Kahler potentials for the MSSM inflation and the spectral index
aa r X i v : . [ h e p - t h ] N ov Preprint typeset in JHEP style - HYPER VERSION
HIP-2007-54/TH
K¨ahler potentials for the MSSM inflationand the spectral index
Sami Nurmi ∗ Department of Physical Sciences, P.O. Box 64, FIN-00014 University ofHelsinki, Finland,Helsinki Institute of Physics, P.O. Box 64, FIN-00014 University of Helsinki,Finland
Abstract:
Recently it has been argued that some of the fine-tuning problems of the MSSMinflation associated with the existence of a saddle point along a flat direction may besolved naturally in a class of supergravity models. Here we extend the analysis andshow that the constraints on the K¨ahler potentials in these models are considerablyrelaxed when the location of the saddle point is treated as a free variable. We alsoexamine the effect of supergravity corrections on inflationary predictions and findthat they can slightly alter the value of the spectral index. As an example, for flatdirection field values | ¯ ϕ | = 1 × − M P we find n ∼ . ... .
94 while the predictionof the MSSM inflation without any corrections is n ∼ . Keywords:
Cosmology, Inflation, Supergravity, MSSM. ∗ E-mail: sami.nurmi@helsinki.fi ontents
1. Introduction 12. The supergravity models 33. The supergravity corrections to inflation 54. The spectral index 75. Conclusions 10A. The supergravity scalar potential 11
1. Introduction
Recently it has been argued that inflation can be realized already within the Mini-mally Supersymmetric Standard Model (MSSM) [1, 2]. In this case the inflaton fieldis a particular gauge invariant combination of squarks and sleptons corresponding toa flat direction of the MSSM. Its couplings to other MSSM degrees of freedom arethus fully determined and at least in principle measurable in laboratory experimentssuch as LHC or a future Linear Collider. This is in sharp contrast with the con-ventional models where the inflaton field is usually taken to be some ad hoc gaugesinglet.As discussed in [1], the phenomenologically acceptable candidates for the inflatonfield are the dimension six flat directions udd and LLe . The potential along theseflat directions can be written to leading order order as V = 12 m | φ | − Aλ | φ | + λ | φ | , (1.1)where | φ | is the absolute value of the field parameterizing the flat direction. Hereand elsewhere in the text we use units where M P ≡
1. The parameters m and A are supersymmetry (SUSY) breaking terms depending on the underlying super-gravity (SUGRA) model and λ is an effective coupling constant associated to thenon-renormalizable operator lifting the flat direction. For a discussion of the properties of flat directions see e.g. [3, 4] and for a review of theircosmological implications, e.g. [5]. – 1 –or generic values of the SUSY breaking parameters, the potential Eq. (1.1) doesnot give rise to inflation. However, if one imposes the condition A = 40 m , (1.2)the potential has a saddle point at | φ | = (cid:16) m √ λ (cid:17) / ≪ . (1.3)Close to the saddle point the potential becomes flat enough to support inflation andcan be expanded as V ≈ m | φ | + 163 m | φ | ( | φ | − | φ | ) . (1.4)If the initial conditions are such that φ ≃ φ , there follows a period of slow rollinflation with a very low scale H inf ∼ −
