SUSY Breaking in Local String/F-Theory Models
R. Blumenhagen, J.P. Conlon, S. Krippendorf, S. Moster, F. Quevedo
aa r X i v : . [ h e p - t h ] J u l Preprint typeset in JHEP style - HYPER VERSION MPP-2009-76OUTP-09/14PDAMTP-2009-48CERN-PH-TH/2009-089SUSY Breaking in Lo al String/F-Theory ModelsR. Blumenhagen , J. P. Conlon , S. Krippendorf , S. Moster , F. Quevedo , Max-Plan k-Institut für Physik, Föhringer Ring 6, 80805 Mün hen, Germany Rudolf Peierls Centre for Theoreti al Physi s, 1 Keble Road, Oxford OX1 3NP, UK DAMTP, University of Cambridge, Wilberfor e Road, Cambridge, CB3 0WA, UK CERN PH-TH, CH 1211, Geneva 23, Switzerland.Abstra t: We investigate bulk moduli stabilisation and supersymmetry breaking in lo alstring/F-theory models where the Standard Model is supported on a del Pezzo surfa e or sin-gularity. Computing the gravity mediated soft terms on the Standard Model brane indu edby bulk supersymmetry breaking in the LARGE volume s enario, we expli itly (cid:28)nd suppres-sions by M s /M P ∼ V − / ompared to M / . This gives rise to several phenomenologi als enarios, depending on the strength of perturbative orre tions to the e(cid:27)e tive a tion andthe sour e of de Sitter lifting, in whi h the soft terms are suppressed by at least M P / V / and may be as small as M P / V . Sin e the gravitino mass is of order M / ∼ M P / V ,for TeV soft terms all these s enarios give a very heavy gravitino ( M / ≥ GeV ) andgeneri ally the lightest moduli (cid:28)eld is also heavy enough ( m ≥
10 TeV ) to avoid the os-mologi al moduli problem. For
TeV soft terms, these s enarios predi t a minimal value ofthe volume to be
V ∼ − in string units, whi h would give a uni(cid:28) ation s ale of order M GUT ∼ M s V / ∼ GeV . The strong suppression of gravity mediated soft terms ouldalso possibly allow a s enario of dominant gauge mediation in the visible se tor but with avery heavy gravitino M / > .Keywords: Supersymmetry Breaking. F-theory. Branes at singularities.ontents1. Introdu tion 22. E(cid:27)e tive (cid:28)eld theories and moduli stabilisation 32.1 Gauge ouplings on the GUT brane 42.2 Moduli Stabilisation 62.2.1 Fixing the non-standard model/GUT y les 62.2.2 Fixing the standard model/GUT y le 72.3 In luding matter (cid:28)elds 82.4 Summary of EFTs 93. Gravity mediated soft terms 93.1 Stru ture of soft terms 103.1.1 Gaugino masses 103.1.2 Squark/Slepton masses 113.1.3 ˆ µ/ ˆ µB -terms 123.1.4 A-terms 123.1.5 Anomaly mediated gaugino masses 133.2 Summary of gravity mediated soft masses 143.3 Uplift and an ellations 144. Consequen es for Supersymmetry Breaking 174.1 Gauge mediated s enarios 174.2 Impli ations for the Cosmologi al Moduli Problem 205. Con lusions 21A. F-Terms 23B. Vanishing D-terms in luding matter 25(cid:21) 1 (cid:21). Introdu tionDuring the last years moduli stabilisation, in parti ular for Type IIB orientifolds on om-pa t Calabi(cid:21)Yau threefolds, has been under intense study and several s enarios have beenproposed. The original example is the KKLT s enario [1℄, where dilaton and omplex stru -ture moduli are (cid:28)xed at tree-level by (cid:29)uxes while Kähler moduli are stabilised via instantongenerated terms in the superpotential W = W + A e − a T s . (1.1)As a generalisation of this, in luding also next-to-leading order orre tions to the Kählerpotential, the volume of the ompa ti(cid:28) ation manifold V an be stabilised at exponentiallylarge values [2,3℄. These large volume minima are quite generi [4℄ and exist whenever thereis a four- y le in the Calabi(cid:21)Yau threefold whi h is shrinkable to zero size, su h that thetotal volume of the spa e remains (cid:28)nite. These supersymmetry breaking Type IIB va uahave been alled the LARGE Volume S enario (LVS). Their phenomenologi al featureswere studied in very mu h detail for the string s ale in the intermediate regime M s ≃ GeV [5, 6℄ leading to
TeV soft terms and intermediate s ales for the neutrino andaxion se tor of the MSSM in the preferred range. For omputing the high s ale soft termsit was assumed that the D7-branes supporting the MSSM gauge and matter (cid:28)elds wrapthe same four- y le supporting also the D3-brane instanton. In [7℄, it was pointed out thatinstanton zero mode ounting a tually forbids su h a s enario and that the D7-branes andinstantoni D3-branes should better wrap distin t four- y les in the underlying Calabi(cid:21)Yaumanifold. In fa t the Kähler moduli asso iated to the sizes of the four- y les wrapped bythe D7-branes an be stabilised by D-terms, often at string s ale size at the boundary ofthe Kähler one. In this sense the MSSM branes are sequestered from the bulk of theCalabi(cid:21)Yau.A way of realising the MSSM gauge and matter (cid:28)elds in the LVS is by studying fra -tional D3-branes at the singular point, dis ussed in [8, 9℄. Various low-energy models werestudied on the (cid:28)rst two del Pezzo surfa es dP and dP , allowing for both GUT-like andextended MSSM s enarios. From the e(cid:27)e tive (cid:28)eld theory point of view, both s enarios aresimilar sin e after stabilising the moduli, both va ua are at or lose to the singular point.A very similar s enario was proposed re ently in the ontext of lo al F-theory modelswith an SU (5) GUT brane. First of all, it was realised that some of the model buildingproblems one had with realising simple GUT groups in orientifold onstru tions [10℄ areni ely re on iled in F-theory models on ellipti ally (cid:28)bered Calabi(cid:21)Yau fourfolds [11(cid:21)18℄.The reason for this substantial improvement is that F-theory is genuinely non-perturbativeand also allows for the appearan e of ex eptional groups E , E , E , whi h are supportedalong a omplex surfa e in the base threefold, over whi h the ellipti (cid:28)ber degeneratesappropriately.1 As a onsequen e, by a further breaking also the spinor representation ofan SO (10) GUT and the top-quark Yukawa ouplings
10 10 5 H in SU (5) GUTs an berealised.1In the Type IIB interpretation, su h lo i support general ( p, q ) seven brane systems, where the extrastates are realised by massless string jun tions. (cid:21) 2 (cid:21)oreover, it was proposed in [13℄ that su h models an allow for an essentially lo altreatment, if there exists a limit in whi h gravity de ouples from the gauge theory on theGUT brane. Geometri ally this means that the spa e transverse to the brane an be omearbitrary large or from a di(cid:27)erent perspe tive that the four- y le the brane is wrapping anshrink to zero size. Su h four- y les are so- alled del Pezzo surfa es, whi h are P blown upat up to eight points. Sin e there exists a limit where gravity de ouples from the physi son the SU (5) brane, one expe ts that for gravity indu ed ouplings on the brane thereexists an M s /M P suppression relative to their general values. This de oupling argumentwas used heavily when studying the further phenomenologi al impli ations of lo al F-theoryGUTs [19, 20℄. In parti ular, it was argued that, sin e gravity/moduli mediated supersym-metry breaking soft terms on the GUT brane should also experien e su h a suppression,gauge mediation ould be ome the dominant sour e. Indeed, under this assumption a veryni e numerology for the soft terms was dedu ed, whi h besides (cid:29)avour universality in ludesa solution to the µ/µB problem and a andidate for the QCD axion solving the strong CP-problem. The gauge mediation was parametrised in the usual way by the non-zero VEV ofa s alar (cid:28)eld h X i = x + θ F mediated to the MSSM by harged messenger (cid:28)elds. Note thatin this s enario the gravitino mass was assumed to be dominantly set by gauge mediation.Therefore, the gravitino was the LSP with a mass of .Sin e both the LARGE volume s enario and lo al F-theory GUTs require the same kindof geometry, i. e. Calabi(cid:21)Yau respe tively base threefolds ontaining del Pezzo surfa es, itis natural to ombine these two set-ups and study, for this on rete moduli stabilisationme hanism in the bulk, the omputable e(cid:27)e ts of gravity mediation for the physi s onthe GUT brane. It is the primary aim of this paper, to ompute for a minimal set-upthese gravity mediated soft terms expli itly and ompare them with the expe tation ofan V − = M s /M P suppression. Indeed, as we will show su h a omputation requires to ompute the soft terms at next-to-leading orders in / V .This paper is organised as follows: In se tion 2 we review the geometri frameworkof lo al GUTs and the LARGE volume approa h to moduli stabilisation that is appli ablein this regime. In se tion 3 we des ribe the omputation of gravity mediated soft terms.We des ribe how the soft terms an