JHEP08(2008)004 July 14, 2008 June 30, 2008 August 1, 2008 Avity Models

JHEP08(2008)004 July 14, 2008 June 30, 2008 August 1, 2008 avity Models. y-localized visi- Accepted: Received: Published: integrated and transparent s can occur while dangerous scussion is a well-controlled finite) boundary-to-boundary ity, eby solving the latter’s tachy- aking, in which the central re- malism of the five-dimensional omaly-mediation, or for string cal supersymmetry, in the pres- in only mild warping, which allows hieved perturbatively. This makes [email protected] Published by Institute of Physics Publishing for SISSA , Supersymmetry Breaking, Extended Supersymmetry, Supergr We study a five-dimensional supergravity model with boundar [email protected] Department of Physics and Astronomy, Johns Hopkins Univers 3400 North Charles St.,E-mail: Baltimore, MD 21218, U.S.A. Minho Son and Raman Sundrum ble sector exhibiting anomaly-mediatedquirements supersymmetry of bre sequestering and radius stabilizationit are possible ac to understandfashion, these mostly various from mechanisms the in higher-dimensionalence a viewpoint. of more Lo visible sector quantumsuperconformal effects, tensor calculus. is enforced The construction by results a the natural for supersymmetry-breaking mediation mechanismgravity of loops ( to co-dominateonic with slepton anomaly-mediation, problem. ther Weloops make the of non-trivial stabilizing check thatstarting fields thi point remain highly fortheory suppressed. considering realizations. other Our generalizations di of an Keywords: Abstract: Anomaly-mediation and sequestering fromhigher-dimensional a viewpoint JHEP08(2008)004 7 8 7 2 6 9 11 12 13 14 17 19 11 14 18 24 25 21 23 – 1 – 2 /Z 1 S 3.1 Parity assignment 3.2 Induced 4D3.3 boundary SUSY multiplets SUSY multiplets in the 4D effective theory 9 5.1 The visible5.2 sector classical action Sequestering 5.3 Visible quantum5.4 dynamics and anomaly The mediation hidden sector and SUSY breaking 6.1 Stabilization and6.2 preserved SUSY Mass of radion 7.1 Supersymmetry breaking 7.2 Relation between compensator 12 and stabilization 20 5. Boundary physics 6. Supersymmetric radius stabilization 7. The SUSY-breaking vacuum 4. Bulk potential for 4D scalars 2. 5D bulk superfields 3. Compactification on Contents 1. Introduction 9. Bulk radiative corrections 10. Numerical estimates A. Convention B. Bulk potential for 4D scalars in superspace notation 27 8. 4D effective theory and sequestering JHEP08(2008)004 . sequestering SB and sequestering ensional supergravity pective, with minimal rder to understand the ve effect. In principle, uestering. We hope that extra-dimensional radius iliar non-supersymmetric maly, and to AMSB and ant but rather subtle Su- t emphasis. However, our um SM. As a result, the g within supergravity theo- ons in a higher-dimensional ne can consider it as either visible” or Standard Model M) or when combined with re, non-perturbative physics fication (with perhaps even rgest), or just as a simplified re we give to tracking local tly the subtlety involves the important role played by the relevant calculations could be the 4D effective field theory etric flavor and CP problems sector, known as h AMSB. The basic 5D model estering connection somewhat nlinear treatment at 5D level. oring, in the simplest possible to achieve a more microscopic, deduced from matching to the ble when applied to either non- is has proven difficult. ions, 5D effective theories allow ible sector and hidden sector are eties of AMSB and sequestering on, for the sake of greater clarity, acetime [1]. Therefore one might – 2 – For perturbative visible sectors, AMSB is also a perturbati One forbidding aspect has been the complexity of higher-dim In the present paper, we study a perturbative 5D model with AM Let us review the central issues of AMSB and sequestering in o sequestering can also take a perturbativelocalized form, when on the separated vis “branes” in aexpect higher-dimensional that sp realistic models of AMSBworked could out be by built means where of all locally Feynman diagrams supersymmetric and setting. classical In soluti could such be a setting understood the most subtl cleanly. Perhaps surprisingly, th with bulk matteroriginal and discussions their of consistentbelow sequestering couplings the worked to compactification scale, at the withhigher-dimensional the quant key theory [5, properties level 6]. being of Thisopaque, makes but the it AMSB-sequ was possiblehas to proceed been in invoked this manner. instabilization, the Even an he issue 4D entangled effective with AMSB. theoryhigher-dimensional, One and in would perturbative like view order of to whatcertainty, is achieve and going as the basis for further developments. which allows onerecourse to to see the them 4Dan operate effective effective theory directly field below from compactification. theorymore the extra of O dimensions, 5D a but where possible pers model. the string fifth Given theory dimension the is complexity compacti of theone full la to string zoom theory construct inthe onto 5D just effective the description essential wemanner, features give new of is mechanisms that AMSB a can and useful functionis basis seq symbiotically not wit for new expl anddiscussion was proposed has in important ref. newsupersymmmetry [7], elements, in with the in somewhat presence differen particularsequestering. of the We the give ca quantum a conformal more explicit, ano complete and fullypath no we will follow in the paper. We begin with a simple and fam 1. Introduction Anomaly-Mediated Supersymmetry Breaking (AMSB) is an eleg persymmetry (SUSY) breaking mediationries mechanism operatin [1, 2]. Itof offers weak scale an supersymmetry, attractive and solutionminimal realistic models to supersymmetric are the extensions possi supersymm ofother the SUSY Standard breaking Model effects (S [3,quantum 4]. conformal Part anomaly of (running thesector, subtlety couplings) is when within the propagating the “ inprecondition a for the supergravity dominance background. of AMSB effects Par in the visible JHEP08(2008)004 ), x ( (1.1) (1.6) (1.8) (1.2) (1.3) (1.7) (1.4) (1.5) δC clearly no 1+ -dependence C µ ≡ ), the coupling ) x ( x ( C δC -dependence cancels C ce can be both exact, uple this theory to 4D metric. dimensions, tational dynamics. Let us contains conformal invari- rdinate transformation the -dependence introduced to ormations, the transformed , D spacetime, t the quantum levelǫ this is . ning gauge coupling, µ . ν 2 ′ ssically ) y varying formal transformations, which us as is familiar in non-abelian mode of the metric, lls theory due to , le , and must couple precisely so dx a µν a ate conformal invariance implies νβ µ C . ′ F ) 2 F ( ) x . , but this symmetry is broken quan- dx ( a µα a ) . µν ′ f µν F 2 F x ) δC η ( ( ) αβ ′ ) → a + the rest )) µν g )) . ′ µν x µ µ F ( x g ( ( ( C µν µν ( 2 3 g η 2 g = 1 f g ( ) → g 1 1 2 2 i β x µC 4 g 1 ν – 3 – ( ( 4 C 2 2 )= − + h ′ g dx − C x ) 4 µ = ( µ ≡ − ( dx L ) background takes the form 1 ′ µν g 2 ) g x g = so as to appears as a spontaneous breaking in Minkowski x − µν ( 4 ( η √ L C C − µν = g = 2 = L ds L Deviations of C about this Vacuum Expectation Value (VEV), At the quantum level, the question is how conformal invarian Classically, this theory is exactly conformallytum invariant mechanically by thegravity running classically, of the coupling. Now let us co warm-up, namely pure non-abelian Yang-Mills theory in flat 4 out of the gravitationallynot coupled so. Yang-Mills action. For example But in a dimensional regularization, in 4 + space, In this way, the breaking of conformal invariance in Yang-Mi as to compensate forfocus any on conformal just this breaking mode. in It the is non-gravi straightforward to see that cla The action nowance enjoys as general a coordinate subgroupMinkowski in invariance, metric the which transforms following as sense. Under a general coo longer cancels out, andbalance multiplies dimensions. the After renormalization renormalization,the exact sca Yang-Mills coordin dynamics in the is dressed up by the couplings to in the usual way. Butmetric for takes a the subgroup form of such coordinate transf being a subgroup of coordinateYang-Mills invariance, theory. and also The anomalo answer is that the scalar (off-shell) This subgroup of coordinate transformationstherefore defines must the be con a symmetry of the action when coupled to the correspond to real spacetime curvature. For small and slowl transforms under conformal transformations, to the Yang-Mills sector is given by Taylor expanding the run JHEP08(2008)004 2 C θ via C (1.9) F (1.11) (1.12) (1.10) Q, V .. c . +h .) The standard 2 α τ W in ) C Cµ uced visible soft terms ( s that the couplings are erpotential is not renor- dependence is dominated n get a non-zero VEV in the visible sector µ loop ter sector, say a hidden sec- onformal anomaly, the soft − d does not provide a solution re they may be highly flavor- 1 ace, taking into account aking, explicit or anomalous, lings. If we are only interested hese results. For example, the idden sectors, such as wavefunction renormalization. vity formalism, the Supercon- rance of f. [8, 9] for other discussions of alizing our Yang-Mills example) ial status to the compensator in classical conformal invariance θτ 6= 0 can only occur in Minkowski nt typically relates these two by t the en we can write the couplings to 2 must be promoted to an off-shell C d F C Z Q , . . 