PROTON RADIUS PUZZLECorso di Laurea Triennale in Matematica Tesi di Laurea Triennale PROTON RADIUS PUZZLE Relatore: Laureando: Prof. Andrea Vacchi Klest Dedja Correlatore: Dott. …

Universit

`

a degli Studi di Udine

Dipartimento di Matematica

Corso di Laurea Triennale in Matematica

Tesi di Laurea Triennale

PROTON RADIUS PUZZLE

Relatore: Laureando:

Prof. Andrea Vacchi Klest Dedja

Correlatore:

Dott. Emiliano Mocchiutti

Anno Accademico 2015-2016

Contents

1 Introduction 1

2 Atomic Structure 3

2.1 Schrodinger’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.1 Time independent Schrodinger’s equation . . . . . . . . . . . . . . 42.1.2 Quantum numbers ` and m` . . . . . . . . . . . . . . . . . . . . . 5

2.2 Hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Spin number s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Corrections to the Bohr energies of hydrogen . . . . . . . . . . . . . . . . 92.3.1 Fine structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.2 Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Hyperfine splitting in hydrogen . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Corrections to the hyperfine splitting . . . . . . . . . . . . . . . . . . . . . 15

3 The Puzzle 19

3.1 The muon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.1 Muonic hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 The Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.1 Proton radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.2 Charge radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.3 Zemach radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 State of art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.1 Motivation for the measurement of the HFS 1S in µH . . . . . . . 31

4 Rydberg’s constant 33

4.1 R1 and �E in hydrogen atoms . . . . . . . . . . . . . . . . . . . . . . . 344.1.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . 344.1.2 Lamb shift measurements in hydrogen . . . . . . . . . . . . . . . . 37

4.2 Determination of R1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 The FAMU experiment 41

5.1 The experimental proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Muonic atoms formation and thermalization . . . . . . . . . . . . . . . . . 43

iii

5.3 Laser requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.4 Muon transfer to higher-Z gas . . . . . . . . . . . . . . . . . . . . . . . . . 445.5 The RIKEL-RAL facility . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

CONTENTS v

Abstract

The proton is along with the neutron and the electron the main subatomic componentof matter and its existence is well know since a long time. Despite this fact, the realvalue of its charge radius rE is still an unsolved problem in physics: di↵erent techniquesgive di↵erent results even if they are independent from each other. Until 2010 the protoncharge radius was thought to be known to about 1% accuracy, however from that yearnew experimental measurements, instead of boosting accuracy, gave largely di↵erentvalues with a discrepancy of 7� standard deviations from the originally accepted one.The explanation for this big discordance is still unclear: it could either be that some

sources of systematic error have not been considered, or even new physics beyond thestandard model needs to be included. Depending on future results our knowledge aboutRydberg constant might also be changed, as some experimental techniques extract itsvalue using the proton radius.

vi CONTENTS

Fundamental constants

Quantity Symbol Value Dimension

Planck’s constant ~ 1.05457⇥ 10�34 J sSpeed of light c 2.99792⇥ 108 ms�1

Mass of electron me 9.10939⇥ 10�31 kgMass of proton mp 1.67262⇥ 10�27 kgUnit charge e 1.60218⇥ 10�19 CPermittivity of space ✏

0

8.85419⇥ 10�12 C2 J�1m�1

Vacuum permeability µ0

1.25664⇥ 10�6 NA�2

Fine structure constant ↵ = e2/4⇡✏0

~c 7.29735⇥ 10�3

Bohr radius a = ~/↵mec 5.29177⇥ 10�11 mRydberg constant R1 = ↵2mec/4⇡~ 1.09737⇥ 107 m�1

Chapter 1

Introduction

Being the “first” and simpler known atom, the knowledge about hydrogen has progressedin parallel with familiarity of the atomic and nuclear structure.

The idea that matter was made of atoms (“ätomoc”) is very old and dates back toDemocritus but for the first real atomic theory we have to wait until the early 1800s,when Dalton used the concept of atoms to explain why elements always react in ratios ofsmall whole numbers. According to Dalton, atoms were the first constituent of matter,elementary, homogeneous and indestructible. Atoms of a given element should havebeen identical in mass and properties and chemical reactions could be interpreted asrearrangements of atoms. The essence of Dalton’s theory is still valid, but scientificprogress has been adding new details to atomic structure. The discovery of the electronand natural radioactivity for example, led to Thompson’s plum pudding model and theatom lost its homogeneity in favor to a more accurate model including subatomic particleslike electrons.

The rate of new experimental discoveries increased rapidly in the beginning of the 20thcentury, leading to Rutherford’s planetary model first and then to Bohr model. It waswith the latter that for the first time electrons were allowed to orbit the nucleus only ina discrete set of orbits, and could jump between these orbits only through absorption orradiation of a photon. This model describes quite accurately hydrogen atoms, and thefor the first time we see some quantum theory, even if still at an early stage [1]. Moreimprovements involving quantum theory were made in the 20s after the introduction ofthe spin quantum number [2] and the formulation of the Heisenberg uncertainty principle.The Bohr model was eventually replaced by an atomic orbital one, where probabilitiesof observing an electron are given according to the fundamental principles of quantumtheory. Upon these bases, a huge theoretical structure was built with the aim to explainnew physics and improve accuracy with experimental results.

Nowadays, we have a good knowledge of nuclear structure: we know the role played bythe fundamental forces inside the nucleus, we know what are protons or neutrons madeof and moreover, a wide variety of other subatomic particles have been discovered. Evenconcerning electronic orbitals, many contributions to their binding energies have beenstudied in detail: However there is still little confidence in the measurement of what

1

2 CHAPTER 1. INTRODUCTION

seems to be a rather “basic” physical quantity: the proton charge radius. When it comesto measuring the Lamb Shift in muonic hydrogen for example, the experimental valuesturns out to be smaller than theoretical predictions and the extrapolated value of theproton charge radius is smaller than what previously measured with traditional methodslike electron scattering or hydrogen spectroscopy. The discrepancy is very large: 7�, butnobody has been able to explain so far the the reason and many hypotheses are still onthe table: from experimental accidents, to e↵ects of unconsidered contributions in thecalculations or to hints for new physics.The FAMU project (Fisica Atomi MUonici) aims at studying the proton structure

using muonic atoms, but instead of the charge radius, the goal of the experiment is themeasurement of the Zemach radius of the proton which can be extracted performing aprecise measurement of the hyperfine splitting of the ground state in muonic hydrogen.This quantity has been already measured using ordinary hydrogen, and a comparisonwith the value extracted from muonic hydrogen may either reinforce or delimit the ProtonRadius Puzzle: a substantial agreement with the value of r

Z

obtained with ordinaryhydrogen could suggest that the explanation of the puzzle may lie in unconsideredmethodology uncertainties of the electron-based experiments, while a big discrepancywould give good reasons to look for new physics beyond the Standard Model [4].

Chapter 2

Atomic Structure

When studying the proton and its properties, the best starting point certainly is thehydrogen atom and the studying of its only surrounding electron. It is relatively easyto describe and predict its behavior thanks to quantum theory applied to Schrodinger’sequation. It takes four so called quantum numbers to completely describe all possiblequantum states of an electron:

n is the so called principal quantum number;

` : is the azimuthal quantum number;

m` : stands for magnetic quantum number;

s : finally, the spin quantum number.

We will see how these four quantum numbers work, how their value a↵ects the energyof each electronic orbital and which rules do electrons have to obey for a quantum stateto be physically acceptable. We anticipatre them here: .

n positive integer;

` 2 {0, 1, ..., (n� 1)};

m` 2 {�`, (�`+ 1), ..., `};

s 2 {�1/2, 1/2}.

Finally, for practical reasons, we will also use the following quantum numbers. They arenot independent from the 4 previous ones, but will be useful in some treatments:

the total angular momentum quantum number j. It parametrizes the total angularmomentum of a given particle expressed as the sum of its orbital and spin angularmomentum. The relative quantum number is defined as the sum of the azimuthaland the spin quantum numbers: j = s+ `;

the F quantum number relative to the electron-proton system: F = j + sp =`+ se + sp.

3

4 CHAPTER 2. ATOMIC STRUCTURE

Selection rules

In spectral phenomena such as the Zeeman e↵ect it becomes evident that transitionsare not observed between all pairs of energy levels. In fact, in Quantum Electrodynamicscertain electron transitions are more likely to take place than others and there exist a setof rules which tells us which are the likely ones and which are not. We will call “allowed”transition those which have a high probability of taking place whilst those which areare less likely are called “forbidden” transitions. This set of rules exists is known asthe selection rules and the number of split components observed in the Zeeman e↵ect isconsistent with:

�` = ±1

�m` = 0,±1(2.0.1)

These are the selection rules for an electric dipole transition and one can say that theoscillating electric field associated with the transitions resembles an oscillating electricdipole. When this is expressed in quantum terms, photon emission is always accompaniedby a change of 1 in the orbital angular momentum quantum number. The magneticquantum number can change by zero or one unit.

2.1 Schrodinger’s equation

When dealing with atomic scales and physics, quantum physics takes over classicalmechanics. Instead of a particle’s equation of motion x(t),1 we will now look for a wavefunction satisfying given initial conditions. The basis of all this is given by Schrodinger’sequation:

i~@ @t

= � ~22m

@2

@x2+ V (2.1.1)

whose solutions (x, t) describe the behavior of a wave, and whose squared modulus| (x, t)|2 gives the probability of finding a particle at position x and time t. Quantitieslike hxi , hpi and hT i have the same physical significance they used to have in classicalmechanics, with the only di↵erence that in most of the cases they have a statisticalinterpretation rather than giving the predicted value of a single outcome.Hereinafter we shall consider only continuous, di↵erentiable and normalizable solutions,

these are the only ones with some physical meaning.

2.1.1 Time independent Schrodinger’s equation

In many cases the potential V turns out to be independent of t and Schrodinger’sequation becomes easier to solve. In fact, we can now consider only separable solutions (x, t) = (x, t)f(t) and obtain all possible (x, t) as linear combination of functions

1we will use x instead of ~x for better readability, even when studying 3-dimensional cases

2.1. SCHRODINGER’S EQUATION 5

of this type. With some simple algebra, we can rewrite Equation (2.1.1) as an ordinarydi↵erentiable equation:

� ~22m

d2

dx2+ V = E , (2.1.2)

where E = i~@ @t is proven to be constant and therefore solutions are easily found. If we

now call:

H(x, p) =p2

2m+ V (x)

and its correspondent Hamiltonian operator:

H = � ~22m

@2

@x2+ V (x)

we get Equation (2.1.1) written in a very elegant way:

H = E . (2.1.3)

Remark. Equation (2.1.3) suggests us that hHi = E and that any measurement of thetotal energy is certain to give the value E.

The generalization to three dimensions is straightforward, Schrodinger’s equation nowsays:

i~@ @t

= H (2.1.4)

where Hamiltonian operator H in three dimensions is now linked to the Laplacianoperator2 and gives:

i~@ @t

= � ~22m

r2 + V . (2.1.5)

Again stationary state solutions are of the form:

(x, t) = (x)e�iEt/~.

