Metastable SUSY Breaking, de Sitter Moduli Stabilisation and Kähler Moduli Inflation
aa r X i v : . [ h e p - t h ] A ug Preprint typeset in JHEP style - HYPER VERSION DAMTP-2009-12CERN-PH-TH/2009-014Metastable SUSY Breaking, de Sitter ModuliStabilisation and Kähler Moduli In(cid:29)ationSven Krippendorf , Fernando Quevedo , DAMTP, Centre for Mathemati al S ien es,Wilberfor e Road, Cambridge, CB3 0WA, United Kingdom CERN PH-TH, CH 1211, Geneva 23, Switzerland.Abstra t: We study the in(cid:29)uen e of anomalous U (1) symmetries and their asso iatedD-terms on the va uum stru ture of global (cid:28)eld theories on e they are oupled to N = 1 supergravity and in the ontext of string ompa ti(cid:28) ations with moduli stabilisation. Inparti ular, we fo us on a IIB string motivated onstru tion of the ISS s enario and examinethe in(cid:29)uen e of one additional U (1) symmetry on the va uum stru ture. We point outthat in the simplest one-Kähler modulus ompa ti(cid:28) ation, the original ISS va uum getsgeneri ally destabilised by a runaway behaviour of the potential in the modulus dire tion.In more general ompa ti(cid:28) ations with several Kähler moduli, we (cid:28)nd a novel realisationof the LARGE volume s enario with D-term uplifting to de Sitter spa e and both D-termand F-term supersymmetry breaking. The stru ture of soft supersymmetry breaking termsis determined in the preferred s enario where the standard model y le is not stabilisednon-perturbatively and found to be (cid:29)avour universal. Our s enario also provides a purelysupersymmetri realisation of Kähler moduli (blow-up and (cid:28)bre) in(cid:29)ation, with similarobservational properties as the original proposals but without the need to in lude an extra(non-SUSY) uplifting term.Keywords: dS va ua in string theory, Strings and branes phenomenology,Supersymmetry Breaking, In(cid:29)ation.ontents1. Introdu tion 22. The ISS model and its embedding into String Theory 33. The One-modulus Case 54. The Two-moduli Case 74.1 Strategy 84.2 Matter (cid:28)eld stabilisation 94.3 Stabilising the Kähler moduli 125. D7 Soft-Terms 156. Kähler moduli in(cid:29)ation 197. 3-Parameter K3 (cid:28)bration and Fibre In(cid:29)ation 217.1 Fibre In(cid:29)ation 238. Constraints for metastable dS va ua in supergravity setups 258.1 Constraints on Anomalous U (1) Gauge Symmetries as Uplifting Potential 269. Con lusions 26A. On stabilising the F-term potential with a onstraint 27B. Next to leading order orre tions to Φ N = 1 supergravity is not positive de(cid:28)nite andtherefore the lo al minima tend to be at negative values of the va uum energy. Theoriginal uplifting me hanism to de Sitter spa e proposed in referen e [1℄ by introdu inganti D3 branes requires the expli it breaking of supersymmetry in the e(cid:27)e tive (cid:28)eldtheory. Further uplifting me hanisms have been proposed [2, 3℄. Despite partialsu ess, there is not at the moment a ompelling me hanism for de Sitter uplifting.The onsideration of D-terms as proposed in [2℄ has to be implemented in on retesetups sin e, even though D-terms add a positive de(cid:28)nite ontribution to the e(cid:27)e tivepotential, it is known that if the F-terms vanish then D-terms also vanish [4℄. Con reteexamples have been provided in (cid:28)eld theoreti al [5℄ and string inspired models [6, 7℄.2. In the past two years there has been a large amount of work on the natural ap-pearan e of metastable supersymmetry breaking va ua in global supersymmetry andin parti ular, the breakdown of global supersymmetry an be a hieved signi(cid:28) antlysimpler. This is the ISS s enario [8℄. Some realisations of this me hanism in stringmodels have also been obtained [9℄. Nevertheless, there is an impli it assumption inthis me hanism, i.e. that the gauge oupling of the orresponding gauge theory is onstant and therefore the dynami al non-perturbative s ale Λ ∼ e − a/g appears as a onstant in the e(cid:27)e tive a tion. In string theory, however, the gauge oupling is (cid:28)elddependent /g ∼ Re T with T the omplex s alar omponent of a hiral super(cid:28)eld orresponding to a losed string modulus. Therefore the exponential dependen e of Λ would tend to give a runaway behaviour to the s alar potential as a fun tion of Re T .3. In the past few years several string theory me hanisms for osmologi al in(cid:29)ation havebeen proposed. The in(cid:29)aton (cid:28)eld orresponds to either an open string mode as inbrane-antibrane [10℄, D3/D7 [11℄, monodromy [12℄ or Wilson line [13℄ in(cid:29)ation or a losed string modulus as in Ra etra k [14,15℄, Kähler moduli [16,17℄, monodromy [18℄or (cid:28)bre [19℄ in(cid:29)ation. In most of these s enarios, the realisation of in(cid:29)ation depends ru ially on the uplifting me hanism for de Sitter moduli stabilisation. Thereforeit tends to go beyond the N = 1 supersymmetri e(cid:27)e tive a tion if the upliftingme hanism is the presen e of anti D3 branes as in KKLT. It would be desirable to(cid:28)nd a string in(cid:29)ation me hanism derived from a fully N = 1 supersymmetri a tion.Re ently in [20℄, general onstraints on this possibility have been found, substantiallyrestri ting the lass of supersymmetri models that an give rise to in(cid:29)ation. Itis a hallenge to (cid:28)nd a on rete stringy realisation of in(cid:29)ation that satis(cid:28)es those onstraints. (cid:21) 2 (cid:21)e address these three issues by onsidering a lass of models that an be realised in terms ofD-brane orientifold onstru tions in (cid:29)uxed Calabi-Yau manifolds. In parti ular we onsidera system of magnetised D7 branes with hiral matter (cid:28)elds. The existen e of anomalous U (1) 's indu es Fayet-Iliopoulos D-terms. Fluxes of RR form give rise to a tunable onstantterm in the superpotential and also the presen e of matter (cid:28)elds provides a non-perturbativesuperpotential. Depending on the number of D-branes this an be realised in the ele tri ormagneti phase of SQCD. The ele tri phase was onsidered in [6℄. Here we will on entrateon the magneti phase. This makes onta t with the ISS models [8℄ and then naturallyaddresses the issue of the runaway behaviour of the s alar potential. We (cid:28)nd that in thesimplest ase of one single Kähler modulus, instead of a metastable va uum, the potentialruns away in the dire tion of the modulus orresponding to the size of the 4- y le wrappedby the magnetised D7 branes, ruining the interesting properties of the ISS me hanism.However, on e we have several Kähler moduli, the situation hanges drasti ally. We (cid:28)ndminima of the s alar potential at (cid:28)nite values of the (cid:28)elds with supersymmetry brokenas in the ISS me hanism. Furthermore, depending on the values of the free parameters,the minima orrespond to either anti de Sitter or de Sitter spa es. In the latter ase, itprovides a natural uplifting me hanism for moduli stabilisation, addressing also the (cid:28)rstissue mentioned above.Even though the presen e of matter (cid:28)elds and D-terms hanges the stru ture of thes alar potential very mu h, we (cid:28)nd remarkably that, on e the matter (cid:28)elds are integratedout, the s alar potential for moduli (cid:28)elds ends up to be very similar, but not identi al, tothe one found in [21, 