0. This simplified model of interest is motivated by orientifolded orbifolds [14,15], and it is given by (setting M P = 1), Here, M 2 P ≃ V/α ′ . The non-perturbative term W N P for the Kähler modulus T is introduced by gaugino condensates in the superpotential W to stabilize the T modulus [16], in which the two terms form the racetrack, with the coefficients a = 2π/N 1 for SU(N 1 ) gauge symmetry and b = 2π/N 2 for SU(N 2 ) gauge symmetry. The flux parameters c i , b i , d i and α ij in W 0 (U i , S) are to be treated as independent (real) random variables with smooth probability distributions that allow the zero values, while the dilation S and the complex structure moduli U i are to be determined dynamically. W 0 (φ) and K H for the Higgs-like φ will be discussed later. The dependence of A and B (also functions of some flux parameters) on U i , S is suppressed. The model also includes the α ′ -correction (the ξ term) to the Kähler potential to lift the supersymmetric solution to de-Sitter space [17,18]. This lifting to dS space is different from the KKLT scenario [16]. The Kähler uplift model has been well studied [6,7,19], and so has the racetrack model [8][9][10]. They are merged into the model where all parameters are replaced by flux parameters to be scanned over [1].
The superpotential W 0 (U i , S) and its supersymmetric solutions (D U i W 0 = D S W 0 = 0) have been studied in some detail [5,7]. Here we simply state that W 0 (U i , S, φ) takes some value W 0 ≡ W 0 (U i , S, φ) = W 0 (U i , S, φ)| sol after the equations have been solved. It turns out that the solved value of W 0 varies little by the Kähler uplift, i.e., solving the equation for T . Our goal here is to show that W 0 is expected to be exponentially small. Here we closely follow Ref. [1], where more details can be found. Validity of a number of approximations taken can also be found in Ref. [5,7].
In the large volume region, the resulting potential may be approximated to, with T = t + iτ , Extrema can be found by imposing ∂ t V = ∂ τ V = 0, where the latter is immediately satisfied for y = 0 (extrema with y = 0 are not minima, see Ref. [1]), while the former yields where the T -dependence of W 0 in the Kähler uplift is negligible [5]. Plugging (2.3) into the Hessian (mass squared) components, and recasting the result in terms of λ(x, y), we find So the stability condition (positive mass squared for both x = at and y = aτ at the extremum) puts a strong constraint on the value of λ = −2V | ext /a 3 AW 0 . Requiring both of them to be positive (hence the extremum is a minimum) gives, and AW 0 < 0. Here we have in mind W 0 > 0 and A < 0, β 1 and x ∼ O(100) respectively. The e −x factor suggests very small λ. The stability condition (2.5) in the large volume approximation takes the form 0 ≤ λ min ≤ λ ≤ λ max . So we see that a positive but small Λ is guaranteed together with the large volume V and β 1. For large x, λ min → λ max so at leading order, and therefore Λ approaches an exponentially small positive value at the large volume (x → ∞) limit. UsingĈ (2.2), one can show that in this limit, where the x 7/2 term is crucial to satisfy the assumption |A|e −x |W 0 | ≪ 1. Using (2.6) and (2.7), we can easily obtain, As it might be expected from the exponential terms in (2.8) and (2.7), the bigger the volume modulus, the smaller W 0 and Λ have to be in order to find a solution. Note that ξ = 0 implies Λ = 0, a property of the no-scale structure in supergravity.
As explained in Ref. [1], we can analyze the probability distribution of the cosmological constant, P (Λ). After randomizing A, B and W 0 , we collect all the classically stable solutions and find that the probability distribution P (Λ) for small positive Λ is approximately given by [1], So for β 1, we see that the diverging behavior of the properly normalized P (Λ) is very peaked as Λ → 0. We see that the expected value of Λ is very sensitive to the value of β. Due to tadpole-cancellation and other constraints in F-theory, we expect the value of N max to be bounded. Ref. [20] finds that N max can easily exceed a hundred. In principle, we should also scan through all allowed values of N in the SU(N) gauge groups, i.e., N = 2, 3, 4, ...N max . It turns out that the divergence of P (Λ) at Λ = 0 is dominated by the most divergent term, i.e., the smallest β, or β = N max /(N max − 1). For simplicity, we shall simply use the smallest allowed β to perform our estimates.