el at O ( M / ) and how it is ne essary to onsider sub-leading orre tions. We (cid:28)nd sub-leading ontributions to soft terms at order O ( M / / √V ) = O (cid:18) M / / M / P (cid:19) . In ertain ir umstan es these ontributions an also an el and we give a setof well posed assumptions when this an o ur. In se tion 4 we dis uss the impli ationsfrom these soft terms for both gauge mediation and the osmologi al moduli problem, andin se tion 5 we on lude.2. E(cid:27)e tive (cid:28)eld theories and moduli stabilisationThe minimal set-up we are investigating in this paper is that we have a threefold with atleast three four- y les, one large y le and two small del Pezzo four- y les, i. e. the threefoldis of the (strong) swiss- heese type. One of the del Pezzos supports the SU (5) /MSSM gaugetheory while the other one an support a D3-brane instanton indu ing a non-perturbative(cid:21) 3 (cid:21)ontribution to the superpotential. Therefore, for the size of the overall volume and the in-stanton four- y le (without any further ontributions) there exists the non-supersymmetri AdS-type LARGE volume minimum. Sin e the GUT brane is lo alised on a del Pezzosurfa e orthogonal to the instantoni del Pezzo and the size of the GUT brane is (cid:28)xedby D-terms at small values, the previous omputations of the gravity indu ed soft termsshould be modi(cid:28)ed. The same al ulation is also ne essary for the ase that the GUT y leis ollapsed at the quiver lo us.There are two basi regimes where the e(cid:27)e tive (cid:28)eld theory (EFT) for light modes isreliable: • All of the 4- y les, in luding the standard model or GUT y le are larger than thestring s ale. This is the geometri regime. • The size of the standard model y le is mu h smaller than the string s ale. It is astandard blow-up mode expanded around its vanishing value orresponding to the delPezzo singularity. Fortunately string theory is under ontrol at the singularity andthe EFT an be safely de(cid:28)ned in an expansion on the blow-up mode.Sin e the D-term onditions tend to prefer a small value of the standard model y le, itis important to understand the physi s in both regimes of validity of EFT. It is lear thatthese are two di(cid:27)erent e(cid:27)e tive (cid:28)eld theories for standard model physi s. But, as we willsee, sin e the standard model y le does not parti ipate in the breaking of supersymmetry,the stru ture of soft breaking terms will be the same in both ases.Let us dis uss the ingredients in some more detail.2.1 Gauge ouplings on the GUT braneLet us re all the set-up for Type IIB respe tively F-theory GUT models, where we usefor on reteness the Type IIB orientifold language of [21, 22℄. We onsider the Type IIBstring ompa ti(cid:28)ed on a ompa t Calabi(cid:21)Yau threefold M modded out by an orientifoldproje tion Ω σ ( − F L . The holomorphi involution is su h that one gets O7- and O3-planes.The base of the orresponding ellipti ally (cid:28)bered four-fold is then B = M /σ . The SU (5) GUT is lo alised on D7-branes wrapping a rigid del Pezzo surfa e D a . The resulting tree-level SU (5) gauge kineti fun tion f SU (5) = πg X + i Θ is simply given by f SU (5) = T a = 12 g s ℓ s Z D a J ∧ J + i Z D a C , (2.1)where g s = e ϕ denotes the string oupling onstant and Vol( D a ) = R D a J ∧ J is the volumeof the del Pezzo surfa e D a .In orientifold models, we a tually get on a sta k of (cid:28)ve D7-branes the Chan(cid:21)Paton gaugegroup U (5) , whi h allows for a non-vanishing gauge (cid:29)ux F a in the diagonal U (1) ⊂ U (5) .Sin e a del Pezzo is rigid and does not even ontain any dis rete Wilson lines, the gaugesymmetry is broken to SU (3) × SU (2) × U (1) Y by a non-trivial U (1) Y gauge (cid:29)ux F Y supported on a two- y le C a ∈ H ( D a , Z ) whi h is trivial in H ( M , Z ) [13,14℄. As explained(cid:21) 4 (cid:21)n [22℄, this way of breaking the SU (5) gauge group leads to a spe i(cid:28) pattern of MSSMgauge ouplings at the uni(cid:28) ation s ale f i = T a − κ i S, i ∈ { , , } , (2.2)with κ = Z D a F a , κ = Z D a F a + F Y + 2 F a F Y (2.3) κ = Z D a F a + ( F Y + 2 F a F Y ) . (2.4)For on reteness we are using these orientifold relations in the following.In the limit that the y le is ollapsed to the singularity, the gauge kineti fun tiontakes a similar form: f i = δ i S + s ik T k , (2.5)where now T k has to be understood as the blow-up modes that resolve the singularity.For Z n singularities δ i is universal; however for more ompli ated singularities δ i an benon-universal. For appli ations to uni(cid:28) ation, we are interested in singularities where thedi(cid:27)erent gauge groups have universal ouplings at the singularity.For both lasses of lo al models the GUT uni(cid:28) ation s ale and string s ale di(cid:27)er signi(cid:28)- antly by a fa tor of the bulk radius. More pre isely, the GUT uni(cid:28) ation s ale M X is givenby M X = RM s , where R ∼ V / is the bulk radius of the Calabi(cid:21)Yau in string units. This an be seen through the Kaplunovsky(cid:21)Louis relation between physi al and holomorphi gauge ouplings, g − a (Φ , ¯Φ , µ ) = Re( f a (Φ)) + ( P r n r T a ( r ) − T a ( G ))8 π ln (cid:18) M P µ (cid:19) + T ( G )8 π ln g − a (Φ , ¯Φ , µ )+ ( P r n r T a ( r ) − T ( G ))16 π ˆ K (Φ , ¯Φ) − X r T a ( r )8 π ln det Z r (Φ , ¯Φ , µ ) . (2.6)Using the IIB Kähler potential ˆ K = − V and the behaviour for lo al models ˆ Z = V − / we obtain g − a ( µ ) − T ( G )8 π ln g − a ( µ ) = Re( f a (Φ)) + β a ln (cid:18) ( RM s ) µ (cid:19) , (2.7)giving e(cid:27)e tive uni(cid:28) ation at RM s . As des ribed in [23(cid:21)25℄, at the string level this depen-den e arises from the presen e of tadpoles that are sour ed in the lo al model but are only an elled globally. This omes from the fa t that the U (1) Y (cid:29)ux that breaks the GUTgroup is on a two- y le that is non-trivial in H ( D a , Z ) and trivial in H ( M , Z ) . Lo allythe U (1) Y (cid:29)ux sour es an RR tadpole, whi h is in fa t absent globally due to the trivialityof the y le. The (cid:28)niteness of threshold orre tions is tied to the absen e of RR tadpoles,but the triviality of C a requires knowledge of the global geometry, leading to the presen eof the s ale RM s . (cid:21) 5 (cid:21).2 Moduli StabilisationSo far we essentially onsidered a lo al part of the overall Calabi(cid:21)Yau geometry where theGUT physi s is lo alised. As has been pointed out in [13℄, supersymmetry breaking on ahidden D-brane and mediation via gauge intera tion to the visible GUT brane might alsopartly allow a ompletely lo al treatment. This presumes of ourse that all possible Plan ks ale suppressed terms are sub-leading.In this paper, we do not postpone these global issues but instead ontinue the quitesu essful investigation of moduli stabilisation in the framework of Type IIB orientifolds.More on retely, we onsider a set-up where the bulk moduli orthogonal to D a are stabilisedby the LARGE volume s enario and ompute the indu ed soft terms on the SU (5) brane,respe tively the del Pezzo singularity onstru tions.2.2.1 Fixing the non-standard model/GUT y lesFor self- onsisten y let us review brie(cid:29)y the main ingredients for the KKLT respe tivelyLARGE volume s enario.At order V − in the large volume expansion, the omplex stru ture moduli and thedilaton are stabilised by a non-trivial G -(cid:29)ux giving rise to a tree-level superpotential ofthe form [26℄ W (cid:29)ux = Z M G ∧ Ω . (2.8)The resulting s alar potential is of the no-s ale type, with the Kähler moduli still (cid:29)atdire tions.In the LARGE volume s enario the no-s ale stru ture is broken by a ombination of α ′ - orre tions to the Kähler potential and a D3-instanton orre tion to the superpotential.Con retely, the Kähler potential in luding α ′ - orre tions [27℄ reads K = − (cid:16) V + ˆ ξ (cid:17) − ln (cid:0) S + ¯ S (cid:1) + K CS , (2.9)where ˆ ξ = ξ/g / s and g s is the string oupling. The resulting inverse Kähler metri for theKähler moduli T a and the axio-dilaton S reads K a ¯ b = − (cid:16) V + ˆ ξ (cid:17) (cid:18) ∂ V ∂τ a ∂τ b (cid:19) − + τ a τ b V − ˆ ξ V − ˆ ξ , K a ¯ S = −