22℄ and provides a new realisation of the LARGE volume s enario ofmoduli stabilisation.Due to the similarity with the original LARGE volume s enario, we derive the stru tureof soft supersymmetry breaking terms and ompare with previous results, we then revisit theKähler moduli and (cid:28)bre in(cid:29)ation s enario in our setup and (cid:28)nd that they are also naturallyrealised, therefore providing a pure N = 1 realisation of these in(cid:29)ationary s enarios.2. The ISS model and its embedding into String TheoryWe want to dis uss the va uum stru ture of the magneti dual within the Seiberg dualitysetup of SQCD whi h arises in the range N C < N F < / N C . As ommonly known in thisphase of SQCD there exists a dual des ription of the low-energy (cid:28)eld theory in terms of aninfrared-free modi(cid:28)ed version of SQCD, the so- alled free magneti dual des ription. It is hara terised by the meson (cid:28)elds Φ , quarks q and anti-quarks p whi h transform under thefollowing symmetry groups: SU ( N F − N C ) SU ( N F ) L SU ( N F ) R U (1) B U (1) A U (1) R q (cid:3) (cid:3) p ¯ (cid:3) (cid:3) − (cid:3) (cid:3) − (2.1)where the only gauge symmetry is the SU ( N F − N C ) symmetry. The Kähler potential is(cid:21) 3 (cid:21)aken to be anoni al and looks like K = | q | + | p | + | Φ | . (2.2)In order to obtain dynami al SUSY breaking one introdu es a mass term. Hen e the mostgeneral invariant superpotential we an write down is W = Λ q Φ pµ + Λ m Φ , (2.3)where m denotes the mass parameter of our theory, Λ the dynami al s ale of our theory, and µ is another parameter that is determined by the duality relations. In the above equationsand in the whole arti le we omit indi es where possible. Please note that the s ale Λ hasto be introdu ed due to dimensional requirements.The metastable va uum solution is given by Φ = 0 and p = q = i √ µm. A derivation an be found in [23℄. The mass term expli itly breaks the U (1) A global symmetry.There have been several realisations of the ISS s enario within string theory in termsof di(cid:27)erent brane on(cid:28)gurations [9℄. Here we are more interested in apturing the mainingredients that a string theory realisation will add to this s enario, namely the fa t thatthe parameters m and Λ have to be dynami al variables. In parti ular Λ in string theoryhaving a non-perturbative origin is of the form Λ ∼ e − aT with T a losed string modulus.One simple expli it realisation is the magneti version of the ase onsidered in [6℄. Thebasi setup is an orientifold model onsisting of a sta k with N C + 1 branes, from whi h onehas a non-vanishing magneti (cid:29)ux leading to U ( N c ) × U (1) gauge theory. The open stringsgoing from the sta k of N c branes to the magnetised one are the elementary hiral (cid:28)elds Q .The anti- hiral (cid:28)elds ˜ Q have their end-points in the sta k of N c branes and the orientifoldimage of the magnetised brane, whereas the (cid:28)elds ρ with endpoints in both images of themagnetised brane are singlets under the non-abelian group but harged under the U (1) .The e(cid:27)e tive (cid:28)eld theory in the ele tri phase ( N F < N c ) was studied in detail in [6℄.Here we will onsider the magneti phase appropriate for N c < N F < N c / for whi hinstead of Q, ˜ Q the fundamental (cid:28)elds are the dual (cid:28)elds whi h we denote q, p and themeson-like (cid:28)eld Φ . The singlet (cid:28)eld ρ will play the role of the mass parameter. Sin egeneri ally the U (1) is anomalous, the modulus T whose real part is the inverse gauge oupling is also harged under the U (1) . The U (1) harges are given in the table. q p ρ Φ e aT U (1) − / − / − The e(cid:27)e tive (cid:28)eld theory will be determined by the superpotential W , the gauge kineti fun tion f and Kähler potential K as follows: the gauge kineti fun tion for the gauge (cid:28)eldson the relevant D7 brane is at leading order f = T s , where T s is the Kähler modulus whosereal part is τ s . The superpotential is of the form W = W + W np (2.4)(cid:21) 4 (cid:21)ith W a (cid:29)ux indu ed superpotential whi h is a onstant after (cid:28)xing the omplex stru turemoduli and the non-perturbative superpotential is taken to be of the moduli dependent ISSform 1: W np = αe − aT s (cid:18) p Φ qµ + Φ ρ (cid:19) , (2.5)Here α, a, µ are onstants. The Kähler potential is K = − V + ξ ) + K matter , (2.6)where V is the volume written in terms of the Kähler moduli τ a and ξ orresponds to theleading order α ′ - orre tions. The hiral matter Kähler potential is only known in a small(cid:28)eld expansion as a fun tion of the Kähler moduli, with the omplex stru ture dependen eunknown, however this is enough for our purposes here. From the analysis in [24℄ we anwrite: K matter = τ ns V / (cid:0) | Φ | + | ρ | + | q | + | p | (cid:1) , (2.7)where n is the modular weight. We an then ask how the stru ture of the va uum forthe ISS s enario is modi(cid:28)ed on e the moduli dependen e of the superpotential and Kählerpotential are in luded.3. The One-modulus CaseWe (cid:28)rst onsider the simplest ase of a one Kähler modulus Calabi-Yau with T = τ + ib, where τ denotes the Einstein frame volume of the 4- y le X and b = R X C . This is similarto the setup presented by Nakayama et al. [25℄ and is onsidered as a stabilisation in thespirit of KKLT [1℄. After dis ussing the setup, we will show why it is not possible to stabilisethe Kähler moduli and brane moduli in a viable regime.The Kähler potential takes the simple form (here the expli it dependen e on g s and M P is in luded sin e they are important for the numeri al estimates): K = − M P log V + ξg / s ! + 1( g s τ ) n (cid:0) | Φ | + | ρ | + | q | + | p | (cid:1) , (3.1)where we kept the modulus weight general and in luded the leading order α ′ − orre tions.The volume simply is assumed to be V = τ / and is in the Einstein frame. The matter(cid:28)elds have mass dimension 1. The superpotential be omes W = g / s M P W + M P g / s e − aT α (cid:18) p Φ qµ + ρ Φ (cid:19) . (3.2)We an now al ulate the D-term potential for the anomalous U (1) with the previouslydis ussed harge assignments. Setting g s = M P = 1 , for simpli ity of notation, the D-term1A full string theory derivation of this non-perturbative superpotential is not yet available (see [25℄ fora previous dis ussion of this system). (cid:21) 5 (cid:21)otential be omes: V D = 12 τ −| ρ | − ( | q | + | p | ) + 2 | Φ | τ n + 1 a (cid:18) − √ τ ξ + τ / ) − n | ρ | + | p | + | q | + | Φ | τ n +1 (cid:19)(cid:19) , (3.3)The F-term potential is al ulated from the standard supergravity formula V F = e KM P (cid:18) D i W D ¯ j ¯ W K − i ¯ j − | W | M P (cid:19) . (3.4)with D i W = ∂ i W + W ∂ i K/M P . If we assume no impli it dependen e of the matter (cid:28)eldson the volume, the leading order ontributions to the F-term potential are given by: e K ∼ (cid:0) τ / + ξ (cid:1) , (3.5) ∂W ∂ ¯ W K − ∼ (cid:18) τ − nτ − n (cid:19) a e − aτ α (cid:12)(cid:12)(cid:12)(cid:12) p Φ qµ + ρ Φ (cid:12)(cid:12)(cid:12)(cid:12) for n > , (3.6) W ∂W ∂ ¯ KK − + . . ∼ W aα τ Re (cid:18) e − aT (cid:18) p Φ qµ + ρ Φ (cid:19)(cid:19) , (3.7) | W | ∂K∂ ¯ KK − ∼ | W | (cid:18) ξ τ / + (1 − n ) | p | + | q | + | ρ | + | Φ | τ n (cid:19) . (3.8)Bearing in mind that the (cid:29)ux parameter W has to be exponentially small in KKLT-likesetups, the key observation for the following dis ussion is the fa t that we have got ahierar hy between the D-term potential and the F-term potential, simply sin e the F-termpotential is exponentially suppressed and the D-term potential is not. Therefore the D-term potential has to vanish almost exa tly, that is it should vanish up to the exponentialsuppression of the F-term potential.Be ause of the D-term dominan e, we are required to address a ompletely di(cid:27)erentstabilisation pro edure ompared to the global (cid:28)eld theory setup of ISS. It is a tually notpossible to integrate out the (cid:28)elds at the minimum of the global SUSY model.The multi-(cid:28)eld minimisation pro ess an be simpli(cid:28)ed (as in the global ase) by notingthat the D-term potential arising from the SU ( N F − N C ) gauge theory an be minimisedby simply requiring the VEV of both type of (cid:28)elds q and p to be the same, sin e: V SU ( N F − N C ) D = g X A (cid:16) Tr q † T A q − Tr pT A p † (cid:17) , (3.9)where T A denotes the generator of the fundamental representation of SU ( N F − N C ) . It an further be seen that the minimum orresponds when both vanish q = p = 0 .2 Sin ethe D-term is dominant the remaining matter (cid:28)elds are related by a ondition of the form | ρ | ∼ | Φ | − b, where b is determined by the leading order FI term. This redu es the2See Appendix C for a detailed dis ussion. (cid:21) 6 (cid:21)otential as a fun tion of, say, | Φ | and τ . As expe ted we do not (cid:28)nd a minimum for largevalues of τ where the potential shows the standard runaway behaviour as illustrated in the(cid:28)gure 1 below. Figure 1: This plot shows theF-term potential in dependen eof the Kähler modulus τ and theremaining matter degree of free-dom Φ . We display small valuesof Φ in order to show the readerthe runaway behaviour, whi his learly shown in the pi ture.Keeping Φ onstant would givethe typi al KKLT situation.For small values of τ lo al minima an be found, but in a regime that does not justifykeeping only the leading order orre tions to K [25℄.4. The Two-moduli CaseSin e the one-modulus ase leads naturally to the runaway behaviour, it is tempting to on lude that string theory realisations of the ISS me hanism do not have a metastablestate but a runaway potential. We will now explore the Swiss heese, two-moduli asewhi h has proven to lead to qualitatively di(cid:27)erent va uum stru ture in the LARGE volumes enario of moduli stabilisation and re onsider moduli stabilisation in our set-up.We will then onsider the geometry of the Calabi-Yau de(cid:28)ned as the surfa e in proje tivespa e P [1 , , , , where the volume is given by V = τ / − τ / , (4.1)where τ denotes the large 4- y le and τ the small one. Then our stringy setup is givenby: K = − M P log τ / − τ / + ξg / s ! + g ns τ n g s τ ( | p | + | q | + | ρ | + | Φ | ) , (4.2) W = M P g / s W + αM P g / ns e − aT (cid:18) p Φ qµ + Φ ρ (cid:19) . (4.3)In the following we will use the harge assignment q p ρ Φ e aT U (1) 1 1 2 − (cid:21) 7 (cid:21)hat is the D7 branes are wrapping the small four- y le and then it is T that gets hargedunder U (1) and not T . Then the D-term potential be omes V D = g s τ (cid:20) | ρ | τ n g − ns τ + ( | q | + | p | ) τ n g − ns τ − | Φ | τ n g − ns τ ++ 1 a M P √ τ (cid:18) V + ξg / s (cid:19) + ( | ρ | + | p | + | q | + | Φ | ) nτ n − g − ns τ , where the overall g s arises due to the transformation from string to Einstein frame. Ingeneral there are two possible harge assignments; one is displayed above, and the otherinter hanges the harges ρ and Φ , meaning that both (cid:28)elds are hanging their role in theD-term stabilisation.The F-term potential is given by the usual supergravity formula (divided by g s whi harises by integrating out the dilaton dependen e) V F = 1 g s e KM P (cid:18) D i W D ¯ j ¯ W K i ¯ j − | W | M P (cid:19) (4.4) = e KM P g s (cid:18) ∂ i W ∂ ¯ j ¯ W K i ¯ j + ∂ i W ∂ ¯ j ¯ KK i ¯ j ¯ WM P + WM P ∂ i K∂ ¯ j ¯ W K i ¯ j + | W | M P ∂ i K∂ ¯ j ¯ KK i ¯ j − | W | M P (cid:19) , where the indi es run over the two Kähler moduli T , T and matter (cid:28)elds p, q, Φ .4.1 StrategyOur aim is now to (cid:28)nd a minimum of the whole potential at large volume to guaranteestability towards unknown higher order orre tions. Therefore we should always keep tra kof the power dependen e in τ whi h orresponds dire tly to the power suppression by thevolume. The results presented are only reliable up to some order in the volume suppression.Let us start by emphasising the following points: We an see that the D-term potentialis suppressed by /τ whereas the F-term potential is at least suppressed by /τ . Thisdire tly implies that the D-term ontribution is a priori leading ompared to the F-termpotential and hen e the global SUSY analysis of ISS will be modi(cid:28)ed. Also, minimisationwith respe t to the matter (cid:28)elds will lead to a moduli dependen e of their VEVs.Minimising the whole potential analyti ally is not possible due to the omplexity ofthe system orresponding to a potential as a fun tion of six omplex (cid:28)elds. Instead, wewill use the suppression with respe t to the volume of every single term as a natural order riterion, sin e this allows us to dis ard various higher order ontributions. Our approa h an be summarised as follows:1. The (cid:28)rst step of our minimisation pro edure will be to (cid:28)nd the dependen e of thematter (cid:28)elds (e.g. χ ) on the large Kähler modulus, at leading order, writing χ = ˜ χ/ V m and determine m for ea h matter (cid:28)eld χ , still not (cid:28)xing the matter (cid:28)elds ompletely(leaving ˜ χ still un(cid:28)xed). (cid:21) 8 (cid:21). After that we an minimise the matter (cid:28)elds ompletely by (cid:28)xing ˜ χ . We then end upwith a s enario whi h looks roughly like the original LARGE volume s enario withan additional uplifting D-term.3. We (cid:28)nally stabilise the potential with respe t to the Kähler moduli numeri ally.4.2 Matter (cid:28)eld stabilisationSimilar to the one-(cid:28)eld ase, the (cid:28)elds p and q an be stabilised at p = q = 0 .3 Then weneed to determine Φ and ρ as fun tions of the volume. We should note at this point thatit is not possible to set any of these two (cid:28)elds to zero, in parti ular ρ = 0 , sin e then the ontribution from ∂W ∂KK − vanishes as well, and we remain with only positive de(cid:28)niteterms in the F-term potential.We would now like to (cid:28)x the impli it dependen e of the matter (cid:28)elds on the Kählermoduli. In order to ensure that there will not be any further leading order orre tion tothis assumption it is ne essary to satisfy one of the following onditions:After (cid:28)xing the quark and antiquark (cid:28)elds the D-term potential looks like: V D = g s τ V / | ρ | (36 aτ + n )18 ag − ns τ − n − | Φ | (18 aτ − n )18 ag − ns τ − n + 3 M P √ τ a V /