It is informative to introduce the value Λ Y , defined as the value of Λ such that the cumulative distribution is Y %, In this sense, Λ 50 is the median. It is interesting that one can find very simple formulae for Λ 10 and Λ 50 as a function of N max , Λ 10 ≃ 10 1.57−1.91Nmax , Λ 50 ≃ 10 −2.61−0.59Nmax (2.10) when N max is large. In Table 1 and 2, we present two cases, namely Λ 10 , Λ 50 matching the observed Λ ∼ 10 −122 .
Setting the median Λ 50 equal to the observed Λ ∼ 10 −122 M 4 P , and recalling that the superpotential has mass dimension 3, Eq.   If we match the observed Λ to Λ 10 , that is, there is only a 10% probability that Λ has a value smaller or equal to the observed value, we find that m drops by less than one order of magnitude.
Although going from one term to two terms (the racetrack) in W N P makes a big difference, including additional non-perturbative terms in W N P makes little quantitative difference. It has already been shown that including additional Kähler moduli does not change the qualitative picture [1]. In this sense, the above order-of-magnitude estimate of m is robust.

The Higgs Sector
Perhaps the simplest way to implement a Higgs-like field in the effective theory for the racetrack Kähler uplift model [1] is via D3-brane separation. Let us briefly review the setup we have in mind. Dynamical flux compactification introduces warp geometry due to branes, O-planes and background fluxes [21]. That is, in Einstein frame, ds 2 10 = e 2w(y) g µν dx µ dx ν + e −2w(y)g mn (y)dy m dy n , where e −4w(y) is the warp factor, the 4-dimensional metric g µν is either Minkowski, AdS or dS, while the 6-dimensional metricg mn is the underlying CY-like metric. A realistic picture envisions a bulk with warped throats attached to it. A D3-brane tends to sit at the bottom of a throat, the geometry of which may be described by a deformed conifold. Let the complexified D3-brane position for the I-th D3-brane be Z i I (i = 1, 2, 3), where we choose the co-ordinate where Z i = 0 at the tip of a particular conifold. However, because of the deformation, the bottom of that deformed (or resolved) conifold ends at r 0 (in theg metric), and so all D3 positions must be |Z i I | ≥ r 0 . While brane positions Z i I , are good Kähler moduli for the effective field theory, the presence of D3-branes leads a redefinition of both the Kähler coordinate T and the Kähler potential. Using Φ i = Z i /2πα ′ , we have [22][23][24] where the sum over i, I are implied, and k(Φ,Φ) is the "little" Kähler potential of the underlying internal metricg which we approximated to δ ij Φ iΦj . Now, let us consider just two D3-branes I = 1, 2 and work in the regime where, for each brane This situation is indeed realized in the large volume limit, t ≫ 1 and even more if branes are located at the bottom of a throat, |Z i 1 |, |Z j 2 | r 0 (i.e., |Φ i 1 |, |Φ i 2 | r 0 /2πα ′ ). In the latter case α ′ |Φ| 2 is minimized, since α ′ |Φ| 2 ≃ (2π) −2 α ′−1 r 2 0 ∼ r 2 0 /R 2 is the smallest it can get (we used the fact that the size of the throat R is typically of string scale R ∼ α ′1/2 ≫ r 0 ). To leading approximation we can thus treat (3.4) as a perturbation, alongside the α ′ correction ξ.

Now, define the triplets
Since these are linear combinations of Φ i 1 , Φ i 2 , they are good Kähler coordinates. φ i corresponds to the D3 separation and we will identify one of the φ i as our Higgs-like field. Considering (3.4), we can expand the Kähler potential (3.2) as where we re-introduce the uplifting ξ term. Here, the scalar fields have mass dimension one (to follow standard notation in phenomenology), as opposed to the dimensionless moduli. K H can easily be extended to more fields if we consider a stack of branes.
To consider a Higgs-like field, let where φ u and φ d transform oppositely under a symmetry (U(1) or SU(2) × U(1)), so the superpotential is invariant. Under spontaneous symmetry breaking (SSB), Goldstone bosons appear in the absence of coupling to gauge fields. For the sake of simplicity, let us restrict now to a single scalar field, φ ≡ φ 1 , and study the resulting effective theory with For now, let us treat c φ , ǫ, µ and ρ are independent flux parameters or functions of flux parameters to be scanned over. We shall set ǫ = 0 to simplify the discussions below.
