Fermionic vacuum polarization in higher-dimensional global monopole spacetime

# Fermionic vacuum polarization in higher-dimensional global monopole spacetime

E. R. Bezerra de Mello
Departamento de Física-CCEN
Universidade Federal da Paraíba
58.059-970, J. Pessoa, PB
C. Postal 5.008
Brazil
E-mail: emello@fisica.ufpb.br
###### Abstract

In this paper we analyse the vacuum polarization effects associated with a massless fermionic field in a higher-dimensional global monopole spacetime in the ”braneworld” scenario. In this context we admit that the our Universe, the bulk, is represented by a flat dimensional brane having a global monopole in a extra transverse three dimensional submanifold. We explicitly calculate the renormalized vacuum average of the energy-momentum tensor, , admitting the global monopole as being a point-like object. We observe that this quantity depends crucially on the value of , and we provide explicit expressions to it for specific values attributed to .

PACS numbers: , ,

## 1. Introduction

Recently the braneworld model has attracted renewed interest. By this scenario our world is represented by a four dimensional sub-manifold, a three-brane, embedded in a higher dimensional spacetime [1]. Braneworlds naturally appear in the string/M theory context and provide a novel setting for discussing phenomenological and cosmological issues related to extra dimensions. The models introduced by Randall and Sundrum are particularly attractive [2, 3]. The corresponding spacetime contains two (RSI), respectively one (RSII), Ricci-flat brane(s) embedded on a five-dimensional Anti-de Sitter (AdS) bulk. It is assumed that all matter fields are confined on the branes and only the gravity propagates in the five dimensional bulk. The idea that matter is confined to a lower dimensional manifold is not a new one. The localization of fermions on a domain wall has been discussed in [4].

The hierarchy problem between the Planck scale and the electroweak one is solved in the RS1 model, if the distance between the two branes is about times the AdS radius. The braneworld model also provide some alternative discussions about one of the most important problem in the modern physics: the cosmological constant problem (see, for instance, Ref. [5]). In this way, the Casimir energy associated with quantum fields which propagates in the bulk obeying specific boundary conditions on the branes may contribute to both, the brane and bulk cosmological constant. The Casimir energy associated with scalar field on the five-dimensional Randal-Sundrum model are calculated in [6]. Surface Casimir densities and induced cosmological constant on the branes are calculated in [7] for a massive scalar quantum field obeying Robin boundary conditions on two parallel brane in a general dimensional anti-de Sitter bulk.

Although topological defects have been first studied in a four-dimensional spacetime [8], they have been considered in spacetimes of higher dimensions in the context of braneworld. In this scenario the defects live in a dimensions submanifold, having their cores on the brane. In this way, domain wall and cosmic string cases have been analyzed in [4] and [9, 10], respectively, considering and . Local and global monopoles have also been analyzed in [11, 12] and [13] to [17], respectively, considering . Specifically in [17] it was shown that if , the energy scale where the gauge symmetry of the global system is spontaneously broken, is smaller than the Planck mass, the seven-dimensional Einstein equations admit a solution which for points outside the global monopole’s core can be expressed by

where is the Minkowski metric, , a parameter smaller than unity, and related with the seven-dimensional Planck mass. In order to be more precise, in [13] the authors have obtained the solution to the Einstein equations considering a general dimensional Minkowski brane worldsheet and a global monopole in the transverse extra dimensions. In this general case the metric is a generalization of (1) with , and . The solid angle associated with the geometry above depends on the parameter and reads , so smaller than the usual one. Consequently in this submanifold there is a solid angle deficit

Composite topological defects have also been first studied in a four-dimensional space-time. Specifically a composite monopole, i.e., a system composed by a local and a global monopoles was analyzed in [18, 19, 20] (for composite strings see [21]). More recently, the composite monopole has been analysed in the braneworld scenario in [22].

