aa r X i v : . [ h e p - ph ] J u l Fermion Flavor in Soft-Wall AdS
Tony Gherghetta a, and Daniel Sword b, a School of Physics, University of Melbourne, Victoria 3010, Australia b School of Physics and Astronomy, University of Minnesota,Minneapolis, Minnesota 55455, USA
Abstract
The formalism for modeling multiple fermion generations in a warped extra di-mension with a soft-wall is presented. A bulk Higgs condensate is responsible forgenerating mass for the zero-mode fermions but leads to additional complexity fromlarge mixing between different flavors. We extend existing single-generation analy-ses by considering new special cases in which analytical solutions can be derived anddiscuss flavor constraints. The general three-generation case is then treated using asimple numerical routine. Assuming anarchic 5D parameters we find a fermion massspectrum resembling the standard model quarks and leptons with highly degeneratecouplings to Kaluza-Klein gauge bosons. This confirms that the soft-wall model hassimilar attractive features as that found in hard-wall models, providing a frameworkto generalize existing phenomenological analyses. E-mail: [email protected] E-mail: [email protected]
Introduction
The warped extra dimension provides an alternative framework in which to address theStandard Model (SM) gauge hierarchy problem [1] and the fermion mass hierarchy [2, 3]. Itis a particularly attractive scenario because, by the AdS/CFT correspondence [4], the five-dimensional (5D) framework is dual to a four-dimensional (4D) strongly-coupled conformalfield theory. This allows the physics of the warped fifth dimension to be recast in terms of4D strong dynamics. In particular, an infrared (IR) brane used to generate the Higgs cutoffscale is interpreted in the 4D dual as the breaking of conformal symmetry at low energyand the generation of a mass-gap by an operator of infinite scaling dimension. While theessential physics is captured in this theoretically idealized situation, it is more natural toexpect operators of finite dimension in the dual theory. This can be achieved in soft-wallmodels in which the “hard-wall” IR brane is replaced by a scalar field (the “dilaton”) whosenontrivial bulk profile corresponds to conformal symmetry breaking in the dual 4D theory.The Standard Model in the soft-wall warped dimension was considered in Ref.[5]. Sincethere is no IR brane, SM fields are necessarily bulk fields, which includes not only the gaugebosons and fermions, but also the Higgs field. Even though the fifth-dimension is semi-infinite, the dilaton does provide a dynamical cutoff to the warped dimension. This leads toa discrete Kaluza-Klein (KK) mass spectrum, but with the distinctive feature that there isa variety of KK spacing between the resonances, including linear Regge-like behavior as inQCD. A bulk Higgs condensate is responsible for breaking electroweak symmetry but causesthe analysis of fermions to be particularly involved. An analytical solution can neverthelessbe obtained in the case of a single fermion generation confirming that the nice features ofhard-wall scenarios, such as fermion mass hierarchies and universal KK gauge couplings, alsoexist with the soft wall [3]. However these features have yet to be confirmed in a completethree-generation soft-wall model.In this paper we present a more comprehensive analysis of bulk fermions in a soft-wallwarped dimension (see also [6, 7]). While new analytical solutions are found for specialcases in the case of a single generation, the complete three-generation case can only betreated numerically. This is because the bulk Higgs condensate causes large mixing betweenfermion flavors in the equations of motion which makes finding analytical solutions nontrivial.Nonetheless, numerical techniques can be used and we present a numerical routine that cansolve the general problem with arbitrary 5D mass parameters. Importantly we find thatstarting with “anarchic” 5D parameters we are able to generate fermion mass hierarchiesand universal couplings to KK gauge bosons, analogous to that found in hard-wall models.This provides a framework to generalize existing phenomenological analyses to include thesoft-wall Standard Model.The layout of this paper is as follows. In Section 2, we review the setup needed tomodel fermions in the soft-wall background and present the fermion equations of motion.In Section 3, we develop the tools needed to solve the equations of motion in the specialcases where the equations can be partially decoupled. One of these cases has been detailedbefore in Refs. [5, 6], while the remaining cases are new. We then show that these additionalanalytical cases can be used to reproduce many of the recent numerical results of [7]. We1onclude this section by discussing flavor changing neutral current processes and by detailingthe couplings of fermions to gauge bosons–an analysis which is easily generalized to otherbulk couplings. In Section 4, we present a very simple, non-iterative routine which can beused to analyze multiple generations of fermions in an arbitrary background. We presentthe full dependence of SM fermion masses on the 5D bulk mass parameters and comparethe results to a typical hard-wall model. The behavior is shown to be very different inthe phenomenologically interesting region of the parameter space, where the bulk SU (2) L doublet and singlet fermions have opposite bulk masses. We then present results for thecase of three fermion generations with substantial mixing between bulk profiles, and findexample spectra resembling the up- and down-type quarks (and charged leptons) in thespirit of Ref.