11 + ξg / s V . (4.5)Starting from the observation that the only negative ontribution in the D-term potential omes from the term in luding Φ (cid:28)elds, it is natural to think that this term will an elthe leading order FI ontribution.4 Can elling the leading order FI-term with the Φ (cid:28)eldimplies the following impli it volume dependen e: | Φ | = | ˜Φ | V / . (4.6)With this assumption we an rewrite the D-term as follows: V D = g s τ V / (cid:18) | ρ | (36 aτ + n )18 ag − ns τ − n (cid:19) + 1 V / √ τ a M P ξg / s V − | ˜Φ | (18 aτ − n )18 ag − ns τ − n + 2 V / √ τ a M P ξg / s V − | ˜Φ | (18 aτ − n )18 ag − ns τ − n (cid:18) | ρ | (36 aτ + n )18 ag − ns τ − n (cid:19) . (4.7)To an el the FI-term we (cid:28)nd: | Φ | = | ˜Φ | V / = 54 M P g − ns τ / − n aτ − n ) V / aτ ≫ = 3 M P g − ns τ / − n a V / . (4.8)This assignment minimises the potential with respe t to Φ at leading order. Higher order orre tions are in prin iple of importan e if we stabilise ρ at a higher order ompared to3See Appendix C for a detailed dis ussion.4The next to leading order FI ontribution is of sub-leading order even ompared to the leading orderF-term potential. (cid:21) 9 (cid:21) . However, we show in the appendix that we an negle t these higher order orre tions atour minimum.Fo using on the mass (cid:28)eld ρ, the (cid:28)rst guess might be that there should be no di(cid:27)eren ein the impli it dependen e between Φ and ρ sin e they o ur ompletely symmetri allyafter setting the quark (cid:28)elds to zero apart from their appearan e in the D-term. In this ase, we end up with the following two possibilities:1. Φ and ρ both get VEVs that are of the same order as the FI ontribution but we haveto an el the D-term to leading order, so we do have to an el the FI-term with Φ .This (cid:28)xes one of the matter (cid:28)elds up to a phase. We then have to minimise the F-term potential on its own but with the following onstraint | ρ | ∼ | Φ | − b, where b isdetermined by the leading order FI part. The stru ture of the F-term looks promisingsin e we (cid:28)nd the exa t same volume suppression as in the LARGE volume s enarioand we an minimise with respe t to the volume. But when we then try to stabilisethe remaining matter (cid:28)eld, it turns out that one annot a hieve it at large values forthe volume. A detailed proof of this statement an be found in appendix A.2. Φ and ρ get VEVs that are larger then the FI-term and they then di(cid:27)er at that "sub-leading" order. The problem now is that the matter (cid:28)elds be ome so massive thatwe annot trust our expansion anymore sin e we have to onsider trans-Plan kiandynami s. In addition, one runs into similar problems as above.Sin e the D-term potential should be suppressed higher than the F-term potential we wouldlike ρ to be higher suppressed than Φ . Hen e we have to give ρ a higher impli it suppressionwith respe t to the Kähler moduli than to Φ . To (cid:28)nd the impli it suppression and havea maningful approximation, we demand that we have a very large volume and then, afterthe full stabilisation pro ess is (cid:28)nished, he k if the assumption is onsistent. This gives uswhi h term should be leading order in τ , namely: ∂W ∂ ¯ W K − e K . (4.9)Sin e the next to leading order ontribution from the FI-term is learly subleading, wehave to suppress the term proportional to ρ at least more than the leading order F-termpotential term. This onstraint implies that the leading order F-term ontribution from theterm above is independent of ρ. Clearly ρ will o ur in the leading order term oming from ∂W ∂ ¯ KK − e K . With this impli ation we an determine the leading order F-term potentialwhi h looks as follows: V F = M P g s τ g ns | ˜Φ | α √ τ e − aτ M P τ n + 4 ag ns W ατ e − aτ Re (˜ ρ ˜Φ) M P τ / m + 3 W (cid:18) ξg / s + nτ n | ˜Φ | g − ns M P (cid:18) − ξg / s τ / (cid:19)(cid:19) τ / , (4.10)(cid:21) 10 (cid:21)here we kept the expli it dependen e on ˜Φ and m re(cid:29)e ts the un ertainty in the impli itsuppression of ˜ ρ with respe t to the large Kähler modulus and worked in the limit aτ ≫ for larity. Terms in luding higher order orre tions in Φ have not been written down butwould not enter at leading order. From the stru ture above, we an see that we annotstabilise the potential with respe t to ˜ ρ with a vanishing D-term sin e the potential givesa runaway behaviour to leading order with respe t to ˜ ρ. In order to avoid any further impli it dependen e on τ , we have to hoose the impli itdependen e in a way that both terms, the D-term ontribution and leading order F-term ontribution from ∂W ∂ ¯ KK − e K have the same τ suppression. Fortunately this an bea hieved by the following impli it dependen e: | ρ | = | ˜ ρ | τ / = | ˜ ρ | V / . (4.11)At leading order we an stabilise with respe t to ˜ ρ as follows: Negle ting higher order orre tions in Φ , we fa e the following leading order potential in the limit aτ ≫ evaluatedat ˜Φ = ˜Φ min : V = V D + V F = 2 τ n − | ˜ ρ | g − ns V / + 3 M P g s α e − aτ aτ n − / V / + 2 √ g / n/ s M P W α √ τ / − n/ e − aτ Re ˜ ρ V / + 3 g s M P W g / s V / ξ + g / s n √ τ a ! . (4.12)The stabilisation with respe t to ˜ ρ goes as follows: Assuming W to be real, we (cid:28)rstobserve that all oe(cid:30) ients are in front of ˜ ρ are real. Di(cid:27)erentiating with respe t to ˜ ρ andits omplex onjugate then dire tly leads to the fa t that ˜ ρ has to be real. We an nowstraightforwardly stabilise ˜ ρ ∂∂ ˜ ρ A (˜ ρ ) (˜ ρ ∗ ) + B (˜ ρ + ˜ ρ ∗ ) , (4.13)where A and B are abbreviations for the oe(cid:30) ients in the leading order Kähler potential.That onstraint implies A ˜ ρ (˜ ρ ∗ ) + B ˜ ρ ∈ R = 2 A ˜ ρ + B . (4.14)This an be solved dire tly for ˜ ρ and we obtain in the limit aτ ≫ ρ min = (cid:18) a (cid:19) / M P g / − n/ s ( − W α ) / τ / − n/ e − aτ / . (4.15)After stabilising ˜ ρ, we end up s hemati ally with the following leading order potential(where we (cid:28)xed n = 1 / for simpli ity): V = g s M P α e − aτ aτ / V / + 2 (cid:16) g / s | ˜ ρ | + √ aM P W α Re (˜ ρ ) τ / e − aτ (cid:17) g / s M P τ / V / + 3 W ξ g / s V / = g s M P Ae − aτ τ / V / − Bτ / e − / aτ V / + C V / ! . (4.16)(cid:21) 11 (cid:21)here we have de(cid:28)ned: A = 3 α a , B = 6 / − (cid:18) (cid:19) / ! a / ( − W α ) / , C = 3 W ξ g / s . (4.17)Due to the minimisation with respe t to ρ we obtain a di(cid:27)erent exponential suppression thanthe standard LARGE volume s enario, but keep a similar stru ture with similar results.We an learly see from the potential that the extremisation with respe t to ˜ ρ givesa minimum sin e the oe(cid:30) ients A and B are positive and at the minimum we have anegative ontribution from this fa tor.4.3 Stabilising the Kähler moduliTo ompare our stabilisation pro edure to the LARGE volume s enario we go one step ba kto the potential before extremising with respe t to ˜ ρ : V = 2 τ n − | ˜ ρ | g / s V / + 3 g s M P α e − aτ aτ n − / V / + 2 √ M P g / s W α √ τ / − n/ e − aτ Re ˜ ρ V / + 3 M P W V / ξg / s + n √ τ a ! . (4.18)This leading order stru ture looks very promising sin e it is very lose to the originalLARGE volume s enario with an additional uplifting term: V LV = g s M P (cid:18) a A √ τ e − aτ V + W Aaτ e − aτ V + W ξ V + V uplift V (cid:19) , (4.19)where we negle ted numeri al onstants for larity. The di(cid:27)erent suppressions with respe tto the volume are re(cid:29)e ted in an overall res aling. Nevertheless, there is a slight di(cid:27)eren ein the power dependen e with regard to the volume in the ∂W ∂ ¯ KK − e K ( / would beperfe t) whi h