Threshold Pion Photoproduction

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THRESHOLDPION

PHOTOPRODUCTION

A ThesisSubmitted to the College of Graduate Studies and Research

in Partial Fulfillment of the Requirementsfor the Degree of Master of Science

in the

Department of Physics andEngineering Physics

University of Saskatchewan

byTerry Glenn Pilling

Saskatoon, SaskatchewanCANADA

Summer, 1998

c1998 T. G. Pilling. All rights reserved.


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In presenting this thesis in partial fulfilment of the requirements for a Postgraduatedegree from the University of Saskatchewan, the author agrees that the Libraries ofthis University may make it freely available for inspection. The author further agreesthat permission for copying of this thesis in any manner, in whole or in part, forscholarly purposes may be granted by the professor or professors who supervised thethesis work or, in their absence, by the Head of the Department or the Dean of theCollege in which the thesis work was done. It is understood that any copying orpublication or use of this thesis or parts thereof for financial gain shall not be allowedwithout the authors written permission. It is also understood that due recognition

shall be given to the author and to the University of Saskatchewan in any scholarlyuse which may be made of any material in the thesis.

Requests for permission to copy or make other use of material in this thesis inwhole or part should be addressed to:

Head of the Department of Physics and Engineering PhysicsUniversity of Saskatchewan

Saskatoon, Saskatchewan S7N 0W0


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Abstract

An effective chiral Lagrangian is used to calculate the near threshold contributions to

single pion photoproduction from nucleons. The effects due to Vector meson exchangeas well as (1232) and N(1440) resonance excitations are also considered. The result-ing observables are then compared with recent experimental data and some resultsof chiral perturbation theory. Good agreement with the data is found for chargedpion production and moderate agreement with the data for neutral pion productionalthough corrections due to rescattering and other loop contributions are neglected.This indicates that the higher order corrections, which may be individually large,are comparatively small when taken as a whole. The effects of varying the reso-nance off-shell parameters are investigated as an estimate of the uncertainty of themodel. It is found that the E0+ multipole is especially sensitive. Disagreement still

exists for the E1+ multipole in neutral pion production. This multipole is found tobe quite sensitive to the off-shell parameters but not enough to allow a fit with datawhile remaining in the accepted ranges for the off-shell parameters. The possibility oftreating the resonances as very heavy particles, and thereby integrating them out ofthe theory, is examined and found to be lacking for the lower mass resonances treatedhere. The possibility still exists that one could treat higher mass resonances in thisway.


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Acknowledgements

I would like to thank my supervisors Dr. Dennis Skopik and Dr. Mohamed Benmer-

rouche. Dr. Skopik invited me to the Saskatchewan Accelerator Laboratory, first asa summer student and then as a graduate student. He has taught me to search foran understanding of the deep physical concepts behind the abstract calculations andhis experience, constant guidance and encouragement, as well as his sense of humourhave made him a great source of inspiration for me from the beginning. Dr. Benmer-rouche has always been available to discuss the intricacies of various calculations andhis knowledge of field theory has been invaluable to the completion of this work. Ourfrequent philosophical discussions have been both interesting and stimulating.

Thanks also to the computing staff and scientists at SAL, who have providedassistance with computing and programming difficulties and have made this work

much more manageable.I am very grateful to my fellow graduate students for our weekly meetings at

the pub. In particular I would like to thank Trevor Fulton and Darren White fortheir valued friendship and our countless interesting discussions and debates. Specialthanks to my officemate Dave Hornidge whose amazing sense of humour and positiveattitude has made our office an enjoyable place to work. It is impossible to overstatehis importance in keeping me motivated and helping me solve problems both work-related and other.

Finally, I must thank my parents Glenn and Linda Pilling, my brother Rick Pillingand my sister Tammy Schock for their love and support throughout.


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Contents

1 Introduction 11.1 Field Theory and the S-matrix . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 The S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Pion Photoproduction from Nucleons . . . . . . . . . . . . . . . . . . 61.4 N(, ) Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Photoproduction Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 11

2 The Kroll Ruderman, PVBorn and Pion pole Terms 132.1 The Pseudovector Coupling Lagrangian . . . . . . . . . . . . . . . . . 132.2 The Kroll-Ruderman Term . . . . . . . . . . . . . . . . . . . . . . . . 142.3 The Born Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 The Pion Pole Term . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 The CGLN Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5.1 Born, Pion Pole and Kroll-Ruderman Amplitudes . . . . . . . 232.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6.1 p 0p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6.2 n 0n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.6.3 The Charged Pion Reactions n p and p +n . . . . 29

3 The N(1440) Resonance 32

3.1 The Lagrangian and Feynman Rules . . . . . . . . . . . . . . . . . . 323.1.1 Coupling Constants . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 The CGLN Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.1 N(1440) Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 The (1232) Resonance 434.1 The (1232) Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . 434.2 The Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3 The CGLN Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3.1 Resonance Amplitudes . . . . . . . . . . . . . . . . . . . . . . 494.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Vector Meson Exchange 605.1 The Lagrangian and Feynman Rules . . . . . . . . . . . . . . . . . . 605.2 The CGLN Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.2.1 Vector Meson Amplitudes . . . . . . . . . . . . . . . . . . . . 645.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64


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6 Integrating Out the Resonance Fields 696.1 The Decoupling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 696.2 Integrating Out the N(1440) Resonance . . . . . . . . . . . . . . . . 706.3 Integrating Out the (1232) Resonance . . . . . . . . . . . . . . . . . 74

7 The (1232) Off-shell Parameters 83

8 Results and Discussion 938.1 Neutral Pion Production . . . . . . . . . . . . . . . . . . . . . . . . . 93

8.1.1 Photoproduction from the Proton . . . . . . . . . . . . . . . . 938.1.2 Photoproduction from the Neutron . . . . . . . . . . . . . . . 106

8.2 Charged Pion Production . . . . . . . . . . . . . . . . . . . . . . . . 109

9 Summary and Conclusions 120

APPENDICES 122

A Units and conventions 122A.1 General Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122A.2 Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125A.3 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127A.4 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

B Multipoles and observables 131

C Introduction to Chiral Perturbation Theory 134C.1 Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135C.2 Chiral Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . 137C.3 Heavy Baryon Chiral Perturbation Theory . . . . . . . . . . . . . . . 138

References 141


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List of Figures

1.1 Tree-level Feynman diagrams for + N + N . . . . . . . . . . . 92.1 The Kroll-Ruderman contribution to + N + N . . . . . . . . . 162.2 The N N vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 The N N vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 PVBorn S-channel diagram . . . . . . . . . . . . . . . . . . . . . . . 202.5 PVBorn U-channel diagram . . . . . . . . . . . . . . . . . . . . . . . 202.6 The vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.7 The pion pole, t-channel diagram . . . . . . . . . . . . . . . . . . . . 222.8 Multipoles for the Born terms in the reaction + p 0 + p. They

are given from top left to bottom right as E0+, M1

, M1+, and E1+ . . 26

2.9 Multipoles for the Born terms in the reaction + n 0 + n. Theyare given from top left to bottom right as E0+, M1, M1+, and E1+ . . 28

2.10 Multipoles for the Born terms in the charged pion reactions. The topfour are production and the bottom four are + production. Theyare given from top left to bottom right as E0+, M1, M1+, and E1+. . 30

2.11 Multipoles for the Born terms in the charged pion reactions. The topfour are production and the bottom four are + production. Theyare given from top left to bottom right as E0+, M1, M1+, and E1+. . 31

3.1 The N R vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 The N R vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 Multipoles for the N(1440) resonance terms in the neutral pion reac-tions. They are given from top left to bottom right as E0+, M1, M1+,and E1+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 Multipoles for the N(1440) resonance terms in the neutral pion reac-tions. They are given from top left to bottom right as E0+, M1, M1+,and E1+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5 Multipoles for the N(1440) resonance terms in the charged pion reac-tions. They are given from top left to bottom right as E0+, M1, M1+,and E1+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6 Multipoles for the N(1440) resonance terms in the charged pion reac-

tions. They are given from top left to bottom right as E0+, M1, M1+,and E1+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1 The N vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2 The N vertices in g1 and g2 coupling . . . . . . . . . . . . . . . . 474.3 S-channel (1232) resonance exchange . . . . . . . . . . . . . . . . . 484.4 U-channel (1232) resonance exchange . . . . . . . . . . . . . . . . . 49


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4.5 Multipoles for the (1232) g1 coupling in the reaction +p 0 +p.They are given from top left to bottom right as E0+, M1, M1+, and E1+ 52

4.6 Multipoles for the (1232) g2 coupling in the reaction +p 0 +p.They are given from top left to bottom right as E0+, M1, M1+, and E1+ 53

4.7 Multipoles for the (1232) g1 coupling in the reaction + n

0 + n. 54

4.8 Multipoles for the (1232) g2 coupling in the reaction + n 0 + n. 554.9 Multipoles for the (1232) g1 coupling in the reaction + p + + n. 564.10 Multipoles for the (1232) g2 coupling in the reaction + p + + n. 574.11 Multipoles for the (1232) g1 coupling in the reaction + n + p. 584.12 Multipoles for the (1232) g2 coupling in the reaction + n + p. 595.1 The V vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2 The V N N vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3 The vector meson exchange contribution . . . . . . . . . . . . . . . . 625.4 Multipoles for the vector meson exchange terms in the reaction +p

