Pion Photo- and Electro-Production from the Nucleon

PION PHOTO- AND ELECTRO-PRODUCTION FROM THE NUCLEON

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

In partial fulﬁllment of the requirements for the degree Doctor of Philosophy

PION PHOTO- AND ELECTRO-PRODUCTION FROM

THE NUCLEON

BY GEORGE LAURENTIU CAIA

has been approved for

the Department of Physics and Astronomy

and the College of Arts and Sciences by

LouisE.Wright

Professor of Physics

Leslie A. Flemming

Dean, College of Arts and Sciences CAIA, GEORGE L. Ph.D. August 2004. Physics Pion Photo- and Electro-production from the Nucleon (117pp.) Director of Dissertation: L. E. Wright

We present a relativistic dynamical model for pion photo- and electro-production on the nucleon. The model uses a manifestly Lorentz-covariant approach based on solving a relativistic coupled-channel scattering equation in the pion-photon channel space. The eﬀect of the pion-nucleon ﬁnal state interaction is thus explicitly included in a way that is consistent with the two-body unitarity — the Watson theorem is obeyed exactly. This work applies the dynamical model for πN scattering of Pascalutsa and Tjon, to virtual photons by introducing electromagnetic form factors in the Born (non-resonant) and resonant terms. Special care is taken to satisfy current conservation for the Born terms by imposing the Ward-Takahashi identity at the nucleon and pion electromagnetic vertices, thereby allowing us to use realistic electromagnetic form factors. We perform a multipole decomposition and analyze the electromagnetic form factors and quadrupole deformation (REM and RSM ) of the ∆(1232). We compare our model with recent data for single polarization observables, for both pion photo- and electro-production.

Approved:

Louis E. Wright Professor of Physics In memory of my father ... Acknowledgements

I would like to express my gratitude for the excellent advice, help and patience that I have received from my advisor Professor Louis E. Wright during the course of my Ph.D. study. I thank Vladimir Pascalutsa for being a very helpful collaborator as well as being a good friend. I gratefully acknowledge the support that I have received from the Department of Physics and Astronomy. I would like to thank my wife for her encouragement, and power of overcoming these long years of both joy and frustration. Her deep love and great patience, helped me to get over all the obstacles and stay focused till the end of my PhD. I thank to my mother and father for their wiseness and power of getting me through the though years of the past; without them I would have not accomplished myself. At last but not at least, I thank my sister and my brother in law who helped me, materially as well as morally, from far away, with their good words and advices. 7

Contents

Abstract 4

Acknowledgements 6

List of Tables 10

List of Figures 11

1 Introduction 13 1.1 History ...... 13 1.2 Background ...... 18

2 Photo- and Electro-production Potential 22 2.1 Born terms ...... 23 2.2 Vector meson exchanges ...... 31 2.3 Resonant terms ...... 32 2.4 Isospin decomposition of the amplitude ...... 34

3 πN Elastic Scattering 38 3.1 Tree Level potential πN matrix ...... 38 3.2 Two-body scattering equation ...... 40 3.3 Renormalization procedure ...... 43

4 Pion Production from Free Nucleons 46 4.1 Coupled channel approach ...... 46 8

4.2 Multipole decomposition ...... 47 4.3 Ingredients of the model ...... 50 4.3.1 Electromagnetic form factors: non-resonant terms ...... 50 4.3.2 Electromagnetic form factors: resonant terms ...... 51 4.3.3 Current conservation ...... 55

4.4 Problem of singularities in Vπγ∗ ...... 56

5 Alternative Approach for Solving the Scattering Equation 60 5.1 Introduction ...... 60 5.2 Exact integration over the azimuthal angle ...... 61 5.3 Spin complications: the πN system ...... 64 5.4 Numerical results ...... 65 5.5 Extension to pion photoproduction ...... 68 5.6 Conclusion of Ch. 5 ...... 70

6 Results and Discussion 71 6.1 Photo-Production (Q2 = 0) Results ...... 71 6.2 Electro-Production (Q2 6= 0) Results ...... 79 6.3 Conclusion ...... 90

APPENDICES 92

A Useful relations 92 A.1 Helicity spinors ...... 92 A.2 Kinematics ...... 93

A.3 Sample Isospin Decomposition of the ΓγNN Vertex ...... 96

B Form Factors Used in the Model 99 B.1 Hadronic form factors ...... 99 B.2 Electromagnetic form factors ...... 100

C Azimuthal Dependence of One Nucleon Exchange 102 9

D Observables in Terms of Multipoles 106 D.1 Deﬁnitions of the obsevables ...... 106 D.2 CGLN amplitudes in terms of multipoles ...... 108 D.3 Response functions in terms of the CGLN amplitudes ...... 109

ρ √ E Integration of GET (|~q|; s) 111

Bibliography 113 10

List of Tables

1.1 Properties of the pions...... 15 1.2 Relations between electromagnetic multipoles and pion production multipoles...... 20

6.1 Couplings/parameters used in our model in the spectator approximation. 72 (3/2) 6.2 Parameters used in the N ↔ ∆ transition form factors to ﬁt M1+ , 2 REM and RSM .(I) ±Q /2W modiﬁcation in the u- and t-channel terms and (II) spectator approximation...... 82 11

List of Figures

1.1 The total photo-absorption cross section of the proton and its decomposition into selected channels as a function of CM photon energy in GeV. Figure adopted from [26]...... 19

2.1 The contributions included in our model...... 23

3.1 Diagrammatic form of a relativistic two-body scattering equation. .. 41 3.2 Phase shifts calculations for πN scattering. Data points are from various partial wave analysis [67]...... 43

5.1 One-nucleon-exchange πN potential...... 65

++ 2 LAB 5.2 Angular dependence for |T 1 1 | at Eπ = 300 MeV. The solid line is 2 2 the full calculation, and the dashed and dotted lines are the resumming of partial terms...... 68

++ 2 CM 5.3 Energy dependence for |T 1 1 | at θπ = π. The lines are deﬁned the − 2 2 same as in Fig. 5.2...... 69

6.1 Multipole calculations. The black line is the real part of the amplitude and the red line is the imaginary part of the amplitude. Data points are from VPI [67]...... 73 6.2 Same as in Fig. 6.1...... 74 6.3 Same as in Fig. 6.1...... 75 6.4 Same as in Fig. 6.1...... 76 12

6.5 Diﬀerential cross sections and single polarization observables T , Σ, and P at Q2 = 0 for γ + p → π+ + n (left) and γ + p → π0 + p (right) at

Tγ = 320 MeV. Experimental data are from Refs.[69]-[73]...... 77 6.6 Total cross section for the diﬀerent γ + N → π + N isospin channels at the photo-production point. The experimental data are from the SAID website [67] compilation...... 78 6.7 Various contributions to the resonant multipoles...... 80

3/2 2 −2 6.8 Im(M1+ )/FD at W = 1.232 GeV where FD = (1 + Q /0.71) is the standard dipole form factor. The data at Q2 = 2.8 and 4.0 (GeV/c)2 is from Ref.[64]; other data are from Refs.[74]-[79]...... 81

2 6.9 REM and RSM at W = 1.232 GeV plotted vs. Q . The data are from Refs.[62]-[66]. The points (∗) are calculations of the Sato-Lee model [23]. 82 6.10 The Q2 dependence of the determined N → ∆ electromagnetic form factors divided by the dipole form factor, at W = 1.232 GeV . .... 83 6.11 Nucleon u-channel (left) and pion t-channel (right) contributions at W = 1.232 GeV for the resonant multipoles. See text for explanations. 85 6.12 Virtual photon differential cross sections on p(e, e0p)π0. The data are from [64, 82]...... 86 6.13 Virtual photon differential cross sections on p(e, e0n)π+ at Q2 = 0.6(GeV/c)2, W = 1.23 MeV and = 0.69 for fixed θ. The data are from [64, 82]...... 87

0 + 2 2 6.14 Beam asymmetry versus φπ for p(e, e n)π at Q = 0.4(GeV/c) , W = 1.22 MeV and for ﬁxed θ. The data points are from [68]...... 88

0 + 0 0 2 2 6.15 σLT 0 versus cos θπ for p(e, e n)π and p(e, e p)π at Q = 0.4(GeV/c) . The data points are from [68]...... 89

A.1 Schematics of the lab frame kinematics pion electro-production. ... 93 Chapter 1 Introduction

1.1 History

One of the major quests of contemporary theoretical physics is the search for a quantum field theoretical description of the fundamental processes. A major achievement was the unification of the electromagnetic and weak force in the 1960s and 1970s in a single, electroweak, theory due to Weinberg [1] and Salam [2]. Later on, Quantum Chromodynamics (QCD), the field theory driving the strong force, which holds together the nuclear particles, comes into play. The last force, which to date seems to 00evade00 a unified description within a quantum field theoretical framework, is the gravitational force. The biggest success of the field theoretical description of subatomic particles is based on the predictive power of the electroweak theory, and in particular on Quantum Electrodynamics (QED). This is due to the smallness of the electromagnetic fine structure constant αem = 1/137, which allows for a perturbative expansion of the theory in powers of αem. This theory, combined with the power of perturbation theory, proved very accurate in predicting experimental data. Unfortunately, this technique does not entirely work in the case of QCD. At sufficiently high energies, QCD exhibits a perturbative behavior similar to QED, the color coupling constant

αs of QCD becomes very small. This weakening at high energies is commonly called 00asymptotic freedom00. Therefore, at these high energies, quarks (which are constrained by the strong color force), can be treated by perturbation theory and this is denoted by perturbative QCD (pQCD). In contrast to the electromagnetic force, the color coupling becomes large at low energies, hence 00confining00 the quarks into the observed hadrons. This fact prevents QCD from being investigated using perturbation 14 theory. A compelling description of QCD in this energy regime can be achieved through Wilson’s lattice gauge theory [3]. He showed a method of quantizing a gauge field theory such as QCD on a discreet four dimensional lattice in Euclidean space- time, while preserving exact gauge invariance. Thus the extraction of qualitative non-perturbative information concerning QCD became possible. However, due to the enormous computational power necessary for the numerical treatment, lattice QCD has only started to be able to describe baryon resonance masses and decay widths (see, e.g. [4, 5]). This gave rise to the necessity of developing effective quark models of hadrons (see, e.g. [6, 7]), which aimed at predicting the properties of the hadrons by reducing the strongly self-interacting multi-quark and gluon systems to an effective two- or three-quark system (denoted by the term constituent quark model). Since these two approaches are far from being able to offer practical solutions to low and intermediate energy scattering reactions, it is necessary to use effective methods for the description of the dynamical structure of these processes. These effective methods account for the inner structure of the baryons by introducing explicit baryon resonance states, whose properties are then extracted by comparison with experimental observables. Therefore the idea of the effective Lagrangian models is to account for the symmetries of the fundamental theory (QCD) by including only effective degrees of freedom instead of quarks. These effective degrees of freedom are modelled by using the properties of known baryons and mesons (which exist as bound quark states). This method has the advantage that it gives more intuitive insight on the dynamics of the reaction and makes the interpretation of the results somewhat easier. On the other hand, the effective Lagrangians and the scattering solutions based on them are involved and matching the constraints of unitarity, gauge invariance and analyticity is not easy. Because of this, many of the existing effective Lagrangian models are not analytic and many of them are not even unitary or gauge invariant (for the electromagnetic case). In the framework of effective Lagrangian models for hadrons, the nucleon and the pion are among the most important particles. The pion plays an important role 15 in mediating the nuclear force and is responsible for the long-range nucleon-nucleon interaction. Its presence can be observed in reactions such as photo- and electro- production on nucleons and nuclei. The general properties of the pions are listed in Table 1.1. Pion photo- and electro-production are very useful tools in investigating

IG J PC mass [MeV] lifetime [s] Decay modes branching ratio [%]

± − − −8 π 1 0 139.57 2.60 · 10 µνµ 100

π0 1− 0−+ 134.98 8.40 · 10−17 γγ 98.8

Table 1.1: Properties of the pions.

the structure of nucleons and nuclei. New high intensity, high duty-factor electron accelerators such as Jeﬀerson Laboratory (Newport News), MAMI (Mainz), ELSA (Bonn), as well as modern laser backscattering facilities such as LEGS (Brookhaven), SPRING-8 (Japan) and GRAAL (Grenoble), permit the experimental investigation of coincidence experiments with various polarization and/or spin observables. The theory of pion photo-production on the nucleon was ﬁrst written in the 1950s. Kroll and Ruderman [8] derived the model independent predictions of the observables in the threshold region, the so called low energy theorems (LET). The general formalism for the process γ + N → π + N was developed by Chew et al. [9] (CGLN amplitudes, see e.g. Appendix D). Fubini et al. [10] extended the predictions of the LET by including the hypothesis of a partially conserved axial current (PCAC). Their model succeeded in describing the threshold amplitudes as

2 a power series in the ratio mπ/mN up to terms of order (mπ/mN ) . For photon energies up to 500 MeV (laboratory frame), the experimental results, coupled with theoretical dispersion calculations [11], allow these results to be tabulated in terms of multipoles as a function of photon energy [12, 13, 14, 15]. In 1969 Peccei [16] first introduced an effective chiral Lagrangian for single pion photo-production that explicitly included a phenomenological πN∆ interaction. The main difference among 16 various effective Lagrangian approaches is the treatment of the ∆ resonance, including the modelling of the mass as well as the energy dependent width. A very simple and effective model for describing pion production processes from the free nucleon was developed in the 70’s by Blomqvist and Laget [17]. It has provided a very good description of the available experimental data up to the first resonance region. They

2 gave a non-relativistic operator expansion up to order (p/mN ) which could be easily incorporated in nuclear physics applications [18, 19]. A different model was developed by Nozawa, Blankleider and Lee [20], where they imposed unitarity by a dynamical model which takes into account off-shell effects in the final πN rescattering. In order to obtain a good fit of the photo-production data they multiplied all of the Born terms by a phenomenological cutoff parameter; as a result a strong dependence of the calculations on this cutoff was observed. Two more recent models are worth mentioning in detail: one is the unitary isobar model developed by Drechsel et al. [21] (known as MAID) and the second is the dynamical model of pion photo- and electro-production constructed by Sato and Lee [22, 23] (known as the SL model). MAID uses the prescriptions of the isobar model of [24, 25] which assumes that the resonance contributions in the relevant multipoles have Breit-Wigner forms.

They explicitly include nucleon resonances such as: P33(1232), P11(1440), D13(1520), 2 ∗ S11(1535), F15(1680) and D33(1700). The Q dependence of the γNN vertices are determined via the corresponding helicity amplitudes. The multipole amplitudes relevant to the resonant contributions written in Breit-Wigner form are:

iφ ΓtotWRe Al±(W ) = Al±fγN (W ) 2 2 fπN (W )CπN , (1.1) WR − W − iWRΓtot ∗ where fπN is the Breit-Wigner factor which describes the decay of the N resonance with total width Γtot and partial width ΓπN . CπN is the isospin factor of the resonance. The factor fγN (W ) is a parametrization of the W dependence of the ∗ γNN vertex beyond the resonance peak and WR is the total energy in the center-of- momentum frame (CM) at the resonance position. The electromagnetic amplitudes

Al± are linear combinations of the usual electromagnetic helicity amplitudes A1/2 17 and A3/2. The non-resonant contributions are described by traditional evaluation of the Feynman diagrams, derived from an eﬀective Lagrangian density. These non- resonant contributions are commonly referred to as the Born terms and are described using a mixed pseudovector-pseudoscalar πNN coupling hence taking in account the consistency of the pseudo-vector coupling with low energy theorems, while the renormalizability of pseudo-scalar coupling implies a better description of the data at high energies (above 500 MeV ). MAID implements both schemes by introducing a gradual transition between them. Their eﬀective Lagrangian for such a hybrid model is written as:

2 2 HM Λ PV ~q0 PS LπNN = 2 2 LπNN + 2 2 LπNN , (1.2) Λ + ~q0 Λ + ~q0 where ~q0 is the asymptotic pion momentum in the πN CM frame and the cut-oﬀ parameter Λ = 450 MeV . The unitarity of the model is implemented via the parameter φ in (1.1) and its role is to adjust the phase of the total multipole (background plus resonance) to the corresponding pion-nucleon scattering phase shift δπN . The latter values are taken from the analysis of the VPI group (SAID program) [67]. In the extension to pion electro-production, MAID assumes the same electromagnetic form factors at each relevant vertex in the Born terms in order to satisfy current conservation. Such a simple and practical model, due to the parameterization of each of the resonant contributions, describes the individual multipoles very well and overall agrees extremely well with the experimental observables. The SL model describes the pion photo- and electro-production in terms of photon and hadron degrees of freedom. They start with the Hamiltonian:

X H = H0 + ΓMB↔B0 (1.3) M,B,B0 where H0 is the free hamiltonian and ΓMB↔B0 describes the absorption and emission of a meson (M) by a baryon (B). Such a Hamiltonian is obtained from a phenomenological Lagrangian for N, ∆, π, ρ, ω and photon ﬁelds. A unitary 18 P transformation is performed in (1.3) up to second order on HI = ΓMB↔B0 to obtain an eﬀective Hamiltonian:

Heff = H0 + vπN + vγπ + ΓπN↔∆ + ΓγN↔∆, (1.4) where vπN and vγπ are the non-resonant πN ↔ πN potentials and non-resonant γN ↔ πN transition, respectively. The ∆ excitation is described by the vertex interactions ΓγN↔∆ and ΓπN↔∆. The non-resonant vγπ consists of the usual pseudovector Born terms, ρ and ω exchanges, and the crossed ∆ term. The idea behind this unitary transformation is to eliminate from the Hamiltonian the

0 unphysical vertex interactions, MB ↔ B with mM + mB < mB0 . The resulting eﬀective Hamiltonian Heff is energy independent and hermitian, hence the unitarity of the resulting amplitude is trivially satisﬁed. The drawback of the model is again the way they satisfy current conservation, that is by assuming the same electromagnetic form factors at each relevant vertex in the Born terms. This dynamical model was used to investigate photo- and electro-production reactions on the nucleon and obtained a very good description of the new data from JLab and MIT-Bates.

1.2 Background

In this section we will introduce to the reader some of the quantities necessary to understand this work. Single pion production above the threshold region is dominated by the excitation of nucleon resonances. Below the two-pion threshold it is practically the only source that contributes to the total photo-absorption cross section, shown in Fig. 1.1. The M1 (magnetic dipole) excitation of the ﬁrst, the E1 (electric dipole) excitation of the second and the E2 (electric quadrupole) excitation of the third resonance region clearly exhibit peaks associated with some of the underlying resonances. It is obvious from Fig. 1.1, that below 500 MeV (photon CM energy), the total cross section is almost entirely dominated by the M1 ∆(1232) excitation. The notation of the multipoles is associated with the electromagnetic nature of the excitation modes induced by the photon. For example El±, Ml±, Sl± (Ll±) are 19

Figure 1.1: The total photo-absorption cross section of the proton and its decomposition into selected channels as a function of CM photon energy in GeV. Figure adopted from [26]. the electric, magnetic and scalar (longitudinal) multipoles, respectively, where the

1 1 ± sign is the abbreviation for J = l ± 2 , while the factor 2 corresponds to the nucleon spin. The ﬁnal state is described by J, the total spin of the initial (ﬁnal) state, and l, the relative orbital momentum of the πN system with parity (−1)l+1. The transverse polarizations λ = ±1 of the photon, lead to electric and magnetic multipole transitions EL and ML; the longitudinal polarization λ = 0 (only for the virtual photon) leads to longitudinal or Coulomb transitions CL. In Table 1.2 the lowest electromagnetic excitation modes and their corresponding states of the πN system are shown. A systematic analysis of pion production processes allows the determination of the multipoles which correspond to each of the nucleon resonances. Experimental extraction of the multipoles provides a set of quantities representing the experimental data to be compared with theoretical models. Several quantities and problems of major interest, which have been the focus of both experimental and theoretical approaches are: 20

Electromagnetic πN system Pion production

L multipole J l multipole

0 C0 1/2 1 L1−

1 E1, C1 1/2 0 E0+, L0+

3/2 2 E2−, L2−

M1 1/2 1 M1−

3/2 1 M1+

2 E2, C2 3/2 1 E1+, L1+

5/2 3 E3−, L3−

M2 3/2 2 M2−

5/2 2 M2+

Table 1.2: Relations between electromagnetic multipoles and pion production multipoles.

2 • the extraction of the M1 and E2 strength at the photon point (GM (Q = 0) 2 and GE(Q = 0));

• the distribution in both W (the total CM energy of the πN system) and Q2

2 µ µ (Q = −qµ · q where q is the four momentum of the virtual photon) space of the magnetic dipole (M1), which is associated directly with the ﬁrst resonance

region (∆(1232 or P33);

• the small admixture of the E2 quadrupole in the ﬁrst resonance region, usually associated with the ∆ quadrupole deformation (the E2/M1 ratio);

• the 00breathing00 mode of the ∆ which is associated with the C2/M1 ratio, when Q2 6= 0 (electro-production)

• extraction of the 00bare00 γ∗N ↔ ∆ electromagnetic form factors and comparison with the predictions from constituent quark models; 21

• direct comparison between experimental and theoretical polarization observables;

• the problem of gauge invariance and various ways of restoring it.

