1.Introduction
The five-dimensional braneworld scenarios require standard model fields to be
confined to the four-dimensional sector of the theory. In particular, when the
electromagnetic field localization is considered, a non-normalizable gauge field is
found 1. This issue has been
addressed in several opportunities, and some proposals have emerged in order to
solve the problem. For instance, in 2, massive bulk vector fields are coupled to the
Randall-Sundrum (RS) brane 3
through a quadratic interaction term, in such a way that the photon is the bounded
state of a vector fields spectrum. On the other hand, in 4, where the brane is generated by a domain wall
solution to the Einstein scalar field system, the coupling between the bulk gauge
fields and a dilaton is required to find a normalizable gauge boson on the thick
brane. In both mechanisms, the localized state generates the standard
electromagnetic interaction. Outside of this framework, other proposals can be found
in Refs. 5-10.
In this paper, we consider the vector field localization on self-gravitating domain
walls via the coupling of the bulk vectors with the scalar field of the wall. We
propose an interaction term defined by the scalar potential of the wall. As a
result, the generalization of the Ghoroku-Nakamura mechanism 2 to thick walls is obtained. We also show that the
four-dimensional degrees of freedom of the bulk vectors are determined by a
supersymmetric quantum mechanics problem where the ground state yields the standard
electrostatic potential on the wall.
We apply the mechanism on the RS scenario and regularized versions of it, and the
so-called singular domain walls 11, where the scalar field interpolates between the lower values
of the scalar potential. Whereas in the first case, standard electromagnetism on the
wall is obtained, in the second one, the electrostatic interaction is the
five-dimensional way.
Finally, we discuss coordinate covariance and gauge the symmetry of the model.
2.Vector field coupled to the wall
Consider the five-dimensional coupled Einstein-scalar field system (Latin and Greek
indices correspond to five and four-dimensions, respectively)
Lgg=12R-12∂aϕ∂aϕ-V(ϕ).
(1)
We are interested in domain wall geometries, i.e., smooth scenarios
where the scalar field ϕ interpolates between the minima of the self-interaction potential
V(ϕ). Besides consider the coupling of the bulk vectors with the scalar field
of the wall, namely
LAg=-14FabFab+23V(ϕ)AaAa-Q2AaJa,
(2)
where Q is the five-dimensional coupling constant between two charged particles
on the wall.
Before moving forward, it is necessary to point out two aspects regarding (2): i) we
will assume that the vector fields do not modify the gravitation of the scenario and
ii) the term V(ϕ)AaAa will not be justified. We want to show that it leads to standard
electromagnetism in the four-dimensional sector of the wall.
In conformal coordinates,
ds2=e2a(z)ημνdxμdxν+dz2,
(3)
from (1), we have
(∂zϕ)2=3(∂za)2+∂z2a
(4)
and
V(ϕ(z))=-323(∂za)2+∂z2ae-2a,
(5)
and from (2), we get
ηγα∂z+∂za∂z+4V(ϕ)e2a3Aa+ηβσηaγ∂βσ2Aa-ηγα∂z+∂za∂aAz=Q2e4αJγ
(6)
and
(ηβσ∂βσ2+4V(ϕ)e2a3)Az-ηβσ∂z∂βAσ=Q2e4aJz.
(7)
We want to study the electromagnetic interaction in the four-dimensional sector of
the five-dimensional spacetime. To do this we will calculate the electrostatic
potential, which can be determined from the following generating functional
WJ=W0exp-i2∫d4xdzg(z)∫d4x´dz´g(z´)×J1a(x,z)Gab(x-x´,z,z´)J2b(x´,z´)
(8)
with Gab in the bulk propagator. In the momentum space, (8) can be written as
WJ=W0exp-i2∫dt ∫dxg(z) ∫dz´g(z´)×∫d3p2π3J~α(-p→,z)G~ab(p→,z,z´)J~b(p→,z´)
(9)
where, tilde indicates four-dimensional Fourier transform. So, if we assume that the
electrostatic sources are at z=z'=0, namely J̃a(p⃗,z)=δ(z)δμaj̃μ(p⃗), we find that the potential is given by
U[j]=∫d3p(2π)3j̃1μ(-p⃗)G̃μν(p⃗,0,0)j̃2ν(p⃗).
