Finite temperature behaviour of the ISS-uplifted KKLT model
aa r X i v : . [ h e p - t h ] M a y DESY-08-012
Finite temperature behaviour ofthe ISS-uplifted KKLT model
Chlo´e Papineau
Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany [email protected]
Abstract
We study the static phase structure of the ISS-KKLT model for moduli stabilisa-tion and uplifting to a zero cosmological constant. Since the supersymmetry breakingsector and the moduli sector are only gravitationally coupled, we expect negligiblequantum effects of the modulus upon the ISS sector, and the other way around.Under this assumption, we show that the ISS fields end up in the metastable vacua.The reason is not only that it is thermally favoured (second order phase transition)compared to the phase transition towards the supersymmetric vacua, but rather thatthe metastable vacua form before the supersymmetric ones. This nice feature is ex-clusively due to the presence of the KKLT sector. We also show that supergravityeffects are negligible around the origin of the field space. Finally, we turn to themodulus sector and show that there is no destabilisation effect coming from the ISSsector. ontents In the last years, quite a large attention has been given to the problem of modulistabilisation, especially concerning the cosmological implications they could have [1].Following an earlier proposal [2], Kachru, Kallosh, Linde and Trivedi (KKLT, [3])have recently provided the first explicit model in which all moduli are fixed. They doso by turning on fluxes in a first step, which fix the complex moduli and the dilaton S , and introducing non-perturbative superpotentials [4] in a second step in orderto stabilise the K¨ahler moduli T . For a more detailed study of the phenomenologyarising from these models, see [5].Unfortunately, the resulting low energy potential for T has an anti-de Sitter vac-uum which needs to be uplifted. The strategy proposed in [3] was to introduce ananti- D D -terms [6], i.e.using a fully supersymmetric sector, but this generically leads to a heavy gravitino .Parallel works have considered an F -term uplifting [7]. This relies on adding a newsector in which supersymmetry is spontaneously broken by some field Φ, F Φ = 0. If Actually this is not the case for the last two references of [6] because the uplift there is mainly realisedby an F -term. his sector and the KKLT setup are decoupled in such a way that the K¨ahler potentialand superpotentials add up, then the uplifting is trivially realised by relating theparameters of both sectors. The last two years, a rather large sample of such modelstogether with their direct phenomenology have been proposed [8, 9, 10].In this paper, we shall focus on the setup developped in [8], where the upliftingsector was chosen to be the Intriligator, Seiberg and Shih model (ISS, [11]). A non-exhaustive list of string realizations of it can be found in Ref. [12]. This dual SQCDmodel is of particular interest since it realises a breaking of supersymmetry in localminima in the squarks direction. Elsewhere in the field space, in the mesons direction,there are supersymmetric vacua, and both locally stable points are separated bya potential barrier. This ensures a long life-time for the SUSY breaking vacua.Hence, one does not have to give up the idea of a global (supersymmetric) minimum.Now, from cosmological considerations, we are led to wonder whether we endedup living in the metastable vacuum or not. And indeed, following the results ofRefs. [13, 14, 15, 16], we do. There, the authors showed that finite temperaturecorrections to the ISS setup favour the fields to go in the metastable vacua ratherthan in the supersymmetric ones. In [14], it was assumed that the fields start inthe supersymmetric phase. Instead, the authors of [15, 16] assumed the startingpoint to be the origin of the field space, which is a minimum at high temperature .This is because the origin of the field space contains the highest number of lightdegrees of freedom and hence maximises the entropy. We shall adopt the sameattitude. In these various studies, it was found that the supersymmetric vacua format a higher temperature than the metastable ones, but the origin is always a localminimum in the mesons direction. Therefore, the phase transition is first ordertowards the supersymmetric vacua. This is thermally disfavoured in comparisonwith the second order phase transition that occurs towards the non-supersymmetricvacua, even though the latter happens at a lower temperature.In this paper, we complete the study done in [16]. We work out the completephase structure of the ISS-KKLT model. As will become clear in the text, the ISSfields do end up in the non-supersymmetric vacua, and the modulus, on the otherhand, is not destabilised by thermal effects, as suggested in [18]. We will show thatin our case, the presence of the modulus sector modifies the thermally corrected ISSpicture in such a way that the metastable vacua form first, and they remain the truevacua of the theory during a certain time. Later on, the supersymmetric vacua form,but the fields have long gone in the SUSY breaking ones. Not until an even lowertemperature are the two vacua degenerated. From that moment on, the fields cantunnel down from the metastable vacua to the supersymmetric vacua.Another feature that was pointed point out in [16] is that the origin of the ISSfield space may no longer be a minimum at high temperature when this sector iscoupled to the KKLT sector. This is due to supergravity, and could have a non-trivial effect on the phase transition. We study in great detail this point and findthat as expected, this displacement is very small.However, let us emphasize that this study is still at the toy model level. We willnot at all address cosmological problems such as the gravitino overproduction that For a review on finite temperature field theory, see [17]. We will develop these points in the following Sections. sually happens when the supersymmetry breaking sector is in thermal equilibrium.Even though the present paper obviously aims at a more realistic application, weleave these investigations for future work.The paper is organised as