2 V θ P l e )+ ∗ C is renormalized at only one loop. In this Q /M F 2 Σ 2 P l Q,C ∗ hid /g ( M Σ 1 – 4 – θ =1+ ∼ F ≡ 4 θW i d 2 C d C F h Z loop Z − ∼ 1 τ + Q V e ∗ Q ) ∗ CC √ µ ( θ Z 4 d Z = However, in supergravity supersymmetry breaking When we pass to supergravity, the scalar mode using flat superspace notation. Local supersymmetry demand L Then there can be additional sources of SUSY breaking felt by even Planck-suppressed couplings between the visible and h by gauge dynamics, the soft terms induced are flavor-blind. space if accompanied by supersymmetrytor. breaking from Then the cancellation mat of the effective cosmological consta exactly superconformally invariant, results of AMSB result by expanding this equation in supersp It is convenient (butformal not Tensor essential) Calculus that [10 – there 13],mode, that is and effectively a to gives the supergra spec exactin superconformal the invariance of propagation its of coup a renormalizablein matter such sector a supergravity (gener Poincaré-invariant background, th C chiral multiplet which compensatesof for the superconformal matter bre sector.Minkowski spacetime It (i.e. Poincaréinvariant VEV), is the only supergravity mode that ca AMSB and analogous phenomena. This is the bosonic equivalent of anomaly-mediation. See re where Σ denotes someare hidden SM-loop chiral suppressed, fields. sinceterms they directly While from typically the the depend AMSB-ind hiddenviolating. on sector If the are such c not, terms and areto present, furthermo the AMSB is supersymmetric subdominant flavor an problem. (Each of the original refs.full [1, combined 2] list contain is some reviewed in but ref. not [4]). all of To the t extent tha language all the remaining running(This is allows us incorporated to into use the holomorphicity to constrain the appea Here, the superpotential is exactlyand cubic in we fields are to mainta workingmalized in and the the field gauge normalization coupling such that the sup JHEP08(2008)004 al visible sector, nd sequestering. For ithout stabilization, the hidden-visible couplings any unsequestered SUSY ts in unacceptable tachy- The highly warped case is e the constraints of local tudy of these sequestering oo much smaller than the echanism of “conformal se- or 5D supergravity [14, 15] arlier AMSB work [5, 6] in ary from SUSY breaking on he 4D effective field theory lism incorporates 5D super- indirect in earlier studies of vely [5]. In the present paper ravity). The stabilization is hen play the role of the 4D is facilitated by the develop- vity theory, so that no direct ise this question: given that mplished by non-perturbative 4D effective field theory, while d matter [14, 15, 18 – 20] and ilzation, by means of massive p effects are far subdominant as proposed earlier in ref. [7] his type has been established te gravity loop effect that can at hidden-visible couplings are with AMSB, and therefore can of having the hidden and visible 28, 29]. The highly warped case rger-Wise mechanism [25]. ension supergravity, but requires hidden sector SUSY breaking [5]. – 5 – = 0, with vanishing AMSB [5]. In other examples, However, this is only true of mildly warped compactifi- C 1 F problem of the MSSM though. µ A final consideration in AMSB is that when applied to the minim A mechanism to further suppress even the Planck-suppressed The issue of radius stabilization is intertwined with AMSB a This still does not solve the 1 cations. In highly warpedto AMSB. compactifications We the will gravity thereforeinteresting loo study because the it mildly is warped AdS/CFT casequestering”, dual here. which [27] avoids to the the complications purelynon-perturbatively of 4D higher-dim strong m couplings in theis hidden also sector interesting [ because a fully sequestered example of t However, the requisite stabilizingthese fields fields are in needed to thethe get AMSB bulk hidden in sector also the boundary, visible ra breaking how sector is bound effects it as thatsupersymmetry, well? they but do not the Of mediate understanding course,AMSB has from any higher been dimensions answer necessarily because should stabilizationgaugino was condensation, involv acco which can only bematching captured to within this the theory from 5Dwe must be adopt performed a perturbati transparentbulk classical hypermultiplet mechanism of fields radius with(see stabil boundary refs. superpotentials, [23, 24]essentially for a supersymmetric related generalization work, of but the in Goldbe absence ofthe superg Minimal Supersymmetric Standardonic Model slepton (MSSM), masses. it resul Planck When scale, the there compactification is scalegenerate an non-tachyonic attractive is visible flavor-blind not scalar and t massesresult UV-fini that in compete a viable spectrum [26]. failure to stabilize results in runaway radius moduli after is known as ”sequestering”. We followsectors the localized original proposal on different 4Dcouplings boundaries are allowed in by a locality 5D [1].then supergra not One induced must also by check integratingissues th out and massive AMSB bulk directly modes.ment in in The the recent s higher-dimensional years context of(with a important 5D developments Superconformal earlier Tensortheir in Calculus couplings f [16, to 17] 4D asconformal boundary well) fields bulk an [21, compensators 19, whosecompensator 22]. reviewed boundary This above. restrictions forma This t which is the more compensator transparent is thanbelow e introduced compactification. only after arriving at t example in the no-scaleeffective model which 4D emerges compensator from a has 5D set-up w JHEP08(2008)004 (2.1) ds by ˙ 3 for the flat , Lorentz sym- , special super- ,..., A ˙ A 0 P K = w energy 4D effective cified, we will use 5D SM (for hiddens sectors ngerous non-flavor-blind ning introduction to the s supersymmetric radius own to provide an attrac- a,b,... calculus, including hyper- matter to the boundaries. ogical constant. It also ex- tive tree-level potential for ese symmetries are gauged arped 5D model will help in erous stabilizing-field loops. nt couplings of the compen- r dynamics. This is the key e sector, thereby establishing avor-blind gravity loops work scusses the possible couplings ample numerical estimates to l symmetry ation of the fifth dimension to a gravity-loop resolution of the l local supersymmetry down to M fields arising from hidden sector ction 8 studies the form of SUSY ~ V is the gravitino. In off-shell for- , sses radiative corrections to visible symmetry. In the Superconformal ˙ 5 for the flat 5D spacetime indices and M M , ˙ 3 ψ U , b , ˙ 2 , M ˙ 1 , , φ ˙ 0 A M = ,f – 6 – M , special conformal symmetry Q , ψ A,B,... 3 for the curved 5D indices. AB M 2 ,ω and internal SU(2) is the and fünfbein ,..., A M e D A M = 0 e µ,ν,... become dependent fields in terms of the other independent fiel 1. 5 for the curved 5D indices. Similarly we will use M ≡ , 3 5 , φ , 2 A M M , 1 , and dilatation , ordinary supersymmetry , ,f S AB AB = 0 M M ω Throughout the remainder of the paper, unless otherwise spe The paper is organized as follows. Section 2 contains a light Throughout this paper, we will use 2 2. 5D bulk superfields Rigid 5D superconformal symmetry consists of translationa M,N,... multiplets and component fieldsmuliplet matter. of 5D Section 3 superconformalan reviews tensor interval, the the orbifold consequent compactific 4D breaking N=1 of supersymmetry higher-dimensiona attheory. the Section two boundaries 44D as VEVs massages Poincaré-invariant into well the a as bulk useful in contribution form.of the to Section bulk lo the 5 supergravity di effec New (including couplings the are compensators) reported,sators and and in care bulk light is of givento to the understanding the conformal AMSB consiste anomaly instabilization. of this the Section 5D visible 7 context. discusses secto SUSY the breaking Section corrections and 6 to the discusse plicitly bulk cancellation connects of the the stabilizing 4D fieldsbreaking effective to transmitted cosmol visible from AMSB. Se the hiddensequestering boundary with negligible to corrections. the visibl SectionSUSY 9 breaking discu due to loopstive of solution bulk to fields. thewith Gravity AMSB loops large tachyonic are D-term slepton kn SUSY problemeffects breaking), of of but loops the we of MS show also stabilizing that estimate successful fields. AMSB the is In da achievable,tachyonic section in slepton particular 10, with problem we and giveThe adequate s appendices suppression contain of some useful dang notation and formulae. finding mildly warped string theoryin constructions conjunction in with which AMSB. fl Planck units, within string theory [30]. We hope that our study of a mildly w metry 4D spacetime indices and symmetry where Tensor Calculus approach toand 5D corresponding supergravity gauge fields [15, are 14, 18], th application of constraints. JHEP08(2008)004 , α αβ ) A M the ρ e α (3.1) (2.5) (3.2) β j (2.2) (2.3) (2.4) =1 2ˆ i d, A for the ,F ij ultiplets 1 ǫ C − T α = =2 2ˆ α i 4 denote the ∗ ζ , α i ,F , form 5D Weyl ˆ α symmetry while − A = ), by the relation, C ,ζ 3 0 = 3 , U generater to break i γ α =2 † ≡ − 2ˆ i ,Y physically consistent α 3 ) ,χ 2 α i α σ even fields are given by ,A ζ 1 2 A ( AB ,Y / − 1 v truction of the action and f 5D hypermultiplets in a α =1 ≡ i 2ˆ Y A ds, α = 1 tiplet, , i.e. ; ¯ e fix SU(2) ζ . 2 = ( , 3 1 ~ , T Y C ) , ,Y , 5 i j i − x . ~σ denotes the SU(2) index while the Ω ,χ ) − · , , i ( , 5 ij ) ) ζ ~ 5 Y AB ˙ 5 α i i x ,Y terms) and γ symmetries. After these gauge fixings the , v i − 3 = M,A , F ( Ω ; α M S i α , C A iσ kj F , , b ǫ – 7 – ,ζ M + ij α i M ik and ,χ A Y → P → P . ˙ , V 5 ( ∗ a ) ) D M, A , we will preserve ) i = M 5 5 ( which is in the Weyl multiplet. 