Study of Barrelet Zeros and Analysis of the B−1 Resonance Photoproduction
Abstract
As an essential first step towards the study of the π−1 hybrid meson, an analysis of the b−1 resonance photoproduction and decay is performed in this work. This thesis is divided into two main parts. In the first chapter, a Monte Carlo simulator is designed to generate events of a proton-pion elastic scattering. Using the simulated data and serving as an initial approach towards the b−1 → ωπ− decay study, a partial-wave analysis of the final state of the system is conducted, first for scalar particles and later introducing spin as a variable. The energy-dependent partial-wave amplitudes, each accompanied by a group of ambiguities, are analytically derived from observable magnitudes using the moment’s Barrelet Zeros. In the second chapter, the theoretical formalism of the γ+ p → b−1 + Δ++ photoproduction is explored. Two mechanisms for the SDMEs extraction are presented and the importance of the reference frame of study is emphasized.
Full text
𝑧 𝑥 𝑦 𝜸 𝒃𝟏 − 𝚫++ 𝝎 𝜽𝝎 𝝋𝝎 𝒑 𝝅− 𝝅𝟎 𝝅+ 𝝅− 𝑧𝐻 𝑥𝐻 𝑦𝐻 𝝋𝑯 𝜽𝑯 Master’sThesis Inter-University Master’sDegree in Nuclear Physics STUDY OF BARRELET ZEROS AND ANALYSIS OF THE b− 1RESONANCE PHOTOPRODUCTION Marta Campos Forn´ es Supervisor: Prof. Vincent Mathieu Institut de Ci` encies del Cosmos, Universitat de Barcelona (ICC-UB, Spain) Academic year 2023/2024
Abstract As an essential first step towards the study of the π− 1hybrid meson, an analysis of the b− 1resonance photoproduction and decay is performed in this work. This thesis is divided into two main parts. In the first chapter, a Monte Carlo simulator is designed to generate events of a proton-pion elastic scattering. Using the simulated data and serving as an initial approach towards the b− 1→ωπ− decay study, a partial-wave analysis of the final state of the system is conducted, first for scalar particles and later introducing spin as a variable. The energy-dependent partial-wave amplitudes, each accompanied by a group of ambiguities, are analytically derived from observable magnitudes using the moment’s Barrelet Zeros. In the second chapter, the theoretical formalism of the γ+p→ b− 1+ ∆++ photoproduction is explored. Two mechanisms for the SDMEs extraction are presented and the importance of the reference frame of study is emphasized.
Acknowledgements I would like to express my heartfelt gratitude to my supervisor, Professor Vincent Mathieu, for his guidance and assistance throughout every stage of this thesis. His willingness to help and his patience have made this work much more accessible to me. I would also like to extend my sincere thanks to the GlueX collaboration for sharing preliminary results with me, which has enabled me to properly frame the motivation for this work. Additionally, I am deeply appreciative of the IAEA for generously funding my Master’s study expenses through their Marie Sklodowska-Curie Fellowship Programme. Finally, but definitely not less important, words can not express how grateful I am to my family, my partner and my friends. They have been my support system throughout this journey and, without their understanding and encouragement, this endeavor would not have been possible.
Contents 1 Introduction and motivation 1 2 Partial-wave analysis 4 2.1 Introduction to the partial-wave analysis . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Introductoryconcepts................................ 4 2.2.1 Four-momentum .............................. 4 2.2.2 Massshell ................................. 5 2.2.3 Mandelstam variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.4 A system’s degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.5 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.6 Amplitudes and moments of a scattering . . . . . . . . . . . . . . . . . . 7 2.3 Datasimulation................................... 9 2.3.1 Eventgeneration .............................. 9 2.3.2 Event acceptance and rejection . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.3 Distributions of accepted events . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Partial-waveanalysis ................................ 13 2.4.1 General outline of the program . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.2 Moments normalization . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.3 BarreletZeros................................ 16 2.4.4 Variation of ℓmax in the SAID data analysis . . . . . . . . . . . . . . . . 18 2.4.5 Errorcomputation ............................. 19 2.5 Results and figures for the SAID model . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Spin inclusion in the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Formalism of the b− 1photoproduction 29 3.1 Theoreticalintroduction............................... 29 3.1.1 Crossingsymmetry............................. 29 3.1.2 Gottfried-Jackson and Helicity frames . . . . . . . . . . . . . . . . . . . 32 3.1.3 Wignerrotations .............................. 34 3.1.4 Lorentz transformations and SDMEs . . . . . . . . . . . . . . . . . . . . 36 3.2 Formalism of the photoproduction . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 Amplitude of the b− 1photoproduction . . . . . . . . . . . . . . . . . . . 39 3.2.2 Instensity of the b− 1photoproduction.................... 42 3.3 Model for the SDMEs calculation . . . . . . . . . . . . . . . . . . . . . . . . . 45 4 Conclusions and outlook 49 Aℓmax =2and ℓmax =1data analysis 53 A.1 BarreletZeros.................................... 53 A.2 Errorcomputation.................................. 54 B Basic model 56 B.1 Distributions of accepted events . . . . . . . . . . . . . . . . . . . . . . . . . . 57 B.2 ResultsandFigures................................. 58 C SAID model 63 D Introductory concepts 65 D.1 WignerD-functions................................. 65 D.2 Helicity ....................................... 65
E Moment expansion for the b− 1photoproduction 67 F SDMEs model 68 F.1 Kinematics and conventions for the spin functions . . . . . . . . . . . . . . . . . 68 F.2 ρ0matrices for the sand tchannels......................... 69 G Breit-Wigner function and change of variable 71
1 INTRODUCTION AND MOTIVATION 1 Introduction and motivation The investigation of the structure of hadrons has been the subject of extensive experimental and theoretical scientific research since the 1950s. Studies carried out in the 1970s and later have been conducted within the framework of quantum chromodynamics (QCD), the fundamental theory that describes the interactions of quarks and gluons. Traditionally, hadrons were classified in terms of their valence quarks, using the quark model. Accordingly, mesons were conceived as simple hadron states consisting of just a quark-antiquark (q¯q) pair. However, as research advanced, some exceptions appeared that did not fit into this definition. One of these exceptions were hybrid mesons, in which the q¯qpair is coupled to the excitation of the gluonic field [1]. In this case, the gluonic field directly affects the quantum numbers of the particle: the total angular momentum J, the parity Pand the charge conjugation C. Several hybrid mesons have spin-exotic quantum numbers, as their JPC values are not possible for conventional mesons. Taking into account that for natural parity states in the quark model: |L−S| ≤ J≤L+S,P=(−1)L+1, and C=(−1)L+S; some spin-exotic quantum numbers are JPC =0−−,0+−,1−+,2+−,3−+, .... Determining the spectrum of these hybrid states, which due to their short lifetimes are also referred to as hybrid meson resonances, highly contributes to the study of the role of gluons in QCD bound states, which makes this research a central interest in hadronic physics [2]. Lattice QCD (LQCD) predicts the existence of hybrid mesons in its principles, with the lighest spin-exotic nonet having JPC =1−+and the lighest state of the nonet being π− 1(with isospin I=−1) [3, 4]. According to recent experimental measurements, π− 1(1600) is the best candidate for the lighest hybrid meson [3]. Additionally, results from LQCD indicate that b− 1π0is the dominant decay mode of π− 1, with a branching fraction of 70% [1, 3]. This prediction is supported by a considerable amount of experimental reports of the π− 1(1600) hybrid meson decaying to b− 1π0 [5, 6, 7, 8]. Photoproduction is an experimental process where particles are produced as a result of the interaction of high energy photons with a proton target. This method is widely used, as the energy and momentum of photons can be controlled and it is able to produce the full hadron spectrum of any quantum numbers. Experimental data from the GlueX experiment 1, hosted by Jefferson Laboratory and focused on searching for exotic meson resonances through photoproduction, indicates that the reaction with the largest cross-section when producing π− 1as a result of photoproduction is γ+p→π− 1+ ∆++ [3]. Gathering all the information from the previous two paragraphs and considering charge conservation, it is easy to infer that the resulting particles in the photoproduction of π− 1will be b− 1+π0+∆++. However, b− 1(1235) is a light scalar meson which also constitutes a resonance, being b− 1→ω+π− its dominant decay mode [9]. Furthermore, ωdecays into 3 pions (ω→π++π0+π−) and ∆++(1232) is a baryon resonance, being ∆++ →π++pits dominant decay mode [9]. 1http://www.gluex.org/ 1
1 INTRODUCTION AND MOTIVATION All the presented resonances and their associated decays constitute the reaction chain represented in the following figure: 𝛾 𝑝 𝑏1 − Δ++ 𝜋− 𝜔 𝜋+ 𝜋− 𝜋0 𝜋+ 𝑝 𝜋1 − 𝜋0 Figure 1: Complete reaction chain resulting from the photoproduction of the π− 1hybrid meson. As it can be deduced from Figure 1, the complexity of this reaction chain deeply complicates the experimental retrieving of information from the detected particles. Before working on the π− 1→b− 1+πdata, we must ensure that we can properly identify the b− 1, which is not a trivial task whatsoever. That is why current research approaches the analysis of this reaction with a slightly simpler approximation. The reaction currently studied dismisses the decay of π− 1into b− 1+π0and assumes γ+p→ π− 1+ ∆++ →b− 1+ ∆++, with all the corresponding subsequent decays. Therefore, the current studied reaction chain is the following: 𝛾 𝑝 𝑏1 − Δ++ 𝜋− 𝜔 𝜋+ 𝜋− 𝜋0 𝜋+ 𝑝 Figure 2: Simplified reaction chain resulting from the photoproduction of a charged b1meson. The omission of π0from the initial photoproduction simplifies the number of daughter particles and allows us to focus on the detection of the b− 1meson in the ωπ−system. The first analysis of the b− 1meson in photoproduction can be found in [10]). The reaction chain presented in Figure 2 will be our subject of study during this Master’s Thesis, which will be divided into two main chapters. Our primary goal in this work will be to establish a base of resources which, in future work, will enable us to extract as much information as possible about the π− 1hybrid meson using experimental data from photoproduction. 2
1 INTRODUCTION AND MOTIVATION Given the large amount of resonances and consequent decays of this reaction chain, the partialwave analysis of a system based on its final state (daughter particles), will be incredibly useful as a tool to retrieve information from previous steps of the reaction chain. Specifically, we will be the most interested in the b− 1meson decay partial-wave analysis. Although ωand ∆++ are also unstable and decay into 3 pions and π+p, respectively, they are very narrow resonances that can be easily identified without the need of partial-wave analysis (Figure 3). As we will justify in section 3.2.1, the b− 1quantum numbers JP=1+correspond to the Sand Dwaves (L=0,2) of the ωπ− system, as the Pwave (L=1) has quantum numbers JP=1−. Since the ωis a vector meson with JP=1−and π−is a pseudoscalar meson with JP=0−, a partial-wave analysis will have to be performed in order to extract b− 1. 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 ) 2 ) (GeV/c 0 π - π + πM( 0 2 4 6 8 10 12 14 16 18 20 3 10× 2 Candidates / 2 MeV/c ) 0 π - π + πGlueX-I Data: M( 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 ) 2 ) (GeV/c + πM(p 0 2 4 6 8 10 12 14 3 10× 2 Candidates / 5 MeV/c ) + πGlueX-I Data: M(p 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 ) 2 ) (GeV/c - πωM( 0 500 1000 1500 2000 2500 3000 3500 2 Candidates / 4.5 MeV/c ) - πωGlueX-I Data: M( Figure 3: Preliminary (pre-published) data from the GlueX experiment. Both the left-hand graph, corresponding to a π−π0π+system, and the center one, corresponding to a π+psystem, show a very distinguishable peak, which clearly facilitates the identification of the ωand ∆++ resonances, respectively. When it comes to the right-hand graph, corresponding to an ωπ−system, the peak is not as clear. In this last case, some interferences coming from resonances with different quantum numbers than b− 1appear, which is why a partial-wave analysis of the system’s final state is needed. Source: courtesy of the GlueX collaboration. The first chapter of this work (2) will be focused on computational partial-wave analysis. We will use Monte Carlo simulations to recreate a simple 2 →2 elastic scattering reaction with spinless particles (section 2.3) and we will analyze its final state (section 2.4) through a method known as “Barrelet Zeros extraction” (subsection 2.4.3). The partial-wave analysis methodology that we are interested in is equivalent in this case and in the study of a resonance decay into 2 daughter particles. The extraction of partial-wave amplitudes is a process plagued with ambiguities, and working with a computationally simulated spinless simple case as the 2 →2 elastic scattering will help us illustrate this problematic and tackle it in a controlled environment. At the end of the chapter, we will also discuss the inclusion of spin as a variable in the presented model and comment on how it would affect the presented data analysis (section 2.6). In the second chapter of this work (3), we will go back to the general picture of the b− 1photopro- duction and we will explore and discuss the formalism of the reaction chain presented in Figure 2. Our main goal throughout the chapter will be to obtain the spin density matrix elements (SDMEs) of the reaction chain (section 3.3), starting by computing the general amplitude of the system (section 3.2.1) and its intensity (section 3.2.2). Finally, we will also explore a theoretical model that will allow us to extract the b− 1SMDEs that we are interested in. It is relevant to mention that all the presented algebraic expressions in this work will be computed considering natural units: c=1, ℏ=1. 3