[8]. We work in a 5D spacetime ( x µ , z ) with conformal coordinate z and metric: ds = e − A ( z ) η MN dx M dx N , (1)where η MN = diag( − , + , + , + , +). In particular we will consider a pure AdS metric, i.e. A ( z ) = log kz with k the AdS curvature scale. The spacetime is defined on the interval z ∈ [ z , ∞ ), where z is the location of the ultraviolet (UV) brane. Though the spacetimeextends to z → ∞ , we have in mind a soft-wall setup in which the dilaton, Φ obtains abackground value and provides a dynamical cutoff to spacetime along the fifth coordinate z .In this scenario, gauge and matter fields are described by the action, S = Z d x √− g e − Φ L , (2)where L is the 5D Lagrangian. While much of our discussion of fermions is valid in general,for the sake of concreteness we will specifically consider a dilaton profile given by:Φ( z ) = ( µz ) , (3)where the soft-wall mass scale µ ∼ L (Ψ R ) which transform as a doublet (singlet) under SU (2) L . It is straightforward to embed our setup in a theory with a bulk custodial SU (2) L × SU (2) R symmetry, but this will not be essential for our discussion. In the absence of Yukawainteractions, the fermion action is given by: S = − Z d x √− g e − Φ (cid:20) (cid:0) ¯Ψ aiL e MA γ A D M Ψ aiL − D M ¯Ψ aiL e MA γ A Ψ aiL (cid:1) + M ijL ¯Ψ aiL Ψ ajL + 12 (cid:0) ¯Ψ iR e MA γ A D M Ψ iR − D M ¯Ψ iR e MA γ A Ψ iR (cid:1) + M ijR ¯Ψ iR Ψ jR (cid:21) , (4)2here e MA = e A δ MA is the vielbein and D M = ∂ M + ω M is the covariant derivative with spinconnection ω M . The index a is an SU (2) label, while i, j are 5D flavor indices.The projections of the Dirac spinors are given by Ψ aiL ± = ± γ Ψ aiL ± and similarly for Ψ iR .Dirichlet conditions are imposed on the fields Ψ iaL − and Ψ iR + at the UV boundary:Ψ aiL − ( x, z ) (cid:12)(cid:12) z = 0 , Ψ iR + ( x, z ) (cid:12)(cid:12) z = 0 . (5)Without bulk Yukawa interactions these boundary conditions give rise to massless chiralfermions from the 4D point of view. These zero-modes can obtain a mass by introducing aYukawa coupling to the Higgs, whose vacuum expectation value (VEV) is z -dependent. TheYukawa interaction contribution to the action is: S Y ukawa = − Z d x √− ge − Φ " λ ij √ k ¯Ψ aiL ( x, z ) H a ( x, z )Ψ jR ( x, z ) + h . c . , ≡ − Z d x √− ge − Φ h m ij ( z ) ¯Ψ iL ( x, z )Ψ jR ( x, z ) + h . c . i , (6)where we have substituted the background value for the Higgs field: H ( x, z ) → H ( z ) = h ( z ) √ (cid:18) (cid:19) , (7)and dropped the SU (2) labels Ψ L ≡ Ψ L . The effective z -dependent bulk mass term arisingfrom the Yukawa interaction is simply: m ij ( z ) ≡ λ ij √ k h ( z ) . (8)To ensure a discrete spectrum of fermion masses, the Higgs VEV must grow faster than themetric factor, e A ( z ) = 1 / ( kz ), decays. Namely, lim z →∞ h ( z ) z → ∞ . (9)Varying the action with respect to ¯Ψ L,R , we find the equations of motion: γ µ ∂ µ ψ iL ± ∓ ∂ ψ iL ∓ + e − A M ijL ψ jL ∓ + e − A m ij ψ jR ∓ = 0 , (10) γ µ ∂ µ ψ iR ± ∓ ∂ ψ iR ∓ + e − A M ijR ψ jR ∓ + e − A m † ij ψ jL ∓ = 0 , (11)where we have defined Ψ = e A +Φ / ψ . This transformation shows that the fermion massspectra do not depend on the presence of the dilaton. Rather, it is the Higgs VEV that setsthe fermion spacing, in contrast to the case of bosonic fields. Other possiblities may also be considered. For example, if lim z →∞ h ( z ) /z → µ >
0, there can exist discretelow-lying modes with a continuous spectrum above a “mass gap”, as in Refs. [9, 6]. ψ L,R ± is assumed to be: ψ iL ± ( x, z ) = X n,α f iα ( n ) L ± ( z ) ψ α ( n ) ± ( x ) , (12) ψ iR ± ( x, z ) = X n,α f iα ( n ) R ± ( z ) ψ α ( n ) ± ( x ) , (13)where γ µ ∂ µ ψ α ( n ) ± = − m αn ψ α ( n ) ∓ (no sum over α ). Similar to the conventions of Ref.[8] we haveintroduced separate Latin and Greek indices labelling the 5D and 4D flavor, respectively.Defining the vectors: f iα ( n ) ± = f iα ( n ) L ± f iα ( n ) R ± ! , (14)allows the equations of motion for the 5D fields to be written in the form: (cid:2) ± ∂ δ ij + M ij (cid:3) f jα ( n ) ± ( z ) = m αn f iα ( n ) ∓ , (15)where the mixing matrix is defined as M = e − A (cid:18) M ijL m ij ( z ) m † ij ( z ) M ijR (cid:19) . (16)Note that α, i, j run from 1 , . . . N F , where N F is the number of fermion generations. Thus M ij is a 2 N F × N F matrix, and Eq.