- Email: [email protected]
Massive particles in five dimensions
V o l u m e 157B, n u m b e r 1
PHYSICS L E T T E R S
4 July 1985
M A S S I V E P A R T I C L E S IN FIVE D I M E N S I O N S E.J. C O P E L A N D Department of Theoretical Physics, University of Newcastle upon Tyne, Newcastle upon Tyne NEI 7RU, UK Received 17 January 1985
We consider a five-dimensional model of the universe with a dynamical extra dimension. Calculations of the ratio of the number density of Kolb and Slansky type pyrgons to that of photons show the model to be unacceptable. However by inserting N matter fields into the original action, it becomes possible to reduce the ratio below the observational bound.
1. Introduction. One of the first successful attempts to unify gravity and a gauge group was in the work of Kaluza [1] and Klein [2]. They showed that general relativity in five dimensions, where the fifth dimension is curled up into an unobservably small circle, in fact contained a local U(1) gauge symmetry arising from the isometry of the fifth dimension. The vacuum geometry determines the effective low energy theory. If the vacuum is M4 X S 1, then five-dimensional gravity reduces to four-dimensional gravity plus electrodynamics. In such a vacuum each field has a harmonic expansion which is just a Fourier series in the extra coordinate, with coefficients that are four-dimensional fields. In addition to possible zero modes corresponding to the low-mass particle spectrum, there is also an infinite sequence of higher modes with masses of the order b (b is the compactification scale). Kolb and Slansky [3], have investigated the cosmological consequences resulting from the particles corresponding to the nonzero modes in the harmonic expansion. They showed that by imposing an additional charge onto these charged pyrgons, then if the pyrgon mass satisfies mq~ "~ mp1, annihilation through pyrgon, anti-pyrgon collisions can be effective enough to reduce the pyrgon density below the present observational bound. For this to occur mq~ < 106 GeVwith the ratio of the pyrgon number density to the photon number density, r = n~/n~ <<,10 -14. By estimating the present mass density due to the pyrgons they find the usual 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
five-dimensional model to be unacceptable, because if b = m~-i1 it would result in r ~ 1. Possible ways of avoiding this catastrophe include letting b vary during the entire evolution of the universe. In a ' q ' o y model" we relax the constraint of constant b, and find that the original problem still arises. A way of circumventing it is found by introducing a large number,N, of matter fields. The effect of this is to rescale the radius b by a factor N 1/2, and the pyrgon mass by a factor N -1/2. For suitably large N it is shown that the ratio n~/n, can in fact be made to fall below observational bounds. By using the S-matrix formalism, production rates are obtained for both the massive pyrgons and photons due to the coupling of the extra dimension with these fields, in the early universe. By treating the extra dimension as a decaying Higgs field, this external field decays into vector bosons.
2. The five-dimensional model. The vacuum M4 X S 1 and the low-energy theory has a higher-dimensional metric of the form: gAB(X,y) =b -1 ( G1/2gtav+AuAvb3 Aub3 Avb3 b 3 )' where we use the conventions of MTW [4]. Upper case Latin letters A, B, C denote five-dimensional indices 0, 1,2, 3, 5; lower case Greek indices run over four dimensions 0, 1,2, 3; whereas lower case Latin 27
Volume 157B, number 1
indices run over three-dimensional spatial values 1,2, 3. In the model, b is the radius of the external dimension and, in fact acts as a Brans-Dicke [5] scalar field. The conformal transformation is the usual Weyl transformation which ensures the coefficient of ~%-gR is constant, where R is the curvature scalar in four dimensions, and g is the determinant o f the fourdimensional metric tensor guy" Introducing G, the four-dimensional newtonian gravitational constant into the metric ensures that the dimensions of the metric are correct. In our example g55 has dimensions L, because y satisfies 0 ~
SG -
16rrG 5
f
- 2/k),
(1)
where G 5 is the five-dimensional newtonian gravitational constant, g5 is the determinant o f the fivedimensional metric,/~ and/~ are the five-dimensional curvature scalar and cosmological constant, respectively. Dimensionally reducing (1) and identifying G 5 = 21rG 3/2, we obtain,
SG =
4 July 1985
PHYSICS LETTERS
1 f do x [R - 3 ( 7 b / b ) 2 16rrG a
- (b3/4,v@)FuvFU~' - 2 ~ . v ~ - b - l l ,
(2)
where Fuu = auA v - bvA u and dv x is the four-volume element. The action for a charged matter field in the higherdimensional spacetime is written as
s° =- 4~_~cf d5xvrS-~[(f~)2+m2~2],
(3)
where 7 is the five-dimensional gradient operator. In the vacuum space M 4 X S 1 , the fields are expanded in a Fourier series in the extra coordinate (y = 21r0) with coefficients that are fields on M4 .