(10)
where Gμν are the four-dimensional components of propagator Gab.
By choosing
Aa=∫d4x'dz'g(z') Gab(x-x';z,z')Jb(x',z'),
(11)
from (6) and (7), in the momentum space, we find a coupled system of equations for
the components of G̃ab given by
ησae-a∂zea∂z-p-2+43Vϕe2aG~abp,z,z´+p-σp-αG~abp, z, z´+ip-σ∂z+∂zaG~zbp,z,z´=Q22π2e-aδbσδ(z-z´)
(12)
and
-p-2+43Vϕe2aG~zbp,z,z´+ip-a∂zG~abp,z,z´=Q22π2e-aδbzδz-z´,
(13)
where p¯α≡ηαβpβ. Now, in (12, 13), the transversal and longitudinal decomposition of the
four-dimensional components of the propagator is considered,
i.e.
G̃αβ=ηαβ-pαpβp¯2G1+pαpβp¯2G2,
(14)
where G1 and G2 satisfy
e-a∂zea∂zG1+-p-2+43V(ϕ)e2aG1=Q22π2e-aδ(z-z´)
(15)
and
e-ap-2∂zea-p-2+43V(ϕ)e2a-1∂2G2+43Vϕe2aG2+e2∂zea∂zG2=Q22π2e-aδ(z-z´)
(16)
On the other hand, for b = z, we find
G~βz=-ipβ-p-2+4Vϕe2az´3∂z,G2
(17)
G~βz=-ipβ-p-2+4Vϕe2az´3∂z,G2
(18)
G~zz=-ipβ-p-2+4Vϕe2az3∂z,G~βz+Q22π2e-az-p-2+4Vϕe2az3δ(z-z´)
(19)
Due to that pαj̃α=0, only the traversal sector of (14) contributes with the electrostatic
potential (10). Therefore, we will focus on finding a solution to (15).
Let us consider a continuous set of states ψm(z) satisfying the Schrödinger like the equation
-∂z2+VQMψm=m2ψm,
(20)
VQM=12∂z2a+14∂za2-43V(ϕ)e2a
(21)
and the following orthonormality and closure relationships
∫-∞∞ψm'*(z)ψm(z)dz=δ(m-m'),
(22)
∫ψm*(z')ψm(z)dm=δ(z'-z).
(23)
Notice that, for V(ϕ) given by (5), the eigenvalues equation (20, 21) can be written as a
supersymmetric quantum mechanics problem,
∂z+52∂za-∂z+52∂zaψm=m2ψm,
(24)
and hence, in the eigenfunctions spectrum, there are no states with negative mass
12,13. On the other hand, the
orthonormality condition (22) is divergent for all m=m', which is a technical problem that can be solved by introducing two
regularity branes at ±zr
14, in such a way that in limit
zr→∞ the initial scenario is recovered. As a consequence of the regulatory
branes, the basis is discretized, and the orthonormality relationships satisfied in
the sense of Kronecker
1zr∫-zrzrψm'*(z)ψm(z)dz=1zrδm'm.
(25)
Now, by expanding G1 in the discrete basis
G1=-Q2∑m1p2+m2ψm*(z')ψm(z)e[(a(z')+a(z)]/2,
(26)
we find that (26) satisfies automatically the Eq. (15) and that the electrostatic
potential, in the coordinates space, between two charged particles q1 and q2, i.e., jiμ(x⃗)=qiδ(x⃗-xi⃗)δ0μ, is determined by
U(r)=Q22πq1q2r(|ψ0(0)|2+∑m>0∞|ψm(0)|2e-mr),
(27)
where r=x2⃗-x1⃗. Notice that, in the presence of regulatory branes, the system resembles
a well of the infinite potential of width 2zr, so it is estimated that m is quantized in units of
1/zr. Thus, for zr→∞, the potential (27) can be written as
Ur=Q22πq1q2rψ0(0)2+κlimzr→∞∫0∞ψm(0)2e-mrzrdm.
