Fermions in gravity and the skyrmion backgrounds in six dimensional brane-worlds

Fermions in gravity and the skyrmion backgrounds in six dimensional brane-worlds

Yuta Kodama    Kento Kokubu    Nobuyuki Sawado Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
July 14, 2019
Abstract

We construct brane solutions in six dimensional Einstein-Skyrme systems. A class of baby skyrmion solutions realize warped compactification of the extra dimensions and gravity localization on the brane for negative bulk cosmological constant. Coupling of the fermions with the brane skyrmions successfully lead to the brane localized fermions. The standard representation of the gamma matrices is used to obtain massive localized modes as well as the massless one. Nonlinear nature of the skyrmions brings richer information for the fermions level structure. In terms of the level crossing picture, emergence of the massive localized modes as well as the zero mode are observed.

pacs:
11.10.Kk, 11.27.+d, 04.50.-h, 12.39.Dc

I Introduction

Theories with extradimensions have been expected to solve the hierarchy problem and cosmological constant problem. Experimentally unobserved extradimensions indicate that the standard model particles and forces are confined to a 3-brane arkani-hamed98 (); randall99-1 (); randall99-2 (). Intensive study has been performed for the RS brane model in five space-time dimensions randall99-1 (); randall99-2 (). In this framework, the exponential warp factor in the metric can generate a large hierarchy of scales. This model, however, requires unstable negative tension branes and the fine-tuning between brane tensions and bulk cosmological constant.

There is hope that higher dimensional brane models than five could evade those problems appeared in five dimensions. In fact brane theories in six dimensions show a very distinct feature towards the fine-tuning and negative tension brane problems. In Refs. carroll03 (); navarro03 (), it was shown that the brane tension merely produces deficit angles in the bulk and hence it can take an arbitrary value without affecting the brane geometry. The model is based on the spontaneous compactification by the bulk magnetic flux. If the compactification manifold is a sphere, two branes have to be introduced with equal tensions. If it is a disk, no second 3-brane is needed. But still the fine-tuning between magnetic flux and the bulk cosmological constant can not be avoided although non-static solutions could be free of any fine-tuning kanti01 ().

Alternatively to the flux compactification in 6 dim., the nonlinear sigma model has been used for compactifications of the extra space dimensions gell-mann84 (); kehagias04 (); rubakov04 (); lee05 (). As in the flux compactification, no second 3-brane is needed if the parameters in the sigma model and bulk space-time are tuned.

Warped compactifications are also possible in six space-time dimensions in the model of topological objects such as defects and solitons. In this context strings cohen99 (); gregory00 (); gherghetta00 (); giovannini01 (); ringeval05 () were investigated, showing that they can realize localization of gravity. Interestingly, if the brane is modeled in such a field theory language, the fine-tuning between bulk and brane parameters required in the case of delta-like branes turns to a tuning of the model parameters ringeval02 ().

The Skyrme model is known to possess soliton solutions called baby skyrmions in two dimensional space piette94 (); piette95 (). In this paper we therefore consider the warped compactification of the two dimensional extra space by the baby skyrmions. We find that in the 6 dim. Einstein-Skyrme systems, static solutions which realize warped compactification exist for negative bulk cosmological constant. Since the solution is regular except at the conical singularity, it has only single 3-brane. Thus no fine-tuning between brane tensions is required. The Skyrme model possess a rich class of stable multi soliton solutions. We find various brane solutions by such multi-solitons.

It should be noted that general considerations in the 6 dim. brane model with bulk scalar fields suggest that the mechanism of regular warped compactification with single positive tension brane is not possible chen00 (). However, the model under consideration is restricted to the bulk scalar field depending only on the radial coordinate in the extra space. The scalar field in the Skyrme model depend not only the radial coordinate but also the angular coordinate to exhibit nontrivial topological structure, which makes possible to realize regular warped compactification.

