Polarization of the fermionic vacuum by a global monopole with finite core

0.26 MB·English

by  E. R. Bezerra de Mello

Polarization of the fermionic vacuum by a global monopole with
finite core
E. R. Bezerra de Mello1∗and A. A. Saharian1,2†
1Departamento de F´ısica-CCEN, Universidade Federal da Para´ıba
7 58.059-970, Caixa Postal 5.008, Jo˜ao Pessoa, PB, Brazil
0
0 2Department of Physics, Yerevan State University,
2
375025 Yerevan, Armenia

n
a
February 7, 2008
J

5
1

1
Abstract
v
3
We study the vacuum polarization effects associated with a massive fermionic field in a
4
1 spacetime produced by a global monopole considering a nontrivial inner structure for it. In
1 the generalcase of the spherically symmetric static core with finite support we evaluate the
0 vacuum expectationvaluesof the energy-momentumtensor andthe fermioniccondensate in
7
the region outside the core. These quantities are presented as the sum of point-like global
0
monopole and core-induced contributions. The asymptotic behavior of the core-induced
/
h vacuumdensitiesareinvestigatedatlargedistancesfromthecore,nearthecoreandforsmall
t
- values of the solid angle corresponding to strong gravitational fields. As an application of
p
generalresultstheflower-potmodelforthemonopole’scoreisconsideredandtheexpectation
e
values inside the core are evaluated.
h
:
v
i
X PACS number(s): 03.07.+k, 98.80.Cq, 11.27.+d
r
a
∗E-mail: emello@fisica.ufpb.br
†E-mail: [email protected]
1
1 Introduction
Symmetrybrakingphasetransitionsintheearlyuniversehaveseveralcosmological consequences
and provide an important link between particle physics and cosmology. In particular, different
types of topological objects may have been formedby thevacuum phasetransitions after Planck
time [1, 2]. These include domain walls, cosmic strings and monopoles. A global monopole is
a spherical symmetric gravitational topological defect created by a phase transition of a system
comprised by self-coupling scalar field, ϕa, whose original global O(3) symmetry is sponta-
neously broken to U(1). The matter fields play the role of an order parameter which outside the
monopole’s core acquires a non-vanishing value. The global monopole was first introduced by
Sokolov and Starobinsky [3]. A few years later, the gravitational effects associated with a global
monopole have been considered in Ref. [4], where the authors have found that for points far
from the monopole’s center, the geometry is similar to the black-hole with a solid angle deficit.
Neglecting themasstermwegetthepoint-likeglobalmonopolespacetimewiththemetrictensor
given by the following line element
ds2 = dt2 dr2 α2r2(dθ2+sin2θdφ2) , (1)
− −
where the parameter α2 is smaller than unity and depends on the energy scale where the sym-
metry is broken. Itis of interest to note thattheeffective metricproducedinsuperfluid3He A
−
by a monopole is described by the line element (1) with the negative angle deficit, α> 1, which
corresponds to the negative mass of the topological object [5]. The quasiparticles in this model
are chiral and massless fermions.
In quantum field theory the non-trivial topology of the global monopole spacetime induces
non-zero vacuum expectation values for physical observables. The quantum effects due to the
point-like global monopole spacetime on the matter fields have been considered for massless
scalar[6]andfermionic[7]fields,respectively. Theinfluenceofthenon-zerotemperatureonthese
polarization effects has been discussed in Ref. [8] for scalar and fermionic fields. Moreover, the
calculation of quantum effects on massless scalar field in a higher dimensional global monopole
spacetime has also been developed in Ref. [9]. The quantum effects of a scalar field induced
by a composite topological defect consisting a cosmic string on a p-dimensional brane and a
(m+1)-dimensional global monopole in the transverse extra dimensions are investigated in Ref.