10 GeV producing primordial perturbationsat the observed level and with the spectral index n ≃ .
92 [1].The success of the MSSM inflation obviously relies on the existence of the saddlepoint. Due to the exceptionally low inflationary scale, the potential needs to beextremely flat to produce large enough primordial perturbations. Consequently, thesaddle point condition Eq. (1.2) must be satisfied with an accuracy of about 10 − [2]. However, as proposed in [6] this apparent fine-tuning problem can be solvednaturally in a class of supergravity models where the K¨ahler potential is chosenin such a manner that the saddle point condition Eq. (1.2) is identically satisfied.In [6] it was found that this can be achieved with K¨ahler potentials that up toquadratic part in | φ | have a fairly natural form encountered in various string theorycompactifications but that also require fixing of some higher order terms. In this workwe show that the constraints on the K¨ahler potentials are considerably relaxed if thelocation of the saddle point is treated as a free variable. In particular, we find thatin order to identically produce the flat potential required by the MSSM inflation, theK¨ahler potential needs to be completely fixed only up to quadratic terms in | φ | andnot to higher orders as in [6]. This considerably extends the class of allowed K¨ahlerpotentials and consequently increases the possibility to find theoretically motivatedmodels that could yield the MSSM inflation.We also discuss the effect of supergravity corrections on inflationary predictions.Although the corrections are suppressed by powers of | φ | , they become significant inthe vicinity of the saddle point Eq. (1.2) where the first and second derivative of the It should be kept in mind though that in the MSSM inflation [1], the flat direction is the onlydynamical degree of freedom during inflation and the moduli fields of the underlying supergravitymodel are thus implicitly assumed to be stabilized before the beginning of inflation. This representsa non-trivial constraint in any realistic supergravity model and might also be a source of additionalfine-tuning, see e.g. [7]. – 2 –eading order potential Eq. (1.1) vanish. We find that in supergravity models wherethe MSSM inflation can be naturally realized, the relevant supergravity corrections tothe inflaton potential will manifest themselves as additional linear terms in Eq. (1.4).This kind of corrections to the MSSM inflation have been discussed in [8, 9] (see also[10] for a discussion on dark matter and the MSSM inflation) without any particularsupergravity motivation and it is well known that they can affect the spectral indexand total number of e-foldings. The difference here is that since the corrections arisefrom a given supergravity model they are not arbitrary but can be exactly calculated.We find that the supergravity corrections generically tend to increase the valueof the spectral index. For the most typical field values [1] of the MSSM inflation | ¯ ϕ | ∼ − , where | ¯ ϕ | denotes value of the canonically normalized field at thesaddle point, it is fairly easy to find K¨ahler potentials that bring the spectral indexclose to the observationally favoured value n = 0 . ± .
015 [11]. For smaller fieldvalues, | ¯ ϕ | . − the corrections become negligible and one recovers the result n ∼ .
92 whereas large field values | ¯ ϕ | & − typically yield too large spectralindex. However it is still possible to choose the K¨ahler potential such that theresulting spectral index is consistent with observations even with large field values.
2. The supergravity models
In [6] it was found that the saddle point condition of the MSSM inflation can besatisfied identically in supergravity models with F-term supersymmmetry breakingand K¨ahler potentials of the form K = X m β m ln( h m + h ∗ m ) + κ Y m ( h m + h ∗ m ) α m | φ | + µ (cid:16) κ Y m ( h m + h ∗ m ) α m (cid:17) | φ | + ν (cid:16) κ Y m ( h m + h ∗ m ) α m (cid:17) | φ | + O ( | φ | ) , (2.1)where h m denote hidden sector fields and κ, β m , α m , µ, ν are constants. The super-potential is taken to be of the form W = ˆ W ( h m ) + ˆ λ ( h m )6 φ , (2.2)and the hidden sector dependent parts are treated as constants. The MSSM inflationis not a generic outcome of all such supergravity models, though, but one needsto place constraints on the parameters of the K¨ahler potential, see [6]. Moreover,the hidden sector or moduli fields h m need to be stabilized before the beginning ofinflation by some mechanism not consistently taken into account here.It turns out that the fairly strict constraints on the K¨ahler potentials found in[6] can be considerably relaxed by allowing the location of the saddle point | φ | toslightly vary from the value given by Eq. (1.2). This is indeed a natural thing to– 3 –o since Eq. (1.2) results from the leading order part of the inflaton potential alonewhile supergravity models typically yield higher order corrections as well. Thereforewe write | φ | as an expansion | φ | = ¯ | φ | (1 + ∆ | φ | + ∆ | φ | + ... ) , (2.3)where ¯ | φ | denotes the leading order part determined by Eq. (1.2) and the terms∆ n | φ | ∼ O ( ¯ | φ | n ) represent yet unfixed higher order degrees of freedom corre-sponding to the higher order terms in the potential.Using Eq. (2.3) and repeating the analysis of [6] one finds that the flat potentialof the MSSM inflation is identically obtained if the parameters in the K¨ahler potentialEq. (2.1) are chosen according to Table (1). The conditions in Table (1) are much less Table 1:
The constraints on the parameters of the K¨ahler potential Eq. (2 .