Study of localization of fermions and gauge fields on topological defects have been extensively studied with co-dimension one kahagias01 (); melfo06 (); ringeval02f (); koley08 () and two randjbar-daemi00 (); libanov01 (); neronov02 (); zhao07 (); randjbar-daemi03 (); parameswaran07 (). In many years before, particle localization on a domain wall in higher dimensional space time was discussedrubakov83 (); akama (). The authors suggested the possibility of localized massless fermions on the 1 dim. kink background in 4+1 space-time with Yukawa-type coupling manner. Localization of chiral fermions on RS scenario is in Ref.kahagias01 (). Analysis for the massive fermionic modes was done by Ringeval et,al. in Ref.ringeval02f (). For co-dimension two, the localization on higher dimensional generalizations of the RS model was studied by the coupling of real scalar fields randjbar-daemi00 (). Many studies have been followed and most of them are based on the Abelian Higgs model coupled with the chiral fermions.

Problem of fermion mass hierarchy was discussed in Ref.arkani-hamed00 (); dvali00 (); libanov01 (); neronov02 () within different mechanisms. Especially, in Ref.libanov01 () the hierarchy between the fermionic generations are explained in terms of multi-winding number solutions of the complex scalar (Higgs) fields. They observed three chiral fermionic zero modes on a topological defect with winding number three and finite masses appear the mixing of those zero modes. Although any brane localization mechanism is absent in their discussion, the idea is promising.

In this article, we employ somewhat different set up: we consider the localization of the fermions on the skyrmion branes and, to treat the massive fermionic modes directly, we use the standard representation of the higher dimensional gamma matrices instead of the chiral one. The fermion localization and the existence of the zero modes are confirmed through the analysis of spectral flow of the one particle state.

This paper is organized as follows. In the next section we describe the Einstein-Skyrme system in six dimensions and derives the coupled equations for the Skyrme and gravitational fields. We derive a class of multi-winding number solutions. We will show some typical numerical solutions. Formulation of the fermions in higher dimensional curved space time is discussed in Sec.III. Coupling of the fermions and the skyrmions is introduced in this section. Conclusion and discussion are given in Sec.IV.

Ii Construction of the baby-skyrmion branes

ii.1 Model

We consider the model of the 6 dim. Einstein-Skyrme system with a bulk cosmological constant coupled to fermions. The action comprises

 S=Sgravity+Sbrane+Sfermion. (1)

Here is the six dimensional Einstein-Hilbert action

 Sgravity = ∫d6x√−g[12κ2R−Λb] (2)

In the parameter , is the six-dimensional Planck mass, denoted the fundamental gravity scale, and is the bulk cosmological constant.

For we use the action of baby-Skyrme model piette94 (); piette95 ()

 Sbrane = ∫d6xLbrane (3)

with

 Lbrane=√−g[F22∂M→ϕ⋅∂M→ϕ+14e2(∂M→ϕ×∂N→ϕ)2 +μ2(1+→n⋅→ϕ)], (4)

where run over and . denotes a triplet of scalar real fields with the constraint . The are the Skyrme model parameters wih the dimension of , , respectively. The first term in (4) is familiar from model. The second term is the analogue of the Skyrme fourth order term in the usual Skyrme model which works as a stabilizer for obtaining the soliton solution. The last term is referred to as a potential term which guarantee the stability of a baby-skyrmion.

The solutions of the model would be characterized by following topological charge in curved space-time

 Q=14π∫d2x→ϕ⋅(∇1→ϕ×∇2→ϕ) (5)

where means the space-time covariant derivative. Let us assume that the matter Skyrme fields depend only on the extra coordinates and impose the hedgehog ansatz

 →ϕ=(sinf(r)cosnθ,sinf(r)sinnθ,cosf(r)). (6)

The function which is often called profile function, has following boundary condition

 f(0)=−(m−1)π,  limr→∞f(r)=π (7)

where is arbitrary integer. This ansatz ensures the topological charge

 Q=n(1−(−1)m)/2. (8)

We consider the maximally symmetric metric with vanishing 4D cosmological constant,

 ds2=B2(r)ημνdxμdxν+dr2+C2(r)dθ2 (9)

where is the Minkowski metric with the signature in our convention and and . This ansatz has been proved to realize warped compactification of the extra dimension in models where branes are represented by global defects olasagasti00 ().

is the action of a fermions coupled with the skyrmions and the warp factors; that would be described in Sec.III.

The general forms of the coupled system of Einstein equations and the equation of motion of the Skyrme model are

 GMN=κ2(−ΛbgMN+TMN), (10) 1√−g∂N(√−gF2→ϕ×∂N→ϕ

where the stress-energy tensor is given by

 TMN=−2δLbraneδgMN+gMNLbrane =F2∂M→ϕ⋅∂N→ϕ+1e2gAB(∂A→ϕ×∂M→ϕ)⋅(∂B→ϕ×∂N→ϕ) +gMNLbrane. (12)

Inserting Eq. (6) into Eq. (4), one obtains the Lagrangian

 Lbrane=−B4~CF4e2[uf′2+n2sin2f~C2+2~μ(1+cosf)]

where we have introduced the dimensionless quantities

 ~xμ=eFxμ, y=eFr, ~C=eFC, ~μ=1eF2μ (14)

and

 u=1+n2sin2f~C2. (15)

The prime denotes derivative with respect to the radial component of the two extra space. The Skyrme field equation is thus

 f′′+(4B′B+~C′~C+u′u)f′ −12u[n2sin2f~C2(1+f′2)+2~μ2sinf]=0 (16)

where

 u′u=n2~C2+n2sin2f[f′sin2f−2~C′~Csin2f]. (17)

Within this ansatz, the components of the stress-energy tensor (12) becomes

 Tμν=−F4e2~B2ημντ0(y), τ0(y)≡u2f′2+n2sin2f2~C2+~μ2(1+cosf) (18) Trr=−F4e2τr(y), τr(y)≡−u2f′2+n2sin2f2~C2+~μ2(1+cosf) (19) Tθθ=−F4~C2τθ(y), τθ(y)≡^u2f′2−n2sin2f2^C2+~μ2(1+cosf) (20)

where

 ^u=1−n2sin2f~C2. (21)

The Einstein equations with bulk cosmological constant are written down in the following form

 3^b′+6^b2+3^b^c+^c′+^c2=−α(~Λb+τ0(y)) (22) 6^b2+4^b^c=−α(~Λb+τr(y)) (23) 4^b′+10^b2=−α(~Λb+τθ(y)) (24)

where is a dimensionless coupling constant and is a dimensionless bulk cosmological constant. Also, we introduce for convenience.

ii.2 Boundary conditions

At infinity, all components of the energy-momentum tensor vanishes and the Einstein equations (22)-(24) are then reduced to

 3^b′+6^b2+3^b^c+^c′+^c2=−α~Λb (25) 6^b2+4^b^c=−α~Λb (26) 4^b′+10^b2=−α~Λb. (27)

The general solution has been obtained in Ref. giovannini01 (); ringeval05 () which is given by

 ^b=pAe52py−e−52pyAe52py+e−52py,^c=5p22^b−32^b (28)

where is an arbitrary constant and

 p=√−α~Λb10. (29)

Since we are interested in regular solutions with warped compactification of the extra-space, the functions and must converge at infinity. This is achieved only when and with the solution

 B→ϵ1e−px,~C→ϵ2e−px (30)

where and are arbitrary constants. Then, the asymptotic form of the metric which realizes warped compactification is given by

 ds2∞ = ϵ1e−2√−α~Λb10yημνdxμdxν (31) + dy2+ϵ2e−2√−α~Λb10ydθ2.

The four-dimensional reduced Planck mass is derived by the coefficient of the four-dimensional Ricci scalar, which can be calculated inserting the metric (9) into the action (2),

 M2pl2∫d4x√−g(4)R(4)=M462∫d6x√−gB−2(r)R(4) =M462∫d4x√−g(4)R(4)∫drdθB2(r)C(r) =2πM462∫drB2(r)C(r)∫d4x√−g(4)R(4)

where the superscript represents a tensor defined on the four-dimensional submanifold. Thus, we find the relation between and as

 M2pl=2πM46∫∞0drB2(r)C(r). (32)

The requirement of gravity localization is equivalent to the finiteness of the four-dimensional Planck mass. For the asymptotic solution (31), the localization is attained.

