Calculation and Modeling of the Neutron's Magnetic Moment

University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange

Masters Theses Graduate School

8-2021

Calculation and modeling of the neutron’s magnetic moment

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Recommended Citation Sharda, Abhyuday, "Calculation and modeling of the neutron’s magnetic moment. " Master's Thesis, University of Tennessee, 2021. https://trace.tennessee.edu/utk_gradthes/6167

This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council:

I am submitting herewith a thesis written by Abhyuday Sharda entitled "Calculation and modeling of the neutron’s magnetic moment." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Physics.

Yuri Kamyshkov, Major Professor

We have read this thesis and recommend its acceptance:

Geoff Greene, Thomas Papenbrock, Yuri Kamyshkov

Accepted for the Council: Dixie L. Thompson

Vice Provost and Dean of the Graduate School

(Original signatures are on file with official studentecor r ds.) Calculation and modeling of the neutron’s magnetic moment

A Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville

Abhyuday Sharda August 2021 Dedication For my little sister, I ﬁnally did it.

ii Acknowledgment I am indebted to my advisor, Dr. Yuri Kamyshkov, who gave me freedom to explore my interests and passions in physics and for always being supportive of my research. I am also deeply grateful to Dr. Geoﬀ Greene, who gave me valuable guidance at key moments during my program to help me towards graduation. A big credit for my program completion goes to him. I would also like to thank Dr. Thomas Papenbrock, who guidance and knowledge I could count upon to get unstuck from any problem. Deep thanks to Dr. Breinig who has always backed me and supported me during some rough times in my program. It was her support I counted on when things seemed bleak. Finally, big thanks to all my friend in the physics department who always went above and beyond to help me whenever I needed it. May I get a chance to repay your kindness someday.

iii Abstract This thesis presents the current state of neutron magnetic moment calculations. It details the development of calculations through history. It also delves into an experiment measuring the neutron magnetic moment. It ex- plores other methods by which calculations can be improved to get a better/ more accurate number. The conclusion is that there are still a lot of areas unexplored in context of the calculation of neutron magnetic moment and areas relevant to be worked upon are detailed.

iv Table of Contents

1 Introduction ...... 1 2 Theoretical Background ...... 3 2.1 Dirac Fermions ...... 3 2.2 Transition Magnetic Moment ...... 4 2.3 Neutron Substructure ...... 4 2.4 Static Quark Model ...... 5 2.4.1 Baryon Wave Functions ...... 5 2.4.2 Baryon Octet ...... 8 2.4.3 Calculation of Neutron Magnetic Moment ...... 8 3 Nature of Magnetic Moment ...... 16 3.1 Nuclear Magnetic Resonance ...... 16 3.2 Measurement of Neutron Magnetic Moment ...... 19 3.3 Nature of Neutron Magnetic Moment ...... 22 3.3.1 Probing the theory ...... 22 3.3.2 Comparison with Experiment ...... 25 3.4 Reflections of Neutrons from Magnetized Mirrors ...... 27 3.4.1 Neutron Reflection Theory ...... 28 3.4.2 Experimental Method ...... 28 4 Probing the Static Quark Model ...... 32 4.1 Review of Baryon Magnetic Moment in PDG ...... 32 4.1.1 General Quark-Model Analysis ...... 32 4.1.2 Configuration Mixing in Baryons ...... 36 4.1.3 Precision Description of the Baryon Octet Magnetic Moments40 5 Deep Inelastic Scattering and Spin Crisis ...... 44 5.1 The Proton Spin Crisis ...... 44 5.2 Deep Inelastic Scattering Experiments ...... 44 5.3 Polarized 3He Neutron Scattering ...... 46 5.4 Results ...... 48 6 Conclusion and Future Work ...... 51

v References ...... 52 Vita ...... 57

vi List of Figures

2.1 Standard Model[38] ...... 6 2.2 The Baryon Octet [4] ...... 9

2.3 ψ12 [4]...... 9

2.4 ψ23 [4] ...... 10

2.5 ψ13[4] ...... 10

2 3.1 Graph of |c2|max as a function of ω, where ω21 is the resonant frequency [39] ...... 18 3.2 Schematic view of spectrometer with detail of proton polarization and detection equipment[3] ...... 20 3.3 Schematic view of spectrometer with detail of neutron polarization and detection equipment[3] ...... 20 3.4 Hyperfine splitting of ground state Hydrogen[40] ...... 26 3.5 Plan view of apparatus for reflection of neutrons from magnetized mirrors[59] ...... 29 3.6 Intensity of filtered neutrons reflected from a magnetized iron mirror. Experimental points for two directions of magnetization, φ = 0◦ and φ = 90◦[59] ...... 31

4.1 Baryon Decuplet[41] ...... 37

5.1 Inside the proton[26] ...... 45 5.2 Layout of the experimental setup. Two independent single-arm spectrometers are shown[47] ...... 47 5.3 The PHENIX detector at RHIC[26] ...... 50

vii List of Tables

4.1 Baryon magnetic moments(in nuclear magnetons) ...... 33 4.2 Comparison of quark magnetic moment predictions without and with conﬁguration mixing ...... 39 4.3 Modiﬁed tensor for the baryon octet ...... 43

viii Chapter 1

Introduction

The concept of spin is a unique enigma in particle physics. There is a lot unknown about the concept of spin, especially the origins of it. A popular way of introducing the abstruseness of this concept is: “Imagine a ball that is spinning except it is not a ball and it is not spinning”. It was this mystery that got me interested in my thesis topic. I used research in the origin and nature of baryon magnetic moment as an avenue to study spin in more detail. The question of the exact origins of baryon magnetic moment has long been an unresolved one. We know that quantum spin causes observable effects like electron configuration in atoms (via Hund’s rule of maximum multiplicity) or magnetic moments in particles. Yet how exactly does spin lead to a magnetic dipole moment is not understood. Probing deeper into calculation of neutron magnetic moment could provide us deeper insight into the nature of spin. The static quark model pro- duces a value that is accurate to 2.7% of the value that we detect in the experiments[5]. The theoretical accuracy 2.7% allows the possibility of considering models where additional non-conventional mechanisms of contributions to the neutron magnetic moment can be introduced with a magnitude up to 1%, e.g. the existence of a transition magnetic moment when a neutron oscillates to a mirror neutron could be considered if there is considerable difference between predicted value and observed value[2]. This thesis is structured as follows. Chapter 2 discusses the theoretical background for magnetic moments of Dirac particles (like neutron) and di-

1 vulges details of the static quark model which is the currently accepted model. Chapter 3 delves into the nature of magnetic moment and early attempts to try to determine the cause of dipole moment of the neutron. Chapter 4 delves into advanced concepts that depart from the static quark model to explain discrepancies and attempts to cover physics ignored by it. Chapter 5 de- scribes how deep inelastic scattering experiments exposes the shortcomings of the static quark model by demonstrating that quarks do not contribute to most of the spin of the nucleon. Chapter 6 concludes the thesis and discusses what can be done in the future using the conclusions and analysis of this research.

2 Chapter 2

Theoretical Background

2.1 Dirac fermions

All particles in nature have a fundamental property called ‘spin’. Spin is often described as “intrinsic angular momentum” where the word ‘intrinsic’ is supposed to hint that we have no clear idea on what it is or it’s none of our business or both[6]. If a fundamental particle has spin 1/2 and finite electrical charge, then it will have a finite magnetic dipole moment represented by µ. A letter in bold −→ font indicates that it is a vector(S ≡ S ). µ is given by: q µ = g S (2.1) 2mc where q: Charge of the particle m: Mass of the particle c: Speed of light S: Spin of the particle. There are two kinds of particles in nature- fermions and bosons. In the standard model, all visible matter made of fermions. Bosons can be described as force carrying particles. All fermions that have spin 1/2 and different from its antiparticles are called Dirac fermions. The electron magnetic moment as defined by the standard model is very precisely calculated and coincides with observed value with extreme accuracy. It is one of the most precisely known quantity in physics[8]. However, recently there has been some disagreement found after analyzing data for the magnetic moment of the muon. The theoretical and observed value disagree after the

3 7th decimal place[1]. This disagreement could be resolved by using a diﬀerent theoretical approach or it might mean new physics in the future. Regardless, one would naively expect similar contributions for quarks in baryons.