spoils the possibility of an exa t analyti al dis ussion of the minimisationas it was possible in the LARGE volume ase. However the di(cid:27)eren e to the original setupis rather small. Therefore the suppression looks to be a bit larger driving the minimum tolarger values. Obviously the adho uplifting ontribution from previous LARGE volume onstru tions gets repla ed by a on rete supersymmetri D-term ontribution.After stabilising the matter (cid:28)elds, we now have a potential whi h only depends on theKähler moduli. Sin e we are not able to al ulate the va uum stru ture analyti ally we(cid:28)nd the minima numeri ally. To be more expli it we will spe ialise to the modulus weight n = 1 / from now on, whi h is for pra ti al reasons only sin e similar results will hold forother weights. After stabilising the matter (cid:28)elds we obtained the leading order potential inthe limit aτ ≫ given by (4.16), whi h an be written in terms of τ , as: V = g s M P Ae − aτ τ / τ / − Be − / aτ τ / τ / + Cτ / ! (4.20)(cid:21) 12 (cid:21)fter (cid:28)xing the matter (cid:28)elds to their value at the minimum, the uplifting term from theD-term seems to disappear sin e we only have three terms in the potential whi h are similarto the original LARGE volume. However the e(cid:27)e t of the D-term potential an be seen inthe hange of the exponential suppression of the term proportional to B whi h orrespondsto the negative ontribution. This means that we have a smaller negative ontribution ompared to the setup without the D-term. Despite the fa t that the D-term does notseem to be present anymore, it a(cid:27)e ts the shape of the potential in exa tly the way anuplifting term does.In general, the shape of the potential after (cid:28)xing the matter (cid:28)elds suggests that thereis a LARGE volume like stru ture with possible D-term lifting. Working stri tly withinthe Kähler one (i.e. τ ≪ τ ), at relatively small values of the volume the term propor-tional to the α ′ orre tions is dominant. In an intermediate range the attenuated negative ontribution proportional to W an be dominant and at large volumes the leading order ontribution will ome from the ∂W ∂ ¯ W K − part. Due to the domination of this at largevolumes, the potential approa hes zero from above in all dire tions. It is in fa t straigtfor-ward to see that when the volume tends to in(cid:28)nity, the negative term annot dominate.This behaviour at large volume is in agreement with the uplifted LARGE volumes enario, where we approa h zero from above. Therefore, by hanging the value of the pa-rameters we may obtain an AdS, almost (cid:29)at or dS minimum or have a runaway behaviour.This is similar to the lifted LARGE volume s enario but di(cid:27)erent from the unlifted onewhere zero is approa hed from below, so we an see the indire t e(cid:27)e t of the D-term to up-lift the minimum. Noti e also that it is di(cid:27)erent from the previous ways of getting de Sitterspa e from a purely supersymmetri potential [26℄, [27℄, [28℄ in whi h the stabilised value ofthe volume was relatively small, whereas here we are obtaining exponentially large volumes.Let us study an illustrative example of these possible va uum stru tures: First of all, weare indeed able to stabilise the Kähler moduli at large values for the large Kähler modulus τ . For instan e as depi ted in (cid:28)gure 2, we are able to stabilise the Kähler moduli with thefollowing naturally hosen parameters: Figure 2: A plot of the numer-i al behaviour of the s alar po-tential with respe t to the Käh-ler moduli showing an AdS min-imum. τ τ V min Φ ρ W a g s α ξ .
03 5 . − . × − g s M P . g / s M P . g / s M P − .
85 0 . . .
10 0 . (cid:21) 13 (cid:21)able 1: Showing the parameter values in the AdS-example.In addition to the usual stabilisation orresponding to an AdS geometry, we now an a hievea stabilisation of the Kähler moduli that orrespond to a dS geometry. Taking the previousexample, this is a hieved by tuning the (cid:29)ux parameter from W = − . to W = − . . The resulting (cid:28)gure is 3 and the numeri al values in this example are summarised in thefollowing table: Figure 3: A plot of the numer-i al behaviour of the s alar po-tential with respe t to the Käh-ler moduli showing a dS mini-mum. τ τ V min Φ ρ W a g s α ξ .
40 5 .
44 1 . × − g s M P . g / s M P . g / s M P − .
25 0 . . .
10 0 . Table 2: Showing the parameter values in the dS-example.Changing the parameters too drasti ally now results in a disappearan e of the minimum asdepi ted in (cid:28)gure 4. Figure 4: A plot of the nu-meri al behaviour of the s alarpotential with respe t to theKähler moduli showing no min-imum.In general we (cid:28)nd a smooth transition from AdS to dS solutions. We may tune the param-eters to obtain almost Minkowski minima 5. The following example 5 shows a onsiderabletuning to a minimal value for the potential of V = 5 . × − . V = 5 . × − . For a full analysis we should he k our analyti al results for the matter (cid:28)elds by visualisingthe potential with respe t to the matter (cid:28)elds as well. In the ase of the dS example fromthe previous se tion we obtain the following (cid:28)gures 6 and 7.Figure 6: A plot of the numeri al behaviour of thes alar potential with respe t to the matter (cid:28)eld ρ . Figure 7: A plot of the numeri al behaviour of thes alar potential with respe t to the matter (cid:28)eld Φ .Both plots support our previous analysis with a minimum in the predi ted region. Inparti ular the orre tion to Φ does not spoil our uplifting me hanism.5. D7 Soft-TermsWe now would like to study the me hanism of supersymmetry breaking within the 2-moduliframework presented in the previous se tions. For this we assume a brane onstru tion on aseparate four y le τ (e.g. through an SU (5) GUT model [30℄) leading at low-energies to aMSSM realisation. The Standard Model y le τ annot be stabilised by non-perturbativee(cid:27)e ts as pointed out in [33℄, however loop e(cid:27)e ts ould stabilise the additional y le [39℄,leaving us for now with a regime of e(cid:27)e tive (cid:28)eld theory ( τ = 0 ). This ansatz is slightlydi(cid:27)erent from the original study of soft-terms in the LARGE volume s enario ( f. [22℄and [29℄), but was re ently dis ussed in [34℄. 6.The volume in this extended Swiss- heese Calabi-Yau an be written as V = τ / − τ / − τ / . The philosophy of our approa h is depi ted in (cid:28)gure 8 below.6We on entrate here on generi D7 brane soft terms without in luding expli tly the me hanism forstabilising the MSSM modulus. We only assume that it is not stabilised at a singular point. A detailedstudy of this onsideration is left for future work.