All of the above problems are addressed in the present work. After the introduction in Chapter 1, we derive the ingredients of the model in Chapter 2, focussing on the driving term of the scattering equation Vπγ∗ and its derivation from the usual Feynman diagrams. In Chapter 3 we introduce the dynamical model of [34] for πN scattering which is used in this work. In Chapter 4 we describe the procedure of dynamically including the ﬁnal state interaction and calculating the scattering matrix

Tπγ∗ . In Chapter 5 we describe an alternative approach for solving two-body integral equations. In Chapter 6 we present the results of our calculations, by ﬁtting the data

2 for the Q dependence of the M1+ multipole, and the E2/M1 and C2/M1 ratios. We leave for the Appendices the derivations of some of the equations as well as some of the conventions, and quote additional relations used in this work. In the Appendices, we also summarize the parameters used in our work, such as the electromagnetic couplings and form factors. Finally, the references are listed. Chapter 2 Photo- and Electro-production Potential

Electromagnetic induced reactions on the free nucleon are usually treated by starting with the tree level diagrams. A photon (real or virtual) incident on a free nucleon couples to the nucleon electromagnetic current causing the target nucleon to emit a pion if the energy of the incident photon is high enough (the threshold energy for the production of a single pion is about 140 MeV ). If the energy of the photon is below approximately 500 MeV , the process is dominated by the one pion production mechanism. In describing this process one must also consider the isospin degrees of freedom since there are four basic single pion production channels:

γ∗ + p → n + π+ (2.1a)

γ∗ + p → p + π0 (2.1b)

γ∗ + n → p + π− (2.1c)

γ∗ + n → n + π0. (2.1d)

In building a model for single pion production, the fundamental ingredient is the scattering matrix at the tree level (since we work in the one photon exchange approximation). It is customary to systematically consider the scattering matrix as being composed of two major contributions: the background (Born) terms and the resonance terms. Hence, the observables measured in an experiment (differential cross-section, beam asymmetry, target polarization, etc.) are a coherent sum of these two major contributions. Before starting to actually build the full model one has to carefully include the significant exchanges (see Fig. 2.1) at the tree level and make sure that the resulting potential (Vπγ∗ ) satisfies current conservation and gauge invariance. 23

N VSJ* = + N

' KR + + + '

+ S + U, Z

Figure 2.1: The contributions included in our model.

In the following we will make a distinction between real photons, when the photon interacts directly with the nucleon, and virtual photons (denoted with an asterisk) when an electron scatters from a free nucleon resulting in the creation of a pion. In the latter case, the electron interacts with the nucleon via a virtual photon (one photon exchange approximation) carrying energy and momentum. This distinction is made in order to understand the similarities between these two apparently diﬀerent processes (pion photo- and electro-production). We will see that real photon pion production is a particular case of virtual photon pion production.

2.1 Born terms

It customary to consider as Born terms the following contributions (refer to Fig. 2.1): direct and crossed nucleon exchanges (s- and u- channel respectively), pion exchanges (t- channel) and the Kroll-Ruderman (contact) term. There are two possible coupling

PS 5 schemes at the πNN vertex: pseudoscalar ( with the structure gπNN γ in the PV 5 Lagrangian) and pseudovector (with the structure gπNN γ ∂/). When pseudo-vector coupling (PV) is used,the Kroll-Rudermann term is required to maintain gauge invariance. It originates from the minimal substitution (∂µ → ∂µ + ieAµ) in the 24

PV Lagrangian LπNN . PV coupling is preferred since it is consistent with low energy theorems (LET ) and current algebra predictions. In the current work we will use PV coupling and we will use the following convention: q is the 4-momentum of the photon, k the 4-momentum of the pion and p(p0) is the the 4-momentum of the incoming (outgoing) nucleon. We will also denote the 00mass squared00 of the virtual photon by

2 µ 2 −Q = qµ · q . For the case of real photons, Q = 0. The electromagnetic transition vertices are calculated from the corresponding eﬀective Lagrangians. In the following we give the both the Lagrangian and the derived electromagnetic transition vertices, and omit the isospin factors (instead we will treat in detail the isospin contribution for each diagram later in this chapter):

• photon coupled to nucleon

– the Lagrangian: µν τ µ κN σ Fµν LγNN = eΨN γ Aµ + i ΨN , (2.2) 2 4mN where

µ ν ν µ Fµν = ∂ A − ∂ A (2.3)

µ is the electromagnetic tensor, A is the photon ﬁeld, ΨN is the nucleon ﬁeld function, and τ/2 is the nucleon isospin operator.

– the vertex function: µν µ µ 2 κN σ qν 2 ΓγNN (q) = e γ F1(Q ) + ı F2(Q ) , (2.4) 2mN • photon coupled to pion

– the Lagrangian:

i µ j† Lγππ = eij3φπ∂ φπ Aµ, (2.5)

i j where φπ and φπ are the pion ﬁeld functions. – the vertex function:

µ 0 0µ µ 2 Γγππ(k , k) = e(k + k )Fπ(Q ), (2.6) 25

• Kroll-Ruderman term

– the Lagrangian:

ıegπNN µ ~ LKR = ΨN γ γ5[~τ × φπ]3ΨN Aµ (2.7) 2mN – the vertex function:

µ egπNN µ 2 m(KR) = γ γ5FA(Q ), (2.8) 2mN

2 where Fi(Q ) are the electromagnetic form factors which take into account the ﬁnite size of the hadrons, and κN is the anomalous contribution to the nucleon’s magnetic moment. In the nonrelativistic limit, the form factors are associated with the charge and magnetization distributions of the corresponding particles. These functions are determined from experiment and their calculation is one of the goals of many models. In addition we also quote the Lorentz structure of the hadronic vertex involved in the Born terms,

• pseudo-vector coupling πNN:

– the Lagrangian:

gπNN µ LπNN = ΨN γ5γ (∂µφπ)τΨN (2.9) 2mN – the vertex function:

gπNN ΓπNN (k) = γ5k/. (2.10) 2mN A major concern in electromagnetic induced reactions is gauge invariance. When

2 the potential matrix Vπγ∗ is calculated on-shell at the photon point (i.e. Q = 0), current conservation (and consequently gauge invariance) is obeyed. However, once we calculate at Q2 6= 0 (i.e. introduce electromagnetic form factors at each vertex where the virtual photon is coupled to), current conservation may be lost. This occurs because while current conservation is intrinsically satisﬁed for both resonant and vector meson exchanges (due to the Lorentz structure of their electromagnetic vertices), current conservation for the Born terms (N s− and u−channel exchanges, 26

π t−channel exchange and the Kroll-Ruderman term) requires cancellation among the various terms. One common solution to this problem, implemented by various authors [21, 23], is to choose all of the electromagnetic form factors (i.e., nucleon, pion and axial form factors) to be the same. This ad-hoc solution is rather convenient, and it is not too bad an approximation at low Q2. However, it is well known from experiment that these various form factors have diﬀerent Q2 dependence at higher Q2. In view of this, we choose a procedure that will permit arbitrary choices of the electromagnetic form factors involved, yet still maintain current conservation, and hence, gauge invariance, in the tree-level Vπγ∗ interaction. This is achieved by imposing the Ward-Takahashi identities on the electromagnetic vertices (Eq. (2.4) and Eq. (2.6)):

µ −1 0 −1 qµ · ΓγNN (q) = e(SN (p ) − SN (p)), (2.11)

µ 0 −1 0 −1 qµ · Γγππ(k , k) = e(∆π (k ) − ∆π (k)), (2.12)

2 2 where SN (p) = 1/(p/ − mN ) is the nucleon propagator and ∆π(p) = 1/(p − mπ) is the pion propagator. It is obvious that relations (Eq. (2.11) and Eq. (2.12)) are fully satisﬁed as long as the particles are on their mass-shell (i.e. p2 = m2) and the form factors involved are set equal to 1 (which is the case for real photons). The extension to electro-production (Q2 6= 0), requiring electromagnetic form factors, results in violation of gauge invariance unless all the form factors are the same. We will illustrate this by the following. Let’s ﬁrst consider the photo-production case with all of the external particles on their mass-shell. The explicit contribution for the amplitudes of each of the Born diagrams is:

• s - channel N exchange

µ µ m(s,N) = ΓπNN (k)SN (p + q)ΓγNN (q) (2.13)

• u - channel N exchange

µ µ 0 m(u,N) = ΓγNN (q)SN (p − q)ΓπNN (k) (2.14) 27

• t - channel π exchange

µ µ m(t,π) = ΓπNN (k − q)∆π(k − q)Γγππ(k, k − q) (2.15)

• Kroll -Rudermann (contact) term

µ egπNN µ 2 m(KR) = γ γ5FA(Q ). (2.16) 2mN

2 In Eq. (2.16) we have introduced the nucleon axial form factor FA(Q ) which equals one when Q2 = 0. Performing the isospin decomposition (see further details later in this chapter) we write down the full Born amplitude for total isospin 1/2:

µ µ µ µ µ m 1 = 3m(s,N) − m(u,N) + 2m(t,π) + 2m(KR). (2.17) (T = 2 )

At Q2 = 0 gauge invariance for this amplitude is checked by using the standard

µ method of building the invariant qµ · m , and conﬁrming that it vanishes. Using Eq. (2.11) and Eq. (2.12) in Eq. (2.17) we get:

µ µ µ µ µ qµ · m 1 = qµ · [3m(s,N) − m(u,N) + 2m(t,π) + 2m(KR)] (T = 2 ) egπNN 5 −1 −1 = {3γ k/SN (p + q)[SN (p + q) − SN (p)] 2mN −1 0 −1 0 0 − [SN (p ) − SN (p − q)]SN (p − q)γ5k/

5 −1 −1 + 2γ (k/ − q/)∆π(k − q)[∆π (k) − ∆π (k − q)] + 2q/γ5} eg = πNN [3γ5k/ − γ5k/ − 2γ5(k/ − q/) + 2q/γ5] = 0. (2.18) 2mN Hence the current is conserved for the Q2 = 0 case and for the on-shell Born terms. For Q2 6= 0 one has to include electromagnetic form factors (see Eq. (2.4), Eq. (2.6) and Eq. (2.16) ). Under these circumstances Eq. (2.18) becomes: eg q · mµ = πNN {γ5k/[F (Q2) − F (Q2)] + γ5q/[F (Q2) − F (Q2)]}, (2.19) µ (T = 1 ) 1 π π A 2 2mN which shows that the on-shell Born amplitude is gauge invariant if the electromagnetic

2 2 2 form factors are chosen equal, i.e. F1(Q ) = Fπ(Q ) = FA(Q ). In reality it is physically reasonable to assume (and experimental data support this assumption) 28 that the electromagnetic form factors are different for each particle and interaction type, which means that in order to preserve gauge invariance of the on-shell Born amplitude, one has to modify the vertex functions in Eq. (2.4) and Eq. (2.6) such that the Ward-Takahashi relations (Eq. (2.11) and Eq. (2.12)) are satisfied also when Q2 6= 0. A modification of the vertex function which is Lorentz covariant and does not affect the Q2 = 0 result is:

µ µν µ µ 2 µ q q/ κN σ qν 2 ΓγNN (q) = e γ + (F1(Q ) − 1) γ − 2 + i F2(Q ) (2.20) q 2mN

qµq · (k + k0) Γµ (k0, k) = e (k0µ + kµ) + (F (Q2) − 1) (kµ + k0µ) − (2.21) γππ π q2

µ 5 µ gπNN µ 5 2 µ 5 q q/γ m(KR) = γ γ + (FA(Q ) − 1) γ γ − 2 (2.22) 2mN q

µν Note that the term involving the anomalous moment (i.e. σ qν) is not aﬀected by µν these considerations since it is gauge invariant by itself (qµσ qν = 0). Unfortunately this is not the whole story since the hadrons involved in the reaction are composite particles and in order to account for their compositeness one should introduce (hadronic) form factors for each of the baryons. These hadronic form factors also serve as cut-oﬀ functions where the rescattering is introduced. The hadronic form factors will also spoil current conservation (the cancellation among graphs (see Eq. (2.18)) would no longer occur). Thus, simply introducing hadronic form factors into the pion and nucleon exchange graphs is not reasonable. One way of introducing hadronic form factors without loss of current conservation is to include the additional contributions which would arise after minimal substitution is made on the hadronic form factors themselves. The problem was solved successfully by many authors, see e.g, [27, 28, 29]. Below I will show in detail how this method applies to the standard electromagnetic vertex functions (γNN and γππ). In the nucleon case one starts with the free Lagrangian:

−1 2 −1 2 L = [f (∂ )ψ](i∂/ − mN )f (∂ )ψ, (2.23) 29 where f(∂2) is the general form of the form factor operator. Using minimal substitution, where ∂µ is replaced by Dµ = ∂µ − ieAµ in Eq. (2.23) and expanding the form factor with a Taylor series up to the terms linear in the electromagnetic −1 2 −1 2 µ µ 2 2 ∼ −1 2 µ µ ∼ ﬁeld (f (D ) = f (∂ − ie(∂µA + Aµ∂ ) − e A ) = f (∂ − ie(∂µA + Aµ∂ )) = −1 2 µ µ −1 2 0 f (∂ ) − ie(∂µA + Aµ∂ )(f (∂ )) ), we get:

−1 2 µ µ −1 2 0 L = [f (∂ ) − ie(∂µA + Aµ∂ )(f (∂ )) ψ]

× (i∂/ − mN + eA/)

−1 2 µ µ −1 2 0 × [f (∂ ) − ie(∂µA + Aµ∂ )(f (∂ )) ψ]. (2.24)

After dropping the nonlinear terms in the vector ﬁeld A and considering just the interaction part of the Lagrangian we get:

−1 2 −1 2 Lint = e(f (∂ )ψ)(A/)(f (∂ )ψ)

µ µ −1 2 0 −1 2 − e[(i∂µA + Aµi∂ )(f (∂ )) ψ](i∂/ − mN )[f (∂ )ψ]

−1 2 µ µ −1 2 0 − e[f (∂ )ψ](i∂/ − mN )[(i∂µA + Aµi∂ )][(f (∂ )) ψ]. (2.25)

To write down the modified vertex we use the standard Feynman rules (i.e. 00wipe off00 the field variables, ψ, ψ, Aµ and replace i∂µ → pµ) and obviously, keep track of the −1 2 −1 2 −1 2 −1 02 order of the ∂µ operator (for example [f (∂ )ψ] → f (p ), [f (∂ )ψ] → f (p ), etc.).

µ 0 −1 02 −1 2 µ ΓγNN (p , p) = ef (p )f (p )γ

0µ µ −1 02 0 0 −1 02 − e(p + p )(f (p )) (p/ − mN )f (p )

0µ µ −1 2 0 −1 02 − e(p + p )(f (p )) (p/ − mN )f (p ) (2.26)

Again using a Taylor expansion to write down the ﬁrst derivative of the form factor (f −1(p2) ∼= f −1(p02) + (p02 − p2)(f −1(p02))0 and similarly for (f −1(p2))0 ) and including the anomalous part of the vertex (which does not take part in this whole logic since it is gauge invariant by itself), Eq. (2.26) becomes: (p0 + p)µqν Γµ (p0, p) = ef −1(p02)f −1(p2) gµν − γ γNN q · (p0 + p) ν 0 µ µν (p + p) −2 02 −1 0 −2 2 −1 eκσ qν −e 0 [f (p )SN (p ) − f (p )SN (p)] + . (2.27) q · (p + p) 2mN 30

Considering the pion case in the same fashion, i.e., starting by gauging the free Lagrangian of the spin−0 particle with form factors included,

−1 2 ∗ µ 2 −1 2 L = [f (∂ )φ ](∂µ∂ − mπ)[f (∂ )φ], (2.28) we get:

µ 0 0 µ −1 02 −1 2 Γγππ(k , k) = e(k + k) [f (k )f (k ) f −1(k02) − f −1(k2) + (f −1(k02)S−1(k0) + f −1(k2)S−1(k)) ]. (2.29) N N k02 − k2 Since the free Lagrangian is modiﬁed by the form factors, the propagators take the following form:

2 2 fN (p ) SN (p) = (2.30a) p/ − mN

2 2 fπ (k ) ∆π(k) = 2 2 . (2.30b) k − mπ As a result, the modified vertices (Eq. (2.27), Eq. (2.29)) and propagators (Eq. (2.30)) satisfy the same Ward Takahashi identities as the unmodified ones. The Kroll Ruderman term does not get modified since we do not introduce hadronic form factors in the contact interaction. Now we should point out that our vertices have undergone two modifications, both of which are taken in account in the final form of the model. The first modification (outlined by Eq. (2.17) to Eq. (2.22)), correctly introduces different electromagnetic form factors without violating (on-shell) current conservation, while the second modification (outlined by Eq. (2.23) to Eq. (2.29)) obeys current conservation even with hadronic form factors. The final forms of the electromagnetic vertices of the nucleon and pion along with the modified Kroll Ruderman term (Eq. (2.22)) are: (p0 + p)µqν Γµ (p0, p) = ef −1(p02)f −1(p2) gµν − γ γNN q · (p0 + p) ν (p0 + p)µ −e [f −2(p02)S−1(p0) − f −2(p2)S−1(p)] q · (p0 + p) N N µ µν −1 02 −1 2 2 µ q q/ ieκσ qν 2 +ef (p )f (p )(F1(Q ) − 1) γ − 2 + F2(Q ) (2.31) q 2mN 31

µ 0 0 µ −1 02 −1 2 Γγππ(k , k) = e(k + k) [f (k )f (k ) f −1(k02) − f −1(k2) +(f −1(k02)S−1(k0) + f −1(k2)S−1(k)) ] N N k02 − k2 qµq · (k + k0) +ef −1(k02)f −1(k2)(F (Q2) − 1) (kµ + k0µ) − . (2.32) π q2 The speciﬁc forms of f −1(k2) for each vertex function are given in the Appendix B.

2.2 Vector meson exchanges

For the vector meson exchanges (refer again to Fig. 2.1) the electromagnetic and hadronic transition vertices, written in momentum space, are:

• photon coupled to vector meson (v = ω, ρ)

– the Lagrangian:

gγπv µαβν i ν ν Lγπv = ∂µAα∂βφπ (δi3ω + ρi ) , (2.33) mπ

where ω and ρi are the ﬁeld functions of the ω and ρ respectively.

– the vertex function:

µα gγπv µαβν 2 Γγπv(q, k) = kβqνFv(Q ) (2.34) mπ • vector meson coupled to nucleon

– the Lagrangian: α ıκv αν ν α α LvNN = gvNN ΨN γ + σ ∂ (ω + τiρi )ΨN (2.35) 2mN – the vertex function: α α κv αν ΓvNN (p) = gvNN γ − σ pν , (2.36) 2mN where k and q in Eq. (2.34) are the 4-momenta of the pion and photon, respectively, and p in Eq. (2.36) is the 4-momentum of the corresponding vector meson. µ and α are the indexes of the electromagnetic ﬁeld and boson ﬁeld respectively, and κv is the tensorial coupling of the meson to the nucleon. Gathering all of the pieces together 32 we can write down the Feynman tree-level amplitudes for these contributions (again omitting the isospin factors).

• t - channel ω/ρ exchange

µ α αβ µβ m(t,v) = ΓvNN (k − q)∆v (k − q)Γγπv(k, k − q), v = (ω, ρ), (2.37)

αβ αβ 2 2 where ∆ = g /[(k − q) − mv] is the vector meson propagator. For the ω/ρ exchanges alone, current conservation is obeyed due to the Lorentz structure of the

µαβν electromagnetic vertex (qµ kβqν = 0), therefore we can directly introduce realistic 2 electromagnetic form factors (Fv(Q )) at the photon vertex, as well as hadronic form factors, without any complications.

2.3 Resonant terms

Clearly nucleon resonances, particularly the ∆(1232), play a very important role in pion photo- and electro-production. We will keep our model as simple as possible by only including the ∆(1232) resonance, although higher resonances could be added.

Lab Thus, we are aiming to reproduce data up to the ﬁrst resonance energy (i.e. Eγ ∼ 400 MeV ). We include both direct and crossed ∆ exchange diagrams. For the electromagnetic transition vertex of the ∆ we implement the vertex suggested by [30], which obeys the fundamental properties of being gauge invariant in both the electromagnetic and high-spin ﬁeld sense. Writing the vertex function in momentum space and omitting the isospin factor we get:

• photon coupled to ∆

– the Lagrangian:

−3e(m∆ + mN ) + LγN∆ = 2 2 ΨN T3 4mN [(m∆ + mN ) − q ] 2 2 µν 2 µν G3(Q ) β µν × G1(Q )GeµνF + G2(Q )γ5GµνF + γ5γ Gβν∂µF , mN (2.38) 33

µν µ ν ν µ where ∆µ is the Rarita-Schwinger ﬁeld function [31], G = ∂ ∆ − ∂ ∆ µν 1 µναβ is the Rarita-Schwinger ﬁeld invariant tensor, Ge = 2 Gαβ, and T3 is the isospin 3/2 operator.

– the vertex function:

αµ −3e(m∆ + mN ) 2 αµβν ΓγN∆(p, q) = 2 2 [G1(Q ) pβqν 2mN [(m∆ + mN ) − q ] 2 αµ α µ 5 + ıG2(Q )(p · qg − q p )γ G (Q2) + ı 3 [q2(pµγα − p/gαµ) + qµ(p/qα − p · qγα)]γ5] mN (2.39)

• ∆ coupled to nucleon

– the Lagrangian:

fπN∆ αβµν a LπN∆ = (∂α∆e β)γ5γµTaΨN ∂νφπ + h.c., (2.40) mπm∆

where Ta is the 3/2 isospin operator.