In previous publication [23], we have analyzed the vacuum polarization effects associated with a massless scalar quantum field in an dimensional bulk spacetime which has the structure of a dimensional Minkowiski brane with a global monopole in the transverse three-dimensional sub-manifold.111Although the physical interesting case corresponds to , we developed our formalism considering as an arbitrary number Specifically we calculated the renormalized vacuum expectation value of the square of the field, , and have showed that this quantity depends crucially on the values attribute to . We also analyzed the structure of the renormalized vacuum expectation value of the energy-momentum tensor, . In order to develop these investigations we calculated the respective Euclidean scalar Green function, .

Continuing in the same line of investigation, in this paper we shall analyze the polarization effect of fermionic vacuum induced by a point-like three-dimensional global monopole embedded in a higher-dimensional bulk in a braneworld scenario. Specifically we shall consider that the bulk has its geometric structure given by the line element (1). Our main objective is to calculate the renormalized vacuum expectation value of the energy-momentum tensor, . Differently from the scalar case, here we obtain a simpler expression to the fermionic Green function and for the energy-momentum tensor, which is obtained in a closed form for an arbitrary parameter angle deficit.

This paper is organized as follows: In section we calculate the Euclidean fermionic propagator associated with a massless field in the background of a point-like global monopole transverse to a flat dimensional brane in a braneworld scenario, considering and . In order to do that we write the general Dirac differential operator and the equation obeyed by the propagator. In section we analyze this function in the coincidence limit and extract all divergencies from it in manifest form. In this way we provide the explicit expressions to the components of the renormalized energy-momentum tensor, and analyze their behavior in some limiting cases. In section , we summarize our most important results. In this paper we use signature , and the definitions: , . We also use untis .

## 2. Spinor Green Function

Before to start the calculation of the spinor Green function in the six-dimensional global monopole spacetime, we shall first review briefly some important properties of the Dirac equation in a flat space.

In a flat six-dimensional space, the Dirac matrices, , are matrices, which can be construct from the four-dimensional ones [24] as shown below [25]:

 Γ(μ)=(0γμγμ0) , Γ(4)=(0iγ5iγ50) , Γ(5)=(0I−I0) , (2)

where , and the identity matrix. It can be easily verified that these matrices obey the Clifford algebra: , for .

In these representation, the matrix is written as:

 Γ(7)=Γ(0)Γ(1) ...Γ(5)=(I00−I) . (3)

This matrix has two chiral eigenstate defined as and . Consequently any six-dimensional fermionic wave-function, , can be decomposed in terms of its chiral components as222It is worth to mention that the six-dimensional chirality does not correspond to four-dimensional chirality. Six-dimensional chiral wave-functions correspond to a four-dimensional wave-functions, which still contain two four-dimensional chiral components eigenstates of .

 Ψ=(Ψ+Ψ−) . (4)

Solutions for the Dirac equation,

 iΓ(M)∂(M)Ψ=MΨ , (5)

with defined chirality can only be possible for . In this way for positive chirality, equation (5) reduces to

 σ(M)∂(M)Ψ+=0 , (6)

being a set of matrix; and for negative chirality it reduces to

 ~σ(M)∂(M)Ψ−=0 , (7)

being now .

In order to write the Dirac equation in the six-dimensional global monopole spacetime, we shall choose the following coordinate system and basis tetrad:

 xA=(t, r,θ, ϕ, x, y)=(xμ, ya) , (8)

where and , and

 eA(M)=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1000000αsinθcosϕcosθcosϕ/r−sinϕ/rsinθ000αsinθsinϕcosθsinϕ/rcosϕ/rsinθ000αcosθ−sinθ/r000000010000001⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ . (9)

The Dirac equation for a massive field in the above coordinate system reads

 i⧸∇Ψ−MΨ=0 , (10)

with the covariant derivative operator given by

 ⧸∇=eA(M)Γ(M)(∂A+ΠA) , (11)

where is the spin connection, given in terms of the flat spacetime Dirac matrices by

 ΠA=−14Γ(M)Γ(N)eC(M)e(N)C;A . (12)

For the above basis tetrad, the only nonzero spin connections are:

 Πθ = i2(1−α)→Σ(8)⋅^ϕ Πϕ = −i2(1−α)sinθ→Σ(8)⋅^θ , (13)

where

 →Σ(8)=(→Σ00→Σ) , →Σ=(→σ00→σ) , (14)

being and the standard unit vectors along the angular directions and the Pauli matrices.

