A.T of ighi
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and M.Shokri
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Department of Nuclear Physics, Faculty of Basic Science,
U IB
CO Abstract
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University of M azandaran,
P .O . Box 47416-95447, Babolsar, Iran
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Generalized split fermion models
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In the split fermion model, fermions are localized in extra dimension by coupling to a scalar. We generalize this formalism to the case of N non-interacting scalars. We also provide exact solutions.
Introduction
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PACS: 11.10.Kk, 11.27.+d, 11.25.Mj Keywords:Extra dimension, Split fermion
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The Standard Model (SM) of particle physics[1 − 3] based on the gauge symmetry group N N SU(3)c SU(2)L U(1)Y has been successful to explain many of the known experimental facts with high precision. But SM is deficient in many respects. For instance there is no explanation in the standard model for the hierarchial structure of the fermion masses. In the split fermion model by invoking an extra spatial dimension the hierarchial structure of fermion masses are explained by localizing fermions in different locations in this extra dimension [4]. Configuration for fermions in the case of an infinite extra dimension has been studied [5]. But it has been found that by using an infinite extra dimension one can not accommodate the observed CP violation in the Kaon system[6]. The authors of Ref. [7] considered a compact extra dimension subject to orbif old boundary conditions S1 /Z2 . But fermion mass ¯ R ΨL are forbidden by Z2 symmetry. Hence the zero modes are localized terms of the form Ψ around one of the orbif old fixed points and the main goal of the split fermion scenario where fermions are localized at different points in the bulk bulk can not be realized. In Refs. [8, 9] it was noticed that while a dirac mass term is prohibited due to orbif old ¯ R ΨL provided the scalar Φ is odd symmetry consideration but a mass term of the formΦΨ ∗
Email:
[email protected]
1
under the Z2 orbif old symmetry is allowed. In these papers a two scalar model has been constructed which provided an explanation for the observed quark masses, mixing and CP violation. The main idea expressed in these paper was to build a localizer which have two nodes in the bulk and two use these points in addition to the fixed points of the orbif olds to localize the fermions. Moreover in split fermion model constraints, arising from flavor changing neutral currents (F CNC)have been studied in [10]. To address this F CNC problem, in another scheme [11] instead of fixed width fermions localized at different points, variable width fermions localized at the same point has been considered. In a previous study [12] we have obtained an exact solution for the two-scalar model of [8, 9]. We studied the problem of the stability of the localizer as well. In the present work we generalize our results to the case where the localizer is consist of N scalar. The plan of this paper is as follows. In section two we describe our model. We comment on the issue of the stability and we provide expressions for the width of the fermion wave function in the extra dimension. In section three we discuss some special three scalar and four scalar models. And finally in section four we presents our conclusions.
2
The Model
In this work we assume a flat background metric, gM N = Diag[+1, −1, −1, −1, −1],where the capital Latin letters M, N, ...run over five dimensions and the Greek letters over the four dimensions. The space coordinates X M = [xµ , w] are decomposed into the 4d(uncompactif ied)subset xµ and the compactif ied direction,w. The space-time of the model is described by an M4 × S1 /Z2 orbif old. The physical region is defined as 0 ≤ u ≤ 1 where u = wL such that L is the size of the extra dimension. Our model consists of N non-interacting scalars Φi , i = 1, ..., N and a fermion field Ψ. The action is S=
Z
¯ M ∂M − dxM [Ψ(iγ
N X
f˜i Φi )Ψ +
i=1
˜ i are real with λ ˜i > 0 Where f˜i and λ If we define the re-scaled fields by √ ψ = LΨ,
N X
1 ˜ i (Φ2 − v˜2 ))]. ( ∂ M Φi ∂M Φi − λ i i 2 i=1
and
ϕ=
Φ . v˜
(1)
(2)
and if we utilize the following dimensionless quantities ai ≡
f˜i fi ≡ q , 2λ˜i
q
˜ i v˜i L, 2λ
Xj ≡ −
aj+1 fj+1 , a1 f1
j = 1, ..., N − 1.
