Unparticle inspired corrections to the gravitational quantum well

Andre Luıs Monteiro Alves

Dissertacao para a obtencao de Grau de Mestre em

Engenharia Fısica Tecnologica

Juri

Presidente: Prof. Doutor Alfredo Barbosa HenriquesOrientador: Prof. Doutor Orfeu Bertolami NetoVogal: Doutor Jorge Tiago Almeida Paramos

Outubro 2010

Acknowledgements

First and foremost I would like to thank my supervisor O. Bertolami for all the valuable help and supportduring this interesting work. I would also like to thank his collaboration and guidance in the article publishedduring the development of this work. I fell like I have learnt with him how to “start” on this adventure ofScience and that gave me a great pleasure.

I also would like to thank my family and friends, specially my mother and my girlfriend, that encouragedme to continue in the worst moments.

Finally I would like to thank Instituto Superior Tecnico for providing facilities where I worked duringthis 5 years of school and where I wrote and developed most of this work.

ii

Resumo

Recentemente foi proposta uma extensao invariante de escala ao modelo padrao (MP) para um patamar deenergia alem dos TeV [1]. Esta extensao poder-se-a manifestar experimentalmente num futuro proximo noacelerador de hadroes, LHC (Large Hadron Collider). Desta implementacao surgem algumas implicacoespeculiares como um espectro de massa contınuo e uma dimensao de escala nao-inteira. Esta extensao foidenominada de fısica das nao-partıculas devido ao vazio de partıculas com uma massa bem definida nestesector.

Neste trabalho sao consideradas correccoes do tipo (RG

r )β, inspiradas em nao-partıculas, ao potencial de

Newton no contexto do poco gravıtico quantico. Um novo espectro de energia e calculado e sao obtidoslimites aos parametros dessas correccoes a partir do conhecimento dos valores proprios de energia do pocogravıtico quantico medidos na experiencia GRANIT com neutroes ultra frios. O material presente nesta tesesegue o trabalho desenvolvido em [2].

Palavras-chave: correccoes inspiradas em nao-partıculas, correccoes ao potencial de Newton, cor-reccoes ao poco gravıtico quantico, experiencia GRANIT, neutroes ultra frios.

iii

Abstract

Quite recently, a scale invariant extension to the standard model (SM) beyond TeV scale, that could bedetected in a near future at the Large Hadron Collider (LHC), was proposed [1]. From this implementationsome peculiar implications arise such as a continuous mass spectrum and a non-integer scaling dimension.This extension was coined unparticle physics due the void of particles with a well defined mass in this sector.

In this work, unparticle inspired corrections of the type (RG

r )β

to the Newtonian potential are consideredin the context of the gravitational quantum well. The new energy spectrum is computed and bounds on theparameters of these corrections are obtained from the knowledge of the energy eigenvalues of the gravitationalquantum well as measured by the GRANIT experiment with ultra-cold neutrons. The material presented inthis thesis follows the work developed in [2].

Keywords: Unparticle inspired corrections, corrections to Newtonian potential, corrections to gravi-tational quantum well (GQW), GRANIT experiment, ultra-cold neutrons.

iv

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiResumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Contents v

List of Tables viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Figures ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Introduction 11.1 The gravitational quantum well (GQW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 GRANIT experiment with ultra-cold neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Unparticles 132.1 Unparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Ungravity and unparticle long range forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Ungravity and unparticle constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Relevant constraints on RTG, RVG and dU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Results 233.1 The modified gravitational quantum well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Conclusions and Outlook 274.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Bibliography 29

v

List of Tables

1.1 Table of zeros of the Airy function, Ai(ξ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Laboratory constraints on power-law potentials based on torsion-balance measurements (68% ofconfidence level) [33]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Laboratory and cosmological constraints (68% confidence level) on λ from the effective potentialdue to vector unparticle exchange [35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Upper bounds on Λ as function of b using GRANIT experimental data and a LED effectivepotential [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 Table of values for β as a function of RG for the first energy level as modified by Eq.(3.5) [2]. . . 24

vii

List of Figures

1.1 The Airy Function, Ai(ξ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Wave functions of neutrons |ψn(z)|2 for n=1,2,3 in the GQW. . . . . . . . . . . . . . . . . . . . . 41.3 Wave functions of the quantum states of neutrons for n=1,2,3,4 in the GRANIT experiment [13]. 61.4 The GRANIT first experimental setup [13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 GRANIT - N(∆h) experimentally measured [13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 GRANIT - F (∆h, vhor) experimentally measured [14]. . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 Allowed region for RTG as a function of dU [32]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Allowed region for ΛU as a function of M∗ for various values of dU [32]. . . . . . . . . . . . . . . 172.3 Upper bounds on RTG as function of du for |∆G/G| ≤ 0.2, 0.15, 0.1, 0.086, 0.036, 0.01 (from top

to bottom)[30]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Allowed region for λ, below the curves, constrained from ∆G/G ≥ −0.036. ΛU = 0.1, 1, 10, 100

TeV were used from the bottom to the top curves. Solid curves are from BBN bounds and thedashed from ISL violation data [33],[30]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Contour plot of Tr − 1 in function of RTG and dU [24]. . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1 First energy eigenvalue E′1 for RG/RE = 1.5, 5, 10, 50, 100 (from top to bottom) as a function ofRG and β as modified by tensor unparticle exchange interactions [2]. . . . . . . . . . . . . . . . . 25

3.2 First energy eigenvalue E′1 for RG/RE = 100, 50, 10, 5, 1.5 (from top to bottom) as a function ofRG and β as modified by vector unparticle exchange interactions [2]. . . . . . . . . . . . . . . . 25

ix

Chapter 1

Introduction

This thesis is devoted to the computation of a new energy spectrum to the gravitational quantum well (GQW)based on a unparticle inspired correction to the Newtonian potential. This particular parametrization of thepotential will introduce new parameters to be constrained, RG and β. We used GRANIT experimental datato set bounds on both parameters. This experiment concerns the measurement of the bouncing heights(quantum states) of ultra-cold neutrons in the Earth’s gravitational field.

1.1 The gravitational quantum well (GQW)

The gravitational quantum well is a well known quantum mechanics problem which solution can be found intextbooks (see e.g. [4, 5]). Although here as well, we will solve this problem once again following [4], just forthe sake of completeness.

Imagine a quantum particle moving in the gravitational field, −→g = −g−→ez , over the Earth’s surface. Wecan immediately approximate the Newtonian potential by the homogeneous gravitational potential becausethe height z is much smaller that the Earth radius, RE . In order to have bound states in the problem weneed to assume that particles are elastically reflected, for example: a horizontal “mirror” is placed at theEarth’s surface, z = 0. Therefore this 2-dimensional problem can be reduced to an 1-dimensional quantummechanical problem in the z-direction given that the particle motion in the horizontal direction is free. Thevertical motion of a bouncing particle at the quantum level is described by Schrodinger eigenvalue problemHψn(z) = Enψn(z) with the Hamiltonian operator

H = − ~2

2m

d2

dz2+mgz . (1.1)

The boundary conditions to this problem are ψn(0) = 0 because of the assumption of elastic reflection atz=0, V (z ≤ 0) = ∞, and ψn(∞) → 0 due to the fact that V = mgz → ∞ as z → ∞. In order to solve thiseigenvalue problem it is convenient to use the abbreviations

2m2g

~2= θ3;

−2m2En~2

= αnθ2 (1.2)

and the following change of variable

ξ = θz + αn (1.3)

where 1θ is a characteristic length equal to 5.89µm, in the case of the neutron, and αn a dimensionless

parameter. Rewriting the eigenvalue problem, Eq.(1.1) one obtains

d2ψndξ2

− ξψn = 0 . (1.4)

1

-10 -8 -6 -4 -2 2Ξ

-0.4

-0.2

0.2

0.4

0.6

AiHΞL

Figure 1.1: The Airy function.