2.1.2 Quantum numbers ` and m`

In order to study the hydrogen atom, we wish to turn to spherical coordinates (r,�, ✓)and take advantage of the spherical symmetry of the Coulombian potential generated bya proton-made nucleus to electrons orbiting around it.One of the first things we need to calculate in these new coordinates is the the Laplacian

and it has the following form:

r2 =1

r2@

@r

✓r2@

@r

◆+

1

r2 sin ✓

@

@✓

✓sin ✓

@

@✓

◆+

1

r2 sin2 ✓

✓@2

@�2

◆. (2.1.6)

The 3-D version of the time-independent Equation (2.1.2) now reads:

2given a C

2 real-valued function in R3, the Laplacian operator is r2 = @

2

@x

2 + @

2

@y

2 + @

2

@z

2


� ~22m

"1

r2@

@r

✓r2@

@r

◆+

1

r2 sin ✓

@

@✓

✓sin ✓

@

@✓

◆+

1

r2 sin2 ✓

✓@2

@�2

◆#+V = E , (2.1.7)

where has now the form (r,�, ✓). As usual, we shall consider only separable solutionsof the form (r,�, ✓) = R(r) Y (�, ✓). Starting from Equation (2.1.7) we get the followingresult:

1

R

d

dr

✓r2

dR

dr

◆� 2mr2

~2 [V (r)� E] = `(`+ 1) (2.1.8)

Thanks to the same reasoning we did in Subsection 2.1.1, we obtain a new separationconstant. We will write it in the form `(` + 1) since ` turns out to be an integer andplays a key role as a quantum number.The last result is already interesting, but further separation is needed for a complete

analysis, we shall therefore consider Y (✓,�) on the form ⇥(✓)�(�). Equation (2.1.8)becomes now:

1

⇥

sin ✓

d

d✓

✓sin ✓

d⇥

d✓

◆�+ `(`+ 1) sin2 ✓ = m2

` = � 1

�

d2�

d�2. (2.1.9)

Again a new constant m2

` is found, and the latter equality tells us that �(�) = eim`

�.Furthermore, because of spherical symmetry we also know that �(�+2⇡) = �(�), givingus e2⇡im` = 1 and therefore m` being an integer. Before we continue we need to introducesome functions which are well known among physicists and mathematicians:

Definition (Legendre polynomial). Legendre polynomials can be defined as the solutionsof the following di↵erential equation:

d

dx

(1� x2)

d

dxPk(x)

�+ k(k + 1)Pk(x) = 0.

Remark. By using multiple di↵erentiation, we can see that the previous condition istantamount to the so called Rodrigues formula:

Pk(x) =1

2kk!

dk

dxk

h(x2 � 1)k

i.

Therefore these functions are polynomials, and an immediate generalization comes withthe following:

Definition (Legendre function). Given integers k and b, we define the associated Legendrefunction as:

P bk(x) = (1� x2)|b|/2

✓d

dx

◆|b|Pk(x).

2.2. HYDROGEN ATOM 7

Please note that in order for the latter definition to make sense, we need n and bto be both integers and |b| k. Focusing now our e↵orts on the left hand side ofequation (2.1.9) we notice that solutions are just the ones on the form:

⇥(✓) = A Pm`

` (cos ✓), (2.1.10)

where Pm`

` is the associated Legendre function with ` and m` as parameters of the abovedefinition. In conclusion ` and m` are the quantum numbers who play the main rolein this equation, and the last remark tells us an important relation between these twonumbers:

m` 2 {±`,±(`� 1), ...,±1, 0}.

2.2 Hydrogen atom

Let’s now focus our analysis on the hydrogen atom. It’s structure makes all treatmentmuch simpler due to many reasons:

hydrogen atoms are essentially a proton with an orbiting electron with mass me.We are expected to solve an easy two-body problem, which is not a di�cult task.Having more electrons would lead to a solving a many-body problem which isnowadays far from being solved in its more general case.

At early stages the hydrogen atom system can be further simplified assuming theproton motionless. We will see how this assumption a↵ects the results but drop itas soon as things become trickier, in Subsection 2.3.1 for example.

Not only the electron motion is a↵ected by other electrons, but also its energy levelsare sensible to the presence of other electrons by the so called electron screeninge↵ect. Unfortunately it is too big to be neglected [6] so when dealing with multipleelectrons it should be included at least as part of a perturbation theory.

All we need now is to find and expression for the potential V . Since the proton ismuch heavier than the electron, we will assume it is motionless and we will neglect the“reduced mass e↵ect” on the system by now. Its potential energy can be deducted byCoulomb’s law:

V (r) = � e2

4⇡✏0

r. (2.2.1)

Putting this potential in Equation (2.1.8) and using u(r) := rR(r) we get:

� ~22me

d2u

dr2+

� e2

4⇡✏0

r+

~2`(`+ 1)

2mer2

�u = Eu. (2.2.2)

Finding u(r) and all possible E-s will lead us to the right track in analyzing the hydrogenspectrum. In particular, we want to be able to find the corresponding E-s given a


potential V . Eventually as the potential takes into account more and more e↵ects andcorrections, energy levels will be a↵ected and their theoretical value will be more andmore in agreement with experimental results. Starting form our “basic” potential first,we make the following substitutions:

⇢ =rp�2meE

~ and ⇢0

=mee2

2⇡✏0

~p�2meE

, (2.2.3)

and the equation becomes:

d2u

d⇢2=

1� ⇢

0

⇢+`(`+ 1)

⇢2

�u. (2.2.4)

The asymptotic behavior of the u(⇢) function suggests using v(⇢) = u(⇢)⇢�`�1e⇢ instead.With the assumption that v(⇢) can be expressed as a power series in ⇢ we can find itscoe�cients aj using Equation (2.2.4) with the newly built v(⇢), finding the expressions

for dvd⇢ and d2v

d⇢2 and finally equating coe�cients of like powers yields the recursive formulafor the aj-s:

aj+1

=2(j + `+ 1)� ⇢

0

(j + 1)(j + 2l + 2)aj . (2.2.5)

We have a0

= 3A, and for large j we have aj ⇡ 2

j

j! . Then

v(⇢) = A1X

j=0

2j

j!⇢j

blows up as an exponential and can’t be normalizable. The only possibility left is thatthe series terminates: a maximal jmax 6= 0 must therefore exist and all the following akmust be equal to 0. In formulas, see Equation (2.2.5), we have 2(jmax + `+ 1)� ⇢

0

= 0and setting

n := jmax + `+ 1, (2.2.6)

we get ⇢0

= 2n. This n is what we’ve called the principal quantum number and it is themost important number in quantum mechanics as it determines the allowed energies Ein Equation (2.2.2). To be more precise, n is the conserved quantity corresponding to

the energy of a quantum system with V (r) = � e2

4⇡✏0r. Together with ⇢

0

, we find indeedone single energy En for each n > 0 thanks to formulas in (2.2.3), in particular the En-sturn out to be:

En = �"me

2~2

✓e2

4⇡✏0

◆2

#1

n2

=E

1

n2

with n = 1, 2, 3, ... (2.2.7)

and the famous value E1

= �13.61 eV is recovered.

3We find the value of the constant A via normalization

2.3. CORRECTIONS TO THE BOHR ENERGIES OF HYDROGEN 9

2.2.1 Spin number s

In addition to orbital angular momentum associated with the motion of the electronaround the nucleus, the electron also carries another form of angular momentum whichhas nothing to do with motion in space, we will say that it carries an intrinsic angularmomentum S called spin angular momentum, in addition to the extrinsic L. Thefundamental relation between s quantum number and S is by definition:

S = ~ps(s+ 1). (2.2.8)

The algebraic theory of spin is the same one used in studying the orbital angularmomentum L and what we get are these fundamental commutation relations:

[Sx, Sy] = i~Sz, [Sy, Sz] = i~Sx, [Sz, Sx] = i~Sy. (2.2.9)

A big di↵erence though, is that unlike L, S eigenvectors can also be half integers. Asa consequence physicists often group elementary particles in fermions with half-integerspins (usually s = 1/2 or 3/2), and bosons with integer spin (mostly s = 0, 1 or 2).

2.3 Corrections to the Bohr energies of hydrogen

In Section 2.2 we took the Hamiltonian to be H = K + V that is:

H = � ~22me

r2 � e2

4⇡✏0

r, (2.3.1)

but improvements are needed. There are indeed many other e↵ects a↵ecting the Hamil-tonian, leading to a variety of di↵erent and even new energy levels and to a consequentsplitting of the so far known spectral lines. As already mentioned, taking into accountthe reduced mass µ instead of m will improve the predicted values by a bit, but there arestill many corrections that can be made to fit theory predictions with experimental data.An important constant 4 at these levels is the fine structure constant :

↵ =e2

4⇡✏0

~c ⇡ 1

137.036.

A part from his important physical meaning and its key role in quantum electrodynamics,↵ turns out to be a useful indicator of the order of magnitude of the corrections to theenergy levels. We will see how all these can be rewritten so that we can appreciate theirbeing proportional to powers of ↵.

2.3.1 Fine structure

The first correction to the Bohr energy levels is represented by Fine Structure: thiskind of structure it is smaller by a factor ↵2 and it can be split in two di↵erent contribu-tions: relativistic correction and spin-orbit coupling e↵ect.

4The value of ↵ varies with energy scale, so by definition we consider the scale of the electron mass,lowest bound for this kind of interaction.


Relativistic correction

The correction a↵ects the kinetic term of the Hamiltonian, in particular we have toredefine the kinetic energy operator5 and momentum as

K =mec2p

1� (v/c)2�mec

2, p =mevp

1� (v/c)2.

Expanding in power series K as a function of p, c and me we get:

K = mec2

2

66641

2

✓p

mec2

◆2

| {z }classical

�1

8

✓p

mec

◆4

+ ...

3

7775(2.3.2)

and the under-brace in the last equation clearly shows that the first relativistic contribution

H 0rel

to the Hamiltonian is H 0rel

= � p4

8m3e

c2 and the correction to En must be:

E1

rel

=⌦H 0

rel

↵= � 1

8m3

ec2

⌦ |p4

↵= � 1

8m3

ec2

⌦p2 |p2

↵(2.3.3)

Which gives, after tedious calculation, the following:

E1

rel

= � (En)2

2mec2

4n

`+ 1/2� 3

�(2.3.4)

Notice that the relativistic correction is a small factor when considering orbiting electrons,being about 2⇥ 105 < ↵�2 times smaller than the energy levels.

Spin-orbit coupling

Since electrons and protons are charged particles moving with respect one to the other,a magnetic field B exerting a torque µe ·B on the “spinning” electron is generated andthe Hamiltonian has to take account of this. To calculate the value of B we will betterconsider the electron’s frame and calculate the resulting magnetic field generated by thespinning proton. Biot-Savart law is su�cient and it gives: B = µ0I

2r . Putting e/T as thee↵ective current and the classical formula L = 2⇡mer2/T leads to:

B =1

4⇡✏0

e

mec2r3L (2.3.5)

due the fact that B and L are parallel vectors. In addition, we have to remember that therelativistic electron’s magnetic moment is µe = � e

me

S and so Equation (2.3.5) becomes 6

H 0soc

=e2

8⇡✏0

1

m2

ec2r3

S · L. (2.3.6)

5for better readability, we have omitted hats over v and some p operators6we have a rescaling to a factor two, known as the Thomas precession

2.3. CORRECTIONS TO THE BOHR ENERGIES OF HYDROGEN 11

Remark. H 0soc

commutes with L2, S2 and with their sum J := S+ L, which we havealready defined as total angular momentum in page 3.