2 PARTIAL-WAVE ANALYSIS 2 Partial-wave analysis 2.1 Introduction to the partial-wave analysis As presented in this work’s introduction, the partial-wave analysis of the ωπ−system is a crucial step to a proper understanding of the b− 1photoproduction. We will dedicate this chapter to the study of the partial-wave analysis of a simple case: the elastic scattering between a pion and a nucleon (proton). This will allow us to make an initial approach to our resonance decay of interest: b− 1→ωπ−, as we will focus on the final state of the scattered system and recover information about the wave function. The analysis of experimental data in the field of particle physics has become an increasingly challenging task. In particle scattering experiments, the study of the final state of the system, done through partial-wave analysis, results in multiple solutions for the partial-wave amplitudes, ambiguity in the general phase of the scattering amplitude, and the need to adjust some analysis parameters depending on the considered energy range. Describing the scattering of 2 particles using a known realistic theoretical model (SAID) [11, 12] and simulating events from it can be immensely beneficial before dealing with real experimental data. Having a realistic model that describes our experiment will allow us to compare the results obtained from the data analysis to the theoretical values, and thus verify their validity and ensure the proper functioning of the analysis program. Before delving into this realistic model, we will start with a simpler version for training purposes. Our main goal throughout this chapter is to construct an optimal tool for analyzing data resulting from a scattering event. In order to fulfill this objective, we will first create a controlled environment and build a Monte Carlo event simulator, from which we will be able to recreate experimental particle scattering events computationally. Afterward, we will work on developing an analysis program that examines the final state of the system by decomposing the scattering amplitude into partial-wave amplitudes aL(L∈ {0, ..., ℓmax}). With the purpose of extracting the energy-dependent values of the partial-wave amplitudes, we will compute the moments HL(L∈ {0, ..., 2ℓmax}) associated with the simulated differential cross-section and, using the obtained values, we will apply the Barrelet Zeros method. Finally, using the calculated Barrelet Zeros, we will recover the partialwave amplitudes. We will develop a version of this program in which we will not consider the spin of the nucleon and focus solely on the angular distribution and energy of the system. Subsequently, we will analyze the inclusion of spin as a study factor in the final state of the system. In this case, we will observe how both the general phase ambiguity of the scattering amplitude and the multiplicity of solutions for the partial-wave amplitudes are doubled. Before digging into the development of the described simulation and analysis programs, we will present a series of introductory concepts that will be highly useful in the central sections of this work. 2.2 Introductory concepts 2.2.1 Four-momentum For a given particle with energy Eand momentum pwe define the following notation and metric for its covariant four-momentum: Pµ=E,px,py,pz=E, p(1) 4
2.3 Data simulation 2 PARTIAL-WAVE ANALYSIS =v ts2+2s m2 π+m2 π2−2s m2 p−2m2 pm2 π+m2 p2 4s−m2 π=(19d) =qs2+m2 π2+m2 p2−2s m2 π−2s m2 p−2m2 pm2 π 2√s=(19e) = λ1/2s,m2 p,m2 π 2√s(19f) Where we have used the definition: λ(a,b,c)=a2+b2+c2−2(ab +bc +ca). After performing the analogous calculation for pf, it is easy to conclude that pi=pf. With the main goal of validating the correct operation of our Monte Carlo simulation, we will apply a series of tests which will ensure the correct calculation of the system’s four-momenta. Let us start by computing the Mandelstam variables using the definitions (3a), (3b) and (3c). The first test we will perform is calculating susing expression (3a) and check if it matches the value of E2 cm, where Ecm is the value computed using the calculated Eπand Ep. The second and last test we will perform consists on the comparison of the cos θrandomly generated value to the one computed with the Mandelstam variables using the following expressions: cos θ= s(t−u)+m2 π−m2 p2 λs,m2 p,m2 π=1+2st λ(s,m2 p,m2 π)(20) This last test concludes the event generation part of our simulation. The next step in the process consists in defining a clear acceptance and rejection criterion for the randomly generated events. 2.3.2 Event acceptance and rejection During the generation of our sample data, we will use two different models and, consequently, two different acceptance-rejection functions. We will call the first and simplest model the “basic model”. The use of this criterion will be a first approximation to the proton-pion scattering simulation, the functions describing our system’s variables are not very complex and, therefore, it will be a great resource to ensure the optimal functioning of our Monte Carlo simulator. We will refer to the second model as the “SAID model” [11, 12]. This acceptance-rejection function is a result of real experimental data modeling and has a much more complicated algebraic expression. Since our focus during this first chapter will be on the study of the most realistic model, the SAID model, both the description and the results obtained from the basic model simulation will be presented in Appendix B. Evidently, the data obtained from each model will be analyzed separately. As it will be presented in section 2.4, our data analysis program will not depend on the distribution that rules our variables of study (sand cos θ), which will allow us to design and work with a single analysis program for both samples of data. Let us consider the following expressions for the system’s differential cross-section and scattering amplitude, respectively, which were presented in subsection 2.2.6 and will be further discussed and analyzed in subection 2.4.1: dσ(s,z) dΩ=1 16π2s|A(s,z)|2(21a) 11
2.3 Data simulation 2 PARTIAL-WAVE ANALYSIS A(s,z)= ℓmax X ℓ=0 (2ℓ+1)aℓ(s)Pℓ(z) (21b) Taking into account that during the event generation ℓmax =3 for both models and substituting in expressions (21a) and (21b), we obtain the following acceptance criterion: dσ(s,z) dΩ=|a0(s)P0(z)+3a1(s)P1(z)+5a2(s)P2(z)+7a3(s)P3(z)|2 16π2s(22) Having set this basis, we can move on to analyzing the SAID model and representing the real and imaginary parts of its partial-wave amplitudes. Given the complexity of the distributions that describe the partial-wave amplitudes, we will not provide the algebraic expressions for them. As a matter of fact, during the Monte Carlo simulation, we do not employ the algebraic expressions either. Instead, we download abundant numerical partial-wave amplitude data from [12] in terms of s. The model is constituted by a set of partial waves for discrete values of the energy: for a given set of energy values {si}i∈N, the values Re aℓ(si),Im aℓ(si)∀ℓ∈ {0, ..., ℓmax}are provided. For the values of swhich are not explicitly included in the data, we interpolate the theoretical partial-wave amplitude values using a straight line between the two closest values of sincluded in the downloaded data. Representing the downloaded data of the real and imaginary parts of each partial-wave amplitude in terms of the squared center-of-mass energy (s=E2 cm) we obtain the following figure: Figure 5: Real and imaginary parts of the partial-wave amplitudes corresponding to the SAID model, with ℓmax =3. 2.3.3 Distributions of accepted events In order to get a general image of the distribution of the variables sand cos θ, we represent the values of these magnitudes for all the accepted events in a histogram form. 12
2.4 Partial-wave analysis 2 PARTIAL-WAVE ANALYSIS We can easily check whether the corresponding theoretical model correctly predicts the obtained histograms. We will represent the theoretical values of each model employing the functions σ(s)=R1 −1 dσ(s,z) dΩdΩand Rsmax smin dσ dΩ(s,z)ds=g(z)over the sand the cos θhistograms, respectively, built using the simulated data. Having performed this comparison between the theoretical and the simulated data, we obtain the following figures for the SAID model: 12345 s = E 2( GeV 2) 0.0 0.2 0.4 0.6 0.8 1.0 Nº of events 1e7 SAID MC simulation (a) Distribution of s. 1.0 0.5 0.0 0.5 1.0 z = cos( ) 0.00 0.25 0.50 0.75 1.00 1.25 Nº of events 1e7 SAID MC simulation (b) Distribution of cos θ. Figure 6: Histograms of our variables of study for the accepted events of the simulation using the SAID model. The dark blue line represents the theoretical data from the model. These distributions have been obtained using a simulation of 108events. The equivalent figures for the basic model can be found in Appendix B. 2.4 Partial-wave analysis Having presented a step-by-step on how we generate the computational data that recreates an elastic proton-pion scattering, in this next section we will thoroughly describe the development of an analysis program for this data. This program will allow us to perform a partial-wave analysis of the scattering simply by studying the final state of the system. Therefore, our goal throughout the program’s development will be to derive the energy-dependent partial-wave amplitudes analytically. We will achieve this through a method known as the Barrelet Zeros, which we will focus on in subsection 2.4.3. In order to carry out this study, we will employ the information contained by the distributions of our 2 variables: s, which represents the squared total energy of the system in the center-of-mass frame, and cos θ, which is the scattering angle. The concepts of scattering amplitude and moments introduced in section 2.2.6 will be essential to comprehend this data analysis. 2.4.1 General outline of the program First, we must properly understand what is our starting point. What information do we possess and how can we best utilize it? As mentioned in this section’s introduction, we will employ the data from the sand z=cos θvariables of the accepted events, as this would typically be the information available when working with real experimental data. Starting from the minimum and maximum values of s, we divide the full variable’s range into nevenly spaced intervals. 13
