Towards the core of the cosmological constant problem

We apply a new self-tuning mechanism to the well-known Kachru-Kallosh-Linde-Trivedi (KKLT) model to address the cosmological constant problem. In this mechanism the cosmological constant $\lambda$ contains a supersymmetry breaking term ${\mathcal E}_{\rm SB}$ besides the usual scalar potential ${\mathcal V}_{\rm scalar}$ of the $N=1$ supergravity, which is distinguished from the usual theories where $\lambda$ is directly identified with ${\mathcal V}_{\rm scalar}$ alone. Also in this mechanism, whether $\lambda$ vanishes or not is basically determined by the tensor structure of the scalar potential density, not by the zero or nonzero values of the scalar potential itself. As a result of this application we find that the natural scenario for the vanishing $\lambda$ of the present universe is to take one of the AdS (rather than dS) vacua of KKLT as the background vacuum of our present universe. This AdS vacuum scenario has more nice properties as compared with dS vacua of the usual flux compctifications. The background vacuum is stable both classically and quantum mechanically (no tunneling instabilities), and the value $\lambda =0$ is also stable against quantum corrections because in this scenario the perturbative corrections of ${\mathcal V}_{\rm scalar}$ and quantum fluctuations $\delta_Q {\hat I}_{\rm brane}^{(NS)} + \delta_Q {\hat I}_{\rm brane}^{(R)}$ on the branes are all gauged away by an automatic cancelation between ${\mathcal V}_{\rm scalar} + \delta_Q {\hat I}_{\rm brane}^{(NS)} + \delta_Q {\hat I}_{\rm brane}^{(R)}$ and ${\mathcal E}_{\rm SB}$.


I. Introduction
One the most mysterious problems in the area of high energy physics including cosmology can be summarized as why the vacuum energy (or the cosmological constant) of our present universe is so small despite that the supersymmetry of our universe is considerably broken. Recently there has been proposed a new mechanism to address the cosmological constant problem in the framework of type IIB supergravity [1], where the four-dimensional cosmological constant λ is forced to vanish by six-dimensional Einstein equation of the transverse sector, and therefore tunes itself to zero as a result. This mechanism is based on the viewpoint that our three-dimensional space is a stack of (visible) BPS D3-branes located at the conifold singularity of the Calabi-Yau threefold, and in this setup λ generally appears as brane − E SB . brane are expected to cancel out when supersymmetry of the brane region is unbroken. (The cancelation at one-loop order on the BPS D3-branes has been proven explicitly in Sec. VIIA of [1] for the case where the three-form fluxes of type IIB theory are turned off.) The last term E SB is a supersymmetry breaking term which originates from a gauge symmetry breaking of the R-R four-form A (4) arising at the quantum level in the brane region. So E SB is an energy scale of the supersymmetry breaking of the brane region, and at the same time it is also an energy scale of the gauge symmetry breaking (or an anomaly) generated by quantum fluctuations. (Finally in (1.1), κ 2 = 1/2M 2 pl where M pl is the four-dimensional Planck scale.) Eq. (1.1) is distinguished from the equation of the usual flux compactifications where λ is simply given by V scalar alone. According to (1.1) λ = 0 does not necessarily imply V scalar = 0 unlike in the usual N = 1 supergravity, for instance, in [2,3].
E SB is kind of an anomaly generated in the brane region, and for the D3-branes located at the conifold singularity of the Calabi-Yau threefold it takes (at one-loop order) the form T and δµ m T (φ) are given by (1.4) and (8.10), respectively) where ǫ 5 = det|ĥ mn | dψ ∧ dθ 1 ∧ dφ 1 ∧ dθ 2 ∧ dφ 2 is the volume-form of the base of the cone in the conifold metric ds 2 = dr 2 + r 2 dΣ 2 1,1 with so the volume of the base of the cone with unit radius is given by Vol(B) = ǫ 5 . Also the integration r 5 drǫ 5 in (1.2) is taken over the brane region, 0 < r < r B , where r B being the thickness of the brane, and the constant δ 0 is given by δ 0 = 6/[r 6 B Vol(B)]. Finally, ρ (1) T sources the supersymmetry breaking of the brane region (which is why E SB is called supersymmetry breaking term. See (8.11).), and it takes the form In the self-tuning mechanism of this paper, whether λ vanishes or not is basically determined -in the six-dimensional internal space -by the tensor structure of the scalar potential density, not by the zero or nonzero values of the scalar potential V scalar itself. Thus in our self-tuning mechanism, whether V scalar vanishes or not is not important unlike in the usual theories where λ = 0 is equivalent to V scalar = 0. In this paper we will apply this mechanism to the well-known scenario of KKLT [3] to address the cosmological constant problem, especially aiming at explaining the (cause of the) vanishing cosmological constant of our present universe. As a result of this application we find -basically in the framework of the type IIB N = 1 supergravity -that the natural scenario for the vanishing λ of our present universe is to take one of the AdS (rather than dS) vacua of KKLT as the background vacuum of the present universe. This AdS vacuum scenario does not suffer from the problematics of the dS vacua of KKLT. The background vacuum is stable both classically and quantum mechanically (i.e., no tunneling instabilities), and the value λ = 0 is perturbatively (radiatively) stable unlike in the usual theories because in the self-tuning mechanism of this paper the perturbative and nonperturbative corrections of V scalar are all gauged away by an automatic cancelation between V scalar + δ QÎ brane and E SB .
II. Scalar potential of KKLT Kachru et al. have shown in the framework of the Klebanov-Strassler (KS) compactifications [4] that one can construct a de Sitter (dS) vacuum (of type IIB theory) with broken supersymmetry if we allow for nonperturbative corrections. They first obtained a supersymmetric anti-de Sitter (AdS) vacuum from the superpotential of the form where W 0 is a tree level contribution arising from the fluxes and does not contain the Kähler modulus ρ. The second term is a nonperturbative correction coming from Euclidean D3-branes [5], or the gaugino condensation in the N = 1 supersymmetric SU(N c ) gauge theory generated by the stack of N c coincident D7-branes wrapping four-cycles in the Calabi-Yau threefold [6]. Since W contains Kähler modulus the no-scale structure of the Lagrangian has been broken and the supersymmetric vacuum is now described by but not necessarily W = 0, where the covariant derivative D a W is defined by D a W = ∂ a W + (∂ a K)W , and where the Kähler potential K is given at the tree level of type IIB by (see [7]) In (2.3), τ is type IIB axion/dilaton and Ω is holomorphic three-form of the Calabi-Yau threefold M 6 From the superpotential W and the Kähler potential K one can construct the scalar potential of the N = 1 supergravity [2,7]: where G ab = ∂ a ∂bK, and a, b are summed all over the complex structure moduli τ i , the axion/dilaton τ , and the Kähler modulus ρ. For the no-scale structure [7,8] in which W = W 0 , (2.4) reduces to where i, j are now summed over τ i and τ , and the superpotential W 0 is given by where F (3) and H (3) are R-R resp. NS-NS three-form field strengths. If we take F (3) and , then the potential (2.5) fixes the moduli at values for which G (3) is imaginary self-dual (ISD) at the tree level [3]. But once the nonperturbative term comes in, G (3) will not be ISD anymore. Turning back to the superpotential (2.1), one can now concentrate only on the Kähler modulus ρ because the no-scale part (2.5) vanishes for the ISD G (3) . The scalar potential is therefore given by and using (2.1) one obtains [3] V scalar = 1 2κ 2 10 e Kτ +Kcs aA 0 e −aσ 2σ 2 where the axion in ρ has been set to zero and σ is defined by ρ = iσ. From (2.7) one finds that the minimum of V scalar takes negative values for the superpotential D a W = 0, and therefore it describes supersymmetric AdS vacua because V scalar is identified with the four-dimensional cosmological constant λ in the usual theories of flux compactifications including KKLT. At the final step of KKLT the AdS minimum is uplifted to a dS minimum by the D3-branes introduced at the tip of the KS throat where the introduction of D3-branes does not violate the tadpole condition. By this process V scalar acquires an additional term where D is given by 1 D ∼ a 4 0 p T 3 /g s where a 0 is the warp factor at the location of D3-branes and p\T 3 is the number\tension of the D3-branes. So in [3] Kachru et al. obtain dS vacua by fine-tuning the constant D so that the minimum of the resulting V scalar becomes very close to zero. III. A self-tuning mechanism for λ 1 Note that our convention is ρ = b √ 2 + ie 4u and the prefactor 1 2κ 2 10 e Kτ +Kcs of (2.8) has been omitted in (2.9).
Though the KKLT is an attractive scenario for the late-time cosmology with a small positive cosmological constant, it has some difficulties as for being a realistic model of our universe. One of the well-known defects of KKLT is related to the problem of tunneling instability, which is basically due to the fact that the scalar potential V dS of the dS minimum at σ = σ m takes positive (though it is very small) values, while V scalar asymptotically vanishes, V scalar | σ→∞ = 0. So the dS vacua of KKLT are only local minima of the potential and they eventually decay into the run away vacuum at σ = ∞ which corresponds to a Minkowski space with a large internal Calabi-Yau volume.
Such an instability is unavoidable as far as V scalar is directly identified with λ because in that case V scalar takes always positive values for the dS vacua and therefore it cannot avoid decaying into the stable vacuum at σ = ∞. This in other words means that the above tunneling instability can be avoided if V scalar can take different values from λ as in (1.1). In (1.1), λ contains an additional term E SB which possesses gauge arbitrariness as mentioned in (1.4). Hence in (1.1) we are allowed to take negative values for V scalar at the minimum σ = σ m in contrast to the case of KKLT, and this does not directly imply a negative λ because the negative values of V scalar can be compensated by E SB so that λ vanishes as a result. In Sec. 3.5 it will be shown -by using type IIB action -that λ is forced to vanish by a self-tuning equation (see (3.41)), and thus any nonzero V scalar (and δ QÎ brane as well) is gauged away by E SB so that λ tunes itself to zero as a result. In this section we will discuss about the basic principle of our self-tuning mechanism descried above, together with brief reviews of some formulas and ideas presented in [1] if necessary for reader's convenience.