– the vertex function:

α fπN∆ αβµν 5 ΓπN∆(k, p) = pβγµγ kν, (2.41) mπm∆ where in Eqs. (2.39) and (2.41) the index α is related to the ∆ field and µ is related to the electromagnetic field, while p, q and k are the 4- momenta of the ∆, photon and pion, respectively. αµ From Eq. (2.39) it is easy to see that ΓγN∆ obeys gauge invariance of high αµ spin fields (i.e. pαΓγN∆ = 0) and gauge invariance of the electromagnetic field αµ rd (i.e. qµΓγN∆ = 0). The 3 coupling in Eq. (2.39) has no contribution for the case of real photons (Q2 = 0), but for virtual photons (Q2 6= 0) determines the longitudinal/Coulomb couplings of the photon to the nucleon. The Feynman tree- level amplitudes for the resonant contribution (again omitting the isospin factors) is:

• s−channel ∆ exchange

µ α αβ µβ m(s,∆) = ΓπN∆(k, p + q)S∆ (p + q)ΓγN∆(p + q, q) (2.42) 34

• u− channel ∆ exchange

µ µα αβ β m(u,∆) = ΓγN∆(p − k, q)S∆ (p − k)ΓπN∆(k, p − k), (2.43) where we have used just the 4- momenta of the incoming nucleon, p, outgoing pion,

3 k, and incoming photon, q. The spin- 2 ﬁeld propagator is given by: αβ 1 αβ 1 α β 1 α β α β S∆ (p∆) = g − γ γ − 2 (p/∆γ p∆ + p∆γ p/∆) . (2.44) p/∆ − m∆ 3 3m∆

2.4 Isospin decomposition of the amplitude

In this section, we present the isospin formalism for the photo-production amplitude. Assuming for the moment that only the meson current inﬂuences the charge symmetry properties of pion photo- and electro-production, i.e. the photon behaves as an isoscalar particle, one obtains cross-section relations as in, e.g., πN → ηN:

σ(γp → π+n) = σ(γn → π−p)

σ(γp → π0p) = σ(γn → π0n).

However it is well known that the photon has a more complicated isospin structure and these relations are not satisﬁed by the experimental observations. As ﬁrst pointed out by Watson [32], the photon is composed of an isoscalar |I,Izi = |0, 0i part and a part which is the third component of an isovector |I,Izi = |1, 0i part. It is because of this mixing that one must carefully treat the isospin degree of freedom in

1 reactions involving photons. In the ﬁnal state, we have particles involving isospin- 2 (the nucleon) and isospin-1 (the pion), therefore they can couple to the following:

1 • total isospin I = 2 : isoscalar photon coupled to nucleon.

1 • total isospin I = 2 : isovector photon coupled to nucleon.

3 • total isospin I = 2 : isosvector photon coupled to nucleon. 35

According to the above statements the following three isospin decompositions are performed in computing the amplitude:

µ µ µ µ m(a) = δa3m(+) + τam(0) + ia3bτbm(−) 1 1 = τ τ mµ + τ mµ + δ − τ τ mµ (2.45) 3 a 3 (1/2) a (0) a3 3 a 3 (3/2) 1 1 1 = τ (1 + τ ) mµ + τ (1 − τ ) mµ + δ − τ τ mµ , 2 a 3 p (1/2) 2 a 3 n (1/2) a3 3 a 3 (3/2) where τ are the usual Pauli matrices, index 0a0 stands for the pion isospin states (π±, π0) and the lower index, in brackets, refers to the total isospin. Some relations among various representations are:

µ µ µ µ µ µ m(3/2) = m(+) − m(−), m(1/2) = m(+) + 2m(−) 1 1 mµ = mµ + mµ , mµ = mµ − mµ . p (1/2) (0) 3 (1/2) n (1/2) (0) 3 (1/2) The above amplitudes are related to the specific reactions of interest by: 2 mµ(γ∗p → π0p) = mµ + mµ = mµ + mµ , (+) (0) 3 (3/2) p (1/2) 2 mµ(γ∗n → π0n) = mµ − mµ = mµ − mµ , (+) (0) 3 (3/2) n (1/2) √ √ 1 mµ(γ∗p → π+n) = 2(mµ + mµ ) = 2 − mµ + mµ , (0) (−) 3 (3/2) p (1/2) √ √ 1 mµ(γ∗n → π−p) = 2(mµ − mµ ) = 2 mµ + mµ . (0) (−) 3 (3/2) n (1/2) In the following I will show as an example how to project the Born amplitudes onto specific isospin states and therefore find the contribution of each exchange to the isospin amplitude. The γNN vertex function for the real photon (for the case Q2 6= 0 the procedure is similar) is usually decomposed as follows:

µν µ † µ ieσ qν ΓγNN (q; κ) = χN γ (1 + τ3) + [(κp + κn) + τ3(κp − κn)] χN , (2.46) 2mN where κN (with N = p, n ) is the anomalous magnetic moment of the nucleon and τ3 is the third component of the isospin matrix operator. Considering the case of the 36 nucleon s−channel exchange we have:

µ,a † 0 µ m(s,N) = χN τau(p )γ5k/SN (p + q)(1 + τ3)γ u(p)χN µν † 0 ieσ qν + χN τau(p )γ5k/SN (p + q)(κS + τ3κV ) u(p)χN 2mN † µ † µ = χN τa(1 + τ3)χN V + χN τa(κS + τ3κV )χN T 1 = χ† τ χ (Vµ + κ Tµ) + χ† τ τ χ 3(Vµ + κ Tµ), (2.47) N a N S N 3 a 3 N V where we have used V µ and T µ as a generic shorthand notation for the vector and tensor components for this diagram, the index 0a0 is the isospin index of the outgoing pion, and κS = κp+κn, and κV = κp−κn. Comparing this last relation with Eq. (2.45), we can exactly identify in which isospin channel this contribution appears. Similarly for the u−channel nucleon exchange using a similar shorthand notation, we obtain:

µ,a † µ µ † µ µ m(u,N) = χN τaχN (V + κST ) + χN τ3τaχN (V + κV T ), (2.48)

1 and after using some properties of the Pauli matrices in Eq. (2.48) (i.e. 2 [τi, τj] =

δij − τjτi) and doing some algebraic manipulations we get: 1 mµ,a = χ† τ χ (Vµ + κ Tµ) + χ† τ τ χ (Vµ + κ Tµ) (u,N) N a N S N 3 a 3 N V 1 + χ† δ − τ τ χ (Vµ + κ Tµ). (2.49) N a3 3 a 3 N V µ µ µ µ In Eq. (2.47) and Eq. (2.49) we identify V +κST and V +κV T as the isoscalar and isovector amplitudes (which arise due to the isospin components) of the photon. The µ isoscalar amplitudes contribute only to m(0) (refer to Eq. (2.45)), while the isovector µ µ amplitudes contribute to both mT =(1/2) and mT =(3/2). For the case of the t−channel exchanges (π, ω, and ρ) and ∆ exchanges, a similar path of calculation is followed. We merely quote the ﬁnal expressions: 1 1 1 mµ,a = χ† [τ , τ ]χ mµ = χ† δ − τ τ (−mµ) + τ τ (2mµ) χ (2.50) (t,π) N 2 3 a N N a3 3 a 3 3 a 3 N

1 1 mµ,a = χ† δ χ mµ = χ† δ − τ τ mµ + τ τ mµ χ . (2.51) (t,ω) N a3 N N a3 3 a 3 3 a 3 N For ρ exchange, due to the fact that it is an isovector particle, it only contributes µ to the m(T =0) amplitude. The Kroll-Ruderman term has the same isospin structure 37 as the π t−channel exchange (see Eq. (2.50)). Using these last considerations and Eq. (2.47)-Eq. (2.51) we are able to determine the contribution of each exchange to the isospin decomposed amplitudes:

µ µ µ µ µ µ µ m 1 = 3m(s,N) − m(u,N) + 2m(t,π) + 2m(KR) + m(t,ω) + 2m(u,∆) (2.52) (T = 2 )

3 1 mµ = 2mµ − mµ − mµ + mµ + mµ + mµ (2.53) (T = 3 ) (u,N) (t,π) (KR) (t,ω) (s,∆) (s,∆) 2 2 2

µ µ µ µ m(T =0) = m(s,N) + m(u,N) + m(t,ρ). (2.54) Chapter 3 πN Elastic Scattering

In this chapter I will brieﬂy summarize the model for πN scattering which we have used for our calculations. The model has been described extensively in [33, 34, 35]. It is based on the solution of a 3-dimensional (equal-time or instantaneous) reduction of the Bethe-Salpeter equation. The driving term in the scattering equation consists of the following tree-level exchanges derived from an eﬀective Lagrangian: N(938),

∗ N (1440), ∆(1232), D13(1520), S11(1535), ρ(770) and σ(550). The driving potential term is regularized by hadronic form factors, and the corresponding cutoﬀ masses, coupling constants, and baryon masses, are ﬁtted to the πN scattering partial-wave data base. The model provides a very good description of the S−, P − and D− wave phase shifts up to 600 MeV pion lab frame kinetic energy.

3.1 Tree Level potential πN matrix

In this section we consider the tree-level potential matrix (driving term in the scattering equation) and the speciﬁc contributions included in the model. The following exchanges are taken into consideration, based on Weinberg’s formulations [36]: Born terms, with nucleon direct and crossed exchanges, the Weinberg-Tomozawa contact term [37], which is represented by a ρ- meson exchange and ∆ direct and crossed exchanges. We proceed similarly as in the case of pion electro-production and quote each vertex function separately (we do not include explicitly the isospin degrees of freedom):

• pion coupled to nucleon:

gπNN ΓπNN (k) = γ5k/ (3.1) 2mN where k is the pion 4-momentum. 39

• ρ- meson coupled to nucleon: α α κρ αν ΓρNN (q) = gρNN γ − σ qν (3.2) 2mN

where q is the ρ 4- momentum.

• ρ-meson coupled to pion:

(k02 − k2)(k0µ + kµ) Γµ (k0, k) = g (k0µ − kµ) + (3.3) ρππ ρππ (k0 + k)2

where k0, k are the 4-momenta of the pions.

• ∆ ↔ πN transition vertex:

α fπN∆ αβµν ΓπN∆(k, p) = pβγµγ5kν (3.4) mπmN

where k and p are the π and ∆ 4-momenta.

In Eq. (3.4) we have introduced the vertex proposed in [38, 39] and referred to as gauge invariant (GI) πN∆ coupling. It is invariant under the Rarita-Schwinger gauge transformation: ∆µ(x) → ∆µ(x) + ∂µ(x), where (x) is a spinor ﬁeld. This coupling does not involve the spin-1/2 components of the ∆ ﬁeld, hence the spin-1/2 background is totally absent from the corresponding ∆-exchange amplitude. Using Eq. (3.4) the ∆-exchange amplitudes are:

2 2 fπN∆ P 3/2 0α β V∆,s−channel = Pαβ (P )k k , (3.5) mπm∆ P/ − m∆

2 2 fπN∆ pu 3/2 α 0β V∆,u−channel = Pαβ (pu)k k , (3.6) mπm∆ p/u − m∆ 0 where pu = P − k − k , and 1 1 P3/2(p) = g − γ γ − (p/γ p + p γ p/) (3.7) αβ αβ 3 α β 3p2 α β α β is the spin-3/2 projection operator. In Eq. (3.5) and Eq. (3.6) the following identity is used: λ σ µµ0 1 µ µ0 ν ρ 2 3/2 p γ g − γ γ 0 p γ = −p P (p). (3.8) µλσα 3 µ νρβ αβ 40

In addition to these contributions a σ-exchange is used to simulate the isoscalar contribution of the correlated two-pion exchange. The corresponding exchange potential is: 2 gσNN gσππ t − 2mπ Vσ = 2 . (3.9) 8πmπ mσ − t ∗ The P11 and S11 N resonances are treated the same way as the nucleon but with different masses, couplings, and, in the case of the S11, different parity. The D13 contribution is included in a similar manner as the ∆ (i.e. the same propagator and interaction vertex), but with different isospin, parity and mass.

3.2 Two-body scattering equation

The starting point of to construct the πN scattering amplitude is the Bethe-Salpeter (BS) equation, shown schematically in Fig. 3.1, Z d4q00 T (q0, q; P ) = V (q0, q; P ) + i V (q0, q00; P ) G(q00; P ) T (q00, q; P ), (3.10) (2π)4 where T is the T −matrix, G is the two-particle propagator, and V is the two-particle- irreducible potential. This is the Bethe-Salpeter equation for the case of elastic scattering of a scalar

00 00 00 00 with mass mπ — the pion — on a spinor with mass mN — the nucleon . We attribute the momenta p, p0 to the nucleon and k, k0 to the pion. The relative 4- momentum of the incoming channel is conveniently deﬁned by q = βp − αk, where the Lorentz scalars α and β are given by:

2 2 α = p · P/s = (s + mN − mπ)/2s ,

2 2 β = k · P/s = (s − mN + mπ)/2s , (3.11) where the total 4-momentum P = p + k = p0 + k0 and s = P 2. Similarly one deﬁnes q0 = βp0 − αk0 and q00 = βp00 − αk00 as the relative 4-momenta of the outgoing and intermediate state, respectively. In terms of these variables, the two-body πN Green’s function of Eq. (3.10) is:

1 (αP + q) · γ + mN G(q; P ) = 2 2 2 2 . (3.12) (βP − q) − mπ + i (αP + q) − mN + i 41 k k'

T = V + V G T p p'

Figure 3.1: Diagrammatic form of a relativistic two-body scattering equation.

Projecting the equation onto the basis of the nucleon helicity spinors (deﬁned in Appendix A), we obtain

ρ0ρ 0 ρ0ρ 0 Tλ0λ(q , q; P ) = Vλ0λ (q , q; P ) Z 4 00 X d q ρ0ρ00 0 00 (ρ00) 00 ρ00ρ 00 + i V 0 00 (q , q ; P ) G (q ; P ) T 00 (q , q; P ), (3.13) 4π3 λ λ λ λ λ00ρ00 where the helicity amplitudes are deﬁned as

ρ0ρ 0 (ρ0) 0 0 (ρ) Tλ0λ(q , q, P ) = (1/4π)u ¯λ0 (αP + q ) T (q , q, P ) uλ (αP + q), (3.14) and analogously for V , while the deﬁning equation for G(ρ) is

(ρ0) 0 0 (ρ) (ρ) u¯λ0 (αP + q) γ G(q; P ) γ uλ (αP + q) = δλ0λ δρ0ρ G (q; P ), (3.15) and hence 1 1 G(±)(q; P ) = √ √ , (3.16) 2 2 q0 + α s ± (EαP +q − i) (β s − q0) − ωβP −q + i

p 2 2 p 2 2 0 0 with Eq = q + mN and ωq = q + mπ. The labels λ(λ ) and ρ(ρ ) are referred to as helicity and ρ−spin projections, respectively. In the equal-time (ET) reductions of the BS equation (see e.g. [40]) one eﬀectively removes the q0 poles from the potential (V ), and exactly treats the poles of the two- particle propagator (G). The removal of the poles from V is usually done by ﬁxing the relative-energy variable q0. The common choice is the covariant form P · q = 0, which reduces in the CM frame to q0 = 0.

ρ0ρ 0 ρ0ρ 0 Tλ0λ(q , q; P ) = Vλ0λ (q , q; P ) Z 3 00 X d q ρ0ρ00 0 00 (ρ00) 00 ρ00ρ 00 + V 0 00 (q , q ; P ) G (q ; P ) T 00 (q , q; P ), (3.17) 4π2 λ λ ET λ λ λ00ρ00 42 where the equal-time two-particle propagator in the CM system is given analytically by the contour integration: Z ∞ (ρ) √ dq0 (ρ) −ρ GET (|q|; s) = 2i G (q; P ) = √ . (3.18) −∞ 2π ωq(−ρ s + Eq + ωq − i) Rotational invariance and parity conservation allows one to perform a partial wave decomposition of Eq. (3.17) using: ρ0ρ 0 X 1 J Jρ0ρ 0 0 T 0 (q , q) = J + D 0 (Ω 0 )T 0 (q , q , q , q; P ), (3.19) λ λ 2 λ λ q ,q λ λ 0 0 0 J Z +1 Jρ0ρ 0 0 ρ0ρ 0 J Tλ0λ (q0, q , q0, q; P0) = d(cos θ)Tλ0λ(q , q)dλ0λ(θ), (3.20) −1 0 where Ωq0q is the solid angle between ~q and ~q. In the CM frame, the oﬀ-shell scattering amplitude can be written in terms of eight scalar amplitudes,

ρ0ρ 0 ρ0 0 ρ0ρ 0 ρ0ρ 0 ρ T± (q , q; P0) = uλ0 (~q )[γ+T+ (q , q; P0) + γ−T− (q , q; P0)]uλ(~q) (3.21)

1 with deﬁnite parity (±), where γ± = 2 (1 ± γ0), hence the following parity conserving amplitude satisﬁes: Z ∞ 0 0 1 X 00 0 00 00 T Jρ ρ = V Jρ ρ + dqq2 Gρ V Jρ ρ T Jρ ρ. (3.22) ± ± π ET ± ± 0 ρ00 In the calculation of the phase-shifts, which are commonly used in comparing with

J++ data, only the on-shell scattering amplitude, with the positive parity (i.e. T± ) is involved. Hence, the deﬁnition of the phase-shifts is: η e2iδl± − 1 f = l± , (3.23) ± 2i 1 where ηl± is the inelasticity and l = J ± 2 is angular momentum of the outgoing π. In the above equation one can identify:

J++ ˆ ˆ fl± =qαT ˆ (E, q,ˆ E, qˆ; P0), (3.24) where r [s − (m − m )2][s − (m + m )2] qˆ = N π N π , (3.25) 4s ˆ p 2 2 and E = P0α = qˆ + mN are the on-shell 3-momentum and energy, respectively, of the πN system in the CM frame. In Fig.3.2 we show some of the predictions of this model for phase shifts. 43

Figure 3.2: Phase shifts calculations for πN scattering. Data points are from various partial wave analysis [67].

3.3 Renormalization procedure

Since there are s-channel singularities in the driving term of Eq. (3.10), a renormalization procedure is employed. To perform such a procedure one starts with separating the potential into two terms V = Vs + Vu, where

0 X 0 Vs(q , q) = ΓB(q )SB(P )ΓB(q), (3.26) B represents the s−channel baryon exchanges, or pole terms (i.e. s−channel N, N ∗, ∆, etc. exchanges), and Vu contains the rest of the diagrams (i.e. u−, and t−channel exchanges), or the non-pole terms. In Eq. (3.26), ΓB represents the corresponding bare, or undressed, vertex function of the baryon B, and SB is the propagator function.

Since VsG is a separable kernel, hence allowing the resumming of these contributions, 44 the amplitude can be written as:

0 X ∗ 0 ∗ ∗ 0 T (q , q) = ΓB0 (q )SB0B(P )ΓB(q) + Tu(q , q), (3.27) B0B where the dressed vertex, Z d4q00 Γ∗ (q) = Γ (q) + i Γ (q00)G(q00)T (q00, q), (3.28) B B 4π3 B u the dressed propagator1, 1 1 ∗ = ∗ δB0B − ΣB0B, (3.29) SB0B SB and the indexes reﬂect the baryon B0 ↔ B mixed states, in the self energy matrix

ΣB0B. The non-pole scattering matrix, Tu, is: Z d4q00 T (q0, q) = V (q0, q) + i V (q0, q00)G(q00)T (q00, q). (3.30) u u 4π3 u u The calculation of the self energy for the intermediate state of the nucleon proceeds as follows: the renormalized spin-1/2 baryon propagator,

S(P/) = [P/ − m − Σren(P/) − i]−1 (3.31) where the renormalized self energy,

ren Σ (P/) = Σ(P/) − Z2(m0 − m) − (1 − Z2)(P/ − m), (3.32) with m0 being the bare mass and Z2 the ﬁeld renormalization constant. Again, using the projection operators γ± onto positive and negative energy-states, we get the positive (S(+)) and negative (S(−)) energy state propagators:

(±) ren −1 S (P0) = [±P0 − m − Σ± + i] , (3.33) with

ren ren ren Σ (P0) = Σ+ (P))γ+ + Σ− (P0)γ−. (3.34)

In the adopted renormalization scheme, several requirements are imposed:

100mixing00 occurs between baryon states B0 ↔ B with the same quantum numbers, such as spin and isospin, and parity. Mixing between baryons with the same spin, isospin but opposite parity occurs due to the presence of intermediate negative energy states. 45

• the renormalized baryon self energy2,

ren Σ± (P0) |P0=±m→ 0 (3.35)

• the ﬁrst derivative of the renormalized baryon self energy,

ren ∂Σ± (P0) |P0=±m → 0 (3.36) ∂P0

• the renormalized vertex (πN ↔ B) is equal to the bare vertex at the renormalization scale µ, deﬁned as the point where all the particles involved

2 2 2 2 2 2 in the vertex are on the mass-shell (k = mπ, p = mN , P = mB). The renormalized vertex3 is deﬁned as:

∗ ΓB(q; P ) = Z1ΓB, (3.37)

∗ where Z1 = ΓB(µ)/ΓB(µ) is the coupling renormalization constant.

Note: The renormalization constants (i.e. m0−bare mass, Z1−coupling renormalization, and Z2−wave function renormalization) are determined in the πN model and are used as input parameters in the pion photo- and electro-production calculation. Extensive discussion of these points can be found in [33, 34].