[10]. Thecombinedvacuumpolarizationeffects bythenon-trivialgeometryofaglobalmonopole
and boundaryconditions imposed on the matter fields are investigated as well. In this direction,
the total Casimir energy associated with massive scalar field inside a spherical region in the
global monopole background has been analyzed in Refs. [11, 12] by using the zeta function
regularization procedure. Scalar Casimir densities induced by spherical boundaries have been
calculated in Refs. [13, 14] to higher dimensional global monopole spacetime by making use of
the generalized Abel-Plana summation formula [15]. More recently, using also this formalism,
a similar analysis for spinor fields obeying MIT bag boundary conditions has been developed in
Refs. [16, 17].
In general, the quantum analysis of matter fields in global monopole spacetime consider this
object as been a point-like one. Because of this fact the renormalized vacuum expectation value
of the energy-momentum tensor presents singularities at the monopole’s center. Of course,
this kind of problem cannot be expected in a realistic model. So a procedure to cure this
divergence is to consider the global monopole as having a non-trivial inner structure. In fact,
in a realistic point of view, the global monopoles have a characteristic core radius determined
by the symmetry braking energy scale where the symmetry of the system is spontaneously
broken. A simplified model for the monopole core is presented in Ref. [18]. In this model
the region inside the core is described by the de Sitter geometry. The vacuum polarization
2
effects associated with a massless scalar field in the region outside the core of this model are
investigated in Ref. [19]. In particular, it has been shown that long-range effects can take place
dueto thenon-trivial core structure. Recently the quantumanalysis of a scalar fieldin ahigher-
dimensional global monopole spacetime considering a general spherically symmetric model for
the core, has been considered in Ref. [20]. In the four-dimensional version of this model the
inducedelectrostatic self-energy andself-forceforachargedparticleareinvestigated inRef. [21].
Continuing in this direction, in the present paper we analyze the effects of global monopole core
on properties of the fermionic quantum vacuum. The most important quantities characterizing
these properties are the vacuum expectation values of the energy-momentum tensor and the
fermionic condensate. Thoughthe correspondingoperators are local, dueto theglobal natureof
the vacuum, the vacuum expectation values describe the global properties of the bulk and carry
an important information about the structure of the defect core. In addition to describing the
physical structure of the quantum field at a given point, the energy-momentum tensor acts as
the source of gravity in the Einstein equations. It therefore plays an important role in modelling
a self-consistent dynamics involving the gravitational field. The problem under consideration is
also of separate interest as an example with gravitational polarization of the fermionic vacuum,
where all calculations can be performed in a closed form. The corresponding results specify the
conditions under which we can ignore the details of the interior structure and approximate the
effect of the global monopole by the idealized model. The exactly solvable models of this type
are useful for testing the validity of various approximations used to deal with more complicated
geometries, in particular in black-hole spacetimes.
The plan of this paper is as follows. Section 2 presents the geometry of the problem and the
eigenfunctions for a massive spinor field. By usingthese eigenfunctions, in Section 3 we evaluate
the vacuum expectation values of the energy-momentum tensor and the fermionic condensate
in the region outside the monopole core. In Section 4 we consider the special case of the flower-
pot model for the core and the vacuum expectation values are investigated in both exterior
and interior regions. Various limiting cases are considered. The main results of the paper
are summarized in Section 5. In Appendix A we discuss the contribution of possible bound
states into the vacuum expectation values of the energy-momentum tensor and show that in
the flower-pot model there are no bound states. Throughout we use the system of units with
~= c = 1.