2.4 Partial-wave analysis 2 PARTIAL-WAVE ANALYSIS Henceforth, we will assign to each of these energy intervals their midpoint value. Let [smin,s1) be our first energy interval. Then, we will assume all the events contained in this energy interval to have the value s=smin+s1 2. We impose this energy division with the objective of defining the differential cross-section of the system for a few fixed energy values, which will be the interval’s midpoints. Therefore, we will have to take into consideration an energy uncertainty of an interval’s midlength in every result that we obtain, even if not represented using horizontal error bars. That is to say, for a given partial-wave amplitude result ai(s), we will consider an energy uncertainty of δs=s1−smin 2. As presented in section 2.2.6, we define the moments as the coefficients necessary to express the system’s differential cross-section as a Fourier-Legendre series: dσ(s,z) dΩ= 2ℓmax X L=0 (2L+1)HL(s)PL(z) (23) Where PL(z) are the angle-dependent Legendre polynomials. As mentioned in previous sections, we have chosen ℓmax =3 for both simulation models. From the definition (23), we can extract the inverse relation: HL(s)=1 2Z1 −1 dσ(s,z) dzPL(z)dz∀L∈ {0, ..., 2ℓmax}(24) This last expression (equation (24)), allows us to calculate the moments for each energy interval. In equation (24), we are able to calculate the desired integral by defining a histogram of z=cos θ for each energy interval. Then, for a fixed svalue and, consequently, for a defined zhistogram, we equate dσ(s,z) dzto the height of each zbin, dzto the width of each zbin, and PL(z) to the Legendre polynomial of degree Lwith zas the mid-value of the corresponding zbin. Technically, by employing this technique we are not calculating the moments, as the magnitudes we compute using the described method depend on the number of accepted events. We will call these calculated magnitudes “simulated moments” ˆ HL(s) and in order to obtain the real moments HL(s), we will have to apply a normalization factor. We will further explore this normalization in section 2.4.2. Before moving on, it is relevant to note that this normalization will provide the correct dimension to our normalized moments HL(s), as the simulated moments ˆ HL(s) will have “events” as units and, consequently, they will be dimensionless. It is also essential to take into account that, unless mentioned otherwise, the moments that appear in all the expressions of this work are the normalized moments HL(s). Once we obtain the results for the 2ℓmax +1 energy-dependent moments HL(s), we are interested in comparing them to the theoretical values, as this allows us to ensure the correct functioning of our analysis program up to this point. Let us compute the relation that will allow us to calculate these theoretical values directly from the partial-wave amplitudes used in the corresponding simulation model. In order to do so, we will need the following property of the Legendre polynomials: Z1 −1 Pℓ(z)Pℓ′(z)PL(z)dz=2 2L+1|CL0 ℓ0;ℓ′0|2(25) Where CL0 ℓ0;ℓ′0are Clebsch-Gordan coefficients. Substituting (21a), (21b), and (25) in (24) we obtain the following expression for the moments in terms of the partial-wave amplitudes: HL(s)=1 16π2s ℓmax X ℓ,ℓ′=0|CL0 ℓ0;ℓ′0|2(2ℓ+1)(2ℓ′+1) 2L+1aℓ(s)a∗ ℓ′(s) (26) 14
2.4 Partial-wave analysis 2 PARTIAL-WAVE ANALYSIS The moment results (calculated using equation (24)) will be presented accompanied by their theoretical values (equation (26)) and by their corresponding uncertainties. We calculate the uncertainties δHLof the normalized moments HLusing the simulated moments ˆ HL. We approximate δˆ HL=pˆ HL, as we consider the moments to be magnitudes ruled by the Poisson distribution. The standard deviation for this type of distribution is √λ, where λis the mean value. In order to obtain δHL, we must simply normalize δˆ HL, which restores the correct units into the magnitude. As mentioned in a previous paragraph, we will further study this normalization in section 2.4.2. The last steps of the program are based on the extraction of roots of the polynomial P(2ℓ+1)aℓ(s)Pℓ(z)= 0, where zis the variable and sis fixed. These roots are known as the Barrelet Zeros and we consider them extremely valuable, because they allow us to establish a direct and simple analytic relation to the partial-wave amplitudes: A(s,z)= ℓmax X ℓ=0 (2ℓ+1)aℓ(s)Pℓ(z)=N(s) ℓmax Y i=1 (z−zi(s))(27) Where zi(s) are the Barrelet Zeros and N(s)∝aℓmax (s). We will further explore the analytic relation between the partial-wave amplitudes and the Barrelet Zeros in subsection 2.4.3. Equivalently to the moments calculation, the Barrelet Zeros are computed for each energy interval individually, which allows us to obtain partial-wave amplitude solutions for each energy value. The key expression that will enable us to extract the Barrelet Zeros is the following: dσ(s,z) dΩ= 2ℓmax X L=0 (2L+1)HL(s)PL(z)=1 16π2s|A(s,z)|2=|N(s)|2 16π2s ℓmax Y i=1 [z−zi(s)]·hz−z∗ i(s)i(28) Since we were able to obtain the values for HL(s) from equation (24), imposing the equality P(2L+1)HL(s)PL(z)=0 will allow us to extract the ℓmax pairs of complex conjugate roots {zi(sk),z∗ i(sk)}(with i∈ {1, . . . , ℓmax}) for each energy bin k. For each energy bin, this family of roots contains both the Barrelet Zeros and their complex conjugates, which we can not distinguish and, consequently, an ambiguity in the partial-wave amplitudes is generated. We will further discuss these ambiguities in subsection 2.4.3. Finally, we compute the values of the energy-dependent partial-wave amplitudes using equation (27) and the extracted Barrelet Zeros. We will present our results with a comparison to the theoretical values of the partial-wave amplitudes used during the corresponding simulation. In this representation, we will also include our solution’s error bars, whose calculation is explored in subsection 2.4.5. This concludes the general outline of our program. In the next subsections, we will further discuss some specifics of the program’s operations. 2.4.2 Moments normalization In this subsection, we will explore the normalization necessary to transform the simulated moments ˆ HL(s) into the real moments HL(s). Given that the theoretical moments are not dimensionless ([Hi]=L2and in units mb) but the partial-wave amplitudes are, it is easy to infer that there is also a hidden (ℏc)2factor in expression (26). As previously mentioned, the program calculates the simulated moments ˆ HLusing the number of accepted events at a specific energy range instead of the normalized differential cross-section. 15
2.4 Partial-wave analysis 2 PARTIAL-WAVE ANALYSIS The obtained values for the moments are proportional to the total number of accepted events and, consequently, the more events we generate, the bigger the moments will be. Therefore, they need to be normalized in order to compare them to our reference theoretical values HL. Realistically, in an experiment, we would need to know its incoming flux to deduce its normalization. However, in our case, we use the model employed to generate the data and apply a simple rate between the theoretical and the simulated differential cross-section: HL(s)=(dσ/dΩ)theo (dσ/dΩ)sim ·ˆ HL(s) (29) Where sim and theo indicate the simulated and theoretical differential cross-sections; respectively. As presented in a section 2.4.1, while the simulated moments ˆ HLhave “events” as a unit and are, consequently, dimensionless, the normalized moments have units mb. Thus, similarly to expression (24), there is a factor (ℏc)2missing in equation (29) due to natural units c=1 and ℏ=1. We calculate both the theoretical and the simulated differential cross-sections employing the following expression: dσ dΩ=Zsmax smin ∞ X L=0 (2L+1)HL(s)PL(z) ds(30) Since we do not use the same number of energy intervals for the theoretical and the simulated values, we integrate with respect to s, where dsis the width of the corresponding energy interval. As previously mentioned, during the results presentation section (section 2.5), we will compare the normalized moments Hito the theoretical ones (directly computed from the model partial-wave amplitudes using relation (26)). 2.4.3 Barrelet Zeros In this subsection, we will develop the analytic expressions needed to extract the Barrelet Zeros for ℓmax =3. Let us start by substituting the ℓmax value in (27). We obtain: A(z)= ℓmax X l=0 (2l+1)aℓPℓ(z)=a0+3a1z+5a2 3z2−1 2+7a3 5z3−3z 2= =35 2a3[(z−z1) (z−z2) (z−z3)]= =35 2a3hz3−(z1+z2+z3)z2+(z1z2+z1z3+z2z3)z−z1z2z3i (31) Observing (31), we can easily establish the relations: −35 2a3(z1+z2+z3)=15 2a2⇒a2=−7 3(z1+z2+z3)a3 35 2a3(z1z2+z1z3+z2z3)=3a1−21 2a3⇒a1=7 6a3[5(z1z2+z1z3+z2z3)+3] −35 2a3(z1z2z3)=a0−5 2a2⇒a0=−35 2a3z1z2z3+z1+z2+z3 3 (32) Following the procedure presented in subsection 2.4.1, let us perform the corresponding moment expansion to the system’s differential cross-section as an intermediate step to impose equality (28): 16
2.4 Partial-wave analysis 2 PARTIAL-WAVE ANALYSIS dσ(s,z) dz= 2ℓmax X L=0 (2L+1)HLPL(z)= =H0+3H1z+5H2 3z2−1 2+7H3 5z3−3z 2+9H4 35z4−30z2+3 8 +11H5 63z5−70z3+15z 8+13H6 231z6−315z4+105z2−5 16 (33) Equation (33) contains a polynomial whose coefficients are known to us, as we were able to compute the moments of the system (H0(s),H1(s),H2(s),H3(s),H4(s),H5(s),H6(s)) using the simulated differential cross-section (equation (24)). When we equate this sixth degree polynomial to 0, we are able to extract 6 roots: z1(s), z∗ 1(s), z2(s), z∗ 2(s), z3(s), z∗ 3(s); being zi(s)∈Cand z∗ i(s) their complex conjugates, with i∈ {1,2,3}. Since the coefficients HL(s) are energy-dependent, the Barrelet Zeros {zi(s),z∗ i(s)}i∈{1,...,ℓmax}will also be specific to each energy interval. When we extract the roots, we can not distinguish which of the values are the original Barrelet Zeros and which ones are their complex conjugates. Therefore, all the possible different assignations generate an ambiguity in the computation of aℓ(s),∀ℓ∈ {0, . . . , ℓmax}. Specifically, the number of solutions obtained for a certain partial-wave amplitude in a fixed energy interval is 2ℓmax which, in our case, 2ℓmax =8. Substituting (31) and (33) in (28) allows us to extract the following relation between the highestorder moment and the highest-order partial-wave amplitude: 13 ·231 16 H6=35 22|a3|2 16π2s⇒|a3(s)|=r429 700H6(s)·16π2s(34) Given that dσ(s,z) dΩ∝|A(s,z)|2(equation (21a)), and considering that the information we obtain experimentally as an observable is dσ(s,z) dΩ, the global phase of A(s,z)will be unknown. In order to solve for the value of a3(s) in equation (34), a global phase will have to be chosen. Since we are only interested in the relative phase between different partial-wave amplitudes, the global phase election can be subject to the most convenient value for the resolution of the presented equations. This accepted arbitrariness in the phase choice is easily justified considering eiϕ 2=1, because then: dσ(s,z) dΩ∝|A(s,z)|2=eiϕA(s,z) 2. As a result of this, we can choose a certain global phase ϕ(s) for each energy value ssuch that for a specific selected value of ℓ∈ {0, . . . , ℓmax}, the corresponding partial-wave amplitude is real and positive instead of complex (aℓ(s)∈R+). In our study, the logical choice to make calculations easier is aℓmax (s)=a3(s)∈R+. Therefore, a3(s)=q429 700 H6(s)·16π2s. Let us analyze the effect of this general phase change on the partial-wave phases. Let ϕ3(s) the theoretical phase of the a3(s) partial-wave amplitude in a fixed energy interval s. We have to apply a phase change of e−iϕ3to A(s,z) if we want a3(s)∈R+: e−iϕ3A(s,z)= a0(s)+3a1(s)z+5a2(s)3z2−1 2+7a3(s)5z3−3z 2!e−iϕ3= =|a0(s)|ei(ϕ0−ϕ3)+3|a1(s)|ei(ϕ1−ϕ3)z+5|a2(s)|ei(ϕ2−ϕ3)3z2−1 2+7|a3(s)|5z3−3z 2 (35) 17
2.4 Partial-wave analysis 2 PARTIAL-WAVE ANALYSIS As expected, we have obtained a3(s)∈R+and a0(s),a1(s),a2(s)∈C, but the differential crosssection is left intact: dσ(s,z) dΩ=1 16π2s|A(s,z)|2=1 16π2se−iϕ3A(s,z) 2. After all of these considerations have helped us obtain a3(s)∀s∈[smin,smax], going back to equations (32), we can obtain the solutions for a0(s),a1(s),a2(s)∀s∈[smin,smax]. Finally, given that the usual consensus is to choose a general phase for the system such that a0∈R+, before representing the obtained results for aℓ(s), ∀ℓ, s, we apply a general phase shift for each energy interval such that a0(s)∈R+. As a general remark, it is relevant to note that, for each energy s, we are able to obtain the values for aℓ(s)∀ℓ∈ {0, ..., ℓmax}using the moments HL(s),L∈ {0, ...2ℓmax}due to the system’s global phase ambiguity. Let us elaborate on this idea. By definition, there are 2ℓmax +1 real moments and ℓmax +1 complex partial-wave amplitudes. We would typically be able to express the ℓmax +1 complex partial-wave amplitudes as 2ℓmax +2 real numbers (1 for each real part and 1 for each imaginary part). However, given the system’s global phase ambiguity, we can choose ℓ∈ {0, ..., ℓmax}such that aℓ(s)∈R+, which allows us to express the ℓmax +1 complex partial-wave amplitudes in terms of 2ℓmax +1 real numbers. Thus, we have the same number of unknowns (2ℓmax +1 real numbers describing the partial-wave amplitudes) as moments. This equality makes the solving for the aℓ(s) values dimensionally possible. 2.4.4 Variation of ℓmax in the SAID data analysis As presented in section 2.2.6, the theoretical equations that allow us to perform the partial-wave analysis are expanded for an infinite number of orbital angular momentum components. Realistically, when analyzing data, a maximum orbital angular momentum (ℓmax) must be chosen. We will consider components of higher orbital angular momentum than the maximum to be negligible in magnitude. In an experimental context, the adequate ℓmax that needs to be chosen for the data analysis is unknown. Therefore, a previous study of the system must be performed and ℓmax must be guessed from the highest non-zero moment HL(s) in each specific energy. Throughout the development of this program, we noticed that we had a similar problem when analyzing the data resulting from the SAID model simulation. We realized that adapting the ℓmax for the data analysis in each energy bin is beneficial for higher accuracy results. Thus, considering the adequate value for ℓmax in A(s,z)=Pℓmax ℓ=0(2ℓ+1)aℓ(s)Pℓ(z) and dσ(s,z) dz=P2ℓmax L=0(2L+1)HL(s)PL(z) for each energy value can positively impact the coherency of the obtained solutions. This may sound confusing, as the real ℓmax is fixed for all energy bins by the number of partialwave amplitudes used during the event simulation. However, the highest order partial-wave amplitudes are very close to 0 for small energies, which causes a problem in the root extraction of the Pℓmax ℓ=0(2ℓ+1)aℓ(s)Pℓ(z)=N(s)Qℓmax i=1(z−zi(s))polynomial (Barrelet Zeros). We know that aℓmax (s)∝N(s) and H2ℓmax (s)∝aℓmax (s) 2, which implies that both N(s) and H2ℓmax (s) will also be close to 0 for low energies. This would cause the roots of P2ℓmax L=0(2L+1)HL(s)PL(z)= N(s)2 16π2sQℓmax i=1(z−zi(s))z−z∗ i(s)to vary in number and in value. Since all of our solutions for the energy-dependent partial-wave amplitudes aℓ(s) are calculated using the extracted Barrelet Zeros, this possible root alteration poses a big problem. Other risks of not following this approach for the data analysis are obtaining enormous error bars for aℓ(s) and including unnecessary ambiguities among our solutions. The divergence of the er- 18