(3)
Then the action is S = S1 + S2 where S1 =
Z
dxµ
Z
0
1
¯ µ ∂µ − du[ψ(iγ
N −1 X f1 a1 1 Xj ϕj+1 ))ψ]. γ5 ∂5 − (ϕ1 − L L j=1
2
(4)
and S2 =
Z
dxµ
Z
1
du
0
N X
v˜i2 2 1 µ 1 a2 [L ∂ ϕi ∂µ ϕi − ∂5 ϕi ∂5 ϕi − i (ϕ2i − 1)2 ]. 2 2 2 i=1 L
(5)
∂ . where ∂5 ≡ ∂u Next we will investigate the functionhi (u) ≡< ϕi > (u). Neglecting the Y ukawa interactions the equation of motion for the scalars in the static case is.
∂52 ϕi − 2a2i ϕi (ϕ2i − 1) = 0,
i = 1, ..., N.
(6)
The solutions can be expressed[13,14] in term of Jacobi elliptic Sn function with modular parameter k (0 ≤ k ≤ 1). They are hi (u) = γi Sn(βi u, ki ), where γi2 =
2ki2 , 1 + ki2
βi2 =
i = 1, ..., N.
2a2i , 1 + ki2
i = 1, ..., N.
(7)
(8)
The function Sn(u) is odd in u and oscillates between 1 and −1. Its period is 4K(k). We consider the region 0 ≤ u ≤ 1. Requiring the period of the oscillatory solution to be 2L we find 1 i = 1, ..., N. (9) ai = (2(1 + ki2 )) 2 Ki (ki ), Next from these functions we construct the localizer g(u) = h1 (u) −
N −1 X
Xj hj+1(u).
(10)
j=1
By a appropriate choice of the parameters of the model this function will have several nodes in the extra dimension. For instance we find that for the case N = 2 the localizer will have two nodes in the bulk if the parameter X1 satisfy: γ1 β1 γ1 < X1 < γ2 β2 γ2
(11)
Previously [12] we have shown that a localizer composed of two non-interacting scalar is stable. It is straightforward to generalize that argument to the case of N non-interacting scalar. Hence we have a stable localizer. The fermion zero mode is obtained from [4, 7, 8, 15, 16] by y(u) ∼ exp[−f1 a1
Z
u
g(w)dw].
(12)
0
Performing the integration and up to normalization factor the fermion zero mode can be expressed as y(u) ∼
N Y
[Dn(βi u, ki ) + ki Cn(β1 u, ki )]fi ,
i=1
3
(13)
Where the Jacobi Elliptic function Cn and Dn are related to Jacobi Sn function by Cn2 u = 1 − Sn2 u,
and
Dn2 u = 1 − k 2 Sn2 u.
(14)
For our model first we consider a situation where the fermion is localized on one of the fixed points of the orbif old. By generalizing our previous result[12] we find that the expression for the width is 1 Γ ∼ qP . (15) N 2 (k ) f k K 2 i i i i i=1
But when the fermions are localized in the bulk the width is
1 , Γ ∼ qP N 2 f k K (k )Cn(β u , k )Dn(β u , k ) 2 i i 0 i i 0 i i=1 i i i
(16)
where u0 denotes the location of the node of the localizer.
3
Special cases
In the previous section we described a general N scalar model. In this section we describe some special cases in detail.
3.1
Three scalar model
From Eq. (10) with N = 3 the localizer is g(u) = h1 (u) − X1 h2 (u) − X2 h3 (u).
(17)
Its behavior is much more complicated than two scalars case because it depends on two parameters X1 and X2 . When X1 >
γ1 (β3 − β1 ) , γ2 (β3 − β2 )
and
X2 =
γ 1 − γ 2 X1 γ3
(18)
the localizer will have three nodes in the bulk. Due to symmetry one of these nodes is located in the midpoint of the orbifolded extra dimension. In Figure 1 such localizer is shown by the dotted curve. The value of the parameters are X1 = 2.97, X2 = −1.997, k1 = 0.95, k2 = 0.99 and k3 = 0.99. However we find that if the value of the parameter X2 is smaller than the value stated in eq. (18) then the localizer will have four nodes in the bulk. Again in Figure 2 such localizer is shown by the dotted curve. The value of the parameters are X1 = 2.5, X2 = −1.6, k1 = 0.95, k2 = 0.99 and k3 = 0.99.
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3.2
Four scalar model
From Eq. (10) with N = 4 the localizer is g(u) = h1 (u) − X1 h2 (u) − X2 h3 (u) − X3 h4 (u).
