A slice of AdS_5 as the large N limit of Seiberg duality
aa r X i v : . [ h e p - t h ] O c t IPPP/10/86 ; DCPT/10/172CERN-PH-TH/2010-234
A slice of AdS as the large N limitof Seiberg duality Steven Abel a,b, and Tony Gherghetta c, a Institute for Particle Physics PhenomenologyDurham University, DH1 3LE, UK b Theory Division, CERN, CH-1211 Geneva 23, Switzerland c School of Physics, University of MelbourneVictoria 3010, Australia
Abstract
A slice of AdS is used to provide a 5D gravitational description of 4D strongly-coupledSeiberg dual gauge theories. An (electric) SU ( N ) gauge theory in the conformalwindow at large N is described by the 5D bulk, while its weakly coupled (magnetic)dual is confined to the IR brane. This framework can be used to construct an N = 1 MSSM on the IR brane, reminiscent of the original Randall-Sundrum model. Inaddition, we use our framework to study strongly-coupled scenarios of supersymmetrybreaking mediated by gauge forces. This leads to a unified scenario that connects theextra-ordinary gauge mediation limit to the gaugino mediation limit in warped space. [email protected] [email protected] Introduction
Seiberg duality [1, 2] is a powerful tool for studying strong dynamics, enabling calcu-lable approaches to otherwise intractable questions, as for example in the discovery ofmetastable minima in the free magnetic phase of supersymmetric quantum chromodynam-ics (SQCD) [3] (ISS). Despite such successes, the phenomenological applications of Seibergduality are somewhat limited, simply by the relatively small number of known examples.An arguably more flexible tool for thinking about strong coupling is the gauge/gravitycorrespondence, namely the existence of gravitational duals for strongly-coupled gaugetheories [4–6]. Such duals are known in certain cases to be equivalent to a cascade ofSeiberg dualities, the prototypical example being an
SU( N + M ) × SU( N ) theory [7, 8],which describes the moduli space of a theory of branes and M fractional branes on thewarped-deformed conifold (see [9, 10] for reviews). Warped geometries are indeed gener-ally characteristic of compactifications with flux [11]. Metastable ISS-like supersymmetrybreaking can be implemented by having various different brane configurations at the endof such throat-like geometries, as in for example [12–15], and can be successfully mediatedas in [16–18].It is natural to suppose that these and similar examples can be approximated by the“slice-of-AdS” Randall-Sundrum scenario [19] (RS1). The RS1 formulation provides anideal framework for discussing the effects of strong coupling without having to use thefull complexity of string theory. Indeed the phenomenology is typically dominated by thelow energy modes, and the throat itself enters mainly through the Kaluza-Klein (KK)spectrum. Moreover the latter tends to be linear, so that certain phenomena can berather universal. RS1 can therefore be a useful approximation because precise knowledgeof the gravitational dual is often unnecessary, aside from its general scale and warping.One example, again in the context of supersymmetry breaking, is gaugino mediation [20–29]. There supersymmetry breaking is mediated to scalars localized on the UV brane viabulk gauge modes. However, the resulting suppression of the scalar masses is relativelyindependent of whether the bulk is flat or warped, and depends only on the separation inthe approximately linear KK spectrum.On the other hand there is an obvious drawback of RS1: it is unclear whether thestrongly coupled 4D system that is supposed to correspond to a given configuration actuallyexists. Because of this it seems interesting to study strongly coupled 4D field theories thathave an RS1-like configuration, namely a period of conformal running with a well behavedand calculable perturbative theory in the infra-red. In particular, we would like to begin with simple N = 1
4D dynamics that has certain properties (such as supersymmetrybreaking), and by taking a strong coupling limit introduce desirable features of extradimensional physics (such as gaugino mediation).1his paper presents a straightforward method based on Seiberg duality by which suchmodels can be constructed. It works as follows. Suppose one is interested in reproducingan RS1 configuration that has particular weakly coupled theories located on the IR brane,and the UV brane. The holographic principle tells us that the UV theory should exist aselementary degrees of freedom in the 4D theory whereas the IR theory should be composite.Hence the IR theory (including any gauge degrees of freedom) can be identified as the free-magnetic dual of an electric/magnetic pair. Assuming for definiteness that the theory is N = 1 , SU ( n ) and vector-like, then there should be F > n fundamental flavours in the IRtheory for it to be IR-free. This theory becomes strongly coupled at some scale Λ IR , abovewhich an electric theory takes over. Many different (indeed an infinity of) electric theoriesflow to this particular magnetic theory. The canonical one is an asymptotically free N = 1 SU ( N ) theory, where N = F − n . However, consider instead an SU ( N ) electric theory with F = ∆ F + F flavours, of which ∆ F have mass Λ IR , with the rest being massless. If wechoose N < F < N , then this theory is in the conformal window. Above the mass-scale Λ IR the theory flows to a fixed point and can enjoy an arbitrarily long period of conformalrunning. At the scale Λ IR however, we integrate out the ∆ F heavy flavours, and the theoryflows to the same weakly coupled magnetic theory in the IR. Moreover the SU ( N ) theoryapproaches strong coupling as F → N . By taking a large N limit and adding massiveand massless flavours to keep the ratio N/F fixed (the Veneziano limit [30]), one canarrange for the conformal fixed point to be at arbitrarily strong coupling. In this limit, wereplace the period of conformal running with a bulk gravitational dual, and arrive at anRS1 configuration, with the global flavour symmetries becoming bulk gauge symmetries,the magnetic theory confined to the IR brane, and any elementary degrees of freedom towhich the theory weakly couples, becoming degrees of freedom on the UV brane. In asense this is simply the shortest possible cascade, in which strong and weak coupling areconnected by a single Seiberg duality.Our bulk is required to be a slice of the gravitational dual of strongly coupled SQCD,whatever that may be. Since the N = 1 conformal theory has an R -symmetry, a goodcandidate for the geometry is six dimensional, namely AdS × S with the R -symmetrycorresponding to translations along the circle, S . A solution of this type for general N, F was constructed in Ref. [31] using the effective action of six-dimensional non-criticalsuperstrings. This specific background can be realized with a stack of D3 branes at the tipof the SL (2 , R ) / U(1) cigar, with the flavours of fundamental and antifundamental beingprovided by space-time filling uncharged D5 branes.It should be noted that this background is not under good theoretical control. Itscurvature is large so the solution will get order one corrections from higher-derivative termsin the effective action. Indeed as argued in Ref. [32] and also in Section 3.3, it is likely thatthere can never be a weakly curved gravitational dual of SQCD in the conformal window.Nevertheless, it is also likely that the effect of the large curvature will be to change theparameters of the solution in Ref. [31] while leaving its general properties intact [33, 34].2s we have said, for phenomenological purposes these general properties can often besufficient.We will present in this paper two applications that illustrate the approach. Followinga summary of the Renormalisation Group (RG) properties of SQCD in the next section,and more details of the configuration in Section 3, we will construct an N = 1 MSSM-on-the-IR-brane type of model, reminiscent of the original RS1 configuration. We beginwith the basic MSSM and find the electric theory by identifying
SU(2) L with Sp(1) , andthen performing an
Sp( N ) type Seiberg duality. In order to be able to put the electrictheory in the conformal window and then take the large N limit, the only modificationwe need to make to the spectrum of the usual MSSM is to include an arbitrary numberof heavy Higgs pairs. The left-handed states are all identified as composite states, as arethe gauge degrees of freedom of SU(2) L . The right-handed fields on the other hand can bean arbitrary mixture of elementary and composite states (with the latter being identifiedas the “mesons” of the Seiberg duality). The remaining SU(3) c × U(1) Y gauge degrees offreedom are bulk modes.The second application is to gauge mediation of supersymmetry breaking. There hasbeen recent interest in 4D models that can reproduce the phenomenology associated withgaugino mediation, namely gaugino masses that are much heavier than scalar masses atthe mediation scale [20–25]. While this set-up is simple to configure in extra-dimensionalmodels, the underlying supersymmetry breaking has to be added by hand. The recentinterest has therefore been in finding models in which the dynamics of supersymmetrybreaking is included [26–29]. Here we begin with the “simplified gauge mediation” scenarioof Murayama and Nomura [35]. We show that taking the large N limit of this theory asdescribed above yields a gaugino mediation model in AdS, with metastable supersymme-try breaking on the IR-brane, matter and messenger fields on the UV brane and gaugefields in the bulk, a scenario that has been considered in a number of phenomenologicalapplications [36, 37]. As in conventional gaugino mediation, the scalar mass squareds aregenerated by (5D) one loop diagrams that are relatively suppressed due to the bulk sepa-ration of matter fields and supersymmetry breaking. We find that (in a certain limit) thesuppression is of the form m i ∼ M λ /b CFT where b CFT is the bulk contribution to the betafunction coefficient, which effectively counts the messenger content of the strongly coupledCFT. This is the strong coupling version of the usual m i ∼ M λ /N mess relation that onefinds in perturbative extra-ordinary gauge mediation [38]. From this “extra-ordinary gaugemediation limit”, we will show how one can go continuously to the opposite extreme ofsupersymmetry breaking in AdS by twisted boundary conditions [23], which we refer to asthe “gaugino mediation limit”. 3 U( N ) SU ( F ) L SU ( F ) R U(1) B U(1) R Q (cid:3) (cid:3) N − NF ˜ Q (cid:3) (cid:3) − N − NF Table 1:
Spectrum and anomaly free charges in
SQCD . We begin by discussing the possible UV completions of SQCD from the Seiberg dualitypoint of view. It is useful to study the flow down from high energies. In the far UV weassume a standard N = 1 supersymmetric QCD theory which we refer to as the “electrictheory”. This is an SU ( N ) theory with F flavours [1,2]. With no superpotential this theoryhas a global SU ( F ) L × SU ( F ) R × U(1) B × U(1) R symmetry. These global symmetries areanomaly free with respect to the gauge symmetry. There is also an anomalous U(1) A symmetry that will be irrelevant for our discussion. The particle content is shown inTable 1.Although the features of the RG flow of these theories will be well known to manyreaders, it is for clarity worth recapping those elements that we need for our discussion .The coupling runs according to the exact NSVZ β -function [39] β π g = 3 N − F (1 − γ Q )1 − Ng π (1)where γ Q is the anomalous dimension of the quarks, given at leading order by γ Q = − g π (cid:18) N − N (cid:19) . (2)Assume that the high energy theory begins in the so-called conformal window, N < F < N . (3)The theory then runs to a fixed point as can be most easily seen when it is just inside theconformal window, F = 3( N − ν ) in the limit where ≤ ν ≪ N . Solving for the vanishingof the beta function gives the anomalous dimension of the quarks to be order ν and a fixedpoint at g ∗ = 8 π N − ν . (4) The behaviour of this theory under RG flow has been described in many excellent texts, for exampleRef. [10]. U( n ) SU ( F ) L SU ( F ) R U(1) B U(1) R q (cid:3) (cid:3) n − nF ˜ q (cid:3) (cid:3) − n − nF ϕ (cid:3) (cid:3) nF Table 2:
