Radiative K\"ahler moduli stabilization

We propose a new type of K\"ahler moduli stabilization mechanisms in type IIB superstring theory on Calabi-Yau manifolds with the positive Euler number. The overall K\"ahler modulus can be perturbatively stabilized by radiative corrections due to sparticles. Its minimum is the anti-de Sitter vacuum, where supersymmetry is broken. We can uplift it to the de Sitter vacuum by introducing anti-D-branes, keeping the modulus stabilized. Although our numerical results depend on the choice of the cutoff scale and degeneracies of sparticles, at any rate there exist the parameter spaces where the masses of Kaluza-Klein and stringy modes are larger than the cutoff scale. Furthermore, this stabilization scenario predicts an ultralight axion.


Introduction
The moduli fields appear ubiquitously in four-dimensional low-energy effective field theory derived from superstring theory on six-dimensional compact space. These fields correspond to a geometrical character of compact space. Unless these moduli fields are stabilized at a high scale, it will lead to the fifth force. Hence, moduli stabilization is one of the major topics in the string cosmology and phenomenology.
So far, moduli stabilization has been well studied in the type IIB superstring theory on Calabi-Yau (CY) manifolds. The closed string moduli are categorized into the dilaton S, complex structure moduli U and Kähler moduli T . 1 Here, we denote the numbers of complex structure moduli and Kähler moduli by h 2,1 and h 1,1 , which correspond to the numbers of threecycles and four-cycles of CY, respectively. Three-form fluxes in the type IIB superstring theory can stabilize the dilaton S and all the complex structure moduli U at the compactification scale [3] and they generically generate the nonvanishing flux-induced superpotential W 0 = W flux [4]. The remaining Kähler moduli are stabilized at an anti-de Sitter minimum in the Kachru-Kallosh-Linde-Trivedi (KKLT) scenario [5] and LARGE volume scenario (LVS) [6] using the non-perturbative superpotential for the Kähler moduli. Such an anti-de Sitter minimum is uplifted to a metastable de Sitter minimum by introducing anti-D3 branes [5]. The F -term uplifting scenario is another way to realize the de Sitter minimum [7,8]. In the KKLT scenario, the stabilization of Kähler moduli is achieved by tuning |W 0 | ≪ M 3 Pl , where M Pl is the reduced Planck mass. On the other hand, even if |W 0 | ∼ O(M 3 Pl ), the LVS works for CY manifolds with the negative Euler number, χ = 2(h 1,1 − h 2,1 ) < 0, i.e., h 2,1 > h 1,1 > 1. 2 Note that stringy modes and Kaluza-Klein modes should be sufficiently heavier than all the moduli fields to justify description of the low-energy effective field theory. In this paper, along the line of Ref. [10], we propose a new type of Kähler moduli stabilization mechanisms in type IIB superstring theory on CY orientifolds with the positive Euler number χ > 0, namely h 1,1 > h 2,1 > 1. In Ref. [11], it was discussed that the overall vomlume modulus can be stabilized perturbatively by one-loop corrections to the Kähler potential. In that scenario, one needs a certain amount of fine-tuning of complex structure moduli to realize the Kähler moduli stabilization. In contrast to Ref. [11], here we focus on the radiative corrections due to the sparticles living on D7-branes wrapping a divisor of CY. Such corrections generate the potential of overall volume modulus through the Coleman-Weinberg potential [12]. Since the soft masses are functions of only the overall volume modulus, the overall Kähler modulus can be perturbatively stabilized at a sufficiently large volume region without tuning W 0 . In this large volume region, the string axion associated with the overall volume modulus remains light and it could be a candidate of dark matter. Other Kähler moduli could be stabilized by the non-perturbative effects and/or moduli-dependent D-terms. (See, e.g. Refs. [13].) We find that stringy and Kaluza-Klein modes are sufficiently heavier than sparticles and Kähler moduli at the anti-de Sitter minimum. We can uplift the anti-de Sitter vacuum to the de Sitter vacuum by introducing anti-D branes, and such uplifting does not change the behavior of the moduli stabilization.
The remaining paper is organized as follows. In Sec. 2, we discuss the stabilization of overall Kähler modulus without uplifting terms in type IIB superstring theory on CY orientifolds with D7-branes. We add the uplifting term in the setup of Sec. 3 to achieve a tiny cosmological constant. We study the modulus potential with the uplifting terms analytically and numerically in Secs. 3.1 and 3.2, respectively. Other Kähler moduli can be stabilized by two scenarios. One is due to moduli-dependent D-terms as discussed in Sec. 3.3, and the other is due to non-perturbative effects as discussed in Sec. 3.4. Also, in Sec. 3.5, we show that the nonperturbative effects generate the mass of ultralight axion associated with the volume modulus. Finally, Sec. 4 is devoted to the conclusion.