follows. Section 2 reviews the zero temperature ISS-KKLT setup in order for the paper to be self-contained. We introduce the main toolsof finite temperature effective potential in Section 3. In Section 4, the relevant tem-peratures and phase transition of the ISS sector are derived assuming the rigid limit(zeroth order in supergravity expansion). In subsection 4.1, we compute the criticaltemperature of the second order phase transition towards the would-be metastablevacua. We give an insight of how the supersymmetric minima form in subsection 4.2.We eventually compute the degeneracy temperature between the non-supersymmetricand the supersymmetric vacua in subsection 4.3. The rigid limit assumption of Sec-tion 4 is verified by working out the supergravity corrections around the origin inSection 5. Section 6 deals with the modulus sector. We show that the temperaturecorrections coming from the thermalised ISS sector do not destabilise the modulus.Finally, we conclude in Section 7 and draw the future directions that seem relevantto us. Let us start by recalling the KKLT construction for moduli stabilisation in the frame-work of type IIB string theory. In [3], the authors used non trivial background fluxes,i.e. non-zero vacuum expectation values for certain field strengths in the internal di-rections, in order to stabilise all complex structure moduli as well as the dilaton.However, the K¨ahler modulus T , which describes the fluctuations of the overall in-ternal volume, cannot be stabilised in this manner. Non-perturbative effects suchas gaugino condensation on D ≪ M P . At low energy, the procedureresults in the following setup K = − (cid:0) T + T (cid:1) , W = W + ae − bT , (2.1)where the constant W is remnant of the stabilisation of all other moduli at thePlanck scale.The model exhibits a supersymmetric minimum D T W = ∂ T W + K T W = 0 at T = T , implying W = − ae − bT ( b (cid:0) T + T (cid:1) ) < , (2.2) h V KKLT i = h e K h K T T D T W D T W − | W | i i = − a b e − b ( T + T )3 (cid:0) T + T (cid:1) < , where K T T = (cid:0) K − (cid:1) T T is the inverse metric for the K¨ahler potential K .As mentionned in the Introduction, the energy can be uplifted to a positive valueby adding a sector in which supersymmetry is spontaneously broken. In [8], the plifting sector was chosen to be the ISS model [11] K = Tr | ϕ | + Tr | e ϕ | + Tr | Φ | , W = h Tr ( e ϕ Φ ϕ ) − hµ Tr Φ . (2.3)This is the magnetic dual of a SUSY-QCD theory with gauge group SU ( N c ). Whenthe number of flavours satisfies N f N c /
2, the electric theory is asymptoticallyfree whereas its dual, with gauge group SU ( N f − N c ), is infrared free.The magnetic fields under consideration are the gauge singlets Φ = (cid:16) Φ ij (cid:17) , whichwe call mesons because they are in one-to-one correspondence with the electricmesons. The quarks ϕ = (cid:0) ϕ ia (cid:1) , and the anti-quarks e ϕ = ( e ϕ ai ) are in the funda-mental and antifundamental representations of SU ( N ). In the rigid supersymmetrylimit, the theory (2.3) has a global symmetry G = SU ( N f ) L × SU ( N f ) R × U (1) B × U (1) ′ × U (1) R which is explicitly broken to SU ( N f ) × U (1) B × U (1) R by the massparameter µ .We denote by N the magnetic number of colours N = N f − N c , which satisfies N f > N . The indices run as i, j = 1 , . . . , N f and a = 1 , . . . , N . For convenience, wewill omit the flavour and colour indices from here on and will just keep in mind thatΦ is an N f × N f matrix, whereas ϕ and e ϕ T are N f × N matrices.The setup (2.3) has supersymmetry breaking solutions ϕ = e ϕ T = (cid:18) µ N (cid:19) , Φ = 0 (2.4)generated by non-vanishing F -terms for the mesons F Φ = h (cid:0) e ϕϕ − µ N f (cid:1) . Noticethat the supersymmetry breaking does not affect the gauge sector since it is drivenby gauge singlets. The corresponding vacuum energy is V min = (cid:12)(cid:12) h µ (cid:12)(cid:12) ( N f − N ) . (2.5)Far away from the origin in the mesons direction, after integrating out the quarks,gaugino condensation produces a non-perturbative superpotential [4] W dyn = N h N f det ΦΛ N f − Nm ! , (2.6)which gives rise to supersymmetric vacua h h Φ i = Λ m ǫ N/ ( N f − N ) N f . (2.7)In the above expressions, Λ m is the dynamical scale of the magnetic theory, and ǫ ≡ µ/ Λ m is a small parameter. The existence of these vacua renders the non-supersymmetric ones (2.4) metastable. Both regions of the ISS field space are sep-arated by a potential barrier. The lifetime of the metastable vacua can be madearbitrarily large by tuning ǫ very small, or equivalently, Λ m very large for µ fixed.We now couple both sectors in the following way K = K (cid:0) T, T (cid:1) + K (cid:0) χ i , ¯ χ ¯ j (cid:1) , W = W ( T ) + W (cid:0) χ i (cid:1) , (2.8) here χ i denote collectively the ISS fields ϕ, e ϕ, Φ.As explained in [8], such a decoupling between the two sectors can be achievedby considering systems of D D SU ( N ) arises froma stack of N D m and the mass parameter µ depend on the dilaton, which was already stabilised athigher energies. The mesons are interpreted as the positions of N f D D D ∼ µ /M P . There, the scalarpotential is well approximated by V (cid:0) χ i , ¯ χ ¯ i , T, T (cid:1) ≃ (cid:0) T + T (cid:1) V ISS (cid:0) χ i , ¯ χ ¯ i (cid:1) + V KKLT (cid:0)
T, T (cid:1) , (2.9)where V ISS is the global supersymmetric (as opposed to supergravity) scalar poten-tial for the ISS sector. However, when computing the critical temperature, we willconsider the expansion (2.9) to be valid at the origin of the field space as well. Wethen explicitly verify it in Section 5.The fine-tuning of the cosmological constant to zero is given by h V i = 0 = ⇒ (cid:12)(cid:12) h µ (cid:12)(cid:12) ( N f − N ) ≃ | W | , (2.10)and illustrated in Figure 1. V(T) 200 T150100
KKLT
Figure 1: The KKLT potential (purple, dashed) and the uplifted ISS-KKLT potential(green, plain). The vev T ≃ T is not modified by the uplifting mechanism. On the other hand, the gravitino mass is m / = h e K | W | i ≃ | W | (cid:0) T + T (cid:1) ≃ a b e − b ( T + T )9 (cid:0) T + T (cid:1) , (2.11) here (2.2) together with the condition bT ≫ .Here and in the following numerical results, we fix a = h = 1 and b = 0 .
