Some problems with reproducing the Standard Model fields and interactions in five-dimensional warped brane world models
aa r X i v : . [ h e p - t h ] F e b Some problems with reproducingthe Standard Model fields and interactions infive-dimensional warped brane world models
Mikhail N. Smolyakov, Igor P. VolobuevSkobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,119991, Moscow, Russia
Abstract
In the present paper we examine, from the purely theoretical point of view and ina model-independent way, the case, when matter, gauge and Higgs fields are allowed topropagate in the bulk of five-dimensional brane world models with compact extra dimen-sion, and the Standard Model fields and their interactions are supposed to be reproducedby the corresponding zero Kaluza-Klein modes. An unexpected result is that in orderto avoid possible pathological behavior in the fermion sector, it is necessary to imposeconstraints on the fermion field Lagrangian. In the case when the fermion zero modesare supposed to be localized at one of the branes, these constraints imply an additionalrelation between the vacuum profile of the Higgs field and the form of the backgroundmetric. Moreover, this relation between the vacuum profile of the Higgs field and theform of the background metric results in the exact reproduction of the gauge boson andfermion sectors of the Standard Model by the corresponding zero mode four-dimensionaleffective theory in all the physically relevant cases, allowed by the absence of pathologies.Meanwhile, deviations from these conditions can lead either back to pathological behaviorin the fermion sector or to a variance between the resulting zero mode four-dimensionaleffective theory and the Standard Model, which, depending on the model at hand, may,in principle, result in constraints putting the theory out of the reach of the present dayexperiments.
Models with extra dimensions have been attracting a great interest during the last fifteenyears. There were many attempts to solve various theoretical problems with the help of extradimensions. A wide branch of multidimensional models is that of brane world models, whichwere proposed in their modern form in [1, 2]. Although some theoretical problems (such as, forexample, the hierarchy problem of gravitational interaction) were successfully solved within theframework of brane world models, realistic theories must also describe all the physical aspectsof our four-dimensional world. In particular, they must correctly reproduce the interactions ofthe Standard Model (SM) particles that have already been tested experimentally.In the original formulation of brane world models the SM fields were supposed to be locatedon a brane (in the Randall-Sundrum model [2], on the TeV brane). Later the idea of braneworlds was joined with the idea that all the fields can propagate in extra dimensions [3] thusgiving rise to the theories with universal extra dimensions, where the matter, gauge and Higgs1elds are allowed to propagate in the bulk of five-dimensional brane world models with compactextra dimension. In this case all these fields posses towers of Kaluza-Klein excitations, their zeromodes being the SM fields. There exist many papers describing how the SM can be embeddedthis way into multidimensional brane worlds and what new effects can be produced in suchtheories.However, there are some finer points that have been missed in the previous studies. Belowwe will discuss them in detail from a purely theoretical point of view and in a mathemat-ically consistent way. In particular, we argue that it is impossible to exactly reproduce bythe zero Kaluza-Klein modes the electroweak gauge boson sector of the SM in the effectivefour-dimensional theory, unless the vacuum profile of the Higgs field in the extra dimensionbehaves like the square root of the inverse warp factor. The same vacuum profile of the Higgsfield (together with extra constraints on the parameters of the five-dimensional fermion fieldLagrangian) is necessary for reproducing the fermion sector of the SM. An unexpected result isthat deviations from these conditions may lead either to pathologies in the fermion sector whenexpanding in Kaluza-Klein modes (for example, in the case admitting consistent localization ofthe fermion zero mode at one of the branes) or to an additional modification of the couplingsof fermions to gauge bosons in the zero mode sector, which may lead, in principle, to severerestrictions on the value of the five-dimensional energy scale (like the constraints coming fromthe zero mode gauge boson sector [4, 5]). Meanwhile, the demand for the extra restrictionon the vacuum profile of the Higgs field, mentioned above, leads to some difficulties, such asthe necessity for an extra fine-tuning. This problem needs to be addressed, at least for betterunderstanding the structure of brane world models.The paper is organized as follows. In Section 2 we consider an illustrative example of bulkgauge fields interacting with the bulk Higgs scalar field and show which conditions should befulfilled in order the gauge fields exactly reproduce the electroweak gauge sector of the SMby the lowest Kaluza-Klein modes. In Section 3 fermions are examined in the same way. InSection 4 we consider interactions between fermions and gauge bosons in a simple theory, wherepathologies in the fermion sector are absent, whereas no extra conditions on the Higgs field areimposed. The obtained results are discussed in the last section.