Spectrum and anomaly free charges in
SQCDDecreasing the number of flavours increases the value of the coupling at the fixed point,until in the opposite limit, where F = ( N + ν ) , it approaches strong coupling: solving forthe vanishing of the beta function there, one finds that γ Q = − νN , (5)and therefore by Eq.(2) we have g ∗ N & π . This result can also be derived from the factthat at a conformal fixed point the R -charges and dimensions of operators O are relatedas dim O = R O : this relation and the definition dim Q = 1 + γ Q give γ Q (cid:18) − NF (cid:19) , (6)the same result as that deduced from the vanishing of the NSVZ beta function in Eq.(1)The theory at the IR fixed point has an equivalent magnetic description. The magneticdual theory (which we will call SQCD) has a gauged SU( n ) symmetry, where n = F − N [1, 2]. Its spectrum is given in Table 2. (Throughout we will denote magnetic superfieldswith small letters and electric superfields with capitals.) The two theories satisfy all theusual tests of anomaly and baryon matching if one adds a superpotential W mag = h qϕ ˜ q. (7)The equation of motion of the elementary meson, ϕ then projects the superfluous compositemeson q ˜ q out of the moduli space of the magnetic theory. Obviously the magnetic theoryis also inside the conformal window, but where the electric description is strongly coupledthe magnetic one is weakly coupled, and vice-versa.Consider the theory with F = ( N + ν ) with N ≫ ν . In the absence of the coupling h the SQCD theory would clearly be no different from the original SQCD one; the magneticdual has F = 3( n − ν ) and hence a fixed point at h ∗ = 0¯ g ∗ = 8 π n − ν . (8) Although the anomalous dimension in Eq.(2) is accurate to one loop, the NSVZ beta function in Eq.(1)is exact. Hence even though the precise form of the anomalous dimension γ Q is unknown, as long as it canapproach arbitrarily close to − , a fixed point will be found. not however the fixed point corresponding to that of the electric theory and indeedany non-zero coupling h precipitates flow to a new fixed point. The dim O = R O argumentpredicts it to be where γ q = 1 − nF ; γ ϕ = − nF . (9)Note that the beta function of the coupling h is β h = h ( γ ϕ + 2 γ q ) which indeed vanishesat this point. The anomalous dimensions themselves are γ q = 116 π (cid:18) F h − g (cid:18) n − n (cid:19)(cid:19) = 1 − nFγ ϕ = 116 π (cid:0) nh (cid:1) = − nF , (10)giving n ( h ′∗ ) π = 12 νF (¯ g ′∗ ) = n ( n + 2 F )4( n −
1) ( h ′∗ ) . (11)When F = 3( n − ν ) and n ≫ ν both fixed points in the magnetic description are atweak coupling. And in the opposite limit where F = ( n + ν ) , both fixed points are atstrong coupling with nh ∗ & π . A numerically evolved example of such flow is shown inFigure 1a: the solid and dashed lines are the flows in the electric and magnetic theoriesdescribed above. The magnetic theory is indeed seen to flow initially towards the unstablefixed point with h ∗ = 0 , g ∗ , before ending up at the stable fixed point h ′∗ , g ′∗ . The fixedpoint values of the couplings indeed obey the above relations.Now let us consider what happens when we add a relevant deformation to the electricmodel – i.e. a new term in the superpotential. If this new term breaks the remaining R -symmetry the theory will flow, either to a new fixed point, or to an IR-free or asymptoticallyfree theory. The simplest example of such a deformation is a quark mass term in the electrictheory: W elec = ( m Q ) ji Q i ˜ Q j , (12)where i, j are flavour indices. Consider the case where m Q is diagonal of the form m Q = m F × F F × F F × F F × F ! , where F + F = F , and where this value is set at some UV scale, Λ UV . This term thenexplicitly breaks both the R -symmetry and the global symmetry down toSU ( F ) L × SU ( F ) R × U(1) B × U(1) R → SU( F ) D × SU( F ) L × SU( F ) R × U(1) B . (13)6t low scales one can integrate out F quark/antiquark pairs, leaving F light flavours.The global symmetry is further broken, but a (different) R -symmetry is recovered: SU( F ) D × SU( F ) L × SU( F ) R × U(1) B → SU( F ) L × SU( F ) R × U(1) B × U(1) R ′ . (14)Thus provided that F > N the theory remains in the conformal window and flows to anew fixed point at stronger coupling than the previous one, corresponding to F flavoursand N colours; this is the solid line in Figure 1b. The scale at which one can integrate outthe heavy quarks is not simply m because the quarks have a dimension (1 − N/F ) whichis different from unity. One finds that the mass of the canonically normalised quarks islarger than the energy scale, when the latter drops below a value Λ IR given by Λ IR = m (cid:18) Λ IR Λ UV (cid:19) − ∆ ϕ , (15)where ∆ ϕ = (1 − N/F ) is the dimension of Q ˜ Q . (Note that at weak coupling when ∆ ϕ =2 + γ ϕ this gives the usual perturbative approximation Λ IR = m (1 − γ ϕ log(Λ IR / Λ UV )) .)(a) (b) (c) - t Λ Π - t Λ Π - t Λ Π Figure 1:
Types of SQCD RG flow with t = log E . The solid and dashed lines are theelectric and magnetic theories and λ = g N or ¯ g n respectively. The undeformed theoriesflow to their conformal fixed points in (a). Upon adding a mass deformation the theoriesflow to new fixed points as in (b) or to IR-free theories as in (c). The magnetic theory wasstarted at small coupling in order to show its evolution towards the unstable fixed point first h ∗ = 0 , g ∗ before ending up at the stable fixed point h ′∗ , g ′∗ . The corresponding deformation in the magnetic theory is a linear meson term thatinduces a Higgsing; the magnetic superpotential becomes W mag = h qϕ ˜ q − hµ ϕ ϕ , (16)where the linear ϕ term corresponds to the quark mass term and has the same R -chargeand flavour structure: µ ϕ = µ F × F F × F F × F F × F ! . F of the quarks to acquire VEVs of the form h q ˜ q i = µ ϕ . (17)The gauge symmetry is Higgsed down to SU( n ) where n = n − F = F − N , and F quarks remain light. (Note that we assume F ≤ n .) One can arrange for the changein the number of degrees of freedom to be at the same energy scale in both theories byappropriately choosing µ . This behaviour is shown in Figure 1b.If one chooses F such that N + 1 < F < N then the behaviour is different again(c.f. Figure 1c). At scales below Λ IR the theory falls out of the conformal window (so tospeak) and into the free magnetic range. The gauge coupling of the electric theory hitsa Landau pole below Λ IR and can be matched onto an IR free magnetic description. Onthe other hand the original magnetic theory flows smoothly (without encountering strongcoupling) to the same IR free theory. This is classic quasi-conformal (walking) behaviour(see Ref. [40] for a recent summary).Note that upon adding the mass deformation the electric and magnetic theories describethe same physics at scales much higher than Λ IR and also in the far IR well below it. Around Λ IR however we see that they are very different. Thus it is important to realise that theseare two physically distinct types of UV completion of a single IR free theory. The firstconsists of a Landau pole followed by a relatively strongly coupled conformal UV phase.The second consists of a smooth transition to a weakly coupled conformal theory in theUV. A third possible type of UV completion would of course be the asymptotically freeelectric Seiberg dual of the IR free theory. One of the three possibilities for the UV completion of the deformed SQCD (with a quarkmass term in the electric theory) is a relatively strongly coupled conformal phase in thelimit of large N . By the AdS/CFT correspondence [4], this phase admits a weakly-coupled5D gravitational description. Furthermore given that the 4D deformed SQCD is onlyconformal below a UV scale, with conformal invariance broken by a mass deformation inthe IR, the dual description must be a slice of AdS . A simple way to mimic these featuresis to introduce a UV brane and an IR brane corresponding to an RS1 scenario [19]. This isa crude approximation to the underlying dynamics of the microscopic theory. Neverthelessas stressed in the Introduction, many of the qualitative features can be understood inthis simplified framework, just as AdS/QCD models are thought to encapsulate certainfeatures of QCD. In this subsection we wish first to sketch out the general features that8uch a description must have, based on known attempts in the literature to constructgravitational duals of strongly coupled SQCD. We then go on to consider the possible 5Dconfigurations that correspond to strongly coupled 4D Seiberg duals.First the general set-up. We take a slice of AdS and place a UV brane at Λ UV and anIR brane at a confinement scale loosely associated with Λ IR (we will discuss this point inmore detail below). In order to accommodate the intrinsic U(1) R symmetry, the dual 5Ddescription must actually be AdS × U(1) . A solution of this type in the conformal windowfor general
N, F was constructed in Ref. [31] from the effective action of six-dimensionalnon-critical superstrings. Even though this solution is subject to order one corrections fromhigher-derivative terms in the effective action, it does provide the required background forthe strongly-coupled deformed SQCD. In the string frame the effective 5D coupling is [31] g = 34 k N F, (18)where k is the AdS curvature scale, so that in the limit N, F ≫ , with the ratio N/F fixed, the theory becomes weakly coupled. The AdS curvature scale is of order the stringscale and in the conformal window the radius of the circle S satisfies < kR S < .As a first approximation we can ignore the S since its radius remains approximatelyconstant, and consider the 5D spacetime x M = ( x µ , z ) with the fifth dimension z com-pactified on an S /Z interval with 3-branes located at z UV = k − ∼ Λ − UV (the UV brane)and z IR ∼ Λ − IR (the IR brane) where z UV /z IR = Λ IR / Λ UV . The AdS metric is written usingconformal coordinates ds = 1( kz ) ( η µν dx µ dx ν + dz ) , (19)where η µν = diag( − + ++) is the 4D Minkowski metric.The 5D field content is dictated by the AdS/CFT dictionary. In the 4D electric theory(SQCD) at strong coupling, corresponding to F = 3 / N + ν ) , each operator O ( x ) corre-sponds to a field Φ( x, z ) in the 5D bulk theory. Furthermore, global symmetries of the 4Dtheory are interpreted as local gauge symmetries in the bulk. Therefore we assume thatthe 5D theory has an SU ( F ) L × SU ( F ) R × U (1) B gauge symmetry. Note that the U (1) R symmetry is associated with the isometry of the S . We are interested in a deformationcorresponding to chiral symmetry breaking. Therefore we introduce bulk fields A aLµ and A aRµ , with a = 1 . . . N corresponding to the vector currents of the quark fields, and a bifun-damental field Φ ij with i, j = 1 . . . F , corresponding to the operator Q i ˜ Q j (in the sense thatthe UV VEV of Φ determines m Q ). The 5D masses m of the bulk fields are determinedby the relation, m = (∆ − p )(∆ + p − k where ∆ is the dimension of the corresponding p -form operator [5, 6]. For a vector current with ∆ = 3 this corresponds to massless 5Dvector fields. The dimensionality of the bulk field corresponding to the squark bilinear Q ˜ Q (whose dimension is − dim( Q ˜ Q ) ) can be deduced from the R -charges of Q in Table 1 to9e ∆ = 4 − − N/F ) so that < ∆ < inside the conformal window. Likewise the fieldcorresponding to the quark bilinear has ∆ = 3 − − N/F ) with < ∆ < . The scalarfield component of Φ has a 5D mass-squared m = − k (1 − N/F )(1 + 3