Setup
Let us consider the stabilization of closed string moduli on the basis of type IIB superstring theory on CY orientifolds with D7-branes. For a general class of CY threefolds, the total Kähler potential at the leading order of α ′ is obtained through Kaluza-Klein reduction of type IIB supergravity action [14,15], where V is the volume of CY manifold in Einstein frame measured by the string length l s = 2π √ α ′ , K(S, U) represents the Kähler potential of dilaton S and complex structure moduli U. The leading α ′ -correction is characterized by where χ is the Euler number of CY and g s is the string coupling [14]. Here, Q (i) a are the matter fields living on D7-branes wrapping the divisor D i ∈ H 4 (CY, Z) and their Kähler metrics Z (i) aā are given by with n a i being the modular weights and T i = τ i + iσ i with i ∈ {1, 2, · · · , h 1,1 + } denotes the Kähler modulus. 3 Also, the gaugino fields couple to the Kähler moduli through the gauge kinetic function To stabilize the complex structure moduli U and dilaton S, we consider the following superpotential induced by imaginary self-dual three-form fluxes [4], By the flux-induced superpotential, all the complex structure moduli and dilaton fields are stabilized at the compactification scale [3]. When W 0 = W flux = 0 at the minima of the complex structure moduli and dilaton, the scalar potential is nonvanishing because of the breaking of the so-called no-scale structure [14] where K T i = ∂ T i K and K T iTj is the inverse of Kähler metric K T iTj = ∂ T i ∂T j K. The nonvanishing F -term of the Kähler modulus generates the soft scalar masses and gaugino masses [16], with evaluated in the large volume region. 4 Here, we evaluate M Pl ≃ −m 3/2 M Pl , which is satisfied in a general class of CY threefolds with a sufficiently large volume. When these sparticles contribute to the radiative corrections, the 1-loop Coleman-Weinberg (CW) potential is given by where M characterizes the mass matrix for canonically normalized bosons and fermions, and with c a b and c f being the multiplicities of bosons Q (i) a 5 and fermions, respectively. Here, we take the limit M 2 /Λ 2 ≪ 1, where our description of low-energy effective theory is valid and the supertrace is defined as for an arbitrary function f . We consider the contribution of ghosts in the gauge 3 µ=0 γ µ ψ µ = 0, where ψ µ and γ µ are the gravitino and four-dimensional gamma-matrices, respectively. (For more details, see Refs. [17].) Obviously, there is no contribution from sfermion fields with n a i = 1. On the other hand, when the visible sector consists of the MSSM with singlets and multi-Higgs doublets, the multiplicities are given by 4 We take W 0 as a real constant for simplicity and discuss the case with n a i = 0 later. 5 We use the same notation of chiral superfields Q a and their lowest components.
where n H and n S denote the numbers of multi-Higgs doublets and singlet fields. Note that the following moduli stabilization is also applicable to the scenario where sparticles living on hidden D7-branes contribute to the radiative corrections. If there exist multiple D7 i -branes wrapping the divisor D i , the matter fields Q (i) a and gauginos living on D7 i -branes contribute to the radiative corrections through the soft masses (7). For sake of simplicity, we discuss the D7-branes wrapping only the divisor D i with volume τ i in the following analysis.
To justify this low-energy effective action, the soft masses and masses of Kähler moduli m T should be smaller than those of Kaluza-Klein (KK) modes, stringy modes, complex structure and dilaton moduli, namely and their typical masses are given by with N > 1 being a parameter determined by three-form fluxes. Note that we have taken the string-frame volume V s = Vg where R is the typical length of CY in string units. Since the cutoff scale is a physical quantity in the non-renormalizable theory, there exist several options for the cutoff scale. For concreteness, we take the cutoff scale as the typical masses of KK modes in the following analysis, since there exists N = 2 supersymmetry above the KK scale. The CW potential is then valid below the cutoff scale. The string loop corrections to the Kähler potential was discussed in Ref. [11], and they can also be interpreted as the 1-loop CW potentials [18]. However, in this paper, we focus on the situation where sparticle contributions dominate over the 1-loop CW potential. 6 Since the following analysis requires that a relatively large number of sparticle contribute to stabilize the CY volume, they could dominate over the string loop corrections to the Kähler potential controlled by the string coupling.