S,o
= -
l fdox
[(7~00) 2 + m 2 ~ / G b - l ( f f O ) 2 ] ,
(5)
whereas the first order field (k = 1) gives,
s ~ = - ~1 f dox [(v~ 1)2 + (V r ~ b - l m 2 + V c~b -3)(41)2]-
(6)
3. Particle production. In (2), (5) and (6), the b parameter can act as a decaying Higgs type field which can be treated as an external field. The Higgs oscillations decay into massive vector bosons and other boson Higgs fields. By associating an external source term Jib] corresponding to the decaying Higgs fields it is possible to evaluate production rates for the massive vector bosons. In general [6], for a field ~ decaying into fields ~ with an action, s[o]
= -
fdox
((TriP) 2 + m2O 2 + J[q~] alp2},
(7)
then J [ ¢ ] = ~7(¢2 - q~2)is the source, 7/is the coupling between • and ~, with ¢0 a stable minimum value of q~. In our model we introduce an oscillating extra dimension which would then correspond to the decaying field,
b(t) = be u ,
(8)
where u(t) = u 0 cos (Mr) and u 0 ,~ 1 so that the size o f the oscillations is kept very small. M is the mass of the decaying Higgs field. Defining r~ 2 = m2vCGb -1 enables (5) to be written as S~o = - 2 1 fdox
((voo)2 + mo2(,o): +Jo tul (,o):), (9)
where Jo [u] = ~2(e-U - 1). The amplitude for the creation o f a ~00if0 pair with momenta k l , k 2 in the background Jo [u] is, T.A.= (klk 2 out l0 in) (10)
Ol(x,y) = ~=_~. ~kk(x)exp(ikO) •
(4)
Here j is a spacetime index and k labels the mass eigenstate. The zero mode (k = 0) field in (3) gives for the coupling between the extra dimension and the ~k0 field, 28
= CqClk 2 i n l e x p ( - i f d o x J o [ u l : ( ~ O ) 2 : ) t O i n ) , where c is a normalisation constant determined by the requirement that the probabilities o f producing any even number of particles must sum to one. Using the definitions ofBirrell and Davies [7] for the matter fields, we obtain for the probability of creating the
Volume 157B, number 1
PHYSICS LETTERS
4 July 1985
pair, using the first term in the perturbation expansion of (10)
for the zero mode case,
P-~k 1,k2 = C2 V l ) o ( - k ° - ko)12
if r~4u 2 "~ M 4 , which is the case since u 0 itself satisfies u 0 "¢ 1.