N/F ) whichapproaches − k at F = N . In the underlying non-critical string theory the vector fieldsrepresent 55 strings propagating on the worldvolume of the F spacetime filling D5 branes,while Φ represents the open string tachyons on the D5 branes.The 5D Lagrangian consists of two parts involving a bulk N = 2 vector supermultiplet ( V, χ ) containing an N = 1 vector supermultiplet, V and chiral supermultiplet χ , and an N = 2 hypermultiplet (Φ , Φ c ) , containing N = 1 chiral supermultiplets Φ , Φ c . It is givenby [41–44] S = Z d x (Z d θ kz ) " Φ e − V Φ † + Φ c e V Φ † c + 2 g (cid:18) ∂ V − kz √ χ + χ † ) (cid:19) + Z d θ (cid:20) g W α W α + 1( kz ) h Φ c (cid:18) D z − ( 32 − c ) 1 z (cid:19) Φ+ δ ( z − z UV ) W UV + δ ( z − z IR ) W IR i + h . c . i) = Z d x √− g (cid:20) −| D Φ | − m | Φ | − g L F L − g R F R + . . . (cid:21) , (20)where D µ Φ = ∂ µ Φ − iA Lµ Φ + iA Rµ Φ with A L,R = A aL,R t a , and c is a bulk mass parameter.Note that at the massless level the orbifold breaks the N = 2 supersymmetry down to N = 1 supersymmetry. We shall choose the Φ superfield to be even under the orbifoldaction, and Φ c to be odd.Inserting the F -term equations, F ∗ = ∂ z φ c − (cid:18)
32 + c (cid:19) z φ c − δ ( z − z UV ) ∂ Φ W UV − δ ( z − z IR ) ∂ Φ W IR ,F ∗ c = − ∂ z φ + (cid:18) − c (cid:19) z φ , (21)into the equations of motion results in a set of well-known bulk solutions for the componentsof Φ at arbitrary momentum, p . The bulk solutions for the zero modes ( p = 0 ) correspondsimply to setting the F -terms to zero [41, 42, 44] φ ( z ) = φ ( z UV ) (cid:18) zz UV (cid:19) − c ,φ c ( z ) = ε ( z ) φ c ( z + UV ) (cid:18) zz UV (cid:19) + c , (22)10here ε ( z ) = sign ( z ) forces φ c to be odd and where φ c and φ are the scalar components of Φ c and Φ respectively. Since we are choosing Φ c to be odd under the orbifolding then wecan just set φ c ( z IR ) = φ c ( z UV ) = δF c ( z IR ) = δF c ( z UV ) = 0 . Demanding the vanishing of thedelta-function contributions to F ∗ gives the additional constraint kz ) + c ∂W∂ Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = z UV = − kz ) + c ∂W∂ Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = z IR . (23)Alternatively this condition can be derived by considering boundaries at z + UV and z − IR . Gen-erally the solutions then have to be consistent with the vanishing of the boundary termsin the variation of the action: δS = Z d x (cid:20) ( kz ) − (cid:18) δF c ∂W∂ Φ c + δF ∂W∂ Φ (cid:19)(cid:21) z = z UV + (cid:20) ( kz ) − (cid:18) δF c ∂W∂ Φ c + δF ∂W∂ Φ (cid:19)(cid:21) z = z IR + 12 (cid:2) ( kz ) − ( δφF c − δφ c F − φ δF c + φ c δF ) (cid:3) z = z UV z = z IR . (24)Setting the δF component to zero with the solutions of Eq.(22) also gives Eq.(23).Using the AdS/CFT relation ∆ = 2 + p m /k with m = ( c + c − / k we find that ∆ = 5 / c . The φ solution in (22) therefore behaves like φ ( z ) ∼ z − ∆ .These general solutions have to be matched to whatever VEV Φ may acquire due to braneinteraction terms. In particular we are interested in deformations of the strongly coupledSQCD by the addition of a quark mass term and a consequent explicit breaking of thechiral symmetry. In the 5D description this corresponds to specifying the UV boundarycondition of the bifundamental field Φ ij , such that √ k ( kz ) − ∆ Φ ij (cid:12)(cid:12)(cid:12) z UV = ( m Q ) ij . (25) We now connect this AdS picture with the underlying Seiberg duality, in particular identi-fying the possible configurations of the bulk and brane theories, the location of the variousdegrees of freedom involved, and the brane superpotentials W UV and W IR .It is natural to suppose in the RS1 context that the composite degrees of freedom(i.e. the weakly coupled magnetic theory) should be placed on the IR brane. Note thatthe magnetic SU( n ) gauge fields are forced to appear there because they must be purely emergent degrees of freedom – they can have no bulk presence since that would implythat a global symmetry of the strongly coupled theory has become a gauge theory atlow energies, contradicting the theorem of Ref. [45]. We have seen that in order for the11nderlying strongly coupled theory to have the required mass term, W ⊃ m Q Q ˜ Q , theremust be an additional bulk meson Φ whose UV boundary value, by the bulk/boundarycorrespondence, acts as a source field for the operator O = Q ˜ Q through the interaction W ⊃ Φ Q ˜ Q. (26)In accord with W elec of the electric Seiberg dual, the value of Φ on the UV brane shouldbe determined by m Q as specified in Eq.(25). Note that as described in Ref. [44], theholographic correspondence in the supersymmetric theory appears as an interaction in thesuperpotential on the UV boundary. Hence the correspondence is between a source andcurrent superfield so that, in particular, the scalar operator O of the CFT couples to the F -term of the source superfield.The fact that m Q is associated with the dynamical degree of freedom Φ in the AdSpicture implies that we should base the rest of the structure on the dual-of-the-dual electrictheory. To see this, let us briefly return to 4D Seiberg duality and rename the meson in themagnetic theory η ≡ Q ˜ Q/ Λ , so that the superpotential is W mag = ηq ˜ q . (We shall henceforthdrop the h -couplings unless we are dealing with them specifically, and shall assume themto be of order unity. We shall also for the moment set the dynamical scales of the electricand magnetic theories to be degenerate, Λ .) On performing the dual of this theory onearrives at an alternative (dual-of-the-dual) electric theory with meson Φ ≡ − q ˜ q/ Λ andsuperpotential coupling W ′ elec = Φ Q ˜ Q − ΛΦ η . (27)The first term of Eq.(27) is of course precisely the required source term of Eq.(26). (Theminus sign comes from the matching of dynamical scales in the electric and magnetictheories, as described below in Section 3.3.) One of the checks of Seiberg duality is thatthis theory flows to the same IR physics as the theory we first started with. Indeed both Φ and η have masses of order Λ . Upon integrating them out one finds η = Q ˜ Q/ Λ , leadingto the same spectrum and zero superpotential as the original theory. However note thatwe may also choose to keep all the degrees of freedom and dualise (yet again) to a second magnetic theory. Now the mesons Φ and η are to be treated as elementary and a newcomposite meson ϕ ≡ Q ˜ Q/ Λ is introduced: the superpotential is W ′ mag = qϕ ˜ q + ΛΦ ϕ − ΛΦ η . (28)Integrating out Φ and ϕ − = √ ( η − ϕ ) leaves us with the magnetic spectrum and superpo-tential ( W mag = qϕ + ˜ q where ϕ + = √ ( η + ϕ ) ) as required. It also identifies the two mesons ϕ and η . 12 , V F ϕ , q , ˜ q , v n η W UV = W IR = qϕ ˜ q − m ϕ Φ ϕm η η (Φ − Φ ) Figure 2:
The first configuration for Seiberg duality in the large N limit. This correspondsto the “usual” case in which there is a massless meson.The bulk gauge symmetry is SU( F ) L × SU( F ) R × U(1) B . The unbroken symmetry on the IR-brane, SU( F ) L × SU( F ) R × U(1) B ,corresponds to the light quarks in the electric dual. The upper F × F flavour block of Φ has VEVs of √ km , and is consequently Higgsed out of the low energy theory by the F ϕ = 0 condition, at a scale h ˆ q ˆ˜ q i = ( √ km ϕ )Λ IR . Φ , V F ϕ , q , ˜ q , v nW UV = 0 W IR = qϕ ˜ q − m ϕ Φ ϕ Figure 3:
The second configuration for Seiberg duality in the large N limit, when there isno massless meson, but a field Φ and a coupling Φ Q ˜ Q in the electric superpotential. As inFigure 2, the bulk meson is the source field for the composite operator O = Q ˜ Q (i.e. itsUV value corresponds to the quark masses). N limit: • Configuration 1 – a massless meson ϕ : In this case, because we have a dynam-ical Φ , we must add a new elementary meson η on the UV brane that has the samequantum numbers as ϕ . The superpotential on the UV brane must be W UV = m η (Φ − Φ ) η , (29)where Φ = m Q √ k ( kz UV ) − ∆ fixes the UV VEV of Φ through the F η = 0 equation ofmotion. The superpotential on the IR brane is of the form W IR = qϕ ˜ q − m ϕ Φ ϕ . (30)Next let us discuss scales. First note that, using ∆ ϕ = − c , the purely 4D relationin Eq.(15) gives Λ IR = m (cid:18) z UV z IR (cid:19) + c , (31)so that the factor that naturally warps the physical scales in this set-up is ( z UV /z IR ) + c .Now, by comparison with Eq.(28), m ϕ ∼ m η are identified with the compositenessscale, but not directly. Indeed because Φ is a 5D field m ϕ and m η have mass dimen-sion / . Furthermore canonical normalisation of the fields on the IR brane leads tothe physical Yukawa coupling being of order unity, hence the normalised brane fieldsare ˆ q = q/ ( kz IR ) , ˆ˜ q = ˜ q/ ( kz IR ) and ˆ ϕ = ϕ/ ( kz IR ) . Thus once the fields are canonicallynormalised, the term m ϕ gives a physical mass that is also naturally warped downby a factor ( z UV /z IR ) + c compared to that generated by m η . (For the most part wewill allow m ϕ and m η to be free parameters.) The relation of m ϕ and m η to thecompositeness scale can be determined by the Higgsing in the SU( F ) L × SU( F ) R flavour-block of the magnetic theory corresponding to the heavy quarks of the elec-tric dual: upon canonically normalising the magnetic quarks and using Eqs.(22), (25)and (31), the F ϕ = 0 condition for W IR gives h ˆ q ˆ˜ q i = ( kz UV ) − ∆ ( √ km ϕ )Λ IR . (32)Thus, since Λ IR is the physical mass of the heavy quarks in the electric dual, we canidentify ( kz UV ) − ∆ √ km ϕ as the compositeness scale, and set its value to be ∼ Λ IR . Furthermore the boundarycondition in Eq.(23) correctly equates the two canonically normalised mesons ˆ ϕ and ˆ η and their masses as (cid:18) z UV z IR (cid:19) + c m ϕ ˆ ϕ = m η ˆ η . (33)14his identifies the heavy mode as being mostly η and the orthogonal massless modeas being mostly ϕ , provided that c > − . Note that if m ϕ = 0 , then η = 0 andwe recover the standard Seiberg picture in which the massless meson is identifiedentirely as ϕ , and η is integrated out. The set-up for this configuration is as shownin Figure 2. • Configuration 2 – no massless meson : In the 4D theory this corresponds tohaving just the source term in the electric superpotential without the η field: W elec = Φ Q ˜ Q. (34)Upon dualizing one finds a magnetic superpotential W mag = qϕ ˜ q + ΛΦ ϕ . (35)Thus both Φ and ϕ gain a mass of order Λ and may be integrated out of the lowenergy theory, leaving no mesons and no superpotential in the magnetic theory. Thegravitational dual has a bulk field for every CFT operator so Φ ↔ Q ˜ Q must bethere by the bulk/boundary correspondence. Hence the bulk configuration, shownin Figure 3, must be as before but with no additional η meson. For the boundarysuperpotentials in the gravitational dual we have W UV = 0 , (36)and W IR = qϕ ˜ q − m ϕ Φ ϕ . (37)As the elementary η and composite ϕ mix, an obvious possible extension is to generalize thepicture and have just a single bulk field ϕ . However by the bulk/boundary correspondence,the VEV of this field on the UV brane would be identified as the source for an operatorin the underlying CFT (i.e. the electric theory). This operator would have to have thesame flavour charges as Φ ∼ q ˜ q but such an operator is not readily available in the electricSeiberg dual (a possible exception being the case F = N + 1 , when the magnetic quarkscan be written as electric baryons). Hence although the physics would be similar to thatof configuration 1, the direct link to strongly coupled 4D Seiberg duals would be lost. In the context of AdS/CFT the flavour symmetries of our strongly coupled 4D theory areweakly gauged, and their RG behaviour (in particular the possibility of Landau poles)is then important. Let us first consider this from a 4D point of view, by returning tothe electric theory with F = ( N + ν ) with N ≫ ν . The SU ( N ) gauge groups are15nside the conformal window, but the (now gauged) flavour groups, SU ( F ) L × SU ( F ) R , see ˆ n f = N/ flavours and ˆ n c = F colours. However the flavour groups are anomalous andmust be cancelled. According to the anomaly matching procedure of ’t Hooft (see below),the flavour anomalies of the electric and magnetic descriptions should both be cancelled bythe same spectator sector. In the present case the coefficients of the SU ( F ) L and SU ( F ) R anomalies are − N and N respectively. Thus the spectator sector could be an additional N fundamentals of SU ( F ) L and N antifundamentals of SU ( F ) R . The total number of SU ( F ) flavours is then ˆ n f = N and colours is ˆ n c = F = (ˆ n f + ν ) . Since ˆ n f < ˆ n c one wouldconclude that a vacuum does not exist for these theories. However one may add additionalstates with masses around Λ IR which