Evaluating the Electromagnetic Form Factors of Light Nuclei a Thesis

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Evaluating the Electromagnetic Form Factors of Light Nuclei

A thesis presented to the faculty of the College of Arts and Sciences of Ohio University

In partial fulﬁllment of the requirements for the degree Master of Science

This thesis titled

Evaluating the Electromagnetic Form Factors of Light Nuclei

by KELLEN J. MURPHY

has been approved for

the Department of Physics and Astronomy and the College of Arts and Sciences by

Daniel Phillips Professor of Physics and Astronomy

Howard Dewald Interim Dean, College of Arts and Sciences 3 Abstract

MURPHY, KELLEN J., M.S., August 2011, Physics Evaluating the Electromagnetic Form Factors of Light Nuclei (60 pp.)

Director of Thesis: Daniel Phillips We present an overview of electromagnetic form factors for light nuclei (A = 1, 2, 3). Our discussion revolves around presenting the technology to evaluate form factors; functions which allow one to encode electromagnetic structure into simple, analytic functions. We evaluate the charge form factor for Deuterium and Helium-3, using well-known numerical techniques, and discuss the application of chiral perturbation theory to form factor calculations. Approved: Daniel Phillips

Professor of Physics and Astronomy 4

To my father. 5 Acknowledgments

I would like to thank Dr. Daniel Phillips for his guidance throughout the course of this project, Matthias Schindler for answering questions when Daniel wasn’t around, and my colleagues for not distracting me too much during the course of this work. 6 Table of Contents

Page

Abstract ...... 3

Dedication ...... 4

Acknowledgments ...... 5

List of Tables ...... 7

List of Figures ...... 8

1 Introduction ...... 9 1.1 Single Nucleon Form Factors ...... 12

2 Chiral Perturbation Theory ...... 15 2.1 QCD and Chiral Symmetry ...... 15 2.2 Transformation Properties ...... 18 2.3 Constructing the OPE Lagrangian ...... 20 2.4 Power Counting ...... 22 2.5 Form Factor Calculations ...... 25 2.6 A Quick Word on the NN Interaction ...... 25

3 Deuterium ...... 27 3.1 The One-Pion Exchange Potential ...... 29 3.2 Deuteron Wavefunctions ...... 31 3.3 Evaluation of F(q2) ...... 35 3.3.1 Determination of Analytic Form of GC ...... 37

4 Helium-3 ...... 45 4.1 Three-Nucleon Potentials ...... 46 4.2 Structure of the 3N Wavefunction ...... 47 4.3 Constructing the Charge Form Factor ...... 49 4.4 Results and Discussion ...... 52

5 Summary and Outlook ...... 55

References ...... 57

Appendix: Monte Carlo Methods ...... 59 7 List of Tables

Table Page

3.1 Comparison of ([8]) results to the results of our code. Good agreement in all quantities indicates that our code works well for generation of wavefunctions. . 34 8 List of Figures

Figure Page

1.1 Diagram schematically showing lowest-order electron-proton elastic scattering (with momentum assignments). The hadronic vertex is left as a large circle because the ﬁne details of the interaction at this vertex are unclear. We encode the ﬁne details of this interaction inside the form factors...... 12

2.1 Tree-level (top diagram) and one-loop order diagrams in χPT...... 24

3.1 Elastic electron-deuteron scattering through exchange of one photon...... 27 3.2 Wavefunctions obtained via integrating in the OPEP. ([20]) ...... 32 3.3 Momentum-space wavefunctions ψ0 and ψ2...... 34 3.4 Position-space wavefunctions u0 and u2 ...... 34 3.5 Tree-level diagram for e−d scattering. The photon interacts with only one nucleon. Two-body eﬀects are neglected...... 35 3.6 Breit frame kinematics for tree-level elastic e−d scattering...... 36 2 1 3.7 Our results for the form factor F(q ) evaluated to q = 100 MeV 0.5 fm− . .. 42 2 ≈ 1 3.8 Our results for the form factor F(q ) evaluated to q = 2 GeV 5 fm− ...... 42 ≈ 3 4.1 Breit frame kinematics for tree-level elastic e− He scattering...... 49 4.2 Charge form factor for Helium-3, evaluated for 10,000 Monte Carlo points for 1 each of 500 values of q in the region q [0, 5]fm− ...... 53 4.3 Charge form factor for Helium-3, evaluated∈ by [19]. Our work closely matches the impulse approximation (IA) data...... 54 9 1 Introduction

Quantum Chromodynamics (QCD), a nonabelian gauge theory of the strong nuclear force, is asymptotically free. This means that as the distance between particles (say, quarks inside a proton) decreases, the strength of the interaction between them becomes small; that is, they behave as if they were free for asymptotic energies. If one were to attempt to pull apart two strongly bound quarks, for example, as the distance between the quarks became large, the binding energy becomes suﬃciently high that we begin producing new particle-antiparticle pairs. This explains the absence of observed color monopoles in nature. It was for their discovery of asymptotic freedom that David Gross, David Politzer, and Frank Wilzcek won the 2004 Nobel Prize in Physics. ([6, 15]) But it is this odd (yet interesting) facet of QCD that causes diﬃculties for the application of the theory to the evaluation of low-energy observables. The usual approach

of applying a perturbation series expansion to the Lagrangian for calculation of observables fails below particle physics energies ( 1 GeV), since it is only at such high ≈ energies that the strong nuclear coupling, αs, becomes sufficiently small that the series expansion in terms of αs converges and perturbation theory gives calculable results. One method of getting around the low-energy problems encountered in perturbative QCD is to employ an effective field theory approach. The goal of an effective field theory is the calculation of meaningful results within some specified region of interest, via expressing the Lagrangian of some “higher” theory in terms of operators which apply only in the regime of interest. Typically such field theories work by developing a perturbation series in some parameter which is small (and hence sufficient for application to

p perturbation theory) only in some limited region, such as Λ , which allows for perturbation series convergence only below some given energy (i.e. Λ). For example, in this work we discuss one such eﬀective theory: Chiral Perturbation

Theory (χPT). χPT is an eﬀective ﬁeld theory constructed as an approximation to QCD 10

which is consistent with the well-known and well-understood chiral symmetry of QCD.

Unlike perturbative QCD (pQCD), which uses quark and gluon ﬁelds as the relevant degrees of freedom, χPT uses nucleon and pion ﬁelds as explicit degrees of freedom. χPT does this, because unlike pQCD, rather than being a perturbation series expansion in the

p strong coupling parameter αs, it is a series expansion in (where Λχ is the scale of Λχ chiral-symmetry breaking in QCD, which is approximately the nucleon mass). This establishes the domain of applicability of the theory. Namely, we expect our results only to hold for momenta suﬃciently below the nucleon mass. We will further discuss Chiral Perturbation Theory in Chapter 2. Our ultimate goal in this thesis is an exploration of the electromagnetic (speciﬁcally,

charge) form factors of light nuclei. We focus principally on Helium-3, however we demonstrate the technology of form factors step by step for A=1 and A=2 nuclei before contemplating the more diﬃcult three-body system. We seek this study of electromagnetic form factors because of the global importance of electron scattering experiments.

Electron-scattering experiments represent a vast and wide-ﬁeld of physics research to date, because of the ﬂexibility and fundamental nature of electron-scattering. Deep Inelastic Scattering (DIS), in particular, has opened up a wide range of new understanding of nuclei. Deep Inelastic Scattering occurs when a high-energy electron interacts with and

breaks apart a nucleus, allowing one to probe the internal structure of hadrons; it has been used to make several advances in the world of nuclear physics. For example, DIS gave the ﬁrst dynamical evidence for the quark substructure of hadrons, and has been used as evidence that the structure of the weak nuclear force (i.e. that responsible for β-decay) has the same nature as that which is predicted by the standard model. ([22]) Furthermore, DIS

and other electron-scattering events open up a wide range of systematic nuclear observables; such basic nuclear properties as magnetic moments, as well as more 11

sophisticated properties such as nuclear response functions (which are roughly the

analogue of form factors in scenarios involving inelastic scattering). Form factors are relevant in the theoretical sense because they allow us to image the charge and current distributions of nuclei. However, before we get ahead of ourselves diving into complicated nuclear systems it is best to start simply. We therefore conclude

this chapter with a discussion of how electromagnetic form factors allow us to parametrize our ignorance of the underlying structure of the nucleons. The things we learn from studying the form factors of single nucleons will be valuable when we must build-up the form factors for larger nuclei. We move to our ﬁrst discussion of compound nuclei in Chapter 3, whence we discuss

the technology used to construct form factors for a simple two-body bound state; the deuteron (2H nucleus). We seek to compute the charge form factor for the deuteron, however, rather than pursue a detailed calculation scheme, we seek to merely demonstrate the technology for form factor calculations. Hence, we treat the deuteron as a simpliﬁed

construct of a proton and a neutron bound in an L = 0 S-wave only. We ﬁnally study form factor calculations in detail in Chapter 4, whence we discuss the details of evaluating the charge form factor for Helium-3. This more complicated problem uses wavefunctions obtained from well-known and well-tested potential models

(AV18+UIX), which we discuss, but without great detail, providing references for more thorough investigation. We then compute the charge form-factor using a very popular numerical technique known as Monte Carlo integration, which is well-suited to the higher-dimensional integrations which we encounter (we discuss the details of Monte Carlo integration in Appendix A).

Now begins our discussion of form factors for single nucleons, as well as a discussion of ways in which these form factors encode all of the electromagnetic structure information into just a few analytic functions. 12

1.1 Single Nucleon Form Factors

Consider for the moment elastic electron-proton scattering. If the proton were structureless, we can obtain the form of the cross-section by looking at the electron-muon elastic diﬀerential cross-section ([7]), and obtain the desired form by merely replacing the

muon mass everywhere with the proton mass:

dσ α2 E θ q2 θ = ′ 2 2 . 2 θ cos 2 sin (1.1) dΩ lab 4E2 sin  E ( 2 − 2M 2)  2    Here M = 938 MeV is the mass of the proton, θ is the scattering angle, and the factor

E 1 ′ = (1.2) E + 2E 2 θ 1 M sin 2 arises from target recoil.

