EPJ manuscript No.
(will be inserted by the editor)
Localization of five-dimensional Elko Spinors on dS/AdS Thick
Branes
Xiang-Nan Zhou1 , Xin-Yuan Ma1 , Zhen-Hua Zhao2 , and Yun-Zhi Du3
1
College of Physics and Information Engineering, Shanxi Normal University, Linfen 041004, People’s Republic of China
Department of Applied Physics, Shandong University of Science and Technology, Qingdao 266590, People’s Republic of China
3
Institute of Theoretical Physics, Datong University, Datong 037009, People’s Republic of China
arXiv:1812.08332v2 [hep-th] 31 Jul 2019
2
Received: date / Revised version: date
Abstract. Different from the Dirac spinor, the localization of a five-dimensional Elko spinor on a brane
with codimension one is very difficult because of the special structure of the Elko spinor. By introducing
the coupling between the Elko spinor and the scalar field generating the brane, we have two mechanisms for
localization of the zero mode of a five-dimensional Elko spinor on a brane. One is the Yukawa-type coupling
and the other is the non-minimal coupling. In this paper, we investigate the localization of the Elko zero
mode on de Sitter and Anti-de Sitter thick branes with the two localization mechanisms, respectively. It
shows that both the mechanisms can realize the localization. The forms of the coupling functions of the
scalar field in the two mechanisms have similar properties respectively and they play similar role for the
localization.
PACS. 11.27.+d Extended classical solutions; cosmic strings, domain walls, texture – 04.50.-h Higherdimensional gravity and other theories of gravity
1 Introduction
There are many classical problems which the Standard Model (SM) can not interpret sufficiently, such as the hierarchy
problem [1,2,3,4,5,6], cosmological problem [7,8,9,10,11,12,13,14] and the nature of dark matter and dark energy
[15,16,17,18]. Because extra dimension and brane-world theories can provide new mechanisms to solve these problems,
they have attracted more and more attention since the famous Arkani-Hamed, Dimopoulos and Dvali (ADD) [1] and
Randall-Sundrum (RS) brane-world models [2,19] were presented. Unlike early models where the brane is a geometric
hypersurface embedded in a higher dimensional spacetime, in a more realistic brane model the brane should have
thickness and inner structure. Such thick branes can be generated by bulk matter fields (mostly scalar fields) [20,21,
22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40], or can be realized in pure gravities [41,42,43] (see Refs.
[29,44] for more detailed introduction about thick brane models).
In the braneworld scenario, an important and interesting issue is to investigate the mechanism by which the KaluzaKlein (KK) modes of various fields could be localized on the brane. These KK modes contain the information of extra
dimensions. Especially, the zero modes of matter fields on the brane stand for the four-dimensional massless particles,
and they can rebuild the four-dimensional SM at low energy. A lot of work about the localization of various matter
fields on branes have been done [45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68].
On the other hand, Elko spinor, which is named from the eigenspinor of the charge conjugation operator, has
attracted continuing interest since it was introduced by Ahluwalia and Grumiller in 2005 [69,70]. It is a new spin-1/2
quantum field which satisfies the Klein-Gordon (KG) equation rather than the Dirac one, and only interacts with
itself, Higg fields and gravity [69,70,71,72,73,74,75,76]. As a candidate of dark matter, it was widely investigated in
particle physics [69,70,71,72], cosmology [77,78,79,80,81,82,83,84,85,86] and mathematical physics [87,88,89,90,91,
92,93].
In Refs. [94,95,96], the localization of the zero mode of a five-dimensional Elko spinor on various Minkowski branes
has been considered. A coupling mechanism should be introduced in order to localize the Elko zero mode on a brane.
¬
The first choice is the Yukawa-type coupling −ηF (φ or R) λ λ between the five-dimensional Elko spinor λ and the
background scalar field φ [94,95] or the Ricci scalar R [96]. Here F is a function of the background scalar field or
Correspondence to: duyzh13@lzu.edu.cn
2
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the Ricci scalar and η is the coupling constant. Recently, another localization mechanism, the non-minimal coupling
f (φ)LElko between the Elko spinor and the background scalar field, has been investigated in Ref. [97]. Here LElko is
the Lagrangian of the Elko spinor and f (φ) is a function of the background scalar field. It has been shown that by
introducing an auxiliary function K(z), the general expression of the Elko zero mode α0 and the scalar function f (φ)
could be obtained. And different forms of K(z) will lead to different solutions of the zero mode and the scalar function.