The differential equation is solved by Bessel functions of the order 13 , J 1

3, [6], and solution that matches the

boundary conditions above, more precisely ψn(∞)→ 0, is given in terms of the the Airy function, Ai(ξ) (seeFig.(1.1)). As follows one can write the solution to the problem in terms of regular Airy functions,

ψn(z) = NnAi(θz + αn) , (1.5)

and use Eq.(1.2) to obtain the energy eigenvalues,

En = −(mgθ

)αn , (1.6)

where the normalization factor, Nn is given by

Nn =

[∫ +∞

0

[Ai(θz + αn)]2dz

]− 12

, (1.7)

αn being the nth zero of the regular Airy function Ai(ξ). It is worthwhile to present a more detailed studyof Ai(ξ). For positive values of ξ, this function may be expressed in terms of the modified Bessel function ofsecond kind, K 1

3, [6],

Ai(ξ) =1

π

√ξ

3K 1

3( 2

3ξ32 ), ξ > 0. (1.8)

The asymptotic behavior of K 13(s)→

√π2se−s for s→∞ allows us to rewrite Eq.(1.8) as

ψn(ξ)→ Nn1

2

√3

πξ−

14 e−

23 ξ

32 for ξ →∞. (1.9)

For negative values of ξ, between classical turning points, more precisely between ξ = αn and ξ = 0, the Airyfunction is represented by Bessel functions [6], and we can write the wave functions for the bound states as

ψn(ξ) =Nn3

√|ξ|{J 1

3( 2

3 |ξ|32 ) + J− 1

3( 2

3 |ξ|32 )} for ξ < 0. (1.10)

and according to the boundary condition that sets ψn(z = 0) = 0 this function must vanish at ξ = αn for

all the n zeros, indeed J 13( 2

3 |αn|32 ) + J− 1

3( 2

3 |αn|32 ) = 0. We present below a table with the first 10 zeros

of the Airy function, which occur for ξ < 0. This wave function shows an oscillatory behavior alreadydepicted in Fig.(1.1). For higher values of |αn| >> 1, the asymptotic relations of Bessel functions are

2

Table 1.1: Table of zeros of Ai(ξ).

Ai(αn) = 0n αn1 -2.3382 -4.0883 -5.5214 -6.7875 -7.9446 -9.0237 -10.0408 -11.0099 -11.93610 -12.829

J 13(s) →

√2πscos(s −

5π12 ) and J− 1

3(s) →

√2πscos(s −

π12 ). This relations lead to an asymptotic relation for

the wave function given by

ψn(ξ)→ Nn√3π|ξ|− 1

4 cos(

23 |ξ|

32 − π

4

)for ξ << 0. (1.11)

Using Eq.(1.11) and ψn(α) = 0, one obtains an eigenvalue condition

2

3|αn|

32 =

(2n− 1

2

)π

2. (1.12)

One can now rewrite the energy spectrum, Eq.(1.6), using the latter eigenvalue condition, in terms of ninstead of αn as follows:

En>>1 = −(mgθ

){(n− 1

4

)3π

8

} 23

for n >> 1. (1.13)

The GQW despite being a quantum mechanical problem allows for a connection to classical mechanics.In classical mechanics, a particle with energy En will bounce in the Earth’s gravitational field at maximumheight zn = En

mg . In quantum mechanics the particle, in this problem, will bounce in discrete levels and the

probability of finding it in an nth quantum state with an energy En at a height z equals the square modulusof its wave functions |ψn(z)|2. This probability will never be formally zero for z > 0 and will have a maximumat the classical turning points1 given by

hn =Enmg

= −αnθ. (1.14)

Using Eq.(1.6), Eq.(1.14) and Table (1.1) one calculates the energies, En, for the 3 lowest quantum statesand their respective classical turning points heights for the neutron

hn = {13.7, 24.0, 32.4, ...} µm, (1.15)

En = {1.407, 2.461, 3.322, ...} peV. (1.16)

We depict in Fig.(1.2) the wave functions of the neutrons |ψn(z)|2 for n=1, 2, 3. In this plot we can seeclearly the maxima of |ψn(z)|2 for the classical turning points. We also see that the probability |ψn(z)|2 offinding particles at the height hn has n maxima and (n− 1) minima with probability zero for each minimumas any standing wave.

1We have a classical turning point when E=V, in this case the particle will basically bounce between the two classical turningpoints because it is in a bound state [7].

3

height z

ÈΨn2HzL

Figure 1.2: Wave functions of neutrons |ψn(z)|2 for n=1,2,3 in the GQW as function of the bouncing heightz. Dashes get longer as n increases.

One can also calculate the energy spectrum of the gravitational quantum well in the quasi-classicalapproximation, or in the WKB approximation [7], using the Bohr-Sommerfeld formula Eq.(1.17) [8, 9], thatworks extremely well for linear potentials. This formula gives the correct result for the first energy levelwithin 1% of accuracy in this particular problem.

1

~

∫ b

0

p(z)dz +π

4= πn n = 1, 2, 3, ... , (1.17)

where p(z) =√

2m(E −mgz) and b is the classical turning point. The energy spectrum obtained is exactlythe same as the one obtained in Eq.(1.13), as expected.

Finally, in this topic, one analyzes the gravitational quantum well in the classical limit, to show whymacroscopic bodies do not bounce at discrete levels in the Earth’s gravitational field. The characteristicscaling of this Schrodinger eigenvalue problem is 1

θ which will depend of the mass of the particle. One

gets smaller characteristic lengths for larger masses ( 1θ ∝ m−

23 ), for example the energy for the neutron

is in peV which gives us a fair idea of the vanishing quantum effects in macroscopic bodies. Applying thesame mathematical framework of the gravitational quantum well to a tennis ball, that weights 57× 10−3Kg,bouncing in floor of a court one obtains a characteristic scaling 1

θ of 5.59× 10−17µm which is astonishinglysmall. Smaller characteristic length means continuous spectrum, classical mechanics. In order to extract anymeaningful information of the wave function Eq.(1.11), that is a rapidly oscillating function, one takes itsaverage square

〈ψn(z)2〉 =

1

2

|Nn|2

3π|ξ|− 1

2 =|Nn|2

6π

1√θz + αn

. (1.18)

The particle achieve its highest height at the classical turning points, hn = −αn

θ , and substituting thisrelation in the last equation and being aware that the modulus under the square root is only of interest inthe distance |z − h|,

〈ψn(z)2〉 ∝ 1√

|z − h|, (1.19)

which is in perfectly agreement with the classical predictions. One knows that 〈ψn(z)2〉dx gives the probability

of observing a particle in the interval dx and this probability is proportional to the time interval of stay dt,Eq.(1.21),

〈ψn(z)2〉dx ∝ dt. (1.20)

Thus, v = dxdt ∝

√|z − h| which is in total agreement with the classical velocity of a free fall body:

v =√

2g|z − h|. (1.21)

4

One can conclude that the tennis ball will follow the classical behavior as expected. One can also concludethat the subtle division between classical and quantum world in this particular case lays on the threshold onechooses to consider that wavelength is small, that is clearly the high-energy limit (for an interesting discussionon this topic, see [10]). As a final remark it interesting to note that, despite its mathematical difficulties, thegravitational quantum well is still capable of giving an intuitive description of what it is happening.

5

1.2 GRANIT experiment with ultra-cold neutrons

The gravitational quantum well was considered a long time ago, however just as an academic exercise ofquantum mechanics. Quantum stationary states of matter exist in nature for the electromagnetic and strongfields. However to observe quantum stationary states created by the gravitational field one needs to imposea “wall” to particles trajectories since the gravitational field itself only confines particles forcing them tofree fall along field lines. Furthermore it is very difficult to measure this quantization due to vanishinglysmall quantum effects for macroscopic bodies, as shown in the previous section. Also the electromagneticinteraction is the dominant one for charged particles. Hence one should use neutral elementary particles witha long lifetime.

A first attempt to discuss the conceptual aspects and measuring methods of a possible experiment usingslow neutrons was proposed in the 1970’s [11], but concrete steps to set up an experiment were only takenin 2000 by Nesvizhevsky and his collaborators [12] at Institute Laue Langevin (ILL) . Finally in 2003 , thesame working group, reported on the evidence of the quantization of vertical motion of ultra-cold neutrons(UCN) bouncing above a mirror in the Earth’s gravitational field [13]. This was the first time that thelowest quantum state of neutrons in the Earth’s gravitational field was observed. A high precision neutrongravitational spectrometer was used specially optimized for this purpose. More results of this experimentwere studied and presented in [14]. It is important to say that GRANIT experiment mimics almost perfectlythe gravitational quantum well (see Fig.(1.3)) described in the previous section and one can understand thisexperiment in its mathematical framework within certain experimental limits.