32 ( S + ¯ S ) ˆ ξ V − ˆ ξ τ a , K S ¯ S = ( S + ¯ S ) V − ˆ ξ V − ˆ ξ . (2.10)For a D3-instanton to generate a ontribution to the superpotential it has to have the rightnumber of zero modes. In fa t an O (1) instanton wrapping a rigid four- y le, whi h doesnot interse t any four- y le arrying D7-branes, has the right number of zero modes to givea ontribution W = W + A e − a T s (2.11)(cid:21) 6 (cid:21)o the superpotential. Here W is the value of the GVW superpotential in the minimumand we impli itly assumed that one an (cid:28)rst integrate out the omplex stru ture moduliand the axio-dilaton multiplet. This is justi(cid:28)ed by observing that the instanton indu eds alar potential is at order O ( V − ) in the LARGE volume expansion and that the Kählermetri for the omplex stru ture and Kähler moduli has a fa torised form (for dis ussionsof integrating out moduli in supergravity see [28(cid:21)31℄).Working in the large y le regime, in the simplest ase, one hooses the volume V ofthe internal spa e to be of swiss- heese form with three Kähler moduli V = ( η b τ b ) / − ( η s τ s ) / − ( η a τ a ) / . (2.12)Here τ b determines the size of the Calabi(cid:21)Yau and the small four- y le of size τ s is wrappedby the D3-brane instanton. The resulting F-term potential at order O ( V − ) reads V F = e K (cid:16) K a ¯ b D a W D ¯ b ¯ W − (cid:12)(cid:12) W (cid:12)(cid:12) (cid:17) = λ ( aA ) √ τ s e − aτ s V − µ a (cid:12)(cid:12) AW (cid:12)(cid:12) τ s e − aτ s V + ν ξ (cid:12)(cid:12) W (cid:12)(cid:12) g / s V + . . . , (2.13)with oe(cid:30) ients λ = g s η / s , µ = 2 g s , ν = , featuring the LARGE volume AdS minimumat V ∼ e aτ s . More details of this minimum are olle ted in appendix A, whi h allows oneto ompute the value of the s alar potential (2.13) in this minimum to be V = − aτ s ξg / s W V M P . (2.14)Clearly it is negative, but due to a an ellation of the leading order terms, it ontains anextra suppression by ( aτ s ) ≃ log( V ) . For a realisti model this negative va uum energy hasto be uplifted to V ≃ . For the omputation of the gravity and anomaly mediated softterms in se tion 3, we will start by negle ting the e(cid:27)e ts of uplifting, but will then onsiderthe ontributions of the uplifting se tor.2.2.2 Fixing the standard model/GUT y leComing ba k to the GUT brane, following the zero mode arguments in [7℄, we assume thatthe GUT branes are wrapping a four- y le of size τ a whi h is (cid:16)orthogonal(cid:17) to the instanton y le. As mentioned, in Type IIB orientifolds we allow for an additional gauge (cid:29)ux F a in the diagonal U (1) a ⊂ U (5) perturbative Chan(cid:21)Paton gauge group. Vanishing of the(cid:16)Fayet(cid:21)Iliopoulos(cid:17) U (1) a D-term onstraint (at order V − ) Z D a J ∧ F a = 0 (2.15)implies that τ a → so that one is driven to the quiver lo us where α ′ - orre tions annotbe ignored. In the EFT the ondition (2.15) is essentially that the (cid:28)eld dependent FI-termvanishes K T a = 0 . Using the Kähler potential in both the geometri and quiver regimes,(cid:21) 7 (cid:21)his ondition show expli itly a dynami al preferen e for a ollapsed y le τ a → . TheF-term of the (cid:28)eld T a is of the form F a = e K / ( W T a + W K T a ) . (2.16)Sin e the superpotential W does not depend on the modulus T a and the D-term onditionimplies K T a = 0 , we an see that this (cid:28)eld does not break supersymmetry, i. e. F a =0 . Noti e that this on lusion will not be modi(cid:28)ed by in luding perturbative and non-perturbative orre tions to the Kähler potential sin e these orre tions will equally modifythe D- and F-terms. Sin e τ a = 0 one (cid:28)nds that also F a = 0 , whi h is a very important on lusion, as it indi ates that the standard model is somehow sequestered from the sour esof supersymmetry breaking.A loophole to this argument is that it impli itly assumes that the standard model(cid:28)elds, harged under the orresponding U (1) , will not get a VEV. Otherwise they would ontribute to the D-terms and an el the ontribution from the FI-term. Even thoughthis is desirable phenomenologi ally to avoid a large s ale breaking of the standard modelsymmetries, su h as olour, it should be the out ome of a al ulation. We illustrate in theappendix in a toy model that this is a tually the ase as long as the soft s alar masses arenot ta hyoni .A dire t onsequen e is that the soft terms on the GUT brane an only be generatedat (cid:16)sub-leading(cid:17) order by F b , F s and F S , i. e. by moduli whi h are sort of sequestered fromthe GUT brane.2.3 In luding matter (cid:28)eldsSo far we have on entrated only on the EFT for moduli (cid:28)elds and their stabilisation. Inorder to study soft-supersymmetry breaking we need to properly introdu e the matter (cid:28)elddependen e in the EFTs in both the geometri and singular y le regimes. The importantterm to be in luded is the matter (cid:28)elds' Kähler potential ˜ K = Z αβ ϕ α ϕ ∗ β + · · · with Z αβ afun tion of the moduli (cid:28)elds.At this state, only the dependen e on τ b and τ s is relevant, as all the other (cid:28)elds do notbreak supersymmetry (to leading order). Z should only depend on τ b , S and the Kählermodulus of the GUT brane τ a , so Z = Z ( τ b , τ a , S ) . The leading order expression for Z wasdetermined in [32℄ with Z ∼ / V / (see also [33℄) whi h applies to both hiral matter atmagnetised D7-branes and to the better understood fra tional D3-branes at singularities.Sin e the α ′ - orre tions to the Kähler potential are ru ial to determine the large volumeva uum, onsisten y requires that these orre tions should also be in luded in the matter(cid:28)eld Kähler potential. Unfortunately these orre tions are not known at present. However,as in the tree-level ase, we are mostly interested on their overall volume dependen e.Let us parametrise the α ′ - orre tions by a so far unknown fun tion f : Z α = k α τ b (cid:18) f (cid:18) Re( S ) τ b (cid:19)(cid:19) . (2.17)The dependen e of f on the variables an only be in the indi ated way in order to have theright power in g s . Now onsider the next-to-leading order orre tion in α ′ to the tree-level(cid:21) 8 (cid:21)esult, whi h, we laim, must be of the form: Z α = k α τ b − δ (cid:18) Re( S ) τ b (cid:19) n + · · · ! , (2.18)with n = 1 , , . . . denoting the ( α ′ ) n order of this term. The question now is at whi horder in ( α ′ ) n the (cid:28)rst orre tion appears. Sin e we are only interested in the orre tionwhi h does not in lude τ a , we an use a s aling argument like in [32℄. Assuming that thephysi al Yukawa ouplings do not depend on the overall volume of the spa e and takinginto a ount the Kähler potential (2.9), the leading order orre tion to the Kähler metri swere shown to s ale as k α τ b . Then it is expe ted that also at next-to-leading order thes alings must mat h, whi h means that also the Kähler metri s are orre ted at order ( α ′ ) .This argument shows that n = 3 is the smallest expe ted orre tion in (2.18) and then Z α = k α τ b (cid:18) − δ (cid:16) Re( S ) τ b (cid:17) / (cid:19) .2.4 Summary of EFTsWe an (cid:28)nally summarise the expressions for the EFTs we are using for the two relevantregimes:1. In the geometri regime the EFT is determined by: K = − (cid:16) V + ˆ ξ (cid:17) − ln (cid:0) S + ¯ S (cid:1) + K CS + Zϕϕ ∗ + · · · , (2.19) W = W + Ae − aT s + W matter , (2.20) f i = T a − κ i S , (2.21)where V = ( η b τ b ) / − ( η s τ s ) / − ( η a τ a ) / and Z = k (cid:16) − δ (Re( S )) / / V (cid:17) / V / .2. In the singular y le (blow-up) regime there is a slight hange in the standard model y le dependen e of K : K = − (cid:16) V + ˆ ξ (cid:17) + α τ a V − ln (cid:0) S + ¯ S (cid:1) + K CS + Zϕϕ ∗ + · · · , (2.22) W = W + Ae − aT s + W matter , (2.23) f = δ i S + s ik T k , (2.24)with now V = ( η b τ b ) / − ( η s τ s ) / and Z = ( β − δ/ V + γτ ma ) / V / with m > .Sin e in both ases the standard model/GUT y le does not break supersymmetry, thestru ture of soft breaking terms will be essentially the same.3. Gravity mediated soft termsAs we have seen, the LARGE volume minimum of the s alar potential breaks supersymme-try, so that this breaking indu es soft supersymmetry breaking terms on the GUT brane.