2.2 Transition Magnetic Moment

In Refs.[56][57] the idea was conjectured that the neutron n can be trans- formed into a sterile neutron n0 that belongs to a hypothetical parallel mirror sector. This mirror sector is an exact copy of the ordinary particle sector with identical fermion content and identical gauge forces, different in the respect that the strong and electroweak forces described by the Standard Model (SM) act only between ordinary particles, and gauge forces of the mirror Standard Model (SM’) act only between mirror particles. The following Hamiltonian can be used to describe the time evolution of the mixed (n, n0) system in the background of uniform magnetic fields B and B0 and the possible presence of ordinary and/or mirror matter[2]:   m + V + µBσ + ηBσ + η0B0σ ˆ Hnn0 =   (2.2) + ηBσ + η0B0σ m0 + V 0 + µ0B0σ where V and V’ stand for n and n0 Fermi potentials induced respectively by ordinary and mirror matter, σ is a set of Pauli matrices, = nn0 is the mass mixing term and η and η0, are the Transition Magnetic moments(TMMs) between n and n0 related respectively to ordinary and mirror magnetic fields. Keeping present experimental limits in mind, the magnitude of nTMM is

−4 −5 suggested to be of the order of 10 − 10 µN [2]. We will probe in upcoming chapters whether the theory of neutron magnetic moment allows room for the existence of proposed nTMM.

2.3 Neutron Substructure

The neutron is a massive fermion found mostly within the atomic nucleus. It has zero charge and a rest energy of ∼939 MeV. The neutron is stable inside a β-stable nucleus but as a free particle, has a mean lifetime of 879 seconds or roughly 15 minutes[5]. The neutron is a constituent particle i.e. it is not an elementary particle. In QCD, a neutron is composed of a sea of

4 further elementary particles like quarks, antiquarks and gluons. A neutron can be classiﬁed as a baryon i.e. a subatomic particle which contains an odd number of valence quarks (at least 3). Quarks are Dirac fermions; they are fundamental constituents of matter that combine together to form neutrons and protons which make up most of the visible matter in the universe[5]. There are 6 kinds of quarks found in nature. The baryon octet that we are going to consider is made up of the three lightest ones- up, down and strange quarks. When magnetic moment of the electrically neutral neutron was detected it was one of the ﬁrst indication of the substructure of the neutron. Because the neutron has charge zero, as an elementary particle it would not have a magnetic moment (look at equation 2.1). The neutron is composed of 3 valence quarks- 2 down quarks and 1 up quark. However, in principle, all 6 quark types can contribute to the magnetic moment of the neutron via “sea quarks”[9].

2.4 Static quark model

2.4.1 Baryon Wave Functions

To calculate wave functions, the angular momentum of the baryon is assumed to come from the combined spin of the 3 valence quarks[4]. Also, we assume that the valence quarks are in ground state for the baryons i.e. l = l0 = 0 (more on this in chapter 5) Quarks are spin 1/2 particles therefore they can form either spin up(↑) or spin down(↓) states. Two quark spins combine to give total spin of either 3/2 or 1/2. Because we are primarily interested in the baryon octet (all of which have total spin 1/2), we will not consider here total spin 3/2 baryon states.

1 1 √ | i ≡ (↑↓ − ↓↑) ↑ / 2 (2.3) 2 2 12 1 −1 √ | i ≡ (↑↓ − ↓↑) ↓ / 2 (2.4) 2 2 12

5 Figure 2.1: Standard Model[38]

6 1 1 √ | i ≡↑ (↑↓ − ↓↑)/ 2 (2.5) 2 2 23 1 −1 √ | i ≡↓ (↑↓ − ↓↑)/ 2 (2.6) 2 2 23

1 1 √ | i ≡ (↑↑↓ − ↓↑↑)/ 2 (2.7) 2 2 13 1 −1 √ | i ≡ (↑↓↓ − ↓↓↑)/ 2 (2.8) 2 2 13

Equations 2.3 and 2.4 provide wave functions that are antisymmetric in particles 1 and 2 while equations 2.5 and 2.6 describe wave functions that are antisymmetric in particles 2 and 3. While equations 2.29 and 2.27 are not independent of equations 2.3-2.6, we will need them to construct the neutron wave function. From the Pauli exclusion principle and quantum mechanics, we know that two fermions cannot have the same wave function. Therefore, to accommo- date a wave functions for 2 fermions (which are indistinguishable particles), we arrive at: 1 ψ(1, 2) = √ [ψα(1)ψβ(2) − ψβ(1)ψα(2)] (2.9) 2 where one particle is in state ψα and one is in ψβ. We notice that the total wave function is antisymmetric. This must be kept in mind when constructing the wave function of baryons. For the wave function of a baryon, we have to consider 4 components of the quarks: Spatial component: Describing the location of 3 quarks Spin: Representing spins of 3 quarks Color: Specifying color of 3 quarks. Assigned “colors” can be red, green or blue. Because all naturally occurring particles are colorless, all the colors must add up to white Flavor: Indicating what combination of u, d and s quark is involved Therefore the wave function can be expressed as:

ψ = ψ(space)ψ(spin)ψ(color)ψ(flavor) (2.10)

7 Constrained by the condition that we must arrive at a colorless conﬁguration, for baryons, the color state is always given by: √ ψ(color) = (rgb − rbg + gbr − grb + brg − bgr)/ 6 (2.11)

The color wave function is the same for all baryons therefore is generally not included. Notice that the above state is antisymmetric under interchange of any 2 quarks. Therefore rest of the wave function has to be symmetric to keep the overall wave function antisymmetric. Because we are assuming ground state for the baryon, we know that ψ(space) is symmetric. Therefore, the product of ψ(spin) and ψ(ﬂavor) has to be symmetric.

2.4.2 Baryon Octet

All the 8 spin 1/2 baryons can be arranged in a figure with decreasing strangeness and so that particles having the same charge are in a straight line. This arrangement forms the baryon octet(figure 2.2). Like the color wave function, we also have to construct flavor wave functions in such a way that antisymmetric under the exchange of 2 quarks. For a wave function antisymmetric in particle 1 and 2, we get wave functions described in figure 2.3. For a wave function antisymmetric in particle 2 and 3, we get wave functions described in figure 2.4. For a wave function antisymmetric in particle 1 and 3, we get wave functions described in figure 2.5.

2.4.3 Calculation of Neutron Magnetic Moment

In the static quark model, net magnetic moment of a baryon is simply the vector sum of magnetic moments of the 3 valence quarks:

µ = µ1 + µ2 + µ3 (2.12)

We now know this model to be inaccurate but this was formulated in the early days of the quark model. It agreed very well with the values that we observed in experiments. The agreement was hailed as a tremendous victory of the quark model. Knowing what we know now, it is very surprising that the static quark model agreed so well with experiments. The reason why the agreement is

8 Figure 2.2: The Baryon Octet [4]

Figure 2.3: ψ12 [4]

9 Figure 2.4: ψ23 [4]

Figure 2.5: ψ13[4]

10 surprising is discussed in chapter 5. And some of the factors contributing to this surprising agreement are discussed and probed in Chapter 4. According to equation 2.1, the magnitude of the magnetic moment for a spin 1/2 particle would be given by(assume g=2): q µ = ~ (2.13) 2mc

To be speciﬁc, this is the value of µz in the spin up state. Substituting value of charges for u, d and s quarks in equation 2.13, we get:

2 e~ µu = (2.14) 3 2muc 1 e~ µd = − (2.15) 3 2mdc 1 e~ µs = − (2.16) 3 2msc

The spin/ﬂavor wave function of a neutron with spin up is given by:

1 1 1 1 |n : i = [ (↑↓↑ − ↓↑↑)(udd − dud) + (↑↑↓ − ↑↓↑)(dud − ddu)+ 2 2 2 2 √ 1 2 (↑↑↓ − ↓↑↑)(udd − ddu)] (2.17) 2 3 In the above equation, we have multiplied the spin component of the wave function of the neutron found in equations 2.3- 2.27 while the ﬂavor components are found in ﬁgures 4-6. We can further solve this equation to simplify it.