(15) represents a coupled system of 4 N F differentialequations for each α . The 4D fermion fields ψ α ( n ) ± ( x ) are canonically normalized by requiringthat: Z ∞ z dz h ( f iα ( n ) L ± ) † f iβ ( m ) L ± + ( f iα ( n ) R ± ) † f iβ ( m ) R ± i = δ nm δ αβ . (17)Note that the index i is to be summed over in this expression. This completes the generaldiscussion of the fermion setup. To obtain the spectrum of fermion masses, the equations ofmotion (15) are solved subject to the boundary conditions (5) and orthonormality conditions(17). The coupled equations (15) cannot be solved analytically except for a few special cases,depending upon the particular form of the Higgs VEV and the relative bulk masses for thefermions. Up to this point, the only solvable cases that have been presented in the literaturehave involved just a single generation with degenerate bulk masses for the fields Ψ L and Ψ R .As it turns out, in AdS space there are additional special cases which allow for the second-order equations of motion to be diagonalized and solved exactly. The collection of solvablecases provides a qualitatively complete picture of fermion behavior in the entire parameterspace. 4ext, we review the single generation case in detail. We begin very generally, emphasizingthat these methods apply to a wide variety of soft-wall models in AdS. We then specialize to aquadratic VEV and solve the equations of motion directly for the special cases. The analyticsolutions allow us to verify the results of the numerical routine we present in Section 4 (aswell as a recent numerical treatment in which the Yukawas are treated perturbatively [7]). For a single generation of fermions, equation (15) becomes:( ± ∂ z + M ) f ( n ) ± ( z ) = m n f ( n ) ∓ ( z ) , (18)where M is a 2 × M = e − A (cid:18) M L m ( z ) m ( z ) M R (cid:19) . (19)The equations for f ( n )+ and f ( n ) − can be decoupled by deriving a second-order equation from(18). The fields f ( n ) ± obey a Schr¨odinger-like equation: (cid:0) − ∂ z + V ± (cid:1) f ( n ) ± = m n f ( n ) ± , (20)where the “potentials” are given by: V ± ( z ) = M ∓ M ′ . (21)The difficulty in solving (20) is due to the fact that the mixing matrix generally cannotbe diagonalized through global transformations of the functions, f ( n ) L,R ± . However, there arespecial cases for which the second-order equations can be decoupled further. They occurwhenever: M L = M R , “degenerate” M L + M R ± ∂ z e A ( z ) = 0 . “split” (22)The “degenerate” case is separable in any background. The “split” cases are separableregardless of the Higgs VEV in AdS , where the split-case condition simply becomes M L + M R ± k = 0.For generic forms of the Higgs VEV, it is most useful to work with transformed fields, g ( n ) ± = g ( n ) L ± g ( n ) R ± ! = U f ( n ) ± ≡ √ (cid:18) − (cid:19) f ( n ) L ± f ( n ) R ± ! . (23)In this basis the equations of motion are given by, (cid:16) ± ∂ z + f M (cid:17) g ( n ) ± ( z ) = m n g ( n ) ∓ ( z ) , (24)5here f M = U † M U = e − A (cid:18) m ( z ) + M L + M R M L − M R M L − M R − m ( z ) + M L + M R (cid:19) , (25)while the boundary conditions (5) become: g ( n ) L ± (cid:12)(cid:12) z = ± g ( n ) R ± (cid:12)(cid:12) z . (26)We may also define transformed potentials, e V ± , in direct analogy with (21). For the degen-erate case, both of the potentials e V + and e V − are simultaneously diagonal in this basis. In thesplit cases, only one of the potentials e V ± will be diagonal. After solving the correspondingpair of decoupled second-order equations, the first-order equations (24) can then be used togenerate the remaining solutions.Below, we consider the degenerate case and one of the two split cases, M L + M R + k = 0,assuming the following form for the Higgs VEV: h ( z ) = ηk / µ z , (27)giving m ( z ) = bk ( µz ) where b = λ η/ √
2, as in [5]. We also parameterize the bulk massesin units of the AdS curvature, M L,R = c L,R k , where c L,R are dimensionless coefficients.
The solution to the degenerate bulk mass case c L = c R = c was presented in detail in [5].The lowest-lying mode was found to be: m ≃ ( bµ Γ( − / | c | ) ( bµ z ) − / | c | for | c | > , bµ π sec cπ − ψ (1 / − c ) − ψ (1 / c ) for | c | < , (28)where ψ is the digamma function. Note that the lowest-lying mode is very light for µz ≪ | c | > / | c | ), while for | c | < / bµ . Here we consider one of the “split” cases, c L + c R + 1 = 0. The other case, c L + c R − e V + for any Higgs VEV in AdS. However, for our choice, h ( z ) ∼ z , the untransformed potential V − happens to be diagonal, V − = c ( c − z + b µ z ( c +1)( c +2) z + b µ z ! , (29)so we will work in this basis.A consistent solution requires that either f ( n ) L − = 0 or f ( n ) R − = 0. This is a peculiarity ofthe particular choice of the Higgs VEV and will not be true for other forms. The result is6hat the full tower of orthogonal solutions is most easily described in terms of two “distinct”KK towers of solutions. The first solution is: f ( n ) L − ( z ) = N ( n ) L − e − bµ z / z c U (cid:18)