Figure 1.3: Wave functions of the quantum states of neutrons for n=1,2,3,4 in the GRANIT experiment [13].

Since part of the results presented in this thesis are based on the data collected in this experiment andits recent improvement, GRANIT 2010 project, we will describe it in great detail in order to provide a fullview of this work.

This experiment makes use of an intense beam of ultra-cold neutrons from ILL nuclear reactor in Grenoble(France), and its basic idea is to let neutrons flow - with a certain horizontal velocity, 〈vhor〉 ∼ 6.5m/s -between a mirror and an absorber/scatterer above. The absorber acts as selector for the vertical velocitycomponent. Then one measures in the detector the number N of transmitted neutrons as a function of

6

the absorber height, ∆h. Clearly this measurement allows to distinguish between a classical dependenceNCL(∆h) for the neutron trajectory or a stepwise quantum-mechanical dependence NQM (∆h) at smallabsorber heights, more precisely, at values of ∆h < h1, the smallest bouncing height corresponding to thefirst energy level. We show the experiment setup in Fig.(1.4).

Figure 1.4: The GRANIT first experimental setup [13].

One provides here some brief information about the experimental apparatus. The gravitational levels areexpected to be formed above the bottom mirrors (6). The mirrors and the absorber as well as the detector(7) are inside a vacuum chamber that has a 30 µm-thick aluminum foil entrance window. The separationbetween the entrance of the vacuum chamber and the neutron guide is approximately 1.5 cm. This separationpermits to mechanically isolate the setup against parasitic vibrations, coming from the neutron guide. Inaddition, the setup is mounted on a standard pressurized anti-vibration table (2). A magnetic shield (8)insures screening of external magnetic field gradients. At the entrance of the vacuum chamber there is anadjustable titanium collimator that, together with the mirrors and the absorber, allow for the selection ofthe neutrons velocity components. In order to ensure a perfect leveling of the granite polish slab (3) asystem of three interconnected pneumatic valves provide a “floating” tabletop where active legs consisting ofpiezotranslators sustain the granite slab. These piezotranslators are connected in a closed loop with preciseinclinometers (11) installed on the granite surface. Length variation of the piezotranslators allows for anautomatic leveling of the granite slab with an absolute precision better than 10 µrad.

After the collimation of the beam, the absorber and the mirror configuration reduce the range of theincoming vertical velocities by an order of magnitude, typical values are 〈vver〉 ∼ 1.7 cm/s. The neutrontransmission between the bottom mirror and the absorber shows no correlation on the inclination angle ofthe complete setup, demonstrating the high uniformity of the phase space of the incoming neutrons. Thisimproves the quality of the experimental data. In particular, it allows for the application, with great success, ofa phenomenological quantum model for the interaction of the neutron with the mirror, the absorber/scattererand the detector and its transmission through the slit, that also depends on the horizontal velocity of theneutrons inside the chamber [14].

The phase space neutron density that corresponds to the lowest quantum states is fairly low. The detectedcount rate ranges from 10−3 to 10−1 s−1 in the “quantum” regime, at low absorber heights, despite the totalneutron flux at the entrance being approximately 106 s−1. For this reason an efficient background suppressionis important. The background due to extremal thermal neutrons is suppressed by “4π shielding” of thedetector with a B4C rubber. The titanium collimator at the entrance inhibits direct view from the neutronguide to the detector so that the trapped neutrons that move into the setup follow parabolic trajectories.The mirrors used in this setup are standard optical glass plates that had been previously tested with smallangle scattering x-rays in order to ensure that all the reflections are specular. The specular reflections ensure

7

that motion parallel and normal to the surface are totally independent allowing to have a continuous neutronspectrum in the horizontal plane and quantum levels for the vertical motion. The absorber/scatterer is aglass plate with high macroscopic flatness and with high micro-roughness (few µm). This glass is coatedwith a anti-reflecting sublayer of Ti-Zr-Gd, 0.2 µm thickness, usually used in polarizers for cold neutrons.Due to this sublayer, neutron can be either adsorbed or scattered mainly non-specularly. The neutron countrate N(∆h) is measured with a cylindrical 3He gaseous detector with approximately 20 cm long and 1.7 cmdiameter and its neutron entrance window, made of aluminum foil, has 12 cm long and 1.5 mm high. Thisentrance acts as a filter for neutrons with 〈vhor〉 < 3.2 m/s although these neutrons had already been removedfrom the spectrum by the windows at the entrance of the vacuum chamber.

In the first part of this experiment only evidence of the first quantum level is found. The number Nof transmitted neutrons through the slit is measured as a function of ∆h, the distance between the mirrorand the absorber, N(∆h). According to classical mechanics, N(∆h) should follow a relation (∆h)3/2, dueto the simultaneous increase in the slit size (power factor 1) and in the range of acceptable vertical velocitycomponents (power factor 1/2). On the other hand, the quantum mechanical dependence must be a stepwisefunction, NQM (∆h), that for large n converge to the classical limit. One depicts in Fig.(1.5) the experimentaldependence of N(∆h) and the fits of classical and quantum models for N(∆h).

Figure 1.5: N(∆h) dependence experimentally measured [13].

The solid line corresponds to the classical dependence, NCL(∆h), the dashed line, the NQM (∆h), and thedotted line is a quantum mechanical simplified dependence, NQM,0(∆h), that accounts for the first quantumlevel h1, and for larger heights that follow the classical dependence. First, one clearly sees that for heightslower than 15 µm the slit is not transparent for neutrons showing a distinct evidence of the first discreteenergy level, not far from the values of h1 Eq.(1.15). One can also conclude that NQM (∆h) (dashed line)fits the experimental points better than the shifted classical formula, the dotted line. Finally it is importantto note that the background, straight solid line with dashed errors bars, is remarkably low for all importantmeasurements. The quantum mechanical dependence NQM (∆h) can be approximate, if one assumes theabsorber capture, independently of its strength, of all neutrons at heights higher than the absorber potentialmg∆h, given by

NQM (∆h) =α

2

∞∑n=1

νn

[1 + erf

(∆h− h′

n√2d

)]. (1.22)

where erf() is the standard error function. This follows from the convolution of the Heaviside function witha Gaussian resolution function with a dispersion larger than d = 2µm and the occupation number νn. Onealso introduces h

′

n = hn+ δ where δ is an upward shift, since the overlap of the wave function of the neutronsand the absorber potential must be sufficient for the absorption of a neutron. Its value is estimated from

8

experimental fitting and is equal 3 µm. Finally hn is obtained using Eq.(1.13) in the classical turning point,

hn =

{9

8 g

[(n− 1

4

)π~m

]2}1/3

. (1.23)

It is evident that the measured data, in particular for ∆h < 20µm, strongly contradicts the classicaldependence NCL(∆h). The measured data is acquired for two different horizontal velocities, approximately6 m/s and 10 m/s, showing that the count rates do not depend on the horizontal velocity and only thevertical motion is important. As we will see later this is not true. It is also tested a reversed geometry wherethe absorber is placed at the bottom and the mirror at the top. The number of absorbed neutrons whencompared with the standard setup geometry, exceeds it by a factor of 20 at a height of 120 µm and by a factorof 50 at height of 60 µm due to the confinement of neutrons in the gravitational field. The absorber efficiencyper collision is on average 90%. It is not possible in this first combination of setup/detector to have a clearobservation of the energy levels mainly due to the lower energy resolution and to the smaller difference inbouncing heights. In addition, experimental precision decrease for higher energies makes a background/signalcomparison even more difficult.

In an ensuing work [14], the experimental resolution was improved by a more accurate mechanism forthe positioning of the scatterer/absorber. Statistical accuracy was also improved by optimizing neutrontransport in front of the experimental installation and by a more efficient use of the beam time allowedthe automatisation of the measurement. The positioning of the absorber was improved allowing a spatialresolution with an uncertainty lower than 3 µm, that corresponds to the estimated sharpness of the neutronswave functions as a function of height as given by Eq.(1.25).