Let us now insert this W 0 (φ) (3.7) and K H (φ) (3.5) into the F-term potential V (2.1) and compute m φ and Λ. First, S, U i , φ are stabilized supersymmetrically, D S W = D U i W = D φ W = 0, and this is followed by the stabilization of T due to the interplay between ξ and non-perturbative terms, when supersymmetry is broken by this uplift to dS vacua. Due to (3.4), to leading order D φ W ≃ ∂ φ W 0 (φ), and this has two solutions : in the absence of SSB, with φ = 0, or in case of spontaneous symmetry breaking (SSB), with φ = −2µ/3ρ = 0 (at leading order).
When φ = 0, it is clear that the analysis of the vacuum energy proceeds exactly as in the previous discussion. Indeed, since t, τ are much lighter than S, U i , φ (whose masses are determined by fluxes), one can focus the attention on the effective theory in T , treating S, U i , φ as already stabilized to their vevs. The only change with respect to the previous discussion is the constant flux parameter in W 0 (U i , S, φ), now replaced by c 1 → c = c 1 + c φ . On the other hand, in order to find the order of magnitude of m φ , we can totally neglect α ′ and non-perturbative corrections. Exploiting the condition (3.4) for the computation of the mass matrix, we see that at leading order (and after canonicalising kinetic terms) Taking into account α ′ and non-perturbative corrections will confer a small mass to t, τ while shifting all other mass values by negligible amounts. This is also a good estimate of the m φ in the SSB case. The moduli masses have been evaluated in Ref. [12]. Both m t and m τ are typically much lighter than φ and are closer to Λ/M 2 P . Without fine-tuning, we see that µ sets the electroweak scale. It should be clear that the same conclusion applies to the two Higgs doublet model (3.6), where in Eq.(3.12),ĉ 2 = c 2 − µ 2 /η after SSB.
In the case of SSB, φ ≃ −2µ/3ρ ∈ R, the situation gets a bit more involved. However, it turns out that, to a good approximation, m 2 φ is still given by (3.8) 2 The potential (2.1) also gets shifted by the K H term. We look for extrema and impose positivity of the Hessian, and find, for x ≫ 1 (see Table 2), with W 0 = W 0 (U, S) + c φ + 4µ 3 /27ρ 2 and We see that the new term gives a negligible contribution to (3.10) if dx 1/2 ≪ 1, that is when φ ∼ µ ρ ≪ 10 −2 M s . Therefore, in this regime, (3.9) collapses to (2.6), and Λ is essentially the same as in the φ = 0 case.
In summary, vacua with φ = 0 and φ = 0 do not differ much. The order of magnitude of m φ is the same (3.8), and it is determined by the scale of µ. The only difference consists in the expression for such W 0 : W 0 = W 0 (U, S) + c φ for φ = 0, and W 0 = W 0 (U, S) + c φ + 4µ 3 /27ρ 2 for φ = 0. Note that smooth probability distributions P (W 0 (U, S)) and P (W 0 (φ)) imply a smooth distribution P (W 0 ). In fact, P (W 0 ) can remain smooth even if both P (W 0 (U, S)) and P (W 0 (φ)) peak at zero. In the absence of fine-tuning, one expects all the flux parameters c, c 2 , b i , d i (each with mass dimension 3) as well as µ 3 to have comparable values. Assuming smooth probability distributions for the flux parameters, P (W 0 ) also has a smooth distribution [7]. In general, this W 0 has a wide range. However, setting Λ 50 equal to the observed value, m 3 = W 0 ∼ 10 −51 M 3 P is required to yield meta-stable solutions.
In the SSB case, there are 3 possible scenarios: (a) m 3 ≃ W 0 (φ) ≃ W 0 (U i , S). Then, µ ∼ ρ 2/3 m, and m φ ∼ ρ 2/3 t −1/2 m. This is also the case if In this case, the two terms (almost) cancel each other, giving the small m. Then, µ ≫ m, and so m φ ≫ m; (c) m 3 ≃ W 0 (U i , S) W 0 (φ). Then, µ ρ 2/3 m, and so m φ ρ 2/3 t −1/2 m. Since we have to scan over all values of the flux parameters, we expect that, in the absence of fine-tuning, µ ∼ m ≃ 10 2 GeV within some orders of magnitude.
In more realistic versions of the model, c φ , µ, ρ are expected to be functions of fluxes and can depend on U i and S [25], in which case, W 0 (φ) and W 0 (U i , S) are actually coupled. In general, coupling different sectors tend to render them to have comparable scales, so statistically, we expect that they have the same magnitude, W 0 (φ) ≃ W 0 (U i , S). In this sense, having m 3 ≡ W 0 (U i , S, φ) ≃ W 0 (φ) ≃ W 0 (U i , S) would be the most likely (and statistically natural) scenario, yielding a natural explanation also for the EW scale µ.