We can conclude, after some calculations involving L · S eigenvalues, that spin-orbitcoupling gives a correction equal to:

E1

soc

=(En)2

mec2

⇢n [j(j + 1)� `(`+ 1)� 3/4]

`(`+ 1/2)(`+ 1)

�. (2.3.7)

Now, summing up the two contributions from Equations (2.3.4) and (2.3.7) and bearingin mind that in this case j = `± 1/2 we find that total contribution due to fine structureis:

E1

fs

=(En)2

2mec2

✓3� 4n

j + 1/2

◆(2.3.8)

and adding this to the Bohr formula (2.2.7) we obtain the grand result for the energylevels of hydrogen including fine structure:

En,j =E

1

n2

1 +

↵2

n2

✓n

j + 1/2� 3

4

◆�. (2.3.9)

This modification of the energy levels of a hydrogen atom due to a combination ofrelativity and spin-orbit coupling is known as fine structure. Let us examine the e↵ect ofthe fine structure energy-shift on these eigenstates for n = 1 and 2:

For n = 1, in the absence of fine structure, there are two degenerate 1S1/2 states.

According to Equation (2.3.8), the fine structure induced energy-shifts of these twostates are the same. Hence fine structure does not break the degeneracy of the two1S

1/2 states of hydrogen.

For n = 2, in the absence of fine structure, there are two 2S1/2 states, two 2P

1/2

states, and four 2P3/2 states, all of which are degenerate. According to (2.3.8), the

fine structure induced energy-shifts of the 2S1/2 and 2P

1/2 states are the same asone another, but are di↵erent from the induced energy-shift of the 2P

3/2 states.Hence, fine structure does not break the degeneracy of the 2S

1/2 and 2P1/2 states

of hydrogen, but does it for the 2P3/2 states.

Figure 2.1: Fine splitting structure between the 2S and 2P orbitals in hydrogen atom.


2.3.2 Lamb Shift

According to the hydrogen Schrodinger Equation (2.2.7) the energy levels of the electronshould depend only on the principal quantum number n. In 1947 though, Willis Lambdiscovered that this was not the case: the 2P

1/2 state is slightly lower than the 2S1/2 state

resulting in a slight shift of the corresponding spectral line. Because of this discovery,this energy shift is named after the famous physicist: the Lamb shift.This shift is associated to the quantization of the Coulomb field and the explanation

of this e↵ect is one big success of Quantum Electrodynamics. At the heart of theprocess there is the exchange force between charges, which interact by the exchangeof photons. There can therefore be a self-interaction of the electron by exchange of aphoton, this “smears out” the electron position over a range of about 0.1 fm and causesa slight weakening of the force on the electron when it is very close to the nucleus. Asa consequence a 2S electron, whose wave function has non-vanishing probability in theorigin, must have a slightly higher in energy than a 2P

1/2 electron.Some heuristic considerations given by [7] give us a value for the di↵erence of the

potential energy of:

h�V i = 4e2

12⇡✏0

e2

4⇡✏0

~c

✓~

mec

◆2 1

8⇡a3ln

4✏0

~ce2

= �↵5mec2 ln

✓~⇡↵mec

◆/6⇡. (2.3.10)

It might seem that such a tiny e↵ect would be insignificant, but in this case that shiftprobed the depths of our understanding of electromagnetic theory. Many measurementshave been performed so far including several techniques, both direct and indirect, and anoverview is given by the following schematic representation:

Figure 2.2: Some recent measurements of the 2S1/2 � 2P

1/2 Lamb shift.

We will study in better detail this important phenomenon, and see how it can help tosolve some questions about sub-atomic physics such as the Proton Radius Puzzle.

2.4 Hyperfine splitting in hydrogen

The idea of the hyperfine structure is that the proton, in addition to the the magneticfield due to its moving in the electron’s frame, also has its own magnetic dipole moment.The hyperfine corrections may be very di�cult to measure in transitions between states

2.4. HYPERFINE SPLITTING IN HYDROGEN 13

of di↵erent n however their contribution becomes significant when dealing with groundstate of the hydrogen atom. We will end up with a perturbation proportional to thedot product of these two spins hSp · Sei. What we are looking for is a correction to theHamiltonian operator, and this is given by:

H 0hf

= �µ ·B, (2.4.1)

where µ is the magnetic moment of the dipole. The magnetic dipoles of proton andelectron are proportional to the respective spins, in particular we agree with the followingresults:

µp =�pe

2mpSp, µe =

��ee2me

Se (2.4.2)

where the �’s are the gyromagnetic 7 ratios. Now, calling r the distance between theelectron and the proton and r its unit vector, we have thanks to some knowledge inelectrodynamics [8] that a dipole µ sets up a magnetic field B according to the formula:

B =µ0

4⇡r3[3(µ · r)r � µ] +

2µ0

3µ�3(r) (2.4.3)

This represents the energy of interaction of the two dipoles, and we can notice that it ismade of two terms. The first one represents the usual field associated with a magneticdipole, while the existence of the second term is explained taking into account the currentloop that goes to zero: the field will be infinite at the origin and this contribution is whatis reflected. In this second term we have to assume, by now, that the proton is point-like.Now, in first order perturbation theory [9], the change in energy of a quantum state is

given by the expectation value of the perturbing Hamiltonian E1

n = h 0

n|H 0| 0

ni. In ourcase, we have to consider Equation (2.4.3) with µ being the electron’s magnetic dipoleµe, so that we can calculate the Hamiltonian:

H 0hf

=µ0

�ee2

8⇡mpme

[3(Sp · r)(Se · r)� Sp · Se]

r3+

µ0

�ee2

3mpmeSp · Se�

3(r). (2.4.4)

Here, the first term gives the energy of the nuclear dipole in the field due to the electronicorbital angular momentum, while the second gives the energy of the ”finite distance”interaction of the nuclear dipole with the field due to the electron spin magnetic moments.We are here assuming that both the proton and the electron are pointlikeNow, for states with ` = 0 such as the ground state, the wave function is spherically

symmetric:

100

(r, ✓,�) =1p⇡a3

e�r/a|si, (2.4.5)

where a = 4⇡✏0

~2/mee2 is the so called Bohr radius. Such symmetric configurationmake the first term disappear when calculating the expectation value of the perturbing

7the g-factor of a proton is di↵erent from the electrons one because of its composite structure, thevalue of the first being 5.59 instead of 2.00


Hamiltonian E0hf

=R ⇤0

H 0

d(r),. The second term is all what is left, so the expectationvalue in particular we obtain:

E0hf

=µ0

�ee2

3mpmehSp · Sei| (0)|2, (2.4.6)

and by substituting Equation (2.4.5) we get the final result:

E0hf

=µ0

�ee2

3⇡mpmea3hSp · Sei. (2.4.7)

Remark. The hSp · Sei term makes us call this kind of coupling a spin-spin one, incontrast with spin-orbit coupling which involved vectors S and L.

The last thing to do now is to evaluate hSp · Sei. We shall recall the relation given byEquation (2.2.8) and equality hSp ·Sei = 1/2

�S

2 � S

2e � S

2p

�, where S := Sp+Se. We

finally get hSp · Sei = 1

4

~2 or �3

4

~2 depending on the spins being either parallel (triplet)or anti-parallel (singlet).This final assessment breaks the spin degeneracy of the ground state, lifting the energy

level for the triplet configuration and depressing the singlet one. The energy gap betweenthese configurations [6] therefore is:

�Ehfs = E0hfs

=µ0

�ee2

3⇡mpmea3(1/4� (�3/4))~2 = 5.88⇥ 10�6eV. (2.4.8)

There are numerous studies on the hyperfine splitting in hydrogen nowadays, and aremotivated by the level of accuracy achieved in recent atomic physics experiments, asthey are able to give model-independent information about nuclear structure with anotherwise unattainable precision, as we will better see in Section 3.2. In addition tothis, we will also see that some kind of upcoming results may also call into question thevalue of physical constants such as the Rydberg’s R1 (Chapter 4). We will thereforededicate more words to this e↵ect on this topic in the following pages. To begin with,it is convenient for us to rewrite expression (2.4.8) as a function of the fine structureconstant ↵ rather than Bohr radius a. We get:

�Ehfs =4�e↵4m2

ec2

3mp(2.4.9)

And to sum up all contributions, Bohr energy levels will be listed in the following table.Orders of magnitude are be included only:

Bohr Energies of order ↵2mec2

Fine Structure of order ↵4mec2

Lamb Shift of order ↵5mec2

Hyperfine splitting of order ↵4m2

ec2/mp

Table 2.1: Hierarchy of corrections to the Bohr energies of hydrogen

2.5. CORRECTIONS TO THE HYPERFINE SPLITTING 15

The resulting energy levels are depicted in the following figure. We can appreciate howeach contribution di↵ers depending also on the a↵ected orbital: some of them are eitherpositive or negative, other increase or decrease depending on some quantum numbers. Inaddition, there are no more degenerate states.

Figure 2.3: Complete structure for the the lowest orbitals in hydrogen atom, not to scale.All degeneracy is now broken.

2.5 Corrections to the hyperfine splitting

The importance of Equation (2.4.9) is huge and will be widely used through this work.Hereinafter we will refer to it as the Fermi Formula, and we will rewrite in a more suitablesystem of units:

�Ehfs =8↵4

3

m2

pm2

l c2

(ml +mp)3

. (2.5.1)

Please note that in this reformulation we are also using ml instead of me as we are nowreferring to leptons in general (electrons and muons)8 rather than electrons only. Allprevious results can in fact be applied to leptons simply by replacing the correspondingmass to the ex-me term. We will be convinced by the genuineness of this statement inthe next chapter.

8We could apply this formalism to tauons as well, but they are too unstable for our purposes


Remark. The wavelength � = 2⇡~c�E of the corresponding emitted photon is 21 cm, falls in

the microwave region and is very well known among astronomers. A positive concurrenceis the fact that spectroscopy has reached a high level of precision at these frequencies and,as continuous progress is being made, we have the possibility to push our limits furtherand further in experimental research. It is thanks to nowadays attained precision that anydiscrepancy between theoretical predictions and experimental data can be immediatelyspotted.

Factors we have considered so far are enough to get a correct result within one percent,so before comparing the most accurate experimental measurements we shall add morecontributions to the hyperfine splitting e↵ect. Only then we are are ready to checkwhether there is good matching with the experimental values. In particular, we will takeinto account the following:

Quantum Electrodynamics correction: we will refer to them as �QED

This term includes higher order quantum-electrodynamical e↵ects and it is knowwith the precision of 1 ppm [10]. Its influence is the greatest among all correctionand it a↵ect E0

hfs

by a O(↵) ⇡ 10�2 factor.

Proton form factor: �FF

Is due to the fact that nuclei can’t be assumed as point-like, so we must takeinto account proton’s internal structure. In particular we cannot assume theproton point-like anymore so the interaction between the proton and the electronmagnetic dipole doesn’t involve | (0)|2 like in (2.4.6). Indeed the proton has afinite size, and the nuclear magnetic moment is distributed over a finite regionaccording to a magnetic moment density distribution ⇢M (r). This e↵ect can betaken into account with the obvious substitution | (0)|2 !

Rr⇢M (r)| (r)|2d3 in

Equation (2.4.6). After some calculations, Zemach and Brodsky showed that thecorrection to hyperfine splitting due to the proton size depends on the behaviour ofthe bound-state wave function close to the origin. This term considers proton’scharge and magnetic moment distribution and the correction and adds up to a10�4 correction as given by the the approximated formula [10]:

�FF = �2 [mlmp ↵ c/~(ml +mp)] hr2Zi

As an anticipation of what we are going to see in in Section 3.2, RZ is the integral ofa combination of charge and magnetic moment distributions of the proton. Simplesubstitution gives us the previous result in the case of an orbiting electron.