X 8(k 1 + k2)/[(27r)3(kOkO)] ,
(11)
where V is the total volume under consideration, A denotes a Fourier transform with respect to time and k 0 satisfies (k0) 2 = k 2 + rn 2 . Summing over all possible momentum states enables the number decay rate to be obtained (12)
4r~2)l/2/(87rM),
(18)
4. Photon production. Using (2) it is possible to evaluate the production rate o f photons due to the coupling of the electromagnetic field with the extra dimension. The electromagnetic part o f the action can be written as, SE M _
F_~~o ~o = Po~o~o/( VT) = c2~4m0u0(M22
1,2 vertex/i, lvertex ~ ~ 4 u 2 / M 4 ,~ 1 ,
1 ~,. fdox(-b3 FuvFUV+j[ulFuvFUV), 647rG~l Z ., (19)
where
M2 > 4 ~ 2 ,
where T is the total time under consideration. Defining;r~2 = G1/2~-l(m 2 + ~ - 2 ) , enables (6) to be written as So 1 = _ 21fjdox ((7 I~1) 2 + rn 2(~1) 2 +J1 [u]( ~1)2}
(13) where
J[u] = (b 3 - ~3) = ~3(e3U _ 1).
(20)
The amplitude for the production of two photons with momenta k 1 , k 2 and polarisation vectors ;k1 , ~k2 is, T.A. = c 1 (k 1 ~klk2 ~2 inlexp(iSEM)[0 in),
(21)
which to first order in perturbation expansion is,
J1 [u] = G1/2b-3(e -3u - 1) + Jo [u] .
(14) T.A. = Cl(klXlk2X2in[
From (14) it follows that to order u 0, ) 1 [ - k ° - k ° ] = [1 + 3 Gl[2/(b3~t2)]Jo [ - k 0 - k O]
i (dOxN[J[u] 167rG3/2 a
X FuvFUV ] l0 in).
(22)
(15) We have used the notation of Birrell and Davies [7] for the gauge fields
and using (11) that 17__,qj1~1 = [1 +3G1/2/(b3rn2)]
3
× [(M 2 - 4rh12)/(M 2 - 4r~2)] 1/2I" ~ o o q j
0 ,
(16)
Aa(x) = ~
~
k h=0
w i t h M 2 > 4r~ 2. The energy decay rate is calculated by integrating (11) over all possible energies k 0, k 0 from which we obtain,
nergy__ 4 2 2 ~oq)o c 2 ~mOuO(M
4r~2)1/2/16~r, ~- M 2 > 4 ff~2 . (17)
Eq. (16) gives the energy decay rate to two pyrgons, with f' replaced by [,Energy. The calculation has involved only the single vertex tree level graph. The contribution of the two vertex graphs needs to be evaluated in order to Validate the perturbation expansion. It is straightforward using, Feynman diagrams to obtain an order of magnitude estimate for the two vertex contribution. We obtain
+
*t~
[akxu~x(x) + akxUkx(X)],
(23)
where uTcx(x) = [2(2~')3k 0]-l/2e~xeikx,
Ikl = k ° ,
and a
#
ga[3ekk ekh = ghx' ' a~h and akh are associated with the usual creation and annihilation operators respectively. By imposing the Lorentz gauge, and using gauge invariance to implement the condition e0x = 0, w e then obtain,
kaea=O,
e0 = 0 .
(24)
Inserting (23) and (24) in (22), leads to the amplitude, 29
Volume 157B, number 1
PHYSICS LETTERS
T.A. = icl/(327rG3/2(kOkO)l/2) X J ' I - k 0 - k0] 6(kl + k2) × [ e *~ *~ I e k2h: *p k-10 k~z~J" 1 k l h l e a*k z k 2 k lPk 2 p - efqh
(25)
From (25) the number decay rate and the energy decay rate to two photons can be obtained, and give, F_.2photons = 9c2b6u~4/a(167rG) 3 ,
(26)
FEnergy = 9c2b6u~)M5/8(167rG) 3 -+2 photons
(27)
Eqs. (27) and (17) give the rate of decay of the Higgs fields into both photons and the first order massive pyrgons. Kolb and Slansky [3] have obtained values for the ratio of the number density of pyrgons present in the universe to the number density of photons (n o/nz). A better expression for the same ratio is obtained from (27) and (17). no I
n~
3G1/2~ 8(167r)2/n4G 3 [I1 + _ _ (M 2 - 4 r ~ 2 ) 1/2. • ~3r~2 ] 9~6M5 (28)
Typical values for the masses of the Higgs fields can be obtained by choosing specific models. One example is the Candelas-Weinberg [8] scenario in which the mass M of the Higgs field is related to the curvature 'at the bottom of the potential well giving, M 2 ~ ~ ft,,
(29)
where X ~ 1/a.