are charged only under the flavour symmetry. Asfar as the Seiberg duality is concerned these states are gauge singlets, and clearly theiradditional contribution to ˆ n f can bring the SU ( F ) L × SU ( F ) R flavour symmetry within itsconformal window as well.What if one instead has a theory where the flavour symmetries are in the IR-free phase,with effectively ˆ n f > n c ? (We shall encounter an example later.) In that case the flavoursymmetries would hit a Landau pole in the UV and another Seiberg duality would berequired. From a 4D viewpoint the original SU ( N ) would in turn become IR-free and leadto a duality cascade involving the flavour symmetry as well (and most likely a duality wallwould ensue). This can be averted by instead adding massive gauge states around the scale Λ IR . These come in N = 2 vector multiplets; they can be decomposed as an N = 1 vectorand an N = 1 chiral superfield both in the adjoint. Hence a single massive gauge multipletcontributes n c to the beta function coefficient and such states can again bring the flavourgroups inside the conformal window.In short therefore, one expects that even from an entirely 4 dimensional point of view,additional massive states can tame the behaviour of the flavour symmetries above thescale Λ IR should we choose to gauge them. In the 5D AdS picture such states correspondto Kaluza-Klein modes of the bulk gauge fields. The net effect of the coupling to the CFT isthat the RG behaviour of the flavour groups becomes logarithmic [46, 47]. A bound resultson the contribution that the flavoured degrees of freedom in the CFT make to the betafunction in order to avoid Landau poles in the flavour couplings below the UV scale [48]: b CFT . πα (Λ IR ) 1log (Λ UV / Λ IR ) . (38)When taking the Veneziano limit we necessarily have b CFT ∼ N and hence this translatesinto an an upper bound on N which depends on α (Λ IR ) . For RS1 type scenarios wherethe flavour symmetries are identified with SM gauge symmetries, and where Λ IR ∼ N . O (10) . In other cases, for example when the bulk gaugesymmetries are not SM gauge symmetries (with the latter being either emergent symmetrieslocated on the IR brane or having very small values in the IR), or in the “little RS”scenario [49], the bound can be greatly relaxed.16hese bounds are generally equivalent to a lower bound on the curvature of any would-be gravitational dual [48]. Indeed the radius of AdS is typically given by R AdS ∼ ℓ s N δ wherefor example δ = 1 / for AdS × S , so that the bound on N immediately translates intoa bound on R AdS . In the sub-critical construction of Ref. [31] which we shall be invoking,we have δ = 0 and then the curvature is quite large anyway, R AdS ∼ √ ℓ s . This is in linewith the general expectation of Ref. [32].This then is the first caveat about the validity of this framework that one must bearin mind. The second is related, and concerns the interpretation of the dynamical scales ofthe field theory. In particular, the rightmost panel in Figure 1 is the 4D field theoreticalsituation whose large N limit we are supposed to be approximating. But, in the figure,both electric and magnetic theories have their Landau poles around the scale Λ IR , andlook to be strongly coupled there: why then are we entitled to place a weakly coupledtheory on the IR brane? The answer can be found in the matching relation for dynamicalscales in Seiberg duality which still permits the Landau poles of the electric and magneticdescriptions to be in different places.To see this, we can return to the 4D theories and “blow up” the running around thescale Λ IR in order to examine what happens there when we match the two descriptions.Let us treat the dynamical scales of the electric theory ( Λ ) and magnetic theory ( ¯Λ ) morecarefully, and temporarily reinstate the coupling h which we have been setting to oneabove. We will need the matching relation for Seiberg duality, which is ¯Λ ¯ b Λ b = ( − ) F − N ˆΛ b +¯ b , (39)expressed in a basis where the quarks of both theories are canonically normalised, andwhere the magnetic superpotential is written W mag = 1ˆΛ Q ˜ Qq ˜ q + m Q Q ˜ Q . (40)In Eq.(39), b = 3 N − F and ¯ b = 3 n − F are the one-loop beta-function coefficients so that πα ( t ) = b ( t − t Λ ) , π ¯ α ( t ) = ¯ b ( t − t ¯Λ ) , (41)where t = log E . For an SQCD theory in the free magnetic phase we have b > and ¯ b < , so that we get a perturbative overlap of the electric and magnetic descriptions when ˆΛ > Λ > ¯Λ . In the interval between Λ and ¯Λ both descriptions are in principle valid.The relation is illustrated graphically in Figure 4. The scale ˆΛ is unknown, nevertheless asshown later in Section 5.1 (see Eq.(75)) one can identify hϕ = Q ˜ Q ˆΛ hµ ϕ = m Q ˆΛ , (42)17 t Λ − t − t ¯Λ − t ˆΛ ¯ α − α − Figure 4:
Representation of the scale matching of Eq. (39) . The parameter t = log( E ) ,so that the IR is to the right in the diagram. The scale ˆΛ is formally where α = − ¯ α , asindicated by the extension of the α − line below the axis. where ϕ is the canonically normalised meson. On dimensional grounds one expects theKähler potential to have terms of the form K ⊃ Q ˜ QQ † ˜ Q † κ Λ ≡ ϕϕ † , (43)where κ is some unknown coefficient. The coupling, h , is related to this coefficient as h = Λ κ ˆΛ . (44)Thus in addition to m Q there are three free (or rather unknown) parameters which we canchoose to be h , Λ and ¯Λ (with ˆΛ and κ being determined by Eqs.(39) and (44)).It is clear from this discussion what must happen near the IR brane when we havethe more complicated situation described above, namely a conformal interval terminatedby quark mass terms that cause the theory to enter the free magnetic phase below theirmass, Λ IR . In that case the behaviour shown in Figure 5 is perfectly acceptable. Above Λ IR the coupling does not run. Below the scale Λ IR the electric coupling quickly becomesnon perturbative. But the magnetic theory can be made arbitrarily weakly coupled inthis region since ¯Λ is a free parameter, irrespective of how strongly coupled is the electrictheory. 18 t Λ − t − t ¯Λ − t Λ IR ¯ α − α − Figure 5:
As in Figure 4 but with the theory entering a conformal phase above the scale Λ IR . Above this scale the couplings do not run. Below the scale Λ IR the electric couplingquickly becomes non perturbative. The “IR brane” is represented by the grey strip in theinterval between Λ IR and Λ . The magnetic theory is weakly coupled in this region. Klebanov and Maldacena considered the 6d non-critical string action and found an AdS × S solution in the presence of F space filling D5 and anti-D5 branes and with ˜ N = O ( N ) unitsof RR flux. Despite the fact that these solutions are relatively strongly curved, it is arguedthat they are a qualitative approximation to the gravitational dual of SQCD. However forour 5D AdS approximation to be valid it is important that the radius of S is small andconstant. This is indeed the case, and actually the more general solutions of this systemcan be seen to exhibit some of the features of the flows we see in the gauge theory. Theansatz for the 6d metric in the string frame is ds = e f dx + dr + e g dθ . (45)Thus the radius of curvature of the S is R S = √ α ′ e g . The evolutions of the scalars isgiven by g ′ = ˜ NF e − g − F ˜ N e g − ˜ N e φ − g φ ′ = − F ˜ N e g + ˜ N e φ − g f ′ = ˜ N e φ − g , (46)19nd where φ is the dilaton. For constant F , the solutions exhibit two typical kinds ofbehaviour. The choice e g =
23 ˜ N F and F e φ = is the AdS × S solution with constant R S . If we perturb away from this solution it is straightforward to see that the theory willflow to a linear dilaton behaviour in the UV (i.e. as r → ∞ ). The dilaton behaves as φ → φ − F ˜ N r with the theory becoming weakly coupled, while for generic starting valuesof g we have e g → ˜ NF (asymptoting as a tanh function), and f → const . This behaviouris clearly seen in Figure 6a where we show g ( r ) and φ ( r ) . The flat region is an AdS × S “slice”, while the left of the plot (the UV) is the linear dilaton region.It is interesting to look at the solutions when F is itself a function of r . This is thegravitational equivalent of integrating out heavy quarks below their mass on the gaugeside: in this case it would correspond to a recombination of the D5/anti-D5 flavour branesin the bulk. We can mimic this by adjusting F by hand at some value of r . The resultingbehaviour is shown in Figure 6b. A change in flavour perturbs the AdS × S solutionto one with a different R S curvature and dilaton, with the AdS curvature remaining thesame. There is an obvious analogy with Figure 1. - r g H r L , Φ H r L - r g H r L , Φ H r L Figure 6:
Flow in the Klebanov-Maldacena solution. Left: the warping g ( r ) (dotted) anddilaton φ ( r ) (solid) for fixed F . Right: g ( r ) (dotted) and dilaton φ ( r ) (solid) for an r -dependent F . As we have already mentioned the electric and magnetic theories of Tables 1 and 2 areanomalous in the flavour symmetries: for example both the the SU ( F ) L and SU ( F ) R anomalies are non-zero. In a purely 4D setting the ‘t Hooft anomaly matching idea assertsthat such flavour anomalies should be the same in both theories; the reasoning is that ifone were to gauge the flavour symmetries one would have to add a spectator sector chargedonly under the flavour symmetries in order to cancel these anomalies. However these new20ectors would be blind to the SU ( N ) gauge group and so insensitive to its behaviour. (Byanalogy, the lepton sector of the Standard Model does not change when QCD confines.)That both electric and magnetic descriptions have the same flavour anomalies is one of themost powerful tests of Seiberg duality in 4D.Now however the situation has changed: the flavour symmetries really are gauged inthe bulk. What, therefore, becomes of anomaly matching – is it even consistent with ourpicture of AdS as the large N limit of Seiberg duality? (See also [37].) In order to addressthis question let us consider what the spectator sector has to be. Because anomaliesdo not exist in 5D, the 4D anomalies are associated with contributions localised at thebranes [50–53]. They can be written √− g A a ( x, z ) = A a UV ( x ) δ ( z − z UV ) + A a IR ( x ) δ ( z − z IR ) , (47)where for i = U V, IR A ai = n i π ε µνκλ tr (cid:20) T a ∂ µ (cid:18) A ν ∂ κ A λ + 12 A ν A κ A λ (cid:19)(cid:21) . (48)Here the gauge fields can be associated with the 4D zero modes up to an overall nor-malisation since their bulk profiles are flat. The constants n i receive the following con-tributions. A left-chiral fundamental fermion localised on the UV (IR) brane contributes n UV = 1 , n IR = 0 ( n UV = 0 , n IR = 1 ). The contribution from a massless left-chiral bulkfermion is n UV = n IR = 1 / . On an S / ( Z × Z ′ ) orbifold heavy modes with parities (+ − ) and ( − +) can give localised anomalies that are equal and opposite on the branes. Thecontribution from a fundamental fermion with parity (+ − ) is n UV = − n IR = 1 / and froma fermion with parity ( − +) , it is n UV = − n IR = − / . Finally there is a contribution froma 5D bulk Chern–Simons term which also gives equal and opposite gauge anomaly termson the branes. In a consistent gauge invariant theory the summed contributions shouldvanish on both boundaries.Now let us interpret the bulk and brane fields in terms of Seiberg duality and discussthe roles that they would play in anomaly matching. Fields on the IR brane clearly belongto the magnetic theory – we shall call this contribution n (mag)IR . Fields in the bulk have botha composite and an elementary component – they are therefore present in both electricand magnetic theories, and moreover they are singlets of both the magnetic gauge group(because that lives on the IR-brane only) and the electric gauge group. They can thereforebe thought of as part of the spectator sector. They contribute the same anomaly at eachbrane: n (bulk)UV = n (bulk)IR = 12 n (bulk) , where n (bulk) is their total contribution to the 4D anomalies. Finally the Chern-Simons andpossibly heavy mode contributions are − n (CS)UV = n (CS)IR = n (CS) .