(28)
where κ is a proportionality constant defined by boundary conditions at
±zr.
For a domain wall spacetime, we expect a zero-mode localized around z = 0, ψ0∼exp(5a/2) in correspondence with standard electromagnetism, and a tower of
unbounded KK modes generating corrections to the Coulomb law.
Next, we will determine the electrostatic potential ([VR]) on three scenarios: RS
brane, regular domain wall, and singular domain wall.
3.Vector fields on RS brane world
The RS scenario 3 in the conformal
coordinates ([3) is determined by
a(z)=-ln1+αz,
(29)
and
V(ϕ(z))=-3443Λ-23τδ(z),
(30)
with τ=6α the brane’s tension and Λ=6α2 the bulk cosmological constant.
In this scenario, the localization mechanism defined by action (2) takes the form
LAg=-14FabFab-12×43Λ-23τδzAaAa-Q2AaJa,
(31)
which is similar to the proposed in Ref. 2: a five-dimensional Proca theory with generic massive terms
in five and four dimensions. However, in (31), the bulk mass of the vector fields
and the four-dimensional coupling parameter are determined by the cosmological
constant and the brane’s tension, respectively.
To evaluate the electrostatic potential (28), from (21) and (29) we obtain
VQM=354α21+αz2-5αδ(z);
(32)
such that the eigenstates of the problem (20, 32) are given by a normalizable zero
mode
ψ0=2α1+αz-5/2,
(33)
and a discrete spectrum of unbounded eigenstates
ψm+=Nmα-1+zJ3ma-1+z+B+Y3mα-1+z,z>0,
(34)
ψm-=Nmα-1+zA-J3ma-1+z+B-Y3mα-1+z,z<0,
(35)
with J and Y, the Bessel functions of the first and
second kind and A, B, and Nm the integration and normalization
constants.
In agreement with 15, each
eigenvalue of (20) is associated with a pair of eigenfunctions, ψmc and ψmd; which, for the particular case (32), where VQM(z)=VQM(-z), corresponds to odd and even eigenstates of the problem (20, 32). Hence,
on the brane, they satisfy
ψmc(0-)=ψmc(0+)=0,
(36)
ddzψmc(0-)-ddzψmc(0+)=0,
(37)
and
ψmd(0-)=ψmd(0+),
(38)
ddzψmd(0-)-ddzψmd(0+)=5αψmd(0);
(39)
while, on the regular branes
ψm±d(±zr)≃0, ddzψm±d(±zr)≃0.
(40)
Therefore, the constants of integration set for ψmc and ψmd can be determined by (36, 37) and (25) in the first case; and by (38,
39) and (25) in the second one. On the other hand, the boundary condition (40)
induces the discretization of the mass, i.e., m∼nπ/zr with n∈Z+.
Thus, for the integration constants, we have
A-c=1, B-c=-B+c, B+c=-J3(m/α)Y3(m/α),
(41)
and
A-d=1+Y3(m/α)J3(m/α)-1, B-d=-B+d,
(42)
B+d=-12Y3mαJ3mα+J2mαY2mα×1+Y32mαJ32mα-1-12J2mαY2mα.
(43)
Finally, on the brane and for m≪α, we find
zrψmd(0)2≃π32mα3+π48mα5+π86mα7,
(44)
such that the electrostatic interaction
U(r)∼1r1+3321(αr)4+541(αr)6+1260431(αr)8,
(45)
is in correspondence with the standard four-dimensional potential for r≫α-1.
4.Regular Domain Walls
Let a(z)be the metric factor of a domain wall corresponding to a regularized version
of the RS brane 16-21. Following , it is possible to estimate at least the order
of corrections as follows. Asymptotically, (where the effects of wall thickness are
negligible) the metric factor of a regularized wall resembles the RS solution:
a(z)∼-ln(1+α|z|). Hence, for z≫α-1 the quantum mechanics potential (21)
VQM→5252+11z2.
(46)
In Ref. 22, it is shown that for a
VQM with asymptotic behavior similar to (46), the density of the state on
the wall is determined by ψm(0)∼(m/α)(5/2)-1 in such a way that, the corrections to the electrostatic potential (28)
go as r-5 and they are negligible compared to the Coulomb term from a critical
radius determined by the parameters of the regular scenario.