1) implied bythe flatness of the inflaton potential. β = P β m α = P α m γ = P α m β m δ = P α m β m − − µ δ − − − − µ − − µ − µ + ν − −
19 2881 − µ − µ − µ + ν − − − − µ − − µ − µ + ν restrictive than those found in [6]. In particular, the parameters µ and ν determiningthe form of the K¨ahler potential are not fixed which considerably extends the classof allowed K¨ahler potentials. They can not be chosen completely at will however,since one needs to see to that real solutions for the conditions on α m and β m in Table(1) exist. A necessary condition for this is to require | δ | ≤ | γ | / , assuming β m ’s tobe negative integers as suggested by the string theory motivated models. Moreover,the parameters µ have to be chosen such that γ < K = − ln (cid:16) Y m ( h m + h ∗ m ) − β m − κ Y m ( h m + h ∗ m ) α m − β m | φ | (cid:17) , (2.4)where the parameters are now subject to the constraints in Table (2). One can check Note that in Table (1) we have not included the case with β = − α = − / µ can not be chosen as a free parameter and for β m ∈ Z − theresulting constraints have no solutions α m ∈ R . – 4 – able 2: The constraints on the parameters of the K¨ahler potential Eq. (2 .
2) implied bythe flatness of the inflaton potential. β = P β m α = P α m γ = P α m β m δ = P α m β m − − δ − − − − − − − − − − − − that solutions for these constraints do exist. As an example, in the case β = 7, α = 0the constraints in Table (2) are satisfied for a choice β m = − , α = 1 , α = α = α = α = − , α = α = 0 . (2.5)Both the logarithmic K¨ahler potentials Eq. (2.4) and the more generic formsEq. (2.1) bear some resemblance to the results appearing in various string theorycompactifications. Up to the quadratic part, the form of Eq. (2.4) is encounterede.g. in Abelian orbifold compactifications of the heterotic string theory [12] and inintersecting D-brane models [13]. Logarithmic K¨ahler potentials on the other handare obtained e.g. in large radius limit of Calabi-Yau compactifications and also in no-scale supergravity models [14] although the results are not precisely of the form givenin Eq. (2.4). However, it is interesting even in its own rights that the saddle pointcondition of the MSSM inflation is satisfied to the required extraordinary precisionwith K¨ahler potentials Eq. (2.4) that can be expressed in terms of a single naturalfunction.
3. The supergravity corrections to inflation
The field φ parameterizing the flat direction has a non-canonical kinetic term due tothe form of the K¨ahler potentials Eq. (2.1). Instead of using φ we therefore switchto the canonically normalized field ϕ ≡ ( κ Y m ( h m + h ∗ m ) α m ) / φ (1 + O ( | φ | )) ≡ ˆ Z / φ (1 + O ( | φ | )) , (3.1)that will be interpreted as the inflaton. Provided the conditions in Table (1) aresatisfied, the inflaton potential in the supergravity models described above identicallybecomes [6] V ( | ϕ | ) = 415 m ϕ | ¯ ϕ | + 163 m ϕ | ¯ ϕ | ( | ϕ | − | ¯ ϕ | ) + ξm ϕ | ¯ ϕ | ( | ϕ | − | ¯ ϕ | ) + . . . , (3.2)– 5 –n the vicinity of the point | ¯ ϕ | ≡ ˆ Z / | ¯ φ | ≡ (cid:16) m ϕ √ λ ϕ (cid:17) / . (3.3)Here m ϕ = m ˆ Z − / , λ ϕ = λ ˆ Z − / and the explicit expressions for them are given inthe Appendix. The first two terms in Eq. (3.2) arise from the leading order part ofthe supergravity scalar potential and, due to the constraints in Table (1), the lowestorder non-vanishing supergravity correction is of the form ξm ϕ | ¯ ϕ | ( | ϕ | − | ¯ ϕ | ) andit is small enough not to spoil the flatness of the potential [6]. The coefficient ξ isdetermined by the o ( | φ | ) part of the K¨ahler potential.Although the small