(cid:21) 9 (cid:21)here are two sour es whi h are relevant here. First, there are of ourse the gravity medi-ated soft terms. However, sin e the GUT brane is sequestered from the non-supersymmetri bulk one might expe t that anomaly mediation is the leading order ontribution. In thisse tion we ompute the gravity mediated soft terms, i. e. the gaugino- and sfermion-massesas well as the µ -, A- and B-terms. Moreover, we ompute the anomaly mediated gauginomasses. Let us emphasise again that the s enario di(cid:27)ers from the usual intermediate s aleLARGE volume s enario in that the string s ale is mu h higher (we assume M s ∼ GeV for onsisten y with uni(cid:28) ation at M X ∼ GeV ), and that the GUT or MSSM branesare wrapping a four- y le ompletely sequestered from the four- y les supporting D3-braneinstantons.First we express the string s ale M s = ( α ′ ) − / in terms of the Plan k s ale and thevolume V of internal Calabi(cid:21)Yau (in Einstein frame and in units of ℓ s = 2 π √ α ′ ) M s = √ π g / s √V M P . (3.1)Thus we obtain M s ≃ GeV and M X ≃ . · GeV for V = O (10 − ) , avalue large enough to trust the V − expansion. Moreover, we immediately realise thatthe LARGE volume expansion parameter is dire tly related to the lo al GUT expansionparameter, i. e. V − / ≃ M s /M P . For omputing the gravitino mass we simply utilise thegeneral formula M / = e K W leading in our ase to M / = g / s | W |√ V M P . (3.2)3.1 Stru ture of soft termsWe are now in a position to ompute ea h of the gravity mediated soft supersymmetrybreaking terms in this lass of s enarios.3.1.1 Gaugino massesFor gravity mediated supersymmetry breaking, the gaugino masses are al ulated as M e G = 12 Re( f i ) F I ∂ I f i (3.3)for i = 3 , , , where for the gauge kineti fun tions we use (2.2) with τ a ≃ due to theD-term onstraint.Sin e the GUT brane is sequestered from the bulk we have F a = 0 and the only ontribution an ome from the dilaton F-term F S = e K / K S ¯ J ¯ F ¯ J . We assume that the F-term ondition for the axio-dilaton F S = 0 is ful(cid:28)lled at leading order. At next-to-leadingorder, there are then only sub-leading ontributions from F S as well as terms from F b addingup to F S = √ γ ξg s W V where γ is an O (1) fa tor (see in the appendix for a more detailedderivation). Thus, the gravity mediation indu ed term for the gaugino masses reads: M e G = 34 √ γ ξg s | W | M P V = 34 γ ξg / s M / V , (3.4)(cid:21) 10 (cid:21)ndependent of the MSSM gauge group fa tor, as the fa tor κ i in (2.2) an els. Here wehave assumed that the D-term (cid:28)xes the size of the GUT four- y le at small volume in stringunits, so that the leading ontribution to Re( f i ) ≃ omes from the gauge (cid:29)ux indu ed orre tion ≃ κ i Re( S ) .3.1.2 Squark/Slepton massesThe s alar masses obtained for gravity mediation of supersymmetry breaking read M e Q = M / + V − F I F ¯ J ∂ I ∂ ¯ J ln Z α , (3.5)where the potential in the minimum V is assumed to be already uplifted so that V ≃ .Computing now the soft-sfermion masses, let us (cid:28)rst dis uss the tree-level term in Z α .In this ase (3.5) redu es to M e Q = M / − ( F b ) τ b , (3.6)where we have negle ted the va uum energy in the minimum. Again, there is a an ellationof the gravitino mass squared with the leading term in ( F b ) . The term quadrati in F S is sub-leading as being of order V − . In the appendix A we ompute the next-to-leadingorder term in F b , whi h reads ( F b ) ≈ τ b (cid:20) M / + 38 aτ s ξg / s (cid:18) aτ s (cid:19) M / V (cid:21) . (3.7)Therefore, one gets for the soft sfermion masses squared M e Q = − aτ s ξg / s (cid:18) aτ s (cid:19) | W | M P V = − aτ s ξg / s M / V , (3.8)whi h at this stage ome out ta hyoni .Next we need to dis uss the higher α ′ - orre tions in (2.18). The term with the highestpower in / V is the one with ( F b ) ∂ b ∂ b log · · · . It is straightforward, that for τ a /τ b ≪ this simpli(cid:28)es to F m F n ∂ m ∂ n log − δ (cid:18) Re Sτ b (cid:19) n + · · · ! ≃ F b F b δn ( n + 2) (Re S ) n τ n +2 b ∼ δg n − s | W | M P V ( n ) . (3.9)Therefore, if there would be orre tions of order n = 1 , , they would dominate over the orre tions in (3.8). It is pre isely the third order orre tions in α ′ whi h ontribute to thesfermion masses at the same order in / V . In luding also the other moduli (cid:28)elds in (3.9),the overall value of the squared s alar masses will then be proportional to δ − ξ/ : M e Q = M / − aτ s ξg / s V + 15( δ − ξ/ g / s V ! . (3.10)(cid:21) 11 (cid:21)herefore, depending on the relative size of these two ontributions one an get ta hy-oni or non-ta hyoni sfermion masses. Moreover, it also shows that for δ = ξ/ there arefurther an ellations taking pla e at this order. This is pre isely the value one expe ts fromthe above mentioned s aling argument of the physi al Yukawa ouplings. Later we will givean argument under whi h quite general assumptions su h an ellations should o ur. Oneof the assumptions will be that really the uplifting se tor is orre tly taken into a ount,whi h leads to a further dependen e of the Kähler metri on a supersymmetry breaking(cid:28)eld. Note that indeed the soft sfermion masses (3.8) are of the same order as the AdSva uum energy (2.14), indi ating that in these omputations the uplift se tor annot benegle ted.3.1.3 ˆ µ/ ˆ µB -termsThe formula for the ˆ µ -term is ˆ µ = (cid:16) e K / µ + M / Z − ¯ F ¯ I ∂ ¯ I Z (cid:17) ( Z H Z H ) − / , (3.11)where µ denotes the supersymmetri µ -parameter, whi h we keep for ompleteness, althoughit an be argued to vanish under very general assumptions [32℄. We assume again the Kählermetri (2.18) for the Higgs (cid:28)elds as well as for Z . Here again, a an ellation of the se ondand the third term o urs. Note, if the µ parameter is not equal to zero, it dominates overthe sub-leading terms stemming from F b . Dropping the fa tors of order one, we are leftwith: ˆ µ ≈ √ g s √ τ b V µ − M e G aτ s . (3.12)The expression for B ˆ µ is more ompli ated: B ˆ µ =( Z H Z H ) − / (cid:18) e K / µ ( F I ∂ I K + F I ∂ I log µ − F I ∂ I log( Z H Z H ) − M / )+ (2 M / + V ) Z − M / ¯ F ¯ I ∂ ¯ I Z + M / F I ( ∂ I Z − Z∂ I log( Z H Z H )) − F ¯ I F J ( ∂ ¯ I ∂ J Z − ( ∂ ¯ I Z ) ∂ J log( Z H Z H )) (cid:19) . (3.13)However, due to the simple Kähler metri and assuming that µ is just an input parameterwithout any moduli dependen e, after a long but straightforward al ulation, the result israther simple: B ˆ µ = − (cid:18) √ g s √ τ b V µ + M / aτ s (cid:19) M e G , (3.14)where we have dropped again the order one onstants k H i and z .3.1.4 A-termsThe A-terms are given by: A αβγ = F I ( ∂ I K ) + F I ∂ I log Y αβγ − F I ∂ I log Z α Z β Z γ . (3.15)(cid:21) 12 (cid:21)he Pe ei(cid:21)Quinn shift-symmetry forbids a dependen e of the holomorphi superpotentialon the axio-dilaton or Kähler moduli, thus the Yukawa ouplings Y αβγ an only depend onthe omplex stru ture moduli and they drop out.There is a an ellation of F b in the remaining two sums and we are left with A αβγ = F s ( ∂ s K ) + F S ( ∂ S K ) . (3.16)As listed in the appendix, F s ( ∂ s K ) is suppressed with respe t to F S ( ∂ S K ) by a fa tor of /aτ s . As we are interested only in orders of magnitude, we keep only the latter term andget as result: A αβγ ≈ F S ( ∂ S K ) = − √ ξg s | W |V M P = − M e G . (3.17)3.1.5 Anomaly mediated gaugino massesLet us also now estimate the anomaly mediated gaugino mass. It is lear that, for su ha sequestered observable se tor, one would have guessed that not gravity mediation butanomaly mediation indu es the leading order soft terms. General formula for all the di(cid:27)erentsoft terms are not available, so that in this se tion we just ompute the anomaly mediatedgaugino masses. The expression for them reads [34℄ (see also [35℄):2 M anom e G = − g π (cid:20) (3 T G − T R ) M / − ( T G − T R )( ∂ I K ) F I − T R d R F I ∂ I log det Z αβ (cid:21) , (3.18)where T G is the Dynkin index of the adjoint representation, normalised to N for SU ( N ) , and T R is the Dynkin index asso iated with the representation R of dimension d R , normalisedto / for the SU ( N ) fundamental.A areful al ulation of F b , worked out in the appendix, reveals that it is proportionalto the gravitino mass at leading order: F b ≈ − τ b M / − τ b aτ s M e G . This leads to a pre ise an ellation of M / in (3.18). The (cid:28)nal expression for the anomaly mediated gaugino massfor a SU ( N ) gauge group is thus M anom e G = − g π (cid:20)(cid:0) N − (cid:1) − aτ s (cid:0) N − (cid:1)(cid:21) M e G . (3.19)Surprisingly, though the gravity mediated ontribution to the gaugino mass is suppressedwith respe t to the gravitino mass by a fa tor of ( M X /M P ) , anomaly mediation is not thedominating sour e for the gaugino mass. It is suppressed by the usual one-loop fa tor withrespe t to the gravity mediated ontribution. One expe ts a similar suppressed behaviourfor the other soft terms, so that anomaly mediation is even sub-leading to gravity mediation.2We use a di(cid:27)erent sign onvention for the F-terms leading to a di(cid:27)erent sign in the se ond and thirdterm in the anomaly mediated mass term than in [34℄.(cid:21) 13 (cid:21).2 Summary of gravity mediated soft massesIn the above omputation of soft terms we have seen that the leading terms an el and thatwe need to in lude higher order orre tions in V − . Sin e this s ale is dire tly orrelated with ζ = M s /M P , we an express these gravity mediated soft terms in terms of the s ales M / and ζ = M s /M P . The results are listed in table 1, where we have set the supersymmetri µ parameter to zero and estimated r π ξ ≃ s χ ( M ) ≃ , and g s ≃ . (3.20)soft-term s ale M e G M / ζ M e Q