Figure 1.1: Diagram schematically showing lowest-order electron-proton elastic scattering (with momentum assignments). The hadronic vertex is left as a large circle because the ﬁne details of the interaction at this vertex are unclear. We encode the ﬁne details of this interaction inside the form factors.

The transition amplitude for this cross section (at lowest-order in α) is given by ([7]):

1 µ 4 T f i = i jµ J d x, (1.3) Z q2 !

µ where q = k k′ is the momentum transfer. The currents j and J represent electron and − µ

proton transition currents, respectively. The electron transition current jµ may be written down from QED:

i(k k) x j = e u(k′) γ u(k) e ′− · , (1.4) µ − µ 13

where k and k′ are the four-momentum of the incoming and outgoing electron respectively. We may guess at the form of the hadronic current from QED, based on what it would be were the proton elementary. We ﬁnd that

µ µ i(p′ p) x JQED = +e u(p′) γ u(p) e − · , (1.5)

where p and p′ are the incoming and outgoing proton four-momentum, respectively, and u (u) represents the Dirac spinor for the outgoing (incoming) electron, with γµ the usual Dirac matrices. To incorporate information about the proton’s structure into the hadronic current we modify the QED vertex factor (γµ ), substituting

κ γµ F (q2) γµ + F (q2) iσµνq . (1.6) → 1 2M 2 ν

This substitution arises because we know that ultimately Jµ must be a four-vector, so we must build our structure-encoding vertex factors from the set of all operators constructable

µ 5 µ from p, p′, q and the Dirac matrices (γ and γ ), such that at the end of the day J transforms as a four-vector under Lorentz boosts. We ﬁnd that there are only two

µ µν independent terms with these properties, γ and iσ qν, which are exactly the terms

appearing within our substitution. The functions F1 and F2 are the “form factors”, or analytic functions which weight the various structure components of the current, that tells

one how much each term matters. (Note that vertex factors involving γ5 are neglected because of parity conservation in electromagnetic processes.) Ultimately, the hadronic current then is

κ µ 2 µ 2 µν i(p′ p) x J = +e u(p′) F1(q ) γ + F2(q ) iσ qν u(p) e − · . (1.7) 2M Here κ is a constant which represents the anomalous magnetic moment of the proton (the diﬀerence between the proton’s gyromagnetic ratio and it’s nuclear g factor). In this, and throughout, we invoke Einstein summation notation, whence internal indices always 14

indicate summation. The form factors F1 and F2 are independent functions of the momentum transfer and are directly related to observables, as we shall see in a moment. Using this form of the hadronic current yields a diﬀerential cross section of the form:

dσ α2 E κ2q2 θ q2 θ = ′ 2 2 2 ( + κ )2 2 , 2 θ F1 2 F2 cos 2 F1 F2 sin (1.8) dΩ lab 4E2 sin  E ( − 4M ! 2 − 2M 2)  2      2 in analogy with Eq. (1). Note the dependence of the second termon(F1 + κF2) . If we let

κq2 G = F + F (1.9) C 1 4M2 2

GM = F1 + κF2, (1.10)

Eq. (8) becomes

dσ α2 E G2 + τG2 θ θ = ′ C M cos2 2τG2 sin2 , (1.11) dΩ  2 2 θ  E 1 + τ 2 − M 2 lab 4E sin 2 ( ! )     with τ = q2/4M 2. Since there are no interference terms (i.e. G G ), these form factors − C M may be regarded as functions closely related to the charge and magnetic moment

distributions of the proton.

As we have learned, the form factors F1 and F2 (or, equivalently, GC and GM) encode all of the electromagnetic structure information for a nucleon into two analytic functions. Though for a single nucleon the constituent charges stem from quarks, we shall see in the

coming chapters how the electromagnetic structure of nuclei (arising from the proton and the neutron inside the nuclei) can be encoded in a similar manner. First though, we now move past this terse theoretical discussion of the form factors for a single nucleon, and discuss the eﬀective ﬁeld theory known as Chiral Perturbation Theory, and it’s application for electromagnetic structure calculations. 15 2 Chiral Perturbation Theory

Effective field theories are approximations to more complicated, fundamental theories which grant access to calculations which are difficult or impossible given the full theory. Generally, they are low-energy approximations to the underlying theory, which maintain particular important facets of that more fundamental theory. Chiral Perturbation

Theory (χPT) is a perfect example of one such theory. Because QCD exhibits a phenomenon known as asymptotic freedom, the low-energy regime of QCD cannot be studied by a perturbative version of QCD which maintains quarks and gluons as the active degrees of freedom. The well known chiral symmetry of QCD (to be discussed in just a moment) allows us to construct an eﬀective ﬁeld theory known as Chiral Perturbation Theory; a theory constructed such that the approximate chiral symmetry of quantum chromodynamics (QCD) is maintained. It is ideal for studying the low-energy behavior of QCD, as instead of quarks and gluons, it features nucleons and light mesons (i.e. pions) as active degrees of freedom. Moreover, we can use χPT to construct matrix elements involving baryons that allows the calculation of static properties of nuclei such as magnetic moments and form factors. In this chapter we present an outline of Chiral Perturbation Theory and its application for electromagnetic structure calculations, following the method of ([18]).

2.1 QCD and Chiral Symmetry

QCD, the theory of the strong force, gives the mathematical model for interactions between quarks (of which there are six ﬂavors: up, down, strange, charm, bottom, and top) and gluons (the strong force mediator particle). Quarks are electromagnetically charged fermions which also carry an additional type of charge intrinsic to strongly interacting particles: color, a unit of charge in an SU(3) space, which we label as red, green, and blue. Gluons, as the strong mediator, carry two units of color (allowing 16

interactions between quarks) and are hence referred to as bicolor objects. One of the most

interesting facets of QCD is that gluons can interact with other gluons. This is not a common trait shared by the photons of QED, as only gauge bosons associated with a non-Abelian gauge theory may self-interact, since noncommutativity within the gauge group leads to self-interaction terms in the Lagrangian.

The fundamental Lagrangian for QCD is:

1 = q i D m q a µ ν,a, (2.1) LQCD f − f f − 4g2 Gµν G f =u,X d, s, c, b, t

where the q f represent the quark ﬁelds of ﬂavor f , and the gauge covariant derivative enters as

µ µ a D = γ Dµ = γ ∂µ + igTaGµ . (2.2) Here, T (a = 1 ... 8) are a set of lineraly independent 3 3 matrices, and Ga are the a × µ (eight) gluon gauge ﬁelds. The tensor

a = ∂ Ga ∂ Ga gf GbGc (2.3) G µν µ ν − ν µ − abc µ ν

a represents the gluon field tensor, and is a function of the gluon gauge fields Gµ, the gluon coupling g, and the QCD structure functions fabc. Two of the quark flavors (bottom and top) are rather heavy compared to the nucleon mass; they are neglected from this discussion because we would like to consider only the largest contributions to further calculations. The strange quark mass ( 80 - 120 MeV) and charm quark mass ( 1 GeV) ≈ ≈ are also sufficiently large that we limit it’s consideration as well; were we to work in an

SU(3) framework and seek to include the effects of these larger mass quarks, we would allow them to enter as diagrams involving hadrons constructed of suck quarks (i.e. we write down kaon diagrams if we seek to study the effect of the strange quark). For this discussion, however, we limit ourselves to only the two lightest quark flavors (up and

down, each with a mass on the order of a few MeV). 17

If we consider a nucleon (say, a proton, with mp = 938.3 MeV), one ﬁnds that the quark content of the particle does not make up suﬃcient mass to yield the proton mass:

mp >> 2mu + md. (2.4)

That is, most of the mass in the nucleon has nothing to do with the mass of the quanta. Since this masses are intrinsically small (compared to the mass of the nucleon) we should be able to let the quark mass terms drop out of our Lagrangian, hence (neglecting flavor indicies) 1 0 = q i D q µν. (2.5) LQCD f f − 4g2 Gµν G fX=u, d When the quark masses vanish in the Lagrangian in this “chiral limit”, if we decompose the quark fields into left- and right-handed fields, by defining the projection operators

1 P = (1 + γ ) P2 = P L 2 5 L L 1 P = (1 γ ) P2 = P R 2 − 5 R R

PL + PR = 1

one finds 1 = qL i D qL + qR i D qR µν. (2.6) LQCD f f f f − 4g2 Gµν G fX=u, d n o That is, the left- and right-handed portions of the Lagrangian decouple: modifying the left-handed fields do not affect the right-handed fields and vice-versa. That the quark-fields decouple in this manner allows us to identify the symmetry of this

Lagrangian as SU(2) SU(2) , where “2” comes from the number of quark ﬂavors L × R present. This is the so-called “chiral symmetry” of QCD. If one does not look at the limit of quark masses vanishing, the introduction of mass terms in Eq. (5) leads to additional terms:

1 = qL i D qL + qR i D qR + qL qR + qR qL µν, (2.7) LQCD f f f f f M f f M f − 4g2 Gµν G fX=u, d n o 18

where

mu 0 =   (2.8) M    0 md    is the quark mass matrix for QCD. Note that introduction of this term introduces explicit SU(2) SU(2) “chiral symmetry breaking”, since the Lagrangian is now no longer L × R separable into left- and right-handed components which remain invariant under left- and right-handed transformations.

