Angular Distributions, Polarization Observables, Spin Density Matrices and Statistical Tensors in Photoproduction of Two Pseudoscalar Mesons off a Nucleon
aa r X i v : . [ h e p - ph ] F e b FLORIDA STATE UNIVERSITYCOLLEGE OF ARTS AND SCIENCESANGULAR DISTRIBUTIONS, POLARIZATION OBSERVABLES, SPIN DENSITYMATRICES AND STATISTICAL TENSORS IN PHOTOPRODUCTION OF TWOPSEUDOSCALAR MESONS OFF A NUCLEONByHIRAM G. MENENDEZ SANTIAGOA Dissertation submitted to theCollege of Arts and Sciencesin partial fulfillment of therequirements for the degree ofDoctor of Philosophy2020Copyright © CKNOWLEDGMENTS
As is the case with any endeavor requiring hard work and dedication, this work would not havebeen possible without the help of some wonderful individuals that have earned my gratitude. Noone has helped me more in the development of this research than my adviser, Dr. Winston Roberts.Without his guidance, completion of this work would have been significantly harder. I would nothave been able to arrive at some of the results in this work if not for his insight and experience. Thefact that we would often both arrive at the same conclusions by thinking in completely differentways was of immense value. But even more so were the times when his insight would cast doubtinto some of the results I would obtain. Having someone always available that could challenge mywork and ideas made this research project much better than it would have been. I am also gratefulfor our many discussions on topics in physics that were, at best, tangentially related to this work,and to those not related in any way to earning my degree.I would like to thank the many wonderful friends that I have met after arriving in Tallahassee,including many who left town long ago to pursue greater things. First, thanks to the members ofthe Friday night board games group: Jonathan Baron, Nellie Speirs Baron, Andrue ChristopherHenderson, Sierra Henderson, Chaille Kent, Stephen Kolar and Taylor Van Winkle. Those nightswere consistently some of the best highlights of my weeks. The games were great, but all of youwere phenomenal. Thanks also to the awesome friends I met at a trip to New Orleans, where webonded in our misery at not being able to sit down and rest our feet unless we bought alcohol wedid not intend to drink: Archishman Ghosh, Vishav Pandhi, Dhevathi Rajagopalan, and GarretDan Vo. We could always on Archis to come up with new ideas for fun things to do. I’m happythat Archis and Dhev had the chance to attend Friday night board games, if only for a brief time.I am also grateful to have had the company of Dhev while working long hours at All Saints Caf´e,which made the work experience so much more enjoyable. I wish to thank my office mate and friendZulkaida Akbar, with whom I also spent many times at All Saints Caf´e. I cherish all the good timesI’ve had with him and I am proud to have helped him obtain his driving license. I am also happyto have helped Dhev obtain his driving license as well. I was lucky to have another group of friendsfrom a different game group: Ryan Baird, Dan Mock, and Alex Parker. I was introduced to manygreat games thanks to them, and I also enjoyed our many conversations. Many thanks to Anishiiihardwaj, one of the first people I met at the physics department, and all the times we spent atSweet Pea Caf´e. Thanks also to Nabin Rijal. I will miss the free beer we got twice a week at PoorPaul’s Poorhouse. Last but not least, I am extremely grateful to have met Weng Ramirez. I neverwould have guessed I would ever visit The Philippines and have such an amazing time. Thank youso much for all you’ve done for me.But I also have to thank those friends who I have met many years before the beginning of mygraduate studies. While I have not been able to physically spend time with them in a long time, thedays I get to contact them are always great. These include Ricardo Arzola, Janet Col´on Castellano,Walter Morales, Kendrick Ng and Juan Carlos S´anchez.I wish to give a huge thanks to the Florida State University Department of Physics, for givingme the chance to pursue this degree despite my very limited experience in the field of physics. Ithas been a wonderful experience to finally get work in this field, which is something I had wantedto do for a long time. I would like to thank the excellent professors that make up this department.Many thanks to Dr. Jorge Piekarewicz, for his excellent lectures on classical electrodynamics andfor his office’s open door policy that led to us having some interesting conversations. Thanks alsoto Dr. Efstratios Manousakis, for his also excellent lectures on quantum mechanics, his carefullyhandwritten lecture notes that I still keep, and for the discussions we’ve had, especially on thefoundations of quantum mechanics. Many thanks to Dr. Simon Capstick, for always being availablein his office to answer any kinds of questions that came to my mind. I also wish to thank Dr. VolkerCrede, for our helpful discussions regarding my research and the experiments being done at JLab.From outside the Department of Physics, I would like to thank Dr. Wei Yang for taking some timeout of his work to serve on my graduate committee. And finally, many thanks to Dr. Don Robson,whose comments regarding my research during my presentation at a nuclear seminar led me tosignificantly expand the scope of this research.I am eternally grateful to the members of my family, among them my sisters Lorena Men´endezand Hiradith Men´endez, and my favorite niece Lorena Gabriela Men´endez, . While we are eachliving our own separate lives far away from each other, it makes me happy to know that I havea group of people in my life that I know I can always count on, no matter the time, place, orcircumstances. ivnd finally, but certainly not least, I give my heartfelt thanks to my mother, Judith IvetteSantiago Rivera. I can say, without a shadow of a doubt, that there has never been anyone inthe world who has helped, taught, worried about and cared about me to the extent that she has.Everything I have accomplished in life, including this work, would not have been possible withoutthe immense influence she has had in shaping my life.I would like to thank the FSU College of Arts and Sciences and the FSU Office of Research formaking this work possible. This work was partially supported by the US Department of Energythrough award DE-SC0002615. v
ABLE OF CONTENTS
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x γN → M M B
307 Polarization Observables 358 Angular Distributions in γN → M ∗ B → M M B , γN → M B ∗ → M M B , and γN → M B ∗ → M M B
479 Polarization Observables in Terms of SDME’s and Statistical Tensors of theQuasi-Two-Body States 5410 Parity Invariance Considerations 6011 Angular Distributions and Polarization Observables When More Than OneQuasi-Two-Body State Channel Contributes 65
12 General Expressions for the Angular Distributions and Non-Recoil PolarizationObservables 76
14 Summary 11115 Conclusions 120AppendixA Changing Phase Space Coordinates 126
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137vii
IST OF TABLES φ ∗ → − φ ∗ . . . . . . . . . . . . . . . . . . . . . . . . . 9613.1 Symmetry of recoil observables under φ ∗ → − φ ∗ when the polarization is measuredalong the x ′ -axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10913.2 Symmetry of recoil observables under φ ∗ → − φ ∗ when the polarization is measuredalong the y ′ -axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10913.3 Symmetry of recoil observables under φ ∗ → − φ ∗ when the polarization is measuredalong the z ′ -axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110viii IST OF FIGURES γN → ππN in the overall center-of-mass frame. Thecoordinate system is defined such that k , p and p ′ lie on the z - x , plane (blue). Con-servation of momentum constrains the three momenta of the final states to lie on aplane (green), shown in the figure with the vector normal to it, ˆ n . Another coordinatesystem with the same y -axis but with its z ′ -axis pointing in the direction of q + q is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Kinematics for the reaction γN → ππN in the overall center-of-mass frame with thecoordinate system defined such that q now lies on the z - x plane (blue). We canarrive at this coordinate system from the one in fig. 3.1 by applying a rotation alongthe normal to the green plane ˆ n . From here we can reach the center-of-mass frame of q and p by boosting in the negative z ′ direction. . . . . . . . . . . . . . . . . . . . . 163.3 Kinematics for the reaction γN → ππN in the pion-pion center-of-mass frame. It isreached from the overall center-of-mass frame shown in fig. 3.1 by boosting along the z ′ direction. The three-momenta of the pions lies on the green plane. Their angulardistribution is given by θ ∗ and φ ∗ , which are the polar and azimuthal angles of q ∗ inthe x ′ - y - z ′ coordinate system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1 The direction of the electric field vector of linearly polarized photons is defined as theangle, β , it makes with respect to the x -axis of the x - y plane. . . . . . . . . . . . . . . 22ix BSTRACT
In two meson photoproduction off a nucleon, for the case in which two of the three final statehadrons are products of the decay of an intermediate resonance, general expressions for its decaydistribution and for polarization observables are derived in a model independent way. These arefunctions of either the spin density matrix elements (SDME’s) or statistical tensors of the resonance,and the angles of its decay products. The expressions are general enough that it also describes caseswhere more than one resonance of arbitrary quantum numbers contributes, including interferenceeffects. They can therefore be used to extract the SDME’s or statistical tensors of the resonancesthat contribute to the reaction. x
HAPTER 1INTRODUCTION
The field of hadronic nuclear physics aims to understand phenomena arising due to the strongnuclear force, and the properties of particles that interact via said force, called hadrons. Its existencewas deduced from the fact that the electromagnetic interaction could not be responsible for bindingprotons and neutrons in an atomic nucleus, since they have positive and neutral charge, respectively.In the electromagnetic interaction, positively charged particles would experience a repulsive force,and neutral particles would not participate in said interaction. It was therefore clear that protonsand neutrons had to also participate in another type of interaction, stronger and shorter in rangethan the electromagnetic one. This interaction came to be known alternatively as the strong nuclearforce, the strong force, or the strong interactions. This section highlights some of the importantdiscoveries that have increased our understanding of it throughout the years after its discovery.In 1932, Werner Heisenberg made the observation that, if the electromagnetic interaction couldsomehow be “turned off”, protons and neutrons would interact via the strong force identically. Infact, if additionally it were the case that the proton and the neutron had identical masses, therewould be no way to distinguish between them. Since the electromagnetic interaction is orders ofmagnitude weaker than the strong one, and since the proton and neutron mass is almost identical, ifall protons in a nucleus were to be replaced with neutrons and vice versa, the mass of the resultingnucleus would be almost identical to the previous one.Heisenberg therefore introduced the concept of an approximate symmetry which came to beknown as isospin [1]. The reason for this name is due to its analogy with the concept of spin. Allthe fundamental particles known at the time have spin 1/2 (in units of ℏ ), and came in two stateswith different spin projection along a coordinate axis:+1/2 and − − SU (2). The nucleonbelongs to its fundamental representation, and is therefore an isospin doublet. Since this is anapproximate symmetry, the members of this doublet (the proton and the neutron) are almostdegenerate in mass. And just as angular momentum is conserved in interactions due to rotationalsymmetry, total isospin is conserved in the strong interactions, but not in the electromagnetic andweak interactions. One example of the usefulness of isospin symmetry is that it can be used torelate the reaction rates of different strong processes that are related by rotations in isospin space.The first hadrons other than the nucleon were discovered in cosmic ray experiments. In these,detectors such as bubble chambers were constructed in order to detect the particles produced whencosmic rays collide with Earth’s atmosphere. It was in these experiments that the charged pionswere discovered in 1947 [2].The neutral pion was discovered three years later [3–7]. The pionswere actually predicted to exist earlier in 1935 by Hideki Yukawa [8], who described it as themediator of the strong interaction among nucleons (in the same way that the photon mediatesthe electromagnetic interactions). He also correctly predicted its mass based on the range of theinteraction.Unlike the nucleon, which has half-integer spin and is therefore a fermion, the pions have integerspin 0 and are therefore bosons. Hadrons that are fermions are called baryons, while those thatare bosons are called mesons. There are three pions with different electric charges that are almostdegenerate in mass: π + , π , and π − . This same year, the kaons were also discovered [9], also havingdifferent electric charges and almost degenerate masses: K , K + , K − , and K .Particle accelerators with energies in the hundreds of MeV were eventually built. This led tothe discovery of many more “fundamental” particles during the 1950’s and 1960’s, such as the Λ baryon [10], the η and η ′ mesons [11–13], the ∆ baryons [14, 15], the Σ baryons [16], etc. Findingpatterns in the properties of these particles (sometimes informally called the “particle zoo”) as away to categorize them into different classes became important.It was found that, in the same way that the states of the nucleon formed an isospin doublet, sotoo could these newly discovered hadrons be assigned to different representations of SU (2) isospin,with each of their members having nearly degenerate masses. The three pion states form an isospintriplet, K and K + form a doublet, K − and K form another doublet, Λ forms a singlet, η and2 ′ each forms a singlet, the Σ baryons form a triplet (Σ + , Σ , and Σ − ), the delta baryons form aquadruplet (∆ ++ , ∆ + , ∆ and ∆ − ), etc.Another property that was used to categorize them came from the observation that some ofthese hadrons had mean lifetimes that were orders of magnitude larger than others. Because of this“strange” behavior, they were given the name strange hadrons. Their long lifetimes were explainedby the introduction of a new quantum number called strangeness [17–20], which is conserved inthe strong and electromagnetic interactions but not in the weak interaction. Therefore, the formerinteractions can produce particle pairs with net strangeness of zero in relatively large amounts. Butsince each particle in the pair carries strangeness, their ground states can only decay via the weakinteraction, which leads them to have longer mean lifetimes.The isospin multiplets that grouped the different hadrons into groups were still too numerous.A more general organization scheme called the eightfold way was developed in 1961 by MurrayGell-Mann and Yuval Ne’eman [21, 22]. In this scheme, the hadrons were organized into larger SU (3) supermultiplets, with its members labeled by their isospin projection ( SU (2) is a subgroupof SU (3)) and strangeness. The name eightfold way is a reference to the fact that SU (3)’s asso-ciated lie algebra is eight-dimensional, i.e., it has eight generators (unlike SU (2), which has 3).The degeneracy in the mass of the members of these multiplets was not as exact as in SU (2)isospin, making SU (3) a less exact symmetry than the former, but still good enough to be useful.Experimental validity for the eightfold way came with the discovery of the omega baryon (Ω − ) in1964 [23].The mesons were organized into a singlet and an octet representation, while the baryons wereorganized into an octet and decuplet representation. But no collection of particles seemed to belongto the fundamental representation, the triplet. It was also known that tensor representations ofgroups can be decomposed into irreducible representations. For example, the representation 3 ⊗ ¯3(the number refers to the dimensions of the representation, while the bar refers to the conjugaterepresentation) can be decomposed into 8 ⊕
1, the octet and singlet representations found in themesons. The representation 3 ⊗ ⊗ ⊕ ⊕ ⊕
1, which contains the decuplet andoctet representations found in the baryons. It was therefore proposed, independently by MurrayGell-Mann and George Zweig in 1964 [24,25], that the hadrons were actually bound states of point-3ike spin-1/2 constituents belonging to the triplet representation. These were called quarks, andthis proposal gave rise to the first quark model.Since the quarks belong to the three-dimensional representation, they come in three types, orflavors, called up, down, and strange ( u , d and s ). Their associated antiquarks (¯ u , ¯ d and ¯ s ) belongto the conjugate of this representation. As their names imply, the up and down quark have zerostrangeness, while the strange quark and strange antiquark have strangeness with opposite sign.The baryons could therefore be organized into an octet and a decuplet because they are boundstates of three quarks (for the antibaryons, three antiquarks). The mesons could be organizedinto a singlet and an octet because they are bound states of a quark and an antiquark. Since thehadrons have integer electric charge, the quarks must therefore have fractional charges (+2/3 forthe up quark, -1/3 for the down and strange quark. The antiquarks, have the opposite charge).This approximate SU (3) symmetry came to be known as flavor symmetry, since it describes thesymmetry between different flavored hadrons.The reason that this symmetry is not exact is in part due to the fact that the quarks havedifferent masses. SU (2) isospin is a better symmetry than SU (3) flavor because the up and downquark masses are very nearly identical. The latter is more badly broken because the strange quarkhas a significantly higher mass than the other two.While this quark model was very successful in describing the properties of the known hadronsat the time, its biggest perceived failure was that quarks were never found in isolation, leadingto discussions about whether quarks were “real” or the quark model was just a mathematicalbookkeeping tool, hiding as of yet unknown physics that would explain the properties of the hadrons.It would not be until the late 1960’s and 1970’s that deep inelastic scattering experiments doneat the Stanford Linear Accelerator Center (SLAC) confirmed that the nucleon was made up offractionally-charged point-like constituents [26, 27].Eventually, three other flavors of quarks were discovered: the charm [28, 29], bottom [30], andtop quarks [31, 32], with electric charges +2/3, -1/3, and +2/3, respectively. This means that thehadrons can, in principle, be organized into SU ( N ) multiplets, with N being the number of quarkflavors (It should be noted, however, that the top quark does not form bound states because itslifetime is too short). However, these three quarks are much higher in mass than the three lightquarks, so the symmetry ends up being much more badly broken.4nother problem came from the observation that the ∆ ++ baryon required the spins of itsthree up quarks to be parallel and have zero total orbital angular momenta. It therefore seemedthat this baryon could not have an antisymmetric wave function as is required of fermions by thePauli exclusion principle. A proposed solution was to postulate the existence of a new quantumnumber, called color [33]. The idea was that the quarks came in three different color states: red,blue, and green. This also offered an explanation for the lack of observation of quarks in isolationby postulating the color hypothesis, which states that the color quantum number in all observedhadrons is described by a totally antisymmetric, color-singlet wave function. Therefore, a red,blue, and green quark would form bound states that have net color “white”, i.e., it would bea color neutral state, while a colored quark and antiquark of the corresponding anti-color couldalso form a color-neutral bound state. Since individual quarks have net color, they can never beobserved in isolation. They can only be confined as color-neutral bound states that manifest asthe hadrons that are observed in experiments. This idea is often called color confinement. Inmathematical language, a new SU (3) color symmetry was postulated in which, in color space,quarks and antiquarks belonged to the fundamental representation and its conjugate, respectively,and that only bound states belonging to the singlet representation in color space (color neutral)are realized in nature.Color confinement also has the implication that the energy of the bound quarks would increaseif they get farther away from each other. Therefore, it was expected that if one of the quarksgets struck in a scattering event and gets pulled apart from the other quarks, energy gained bythe system due to their strong interactions may be enough for new quark-antiquark pairs to becreated from the vacuum, which could then form new hadrons with the outgoing quark and theother remaining quarks.While more experiments were providing clues that would hopefully lead to a general theoryof the strong interactions, the electromagnetic interaction at the quantum level was much betterunderstood. It’s theory was called quantum electrodynamics (QED), which is a field theory offermions and the photon. In addition to having the required global Lorentz symmetry, the U(1)global symmetry already present in the Dirac equation that described relativistic spin-1/2 fermionswas promoted to a local U(1) gauge symmetry. Local symmetries require the presence of masslessvector gauge bosons belonging to the adjoint representation of the symmetry group. The adjoint5epresentation of U(1) is one dimensional, so only one vector gauge boson is needed, the photon.Gauge bosons are said to be the force carriers that mediate the interaction. Therefore, interactionsamong particles can emerge from the requirement of local gauge invariance.Since QED was extremely successful in correctly calculating known experimental quantities tovery high precision (such as the fine structure constant), the idea to formulate the strong interactionsas a local gauge theory eventually emerged and the SU (3) color symmetry was promoted to a localgauge symmetry [34,35]. The group SU (3) is non-abelian, unlike U(1) which is abelian. This madethe field theory of the strong interactions more complicated than QED, and had properties notfound in the latter. For example, since the adjoint representation of SU (3) is eight dimensional,the gauge bosons of the theory belong to a color octet. The theory therefore had 8 gauge bosonswith different color charges, called gluons, in contrast to the single one in QED. Since the gluonsthemselves have a color charge, they can interact directly with each other. By contrast, photonscannot directly interact with each other since they lack an electric charge. This theory of the stronginteractions was given the name quantum chromodynamics (QCD).QCD has a property called asymptotic freedom [36, 37], by which the strength of the stronginteraction is less at higher momentum transfers (or shorter distance scales). This is due to the factthat the coupling strength between quarks and gluons receives quantum corrections that dependon the interaction energy, giving rise to a renormalized coupling. Unlike in QED, in QCD therenormalized coupling becomes smaller at larger energies. This means that in high energy collisions,the quarks and gluons barely interact with each other. Therefore, calculations using the techniquesof perturbation theory can be applied in this energy regime in QCD to make precise predictions.These calculations have been put to the test in many experiments, and were very successful indescribing experimental data.Due to the immense success of QCD, it is nowadays widely considered the correct theory ofthe strong interactions. However, perturbation theory cannot be applied to low-energy phenomenain the non-perturbative energy regime, where the interaction strength becomes large. Therefore,perturbative QCD is incapable of describing a lot of phenomena of interest, such as the hadronspectrum, the nature of color confinement, the hadronization process (in which individual quarksscattered during reactions form hadrons), and the