3. Wealso set N f = 7 and N = 2. Asking for a TeV range gravitino mass and imposing(2.10), the parameters are found to be T ≃ , | W | ≃ (cid:0) − − − (cid:1) M P , µ ≃ − · − M P . (2.12)When needed, we will also take a coupling constant g = 0 .
1. Since the ISS gaugesector lives on D g ( M P ) ∼ π/ Re S and run this valuedown to a scale of order µ . However, the dilaton vev strongly depends on the UVcompletion of the model. Though we believe this point to be crucial, it goes farbeyond the aim of the present work. We also consider the scale Λ m ≃ M P . Letus recall that this scale is a Landau pole, which is not physical, and it is therefore notsurprising to have Λ m higher than the Planck scale. The previous value correspondsto g ( M P ) = 0 . The general one-loop effective potential including finite temperature effects can besplit into different contributions [19] V eff ( χ i , T ) = V ( χ i , T ) + V ( χ i , T ) + V Θ1 ( χ i , T ) , (3.1)where V = e K n K T T D T W D T W + K i ¯ j D i W D ¯ j W − | W | o (3.2)is the tree-level supergravity potential, and K i ¯ j is the inverse metric for the K¨ahlerpotential K .The potential V is the usual one-loop temperature independent Coleman-Weinbergeffective potential [20], and V Θ1 is the finite temperature contribution V Θ1 = Θ π (X B n B Z ∞ dx x ln (cid:16) − e − √ x + M B / Θ (cid:17) − X F n F Z ∞ dx x ln (cid:16) e − √ x + M F / Θ (cid:17)) . (3.3)Here n B ( n F ) are the bosonic (fermionic) degrees of freedom, and M B ( M F ) are thebosonic (fermionic) field-dependent mass eigenvalues. One can immediately see thatfinite temperature corrections break supersymmetry.The potential (3.3) may be expanded at high temperature, Θ ≫ M B , M F , V Θ1 ≃ − π Θ (cid:18) n B + 78 n F (cid:19) + Θ (cid:0) M v + Tr M f + Tr M s (cid:1) + . . . , (3.4) As usual, we assume that the uplift of the modulus potential does not substantially modify its vev.This can be easily verified graphically (Figure 1). here M x are the mass matrices for vectors, fermions, and scalars, expressed interms of the fields. The trace Tr M f is summed over Weyl fermions.In general, one should use the following supergravity formulæ for the mass ma-trices in the presence of a non-canonical K¨ahler potential [21]Tr M f = h e G h K AB K CD ( ∇ A G C + G A G C ) (cid:0) ∇ B G D + G B G D (cid:1) − i i , (3.5)and Tr M s = 2 h K AB ∂ V ∂χ A ∂ ¯ χ B i . (3.6)In the above expressions, χ A represent the scalar fields in thermal equilibrium. Theterm − G = K +ln | W | is the supergravity K¨ahler invariant potential, andwe also introduced G A = ∂G/∂χ A and ∇ A G B = G AB − Γ CAB G C , with the connectionΓ CAB = K CD ∂ A K BD . (3.7)However, as briefly mentionned in the Introduction, we will be concerned withthese general results in Section 5 when we explicitly calculate how supergravity to-gether with temperature effects displace the minimum from the origin in the mesonsdirection, and in Section 6 when we study the destabilisation of the modulus. Sinceboth T and Φ are singlets under SU ( N ), the gauge bosons contribution will not berelevant when computing the derivatives of the effective potential (3.4). This is whywe did not write Tr M v here above.At the origin, we keep the ISS sector at the rigid level. When computing the finitetemperature corrections there, we shall use the results of global supersymmetry3 Tr M v = 6 h D αi D α i i , Tr M f = h F ij F ij i + 4 h D αi D α i i , (3.8)Tr M s = 2 h F ij F ij i + 2 h D αi D α i i , where, as usual, F ij = ∂ W/∂χ i ∂χ j and D αi = ∂D α /∂χ i . Here again, χ i representthe scalar fields associated with ϕ , e ϕ and Φ. One should be aware that the tracesabove run over the flavour and colour indices as well. The index α labels the adjointrepresentation of SU ( N ).Let us now turn to the main part of this paper, namely the phase structure ofthe model (2.8) once finite temperature corrections are included. When the Universe cools down, the ISS fields end up in the non-supersymmetricvacua, as studied in [13, 14, 15, 16]. In this section, we show that the picture isnot drastically modified when we add the modulus sector. However, this schemeis valid only if we consider the KKLT sector to be classical, which means that we ssume the modulus to be already lying in its minimum T = T and we neglect itsquantum corrections to the ISS sector. In turn, Section 6 deals with the eventualityof a modulus destabilisation by temperature.Finite temperature effects are to restore all symmetries. At sufficiently hightemperature, all the fields sit at the origin of the ISS field space. As we shall see,when the temperature lowers, the potential starts to exhibit a tachyonic directiontowards the non-supersymmetric vacua, which form first. At a lower temperature,the would-be supersymmetric vacua form, but the origin remains a local minimumin the mesons direction (saddle point). Therefore, the origin and these new minimaare separated by a barrier. In what follows, we focus on the behaviour of the potential at the origin of the fieldspace. The symmetry restoration due to finite temperature appears when the tachy-onic tree level masses are compensated by the thermal masses (second derivativesof the potential (3.3)) at the origin. This is also a good reason to keep the lowestorder in supergravity around the origin. Indeed, even though the corrections to thepotential are negligible, supergravity effects could have a non-trivial impact on itsderivatives and one typically has to take them into account. However, in the case ofthe critical temperature, and motivated by the results of [16], even if the exact loca-tion of the origin may vary with temperature and supergravity, the moment when thecurvature of the potential at the origin becomes negative should not be drasticallyaffected by supergravity effects. This, obviously, assumes that the origin is indeed aminimum at high temperature, as we will show in Section 5.We use the high temperature expansion (3.4) because the tree level masses areof order h µ / (cid:0) T + T (cid:1) , see for instance (2.12). We follow the standard procedure[19] but there is no need to shift the fields here since we work at the origin of thefield space.The traces (3.8) expressed in terms of the fields are easily calculated from thesuperpotential (2.3). We find3 Tr M v = 3 g N − N Tr | ϕ | + Tr | e ϕ | (cid:0) T + T (cid:1) , (4.1)Tr M f = 2 (cid:18) h N f + g N − N (cid:19) Tr | ϕ | + Tr | e ϕ | (cid:0) T + T (cid:1) + 2 h N Tr | Φ | (cid:0) T + T (cid:1) , Tr M s = (cid:18) h N f + g N − N (cid:19) Tr | ϕ | + Tr | e ϕ | (cid:0) T + T (cid:1) + 4 h N Tr | Φ | (cid:0) T + T (cid:1) , where g is the coupling constant. It follows that the potential (3.4) reads V Θ1 = Θ (cid:0) T + T (cid:1) (cid:26)(cid:18) h N f + g N − N (cid:19) h Tr | ϕ | + Tr | e ϕ | i + h N Tr | Φ | (cid:27) , (4.2)where we dropped the constant term ∝ Θ in (3.4) since it is not relevant for thecomputation of the critical temperature. e compare the scalar thermal masses in (4.2) to the tree level masses at theorigin. The latter are ± h µ / (cid:0) T + T (cid:1) or ± h µ ∗ / (cid:0) T + T (cid:1) for the squarks and0 for the mesons, as easily seen from the superpotential (2.3) and from the expansion(2.9). The thermal mass matrix is diagonal and positive definite, while the classicalmass matrix is anti-diagonal. We ask for the determinant of the whole squared massmatrix to be zero at the critical temperature Θ c . This means that all the eigenvaluesare positive above the critical temperature, while below Θ c , tachyonic directionsappear in the potential towards the would-be metastable vacua. From (4.2), we getΘ c = 4 (cid:12)(cid:12) µ (cid:12)(cid:12) N f + g h N − N , (4.3)which is in agreement with the critical temperature derived in [16] in the rigid limit.As computed there, this is only slightly modified by supergravity corrections.The critical temperature (4.3) is of order µ and the tree-level masses are oforder hµ / (cid:0) T + T (cid:1) , with T ∼ m / = 1TeV, we find that Θ c ≃ · − M P . Notice that the critical temperature does notdepend on the modulus, as one could have expected from the expansion (2.9). Thisis a crucial point since it is the reason why the would-be metastable vacua form first,as we show now. Having computed the critical temperature does not yet ensure that the ISS fieldsactually go in the metastable vacua. In this section, we turn to the mesons directionand work out the temperature Θ susy at which the SUSY preserving vacua appear. Inparticular, we want to know if they are already formed when Θ = Θ c . In order toachieve this, we ask for the mesons to be away from the origin and integrate out theheavy quarks. The low energy theory is then pure Yang-Mills, it is strongly coupledin the IR and gaugino condensation [4] produces the non-perturbative term (2.6)which gives rise to the supersymmetric vacua W NP = N A (det Φ) /N , (4.4)with A = h ν Λ − ν +3 m and ν = N f /N .At zero temperature, the vacua are b Φ = (cid:0) A − hµ (cid:1) / ( ν − N f and the quarksmasses are m ϕ, e ϕ = h b Φ / (cid:0) T + T (cid:1) / .We decompose the mesons into a classical background b Φ and a quantum field φ as follows Φ = b Φ 1I N f + φ . (4.5)Expanding the total superpotential h Tr ( e ϕ Φ ϕ ) − hµ Tr Φ + W NP according to the bove decomposition, we get W = (cid:16) N A b Φ ν − − hµ N f (cid:17) b Φ + (cid:16) A b Φ ν − − hµ (cid:17) Tr φ + h b ΦTr ( e ϕϕ )+ h Tr ( e ϕφϕ ) + 12 A b Φ ν − ( (Tr φ ) N − Tr φ ) . (4.6)Keeping the quadratic order in φ is sufficient because, following the standard proce-dure [19], we express the masses in terms of the classical field b Φ and hence higherpowers in φ are not relevant. Notice also that the quarks ϕ and e ϕ should not bepresent in W since they have been integrated out. This point will become clear aswe advance in the computation.At this stage, we would like to emphasize that working out the whole finite tem-perature corrected potential in the context of supergravity drives a lot of technicalcomplications. For the sake of clarity, in order to sketch the mechanism that hap-pens around the SUSY vacua, we will again consider the rigid limit. We believe thatsupergravity corrections do not strongly modify the following results.If the quarks are integrated out, it means that the temperatures we consider areΘ ≪ h b Φ / (cid:0) T + T (cid:1) / . On the other hand, using (3.8), we find that the masses ofthe mesons are Tr M f + Tr M s = 3 (cid:0) N f − ν + ν (cid:1) A b Φ ν − (cid:0) T + T (cid:1) , (4.7)which is in agreement with [15]. The high temperature expansion (3.4) is thus legit-imate and one finds that the effective potential is V Θ1 = − C Θ + (cid:16) N f − ν + ν (cid:17) A b Φ ν − Θ (cid:0) T + T (cid:1) . (4.8)From this expression, we see that the thermal contribution to the mesonic massmatrix, namely the second derivative of (4.8) with respect to b Φ, is diagonal andpositive definite. Therefore there is no way that this contribution can lead to adestruction of the SUSY vacua and hence to a “critical” temperature. In otherwords, the masses are already positive at zero temperature and thus the origin andthe supersymmetric vacua are separated by a barrier. When temperature effects areincluded, only a first order phase transition can happen.Moreover, the contribution (4.8) is much smaller than that of the tree-level masses m φ ∼ A b Φ ν − / (cid:0) T + T (cid:1) / by assumption . Even if we used a high temperature ex-pansion, recall that the thermal mass is proportional to the quartic self-coupling ofthe mesons. Since it arises from non-perturbative effects, it is unnaturally small.Even though the squarks have been integrated out at tree level, their effect in theloops may be important. However, due to their large masses, the high temperature Since T ≃ ≪ h b Φ / (cid:0) T + T (cid:1) / implies that Θ is even smaller than h b Φ. xpansion can not be used. From (3.3), we derive a low temperature expansion V Θ1 = − Θ / π / ( X B n B M / B e − M B / Θ + X F n F M / F e − M F / Θ ) , (4.9)with M B, F ≫ Θ.At the leading order, the squarks mass matrix is almost diagonal and its eigen-values are h b Φ / (cid:0) T + T (cid:1) . Recall this sector is supersymmetric, so the fermionicand bosonic degrees of freedom give the same contribution. Using (4.9), we find V Θ1 = − N N f π / h b Φ (cid:0) T + T (cid:1) / ! / Θ / exp ( − h b ΦΘ (cid:0) T + T (cid:1) / ) . (4.10)This consists of a negative contribution which, together with the tree level potentialdeduced from (4.6) and with the effective potential (4.8), gives the total potential.One then has to consider the system ∂V tot /∂ b Φ = 0 = ∂ V tot /∂ b Φ . Solving it brings us to knowing the temperature Θ susy and the corresponding vev b Φ (Θ susy ). However, it turns out to be very hard to solve and we approximate b Φ = b Φ at all temperatures. We concentrate on the second equation of the expression aboveand find Θ susy = h b Φ B (cid:0) T + T (cid:1) / , (4.11)with B = − ln (cid:16) N f − ν + ν (cid:17) π / h N N f A b Φ ν − (cid:0) T + T (cid:1) > . For B ≫
1, we fulfill the consistency condition that the squarks are integrated outat tree level.
Numerical results
As we already explained at the end of Section 2, the dynamical scale Λ m of the theoryrelies on the UV completion of our model. Since it is not the object of this work, wechoose to consider the case of a half-unit gauge coupling at the Planck scale. ThenΛ m = M P e − π/ (3 N − N f ) g ( M P ) is approximately 10 M P . Recall that N f > N sothat the argument of the exponential is positive.For this value and the rest of the parameters given by (2.12), we find the followingresults b Φ ≃ · − , m ϕ, e ϕ ≃ · − , m φ ≃ · − (4.12)for the vev of the mesons and for the tree level masses, and B ≃ , Θ susy ≃ · − or the temperature. All these results, except for B , are expressed in units of M P .Notice that B is larger than one.The main conclusion is that the SUSY vacua form at a temperature which issmaller than the critical temperature (4.3). Obviously, this result depends on thechoice of Λ m and we emphasize, again, that a closer study of the UV physics of ourmodel is required. However, we find numerically that the constant B is negative forΛ m smaller than 10 which corresponds to a gauge coupling of 0 . m in this range yields10 Λ m = ⇒ . · − m ϕ, e ϕ . · − . (4.13)Hence, even for very high values of Λ m , the upper bound under which one canintegrate out the squarks is only slightly above the critical temperature (4.3), andthe corresponding SUSY temperature is 3 · − M P . Θ c .It is clear that the major cause of such an effect is the explicit dependence of theSUSY temperature on the modulus. This pushes the tree level squarks masses tovery low values compared to the original ISS scenario.Another result that we were able to derive numerically is that already oncethe squarks are integrated out, their contribution (4.10) is very small comparedto the tree-level one (4.6). This means that whenever one can consider the non-perturbatively generated superpotential (4.4), then the vacua are already there. Assuch, the SUSY temperature (4.11) does not really make sense, and we are moreencline to rely on the evaluation of the squarks masses m ϕ, e ϕ as in (4.13). Also, sincethese are tree level masses, they do not depend on b Φ (Θ) and thus are not biased byour approximations.The conclusion is unchanged : the supersymmetric vacua form after the would-bemetastable ones, and this is due to the presence of the modulus.