2 Model and the eigenfunctions for the spinor field
In this section we analyze the eigenfunctions for a massive fermionic field in background of the
global monopole geometry considering a nontrivial inner structure to the latter. The explicit
expression for the metric tensor in the region inside the monopole core is unknown and here
we consider a general model for a four-dimensional global monopole with a core of radius a,
assuming that the geometry of the spacetime is described by two distinct metric tensors in the
regions inside and outside the core. Adopting this model we can learn under which conditions
we can ignore the specific details for its core and consider the monopole as being a point-like
object. In spherical coordinates we will consider the corresponding line element in the interior
region, r < a, with the form
ds2 = e2u(r)dt2 e2v(r)dr2 e2h(r)(dθ2+sin2θdφ2) . (2)
− −
In the region outside, r > a, the metric tensor is given by the line element (1), where the
parameter α codifies the presence of the global monopole. For an idealized point-like global
monopole the geometry is described by line element (1) for all values of the radial coordinate
and it has singularity at the origin.
3
In Eq. (2) the functions u(r), v(r), h(r) are continuous at the core boundary, consequently
they satisfy the conditions
u(a) = v(a) = 0, h(a) = ln(αa) . (3)
If there is no surface energy-momentum tensor on the bounding surface r = a, the radial
derivatives of these functions are continuous as well. When the surface energy-momentum
tensor is present the junctions in the radial derivatives of the components of the metric tensor
are expressed in terms of the surface energy density and stresses1. Introducing a new radial
coordinate r˜ = eh(r) with the core center at r˜ = 0, the angular part of line element (2) is
written in the standard Minkowskian form. However, with this choice, in general, we will
obtain non-standard form of the angular part in the exterior line element (1). In the model
under consideration we will assume that inside the core the spacetime geometry is regular. In
particular, from the regularity of the interior geometry at the core center one has the conditions
u(r),v(r) 0 and h(r) lnr˜for r˜ 0.
→ ∼ →
We are interested in quantum effects of a spinor field propagating on background of the
spacetime described by line elements (1) and (2). The dynamics of the massive spinor field in
curved spacetime is governed by the Dirac equation
iγµ ψ Mψ = 0 , (4)
µ
∇ −
with the covariant derivative operator
= ∂ +Γ . (5)
µ µ µ
∇
Here γµ = eµ γ(a) are the Dirac matrices in curved spacetime, and Γ is the spin connection
(a) µ
given in terms of the flat γ matrices by the relation
1
Γ = γ(a)γ(b)eν e . (6)
µ 4 (a) (b)ν;µ
In the equations above eµ is the vierbein satisfying the condition eµ eν ηab = gµν.
(a) (a) (b)
In the region inside the monopole core we choose the basis tetrad
e−u(r) 0 0 0
0 e−v(r)sinθcosϕ e−h(r)cosθcosϕ e−h(r)sinϕ/sinθ
µ
e =  −  , (7)
(a) 0 e−v(r)sinθsinϕ e−h(r)cosθsinϕ e−h(r)cosϕ/sinθ
 0 e−v(r)cosθ e−h(r)sinθ 0 
 − 
 
where the rows of the matrix are specified by the index a and the columns by the index µ. By
using Eq. (7), for the non-vanishing components of the spin connection we find
1
Γ = u′eu−vγ(0)~γ rˆ,
0
2 ·
i
Γ = (1 h′eh−v)Σ~ ϕˆ , (8)
2
2 − ·
i
Γ = sinθ(1 h′eh−v)Σ~ θˆ,
3
−2 − ·
where the prime denotes derivative with respect to the radial coordinate, rˆ , θˆ and ϕˆ are the
standard unit vectors along the three spatial directions in spherical coordinates, and
~σ 0
Σ~ = , (9)
0 ~σ
(cid:18) (cid:19)
1This point will be analyzed in more detail at the end of the nextsection
4
with ~σ = (σ ,σ ,σ ) being the Pauli matrices. From the obtained results, for the combination
1 2 3
entering in the Dirac equation we have
u′e−v
γµΓ = ~γ rˆ e−h(1 h′eh−v)~γ rˆ. (10)
µ
2 · − − ·
After some intermediate steps, the Dirac equation reads
ie−uγ(0)∂ ψ+ie−u/2−v−hγ(r)∂ (eu/2+hψ) ie−hγ(r)(Σ~ L~ +1)ψ Mψ = 0 , (11)
t r
− · −
with L~ being the standard angular momentum operator.