The distance between the mirror and the scatterer ∆h was measured using a capacitor method. A fewmetallic electrodes were applied onto the glass flat mirror and the absorber was coated with a metallic layerin the lower part. The capacitors formed between these electrodes and the metallic coating of the absorberwere connected in RC electric chains which defined oscillation frequencies for the relaxation RC-generators.All of theses oscillation frequencies were connected to a well-reproducible distance ∆h and could be ratherprecisely approximated by a second-power polynomial in the range 5 µm < ∆h < 1 mm. The calibration ofthis mechanism allows to define an absolute height with an accuracy of ∼ 0.5 µm. However due to electricproblems and parasitic capacitances the uncertainty in the scatterer height is of 5− 7%. The correspondinguncertainty in ∆h is equal to ± 1.0µm at ∆h ∼ 15µm and ± 1.6µm at ∆h ∼ 25µm.

A new model to estimate the count rate of neutrons that reach the detector was developed in the secondwork of the group [14]. The count rate or the transmitted flux of neutron was considered as a functionof the slit height ∆h and vhor, F (∆h, vhor). This was developed because the spatial resolution of thisspectrometers strongly depends on: 1) The finite sharpness of the asymptotic decrease of the Airy functionsdescribing the wave functions of neutrons and, in addition, the resolution is limited by the finite time ofobservations of neutrons in the spectrometer; 2) Not fully understand scatterer-neutron interaction thatintroduce uncertainties in the model, mainly in the neutron wave functions deformations due to the scatterer;3) Finite width of the spectrum of the horizontal neutron velocity. This model takes into account the finitesharpness of the gravitational barrier penetrability as a function of the scatterer height and uses the idea ofnuclear α−decay which states that the decay rate of such a nucleus is equal to the penetrability of the nuclearpotential barrier for an α−particle multiplied by the frequency of collisions of that particle with the nuclearbarrier. The model uses the non-disturbed probability density, integrated square of Eq.(1.9), to estimatethe probability of tunneling of the neutron into the classical forbidden region, ∆h > hn. It assumes theprobability of tunneling is equal to 1 for ∆h ≤ hn. The rate decay (τabsn )−1 results from the product of thecollision frequency ωn, estimated using Eq.(1.13) divided by ~, with the probability of tunneling of neutroninto the scatterer . In order to relate the probability of decay with the vhor it is considered that during thepassage time, τpass, along the slit with length L, the n-th neutron quantum state partially decays and theprobability of neutron transmission Pn(∆h, vhor) is given by

Pn(∆h, vhor) = exp

(−τpass(vhor)

τabsn

). (1.24)

9

Finally, the transmitted neutron flux for several quantum states taking in account the horizontal velocity isequal to:

F (∆h, vhor) =∑n

Fn(∆h, vhor) = F0

∑n

νn exp

(−α L

vhor· 3

√mπ2g2

3~(n− 14 )× (1.25){

exp(− 4

3 ((∆h− hn)θ)32

), ∆h > hn

1, ∆h ≤ hn

),

where νn is the population of the n-th quantum state, F0 is the normalization constant and α is the coefficientresponsible for a finite scatterer efficiency. The sharpness of F (∆h, vhor) on ∆h in the range ∆h > hn definesthe possible accuracy to identify the different quantum states of neutrons in the measurement of the totalflux; hence, it defines the best achievable spatial resolution of the spectrometer. It is estimated that the totalmodel uncertainty could produce a systematic shift in hn of about ±1.5µm.

The experimental dependence of the neutron flux through the slit between the mirror and the scatterer isdepicted in Fig.(1.6), together with the classical dependence (solid black line), quantum model dependenceF (∆h, vhor) (in red) and the dotted line represent the simplified quantum relation NQM,0(∆h). The circlepoints show the results obtained with copper coated and Ti-Zr-Gd coated rough scatterers. The averagehorizontal velocity component for this measurements is 4.9 m/s. Longer observation times for neutrons inthe slit allow “sharper” measurements. The solid red curve (1.25) in Fig.(1.6) approximates the experimental

Figure 1.6: F (∆h, vhor) dependence experimentally measured [14].

data with χ2 = 0.9. In this calculation all the critical heights correspond to (1.15) except the two lowestquantum heights h1 and h2 that were fitted from the experimental data. The quantum state populationswere assumed to be equal to each other apart from the lowest quantum state population ν1 that is calculatedfrom the experimental data. The scatterer efficiency α was also calculated from the experimental data. Afteradding the statistical and systematic uncertainties the average values of the two lowest critical heights are

hexp1 = 12.2± 1.8(syst)± 0.7(stat)(µm), (1.26)

hexp2 = 21.6± 2.2(syst)± 0.7(stat)(µm).

These values match the theoretical values of Eq.(1.15) within 25%. It is important to note that the experi-mental values obtained for h1 and h2 are slightly low since the correction for the deformation of the neutronwave functions is not taken into account explicitly. It also relevant that several measurements reveal that asthe values of vhor increase the hn measured decrease systematically. In this experiment spatial resolution is

10

close to its theoretical limit, defined by the sharpness of neutron wave functions as a function of height aswe have already mention before. A significant increase in the accuracy could be achieved by a longer storageof neutrons, which lifetime is τ = (885.7 ∼ 0.8)s [15], and by the measurement of the resonant transitionfrequencies between them [14].

Recent improvements in the GRANIT experiment based on a development of a second-generation gravi-tational spectrometer are excepted to allow for extended periods of storage of the neutrons in a selected grav-itational quantum state using a closed specular trap [16]. This improvement will enable the measurements ofresonant transitions, between quantum states, using a magnetic inductor as proposed in [17]. Furthermore,upgraded GRANIT will also benefit from a dedicated 4He ultra-cold neutron source that injects into thespectrometer ultra cold-neutrons with a precise phase space density, which helps to control vhor. Indeed, anew model was built to deal with the interaction between neutrons and the scatterer (rough waveguide) [18].

The first experiments with this new spectrometer were scheduled to start in 2009. In February 2010 atconference in Les Houches [19], new experiments were proposed and our results [2] were presented by O.Bertolami. A short article [3] is to be publish in the context of this conference.

As a final note, the GRANIT setup offers the opportunity to confront observations with various theoreticalmodels such as noncomutative geometry [20], existence of an intrinsic minimal length [21], the presence ofextra dimensions contributions [22], the effect of Yukawa type interactions [23] and quite recently, ungravityinspired corrections to the Lane-Emden equation [24] and to the gravitational quantum well [2].

11

Chapter 2

Unparticles

2.1 Unparticles

In 2007, an interesting extension of the standard model (SM) beyond the TeV scale was proposed by Georgi[1]. This extension aims to implement scale invariance in the SM. Particles obey to the dispersion relation

ω2 = c2 k2 + m2c4

~2 (2.1)

that is not space-time scale invariant, unless for the trivial case of photons that have vanishing mass. Thereforeto create this scale invariant sector it requires new scale invariant objects named unparticles. These objectsare not governed by a dispersion relation but rather by a rescaling parameter, dU . Thus, the implementationof a scale invariant sector in the SM requires considering an additional set of fields, the Banks-Zaks (BZ)fields, with a non-trivial IR fixed point. The interaction between SM and BZ fields is mediated by particleswith a large mass, MU . This nonrenormalizable coupling, below MU scale, can be written as

LBZ =1

MkU

OSMOBZ , (2.2)

where OSM denotes an operator with mass dimension dSM constructed from SM fields and OBZ is an operatorwith mass dimension dBZ constructed from the BZ fields. At an energy scale, ΛU & 1 TeV, the BZ operatorsmutate into unparticles operators (OU ), with non-integer scaling dimension dU , such that:

LU =CUΛdBZ−dU

U

MkU

OSMOU , (2.3)

where CU is a coefficient function. This dimensional transmutation is caused by renormalizable couplings ofthe BZ fields at this energy scale that allow for the emergence of scale invariance in this sector. One canlook at Eq.(2.3) as an effective field theory interaction, below ΛU . The effect of fields with nontrivial scalingis to generate “new” forces. The most important effects arise from a single interaction of the type (2.3) insome SM processes. To calculate the probability for such a process, we need to evaluate the density of finalstates for unparticles. Let us assume that OU is a Lorentz scalar and consider the vacuum matrix elementrepresenting the creation of unparticles

〈0|OU (x)O†U (0)|0〉 =

∫e−iPx|〈0|OU (0)|P 〉|2ρ(P 2)

d4 P

(2π)4, (2.4)

where |P 〉 is the unparticle 4-momentum P ν produced from the vacuum by OU and ρ(P 2) is the densityof phase space in the final state. Knowing that the unparticle operator have dimension dU automaticallyinsures that the spectral density is given by

|〈0|OU (0)|P 〉|2ρ(P 2) = AdU θ(P0)θ(P 2)(P 2)dU−2 , (2.5)

13

because of scale invariance of the matrix element (2.4) in this sector and also because step functions θ areinvariant. The phase space for unparticles should be identified (the left-hand side of (2.5)) with the phasespace for n massless particles that can be written explicitly in the following way:

(2π)4δ4

(P −

n∑i=1

pi

)n∏i=1

δ(p2i )θ(p

0i )

d4 pi(2π)3

= Anθ(P0)θ(P 2)(P 2)n−2 , (2.6)

where

An =16π5/2

(2π)2n

Γ(n+ 12 )

Γ(n− 1)Γ(2n). (2.7)

This identification is deeper and we can resume it as [1]:

“Unparticle stuff with scale dimension dU looks like (2.8)

a non-integral number of invisible particles.”