(cid:21) 15 (cid:21)igure 8: In the simplest modelthe geometry of the Calabi-Yau isparametrised by three 4 y les. τ is the large y le; τ , τ denote thesmall (blow-up) y les. We assumethe MSSM to be lo alised at τ andour ISS brane setup is lo alised on τ . We start with the following general e(cid:27)e tive supergravity ansatz: K = − M P log (cid:16) V + ξs / (cid:17) − log ( s ) + τ n τ (cid:0) | p | + | q | + | ρ | + | Φ | (cid:1) + ˜ K i | υ i | , (5.1) W = M P g / s W + M P g / ns αe − aT (cid:18) p Φ qµ + ρ Φ (cid:19) + W MSSM , (5.2)where we reintrodu ed the dilaton dependen e s ; υ denotes one of the hiral super(cid:28)eldswithin the MSSM whose superpotential is given by W MSSM . For now, the orre t form ofthe moduli weights in the Kähler potential is not of major importan e, and we keep ourstudy as general as possible.A priori we an say that the s heme of supersymmetry breaking is gravity (moduli)mediated. The amount of gauge mediated ontribution an be seen in the fa t that themasses asso iated with the hidden se tor matter (cid:28)elds ρ and Φ are larger or of the order ofthe gravitino mass (see the appendix for an estimate of the masses asso iated to our model).In due ourse we will establish the same or even stronger ontributions to the modulimediated soft masses ompared to the LVS and we therefore an negle t ontributionsarising from anomaly mediation.We follow the standard me hanism of al ulating soft supersymmetry breaking termsas for example used and des ribed in [35℄.Gravitino and Gaugino MassesThe gravitino mass is found to have the standard volume suppression: m / = e K M P | W | M P = M P | W | g / s V E . (5.3)where the subs ript E is added to larify that this is the Einstein-frame volume.The F-terms needed for the remaining soft-terms are al ulated in detail in appendixF. Gaugino masses are in general given by M a = 12 (Re f a ) − F m ∂ m f a . (5.4)(cid:21) 16 (cid:21)ssuming our standard model is onstru ted via D7 branes, the gauge kineti fun tion is f D = T , (5.5)Using those results, we (cid:28)nd for the D7 brane gaugino masses the following mass: M D ∼ W M P g / s V E = m / . (5.6)Regarding the volume suppression of the gaugino masses, the results oin ide with theLARGE volume analysis. The additional suppression of the gaugino masses due to a nextto leading order an ellation in the F-terms (dis ussed in [36℄) does not o ur sin e τ isnot stabilised by non-perturbative e(cid:27)e ts.S alar MassesS alar masses an be found by the following formula m i = m / − F m ¯ F ¯ n ∂ m ∂ ¯ n log ˜ K i , (5.7)where we negle t ontributions from the va uum energy.7 To al ulate the s alar masseswe have to spe ify the matter metri ˜ K i . However we an start with the following generalansatz ˜ K i ∼ τ b s c τ d . (5.8)This gives the following leading order results ∂ τ ∂ τ log ˜ K i = d τ , (5.9) ∂ τ ∂ τ log ˜ K i = − b τ , (5.10) ∂ s ∂ s log ˜ K i = − c s . (5.11)Using the results for the F-terms derived in the appendix, we (cid:28)nd the following results: ( F τ ) ∂ τ ∂ τ log ˜ K i ∼ dW M P g s V (5.12) ( F τ ) ∂ τ ∂ τ log ˜ K i ∼ − bM P W g s V (5.13) ( F s ) ∂ s ∂ s log ˜ K i ∼ − cM P W g s V (5.14)We an dire tly see that the ontribution arising from the dilaton is negligibly small. Inaddition, we (cid:28)nd that the leading order term proportional to the gravitino mass vanishesif d − b = 1 . If this relation is satis(cid:28)ed, we have to identify the next to leading order7Here the addition of D-terms does not hange the formula for s alar masses sin e the SM (cid:28)elds arenot harged under the anomalous U (1) in the hidden se tor. A general formula was presented for examplein [37℄. (cid:21) 17 (cid:21)ontribution. Unlike in the original LARGE volume analysis, there is no sub-leading an- ellation in the F-terms [34℄ for the Kähler moduli sin e the e − aτ term in (4.20) omesfrom ∂ φ W ∂ φ W K φφ rather than ∂ τ W ∂ τ W K τ τ and the D-term uplifting ontribution re-quire a slightly di(cid:27)erent stabilisation as dis ussed in the previous se tion. This determinesthe sub-leading ontribution (see appendix for further details) to ome from the mixing ofthe non-perturbative leading order ontribution to the F-term and the other leading orderF-term ontribution. We obtain the interesting result: F τ F τ np ∂ τ ∂ τ log ˜ K i ∼ dW g ms ˜ ρ ˜Φ τ e − aτ V τ / , (5.15) F τ F τ np ∂ τ ∂ τ log ˜ K i ∼ − bW g ms ˜ ρ ˜Φ τ e − aτ V τ / . (5.16)Evaluating these ontributions at the LARGE volume minimum ( e aτ ∼ V x ) gives thefollowing ommon volume suppression: F τ i F τ i np ∂ τ i ∂ τ i log ˜ K i ∼ d − b V + x . (5.17)Overall we (cid:28)nd the general expression for the s alar masses to be m i = (1 − d + b ) m / − d − b ) W g ms ˜ ρ ˜Φ τ e − aτ V τ / , (5.18)where the se ond term is subleading sin e it is higher suppressed with respe t to the volume.For example, taking d = 1 and b = 1 / as in previous se tions and in the analysis ofsoft terms in the LVS [29℄, we obtain no an ellation at leading order and (cid:28)nd the followings alar masses: m i = 13 m / + higher order corrections . (5.19)A-termsThe A-terms are given by A ijk = F m ( ∂ m K + ∂ m log Y ijk − ∂ m log ˜ K i ˜ K j ˜ K k ) (5.20)Assuming a onstant Y ijk we an estimate the A-terms to be given by the following expres-sion at leading order: A iii = − τ √ g s | W | M P V (cid:18) τ − d τ (cid:19) − τ √ g s | W | M P V (cid:18) √ τ V + 3 b τ (cid:19) + 2 τ √ g s | W | M P V (cid:18) √ τ V (cid:19) = − m / (1 − d + b ) + h. o. , (5.21)whi h are universal and of the order of the gravitino mass.(cid:21) 18 (cid:21)hort SummaryOverall we (cid:28)nd three di(cid:27)erent s enarios a ording to the value of d − b and τ :1. If d − b = 1 the soft terms are of the order the gravitino mass.2. If d − b = 1 the gaugino masses are of order of the gravitino mass but s alar massesand A -terms are mu h more suppressed.In all ases the soft terms are universal [29, 38℄.We an ompare these results with the original LARGE volume analysis from [29℄where an elations in the F-terms o urred due to the minimization of the small y les vianon-perturbative e(cid:27)e ts and the soft-terms where found to be: M i ∼ m / log m / , m i = nM i , A αβγ = − nM i . The additional small suppression of the soft-terms ompared to the gravitino mass foundin the original LARGE volume analysis is not present here and this feature distinguishesthe stru ture of soft-terms from both s enarios.The di(cid:27)eren e to the orresponding 3- y le set-up in the LVS [34℄ is that the slight dif-feren e in the potential (D-term uplifting ontribution espe ially) renders the s alar massesnot a(cid:27)e ted by sub-leading orre tions in the F-term or the Kähler metri . A detailedquantitative analysis of this s enario is out of the s ope of this arti le.6. Kähler moduli in(cid:29)ationIn the previous se tions we des ribed a me hanism that allows us to obtain dS va ua withina fully N = 1 supersymmetri a tion. We now would like to show how to implement themodel of Kähler moduli in(cid:29)ation [16℄ within this framework.The starting point of the dis ussion is the modi(cid:28) ation of the original geometry of P [1 , , , , by adding an additional small 4- y le. The volume is modi(cid:28)ed to V = τ / − τ / − τ / . (6.1)On that additional 4- y le we assume a non-perturbative e(cid:27)e t Be − bT . The Kähler andsuperpotential are hanged to: K = − M P log V + ξg / s ! + g ns τ n g s τ ( | p | + | q | + | ρ | + | Φ | ) , (6.2) W = M P g / s W + M P g / ns e − aT α ( p Φ qµ + Φ ρ ) + M P g / s Be − bT . (6.3)This hange alters the leading order potential to: V = V old + g s M P (cid:18) bB ) √ τ e − bτ V + 4 W bBτ e − bτ V (cid:19) , (6.4)(cid:21) 19 (cid:21)here V old is the potential dis ussed in previous se tions. The old potential is independentof τ and we an hen e stabilise the new part with respe t to τ at onstant