0

+p. They are given from top left to bottom right as E0+, M1, M1+,and E1+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.5 Multipoles for the vector meson exchange terms in the reaction +n

0 +n. They are given from top left to bottom right as E0+, M1, M1+,and E1+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.6 Multipoles for the vector meson exchange terms in the reaction +p ++n. They are given from top left to bottom right as E0+, M1, M1+,and E1+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.7 Multipoles for the vector meson exchange terms in the reaction +n +p. They are given from top left to bottom right as E0+, M1, M1+,and E1+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.1 Multipoles for the reaction + p 0 + p with the integrated outN(1440) resonance terms. . . . . . . . . . . . . . . . . . . . . . . . . 74

6.2 Multipoles for the integrated out (1232) g1 coupling in the reaction+p 0 +p. The solid line gives the explicit amplitude of Chapter 4and the dashed line gives the amplitude with the resonance integratedout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.3 Multipoles for the integrated out (1232) g2 coupling in the reaction+p 0 +p. The solid line gives the explicit amplitude of Chapter 4and the dashed line gives the amplitude with the resonance integrated

out. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.4 Multipoles for the integrated out (1232) g1 coupling in the reaction

+n 0+n. The solid line gives the explicit amplitude of Chapter 4and the dashed line gives the amplitude with the resonance integratedout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79


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6.5 Multipoles for the integrated out (1232) g2 coupling in the reaction+n 0+n. The solid line gives the explicit amplitude of Chapter 4and the dashed line gives the amplitude with the resonance integratedout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.6 Multipoles for the integrated out (1232) g1 coupling in the reaction

+p + + n (the g2 integrated out multipoles are zero). The solidline gives the explicit amplitude of Chapter 4 and the dashed line givesthe amplitude with the resonance integrated out. . . . . . . . . . . . 81

6.7 Multipoles for the integrated out (1232) g1 coupling in the reaction+ n +p (the g2 integrated out multipoles are zero). The solidline gives the explicit amplitude of Chapter 4 and the dashed line givesthe amplitude with the resonance integrated out. . . . . . . . . . . . 82

7.1 The total cross section for +p 0 +p. The solid line uses the Born,N(1440), Vector Mesons and the (1232) where we use the previous

values for the off-shell parameters (Chapter 4). The thick dashed line(almost identical) is the same graph with the above values for the off-shell parameters (see Table 7.1). The squares are data from SAL [1]. . 84

7.2 Multipoles for the (1232) g1 coupling in the reaction +p 0 +p.They are given from top left to bottom right as E0+, M1, M1+, andE1+. The calculation of chapter 4 (solid line) is given for comparisonwith the present calculation (dashed line) using the modifed off-shellparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.3 Multipoles for the (1232) g2 coupling in the reaction +p 0 +p.They are given from top left to bottom right as E0+, M1, M1+, andE1+. The calculation of chapter 4 (solid line) is given for comparisonwith the present calculation (dashed line) using the modifed off-shellparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.4 Multipoles for the (1232) g1 coupling in the reaction + n 0 + n.The calculation of chapter 4 (solid line) is given for comparison with thepresent calculation (dashed line) using the modifed off-shell parameters. 87

7.5 Multipoles for the (1232) g2 coupling in the reaction + n 0 + n.The calculation of chapter 4 (solid line) is given for comparison with thepresent calculation (dashed line) using the modifed off-shell parameters. 88

7.6 Multipoles for the (1232) g1 coupling in the reaction +p + + n.The calculation of chapter 4 (solid line) is given for comparison with the

present calculation (dashed line) using the modifed off-shell parameters. 897.7 Multipoles for the (1232) g2 coupling in the reaction +p + + n.

The calculation of chapter 4 (solid line) is given for comparison with thepresent calculation (dashed line) using the modifed off-shell parameters. 90

7.8 Multipoles for the (1232) g1 coupling in the reaction + n +p.The calculation of chapter 4 (solid line) is given for comparison with thepresent calculation (dashed line) using the modifed off-shell parameters. 91


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7.9 Multipoles for the (1232) g2 coupling in the reaction + n +p.The calculation of chapter 4 (solid line) is given for comparison with thepresent calculation (dashed line) using the modifed off-shell parameters. 92

8.1 The total cross section for + p

0 + p. The solid line uses the

Born, N(1440), Vector Mesons and the (1232), the circles are datafrom [2], the squares are from [3], the triangles are from [4] and thestars are from [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

8.2 This figure shows an expanded view of the previous figure with energyranging to 170 MeV to take full advantage of the SAL data. . . . . . 95

8.3 The contributions to F0 due to each of the channels separately and thetotal given by the solid line. The data are taken from J. C. Bergstromet al.(1997) [4]. It is interesting to note the cancellation of the energydependence between the (1232) and the Born terms. . . . . . . . . . 96

8.4 The above Feynman diagrams are examples of the 1-loop diagrams,

required by unitarity, in the reactions + N + N. . . . . . . . . 988.5 P-waves and E0+ multipole for the reaction +p 0 +p. The dataare taken from J. C. Bergstrom et al.(1997) [4]. . . . . . . . . . . . . 98

8.6 Differential cross sections for the reaction + p 0 + p. . . . . . . 998.7 Differential cross sections for the reaction + p 0 + p. . . . . . . 1008.8 Differential cross sections for the reaction + p 0 + p. . . . . . . 1018.9 Differential cross sections for the reaction + p 0 + p. . . . . . . 1028.10 Differential cross sections for the reaction + p 0 + p. . . . . . . 1038.11 Differential cross sections for the reaction + p 0 + p. . . . . . . 1048.12 Total p(, 0)p multipoles (solid line) compared with dispersion rela-

tions [6] (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.13 P-waves, E0+ multipole and cross section for the reaction + n 0 +n.1078.14 Total n(, 0)n multipoles (solid line) compared with dispersion rela-

tions [6] (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.15 Total cross section for the reaction +p + + n. The data are taken

from Adamovich et al. [7] . . . . . . . . . . . . . . . . . . . . . . . . 1108.16 Differential cross sections for the reaction + p + + n. The data

are taken from Adamovich et al. [8]. . . . . . . . . . . . . . . . . . . 1118.17 P-waves E0+ multipole for the reaction + p + + n. . . . . . . . 1128.18 E0+ multipole (solid line) and dispersion relation result [6] (dashed

line) for the reaction p(, +)n. . . . . . . . . . . . . . . . . . . . . . 113

8.19 Total cross section for the reaction + n + p. The data arecalculated by Legendre polynomial fits to the angular distribution ofHutcheon et al. [9] and from Adamovich et al. [8] . . . . . . . . . . . 114

8.20 Differential cross sections for the reaction + n + p. The dataare taken from Hutcheon et al. [9]. . . . . . . . . . . . . . . . . . . . 115

8.21 P-waves E0+ multipole for the reaction + n + p. . . . . . . . 116


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8.22 E0+ multipole (solid line) and dispersion relation result [6] (dashedline) for the reaction n(, )p. . . . . . . . . . . . . . . . . . . . . . 117


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Chapter 1

Introduction

Photoproduction of mesons (pions) has been an important tool in the study of thestrong interaction as far back as the first accelerators and cosmic ray experiments. Thereason for this is the relative ease with which one can extract valuable experimentalinformation and also test theoretically the low energy effective theories resulting fromthe fundamental Lagrangians of the standard model. In this thesis we will be studyinga fully relativistic effective theory, based on the linear sigma model, in which weuse chirally symmetric Lagrangians to model the pion, the nucleon and the nucleonresonances.

Recently, there has been revived interest in pion photoproduction from nucleonsdue to the advent of more precise data and more effective theoretical treatments of

pion production at low energies such as chiral perturbation theory (CHPT). Usingchiral perturbation theory, corrections can be derived to the low energy effectivetheory presented in this thesis. They are found to be larger than expected. Thiscould indicate that the expansion is not rapidly converging for certain observables.These corrections are due only to the higher order loop diagrams involving pionsand nucleons and do not contain vector mesons or nucleon resonances. This impliesthat the theory presented in the following chapters will not be adequate to describethe experimental data at tree level, since the corrections occur at higher order. Themystery (or accident) is that the theory presented here is not far from the experimentaldata at tree level when the exchange of baryon resonances and heavier vector mesons

are included. This may suggest that the loop corrections to the LET, although beingsignificant individually, may cancel each other somewhat when taken as a whole andresult in only a small net correction. In the following chapters we will explore thisquestion and others by detailing the calculation of the tree level result (the LET)along with fairly significant corrections due to model dependent resonance and mesonexchange. We will make comparisons with recent experiments as well as some of theCHPT results which contain the added loop corrections mentioned above.

