Estimate for the size of the compactification radius of a one extra dimension universe

Abstract

In this work, we use the Casimir effect to probe the existence of one extra dimension. We begin by evaluating the Casimir pressure between two plates in a M4 × S¹ manifold, and then use an appropriate statistical analysis in order to compare the theoretical expression with a recent experimental data and set bounds for the compactification radius.

Casimir effect; Kaluza-Klein theory

Estimate for the size of the compactification radius of a one extra dimension universe

F. Pascoal^I; L. F. A. Oliveira^II; F. S. S. Rosa^III; C. Farina^IV

^IDepartamento de Física, Universidade Federal de São Carlos, Via Washington Luis, km 235. São Carlos, 13565-905, SP, Brazil

^IICENPES, Petróleo Brasileiro S.A., Av. Horácio de Macedo, 950, P 20, S 1082, Ilha do Fundão, RJ, 21941-915, Brazil

^IIITheoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87544, USA

^IVInstituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21941-972, Rio de Janeiro, RJ, Brazil

ABSTRACT

In this work, we use the Casimir effect to probe the existence of one extra dimension. We begin by evaluating the Casimir pressure between two plates in a M⁴ × S¹ manifold, and then use an appropriate statistical analysis in order to compare the theoretical expression with a recent experimental data and set bounds for the compactification radius.

Keywords: Casimir effect; Kaluza-Klein theory

1. INTRODUCTION

In a broad sense, it is fair to say that the search for unification is the greatest enterprise of theoretical physics. It started a long time ago, when Sir Isaac Newton showed that celestial and terrestrial mechanics could be described by the same laws, and reached one of its highest peaks in the second half of the nineteenth century, when electricity, magnetism and optics were all gathered into Maxwell equations.

The quest for unification continued, and, in a historical paper T. Kaluza [1] managed to combine classical electromagnetism and gravitation into a single, very elegant scheme. The downside was that his theory required an extra spacial dimension, for which there was no evidence whatsoever. Some years later, O. Klein pushed the idea a little further [2], proposing, among other things, a circular topology of a very tiny radius for the extra dimension, maybe at the Planck scale region. Although it presented a great unification appeal, the Kaluza-Klein idea has been left aside for several decades. Only in the mid-seventies, due to the birth of supergravity theory [3], the extra dimensions came back to the theoretical physics scenario. As supergravity also had its own problems, it seemed that the subject would be washed out again, but, less than a decade later, the advent of string and superstring theories [4] made it a cornerstone in extremely high energy physics. Nowadays, with the development of M-theory [5] and some associated ideas, like the cosmology of branes [6], it might even be said that extra dimensions are almost a commonplace in modern high-energy physics.

1.1. The hierarchy problem

If our universe indeed has extra dimensions, then a lot of intriguing facts should readily come into play. A major development regarding this issue consists in the alternative approaches to the so called hierarchy problem, which stands unclear despite all the efforts carried out over the last thirty years [7]. In most of the extra-dimensional models, the additional dimensions are tightly curled up in a small volume, explaining thus how they have evaded our perception so far. Initially it was thought that small volume should mean Planck-scale sized volume, but now it is conceded that some extra dimensions may be as large as a human cell, standing at the micrometer scale [8]. Well, how the hierarchy problem fits in this picture? In order to answer it, let us consider a N-dimensional space-time R in which 4 dimensions are large and n = N - 4 are compactified, containing in addition a small mass m at a given point P. For regions that are faraway from P, at least compared to the compactification radius r_c¹ everything should be as if the universe were four-dimensional, so the gravitational interaction is the observed newtonian field

where we took = c = 1 and identified the Planck mass M₄ 1.2 ·10¹⁹eV. When we go to opposite limit (r << r_c) it is not possible to ignore the existence of the extra dimensions anymore, but, from the N-dimensional Gauss law and some dimensional analysis we get