Six-dimensional Einstein equation
In the string frame the type IIB action is given by where φ is the dilaton with e φ = g s eφ, and . Among these field strengths,F (5) is self-dual and the ansatz is given byF In addition to this we have the local terms where G µν is a pullback of the target space metric G M N to the four-dimensional brane world. Also T (φ) represents the tension of the D3-brane and at the tree level it is given by T (φ) = T 3 e −φ . But at the quantum level it becomes T (φ) = T 3 e −φ + ρ vac (φ), where ρ vac (φ) represents quantum correction terms and it is identified with NS-NS sector vacuum energy density of the three-dimensional space. Similarly, µ(φ) is simply µ(φ) = µ 3 at the tree level. But it turns into µ(φ) = µ 3 + δµ(φ) at the quantum level, where δµ(φ) is an R-R counterpart of ρ vac (φ) representing R-R sector vacuum energy density of the three-dimensional space. Upon reduction where µ, ν = (0, 1, 2, 3), m, n = (5, · · · , 10), the type IIB action (3.1) reduces into where L F is given by 6) and the topological term comes from the Chern-Simons term e φ A (4) ∧ G (3) ∧Ḡ (3) , which does not involve any moduli (except the dilaton τ ) or the metric. From (3.5) the six-dimensional action defined on the internal space can be written as thus on the brane, β = 4λ for the maximally symmetric spacetime. Varying (3.7) with respect to δh mn , one obtains where the energy momentum tensor T mn is defined by In (3.9), R mn and R 6 vanish at the classical level because the internal Calabi-Yau is Ricci-flat. But at the quantum level, h mn acquires correction terms in the equations of motion, 11) in our perturbation scheme (see (7.13)). Hence in (3.9)(and also in what follows) we are not allowed to take R mn (h mn ) = R 6 (h mn ) = 0, though we have R mn (h

Four-dimensional cosmological constant λ
The four-dimensional action defined on the external space can be obtained by rewriting (3.5) as where 2κ 2 ≡ 2κ 2 10 g 2 s / d 6 y √ h 6 eφ −2B , andÎ bulk is defined bŷ Adding (3.3) to (3.12) one can show that the total action I IIB + I brane can be written in the form where the cosmological constant λ is defined by Turning back to (3.6) we see that the Lagrangian L F can be written as where K mn and V , the kinetic and potential parts of the Lagrangian, take respectively the forms K mn = a,b F ab (φ c )∂ m φ a ∂ n φ b and V = V (φ a , h mn ), where φ a 's represent the six-dimensional scalar fields such asφ, B, A (0) and ξ etc. Namely in (3.16), while V involves h mn , K mn does not. Also in (3.16), V is related to V scalar by the equation and thus for the no-scale structure the potential density V arising from the fluxes is identified with − g 2 (see (4.3)) in the case of type IIB action. Now we substitute (3.10) -with L F given by (3.16) -into (3.9) and contract the indices m and n. Then we obtain 18) where N defined by N ≡ h mn ∂ ∂h mn . Again, we are not allowed to take R 6 = 0 in (3.18) because h mn in R 6 (and other fields in (3.18) as well) contains correction terms coming from quantum effects. But integrating (3.18) and using (3.13) one finds that Also substituting (3.19) into (3.15) (and using β = 4λ) one finally obtains which is now independent of R 6 (h mn ).

Self-tuning equation for λ
Now we proceed to obtain a self-tuning equation for λ, which is one of the main points of this paper. First, we substitute L F in (3.18) into (3.10) to get Next, substitute (3.21) into (3.9) and contract m and n. Then we obtain Let us repeat the same procedure again. Substitute (3.22) into (3.18) to obtain Next, substitute (3.23) into (3.10) to obtain Finally, substitute (3.24) back into (3.9) and contract m and n. We obtain β = 1 6 (3.25) (3.22) and (3.25) suggest that β always contains the operators (N − 1) and (N − 3) in common. We prove this as follows. First, we observe that β's in (3.22) and (3.25) both take the form where b 0 is a constant and Π(N ) is an operator of the form where n i are integers. So we start by assuming that β always appears in the form (3.26). Now we substitute (3.26) into (3.18) to obtain Next, substitute (3.28) into (3.10) to obtain (3.29) Finally, substitute (3.29) back into (3.9) and contract m and n. Then we obtain Note that (3.30) takes the form (3.26) again, which ensures that the prerequisite assumption (3.26) on β is valid. Also (3.30) shows that β always contains (N − 1) and (N − 3) acting on V , which proves the proposition. We obtained (3.30) starting from the Einstein equation (3.9). But (3.30) does not include R 6 of (the perturbed) h mn because it has canceled out during the process of obtaining (3.30). This suggests that λ (recall that β = 4λ) may not be affected by the internal curvature or the geometry of the six-dimensional space even at the quantum level at least in the supergravity framework. Besides this, (3.30) suggests a very important fact. According to (3.30), whether λ vanishes or not is entirely determined by the tensor structure of V , not by any other factors like zero or nonzero values of the scalar potential V scalar etc. We will be back to this point in Sec. 3.5.