momentum, position, or spin distributions of the6uarks that form the hadrons. The phenomena of color confinement has also not been able to beproven analytically from QCD.The only known way to extract information about low-energy phenomena from first-principlesQCD calculations is from a formalism known as lattice QCD (LQCD) [38–40], in which spacetimeis discretized to form a lattice in order to perform numerical computations. For example, evidencefor color confinement has been gathered from LQCD [41, 42]. The drawback of this method is theimmense amount of time and computational resources needed. Many techniques have been used tocut down the computation time to manageable levels, such as using pion masses that are heavierthan their real values as input, but at the cost of having less accurate results. But advances inin computer technology and computational techniques have allowed more realistic computations tobe performed. Despite these drawbacks, LQCD has been very successful in describing the hadronspectrum, correctly predicting, with some uncertainty, the masses of the known resonances [43–48].We can therefore distinguish between the short-distance perturbative regime, where stronginteraction physics is best described in terms of nearly free, point-like quarks and gluons, and thelong-distance non-perturbative regime, where it is best described in terms of hadrons. For manyof the long-distance phenomena of interest, there is often no known way to obtain general analyticexpressions for quantities of interest from QCD. For example, while the momentum distribution ofthe quarks and gluons in the hadrons can be measured from experiments [49–52], there is no knownway to find general expressions for them from QCD.Physicists have therefore relied on models to describe the long-distance physics of hadrons, eachwith varying degrees of success and range of applicability. For example, starting in the 1950’s, avariety of quark models have been used that have given good predictions to the hadron spectraand other properties such as couplings and decay rates [53, 54]. Many fall under the class knownas potential quark models, in which the interaction between constituent quarks is described witha non-local potential. Many of the potentials used in these models were in fact inspired by QCD,i.e., known properties of QCD were used as guidance in developing them. Early quark models werenon-relativistic, but eventually ones that did incorporate special relativity were developed.There are also low-energy effective descriptions of QCD. These come about when the energyscale of the interactions being studied is much smaller than some large energy scale that describessome short distance phenomena. From QCD, perturbative expansions can be done with respect to7he ratio of these energy scales, and the techniques of perturbation theory can be used to makecalculations. These are called effective field theories. One of them is heavy quark effective theory(HQET) [55,56], which gives an exact description in the limit where the mass of one of the quarks inthe hadron goes to infinity. It has been used to give good descriptions of hadrons which contain atleast one charm or bottom quark. Another such effective field theory is chiral perturbation theory( χ PT) [57, 58] which gives an exact description in the limit where the quark masses go to zero.In summary, since the discovery of the nuclear strong interactions responsible for the bindingof protons and neutrons into nuclei at the center of atoms, experimental and theoretical effortsthroughout the years have gradually increased our understanding of it, leading to the developmentof QCD, the theory of the strong interactions. Despite this, many aspects of the strong interactionare not well understood, especially in the low energy non-perturbative regime. This thesis presentsmy theoretical efforts in furthering our understanding of the strong interactions in this regime.8
HAPTER 2MOTIVATION
One area of experimental research that was fundamental in furthering our understanding of thestrong interaction is hadron spectroscopy, which is concerned with searches of resonances arisingin scattering experiments, and establishing their mass and decay widths. A compilation of theknown resonances and their respective masses and widths have been compiled in the Review ofParticle Physics publication by the Particle Data Group (PDG) collaboration [59]. Predictions forthe hadron spectrum have been extracted from constituent quark models since their developmentin the 1960’s. Many of the resonances predicted by these models have been found in experiments,and their calculated properties (masses, widths, couplings, etc.) have matched the measured valueswith a good degree of success. Years later, LQCD has also been able to make predictions about thehadron spectrum and, again, many of the predicted resonances were found in experiments, withthe calculated masses roughly matching those measured in experiments.In the case of baryon spectroscopy, much of the information that has been gathered on the low-lying non-strange and strange baryon resonances have been obtained from πN and ¯ KN scattering[60]. This energy region is complicated, with numerous overlapping and broad resonances, makingtheir identification difficult. Thus, measurements of the total and differential scattering crosssections are not enough to disentangle the contributions of the many resonances to the scatteringamplitude.To remedy this, Partial Wave Analysis (PWA) has been employed as a tool in analyzing theexperimental data [61–63], where the scattering amplitude is decomposed into a sum of partialwaves of definite orbital angular momenta, total isospin, and total angular momenta. This helpsin the search for resonances because each one of a particular spin, parity, and total isospin receivescontributions from a single πN partial wave. The goal of these analyses is to extract the partial waveamplitudes from experimental data, since their behavior as a function of the kinematic variables(center-of-mass energy, momentum transfer, etc.) can be interpreted in terms of the resonances thatcontribute to the scattering process, thus facilitating the identification of resonant contributions.9artial wave amplitudes can be found if the helicity amplitudes of the scattering process are known,which are the scattering amplitudes for processes in which the spin state of the particles involvedis known. PWA’s therefore require polarization experiments to be carried out, which make useof polarized beams and/or targets, and may be able to measure the polarization of the reactionproducts. The observables of interest that are used to extract the amplitudes are called polarizationobservables [64, 65], which is one of the main topics of this work.As more resonances were discovered, it became clear that, while many of the resonances pre-dicted by constituent quark models were found, there were also a significant number that werenot [54, 66–72]. LQCD also predicts an overabundance of resonances [73]. This is not necessarilya failure of quark models. One possible proposed solution is that some of the degrees of freedomwithin the quark model are “frozen”, leading to a reduction in the number of resonances. This canhappen if two of the constituent quarks in the baryon behaves as a collective unit. The baryonwill then effectively be a quark-diquark pair [74–77], and the reduction in the number of degreesof freedom would reduce the number of expected baryon resonances. While early formulationof “static” quark-diquark models have recently been excluded by experiments, “dynamic” quark-diquark models have also been proposed as a possible solution. However, even with this reduction,the quark-diquark model still predicts more resonances that have been observed.Another possible solution is that the missing/undiscovered resonances couple weakly to theproduction channel ( πN ). This hypothesis has been verified in quark model calculations [69,78–81].These same models suggest that many of these missing resonances couple strongly to channels suchas π ∆ and ρN (as of yet, LQCD calculations for strong decay couplings of exited baryons have notbeen carried out).Due to this, experiments using different production mechanisms, such as photoproduction andelectroproduction, have been proposed. These have been carried out in a number of facilities [82],such as CEBAF, ESRF, MAMI, ELSA, SPring-8 and ELPH. At CEBAF, for example, polarizedelectron and photon beams are produced [83], and its CLAS detector [84] contains a polarizedtarget (FroST) [85]. It is, however, not able to directly measure the polarization of the reactionproducts (but the polarization of weakly decaying particles, such as the Λ baryon, can be deducedfrom the angular distribution of their decay products).10ome reactions of recent interest have been single and double pion photoproduction, γN → πN and γN → ππN , where the photon and/or the target nucleon are polarized. Since it is wellknown that the latter reaction has contributions from quasi-two-body states, reactions such as γN → ρN → ππN and γN → π ∆ → ππN have also been of interest [86]. In these reactions, oneof the hadrons in the quasi-two-body state undergoes a two-body decay (in the previous examples, ρ → ππ and ∆ → πN ). It has often been of interest to measure the spin properties of this decayinghadron.In scattering experiments, the beam and target are made up of a large number of particles withspin. The spin state of every single particle that is a member of the beam or target is not known.Rather, only partial information is ever known: the expectation value of the spin projection amongall the particles, also known as the polarization of the system. Therefore, the beam and target arenot pure quantum states, but rather statistical ensembles of single particle spin states, with eachmember of the ensemble having a probability of being realized. This is called a mixed quantum state,and it cannot be described by a state vector, but rather by a density matrix [87, 88]. The densitymatrix formalism will be discussed in section 4. All spin information about the beam, target, andrecoiling particles is therefore contained in their respective spin density matrix. Experiments havecontrol over the spin state of the beam and target, which means the spin density matrix elements(SDME’s) of the spin and target will be known. For an initial spin configuration, the dynamics,i.e., the helicity amplitudes of the process will determine the SDME’s of the recoil baryon.Spin density matrices are not the only ways to represent the spin of a mixed state. Alternatively,the expectation values of the polarization operators, called statistical tensors, can also be used[87]. While recently in the literature the topic of polarization observables has almost always beendiscussed in terms of the SDME’s, this thesis will utilize both representations. This is because usingstatistical tensors instead of SDME’s offer many advantages that will be discussed throughout thisthesis. Their basic properties are discussed in section 5.Recently there has been interest in so-called complete experiments. These are scattering exper-iments in which enough polarization measurements are made to be able to extract all of the helicityamplitudes of the process (except for an overall phase). These types of measurements are calledpolarization observables, and will be discussed in section 7.11chieving complete experiments involves performing various scattering experiments where thebeam and/or target is polarized in some chosen direction, and may also involve measuring therecoil baryon’s polarization [89–91]. These allow for the measurement of polarization observablesat different kinematic points (scattering angle, momentum transfer, etc.), which can then be usedto extract the SDME’s of the recoil baryon for the chosen configuration of the beam and targetpolarizations. Since the values of these SDME’s are determined by the helicity amplitudes, theycan in turn be used to extract them.One method that has been used is to express the angular distribution of the decay productsof the resonance forming part of the quasi-two-body state as functions of its SDME’s [92–96]. Forexample, the following expression for the decay distribution of a photoproduced vector meson hasappeared in the literature, often in slightly different form, W ( θ ∗ , φ ∗ ; V ) = ρ ( V ) cos θ ∗ + 12 ( ρ ( V ) + ρ − − ( V )) sin θ ∗ − √ ℜ [ ρ ( V ) − ρ − ( V )] sin 2 θ ∗ cos φ ∗ + 1 √ ℑ [ ρ ( V ) − ρ − ( V )] sin 2 θ ∗ sin φ ∗ − ℜ [ ρ − ( V )] sin θ ∗ cos 2 φ ∗ + ℑ [ ρ − ( V )] sin θ ∗ sin 2 φ ∗ , (2.1)while the following expression for the angular distribution of a spin-3/2 baryon has also appeared,also often in slightly different form, W ( θ ∗ , φ ∗ ; 3 /
2) = 58 ( ρ (3 /
2) + ρ − − (3 / θ ∗ )+ 34 ( ρ (3 /
2) + ρ − − (3 / θ ∗ − √ ℜ [ ρ (3 / − ρ − − (3 / θ ∗ cos φ ∗ + √ ℑ [ ρ (3 / − ρ − − (3 / θ ∗ sin φ ∗ − √ ℜ [ ρ − (3 /
2) + ρ − (3 / θ ∗ cos 2 φ ∗ + √ ℑ [ ρ − (3 /
2) + ρ − (3 / θ ∗ sin 2 φ ∗ . (2.2)The angles θ ∗ and φ ∗ are the polar and azimuthal angles, respectively, of the three-momentumvector of one of the decay products, defined in the rest frame of the decaying resonance (in thisframe, the other decay product has opposite three-momentum). The ρ ( V )’s and the ρ (3 / γN → ππN,γN → πηN,γN → ηηN,γN → K ¯ KN,γN → πKY,γN → KK Ξ , (2.3)13 HAPTER 3KINEMATICS
Our reaction of interest is of the general form γN → M M B, (3.1)where N is the target nucleon, M and M are spin-0 mesons, and B is a spin-1/2 baryon. Anexample of a process of this form that is being studied is double-pion photoproduction, γN → ππN. (3.2)We will consider three different types of channels, which we will call pathways A, B, and C. Thegeneral form of pathway A is γN → M ∗ B → M M B, (3.3)in which a meson resonance or arbitrary spin M ∗ decays into M and M . The form of pathway Bis γN → M B ∗ → M M B, (3.4)in which a baryon resonance of arbitrary spin B ∗ decays into M and B . Pathway C has the form γN → M B ∗ → M M B, (3.5)in which a baryon resonance of arbitrary spin B ∗ decays into M and B .An example of a contribution to double-pion photoproduction of the form of pathway A is vectormeson photoproduction, such as the photoproduction of a ρ meson, γN → ρN → π π N. (3.6)The subscripts on the pions are used to indicate that each is described by different scatteringangles. An example of a contribution to the same process of the form of path B and path C is thephotoproduction of a spin-3/2 baryon, such as the photoproduction of a ∆ baryon, γN → π ∆ → π π N,γN → π ∆ → π π N. (3.7)14 ( γ ) p ( N ) zy x z ′ yx ′ p ′ ( N ′ ) q + q q ( π ) q ( π ) θ ˆ n Figure 3.1: Kinematics for the reaction γN → ππN in the overall center-of-mass frame.The coordinate system is defined such that k , p and p ′ lie on the z - x , plane (blue).Conservation of momentum constrains the three momenta of the final states to lie on aplane (green), shown in the figure with the vector normal to it, ˆ n . Another coordinatesystem with the same y -axis but with its z ′ -axis pointing in the direction of q + q is alsoshown.Note how for the case of baryon photoproduction where M and M are the same meson, bothpathways B and C will contribute, since the baryon can decay into either of the two mesons.Energy-momentum conservation requires that k + p = q + q + p ′ , (3.8)where k , p and p ′ are the four-vectors of the photon, target nucleon and recoil baryon B , respectively,while q and q are the four-vectors of the final state mesons M and M .The coordinate system that will be used in the overall center of mass frame will depend on thepathway of interest. For pathway A it is (see fig. 3.1): ˆ z = k | k | , ˆ y = k × q | k × q | , ˆ x = ˆ y × ˆ z , (3.9)where k is the spatial part of k and q is the spatial part of the four-momentum q = q + q . (3.10)15 ( γ ) p ( N ) zy x z ′ yx ′ q + p ′ q ( π ) p ′ ( N ′ ) θ ˆ nq ( π ) Figure 3.2: Kinematics for the reaction γN → ππN in the overall center-of-mass framewith the coordinate system defined such that q now lies on the z - x plane (blue). We canarrive at this coordinate system from the one in fig. 3.1 by applying a rotation along thenormal to the green plane ˆ n . From here we can reach the center-of-mass frame of q and p by boosting in the negative z ′ direction.Note that q is the four-vector of the meson resonance M ∗ . The z - x -plane is called the reactionplane, since the hadrons involved in the initial two-body reaction γN → M ∗ B will lie on this plane.This is shown as the blue plane in fig. 3.1.The coordinate system in the center-of-mass frame for pathways B and C are, respectively, ˆ z = k | k | , ˆ y = k × q | k × q | , ˆ x = ˆ y × ˆ z , q ≡ q + p ′ , (3.11)and ˆ z = k | k | , ˆ y = k × q | k × q | , ˆ x = ˆ y × ˆ z , q ≡ q + p ′ . (3.12)Therefore, in these last two coordinate systems, it is the the baryon B and one of the mesons, M or M , that define the reaction plane.Conservation of momentum restricts the momenta q , q and p ′ to lie on a plane. For pathwayA, this is shown as the green plane in fig. 3.1. To change from each of the three coordinate systems16n eqns. (3.9), (3.11) and (3.12), a rotation around the vector normal to that plane, ˆ n , can beapplied, as shown in fig. 3.2.For simplicity, we will use the coordinate system described in eqn. (3.9) (shown in fig. 3.1) forthe rest of this section. In it, the momentum of the baryon B , p ′ , is on the reaction plane. Thefour-vectors in the center of mass frame are k = (cid:0) | k | , , , | k | (cid:1) , q = (cid:0) E q , | q | sin θ, , | q | cos θ (cid:1) ,p = (cid:0) √ s − | k | , , , −| k | (cid:1) , p ′ = (cid:0) E p ′ , −| q | sin θ, , −| q | cos θ (cid:1) , (3.13)where √ s is the center of mass energy, while E q , E p ′ and | q | are E q = s + s M M − m B √ s ,E p ′ = s − s M M + m B √ s , | q | = λ / ( s, s M M , m B )2 √ s ,s M ,M = ( q + q ) , (3.14)where m B is the mass of the recoil baryon, s M M is the square of the invariant mass of thetwo-pseudoscalar-meson system in the final state, and λ ( x, y, z ) = x + y + z − xy − yz − xz, (3.15)is the K¨all´en function. The angle θ between k and q is called the scattering angle,cos( θ ) = k | k | · q | q | . (3.16)Since two of the final state particles in the reaction will come from a decay, it is convenientto describe another coordinate system with its z -axis in the direction of the decaying particle’sthree-momentum, and its y -axis perpendicular to this three-vector and ˆ k . For example, if M and M come from the decay of M ∗ , we define the this coordinate system, ˆ z ′ = q | q | , ˆ y ′ = k × q k × q = ˆ y , ˆ x ′ = ˆ y ′ × ˆ z ′ , (3.17)which we will call the primed coordinate system, and it will be used in the rest frame of the decayingresonance. In this frame, the four-momenta of the decay products q ∗ and q ∗ are q ∗ = (cid:0) E ∗ q , | q ∗ | sin θ ∗ cos φ ∗ , | q ∗ | sin θ ∗ sin φ ∗ , | q ∗ | cos θ ∗ (cid:1) ,q ∗ = (cid:0) E ∗ q , −| q ∗ | sin θ ∗ cos φ ∗ , −| q ∗ | sin θ ∗ sin φ ∗ , −| q ∗ | cos θ ∗ (cid:1) ,p ′∗ = (cid:0) E ∗ p ′ , −| p ′∗ | sin θ, , −| p ′∗ | cos θ (cid:1) , (3.18)17 ( γ ) p ( N ) zy x z ′ yx ′ φ ∗ θ ∗ q ∗ ( π ) q ∗ ( π ) θ ˆ np ′∗ ( N ′ ) Figure 3.3: Kinematics for the reaction γN → ππN in the pion-pion center-of-mass frame.It is reached from the overall center-of-mass frame shown in fig. 3.1 by boosting alongthe z ′ direction. The three-momenta of the pions lies on the green plane. Their angulardistribution is given by θ ∗ and φ ∗ , which are the polar and azimuthal angles of q ∗ in the x ′ - y - z ′ coordinate system.where E ∗ q = s M M + m M − m M √ s M M ,E ∗ q = s M M − m M + m M √ s M M ,E ∗ p ′ = s − s M M − m B √ s M M , | q ∗ | = λ / ( s M M , m M , m M )2 √ s M M , | p ′∗ | = λ / ( s, s M M , m B )2 √ s M M , (3.19) m M ( m M ) is the mass of the M ( M ) final state meson and the decay angles θ ∗ and φ ∗ are thepolar and azimuthal angles of the unit vector ˆ π = q ∗ | q ∗ | with respect to the primed coordinatesystem, cos( θ ∗ ) = ˆ π · ˆ z ′ , cos( φ ∗ ) = ˆ y ′ · ( ˆ z ′ × ˆ π ) | ˆ z ′ × ˆ π | , sin( φ ∗ ) = − ˆ x ′ · ( ˆ z ′ × ˆ π ) | ˆ z ′ × ˆ π | . (3.20)This coordinate system is shown in fig. 3.3. Note how the angle φ ∗ is also the angle betweenthe plane containing the vectors q ∗ , q ∗ and p ′∗ (shown as the green plane), and the reaction plane(shown as the blue plane). 18 HAPTER 4THE SPIN DENSITY MATRIX
The spin state of a quantum system is represented by a state vector belonging to a Hilbert space,and is in general a coherent superposition of states from a chosen basis. For example, the statevector of a massive one particle system with spin S can be represented as a linear superposition of2 S + 1 orthogonal states. These can be chosen to be the eigenstates of the spin projection operatoralong the z -axis ˆ S z , each with eigenvalue s m , | Ψ( S ) i = S X m = − S ψ zm | s m , z i , (4.1)where m labels the eigenvalues and ψ zm are complex amplitudes. These are also called pure states.The state vector contains all of the information that can be known about the system: if thevalues of all ψ zm are known (up to an overall phase), complete information about the system isknown. In particular, the expectation values of any spin observable, with associated hermitianoperator ˆ S O , can be found from h S O i = h Ψ | ˆ S O | Ψ ih Ψ | Ψ i , (4.2)where usually the state vector is normalized so that h Ψ | Ψ i = 1.However, one could have a more complicated system, such as a beam or target used in a scat-tering experiment made up of a large number of particles with spin. The state of this system wouldhave to include the amplitudes ψ zm of all of the particles in the beam, making it too complicated,and this level of fine detail is usually not known or of interest. Instead, having only partial in-formation on the system is of interest, such as the expectation value of an observable, e.g., a spinprojection, of all the particles as a whole. The system is therefore considered a statistical ensembleof many possible pure states | Ψ i i , where the i -th state has a probability ω i of being realized duringa measurement (such that P i ω i = 1). These types of systems are called mixed states, and cannotbe represented by a state vector. 19he expectation values of mixed states can be expressed in terms of a weighted average of theexpectation values of every member of the ensemble, h S O i = P i ω i h Ψ i | ˆ S O | Ψ i i P i ω i h Ψ i | Ψ i i , (4.3)where S O is some spin observable, such as a spin projection. This can be expressed as h S O i = Tr h(cid:16) P i ω i | Ψ i i h Ψ i | (cid:17) ˆ S O i Tr h P i ω i | Ψ i i h Ψ i | i . (4.4)If we define the density matrix of the system asˆ ρ = X i ω i | Ψ i i h Ψ i | , (4.5)we can rewrite the expectation value, h S O i = Tr h ˆ ρ ˆ S O i Tr (cid:2) ˆ ρ (cid:3) . (4.6)If the states in the ensemble are normalized,Tr (cid:2) ˆ ρ (cid:3) = 1 when h Ψ i | Ψ i i = 1 for all i. (4.7)A mixed state is therefore described by a density matrix, since it contains all the information ofinterest that can be known about the system. It can be used to calculate the expectation values ofany observable using eqn. (4.6). If the observables of interest are related to spin, as in this thesis,it is customarily called the spin density matrix. The spin state of these mixed states is completelyknown if all spin density matrix elements (SDME’s) are known (up to an overall phase).The dimensions of the spin density matrix depend on the dimensions of the Hilbert space of thestates in the ensemble. If the states in the ensemble are made up of massive particles of spin S , thespin density matrix will have dimensions of (2 S +1) × (2 S +1). For example, if the ensemble is madeup of spin-1/2 particles, whose Hilbert space is spanned by a basis of two orthogonal states (stateswith spin projections of 1 / − / z -axis, for example), the spin density matrix ofthe system will be 2 ×