2 ,v ij i 2 x x j 3 µ ( ( , ψ Y /Z Y ζ 1 correspond to the parity even, odd respectively [22]. , i.e. for the bosonic and fermionic fields, AB A 1 ,V A M /Z 2 v 2 e − S , 1 = ( Z , 1 , is related to the isovector, 5 S ij ij ,V Y µ Y ,b = +1 appears within a seperate supermultiplet from graviton fiel and the parity assignment can be carried out consistently by − 5 P R M ,ψ A (1) + is the 5D charge conjugation matrix. µ U C ,ψ 2) index. As the above indices imply, we will treat the hyperm 2 correspond to the unphysical compensator while ˙ 5 5 , , ,e (2 a µ e = 1 Usp down to α R The details of the supersymmetry transformations, the cons We summarize the parity assignments of the fields. The parity The SU(2) tensor, e.g. The hypermultiplet compensator is needed in order to derive =+1:( P the gauge fixing can be found in refs. [15,3. 14, 18]. Compactification on where the eigenvalues 3.1 Parity assignment Regarding the orbifolding on and it satisfies hermiticity SU(2) transformation law under the fermions where for the scalars (same reality condition for malism the independent gauge fields along with auxiliary fiel multiplet given by The system contains threecompact notation, more supermultiplets: two types o where, as the complex quantities and impose the reality constraint denotes physical hypermultiplet, and a 5D central charge vector mul superconformal gravity invariant action andcentral is charge used vector to gaug multiplet fixes The “graviphoton” but in the action it mixes with theory reduces to 5D Poincarésupergravity. The JHEP08(2008)004 is n (3.5) (3.6) (3.3) (3.7) (3.4) , ) α =2 . 2ˆ M, i ) + ,F χ M 1 5 ( − ˙ 5 Γ a α =1 i 4 2ˆ ˙ i 5 . µ = 2 corresponds to ) ,F − ˆ 1 R ˆ α + α . − − 4 1 ,ζ α etry. The 4D supercon- =1 R Ω i 2ˆ . ˙ 5 + 2 3 is therefore important to α , r of N=1 chiral multiplets =1 a wn to 4D superconformal A 2ˆ i ) ˙ v 5 ollowing 4D superconformal = − ) a symmetries this corresponds ˆ h [22]. ,A D α =1 e fields of 4D superconformal 1 Ω n 2ˆ i ersymmetry down to 4D N=1 V iγ ( − + + A ˙ , 5 α ˙ ¯ =2 ) 5 1 χ ) in the right hand side in eq. (3.5) are i 2ˆ a + 2 bc ˆ µ − 1 2 D ˆ Q A ˆ , via f α ( R ; + =1 , ˙ 2ˆ 5 i 3 + R 1 2 µ µ Ω )+ ˙ ˆ ˆ 5 φ R α ,Y + =1 ˆ , , iF 2ˆ D i + M ˙ 1 5 orbifolding [22], i.e. 2 ˙ ab 5 c − Ω a µ 2 v , − α , iF v b µ 2ˆ 0) and has the lowest component given by α ˆ , ,ω , /Z ζ D 2ˆ A + -charge), 1 µ ; M, ζ R − ˙ ˙ 5 5 – 8 – ) − S R R a a P , a i ˆ v Q µ P F ,χ abc 2 ( i ˙ 5 ab , b 2 µ (2 ) = (1 − µ ˙ ǫ 5 + 2 , + − ,v a ˆ 1 1 6 ˙ R µ 5 , 3 v 5 5 a − w,n ˆ α α Γ − , ψ F =2 =2 ,V + i is given in appendix A. 2ˆ i i 2ˆ , a µ 2 , denote Weyl weights, chiral weights respectively and − 2 1 2 5 e A A 1 ) µ ( at the boundary (i.e. it satisfies local SUSY transformation M Y n 2) under the C + Q 3 2 ,V / ( , 1 4 , 5 ˙ 5 3 + can also couple at the boundary via a parity even real general 1 a w , µ ,b − ˆ 1 : ( 2 Y R ψ M , / 3 ˙ 5 2 5+ 5 − ) , a A Γ Y ˙ 5 ,ψ v − ˙ 5 =+1: ( = + µ a − µ ˆ Ω v D γ µ ¯ 5 P P ψ 5 2 ) = (3 + Γ i ,ψ 2 Γ i a 5 3 − µ 2 − − ,e V w,n − . Note that ˙ ( 5 µ a µ M µ ˙ e 2 5 3 4 f φ M ˆ M, D = 1 corresponds to the compensator hypermultiplet and ˆ = = = 1 :( a α µ µ µ − ˆ ˆ = a f φ The graviphoton Each 5D hypermultiplet is decomposed into a vector-like pai The consistent parity assignment uniquely determines the f = The definitions of the supercovariant curvatures such as V 3 P related to the standard chiral weights (or given in ref. [22]. The definition of Γ This boundary restricted multipletsymmetry corresponds [10 – to 13]. the gaug while the odd fields are It has a superconformal“dilaton” weight ( with weight ( type 4D multiplet made from its gauge invariant field strengt where ˆ the physical hypermultiplet. Thesupersymmetry. orbifolding breaks 5D sup 3.2 Induced 4D boundary SUSYThe multiplets orbifolding breaks 5Dgauge superconformal symmetry. After gauge gauge symmetry fixingto the do breaking extra 5D superconformal localformal supersymmetry symmetry to N=1 constrains 4D anydetermine local boundary how supersymm bulk action fields terms decompose and under it this symmetry. Weyl (or gravity) multiplets law of 4D superconformal gravity) [22], where JHEP08(2008)004 - 5 x (4.1) (3.9) (3.8) . − 5 ψ is the length of 5 Γ M ) + πr 2 5 ¯ χ i 4 iV + + . While we do not wish ˙ 5 1 ϕ 5 5 , where ariant ansatz into the bulk metry of the effective field V ef o be discussed) as well as a D superspace notation (see ( . e” /r i ion 3.2, except for being l necessarily arise below as a warped metric ncorporating the zero-modes s in the 4D Poincaréinvariant m 5D bulk action [18]. The 4D Weyl multiplet zero-mode 2 − . ckground in which the visible ite the locally supersymmetric l for all 4D scalars, even when − D vectors and tensors. In this ˙ 2 5 5 − e , -independent (hence parity-even) )Ω dϕ − 5 5 2 2 x Ω ) r 5 ˙ 5 a Γ − appearing within a 4D radion general ˙ v 5 5 ( ν e ˙ 5 5 (1+Γ e − dx 5 + 4 µ C− iψ 1 4 and the 4D Weyl (gravity) multiplet. These dx M – 9 – − µν , 5 η − ) + 2 ) 5 , 2 C,H ϕ i ψ 5 ( ψ a 4 ˙ σ 5 iY v b 2 − 2 v e , b + 5 − γ 1 = , 5 1 iA 2 5 Y ( V ˙ 5 5 − ds 2 e , + 2Γ 2 2 5 M ˙ 5 5 V − 5+ e 2 φ − , = + 2 − 5 ˙ 5 5 Σ e i iψ 2 + 2 χ − [22], − 5 , Γ 5 ˙ 5 5 i 4 e A = and The effective potential from plugging in the 4D Poincaréinv In addition, we should include boundary-localized fields (t ˙ 5 5 W e We are taking the fifth dimensional coordinate to be the “angl action can be economically written in compact flat (global) 4 to consider a highly warpedbackreaction spacetime, to a the mild stabilizing warp fields. factor wil In general, both these forms4D of radion effective field Lagrangian. are required to wr 4. Bulk potential for 4D scalars In order to determinesector the fields propagate supergravity we and must determine stabilization thevacuum. VEVs ba of This all requires field usthese to minimize scalars the arise classical fromsection potentia extra-dimensional we components summarize of 5 themost contribution general 4D to Poincaréinvariant geometry this is potential given by fro the 3.3 SUSY multiplets in the 4DAfter compactification, effective most theory field modes get masses of order 1 the extra dimension. The approximate zero-modes are fields. The most obvious of these are are the analogs of the boundary-induced fields in the subsect independent as opposed to restrictedforms to the the boundaries. gauge fields The theory of below the the preserved compactification 4D scale. superconformal sym 4D radion multiplet. One canof form a radion chiral multiplet i The 4D effective theory also has the zero-mode multiplet [22], JHEP08(2008)004 , ,

is 2 2 F ) ¯ 0. H 3 . 5 A c ˜ F V . (4.5) (4.2) (4.3) (4.4) A σ 1 2 ≡ e ϕ< 2 + )+h θ C ∗ . ¯ c , C ¯ + . H 3 2 5 ¯ 1 for ˜ V H ∂ A ∗ − +h m = ¯ 3 C ≡ 2 α ¯ 2 √ H is odd while ˜ ¯ − W − H ontraint of the 5D 5 ), α ∂ 0 and ′ , , C ¯ ˜ 4 2 He 5 C W ¯ in the ﬁrst and second H ∂ H . A − ) ∗ ) Σ F 4 ϕ> 5 D σ ’s denote those of the 4D C H i 2 ≡ 7): the scalar components and e nalysis. While the straight 2 ˜ . V F 2 C 2 + θ − | H m imA work 3 ¯ ¯ nsider the hypermultiplet mass + ¯ 2 C H H √ | + 5 een 5 e H 5 ↔ in the above equations are conve- 1 2 ∂ ∗ ∇ rspace for 5D action. ∂ = must also be hypermultiplet charge ∗ ¯ H C terms and other 5D ﬁelds, are given − C ¯ , H m 2 − = ( | − C F ∗ (see eq. (B.11) in appendix B). The + , ¯ ¯ H ¯ ,H H ¯ C 3 | H H H 5 , ¯ ¯ 5 D , C C Y 2 1 2 , ∇ Σ) 3 H F ∇ 5 A ∗ σ . − +2 √ ) is equals to +1 for e A ) in eq. (3.7), e.g. ∗ im 3 5 2 H 2 2 ϕ 2 | – 10 – ( | i ( θ 1 V ¯ 3 M σ/ C ǫ H , 5 ¯ 3 | − + 2 5 e ) H CC , . Also note that under parity + | α 5 =2 2 V − iV 5 α

i 2ˆ 2 + M ¯ ↔ θ ∂ C / r ) + ) A 2 + A 3 | ∗ 2 = 1 2 , = ≡ terms of eq. (3.7)). Similarly imA | C T . Note that ( + 1 M | ¯ α ¯ α HY ϕ =2 Y C − 2 + ), where F 2ˆ i | H ˜ | = W 5 ϕ T A + 2 ( ∂ ( H 3 M , T | ǫ 2 mH σ 1 2 6 2 0 T , 3 M A σ / = ( − ) ( 2 F m ′ +(2 θ e , σ D C 2 H e ) θ e + 1 = r 5 r d σ 2 4 σ 2 2 σ D θ | d 4 ∇ e 4 , m σ Z e ¯ ( e C 2 Z + ˙ 5 C e + + + ∂ F T 3 2 − | σ ¯ θ 2 √ e 2 dϕ | term, as functions of the original 5D ( (Similarly for the 2 π θ 2 θ 0 C Σ= ¯ 1 2 θ D Z i − 2 + A 2 −| θ − M is simply a Lagrange multiplier which forces the following c 2 − 1 M C -dependent, ≡ = ′ ϕ ¯ C = = 2( = = = C The covariant derivative in eq. (4.2) is defined as See refs. [34, 35] for some early development of using 4D supe 2 bulk ˜ ˜ V T denotes F-term appearing in the 5D Lagrangian the curly 4 C V V A − even. Therefore, in the orbifoldedto theory be we are forced to co Note that in 5D supergravity a hypermultiplet mass appendix B for the derivation. See refs. [31 – 33] for related for the central charge gauge boson, in terms of the “fake” flat superspace multiplets, These fields are all functions of and equations of eq. (4.3) is covenient renaming of nient renaming of theof 4D N=1 these hypermultiplets fields given are by related eq. (3. to in appendix B. WeF will not require these expressions for our a The terms and chiral supermultiplets. superconformal gravity (see ref. [18] for the relation betw JHEP08(2008)004 M (5.3) (5.1) (5.2) VEVs of . ) to be inde- -dependent ) ′ θ ¯ C V τ , d combinations ¯ ∗ C H, ′ 0), and ¯ , H H, 5 ( vis it dependence of as well as some gauge , ∂ symmetry are formed ′ W use all the Q ¯ )= H ned to have superconfor- ′ 5 ¯ C m , , ∂ ′ ltipet fields to aid in radius ∗ l weight (0 symmetry have not yet been ′ plings ¯ folding has already broken 5D he vanishing of any boundary H ¯ rfields, C s in the standard fashion, but , form the 4D N=1 supersymme- . U ′ 5 servation of 4D supersymmetry. can not couple gauge invariantly 2 H , ∂ vis ′ ( W L ¯ iY C ) 5 , + π .c. 3 , ∂ 1 and / ∗ − ′ r 1 Y ) Σ ϕ ∗ ( ≡ )+h ,H ¯ ′ δ H Y ¯ x H – 11 – 5 are order parameters of the 5D supersymemtry CH d 2 π / ,V,H J F and . . ∗ 1 Z c . 