14 + c − m n bµ ,
12 + c, bµ z (cid:19) , (30) f ( n ) R + ( z ) = bµ zm n f ( n ) L − , (31) f ( n ) L + ( z ) = 1 m n (cid:16) cz f ( n ) L − − f ( n ) ′ L − (cid:17) , (32) f ( n ) R − ( z ) = 0 , (33)where N ( n ) L − is a normalization constant and U ( a, b, y ) is the Tricomi confluent hypergeometricfunction. For this tower, the boundary conditions f ( n ) R + (cid:12)(cid:12) z = 0 and f ( n ) L − (cid:12)(cid:12) z = 0 are equivalent.There is only a single orthonormality condition, Z ∞ z dz f ( n ) L − f ( m ) L − = δ nm , (34)which in fact implies the correct orthonormality condition for the remaining fields, Z ∞ z dz h f ( n ) L + f ( m ) L + + f ( n ) R + f ( m ) R + i = δ nm . (35)The other KK tower is given by: f ( n ) R − ( z ) = N ( n ) R − e − bµ z / z c U (cid:18)
54 + c − m n bµ ,
52 + c, bµ z (cid:19) , (36) f ( n ) L + ( z ) = bµ zm n f ( n ) R − , (37) f ( n ) R + ( z ) = − m n (cid:18) cz f ( n ) R − + f ( n ) ′ R − (cid:19) , (38) f ( n ) L − ( z ) = 0 , (39)where N ( n ) R − is a normalization constant. The spectrum for this tower is found by imposingthe boundary condition f ( n ) R + (cid:12)(cid:12) z = 0, while the normalization condition is, Z ∞ z dz f ( n ) R − f ( m ) R − = δ nm . (40)The lowest lying mode of the second tower is very light and approximating the mass re-quires some care. For m ≪ bµ , we can expand the functions using techniques of so-calledboundary perturbation theory of quantum mechanics [10, 11]. The function f ( n ) R − obeys theSchr¨odinger-like equation, − ∂ z f ( n ) R − + (cid:20) ( c + 1)( c + 2) z + b µ z (cid:21) f ( n ) R − = m n f ( n ) R − . (41)7or small m , we can write f (0) R − as a product of the zero-mode solution and a correction: f (0) R − = ζ ( z ) F ( z ) , (42)where ζ ( z ) satisfies the zero-mode equation − ∂ z ζ + (cid:20) ( c + 1)( c + 2) z + b µ z (cid:21) ζ = 0 . (43)The solution may be written as, ζ ( z ) = N (0) R − z / K ν ( bµ z / , (44)where N (0) R − is a constant, K ν is the modified Bessel function and ν = 3 / c/
2. The function F ( z ) obeys the second-order equation, (cid:2) ζ − ∂ z ( ζ ∂ z ) + m (cid:3) F ( z ) = 0 , (45)and may be expanded in powers of m as, f (0) R − ≃ ζ ( z ) (cid:20) m Z zz dz ′ ζ − Z ∞ z ′ dz ′′ ζ + O ( m ) (cid:21) . (46)Such an expansion has also been used in Ref.[12] to approximate wavefunctions in soft-wallmodels. In contrast, here we are using it to solve the boundary value problem. The UVboundary condition, f (0) R + (cid:12)(cid:12)(cid:12) z = (cid:18) f (0) ′ R − + 1 + cz f (0) R − (cid:19) (cid:12)(cid:12)(cid:12) z = 0 , (47)results in the following approximate expression for m : m ≃ K − ν ( bµ z / K ν ( bµ z / I ( z ) z , (48)where I ( z ) ≡ Z ∞ z dz ′ ζ ( z ′ ) . (49)The expression (48) can now be expanded for small z . For c > − / m ≃ c/ / c/ / (cid:18) b µ z (cid:19) "(cid:18) bµ z (cid:19) c − / Γ ( − c/ /
4) + Γ ( c/ − / . (50)In the limit c ≫ /
2, this expression simplifies further to m ≃ (cid:18) c − / (cid:19) b µ z . (51)8 - c L m Ž n Figure 1: The first several masses in the split case plotted as a function of c L . The separateKK towers (dashed and solid lines) coincide for large negative and positive values of c L .Here, ˜ m n = m n /bµ and µz ≪ m min = ( µz ) bµ .For c ≪ − /
2, the above expansions are poor approximations because the mass becomes O ( bµ ). To deal with this regime, we can apply mathematical techniques from “supersym-metric quantum mechanics” to determine the mass [13]. Consider a quantum mechanicalsystem for which the Hamiltonian may be factorized as:[ − ∂ z + W ( z )] [ ∂ z + W ( z )] ψ = m n ψ. (52)The “superpotential,” W ( z ) = 1 + cz + bµ z, (53)gives rise to the “ordinary” potential for the function ψ : V ( z ) = W − W ′ = ( c + 1)( c + 2) z + b µ z + (2 c + 1) bµ . (54)It is clear from (52) that there exists a zero mode solution, ψ ∼ e − R W , with boundaryconditions that are given trivially by the equations of motion. In the limit µz →
0, however,this is equivalent to the boundary condition (47). Since the potential in (41) is equivalent to(54) up to a constant shift of the reference potential, we can conclude then that the solution ψ ∼ e − R W is in fact a good approximation for f (0) R − and that, m ≃ ( − − c ) bµ . (55)9his is clearly only valid when c < − /
2. We have checked that the expressions (51) and(55) match the exact results well in this region. The full spectrum is plotted in Fig. 1. Thedistinctive KK tower structure of the split case suggests the possibility of novel KK physicsunlike that found in hard-wall models and may be interesting to study in other soft-wallbulk Higgs models as well.