2.2 Transformation Properties

Our goal is to now brieﬂy demonstrate the technology for constructing the

Lagrangian of χPT, but before we may do this, we must explore the way the relevant ﬁelds transform under operations of the various left- and right-handed transformations, which we subsume in the so-called “chiral symmetry group,” G. We move to express the Lagrangian in terms of the low energy degrees of freedom of QCD. Working in an SU(2)

representation of the chiral symmetry group,

G = SU(2) SU(2) , (2.9) L × R these degrees of freedom are nucleons and pions. Furthermore, we require that this explicit chiral symmetry of the Lagrangian be maintained as one invokes the chiral limit

m m 0. (2.10) u ≈ d ≈ To consider transformation properties of the dynamical (pion) ﬁelds, we introduce them in terms of the following:

3 π0 √2π+ φ(x) = τi φi(x) =   . (2.11)  0  Xi=1  √2π− π   −    We then further encode these ﬁelds into the unitary matrix  φ(x) U = exp i , (2.12) F0 19

where F 93 MeV is a free parameter, related to the decay π+ µ+ ν , and is referred to 0 ≈ → µ as the “pion decay constant in the chiral limit.” ([18]) The symmetry group G deﬁned in Eq. (9) has elements

g = L, R G, (2.13) { } ∈ under which U transforms as:

U RU L† U† LU† R† → →

∂ U R ∂ U L† ∂ U † L ∂ U † R† µ → µ µ → µ with a requirement that in constructing the chiral Lagrangian, it must remain invariant under these transformations.

Thus far, we have limited the discussion to pions, but we now turn turn our attention to nucleons. Because the proton and neutron form an isospin doublet, we introduce in the same fashion the nucleon spinor: p Ψ =   . (2.14)  n      This spinor will enter the Lagrangian in any term which deﬁnes nucleon properties (i.e. the nucleon mass term: mNΨΨ) or interactions (such as the nucleon-pion interaction terms, which go like λΨφΨ). The question to consider now is that of “how does the nucleon spinor transform under operations in the symmetry group G?” Consideration of the multiplet

U Φ   , (2.15) ≡  Ψ      one ﬁnds ([18]) that the multiplet transforms as 

RUL† Φ Φ′ =   , (2.16) →    K(L, R, U)Ψ      20 where the nucleon transformation matrix K is deﬁned as

1 − K(L, R, U) = √RUL† R √U. (2.17) The most important thing to note is not the overall structure of the nucleon transformation matrix, but that this transformation depends explicitly on the pion spinor U. It is more readily expressed in terms of the reduced pion spinor, u;

u(x) = U(x) u′(x) = √U = √RUL , (2.18) → ′ † p whence Eq. (17) takes the form:

1 K(L, R, U) = u′− (x)Ru(x). (2.19)

2.3 Constructing the OPE Lagrangian

We seek now to write down the form of the one-pion exchange (OPE) Lagrangian; that Lagrangian which contains terms of up to one pion field (and a single nucleon field). We do this to illustrate a basic principle: the terms which appear in the Lagrangian for chiral perturbation theory are fundamentally constructed to be consistent with the desired symmetry (in this case, that defined by G). We already have a representation for the desired objects of interest (the nucleon and pion fields) and we know something about how they transform under the desired symmetry group, as illustrated above. We will take the information we have gleaned about the transformation properties of the fields, together with the fact that fundamentally we are seeking a power series expansion in the most general effective Lagrangian which is parametrized by

• counting the number of pion ﬁelds, and by

• counting the number of derivatives of the ﬁelds.

Using this expansion we construct the simplest of our Lagrangian terms, those with a single pion ﬁeld and the fewest number of derivatives. 21

For these transformations, given the characteristic nature of the transformation, we

must introduce a covariant derivative which transforms in the same manner as the ﬁeld:

D Ψ K(L, R, U) D Ψ. (2.20) µ → µ

We a priori expect that the covariant derivative will contain U(x) and U†(x) (invariably, as the reduced ﬁeld u(x) = √U(x) and its adjoint). We also expect that derivatives of the U will enter (since K is a function of L, R, and U). The covariant derivative which maintains this requirement (for the nucleon doublet) is given by ([18]):

D Ψ = ∂ + Γ iv(s) Ψ, (2.21) µ µ µ − µ where the connection term is

1 Γ = u† ∂ ir u + u ∂ il u† . (2.22) µ 2 µ − µ µ − µ h i

These expressions hold for nucleons in the presence of external ﬁelds rµ (right-handed), lµ

(s) (left-handed), and vµ (singlet vector field). We expand the Lagrangian in terms of number of fields and number of derivatives of fields, requiring that the Lagrangian be invariant under the transformation of the nucleon spinor defined above, and of the pion spinor defined in the previous section. In doing this, one can pick out the terms in the net Lagrangian which correspond to the most general effective πN Lagrangian. This Lagrangian describes processes involving a single nucleon in the initial and final state, and hence has the structure ΨOˆ Ψ. Here Oˆ is an operator defined in the space of Eq. (15) above, which transforms under our chiral symmetry G as

Oˆ K Oˆ K†. (2.23) →

The most general Lagrangian which meets these requirements, at lowest order (i.e. with the fewest number of derivatives) is

g (1) A µ πN = Ψ i D mN + γ γ5uµ Ψ. (2.24) L − 2 22

This Lagrangian contains two parameters not deﬁned by chiral symmetry, the nucleon

mass mN and the axial coupling parameter gA. Note that in Eq. (24) above, we deﬁne uµ in terms of the pion ﬁelds and their derivatives:

uµ = iu†DµUu† = uµ†. (2.25)

2.4 Power Counting

We seek to discuss the application of chiral perturbation theory to the evaluation of electromagnetic form factors, and hence now turn to a discussion of the relevant diagrams which contribute to such calculations, up to a rather low order in the perturbation theory expansion. To do this, invoke the power counting scheme of Weinberg ([18]); we do this

because in order to calculate anything in an effective field theory, one must generally write down all possible terms in the Lagrangian compatible with the desired symmetry. In this case, we already know our symmetry (SU(2) SU(2) ), and have even written down L × R one example Lagrangian, before defining how the power counting plays out in general. First, we need some method to organize the most general effective Lagrangian such that we assess the importance of each term, and order the terms such that the largest contributions correspond to lowest order in the expansion, and so on. To do this, we organize our Lagrangian by successive, increasing numbers of derivatives (of the Goldstone Boson, or pion, fields) which appear in the Lagrangian, hence

(1) (2) (3) ﬀ + + + ... (2.26) Le ≈L L L where (N) represents a Lagrangian with N derivatives (or matching power) in its terms. L We continue the method of ([18]) in outlining the power counting scheme:

• Weinberg’s scheme analyzes the behavior of a given diagram under a scaling of spacetime, which is linear for all external momenta (pµ tpµ, µ = 1, 2, 3) and a → quadratic rescaling of masses (m t2m). → 23

• A loop integration contributes a factor of p4.

2 • Pion propagators contribute p− .

1 • Fermion propagators contribute p− .

• Vertices arising from (k) contribute pk. While in general those arising from (2k) LπN Lπ contribute p2k.

Under this scheme, the “chiral dimension”, D, of a given diagram is deﬁned (in four-dimensions) by

∞ ∞ D = 2N + 2 E + 2(k 1) (2k) + (k 1) (k), (2.27) L − N − Nπ − NπN Xk=1 Xk=1 where

• (2k) is the number of vertices in (2k), Nπ Lπ

• (k) is the number of vertices in (k) , NπN LπN

• EN is the number of external nucleon lines,

and

• NL is the number of independent loop momenta.

We may illustrate the chiral dimension of some diagrams by choosing now the diagrams which are relevant for evaluation of electromagnetic form factors. We choose to work in

terms of the second order (k = 2) in χPT, hence we choose the most general eﬀective

(1) (2) Lagrangian with the fewest number of derivatives (i.e. ﬀ = + ). For the top Le LπN Lπ

diagram, the number of loops NL = 0, the number of external nucleon lines is EN = 1, the number of vertices from (2) is (2) = 0, and the number of vertices from (1) is (1) = 1, Lπ Nπ LπN NπN hence the order of the ﬁrst diagram is D = 1. The remaining diagrams in Figure 2.1 are all 24

Figure 2.1: Tree-level (top diagram) and one-loop order diagrams in χPT.

suppressed by the presence of an additional loop and hence have D = 3. (Note that D = 1 in the top diagram is for the charge form factor calculation. If you were to attempt

evaluation of the magnetic form factor, one would pick up and addition factor of p/mN in

the spatial components of Jµ, hence the diagram would be D = 2). For an eﬀective theory in small mass and momenta, the scaling parameter t is less than one (0 < t < 1), and the amplitude for a process described in such a theory is suppressed by

(tp, t2 m) = tD (p, m). (2.28) M M 25

Hence with higher and higher chiral dimension, diagrams become further suppressed; we

can see that the term with lowest chiral dimension indeed corresponds to the tree level diagram at the top of Figure 2.1.

2.5 Form Factor Calculations

If one were to begin evaluation of the electromagnetic form factors for, say, the single nucleon, we can see that at lowest order the diagram is the same as the top diagram in Figure 2.1. However, contributions at next order involve pion loops at the nucleon photon interaction vertices. These loop diagrams require, in addition to (1) , above, second order LπN (in the number of derivatives) Lagrangians:

(1) (2) ﬀ = + , (2.29) Le LπN Lπ where 2 2 (2) F0 µ F0 = Tr D U(D U)† + Tr χU† + Uχ† . (2.30) Lπ 4 µ 2 n o n o In all of these Lagrangians,

uµ = iu†DµUu† = uµ†, (2.31) and

mu 0 χ = 2B   . (2.32)    0 md      One could similarly extend the calculation to even higher powers in the perturbative expansion, doing the same steps to construct the Lagrangians (expanding and power counting), however, we merely wish to illustrate the point.