3 5 − = )+ V ∗ Q,Q Q +h ( ( vis α f S vis HH 3 V W / W 2 2 α ) / 2 ∗ 1 W C ) ( − Q CC σ ∗ ( ( 3 ¯ C σ 2 θ e θτ ¯ C 2 2 . These dynamical fields couple to the 4D Poincaréinvariant d d θ e -dependent “radion” multiplet V + 4 ˙ 5 5 ∗ d Z Z 0). For this purpose it has been convenient to define the prime e , + + Z CC ( = = which means that the scale invariance and SU(2) vis ′ L ¯ C V It is convenient to write in the flat superspace notation beca Note that the bulk potential given by eq. (4.2) has the explic Here, we have written the most general term Kähler constrai to the boundary as explained in ref. [22]. Note that the supermultiplets try order parameters.auxiliary In fields) particular, in their a vanishingNote classical (and that solution t 5D guarantees auxiliary the pre fields such as We consider the visible sector consisting of N=1 chiral supe components of eq. (4.3) as well as the boundary-induced bulk fields. not the 4D subalgebra.supersymmetry. These are less informative since orbi and of the boundary-induced bulk fields with zero superconforma We have explicitly writtenstabilization. a superpotential Note that for we the have hypermu chosen these protected cou gauge-fixed. 5. Boundary physics Boundary action termsstraightforwardly invariant by under writing the 4Dusing N=1 superconformal the boundary-induced 4D invariant 4D local gravitational fields. super 5.1 The visible sector classicalThe action visible action to be added to the bulk action takes the for pendent of the bulk fields, as is technically natural. mal weight (2 JHEP08(2008)004 (5.4) (5.5) (5.6) .c. .c. y renormalize the visible .c. visible sector fields comes 2) compensator definition n superconformal gravity if is aspect we will keep only / 3 )+h , )+h )+h 2 3 ed by the requirement of main- r radius stabilization, the VEVs ymmetry breaking components. 3 / ) r fields effectively propagate in CH π λQ λQ , V J Q . . ∗ + c V + . . . . . e 2 c 2 )+ c . ∗ . Q Q Q,Q Q +h Q 3 ( ( / Q 3 +h α 2 f +h / vis m 2 3 α C α ( ) / W W ∗ 2 σ 2 α W ) e W 2 ∗ C α – 12 – α in order to get the more familiar 4D compensator W Q C Q ) CC σ ( 3 W ( 3 V W / m Q CC σ e 2 ( σ ( ( 3 ∗ 2 θ e θτ θ θτ C σ 2 2 2 2 2 θ e θτ compensates for the breaking of superconformal invari- d d θ e d d θQ 2 2 4 → 3 4 d d θ e / d Z Z d Z Z 4 2 C d Z Z + + C Z + + Z to arrive at σ + + Z ≈ e ≈ Q = vis vis → L vis L Q L 3 / 2 C σ e 1). However we will stick to our weight (3 , . This feature is sequestering. C F The visible action appears very similar to purely 4D action i inherited from the bulk theory. 5.2 Sequestering In section 8 we will show theof very all important the result primed that, fields afte inThat eq. is, (5.3) have negligible these 4D and VEVsKähler supers superpotentials. are essentially Therefore, pureonly an numbers the induced-compensator which visible background, merel secto one makes the field redefinition of weight (1 In particular the onlyfrom 4D supersymmetry breaking felt by the 5.3 Visible quantum dynamics and anomalyThe mediation compensator-dependence of the visibletaining action exact is superconformal dictat gaugethe symmetry. renormalizable To terms focus of on the th visible sector fields. Now redefine In this form it is clear that ance due to visible masses. JHEP08(2008)004 , 4 M (5.7) (5.8) (5.9) 2 Λ ∼ eaking or c ulk fields possible in . . dary directly in the 5D c . . . ive hidden sector dynam- gs also violates supercon- c g within the primed bulk . 4D effective gravitino. .c. nd all hidden masses are of reaking at the intermediate o they play a negligible role. paper. +h aly mediated supersymmetry se will have far weaker effects icate issue of sequestering and α redominantly, this cancellation o be at the hidden boundary, ies that we can treat as sources )+h Q )+h stabilization, while the constant W 3 V , α e CH ∗ λQ 0 W hid Q ) with the visible sector fields replaced J vacuum energy. Therefore 3 + L / + 2 ) 2 4 3 vis / 2 ϕ Q r L C 1 ( 3 ) σ δ / ∗ cC 2 can be taken to be unity at any point in the ( x µ e . At this level, the dominant contributions to C . 5 ( θ ) CC σ d 2 C ϕ ( – 13 – e ( hid d σ F σ loop S Z also compensates for this, Q e − Z µ e 1 m = C ( + θ θτ 4 , that we consider. However, we will have to consider 2 2 hid will be adjusted to cancel the 4D effective cosmological d d Λ θ Z J S 4 c − d Z Z = + + Z ≈ hid V vis − L GeV. The warp factor 5 . 11 takes the entirely analogous form to is the effective 4D Planck scale. 10 4 hid ∼ M L = 0) = 0. We assume that all hidden sector VEVs (whether SUSY br However, at the quantum level the running of visible couplin The bulk couplings are only a slight perturbation to the mass There are also a variety of other couplings of hidden VEVs to b This demonstration of anomaly mediation on the visible boun , essentially Λ-scale or smaller sources for bulk fields. The ϕ hid ( the effective potential for 4D scalars is given by formal invariance and it is vital that As reviewed in Introduction thisbreaking leads in to the the pattern visible of sector, anom seeded by ics. Therefore the hidden VEVsfor are the effectively given bulk quantit fields via their coupling in preserving) are at most oforder order Λ Λ or to above, except the for appropriate a power, a massless Goldstino eaten by the S where such couplings more carefully when we are discussing the del visible SUSY breaking, in section 8. We will find that there to where extra-dimensional interval. Weσ will choose this location t Again the hypermultiplet superpotential willsuperpotential aid in with radius coefficient than the other sources, such as set-up (once we demonstratefields, the as done suppression in of section SUSY 8), breakin is5.4 one of The the hidden main sector results and ofThe SUSY this hidden breaking sector action takes the form constant when supergravity effects are taken into account. P is against the positive supersymmetry breaking Λ by some hidden sectorscale fields Λ responsible for supersymmetry b JHEP08(2008)004 (6.4) (6.3) (6.1) (6.2) . ) . c . , is also uniquely to the boundaries 4 . We are consider- 0 +h M H which is a supersym- eaking in section 7.1. m c CH π he size of extra-dimension cation scale is determined . (6.1) [7, 36]. J ffective potential, the form ) erturbative setup for the su- k mass, π plings of ll that we are working in a 5D − 36]. However our treatment will e SUSY breaking hidden sector, oduced in ref. [7], and studied in tion will relate it to SUSY break- ϕ ( . , δ 2 1 πr | − 0 − 0 m J | 2 0 π − CH J J e πr 0 πr . 3 0 J ln − ) m = 1 ϕ ( 2 4 0 – 14 – δ , whose effect is to cancel the 4D SUSY breaking ∼ ( c and 5D hypermultiplet mass m M σ π 3 ∼ J θ e 2 radion 1 is given by eq. (4.2). 2 πr d m Z bulk V + bulk V . We are dropping the constant superpotential 0 J 1.), dϕ ≡ Z 5 − M = eff V greater than , and so we only include it when considering supersymmetry br 4 − π The stabilization mechanism of radius modulus determines t After performing superspace integration of the above full e J That is, in this sectionand the we constant are hidden neglecting superpotential, visible sector, th 6.1 Stabilization and preserved SUSY The stabilization of radius modulus is carried out by the cou This will be discussed in subsection 6.2. ing 6. Supersymmetric radius stabilization We will study the supersymmetric radius stabilization intr the formalism of 5D superconformal tensorbe calculus somewhat in different. ref. Natural [ expansionpersymmetric parameters radius in stabilization our are p metric coupling, but the 4D cosmologicaling constant Λ cancella as will be discussed in detail in subsection 6.1. The 4D Planc related to the sizePlanck of unit, extra-dimension by the relation (Reca The canonical mass ofto the be radius roughly modulus below the compactifi in terms of these 5D parameters, i.e. as well as 5D mass of the hypermultiplet as was indicated in eq vacuum energy. The form of JHEP08(2008)004 , ′ . . C c c . . (6.5) (6.7) (6.6) . +h +h . c . ) ) , π π when these − − 3 5 ∗ )+h ϕ ϕ ¯ C ( V ( H , 5 3 ∗ 5 T ∂ Cδ Hδ ∗ F rV ∗ π π ¯ 2 1 2 C J J C 2 ∗ 2 1 T − . − s are given in terms of )+ )+ c F 2 . | ϕ π C 2 1 )+ ( 5 Y ϕ | eld configuration preserves . − ∂ ( +h c r ∗ . , Cδ . . )+ 6 ϕ c 0 2 ) . C ( | π J Hδ 2 2 − | 0 ¯ H +h H − − 2 J Cδ ¯ − | | H ¯ 1 2 π 1 2 ¯ ¯ ϕ H )+h , H H ( J C ¯ 5 − − | C ∗ C + |F 5 mT ¯ 2 ∇ C r 2 F | Hδ ∂ , | ∗ )+ σ ∗ + π C mT ¯ H 2 + 5 ϕ ¯ H H | J C σ ( | σ ∂ ( 2 − ∗ − 5 | 5 + 2 2 1 3 2 σ ¯ − ∂ ∂ ¯ ∗ H H C Cδ 5 H ) 2 3 − , 2 3 ∗ + T + ∂ 0 F 5 |F ϕ mrD 2 J 2 3 F 5 + ¯ – 15 – -dependent components of eq. (4.3) as well as ( C | r + C ∇ 5 ∂ 5 θ 2 1 ∗ ¯ ∗ 5 + − C ∂ ∂ + ¯ − 5 ∂ C Hδ )+ ∗ are given by eq. (4.4). We also integrate out H − ¯ 2 ∂ ∗ ¯ T 0 , H ( ∗ σ | H − | + C 2 3 5 J − ¯ ¯ ∗ 5 ) F C . H F H 2 ∗ ∂ T − ∗ c T σ 5 2 1 ( + . ¯ + r 5 |F F ∗ H − F ∗ = 1 irV ∇ ∗ r ∂ D ∗ 1 2 ¯ ¯ − mT ( C H 2 2 ¯ 3 ¯ H + , | C +h H C ∗ mT T ∗ C T 5 + − 2 1 − − 1 2 C − H T F | F + 5 ∇ σ σ ) ∗ 2 ¯ 2 H − F 2 1 − | HY ∗ 2 5 5 2 1 | σ σ ∇ F ¯ C ∂ ∂ ¯ H 5 5 H ¯ rD ¯ H C 3 3 2 2 ∂ ∂ H C |F ¯ 3 2 H H + 2 1 r F 3 3 2 2 F mrH + + − | 2 F mH F − H + 5 5 2 + + | 5 +2 − + − + +2 + 2 ∂ ∂ 5 5 ¯ ¯ H σ H , ∇ H ∂ ∂ 4 | ∗ ( H r r 1 1 m mH ( mH dϕ e form the 4D N=1 supersymmetry order parameters. Therefore, i r r 1 1 1 3 4 3 1 − − − Z Y ======∗ 3 ¯ 0= ¯ ∗ ∗ ∗ ∗ C C 5 H H D Y V F F F F eff and V 3 5 − supersymmetry order parameters4D vanish N=1 the supersymmetry. correspondingother fi These fields supersymmetry by their order equations parameter of motion from eq. (6.5), As was discussed in section 4, all the in terms of 4D auxiliary fields is given by V appearing in eq. (4.2), resulting in the constraint where the definitions of JHEP08(2008)004 (6.8) (6.9) (6.12) (6.15) (6.14) (6.13) (6.10) (6.11) = 0) = 0. ϕ . Solving for ( r . σ 2 | ¯ H can be made real by | 5 1 2 π by has a non-vanishing field , J Note that the solutions of 0 1+ (0)) ¯ q. (6.6) also satisfy the full J H , σ ions, lly satisfied. The warp factor | is just equal to − , ) ) C σ ϕ | T ( given by eq. (6.7) [37]. 5 ) . σ ng fields as is clear in the form of ∂ ϑ = 0 ( 2 C =0 as mentioned above) and the ( | 5 3 2 3 , ¯ 2 3 5 ∂ ¯ C , + H ( | 0 V | ) i ϕ ) | ϕ ∗ . ( ∼ = )+ T + = 0. δ 0 T | 2 . | 1 3 ¯ D m Tπ C C + ¯ e 0 | H ) | = ) = 0 m = 0 T ) )= σ ( − (0) ϕ 5 σ Y ϑ e ( 5 ∂ – 16 – m 5 δ C x, ϕ . The imaginary part of the above equation gives ( π is determined by our convention, ∂ 2 ( ∂ 6 = ) functions inside 1 0 3 2 J iϑ 2 )) ϕ J 3( 2 σ e H ( − | | ϕ ǫ − ) + ( ¯ − C,T ǫ C C | ϕ = 0 | ( | 2 ( ) J | C ǫ σ | σ ¯ . Therefore, the VEV of = ¯ we can neglect warp factor. 