1 1 1 |n : i = √ [udd(↑↓↑ + ↑↑↓ −2 ↓↑↑) − dud(2 ↑↓↑ − ↓↑↑ − ↑↑↓) 2 2 3 2 − ddu(2 ↑↑↓ − ↑↓↑ − ↓↑↑)] (2.18)

1 1 1 |n : i = √ [udd(↑↓↑ + ↑↑↓ −2 ↓↑↑) + dud(−2 ↑↓↑ + ↓↑↑ + ↑↑↓) 2 2 3 2 + ddu(−2 ↑↑↓ + ↑↓↑ + ↓↑↑)] (2.19) √ 1 1 1 1 2 |n : i = √ u(↑)d(↓)d(↑)+ √ u(↑)d(↓)d(↓)− u(↓)d(↑)d(↑)]+permutations 2 2 3 2 3 2 3 (2.20)

11 The magnetic moment of the neutron (or any baryon) can be calculated by taking the expectation value of the sum of its quark moments:

µn = hn ↑ |(µ1 + µ2 + µ3)z|n ↑i (2.21)

3 2 X µn = hn ↑ |µi(Si)z|n ↑i (2.22) ~ i=1

µ = [µ ~ + µ (−~) + µ (~)] |u(↑)d(↓)d(↑)i (2.23) n u 2 d 2 d 2 Substituting values from equations 2.14-2.16, for the ﬁrst term of the wave function(from 2.20), we get:

1 2 2 ~ ~ ~ ( √ ) hu(↑)d(↓)d(↑)|µu + µd(− ) + µd( )|u(↑)d(↓)d(↑)i (2.24) 3 2 ~ 2 2 2 Because the ﬂavor and spin eigenstates form complete orthonormal bases, u2 =↑2= 1 1 µ [µ − µ + µ ] = u (2.25) 18 u d d 18 Similarly for the second term:

1 2 2 ~ ~ ~ ( √ ) hu(↑)d(↑)d(↓)|µu + µd(− ) + µd( )|u(↑)d(↑)d(↓)i (2.26) 3 2 ~ 2 2 2 Solving this equation, we again get: µ u (2.27) 18 Lastly for the third term of the wave function: √ 2 2 2 −~ ~ ~ ( ) hu(↓)d(↑)d(↑)|µu + µd + µd |u(↑)d(↑)d(↓)i (2.28) 3 ~ 2 2 2 2 (2µ − µ ) (2.29) 9 d u Instead of computing all 9 terms, the others of which are simply permutations, we add up equations 2.25, 2.27 and 2.29 and multiply the sum by 3 to get magnetic moment of the neutron 2 µ µ µ = 3[ (2µ − µ ) + u + u ] n 9 d u 18 18 2 2 µ µ = 2µ − µ + u + (2.30) 3 d 3 u 6 6 4 µ = µ − u 3 d 3 To calculate numbers, we need the actual mass of the quarks(recall 2.14 and 2.15). And here we make an approximation. While the rest energy for up

12 current quark is ∼2.3 MeV/c2 and down quark is ∼4.8 MeV/c2, we do not use these masses for said calculation. Most of the mass of the neutron (99%) is due to the ﬁeld energy of gluons. However, when calculating magnetic moment for the static quark model, we use constituent mass for our calculations. The constituent masses are

2 2 mu = md = 336MeV/c and ms = 510MeV/c [33] Constituent mass is obtained when we take into account eﬀects of “con- ﬁnement”. We can estimate the constituent mass value from the Heisenberg uncertainty principle[32]:

∆x∆px ∼ ~ (2.31)

As the quark is conﬁned within the proton radius of about a Fermi(10−15m):

∆x ∼ 10−15m (2.32)

Using natural units, where

1GeV −1 = 0.197 × 10−15m (2.33)

We get:

−16 ∆px ∼ ~/∆x ∼ 6.58 × 10 × 0.197eV GeV s

We also know that 1GeV −1 = 6.58 × 10−25s (2.34) ( −16 ((−(1 6.58 × 0.197 × 10 eV (GeV((( GeV ∆p ∼ x 6.58 × 10−25 ∼ 0.197 × 109eV

∼ 197MeV (2.35)

In 3 dimensions, q 2 2 2 ∆p ∼ (∆px) + (∆py) + (∆pz) (2.36) √ ∆p ∼ 3∆px ∼ 336MeV (2.37)

Because mq ∆p, we have: q 2 2 ∆E = (∆p) + mq ∼ ∆p (2.38)

It is this energy which is usually termed as “constituent mass”

13 For the strange quark, we get the constituent mass by applying the MIT bag model[34], which gives us the following formula for a single quark of mass m conﬁned to a sphere of radius R[33]:

ω = (m2 + x2/r2)1/2 (2.39) where ω ≡ constituent mass and x is a dimensionless number equal to 2.5[4]. For the strange quark with rest energy of ∼ 100 MeV, assuming a radius of 1fm: s 2.5 2 ω ∼ 10000 + 1/200 √ ω ∼ 260000

ω ∼ 540MeV (2.40)

Plugging these numbers in equations 2.14 and 2.15 and in 2.30, we get:

µn = −1.86µN (2.41)

e~ where µN is nuclear magneton ( ). 2mpc

This number is reasonably close to the experimentally tested value of µn =

−1.91µN . It has a diﬀerence of 2.7% which is impressive considering the uncertainty in quark masses and other assumptions that we will address in later chapters. An even more accurate prediction is obtained if we take ratios, where static quark model predicts the magnetic moment of the proton(µp) to be 1 µ = (4µ − µ ) (2.42) p 3 u d If we take the ratio between the predicted magnetic moment (equations 2.30 and 2.42), then we get: µ 2 n = − (2.43) µp 3 which agrees remarkably well with the experimental value −0.684[5].The results of this calculation are encouraging, but the masses of the up or down quarks were assumed to be 1/3 the mass of a nucleon. The masses of the quarks are actually only about 1% that of a nucleon. The discrepancy stems from the complexity of the Standard Model for nucleons, where most of their mass originates in the gluon ﬁelds, virtual particles, and their associated energy that are essential aspects of the strong force.

14 Furthermore, the complex system of quarks and gluons that constitute a neutron requires a relativistic treatment (Chapter 5). As we shall see, this agreement is deceptively close. Considering the incorrect assumptions the static quark model makes (Chapter 4), this agreement is not a true indicator of the accuracy of this model.

15 Chapter 3

Nature of Magnetic Moment

3.1 Nuclear Magnetic Resonance

Soon after the discovery of the neutron in 1932, there was evidence that it had a non-zero magnetic moment. By 1934 groups led by Otto Stern and I. I. Rabi in New York had independently measured the magnetic moments of the proton and deuteron[12][37][36]. Since a deuteron is composed of a proton and a neutron with aligned spins, the neutron’s magnetic moment could be inferred by subtracting the deuteron and proton magnetic moments. The discovery by Kellog, Rabi, Ramsey, and Zacharias[43][44] that the deuteron has an electric quadrupole moment indicated that the deuteron could not

3 be in a pure S1 state but must have some D-state admixture[45]. Thus the additivity of neutron and proton moments in the deuteron could not be exact. The ﬁrst actual measurement of µn was reported by Alvarez and Bloch in 1940[42]. They produced neutrons by deuteron bombardment of Be and used the neutrons in a single-coil Rabi-type magnetic resonance apparatus[3]. By 1939, Rabi had developed a method to measure magnetic moment using nuclear magnetic resonance. We will take a look at the underlying quantum mechanical principles behind his technique. The negative value for the neutron magnetic moment could not be explained and was unexpected. It would remain a puzzle till the development of the static quark model in the 1960s. Consider the exactly solvable problem of a two-level system with a sinu-

16 soidal oscillating potential:

Hˆ0 = E1 |1i h1| + E2 |2i h2| (3.1) Vˆ (t) = γeiωt |1i h2| + γe−iωt |2i h1| If only the level |1i is occupied at t = 0,

c1(0) = 1, c2(0) = 0, (3.2) then the probability of being found in each of the two states is given by Rabi’s formula[13]: 2 2 2 2 2 γ /~ 2 γ (ω − ω21) 1/2 |c2(t)| = 2 2 2 sin {[ 2 + ] t} (3.3) γ /~ + (ω − ω21) /4 ~ 4 2 2 |c1(t)| = 1 − |c2(t)| (3.4) where (E2 − E1) ω21 ≡ (3.5) ~ 2 We see that the probability |c2(t)| for ﬁnding the system in the upper state |2i exhibits an oscillatory time-dependence with angular frequency, two times that of s γ2 (ω − ω )2 Ω = + 21 (3.6) ~2 4

(E2−E1) Consider equations 3.3 and 3.6 at resonance condition(ω ' ω21 = ): ~ γ ω = ω12, Ω = (3.7) ~ Now consider a spin 1/2 system (bound electron) subjected to a time- dependent uniform magnetic ﬁeld in the z-direction and to a time-dependent magnetic ﬁeld rotating in the xy-plane:

B = B0zˆ + B1(cos ωtxˆ + sin ωtyˆ) (3.8) with B0 and B1 constant. The uniform t-independent Zeeman term corresponds to Hˆ0, while the t-dependent term corresponds to Vˆ . For e µˆ = Sˆ (3.9) mec we have e~B0 Hˆ0 = (|+i h+| − |−i h−|) (3.10) 2mec e B Vˆ (t) = ~ 1 [cos(ωt)(|+i h−| + |−i h+|) + sin(ωt)(−i |+i h−| + i |−i h+|)] 2mec (3.11)

17 2 Figure 3.1: Graph of |c2|max as a function of ω, where ω21 is the resonant frequency [39]

18 One can identify |+i ⇐⇒ |1i (aligned with the magnetic ﬁeld) and |−i ⇐⇒ |2i (anti-aligned with the magnetic ﬁeld). The angular frequency character- istic of the two-state system is:

|e|B0 ω21 = (3.12) mec which is just the spin-precession frequency for the B1 = 0 problem. After making the identification: −e B ~ 1 → γ, ω → ω (3.13) 2mec 2 We see that |c1| is the probability of finding a particle aligned with the 2 magnetic field while |c2| is the probability of finding a particle anti-aligned 2 with the magnetic field. In figure 3.1, we note that |c2| = 1 indicates that complete polarization has been achieved of our sample. Also note that our time dependent problem is precisely of the form (3.1). The resonance condition is satisfied whenever the frequency of the rotating magnetic field coincides with the spin-precession frequency (3.12). This resonance problem is how we make very precise measurements of the magnetic moment. The current value of magnetic moment is known to a very high accuracy: µn = -1.9130427 ± 0.0000005 µN [5]

3.2 Measurement of Neutron Magnetic Moment

In this section, we will briefly discuss an experiment to measure the neutron magnetic moment using the principles discussed in the preceding section. This experiment was performed by G.L. Greene et al[3]. In this experiment, the Larmor-precession frequencies for neutron and proton in the same magnetic field were compared. The experimental apparatus consisted of a separated-oscillatory-field magnetic resonance spectrometer, capable of measuring the neutron and proton Larmor frequencies almost si- multaneously. Two different guide tubes were used in the spectrometer: Middle Tube: Conducts either neutrons or water through the spectrometer Outer Tube: Conducts only water and serves as permanent field monitor

19 Figure 3.2: Schematic view of spectrometer with detail of proton polarization and detection equipment[3]

Figure 3.3: Schematic view of spectrometer with detail of neutron polarization and detection equipment[3]

20 As the middle tube conducted neutrons, a nearly simultaneous determination of the neutron resonance frequency in the middle tube and the proton resonance frequency in the outer tube was made. When the middle tube was ﬁlled with ﬂowing water, it was possible to make a nearly simultaneous measurement determination of the proton resonance frequency in both tubes. These two comparisons can be expressed as the following ratios:

ωnm R1 ≡ (3.14) ωpo

ωpo R2 ≡ (3.15) ωpm where ωnm is the neutron resonance frequency in the middle tube and ωpm,

ωpo are the proton resonance frequencies in the middle and outer tubes, respectively. The following reasonable assumptions are made:

1. The ratio of the average ﬁeld in the outer tube to the average ﬁeld in the middle tube is constant in time

2. The presence or absence of water in the middle tube does not aﬀect the ﬁeld average over the outer tube

3. The protons and neutrons take the ﬁeld average over the middle tube in the same way

The above assumptions allow one to arrive at the following expression:

ωn = R1R2 (3.16) ωp(cyl, H2O, θ) where ωp(cyl, H2O, θ) represents the resonance frequency for protons in a cylindrical sample of water at temperature θ.

◦ ◦ 0 µn ωn ωp(cyl, H2O, 22 ) ωp(sph, H2O, 22 ) µp = ◦ ◦ ◦ (3.17) µB ωp(cyl, H2O, 22 ) ωp(sph, H2O, 22 ) ωp(sph, H2O, 35 ) µB 0 where the notation “sph” denotes a spherical sample and µp is the eﬀective ◦ proton moment in a spherical sample of H2O at 35 C as measured by Philips, Cook, and Kleppner[50], and the sign is that determined by Rogers and Staub[51]. The ﬁrst ratio is the experimental result arrived at by the authors. The second ratio is given from the data summarized by Pople et al[52]. The third

21 ratio is taken from the results of Hindman[53] as interpreted in a footnote in Philips et al[50]. Using these results, the neutron magnetic moment in Bohr magnetons is given by: µ n = −1.04187564(26) × 10−3 (3.18) µB the above result can also be expressed in terms of the nuclear magneton µN . However, in this case the accuracy is slightly degraded due to an uncertainty in the electron to proton mass ratio. Using the value of m/M obtained by combining results from Philips et al[50] and Cohen and Taylor[54]: µ n = −1.91304184(88) (3.19) µN

3.3 Nature of Neutron Magnetic Moment

A magnetic dipole moment can be generated by two possible mechanisms[14]. One way is by a small loop of electric current, called an “Amp`erian”magnetic dipole. Another way is by a pair of magnetic monopoles of opposite magnetic charge, bound together in some way, called a “Gilbertian” magnetic dipole. In 1930, Enrico Fermi demonstrated that magnetic moments of nuclei are Amperian in nature[15]. It was later shown that magnetic moment of all elementary particles are Amperian in nature. For neutrons, scattering from ferromagnetic materials were able to provide proof to demonstrate that the neutron magnetic moment is Amperian in nature.

3.3.1 Probing the Theory

The magnetic ﬁeld of a magnetic monopole −→m, located at origin of coordi- nates, is given by[6]: 3ˆr(ˆr · m) − m B (r) = (3.20) 1 r3 where r is distance from origin Equation 3.20 is valid at distances large compared to distribution of magnetic charges producing −→m. The interaction of magnetic moment of electron and nuclear magnetic moment is given by:

H = −µe · B1(r) (3.21)

22 Putting 3.20 in 3.21(where m ≡ µn): 1 H = [µ · µ − 3(ˆr · µ )(ˆr · µ )] (3.22) r3 e n e n Interaction of nucleus’ magnetic moment with magnetic field of orbiting electron is given by: eL B (0) = (3.23) orbital mcr3 e 1 H = − L · µ (3.24) hfs mc r3 n Equations 3.23 and 3.24 account for magnetic part of hyperfine structure of atomic states with finite electronic orbital angular momentum. In first-order perturbation theory, the energy shift caused by this localized interaction is Z † 3 ∆Ehfs = − ψ (r)µe · BN (r)ψ(r)d r (3.25) r

23 The integral over solid angle is evidently a vector in the directionr ˆ0 (because that is the only direction that survives the integration over the directions of rˆ). It is straight forward to show that: Z rˆ 4π 0 r< dΩ 0 = rˆ 2 (3.31) r=R |r − r | 3 r> 0 where r<(r>) is the smaller (larger) of r’ and R. By assumption, r < R where J(r0) 6= 0. Thus equation 3.30 becomes Z 4π Z B(r)d3r = r0 × J(r0)d3r0 (3.32) r

0 ρM (r) and the resulting magnetic ﬁeld B (r). Then in complete analogy 0 0 with electrostatics one has B = −∇φM and Z ρ (r0) φ0 (r) = M d3r0 (3.35) M |r − r0| thus Z Z Z 3 0 3 2 0 B(r)d r = − ∇φM d r = −R rφˆ M dΩ (3.36) r

0 0 Again, r = r<,,R = r> and the integral of B is Z Z 3 4π 0 0 3 0 B(r)d r = − r ρM (r )d r (3.38) r

24 Thus one ﬁnds, in contrast to equation 3.34, Z 4π B(r)d3r = − m0 (3.40) r

3.3.2 Comparison with Experiment

The hyperﬁne energy shift(3.26) for s-states in one-electron atoms can evidently be written as 8π ∆E = − λ < µ · µ > |ψ(0)|2 (3.41) hfs 3 e N The above equation can be written as 8π ∆E = λ|µ ||µ | <σ · σ > |ψ(0)|2 (3.42) 3 e N e N where σ’s are Pauli Spin operators. We know that < σe · σN >= +1 for triplet and −3 for singlet, the energy splitting becomes 32π ∆E = |λ||µ ||µ ||ψ(0)|2 (3.43) 3 e N where 3 2 Z |ψ(0)| = 3 3 πa0n e~ |µe| = 2mec e~ µn = 2mpc

here nuclear moment is expressed in units of nuclear magneton(µN ). The energy splitting now becomes: 3 8|λ| |µN | me Z 4 2 ∆E = meα c (3.44) 3 µn mp n