2.6 A Quick Word on the NN Interaction

The Lagrangians derived from χPT actually give potentials corresponding to eﬀective ﬁeld theory analogues of the potentials one encounters in using an empirical approach to 26

evaluate the form factors. We can construct chiral potentials which would be the

analogues of, say, the AV18+UIX potential we use for Helium-3 in this work. In doing so one can approach the evaluation of form-factors in a slightly diﬀerent light. The current picture of the nucleon-nucleon force is one based on meson-exchange, which we discuss in Chapter 4. The derivation of the potentials from Chiral Perturbation

Theory provides an alternative to this approach which has been used to great success. ([5]) 27 3 Deuterium

The deuteron (2H nucleus) is an isospin singlet bound state of a single proton and a single neutron. It is loosely bound, with binding energy (B 2.25 MeV) much smaller d ≈ than the average binding between a pair of nucleons in other stable nuclei.([24]) Different aspects of the underlying nuclear structure of the deuteron can be obtained through analysis of scattering of various probes at different energy scales. As a relevant example, consider elastic electron-deuteron scattering. The tree-level diagram for such a process involves exchange of a single virtual photon. The figure (3.1) illustrate an example diagram, but note that photon-deuteron vertex here would include many possible interactions, i.e. the photon interacting with one nucleon, the photon interacting with some particle being exchanged between two nucleons, etc.

Figure 3.1: Elastic electron-deuteron scattering through exchange of one photon.

• When the exchanged photon’s momentum-transfer is approximately 45 MeV

(roughly 6/rm, where rm is the matter radius of the deuteron), we are able to discern that the deuteron is not elementary.

• At momentum-transfers on the order of the pion mass (m 100 MeV) we begin to π ≈ resolve the short-distance ( 2 fm) structure of the deuteron wavefunctions. ≈ 28

• And at momentum-transfers of roughly the nucleon mass (m 1 GeV) we begin to N ≈ resolve the short distance structure of the nucleons.

The deuteron is an isospin singlet (I = 0) boson composed of two nucleons whose total spin is J = 1. This tells us that the deuteron could have two channels: an S-wave with L = 0 and a D-wave with L = 2. Further evidence that there is a nonzero orbital angular momentum channel arises from the fact that the deuteron has a nonzero quadrupole

2 moment. It is known from experiment ([1]) that Qd = 0.2859(3) fm , implying a spherically asymmetric wavefunction. Because an L = 0 S-wave is always spherically symmetric, there must be a higher angular-momentum channel in addition to the S-wave. ([3])

As in the case of the proton, the deuteron has both an electric charge form factor (GC) and a magnetic-moment form factor (GM). Furthermore, the nonzero electric quadrupole moment implies the presence of GQ, the electric quadrupole form factor. In this work we

2 consider elastic electron scattering oﬀ of the deuteron in order to compute GC(q ) only. Our calculations consider only single photon interactions. Two photon-exchange

corrections will be further suppressed by a factor of α. First, we discuss the One-Pion Exchange Potential (OPEP), using it as an example to illustrate a method for determining deuteron wavefunctions. We then discuss a more sophisticated potential model, the Charge-Dependent Bonn Potential (CD Bonn), which

we use to construct the deuteron charge form factor GC, in an approximation where the S-channel is the only contributing channel. Because the S-channel gives the dominant (95%) contribution, this is tantamount to approximating the deuteron as a spin-zero boson, hence neglecting higher orbital-angular momentum interaction terms. This calculation is

not meant to be a deﬁnitive calculation of the form factor, but rather an illustrative example of the technology used to construct form factors for light nuclei. For a recent accurate theoretical description of the deuteron form factors, see [20]. 29

3.1 The One-Pion Exchange Potential

We work in an energy regime where we expect the long-range form of the nucleon-nucleon potential to be dominated by exchange of the lightest strongly interacting particle, the pion. One may construct a perturbation series expansion in the Lagrangian for

this system, keeping only terms up to those corresponding to single-pion exchange. In this approximation the wavefunction, M P , for the deuteron can be expressed in | i a non-relativistic decomposition ([12]):

d3 p M P = P p u˜ L(p) (LmL S mS J M) pˆ LmL S mS (3.1) | i Z (2π)3 | i | h | i| i XL,S mX1,m2 where (Lm S m J M) are the Clebsch-Gordon coeﬃcients, pˆ Lm are spherical L S | h | L i harmonics, and the radial components enter as spherical Bessel transforms of the radial functions:

L u˜ L(p) = 4πi dr jL(pr) ruL(r) (3.2) Z and furthermore where L 0, 2 . ∈{ }

One method for the determination of the wavefunctions uL is by solving the usual nonrelativistic radial Sch¨odinger equation:

d2u (r) l(l + 1) l + u (r) + MV (r) u (r) = γ2u (r) (3.3) − dr2 r2 l ll′ l′ − l lX′=0,2 30

where γ √MB 1, with B the binding energy of the deuteron. Note that there is a sum ≈ d d over potentials which mix S- and D-state wavefunctions; these potentials are of the form:

mπr 2 e− V00 = mπ fπNN (3.5) − mπr e mπr 3 3 = = 2 − + + V02 V20 mπ fπNN 1 2 (3.6) − mπr mπr (mπr) ! e mπr 6 6 = + 2 − + + , V22 mπ fπNN 1 2 (3.7) mπr mπr (mπr) ! where g2 m2 = A π fπNN 2 (3.8) 16 π fπ is the pion-nucleon-nucleon coupling obtained from the Goldberger-Treiman relation ([14]). This is related to the so-called “pion decay constant”, f 92 MeV, a numerical π ≈ quantity arising from charged pion decay.2 To determine the wavefunctions completely requires solving this set of coupled second-order diﬀerential equations (Eq. 3) for the

wavefunctions u(r) = u0(r) and w(r) = u2(r). For boundary conditions, assuming that at some large radius (compared to the scale deﬁned by the scale of the potential; in this case r = 20fmisasuﬃciently large choice), all potentials vanish (i.e. all particles stop interacting), yields

d2u (r) l(l + 1) l + u (r) = γ2u (r). (3.9) − dr2 r2 l − l 1 The full γ is actually somewhat more sophisticated than this:

2 4M2 M2 M2 M2 M2 2 p n − d − p − n γ 2 , (3.4) ≡ 4Md

however, in the approximation that Mn Mp and Md = Mp + Mn, we recover the simpler version. 2 ≈ This is the physical value of the pion-decay constant. In chapter 2, when referencing F0 we were discussing the pion-decay constant in the chiral limit. 31

The solution to this diﬀerential equation is easily determined using analytic techniques in

this asymptotic region. The form of the wavefunctions is:

γr u(r) = AS e− (3.10)

γr 3 3 w(r) = η AS e− 1 + + (3.11) γr (γr)2 ! where η is the Asymptotic D/S-Ratio:

η = AD/AS . (3.12)

The wavefunctions u(r) and w(r) are presented in Figure 3.2. The S-wave amplitude, AS , is set as an overall normalization requiring

dr u2 + w2 = 1. (3.13) Z h i The ﬁgure (3.2) illustrates the deuteron wavefunctions evaluated from the OPE potential.

The oscillations near zero are caused by out method of integrating in from inﬁnity to ﬁnd the solution, and not of an indication of a breakdown in the model. However, we circumvent the problem of treating the one-pion exchange potential at zero by selecting an alternate method of constructing the deuteron wavefunctions; one which is fundamentally based on meson-exchange, but which uses a large space of operators to construct the potential than the simple OPE model.

3.2 Deuteron Wavefunctions

Evaluating the electromagnetic form factors for the deuteron requires wavefunctions which are more accurate than those generated via integrating the Schr¨odinger equation with the one-pion exchange potential. We generate these more accurate wavefunctions using the Charge-Dependent Bonn Potential (CD-Bonn) following the procedure of ([8]). The charge-dependent Bonn nucleon-nucleon potential is fundamentally based on meson exchange. It stems from the comprehensive ﬁeld-theoretic construction of the 32

u(r) w(r)

0.4

0.2

-0.2 0 5 10 15 20 25 r (fm) Figure 3.2: Wavefunctions obtained via integrating in the OPEP. ([20])

nucleon-nucleon force developed at the University of Bonn in the 1980’s. The Bonn Full Model was a necessary development to test the validity of the meson-exchange concept of nuclear forces and to determine the extent of that validity. ([9]) The charge-dependent Bonn model is a nucleon-nucleon potential which reproduces important charge-dependent

characteristics of the Bonn Full Model while avoiding the problems which arise in application of the Bonn Full Model. (By deﬁnition, charge independence is invariance under any rotation in isospin space, with violation of this symmetry referred to as charge-independence breaking.) Because the charge-dependence of the Bonn Full Model

is reproduced accurately, it is called the Charge Dependent Bonn NN Potential. The CD-Bonn potential includes meson exchange terms for all non-strange mesons with mass

less than the nucleon mass (mN), with the largest contributions occurring due to pions. The potential models are ﬁt to pp-, nn-, and np-scattering data, and are adjusted to deliver the empirical deuteron binding energy. ([8])

The deuteron wave functions in momentum-space we denote by ψL(p), with subscript L indicating the orbital angular momentum of the state, i.e. for the S-wave L = 0 and for 33

the D-wave L = 2. From ([8]), we ﬁnd a closed expansion form for these functions:

2 11 C ψ (p) = j (3.14) 0 rπ p2 + m2 Xj=1 j 2 11 D ψ (p) = j (3.15) 2 rπ p2 + m2 Xj=1 j

with C j and D j ﬁt coeﬃcients obtained from ([8]). The mass terms are

m = γ + ( j 1)m , (3.16) j − 0

1 with m0 = 0.9 fm− and

2 1 γ = 0.2315380 fm− , (3.17) a parameter closely related to the binding energy of the deuteron (γ2 MB ). We can ≈ d

obtain the position-space wavefunctions uL(r) via Bessel transform:

uL(r) 2 ∞ 2 = dp p jL(pr)ψL(p) (3.18) r rπ Z0 with jL(pr) the usual spherical Bessel functions. To test the accuracy of our wavefunction generation routines we evaluate several quantities of relevance for comparison to values presented in ([8]), including the normalization condition

∞ 2 2 N = dr u0(r) + u2(r) , (3.19) Z0 n o the matter radius 1/2 1 ∞ r = dr u2(r) + u2(r) , (3.20) d 2 0 2 (Z0 ) the quadrupole moment

1 ∞ Q = dr r2 u (r) 2 √2 u (r) u (r) , (3.21) d 20 2 0 − 2 Z0 h i the D-state probability

∞ 2 PD = dr u2(r) (3.22) Z0 { } 34 and the asymptotic D/S-ratio, η. Our results in comparison to [8] are presented in Table I.