Thus, a non-minimal coupling can also provide the possibility of localizing the Elko zero mode. By now, all of the
investigations about the localization of a five-dimensional Elko spinor were concerned with Minkowski branes. As we
know, the properties of de Sitter (dS) and Anti-de Sitter (AdS) branes are very different from those of Minkowski
branes, and the results of the localization of matter fields on dS and AdS branes are different compared to those on
Minkowski branes. Thus, the localization of a five-dimensional Elko spinor on dS and AdS branes is an interesting
topic. At the same time, there are two kinds of localization mechanisms. What are the differences and similarities
between them also attract our great interest. We believe that investigating the differences and similarities between
them will be helpful to further explore the new localization mechanism and expand the possibility of localizing fields.
Therefore, in this paper, we will investigate this localization problem with the above two mentioned mechanisms. It
will be shown that both localization mechanisms will work for dS and AdS thick brane models and the functions F (φ)
and f (φ) play a similar role for the localization.
This paper is organized as follows. We first review the Yukawa-type and non-minimal couplings in Sec. 2. The
localization of the zero mode of a five-dimensional Elko spinor on the dS and AdS thick branes is investigated with
the two mechanisms in Sec. 3. Then in Sec. 4, we consider the localization of the Elko zero mode on another AdS thick
brane. Finally, a brief conclusion is given in Sec. 5.
2 Review of localization mechanisms
In this section, we review the two localization mechanisms for a five-dimensional Elko spinor in a thick brane model,
namely, the Yukawa-like coupling and the non-minimal coupling between the Elko spinor and the background scalar
field generating the thick brane.
The line-element is generally assumed as
ds2 = e2A(y) ĝµν dxµ dxν + dy 2 ,
(1)
where the warp factor e2A(y) is a function of the extra dimension y. By performing the following coordinate transformation
dz = e−A(y) dy,
(2)
ds2 = e2A (ĝµν dxµ dxν + dz 2 ),
(3)
the metric (1) is transformed as
which is more convenient for discussing the localization of gravity and various matter fields.
2.1 Yukawa-like coupling
Firstly, we start with the action of a five-dimensional massless Elko spinor
Z
√
S = d5 x −gLElko ,
¬
¬
¬
1
LElko = − g MN DM λ DN λ + DN λ DM λ − ηF (φ) λ λ,
4
(4)
where the last term is the Yukawa-like coupling with F (φ) a function of the background scalar field φ, and η is
the coupling constant. In this paper, M, N, · · · = 0, 1, 2, 3, 5 and µ, ν, · · · = 0, 1, 2, 3 denote the five-dimensional and
four-dimensional spacetime indices, respectively. The covariant derivatives are
¬
¬
¬
DM λ = (∂M + ΩM )λ, DM λ = ∂M λ − λ ΩM ,
(5)
where the spin connection ΩM is defined as
ΩM = −
S ĀB̄ =
i
P
− eB̄ N ∂M eĀN S ĀB̄ ,
eĀP eB̄ N ΓMN
2
i Ā B̄
[γ , γ ].
4
(6)
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3
Here eĀM is the vierbein and satisfies the orthonormality relation gMN = eĀM eB̄N ηĀB̄ . Ā, B̄ · · · = 0, 1, 2, 3, 5 stand for
the five-dimensional local Lorentz indices. So the non-vanishing components of the spin connection ΩM are
Ωµ =
1
∂z Aγµ γ5 + Ω̂µ .
2
(7)
Here γµ and γ5 are the four-dimensional gamma matrixes on the brane, and they satisfy {γ µ , γ ν } = 2ĝ µν .
Then the equation of motion for the Elko spinor coupled with the scalar field is read as
√
1
√ DM ( −gg MN DN λ) − 2ηF (φ)λ = 0.
−g
(8)
By considering the metric (3) and using the non-vanishing components of the spin connection (7), the above equation
can be rewritten as
p
1 2
1
√ D̂µ ( −ĝĝ µν D̂ν λ) + − A′ ĝ µν γµ γν λ
4
−ĝ
p
1 ′ 1
+ A √ D̂µ ( −ĝĝ µν γν γ5 λ) + ĝ µν γµ γ5 D̂ν λ
2
−ĝ
−3A
3A
+e
∂z (e ∂z λ) − 2ηe2A F (φ)λ = 0.
(9)
Here ĝµν is the induced metric on the brane, and D̂µ λ = (∂µ + Ω̂µ )λ with Ω̂µ the spin connection constructed by the
induced metric ĝµν . From D̂µ êaν = 0, we can get D̂µ ĝ λρ = D̂µ (êaλ êaρ ) = 0. Thus, the above equation can be simplified
as
p
1
2
√ D̂µ ( −ĝĝ µν D̂ν λ) −A′ γ 5 γ µ D̂µ λ−A′ λ+e−3A ∂z (e3A ∂z λ) − 2ηe2A F (φ)λ = 0.