Recently, the possibility of modeling fermions by introducing non-constant bulk Dirac massterms has been considered in Ref.[7]. For a single generation setup with quadratic bulk massterms, the equations of motion are the same as (18), but now the mixing matrix (16) can bewritten effectively as, M = e − A (cid:18) c L k + c L kµ z bkµ z bkµ z c R k + c R kµ z (cid:19) , (56)where c L,R , c L,R are constant coefficients. The effect of this non-constant bulk mass is thatnormalizable zero-modes persist (depending on the choice of the signs of c L,R ) even in thelimit b →
0. For small values of b , the spectrum may be found by treating the bulk Yukawainteraction as a perturbation on the b = 0 solutions.Such an approach can be related to ours in some cases. For example, in the case ofdegenerate constant mass pieces, c L = c R , global unitary transformations may still be usedto diagonalize the mass matrix when the bulk masses have the same functional form as theHiggs VEV. Thus, the introduction of non-constant bulk masses can be viewed as effectivelychanging the boundary conditions on the fields in such cases.It is interesting to note that the case considered in [7] is similar to our “split” case. Inparticular, they examine c L = − c R and c L = − c R in detail. In the slightly different splitconfiguration for the constant pieces of the bulk mass, c L = ± − c R , analytical solutionscan be obtained in a similar fashion to our earlier analysis. For c L = − − c R , we find thelowest lying KK tower to be: f ( n ) R − ( z ) = N ( n ) R − e − ˜ bµ z / z c L U (cid:18)
54 + c L − ˜ m n bµ ,
52 + c L , ˜ bµ z (cid:19) , (57) f ( n ) L + ( z ) = bµ zm n f ( n ) R − , (58) f ( n ) R + ( z ) = − m n (cid:18) f ( n ) ′ R − + 1 + c L z f ( n ) R − + c L µ zf ( n ) R − (cid:19) , (59) f ( n ) L − ( z ) = 0 , (60)where we have defined effective parameters, ˜ b = ( c L ) + b , and ˜ m n = m n − c L (2 c L + 1) tomake the comparison with (36)-(39) clear. Note that there remains a lower bound on themass for c L ≫ /
2. We have checked that this solution describes the large c L behavior forthe case considered in Ref.[7], c L = − c R . We expect that all of the basic features of thenon-constant bulk mass model should be contained within our exact solutions.10enerically the split and degenerate cases allow one to find the spectrum exactly bysolving a set of decoupled second-order equations. Even when the equations cannot be solvedexactly, approximate methods such as those we have described above may be employed.Additionally, as our numerical results will verify, one can expect the behavior in these specialcases to provide a complete qualitative picture of the full parameter space dependence. Of significant interest in models involving extra dimensions is the coupling of fermions tothe KK gauge bosons. When the fermions are localized at different points along the ex-tra dimension, they can obtain non-universal couplings to the excited gauge bosons. Suchnon-universality will generically lead to large contributions to flavor physics observables,providing very stringent lower bounds on the allowed KK scale [14].In hard-wall models, the couplings can become universal for certain regions of the pa-rameter space, resulting in a GIM-like suppression of flavor changing neutral currents [3],thereby greatly lowering the bound on the allowed KK scale. We therefore would like to seeif a similar effect is present in the soft-wall case. Moreover, we would like to develop ourformalism in such a way that multiple fermion generations can be incorporated.The couplings of the zero mode fermions to the KK gauge bosons are found to be: g αβ ( n ) ± = g Z ∞ z dz f ( n ) A h ( f iα (0) L ± ) † f iβ (0) L ± + ( f iα (0) R ± ) † f iβ (0) R ± i , (61)where f ( n ) A is the gauge boson profile along the extra dimension. The gauge boson profilesarising from a quadratic dilaton (3) were derived in [5]. Here we simply use the results. Thezero-mode couplings g αβ (0) ± ≡ gδ αβ remain universal due to the orthonormality condition(17) and the flat zero mode gauge boson profile. This is because the dilaton factor explicitlycancels and plays no role.The degenerate single-generation case was considered in [5], where it was found that onlyone of the couplings, g + or g − , can become universal due to the opposite localizations ofthe fermion modes. In Ref [7], it was seen that opposite constant and non-constant bulkmasses led to universal couplings for both g + and g − . This happens as well for the split casesolutions. We have plotted the couplings for this case in Figs. 2 and 3. We find that thecouplings g + and g − become universal simultaneously whenever c ≫ / m K arising from non-universal couplings.The effective 4D Lagrangian contains operators that are suppressed by the squared massesof the KK gauge bosons mediating the strangeness-changing transitions (∆ S = 2): L ∆ S =2 ⊇ ∞ X n =1 M n h ¯ d αL ˜ g αβ ( n )+ γ µ d βL + ¯ d αR ˜ g αβ ( n ) − γ µ d βR + h . c . i , (62)11 - c L - - - - g + n (cid:144) g Figure 2: The ratio g n + /g for n = 1 (solid), n = 2 (dashed), and n = 3 (dotted) KK gaugemodes coupled to the zero-mode fermion in the split case, as calculated using (61). - - c L - - - - g - n (cid:144) g Figure 3: The ratio g n − /g for n = 1 (solid), n = 2 (dashed), and n = 3 (dotted) KK gaugemodes coupled to the zero-mode fermion in the split case, as calculated using (61).12here the sum is over the gauge boson KK modes with KK masses M n , and ˜ g αβ ( n ) ± = V dL,R g αβ ( n ) ± V d † L,R with V dL,R generic unitary matrices [14]. Thus, in the presence of non-degenerate couplings to the bulk KK gauge bosons, bounds from flavor experiments maybe interpreted as a lower bound on the KK scale.The key point is that the total amount of suppression in (62) depends upon the spacingof the KK tower. In a hard-wall model, for example, m n ∼ n M KK , where M KK is theKK mass scale. This compares with the soft-wall scenario where it would seem to imply aproblem, because the squared mass trajectories grow generically as m n ∼ nM KK (indeed,this spacing was the