Let us take a five-dimensional space-time with the coordinates x M = { x µ , z } , M = 0 , , , , S /Z , which can be representedas the circle with the coordinate − L ≤ z ≤ L and the points − z and z identified. In whatfollows, we will use the notation x for the coordinates x µ . We consider the following standardform of the background metric, which is often used in brane world models: ds = e σ ( z ) η µν dx µ dx ν − dz . (1)This metric is assumed to correspond to a regular brane world model, i.e. it is a solution toequations of motion for five-dimensional gravity, two branes with tension and, for example, astabilizing bulk scalar field. We do not specify the explicit form of the solution for σ ( z ).2e start with the gauge fields and choose the following action of an SU (2) × U (1) gaugeinvariant model in this background: S = Z d xdz √ g (cid:18) − ξ F a,MN F aMN − ξ B MN B MN + g MN ( D M H ) † D N H − V ( H † H ) (cid:19) , (2)where F aMN = ∂ M A aN − ∂ N A aM + gǫ abc A bM A cN , (3) B MN = ∂ M B N − ∂ N B M , (4) D M H = (cid:18) ∂ M − ig τ a A aM − i g ′ B M (cid:19) H (5)and the fields satisfy the orbifold symmetry conditions A aµ ( x, − z ) = A aµ ( x, z ), A a ( x, − z ) = − A a ( x, z ), B µ ( x, − z ) = B µ ( x, z ), B ( x, − z ) = − B ( x, z ), H ( x, − z ) = H ( x, z ). Here ξ = √ L isa constant, which is introduced for convenience and chosen so that the dimension of the bulkgauge fields is mass. The scalar field potential can include brane-localized terms of the form λ ( H † H ) δ ( z ) and λ ( H † H ) δ ( z − L ). It is easy to see that action (2), which has a rather standardform, resembles the bosonic sector of the electroweak part of the ordinary four-dimensional SM.This action gives rise to the equations of motion for the gauge and the Higgs fields thatlook like ∇ N F a,MN + gǫ abc A bN F c,MN + i gξ (cid:18)(cid:0) D M H (cid:1) † τ a H − H † τ a D M H (cid:19) = 0 , (6) ∇ N B MN + i g ′ ξ (cid:16)(cid:0) D M H (cid:1) † H − H † D M H (cid:17) = 0 , (7) ∇ M D M H − g MN (cid:18) ig τ a A aM + i g ′ B M (cid:19) D N H + dVd ( H † H ) H = 0 , (8) ∇ M denoting the covariant derivative with respect to metric (1).Let us consider the vacuum solution for these fields. The vacuum solution, breaking thegauge group SU (2) × U (1) to U (1) em , leaving the Poincare invariance in four-dimensionalspace-time intact and satisfying equations (6), (7), can be taken in the form A aM ≡ , B M ≡ , H ≡ v ( z ) √ ! , (9)where v ( z ) is a real function. It is not difficult to understand that in the general case thevacuum solution for the Higgs field v ( z ) may depend on the coordinate of the extra dimension.Of course, the scalar field potential must provide for such a solution to equation (8). At thispoint we do not specify the explicit form of v ( z ).Now let us turn to examining the excitations in the model at hand. Below we will beinterested in the behavior of only the four-vector components of the five-dimensional gaugefields, whose zero modes must play the role of the SM gauge bosons. For this reason, fromhere on we retain only these components of the gauge fields and drop the components A a , B
3f the vector fields and the fluctuations of the Higgs field. From action (2) it is easy to get thefollowing effective action for the four-vector components of the five-dimensional gauge fields: S eff = Z d xdz (cid:18) − ξ η µν η αβ F aµα F aνβ + e σ ξ η µν ∂ A aµ ∂ A aν − ξ η µν η αβ B µα B νβ (10)+ e σ ξ η µν ∂ B µ ∂ B ν + e σ η µν H † (cid:18) g τ a A aµ + g ′ B µ (cid:19) (cid:18) g τ a A aν + g ′ B ν (cid:19) H (cid:19) , where we have also dropped the terms containing only the vacuum configuration of the Higgsfield.Now we are ready to perform the Kaluza-Klein mode decomposition. First, using thestandard redefinition Z µ = 1 p g + g ′ (cid:0) gA µ − g ′ B µ (cid:1) , A µ = 1 p g + g ′ (cid:0) gB µ + g ′ A µ (cid:1) , W ± µ = 1 √ (cid:0) A µ ∓ iA µ (cid:1) , (11)we can pass to the physical degrees of freedom of the theory. Next, let us consider only thequadratic part of effective action (10) in terms of these new fields. It takes the form S eff = Z d xdz (cid:18) − ξ η µν η αβ W + µα W − νβ − ξ η µν η αβ F µα F νβ − ξ η µν η αβ Z µα Z νβ (12)+ e σ ξ η µν ∂ W + µ ∂ W − ν + e σ ξ η µν ∂ A µ ∂ A ν + e σ ξ η µν ∂ Z µ ∂ Z ν + e σ v ( z ) g η µν W + µ W − ν + e σ v ( z ) g + g ′ η µν Z µ Z ν (cid:19) , where W ± µν = ∂ µ W ± ν − ∂ ν W ± µ , F µν = ∂ µ A ν − ∂ ν A µ , Z µν = ∂ µ Z ν − ∂ ν Z µ . The equations for thewave functions and the masses of the Kaluza-Klein modes are − m W,n f W,n − ∂ ( e σ ∂ f W,n ) + g ξ e σ v ( z ) f W,n = 0 , (13) − m Z,n f Z,n − ∂ ( e σ ∂ f Z,n ) + g + g ′ ξ e σ v ( z ) f Z,n = 0 , (14) − m A,n f A,n − ∂ ( e σ ∂ f A,n ) = 0 . (15)where the subscript n denotes the number of the corresponding Kaluza-Klein mode. As usual,the lowest (zero) Kaluza-Klein modes of the fields are supposed to correspond to the four-dimensional SM particles. So, below we will focus only on the zero modes.It follows from (15) that the solution for the lowest mode of the field A µ (the photon)is m A, = 0 and f A, ( z ) ≡ const, i.e. its wave function does not depend on the coordinateof the extra dimension. This is an important result, which provides the universality of theelectromagnetic charge [6]. But, as one sees from (13) and (14), in the general case it is notso for the zero modes of the fields W µ and Z µ , which correspond to the SM massive gaugebosons. The latter has the following well-known consequences. Indeed, in the SM the self-coupling of massive gauge bosons comes from the term F a,µν F aµν and the corresponding couplingconstants are defined only by the structure of the gauge group. In the five-dimensional caseunder consideration the self-coupling terms also come from the same term of (10), but now the4orresponding coupling constants are also defined by the overlap integrals over the coordinate z ,which include the wave functions f W, ( z ) and f Z, ( z ). The only case, when the zero mode sectorof the model automatically completely coincides with