5 H,C, 5 5 H | ∂ ∂ C ∂ ) π H, C ( )= ∗ , , to match the delta functions J F 2 3 J T given by eq. (6.7) (with H 0= + x, ϕ + ( C 2 | ¯ T H ( C = 0 equation of eq. (6.6) to be | m 5 ∂ ( D )) is used to treat the 2 4 1 1 π = 0 equation of eq. (6.6) implies − = Y given by eq. (6.9), the real part of eq. (6.13) reduces to ϕ 0= ( δ ¯ H was parameterized as C The solution of the second equation of eq. (6.6) is then given The identity, 5 is determined by using eq. (6.10) gives us eq. (6.1). Taking into account that The third equation of eq. (6.6) gives rise to Note that the integration constant of In the remainder of this section we will solve the set of equat that gives rise tothe 4D set N=1 of supersymmetric equations fieldset of configuration. of motion second-order given differential by equationseq. plugging of (6.5). eq. the propagati Further, (6.8) we in will e seek a solution with and, to the leading order of ( similarly for solution of Using the constraint of rise to where since to the leading order of absorbing their phases into configuration, r The first and fourth equationsσ of eq. (6.6) are now automatica JHEP08(2008)004 , 0 C (6.16) (6.17) (6.19) (6.20) (6.18) terms, and F 0 H , , T 2 2 and they can be | = integer). The 2 0 ) n r 1 3 5 rD H | V 3 2 as is clear in the 4D ) ∼ ( ) 0 ∗ r 2 − T H 2 e theory below the com- ymmetrically, i.e. subject | + + )=0( ¯ T C 2 ϕ ( | ymmetry order parameters, ( 6) supersymmetrically with last equation of eq. (6.6) is 0 olutions that satisfy the full t tree level. e 4D effective theory of these δ adion are given by the well- and |F Y he zero modes represented by m | r r − 2 +1 . e 6 n . δT c . 2 . nd (6.14). | − ) − − ϕ . . )) | 2 | c ∗ T T ϕ . (1 +h R C 0 ( F 0 ǫ 0 m ∗ |F 1 C +h m − ¯ . Note that supersymmetrically the sta- r g πr H 0 e 2 . 2 ∗ T Y ) − H 0 x ) + − as supermultiplets, including their √ ( T = 0 H 2 2 0 | | 0 mH 0 − T 0 ¯ H m H eff – 17 – , + C = − | V 0 |F rπJ e ) T C 0 ∗ r π )= F J T m ¯ + H radion -integration, the 4D effective theory for − 2 + x, ϕ | L ϕ ( 0 + ( T H J mH ( 2 H ( | |F + , the hypermultiplets have a conserved hypermultiplet num- θ − Y r 2 | J d θ 4 πr σ , is given by d 4 Z 6 π J − − Z dϕ e ≈ Z eff integration was absorbed into V = − in this case while the other fields get masses of order − ϕ eff 0 V C from and π 0 = 0 (In this limit these light fields become massless), e.g. In this way we have solved all of equations of eq. (6.6) supers Since the full effective potential is quadratic in the supers H ,π , 0 effective superpotential in eq.known no-scale (6.19). structure [38], The kinetic term of r bilization gives rise to a supersymmetric Dirac mass for the resulting vacuum energyset on of the equations locally in supersymmetric eq. s (6.8) identically vanishes, 6.2 Mass of radion The mass of radionpactification can scale. be The easily 4D estimatedT effective from theory the is described 4D by effectiv t above equation is automaticallyalso satisfied automatically by satisfied by eq. eqs. (6.12). (6.9), (6.11), The (6.12) a to eq. (6.8). i.e. In the second equality, we have used an identity, ( integrated out in a supersymmetriclight way. fields, In order we to solve deriveJ the th equations of motion given by eq. (6. Here, into eq. (6.4) and performing the ber, so that they can not correct the radion effective action a At leading (zeroth) order in to the leading order of After plugging in eq. (6.18) as well as JHEP08(2008)004 4 to Λ c (7.3) (7.1) (7.2) ∼ , only (6.22) (6.21) (6.23) is radion 0 m H ∼ , . The constant c l transformation, . δT . . 2 | 6.10) and (6.11), the the mass of the scalar ant superpotential δT | in the bulk from the 5D eff r ation scale ˜ e deviation of the VEV of V 0 0 2 | the radion is given by, − H 0 πm tural expansion parameters in 2 J 2 limits). | ) , − e fluctuations 3 0 r 2 . e ∗ 0 r µ m r ∂ − H 0 2 2 | ( ∗ 0 1 0 2 4 J . πm J 2 | 0 + 0 M . − ) 2 4 3 m e πr | 2 rJ 2 4 3 0 1 2 0 | c 0 eff − c + 2 | M m V H | 1 ˜ )) R 2 πm | ∼ 2 3 2 ϕ 4 usefully parameterizes supersymmetry breaking. 0 – 18 – ( 4 2 ∼ J R− ǫ c and the constant superpotential | M ≈ ( Λ 0 r 0 πr 2 0 − as can be seen in eq. (6.22). Given the form of the H , m − c 0 ∼− ( , πm 2 radion ˜ g H , is given by the VEV of π g C 6 J m − C 11 − F eff /C √ p V C 2 2 F = = 4 r r 4 is related to the supersymmetry breaking vacuum energy 4 4 2 2 0 π π M M , is of order 4 4 ,m radion H ≈ = π L J eff ˜ V ≪ c , i.e. (as well as the light fields of the visible and hidden sectors) µν 0 ˜ g . It does not mix with C 2 4 πr δT M 2 The radion has its kinetic term in the Einstein frame via a Wey This implies that the 4D effective theory, below the stabiliz The “seed” of AMSB, = , the zero mode of 0 µν We can get theradion physical mass of the Dirac state by just getting kinetic term of the radion in eq. (6.21), the physical mass of Expanding about supersymmetricEinstein-frame effective VEVs, potential to following quadratic order from in th eqs. ( g It induces a deviationH from the supersymmetric VEVs. e.g. th as will be discussed in section 7.2. This recovers the results of refs. [7, 36] (in their unwarped In this sense the constant superpotential This will be discussed in section 7.1. perspective, contains superpotential 7. The SUSY-breaking vacuum In order to discuss the supersymmetry breaking we add a const by the 4D cosmological constant cancellation, our effective potential givenour by eq. perturbative (6.4). setup now Therefore, are the na JHEP08(2008)004 to as and c 0 e 4D (7.9) (7.8) (7.7) (7.4) (7.5) (7.6) 0 H H , T 2 | 0 rameterize : H θ | gives the minimum, ) . | ) ction 6, must be pro- ) . ∗ 4 0 . e light fields aívratvacuum). caréinvariant c T 0 . H | + . rJ + Λ T c ( . c 0 | +h 0 introduced in eq. (5.9) (and . 0 0 m 4 c e constant superpotential H πm +h − | H ect to e 0 0 } c 11 12 + Λ m ) − m | . 2 0 + 0 c | ≈ . J (1 J 0 ∗ 0 . 0 || c δT 4 . H | c . H 1 | ) +h m 2 0 2 2 4 ) 4 | 2 T ( 2 2 c m 0 Y c | M 2 − 0 ( − + m | at 2 0 and Λ 2 − H 2 | 11 24 | | J e 0 0 | 0 c 0 π 4 ∼ O – 19 – ≈ H C H J | πr | | 4 0 rπJ ) r r − 0 ∗ δT 2 0 Λ + Λ 2 0 0 0 πm T m 2 m 2 J m | m 2 ( 2 − c + | πr | 2 | { e 0 0 + ( 0 T J J | ( 2 − | | 11 24 π π 1 is given by θC − Y 2 | . Noting this, after performing superspace integration, th 2 6 . 6 d 11 θ + c c 11 ≈− 4 4 πr has only a quadratic term, at the minimum of the 4D effective d ≈ Λ Z 6 ≈ min − − − Z eff eff δT V V V ≈ the eff V 2 c − , and minimize the 4D effective potential with respect to iθ e | 0 H | The approximate 4D effective potential of the zero modes of th Any deviation from the supersymmetric VEVs, discussed in se Now, in order to analyze the remaining part of eq. (7.5), we pa = is now given by which ensures that we are correctly expanding about a 4D Poin 0 0 4 The cancellation of the 4D cosmological constant relates th 7.1 Supersymmetry breaking We now consider the effects of the constant superpotential Finally, minimizing the above effective potential with resp Λ C This is just eq. (6.19) with the addition of Since to order potential of eq. (7.5) H portional to some power of the positive vacuum energy, effective potential to order JHEP08(2008)004 by c H (7.13) (7.12) (7.11) (7.10) to the ) π vanishes, /C − 0 C ϕ . H ( F δ ) is order are determined π ) 2 J c , π , ¯ x, ϕ } ( ) − ¯ h − (therefore, singular- c π ) ϕ , H ϕ ( ) − ) by parameterizing as ( . F . The above expression ) ϕ , ) . Hδ ( . (7.8), the 5 dimensional Hδ x,π C x, ϕ π ∗ 0 π ( 0 h the VEV of ( 2 2 F J x,π c Cδ J H − ( H given by π ∗ 0 − π ϕ ), ¯ J , one can obtain the expressions C + J 4 ssion), e.g. the singular parts of ide ( J ) 0 π ¯ C . 2 C ϕ given by eq. (6.12), J − ( m Cδ )+ x, ϕ ( π σ 2 1 ϕ σ ¯ h ( J ) (see eq. (7.8)), ¯ σ 2 Hδ 5 )= )= c 5 0 ( ∂ )) 2 ∂ Cδ J 3 2 )+ ϕ 0 3 2 x,π x,π ( J ϕ ( ( ǫ + + ( ∼ O ¯ ( c + ¯ h is given by eq. (6.7). c − 5 0 5 ∂ ))¯ Cδ ). The boundary values of ¯ h ∂ C , H 0 π ∼− – 20 – )) J , 0) ¯ ∗ , gets a non-trivial contribution only via the VEV r π − C x, ϕ − ( ¯ 0) x, H 2 ϕ − c | ( − /C ∗ ( ¯ )¯ H x, ¯ δ ϕ C H ϕ ( H ( H ( F 0 δ − ǫ C 4 mH from the eq. (6.5) imply J and the warp factor − | 0 ) − 2 ≡ ∗ ¯ J 2 − ϕ C | mT ) ¯ ( ) H , ∼− δ ϕ mT H − , ( rC ¯ H 2( δ 2 C + σ H x, ϕ 0) = 0) = C − | ( F 5 2( σ 2 5 ¯ { ∂ x, x, | C 5 5 ( ∂ ( 3 2 ¯ ∂ ¯ c C ∂ ¯ h σ ), | 2 3 4 r σ + 2 e 4 r 5 e + 2 becomes discontinuous at the boundary. − x, ϕ ∂ 5 ( − − ¯ ∂ C ¯ as is indicated by the fourth equation of eq. (6.6)) ¯ h H in terms of the non-auxiliary fields. In particular, ) T ϕ r r 0= 0= ( ¯ F H H 2 2 ǫ ¯ mH C,T ≡ − − + ) ¯ H,C, = Given the profiles of Taking into account that the VEV of It is convenient to work with parity even fields H, ∗ ∗ C x, ϕ F ( C F ¯ the parity odd eq. (7.12). Note that unlike the supersymmetric case in whic leading order of our expansion parameters is approximately A similar result wasimplies that derived the in “seed” ref. of AMSB, [7] from 4D EFT viewpoint ities inside of 7.2 Relation between compensator and stabilization After some algebra, using the first five equations in eq. (6.6) of a stabilizing field in the bulk. Using the result given by eq H These boundary values also remove all the singularities ins Solving the above equations gives rise to The constraint of the lowest component of by their equations of motionthe (see equations ref. of [24] motion for of related discu JHEP08(2008)004 5 )= ϕ (8.2) (8.1) ( δ (7.14) +1 n 2 )) ϕ ( ǫ ( and/or hidden fields, 0 t been previously done. e field theory below the low. This is the content n fields. The danger is in C t the organizing power of this. Therefore it makes remarkable therefore that and its highly constrained abilization