In chapter 1 we will show the general technique involved in quantum field theorycalculations with effective Lagrangians, and further discuss the photoproduction ofcharged and neutral pions in this context. We outline the multipole analysis that is

of common use in this field and discuss the appropriate kinematics for single pionphotoproduction. This is intended as an introduction to the formalism and the toolsthat we will be using in later chapters and can easily be omitted by the readeralready familiar with the techniques. Chapter 2 contains the calculation of the lowenergy theorem (LET) given by the tree level Born and Kroll-Ruderman terms (seeFigure 1.1). Following this, in Chapters 3, 4 and 5, the calculations of corrections tothe LET due to the short lived N(1440) resonance, the (1232) resonance and thet-channel exchange of the (770) and (783) vector mesons are given. In Chapter 6 it

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is shown how the resonances can be viewed as mass corrections to the vertices of lowerorder diagrams by treating them as heavy static sources and thereby integrating themout of the theory. It is interesting to estimate the accuracy of this technique for the lowlying resonances, since it follows that one could eventually include contributions fromall possible resonance excitations in a simple way. A similar technique is employed in

heavy baryon CHPT and it is useful to see the differences involved from our treatmentof them as explicit degrees of freedom. In chapter 7 we discuss the dependence of the(1232) resonance amplitudes on the off-shell parameters contained in the resonanceLagrangian. Chapter 8 gives a comparison of our results with experimental datafor both neutral and charged pion production reactions, concentrating mainly onthe reaction + p 0 + p, since this has been the most actively studied anddebated reaction recently. We conclude this thesis in Chapter 9 with a summary ofour findings. Our definitions and conventions are defined in Appendix A and ourformalism in Appendix B. Appendix C gives a brief introduction to CHPT.

1.1 Field Theory and the S-matrix

We would like to predict physical observables measured in the lab from basic sym-metries of nature. An elegant way of doing this has been developed in field theory.Quantum field theory (QFT) has been found to be very successful in describing thephysical interactions between the fundamental particles in nature. In QFT one writesthe particles as fields composed of direct products of vectors in 4-dimensional space-time and quantized vectors in other spaces, such as flavour space, which describes theisospin of the particle. In this way, particles which have traditionally been thoughtof as individuals can be grouped in flavour space as different components of the same

particle. For example, the proton and neutron can be thought of as the up anddown components of a more general particle called a nucleon. Similarly, the pionsform an iso-triplet. These particles themselves are known to be different combinationsof even more elementary particles called quarks. The quarks can be grouped into asingle vector in flavour space with the individual quarks forming the basis vectorswhich can be rotated into one another through interactions. The model of particlesand interactions between them, called the Standard Model, has been very successfuland thus we have a large amount of faith in its predictions of experimental mea-surements. At the root of the standard model is a theory called chromodynamics,which explains the motion and interactions of the quarks and gluons, believed (at

present) to be the fundamental point particles from which all (hadronic) matter ismade. In quantum chromodynamics (QCD), the force between the elementary parti-cles (quarks) is carried by massless spin one gauge bosons, just like in electrodynamics(QED). In QED the electromagnetic force between particles with electric charge iscarried by photons, and likewise in QCD the strong force between particles with coloris carried by gluons. Since quarks have electromagnetic charge in addition to colorthey will also interact with each other and other charged particles via photons. Thestrong force is completely independent of electric charge so that the quark can berepresented as a vector in flavour (charge) space. Each flavour of quark is the same

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particle as seen by a gluon but is a different particle as seen by a photon. Althoughthe matter in the macroscopic world is in the form of bound states of these quarks,the calculation of processes involving these bound states is prohibitively complicatedin QCD. It is therefore imperative that we develop theories which model QCD at lowenergies. In the present thesis, we will be doing exactly this, modeling the strong

force through interactions with the quark composites called pions mentioned above.Pions are made of quark-antiquark pairs and therefore interact strongly with nuclei.In fact, the force between separate nucleons can be effectively modeled by the ex-change of virtual mesons. We will be doing this through the use of effective fieldtheory (see Section 1.3). Effective field theories are low energy approximations toarbitrarily high energy physics, in that in the low energy, long wavelength region, ourprobes cannot resolve the internal structure of nucleons and we can therefore treatthe nucleons as point-like elementary particles and simply use structure functions tomodel the asymptotic effects of the nucleon substructure. To develop an effectivetheory, we first introduce a momentum cutoff, meaning that we are deciding to treat

the effects of physics occurring above the cutoff as local (i.e point-like). We thenadd local interactions to the Lagrangian which mimic the effects of the true shortdistance physics. Since a probe of wavelength is insensitive to details of structureat distances d we must renormalize or account for the effects of these small struc-tures without explicitly including them. Renormalization is used when the particlesare embedded in a background space (very short distance structure) and to calculatethe real physical properties of the particle from the measured properties one mustsubtract the effects of the background. For example, the mass of a particle travelinginside a material (renormalized mass) will be different than if it were measured out-side of the material (bare mass) because of (possibly unobservable) interactions with

the material. In field theory, all particles are traveling in a background space-timeand since there is no way to remove the particle from this background, we alwaysmeasure the renormalized mass, and the bare properties, which are the parameters ofthe theory, are unobservable. A complicated current source of size d that generatesradiation with wavelengths d is accurately modeled by a sum of point-like mul-tipole currents (E1, M1, etc.). It is simpler to treat the source as a sum of multipolesthan to deal with the true current directly, because usually only the first few multi-poles are needed for sufficient accuracy. The multipole expansion is a simple exampleof a renormalization analysis [10].

To calculate real physical observables from our effective theory, we need to calcu-late the scattering matrix. The scattering matrix is a measure of the probability fora given interaction to occur, and it allows one to extract the predictions of a theoryin order to test them experimentally. One of the methods used to find the scatteringmatrix (S-matrix) from the interaction Lagrangian describing a scattering process isthe Feynman path integral [10, 11].

In the path integral method, one writes down a generating functional in termsof the complete Lagrangian for a theory and uses functional differentiation to find

For example, we will treat the nucleon as a point particle in our effective theory, since at lowenergies the photon probe that we are using cannot see the quarks residing inside it.

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the Feynman rules of the theory. These Feynman rules are then put together tocorrespond to whatever scattering process (diagram) we are considering. The resultis proportional to the S-matrix, or equivalently the M-matrix (scattering amplitude),and contains all of the physics of the interaction. The M-matrix will be described ingreater detail in the next chapter. Once accustomed to how the method works, one

can almost write down the Feynman rules by inspection directly from the Lagrangian.

1.1.1 The S-matrix

In order to make the above comments more quantitative we will derive the generatingfunctional for photon-pion-nucleon interactions and describe how it can be used tofind the Feynman rules.

The generating functional is defined as the integral of all possible field configu-rations in space-time weighted by the exponential of the action. For our fields ofinterest, it is written as

Z[a, b, j, Jc] =

DNDNDA D exp

i

d4x

L0 + Naa+ bNb + j

A + Jcc + Lint

(1.1)

where we use lower case Latin indices, a, b and c as isospin indices for the nucleonfields N and N and the pion field respectively. Lower case Greek indices areLorentz indices ( is the Lorentz index for the photon field A). L0 is the sum ofall free particle Lagrangians and Lint is the interaction Lagrangian. The nucleonsource currents and as well as the nucleon fields are now Grassmann numbers

that obey anticommutation relations whereas the pion and photon fields and currentsare commuting.We now define the n-point function (or n-point Greens function) to be the vacuum

expectation value of the time-ordered product of fields at n space-time points x1,...,xnas follows

< 0|T((x1) (xn)) |0 >

1

i

J(xn)

1

i

J(x1)

Z[J]|J=0 (1.2)

where Z[J] is the generating functional (1.1) and the fields (xi) can be any of the , N , N or A fields. In particular for the N(, ) vertex we would write

< 0|TNa(x4)c(x3)Nb(x2)A(x1) |0 >=1

i

a(x4)

1

i

Jc(x3)

1

i

b(x2)

1

i

j(x1)

Z|J=j===0 (1.3)

where we have explicitly indicated the isospin indices. We see by the form of thegenerating functional (1.1) that an application of a particular functional derivativewill bring down a factor of the corresponding field. In this way we can construct anypolynomial in the fields by merely acting on the generating functional with functional

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derivatives. In particular, we can expand a given interaction Lagrangian as a polyno-mial in the fields and then re-write it in terms of the functional derivatives. We cantherefore re-write Z[J] as

Z =

1

Nexp

i

d

4

xLint(

a(x) ,

b(x) ,

j(x) ,

Jc(x))

Z0 (1.4)

where Z0 is the remaining terms in (1.1) after removing the interaction part. We havedivided by N = Z|J=j===0, which has the effect of cancelling all of the so-calledvacuum bubble diagrams which have no external lines and are hence unobservable.To use this expression one applies well known methods [12, 13] to reduce the free fieldgenerating functional Z0 (which is simply a product of Gaussians) into the followingform

Z0 = expi d4x d4y a(x)S

abF (x y)b(y) +

1

2j(x)D

F (x y)j(y)

+1

2Ja(x)

abF (x y)Jb(y)

(1.5)

where the Feynman propagators have been defined in Appendix A. Finally we formthe generator of connected graphs by writing Ziln(Z), which removes all of thediagrams that are disconnected.

In the path integral method, one uses this generating functional to find the prop-agator, the vertex functions and, as a result, the Feynman rules of the theory. Ratherthan going through the renormalization analysis that is involved in finding thesefunctions we will simply refer the interested reader to the many texts discussing it

[12, 13, 11].