Gauge symmetry breaking
brane at the quantum level We see from (3.3) that I brane consists of two (NS-NS and R-R) parts. Among these two parts, the second one represents an electric coupling of D3-branes to the R-R fourform A (4) , and it can be rewritten as where J µ 0 µ 1 µ 2 µ 3 is the world volume current density of the D3-brane, At the classical level J µ 0 µ 1 µ 2 µ 3 is just a solitonic current density, J µ 0 µ 1 µ 2 µ 3 sol , representing classical world volume dynamics of the D3-brane. In that case X µ (x)'s in (3.32) stand for the classical fields, X µ cl (x), defined on the world volume of the D3-brane, and for the embedding X µ cl (x) = x µ , J 0123 sol is simply µ 3 . At the quantum level, however, X µ (x)'s include fluctuations X µ ′ , X µ = X µ cl + X µ ′ . Since X µ ′ 's are fluctuations of the open string degrees of freedom, they correspond to the fluctuations of the standard model fields with support on the D3-brane. Due to these fluctuations J µ 0 µ 1 µ 2 µ 3 acquires an additional term, > represents quantum corrections arising from the fluctuations (of the standard model fields with support) on the D3-brane. Denoting J 0123 sol and < χ 0123 vac >, respectively, by µ 3 and δµ(φ), one can rewrite (3.31) as where we have used and the normalization convention d 6 y √ h 6 δ 6 (y) = 1 of the six-dimensional delta function. (3.33) is just the second term of (3.3), and where µ(φ) = µ 3 + δµ(φ) as before.
(2) Gauge symmetry breaking Going back to the classical level, the second term of (3.3) is invariant under the gauge transformation A (4) → A (4) + δA (4) with δA (4) ∂Σ is the boundary of the four-dimensional spacetime which, however, is assumed to have no boundary. But once we go up to quantum level, I >= 0. So the total J µ 0 µ 1 µ 2 µ 3 is not locally conserved at the quantum level, and the gauge transformation brane . Integrating by part one obtains from (3.31) that which generally takes nonzero values because so does < ∂ µ 0 χ µ 0 µ 1 µ 2 µ 3 vac >.
In addition to (3.35) there is another important variation of I (R) brane which plays a crucial role in our self-tuning mechanism. To find its explicit form, rewrite δ G I (3) and take an ansatz [1] where F (y) is an arbitrary function of the internal coordinates y m . (3.36) is the most appropriate ansatz which accords with (3.34) and therefore respects the Poincaré symmetry of our four-dimensional spacetime. Once we take which, at the classical level, vanishes for the embedding X µ cl (x) = x µ because ∂Y m cl /∂x α 0 = 0. So the nonzero contribution to J m123 comes from the quantum excitations < χ m123 vac >. Denoting < χ m123 vac > by δµ m T (φ) (and omitting the second term) one can rewrite (3.37) as where f m (y)'s are arbitrary functions of y m , representing local gauge parameters.

Brane action densityÎ brane
From (3.3) and (3.39) (or from (8.1)) one finds that at the quantum level the brane action consists of various parts, (3.40) Among these terms, I (N S) brane (tree) and I (R) brane (tree) are the tree level actions, and they always cancel out by field equations for the BPS D3-branes (see, for instance, Sec VI.C of [1] for this). The correction terms δ Q I (N S) brane and δ Q I (R) brane arise from ρ vac (φ) and δµ(φ), and they represent quantum fluctuations (of the gravitational and standard model fields with support) on the D3-brane. So δ Q I brane correspond to the gravitational plus electroweak and QCD vacuum energies of the standard model configurations of the brane region. These two terms are also conjectured to cancel out to all orders of perturbations in supersymmetric theories, though they do not when supersymmtry is broken. (The cancelation at one-loop order on the BPS D3-branes has been proven explicitly for the case G (3) = 0 in Sec. VIIA of [1].) In our self-tuning mechanism, however, it is not important whether such a cancelation occurs or not, as we will see in what follows.
Using β = 4λ, one can rewrite (3.30) as where V n represents a class of potential densities satisfying and in both cases λ vanishes by (3.41). In our scenario proposed in Sec. 5.2, the background vacuum of our present universe is identified with one of AdS vacua of KKLT, and in Secs. IV and VI it will be shown that these AdS vacua basically belong to V 3 , V AdS ∈ V 3 . So in our case λ is essentially given by (3.44). Let us go back to (3.40). We have seen in Sec. 3.4 that the last term δ GÎ represents the magnitude of gauge symmetry breaking ofÎ brane (caused by an anomaly < ∂ m χ m123 vac > = 0) arising at the quantum level, where < χ m123 > are quantum excitations on the branes with components along the transverse directions of the D3-branes. In [1], it was shown that δ GÎ from (3.39) and (3.45). Note that E SB contains arbitrary gauge parameters f m (y). This implies that E SB possesses gauge arbitrariness. Due to this property of E SB , any nonzero values of V scalar and δ QÎ brane in (1.1) can be gauged away by this E SB , and as a result λ vanishes (by (3.41)) as long as the potential density V satisfies V ∈ V n with n = 1 or 3.
The above self-tuning mechanism is distinguished from the theories where λ is directly identified with V scalar . In those theories including KKLT, λ is always unstable under perturbative (radiative) corrections because so is V scalar . Also the dS vacua necessarily imply V scalar > 0, which leads to a tunneling instability as mentioned in the opening paragraph of this section. But in our self-tuning mechanism described above, these are not to be the cases anymore. λ can vanish by (3.41) regardless of whether V scalar in (1.1) vanishes or not. So we can take V scalar < 0 (while maintaining λ = 0) to avoid the tunneling instability. Also the self-tuning λ = 0 is always stable against radiative corrections. Any nonzero contributions to V scalar and quantum fluctuations on the branes are forced to be gauged away by (3.41) once V satisfies V ∈ V n with n = 1 or 3, and the self-tuning λ = 0 is automatically achieved in our self-tuning mechanism of this paper. Hence in the following sections we will mainly check if our background configurations really satisfy V ∈ V n with n = 1 or 3.

IV. AdS vacua of KKLT and gravitino mass
In no-scale structure (and in the ISD background) λ trivially vanishes from (3.41) because potential densities arising from ISD fluxes all vanish. But once the no-scale structure is broken by nonperturbative effects as in AdS vacua of KKLT, the potential density does not vanish anymore because G (3) now acquires imaginary anti self-dual (IASD) components due to the presence of nonperturbative terms in the superpotential W . In general, the scalar potential (2.4) receives nontrivial contributions from both perturbative and nonperturbative corrections of the superpotential and the Kähler potential. More explicitly, while the superpotential receives only nonperturbative corrections W = W tree + W np as in (2.1) [9], the Kähler potential receives both perturbative and nonperturbative corrections K = K tree + K p + K np . So in order to maintain λ = 0, the density of scalar potential must remain to satisfy (3.42) with n = 1, 3 even under these corrections. In this section we want to check if the AdS vacua of KKLT belong to V n with n = 1, 3, or not.