Therefore we can also identify An with AdU replacing n→ dU , since Γ(dU ) is well defined function for a realpositive argument. This is just a convention, as a different definition could be absorbed in the coefficientfunction CU .

In the processes involving unparticles one can consider, OU as a scalar, a vector, a tensor or even aspinor. Signatures for colliders [25, 26] as well as other phenomenological aspects have been investigated inRefs.[24, 27, 28, 29, 30, 31].

2.2 Ungravity and unparticle long range forces

The exchange of unparticles can give rise to forces which lead to deviations from the inverse square law (ISL)due to the anomalous scaling dimension, dU , of the unparticle propagator. Indeed, if OU is a rank-two tensorit could couple to the stress-energy tensor Tµν and this gives raise to a modification of Newtonian gravityleading to an effective potential. If one consider an effective interaction of the type

1

M∗ΛdU−1U

√g TµνOUµν , (2.9)

where M∗ = ΛU

(MU

ΛU

)dUV

, and here dBZ ≈ dUV (weak coupling). This interaction leads to an effective

potential and considering effects of unparticle exchange only to the lowest order, in the non-relativistic limit,and for dU 6= 1, [32]:

V (r) = −GNMm

r

[1 +

(RTGr

)2dU−2], (2.10)

whereGN is the Newton’s gravitational constant andRTG is the characteristic length for which this “ungravity”interactions become significant and is defined as

RTG =1

πΛU

(MPl

M∗

)1/(dU−1)

× (2.11)[2(2− α)

π

Γ(dU + 12 )Γ(dU − 1

2 )

Γ(2dU )

]1/(2dU−2)

,

where MPl = 1.22×1019GeV is the Planck mass and α is a constant dependent on the type of the propagatorin question (α = 1 for the graviton case). The case du < 1 leads to forces which fall less steeply than gravity,which for large distances can be directly tested experimentally and easily ruled out1 based on “fifth force”type experiments [33]. Therefore the case du > 1 is apparently a more relevant case to consider because itleads to an effective potential that falls off faster than 1/r. Conformal invariance constraints require that

1We will see later that dU < 1 leads to a correction of the Newtonian potential that is allowed if dU ∼ 1.

14

dU > 1 + s, where s is the spin of the operator, s = 0 for a scalar for example, otherwise OU is not gauge-invariant, neither the theory is unitary [34]. However, short distance ungravity contribution seems to bequite constrained by sub-millimeter tests of the ISL [33] (we will come back to this point in the next section).Furthermore, we also need to constrain RTG and regarding that some limits will be presented in the nextsection. In particular, in our work, we consider an ungravity inspired model such that the source is Earthand bounds are imposed on RTG based on that.

We can also consider a vector unparticle exchange [35]. Considering a coupling between a vector unparticleand a baryonic (or leptonic) current Jµ of the type

λ

ΛdU−1U

JµOµU , (2.12)

where λ is dimensionless constant somewhat similar to the one used in Eq.(2.9), we obtain an effectivepotential that combined with the gravitational potential leads to [35]:

V (r) = −GNMm

r

[1−

(RVGr

)2du−2], (2.13)

where RVG is slightly different from RTG, but is clearly the characteristic length scale of this vector exchangedefined as:

RVG =1

ΛU

(λMPl

ζ

)1/(dU−1)

× (2.14)[1

2π2dU

Γ(dU + 12 )Γ(dU − 1

2 )

Γ(2dU )

]1/(2dU−2)

,

where ζ = 931.3 MeV is the atomic mass unit. Notice that the vector contribution leads to a subtractivecontribution to the potential given the repulsive nature of the vector unparticle exchange. Similarly to thework in Ref.[32], stringent bounds on RVG were obtained in [35] using data from [33] for power-law corrections.The results concerning these bounds will be presented in the next section.

It is important to stress at this point that the modifications of gravity presented above differ from thoseintroduced by extra dimension models in many aspects. Contrasting with unparticle corrections, extradimensions Arkani-Hamed-Dimopoulos-Dvali models [36] parametrize the potential in a way that it falls offexponentially at large distances r/R > 1, where R is the compactification length scale, while at r/R � 1the potential exhibits a power-law dependence of type 1/rn+1 where n is an integer. As we have seen thecorrections from ungravity to the effective potential are scale invariant dependences for r of the form 1/r2dU

−1,valid at both short and large distances, where dU is a non-integer scaling dimension. Moreover, for example,the extra dimensions model presented in [36] only allows for a power-law modification of the type 1/ra witha ≥ 3, in order to match solar system physics. The case of warped extra dimensions models [37, 38] also leadto corrections that are rather different from the ones presented above.

We can conclude saying that it is feasible to distinguish experimentally between extra dimension modelsand the ungravity corrections to the Newtonian potential. Finally, fractional modifications of the ISL werestudied by [39] showing that they lead to strong coupling effects.

2.3 Ungravity and unparticle constraints

Ungravity and unparticle long range forces generate power-law modifications to the Newtonian potentialthat affect mainly the sub-millimeter range and due to that some work has been done to constrain the freeparameters of this theory, dU , ΛU , M∗ and RT,VG .

We review here bounds obtained, from previous work, on RT,VG and dU from fifth force laboratory exper-iments [32, 33, 35], cosmological [30, 35] and astrophysical [24] arguments. These results will help the reader

15

to put in perspective the work of the next section where bounds on dU are obtained, in the context of aunparticle inspired model, using GRANIT experimental data [2]. The typical scale of GRANIT experimentis the µm.

In torsion-balance experiments one searches for power-law modifications of the inverse square law (ISL)for 0.05 mm and no significant deviations have been found [33]. More specifically, using the data from [40]and the effective potential

V (r) = −GNMm

r

[1 + βk

(1mm

r

)k−1], (2.15)

where one can identify dU = (k+ 1)/2 and RTG = β1/(k−1)k , positive corrections to the ISL are constrained to

the values presented in the following table.

Table 2.1: Laboratory constraints on power-law potentials based on torsion-balance measurements (68% ofconfidence level) [33].

k |βk| [33] |βk| (previous works)2 4.5× 10−4 1.3× 10−3 [41]3 1.3× 10−4 2.8× 10−3 [42]4 4.9× 10−5 2.9× 10−3 [42]5 1.5× 10−5 2.3× 10−3 [42]

Later, Goldberg and Nath [32], using an extrapolation of data of Table (2.1), obtained an upper limit onRTG as a function of dU based on the effective potential (2.10). Results are depicted below, Fig.(2.1). Moreover,

Figure 2.1: Allowed region for RTG as a function of dU [32].

using the constraints obtained on RTG and following Eq.(2.11), it was also presented a plot of the allowedregion of the ΛU −M∗ parameter space. We depict these results in Fig.(2.2) where the excluded parameterregions are below the lines. We can see from Fig.(2.2) that lower limit imposed on ΛU is approximately10−12 TeV for values of M∗ ' MPl. This alone may indicate that one should have M∗ < MPl or eventhat we should search regions where dU ∼ 1 in order to obtain more stringent bounds. As described above,

16

Figure 2.2: Allowed region for ΛU as a function of M∗ for various values of dU [32].

unparticles exchange yields contributions to the ISL that correspond to non-integers of β = 2dU − 2. Asit will be seen, the new results presented in this work that are based on GRANIT experimental data areparticularly sensitive to values of β <∼ 0, for a specific set of ranges of RT,VG .