2. For massless vector bosons, such as the photon, it will always be a 2 × ρ † = ˆ ρ, (4.8)as is required to guarantee that expectation values of experimental observables are real.As an example, the spin density matrix of a spin-1/2 particle (such as a nucleon), or a masslessparticle with spin (such as a photon), will have the general formˆ ρ = (cid:18) ρ ρ ρ ρ (cid:19) = (cid:18) ρ ρ ρ ∗ ρ (cid:19) . (4.9)Note how the hermiticity condition of eqn. (4.8) reduces the number of independent parametersneeded to fully specify a complex 2 × ~ Λ = Λ x Λ y Λ z = 1 S h S x ih S y ih S z i , (4.10)where S is total spin and the S i ’s are the expectation values of the spin projections along the i -axescalculated from eqn. (4.6). The factor of S is a convention used to guarantee that the magnitude ofthis vector is at most 1. This vector is called the polarization vector, or the degree of polarization.Each of the components could be interpreted asΛ i = n + i − n − i n , (4.11)where n + i is the number of particles in the ensemble with positive spin projection along the i -axis,vice versa for n − i , and n is the total number of particles in the ensemble.In the case of the photon, the polarization vector can alternatively be expressed asΛ x ( γ ) = δ l cos 2 β, Λ y ( γ ) = δ l sin 2 β, Λ z ( γ ) = δ ⊙ , (4.12)where δ l and δ ⊙ are the degrees of linear and circular polarization, respectively, and β is the anglethat the photon’s electric field vector makes with respect to the x -axis on the x - y plane, shown infig. 4.1. For linearly polarized photons, δ l = 1 and δ ⊙ = 0, while for circularly polarized photons,21 E p ( N ) k ( γ ) β x Figure 4.1: The direction of the electric field vector of linearly polarized photons is definedas the angle, β , it makes with respect to the x -axis of the x - y plane. δ l = 0 and δ ⊙ = 1 (If both are non-zero, the photon is said to be elliptically polarized, but suchcases will not be considered in this thesis).We say that a system is fully polarized if all the states in the ensemble are the same. A fullypolarized state is therefore a pure state. Its spin density matrix is given byˆ ρ pol = | Ψ i h Ψ | , (4.13)were every member of the ensemble is in state | Ψ i . As an example, we will write the density matrixfor fully polarized spin-1/2 particles. We will choose the eigenstates of the operator for the spinprojection along the z -axis, ˆ S z = 12 (cid:18) − (cid:19) , (4.14)with basis states, | + , z i = (cid:18) (cid:19) , |− , z i = (cid:18) (cid:19) . (4.15)The spin density matrices for a state where all particles have spin projection of +1 / z -axis, and one where all particles have spin projection of − / ρ pol (+ , z ) = (cid:18) (cid:19) , ˆ ρ pol ( − , z ) = (cid:18) (cid:19) . (4.16)The polarization vectors of these states are ~ Λ(+ , z ) = , ~ Λ( − , z ) = − . (4.17)We therefore say that are states are fully polarized along the z -axis.22s another example, we could have every state in the ensemble be an eigenstates of the operatorfor spin projection along the y -axis, ˆ S y = 12 (cid:18) − ii (cid:19) , (4.18)with corresponding eigenstates | + , y i = 1 √ (cid:18) i (cid:19) , |− , y i = 1 √ (cid:18) − i (cid:19) . (4.19)The density matrix for a polarized system with spin projection along the y -axis of 1 / − / ρ pol (+ , y ) = 12 (cid:18) − ii (cid:19) , ˆ ρ pol ( − , y ) = 12 (cid:18) i − i (cid:19) . (4.20)The polarization vectors for these states are ~ Λ(+ , y ) = , ~ Λ( − , y ) = − . (4.21)A state can in general be fully polarized along any axis. The state vector of a spin-1/2 particlewith spin projection +1 / ω = sin θ cos φ sin θ sin φ cos θ , (4.22)described by polar and azimuthal angles θ and φ , is given by | ψ ( ˆ ω ) , z i = (cid:18) cos( θ/ θ/ e iφ (cid:19) (4.23)in the z basis. Therefore, the general expression (up to a phase) for the normalized spin densitymatrix of a spin-1/2 system fully polarized along the ˆ ω ( θ, φ ) direction isˆ ρ pol ( ˆ ω ) = (cid:18) cos ( θ/
2) cos( θ/
2) sin( θ/ e − iφ cos( θ/
2) sin( θ/ e iφ sin ( θ/ (cid:19) . (4.24)In the case of a fully polarized system, the polarization vector is equal to ˆ ω , ~ Λ = sin θ cos φ sin θ sin φ cos θ . (4.25)23ote how only two parameters are needed to describe the previous polarization vector, not three.This is because for systems that are fully polarized, the magnitude of its polarization vector equals1, | ~ Λ pol | = Λ x + Λ y + Λ z = 1 (4.26)(assuming the spin density matrix’s trace is normalized to 1).For example, the state vector of a spin-1/2 particle with spin projection +1 / θ = 60 ◦ and φ = 90 ◦ is | ψ i = √ i ! , (4.27)has density matrix ˆ ρ ( ψ ) = − i √ i √
34 14 ! , (4.28)and polarization vector ~ Λ( ψ ) = √ . (4.29)A necessary and sufficient condition for a density matrix to describe a fully polarized state isˆ ρ = ˆ ρ pol , (4.30)as can be verified for the density matrices in the previous examples. The density matrix of a fullypolarized system is therefore a projection operator.A mixed spin state is said to be fully unpolarized if the members of the ensemble have equalprobability to be in any one of the states that forms a complete basis. Using an arbitrary set ofbasis states, | m i , where each has the same probability N of being realized in the ensemble, thedensity matrix of an unpolarized state isˆ ρ = X m ω m | m i h m | = N (cid:16) X m | m i h m | (cid:17) = N ˆ I, (4.31)where the last equality follows from the completeness property of basis states, and ˆ I is the identitymatrix. Therefore, the spin density matrix of fully unpolarized states is always proportional to theidentity matrix, regardless of the chosen basis. When the trace of a spin- S spin density matrix isnormalized to 1, N = 1 / (2 S + 1). 24or example, in a beam of spin-1/2 particles in which the spin projection along the z -axis ofhalf of them is 1 / − /
2, its density matrix is ρ unpol = 12 (cid:16) | + , z i h + , z | + |− , z i h− , z | (cid:17) ,ρ unpol = 12 (cid:18) (cid:19) . (4.32)It’s polarization vector is ~ Λ unpol = . (4.33)A partially polarized state will have a polarization vector with magnitude between 0 and 1,0 < | ~ Λ | < . (4.34)A pure spin S state, | ψ S i , transforms under rotations as | ψ ′ Sm i = ˆ D ( α, β, γ ) | ψ Sm i ,ψ ′ Sm = S X m ′ = − S D Smm ′ ( α, β, γ ) ψ Sm ′ ,ψ Sm ≡ h S, m | ψ S i , (4.35)where the D Smm ′ ’s are the well known Wigner D -functions and α , β , and γ are the Euler anglesdescribing the rotation. Therefore, it can be shown from eqn. (4.5) that the spin density matrix ofa spin- S system transforms under rotations asˆ ρ ′ ( S ) = ˆ D S ( α, β, γ )ˆ ρ ( S ) ˆ D S † ( α, β, γ ) ,ρ ′ mm ′ ( S ) = S X n,n ′ = − S D Smn ( α, β, γ ) ρ nn ′ ( S ) D S ∗ m ′ n ′ ( α, β, γ ) . (4.36)This previous equation and the definition of the spin density matrix in eqn. (4.5) can be usedto show, for example, that any spin density matrix for a polarized state (like the one in eqn. (4.28))can be transformed by rotations into the ones in eqn. (4.16). It can also be used to show that afully unpolarized state will take the form in eqn. (4.32) in any basis, i.e., the identity matrix isinvariant under rotations of the coordinate system.Since complex numbers are described by two real numbers, a spin density matrix is described byhas 2 × (2 S + 1) × (2 S + 1) real numbers. But they are also hermitian matrices, so only half of them,(2 S + 1) , will be independent (one less if the matrix is normalized to have a trace of one). For25xample, the spin density matrix of a spin-1/2 particle will have 4 independent parameters, whilea massive spin-1 particle will have 9 (massless vector bosons will have 4 independent parameters,since their spin density matrix will always be 2 × ~ Λ. 26
HAPTER 5STATISTICAL TENSORS
Since only some of the elements of a given spin density matrix are independent, it is only necessaryto specify the real and imaginary parts of these independent elements to completely define thematrix. However, it is possible to specify instead a different set of quantities, each one being alinear combination of the independent matrix elements, such that they transform under differentrepresentations of the rotation group. For example, the spin density matrix for a massive spin-1particle, described by 9 independent quantities, can be written asˆ ρ ( S = 1) = 13 N ( ˆ I + X i v i ˆ S i + X i,j q ij ˆ Q ij ) ,i, j ∈ { x, y, z } , (5.1)where ˆ I is the 3 × S i ’s are the Cartesian components of the spin operator,and the ˆ Q ij ’s are the Cartesian components of the quadrupole operator,ˆ Q ij ≡
12 ( ˆ S i ˆ S j + ˆ S j ˆ S i − δ ij ˆ I ) , ˆ Q ij = ˆ Q ji , X i ˆ Q ii = 0 . (5.2)From the form of the spin density matrix in eqn. (5.1), along with eqns. (4.6), (4.10), and (5.2), itcan be shown that v i is equal to the polarization vector, v i = Λ i . (5.3) N is a rotational scalar, while the v i ’s and the Q ij transform under tensor representations ofthe rotation group, v i ′ = X j R ij ( ~ω ) v j ,q ij ′ = X kl R ik ( ~ω ) R jl ( ~ω ) q kl , (5.4)where the R ij ’s are the elements of the three dimensional representation of the rotation group, fora rotation along the axis pointing in the direction of ~ω by an angle | ~ω | . Note that since the q ij ’s27re elements of a traceless, symmetric matrix, only 5 of its elements are independent. Therefore, N , the v i ’s and the q ij ’s together form 9 independent quantities which completely specify the spindensity matrix. These quantities are the eigenvalues of hermitian operators (the ˆ S i ’s and ˆ Q ij ’s)and as such, are purely real, i.e., they are experimental observables.As seen in eqn. (5.4), q ij does not transform under an irreducible representation of the rotationgroup. It is therefore convenient to decompose the spin density matrix as in eqn. (5.1) but intomatrices that transform under irreducible representation,ˆ ρ ( S ) = S X L =0 L X M = − L t LM ( S ) ˆ T † LM ( S ) , (5.5)where the matrices ˆ T LM ( S )’s are called the polarization operators and the t LM ( S )’s are called thestatistical tensors of rank L . The matrix elements of the ˆ T LM ( S )’s are defined as[ T LM ( S )] mm ′ = ( − S − m ′ C LMSm ; S − m ′ , (5.6)where the C LMSm ; S − m ′ ’s are the well known Clebsch-Gordan coefficients (the matrix elements aretherefore real). The t LM ’s are the expectation values of the polarization operators, t LM ( S ) = h ˆ T LM ( S ) i = Tr[ˆ ρ ( S ) ˆ T LM ( S )] = S X m,m ′ = − S ( − S − m ρ mm ′ ( S ) C LMsm ′ ; s − m . (5.7)The inverse relation of eqn. (5.7) is ρ mm ′ ( S ) = S X L =0 L X M = − L ( − S − m t LM ( S ) C LMSm ′ ; S − m . (5.8)Under rotations, the t LM ( S )’s transform as t ′ LM ( S ) = L X M = − L D LMM ′ ( ~ω ) t LM ′ ( S ) , (5.9)where the D LMM ′ ( ~ω ) are the elements of the Wigner D -matrix, called the Wigner D -functions. TheWigner- D matrices are the 2 L + 1 dimensional irreducible representations of the rotation group.Note that, unlike the Cartesian tensors ( v i , q ij , and other higher rank tensors), the t LM ’s arecomplex quantities. However, with the standard Clebsch-Gordan phase convention, we find t ∗ LM ( S ) = ( − M t L − M ( S ) , (5.10)28o that the total number of independent parameters remains the same. Written in terms of thestatistical tensors, the total number of independent parameters N p required to describe the spinstate of a spin- S particle is N p = S X L =0 (2 L + 1) . (5.11)While in the Cartesian rotation operators R ij the i ’s and j ’s label Cartesian components ( x , y and z ), the M ’s of the tensor operators label spherical components, i.e., the polarization operators arespherical tensors operators. The Wigner-Eckart theorem therefore applies to them, h jm | ˆ T LM | j ′ m ′ i = C jmj ′ m ′ ; LM h j || ˆ T L || j ′ i , (5.12)where h jm | and | j ′ m ′ i are angular momentum eigenstates, and the factor h j || ˆ T L || j ′ i is a quantitythat does not depend on M , m , or m ′ , called the reduced matrix element. By writing the spindensity matrix in terms of tensor operators, rotational symmetry can be exploited to simplify thecalculation of matrix elements by using the Wigner-Eckart theorem.In summary, there are two equivalent ways to describe the spin state of a statistical ensembleof states: either by specifying the real and imaginary parts of all independent elements of the spindensity matrix, or by specifying the values of the statistical tensors. The two are related by thelinear relations in eqns. (5.7) and (5.8), so it is easy to rewrite any equation involving the matrixelements in terms of the the tensors, and vice versa. The advantage of the tensors is that theyhave good transformation properties under rotations, and can be used to take advantage of therotational symmetry of the system of interest via the Wigner-Eckart theorem.29 HAPTER 6SDME’s AND HELICITY AMPLITUDES IN γN → M M B The quantum states describing the system before and after the scattering process are related viathe transition matrix ˆ M , | Ψ i ( M M B ) i = ˆ M | Ψ i ( γN ) i , (6.1)where | Ψ i ( γN ) i and | Ψ i ( M M B ) i are the state vectors of the initial and final systems, γN and M M B , and i labels the different members of the statistical ensemble of the mixed state.The initial states in the ensemble consist of a beam ( γ ) and target (N). Each will be a purestate consisting of a photon and a nucleon. Since they are uncorrelated, the state vector of thissystem is a tensor product of the state vectors of the individual subsystems, | Ψ i ( γN ) i = | Ψ i ( γ ) i ⊗ | Ψ i ( N ) i . (6.2)In this thesis, we will always describe spin states using the helicity basis, in which the quantizationaxis is chosen to be in the same direction as the momentum of the particle. Since both the beamand target have two possible spin projections ( ± ± / | Ψ i ( γN ) i = ψ i ( γN ) ψ i - ( γN ) ψ i - ( γN ) ψ i - - ( γN ) , (6.3)where ψ ijk ( γN ) is the probability amplitudes for the i -th member of the ensemble to have helicities j and k for the bean and nucleon, respectively. Similarly, the final state is | Ψ i ( M M B ) i = ψ i ( M M B ; θ ) ψ i - ( M M B ; θ ) ! . (6.4)The subindices refer only to the spin projection of the baryon B , since M and M are spinless.These two amplitudes are also functions of the scattering angle θ . Since the initial state is given30y a 4 × ×
1, the transition matrix is 2 × M ( θ ) = M ;1 ( θ ) M ;1 - ( θ ) M ; - ( θ ) M ; - - ( θ ) M - ;1 ( θ ) M - ;1 - ( θ ) M - ; - ( θ ) M - ; - - ( θ ) ! . (6.5)The elements of the transition matrix are known as the helicity amplitudes. They are the probabilityamplitudes for an initial state to transition into a particular final state. For example, M ;1 ( θ ) isthe probability amplitude for an initial state with helicities 1 and 1 / M M B state with recoil baryon helicity 1 / θ (defined in section 3).As mentioned in section 2, a complete experiment is one in which enough measurements canbe made to be able to extract the values of all helicity amplitudes (except an overall phase) atdifferent kinematic points (different values of θ and other kinematic variables). The dependenceon the kinematic variables of these amplitudes can be interpreted in terms of the resonances thatcontribute to the scattering process.Since the beam, target, and recoil baryon are mixed states represented by density matrices,it is necessary to find how the SDME’s are related to the helicity amplitudes. The states in theensemble are product states, which means the spin density matrix of the ensemble is also a tensorproduct of two spin density matrices, ˆ ρ ( γN ) = ˆ ρ ( γ ) ⊗ ˆ ρ ( N ) . (6.6)Since the pure states in the ensemble shown in eqn. (6.3) are 4 × × ρ ( γN ) = ρ ;1 ( γN ) ρ ;1 - ( γN ) ρ ; - ( γN ) ρ ; - - ( γN ) ρ - ;1 ( γN ) ρ - ;1 - ( γN ) ρ - ; - ( γN ) ρ - ; - - ( γN ) ρ - ;1 ( γN ) ρ - ;1 - ( γN ) ρ - ; - ( γN ) ρ - ; - - ( γN ) ρ - - ;1 ( γN ) ρ - - ;1 - ( γN ) ρ - - ; - ( γN ) ρ - - ; - - ( γN ) . (6.7)Note that the “row” and “column” indices are actually made up of two indices: one for the helicityof the beam, and one for the helicity of the target. Combining the definition of spin density matricesshown in eqn. (4.5) with eqn. (6.1), we can see that the relation between the spin density matricesof the initial and final state is ˆ ρ ( M M B ) = ˆ M ˆ ρ ( γN ) ˆ M † . (6.8)31he spin density matrix that describes the ensemble of the final three hadrons will therefore be a2 × ρ ( M M B ) = ρ ; ( M M B ; θ ) ρ ; - ( M M B ; θ ) ρ - ; ( M M B ; θ ) ρ - ; - ( M M B ; θ ) ! . (6.9)As we saw in eqn. (5.1), every spin density matrix can be written as linear combination of aparticular set of matrices. Since the spin density matrix of both the beam and the target are 2 × S i ( γ ) = ˆ σ i , ˆ S i ( N ) = 12 ˆ σ i , (6.10)where i is a Cartesian component and the ˆ σ i ’s are the well known Pauli matrices,ˆ σ x = (cid:18) (cid:19) , ˆ σ y = (cid:18) − ii (cid:19) , ˆ σ z = (cid:18) − (cid:19) . (6.11)Therefore, the spin density matrix can be expressed asˆ ρ ( γN ) = ˆ ρ ( γ ) ⊗ ˆ ρ ( N )= 12 ( ˆ I + X i Λ i ( γ ) ˆ σ i ) ⊗
12 ( ˆ I + X i Λ i ( N ) ˆ σ i )= 14 ( ˆ I ⊗ ˆ I + X i Λ i ( γ ) ˆ σ i ⊗ ˆ I + X i Λ i ( N ) I ⊗ ˆ σ i + X i,j Λ i ( γ ) Λ j ( N ) ˆ σ i ⊗ ˆ σ j ) ,i, j ∈ { x, y, z } . (6.12)The Λ i ( N ) ’s and Λ i ( γ ) ’s are the components of the polarization vector of the photon and nucleon,respectively. The prefactor of 1 / ρ ] = 1. Since the indices i and j run over three values,the density matrix has been decomposed into a linear combination of 16 matrices. Each of thesematrices is a tensor product of two 2 × ×
4. Anexample of one of these matrices isˆ I = (cid:18) (cid:19) , ˆ σ z = (cid:18) − (cid:19) , ˆ I ⊗ ˆ σ z = (cid:18) (cid:19) ⊗ ˆ σ z = (cid:18) ˆ σ z
00 ˆ σ z (cid:19) = − − . (6.13)32nother example is ˆ σ x = (cid:18) (cid:19) , ˆ σ y = (cid:18) − ii (cid:19) , ˆ σ x ⊗ ˆ σ y = (cid:18) (cid:19) ⊗ ˆ σ y = (cid:18) σ y ˆ σ y (cid:19) = − i i − i i . (6.14)Combining eqns. (6.8) and (6.12) gives us the spin density matrix of the recoiling baryon, as asum of 16 2 × ρ ( B ) = ˆ ρ + X i Λ i ( γ ) ˆ ρ i ( γ ) + X i Λ i ( N ) ˆ ρ i ( N ) + X i,j Λ i ( γ ) Λ j ( N ) ˆ ρ ij ( γN ) , (6.15)where the 16 matrices are defined as ˆ ρ ( B ) = ˆ M ( ˆ I ⊗ ˆ I ) ˆ M † , ˆ ρ i ( γ ) ( B ) = ˆ M (ˆ σ i ⊗ ˆ I ) ˆ M † , ˆ ρ i ( N ) ( B ) = ˆ M ( ˆ I ⊗ ˆ σ i ) ˆ M † , ˆ ρ ij ( γN ) ( B ) = ˆ M (ˆ σ i ⊗ ˆ σ j ) ˆ M † . (6.16)In terms of the matrix elements,the expression is ρ λ B λ ′ B ( B ) = 14 X λ γ λ N M λ B ; λ γ λ N M ∗ λ ′ B ; λ γ λ N ,ρ i ( γ ) λ B λ ′ B ( B ) = 14 X λ γ λ ′ γ λ N M λ B ; λ γ λ N σ iλ γ λ ′ γ M ∗ λ ′ B ; λ ′ γ λ N , (6.17) ρ i ( N ) λ B λ ′ B ( B ) = 14 X λ γ λ N λ ′ N M λ B ; λ γ λ N σ iλ N λ ′ N M ∗ λ ′ B ; λ γ λ ′ N ,ρ ij ( γN ) λ B λ ′ B ( B ) = 14 X λ γ λ ′ γ λ N λ ′ N M λ B ; λ γ λ N σ iλ γ λ ′ γ σ jλ N λ ′ N M ∗ λ ′ B ; λ ′ γ λ ′ N . These can be further simplified by using the explicit forms of the Pauli matrices in the z basisfound in eqn. (6.11). For example, ρ x ( γ ) λ B λ ′ B ( B ) = 14 X λ γ λ N M λ B ; λ N λ γ M ∗ λ ′ B ; λ N − λ γ , y ( γ ) λ B λ ′ B ( B ) = − i X λ N λ γ λ γ M λ B ; λ N λ γ M ∗ λ ′ B ; λ N − λ γ , (6.18) ρ z ( γ ) λ B λ ′ B ( B ) = 14 X λ N λ γ λ γ M λ B ; λ N λ γ M ∗ λ ′ B ; λ N λ γ . These expression are general for the spin density matrix of any spin-1/2 final state whose initialstate has two particles, each with two possible spin projections.Note that each of the elements of these 16 matrices do not depend on the initial polarizationof the beam and target. They are, however, functions of the scattering angle θ . Since they donot depend on the initial polarizations, they only contain information about the dynamics of thereaction. This can be seen from eqn. (6.17), which shows that the elements of these 16 matricesdepend only on the helicity amplitudes.Since each of the 16 matrices in the expansion of ˆ ρ ( B ) are 4 × HAPTER 7POLARIZATION OBSERVABLES
A question that naturally arises is: How can the polarization state of the reaction products in ascattering experiment be determined for a given polarization of the beam and target?, i.e., whichexperimental quantities should be measured in order to determine the independent parametersdiscussed in the previous section? These quantities are the polarization observables, which will bedefined later in this section.The way to extract them experimentally is to define 64 quantities known as the polarizationobservables, which are linear combinations of the 64 independent parameters. We can thereforedetermine the SMDE’s of the 16 matrices by measuring the observables. Note that while the densitymatrix ˆ ρ ( B ) has only four independent parameters, these are functions of the initial polarizations ~ Λ ( γ ) and ~ Λ ( N ) . This dependence on the initial polarization was factored out in eqn. (6.15) so thatthe SDME’s of the 16 expansion matrices only depend on the helicity amplitudes. The polarizationobservables can only be used to extract the SDME’s of these expansion matrices.The observables are categorized as either single, double, or triple polarization observables,based on how many of the particles with spin (beam, target, and recoil baryon) are involved in themeasurement. We will first show their definitions in terms of the SDME’s and we will later showhow they relate to polarized cross sections. The unpolarized observable is I = Tr[ˆ ρ ] . (7.1)The three single polarization beam observables are I I c = Tr[ˆ ρ x ( γ ) ] ,I I s = Tr[ˆ ρ y ( γ ) ] ,I I ⊙ = Tr[ˆ ρ z ( γ ) ] . (7.2)The three single polarization target observables are I P i = Tr[ˆ ρ i ( N ) ] . (7.3)35he three single polarization recoil observables are I P i ′ = Tr[ˆ ρ ˆ σ i ] . (7.4)More generally, the spin operator is used in place of ˆ σ i ’s for higher spin particles. The 9 beam-targetdouble polarization observables are I P ci = Tr[ˆ ρ xi ( γN ) ] ,I P si = Tr[ˆ ρ yi ( γN ) ] ,I P ⊙ i = Tr[ˆ ρ zi ( γN ) ] . (7.5)The 9 beam-recoil double polarization observables are I P ci ′ = Tr[ˆ ρ x ( γ ) ˆ σ i ] ,I P si ′ = Tr[ˆ ρ y ( γ ) ˆ σ i ] ,I P ⊙ i ′ = Tr[ˆ ρ z ( γ ) ˆ σ i ] . (7.6)The 9 target-recoil double polarization observables are I O αβ ′ = Tr[ˆ ρ α ( N ) ˆ σ β ′ ] . (7.7)The 27 triple polarization observables are I O cαβ ′ = Tr[ˆ ρ xα ( γN ) ˆ σ β ′ ] ,I O sαβ ′ = Tr[ˆ ρ yα ( γN ) ˆ σ β ′ ] ,I O ⊙ αβ ′ = Tr[ˆ ρ zα ( γN ) ˆ σ β ′ ] . (7.8)To measure the observables, we need to be able to relate them to differential cross sections.These are in turn related to the scattering amplitude, which is given in terms of the scattered state | Ψ i and the state found after a measurement | Φ i by, σ ∝ | h Φ | Ψ i | . (7.9)While the symbol σ is usually used for total cross sections, in order to simplify the notation we willtake it to refer to the differential cross section (Remember that the SDME’s are not only functionsof the spin projections, but also of the scattering angle θ ). If the scattered state is instead a mixedstate, it is not represented by the state vector | Ψ i but rather by a density matrix ˆ ρ (Ψ). It can36e shown from the definition of the density matrix in eqn. (4.5) that the probability of finding amember of the statistical ensemble in state Φ after a measurement is given by σ ∝ h Φ | ˆ ρ (Ψ) | Φ i . (7.10)We will soon show that the polarization observables are related to ratios of sums of cross sections,which means that the proportionality constant that turns eqns. (7.9) and (7.10) into an equalityis irrelevant. Therefore, for the rest of this thesis we will instead use σ ∝ h Φ | ˆ ρ (Ψ) | Φ i ⇒ σ = h Φ | ˆ ρ (Ψ) | Φ i . (7.11)Since we can now relate matrix elements of density matrices to cross sections, we can relate theobservables defined in eqns. (7.1) to (7.8) to cross sections. The expression for the spin densitymatrix of the recoil baryon shown in eqn. (6.15) depends on the state of the beam and targetthrough the quantities ~ Λ ( γ ) and ~ Λ ( N ) . We will therefore write it asˆ ρ ( B ) ≡ ˆ ρ ( ~ Λ ( γ ) , ~ Λ ( N ) ) . (7.12)Each of the polarization vectors in the previous equation have three components, so the spin densitymatrix of the recoil baryon requires six quantities to specify the spin state of the beam and targetthat gave rise to it. However, in the rest of this section we will assume that the experiments willbe set up such that the beam and target are either polarized along one of the three Cartesiancoordinate axes ( x , y or z ), or unpolarized. Therefore, it will depend on two quantities: the twodegrees of polarization of the beam and target along an axis. From eqn. (7.11) we see that the crosssection for measuring the recoil nucleon with spin projection λ i along the i -axis from a reaction inwhich the photon beam and target nucleon are fully polarized along the j - and k -axis respectively,is given by σ i ; jk ( ± ; ± , ± ) = h λ i = ± | ˆ ρ (Λ j ( γ ) = ± , Λ k ( N ) = ± | λ i = ± i . (7.13)This gives us eight possible polarized cross sections.For example, the cross section for the recoil nucleon to be measured with spin projection − / y -axis when the beam is polarized with spin projection − z -axis (circularlypolarized beam) and the target is polarized with spin projection 1 / x -axis is σ y ; zx ( − ; − , +) = h λ y = − | ˆ ρ (Λ z ( γ ) = − , Λ x ( N ) = +1) | λ y = − i . (7.14)37f the beam or target are unpolarized, Λ i = 0 for the corresponding hadron. By using this identitythat can be shown from eqn. (6.15),ˆ ρ ( ~ Λ ( γ ) , Λ z ( N ) = 0) = 12 (cid:16) ˆ ρ ( ~ Λ ( γ ) , Λ z ( N ) = 1) + ˆ ρ ( ~ Λ ( γ ) , Λ z ( N ) = − (cid:17) , (7.15)we can show that, if in the expression shown in eqn. (7.14) the target baryon was instead unpolar-ized, the cross section would be σ y ; zz ( − ; − ,