2This means that the dressed particle propagator is equal to the bare propagator. 3 The multiplicative renormalization is used since it easily maintains unitarity. Chapter 4 Pion Production from Free Nucleons

4.1 Coupled channel approach

Based on the unitarity dynamics of the πN scattering model presented in Chapter 3, it is possible to approach the electromagnetic induced reactions in a way which satisfies the unitary dynamics for the entire photo-pion channel space. This is achieved by setting up the following coupled-channel equations:     Tππ Tπγ∗ Vππ Vπγ∗       =   Tγ∗π Tγ∗γ∗ Vγ∗π Vγ∗γ∗       Vππ Vπγ∗ Gπ 0 Tππ Tπγ∗       +       ,(4.1) Vγ∗π Vγ∗γ∗ 0 Gγ∗ Tγ∗π Tγ∗γ∗ where T and V are the amplitudes and driving potentials of the πN scattering (ππ), pion electro-production (γ∗π), absorption (πγ∗), and the nucleon Compton effect (γ∗γ∗), respectively. The coupled equations simplify considerably if one only treats the electromagnetic coupling to first order:

Tππ = Vππ + VππGπTππ,

Tγ∗π = Vγ∗π + Vγ∗πGπTππ, (4.2)

Tπγ∗ = Vπγ∗ + TππGπVπγ∗ ,

Tγ∗γ∗ = Vγ∗γ∗ + Vγ∗πGπTπγ∗ .

Since the coupled equations are only solved up to ﬁrst order in the electromagnetic coupling e, two body unitarity is only preserved to this order. The great advantage of this approximation is that the scattering equation has to be solved iteratively only 47 for the πN scattering amplitude ( as discussed in Ch. 3) and then one can evaluate the electromagnetic amplitudes in a one loop calculation.

Our model for the pion production potential (i.e., the driving term Vπγ∗ in Eq. (4.1)), is given in Ch.2 and includes the following tree-level contributions: N direct and crossed terms, t- channel π, ω and ρ exchanges, the Kroll-Ruderman (contact) term, and the direct and crossed ∆ terms (see Fig. 2.1). The model for the

πN driving term (i.e. Vππ) has been described in detail in Ch.3. The main effect of including rescattering as in Eq. (4.1), is the dressing of the pole contributions in Vπγ∗ (as shown in [33]). As a result, both N- and ∆-pole contributions are constructed using the 00bare00 parameters extracted in the πN model of [35]. Furthermore, in view of the 2-body unitarity of πN scattering, the hadronic parameters (i.e. cut-offs, coupling constants, etc.) are fixed by the analysis of πN scattering.

4.2 Multipole decomposition

Since a vector particle, such as the (virtual) photon, carries an intrinsic spin of 1, the total spin J and parity are not suﬃcient to describe a γN helicity state. The

1 3 ∗ additional characteristic is introduced as the total helicity λ = λγ − λN = 0, ± 2 , ± 2 . The parity conserving amplitudes for pion electro-production represent the transition amplitude from the γ∗N partial-wave state (see e.g. [41, 42]) |J, +λi ∓ |J, −λi |J, λ; ±i = √ (4.3) 2 to the πN partial-wave state |J, +λ0i ∓ |J, −λ0i |J, λ0; ±i = √ , (4.4) 2

0 1 where the total helicity of ﬁnal state λ = ± 2 (since the pion is a spin−0 particle). J± 1 The parity of these states is P = (−1) 2 . Hence, the 6 independent two-particle 48

J± 0 J helicity state amplitudes of total spin J are (Tλ0λ = hJ, λ ; ±|T |J, λ; ±i):

J± ∗ J J T 1 3 (πN ← Nγ ) = T+ 1 + 3 ∓ T+ 1 − 3 2 2 2 2 2 2 J± ∗ J J T 1 1 (πN ← Nγ ) = T+ 1 + 1 ∓ T+ 1 − 1 (4.5) 2 2 2 2 2 2 J± ∗ J J T 1 (πN ← Nγ ) = T+ 1 +0 ∓ T+ 1 −0. 2 0 2 2

In the literature it is common to describe the photon ﬁeld in terms of the classical electrodynamics quantities of magnetic (M), electric (E), and scalar (longitudinal) (S) photon states. The ﬁrst two are transversely polarized with respect to the three- momentum q of the photon and involve combinations of the polarization vectors µ (q) = √∓1 (0, 1, ±i, 0), while the last one is longitudinally polarized (only exists for a ±1 2 µ 1 virtual photon) and hence is proportional to 0 (q) = √ (|~q|, 0, 0, ω). This multipole Q2 decomposition is also the one that is commonly used for the experimental partial-

∗ wave decomposition of γ N → πN. Denoting the total spin of the photon state by jγ L and the photon angular momentum by lγ (with jγ = lγ 1), then one can construct a direct relation between the two-particle helicity states |J, λ; ±i and the magnetic, electric and scalar photon nucleon states [43, 44]: 1 1 p 1 p 3 |J = jγ + ,M(E)i = ∓p jγ|J, ; ±i + jγ + 2|J, ; ±i 2 2(jγ + 1) 2 2 1 1 p 1 p 3 |J = jγ − ,M(E)i = ∓p jγ + 1|J, ; ∓i − jγ − 1|J, ; ∓i 2 2(jγ + 1) 2 2 1 |J = j ± ,Si = ±|J, 0; ∓i. (4.6) γ 2 Sandwiching the interaction matrix T between the multipole states (4.6) and the πN of deﬁnite parity helicity states (4.4) (see [45]), considering relations (4.5) and using the relation between lπ = l (the orbital momentum of the πN system) and jγ, the multipole amplitudes for the transition to a pion nucleon helicity state are: √ r ! 2 J+ l + 2 J+ Ml+ = T 1 1 + T 1 3 (4.7a) 4(l + 1) 2 2 l 2 2

√ r ! 2 J− l − 1 J− Ml− = −T 1 1 + T 1 3 (4.7b) 4l 2 2 l + 1 2 2 49 √ r ! 2 J+ l J+ El+ = T 1 1 − T 1 3 (4.7c) 4(l + 1) 2 2 l + 2 2 2

√ r ! 2 J− l − 1 J− El− = T 1 1 + T 1 3 (4.7d) 4l 2 2 l + 1 2 2

1 S = − T J+ (4.7e) l+ 1 0 2(l + 1) 2

1 S = − T J−. (4.7f) l− 1 0 2l 2 The amplitudes in Eq. (4.7) are calculated as follows:

• calculate the helicity amplitudes of the driving term Vπγ∗

ρ0ρ 0 ρ0 0 µ ρ Vλ0λ+σ(~q , ~q) = uλ0 (~q )(σ · Vµ) uλ(~q) (4.8)

where σ = ±1, 0 is the polarization of the photon; for the other indexes see Appendix A. The relative three momentum of the incoming and outgoing channel are ~q 0, ~q (in the equal time approximation the relative three momentum is equal to the CM three momentum of the particles).

• second, integrate out the θ dependence (i.e. calculate the corresponding partial wave)

Z +1 ρ0ρJ 0 J ρ0ρ 0 Vλ0λ+σ(|~q |, |~q|) = d(cos θ)dλ0λ+σ(θ)Vλ0λ+σ(~q , ~q). (4.9) −1

• third, calculate the parity conserving amplitudes for the helicity states of total spin J according to Eq. (4.5)

ρ0ρ,J± 0 ρ0ρJ 0 ρ0ρJ 0 Vλ0λ+σ (|~q |, |~q|) = Vλ0λ+σ(|~q |, |~q|) ∓ Vλ0,−λ−σ(|~q |, |~q|). (4.10) 50

• fourth, after storing the above amplitudes for asymptotic values of |~q| and a range of (half-shell) values |~q 00|, solve in one integration the dynamical scattering equation

Z ∞ ++,J± 0 ++,J± 0 X 00 002 T 1 (|~q |, |~q|) = V 1 (|~q |, |~q|) + dq q × 2 ,λ+σ 2 ,λ+σ ρ00λ00 0 +ρ00,J± 0 00 ρ00 00 ρ00+,J± 00 × T 1 00 (|~q |, |~q |)|πN→πN G (|~q |)Vλ00λ+σ (|~q |, |~q|) 2 λ (4.11)

where G±(|~q|) was deﬁned in Eq. (3.18). Notice that we consider only the particle states since for calculating the multipoles or other external observables one considers only the on-shell values of the scattering matrix.

00 Note that Eq. (4.11) has a singularity at the on-shell value for |~q | = |~qon−shell|. We will show in the Appendix E how to deal with this problem. This procedure allows to calculate the various multipoles which can be compared to those extracted from the experiment.

4.3 Ingredients of the model

4.3.1 Electromagnetic form factors: non-resonant terms

The procedure introduced in Section 2.1 permits us to use the experimentally determined form factors in the Born terms and we parameterize them so that they reproduce the experimental values as closely as possible. Since many of the form factors have a dipole form–at least asymptotically according to perturbative QCD– we introduce the function Q2 −2 F (Q2, Λ) = 1 + , (4.12) D Λ where Λ and Q2 are given in units of (GeV/c)2. For the proton electric form factor we used the data from [46] which we parameterized by

p 2 2 −Q2 2 GE(Q ) = (1 + Q e )FD(Q , 0.4), (4.13) 51 which is valid for Q2 up to 4 (GeV/c)2. For the neutron electric form factor we used the Galster parameterization [47, 48] (given in detail in Appendix B); for the magnetic form factors for both proton and neutron we have used the dipole parameterization

p/n 2 2 GM (Q ) = µp/nFD(Q , 0.71). (4.14)

(see Appendix A for the ΓγNN vertex isospin decomposition). The pion charge form factor is of the monopole form and has been determined to be:

Q2 −1 F (Q2) = 1 + , (4.15) π 0.45 while for the axial form factor we use:

2 2 FA(Q ) = FD(Q , 0.9). (4.16)

In the case of the vector meson exchanges (ρ/ω), we have used the theoretical prediction given in [49]

1 + Q2 F = , (4.17) ρ/ω 1 + 3.04Q2 + 2.42Q4 + 0.36Q6 where once again Q2 is given in (GeV/c)2. From our point of view, all of these form factors have been determined from previous experiments.

4.3.2 Electromagnetic form factors: resonant terms

In Ch. 2 we introduced the form of the Γγ∆N vertex function (see Eq. (2.39)) which we use in our computer program. The form factors in Eq. (2.39) are similar to the Dirac- Pauli form factors for the nucleon, whereas the Sachs form factors are more closely connected to the various multipoles contributing to the process. Jones and Scadron

[50] introduced two covariant decompositions of the Γγ∆N vertex function using these two diﬀerent types of form factors and then constructed equivalence relations between the form factors of these two representations. Firstly, they give the Dirac-Pauli form,

αµ 2 αµ 2 αµ 2 αµ Γγ∆N (p, q) = G1(Q )K1 + G2(Q )K2 + G3(Q )K3 , (4.18) 52

1 where P = 2 (p∆ + pN ) and

αµ α µ αµ 5 K1 = (q γ − q/g )γ ,

αµ α µ αµ 5 K2 = (q P − q · P g )γ ,

αµ α µ 2 αµ 5 K3 = (q q − q g )γ .

Then, they give the Sachs form,

αµ 2 αµ 2 αµ 2 αµ Γγ∆N (p, q) = GM (Q )KM + GE(Q )KE + GC (Q )KC , (4.19) where,

αµβν αµ −3ε Pβqν m∆ + mN KM = 2 2 · (m∆ + mN ) − q 2mN αµβν αµ 3ε Pβqν m∆ + mN KE = 2 2 · (m∆ + mN ) − q 2mN ασβν µσρτ 5 6ε Pβqνε p∆,ρqτ γ m∆ + mN − 2 2 2 2 · [(m∆ + mN ) − q ][(m∆ − mN ) − q ] mN α 2 µ µ 5 αµ −3q (q P − q · P q )γ m∆ + mN KC = 2 2 2 2 · . [(m∆ + mN ) − q ][(m∆ − mN ) − q ] mN

2 2 At the photon point Q = −q = 0, only KM and KE give contributions (since µ µ ε± · qµ = 0 and ε0 = (1, 0, 0, 1), i.e. real photons are purely transverse), therefore one gets:

αµ αµ αµ Γγ∆N (p, q) = GM (0)KM + GE(0)KE , (4.20) with:

αµβν −3ε p∆,βqν KM = 2mN (m∆ + mN )

αµβν ασβν µσρτ 5 3ε p∆,βqν 6ε p∆,βqνε p∆,ρqτ γ KE = − 2 , 2mN (m∆ + mN ) mN (m∆ + mN )(m∆ − mN ) Here we have used the fact that: 1 εαµβνP q = εαµβν p − q q = εαµβνp q . β ν ∆,β 2 β ν ∆,β ν The result in Eq. (4.20) is identical with Vanderhaeghen’s choice [51]. 53

We follow the same procedure as in [50] and use the 00∆ pole equivalence00 as in αµ [38] (i.e., the vertex ΓγN∆ is contracted with the free Rarita-Schwinger vector spinor 2 2 ψµ(p∆), where p∆ = m∆ is the on-shell 4-momentum) to ﬁnd the equivalence between the Gi (i = 1, 2, 3) in Eq. (2.39) and Gα in Eq. (4.19)(α = M,E,C), where M, E, and C refer to magnetic, electric and Coulomb form factors of the Sachs type. The 3/2 3/2 3/2 N −∆ contributions to the various multipoles such as M1+ , E1+ , and S1+ are almost 2 2 2 directly proportional to GM (Q ), GE(Q ), and GC (Q ) respectively. Using this procedure with Eq. (2.39) and Eq. (4.18), it is easy to get the following relations: 1 (p · qgαµ − qαpµ )γ5 = − Kαµ + Kαµ (4.21) ∆ ∆ 2 2 3

2 µ α αµ µ α α 5 αµ α β βµ [q (p∆γ − p/g ) + q (p/∆q − p∆ · qγ )]γ = p/∆K3 − γ p∆ ·K3 . (4.22)

Using the relations from Scadron we obtain:

2 αµ αµ 2 αµ αµ [(2q · p∆ − q )(KM + KE ) + 2(2m∆ − q · p∆)KC ] K2 = 2 2 2 (4.23) 4Λ[m∆q − (q · p∆) ]

2 αµ αµ αµ αµ [q (KM + KE ) + 2q · p∆KC ] K3 = 2 2 2 , (4.24) 2Λ[m∆q − (q · p∆) ] where

−1 mN + m∆ 2 2 2 2 Λ = −3∆ , ∆ = [(m∆ + mN ) − q ][(m∆ − mN ) − q ], (4.25) mN and we have used on-shell Dirac algebra for the Delta. Using Eq. (4.21)-Eq. (4.24) in Eq. (2.39) we get: αµ αµ αµ G2 p/∆G3 αµ G3 α β βµ ΓγN∆ = G1KM + Ξ G2K2 + − K3 + γ p∆ ·K3 (4.26) 2 mN mN where

3(m∆ + mN ) Ξ = 2 2 . (4.27) 2mN [(m∆ + mN ) − q ]

3 This relation, when using free spin- 2 ﬁeld equations,

α ψα(p∆) · γ = 0 and p/∆ψα(p∆) = m∆ψα(p∆), (4.28) 54 becomes, αµ αµ αµ G2 p/∆G3 αµ G3 α β βµ ψαΓγN∆ = G1KM + Ξ G2K2 + − K3 + γ p∆ ·K3 2 mN mN αµ αµ αµ = ψα(GM KM + GEKE + GC KC ). (4.29)

In Eq. (4.29) we imposed the 00pole equivalence00 between the standard vertex (Eq. (4.19)) and our gauge invariant vertex. From Eq. (4.29), after tedious but simple algebra, we can ﬁnd a direct relationship among the Sachs decomposition form factors (GM , GE and GC ) and the gauge invariant vertex form factors (G1, G2 and G3). Namely,

2 2 2 G1(Q ) = GM (Q ) − GE(Q ) (4.30)

2 2 2 2 2 2 2 2[−Q − mN + m∆]GE(Q ) + 2Q GC (Q ) G2(Q ) = 2 2 (4.31) Q + (m∆ − mN )

2 2 2 2 2 2 2 mN [4m∆GE(Q ) − (−Q − mN + m∆)GC (Q )] G3(Q ) = 2 2 . (4.32) m∆(Q + (m∆ − mN ) ) 2 In Eq. (4.30)-(4.32) we have used the following kinematic relation: 2q · p∆ = m∆ − 2 2 2 2 mN + q . It is easy to see that at Q = −q = 0 we get:

G1(0) = GM (0) − GE(0) (4.33)

2(m∆ + mN ) G2(0) = GE(0) (4.34) m∆ − mN

4m∆mN mN (m∆ + mN ) G3(0) = 2 GE(0) − GC (0). (4.35) (m∆ − mN ) m∆(m∆ − mN )

The N ↔ ∆ transition form factors include a strength gα (α = M,E,C) with gα = 2 2 Gα(Q = 0) and Q dependence. We ﬁnd it convenient to represent these form factors by the parameterization:

2 2 2 Q − Q 2 Gα(Q ) = gα 1 + e bα FD(Q , cα), (4.36) aα where the parameters will be determined by ﬁtting the experimental data. The parameters α = a, b, c will be given in units of (GeV/c)2 and all of these form factors are bare in the sense that the πN rescattering dresses them. 55 4.3.3 Current conservation

From Eq. (4.2), regardless of the vertex modifications described in Ch. 2, one can see that the Born terms in Vπγ∗ preserve the current only if the outgoing particles (i.e., π and N) are on-shell. This is no longer the case if the final state interaction is included dynamically as in Eq. (4.1) since it requires that the π and N following the photon vertex be off-shell. Current conservation can be restored for this situation if the photon is minimally substituted in the full (dressed) vertex (see Eq. (3.28)) or more plainly said, if the photon is coupled to all of the internal and external lines of the dressed vertex ΓπNN . The calculation of the resulting amplitude would be a very complicated task, which is beyond the coupled scattering equation implemented in our work. There is a simple, and often used prescription, to eliminate this problem. One modifies the current by writing:

q · J J 0 = J − n , (4.37) µ µ n · q µ where nµ is an arbitrary 4−vector. From Eq. (4.37) one can see that the new current 0 (Jµ) satisﬁes current conservation as follows: q · J q · J 0 = q · J − q · n = q · J − q · J = 0. q · n

We choose nµ = qµ and if we use the standard choice of the photon 4−momentum q = (ω, 0, 0, ~q), i.e. the 3−momentum of the photon is deﬁned to be along z−axis in the CM frame, then we have: ωJ − |~q|J ω J 0 = J ,J 0 = J − 0 3 ω = (|~q|J − ωJ ), 1 1 0 0 ω2 − |~q|2 Q2 0 3 ωJ − |~q|J |~q| J 0 = J ,J 0 = J − 0 3 |~q| = (|~q|J − ωJ ). (4.38) 2 2 3 3 ω2 − |~q|2 Q2 0 3

0 0 From Eq. (4.38) one sees that J3 = (|~q|/ω)J0 as required by current conservation. We must point out that 00global ﬁxing00 of the current in Eq. (4.37) is necessary to

fix the whole off-shell potential matrix Vπγ∗ since the method of fixing introduced in Ch. 2 does not restore current conservation to the off-shell elements of the matrix. 56

Thus, even if it looks redundant at 00ﬁrst glance00, it is necessary to 00adjust00 the electromagnetic vertices, since the driving terms in Eq. (4.1) must be on-shell when deﬁning the multipoles and consequently the observables. Instead of performing a

00 00 global ﬁxing at the end of the calculations for the resulting Tπγ∗ matrix, we do this for each of the elements of the scattering matrix.