To find the vacuum expectation value (VEV) of the energy-momentum tensor we need the
corresponding eigenfunctions. For the problem under consideration there are two types of
eigenfunctions with different parities which we will distinguish by the index σ = 0,1. These
functions are specified by the total angular momentum j = 1/2, 3/2, ... and its projection
m = j, j +1, ..., j. Assuming the time dependence in the form e−iωt, the four-component
− −
spinor fields specified by the set of quantum numbers β = (σkjm) with k2 = ω2 M2, can be
−
written in terms of two-component ones as
f (r)Ω (θ,ϕ)
ψ = e−iωt β jlσm , (12)
β n g (r)Ω (θ,ϕ)
σ β jl′m
(cid:18) σ (cid:19)
where
n n
n =( 1)σ, l = j σ, l′ = j + σ, (13)
σ − σ − 2 σ 2
and Ω (θ,ϕ) are the standard spinor spherical harmonics (see, for instance, Ref. [22]). The
jlσm
latter are eigenfunctions of the operator =~σ L~ +I as shown below:
K ·
Ω = κ Ω , κ = n (j +1/2). (14)
K jlσm − σ jlσm σ − σ
Note that we have the relation
Ωjlσ′m = ilσ−lσ′ (nˆ·~σ)Ωjlσm . (15)
Substituting the function ψ into the Dirac equation above, and using for the flat Dirac
β
matrices the representation given in Ref. [22], for the radial functions we obtain a set of two
coupled linear differential equations:
f′ +(u′/2+h′+κ ev−h)f = (M +e−uω)evg , (16)
β σ β β
g′ +(u′/2+h′ κ ev−h)g = (M e−uω)evf . (17)
β − σ β − β
In the region r > a for the radial functions we have the solutions
1
(ex)
f (r) = f (r) = [c J (kr)+c Y (kr)], (18)
β β √r 1 νσ 2 νσ
1 n k
(ex) σ
g (r) = g (r)= [c J (kr)+c Y (kr)], (19)
β β −√rω+M 1 νσ+nσ 2 νσ+nσ
where J (x) and Y (x) are the Bessel and Neumann functions of the order
νσ νσ
j +1/2 n
σ
ν = . (20)
σ
α − 2
5
By taking into account relation (15), the corresponding eigenfunctions are written in the form
(ex)
f (r)Ω (θ,ϕ)
ψ(ex) = e−iωt β jlσm . (21)
β (ex)
−igβ (r)(nˆ·~σ)Ωjlσm(θ,ϕ) !
In Eqs. (18) and (19), the integration constants c and c are determined from the matching
1 2
condition with the interior solution.
The regular solution to Eqs. (16), (17) in the interior region, r < a, we will denote by
n k
σ
f (r) = R (r,k), g (r) = R (r,k). (22)
β 1,nσ β −ω+M 2,nσ
Near the core center, r˜ 0, these functions behave like R ∝ r˜j+(1−nσ)/2 and R ∝
→ 1,nσ 2,nσ
r˜j+(1+nσ)/2, where the radial coordinate r˜ is introduced in the paragraph after formula (3).