the electroweak gauge boson sector of theSM, is the one, where the wave functions f W, ( z ) and f Z, ( z ) do not depend on the coordinateof the extra dimension. In this case the self-coupling constants of the massive gauge bosons aredefined in terms of the constants g and g ′ exactly in the same way as in the ordinary SM. Theindependence of the wave functions f W, ( z ) and f Z, ( z ) on the coordinate of the extra dimensioncan be achieved only when e σ v ( z ) ≡ const, i.e., when v ( z ) ≡ ξ ˜ ve − σ , (16)where ˜ v is a constant of dimension M . For the choice (16), the masses of the zero mode gaugebosons are given by m W, = g ˜ v , m Z, = p g + g ′ ˜ v . (17)Thus, in the case under consideration ˜ v must coincide with the Higgs field vacuum expectationvalue of the SM.Of course, the results presented above are rather trivial. Moreover, it is well known that inthe general case a modification of the shapes of the zero mode gauge boson wave functions hasan influence on the electroweak observables, this problem was discussed in detail in [4, 5]. It isshown in these papers that, for example, in the case of the Randall-Sundrum model [2] such amodification leads to restrictions on the value of the five-dimensional energy scale, which putthe theory out of the reach of the present day experiments. For the choice (16) the lowest modefour-dimensional effective theory exactly reproduces the electroweak gauge boson sector of theSM, thus imposing no restrictions on the value of the five-dimensional energy scale. Meanwhile,one can imagine that there exists a profile for the Higgs vacuum solution, which differs from(16) but provides somehow the necessary values of the zero mode gauge boson masses and self-coupling constants with a good accuracy. Unfortunately, the situation becomes more involved,when one comes to fermions. It is well known that, since there is no chirality in five-dimensional space-time, in order toobtain a nonzero mass term for the zero Kaluza-Klein fermion mode via the Higgs mechanismit is necessary to take two five-dimensional spinor fields (see, for example, [7, 8, 9]) satisfyingthe orbifold symmetry conditions Ψ ( x, − z ) = γ Ψ ( x, z ) , (18)Ψ ( x, − z ) = − γ Ψ ( x, z ) . (19)Thus, as a simple example, we consider a model with the action of the most general form S = Z d xdz √ g (cid:0) E MN i ¯Ψ Γ N ∇ M Ψ + E MN i ¯Ψ Γ N ∇ M Ψ (20) − F ( z ) ¯Ψ Ψ − F ( z ) ¯Ψ Ψ − G ( z ) (cid:0) ¯Ψ Ψ + ¯Ψ Ψ (cid:1)(cid:1) , M, N = 0 , , , ,
5, Γ µ = γ µ , Γ = iγ , ∇ M is the covariant derivative containing the spinconnection, E MN is the vielbein, F , ( z ) and G ( z ) are some functions satisfying the symmetryconditions F , ( − z ) = − F , ( z ) and G ( − z ) = G ( z ). For the case of metric (1) action (20) canbe rewritten in the form (see, for example, [9, 10] for the explicit form of the vielbein and spinconnections) S = Z d xdze σ (cid:0) e − σ i ¯Ψ γ µ ∂ µ Ψ − ¯Ψ γ ( ∂ + 2 σ ′ ) Ψ − F ( z ) ¯Ψ Ψ (21)+ e − σ i ¯Ψ γ µ ∂ µ Ψ − ¯Ψ γ ( ∂ + 2 σ ′ ) Ψ − F ( z ) ¯Ψ Ψ − G ( z ) (cid:0) ¯Ψ Ψ + ¯Ψ Ψ (cid:1)(cid:1) , where ′ = ∂ . The equations of motion, following from this action, take the form e − σ iγ µ ∂ µ Ψ − γ ( ∂ + 2 σ ′ ) Ψ − F ( z )Ψ − G ( z )Ψ = 0 , (22) e − σ iγ µ ∂ µ Ψ − γ ( ∂ + 2 σ ′ ) Ψ − F ( z )Ψ − G ( z )Ψ = 0 . (23)Suppose that G ( z ) ≡
0. In this case there always exists the solutionΨ = C f exp − z Z F ( y ) dy − σ ( z ) ψ L ( x ) , iγ µ ∂ µ ψ L = 0 , γ ψ L = ψ L , (24)where C f is a normalization constant, describing a massless four-dimensional fermion. Ananalogous solution exists for the field Ψ (but with a right-handed four-dimensional fermion).The latter clearly indicates that the existence of only one five-dimensional fermion is not enoughto provide a massive four-dimensional lowest mode. This also indicates that it is the term with G ( z ) G ( z ) should be somehow connected with the five-dimensional Higgs field. It isnatural to take this function as G ( z ) ≡ hv ( z ) , (25)where h is a coupling constant of dimension M − . Such a construction may arise, when oneconsiders the standard Higgs mechanism in the bulk after the spontaneous symmetry breaking,leading to (9), whereas the zero modes of the fields Ψ , Ψ are supposed to represent a massivelepton (for example, the electron).It should be noted that the “localizing” functions F ( z ) and F ( z ) are not connected withthe Higgs field in the general case. Meanwhile, the corresponding terms are not forbidden and,according to (24), they are responsible for the localization of the lowest fermion Kaluza-Kleinmodes. In fact, the form of the terms with the functions F ( z ) and F ( z ) in (20) is the only onesuitable for the localization of the fermion zero modes in a consistent field-theoretical manner,though the origin of the localizing functions can be different. In the context of multidimensionalmodels, such a mechanism was proposed in [11] for the theory with one infinite extra dimension,in which the localizing function was just a profile of the topological soliton modeling a domainwall. An analogous mechanism (but with another form of the localizing function) was usedin [12], where fermion fields were also supposed to be confined to a thick domain wall in flatextra dimensions, but different fermions were localized at different points of the wall. In five-dimensional brane world models the localizing functions are usually not explicitly connected6ith something like a domain wall, very often the corresponding terms have the form similar tothe standard fermion mass term, but with an antisymmetric “mass” according to the orbifoldsymmetry, i.e., F ( z ) = C sign( z ). The value of the constant C defines at which brane thefermion zero mode is localized and what is the width of its wave function [13, 14, 15].Now let us recall the ordinary four-dimensional free spinor field satisfying the Dirac equation.It is well known that each component of this field satisfies the Klein-Gordon equation, which isa second-order differential equation. Of course, all the components of the spinor field are