116 log ζ M / ζ ˆ µ -term
18 log ζ M e G B ˆ µ -term
14 log ζ M / M e G A -term − M e G Table 1: Classi al gravity mediated soft terms for a naïve omputation of soft terms. Herethe expansion parameter is ζ = M s /M P . We have assumed the supersymmetri µ -term tovanish [32℄.All soft terms in table 1 are suppressed by ( M s /M P ) relative to the naïve expe tation M n / with n = 1 , depending on the mass-dimension. This expli itly demonstrates thatgravity e(cid:27)e ts from the bulk are suppressed on the shrinkable GUT y les, whi h is themain assumption of the lo al F-theory GUTs.However, as seen in the text in ertain ases there an be more an ellations leadingto even higher suppressions. Indeed so far we have negle ted the uplift se tor, but haveseen that the sfermion masses are a tually of the same order of magnitude as the uplift sothat it should better not be negle ted. We now dis uss under whi h well-posed assumptionsfurther an ellations are present.3.3 Uplift and an ellationsIn the last se tion we have omputed the gravity indu ed soft terms on the GUT brane.As we have explained, the omputation relies on assumptions about the expansions of thematter metri s at higher orders in α ′ . While su h orre tions must surely be present, itis di(cid:30) ult to know the pre ise form of these orre tions. We have expli itly seen for thesfermion masses that these orre tions ontribute at the same order in / V as the next-to-leading order ontributions from F b . Indeed, as seen in eq. (3.10) there an potentially(cid:21) 14 (cid:21)e further an ellations at this order. We have also omputed the soft terms under the as-sumption of V = 0 , but have not taken into a ount the ontribution of the supersymmetrybreaking from the uplifting se tor to the soft terms. To onsider these possibilities, let usargue in this se tion, how one an arrive at quite general statements by making some wellposed assumptions and exploiting the onsequen es of using the supergravity formalism.Re all that the physi al Yukawas are given by ˆ Y αβγ = e K / Y αβγ p Z α Z β Z γ . (3.21)The shift-symmetries of the Kähler moduli imply that they do not appear perturbativelyin the superpotential Yukawa ouplings Y αβγ . Let us make the assumption that the phys-i al Yukawas, being lo al renormalisable ouplings, do not depend on the (cid:28)elds breakingsupersymmetry. This in ludes the volume and also the hidden se tor (cid:28)elds that are respon-sible for uplifting and giving vanishing osmologi al onstant. We also assume pure F-termuplifting.Su h supersymmetry breaking (cid:28)elds appear in the overall Kähler potential, and the on-straints of holomorphy then imply that in order for the physi al Yukawas to be independentof su h (cid:28)elds, Z α = e K / . (3.22)Note that this in ludes the tree-level behaviour of lo al matter (cid:28)elds, Z α ∼ V / ∼ T b + ¯ T b ) .In this ase it follows that m Q = V + M / − F m ¯ F ¯ n ∂ m ∂ ¯ n ln Z α = V + M / − F m ¯ F ¯ n K m ¯ n V = 0 , (3.23)for the ase of vanishing osmologi al onstant.The A-terms also vanish under this assumption. In this ase the A-terms an be mostintuitively written as A αβγ Y αβγ = F I ∂ I ˆ Y αβγ , (3.24)with ˆ Y αβγ the physi al Yukawa ouplings. So it immediately follows that if the physi alYukawa ouplings do not depend on the (cid:28)elds breaking supersymmetry, the A-terms allvanish.The anomaly mediated ontribution for gaugino masses gives M anom e G = b a π M / − ( P r n r T a ( r ) − T ( G ))16 π F m ∂ m K (Φ , ¯Φ) + X r n r T a ( r )8 π F m ∂ m ln Z r (Φ , ¯Φ)= b a π M / − ( P r n r T a ( r ) − T ( G ))16 π F m ∂ m K (Φ , ¯Φ)3= b a π (cid:18) M / − F m ∂ m K (cid:19) , (3.25)where we have used Z = e K / . The size of the anomaly mediated ontributions to gauginomasses then depends on the size of M / − F m ∂ m K . The no-s ale an ellation for τ b implies(cid:21) 15 (cid:21)he O ( V − ) terms an el with non-vanishing terms at O ( V − ) . However (3.25) also in ludesthe hidden uplifting se tor, whi h must have K φ ¯ φ F φ F ¯ φ ∼ V (in order to uplift the va uumenergy to Minkowski). At this level we therefore annot rule out that F φ ∂ φ K ∼ V − / ,giving gaugino masses of order g π V / .For the µ -term, we obtain ˆ µ = e K / µ + ( M / − F I ∂ I K ) . (3.26)For the B-term, we have (assuming no moduli dependen e in µ ) ( Bµ ) = ( Z H Z H ) − / e K / µ (cid:0) F I ∂ I K − F I ∂ I ln( Z H Z H ) − M / (cid:1) + (2 M / + V ) Z − M / F ¯ I ∂ ¯ I Z + M / F I ( ∂ I Z − Z∂ I ln ( Z H Z H )) − F I F ¯ J ( ∂ ¯ I ∂ J Z − ( ∂ ¯ I Z ) ∂ J ln ( Z H Z H )) ! . (3.27)If we take Z = Z H = Z H = e K / then we eventually obtain Bµ = e K / µ (cid:18) F I ∂ I K − M / (cid:19) + (cid:12)(cid:12)(cid:12)(cid:12) F I ∂ I K − M / (cid:12)(cid:12)(cid:12)(cid:12) . (3.28)This implies that the µ - and B-terms involve the same expression as appeared in the anomalymediated expression (3.25) and that the µ - and Bµ -term are of the same order as requiredfor su essful ele troweak symmetry breaking.As we do not urrently know the form of α ′ - orre tions to the matter metri s, we donot know whether the form Z = e K / is orre t. However it is a natural hoi e in the sensethat it simply says that the physi al Yukawa ouplings, being lo al, do not depend on thevalue of bulk (cid:28)elds. In the ontext of the ζ (3) χ ( M ) α ′ - orre tion that entered the modulistabilisation, this is equivalent to the statement that physi al Yukawa ouplings do notalter if you perform a onifold transition in the bulk (whi h alters the Calabi(cid:21)Yau Eulernumber).The advantage of phrasing the omputation in this way is that we an say that moduligenerate soft s alar masses to the extent to whi h the physi al Yukawa ouplings depend onthe moduli. While not straightforward, it is in prin iple easier to ompute the dependen eof physi al Yukawa ouplings on the moduli. String CFT omputations give the dire tlyphysi al ouplings and therefore one ould analyse for ertain lo al models (for example fora sta k of D3-branes at an orbifold singularity in a ompa t spa e) whether the physi al ouplings do depend on the volume through a dire t vertex operator string omputation.We an also use (3.23) to ompute the minimal value of the soft s alar masses. The om-plete an ellation in (3.23) arose from the assumption that the physi al Yukawa ouplingshas no dependen e on all (cid:28)elds with non-zero F-terms. However we know this statement isnot true. The dilaton has an irredu ible F-term of O ( V − ) and enters the physi al Yukawas.This provides a minimal value for the s ale of the physi al Yukawa ouplings.(cid:21) 16 (cid:21). Consequen es for Supersymmetry BreakingIn this previous se tion we have seen that both gravity and anomaly mediated ontributionsto soft terms o ur at levels far lower than naïve expe tation. This gives novel phenomeno-logi al onsequen es for various aspe ts of supersymmetry breaking, whi h we now dis uss.4.1 Gauge mediated s enariosIn lo al F-theory models an interesting proposal was made assuming a model of gaugemediation to dominate supersymmetry breaking in the observable se tor. This is veryinteresting sin e it in orporates the positive properties of gauge mediation, su h as positivesquared s alar masses and (cid:29)avour universality and yet address its problems, su h as the µ/Bµ problem. This proposal though requires the following impli it assumptions:1. The me hanism responsible for moduli stabilisation, whi h was not onsidered, (cid:28)xesmoduli at a high mass and de ouples from supersymmetry breaking.2. Introdu e a new matter se tor that breaks supersymmetry dynami ally and a set ofmessengers that ommuni ate this breaking to the standard model (cid:28)elds.3. An anomalous U (1) was proposed to ommuni ate both se tors and address the µ/Bµ problem of gauge mediation. The anomalous U (1) is naturally as heavy as the strings ale but has low-energy impli ations after being integrated out.These onditions look at (cid:28)rst sight too strong and unnatural. A hieving moduli stabil-isation without supersymmetry breaking and small osmologi al onstant is a very strongassumption not realised in any of the moduli stabilisation s enarios so far. It is knownthat a supersymmetri va uum in supergravity, su h as in KKLT before the uplifting, isnaturally anti de Sitter sin e in that ase the va uum energy is V = − M / M P , whi h isvery large unless the superpotential is tuned in su h a way that it almost vanishes at thesupersymmetri minimum. Also a positive osmologi al onstant has to be indu ed aftersupersymmetry breaking. If the lo al supersymmetry breaking is responsible for this liftingthen