The wavefunctions generated by Eqs. (18) and (19) are presented in Figures 3.3 and 3.4.

Table 3.1: Comparison of ([8]) results to the results of our code. Good agreement in all quantities indicates that our code works well for generation of wavefunctions.

Deuteron Properties

Quantity [8] Our Results

N 1 0.999901

rd (fm) 1.966 1.960915

2 Qd (fm ) 0.270 0.270013

PD 0.0485 0.048562 η 0.0256 0.025813

ψ (p) 0 u (r) ψ 0.5 0 2(p) 10 u2(r)

) 0.4 ) -1/2 -1/2 fm 0.3 fm ( L 5 ( L ψ u 0.2

0.1 0 0 1 2 3 4 5 0 -1 0 5 10 15 20 p (fm ) r (fm) Figure 3.3: Momentum-space wave- Figure 3.4: Position-space wavefunc- functions ψ0 and ψ2. tions u0 and u2 35

3.3 Evaluation of F(q2)

The charge form factor, F(q2), provides detailed information about the charge

distribution of the deuteron. Schematically, the e−d scattering process can be envisaged as follows. In Figure 3.5, a deuteron enters (say, from the right), and undergoes some

Figure 3.5: Tree-level diagram for e−d scattering. The photon interacts with only one nucleon. Two-body eﬀects are neglected.

transformation which decouples the nucleons (represented by the black region). Then, the

electron scatters off of one of the constituent nucleons (N1 or N2), via exchange of virtual photon (only photon shown) carrying momentum q. The nucleons, one of which now has a different momentum, then recombine into a deuteron. If we were to naively count the number of diagrams (and hence, integrals) contributing to the total calculation we obtain two; one for the photon striking the neutron (in Fig. 3.5 above N1 would be the neutron and N2 the proton) and another for the photon striking the proton (N1 a proton, N2 the neutron). However, as we shall see, the contribution from both diagrams can be evaluated 36 from one integral. There are, of course, other diagrams corresponding to higher-order terms in the χPT series expansion. These higher order terms correspond to various exchange mechanisms between the two nucleons and here are neglected for the time being because the tree-level one-body (above) diagrams contribute most significantly to the form factors.

We work the calculation in the Breit Frame (or ”Brick Wall“ Frame) which deﬁnes nucleon momenta as follows in Figure 3.6. Initial and ﬁnal deuteron momenta are

Figure 3.6: Breit frame kinematics for tree-level elastic e−d scattering.

P P represented by and ′ respectively, with pd (pd′ ) the initial (ﬁnal) relative nucleon momenta. Characteristic observables for this process (such as cross section, etc.) are related to the invariant amplitude for this process, . This amplitude is related to the Med transition amplitude for electron-deuteron scattering by

3 (3) P′ k′ T P k = i(2π) δ (P P′ + q) . (3.23) h | ed| i − − Med 37

This transition amplitude may be evaluated as

3 3 d pd d pd′ Ted = Ψ(P′) p′ k′ p′ TeN k p1 pd Ψ(P) . (3.24) Z (2π)3 Z (2π)3 h | d ih 1 | | ih | i

The relationship between the ﬁnal state ( f ) to initial state (i) transition amplitude T f i and the invariant amplitude is well-established (see [7]), and merely a generalization of Eq. M (24);

3 (3) T = i(2π) δ (p p′ + k k′) . (3.25) f i − − − M Hence in Eq. (23) above we may readily replace the elastic electron-nucleon transition amplitude (T ) with the invariant amplitude for this same process via substitution eN MeN deﬁned by Eq. (24). The invariant amplitude for elastic electron-nucleon scattering is well known,

2 e 2 µ κ 2 µν = u(k′) γ u(k) u(p′) F (q ) γ + F (q ) iσ q u(p) . (3.26) MeN q2 µ 1 2M 2 ν n o We seek to use this invariant amplitude to arrive at the the form of the charge form factor for the deuteron. As we shall see, this invariant amplitude enters as

∞ 3 GC = ψ p′ eN p ψ d p. (3.27) Z0 h | iM h | i

Let us arrive at this form of GC now.

3.3.1 Determination of Analytic Form of GC

Substitution of the form of Eq. (24) in for TeN in Eq. (25), gives

3 3 3 (3) d pd d pd′ q Ted = i(2π) δ (P P′ + q) Ψ(P′) pd + pd Ψ(P) − − Z (2π)3 Z (2π)3 h | 2 ih | i P P (3) ′ q eN δ − + ×M 2 2 ! 3 (3) i(2π) δ (P P′ + q) , ≡ − − Med hence 3 3 P P d pd d pd′ q (3) ′ q ed = Ψ(P′) pd + pd Ψ(P) eN δ − + . (3.28) M Z (2π)3 Z (2π)3 h | 2 ih | iM 2 2! 38

However, the forms of and are speciﬁed by electrodynamics: for single photon MeN Med exchange, in general e2 − jlep Jµ, (3.29) MX ≡ q2 µ X where X denotes the baryonic current in a general process (i.e. ed-scattering, eN-scattering). Note that the leptonic current for both ed and eN scattering is the same, − − and like the overall factor (e/q)2, cancels in Eq. (29). Let us discuss now, the relevant − µ µ currents: Jed and JeN . For eN-scattering, the form of the current comes by examining the relevant Dirac bilinears in the current: P P µ ′ 2 µ κ 2 µν JeN = u + pd′ F1(q ) γ + F2(q ) iσ qν u + pd . (3.30) 2 ! 2M 2 !

If we examine this current at q = 0 to examine it’s form in a simple setting, F1(0) = 1 and

F2(0) = 0 (i.e. elementary nucleons), so after some algebra we ﬁnd that it reduces greatly: P µ + pd µ = 2 JeN,proton , (3.31) mN for interactions on the proton. This is because, in general ([7]), pµ u(p)γµ u(p) = . (3.32) m There are inherently two cases with elastic electron-deuteron scattering: one case where the incident virtual photon scatters oﬀ of the neutron, and another when the incident virtual photon strikes the proton. At this point, our momentum in equation Eq. (32) holds

only for the proton, that is, Eq. (32) represents the proton interaction current. For the interaction on the neutron, since the neutron carries momenta P p , the neutron 2 − d interaction current reads P µ pd µ = 2 − JeN,neutron , (3.33) mN where for simplicity we for the moment have assumed the charge of the neutron and the proton to be the same, a fact we shall correct at the end of the calculation by assuming a 39

particular parameterization of the relevant nucleon’s form factor. This enable us to

demonstrate the vector nature of the current, and that the isovector structures cancel out in summing the proton and neutron terms, while not worrying about what constant is involved with each current until the necessary point in the calculation.

µ For ed-scattering, let us again examine the form of the current Jed. Because we treat the deuteron, in this case, as a spin-0 boson, the form of the current stems directly from the electrodynamics of spin-0 particles ([7]):

P P µ µ ( + ′) 2 Jed = F(q ). (3.34) mN

Note that there is now a form factor, tentatively called F(q2) that is NOT a priori ﬁxed at

one. This form factor is for the interaction on deuterium and will become (in just a

moment), GC. At q = 0, however, P′ = P, so the form of Eq.(34) simpliﬁes;

µ µ 2 (P) Jed = F(0). (3.35) mN

With these deﬁnitions of the interaction currents we ﬁnd that Eq. (29) becomes:

2 (P)µ d3 p 2 = = d Ψ P Ψ P F(q 0) 2 3 ( ′) pd pd ( ) (3.36) mN Z (2π) h | ih | i P µ ( pd) 2 − , neutron, mN  P µ ( +pd) ×  2 , proton.  mN   The overall factor of 2 on the RHS comes from the momentum-tra nsfer delta function in Eq. (29).3

3 This is because 1 δ (a x) = δ (x) , (3.37) a | | hence, P P (3) ′ q (3) δ − + = 2 δ P P′ + q . (3.38) 2 2 ! − 40

The charge form factor necessarily stems from the zeroth component of these

currents. Summing the proton and the neutron interactions, we ﬁnd:

2 (P)µ d3 p Pµ 2 = = d Ψ P Ψ P . F(q 0) 2 3 ( ′) pd pd ( ) (3.39) mN Z (2π) h | ih | i × mN

As you can see, the prefactor on the LHS cancels all of the odd factors in the RHS, leaving

a straightforward integral (eﬀectively, a normalization) for F(q2 = 0). We extend this simpliﬁed example to nonzero q, by showing that the form of the current on the RHS cancels the form of the current on the LHS, i.e. the vector structure of the LHS matches the vector structure of the RHS exactly. But what of the case for q , 0? In this case we construct the equality of currents from

Eq. (31) and Eq. (35), above:

(P + P )µ d3 p q ′ 2 = d Ψ P + Ψ P F(q ) 2 3 ( ′) pd pd ( ) (3.40) mN Z (2π) h | 2 ih | i

P′ p 2 µ κ p 2 µν P proton: u + pd′ F1 (q ) γ + F2 (q ) iσ qν u + pd  2 2mN 2 P h i P ×  ′ n 2 µ κ n 2 µν neutron:  u p′ F (q ) γ + F (q ) iσ qν u pd  2 − d 1 2mN 2 2 −  h i  We expand the terms within the square brackets in Eq. (41), in powers of p , and in mN

the sum of neutron and proton components, at lowest order all terms proportional to F2 vanish. We then ﬁnd, after invoking the point parameterization for the neutron form factor

n and letting F1 = 0:

(P + P )µ (P + P )µ d3 p q ′ 2 = ′ p 2 d Ψ P + Ψ P F(q ) 2 F1 (q ) 3 ( ′) pd pd ( ) . (3.41) mN 2mN Z (2π) h | 2 ih | i

p 2 If we further assume the point parametrization of the proton form factor F1 (q ) = 1, and we cancel out the equivalent vector structures on the left- and right-hand sides, we are left with a ﬁnal closed expression for F(q2):

3 2 d pd q F(q ) = Ψ(P′) pd + pd Ψ(P) (3.42) Z (2π)3 h | 2 ih | i 41

or, expressed in terms of wavefunctions:

1 ∞ q 2 = Ω 2 ψ + ψ . F(q ) 3 d dp p †(p ) (p) (3.43) (2π) Z Z0 2

where p ψ = ψ(p) is the deuteron wavefunction in momentum space, obtained via the h | i method outlined previously.