AdS vacua of KKLT
The scalar potential arising from the fluxes can be obtained from the G mnpḠ mnp term of the action (see (3.6)). Rewrite the G mnpḠ mnp term as 2 (see [7]) 10 Imτ . The scalar potential (arising from the fluxes) is defined by the second term of (4.1) as Since V no−scale is given as to be ∝ G IASD , it (and its density as well) vanishes in the ISD compactifications where the superpotential is simply given by (2.6). But once the nonperturbative term is added as in (2.1), G (3) can not remain ISD anymore. The unbroken supersymmetry DW = 0 requires that G (3) must also contain (1,2) and (3,0) components (see for instance [10]) in addition to the ISD components. Hence in this case (4.3) receives nonzero contributions from these fluxes.
The nonzero density of (4.3), however, satisfies V no−scale ∈ V 3 , so it does not contribute to λ in (3.41). But (4.3) is only referred to the no-scale type potential (2.5). In the AdS vacua of KKLT there is another important contribution to V scalar coming from the nonperturbative superpotential (2.1). Namely from (2.7) one obtains under D ρ W = 0, which reduces to (2.8) by (2.1). (4.4) is associated with the nonperturbative correction because W in (4.4) contains the term A 0 e −aσ . The nonperturbative term can arise for instance from the gaugino condensation on D7-branes wrapping a four-cycle of the internal space [6]. In the heterotic string theory the three-form structure of the potential density with a gaugino condensation < trλΓ mnp λ > is manifest in the action [11], So the potential density associated with (4.5) obviously belongs to V 3 . In the case of type IIB theory, however, the structure of the potential density associated with (4.4) is not quite obvious at this point, and we need some procedure to find it out. Since G (3) contains both ISD and IASD components in the AdS vacua of KKLT, we decompose G (3) as where Ω is the holomorphic (3,0)-form and χ I denotes the basis of H (2,1) . Then using (4.6) one can express W 0 in (2.6) as On the other hand, W satisfies at the supersymmetric (AdS) minimum σ = σ m of KKLT, where the index I labels the complex structure moduli. From (2.3) and the definition of the covariant derivative D I one finds that (4.8) requires W to take the form whereᾱ can depend on τ , but not on the complex structure moduli. Substituting (4.7) and (4.9) into (2.1) also gives whereb 0 =ᾱ −ᾱ 0 . The above equations show that W , W 0 and the nonperturbative term A 0 e −aσm all have the same structure ∝ Ω ∧Ω at the supersymmetric minimum of KKLT. From (4.7), (4.9) and (2.6) W can now be written as where c =ᾱ/ᾱ 0 . 3 (4.11) shows that W is a superpotential generated by an effective three-form flux G eff (3) , where the nonperturbative effect is merged with G (3) to form G eff (3) .
So at least at the supersymmetric minimum the effect of the nonperturbative term A 0 e iaρ in W is to change G (3) into a new flux G eff (3) which also has the three-form structure like the original G (3) .

Gravitino mass
The scalar potential arising from (2.1) does not vanish at the supersymmetric minimum of the potential. In general it is proportional to |W 0 | 2 , or more precisely, at the extremum of the potential [9,12,13]. Indeed at the AdS minimum D ρ W = 0 of KKLT, the coefficient A 0 is given by and therefore the scalar potential (2.8) becomes proportional to e K |W 0 | 2 there, where K cs , K τ are the Kähler potentials for the complex structure moduli and the axion/dilaton, respectively.
(4.15) shows that W 0 necessarily takes nonzero values in the presence of the nonperturbative correction A 0 e −aσm . The nonzeroness of W 0 implies that G (3) must contain (0, 3) component, and in the presence of this component the gravitino generally acquires nonzero mass m 3/2 from the G (3) flux. The gravitino mass term of the reduced action for the type IIB theory can be obtained through the decomposition where Ψ µ /ψ µ are the ten/four-dimensional gravitini, respectively, and η is a six-dimensional killing spinor satisfying γīη = 0, where γī is the six-dimensional Dirac matrix represented in the complex basis. In the real basis of the Calabi-Yau one obtains [13] where m 3/2 is identified as Since all components except η + γījkη * (= Ω¯ijk/ Ω ) of η + γ mnp η * vanish by γīη = 0 in the complex basis, only the (0, 3) piece of G (3) contributes to m 3/2 .
The result V AdS ∈ V 3 is not affected by the perturbative and nonperturbative corrections K = K tree + K p + K np and W = W tree + W np . The corrections K p , K np in K act only as multiplicative factors e Kp , e Knp in (4.13) (or in (4.14)), and on the other hand the corrections W np in W have already been taken into account in our discussions (namely in (2.1) and (6.3)). So the structure (4.13) of V AdS , and consequently the result V AdS ∈ V 3 does not change by K = K tree + K p + K np and W = W tree + W np .

V. dS vacua of KKLT and AdS vacuum scenario
Now we have to introduce anti-D3-branes at the end of the KS throat to obtain dS vacua. Introduction of anti-D3-branes induces an additional term δV scalar (see (2.9)) to the scalar potential as anticipated from the analysis of [14]. Thus the scalar potential after introducing anti-D3-branes must be the sum of (2.8) and (2.9), where V scalar in (2.8) (namely, V AdS ) has already been verified to respects (3.42) with n = 3 at the supersymmetric minimum. So the next procedure will be to check what happens to the structure of the potential density after adding (2.9) to the nonperturbative potential (2.8). Does the sum of these two potentials still respects (3.42) at the dS minima? As an answer to this question, we will first show in Sec. 5.1 that in general δV scalar (and consequently the sum of (2.8) and (2.9)) does not respect (3.42). This means that the density of δV scalar ( ≡ V D3 ) caused by an introduction of anti-D3-branes makes a nonzero contribution to λ in (3.41), and consequently λ of the dS vacua described by (2.8) plus (2.9) may not be fine-tuned to vanish unlike in the scenario of the original KKLT. Hence in the second part of this section (Sec. 5.2) we will propose an alternative scenario for the vanishing λ of the present universe. This alternative scenario uses AdS, instead of dS, vacua of KKLT, and it does not suffer from the problematics of the dS vacua of KKLT.

δV scalar due to D3-branes
In [14] the dynamics of anti-D3-branes is described by a Dirac-Born-Infeld (DBI) plus Chern-Simons (CS) world volume action for the NS5-brane due to technical difficulties in obtaining DBI action for the pure anti-D3-brane in the KS background geometry. In this S-dual description the anti-D3-branes are described by NS5-branes wrapping S 2 inside the A-cycle of the conifold geometry. At the apex of the conifold the metric becomes where a 0 and R 0 are constants, and the world volume action for the NS5-brane of type IIB theory takes the form (see [14] or [15]) where hmn is a two-dimensional metric induced along S 2 of the A-cycle, and 2πF (2) = 2πF (2) − A (2) with F (2) = dA a two-form field strength of the world volume gauge field of the NS5-brane. In (5.2) F (2) is assumed to satisfy so that the NS5-brane carries D3 charge p. R-R two-form A (2) is also assumed to satisfy which follows from the well-known R-R flux quantization A F (3) = 4π 2 M. The DBI part of (5.2) contains an internal metric hmn because (5.2) is an world volume action for the NS5-brane rather than genuine D3-brane. But using (5.3) and (5.4), one finds that (5.2) turns into which is typical of the world volume action for the D3/D3-branes. In (5.5), L D3 (ψ) can be written, upon takingψ = 0, in the form where M is related with hmn by the integral S 2 d 2 y det(hmn + 2πg s F (2) ) = 4π 2 Mg sV (ψ) , andV (ψ) ≃ p/M for ψ ≪ 1. The scalar potential for the anti-D3-branes can be read from L D3 (ψ) in (5.6) and it turns out to take the form (2.9). 4 Turning back to (5.5), L D3 (ψ) is given as a function of ψ and anti-D3-branes correspond to ψ = 0. But in the KS geometry, ψ = 0 is not a stable or a metastable point of the potential, and hence in the S-dual description the anti-D3-branes are necessarily described by the NS5-branes which occupy S 2 of the A-cycle in the internal space. So L D3 (ψ) in (5.5) necessarily contains the two-dimensional internal metric hmn implicitly in the form d 2 y det(hmn + 2πg s F (2) ) (see (5.6) and (5.7)), and we infer that the potential density V D3 will be of the form