4.4 Problem of singularities in Vπγ∗

The virtuality of the photon poses a serious problem in the case when one calculates

2 the oﬀ-shell elements of the matrix Vπγ∗ (i.e. Q 6= 0). Namely, singularities are encountered in the integration path when the one-loop integration is calculated in Eq. (4.1). In the following we will show how this happens, and give a prescription for avoiding this problem for certain ranges of values for Q2 and W . Fortunately the problem only arises in the t- and u-channel terms. Let’s start with the structure of the denominator of the propagator, encountered in the case of the u−channel exchange (see Eq. (2.14)), which in the oﬀ-shell case, is:

p/ − k/ + mN SN (p − k) = 2 00 2 (4.39) u(Q , W, |~q |, cos(θ)) − mN where 2 2 00 2 2 ~ 2 u(Q , W, ~q , cos θ)) − mN = (EN − k0) − ~p − k − mN √ q 2 002 ~ 00 2 2 = EN − β s + q0 − (~p + βP − ~q ) + mN , (4.40) √ where mN , mπ are the masses of the nucleon and pion, respectively, and W = s = µ Pµ · P is the total CM energy of the system. In Eq. (4.40) I have made use of the µ fact that the incoming nucleon has 4-momentum deﬁned by p = (EN , ~p) (it is fully µ ~ on-shell), and the outgoing π has 4-momentum k = (k0, k), which written in terms √ 00 of the relative 4-momentum of the outgoing channel has the energy k0 = β s − q0 and the 3-momentum ~k = βP~ − ~q 00 (recall the deﬁnitions of the scalars α and β in Eq. (3.11)). I have also assumed that all the kinematics are in the CM frame (i.e., √ P µ = (P 0, 0) = ( s, 0)). In this frame, and from kinematics considerations (see 57 further details in Appendix A), the asymptotic energies of the incomming nucleon and outgoing pion are: W 2 − m2 + m2 ω = N π , (4.41) π 2W

W 2 + m2 + Q2 E = N , (4.42) N 2W

p 2 2 and the on-shell 3-momentum of the incoming nucleon is |~p| = EN − mN . The whole problem of poles arises when the denominator of the propagator described by the function

q q 00 00 2 2 00 00 2 2 f = EN − ωπ + q0 − (~p + ~q ) + mN EN − ωπ + q0 + (~p + ~q ) + mN , (4.43)

00 goes through zeroes. Note that in the equal time approximation q0 = 0. The relative asymptotic energy of the nucleon-pion system is given by: 2m2 − m2 + Q2 E − ω = N π , (4.44) N π 2W

00 while the square root term ranges from mN to ∞ as the integration variable |~q | ranges over all possible values. Since the minimum value of W is mN + mπ, there the function f has no zeroes for Q2 = 0. (Note that it is the ﬁrst term in f which could possibly have a zero, but for Q2 = 0 it is always negative.) However, as Q2 increases

2 2 and EN − ωπ becomes larger (by the addition of Q /2W as compared to the Q = 0 case), the ﬁrst term in f can vanish. Furthermore, it vanishes at the smallest value of Q2/2W when ~p and ~q 00 are antiparallel and equal in magnitude. Clearly these singularities, which arise from the negative mass squared of the

00 2 virtual photon, can be avoided by choosing the relative energy q0 = −Q /2W . This choice has the advantage that the propagators in the u-channel are not modiﬁed at the photon point. Similar considerations also apply for the case of the u−channel ∆ exchange, t−channel π, ρ, ω exchanges, as well as for the case of the hadronic form factors, 58 such as the pion (monopole), and the rho and omega (one boson) form factors (see

00 Appendix B). The same choice for q0 works for the u-channel ∆ exchange while for 00 2 the t-channel terms and hadronic form factors, one must choose q0 = +Q /2W . Notice that the direct contributions (s−channel) do not get affected, hence there is no need for this adjustment. Obviously, one should ask the question of how much does this approximation affect the Q2 dependence of the calculated observables. The non- pole off-shell contributions usually have small contributions to Tπγ∗ and specifically small contributions to the major multipoles such as M1+, E1+, S1+, and E0+. The first three are almost completely determined by the direct ∆-exchange, while the latter is dependent of the type of πNN coupling, hence depending mostly on the direct nucleon exchange. The main objection to this ad-hoc approximation is that it violates current conservation even at the on-shell values of the potential matrix

(Vπγ∗ ). Nevertheless, this problem is ﬁxed by the global restoration of the current, described in the previous section. However, since one of the main goals of this work is to construct a gauge invariant current, at least for the on-shell tree level, this choice was considered unacceptable. Note that the calculation of the observables with this prescription ﬁts the available data quite well. In view of these considerations, we consider an alternative approach to avoid these poles [52], namely what is known as the spectator approximation. This approximation assumes the following:

(1) In the u−channel propagator set the outgoing pion on its mass shell, which gives for the outgoing nucleon the following energy:

p 002 2 EN 0 = W − ~q + mπ. (4.45)

Under the spectator approximation the ﬁrst term in Eq. (4.43) becomes equal

p 002 2 p 00 2 2 to W − 2 ~q + mπ − (~p + ~q ) + mN , which remains negative for all values of |~q 00|.

(2) In the t−channel propagator set the outgoing nucleon on its mass shell, which 59

gives for the outgoing pion the following energy:

q 002 2 ωπ = W − ~q + mN , (4.46)

which also avoids the singularities in the t-channel terms.

An advantage of the spectator approximation is that the on-shell potential matrix

(Vπγ∗ ) does not violate current conservation, since this approximation, at the tree level, corresponds to the asymptotic kinematics. Note that as in our alternative approach, the values of the energies introduced in Eq. (4.45) and Eq. (4.46) are introduced only in the denominator of the corresponding propagators as well as in the t− channel one boson exchange form factors. The remaining structure in the current does use the old values for the energies (see Appendix A). The main disadvantage of the spectator approximation is that it modifies the off shell behavior even at the photon point, where no anomalous singularities arise in the u-channel and t-channel terms. In order to describe the Q2 = 0 data in the spectator approximation, it is necessary to refit some of the electromagnetic couplings. We have done this and all the results presented in Ch. 6 have been calculated using the spectator approximation. Chapter 5 Alternative Approach for Solving the Scattering Equation

We consider two-body integral equations and show how they can be dimensionally reduced by integrating exactly over the azimuthal angle of the intermediate momentum. Numerical solution of the resulting equation is feasible without employing a partial- wave expansion. We illustrate this procedure for the Bethe-Salpeter equation for pion- nucleon scattering and give explicit details for the one-nucleon-exchange term in the potential. Finally, we show how this method can be applied to pion photoproduction from the nucleon with πN rescattering being treated so as to maintain unitarity to ﬁrst order in the electromagnetic coupling. The procedure for removing the azimuthal angle dependence becomes increasingly complex as the spin of the particles involved increases. This chapter is based on the results which appeared in [53].

5.1 Introduction

In cases when solving the Lippmann-Schwinger or Bethe-Salpeter type of equation is numerically involved, one often resorts to a partial-wave decomposition (PWD) in the center-of-mass (CM) frame. In doing so one can exploit the spherical symmetry of the interaction and perform the integration over the two-dimensional solid angle of the intermediate momentum analytically. While this reduces the equation’s dimension by two, one has to deal with summing the partial-wave series, and hence this procedure is beneﬁcial when only a few partial waves dominate. In the case when many partial waves must be taken into account, when restriction to the CM frame is not desirable, or when the potential is not spherically-symmetric, the partial-wave expansion is not helpful and one has to face the complexity of three- or four-dimensional integral equations. 61

Fortunately, as had been noted by Glöckle and collaborators [54, 55] in the context of the nucleon-nucleon (NN) interaction, the dependence on the intermediate momentum azimuthal angle factorizes, and the integration can still be performed analytically without employing any kind of expansion or truncation. While this procedure has been successfully applied a number of times to the NN situation [55, 56, 57], here we would like to examine general conditions which potentials must satisfy to factorize the azimuthal integration. We then apply it to solve a specific example of relativistic potential scattering in the pion-nucleon (πN) system and compare with the usual method of using the partial-wave expansion. In Section 5.2 we give the general requirements on the potential that allow one to remove the azimuthal angle dependence in the integral equation. In Section 5.3 we focus on the Bethe-Salpeter equation for πN scattering with the one-nucleon-exchange potential and show in detail how the azimuthal-angle dependence can be integrated out in this case. Furthermore, in Section 5.4, we solve the πN scattering equation using a quasipotential approximation and compare the solution to the one obtained using the partial-wave expansion. In Section 5.5 we examine an extension of this approach to the calculation of pion electro-production from the nucleon including the πN final state interaction. Our conclusions regarding this approach are summarized in Section 5.6.

5.2 Exact integration over the azimuthal angle

The starting point in calculating observables of a two-body scattering process is an equation for the scattering amplitude. We shall assume relativistic scattering, in which case the equation is a 4-dimensional integral equation of the Bethe- Salpeter type (Eq. (3.10)). Moreover, throughout this chapter, q, q00, q0 stand for the relative 4-momenta of the incoming/intermediate/outgoing channel, while P = p + k = p0 + k0 = p00 + k00 is the total 4-momentum with k, k00, k0 and p, p00, p0 the incoming/intermediate/outgoing momenta of particle one and particle two, respectively. 62

In order to investigate the conditions under which Eq. (3.10) can be integrated over the intermediate azimuthal angle, we work in the helicity basis and only display the dependence on the azimuthal angle and helicity: 2π Z 00 0 0 X dϕ 0 00 00 00 Tλ0λ(ϕ , ϕ) = Vλ0λ(ϕ , ϕ) + Vλ0λ00 (ϕ , ϕ ) G(ϕ ) Tλ00λ(ϕ , ϕ) . (5.1) 00 2π λ 0 An important point here is that the two-particle propagator G can always be made independent of the intermediate angle ϕ00 by choosing the total three-momentum along the z-axis, i.e., choosing the co-linear frame: P = (P0, 0, 0,P3). Furthermore, we shall observe that in the case when only spin-0 and spin-1/2 particles are involved, the azimuthal-angle dependence of the fully oﬀ-shell potential 1 in the co-linear frame is given as follows:

0 −iλ0ϕ0 0 iλϕ Vλ0λ(ϕ , ϕ) = e vλ0λ(ϕ − ϕ) e , (5.2) where λ and λ0 stand for the combined helicities of the initial and ﬁnal state, respectively. The half-oﬀ-shell potential then takes a very simple form:

0 0 0 −i(λ −λ)ϕ Vλ0λ(ϕ , ϕ)|half−oﬀ−shell = e vλ0λ(0) , (5.3) half−oﬀ−shell where λ is the helicity of the on-shell state. It is in this case, when conditions (5.2) and (5.3) are met, that the exact integration over the azimuthal-angle can readily be done. First, by using Eq. (5.2) in Eq. (5.1), we see that the azimuthal dependence of the t-matrix is given by:

0 −iλ0ϕ0 0 iλϕ Tλ0λ(ϕ , ϕ) = e tλ0λ(ϕ − ϕ) e . (5.4)

Since v and t only depend on the diﬀerence ϕ0 − ϕ, we can expand them in a simple Fourier series:

X (m) imφ X (m) imφ vλ0λ(φ) = vλ0λ e , tλ0λ(φ) = tλ0λ e . (5.5) m m 1In general, we deal with the fully off-shell situation, that is when both the initial and final states are off the mass (or energy, in the non-relativistic case) shell. The situation when either the initial or the final state is on-shell is referred to as the half-off-shell case, and it is well known that one only needs the half-off-shell result to solve the integral equation. 63

It is straightforward to show that their Fourier transforms, 2π 2π Z Z (m) dφ −imφ (m) dφ −imφ v 0 = v 0 (φ) e , t 0 = t 0 (φ) e , (5.6) λ λ 2π λ λ λ λ 2π λ λ 0 0 satisfy the following equation which does not involve the ϕ-integration:

(m) (m) X (m) (m) tλ0λ = vλ0λ + vλ0λ00 G tλ00λ . (5.7) λ00 In principle, m runs to infinity and so we have an infinite number of equations to solve even though they are not coupled. Fortunately, since only the half-off- shell potential is needed to solve the equations and it obeys condition (5.3), the corresponding Fourier transform is non-vanishing only for m = −λ:

(m) vλ0λ = δ−λm vλ0λ(0)|half−oﬀ−shell . (5.8) half−oﬀ−shell The scalar system is the simplest one where this procedure can be demonstrated. In that case the potential is a scalar function of scalar products of the relevant 4- momenta:

V (q0, q; P ) = V (q · q0,P · q, P · q0, q2, q02,P 2). (5.9)

0 Given q = (q0, |q| sin θ cos ϕ, |q| sin θ sin ϕ, |q| cos θ) and similarly for q , we easily convince ourselves that, in the co-linear frame, the azimuthal dependence enters only through the product:

0 0 0 0 0 0 q · q = q0q0 − |q| |q | [cos θ cos θ + sin θ sin θ cos(ϕ − ϕ)] , (5.10) and hence it is of the necessary form given in Eq. (5.2). Furthermore, in the half- off-shell case, the momentum of the on-shell state, say q, can always be chosen along the z-axis, i.e., such that θ = 0. Hence the half-off-shell potential is independent of azimuthal angles, which fulfills condition (5.3) for the spinless case. The two-particle propagator G(q; P ) = G(P · q, q2,P 2) is of course independent of ϕ in the co-linear frame. Once we have found that conditions (5.2) and (5.3) are satisfied, while G is independent of ϕ, the integration over ϕ can be done immediately. We will now show this more explicitly for the more complicated case of a scalar-spinor system. 64 5.3 Spin complications: the πN system

The most general Lorentz structure of the fully oﬀ-shell potential in the helicity basis can be written in the form2:

ρ0ρ 0 ρ0 0 h ρ0ρ ρ0ρ 0 ρ0ρ ρ0ρ 0 i ρ Vλ0λ00 (q , q; P ) =u ¯λ0 (αP + q ) A1 + A2 γ + (A3 + A4 γ ) γ · P uλ(αP + q), (5.11)

where Ai are scalar functions of the dot-products of the relevant momenta, i.e.,

0 0 2 02 2 Ai = Ai(q · q ,P · q, P · q , q , q ,P ). (5.12)

Considering the dependence of these functions on the azimuthal angles of q and q0, we see that — in the co-linear frame — it is given by the diﬀerence ϕ0 − ϕ, for the reason described below Eq. (5.9). The rest of the ϕ-dependence resides in the nucleon spinors. According to Eq. (5.11), in the co-linear frame we need to consider

† 0 0 † 0 0 only χλ0 (Θ , φ )χλ(Θ, φ) and χλ0 (Θ , φ )σ3χλ(Θ, φ) where χ’s are the Pauli spinors (cf. Appendix A), and Θ, ϕ and Θ0, ϕ0 deﬁne the orientation of αP + q and αP + q0, respectively. Since,

† 0 0 −iλ0ϕ0 hX 1/2 0 1/2 iλ00(ϕ0−ϕ)i iλϕ χ 0 (Θ , ϕ ) χλ(Θ, ϕ) = e d 0 00 (Θ ) d 00 (Θ)e e , (5.13) λ λ00 λ λ λλ † 0 0 −iλ0ϕ0 hX 1/2−λ00 1/2 0 1/2 iλ00(ϕ0−ϕ)i iλϕ χ 0 (Θ , ϕ ) σ3 χλ(Θ, ϕ) = e (−1) d 0 00 (Θ ) d 00 (Θ)e e , λ λ00 λ λ λλ (5.14) we observe that the ϕ-dependence of these elements is of the desired form in Eq. (5.2). For the half-oﬀ-shell situation, where we can choose θ = 0 (hence Θ = 0 in the co- 1/2 linear frame) and use dλλ00 (0) = δλλ00 , we ﬁnd the form,

† 0 0 −i(λ0−λ)ϕ0 1/2 0 χλ0 (Θ , ϕ ) χλ(0, ϕ) = e dλ0λ(Θ ), (5.15)

† 0 0 −i(λ0−λ)ϕ0 1/2−λ 1/2 0 χλ0 (Θ , ϕ ) σ3 χλ(0, ϕ) = e (−1) dλ0λ(Θ ), (5.16) 2To bring a general expression to this form we use properties of the Dirac spinors, such as:

ρ 0 ρ (γ · q − mN ) uλ(q) = (q0 − ρEq) γ uλ(q). 65

k p'

p k'

Figure 5.1: One-nucleon-exchange πN potential. which obeys the necessary half-oﬀ-shell condition Eq. (5.3). Therefore, we have demonstrated that the azimuthal-angle dependence of a pion- nucleon potential in the co-linear frame always satisﬁes conditions (5.2) and (5.3). It is also apparent from Eq. (3.16) that the two-particle Green’s function does not have any azimuthal dependence in that frame. Thus the integration over ϕ can exactly be done in the Bethe-Salpeter equation for the πN system by means of the procedure of Sec. 5.2. Similar arguments apply in the case when both particles have spin 1/2, e.g., nucleon-nucleon (NN) scattering. It should also be noted that in this case the

0 0 0 potential satisﬁes conditions (5.2) and (5.3) with λ = λ1 − λ2, λ = λ1 − λ2. In other words, the helicities of the two particles must be combined.

5.4 Numerical results

The standard route to solution of a potential scattering equation such as Eq. (3.13) is to decompose it into an infinite set of equations for partial-wave amplitudes, see e.g. [41, 42]. The advantage of doing a partial wave decomposition is that the equation for each partial wave is of two lesser dimensions than the original equation, while the partial-wave series is usually rapidly converging, hence only the first few partial-wave amplitudes need to be determined for. On the other hand, solving for the full amplitude directly has its own important benefits. If the exact azimuthal-angle integration can be done a priori, the numerical feasibility of this approach becomes comparable to the PWD method. 66

In this section we would like to compare the two methods for the example of solving a relativistic equation for the πN system. For our toy-calculation potential we take the one-nucleon exchange, Fig. 5.1, and use the instantaneous approximation, thus neglecting retardation eﬀects in the potential. The latter approximation allows us to perform the relative-energy (q0) integration such that we are left with a relativistic 3-dimensional Salpeter equation as shown in Section 3.3. The 3-dimensional equation for πN (Eq. (3.17)) has been described in detail and solved using a PWD in the CM system by Pascalutsa and Tjon [33, 34, 58]. We, on the other hand, solve this equation by using the framework of the two previous sections to reduce the ϕ-integration analytically and to solve numerically the resulting 2-dimensional integral equation for the m-th Fourier component of the full amplitude: Z ∞ 00 Z π (m)ρ0ρ 0 0 (m)ρ0ρ 0 0 X d|q | 00 2 00 t 0 (|q |, θ , |q|, θ) = v 0 (|q |, θ , |q|, θ) + |q | dθ λ λ λ λ, 2π λ00ρ00 0 0 (m)ρ00ρ0 0 0 00 00 (ρ00) 00 (m)ρ00ρ 00 00 × vλ00λ0 (|q |, θ , |q |, θ ) GET (|q |) tλ00λ (|q |, θ , |q|, θ), (5.17) where, without loss of generality, we have also assumed the CM frame. The explicit form of the Fourier transform of the one-nucleon-exchange potential is worked out in Appendix C. Let us emphasize that it is necessary to solve for only one of the Fourier components (either m = −1/2 or m = 1/2), the other ones either vanish or can be obtained by symmetry relations due to parity and time-reversal invariance. We solve Eq. (5.17) by the Pad´eapproximants as in Refs. [33, 58], thus maintaining exact elastic unitarity. The numerical integrations are performed by the Gauss- Legendre method. The integral over |q00| in Eq. (5.17) contains the cut singularity

00 p 2 2 at |q | = [s − (mN − mπ) ][s − (mN + mπ) ]/4s ≡ qˆ, which is handled by the well- known identity: Z ∞ f(|q|) Z ∞ f(|q|) d|q| = P d|q| − iπf(ˆq), (5.18) 0 |q| − qˆ + i 0 |q| − qˆ where P denotes the principal-value integral. When computing the latter, the integration region is divided into two intervals: |q| ∈ [0, 2ˆq], and |q| ∈ (2ˆq, ∞). 67

The Gaussian points are then distributed separately for each interval to make use of the property that an even number of Gaussian points fall symmetrically with respect to the middle of the interval, hence the singularity in the middle of the first interval is avoided. The polar angle integration is straightforward for both the principal value term and the imaginary contribution. We find it sufficient to use 16 Gauss points for the momentum integration and 8 points for the polar-angle integration. Upon increasing the number of points to 32 and 16 respectively, the results change by less than 0.5% in the considered energy range. In all cases we found that 6 iterations combined with the use of Padéapproximants works extremely well. After we solve Eq. (5.17) to find the full πN T -matrix, we can of course also find the partial wave amplitudes: Z π Jρ0ρ 0 ρ0ρ 0 J Tλ0λ (|q |, |q|) = dθ Tλ0λ(|q |, |q|, θ) dλ0λ(θ), (5.19) 0 where θ is the angle between q and q0. We then investigate the convergence of the partial wave series:

ρ0ρ 0 X 1 Jρ0ρ 0 J Tλ0λ(|q |, |q|, θ) = J + 2 Tλ0λ (|q |, |q|) dλ0λ(θ) . (5.20) J

ρ0ρ 2 In particular, in Fig. 5.2 and Fig. 5.3 we plot the on-shell values of |Tλ0λ| compared with the truncation of the partial-wave series for 3 terms and 5 terms (i.e., J =

1 5 1 9 2 ,..., 2 and J = 2 ,..., 2 respectively). In order to compare the computational efficiency of the two methods, we compare the number of partial waves needed to achieve convergence in the PWD method with the number of Gauss points for the polar-angle integration which appear in the “w/o PWD” method. The figures show that the effect of truncations of the partial-wave series increases with angle (Fig. 5.2) and the energy of the incoming π (Fig. 5.3). In our particular case of one-nucleon exchange, computing 5 or more partial wave amplitudes is sufficient to reproduce the full result to a 1% accuracy in a broad energy domain. Thus, in this case, the efficiency of the two methods is comparable, since we need 5 multipoles versus 8 Gauss points in the polar-angle integration. 68

1.0 full no. of terms=3 no. of terms=5 0.8 ] 2 -

N 0.6 m [

2 | 2 / 1

2 0.4 / 1 T |

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 c.m. TS [S]

++ 2 LAB Figure 5.2: Angular dependence for |T 1 1 | at Eπ = 300 MeV. The solid line is the full 2 2 calculation, and the dashed and dotted lines are the resumming of partial terms.

It is important to emphasize that the ability to do the azimuthal-angle integration analytically is necessary to achieve comparable eﬃciency. We have checked that it usually takes at least 16 Gaussian points for the azimuthal integration, which slows down the calculation by more than an order of magnitude.