As an illustration, let us consider a W (φ) that depends on S. Since the dilaton S dictates all couplings (closed string coupling goes like 1/S so open string coupling ρ ∝ 1/ √ S), let us consider the simple case where (ignoring some order one numerical factors), where we have substituted in the vev for φ. Now we can solve the supersymmetric equations for U i and S, where W 0 (U i , S, φ) is now given by W 0 (U i , S) in Eq.(2.1) with c 1 → c = c 1 + c φ and c 2 →ĉ 2 = c 2 − µ 3 SSB (3.11) and no change in c 2 in the absence of SSB. In the α ij = 0 case (see Ref. [12] for the α ij = 0 case), the supersymmetric solution of W (U i , S) has been solved for real flux parameters [5,7]. For example, for the h 2,1 = 2 complex structure moduli case, one finds that and φ ≃ −µ c/ĉ 2 . Here s = Re(S) = c/ĉ 2 > 1 to stay in the weak coupling approximation. To satisfy Eq.(2.11), we can take W 0 ∼ c. Since couplings in the standard model is small but not vanishingly small, s 1, which implies W 0 ∼ĉ 2 . Without fine-tuning, Eq.(3.11) suggests µ ∼ m and this combined with Eq.(2.11) yields Eq.(1.1).
Again, the uncertainty of µ is hard to estimate: µ ≪ m if c ≫ µ 3 , and if (b i −sd i ) → 0, c and c 2 and so µ 3 can be much bigger, i.e., µ ≫ m. For h 2,1 > 2, W 0 (U i , S, φ) ∝ (c + sĉ 2 )Π(b i − sd i ), and s → c/ĉ 2 if any of the factor (b i − sd i ) → 0. For more non-trivial couplings within W 0 (U i , S, φ), the analysis becomes more complicated. To get a better determination of µ with respect to m, we need to determine the explicit functional forms of the flux parameters and their dependence on the moduli.

Discussions and Summary
Both the Kähler uplift model and the racetrack model are well studied in string phenomenology, so putting them together is natural [1]. The string theory landscape, generated by treating all parameters not as free parameters but as flux parameters or functions of flux parameters that we scan over, is crucial in allowing the statistical preference for an exponentially small Λ and a Higgs-like scale to emerge.
It is shown that a very large fraction of the classically stable vacua have an exponentially small positive Λ, so it is likely that our universe ends in a vacuum with an exponentially small positive Λ. Since this property is probabilistic, we cannot determine the precise value of Λ, but if we let the median Λ 50 to match the observed Λ, we find that a mass scale of m ∼ 100 GeV emerges. While the dilaton S, the complex structure moduli U i and Kähler modulus T are all closed string modes, a Higgs-like scalar open string mode φ is introduced so its mass scale matches this intermediate scale, µ ∼ m ∼ 100 GeV.
In scanning over all values of the flux parameters (or functions of flux parameters, like W 0 ), we note that only tiny ranges of some of them lead to classically stable vacua. That is, large patches of the landscape have no meta-stable solutions. In particular, for N max ≫ 2, only a tiny range of around a very small W 0 contributes to the solutions. So, replacing a flat probability distribution P (W 0 ) by a peaked (at W 0 = 0) distribution does not make much of a difference to P (Λ). On the other hand, it remains to be seen whether the "dense discretuum" is dense enough to accommodate such a small W 0 . At times, a soft Higgs term or a D-term is present to the effective V in supersymmetric electroweak phenomenology. Such a term must couple to the closed string sector, in particular the moduli as well as S. It will be interesting to see what is the resulting probability distribution P (Λ) and the Higgs scale. Based on rather limited studies so far, it seems that more couplings with more fields tend to enhance the peaking of P (Λ). Obviously more investigations are important. We believe that the search for the standard model of electroweak and QCD should only take place in regions of the landscape where Λ is exponentially small.
How about other scales in nature ? Another scale presumably is the inflaton scale in the inflationary universe scenario. Recent data indicates that the brane inflation model, natural in string theory, provides an excellent fit to the observation [26]. Another possible scale is the fuzzy dark matter model, where the dark matter is due to a scalar field with mass of order of 10 −22 eV. It turns out that some moduli in the string theory model [12], in particular an axionic mode [27], will fit in nicely. In short, the string theory landscape with no parameter is a fruitful playground to explore nature.