The infinite sequence of higher modes have masses of order b and it is assumed b is of order G 1/2 = mp1 -1 • Recalling that r~ 0 corresponds to the low-mass particle spectrum (28) can be expanded to give the photon density equaling the density of charged spin 2 pyrgons with masses of order mp1. This result is also predicted by Kolb and Slansky [3], and is of course far too high. A possible method of overcoming this problem is to include a large number N of matter fields into the original higher-dimensional action. In such scenarios [8], the masses of the particles are rescaled, because the radius is increased by a factor N 1/2, and ft, is reduced by a factor N. Thus in (28),
~2 ~ Tn2/v/~,
~,2 t^r3/2 . (30) M 2 ..,.M2/N . . . .~21 _+,,,its,
Substituting (30) into (29) leads to the ratio n o/n.~ being reduced by a factor o f N 3. In ref. [3], an upper 30
4 July 1985
bound of less than 10 -14 today was estimated. For this to occur in this scenario,N must be of the order of 105. The same number o f matter fields have also been demanded by Candelas and Weinberg [8], and by Toms [9], although in ref. [9] the extra fields were required to obtain realistic results when an induced Einstein-Maxwell action is obtained from quantum corrections due to matter fields on a fivedimensional Kaluza-Klein background.
5. Curvature effects. We have presented results based essentially on flat spacetime calculations. The reason being that particle production in very early universe scenarios [5] occurs at the bottom of the potential where curvature effects are negligible. This can be shown to be the case here. The radius of curvature of the spacetime is given by H -1 , where H is Hubble's constant. It can be shown that H 2 ~ fi, u02 which can be made arbitrarily small by choosing u 2 ~ 1. This is the condition that must be satisfied for the original S-matrix calculation to be valid. 6. Conclusion. Kolb and Slansky effectively showed that five-dimensional models with a static extra dimension were unacceptable due to the relatively large ratio predicted for n O/n~. We have shown that by introducing a large number of matter fields, and a dynamical extra dimension, this ratio can be reduced to physically acceptable levels. Maeda [ 10], emphasised the drastic consequences due to particle creation in the higher dimensions. He claims the production would be such as to isotropise the spacetime thus making the process of cosmological dimensional reduction invalid. A more realistic calculation would be to investigate particle creation in these models at nonzero temperatures, as temperature effects are very important in early universe scenarios. These problems are currently being investigated. I am grateful to Dr. I.G. Moss for helpful discussions and to Professor P.C.W. Davies and Dr. D.J. Toms for critically reading the manuscript. I would like to thank the Science and Engineering Research Council for financial support.
Volume 157B, number 1
PHYSICS LETTERS
References [ 1 ] Th. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin, Math. Phys. K1 (1921) 966. [2] O. Klein, Z. Phys. 37 (1926) 895. [ 3 ] E .W. Kolb and R. Slansky, Phys. Lett. 135 B (1984) 378. [4] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (Freeman, San Fransisco, 1973).
4 July 1985
[5] C. Brans and R.H. Dicke, Phys. Rev. 124 (1961) 95. [6] I.G. Moss, PhD Thesis, unpublished. [7] N.D. Birrell and P.C.W. Davies, Quantum fields in curved space (Cambridge U.P., London, 1982). [8] P. Candelas and S. Weinberg, Nucl. Phys. B237 (1984) 397. [9] D.J. Toms, Phys. Lett. 129B (1983) 31. [10] K. Maeda, Effect of particle creation on Kaluza-Klein cosmologies, preprint.