21s we have said the total contribution must vanish at each boundary. Thus at the IRboundary we have to satisfy n (mag) + 12 n (bulk) + n (CS) = 0 . (49)The interpretation in terms of ’t Hooft anomaly matching of the 4D field theories is thata spectator sector added to cancel the anomalies of the magnetic theory would have totalanomaly n (spec-mag) = 12 n (bulk) + n (CS) . (50)On the other hand the UV brane has total anomaly n (brane)UV + 12 n (bulk) − n (CS) = 0 , (51)where n (brane)UV are the contributions from whatever degrees of freedom are localised on theUV brane. In order to cancel anomaly contributions here we require n (brane)UV + n (bulk) = n (spec-mag) . (52)But the left hand side of this equation is the total contribution to the flavour anomalycoming from the elementary degrees of freedom that couple to the CFT. This is nothingother than the spectator sector as seen by the underlying electric theory, n (spec-elec) . Thusthe anomaly matching condition (i.e. n (spec-elec) = n (spec-mag) ) is satisfied: the same spectatorsector can serve to cancel anomalies in both the electric and magnetic theories. Notethat the essential features of the bulk physics required for this to work are that a) thecontributions from the bulk modes are the same at each brane and b) the contributionsfrom the Chern-Simons terms and the heavy modes are equal and opposite.As an explicit example consider the anomalies when there is a single bulk meson field Φ . In total this contributes − F to the SU ( F ) L anomalies (and the negative to the SU ( F ) R anomaly), with one half appearing at each boundary. The magnetic theory on the IRbrane has a contribution of N . Hence a would-be spectator sector (including Φ ) that couldcancel the anomaly would have to contribute − N . However the Chern-Simons contributionhas to be F − N at the IR-brane. On the UV brane the net anomaly from Φ and theChern-Simons term is N − F = − n so we have to add additional elementary fields herethat contribute n to the SU ( F ) L anomaly. From a 4D perspective the total contributionfrom the elementary degrees of freedom (i.e. the spectator sector) is then this plus thecontribution from Φ , giving the same − N that we required for the magnetic theory. Inthis example the elementary sector could be an additional n fundamentals of SU ( F ) L and n antifundamentals of SU ( F ) R . 22 The MSSM in RS1
We now turn to applications of the proposed “large N ” limit of Seiberg duality, the firstbeing an explicit realisation of the Randall-Sundrum (RS1) idea, that the particles of theStandard Model arise as composite degrees of freedom on the IR brane of a slice of AdS.Of course in their original proposal Randall and Sundrum were interested in protecting theHiggs mass in non-supersymmetric scenarios. Since we are working in Seiberg duality wewill consider the N = 1 supersymmetric MSSM instead. (This model is morally similar tothe super-technicolour models of Ref. [54].)In Table 3 we show the MSSM theory. This is the theory that we wish to put on the IRbrane of a slice of AdS. The advantage of our approach is that we can establish the set-upworking mostly in 4D field theory. Following the discussion of the previous sections wewill do this by deriving the MSSM as the magnetic dual of an electric theory. Additionalstates will then be integrated into the latter in order to make the electric phase conformalabove some mass scale, and then the “large N ” limit will be taken. This last step requiressome particle content in the magnetic theory that can be adjusted – this content is anarbitrary number of n h Higgs pairs, with n h = 1 corresponding to the usual MSSM. Thegauge group that we will dualize is SU(2) L ≃ Sp(1) .We assume for concreteness that the Higgs fields, being in vector-like pairs, have ageneric range of mass terms, and that the lightest Higgs is the only one that ends up witha VEV (i.e. this is the one that plays the role of the usual MSSM Higgs). One can envisagemore complicated cases in which the electroweak breaking is distributed amongst the Higgspairs. At low scales, the theory runs precisely as the MSSM, with an asymptotically free
SU(3) and a mildly positive beta function for
Sp(1) : the
Sp(1) group sees an effective flavournumber f Sp(1) = 7 , and the beta function coefficient is given by ¯ b Sp( m ) = 3( m +1) − f Sp( m ) , sothat ¯ b Sp(1) = − as usual. At energy scales above the masses of the additional Higgs fields,those states can be “integrated in” to the theory and start to contribute to the running aswell. Eventually we have f Sp(1) = 6 + n h . The SU(3) running continues unchanged butthe beta function of
Sp(1) becomes more negative, eventually reaching ¯ b Sp(1) = − (1 + n h ) .Therefore we can expect the theory to reach a Landau pole at some scale Λ , above whichan electric description takes over.In order to get to an electric description we use the Seiberg duality for Sp( m ) groups;the electric gauge group is Sp( M ) where M = f Sp( m ) − ( m + 2) = 3 + n h . (53)The content of the theory is shown in Table 4. In this section we will use the conventionthat additional Higgs-like states with R -parity R p = 1 are denoted either with Φ or ϕ , witha suffix to indicate which SM field they resemble charge-wise.23he R -parity charges in the electric theory allow the following superpotential (we setall couplings to one and suppress generation indices to avoid clutter): W elec = Φ E LL + Φ D QL + Φ D c QQ + Φ S QQ + Φ σ H U H D + µ σ Φ σ . (54)This theory has a number of features that we should comment on. First the charges ofthe Higgses, left-handed quarks and left-handed leptons are the opposite of the charges oftheir magnetic counterparts. This is because the gauge groups other than Sp(1) are playingthe role of “flavour” in the Seiberg duality, and as usual those charges must be reversed.Second, note that the right handed fields of the magnetic theory have no counterpart inthe electric theory: in fact those fields are composite bound state “mesons” of the electrictheory. In addition there are some new states charged under
SU(3) and
U(1) Y . These canbe identified as the composite “mesons” of the magnetic theory (the term “meson” of coursealways referring to mesons of the Sp groups): Φ E ↔ ll Φ D ↔ lq Φ D c ↔ q [ i q j ] Φ S ↔ q { i q j } Φ σ ↔ h u h d . (55)The trilinear terms in Eq.(54) are simply the superpotential terms associated with thatidentification (c.f. Eq.(7)). Finally the linear singlet term in Eq.(54) corresponds to theHiggs mass terms of the magnetic theory: the masses are ∼ µ σ / Λ where Λ is the Landaupole scale of the magnetic theory, hence we require that µ σ < Λ . (56)It is instructive to confirm that the electric theory we have just described does indeedflow to the MSSM (augmented by extra Higgs fields) of Table 3. (As we have presented it,this is in fact nothing other than the test that the dual-of-a-dual gives back the originaltheory [2].) Upon performing the Seiberg duality of the electric theory we find the spectrumof Table 5. As we have already said, the block of right-handed MSSM fields are boundstate “mesons” of the electric theory. The identification is e ci ↔ L i H D ν ci ↔ L i H U d ci ↔ Q i H D u ci ↔ Q i H U . (57)24 U(3) Sp(1) U(1) Y R p q i (cid:3) (cid:3) − l i (cid:3) − − n h × h u (cid:3) n h × h d (cid:3) − e ci − ν ci − d ci (cid:3) − u ci (cid:3) − − Table 3:
The MSSM spectrum augmented by n h − additional massive Higgs pairs. Theindex i = 1 . . . is the usual generation index. SU(3) Sp(3 + n h ) U(1) Y R p Q i (cid:3) (cid:3) − − L i (cid:3) − n h × H U (cid:3) − n h × H D (cid:3) × Φ E − × Φ D c (cid:3) × Φ S × Φ D (cid:3) − singlets Φ σ Table 4:
The electric dual theory.
The third block of states are also composite “mesons” of the magnetic theory: × ϕ e c ↔ LL × ϕ d ↔ Q [ i Q j ] × ϕ s c ↔ Q { i Q j } × ϕ d c ↔ QL singlets ϕ σ ↔ H U H D . (58)The superpotential of this theory is W mag = Λ(Φ E ϕ e c + Φ D ϕ d c + Φ D c ϕ d + Φ S ϕ s c + Φ σ ϕ σ ) + µ σ Φ σ + e c lh d + ν c lh u + d c qh d + u c qh u + ϕ σ h u h d . (59)25 U(3) Sp(1) U(1) Y R p q i (cid:3) (cid:3) − l i (cid:3) − − n h × h u (cid:3) n h × h d (cid:3) − e ci − ν ci − d ci (cid:3) − u ci (cid:3) − − × ϕ e c × ϕ d (cid:3) − × ϕ s c ¯ 1 − × ϕ d c (cid:3) singlets ϕ σ × Φ E − × Φ D c (cid:3) × Φ S × Φ D (cid:3) − singlets Φ σ Table 5:
Flowing down from the electric theory. This magnetic theory is arrived at bydualizing the electric theory of Table 4. One finds mass terms in the superpotential for thestates in the last two blocks, and they can be integrated out to arrive at the MSSM spectrumof Table 3.