Note that the functions R (r,k) and n kR (r, k)/(M ω) are solutions of the same
1,nσ σ 2,−nσ − −
equation and, hence,
R (r, k) = const R (r,k). (23)
2,−nσ − · 1,nσ
The interior eigenfunctions have the form
R (r,k)Ω (θ,ϕ)
ψ(in) = e−iωt 1,nσ jlσm . (24)
β inσk R (r,k)(nˆ ~σ)Ω (θ,ϕ)
(cid:18) ω+M 2,nσ · jlσm (cid:19)
From the continuity condition of the eigenfunctions on the surface r = a, for the coefficients c
1
and c in Eq. (18) one finds
2
π
c = n ka3/2[R (a,k)Y (ka) R (a,k)Y (ka)], (25)
1 −2 σ 1,nσ νσ+nσ − 2,nσ νσ
π
c = n ka3/2[R (a,k)J (ka) R (a,k)J (ka)]. (26)
2 2 σ 1,nσ νσ+nσ − 2,nσ νσ
Note that from Eqs. (16), (17), for the values of the interior radial functions on the boundary
of the core we have the following relations
n kR (a,k) = R′ (a,k)+ u′/2+h′ n λ/a R (a,k), (27)
− σ 2,nσ 1,nσ a a − σ 1,nσ
n kR (a,k) = R′ (a,k)+ u′/2+h′ +n λ/a R (a,k), (28)
σ 1,nσ 2,nσ (cid:0) a a σ (cid:1) 2,nσ
where R′ (a,k) = ∂R (r,k)/∂r , u′ = du(cid:0)/dr , h′ = dh/d(cid:1)r , and we have intro-
j,nσ j,nσ |r=a a |r=a a |r=a
duced the notation
λ = (j +1/2)/α. (29)
Substituting the expressions for the coefficients c and c into the formulae for the radial
1 2
eigenfunctions in the exterior region, one finds
πR (a,k)e−iωt g (ka,kr)Ω
ψ(ex) = 1,nσ νσ,0 jlm , (30)
β 2 r/a (cid:18) ωin+σMk gνσ,nσ(ka,kr)(nˆ ·~σ)Ωjlm (cid:19)
with the notation p
g (x,y) = Y˜ (x)J (y) J˜ (x)Y (y), p = 0,n . (31)
νσ,p νσ νσ+p − νσ νσ+p σ
Here and in what follows, for a function F (x) with F = J,Y we usethe tilded notation defined
νσ
by the formula
F˜ (ka) = n ka[F (ka) F (ka)R (a,k)/R (a,k)]
νσ − σ νσ+nσ − νσ 2,nσ 1,nσ
R′ (a,k) 1 1
= kaF′ (ka) a 1,nσ + au′ +ah′ F (ka). (32)
νσ − R (a,k) 2 a a− 2 νσ
(cid:20) 1,nσ (cid:21)
6
In deriving the second form we have used the relation (27) and the recurrence formula for the
cylindrical functions.
The eigenfunctions are normalized by the condition
d3x√γψβ+ψβ′ = δββ′, (33)
Z
where γ is the determinant for the spatial metric and δ is understood as the Kronecker delta
ββ′
symbol for the discrete components of the collective index β and as the Dirac delta function for
thecontinuousones. Asthenormalization integral isdivergentforβ = β′,themaincontribution
comes from large values r. We may replace the cylindrical functions with the argument kr in
Eq. (30) by the corresponding asymptotic expressions. In this way one finds
2k(M +ω) −1
R2 (a,k) = J˜2 (ka)+Y˜2 (ka) . (34)
1,nσ π2α2aω νσ νσ
h i
This relation determines the normalization coefficient for the interior region. In addition to the
modes with real k, modes with purely imaginary k can be present. These modes correspond to
the boundstates. The eigenfunctions for the boundstates and their normalization are discussed
in Appendix A.
3 Vacuum expectation values in the exterior region
In this section we consider the VEVs for the energy-momentum tensor and the fermionic con-
densate in the region outside the global monopole core. We expand the field operator in terms
(+) (−)
of the complete set of eigenfunctions ψ ,ψ :
{ β β }
ψˆ= [aˆ ψ(+) +ˆb+ψ(−)], (35)
β β β β
β
X
where aˆ is the annihilation operator for particles, andˆb+ is the creation operator for antiparti-
β β
(+) (−)
cles, ψ = ψ , for ω > 0 and ψ = ψ , for ω < 0. In order to find the VEV for the operator
β β β β
of the energy-momentum tensor, we substitute the expansion (35) and the analog expansion for
the operator ψˆ¯ into the corresponding expression for spinor fields,
T ψˆ¯,ψˆ = i[ψˆ¯γ ψˆ ( ψˆ¯)γ ψˆ] . (36)
µν{ } 2 (µ∇ν) − ∇(µ ν)
By making use of the standard anticommutation relations for the annihilation and creation
operators, for the VEV one finds the following mode-sum formula
0T 0 = T ψ¯(−)(x),ψ(−)(x) , (37)
h | µν| i µν{ β β }
β
X
where 0 is the amplitude for the corresponding vacuum state.