2.4 Partial-wave analysis 2 PARTIAL-WAVE ANALYSIS ror bars stems from the same analytic limitation as the one presented in the preceding paragraph. When it comes to the ambiguities, since the number of solutions obtained for each aℓ(s) is 2ℓmax for each energy value, the lower the maximum orbital momentum used for the data analysis is, the less ambiguities we will obtain among our results. Thus, for each energy bin, it is worth assessing the effect of the highest-order partial-wave amplitudes over the rest of the solutions. For those energy values where the highest-order amplitudes are very close to 0, it will be best to operate with a lower ℓmax in the data analysis. As it will be proven in a latter section where results are presented (section 2.5), this adaptation of ℓmax in terms of the energy sdoes not have a significant impact on the solutions’ deviation from their theoretical value. In fact, it improves the solutions’ accuracy because, as mentioned earlier, this mechanism reduces the risk of encountering errors during the Barrelet Zeros extraction. The choice of the appropriate ℓmax for each energy bin is performed in terms of the value of H2ℓmax (s). We establish a threshold relative to its maximum value (1.5%) and, if the value of H2ℓmax (s) for a certain energy sis inferior to that threshold, we lower ℓmax to ℓ′ max =ℓmax −1, resulting in H(2ℓmax−2)(s)=H2ℓ′ max (s) being the new highest-order moment in the data analysis. We repeat this procedure iteratively until we find an ℓ′′ max such that H(2ℓ′′ max)overcomes its relative threshold. The expressions presented in the previous subsection (2.4.3) are completely ℓmax-dependent, which implies that they are only valid for ℓmax =3. Therefore, new expressions must be developed specifically for ℓmax =2 and for ℓmax =1. We will not consider ℓmax =0, as having studied the SAID model we have concluded that H2(s) overcomes the established threshold ∀s. These new expressions needed for ℓmax =1,2 are presented in Appendix A.1. 2.4.5 Error computation As in the case of section 2.4.3, the algebraic formalism necessary to calculate the errors associated to the partial-wave amplitude results depends on ℓmax. However, the general process will be the same ∀ℓmax. Starting from equation (26), which establishes the relation between the system’s moments and the partial-wave amplitudes, we substitute it ∀Hi, with i∈ {0, ..., 2ℓmax}and we express the partialwave amplitudes in term of Re aj, Im aj,∀j∈ {0, ..., ℓmax}. Then, having established the corresponding relations, we employ the following error propagation equation: δHi= ℓmax X j=0 ∂Hi ∂Re ajδRe aj+ ℓmax X j=0 ∂Hi ∂Im ajδIm aj∀i∈ {0, ..., 2ℓmax}(36) For each energy bin, the error calculation is performed considering the specific general phase such that a0∈R+, which implies Re (a0)=a0and Im (a0)=0. Certain expressions presented during the partial-wave amplitudes calculation (section 2.4.3) assume a different general phase for the system (aℓmax ∈R+). Therefore, before moving on to error calculation, we will perform the general phase change e−iΦ0, where Φ0is the phase of a0when aℓmax ∈R+. Employing the relations between errors obtained from (36), we can build a matrix Mof dimensions (2ℓmax +1)×(2ℓmax +1)such that: 19
2.4 Partial-wave analysis 2 PARTIAL-WAVE ANALYSIS (16π2s) δH0 ... δH2ℓmax−1 δH2ℓmax =M· δa0 ... δRe aℓmax δIm aℓmax (37) Then, we can compute the inverse of the matrix Mand obtain δRe aj,δIm aj,∀j∈ {0, ..., ℓmax} in terms of Hi∀i∈ {0, ..., 2ℓmax}: δa0 ... δRe aℓmax δIm aℓmax =(16π2s)·M−1· δH0 ... δH2ℓmax−1 δH2ℓmax ≃(16π2s)·M−1· √H0 ... pH2ℓmax−1 pH2ℓmax (38) In this last expression, we have approximated δHLas the normalization of the corresponding simulated moment uncertainty δˆ HL=pˆ HL, as we further justified in section 2.4.1. Let’s proceed to apply the described process for ℓmax =3. Starting from equation (26), we obtain the expressions: 16π2sH0(s)=|a0(s)|2+3|a1(s)|2+5|a2(s)|2= =a2 0+3 Re (a1)2+3 Im (a1)2+5 Re (a2)2+5 Im (a2)2+ +7 Re (a3)2+7 Im (a3)2 16π2sH1(s)=a0(s)a∗ 1(s)+a∗ 0(s)a1(s)+2[a1(s)a∗ 2(s)+a∗ 1(s)a2(s)] = =2a0Re (a1)+4 Re (a1)Re (a2)−4 Im (a1)Im (a2)+ +6 Re (a2)Re (a3)−6 Im (a2)Im (a3) 16π2sH2(s)=a0(s)a∗ 2(s)+a∗ 0(s)a2(s)+6 5|a1(s)|2+10 7|a2(s)|2= =2a0Re (a2)+6 5Re (a1)2+6 5Im (a1)2+18 5Re (a1)Re (a3) −18 5Im (a1)Im (a3)+10 7Re (a2)2+10 7Im (a2)2+ +28 15 Re (a3)2+28 15 Im (a3)2 16π2sH3(s)=9 7[a1(s)a∗ 2(s)+a∗ 1(s)a2(s)] = =2a0Re (a3)+18 7Re (a1)Re (a2)−18 7Im (a1)Im (a2)+ +8 3Re (a2)Re (a3)−8 3Im (a2)Im (a3) 16π2sH4(s)=4 3a1a∗ 3+a∗ 1a3+10 7|a2(s)|2+14 11 |a3(s)|2= =8 3Re (a1)Re (a3)−8 3Im (a1)Im (a3)+10 7Re (a2)2+ +10 7Im (a2)2+14 11 Re (a3)2+14 11 Im (a3)2 16π2sH5(s)=50 33 [a2(s)a∗ 3(s)+a∗ 2(s)a3(s)] = =100 33 Re (a2)Re (a3)−100 33 Im (a2)Im (a3) 16π2sH6(s)=700 429 |a3(s)|2=700 429 Re (a3)2+700 429 Im (a3)2 (39) 20
2.6 Spin inclusion in the model 2 PARTIAL-WAVE ANALYSIS In an experimental setting, only the events where the polarization of the target is perpendicular to the scattering plane are selected [14, 15, 16]. In a laboratory frame, the target’s polarization will be fixed, but the scattering plane will vary. This explains why polarization in our target significantly reduces the amount of experimental data we are able to collect, as we only save a small percentage of the generated events. In order to select the desired experimental events, a magnetic field is employed. When taking a measurement where half of the time the initial protons have a spin of +1 2(magnetic field fixed in direction ˆ B) and the other half −1 2(magnetic field fixed in direction −ˆ B), we can eliminate the non-polarized part of expression (47). Therefore: ( δ· P)dσ dΩ=δy 8π2sIm A+A∗ −(48) In conducting an experiment, we collect data from the measurement of observable magnitudes. In our case, these magnitudes are the differential cross-section and the polarized differential crosssection. Diagonalizing the algebraic expressions relating these magnitudes to amplitudes A+and A−(equations 42 and 48) will simplify the computation of A+and A−. It is important to note that this diagonalization is only essential when δy,0. We can obtain the studied equations in a diagonalized form by applying a change of variable: (T+=A++iA− T−=A+−iA−⇒(A+=1 2(T++T−) A−=i 2(T−−T+)(49) Applying equation (49) to expressions 42 and 48, we obtain the following diagonalized forms: dσ dΩ=1 16π2sh|A+|2+|A−|2i=1 64π2s(T++T−)T∗ ++T∗ −+(T−−T+)T∗ −−T∗ += =1 64π2s|T+|2+T∗ +T−+T+T∗ −+|T−|2+|T+|2+|T−|2−T+T∗ −−T∗ +T−=(50) =1 32π2s|T+|2+|T−|2 ( δ· P)dσ dΩ=δy 8π2sIm A+A∗ −=δy 32π2sIm i(T++T−)T∗ −−T∗ += =δy 32π2sRe (T++T−)T∗ −−T∗ +=(51) =δy 32π2sRe T+T∗ −−|T+|2+|T−|2+T−T∗ += =δy 32π2sRe h|T−|2−|T+|2+2iIm T+T∗ −i=δy 32π2s|T−|2−|T+|2 Observing the previous expressions, we can conclude that with polarization, we have access to a linear combination of the square modulus of two independent functions (two independent amplitudes). Therefore, in order to extract the partial-wave amplitudes for a system with a polarized target where we have only kept the perpendicular polarization, we simply have to repeat twice the treatment presented and developed in section 2.4, one for each independent amplitude. Having performed this brief analysis on the effect of the inclusion of spin as a variable in our model, we close the first chapter of this work. Before moving on to a completely new chapter, we must recapitulate and go over the main goals we have reached thus far. In this chapter, we have achieved a good comprehension of the partial waves extraction from the differential cross-section 27
2.6 Spin inclusion in the model 2 PARTIAL-WAVE ANALYSIS of a 2 scalar particle production. In this last section (2.6), we have also included spin and polarization as observables in our model. Overall, we have shown that for a proton-pion system, the partial waves can be extracted from a set of observables. As previously stated, this study on the data analysis of a simple scattering case (proton-pion) constitutes an incredibly useful resource when it comes to the partial-wave analysis of a resonance decaying into 2 daughter particles. Both the development of a data analysis program and the exploration of the inclusion of spin bring us a step closer to a better understanding of the photoproduction of exotic mesons. Specifically, we have set a first group of tools that will be of great help for retrieving information about the experimentally photoproduced b− 1resonance that decays into an ω+π. In order to share the findings discussed in this chapter, I delivered a poster presentation at the 10th International Conference on Quarks and Nuclear Physics in July 2024 [17]. This provided me with the opportunity to interact with peers and experts in the field and receive valuable feedback. 28
3 FORMALISM OF THE b− 1PHOTOPRODUCTION 3 Formalism of the b− 1photoproduction reaction chain In this second chapter, we will explore the formalism of the reaction chain resulting from the photoproduction of an ωπ−coupling. Specifically, we will concentrate on the coupling with quantum numbers JP(ωπ−)=1+, which corresponds to the b− 1(1235) meson resonance (Figure 2). We will consider the ∆++(1232) baryon as the recoiling particle in the initial photoproduction reaction γ+p→b− 1+ ∆++, whose decay results into a pion and a proton. Additionally, this analysis will also include the decay of the ωmeson into 3 pions. As stated in this work’s introduction, this reaction chain is studied as an important step towards the analysis of the π− 1hybrid meson photoproduction process. Before tackling the π− 1→b− 1π0decay, it is essential to establish that we can reliably extract the b− 1from the ωπ−spectrum. Although the reduction in the number of final daughter particles facilitates the retrieval of information from the initial reaction, throughout this chapter we will observe that, by no means, this study becomes trivial. The analysis of the simplified reaction chain (Figure 2) still poses a challenge. Our main goal throughout this chapter will be to calculate the Spin Density Matrix Elements (SDMEs) corresponding to the b− 1meson photoproduced during the process under study, for which we will present a theoretical computation model at the end of this chapter (section 3.3). We will also compute the general amplitude of the system (section 3.2.1) and calculate its intensity (section 3.2.2), for which we will additionally perform a moment expansion (Appendix E). The derived expressions for the system’s amplitude and intensity will allow us to take another approach to the SDMEs calculation, this time more oriented towards data simulation and the observable magnitudes that can be measured in experiments. In order to undertake our challenge with a solid base, we will preface it with a theoretical introduction, where we will go over a series of basic concepts which will be essential for our analysis. 3.1 Theoretical introduction In the following subsections, we will present a series of definitions that we will require while developing the b− 1meson photoproduction formalism. There are two introductory topics that we will consider the reader to be familiar with: helicity and Wigner D-functions. Nonetheless, they will be thoroughly discussed in Appendices D.1 and D.2. As a brief presentation to the concept of helicity, we will provide its definition before moving on to other introductory subjects. For a particle at rest, its spin is quantized using the zaxis. However, when the particle’s momentum is not null, both the zaxis and its momentum direction can be used to quantize its spin. The projection of a particle’s spin onto its momentum direction is called helicity and it was introduced for the first time by Jacob and Wicks [18]. The advantage of helicity over the spin’s 3rd component is that helicity is invariant under rotations and Lorentz boosts in the direction of the particle’s momentum. As mentioned in the previous paragraph, a further exploration of this topic is performed in Appendix D.2. 3.1.1 Crossing symmetry 1 2 𝑎 𝑏 Figure 10: General diagram of a scattering reaction. For a given scattering process, all Feynman diagrams containing a single force carrier can be classified according to the fourmomentum of that particle. Considering the general diagram illustrated in Figure 10, the squared four-momentum of this mediator 29