notindependent — with the help of the initial Dirac equation one can restore, for example, the two-component spinor ψ R using a solution for the two-component spinor ψ L , where the componentsof ψ L are supposed to be independent and to satisfy the Klein-Gordon equation. The Klein-Gordon equation is known to have no pathologies, so one can be sure that the whole theoryis consistent. The five-dimensional fields Ψ and Ψ satisfying equations (22), (23) should beconsidered as free fields as well, because they are coupled only to the vacuum configurations ofthe Higgs and gravity fields. Therefore, one expects that in a consistent theory each component of the five-dimensional spinor fields Ψ and Ψ (or at least of their linear combinations) alsosatisfies a five-dimensional second-order differential equation, which contains derivatives inthe four-dimensional coordinates only in the form (cid:3) = η µν ∂ µ ∂ ν , otherwise one may expect theappearance of various pathologies when expanding in Kaluza-Klein modes (which is, in fact, thefirst step in examining the four-dimensional effective theory), an example of such a pathologicalbehavior will be presented below. The latter is not good taking into account the fact that thecomponents of the five-dimensional spinors make up four-dimensional fermion fields (just likehow ψ L and ψ R make up a four-dimensional four-component spinor field), see, for example,[8, 16]. So, let us try to obtain the corresponding second-order differential equations. From(22) and (23) it is not difficult to obtain: − (cid:3) Ψ + e σ ( ∂ + 2 σ ′ ) e σ ( ∂ + 2 σ ′ )Ψ + e σ ∂ ( e σ F ( z )) γ Ψ − e σ ( F ( z ) + h v ( z ))Ψ (26)+ he σ ∂ ( e σ v ( z )) γ Ψ − he σ v ( z ) ( F ( z ) + F ( z )) Ψ = 0 , − (cid:3) Ψ + e σ ( ∂ + 2 σ ′ ) e σ ( ∂ + 2 σ ′ )Ψ + e σ ∂ ( e σ F ( z )) γ Ψ − e σ ( F ( z ) + h v ( z ))Ψ (27)+ he σ ∂ ( e σ v ( z )) γ Ψ − he σ v ( z ) ( F ( z ) + F ( z )) Ψ = 0 . From (26) and (27) one can see that formally the equations for the components of the fieldsΨ and Ψ do not decouple. It turns out that in the general case we can not obtain second-order differential equations for each component of the fields Ψ and Ψ (or of their linearcombinations) separately, as it happens in the ordinary four-dimensional theory, except severalspecial cases. The first obvious exception is when the following conditions fulfill: F ( z ) ≡ − F ( z ) , (28) ∂ ( e σ v ( z )) ≡ . (29)The second condition completely coincides with (16). Introducing the dimensionless couplingconstant ˜ h = hξ and taking into account (28) and (29) we can rewrite equations (26) and (27)as − (cid:3) Ψ + e σ ( ∂ + 2 σ ′ ) e σ ( ∂ + 2 σ ′ )Ψ + e σ ∂ ( e σ F ) γ Ψ − ( e σ F + ˜ h ˜ v )Ψ = 0 , (30) − (cid:3) Ψ + e σ ( ∂ + 2 σ ′ ) e σ ( ∂ + 2 σ ′ )Ψ − e σ ∂ ( e σ F ) γ Ψ − ( e σ F + ˜ h ˜ v )Ψ = 0 , (31)7here F ( z ) ≡ F ( z ) ≡ − F ( z ), which indeed lead to the second-order differential equations foreach component of the fields Ψ and Ψ separately.The solution to these equations, corresponding to the zero mode, has the form (it can alsobe easily obtained from (22), (23))Ψ = C f exp − z Z F ( y ) dy − σ ( z ) ψ L ( x ) , iγ µ ∂ µ ψ L − ˜ h ˜ vψ R = 0 , γ ψ L = ψ L , (32)Ψ = C f exp − z Z F ( y ) dy − σ ( z ) ψ R ( x ) , iγ µ ∂ µ ψ R − ˜ h ˜ vψ L = 0 , γ ψ R = − ψ R , (33)where again C f is a normalization constant. This solution indeed corresponds to the lowestmode, see Appendix A for details. It is clear that the fields ψ L and ψ R are localized in the vicin-ity of the same point in the extra dimension. Taken together they make up a four-dimensionalDirac fermion with mass ˜ h ˜ v . As for the physical degrees of freedom corresponding to higherKaluza-Klein fermion modes, for the case of equations (30), (31) they can be examined exactlyin the same way as it was made in [16] for the model with infinite extra dimension.It should be also mentioned that the fermion action exactly of form (21) with conditions(28) and (29) (but in other notations) was considered in [9] for examining discrete symmetriesin brane world models.It is interesting to note that if the localizing functions F ( z ) have one and the same form forall fermion fields in the theory (leptons, quarks), then the wave functions of the zero modes alsohave the same form for different fermions regardless of the four-dimensional mass of the mode(see (32), (33)). In this case the coupling constants of fermions to gauge bosons in the zero modesector appear to be exactly the same as in the SM. This happens because the wave functionsof all the zero mode gauge bosons do not depend on the coordinate of the extra dimension ifrelation (29) holds (see Section 2) and all the corresponding vertices (even those containingtwo different four-dimensional fermions) in fact contain the integral of the same fermion wavefunction squared, and this integral is equal to unity due to the normalization conditions.The question arises, whether there are other exceptions in equations (26), (27), leading tosecond-order differential equations of motion for any form of v ( z )? We found another simpleexception for the general case (in principle, there are more exceptions, but they seem to bevery unnatural, see Appendix B for a detailed discussion of the decoupling of equations (26),(27)), which is, in fact, rather obvious and follows even from the form of equations (26), (27).Namely, if the relation F ( z ) ≡ F ( z ) (34)is fulfilled, then one can simply add and subtract equations (26), (27) to obtain two independentsecond-order differential equations for the combinations Ψ + Ψ and Ψ − Ψ , which look likethey should not lead to any pathologies.Relation (34) seems to be rather unphysical (indeed, a consistent localization of the zeromodes of fermion fields demands (28), see also, for example, [7, 9, 17]; while it is unclearwhat could be the physical motivation for the condition F ( z ) ≡ F ( z ) F ( z ) ≡ F ( z ) ≡