supergravity correction ξm ϕ | ¯ ϕ | ( | ϕ | − | ¯ ϕ | ) does not invali-date the success of the MSSM inflation, it may still be significant at the early stagesof the inflationary period where the slope arising from the leading order potential isvery small. Indeed, small linear corrections like this have been considered withoutany particular supergravity motivation in [8, 9] and it has been shown that they willaffect the resulting spectral index. The difference in our analysis is that the correc-tions are not arbitrary but arise from the supergravity model and are thus completelyspecified. In the particular supergravity models considered here, the linear term inEq. (3.2) is also the only relevant correction to the leading order potential since thehigher order supergravity corrections are too small to leave any observable imprints[6]. The inflationary properties of the potential Eq. (3.2) can be straightforwardlyanalyzed [8, 9] using the standard slow-roll approximation. If the field starts atrest close to | ¯ ϕ | , there follows a period of inflation with the amplitude of curvatureperturbation given by P / R ≈ √ π m ϕ ξ | ¯ ϕ | h (cid:16) arctan (cid:0) √ ξ | ¯ ϕ | (cid:1) − p ξ | ¯ ϕ | N ∗ (cid:17)i − (3.4) ≈ P / R ( ξ = 0) (cid:16) − N ∗ ξ | ¯ ϕ | + O ( ξ | ¯ ϕ | ) (cid:17) , and the spectral index by n ≈ − p ξ | ¯ ϕ | tan (cid:16) arctan (cid:0) √ ξ | ¯ ϕ | (cid:1) − p ξ | ¯ ϕ | N ∗ (cid:17) (3.5) ≈ n ( ξ = 0) + 300 N ∗ ξ | ¯ ϕ | + O ( ξ | ¯ ϕ | ) . By taking ξ = 0 one recovers the results of [1] for the MSSM inflation without anycorrections. Here N ∗ ∼
50 is the number of e-foldings after the observable scales exitthe horizon and we have assumed that the end of inflation is determined by | η | = (cid:12)(cid:12)(cid:12) V ′′ V (cid:12)(cid:12)(cid:12) ∼ ⇒ | ¯ ϕ | − | ϕ end || ¯ ϕ | ≈ . (3.6) The expressions for ξ < ξ > ξ ’s with negative ones in the end results. – 6 –he magnitude of the supergravity corrections in Eqs. (3.4), (3.5) is determinedby the term ξ | ¯ ϕ | and assuming N ∗ ∼
50 they become significant for √ ξ | ¯ ϕ | & × − . In the supergravity models considered here one finds | ξ | . O (100) andfor the field values | ¯ ϕ | ∼ − typical in the MSSM inflation [1, 2], the correctionscan thus become important. On the other hand, the corrections can always bemade negligible by taking the field values to be small enough | ¯ ϕ | . − , whichcorresponds to | λ ϕ | ≫ m ϕ is regarded as an adjustable parameter, the supergravitycorrections can be seen as modifications of the spectral index alone. This is becausethe amplitude of perturbations Eq. (3.4) depends explicitly on m ϕ while the spectralEq. (3.5) index does not. By slightly changing the value of m ϕ , the amplitude canthus be kept fixed while varying the spectral index. However, besides the spectralindex, the supergravity corrections also affect the total number of e-foldings tendingto make the inflationary period shorter for ξ > ξ > N tot ≈ √ ξ | ¯ ϕ | (cid:16) arctan (cid:16) √ ξ | ¯ ϕ | (cid:17) − arctan (cid:16) √ ξ | ¯ ϕ | − | ϕ in || ¯ ϕ | (cid:17)(cid:17) , (3.7)which becomes strongly dependent on the initial value of the field | ϕ in | when thesupergravity corrections get large and, unlike in the ξ = 0 case, the initial conditions can not be explained by a period of eternal inflation even in principle since theclassical force always overcomes the quantum effects for | ξ | & − | λ ϕ | . Requiringsufficiently long period of inflation N tot &
50, Eq. (3.7) yields an absolute upperbound √ ξ | ¯ ϕ | . × − for the allowed magnitude of supergravity correctionsand using Eq. (3.5) this implies n .