31
PHYSICS L E T T E R S
4 July 1985
M A S S I V E P A R T I C L E S IN FIVE D I M E N S I O N S E.J. C O P E L A N D Department of Theoretical Physics, University of Newcastle upon Tyne, Newcastle upon Tyne NEI 7RU, UK Received 17 January 1985
We consider a five-dimensional model of the universe with a dynamical extra dimension. Calculations of the ratio of the number density of Kolb and Slansky type pyrgons to that of photons show the model to be unacceptable. However by inserting N matter fields into the original action, it becomes possible to reduce the ratio below the observational bound.
1. Introduction. One of the first successful attempts to unify gravity and a gauge group was in the work of Kaluza [1] and Klein [2]. They showed that general relativity in five dimensions, where the fifth dimension is curled up into an unobservably small circle, in fact contained a local U(1) gauge symmetry arising from the isometry of the fifth dimension. The vacuum geometry determines the effective low energy theory. If the vacuum is M4 X S 1, then five-dimensional gravity reduces to four-dimensional gravity plus electrodynamics. In such a vacuum each field has a harmonic expansion which is just a Fourier series in the extra coordinate, with coefficients that are four-dimensional fields. In addition to possible zero modes corresponding to the low-mass particle spectrum, there is also an infinite sequence of higher modes with masses of the order b (b is the compactification scale). Kolb and Slansky [3], have investigated the cosmological consequences resulting from the particles corresponding to the nonzero modes in the harmonic expansion. They showed that by imposing an additional charge onto these charged pyrgons, then if the pyrgon mass satisfies mq~ "~ mp1, annihilation through pyrgon, anti-pyrgon collisions can be effective enough to reduce the pyrgon density below the present observational bound. For this to occur mq~ < 106 GeVwith the ratio of the pyrgon number density to the photon number density, r = n~/n~ <<,10 -14. By estimating the present mass density due to the pyrgons they find the usual 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
five-dimensional model to be unacceptable, because if b = m~-i1 it would result in r ~ 1. Possible ways of avoiding this catastrophe include letting b vary during the entire evolution of the universe. In a ' q ' o y model" we relax the constraint of constant b, and find that the original problem still arises. A way of circumventing it is found by introducing a large number,N, of matter fields. The effect of this is to rescale the radius b by a factor N 1/2, and the pyrgon mass by a factor N -1/2. For suitably large N it is shown that the ratio n~/n, can in fact be made to fall below observational bounds. By using the S-matrix formalism, production rates are obtained for both the massive pyrgons and photons due to the coupling of the extra dimension with these fields, in the early universe. By treating the extra dimension as a decaying Higgs field, this external field decays into vector bosons.
2. The five-dimensional model. The vacuum M4 X S 1 and the low-energy theory has a higher-dimensional metric of the form: gAB(X,y) =b -1 ( G1/2gtav+AuAvb3 Aub3 Avb3 b 3 )' where we use the conventions of MTW [4]. Upper case Latin letters A, B, C denote five-dimensional indices 0, 1,2, 3, 5; lower case Greek indices run over four dimensions 0, 1,2, 3; whereas lower case Latin 27
Volume 157B, number 1
indices run over three-dimensional spatial values 1,2, 3. In the model, b is the radius of the external dimension and, in fact acts as a Brans-Dicke [5] scalar field. The conformal transformation is the usual Weyl transformation which ensures the coefficient of ~%-gR is constant, where R is the curvature scalar in four dimensions, and g is the determinant o f the fourdimensional metric tensor guy" Introducing G, the four-dimensional newtonian gravitational constant into the metric ensures that the dimensions of the metric are correct. In our example g55 has dimensions L, because y satisfies 0 ~
SG -
16rrG 5
f
- 2/k),
(1)
where G 5 is the five-dimensional newtonian gravitational constant, g5 is the determinant o f the fivedimensional metric,/~ and/~ are the five-dimensional curvature scalar and cosmological constant, respectively. Dimensionally reducing (1) and identifying G 5 = 21rG 3/2, we obtain,
SG =
4 July 1985
PHYSICS LETTERS
1 f do x [R - 3 ( 7 b / b ) 2 16rrG a
- (b3/4,v@)FuvFU~' - 2 ~ . v ~ - b - l l ,
(2)
where Fuu = auA v - bvA u and dv x is the four-volume element. The action for a charged matter field in the higherdimensional spacetime is written as
s° =- 4~_~cf d5xvrS-~[(f~)2+m2~2],
(3)
where 7 is the five-dimensional gradient operator. In the vacuum space M 4 X S 1 , the fields are expanded in a Fourier series in the extra coordinate (y = 21r0) with coefficients that are fields on M4 .