The fermionic propagator obeys the following differential equation:

 (i⧸∇−M)SF(x,x′)=1√−gδ(6)(x−x′)I(8) , (15)

where and is the identity matrix. This propagator is a bispinor, i.e., it transforms as at point and as at .

If a bispinor satisfies the differential equation below

 (□−M2−14R)DF(x,x′)=−1√−gδ(6)(x−x′)I(8) , (16)

with generalized d’Alembertian operator given by

 □=gMN∇M∇M=gMN(∂M∇N+ΠM∇N−{SMN}∇S) , (17)

the spinor Feynman propagator can be written as

 SF(x′,x)=(i⧸∇+M)DF(x′,x) . (18)

Now, after this brief review about the calculation of spinor Feynman propagator, let considering the six-dimensional global monopole spacetime (1), where the scalar curvature . Choosing the basis tetrad (9), we obtain, after some intermediate steps, that

 K=□−14R=−∂2t+α2(∂2r+2r∂r)−→L2r2+∂2x+∂2y−(1−α)r2(1+→Σ(8)⋅→L) , (19)

being the ordinary angular momentum operator.

The system that we shall consider consists of a massless positive chiral fields. In this way Eq. (10) can be written in terms of a matrix differential equation

 ⧸DΨ+=0 , (20)

with

 ⧸D=γ0∂t+αγr∂r−1rγr(→Σ⋅→L+1)+iγ5∂x−I∂y+αrγr , (21)

being .

The Feynman four-component propagator obeys the equation

 i⧸DSF(x,x′)=1√−gδ(6)(x−x′)I , (22)

and can be written in terms of the bispinor by

 SF(x,x′)=i⧸DGF(x′,x) , (23)

where now obeys the differential equation

 ¯KGF(x,x′)=−1√−gδ(6)(x−x′)I , (24)

with

 ¯K=−∂2t+α2(∂2r+2r∂r)−→L2r2+∂2x+∂2y−(1−α)r2(1+→Σ⋅→L) . (25)

The vacuum average value for the energy-momentum tensor can be expressed in terms of the Euclidean Green function. It is related with the ordinary Feynman Green function [26] by the relation , where . In the following we shall consider the Euclidean Green function.

In order to find a solution for the bispinor , we shall obtain the solution for the eigenvalue equation

 ¯KEΦλ(x)=−λ2Φλ(x) , (26)

with , so we can write

 GE(x,x′)=∑λΦλ(x)Φ†λ(x′)λ2 . (27)

Due to fact that our operator (19) is self-adjoint, the set of its eigenfunctions constitutes a basis for the Hilbert space associated with four-component spinors. Moreover, because operator is a parity even operator, its eigenfunctions present a defined parity, so the normalized eigenfunctions can be written as:

 Φ(σ)λ(x) = e−ikx(2π)3/2√αpr⎛⎜⎝Jνσ(pr)φ(σ)j,mj(θ,ϕ)inσJνσ+nσ(pr)^r⋅→σφ(σ)j,mj(θ,ϕ)⎞⎟⎠ , (28) λ2 = k2+α2p2 , (29)

where , and represents the cylindrical Bessel function of order

 νσ=j+1/2α−nσ2 , with nσ=(−1)σ , σ=0, 1 . (30)

These functions are specified by the set of quantum number , where , , denotes the value of the total angular quantum number, determines its projection. specifies two types of eigenfunctions with different parities corresponding to being the orbital quantum number. In (28), are the spinor spherical harmonics which are eigenfunctions of the operators and as shown below:

 →L2φ(σ)j,mj = l(l+1)φ(σ)j,mj , (31) →σ⋅→Lφ(σ)j,mj = −(1+κ(σ))φ(σ)j,mj , (32)

with and . Explicit form of above standard function are given in Ref. [24], for example.