5.Singular Domain Walls
The domain walls are understood as solutions to the coupled Einstein-scalar field
system where ϕ interpolates between the minimums of the scalar potential. However,
there is another family of walls where V(ϕ) does not have minimums, but ϕ interpolates among the lower values of the scalar potential. These
scenarios are called singular domain walls, and like standard walls, they can locate
gravity 11.
Next, let us explore the four-dimensional effective behavior of the vector field on
the singular scenario reported in wherein in conformal coordinate (3), the warp
factor, the scalar field, and potential are given by
a(z)=-lncosh(αz), α>0.
(47)
ϕ=32 αz ,
(48)
and
V(ϕ)=3 α2 1-34 cosh22/3ϕ .
(49)
Notice that the scalar field interpolates between ±∞, and the scalar potential has a maximum in ϕ=0. On the other hand, the scalar curvature for this geometry is determined
by
R=14α21-32cosh(2αz)
(50)
which diverges for z→±∞. Thus, the solution represents a wall embedded in a five-dimensional
spacetime that interpolates between two subspace with naked singularities in the
horizon.
To calculate the electrostatic potential (28), the density of states associated to
(20) with
VQM=m0¯2-75m0¯2cosh-2(αz) , m0¯≡52α ,
(51)
is required. In this case, we can see that the zero mode is separated from continuous
modes by a mass gap defined by m0¯.
By considering the change of variable ξ=αz, the equation (20, 51) takes the form of a Schrödinger equation with a
potential of Pöschl-Teller ,
-12 d2dξ2ψ(ξ)-358cosh-2(ξ)ψ(ξ)=Eψ(ξ) ,
(52)
where
E=258m2m0¯2-1,
(53)
with bounded states determined by
E0=-258 , ψ0=N0cosh-5/2(ξ) ,
(54)
and
E1=-98 , ψ1=N1cosh-5/2(ξ) sinh(ξ),
(55)
such that
m02=0, m12=1625 m0¯2, N02=1615πm0¯.
(56)
Regarding the free states, under the change of variable
u=tanh(ξ),
(57)
the differential equation (52) takes the form of a Legendre equation
ddu1-u2dduψ+354+2E21-u2ψ=0 ,
(58)
whose solutions are the associated Legendre functions of the first and second kind,
of 5/2 grade and order μ=±-2E, given by
P52μu=1Γ(1-u)1+u1-uμ/2×2F1-52,72;1-μ;1-u2,
(59)
Q52μu=πΓ(72+μ)6eiπμu2-1μ/2u72+μ×2F174+μ2,94+μ2;4;1u2
(60)
The functions P5/2μ and Q5/2μ are orthogonal for |1-u|<2 and |u|>1, respectively; in particular, the change (57) satisfies |1-u|<2. Hence
ψmu=Nm2AmΓ-P52μu+Γ+P52-μ(u)-iΓ-P52μu-Γ+P52μ(u)
(61)
where Γ-=Γ(1-μ) and Γ+=Γ(1+μ).
The solution is doubly degenerate and satisfies the following boundary conditions
ψmc(0-)=ψmc(0+)=0 , ψm'c(0-)=ψm'c(0+) ,
(62)
ψmd(0-)=ψmd(0+) , ψm'd(0-)=ψm'd(0+).
(63)
Therefore, the density of the state in z = 0 is determined by
zrψmd(0)2=Γ-Γ+P5/2μ(0)P5/2-μ(0) .
(64)
Now, the integral in the electrostatic potential saturates for m≫m0¯, and in this case, the states ([Modomfull]) reduces to
zrψmd(0)2≃1-4916 m0¯2m2 .
(65)
By substituting the modes (54), (55) y (65) in (28), we obtain
U(r)∼1r-3316-4916m0¯rlnm0¯r+1-m0¯rm0¯r
(66)
where the dominant term for r≪α-1 is the order of r-2 such that the electrostatic interaction
on the four-dimensional sector of the scenario is defined for a five-dimensional
potential.