dS vacua of KKLT and an alternative scenario
In KKLT, the scalar potential is so adjusted that the two terms (2.8) and (2.9) cancel out at the dS minima. So the scalar potentials of the dS vacua almost vanish at their dS minima σ = σ m , and we can write where ǫ is an arbitrarily small positive constant of order ∼ 10 −120 /O(V AdS ). If V dS in (5.9) can take sufficiently small values, the corresponding λ will also be very small, and we may take one of the dS vacua of KKLT as the background vacuum of our present universe. However, this is true only in the traditional KKLT. According to our previous discussions it is not quite obvious whether such fine-tuning is really possible. The superpotentials of dS vacua do not satisfy DW = 0 at the dS minimum because the introduction of anti-D3-branes breaks the supersymmetry slightly. Thus the superpotential for the dS vacua of KKLT does not have the structure (4.9) (or (4.11)), and consequently V dS of dS vacua cannot be written in the form (4.13), which suggests that the corresponding V dS necessarily makes a nonzero contribution to λ in (3.41). Indeed in Sec. 5.1, we have seen that V D3 associated with (2.9) does not respect (3.42), which implies that δV scalar arising from anti-D3-branes makes nonzero contributions to λ in (3.41). In the case of AdS vacua, however, V AdS belongs to V 3 as shown in Sec.IV and it does not contribute to λ as opposed to the case of V D3 . So these things make us to doubt that λ of dS vacua can be really fine-tuned to vanish by adding V AdS and δV scalar in (2.8) and (2.9). Indeed, even when we accept the possibility of this fine-tuning, it is preserved only at the tree level. The problem is that once the perturbations enter, there is no way to make V dS remain to be of order ∼ 10 −120 in the units of Planck density. Namely the fine-tuning is perturbatively unstable.
The dS vacua of KKLT have some significant defects as for being a realistic model of our present universe, as listed below.
• The most significant defect of the KKLT scenario is that the cosmological constant λ is perturbatively (radiatively) unstable. In the usual flux compactifications including KKLT, V scalar , and therefore λ receives important corrections from g sperturbations. Besides this, λ is also affected by quantum fluctuations of NS-NS and R-R degrees of freedom with support on the D3-branes.
• dS vacua of KKLT have the well-known tunneling instability. Since dS vacua of KKLT have positive values of V scalar at their dS minima, they eventually decay into the stable minimum at σ = ∞.
• The dS vacuum scenario of KKLT suffers from another important tunneling instability. We have seen that dS vacua of KKLT are caused by anti-D3-branes. But these nonsupersymmetric configurations with anti-D3-branes are stable only classically. In the NS5-brane description [14] of anti-D3-brane, they decay by quantum tunneling, and roll down the potential until they reduce to a supersymmetric configuration with D3-branes located at the north pole ψ = π of the A-cycle.
• In order to obtain a late-time cosmology describing our present universe with nearly-vanishing cosmological constant we need to fine-tune the coefficient D in (2.9), together with the fluxes which are responsible for the compactifications of the internal dimensions. But both D and the fluxes vary discretely, and can not be easily fine-tuned with arbitrary precision. 5 These problems of dS vacua are unavoidable in the framework of the ordinary theories where V scalar is directly identified with λ. But once we accept the self-tuning mechanism of this paper the above problems can be solved immediately. The simplest, and perhaps the most natural scenario using KKLT is to take one of the AdS vacua of KKLT as the background vacuum of our present universe. (Again, recall that V scalar < 0 does not necessarily imply λ < 0 due to (1.1).) These AdS vacua can substitute for the dS vacua of KKLT in the framework where λ is given by (1.1), and they do not suffer from the above problems. The configuration of AdS vacuum with certain numbers of D3-branes may be identified with the supersymmetric (stable) minimum at ψ = π in the brane/flux annihilation description in [14]. Namely the nonsupersymmetric configuration of p anti-D3-branes rolls down − via tunneling and a classical process at some early stage during or after inflation − the potential to the north pole ψ = π (the stable minimum) to form a supersymmetric configuration with M − p D3-branes, which is now identified with the present stage of our universe. In this scenario the supersymmetry breaking of the brane region is basically generated by E SB , not by D3-branes (see Sec.VIII).
The AdS vacuum scenario does not suffer from the problems listed above.
• The AdS vacua of KKLT do not have the tunneling instability because V scalar of AdS vacua take negative values as opposed to the dS vacua.
• The supersymmetric configuration with no D3-branes corresponds to the stable minimum of the brane/flux annihilation description in [14]. There is no other minimum (or minima) to decay into.
• There is no any parameter or coefficient to be fine-tuned in the AdS vacuum scenario. λ = 0 is automatically achieved by the cancelation between V scalar (plus δ QÎ brane ) and E SB forced by (3.41).
• Most of all, the AdS vacuum scenario does not suffer from the radiative instability. Any contributions to V scalar coming from g s -perturbations and quantum fluctuations on the visible D3-branes are all gauged away by E SB and as a result λ = 0 is always preserved.
The dS vacua with anti-D3-branes might be suitable for the description of early universe including inflation, rather than the present universe with vanishing λ. The anti-D3-branes are indispensable in the brane-antibrane inflation scenario [16,17] because the potential for the inflation (inflation potential) is generated by the brane-antibrane interaction. Also in the inflationary era the coefficient D in (2.9) − and therefore ǫ in (5.9) − does not need to be fine-tuned. Entire V dS in (5.9) can contribute, together with the potential generated by the brane-antibrane interaction, to λ to make it positive. But these nonsupersymmetric dS vacua with anti-D3-branes are only metastable, and hence must eventually decay into the supersymmetric AdS vacua describing our present universe in the AdS vacuum scenario.