scales. We are choice of couplings to the gligible corrections, leading imal transparency but have . rained enough that a rather lus. We will show that this 4D multiplets are the visible primed bulk fields induced on 1 that are communicating SUSY y fields, including in particular ional or gauge fields one might − cal pattern of AMSB. That other erms. This visible SUSY breaking ∗ ∗ π 0 , J J Λ ln ≫ 0 Q X, V m e ∗ πr ∗ ∗ c 2 radion 2 c 2 2 m – 21 – θQ 4 )) )) d ≫ ϕ ϕ ( ( ǫ ǫ Z ( ( /r 1 − − = = . If sequestering fails, that is if the prime fields in section 0 C C C F =integer) gives rise to a non-vanishing contribution while n )( ϕ ( δ n 2 6 )) ϕ ( ǫ is given by Let us begin by considering couplings which do involve hidde Recall that ( /C 6 C and will later showsense to samples study of the parameterradion effects inputs stabilization of that scale. SUSY achieve fields, breaking Below the within hidden this a sector 4D scale fields,the effectiv and the compensator 4D zero-mode, only effective light supergravit constrained by the effectivecannot 4D happen without superconformal great tensor suppression. calcu termsKähler of the form Eq. (7.13) shows us thatbreaking on-shell to it the is visible thehave sector. stabilizing expected Since fields their these couplings aregeneral to not pattern gravitat visible of fields visiblethe to SUSY SUSY be breaking breaking unconst is wouldcouplings communicated to dominately result. the via visible eq. It sector,forms (7.13) resulting is of in SUSY the very breakingof speci coupling sequestering. are subdominant is argued be 8. 4D effective theory and sequestering Here, we demonstrate sequestering atthe tree visible level, boundary namely have the onlyto supersymmetric the VEVs, dominance with of ne We AMSB would as like discussed ideally to in donot section this managed 5. in to a This do direct hasthe this, 5D no 4D analysis and for effective so max we fieldconsidering instead theory in we below this will the paper partly a compactification exploi hierarchy and of st scales, acquire non-negligible SUSY breakingvisible fields VEVs in eq. then (5.2), they clearly,must can by induce be new visible accounted soft t for in the 4D EFT by visible couplings to F 0. JHEP08(2008)004 ′ , ′ , 0) th V ¯ C , δ δ (8.5) (8.3) (8.4) , δC weight (2 will certainly . SUSY VEVs ′ X ing. Indeed X ¯ × H ′ 5 ign ss-squared of order the arbitrary coeffi- can have dimension to be chiral in these e not constrained like s in he tree level potential, . If the hidden sector X bulk vector multiplets ary restricted X . 2 | D θXδH emaining bulk fields are s are spatially separated 3 θX δ∂ 4 X / s for the 4D induced bulk 2 1 d is necessarily a composite d ) rough integrating out some |F idden boundary, the leading , giving a chiral result. That ate visible couplings to non- 2 Z | . idered Z , ) arise from couplings Kähler X X is not mediated to the visible ¯ r couplings are of the form, H , g, if such a coupling arises from | in eq. (8.3). But this coupling is ′ X 2 / D X ” to indicate the one which connects 1 δ D − ). The soft visible mass-squareds will 3 2 | θX δH 2 (Λ H SUSY VEVs d | SUSY VEVs = 0, so M 2 × ′ Z only applies to × / ′ ∼ O is given by ′ 1 ¯ , θ V ′ ¯ ′ 2 – 22 – C X δM − V d ¯ 5 C F 2 5 R | 1, θXδ ¯ 4 C | d ≡ θXδ∂ ′ + 4 Z d , 2 θX δ∂ | M 2 Z d C , | ( Z . At tree level precisely one of these bulk fields contracts wi = gauge invariants begin at dimension 2, ′ hid X S M . 6 SUSY VEVs θX δC, × 2 SUSY VEVs ′ d ¯ -dependence so that the ). This is the “going rate” for worrying about non-sequester × H has dimension at least 3, µ Z 4 5 x X (Λ is non-chiral in eq. (8.2), then there will be soft visible ma is a composite of hidden sector fields. For simplicity, we ass F are 4D chiral multiplets and since we are only interested in t where the lowest component of , because the scalar components of such vector multiplets ar ′ ′ θXδC θXδ∂ 4 4 X ∼ O ¯ X X H to bulk fields in d d δ X δM Let us look at the special features of such terms. Since bound If , . Since non-chiral Z Z ′ X D X X sector. It isbeyond the very “graviphoton” important multiplet, to thenchiral we note can that really if medi we had had further where so the compensator is unnecessary. Hidden SUSY breaking VEV By the constraint of eq. (4.5), D 4, contribute to visible SUSY breaking,matching and to spoil 5D. sequesterin Butin does 5D, it? effective couplings Sincebulk like the fields the that hidden one couple and above tocontributions visible both can at field boundaries. only tree-level arise Focussing (we on th of the deal h with loops in section 9 δH to the visible sector. Then the general set of allowed Kähle is non-chiral and one might worry that it couples to Since the SUSY VEVscients are of pure these numbers couplings. they can be subsumed into a Feynman diagram thatset ends to on their supersymmetric thesuperfields VEVs. visible defined in We sector, subsection will 5.1, while use and the the put a primed r prefix basi “ we can drop The soft visible mass-squareds from eq. (8.2) are then order cases, and coupled as is, without loss of generality we could just as well have cons does not contain hidden gauge symmetry singlets, then field and therefore be order Λ D JHEP08(2008)004 is Q (8.7) (8.6) has unit is source Q ., , compared to c . ., c . +h 2 radion to the visible sector, 2 α +h h we are within our ing sequestering, but 0 /m 2 α W 2 C m, namely the negative , so that this source of 2 ttractive AMSB solution 1 W g rmal tensor calculus, the 2 a superderivatives are sup- 1 θ ed by other positive flavor- g ctive field theorists. But a 2 heck this precondition. See ing to the 5D theory. Rather radion rm with zero weight for θ d is the standard version of the ector will feel no SUSY break- aréinvariant VEV of the 4D 2 e would have gotten soft terms , that is no breaking of seques- m ons. However, there are other types 0 d ionless Λ 0 Z plings of in classical conformal invariance. 1). C C , so as to adequately suppress its , to check. By 4D superconformal Z 7 F )+ − Q )+ ( 0 in the effective theory. Here W 3 0 0 Q,C C ( θC the suppression of visible soft mass-squareds 2 d θW – 23 – 2 Z d radion + suppresses non-sequestered contributions is the main Z m Q compared to going rate corresponds to a dimensionless + V . In this paper, we take the simple approach of ensuring 2 e Q † 2 radion V Q e 2 † /m 2 radion | 2 0 in terms of the old. Note that C θQ /m | 4 0 2 θ d 4 d Z .) Again, we will find this suppression is enough to neglect th Q/C 4 Z = is Λ is very small, numerically of order 10 = eff ∝ Q L eff 2 | L 0 2 radion C |F /m 2 We have shown that we can neglect hidden sector fields in check Since the cutoff of 4D EFT is We see that any classically conformally invariant visible s This observation that Λ . Thus we have a “special bulk content” in this sense, althoug ′ that we have demonstrated, bysuppression Λ of at least Λ that Λ this still leaves effectivegauge visible-compensator symmetry couplings ofleading low tree energy level supergravity couplingssupergravity in of fields, the must the simply superconfo visible be sector given by to a Poinc contributions compared with AMSB byto enough the so supersymmetric to flavor retain problem. the a rights to choose thestring light theory particle construction content alongref. of [39, these the 30] lines bulk for would related as have discussion effe to of this c point. M that might arise in thethan 4D effective doing theory such upon a careful matching,pressed match we by again place powers crude ofvisible bounds: soft at extr mass-squareds least is also the suppressed cutoff by the of dimens the 4D EFT, where the new where the superpotential is exactly cubicThis in fields form to mainta manifests the compensating role of tering. But we must consider higher (super-)derivative cou superconformal weight. The corresponding more standard fo ing at tree-level from supergravity VEVs, in particular effective 4D compensator with superconformal weight (1 the going rate. (Ifof the order superderivatives were not required w of visible SUSY breaking compared to AMSB. value added by the effective theory organization of correcti 9. Bulk radiative corrections The spectrum of pure AMSBmass applied squared to MSSM of has the aconserving sleptons. serious contributions proble However, generated this by can bulk be gravity compensat loops. given by JHEP08(2008)004 (9.1) (9.3) (9.2) (10.2) (10.1) (10.3) ction into ). Thus the discusssed in . πr ) = π T − ϕ ( ith rmultiplet loops in our breaking models with a δ between the hidden and y a “KK regularization” t size and sign to correct ∗ . divergent result one would . . In the 4D effective theory 2 4 4 2 ant, so as not to reintroduce HH 4 r scalar masses was calculated mply estimate the hypermul- 4 ration between the visible and le in magnitude. For example, ∗ j Λ . ion from the 4D effective low . given by eq. (9.1) can compen- M Λ M /πr 3 Q 2 7 4 i ) 4 2 0 e size of the extra-dimension to be s sufficient that roughly [28] avor-violating, are estimated to be − 1 Λ Q πr M m πr ( 10 ij 5 2 c 2 4 ) − 4 × Λ πr M 3 10 πr 0 )+ 0 2 . m × ϕ . π m 2 6 1 ( 2 3 − 16 − e ∼ e & Hδ ∗ ∼ 4 − – 24 – πr H violating (1 2 ¯ θ − 2 2 gravity π 1 θ gravity ) ) 4 16 Λ 2 vis flavor 2 vis ∼ ) m m θ ( ( 4 2 vis 2 vis d m m ( Z dϕ Z -term [26]. x D 4 d is not canonically normalized, but we have taken its wave fun represents the supersymmetry breaking vacuum energy as was Z 0 4 H . This contribution is positive in a class of a supersymmetry The flavor-blind result of the gravity loops to visible secto In an analogous fashion, hypermultiplet loops also mediate in eq. (9.2) is replaced by its zero mode given by eq. (6.18) (w /πr hypermultiplet induced visible scalar masses, in general fl in ref. [40, 41, 26], Note that Λ dominant hidden of bulk loops that can violate flavor symmetry such as the hype model. We must ensureslepton that the masses gravity while loops hypermultipletsupersymmetric can loops be flavor are problem. of far the subdomin righ Saturating eq. (10.1) using eq. (9.1) roughly determines th visible sectors via the couplings in the action, section 5.4. The result ishidden UV-finite sectors. because of We theobtain can finite sepa in view a this purely result1 4D as theory, the but quadratically with the UV-cutoff replaced b Note that To adequately suppress flavor-violating contributions it i account. We expect this to be a conservative estimate. 