S,o
= -
l fdox
[(7~00) 2 + m 2 ~ / G b - l ( f f O ) 2 ] ,
(5)
whereas the first order field (k = 1) gives,
s ~ = - ~1 f dox [(v~ 1)2 + (V r ~ b - l m 2 + V c~b -3)(41)2]-
(6)
3. Particle production. In (2), (5) and (6), the b parameter can act as a decaying Higgs type field which can be treated as an external field. The Higgs oscillations decay into massive vector bosons and other boson Higgs fields. By associating an external source term Jib] corresponding to the decaying Higgs fields it is possible to evaluate production rates for the massive vector bosons. In general [6], for a field ~ decaying into fields ~ with an action, s[o]
= -
fdox
((TriP) 2 + m2O 2 + J[q~] alp2},
(7)
then J [ ¢ ] = ~7(¢2 - q~2)is the source, 7/is the coupling between • and ~, with ¢0 a stable minimum value of q~. In our model we introduce an oscillating extra dimension which would then correspond to the decaying field,
b(t) = be u ,
(8)
where u(t) = u 0 cos (Mr) and u 0 ,~ 1 so that the size o f the oscillations is kept very small. M is the mass of the decaying Higgs field. Defining r~ 2 = m2vCGb -1 enables (5) to be written as S~o = - 2 1 fdox
((voo)2 + mo2(,o): +Jo tul (,o):), (9)
where Jo [u] = ~2(e-U - 1). The amplitude for the creation o f a ~00if0 pair with momenta k l , k 2 in the background Jo [u] is, T.A.= (klk 2 out l0 in) (10)
Ol(x,y) = ~=_~. ~kk(x)exp(ikO) •
(4)
Here j is a spacetime index and k labels the mass eigenstate. The zero mode (k = 0) field in (3) gives for the coupling between the extra dimension and the ~k0 field, 28
= CqClk 2 i n l e x p ( - i f d o x J o [ u l : ( ~ O ) 2 : ) t O i n ) , where c is a normalisation constant determined by the requirement that the probabilities o f producing any even number of particles must sum to one. Using the definitions ofBirrell and Davies [7] for the matter fields, we obtain for the probability of creating the
Volume 157B, number 1
PHYSICS LETTERS
4 July 1985
pair, using the first term in the perturbation expansion of (10)
for the zero mode case,
P-~k 1,k2 = C2 V l ) o ( - k ° - ko)12
if r~4u 2 "~ M 4 , which is the case since u 0 itself satisfies u 0 "¢ 1.
