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arXiv:gr-qc/0605125v1 24 May 2006

Exploring Extra Dimensions in Spectroscopy Experiments

Feng Luo∗and Hongya Liu†

Department of Physics, Dalian University of Technology, Dalian, 116024, P. R. China

Abstract

We propose an idea in spectroscopy to search for extra spatial dimensions as well as to detect

the possible deviation from Newton’s inverse-square law at small scale, and we take high-Z hydro-

genic systems and muonic atoms as illustrations. The relevant experiments might help to explore

more than two extra dimensions scenario in ADD’s brane world model and to set constraints for

fundamental parameters such as the size of extra dimensions.

PACS numbers: 04.80.-y, 11.10.Kk, 32.30.-r, 36.10.-k

Keywords: Precision spectroscopy; Extra dimensions; Newton’s inverse-square law.

∗fluo@student.dlut.edu.cn

†hyliu@dlut.edu.cn

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I. INTRODUCTION

The possibility that there exists more than three spatial dimensions in nature has aroused

physicists’ interests for many years. Recently, the studying focus of theories about ex-

tra dimensions has shifted to the “brane world” scenario.In contrast to the original

string/superstring theories in which extra dimensions are assumed to be as small as Planck

scale (∼ 10−33cm) and hence the detection of these tiny dimensions is hopeless, extra di-

mensions in brane world scenario can be large or even infinite (for a review, see e.g., [1]), so

that the search for such hidden dimensions becomes much more encouraging.

Among various brane world models, the one proposed by ADD (N. Arkani-Hamed, S.

Dimopoulos and G. Dvali) is especially interesting [2], [3], [4], and its implications in accel-

erator physics, astrophysics and cosmology have been widely studied (see, for example, [5],

[6], [7]). Perhaps the most distinctive character of this model is it predicts that the familiar

Newton’s inverse-square law (ISL) would fail below the size of extra dimensions. The reason

is as follows: to solve the hierarchy problem in particle physics, that is, the unnatural huge

energy gap between Planck mass Mpl ∼ 1019Gev and electroweak mass mEW ∼ 103Gev,

ADD assume the four dimensional Mplis not a fundamental scale, but an induced one from

the (4 + n) dimensional Planck mass Mpl(4+n)through

M2

pl∼ M2+n

pl(4+n)Rn,(c = 1,? = 1),

(1)

where n and R are the number and radius of the extra dimensions respectively. For

Mpl(4+n)= 1Tev, this equation is equivalent to

R ∼ 10−17+32

ncm.

(2)

They further assume Mpl(4+n) is around the scale of mEW, then the hierarchy becomes

trivial. Considering within brane world scenario gravitational field (with possible exception

of some hypothetic very weakly coupled fields) is the only field that can propagate in extra

dimensions, a straightforward use of (4 + n) dimensional Gauss’ law infers that the usual

ISL of gravitational force would change to a much stronger one at small scale as

F ∝

1

rn+2,for r ≪ R,

(3)

and it recovers to ordinary ISL for r ≫ R. While the n = 1 case, for which R ∼ 1012m from

Eq.(2), is excluded from planetary motion observations, other possibilities of n cannot be

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boldly ruled out. Particularly, the n = 2 case implies sub-millimeter extra dimensions, while

the experimental conditions for testing Newton’s ISL by torsion pendulum method was just

about to be available when ADD’s proposal appeared, therefore, many people have devoted

to the search of deviation from ISL as well as extra dimensions during the past few years [8],

[9], [10], [11]. By now, there is no reported deviation from ISL and the parameter Mpl(4+n)

has been constrained to larger than several Tev by torsion-balance experiments ([11], [6],

including also constraints from accelerators and astrophysics etc.).

Notice that Eq.(2) results in smaller R for larger n. For n = 3, one obtains R ∼ 10−7cm.

Direct measurement of gravitational force using torsion pendulum at such small scale is far

beyond the reach of current and foreseeable future experimental ability. (Note, however,

that the scales of different extra dimensions are not guaranteed to be the same, so the

deviation from ISL may also appear in micron range even for n > 2.) Considering the

limitation of torsion-balance experiments and the importance of detection of the possible

deviation from ISL as well as exploring extra dimensions, searching for other alternative

experimental methods is worthwhile. It is well know that spectroscopy is one of the most

precise and thoroughly studied fields in physics, so we wonder whether clues about deviation

from ISL and extra dimensions can be found in such field. The idea is quite simple: since

gravity may become much stronger at small scale, then for small scale systems, such as

atoms, ions and even subatoms, effects of the original safely neglected gravity may actually

be large enough and able to show itself in spectroscopic spectra. Because of the precision of

spectroscopy, such experiments may help to set constraints of Mpl(4+n)and R, or some other

model depended parameters. Moreover, these spectroscopic experiments may be capable of

dealing with the n = 3 case and detect the possible deviation from ISL down to nanometer

range.