1.2 Observables

We would like to use all of this analysis to find real physical observables and therefore,as mentioned above, we need to form the S-matrix. First we find the Greens functionfor the theory using (1.3) and then remove the external legs by multiplying by theinverse propagator for each leg. Finally, we multiply by the external free particlewave functions. All of this can be accomplished through the reduction formula forthe S-matrix as follows

Sf i =

dx1...dx4e

iqx2eipx3U

s

(p)Kx2

Dx3

(x1...x4)Dx4

Px1 U

s(p)eipx4eikx1

(1.6)

where (x1...x4) is the 4-point Greens function defined in (1.3),Kx,

Dx and

Px are the

Klein-Gordon, Dirac and Photon operators (inverse propagators), is the photon po-

larization vector and Us

(p) and Us(p) are the Dirac 4-spinors defined in Appendix A.In what follows, rather than proceeding with the manipulations involved in (1.1)

to find the Greens function, we will merely use the respective Feynman rules that

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have been derived from each interaction Lagrangian and form the Greens functiondirectly.

In calculating the S-matrix for a real process it is convenient to write it as aperturbation expansion in powers of the coupling constant for a given interactionLagrangian, and then evaluate only the first few terms in this expansion. It is assumed

that the terms in the series diminish in strength rapidly as the power of the couplingconstant increases. The expansion is written as

< f|S|i >= Sf i =< f|T(ei

d4xLint(x))|i > (1.7)

where the T indicates the time-ordered product (ensuring causality).When the vertex function has been found for a given process, the S-matrix [14]

can be written as

Sf i = f i + i(2)44 (Pf + q k Pi)Mf i (1.8)

The scattering cross-section for the process p1 + p2 f is given by

d(a1 + a2 f) = (p1 + p2 f)Ji

dNf (1.9)

where = (2)44 (p1 + p2

Pf) |Mf i|2 is the transition probability per unit time,dNf is the density of final states, and the flux factor Ji can be written in the centreof mass system (CM) as

Ji = 2p012p

02|

p1p01 p2

p02| = 4|p1p02 p2p01| = 4|p1|(p01 + p02)

= 4[(p1 p2)2 m21m22]1

2 .

(1.10)

Since Ji is expressed in terms of a Lorentz-invariant quantity, it is valid in any refer-ence frame. The cross section is then given by

d =(2)44 (p1 + p2

Pf)

4[(p1 p2)2 m21m22]1

2

|Mf i|2bosonsj=1d3pj

(2)32p0jfermionsl=1

d3pl Ml(2)3p0l

S (1.11)

where, in this case S = i1

mi!is the symmetry factor associated with mi identical

particles in the final state.Now that we have found the S-matrix for a given process we form the M-matrix,

the scattering amplitudes and the various physical observables. We will do this inChapter 2 for single pion photoproduction.

1.3 Pion Photoproduction from Nucleons

Photoproduction of mesons plays a significant role in the study of the hadrons andtherefore the nature of strong interactions. Familiarity with electrodynamics and

The production of mesons while using photons as a projectile.

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thus the electromagnetic interaction between hadrons results in a very good tool forthese investigations and for providing important information about the structure ofmatter over short distances. It is especially so due to the interaction coupling being small enough that we need only compute the lowest orders of a perturbationexpansion in it.

Pion photoproduction near threshold has been studied by many groups of boththeoretical [15, 16, 17, 18, 19, 20, 21] and experimental physicists [2, 3, 4, 5, 22, 23].The reason for this is relative ease with which both calculations based on theories, andexperiments to test those calculations, can be done. Hence extensive experimentaldata has been accumulated on photoproduction of charged and neutral pions fromnucleons and light nuclei in the energy region from threshold to about 500 MeV.

These experiments have shown that nucleons have finite size and a polarizability,which indicates that they do indeed have substructure. As well, photoproduction ofmesons has also led to the discovery of nucleon resonances like the and N isobarsand they measured the effects of heavier vector mesons as well.

For the reactions + N + N there are two particles in the final state sothat a measurement of the angle of scattering along with the energy of one of thefinal particles will determine the energy of the incoming gamma ray. Alternatively, aphoton tagger can be used to match each photon with the electron that produced it,allowing the photon energy to be determined from the energy of the correspondingelectron. In a typical experiment, detectors are set up around a target of protonsand neutrons, such as liquid hydrogen or deuterium. The detectors can be designedto measure the energy distribution of the charged pions or the recoil particle at afixed scattering angle, or they can measure the angular distribution at a fixed energy.Neutral particles can be detected via their decay products as in the case of photons

from the 0

2 decay.The radiation field is usually expanded into multipoles, or states of definite angularmomentum and parity. The photon has intrinsic spin 1, and hence the total angularmomentum j and the orbital angular momentum l of a multipole are related byj = l, l 1. The parity of the photon field ()l can be either even or odd and we cantherefore separate the photon field into two types, those that have parity ()j, calledelectric multipoles (denoted Ej ), and those that have parity ()j+1, called magneticmultipoles (denoted Mj). Assuming the initial nucleon is in an S1

2state (l = 0, s = 1

2)

we have final N states for j = 1, 2 as shown in Table 1.1.In the past few decades there has been theoretical development of model inde-

pendent predictions from the fundamental principles of physics that can be directlycompared to experiment.

In the 1950s, Kroll and Ruderman [24] were the first to derive model-independentpredictions in the threshold region (called low energy theorems (LETs)), by applyinggauge and Lorentz invariance to the reaction + N + N. The general formalism

We refer here to the incident photon energy in the laboratory frame.This column gives the usual notation of photopion physics, El and Ml, where the E and M

denote the incident photon type and l denotes the total angular momentum l 12 of the final statepion (l = 0, 1,...) and nucleon (s = 1

2).

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Table 1.1: Multipoles for j = 1 and j = 2Multipole j Parity Final State Photopion Notation

E1 1 odd S12

, D 32

E0+, E2M1 1 even P1

2, P3

2M1, M1+

E2 2 even P32 , F5

2 E1+, E3M2 2 odd D 3

2, D 5

2M2, M2+

for this process was developed by Chew, Goldberger, Nambu and Low [19],[20] (CGLNamplitudes).

In 1965 Fubini et al. [25] extended earlier predictions of LETs by including thehypothesis of partially conserved axial current (PCAC). In this way they succeededin describing the threshold amplitude in a power series in the ratio = m

Mup to

terms of order 2.Our Lagrangian, which will eventually include some of the nucleon resonances as

explicit degrees of freedom, is an effective Lagrangian, as mentioned above. It usesthe asymptotic fields (nucleons, pions, resonances) as the fundamental entities ratherthan a fundamental Lagrangian such as that of QCD (or QED) which describes thequark (electron) fields and gauge field interactions. Using such an effective Lagrangianprovides a much simpler method of modeling reality, especially in lower energy, longwavelength regions where the effect of the individual quarks is hidden inside thecomposite particles. The couplings and other parameters of the theory are assumedconstant over a small energy range near pion production threshold and are fixedphenomenologically. The resulting theory can then be used to make predictions.

We begin by developing the Lagrangians which describe the interactions betweenpions, nuclei and photons as well as the contributions due to the nucleon resonances and N, and vector meson ( and ) exchange currents.

In the next chapter we will use gauge invariant chiral Lagrangians to calculate theCGLN amplitudes (1.22), and the multipoles, in a standard way (see Appendix B).We will begin by examining the CGLN amplitudes for the nucleon Born terms, Vectormeson, Delta resonance exchange and Roper resonance exchange. We then compareour findings with experiments performed at SAL [4, 23] and Mainz [22].

The tree-level contributions to the process of photoproduction are shown in Fig-ure 1.1. In particular, the Born terms with nucleon and pion pole terms (singularity

for m 0) and the seagull or Kroll-Ruderman term as well as resonance contribu-tions in the s-channel (nucleon resonances N and ) and in the t-channel (heaviermesons, and ) are shown.

Of the tree-level diagrams, the only one-particle irreducible (1PI) graph is theKroll-Ruderman graph. 1PI graphs are those Feynmann diagrams that cannot be

The conventional use of the term LET refers to model independent low energy predictions. Wewarn the reader however that our full Lagrangian has the Kroll-Ruderman LET as a starting pointbut the treatment of the resonance fields in Chapters 3 and 4 is not strictly model-independent andit is therefore not a LET in this sense.

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Figure 1.1: Tree-level Feynman diagrams for + N + N

separated into two different diagrams by cutting a single line. To calculate the re-duciblegraphs, like the Born terms, we merely have to calculate two separate 3-pointfunctions instead of a 4-point function and put the results together by inserting apropagator, corresponding to the cut line, between the two vertex rules.

1.4 N(, ) Kinematics

In this section we work out the detailed kinematics of the + N + N reaction.The notation we use is as in Appendix A; in particular, we define the energy and

momenta of the participant fields as Ep = p

0

=

p

2

M2

where we use the symbolsPi and Pf for the 4-momenta of the initial and final nucleon fields respectively, q forthe produced pion field and k for the incoming photon field.

In the lab frame, we can set the 3-momentum of the initial nucleon to zero, sinceany motion that it does have should be negligible. Because it is a two body reaction,we can define a plane by the incoming photon momentum k, the scattered pionmomentum q and the recoil momentum of the nucleon Pf. The scattering angle is then defined as the angle subtending the incident photon and scattered pionmomentum.

We can write the conservation of 4-momenta in the following way

Pi + k = Pf + qPi + k W

(Pi + k)2 = W2

M2i + 2P0

i k0 2Pi k = W2M2i + 2Mik0 = W

2

since Pi = (Mi, 0) in the lab frame. We have (with Mi = Mf = M)

W =

M2 + 2Mk0 (1.12)

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Since W2 = s is an invariant Mandelstam variable, we can evaluate it in any frame wechoose, so we will choose the center of momentum frame. In the center of momentumframe the initial and final 3-momenta are related by Pi = k and Pf = q, wherethe asterisk denotes the center of momentum frame. So, we can write W2 as

s = (P0i + k0)2 (Pi + k)2= (P0i + k

0)2

= (P0f + q0)2.