6.Discussion
In this paper, the mechanics of Ghoroku-Nakamura 2 for confining vector fields on the RS brane was
extended to self-gravitating domain walls in a five-dimensional bulk.
This was achieved by considering the coupling of the vector boson with the scalar
field of the wall. We found that the four-dimensional degrees of freedom of the
vector field can be expanded in terms of a basis of eigenstates of the Schrödinger
equation, free of tachyonic states. In the electrostatic potential, the ground state
defines the Coulomb interaction on the wall, while the massive state density
generates corrections to the potential.
When mechanism (2) is applied on the RS brane, the bulk cosmological constant plays
the role of mass for the vector field while the brane tension defines the parameter
coupling with the wall. The corrections generated by the massive states in the
electrostatic potential (28) are exponentially suppressed, and the standard
electromagnetism can be recovered on the brane from a critical radius similar to the
gravitational case.
In the case of thick brane, the mechanism was applied on regularized versions of the
RS scenario and singular walls. In the first case, in agreement with the Csaki
theorem 22, we find that
deviations to the Coulomb law are the order of r-5. In the second case,
the deviations are not negligible, and the electrostatic interaction goes to
r-2 on the four-dimensional sector of the wall.
Notice that theory (2) is covariant but not gauge-invariant due to the presence of
the quadratic term in the action. This can also be seen in the four-dimensional
effective theory (see Appendix),
LA(4)=-14fαβ2+∑n-14fαβn2-12mnaμn+∂μα5n2+4∑p,qlimzr→∞∫-zrzrdz∂zaψqφp∂μa5paμq+23∑p,qmpmqlimzr→∞∫-zrzrdz e2αV(ϕ(z))ψpψqa5pa5q
(67)
with
fαβ=∂αaβ-∂βaα, fαβn=∂αaβn-∂βaαn,
(68)
where we have considered Ab→e-a/2Ab and
Aμ(x,z)=aμ(x) ψ0(z)+∑n≠0aμn(x) ψn(z),
(69)
A5(x,z)=∑p≠0a5p(x) φp(z),
(70)
such that φn=(-∂z+5/2 ∂za)ψn/mn is the supersymmetric partner of ψn for mn≠0
12. In (67), the first two terms
correspond to the Maxwell action for aμ and the Stueckelberg action for the massive vector aμn. The absence of gauge symmetry is due to the final two terms; the second
to last is a term of interaction between aμp and a5n via the dynamics of the scalar modes ∂μa5n, and the last one corresponds to massive terms for (a5n)2 when n = p and to interaction terms
between a5n and a5p when n≠p.
In a forthcoming paper, we look forward to reporting a gauge-invariant generalization
of (2) and justify the term of interaction between vector fields.
This work was supported by IDI-ESPOCH under the project FCPI167.
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Appendix
A. The eigenvalues problem
To find the four-dimensional effective theory (67), we have considered
∫-∞∞dz∫d4x LA ⟶ limzr→∞∫-zrzrdz∫d4x LA,
(A.1)
where the spacetime is bounded by two regulatory branes, each with negative
tension and located at ±zr. Under this approach, it is possible to expand the components of
Ab in terms of a discrete base of functions with support along the
extra dimension.
In this sense, consider the operators
Q=∂z+52a' and Q+=-∂z+52a'.
(A.2)
If ψn is the set of states of the eigenvalues problem
QQ+ψn=mn2ψn,Q+ψn|±zr=0,n=0,1,2,3,…
(A.3)
with mn2≥0 (because the differential operator is factorizable) then, φn is the set of states of the problem
Q+Qφn=mn2φn, n=1,2,3,…
(A.4)
where φn=Q+ψn/mn for all mn≠0. Hence, ψn y φn always come in pairs, except for the zero modes of ψn (see Ref. for details). On the other hand, the orthonormality
relationship for each discrete set of functions is determined by
∫-zrzrdz ψnψp=δnp and ∫-zrzrdz φnφp=δnp.
(A.5)
For a five-dimensional vector field 𝐴 𝑏 , the components expand as indicated in
(69) and (70) where ψ0(z)∼ e5a(z)/2.