Finally, in this paragraph we compute the degeneracy temperature Θ deg at which itbecomes possible for the fields to go from the metastable vacua to the supersymmetricones. This temperature is defined as the moment when both vacua have the sameenergy.The total number of degrees of freedom in the non-supersymmetric vacua is( N f + N ) −
1. The vacuum energy there is h V i| meta = − π Θ h ( N f + N ) − i + (cid:12)(cid:12) h µ (cid:12)(cid:12) ( N f − N ) (cid:0) T + T (cid:1) , where we did not account for the KKLT energy since it is constant over the wholeISS field space.In the last paragraph, we showed that the squarks can be totally neglected inthe supersymmetric vacua. Therefore only the finite (high) temperature correctioncoming from the mesons is relevant. Recalling that these vacua have zero energy at ree-level, using (4.8) and assuming again that b Φ = b Φ , one finds h V i| susy = − π Θ N f + (cid:16) N f − ν + ν (cid:17) A b Φ ν − Θ (cid:0) T + T (cid:1) . Actually, it is easily seen from our numerical results (4.12) that the last term in theabove expression is negligible. To good approximation, the degeneracy temperatureis thus given by Θ ≃ s N f π (2 N N f + N − (cid:12)(cid:12) hµ (cid:12)(cid:12) ( T + ¯ T ) / . (4.14)Using our parameters, one finds numerically Θ deg ≃ · − M P . As before, thedegeneracy temperature explicitly depends on the modulus, reason why it is so lowcompared to the critical temperature. We believe that this is a major improvementover the case of an isolated ISS sector. As was already noted in [8], the presence of themodulus enhances the lifetime of the ISS metastable vacua. We confirm this resulthere by showing that the supersymmetric vacua actually become the true vacua ofthe theory only at relatively late times. The computations of Section 4 have assumed that the origin of the ISS field space is aminimum of the potential at high temperature. However, as pointed out in [16], thisis not as straightforward once one includes supergravity. Consider for instance thecross term K T T K T W ∂ T W in the supergravity potential (3.2). It contains a linearterm in Φ which contributes as a constant to the equation ∂ Φ V = 0, and producesa displacement from the origin. Generical temperature corrections contain similarterms and one has to work out the full supergravity plus temperature correctedpotential and solve for a minimum around the origin. This is an important pointbecause, even though unexpected, the displacement could be large enough to spoilthe phase transition towards the supersymmetry breaking vacua.From the superpotential and K¨ahler potential (2.3), one can see that only termsof at least quadratic order ∼ ϕ , e ϕ , ϕ e ϕ can appear in the scalar potential. Con-sequently, the origin ϕ = e ϕ = 0 is always a solution to the extremum equations ∂ ϕ V = 0 = ∂ e ϕ V . In what follows, we concentrate on the equation ∂ Φ V eff = 0 in thebackground ϕ = e ϕ = 0 (here V eff stands for the full potential defined in (3.1)). This result is in slight disagreement with [15]. First of all, they computed the degeneracy temperaturefrom the origin to the supersymmetric vacua. Indeed, when the KKLT sector is not present, the latterform before the metastable vacua. However, by dropping the T -dependence and replacing N f by N f − N in the prefactor of (4.14), we do not find exactly their result. This is due to the fact that they use a hightemperature expansion even for the squarks in the supersymmetric vacua, which results in dropping the2 N N f in the denominator. he tree-level scalar potential V = e K h K T T D T W D T W + K i ¯ j D i W D ¯ j W − | W | i (5.1)receives temperature corrections given by (3.4), where the mass matrices squared(3.5) and (3.6) can be developped using the semi-canonical K¨ahler potential (2.8)Tr M f = h e G h K i ¯ k K j ¯ l ( G ij + G i G j ) ( G ¯ k ¯ l + G ¯ k G ¯ l ) − i i , (5.2)and Tr M s = h K i ¯ j ∂ V ∂χ i ∂ ¯ χ ¯ j i . (5.3)The new minimum at high temperature satisfies ∂V ∂ Φ + Θ ∂∂ Φ (cid:8) Tr M f + Tr M s (cid:9) (cid:12)(cid:12)(cid:12)(cid:12) ϕ = e ϕ =0 = 0 . (5.4)Let us start with the zero-temperature potential (5.1). Differentiating with re-spect to Φ yields ∂ Φ V = e K h K T T (cid:8) ( D T W K Φ + W Φ K T ) W T + K Φ W T K T W (cid:9) + K i ¯ j D ¯ j W ( K Φ D i W + K i W Φ + W i Φ ) + W D Φ W i . Since we expect the displacement h Φ i to be small, it is sufficient to keep the linearorder in Φ. One gets h ∂ Φ V i = e K h Φ n K T T (cid:0) D T W W T + W T K T W (cid:1) + (cid:12)(cid:12) h µ (cid:12)(cid:12) N f + | W | o − hµ N f n K T T K T W T + W − hµ ∗ Tr Φ oi , (5.5)where K and W are the pure KKLT potentials defined in (2.1).It is a long but straightforward computation to derive the other two contributionsin (5.4) ; some steps are given in the Appendix A for the interested reader.The general solution to the linearised equation (5.4) is of the formΦ (cid:0) T, T , Θ (cid:1) = hµ (cid:18) A + B Θ C + D Θ (cid:19) · N f , (5.6)where A, B, C, D are functions of T and T only, and given in (A.6).Figure 2 shows the behaviour of (5.6) with respect to temperature for T = T . Itis of some relevance to consider two different situations. For instance, the gravitinomass (2.11) fixes all the parameters, since the relation (2.2) between T and W on the one hand, and the zero cosmological constant (2.10) on the other hand areconditions of our model.We choose to consider m / = 1 TeV (blue, dashed line) and m / = 100 GeV (red,plain line) as an example. In both cases, as expected, the origin is the only vacuumat very high temperature. One can already approximate Φ ∼ ∼ − M P for /2 ΘΦ(Θ) −7 −7 =1.10 Θ = 4.10 c Θ c = 2.4 10 −12 Φ(Θ ) c −12 = 0.2 10 c Φ(Θ ) m = 100 GeVm = 1 TeV