5.5 Extension to pion photoproduction

Our procedure for performing the analytic ϕ-integration is applicable in the photo- or electro-meson production to first order in the electromagnetic coupling. Here we describe the extension to the case of π photoproduction within a simple final-state- interaction model [33, 59]. In solving the photoproduction scattering equation we calculate first Vππ as described for πN scattering. We then iterate in the following manner:

Tπγ = Vπγ + VππGπVπγ + VππGπVππGπVπγ + ..., (5.21) 69

5 full no. of terms=3 4 no. of terms=5 ] 2 - N m

[ 3

2 | 2 / 1

2 / 1 -

T 2 |

0 0 200 400 600 800 1000 LAB ES [MeV]

++ 2 CM Figure 5.3: Energy dependence for |T 1 1 | at θπ = π. The lines are deﬁned the same as − 2 2 in Fig. 5.2.

where we have used Tπγ = Tγπ from time-reversal invariance. This solution procedure is obviously suitable for our case since the half-oﬀ-shell

Vπγ has a simple azimuthal angle dependence similar to the case of Vππ (see Eq. (C.7)). The reduced kernel (see Eq. (C.11)) has two terms rather than the one term in the πN case due to the 00complication00 of having to couple a spin 1 photon to spin 1/2 as opposed to coupling a spin 0 meson to spin 1/2. For example, if one considers the nucleon u-channel exchange (compare to the πN case in Eq. (C.2)) the half-oﬀ-shell photoproduction potential can be written as:

ρ0ρ 0 ρ0ρ 0 0 0 −i(λ0−λ−σ)φ0 ρ0ρ 0 0 0 −i(λ0+λ)φ0 Vλ0λσ(q , q) = 1Vλ0λσ(q0, |q , q0, |q|, θ )e +2Vλ0λσ(q0, |q , q0, |q|, θ ) e (5.22) where σ = ±1 represents the helicity of the incoming photon. One sees that when Eq. (5.22) is iterated in Eq. (5.21), two de-coupled scattering equations are obtained (each corresponding to 1V or 2V ). For each of these equations, one can show that the corresponding ϕ0 dependence re-appears after doing the ϕ00 70 integration, and therefore once again we can perform the azimuthal-angle integration analytically. As in the πN case, the resulting 00reduced00 kernels obey 2-D integral equations. As a check of our procedures, we calculated the u-channel contribution to pion photoproduction using the analytic azimuthal-angle integration along with 2- D numerical integration and compared to the results of Refs. [33, 59] obtained using the multipole expansion. At Eγ of 300 MeV with ﬁve multipoles we found agreement to better than 1% over a wide angular range.

5.6 Conclusion of Ch. 5

In recent years Glöckle and collaborators [54, 55] introduced a method which greatly simplifies the numerical integration of two-body scattering equations without performing the partial-wave expansion. The method exploits a certain azimuthal symmetry of the potential thus allowing exact integration of the azimuthal dependence. In this chapter we have established the general form of the azimuthal-dependence of the kernel, which allows for this procedure to go through. We have argued that these conditions are generally applicable to any system of spin 0 and/or spin 1/2 particles. We have applied this method to the case of π + N → π + N and γ + N → π + N scattering. With some extra effort it can be applied to higher-spin systems, however the procedure becomes increasingly complex with the increase of the spin of the involved particles. We have successfully applied the method to pion photo- and electro-production from the nucleon, however only to leading order in the electromagnetic coupling. Even though we have used the Salpeter equation for numerical exercises, the method can of course be applied to the full 4-D Bethe-Salpeter equation, which for the πN system has so far been solved in partial waves only [60, 61]. Performing the azimuthal-angle integration analytically greatly facilitates finding the full solution and makes the numerical feasibility of this approach comparable to finding the solution using the partial-wave expansion. Chapter 6 Results and Discussion

In recent years, high precision data, including polarization observables, have been obtained in the ∆ region for pion photo-production at LEGS [62] and Mainz [63] and for electro-production at JLab [64, 65], and MIT-Bates [66]. These data allow us to investigate more precisely the electromagnetic excitation of the ∆ resonance, as well as the extraction of the of the functional form of the electromagnetic form factors of the ∆.

6.1 Photo-Production (Q2 = 0) Results

In Ref. [35] a dynamical covariant description of πN scattering was developed and later extended to include real photons in [33]. A very good description of the pion

00 00 photo-production multipoles, as well as the REM ratio was obtained. In Fig. 6.1- Fig. 6.4 we show the multipole calculations for l ≤ 2. The red lines represent the imaginary part of the multipole amplitude while the black line is the real part.

LAB Though the plots are shown up to Tγ ≈ 650 MeV , we must point out that our model, due to the 00limitations00 of including just the first resonance ∆, is expected to be 00valid00 only up to photon energies of ≈ 450 MeV . The model results are compared with the analysis of the VPI group [67]. The electromagnetic coupling parameters used in the calculations are given in Table 6.1. The hadronic parameters are fixed in the πN (re)scattering process through the unitarity construction of the model. The Q2 = 0 results were used to determine four free parameters of the model (i.e. gωNN , GM (0), GE(0) and the extrapolated value of GC (0)). Hence, certain multipoles are suited for fixing coupling constants of the contributions which are sensitive to the corresponding multipoles. For example, non-resonant multipoles (1/2) (1/2) such as p,nM1+ and p,nE1+ , are independent of the gπNN coupling, but especially 72

(3/2) (3/2) sensitive to the ω exchange contribution. The resonant multipoles M1+ , E1+ and (3/2) S1+ are extremely sensitive to the corresponding ∆ contribution, GM (0), GE(0) and

GC (0), respectively. For example GM (0) was adjusted to reproduce the maximum (3/2) value of Im{M1+ } ≈ 38 located at the ∆ resonance peak (W ≈ 1.232 GeV ). After

fixing GM (0) and gωNN , we fitted the value of GE(0) such that the experimentally 2 extracted value of the quadrupole admixture REM (Q = 0) ≈ −2.7% was reproduced. Other multipoles are relatively insensitive to the ∆-isobar contributions due to our choice of the N ↔ ∆ electromagnetic transition vertex, which is gauge invariant from both the high-spin and electromagnetic field points of view (i.e., the spin−1/2 background is absent). From Figs. 6.1-6.4 one can see that very good agreement is (1/2) obtained, except for some multipole amplitudes, namely p,nM1− , which have the same spin, isospin and parity numbers as the P11(1440) Roper resonance. We did not explicitly include this nucleon resonance in our model and we believe this is the cause (1/2) for such a disagreement. In the case of p,nE0+ the discrepancies could be caused by the fact that a pseudo-vector πNN coupling was used, and as pointed out by [21], a mixed (hybrid model, see Eq. (1.2)) pseudoscalar-pseudovector coupling would give a good fit for these non-resonant multipoles. For higher l multipole amplitudes (see Figs. 6.3, 6.4) such as 2+ and 3−, the fit is not very good since in our model we do not include explicitly higher mass resonances, such as D13(1520), S11(1535), F15(1680) and D33(1700). However they are shown in Fig. 6.3 and Fig. 6.4 for completeness. Note that the ∆ electromagnetic couplings given in Table 6.1 are for the spectator

GM (0) GE(0) GC (0) gγωπ gγρπ gωNN κω/gωNN gρNN κρ/gρNN

3.1 0.048 −0.18 0.313 gγωπ/3 11.2 −0.12 2.66 1.8

Table 6.1: Couplings/parameters used in our model in the spectator approximation.

approximation, discussed in Ch.4. Using the modified equal time approximation the fits given for the multipoles are essentially the same with the exception that different

γN ↔ ∆ couplings are used: GM (0) = 2.67, GE(0) = 0.112, and GC (0) = −0.38. 73

6 15 pM (1/2) [10-3/m ] pE (1/2) [10-3/m ] 1- π 0+ π

10 2

0 5

-2

-4 0 150 200 250 300 350 400 450 500 550 600 650 150 200 250 300 350 400 450 500 550 600 650

4 0 nM (1/2) [10-3/m ] 1- π

(1/2) -3 2 nE [10 /m ] 0+ π -5

-10

-2

-15 -4

-6 -20 150 200 250 300 350 400 450 500 550 600 650 150 200 250 300 350 400 450 500 550 600 650

5 10

(3/2) -3 M (3/2) [10-3/m ] E [10 /m ] 1- π 0+ π

-10

-5

-20

-10 -30 150 200 250 300 350 400 450 500 550 600 650 150 200 250 300 350 400 450 500 550 600 650 T (LAB) [MeV] T (LAB) [MeV] γ γ

Figure 6.1: Multipole calculations. The black line is the real part of the amplitude and the red line is the imaginary part of the amplitude. Data points are from VPI [67]. 74

1 3 (1/2) -3 (1/2) -3 pE [10 /m ] pM [10 /m ] 1+ π 1+ π 0 2

-1

-2

0 -3

-4 -1 150 200 250 300 350 400 450 500 550 600 650 150 200 250 300 350 400 450 500 550 600 650

5 1 (1/2) -3 nM [10 /m ] nE (1/2) [10-3/m ] 1+ π 1+ π 4 0

2 -1

1 -2

-1 -3 150 200 250 300 350 400 450 500 550 600 650 150 200 250 300 350 400 450 500 550 600 650

2 (3/2) -3 40 M [10 /m ] E (3/2) [10-3/m ] 1+ π 1+ π 1 30

0 20

-1

-2 0

-10 -3

-20 -4 150 200 250 300 350 400 450 500 550 600 650 150 200 250 300 350 400 450 500 550 600 650

(LAB) T (LAB) [MeV] T [MeV] γ γ

Figure 6.2: Same as in Fig. 6.1. 75

6 2 (1/2) -3 pE [10 /m ] pM (1/2) [10-3/m ] 2- π 2- π

4 1

2 0

0 -1

-2 -2 150 200 250 300 350 400 450 500 550 600 650 150 200 250 300 350 400 450 500 550 600 650

2 1 (1/2) -3 nE [10 /m ] (3/2) -3 2- π E [10 /m ] 2+ π

-2

-1 -4

-6 -2 150 200 250 300 350 400 450 500 550 600 650 150 200 250 300 350 400 450 500 550 600 650

2 2 (3/2) -3 E [10 /m ] pE (1/2) [10-3/m ] 2- π 3- π

1 -2

-4 0

-6

-8 -1 150 200 250 300 350 400 450 500 550 600 650 150 200 250 300 350 400 450 500 550 600 650 T (LAB) [MeV] T (LAB) [MeV] γ γ

Figure 6.3: Same as in Fig. 6.1. 76

2 1 pE (1/2) [10-3/m ] nE (1/2) [10-3/m ] 3- π 3- π

1 0

0 -1

-1 -2 150 200 250 300 350 400 450 500 550 600 650 150 200 250 300 350 400 450 500 550 600 650 (LAB) T [MeV] T (LAB) [MeV] γ γ

Figure 6.4: Same as in Fig. 6.1.

In Fig. 6.5 we show the diﬀerential cross section and several single polarization observables for real photons. Two reaction channels are shown, γ + p → π+ + n and

0 LAB γ + p → π + p, for a laboratory photon energy of Tγ = 0.32 GeV (W ≈ 1.232 GeV). The single polarization observables such as the beam asymmetry (Σ), target asymmetry (T ), and recoil polarization (P ), are calculated using relations introduced in Appendix D. We sum multipoles up to l = 2+ and l = 3−. Higher multipoles would have very small effects within the energy range of the first resonance (P33(1232)), where our model is designed to work. One can see that the agreement is fairly good, especially for the neutral channel, where the resonant contribution dominates. For the charged channel a slight disagreement is apparent at forward angles, which also occurs, at various other photon energies. In Fig. 6.6 we show our calculations for the total cross section for all pion photo-production channels. This observable is not sensitive to the various contributions included in our model (it is directly proportional only to the transverse response function (see Appendix D). The figures exhibit very

LAB clearly the ∆ resonance peak at Tγ ≈ 320 MeV. 77

(+) (0) γ+p->π +n γ+p->π +p LAB LAB T =320 MeV T =320 MeV γ γ 25 35

30 20 ] ] 25 sr sr 15

b/ 20 b/ µ µ

[ [ 15 Ω

Ω 10 /d /d 10 σ σ d d 5 5

0 0 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 CM CM θ θ π π

1.0 0.5

0.5

0.0

T T

0.0 -0.5 0 20 40 60 80 100 120 140 160 180 CM 0 20 40 60 80 100 120 140 160 180 θ CM π θ π 0.5 0.6

0.4

0.4 0.3

Σ Σ 0.2 0.2 0.1

0.0 0.0 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 CM CM θ θ π π

0.2

0.5 P P

0.0

0.0 -0.2 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 CM CM θ θ π π

Figure 6.5: Diﬀerential cross sections and single polarization observables T , Σ, and P at 2 + 0 Q = 0 for γ +p → π +n (left) and γ +p → π +p (right) at Tγ = 320 MeV. Experimental data are from Refs.[69]-[73]. 78

(-) (+) γ+n->π +p γ+p->π +n

300

200

100 b] µ [ σ

0 (0) (0) γ+p->π +p γ+n->π +n

300

200

100

0 200 300 400 500 600 200 300 400 500 600 T LAB [MeV] γ

Figure 6.6: Total cross section for the diﬀerent γ + N → π + N isospin channels at the photo-production point. The experimental data are from the SAID website [67] compilation. 79 6.2 Electro-Production (Q2 =6 0) Results

With all coupling constants determined by our fits to pion photo-production, we now compare our model to pion electro-production. The only additional ingredients needed are the various electromagnetic form factors. As noted earlier, the form factors associated with the Born terms are quite well determined up to Q2 values greater than 5 (GeV/c)2, and we have taken a recent theoretical model for the vector meson form factors [49]. We have adopted the position that all of these form factors (many of which have been fitted to experiment) are fixed and that we can use our model to

2 2 investigate the Q dependence of the N → ∆ electromagnetic form factors GM (Q ), 2 2 00 00 GE(Q ) and GC (Q ). We can consider these as bare form factors in that they enter our model at the tree level and the various multipoles which depend on them become 00dressed00 by the rescattering. In Fig. 6.7 we show various contributions of the included exchanges to the calculations of the resonant multipoles at the resonance position W ≈ 1.23 GeV. At this kinematic point, the real part of the amplitude becomes zero. Hence, in the ﬁgure we show just the imaginary components of the multipoles. The calculations shown are performed using the full rescattering (i.e., Tππ in Eq. (4.11) includes all the contributions from the πN model) and in the Vπγ∗ amplitude we have included just the corresponding contribution (quoted in the legend of Fig. 6.7). Note that in

2 2 the M1+ amplitude for Q ≤ 0.5 (GeV/c) , the Born and ∆ terms have almost the same strength and contribute about 90% of the total amplitude (the vector meson exchanges contribute the remaining 10%). This is rather in contrast with the common belief that the ∆ magnetic dipole transition (M1+) is dominated by ∆ exchange at low Q2. Above 0.5 (GeV/c)2 the ∆ does dominate and one can say that the N → ∆

2 magnetic form factor, GM (Q ), determines the behavior of this amplitude. In the case of the E1+ multipole, the matter is even more complicated. One can see from the ﬁgure that the Born and ∆ contributions have opposite signs and even at higher Q2, the Born terms are a very large contribution and aﬀect the Q2 evolution. Around Q2 ≈ 3.5 (GeV/c)2 the ∆ + ρ + ω terms become dominant and overcome the Born 80

1 E (3/2) [10-3/m ] 1+ π

-1

-2

40 M (3/2) [10-3/m ] 1+ π

S (3/2) [10-3/m ] 1+ π 0

-2 full Born ρ+ω -4 ∆ 01234 Q2 [(GeV/c)2]

Figure 6.7: Various contributions to the resonant multipoles. terms. This sensitive cancellation among resonant and non-resonant terms results

2 in a small amplitude for the E1+ and makes determination of GE(Q ) rather model dependent. For the scalar multipole S1+, the Born terms dominate all the way up to Q2 ≈ 1.5 (GeV/c)2, while above that the ∆ contribution becomes larger. Notice that the ∆ contribution to this multipole is almost 00ﬂat00 from small Q2 ≈ 0.1 (GeV/c)2 up

(3/2) 2 to high values. In Fig. 6.8 we show our calculation for the M1+ /FD(Q ) multipole 2 at W = 1232 MeV using the bare GM (Q ) form factor given in Eq. (4.36) in Ch.4. The ﬁtting of this resonant multipole was to determine the parameters corresponding

2 to the bare form factor GM (Q ) (see Table 6.2). 81

200 M (3/2)/F [10-3/m ] 1+ D π 150

100 Alder Baetzner Bartel Frolov 50 Stein Stoler Stuart full 0 01234 Q2 [(GeV/c)2]

3/2 2 −2 Figure 6.8: Im(M1+ )/FD at W = 1.232 GeV where FD = (1+Q /0.71) is the standard dipole form factor. The data at Q2 = 2.8 and 4.0 (GeV/c)2 is from Ref.[64]; other data are from Refs.[74]-[79].

The focus of many calculations [21, 23] and experiments [64] was the extraction

2 of REM = Im(E1+)/Im(M1+) and RSM = Im(S1+)/Im(M1+) behavior at Q 6= 0. They were motivated by the possibility of determining the range of the momentum transfer where perturbative QCD (pQCD) would become applicable. In the limit

2 Q → ∞, pQCD predicts REM → 100% while RSM → const. Experiments performed recently give a non-vanishing ratio REM with values lying between −2.5% [62] and −3.0% [63] at Q2 = 0. This is an indication of a deformed ∆, or more exactly, an admixture of a D-state in the ∆ quark wave function. Consequently, important information about D-state components of the nucleon and ∆ wave functions may be extracted from the study of REM and RSM . In Fig. 6.9 are shown our extracted 2 values for REM and RSM at W = 1.232 GeV plotted vs. Q . The main diﬀerence between our calculation and the recent analysis from Ref. [64] is that our values of

2 REM show a clear tendency to cross zero and change sign as Q increases. This is in contrast to the recent analysis [64] of the data at Q2 = 2.8 and 4.0 (GeV/c)2 and also with Sato and Lee [23] calculations which concluded that REM would stay negative and tend toward more negative values with increasing Q2. However, our results for

REM agree with the calculations of Kamalov et al. [80] and Aznauryan [81] which also 82

5 10 R [%] R [%] EM SM JLAB-CLAS JLAB-HALL C Bates SL model 0 0 model

-5 -10 JLAB-CLAS JLAB-HALL C LEGS MAMI BATES SL model model -10 -20 0123401234 2 2 2 2 Q [(GeV/c) ] Q [(G eV/c) ]

2 Figure 6.9: REM and RSM at W = 1.232 GeV plotted vs. Q . The data are from Refs.[62]-[66]. The points (∗) are calculations of the Sato-Lee model [23].

2 2 predict a change of sign for REM below Q = 4.0 (GeV/c) . Additional experiments and extractions of multipoles at higher Q2 are needed to settle this point. We also

2 note that our results indicate that the absolute value of RSM is increasing with Q .

ΛI AI BI ΛII AII BII

2 GM (Q ) 0.71 1.21 1.40 0.60 1.02 1.20

2 GE(Q ) 0.50 −1.10 2.00 0.50 −1.10 2.00

2 GC (Q ) 0.82 −0.90 1.00 0.78 −0.90 1.00

(3/2) Table 6.2: Parameters used in the N ↔ ∆ transition form factors to ﬁt M1+ , REM 2 and RSM .(I) ±Q /2W modiﬁcation in the u- and t-channel terms and (II) spectator approximation.

In Table 6.2 are shown the parameters we used in the N ↔ ∆ form factors using the spectator approximation (II), while (I) denotes the same parameters using our modiﬁcation (see. Eq. (4.36) and Ch. 4). In Fig. 6.10 we plot the Q2 dependence of the determined electromagnetic form factors in comparison with the standard dipole 83

2.0 G (Q2)/F (Q2), i=M, E, C i D 2 1.5 spectator Q /2W

1.0

0.5

0.0 01234 Q2 [(GeV/c)2]

Figure 6.10: The Q2 dependence of the determined N → ∆ electromagnetic form factors divided by the dipole form factor, at W = 1.232 GeV .

2 form factor FD(Q ). The solid lines are the form factors determined in the spectator approximation, while the dashed lines are the same form factors determined using

2 our equal time modiﬁcation. Note that GM (Q ), in both parameterizations, is overall harder than the dipole at low Q2, but it tends toward the dipole values at higher Q2.