| i
Substituting the eigenfunctions (30) into the mode-sum formula (37), using the formula
j
2j +1
Ω (θ,ϕ)2 = , (38)
| jlσm | 4π
m=−j
X
7
and relation (34), the VEV of the energy-momentum tensor is presented in the form (no sum-
mation over µ)
δν ∞ ∞ f(µ) [x,g (x,xr/a)]
0Tν 0 = µ (2j +1) dx σνσ νσ,p . (39)
h | µ| i 8πα2a3r J˜2 (x)+Y˜2(x)
j=1/2 σ=0,1Z0 νσ νσ
X X
In formula (39) we use the notations
f(0) [x,g (x,y)] = x ( x2+M2a2 Ma)g2 (x,y)
σνσ νσ,p − − νσ,0
+(h xp2+M2a2+Ma)g2 (x,y) , (40)
νσ,nσ
px3 i
f(1) [x,g (x,y)] = g2 (x,y)+g2 (x,y) 2f(2) [x,g (x,y)],(41)
σνσ νσ,p √x2+M2a2 νσ,0 νσ,nσ − σνσ νσ,p
x3(2ν +n )(cid:2) (cid:3)
f(µ) [x,g (x,y)] = σ σ g (x,y)g (x,y), µ = 2,3, (42)
σνσ νσ,p 2y√x2+M2a2 νσ,0 νσ,nσ
and the function g (x,y) is defined by Eq. (31). The expression on the right of Eq. (39) is
νσ,p
divergent. It may beregularized introducing a cutoff function ψ (ω) with the cutting parameter
µ
µ which makes the divergent expressions finite and satisfies the condition ψ (ω) 1 for µ 0.
µ
→ →
After the renormalization the cutoff function is removed by taking the limit µ 0. In the
→
discussion below we will implicitly assume that the corresponding expressions are regularized.
The parts in the VEVs induced by the non-trivial structure of the core are finite and do not
depend on the regularization scheme used.
To find the part in the VEV of the energy-momentum tensor induced by the non-trivial core
structure,wesubtractthecorrespondingcomponents forthepoint-like monopolegeometry. The
latter are given by the expressions which are obtained from Eq. (39) replacing the integrand by
(µ)
f [x,J (xr/a)] (see Ref. [16]). In order to evaluate the corresponding difference we use the
σνσ νσ
relation
f(µ) [x,g (x,xr/a)] 1 J˜ (x)
σνσ νσ,p f(µ) [x,J (xr/a)] = νσ f(µ) x,H(s)(xr/a) , (43)
J˜2 (x)+Y˜2(x) − σνσ νσ −2 H˜(s)(x) σνσ νσ
νσ νσ sX=1,2 νσ h i
(s)
where H (z), s = 1,2, are the Hankel functions. This allows to present the vacuum energy-
ν
momentum tensor components in the form
0Tν 0 = 0 Tν 0 + Tν , (44)
h | µ| i h m| µ| mi h µic
where 0 Tν 0 is the VEV for the point-like monopole and the part (no summation over µ)
h m| µ| mi
δν ∞ ∞ J˜ (x)
Tν = − µ (2j +1) dx νσ f(µ) x,H(s)(xr/a) , (45)
h µic 16πα2a3r jX=1/2 σX=0,1sX=1,2Z0 H˜ν(sσ)(x) σνσ h νσ i
is induced by the non-trivial core structure. In formula (45), the integrand of the s = 1 (s = 2)
term exponentially decreases in the upper (lower) half of the complex plane x. Consequently,
we rotate the integration contour on the right of this formula by the angle π/2 for s = 1 and by
the angle π/2 for s = 2. This leads to the representation
−
iδν ∞ ∞
Tν = − µ (2j +1) dx η
h µic 16πα2a3r s
j=1/2 σ=0,1Z0 s=1,2
X X X
J˜ (eηsπi/2x)
νσ f(µ) eηsπi/2x,H(s)(eηsπi/2xr/a) , (46)
×H˜ν(sσ)(eηsπi/2x) σνσ h νσ i
8
where η = ( 1)s+1.