Proton polarizability: �POL

It’s the most di�cult term to calculate with precision, but luckily it is small. Itincorporates the corrections due to the fact that the proton charge and magneticmoment distributions are polarized by the orbiting lepton. It can be shown [10]

2.5. CORRECTIONS TO THE HYPERFINE SPLITTING 17

that in first approximation �POL is proportional to the orbiting particle mass ml

and an upper bound of the correction is given by

|�POL| 4⇥ 10�6ml/me

There are other contributions such as �WK (due to weak interaction), �DIR (relativisticcorrections) and �V P (hadronic and muonic polarization) but are either too small oralready known with su�cient accuracy that pose no issue to the proton radius puzzle.Equation (2.5.1) with the significant contributions becomes now

�Ehfs =8↵4

3

m2

pm2

l c2

(ml +mp)3

�1 + �QED + �FF + �POL

�(2.5.2)

and a summary of these is given here:

e

� Hydrogen:Order of magnitude 9 relative known error

�Ehfs 1 10�7

�QED O(↵) 10�6

�FF 10�4 2 %�POL < 10�6 100%


The latter formula is an important milestone as it includes all bigger contributions to theatomic energy levels. Other �-s such as the �HV P (hadron vacuum polarization) thoughwidely cited in literature, are too small to a↵ect results at this level of precision. We havenow a su�ciently complete theoretical building which can be tested with experimentaldata. Many surprises are waiting behind the corner if we just focus on muonic atoms butin order to do this, we need to know something more about these muons.

9With respect to the main term �E

hfs


Chapter 3

The Puzzle

3.1 The muon

The muon is an elementary particle similar to the electron with the same electric chargeand spin but with a much greater mass. It is part of the lepton family and interacts withmatter in the same way than electron do, with mass being the only tangible di↵erence.As is the case with other leptons, the muon is not believed to have any sub-structure.It’s not a stable particle having a mean lifetime of 2.2 s but this value is long enoughfor experimental purposes 1.

1Muons traveling at relativistic speed can cover big distances before they decay, and are thereforeeasy to detect

19

20 CHAPTER 3. THE PUZZLE

The following table gives a short insight, with CODATA [5] values

Electron Muon

Mass 0.511MeV/c2 105.7MeV/c2

Generation First SecondElectric charge �e �eMean lifetime stable 2.2 µs

Interacting forcesGravity, Weak Gravity, WeakElectromagnetic Electromagnetic

Table 3.1: A comparison between muon and electron

3.1.1 Muonic hydrogen

At atomic levels the most relevant force is the electromagnetic one, we therefore expectthat muons can orbit around nuclei just like electrons do thanks to electromagneticattraction. This is what happens in the so called muonic atoms, where not only electrons,but also muons can be found orbiting around the nucleus. We will be interested inmuonic hydrogen, a two-body system made of a proton and an orbiting muon. Theseexotic atoms really can exist in nature but are not stable because of the muon, whichcan disappear either because of muonic decay or by nuclear capture.Another peculiar aspect of muonic hydrogen (briefly called µp) is the fact that muons

have a mass about 207 times bigger than electrons so they need bigger attractivecoulombian forces to stay in orbit. As a result, they also orbit about 207 times closerthan an electron would do in the same quantum state. A schematic view is given by thefollowing figure:

Figure 3.1: Comparison, not to scale, between classic hydrogen (e�p) and muonichydrogen (µp)

A part from this peculiar aspects, muonic hydrogen behaves like classic hydrogen, andbecause easy to produce and detect in laboratories and Nevertheless muonic hydrogenhas many experimental advantages

3.1. THE MUON 21

Just like we did in Section 2.2 we will treat the muonic hydrogen case only. We willrefer to it briefly as µp as opposed to the classical hydrogen atom e�p.

Muonic hydrogen hyperfine structure

Since the muon appears to behave just like a heavier electron, the Fermi Formula asin (2.5.2) has no reason not to be valid even when considering muons instead of electrons,given that we properly substitute muonic mass mµ to the me and recalculate reducedmasses. What happens then, for example, is that all energy levels are shifted to di↵erentorders of magnitude mixing up the balance of power from the electron case.We are interested in the hyperfine splitting contributions, and we can see that di↵erent

contributions � are rescaled to di↵erent factors, and some of those who could be considerednegligible become now significant. Let’s see in detail how:

�E

hfs, the overall term, is boosted by a (mµ/me)2 factor. In particular the energy

gap jumps from a mere 5.88⇥ 10�6eV to 0.183eV.

�

QED remains a O(↵) correction. Quantum-electrodynamics corrections dependindeed on the ↵ constant only.

�

FF was observed to be linearly depend on the mass. When dealing with muons,we therefore obtain a rescaling factor of mµ/me

�

POL similarly, boosted by a mµ/me factor.

Again a table will help us to visualize the situation in the case of a muonic atom:

µ

� Hydrogen:Order of magnitude relative known error

�Ehfs 10�7

�QED O(↵) 10�6

�FF 10�2 2 %�POL < 10�3 100%


We can see how in the muonic case corrections are relatively big and measurementsin muonic atoms can detect if something doesn’t behave “as expected”. After takinginto account all possible corrections (as we are doing in our treatment) we can saythat su�ciently precise measurements of the �Ehfs in classic hydrogen and in µp arecomplementary to each other. We therefore expect that the extrapolated value of anyphysical quantity related to energy gaps between orbitals, no matter if electronic ormuonic, should give the same result.In reality this doesn’t happen, as in recent years di↵erent techniques have given

incompatible values for one physical quantity in particular: the proton’s charge radius.The repeated failure to match values from di↵erent experiments using electrons or muonshas given raise to what scientists now call the Proton Radius Puzzle.


3.2 The Puzzle

The Proton Radius Puzzle is a prime problem in proton structure physics and itrose with the necessity of measuring its radius with bigger accuracy so that QuantumElectrodynamic could be tested. Starting points for many QED tests are indeed high-precision measurement of hydrogen splitting lines and the most considered transitions atpresent are the Lamb Shift and the Hyperfine splitting. Hyperfine splitting in particular,is one of the most accurately measured quantities, with a relative experimental uncertaintyless than 10�12. This quantity is tiny at the point that theoretical predictions based onQED, although a↵ecting only a small fraction of �E, can be directly verified. There arethough, many other reasons behind the uncertainties about theoretical models: firstlythere are computational di�culties with higher order terms. Secondly, we still have alimiting precision of fundamental constants such as R1, me/mp and ↵ .But how does the Proton Radius Puzzle raise? To state the problem is simple: if

we measure the proton charge radius using electrons and then we measure it usingmuons, we get incompatibly di↵erent results without finding any possible explanation.Di↵erent techniques for each of these two branches have already been tested, in particular,measurements made using electrons come either from electron-proton scattering orfrom atomic spectroscopy. The proton charge radius obtained from di↵erent electronicmeasurements substantially agree, and give a result with an uncertainly of order 0.6%. Onthe muonic side there is only one set of measurements, and the proton size is derived fromthe muonic hydrogen 2S�2P line, the Lamb shift. Other experiments are being developedin recent years thanks to the increasing curiosity among the scientific community aboutthis Puzzle. The muon-based experiments should give more accurate results thanks tomuon’s larger mass: orbiting much closer to the nucleus we have bigger contributionsto the energy levels due to protons’ structure. The result extracted from the muonic2S � 2P experiment though, besides being a tenfold more precise, is an astonishing 4%smaller than normal hydrogen. This, translated to statistical relevance gives a huge 7�deviation from the previously accepted results.We will begin with a brief description of the electron based experiments and followed

by some remarks about how the muonic Lamb measurement which has been carried outto give the charge radius. Afterwards we will cite some of the theory calculations used tointerpret the experiments. Much work has been done checking the corrections involvedin pulling out the charge radius from the muonic experiment, but no correction is able tojustify the result that has been found. The importance of the problem has led to a lotof experimental activity, and we will describe some of them and o↵er some concludingremarks.

3.2.1 Proton radius

Defining a radius for a proton is not as simple as expected. Rather than just being amicroscopical “billiard ball”, the proton does not have definite boundaries and shouldrather be interpreted as dense cloud with its own charge and magnetic moment. Howeverwe can still make a model of the proton as being a sphere of positive charge for the

3.2. THE PUZZLE 23

interpretation of electron scattering experiments: because there is no definite boundaryto the nucleus, the electrons will “see” a range of cross-sections, but by taking the meanwe obtain a single value for the radius. The qualification of “rms” for root mean squareradius arises because the nuclear cross section determining for electron scattering isproportional to the square modulus of the radius.We will use two di↵erent definition of proton radius. The first one is the so called

charge radius rE and is the nearest to the intuitive definition of the radius. The second isthe Zemach radius rZ and involves both charge and magnetic distribution of the proton.

Remark. The rZ and rE are two di↵erent quantities and they describe di↵erent propertiesof the proton’s structure, rE involving electric charge whilst rZ includes magnetic momentdistribution as well. Nevertheless their values are related: if anomalies are detected whenmeasuring one of the radii, we expect to spot them when measuring the other quantityas well.

3.2.2 Charge radius

Proton’s root mean square charge radius is defined as

r2E =

Zr2⇢E(r) d

3r (3.2.1)

where ⇢E(r) is the normalized charge density of the proton. Most of published data havebeen analyzed by means of a two- or three-parameter Fermi charge distribution of theform:

⇢E(r) = ⇢0

⇢1 + exp

4 ln 3

r � c

t

��1

. (3.2.2)

Here we are using r instead of r by assuming spherical symmetry, while the role fornormalization is played by ⇢

0

: it is supposed to be such thatR⇢E(r)4⇡r2dr = e.

The numerous experiments during the last decades have given the opportunity to benefitfrom many independent results, but not all of them are compatible. Measurements ofthe proton radius using electrons, either by electron-proton scattering or by carefulmeasurements of atomic energy levels do not originate the puzzle buy itself as they arein agreement, but when muon based measurement appeared, they showed incompatiblevalues for rE and the problem arose. Nowadays the state of art as it concerns rEmeasurements in e�p is the following:

Electron Scattering

This kind of technique is a fundamental tool to investigate the properties of an object.Indeed in his famous experiment Rutherford studied the ↵ particles scattered by goldnuclei and proved that the positive charge in an atom was concentrated in a much smallervolume than the atom itself and was even able to put an upper limit on the radius of thegold nucleus of 34 fm which also gives the correct order of magnitude.