its e(cid:27)e t should not have been negle ted for moduli stabilisation in the (cid:28)rst pla e. Fi-nally, it is not onsistent to onsider the low-energy e(cid:27)e ts of a very heavy anomalous U (1) without also in luding the e(cid:27)e ts of the moduli (cid:28)elds whi h are generi ally mu h lighterthan the string or ompa ti(cid:28) ation s ale. In parti ular the Fayet(cid:21)Iliopoulos term of theanomalous U (1) is a fun tion of the moduli.Nevertheless, our expli it results here show that a s enario similar to this may notbe impossible to realise. The main point is that although moduli are stabilised at a non-supersymmetri point, the breaking of supersymmetry is suppressed by inverse powersof the volume or equivalently by powers of M s /M P . This makes the (cid:28)rst point aboveapproximately orre t. The se ond point still has to be assumed as in all models of gaugemediation and requires an expli it realisation. Here the relevant observation is to omparethe strength of gauge mediation F X /x to the strength of gravity mediation whi h is usuallytaken to be M / . However as we have seen the proper omparison is between F X /x with thesize of the gravity mediation soft breaking terms whi h are mu h smaller than the gravitino(cid:21) 17 (cid:21)ass. Regarding the third point an expli it analysis should be performed in whi h both theanomalous U (1) and the moduli are taken into a ount in the pro ess of moduli stabilisationand supersymmetry breaking.Very similar to the re ently dis ussed lo al F-theory models, we may expand our modelassuming that there exists a sour e for gauge mediation, whi h is parametrised by theva uum expe tation values of a s alar (cid:28)eld h X i = x + θ F X . This supersymmetry breakinghappens in a se tor hidden from the GUT brane and is being mediated by messenger (cid:28)elds,whi h are harged under the GUT gauge group. In order not to spoil gauge ouplinguni(cid:28) ation, this is generi ally assumed to be a ve tor-like pair in the representationof SU (5) . For our purposes, in this paper we won't present a viable dynami al stringyrealisation of this supersymmetry breaking, but just assume that there exists an extrase tor, whi h stabilises the new moduli su h that just the (cid:28)eld F X develops a non-zero VEVwithout spoiling the LARGE volume minimum for the bulk moduli. This is learly a strongassumption, as a dynami al realisation of gauge mediation is known to be hallenging [36(cid:21)38℄. We will omment more on this towards the end of this se tion.The gauge mediated gaugino and sfermion masses are of order M gauge e Q ∼ M gauge e G = α X π F X x , (4.1)where the α X / π prefa tor is due to the fa t that these masses are indu ed via a one-loope(cid:27)e t for the gauginos and via a two-loop diagram for the sfermions. Note that theseformulae used a anoni al normalised super(cid:28)eld X .Now, we would like these gauge mediated soft masses to dominate the gravity mediatedones. In parti ular, we want the gauge mediated sfermion masses to dominate over thegravity mediated ones. To get a (cid:28)rst impression of the numerology we get, we also imposethe strong onstraint that the supersymmetry breaking F X already uplifts the negativeva uum energy (2.14) of the LARGE volume minimum. We therefore require F X M P ≃ M /
16 log (cid:16) M P M s (cid:17) M s M P , (4.2)where M s is the string s ale, leading to the relation F X ≃ r log (cid:16) M P M s (cid:17) M / M s . (4.3)Requiring now that M gauge e Q > | M grav e Q | leads to the moderate bound x < α X π M P ≃ GeV , (4.4)where we used the relation (3.8). If there is a further suppression, i. e. M e Q ≃ M / / V ,then this bound be omes even more relaxed. For solving the hierar hy problem, one alsoneeds F X /x ≃ GeV . On e one has spe i(cid:28)ed the favourite values for x and F X , one anuse (4.3) to determine the value of the gravitino mass, whi h we would like to stress willbe gravity-dominated. Let us dis uss two examples.(cid:21) 18 (cid:21) In the lo al F-theory models, it was argued that the best values are x ≃ GeV , F X ≃ GeV (4.5)whi h lead to M / ≃ , whi h needs a ertain amount of tuning of W . However,the light modulus τ b has a mass of the order M τ b ≃ M / M s M P , (4.6)whi h in this ase gives M τ b ≃ . For su h a light modulus, we expe t to fa ethe osmologi al modulus problem (CMP). • Let us now require that the light modulus avoids the CMP by having a mass M τ b ≃
100 TeV . Then a ording to (4.6), the gravity mediated gravitino mass has to be ofthe order M / ≃ TeV . Using (4.3), this leads to F X ≃ GeV . For gaugemediated soft masses of the order
500 GeV , we therefore get x ≃ · GeV , whi his slightly beyond the stronger limit (4.4). For further suppression of the sfermionmasses there is no problem. • In the (cid:28)rst ase one ould ameliorate this problem by allowing for a ertain tuning ofthe Higgs mass, so that the supersymmetry breaking s ale for the visible se tor an belarger than
500 GeV . Let us still have F X ≃ GeV to avoid the CMP and require x ≃ · GeV to satisfy the onstraint (4.4) for gauge mediation dominan e. Thenthe gauge mediated soft masses are of the order
50 TeV .Finally, let us dis uss in whi h way this simple model of gauge mediation needs tobe improved in order to show that it an really be embedded into string theory. As wealready mentioned, we did not dynami ally explain where the SUSY breaking (cid:28)eld X getsits VEV from. Re ently, various kinds of models have been suggested, whi h, we think, sofar are not ompletely onvin ing from a string theory point of view. One promising modelis the so- alled Fayet(cid:21)Polonyi model. It ombines an anomalous Pe ei(cid:21)Quinn symmetrywith a linear superpotential in X generated by another D3-instanton wrapping a del Pezzosurfa e of size T FP. This gives rise both to a D-term potential with a T FP dependent Fayet(cid:21)Iliopoulos term and an F-term potential form the linear superpotential. Note, that thelatter also depends on T FP. Now, also taking the Kähler potentials into a ount one has toshow that dynami ally really supersymmetry an be broken in su h a way that the desiredvalues for x and F X arise.3 Moreover, one expe ts that also F T FP = 0 , whi h gives anothersour e of supersymmetry breaking. Finally, one has to ensure that the moduli stabilisationin the bulk, i. e. of the τ b and τ s moduli and the moduli stabilisation of the lo al X and T FP moduli de ouple. All these hallenging questions are beyond the s ope of this paper.3It was shown in [37℄, that this model with a simple hoi e of the Kähler potential a tually still possessupersymmetri minima. (cid:21) 19 (cid:21).2 Impli ations for the Cosmologi al Moduli ProblemLet us (cid:28)nish this se tion with some omments about the osmologi al moduli problem [39(cid:21)41℄. The osmologi al moduli problem refers to the existen e of late de aying moduli, withmass omparable to the gravitino. The moduli are expe ted to be displa ed from theirminimum during the in(cid:29)ationary epo h, subsequently os illating about their minimum andred-shifting as matter. The lifetime of su h moduli is τ ∼ M P m φ ≫ for m φ . .Moduli ome to dominate the energy density of the universe, but if they de ay too latethen they fail to reheat the universe to temperatures su(cid:30) ient for nu leosynthesis. In someways the moduli problem is the most severe problem fa ing low-energy supersymmetry asit is very di(cid:30) ult to onstru t a viable osmology with su h long-lived moduli.There are various possible approa hes to this problem. In the absen e of moduli-(cid:28)xingme hanisms, it may have been hoped that one ould stabilise the moduli at s ales far abovethe gravitino mass. The more that has been learned about moduli stabilisation the lessplausible this s enario has be ome.The results in this paper suggest a novel approa h to this problem. One of the propertiesof lo al LARGE volume GUTs with D-term stabilisation is that the soft terms appearat a s ale hierar hi ally smaller than the gravitino mass. Depending on the extent of an ellations, we have seen that soft terms appear at an order not larger than m soft ∼ M / / M / P ,in the ase when the dilaton F-term is responsible for uplifting. In all other ases gauginomasses will be further suppressed, with at least an extra loop fa tor as in anomaly mediation,and possibly even as far as m soft ∼ M / M P . For the two extreme ases the gravitino massappropriate to TeV soft terms is m soft ∼ M / / M / P −→ M / ∼ GeV m soft ∼ M / M P −→ M / ∼ GeV . (4.7)Instead of solving the moduli problem by making the moduli heavy and keeping soft terms omparable to the gravitino mass, this suggests making the gravitino heavy and having softterms mu h lighter than the gravitino mass.In the LARGE volume models the volume modulus T b is relatively light and has a mass m T b ∼ M / / M / P , while all other moduli have masses omparable to M / . In the (cid:28)rst ase listedabove, with a gravitino mass of around GeV , the volume modulus has m ∼ andstill poses osmologi al problems. However in the other ases m T b is su(cid:30) iently large tode ay before nu leosynthesis. In the ase of maximal suppression, with M / ∼ GeV ,then we have m T b ∼ GeV with no osmologi al problems. In all ases the other moduli(for example dilaton and omplex stru ture moduli) have masses omparable to the gravitinomass and de ay very rapidly.It would also be interesting to study whether these suppressed soft terms would a(cid:27)e tthe thermal behaviour of the LARGE volume models studied in [42℄.