Let us consider the asymptotic solutions for skyrmions. we can write

 f(y)=¯f+δf(y), (33)

where for , . The linearized field equations are given by

 δf′′−5pδf′−~μδf=0. (34)

Assuming that falls off exponentially, one obtains for

 δf(y)→fce−qy    with    q=√25p2+4~μ−5p2 (35)

where is an arbitrary constant.

Following regularity of the geometry at the center of the defect are imposed

 B′(0)=0,  C(0)=0,  C′(0)=1 (36)

and we can arbitarily fix . Boundary conditions for the warp factors and the profile function at the origin are determined by expanding them around the origin. For the different topological sectors, the first few terms are schematically written down as

 f(y)=−(m−1)π+f(n)(0)yn+O(yn+1) (37) b(y)=By+O(y3) (38) ~C(y)=y+Cy3+O(y5) (39)

where

 (m,n)=(1,1) B=−α4(~Λb+2~μ−12f′(0)4), (40) (m,n)=(1,2) B=−α4(~Λb+2~μ),  C=α12(~Λb+2~μ) (41) (m,n)=(2,1) B=−α4(~Λb−12f′(0)4), C=α12(~Λb−2f′(0)2−52f′(0)4) (42) (m,n)=(2,2) B=−α4~Λb,  C=α12~Λb. (43)

Thus one finds that the only or is the free parameter vicinity of the orgin.

Consider linear combinations of Eqs.(22)-(24), we obtain

 ^b′+4^b2+^b^c=−12α~Λb+α4(τr+τθ), (44) 4^b^c+^c′+^c2=−12α~Λb+α4(4τ0+τr−3τθ). (45)

Integrating Eqs.(44),(45) from zero to , we get

 B3(xc)B′(xc)~C(xc) =−α2~Λb∫xc0B4~Cdx−α4(μr+μθ), (46) B4(xc)~C′(xc) =1−α2~Λb∫xc0B4~Cdx−α4(4μ0+μr−μθ). (47)

(46) is the six-dimensional analogue of the relation determining the Tolman mass whereas Eq.(47) is the generalization of the relation giving the angular deficit. Combining these the following relations are obtained in the

 α∫∞0B4n2sin2f~C(1+f′2)dx=1 (48)

or

 α∫∞0B4[n2sin2f~C+2~Λb~C+2~μ~C(1−cosf)]dx=1.

These conditions are used for checking the numerical accuracy of our calculations.

ii.3 Numerical solutions

The equations (16),(22)-(24) should be solved numerically since they are highly nonlinear. The simple technique to solve the Einstein-Skyrme equations are the shooting method combined with the 4th order Runge-Kutta forward integration shiiki05 (). However, a unique set of boundary conditions at produces 2 distinct solutions, one of which grows exponentially and the other decays exponentially as . This causes instability of solutions when the forward integration is performed. Instead, we use a backward integration following Refs. giovannini01 () where the 6 dim. vortex-like regular brane solutions were constructed. The backward integration method requires a set of boundary conditions at infinity. We, however, truncate and take the distance far enough from the origin so that the Skyrme profile would fall off before it reaches . The set of boundary conditions at produces a unique solution which satisfies the boundary conditions at and hence it is numerically stable. We present our typical numerical results in Figs.[1,2].

Iii Fermions

iii.1 Basic formalism

The action of the fermions coupled with the Skyrme field in a Yukawa coupling manner can be written as

 Sfermion=∫d6xLfermion (50)

with

The six-dimensional gamma matrices are defined with the help the vielbein and those of the flat-space ,i.e., . The covariant derivative is defined as

 DA:=12↔∂A+12ω^a^bAσ^a^b (52)

where are the spin connection with generators . The simbol implies that . Here are the six-dimensional space-time index and corresponds to the flat tangent six-dimensional Minkowski space. The vielbein is defined through . The definition which was introduced in Refs.neronov02 (); randjbar-daemi03 () is simply defined as

 e^aμ=B(r)δ^aμ,  μ=0,⋯,3, e^rr=1,  e^θθ=C(r). (53)