It is then straightforward to deduce the behavior of the gravitational field

where Sⁿ is the appropriate n-sphere and stands for its n-area. Let us notice that we had to choose a tiny n-sphere to apply the Gauss law, or we would not be in the r << r_c regime. But there is nothing fundamental about this choice, and we may as well use the Gauss law (2) in order to find the behavior of the gravitational field for large distances. We shall merely quote the result

and refer the reader to the bibliography [9] for more details. Now, comparing (1) and (4), we find the following constraint relation

which shows that we may look at M₄ as an effective Planck mass, depending fundamentally on the true Planck mass M_4+n and the compactification radius r_c. That is a very interesting relation from the perspective of the hierarchy problem, because it allows for the effective mass M₄ that we observe to be huge even when the true mass M_4+n is not that big. Just to put some numbers, let us consider r_c 1 µm (r_c)^-1 0.18 eV, which is just below the lower bound for the experimental validity of newtonian gravitation [10]. This automatically gives the following values for M_4+n

and so on. Knowing that the electroweak scale M_EW is about 100 GeV, we conclude that an universe with just one extra dimension is definitely not the best case scenario. This does not mean that the reduction of the Planck mass by nine orders of magnitude is not quite something, but only that a ratio of M₅/M_EW 10⁸ still leaves a great desert ahead. However, despite this partial frustration in solving the hierarchy problem, we will proceed with just one extra dimension, for the plain reason that it is the simplest model to work with from both the theoretical and statistical perspectives. Last but not least, it is important to say that in more sophisticated models it is possible to deal with the hierarchy issue in a 5-dimensional picture, some of which are enjoying great successnowadays [11].

1.2. The Casimir effect

As the Casimir effect [12, 13] has a strong dependence with the space-time dimensionality, the Casimir force experiments [14, 15] may be a powerful tool to detect the existence of extra dimensions. In a recent paper, Poppenhaeger et al. [16] carried out a calculation in order to set bounds for the size of an hypothetic extra dimension. They conclude that their modified expression for the Casimir force with one extra dimension is with the experimental data of M. Sparnaay [17], as long as the upper limit for the compactification radius is at the nanometer range. Although we find these results very interesting, we would like to stress that the data present in [17] may be inadequate for such estimations, due to its lack of precision ². It is then a natural step to replace [17] for some more sophisticated experiments, which is precisely the purpose of this work.

We begin by evaluating the Casimir pressure between two plates in a hypothetical universe with a M⁴ × S¹ topology. We use the standard mode summation formula for the Casimir effect, and the calculations are carried out within the analytical regularization scheme, which is closely related to some generalized zeta functions. The result for the Casimir energy and pressure show an explicit dependence on the distance between the plates and on the S¹ radius, as they should. As our final task, we use some recent experimental data [15] and do a proper statistical analysis in order to set limits for the values of the compactification radius.

2. THE CASIMIR EFFECT IN A M4 × S1 SPACETIME

Let us begin by writing the line element of the M⁴ × S¹ universe

where r is the S¹ radius. Due to the simplicity of this metric, the field equations in this manifold are essentially the same as the minkovskian ones. This holds in particular for the massless vectorial field, and so we have

where

In the radiation gauge we may write

and so the field equation may be recast into

Let us assume that the conducting plates are at the planes x = 0 and x = a. This setup leads to the following boundary conditions (BC)

The S¹ topology also imposes a periodicity condition for the electromagnetic field

Now we have to solve equation (11) constrained by conditions (12) e (13). That is a straightforward task, so we merely quote the eigenmodes and the eigenfrequencies

where the fields amplitudes are related by

as a consequence of the gauge condition (10).

The Casimir energy of the electromagnetic field in a M⁴ × S¹ universe is given by the sum of allowed modes

where p is the number of possible polarizations of the photon (p = 3 in this case). The previous expression is purely formal, since its r.h.s. is infinite. So, in order to proceed, we introduce a cut-off parameter s in (16). Then

Performing the integral in k_|| we arrive at

Let us now recall the definition of the Epstein functions, and, as a particular case, the Riemann zeta function [19]

By using these definitions, we may recast expression (18) into

The Epstein functions have a well known analytical continuation, which were thoroughly studied in [19], among other references. As a more detailed discussion of that matter would take us too far afield, let us merely quote the analytic continuation of the Epstein function E₂(s; a₁, a₂)

where K_η(x) stands for the modified Bessel function. The reflection formula for the Riemann zeta function will also be very useful

It is now a straightforward matter put (20) into the form

and, in the limit of s → 0, we get

Due to renormalization issues, we now have to evaluate the Casimir energy of the region defined by the plates, but with no plates whatsoever. This calculation is analogous to the one leading to (24), so we merely state the result

Then, subtracting this term from (24), we finally obtain the Casimir energy for the M⁴ × S¹ with Dirichlet plates

The first thing to be stressed about the previous result is that it precisely coincides with the expression found on [16], although we have derived it in a more clear and pedagogic way. Now, if we want to make some comparison with the experiments, we need an expression for the Casimir pressure. Fortunately, the relation between the Casimir energy and pressure is a simple one

where we used some recurrence relations between the modified Bessel functions [20]. If we now make p = 2 in expressions (26) and (27) and take the limiting case of r → 0, we will get respectively the standard Casimir energy and pressure obtained in [12].