0, which means that all the fermion fields can freely propagate in the8ulk. Thus, to examine this case in more detail let us simplify the task and take σ ( z ) ≡ F ( z ) ≡ F ( z ) ≡ v ( z ) const (for the case σ ( z ) ≡ − (cid:3) (Ψ + Ψ ) + ∂ (Ψ + Ψ ) − h v (Ψ + Ψ ) + hv ′ γ (Ψ + Ψ ) = 0 , (35) − (cid:3) (Ψ − Ψ ) + ∂ (Ψ − Ψ ) − h v (Ψ − Ψ ) − hv ′ γ (Ψ − Ψ ) = 0 . (36)Using these equations it is not difficult to show that, according to the orbifold symmetryconditions (18), (19), the fields Ψ and Ψ can be decomposed into the Kaluza-Klein modes asΨ = X n (cid:0) f n + ( z ) ψ nL ( x ) − f n − ( z ) ψ nR ( x ) (cid:1) , (37)Ψ = X n (cid:0) f n − ( z ) ψ nL ( x ) + f n + ( z ) ψ nR ( x ) (cid:1) , (38)where γ ψ L = ψ L , γ ψ R = − ψ R , f n + ( z ) = f n ( z ) + f n ( − z ) , f n − ( z ) = f n ( z ) − f n ( − z ) (39)and the function f n ( z ) is a periodic continuously differentiable solution to the equation m n f n + ∂ f n − h v f n + hv ′ f n = 0 (40)in the interval [ − L, L ], corresponding to the eigenvalue m n (recall that v ( − z ) = v ( z )). Accord-ing to the general theory [18], the functions f n ( z ) make up an orthonormal set of eigenfunctionsfor equation (40), the lowest eigenvalue m being simple (which means that we get only onefermion with mass m in the effective four-dimensional theory). Moreover, it is not difficultto show that m > and Ψ in (37), (38) dif-fers from that in (32), (33). This difference leads to certain problems when taking into accountthe interactions with the gauge fields. This issue will be discussed in the next section.What can happen when the decoupling of the equations of motion for the components ofthe fermion fields seems to be impossible, at least in the standard way (like the one presentedin Appendix B)? To show it, let us again simplify the task and consider the case σ ( z ) ≡ − (cid:3) Ψ + ∂ Ψ + F ′ γ Ψ − (cid:0) F + h v (cid:1) Ψ + hv ′ γ Ψ − hv ( F + F ) Ψ = 0 , (41) − (cid:3) Ψ + ∂ Ψ + F ′ γ Ψ − (cid:0) F + h v (cid:1) Ψ + hv ′ γ Ψ − hv ( F + F ) Ψ = 0 . (42)The equation for, say, the first component of the field Ψ can be easily obtained and turns outto be the fourth-order differential equation (cid:2) (cid:3) − ∂ − F ′ + F + h v (cid:3) hv ( F + F ) − hv ′ (cid:2) (cid:3) − ∂ − F ′ + F + h v (cid:3) Ψ (1)1 (43) − ( hv ( F + F ) − hv ′ ) Ψ (1)1 = 0 . Analogous equations can be obtained for the other components of the field Ψ and for thecomponents of the field Ψ (though, as in the four-dimensional case, not all the components9f the fields Ψ and Ψ are completely independent, some of them can be expressed throughthe other: for example, the field Ψ can be restored from the field Ψ with the help of initialequation (22)). It is obvious that even in the flat case σ ( z ) ≡ v ( z ) ≡
0, so inprinciple one can expect an increase of the number even of the zero modes in addition to otherpossible pathologies (i.e., one could imagine that there would appear, say, two electrons in theeffective four-dimensional theory). We have failed to solve such fourth-order equations of motionanalytically even in the simplest cases and we have not an explicit example which could supportthis statement, however, such a possibility is not excluded by the general reasonings. Thus, inour opinion, one should avoid the appearance of such fourth-order differential equations whenconstructing multidimensional models, at least to be sure that the resulting theory is devoidof any pathologies and the physical degrees of freedom can be isolated in a mathematicallyconsistent way using the well-developed theory of second-order differential equations.An important comment is in order here. In many brane world models the expansion inthe Kaluza-Klein modes for the fields Ψ and Ψ is performed without taking into account theinteraction with the Higgs field (indeed, in the case v ( z ) ≡ v ( z ) ≡ v ( z ) v ( z ) ˜ ve − σ (including the generalized functions like the delta-function). It is clear that, inprinciple, the resulting fields do not represent the physical degrees of freedom of the theory andcannot be used for consistent calculations, which poses the question about the diagonalizationof the mass matrix in the effective four-dimensional theory by algebraic (maybe perturbative)methods. However, as it was noted above, in the general case the systems described by fourth-order equations of motion like the one in (43) have more degrees of freedom than the systemsdescribed by second-order equations of motion like the one in (26) with v ( z ) ≡
0. The lattermakes the use of the perturbation theory and the subsequent diagonalization of the mass ma-trix questionable, because in the general case this mass matrix, obtained using the solutions ofequations (26) and (27) with v ( z ) ≡
0, may not include all the degrees of freedom (includingpossible pathological modes) described by equations (26) and (27) with v ( z )
0. We notethat it is a nonperturbative effect, which may appear no matter what is the relation betweenthe energy scale of the Higgs field and the typical energy scale of five-dimensional theory. Insome sense this situation is similar to the case of U (1) massless gauge field: if one adds themass term to the action, the third degree of freedom arises, no matter how small the mass of There is a simple algebraic example, demonstrating an analogous nonperturbative effect. The equation x − αx + x − | α | ≪ x ≈ − α indeed can be obtained perturbatively, whereas it is not so for the other root x ≈ − α − α . v ( z ) ≡ To demonstrate in a simple way, how possible latent problems can pop up in the four-di-mensional effective theory, corresponding to equations (35) and (36), let us consider a five-dimensional action, describing fermion fields minimally coupled to the SU (2) × U (1) gaugefields in the flat ( σ ( z ) ≡