(10)
−ĝ
Next, we introduce the following KK decomposition
X
(n)
(n)
αn (z)ς± (x) + αn (z)τ± (x)
λ± = e−3A/2
n
= e−3A/2
X
αn (z)λ̂n± (x).
(11)
n
(n)
n
For simplicity, we omit the ± subscript for the αn functions in the following. ς±
(x) and τ± (x) are linear independant
four-dimensional Elko spinors, and they satisfy
γ µ D̂µ ς± (x) = ∓iς∓ (x), γ µ D̂µ τ± (x) = ±iτ∓ (x),
5
γ ς± (x) = ±τ∓ (x),
5
γ τ± (x) = ∓ς∓ (x).
(12)
(13)
And the four-dimensional Elko spinor λ̂n should satisfy the K-G equation:
p
1
√ D̂µ ( −ĝĝ µν D̂ν λ̂n ) = m2n λ̂n
−ĝ
(14)
with mn the mass of the Elko spinor on the brane. Thus, we can get the following equations of motion for the Elko
KK modes αn :
α′′n −
3 ′′ 13 ′ 2
A + (A ) + 2ηe2A F (φ) − m2n + imn A′ αn = 0.
2
4
(15)
For the purpose of getting the action of the four-dimensional massless and massive Elko spinors from the action of a
five-dimensional massless Elko spinor with Yukawa-like coupling:
Z
¬
¬
¬
1 MN
5 √
SElko = d x −g − g
(DM λ DN λ + DN λ DM λ) − ηF (φ) λ λ
4
Z
¬
¬
ˆn n
ˆn
ˆn
1 µν ˆ ¬
1X
n
2
n
4
ˆ
ˆ
(16)
d x ĝ (Dµ λ D̂ν λ̂ + Dν λ Dµ λ̂ ) + mn λ λ̂ ,
=−
2 n
2
4
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we should introduce the following orthonormality condition for αn :
Z
α∗n αm dz = δnm .
(17)
For the Elko zero mode (m0 = 0), Eq. (15) reads
[−∂z2 + V0Y (z)]α0 (z) = 0,
(18)
where
V0Y (z) =
3 ′′ 13 ′ 2
A + A + 2ηe2A F (φ).
2
4
(19)
For this case, the orthonormality condition is given by
Z
α∗0 α0 dz = 1.
(20)
As we show in our previous work [94], there exist many similarities between the Elko field and the scalar field. For
a five-dimensional free massless scalar field, the Schrödinger-like equation for the scalar zero mode h0 [49,94] can be
read as
3
9 2
[−∂z2 + VΦ ]h0 = [−∂z2 + A′′ + A′ ]h0
2 4
3
3 ′
−∂z + A′ h0
= ∂z + A
2
2
= 0.
(21)
3
The solution is given by h0 (z) ∝ e 2 A(z) and it satisfies the orthonormality relation for any brane embedded in a fivedimensional Anti-de Sitter (AdS) spacetime. However, the effective potential V0 for the five-dimensional free massless
Elko spinor is [94]
V0 (z) =
3 ′′ 13 ′2
3
9
A + A = A′′ + A′2 + A′2 .
2
4
2
4
(22)
The additional term A′2 prevents the localization of the zero mode.
When the Yukawa-like coupling is introduced, the coefficient numbers of A′′ and A′2 can be regulated:
3 ′′ 13 ′ 2
A + A + 2ηe2A F (φ) = (pA′ )′ + (pA′ )2 ,
2
4
(23)
where p is a real constant. From Eq. (23), the form of F (φ) can be got
1
F (φ) = − e−2A
2η
13
3
′′
2
′2
A + p −
A .
p−
2
4
(24)
Then, Eq. (18) can be rewritten as
[−∂z2 + V0Y ]α0 = [−∂z2 + pA′′ + p2 A′2 ]α0
= [∂z + pA′ ] [−∂z + pA′ ] α0
= 0,
(25)
α0 (z) ∝ epA(z) .
(26)
and the Elko zero mode reads
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5
2.2 Non-minimal coupling
On the other hand, for the non-minimal coupling, the action could be written as
Z
√
S = d5 x −gf (φ)LElko ,
¬
¬
1
LElko = − g MN DM λ DN λ + DN λ DM λ .