(19)
Let us rewrite the above expression as g(u) = g1 (u) + Ag2 (u),
(20)
g2 (u) = h4 (u) − Bh3 (u),
(21)
where and
X2 . X3 To investigate the nodes of localizer in the bulk. We introduce the parameters A = −X3 ,
η1 =
γ1 β1 , γ2 β2
and ηB =
η2 =
γ4 , γ3
B=−
ζ1 =
X1 γ2 β2 − γ1 β1 , γ4 β4 − Bγ3 β3
γ1 γ2
and
ζB =
γ 1 − γ 2 X1 . γ3 B − γ4
ζ2 =
γ4 β4 , γ3 β3
(22)
(23)
(24)
The condition that each of two sub-localizers g1 (u) and g2 (u) have two nodes in the bulk is η1 < X1 < ζ1 ,
η2 < B < ζ2 .
(25)
In addition if ηB < A < ζB then the localizer will have four nodes in the bulk. In figure 3 such localizer is shown. The value of the parameters for this figure are X1 = 0.65, X2 = 1.25,X3 = −1.0. k1 = 0.95, k2 = 0.99, k3 = 0.99 and k4 = 0.99.
4
Conclusions
We have utilized N non-interacting scalars. It is possible to envision models with coupling between these scalars, but then it would be very difficult to investigate the issue of stability of these models. Also for simplicity we only considered the quadratic and quartic terms for the scalars. In other cases it is not easy to obtain closed analytical solution for the localizer. The configurations of the charged lepton in the compact extra dimension has not been reported previously. We plan to utilize the formalism presented in this paper to study the charged leptons configurations in the extra dimension.
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References [1] S. L. Glashow, Nucl. Phys. 22, 579 (1961). [2] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967). [3] A. Salam, in: Elementary Particle Physics, N. Svarthholm. ed, (Almquist and Wicksell, Stockholm), 367 (1968). [4] N. Arkani-Hamed and M. Schmaltz, Phys. Rev. D, 61, 033005 (2000). [5] E. A. Mirabelli and M. Schmaltz, Phys. Rev. D 61, 113011 (2000). [6] G. C. Branco, A. de Gouvea and M. N. Rebelo, Phys. Lett. B, 506, 115 (2001). [7] D. E. Kaplan and T. M. Tait, JHEP, 111, 051 (2001). [8] Y. Grossman and G. Perez, Phys. Rev. D, 67, 015011 (2003). [9] Y. Grossman and G. Perez, Pranama, 62, 733 (2004). [10] B. Lillie and J. L. Hewett, Phys. Rev. D, 68, 116002 (2003). [11] B. Lillie, JHEP, 0312, 030 (2003). [12] A. Tofighi and M. Moazzen, Int. J. Theor. Phys. 50, 1709 (2011). [13] N. S. Manton and T. M. Samols, Phys. Lett. B, 207, 179(1988). [14] H. T. Cho, Phys. Rev. D, 72, 056010 (2005). [15] H. Georgi, A. K. Grant and G. Hailu, Phys. Rev. D, 61, 064027 (2001). [16] B. Grzadkowski and M. Toharia, Nucl. Phys. B, 686, 165 (2004).
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FIGURE CAPTIONS Figure 1: The typical shapes of the scalar vevs and the fermion wave functions. The the solid curve corresponds to the fermion wave functions, y(u) [eq. (13)]. The dotted curve correspond to the effective scalars vev, g(u) [eq. (17)]. This localizer is composed of three scalar and has three nodes in the bulk. The relevant parameters are specified in the text . Figure 2: The typical shapes of the scalar vevs and the fermion wave functions. The the solid curve corresponds to the fermion wave functions, y(u) [eq. (13)]. The dotted curve correspond to the effective scalars vev, g(u) [eq. (17)]. This localizer is composed of three scalars but has four nodes in the bulk. The relevant parameters are specified in the text. Figure 3: The typical shapes of the scalar vevs and the fermion wave functions. The the solid curve corresponds to the fermion wave functions, y(u) [eq. (13)]. The dotted curve correspond to the effective scalars vev, g(u) [eq. (19)]. This localizer is composed of four scalars and has four nodes in the bulk. The relevant parameters are specified in the text .
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Figure 1
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Figure 2
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Figure 3
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