X 8(k 1 + k2)/[(27r)3(kOkO)] ,
(11)
where V is the total volume under consideration, A denotes a Fourier transform with respect to time and k 0 satisfies (k0) 2 = k 2 + rn 2 . Summing over all possible momentum states enables the number decay rate to be obtained (12)
4r~2)l/2/(87rM),
(18)
4. Photon production. Using (2) it is possible to evaluate the production rate o f photons due to the coupling of the electromagnetic field with the extra dimension. The electromagnetic part o f the action can be written as, SE M _
F_~~o ~o = Po~o~o/( VT) = c2~4m0u0(M22
1,2 vertex/i, lvertex ~ ~ 4 u 2 / M 4 ,~ 1 ,
1 ~,. fdox(-b3 FuvFUV+j[ulFuvFUV), 647rG~l Z ., (19)
where
M2 > 4 ~ 2 ,
where T is the total time under consideration. Defining;r~2 = G1/2~-l(m 2 + ~ - 2 ) , enables (6) to be written as So 1 = _ 21fjdox ((7 I~1) 2 + rn 2(~1) 2 +J1 [u]( ~1)2}
(13) where
J[u] = (b 3 - ~3) = ~3(e3U _ 1).
(20)
The amplitude for the production of two photons with momenta k 1 , k 2 and polarisation vectors ;k1 , ~k2 is, T.A. = c 1 (k 1 ~klk2 ~2 inlexp(iSEM)[0 in),
(21)
which to first order in perturbation expansion is,
J1 [u] = G1/2b-3(e -3u - 1) + Jo [u] .
(14) T.A. = Cl(klXlk2X2in[
From (14) it follows that to order u 0, ) 1 [ - k ° - k ° ] = [1 + 3 Gl[2/(b3~t2)]Jo [ - k 0 - k O]
i (dOxN[J[u] 167rG3/2 a
X FuvFUV ] l0 in).
(22)
(15) We have used the notation of Birrell and Davies [7] for the gauge fields
and using (11) that 17__,qj1~1 = [1 +3G1/2/(b3rn2)]
3
× [(M 2 - 4rh12)/(M 2 - 4r~2)] 1/2I" ~ o o q j
0 ,
(16)
Aa(x) = ~
~
k h=0
w i t h M 2 > 4r~ 2. The energy decay rate is calculated by integrating (11) over all possible energies k 0, k 0 from which we obtain,
nergy__ 4 2 2 ~oq)o c 2 ~mOuO(M
4r~2)1/2/16~r, ~- M 2 > 4 ff~2 . (17)
Eq. (16) gives the energy decay rate to two pyrgons, with f' replaced by [,Energy. The calculation has involved only the single vertex tree level graph. The contribution of the two vertex graphs needs to be evaluated in order to Validate the perturbation expansion. It is straightforward using, Feynman diagrams to obtain an order of magnitude estimate for the two vertex contribution. We obtain
+
*t~
[akxu~x(x) + akxUkx(X)],
(23)
where uTcx(x) = [2(2~')3k 0]-l/2e~xeikx,
Ikl = k ° ,
and a
#
ga[3ekk ekh = ghx' ' a~h and akh are associated with the usual creation and annihilation operators respectively. By imposing the Lorentz gauge, and using gauge invariance to implement the condition e0x = 0, w e then obtain,
kaea=O,
e0 = 0 .
(24)
Inserting (23) and (24) in (22), leads to the amplitude, 29
Volume 157B, number 1
PHYSICS LETTERS
T.A. = icl/(327rG3/2(kOkO)l/2) X J ' I - k 0 - k0] 6(kl + k2) × [ e *~ *~ I e k2h: *p k-10 k~z~J" 1 k l h l e a*k z k 2 k lPk 2 p - efqh
(25)
From (25) the number decay rate and the energy decay rate to two photons can be obtained, and give, F_.2photons = 9c2b6u~4/a(167rG) 3 ,
(26)
FEnergy = 9c2b6u~)M5/8(167rG) 3 -+2 photons
(27)
Eqs. (27) and (17) give the rate of decay of the Higgs fields into both photons and the first order massive pyrgons. Kolb and Slansky [3] have obtained values for the ratio of the number density of pyrgons present in the universe to the number density of photons (n o/nz). A better expression for the same ratio is obtained from (27) and (17). no I
n~
3G1/2~ 8(167r)2/n4G 3 [I1 + _ _ (M 2 - 4 r ~ 2 ) 1/2. • ~3r~2 ] 9~6M5 (28)
Typical values for the masses of the Higgs fields can be obtained by choosing specific models. One example is the Candelas-Weinberg [8] scenario in which the mass M of the Higgs field is related to the curvature 'at the bottom of the potential well giving, M 2 ~ ~ ft,,
(29)
where X ~ 1/a.