They are the definitions which produce the gamma matrices parallel to the polar coordinate . Thus, special care is needed to explore the conserved quantities like angular momenta from the corresponding hamiltonian. Non-vanishing components of the corresponding spin connection are found to be

 ω^μ^4μ=δ^μμB′(r),  ω^4^55=−C′(r) (54)

where . The Dirac equation takes the form

 [i1Bδμ^μγ^μ∂μ+iγ^4(∂r+2B′B+C′2C) +iγ^51C∂θ−M→τ⋅→ϕ]Ψ=0. (55)

 H=−iγ^0γ^4(∂r+2B′B+C′2C)−iγ^0γ^51C∂θ+γ^0M→τ⋅→ϕ.

The equation (55) can be solved numerically in terms of, for example, the shooting method. At present study, however, we will treat the problem by somewhat different way. According to the Rayleigh-Ritz variational method bransden (), the upper bound of the spectrum can be obtained from the secular equation;

 det(A−ϵB)=0 (57)

where

 Aij=∫d3xφ†iHφj,  Bij=∫d3xφ†iφj

where is some complete set of the plane-wave spinor basis. For , the spectrum becomes exact. Eq.(57) can be solved numerically. For simplicity, we are to construct plane-wave basis in the flat space-time, i.e.,, as . However, no corresponding flat Hamiltonian subject to Eq.(LABEL:hamiltonian8_old) exists; thus it is not easy task to construct the plane-wave basis.

We employ a different form of the vielbein which was used at, e.g., Ref.zhao07 (), that is

 e^aμ=B(r)δ^aμ,  μ=0,⋯,3, e^4r=cosθ,  e^5r=sinθ, e^4θ=C(r)sinθ,  e^5θ=C(r)cosθ. (58)

The new definition produces the gamma matrices parallel to the Cartesian coordinate . Non-vanishing components of the corresponding spin connections are then

 ω^μ^4μ=δ^μμB′(r)cosθ,  ω^μ^5μ=δ^μμB′(r)sinθ, ω^4^5θ=1−C′(r). (59)

The Dirac equation is now

 [i1Bδμ^μγ^μ∂μ+i(cosθγ^4+sinθγ^5)(∂r+2B′B−1−C′2C) −i(sinθγ^4−cosθγ^5)1C∂θ−M→τ⋅→ϕ]Ψ=0. (60)

The Dirac gamma matrices should satisfy the anti-commutation relations and there are the candidates preserving such Clifford algebra. In most of previous studies in 6 dim., their analyses are based on the localization on the Abelian vortex and the chiral representation of gamma matrices. In this representation, the spinor can be expanded into the right and the left components. The zero mode appears as a eigenstate of right or left component. For the massive modes, they can be estimated from the mixing of both components. To treat the massive fermionic modes directly, we employ the standard representation of the higher dimensional gamma matrices instead of the chiral one. Since the eigenvalues depend on the properties of the background brane solutions, zero modes should appear as a special case of the massive modes. The standard representation of the gamma matrices in 6 dim. can be defined as

 γ^4:=(0−iI4−iI40),γ^5:=(0−I4I40) (61)

where means the identity matrix of the dimension . The six dimensional spinor can be decomposed into the four dimensional and the extra space-time components

 Ψ(xμ,r,θ)=ψ(xμ)(U(r,θ)V(r,θ)). (62)

Here the four dimensional part is the solution of the corresponding Dirac equation on the brane

 i¯γμ∂μψ=wψ (63)

in which the eigenvalues indicate an 4 dim. effective mass of the fermions. Substituting the ansatz (62) and (63) into the Dirac equation (55) yields the equations for

 −Be−iθ(∂y−i∂θ~C+Ay)V+Bm→τ⋅→ϕU=~wU Beiθ(∂y+i∂θ~C+Ay)U−Bm→τ⋅→ϕV=~wV (64)

where . Here we have introduced the dimensionless coupling constant (and ). In order to eliminate such induced potential from the equations, it is convenient to replace the eigenfunctions into  randjbar-daemi03 () by

 (UV) =exp[−2lnB(y)−12ln~C(y)+12∫ydy′~C(y′)](uv).