VI. Open string moduli
In the AdS vacuum scenario the supersymmetric configuration at ψ = π contains D3-branes in the KS throat. Introduction of D3/D3-branes generally induces a scalar potential coming from the DBI plus CS action. For instance in KKLT, an introduction of anti-D3-branes induces an additional term (2.9) to the scalar potential as we have already seen. Also the potential for the D3-branes, which vanishes in the ISD compactifications, acquires nonzero contributions once the background turns into IASD because in this background the IASD fluxes become a source for the scalar potential of the D3branes. Besides this, the presence of D3/D3-branes also yields open string moduli such as locations of the branes in the compact space. Thus we need to check if all these contributions to the scalar potential also respect (3.42) with n = 1, 3 to make λ vanish. In this section we want to check the contribution coming from the open string moduli of the D3-branes, and then in the next section we will consider the D3-brane potential in the presence of the IASD fluxes. In our discussions of this section we will consider the general case where the nonperturbative vacua without branes are basically given by the AdS type vacua, rather than dS, of KKLT according to the discussion of the previous section. So we do not have D3-branes in our configurations.
Suppose that we have a single (or a stack of) D3-brane(s) in the six-dimensional compact space for simplicity. In the presence of a D3-brane the Kähler modulus 6 ρ acquires an additional term k(Y,Ȳ ) [18]: where the three complex scalars Y α , α = 1, 2, 3, in k(Y,Ȳ ) represent the location of the D3-brane. 7 The Kähler potential for this Kähler modulus is therefore Besides this, the nonperturbative superpotential (2.1) also changes in the presence of D3-brane into the form [18] So the supersymmetric vacua must satisfy and from these two equations one obtains (6.6) (6.6) guarantees that (6.4) and (6.5) are not inconsistent to each other as far as it admits a solution. Now we can show that the potential density associated with the superpotential (6.3) belongs to V 3 . W in (6.3) differs from W in (2.1) only in that e iaρ is replaced by e iaρ −ζ(Y ) , and in (6.3) the complex structure moduli are contained only in A 0 as before. So (4.8) still requires W to take the form (4.9) only except thatᾱ now depends on both τ and Y α , instead of τ alone. Similarly we obtain through the same discussions as in Sec. 4.1, and also we obtain (4.11) with G eff (0,3) = c G (0,3) where c now depends on τ and Y (see, however, footnote 7). After all, the scalar potential associated with (6.3) can be written as (4.13) again, and therefore the potential density of this case also belongs to V 3 as in Sec. 4.1.
One can reaffirm the above result as follows. Substituting (6.3) into (6.4) gives (see (6.1)), we obtain (6.10) (6.10) is precisely of the same form as (4.15) except e aσm is replaced by e −iaρ+ζ(Y ) | m . So the scalar potential associated with (6.3) will take the form (4.14) again by (6.10) at the supersymmetric minimum, and by the same discussion as in Sec. 4.2 one finds that the potential density associated with (6.3) also belongs to V 3 as before.

VII. D3-brane potential
In the ISD compactifications − and in the absence of branes − λ trivially vanishes from (3.41) because potential density arising from the fluxes vanishes in the ISD background. But once the perturbations come into the theory, V scalar does not vanish anymore because in this case G (3) aquires IASD components, and besides this the IASD fluxes induce a potential for the D3-branes because they become a dominant source in the equation of motion for the D3-brane potential. In [19], it was shown that there exist three distinct types of closed, IASD three-form fluxes which induce the D3-brane potential.
Among these fluxes the simplest one is the type I flux which contains only G (1,2) , the IASD G (3) of Hodge type (1,2). Compared with other two types of fluxes, the type I flux is of particular importance because the other two contain non-primitive (2, 1) which is forbidden in a compact Calabi-Yau space. Besides this, it was also shown in [19] that there is a holographic correspondence between perturbations of supergravity solution by the type I flux and superpotential perturbations of the conformal field theory. In this correspondence the scalar potential for a probe D3-brane in the conifold geometry precisely matches the scalar potential computed in the gauge theory with superpotential W , and the scalar potential for a D3-brane in the conifold geometry is reproduced by the G (1,2) flux.

D3/D3-brane potentials in the string frame
The D3/D3-brane potentials follow from the DBI plus CS action (3.3) with T (φ)\µ(φ) replaced by T 3 e −φ \µ 3 . In string frame it is given by For the given compactification (3.4), I D3/D3 becomes where Φ ± are defined by 1 Here we ignored the kinetic terms of the D3/D3-brane actions because we assumed that the D3/D3-branes are all fixed at some points of the compact space. According to (7.4), Φ − (y) vanishes in the ISD background if the (bulk) supersymmetry is unbroken (see Sec. V of [1]), and therefore D3-branes feel no potential in this case. But once the higher order perturbations come into the theory, the situation changes. Because higher order terms of G (3) generally contain IASD components, and these components become a dominant source in the equation of motion for Φ − (y) [18,19], the D3-branes certainly feel a potential arising from the higher order terms of Φ − (y). In this section we will show that this D3-brane potential arising from the IASD flux perturbations also respects the condition (3.42), but this time not with n = 3, but with n = 1. Namely V D3 ∈ V 1 .