Although we have developed this formalism for a six-dimensional spacetime, it can be adapted to a four and five dimensional space. The reason resides in the representations for the Dirac matrices in these dimensions. For four dimensions a irreducible representation for the flat Dirac matrices is the well known ones, so we may use . As a consequence, in the corresponding analysis we have to discard the derivative with respect to the coordinates and in the operator . For a five dimensions a possible representation for the flat Dirac matrices is also which can be given by . In this case the coordinate should be discarded. So on basis of these arguments it is possible to generalize the eigenfunction of the operator as:

 Φ(σ)λ(x) = e−ikx(2π)n/2√αpr⎛⎜⎝Jνσ(pr)φ(σ)j,mj(θ,ϕ)inσJνσ+nσ(pr)^r⋅→σφ(σ)j,mj(θ,ϕ)⎞⎟⎠ , (33)

with . From now on, we shall use the above eigenfunction and make applications for specific values attributed to .

Now we are in position to obtain the bispinor , which is given by

 GE(x,x′)=∫∞0 ds∫dnk∫∞0 dp∑σ,j,mjΦ(σ)λ(x)Φ(σ)†λ(x′) e−sλ2 . (34)

Finally substituting (33) into (34) with , we obtain with the help of [27] a closed expression to the Euclidean Green function:

 GE(x′,x)=1(2πrr′)n+12(−α2sinhu)n−12∑j,mj(Fj,mj(coshu,Ω,Ω′)00Fj,mj(coshu,Ω,Ω′)) ,

where

 Fj,mj(coshu,Ω,Ω′)=Qn−12ν0−1/2(coshu)C(0)j,mj(Ω,Ω′)+Qn−12ν1−1/2(coshu)C(1)j,mj(Ω,Ω′) , (36)

being

 coshu=r2+r′2+α2(Δx)22rr′≥1 , (Δx)2=¯ηab(x−x′)a(x−x′)b , (37)

and a matrix. is the associated Legendre function. We can express this function in terms of hypergeometric functions [27] by:

 Qλν(coshu) = eiλπ2λ√π Γ(ν+λ+1)Γ(ν+3/2)e−(ν+λ+1)u(1−e−2u)λ+1/2(sinhu)λ× (38) F(λ+1/2 , −λ+1/2 ; ν+3/2 ; 11−e2u) .

In this analysis we identify the parameter with and take . So the relevant hypergeometric function is

 F(n2 , −n−22 ; νσ+1 ; 11−e2u) .

If is a even number, this function becomes a polynomial of degree ; however being an odd number, this function is an infinite series.

Taking , and ; however for the first index the angular quantum number assumes crescent integer number starting from zero, i.e., , while for the second index . For the only term that contributes to (2.) in the summation is . For both associated Legendre functions contribute, . Using the explicit expressions for the spinor spherical harmonics, and using the addition theorem for the spherical harmonics, it is possible to express (2.) by:

 GE(x′,x)=1(2πrr′)n+1214π(−1sinhu)n−12∑l≥0(2l+1)Qn−12l(coshu)Pl(cosγ)I . (39)

Now taking and using the sum of Legendre functions and polynomials [27], we obtain standard expressions for the Green function. So:

• For , we have

 GE(x′,x)=14π21(x′−x)2I . (40)
• For , we have

 GE(x′,x)=18π21(x′−x)3I . (41)
• For , we have

 GE(x′,x)=14π31(x′−x)4I . (42)

After this application of our formalism, let us return to higher dimensional global monopole spacetime. In this space the fermionic Green function, , can be given by applying the Dirac operator on the bispinor (2.) according to (23).