Stabilization of overall volume modulus
In the following analysis, we first consider the large volume region V → ∞, where τ i → ∞ with i = 1, 2, · · · , h 1,1 for all of the divisors. (In the LVS, some of the divisor volumes are not extremely large in string units [6].) When there exist the D-brane instanton effects or gaugino condensation on D7-branes, the non-perturbative superpotential for the Kähler moduli is generated as where A i are functions of the complex structure moduli and a i = 2π for the brane instanton and a i = 2π/N for the gaugino condensation on N stacks of D7-branes wrapping the divisor with volume τ i . In the moduli space τ i → ∞, these non-perturbative effects are negligible and the total scalar potential consists of the leading α ′ -corrections and radiative corrections. We come back to these non-perturbative effects in Secs. 3.3-3.5.
Including the radiative corrections due to sfermions and gauginos, we can write the scalar potential by It turns out that the F -term scalar potential is a function of only the overall volume V. In the large volume regime V ≫ 1, the above scalar potential is further approximated as Let us fix the cutoff scale as the KK scale and consider the following units and then the dimensional quantities are rewritten as In units of m KK , the approximated scalar potential reduces to From this scalar potential, we aim to find the minimum of V in the large volume regime V ≫ 1. The extremal condition of modulus field ∂ V V F = 0 in the large volume limit, is satisfied at in string units. In particular, when the CY volume is dominated by a single Kähler modulus V ≃ κ(T +T ) 3/2 with κ being a real positive constant, the value of overall volume modulus is evaluated as where T ≡ τ + iσ. We find that the ratio c 1 /ξ should be negative and larger than 10 2−3 such that the Kähler moduli space resides in the physical domain. Furthermore, the positivity of requires that c 1 should be positive. That indicates the negative ξ from Eq. (23). Obviously, the sfermions with n a i = 1 have no contribution in the potential. Thus, it turns out that the case with n a i = 1 for all sfermions is prohibited, because such a case corresponds to the negative c 1 from Eq. (10). In addition, the potential energy becomes negative at this minimum, namely anti-de Sitter minimum, where ξ is negative. At this minimum, the mass scales of typical modes become Especially, when the CY volume is dominated by a single Kähler modulus, the mass scale of canonically-normalized overall Kähler modulus is In this way, we find that the volume modulus can be perturbatively stabilized at the anti-de Sitter minimum, where four-dimensional supergravity description is reliable. The nonperturbative effects for the overall Kähler modulus are suppressed enough by the value of Kähler modulus, e −2πτ ≪ 1, but it generates the potential of remaining massless axion as discussed in Sec. 3.5. However, the negative ξ indicates that we have to consider the CY threefolds with the positive Euler number, namely a large number of Kähler moduli compared with that of complex structure moduli 7 . (In the LVS, the negative Euler number is required to realize the large CY volume.) Since such CY threefolds with the positive Euler number account for half of the whole CY threefolds in the sense of mirror symmetry, the CYs with the positive Euler number are not so restricted manifolds, but generic. The O7-plane contribution will also help us to change the number of CY Euler number in a weak coupling limit [21]. In Secs. 3.3 and 3.4, we discuss the stabilization of other Kähler moduli. Although for a generic value of e K(S,U ) /2 |Ŵ 0 |, the overall volume modulus can be stabilized at the anti-de Sitter minimum in a way similar to the LVS, we require the relatively large c 1 to realize the large volume of CY, e.g., c 1 ≃ O(10 2 ) for |ξ| ≃ O(10 −1 ). Such a situation is easily realized by the models with multi-Higgs doublets and singlets via Eq. (13), and such models often appear from concrete string model constructions.
(For the string model building, see e.g. Refs. [22,23].) Finally, we comment on the case with no leading α ′ -corrections, which corresponds to the CY manifold with the vanishing Euler number, i.e., χ = 0. Then, the scalar potential consists of only the CW potential, with It can be rewritten in units of Λ = m KK = 1, in the large volume regime V ≫ 1 and e K(S,U ) /2Ŵ 0 ∼ O(1). When c 2 is almost the same order of c 1 , we cannot find the minimum with a sufficiently large CY volume.

