Corrections to the fine structure constant in higher dimensional global monopole spacetime

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by  Geusa de A. Marques

Corrections to the fine structure constant in higher dimensional global monopole
spacetime
Geusa de A. Marques1∗and V. B. Bezerra2∗∗.
1Departamento de F´ısica,
Universidade Federal de Campina Grande
58109-790, Campina Grande, Pb, Brazil.
2Departamento de F´ısica,
Universidade Federal da Para´ıba
58051-970, Jo˜ao Pessoa, Pb, Brazil.
(Dated: February 1, 2008)
InthispaperweusetheGeneralizedUncertaintyPrinciple(GUP)toobtainthecorrectionstothe
finestructureconstantin (D+1)-dimensionalglobal monopolespacetime. Theresultisparticular-
8 ized to D-dimensional spacetime. We also discuss the particular case D = 3 corresponding to the
0
(3+1)-dimensional global monopole spacetime.
0
2

n 1. INTRODUCTION structureoftheglobalmonopolebutpreservingthesolid
a angledeficit. Itisdescribedbythefollowinglineelement
J
The Standard Cosmological Model predicts that the
4 dr2
Universe has experienced a chain of phase transitions ds2 = c2b2dt2+ +r2 dθ2+sin2θdϕ2 , (1.5)
] with spontaneous symmetry breaking. During this pro- − 3 b23
c cess the so called topological defects arised, as for ex- (cid:0) (cid:1)
q where b2 = 1 8πGη2, with the subindex indicating the
- ample, monopoles, strings and domain walls[1]. In par- dimensio3n of−spacce4. The scalar curvature is given by
r ticular, global monopoles appear in models with broken
g global SO(3) symmetry[2]. The model consists of the R = 1−b23. Therefore, this spacetime is not flat, but
[ r2b23
Higgs triplet of scalar fields with Lagrangian surprisingly there is no Newtonian potential associated