In the proton’s case though, a bigger precision is needed and ↵ particles can’t be usedas they also interact via strong force with protons. On the other hand, electrons are muchsuited for this kind of experiment as they are point-like, immune to strong and weakforce influence, and the scattering phenomenon can be considered elastic. We only haveto be aware that we need a spatial resolution of ⇡ 1 fm which corresponds to energies ofthe order of 200MeV. At these ranges electrons have a relativistic behavior so more careis needed when implementing theoretical analysis. The relativistic expression of the crosssection for elastic scattering o↵ a static target is due to Mott [12]

d�

d⌦

Mott

=

✓Z↵

2E

◆2 (1� � sin2 ✓/2)

sin4 ✓/2, (3.2.3)

where � stands for the incident electron’s velocity v divided by c. After that, a full QEDtreatment is needed to obtain the form factors from the measurements and experimentalresults give the following estimate:

rE = 0.897(18) fm. (3.2.4)

Spectroscopic analysis

Spectroscopy of atomic hydrogen has been fundamental for the development of modernphysics since the dawn of quantum theory more than a century ago. The discovery ofdeviations of the energy levels from the ones expected first showed the e↵ects of QED andsince the observation of this e↵ect, QED has reached an impressive level of accuracy in thecalculation of the hydrogen levels: as an example, the 1S � 2S transition in now knownwith an associated error of 4⇥1015 [13]. With such a level of accuracy, QED can be testedcomparing the results of the calculation of the hydrogen levels with the experimentalvalues. However, the precision of the QED test is limited by two parameters that enterin the calculation: the Rydberg constant R1 and the root-mean-squared radius of theproton. Thus, one can either obtain these parameters in di↵erent ways and use themto test QED with experimental measurements, or extract them from the measurementsassuming QED is correct. Both these approaches are being studied by di↵erent groupsof research. The first approach is based on precise measurements of the R1 constantand involves theory behind Lamb shift in particular. There are no particular issue to thetheoretical apparatus, and parallel improvements on the precision for this constant havebeen made all over the last five decades reaching nowadays’ impressive 10�11 relativeuncertainty. However, more accuracy is needed if we want to test QED theory. Untilnow in fact, we have only been exploiting the second option: we’ve been calculating R1assuming QED correct. By doing though, we are prevented from reversing the analysisand test QED, unless we shift to di↵erent techniques for the R1 measurement. We willsee the state of art concerning Rydberg’s constant later in chapter 4.Let’s take a step back and consider the second approach, extract the Rydberg constant

and the proton charge radius from spectroscopic measurements, assuming QED correct.For the purpose, one needs at least two transition frequencies in hydrogen and given this,

3.2. THE PUZZLE 25

the energy of the S-states in hydrogen can be approximated [15] as:

EnS ⇡ �R1n2

+L1S

n3

, (3.2.5)

where L1S is the Lamb shift in the 1S state; typically one uses the 2S � 8S transition to

extract the Rydberg constant and the 1S � 2S transition to determine rE . This choiceis motivated by the fact that for high n the Lamb shift contribution is smaller due tothe n3 scaling so we are allowed an easier determination of the Rydberg constant, whilethe 1S � 2S transition contains the maximal 1S Lamb shift and is therefore maximallysensitive to the proton charge radius.For this kind of measurements an outstanding experimental accuracy is needed. An

example of the experimental strategy for the precise determination of the 1S � 2Stransition energy can be found in [13]. Thanks to this procedure a value of the transitionfrequency between the 1S and the 2S state with a fractional uncertainty of 4.2⇥ 1015

was extracted and the values of the proton charge radius rE could be evaluated as:

rE = 0.877(7) fm, (3.2.6)

which is in agreement with data obtained by electron scattering and provides a slightlybetter accuracy. This result is still used as a standard by the Committee on Data forScience and Technology (CODATA).

µp Lamb shift

A rE determination from an independent source was (and still is) highly desired toenhance the precision of the test on bond state QED, limited by the large uncertainty onthe proton charge radius. Since its first discovery in 1947 the Lamb shift in hydrogen orhydrogen-like atoms is the most precisely measured e↵ect of this kind. Muonic hydrogenthen, is even a better good system to study the proton structure since, given the largermass of the muon, the Bohr radius for µp is about 207 times smaller. The ratio betweenthe reduced masses of ordinary and muonic hydrogen is ⇡ 186, thus the binding energiesof µp are ⇡ 186 times larger than the ones of ordinary hydrogen, falling in the keVregion. The first conclusion is that we expect we can reach more accurate measurementsexploiting muon’s physical properties and all these favorable conditions (compare alsoTable 2.2 with Table 3.2).In particular, the nS states of muonic hydrogen overlap considerably with the nuclear

volume, and as a consequence their energy is much more sensitive to the proton structure.The corrections due to the finite size of the proton represent about 1.8% of the Lambshift for n = 2, about two orders of magnitude more than in the e�p atom [4]. Thus,the measurement of the 2S � 2P Lamb shift in muonic hydrogen is an excellent toolto investigate the proton structure. Such a measurement was performed at the protonaccelerator of the Paul Scherrer Institute (PSI) in Switzerland and the correspondingtransition energy was measured to be �Eexp = 206.2949(32)meV. This has to becompared with the theoretical value taking in account all contributions being equal to


�Eth = 209.9779(49)� 5.2262r2E + 0.0347r3E . The most recent [14] extrapolated valuefor rE with this technique at PSI is equal to:

rE = 0.840 87(39) fm. (3.2.7)

Note how this result is one order of magnitude more precise, but is nowhere in agreementwith the previous e�p-based techniques by a huge 7� discrepancy. The reasons behindthis discrepancy are not clear and many hypotheses are still on the table:

Electronic scattering and spectroscopy might be wrong because of unpredictedsystematic errors. It’s an ambitious statement as we would call into questionyears and years of studies from independent teams. In addition, should both thesetwo techniques be wrong, their agreement around one result would be just a bigcoincidence. On the other hand, it is historical that measured values of constantsvary dramatically before converging to their “correct” values. The 7� discrepancyhowever, has very little precedents. Few-sigma corrections are more common, aswe can see in the case of some physical quantities concerning the neutron.

Figure 3.2: Neutron’s mean lifetime and mass measurements over the decades. Jumps ofup to 2 or 3� in measurements are common events in experimental physics as systematicerrors may be hard to predict.

Related to the electronic scattering, one source of error might come from thecorrelation constant between the Rydberg constant r1 and the rE according toCODATA. A new determination of R1 at a level of a few parts in 1012 would helpin shedding new light on the discrepancy between e�p and µp results.

There might be something inaccurate in Quantum Electrodynamics (see �QED

term ). Either some contributions have been underestimated or the theoreticalbuilding around QED is incomplete. The first hypothesis is very unlikely as suchdiscrepancies are too big not to be already being identified. The second one is evenmore ambitious, as QED has been proving its reliability for decades, but is at thesame time fascinating as it would lead to new theories beyond the Standard Model.

3.2. THE PUZZLE 27

Discrepancies can be due to unknown sources of error inside the �POL and �FF

terms. Again, this conjecture sounds a little unlikely nowadays, but can’t bedropped.

The most intriguing hypothesis is that the electron and the muon interact di↵erentlywith the proton and this would require physics beyond the Standard Model thatviolates the principle of lepton universality. This possibility attracts discrete interestsince the recent measurements of the anomalous magnetic moment of the muons(gµ � 2)/2 shows a discrepancy with the theory that, although very small, has astatistical significance of 3.6�. This di↵erence, as well as the proton radius puzzle,could be due to a new kind of interaction that acts di↵erently on the muon and onthe electron.

3.2.3 Zemach radius

The proton Zemach radius rZ is defined as the first moment of co-involution of spatialcharge ⇢E(r) and magnetic moment distributions ⇢M (r). In formulas

rZ =

Z ✓Z⇢E(r

0)⇢(r � r0)d3r03◆rd3r. (3.2.8)

This Zemach radius can be determined by measuring the hyperfine splitting energies inhydrogen-like atoms, we therefore need a deep understanding of Laser spectroscopy inmuonic hydrogen.The last remark opens us a a new path for further researches: switching our experimental

e↵orts towards rZ measurements gives us indirect feedback about rE , confirming ordisproving some of the results. Measurements based on di↵erent techniques will help usto draw an even more complete picture of the situation. We can indeed extract the rZboth from scattering and spectroscopy experiments from e�p just like the rE radius, orvia µp lamb shift or hyperfine splitting spectroscopy. Many electron-based experimentshave been settled so far and we will summarize them in the following discussion.

e

�p hyperfine splitting 1S

The first experiment of this kind dates back to 2003 in the LCAR-IRSAMC laboratory,see [16]. Here QED is assumed to be correct and therefore we can extract a value forrZ studying its role played in the �POL term which, together with other uncertainties,depends also on the Zemach radius. The procedure basically inverts the relation of theFermi Formula (2.5.2) and makes use of data and uncertainties such as in Table 2.2. Acomparison with experimental data �Ehfs will lead to rZ :

rZ = �⇣�hfs

exp/EF � 1� �QED � �FF � �POL

⌘/2µep↵, (3.2.9)

where µep stands for the reduced mass of the electron-proton system. This eventuallyverifies the credibility of the theoretical evaluation of proton polarizability e↵ects and


gives a value for the proton Zemach radius of:

rZ = 1.040(16) fm. (3.2.10)

The same method is followed by Volotka [17] and Distler [18] on their own papers,and the results are in agreement with rZ estimated being 1.045(16) fm and 1.045(6) fmrespectively.

Electron scattering

Completely di↵erent is Friar and Sick’s apporach. In their article [19] they calculateproton’s rZ using the world data on elastic electron–proton and electron–deuteronscattering. Their result though, is not compatible as the value for rZ turns out to be

rZ = 1.086(12) fm, (3.2.11)

significantly bigger than the three previous ones.

Being these results anyway still unclear and not su�ciently accurate for a QED test,we can switch to muon-based experiments and eventually get an independent result withhigher accuracy. Just like we did in Section 3.2.2, we will seek for independent resultsbased on di↵erent techniques, in particular we will see rZ has been extrapolated startingfrom measurements of the hyperfine splitting 2S in µp in the PSI laboratory in recentyears.

µp hyperfine splitting 2S

In addition to that, there are other advantages due to the fact that there are twomeasured transitions, so other e↵ects can be canceled out when taking out the di↵erence.

The situation is well described by Figure 4.1, and if we call �Efs

and �Els

the energytransitions due to fine structure and Lamb shift respectively, we have the following:

3.2. THE PUZZLE 29

Figure 3.3: The 2S and 2P energy levels. Two measured transitions are shown, and theinteraction between the 2P (F = 1) states is included with the � term. Note that in themuonic case, unlike classical hydrogen, the 2S

1/2 has less energy than both 2P1/2 and

2P3/2 states.

2⇡~⌫t = E⇣2PF=2

3/2

⌘� E

⇣2SF=1

1/2

⌘

= �Els

+�Efs

+3

8�E

2P3/2

hfs

� 1

4�E

hfs

(3.2.12)

2⇡~⌫s = E⇣2PF=1

3/2

⌘� E

⇣2SF=0

1/2

⌘

= �Els

+�Efs

� 5

8�E

2P3/2

hfs

+ � +3

4�E

hfs

(3.2.13)

Now, for the fine structure splitting values from [21] are followed, where �E2P3/2

hfs

=8.352 082meV and � = 0.144 56meV respectively. After this, a list of contributions toµp Lamb shift is included and analyzed. Note that the same correction-listing approachcan be useful do detect anomalies for the rE result in muonic Lamb shift. The majorterms of this turn out to be:

One-loop electron-positron vacuum polarization, shortly called “eVP”. The physicalinterpretation of this phenomenon is the creation of virtual pairs of electrons andpositrons.

Finite extension of the proton charge, proportional to r2Eµ3

µp. In this case, theproton cannot be considered point-like.


Kallen-Sabry two-loop eVP diagram contribution.

Muon one-loop self energy + one-loop muon-antimuon vacuum polarization “µVP”.

Other contributions are smaller and can hardly explain discrepancies for the rE radius,on the other hand they must be take into account when improving precision to the rZmeasurement for this kind of experiments; a complete list of these is included in [20]. Wewill limit ourselves in stating that the relation between �Ehfs

2S(th) and the Zemach radiusis given by the following:

�Ehfs

2S(th) = 22.9843(30)� 0.1621(10)rZ (3.2.14)

Now, the energy shift �Ehfs

2S(exp) has been measured to be 22.8089(51)meV, and the

previous equation therefore gives a vale for rZ being equal to 1.082(37) fm.As we can see, this result is not accurate enough to distinguish from the older in-

compatible ones, the main responsible of these uncertainties being the last term, soimprovements are needed towards that direction. On the other hand, this experimentopens new ways of exploring the proton properties, with µp this time playing the mainrole on the stage and upcoming measurements can shed new light on the causes of thisdisagreements. Depending on the results, some hypothesis listed on page 26 may becomestronger candidates while others may faint or definitely be abandoned.