Numerical estimation
In this section, we numerically analyze the stabilization of moduli fields. The unapproximated scalar potential of the overall volume modulus (17) in units of Λ = m KK is drawn as functions 7 Our result is similar to Ref. [20], where the higher-derivative corrections are taken into account. At this vacuum in Fig. 1, the CY volume is determined by In addition, the mass scale of canonically-normalized overall Kähler modulus is when the volume of CY is dominated by the single Kähler modulus T = τ + iσ. Hence, even when we involve the next leading terms in the scalar potential, which are volume-suppressed, the numerical result is in agreement with the analytical one in Sec. 2.2.

Kähler moduli stabilization with uplifting
In this section, we introduce the uplifting term to achieve a tiny cosmological constant. In particular, we focus on the anti-D3 branes to uplift the anti-de Sitter minimum to the de Sitter minimum.

Analytical estimation
When there exist anti-D3 branes at certain moduli space of CY manifold, the uplifting term is given by where ǫ is a real constant suppressed by the warp factor. Hence, the total scalar potential is a sum of V F in Eq. (17) and V up . Similar to the analysis in Sec. 2, we aim to find the minimum of modulus field analytically. In the large volume regime, the above scalar potential in units of m KK is simplified as from which the minimum of overall volume modulus is determined by solving ∂ V V = 0, and ǫ is chosen such that V ≃ 0, In particular, when the CY volume is dominated by a single Kähler modulus V ≃ κ(T +T ) 3/2 with κ being a real positive constant, the modulus value at this minimum is evaluated as It turns out that the deviation from the SUSY-breaking anti-de Sitter minimum to the uplifted minimum is estimated as Our scalar potential still resides in the supergravity-controlled regime, since the mass scales of typical modes have a desirable hierarchical structure, Furthermore, when the CY volume is dominated by a single Kähler modulus, the mass scale of canonically-normalized overall Kähler modulus is We find that the volume modulus is still stabilized at the supergravity-reliable de Sitter minimum, even after the anti-de Sitter minimum is uplifted to the de Sitter vacuum. By setting e K(S,U ) /2 |Ŵ 0 | = 1, g s = 0.1, and ξ = −0.1, we list the volume of CY, the typical scales of gravitino, moduli fields, KK and stringy modes for several values of c 1 in Table 1 Table 1: The volume of CY (V) and divisor (τ ), and typical scales of several modes, setting e K(S,U ) /2 |Ŵ 0 | = 1, g s = 0.1, and ξ = −0.1.