(cid:21) 20 (cid:21). Con lusionsIn this paper we have studied the stru ture of gravity mediated soft terms that arise when ombining LARGE volume moduli stabilisation with lo al GUTs and D-term stabilisationof the y le supporting the GUT brane.We (cid:28)nd that the modulus determining the size of the standard model y le does notbreak supersymmetry and therefore the s ale of gravity mediated soft terms is highlysuppressed ompared to the gravitino mass. Both (cid:16)standard(cid:17) gravity mediated terms of O ( M / ) and also known anomaly mediated terms of O ( g M / / π ) vanish. The (cid:28)rstnon-zero terms appear to arise at O (cid:16) M / √V (cid:17) ≃ M / / M / P . However it is possible that additional an ellations o ur and suppress the soft terms even further than this down to O (cid:18) M / M P (cid:19) .The appearan e of these further an ellations is related to the (in)dependen e of the phys-i al Yukawa ouplings on the (cid:28)elds breaking supersymmetry.The an ellation of ontributions to the soft masses of order M / introdu es severalsubtleties. In parti ular, as the soft terms o ur at a s ale parametri ally smaller thanthe gravitino mass e(cid:27)e ts whi h are normally negligible be ome important. We have triedto in lude all known e(cid:27)e ts and have given general arguments as to when an ellationswill take pla e. Nonetheless, it is important to look for any further possible ontributionsto soft terms whi h ould possibly be dangerous. In this respe t one would ideally likea dire t stringy omputation of soft terms that would bypass the need to go through thesupergravity formalism.The suppression of soft terms relative to the gravitino mass opens new avenues forthinking about the osmologi al moduli problem. Rather than the traditional approa h ofmaking the moduli heavy while keeping the gravitino and soft terms around a TeV , thisopens the possibility of having the moduli and gravitino mu h heavier than a
TeV whilestill maintaining
TeV s ale soft terms.If the gravitational soft terms are of the order M / / M / P , the volume modulus howeverremains a problem in the LARGE volume s enario as its mass is mu h lighter than thegravitino mass and would be the same order as the soft terms. If further an ellations o urand the soft terms are of order M / M P , then the volume modulus eases to be a osmologi alproblem.Several s enarios regarding gravity and anomaly mediation are possible and whi h ofthese is a tually realised may be model dependent. The main possibilities are: • If the F-term of the dilaton (cid:28)eld is responsible for the uplifting to de Sitter spa e,then F S ∼ V − / and all the soft masses are of order M P V / ∼ M / √V . This is of thesame order as the mass of the lightest modulus, the volume modulus, and this (cid:28)eldremains dangerous for the osmologi al moduli problem. • If any other (cid:28)eld is responsible for the de Sitter uplifting, the dilaton indu es gravitymediated gaugino masses of order M P V or from anomaly mediation, barring any further an ellation, of order α M P V / where α is a loop fa tor. In both of these ases, identifying(cid:21) 21 (cid:21)he gaugino masses with the TeV s ale, the osmologi al moduli problem is absentsin e the volume modulus would be at least as heavy as
10 TeV . • For ea h of the two ases of the previous item, gravity mediated s alar masses, ifnot ta hyoni , are of order M P V / and therefore hierar hi ally heavier than the gauginomasses, indi ating a minor version of split supersymmetry [43(cid:21)45℄. However if we haveperfe t sequestering in the sense that physi al Yukawa ouplings do not depend onthe Kähler moduli (cid:28)elds that break supersymmetry, su h terms will an el. Howevers alar masses will always re eive a ontribution from the dilaton F-term at order M P V . • Sin e leading order gravity and anomaly mediation ontributions to the soft termsare suppressed, then other e(cid:27)e ts have to be onsidered. In parti ular string loop orre tions ould be relevant, e. g. as in [46, 47℄, (giving potential ontributions tos alar masses of order M P V / [48, 49℄) but also a novel s enario may be on eived inwhi h the main sour e of supersymmetry breaking for the observable se tor is gaugemediation, however the gravitino mass remains very large and unlike previous modelsof gauge mediation, the LSP is no longer the gravitino but an be a more standardneutralino.Even though there are several s enarios, we an still extra t some general on lusionsfrom this analysis. First, as emphasised in [50℄, the e(cid:27)e ts of the de Sitter uplifting playan important rle on the soft breaking terms. This is unlike previous s enarios based onthe LARGE volume in whi h they were negligible. Se ond, in all s enarios the gravitinomass is mu h heavier than the TeV s ale M / ≥ GeV whi h relaxes the osmologi alproblems asso iated to low-energy supersymmetry. Generi ally (ex ept in the ase that thedilaton is responsible for uplifting) the lightest modulus is heavier than the soft terms andtherefore osmologi ally harmless also.Finally we point out that even though there are several an ellations that redu e thevalue of the volume to have the
TeV s ale, there is a minimum value of the volume that an be extra ted from this analysis. Namely, the universal sour e of gaugino masses dueto the dilaton dependen e of the gauge kineti fun tion, implies that the gaugino masses annot be smaller than M P V . The same limit appears for s alar masses for the ase ofperfe t sequestering ( Z = e K / ). This provides a bound for the size of the overall volume V ∼ − in string units whi h orresponds to a string s ale of order M s ∼ GeV .Combining this with the re ent result [23℄ that in lo al models the GUT uni(cid:28) ation s ale isgiven by M GUT ∼ M s V / this gives a uni(cid:28) ation s ale of the same order as the one expe tedfor supersymmetri GUT models from LEP pre ision results of M GUT ∼ GeV . If thiss enario is a tually realised it would provide an example in whi h a string model addressessimultaneously the two positive properties of the MSSM, namely the full hierar hy problem,without tuning, and obtaining the preferred s ale of gauge uni(cid:28) ation.Furthermore, this value of the volume is of the order of magnitude preferred by modelsof in(cid:29)ation in order for the in(cid:29)aton to give rise to density perturbations of the right ampli-tude, normalised by COBE. In parti ular a volume
V ∼ − was needed to a hieve(cid:21) 22 (cid:21)ähler moduli in(cid:29)ation [51℄. It also ameliorates the gravitino mass problem pointed outin [52, 53℄.We onsider our results bring loser lo al string/F-theory models to honest-to-Godstring ompa ti(cid:28) ations sin e we in orporate the main properties of su h models regardingsupersymmetry breaking and moduli stabilisation. Many questions remain open. Con reteexamples where the an ellations illustrated here are realised, in luding an uplifting term,loop orre tions, et . are desirable. The presen e of su h sub-leading ontributions tosoft terms an be re ast in the presen e of orre tions to the physi al Yukawa ouplings.Spe i(cid:28) ally, the s ale of the soft terms an be related to the extent to whi h the (lo al)physi al Yukawa ouplings depend on the (bulk) supersymmetry breaking (cid:28)elds. In thelimit of perfe t sequestering the Kähler moduli ontribution to soft masses vanish. Itmay be possible to study this issue more pre isely using the te hniques of orbifold CFT.Furthermore, for F-theory onstru tions, even though in general they are treated in a waysimilar to orientifold onstru tions, the 4D e(cid:27)e tive (cid:28)eld theory for F-theory models isless under ontrol. In parti ular the α ′ - orre tions whi h are ru ial in the large