25 Figure 3.4: Hyperﬁne splitting of ground state Hydrogen[40]

26 For atomic hydrogen, µp = 2.7928µN . ∆E The ground state energy splitting corresponds to a frequency( h ) of 1421|λ| MHz. When we compare this value with experiment, we find that the actual energy difference corresponds to ν = 1420.40575 MHz (This value is known with a high precision). This corresponds almost perfectly to λ = 1 or magnetic dipole moment due to circulating currents. Thus establishing beyond a shadow of doubt that magnetic dipole moments of nuclei are not due to isolated magnetic monopoles. Another example of a nucleus may be considered, namely 3He. It consists of a pair of protons and a neutron, all assumed in s-states. The protons are in a singlet spin state; the angular momentum and magnetic moment of 3He are presumably caused solely by the neutron. The measured magnetic moment(−2.1275µN ) is, however, different from the neutron’s moment

(−1.91315µN ) by about 11%, showing that non-central forces and exchange currents modify the simple expectation somewhat[6]. For the singly-ionized ground state, the hyperﬁne interval is observed to be νobs = 8665.649867 ± 0.000010 MHz[46]. With the known magnetic moment, equation 3.44 yields an anticipated value of ν = 8660.9|λ| MHz. At the level of choosing between λ = 1 and λ = −1/2, 3He shows that a bound neutron’s magnetism, as well as the 11% contribution from orbital motion and exchange currents, is caused by circulating currents. While we have established that magnetic moment of the neutron is due to circulating currents, it is very diﬃcult in visualising steady currents inside the dynamic baryon. There is no classical analogy for this phenomenon and remains an obscure quantum mechanical phenomenon.

3.4 Reﬂections of Neutrons from Magnetized Mirrors

D.J. Hughes and M.T. Burgy provided ﬁrst experimental proof of the nature of the neutron magnetic moment[59]. They studied intensity of reﬂected neutrons from a magnetized iron mirror.

27 3.4.1 Neutron Reﬂection Theory

For a mirror of a single element with small neutron absorption the critical angle θ is given by: c r Na θ = λ (3.45) c π where λ: Neutron wavelength N: Density of nuclei a: Average coherent scattering amplitude As the neutrons undergo reﬂection from a magnetized iron mirror, they ex- perience a two-valued potential, ±µB. It follows that there are just two possibilities for the index of refraction for neutrons in iron[62][63]: λ2Na µB n2 = 1 − ± (3.46) π E where E is the neutron energy. At the time, the hypothesis that magnetic moments of fundamental particles were due to presence of magnetic monopoles was initially championed by Bloch, among others[61] while circulating currents as the underlying cause(termed by the authors as “dirac formulation”) was an hypothesis used by Schwinger, among others[60]. The Dirac interaction, led to equation 3.46 while the Bloch interaction led to the same formula but B replaced with H(magnetic ﬁeld strength). One can thus experimentally measure the critical angle and therefore get information about the nature of magnetic interaction.

3.4.2 Experimental Method

Slow neutrons were attempted to be scattered from a magnetic mirror with the same polarization as the neutrons. Owing to the dependence of scattering amplitude a, a BeO filter was used to achieve a high intensity distribution of energetic neutrons with a sharp cutoff on the short wavelength side. The scattering of polycrystalline BeO is exceedingly small for neutrons of wavelength greater than 4.4A(twice the largest lattice spacing in BeO) hence a block of BeO surrounded by Cd, as shown in figure 3.5, acts as a neutron filter. The neutrons of wavelength less than 4.4A are scattered in the block

28 Figure 3.5: Plan view of apparatus for reﬂection of neutrons from magnetized mirrors[59]

29 and captured by Cd while those of longer wavelength are transmitted almost completely[59]. The incident angle, θ, is measured in terms of the angle, 2θ between the reflected and the direct beam. The intensity of the reflected beam is measured as θ is increased by moving the detector so as to keep it at an angle of 2θ to the direct beam. The intensity measured in this way will at first increase linearly with θ, because the mirror intercepts more of the incident beam as θ increases. After θc is passed, the intensity will drop as neutrons of wavelength greater than 4.4A are successively eliminated from the reflected beam[59]. The critical angles for this experiment are given by: r λ2Na µ(H + 4πCM) θ = ± f(φ) (3.47) c π E where f(φ) is a function causing the critical angle to depend on φ, the angle between neutron propagation direction and the the magnetic field. M is the density of magnetic dipole moments of the mirror The constant C(called Bloch’s constant) would be zero for the Gilbertian dipole and unity for the Amperian dipole. As the slow neutrons were scattered from magnetized iron mirrors, it was anticipated that if C=0, there would be no magnetic term because H is continuous across the boundary of the mirror i.e. field configuration H inside the magnet is the same as the external field[58]. However, if C = 1, then there would be two separated critical angles because of the term 4πM, causing a field discontinuity according to equation 3.47. The results of the experiment performed are given in figure 3.6. One can clearly see that the points agree quite well with the C = 1 curve, thus showing that the effective field in the iron is B instead of H and proving the Amperian model is correct for both the magnetic moment of the neutron as well as microscopic nature of magnetism.

30 Figure 3.6: Intensity of ﬁltered neutrons reﬂected from a magnetized iron mirror. Experimental points for two directions of magnetization, φ = 0◦ and φ = 90◦[59]

31 Chapter 4

Probing the Static Quark Model

4.1 Review of Baryon Magnetic Moment in PDG

As we have discussed in Chapter 2, magnetic moments of baryons can be calculated using quark-model wave functions also known as the static quark model. However, the static quark model makes certain assumptions to calculate the wave functions and magnetic moments. The particle data group (PDG), in a document dedicated speciﬁcally to baryon magnetic moments, lists a number of references that are used to arrive at the current understanding of baryon magnetic moments[16]. In an eﬀort to make a better model, it lists a number of references to papers further studying and probing magnetic moments. In the upcoming sections, we will be studying and unpacking 3 of these papers to better understand the recent advancements in theoretical calculation of baryon magnetic moments.

4.1.1 General Quark-Model Analysis

In this paper[17], sum rules and other parametrizations are used to test var- ious quark-model assumptions against the measured baryon magnetic moments. We add up diﬀerent baryons and compare their magnetic moments. Ex:

Σ+ − Σ− + Ξ− − Ξ0 = p − n = 4.70(4.04 ± 0.07) (4.1)

In the above equation, if one writes out a wave function (as dependent on quarks) of mentioned baryons, that should turn out to be equivalent. The fact that they don’t, is an indication that there is something that missing when using the static quark model to calculate baryon magnetic moment.

32 Table 4.1: Baryon magnetic moments(in nuclear magnetons) Baryon Experiment Static quark model p 2.79 2.68(5) n -1.91 -1.79(6) Λ0 -0.61 ±0.01 -0.62(0) Σ+ 2.38 ±0.02 2.58(16) Σ0 0.83 Σ− -1.1 ±0.05 -1 Ξ0 -1.25 ±0.01 -1.38(6) u 1.79 ±0.07 d -0.89 ±0.04 s -0.58 ±0.08

33 Table 4.1 lists values for baryons

The quark model prediction for baryon moments are: 4u − d0 p = (4.2) 3 4d − u0 n = (4.3) 3 4u − s0 Σ+ = (4.4) 3 4d − s0 Σ− = (4.5) 3 4s − u0 Ξ0 = (4.6) 3 4s − d0 Ξ− = (4.7) 3

In this chapter instead of writing magnetic moment of a baryon as µB, it will be represented as B i.e. µp ≡ p. The derivation of all these wave functions is done in the same manner as demonstrated in Chapter 2. The magnetic moment contribution of the unlike quark in the baryon is primed (u0, d0, s0). The guiding principle behind sum rules is that it can be probed how dominant nonstatic effects are. There are 3 major kinds of nonstatic effects identified by the author[17]:

1. Orbital Contributions: Small admixtures of higher orbital angular momenta in the baryons will alter the predictions of the static quark model. Recall that in our calculations, we assumed ground state for the baryon i.e. l = 0

2. Exchange Currents: Meson exchange or quark-antiquark pairs

3. Relativistic Eﬀects: Our calculations are based on the assumption that

v 2 inside the baryon, ( c ) << 1

As a few sum rules are probed, it is important to to keep in mind that one cannot just add up any baryons ad hoc. The sum rules in this paper form linear combinations of baryon moments that cancel out exchange current contributions as well, so that they depend only on relative quark charges[18]. Now the author considers a sum rule that gives a value for the strange quark and compares it with the value that is known experimentally

−Σ+ − 2Σ− (4.8)