This allows us to write P0i = W k0 and P0f = W q0. The first of these leads to

(P0i )2 = W2 + (k0)2 2W k0

|Pi |2 + M2 = W2 + |k|2 2W k02W k0 = W2 M2

which gives

k0 =W2 M2

2W(1.13)

for the photon center of momentum energy. The second relation leads to

(P0f )2 = W2 + (q0)2 2W q0

|Pf|2 + M2 = W2 + m2 + |q|2 2W q02W q0 = W2 + m2 M2

which gives

q0 = W2 + m2 M22W

(1.14)

for the pion center of momentum energy. Using these two expressions we can nowfind the center of momentum energy for the initial and final nucleons respectively as

P0i =W2 + M2

2W

P0f =W2 m2 + M2

2W.

(1.15)

The 3-momenta in the CM frame in terms of the invariant mass are found similarly

to be

|Pi | = |k| =W2 M2

2W

|Pf| = |q| =

[(WM)2 m2] [(W + M)2 m2]2W

.

(1.16)

The Mandelstam variable s W2 is usually called the invariant mass when written in this waysince s is a frame-independent Lorentz invariant.

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At pion photoproduction threshold, q = Pf = Pi = 0 we have

s = (Pi + k)2 = (Mi + k)

2 = M2i + 2Mik0

(Pf + q)2 = M2i + 2Mik

0

M2

f + m2

+ 2P0

q0

2Pf q = M2

i + 2Mik0

M2f + m2 + 2Mfm = M

2i + 2Mik

0

(M2f M2i ) + m2 + 2Mfm = 2Mik0

which gives k0 as

k0 =(M2f M2i ) + m2 + 2Mfm

2Mi. (1.17)

For Mi = Mf = M being the average nucleon mass, this gives the threshold photonlab energy as

k

0

thr =

m(2M + m)

2M . (1.18)The threshold numerical values are given in Table 1.2.

Table 1.2: Incident photon lab energy and invariant mass at pion threshold.Reaction Threshhold k0thr (MeV) Invariant mass Wthr (MeV)

+ p + + n 151.437 1079.14+ n + p 148.452 1077.84+ p 0 + p 144.685 1074.54+ n 0 + n 144.672 1073.25

+ N

+ N 149.943 1078.49

where we have used the masses tabulated in Appendix A.

1.5 Photoproduction Amplitudes

The M-matrix element Mf i is given as a linear combination of the independentLorentz invariants Mi

iM

f i = j

Aj Uf(Pf, sf)MjUi(p, si)

=

j

Aj (s,t,u)Mf i

j .(1.19)

This value corresponds to the charged pion reactions where we use the average nucleon mass inthe calculations.

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where the Ms are given by

M1 = 5 k M2 = 25 (Pi Pf k Pf Pi k)M3 = 5 ((Pi

Pf)

k

(Pi

Pf)

k

)

M4 = 5 ((Pi + Pf) k (Pi + Pf) k ) 2MM1

(1.20)

A nice derivation of the above is found in [15], which is begun by decomposing themost general Lorentz invariant pseudovector into a linear combination of eight basicpseudovectors. Current conservation and the transversality condition for the photonare then used to reduce them to six Ms, two of which are only applicable to elec-troproduction which, upon setting k2 = 0 for real photons, leaves us the above four(1.20).

The dynamics of the process are therefore contained in the four scalar amplitudesAi, which depend only on the coupling constants and the Mandelstam variables (in

CM frame)

s = (Pi + k)2

t = (k q)2u = (Pf q)2 .

(1.21)

The isospin decomposition of the invariant amplitudes is

Aj(s,t,u) = A(+)j

{, 3}2

+ A()j

[, 3]

2+ A

(0)j (1.22)

which are related to the isospin amplitudes by

A(+)j =

A(1)j + A(3)j

3

A()j =

A(1)j A(3)j

3.

(1.23)

The amplitudes for specific reactions can be expressed in terms of the isospinamplitudes as

A(p+n)j =

2 A

(0)j + A

()j

A(np)

j =2

A(0)j A()j

A

(p0p)j = A

(+)j + A

(0)j

A(n0n)j = A

(+)j A(0)j .

(1.24)

Once these amplitudes have been found, we can proceed by finding the multipoles,cross section and polarization observables. The formalism for doing this, using theinvariant amplitudes above, is given in Appendix B.

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Chapter 2

The Kroll Ruderman, PVBorn and Pion pole Terms

2.1 The Pseudovector Coupling Lagrangian

The pion is a Lorentz pseudoscalar, iso-vector field. It is grouped as a triplet inisotopic spin space as mentioned in Chapter 1 and in Appendix A (A.23). Its pseu-doscalar nature is known by the fact that it exhibits odd parity.

A Lagrangian for any interaction must be a true scalar, since it is related to theenergy of the field (which is a scalar quantity). Therefore any interaction with pionsmust have an even number of odd parity objects in it so that the overall parity of theLagrangian is even. The nucleon is a Dirac spinor, Lorentz scalar and a doublet iniso-spin space whose up and down components are the proton and neutron fields

respectively. To build a Lagrangian governing the interactions of pions and nuclei wemust have all of the Lorentz, Dirac spin and isospin indices contracted as well as evenoverall parity.

Interactions between photons and nuclei come about naturally when one requiresthe free Dirac Lagrangian to be invariant under U(1) symmetry. We will use thisprocess, which is brought about by minimal coupling, when we want to add photoninteractions to a particular Lagrangian.

For pions, the same technique is used, only this time we require the free nucleonLagrangian to be invariant under chiral SU(2) gauge transformations where the chi-ral nature of the transformations leads naturally to an interaction Lagrangian with

pions (called the sigma model) which has overall even parity. Using these constraintswe can devise two Lagrangians that describe nucleon-pion interactions, one involvesderivatives of pion fields and is hence called pseudovector coupling and the other,called pseudoscalar coupling, doesnt contain derivatives. We will use the pseudovec-tor Lagrangian given by

LP VN N =fm

N 5iN i, (2.1)

since it is the usual choice in the realm of chiral symmetry and can more easily beadapted to photon interactions via minimal substitution via a covariant derivative.

We realize that minimal coupling applies only to point-like Dirac particles and it is wellknown that nucleons are not pointlike, but contain substructure. For our purposesnear threshold the incident photon energy is small and the long photon wavelengthis unable to resolve quark substructure, therefore the point-like approximation is a

U(1) symmetry corresponds to phase transformations and therefore it is not surprising that aninteraction term with photons comes about, since the effect of an electromagnetic potential on acharged particle is a phase shift and hence for the Hamiltonian, which is essentially the energy, to beinvariant under these phase shifts we include interactions with the field that is adding or removingenergy, namely the photon.

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good one and we can model the substructure via form factors. We also note thatthe pseudovector Lagrangian gives a non-renormalizable theory since its index ofdivergence is 1 (due to the derivative coupling). Effective theories differ fromfundamental theories in that renormalization is not an issue, so we do not need toconcern ourselves with the non-renormalizability of our theory. When we want to

move to a region away from threshold we can move to a new effective theory (seeAppendix C).

2.2 The Kroll-Ruderman Term

Let us begin with the Kroll-Ruderman diagram which has no internal propagators.We form the interaction Lagrangian LNN and the 4-point function as defined inChapter 1 (1.3). The Lagrangian LP VN N contains the product where the isovectors and are defined in a Cartesian basis. In order to facilitate the physical pion fieldswe will express this in an isospin basis defined by

+ 12

(1 + i2)

12

(1 i2)0 3+ 1

2(1 + i2)

12

(1 i2)

(2.2)

where + and create + and fields respectively and the + and are theisospin raising and lowering operators for the nucleon fields. The isospin dot productthen becomes

= 11 + 22 + 33=

2 (+ + +) + 30.(2.3)

The index of divergence (which is equal to the negative of the dimension of the coupling constant)is given by I = 1

2(d2)B + 1

2(d1)Fd + D, where B, F and D are the number of external bosons,

fermions and the number of derivatives respectively. If I > 0 the theory is non-renormalizable andthe coupling constant has negative (mass) dimension. I = 0 gives a renormalizable theory with adimensionless coupling constant and I < 0 gives a super-renormalizable theory, the coupling constanthaving positive dimension. The superficial degree of divergence of a diagram is given in terms ofthe indices of divergences of each of its vertices (i) via = 4 B 3

2F +

i Ii (see for example

Reference [12]).

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Now we perform the minimal substitution iqA where q is the charge ofthe pion. Keeping only the terms that contain a photon field gives

ie

2 (+ +) A

=

ie1

2(2i12

2i21) A

= ie 12

b, 3

bA

=ie

2

3,

b

bA.