Numerical estimation
Next, let us numerically estimate the unapproximated scalar potential V = V F + V up setting the same parameters as ones in Sec. 2.3 and under which the scalar potential with the uplifting term is drawn in Fig. 2.
In the left panel in Fig. 2, the CY volume is determined as and in the right panel, the vacuum expectation value of overall Kähler modulus is The mass scales of typical modes become and in addition the mass scale of canonically-normalized overall Kähler modulus is Hence, even when we involve the next leading terms in the scalar potential, which are volumesuppressed, the numerical result is in agreement with the analytical one.

Inclusion of the D-terms
So far, we have not considered the non-perturbative effects for the Kähler moduli, since those are negligible in the large volume regime V → ∞, where τ i → ∞ for all the volumes of divisors.
To stabilize the Kähler moduli fields expect for the overall volume modulus, we consider the D-term potential induced by the Fayet-Iliopoulos (FI) term [13]. When the anomalous U(1) symmetries appear on D7 i -branes wrapping the divisors τ i , the D-term potential is written by where f D7 i are the gauge kinetic functions of D7 i -branes, q represent the gauge fluxes and φ m with U(1) charges q are the canonically normalized matter fields living on D7 i -branes. In addition to these D7-brane contributions, anomalous U(1) symmetries on the fractional D3-branes located at the singularities of CY also induce the D-terms where f D3 k are the gauge kinetic functions of D3 k -branes with k being the number of singularities, q (D3 k ) T k represent the U(1) charges of moduli fields and ϕ m with U(1) charges q (D3) ϕ m are the canonically normalized matter fields living on the fractional D3 k -branes. In both cases, the anomalous U(1) gauge bosons eat the linear combination of string axions through the Stückelberg couplings and become massive at the compactification scale. In order not to spoil the stabilization of F -term potential, we require the vanishing D-term potential, which fix the linear combination of closed string moduli and matter fields (open string moduli).
In addition, we consider the F -term potential of matter fields, where Φ denotes φ m and ϕ m , m 2 Φ , A Φ , λ Φ are T i -dependent functions. When Φ develops a nonvanishing value, the D-term and F -term contributions are possible to stabilize all the moduli fields except for the overall volume modulus, although it depends on a topology of CY manifold. We leave an explicit moduli stabilization for a future work, since it is difficult to analyze the topology of CY threefold with huge h 1,1 . (For the model building in the LVS with small h 1,1 , see Ref. [24].)

Inclusion of the non-perturbative effects
In contrast to the previous section, we study alternative stabilization scenario for Kähler moduli other than the overall volume modulus. We consider the large volume regime V → ∞, but the volume of one divisor τ s is not extremely larger than the other Kähler moduli τ i , namely τ i ≫ τ s > 1 in string units. Note that we focus on the Kähler cone where τ s is larger than the string length to suppress the KK and stringy corrections.
Let us assume the non-pertrubative superpotential with respect to τ s , where W 0 = W flux , A s and a s are the constants depending on the origin of non-perturbative effects, e.g., a s = 2π for the brane instanton and a s = 2π/N for the gaugino condensation on N stacks of D7-branes wrapping the divisor with volume τ s . To simplify the following analysis, we take both the W 0 and A s as real constants. From the Kähler potential (1) and superpotential (54), the F -term scalar potential is simply given by in the large volume regime, where D T i W = W T i + K T i W with W T i = ∂ T i W and the CY volume is taken as V = ijk κ ijk 6 t i t j t k with κ ijk being the intersection numbers among the twocycles t i of CY. Now, the imaginary part of T s is set as its minimum and we use K T iTj KT j = −2τ i +O(V −1 ) and K T iTj = −4Vκ ijk t k +4τ i τ j +O(V −1 ) [25], where the divisor volume is defined as τ i = 1 2 κ ijk t j t k . Note that the first term in Eq. (55) is positive, since it is originating from the positive definite term. The simplified above scalar potential is a well-known form as discussed in the LVS, where the coefficient of α ′ -correction ξ is positive, namely h 2,1 > h 1,1 > 1. In LVS, the "small" modulus τ s and volume modulus V are stabilized at because of the positivity of ξ.
On the contrary, in this paper, we proceed to discuss the opposite sign of ξ, namely h 1,1 ≥ h 2,1 > 1. The scalar potential is given by a sum of the F -term and leading radiative corrections, where A s =Â s M 3 Pl and W =Ŵ M 3 Pl . In units of Λ = m KK = √ πM Pl /V 2/3 = 1, the scalar potential reduces to The extremal condition of τ s reads with W 0 =Ŵ 0 M 3 Pl , where we use ∂t j /∂τ s < 1 and a s τ s ≫ 1 to suppress the higher instanton effects. In this way, we obtain Another extremal condition of V gives rise to whereξ Consequently, the volume modulus is stabilized at