0) space-time: S = Z d xdz (cid:16) i ¯ˆΨ Γ M D M ˆΨ + i ¯Ψ Γ M D M Ψ − √ h h(cid:16) ¯ˆΨ H (cid:17) Ψ + h.c. i(cid:17) . (44)Here the SU (2) doublet, constructed from five-dimensional spinors, is denoted byˆΨ = Ψ ν Ψ ψ ! , ¯ˆΨ = (cid:16) ¯Ψ ν , ¯Ψ ψ (cid:17) (45)and the five-dimensional SU (2) singlet is denoted by Ψ . The covariant derivatives are definedby D M ˆΨ = (cid:18) ∂ M − ig τ a A aM + i g ′ B M (cid:19) ˆΨ , (46) D M Ψ = ( ∂ M + ig ′ B M ) Ψ . (47)The vacuum solution for the Higgs field is supposed to have the form H ≡ v ( z ) √ ! (48)with v ( z ) const.First, let us consider the free theory. We will be interested only in the lowest mode sector,so we neglect all the higher Kaluza-Klein modes of gauge and fermion fields. According to(35)–(40), we can represent the fermion zero modes as (below we will omit the superscript “0”for the zero ( n = 0) modes of the fields)ˆΨ ( x, z ) = (cid:18) ξν L ( x ) f + ( z ) ψ L ( x ) − f − ( z ) ψ R ( x ) (cid:19) , Ψ ( x, z ) = f − ( z ) ψ L ( x ) + f + ( z ) ψ R ( x ) , (49)where ν L = γ ν L . The factor ξ = √ L is just the canonically normalized wave function ofthe zero mode of the field Ψ ν (since this field does not interact with the vacuum solution of11he Higgs field H , in the flat background and with F ( z ) ≡ F ( z ) ≡ S = Z d x (cid:0) i ¯ ν L γ µ ∂ µ ν L + i ¯ ψγ µ ∂ µ ψ − m ¯ ψψ (cid:1) , (50)where ψ = ψ L + ψ R . In order to have the canonically normalized kinetic term of the field ψ in(50), the condition a + b = 1 , a = Z dzf ( z ) , b = Z dzf − ( z ) . (51)must be fulfilled. The mass m is the eigenvalue of the problem (40), corresponding to theeigenfunction f ( z ). By tuning the coupling constant h one can, in principle, get the desiredvalue of the mass m .Now let us turn to examining the interactions with the gauge bosons. In order to isolate theeffects caused only by the fermions, below we choose the following ansatz for the zero modesof the gauge fields: A aµ ( x, z ) ≡ A aµ ( x ) , B µ ( x, z ) ≡ B µ ( x ); (52)for simplicity we will also drop the components A a ( x, z ) and B ( x, z ) of these fields. We do notdiscuss here possible ways for obtaining this ansatz in a consistent way. For example, one maysimply imagine that there exists a second Higgs field, interacting with the gauge fields only,which provides the necessary forms of the corresponding wave functions.Passing to the physical degrees of freedom (11), using (49), (52) and integrating over the co-ordinate of the extra dimension, we can obtain the effective four-dimensional action, describingthe interaction of the zero mode fermions with the gauge bosons. We do not present the explicitcalculations here, they are straightforward. It is not difficult to show that the electromagneticcoupling constant appears to be the same as in the SM (from here and below “the same as inthe SM” means that it can be expressed through the constants g and g ′ exactly in the sameway as it happens in the SM). The coupling constant of the interaction with the charged gaugebosons is found to be g √ L Z dzf + ( z ) (53)instead of g in the SM (recall that the wave function, corresponding to the field ν L , is just √ L ). As for the interaction of the field ψ with the neutral gauge boson Z , the vector couplingconstant appears to be the same as in the SM, whereas the axial coupling constant has theform g A = g SMA ( a − b ) . (54)It is clear that in the general case c = √ L R dzf + ( z ) = 1 and, according to (51), a − b = 1.Meanwhile, in order to get rid of the difference with the well known parameters of the SM, In the derivation of action (50) it is convenient to use the equations f ′ + + hvf − = − mf − , f ′− + hvf + = mf + , which are fulfilled if equation (40) holds. c = 1 and a − b = 1 (or the values which are close to unity with a goodaccuracy). The latter can be achieved if b = 0, which means that f ( z ) ≡ const; in this case c = 1 too. But, according to (40), the condition f ( z ) ≡ const means that v ( z ) ≡ const, whichagain corresponds to (29) with σ ( z ) ≡
0. Thus, the farther v ( z ) from a constant is, the fartherthe values of the corresponding coupling constants are from those of the SM.It is obvious that for the non-flat case σ ( z ) ∂ ( e σ v ( z )) σ ( z ) ≡ v ( z ) from ∼ e − σ is, the fartherthe values of the corresponding coupling constants are from those of the SM (note that in arealistic theory with ∂ ( e σ v ( z )) v ( z ) ≡ const may lead,in principle, either to an unacceptable theory or put it out of the reach of the present dayexperiments.As a final remark to this section, let us draw an analogy with the case of gauge fields.As it was mentioned in Section 2, the modification of the shapes of the gauge boson wavefunctions due to the interaction with the vacuum solution of the Higgs field may affect the four-dimensional effective theory considerably [4, 5]. In the most cases the interaction of fermionfields with the vacuum solution of the Higgs field affects the corresponding effective theoryeither in an analogous way (by a modification of the zero mode wave functions and the chiralstructure), or even more dramatically, leading to pathologies like those discussed in Section 3.The latter clearly indicates that the interaction of fermion fields with the vacuum solution ofthe Higgs field should be treated much more carefully than it is usually done. As it was demonstrated in the previous sections, the only obvious possibility to automati-cally get a self-consistent, from the theoretical point of view, four-dimensional SM in a five-dimensional brane world model (i.e., without possible pathologies in the free theory and withthe correct couplings) is to have a vacuum solution of form (16) for the Higgs field together with(28). Condition (28) corresponds, in fact, both to the case of localized fermion zero modes andto the case when the fermion fields can freely propagate in the bulk (if F ( z ) ≡ F ( z ) ≡ Itis not difficult to check that in this case the zero mode fermion and gauge boson sectors of theresulting effective theory indeed exactly reproduce those of the SM, including the interactions,at least for the case of the standard form of five-dimensional gauge invariant action (of course,if the localizing functions F ( z ) are one and the same for all the fields, corresponding to different This reasoning is valid only for the case F ( z ) ≡ F ( z ) ≡
0. If F ( z ) ≡ F ( z )