Evaluating the integral Eq. (41) is straightforward, with one complication of note. Notice that the integral is a function of the vectorially added momentum p + q/2. Hence we may not integrate over solid angle and merely pick up a factor of 4π. We must expand the vector sum:

3 q 3 2 2 d p ψ† p + ψ( p ) = d p ψ† p + q/2 + p q ψ(p) | 2 | | | | | | | · Z Z = d3 p ψ p 2 + q/2 2 + p q cos θ ψ( p ) | | | | | || | | | Z 1 2 2 q 2 = 2π dp p ψ0( p ) dx ψ0∗ p + + pq x , Z | | | | Z 1  r| | | 2| | |  −     resulting in a two-dimensional integral in p and the dimensionless parameter x. Note that in the last term we have replaced ψ ψ because we are only evaluating the form factor → 0 for the S-wave deuteron state. This ﬁnal closed form of (42):

1 2 2 2 q 2 F(q ) = 2π dp p ψ0( p ) dx ψ0∗ p + + p q x (3.44) Z | | | | Z 1  r| | | 2 | | || |  −     we may evaluate using any number of numerical techniques. For this work, we constructed a multidimensional Gauss-Legendre routine for evaluation of (44), which is a 2D implementation of the 1D qgaus routine from ([16]) using Gauss-Legendre points and weights generated via routine from ([13]). Figure 3.7

shows the result of our code for all values of q < 100 MeV. Note that the form factor G | | C is approximately linear with q2 for small q2. 42

0.9

0.8

F(q) 0.7

0.6 q = 100 MeV

0.5 0 0.05 0.1 0.15 0.2 0.25 2 -2 q (fm )

2 1 Figure 3.7: Our results for the form factor F(q ) evaluated to q = 100 MeV 0.5 fm− . ≈

The numerical results that we obtained are compared against a routine from Daniel Phillips which calculates the charge form factor using Heavy Baryon Chiral Perturbation theory. We compare our result (which is evaluating the form factor using only the S-wave) against both an S-wave only calculation, as well as a calculation using the full deuteron wavefunction (S- and D-wave).

FC(q)

KM DP (HBχPT @LO / S-Wave Only) 0.8 DP (HBχPT @LO / S- & D-Wave)

0.6

0.4

0.2

0 0 1 2 3 4 5 -1/2 q (fm )

2 1 Figure 3.8: Our results for the form factor F(q ) evaluated to q = 2 GeV 5 fm− . ≈ 43

As you can see, the general shape of our curve is consistent with the work from

([13]). That there are inconsistencies in the precise deuteron charge form factor are acceptable; the largest source of diﬀerence between our results and the results of D. Phillips are that diﬀerent potential models are invoked in each calculation, however, that there is general agreement is a testament to the applicability of the procedure.

Furthermore, in no way is this calculation meant to be deﬁnitive, but merely to demonstrate the technology by which form factor calculations are done. Another check of consistency for our code is by comparing the result of the form factor versus q2 in the low q region. In this region, the nature of the form factor is

1 F(q2) 1 r2 q2. (3.45) ≈ − 6h di

Fitting function linear in q2 to our data in the region 0 < q2 < 0.05, we ﬁnd that a function of the form F(q2) = 0.95071 0.64081 q2 (3.46) − × ﬁts the data with a standard deviation of 0.0044. This corresponds to a squared matter ± radius expectation value of r2 = 3.84486 fm2, (3.47) h di which may be compared to the matter radius obtained from Eq.(21) (c.f. Table I). As we see,

rd = 1.9608fm (3.48) matches our theoretically obtained result (as well as the empirical value established by ([8])) extremely well. This calculation demonstrates the viability of evaluating electromagnetic form factors

for a nuclear system given a particular potential model. We constructed this as a simple example of how to go about evaluating electromagnetic form factors in a simple setting. In 44 the next chapter, we move beyond simple constructs (such as deuterons bearing S-waves only) and evaluate the charge form factor of the three-body helium-3 system. 45 4 Helium-3

Moving into the more-interesting three-body sector, we seek to study the electromagnetic properties of three-body nuclei, namely 3He. Helium-3 is a nonradioactive isotope of Helium whose nucleus consists of two protons and one neutron. We are motivated to study Helium-3 for various reasons, however, two stand out as of principal concern to the low-energy nuclear theorist:

• 3He is the ideal test bed for probing the three-nucleon (3N) force. So long as a largely nonrelativistic approach applies1, the nucleon-nucleon forces (NN) are well

described by current theoretical models, hence we can with a large degree of certainty account for 2N-force eﬀects in the 3N scattering data. By also ﬁtting additional observables to this data, such as the binding energy of the 3He nucleus, we are able to unlock parameters in the 3N-force models with some degree of

accuracy. ([21]).

• 3He is also an ideal laboratory for probing neutron properties, since Helium-3 approximately carries net spin which is equivalent to the neutron’s (i.e. the two protons, being fermions, when in the net L = 0 state, must have opposite spin).

We begin with a discussion of the various 3N potentials, focusing particularly on the Argonne v18 NN potential with Urbana IX 3N terms (AV18+UIX). Following that, a brief discussion on the structure of the 3N wavefunctions as generated via these potentials. We then discuss and calculate the charge form factor for 3He, utilizing Monte Carlo integration.

1 This requires that the 3-momentum of the virtual photon incident upon the nucleus is such that the three-nucleon center of mass frame energy in the outgoing conﬁguration remains suﬃciently below the pion production threshold and that the total three-nucleon momentum remains below the nucleon mass. 46

4.1 Three-Nucleon Potentials

There are various possible choices for potential model from which we can calculate the Helium-3 wavefunction. For this work, our evaluation of the Helium-3 charge form factor utilizes wavefunctions derived from the AV18+UIX 3N-system potential model,

obtained from Andreas Nogga ([11]).

The Argonne v18 NN potential contains two principle components: a charge-independent part with 14 operator elements (an updated version of the AV14 NN potential), as well as a charge independence breaking part with four additional operators, which includes a complete1 speciﬁcation of the electromagnetic interaction. The potential

has free parameters that are ﬁt to np- and pp-scattering data (potential models which are ﬁt to np-scattering data alone have been historically bad at predicting pp-scattering observables) as well as to low-energy nn-scattering properties and deuteron parameters. ([23]) The form of this potential is a sum of an electromagnetic part, a one-pion exchange

part, and a phenomenological part which is expressed, as in the AV14 NN potential, as a sum of central, tensor, L2, spin-orbit, and quadratic spin-orbit terms. Each term in the

complete potential model is dependent upon S , T, and Tz states with intermediate- and short-range characteristics embodied principally within the phenomenological part of the

potential. The Urbana IX 3N force extends the formulation of the potential to include explicit 3N interaction terms to give more accurate results in 3N-sector calculations than AV18 can give alone. The AV18 potential model alone is suﬃcient to construct a model of the three nucleon system, however, nuclei modeled using only NN potentials tend to be underbound (i.e. nontrivial many-nucleon interactions are expected to provide the missing binding energy). Using the Urbana IX three-nucleon interactions, one includes a two-pion exchange contribution (with strength set to reproduce the binding energies of 3He and

1 Includes Coulomb, Darwin-Foldy, vacuum-polarization, and magnetic moment terms. 47

4He), as well as a phenomenological spin- and isospin-independent interaction with a strength adjusted such that the empirical equilibrium density of nuclear matter is obtained. ([17])

4.2 Structure of the 3N Wavefunction

Our wavefunctions are generated (via a code by Andreas Nogga) using the Argonne

v18 NN interaction with Urbana IX 3N interaction terms. These functions are decomposed such that we consider Helium-3 as a 2 + 1-body system, where we treat two nucleons as a bound NN pair (referred to as the “12-system”), upon which interactions occur, while a third nucleon spectator (the “3-system”) is unaﬀected. The wavefunctions are required to be antisymmetric, normalized, and projected onto the jj-coupling basis. In this basis, a Helium-3 state is speciﬁed by:

p p α = p p (l s ) j (l s ) j ( j j ) J M (t t ) T M , (4.1) | 12 3 i | 12 3i×| 12 12 12 3 3 3 12 3 Ji×| 12 3 T i

where p = p and p = p are the magnitudes of the Jacobi momenta of the Helium-3 12 | 12| 3 | 3| system,2 deﬁned as:

1 p = (k k ) (4.2) 12 2 2 − 1 1 p = (2 k k k ) (4.3) 3 3 3 − 1 − 2 where kn is the momentum of the n-th nucleon. Since the spectator particle is a simple