## 3. Vacuum Average of the Energy-Momentum Tensor

In this section we shall calculate in a explicit way, the renormalized vacuum expectation value (VEV) of the energy-momentum tensor, . Because the metric tensor does not depend of any dimensional parameter, and also because we are working with natural units system, we can infer that the VEV of the energy-momentum tensor depends only on the radial coordinate . Moreover, due to the nonvanishing of Riemann and Ricci tensor, and scalar curvature of the spacetime, this VEV also depends on the arbitrary mass scale, , introduced by the renormalization prescription. So, by dimensional analysis it is expected that

 ⟨TAB⟩Ren.=1(√4πr)n+3(FAB+GABln(μr/α)) , (43)

where the tensors and depends only on the parameter . Because the presence of the arbitrary cutoff scale, , there is an ambiguity in the definition of (43). Finally for a spacetime of odd dimension, i.e., for a even value of , . Obviously the tensors are diagonal and due to the spherical symmetry of the problem we should have and .

The renormalized VEV of the energy-momentum tensor must be conserved,

 ∇A⟨TAB⟩Ren.=0 , (44)

and provide the correct trace anomaly for spacetime of even dimension [28]:

 ⟨TAA⟩Ren.=1(4π)n+32Tr(a(n+3)/2)=4T(√4πr)n+3 . (45)

As we shall see, taking into account these informations, it is possible to express all components and in terms of the zero-zero ones, and , and the trace

Using the point-splitting procedure [26], the VEV of the energy-momentum tensor for this four-component spinor Feynman propagator, compatible with the eight-component one, has the following form:

 (46)

Because the dependence of the fermionic Green function on the time variable, the zero-zero component of the energy-momentum tensor reads:

 ⟨T00(x)⟩=limx′→xTr γ0∂0SF(x,x′) , (47)

which can be expressed by

 ⟨T00(x)⟩=−ilimx′→x∂2t TrGF(x′,x)=−limx′→x∂2τ TrGE(x,x′) . (48)

In the obtainment of the above expression we have first taken in the Green function, (2.) and (36), the coincidence limit of the angular variable, , and sum over . This makes this function proportional to the unit matrix . So, only the term with time derivative in (21) provides a nonvanishing contribution for the zero-zero component of the energy-momentum tensor.

Now after these brief comments about general properties of the VEV of the energy-momentum tensor, we shall start the explicit calculation of this quantity, specifying the values for .

### 3..1 Case n=1

For , the bulk spacetime corresponds to the four-dimensional global monopole spacetime [29]. For this case the calculations of the vacuum polarization effects associated with massless two-component spinor field on this spacetime has been developed in [30] long time ago. More recently some Casimir densities associated with four-component massive fermionic field obeying the MIT bag boundary condition on, one spherical spherical shell, and two concentric spherical shells, in the global monopole spacetime have been analyzed in [31] and [32], respectively.

### 3..2 Case n=2

The case is a new one, it corresponds to a two-dimensional Minkowiski brane with a global monopole on the transverse sub-manifold. For this case the associated Legendre function in (36) assumes a very simple expression

 Q1/2νσ−1/2(coshu)=i√π2e−νσ√sinhu . (49)

Taking and into (2.), summing over and using the above expression for the Legendre function, it is possible to develop the sum over , which is a geometric series, we obtain a closed formula for the Euclidean Green function:

 GE(x′,x)=α64π2r31sinh(u/2)1sinh2(u/2α)I , (50)

with .

The above Green function is divergent in the coincidence limit, . We can verify its singular behavior by expanding (50) in power series of :

 GE(x′,x) = [18π21(Δx)3−1−α296π2r21(Δx)+1−α41920π2r4(Δx) (51) + 31α6−21α2−10483840π2r6(Δx)3+O((Δx)5)]I .