0) = h λ y = − | ˆ ρ (Λ z ( γ ) = − , Λ z ( N ) = 0) | λ y = − i = h λ y = − | (cid:16) ˆ ρ (Λ z ( γ ) = − , Λ z ( N ) = 1)+ ˆ ρ (Λ z ( γ ) = − , Λ z ( N ) = − (cid:17) | λ y = − i = 12 h h λ y = − | ˆ ρ (Λ z ( γ ) = − , Λ z ( N ) = 1) | λ y = − i + h λ y = − | ˆ ρ (Λ z ( γ ) = − , Λ z ( N ) = − | λ y = − i i = 12 (cid:2) σ y ; zz ( − ; − , +) + σ y ; zz ( − ; − , − ) (cid:3) . (7.16)The last line of the previous expression shows that a cross section with an unpolarized beam ortarget is equal to the average of the polarized cross sections with the two possible orthogonal spinstates in some basis. Similarly, if the polarization of the recoil nucleon is not measured, its crosssection is equal to the sum of the cross sections with all possible polarizations in some basis. Forexample, if the photon has spin projection − z -axis, the target has spin projection +1 / x -axis and the recoil polarization is not measured, the cross section is σ z ; zx (0; − , +) = σ z ; zx (+; − , +) + σ z ; zx ( − ; − , +)= h λ z = | ˆ ρ (Λ z ( γ ) = − , Λ x ( N ) = 1) | λ z = i + h λ z = − | ˆ ρ (Λ z ( γ ) = − , Λ x ( N ) = 1) | λ z = − i = Tr[ˆ ρ (Λ z ( γ ) = − , Λ x ( N ) = 1)] . (7.17)Therefore, the trace of the spin density matrix is taken when the recoiling particle’s polarization isnot measured.We can now write the observables in terms of polarized differential cross sections. For theobservable I , we get I ≡ Tr[ ρ ]= Tr[ˆ ρ (Λ z ( γ ) = 0 , Λ z ( N ) = 0)]= σ z ; zz (0; 0 , . (7.18)38n other words, this observable is simply the unpolarized cross section (up to a spin-independentconstant). Written in terms of polarized cross sections, I = 14 [ σ z ; zz (+; + , +) + σ z ; zz (+; + , − ) + σ z ; zz ( − ; + , +) + σ z ; zz ( − ; + , − )+ σ z ; zz (+; − , +) + σ z ; zz (+; − , − ) + σ z ; zz ( − ; − , +) + σ z ; zz ( − ; − , − )]= 14 σ , (7.19)where we have defined the quantity inside the brackets on the first line as σ (the factor of comesfrom averaging over the initial polarizations for the photon and the target nucleon).To find expressions for the beam or target polarization observables, we need to be able to findthe 16 matrices ˆ ρ i from ˆ ρ ( ~ Λ ( γ ) , ~ Λ ( N ) ). To find ˆ ρ x ( N ) , for example, we use eqn. (6.15) to find,ˆ ρ x = 12 (cid:16) ˆ ρ (Λ x ( γ ) = 0 , Λ x ( N ) = 1) − ˆ ρ (Λ x ( γ ) = 0 , Λ x ( N ) = − (cid:17) . (7.20)Therefore, the observable P x is given by P x ≡ Tr[ˆ ρ x ( N ) ]Tr[ˆ ρ ]= (cid:8) Tr[ˆ ρ (Λ z ( γ ) = 0 , Λ x ( N ) = 1)] − Tr[ˆ ρ (Λ z ( γ ) = 0 , Λ x ( N )] = − (cid:9) (cid:8) Tr[ˆ ρ (Λ z ( γ ) = 0 , Λ x ( N ) = 1)] + Tr[ˆ ρ (Λ z ( γ ) = 0 , Λ x ( N ) = − (cid:9) = σ z ; zx (0; 0 , +) − σ z ; zx (0; 0 , − ) σ z ; zx (0; 0 , +) + σ z ; zx (0; 0 , − ) . (7.21)This shows that the target polarization observables are simply the asymmetry in the polarizedcross sections along some axis, normalized over the unpolarized cross section, at each value ofthe scattering angle. Therefore, in order to measure this observable in the lab, the experimenterwould have to run the experiment with the target polarized along the positive x -axis and measurethe scattering cross section without measuring the spin of the recoil nucleon, and then repeat themeasurement with the experiment set with the target polarized along the negative x -axis. Thedifference between these two quantities divided by their sum gives you the observable. If writtenin terms of all the polarized cross sections, we get, P x = (cid:2) σ z ; zx (+; + , +) + σ z ; zx (+; − , +) + σ z ; zx ( − ; + , +) + σ z ; zx ( − ; − , +) − σ z ; zx (+; + , − ) − σ z ; zx (+; − , − ) − σ z ; zx ( − ; + , − ) − σ z ; zx ( − ; − , − ) (cid:3), σ . (7.22)In general, every single observable will be a linear combination of all of the eight polarized crosssections normalized over the unpolarized cross section. For each observable, the relative minussigns among the eight terms will be different. 39o find the expressions for the recoil observables, we need to use the identityTr[ ˆ A ˆ σ z ] = h + , z | ˆ A | + , z i − h− , z | ˆ A |− , z i , (7.23)where ˆ A is any 2 × P z ′ can be expressed as P z ′ ≡ Tr[ˆ ρ ˆ σ z ]Tr[ˆ ρ ]= h + , z | ˆ ρ (Λ z ( γ ) = 0 , Λ z ( N ) = 0) | + , z i − h− , z | ˆ ρ (Λ z ( γ ) = 0 , Λ z ( N ) = 0) |− , z i Tr[ˆ ρ (Λ z ( γ ) = 0 , Λ z ( N ) = 0]= σ z ; zz (+; 0 , − σ z ; zz ( − ; 0 , σ z ; zz (+; 0 ,
0) + σ z ; zz ( − ; 0 , . (7.24)Once again, we get an asymmetry in the cross sections of measuring the recoil nucleon along someaxis, normalized over the unpolarized cross section.In order to find the expression for P x ′ or P y ′ , we need these relations, which are correct up toan irrelevant phase factor,ˆ σ x = ˆ R y (cid:0) π (cid:1) ˆ σ z ˆ R y (cid:0) - π (cid:1) , ˆ σ y = ˆ R x (cid:0) - π (cid:1) ˆ σ z ˆ R x (cid:0) π (cid:1) , ˆ R y (cid:0) π (cid:1) |± , z i = |± , x i , ˆ R x (cid:0) - π (cid:1) |± , z i = |± , x i , (7.25)where ˆ R i (cid:0) ± π (cid:1) is the rotation matrix around the i -axis by ± π/ P x is therefore P x ′ ≡ Tr[ˆ ρ ˆ σ x ]Tr[ˆ ρ ]= Tr[ˆ ρ ˆ R y (cid:0) π (cid:1) ˆ σ z ˆ R y (cid:0) - π (cid:1) ]Tr[ˆ ρ ]= Tr[ ˆ R y (cid:0) - π (cid:1) ˆ ρ ˆ R y (cid:0) π (cid:1) ˆ σ z ]Tr[ˆ ρ ] (7.26)= h + , z | ˆ R y (cid:0) - π (cid:1) ˆ ρ ˆ R y (cid:0) π (cid:1) | + , z i − h− , z | ˆ R y (cid:0) - π (cid:1) ˆ ρ ˆ R y (cid:0) π (cid:1) |− , z ) i Tr[ˆ ρ ]= h + , x | ˆ ρ | + , x i − h− , x | ˆ ρ |− , x ) i Tr[ˆ ρ ]= σ x ; zz (+; 0 , − σ x ; zz ( − ; 0 , σ x ; zz (+; 0 ,
0) + σ x ; zz ( − ; 0 , , where in the third line of the previous equation we used the cyclic property of the trace. Onceagain, we get an asymmetry, this time for measuring the recoil baryon’s spin along the x -axis.40or a double polarization observable, we find (using the beam-target observable P sx as an ex-ample), P sx ≡ Tr[ˆ ρ yx ]Tr[ˆ ρ ] , = 12 (cid:26) (cid:16) Tr[ˆ ρ (Λ y ( γ ) = 1 , Λ x ( N ) = 1)] − Tr[ˆ ρ (Λ y ( γ ) = 1 , Λ x ( N ) = − (cid:17) − (cid:16) Tr[ˆ ρ (Λ y ( γ ) = − , Λ x ( N ) = 1)] − Tr[ˆ ρ (Λ y ( γ ) = − , Λ x ( N ) = − (cid:17)(cid:27), Tr[ˆ ρ ] (7.27)= (cid:26) [ σ z ; yx (0; + , +) − σ z ; yx (0; + , − )] − [ σ z ; yx (0; − , +) − σ z ; yx (0; − , − )] (cid:27) [ σ z ; yx (0; + , +) + σ z ; yx (0; + , − ) + σ z ; yx (0; − , +) + σ z ; yx (0; − , − )]= σ z ; yx (0; + , +) − σ z ; yx (0; + , − ) − σ z ; yx (0; − , +) + σ z ; yx (0; − , − ) σ z ; yx (0; + , +) + σ z ; yx (0; + , − ) + σ z ; yx (0; − , +) + σ z ; yx (0; − , − ) . Note from the third line of the last equation that a double polarization observable is an asymmetryof an asymmetry, i.e., an asymmetry in the target is measured for each of the two values of thebeam polarization, and then the asymmetry between these two values is taken.Recall that eqn. (6.17) shows that the 64 SDME’s are equal to bilinear combinations of thehelicity amplitudes in eqn. (6.5). The observables can therefore also be expressed in terms of saidamplitudes. As such, measurements of observables can be used to extract the helicity amplitudesof the process at different kinematic points. As previously mentioned, the behaviour of theseamplitudes as a function of the kinematic variables is needed to carry out the PWA’s, which couldestablish the presence of the resonances contributing to the reactions. Eqn. (6.5) shows there are8 complex amplitudes in double pion photoproduction, for a total 16 parameters. Since a statevector describes a quantum state up to an overall phase factor, it should come as no surprise that15 parameters can be extracted from the measurement of the observables.Substituting the expressions in eqn. (6.17) into the definitions of the observables in eqns. (7.1)-(7.8) allows us to write the observables in terms of the helicity amplitudes. The unpolarized crosssection is I ≡ Tr[ ρ ] = 14 (cid:16) | M + , ++ | + | M + , − + | + | M + , + − | + | M + , −− | + | M − , ++ | + | M − , − + | + | M − , + − | + | M − , −− | (cid:17) ,σ = | M + , ++ | + | M + , − + | + | M + , + − | + | M + , −− | (7.28)+ | M − , ++ | + | M − , − + | + | M − , + − | + | M − , −− | , I ⊙ ≡ Tr[ ρ z ( γ ) ]Tr[ ρ ] = (cid:18)h | M + , ++ | + | M + , − + | + | M − , ++ | + | M − , − + | i − h | M + , + − | + | M + , −− | + | M − , + − | + | M − , −− | i(cid:19), σ , (7.29) I c ≡ Tr[ˆ ρ x ( γ ) ]Tr[ˆ ρ ] , = 2 ℜ h M +;++ M ∗ +;+ − + M +; − + M ∗ +; −− + M − ;++ M ∗− ;+ − + M − ; − + M ∗− ; −− i, σ , (7.30) I s ≡ Tr[ˆ ρ y ( γ ) ]Tr[ˆ ρ ] , = 2 ℑ h M +;++ M ∗ +;+ − + M +; − + M ∗ +; −− + M − ;++ M ∗− ;+ − + M − ; − + M ∗− ; −− i, σ . (7.31)Note that all observables are therefore the ratio of a bilinear sum of the transition amplitudes andthe unpolarized cross section.Eqn. (6.15) shows the spin density matrix of the nucleon ˆ ρ ( B ) is a function of 16 expansion ma-trices. But since the observables are functions of the SDME’s of these matrices, it can alternativelybe expressed in terms of the observables. To show this, we will need these identities,Tr[ˆ σ i ] = 0 , Tr[ˆ σ i ˆ σ j ] = δ ij . (7.32)Since any 2 × × ρ x ( N ) as ˆ ρ x ( N ) = A + X i B xj ˆ σ j , (7.33)where A and the B xj ’s are as-of-yet-unknown expansion coefficients (the first term has an impliedfactor of a 2 × A , we take the trace on both sides of the equation, I Tr[ˆ ρ x ( N ) ] = 2 I A = P x , → A = 12 P x I (7.34)42The factor of 2 comes from taking the trace of the identity matrix). To find the B xj ’s, we multiplyboth sides by a Pauli matrix and take the trace, I Tr[ˆ ρ x ( N ) ˆ σ i ] = 2 I B xi = O xi ′ , → B xi = 12 O xi ′ I . (7.35)Therefore, ˆ ρ x ( N ) = P x + X i O xi ′ ˆ σ i . (7.36)Applying this derivation to all 16 matrices, we findˆ ρ = 12 (cid:16) I + X i ′ P i ′ ˆ σ i ′ (cid:17) , ˆ ρ i ( N ) = 12 I (cid:16) P i + X β ′ O iβ ′ ˆ σ β ′ (cid:17) , ˆ ρ x ( γ ) = 12 I (cid:16) I c + X i ′ P ci ′ ˆ σ i ′ (cid:17) , ˆ ρ y ( γ ) = 12 I (cid:16) I s + X i ′ P si ′ ˆ σ i ′ (cid:17) , ˆ ρ z ( γ ) = 12 I (cid:16) I ⊙ + X i ′ P ⊙ i ′ ˆ σ i ′ (cid:17) , ˆ ρ xi ( γN ) = 12 I (cid:16) P ci + X β ′ O ciβ ′ ˆ σ β ′ (cid:17) , ˆ ρ yi ( γN ) = 12 I (cid:16) P si + X β ′ O siβ ′ ˆ σ β ′ (cid:17) , ˆ ρ zi ( γN ) = 12 I (cid:16) P ⊙ i + X β ′ O ⊙ iβ ′ ˆ σ β ′ (cid:17) . (7.37)By substituting these expressions into eqn. (6.15) we getˆ ρ ( B ) I (cid:0) ~ Λ ( γ ) , ~ Λ ( N ) (cid:1) = 12 I ((cid:16) X i Λ i ( N ) P i + X i ′ ˆ σ i ′ P i ′ + X α,β ′ Λ α ( N ) ˆ σ β ′ O αβ ′ (cid:17) + δ ⊙ (cid:16) I ⊙ + X i Λ i ( N ) P ⊙ i + X i ′ ˆ σ i ′ P ⊙ i ′ + X α,β ′ Λ α ( N ) ˆ σ β ′ O ⊙ αβ ′ (cid:17) (7.38)+ δ l (cid:20) sin 2 β (cid:16) I s + X i Λ i ( N ) P si + X i ′ ˆ σ i ′ P si ′ + X α,β ′ Λ α ( N ) ˆ σ β ′ O sαβ ′ (cid:17) + cos 2 β (cid:16) I c + X i Λ i ( N ) P ci + X i ′ ˆ σ i ′ P ci ′ + X α,β ′ Λ α ( N ) ˆ σ β ′ O cαβ ′ (cid:17)(cid:21)) , ρ ( B ) is normalized so that its trace is equal to 1, I isthe reaction rate when the spin of the recoil nucleon is not measured and is a function of the initialpolarizations ~ Λ ( γ ) and ~ Λ ( N ) .As established previously, the spin density matrix ˆ ρ ( B ), can also be expressed in terms of thestatistical tensors instead of the SDME’s of the 16 matrices in eqn. (6.15). By combining thisequation with eqn. (5.5), we getˆ ρ ( B ) = X L =0 L X M = − L ( t LM + X i Λ i ( γ ) t i ( γ ) LM + X i Λ i ( N ) t i ( N ) LM + X i,j Λ i ( γ ) Λ j ( N ) t ij ( γN ) LM ) ˆ T † LM , (7.39)where, for example, t i ( γ ) LM = / X m,m ′ = − / ( − − m ρ i ( γ ) mm ′ C LM m ′ − m (7.40)(The sum goes over − / / B is spin-1/2). The inverse relationof eqn. (7.40) is ρ i ( γ ) mm ′ = X L =0 L X M = − L t i ( γ ) LM [ ˆ T LM ] m ′ m , = X L =0 L X M = − L ( − − m ′ t i ( γ ) LM C LM m ′ ; − m . (7.41)By comparing eqns. (7.39) and (5.5), we see that just as the spin density matrix can be expandedinto a linear combination of 16 other matrices, the statistical tensors can also be expanded into alinear combination of 16 other tensors, t LM = t LM + X i Λ i ( γ ) t i ( γ ) LM + X i Λ i ( N ) t i ( N ) LM + X ij Λ i ( γ ) Λ j ( N ) t ij ( γN ) LM . (7.42)To find the expression relating the polarization observables to these statistical tensors, all thatis needed is to combine equation (7.41) with eqns. (7.1) through (7.8) and use the trace propertyof the polarization operators, Tr[ ˆ T LM ( S )] = √ S + 1 δ L δ M . (7.43)The unpolarized observable is I = √ t . (7.44)44he three beam single polarization observables are given by I I c = √ t x ( γ )00 ,I I s = √ t y ( γ )00 ,I I ⊙ = √ t z ( γ )00 . (7.45)All of the observables that do not involve the recoil baryon will have the same form as eqns. (7.44)and (7.45), with the only difference among them being the superscript of t ( x ( N ), z ( γ ), y ( γN ),etc). To find the expressions for the recoil polarizations, we need to express the Pauli matrices interms of the spherical components of the recoil baryon’s spin operator,ˆ σ z =2 ˆ S = √ T , ˆ σ x =2( ˆ S − − ˆ S +1 ) = √
2( ˆ T − − ˆ T ) , ˆ σ y = i
2( ˆ S − + ˆ S +1 ) = i √
2( ˆ T − + ˆ T ) , (7.46)so that we can we can exploit the propertyTr[ ˆ T L M ˆ T L M ] = ( − M δ L L δ M − M . (7.47)Using eqns. (7.46), (7.47) and (5.10) along with the equations for the observables that involve therecoil baryon from eqns. (7.1) to (7.8), we get for example I P z ′ = √ t ,I P x ′ = √ t − t − ) = 2 √ ℜ [ t ] ,I P y ′ = − i √ t + t − ) = 2 √ ℑ [ t ] , (7.48)where in the last two lines the hermiticity property of the tensors shown in eqn. (5.10) was used. Allobservables that involve the recoil baryon will have this form. The ones involving the z -projectionof the recoil baryon will be proportional to a t , while those involving the x - and y -projections ofthe recoil baryon are proportional to the real and imaginary parts of a t , respectively. The onlydifference among them will be the superscript of the tensor. For example, the observable I O zx ′ = 2 √ ℜ [ t z ( N )11 ] (7.49)has a z ( N ) superscript because it involves the z -projection of the target, and is proportional to thereal part of the tensor because it involves the x -projection of the recoil baryon.45or comparison, the same observables in eqns. (7.45) and (7.48) in terms of the SDME’s are, I I c = ρ x ( γ )
12 12 + ρ x ( γ ) - - ,I I s = ρ y ( γ )
12 12 + ρ y ( γ ) - - ,I I ⊙ = ρ z ( γ )
12 12 + ρ z ( γ ) - - , (7.50)and I P z ′ = ρ
12 12 − ρ - - ,I P x ′ = ρ - + ρ -
12 12 = 2 ℜ h ρ - i ,I P y ′ = i (cid:18) ρ - − ρ -
12 12 (cid:19) = − ℑ h ρ - i . (7.51)While the SDME’s and the statistical tensors contain the same information, it can be seen from thiscomparison that expressions for the observables in terms of the tensors are simpler than the onesin terms of the SDME’s. This, in addition to the fact that the tensors transform under irreduciblerepresentations of the rotation group (the well known Wigner D -matrices) and the fact that itallows the use of the Wigner-Eckart theorem shown in eqn. (5.12) in theoretic calculations, makesthem a more natural choice in describing the observables. Nevertheless, since the use of SDME’s iswidespread in the literature, we will express all results in this thesis in terms of both the SDME’sand the statistical tensors. 46 HAPTER 8ANGULAR DISTRIBUTIONS IN γN → M ∗ B → M M B , γN → M B ∗ → M M B , AND γN → M B ∗ → M M B The previous section dealt with polarization observables in photoproduction reactions withoutconsidering the reaction mechanism. Since it is well known that many of the contributions to thisreaction are quasi-two-body states, we will consider them in the rest of this thesis. As mentioned insection 3, the three possible channels are γN → M ∗ B → M M B , γN → M B ∗ → M M B , and γN → M B ∗ → M M B , where M , M and M ∗ are mesons while B and B ∗ are baryons. Thegoal of this section is to find an expressions that relates the decay distribution and the polarizationobservables of the photoproduction reaction to the SDME’s or statistical tensors of the decayinghadron ( M ′ or B ′ ). For now, we will make the assumption that only one such quasi-two-body statecontributes to the reaction. The examples we’ll consider are the reactions γN → V N → M M N ,where V is a vector meson, γN → M B / → M M N , where B / is a spin-3/2 baryon, and γN → M B / → M M N . This procedure is also laid out in a somewhat different mathematicallanguage in refs. [92, 93].The V B system has a massive spin-1 hadron (3 orthogonal spin states) and a spin-1/2 hadron(2 orthogonal states). Therefore, the state vector in the ensemble can be represented by a 6 × | Ψ i ( V B ) i = ψ i ( V B ) ψ i - ( V B ) ψ i ( V B ) ψ i - ( V B ) ψ i - ( V B ) ψ i - - ( V B ) , (8.1)47here i labels a members of the statistical ensemble in the mixed state. Since the initial stateshown in eqn. (6.3) is represented by a 4 × × M ( θ ; γN → V B ) = M ;1 ( θ ) M ;1 - ( θ ) M ; - ( θ ) M ; - - ( θ ) M - ;1 ( θ ) M - ;1 - ( θ ) M - ; - ( θ ) M - ; - - ( θ ) M ;1 ( θ ) M ;1 - ( θ ) M ; - ( θ ) M ; - - ( θ ) M - ;1 ( θ ) M - ;1 - ( θ ) M - ; - ( θ ) M - ; - - ( θ ) M - ;1 ( θ ) M - ;1 - ( θ ) M - ; - ( θ ) M - ; - - ( θ ) M - - ;1 ( θ ) M - - ;1 - ( θ ) M - - ; - ( θ ) M - - ; - - ( θ ) . (8.2)From the definition of the density matrix in eqn. (4.5), the density matrix of the V B system is6 × V B system fromthat of the initial state by, ˆ ρ ( V B ) = ˆ M ˆ ρ ( γN ) ˆ M † , (8.3)or, in terms of the individual matrix elements, ρ ( V B ) λ V λ B ; λ ′ V λ ′ B = X λ γ λ N λ ′ γ λ ′ N M λ V λ B ,λ γ λ N ρ λ γ λ N ; λ ′ γ λ ′ N ( γN ) M ∗ λ ′ V λ ′ B ; λ ′ γ λ ′ N , (8.4)where the λ γ , λ N , λ V , and λ B are the helicities of the beam, target, vector meson, and recoilnucleon respectively. Next, the state will transition once again, this time from the V B state to the M M B state through the decay of the vector meson into M and M . The transition matrix forthe two body decay of an arbitrary particle at rest of spin S , A → BC , is proportional to a Wigner D -matrix, M λ B λ C ; λ A ∝ D S ∗ λ A ( λ B − λ C ) . (8.5)Since the B subsystem is a spectator during the decay of the vector meson, the transition matrixis therefore the tensor product of two transition matrices, one acting on the V subsystem and theother on the B subsystem, ˆ M ( θ ∗ , φ ∗ ; V B → M M B ) = ˆ D † ( θ ∗ , φ ∗ ) ⊗ ˆ I, (8.6)where ˆ I is the identity matrix, and ˆ D † is the 1 × D † = c (cid:0) D ∗ ( θ ∗ , φ ∗ ) D ∗ ( θ ∗ , φ ∗ ) D ∗− ( θ ∗ , φ ∗ ) (cid:1) = c (cid:16) − e − iφ ∗ sin( θ ∗ ) √ cos( θ ∗ ) e iφ ∗ sin( θ ∗ ) √ (cid:17) . (8.7)48he D ∗ λ V are complex conjugates of the Wigner D -functions, which are functions of the decayangles θ ∗ and φ ∗ defined in section 3, and c is a proportionality constant that won’t be relevant inthese discussions, and so will be dropped for the rest of this thesis. The state of the B subsystemis being multiplied by the identity matrix because the final nucleon does not transition into newparticles (it does not decay). We can use the transition matrix shown in eqn. (8.6) to find the spindensity matrix of the M M B system,ˆ ρ ( M M B ) = ( ˆ D † ⊗ ˆ I )ˆ ρ ( V B )( ˆ D ⊗ ˆ I ) , (8.8)or, using index notation, ρ λ B λ ′ B ( M M B ) = X λ V λ ′ V D ∗ λ V ( θ ∗ , φ ∗ ) ρ λ V λ B ; λ ′ V λ ′ B ( BV ) D λ ′ V ( θ ∗ , φ ∗ ) . (8.9)We showed in eqn. (7.11) that the trace of the spin density matrix is proportional to the crosssection. We therefore take the trace of the previous equation,Tr[ˆ ρ ( M M B )] = X λ B ρ λ B ,λ B ( M M B )= X λ V λ ′ V D ∗ λ V ( θ ∗ , φ ∗ ) h X λ B ρ λ V λ B ; λ ′ V λ B ( V B ) i D λ ′ V ( θ ∗ , φ ∗ ) . (8.10)The expression in square brackets in the second line of the previous equation is called a partialtrace over the spin density matrix, h X λ B ρ ( V B ) λ V λ B ; λ ′ V λ B i ≡ ρ λ V λ ′ V ( V ) , Tr B [ˆ ρ ( V B )] ≡ ˆ ρ ( V ) . (8.11)The subscript B on the second line of the previous equation means that we are taking the traceonly over the indices of the recoil baryon B . In terms of experiments, taking the trace over recoilbaryon indices means that its spin is not being measured. This removes the spin information ofthe recoil baryon that the density matrix contains. What we end up with is therefore consideredthe spin density matrix of the vector meson V subsystem. We can then consider only the processof the decay of the V into M and M . Equation (8.10) therefore simplifies toTr[ˆ ρ ( M M )] = X λ V λ ′ V D ∗ λ V ( θ ∗ , φ ∗ ) ρ λ V ,λ ′ V ( V ) D λ ′ V ( θ ∗ , φ ∗ ) ≡ W ( θ ∗ , φ ∗ ; V ) , (8.12)49note that since M and M are spinless, its density matrix ˆ ρ ( M M ) is just a real number. Whilein general a trace must be taken, in this example there is technically no need to take a trace). Sincethe trace is proportional to the cross section of a decay process, and since its angular dependenceis known, we can also call it the angular distribution of the decay products M and M , which wedesignate as W ( θ ∗ , φ ∗ ; V ). Using the explicit forms of the Wigner D -functions, the decay rate is W ( θ ∗ , φ ∗ ; V ) = ρ ( V ) cos θ ∗ + 12 ( ρ ( V ) + ρ − − ( V )) sin θ ∗ − √ ℜ [ ρ ( V ) − ρ − ( V )] sin 2 θ ∗ cos φ ∗ + 1 √ ℑ [ ρ ( V ) − ρ − ( V )] sin 2 θ ∗ sin φ ∗ − ℜ [ ρ − ( V )] sin θ ∗ cos 2 φ ∗ + ℑ [ ρ − ( V )] sin θ ∗ sin 2 φ ∗ , (8.13)where we have used the hermiticity condition of the density matrix in eqn. (4.8) to express W onlyin terms of the independent elements. Note that, since the diagonal elements are purely real, thedistribution is real, as expected.In the case of the γN → M B ∗ → M M B / , we can also find the distribution of the decayproducts using the same procedure. This time the γN systems in the ensemble transition into a M B ′ state with spin state vector | Ψ i ( M B / ) i = ψ i ( M B / ) ψ i ( M B / ) ψ i − ( M B / ) ψ i − ( M B / ) . (8.14)Having four possible spin states, the spin transition matrix is therefore 4 × M ( θ ; γN → M B / ) = M ;1 ( θ ) M ;1 - ( θ ) M ; - ( θ ) M ; - - ( θ ) M ;1 ( θ ) M ;1 - ( θ ) M ; - ( θ ) M ; - - ( θ ) M - ;1 ( θ ) M - ;1 - ( θ ) M - ; - ( θ ) M - ; - - ( θ ) M - ;1 ( θ ) M - ;1 - ( θ ) M - ; - ( θ ) M - ; - - ( θ ) . (8.15)Just as in the example of a decaying vector meson, we can write the spin density matrix of the M M B system in terms of the spin density matrix of the M B / system,ˆ ρ ( M M B ) = ˆ D / † ˆ ρ ( M B / ) ˆ D / . (8.16)Since M is spinless and only a spectator in the decay of B / , we can instead write it asˆ ρ ( M B ) = ˆ D / † ˆ ρ ( B / ) ˆ D / , (8.17)50here we call ˆ ρ ( B / ) the spin density matrix of the baryon subsystem B / . Using index notation,it can be expressed as ρ λ B λ ′ B ( M B ) = X λ B / λ ′ B / D / ∗ λ B / λ B ( θ ∗ , φ ∗ ) ρ λ B / λ ′ B / ( B / ) D / λ ′ B / λ ′ B ( θ ∗ , φ ∗ ) , (8.18)where the transition matrix for the decay is nowˆ D / † = c D / ∗
32 12 ( θ ∗ , φ ∗ ) D / ∗
12 12 ( θ ∗ , φ ∗ ) D / ∗−
12 12 ( θ ∗ , φ ∗ ) D / ∗−
32 12 ( θ ∗ , φ ∗ ) D / ∗− − ( θ ∗ , φ ∗ ) D / ∗− − ( θ ∗ , φ ∗ ) D / ∗− − ( θ ∗ , φ ∗ ) D / ∗− − ( θ ∗ , φ ∗ ) , (8.19)where c is a proportionality constant. This matrix in terms of the explicit expressions for theWigner D -functions is too large to be displayed in this article. Just as in the example of thedecay of the vector meson V , taking the trace of the matrix in eqn. (8.18) and using the explicitexpressions for the Wigner D -functions gives us the angular distribution of the decay products, M and B ′ , W ( θ ∗ , φ ∗ ; B / ) = 58 ( ρ ( B / ) + ρ − − ( B / ))(1 + 35 cos 2 θ ∗ )+ 34 ( ρ ( B / ) + ρ − − ( B / )) sin θ ∗ − √ ℜ [ ρ ( B / ) − ρ − − ( B / )] sin 2 θ ∗ cos φ ∗ + √ ℑ [ ρ ( B / ) − ρ − − ( B / )] sin 2 θ ∗ sin φ ∗ − √ ℜ [ ρ − ( B / ) + ρ − ( B / )] sin θ ∗ cos 2 φ ∗ + √ ℑ [ ρ − ( B / ) + ρ − ( B / )] sin θ ∗ sin 2 φ ∗ . (8.20)For the case of the reaction γN → M B / → M M B , the same procedure just described applies.The only difference is that in this case the variables θ ∗ and φ ∗ describe different angles, becausethey describe the decay distribution of M and B instead of M and B .As mentioned in section 3, the coordinate system used to define the angles θ ∗ and φ ∗ is differentfor each of the three pathways shown in eqns. (3.3), (3.4) and (3.5). The z ′ -axis is always definedas pointing in the opposite direction as the three-momentum of the spectator hadron, and the θ ∗ and φ ∗ are the polar and azimuthal angle of one of the decaying hadrons with respect to this axis,as shown in fig. 3.3. 51s we have seen in eqn. (5.5) and (5.6), we can express spin density matrix elements in termsof statistical tensors. In terms of these, the angular distribution for the decay of a V meson is givenby W ( θ ∗ , φ ∗ ; V ) = 1 √ t ( V ) − √ t ( V )(1 + 3 cos 2 θ ∗ )+ ℜ [ t ( V )] sin 2 θ ∗ cos φ ∗ + ℑ [ t ( V )] sin 2 θ ∗ sin φ ∗ − ℜ [ t ( V )] sin θ ∗ cos 2 φ ∗ − ℑ [ t ( V )] sin θ sin 2 φ ∗ , (8.21)and the angular distribution for the decay of a B / baryon is given by W ( θ ∗ , φ ∗ ; B / ) = 12 t (3 / − t (3 / θ ∗ )+ 12 r ℜ [ t (3 / θ ∗ cos φ ∗ + 12 r ℑ [ t (3 / θ ∗ sin φ ∗ − r ℜ [ t (3 / θ ∗ cos 2 φ ∗ − r ℑ [ t (3 / θ ∗ sin 2 φ ∗ . (8.22)As can be seen from these last two equations, only statistical tensors of even rank appear in theexpressions for the distribution (rank 1 tensors don’t appear in the expression for the distributionof a V meson, while rank 1 and 3 tensors don’t appear in the expression for the distribution of the B / baryon). As long as the decay process conserves parity, this property holds true even wheninterference effects of different resonances are taken into account, as we will show in section 12.Note how the functions of the angles multiplied by each of the tensors in the sum of the lasttwo equations is a spherical harmonic, W ( θ ∗ , φ ∗ ; V ) = 2 r π t ( V ) Y ( θ ∗ , φ ∗ ) − r π t ( V ) Y ( θ ∗ , φ ∗ ) − r π ℜ [ t ( V )] ℜ [ Y ( θ ∗ , φ ∗ )] + 4 r π ℑ [ t ( V )] ℑ [ Y ( θ ∗ , φ ∗ )] − r π ℜ [ t ( V )] ℜ [ Y ( θ ∗ , φ ∗ )] + 4 r π ℑ [ t ( V )] ℑ [ Y ( θ ∗ , φ ∗ )] , (8.23)and W ( θ ∗ , φ ∗ ; B / ) = √ πt (3 / Y ( θ ∗ , φ ∗ ) − r π t (3 / Y ( θ ∗ , φ ∗ ) − r π ℜ [ t (3 / ℜ [ Y ( θ ∗ , φ ∗ )] + 2 r π ℑ [ t (3 / ℑ [ Y ( θ ∗ , φ ∗ )] (8.24) − r π ℜ [ t (3 / ℜ [ Y ( θ ∗ , φ ∗ )] + 2 r π ℑ [ t (3 / ℑ [ Y ( θ ∗ , φ ∗ )] . L and M are proportional to thewell known spherical harmonics with same L and M . By contrast, in the expressions in terms ofthe SDME’s in eqns. (8.13) and (8.20), the quantities multiplied by the angular functions are linearcombinations of SDME’s, with no obvious connection between them and the angular function thatmultiply them. In section 12 we will show that this is in general true for any quasi-two-body statein which the decay is parity conserving. 53 HAPTER 9POLARIZATION OBSERVABLES IN TERMS OFSDME’S AND STATISTICAL TENSORS OF THEQUASI-TWO-BODY STATES