4
(27)
Here f (φ) is a function of the background scalar field φ, which is only a function of the extra dimension z (or y). From
the action (27) the following equation of motion can be got
√
1
√
DM ( −gf (φ)g MN DN λ) = 0.
−gf (φ)
(28)
By considering the metric (3) and using the non-vanishing components of the spin connection (7), we can rewrite Eq.
(28) as:
p
1 2
1
√ D̂µ ( −ĝĝ µν D̂ν λ) + − A′ ĝ µν γµ γν λ
4
−ĝ
p
1 ′ 1
+ A √ D̂µ ( −ĝĝ µν γν γ5 λ) + ĝ µν γµ γ5 D̂ν λ
2
−ĝ
−3A −1
3A
+e
f (φ)∂z (e f (φ)∂z λ)
p
1
2
= √ D̂µ ( −ĝĝ µν D̂ν λ) −A′ γ 5 γ µ D̂µ λ−A′ λ
−ĝ
+ e−3A f −1 (φ)∂z (e3A f (φ)∂z λ) = 0.
In this case, we introduce the following KK decomposition:
X
1
(n)
(n)
αn (z)ς± (x) + αn (z)τ± (x)
λ± = e−3A/2 f (φ)− 2
(29)
n
=e
−3A/2
− 21
f (φ)
X
αn (z)λ̂n± (x).
(30)
n
(n)
(n)
(n)
(n)
By noticing the linear independance of the ς+ and τ+ (ς− and τ− ) and the K-G equation of the four-dimenional
Elko spinor, the equation of motion of the KK mode αn can be got
3
1
1 −2
f (φ)f ′2 (φ) + A′ f −1 (φ)f ′ (φ) + f −1 (φ)f ′′ (φ)
4
2
2
3 ′′ 13 ′ 2
2
′
+ A + (A ) − mn + imn A αn = 0.
2
4
α′′n − −
(31)
For the Yukawa-like coupling case, by introducing the orthonormality conditions (17), we can get the action of the
four-dimenional massless and massive Elko spinors from the action (27):
Z
¬
¬
√
1
d5 x −gf (φ)g MN (DM λ DN λ + DN λ DM λ)
SElko = −
4
Z
n
¬
¬
ˆn
ˆn
1 µν ˆ ¬
1X
n
n
2ˆ
n
4
ˆ
(32)
d x ĝ (Dµ λ D̂ν λ̂ + D̂ν λ Dµ λ̂ ) + mn λ λ̂ .
=−
2 n
2
For the Elko zero mode with mn = 0, Eq. (31) is simplified as
[−∂z2 + V0N (z)]α0 (z) = 0,
(33)
1
3
1
3
13
V0N (z) = − f −2 (φ)f ′2 (φ) + A′ f −1 (φ)f ′ (φ) + f −1 (φ)f ′′ (φ) + A′′ + A′2 ,
4
2
2
2
4
(34)
where the effective potential V0N is given by
6
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and the Elko zero mode α0 (z) satisfies the orthonormality condition (20). By introducing three new functions B(z),
C(z) and D(z) satisfying
A′2
C′
f ′ (φ)
= −3A′ +
−C−
,
f (φ)
C
C
3
1
∂z D(z) = A′ + B + C,
2
2
B(z) =
(35)
(36)
the effective potential (34) reads
1 2 3 ′
1
3
13
B + A B + B ′ + A′′ + A′2
4
2
2
2
4
= D′′ + D′2 ,
V0N (z) =
(37)
and Eq. (33) can be reduced as
[−∂z2 + V0N ]α0 = [−∂z2 + D′′ + D′2 ]α0
= [∂z + D′ ] [−∂z + D′ ] α0
= 0.
(38)
In addition, it will be convenient to define a new function K(z):
K(z) ≡
C′
− C.
C
(39)
It should be noticed that the form of K(z) is arbitrary, and the forms of C(z) and B(z) are determined by the warp
factor and any given K(z). Now it is easy to get the zero mode α0 :
α0 (z) ∝ eD(z)
Z z ′2
A
C′
1
= exp
−
+ C dz̄
2 0
C
C
Z z ′2
A
1
− K dz̄
= exp
2 0
C
Z z ′2
Z
1 z
1
A
= exp
dz̄ exp −
Kdz̄ ,
2 0 C
2 0
(40)
and the form of f (φ):
f (φ(z)) = e
Rz
0
B(z̄)dz̄
z
A′2
C′
−3A′ +
dz̄
−C −
C
C
0
Z z ′2
A
= exp −3A − 2 ln |C| +
+ K dz̄
C
0
Z z
Z z ′2
A
dz̄ exp
Kdz̄ .