µH hyperfine splitting 1S

Measuring �Ehfs

1S(exp) for example is not only a new interesting path on its own, but

combined with what we’ve seen so far with its “cousin” �Ehfs

2S(exp) we can achieve betterresults as some contributions due to structure cancel out when taking the di↵erence�Ehfs

1S(exp) � 8 �Ehfs

2S(exp). In particular we know that the leading nuclear structurecontributions are determined by two photon exchanges with high momentum transferand therefore cancel out when calculating the di↵erence. The ab initio calculations ofthis di↵erence in light hydrogen-like atoms ensure a much higher absolute accuracy thancalculations of the hyperfine splitting separately for the states in this di↵erence. Thestate of art can be summarized by the following tables:

Technique rE (fm)

e�p scattering 0.897(18)Spectr. analysis 2 0.877(7)µp Lamb shift 0.8409(4)

Technique rZ (fm)

e�p HFS (1S) 3 1.037(16)e�p HFS (1S) 1.045(16)e�p scattering 4 1.045(6)µp HFS (2S) 1.082(37)

Table 3.3: Summary of the di↵erent values for rE and rZ obtained so far with di↵erenttechniques.

2See Parthey et al, see [13]3according to Dupays in [16] and Volotka [17] respectively4see Distler in [18]

3.3. STATE OF ART 31

Remark. As said before, the proton structure correction �FF in muonic hydrogen isenhanced (compared to hydrogen) by a factor of ⇡ 207. Therefore, a measurement ofHFS in µp (1S) can not be a good test of QED since QED e↵ects are overshadowed bythe proton structure corrections. In addition to this we have to underline that in bothclassical and muonic hydrogen, the proton structure correction �FF is dominated by twoindependent terms: the Zemach term and the polarizability term �POL

3.3 State of art

The unexpected results of the Lamb Shift experiment at PSI on the proton chargeradius [3] immediately raised great sensation in the scientific community. The discrepancybetween the value obtained from the Lamb shift in muonic hydrogen and the one calculatedby CODATA is huge and has not been explained yet, although several hypotheses havebeen made since the first days after the publication of the results in 2010.Possible origins of this disagreement have been listed in these years and have already

been mentioned on page 26. The last one in particular involves physics beyond theStandard Model and some of its aspects can be tested by studying the role played by thephoton-photon exchange contribution. Alternatively possible di↵erences between e�pand µp interaction can be spotted from the comparison of the Zemach radius rZ . Theextraction of the proton Zemach radius from the measurement of HFS in the ground stateof muonic hydrogen is the goal of the FAMU collaboration and this section is devoted tothe description of the experimental method developed for this measurement.

3.3.1 Motivation for the measurement of the HFS 1S in µH

As in the case of the Lamb shift, the hyperfine splitting of the ground state of ordinaryand muonic hydrogen shows very di↵erent characteristics. The experimental value of thehyperfine splitting for example, is known with a relative uncertainty smaller than 1012

while the theoretical predictions are less accurate. Assuming the di↵erent correction tobe accurate one can obtain the Zemach radius and compare it to other values based ondi↵erent proton form factors fits. We’ve already shown (see Table 3.3) how the Lambshift experiment renewed interest in the evaluation of the proton Zemach radius from ameasurement of the hyperfine splitting in muonic hydrogen, as a comparison between theZemach radius extracted from e�p and µp could either reinforce or delimit the protonradius puzzle: a substantial agreement with the value of rZ obtained with ordinaryhydrogen could suggest that the explanation of the puzzle may lie in unconsideredmethodology uncertainties, while a big discrepancy would give good reasons to look fornew physics beyond the Standard Model.The theoretical uncertainty on �Ehfs

nµn(th) is mainly due to the uncertainty on theestimation of the Zemach and polarizability corrections, which have to be added to notyet calculated higher order radiative terms and to those small extra corrections cited onpage 17. The accuracy of the current value of may be improved by a measurement of thehyperfine splitting in the µp ground state if the experimental uncertainty is kept below


nowadays uncertainty of 10�4. We have to be aware of the fact that, the polarizabilitycorrection is not related to a single physical parameter like the Zemach term, but isexpressed in terms of the polarized structure functions of the proton that introducemodel-dependent parameters. Note that the experiment held in the PSI laboratories (seepage 28) has a relative uncertainty of 4% which is too large for any useful comparisonbetween di↵erent experimental values and theoretical predictions.In order to provide helpful data to solve the proton radius puzzle we need a relative

uncertainty below 0.5%, thus a big jump is needed in order to attain this kind of precision.Luckily, the outstanding progress in the development of tunable laser systems in the farinfrared region, together with the development of a suitable experimental strategy inrecent years, provide a strong motivation for the accurate determination of the Zemachradius. The FAMU has been conceived in order to exploit these recent technologicalevolution and aims at reaching the needed experimental precision.

Chapter 4

Rydberg’s constant

The Rydberg constant, is a physical constant relating to atomic spectra in the science ofspectroscopy. It is not a fundamental constant (see Table on page vi) but has neverthelessa great significance in physics. The Rydberg constant represents in fact the limitingvalue of the highest wave number (or frequency) of any photon that can be emitted fromthe hydrogen atom. Alternatively, switching to absorption spectra, it stands for thewave number of the lowest-energy photon capable of ionizing the hydrogen atom from itsground state. Now keeping in mind Planck’s law we can predict some hydrogen spectrumlines in terms of R1 with good accuracy:

1

�= R1

1

n2

i

� 1

n2

f

!(4.0.1)

where � is the wavelength of the emitted (or absorbed) photon while ni and nf stand forthe principal quantum number of the initial and final state respectively.This relation, albeit approximate, suggests that hydrogen spectroscopy is a powerful

tool for extrapolating the the Rydberg constant. We can expect then that progressesmade in spectroscopy are naturally followed by improvements in the calculation of R1.Nowadays in fact, thanks to techniques based on tunable lasers or nonlinear Doppler freespectroscopy, we have reached fractional accuracy up to 10�14, and many other opticalfrequencies are know within one part over 1011 [23]. In parallel, Rydberg constant hasbeen measured with better and better accuracy, and as things stand today, has a relativeuncertainty of 6.6⇥ 10�12. This larger uncertainty compared to the ones in hydrogenspectroscopy is essentially due to theoretical issues.

33

34 CHAPTER 4. RYDBERG’S CONSTANT

Figure 4.1: History of the relative precision of the Rydberg constant from 1920 to present.

4.1 R1 and �E in hydrogen atoms

Starting form Bohr model and following the same approach we did in Sections 2.2 - 2.5,we will build a theoretical background for the Rydberg constant, looking for relationsbetween R1 and corrections to the hydrogen energy levels. After the list is complete,we will be able to establish which contributions need better measurements in order toimprove our knowledge on R1 and which kind of progress are we expecting from thefuture.

4.1.1 Theoretical background

According to Bohr model, Rydberg constant is known as a function of the fine structure↵, Planck (reduced) constant ~, speed of light c and electron mass me and the relation isvery simple:

R1 =↵2mec

4⇡~ (4.1.1)

Besides being just an approximate form, Equation (4.1.1) is very useful for adjustmentsof the fundamental constants. Physicists often are interested in using the value of R1 inorder to deduce either ↵ or the ~/me ratio.As we know though, hydrogen energy levels do not depend solely on the principal

quantum number n, but also on several other corrections. Conventionally, hydrogenenergies are expressed as sums of these three terms:

En,`,j = EDirac

n,j + ERecoil

n + Ln,`,j (4.1.2)

where:

4.1. R1 AND �E IN HYDROGEN ATOMS 35

The first term stands for the energy given by the Dirac equation of a particle of aparticle, also taking into account the reduced mass e↵ect of the e�p system.

The second indicates the first relativistic correction due to recoil of the proton. Itdepends on the principal quantum number n only.

Finally, we have the Lamb shift term. It depends on all n, ` and j and its calculationis very di�cult [23], we will see some of the contributions in detail and list theothers. A part from these theoretical issues, another problem of the evaluation ofthe Ln,`,j is given by the fact that it is obtained as a power series of ↵, Z↵, me/mp

and rE , but as seen in Section 3.2 these have some limiting uncertainties.

Figure 4.2: Energy levels in hydrogen for successive steps of the theory.

Recoil corrections

As the second term includes corrections due to recoil of the proton, we expect adependence on the reduced mass of the system µep and this is exactly what happens.Expressing the contribution in power series, the first one turns out to be:

ERecoil

n = �µ2

epc2

me +mp

(Z↵)4

8n4

. (4.1.3)

For the 1S orbital (n = 1) the correction is 10�8 times the ionization energy of hydrogen.For better precision (remember we know frequencies of spectroscopic lines up to 14digits) we need higher order terms. The first two vary as mec2(me/mp)2(Z↵)4 andmec2(me/mp)(Z↵)5 respectively and have an exact expression. The following one is pro-portional to mec2(me/mp)(Z↵)6 and has already been calculated by several authors[23].


The overall uncertainty thanks to these additional terms is shrunk to one part over 1014,su�ciently good for our purposes.

Lamb shift corrections

Corrections due to QED are the main contributions to the Lamb Shift and those whoa↵ect the Lamb shift have already been listed on page 29 for the muonic case. Mostof them are as useful for the current treatment as well: the vacuum polarization e↵ect“eVP” is an example.

A similar QED correction is due to the residual energy of the electromagnetic field, wherethe electron is induced to fluctuations in the vacuum field. These fluctuations modify theCoulombian potential seen by the electron and reduces binding energy sensitively in theS states, resulting in what we call a self energy correction. The lowest order correctionfor this involves single-loop Feynman diagrams and is expressed by:

ESE

=↵(Z↵)4

⇡n3

F (Z↵)mec2, (4.1.4)

where F (Z↵) is a function in terms of Z↵, ln(Z↵) and similar, there is good theoreticalagreement on the coe�cients.Similarly, we take into account the vacuum polarization correction, described on page 29,

and the radiative corrections, a result of the emission and absorption of pairs of virtualphotons whose formula is similar to (4.1.4), but with an extra ↵/⇡2 factor..Other corrections are mostly higher-order Feynman diagrams such as the three-loop

radiative correction and the µ+µ� or ⌧+⌧� virtual pairs formation and are negligible.On the other side, nuclear size plays an important role when dealing with high precisionspectroscopy. In the e�p case it is given by:

ENucl. Size

=2µ3

ep

3m3

e

↵2mec2

n3

⇣↵rEmec

~

⌘2

(4.1.5)

The following table, drawn up by Biraben [23], gives us a clear overview:

4.1. R1 AND �E IN HYDROGEN ATOMS 37

Term of the Lamb Shift Value for the 1S level Uncertainties(in kHz) 1 (in kHz)

Self-energy (one-loop) 8 383 339.466 0.083Vacuum polarization (one-loop) �214 816.607 0.005Recoil corrections 2401.782 0.010Proton size 1253.000 50Two-loop corrections 731.000 3.300Radiative recoil corrections �12.321 0.740Vacuum polarization (muon) �5.068 < 0.001Vacuum polarization (hadron) �3.401 0.076Proton self-energy 4.618 0.160Three-loop corrections 1.800 1.000Nuclear size corrections to SE and VP �0.149 0.011Proton polarization �0.070 0.0131S Lamb shift 8 172 894.000 51

Table 4.1: Summary of the calculation of the 1S Lamb Shift. We can notice that themain uncertainties are due to proton size corrections and two-loop corrections.