original motivation for studying the soft-wall [15]). While the sum of1 /n diverges as n → ∞ , we should of course truncate the sum at some high energy cutoff.Nevertheless, the naive implication is that the constraints on soft-wall models should beconsiderably tighter.However, this argument ignores the fact that the gauge bosons become increasingly IRlocalized with increasing mode number n . Thus, any off-diagonal terms in the gauge couplingmatrices are further suppressed for large n . By performing a numerical fit using the firstseveral dozen gauge boson modes and our split case solutions, we find that the couplings falloff as n − . to a very good approximation in the region where the couplings are independentof localization. This implies that the terms in the sum (62) grow as n − . . All other thingsbeing equal, this implies that the constraints from flavor physics are roughly a factor of twomore stringent in this model than in hard-wall models.While this presents no great problem for the model with a quadratic dilaton, for ageneric power law behavior in the dilaton, Φ ∼ z α , the spectrum of gauge bosons growsas m n ∼ n − /α M KK [5]. This means that for less steep potentials, even tiny amountsof non-degeneracy among the bulk couplings has potentially severe implications for flavorphysics. The analytical solutions that we have presented are of limited use, and instead we would liketo solve the full fermion mass problem including flavor. Our goal is to solve the eigenvalueproblem (15) with mixed boundary conditions. The “initial conditions” (5) specify half ofthe boundary values at the UV brane or, equivalently, half of the integration constants forthe system. The remaining constants of integration are fixed by the normalization conditions(17), which can only be satisfied if the eigenvalue, m αn , has been chosen correctly.We convert the problem to an initial value one by extending the shooting method tolinear boundary value problems of arbitrary order [16]. The solutions to (15) may be writtenas: f iα ( n ) ( z ) = U ( m αn ; z, z ) ij f jα ( n ) ( z ) , (63)where the propagator U ( m αn ; z, z ) is a linear operator and the f iα ( n ) ( z ) are 4 N F × N F matrix-valued functions for N F fermion generations.13 egeneratesplit - - c L - - - - - H m (cid:144) b Μ L Figure 4: Lowest lying masses for the “degenerate” ( c = c L = c R ) and “split” ( c = c L = − − c R ) cases, where the solid lines are determined in Ref.[5] (degenerate) and from (48)(split). The dots represent values obtained using the numerical method of Section 4.1.The matrix elements of U may be found by integrating a set of 4 N F linearly independentbasis vectors that span the space of initial values, f iα ( n ) ( z ), and inverting (63). The initialvalues that lead to normalizable solutions correspond to eigenvectors of U ( m αn ; z, z ) withvanishing eigenvalues in the limit z → ∞ . There are generally 2 N F such eigenvectors.Numerically, we can estimate the values of these vectors by considering the eigenvectorsof U ( m αn ; z , z ), where our cutoff satisfies z ≫ µ − . In practice, results are much morereliable if one starts the forward integration from some intermediate range z ∗ ∼ µ − andthen integrates the normalizable modes back to z . Variations on this theme can be explored.We scan over m αn , at each point integrating the system from a set of initial values so as toreconstruct the normalizable solutions. If there exists a linear combination of the solutionsthat matches the boundary conditions (5), then m αn is a solution to the system. To determinewhen this occurs, we define a merit function as the absolute value of the determinant of amatrix and search for a minimum. The matrix we use has columns formed by projectingout of the normalizable initial value vectors those components that are not restricted by theboundary conditions. These projected column vectors must be linearly dependent in orderto satisfy the boundary conditions of the problem.When the hierarchy between µ and z − is very large, increasingly high precision is nec-essary to achieve reliable results. Iterative methods may be better suited to the problem insuch cases. Our primary goal is to highlight the differences between fermions in soft-wall andhard-wall scenarios, and the speed and simplicity of this technique are its chief advantages.For this reason, we have limited our attention to a modest hierarchy.14 - - - - - - - - - - - - - - - - - - - c L c R Figure 5: Contours of log ( m / √ bµ ) for the lowest lying masses in our soft-wall setup with √ bµz = 10 − . We first present results for a single generation of fermions, as this case illustrates the essentialfeatures of the fermion mass behavior in the soft-wall, and allows us to compare our numericalresults with the analytical cases in the appropriate limits as well as to a typical hard-wallsetup. Assume the following values of the parameters: µ = 1 TeV; µz = 10 − ; b = 1 . (64)In Figure 4, we compare the numerical results to the analytical results from Section 3.1where it can be seen that the two methods agree very well. Next in Figure 5 we plot thefermion mass contours to show the full dependence on the parameters c L and c R . The shapeof the plot is easily understood from the analytical results. The numerical solution smoothlyinterpolates between the solutions along the lines c L = c R and c L = ± − c R . Because asimilar analysis can be repeated for other Higgs VEVs, this provides a natural way to beginstudying the qualitative aspects of other models in AdS as well.We can compare the soft-wall behavior with a typical hard-wall setup. In Figure 6 weprovide the corresponding contour plot for a hard-wall model in which the SM fermion massesare simply proportional to the values of the wavefunctions at z = 1 /µ . The most strikingdifference between the plots occurs in the lower right-hand corner. This is the region where c L > / c R < − /