= e−3A C −2 exp
C
0
0
= exp
Z
(41)
It is not difficult to check that the orthonormality condition (20) always requires that K(z) is an odd function and
positive as z > 0. Here Eqs. (40) and (41) give the general expressions of the zero mode α0 and the function f (φ)
because the function C(z) can be expressed by the function K(z) according to Eq. (39):
Rz
C(z) =
C1 −
e1
Rz
1
K(z̄)dz̄
R ẑ
K(z̄)dz̄
1
e
dẑ
,
(42)
where C1 is an arbitrary parameter. As we showed in our previous work, the role of K(z) is similar to the auxiliary
superpotential W (φ), which is introduced in order to solve the Einstein equations in thick brane models. For a given
K(z) the zero mode α0 is obtained by integrating Eq. (40). Then, the scalar field function f (φ(z)) is determined by
integrating Eq. (41). For different forms of K(z), there exist different configurations of the zero mode α0 and function
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7
f (φ). It gives us more choices and possibilities to study the localization of the Elko zero mode on the branes. Next we
will consider the localization of the Elko zero mode with this two kinds of couplings on dS/AdS thick branes.
3 Localization of Elko zero mode on dS/AdS thick branes
In this section, we investigate the localization of the Elko zero mode with two kinds of couplings on single-scalar-field
generated dS/AdS thick branes [26,66]. The system is described by the action
Z
√
M5
1
R − ∂M φ∂ M φ − V (φ) ,
(43)
S = d5 x −g
4
2
where R is the five-dimensional scalar curvature and V (φ) is the potential of the scalar field. For convenience, the
fundamental mass scale M5 is set to 1. The line element is described by Eq. (1) and the induced metrics ĝµν on the
branes read
−dt2 + e2βt (dx21 + dx22 + dx23 ) dS4 brane,
(44)
ĝµν =
−2βx3
e
(−dt2 + dx21 + dx22 ) + dx23 AdS4 brane.
Here the parameter β is related to the the four-dimensional cosmological constant of the dS4 or AdS4 brane by
Λ4 = 3β 2 or Λ4 = −3β 2 [31,56,66]. By introducing the following scalar potential
V (φ) =
3 2
a (1 + Λ4 ) 1 + (1 + 3s)Λ4 cosh2 (bφ)
4
−3a2 (1 + Λ4 )2 sinh2 (bφ),
(45)
a brane solution can be obtained [26,66]:
1
A(y) = − ln[sa2 (1 + Λ4 ) sec2 ȳ],
2
1
φ(y) = arcsinh(tan ȳ),
b
(46)
(47)
q
2(1+Λ4 )
. Note that the thick
where ȳ ≡ a(1 + Λ4 )y. The parameters a, s and b are real with s ∈ (0, 1] and b = 3(1+(1+s)Λ
4)
π
π
|, | 2a(1+Λ
| . By performing the coordinate transformation (2), we
brane is extended in the range y ∈ −| 2a(1+Λ
4)
4)
can get
h
πi
1
2 arctan ehz −
(48)
y=
a(1 + Λ4 )
2
q
4
with h ≡ 1+Λ
s . It should be noticed that the range of the coordinate z will trend to infinite. By substituting the
relation (48) into the solution (46) and (47), we can obtain the warp factor and scalar field in the coordinate z [66]:
1
A(z) = − ln a2 s(1 + Λ4 ) cosh2 (hz) ,
2
h
1
φ(z) = arcsinh [sinh(hz)] = z.
b
b
(49)
(50)
The warp factor e2A(z) is convergent at boundary. When Λ4 = 0, the above solution reduces to the flat brane one.
3.1 Yukawa-like coupling
Firstly, we consider the Yukawa-like coupling mechanism for the localization of the Elko zero mode. According to Eqs.
(23)-(26), (49) and (50), the Elko zero mode, the function F (φ) and the effective potential V0Y are given by
p
α0 ∝ epA(z) = (a2 s(1 + Λ4 ))− 2 sechp (hz),
h2 a2 s
(1 + Λ4 ) 25 − 4p(2 + p) + (4p2 − 13) cosh(2bφ) ,
F (φ) = −
16η
Y
′′
V0 = pA + p2 A′2 = −ph2 sech2 (hz) + p2 h2 tanh2 (hz).