volume. Weobtain in the limit bT ≫ bBe − bT = W √ τ V . (6.5)Plugging this value into the original potential gives the following ontribution at the ex-tremal value for τ : V = V old − W ( τ ) / V . (6.6)Sin e the ontribution from the potential in luding τ is negative, we have to be at aminimum with respe t to τ . Having a negative ontribution from the in(cid:29)ationary potentialat the minimum then also requires to stabilize the potential at a (cid:16)large(cid:17) positive value andnot at an almost Minkowski minimum. The assumption that V an be taken to be onstantduring in(cid:29)ation is justi(cid:28)ed in the limit b ≫ a as the e(cid:27)e t of the additional y le τ towardsthe potential is negligibly small ompared to the old potential. Hen e we end up with thesame potential as in the Kähler moduli in(cid:29)ation but with the advantage that we an useour new me hanism to stabilise the volume at a positive va uum energy.Following the analysis of the original paper [16℄, we an now al ulate the in(cid:29)ationary hara teristi s of this model. Our in(cid:29)ationary potential is given by V = V old + 4 τ W bBe − bτ V , (6.7)where we negle t the higher suppressed and hen e irrelevant term in luding the doubleexponential. The physi ally relevant parameter is the anoni ally normalised version of τ whi h is found to be at leading order τ c = r τ M P ( τ ) / ( τ ) / , (6.8)where the exponent " " denotes that the value is taken at the minimum. Rewriting thepotential in terms of the anoni ally normalised (cid:28)eld τ c shows the exponential suppressionwith respe t to the volume V = V + 4 W bBM P V (cid:18) V (cid:19) / ( τ c ) / exp " − b (cid:18) V (cid:19) / ( τ c ) / M P . (6.9)From this potential we (cid:28)nd the following slow-roll parameters: ǫ = M P (cid:18) V ′ V (cid:19) = 32( W bB ) V V τ / (1 − bτ ) e − bτ , (6.10) η = M P V ′′ V = 4 W bB √ τ V V (1 − bτ + 4( bτ ) ) e − bτ , (6.11) ξ = M P V ′ V ′′′ V = − W bB ) V V τ (1 − bτ )(1 + 11 bτ − bτ ) + 8( bτ ) ) e − bτ , (6.12)where the derivatives are taken with respe t to the anoni ally normalised (cid:28)elds. In theseparameters the only small di(cid:27)eren e is that we have not spe i(cid:28)ed V yet, whi h will be of a(cid:21) 20 (cid:21)imilar form ompared to the LARGE volume s enario. From the slow-roll parameters one an determine the spe tral index and its running as n − η − ǫ + O ( ξ ) , (6.13) dnd ln k = 16 ǫη − ǫ − ξ . (6.14)In analogy to the original dis ussion of Kähler moduli in(cid:29)ation we (cid:28)nd the number ofe-foldings to be given by N e = Z φφ end VV ′ dφ = 3 V W bB Z τ τ end3 e bτ √ τ (1 − bτ ) dτ . (6.15)To mat h the COBE normalisation for the density (cid:29)u tuations δ H = 1 . × − we haveto satisfy the following onstraint V / M P V ′ = 5 . × − , (6.16)where the potential is evaluated at the horizon exit, whi h means N e = 50 − e-foldingsbefore the end of in(cid:29)ation. This endows us with a onstraint to determine the ontributionfrom the original potential V . In general the only modi(cid:28) ation on the model of Kählermoduli in(cid:29)ation is given by the fa t that the old LARGE volume ontribution is repla edby the new potential whi h was developed and dis ussed in the previous se tions. It istherefore most likely that a on rete al ulation, whi h we have not ompleted yet, willgive the same numeri al results as in the original Kähler moduli s enario, whi h are givenby: • The tensor-to-s alar ratio was found to be r ∼ ǫ . (6.17) • For − e-foldings, the model gives rise to the following hara teristi s: . < n < . , (6.18) − . < dnd ln k < − . , (6.19) < | r | < − , (6.20) l s ≤ V ≤ l s . (6.21)7. 3-Parameter K3 (cid:28)bration and Fibre In(cid:29)ationTo show the generality of the uplifting me hanism, we now onsider the example of a 3-parameter K3 (cid:28)bration whi h allows Kähler moduli stabilisation at LARGE volume ( f. [39℄and [19℄). We start with the same expression for the volume as in the mentioned arti les V = α √ τ ( τ − βτ ) − γτ / . (7.1)(cid:21) 21 (cid:21)t was shown that the ombination α √ τ ( τ − βτ ) plays the rle of the exponentiallydominating volume as in the original LARGE volume s enario. τ plays the role of ablow-up and is ru ial for the existen e of a stable minimum at LARGE volume. Takingonly the s alar potential, one still remains with one (cid:29)at dire tion, orresponding essentiallyto τ . It was shown in [39℄ that this runaway behaviour an be stabilised by onsideringloop- orre tions to the potential.To embed our uplifting s enario in this K3 setup, we need to know where (i.e. onwhi h y le) non-perturbative e(cid:27)e ts are required. To keep the same stru ture of modulistabilisation it is ne essary to pla e our brane setup on the blow-up y le τ . Sin e both ofthe other y les have to be too large and the non-perturbative e(cid:27)e ts are hen e negligiblysmall on those y les.Our setup in terms of the Kähler and superpotential then looks like K = − M P log V + ξg / s ! + τ m g − ms V / (cid:0) | p | + | q | + | ρ | + | Φ | (cid:1) , (7.2) W = g / s M P W + M P g / ms δe − aT (cid:18) p Φ qµ + ρ Φ (cid:19) . (7.3)Wrapping around the 4- y le parametrised by τ gives us the following leading order stru -ture in the D-term potential: V D = 12 τ (cid:18) τ m V / (cid:0) q ρ | ρ | + q Φ | Φ | (cid:1) + Q τ (cid:18) √ τ V + ξ ) + mτ m − ( | ρ | + | Φ | ) V / (cid:19)(cid:19) , (7.4)where we dire tly set the quark (cid:28)elds to zero. The stru ture of the D-term potential isessentially the same as in the analysis of the P [1 , , , , geometry from the previous se tion.This not only allows us to set the quark (cid:28)elds to zero but also to an el the FI-term withthe least suppressed Φ − (cid:28)eld.Hen e obtain the following impli it dependen e of Φ on the large Kähler moduli | Φ | = | ˜Φ | V / = Q τ V / τ − m q Φ ( V + ξ ) (1 + mQ τ q Φ τ ) − ≈ Q τ τ − m q Φ V / . (7.5)As in the Swiss- heese ase we an only assume at this stage that ρ is impli itly highersuppressed than Φ with respe t to the Kähler moduli.With this assumption and the knowledge from the previous study in the Swiss- heese ase we an now estimate the leading order ontributions to the F-term potential: e K ∼ V (7.6) ∂W ∂ ¯ W K − ∼ g ms M P δ e − aτ V / | ˜Φ | τ m (7.7) ∂W ∂ ¯ KK − ¯ W + . . ∼ W g / ms M P δ Re ˜ ρ ˜Φ e − aτ γ V / τ ( m + aτ ) (cid:16) g s M P γ V / τ / + 2 g ms mα √ τ ( βτ − τ ) τ m | ˜Φ | V / (cid:17) V / n (7.8) ∂K∂ ¯ KK − ∼ | W | + W Aξ + B | ˜Φ | τ / b , (7.9)(cid:21) 22 (cid:21)here the oe(cid:30) ients A and B are introdu ed to keep the overall stru ture feasible. τ b denotes the power suppression with respe t to the large Kähler moduli.We an now determine the impli it dependen e of ρ on the Kähler moduli in the sameway as before to be n = 5 / . This enables us to integrate out the matter (cid:28)elds ompletelyand we end up with the following potential with respe t to the Kähler moduli V = Aτ / − m e − aτ V / − Be − aτ τ / − m V + + C V . (7.10)Let us ompare this potential with the original 3 parameter K3 potential whi h was al u-lated in [39℄ to be given by V = A √ τ e − aτ V − Bτ e − aτ V − C V . (7.11)The omparison leads to exa tly the same results as in the P [1 , , , , analysis from theprevious se tion: • We have a marginally