Figure 2: Evolution of the mesons vev with the temperature for two values of the gravitinomass. When the temperature hits its critical value, the minimum at Φ turns into a saddlepoint. the light gravitino case. The surprise comes from the fact that the minimum fadesaway from the origin very fast when the temperature lowers down, and this happenswhile the high temperature expansion is still valid. However it could very well bethat keeping the linear order in Φ is no longer a good approximation there.The critical temperature (4.3) developped in Paragraph 4.1 depends on µ andthus on the gravitino mass. From Figure 2, it is clear that the mesons still have avery small value at the critical temperature, for both cases we considered. Therefore,the phase transition towards the non-supersymmetric vacua, in the squarks direction,will not be affected by the displacement. A problem would have arised if the mesonsvev had been too high (at the critical temperature), forcing us to take into accountthe non-perturbative superpotential (4.4).Moreover, the system with a light gravitino remains around the origin duringa longer time, ensuring even more the phase transition. Indeed, one could find aset of parameters matching our two conditions (existence of a minimum for T , zerocosmological constant) for any gravitino mass. In the case of a substantially heaviergravitino, not only the volume modulus would have a too small vev, but the phasetransition towards the would-be metastable vacua would be spoiled. We concludethat, even though it is not a very strong effect, our model seems to prefer a lightgravitino. Up to now, we have been considering that the modulus T was sitting in its minimum.In this section, we shall derive the condition under which this is valid at the typicalISS temperatures.Let us recall that since the moduli are only gravitationally coupled to the thermal ath, their interaction rate isΓ ≃ Θ M P ≪ H ≃ Θ M P . As such, the moduli potential is not in thermal equilibrium. However, indirect tem-perature corrections coming from other sectors could destabilise a modulus becausethey would result in an extra source of uplifting.For instance, in [18], the authors studied the maximal (or critical) temperaturebeyond which a minimum generated by non-perturbative effects would be destroyed.Assuming that the visible sector lives on D g ∼ / Re S . This implies that the dilaton potentialis thermally perturbed through the gauge coupling. Typically, these effects destroythe minimum if they compensate the barrier between the metastable vacuum atRe S ∼ & p m / M P ≃ − M P for a gravitino in the TeV range. Thesame could happen to the T -modulus if the visible sector lives on D W ∼ e − bS , but rather by non-trivial background fluxes W ∼ m + nS . Whereasgaugino condensation takes place at a scale Λ ≪ M P , resulting in a low mass for themodulus ( T or S according to the model), a stabilisation by fluxes happens at highenergy ∼ M P . The dilaton is then heavy enough not to be affected by temperature,and we can simply decouple it at low energy, as in the zero temperature theory. Inwhat follows, we assume that this is the case, i.e. that the visible sector does live on D T modulus potential receives temperature corrections fromthe ISS sector. If there exists a critical temperature above which the potential isdestabilised, we assume it to be higher than the temperatures computed in Section4. In this case, the ISS fields are at the origin, with the mesons slightly displaced,eq. (5.6).We define the destabilisation temperature Θ d and the corresponding value T d forthe modulus as the point where the minimum turns into a saddle point : ∂V eff ∂T ( T d , Θ d ) = 0 = ∂ V eff ∂T ( T d , Θ d ) , (6.1)where V eff is the effective potential (3.1).The computation follows similar steps as in Section 5 and Appendix A. Wesimply give here the result for the effective potential at linear order in the mesonsdisplacement V = V KKLT + e K (cid:2)(cid:12)(cid:12) h µ (cid:12)(cid:12) N f − hµ A Tr Φ − hµ ∗ A ∗ Tr Φ (cid:3) , where A (cid:0) T, T (cid:1) = K T T K T W T + W was also defined in the Appendix A.The traces of the mass matrices (5.2) and (5.3) areTr M f = − e K h | W | − hµ W Tr Φ − hµ ∗ W Tr Φ i , Since we expect the destabilisation temperature to be very high, the linear approximation made inSection 5 is even more valid as one can convince oneself from Figure 2. ndTr M s = 2 e K h N f ( N f + 2 N ) n K T T D T W D T W − | W | o + { N f ( N f + 2 N ) } (cid:12)(cid:12) h µ (cid:12)(cid:12) N f − (cid:16) hµ Tr Φ { N f ( N f + 2 N ) + 1 } (cid:16) K T T K T W T + 2 W (cid:17) + h . c . (cid:17)i . This expression is easily implemented in a Mathematica routine in order to solvethe system (6.1). m = 1 TeV100 110 T eff
Figure 3: Non-destabilisation of the T modulus at high temperature. As a result, the one-loop effective potential is shown in Figure 3. We point out thatthe constant term − C Θ in (3.4) has not been included for graphical convenience.One can see that there is no destabilisation of the modulus at all, and indeed thesystem (6.1) turns out to be non-solvable.We already argued in Section 5 that the parameters are fixed by the gravitinomass. In Figure 3, we took m / = 1 TeV ; the rest of the parameters is the same,namely a = 1, h = 1, and b = 0 .