In a previous paper [12], we did some calculations based on one-proton one-lepton systems,

that is, hydrogen atom, muonic hydrogen etc., and found that the corrections from the

strengthened gravity for the ground state binding energy of these systems are many orders

of magnitude larger than the ones from the exact ISL gravity. Now we would like further

develop our work to high-Z hydrogenic systems (one electron) and muonic atoms, which are

particular interesting in precision spectroscopy since their important role in testing quantum

electrodynamics (QED) (see, for example, [13], [14], [15], [16]). The effects of gravity in such

systems are much larger and hence may be more encouraging for our proposal. In this sense,

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these systems may be not only suitable to test the Standard Model of particle physics, but

also help to explore scenarios beyond the Standard Model.

II.GRAVITY CORRECTIONS IN HIGH-Z HYDROGENIC SYSTEMS

Even within ADD’s framework where gravity becomes much stronger at small scale,

gravity is still far smaller compared to electromagnetic force in high-Z hydrogenic systems,

so we can conveniently treat it as a perturbation. Also, for estimating gravity effects from

the view of orders of magnitude, we just need to consider corrections from the leading

Schrodinger term.

As an illustration, we will only perform calculations for the ground state energy level,

which is also the simplest case. Clearly, correction in this energy level is the largest one

because of the smallest Bohr radius, which serves as the mean distance between the gravi-

tational sources — the atomic nucleus and the electron.

The first-order correction of the ground state energy level is written as

∆E =

?∞

0

?2π

0

?π

0

Ψ∗

100ˆV (r)Ψ100r2sinθdθdφdr,

(4)

where Ψ100= π−1

2a−3

2e−r

a is the ground state wave function of the high-Z hydrogenic system,

a =

?2

me2Zis the first Bohr radius, m =

mlM

ml+Mis the reduced mass, mland M are the masses

of electron and atomic nucleus respectively. The gravitational potential is given as

ˆV (r) =

−G(4+n)mlM

rn+1

−G4mlM

r

−G4mlM

,r ≪ R

(1 + αe−r

λ), r ∼ R

r

,r ≫ R

,

(5)

where the (4+n) dimensional Newton’s constant is G(4+n)∼ RnG4, and the four dimensional

Newton’s constant is G4= M−2

pl. The second line of Eq.(5) takes the Yukawa type, in which

the parameters α ∼ n and λ ∼ R. In above expressions, we have neglected constants of order

unity which depends on the specific compactification forms of extra dimensions, and we have

assumed that the sizes of all the extra dimensions are the same. More detail expressions can

be seen in [10], [12].

From Eq.(4) and the first line of Eq.(5), one can notice that the integral diverges as r → 0

for n ≥ 2. Considering the atomic nucleus is not point like, we introduces a safe cutoff value

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rmwith the atomic nucleus size for the lower limit of the integral. A convenient expression

of rmis rm= r0A

1

3, where A is the mass number of the atomic nucleus and r0is of size

∼ 10−13cm. Also note that such cutoff leads to a relatively conservative estimation of the

gravity effects.

Neglecting coefficients of order unity, the above equations give

∆E ∼

−G4mlMa−1,n = 0

−M−3

pl(4+n)mlMa−2,

−M−4

n = 1

pl(4+n)mlMa−3,

−M−5

n = 2

pl(4+n)mlMa−3r−1

−M−6

m, n = 3

pl(4+n)mlMa−3r−2

m, n = 4

.

(6)

Since rm ∼ 10−12cm and M−1

pl(4+n)∼ 10−17cm for Mpl(4+n) ∼ 1Tev, then for n ≥ 4, the

corrections are much smaller than the ones for n = 2 and n = 3, which are the cases of our

interest.

In fact, for n = 2 and n = 3, ∆E ∝ AZ3and ∆E ∝ A

2

3Z3respectively. Therefore,

it is clear that the corrections for high-Z hydrogenic systems are much larger compared to

the ones for hydrogen atom. Convert ∆E to corrections in terms of frequency ∆ν through

∆E = h∆ν. For a specific system, say207Pb81+, ∆ν ∼ 100Hz for n = 2 and 10−5Hz

for n = 3, while the corresponding corrections in hydrogen atom are 10−8Hz and 10−13Hz

respectively. Also note that the exact ISL, that is, the n = 0 case gives a correction as small

as ∆ν ∼ 10−24Hz for hydrogen atom.

The above calculations also suit to muonic atoms (one muon in the atoms or ions, not

confined to hydrogen-like), since the ground state muonic orbit is far interior compare to the

first electronic Bohr orbit and hence the muon sees approximately a pure Coulomb potential

from the atomic nucleus. In this case, what we need to do is just assigning mlin Eq.(6) the

mass of muon, which is about 200 times larger than electron. Notice mlalso appears in the

expression of a, such substitution further results in several orders of magnitude increase of

∆ν. For207Pb muonic atom, ∆ν ∼ 109Hz for n = 2 and 104Hz for n = 3. For a medium-Z

muonic atom, e.g.,40Ca muonic atom, ∆ν ∼ 106Hz for n = 2 and 102Hz for n = 3.

The sensitivity of ∆ν to Mpl(4+n)and the corresponding R (notice Eq.(1)) is easily seen

from Eq.(6), since ∆ν ∝ M−4

pl(4+n)and ∆ν ∝ M−5

pl(4+n)for n = 2 and n = 3 respectively.

Fig. 1 show the40Ca muonic atom case for n = 2, and Fig. 2 show the207Pb muonic atom

case for n = 3.

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