4. The spectral index
As a simple example we first discuss the supergravity corrections that arise from thelogarithmic K¨ahler potentials defined by Eq. (2.4) and Table (2). In this case thesupergravity corrections and in particular the parameter ξ in Eq. (3.2) are completelydetermined since the K¨ahler potential is known to all orders in | φ | . Using standardsupergravity formulae it is then straightforward to work out the explicit expressionsfor ξ in each of the cases of Table (2), see the Appendix. The results are shownin Table (3). Assuming β m to be negative integers, the constraints in Table (2)imply ǫ ≡ P α m /β m ∼ − | ξ | .
1. As discussed above, For a discussion on initial conditions, see [15]. – 7 – able 3:
The coefficient ξ of the lowest order non-vanishing supergravity correctionsarising from the K¨ahler potentials defined by Eq. (2.4) and Table (2). Here ǫ ≡ P α m /β m and for simplicity we have given the expressions for ξ to the precision of two digits. β = P β m α = P α m γ = P α m β m δ = P α m β m ξ − − δ − − − − − × (1 .
16 + ǫ ) − − − − − . × (1 .
00 + ǫ ) − − − − − . × (1 .
16 + ǫ )the supergravity corrections to the spectral index Eq. (3.5) become significant for √ ξ | ¯ ϕ | & × − and for | ξ | . | ¯ ϕ | & − . For the typical fieldvalues of the MSSM inflation | ¯ ϕ | ∼ − the corrections are thus negligible and werecover the standard result n ∼ .
92 for the spectral index of the MSSM inflation.In the case β = − α = 0 of Table (2) this actually holds for any field valuessince the coefficient ξ vanishes identically.To discuss the supergravity corrections with the more generic K¨ahler potentialsdefined by Eq. (2.1) and Table (1), we first need to determine the potential up to | φ | . The most natural extension of Eq. (2.1) is to write K = X m β m ln( h m + h ∗ m ) + κ Y m ( h m + h ∗ m ) α m | φ | + µ (cid:16) κ Y m ( h m + h ∗ m ) α m (cid:17) | φ | + ν (cid:16) κ Y m ( h m + h ∗ m ) α m (cid:17) | φ | + ρ (cid:16) κ Y m ( h m + h ∗ m ) α m (cid:17) | φ | + O ( | φ | ) , (4.1)where ρ is a free constant. The coefficient ξ can again be straightforwardly computedin the different cases of Table (1) and the result will be of the form ξ = ξ ( µ, νρ, ǫ ),where ǫ ≡ P α m /β m . The explicit expressions are given in the Appendix and bysubstituting them into Eq. (3.5) one readily finds the resulting spectral index.Just like with the simple logarithmic K¨ahler potentials discussed above, ξ van-ishes identically in the case β = − , α = 0 of Table (1) and in this particular casethe supergravity corrections are thus absent. However, the situation is more com-plicated in the other cases of Table (1) as can be seen in Fig. (1) below. There thedependence of the spectral index Eq. (3.5) on the parameters of the K¨ahler potentialEq. (4.1) is illustrated for different field values | ¯ ϕ | . In Fig. (1) we have shown onlythe values of µ and ν for which the necessary conditions for the existence of solutionsfor the constraints in Table (1), discussed in Section 2, are satisfied. We have alsofixed the parameters ǫ and ρ appearing in the expression of ξ , but changing theirvalues will not significantly alter the qualitative behaviour of the spectral index.Fig. (1) clearly shows that the supergravity corrections typically tend to bringthe spectral index above the value n ∼ .
92 that corresponds to the MSSM inflation– 8 – = − , α = − / , | ¯ ϕ | = 1 × − µν −0.2 0 0.2 0.4 0.6 0.8−0.200.20.40.60.8 β = − , α = − / , | ¯ ϕ | = 1 × − µν µν β = − , α = − , | ¯ ϕ | = 1 × − −0.2 0 0.2 0.4 0.6 0.8−0.200.20.40.60.8 β = − , α = − / , | ¯ ϕ | = 3 × − µν −0.2 0 0.2 0.4 0.6 0.8−0.200.20.40.60.8 β = − , α = − / , | ¯ ϕ | = 3 × − µν β = − , α = − , | ¯ ϕ | = 3 × − µν −0.2 0 0.2 0.4 0.6 0.8−0.200.20.40.60.8 Figure 1:
The values of the spectral index in the different cases of Table (1). The case β = − , α = 0 is trivial with n = 0 .