The infinite sequence of higher modes have masses of order b and it is assumed b is of order G 1/2 = mp1 -1 • Recalling that r~ 0 corresponds to the low-mass particle spectrum (28) can be expanded to give the photon density equaling the density of charged spin 2 pyrgons with masses of order mp1. This result is also predicted by Kolb and Slansky [3], and is of course far too high. A possible method of overcoming this problem is to include a large number N of matter fields into the original higher-dimensional action. In such scenarios [8], the masses of the particles are rescaled, because the radius is increased by a factor N 1/2, and ft, is reduced by a factor N. Thus in (28),
~2 ~ Tn2/v/~,
~,2 t^r3/2 . (30) M 2 ..,.M2/N . . . .~21 _+,,,its,
Substituting (30) into (29) leads to the ratio n o/n.~ being reduced by a factor o f N 3. In ref. [3], an upper 30
4 July 1985
bound of less than 10 -14 today was estimated. For this to occur in this scenario,N must be of the order of 105. The same number o f matter fields have also been demanded by Candelas and Weinberg [8], and by Toms [9], although in ref. [9] the extra fields were required to obtain realistic results when an induced Einstein-Maxwell action is obtained from quantum corrections due to matter fields on a fivedimensional Kaluza-Klein background.
5. Curvature effects. We have presented results based essentially on flat spacetime calculations. The reason being that particle production in very early universe scenarios [5] occurs at the bottom of the potential where curvature effects are negligible. This can be shown to be the case here. The radius of curvature of the spacetime is given by H -1 , where H is Hubble's constant. It can be shown that H 2 ~ fi, u02 which can be made arbitrarily small by choosing u 2 ~ 1. This is the condition that must be satisfied for the original S-matrix calculation to be valid. 6. Conclusion. Kolb and Slansky effectively showed that five-dimensional models with a static extra dimension were unacceptable due to the relatively large ratio predicted for n O/n~. We have shown that by introducing a large number of matter fields, and a dynamical extra dimension, this ratio can be reduced to physically acceptable levels. Maeda [ 10], emphasised the drastic consequences due to particle creation in the higher dimensions. He claims the production would be such as to isotropise the spacetime thus making the process of cosmological dimensional reduction invalid. A more realistic calculation would be to investigate particle creation in these models at nonzero temperatures, as temperature effects are very important in early universe scenarios. These problems are currently being investigated. I am grateful to Dr. I.G. Moss for helpful discussions and to Professor P.C.W. Davies and Dr. D.J. Toms for critically reading the manuscript. I would like to thank the Science and Engineering Research Council for financial support.
Volume 157B, number 1
PHYSICS LETTERS
References [ 1 ] Th. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin, Math. Phys. K1 (1921) 966. [2] O. Klein, Z. Phys. 37 (1926) 895. [ 3 ] E .W. Kolb and R. Slansky, Phys. Lett. 135 B (1984) 378. [4] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (Freeman, San Fransisco, 1973).
4 July 1985
[5] C. Brans and R.H. Dicke, Phys. Rev. 124 (1961) 95. [6] I.G. Moss, PhD Thesis, unpublished. [7] N.D. Birrell and P.C.W. Davies, Quantum fields in curved space (Cambridge U.P., London, 1982). [8] P. Candelas and S. Weinberg, Nucl. Phys. B237 (1984) 397. [9] D.J. Toms, Phys. Lett. 129B (1983) 31. [10] K. Maeda, Effect of particle creation on Kaluza-Klein cosmologies, preprint.
31