As we can see, many contributions are involved and the situation seems very entangled,but the Lamb shift is such an important term when it comes to determine R1 that itdeserves this kind of e↵ort and investigation.

4.1.2 Lamb shift measurements in hydrogen

We have already mentioned some of the most recent results of measurements of theLamb shift in Section 2.3.2, we will now see how some of these values have been obtainedand how do they possibly lead to adjustments of the fundamental constants and tothe Rydberg constant. We will now make a list of the principal Lamb shift-relatedmeasurements in recent years:

Direct 2S1/2 � 2P1/2: there are several measurement sof this kind, starting fromthe historic Lamb-Retherford one in 1947 to the most precise by Lundeen andPipkin in 1981, as reported in [24], with a value of 1057.845(9)MHz.

2S1/2/2P1/2-lifetime ratio: indirect measurements are in agreement with theprevious value. Pal’chikov, Sokolov and Yakovlev’s one is particularly precise, givinga value of 1057.8514(19)MHz. This was extracted thanks to atomic interferometry,where the ratio between 2S

1/2 and 2P1/2 lifetime. There is a controversy over the

theoretical value of this lifetime value [23].

1S1/2 � 2S1/2 Doppler free: this transition was poposed first proposed byVasilenko et al. [23]. The princicple is to pace an atom in a stationary wave

1In this section we will talk about the emitted/absorbed frequencies instead of energy shifts. Theconversion is 1 eV to e/2⇡~ ⇡ 2.4180⇥ 1014 Hz.


resulting from the sum of two counter-propagating laser beams with the samefrequency. There are big advantages for this kind of experimental set-up as thetotal momentum of the absorbed photons is zero and we have a cancellation of thefirst-order Doppler e↵ect and recoil e↵ect. On the other hand, precise experimentswith this technique are still out of reach because of the UV wavelength of thetransition.

Improvements have nevertheless been made thanks to the creation of a complexfrequency chain which links the 1S�2S transition to optical frequency measurements.The obtained value is a super-precise:

⌫1S�2S = 2466 061 413 187 074(34)Hz (4.1.6)

2S1/2 � nS/nD transitions: these kind of measurements are complementary tothe 1S � 2S frequency measurements as the Lamb shift of the 2S level is alreadywell-known. Transitions like the 2S�8S/D fall in the near infrared and and counterpropagating laser beams do their work just like in Vasilenko’s proposal.

4.2 Determination of R1

The Rydberg constant can be deduced in several ways. One example makes use ofhydrogen-based data together with the ↵ constant and the me/mp ratio. Alternatively,we can perform global adjustments concerning all fundamental constants, we will obtainbetter precision, but the mixing of experimental and theoretical results prevents us fromhaving a view of which are the most important input data.

R1 from 1S - 2S energy split in hydrogen

A straightforward method consists on extracting R1 from Equation (4.1.2) since weknow the exact expression for the EDirac

n,j ERecoil

n terms, and we can write them as functionsof the Rydberg constant as well:

En,`,j = 2⇡an,j~cR1 + Ln,`,j . (4.2.1)

Here, the coe�cient 2 an,j as a function of ↵ and me/mp is known. And R1 is deducedwith a relative uncertainty of 1.4 ⇥ 10�11 which is mainly due to the measurementuncertainty of the 2S

1/2 � 8D5/2 frequency splitting. It is noticeable that with this

proceeding we are able to deduce R1 without measuring proton’s charge radius rE .Slightly better accuracy is attained when measuring a similar range of frequencies indeuterium, getting R1 = 10 973 731.568 54(10)m�1 with a relative uncertainty of lessthan one part over 1011. Other experiments which can measure the Rydberg constantbypassing inter-dependence with rE , using for example very high-lying states whereproton size is negligible, are being considered. An experiment concerning this possibilityis underway at the National Institute of Science and Technology (USA) [25].

2Recalling Equation (2.2.7) we should remember that an,j

⇡ 1/n2

4.2. DETERMINATION OF R1 39

R1 involving rE

As discussed earlier in sections 3.2.2 and 4.2, the Rydberg constant and the protoncharge radius are determined in the same hydrogen spectroscopy experiments. So farwe know the Rydberg constant to about 5 parts over 1012 and the proton charge radiusto a bit better than 1% when averaging electron-based experiments. Since these twoquantities are though (R1 can be extrapolated using rE), di↵erent rE-s give di↵erentvalues for R1. In particular the 7� shift in the proton charge radius translates into a 4�discrepancy for Rydberg constant.A solution to this problem can be For the small transition splitting measurements in

fact, the Rydberg constant is accurately known enough from other sources that we cansubtract its contribution from �E and obtain a value for rE . For big transitions on theother hand, the rE term is tiny compared to the main term and the Rydberg constantand is not independently known well enough to accurately obtain the rE term from asingle measurement. A solution would be measuring two such splittings and solve theelementary pair of simultaneous equations to obtain both R1 and rE .Shifting also in the field of muonic experiments has therefore a double importance and

significance, since it would bring progress both to the solution of the Proton RadiusPuzzle, and to an independent determination of the Rydberg constant.


Chapter 5

The FAMU experiment

As things stand, one of the most promising measurement, and at the same time abig missing piece concerning muonic measurements, is the µp 1S hyperfine splitting.Although several experimental methods have been proposed in the last twenty years [11],this kind of experiment has never been performed, but the recent results from themeasurement of the Lamb shift in muonic hydrogen are giving extra motivation tointensify the experimental e↵orts in this direction. The goal of the FAMU project is toextract the Zemach radius of the proton from the measurement of �Ehfs

µ1 with a relativeprecision below 0.5%. This kind of precision will allow useful comparison with the othervalues in Table 3.3 unlike those results extrapolated from muonic hyperfine splitting ofthe 2S level.Optimism towards this new approach comes again from technological developement of

tunable infrared lasers as they o↵er a wide range of new experimental proposals. In our�Ehfs

µ1 case, the experimental method was proposed by Adamczak in [22] and combineselementary particles with laser spectroscopy techniques and is based on the reaching aresonance e↵ect with tunable lasers.

5.1 The experimental proposal

The proposal is based on an interesting chain of processes: a muon beam hitting ahydrogen gas target forms muonic hydrogen atoms. These atoms are quickly de-excitedto the ground para-state (F = 0), but after after absorbing a photon of the resonancehyperfine splitting energy �Ehfs ⇡ 0.182 eV, they are excited to the (F = 1) ortho-spinstate. Being these atoms in contact with other molecules, they are quickly de-excited insubsequent collisions with the surrounding H

2

. At the exit of the collision the muonicatom is accelerated by some 2/3 (⇡ 120 eV) of the excitation energy �Ehfs, which it takesaway as kinetic energy. The number of muon transfer events depends on the number ofspin-flip events, so the first challenge is to detect these “kicked” µp-s exploiting the extra120 eV kinetic energy. An essential condition for the succeeding of this experimentalproject is to find a reaction depending on µp velocity, this kind of di�culty can be easilyovercome, but some methods are better than others when exploring their feasibility.

41

42 CHAPTER 5. THE FAMU EXPERIMENT

The original idea of Bakalov [11] for example, was to observe the di↵usion of µp atomsin a small volume studying the number of muon-transfer events on the target wallsfilled with gold or other big Z elements, sensible to “muon bombing” (we have a X-rayemission). A big drawback of this approach though, is the fact that the the transitionprobability of these spin-flips is too small to become statistically relevant because of poorlaser radiation. The impossibility to embed the target in a multipass optical cavity inorder to amplify laser e�ciency does the rest, leaving this method practically inapplicable.Di�culties should be overcomed with a most recent approach, based on a second idea

in [22], where study of the muon-transfer events (briefly µTEs) from muonic hydrogento another higher-Z gas was proposed instead of a detection based on target walls. Agas candidate should exhibit a sensitive energy dependence in the muon-transfer crosssection at these low energies so that resonance e↵ects can be observed, as it is the casefor oxygen and argon, for example. Using small quantities of these gases together withhydrogen in the chamber allows an e�cient detection of the X-ray emission from muonsand we can also build a multipass cavity for the chamber, dramatically increasing thefeasibility of the experiment, as we will see in Section 5.3. A schematic representation ofthis second approach is given by the following picture:

Figure 5.1: The FAMU set up. Here an example with a mixture of hydrogen and oxygen.

We will now analyze step by step the technological requirements and possible issues of

5.2. MUONIC ATOMS FORMATION AND THERMALIZATION 43

the latter proposal. Needless to say, a symbiosis between theoretical calculations andMonte Carlo simulation was mandatory to get this huge amount of results.

5.2 Muonic atoms formation and thermalization

As soon as muons are shot inside the hydrogen chamber, the process of µp formationand thermalization begins and it is necessary to know what happens at these stage. Anestimate of the needed time for these processes to complete s also needed, so that muonbeams can be sent at a proper rate and collect good statistics in reasonable time.The muon beam interacts with matter inside the chamber is initially scattered from

atom to atom, gradually losing its energy until it decays or is captured into an externalatomic orbit of an atom. The energy released in the atomic capture process is transferredto Auger electrons that are emitted from the atom. The characteristic cross sectionfor the atomic muon capture depends not only on Z, but also on some features of theatomic structure such as the number of loosely bound electrons [26]. In the hydrogencase though, things become much simpler. We know that typically, muonic hydrogen isformed at n ⇡ 14 energy levels since it’s there where the optimum overlap of the boundmuon and electron wave functions occurs. The formation of these highly excited atoms isfollowed by a number of transitions, also called muon cascade, down to the 1S groundstate (⇡ 99%) or to the 2S meta-stable state. Moreover, for a pressure of the gas targetlarge enough ( > 10 atm) the whole cascade process takes no more than about 1 ns.A part from this, there are other motivations that encourage using a large target

pressure as a solution to having enough muons stopped in the gas target. From somesimulation [27] in fact, it turns out that the needed time window for µp atoms to becompletely thermalized and depolarized is:

�t[ns] ⇡ 20T [K]

P [atm], (5.2.1)

where square brackets indicate units of measurement. Again, higher pressures turn outto be a better choice since as faster thermalization means that there is less muonic decayby the end of the process.

5.3 Laser requirements

Once the muonic atoms have been thermalized and depolarized, a laser with a frequencytuned in the HFS region is sent into the target. The laser must be tunable to wavelengthsnear to 6.8 m so that resonance e↵ect can be detected. The probability P for a laser tospin-flip muonic atoms was evaluated in [29] as a function of laser power, section areaand temperature:

P ⇡ 8⇥ 10�5 · E[J]

A[m2]pT [K]

. (5.3.1)

Nowadays technology o↵ers tunable lasers with a power of 0.25mJ, focused in a 1 cm2

area, so the probability results to be P ⇡ 1.2⇥ 10�5 at room temperature. As we can


see this probability is definitely too small for the proposed experiment: even if transitionprobability may be increased by lowering the temperature of the target, T should bekept higher than 10K to avoid the formation of unwanted molecular µHe+

2

ions. Thissort of temperatures would result though in an increase of a ⇡ 5.5 factor, but this is stilltoo small for our purposes and even if it wasn’t the case, these huge refrigerating e↵ortswould be very expensive.The e�ciency of this process may be substantially increased by squeezing the laser

beam reducing its cross section and placing the target atoms within a multipass cavitywith a high number k of reflections. Using similar cavities to the one used in the Lambshift experiment (k ⇡ 2⇥ 103) [3] it is possible to reach a transition probability of about12%, much more adequate for the goal of this experiment.

Figure 5.2: Mirror positioning must be very accurate in order to amplify laser’s illumina-tion and e�ciency.