s
−
First of all let us consider the part of the integral over the interval (0,Ma). By using the
standard properties of the Hankel functions it can be easily seen that in this interval one has
eiπλf(µ) eπi/2x,H(1)(eπi/2xr/a) = e−iπλf(µ) e−πi/2x,H(2)(e−πi/2xr/a) . (47)
σνσ νσ σνσ νσ
h i h i
Further, from equations (16) and (17) it follows that the interior solution can be written as
R (r,k) = const R(r,ω). On the base of this observation for the combination entering into
1,nσ ·
the tilted notations in Eq. (46) one finds
R′ (a,eπi/2x/a) R′ (a,e−πi/2x/a)
1,nσ = 1,nσ , 0 6 x6 Ma. (48)
R (a,eπi/2x/a) R (a,e−πi/2x/a)
1,nσ 1,nσ
Now, it can be seen that for these values x one has
J˜ (eπi/2x) J˜ (e−πi/2x)
e−iπλ νσ = eiπλ νσ . (49)
H˜(1)(eπi/2x) H˜(2)(e−πi/2x)
νσ νσ
Combining relations (47) and (49), we see that the part of the integral in Eq. (46) over the
interval (0,Ma) vanishes.
To simplify the part of the integral over the interval (Ma, ), we note that for the functions
∞
with different parities the following relation takes place:
eiπλf(µ) eπi/2x,H(1)(eπi/2xr/a) = e−iπλf(µ) e−πi/2x,H(2)(e−πi/2xr/a) , (50)
σνσ νσ − σ′νσ′ νσ′
h i h i
where σ,σ′ = 0,1, and σ = σ′. Further we note that the functions R (r, k) and R (r,k)
6 2,nσ′ − 1,nσ
satisfy the same equation and, hence, R (r, k) = const R (r,k). By using relations (27)
2,nσ′ − · 1,nσ
and (28), now it can be seen that
R (a,ke∓iπ/2) R (a,ke±iπ/2)
2,nσ′ = 1,nσ . (51)
R (a,ke∓iπ/2) R (a,ke±iπ/2)
1,nσ′ 2,nσ
From this formula, by taking into account that ν = ν +n , we find the relation
σ′ σ σ
J˜ (eπi/2x) J˜ (e−πi/2x)
e−iπλ νσ = eiπλ νσ′ . (52)
H˜(1)(eπi/2x) H˜(2)(e−πi/2x)
νσ νσ′
Combining formulae (50) and (52), we see that the different parities give the same contribution
to the VEV of the energy-momentum tensor.