 H(uv)=~w(uv) (65) H:=B⎛⎜⎝m→τ⋅→ϕ−e−iθ(∂y−i∂θ~C)eiθ(∂y+i∂θ~C)−m→τ⋅→ϕ⎞⎟⎠. (66)

is regarded as a effective Hamiltonian of the model. If background is flat (i.e.,), the corresponding Hamiltonian easily reads

 Hflat=−γ^6γ^4∂4−γ^6γ^5∂5+γ^6m→τ⋅→ϕ (67)

where the partial derivatives in Cartesian coordinate are defined as . For obtaining the concise form (67), we introduce and use the additional component of the gamma matrix . One easily confirms that commutes with “grandspin”

 K3:=l3+γ^62+nτ32,  l3:=x4p5−x5p4 (68)

and also “time-reversal operator”

 T:=iγ^4⊗τ2C (69)

where is the charge conjugation operator. We emphasize that these operators also commute with the Hamiltonian in the curved space-time. As a consequence, eigenstates are specified by the magnitude of the grandspin,i.e.,

 K3=0,±1,±2,±3⋯for odd n K3=±12,±32,±52,⋯for even n. (70)

Since the Hamiltonian is invariant under time reverse, the states of are degenerate in energy.

In order to treat the eigenproblem (65), we employ the method which was originally proposed by Kahana-Ripka kahana () for solving the Dirac equation with non-linear chiral background. We construct the plane-wave basis in large circular box with radius as a set of eigenstates of the flat, unperturbed () Hamiltonian i.e., . The solutions are

 (uv)0,up =Mki(ω+ϵkiJp−1(kiy)ei(p−1)θω−ϵkiJp(kiy)eipθ)⊗(10) (uv)0,down=Nli(ω−ϵliJq(liy)eiqθω+ϵliJq+1(liy)ei(q+1)θ)⊗(01)

with

 Mki=[2πD2ϵkiϵki+m(jp−1(kiD))2]−1/2 Nli=[2πD2ϵliϵli+m(jq+1(liD))2]−1/2

and . The momenta are discretized by the boundary conditions

 Jp(kiD)=0,  Jq(liD)=0 (72)

where . The orthogonality of the basis is then satisfied by

 ∫D0drrJν(kir)Jν(kjr)=∫D0drrJν±1(kir)Jν±1(kjr) =δijD22[Jν±1(kiD)]2,  ν:=K3±12∓n2. (73)

Expanding the eigenstates of Eq.(65) in terms of the plane-wave basis, the eigenproblem reduces to the symmetric matrix diagonalization problem. A special care is taken on the estimation of the matrix element from the Hamiltonian (66). In order to hold the Hermiticity of the matrix, the following differential rule is imposed

 ⟨ψ|↔∂y|ϕ⟩=∫dydθ~C(y)12[ψ†∂yϕ−(∂yψ†)ϕ] (74)

Once the desired eigenfunctions are obtained, angular averaged fermion densities on the brane can be estimated as follows

 R(y)=N(y)ρ(y) (75) ρ(y):=∫dθ[u†(y,θ)u(y,θ)+v†(y,θ)v(y,θ)] (76)

where

 N(y)=exp[−4lnB(y)+∫ydy′~C(y′)]. (77)

For a numerical convenience, we divide the extra space-time into two domains. We introduce an effective size of brane that the stress energy tensors are finite inside. Outside this domain, no brane exists and only the warping of the geometry is effective. We assume that the fermions we should observe in our 4D space-time are perfectly trapped inside and never leak outside. We set the size as and tentatively choose the value , which is larger than the distribution of the stress energy tensors and the topological charge (see Fig.2). Of course outside of the domain is not Minkowski space so that the fermions at the boundary feel the effects. The numerical results necessarily depend on the choice of . This crude approximation works well if the effects of the geometry do not affect to the localized modes of the fermions.

For the numerical analysis except for the results of Figs.[9,10], we use the brane solutions with following parameter set for the background fields

 (m,n)=(1,1): α=0.5, ~Λb=−0.1, ~μ=0.220525915, f′(