The first set of terms are masses of order Λ so these fields can be integrated out, whereuponwe recover the original spectrum of Table 5 and the MSSM superpotential W mag = µ σ Λ h u h d + e c lh d + ν c lh u + d c qh d + u c qh u . (60)The first term gives as required the set of masses for the extra Higgs fields (includingone for the lightest field that would correspond to the usual “ µ -term” of the MSSM). Thetrilinear terms are the usual set of Yukawa couplings: in Seiberg duality we have the addedbonus that these interactions necessarily arise because the right-handed fields are compositemesons. Naturally there is then the question of how one could explain the hierarchy thatwe observe in these interactions. This is outside the scope of the present work but is surelyan interesting topic for future study. 26aving found a candidate electric dual for the MSSM, all that remains is to take thelarge M limit and propose a gravitational dual. In this case we first add into the electrictheory the required states to bring it into the conformal window of the Sp group. The freemagnetic window is given by M + 2 < f Sp( M ) ≤
32 ( M + 1) , (61)and the conformal window by
32 ( M + 1) < f Sp( M ) < M + 1) . (62)Initially (i.e. for the theory of Table 4) the electric theory has M = n h + 3 and f Sp( M ) = n h + 6 which, as expected, places it within the free magnetic range for any value of n h > .Now let us add some additional Higgs states but with masses Λ IR . Below Λ IR these fieldsare integrated out and the theory enters the free magnetic phase described above; thus asdiscussed in section 2 it is the mass Λ IR which generates the “IR-brane”. We will add anextra n ′ h pairs of Higgses H ′ U and H ′ D so that above Λ IR we have f Sp( M ) = n ′ h + n h + 6 .In order to bring the electric theory into the conformal window above Λ IR we can define aparameter ν such that n ′ h = 12 ( n h + 3 ν ) , (63)so that f Sp( M ) = 32 ( M + 1 + ν ) . (64)We are then free to take the large M limit. In this case sending n h → ∞ but keeping ν = O (1) formally gives a parametrically strongly coupled conformal Sp( M ) . Of course noneof this changes the magnetic theory which remains the MSSM with some extra Higgses.Finally we should discuss the running of the SU(3) and U(1) Y groups. The electrictheory has a large gauge group Sp( n h + 3) . From the point of view of the SU(3) groupthis provides a large number of additional flavours. In fact the effective number of flavourscontributing to the
SU(3) beta function is f SU(3) = 3( n h + 11) . (65)Clearly the SU(3) group is in principle now highly IR-free in the large n h limit (and evenwhen n h = 1 ). Usually one would expect the theory to exhibit some sort of cascadebehaviour above Λ IR with SU(3) hitting a Landau pole. In fact the situation is the oneoutlined in Section 3.3. In a conformal phase these beta functions are tamed by additionalmassive modes charged only under
SU(3) (or
U(1) Y ) appearing at the scale Λ . From theRS point of view these states appear automatically as the low-lying Kaluza-Klein modesof the bulk gauge and matter fields. As explained, the end result is a logarithmic runningand a bound on the value of M . 27aving made this caveat we can now propose the entire gravitational dual of this theoryat large M. The set-up is shown in Figure 7. The SU(2) L group is emergent and so mustappear on the IR brane. From the perspective of the strongly coupled theory, the remaininggauge symmetries are flavour symmetries and have to appear in the bulk. From Section3, it is clear that all the magnetic “quarks” denoted generically by q (i.e. q = q, l, h u , h d )and every low energy “matter” meson of the Sp groups with R-parity − , denoted by ϕ − (i.e. ϕ − = e c , ν c , d c , u c ), also appear on the IR brane. Moreover as in the field theory W IR ⊃ q ϕ − q on the IR-brane gives the required MSSM superpotential terms.However there is a further modification required in the gravitational dual because thereare bulk fields corresponding to every composite operator of the CFT. Therefore the bulkcontains not only the original R p = +1 fields as above denoted generically as Φ + , Φ + = Φ E ↔ ll Φ D ↔ lq Φ D c ↔ q [ i q j ] Φ S ↔ q { i q j } Φ σ ↔ h u h d , but also the R p = − bulk fields that did not appear in the field theory, denoted genericallyby Φ − : Φ − = E ↔ lh d ν ↔ lh u D ↔ qh d U ↔ qh u . As we saw in section 3 the two types of meson are distinguished by the fact that the latterare in configuration 1, with a corresponding set of matter fields (i.e. η − = ˆ e c , ˆ ν c , ˆ d c , ˆ u c )coupling to Φ − on the UV brane, whereas the former with no corresponding η + fields onthe UV brane, are in configuration 2. The superpotentials are then given by W UV = m η η − Φ − ,W IR = q ϕ − q + q ϕ + q − m ϕ Φ − ϕ − − m ϕ Φ + ϕ + , (66)where the trilinear couplings automatically contain all the terms consistent with R -parityconservation. Recall that as in the field theory all the mesons without a UV counterpart(i.e. the ϕ + ’s) are massive and can be integrated out of the low energy theory, whereasthose with a UV counterpart (i.e. the ϕ − ’s) leave a light linear combination of η − and ϕ − in the low energy theory. Since the wave-function of the Φ − can be warped, the remaininglight states could be mostly ϕ − or mostly η − . Hence the final low energy right-handedfields, e c ′ , ν c ′ , d c ′ , u c ′ can naturally have different degrees of compositeness, while everythingcharged under SU(2) L must necessarily be entirely composite.28 + Φ − V SU(3) c × U(1) Y q ≡ left-matter+higgs ϕ − ≡ comp. right-matter ϕ + ≡ charged heavy states v SU(2) L η − ≡ elem.right-matter W UV = m η η − Φ − W IR = q ϕ − q + q ϕ + q − m ϕ Φ − ϕ − − m ϕ Φ + ϕ + Figure 7:
The configuration for the MSSM in RS1 that naturally arises by consideringSeiberg duality of
SU(2) L ≃ Sp(1) . The left-handed matter and Higgs fields are identified as“quarks” of the
Sp(1) , while a linear combination of the composite “mesons” and elementaryfields form the light right-handed matter fields.
Our second application is to supersymmetry breaking on the IR brane and its media-tion. The “large N ” limit of Seiberg duality will clearly yield a version of the metastablesupersymmetry breaking mechanism of Intriligator, Seiberg and Shih [3], but in an RS1configuration similar to those discussed in Refs. [36,37]. (A string configuration that corre-sponds to this case was presented in Ref. [55]. This case is morally similar to the metastablesuperconformal models of Ref. [56].)The supersymmetry breaking is a feature of the magnetic theory and so one expectsit to appear on the IR-brane. Thus proposals for gauge-mediation that were discussed inthe context of RS1 should also be applicable to our strongly coupled configuration. Oneparticular application that we would like to revisit is gauge mediation with gaugino massesthat are dominant over scalar ones. In the context of extraordinary gauge mediation [38]this corresponds to increasing the “effective number of messengers”, and a region of pa-rameter space that naïvely corresponds to strong coupling. Calculable and explicit modelshave long been known in the context of extra dimensional models [20–26, 43]. Interest hasbeen revived recently in 4D models that can achieve the same kind of screening of scalarmass contributions in for example Refs. [26–29]. Here we shall be using the large N limitof the simple perturbative gauge mediation model of Ref. [35] in order to achieve the sameeffect. 29irst let us look at the ISS supersymmetry breaking sector and briefly review the modelfor comparison. Ref. [3] worked in the free-magnetic phase N ≥ F ≥ N +1 and noted thatthe classical superpotential W mag in Eq.(16) is of the O’Raifeartaigh type. Supersymmetrybreaking occurs because of the so-called rank condition: F ϕ ij = ˜ q j q i − µ ϕ δ ji = 0 , (67)can only be satisfied for a rank- n submatrix of the F ϕ where i, j are flavour indices. Theheight of the potential at the metastable minimum is then given by V + (0) = N | µ ϕ | , (68)where for ease of notation we are again setting the coupling h = 1 . The supersymmetricminima in the magnetic theory are located by allowing ϕ to develop a vev. The q and ˜ q fields acquire masses of h ϕ i and can be integrated out, whereupon one recovers a pureSU ( n ) Yang-Mills theory with a nonperturbative contribution to the superpotential of theform W (dyn)IR = n (cid:18) det ϕ ¯Λ F − n (cid:19) n . (69)This leads to N nonperturbatively generated SUSY preserving minima at h ϕ ji i = µ ϕ ǫ − FN δ ji , (70)where ǫ = µ ϕ / ¯Λ , in accord with the Witten index theorem. The minima can be made toappear far from the origin if ǫ is small and N > F , the condition for the magnetic theoryto be IR-free. The positions of the minima are bounded by the Landau pole such that theyare always in the region of validity of the macroscopic theory.Now for the holographic version. Following the discussion in Section 2, we work downfrom the electric theory. In contrast to Ref. [3] we begin in the conformal window but withthe global flavour symmetry explicitly broken by relevant mass-terms as SU ( F ) L × SU ( F ) R × U(1) B × U(1) R → SU( F ) D × SU( F ) D × SU( F ) D × U(1) B , (71) where F + F + F = F , and where we choose N < F ≤ NN + 1 ≤ ( F + F ) ≤ N . (72)We also add to the electric spectrum a superfield Φ that transforms as an adjoint underthe SU( F ) D symmetry. The superpotential of the electric theory is (we set all Yukawacouplings to one) W elec = m Q Q ˜ Q + Q Φ ˜ Q − µ Φ , (73)30here m Q = m F × F m F × F
00 0 m F × F , and where m ≫ m ≫ m . Initially the model behaves according to the discussion ofSection 2; that is at the scale Λ IR = (cid:0) m Λ UV ∆ ϕ − (cid:1) ϕ − , (74)we can integrate out the F heavy quark states. The theory can be dualized to a weaklycoupled and IR-free magnetic phase with F + F flavours and a dynamical scale Λ ∼ Λ IR .However the magnetic superpotential is W mag = qϕ ˜ q − µ ϕ ϕ + Λ ϕ Φ − µ Φ , (75)where µ ϕ = µ F × F F × F F × F µ F × F ! . The flavour contractions are self-evident: for example the contraction ϕ Φ can involve onlythose elements of ϕ in the F × F upper block. As this is a mass term, the F × F blocksof flavour adjoints may be integrated out supersymmetrically near the Landau pole scale Λ to leave a superpotential W ′ mag = µ Λ q ˜ q + q ϕ ˜ q + q ϕ ˜ q + q ϕ ˜ q − µ ϕ , (76)where we now indicate the flavour blocks with indices. Finally, assuming that the mass for q and ˜ q dominates, i.e. that µ / Λ ≫ µ ϕ , we may integrate out these fields as well, tofind W ′′ mag = q ϕ ˜ q − µ ϕ . (77)Since no gauge symmetry has been broken so far, this is an SU( F + F − N ) O’Raighfeartaightheory that has F flavours of quarks. We may now take a large N limit. In this case,remaining inside the correct ranges of flavours means that F + F + F > N and F + F ≥ N + 1 also become large. However there is no such constraint on either F or n = F + F − N which may both be of order unity. Hence the IR theory can be weaklycoupled. Note that this type of SUSY breaking could be inserted directly into the Higgssector of the MSSM model in the previous subsection. The metastable SUSY breaking of the previous section lends itself to an RS1 implemen-tation of the “simplified” gauge mediation scenario discussed in Ref. [35]. The result is31 holographic version of general gauge mediation [57]. To briefly recap, the 4D pictureis as follows. Suppose that the supersymmetry breaking sector (i.e. the