In order to obtain a finite and well defined result to the VEV of the zero-zero component of the energy-momentum tensor, we must extract all divergent terms in the coincidence limit after applying the second time derivative. In order to do that in a manifest form, we subtract from the Green function the Hadamard one. In [28] is given, for any dimensional spacetime, the general formal expression for the Hadamard function. For a five-dimensional spacetime it reads

 GH(x′,x)=Δ1/2(x′,x)16π2√21σ3/2(x′,x)[a0(x,′,x)+a1(x′,x)σ(x′,x)−a2(x′,x)σ2(x′,x)] . (52)

For this manifold, , the Van Vleck-Morette determinant, and the coefficients , are given below:

 Δ=1 , a0=I , a1=−1−α26r2I  and a2=−1−α460r4I . (53)

The one-half of the geodesic distance for and , is . So, we can see that the singular behavior for the Green function, Eq. (51), after taking its second time derivative has the same structure as given by the Hadamard function

 GH(x′,x)=[18π21(Δx)3−1−α296π2r21(Δx)+1−α41920π2r4(Δx)]I . (54)

So, on basis of this fact we can see that the renormalized VEV of zero-zero component of the energy-momentum vanishes, i.e.,

 ⟨T00(x)⟩Ren.=limx′→x∂2τ Tr[GE(x′,x)−GH(x′,x)]=0 . (55)

For an odd dimensional spacetime there is no trace anomaly, i.e., . Writing , we have that the sum of all these components vanishes; moreover, the geometric structure of the brane section of this five-dimensional spacetime is Minkowiski-type; consequently the Green function and the Hadamard one depend on the variables on the two dimensional brane by . By the definition of the VEV of the energy-momentum tensor, Eq. (46), we have

 ⟨T44(x)⟩=limx′→xTr∂2xGE(x,x′) . (56)

So, we can infer that . In this way we have , being both zero. The conservation condition, , provides , and , . So on basis of all these informations we conclude that all components of the tensor are zero, consequently,

 ⟨TAA⟩Ren.=0 . (57)

### 3..3 Case n=3

The case corresponds to a flat three-dimensional brane transverse to a three-dimensional global monopole space. The respective Euclidean Green function is given in terms of an infinite sum of the associated Legendre function with . Taking the angular coincide limit, , the sum below, given in (2.), can be expressed as:

 S0=∑j,mjQ1ν0−1/2(coshu)C(0)j,mj(Ω,Ω)=14π∑l≥0(l+1)Q1l+1α−1(coshu)I(2) (58)

and for

 S1=∑j,mjQ1ν1−1/2(coshu)C(1)j,mj(Ω,Ω)=14π∑l≥1lQ1lα(coshu)I(2) . (59)

Using , and the integral representation below for the Legendre function

 Qν(coshu)=1√2∫∞u dt e−(ν+1/2)t√cosht−coshu , (60)

it is possible to develop the sums over in (58) and (59) getting:

 S0(coshu)=I(2)sinhu16π√2ddz[∫∞arccoshzdt et/2√cosht−z1sinh2(t/2α)]∣z=coshu (61)

and

 S1(coshu)=I(2)sinhu16π√2ddz[∫∞arccoshzdt e−t/2√cosht−z1sinh2(t/2α)]∣z=coshu . (62)

Substituting the sum into (2.), we can express the Euclidean Green function by

 GE(x′,x)=−I32π3α2(r′r)2ddz⎡⎢ ⎢ ⎢ ⎢⎣∫∞√z−1dy√y2+1−z1sinh2(arcsinh(y/√2)α)⎤⎥ ⎥ ⎥ ⎥⎦∣z=coshu . (63)

where we have introduced a new variable .

The VEV of the zero-zero component of the energy-momentum tensor can be formally given by substituting (63) into (48). Because this procedure provides a divergent result, we must renormalize it by extracting all the divergent terms. In what follows we shall develop a procedure to extract the divercencies in a manifest form. Let us consider the integral inside the bracket of (63) and write it as:

 Iα(z) = I1(z)+I2(z)=∫1√z−1dy√y2+1−z1sinh2(arcsinh(y/√2)α) (64) + ∫∞1dy√y2+1−z1sinh2(arcsinh(y/√2)α) .

Subtracting and adding into the integrand of the first four terms of power series of the function

 1sinh2(arcsinh(y/√2)α)=2α2y2+(α2−1)3−(α4−1)30α2y2+(31α6−21α2−10)3780α