The interaction√g Tµ;µO

U/M∗ΛdU−1U can also be constrained by high energy scattering processes [32].

Consider the process (ff → scalar unparticle) that gives Feynman amplitude M = mu(p1)v(p2)/M∗ΛdU−1U ,

where m is the mass of the fermion f and p1, p2 are the incident momenta. Following the notation of Ref.[1], the cross section of this process can be written as

σ(ff → U) = AdU1

4s

(m

M∗

)2(√s

ΛU

)2dU−2

. (2.16)

The annihilation to the unparticle proceeds through a single partial s-wave making the cross section boundedby unitarity, σ < 16π/s. So one can rewrite this inequity using the explicit value of AdU and this gives anupper bound on the energy in order the keep LU compatible with a unitary theory [43]:

√s <

1

R∗

(√64π

AdU

MPl

m

)1/dU−1

, (2.17)

where we have express the unitary in terms of the quantity

R∗ ≡(

1

ΛU

)(MPl

M∗

)1/dU−1

∝ RTG. (2.18)

This means that we can write the upper bound on RTG, see Fig.(2.1), in terms of R∗. For the region 2 < dU < 3the following parametrization Rmax∗ ' [0.5 + 1.75(dU − 2)] × 1012 GeV −1 is used. Moreover, we can saythat R∗ must lie in the range 0.1 Rmax∗ < R∗ < Rmax∗ if one consider the exchange of scalar unparticleconsistent with recent ISL experimental limits [32]. If one inserts this restriction in Eq.(2.17) and considerthe worst-case scenario, with the fermion being the top quark, we obtain that unitary is not violated up to1.2 TeV for 2 < dU < 2.3. For a lower fermion mass the limit range is relaxed. Even requiring compatibilitywith perturbative QCD for the light quarks, then 2 < dU < 2.2 for

√smax ' 1.2 TeV . This shows that we

can maintain compatibility of scale invariance with both high and low energy constraints and at the same

17

time not to exclude the possibility of measuring ungravity corrections to Newton’s law in future generationof gravitational experiments.

It also of interest to obtain bounds for M∗. From Eq.(2.18) and the inequality (2.17) [44],

M∗ >

(m

ΛU

)√AdU64π

(√s)dU−1. (2.19)

Therefore from the bound on√smax ' 1.2 TeV :

M∗ >

(m

ΛU

)√AdU64π

(1.2× 1012)dU−1. (2.20)

Finally considering the range 2 < dU < 2.2 and making the approximation 1.2dU−1 ≈ 1.2 and AdU ≈ 4×10−4,we obtain a bound condition for M∗ that depends only on the mass of the fermion and on IR energy fixedpoint,

M∗ >∼ m(

2.4× 10−2

ΛU/TeV

)√1

64π. (2.21)

However the lower bounds obtained by this method are relaxed if we compare with Fig.(2.2). The tightestlower constraint obtained is 9.0× 10−4 for (M∗/MPl) for the case of the top quark and ΛU =1 TeV.

Let us consider now the constraints imposed on long range forces resulting from the coupling of vectorunparticles [35]. In Ref.[35], the term βk of potential (2.15) was compared with the correction term ofthe effective potential (2.13) and bounds were imposed on the coupling constant λ for the baryon (lepton)current Jµ. The range of interest is 1 < dU < 2. The limits were derived on the coupling λ, for k = 2and for k = 3 and we have the relation dU = (k + 1)/2. Since [33] also provide constraints on the Yukawapotential generated by exchange of a scalar or vector bosons, it was also possible the derive a limit for thecase k ≈ 1 → dU ≈ 1. The reason for this is that for small enough boson masses m, the experimentaldistance r is much smaller than the Compton wavelength, ~/mc, and in this limit Yukawa potential can beapproximated by 1/r. The bounds on λ for the rest of non-integer values of k were interpolated linearly, notaffecting the order of magnitude of the bounds obtained. These constraints on λ for Jµ (baryon current) arepresented in Table (2.2). The limits of |βk| shown result from direct calculation, as it was mention above,and linear interpolation based on the values of Table (2.1). The values presented here are proportional toΛ2−2dUU as one can easily check in the expressions for RVG, Eq.(2.14) and Eq.(2.13).

Table 2.2: Laboratory and cosmological constraints (68% confidence level) on λ from the effective potentialdue to vector unparticle exchange [35].

dU |βk| |λ| |λ| (SN)1 1.8× 10−2 3.9× 10−20 3.0× 10−11

1.25 9.1× 10−3 7.5× 10−17 4.1× 10−10

1.5 4.5× 10−4 4.3× 10−14 5.5× 10−9

1.75 2.9× 10−4 8.8× 10−11 7.4× 10−8

2.0 1.3× 10−4 1.5× 10−7 1.0× 10−6

It was also included, in this table, results from supernova constraints on λ (SN), coming from the too-rapidcooling of supernovae induced by unparticles [45, 46]. The constraint on λ follows the relation

λ

(ΛU

30 MeV

)1−dU< 3× 10−11. (2.22)

This relation sets reliable limits because if scale invariance is broken at an intermediate scale between thethermal wavelength of a supernovae, roughly inverse MeV, and the sub millimeter [33], these limits are still

18

applicable. The constraints presented in Table (2.2), are quite stringent on the range of dU . These limits mayalso indicate that scale invariance is only an approximation. We should also notice that forces that couplelinearly to Jµ, which extend over a long range, are even more tightly constrained if they deviate significantlyfrom 1/r since these would affect Newtonian orbits.

GRANIT experimental data (1.26), was used to constrain Λ, with the same meaning as RTG, in thecontext of large extra dimension models (LED) [22]. Usually, LED corrections to the Newtonian potential areparametrized by a exponential function of the distance. But another approach to solve the hierarchy problembased on braneworld models proposed by [47] consists in allowing the extra dimensions to be infinite. Thistype of correction leads to modifications of the Newton’s law that, at large distances, are given by [48]:

V (r) = −GNMm

r

(1 +

kbrb

), r � Λ = b

√kb, (2.23)

where Λ is the typical length scale at which corrections due to the infinite extra dimensions become relevant.Despite the different theoretical backgrounds, constraints on RTG and dU coming from an ungravity effectivepotential can be compared with the parameters in (2.23). A different approach to GRANIT data and to themodified gravitational quantum well was used if we compare this work with Ref. [2]. To proceed with thiscomparison, the following numerical identification has to be made: b = 2dU − 2 and Λ = RTG. Results arepresented below in Table (2.3.

Table 2.3: Upper bounds on Λ as function of b using GRANIT experimental data and a LED effectivepotential [22].

b 1 2 3 4|Λ(m)| < 8.13× 104 4.34 0.076 0.009

Bounds on RTG can also be derived from the Big-Bang nucleosynthesis (BBN) [30]. The limits on RTG arederived from the variation of the gravitational coupling G, at the time of BBN. This can be written, usingthe effective ungavity potential (2.10), as:∣∣∣∣∆GG

∣∣∣∣ ≡ ∣∣∣∣G(r)−GNGN

∣∣∣∣ = (2dU − 1)

(RTGr

)2dU−2

. (2.24)

The effect of a varying G on BBN is set by the expansion rate of the universe, H ≡ dln(a)dt ∝

√G, where a is

the scale factor, that determines the neutron-proton density fraction at freeze-out and the efficiency of 2Heburning at nucleosynthesis. The typical constraints on the variation of G are [49]:

− 0.036 ≤ ∆G

G≤ 0.086, (2.25)

at 95% confidence level. The values YP = 0.250±0.003 and 2He/H = 2.87+0.22−0.021 have been used for the

4He mass fraction and the normalized deuterium number density, respectively. The bounds imposed aredominated by the effect on 4He. The distance r used in the Eq.(2.24) is typical distance between nucleonsthat is approximately 1 m and corresponds to the mean free paths of neutron and proton by the time ofBBN. We depict below the plot with upper bounds on RTG as a function of dU , Fig.(2.3).