By taking into account this result and introducing the modified Bessel functions, for the
core-induced part in the VEV we find (no summation over µ)
δν ∞ ∞ x3 I¯(ηs,0) (ax)
Tν = µ l dx l/α−1/2 F(µ,ηs) x,K (xr) . (53)
h µic 2π2α2r Xl=1 ZM √x2−M2 sX=1,2K¯l(/ηαs,−0)1/2(ax) l/α−1/2(cid:2) l/α−1/2 (cid:3)
Here for a given function f (y) we use the notations
ν
M2 iη M
F(0,ηs)[x,f (y)] = 1 1+ s f2(y)
ν ν x2 − √x2 M2 ν
(cid:18) (cid:19)(cid:20)(cid:18) − (cid:19)
iη M
1 s f2 (y) , (54)
− − √x2 M2 ν+1
(cid:18) − (cid:19) (cid:21)
2ν +1
F(1,ηs)[x,f (y)] = f2(y) f2 (y) λ f (y)f (y), (55)
ν ν ν − ν+1 − f y ν ν+1
2ν +1
F(µ,ηs)[x,f (y)] = λ f (y)f (y), µ = 2,3, (56)
ν ν f 2y ν ν+1
9
with λ = 1 (the function F(µ,ηs)[x,I (y)] with λ = 1 will be used below), and the barred
K ν ν I
−
notation is defined by the formula
R′ (a,eηsπi/2x/a)
f¯(ηs,σ)(x) = xf′(x) a 1,nσ +au′/2+ah′ 1/2 f(x). (57)
−" R1,nσ(a,eηsπi/2x/a) a a − #
It can be checked that the core-induced part in the VEV of the energy-momentum tensor obeys
the continuity equation Tν = 0, which for the geometry under consideration takes the form
h µic;ν
d
r T1 +2 T1 T2 = 0. (58)
drh 1ic h 1ic−h 2ic
(cid:0) (cid:1)
In addition, for a massless spinor field this part is traceless and the trace anomaly is contained
in the point-like monopole part only.
The fermionic condensate can be found from the formula for the VEV of the energy-
momentum tensor by making use of the relation Tµ = Mψ¯ψ. The condensate is presented
µ
in the form of the sum of idealized point-like monopole and core-induced parts:
0ψ¯ψ 0 = 0 ψ¯ψ 0 + ψ¯ψ . (59)
m m c
h | | i h | | i h i
By taking into account formula (53) for the components of the energy-momentum tensor, for
the part coming from the non-trivial core structure one finds
1 ∞ ∞ I¯(ηs,0) (ax)
ψ¯ψ = l dxx l/α−1/2
h ic 2π2α2r Xl=1 ZM sX=1,2K¯l(/ηαs−,01)/2(ax)
M M
iη K2 (xr) +iη K2 (xr) .(60)
× √x2 M2 − s l/α−1/2 − √x2 M2 s l/α+1/2
(cid:20)(cid:18) − (cid:19) (cid:18) − (cid:19) (cid:21)
This formula may also be derived directly from the mode-sum formula 0ψ¯ψ 0 = ψ¯(−)ψ(−)
h | | i β β β
withtheeigenfunctions (21). Inderivingformulae(53)and(60)wehave assumedthatnobound
P
states exist. In Appendix A we show that these formulae are valid also in the case when the
bound states are present.
For r > a the core-induced parts in the VEVs of the energy-momentum tensor and the
fermioniccondensate,givenbyEqs. (53)and(60),arefiniteandtherenormalizationisnecessary
for the point-like monopole parts only. Of course, we could expect this result as in the region
r > a the local geometry is the same in both models and, hence, the divergences are the same
as well.
As it has been already mentioned, if there is no surface energy-momentum tensor on the
bounding surface r = a, then the radial derivatives of the metric are continuous and, hence,
in Eqs. (32) and (57) one has u′ = 0, h′ = 1/a. In models with an additional infinitely
a a
thin spherical shell located at r = a, these quantities are related to the components of the
corresponding surface energy-momentum tensor τ . Denoting by nµ the normal to the shell
µν
normalized by the condition n nµ = 1and assumingthat it points into the bulk on both sides,
µ
−
from the Israel matching conditions one has
K Kh = 8πGτ . (61)
µν µν µν
{ − }
Inthisformulathecurlybrackets denotesummationover eachsideoftheshell,h = g +n n
µν µν µ ν
is the induced metric on the shell, K = hρhδ n its extrinsic curvature and K = Kµ. For
µν µ ν∇ρ δ µ
10