3. The constants W and µ are fixed by equations(2.2) and (2.10), and the solution Φ ( T, Θ) was derived in the Appendix A. Thiscomputation assumes that the value of T at the minimum does not vary too much,which is cross-checked on Figure 3.In [22], the authors worked out the phase structure of the O’KKLT model [10],which can be viewed as a simplified version of our model. Although they assumed themodulus to be in thermal equilibrium, it was found there that it is not destabilisedby thermal corrections. In this perspective, we recover their result as the limit inwhich the thermal contribution of T is negligible, which is indeed the case of interestfor an expanding Universe. Following earlier work [13, 15, 16], we have studied in great detail the static phasestructure of the ISS-KKLT model when thermal corrections are considered. e are now able to give the complete picture of its thermal evolution. At veryhigh temperature, Θ ≫ Θ c , the ISS fields are at the origin because this is the pointwhere the entropy is maximised. At these temperatures, the modulus T is alreadystabilised (Fig. 3). Once it lies in its minimum, we can consider it to be static andneglect its quantum corrections to the ISS fields. Then, as the Universe cools down,the ISS fields start being driven away from the origin (Fig. 2), but they are still veryclose to it when the temperature hits its critical value Θ c ∼ − M P . A second orderphase transition takes place towards the would-be metastable vacua which at thisstage are the true vacua of the theory. At a lower temperature, the supersymmetrypreserving vacua form. They are separated from the origin by a barrier. Therefore,even if one enhances the dynamical scale Λ m in such a way that these vacua formfirst, it would consist of a first order phase transition and would thus be thermallydisfavoured. For our parameters, however, the non-supersymmetric vacua form first.At a temperature Θ deg ∼ − M P , the supersymmetric vacua become the globalvacua of the model and from that moment on, the ISS fields can tunnel from themetastable vacua to the supersymmetric ones.Even though we tried to give a complete and quantitative study of the model,there are still challenges that deserve further attention. First of all, we showed that,if the visible sector lives on D T is not destabilised by finitetemperature corrections coming from the ISS sector. This assumes that the sectorresponsible for the stabilisation of T is out of thermal equilibrium. Another limitingpoint that we have not treated is the dynamical evolution of the system, especially inthe modulus sector. Indeed, the potential generated for a modulus is generically sosteep that it seems very unlikely that the field will actually end up in the minimum,and not overshoot the barrier towards the runaway minimum (this effect is known asthe Brustein-Steinhardt problem [23, 1]). Both issues have been recently addressedin [24]. Based on the conclusions of [18], the authors have studied the conditionsunder which a stabilising sector (in their case, a SUSY-QCD) in thermal equilibriumcan lead to a destabilisation of the modulus at some temperature. They developpedthe whole set of dynamical equations when the stabilising sector is included in thethermal fluid, and constrained the initial conditions for the rolling modulus to reachits minimum. Their conclusion is that there is a region of initial conditions which leadto a stabilisation of the modulus. The allowed region is slightly reduced compared tothe case where temperature corrections are not considered, but this is not a dramaticeffect. We believe that these conclusions can be applied to our case - actually theauthors of [24] do study the KKLT setup - knowing that on the other hand wehave showed that the temperature contribution coming from the ISS sector does notdestabilise T . However, we think that a closer evaluation of the dynamics of ourmodel needs to be done. In particular, thermal fluctuations around the origin mightbe very important.Another interesting direction is inflation. It has been a big challenge for quite awhile to combine inflation with string-inspired supergravity models : see for example[10, 25] and [26] for a review. Here, the coupling of the ISS-flaton [27] to supergravityas in the ISS-KKLT setup could be of particular interest [28]. I thank Z. Lalak and S. Pokorski for bringing my attention on this problem. cknowledgments I very warmly thank W. Buchm¨uller, E. Dudas, M. Endo, Y. Mambrini, M. Postmaand A. Romagnoni for enlightening discussions, support and proofreading during thecompletion of this article. I also thank Z. Lalak and S. Pokorski for many discussionson the dynamics of the modulus. Part of this work was done when I was a PhD stu-dent at the LPT, Universit´e Orsay-Paris XI and at the CPHT, ´Ecole Polytechnique,France.
A Expression of the displacement of the mesons
In this appendix, we derive the displacement Φ of the mesons in terms of the modulus T and the temperature Θ as given in (5.6). As already sketched in Section 5, wehave to solve the equation ∂V ∂ Φ + Θ ∂∂ Φ (cid:2) Tr M f + Tr M s (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) ϕ = e ϕ =0 = 0 , (A.1)where V , Tr M f and Tr M s were respectively defined in (5.1), (5.2) and (5.3).Keeping the linear order in Φ, it is easy to show that the tree-level (and thustemperature independent) contribution to the displacement is h ∂ Φ V i = h e K h Φ n K T T (cid:0) D T W W T + W T K T W (cid:1) + (cid:12)(cid:12) h µ (cid:12)(cid:12) N f + | W | o − hµ N f n K T T K T W T + W − hµ ∗ Tr Φ oi i , (A.2)We now turn to the fermion mass matrix and compute ∂ Φ h Tr M f i . The firstterm in (5.2) gives the following contribution at the linear order h ∂ Φ n e G K i ¯ k K j ¯ l ( G ij + G i G j ) ( G ¯ k ¯ l + G ¯ k G ¯ l ) o i = h e K (cid:2) Φ (cid:8) h (cid:0) N + (cid:12)(cid:12) µ (cid:12)(cid:12) N f (cid:1) (cid:9) + 1I N f (cid:12)(cid:12) h µ (cid:12)(cid:12) Tr Φ (cid:3) i . (A.3)The last term is 2 e G which simply gives h ∂ Φ (cid:0) − e G (cid:1) i = −h e K h Φ | W | − hµ N f (cid:0) W − hµ ∗ Tr Φ (cid:1)i i . (A.4)All together, (A.3) and (A.4) give the contribution ∂ Φ h Tr M f i in equation (A.1).The trace of the scalar mass matrix squared is given in (5.3) and needs the sametreatment as before : h ∂ Φ (cid:18) K i ¯ j ∂ V ∂χ i ∂ ¯ χ ¯ j (cid:19) i . owever, with some patience, one can get the following result for this contribution h e K h Φ n (4 + 2 N f ( N f + 2 N )) · (cid:12)(cid:12) h µ (cid:12)(cid:12) N f + 2 h N + (1 + 2 N f ( N f + 2 N )) (cid:16) K T T D T W D T W − | W | (cid:17)o − hµ N f n − N f ( N f + 2 N )) hµ ∗ Tr Φ (A.5)+ (1 + 2 N f ( N f + 2 N )) (cid:16) K T T K T D T W − W (cid:17)oi i , where we used the fact that K i ¯ j K i ¯ j = 2 N f ( N f + 2 N ), which is a trace over the ISSscalar fields.From all these results, it is clear that the linearised solution takes the form Φ =Φ N f which implies that 1I N f Tr Φ = N f Φ .Eventually, plugging the different contributions into (A.1), the displacement ofthe mesons takes the formΦ( T, T ,
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