3.1 Theoretical introduction 3 FORMALISM OF THE b− 1PHOTOPRODUCTION particle will correspond to one of the previously defined Mandelstam variables s,tor u(section 2.2.3): s=(Pa+Pb)µ(Pa+Pb)µt=(Pa−P1)µ(Pa−P1)µu=(Pa−P2)µ(Pa−P2)µ(52) Depending on the value of this squared four-momentum, the Feynman diagram will correspond to an s-channel, a t-channel, or a u-channel interaction, respectively. The blue circle in Figure 10 represents a general scattering interaction, which can be understood either as the exchange of a particle or as the momentary formation of an intermediate particle and its fast decay. The value of the squared four-momentum of the mediator particle in the scattering will depend on the chosen direction for the timeline in Figure 10. This squared four-momentum will match the system’s total energy in the corresponding center-of-mass reference frame (CM frame). Therefore, the s,tand uchannels will be characterized by their CM frames. 1 2 𝑎 𝑏 𝒔-𝒄𝒉𝒂𝒏𝒏𝒆𝒍 𝒕-𝒄𝒉𝒂𝒏𝒏𝒆𝒍 Figure 11: Timeline orientations for the sand the tchannel in a general scattering diagram. When the timeline in Figure 10 is taken from right to left (Figure 11), the studied reaction is a+b→ 1+2. In this case, the system’s total energy in the CM frame is (Pa+Pb)µ(Pa+Pb)µ, which equals the sMandelstam variable defined in equation (52). This kind of interaction is called an s-channel interaction. When the timeline in Figure 10 is taken from top to bottom (Figure 11), we are considering the reaction a+¯ 1→¯ b+2. Similarly to the previous case, the system’s total energy in the CM’s frame is (Pa−P1)µ(Pa−P1)µ, which equals the tMandelstam variable defined in equation (52). This kind of interaction is called a t-channel interaction. We will also refer to this interaction as the crossed channel interaction. The u-channel interaction (a+¯ 2→1+¯ b) will not be relevant in our case and, therefore, we will focus on the study of the sand the t-channels. A good understanding of these channels and their relation will be of great help when studying the photoproduction of our ωπ−system and defining the reference frames for its analysis. Henceforth, we will assume a left-to-right timeline orientation in all the presented Feynman diagrams. Let us consider the s-channel and t-channel reactions portrayed in Figure 12, as well as their respective CM reference frames. Crossing symmetry is a property of scattering amplitudes which states that particles are indistinguishable from antiparticles traveling back in time. As a consequence of crossing symmetry, we can use the same scattering amplitude function A(s,t)to describe both reactions presented in Figure 12. This property is simultaneously surprising and extremely useful because, at first glance, the reactions presented in Figure 12 seem completely different from a physics point of view. However, their physical regions do not overlap, as the domains of the variables sand tdiffer for each channel. When s>(ma+mb)2and t<0, the function A(s,t)will describe the reaction of the s-channel (Figure 12). On the contrary, when s<0 and t>ma+m¯ 12=(ma+m1)2,A(s,t)will describe the reaction of the t-channel (Figure 12). 30
3.1 Theoretical introduction 3 FORMALISM OF THE b− 1PHOTOPRODUCTION 𝒂 𝒃 𝟐 𝟏 t-𝒄𝒉𝒂𝒏𝒏𝒆𝒍 𝒇𝒓𝒂𝒎𝒆 𝑏 2 𝑎 1 𝒕-𝒄𝒉𝒂𝒏𝒏𝒆𝒍 𝑎 + 1 → 𝑏 + 2 1 2 𝑎 𝑏 𝒔-𝒄𝒉𝒂𝒏𝒏𝒆𝒍 𝑎 + 𝑏 → 1 + 2 𝒂 𝟏 𝟐 𝒃 𝒔-𝒄𝒉𝒂𝒏𝒏𝒆𝒍 𝒇𝒓𝒂𝒎𝒆 Figure 12: General scattering interactions in the sand the tchannels. For each case, the corresponding CM frame has also been represented below the reaction. We can represent the physical regions of the s,t, and uchannels employing the Mandelstam- Kibble plot (Figure 13). As stated in the previous paragraph, the regions do not overlap. The Mandelstam-Kibble plot is a simple way of characterizing the s,t, and uchannels in terms of the domains of the s,t, and uMandelstam variables. 𝑠-𝑐ℎ𝑎𝑛𝑛𝑒𝑙 𝑡-𝑐ℎ𝑎𝑛𝑛𝑒𝑙 𝑢-𝑐ℎ𝑎𝑛𝑛𝑒𝑙 𝑠 = 0 + − Figure 13: Mandelstam-Kibble plot. The physical regions for the s,t, and uchannels are painted in different colors. The black lines represent the s,t, and uMandelstam variables axes. The possibility of utilizing the same diagram (Figure 10) to describe the s,t, and uchannels is also a direct consequence of the crossing symmetry. Now, let us explore a specific example. We will focus on the reaction γp→b− 1∆++ in the s-channel and study its crossing reaction in the t-channel (Figure 14). 31
3.1 Theoretical introduction 3 FORMALISM OF THE b− 1PHOTOPRODUCTION 𝜸 𝒑 𝚫++ 𝒃𝟏 + 𝒕-𝒄𝒉𝒂𝒏𝒏𝒆𝒍 𝒇𝒓𝒂𝒎𝒆 𝛾 𝑏1 + 𝑝 Δ++ 𝒕-𝒄𝒉𝒂𝒏𝒏𝒆𝒍 𝜋+ 𝛾 + 𝑏1 + → 𝑝 + Δ++ 𝜸 𝒃𝟏 − 𝚫++ 𝒑 𝒔-𝒄𝒉𝒂𝒏𝒏𝒆𝒍 𝒇𝒓𝒂𝒎𝒆 𝛾 𝑝 𝑏1 − 𝒔-𝒄𝒉𝒂𝒏𝒏𝒆𝒍 Δ++ 𝜋+ 𝛾 + 𝑝 → 𝑏1 −+ Δ++ Figure 14: Interactions in the sand the tchannels for a specific scattering. For each case, the corresponding CM frame has also been represented below the reaction. Since b− 1is a meson, we can assume that b+ 1≡b− 1, which is reflected in the t-channel representation. The t-channel frame has a specific characteristic that allows us to work in it more easily than in the s-channel. The photon and the b+ 1meson merge into a pion, whose quantum numbers correspond to |J,M⟩π+=|0,0⟩, and become an antiproton ( ¯p) and a ∆++ baryon, respectively. Given that the angular momentum must be conserved throughout the reaction, we can conclude: λγ=λb1and λp=λ∆++ , which significantly simplifies the reaction’s study in the t-channel. In order to properly understand why we can apply the crossing symmetry in this specific example and why we can study the s-channel reaction through the analysis of the t-channel reaction, we must analyze the domains of the sand tMandelstam variables for each case and their position in the Mandelstam-Kibble plot (Figure 13). An interaction between a γand a pin the s-channel explained through a pion exchange (Figure 14) is associated with a reaction with a high svalue and t∼0, with t<0. When it comes to the interaction between a γand a b+ 1meson explained through a pion resonance and the subsequent decay into a ¯pand a ∆++ baryon in the t-channel (Figure 14), the expected values for the Mandelstam variables are s<0 and t∼m2 π. Taking the Mandelstam variables’ values into account, we observe that these two reactions are located pretty close to each other in the Mandelstam-Kibble plot (Figure 13), which explains why we can apply the crossing symmetry property. As previously mentioned, the study of the t-channel in the reaction will help us better understand the choice of the Gottfried-Jackson frame (Section 3.1.2) for the reaction’s analysis. 3.1.2 Gottfried-Jackson and Helicity frames The Gottfried-Jackson (GJ) frame [19] is a reference frame typically used during the study of meson and baryon photoproduction reactions, as well as the analysis of their decay into two or more daughter particles. Resonances are always studied in their rest frame, where the sum of momenta of their daughter particles is 0. The angle distribution of the daughter particles in this frame is closely related to the resonance’s spin, which we are interested in studying. Since the GJ frame is measured in the rest 32
3.1 Theoretical introduction 3 FORMALISM OF THE b− 1PHOTOPRODUCTION frame of the produced particle of study, we consider it ideal for resonances. The yaxis of the GJ frame is defined to be normal to the production plane and the zaxis is taken in the direction of one of the initial colliding particles. The choice of which initial colliding particle will depend on the t-channel. For a given scattering process a+b→1+2 in the s-channel whose crossed reaction in the t-channel is a+¯ 1→¯ b+2, the zaxis will be taken in the direction of particle a(b) if the studied photoproduced particle is 1 (2). 𝜸 𝒃𝟏 − 𝚫++ 𝒑 𝑪𝒆𝒏𝒕𝒆𝒓𝒐𝒇-𝒎𝒂𝒔𝒔 𝒇𝒓𝒂𝒎𝒆 𝑧𝐶𝑀 𝑥𝐶𝑀 𝑦𝐶𝑀 Figure 15: Center-of-mass reference frame for the specific reaction γ+p→ b− 1+ ∆++. We can define the frame’s axes in terms of the center-of- mass (Figure 15) momentum of the photoproduced meson/baryon ( pM/B): ˆxGJ =ˆyGJ ׈zGJ ˆyGJ = pI× pM/B pI× pM/B ˆzGJ = pI pI (53) Where pIcorresponds to the center-of-mass momentum of the incident particle chosen as a reference for the definition of the z-axis. Starting from the t-channel frame, we can move to the GJ frame through a Lorentz boost in the opposite direction of the momentum of our particle of study. Let us study this frame transition considering our specific reaction γ+p→b− 1+ ∆++ in the s-channel. As justified in Section 3.1.1, the corresponding reaction in the t-channel will be γ+b+ 1→¯p+ ∆++, because b− 1=b+ 1. In this case, the studied mesonic resonance will be b− 1. Therefore, we will refer to the Gottfried-Jackson frame as the GJ-b− 1rest frame. 𝜸 𝒑 𝚫++ 𝒃𝟏 + 𝑡-𝑐ℎ𝑎𝑛𝑛𝑒𝑙 Boost in the direction of −𝒑𝒃𝟏 − 𝑧𝐺𝐽 𝑥𝐺𝐽 𝑦𝐺𝐽 𝜸 𝒃𝟏 − 𝚫++ 𝒑 𝐺𝑜𝑡𝑡𝑓𝑟𝑖𝑒𝑑-𝐽𝑎𝑐𝑘𝑠𝑜𝑛 𝑓𝑟𝑎𝑚𝑒 Figure 16: Transition from the t-channel frame to the Gottfried-Jackson frame on the photoproduction reaction γ+p→b− 1+ ∆++ in the s-channel. We interpret particles as their corresponding antiparticles with opposite direction of momentum, which is reflected in this figure for particle b− 1 and its antiparticle b+ 1. Considering the decay of the b− 1meson into an ωπ−pair, we obtain the general outline in the GJ-b− 1 rest frame illustrated in Figure 17. 33
3.1 Theoretical introduction 3 FORMALISM OF THE b− 1PHOTOPRODUCTION 𝑧𝐺𝐽 𝑥𝐺𝐽 𝑦𝐺𝐽 𝜸 𝒃𝟏 − 𝚫++ 𝝎 𝒑 𝝅− 𝐺𝑜𝑡𝑡𝑓𝑟𝑖𝑒𝑑-𝐽𝑎𝑐𝑘𝑠𝑜𝑛 𝑓𝑟𝑎𝑚𝑒 Figure 17: Photoproduction of the b− 1meson in the GJ-b− 1rest frame. The decay of b− 1into a ωπ− pair is also represented. Since the Lorentz boost needed to transition from the t-channel frame to the GJ-frame is in the direction of − pb− 1= pb+ 1, it is relevant to note that the helicity of the b− 1meson is conserved: λt b1=λGJ b1(property of helicity justified in Appendix D.2). As it will be presented in section 3.1.4, the t-channel b− 1Spin Density Matrix Elements (SDMEs) are equivalent to the GJ frame ones. This equality will allow us to extract the SDMEs from the b− 1decay in the Gottfried-Jackson frame and immediately gain access to the t-channel ones. Another commonly used reference frame in the study of photoproduction reactions is the Helicity frame. This frame is also measured in the rest frame of the produced particle of study and, while the xand yaxes are the same as in the Gottfried-Jackson frame, the zaxis differs: ˆxH=ˆyGJ ׈zGJ ˆyH= pI× pM/B pI× pM/B ˆzH= pM/B pM/B (54) Where we have used the same notation as in (53). It is relevant to point out that, while the Gottfried-Jackson frame is accessible through the t-channel, the Helicity frame is accessible through the s-channel. In order to move from the s-channel to the Helicity frame, a boost in the opposite direction of the momentum of the photoproduced particle under study must be performed. In our case, a Lorentz boost in the direction of − pb− 1. There is only a difference in rotation between the Helicity and the Gottfried-Jackson frame but, despite this, as we will prove in the next sections, this rotation caused by the difference in the choice of the zaxis makes a big difference. During the rest of this thesis, we will mainly work in the Gottfried-Jackson frame. In an experimental setting, it is more convenient to work within a reference frame where the studied photoproduced particle (the b− 1meson in our case) is at rest, rather than directly operating in the tor schannel frames, which are not accessible experimentally. The transition from the laboratory reference frame to the GJ-b− 1rest frame only requires a boost, which makes it a very convenient frame to work in. Choosing the GJ-b− 1rest frame makes the experimentally obtained SDMEs only valid in this same reference frame. Therefore, the SDMEs will have to undergo transformations in order to be valid in other reference frames. In the following two subsections, we will explore these transformations. 3.1.3 Wigner rotations Before moving forward in the study of the effects of Lorentz transformations over SDMEs, we will introduce the concept of the Wigner rotation. Let us start by defining a particle’s rest state as 34