10. Numerical estimates The bulk radiative corrections fromsate the the supergravity negative fields slepton mass-squaredit if is they sufficient are if comparab roughly H Modifying the highly warpedtiplet estimates loop of ref. effect [7],energy as we theory, the will with quadratically si the resulting divergent UV-cutoff contribut replaced by 1 JHEP08(2008)004 ˙ 5 = iγ (A.3) (A.2) (A.1) ≡ (10.4) (10.5) can be 5 0 , can be GeV, the J 0 a, b, . . . , 18 m πr 10 ∼ . and 1 0 − πr = 0) = 0). m , is defined as Γ 10 ϕ = 5 ience Foundation grant discussions and advice. ( disciplinary Physics and × σ 4 drew E. Blechman, David a simple way by requiring xed 7 . . M ) , ns and technical assistance. 3 for the curved 4D indices. 5 ion of the metric is . Γ ! = 1 10.2). i 0 1) − π ,..., i − 0 ˙ 5 for the flat 5D spacetime indices o eq. (10.1), that is the suppression − , (1 . , 1 1 2 3 = 0 ˙

3 , J − , − 3 ˙ 2 = = , − , 10 1 ˙ 5 ˙ L 1 , 10 − × P ˙ 1 for the above choice of parameters (Recall 0 , 4 µ,ν,... × , the 5D hypermultiplet mass, . 1 ∼ = 2 9 , which is the key to ensuring the suppression of − . πr σ 4 , γ , , – 25 – ∼ e − ) ! 5 = = 1 µ 10 0 0 2 σ ). The gamma ’ﬁve’ matrix, Γ 4 × radion 2 ~σ µ . A,B,... M 0 ¯ 1 σ − m (1+Γ , J = diag(1 ,

GeV in order to satisfy eq. (10.2) with our sequestering 2 1 1 1 ∼ − − 15 = AB = πr η 10 0 µ 10 R = ( γ m 5 for the curved 5D indices. Similarly we will use × P 2 ¯ , µ ∼ 5 − 3 and using . σ e , 2 , πr . Given eq. (10.3) and 4D Planck mass = 7 1 , 0 ) and m , ~σ 2 radion = 0 1 − /m 2 Λ = ( µ is determined by eq. (6.12) with the boundary condition, ∼ ˙ σ 3 for the flat 4D spacetime indices and σ M,N,... As was discussed in section 8, sequestering is guaranteed in A possible viable choice of parameters is: ,..., ˙ the radion to be heavy, adjusted in eq. (9.3) to satisfy the constraint given by eq. ( Given the value of factor radion mass in a 5D Planck unit is given by The expression of the radion mass is given by eq. (6.23). For fi adjusted to get the value given by eq. (10.4). This choice gives rise to hypermultiplet loop contributions to eq. (10.2)of relative eq. t (9.3) relative to eq. (9.1). Note that that Acknowledgments We would like to thank JonathanE. Bagger, Dmitry Kaplan, V. Taichiro Belyaev, Kugo, An M.S. Markus especially A. thanks Luty, GeroThe Keisuke von research Ohashi Gersdorff of for for R.S. manyNSF-PHY-0401513 and discussio and M.S. by was the supportedAstrophysics Center. Johns by Hopkins the National Theoretical Sc Inter A. Convention Throughout the paper, we will use The gamma matrices are where and We mainly follow the convention of ref. [18, 22]. The convent and the projection operaters are defined as 0 JHEP08(2008)004 . F βα η (A.7) (A.4) (A.5) (A.6) (A.8) (A.9) (A.11) (A.10) (A.12) = , αβ symmetry η ! ¯ C C . U to unphysical = = 2 . symmetry. The + plectic-Majorana ¯ n 1 ζ =2 =1 . A =2 =2 R − − α i α i γ ! A A − (1) − ζ ¯ ∗ · · · − ζ U ∗ ζ . ¯ 2 appearing in the kinetic C − C A ! = α −

γ tiplet is given by β ∗ , by the relation, C i 1 2 = ¯ ∗ d C similarly for the straight = ¯ ζ F ij A F γ − Y =1 p =1 )= =2 α i , =1 ¯ α i 2 R C C 1) . A ζ . The reality constraint of the , F F A , − − α i j i βα ( ¯ +

ζ C δ . ρ ~σ T 1 L β j · + = = ! ζ α ζ 2 ∝ A X ( ζ 2 2 ~ Y perms i , , 1 0 i ij matrix must be symmetric, or ! ǫ 2 = − 1 , =1 =1 = n = η = α α i 0 ′ i 2 = 1 =1 0 = 1. The summation convention is from the γ ≡ 1 ¯ ζ kj , appears in the mass terms of the 5D hyper- α i ∗ † ] ǫ ) α i

– 26 – A 12 n α αβ ik ǫ A , A , A ζ η = ( Y γ , ζ ! . The indices are raised or lowered according to the ! = α i β ¯ ≡ = + − . ! ≡− H H χ · · · ¯ d ζ ζ ∗ i 12 i j α + + ) 2 i α ǫ ¯ ¯ ¯ ζ ζ = = ζ ψ

Y A ij A . We parameterize the scalar ﬁelds and F-temrs as

matrix must vanish not to break γ Y ≡ = 1 =4 =3 ji ) is related to SU(2) tensor, =2 =2 ǫ η = 2 α i α i A 3 [ j = ( ¯ A A 2 L γ ψχ A ζ ,Y ij ∗ 2 = ∗ , ζ , F tenser is = ¯ + Y H n i H ǫ ! ! ,Y 1 R − A 1 A ∗ − ζ H ¯ = ¯ ··· ∗ + ζ H Y = 2 F corresponds to the physical hypermultipelt while ζ = F − A − 2 =3 1 and ζ =1

=4 become 4D Majorana fermions. The matrix = ( α i 1 =1 A ¯ α i j H H ′ γ A = ~ ζ Y F F A A 5 1 ij

Γ ζ ǫ i = = is the 5D charge conjugation matrix. We parameterize the Sym = 4 4 denotes the 5D compensator and the gauge fixing of the SU(2) i , , and C A =3 =3 C ζ α α i i A F The antisymmetrized gamma matrices are The northwest to southeast, i.e. The reality constraintterms) of is the scalar of the hypermultiplet ( The convention of the rules, where Symplectic-Majorana spinors is given by in a bulk theory is done by choosing spinors, satisfying eq. (A.8), as compensator. Another type of matrix, 4D Majorana fermions can be obtained out of terms, eqs. (B.1) ,(B.2) in the appendix B, of the 5D hypermul One notes that multiplet (see eq. (B.2) in appendix B ). The The and satisfies hermiticity isovector, e.g. The diagonal components of JHEP08(2008)004 (B.3) (B.1) (B.2) ¯ C . 5 D , ) ∗ = 1 supersym- i α M ¯ N C 5 A 5 y down to 4D su- N α β D d + 2 M∂ 2 β i 2 2 5 | A ∂ A − ¯ H ( ′ ≡ F formal gravity theory on C C 0 )) 5 i α 2 } k scalar potential given by N − | | + ) D A 0 N ∗ ∗ ¯ 2 ¯ α i i α H | j β tions of 4D N M or 4D scalars from eq. (4.7) in C H A A as well as the full details of the H 5 A 5 ¯ 2 ∗ 0 − | H F ij − ) C = D ¯ αβ 2 F 0 Y | ) 2 2 M∂ 00 − | ij 0 5 − 0 H ) since our set up has only one central A N , 2 − | gt | ∗ 0 ( Y gMt ( ¯ 3 βi ¯ C ¯ ( Y i α H H M ∂ 00 ¯ 1 4 α i F 5 A Y ∗ | A H 2 3 N A 0 M, , D F ij αβ + 1 2 5 + ) ∗ − Y + 2 ∗ βi ∂ 2 − | ¯ . N − | , 1 H − i ~ α Y to unprimed A Y 5 ¯ 5 ˜ − C ¯ H 5 · ∗ βi F + 2 ′ – 27 – H ∂ βi | ij γβ F D ¯ 5 α ¯ C i V A j α H . imMη A H M ~ b Y η ˜ + {| ∗ (instead of F γ β ij F α γ A 5 ) αβ 2 ) = = − 5 | + 2 ) ) V H 0 M 1 Y + 3 0 5 5 6 i α 5 ∇ − βi H 2 + 2 | H α A d A gt β A H ( M 5 − A ( 5 ( 5 5 d 5 ¯ −N gMt M M H 5 5 to refer to same field (see ref. [19] for similar discussion). ( D A A ∇ αβ βi 5 A ¯ ∗ F has been made.) to refer to auxiliary field that belongs to the central charge ) + 2 HY ¯ R A 5 α i ( ij αγ A 0 − 2 3 − H ) 5 A 0 A H imM ij − ij 5 0 1 gt 2 5 ( t 1 A N R V im Y ( M i α ( + D , b 2 1 ′ ∇ 2 2 m m 2 1 3 A 2 2 C i mM M α i − 5 + + − gMt → − − M F ∂ − − + 2 + + + 4 = = m = ( = = β i i α α i α i N dϕ e ˜ A A A F dϕ e 5 2 α β ) Z ) 0 ∇ 0 Z = = and the orbifolding breaks 5D superconformal gauge symmetr gMt gMt 2 ( ( bulk bulk V /Z V 1 − Since we are interested in working in an orbifolded supercon − S B. Bulk potential for 4D scalars inWe collect superspace only notation terms contributing to the scalar potential f ref. [18] (Note that where Note that instead of using vector multiplet, we used Throughout the paper, we have used charge vector multiplet. The relation of bulk action is given in the same ref. [18]. an perconformal gague symmetry, weeq. will (B.1) re-write in the termsmetry. above of We bul the follow the fields convention transforming given like in representa appendix A. JHEP08(2008)004 .

2 5 ) . c 3 iV . , i.e. (B.4) (B.7) (B.8) (B.9) (B.6) (B.5) ˜ V + . (B.10) m ). It is 1 2 1 c 5 . )+h M V + . ¯ ~ V ∗ C 3 ≡ +h 5 ¯ H ∂ 5 M α ˜ ∗ V M 2 V ˜ ) ¯ m , C W 2 is given by − 5 3 2 2 α 2 M , √ 2 ˙ ˜ 5 = iV − iY W − ˆ A 2 D C + + Σ . 5 ¯ 1 A ) (same for 1 − He i ) 5 ∂ 2 3 Y 2 ∗ V σ . | − ˙ ( 5 + , , C i ¯ ,Y or is easily determined by ≡ . ∂ H H Σ 2 2 2 ∗ ∗ | ¯ ) i C ˜ V 2 ¯ Y − − − 2 H H ) can be economically written ) + 5 ( m ,Y ∗ ∗ 3 ¯ ↔ ∂ 5 5 3 σ , 1 σ 2 H as 2 | ˙ 5 5 √ Y 2 Y 2 iV iV , e ∂ C ≡− , e H ) ∇ 1 | ∗ 5 dϕ ∗ , . + − − − = 2 2 ) = ( ∗ ∗ ¯ H + r H , V Ω ¯ ¯ − ¯ H H + 5 ( 2 5 H V 2 C C − , , ~ ) ) | Y + ∗ ∗ − 1 Γ ∗ 5 5 σ 5 5 ¯ µ )Ω ˙ 5 5 C ˙ 2 ¯ 5 a ν 5 H Y C e Σ iV iV e ∂ 5 2 H , ¯ C σ given in eq. (A.11) the , D dx − | σ A ∇ − + − i mA i mA µ 2 im + 4 2 ∗ α e | V / β ¯ +2 3 5 C C 3 + − σ dx d H ∗ ) ) (1+Γ C e e M V ( – 28 – + 2 3 3 3 3 = 4 (2 → 5 5 5 5 − 3 3 , µν − 5 5 5 −| 5 5 ≡ gives rise to one constraint, µ ≡ 5 η ↔ CC V iV iV iV iV a ∂ σ ′ in eq. (B.3) are given by (see refs. [31, 32] for related ) iV ) , appearing in eq. (B.2) was absorbed into R 2 iψ e ) 2 C i ψ η + 5 3 − − − − H 1 2 V e + 4 = 2( 2 5 5 5 5 ) F R | D M − = ∗ ∂ ∂ ∂ ∂ = σ i α σ r , ( ) + 2 ¯ Y 2 T HY | 2 2 A H , e σ , e 43 + 2 α β 4 = ( = ( = ( = ( + σ/ ds H η + M e d iY 2 ζ ¯ T ¯ β i σ C C mH 6 Ω , e 2 H H ( A 3 = 5 5 + 5 2 = 5 5 A ( − 2 1 σ/ ′ 1 e D D σ/ D D θ e σ +(2 C iA ≡ 34 e 2 2 Y r r η 2 ( d 2 − ˙ σ σ H, e , 5 5 θe 4 4 ( e A 4 µ Z e e σ 2 d M A 2 ˙ 5 5 ( / + + + − Z e 3 σ e e → → → dϕ denotes the inner product of isovectors Z Σ V H i . Using the choice for the matrix − ~ Y 2 · m = − ” and the 5D curvature term ~ e Y On the 5D warped spacetime metric, → bulk V − The covariant derivatives are given by The convenient to complexify parity even components The warp factor-dependence inobserving a that basis under with the the Weyl transformation, unit warp fact The matrix elements, discussions) the fields transform like [31] The effective potential from thein bulk action compact given flat by (global) eq. 4D (B.3 superspace notation, the “ One notes the equation of motion of m η JHEP08(2008)004 ), , ¯ ¯ H , H F (B.13) (B.12) (B.11) σ B 557 e 2 and ]; θ + H ¯ , H , , = . ¯ H H Nucl. Phys. C ) ¯ ) H 2 , ¯ 5 C ) 2 5 , ) 2 ¯ terms of B.10) has the explicit 5 H . 