5.4 Muon transfer to higher-Z gas

Once we have solved problems raised in Section 5.3 we have a su�ciently big numberof muonic hydrogen atoms excited to the triplet state.When a muon is stopped in hydrogen, it forms a muonic hydrogen atom that is quickly

de-excited and thermalized as described in the previous section. If in the hydrogen targetthere are small contamination of another gas G, the muonic hydrogen can disappear by:

1) muon decay, with an associated rate we will call �0

, equal to the inverse of muonmean lifetime ⌧µ

2) muon transfer to the G gas with a rate �pG

depending on the gas and of its pressure.

In order to compare these rates, they are usually normalized to the liquid hydrogendensity ⇢

lh

. Hence, lifetime ⌧ = ��1 of the muonic hydrogen atom under the particularcondition of pressure, temperature and contamination of the experiment has more decaychannels than the muon itself, resulting shorter. We will consider the formula proposed

5.5. THE RIKEL-RAL FACILITY 45

by Werthmuller in [31]⌧�1 = � = �

0

+ �cG

�pG

(5.4.1)

where the muonic capture rate coe�cient �pG

is multiplied the atomic density of thegas mixture � (normalized to the density of liquid hydrogen), while c

G

is the atomicconcentration of the G element. The characteristic X-rays of the G gas are emittedimmediately after both a direct atomic capture of a muon or transfer of a muon from aµp atom, but we are interested on the events of the second kind only. However, with aproperly delayed time window, we are able to consider only those X-rays due to the muontransfer we’re interested in. Their time distribution N�,G(t) is expected to follow theevolution of the number of muonic hydrogen atoms Nµp(t), the latter being proportionalto what is left of a exponential decay. In formulas:

N�G(t) / Nµp(t) / N0

µpe��t (5.4.2)

where N0

µp is the initial number of muonic hydrogen atoms, and � is the lifetime parameterseen in the previous equation.Now, the sensitive energy dependence we mentioned in Section 5.1 changes the � pa-

rameter. The di↵erence between the two exponential shapes can be spotted provided thatwe have enough µp (1S) atoms during a su�ciently long time interval starting after thepeak of the prompt characteristic X-rays which follow the direct atomic muon capture ofthe G gas. results from simulations in [27] show that the higher the concentration of O

2

,the faster the depopulation of µp is. Moreover, other Monte Carlo simulation for di↵erentO

2

concentrations suggest that the statistical uncertainty of measurements of the muontransfer rate can be drastically reduced by choosing the right gas concentration c

G

.

5.5 The RIKEL-RAL facility

Running through the demanding list of technical requirements, we’ve seen that a pulsedmuon beam is needed to perform this kind of measurements. Such a requirement can besatisfied only by two facilities in the world: the muon source located at the RutherfordAppleton Laboratory RAL in UK, and the MUSE muon science facility, which is partof the Japan Proton Accelerator Research Complex. The RIKEN-RAL for example,hosts ISIS, a synchrotron able to accelerate protons up to an energy of 800MeV that areused to produce intense pulsed muon and neutron beams. It is there where the FAMUcollaboration tested the feasibility of the measurement described in the previous chapterusing a preliminary version of the setup programmed for the final experiment.

Muon beam

The muon beam delivered to the experimental ports has a 50Hz double-pulse structure,emitting a flux of about 8⇥ 104 muons per second. Each pulse is about 70 ns long andthe time between the two pulses is 320 ns. The distance between the second pulse andthe first pulse of the following cycle is therefore some 20ms, far long enough for our


purposes, but not still short enough to get significant statistics in a reasonable amountof time. The muon beam structure can comfortably be visualized with the next figure:

Figure 5.3: Time structure of the pulsed muon beam.

The momentum of the muon beam of the RIKEN–RAL facility can vary in the 20–120MeV/c range. To obtain the largest achievable number of muonic hydrogen atoms, atuning of the beam energy was needed. The simulations of the muon interaction in thetarget indicate that the best energy is the 60–64 MeV/c range. This estimation provedto be correct as the maximum number of X-rays with an energy corresponding to the2P � 1S transition in muonic oxygen occurred at 61 MeV/c, hence this is considered theoptimal energy for the data acquisition.In order to perform the experiment and all preliminary studies, a big set of facilities

are necessary inside the RAL laboratory:

Gas Target

The gas target is the core of the apparatus, where muons stop and form excited atomsthat will produce X-rays during their subsequent de-excitation. The gas target used in the2014 beam test consisted of an aluminum cylindrical vessel filled with high pressure gas.The vessel was built in the form of a ↵125mm ⇥260mm cylinder with an inner volumeof 2.8 litres [4]. The filling gas consisted in high-purity gases provided by CK ProductsL.t.d., in particular pure hydrogen and a special mixtures of 4% CO

2

or 2% of argonhave been used for the 2014 set-up experiments and not oxygen for safety requirements.Mixtures with oxygen traces (0.05%, 0.3% or 1%) are being used in these months for thefollowing step and data precision is expected to improve.

The hodoscope

For the preliminary studies, a scintillating fiber hodoscope was built in order to obtaininformation about the beam size, position and timing. Even if each pixel acted as abinary device, the SiPM was able to provide an analog information, with an output signalbeing the sum of the signals of all the pixels and thus proportional to the light intensity.Pre-tests showed that the muon beam of the RIKEL-RAL facility caused the SiPMs

saturation hence the second parte of the program expects more powerful hodoscope. Inits upgraded 2014 version, the hodoscope was made of two arrays of 3mm square shaped

5.5. THE RIKEL-RAL FACILITY 47

Bicron BCF12 scintillating fibers arranged in planes with active area about 10⇥ 10 cm2.Each fiber was read at one end by a 3⇥ 3mm2 RGB Advansid SiPM, for a total of 64read-out channels [28].The electronics has been replaced as well, so that a larger dynamic range could be

achieved, allowing better handling with signals corresponding to about 70 muons for eachfiber.

LaBr3

:Ce and HPGe scintillating detectors

The nature of the observation of the characteristic X-ray spectrum of muonic atoms tomeasure the muon transfer rate puts stringent requirements on the detection system. Onone hand, the detector must have a good energy resolution in order to separate di↵erentlines also in the relatively low 60-500 keV energy range. On the other, the detectionsystem has to be fast enough to reduce pile-up events down to a reasonable rate. Ofcourse the pile-up rate can also be reduced by placing the detector at a larger distancefrom the target so that event rates become smaller, but this would imply the need of alarger detection area and the consequent increase of the number of read-out channelsand costs. The requirements on the cost and timing performance identify two kinds ofX-ray detectors: LaBr

3

:Ce and HPGe.The LaBr

3

:Ce detectors are the core of the detector system and are used along all theanalysis. They consist of a transparent scintillator material that permits signal samplingevery 2 ns, fast emission and excellent linearity. It’s decay time (⌧ = 16ns) is the shortestcompared to all inorganic scintillators materials and it also has the best energy resolution( 7%). One of the main drawbacks of LaBr

3

:Ce though, apart from the large cost, isthe presence of a radioactive background mainly due to the presence of 138La, a naturallyoccurring radioisotope of Lanthanum, and in order to improve experimental precision wehave to take into account this phenomenon as well.The second set of detectors consists on HPGe (High Purity Germanium) ones. These

are not ideal, although they present the best performance in terms of energy resolutionin the 3� 800 keV energy range and the signal is sampled every 10 ns, they consist of acylindrical crystal with a 11mm diameter and a 7mm height. The detectors can outputboth pre-amplified waves that display the signal as it is read, or shaped waves that canbe easily analyzed via software but lose information about overlapping events. In currentanalysis their role is to detect any contamination in the gas target and cross check thelanthanum bromides data [29].

Data Acquisition and Analysis

The framework used for data elaboration is ROOT [30], a collection of powerfulsoftware written at the CERN 1 specifically for experimental data analysis. Thanks toits numerous features, we can chose among a huge amount of operations for big dataprocessing, statistical analysis, visualization and storage.

1See also http://root.cern.ch/


Not only data analysis is demanding: data acquisition must follow a rational procedure.Data produced by detectors at every muon beam shot is huge indeed and has to beprocessed as e�ciently as possible in order to get valuable statistics in an acceptableamount of time. Data acquisition and procession is therefore a key aspect of this kind ofprojects and therefore has a very complex structure, as we can see in the following figure:

Figure 5.4: Scheme of the data acquisition during the 2014 test.

Most of the data is produced by the LaBr3

:Ce scintillating detectors so a big, whosesignals are digitized and sent to the DAQ PC and stored. The digitizer, triggered by theFAMU control board, sampled the output signal every 2 ns in a time window of 5 s. Thesignal of the GLP HPGe has to go through a stage of amplification before it is amplifiedand shaped by an amplifier with a 6 s time constant. Both the shaped signal and theone at the output of the pre-amplifier stage are digitized.The first step in the analysis process is the identification of the pulses in the waveform.

A signal is accepted as an event when its increasing rate per time unit is bigger thana certain value empirically determined. Once it is “accepted”, we calculate its energythrough a fitting function raising as a Gaussian and decreasing as an exponential. Theintegral of the latter, properly rescaled, gives us the energy in keV.From these events, those with smaller pulse height are discarded in order to exclude

fake low energy events actually due to other fluctuations. After this, other sort of filtersare needed. For example, clearly when one of the detectors records a high energy eventa spike will appear in all other detectors. This is what we call “pick-up noise” and weare able to cancel it given the fact that these spikes decay much faster than true pulsescoming from the scintillating detectors.The overall “noise-cleaning” procedure is ended, but LaBr

3

scintillating detectors alsohave to be carefully calibrated and retested. Since we are interested not only in the

5.6. CONCLUSIONS 49

number of events but in their energy as well, we have to develop tools able to extrapolateenergy deposited in the detector at each event by looking at the height and the integralof any pulse. There are though issues are due to the non-proportionality between theenergy of an event and the detector response. A second-order polynomial fit was used tocalibrate the detectors and keep non-linearity e↵ects below 1% [4].There are of course, many other aspects that FAMU has to deal with and the choice of

giving only some general overview is due to the huge complexity of the experiment.

5.6 Conclusions

The Proton Radius Puzzle is one of the most intriguing problems in modern physicand the importance of studying the Puzzle from the muonic side has already beenunderlined. Any upcoming result may lead to wide range of consequences and the FAMUproject should have an active role in this scenario. A comparison of the results fromthe two muonic hydrogen experiments is indeed strictly necessary as it will a↵ect futureinvestments and e↵orts towards, hopefully, a decisive conclusion. Muonic experimentsin fact not only give far more precise results but also are quite straightforward toanalyze: most of the di�culties regard the experimental set-up functioning. As said,this sort of measurements may either reinforce or delimit the Proton Radius Puzzle: asubstantial agreement with the value of r

Z

from ordinary hydrogen could suggest thatthe explanation of the puzzle may lie in unconsidered methodology uncertainties, while abigger discrepancy on the other side would give good reasons to look for new physicsbeyond the Standard Model [4].The aim of the FAMU is to measure rZ with a new independent technique, so that

we can make some progress towards the solution of the Puzzle. After some preliminaryresults from 2014, progresses are being made both as it concerns scintillating crystals,laser system and detectors. Theoretical work on atomic dynamics and on muon transferis still going on and might give a big hand for the successful outcome of the project.New data sets has been collected in 2016 and are now being analyzed, new results andguidelines for the following set-ups are expected for this year.Another interesting aspect is related to the Rydberg constant, whose value is now

known up to the 12th figure but improvements on the precision are expected as soon asknowledge about the proton radius increases.


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