3.1 Theoretical introduction 3 FORMALISM OF THE b− 1PHOTOPRODUCTION | 0,jλ⟩=|jλ⟩, where jis the particle’s spin and λis the particle’s 3rd component of spin. If we consider mto be the particle’s mass, we can define its rest four-momentum as Kν 0=m, 0. We define the Lorentz group as the group of all Lorentz transformations of Minkowski spacetime. The subgroups of the Lorentz group whose transformation leave the four-momentum of a given particle invariant are called Wigner’s little groups. The Wigner little group for massive particles is S O(3) and, therefore, the specific Lorentz transformations which leave Kν 0invariant are called rotations. In the case of massless particles, their Wigner little group is IS O(2) and, in consequence, the type of transformations that leave their four-momenta invariant are phase changes in their equation of state [20]. For any four-momentum Kν=Ek, k, we can define a Lorentz transformation Λ0 kthat brings the standard momentum Kν 0to Kν. That is to say UhΛ0 kiKν 0=Kν, transformation illustrated in Figure 18. This allows us to define the general state |Kν,j⟩in terms of |Kν 0,j⟩: |Kν,j⟩=UhΛ0 ki|Kν 0,j⟩(55) Now, let us study the effect of a general Lorentz transformation Λover the general state |Kν,j⟩: U[Λ]|Kν,j⟩=U[Λ]UhΛ0 ki|Kν 0,j⟩=(56a) =UhΛ0Λ kiUΛ0Λ k−1U[Λ]UhΛ0 ki|Kν 0,j⟩=(56b) =UhΛ0Λ kiUΛ0Λ k−1ΛΛ0 k|Kν 0,j⟩(56c) If we apply the Lorentz transformation Λ0Λ k−1ΛΛ0 kto the four-momentum Kν 0, we obtain Kν 0: Λ0Λ k−1ΛΛ0 kKν 0= Λ0Λ k−1ΛKν=Kν 0(57) Consequently, we can conclude that the Lorentz transformation Λ0Λ k−1ΛΛ0 kis a rotation in the particle’s rest frame that moves the direction of kto the direction of k′= Λ k. This rotation is known as the Wigner rotation or Thomas precession and has an associated angle ω, the Wigner angle. A visual representation of this rotation WΛ, kis presented in Figure 18: 𝑘 Λ𝑘 0 Λ Λ0(𝑘) Λ0(Λ𝑘) −1 𝑊 Λ, 𝑘 =Λ0(Λ𝑘) −1Λ Λ0(𝑘) 𝑊 Λ, 𝑘 Figure 18: Representation of the Wigner rotation (WΛ, k), also known as the Thomas precession. This rotation is presented as a composition of transformations that form a cycle and end up leaving Kν 0=(m, 0) invariant. 35
3.1 Theoretical introduction 3 FORMALISM OF THE b− 1PHOTOPRODUCTION It is interesting to note that, even though Kν 0is invariant under the Wigner rotation, the particle’s spin changes: U[Λ]|Kν,j⟩=UhΛ0Λ ki|Kν 0,j′⟩(58) In order to properly define ωusing a covariant representation, we must first introduce some notation. Let Aµ=ma, 0be the four-momentum of a particle of mass maat rest in the reference frame where Kνis defined and let Bµ=mb, 0be the four-momentum of a particle of mass mbat rest in the reference frame where ΛKν=E′ k, k′=E′ k,Λ kis defined. Now, let us respectively boost Aµ,Bµ, and Kνto the reference frame where Kν 0is defined in. We obtain, respectively, Aµ′=Ea, a,Bµ′=Eb, b, and Kν 0=m, 0. Since we had defined ωas the angle between kand k′in the particle’s rest frame, it will also be the angle between aand bin this same reference frame. Having presented the necessary elements, we can define the Wigner angle using the following expression: cos ω=Kν 0A′ µKν 0B′ µ−m2Aµ′B′ µ rKν 0A′ µ2−m2Aµ′A′ µKν 0B′ µ2−m2Bµ′B′ µ (59) Let us prove that this definition holds considering the Aµ′,Bµ′, and Kν 0notation: Kν 0A′ µ2−m2Aµ′A′ µ=(m·Ea)2−m2·m2 A=m2·E2 a−m2 a=m2· a 2(60) Equivalently: Kν 0B′ µ2−m2Bµ′B′ µ=m2· b 2(61) Developing the numerator of equation (59): Kν 0A′ µKν 0B′ µ−m2Aµ′B′ µ=m2·Ea·Eb−m2·Ea·Eb− a· b(62a) =m2·Ea·Eb−m2·Ea·Eb− a· bcos ω=(62b) =m2· a bcos ω(62c) Gathering all the deduced expressions and substituting at 59: cos ω= m2· a bcos ω rm2· a 2m2· b 2 = m2· a· bcos ω m2· a b =cos ω(63) □ 3.1.4 Lorentz transformations and SDMEs Having presented Wigner rotations and some of their properties, let us finally focus on the effect of Lorentz transformations over Spin Density Matrix Elements (SDMEs). Let Λbe the Lorentz transformation needed to transition from the s-channel frame of the γ+p→ b− 1+ ∆++ reaction into the t-channel frame of its corresponding reaction (γ+b+ 1→¯p+ ∆++). Let 36
3.2 Formalism of the photoproduction 3 FORMALISM OF THE b− 1PHOTOPRODUCTION Defining: IαΩω,ΩH,Ωp,Φ=I0Ωω,ΩH,Ωp,Φ, IΩω,ΩH,Ωp,Φ (82a) ρα Λ,Λ′;λ∆,λ′ ∆ (s,t)=1 2X λγ,λ′ γX λ1 Vλγ,Λ;λ1,λ∆(s,t)σα λγ,λ′ γV∗ λ′ γ,Λ′;λ1,λ′ ∆ (s,t)(82b) Where σα=I, σand substituting 76 in 81, we obtain: IαΩω,ΩH,Ωp=3κ 4π2X λγ,λ′ γ,λ1,λ2 1 X Λ=−1 3 2 X λ∆=−3 2 Vλγ,Λ;λ1,λ∆(s,t) 1 X λ=−1 FλD1∗ Λ,λ (Ωω)Yλ1(ΩH)G˜ Fλ2D 3 2∗ λ∆,λ2Ωp· ·σα λγ,λ′ γ· 1 X Λ′=−1 3 2 X λ′ ∆=−3 2 V∗ λ′ γ,Λ′;λ1,λ′ ∆ (s,t) 1 X λ′=−1 F∗ λ′D1 Λ′,λ′(Ωω)Y∗ λ′1(ΩH)G∗˜ F∗ λ2D 3 2 λ′ ∆,λ2Ωp = =3κ 2π2X λ∆,λ′ ∆X Λ,Λ′X λ,λ′X λ2 ρα Λ,Λ′;λ∆,λ′ ∆ (s,t)·FλF∗ λ′D1∗ Λ,λ (Ωω)D1 Λ′,λ′(Ωω)· ·Yλ1(ΩH)Y∗ λ′1(ΩH)|G|2|˜ F|2D 3 2∗ λ∆,λ2ΩpD 3 2 λ′ ∆,λ2Ωp(83) We will develop further the elements highlighted in equation (83). Before doing so, let us start by listing 2 properties of the Wigner D-functions, an equality between Wigner D-functions and spherical harmonics and a relation held between Clebsch-Gordan coefficients: Dj m,n(α, β, γ)=ϵDj∗ −m−n(α, β, γ)=(−1)n−mDj∗ −m−n(α, β, γ)(84a) Dj1 m1,n1(α, β, γ)Dj2 m2,n2(α, β, γ)= j1+j2 X L=|j1−j2| CL,m1+m2 j1,m1;j2,m2CL,n1+n2 j1,n1;j2,n2DL m1+m2,n1+n2(α, β, γ)(84b) Dl m,0(α, β, γ)=r4π 2l+1Y∗ lm (β, α)⇒Y∗ lm (β, α)=r2l+1 4πDl m,0(α, β, γ)(84c) CJ,M j1,m1;j2,m2=⟨j1,m1,j2,m2|J,M⟩=(−1)j2+m2s2J+1 2j1+1⟨j2,−m2,J,M|j1,m1⟩= =(−1)j2+m2s2J+1 2j1+1Cj1,m1 j2,−m2;J,M (84d) Moving forward, we will employ the colors used to highlight different parts of equation (83) to define equivalent analytic expressions. That is to say, all algebraic expressions highlighted using the same color are equal. 43
3.2 Formalism of the photoproduction 3 FORMALISM OF THE b− 1PHOTOPRODUCTION Using properties (84a) and (84b), we can define the coefficients cλ∆,λ′ ∆Ωp, which will allow us to express the system’s intensity in a simpler manner: cλ∆,λ′ ∆Ωp= 1/2 X λ2=−1/2 D 3 2∗ λ∆,λ2ΩpD 3 2 λ′ ∆,λ2Ωp= 1/2 X λ2=−1/2 D 3 2∗ λ∆,λ2Ωp(−1)λ2−λ′ ∆D 3 2∗ −λ′ ∆,−λ2Ωp= = 1/2 X λ2=−1/2 (−1)λ2−λ′ ∆ 3 X L′=0 CL′,λ∆−λ′ ∆ 3 2,λ∆;3 2,−λ′ ∆ CL′,0 3 2,λ2;3 2,−λ2 DL′∗ λ∆−λ′ ∆,0Ωp(85) Next, we can apply properties (84a), (84b), and (84c) to reduce the Wigner D-functions associated to the b− 1meson decay: D1∗ Λ,λ (Ωω)D1 Λ′,λ′(Ωω)=D1∗ Λ,λ (Ωω)·(−1)λ′−Λ′D1∗ −Λ′,−λ′(Ωω)= =(−1)λ′−Λ′ 2 X L=0 CL,Λ−Λ′ 1,Λ;1,−Λ′CL,λ−λ′ 1,λ;1,−λ′DL∗ Λ−Λ′,λ−λ′(Ωω)= =(−1)λ′−Λ′X L (−1)1−Λ′r2L+1 3C1,Λ 1,Λ′;L,Λ−Λ′(−1)1−λ′r2L+1 3C1,λ 1,λ′;L,λ−λ′DL∗ Λ−Λ′,λ−λ′(Ωω)= =(−1)λ′−Λ′−Λ′−λ′X L 2L+1 3!C1,Λ 1,Λ′;L,Λ−Λ′C1,λ 1,λ′;L,λ−λ′DL∗ Λ−Λ′,λ−λ′(Ωω)= =X L 2L+1 3!C1,Λ 1,Λ′;L,Λ−Λ′C1,λ 1,λ′;L,λ−λ′DL∗ Λ−Λ′,λ−λ′(Ωω) (86) Applying relation 84c and the equality deduced in equation (86), we can also reduce the pair of spherical harmonics associated to the ωmeson decay into 3 pions: Yλ1(ΩH)Y∗ λ′1(ΩH)=r3 4πD1∗ λ,0(ΩH)r3 4πD1 λ′,0(ΩH)= =3 4πD1∗ λ,0(ΩH)D1 λ′,0(ΩH)= = 2 X ℓ=0 2ℓ+1 4π!C1,λ 1,λ′;ℓ,λ−λ′C1,0 1,0;ℓ,0Dℓ∗ λ−λ′,0(ΩH) (87) Gathering all the pieces together and substituting in equation (83), we obtain: IαΩω,Ωp,ΩH=3κ 2π2X λ∆,λ′ ∆X Λ,Λ′X λ,λ′ cλ∆,λ′ ∆Ωp·ρα Λ,Λ′;λ∆,λ′ ∆ (s,t)· ·|G|2FλF∗ λ′X ℓ 2ℓ+1 4π!C1,λ 1,λ′;ℓ,λ−λ′C1,0 1,0;ℓ,0Dℓ∗ λ−λ′,0(ΩH)· ·˜ F 2X L 2L+1 3!C1,Λ 1,Λ′;L,Λ−Λ′C1,λ 1,λ′;L,λ−λ′DL∗ Λ−Λ′,λ−λ′(Ωω) (88) 44
3.3 Model for the SDMEs calculation 3 FORMALISM OF THE b− 1PHOTOPRODUCTION Applying the variable changes M= Λ −Λ′and m=λ−λ′, the previous expression transforms into: IαΩω,Ωp,ΩH=3κ 2π2X λ∆,λ′ ∆X Λ,λ X m,M cλ∆,λ′ ∆Ωp·ρα Λ,Λ−M;λ∆,λ′ ∆ (s,t)· ·|G|2FλF∗ λ−mX ℓ 2ℓ+1 4π!C1,λ 1,λ−m;ℓ,mC1,0 1,0;ℓ,0Dℓ∗ m,0(ΩH)· ·˜ F 2X L 2L+1 3!C1,Λ 1,Λ−M;L,MC1,λ 1,λ−m;L,mDL∗ M,m(Ωω) (89) Integrating the final intensity expression (89) over the normal of the 3πplane (ΩH,ωdecay) and taking into account the ortogonality of the spherical harmonics Rπ θH=0R2π ϕH=0Yλ1Y∗ λ′1dΩ = δλ,λ−m= δλ,λ−m=δm,0: IαΩω,Ωp=3κ 2π2X λ∆,λ′ ∆X Λ,λ X M cλ∆,λ′ ∆Ωpρα Λ,Λ−M;λ∆,λ′ ∆ (s,t)· ·|GFλ˜ F|2X L 2L+1 3!C1,Λ 1,Λ−M;L,MC1,λ 1,λ;L,0DL∗ M,0(Ωω) (90) From here, we will integrate the previous expression over Ωpto solely focus on the b− 1→ωπ− decay. In order to do so, we must take into account that Rcλ∆,λ′ ∆ΩpdΩp=2πδλ∆,λ′ ∆: Iα(Ωω)=3κ π|GFλ˜ F|2X λ∆,Λ,λ X M ρα Λ,Λ−M(s,t)·X L 2L+1 3!C1,Λ 1,Λ−M;L,MC1,λ 1,λ;L,0DL∗ M,0(Ωω)(91) Equation (91) provides us with a final expression for the intensity of the reaction chain. This same expression proves that the b− 1SDMEs (ρα Λ,Λ−M) can be extracted from the angular decomposition of the ωπ−system (an observable magnitude). When the studied resonance is not a clear peak and contains different couplings, a moment expansion can be performed for the system’s intensity. This way, the moments, which act as coefficients for the different angular momentum components of the system, can easily be extracted from observable magnitudes and provide an understanding of the significant partial waves. This intensity moment expansion is studied in Appendix E. 3.3 Model for the SDMEs calculation In this subsection, we will present a theoretical model that enables us to calculate the b− 1SDMEs in the t-channel, which we will compute for a fixed svalue and a specific tinterval. Applying the relation between the b− 1SDMEs in different reference frames presented in section 3.1.4, we will also obtain the corresponding SDMEs in the s-channel. Finally, we will compare the results obtained for the sand tchannels and reflect on our choice of reference frame for the b− 1photopro- duction and decay. The employed model will be extracted from [22] and [23]. In this model, we will explain the reaction γ+b+ 1→¯p+ ∆++ in the t-channel as the exchange of a pion. The pion exchange contribution to the interaction dominates the large energy, small angle regions. However, it is important to keep in mind that, although this model will only consider the πexchange, there are other production mechanisms. 45
3.3 Model for the SDMEs calculation 3 FORMALISM OF THE b− 1PHOTOPRODUCTION Let us first consider the interaction between a axial-vector b1, a photon and a pseudoscalar πin this same channel: ⟨γb1|π⟩=gb1 mb1 ϵb1 µϵγ νhpµ γpν b−(pγ·pb)gµνi(92) The coupling constant gb1is normalized by mb1, simply to make it dimensionless. We will obtain gb1’s value from the b− 1meson radiative decay width Γ(b− 1→γπ−). The presented expression (92) is equivalent to equation (6) of [22]. Now, let us consider the following interaction between the ¯p, the ∆and the pion, also in the t-channel: ⟨π|¯p∆⟩=fπN∆ mπ ¯uα(p¯p−p∆)αv(93) Equivalently to the previous case, the coupling constant fπN∆will be extracted from the Delta baryon decay width Γ(∆++ →p). Thanks to the description of the presented interactions, we can write the amplitudes of a pion exchange in the t-channel in the following way: Vt λγ,Λ;λ1λ∆=⟨γb1|π⟩1 m2 π−t⟨π|¯p∆⟩(94) = gb1 mb1 fπN∆ mπ!ϵb1 µϵγ νhpµ γpν b−(pγ·pb)gµνi¯uα(p¯p−p∆)αv(95) In order to obtain a calculable expression for Vt λγ,Λ;λ1λ∆, the kinematics and conventions for the spin functions must be studied. This is done in Appendix F. Substituting the expressions presented in Appendix F: Vt λγ,Λ;λ1λ∆=−gb1 mb1 fπN∆ mπ δλγ,Λ 2 m2 b1−t m2 π−t ×δλ1,Λ∆r2 3 √t m∆ qtq(Et p−mp)(Et ∆+m∆)+q(Et p+mp)(Et ∆−m∆)(96) The only elements left to calculate before being able to compute the b− 1SDMEs are the gb1and fπN∆coupling constants. As previously mentioned, we will use the b− 1and ∆++ decay widths to do so. The following decay widths have been determined experimentally [9]: Γ(b− 1→γπ−)=0.227 MeV Γ(∆++ →pπ+)=116 MeV (97) The general formula for the decay of a particle of spin J, mass Mdecaying into particles of masses m1and m2is: Γ = 1 2J+1 p 8πM2X Λ,λ1,λ2|MΛ,λ1,λ2|2p=λ1/2(M2,m2 1,m2 2) 2M(98) In the rest frame of b− 1, we can calculate the amplitude of the decay b− 1( 0,Λ)→γ( p, λγ)+π(− p), with ω, the angle between the photon momentum and the z-axis used to define Λ. Using the interaction (92): MΛ,λγ(ω)=gb1pλ2 γd1 Λ,λγ(ω)p= m2 b1−m2 π 2mb1 (99) 46
3.3 Model for the SDMEs calculation 3 FORMALISM OF THE b− 1PHOTOPRODUCTION Similarly to the previous case, we can compute the amplitude for the decay ∆( 0,Λ)→p( p, λp)+ π(− p) in the rest frame of the ∆, with ω′, the angle between the nucleon momentum and the z-axis MΛ,λp(ω′)=fπN∆ mπ ¯uα Λ(p¯p−p∆)αuλN(pN) (100a) =fπN∆ mπ pq(m∆+mN)2−m2 π(2λp)r2 3d 3 2 Λ,λp(ω′) (100b) Substituting equation (97) and the corresponding invariant amplitudes in (98), we obtain: Γ(b− 1→γπ−)=mb1 g2 b1 12π m2 b1−m2 π 2m2 b1 3 =0.227 MeV (101a) ⇒gb1=0.241 Γ(∆++ →pπ+)=f2 πN∆ 24π λ3/2(m2 ∆,m2 p,m2 π) 8m5 ∆m2 πh(m∆+mp)2−m2 πi=116 MeV (101b) ⇒fπN∆=2.07 Where equation (101b) is equivalent to expression (2) in [23]. In this same bibliographic source, the obtained value for the fπN∆coupling constant is fπN∆=2.14 ∼2.18, which confirms the validity of our obtained value fπN∆=2.07. In [22], a similar model to the one presented in this section is studied and the value gπγb=0.086 is derived, where e·gπγb mπ=gb1 mb1 . Therefore, according to [22], gb1=e·gπγb mπmb1=0.236, which is also compatible with our value gb1=0.241. We can finally focus on the SDMEs calculation. Recovering the general definition (66), we obtain the following for the non-normalized b− 1t-channel SDMEs: ρ0,t ΛΛ′=1 2X λγ,λ1,λ∆ Vλγ,Λ;λ1,λ∆V∗ λγ,Λ′;λ1,λ∆=1 2X λγ,λ1,λ∆ k(t)2δλγ,Λδλγ,Λ′δλ1,λ∆δλ1,λ∆=(102a) =X λγ k(t)2δλγ,Λδλγ,Λ′=k(t)2δΛ,Λ′(102b) Where: k(t)=−gb1 mb1 fπN∆ 2mπ m2 b1−t m2 π−t×r2 3 √t m∆ qtq(Et p−mp)(Et ∆+m∆)+q(Et p+mp)(Et ∆−m∆)(103) Before representing the SDMEs, we will normalize them. Although k(t) will disappear in the SDMEs normalization, it is very useful to have its expression when considering a contribution different to the pion exchange. Since the expansion of this model to other contributions to the interaction is one of the outlooks of this work, it is interesting to have this expression in mind. On the other hand, recovering the relation between the sand the tchannels SDMEs (equation (69)): ρ0,s λb,λ′ b =X µb,µ′ b d1 µb,λb(ωb)ρ0,t µb,µ′ b d1 µ′ b,λ′ b (ωb) (104) 47