Using the same constraints on the variation of the coupling G and the modified Newtonian potential dueto the coupling of vector unparticle with baryonic currents, Eq.(2.13), upper limits on λ as a function of dUwere also derived. We depict the plot below for the allowed region for λ on the parameter space as a functionof dU , Fig.(2.4). It was added to the plot data from Table (2.1).

In at least one work [24], unparticle inspired corrections, for dU . 1 are obtained from astrophysicalconstraints. The regions dU . 1 for RTG > RSun and dU & 1 for RTG < RSun were considered. Limits like

19

Figure 2.3: Upper bounds on RTG as function of du for |∆G/G| ≤ 0.2, 0.15, 0.1, 0.086, 0.036, 0.01 (fromtop to bottom)[30].

Figure 2.4: Allowed region for λ, below the curves, constrained from ∆G/G ≥ −0.036. ΛU = 0.1, 1, 10, 100TeV were used from the bottom to the top curves. Solid curves are from BBN bounds and the dashed fromISL violation data [33], [30].

dU = 0.94, RTG = 6×1011 m are obtained using a modified Lane-Emden equation where the potential (2.10) isinserted. We show below in a plot the bounds obtained from the uncertainty in the Sun’s central temperature|Tr−1| ≤ 0.06, where the curves were drawn from the numerical solution of a modified Lane-Emden equation[24].

The work [24] discussed for the first time the case dU . 1, that is usually regarded as unfeasible. However,that is a misconception, because if one considers the case where dU . 1, the correction to the Newtonian

20

Figure 2.5: Contour plot of Tr − 1 in function of RTG and dU [24].

potential can be expanded in dU , to the first order, leading to corrections of the type:

(r/RT,VG )2−2dU ≈ 1 + 2(dU − 1) ln

(r

RT,VG

). (2.26)

One can clearly see that for small negative values of (2−2dU ), this yields to a logarithmic correction linearlysuppressed by the smallness of (dU − 1). Therefore for small enough negative values of (2 − 2dU ) thiscorrection is theoretically acceptable and should be tested. The work developed in [2] partially answers thislast question, on how large the negative exponent could be.

2.4 Relevant constraints on RTG, RV

G and dU

In the previous section we have been able to present different bounds on RT,VG and dU [22, 30, 32, 33, 35].We had also presented some bounds on M∗ and ΛU for the sake of completeness, in order to give the readera full overview of topic. However these are only marginally relevant for the discussion that we will developin the next chapter.

Since the GRANIT experiment is sensitive to corrections to the Newtonian potential with dU ∼ 1, weare more interested in the results presented above for dU . 1. Hence, the most interesting bound for us isRTG = 6×1011 m for dU ≈ 0.94 [24]. In our work, we will use values of RT,VG that are multiples of the Earth’sradius, RE .

In what follows we shall present the results obtained in Ref. [2], where unparticle inspired modificationsto the gravitational quantum well are confronted with GRANIT experimental data.

21

Chapter 3

Results

3.1 The modified gravitational quantum well

In this section of our work, we consider the linear approximation of potentials, Eqs. (2.10) and (2.13), ascorrections to the Newtonian potential [2]. One assumes unparticle inspired corrections acting on scalesaway from the range the unparticle modifications were originally envisaged, Λ & 1 TeV , as discussed in theprevious section.

In order to maximize the effect of the unparticle inspired modifications one assumes that RT,VG & RE ,where RE is Earth’s radius. These assumptions allow to turn the corrections to the Newtonian potential intocorrections on the gravitational quantum well (GQW) spectrum as measured by the GRANIT experiment[13]. As it will be seen, available data allow for obtaining bounds for parameters RTG, RVG and β = 2du − 2.

One analyzes now the energy spectrum of the unparticle inspired potential

V ′(r) = −GNMm

r

[1±

(RGr

)β]RG > RE , (3.1)

where one assumes that RTG = RVG = RG1. Expanding this potential around RE for r = RE + z and keeping

only the linear term in z, one obtains:

V ′(r = RE + z) = V ′(RE) +

(dV ′(r)

dr

)r=RE

z +O(z2), (3.2)

which is the new potential for the Schrodinger eigenvalue problem H ′ψn(z) = E′nψn(z). However, in thelimit, that this correction is zero at z = 0, one should recover the energy spectrum given by Eq.(1.6), thus:

H ′ = − ~2

2m

d2

dz2+

(dV ′(RE)

dr

)z . (3.3)

Note that the eigenfunctions are not changed by the modified potential that is also linear in z. Thus,rewriting the energy spectrum with respect to the corrections to the potential V = mgz, the unparticleinspired modification amounts to the change:

g → g′ = g

[1± (β + 1)

(RGRE

)β], (3.4)

where the plus sign denotes tensor unparticle exchange, while the minus sign vector unparticle exchange.Therefore, we can write for the energy eigenvalues of the modified GQW as

E′n = −(mg′

θ

)αn. (3.5)

1 Even in the sub-millimeter range we have that RTG ∼ RV

G , see bounds in [33, 35], so the latter identification is clearlyallowed in a first order approximation.

23

Through Eq.(3.5) one can constrain β for fixed values of RG using the available GRANIT data. One intersectsthe values of Eq.(3.5) with the maximum and minimum allowed energies obtained by the classical turningpoint equation, (1.14) using hexp1 ±∆hexp1 , Eq.(1.26), which is the measured bouncing height for first quantumlevel.

3.2 Results

Next one analyzes the effect of the unparticle corrections to the GQW energy spectrum for different valuesof RG according to the choice exhibited in Table (3.1).

For RG > RE , the source of the new interaction is the whole Earth. Thus, using Eq. (3.5), and GRANITdata we depict in Figs. (3.1) and (3.2) the contributions of tensor and vector unparticle exchange, respec-tively. One tests five different values of RG, and vary continuously β within the range that matches theexperimental measurement of the first energy level E1 ±∆E1. The intersection with the experimental errorbars determines the upper bounds for β. The results are summarized in Table (3.1).

For RG < RE , form factors must be introduced to account for the fact that only a shell of radius RGaround the experiment is relevant. From computations that can be found in Ref. [22], the resulting correctionsto the ISL are too small and yield no significant bounds on the parameters RG and β. See bounds in Table(2.3), in the previous section.

Table 3.1: Table of values for β as a function of RG for the first energy level as modified by Eq.(3.5). Valuesof β greater than the ones indicated are excluded [2].

(RG/RE) βV ector Tensor

100 -0.158487 -0.37823650 -0.179961 -0.42518410 -0.260721 -0.5829925 -0.320562 -0.676435

1.5 -0.502710 -0.846465

The results shown in Table (3.1) indicate that (RG/RE) should be smaller then 1 due to the negativevalues of β obtained. However since these bounds are only valid for RG > RE , as they were derived usingEq.(3.1), β must be regarded as a truly negative exponent. As one can easily see β <∼ 0 corresponds todU <∼ 1 which is an excluded region for unparticle models due to non-unitiray pathologies [34] and becauseof the misconception that it will affect largely the long range behavior of the Newtonian potential falling offslower than 1/r leading to an undesirable effect. Nevertheless, as pointed out in [24], this parameter regioncan be allowed and the deviations from 1/r are heavily suppressed by the relation (2.26). Hence, this workfits perfectly the effort of setting bounds on RG and dU for dU <∼ 1.

24

-1.0 -0.8 -0.6 -0.4 -0.2 0.00.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Β

EHpe

VL

Figure 3.1: First energy eigenvalue E′1 for RG/RE = 1.5, 5, 10, 50, 100 (from top to bottom) as a function ofRG and β as modified by tensor unparticle exchange interactions. The grey region corresponds to the exper-imental results E1 ±∆E1 and limits the allowed RG/RE − β parameter space. The dashed line correspondsto the first energy level of the unperturbed gravitational quantum well, Eq.(1.16) [2].

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.00.0

0.5

1.0

1.5

Β

EHpe

VL

Figure 3.2: First energy eigenvalue E′1 for RG/RE = 100, 50, 10, 5, 1.5 (from top to bottom) as a function ofRG and β as modified by vector unparticle exchange interactions. The grey region corresponds to the exper-imental results E1 ±∆E1 and limits the allowed RG/RE − β parameter space. The dashed line correspondsto the first energy level of the unperturbed gravitational quantum well, Eq.(1.16) [2].