(51)
(52)
(53)
8
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Now the orthonormality condition reads
Z
∝
=
Z
α∗0 α0 dz =
Z
α20 dz
(a2 s(1 + Λ4 ))−p sech2p (hz)dz
4p 2
(a s(1 + Λ4 ))−p 2 F1 (p, 2p; 1 + p; −1) < ∞.
hp
(54)
It requires p > 0. In Fig. 1 we plot the shapes of the zero mode, the effective potential V0Y (z) and the function F (φ),
which show that the Elko zero mode can be localized on the brane. Therefore, the Yukawa-like coupling mechanism
can be successfully used to localize the zero mode of the Elko spinor on the dS/AdS thick brane. We can find that
the effective potential V0Y (z) is a PT potential. The shape of F (φ) has a minimum around φ = 0 and diverges when
φ → ∞. As φ → ∞ the boundary values of the warp factor e2A and F (φ) are just opposite because there exists a
factor e−2A in the expression of F (φ) (24).
Fig. 1. The shapes of the Elko zero mode α0 (z) (51) (thick line), the effective potential V0Y (z) (dashed line) are drawn on the
left and the shape of function F (φ) (52) (right) is drawn on the right. The parameters are set to h = b = p = η = a2 s(1+Λ4 ) = 1.
3.2 Non-minimal coupling
Next, we focus on the non-minimal coupling mechanism. As shown in our previous work [97], there exist different
configurations of the Elko zero mode and f (φ) for different choices of K(z). In this paper we will consider two kinds
of K(z) and investigate the localization of the Elko zero mode (40).
3.2.1 K(z) = −kA′
Firstly, it is a natural choice to consider K(z) = −kA′ with k a positive constant. It is easy to get
1
C(z) =
C1 cosh(hz)−k −
Here F (z) = 2 F1
√
coth(hz)F (z) − sinh2 (hz)
h+hk
.
(55)
2
1 1+k 3+k
2 , 2 ; 2 ; cosh (hz)
. Especially, when k = 1, the form of C(z) will be reduced to
C(z) =
h cosh(hz)
.
hC1 − sinh(hz)
(56)
Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle
9
Here C1 is an arbitrary constant and we always let C1 = 0. Thus, the zero mode is rewritten as
α0 (z) ∝ eD(z)
Z z ′2
Z
1 z
1
A
dz̄ exp −
= exp
Kdz̄
2 0 C
2 0
1
∝ sech(hz)exp − sech2 (hz) .
4
(57)
It is easy to check that the orthonormality condition
Z
Z
∗
α0 α0 dz = α20 dz
Z
1
∝ sech2 (hz̄)exp[− sech2 (hz̄)]dz
2
√
√
(58)
= 2 2F ( 2/2) < ∞
√
can be satisfied. Here, F (z) gives the Dawson integral and F ( 2/2) = 0.512496. For this case, the effective potential
V0N (z) is given by
V0N (z) =
h2
(−2 − 5 cosh(2hz) − 10 cosh(4hz) + cosh(6hz)) sech6 (hz).
32
(59)
We plot the shapes of the zero mode and the effective potential in Fig. 2, from which we can see that the zero mode
(57) is localized on the brane and the effective potential is a PT-like potential. The function f (φ) reads
a4 s2
1
3
2
2
2
f (φ(z)) = 2 (1 + Λ4 ) cosh (bφ) tanh (bφ) exp − sech (bφ) ,
(60)
h
2
and it is plotted in Fig. (2) with h = b = a2 s(1 + Λ4) = 1. It is obvious that the shape of f (φ) is similar to F (φ) in the
previous subsection, and it has a minimum at the point of φ = 0 and diverges at infinity. The boundary behaviour of
f (φ) is also opposite to the warp factor, because there exists a factor e−3A in the expression of f (φ) (41).
Fig. 2. The shapes of the Elko zero mode α0 (z) (57) (thick line), the effective potential V0N (z) (59) (dashed line) are drawn on
the left and the shape of function f (φ) (60) (right) is drawn on the right. The parameters are set to h = b = a2 s(1 + Λ4 ) = 1.
3.2.2 K(z) = kφ
Another natural choice for K(z) is K(z) = kφ = k hb z with positive k, for which the form of C(z) is
q
C(z) = −
2k̄
π
Erfi
q
k̄
k̄
2z
2
e 2 z ,
(61)
10
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where k̄ ≡ k hb and Erfi
q
k̄
2z
is the imaginary error function. Thus the zero mode reads
1
α0 (z) ∝ exp − h2
2
r
π
k̄
L(z) − z 2 .
4
2k̄
(62)
where
L(z) ≡
Z
0
z
Erfi
r
k̄
z̄
2
!
tanh2 (hz̄)exp
−k̄z̄ 2
dz̄.