0, there should be deviationsof the coupling constants from those of the SM even for ∂ ( e σ v ( z )) ≡ θ , seeAppendix B. As it was shown in Appendix B, there may exist other exceptions in equations (26), (27), leading to second-order differential equations of motion. But, according to the results presented above, it is improbable that suchunnatural cases could lead to a completely acceptable effective theory. and Ψ decouple, which provides the correct chiral struc-ture of the corresponding lowest Kaluza-Klein modes. Deviations from these conditions maylead either to pathologies or to a variance between the resulting zero mode four-dimensionaleffective theory and the SM. The latter may result in severe constraints on the parameters offive-dimensional theory and put it, in principle, out of the reach of the present day experimentsin full analogy with how in happens in the gauge boson sector [4, 5].It should be noted that the exponential profile of the vacuum solution for the Higgs field,leading to its localization near the TeV brane in the Randall-Sundrum model [2], was discussedearlier [15]. Our results demonstrate that in order to have a possibility to localize the zeromodes of fermion fields in a consistent way (using an appropriate form of the function F ( z ) in(30) and (31)), the profile of the vacuum solution of the Higgs field should have exactly theform (16).Unfortunately, the restriction on the vacuum profile of the Higgs field poses several problemsfor the case of localized fermion zero modes (i.e., when F ( z ) σ ( z ) = − k | z | and without taking into account the backreaction of the Higgs field on thebackground metric, the fine-tuned bulk Higgs potential should have the form V ( H † H ) = − k H † H to get the vacuum solution (16). Of course, one should also add fine-tuned brane-localizedpotentials, including a term specifying the value of the constant ˜ v (at least on one of thebranes). If one takes more realistic cases of stabilized models, in which the warp factors have amore complicated form (like the one in [20]), the form of the Higgs scalar field potential appearsto be such that it can not be represented in an analytical form. Of course, such a situationlooks unnatural, at least in the absence of a symmetry that can support such a fine-tuning ofthe Higgs potential. Moreover, backreaction of the Higgs field affects the background metric,whereas quantum corrections modify the scalar field potential and, consequently, the vacuumsolution for the Higgs field. Both effects lead to breakdown of the fine-tuned relation betweenthe warp factor and the vacuum solution for the Higgs field.Second, in a scenario with localized zero modes of fermion fields the Higgs field and thestabilizing scalar field cannot be unified, as it was proposed in [21]. Indeed, a consistent stabi-lization mechanism (like the one proposed in [20]) is based on fixing the values of the stabilizingscalar fields on the branes (at the points z = 0 and z = L ). On the other hand, warped braneworld models are interesting if the function e σ has exponentially different values on the branes.The latter means that by taking the Higgs field as the stabilizing field in such a theory, oneintroduces a new hierarchy into the model (because v (0) ≪ v ( L )). For example, the Randall-Sundrum model [2] was proposed to solve the hierarchy problem of gravitational interaction, so14t also looks unnatural to add an extra hierarchy into such a model. Moreover, in order to geta massive radion, one should take into account the backreaction of the stabilizing field on thebackground metric [20]. But if the stabilizing field is the Higgs field with vacuum solution (16),then the range of allowed scalar field potentials and warp factors narrows considerably (whichclearly follows from the self-consistent system of equations for the background configuration ofthe metric and the stabilizing scalar field, which can be found in [20]).One may suppose that if at least the fermion fields are located exactly on the brane, thenthe Higgs field can also be located on the brane, which looks as a solution to the problem (ofcourse, if we do not take into account the gauge fields, see [4]). However, the only realisticfield-theoretical mechanism of fermion localization, which can be used for calculations, is basedon the idea that initially the fermion fields propagate in the whole five-dimensional space-time,whereas only the lowest modes appear to be localized on the brane due to an interaction withsome defect (for example, with a domain wall like in [11]), see the discussion in Section 3. Thisis exactly the situation, which is realized in equations (32), (33), so one can take an appropriateform of the function F ( z ) to make the width of the wave function of the localized fermion assmall as necessary. Meanwhile, the profile of the Higgs field does not depend on the form of thefunction F ( z ), so even for an extremely narrow wave function of a localized mode (which takensquared can be even approximated by the delta-function for calculations) the “right” profile ofthe Higgs field, which does not lead to fourth-order differential equations, should still have theform (16). The latter poses a question whether there exists a field-theoretical mechanism offermion localization, leaving more freedom for the choice of a vacuum profile of the Higgs fieldin the extra dimension.It should be noted that the only obvious exception is the model with the flat five-dimensionalbackground metric like the one proposed in [3], for which the four-dimensional SM can beconstructed from a five-dimensional theory without unnatural fine-tunings and restrictions. Insuch a case the vacuum solution for the Higgs field must be just a constant, which admits avariety of scalar field potentials including the standard Higgs potential (but leaving unsolvedthe problem of the stabilization of the extra dimension size). Acknowledgements
The authors are grateful to E. Boos and S. Keizerov for discussions and useful remarks. Thework was supported by grant 14-12-00363 of Russian Science Foundation.