2 2 GE(Q ) and GC (Q ) start much softer than the dipole, especially GE, and then at 2 2 higher Q tends to become a fraction of FD, as opposed to GC (Q ) which becomes 2 2 harder again at high Q . GE has the same Q dependence in both calculations, while 2 the Q dependence of GC is only slightly modified, being somewhat softer in the spectator approximation. Note that in all cases the overall trend of the N ↔ ∆ transition form factors as a function of Q2 is similar. We can consider the differences as a measure of the uncertainty of our model. We have been trying to determine quantitatively the effect of the two approaches used in avoiding the singularities in the u- and t-channel terms. Why do the coupling constants have to be modified from one approach to the other? The method of avoiding the singularities affects only the u- and t-channel contributions. Hence, in Fig. 6.11 we plot our calculations of the resonant multipoles with the full rescattering model, but in the driving term Vπγ∗ we include just the pion pole (right panel) 84 contribution (t-channel pion exchange) or just the nucleon u-channel exchange (left panel). The t-channel contribution to the M1+ multipole is weakly affected by this modification, while the u-channel changes significantly. This shift results in GM (0) 2 changing from 2.67 to 3.10. The contribution to the E1+ multipole at low Q between model (I) and model (II) is quite large, resulting in GE(0) changing from 0.112 to 0.048. The differences in the two models at higher Q2 are not so large since the effects in the t- and u-channel terms partially cancel. The largest effect is in the S1+ multipole. The u-channel contribution varies significantly at low Q2 between models (I) and (II), while at larger Q2 it is the t-channel contribution which varies. Using

2 the Q = 0 results we had to change GC (0) from -0.38 to -0.18. Depending on the size of the u- and t-channel contributions to the various multipoles, the values at Q2 = 0 can change significantly and the extracted Q2 dependence for GM , GE and GC may change from model (I) to model (II). Fortunately, the overall Q2 dependence as shown in Fig. 6.10 is not changed very much. The nucleon exchange term has a larger effect in the M1+ than the pion pole, in fact, the u-channel change leads to a modification of the shape of the fitted

2 2 GM (Q ). In the case of the E1+ quadrupole, the Q dependence is not affected in 2 2 any of the instances, hence we did not have to refit the Q dependence of GE(Q ). 2 2 The Q dependence of GC (Q ) had to be slightly modified since we see an overall strengthening of both the pion pole and nucleon crossed terms (above Q2 ≈ 1

2 2 (GeV/c) ) from model (II) to model (I), hence we had to make GC (Q ) slightly harder from model (II) to model (I). We did not plot the effects of these two models in the ∆ crossed term but we analyzed them, and as expected, they did not play an important role in the resonant multipoles (most of the ∆ contribution to the resonant multipoles comes from the direct and the Born terms). After obtaining a good fit to the major multipoles by determining the Q2 dependence of the N − ∆ form factors up to Q2 = 4 (GeV/c)2, we calculated the virtual photon differential cross sections, spanning a wide range of Q2 values and also at various W values. Calculations are performed using the relations between 85

0.5 E (3/2) [10-3/m ] E (3/2) [10-3/m ] 1+ π 1+ π 0.0

-0.5

-1.0

-1.5 20 M (3/2) [10-3/m ] M (3/2) [10-3/m ] 1+ π 1+ π

0 (3/2) -3 S [10 /m ] S (3/2) [10-3/m ] 1+ π 1+ π

-0.5

-1.0 N u-channel exchange π t-channel exchange spectator approximation spectator approximation 2 Q /2W modification Q2/2W modification 0123 01234 Q2 [(GeV/c)2] Q2 [(GeV/c)2]

Figure 6.11: Nucleon u-channel (left) and pion t-channel (right) contributions at W = 1.232 GeV for the resonant multipoles. See text for explanations. multipole amplitudes and various observables, quoted in Appendix D. Examples of such calculations and comparison with the data, are shown in Fig. 6.12 and Fig. 6.13 for angular distributions of the virtual photon cross section, around the resonance energy and for diﬀerent kinematic regimes, for neutral and charged channel, respectively. The calculations contain no adjustable parameters other than the ﬁts to the form factors discussed above. Note again that the model does very well overall, particularly for the charged channel, with a slight disagreement at forward angles for the neutral channel (recall the same note for the Q2 = 0 case). 86

30 30 ] r s b/

µ 20 20 [

Ω /d

σ 10 10 d Q2=0.4 (GeV/c)2, Q2=0.638 (GeV/c)2, CM 0 CM 0 W=1.2 GeV, φ =105 , ε=0.842 W=1.212 GeV, φ =45 , ε=0.870 π π 0 0

20 10 ] r s b/

[ 10 5 Ω /d

σ 2 2 2 2 d Q =0.75 (GeV/c) , Q =0.9 (GeV/c) , CM 0 CM 0 W=1.22 GeV, φ =15 , ε=0.6 W=1.220 GeV, φ =75 , ε=0.676 π 0 π 0 15 10 ] r 10 b/s µ

[ 5 Ω

/d 5 σ

d 2 2 Q2=1.15 (GeV/c)2, Q =1.45(GeV/c) , CM 0 CM 0 W=1.22 GeV, φ =105 , ε=0.77 W=1.22 GeV, φ =45 , ε=0.689 0 π 0 π 1.2

1.0 0.4 ] r s 0.8 b/ µ [ 0.6

Ω 0.2

/d 0.4 σ

d 2 2 0.2 Q2=2.8(GeV/c)2, Q =4(GeV/c) , CM 0 CM 0 W=1.235 GeV, φ =19 , ε=0.56 W=1.235 GeV, φ =45 , ε=0.5 0.0 π 0.0 π 0 60 120 180 0 60 120 180 CM CM θ θ π π

Figure 6.12: Virtual photon diﬀerential cross sections on p(e, e0p)π0. The data are from [64, 82]. 87

30 o o cosθ=0.924, θ=22.48 cosθ=0.793, θ=37.53 25 ]

sr 20 b/ µ

[ 15

Ω 10 /d σ

d 5

300 o o cosθ=0.609, θ=52.48 cosθ=0.383, θ=67.48 25 ]

sr 20 b/ µ

[ 15

Ω 10 /d σ

d 5

300 o o 25 cosθ=0.131, θ=82.47 cosθ=-0.131, θ=97.53 ]

sr 20 b/ µ

[ 15

Ω 10 /d σ

d 5

300 o o cosθ=-0.383, θ=112.52 cosθ=-0.609, θ=127.52 25 ]

sr 20 b/ µ

[ 15

Ω 10 /d σ

d 5 0 0 60 120 180 240 300 0 60 120 180 240 300 φ φ

Figure 6.13: Virtual photon diﬀerential cross sections on p(e, e0n)π+ at Q2 = 0.6(GeV/c)2, W = 1.23 MeV and = 0.69 for ﬁxed θ. The data are from [64, 82]. 88

o o θ=29 ; cosθ=0.875 θ=51 ; cosθ=0.625 0.08

0.00

-0.08

o o θ=68 ; cosθ=0.375 θ=83 ; cosθ=0.125 0.08

0.00

-0.08

' LT

A o o θ=97 ; cosθ=-0.125 θ=112 ; cosθ=-0.375 0.08

0.00

-0.08

o o θ=129 ; cosθ=-0.625 θ=151 ; cosθ=-0.875 0.08

0.00

-0.08

90 180 270 90 180 270 φ φ π π

0 + 2 2 Figure 6.14: Beam asymmetry versus φπ for p(e, e n)π at Q = 0.4(GeV/c) , W = 1.22 MeV and for ﬁxed θ. The data points are from [68].

In Fig. 6.14 and Fig. 6.15 we plot the asymmetry ALT 0 and the longitudinal- transverse polarized structure function σLT 0 , respectively. The relationship between these two quantities, following the experimentalists conventions, is as follows: p 2L(1 − )RLT 0 sin φπ ALT 0 = p , (6.1) RT + LRL + 2L(1 + )RTL cos φπ + RTT cos 2φπ with Ri (i = T, L, LT, T T ), L and deﬁned in Appendix D. Therefore the transverse polarized structure function shown in Fig. 6.15 is: σLT 0 = RLT 0 / sin θπ, where

∗ ∗ ∗ ∗ ∗ ∗ RLT 0 = − sin θπIm{(F2 + F3 + cos θπF4 )F5 + (F1 + F4 + cos θπF3 )F6}, (6.2) where Fi with i = 1,..., 6, are the CGLN amplitudes [9] which are deﬁned in Appendix D. We did compare our calculations with some preliminary results from 89

* 0 W=1.22; γ*+p->n+π+ 5 W=1.22; γ +p->p+π

* 0 W=1.18; γ +p->p+π W=1.18; γ*+p->n+π+ 5 b]

µ [ LT' σ

* 0 * + 5 W=1.26; γ +p->p+π W=1.26; γ +p->n+π

-0.5 0.0 0.5 -0.5 0.0 0.5 cos(θ ) cos(θ ) π π

0 + 0 0 2 2 Figure 6.15: σLT 0 versus cos θπ for p(e, e n)π and p(e, e p)π at Q = 0.4(GeV/c) . The data points are from [68].

CLAS [68] and observed a fairly good agreement. This asymmetry measurement is important in determining the t−channel pion pole and contact Born terms contributions in π+n, which otherwise are weak in the π0p channel. Measurements of beam asymmetry ALT 0 where made only for the neutral channel, but since in this reaction the non-resonant amplitude strongly interferes with the imaginary part of (3/2) the dominant ∆(1232) M1+ it is diﬃcult to extract the information about the non- resonant contribution to the observables. The resulting amplitude for π0p is strongly dependent on the rescattering correction and largely dependent on the model (way the model generates the width of the resonance).

+ The calculated angular distribution of σLT 0 for the π n channel show a strong forward peaking for energies around the ∆, in contrast to π0p channel which shows backward peaking. 90 6.3 Conclusion

In summary, we have extended the dynamical model developed in [33, 35] to calculate both pion photo- and electro-production reactions from the nucleon, from threshold up to the ﬁrst resonance region. The parameters of the model were ﬁrst determined

2 at Q = 0 (or ∼ 0 for GC ) by obtaining a good fit for the most of the multipoles up to l = 2, thereby fixing the few parameters of the model (i.e. gωNN , GM (0), GE(0)) and GC (0). For investigating pion electro-production, we were able to use realistic electromagnetic form factors at each photon vertex by carefully treating the problem of gauge invariance for the Born terms. For the resonant terms, we parameterized the bare N − ∆ transition form factors such that the pQCD high Q2 constraint is met, and by fitting the extracted resonance multipoles, determined the strength and

2 2 Q dependence of GM , GE, and GC up to 4 (GeV/c) . Our extracted values for 2 REM at Q 6= 0 are still very far from the pQCD prediction of 100%. However, in contrast to previous results using dynamical models [23] and analysis [64], we ﬁnd that

REM starts from a very small negative value at the photon point and shows a strong tendency to cross zero and change sign at about Q2 ∼= 2 (GeV/c)2. The calculation of the virtual photon cross sections for a large kinematic regime for p(e, e0p)π0 shows very good agreement with recent data. The major theoretical uncertainty in our model is the treatment of the u- and

2 t-channel terms in the potential Vπγ∗ . For large Q (i.e., large negative mass squared of the virtual photon), non-physical singularities occur in these terms when they go off shell in solving the scattering integral equation. We found two approximate ways of solving this problem. The first was the spectator approximation for these diagrams and the second was to modify the relative energy in these diagrams by ±Q2/2W . We chose to use the spectator approximation because it preserved current conservation at the tree level, although it did require quite different values of GE(0) as compared to equal time fits to pion photo-production [33]. The second approximation has the advantage that the photon point results did not have to be refitted, but it does violate current conservation at the tree level. It is interesting to note that the Q2 91 dependence in GM , GE and GC does not vary that much between these two methods of regularizing the u- and t-channel terms. Within the theoretical uncertainty of how to regularize the u- and t-channel terms for pion electro-production, this model provides an overall good prediction of the available data up through the first resonance region. This model should prove useful to the experimental community in extracting N → ∆ transition properties from their data. To improve our results at higher energy, we need to include the coupling with

∗ higher mass N states, among which the Roper (P11(1440)) would be the first to be considered. The inclusion of this nucleon resonance would almost certainly improve 1/2 the fit to certain multipoles, even in the first resonance region, such as pM1− and 1/2 nM1− , and consequently the reproduction of the data for various observables. 92

Appendix A Useful relations

A.1 Helicity spinors

We deﬁne the four-component nucleon helicity spinors as follows:   s E + m 1 (+) p N   uλ (|~p|) = ⊗ χλ(θ, ϕ) (A.1) 2Ep  2λ|~p|  Ep+mN

  s −2λ|~p| E + m u(−)(|~p|) = p N  Ep+mN  ⊗ χ (θ, ϕ), (A.2) λ 2E   λ p 1

p 2 2 where λ = ±1/2 is the helicity, Ep = |~p| + mN is the energy, θ and ϕ are the spherical angles of the 3-momentum |~p|, and χλ is the two-component Pauli spinor. The positive- and negative-energy nucleon spinors satisfy the following orthogonality and completeness conditions:

†(ρ) (ρ0) uλ (p) uλ0 (p) = δρρ0 δλλ0 , (A.3) X (ρ) †(ρ) uλ (p) uλ (p) = 1, (A.4) ρ, λ where ρ is related to the positive (ρ = +1) and negative (ρ = −1) energy states of the spin 1/2 particle. The Pauli spinors along the z-axis are given by:     1 0     χ1/2(0) =   , χ−1/2(0) =   , 0 1 while along an arbitrary direction θ, ϕ they can be obtained using the Wigner rotation functions:

X 1/2 i(λ−λ0)ϕ χλ(θ, ϕ) = d 0 (θ) e χλ0 (0), λ0 λλ 93 or explicitly,     cos(θ/2) −e−iϕ sin(θ/2)     χ1/2(θ, ϕ) =   , χ−1/2(θ, ϕ) =   . eiϕ sin(θ/2) cos(θ/2)

A.2 Kinematics

Here we will deal with the kinematics of the electro-production of the pion and deduce relations among the 4-momenta of the associated particles. We attach an upper index

LAB LAB kπ=(ωπ kπ ) φπ kF, εF

θπ LAB LAB LAB q =(ωγ ,q ) z

kI, εI p′

Pion electroproduction in Lab frame kI (kF) –incoming(outgoing) electron LAB LAB LAB q =(ωγ ,q )–exchanged photon p(p′)- incoming(outgoing) nucleon LAB LAB kπ=(ωπ kπ ) – momentum of the pion

Figure A.1: Schematics of the lab frame kinematics pion electro-production.

whenever necessary to reﬂect the CM or LAB frame. The present work is done in the CM frame so in the end all the relations will be in CM frame. We denote by 94 k = (, k)(k0 = (0, k0)) the 4-momentum of the incoming (outgoing) electron in the

CM CM CM LAB LAB LAB laboratory frame, q = (ωγ , q )(q = (ωγ , q )) the 4-momentum of the virtual photon in the CM (LAB) frame (it is obvious that (qLAB)2 = (qCM )2 = q2 is

0 Lorentz invariant), pN = (E, pN )(pN 0 = (E , pN 0 )) the 4-momentum of the incoming

(outgoing) nucleon in CM frame and kπ = (ωπ, kπ) is the 4-momentum of the outgoing pion in the CM frame. The following deduction is made under the assumption that

2 2 the incoming (target) nucleon is on its mass shell (i.e. pN = mN ) and that we are in the energy range where the mass of the electron, me ≈ 0, which obviously implies that |k| ≈ and |k0| ≈ 0. Therefore one can directly write:

2 0 2 2 0 0 q = (k − k ) = 2me − 2k · k ≈ −2k · k θ = −20 + 2k0 · k = −20(1 − cos θ ) = −40 sin e , (A.5) e 2 where θe is the angle between the incoming and outgoing electron trajectories. Using the facts that the 3-momentum is conserved at the leptonic vertex (e → γ + e0) and the incoming electron path deﬁnes the 0z axis, we can easily deduce the following relations:

LAB 0 |q | cos θγ + |k | cos θe = |k|

LAB 0 |q | sin θγ = −|k | sin θe. (A.6)

From Eq. (A.6) we obtain:

LAB 2 02 2 0 |q | = + − 2 cos θe. (A.7)

2 LAB 2 LAB 2 Using Eq. (A.5), Eq. (A.7) and q = (ωγ ) −(q ) , we get (energy conservation):

LAB 0 ωγ = − . (A.8)

The total 4-momentum of the reaction γ∗ + N → N 0 + π (i.e. ignoring the leptonic 00leg00) expressed in the Mandelstam variable s, can be written in the lab frame (under the assumption that the target nucleon is at rest):

2 LAB LAB 2 2 2 LAB s = W = (q + pN ) = mN + q + 2mN ωγ , (A.9) 95

LAB from which we get the following relation for ωγ ,

2 2 2 LAB W − q − mN ωγ = . (A.10) 2mN Rewriting the total 4-momentum in the CM frame (s is a Lorentz scalar) we get:

2 CM 2 2 2 CM CM 2 s = W = (q + pN ) = q + mN + 2Eωγ + 2|q |

2 2 CM CM 2 2 = q + mN + 2Eωγ + 2[(ωγ ) − q ] q 2 2 CM CM 2 2 2 CM 2 2 = q + mN + 2ωγ (ωγ ) − q + mN + 2[(ωγ ) − q ]. (A.11)

CM Solving Eq. (A.11) for ωγ we get: W 2 + q2 − m2 ωCM = N . (A.12) γ 2W It is easy to get the 3-momentum of the virtual photon in the CM frame (and obviously the 3-momentum of the incoming nucleon):

CM 2 CM 2 2 |q | = (ωγ ) − q . (A.13)

Using Eq. (A.13), the energy of the incoming nucleon in the CM frame is: W 2 − q2 + m2 E = W − ωCM = N . (A.14) N γ 2W For the π emitting vertex, we write in a similar way in the CM frame:

2 2 2 2 s = W = (pN 0 + kπ) = mN + mπ + 2pN 0 · kπ

2 2 2 = mN + mπ + 2ωπEN 0 + 2|kπ|

2 2 2 2 = mN + mπ + 2ωπ(W − ωπ) + 2(ωπ − mπ)

2 2 = mN − mπ + 2ωπW.

From this we obtain the CM energy of the outgoing π and nucleon N 0, W 2 + m2 − m2 ω = π N (A.15) π 2W

2 2 2 W − mπ + mN E 0 = W − ω = , (A.16) N π 2W 96 and obviously the CM 3-momentum of both π and N 0,

2 2 2 |kπ| = ωπ − mπ. (A.17)

One should also remember that in order to have a complete determination of the outgoing channel in the CM, there is need for speciﬁcation of the CM frame angle of the outgoing π (i.e. θπ). This angle, together with the previous quantities, gives a complete determination of all kinematic invariants (in the CM frame):

2 CM 2 u = (pN − kπ) = (pN 0 − q )

2 2 2 = mN + mπ − 2Eωπ − 2|kπ| cos θπ

2 2 CM CM 2 = mN + q − 2Eωγ − 2|q | cos θπ, (A.18)

2 CM 2 t = (pN 0 − pN ) = (q − kπ)

2 0 CM = 2mN − 2EE + 2|q ||kπ| cos θπ

2 2 CM CM = q + mπ − 2ωγ ωπ + 2|q ||kπ| cos θπ. (A.19)

After all of these considerations we have completely determined the kinematics of the

0 0 0 process e + N → e + N + π using just the known quantities: , , θe and θπ, along with the pion and nucleon rest masses.

A.3 Sample Isospin Decomposition of the ΓγNN Vertex

00 00 We will start with the standard electromagnetic vertex ΓγNN since the form factors are normally introduced in this form, but elsewhere we will be concerned with gauge invariance and will introduce a modiﬁed vertex to preserve current conservation.

µν 2 µ N 2 iσ qν N 2 ΓγNN (Q ) = e γ F1 (Q ) + κN F2 (Q ) , (A.20) 2mN

2 2 N 2 where Q = −q is the negative mass squared of the virtual photon, F1 (Q ) and N 2 F2 (Q ) are the form factors which account for the electromagnetic structure of the 97 nucleon, and κN is the anomalous magnetic moment of the nucleon. The index N corresponds to either proton (p) or neutron (n). It customary to perform a linear transformation of these functions into the Sachs form factors such that:

N 2 N 2 N 2 GE (Q ) = F1 (Q ) − τκN F2 (Q ) (A.21a)

N 2 N 2 N 2 GM (Q ) = F1 (Q ) + κN F2 (Q ), (A.21b)

2 2 where τ = Q /4mN . Inverting Eq. (A.21) one obtains: N 2 N 2 N 2 GM (Q ) − GE (Q ) F1 (Q ) = (A.22a) κN (1 + τ)

GN (Q2) + τGN (Q2) F N (Q2) = E M , (A.22b) 2 1 + τ N 2 N 2 where GE (Q ) and GM (Q ) are the electric and magnetic distributions of the nucleon, respectively. This translation is more intuitive since the Fourier transforms of GE and

GM are associated with the usual (classical) charge and magnetic distributions. In the limit Q2 → 0 one has:

p 2 p 2 GE(Q = 0) = 1,GM (Q = 0) = 2.793

n 2 n 2 GE(Q = 0) = 0,GM (Q = 0) = −1.91, where µp = 2.793 = 1 + 1.793 = 1 + κp and µn = −1.91 = κn are the proton and neutron magnetic moments (in Bohr magneton units), respectively. Experimental p 2 ∼ 2 p 2 ∼ 2 data conﬁrm that for the proton GE(Q ) = GD(Q ) and GM (Q ) = µpGD(Q ) −2 2 2 2 Q2 for Q ≤ 1 (GeV/c) , where GD(Q ) = 1 + Λ is the dipole form factor. Experimentally it is determined that Λ = 0.71 (GeV/c)2. For the case of the neutron, the electric Sachs form factor is harder to extract and it is generally considered n 2 ∼ that GE(Q ) = 0. The neutron magnetic form factor is also given by the dipole n 2 ∼ 2 distribution, GM (Q ) = µnGD(Q ). It is useful to perform a isospin decomposition of the vertex function (Eq. (A.20)) such that:

2 µ h (s) 2 (v) 2 i ΓγNN (Q ) = eγ F1 (Q ) + τ3F1 (Q ) µν iσ qν h (s) 2 (v) 2 i + e F2 (Q ) + τ3F2 (Q ) , (A.24) 2mN 98 where τ3 is the isospin projection of the nucleon and,

(s) (p) (n) (v) (p) (n) F1 = F1 + F1 ,F1 = F1 − F1 (A.25a)

(s) (p) (n) (v) (p) (n) F2 = κpF2 + κnF2 ,F2 = κpF2 − κnF2 , (A.25b) where (s) and (v) refer to scalar and vector terms respectively. Eq. (A.25) in the limit Q2 → 0 becomes:

(s) (v) F1 (0) = F1 (0) = 1 (A.26a)

(s) F2 (0) = µp + µn − 1 = κp + κn, (v) F2 (0) = µp − µn − 1 = κp − κn. (A.26b)

00 00 For the real γN → N transition, we can use the fact that ΓγNN is sandwiched between isospinors, in conjunction with the following properties of the isospinors:

† † † χpτ3χp = 1/2 for the proton, χnτ3χn = −1/2 for the neutron and χN 0 χN = δN 0N /2 to write: 1 χ† Γ (0)χ = eχ† {γµ (1 + τ ) N γNN N N 2 3 µν iσ qν 1 + [(κp + κn) + τ3(κp − κn)]}χN (A.27) 2mN 2 Then Eq. (A.27) for γp → p reduces to, µν † µ iσ qν χN ΓγNN (0)χN = e γ + κp , (A.28) 2mN and for γn → n reduces to, µν † iσ qν χN ΓγNN (0)χN = e κn . (A.29) 2mN Eq. (A.25) can be written in a general form as: τ [1 + τ(µ + µ )] F (s)(Q2) = 1 + (κ + κ ) G (Q2) = p n G (Q2) (A.30a) 1 1 + τ p n D 1 + τ D 1 (µ + µ − 1) F (s)(Q2) = (κ + κ )G (Q2) = p n G (Q2) (A.30b) 2 1 + τ p n D 1 + τ D τ [1 + τ(µ − µ )] F (v)(Q2) = 1 + (κ − κ ) G (Q2) = p n G (Q2) (A.30c) 1 1 + τ p n D 1 + τ D 1 (µ − µ − 1) F (v)(Q2) = (κ − κ )G (Q2) = p n G (Q2). (A.30d) 2 1 + τ p n D 1 + τ D 99

Appendix B Form Factors Used in the Model

The high-momentum behavior of the one hadron exchange potential is commonly regulated using oﬀ-shell form factors introduced at the vertices. The physical meaning of these form factors is that they reﬂect the spatial distributions of the hadrons. In this appendix we summarize all the form factors used in the model for future easy reference.