Given the assumption that the photoproduction process goes through a quasi-two-body state, thepolarization observables can be expressed in terms of the SDME’s or statistical tensors of theresonances that decay. We will first examine this for for pathway A, using the example of adecaying vector meson. Afterwards, we will examine this for pathways B and C for the example ofa decaying spin-3/2 baryon.We have shown in equation (6.15) how the spin density matrix of the final M M B systemcan be decomposed into a sum of 16 matrices. But this can be done for any spin density matrix,including the one for the vector meson V,ˆ ρ ( V ) = ρ + X i Λ i ( γ ) ˆ ρ i ( γ ) ( V ) + X i Λ i ( N ) ˆ ρ i ( N ) ( V ) + X i,j Λ i ( γ ) Λ j ( N ) ˆ ρ ij ( γN ) ( V ) , (9.1)where the SDME’s of the 16 matrices are ρ λ V λ ′ V ( V ) = 14 X λ γ λ N λ B M λ V λ B ; λ γ λ N M ∗ λ ′ V λ B ; λ γ λ N ,ρ i ( γ ) λ V λ ′ V ( V ) = 14 X λ γ λ ′ γ λ N λ B M λ V λ B ; λ γ λ N σ iλ γ λ ′ γ M ∗ λ ′ V λ B ; λ ′ γ λ N ,ρ i ( N ) λ V λ ′ V ( V ) = 14 X λ N λ ′ N λ γ λ B M λ V λ B ; λ γ λ N σ iλ N λ ′ N M ∗ λ ′ V λ B ; λ γ λ ′ N ,ρ ij ( γN ) λ V λ ′ V ( V ) = 14 X λ γ λ ′ γ λ N λ ′ N λ B M λ V λ B ; λ γ λ N σ iλ γ λ ′ γ σ jλ N λ ′ N M ∗ λ ′ V λ B ; λ ′ γ λ ′ N . (9.2)Note how the λ B indices on the right-hand sides of the equations are summed over, since the partialtrace was taken. We can use the explicit forms of the Pauli matrices to simplify the expression54urther. If we label their elements with the integers − σ xm,m ′ = δ m, − m ′ ,σ ym,m ′ = − imδ m, − m ′ , (9.3) σ zm,m ′ = − mδ m,m ′ ,m = λ γ or m = 2 λ N , where i is the imaginary number. Examples of some of these matrices when the previous equationis applied are ρ x ( γ ) λ V λ ′ V ( V ) = 14 X λ γ λ N λ B M λ V λ B ; λ γ λ N M ∗ λ ′ V λ B ; − λ γ λ N ,ρ y ( γ ) λ V λ ′ V ( V ) = − i X λ γ λ N λ B λ γ M λ V λ B ; λ N λ γ M ∗ λ ′ V λ B ; − λ γ λ N ,ρ z ( γ ) λ V λ ′ V ( V ) = 14 X λ γ λ N λ B λ γ M λ V λ B ; λ γ λ N M ∗ λ ′ V λ B ; λ γ λ N . (9.4)By substituting eqn. (9.1) into eqn. (8.12), we see that the distribution W can also be decom-posed into 16 terms, W ( θ ∗ , φ ∗ ; V ) = W ( θ ∗ , φ ∗ ; V ) + X i Λ i ( γ ) W i ( γ ) ( θ ∗ , φ ∗ ; V ) + X i Λ i ( N ) W i ( N ) ( θ ∗ , φ ∗ ; V )+ X i,j Λ i ( γ ) Λ j ( N ) W ij ( γN ) ( θ ∗ , φ ∗ ; V ) . (9.5)This last equation is valid not just for vector mesons, but for any kind of decaying particle. The W i ’s are given by W ( θ ∗ , φ ∗ ; V ) = X λ V λ ′ V D ∗ λ V ( θ ∗ , φ ∗ ) ρ λ V ,λ ′ V ( V ) D λ ′ V ( θ ∗ , φ ∗ ) ,W i ( γ ) ( θ ∗ , φ ∗ ; V ) = X λ V λ ′ V D ∗ λ V ( θ ∗ , φ ∗ ) ρ i ( γ ) λ V ,λ ′ V ( V ) D λ ′ V ( θ ∗ , φ ∗ ) ,W i ( N ) ( θ ∗ , φ ∗ ; V ) = X λ V λ ′ V D ∗ λ V ( θ ∗ , φ ∗ ) ρ i ( N ) λ V ,λ ′ V ( V ) D λ ′ V ( θ ∗ , φ ∗ ) ,W ij ( γN ) ( θ ∗ , φ ∗ ; V ) = X λ V λ ′ V D ∗ λ V ( θ ∗ , φ ∗ ) ρ ij ( γN ) λ V ,λ ′ V ( V ) D λ ′ V ( θ ∗ , φ ∗ ) . (9.6)55ach of these W i ’s will have exactly the same mathematical form as W in eqn. (8.13) after thesubstitution ρ λ V λ ′ V → ρ iλ V λ ′ V . For example, W y ( γ ) takes the form W y ( γ ) ( θ ∗ , φ ∗ ; V ) = ρ y ( γ )00 ( V ) cos θ ∗ + 12 ( ρ y ( γ )11 ( V ) + ρ y ( γ ) − − ( V )) sin θ ∗ − √ ℜ [ ρ y ( γ )10 ( V ) − ρ y ( γ )0 − ( V )] sin 2 θ ∗ cos φ ∗ + 1 √ ℑ [ ρ y ( γ )10 ( V ) − ρ y ( γ )0 − ( V )] sin 2 θ ∗ sin φ ∗ − ℜ [ ρ y ( γ )1 − ( V )] sin θ ∗ cos 2 φ ∗ + ℑ [ ρ y ( γ )( V )1 − ( V )] sin θ ∗ sin 2 φ ∗ . (9.7)In eqn. (7.42) we showed how statistical tensors can also be decomposed into other 16 statisticaltensors. The equations that relates the 16 SDME’s to the 16 tensors were shown in eqns. (7.40)and (7.41). To find the W i ’s in terms of these tensors, we simply substitute eqn. (7.41) into thedefinition of the W i ’s in eqn. (9.6) to get, for example, W y ( γ ) ( θ ∗ , φ ∗ ; V ) = 1 √ t y ( γ )00 ( V ) − √ t y ( γ )20 ( V )(1 + 3 cos 2 θ ∗ )+ ℜ [ t y ( γ )21 ( V )] sin 2 θ ∗ cos φ ∗ − ℑ [ t y ( γ )21 ( V )] sin 2 θ ∗ sin φ ∗ − ℜ [ t y ( γ )22 ( V )] sin θ ∗ cos 2 φ ∗ + ℑ [ t y ( γ )22 ( V )] sin θ sin 2 φ ∗ . (9.8)Note how it has the same mathematical form as the the expression for W in terms of the tensorsshown in eqn. (8.21) except that now the tensors have the appropriate superscript.To find how these 16 W i ’s are related to the polarization observables, we use eqn. (7.38) andtake the trace on both sides. The terms proportional to the Pauli matrices, σ i , on the right-handside will vanish. The left-hand side is the definition of the distribution W , W ≡ Tr[ˆ ρ ( M M B )] . (9.9)After taking said trace and comparing it with eqn. (9.5), we get W ( θ ∗ , φ ∗ ; V ) = I ( θ ∗ , φ ∗ ) ,W x ( γ ) ( θ ∗ , φ ∗ ; V ) = I cos 2 βI c ( θ ∗ , φ ∗ ) ,W y ( γ ) ( θ ∗ , φ ∗ ; V ) = I sin 2 βI s ( θ ∗ , φ ∗ ) , (9.10) W z ( γ ) ( θ ∗ , φ ∗ ; V ) = I I ⊙ ( θ ∗ , φ ∗ ) ,W i ( N ) ( θ ∗ , φ ∗ ; V ) = I P i ( θ ∗ , φ ∗ ) ,W ix ( γN ) ( θ ∗ , φ ∗ ; V ) = I cos 2 βP ci ( θ ∗ , φ ∗ ) , iy ( γN ) ( θ ∗ , φ ∗ ; V ) = I sin 2 βP si ( θ ∗ , φ ∗ ) ,W iz ( γN ) ( θ ∗ , φ ∗ ; V ) = I P ⊙ i ( θ ∗ , φ ∗ ) . This last equation is not only valid for the decay of vector mesons, but also for any type of decayingparticle. In deriving these expressions, a sum over the spin indices of the recoil baryon baryon wasdone. This represents the fact that the polarization of the recoiling particle is not measured. Assuch, the W i ’s are related to the observables that do not involve the recoil baryon.For the cases of pathways B and C with a decaying spin-3/2 baryon, γN → M B / → M M B ,and γN → M B / → M M B , we follow the same procedure and decompose the decaying baryon’sspin density matrix into 16 matrices,ˆ ρ ( B / ) = ρ + X i Λ i ( γ ) ˆ ρ i ( γ ) ( B / )+ X i Λ i ( N ) ˆ ρ i ( N ) ( B / ) + X i,j Λ i ( γ ) Λ j ( N ) ˆ ρ ij ( γN ) ( B / ) , (9.11)where ρ λ B / λ ′ B / ( B / ) = 14 X λ γ λ N M λ B / ; λ γ λ N M ∗ λ ′ B / ; λ γ λ N ,ρ i ( γ ) λ B / λ ′ B / ( B / ) = 14 X λ γ λ ′ γ λ N M λ B / ; λ γ λ N σ iλ γ λ ′ γ M ∗ λ ′ B / ; λ ′ γ λ N , (9.12) ρ i ( N ) λ B / λ ′ B / ( B / ) = 14 X λ γ λ N λ ′ N M λ B / ; λ γ λ N σ iλ N λ ′ N M ∗ λ ′ B / ; λ γ λ ′ N ,ρ ij ( γN ) λ B / λ ′ B / ( B / ) = 14 X λ γ λ ′ γ λ N λ ′ N M λ B / ; λ γ λ N σ iλ γ λ ′ γ σ jλ N λ ′ N M ∗ λ ′ B / ; λ ′ γ λ ′ N , (compare with equation (9.2) for the vector meson V). The angular distribution of its decay productscan again be decomposed into 16 terms, W ( θ ∗ , φ ∗ ; B / ) = W ( θ ∗ , φ ∗ ; B / ) + X i Λ i ( γ ) W i ( γ ) ( θ ∗ , φ ∗ ; B / ) (9.13)+ X i Λ i ( N ) W i ( N ) ( θ ∗ , φ ∗ ; B / ) + X i,j Λ i ( γ ) Λ j ( N ) W ij ( Nγ ) ( θ ∗ , φ ∗ ; B / ) . where W ( θ ∗ , φ ∗ ; B / ) = X λ B / λ ′ B / λ B D / ∗ λ B / λ B ( θ ∗ , φ ∗ ) ρ λ B / ,λ ′ B / ( B / ) D / λ ′ B / λ B ( θ ∗ , φ ∗ ) , i ( γ ) ( θ ∗ , φ ∗ ; B / ) = X λ B / λ ′ B / λ B D / ∗ λ B / λ B ( θ ∗ , φ ∗ ) ρ i ( γ ) λ B / ,λ ′ B / ( B / ) D / λ ′ B / λ B ( θ ∗ , φ ∗ ) , (9.14) W i ( N ) ( θ ∗ , φ ∗ ; B / ) = X λ B / λ ′ B / λ B D / ∗ λ B / λ B ( θ ∗ , φ ∗ ) ρ i ( N ) λ B / ,λ ′ B / ( B / ) D / λ ′ B / λ B ( θ ∗ , φ ∗ ) ,W ij ( γN ) ( θ ∗ , φ ∗ ; B / ) = X λ B / λ ′ B / λ B D / ∗ λ B / λ B ( θ ∗ , φ ∗ ) ρ ij ( γN ) λ B / ,λ ′ B / ( B / ) D / λ ′ B / λ B ( θ ∗ , φ ∗ )(note the sum over the index λ B , since we are taking a trace). Each of these W i ’s will have exactlythe same mathematical form as W in eqn. (8.20) after the substitution ρ λ B / ,λ ′ B / ( B / ) → ρ iλ B / ,λ ′ B / ( B / ). For example, W y ( γ ) is given by W y ( γ ) ( θ ∗ , φ ∗ ; B / ) = 58 ( ρ y ( γ )11 ( B / ) + ρ y ( γ ) − − ( B / ))(1 + 35 cos 2 θ ∗ )+ 34 ( ρ y ( γ )33 ( B / ) + ρ y ( γ ) − − ( B / )) sin θ ∗ (9.15) − √ ℜ [ ρ y ( γ )31 ( B / ) − ρ y ( γ ) − − ( B / )] sin 2 θ ∗ cos φ ∗ + √ ℑ [ ρ y ( γ )31 ( B / ) − ρ y ( γ ) − − ( B / )] sin 2 θ ∗ sin φ ∗ − √ ℜ [ ρ y ( γ )3 − ( B / ) + ρ y ( γ )1 − ( B / )] sin θ ∗ cos 2 φ ∗ + √ ℑ [ ρ y ( γ )3 − ( B / ) + ρ y ( γ )1 − ( B / )] sin θ ∗ sin 2 φ ∗ . In terms of statistical tensors, this distribution is W y ( γ ) ( θ ∗ , φ ∗ ; B / ) = t y ( γ )00 ( B / ) − t y ( γ )20 ( B / )(1 + 3 cos 2 θ ∗ )+ r ℜ [ t y ( γ )21 ( B / )] sin 2 θ ∗ cos φ ∗ − r ℑ [ t y ( γ )21 ( B / )] sin 2 θ ∗ sin φ ∗ (9.16) − r ℜ [ t y ( γ )22 ( B / )] sin θ ∗ cos 2 φ ∗ + 12 r ℑ [ t y ( γ )22 ( B / )] sin θ ∗ sin 2 φ ∗ . The polarization observables for this case are again given by eqn. (9.10), W ( θ ∗ , φ ∗ ; B / ) = I ( θ ∗ , φ ∗ ) , i ( N ) ( θ ∗ , φ ∗ ; B / ) = I P i ( θ ∗ , φ ∗ ) ,W x ( γ ) ( θ ∗ , φ ∗ ; B / ) = I cos 2 βI c ( θ ∗ , φ ∗ ) ,W y ( γ ) ( θ ∗ , φ ∗ ; B / ) = I sin 2 βI s ( θ ∗ , φ ∗ ) , (9.17) W z ( γ ) ( θ ∗ , φ ∗ ; B / ) = I I ⊙ ( θ ∗ , φ ∗ ) ,W ix ( γN ) ( θ ∗ , φ ∗ ; B / ) = I cos 2 βP ic ( θ ∗ , φ ∗ ) ,W iy ( γN ) ( θ ∗ , φ ∗ ; B / ) = I sin 2 βP is ( θ ∗ , φ ∗ ) ,W iz ( γN ) ( θ ∗ , φ ∗ ; B / ) = I P i ⊙ ( θ ∗ , φ ∗ ) . Note that while the dependence on the scattering angle θ is contained inside of the SDME’sor statistical tensors of the decaying hadron, their θ ∗ and φ ∗ dependence is not. Rather, they arefully contained in the trigonometric functions in eqns. (9.7), (9.15), (9.10) and (9.17). For theexpressions involving the SDME’s, these trigonometric functions are made up of two multipliedelements of Wigner D -functions. For those involving the statistical tensors, the trigonometricfunctions are spherical harmonics. Since the dependence of the distribution and observables onthese angles is known, the expressions derived here can be used to apply fits to the data withthe SDME’s or statistical tensors as fit parameters. Note, however, that so far we have made theassumption that only one quasi-two-body state process contributes to the process. We will laterconsider the case of multiple channels contributing.59 HAPTER 10PARITY INVARIANCE CONSIDERATIONS
Parity is conserved in electromagnetic and strong interactions. Therefore, the transition amplitudesin the reactions we are discussing will be invariant under parity transformations (up to a phasefactor).For the case of two body scattering such as πN → M B (where M is a pseudoscalar meson and B is a baryon), parity conservation leads to M − λ B ; − λ N ( θ ) = ( − λ N − λ B M λ B ; λ N ( θ ) , (10.1)and reduces the number of independent transition amplitudes from 4 to 2.For a three body scattering process such as πN → M M B (where both M and M arepseudoscalar mesons) the relationship is M − λ B ; − λ N ( θ ; θ ∗ , φ ∗ ) = ( − λ N − λ B M λ B ; λ N ( θ ; θ ∗ , π − φ ∗ ) . (10.2)This relates two amplitudes at different kinematic points. Therefore, for the case of a 3-body finalstate, parity conservation cannot be used to reduce the number of independent amplitudes at asingle kinematic point. For γN → M M B we have M − λ B ; − λ γ − λ N ( θ ; θ ∗ , φ ∗ ) = ( − λ γ − λ N + λ B M λ B ; λ γ λ N ( θ ; θ ∗ , π − φ ∗ ) . (10.3)For the general case of two-body scattering with arbitrary spin, A B → a b , the relations are M − λ a − λ b ; − λ A − λ B ( θ ) = η a η b η A η B ( − ( λ b − λ a ) − ( λ B − λ A ) M λ a λ b ; λ A λ B ( θ ) , (10.4)where the λ ’s and η ’s are the helicities and intrinsic parities of the particles, respectively.Since we are assuming in this thesis that the photoproduction reaction occurs through a quasi-two-body state, we can apply this last equation to the reactions γN → V B , γN → M B / and γN → M B / , M − λ V − λ B ; − λ γ − λ N ( θ ) = ( − ( λ V − λ B ) − ( λ γ − λ N ) M λ V λ B ; λ γ λ N ( θ ) ,M − λ B / ; − λ γ − λ N ( θ ) = ( − − λ B / − ( λ γ − λ N ) M λ B / ,λ γ λ N ( θ ) . (10.5)60or the V meson, these relations can be used with eqn. (9.4) in order to give ρ − λ V , − λ ′ V ( V ) = ( − λ V + λ ′ V ρ λ V ,λ ′ V ( V ) ,ρ i ( γ ) − λ V , − λ ′ V ( V ) = ζ ( γ ) i ( V )( − λ V + λ ′ V ρ i ( γ ) λ V ,λ ′ V ( V ) ,ρ i ( N ) − λ V , − λ ′ V ( V ) = ζ ( N ) i ( V )( − λ V + λ ′ V ρ i ( N ) λ V ,λ ′ V ( V ) ,ρ ij ( γN ) − λ V , − λ ′ V ( V ) = ζ ( γ ) i ( V ) ζ ( N ) j ( V )( − λ V + λ ′ V ρ ij ( γN ) λ V ,λ ′ V ( V ) ,ζ ( γ ) i ( V ) = ( i = y, z, − i = x,ζ ( N ) i ( V ) = ( i = x, − i = y, z. (10.6)The values for the ζ i ( γ ) ( V )’s and ζ i ( N ) ( V )’s are due to the values of the intrinsic parities of theparticles involved in the interaction and from the fact that ( − λ γ = 1, ( − λ N = −
1, and( − λ B = − λ γ (integer), λ N (half-integer), and λ B (half-integer). For the B / baryon,applying these parity relations to eqn. (9.12) gives ρ − λ B / , − λ ′ B / ( B / ) = ( − λ B / + λ ′ B / ρ λ B / ,λ ′ B / ( B / ) ,ρ i ( γ ) − λ B / , − λ ′ B / ( B / ) = ζ ( γ ) i ( B / )( − λ B / + λ ′ B / ρ i ( γ ) λ B / ,λ ′ B / ( B / ) ,ρ i ( N ) − λ B / , − λ ′ B / ( B / ) = ζ ( N ) i ( B / )( − λ B / + λ ′ B / ρ i ( N ) λ B / ,λ ′ B / ( B / ) , (10.7) ρ ij ( γN ) − λ B / , − λ ′ B / ( B / ) = ζ ( γ ) i ( B / ) ζ ( N ) j ( B / )( − λ B / + λ ′ B / ρ ij ( γN ) λ B / ,λ ′ B / ( B / ) ,ζ ( γ ) i ( B / ) = ( i = x, − i = y, z,ζ ( N ) i ( B / ) = ( i = y, z − i = x. Note that for the case of the B / baryon, the values of the ζ i ’s are different from those of the V meson by a factor of −
1. This is because the derivation does not involve the quantity ( − λ B since the hadron in the quasi-two-body state that accompanies the baryon B / is spinless. Wetherefore conclude that, aside from the factor of ( − λ V + λ ′ V in the case of the V meson and a factorof ( − λ B / + λ ′ B / in the case of the B / baryon, 8 of the 16 matrices will get an extra factor of − − W i ’s in eqns. (9.6) and (9.14).61hile eqns. (10.6) and (10.7) were derived for the case of a decaying vector meson and adecaying spin-3/2 baryon, respectively, the values of the ζ factors in eqn. (10.6) will be the samefor any meson of any spin and intrinsic parity, while those in eqn. (10.7) will be the same for anybaryon of any spin and intrinsic parity. The reason it is independent of intrinsic parity is becausewhen the parity relations in eqn. (10.4) are applied in eqn. (6.8), each of the two transition matricesgive you one factor of the same intrinsic parity. Since the only possible values are 1 and −
1, theoverall factor will always be 1 regardless of intrinsic parity. The ζ factor also depends on whetherthe decaying resonance is a baryon or meson, but not on its spin. For a quasi-two-body stateconsisting of a meson and a baryon of arbitrary spins M B , if the meson decays the derivation givesyou factors of ( − λ B . But since λ B is a multiple of 1/2, this factor will always be − − λ M .But since λ M is a multiple of 1, this factor will always be 1 regardless of the spin.Using eqn. (5.7) along with the parity relations in eqns. (10.6) and (10.7) leads to t iLM = ζ i ( − L + M +2 S t iL − M , (10.8)for the statistical tensors.The constraint of parity invariance in the production reaction will reduce the total number ofindependent parameters that describe the 16 matrices in eqns. (9.1) and (9.11), which means theexpressions for the W i ’s will simplify. Take for example the expression for W y ( γ ) ( θ ∗ , φ ∗ ; V ) in eqn.(9.7). By using the relations in eqn. (10.6) and substituting into these equations, we see that all ofthe terms that are proportional to sine functions of φ ∗ vanish. Since φ ∗ then only appears insideof cosine functions, W y ( γ ) ( θ ∗ , φ ∗ ; V ) is an even function of φ ∗ , W y ( γ ) ( θ ∗ , φ ∗ ; V ) = ρ y ( γ )11 ( V ) sin θ ∗ + ρ y ( γ )00 ( V ) cos θ ∗ (10.9) − √ ℜ ρ y ( γ )10 ( V ) sin 2 θ ∗ cos φ ∗ (10.10) − ℜ ρ y ( γ )1 − ( V ) sin θ ∗ cos 2 φ ∗ (the parity relations imply that ρ y ( γ )1 − ( V ) is purely real). We can also see that, for the case of W x ( γ ) ( θ ∗ , φ ∗ ), it is the terms proportional to cosine functions of φ ∗ that vanish (as well as the firsttwo terms). Since the φ ∗ then only appears inside of sine functions, W x ( γ ) ( θ ∗ , φ ∗ ; V ) is an oddfunction of φ ∗ , W x ( γ ) ( θ ∗ , φ ∗ ; V ) = √ ℑ ρ x ( γ )10 ( V ) sin 2 θ ∗ sin φ ∗ (10.11)62 ℑ ρ x ( γ )1 − ( V ) sin θ ∗ sin 2 φ ∗ (the parity relations imply that ρ x ( γ )1 − ( V ) is purely imaginary). We get the same results, as weshould, if we express the W i ’s in terms of statistical tensors instead. By using the parity andhermiticity relations of the tensors, eqns. (10.8) and (5.10), in eqn. (9.8), we get the followingsimplified expression, W y ( γ ) ( θ ∗ , φ ∗ ; V ) = 1 √ t y ( γ )00 − √ t y ( γ )20 (1 + 3 cos 2 θ ∗ )+ ℜ [ t y ( γ )21 ] sin 2 θ ∗ cos φ ∗ − ℜ [ t y ( γ )22 ] sin θ ∗ cos 2 φ ∗ , (10.12)(parity invariance and hermiticity implies that t y ( γ )21 and t y ( γ )22 are purely real). In the same way, W x ( γ ) in terms of the statistical tensors is simplified when parity is conserved, W x ( γ ) ( θ ∗ , φ ∗ ; V ) = −ℑ [ t x ( γ )21 ] sin 2 θ ∗ sin φ ∗ + ℑ [ t x ( γ )22 ] sin θ sin 2 φ ∗ , (10.13)(parity invariance and hermiticity implies that t x ( γ )21 and t x ( γ )22 are purely imaginary, while t x ( γ )00 and t x ( γ )20 are equal to zero).For the case of the B / baryon, it is W y ( γ ) ( θ ∗ , φ ∗ ; B / ) that is an odd function of φ ∗ , while itis W x ( γ ) ( θ ∗ , φ ∗ ; B / ) that is an even function of φ ∗ , W y ( γ ) ( θ ∗ , φ ∗ ; B / ) = √ ℑ ρ y ( γ )10 ( B / ) sin 2 θ ∗ sin φ ∗ + ℑ ρ y ( γ )1 − ( B / ) sin θ ∗ sin 2 φ ∗ , (10.14) W x ( γ ) ( θ ∗ , φ ∗ ; B / ) = ρ x ( γ )11 ( B / ) sin θ ∗ + ρ x ( γ )00 ( B / ) cos θ ∗ − √ ℜ ρ x ( γ )10 ( B / ) sin 2 θ ∗ cos φ ∗ − ℜ ρ x ( γ )1 − ( B / ) sin θ cos 2 φ ∗ . (10.15)In terms of statistical tensors (again, by using the parity and hermiticity relations, eqns. (10.8)and (5.10)), W y ( γ ) ( θ ∗ , φ ∗ ; B / ) = − r ℑ [ t y ( γ )21 ( B / )] sin 2 θ ∗ sin φ ∗ + 12 r ℑ [ t y ( γ )22 ( B / )] sin θ ∗ sin 2 φ ∗ , (10.16)63nd W x ( γ ) ( θ ∗ , φ ∗ ; B / ) = t x ( γ )00 ( B / ) − t x ( γ )20 ( B / )(1 + 3 cos 2 θ ∗ )+ r ℜ [ t x ( γ )21 ( B / )] sin 2 θ ∗ cos φ ∗ − r ℜ [ t x ( γ )22 ( B / )] sin θ ∗ cos 2 φ ∗ . (10.17)We will later show that it is in general true that once the parity constraints on the productionreaction are applied, each of the W i ’s and, therefore, the observables, will be either even or oddfunctions of φ ∗ . 64 HAPTER 11ANGULAR DISTRIBUTIONS ANDPOLARIZATION OBSERVABLES WHEN MORETHAN ONE QUASI-TWO-BODY STATE CHANNELCONTRIBUTES
To examine the case when more than one quasi-two-body state contributes to the photoproduc-tion reaction, we will first examine the example of three specific channels, γN → SB → M M B,γN → V B → M M B,γN → T B → M M B, (11.1)where S , V and T are scalar, vector, and tensor (spin-2) mesons.The main difference from the single-channel case is that in the multi-channel case the γN statetransitions into a linear superposition of the SB , V B , and
T B states, | Ψ( SB, V B, T B ) i = | Ψ( SB ) i + | Ψ( V B ) i + | Ψ( T B ) i . (11.2)In matrix form, this is | Ψ( BS, BV, BT ) i = ψ ( SB ) ψ - ( SB ) ψ ( V B ) ψ − ( V B ) ψ ( V B )... ψ - - ( V B ) ψ ( T B ) ψ − ( T B ) ψ ( T B )... ψ - - ( T B ) , (11.3)65here ψ λ S λ B ( SB ) is the probability amplitude for the state in the ensemble to be found in a SB state with respective spin projections λ S and λ B , and similarly for ψ λ V λ B ( V B ) and ψ λ T λ B ( T B ) .The first 2 entries in this vector are the amplitudes corresponding to the BS state, the next 6correspond to the BV state, and the last 10 correspond to the BT state, making its Hilbert space18-dimensional. Its spin density matrix will therefore be represented by a 18 × M = M SB ; γN ) M −
12 ( SB ; γN ) M − SB ; γN ) M − −
12 ( SB ; γN ) M −
12 ;1 12 ( SB ; γN ) M −
12 ;1 −
12 ( SB ; γN ) M −
12 ; − SB ; γN ) M −
12 ; − −
12 ( SB ; γN ) M V B ; γN ) M −
12 (
V B ; γN ) M − V B ; γN ) M − −
12 (
V B ; γN ) M −
12 ;1 12 (
V B ; γN ) M −
12 ;1 −
12 (
V B ; γN ) M −
12 ; − V B ; γN ) M −
12 ; − −
12 (
V B ; γN ) ... M − −
12 ;1 12 (
V B ; γN ) M − −
12 ;1 −
12 (
V B ; γN ) M − −
12 ; − V B ; γN ) M − −
12 ; − −
12 (
V B ; γN ) M TB ; γN ) M −
12 ( TB ; γN ) M − TB ; γN ) M − −
12 ( TB ; γN ) M −
12 ;1 12 ( TB ; γN ) M −
12 ;1 −
12 ( TB ; γN ) M −
12 ; − TB ; γN ) M −
12 ; − −
12 ( TB ; γN ) ... M − −
12 ;1 12 ( TB ; γN ) M − −
12 ;1 −
12 ( TB ; γN ) M − −
12 ; − TB ; γN ) M − −
12 ; − −
12 ( TB ; γN ) . (11.4)In the previous matrix, the first two rows describe the transition of the initial system into the SB system (first two entries in state vector in eqn. (11.3)). The next six rows describe the transitioninto the V B system (next six entries in the state vector), and the last ten rows describe thetransition into the
T B system (last ten entries in the state vector). Since this state vector is 18 × × ρ ( S, V, T ) = ˆ ρ ( S ) ˆ ξ ( S, V ) ˆ η ( S, T )ˆ ξ † ( V, S ) ˆ ρ ( V ) ˆ χ ( V, T )ˆ η † ( T, S ) ˆ χ † ( T, V ) ˆ ρ ( T ) . (11.5)The diagonal matrices are the ones that are obtained when the reaction is single-channel and nointerference takes place. We will refer to the off-diagonal submatrices as interference matrices. Wecan also write it asˆ ρ ( S, V, T ) = ˆ ρ ( S ) + ˆ ρ ( V ) + ˆ ρ ( T ) + (cid:2) ˆ ξ ( S, V ) + ˆ η ( S, T ) + ˆ χ ( V, T ) + h.c. (cid:3) , (11.6)66here the matrix elements of the interference matrices are ξ ( S, V ) λ V ≡ X λ γ λ N λ ′ γ λ ′ N λ B M λ B ; λ γ λ N ρ ( γN ) λ γ λ N ; λ ′ γ λ ′ N M ∗ λ V λ B ; λ ′ γ λ ′ N ,η ( S, T ) λ T ≡ X λ γ λ N λ ′ γ λ ′ N λ B M λ B ; λ γ λ N ρ ( γN ) λ γ λ N ; λ ′ γ λ ′ N M ∗ λ T λ B ; λ ′ γ λ ′ N ,χ ( V, T ) λ V λ T ≡ X λ γ λ N λ ′ γ λ ′ N λ B M λ V λ B ; λ γ λ N ρ ( γN ) λ γ λ N ; λ ′ γ λ ′ N M ∗ λ T λ B ; λ ′ γ λ ′ N , (11.7)(note how there is a sum over λ B , indicating that the partial trace over the recoil baryon has beentaken). These interference matrices therefore have “mixed” indices: their “row” and “column”indices refer to a different meson. Since these mesons have different spin, these matrices are notsquare: ˆ ξ is 1 ×
3, ˆ η is 1 ×
5, and ˆ χ is 3 × M M , W ( θ ∗ , φ ∗ ; S, V, B ) = ˆ D † ( θ ∗ , φ ∗ )ˆ ρ ( S, V, T ) ˆ D ( θ ∗ , φ ∗ ) . (11.8)This time, the form of the transition matrix ˆ D † isˆ D = D ( θ ∗ , φ ∗ ) D ( θ ∗ , φ ∗ ) D ( θ ∗ , φ ∗ ) D − ( θ ∗ , φ ∗ ) D ( θ ∗ , φ ∗ ) D ( θ ∗ , φ ∗ )... D − ( θ ∗ , φ ∗ ) . (11.9)The distribution can be written as W ( θ ∗ , φ ∗ ; S, V, T ) = W ( S ) ( θ ∗ , φ ∗ ) + W ( V ) ( θ ∗ , φ ∗ ) + W ( T ) ( θ ∗ , φ ∗ ) + W ( SV ) ( θ ∗ , φ ∗ )+ W ( ST ) ( θ ∗ , φ ∗ ) + W ( T V ) ( θ ∗ , φ ∗ ) . (11.10)In this expression W ( S ) is the contribution of a scalar intermediate meson, W ( V ) arise from a vectormeson, and W ( T ) arises from a meson of spin 2. W ( SV ) arises from the interference between a scalarand vector meson, W ( ST ) comes from the interference between a scalar and a tensor, and W ( T V ) arises from interference between a vector and tensor meson. The explicit forms are67 ( S ) ( θ ∗ , φ ∗ ) = ρ ( S ) , (11.11) W ( V ) ( θ ∗ , φ ∗ ) = ρ ( V )2 + ρ ( V ) + ρ − − ( V )4 + (cid:18) ρ ( V )2 − ρ ( V ) + ρ − − ( V )4 (cid:19) cos(2 θ ∗ )+ 12 (cid:18) ℑ [ ρ − ( V )] sin(2 φ ∗ ) − ℜ [ ρ − ( V )] cos(2 φ ∗ ) (cid:19)(cid:18) − cos(2 θ ∗ ) (cid:19) (11.12)+ sin(2 θ ∗ ) √ (cid:18) ℜ [ ρ − ( V ) − ρ ( V )] cos( φ ∗ ) + ℑ [ ρ ( V ) − ρ − ( V )] sin( φ ∗ ) (cid:19) ,W ( T ) ( θ ∗ , φ ∗ ) = 11 + 12 cos(2 θ ∗ ) + 9 cos(4 θ ∗ )32 ρ ( T ) + (cid:18) − cos(4 θ ∗ ) (cid:19)(cid:18) (cid:0) ρ ( T ) + ρ − − ( T ) (cid:1) − (cid:16) ℜ [ ρ − ( T )] cos(2 φ ∗ ) − ℑ [ ρ − ( T )] sin(2 φ ∗ ) (cid:17)(cid:19) + 364 (cid:18) ρ ( T ) + ρ − − ( T ) (cid:19)(cid:18) − θ ∗ ) + cos(4 θ ∗ ) (cid:19) + 116 r (cid:18) ℜ [ ρ − ( T ) + ρ ( T )] cos(2 φ ∗ ) − ℑ [ ρ − ( T ) + ρ ( T )] sin(2 φ ∗ ) (cid:19)(cid:18) θ ∗ ) − θ ∗ ) + 1 (cid:19) − (cid:18) ℜ [ ρ − ( T )] cos(4 φ ∗ ) − ℑ [ ρ − ( T )] sin(4 φ ∗ ) (cid:19)(cid:18) − θ ∗ ) + cos(4 θ ∗ ) (cid:19) − r (cid:18) ℜ [ ρ − ( T ) − ρ ( T )] cos(2 φ ∗ ) (11.13) − ℑ [ ρ − ( T ) − ρ ( T )] sin(2 φ ∗ ) (cid:19)(cid:18) θ ∗ ) + 3 sin(4 θ ∗ ) (cid:19) + 316 (cid:19)(cid:18) ℜ [ ρ − ( T ) − ρ − ( T )] cos(3 φ ∗ ) − ℑ [ ρ − ( T ) − ρ − ( T )] sin(3 φ ∗ ) (cid:19) + (cid:18) ℜ [ ρ − − ( T ) − ρ ( T )] cos( φ ∗ ) − ℑ [ ρ − − ( T ) − ρ ( T )] sin( φ ∗ ) (cid:19) × (cid:18) θ ∗ ) − sin(4 θ ∗ ) (cid:19) ,W ( SV ) ( θ ∗ , φ ∗ ) = 2 ℜ [ ξ ] cos( θ ∗ )+ √ θ ∗ ) (cid:18) ℜ [ ξ − − ξ ] cos( φ ∗ ) − ℑ [ ξ + ξ − ] sin( φ ∗ ) (cid:19) , (11.14)68 ( ST ) ( θ ∗ , φ ∗ ) = 14 ℜ [ η ] (cid:18) θ ∗ ) (cid:19) + r