2 iV 5 iV 5 H given by eq. (B.10) with iV hep-ph/9906296 F , + iV F + ]; 5 σ 1 . (B.1) and (B.3), so that + 5 1 e + 5 imA 1 V 2 5 i V 1 V ( 5 θ y reproduce all the component i 2 ( V + i V ( + i ( Gaugino mass without singlets + ¯ i , H + H ) 5 + ≡− H + C ∂ (1999) 015 [ = ¯ F T H M ¯ F Realistic anomaly-mediated C = σ F F F e 08 5 ( 5 ¯ H ˙ 5 A 5 , 5 A 5 ,H ∂ hep-ph/9905390 D , A 5 i ¯ A . ∇ i C 3 M − M A ) i JHEP F i M √ 2 3 M + | σ i terms and other 5D fields as , + 2 M e Y ¯ – 29 – r + H 2 r + 2 σ/ F | H , ]. θ r − 5 r σ 2 Heavy thresholds, slepton masses and the mu term in i i e r θ e + / i ¯ (2000) 001 [ + i ) was made (same for 5D C + ≡ ≡ = ∗ imA + ¯ ∗ + 1 + = H α ¯ 04 ∗ which means that we have not yet gauge-fixed the scale ¯ D H 2 ∗ C ¯ − ˜ | C σ W σ σ Out of this world supersymmetry breaking ¯ H, C ( 5 H H 5 ]. | σH , T σC ∂ 5 2 ∂ 5 5 2 1 ∂ T , e 3 2 3 2 JHEP √ ∂ ∂ / ) and F , 3 2 3 2 = − , σ − symmetry. hep-ph/9810442 D e ) ∗ + 1 → + ∗ + σ 2 M H ¯ ¯ 2 2 U ∗ 5 D θ ∗ C H | ) e 5 5 σ ¯ ( ¯ ∇ H C , C 2 + ∂ ∂ H ˙ 5 5 5 e ∂ C ∂ − ∂ − T 3 2 − | H, F ¯ θ 2 √ σ 2 | ( hep-th/9810155 e r r r r 1 1 1 1 (1998) 027 [ θ 2 2 C Σ= ¯ θ θ i − 2 = = = = 12 2 −| θ + ¯ M ¯ C C H H − 1 2 M C terms are defined in terms of 5D F F F F = = = = 2( = F JHEP (1999) 79 [ supersymmetry breaking E. Katz, Y. Shadmianomaly and mediated Y. supersymmetry Shirman, breaking 2 By performing superspace integration of the bulk potential ˜ C ˜ V T V A [2] G.F. Giudice, M.A. Luty, H. Murayama and R. Rattazzi, [1] L. Randall and R. Sundrum, [3] Z. Chacko, M.A. Luty, I. Maksymyk and E. Ponton, invariance and SU(2) References the expressions given by eq.terms (B.11) and in (B.12), one eq. candependence (B.3). easil of dilaton Note that the bulk potential given by eq. ( compared to originalphysical component hypermultiplet bulk has a scalar canonical potential normalization. of eqs Note that the ( in terms of the “fake” flat superspace multiplets, The covariant derivatives are defined as The JHEP08(2008)004 ]; D , 05 ng , Phys. 03 Nucl. Phys. , , , 05 Phys. , sions JHEP JHEP , Phys. Rev. , (1978) 3179; JHEP (2000) 048 , ]. , 09 supergravity hep-ph/0003187 ]; D 17 (1985) 235. (2008) 105020 =1 JHEP 73 N , D 77 Viable ultraviolet-insensitive ]; (2001) 164 [ ]; Phys. Rev. hep-ph/0112210 ]. , hep-ph/0006116 B 592 Simplifications of conformal Phys. Rev. Poincaréand conformal supergravity (1981) 189. Superconformal unified field theory , ]. 68 Prog. Theor. Phys. (1978) 430. hep-ph/0012103 , (2002) 231802 [ hep-ph/9908240 (2000) 151 [ hep-th/9910202 Nucl. Phys. (1979) 3592. 88 Cosmological constants as messenger between , – 30 – RG-invariant sum rule in a generalization of B 138 ]; Anomalies, topological invariants and the Gauss-Bonnet Phys. Rept. B 491 D 19 , ]. (2001) 041 [ hep-th/0312148 (1999) 1181 [ 02 superconformal tensor calculus: multiplets with external (1979) 3166; ona´sprrvt asPoincarésupergravity broken superconformal gravity Gauge/anomaly Syzygy and generalized brane world models of Gaugino-assisted anomaly mediation R-parity violating anomaly mediated supersymmetry breaki Sparticle masses from the superconformal anomaly Hierarchy stabilization in warped supersymmetry Radius stabilization and anomaly-mediated supersymmetry ]. ]. ]; Nucl. Phys. 102 =1 (2000) 035008 [ Realistic anomaly mediation with bulk gauge fields Gauge theory of the conformal and superconformal group , R-symmetry, Yukawa textures and anomaly mediated Comments on quantum effects in supergravity theories JHEP Phys. Lett. Phys. Rev. Lett. Conformal and Poincarétensor calculi in Supersymmetric radius stabilization in warped extra dimen D 19 Phys. Rev. N , , , Supergravity Properties of conformal supergravity , hep-ph/0003222 D 62 ]. hep-th/0012158 (2004) 025002 [ ]; On anomaly mediated SUSY breaking (1977) 1109; (1983) 49; Phys. Rev. hep-th/0701023 hep-ph/9903448 hep-ph/0112172 , (1977) 304; (1978) 54; D 70 39 (2000) 017 [ Phys. Rev. , Prog. Theor. Phys. 06 , B 226 B 69 B 76 (2001) 065012 [ hep-ph/0009195 arXiv:0801.0578 Lett. (2007) 040 [ Phys. Rev. M. Kaku and P.K. Townsend, K.I. Izawa, Y. Nomura andbranes T. Yanagida, models with closed algebras M. Kaku and P. Van Nieuwenhuizen, theorem in supergravity Rev. Lett. Lett. supergravity M. Kaku, P.K. Townsend and P. Van Nieuwenhuizen, [ [ N. Arkani-Hamed, D.E. Kaplan, H. Murayama and Y. Nomura, supersymmetry breaking I. Jack and D.R.T. Jones, supersymmetry breaking (1999) 013 [ D.E. Kaplan and G.D. Kribs, A.E. Nelson and N.J.supersymmetry Weiner, breaking Z. Chacko and M.A. Luty, Lorentz indices and spinor derivative operators Phys. M. Carena, K. Huituanomaly and T. mediated SUSY Kobayashi, breaking models B.C. Allanach and A. Dedes, JHEP (2002) 047 [ 64 breaking [8] M. Dine and N. Seiberg, [9] S.P. de Alwis, [7] N. Maru and N. Okada, [6] M.A. Luty and R. Sundrum, [5] M.A. Luty and R. Sundrum, [4] A. Pomarol and R. Rattazzi, [11] S. Ferrara, M.T. Grisaru and P. van Nieuwenhuizen, [10] M. Kaku, P.K. Towsend and P. Van Nieuwenhuizen, [12] T. Kugo and S. Uehara, [13] P. Van Nieuwenhuizen, JHEP08(2008)004 , 104 D 83 ]; N s, Phys. , ]. (2000) Adv. JHEP Prog. , (2007) , , Large- 08 ]. (2000) 267 10 Phys. Rev. Prog. Theor. , (1998) 253 Prog. Theor. JHEP , , , 2 B 570 JHEP Phys. Rev. Lett. D hep-th/9711200 , 5 , ravity ic brane world Prog. Theor. Phys. hep-th/9905111 , ]. D (2001) 056003 6 oguri and Y. Oz, hep-ph/0505126 Exponentially small ]; Nucl. Phys. ]. from , D 63 er, (1999) 1113] [ D (2000) 183 [ 5 38 ]. (2005) 260 [ 323 matter with and without actions hep-th/0403009 Gauge theory correlators from non-critical Adv. Theor. Math. Phys. Phys. Rev. ]. =5 , , B 624 hep-th/9802109 D D-type supersymmetry breaking and ]. Sequestering in string theory hep-th/0111231 , supergravity coupled to matter-Yang-Mills system – 31 – Phys. Rept. ]. ]. =2 , =5 N hep-ph/0010288 ]. Off-shell formulation of supergravity on orbifold (2004) 086002 [ off-shell supergravity in five dimensions Modulus stabilization with bulk fields D Int. J. Theor. Phys. Phys. Lett. (1998) 105 [ , ]. Radius stabilization in a supersymmetric warped limit of superconformal field theories and supergravity =2 hep-th/0106051 (2003) 899. ]. Supersymmetry breaking and composite extra dimensions ]. D 70 (2003) 045007 [ N N Anomaly mediated supersymmetry breaking in four dimension hep-th/0105137 51 Superconformal tensor calculus in five dimensions B 428 Superconformal tensor calculus on orbifold in Off-shell Supergravity tensor calculus in (2001) 323 [ hep-th/0203276 hep-th/0104130 (1998) 231 [ ]. ]. D 67 Superconformal 2 105 Gauged hep-th/0009083 (2001) 671 [ ]. ]; ]; The large- Phys. Rev. , hep-th/0205230 Phys. Lett. 106 hep-th/9907447 , hep-ph/0006231 Supersymmetric brane world scenarios from off-shell superg Off-shell supergravity in five-dimensions and supersymmetr Minimal off-shell supergravity in five dimensions (2002) 066004 [ Anti-de Sitter space and holography (2002) 203 [ (2001) 221 [ Fortschr. Phys. Phys. Rev. , , 108 106 D 65 hep-th/0703105 hep-th/9909144 (2002) 045 [ (2001) 024024 [ hep-ph/9911421 hep-th/9802150 hep-th/9907082 013 [ naturally supersymmetry breaking from extra dimensions [ compactification 10 Phys. scenarios 64 [ For a review see O. Aharony, S.S. Gubser, J.M. Maldacena, H. O (1999) 4922 [ Prog. Theor. Phys. field theories, string theory and gravity brane-to-brane gravity mediation Theor. Phys. Rev. Theor. Math. Phys. 016 [ [ S.S. Gubser, I.R. Klebanov andstring A.M. theory Polyakov, E. Witten, Phys. (2000) 835 [ [25] W.D. Goldberger and M.B. Wise, [30] S. Kachru, L. McAllister and R. Sundrum, [23] N. Arkani-Hamed, L.J. Hall, D. Tucker-Smith and N. Wein [21] M. Zucker, [24] M. Eto, N. Maru and N. Sakai, [22] T. Kugo and K. Ohashi, [20] E. Bergshoeff et al., [28] M.A. Luty and R. Sundrum, [26] T. Gregoire, R. Rattazzi and C.A. Scrucca, [29] M. Luty and R. Sundrum, [18] T. Kugo and K. Ohashi, [27] J.M. Maldacena, [19] T. Fujita, T. Kugo and K. Ohashi, [17] M. Zucker, [16] M. Zucker, [15] T. Fujita and K. Ohashi, [14] T. Kugo and K. Ohashi, JHEP08(2008)004 ]. D =1 4 , (2005) N Phys. D Phys. , ]. 5 , try D D 71 5 ]. hep-th/0602173 Phys. Rev. , hep-th/0408138 ]. action ]. superfield reduction of (2006) 222 [ ]. D D 5 Superfield approach to hep-th/0305184 , ,4 supergravity on general warped (2005) 141 [ Brane world SUSY breaking The road to no scale supergravity B 751 D 5 ]. hep-th/0012163 Higher dimensional supersymmetry in ]. B 709 Supersymmetric radion in the ]. Brane world SUSY breaking from string/M (2003) 171 [ hep-th/0106256 ]; Brane to brane gravity mediation of A very light gravitino in a no scale model hep-th/0305169 Nucl. Phys. – 32 – , (2001) 057 [ B 674 Nucl. Phys. 08 , hep-th/0101233 hep-th/0408224 hep-th/0201256 (2001) 105025 [ JHEP (2004) 025008 [ Supersymmetric theories with compact extra dimensions in hep-th/0111235 Superfield description of Dynamical radion superfield in , ]. Nucl. Phys. , D 64 Supergravity loop contributions to brane world supersymme (2002) 055 [ D 70 (2004) 013 [ 03 (2002) 036 [ 10 (1987) 1. 03 (1984) 99; 145 JHEP Phys. Rev. (2002) 105011 [ , JHEP , Phys. Rev. , hep-th/0501183 , JHEP , B 147 D 65 superspace 105010 [ Randall-Sundrum scenario orbifold SUGRA and heterotic M-theory superfields breaking conformal SUGRA and the radion theory Lett. For a review see A.B.Phys. Lahanas Rept. and D.V. Nanopoulos, geometry Rev. supersymmetry breaking [35] D. Marti and A. Pomarol, [37] J. Bagger, D. Nemeschansky and R.-J. Zhang, [34] N. Arkani-Hamed, T. Gregoire and J.G. Wacker, [31] F. Paccetti Correia, M.G. Schmidt and Z. Tavartkiladze [33] F. Paccetti Correia, M.G. Schmidt and Z. Tavartkiladze [38] J. Ellis, K. Enqvist and D.V. Nanopoulos, [41] I.L. Buchbinder et al., [36] H. Abe and Y. Sakamura, [32] H. Abe and Y. Sakamura, [39] A. Anisimov, M. Dine, M. Graesser and S.D. Thomas, [40] R. Rattazzi, C.A. Scrucca and A. Strumia,