2. 15 - - - - - - - - - - - - - - - - - - - - c L c R Figure 6: Contours of log ( m / √ bµ ) for the lowest lying masses in a typical hard-wall setupwith √ bµz = 10 − .The hard-wall case is characterized by a steep dependence on the bulk mass in this region,where the wavefunctions are proportional to f (0) L + ∼ z − c L and f (0) R − ∼ z c R . For z ≪ µ − , thenormalization constants become vanishingly small: N (0) L,R ∼ z − / ± c L,R . (65)Thus, the values of the functions in the IR at z = µ − are additionally suppressed. This isthe well-known mechanism for generating SM mass hierarchies in Randall-Sundrum scenarioswith bulk fields [2, 3]. For the soft-wall case, however, we can see the lower bound on themass in this region, m ∼ ( µz ) µ, (66)as indicated by the approximate expression (51). This can be understood be noting thatthe normalization (17) involves the sum of two types of fermion contributions which aregenerically not simultaneously suppressed. Next we aim to provide concrete numerical examples involving three generations of fermionsthat fully take into account the 5D flavor mixing to show that the attractive features of thesoft-wall are maintained. For multiple generations, there are three matrices that parame-terize the fermions: two bulk mass matrices M L and M R , and the bulk Yukawa matrix, λ .16e take the action (4) to be written in an arbitrary basis, for example, the CKM basis.Absent some symmetry, there is no reason to expect any structure relating the entries of thevarious bulk parameter matrices. We generically expect that the entries of each matrix areall of order unity (in units of the AdS curvature scale, k ), and that the various matrices aremisaligned. There is of course some basis in which both M L and M R are diagonal. Thus, by“misaligned,” we mean that this basis is distinct from the one in which the Yukawa matrixis diagonal. Indeed, the typical approach is to work in this basis, treating the Yukawa inter-actions as perturbations. Such an approach has been used in both hard-wall [2, 3, 17, 8, 18]and even very recently in soft-wall setups [7].In Ref.[7], it was found that one needed to include the first several ( ∼
10) KK modes inorder to achieve reliable results in such a perturbative expansion when including only a singlegeneration. At such a point, the analysis is essentially a numerical exercise. In our view,it is advantageous to include the entire KK tower in the numerical formulation whereverpossible. In other words it may make the most sense to simply solve the equations of motion(15), which guarantee the orthogonality of the eigenfunctions due to the hermiticity of themixing matrix.We expect that all other interactions may be treated reliably as perturbations. Thisis because the Higgs grows unbounded in the IR where it is the dominant contribution tothe fermion equations of motion. Other observables may thus be calculated using the usualwavefunction overlap approximation. As an application, we will calculate the couplings toexcited gauge bosons for examples involving three generations.We do not attempt to set precise bounds on soft-wall models here, as doing so goessignificantly beyond the scope of this work. Electroweak and flavor constraints have beendiscussed in the context of soft-wall models in Refs. [5, 7]. Detailed analyses in varioushard-wall scenarios can be found in [14, 2, 3, 17, 19, 18] and references therein.However, we will require that the eigenvalues of the bulk mass matrices satisfy m iL & k/ m iR . − m iL in order to get nearly degenerate gauge couplings. Because of the lowerbound on the fermion masses at m ∼ ( µz ) µ in this region, it is clear that the hierarchyconsidered above, µz = 10 − will be inadequate for generating MeV scale masses when µ = 1 TeV, and will only be possible for µz . − . Thus we again assume a quadraticHiggs VEV, h ( z ) = ηk / µ z , and the following for our input parameters: µ = 1 TeV; µz = 10 − . (67)Dealing with much larger hierarchies presents significant numerical challenges. However, thequalitative results of such an analysis should not be substantially different from the resultspresented here.First, we present an example resembling down-type quarks (or charged leptons). Forsimplicity, we take the entries of M L to be nearly degenerate, but we allow for large non-degeneracy in the matrix M R as well as in the Yukawa matrix. Specifically, we consider, M L k = . − .
020 0 . − .
020 0 . . .
024 0 . . , M R k = − . − . − . − . − . − . − . − .
354 0 . , H z L (cid:144) z F -Α Figure 7: The down-type fermion bulk profiles F α − ( z ) (in units of √ µ ) for the first generation(solid), second generation (dashed) and third generation (dotted) showing the overlap withthe Higgs VEV h ( z ) (in units of µ / √ k with η = 1). η √ λ = .
422 0 . − . − .
007 0 .
928 0 . . − .
327 0 . . (68)We find a spectrum of masses resembling the down-type quarks (or charged leptons): m α = 0 .
57 MeV , .
08 MeV , .
310 GeV . (69)The fermion mass hierarchy is clearly obtained, but due to the complexity of the numericalprocedure we do not match the SM masses exactly, and postpone a more detailed analysisfor future work. The fermion bulk profiles, F α − ( z ) = q ( f iα (0) L − ) † f iα (0) L − + ( f iα (0) R − ) † f iα (0) R − areplotted in Figure 7. The fermion profile overlap with the Higgs VEV, h ( z ) leads to thefermion mass hierarchy. The corresponding bulk profiles, F α + ( z ) are not plotted because theprofile differences between the flavors are not as pronounced. This is due to our choice ofUV boundary conditions and bulk masses (68).From expression (61), we can calculate the coupling of the zero mode fermions to theKK gauge bosons (i.e. gluons). The result is a matrix whose off-diagonal entries contributeto flavor violation. We obtain the following results for the first two KK gauge coupling18 H z L (cid:144) z F -Α Figure 8: The up-type fermion bulk profiles F α − ( z ) (in units of √ µ ) for the first generation(solid), second generation (dashed) and third generation (dotted) showing the overlap withthe Higgs VEV h ( z ) (in units of µ / √ k with η = 1).matrices, normalized to the coupling to the massless gauge boson: g (1)+ g = .
186 10 − − − .
187 2 × − − × − . , g (2)+ g = .
140 10 − − − .