Appendix A: Mass of the lowest localized fermion mode
Let us take equation (30) and substitute Ψ ( x, z ) = ψ L ( x ) f L ( z ) with (cid:3) ψ L + µ ψ L = 0 and γ ψ L = ψ L into it. We get( µ − ˜ h ˜ v ) f L + e σ ( ∂ + 2 σ ′ ) e σ ( ∂ + 2 σ ′ ) f L + e σ ∂ ( e σ F ) f L − e σ F f L = 0 . (55)15ultiplying this equation by e σ f L , integrating over the coordinate of the extra dimension z and performing integration by parts in two terms, we arrive to the following equality:( µ − ˜ h ˜ v ) Z dze σ f L = Z dze σ ( f ′ L + 2 σ ′ f L + F f L ) . (56)Since both integrals are nonnegative, we get µ − ˜ h ˜ v ≥
0, which means that the lowestmode indeed has mass ˜ h ˜ v . A completely analogous procedure can be performed for the othersubstitution Ψ ( x, z ) = ψ R ( x ) f R ( z ) with (cid:3) ψ R + µ ψ R = 0 and γ ψ R = − ψ R , as well as for thefield Ψ . Appendix B: Decoupling of the equations of motion forfermions
Let us consider equations (26) and (27) for the left-handed fermions such that Ψ L = γ Ψ L ,Ψ L = γ Ψ L . These equations can be rewritten in the matrix form asˆ L ( x, z ) (cid:18) Ψ L Ψ L (cid:19) + ˆ M ( z ) (cid:18) Ψ L Ψ L (cid:19) = 0 , (57)where ˆ L ( x, z ) is a diagonal operator with equal diagonal elements, which includes derivatives,and the matrix ˆ M ( z ) looks like ˆ M ( z ) = (cid:18) q ( z ) p ( z ) p ( z ) q ( z ) (cid:19) (58)with q ( z ) = e σ ∂ ( e σ F ( z )) − e σ F ( z ) , (59) q ( z ) = e σ ∂ ( e σ F ( z )) − e σ F ( z ) , (60) p ( z ) = he σ ∂ ( e σ v ( z )) − he σ v ( z ) ( F ( z ) + F ( z )) . (61)The form of equation (57) suggests that the decoupling of the equations of motion for thefermion fields is equivalent to the diagonalization of the matrix ˆ M ( z ), so the question is howto diagonalize the matrix ˆ M ( z ) except for the obvious case p ( z ) ≡
0. Since this matrix issymmetric, it can be diagonalized with the help of a rotation matrix ˆ R such thatˆ R T ˆ M ˆ R = diag( λ , λ ) , ˆ R = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) . (62)The eigenvalues of the matrix ˆ M ( z ) can be easily found by the standard procedure and takethe form λ , ( z ) = q + q ± p ( q − q ) + 4 p . (63)A very important point is that the rotation angle θ should not depend on the coordinate of theextra dimension z , otherwise the rotation matrix ˆ R would not pass through the operator ˆ L ,16hich contains derivatives. Since it is the eigenvectors of the matrix ˆ M that form the rotationmatrix ˆ R , it is necessary to find conditions under which these eigenvectors do not depend onthe coordinate of the extra dimension. From the equation, say, for the first eigenvalue andeigenvector ˆ M ( z ) (cid:18) cos θ sin θ (cid:19) = λ ( z ) (cid:18) cos θ sin θ (cid:19) (64)we can easily get tan θ = q − q + p ( q − q ) + 4 p p . (65)It is clear that in the general case the angle θ depends on the coordinate of the extra dimension.An obvious exception is in general q ( z ) ≡ q ( z ).Thus, according to the results, presented above, in the general case we have two conditionsfor the left-handed fermions, for which the mixing matrix ˆ M is either diagonal or can bediagonalized in the standard way. They are ∂ ( e σ v ) = e σ v ( F + F ) , (66) ∂ ( e σ ( F − F )) = e σ ( F − F ) . (67)The procedure, completely analogous to the one presenter above, can be also performed forthe right-handed fermions (Ψ R = − γ Ψ R , Ψ R = − γ Ψ R ), leading to ∂ ( e σ v ) = − e σ v ( F + F ) , (68) ∂ ( e σ ( F − F )) = − e σ ( F − F ) . (69)The most general and simple model-independent conditions, following from (66)–(69), are just(28), (29) or (34), the latter case giving θ = ± π , which corresponds to the combinations of thefields in (35), (36). Of course, in principle one may consider other combinations of conditions(66)–(69): (66) together with (69) or (67) together with (68); or there may exist other specificchoices of the functions F ( z ), F ( z ) and v ( z ) leaving the rotation angle θ independent on thecoordinate of the extra dimension. But such cases seem to be much more unnatural, while theyalso lead to relations between the functions F ( z ), F ( z ), v ( z ) and σ ( z ). References [1] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B (1998) 263.[2] L. Randall and R. Sundrum, Phys. Rev. Lett. (1999) 3370.[3] T. Appelquist, H.C. Cheng and B.A. Dobrescu, Phys. Rev. D (2001) 035002.[4] C. Csaki, J. Erlich and J. Terning, Phys. Rev. D (2002) 064021.[5] G. Burdman, Phys. Rev. D (2002) 076003.[6] V.A. Rubakov, Phys. Usp. (2001) 871. 177] S.L. Dubovsky, V.A. Rubakov and P.G. Tinyakov, Phys. Rev. D (2000) 105011.[8] C. Macesanu, Int. J. Mod. Phys. A (2006) 2259.[9] R. Casadio and A. Gruppuso, Phys. Rev. D (2001) 025020.[10] S. Chang, J. Hisano, H. Nakano, N. Okada and M. Yamaguchi, Phys. Rev. D (2000)084025.[11] V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. B (1983) 136.[12] N. Arkani-Hamed and M. Schmaltz, Phys. Rev. D (2000) 033005.[13] Y. Grossman and M. Neubert, Phys. Lett. B (2000) 361.[14] T. Gherghetta and A. Pomarol, Nucl. Phys. B (2000) 141.[15] S.J. Huber and Q. Shafi, Phys. Lett. B (2001) 256.[16] M.N. Smolyakov, Phys. Rev. D (2012) 045036.[17] A.A. Andrianov, V.A. Andrianov, P. Giacconi and R. Soldati, JHEP (2003) 063.[18] E.A. Coddington and N. Levinson, “Theory of Ordinary Differential Equations”, McGraw-Hill, New York, 1955.[19] R.P. Woodard, Lect. Notes Phys. (2007) 403.[20] O. DeWolfe, D.Z. Freedman, S.S. Gubser and A. Karch, Phys. Rev. D (2000) 046008.[21] M. Geller, S. Bar-Shalom and A. Soni, Phys. Rev. D89