3.3 Model for the SDMEs calculation 3 FORMALISM OF THE b− 1PHOTOPRODUCTION Where: cos ωb=s−u 4ptqt +m2 b 4t m2 ∆−m2 p ptqt (105) Employing the presented expressions for the sand tchannels SDMEs, we obtain the following results: 0.0 0.5 1.0 1.5 t ( GeV 2) 0.5 0.0 0.5 0, t 1 1 0.0 0.5 1.0 1.5 t ( GeV 2) 0.5 0.0 0.5 0, t 1 1 0.0 0.5 1.0 1.5 t ( GeV 2) 0.5 0.0 0.5 0, t 10 0.0 0.5 1.0 1.5 t ( GeV 2) 2 1 0 1 2 0, s 1 1 0.0 0.5 1.0 1.5 t ( GeV 2) 2 1 0 1 2 0, s 1 1 0.0 0.5 1.0 1.5 t ( GeV 2) 2 1 0 1 2 0, s 10 Figure 21: Comparison between the Spin Density Matrix Elements of the b− 1meson in the schannel and the t-channel as a function of the momentum transfer squared −t. The full ρ0matrices for each channel are attached in Appendix F. The trace of the ρ0matrix has been normalized, meaning that we have imposed ρ0 −1−1+ρ0 00+ρ0 11 =1 and, consequently, have applied a normalization factor 1 Nfor all the elements of the matrix. Trace is invariant under rotation, which tells us that, having applied the trace normalization in the tchannel, we can calculate the SDMEs for the s-channel using (69) and the obtained results will already be normalized. As mentioned in a previous section, the s-channel b− 1SDMEs are equivalent to the ones in the Helicity frame, and the t-channel b− 1SDMEs are equivalent to the ones in the Gottfried-Jackson frame. Therefore, Figure 21 can also be considered as a comparison between the GJ and the Helicity frame SDMEs of the b− 1meson. This comparison illustrates the importance of choice for the zaxis definition when deciding on what reference frame to work and develop our b− 1photoproduc- tion and decay study, as the SDMEs obtained for the GJ frame are much simpler and easier to work with than the ones obtained for the Helicity frame. At first, in previous sections, the prioritization of the GJ frame over the Helicity frame might have seemed arbitrary, but these results confirm the correct reasoning behind our choice. With this reflection on the importance of choice for the reference frame of study, we conclude our second chapter and we move on to the final section of this thesis, which presents the conclusions drawn from the completed work and discusses the possible outlook for future research in this field. 48
4 CONCLUSIONS AND OUTLOOK 4 Conclusions and outlook At the beginning of this work, we established as the main objective of this thesis to set a base of tools that, in the future, would contribute to extracting as much information as possible from the π− 1hybrid meson experimental photoproduction. As a first approach to it, we have focused on the study of the b− 1meson, one of the daughter particles resulting from the π− 1decay. In this final section, we will summarize our achievements and present an outlook on the possible next steps of this work. The first chapter of this thesis consisted of the development of a partial-wave analysis program that would allow us to properly identify the b− 1resonance through the study of its daughter particles’ states (ωand π−). We constructed our program using a test case, the elastic scattering between a proton and a pion, for which we performed a Monte Carlo event simulation using 2 different models, a basic and a realistic one [11]. As an initial approach, we did not consider spin as a variable, we treated all particles as scalar particles. We calculated the moments of the system using the energy and angular-dependent differential cross-section (section 2.4.1), extracted the Barrelet Zeros for the different values of energy studied (section 2.4.3), and, finally obtained the desired energy-dependent partial-wave amplitudes (section 2.5), each of them accompanied by a group of ambiguities. We ensured the correct functioning of our analysis program through various checks along the way and the final comparison between the obtained results and the theoretical partial-wave amplitudes of the corresponding model. As a final addition to this chapter, we included spin as a variable of the system, explored its effect on the differential cross-section, and reflected on its influence over the number of ambiguities obtained (section 2.6). Overall, both considering spin and not taking it into account, we proved the possibility of extracting the system’s partial-wave amplitudes from a set of observables. In the second chapter, we focused on the theoretical formalism of the full reaction chain associated with the photoproduction of the b− 1meson (Figure 2). We took 2 different approaches to this study but, in both cases, the main objective was the calculation of the Spin Density Matrix Elements (SDMEs), which provide us with extremely valuable information about the system’s quantum states. For the first approach, we defined a series of adequate reference frames for different phases of the reaction chain and computed an SDME-dependent expression for the reaction’s total amplitude (section 3.2.1), as well as for the system’s intensity (section 3.2.2). Additionally, we performed a moment expansion on the reaction’s intensity (Appendix E), which granted us access to the moments of the system, which constitute a finite set of observable magnitudes from which we can extract angular information about the studied reaction chain. This first approach was concentrated on the experimental extraction of the SDMEs. On the other hand, the second approach was centered around a theoretical prediction of the SDMEs. For the second approach, an SDME-calculation model depending mostly on theoretical data was presented (section 3.3). In this part, we showed the importance of choice for the reference frame where we study our system by comparing the complexity and results of the SDMEs on the sand the tchannels. Equivalently, we presented the SDME calculations for the Gottfried-Jackson frame and the Helicity frame, and concluded the GJ to be the simplest and most accessible. 49
4 CONCLUSIONS AND OUTLOOK All in all, the development of abilities required to complete this thesis has been quite diverse. Ranging from computational competencies necessary to construct the first chapter’s simulation and analysis program, to more profound theoretical deepening, which has been essential to comprehend this work’s second chapter properly. This thesis constitutes a bridge between experimental and theoretical hadron physics, acknowledging at all times the laboratory-accessible reference frames and reactions, and building a connection with the applicable theoretical resources and relations. The study of the b− 1meson photoproduction and decay can be considered a puzzle that requires the assembly of many pieces. In this work, we have selected a few of these pieces and concentrated on tackling the challenges that they present. However, there is still a considerable portion of the puzzle that needs to be solved. This brings us to present an outlook on this specific line of research. One of the most logical next steps would consist on the development of a computational simulation where both spin and detector characteristics as intrinsic and geometric efficiency are taken into account. The best way to initially approach this would be to only consider the decay of b− 1 and assume ωand ∆++ to be stable particles. When this first step was completed, the inclusion of the angular distribution for the ωand ∆++ decays would be essential to advance in the correct direction. Another possible continuation for this work would be the development of an equivalent SDME- computation model to the one presented in section 3.3, but considering another production mechanism, specifically the exchange of a vector particle instead of a pion, which constitutes a scalar one. Finally, as presented in this thesis motivation, the ultimate goal would be to expand all the developed work to the real π− 1hybrid meson photoproduction and decay, including π0and b− 1as its daughter particles. As illustrated in this thesis, the analysis of unstable meson’s photoproduction and consequent decay is a complex and long line of work. The developments presented in this document are just the beginning of the study for a specific hybrid meson, the π− 1. There are still numerous pieces left to complete the puzzle and hadron physics impatiently awaits our contributions. 50
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B.2 Results and Figures B BASIC MODEL With the main objective of improving the visibility of the ambiguities for the obtained solutions, we will represent the same graphs presented in 24, but excluding the error bars. 12345 s = E 2( GeV 2) 0.6 0.8 1.0 1.2 1.4 a 0 SAID MC simulation (a) Partial-wave amplitude a0. 12345 s = E 2( GeV 2) 5 0 5 Re( a 1 ) 1e 1 SAID MC simulation (b) Real part of a1. 12345 s = E 2( GeV 2) 0 1 2 3 4 Im( a 1 ) 1e 1 SAID MC simulation (c) Imaginary part of a1. 12345 s = E 2( GeV 2) 2 0 2 4 6 Re( a 2 ) 1e 1 SAID MC simulation (d) Real part of a2. 12345 s = E 2( GeV 2) 6 4 2 0 Im( a 2 ) 1e 1 SAID MC simulation (e) Imaginary part of a2. 12345 s = E 2( GeV 2) 5.0 2.5 0.0 2.5 5.0 Re( a 3 ) 1e 1 SAID MC simulation (f) Real part of a3. 12345 s = E 2( GeV 2) 0.0 0.2 0.4 0.6 0.8 1.0 Im( a 3 ) SAID MC simulation (g) Imaginary part of a3. Figure 25: Values of the real and imaginary parts for the partial-wave amplitudes aiof the basic model in terms of s=E2 cm. The blue lines represent the theoretical model values after applying the change in the general phase of the system such that a0∈R+. The values obtained from the Monte-Carlo simulations are represented using different colored triangular markers, which allows us to show the ambiguity in the solution obtained for each energy bin. The error bars have not been added for visbility purposes. The solutions presented in this figure have been obtained using a simulation of 108events. 59
B.2 Results and Figures B BASIC MODEL 12345 s = E 2( GeV 2) 1 2 3 4 H 0 1e 2 SAID MC simulation (a) H0moment. 12345 s = E 2( GeV 2) 1.0 0.5 0.0 0.5 H 1 1e 2 SAID MC simulation (b) H1moment. 12345 s = E 2( GeV 2) 0.25 0.50 0.75 1.00 1.25 1.50 H 2 1e 2 SAID MC simulation (c) H2moment. 12345 s = E 2( GeV 2) 5.0 2.5 0.0 2.5 5.0 H 3 1e 3 SAID MC simulation (d) H3moment. 12345 s = E 2( GeV 2) 2 4 6 8 H 4 1e 3 SAID MC simulation (e) H4moment. 12345 s = E 2( GeV 2) 4 2 0 2 4 H 5 1e 3 SAID MC simulation (f) H5moment. 12345 s = E 2( GeV 2) 0 2 4 6 H 6 1e 3 SAID MC simulation (g) H6moment. Figure 26: Values if the Hivalues of the basic model in terms of s=E2 cm. The blue lines represent the theoretical model values. The values obtained from the Monte-Carlo simulations are represented using green triangular markers. The solutions presented in this figure have been obtained using a simulation of 108events. 60
B.2 Results and Figures B BASIC MODEL 12345 s = E 2( GeV 2) 2 1 0 1 2 3 | a 0| SAID MC simulation (a) Modulus of a0. 12345 s = E 2( GeV 2) 4 2 0 2 4 Arg( a 0 ) SAID MC simulation (b) Phase of a0. 12345 s = E 2( GeV 2) 0.5 0.0 0.5 1.0 1.5 | a 1| SAID MC simulation (c) Modulus of a1. 12345 s = E 2( GeV 2) 7.5 5.0 2.5 0.0 2.5 5.0 Arg( a 1 ) SAID MC simulation (d) Phase of a1. 12345 s = E 2( GeV 2) 0.0 0.5 1.0 1.5 | a 2| SAID MC simulation (e) Modulus of a2. 12345 s = E 2( GeV 2) 2 0 2 Arg( a 2 ) SAID MC simulation (f) Phase of a2. 12345 s = E 2( GeV 2) 0 1 2 | a 3| SAID MC simulation (g) Modulus of a3. 12345 s = E 2( GeV 2) 4 2 0 2 4 Arg( a 3 ) SAID MC simulation (h) Phase of a3. Figure 27: Values of the modulus and phase of the partial-wave amplitudes aiof the basic model in terms of s=E2 cm. The blue lines represent the theoretical model values after applying the change in the general phase of the system such that a0∈R+. The values obtained from the Monte Carlo simulations are represented using different colored triangular markers, which allows us to show the ambiguity in the solution obtained for each energy bin. The solutions presented in this figure have been obtained using a simulation of 108events. For visibility purposes, we will represent the graphs presented in 27, but excluding the error bars. 61