138 10 − − − . ; g (1) − g = . ≈ ≈ ≈ .
188 10 − ≈ − . , g (2) − g = . ≈ ≈ ≈ .
139 10 − ≈ − . . (70)This behavior is maintained for higher modes as well. For this choice of parameters, the verynearly degenerate couplings imply that µ of order a few TeV will be consistent with flavorconstraints [19, 18, 6]. Note that we have assumed no contributions to CP violation. Thusthe soft-wall model can accommodate the fermion mass hierarchy with large bulk mixingand small flavor violation.The up-type quarks are only moderately more sensitive to the presence of the top quarkwhen large bulk mixing is allowed. For the choices (67) we obtain M L k = . − .
005 0 . − .
005 0 .
785 0 . .
017 0 .
066 0 . , M R k = − . − . − . − . − . − . − . − . − . ,η √ λ = . − . − . − .
079 0 . − . − . − .
321 1 . , (71)19hich gives rise to the following mass spectrum: m α = 2 .
10 MeV , . , . . (72)Again we see that the correct fermion mass hierarchy can be obtained. The fermion bulkprofiles, F α − ( z ) are plotted in Figure 8. The fermion profile overlap with the Higgs VEV, h ( z )leads to the fermion mass hierarchy. Similarly to the down-type fermions, the correspondingup-type bulk profiles F α + ( z ) are not plotted because the profile differences are negligible dueto the choice of UV boundary conditions and bulk masses (71). The gauge couplings arenearly universal among the first two generations: g (1)+ g = .
186 2 × − × − × − .
185 10 − × − − − . , g (2)+ g = .
140 10 − − − .
139 10 − − − − . ; g (1) − g = .
188 2 × − × − × − .
183 10 − × − − − . , g (2) − g = .
140 10 − × − − .
137 2 × − × − × − − . . (73)Constraints from top quark physics are significantly weaker, so this is not expected to affectthe bound on µ . We have presented a variety of tools useful for studying fermion physics in soft-wall back-grounds, focusing heavily on the treatment of fermion masses. The equations of motion arenon-trivial to solve and generically require numerical techniques. However, we have doc-umented several special cases for which it is possible to decouple the equations of motion.These cases serve as useful examples for qualitatively understanding the full parameter spacebehavior, as they illuminate independent “axes” of the parameter space along which fermionbehavior can be understood in detail. The utility of our approach is due not only to the factthat it effectively reduces the problem to solving a set of one-dimensional Schr¨odinger-likeequations, for which many theoretical and numerical tools have been created, but also tothe fact that it applies to any soft-wall model in AdS space. This opens up the possibility ofanalyzing fermions in a wide variety of Higgs models. For example, it should be possible toanalyze fermion physics in unHiggs scenarios, such as that considered in Ref.[9], or to studyother power-law Higgs behavior, as has been examined in the degenerate case in Refs.[5, 7].Furthermore, we have outlined methods for calculating fermion masses and wavefunc-tions in an arbitrary background with multiple flavors and arbitrary bulk parameters. Theformalism maintains the orthogonality of the KK tower, making it particularly useful forstudying the experimental consequences of soft-wall models with additional bulk fields. Forexample, we showed explicitly how to calculate the fermionic couplings to KK gauge bosons.The off-diagonal entries in the coupling matrix are directly related to the amplitudes of flavor20hanging neutral current processes. Moreover, we argued that the experimental constraintson new sources of flavor violation are generically tighter in soft-wall models than in hard-wallmodels due to the smaller spacing of the KK resonances. While the tightening is not tooconstraining in a model with a quadratic dilaton, for an IR cut-off growing much less quicklythan z , the constraints can become severe for model building.We described a very simple numerical technique for calculating fermion spectra, whichwe used to show the full parameter space dependence of SM fermion masses on the bulkmass parameters. The technique maintains the attractive features of the formalism, suchas an orthogonal KK tower and extends naturally to incorporate several generations offermions. Thus, the solutions allow for the straightforward calculation and interpretation ofnew physics observables.Both the analytical and numerical results suggest the potential for rich collider physicsthat is substantially different from that obtained in hard-wall models. While our results werefor a particular choice of a bulk Higgs VEV, they demonstrate that a soft-wall backgroundcan lead to a distinctive phenomenology. For example, our results indicate the presence ofa lower bound on fermion masses in a large area of the parameter space, suggestive of aseesaw-like mechanism. This could easily be implemented to explain neutrino masses in ascenario where the hierarchy between k and µ is of order the GUT scale. By introducing threeadditional right handed neutrino fields, a fairly random difference between the bulk massesof the right-handed neutrinos and charged leptons could naturally lead to light neutrinomasses. This would be an interesting variation on the ideas that are well-known in thehard-wall picture (cf. [2, 17, 20]).Even with these differences, the essential and attractive features of the hard-wall can beretained in our model. We presented results for three generations of fermions with anarchic5D parameters that reveal standard model-like particle masses and GIM-like suppression ofKK gauge boson mediated flavor changing neutral currents. We argued that this implied afairly modest bound on the KK scale. A more general analysis of flavor physics bounds willlead to stringent constraints in the soft-wall model. Having developed the tools needed toexamine electroweak and flavor physics in full detail, a more detailed study can be now beundertaken. Acknowledgements
We thank Brian Batell, Thomas Kelley, Arkady Vainshtein, and Mikhail Voloshin for helpfuldiscussions. The work of T.G. is supported by the Australian Research Council while thatof D.S. is supported by a Fellowship from the School of Physics and Astronomy at theUniversity of Minnesota.
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