higher suppression with respe t to the overall volume arisingfrom the hange in the leading order F-terms. In addition we have a slightly largersuppression with respe t to the volume in the se ond term, due to the mat hing of ρ D-term and F-term ontributions with powers of the large Kähler moduli. • In the same term we have the only other di(cid:27)eren e in the exponential suppressionarising after integrating out ρ. • Despite missing the analyti minimisation with respe t to the volume by this slight hange, we an still approximate the minimisation. We then see that we have thetypi al exponential hierar hy between the volume and the blow-up Kähler modulus. • To stabilise the remaining large Kähler modulus we an still use loop orre tions asdis ussed in [39℄.Due to the same hange in the potential, we assume at this stage that it is possible tostabilise the Kähler moduli in the same fashion with the additional uplifting as in theSwiss- heese ase.7.1 Fibre In(cid:29)ationFollowing the su essful embedding of D-term uplifting to the 3-Parameter K3 (cid:28)bration itis natural to raise the question whether we an embed (cid:28)bre in(cid:29)ation [19℄ into this upliftings enario. In order to realise the proposal of (cid:28)bre in(cid:29)ation we need to wrap branes aroundthe two large y les τ and τ . These additional branes do not interse t with the blow-up y le and do not reate additional matter (cid:28)eld ontent at the interse tions. To obtain thepotential of (cid:28)bre in(cid:29)ation, one has to study the loop orre tions ( g s ) to the s alar potential.As shown in se tion 3.1.2 of [19℄ the leading order string-loop orre tions are then given bythe following ontributions: (cid:21) 23 (cid:21) From the branes wrapping the 4- y le τ : δV KK( g s ) ,τ = g s W ( C KK1 ) τ V . (7.12) • From the branes wrapping the 4- y le τ : δV KK( g s ) ,τ = 2 g s W ( C KK2 ) τ V . (7.13) • From the interse tions of the two sta ks of branes around τ and τ : δV KK( g s ) ,τ τ = − C W √ τ W V . (7.14) • From the branes wrapping the blow-up y le: δV KK( g s ) ,τ = g s W ( C KK3 ) √ τ V . (7.15)This ontribution does not depend on the "(cid:29)at"-dire tion τ and an be hen e under-stood as a subleading orre tion in the α ′ - orre tions.With those results, we an establish our additional in(cid:29)ationary potential on top of theleading order s alar potential of the previous se tion: V = Aτ / − m e − aτ V / − Be − aτ τ / − m V + + C V + V inf (7.16) = Aτ / − m e − aτ V / − Be − aτ τ / − m V + + C V + W V (cid:18) Dτ − E V√ τ + F τ V (cid:19) . (7.17)We see that as in the dis ussion of Kähler moduli in(cid:29)ation both parts of the potential simplyde ouple. Exa tly as in the original dis ussion of (cid:28)bre in(cid:29)ation it is possible to stabilise thevolume in a (cid:28)rst step and then look at the τ dire tion as the in(cid:29)ationary dire tion. We antherefore say that it is straight forward to embed (cid:28)bre in(cid:29)ation into the uplifted s enarioand under the assumption of a onstant volume the analysis of in(cid:29)ationary parameters willgive exa tly the same results. Although a multi-(cid:28)eld analysis (i.e. taking the volume to benon- onstant) is out of the s ope of this arti le, we an still omment on the stability of ouruplifted s enario with regard to variations of the volume. It was observed in the analysis ofthe (cid:28)bre in(cid:29)ation model that it is ne essary to avoid a runaway in the volume to introdu ean uplifting term proportional to / V / . In our s enario, we exa tly have su h an upliftingterm before integrating out ρ, sin e, as previously dis ussed, the uplifting is aused by V up ≈ | ˜ ρ | V / = | ρ | V / . (7.18)We therefore expe t that the multi-(cid:28)eld analysis would give exa tly the same results as inthe original analysis. (cid:21) 24 (cid:21). Constraints for metastable dS va ua in supergravity setupsIn re ent years it was studied under whi h general onstraints it is possible to obtainmetastable dS va ua and/or in(cid:29)ation in a general supergravity framework [20,40(cid:21)42℄. Sin eour onstru tion from previous se tions is presented in a fully N = 1 supergravity frameworkwe would like to omment on why our approa h satis(cid:28)es these onstraints.These studies fall into the following two ategories:1. The (cid:28)rst type of onstraints was developed by Gomez-Reino and S ru a [40, 42℄.Subje t to a vanishing osmologi al onstant, they developed ne essary but not su(cid:30)- ient onstraints for the existen e of va ua in a general supergravity setup onsistingof D-term and F-term potential.2. In more re ent papers Covi et al. [20,41℄ studied expli itly the possibilities for variousstring ompa ti(cid:28) ations to obtain metastable dS minima and in(cid:29)ation. However, the onstraints used here do not assume a vanishing osmologi al onstant but do notin lude a D-term potential.In order to avoid a long al ulation and introdu tion of terminology to the reader we simplywould like to argue why our approa h falls into the ategory of string models dis ussedin [20, 41℄ and why this also allows us to satisfy the modi(cid:28)ed onstraints after the in lusionof D-term potentials:First of all, our model heavily relies on both omponents of the s alar potential, D-termand F-term. Sin e we are in prin iple able to tune the minimum of our potential to zero osmologi al onstant, we assume that we an use the onstraints developed in the (cid:28)rstseries of papers by Gomez-Reino and S ru a. The onstraints we have to satisfy are: f i f i + d a d a = 1 , (8.1) R i ¯ jp ¯ q f i f ¯ j f p f ¯ q ≤
23 + 23 ( M ab /m / − h ab ) d a d b + 2 h cd h aci h bd ¯ j f i f ¯ j d a d b − (2 h ab h cd − h iab h cdi ) d a d b d c d d + r Q abc m / d a d b d c , (8.2)where f i and d a are the F-term or respe
tively D-term res
aled by /m / and R i ¯ jp ¯ q denotesthe Riemann tensor with respe
t to the Kähler metri
. h ab is the gauge kineti
fun
tionand Q abc is the variation of the gauge kineti
fun
tion with respe
t to the generators of thegauge symmetries.8Negle
ting the D-term for the moment, the se
ond
onstraint simpli(cid:28)es to R i ¯ jp ¯ q f i f ¯ j f p f ¯ q ≤ , (8.3)whi
h was renamed in the se
ond series of papers to σ = R i ¯ jp ¯ q f i f ¯ j f p f ¯ q − > . For varioustypes of string
ompa
ti(cid:28)
ations, the value of σ was determined in the work by Covi et al. In8A more detailed explanation of the quantities in those
onstraints and a derivation of those
an befound in the original papers. (cid:21) 25 (cid:21)arti
ular it was shown that for no-s
ale models with in
luded α ′ − orre
tions, it is alwayspossible to satisfy this
onstraint for a vanishing
osmologi
al
onstant (
f. equation (4.33)in [41℄).In
luding D-terms
an even alleviate this problem, as shown in [40, 42℄ sin
e one
anres
ale the F-terms and the
urvature in su
h a way that the dependen
e on the D-termsseems to disappear: δ I ¯ J z I z ¯ J = 1 , (8.4) ˜ R I ¯ JP ¯ Q z I z ¯ J z P z ¯ Q ≤ , (8.5)where z I = f I / q − P A d A . The
hange in the
urvature
an be evaluated in parti
ularlimits of the relation between gaugino and gravitino mass. In the terminology of thosepapers, we are working in the regime of the (cid:16)light ve
tor limit(cid:17) sin
e g U (1) M U (1) / m /