32 sin(2 θ ∗ ) (cid:18) ℜ [ η − − η ] cos( φ ∗ ) − ℑ [ η + η − ] sin( φ ∗ ) (cid:19) (11.15)+ 12 r (cid:18) − cos(2 θ ∗ ) (cid:19)(cid:18) ℜ [ η + η − ] cos(2 φ ∗ ) + ℑ [ η − η − ] sin(2 φ ∗ ) (cid:19) , and W ( V T ) ( θ ∗ , φ ∗ ) = 14 ℜ [ χ ](5 cos( θ ∗ ) + 3 cos(3 θ ∗ ))+ (cos( θ ∗ ) − cos(3 θ ∗ )) ( √ (cid:18) ℜ [ χ + χ − − ]+ ℜ [ χ − − χ − ] cos(2 φ ∗ ) − ℑ [ χ − + χ − ] sin(2 φ ∗ ) (cid:19) + 14 r (cid:18) ℜ [ χ + χ − ] cos(2 φ ∗ ) − ℑ [ χ − χ − ] sin(2 φ ∗ ) (cid:19)) + 14 √ (cid:18) θ ∗ ) − sin( θ ∗ ) (cid:19)(cid:18) ℜ [ χ − χ − ] cos( φ ∗ )) (11.16)+ ℑ [ χ + χ − ] sin( φ ∗ ) (cid:19) + 12 r (cid:18) sin( θ ∗ ) + sin(3 θ ∗ ) (cid:19)(cid:18) ℜ [ χ − − χ ] cos( φ ∗ ) − ℑ [ χ + χ − ] sin( φ ∗ ) (cid:19) + √ (cid:18) θ ∗ ) − sin(3 θ ∗ ) (cid:19)(cid:18) ℜ [ χ − − χ − ] cos(3 φ ∗ )+ ℑ [ χ − + χ − ] sin(3 φ ∗ )+ ℜ [ χ − − − χ ] cos( φ ∗ ) − ℑ [ χ − − + χ ] sin( φ ∗ ) (cid:19) . We will now examine the case for the decay of photoproduced baryons in the reactions γN → M B / → M M B and γN → M B / → M M B under the assumption that three differentbaryons in the intermediate quasi-two-body state contribute of spins 1 /
2, 3 /
2, and 5 / ρ ( S ) = ˆ ρ (1 /
2) ˆ ξ ˆ η ˆ ξ † ˆ ρ (3 /
2) ˆ χ ˆ η † ˆ χ † ˆ ρ (5 / . (11.17)ˆ ρ (1 / ρ (3 / ρ (5 /
2) are the spin density matrices that the spin-1 /
2, 3 /
2, and 5 / /
2, 3 /
2, and 5 / ξ corresponds to the spin-1/2 baryon, while its right index correspondsto the spin-3/2 baryon. For the matrix ˆ η , the left index corresponds to the spin-1/2 baryon andthe right index to the spin-5/2 baryon. Finally, for the matrix ˆ χ the left index corresponds to thespin-3/2 baryon and the right index corresponds to the spin-5/2 baryons. The matrices with thedaggers are the conjugate transpose of their respective matrix.The distribution can be written as W ( θ ∗ , φ ∗ ; 1 / , / , /
2) = W (1 / ( θ ∗ , φ ∗ ) + W (3 / ( θ ∗ , φ ∗ ) + W (5 / ( θ ∗ , φ ∗ )+ W (1 / , / ( θ ∗ , φ ∗ ) + W (1 / , / ( θ ∗ , φ ∗ ) + W (3 / , / ( θ ∗ , φ ∗ ) . (11.18)In this expression W (1 / is the contribution of a spin-1 / W (3 / arises froma baryon with spin 3/2, and W (5 / arises from a baryon with spin 5/2. W (1 / , / arises fromthe interference between the baryons with spin 1/2 and 3/2, W (1 / , / comes from the interferencebetween baryons with spin 1/2 and spin 5/2, and W (3 / , / arises from interference between baryonswith spin 3/2 and spin 5/2. The explicit forms are W (1 / ( θ ∗ , φ ∗ ) = ρ (1 /
2) + ρ − − (1 / , (11.19) W (3 / ( θ ∗ , φ ∗ ) = 18 ( (cid:18) − cos(2 θ ∗ ) (cid:19)(cid:18) ρ (3 /
2) + ρ − − (3 / (cid:19) + (cid:18) θ ∗ ) (cid:19)(cid:18) ρ (3 /
2) + ρ − − (3 / (cid:19)) (11.20) − √ (cid:18) − cos(2 θ ∗ ) (cid:19)(cid:18) ℜ [ ρ − (3 /
2) + ρ − (3 / φ ∗ ) − ℑ [ ρ − (3 /
2) + ρ − (3 / φ ∗ ) (cid:19) √
32 sin(2 θ ∗ ) (cid:18) ℜ [ ρ − − (3 / − ρ (3 / φ ∗ ) − ℑ [ ρ − − (3 / − ρ (3 / φ ∗ ) (cid:19) ,W (5 / ( θ ∗ , φ ∗ ) = 132 (cid:18) ρ (5 /
2) + ρ − − (5 / (cid:19)(cid:18)
12 + 15 cos(2 θ ∗ ) + 5 cos(4 θ ∗ ) (cid:19) + 164 (cid:18) ρ (5 /
2) + ρ − − (5 / (cid:19)(cid:18) − θ ∗ ) −
15 cos(4 θ ∗ ) (cid:19) + 564 (cid:18) ρ (5 /
2) + ρ − − (5 / (cid:19)(cid:18) − θ ∗ ) + cos(4 θ ∗ ) (cid:19) + √ (cid:18) θ ∗ ) − sin(4 θ ∗ ) (cid:19) × ( ℜ [ ρ − − (5 / − ρ (5 / φ ∗ ) − ℑ [ ρ − − (5 / − ρ (5 / φ ∗ )+ 1 √ (cid:18) ℜ [ ρ − (5 / − ρ − (5 / φ ∗ ) − ℑ [ ρ − (5 / − ρ − (5 / φ ∗ ) (cid:19)) (11.21)+ √ (cid:18) − θ ∗ ) + cos(4 θ ∗ ) (cid:19)(cid:18) ℜ [ ρ − (5 /
2) + ρ − (5 / φ ∗ ) − ℑ [ ρ − (5 /
2) + ρ − (5 / φ ∗ ) (cid:19) + 116 r (cid:18) − θ ∗ ) − θ ∗ ) (cid:19)(cid:18) ℜ [ ρ (5 /
2) + ρ − − (5 / φ ∗ ) − ℑ [ ρ (5 /
2) + ρ − − (5 / φ ∗ ) (cid:19) + 18 √ (cid:18) θ ∗ ) − θ ∗ ) (cid:19)(cid:18) ℜ [ ρ − − (5 / − ρ (5 / φ ∗ ) − ℑ [ ρ − − (5 / − ρ (5 / φ ∗ ) (cid:19) + 116 √ (cid:18) − θ ∗ ) − θ ∗ ) (cid:19)(cid:18) ℜ [ ρ − (5 / − ρ − (5 / φ ∗ ) − ℑ [ ρ − (5 / − ρ − (5 / φ ∗ ) (cid:19) ,W (1 / , / ( θ ∗ , φ ∗ ) = 2 ℜ [ ξ + ξ − − ] cos( θ ∗ )+ sin( θ ∗ ) ((cid:18) ℜ [ ξ − − ξ − ] + √ ℜ [ ξ − − − ξ ] (cid:19) cos( φ ∗ )71 (cid:18) ℑ [ ξ − + ξ − ] + √ ℑ [ ξ − − + ξ ] (cid:19) sin( φ ∗ ) ) , (11.22) W (1 / , / ( θ ∗ , φ ∗ ) = 12 ℜ [ η + η − − ] (cid:18) θ ∗ ) (cid:19) + 12 √ (cid:18) − cos(2 θ ∗ ) (cid:19)((cid:18) ℜ [ η − + η − ] + √ ℜ [ η + η − − ] (cid:19) cos(2 φ ∗ ) (cid:18) ℑ [ η − − η − ] + √ ℑ [ η − η − − ] (cid:19) sin(2 φ ∗ ) ) (11.23)+ sin(2 θ ∗ ) ((cid:18) ℜ [ η − − η − ] + √ ℜ [ η − − − η ] (cid:19) cos( φ ∗ ) − (cid:18) ℑ [ η − + η − ] + √ ℑ [ η − − + η ] (cid:19) sin( φ ∗ ) ) , and W (3 / , / ( θ ∗ , φ ∗ ) = 18 √ (cid:18) θ ∗ ) − sin(3 θ ∗ ) (cid:19)( √ (cid:18) ℜ [ χ − − − χ ] cos( φ ∗ ) − ℑ [ χ − − + χ ] sin( φ ∗ ) (cid:19) + (cid:18) √ ℜ [ χ − − χ − ] + √ ℜ [ χ − − χ − ] (cid:19) cos(3 φ ∗ )+ (cid:18) √ ℑ [ χ − + χ − ] + √ ℑ [ χ − + χ − ] (cid:19) sin(3 φ ∗ ) ) + 18 (cid:18) cos( θ ∗ ) − cos(3 θ ∗ ) (cid:19)( √ ℜ [ χ + χ − − ]+ (cid:18) √ ℜ [ χ + χ − − ] − √ ℜ [ χ − + χ − ] + √ ℜ [ χ − + χ − ] (cid:19) cos(2 φ ∗ )+ (cid:18) √ ℑ [ χ − χ − − ] − √ ℑ [ χ − − χ − ] (11.24)+ √ ℑ [ χ − − χ − ] (cid:19) sin(2 φ ∗ ) ) + 12 (cid:18) θ ∗ ) + cos(3 θ ∗ ) (cid:19) ℜ [ χ + χ − − ]+ 18 (cid:18) θ ∗ ) + sin(3 θ ∗ ) (cid:19)(cid:18) ℜ [ χ − − χ − ] cos( φ ∗ )72 ℑ [ χ − + χ − ] sin( φ ∗ ) (cid:19) + 18 √ (cid:18)
11 sin( θ ∗ ) + 7 sin(3 θ ∗ ) (cid:19)(cid:18) ℜ [ χ − − − χ ] cos( φ ∗ ) − ℑ [ χ − − + χ ] sin( φ ∗ ) (cid:19) + √ (cid:18) θ ∗ ) − θ ∗ ) (cid:19)(cid:18) ℜ [ χ − − − χ ] cos( φ ∗ )+ ℑ [ χ − − + χ ] sin( φ ∗ ) (cid:19) . There is one important aspect to consider in the case of decaying intermediate state baryons.Unlike the case of a decaying meson, if M and M are the same meson, then an individual baryonresonance has two possible decay channels: B / → M B , and B / → M B . This means that thedecay should be considered as an interference of two different states, each with its own individualspin density matrix. Therefore, the density matrix of a single decaying baryon will have this form,ˆ ρ ( B / ) = ˆ ρ ( B B3 / ) ˆ ξ ( B B3 / , B C3 / )ˆ ξ † ( B B3 / , B C3 / ) ˆ ρ ( B C3 / ) ! . (11.25)The diagonal spin density submatrices describe the spin state of the baryon resonance in pathwaysB and C. The off-diagonal submatrices is the interference matrix between the same baryon inpathways B and C. Only minor modifications are needed for the case of interference of intermediate quasi-two-bodystates where some of them contain decaying mesons and some of them contain decaying baryons.To illustrate this, we will use as an example the case where two states interfere: one intermediatestate has a decaying spin-1 meson V , and the other has a decaying spin-3/2 baryon B / , γN → V B → M M B,γN → M B / → M M B (11.26)(for simplicity, we will only consider pathways A and B). In the distribution, there will be a set ofterms that contain the SDME’s of the interference submatrix ˆ ξ , W ( V, B / ) = X λ V λ B / λ B D ∗ λ V ( θ ∗ A , φ ∗ A ) ξ λ V λ B ;0 λ B / D / λ B / λ B ( θ ∗ B , φ ∗ B ) . (11.27)73ote that the angles in the arguments of the two Wigner- D are different. This is due to the factthat the angles for the two decaying resonances are defined in terms of different coordinate systems.Expressions relating these different angles will be given in appendix A.In the cases in which only mesons or only baryons interfered, we could take the partial traceover the spectator hadron to get the spin density matrix of the decaying hadron. But in this case,the helicity indices of the interference matrix have different spectator hadrons in the “row” and“column” indices: the baryon B and the meson M (since M is spinless, its helicity index is equalto 0). Therefore, we cannot define a partial trace on this interference matrix since the λ B helicityindex appears on both this matrix and on one of the Wigner- D functions. In this case, one optionis to write the expression for the distribution in terms of this matrix without taking the partialtrace. Another is to decompose the matrix into a sum of spin density matrices for the V B systeminto eigenstates of total spin, which can take on values of 1 / / ξ λ V λ B ;0 λ B / = X e S, f M C e S f M λ V ; λ B e ξ (cid:18) e S, (cid:19) f Mλ B / = C ( λ V + λ B )1 λ V ; λ B e ξ (cid:18) , (cid:19) ( λ V + λ B ) λ B / (11.28)+ C ( λ V + λ B )1 λ V ; λ B e ξ (cid:18) , (cid:19) ( λ V + λ B ) λ B / . In the previous expression, ˆ e ξ (cid:0) , (cid:1) and ˆ e ξ (cid:0) , (cid:1) are interference spin density matrices between B / and the V B system in its S = 1 / B / and the V B system in its S = 3 / V B system (which is set to f M = λ V + λ B from conservation of angular momentum), while the rightindex labels the spin projection of the B / baryon. The expression for the distribution in equationwill therefore contain terms of this form, W = · · · + X f Mλ B / λ B λ V D ∗ λ V ( θ ∗ A , φ ∗ A ) C / f M λ V ;1 / λ B e ξ (cid:18) , (cid:19) f Mλ B / D / λ B / λ B ( θ ∗ B , φ ∗ B )+ X f Mλ B / λ B λ V D ∗ λ V ( θ ∗ A , φ ∗ A ) C / f M λ V ;1 / λ B e ξ (cid:18) , (cid:19) f Mλ B / D / λ B / λ B ( θ ∗ B , φ ∗ B ) + · · · (11.29)= · · · + X f M,λ B / e D / ∗ f Mλ B ( θ ∗ A , φ ∗ A ) e ξ (cid:18) , (cid:19) f Mλ B / D / λ B / λ B ( θ ∗ B , φ ∗ B )74 X f M,λ B / e D / ∗ f Mλ B ( θ ∗ A , φ ∗ A ) e ξ (cid:18) , (cid:19) f Mλ B / D / λ B / λ B ( θ ∗ B , φ ∗ B ) + · · · , where e D e S ∗ f Mλ B ( θ ∗ , φ ∗ ) ≡ X λ V D ∗ λ V ( θ ∗ , φ ∗ ) C e S f M / λ B , λ V with e S = 1 / , / . (11.30)The quantity defined in the previous equation can be considered a generalized Wigner- D function,and also a decay amplitudes for the e S = 1 / e S = 3 / V B system todecay into B . Using the following properties of the Wigner- D functions and of the Clebsch-Gordancoefficients, D ∗ S − λ − λ ′ = ( − ( λ − λ ′ ) D Sλλ ′ ,C cγaα ; bβ = ( − a − α r c + 12 b + 1 C bβcγ ; a − α , (11.31) γ = α + β, the following property can be shown, e D ∗ e S − f M − λ N = ( − e S − f M + S R + − λ N e D e S f Mλ N , = ( − e S − f M + − λ N e D e S f Mλ N (for S R = 1) , (11.32)where S R is the spin of the meson resonance that decays (a ρ meson, in this example). This propertywill be used later in this article. 75 HAPTER 12GENERAL EXPRESSIONS FOR THE ANGULARDISTRIBUTIONS AND NON-RECOILPOLARIZATION OBSERVABLES
We will now derive the general expression for the angular distribution W when an arbitrary num-ber of quasi-two-body states of arbitrary spin contribute to the reaction, with interference effectsincluded. As we have seen, the distribution will be a sum of different terms that contain different subma-trices, W ( θ ∗ , φ ∗ ) = X i Tr[ ˆ D S i † ( θ ∗ , φ ∗ )ˆ ρ ( S i ) ˆ D S i ( θ ∗ , φ ∗ )] + X i>j (cid:26) Tr[ ˆ D S i † ( θ ∗ , φ ∗ ) ˆ ξ ( S i , S j ) ˆ D S j ( θ ∗ , φ ∗ )]+ Tr[ ˆ D S j † ( θ ∗ , φ ∗ ) ˆ ξ ( S j , S i ) ˆ D S i ( θ ∗ , φ ∗ )] (cid:27) . (12.1)The indices i and j are used to label the spins of all of the contributing intermediate state hadronscontributing to the reaction. The ˆ ρ ( S i )’s are the spin density matrices of the intermediate states.The ˆ ξ ( S i , S j ) are the interference spin density matrices between the hadrons labeled by the indices i and j . Note that the sum goes over i > j terms so as to not overcount the interfering hadronpairs. The ˆ D S i are the Wigner- D matrices.By using the hermiticity condition of the interference submatrices,ˆ ξ ( S j , S i ) = ˆ ξ † ( S i , S j ) , (12.2)we can rewrite the second term inside of the brackets in the previous expression asTr[ ˆ D S j † ˆ ξ ( S j , S i ) ˆ D S i ] = Tr[ ˆ D S j † ˆ ξ † ( S i , S j ) ˆ D S i † ] , (12.3)76 (cid:16) Tr[ ˆ D S i ˆ ξ ( S i , S j ) ˆ D S j † ] (cid:17) ∗ . Equation 12.1 can therefore be written as W = X i Tr[ ˆ D S i † ˆ ρ ( S i ) ˆ D S i ] + X i>j (cid:26) Tr[ ˆ D S i † ˆ ξ ( S i , S j ) ˆ D S j ]+ (cid:16) Tr[ ˆ D S i ˆ ξ ( S i , S j ) ˆ D S j ] (cid:17) ∗ (cid:27) (12.4)= X i Tr[ ˆ D S i † ˆ ρ ( S i ) ˆ D S i ] + X i>j ℜ Tr[ ˆ D S i † ˆ ξ ( S i , S j ) ˆ D S j ] . Written using index notation, the expression is W = X i X λ,λ ′ ,η D S i ∗ λ,η ρ λ,λ ′ ( S i ) D S i λ ′ ,η + X i>j X λ,λ ′ ,η ℜ h D ∗ S i λ,η ξ λ,λ ′ ( S i , S j ) D S j λ ′ ,η i . (12.5)Since the diagonal matrices are hermitian, not all of its elements are independent. We cantherefore simplify the expression by writing it only in terms of the independent elements, whichare the diagonal and top diagonal ones ( λ = λ ′ and λ ′ > λ , respectively). The first term in the lastequation can therefore be written as X i X λ,λ ′ ,η D ∗ S i λ,η ρ λ,λ ′ ( S i ) D S i λ ′ ,η = X λ,η D ∗ S i λ,η ρ λ,λ ( S i ) D S i λ,η + X λ ′ >λ,η n D ∗ S i λ,η ρ λ,λ ′ ( S i ) D S i λ ′ ,η + D ∗ S i λ ′ ,η ρ λ ′ ,λ ( S i ) D S i λ,η o . (12.6)The first term contains the diagonal terms. Inside the brackets, the first term is a sum over topdiagonal elements, and the second is a sum over the bottom diagonal elements. Using the hermitianproperty of the diagonal matrix, ˆ ρ ( S i ) = ˆ ρ † ( S i ) ,ρ ( S i ) λ ′ ,λ = ρ ∗ ( S i ) λ,λ ′ , (12.7)the expression inside the brackets becomes D ∗ S i λ,η ρ λ,λ ′ ( S i ) D S i λ ′ ,η + D ∗ S i λ ′ ,η ρ λ ′ ,λ ( S i ) D S i λ,η = D ∗ S i λ,η ρ λ,λ ′ ( S i ) D S i λ ′ ,η + (cid:16) D ∗ S i λ,η ρ λ,λ ′ ( S i ) D S i λ ′ ,η (cid:17) ∗ = 2 ℜ [ D ∗ S i λ,η ρ λ,λ ′ ( S i ) D S i λ ′ ,η ] . (12.8)Substituting this last equation into eqn. (12.5), we get W = X i ( X λ,η D ∗ S i λ,η ρ λ,λ ( S i ) D S i λ,η + X λ>λ ′ η ℜ h D ∗ S i λ,η ρ λ,λ ′ ( S i ) D S i λ ′ ,η i) + X i>j X λ,λ ′ ,η ℜ h D ∗ S i λ,η ξ λ,λ ′ ( S i , S j ) D S j λ ′ ,η i . (12.9)77he first term in the brackets is a sum over diagonal elements, which are real. Therefore, thisexpression makes explicit the fact that the distribution is a real quantity.The φ ∗ dependence of the distribution occurs only in the Wigner- D functions and in a verysimple way, D Sλ,λ ′ ( θ ∗ , φ ∗ ) = e − iλφ ∗ d Sλ,λ ′ ( θ ∗ ) , (12.10)where these Wigner d -functions are well known. Substituting eqn. (12.10) into (12.9) and usingthe following identity for complex numbers, ℜ [ z z ] = ℜ [ z ] ℜ [ z ] − ℑ [ z ] ℑ [ z ] , (12.11)we get, W ( θ ∗ , φ ∗ ) = X i ( X λ e d S i λ,λ ( θ ∗ ) ρ λ,λ ( S i ) + X λ>λ ′ e d S i λ,λ ′ ( θ ∗ ) ℜ [ ρ λ,λ ′ ( S i )] cos[( λ − λ ′ ) φ ∗ )] − ℑ [ ρ λ,λ ′ ( S i )] sin[( λ − λ ′ ) φ ∗ )] !) + X i>j X λ,λ ′ e d S i ,S j λ,λ ′ ( θ ∗ ) ℜ [ ξ λ,λ ′ ( S i , S j )] cos[( λ − λ ′ ) φ ∗ )] − ℑ [ ξ λ,λ ′ ( S i , S j )] sin[( λ − λ ′ ) φ ∗ )] ! (12.12)where e d S i λ,λ ′ ( θ ∗ ) ≡ X η d S i η,λ ( θ ∗ ) d S i η,λ ′ ( θ ∗ ) , e d S i ,S j λ,λ ′ ( θ ∗ ) ≡ X η d S i η,λ ( θ ∗ ) d S j η,λ ′ ( θ ∗ ) . (12.13)Therefore, the entire dependence on θ ∗ is contained in the functions e d S i λ,λ ′ ( θ ∗ ) and e d S i ,S j λ,λ ′ ( θ ∗ ), whichare simply bi-linear combinations of the well known d -functions, while the entire φ ∗ dependenceis only contained in cosine and sine functions. Every term in the expression for W will either beproportional to a cosine or sine function of φ ∗ , or be independent of it.Eqn. (12.12) is general for any photoproduction reaction with two mesons in the final statethat has any number of quasi-two-body intermediate states of arbitrary spin contributing to theprocess. If we apply it to the case where only one intermediate state contributes (i.e., the index i runs over only one value and the term with the sum over both i and j vanishes) with a decaying78ector meson V ( S i = 1), we recover eqn. (8.13). If the intermediate states contains instead adecaying spin-3/2 baryon B / ( S i = 3 / Since the decomposition of the spin density matrix into 16 matrices shown in eqn. (6.15)is general for any final state whose initial state has two particles each with two possible spinprojections, and since it can be shown to also apply to interefence matrices, the matrices in eqn.(12.12) can therefore also be decomposed. This will give us an expression of the form shown ineqn. (9.5), with each W i given by W i ( θ ∗ , φ ∗ ) = X k ( X λ e d S k λ,λ ( θ ∗ ) ρ iλ,λ ( S k ) + X λ>λ ′ e d S k λ,λ ′ ( θ ∗ ) ℜ [ ρ iλ,λ ′ ( S k )] cos[( λ − λ ′ ) φ ∗ )] − ℑ [ ρ iλ,λ ′ ( S k )] sin[( λ − λ ′ ) φ ∗ )] !)X l>k X λ,λ ′ e d S k ,S l λ,λ ′ ( θ ∗ ) ℜ [ ξ iλ,λ ′ ( S k , S l )] cos[( λ − λ ′ ) φ ∗ )] − ℑ [ ξ iλ,λ ′ ( S k , S l )] sin[( λ − λ ′ ) φ ∗ )] ! , (12.14)the matrix elements are given by ρ λλ ′ ( S k ) = 14 X λ γ λ N λ H M λλ H ; λ N λ γ M ∗ λ ′ λ H ; λ N λ γ ,ρ i ( γ ) λλ ′ ( S k ) = 14 X λ N λ H λ ′ γ λ γ M λλ H ; λ N λ γ σ iλ γ λ ′ γ M ∗ λ ′ λ H ; λ N λ ′ γ ,ρ i ( N ) λλ ′ ( S k ) = 14 X λ N λ ′ N λ γ λ H M λλ H ; λ N λ γ σ iλ N λ ′ N M ∗ λ ′ λ H ; λ ′ N λ γ ,ρ ij ( γN ) λλ ′ ( S k ) = 14 X λ γ λ ′ γ λ N λ ′ N λ H M λλ H ; λ N λ γ σ iλ γ λ ′ γ σ jλ N λ ′ N M ∗ λ ′ λ H ; λ ′ N λ ′ γ , (12.15)and ξ λλ ′ ( S k , S l ) = 14 X λ γ λ N λ H M λλ H ; λ N λ γ M ∗ λ ′ λ H ; λ N λ γ , i ( γ ) λλ ′ ( S k , S l ) = 14 X λ N λ H λ γ λ ′ γ M λλ H ; λ N λ γ σ iλ γ λ ′ γ M ∗ λ ′ λ H ; λ N λ ′ γ , (12.16) ξ i ( N ) λλ ′ ( S k , S l ) = 14 X λ N λ ′ N λ γ λ H M λλ H ; λ N λ γ σ iλ N λ ′ N M ∗ λ ′ λ H ; λ ′ N λ γ ,ξ ij ( γN ) λλ ′ ( S k , S l ) = 14 X λ γ λ ′ γ λ N λ ′ N λ H M λλ H ; λ ′ N λ ′ γ σ iλ ′ γ λ γ σ jλ ′ N λ N M ∗ λ ′ λ H ; λ N λ γ , and λ H is the helicity of the spectator hadron, the one that does not decay, either a meson or abaryon. In the case of the interference matrices ˆ ξ , λ and λ ′ are the helicities of the two hadronsthat are interfering, whose spins are S k and S l . We can use the expressions for the Pauli matricesin terms of the Kronecker delta in eqn. (9.3) to simplify the previous expressions. For simplicity,we will show the simplified expressions only for the interference matrices, with the understandingthat the only difference between them and the non-interference matrices is that in the latter, theindices λ and λ ′ refer both to helicities of a single hadron. The simplified expression for the densitymatrix related to the unpolarized cross section is ξ λλ ′ ( S k , S l ) = 14 X λ γ λ N λ H M λλ H ; λ γ λ N M ∗ λ ′ λ H ; λ γ λ N . (12.17)The ones for the density matrices related to the beam observables are ξ x ( γ ) λλ ′ ( S k , S l ) = 14 X λ γ λ N λ H M λλ H ; λ γ λ N M ∗ λ ′ λ H ; − λ γ λ N ,ξ y ( γ ) λλ ′ ( S k , S l ) = − i X λ γ λ N λ H λ γ M λλ H ; λ N λ γ M ∗ λ ′ λ H ; − λ γ λ N ,ξ z ( γ ) λλ ′ ( S k , S l ) = 14 X λ γ λ N λ H λ γ M λλ H ; λ γ λ N M ∗ λ ′ λ H ; λ γ λ N , (12.18)and the ones related to the target observables are ξ x ( N ) λλ ′ ( S k , S l ) = 14 X λ γ λ N λ H M λλ H ; λ γ λ N M ∗ λ ′ λ H ; λ γ − λ N ,ξ y ( N ) λλ ′ ( S k , S l ) = − i X λ γ λ N λ H λ N M λλ H ; λ N λ γ M ∗ λ ′ λ H ; λ γ − λ N , (12.19)80 z ( N ) λλ ′ ( S k , S l ) = 14 X λ γ λ N λ H λ N M λλ H ; λ γ λ N M ∗ λ ′ λ H ; λ γ λ N . We will not show the expressions for the ones related to the beam/target double polarizationobservables, but it is easy to see from the pattern of the previous expressions what they wouldlook like. The first pattern is that matrices with superscript of 2 and 3 will have an extra factor of − λ γ or − λ N in the summation, the former for those with ( γ ) in the superscript and the latter forthose with ( N ) in the superscript. Additionally, those with a superscript of 2 will have an extrafactor of i . The second pattern is that matrices with superscript 1 and 2 will have the indices λ γ or λ N with opposite signs in each of the helicity amplitudes, the former for those with ( γ ) in thesuperscript and the latter for those with ( N ) in the superscript. These patterns will be importantin deriving parity relations among these matrix elements. These in turn will be used to show whichof the polarization observables will be even and which will be odd in φ ∗ .To relate the W i ’s in eqn. (12.14) to the polarization observables we follow the same procedureused in section 9 for the case of vector meson decays. To summarize, the trace is taken on bothsides of eqn. (7.38). The terms on the right-hand side proportional to the Pauli matrices, ˆ σ i ,vanish, while the left-hand side is equal to W . This expression is then compared to eqn. (9.5) toshow that each W i is proportional to a polarization observable. We reproduce the results here forconvenience, W ( θ ∗ , φ ∗ ) = I ( θ ∗ , φ ∗ ) ,W i ( N ) ( θ ∗ , φ ∗ ) = I P i ( θ ∗ , φ ∗ ) ,W x ( γ ) ( θ ∗ , φ ∗ ) = I cos 2 βI c ( θ ∗ , φ ∗ ) ,W y ( γ ) ( θ ∗ , φ ∗ ) = I sin 2 βI s ( θ ∗ , φ ∗ ) ,W z ( γ ) ( θ ∗ , φ ∗ ) = I I ⊙ ( θ ∗ , φ ∗ ) ,W ix ( γN ) ( θ ∗ , φ ∗ ) = I cos 2 βP ic ( θ ∗ , φ ∗ ) ,W iy ( γN ) ( θ ∗ , φ ∗ ) = I sin 2 βP is ( θ ∗ , φ ∗ ) ,W iz ( γN ) ( θ ∗ , φ ∗ ) = I P i ⊙ ( θ ∗ , φ ∗ ) . (12.20)81 If the reaction is parity conserving, the helicity amplitudes can be related by equation (10.4).When this is applied to eqns. (12.17) to (12.19), we get that the parity relations of the 16 matriceswill have the form ξ − λ − λ ′ = ζ ( − λ + λ ′ ξ λλ ′ ,ζ ∈ { , − } . (12.21)The value of ζ depends on three factors: its superscript (e.g., x ( γ ) , z ( N ) , zy ( γN ), etc.), the intrin-sic parities of the two unstable hadrons that are interfering, and on whether the density matrixdescribes the interference between mesons or baryons. We will now derive the previous equationand the conditions that determine the value of ζ .When the parity relations are applied to eqns. (12.17) to (12.19), four factors will appear, eachequal to either 1 or −
1. Two of them are related to the properties of the hadrons involved, whilethe other two are related to the superscript of the 16 matrices.The first factor has to do with the intrinsic parities of the hadrons involved in the reaction.When the parity relations are applied to eqns. (12.17) to (12.19), this factor appears: ξ − λ − λ ′ ∝ ηη ′ ( η H ) ( η γ η N ) − ξ λλ ′ = ηη ′ ξ λλ ′ , (12.22)where η and η ′ are the intrinsic parities of the two unstable resonances that are interfering, and H labels the spectator hadron. The factors that are raised to the power of two are equal to onebecause all η = ±
1. For the non-interference matrices η = η ′ , so that the factor ηη ′ in the previousequation will equal 1. This last equation tells us that, when the parity relations of the helicityamplitudes are applied in eqns. (12.17) to (12.19), the overall expression may gain a factor of − − ξ − λ − λ ′ ∝ ( − ( λ + λ ′ − λ H ) ξ λλ ′ = ± ( − ( λ + λ ′ ) ξ λλ ′ (12.23)where λ H is the helicity of the spectator hadron. If this hadron is a baryon, the expression gets aminus sign because 2 λ H is twice a half-integer, which is always an odd number. If it is a meson,82he expression does not get the minus sign because 2 λ H is twice an integer, which is always aneven number. Since the unstable resonance will be a meson (baryon) when the spectator hadronis a baryon (meson), we can equivalently say that the density matrix describing mesons will get afactor of −
1, while those describing baryons will not.The third factor depends on which helicity indices on the right-hand side of eqns. (12.17) to(12.19) have minus signs. For example, all of the spin density matrices that have superscripts of 0or z will have the factor (the symbol “ | ” represents the word “or”) ξ | z − λ − λ ′ ∝ ( − λ γ − λ N ) ξ | zλλ ′ = − ξ | zλλ ′ , (12.24)those with superscripts of x or y with ( γ ) will have ξ x | y ( γ ) − λ − λ ′ ∝ ( − − λ N ξ x | y ( γ ) λλ ′ = − ξ x | y ( γ ) λλ ′ , (12.25)and those with superscripts of x or y with ( N ) will have ξ x | y ( N ) − λ − λ ′ ∝ ( − λ γ ξ x | y ( N ) λλ ′ = ξ x | y ( N ) λλ ′ . (12.26)Eqn. (12.24) gets a − λ γ − λ N ) is twice the difference of a integer with a half-integer,which is always an odd integer. Eqn. (12.25) gets a − − λ N is twice a half-integer, whichis always an odd integer. Eqn. (12.26) gets a 1 because 2 λ γ is twice an integer, which is alwaysan even integer. For the density matrices that have ( γN ) in their superscript, both factors thatappear in the associated matrices with superscript ( γ ) and ( N ) appear. For example, ξ xz ( γN ) − λ − λ ′ ∝ − ξ xz ( γN ) λλ ′ (12.27)because ξ x ( γ ) would get a factor of − ξ z ( N ) would get a factor of 1.The fourth factor is related to whether the expression on the right-hand side of eqns. (12.17)to (12.19) have a factor of λ γ or λ N in the summation. This follows from this identity, which isvalid for any arbitrary function of an index f ( λ ), X λ λf ( − λ ) = − X λ λf ( λ ) . (12.28)The transformation λ → − λ can be done because λ is a dummy index. Therefore, all matriceswith superscript 2 or 3 will get a factor of −