1 1 single-nucleon state, we fix s3 = 2 and t3 = 2 . Furthermore, the total spin of the Helium-3 1 3 nucleus gets fixed as J = 2 , as we know that He is a fermion, and the isospin nature of = 1 = + 1 Helium-3 fixes T 2 and MT 2 . Thus, explicit specification of a given channel of the

Helium-3 nucleus requires only 7 quantum numbers as inputs: l12, s12, j12, l3, j3, M, and t . It is this set which we deﬁne as eﬀectively specifying α . 12 | i 2 That the Jacobi momenta are three-dimensional while the wavefunctions depend only upon their magnitudes give rise to the introduction of the spherical harmonics encountered later on in this section. 48

The Helium-3 nucleus contains two protons and one neutron. Because our calculation separates out a “pair” and a “spectator” we must account for the various permutations of these pairs. In general, if we deﬁne the nucleons as Nucleon 1, Nucleon 2, and Nucleon 3 (this nomenclature is not dependent on which is a proton and which is a neutron), we can construct the wavefunction for the full Helium-3 nucleus by summing over the various permutations of 1, 2, 3 in our wavefunction. Hence, the net 3He wavefunction is { }

Ψ(p , p ) = (1 + P P + P P ) ψ(p , p ) , (4.4) | 12 3 i 12 23 13 23 | 12 3 i where ψ is a wavefunction for one speciﬁcation of nucleons 1, 2, and 3. The net 3He | i 1 wavefunctions are dominated by the channel in which the 12-pair is a pp-pair in the S0 state; i.e. when the neutron is the spectator and the nucleus takes on the spin characteristics of the neutron. When the neutron happens to not be the spectator particle (based on whatever particular declaration of the channel we are at in our sums) the strength of the wavefunction is spread over many partial waves. This fact is accounted for in our 3He wavefunction code from Andreas Nogga. The Helium-3 wavefunctions are expressed in terms of the jj-coupling basis we

deﬁned earlier as

s12, ms12 p12 p3; s12ms12 s3ms3 Ψ (JMJ )(T MT ) = ϕ(p12, p3, α) j , m (p ˆ12) h | i Y 12 j12 Xα t12X, mt12 s12X, ms12 Xms3 s3, ms3 j , m (p ˆ3) t12 mt12 t3 mt3 , (4.5) ×Y 3 j3 × | i| i

where our indication of the sums is over all allowed values for that particular quantity, and

the functions sms are deﬁned as: Y jm j

sms = (lml, s, ms j, m) Yl, m (p ˆ) s, ms , (4.6) Y jm j | l | i

and the functions ϕ come from Andreas Nogga and are deﬁned as:

ϕ(p , p , α) = j , m , j , m J, M t , m , t , m T, M p p α Ψ . (4.7) 12 3 12 j12 3 j3 | J 12 t12 3 t3 | T h 12 3 | i 49

The parentheticals in Eqs. (4) and (5) are Clebsch-Gordan coeﬃcients, and the deﬁnition of α stems from our discussion of the jj-coupling basis (above): | i

α = α α = (l s ) j (l s ) j ( j j ) J M (t t ) T M . (4.8) | i | Ji×| T i | 12 12 12 3 3 3 12 3 Ji×| 12 3 T i

With these wavefunctions we may now move forward with our calculation of the 3He charge form factor.

4.3 Constructing the Charge Form Factor

We begin our construction of the Helium-3 charge form factor by examining the Breit Frame kinematics of the Helium-3 nucleus. The momentum of each individual nuclei in the ﬁnal and initial state is expressed in Figure 4.1. The initial and ﬁnal momenta of the

3 Figure 4.1: Breit frame kinematics for tree-level elastic e− He scattering.

3 He nucleus shall be represented by P and P′, respectively, with P and P′ related by

P′ = P + q. (4.9) 50

Here, q is deﬁned as the 3-momentum transfer of the system, incident along the +z-axis:

q = (0, 0, q) . (4.10)

The transition amplitude for elastic e 3He - scattering is related to the invariant amplitude for such a process via

3 (3) T 3 = i (2π) δ (P P′ + q) 3 . (4.11) e He − − Me He

As with the deuteron, we may directly evaluate this transition amplitude:

3 3 3 3 d p12 d p12′ d p3 d p3′ Te 3He = ψ(P′) p′ , p′ Z (2π)3 Z (2π)3 Z (2π)3 Z (2π)3 h | 12 3 i P P p3′ ′ p3 p′ + , k′ Tˆ p , k × h 12 − 2 3 | eN | 12 − 2 − 3 i P P 3 (3) p3 p3′ (2π) δ ( p + + p′ + ) × − 12 − 2 3 12 2 − 3 P P 3 (3) (2π) δ (p + p′ ) p , p ψ(P) , (4.12) × 3 3 − 3 − 3 h 12 3 | i which reduces (after quite a bit of algebra) to:

1 3 3 q q e 3He = d p12 d p3 ψ(P′) p12 + , p3 eN p12, p3 ψ(P) , (4.13) M (2π)6 Z Z h | 2 − 3 iM h | i where once again is the elastic electron-nucleon scattering invariant amplitude. MeN We now turn to the form of the currents. From our earlier work on deuterium, we posit the form of the invariant amplitude for : MeN 1 = e2 jleptons Jµ . (4.14) MeN − µ q2 eN

The form of the current for elastic electron-3He scattering is also straightforward to write

1 down, since Helium-3 is a spin- 2 fermion as well:

2 leptons 1 µ 3 = e j J . (4.15) Me He − µ q2 e3He 51

The leptonic parts of each of these amplitudes are completely equivalent, so Eq. (13)

simpliﬁes to:

µ 1 3 3 q q µ J = d p12 d p3 ψ(P′) p12 , p3 + J p12, p3 ψ(P) . (4.16) e 3He (2π)6 Z Z h | − 2 3 i eN h | i

So, we must now turn our attention to the structure of the relevant currents. The structure of the single-nucleon current comes straight from Eq. (7) of Chapter 1:

µ 2 µ κ 2 µν JeN = u(p′) F1(q )γ + F2(q )iσ qν u(p), (4.17) " 2 MN # where we here deﬁne the momenta of the interacting individual nucleon as in Figure 4.1:

p P p = p 3 + (4.18) 12 − 2 3 P p3′ ′ p′ = p′ + . (4.19) 12 − 2 3

Furthermore, we can immediately write down the form of the 3He current as well:

3 κ 3 µ = He 2 γµ + He 2 σµν , Je3He u(p′) F1 (q ) F2 (q )i qν u(p) (4.20) " 6 MN #

since again, 3He is a fermionic system. Expanding Eq. (17) and Eq. (20) in powers of p MN and keeping only the highest order terms (terms O( p ) end up canceling, so we end up MN p2 only neglecting terms O( 2 ) and above), we ﬁnd that: MN

3 3 3He 2 2 d p12 d p3 q q F (q ) = F1(q ) ψ(P′) p12 + , p3 p12, p3 ψ(P) . (4.21) 1 Z (2π)3 Z (2π)3 h | 2 − 3 ih | i

Up to this point in the derivation we have neglected isospin and spin summations, arising from the spin speciﬁcations in the α and the isospin speciﬁcations in the α . The full | Ji | T i integral which needs to be evaluated, including the isospin summations, is:

3 3 3He 2 2 d p12 d p3 F (q ) = F1(q ) ψ(P′) p˜12 p˜3 α (4.22) 1 Z (2π)3 Z (2π)3 h | i s12,mXs12 ,ms3 mtX12 ,mt3 Xα, α′ q q p˜ , p˜ ; α p + , p p , p p , p ; α′ p , p ; α′ ψ(P) . × h 12 3 | 12 2 3 − 3ih 12 3|| 12| | 3| i h| 12| | 3| | i 52

However, we may further reduce the expressions in the center of the wavefunction kets

down, so that our ﬁnal expression (for coding) reads, after replacing each three-momenta with an integral over the magnitude of the momenta and angles:

3He 2 F = F1(q ) dΩ12 dΩ3 dp12 dp3 (4.23) 1 Z Z Z Z s12,mXs12 ,ms3 mtX12 ,mt3 α,X α′

ϕ†(p ˜ p˜ α) G† (θ , φ , θ , φ ) G (θ′ , φ′ , θ′ , φ′ ) ϕ(p , p , α′), × 12 3 α 12 12 3 3 α′ 12 12 3 3 12 3 where the functions ϕ are the wavefunctions we obtained from Andreas Nogga, and where

q2 q p˜ = p2 + p q cos θ + = p + (4.24) 12 r 12 12 12 4 12 2

2 q2 q p˜ = p2 p q cos θ + = p (4.25) 3 r 3 − 3 3 3 9 3 − 3

throughout. The functions

Gα(θ12,φ12,θ3,φ3) = j , m (θ12,φ12) j , m (θ3,φ3) j12m j j3m j JMJ (4.26) Y 12 j12 Y 3 j3 12 3 | simplify the coding scheme for evaluating the form factor, as we may construct a separate function call to evaluate and test before running the full calculation. Note that the are Y jm j deﬁned by Eq. (4) of this chapter, and that as in Eq. (4) the parenthetical term represents a

Clebsch-Gordan coeﬃcient. Because of the appearance of spherical harmonics in Gα, we

ﬁrst tested the orthogonality of the Gα before proceeding with the full calculation.

4.4 Results and Discussion

We evaluate the integral (Eq. (24)) using a Monte Carlo integration routine which evaluates the integral over a uniformly distributed set of 10,000 six-dimensional points (p , p [0, 5], θ ,θ [0, π], and φ ,φ [0, 2π]). Our results are illustrated in Figure 12 3 ∈ 12 3 ∈ 12 3 ∈ 4.2.