34 4u − s0 4d − s0 − − 2 3 3 4u s0 8d 2s0 − + − + 3 3 3 3 4 s0 − (u + 2d) 3 Assuming u = −2d, just s0 remains. However, putting in values for equation 4.8 gives

−2.38 − 2(1.1) = −0.18µN

The above sum rule clearly disagrees with the value of s (or µs) that quark model gives us. Plugging in values in eq 2.15, Dirac moment gives s0 =

−0.59µN For the second sum rule, the author looks at the down quark:

d0 = Ξ0 − Ξ− = p + 2n = −1.03(−0.56 ± 0.04) (4.9)

In this equation, the hyperon (hyperon is any baryon having at least 1 s quark but no c, b or t quark) sum is given in the brackets while the nucleon sum is given outside it. In this sum rule, it is observed that despite both hyperons being the Ξ baryon, we get a significant difference from the theoretical value of µd = −0.89 ± 0.04. For the third sum rule, the author looks at the difference between s and d quark: s0 − d0 = 3(p − Σ+) = 1.23 ± 0.06 (4.10) 4u − d0 4u − s0 3 − 3 3 0 0 4u − d − 4u + s

s0 − d0 = 1.23 ± 0.06

This sum rule cancels out the similar u-quark contribution in the p and Σ+. This is a huge difference for quarks having the same charge even when the different masses are taken into account. This difference is so significant that large nonsymmetric(baryon dependent), nonstatic effects would be required to produce it. The conclusion that is reached from these sum rules is that nonstatic magnetic moment effects are large and highly baryon dependent, thus demonstrating clearly an incorrect assumption of the static quark model[17]. These

35 sum rules provide an eﬀective method in isolating nonstatic eﬀect that may be cancelling each other out if considering the magnetic moment of just one baryon.

4.1.2 Conﬁguration Mixing in Baryons

This paper[19] studies in detail the assumption of quark model which says that the constituent quarks are assumed to be in ground state i.e. they are assumed to have zero orbital angular momentum. The authors note that small admixtures of higher orbital angular momenta in the baryons will alter the predictions of the static quark model. It is argued that we can expect such admixtures to be more important for the Σ and Ξ hyperons than for the nucleons. The reason is that Σ and Ξ can mix with both octets and decimets of SU(3), while nucleons can mix only with octets(since ∆ baryons

3 have I= 2). Further, the authors claims that we can expect mixings in Σ and Ξ hyperons to be similar to one another, since both are allowed by isospin to mix with decimets as well as with octets. These qualitative arguments can be expressed using the following equations

µp = (A − B)µu + Bµd (4.11)

µn = (A − B)µd + Bµu (4.12)

0 0 µΣ+ = (A − B )µu + Bµs (4.13)

0 0 0 µΣ− = (A − B )µd + B µs (4.14)

0 0 0 µΞ0 = (A − B )µs + B µu (4.15)

0 0 0 µΞ− = (A − B )µs + B µd (4.16)

Again for ease of notation lets depict µX as X. Eliminating A’ and B’ from equations 4.13-4.16, the following 2 relations can be obtained: u − s Σ+ − Ξ0 = (4.17) d − s Σ− − Ξ− u + s Σ+ + Ξ0 = (4.18) d + s Σ− + Ξ− The equations can be further rearranged to get: u Σ+(Σ+ − Σ−) − Ξ0(Ξ0 − Ξ−) = (4.19) d Σ−(Σ+ − Σ−) − Ξ−(Ξ0 − Ξ−)

36 Figure 4.1: Baryon Decuplet[41]

37 Using values for hyperon moments in Table 4.1, the following ratio is obtained: u = −1.75 ± 0.05 (4.20) d u According to Dirac moments and the static quark model, d is expected to be −2, thus indicating a small but noticeable departure. Similarly, equations 4.17 and 4.22 can be rearranged to get s = 0.68 ± 0.02 (4.21) d This ratio, according to Dirac moments, should be 0.63. The configuration model asserts that the magnetic moment of the strange is more negative than predicted by the static quark model. This deduction means that the configuration mixing model predicts a more negative value for Ω−(which consists of 3 strange quarks). For further consideration, one can assume total orbital angular momentum of the baryon to be zero(Lq1q2q3 = 0) but finite orbital angular momenta between quark pairs(lq1q2 ). Lq1q2q3 = 0 implies that[19]:

A = A0 = 1 (4.22)

Using equations 4.11-4.16 and 4.22, we can get the values elaborated in table 4.2. We can use values from table 4.1 to get B and B’. Plugging in values in equation 4.12:

−1.91 = (1 − B)(−1.1) + B(1.98)

−1.91 = −1.1 + 1.1B + 1.98B

−0.81 = 3.08B 0.81 B = − = −0.263 (4.23) 3.08 Similarly, plugging values in equation 4.15:

−1.25 = (1 − B0)(−0.753) + B0(1.98)

−1.25 = −0.753 + 0.753B0 + 1.98B0

0.497 = 2.733B0 0.497 B0 = − = −0.182 (4.24) 2.733

38 Table 4.2: Comparison of quark magnetic moment predictions without and with conﬁguration mixing

Quark µtheory(no mixing) µtheory(mixing)

µu 1.852 1.983 ± 0.026

µd −0.972 −1.103 ± 0.026

µs −0.613 −0.753 ± 0.031

39 • The static quark model predicts that B=B’=−1/3. This also gives quan- titative proof that nucleons are much less mixed than hyperons.

• This also explains of the Ξ0 and Ξ− moments being much closer to one another experimentally than in the naive picture. The u and d contribute less to the Ξ0 and Ξ− (since B’ = −0.18) than in the naive model, where B’ = −1/3

Conﬁguration mixing model predicts that the magnetic moment for the strange quark would be eﬀectively more negative. Therefore, its prediction for mag-

− netic moment of Ω (which consists of 3 strange quarks) is −2.26µN , which is signiﬁcantly more negative than the quark model prediction(−1.84µN ).

The experimental value is: µΩ− = −2.02 ± 0.05µN [5]. While this experimental data fails to vindicate the theory of higher angular momenta in baryons, the measured value is in line with the prediction that magnetic moment of Ω− would be more negative than the value predicted by the static quark model. These calculations coupled with the latest lattice QCD data[20], indicates that there is a non negligible portion of the nucleon’s spin(and therefore magnetic moment) comes from quarks’ orbital angular momentum.

4.1.3 Precision Description of the Baryon Octet Mag- netic Moments

This paper[21] presents a phenomenological approach to ﬁxing the discrepancies between the predicted and measured values of the magnetic moments of the baryon octet. The disagreement as it stands will be quantized later in the thesis. In this section, it shall be demonstrated that the added SU(3) tensor term having isoscalar and isovector parts added to the quark-model expression reduces the χ2 per degree of freedom for the data. The magnetic moment operator can therefore be given as:

X µB = ai(B)µi + Tδ(B)δ (4.25) i=u,d,s where µi are individual quark contributions and Tδ is a single tensor with the strength parameter δ. ai are determined by the wave function of the baryon

40 as given in equations 4.2-4.7.

The strength parameter is given by:

δ = h8||Tδ||8i (4.26) where δ is calculated by evaluating the matrix elements of κ(anomalous magnetic moment operator) between states of baryon 8[64]. Anomalous magnetic moment operator is given by: X κ = TI,0,0 (4.27) µ,I where the sum on tensor operators is extended over the allowable irreducible representations of SU(3)[65].

If δ is set to zero, the quark magnetic moments can be determined µu, µd and µs by minimizing: X µB(expt) − µB χ2 = (4.28) ∆µ B B(expt) The χ2 test is a good indicator of quantizing the departure between predicted values and experimental values. There are 8 data points from the octets and

2 3 parameters(µu, µd and µs). Therefore the degrees of freedom are 5. χ for octet baryon’s magnetic moments is 402. This large χ2 has been termed a “baryon octet moment puzzle” in the CERN courier. The poor ﬁt to data suggests failure of static quark model. We know that the quarks interact with each other via gluons and also there is a sea of quark-antiquark pairs within a baryon. In a simplistic manner, Tδ(B) is expected to play such a role. This modiﬁed tensor can also be written in terms of isospin:

Tδ = I(2I − 1) + 4I3(I − 1) (4.29) where I and I3 are total baryon isospin and its third component respectively.

Solving Tδ for the neutron: 1 1 −1 1 2 × − 1 + 4 − 1 2 2 2 2 ¨ 1 ¨¨ 1 1 (1¨− 1) + 4 × × = +1 (4.30) ¨2¨ 2 2 Similarly we can solve for other baryons to get values in table 4.3.