Equation of motion for Φ −
The equation of motion for Φ − may be obtained from the field equations for χ 1/2 and ξ, among which the latter follows from the field equation for A (4) . The field equation for A (4) can be obtained from the three terms in the actions (3.1) and (3.3), where we have rewritten theF 2 (5) term in (3.1) as 1 8κ 2 10 F (5) ∧ * F (5) for convenience, and replaced µ 3 → µ 3 2 which is necessary to obtain correct equation for the self-dual field A (4) (see for instance [13] or [20] for this). We obtain from (7.5) which, by (3.2), reduces to The field equation for χ 1/2 , on the other hand, can be obtained from (3.6) plus the topological term (7.8) Varying the action with respect to B we obtain Finally the equation of motion for Φ − can be obtained by subtracting (7.7) from (7.9). Upon setting µ 3 = T 3 , we obtain (7.10) (7.10) is the string frame version of Eq. (2.8) of [19], and they coincide if we replace χ 1/2 by e 4A , and h mn by e −φ/2 h mn .
Fortunately, the explicit solutions for the flux perturbations on arbitrary Calabi-Yau cones have been thoroughly studied in [19]. According to the computations of [19] there exist three distinct types of closed, IASD three-forms, (1) type I : (1, 2) flux Λ I = ∇∇f 1 ·Ω , (7.19) (2)type II : (2, 1) NP + (1, 2) flux (3)type III : (3, 0) + (2, 1) NP + (1, 2) flux where f 1 , f 2 , f 3 are harmonic functions and ∇ α denotes the covariant derivative with respect to the Kähler metric h αβ = ∂ α ∂βk. Also k α = h αβ ∇βk, ωᾱ β = Ωᾱ βγ k γ and h = 3f 3 + k α ∂ α f 3 etc. Among these fluxes the type I flux is of particular importance since its contribution to Φ − is dominant over the other two in the neighborhood of y = 0 where the visible D3-branes are located (we will see this soon). Also the type II and III fluxes contain non-primitive (2, 1) which is forbidden in a compact Calabi-Yau space. 9 The potential Φ − due to the type I flux is found to be [19] which is an F -term potential due to the superpotential perturbations of the form d 2 θ △W with △W ∼ f 1 . (7.22) suggests that the potential density V D3 induced by the type I flux belongs to V 1 . From (7.3) and (3.17) V D3 can be written as and since Φ − in (7.22) contains a single h mn in the real basis, (7.23) shows that V D3 ∈ V 1 . So V D3 induced by the type I flux does not contribute to V scalar in λ (see (3.43)).
7.4 D3-branes located at y = 0 In (7.23) V D3 is proportional to Φ − (0) instead of Φ − (y), which is due to the fact that in the AdS vacuum scenario of Sec. 5.2 our D3-branes are not mobile branes anymore because we are considering the present (not inflationary) stage of our universe. In our descriptions of the present universe (a stack of visible sector) D3-branes are assumed to be fixed at the apex y = 0 of the Calabi-Yau cones, and therefore we have the deltafunction δ 6 (y) in (7.23), and also Φ − (0) instead of Φ − (y). The presence of delta-function (or having Φ − (0) instead of Φ − (y)) in V D3 enables us to ignore the whole contributions (not just only the type I) to V D3 arising from the above three types of flux perturbations. This can be shown as follows.
The field equation (7.18) can be solved by using the green function method. Again in [19] 10 it was found that the resulting spectrum of Φ − can be written as where h (δ i ,δ j ) (Ψ) are angular wave functions which are related to the harmonics Y LM (Ψ) of the unperturbed Laplacian and △(δ i , δ j ) are radial scaling dimensions defined by where δ i and δ j are the scaling dimensions of the fluxes Λ i and Λ j . The smallest value of △ is obtained from a square of δ = 5 2 chiral mode of the type I flux, for which Φ − is linear in r, Φ − ∝ r. The other smallest scaling dimensions (including the above △ = 1 of δ 1 = δ 2 = 5 2 ) of the flux-induced potential are which shows that the contribution of the type I flux to Φ − is dominant over the other two as r → 0. In any case, every term in (7.24) vanishes at r = 0 for any △(δ i , δ j ), and so does Φ − (0) in (7.23) as well. This implies that the contributions of the other two can be also ignored − despite that they do not belong to V n with n = 1, 3 − since Φ − (0), and therefore V D3 itself vanishes for the D3-branes fixed at r = 0 of the Calabi-Yau cones. Apart from this, the potential Φ − can also include harmonic functions on the cones as the homogeneous solution to (7.18). The contributions of these harmonic functions, however, can be also ignored for the D3-branes fixed at r = 0. The harmonic expansion performed on the conifold takes the form [21] f (r, Ψ) = where c LM are constant coefficients and the radial scaling dimensions △ f (L) take the values of △ f (L) = 3 2 , 2, 2, 3, √ 28 − 2, · · · . (7.28) Since △ f (L) are all positive, all terms in (7.27) vanish at r = 0, and therefore we can ignore the contributions of these harmonic functions to the potential V D3 as well.
In addition to these terms there might be a constant term, which is the trivial solution to the Laplace equation ∇ 2 f = 0. This constant term does not vanish at r = 0. However, it might be irrelevant to our configurations that do not contain anti-D3-branes.
The constant term appears in the perturbative expansion of the anti-D3-brane potential T 3 Φ + (r; r 0 ) (≡ V D3/D3 (r)) (see [17]). Since mobile D3-branes affect Φ + perturbatively, V D3/D3 depends on the D3-brane position r, and it serves as a potential for the D3-brane. In this expansion of V D3/D3 (r) the constant term appears as the unperturbed potential energy of the anti-D3-branes fixed at r = r 0 , and therefore it must vanish for the configurations which do not contain anti-D3-branes. In a word, those terms (including harmonic functions) arising from the Coulomb interaction V D3/D3 between D3-branes and anti-D3-branes must all be excluded from V D3 because they are irrelevant to our AdS vacuum scenario for the present universe in which anti-D3-branes are not involved at all. See Sec. 5.2.