923 and is thus not included. The curves with labelsdenote countour lines while the boundary curves describe the region where the necessaryconditions for the existence of solutions for Table (1) are satisfied. Here the parameters ρ and ǫ in the K¨ahler potential Eq. (4.1) are chosen as ρ = 1 / ǫ = γ . without any corrections. In the case β = − , α = − / n ∼ .
92 like in the case β = − , α = 0. In the other cases of Table (1), the correctionsare larger and the range of possible values of the spectral index is highly dependenton the field value | ¯ ϕ | . For a typical choice | ¯ ϕ | = 1 × − shown in the upper panelof Fig. (1) the spectral index depends rather weakly on the parameters µ and ν ofthe K¨ahler potential and in the region shown in Fig. (1) we find n ∼ . ... .
94. Forlarger field values the corrections rapidly become larger and the parameters µ and ν need to be chosen more carefully in order to obtain a spectral index consistent withobservations. This is demonstrated in the lower panel of Fig. (1) for | ¯ ϕ | = 3 × − .On the other hand, as discussed in Section 3, for small enough field values | ¯ ϕ | . − the supergravity corrections become negligible and we recover n ∼ .
92 in all thecases of Table (1). – 9 – . Conclusions
In this work we have discussed the supergravity origin of the MSSM inflation [1, 2]extending the analysis of [6]. We have shown that the MSSM inflation can be realizedin supergravity models with K¨ahler potentials of the simple form K = X m β m ln( h m + h ∗ m ) + κ Y m ( h m + h ∗ m ) α m | φ | + µ (cid:16) κ Y m ( h m + h ∗ m ) α m (cid:17) | φ | + ν (cid:16) κ Y m ( h m + h ∗ m ) α m (cid:17) | φ | + O ( | φ | ) , (5.1)that at least up to the quadratic part closely resemble the results found in variousstring theory compactifications [12, 13]. The flatness of the inflaton potential is anatural outcome of such supergravity models provided the parameters β m and α m inthe K¨ahler potential are appropriately chosen, see Table (1) in the text. However,unlike in the result found in [6], we have shown that it is not necessary to completelyfix the coefficients µ and ν . This considerably extends the class of allowed K¨ahlerpotentials and thus increases the possibility to find realistic supergravity models thatwould yield the MSSM inflation. We wish to emphasize though that in consideringthe MSSM inflation [1, 2] driven by a single degree of freedom, we are implicitlyassuming the moduli fields of the supergravity model to be stabilized by some mech-anism before the beginning of inflation. This represents a non-trivial assumptionand should be discussed separately in the context of any realistic model to make theanalysis complete [7].We have also examined the possibility that the underlying supergravity modelwould not yield exactly the MSSM inflation proposed in [1, 2] but a slightly modi-fied model of the type [8, 9]. In this case the supergravity corrections cause smalldeviations from the saddle point condition of the MSSM inflation and thus affect theinflationary predictions, mainly the spectral index. The magnitude of the correctionsdepends both on the parameters in the K¨ahler potential and on the field value | ¯ ϕ | but they typically tend to bring the spectral index above the value n ∼ .
92 thatcorresponds to the MSSM inflation without any corrections [1, 2]. As an example,for a natural choice | ¯ ϕ | = 1 × − one finds n ∼ . ... .
94 if the coefficients µ and ν in the K¨ahler potential are taken to be less than unity. The range of possi-ble values becomes larger for larger field values but it is still possible to choose theK¨ahler potential such that the spectral index is consistent with observations. If thefield values are small enough | ¯ ϕ | . − the corrections become negligible and wealways recover the result n ∼ . Acknowledgments
The author wishes to thank Kari Enqvist, Jaydeep Majumder and Lotta Metherfor useful discussions. The author is supported by the Graduate School in Par-ticle and Nuclear Physic. This work was also partially supported by the EU 6thFramework Marie Curie Research and Training network “UniverseNet” (MRTN-CT-2006-035863) .