In table 4.3, the authors have made an ad hoc change in the sign of Tδ for Σ− to reduce the discrepancy. This is the reason for calling it a “modiﬁed tensor”.

41 With 4 parameters, χ2 is calculated again and now found to be 5.12 which is signiﬁcantly better than what the static quark model without the tensor was giving us[21]. While this approach is purely phenomenological, the origin of this tensor is an interesting problem but the authors don’t suggest any solution of it in the paper.

42 Table 4.3: Modiﬁed tensor for the baryon octet

Baryon Modiﬁed Tδ p -1 n 1 Λ0 0 Σ+ 1 Σ0 1 Σ− 1 Ξ0 -1 Ξ− 1

43 Chapter 5

Deep Inelastic Scattering and Spin Crisis

5.1 The Proton Spin Crisis

As we have explored in Chapter 2, when the static quark model was formulated, it was assumed that the entirety of the baryon’s spin was entirely due to the spin of valence quarks. Two of the three spin 1/2 quarks will align with the spin of the proton, the third one is anti-aligned, and quite simply, the three quarks account for the spin 1/2 of the proton. It was the European Muon Collaboration(EMC) at CERN in the late 1980’s that found the contribution of the quark spins to the spin of the nucleon to be small and actually consistent with zero within margin of error[22]. This led to the “spin crisis” where physicists struggled to explain where the intrinsic angular momentum of the spin 1/2 nucleon comes from[23].

5.2 Deep Inelastic Scattering Experiments

Deep inelastic scattering(DIS) is an extension of Rutherford scattering to much higher energies of the scattering particle and thus to much ﬁner resolution of the components of the nuclei. The basic principle of experiments to reveal spin structure of the nucleon is to use DIS of longitudinally polarized leptons (muons in this case) oﬀ polarized protons, which are usually embed- ded in a gaseous or solid target. The polarized lepton will interact with a quark inside the polarized nucleon via the exchange of a spin 1 virtual photon. This photon can only be absorbed by the quark when the spins of the quark

44 Figure 5.1: Inside the proton[26]

45 and the photon are anti-aligned due to angular momentum conservation[23]. The cross section for this process can be measured for the two cases where the spin of the leptons and the nucleons is aligned and anti-aligned, and depend- ing on which cross section turns out to be bigger, one can learn if the quark spins tend to be aligned or anti-aligned with the spin of the nucleon. This cross section diﬀerence is usually measured as a function of Q2, the negative four-momentum squared of the virtual photon, and of x, the fraction of the momentum of the (fast) proton carried by the struck quark, and it is given

2 in terms of the polarized structure function g1(x,Q ). Using different polarized targets like hydrogen, deuterium, and 3He, one can even disentangle the different quark flavors[24].

3 5.3 Polarized He Neutron Scattering

Neutron scattering is a bit tougher than proton scattering because there are no free neutrons available. A Polarized 3He nucleus is regarded as a good model of a polarized neutron for deep inelastic scattering. As we have discussed, the 3He wave function is primarily in an S-state in which the two protons pair with opposite spins due to Pauli exclusion principle, leaving the neutron spin as the dominant contribution to spin-dependent scattering. A primary assumption in all studies is that isospin symmetry is valid. This assumption implies that the only diﬀerence between a proton and a neutron is that a proton has an up quark, which is equivalent to a down quark in the neutron. If this is true, then the total quark contribution is the same for a proton and a neutron[47]. The neutron scattering experiments are done, in part, to study the polar-

2 ized structure function g1(x, Q ). It can be interpreted as: 1 X g (x) = e2∆q (x) (5.1) 1 2 i i i where

+ + − − ∆qi(x) = qi (x) + qi (x) − qi (x) − qi (x) (5.2)

+ − and where qi (x) (qi (x)) is the density of quarks of ﬂavor i with helicity parallel (antiparallel) to the nucleon spin. This interpretation of g1(x) can be

46 Figure 5.2: Layout of the experimental setup. Two independent single-arm spectrometers are shown[47]

47 understood from the fact that a virtual photon with spin projection +1 can only be absorbed by a quark with spin projection 1/2, and vice versa. 0x0 is the Bjorken scaling variable, given by: Q2 x = (5.3) 2p · q where p and q are the four-vectors of the target nucleon and exchanged boson respectively. In a polarized 3He scattering experiment at SLAC[48], the spin structure function of the neutron was determined over the range 0.03 < x < 0.6. The total event sample amounted to ∼4×108 electrons collected in two single- arm magnetic spectrometers[49], at horizontal scattering angles of 4.5◦ and 7◦(ﬁgure 5.2). The integral of the spin structure function over the measured range of x is: Z 0.6 n g1 (x)dx = −0.019 ± 0.007(stat) ± 0.006(syst) (5.4) 0.03 at an average of Q2 of 2 GeV/c2. The results from this experiment in con- junction with the weak coupling constants from baryon decay[55] can be used to extract the integral over the quark spin distributions from the Quark- Parton model. The results yield ∆u = 0.93 ± 0.06, ∆d = −0.35 ± 0.04, and ∆s = −0.01 ± 0.06. These results imply that the total quark contribution to the nucleon spin (∆u + ∆d + ∆s) is 0.57 ± 0.11. Thus, the quarks contribute approximately one-half of the nucleon spin, and the strange sea contribution is small. Orbital angular momentum and the spin of the gluons may account for the remaining nucleon spin as we have discussed previously.

5.4 Results

The PHENIX collaboration at the Relativistic Heavy Ion Collider(RHIC) published results conﬁrming that most of the gluons’ contribution to the proton spin comes from the gluons with relatively low momentum[25]. To probe the gluons’ role, RHIC physicists collide two beams of protons with their spins aligned in the same direction, and then with the polarization of one beam ﬂipped so the spins are “antialigned”. The PHENIX detector mea- sures the number of particles called pions that come out of the collision zone

48 perpendicular to the colliding beams under these two conditions. Any diﬀer- ence observed in the production of these pions between the two conditions is an indication of how much the gluons’ spins are aligned with, and therefore contribute to, the spin of the proton[26]. The reason for this is that density of gluons increases very rapidly for very low momentum fractions and therefore low momentum gluons far outnum- ber gluons with high momentum. Lattice QCD calculations[27] along with results by the HERMES Collaboration at DESY[27] and the COMPASS Col- laboration at CERN[28] show that the valence quarks contribute about 30% to the spin of the nucleon, while 50% of the proton’s spin is due to gluon polarization and the other 20% can be assumed to be due to orbital angular momentum of the quarks.

49 Figure 5.3: The PHENIX detector at RHIC[26]

50 Chapter 6

Conclusion and Future Work

So in conclusion we reviewed the calculated neutron magnetic moment and compared it with the measured value. We saw that the error(2.7%) is large enough that the possibility of nn0 oscillation giving rise to a transition magnetic moment cannot be ruled out. The transition magnetic moment is of the

−4 −5 order of 10 − 10 µN and is not excluded[2]. Also, as we saw, the origin of magnetic moment of the nucleon is very tricky to understand because of its complicated internal structure and the underlying physics that is yet to be completely understood. What is equally surprising as the disagreements are the agreements. After probing the assumptions of the static quark model and ﬁnding them drastically inaccurate, it remains a puzzle why the static quark model is able to make predictions with a relatively high degree of accuracy. There are other aspects of this topic which can be better studied in the future. Lattice QCD provides a tool that can be used to calculate the baryon magnetic moment with increasing precision[10, 11]. On the experimental side, more accurate information will hopefully follow from the electron-ion collider that is proposed for construction at Brookhaven National Lab. This project would allow one to measure the spin contribution from gluons carrying as little as 0.1% of the proton’s momentum (a 50-fold improvement over present experiments), as well as to make detailed studies of observables dependent on quark and gluon orbital angular momentum[35].

51 References

[1] B. Abi et al. “Measurement of the positive muon anoma-lous magnetic moment to 0.46 ppm”. Phys. Rev. Lett., 126:141801 (2021)

[2] Z. Berezhiani, R. Biondi, Y. Kamyshkov and L Varriano arXiv:1812.11141v2 [nucl-th] (2019)

[3] G. L. Greene, N. F. Ramsey, W. Mampe, J. M. Pendlebury, K. Smith, W. B. Dress, P. D. Miller and P. Perrin, Phys. Rev. D 20, 2139-2153 (1979) doi:10.1103/PhysRevD.20.2139

[4] David Griﬃth. “Introduction to Elementary Particles”. Wiley-VCH (2008)

[5] M. Tanabashi et al (Particle Data Group) “Particle Physics Booklet”, Phys. Rev. D 98, 030001 (2018)

[6] J.D. Jackson. “The nature of intrinsic magnetic dipole moments”. CERN. 77-17: 1–25 (1977)

[7] Dirac, Paul A.M. “Principles of Quantum Mechanics”. Oxford University Press.