2
(63)
It can
h be iverified that the function L(z) approaches a constant as |z| → ∞, which shows that α0 (|z| → ∞) ∝
exp − k̄4 z 2 and its orthonormality condition can be satisfied. Thus, the zero mode (62) can be localized on the brane,
see Fig. 3. The effective potential V0N (z) and the corresponding function f (φ) have slightly complex forms and we
only show their shapes in Fig. 3. Here, we can find the effective potential is an infinite deep potential instead of a PT
one. However, the shape of f (φ) is still similar to the ones in the previous two subsections. Therefore, the functions
F (φ) and f (φ) have similar properties and play similar roles, although they appear in different places in the actions.
Fig. 3. The shapes of the Elko zero mode α0 (z) (62) (thick line), the effective potential V0N (z) (dashed line) are drawn on the
left and the shape of function f (φ) (right) is drawn on the right. The parameters are set to k̄ = h = b = a2 s(1 + Λ4 ) = 1.
4 Localization of Elko zero mode on AdS thick brane with divergent warp factor
In this section, we will consider another kind of AdS thick brane model and investigate the localization of the Elko
zero mode with two kinds of couplings. The warp factor in previous section is convergent at the boundaries of the
extra dimension, but the one in this section will be diverge, for which the zero mode of a five-dimensional free scalar
field can not be localized on this brane [56]. This difference will bring us some different and interesting results. The
action of this system reads [56]
Z
1 MN
1
5 √
∂M φ∂N φ − V (φ) ,
(64)
S = d x −g R − g
2
2
where R is the five-dimensional scalar curvature. Note that M5 is set to 2 here. The metric is described by (3) and
the induced metric is ĝµν = e−2βx3 (−dt2 + dx21 + dx22 ) + dx23 with Λ4 = −3β 2 . For the scalar potential
3(1 + 3δ)β 2
φ
2(1−δ)
V (φ) = −
cosh
,
(65)
2δ
φ0
a thick AdS brane solution was given in Refs. [24,56]:
A(z) = −δ ln cos
β
z
δ
,
β
z
φ(z) = φ0 arcsinh tan
δ
(66)
(67)
Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle
11
with
φ0 ≡
p
3δ(δ − 1).
(68)
δπ
and the parameter δ satisfies δ > 1 or δ < 0. It
Here, the range of the extra dimension is −zb ≤ z ≤ zb with zb = 2β
was found that only when δ > 1, there exists a thick 3-brane which localizes at |z| ≈ 0 [24,56]. Thus, we only consider
the case of δ > 1. In this case, the warp factor e2A(z) is diverge at the boundaries z = ±zb .
4.1 Yukawa-like coupling
With the Yukawa-like coupling and by substituting Eqs. (66) and (67) into (23)-(26),the Elko zero mode α0 , the
function F (φ) and the effective potential V0Y read
β
α0 ∝ epA(z) = cos−pδ
z ,
δ
β2
φ
φ
F (φ) =
6 − 4p + (6 + 13δ − 4p(1 + pδ)) sinh2
cosh−2δ
,
8δβ
φ0
φ0
β
pβ 2
β
V0Y = pA′′ + p2 A′2 =
sec2
z + p2 β 2 tan2
z .
δ
δ
δ
And the orthonormality condition requires pδ < 0:
Z
∝
Z
α∗0 α0 dz
δπ
2β
− δπ
2β
=
cos−2pδ
Z
(69)
(70)
(71)
α20 dz
β
z dz
δ
√
πΓ ( 21 − pδ)
< ∞.
=
Γ (1 − pδ)
(72)
Therefore, the zero mode can be localized on this AdS thick brane for any negative p by introducing the Yukawa-like
coupling mechanism. We plot the zero mode, the effective potential V0Y (z) and the function F (φ) in Fig. 4. The
effective potential is an infinite deep potential. And unlike previous model, here, F (φ) has the shape of a volcano, this
is because the boundary behavior of the warp factor is changed compared to the previous section.
Fig. 4. The shapes of the Elko zero mode α0 (z) (69) (thick line), the effective potential V0Y (z) (71) (dashed line) are drawn
on the left and the shape of function F (φ) (70) (right) is drawn on the right. For visibility, here α0 (z) has been magnified 100
times. The parameters are set to η = −1, δ = β = 2, and p = −5.
12
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4.2 Non-minimal coupling
Finally, we turn to the non-minimal coupling mechanism by considering K(z) = kA′ with k a positive constant. Note
that the sign in front of k is just opposite to the one in the previous section. When k = δ1 , the function C(z) in (42)
has the following form
k̄ sec(k̄z)
C(z) =
ln
where k̄ ≡
β
δ
k̄
cos( k̄
2 z )−sin( 2 z )
k̄
cos( k̄
2 z )+sin( 2 z )
,
and the parameter C1 is set to be zero. Then the zero mode reads
Z z
cos k̄2 z̄ − sin k̄2 z̄
1
1
dz̄ .