B.1 Hadronic form factors

Hadronic form factors are introduced for each of the particles involved in the vertex function. For the pion we use the monopole form factor:

2 2 2 Λπ − mπ fπ(k ) = 2 2 . (B.1) Λπ − k For the σ− exchange (used only in πN scattering), ρ− (used in both πN and electro- production) and ω− (used only in electro-production), we use the one-boson-exchange form factor:

2 Λσ,ρ,ω fσ,ρ,ω(t) = 2 . (B.2) Λσ,ρ,ω − t For the baryons we use the form factor of [83]:

4 2 2 2ΛN fN (p ) = 4 2 2 2 . (B.3) 2ΛN + (p − mN ) For each pion, an additional form factor is introduced:

4 Λπ fRegge(q, s) = 4 . (B.4) Λπ + s|~q| The last form factor (Eq. (B.4)) is motivated by considering the eﬀect of the higher- mass states on the high-energy behavior of the πN propagator, GπN (see Eq. (3.12)). 100 B.2 Electromagnetic form factors

With the risk of being redundant, we summarize the electromagnetic form factors used in the model (see also Ch.4). The 00standard00 dipole form factor is given by: Q2 −2 F (Q2, Λ) = 1 + , (B.5) D Λ where Λ and Q2 are given in units of (GeV/c)2. We used the following proton electric form factor that was ﬁtted to the data for the range 0 ≤ Q2 ≤ 4 (GeV/c)2 [46]:

p 2 2 −Q2 2 GE(Q ) = (1 + Q e )FD(Q , 0.4). (B.6)

For the neutron electric form factor we used the Galster parameterization [47, 48]: −µ (0.895 ± 0.039)τ Gn (Q2) = n F (Q2, 0.71), (B.7) E 1 + (3.69 ± 0.40)τ D 2 2 where τ = Q /4mN . For the magnetic form factors for both the proton and neutron we have used the dipole parameterization:

p/n 2 2 GE (Q ) = µp/nFD(Q , 0.71), (B.8)

(see Appendix A for the ΓγNN vertex isospin decomposition). For the pion form factor we used the monopole form factor: Q2 −1 F (Q2) = 1 + , (B.9) π 0.45 and for the axial form factor we used,

2 2 FA(Q ) = FD(Q , 0.9). (B.10)

For the case of the vector meson exchanges (ρ/ω) we have used: 1 + Q2 F = , (B.11) ρ/ω 1 + 3.04Q2 + 2.42Q4 + 0.36Q6 where the numerical parameters are given in corresponding units such that the form factor has no units. For the N → ∆ transition form factors we determined the following form factors in the spectator approximation (see Ch. 4): 101

• Magnetic form factor:

2 2 2 Q − Q 2 G (Q ) = 1 + e 1.02 F (Q , 0.6) (B.12) M 1.2 D

• Electric form factor:

2 2 2 Q − Q 2 G (Q ) = 1 − e 2.0 F (Q , 0.5) (B.13) E 1.1 D

• Coulomb form factor:

2 2 2 Q − Q 2 G (Q ) = 1 − e 1.1 F (Q , 0.78). (B.14) C 0.9 D

In these relations all the numerical constants are given in units of (GeV/c)2. 102

Appendix C Azimuthal Dependence of One Nucleon Exchange

As an example of the azimuthal angle dependence and how to integrate over it explicitly, we consider the u-channel nucleon exchange potential given by the graph in Fig. 5.1 and the following expression,

2 0 0 gπNN 0 5 (α − β)P/ + (q/ + q/ ) + mN 5 V (q , q; P ) = 2 (βP/ − q/ )γ 0 2 2 γ (βP/ − q/)],(C.1) 4mN [(α − β)P + q + q ] − mN + iε where α and β are deﬁned in Eq. (3.11). For simplicity we choose the CM frame, where the potential in the helicity basis takes the form:

2 ρ0ρ00 0 00 gπNN 1 λ0 00 ρ0ρ00 ρ0ρ00 0 λ00 00 Vλ0λ00 (q , q ) = 2 2 u¯ρ0 (q )[M1 1 + M2 γ ]uρ00 (q ), (C.2) 16πmN u − mN with

ρ0ρ00 0 00 √ 0 00 0 00 M1 (p , p ) = mN [2 s(p0 + p0) − s − 2p · p

02 002 2 0 0 00 00 + p + p + mN + (p0 − ρ Ep0 )(p0 − ρ Ep00 )

√ 0 0 00 00 − s(p0 − ρ Ep0 + p0 − ρ Ep00 )] (C.3)

ρ0ρ00 0 00 √ √ 0 00 0 00 M2 (p , p ) = s[2 s(p0 + p0) − s − 2p · p

02 002 2 0 0 00 00 − p − p − 3mN − (p0 − ρ Ep0 )(p0 − ρ Ep00 )]

02 2 00 00 002 2 0 0 + (p + mN )(p0 − ρ Ep00 ) + (p + mN )(p0 − ρ Ep0 ). (C.4)

The azimuthal dependence φ00 arises from Dirac spinors and from various scalar products involving the four vector q00. Choosing the vector part of the total momentum P to be along the z-axis (or to be zero in the CM frame) allows the 103

φ00 dependence for the fully oﬀ-shell potential, to be displayed in the form:

2 ρ0ρ00 0 00 gπNN 0 00 0 00 Vλ0λ00 (q , q ) = 2 Nq0q00 Ωλ0λ00 (θ , θ , φ , φ ) 16πmN N ρ0ρ00 0 0 00 00 0 00 n 00 0 X Rλ0λ00,n(q0, |q |, q0 , |q |, θ , θ )cos (φ − φ ) , (C.5) d (q0 , |q0|, q00, |q00|, θ0, θ00) + d (q0 , |q0|, q00, |q00|, θ0, θ00) cos(φ00 − φ0) n=0 1 0 0 2 0 0

ρ0ρ00 where Rλ0λ00,n, Np0p00 , and di are factors which depend on the type of diagram and exchanged particle, but are independent of the azimuthal angle. The quantities,

1/2 −iλ0φ0 X 1/2 0 1/2 00 im(φ0−φ00) iλ00φ00 Ωλ0λ00 = e dλ0m(θ )dλ00m(θ )e e , m=−1/2

q Nq0q00 = (Eq0 + mN )(Eq00 + mN )/4Eq0 Eq00 are factors which result from the helicity spinors. In Eq. (C.5) we have employed the usual trigonometric relation between two arbitrary directions deﬁned by q00 and q0:

00 0 00 0 00 0 cos Θq00q0 = cos θ cos θ + sin θ sin θ cos(φ − φ ). (C.6)

It is easy to see that the fully oﬀ-shell potential in Eq. (C.5) has the azimuthal dependence of Eq. (5.2). Furthermore, in iterating Eq. (3.13), the quantization axis is deﬁned by the on-shell relative momentum q (i.e. θ = 0), hence Eq. (C.6) reduces

0 to cos Θq0,q = cos θ , therefore the half-oﬀ-shell potential reduces to:

ρ0ρ 0 ρ0ρ 0 0 0 −i(λ0−λ)φ0 Vλ0λ (q , q) = vλ0λ(q0, |q |, q0, |q|, θ )e , (C.7) which is of the form of the result in Eq. (5.3). Therefore, the azimuthal angle dependence can be removed from the Bethe-Salpeter equation for this case. We achieved this result by explicitly displaying the azimuthal angle dependence and

3 0 ρ0ρ aligning P with the z-axis so that only γ and γ appear in Vλ0λ . The presence of γ1 or γ2 would introduce additional azimuthal angle dependence in the spinor matrix elements and make the algebra much more complicated. For the u-channel nucleon exchange the coeﬃcients di are:

0 00 02 002 √ 0 00 0 00 d1(p , p ) = p + p + s − 2 s(p0 + p0) + 2p0p0

0 00 0 00 2 − 2|p ||p | cos θ cos θ − mN (C.8) 104

0 00 0 00 0 00 d2(p , p ) = −2|p ||p | sin θ sin θ . (C.9)

From these relations one can exactly identify the angular dependence of the potential given in Eq. (C.1) in the 4-vector product:

0 00 0 00 0 00 p · p = p0p0 − p · p

0 00 0 00 0 00 0 00 0 00 0 00 = p0p0 − |p ||p | cos θ cos θ − |p ||p | sin θ sin θ cos(φ − φ ). (C.10)

The relative momenta q0 and q00, deﬁned in Ch. 5, are to be introduced in Eq. (C.3) - Eq. (C.10) by p0 = αP + q0 and p00 = αP + q00, where

1/2 N ρ00ρ0 0 00 00 0 X X ρ0ρ00 1/2 0 1/2 00 v¯λ00λ0,λ(|q |, |q |, θ , θ ) = Rλ0λ00,ndλ0m(θ )dλ00m(θ ) × m=−1/2 n=0 Z 2π dxcosnx × ei(λ−m)x. (C.11) 0 d1 + d2 cos x After applying standard trigonometric manipulations:     2n − 1    cos(2n − 1)θ +   cos(2n − 3)θ + ...      1  1  cos2n−1 θ =   2n−2     2    2n − 1   ... +   cos θ      n − 1 and     2n        cos 2nθ + cos(2n − 2)θ + ...      1 2n 1  1  cos2n θ =   +   , 2n   2n−1     2 2   n  2n   ... +   cos 2θ      n − 1 the integral over the azimuthal angle of the intermediate momentum in Eq. (C.11) can be reduced to integrals of the following type: Z 2π dφ cos(mφ)einφ Z 2π dφ cos(mφ) cos(nφ) Im,n = = . (C.12) 0 1 + a cos φ 0 1 + a cos φ 105

For values |a| < 1, this deﬁnite integral can be evaluated analytically to obtain: " # π b − 1m+n b − 1|m−n| I = + . (C.13) m,n b a a √ where b = 1 − a2. The results given above in Eq. (C.11) work for all standard particle exchanges in the s, t, or u channels. Furthermore, it should be noted that additional azimuthal angle dependencies introduced by various form factors can easily be handled by simple algebraic methods. The maximum power of cosx needed for a particular diagram may increase (for example, N=2 for u-channel ∆ exchange). In addition,v ¯ will, in general, contain a sum of various terms corresponding to each diagram included. However, all of these terms can be evaluated using Eq. (C.13). In addition as noted earlier, this procedure is not at all aﬀected by the equal time approximation and can be applied in the same manner to the full 4-D Bethe-Salpeter equation. 106

Appendix D Observables in Terms of Multipoles

The relations quoted in this appendix are based on the well known paper of Drechsel and Tiator [26]. Throughout this work we follow their conventions.

D.1 Deﬁnitions of the obsevables

The virtuality of the 00exchanged00 photon is described by the degree of transverse polarization, 1 = 2 , (D.1) 2|q| 2 θe 1 + Q2 tan 2 where q (the three momentum of the photon) and θe (the angle of the outgoing electron) are measured in the laboratory frame. The degree of longitudinal polarization of the virtual photon is given by: Q2 = . (D.2) L ω2 The electron scattering coincidence diﬀerential cross section can be written as: dσ dσ = Γ v , (D.3) dF dΩF dΩπ dΩπ where the CM virtual photon diﬀerential cross section (this is the unpolarized cross section, i.e. unpolarized beam, target, and recoil) is given by:

dσv |k| p = CM RT + LRL + 2L(1 + )RTL cos φπ + RTT cos 2φπ , (D.4) dΩπ kγ and the ﬂux of the virtual photon ﬁeld is:

α F kγ 1 Γ = 2 2 . (D.5) 2π I Q 1 − 2 2 Here the photon equivalent energy, kγ = (W − mN )/2mN , is the laboratory energy necessary for a real photon to excite a hadronic system with CM energy W . k is 107

CM the 3−momentum of the outgoing pion in the CM frame and kγ = (mN /W )kγ is the photon equivalent energy in the CM frame. The response functions Ri 2 (i = T, L, T L, T T ) are functions of Q , W and θπ and will be given in terms of the CGLN amplitudes later in this appendix. Note that in Eq. (D.4) we do not consider the degree of polarization of the incoming (outgoing) electron. For a complete determination of all the polarization degrees of freedom, one should take in account additional terms in Eq. (D.4) (see [26]). We need to point out that some of the later

00 00 experimental data have slightly different definitions of RL and RTL, i.e. they drop 00 00 the factor of longitudinal polarization L and absorb it into the definitions of these functions. We use this latter definition in actual calculations. The differential cross section for the reactions induced by real photons is obtained from (D.3) and (D.4) if Γ is replaced with the flux factor for real photons, if all

CM longitudinal currents are dropped, and kγ is replaced by the energy of the real photon in the CM frame. Thus the diﬀerential cross section for real photon is given as follows: dσ |k| = (RT + RTT ) , (D.6) dΩπ ωγ 2 2 where ωγ = (W − mN )/2W is the energy of the photon in the CM frame. The total cross section or inclusive cross section is an incoherent sum of the multipoles and can be written as:

σv = σT + LσL, (D.7) where Z Z ∞ dσT |k| X σT = dΩπ = RT dΩπ = 2π CM × dΩπ k γ l=0 2 2 2 2 2 × (l + 1) [(l + 2)(|El+| + |Ml+1,−| ) + l(|Ml+| + |El+1,−| )], (D.8a)

Z Z ∞ dσL |k| X σL = dΩπ = RT dΩπ = 4π CM × dΩπ k γ l=0 3 2 2 × (l + 1) (|Ll+| + |Ll+1,−| ). (D.8b)

It is common to deﬁne the single polarization observables such as: 108

• polarized photon asymmetry

RTT Σ(θπ) = − (D.9) RT

• polarized target asymmetry

RT (ni) T (θπ) = (D.10) RT

• the recoil nucleon polarization

RT (nf ) P (θπ) = , (D.11) RT

where ni and nf refer to the polarization of the nucleon in the initial and final state, respectively (see Appendix D.3). NOTE: The five-fold differential cross section is usually evaluated in a two-step procedure. First the leptonic current is calculated in the laboratory frame, which translates into the expression for the flux factor in Eq. (D.5). Then the hadronic current is calculated in the CM frame, which translates into the expression for the virtual photon differential cross section in Eq. (D.4).

D.2 CGLN amplitudes in terms of multipoles

For the analysis of the experimental data and also for the purpose of studying individual resonances, pion photo- and electro-production amplitudes are expressed in terms of three types of multipoles, as shown in Ch. 4 (this is because we use current conservation to relate the scalar and longitudinal multipoles). This decomposition allows us to calculate the more customary CGLN amplitudes [9, 11] in terms of the multipoles using:

X 0 0 F1 = {(lMl+ + El+)Pl+1 + [(l + 1)Ml− + El−]Pl−1} (D.12a) l≥0 109

X 0 F2 = [(l + 1)Ml+ + lMl−]Pl (D.12b) l≥1

X 00 00 F3 = [(El+ − Ml+)Pl+1 + (El− + Ml−)Pl−1] (D.12c) l≥1

X 00 F4 = (Ml+ − El+ − Ml− − El−)Pl (D.12d) l≥2

X 0 0 F5 = [(l + 1)Ll+Pl+1 − lLl−Pl−1] (D.12e) l≥0

X 0 F6 = [lLl− − (l + 1)Ll+]Pl , (D.12f) l≥1

0 00 where Pl , Pl are the ﬁrst and the second order derivatives, respectively, of the Legendre polynomials; they are functions of the polar angle of the pion in the CM

CM frame, θ = θπ . Note that in the literature, the scalar transitions are often described by Sl± multipoles, which correspond to the decomposition of the amplitudes F7 and

F8. They are connected to the longitudinal ones by Sl± = |k|Ll±/ω through current conservation. It is this that allows us to use only six CGLN amplitudes in expressing the observables for pion electro-production.

D.3 Response functions in terms of the CGLN amplitudes

In this appendix we give the expressions for the response functions in terms of the

CGLN amplitudes. The angle is the pion CM frame angle, i.e. θ = θπ. 1 R = |F |2 + |F |2 + sin2 θ(|F |2 + |F |2) T 1 2 2 3 4 ∗ 2 ∗ ∗ ∗ − Re{2 cos θF1 F2 − sin θ(F1 F4 + F2 F3 + cos θF3 F4)} (D.13a)

2 2 ∗ RL = |F5| + |F6| + cos θIm{F5 F6} (D.13b)

∗ ∗ ∗ ∗ ∗ ∗ RTL = − sin θRe{(F2 + F3 + cos θF4 )F5 + (F1 + F4 + cos θF3 )F6} (D.13c) 110 1 R = sin2 θ (|F |2 + |F |2) + Re{F ∗F + F ∗F + cos θF ∗F } (D.13d) TT 2 3 4 1 4 2 3 3 4

∗ ∗ ∗ ∗ 2 ∗ RT (ni) = sin θIm{F1 F3 − F2 F4 + cos θ(F1 F4 − F2 F3) − sin θF3 F4} (D.13e)

∗ ∗ ∗ RT (nf ) = − sin θIm{2F1 F2 + F1 F3 − F2 F4 +

∗ ∗ 2 ∗ + cos θ(F1 F4 − F2 F3) − sin θF3 F4} (D.13f)

In the above equations the 00∗00 symbol shows the algebraic operation of complex conjugation for the corresponding amplitude. There are many other polarization observables related to the polarizations of either the beam/recoil nucleon or beam/target nucleon, which can be written down in terms of the CGLN amplitudes (see [26]), but in our work these are the only ones which we actually calculated. 111

Appendix E ρ √ Integration of GET (|~q|; s)

In this appendix we perform in detail the integration through the pole of the 2−body scattering equation (e.g. Eq. (4.11)). Starting with Eq. (3.18) and performing several algebraic manipulations, after ignoring the small quantity i, we arrive at the form: √ √ −ρ −ρ s + E − ω Gρ(|~q|; s) = √ q q , (E.1) 2 2 ωq (−ρ s + Eq) − ωq 2 2 2 where the relative three momentum q = |~q| (integration variable), Eq = q + mN and 2 2 2 ωq = q + mπ. Remembering the deﬁnitions for the Lorentz scalars α and β (see Eq. (3.11)) we can rewrite the denominator in Eq. (E.1): √ √ √ s + m2 − m2 s + E2 − 2ρE s − ω2 = 2 s √N π − ρE q q q 2 s q √ = 2 s(EN − ρEq), (E.2) where we have used the fact that the asymptotic (on-shell) value of the nucleon energy

√ √ 2 2 √ (see Appendix A) EN = α s = ( s + mN − mπ)(2 s). Therefore Eq. (E.1) can be written as: √ √ √ s − ρE + ρω 1 s − ρE + ρω E + ρE ρ √ q q √ q q N q G (q; s) = = 2 2 2 sωq EN − ρEq 2 sωq E − E √ √ N q s − ρE + ρω E + ρE f ρ(q; s) √ q q N q = 2 2 = , (E.3) 2 sωq p − q q − p 2 2 2 where we have used the energy of the nucleon (EN = p + mN ) and √ √ s − ρE + ρω E + ρE f ρ(q; s) = − √ q q N q . (E.4) 2 sωq p + q Ignoring the other variables under the integral, such as the helicities and ρ−spinors in both V and T , and since the pole occurs in the physical region, only in Gρ(q) we can symbolically write: √ Z ∞ 0 00 ρ00 00 00 00 002 V (q , q )f (q ; s)T (q , q) dq q 00 0 = 0 q − q + i Z ∞ 00 00 002 F(q ) 02 0 0 ρ00 0 √ 0 = P dq q 00 0 − iπq V (q , q )f (q ; s)T (q , q). (E.5) 0 q − q 112 √ From Eq. (E.4), using s = EN 0 + ωπ0 = EN + ωγ∗ (ωπ), we obtain: √ √ s − E 0 + ω 0 E 0 + E 0 E 0 f (+)(q0; s) = − √ N π N N = − √N , 0 0 2 sω 0 2q q s √ π √ s − E 0 + ω 0 E 0 − E 0 (−) 0 √ N π N N f (q ; s) = − 0 = 0, (E.6) 2 sωπ0 2q

2 02 2 where EN 0 = q + mN . Gathering these together and plugging back into Eq. (E.5) we obtain the expression of the imaginary part resulting from scattering:

0 00 √ q E 0 − iπq02V (q0, q0)f ρ (q0; s)T (q0, q) = iπ √N V (q0, q0)T (q0, q). (E.7) s Note the fact that the imaginary part does not depend on the anti-particle states (see (E.6)) and, as expected, is determined by the on-shell matrix elements of the potential matrix. This is one of the common choices in the K−matrix approximation of the scattering equation. One neglects the ﬁrst term in Eq. (E.5) and takes in consideration, for unitarity restoration, just the second term (E.7). 113

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