25

Chapter 4

Conclusions and Outlook

4.1 Conclusions

In this work, we have considered unparticle inspired modifications to the ISL of the form of Eq. (3.1). Thesecorrections affect the GQW spectrum according to Eqs. (3.4) and (3.5) and, as discussed in the text, theGRANIT experiment is particularly sensitive to β <∼ 0, which allows to scrutinize a region in the parameterspace that is complementary to one tested by torsion-balance experiments [33]. On the other hand, we addan important contribution to the work developed in Ref.[24], setting negative upper bounds on the exponentβ for dU <∼ 1 in a different context.

We find also that depending on the parameters RG and β, corrections for E′1 when compared with E1 canbe around 40% for vector exchange and 10% for tensor exchange. The difference between vector and tensorcorrections is mainly due to the asymmetric error bars for (hexp), as provided by the GRANIT experimentfor RG > RE , with respect to (h) of the unperturbed gravitational quantum well, Eq.(1.15).

Finally, our results show that certain combinations of values of RG and β are already excluded by theavailable data and clearly an improvement on the GRANIT experiment may allow one to exclude entirelycertain ranges of unparticle-like contributions to the GQW. Given that the improvement of the GRANITspectrometer is on the way (see e.g., [19]), prospects to further test unparticles-type corrections to the GQW,as analyzed here, are rather realistic. Furthermore, these bounds were obtained in the context of quantummechanics and for distances h smaller than 50 µm adding a new range of limits for dU and RG to previousdiscussions [24, 30, 32, 35].

27

Bibliography

[1] H. Georgi, Phys. Rev. Lett. 98, 221601 (2007), arXiv: 0703260[hep-ph], Phys. Lett. B 650, 275(2007), arXiv:0704.2457v2 [hep-ph].

[2] A. Alves and O. Bertolami Phys. Rev. D 82, 047501 (2010), arXiv:1007.2785 [gr-qc].

[3] A. Alves and O. Bertolami , GRANIT 2010 - Ecole de Physique des Houches 14-19 February 2010,Les Houches, France, GRANIT and Unparticle inspired corrections to the Gravitational Quantum Well(2010) (to be published)

[4] S. Flugge, Practical Quantum Mechanics, (Berlin, Springer-Verlag 1971), Problem 40.

[5] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory, (Pergamon, London,1965), p. 74.

[6] M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions, (Dover Publ., New York, 1972).

[7] See, e.g. D. J. Griffths, Introduction to Quantum Mechanics (Pearson, Prentice Hall, New Jersey,2005), 2nd edition.

[8] V. V. Nesvizhevsky and K.V. Protasov, Class. Quantum Gravity 21, L4577 (2004), arXiv: hep-ph/0041179.

[9] H. Wallis, J. Dalibard and C. Cohen-Tannoudji, Appl. Phys. B 54, 407 (1992).

[10] O. Bertolami and J. G. Rosa, Phys. Lett. B 663, 111 (2006), arXiv:hep-ph/0508023.

[11] V. I. Luschikov and A. I . Frank, JETP Lett. 28, 559 (1978).

[12] V. V. Nesvizhevsky et al., Nucl. Instrum. Methods Phys. Res. 440 A, 754 (2000).

[13] V. V. Nesvizhevsky et al., Nature (London) 415, 297 (2002); Phys. Rev. D 67, 102002 (2003),arXiv: hep-ph/0306198.

[14] V. V. Nesvizhevsky et al., The European Phys. J. C 40, 479 (2005), arXiv:hep-ph/0502081.

[15] Particle Data Group, Particle Physics Booklet (CERN, Geneva, 2008).

[16] V. V. Nesvizhevsky et al., (2007), arXiv:0708.2541v1 [quant-ph].

[17] V. V. Nesvizhevsky et al., (2009), arXiv:0902.0156v2 [physics.ins-det].

[18] A. E. Meyerovich and V. V. Nesvizhevsky, Phys. Rev. A 73, 063616 (2006), arXiv:quant-ph/0603203v1.

[19] http://lpsc.in2p3.fr/Indico/conferenceDisplay.py?confId=371

[20] O. Bertolami, J.G. Rosa, C. M. L. de Aragao, P. Catorina, and D. Zappala, Phys. Rev. D 72, 025010(2005), arXiv: hep-th/0505064; Mod. Phys. Lett. A 21, 795 (2006), arXiv: hep-th/0509207.

[21] F. Brau and F. Buisseret , Phys. Rev. D 74, 036002 (2006), arXiv: hep-th/ 0605183.

29

[22] F. Buisseret, B. Silvestre-Brac and V. Mathieu, Class. Quantum Gravity 24, 855 (2007), arXiv:hep-ph/0701138.

[23] O. Bertolami and F. M. Nunes, Class. Quantum Gravity 20, L61 (2003), arXiv: hep-ph/0204284.

[24] O. Bertolami, J. Paramos and P. Santos, Phys. Rev. D 80, 022001 (2009), arXiv:0905.1602v2 [astro-ph.SR].

[25] K. Cheng, W.-Y Keung, and T.-C. Yuan, Phys. Rev. Lett. 99, 051803 (2007), arXiv: 0704.2588[hep-ph].

[26] K. Cheng, W.-Y Keung, and T.-C. Yuan, Phys. Rev. D 76, 055003 (2007), arXiv: 0706.3155 [hep-ph].

[27] A. Lenz, Phys. Rev. D 76, 065006 (2007), arXiv:0705.3518 [hep-ph].

[28] N. Greiner, Phys. Lett. B 653, 75 (2007), arXiv: 0707.1535 [hep-ph].

[29] Y. Liao and J. Y. Liu, Phys. Rev. Lett. 99, 191804 (2007), arXiv:0706.1284 [hep-ph].

[30] O. Bertolami and N. M. C. Santos, Phys. Rev. D 79, 127702 (2009), arXiv:0905.1852v2 [hep-ph].

[31] O. Bertolami and J. Paramos, arXiv:0910.3413 [gr-qc]; arXiv:1006.3094 [gr-qc].

[32] H. Goldberg and P. Nath, Phys. Rev. Lett. 100, 031803 (2008), arXiv:0706.3898 [hep-ph].

[33] E. G. Adelberger et al., Phys. Rev. Lett. 98, 131104 (2007), arXiv:hep-ph/0611223.

[34] G. Mack, Commun. Math. Phys. 55, 1 (1977).

[35] N. G. Deshpande, S. D. H. Hsu and J. Jiang, Phys. Lett. B 659, 888 (2008), arXiv:0708.2735 [hep-ph].

[36] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 429, 263 (1998), arXiv:hep-ph/9803315, I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 436, 257(1998), arXiv:hep-ph/9804398.

[37] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999), arXiv:hep-ph/9905221.

[38] D. J. H Chung, L.L. Everett and H. Davoudiasl, Phys. Rev. D 64, 065002 (2001), arXiv:hep-ph/0010103.

[39] G. Dvali, New J. Phys. 8, 326 (2001).

[40] D. J. Kapner et al., Phys. Rev. Lett. 98, 021101 (2007), arXiv:hep-ph/0611184 .

[41] R. Spero et al., Phys. Rev. Lett. 44, 1645 (1980); J. K. Hokins et al., Phys. Rev. D 32, 3084 (1985).

[42] C. D. Hoyle et al., Phys. Rev. D 70, 042004 (2001), arXiv:hep-ph/0405262 .

[43] N. Greiner, Phys. Lett. B 653, 75 (2007), arXiv:0705.3518v2 [hep-ph].

[44] P. Santos, Astrophysical constraints on Ungravity inspired models, Master thesis, IST/UTL (2009)

[45] H. Davoudiasl, Phys. Rev. Lett. 99, 141301 (2007), arXiv:0705.3636.

[46] S. Hannestad, G. Raffelt and Y. Y. Y Wong, Phys. Rev. D 76, 1217011 (2007), arXiv:0708.1404[hep-ph].

[47] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999), arXiv:hep-th/9906064.

[48] K. A. Bronnikov, et al., Gen. Relativity and Gravitation 38, 1215 (2006), arXiv:gr-qc/0601114.

[49] F. Iocco et al., Phys. Rept. 472, 1 (2009), arXiv:0809.0631[astro-ph].

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