A. The supergravity scalar potential
The supergravity scalar potential is written as V = e K | W | (cid:16) K M ¯ N ( K M K ¯ N + W M W ∗ ¯ N | W | + K M W ∗ ¯ N W ∗ + K ¯ N W M W ) − (cid:17) , (A.1)where K M ¯ N is the inverse of the K¨ahler metric K M ¯ N , and the lower indices denotederivatives with respect to fields. The potential for the flat direction φ is found bysubstituting the K¨ahler and superpotentials, given by Eqs. (4.1) and (2.2) respec-tively, into Eq. (A.1).If the conditions in Table (1) are satisfied and the hidden sector dependent partsof the superpotential are treated as constants, the potential can be expanded as inEq. (3.2) and the explicit expressions for m ϕ , λ ϕ in Eqs. (3.2), (3.3) read m ϕ = 2 e ˆ K | ˆ W | ( α − β −
2) (A.2) λ ϕ = e ˆ K/ ˆ Z − | ˆ λ | , (A.3)(A.4)where we have denoted ˆ K ≡ X m β m ln( h m + h ∗ m ) (A.5)ˆ Z ≡ κ Y m ( h m + h ∗ m ) α m . (A.6)– 11 –he coefficients ξ in Eq. (3.2) in the four different cases of Table (1) are given by ξ = 0 (A.7) ξ = − − µ − µ + 37669 ν (A.8)+ µ ( − ν ) − ρ − ǫξ = 37058362329500 − µ − µ + 4841425 ν (A.9)+ µ ( − ν ) − ρ − ǫξ = − − µ − µ + 516125 ν + µ ( − ν ) − ρ − ǫ (A.10)where the subindices refer to the rows of Table (1). References [1] R. Allahverdi, K. Enqvist, J. Garcia-Bellido, and A. Mazumdar, Phys. Rev. Lett. :191304, 2006 [arXiv:hep-ph/0605035].[2] R. Allahverdi, K. Enqvist, J. Garcia-Bellido, A. Jokinen, and A. Mazumdar,arXiv:hep-ph/0610134.[3] M. Dine, L. Randall and S. Thomas, Phys. Rev. Lett. :398, 1995[arXiv:hep-ph/9503303]; Nucl.Phys.B :291, 1996 [arXiv:hep-ph/9507453].[4] T. Gherghetta, C. F. Kolda and S. P. Martin, Nucl. Phys. B :37, 1996[arXiv:hep-ph/9510370].[5] K. Enqvist and A. Mazumdar, Phys. Rept. :99, 2003 [arXiv:hep-ph/0209244].[6] K. Enqvist, L. Mether and S. Nurmi, [arXiv:0706.2355].[7] Z. Lalak and K. Turzynski, arXiv:0710.0613 [hep-th].[8] J. C. Bueno Sanchez, D. Lyth and K. Dimopoulos, JCAP :015, 2007[arXiv:hep-ph/0608299].[9] R. Allahverdi and A. Mazumdar, arXiv:hep-ph/0610069.[10] R. Allahverdi, B. Dutta and A. Mazumdar, Phys. Rev. D (2007) 075018[arXiv:hep-ph/0702112].[11] D. N. Spergel et al. [WMAP Collaboration], arXiv:astro-ph/0603449.[12] See e.g. L. E. Ibanez and D. Lust, Nucl. Phys. B , 305 (1992)[arXiv:hep-th/9202046] and references therein. – 12 –
13] See e.g. D. Lust, S. Reffert and S. Stieberger, Nucl. Phys. B (2005) 264[arXiv:hep-th/0410074] and references therein.[14] For a review of no-scale models, see A. B. Lahanas and D. N. Nanopoulos, Phys.Rept. :1, 1987.[15] R. Allahverdi, A. R. Frey and A. Mazumdar, Phys. Rev. D (2007) 026001[arXiv:hep-th/0701233].(2007) 026001[arXiv:hep-th/0701233].