3. ESTIMATE OF THE COMPACTIFICATION RADIUS

The plane geometry is by far the simplest to work with in theoretical calculations, but unfortunately the situation is not so friendly from the experimental point of view. A good measurement of the Casimir force between two plates requires, among other things, a high degree of parallelism between the plates, which is very difficult to sustain throughout the course of the experiment. Due to these parallelism problems, the most popular setup nowadays for measuring the Casimir effect is the sphere-plate configuration [14], for which very precise measurements were reported. There is, however, at least one modern experiment designed to detect the Casimir force between parallel plates [15], and due to its relevance for us we feel that it is important to describe it a little further.

The apparatus itself used in that experiment is very interesting. The two parallel plates are simulated by the opposing faces two silicon beams. One of these beams is rigidly connected to a frame, in a such a way to provide an accurate control of the distance between the two beams. The other beam is a thin cantilever that plays the part of a resonator, since it is free to oscillate around its clamping point. The apparatus is designed to measure the square plates oscillating frequency shift (Δν²), that is related to the Casimir pressure in the following way [15]

where m_eff is the effective mass of the resonator.

Substituting (27) in the previous expression, we get

Now that we have a theoretical expression of Δν² as a function of a and r, we will fit r using the least square method and the experimental data of [15]. As we are fitting just one parameter, we can estimate the best value for r from the graph on Fig. 1 just by looking for the value of r that leads to a minimum value of χ².

Our fit for the compactification radius produced the value of

nm, and the uncertainties on r give the upper and lower bounds for this radius³. In a successful fit, the minimum value of χ² should coincide, approximately, with the number of degrees of freedom of the fit. As in this case we have 8 degrees of freedom⁴ and the minimum for χ² turned out to be 18.6, we can state that no good agreement was obtained between the theoretical model and the experimental data.

4. CONCLUSION

In this article, we have used the Casimir effect to probe the existence of one extra dimension. We started by evaluating the Casimir pressure between two perfect conducting plates living in a 4+1 universe, given in (27), where the extra dimension is compactified in a S¹ topology. In order to set bounds for the compactification radius, we proceeded to the comparison of this result with the experimental data of [15], and, after an appropriate statistical analysis, this procedure showed that the best value for the compactification radius is below approximately 120nm.

We know that the results for the Minkowski space-time are in close agreement with the experimental data. In order to be consistent with this picture, the extra compactified dimension should contribute as a small perturbation to the four-dimensional result, but, as we have seen, this is not the case. Among other things, the extra dimension led to a new polarization degree for the electromagnetic field, which essentially bumped the M⁴ result by a factor of approx. 3/2, that is not small. It is important to say that this new polarization freedom does not allow the r → 0 limit to be taken carelessly, for it represents the transition from M⁴ × S¹ to M⁴, in which a polarization degree is discontinuously lost.

We finish by saying that there are other corrections to the Casimir effect, such as finite conductivity and finite temperature contributions [13, 21], that we have not taken into account and may completely overwhelm any extra dimensional effects. Besides that, there is the roughness of the plate material [22] and possibly some edge effects [23], which, if necessary, should also be considered. Hence, in a more rigorous approach, these influences should be taken into account, and the comparison should be made with very accurate experiments.

5. ACKNOWLEDGMENTS

F.P. would like to thank CNPq and L.F.A.O. also acknowledges CNPq for financial support. F.S.S.R. is grateful to both CNPq and FAPERJ for partial financial support and C.F. thanks CNPq for partial financial support.

[1] T. Kaluza, Zum Unitatsproblem der Physik, Sitz. Preuss. Akad. Wiss. Phys. Math. K 1, 996 (1921).

[2] O. Klein, Quantentheorie und funfdimensionale Relativitatstheorie, Z. Phys. 37, 895 (1926).

[3] D. Z. Freedman, P. van Nieuwenhuizen, and S. Ferrara, Phys. Rev. D 13, 3214 (1976); E. Cremmer, B. Julia, and J. Scherk, Phys. Lett. B 76, 409 (1978).