1. If the matrix has two indices in its superscript,83ou multiply the factors that would have appeared for the two associated matrices that have thoseindices. For example, the matrix ξ xz ( γN ) would gets a factor of − ξ x ( γ ) would get a factorof 1 while ξ z ( N ) would get a factor of − ξ − λ, − λ ′ = ¯ η ( − S ∗ +1 ζ ( − λ + λ ′ ξ λ,λ ′ ,ξ i ( γ ) − λ, − λ ′ = ¯ η ( − S ∗ +1 ζ ( γ ) i ( − λ + λ ′ ξ i ( γ ) λ,λ ′ , (12.29) ξ i ( N ) − λ, − λ ′ = ¯ η ( − S ∗ +1 ζ ( N ) i ( − λ + λ ′ ξ i ( N ) λ,λ ′ ,ξ ij ( γN ) − λ, − λ ′ = ¯ η ( − S ∗ +1 ζ ( γ ) i ζ ( N ) j ( − λ + λ ′ ξ ij ( γN ) λ,λ ′ , ¯ η ≡ ηη ′ , where ηη ′ are the intrinsic parities of the two unstable hadrons that interfere. Therefore, ¯ η willbe called the intrinsic parity factor. For non-interference spin density matrices, this factor willalways equals 1. S ∗ is the spin of either one of the interfering hadrons. Since they will both haveeither integer or half-integer spin, it does not matter which one is used. Since the factor ( − S ∗ +1 depends on whether the unstable hadron is a fermion or a boson, we will call this the statisticsfactor. The remaining factor depends only on the type of matrix, and are given by ζ = − ,ζ i ( γ ) = ( , if i = y, z − , if i = x ,ζ i ( N ) = ( , if i = x, z − , if i = y. . (12.30)We will therefore call this the SDME factor. As an example, suppose we want to find the par-ity relations for the non-interference spin density matrix ρ z ( γ ) λλ ′ ( S = 2), for the decay of a spin-2pseudoscalar meson. In this case, ¯ η = 1 because it is a non-interference matrix, ( − S ∗ +1 = − S ∗ = 2, and ζ z ( γ ) = 1 from eqn. (12.30). The parity relation will therefore be ρ z ( γ ) − λ − λ ′ ( S = 2) = − ( − λ + λ ′ ρ z ( γ ) − λ − λ ′ ( S = 2) . (12.31)84 We will now show why when the reaction is parity conserving only terms that are proportionalto either cosine or sine functions will appear in the expression. From the definitions of the e d S i and e d S i ,S j in eqn. (12.13) it can be shown that e d S i λλ ′ = e d S i λ ′ λ e d S i ,S j λλ ′ = e d S i ,S j λ ′ λ . (12.32)Also, the property d S i − η − λ = ( − η − λ d S i ηλ , (12.33)can be used to show that, e d S i λλ ′ = ( − S i ( − λ + λ ′ e d S i − λ − λ ′ , e d S i ,S j λλ ′ = ( − S i ( − λ + λ ′ e d S i ,S j − λ − λ ′ . (12.34)These last equations along with eqns. (12.29), the hermiticity condition in eqn. (4.8), the evenand odd property of the trigonometric functions, and the following properties from the real andimaginary part functions, ℜ [ z ∗ ] = ℜ [ z ] , ℑ [ z ∗ ] = −ℑ [ z ] ,z ∈ C , (12.35)can be used to shown that e d S k λλ ′ ( θ ∗ ) ℜ [ ρ iλλ ′ ( S k )] cos[( λ − λ ′ ) φ ∗ ] = ζ i e d S k − λ ′ − λ ( θ ∗ ) ℜ [ ρ i − λ ′ − λ ( S i )] cos[( − λ ′ + λ ) φ ∗ ] , e d S k λλ ′ ( θ ∗ ) ℑ [ ρ iλλ ′ ( S k )] sin[( λ − λ ′ ) φ ∗ ] = − ζ i e d S i − λ ′ − λ ( θ ∗ ) ℑ [ ρ − λ ′ − λ ( S k )] sin[( − λ ′ + λ ) φ ∗ ] . (12.36)Since ζ i = ±
1, this means that applying the parity relations to each W i will give you either a plusor minus sign to each of the terms proportional to the real part function and the opposite sign toeach of the terms proportional to the imaginary part function. The terms that get the minus signswill always cancel with another term, and therefore either all terms will be proportional to thecosine function or all terms will be proportional to the sine function will vanish. For example, theexpression for W x ( γ ) for a reaction with a vector meson V intermediate state with corresponding85pin density matrix ˆ ρ x ( γ ) ( V ) will contain the following terms, W x ( γ ) = · · · + (cid:18) e d ℜ h ρ x ( γ )10 ( V ) i + e d − ℜ h ρ x ( γ )0 − ( V ) i(cid:19) cos( φ ∗ ) − (cid:18) e d ℑ h ρ x ( γ )10 ( V ) i + e d − ℑ h ρ x ( γ )0 − ( V ) i(cid:19) sin( φ ∗ ) + · · · , = · · · + (cid:18) e d ℜ h ρ x ( γ )10 ( V ) i + e d ℜ h ρ x ( γ )10 ( V ) i(cid:19) cos( φ ∗ ) − (cid:18) e d ℑ h ρ x ( γ )10 ( V ) i − e d ℑ h ρ x ( γ )10 ( V ) i(cid:19) sin( φ ∗ ) + · · · , = · · · + 2 e d ℜ h ρ x ( γ )10 ( V ) i cos( φ ∗ ) + · · · . (12.37)For the terms that contain the interference matrices, the property that the cosine function is evenwhile the sine function is odd can be used to show this identity, e d S i ,S j λλ ′ ( θ ∗ ) ℜ [ ξ λλ ′ ( S i , S j )] cos[( λ − λ ′ ) φ ∗ ] = ± e d S i ,S j − λ − λ ′ ( θ ∗ ) ℜ [ ξ − λ − λ ′ ( S i , S j )] cos[( − λ + λ ′ ) φ ∗ ] , e d S i ,S j λλ ′ ( θ ∗ ) ℑ [ ξ λλ ′ ( S i , S j )] sin[( λ − λ ′ ) φ ∗ ] = ∓ e d S i ,S j − λ − λ ′ ( θ ∗ ) ℑ [ ξ − λ − λ ′ ( S i , S j )] sin[( − λ + λ ′ ) φ ∗ ] , (12.38)which imply that for the terms proportional to the interference matrices, once again either all theterms proportional to the cosine functions vanish or all the terms proportional to the sine functionsvanish. For example, the expression for W x ( γ ) for a reaction with a vector meson V resonanceinterfering with a spin-2 meson T resonance with corresponding spin density matrix ˆ ξ x ( γ ) ( V, T ) willhave the following terms, W x ( γ ) = · · · + (cid:18) e d , − ℜ h ξ x ( γ )1 − ( V, T ) i + e d , − ℜ h ξ x ( γ ) − ( V, T ) i(cid:19) cos(3 φ ∗ ) − (cid:18) e d , − ℑ h ξ x ( γ )1 − ( V, T ) i − e d , − ℑ h ξ x ( γ ) − ( V, T ) i(cid:19) sin(3 φ ∗ ) + · · · , = · · · + (cid:18) e d , − ℜ h ξ x ( γ )1 − ( V, T ) i + e d , − ℜ h ξ x ( γ )1 − ( V, T ) i(cid:19) cos(3 φ ∗ ) − (cid:18) e d , − ℑ h ξ x ( γ )1 − ( V, T ) i − e d , − ℑ h ξ x ( γ )1 − ( V, T ) i(cid:19) sin(3 φ ∗ ) + · · · , = · · · + 2 e d , − ℜ h ξ x ( γ )1 − ( ρ ) i cos(3 φ ∗ ) + · · · . (12.39) We can also write the most general expression for the distribution, eqn. (12.12), in terms of thestatistical tensors. First, we need to note that just like a spin density matrix can be decomposed into86 sum of polarization operators as shown in eqn. (5.5), the same can be done for the interferencespin density matrix between two hadrons of spins S i and S j ,ˆ ξ ( S i , S j ) = L = S i + S j X L = | S i − S j | L X M = − L τ LM ( S i , S j ) ˆ T † LM ( S j , S i ) . (12.40)The matrix elements of the interference polarization tensors, ˆ T LM ( S i , S j ), are now defined as h T LM ( S i , S j ) i m,m ′ ≡ ( − S j − m ′ C LMS i ,m ; S j , − m ′ , (12.41)and the τ LM ( S i , S j )’s are the interference statistical tensors between two particles of spin S i and S j , τ LM ( S i , S j ) = D ˆ ξ ( S i , S j ) ˆ T LM ( S j , S i ) E = Tr[ ˆ ξ ( S i , S j ) ˆ T LM ( S j , S i )]= S i X m = − S i S j X m ′ = − S j ( − S i − m ξ m,m ′ ( S i , S j ) C LMS j ,m ′ ; S i , − m . (12.42)The inverse relation of the previous equation is ξ mm ′ ( S i , S j ) = S i + S j X L = −| S i − S j | M = L X M = − L τ LM ( S i , S j )( − S j − m C LMS j m ′ ; S i − m . (12.43)Substituting the expressions for spin density matrices in terms of statistical tensors, eqns. (5.5)and (12.40), into equation (12.1), we get W ( θ ∗ , φ ∗ ) = X i S i X L =0 L X M = − L t LM ( S i ) Tr[ ˆ D † S i ˆ T † LM ( S i ) ˆ D S i ]+ X i>j S i + S j X L = | S i − S j | L X M = − L n τ LM ( S i , S j ) Tr[ ˆ D † S i ˆ T † LM ( S j , S i ) ˆ D S j ]+ τ LM ( S j , S i ) Tr[ ˆ D † S j ˆ T † LM ( S i , S j ) ˆ D S i ] o . (12.44)This expression can be further simplified by using certain properties of the polarization operators,ˆ T LM ( S i ) and ˆ T LM ( S i , S j ), and of the statistical tensors, t LM ( S i ) and τ LM ( S i , S j ). By using theseproperties of the Clebsch-Gordan coefficients, C LMSm ; S ′ m ′ = ( − S + S ′ − L C LMS ′ m ′ ; Sm ,C LMSm ; S ′ m ′ = ( − S + S ′ − L C L − MS − m ; S ′ − m ′ , (12.45)87hese properties of the elements of the polarization operators can be shown,[ T LM ( S, S ′ )] mm ′ = ( − M + S − S ′ [ T L − M ( S ′ , S )] m ′ m , [ T LM ( S, S ′ )] mm ′ = ( − L + S − S ′ [ T L − M ( S, S ′ )] − m − m ′ , [ T LM ( S, S ′ )] mm ′ = ( − L + M [ T LM ( S ′ , S )] − m ′ − m . (12.46)They can also be used to show a generalized hermiticity property of the interference statisticaltensors, τ LM ( S, S ′ ) = ( − S − S ′ + M τ ∗ L − M ( S ′ , S ) . (12.47)We can use these last two equations to rewrite the second term in the brackets in eqn. (12.44), L X M = − L τ LM ( S j , S i ) Tr[ ˆ D † S j ˆ T † LM ( S i , S j ) ˆ D S i ]= L X M = − L (cid:16) τ LM ( S i , S j ) Tr[ ˆ D † S i ˆ T † LM ( S j , S i ) ˆ D S j ] (cid:17) ∗ , (12.48)where we have used the fact that, since the index M is a dummy index, when all of them in theexpression are replaced by − M the summation remains the same. Eqn. (12.44) therefore becomes W ( θ ∗ , φ ∗ ) = X i S i X L =0 L X M = − L t LM ( S i ) Tr[ ˆ D † S i ˆ T † LM ( S i ) ˆ D S i ]+ X i>j S i + S j X L = | S i − S j | L X M = − L ℜ n τ LM ( S i , S j ) Tr[ ˆ D † S i ˆ T † LM ( S j , S i ) ˆ D S j ] o . (12.49)We will now show how the two traces found in the previous equation can be further simplified.We will show it for the trace in the second term, since the same derivation applies to the one in thefirst term but with S i = S j . In index notation, and for recoil baryon B having spin s , that trace isTr[ ˆ D † S i ˆ T † LM ( S j , S i ) ˆ D S j ] = S i X m = − S i S j X m ′ = − S j s X n = − s ( − S i − m D ∗ S i m,n C LMS j ,m ′ ; S i , − m D S j m ′ ,n . (12.50)Certain properties of the Wigner- D functions and the Clebsch-Gordan coefficients will be used.First, D Sm,m ′ = ( − m ′ − m D ∗ S − m, − m ′ . (12.51)Second is the Clebsch-Gordan series for the Wigner- D functions, D S m ,n D S m ,n = J = S + S X J = | S − S | J X M,N = − J C JMS ,m ; S ,m D JM,N C JNS ,n ; S ,n . (12.52)88hird is the orthogonality condition of the Clebsch-Gordan coefficients, j X m = − j j X m = − j C JMj ,m ; j ,m C J ′ M ′ j ,m ; j ,m = δ JJ ′ δ MM ′ . (12.53)Finally, a relation between the Wigner- D functions and the spherical harmonics, D LM, ( θ, φ ) = r π L + 1 ( − M Y L, − M ( θ, φ ) . (12.54)Using these last four equations, we can rewrite eqn. (12.49) in the simplified form W ( θ ∗ , φ ∗ ; s ) = X i S i X L =0 L X M = − L κ L ( S i ; s )( − M t LM ( S i ) Y L, − M ( θ ∗ , φ ∗ )+ X i>j S i + S j X L = | S i − S j | L X M = − L κ L ( S i , S j ; s ) ℜ n ( − M τ LM ( S i , S j ) Y L, − M ( θ ∗ , φ ∗ ) o , (12.55)where κ L ( S i ; s ) ≡ r π L + 1 Tr[ ˆ T † L ( S i )]= r π L + 1 s X m = − s ( − S i − m C L S i ,m ; S i , − m ,κ L ( S i , S j ; s ) ≡ r π L + 1 Tr[ ˆ T † L ( S j , S i )]= r π L + 1 s X m = − s ( − S i − m C L S j ,m ; S i , − m . (12.56)This last equation shows that, when expressing the distribution W in terms of the statistical tensors,it acquires a very simple form. First, every single contributing hadron of spin S i will contributea sum over the well known spherical harmonics for 0 ≤ L ≤ S i , multiplied by the statisticaltensors of the corresponding values for L and M . The entire angular dependence for the decayof the hadron, θ ∗ and φ ∗ , is contained in them. The prefactor for each term, κ L ( S i ; s ), dependsonly on the spin of the decaying hadron S i , and the spin of the daughter hadron s . As seen intheir definition in eqn. (12.56), they are easily calculated for any combination of S i and s , sincethey are a linear combination of the well known Clebsch-Gordan coefficients. Second, every pairof contributing hadrons with spins S i and S j will also contribute a linear combination of sphericalharmonics, this time for | S i − S j | ≤ L ≤ S i + S j . The prefactor, κ L ( S i , S j ; s ), is now a function of89he spins of the interfering hadrons and the spin of the daughter hadron, but has a similar simpledefinition form as the previous one.Note that this expression makes explicit the fact that distribution is a rotational scalar, sincethe contractions P M ( − M t LM Y L − M and P M ( − M τ LM Y L − M are rotational invariants, X M ( − M t LM Y L − M −→ X M,M ′ ,M ′′ ( − M D LMM ′ D L − M − M ′′ t LM ′ Y L − M ′′ = X M,M ′ ,M ′′ ( − M + M ′′ − M D LMM ′ D L ∗ MM ′′ t LM ′ Y L − M ′′ (12.57)= X M ′ ,M ′′ ( − M ′′ δ M ′ M ′′ t LM ′ Y L − M ′′ −→ X M ( − M t LM Y L − M . One property should be noted about the coefficients κ L ( S i ; s ). By using the properties of theClebsch-Gordan coefficients in eqn. (12.45), and using the fact that ( − S i − m )=1 for all S i and m (since S i and m are both integer or half-integer, 2( S i − m ) is twice an integer number, which isalways an even number), we can rewrite κ L ( S i ; s ) as s X m = − s ( − S i − m C L S i ,m ; S i , − m = s X m = − s ( − S i − m + m − m +2 S i − L C L S i , − m ; S i ,m = s X m = − s ( − S i + m ( − S i − m ) ( − − L C L S i , − m ; S i ,m = ( − L s X m = − s ( − S i + m C L S i , − m ; S i ,m . (12.58)In the last line, we used the fact that ( − − L = ( − L since L is an integer. Using the last equation,we can see that κ L ( S i ; s ) has the property κ L ( S i ; s ) = ( − L κ L ( S i ; s ) ⇒ κ L ( S i ; s ) = 0 for L odd . (12.59)Therefore, in the first term in eqn. (12.55), only the terms with even L appear. Applying the sameprocedure to κ L ( S i , S j ; s ), it can be shown that for the interference tensors τ LM , the L odd termsvanish only if S i = S j .This shows that not all tensors will appear on the expressions for the angular distribution andpolarization observables. This may seem confusing at first, because this would seem to imply that90he general expressions for the decay distribution in terms of the SDME’s has more independentparameters than those in terms of the statistical tensors. However, upon closer inspection, wenotice that when expressed in terms of the SDME’s, the coefficients of each of the trigonometricfunctions in the sum are actually specific linear combination of the SDME’s. For example, in thecase of vector meson decay, its density matrix has a total of 9 parameters. When expressed interms of the statistical tensors in eqn. (8.23), only 6 independent parameters appear. But notethat when expressed in terms of the SDME’s as shown in eqn. (2.1), there are only 6 coefficientsbecause 3 of them are linear combinations of two SDME’s. This is another reason that shows howusing statistical tensors to represent the spin state of a mixed quantum state is more natural thanwith spin density matrices.The last step needed to simplify the general expression in eqn. (12.55) is to use the hermiticitycondition of the non-interference tensors shown in eqn. (5.10), that shows that these tensors arenot all independent. We first split the sum of the non-interference tensors, and then apply thehermiticity condition, and the Spherical Harmonic identity, Y LM = ( − M Y ∗ L − M , (12.60)and the identity for complex numbers shown in eqn. (12.11) to get L X M = − L ( − M t LM ( S i ) Y L, − M ( θ ∗ , φ ∗ ) = t L ( S i ) Y L, ( θ ∗ , φ ∗ )+ L X M> (cid:26) ( − M t LM ( S i ) Y L, − M ( θ ∗ , φ ∗ ) + ( − − M t L − M ( S i ) Y L,M ( θ ∗ , φ ∗ ) (cid:27) = t L ( S i ) Y L, ( θ ∗ , φ ∗ ) + L X M> (cid:26) ( − M t LM ( S i ) Y L, − M ( θ ∗ , φ ∗ )+ h ( − M t LM ( S i ) Y L, − M ( θ ∗ , φ ∗ ) i ∗ (cid:27) = t L ( S i ) Y L, ( θ ∗ , φ ∗ ) + 2 L X M> ℜ h ( − M t LM ( S i ) Y L, − M ( θ ∗ , φ ∗ ) i = t L ( S i ) Y L, ( θ ∗ , φ ∗ ) + 2 L X M> ( − M (cid:18) ℜ h t LM ( S i ) i ℜ h Y L − M ( θ ∗ , φ ∗ ) i − ℑ h t LM ( S i ) i ℑ h Y L − M ( θ ∗ , φ ∗ ) i(cid:19) . (12.61)91ote that the hermiticity condition of the tensor implies that t L is purely real. The previousexpression is therefore manifestly purely real, as expected. The general expression therefore becomes W ( θ ∗ , φ ∗ ; s ) = X i S i X L =0 L X M ≥ κ L ( S i ; s )( − M (cid:18) ℜ h t LM ( S i ) i ℜ h Y L − M ( θ ∗ , φ ∗ ) i − ℑ h t LM ( S i ) i ℑ h Y L − M ( θ ∗ , φ ∗ ) i(cid:19) + X i>j S i + S j X L = | S i − S j | L X M = − L κ L ( S i , S j ; s )( − M (cid:18) ℜ h τ LM ( S i , S j ) i ℜ h Y L, − M ( θ ∗ , φ ∗ ) i − ℑ h τ LM ( S i , S j ) i ℑ h Y L, − M ( θ ∗ , φ ∗ ) i(cid:19) , (12.62)where we have also used the complex numbers identity in eqn. (12.11) to rewrite the expression inparenthesis in the second term in terms of real and imaginary parts of τ . To simplify the notation,we included t L inside the real and imaginary parts functions so that the sum can go over M ≥ ℑ [ t L ] = 0. Note that in the second term of the previous equationthe hermiticity relation for the interference tensor in eqn. (12.47) cannot be used to reduce thenumber of terms in the sum, because it relates τ ( S i , S j ) and τ ( S j , S i ), which are different tensors.Therefore, for the interference tensors the summation over M goes from − L to L . If we want theexpression in terms of sine and cosine functions of φ ∗ , we can write the spherical harmonics interms of the associated Legendre polynomials, Y LM ( θ ∗ , φ ∗ ) = s (2 L + 1)( L − M )!4 π ( L + M )! P ML (cos θ ∗ ) e iMφ ∗ = s (2 L + 1)( L − M )!4 π ( L + M )! P ML (cos θ ∗ )(cos( M φ ∗ ) + i sin( M φ ∗ )) . (12.63)Note how the entire φ ∗ dependence is contained in the cosine and sine function. With this, we getfor the general expression, W ( θ ∗ , φ ∗ ; s ) = X i S i X L =0 L X M ≥ e κ LM ( S i ; s ) P ML (cos θ ∗ ) (cid:18) ℜ h t LM ( S i ) i cos( M φ ∗ )+ ℑ h t LM ( S i ) i sin( M φ ∗ ) (cid:19) + X i>j S i + S j X L = | S i − S j | L X M = − L e κ LM ( S i , S j ; s ) P ML (cos θ ∗ ) (cid:18) ℜ h τ LM ( S i , S j ) i cos( M φ ∗ )+ ℑ h τ LM ( S i , S j ) i sin( M φ ∗ ) (cid:19) , (12.64)92here e κ LM ( S i ; s ) ≡ s ( L − M )!( L + M )! s X m = − s ( − S i − m C L S j m ; S i − m , e κ LM ( S i , S j ; s ) ≡ s ( L − M )!( L + M )! s X m = − s ( − S i − m C L S j m ; S i − m , (12.65)and we eliminated the factor of ( − M by using the identity P − ML (cos θ ∗ ) = ( − M P ML (cos θ ∗ ) . (12.66) The general expression for the 16 W i ’s shown in eqn. (12.14) can also be expressed in terms ofthe statistical tensors. The decomposition of the statistical tensors t in terms or 16 other tensorsshown in eqn. (7.42) also apply to the interference tensors τ . The relationship between the 16tensors to the 16 matrices has the same form as eqn. (12.43), but with a superscript to label whichmember of the 16 matrices or tensors it refers to, τ iLM ( S i , S j ) = D ˆ ξ i ( S i , S j ) ˆ T LM ( S j , S i ) E = Tr[ ˆ ξ i ( S i , S j ) ˆ T LM ( S j , S i )]= S i X m = − S i S j X m ′ = − S j ( − S i − m ξ im,m ′ ( S i , S j ) C LMS j ,m ′ ; S i , − m . (12.67)The inverse relation is ξ imm ′ ( S i , S j ) = S i + S j X L = −| S i − S j | M = L X M = − L τ iLM ( S i , S j )( − S i − m C LMS j m ′ ; S i − m . (12.68)By substituting this last expression into equation eqn. (12.14) we get the most general expressionfor the W i ’s in terms or the statistical tensors, which has the same form as the most general93xpression of the decay angular distribution in eqn. (12.64) but with the appropriate superscript, W i ( θ ∗ , φ ∗ ; s ) = X i S i X L =0 L X M ≥ e κ L ( S i ; s ) P ML (cos θ ∗ ) (cid:18) ℜ h t iLM ( S i ) i cos( M φ ∗ )+ ℑ h t iLM ( S i ) i sin( M φ ∗ ) (cid:19) + X i>j S i + S j X L = | S i − S j | L X M = − L e κ L ( S i , S j ; s ) P ML (cos θ ∗ ) (cid:18) ℜ h τ iLM ( S i , S j ) i cos( M φ ∗ )+ ℑ h τ iLM ( S i , S j ) i sin( M φ ∗ ) (cid:19) . (12.69)Remember that, as shown in eqn. (12.20), the W i are proportional to polarization observables. To derive the parity relations of the statistical tensors, we need to use the parity relations of theSDME’s shown in eqn. (12.29), the relationship between the statistical tensors and the SDME’sshown in eqn. (12.67), and the properties of the Clebsch-Gordan coefficients shown in eqn. (12.45).With these, the parity properties of the statistical tensors can be shown to be t kLM ( S i ) = ¯ η ( − S i +1 ζ k ( − L + M +2 S i t kL − M ( S i ) ,τ kLM ( S i , S j ) = ¯ η ( − S i +1 ζ k ( − L + M + S i + S j τ kL − M ( S i , S j ) . (12.70)This relations can be simplified by using the following properties,( − S i ( − S i = ( − S i = 1 , ( − S i ( − S i + S j = ( − S i +1 ( − − S i − S j = ( − S i − S j . (12.71)These properties follow because S i and S j are either both integer or both half-integer. Therefore,4 S i is an even integer, S i + S j is an integer, and ( − n = ( − − n for any integer n . The parityrelations of the statistical tensors are t kLM ( S i ) = − ¯ ηζ k ( − L + M t kL − M ( S i ) ,τ kLM ( S i , S j ) = − ¯ ηζ k ( − L + M + S i − S j τ kL − M ( S i , S j ) , (12.72)where the expressions for the ζ k were given in eqn. (12.30), and ¯ η is the multiplication of the twointrinsic parities of the interfering hadrons. 94 Just as in the case of the SDME’s, when the scattering reaction is parity conserving, theexpressions for the observables can be further simplified. Specifically, each of the W i ’s shown ineqn. (12.69) will be either an even or odd function of the variable φ ∗ (i.e., all the terms proportionalto either cos( M φ ∗ ) or sin( M φ ∗ ) vanish).In the case of the non-interference tensors, parity conservation in the scattering process impliesthat each of the 16 tensors will be either purely real or purely imaginary. All that needs to be doneto show this is use the hermiticity condition shown in eqn. (5.10) along with the parity relationshown in eqn. (12.72), t kLM ( S i ) = ( − M t k ∗ L − M ( S i ) = − − L ζ k t k ∗ LM ( S i ) ,t kLM ( S i ) = − ζ k t k ∗ LM ( S i ) (12.73)The factor of ( − L vanishes because only even terms appear in the sum for the non-interferencetensors, as shown in eqn. (12.59). For the tensors of the type t kl ( γN ) , we instead have t kl ( γN ) LM ( S i ) = − ζ k ζ kl t k ( γN ) ∗ LM ( S i ) . (12.74)Since ζ k = ±
1, the right-hand side of the second line of eqn. (12.73) will have a factor of either1 or −
1. This factor depends only on the value of the superscript k , not on the spin or the statisticsof the decaying hadron. From the general property of complex numbers shown in eqn. (12.35), ifthis overall coefficient equals 1 then t k ( S i ) is purely real, and if the coefficient is negative then itwill be purely imaginary. The values of ζ k shown in eqn. (12.30) can be used to conclude that ℜ [ t k ] = 0 , if ζ k = 1 → Observable odd in φ ∗ , ℑ [ t k ] = 0 , if ζ k = − → Observable even in φ ∗ , ℜ [ t kl ] = 0 , if ζ k ζ l = 1 → Observable odd in φ ∗ , ℑ [ t kl ] = 0 , if ζ k ζ l = − → Observable even in φ ∗ . (12.75)Since the cosine functions of φ ∗ are multiplied by real parts of tensors, while the sine functionsare multiplied by imaginary parts, we conclude that the observable associated with t k or t kl willbe even (odd) if ζ k or ζ kl equals − φ ∗ → − φ ∗ .Symmetry of observables under φ ∗ → − φ ∗ Even Odd I I I s I I c I I ⊙ I P y I P x I P cy I P z I P sx I P cx I P sz I P cz I P ⊙ x I P sy I P ⊙ z I P ⊙ y φ ∗ → − φ ∗ for the unpolarized and single polarization observables. We can derive similar results forthe interference tensors τ , but using a different procedure. To do so, we need to make use of someproperties of κ L , P ML (cos θ ∗ ), cos( M φ ∗ ), sin( M φ ∗ ), and the parity relations for the interferencetensors shown in the second line of eqn. (12.72). First, in the general expression for the W i ’sshown in eqn. (12.69), the summation over M in the second term is split into two parts: a sumover the M >
M < W i ( θ ∗ , φ ∗ ) = · · · + e κ L ( S i , S j ; s ) P L (cos θ ∗ ) ℜ h τ iL ( S i , S j ) i + e κ L ( S i , S j ; s ) L X M> (cid:18) P ML (cos θ ∗ ) ℜ h τ iLM ( S i , S j ) i cos( M φ ∗ ) (12.76)+ P − ML (cos θ ∗ ) ℜ h τ iL − M ( S i , S j ) i cos( − M φ ∗ ) (cid:19) + · · · . Four properties need to be used to simplify this last equation. First, using the properties of theClebsch-Gordan coefficients shown in equation (12.45), it can be shown that κ L ( S i , S j ) = ( − L + S i − S j κ L ( S i , S j ) . (12.77)Second, the property of the associated Legendre polynomials shown in eqn. (12.66). Third,cos( − M φ ∗ ) = cos( M φ ∗ ) . (12.78)96nd fourth, the parity property shown in the second line of eqn. (12.72). Combining all theseproperties, eqn. (12.76) becomes W i ( θ ∗ , φ ∗ ) = · · · + e κ L ( S i , S j ; s ) P L (cos θ ∗ ) ℜ h τ iL ( S i , S j ) i + e κ L ( S i , S j ; s ) L X M> (cid:18) P ML (cos θ ∗ ) ℜ h τ iLM ( S i , S j ) i cos( M φ ∗ ) (12.79) − ηζ i P ML (cos θ ∗ ) ℜ h τ iLM ( S i , S j ) i cos( M φ ∗ ) (cid:19) + · · · Note the appearance of the factor − ¯ ηζ i in the second term inside of the parenthesis. This sameprocedure can be used with the terms proportional to the imaginary parts of the tensors, but sincesin( − M φ ∗ ) = − sin( − M φ ∗ ) , (12.80)we get instead W i ( θ ∗ , φ ∗ ) = · · · + e κ L ( S i , S j ; s ) L X M> (cid:18) P ML (cos θ ∗ )( θ ∗ ) ℑ h τ iLM ( S i , S j ) i sin( M φ ∗ )+ ηζ i P ML (cos θ ∗ ) ℑ h τ iLM ( S i , S j ) i sin( M φ ∗ ) (cid:19) + · · · . (12.81)The term with ℑ [ τ L ] vanishes because sin(0) = 0. Note how we get an extra minus sign, so thatnow the factor is ηζ i . The only possible values of this factor are 1 and −
1. If it is equal to 1,the expression inside of the parenthesis in eqn. (12.79) vanishes and the observable will contain noterms proportional to cosines of φ ∗ . If the factor instead equals −
1, the expression inside of theparenthesis in eqn. (12.81) vanishes and the observable will contain no terms proportional to sinesof φ ∗ . This shows that once parity conservation is taken into account, the observables will be eithereven or odd in φ ∗ . Also, for the term with ℜ [ τ iL ( S i , S j )], after applying all identities, e κ L ( S i , S j ) P ML (cos θ ∗ )) ℜ h τ iL ( S i , S j ) i = − e ηζ i κ L ( S i , S j ) P ML (cos θ ∗ ) ℜ h τ iL ( S i , S j ) i . (12.82)From this we conclude that when the factor ηζ i = 1, this term vanishes (it does not necessarilyimply that ℜ [ τ L ] = 0). As we’ve mentioned, when ηζ i = 1, all terms proportional to cos( M φ ∗ )vanish. We therefore conclude that the term with ℜ [ τ L ] only appears when the observable is evenin φ ∗ .The value of the factor ηζ i only depends on two things. One is the intrinsic parity of theinteracting hadrons. If their intrinsic parities are not the same, η = −