Ultralight axion
Finally, we take a closer look at the mass of axion associated with the overall volume modulus.
To simplify our analysis, we take into account the CY volume dominated by the single Kähler modulus T , i.e., V = κ(T +T ) 3/2 with κ being a positive real constant. When there exist the D-brane instanton effects or gaugino condensation on D7-branes, the axion potential can be extracted from the superpotential where W 0 = W flux and A (T ) are assumed to be real constants, and n = 1 for the brane instanton and n = N for the gaugino condensation on N stacks of D7-branes wrapping the divisor with volume τ . Such a non-perturbative superpotential term generates the axion potential, where T = τ + iσ. After canonically normalizing the axion θ = √ 2K TT σ, we obtain the tiny mass of axion and its decay constant, where the CY volume and gravitino mass are given in Eqs. (39) and (43). Thus, large volume of CY results in the ultralight axion associated with the volume modulus in a way similar to the LVS. Although the axion mass highly depends on the value of n, we list the typical axion mass in Table 2 It is interesting to discuss the astrophysical and cosmological physics of such an ultralight axion, which will be studied in a separate work.

Conclusion
We have discussed the stabilization of Kähler moduli using the leading α ′ -corrections and the radiative corrections due to the sparticles within the framework of the type IIB superstring theory on Calabi-Yau orientifolds with D7-branes. When all the volumes of the divisors in CY threefold are sufficiently large in string units, the non-perturbative effects for the Kähler moduli are suppressed enough compared with α ′ -corrections and the radiative corrections. We find that these perturbative corrections give rise to the stabilization of the overall Kähler modulus for a general class of CY threefolds, only if the Euler number of CY is positive. Since the volume of CY scales as the number of sparticles, we require that the relatively large number of sparticles contributes to the Coleman-Weinberg potential through the radiative corrections to achieve the large CY volume. Such a large number of sparticle contributions could dominate over the string loop corrections to the Kähler potential discussed in Ref. [11] where one needs a certain amount of fine-tuning of the complex structure moduli to realize the large volume of CY. Furthermore, our scenario of moduli stabilization does not require the tuning of flux-induced superpotential in contrast to the KKLT scenario. The Kaluza-Klein and stringy modes are sufficiently heavier than Kähler moduli and sparticles. In this reason, the low-energy effective action is controllable in the four-dimensional N = 1 supergravity. The vacuum in our scenario of moduli stabilization is the ant-de Sitter minimum. The structure of moduli stabilization is still maintained even after we uplift the anti-de Sitter minimum to de Sitter vacuum by introducing anti-D3 branes.
Note that a requirement of the positive Euler number is different from the conventional LARGE volume scenario, but CY threefolds with the positive Euler number are accounted for half of the CY threefolds in the sense of mirror symmetry. However, the positive Euler number of CY threefold indicates that we have to take into account a lot of Kähler moduli compared with the complex structure moduli. We expect that the D-term contribution from fractional D3-branes and magnetized D7-branes would lead to the stabilization of these Kähler moduli except for the overall volume modulus as discussed in Sec. 3.3. The non-perturbative effects also allow us to stabilize some Kähler moduli without spoiling the stabilization of volume modulus as shown in Sec. 3.4. In this moduli stabilization scenario, the axion associated with the volume modulus remains light and it could be a target of astrophysical and cosmological observations. It is interesting to show explicit moduli stabilization in a detailed setup, but we leave it for a future work.