There are several checks that we can perform against this calculation. The most basic check to perform is that the value of the integral when q = 0 returns the value of the 53

0.1 (q) 0.01 3He 1 F

0.001

0.0001 0 1 2 3 4 5 -1 q (fm )

Figure 4.2: Charge form factor for Helium-3, evaluated for 10,000 Monte Carlo points for 1 each of 500 values of q in the region q [0, 5]fm− . ∈ normalization. We performed the integral for all channels with j 2, and we find that 12 ≤ our normalization for these channels when evaluated independently (through the code from Andreas Nogga) is 0.98660. Our code, evaluated through 50 iterations of the q value in the integral, each with 10,000 MC points, returns a value of0.96586 0.0994236. The ± somewhat larger fluctuations are accounted for by the error in the Monte Carlo routine. The fluctuations are on the order of 1-2%, which is within an acceptable tolerance for a flatly distributed set of Monte Carlo points. The second most reassuring check on our data is that we can once again evaluate the matter radius, as we did for deuterium. We determine this by plotting the data against q2 and using the fact that the analytic form of the form factor at low-q is

3 1 F He = 1 r2 q2. (4.27) 1 − 6h mi 54

Performing this calculation for 3He, we ﬁnd that the matter radius is

r = 1.698805 0.077877fm, (4.28) m ± which is in agreement with the empirical value of 1.77 fm. ([4]) The last check is more of

Figure 4.3: Charge form factor for Helium-3, evaluated by [19]. Our work closely matches the impulse approximation (IA) data. an encouraging glance, which is that the general shape of our data matches quite well the data for the calculation of the 3He charge form factor by ([19]) in the impulse approximation. 55 5 Summary and Outlook

As we have seen, the evaluation of electromagnetic form factors of light nuclei requires understanding the ways in which the electromagnetic current densities of nuclear systems can be constructed and eventually evaluated as a matrix element between nucleon states. We have explored in some detail how electromagnetic form factors for individual nucleons can be constructed, and we have seen the technology for evaluating the charge form factors of Helium-3 and Deuterium. We have also pursued a study of chiral perturbation theory and discussed how chiral perturbation theory provides a valid alternative to current phenomenological calculations of nuclear properties such as electromagnetic form factors. The use of effective field theories for modeling light nuclei is well established ([2], [19]) and is proven successful ([20]). There is significant work which we wish to do to continue our study of this topic.

First and foremost, we believe that by adopting an alternative integrate scheme we will be able to increase the accuracy of our Helium-3 calculation by a factor of 10, were we to choose an adaptive Monte Carlo method which weights the selection of random points by a function which closely mimics our form factor. Because, invariably, as q 0, the form → factor integral gives a net result of a normalization of the Helium-3 wavefunctions, we believe that choosing the integrand of the normalization integral as a weighting function would significantly decrease the effect of Monte Carlo fluctuations in the data and give better convergence to data. Note that in this method, we are essentially adopting the normalization as a fit parameter and are guaranteed to get unity at q2 = 0. As another method for improving our calculation, we could test our calculation against various parametrization of the single nucleon form factors, which appear in the results for both deuterium and helium-3. We chose to use the point parametrization of the 56

single nucleon form factors, wherein the q = 0 behavior of the form factor

p 2 F1 (q = 0) = 1

n 2 F1(q = 0) = 0,

is assumed to persist for all q. In reality, the single-nucleon form factor does not maintain this behavior (a direct result of underlying quark structure of the baryons), and using

diﬀerent parametrizations of the form factors (such as that from [10]) we can compare the eﬀect of electromagnetic structure within the constituent nucleons on the whole. Aside from further improving the calculations in place, we would also like to further explore using Chiral Perturbation Theory to evaluate electromagnetic form factors, by

expanding on our existing work and calculating the eﬀects of pion loop contributions on the form factor and associated meson-exchange corrections to the current operator that is used inside nuclei. We expect that meson exchange corrections become extremely

1 important in the larger q range (i.e. q > 3 fm− ), where the contributions become of the order 10%. Hence, to understand the electromagnetic form factors in higher regimes of ≈ interest, we must further our work with regard to chiral eﬀective theory. 57 References

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[2] D. Choudhury. Investigating Neutron Polarizabilities and NN Scattering in Heavy-Baryon Chiral Perturbation Theory. 2006.

[3] R. F. Code and N. F. Ramsey. Molecular-beam magnetic resonance studies of hd and d2. Phys. Rev. A, 4(5):1945–1959, Nov 1971. doi: 10.1103/PhysRevA.4.1945.

[4] H. Collard, R. Hofstadter, A. Johansson, R. Parks, M. Ryneveld, A. Walker, M. R. Yearian, R. B. Day, and R. T. Wagner. Electron scattering in tritium and he3. Phys. Rev. Lett., 11(3):132–134, Aug 1963. doi: 10.1103/PhysRevLett.11.132.

[5] Evgeny Epelbaum. Few-nucleon forces and systems in chiral eﬀective ﬁeld theory. Prog. Part. Nucl. Phys., 57:654–741, 2006. doi: 10.1016/j.ppnp.2005.09.002.

[6] David J. Gross and Frank Wilczek. Ultraviolet behavior of non-abelian gauge theories. Phys. Rev. Lett., 30(26):1343–1346, Jun 1973. doi: 10.1103/PhysRevLett.30.1343.

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[8] R. Machleidt. The high-precision, charge-dependent Bonn nucleon-nucleon potential (CD-Bonn). Phys. Rev., C63:024001, 2001. doi: 10.1103/PhysRevC.63.024001.

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[11] A. Nogga, A. Kievsky, H. Kamada, W. Gl¨ockle, L. E. Marcucci, S. Rosati, and M. Viviani. Three-nucleon bound states using realistic potential models. Phys. Rev. C, 67(3):034004, Mar 2003. doi: 10.1103/PhysRevC.67.034004.

[12] D. R. Phillips and T. D. Cohen. Deuteron electromagnetic properties and the viability of eﬀective ﬁeld theory methods in the two-nucleon system. Nucl. Phys., A668:45–82, 2000. doi: 10.1016/S0375-9474(99)00422-4.

[13] Daniel R. Phillips. personal communication, 2007. 58

[14] L. Platter and D. R. Phillips. Deuteron matrix elements in chiral eﬀective theory at leading order. Phys. Lett., B641:164–170, 2006. doi: 10.1016/j.physletb.2006.08.053.

[15] H. David Politzer. Reliable perturbative results for strong interactions? Phys. Rev. Lett., 30(26):1346–1349, Jun 1973. doi: 10.1103/PhysRevLett.30.1346.

[16] W. Press, S. Teukolsky, W. Vetterling, and B. Flannery. Numerical Recipes in Fortran 77, Second Edition. Cambridge University Press, 1986.

[17] B. S. Pudliner, V. R. Pandharipande, J. Carlson, and Robert B. Wiringa. Quantum Monte Carlo calculations of A 6 nuclei. Phys. Rev. Lett., 74:4396–4399, 1995. doi: 10.1103/PhysRevLett.74.4396.≤

[18] Stefan Scherer and Matthias R. Schindler. A chiral perturbation theory primer. 2005.

[19] R. Schiavilla, V. R. Pandharipande, and D. O. Riska. Magnetic form factors of the trinucleons. Phys. Rev. C, 40(5):2294–2309, Nov 1989. doi: 10.1103/PhysRevC.40.2294.

[20] Manuel Pavon Valderrama, A. Nogga, Enrique Ruiz Arriola, and Daniel R. Phillips. Deuteron form factors in chiral eﬀective theory: regulator-independent results and the role of two-pion exchange. Eur. Phys. J., A36:315–328, 2008. doi: 10.1140/epja/i2007-10581-4.

[21] W. Gloeckle, J. Golak, R. Skibinski, H. Witala, H. Kamada, and A. Nogga. Electron scattering on helium 3 a playground to test nuclear dynamics. The European Physical Journal A, 21(2):335–348, aug 2004. doi: 10.1140/epja/i2004-10001-5. URL http://dx.doi.org/doi/10.1140/epja/i2004-10001-5.

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[24] S. Wong. Introductory Nuclear Physics. Wiley, 1998. 59 Appendix: Monte Carlo Methods

For the evaluation of the Helium-3 charge form factor, we invoked the use of Monte Carlo integration, because we found ourselves in a situation where we have a large-dimensional integral (large meaning greater than one or two), which we needed to evaluate quickly. Using the method of Gaussian quadrature (which we invoked in the case

of deuterium) would have given perfectly reasonable results but would have take much much longer to compute (as you need to evaluate through loops over six degrees of freedom). This was simpliﬁed with the Monte Carlo algorithm we employed, however, because one can just as easily pick six numbers randomly as you can two numbers, and the time

requirements for higher dimensions do not scale linearly as with other quadrature routines. Here we outline the basic principles of Monte Carlo integration. Fundamentally, the Monte Carlo routine works by choosing a random sample of numbers in the domain of interest and evaluating the function which is desired to be

integrated at each of these numbers. For example, if we wish to integrate the function f (x, y) over a volume V in cartesian space, we seek to evaluate:

x2 y2 I = dx dy f (x, y). (A.1) Zx1 Zy1 over the volume V bounded by x [x , x ], y [y , y ]. ∈ 1 2 ∈ 1 2 A ﬂat-weighted Monte Carlo algorithm will select points uniformly over this volume. If we choose a sample size of n points, then we may estimate the value of this integral by 1 n I f (x , y ) V, (A.2) est ≈ n i i × Xi=1 where V is the volume enclosed. The estimated error in the Monte Carlo routine is given

by the standard deviation of the estimate:

2 V n 1 n σ = v f (x′, y′) f (xi, yi) . (A.3) n tu  i i − n  Xi =1 Xi=1 ′     60

1 = For large n, the error scales like √n , hence we chose for our calculations n 10, 000 so as to acheive approximately 1% error due to Monte Carlo ﬂuctuations. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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