(2.4)

The NN Lagrangian then becomes

LNN = iefm

N 51

2

3,

b

N bA. (2.5)

To make manipulations simpler we will define

Cb = iefm5 1

2

3, b

. (2.6)

We notice that the Kroll-Ruderman diagram is a low energy diagram and a theorybuilt with this interaction is non-renormalizable in the usual sense, since the superfi-cial degree of divergence is equal to 1 in 4 dimensions. Therefore the coupling constanthas negative dimension, much like what one gets after integrating a heavy internalparticle out of a theory leaving a mass factor in the denominator (see Chapter 6).Divergences result in effective theories when we form a point interaction between twofermions and two bosons (which is nonrenormalizable) from the some renormalizableinteraction which has the same external statesfor example, a diagram with an in-

ternal fermion resonance and two vertices, each having index of divergence of zero,being approximated at low energy by a single vertex with the resonance integratedout. This leaves an index of divergence of 1 and adds a negative mass dimension tothe coupling. This does not pose a problem presently since we are using it as a modelat very low energies. Therefore, the vertex only appears at tree-level and renormal-ization is not an issue (the calculation of loop corrections does cause a problem andthis will be further discussed in Appendix C).

We would now like to use the above Lagrangian (2.5) to form the 4-point Greensfunction for the Kroll-Ruderman diagram (see Figure 2.1). The 4-point function isgiven by

(x1, x2, x3, x4) = (x1)

j(x2)

Jc(x3)

(x4)

Z[J,j,,]|J=j===0, (2.7)

where the nucleon, pion and photon currents are respectively , Jc and j and thegenerating functional Z[J,j,,] is defined as

Z[J,j,,] =exp

iLint(z)dzZ0[J,j,,]

iLint(z)dzZ0[J,j,,]|J=j===0

=1

Nexp

i

Lint(z)dz

Z0[J,j,,]

(2.8)

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Figure 2.1: The Kroll-Ruderman contribution to + N + N

and

Z0 = expi dxdy (x)S(x y)(y) +1

2Ja(x)

abF (x y)Jb(y)

+1

2j(x)D

(x y)j(y)

, (2.9)

with the propagators S(x y), (x y) and D(x y) derived from the 2-pointfunctions which are given in Appendix A. The interaction Lagrangian in (2.8) iswritten as an operator by replacing the fields by functional derivatives acting on Z0.

L =1

i

(z)

Cb

1

i

(z)

1

i

Jb(z)

1

i

j (z)

. (2.10)

Now we expand the interaction Lagrangian in the generating functional (2.8)keeping only the first order term to get the Kroll-Ruderman contribution.

Z(1) =iN

dz

(z)Cb

(z)

Jb(z)

j (z)Z0. (2.11)

We can perform the functional differentiations of Z0 by usingJ(x)J(z)

= (x z).This gives, for example,

j(z)Z0 = Z0

j(z) i2 dxdy j(x)D

(x y)j(y)= Z0

i2

dxdy

g(x z)D(x y)j(y)

+ j(x)D(x y)g(y z)

= Z0

i

dxr j(xr)D

(xr z)

.

(2.12)

16


28/154

Similarly for the other functional derivatives we get

Jb(z)Z0 = Z0

i

dxsJa(xs)

abF (xs z)

(z) Z0 = Z0idxtSF(xt z)(xt)

(z)Z0 = Z0

i

dxu(xu)SF(xu z)

(2.13)

where we have used the Grassmann nature of the fermion fields and currents whichgives a minus sign in (2.13). Equation (2.11) now becomes

Z(1) =iN

dz dxr dxs dxt dxu

(i(xu)SF(xu z)) Cb (iSF(xt z)(xt))

iJa(xs)

abF (xs

z) (

ij(xr)D(xr

z))Z0. (2.14)

Now using the formula for the 4-point function (2.7) we have

=iZ0

N

dz dxr dxs dxt dxu

(i(xu x4)SF(xu z)) Cb (iSF(xt z))

(xt x1)iac(xs x3)abF (xs z) ig(xr x2)D(xr z)

. (2.15)

Setting J = j = = = 0 sets Z0 = N, and then completing the xr, xs, xt and xuintegrations leaves

= i

dz [iSF(x4 z)CciSF(x1 z)iF(x3 z)iD(x2 z)] . (2.16)

We now use the reduction formula [13] to remove the external propagators, leavinga momentum conserving delta function. Also attaching the initial and final nucleonspinors and the photon polarization vector gives the M-matrix as

i(2)4(Pf + q Pi k)Mf i = i(2)4(Pf + q Pi k)U(Pf)CcU(Pi)= i(2)4(Pf + q Pi k)U(Pf) ief

m5

1

2[3, c]

U(Pi). (2.17)

The S-matrix element can be formed by adding certain factors for each externalparticle from the definitions of the fields (Section A.3), giving

Sf i = i(2)4(Pf + q Pi k)

MPfMPi

(2)124P0i P0

fk0q0

Mf i. (2.18)

Now that we have the M-matrix for the Kroll-Ruderman diagram we could continueto find the CGLN amplitudes and then the observables. Instead we will continue andfind the M-matrices corresponding to the other tree level diagrams and put everythingtogether at the end.

17


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2.3 The Born Terms

In this section we will calculate the Feynman rules corresponding to the two verticesshown in Figures 2.2 and 2.3. Once we have the Feynman rules then it is quitestraightforward to use them in forming the M-matrix for composite diagrams, as we

will soon show.

Figure 2.2: The N N vertex

Figure 2.3: The N N vertex

In addition to the pseudovector Lagrangian used in our derivation of the Kroll-Ruderman Lagrangian in the above Section (2.1), we now need a Lagrangian thatdescribes the interaction between pions and nucleons. In order to facilitate the quarksubstructure of the nucleon we include anomalous moments as follows

LN N = eNQN A e4M

N KNNF

N CN A + N CN F(2.19)

where Q =1+32

is the nucleon charge operator, =

i2{, }, F is the usual

electromagnetic field strength tensor and KN =

Ks+Kv32

is the anomalous magnetic

moment of the nucleon ( Ks + Kv = kp = 1.793 and Ks Kv = kn = 1.913 are theproton and neutron anomalous magnetic moments respectively). If we recognize e

2M

as the nuclear magneton we can see that this term represents the magnetic momentinteraction (due to internal quark currents) between the nucleons and the electromag-netic field. We can check the validity of this by noting that the magnetic moment

18


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interaction term is of the form given in Appendix A (A.19) where we have the Diracbilinear combination

= eQA e4M

KNF (2.20)

which can be reduced to two component form using the expressions given in Ap-

pendix A to give (keeping only the terms at leading order in1

M)

sM(Pf, Pi)s = seKN4M

kijk Fijs,

= seKN4M

kijk (iAj jAi)s,

= seKN4M

kijk (2iAj)s,

= seKN2M

(A)s,= s (

N

B) s,

(2.21)

from which we see that it is indeed a magnetic moment interaction with nucleonmagnetic moment given by N =

eKN2M

.The 3-point function is given by

N N =1

i3

(x1)

j(x2)

(x3)Z|j===0. (2.22)

For our diagram of interest we only need the linear term in the expansion of Z,leaving

Z =i

Ndz1

i

(z)C

1

i

(z)

1

i

j(z)

1i

(z)C

1

i

(z)

ik

1

i

j(z) ik1

i

j(z)

Z0 (2.23)

where we have used the Fourier transform of the photon field to make the replacementF i(kA kA).

We now insert this into the 3-point function giving

N N =1i2

dz

SF(x3 z)CSF(x1 z)D(x2 z)

SF(x3 z)CSF(x1 z) ikD(x2 z) ikD(x2 z) (2.24)and removing the g and g from the photon propagators gives

N N =

dz SF(x3 z)D(x2 z) [C C(ik) + C(ik)] SF(x1 z) (2.25)

and changing the dummy index in the last term gives

N N =

dz SF(x3 z)D(x2 z) [C (C C) (ik)] SF(x1 z), (2.26)

19


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with the antisymmetry of we have C = C and our 3 point function is then

N N =

dz SF(x3 z)D(x2 z) [C + 2C(ik)] SF(x1 z). (2.27)

Now removing the factors of iSF(x1z), iD(x2z) and iSF(x3z) as well as addingthe momentum conserving factor of (2)44(Pf Pi k) gives the vertex Feynmanrule

i(N N) = i(2)44(Pf Pi k) [C + 2iCk]

= i(2)44(Pf Pi k)eQ ie

2MKNk

.

(2.28)

Similarly, for the pseudovector pion-nucleon Lagrangian (2.1), we have the follow-ing vertex Feynman rule

i(N N) = i(2)44(Pf + q Pi)

ifm

q5c

. (2.29)

With the above two Feynman rules (2.28, 2.29) we can form the S- and U-channelmatrix elements. The S- and U-channel diagrams are shown in Figures 2.4 and 2.5below.

Figure 2.4: PVBorn S-channel diagram

Figure 2.5: PVBorn U-channel diagram

20


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The M-matrix element for the S-channel diagram is given by

i(2)4(Pi + k Pf q)Msf i = U(Pf)

d4r

(2)4i(N N)iSF(r)i(N N)

U(Pi)

(2.30)

which, upon insertion of the propagator and vertex rules becomes

i(2)4(Pi + k Pf q)M(s)f i = U(Pf)

d4r

(2)4

i(2)44(Pf + q r) if

mq5c

ir + MsM2

e

1 + 3

2

ie

2MKNk

i(2)4(r Pi k)U(Pi) (2.31)

and after performing the integration over the momentum of the internal line, usingone of the momentum conserving delta functions leaves

i

M(s)f i = U(Pf)

efm(sM

2

) q5(

Pi+

k + M) c 1 + 3

2

c

KN

2M

k U(Pi).