volumes enario, need to be omputed for F-theory ompa ti(cid:28) ations.A knowledgementsWe gratefully a knowledge dis ussions with C. P. Burgess, B. Campbell, K. Choi, M. Ci- oli, M. Dolan, T. Grimm, L. Ibáñez, D. Lüst, A. Maharana, F. Mar hesano, E. Palti,E. Plaus hinn, M. S hmidt-Sommerfeld, A. Uranga, G. Villadoro, T. Weigand and E. Wit-ten. FQ wants to parti ularly thank S. de Alwis for many enlightning dis ussions on relatedsubje ts. RB would like to thank the Galileo Galilei Institute for Theoreti al Physi s forhospitality and the INFN for partial support during the ompletion of this work. SLK wouldlike to thank the CERN Theory group for hospitality at the early stages of this proje t.JC is grateful to the Royal So iety for a University Resear h Fellowship.A. F-TermsAs there is a an ellation at leading order taking pla e in the al ulation of various softterms, a areful large volume expansion up to the next-to-leading order has to be performed.Let us start with the expressions for e − aτ s and τ / s in the minimum, whi h will be neededlater.For this purpose, onsider the s alar potential (2.13). Upon minimising it with respe tto the two independent variables τ s and V , we get two expression: First, from the ondition ∂V F ∂τ s = 0 , it follows: e − aτ s = µλ | W | aA V √ τ s (1 − aτ s )( − a + τ s ) . (A.1)After developing the denominator in powers of / ( aτ s ) and inserting the expressions for µ and λ we get e − aτ s ≈ η / s aA √ τ s W V (cid:18) − aτ s (cid:19) . (A.2)(cid:21) 23 (cid:21)he se ond expression arises upon solving ∂V F ∂ V = 0 for τ / s and thereby using (A.2). Theresult is τ / s ≈ ˆ ξ η / s (cid:18) aτ s (cid:19) . (A.3)Another approximation needed in the following is: K ab ( ∂ b K ) = − V + V ˆ ξ + 4 ˆ ξ V − ˆ ξ )( V + ˆ ξ ) τ a ≈ − τ a −
32 ˆ ξ τ a V , (A.4)where the sum runs only over Kähler moduli. The (cid:28)rst equality an be derived using theexpressions for the Kähler metri and the derivatives of the Kähler potential with respe tto the moduli in terms of two- y le volumes t a instead of four- y le volumes τ a (see [2, 54℄).We are now in a position to al ulate F b : F b = e K / K bJ D J W = e K / (cid:16) K bτ j ( ∂ τ j K ) W + K bs ( ∂ s W ) + K bS D S W (cid:17) . (A.5)The term involving D S W turns out to be sub-leading in the V − expansion and an benegle ted (see below). The derivative of the superpotential with respe t to T s undergoes asign-(cid:29)ip due to the minimisation with respe t to the orresponding axion as argued in [2℄.Using the approximations (A.2), (A.3) and (A.4), one easily gets: F b = − τ b √ g s √ W V − √ τ b aτ s (cid:18) aτ s (cid:19) W V + O ( V − ) , (A.6)or with the expressions for the gravitino- and gaugino-mass (3.2) and (3.4) inserted: F b = − τ b M / − τ b aτ s (cid:18) aτ s (cid:19) M e G + O ( V − ) . (A.7)From (A.2), it an be derived that aτ s ≈ ln V ≈ . Thus, for the sake of shorter formulae,one may also negle t the se ond term in the parenthesis: F b ≈ − τ b M / − τ b aτ s M e G (A.8)Next, we want to al ulate F S = e K / K S ¯ J D ¯ J ¯ W . Here, a subtlety arises on erning D S W = ∂ S W + W ( ∂ S K ) : the Kähler potential depends on the dilaton not only in theusual way via − ln( S + ¯ S ) , but there is also a ontribution in the α ′ - orre tion in theKähler moduli part. Thus, ∂ S K has V − orre tions: D S W ≈ ∂ S W − g s W − ξg / s W V + O ( V − ) . (A.9)Also as a onsequen e of the α ′ - orre tions, the minimum of the s alar potential for thedilaton is shifted away from the supersymmetri lo us D S W = 0 at order V − . In orderto determine the new minimum, one would have to minimise the full potential, beforeintegrating out the dilaton. However, sin e we do not have an expli it model with a full(cid:21) 24 (cid:21)ux se tor, in order to apture this e(cid:27)e t, we assume that the two leading order termsin (A.9) an el and keep only the next-to-leading order terms in the V − expansion. Theexpression we get in this way has ertainly the orre t order in V and we in lude an orderone onstant γ in the (cid:28)nal result omprising the un ertainty about the true lo ation of thenew minimum. D S W ≈ − γ ′ ξg / s W V . (A.10)In the sum over D I W in the dilaton F-Term F S = e K / K SJ D J W , there are (cid:28)nally two ontributions at order V − : one from K Sb F b and one from K SS F S . The result reads: F S ≈ √ γ ξg s W V (A.11)B. Vanishing D-terms in luding matterWe will onsider in this appendix a on rete example with a generi D-term in luding notonly the (cid:28)eld dependent FI-term but also a harged matter (cid:28)eld. In general vanishing D-terms do not imply vanishing FI-term but a an ellation between the two terms entering theD-term potential. We argue here (following [55℄) that on e soft supersymmetry breakingterms are in luded, as long as the square of the s alar masses is positive the minimum ofthe s alar potential is for vanishing both matter (cid:28)eld VEV and FI-term.Sin e in lo al models the standard model y le is a del Pezzo surfa e that an andusually prefers to shrink to small size, it is dangerous to work in the regime where the y lesize is larger than the string s ale. Even though at sizes of the order of the string s ale thespe trum and ouplings of the model are not understood, the regime lose to a del Pezzosingularity is under a mu h better ontrol, the spe trum is determined by the extendedquiver diagrams and the low-energy e(cid:27)e tive theory an be reliably used in an expansionin the small blow-up mode.This e(cid:27)e tive (cid:28)eld theory has been re ently dis ussed in [9℄. We start with the sameba kground geometry as before in luding one large τ and two small y les τ , τ . On therigid y le τ we have the standard non-perturbative e(cid:27)e t. Being at the singular lo us for τ , the e(cid:27)e tive (cid:28)eld theory an be approximated by the following supergravity set-up: K = − V + ˆ ξ ) + ατ V + Z | ϕ | ,W = W + Ae − aT ,f = d T + S , (B.1)where ϕ denotes a matter (cid:28)eld that is harged under an anomalous U (1) on the standardmodel y le, as is the y le volume itself. As dis ussed in [9℄, the e(cid:27)e tive theory for τ di(cid:27)ers from the standard treatment for relatively large values of τ sin e we are working lose to the singularity. The anomalous U (1) generates a D-term potential with a Fayet(cid:21)Iliopoulos term: V D = 12( dτ + s ) (cid:18) Q ϕ Z | ϕ | + Q τ τ V (cid:19) . (B.2)(cid:21) 25 (cid:21)he matter metri Z is taken to have the general form Z = 1 V / (cid:18) β + γτ λ − δ V (cid:19) , (B.3)where the onstants β , δ an in prin iple depend on the dilaton and omplex stru turemoduli.The D-term potential determines the size of τ and implies τ ∼ | ϕ | V / . (B.4)For a vanishing VEV of ϕ this implies as previously τ = 0 . Expanding around ϕ = 0 , thes alar potential is given by the standard LARGE volume potential and at next-to-leadingorder by a ontribution quadrati in ϕ : V = 1( V + ξ ) | aA | √ τ τ / e − aτ − W aAτ e − aτ + W ξ τ / + Y − β | ϕ | τ | aA | √ τ τ / e − aτ − W aAτ e − aτ + 9 W (5 δβ + 2 ξ )4 τ / !! + β | ϕ | τ ( V + ξ ) | aA | √ τ τ / e − aτ − W aAτ e − aτ + 3 W ξ τ / + Y ! , (B.5)where the last term arises from the expansion of e K and Y denotes the F-term upliftingterm, whi h allows for a stabilisation at zero va uum energy.With zero va uum energy, the mass of ϕ is given by m ϕ = K − ϕϕ − β τ | aA | √ τ τ / e − aτ − W aAτ e − aτ + 9 W (cid:16) ξ − δβ (cid:17) τ / = − τ V min + 45 W (cid:16) ξ − δβ (cid:17) τ / ≈ W ( δβ − ξ )4 τ / . (B.6)Di(cid:27)erent ratios of δ/β allow for ta hyoni , zero or positive masses at this order. In parti -ular: δβ < ξ ta hyoni , = ξ zero, > ξ positive. (B.7)With respe t to the matter metri the ondition δβ = ξ an be understood as follows: The ase of vanishing masses orresponds to the following matter metri : Z = β V / (cid:18) − ξ V (cid:19) ≈ β ( V + ξ ) / = βe K / , (B.8)whi h is the ondition found in se tion 3.3 for extreme sequestering and an ellation of s alarmasses at the / V / level. Without the uplifting term, the e(cid:27)e t of the term arising from(cid:21) 26 (cid:21)he expansion of e K is generally sub-leading to the other ontribution sin e it is suppressedwith /aτ .For positive s alar masses we an learly see that ombining the term m ϕ ϕ with theD-term potential, both the VEV of ϕ and the FI-term vanish at the minimum as desired.For the ta hyoni ase this would indi ate as usual that at the minimum the s alar (cid:28)eldand the FI-term would be non-vanishing. If ϕ is a (cid:28)eld harged under the standard modelgauge group this is undesirable sin e it would break the standard model symmetries at highenergies. If the ondition δβ = ξ3