B.2 Results and Figures B BASIC MODEL 12345 s = E 2( GeV 2) 0.6 0.8 1.0 1.2 1.4 | a 0| SAID MC simulation (a) Modulus of a0. 12345 s = E 2( GeV 2) 4 2 0 2 4 Arg( a 0 ) 1e 2 SAID MC simulation (b) Phase of a0. 12345 s = E 2( GeV 2) 0 2 4 6 8 | a 1| 1e 1 SAID MC simulation (c) Modulus of a1. 12345 s = E 2( GeV 2) 2 0 2 Arg( a 1 ) SAID MC simulation (d) Phase of a1. 12345 s = E 2( GeV 2) 2 3 4 5 6 7 | a 2| 1e 1 SAID MC simulation (e) Modulus of a2. 12345 s = E 2( GeV 2) 2 0 2 Arg( a 2 ) SAID MC simulation (f) Phase of a2. 12345 s = E 2( GeV 2) 0.0 0.2 0.4 0.6 0.8 1.0 | a 3| SAID MC simulation (g) Modulus of a3. 12345 s = E 2( GeV 2) 2 0 2 Arg( a 3 ) SAID MC simulation (h) Phase of a3. Figure 28: Values of the modulus and phase of the partial-wave amplitudes aiof the basic model in terms of s=E2 cm. The blue lines represent the theoretical model values after applying the change in the general phase of the system such that a0∈R+. The values obtained from the Monte Carlo simulations are represented using different colored triangular markers, which allows us to show the ambiguity in the solution obtained for each energy bin. It’s relevant to mention that the number of ambiguities is directly related to the ℓmax used during the data analysis in each specific energy bin: namb =2ℓmax . The error bars have not been added for visibility purposes. The solutions presented in this figure have been obtained using a simulation of 108events. 62
C SAID MODEL C SAID model 12345 s = E 2( GeV 2) 0.0 0.5 1.0 1.5 | a 0| SAID MC simulation (a) Modulus of a0. 12345 s = E 2( GeV 2) 1 0 1 Arg( a 0 ) SAID MC simulation (b) Phase of a0. 12345 s = E 2( GeV 2) 0.5 0.0 0.5 1.0 | a 1| SAID MC simulation (c) Modulus of a1. 12345 s = E 2( GeV 2) 4 2 0 2 4 Arg( a 1 ) SAID MC simulation (d) Phase of a1. 12345 s = E 2( GeV 2) 0.5 0.0 0.5 1.0 | a 2| SAID MC simulation (e) Modulus of a2. 12345 s = E 2( GeV 2) 3 2 1 0 1 2 Arg( a 2 ) SAID MC simulation (f) Phase of a2. 12345 s = E 2( GeV 2) 1.0 0.5 0.0 0.5 1.0 | a 3| SAID MC simulation (g) Modulus of a3. 12345 s = E 2( GeV 2) 5.0 2.5 0.0 2.5 5.0 Arg( a 3 ) SAID MC simulation (h) Phase of a3. Figure 29: Values of the modulus and phase of the partial-wave amplitudes aiof the SAID model in terms of s=E2 cm. The blue lines represent the theoretical model values after applying the change in the general phase of the system such that a0∈R+. The values obtained from the Monte Carlo simulations are represented using different colored triangular markers, which allows us to show the ambiguity in the solution obtained for each energy bin. The solutions presented in this figure have been obtained using a simulation of 108events. For visibility purposes, we will represent the graphs presented in 29, but excluding the error bars. 63
C SAID MODEL 12345 s = E 2( GeV 2) 2 4 6 | a 0| 1e 1 SAID MC simulation (a) Modulus of a0. 12345 s = E 2( GeV 2) 4 2 0 2 4 Arg( a 0 ) 1e 2 SAID MC simulation (b) Phase of a0. 12345 s = E 2( GeV 2) 0.2 0.4 0.6 0.8 1.0 | a 1| SAID MC simulation (c) Modulus of a1. 12345 s = E 2( GeV 2) 3 2 1 0 1 Arg( a 1 ) SAID MC simulation (d) Phase of a1. 12345 s = E 2( GeV 2) 0 1 2 3 | a 2| 1e 1 SAID MC simulation (e) Modulus of a2. 12345 s = E 2( GeV 2) 3 2 1 0 1 Arg( a 2 ) SAID MC simulation (f) Phase of a2. 12345 s = E 2( GeV 2) 0.0 0.5 1.0 1.5 2.0 | a 3| 1e 1 SAID MC simulation (g) Modulus of a3. 12345 s = E 2( GeV 2) 3 2 1 0 1 Arg( a 3 ) SAID MC simulation (h) Phase of a3. Figure 30: Values of the modulus and phase of the partial-wave amplitudes aiof the SAID model in terms of s=E2 cm. The blue lines represent the theoretical model values after applying the change in the general phase of the system such that a0∈R+. The values obtained from the Monte Carlo simulations are represented using different colored triangular markers, which allows us to show the ambiguity in the solution obtained for each energy bin. It’s relevant to mention that the number of ambiguities is directly related to the ℓmax used during the data analysis in each specific energy bin: namb =2ℓmax . However, in this case only 2ℓmax−1solutions can be observed for each energy value, as we have chosen not to plot the complex conjugates of the already represented solutions. The error bars have not been added for visibility purposes. The solutions presented in this figure have been obtained using a simulation of 108events. 64
D INTRODUCTORY CONCEPTS D Introductory concepts D.1 Wigner D-functions The Wigner D-matrices are unitary square matrices that constitute the rotation operators in the basis of angular momentum eigenstates (eigenstates of Jx,Jy,Jz). They depend on the Euler angles (α, β, γ) and we can define their elements, the Wigner D-functions, as follows: Dj m′,m(α, β, γ)=⟨jm′|R(α, β, γ)|jm⟩=e−im′αdj m′,m(β)e−imγ(120) Where Ris a rotation operator defined as R(α, β, γ)=e−iαJxe−iβJye−iγJzand dj m′,m(β) are the elements of the Wigner d-matrices, also known as the Wigner d-functions, which are defined as: dj m′,m(β)=⟨jm′|e−iβJy|jm⟩ =(j+m′)!( j−m′)!( j+m)!( j−m)!1 2 smax X s=smin (−1)m′−m+scos β 22j+m−m′−2ssin β 2m′−m+2s (j−m−s)!s!(m′−m+s)!( j−m′−s)! (121) Where smin =max{0,m−m′},smax =min{j+m,j−m′}. The Wigner D-functions possess certain properties that make them quite useful. They form an orthogonal set over the Euler angles and satisfy the following symmetry relation: Dj m′,m(α, β, γ)=(−1)m′−me−2mα−2m′γDj −m′,−m(α, β, γ) (122) For specific values of j,mand m′(with m,m′∈ {−j, ..., j}), the Wigner D-functions can be written explicitly. Let’s give some examples considering the following Euler angles (α, β, γ)=(Φ, θ, −Φ): D0 0,0(Φ, θ, −Φ)=1D1 1,0(Φ, θ, −Φ)=−1 √2sin θe−iΦ D1 0,0(Φ, θ, −Φ)=cos θD1 −1,0(Φ, θ, −Φ)=1 √2sin θeiΦ (123) D.2 Helicity In order to properly understand the concept of helicity, we must first study a single particle’s states and spin quantum numbers in different reference frames. Given a single particle, let |jm⟩be the state of that particle in its rest frame, where jis the particle’s spin and mis the spin’s 3rd component. We will refer to this state as its ”rest state”. Applying a Lorentz boost to that state, we can define the canonical state of a single particle with spin j, 3rd component of spin mand momentum pas | p,jm⟩. Having presented these definitions, we are ready to describe a free particle’s “helicity state”. We can construct the helicity states in two different ways, both starting from a rest state |jλ⟩: a) Starting from a rest state |jλ⟩, one may apply a rotation ˚ Rso that the spin quantization axis (zaxis) is positioned along the pdirection and then boost the rest state along the pto obtain the helicity state | p,jλ⟩. b) Alternatively, one may first boost the rest state along the zaxis and then apply a rotation ˚ R to obtain the helicity state | p,jλ⟩. 65
D.2 Helicity D INTRODUCTORY CONCEPTS Therefore, by definition, the helicity quantum number λis the projection of spin onto the direction of the particle’s momentum p:λ= S p S| p|. It is also the projection of the total angular momentum Jalong ˆp= p | p|, because the orbital angular momentum L= r× pis perpendicular to ˆpand, consequently its projection onto this direction is 0. We will commonly refer to the helicity quantum number as simply “helicity”. The rotational invariance and the invariance under Lorentz boosts (“pure” Lorentz transformations) of helicity will be extremely useful in our study of the b− 1meson photoproduction. Since these two characteristics will be essential to progress in our system’s formalism, let us explore them deeper and properly justify their validity. We can represent all Lorentz transformations as unitary operators. In the case of a general Lorentz transformation Λ, we will refer to its unitary operator as U[Λ]. For specific cases such as boosts Land rotations R, we will represent their unitary operator as U[L]and U[R], respectively. Given two arbitrary Lorentz transformations Λ1,Λ2, when working with bosons their multiplication is preserved: U[Λ2Λ1]=U[Λ2]U[Λ1](124) Let L βbe a pure Lorentz transformation (boost) on the arbitrary direction β. Then: L β=R(ϕ, θ, 0)Lz(β)R−1(ϕ, θ, 0)(125) Where Lz(β)is a pure Lorentz transformation along the zaxis with velocity βand R(ϕ, θ, 0)is the rotation which takes the zaxis into the direction of βwith spherical angles (ϕ, θ):ˆ β=R(ϕ, θ, 0)ˆz. From the relation 125, we obtain: UL p=Uh˚ R(ϕ, θ, 0)iULz(p)U−1h˚ R(ϕ, θ, 0)i(126) Where ˚ Ris the rotation that takes the zaxis into the direction of pspherical angles (θ, ϕ): ˆp= ˚ R(ϕ, θ, 0)ˆz. Now, we can formally define helicity states using the previously presented construction and the newly introduced nomenclature: | p,jλ⟩=|ϕ, θ, p,jλ⟩=UL pUh˚ R(ϕ, θ, 0)i|jλ⟩=Uh˚ R(ϕ, θ, 0)iULz(p)|jλ⟩(127) It is worth noting that there is a big difference between L pand Lz(p), as the first operator is a much more complicated mathematical object and possesses an angular dependence. Having presented all of these introductory definitions, we can study the effect of a rotation Rover the helicity value. Applying the definition 127: U[R]| p,jλ⟩=UhR˚ RiULz|jλ⟩=|R p,jλ⟩(128) Equivalently, we can study the effect of a Lorentz boost L′which takes pinto p′, with p′|| p, over the helicity value. The invariance of λunder L′may be seen by: UL′| p,jλ⟩=UL′UL pUh˚ Ri|jλ⟩=UL p′Uh˚ Ri|jλ⟩=| p′,jλ⟩(129) We have proven that, as previously stated, the helicity quantum number remains invariant under rotations and Lorentz boosts along the pdirection. 66
E MOMENT EXPANSION FOR THE b− 1PHOTOPRODUCTION E Moment expansion for the b− 1photoproduction In subsection 3.2.2, we have made an effort to reduce the product of 2 Wigner D-functions and 2 spherical harmonics, respectively, into lineal combinations of Wigner D-functions. These calculations might have seemed arbitrary, but they will be extremely helpful in the development of the moment expansion of our reaction chain’s intensity. It is relevant to note that the development that will be introduced in this Appendix is similar to the ones presented in [10, 21, 33] Similarly to the simpler case presented in section 2.2.6, the moment expansion is a Fourier decomposition of the system’s differential cross-section. In this case, instead of using Legendre polynomials as expansion functions, we will employ pairs of Wigner D-functions: DL∗ Mm (Ωω)Dl∗ m0(ΩH), each element of the product pair corresponding to the angular description of the b− 1and the ω decays, respectively. We start working from expression (89) and, as a way of simplifying our system, we assume ∆++ to be stable, We integrate with respect to Ωpand we obtain: Iα(Ωω,ΩH)=3κX Λ,λ X m,M ρα Λ,Λ−M(s,t)·|G|2FλF∗ λ′˜ F 2· ·X ℓ 2ℓ+1 4π!C1,λ 1,λ−m;ℓ,mC1,0 1,0;ℓ,0Dℓ∗ m,0(ΩH)· ·X L 2L+1 3!C1,Λ 1,Λ−M;L,MC1,λ 1,λ−m;L,mDL∗ M,m(Ωω) (130) As a tool to simplify the previous expression, we define the multipole parameters [21]: tα∗ LM =X Λ ρα Λ,Λ−MC1,Λ 1,Λ−M;L,M(131a) fLℓm=X λ FλF∗ λ−mC1,λ 1,λ−m;L,mC1,λ 1,λ−m;ℓ,m(131b) Finally, we are able to define the moments: Hα(LMlm)=3κG˜ F 2tα∗ LM fLℓmC1,0 1,0;ℓ,0(132) And expand the system’s intensity in terms of the proposed moments: Iα(Ωω,ΩH)=X L,MX l,m 2l+1 4π! 2L+1 3!Hα(LMlm)DL∗ Mm (Ωω)Dl∗ m0(ΩH)(133) It is relevant to note how convenient having this expansion is when analyzing experimental data, as these moments Hα(LMlm) are observable magnitudes and constitute a finite set of values that provide us with all of the angular information of our studied reaction chain (Figure 2). 67
F SDMES MODEL F Spin Density Matrix Element calculation model F.1 Kinematics and conventions for the spin functions In this Appendix, we will present a series of introductory definitions necessari to calculate Vt λγ,Λ;λ1λ∆. The model is written in the t-channel (section 3.1.1), the center-of-mass frame of the crossedchannel reaction γb+ 1→¯p∆++. In this channel, both the photon and the b+ 1are moving along the zdirection, the photon in the positive direction and the b+ 1in the negative one. The b+ 1meson has three spin components (λb∈ {−1,0,1}) but the photon has only two (λγ∈ {−1,1}). Their polarization vectors are given by: ϵ(pγ, λγ)=−λγ √20,+1, λγi,0(134a) ϵ(pb, λb)=−λb √2(0,−1, λbi,0)+1−λ2 b mb1pt,Et bsin θt,0,Et bcos θt(134b) It is easy to see that the defined polarization vectors are orthogonal to the particles’ four-momentums (ϵ(p, λ)·p=0), since: pµ γ=(pt,0,0,+pt)pt=t−m2 b 2√t(135a) pµ b=(Eb,0,0,−pt)Eb=t+m2 b 2√t(135b) Considering that the ¯pand the ∆++ form an angle θtwith the zaxis, we can define their fourmomenta: pµ ¯p=Et p,+qtsin θt,0,+qtcos θtqt=λ1/2(t,m2 ∆,m2 p) 2√t(136a) pµ ∆=Et ∆,−qtsin θt,0,−qtcos θtEt p/∆=t±m2 p∓m2 ∆ 2√t(136b) For completeness, we will also provide the following expression for the t-channel scattering angle: cos θt=s−u 4ptqt +m2 b 4t m2 ∆−m2 p ptqt (137) For the anti-proton, we need the spinors: v(p¯p, λp= +1/2) =−βp αpχ− −χ−!χ−= −sin θt cos θt!αp=qt/(Et p+mp) (138a) v(p¯p, λp=−1/2) =−βp αpχ+ χ+!χ+= cos θt sin θt!βp=qEt p+mp(138b) For the ∆, we need to construct a spin-3/2 function by coupling a spin-1 function and a spin-1/2 one with a Clebsch-Gordan coefficient: ¯uµ(p∆, λ∆)=X λ,λ′ C 3 2λ∆ 1λ;1 2λ′ϵµ∗(p∆, λ)¯u(p∆, λ′) (139) 68