(2.32)Similarly for the U-channel we have

iM(u)f i = U(Pf)ef

m(uM2)

1 + 32

c KN2M

c k

(Pf k+M) q5U(Pi). (2.33)

2.4 The Pion Pole Term

To represent pion-photon interactions, which will arise to leading order in the T-channel pion pole diagram, we use the vector current derived from SU(2) invariance

of the sigma model Lagrangian.

L = eV3 A= e3ababA= e (+ +) A

(2.34)

Note that the minus sign between the two terms in (2.34) is necessary or the La-grangian would be equivalent to zero for real photons. To see this, integrate by partsand use the gauge condition k

= 0 for photons of 4-momentum k and polarizationvector . This interaction Lagrangian leads to the Feynman diagram in Figure 2.6.

The usual procedure gives the Feynman rule asi() = i(2)

44(q k q) [ie3ac(q + q)] . (2.35)The M-matrix element for the T-channel diagram shown in Figure 2.7 is found to be

i(2)4(Pi + k Pf q)M(t)f i = U(Pf)

d4r

(2)4i()iF(r)i(N N)

U(Pi)

= (2)4(Pi + k Pf q)U(Pf) iefm

3cbb2q ktm2

(q k)5U(Pi). (2.36)

21


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Figure 2.6: The vertex

The gauge condition for transverse photons k = 0 reduces the M-matrix to

iM(t)f i = U(Pf)2ief

m(t

m2)3cbbq (q k)5U(Pi). (2.37)

Figure 2.7: The pion pole, t-channel diagram

2.5 The CGLN Amplitudes

In this section we will show the formalism involved in combining the above M-matrix

elements for the Born S- (2.32) and U- (2.33) channels, the Kroll-Ruderman channel(2.17) and the Pion pole channel (2.37) to form the CGLN amplitudes. These ampli-tudes will then be used in the following section to find the various photoproductionobservables.

The CGLN amplitudes are defined from the M-matrix as follows:

iMf i = U(Pf)4

=1

A(s,t,u)M U(Pi). (2.38)

22


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where the M are the CGLN basis defined in (1.20) and A(s,t,u) are the CGLNamplitudes and are functions of the invariant Mandelstam variables:

s = (Pi + k)2 t = (q k)2 u = (Pf k)2

= 2Pi

k + M2 =

2q

k + m2 =

2Pf

k + M2 (2.39)

Beginning with the expressions for the M-matrices, we separate out the isospinparts by using the commutation relations of the Pauli matrices (A.7, A.8) and thenreduce the remaining factors that depend on the Dirac gamma matrices by using theircommutation relations and the Dirac equation (A.18). For example, terms like

q5(Pi+ k + M)

which is found in the S-channel electric part of the total M-matrix can be reduced to

5[sM2] + 4MPi 2M kby commuting the Pi to the right and using the Dirac equation. This, combined withthe corresponding term in the U-channel

(Pf k + M) q5 5

[uM2] 4M Pf + 2M k

leads to the S + U channel electric M-matrix as

iM(s+u)

f i = U(Pf)ef

m

c1 + 3

2 4M

(sM2)(uM2) M2+ 2M

1

sM2 +1

uM2

M1

i3cbb

uM2 .5(uM2) 4M5Pf + 2M 5 k

U(Pi) (2.40)

2.5.1 Born, Pion Pole and Kroll-Ruderman Amplitudes

Continuing in the above fashion with the matrix elements for the other channels givesthe following expressions for the amplitudes.

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A1 =efm

cKN

M 1

2[c, 3]

kvM

+2M

uM2

+ 2M c1 + 3

2 1

sM2 +1

uM2A2 =

efm

c

1 + 3

2

4M

(sM2)(uM2)

+1

2[c, 3]

4M

(sM2)(tM2)

A3 =ef

m

cKN

1

sM2 1

uM2

+1

2[c, 3]

2kvuM2

A4 =ef

m

cKN

1

s

M2

+1

u

M2

1

2[c, 3]

2kvu

M2

.

(2.41)

These expressions separated into the isospin basis

Aj = A(0)j c + A

(+)j

1

2{c, 3}+ A()j

1

2[c, 3] , (2.42)

give the amplitudes for the various reactions as follows

A(0)1 =

ef

m

ksM

+M

sM2 +M

uM2

A(0)2 =ef

m

2M

(sM2)(uM2)A

(0)3 =

ef

m

ks

sM2 ks

uM2

A(0)4 = ef

m

ks

sM2 +ks

uM2

(2.43)

A(+)1 =

ef

m

kvM

+M

sM2 +M

uM2

A(+)2 =

ef

m

2M

(sM2)(uM2)A

(+)3 =

ef

m

kv

sM2 kv

uM2

A(+)4 =

ef

m

kv

sM2 +kv

uM2

(2.44)

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A()1 =

ef

m

M

sM2 M

uM2

A()2 =

ef

m

2M

(sM2)(uM2) +4M

(uM2)(tm2)

A()

3 = ef

m kv

sM2 +kv

uM2

A()4 =

ef

m

kv

sM2 kv

uM2

.

(2.45)

The values used for the couplings and anomalous magnetic moments are given inTable 2.1.

Table 2.1: Couplings used for the Born and pion pole termsCoupling Numerical value

e2

41

137.036f

m7.134 GeV1

kn -1.91

kp 1.79Ks -0.06Kv 1.85

2.6 Results

In this section we will examine the various observables for each type of photopro-duction reaction. The expressions for the amplitudes can be reduced in the center ofmomentum (CM) frame as follows

U(Pf)4

=1

A(s,t,u)M U(Pi) =4W

MfFi (2.46)

where the i and f are the initial and final nucleon Pauli spinors respectively, W =

s = (Ei + k0) is the invariant mass, and with

F= iF1 + F2( q)

(k )

+ iF3( k) (q ) + iF4( q) (q ) , (2.47)

one can calculate the multipoles and observables. The formulae for doing so can befound in Appendix B.

We now graph the multipoles for each of the separate charge channels in pionphotoproduction. Certain combinations of the multipoles, called the p-waves (B.13),have been found useful in comparing the energy dependence of the multipoles with

The anomalous magnetic couplings are all given with units of nuclear magnetons.

25


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the experimental results. We will wait until Chapter 8 to graph the p-waves but wewill give the low energy expression for them below, along with a comparison withchiral perturbation theory.

2.6.1 p 0p

145.0 150.0 155.0 160.0 165.0 170.0

E(MeV)

-7.00

-6.75

-6.50

-6.25

-6.00

-5.75

-5.50

M1-

(10-3qk/m

+

3)

p(,0)p

Born M1-

Multipole

145.0 150.0 155.0 160.0 165.0 170.0

E(MeV)

0.030

0.035

0.040

0.045

E1+

(10-3qk/m

+

3)

p(,0)p

Born E1+

Multipole

145.0 150.0 155.0 160.0 165.0 170.0

E(MeV)

-2.50

-2.45

-2.40

-2.35

-2.30

E0+

(10-3/m

+)

p(,0)p

Born E0+ Multipole

145.0 150.0 155.0 160.0 165.0 170.0

E(MeV)

2.80

2.90

3.00

3.10

3.20

3.30

3.40

M1

+(10-3qk/m

+

3)

p(,0)p

Born M1+

Multipole

Figure 2.8: Multipoles for the Born terms in the reaction + p 0 + p. They aregiven from top left to bottom right as E0+, M1, M1+, and E1+

For the following reasons, neutral pion photoproduction from protons is perhapsthe most interesting reaction that has been recently studied. In the late 1980s exper-

imental groups in Mainz [26] and Saclay [3] showed a large discrepancy between theirrespective measured values for the E0+ multipole

and the result due to the tree levelBorn terms given here (the so-called low energy theorem). This discrepancy was dis-comforting, since the low energy theorem is based only on symmetry principles such asLorentz and gauge invariance, with the coupling constants set phenomenologically viadecay widths. The theory is therefore model independent, and a discrepancy would

The E0+ multipole is also called the electric dipole amplitude as well as the slope of the differ-ential cross section at threshold (see B.11).

26


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mean that there are large contributions being neglected. Firstly, these corrections aremodel dependent and secondly, they are highly virtual (far off-shell) and thereforeshouldnt be strong contributors at threshold. Fortunately, it was shown [27] thatthe breakdown of the LET was not as bad as was suspected, although a smaller butsignificant discrepancy still exists.

The currently accepted value for the electric dipole amplitude is E0+ = 1.33 0.08 103/m+ (we will henceforth suppress the natural units) from MAMI [2] andSAL [4], and this indeed differs significantly from the value derived in the presentchapter (see Figure 2.8) of E0+ = 2.458.

We will show in later chapters that this discrepancy begins to disappear when oneincludes contributions due to other interactions. An interesting consequence of thisproblem has been the opportunity to test the corrections due to chiral perturbationtheory (CHPT) [28, 29], which is another effective field theory of the standard model.A cursory introduction to CHPT is given in Appendix C. Their recent results, whichshow a close agreement with the data, will be discussed further in Chapter 8. For

now we will merely compare the expression for E0+ given by CHPT (including theircorrections) with the one we derive using the amplitudes given in Section 2.5.1 alongwith the formalism given in Appendix B. The threshold contribution to E0+ due tothe above Born terms is given by

Ethr0+ = egN8M

1 1

2(3 + kp) +O(2)

= 2.261 10

3

m+, (2.48)

where we have used the relation fmpi

= gN2M

to relate the coupling constant from our

expression to that of the CHPT expression [30] which is

Ethr0+ = egN8M

1 1

2(3 + kp)

M

4F

2 +O(2)

= 0.935

103

m+. (2.49)

We immediately notice that the correction to our result occurs at order and isdependent on the pion decay constant F = 92.4 MeV.