SU( F ) sectorabove) contains no direct connection with the Standard Model gauge groups, G SM , butthat there is an additional pair of messenger fields f, ˜ f that are charged under G SM . Theauthors of Ref. [35] argued that one can expect higher order operators to be generated inthe underlying electric theory of the form W elec ⊃ ( Q ˜ Q )( f ˜ f ) M X + m f f ˜ f , (78)where m f is the messenger mass and M X is the scale of underlying physics, namely themass scale of new modes in the theory that are exchanged between the messengers andthe strongly coupled ISS sector. For convenience we are now (and will henceforth) dropthe indices that identify this as the SU( F ) block. In the low energy theory the SUSYbreaking and mediation part of the superpotential becomes W mag ⊃ Λ M X ϕ f ˜ f + m f f ˜ f + qϕ ˜ q − µ ϕ ϕ . (79)As noted in Ref. [35], the first term is precisely the usual spurion interaction of ordinarygauge mediation. However the effective coupling Λ M X can be very small since generally oneexpects Λ ≪ M X . The advantage of this suppression is that the R -symmetry breakingin the theory is under strict control. Of course the terms in Eq.(79) do explicitly break R -symmetry since ϕ has R -charge 2 (it appears with a linear term in the rest of thesuperpotential), however it is still approximately conserved because of the smallness of thecoupling to the spurion. An equivalent statement (as prescribed by Ref. [58]) is that a newglobal SUSY preserving minimum is introduced but that it is so far away in field spacethat it could never disrupt the metastability of the SUSY breaking ISS sector. Indeed itis clear that the linear ϕ term can be set to zero if h f ˜ f i = − µ ϕ M X / Λ , but this can bemuch larger than the scale Λ , making it irrelevant to physics in the magnetic theory. Thephenomenology of these models is similar to that of conventional gauge mediation (withthe main difference being that the NLSP decay length is parametrically longer [59]).The AdS equivalent of this type of mediation is as shown in Figure 8. We begin withthe ISS configuration of the previous section but add the elementary messenger fields onthe UV brane. As we have seen an additional η meson is required on the UV brane andthis will in general mix with the ϕ meson through its couplings to the bulk field Φ . We willimplicitly assume – in order to justify having bulk gauge bosons – that some of the fields inthe non-supersymmetry breaking sector (i.e. the SU( F ) × SU( F ) sector above) also coupleto the Standard Model gauge groups, but that only the messenger field couples to η . Inaddition we require R -symmetry to be broken in the UV theory which is represented by anexplicit mass term for the messengers. The brane superpotentials are then by comparisonwith Eqs.(78) and (79) given by W UV = ηf ˜ f + m f f ˜ f + m η η (Φ − Φ ) W IR = qϕ ˜ q − m ϕ ϕ Φ . (80)32s we saw this is the theory that remains after all of the confining physics described abovehas taken place, so that q , ˜ q and ϕ represent the fields in the low energy F × F flavourblock. Without the Φ term the anomaly-free R -symmetry of Table 2 is unbroken. Asrequired by comparison to the ISS model, the Φ term induces an expectation value for Φ on the boundary (equivalent to m Q in the underlying strongly coupled QCD theoryby the bulk/boundary correspondence) because of the | ∂W UV /∂η | term in the effectivepotential. (Note that, as we shall see in a moment, we cannot just set ∂W UV /∂η = 0 because supersymmetry is broken.) This leaves a residual but anomalous R -symmetry.The mass term for the gauginos then breaks the R -symmetry entirely on the UV boundaryas in Ref. [35], but the IR brane retains it. In this way the gravitational dual descriptionmakes it geometrically explicit that the approximate R -symmetry of the IR theory is anemergent phenomenon .Before presenting precise details, let us describe how we expect the suppressed medi-ation of Ref. [35] to operate in the gravitational dual description. As we have said, theboundary terms break supersymmetry and enforce a linear combination of the 4D η and ϕ fields to zero. To see how this happens it is useful to temporarily disregard the effect ofwarping and consider the 4D theory whose superpotential is simply the sum of W IR and W UV : W = ηf ˜ f + m f f ˜ f + m η η (Φ − Φ ) + qϕ ˜ q − m ϕ ϕ Φ . (81)One can use the residual flavour symmetry to diagonalise the problem, and it is theneasy to see that n diagonal components of all the F -terms can be set to zero by choosing Φ ii = Φ = ˜ q i q i /m ϕ for i = 1 . . . n . The remaining N contributions to the potential are V ⊃ F X i = n +1 m η (Φ ii − Φ ) + m ϕ Φ ii , (82)where to avoid confusion we are using the same symbol for the superfield and its scalarcomponent. Defining a mixing angle tan ϑ = m ϕ m η , (83)the metastable ISS minimum occurs at Φ ii = cos ϑ Φ ; V + (0) = N Φ m η sin ϑ , (84)with the effective F -terms for ϕ and η being given by µ ϕ = sin ϑ cos ϑ m η Φ µ η = sin ϑ m η Φ , (85) See Ref. [60] for an alternative example of this phenomenon. ϕ ′ = cos ϑ ϕ + sin ϑ η . (86)The low energy theory is then W mag = sin ϑ ϕ ′ f ˜ f + m f f ˜ f + qϕ ′ ˜ q + µ ϕ ϕ ′ . (87)Thus this purely perturbative 4D model is, at energies below m η , essentially the configura-tion of [35] and the gauge mediation is standard. In particular note that the field Φ nevercontributes to supersymmetry breaking.Now consider the strongly coupled theory modelled by a slice of AdS, depicted in Fig.8.The supersymmetry breaking sector is similar to the model above, but altered in two ways.First, the field Φ is now a bulk field and generally has a profile that warps down the effective µ ϕ . Second, the superpotentials themselves get an overall warp factor, which changes therelative sizes of the supersymmetry breaking contributions to the potential on the twobranes. Nevertheless, some coarse aspects of the 4D model above carry over. For examplethe bulk meson Φ never contributes to the supersymmetry breaking, which is insteadshared between the two branes. In the limit of large m η the supersymmetry breaking willall be on the IR brane. Moreover the interesting 5D dynamics all happens above the scale Λ IR . Below this scale the theory behaves like the 4D one described above, modulo thewarping of parameters. Therefore, if we choose a low messenger mass m f < Λ IR , thenthe mediation sector is oblivous to the strong coupling and the low energy phenomenologyclosely resembles that of Ref. [35]. The gaugino mass comes from the usual one-loopdiagram and one finds the usual 4D result: M λ = sin ϑ α π tr ( F ϕ ′ ) m f ≈ sin ϑ α π N µ ϕ m f , (88)in the limit of small ϑ . The scalar masses, being given by two loop diagrams, are similarin magnitude and, as in the 4D theory above, the phenomenology is similar to that ofordinary gauge mediation.However new 5D effects will occur if we choose m f ≫ Λ IR . The scale m f then definesa resolution scale much smaller than the typical length scale corresponding to the Kaluza-Klein separation. The loop integrals that contribute to supersymmetry breaking are theneffected by the localization of supersymmetry breaking on the IR brane. The net resultis a suppression of the scalar masses with respect to the gaugino masses which are stillgiven by Eq.(88). Naively one expects the suppression factor to be given by at least anextra loop factor for the scalars while the gauginos are from the AdS viewpoint a tree-leveleffect. This is nothing other than an AdS form of gaugino mediation very similar to thatin Ref. [23]. It is remarkable that via the AdS/CFT correspondence, the simple model ofRef. [35] becomes a straightforward implementation of general gauge mediation [57]! (Note34 , V F ϕ , q , ˜ q , v n η, f, ˜ f W UV = ηf ˜ f + m f f ˜ f + m η η (Φ − Φ ) W IR = qϕ ˜ q − m ϕ Φ ϕ Figure 8:
The configuration for “simplified” gauge mediation (c.f. Ref [35]). Into theproposed gravitational dual of SQCD we add messenger fields f, ˜ f on the UV brane thatinteract with the bulk meson. The latter provides the heavy (KK) modes that generate theeffective messenger/spurion coupling in the low energy theory. that the scalar mass-squareds in Ref. [23] indeed conform to the general sum-rules derivedin Ref. [57].)After this long heuristic discussion let us now present some precise details. The scalesinvolved are Λ IR ∼ z − IR for the IR physics and Λ UV ∼ k = z − UV for the UV physics, withthe cuts-off being related as Λ IR = Λ UV z UV /z IR . First the effect of the warping on thesupersymmetry breaking: using the bulk solution for the massless modes given in Eq.(22),and taking into account the warp factors in the canonical normalisation of ϕ , the lowenergy potential written in terms of Φ UV = Φ ( z UV ) is V ⊃ F X i = n +1 m η (Φ UV − Φ ) + (cid:18) z IR z UV (cid:19) − (1+2 c ) m ϕ Φ UV , (89)and thus our mixing angle is tan ϑ = (cid:18) z UV z IR (cid:19) + c m ϕ m η ≈ (cid:18) z UV z IR (cid:19) + c , (90)where the last relation is for the choice of m ϕ ≈ m η . (The power of ( + c ) reflectsthe relation between the canonically normalised fields in Eq.(33).) Thus, when c > − tan ϑ ≪ and the supersymmetry breaking is pushed to the IR brane. Note that if onthe other hand c < − then tan ϑ ≫ . The supersymmetry breaking then naively looksto be completely localized on the UV brane, with V + = N Φ m η , but in this case there isno metastability. Instead the non-perturbative terms discussed in Ref. [3] introduce globalsupersymmetric minima at a distance less than Λ IR away in field space, and so the Euclidean35unnelling action is of order S E ∼ π ϕ min /V + ≪ : the supersymmetry breaking has tobe an IR effect.We can model the supersymmetry breaking gaugino masses with local F terms on thebranes. The effective operators are given by W ⊃ a UV g Λ UV ηW α W α δ ( z − z UV ) + a IR g Λ IR ϕW α W α δ ( z − z IR ) , (91)where a UV , a IR are constants. To determine the value of the coefficients we need the 5Dpropagators which can be found in Refs. [23, 41, 42, 61]. The bulk gaugino propagator hasa denominator of the form ˜ J ( ipz IR ) ˜ H ( ipz UV ) − ˜ H ( ipz IR ) ˜ J ( ipz UV ) , (92)where H are Hankel functions of the first kind of order 1, the tilde modification is of theform ˜ J α ( w ) = ( − r + s/ − J α ( w ) + wJ α − , (93)where s = 1 for gauginos, and where the values of r for UV and IR branes respectively are r UV = −
12 + ipz UV a UV F η UV ; r IR = −