4 to it.
v = 1∂ φa∂µφa 1λ φaφa η2 2, (1.1) This metric can be written in a different form if we
8 L −2 µ − 4 − make the following change in time coordinate t t/b .
3 (cid:0) (cid:1) As a result, it turns into → 3
2 where a=1,2,3.
1 The gravitational field corresponding to a global
9. monopoleisrepresentedbythefollowingsphericallysym- dr2
0 metric line element[2] ds2 =−c2dt2+ b2 +r2 dθ2+sin2θdϕ2 . (1.6)
7 3
(cid:0) (cid:1)
0 ds2 = B(r)dt2+A(r)dr2+r2 dθ2+sin2θdϕ2 , (1.2) The main characteristic of this spacetime is the
v: with the−functions A and B give(cid:0)n by (cid:1) existence of a solid angle deficit 4π(1 − b23) whose
i typical value in the framework of Grand Unified
X
Theory(GUT)(η2c2L 1016GeV) is proportional
ar B =A−1 =1−8πGcη42 − 2GrMc2(r), (1.3) 8πcG4η2 ∼10−5. pl ∼
Taking into account that our proposal is to consider
where the parameter η2 is related with the linear energy thehigherdimensionalglobalmonopolespacetime,letus
density. write the generalization of the metric given by eq.(1.6),
The scale of the monopole’s core is defined by the for a (D +1)-dimensional spacetime. In this case, the
Compton wavelength of Higgs bosons, ̺ λ−1/2η−1. Euclidean version of the global monopole line element is
∼
The parameter M = limr→∞M(r) which character- given by[4]
izes the monopole’s mass, assumes values in the interval
0 8πη2 <1 and is given, approximately, by[3]
≤ ds2 =dτ2+ dr2 +r2dΩ2 , (1.7)
6πη b2 D−1
D
M . (1.4)
≈−√λ where D 3 and b is a parameter which codifies the
D
≥
presence of a (D+1)-dimensionalglobalmonopole. The
Therefore, asymptotically the spacetime of a global
monopole corresponds to the Schwarzschild spacetime coordinates are (τ,r,θ1,θ2,...,θD−2,φ). In this case, the
solidangleassociatedtoahyperspherewithunityradius
with negative mass and an additional solid angle deficit
32πc24Gη2. The spacetime of a point-like monopole is ob- is ΩbD = b2D2Γπ(DD2/)2 which is smaller than the solid angle
tainedfromtheabovemetricbydisregardingtheinternal associated to a hypersphere embbeded in flat spacetime.
2
2. CORRECTIONS TO THE FINE STRUCTURE monopole. For D = 3 and b = 1, eq.(2.12) turns into
3
CONSTANT IN HIGHER DIMENSIONS α=e2/4πǫ~c 1/137
≈
The so-called Generalized Uncertainty Principle[7] is
In order to obtain the fine structure constant in the expected to be satisfied when quantum gravitational
(D +1)-dimensional global monopole spacetime, let us effects are important. The interest in this Principle
consideraHydrogenatominthisbackground. Thus,the has been motivated by studies in string theory[8] and
electrostatic force on the electron is given by gravity[7]. In this context, a minimal length scale is ex-
pected to be of the order of Planck length L . This fact
p
leads to corrections to the usual Heisenberg Uncertainty
1 e2 S(b e2
F~ = + D rˆ , Principle,resulting in the GeneralizedUncertainty Prin-
totD (cid:18)−ΩD−1ǫD,0rD−1 ΩD−1ǫD,0rD−1(cid:19) D ciple, which can be expressed as
(2.8)
where the first term correspondsto the generalizationof
the usual Coulomb force to a D-dimensional space. The ∆p 2
sfeeactounrdesteorfmthaerisspesacferotimmethaendgecoomrreetsrpicoanldasndtotoapgoelongericaal-l ∆xi∆pi ≥~"1+β2L2p,D(cid:18) ~i(cid:19) #. (2.13)
izationofaresultobtainedintheframeworkofa(3+1)-
whereβ isanumericalfactoroforderunitythatdepends
dimensionalglobalmonopole[5]. ThefactorS(b )isalso
D
on the specific model and L is the Planck length in
a generalization of a previous result[5] and is given by p,D
D dimensions,whichisgivenbyL =(~G /c3)1/D−1.
p,D D
Thus,inathreedimensionalspaceeq.(2.13)implies that
1 ∞ π2∆ n the lower bound on the length scale is of the order of
S(bD)= 2nX=1(cid:0) (n!)D2(cid:1) |B2n|(cid:0)1−2−2n(cid:1) (2.9) pPllaaynscka’ns ilmenpgotrht,anLtpro=le as(Ga~fu/cn3d)am≈e1n0ta−l3s3ccamle,.and this
p
with ∆ =1 b2 and B being the Bernoulli numbers. Thus,fromeq.(2.13)wecanobtaintheuncertaintyfor
D − D n the momentum which is given by the following relation
Thus, we can obtain the Bohr radius in this background
which can be written as
∆p ∆x 4β2L2
i i p,D
= 1 1 . (2.14)
r = b2DΩD−1ǫD,0~2 4−1D ~ 2βL2p,D  −s − ∆x2i 
B,D me2  
(cid:18) (cid:19)
(1 S(b ))4−D, (2.10) Now, let us consider that the uncertainty in position
× − D is equal to the Bohr radius given by eq.(2.10). Thus, we
can write the following relation
where m and e are the mass and charge of the electron,
respectively. Note that when D = 3 and b = 1, which
3
means that the global monopole is absent,we obtain the
1
so-calledBohr radius rB,3 = 4πmǫe02~2 =0.0529nmin three ∆xi = b2DΩD−1ǫD,0Mp3,DGD 4−D
dimensional space. This expression for the Bohr radius Lp,D me2 !
takes into accountthat the usualquantizationof the an-
gular momentum is now given by L = n~. (1 S(bD))4−D, (2.15)
n,D bD × −
The fine structure constant in D-dimensional space
where M = (~D−2/cD−4G )1/D−1is the Planck’s
was obtained recently[6] and is given by the following p,D D
mass in D dimensional space.
expression
Comparingthe GeneralizedUncertainty Principle and
αD =~2−DeD−1[ΩD−1ǫD,0](1−D)/2cD−4G(D3−D)/2 the Heisenberg Uncertainty Principle, we conclude that
(2.11) the first one can be written, formally, as
where ǫD,0 is the permittivity constant and ΩD−1 =
2πD/2 is the solid angle associated with a hypersphere
Γ(D) ∆x ∆p ~ , (2.16)
2 i i eff
in D dimensions. From this result, we can obtain a cor- ≥
responding generalized one in the sense that the global
where
monopole is present. This can be written as
∆p 2
~ =~ 1+β2L2 i . (2.17)
αˆD =~2−DeD−1[b2DΩD−1ǫD,0](1−D)/2cD−4G(D3−D)/2, eff " p,D(cid:18) ~ (cid:19) #
(2.12)
where the parameter b2 was introduced to take into ac- Substituting eq.(2.15) into (2.14) and the result into
D
count the presence of the (D + 1)-dimensional global (2.17), we get
3
2
~ 1+ 1 b2DΩD−1ǫD,0Mp3,DGD 4−D (1 S(b ))2(4−D)
4β2 me2 − D
~ = (cid:20) (cid:16) (cid:17) 2 . (2.18)
eff 2
1 1 4β2 me2 ǫ M3 G 4−D
× −r − b2DΩD−1(1−S(bD))2(4−D) D,0 pD D ! 
(cid:16) (cid:17)