The p-wave multipoles are given in this LET at threshold (144.67 MeV) by

1

|q|P1,thr =egN

8M2

1 + kp +

1 kp

2

+O(2)

= 10.073 103

|q||k|m+3

,

1

|q|P2,thr =egN

8M21 kp + 2 [3 + kp] + O(2) = 9.826 103 |q||k|m+3 ,

(2.50)

whereas CHPT [31] gives corrections leading to

1

|q|P1,thr =egN

8M2

1 + kp +

1 kp

2+

g2N(10 3)48

+O(2)

= 10.32 103

|q||k|m+3

,

1

|q|P2,thr =egN

8M2

1 kp +

2

3 + kp g

2N

12

+O(2)

= 11.00 103 |q||k|

m+3.

(2.51)

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Comparisons of CHPT and experiment with the full theory including resonanceexchange will be given in Chapter 8, where we will discuss total and differential crosssections and the p-wave multipole combinations.

2.6.2 n 0n

145.00 150.00 155.00 160.00 165.00 170.00

E(MeV)

-5.00

-4.80

-4.60

-4.40

-4.20

-4.00

M1-

(10-3qk/m

+

3)

n(,0)n

Born M1- Multipole

145.0 150.0 155.0 160.0 165.0 170.0

E(MeV)

-0.00080

-0.00060

-0.00040

-0.00020

0.00000

E1+

(10-3qk/m

+

3)

n(,0)n

Born E1+

Multipole

145.0 150.0 155.0 160.0 165.0 170.0

E(MeV)

0.35

0.40

0.45

0.50

0.55

0.60

E0+

(10-3/m

+)

n(,0)n

Born E0+ Multipole

145.0 150.0 155.0 160.0 165.0 170.0

E(MeV)

1.90

2.00

2.10

2.20

2.30

2.40

M1

+(10-3qk/m

+

3)

n(,0)n

Born M1+

Multipole

Figure 2.9: Multipoles for the Born terms in the reaction + n 0 + n. They aregiven from top left to bottom right as E0+, M1, M1+, and E1+

Neutral pion photoproduction from the neutron is an important reaction, becauseit allows us to test the isospin symmetry of the strong interaction. The problem with

this reaction experimentally is that it is impossible to obtain free neutron targets.Therefore in order to study the photonuclear properties of the neutron we are

forced to consider charged pion production in the inverse reaction as discussed below,or we can use more complex targets such as the deuteron and then extract the protonamplitude by making some assumptions about the system.

The multipoles for this reaction due to the Born terms are shown in Figure 2.9.Comparing these curves with those of the previous section we can see a similar

energy dependence, but the numbers are smaller. This is due to the fact that only the

28


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terms involving the anomalous magnetic moments survive, due to the lack of neutroncharge (see Equations 1.24, 2.43 and 2.44).

Our expressions for the p-waves at threshold (144.69 MeV) are as follows

1

|q|P1,thr =egN

8M2kn +

2 kn +O(2

)

= 7.055 103

|q

||k

|m+3 ,1

|q|P2,thr =egN

8M2

kn

2kn +O(2)

= 7.055 103 |q||k|

m+3.

(2.52)

However, CHPT [31] gives corrections leading to

1

|q|P1,thr =egN

8M2

kn +

2

kn +

g2N(10 3)48

+O(2)

= 7.393 103

|q||k|m+3

,

1

|q

|

P2,thr =egN

8M2

kn

2

kn +

g2N12

+O(2)

= 8.360 103 |q||k|

m+3.

(2.53)

2.6.3 The Charged Pion Reactions n p and p +n

The charged pion reactions differ significantly in magnitude from the neutral pionreactions due to the Kroll-Ruderman term which doesnt contribute to the neutralpion reactions. The multipoles are given in Figures 2.10 and 2.11.

Experiments involving charged pions can be accomplished in two ways. Firstly, inthe case of a proton target, one can direct a photon beam and study the photopro-duced pions. Secondly, one can form a beam out of the charged pions, as is commonlydone in the so-called meson factory accelerators, and use it to produce photons inthe inverse process. This can be directly related, by symmetry considerations, to theprocess studied here.

29


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150.0 155.0 160.0 165.0 170.0

E(MeV)

4.0

4.5

5.0

5.5

6.0

M1-

(10-3qk/m

+

3)

p(,+)n

Born M1-

Multipole

150.0 155.0 160.0 165.0 170.0

E(MeV)

4.0

4.5

5.0

5.5

6.0

E1+

(10-3qk/m

+

3)

p(,+)n

Born E1+ Multipole

150.0 155.0 160.0 165.0 170.0

E(MeV)

24.0

25.0

26.0

27.0

28.0

E0+

(10-3/m

+)

p(,+)n

Born E0+

Multipole

150.0 155.0 160.0 165.0 170.0

E(MeV)

-9.0

-8.5

-8.0

-7.5

-7.0

M1+

(10-3qk/m

+

3)

p(,+)n

Born M1+

Multipole

Figure 2.10: Multipoles for the Born terms in the charged pion reactions. The topfour are production and the bottom four are + production. They are given fromtop left to bottom right as E0+, M1, M1+, and E1+.

30


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150.0 155.0 160.0 165.0 170.0

E(MeV)

-8.0

-7.5

-7.0

-6.5

-6.0

M1-

(10-3qk/m

+

3)

n(,-)p

Born M1-

Multipole

150.0 155.0 160.0 165.0 170.0

E(MeV)

-5.5

-5.0

-4.5

-4.0

-3.5

E1+

(10-3qk/m

+

3)

n(,-)p

Born E1+ Multipole

150.0 155.0 160.0 165.0 170.0

E(MeV)

-32.0

-31.5

-31.0

-30.5

-30.0

-29.5

-29.0

E0+

(10-3/m

+)

n(,-)p

Born E0+

Multipole

150.0 155.0 160.0 165.0 170.0

E(MeV)

8.0

8.5

9.0

9.5

10.0

M1+

(10-3qk/m

+

3)

n(,-)p

Born M1+

Multipole

Figure 2.11: Multipoles for the Born terms in the charged pion reactions. The topfour are production and the bottom four are + production. They are given fromtop left to bottom right as E0+, M1, M1+, and E1+.

31


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Chapter 3

The N(1440) Resonance

In this chapter we will calculate the corrections to the Born terms given in Chapter 2which are due to the excitation of the nucleon into a virtual resonance state. Theresonance is too massive to be produced as a real (on-shell) particle at near-thresholdenergies and it is only felt as an internal, very short lived, bound state of the pionand the nucleon. The excitation of a nucleon into a resonance can occur in variousspin states (see Table 1.1) and we will discuss two of them, namely the N(1440) (P11)resonance in the present chapter and the more complicated (1232) (P33) in the next.

3.1 The Lagrangian and Feynman Rules

The N(1440) resonance is a spin- 12

, isospin-12

field with a Lagrangian given by

LN R = fN Rm

N 5aRa + H.c. (3.1)

for the vertex shown in Figure 3.1, and

LN R = e2(M + MR)

R(kSR + kVR 3)N F

+ H.c. (3.2)

for the the vertex shown in Figure 3.2.

These Lagrangians are seen to be the same as LN N (2.1) and LN N (2.19) exceptfor the coupling constants and anomalous magnetic moment. Notice that the electriccoupling in (2.19) does not contribute to the resonance Lagrangian. This is due to thelack of gauge invariance. To see this we make the replacement k in the M-matrixelement as follows

iMf i = ieU(R)QU(Pi)= ieU(R)Q kU(Pi)= ieU(R)Q (R Pi) U(Pi)=

ieU(R)Q (MR

M) U(Pi)

= 0

(3.3)

where we have used 4-momentum conservation in the third line and the Dirac equationin the fourth line above. We can immediately see that the lack of gauge invarianceis due to the mass splitting between the resonance and the nucleon. Since this massdifference was not present in (2.19) the interaction remained gauge invariant.

The N(1440) resonance is also sometimes referred to as the Roper resonance.

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Figure 3.1: The N R vertex

Figure 3.2: The N R vertex

The propagator for the N(1440) resonance is the same Dirac propagator as forany spin-1

2particle and the Feynman rules are derived in the same manner as the

other channels. They are given by

i(RN) = i(2)44(Pf + q R)

ifN Rm

5cq

,

i(N R) = i(2)44(R Pi k) i

e(kSR + kVR 3)

2(M + MR)(g

k

gk) .(3.4)

3.1.1 Coupling Constants

The 3-point function which gave us the vertex rules can be used to find the decayamplitudes of the N(1440) as per the process N + N or N + N. Thesepartial decay widths can then be compared with experimental values and used to setthe numerical values of our coupling constants and anomalous magnetic moments.

Beginning with the decay into a photon and a nucleon, shown in Figure 3.2, we

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form the S-matrix for the process as

Sf i =

MMR

(2)92R0k0P0fU(Pf)(RN)

U(R) (3.5)

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