[4] M. B. Green, J. H. Schwarz, and E. Witten, Supersting Theory, vols. 1-2, Cambridge Univ. Press, Cambridge (1987).

[5] P. Townsend, hep-th/9612121.

[6] P. Binetruy, C. Deffayet, and D. Langlois, Nucl. Phys. B 565, 269 (2000).

[7] M. Peskin and D. Schoroeder, An Introduction to Quantum Field Theory, Perseus Books Publishing (1995).

[8] S. Nussinov and R. Schrock, Phys. Rev. D 59, 105002 (1999) and references therein.

[9] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Rev. D 59, 086004 (1999).

[10] D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gundlach, B. R. Heckel, C. D. Hoyle, and H. E. Swanson, Phys. Rev. Lett. 98, 021101 (2007).

[11] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999).

[12] H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948).

[13] For recent reviews on the Casimir effect, see M. Bordag, U. Mohideen, and V. M. Mostepanenko, Phys. Rep. 353, 1 (2001); K. A. Milton, J. Phys. A 24, R209 (2004); S. K. Lamoreaux, Rep. Prog. Phys. 68, 201 (2005).

[14] S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997); U. Mohideen and A. Roy, Phys. Rev. Lett. 81, 4549 (1998); H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F. Capasso, Science 291, 1941 (2001); R. S. Decca, D. Lopez, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, and V. M. Mostepanenko, Ann. Phys. 318, 37 (2005).

[15] G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, Phys. Rev. Lett. 88, 041804 (2002); G. Bressi, G. Carugno, A. Galvani, R. Onofrio, G. Ruoso, and F. Veronese, Class. Quantum Grav. 18, 3943 (2001).

[16] K. Poppenhaeger, S. Hossenfelder, S. Hofmann, and M. Bleicher, Phys. Lett. B 582, 1 (2004).

[17] M. J. Sparnaay, Physica 24, 751 (1958).

[18] A. Romeo and A. A. Saharian, J. Phys A: Math. Gen. 35, 1297 (2002).

[19] E. Elizalde, Ten Physical Applications of Spectral Zeta Functions, (Springer-Verlag, Berlin, 1995); see also K. Kirsten, J. Math. Phys. 35, 459 (1994).

[20] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover (1964).

[21] I. Brevik, S. A. Ellingsen, and K. A. Milton, New J. Phys. 8, 236 (2006); V. M. Mostepanenko et al., J. Phys. A 39, 6589 (2006); G. L. Klimchitskaya and V. M. Mostepanenko, Contemp. Phys. 47, 131 (2006); M. Brown-Hayes et al., J. Phys. A 39, 6195 (2006); A. Lambrecht, V. V. Nesvizhevsky, R. Onofrio, and S. Reynaud, Class. Quantum Grav. 22, 5397 (2005).

[22] P. A. Maia Neto, A. Lambrecht, and S. Reynaud, Phys. Rev. A 72, 021115 (2005); C. Genet, A. Lambrecht, P. A. Maia Neto, and S. Reynaud, Europhys. Lett. 62, 484 (2003); G. L. Klimchitskaya, A. Roy, U. Mohideen, and V. M. Mostepanenko, Phys. Rev. A 60, 3487 (1999).

[23] H. Gies and K. Klingmueller, Phys. Rev. Lett. 97, 220405 (2006); H. Gies and K. Klingmueller, Phys. Rev. D 74, 045002 (2006).

(Received on 19 August, 2008)

1

We are tacitly assuming that all the curled up dimensions are roughly of the same "size", characterized by

r_c

2

As a matter of fact, all that Sparnaay could conclude from his experiment was that the Casimir force could not be ruled out, or, in his words: The observed attractions do not contradict Casimir theoretical prediction. [].

3

As usual, the uncertainties were obtained by searching the two values of

r that produce χ

² = 19.6 (minimum value plus one). Since a radius can not be negative, we imposed a vanishing lower bound.

4

The degree of freedom of a fit is defined as being the subtraction of the number of experimental points used in the fit by the number of adjusted variables. In this case, we have 9 experimental points and one adjusted variable, which gives the degree of freedom aforementioned.

Publication Dates

• Publication in this collection
  19 Jan 2009
• Date of issue
  Dec 2008

History

• Received
  19 Aug 2008