If we consider a globalmonopole in the GrandUnified which reduces to
Theory framework, the factor bD should be approx-
(1−SD) e2
imately equal to unity, as we can see for the particular αˆ = (1 9.53 10−50β2), (2.22)
case D = 3, in which situation we have b3 = 0.9999, 3,eff 4πǫ0~c − ×
S(b )=0.0024and 1 1. Therefore,we can use
3 (1−S3)2b43 ≈ ina(3+1)-dimensionalglobalmonopolespacetime. This
the term mΩe2D−1ǫD,0Mp3,DGD,which is much less than resultshowsthatthe value of the fine structure constant
one, as the expansion parameter. Thus, we can expand is very close to the one obtained in three-dimensional
eq.(2.18) in terms of this quantity, which gives us the Minkowski flat space.
following result
2
me2 4−D 3. CONCLUDING REMARKS
~ ~ 1+β2 (2.19)
eff ≃  b2DΩD−1ǫD,0Mp3,DGD!
The correctionsto the Planckconstantandto the fine
(1 S(b ))−2(4−D) . structure constant are very close to the ones obtained in
D
× −
a D-dimensional flat spacetime. The difference between
i
Note that in the absence of a globalmonopole, the re- ~eff and~ in relationto ~ is ofthe orderof10−49, when
sultgivenbyeq.(2.19)correspondstotheresultobtained we consider the presence of a global monopole as well
by Nasseri[6] in a D-dimensional space. Applying this as in the absence of this topological defect. Therefore,
result given by eq.(2.19) to a (3+1)-dimensional global the values of the fine structure constantevaluated in the
monopole spacetime, we get framework of the Generalized Uncertainty Principle and
inthepresenceofa(D+1)-dimensionalglobalmonopole
~ ~(1+9.53 10−50β2). (2.20) orintheabsenceofthistopologicaldefect,areveryclose
eff
≃ ×
tothevalueofthefinestructureconstantobtainedinthe
Combining eqs.(2.12) and (2.19), we find that the framework of the Heisenberg Uncertainty Principle.
structure constant in a (D + 1)-dimensional global Acknowledgments We would like to thank
monopole spacetime is given by the expression Conselho Nacional de Desenvolvimento Cient´ıfico e
Tecnol´ogico(CNPq),FAPESQ-PB/CNPq-PRONEXand
αˆD,eff =~2ef−fDeD−1[b2DΩD−1ǫD,0](1−D)/2cD−4G(D3−D)/2, FAPES-ES/CNPq-PRONEX for the partial financial
(2.21) support.
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∗∗
1345 (1997). Electronic address: valdir@fisica.ufpb.br