[8] B. Odom, D. Hanneke, B. D’Urso, G. Gabrielse. “New Mea- surement of the Electron Magnetic Moment Using a One- Electron Quantum Cyclotron”. Phys. Rev. Lett. 97 (3): 030801. doi:10.1103/physrevlett.97.030801.emm. (2006)

[9] K. Rith, A. Sch¨afer.“The Mystery of Nucleon Spin”. Scientiﬁc American (1999)

[10] A. Parre˜noet al., Phys. Rev. D 95, 114513 (2017)

[11] C. Bernard., Phys. Rev. Lett. 49, 1076 (1982)

52 [12] I.I. Rabi, J.M. Kellogg, J.R. Zacharias. “The magnetic moment of the proton”. Physical Review. 46 (3): 157–163. (1934)

[13] R. Paschotta. “Encyclopedia of Laser Physics and Technology”. Wiley- VCH (2008)

[14] Wikipedia: Magnetic moment.

https://en.wikipedia.org/wiki/MagneticmomentT worepresentationsofthecauseofthemagneticmoment

[15] E. Fermi. “Uber die magnetischen Momente der Atomkerne”. Z. Phys. 60 (5–6): 320–333. doi:10.1007/bf01339933 (1930)

[16] C.G. Wohl. “Baryon Magnetic Moments”. https://pdg.lbl.gov/2019/reviews/rpp2018-rev-baryon-magnetic-moment.pdf

[17] J. Franklin. “General quark-model analysis of baryon magnetic moments” Phys. Rev. D29, 2648 (1984)

[18] J. Franklin, Phys Rev. 182, 1607 (1969)

[19] L. Brekke and J.L. Rosner, Comm. Nucl. Part. Phys. 18, 83 (1988)

[20] HERMES Collaboration, A. Airapetian et al., Phys.Rev.D75:012007 (2007)

[21] S.K. Gupta and S.B. Khadkikar, Phys. Rev. D36, 307 (1987)

[22] European Muon Collaboration, J. Ashman et al., Phys.Lett.B206:364 (1988)

[23] F. Ellinghaus. “Revealing the Spin Structure of the Nucleon” https://www.bnl.gov/rhic/news/042208/story3.asp

[24] J. Adam, Phys. Rev. D 99, 051102(R) (2019)

[25] Phys. Rev. D 93, 011501(R) (2016)

[26] K.M. Walsh. “Physicists zoom in on gluons’ contribution to proton spin” https://www.bnl.gov/newsroom/news.php?a=26123

[27] HERMES Collaboration, A. Airapetian et al., Phys.Rev.D75:012007 (2007)

53 [28] COMPASS Collaboration, V. Yu. Alexakhin et al., Phys.Lett.B647:8 (2007)

[29] Wikipedia: Neutron Magnetic moment.

https://en.wikipedia.org/wiki/Neutronmagneticmoment

[30] Wikipedia: Electron Magnetic moment.

https://en.wikipedia.org/wiki/Electronmagneticmoment

[31] M. Thomson “Modern Particle Physics”. Cambridge University Press (2013)

[32] B.R. Holstein. American Journal of Physics, 63, 14 (1995)

[33] F.E. Close. “An Introduction to Quarks and Partons”. Academic Press (1979)

[34] Chodos et al., 1974a,b

[35] S.D. Bass. “Spinning Gluons in the Proton”. https://physics.aps.org/articles/v10/23

[36] Esterman, I.; Stern, O. (1934). “Magnetic moment of the deuton”. Physical Review. 45 (10): 761(A109). Bibcode:1934PhRv...45..739S. doi:10.1103/PhysRev.45.739. Retrieved May 9, 2015.

[37] Rabi, I.I.; Kellogg, J.M.; Zacharias, J.R. (1934). “The magnetic moment of the deuton”. Physical Review. 46 (3): 163–165. Bib- code:1934PhRv...46..163R. doi:10.1103/physrev.46.163.

[38] Wikipedia: Standard Model.

https://en.wikipedia.org/wiki/StandardM odel

[39] J.J. Sakurai “Modern Quantum Mechanics”. Cambridge University Press

[40] Gonzalez, Arcadi. (2014). Constraining the Epoch of Cosmic Reionisa- tion.

[41] Alexandrou, Constantia Baron, R. Carbonell, Jaume Drach, Vincent Guichon, P. Jansen, K. Korzec, Tomasz P`ene,O.. (2009). Low-lying baryon spectrum with two dynamical twisted mass fermions. Phys. Rev. D. 80. 10.1103/PhysRevD.80.114503.

54 [42] L. Alverez and F. Bloch, Phys. Rev. 57, 111 (1940)

[43] J. M. B. Kellog, I. I. Rabi, N. F Ramsey, and J. R. Zacharias, Phys. Bev. 55, 318 (1939)

[44] J. M. B. Kellog, I. I. Babi, N. F. Ramsey, and J. B. Zacharias, Phys. Rev. 57, 677 (1940)

[45] N. F. Ramsey, Molecular Beams (Oxford University, New York, 1956), pp. 319-323

[46] H.A. Schuessler, E.N. Fortson and H.G. Dehmelt, Phs. Rev. 187, 5 (1969)

[47] E. W. Hughes, R. Voss “Spin Structure Functions”. Annual Review of Nuclear and Particle Science 49:1, 303-339 (1999)

[48] P. L. Anthony (1993), Determination of the neutron spin structure function. Physical Review Letters. 71. 959-962. 10.1103/PhysRevLett.71.959.

[49] G. G. Petratos et. al., Report No. SLAC-PUB-5678 (1991)

[50] W.Philips, W.E.Cook, and D.Kieppner, Phys. Rev. Lett. 35, 1619 (1975)

[51] E. H. Rogers and H. B. Staub, Phys. Rev. 76, 980 (1949)

[52] J. A. Pople, W. G. Schneider, and H. J. Bernstein, High Resolution Nuclear Magnetic Resonance (McGraw-Hill, New York, 1959)

[53] J. C. Hindman, J. Chem. Phys. 44, 4582 (1966)

[54] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973)

[55] R. L. Jaﬀe and A. V. Manohar, Nucl. Phys. B337, 509 (1990)

[56] Z. Berezhiani and L. Bento, “Neutron–mirror neutron oscillations: How fast might they be?,” Phys. Rev. Lett. 96, 081801 (2006) [hep- ph/0507031]; Z. Berezhiani and L. Bento, “Fast neutron–Mirror neutron oscillation and ultra high energy cosmic rays,” Phys. Lett. B 635, 253 (2006) [hep-ph/0602227]

[57] Z. Berezhiani, “More about neutron–mirror neutron oscillation,” Eur. Phys. J. C 64, 421 (2009) [arXiv:0804.2088 [hep-ph]]

55 [58] Mezei, F. (1986). “La Nouvelle Vague in Polarized Neutron Scat- tering”. Physica. 137B (1): 295–308. Bibcode:1986PhyBC.137..295M. doi:10.1016/0378-4363(86)90335-9.

[59] Hughes, D. J.; Burgy, M. T. (1951). “Reﬂection of neutrons from magnetized mirrors”. Physical Review. 81 (4): 498–506. Bib- code:1951PhRv...81..498H. doi:10.1103/physrev.81.498.

[60] J. Schwinger, Phys. Rev. 51, 544 1937

[61] F. Bioch, Phys. Rev. 50, 259 (1936);51, 994 1937

[62] Halpern, Hamermesh, and Johnson, Phys. Rev. 59, 981 (1941)

[63] A. Achieser and J. Pomeranchuck, J. Exper. Theor. Phys. U.S.S.R. 18, 475 1948

[64] T. P. Cheng and H. Pagels, Phys. Rev. 172, 1635 (1968)

[65] R.B. Teese, Phys. Rev. D 24, 1413 (1981)

56 Vita

Abhyuday Sharda was born in Delhi, India to parents Namita and Apoorv Sharda. He graduated from St. Columba’s School, New Delhi in 2012. He graduated with a B.Tech. in Nanotechnology from Amity University, Noida in 2016. He graduated with a masters in Physics from The University of Tennessee, Knoxville, in Summer, 2021.