12 + ipz UV a IR F ϕ IR . (94)It is straightforward to extract the pole of the propagator by taking the pz IR , pz UV → limit, which gives a gaugino mass of M λ = z − UV Λ − UV z IR /z UV ) ( a IR F ϕ − a UV F η ) . (95)As usual, the effect of the warping is to scale down the masses in the operator on theIR-brane by a factor Λ IR / Λ UV . Also note that no gaugino mass results if a UV F η is equal to a IR F ϕ : in the F → ∞ limit this would correspond to having Dirichlet boundary conditionson both branes.For simplicity we shall henceforth neglect the small non-zero F -term that is inducedon the UV brane, setting a UV = 0 and focus on the IR-brane contribution. By comparingwith Eq.(88) and using the relation g k = g log( z IR /z UV ) , we determine a IR , to find thatthe gaugino mass is equivalent to an IR-brane localized term of the form W ⊃ sin ϑ z IR z UV ϕ π m f W α W α δ ( z − z IR ) . (96)Let us now compute the scalar mass-squared terms to see the suppression . In order to dothis we have to evaluate the one-loop contributions with bulk gauge fields in the loop. Themass-squared terms are given by m i = 4 g C ( R i )Π , (97) Note that much of this discussion is valid for RS1 models with F -terms on the boundary in general. C ( R i ) is the quadratic Casimir of the representation R i and Π = 1 k log( z IR /z UV ) Z d p (2 π ) [ G V ( p, z UV ) − G F ( p, z UV )] . (98)The gaugino, G F and gauge boson, G V propagators (which can be found in Ref. [23]) areevaluated at z UV , since the external squark fields are assumed to be localized on the UVbrane. For the gauge bosons we obtain the explicit expression G V ( p, z UV ) = 1 ip J ( ipz UV ) Y ( ipz IR ) − J ( ipz IR ) Y ( ipz UV ) J ( ipz UV ) Y ( ipz IR ) − J ( ipz IR ) Y ( ipz UV ) , (99)while the gaugino propagator is given by G F = 1 ip J ( ipz UV ) Y ( ipz IR ) − J ( ipz IR ) Y ( ipz UV ) − ξ ( J ( ipz UV ) Y ( ipz IR ) − J ( ipz IR ) Y ( ipz UV ) ) J ( ipz UV ) Y ( ipz IR ) − J ( ipz IR ) Y ( ipz UV ) − ξ ( J ( ipz UV ) Y ( ipz IR ) − J ( ipz IR ) Y ( ipz UV ) ) , (100) where ξ = a IR F ϕ IR z UV z IR parametrizes the amount of supersymmetry breaking on the IR brane.Note that in the ξ → ∞ ( F ϕ → ∞ ) limit the gaugino wave-function is completely repelledfrom the IR-brane by the non-zero F -term, and one recovers the Green function for twistedboundary conditions, with Neumann (Dirichlet) boundary conditions on the UV (IR)-brane [23], i.e. G F F ϕ → ∞ = 1 ip J ( ipz IR ) Y ( ipz UV ) − J ( ipz UV ) Y ( ipz IR ) J ( ipz IR ) Y ( ipz UV ) − J ( ipz UV ) Y ( ipz IR ) . (101)The gaugino mass in this case is pure Dirac, and there are no divergences in Π . We referto this as the “gaugino mediation limit” . However we are interested in the case when F ϕ is finite. In order to treat this more general case we use the following simplified expressionfor the propagator difference: G V ( p, z UV ) − G F ( p, z UV ) = ip ξz IR z UV + iξ Γ ) , (102)where Γ = I ( pz UV ) K ( pz IR ) − I ( pz IR ) K ( pz UV ) , Γ = I ( pz UV ) K ( pz IR ) + I ( pz IR ) K ( pz UV ) . (103)The I, K functions are modified Bessel functions and therefore Γ , are real valued. Using(102) we obtain (with x = pz IR ). i Π = − z − IR π log ( z IR /z UV ) Z ∞ dx iξ (Γ + iξ Γ ) . (104) In order to calculate these well-known results using current correlators, one would have to use theextended formalism of Ref. [62] which includes Dirac masses. The purely Majorana piece gets exponentiallysuppressed as in [63]. i from the Wick rotation of p → ip in d p has been placed on the LHS of (104). Inthe limit ξ → ∞ one obtains (using log( z IR /z UV ) = 34 . ) i Π = − z − IR π log( z IR /z UV ) Z ∞ dx Γ ≈ (0 . z − IR . (105)This corresponds to the real part of i Π and reproduces the twisted boundary conditionresult in Ref. [23]. Therefore the scalar mass-squared for finite ξ can be obtained byconsidering the real part of i Π . Using (104) we find ℜ [ i Π] = − z − IR π log ( z IR /z UV ) Z ∞ dx Γ Γ ξ (Γ + ξ Γ ) . (106)In the limit ξ → we find that ℜ [ i Π] ∝ ξ as one would expect in normal gauge mediation.The ratio of the scalar masses to the gaugino masses can be parameterised by γ such that Π = γ π M λ . (107)(Numerically the twisted boundary condition result is equivalent to γ = 1 . .) In the ξ → limit we have γ ≃ − (log( z IR /z UV )) Z ∞ dx Γ Γ . (108)A part of this ratio comes from the RG running contribution of the Majorana gauginomasses to the scalar mass-squareds. Therefore, as one would expect, the integral (108)is logarithmically divergent when M λ = 0 . In order to find the remaining piece we cancompare γ with the complete field theory expression for the contribution to the mass-squareds from each gauge factor (neglecting the running of the gauge couplings) [64]: Π a ( µ ) ≈ Π a ( Q ) + log (cid:18) Qµ (cid:19) M λ a π . (109)The logarithmic piece in the integral for γ exactly reproduces this RG running. Subtractingthis piece, we find that in the large log ( z IR /z UV ) limit the remaining finite contribution to Π( Q ) is given by lim z IR /z UV → ∞ [¯ γ ] = 12 log( z IR /z UV ) . (110)Numerically, this approximation is accurate to a few percent for log ( z IR /z UV ) = 34 . say.At first sight the apparent increase of ¯ γ with log ( z IR /z UV ) is a bit puzzling since heuristicallyone expects the supersymmetry mediation to scalars to tend to a constant, but actuallythis relation just reflects the “messenger content” in the bulk. Indeed this limit togetherwith the AdS/CFT relation g k = 8 π /b CFT (c.f. Eq.(38)) gives m i = X a C a b CFT M λ a . (111)38his is the AdS/CFT equivalent to the perturbative extra-ordinary gauge mediation re-lation where the contribution to the beta function coefficient is given by the number ofmessengers N mess , and where we have M λ ∼ m i N mess [38]. Here in the Veneziano limit theCFT contribution to the beta functions coefficients is instead b CFT ∼ N .This result was in the ξ → limit, which we will refer to as the “extra-ordinary gaugemediation limit”. However one can determine γ and extract ¯ γ numerically for the generalcase as we increase ξ . In order to do this we first note that when ξ = 0 the integral isno longer divergent. This is to be expected since the gaugino is massive and therefore thelogarithmic divergence has a natural IR cut-off about the gaugino mass scale. Indeed as x → the integrand in Eq.(106) can be approximated by log( z IR /z UV ) Γ Γ ξ (Γ + ξ Γ ) → x ξ (cid:16) log ( z IR /z UV ) + ξ x (cid:17) , (112)and this function is peaked at x ≡ pz IR = ξ/ log( z IR /z UV ) , that is precisely where p = M λ .The integrand and its approximation are shown in Figure 9. The main feature of the “extra-ordinary gauge mediation limit” is that in this region the gaugino pole is well separated,so that one can define a “messenger scale”, Q mess , below which the contribution to themass-squared integral is well described by the log( Q/µ ) piece in Eq.(109). In order tofind ¯ γ for arbitrary ξ therefore, we can divide the integral into two regions, with ¯ γ beingidentified with the contributions from above the scale Q mess where the two curves diverge.We can use the local minimum to define this point, whose location is well approximatedby the value of the gaugino mass in the gaugino mediation (large ξ ) limit (i.e. it is at0.24 TeV for the values chosen above). This procedure ceases to be meaningful for largevalues of ξ because the pole “melts” into the main contribution. At this point extraordinarygauge mediation behaviour goes over to gaugino mediation behaviour as in Ref. [23]. Thenumerically evaluated γ and ¯ γ (in the extra-ordinary GM region) are shown in Fig. 10. Figure 9:
Contribution to the scalar mass-squareds with momentum in units of Λ IR : thefull Green’s function (upper curve) and the gaugino pole approximation (lower curve). Extra-Ordinary GM Gaugino mediation γ ¯ γ Figure 10:
The parameter γ = 8 π Π /M λ , varying continuously from extraordinary gaugemediation to gaugino mediation behaviour as the relative supersymmetry breaking on theIR brane, ξ , increases. The ¯ γ line, representing the mass-squared value “at the messengerscale”, is the contribution with the gaugino RG term removed, relevant in the small ξ limit. As one final remark, it is worth highlighting the restricted form of general gauge me-diation that one derives from this model. It is by now well known that the most generalconfiguration for gauge mediation allows six independent parameters (assuming no CP vi-olating phases in the gaugino sector), three for the gaugino mass terms, and three for thesquarks [57]. There are five squark masses in total so this requires two sum rules, m Q − m U + m D − m L + m E = 02 m Q − m U − m D − m L + m E = 0 . (113)The squark masses derived here and in Ref. [23] (which are realisations of general gaugemediation in AdS) of course have to satisfy these rules. However there are only four freeparameters for the models discussed here, not six. Assuming that the gaugino masses aredriven by couplings to different F -terms or possibly different couplings to the same F -term,then they can be free parameters, however the mediation to the sfermions is only a functionof the AdS geometry and the suppression is the same for all the Standard Model gaugefactors. Therefore the pattern of soft-supersymmetry breaking can be written in terms of40he gluino mass M and three sfermion mass-squared parameters Π i =1 ... as follows: M = r Π Π M = r Π Π M m Q = 43 α Π + 34 α Π + 160 α Π m U = 43 α Π + 415 α Π m D = 43 α Π + 115 α Π m L = 34 α Π + 320 α Π m E = 35 α Π . (114) We have examined Randall-Sundrum (RS1) like configurations in strongly coupled 4D N = 1 supersymmetric field theory. By taking a large N (Veneziano) limit and combiningit with a Seiberg duality, we showed how one can construct a model in which a conformalphase with relevant operators (specifically quark mass terms) flows to a weakly coupledfree-magnetic phase. The bulk of these theories is approximated by the construction ofKlebanov and Maldacena [31]. The magnetic theory, including its gauge fields, lives entirelyon the IR brane as emergent degrees of freedom.We showed how this construction can be used to derive an RS1 version of the MSSMin which the SU(2) L gauge group is emergent. The SU(3) c and hypercharge gauge bosonsare bulk degrees of freedom and correspond to part of the “flavour” symmetries of theSeiberg duality. The right-handed fields are predicted to be entirely elementary, whereasthe left-handed fields are predicted to be a mixture of elementary and composite degreesof freedom. (The latter are identified as the mesons of the Seiberg duality.)We also showed how gaugino mediation can be implemented, by beginning with theMurayama-Nomura model of gauge mediation in Ref. [35] and taking its large N limit in thespecified manner. The metastable supersymmetry breaking of Ref. [3], being an emergentphenomenon, appears on the IR brane, while the matter fields and messenger fields (beingelementary degrees of freedom in the model) are on the UV brane. The Standard Modelgauge fields are bulk degrees of freedom and therefore gauginos get masses at leading order,whereas the sfermion mass-squareds, which have to be transmitted through the bulk, aresuppressed. The result is an AdS version of extra-dimensional gauge mediation. By varyingparameters, the pattern of supersymmetry breaking can be taken from extra-ordinary gauge41ediation (i.e. equivalent to a large number of messengers that are integrated out belowa typical mass scale) to AdS gaugino mediation similar to that of Ref. [23]. Due to theuniversal nature of the mediation, the model corresponds to general gauge mediation (withadditional Dirac gaugino masses) but with only four free parameters. Acknowledgements
We thank Mark Goodsell and Jose Santiago for discussions. SAA gratefully acknowledgesthe Royal Society and the University of Melbourne, and TG the IPPP Durham, for supportand hospitality. We also thank Yann Mambrini and Orsay, where part of this work wasdone. SAA is supported by a Leverhulme Senior Research Fellowship and the EuropeanRTN grant “Unification in the LHC era” (PITN-GA-2009-237920). TG is supported by theAustralian Research Council.
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