α0 (z) ∝ cos 2 (k̄z) exp k̄δ 2
sin(k̄z̄) tan(k̄ z̄) ln
2
0
cos k̄2 z̄ + sin k̄2 z̄
(73)
(74)
It is easy
R to check that the above α0 (z) vanishes when z → zb and the orthonormality condition for the Elko zero
mode α∗0 α0 dz = 1 can be satisfied. Therefore, the zero mode is localized on the brane (see Fig. 5). The effective
potential is an infinite deep one and the function f (φ) has a closed form but we only show their shapes in Fig. 5. It
can be found that all of the functions F (φ) and f (φ) have a minimum round φ = 0 in both thick brane models.
Fig. 5. The shapes of the Elko zero mode α0 (z) (74) (thick line), the effective potential V0N (z) (dashed line) are drawn on the
left and the shape of function f (φ) (right) is drawn on the right. For visibility, here α0 (z) has been magnified 5 times. The
parameters are set to β = δ = 2.
5 Conclusion and discussion
In this paper, we introduced two localization mechanisms to investigate the localization of the zero mode of a fivedimensional Elko spinor on dS/AdS thick branes. Firstly, we reviewed the two localization mechanisms, i.e, the
Yukawa-type coupling and the non-minimal coupling. It showed that in order to obtain the Elko zero mode on a
brane, the form of
coupling mechanism is determined by the warped factor (i.e., F (φ) =
F (φ) in the Yukawa-type
′2
1 −2A
e
p − 23 A′′ + p2 − 13
A
),
and
the
function f (φ) in the non-minimal coupling mechanism is determined
− 2η
4
by introducing an auxiliary function K(z). Then, we considered two kinds of curved thick brane models and investigated
the localization of the Elko zero mode with the two kinds of localization mechanisms.
In the first brane model, we considered the dS/AdS thick brane generated by a single scalar field. The results
showed that for the Yukawa-type coupling, the zero mode can be localized on the brane under the condition p > 0.
It is interesting to note that the factor e−2A in the expression of F (φ) leads to the result that the boundary behavior
of F (φ) is opposite to the warp factor e2A . Thus the shape of F (φ) diverges when φ → ∞ because of the convergent
warp factor. For the non-minimal coupling mechanism, by introducing two different forms of the auxiliary function
K(z), the zero mode can be confined on the brane. And the coupling functions f (φ) with two different forms of K(z)
are similar to the F (φ) for the Yukawa-type coupling in this brane system. Therefore, these two coupling functions
Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle
13
have similar properties and play similar roles although they appear in different places in the action of five-dimensional
Elko spinor. This result may help us to explore a new localization mechanism and expand the possibility of localizing
Elko spinor.
Next, another AdS thick brane with divergent warp factor was considered. For the Yukawa-type coupling case, the
zero mode can be localized on this AdS thick brane for any negative p, which is just opposite to the condition in the
previous thick brane model. Because of the divergent warp factor in this brane system, the shape of F (φ) seems to be
a volcano. For the non-minimal coupling case with the given auxiliary function K(z), we obtained the localized zero
mode on the brane and the coupling function f (φ). The function f (φ) still has a closed form. An interesting result is
that both F (φ) and f (φ) always have a minimum around φ = 0 no matter what the warp factor is.
In this paper, we gave the expressions of the coupling functions in order to localize the zero mode of a fivedimensional Elko spinor on curved thick branes. We found that the effective potential of the Schrödinger-like equation
satisfied by the zero mode is a PT potential or an infinite deep potential. However, it should be noticed that the
equation satisfied by the massive KK modes is a complex one, which is quite different from that satisfied by the zero
mode. Thus the effective potential function in this paper is not applicable for the massive KK modes. And we can not
judge whether there exists a massive KK mode although the effective potential is PT-like or infinite. In the future, we
will investigate the localization of the massive Elko KK modes on different thick branes.
Acknowledgments
The authors thank Professor Yu-Xiao Liu for his kind help. This work was supported in part by the National Natural
Science Foundation of China (Grant Nos. 11305095, 11647016 and 11705106), the Natural Science Foundation of
Shandong Province, China (Grant No. ZR2013AQ016), and the Scientic Research Foundation of Shandong University
of Science and Technology for Recruited Talents (Grant No. 2013RCJJ026).
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