VIII. Summary and Discussion
In an attempt to address the cosmological constant problem (especially aiming at explaining the fine-tuning λ = 0 of our present universe) we have considered a new type of self-tuning mechanism whose basic principle has already been presented in [1]. The main point of this self-tuning mechanism can be summarized as • Whether λ vanishes or not is basically determined (in the six-dimensional internal space) by the tensor structure of the scalar potential density V , not by the zero or 11 Q 0 in χ 1/2 will cancel with 2κ 2 10 g s T 3 in V D3 (see (7.23)) in the self-tuning equation (3.41). 12 The compactifications with F (3) = H (3) = 0 are good approximations in the AdS vacuum scenario because in KKLT the superpotential W 0 (and therefore G (3) ) is only of an order ∼ 10 −4 , instead of ∼ O(1). nonzero values of the scalar potential V scalar itself. If the density of V scalar belongs to one of V n with n = 1, 3, then λ is forced to be fine-tuned to vanish regardless of whether V scalar vanishes or not.
• In the new self-tuning mechanism λ contains an exceptional term E SB , and where E SB has its own gauge arbitrariness. So any nonzero V scalar and quantum fluctuations δ QÎ brane on the branes can be gauged away by this E SB so that λ vanishes as a result. The cancelation between V scalar + δ QÎ brane and E SB automatically achieved by a self-tuning equation (3.41) once the density of V scalar satisfies V ∈ V n with n = 1, 3 as stated above.
• Hence in the new self-tuning mechanism the self-tuning λ = 0 is radiatively stable.
Any contributions to V scalar coming from g s -perturbation and quantum fluctuations on the D3-branes are all gauged away by E SB , and as a result λ = 0 is always preserved.
We applied the above self-tuning mechanism to the well-known scenario of KKLT to obtain a realistic model of our present universe with nearly vanishing cosmological constant. As a result of this application we found that the simplest, and perhaps the most natural scenario which can avoid the problematics of KKLT is to take one of the AdS (instead of dS) vacua of KKLT as the background vacuum of our present universe. These AdS vacua are stable both classically and quantum mechanically. They do not suffer from the tunneling instabilities unlike the dS vacua of KKLT. This AdS vacuum scenario suggests that the F-term upliftings in the literature [3,22] are basically unnecessary in obtaining a vanishing (or a nearly-vanishing) cosmological constant. The vanishing λ is automatically achieved by the self-tuning equation (3.41) and gauge arbitrariness of E SB contained in (1.1). So the validity of this scenario (or the self-tuning mechanism of this paper) is established by the two unusual equations (1.1) and (3.41).
The first equation (1.1) suggests that the cosmological constant λ is not simply given by a scalar potential V scalar alone. According to (1.1), λ contains an additional term, the supersymmetry breaking term E SB , which possesses its own gauge arbitrariness. Hence in our case λ = 0 does not necessarily imply V scalar = 0, and AdS vacua with V scalar < 0 are not inconsistent with λ = 0 unlike in the theories where λ is directly identified with V scalar . In our self-tuning mechanism λ is generally given by (3.20). But in the AdS vacuum scenario proposed in Sec. 5.2, V is basically given by V AdS , and in this case (3.20) reduces (upon using (3.17)) back to (1.1) by (3.40) and (3.45) because V AdS ∈ V 3 and therefore (N − 1)V AdS = 2V AdS . This result does not change even when we add V 1 (for instance, V D3 due to Φ − in (7.22)) to V because (N − 1)V 1 simply vanishes.
Together with (1.1), the second equation (3.41) suggests that whether λ vanishes or not is basically determined by the tensor structure of the potential density V , not by the zero or nonzero values of V scalar itself. (3.41) leads to the self-tuning λ = 0 once our V belongs to one of the class V n with n = 1, 3. We have shown that the AdS vacua of KKLT including open string moduli of D3-branes belong to V 3 . So λ of our present universe must tune itself to zero in the AdS vacuum scenario of Sec. 5.2.
The negative values of V scalar of the AdS vacua are gauged away by E SB in (1.1), and λ = 0 is automatically achieved by (3.41). This self-tuning process is not affected by the perturbations K = K tree + K p + K np and W = W tree + W np because these perturbations do not change the tensor structure of V (see the last paragraph of Sec. IV). Thus the whole radiative corrections of V scalar are also gauged away by (3.41), and λ = 0 remains stable against these corrections in the self-tuning mechanism of this paper.
The background vacua of the AdS vacuum scenario are supersymmetric, and therefore stable unlike the dS vacua of KKLT. In the descriptions in [14] the dS vacua necessarily involve the anti-D3-branes. So the dS vacua are not supersymmetric and they are stable only classically at most. The dS vacua of KKLT must eventually decay into the supersymmetric configurations of AdS vacua by the brane/flux annihilation process of [14], and this strongly suggests that the AdS vacuum scenario might be more natural description of our present universe as compared with the dS vacua of KKLT.
In the AdS vacuum scenario the supersymmetry is basically broken by δ GÎ where T (φ) and µ(φ) are given by because we are now taking quantum fluctuations on the branes into account. The last term of (8.1) occurs as a result of the gauge symmetry breaking of A (4) arising at the quantum level. The substance of this term is a vacuum energy density (of the brane region) arising from the quantum excitations with components along the transverse directions to the D3-branes, and it plays very important roles in the supersymmetry breaking of the brane region and in the process of self-tuning λ = 0. The supersymmetry transformations of the fermi fields of type IIB supergravity are [23] and these δχ φ and δψ m vanish for (7.29) and constant φ [24]. Hence in the approximation F (3) = H (3) = 0, the supersymmetry is unbroken when φ is constant [1]. Now consider the field equation for φ. Using (8.1) we obtain from a linear combination of the field equations forφ and B. In the given approximation (8.7) reduces to ∇ 2 φ = 0 (8.8) in the bulk region, and therefore the bulk supersymmetry remains unbroken because (8.8) admits constant solutions. In the brane region, however, (8.7) reduces to where c 0 = 2κ 2 10 g s δ 0 (see eq. (6.12) of [1]), and ρ vac , δµ, δµ m T are expanded respectively as In [1] it was shown (up to one-loop level) that all but the last term in (8.9) cancel out for µ 3 = T 3 and µ (1) = ρ (0) which are required by consistency equations (see Sec. VIC of [1]), and we are left with and similarly we obtainÎ brane = δ 0 r 5 drǫ 5 ρ is expected to cancel out and λ is simply given by where T . (8.15) Q T total represents the total vacuum energy (per unit volume of the four-dimensional spacetime) of the brane region which originated from the excitations with components along the transverse directions to the D3-branes. Now the point of the cosmological constant problem can be summarized as whether we can find a nonzero function ρ Now we turn on the three-form fluxes G (3) to obtain a full description of the supersymmetry breaking in our AdS vacuum scenario. In the ISD compactifications (Φ − = G + mnp = 0), the dilaton φ still satisfies (8.8) and (8.11) even in the presence of nonzero G (3) . However, these ISD compactifications are not relevant to our AdS vacuum state because in the AdS minimum, the unbroken supersymmetry DW = 0 requires that G (3) must also contain IASD (1,2) and (3,0) in addition to the ISD (2, 1) and (0, 3) (see [10]). These IASD components of the AdS background are entirely due to the nonperturbative corrections of the superpotential, and they have nothing to do with the perturbative corrections which also give rise to the IASD components of G (3) and Φ − . In any case, the IASD terms with G + mnpḠ +mnp or Φ − acquire nonzero values from both perturbative and nonperturbative corrections, and they are now involved − together with the terms caused by δ GÎ T for a moment, then we expect the solution to the equations of motion becomes a supersymmetric solution satisfying δχ φ = δψ m = 0. 13 Namely the supersymmetry of the AdS background is simply given by δχ φ = δψ m = 0.
Let us finally turn to the situation where the perturbative corrections and supersymmetry breaking generated by E SB are both taken into account. In this case δχ φ and δψ m fail to vanish since they now acquire the terms coming from these perturbations and supersymmetry breaking, and consequently the supersymmetries of the brane and bulk regions are both broken. In the case of V scalar , however, the situation is a little different. In the AdS minima of KKLT the scalar potential V scalar receives contributions both from perturbative and nonperturbative corrections. Hence in KKLT, V scalar already takes nonzero values even when perturbations and supersymmetry breaking are not considered in the theory yet. So KKLT obtained AdS vacua with nonzero (negative) values of λ even at the tree level as we already know. But in our AdS vacuum scenario, any nonzero contributions to V scalar coming from perturbative and nonperturbative corrections, and also the contributions coming from the IASD fluxes described above are all compensated by E SB in (1.1), and λ = 0 is always preserved even when supersymmetry of the system is broken by the perturbations and supersymmetry breaking term E SB .
In (1.2), we decompose ρ T intoρ (1) T + δρ (1) T to get E SB →Ẽ SB + δE SB , whereẼ SB and δE SB areẼ SB = −δ 0 r 5 drǫ 5ρ T and δρ (1) T in E SB are arbitrary because they contain six arbitrary gauge parameters 13 Supersymmetric solutions of type II theories have been discussed, for instance, in [23] (also see the last paper of [11]. brane in (1.1) is automatic by the self-tuning λ = 0 as required by (3.41), andQ T total now plays the role of Q T total as one can see from (8.14) and (8.17). So if we want a nonsupersymmetric theory with λ = 0, we may need to find a nonzero functionρ (1) T satisfyingQ T total = 0 as in the case of G (3) = 0. But still, it may also be possible to take simply because this E SB would be enough to break the supersymmetry of the system sufficiently.

Concluding remarks
So far we have considered a new type of self-tuning mechanism to address the cosmological constant problem, especially aiming at explaining the fine-tuning λ = 0 of our present universe. But more precisely, λ of our present universe is known to take a positive value though it is very small. So the next step of the project would be this issue of identifying small positive λ of our present universe.
In this paper we have considered a theory based on the type IIB supergravity, and from this supergravity action we obtained a result that λ must vanish precisely as far as the density of V scalar satisfies a certain condition that V ∈ V n with n = 1, 3. We have also shown that this result λ = 0 is stable against g s -perturbations. But full string theories require the action to admit α ′ -corrections which are usually higher order in derivatives, and due to these correction terms the self-tuning equation (3.41) may be modified into the corrected form. Also α ′ -corrections of the type I or the heterotic theory contain the terms which do not satisfy V ∈ V n with n = 1 or 3, and in such cases (3.41) requires that λ must take nonzero values. Namely if we take the stringy (or any other) effects which have not been considered in this paper into account, we may expect a result with nonvanishing λ. But still, once λ is determined by (the modified) (3.41), these nonzero values of λ will be also stable against quantum corrections as in the case λ = 0 because (3.41) is based on the self-tuning mechanism where the perturbative corrections of V scalar and quantum fluctuations on the branes are all gauged away by E SB in (1.1). So the result obtained from (3.41) needs to be distinguished from the nonvanishing λ due to α ′ -corrections in the literature [25] in this sense.
In any case, if some convincing values of λ can be obtained from the modified (3.41) with α ′ -corrections, then we may say that the nonzeroness of λ of the present universe is essentially due to the stringy effect of the string theory because λ vanishes in the absence of α ′ -corrections and this result was not affected by the g s -perturbations in the self-tuning mechanism of this paper. Also any nonzero values of λ suggested by (3.41) will be highly suppressed again by the factor χ 1/2 as stated in the last paragraph of Sec. 7.4. Namely λ suggested by (3.41) would be very small anyway.