1. It otherwise equals 1. The97ther is which of the 16 tensors is related to the observables, since it determines the value of ζ i ,which are shown in eqn. (12.30).To summarize, once parity is taken into account, the expressions for the observables will eitherbe W i ( θ ∗ , φ ∗ ; s ) = X i S i X L =0 L X M ≥ e κ L ( S i ; s ) P ML (cos θ ∗ ) ℜ h t iLM ( S i ) i cos( M φ ∗ )+ X i>j S i + S j X L = | S i − S j | L X M ≥ e κ L ( S i , S j ; s ) P ML (cos θ ∗ )( − M ℜ h τ iLM ( S i , S j ) i cos( M φ ∗ ) , (12.83)or W i ( θ ∗ , φ ∗ ; s ) = X i S i X L =0 L X M ≥ e κ L ( S i ; s )( − M P ML (cos θ ∗ ) ℑ h t iLM ( S i ) i sin( M φ ∗ )+ X i>j S i + S j X L = | S i − S j | L X M ≥ e κ L ( S i , S j ; s ) P ML (cos θ ∗ )( − M ℑ h τ iLM ( S i , S j ) i sin( M φ ∗ ) (12.84)(we included the M = 0 term in the sum, even though it vanishes). Parity conservation thereforegreatly reduces the number of independent statistical tensors, since all M < η = −
1. However,we have to remember that the parity relations of transition matrices first shown in equation eqn.(10.4) also applies to the transition matrix of the decay process (which, as shown, is proportionalto a Wigner- D matrix). Therefore, under a parity transformation, both the transition matrix forthe scattering process and the transition matrix for the decay process will produce a −
1, the netresult being a 1. This shows that the value of η ends up playing no role in determining whether anobservable is even or odd in φ ∗ . 98 HAPTER 13GENERAL EXPRESSIONS FOR THE RECOILOBSERVABLES
We will now derive expressions for the recoil baryon polarization observables for the case ofdecaying baryon resonances in γN → M B ∗ → M M B . The polarization observables involving therecoil baryon in terms of the 16 expansion spin density matrices are given in eqns. (7.4), (7.6), (7.7),and (7.8) (for spin higher than 1/2, the appropriate spin operator, S x , S y or S z , is used insteadof the Pauli matrices). These equations show that in order to calculate them, the 16 expansionmatrices for the spin density matrix of the recoil baryon shown in eqn. (6.15) must be multipliedby one of the three spin operators and then the trace must be taken. In this chapter, in order torefer to a generic polarization observable involving the recoil baryon, we will use the definition O ij ≡ Tr[ˆ ρ i ( B ) ˆ S j ] . (13.1)The superscript i is used to identify which of the 16 matrices in the expansion is being used, so itcan take values such as x ( γ ), y ( N ), z ( γN ), etc. The index j identifies which of the spin operatorsis used, so it takes values of x , y or z .We want expressions for the recoil observables in terms of the SDME’s or statistical tensors ofthe decaying hadron, not those of the recoil baryon. We therefore need to use the equation thatrelates these two spin density matrices, which was shown in eqn. (8.17) for spin-3/2 resonances,but have the same form for higher-spin baryons, and it is also valid for any of the 16 matrices inthe expansion, ˆ ξ i ( B ) = ˆ D † ˆ ξ i ( B ∗ ) ˆ D † (13.2)(we have omitted explicitly showing that the spin density matrix on the left-had side of the equationdepends on the meson M since it is spinless) We take ˆ ξ to refer to either a non-interference matrixor an interference matrix. Two examples are O y ≡ P y ′ = Tr[ˆ ρ ˆ S y ] , (13.3)99 z ( γ ) x ≡ P ⊙ x ′ = Tr[ˆ ρ z ( γ ) ˆ S x ] . We will derive the expressions in terms of the statistical tensors instead, since the final expres-sions will acquire a simpler form. If expressions in terms of SDME’s are required, they are relatedto the statistical tensors by eqns. (5.7), (5.8), (12.42), (12.43). First, the spin density matrix ˆ ξ i ( B ∗ )is expressed in terms of the statistical tensors using eqns. (5.8) and (12.43). Next, the two Wigner D -matrices in eqn. (13.2) are expressed in terms of only one using the Clebsch-Gordan series shownin equation (12.52). When the resulting expression is multiplied by a spin operator and then thetrace is taken, we get O ij ( θ ∗ , φ ∗ ) = X k S k X L =0 L X M,N = − L t iLM ( S i ) D LMN
Tr[ ˆ T † NM ( S k ) ˆ S j ]+ X k>l S k + S l X L = | S i − S j | L X M,N = − L n τ iLM ( S k , S l ) D LMN
Tr[ ˆ T † LN ( S l , S k ) ˆ S j ]+ τ iL − M ( S l , S k ) D L − M − N Tr[ ˆ T † L − N ( S k , S l ) ˆ S j ] o . (13.4)The sum over M in the second term in the expression inside of the curly brackets of the previousequation was written as − M for later convenience (remember that M is a dummy index). Since thepolarization tensors T LM are spherical operators, writing the spin operator S j in terms of sphericaloperators will help in finding a more simplified version of the previous expression. The definitionsof the spin operators in the spherical basis areˆ S +1 = − √ S x + i ˆ S y ) , ˆ S − = 1 √ S x − i ˆ S y ) , ˆ S = ˆ S z . (13.5)The inverse relations are ˆ S x = 1 √ S − − ˆ S ) , ˆ S y = i √ S − + ˆ S ) , ˆ S z = ˆ S . (13.6)Note that since the spin operators are hermitian (their expectation values are real), eqn. (13.5)implies that in the spherical basis the operators are also hermitian,ˆ S † +1 = ˆ S +1 S †− = ˆ S − (13.7)ˆ S † = ˆ S In terms of polarization operators, the spin operators in the spherical basis are given byˆ S M = r S ( S + 1)(2 S + 1)3 ˆ T M . (13.8)By using eqns. (12.46), (12.47) and (12.51), it can be shown that t iL − M ( S k ) D L − M − N Tr[ ˆ T † L − N ( S k ) ˆ S j ] = (cid:16) t iLM ( S k ) D LMN
Tr[ ˆ T † LN ( S k ) ˆ S j ] (cid:17) ∗ ,τ iL − M ( S l , S k ) D L − M − N Tr[ ˆ T † L − N ( S l , S k ) ˆ S j ] = (cid:16) τ iLM ( S k , S l ) D LMN
Tr[ ˆ T † LN ( S l , S k ) ˆ S j ] (cid:17) ∗ . (13.9)These two equations can be used to simplify eqn. (13.4), O ij ( θ ∗ , φ ∗ ) = X k S k X L =0 L X M,N ≥ ℜ n t iLM ( S i ) D LMN
Tr[ ˆ T † LN ( S k ) ˆ S j ] o + X k>l S k + S l X L = | S k − S l | L X M,N = − L ℜ n τ iLM ( S k , S l ) D LMN
Tr[ ˆ T † LN ( S l , S k ) ˆ S j ] o , (13.10)which shows explicitly that the observable is real. In the previous equation, the sum over M and N in the first term goes only over values greater or equal to 0 because the first line of eqn. (13.9) usedto simplify eqn. (13.4) shows that the M, N <
M, N > t iLM (or τ iLM ), D LMN and ˆ S j . We use the real part function identityshown in eqn. (12.11) to rewrite the real part function as s X m,m ′ = − s ℜ n τ iLM D LMN ( ˆ S j ) mm ′ o(cid:0) ˆ T † LN (cid:1) m ′ m = s X m,m ′ = − s n ℜ h τ iLM D LMN i ℜ h ( ˆ S j ) mm ′ i − ℑ h τ iLM D LMN i ℑ h ( ˆ S j ) mm ′ io(cid:0) ˆ T † LN (cid:1) m ′ m , (13.11)where we are now using index notation to express the matrix multiplications. Note that (cid:0) ˆ T † LN (cid:1) m ′ m is purely real. Next, we notice that only one of the two terms in the previous equation in non-vanishing because, from eqns. (13.6) and (13.8), ( ˆ S j ) mm ′ is either purely real or purely imaginary.101y also using the complex numbers identity in eqn. (12.11) along with ℑ [ z z ] = ℜ [ z ] ℑ [ z ] + ℑ [ z ] ℜ [ z ] , (13.12)eqn. (13.11) simplifies to s X m,m ′ = − s ℜ n τ iLM D LMN ( ˆ S j ) mm ′ o(cid:0) ˆ T † LN (cid:1) m ′ m = s X m,m ′ = − s d LMN ( θ ∗ ) (cid:0) ˆ T † LN (cid:1) mm ′ ℜ h(cid:0) ˆ S j (cid:1) m ′ m i(cid:18) ℜ h τ LM i cos( M φ ∗ ) + ℑ h τ LM i sin( M φ ∗ ) (cid:19) , for j = x, z, (13.13)or s X m,m ′ = − s ℜ n τ iLM D LMN ( ˆ S j ) mm ′ o(cid:0) ˆ T † LN (cid:1) m ′ m = s X m,m ′ = − s d LMN ( θ ∗ ) (cid:0) ˆ T † LN (cid:1) mm ′ ℑ h(cid:0) ˆ S j (cid:1) m ′ m i(cid:18) ℑ h τ LM i cos( M φ ∗ ) − ℜ h τ LM i sin( M φ ∗ ) (cid:19) , for j = y. (13.14)To further simplify, we will write the spin operator (cid:0) ˆ S j (cid:1) m ′ m in terms of polarization operators usingeqn. (13.8). Since each of the three spin operators have a different form when written in terms ofthe polarization operators, we will continue the derivation using (cid:0) ˆ S y (cid:1) m ′ m , but the other two arederived in a similar way. We get, L X N = − L s X m,m ′ = − s d LMN ( θ ∗ ) (cid:2) ˆ T † LN ( S l , S k ) (cid:3) mm ′ ℑ n(cid:2) ˆ S j (cid:3) m ′ m o = r s ( s + 1)(2 s + 1)6 L X N = − L s X m,m ′ = − s d LMN ( θ ∗ ) (cid:2) ˆ T † LN ( S l , S k ) (cid:3) mm ′ (cid:18)(cid:2) ˆ T − ( s ) (cid:3) m ′ m + (cid:2) ˆ T ( s ) (cid:3) m ′ m (cid:19) = r s ( s + 1)(2 s + 1)6 L X N = − L s X m,m ′ = − s d LMN ( θ ∗ ) (13.15)( − S k − m C LNS l m ′ ; S k − m (cid:18) ( − s − m C − sm ′ ; s − m + ( − s − m C sm ′ ; s − m (cid:19) r s ( s + 1)(2 s + 1)6 s X m,m ′ = − s (cid:18) d LM − ( θ ∗ )( − S k − s C L − S l m ′ ; S k − m C − sm ′ ; s − m + d LM ( θ ∗ )( − S k − s C L S l m ′ ; S k − m C sm ′ ; s − m (cid:19) , where in the second line we used the definitions of the polarization operators shown in eqns. (5.6)and (12.41), and in the last time only a N = − N = 1 term survive because N = m ′ − m = − N = m ′ − m = 1 in another from the properties of the Clebsch-Gordan coefficients.Next, we use the properties shown in the second line of eqn. (12.45) in order to factor out the twoClebsch-Gordan coefficients in the two terms inside of the parenthesis of the last equation, s X m,m ′ = − s ( − S k − s C L − S l m ′ ; S k − m C − sm ′ ; s − m =( − S l + S k +2 s − L +1 s X m,m ′ = − s ( − S k − s C L S l m ′ ; S k − m C sm ′ ; s − m , (13.16)where on the right-hand side of the previous equation we relabeled the dummy indices: − m ′ → m ′ and m → − m . We will soon prove that, for the non-interference tensors, only the odd L termswill appear in the final answer. Therefore, ( − − L +1 equals 1 for the non-interference terms. Also, s is half-integer because the decaying resonance is a baryon. This means that ( − s equals − S k is equal to S l , which includes the non-interference tensors, the factor ( − S k + S l +1 is equal to 1 because S k + S l equals an odd number,since S k and S l are half-integer.This gives us L X N = − L s X m,m ′ = − s d LMN ( θ ∗ ) (cid:2) ˆ T † LN ( S l , S k ) (cid:3) mm ′ ℑ n(cid:2) ˆ S j (cid:3) m ′ m o = r s ( s + 1)(2 s + 1)6 s X m,m ′ = − s ( − S k − s C L S l m ′ ; S k − m C sm ′ ; s − m × (cid:18) ( − S k + S l − L d LM − ( θ ∗ ) + d LM ( θ ∗ ) (cid:19) . (13.17)With these simplifications we are ready to write the final forms of the recoil observables. These are O ix ( θ ∗ , φ ∗ ) = X k S k X L =0 L X M = − L κ L ( S k ; s ) (cid:18) d LM − ( θ ∗ ) − d LM ( θ ∗ ) (cid:19) × (cid:18) ℜ h t iLM ( S k ) i cos( M φ ∗ ) + ℑ h t iLM ( S k ) i sin( M φ ∗ ) (cid:19) (13.18)103 X k>l S k + S l X L = | S k − S l | L X M = − L κ L ( S k , S l ; s ) (cid:18) χ L ( S k , S l ) d LM − ( θ ∗ ) − d LM ( θ ∗ ) (cid:19) × (cid:18) ℜ h τ iLM ( S k , S l ) i cos( M φ ∗ ) + ℑ h τ iLM ( S k , S l ) i sin( M φ ∗ ) (cid:19) ,O iy ( θ ∗ , φ ∗ ) = X k S k X L =0 L X M = − L κ L ( S k ; s ) (cid:18) d LM − ( θ ∗ ) + d LM ( θ ∗ ) (cid:19) × (cid:18) ℑ h t iLM ( S k ) i cos( M φ ∗ ) − ℜ h t iLM ( S k ) i sin( M φ ∗ ) (cid:19) (13.19)+ X k>l S k + S l X L = | S k − S l | L X M = − L κ L ( S k , S l ; s ) (cid:18) χ L ( S k , S l ) d LM − ( θ ∗ ) + d LM ( θ ∗ ) (cid:19) × (cid:18) ℑ h τ iLM ( S k , S l ) i cos( M φ ∗ ) − ℜ h τ iLM ( S k , S l ) i sin( M φ ∗ ) (cid:19) , and O iz ( θ ∗ , φ ∗ ) = X k S k X L =0 L X M = − L κ L ( S k ; s ) d LM (cos θ ∗ ) × (cid:18) ℜ h t iLM ( S k ) i cos( M φ ∗ ) + ℑ h t iLM ( S k ) i sin( M φ ∗ ) (cid:19) (13.20)+ X k>l S k + S l X L = | S k − S l | L X M = − L κ L ( S k , S l ; s ) d LM (cos θ ∗ ) × (cid:18) ℜ h τ iLM ( S k , S l ) i cos( M φ ∗ ) + ℑ h τ iLM ( S k , S l ) i sin( M φ ∗ ) (cid:19) , where we have defined κ L ( S k , S l ; s ) ≡ r s ( s + 1)(2 s + 1)3 s X m,m ′ = − s ( − S k − s C L S l m ′ ; S k − m C sm ′ ; s − m ,κ L ( S k , S l ; s ) ≡ r s ( s + 1)(2 s + 1)3 s X m,m ′ = − s ( − S k − s C L S l m ′ ; S k − m C sm ′ ; s − m , (13.21) κ L ( S k ; s ) ≡ κ L ( S k , S k ; s ) ,κ L ( S k ; s ) ≡ κ L ( S k , S k ; s ) , and χ L ( S k , S l ) ≡ ( − S k + S l − L = ± . (13.22)104he phase factor χ L ( S k , S l ) shows that, depending on the value of L , the factors with the twoWigner d-functions may have a relative minus sign or not. Each of the three general expressionsfor the observables contains a sum over non-interference tensors and one over interference tensors.Each of these two summations is made up of three factors. One of them is the factor κ L or κ L ,which has no angular dependence and can be easily found from Clebsch-Gordan coefficients. Thesecond is is a sum of two Wigner d-functions in the case of O ix and O iy or one Wigner-d function inthe case of O iz , which contain the dependence on θ ∗ . The third factor contains the φ ∗ dependence,and is made up of a sum over cosines of φ ∗ and a sum over sines of φ ∗ . In O ix and O iz , the termswith cosine functions are proportional to the real parts of the statistical tensors and the terms withsine functions are proportional to the imaginary parts of the tensors. In O iy , the terms with cosinefunctions are proportional to the imaginary parts of the statistical tensors, while the terms withsine functions are proportional to the real parts of the statistical tensors.By using the properties of the Clebsch-Gordan coefficients in the second line of eqn. (12.45), itcan be shown that κ L ( S ; s ) = ( − L +1 κ L ( S ; s ) κ L ( S ; s ) = ( − L +1 κ L ( S ; s ) . (13.23)This implies that for the non-interference statistical tensors, only the terms with odd L are non-vanishing. In comparison, in the polarization observables that don’t involve the recoil baryon, it isthe even L terms that are non-vanishing for the non-interference statistical tensors.As an example, if we apply eqn. (13.19) to the observable P ⊙ y ′ for the decay of a spin-3/2 baryon,we get P ⊙ y ′ = 140 ( h − cos(2 θ ) ih cos(3 φ ) ℑ [ t z ( γ )33 ] − sin(3 φ ) ℜ [ t z ( γ )33 ] i + √ h θ ) + 3 ih cos( φ ) ℑ [ t z ( γ )31 ] − sin( φ ) ℜ [ t z ( γ )31 ] i (13.24)+ 10 √ θ ) h cos(2 φ ) ℑ [ t z ( γ )32 ] − sin(2 φ ) ℜ [ t z ( γ )32 ) i + 8 √ h sin( φ ) ℜ [ t z ( γ )11 ] − cos( φ ) ℑ [ t z ( γ )11 ] i) These expressions are valid for the case of a decaying baryon resonance, but not for the caseof a decaying meson resonance, γN → M ∗ B → M M B . The reason is because expressions fordistributions and observables have been given in terms of the SDME’s and statistical tensors of the105ecaying meson subsystem, and as shown in eqn. (8.11), these are found from the partial trace ofthe density matrix of entire M ∗ B system, which requires a sum over the recoil baryon B ’s helicityindices. When this partial trace is taken, we are unable to multiply by the spin operator of therecoil baryon, as is required by eqn. (13.1). When the partial trace is taken, we obtain the spindensity matrix of the meson subsystem, which contains no information on the spin state of therecoil baryon. But the recoil observable requires a measurement of the recoil baryon’s polarization,which means the expression should reference the spin state of this baryon. Therefore, the recoilobservables can’t be expressed in terms of the spin density matrix of the meson resonance.Unlike the density matrix of the beam-target system, which can be written in terms of a tensorproduct of two spin density matrix as shown in eqn. (6.6), the spin density matrix of the M ∗ B system cannot. It can be seen from the righ-hand side of eqn. (8.4) for the spin density matrixof the quasi-two-body state that it cannot be factored into two separate spin density matrices.This indicates that when this state is produced in the reaction, the two subsystems are correlated.Therefore, the measuring process of the recoil baryon’s polarization cannot be described indepen-dently from the decay of the meson resonance. The recoil observables in this case would necessarilyhave to be expressed in terms of the SDME’s or statistical tensors of the full M ∗ B system. For parity conserving reactions the recoil polarization observables, just like the other polariza-tion observables, will be either even or odd in φ ∗ . For the non-interference tensors, we use (5.10),(12.72) and find t iLM ( S k ) = ( − M t i ∗ L − M ( S k ) = − − L ζ i t i ∗ LM ( S k ) ,t iLM ( S k ) = ζ i t i ∗ LM ( S k ) . (13.25)As was mentioned in this chapter, ( − L = − L terms are non-vanishing. Compare this to all other non-recoil polarization observables, whereit is the even L terms that are non-vanishing. For the tensors of type t ij ( γN ) LM , we have instead t ij ( γN ) LM ( S k ) = ζ i ζ j t ij ( γN ) ∗ LM ( S k ) . (13.26)106qn. (12.30) therefore implies ℜ [ t k ] = 0 , if ζ k = − → Observable odd in φ ∗ for O ix and O iz , even in φ ∗ in O iy , ℑ [ t k ] = 0 , if ζ k = 1 → Observable odd in φ ∗ for O ix and O iz , even in φ ∗ in O iy , ℜ [ t kl ] = 0 , if ζ k ζ l = − → Observable odd in φ ∗ for O ix and O iz , even in φ ∗ in O iy , , ℑ [ t kl ] = 0 , if ζ k ζ l = 1 → Observable odd in φ ∗ for O ix and O iz , even in φ ∗ in O iy , . (13.27)Deriving these symmetries for the interference statistical tensors is only slightly more compli-cated. The parity properties can be used to show that in the sum over M for the interferencestatistical tensors, the terms with M <
M >
0. From the propertiesof the Clebsh-Gordan coefficients in the second line of eqn. (12.45), it can be shown that κ L ( S k , S l ; s ) = ( − S k + S l + L κ L ( S k , S l ; s ) κ L ( S k , S l ; s ) = ( − S k + S l + L κ L ( S k , S l ; s ) .. (13.28)Using this property along with the definition of χ L shown in eqn. (13.22), the property of theWigner d-functions, d L − M − N = ( − M − N d LMN , (13.29)the parity properties of the statistical tensors shown in eqn. (12.72) (we ignore the factor of ¯ η because another parity factor comes from the decay amplitude) and the even or odd property ofthe cosine and sine functions shown in eqns. (12.78) and (12.80), it can be shown that in theexpression for O ix we get κ L ( S k , S l ; s ) (cid:18) χ L ( S k , S l ; s ) d L − M − ( θ ∗ ) − d L − M ( θ ∗ ) (cid:19) × (cid:18) ℜ h τ iL − M ( S k , S l ) i cos( − M φ ∗ ) + ℑ h τ iL − M ( S k , S l ) i sin( − M φ ∗ ) (cid:19) = ζ i κ L ( S k , S l ; s ) (cid:18) χ L ( S k , S l ; s ) d LM − ( θ ∗ ) − d LM ( θ ∗ ) (cid:19) × (cid:18) ℜ h τ iLM ( S k , S l ) i cos( M φ ∗ ) − ℑ h τ iLM ( S k , S l ) i sin( M φ ∗ ) (cid:19) , (13.30)in the expression for O iy we get κ L ( S k , S l ; s ) (cid:18) χ L ( S k , S l ; s ) d L − M − ( θ ∗ ) + d L − M ( θ ∗ ) (cid:19) (cid:18) ℑ h τ iL − M ( S k , S l ) i cos( − M φ ∗ ) − ℜ h τ iL − M ( S k , S l ) i sin( − M φ ∗ ) (cid:19) = (13.31) − ζ i κ L ( S k , S l ; s ) (cid:18) χ L ( S k , S l ; s ) d LM − ( θ ∗ ) + d LM ( θ ∗ ) (cid:19) × (cid:18) − ℑ h τ iLM ( S k , S l ) i cos( M φ ∗ ) − ℜ h τ iLM ( S k , S l ) i sin( M φ ∗ ) (cid:19) , and in the expression for O iz we get κ L ( S k , S l ; s ) d L − M ( θ ∗ ) × (cid:18) ℜ h τ iL − M ( S k , S l ) i cos( − M φ ∗ ) + ℑ h τ iL − M ( S k , S l ) i sin( − M φ ∗ ) (cid:19) = ζ i κ L ( S k , S l ; s ) d LM ( θ ∗ ) × (cid:18) ℜ h τ iLM ( S k , S l ) i cos( M φ ∗ ) − ℑ h τ iLM ( S k , S l ) i sin( M φ ∗ ) (cid:19) . (13.32)In other words two things happen. First, in all three observables the terms proportional to sinefunctions of φ ∗ get a minus sign. And second, in O ix and O iz the expression gets an overall factorof ζ i while in O iy the expression gets an overall factor of − ζ i (for tensors of type τ ij ( γN ) LM , we insteadget factors of ζ i ζ j or − ζ i ζ j ). This allows us to reach the same conclusions we reached for thenon-interference tensors in eqn. (13.27). By using the values of ζ i shown in eqn. (12.30), we cansee for example that (using our definition of O j ≡ P j ′ ) P x ′ , P z ′ : odd in φ ∗ ,P y ′ : even in φ ∗ . (13.33)The symmetry properties of the recoil polarization observables are shown in tables 13.1, 13.2 and13.3.Notice that the symmetries derived in this chapter show that recoil observables can be usedto extract those SDME’s or statistical tensors from fits that cannot be found from the non-recoilobservables. For example, it was shown in chapter 12 that the non-interference statistical tensorswith odd L cannot be found from fits to non-recoil observables because they do not appear in theexpressions for those observables. However, in recoil observables, it is instead the odd L termsthat appear in the expressions, so these observables can be used to extract them. The same thinghappens for the interference tensors. For example, if we wanted to extract the tensors ℜ [ τ x ( N ) ], ℑ [ τ x ( N ) ], ℜ [ τ y ( N ) ], ℑ [ τ y ( N ) ], ℜ [ τ z ( N ) ] and ℑ [ τ z ( N ) ] from the non-recoil observables, we would haveto measure the observables P x , P y and P z . But since P x and P z are odd while P y is even, the108able 13.1: Symmetry of recoil observables under φ ∗ → − φ ∗ when the polarization ismeasured along the x ′ -axis.Symmetry of observables under φ ∗ → − φ ∗ : Measurement along x ′ Even Odd I P sx ′ I P x ′ I P ⊙ x ′ I P cx ′ I O xx ′ I O yx ′ I O zx ′ I O cyx ′ I O cxx ′ I O sxx ′ I O czx ′ I O szx ′ I O syx ′ I O ⊙ xx ′ I O ⊙ yx ′ I O ⊙ zx ′ Table 13.2: Symmetry of recoil observables under φ ∗ → − φ ∗ when the polarization ismeasured along the y ′ -axis.Symmetry of observables under φ ∗ → − φ ∗ : Measurement along y ′ Even Odd I P y ′ I P sy ′ I P cy ′ I P ⊙ y ′ I O yy ′ I O xy ′ I O cyy ′ I O zy ′ I O sxy ′ I O cxy ′ I O szy ′ I O czy ′ I O ⊙ xy ′ I O syy ′ I O ⊙ zy ′ I O ⊙ yy ′ φ ∗ → − φ ∗ when the polarization ismeasured along the z ′ -axis. Symmetry of observables under φ ∗ → − φ ∗ : Measurement along z ′ Even Odd I P sz ′ I P z ′ I P ⊙ z ′ I P cz ′ I O xz ′ I O yz ′ I O zz ′ I O cyz ′ I O cxz ′ I O sxz ′ I O czz ′ I O szz ′ I O syz ′ I O ⊙ xz ′ I O ⊙ yz ′ I O ⊙ zz ′ tensors ℜ [ τ x ( N ) ], ℑ [ τ y ( N ) ] and ℜ [ τ z ( N ) ] do not appear in the expressions and therefore cannot beextracted. However, ℜ [ τ x ( N ) ] will appear in the expressions for O xx ′ , O xy ′ and O xz ′ , ℑ [ τ y ( N ) ] willappear in the expressions for O yx ′ , O yy ′ and O yz ′ , and ℜ [ τ z ( N ) ] will appear in the expressions for O zx ′ , O zy ′ , and O zz ′ . 110 HAPTER 14SUMMARY
For convenience, we will reproduce many of the important expressions derived in this chapter. Inthis thesis, we have derived general versions of expressions that have previously appeared in theliterature for the decay distribution of a photoproduced vector meson, and for the decay distributionof a photoproduced spin-3/2 baryon, in terms of their SDME’s, W ( θ ∗ , φ ∗ ; V ) = ρ ( V ) cos θ ∗ + 12 ( ρ ( V ) + ρ ( V ) − − ) sin θ ∗ − √ ℜ [ ρ ( V ) − ρ ( V ) − ] sin 2 θ ∗ cos φ ∗ + 1 √ ℑ [ ρ ( V ) − ρ ( V ) − ] sin 2 θ ∗ sin φ ∗ − ℜ [ ρ ( V ) − ] sin θ ∗ cos 2 φ ∗ + ℑ [ ρ ( V ) − ] sin θ ∗ sin 2 φ ∗ , (14.1)and W ( θ ∗ , φ ∗ ; 3 /
2) = 58 ( ρ (3 /
2) + ρ − − (3 / θ ∗ )+ 34 ( ρ (3 /
2) + ρ − − (3 / θ ∗ − √ ℜ [ ρ (3 / − ρ − − (3 / θ ∗ cos φ ∗ + √ ℑ [ ρ (3 / − ρ − − (3 / θ ∗ sin φ ∗ − √ ℜ [ ρ − (3 /
2) + ρ − (3 / θ ∗ cos 2 φ ∗ + √ ℑ [ ρ − (3 /
2) + ρ − (3 / θ ∗ sin 2 φ ∗ . (14.2)The expression in terms of the spin density matrix was shown in eqn. (12.12), W ( θ ∗ , φ ∗ ) = X i ( X λ e d S i λ,λ ( θ ∗ ) ρ λ,λ ( S i ) + X λ>λ ′ e d S i λ,λ ′ ( θ ∗ ) ℜ [ ρ λ,λ ′ ( S i )] cos[( λ − λ ′ ) φ ∗ )] − ℑ [ ρ λ,λ ′ ( S i )] sin[( λ − λ ′ ) φ ∗ )] !) (14.3)+ X i>j X λ,λ ′ e d S i ,S j λ,λ ′ ( θ ∗ ) ℜ [ ξ λ,λ ′ ( S i , S j )] cos[( λ − λ ′ ) φ ∗ )]111 ℑ [ ξ λ,λ ′ ( S i , S j )] sin[( λ − λ ′ ) φ ∗ )] ! where e d S i λ,λ ′ ( θ ∗ ) ≡ X η d S i η,λ ( θ ∗ ) d S i η,λ ′ ( θ ∗ ) , e d S i ,S j λ,λ ′ ( θ ∗ ) ≡ X η d S i η,λ ( θ ∗ ) d S j η,λ ′ ( θ ∗ ) . (14.4)Expressions for the polarization observables were also derived. Since the matrices ˆ ρ and ˆ ξ canbe decomposed into 16 matrices,ˆ ρ ( S k ) = ˆ ρ + X i =1 Λ i ( N ) ˆ ρ i ( N ) + X i =1 Λ i ( γ ) ˆ ρ i ( γ ) + X i,j =1 Λ i ( N ) Λ j ( γ ) ˆ ρ ij ( Nγ ) , ˆ ξ ( S k , S l ) = ˆ ξ + X i =1 Λ i ( N ) ˆ ξ i ( N ) + X i =1 Λ i ( γ ) ˆ ξ i ( γ ) + X i,j =1 Λ i ( N ) Λ j ( γ ) ˆ ξ ij ( Nγ ) , (14.5) W can also be decomposed into 16 terms, W ( θ ∗ , φ ∗ ) = W ( θ ∗ , φ ∗ ) + X i Λ i ( γ ) W i ( γ ) ( θ ∗ , φ ∗ )+ X i Λ i ( N ) W i ( N ) ( θ ∗ , φ ∗ ) + X i,j Λ i ( γ ) Λ j ( N ) W ij ( γN ) ( θ ∗ , φ ∗ ) . (14.6)The forms of the W i ’s are same as W in eqn. (14.3) but with the density matrices substituted bythe appropriate matrix in the decomposition in eqn. (14.5), W i ( θ ∗ , φ ∗ ) = X j ( X λ e d S j λ,λ ( θ ∗ ) ρ iλ,λ ( S j ) + X λ>λ ′ e d S j λ,λ ′ ( θ ∗ ) ℜ [ ρ iλ,λ ′ ( S j )] cos[( λ − λ ′ ) φ ∗ )] − ℑ [ ρ iλ,λ ′ ( S j )] sin[( λ − λ ′ ) φ ∗ )] !)X j>k X λ,λ ′ e d S j ,S k λ,λ ′ ( θ ∗ ) ℜ [ ξ iλ,λ ′ ( S j , S k )] cos[( λ − λ ′ ) φ ∗ )] − ℑ [ ξ iλ,λ ′ ( S j , S k )] sin[( λ − λ ′ ) φ ∗ )] ! . (14.7)These W i ’s are equal to the 16 polarization observables involving only the beam and/or target, W ( θ ∗ , φ ∗ ) = I ( θ ∗ , φ ∗ ) ,W i ( N ) ( θ ∗ , φ ∗ ) = I P i ( θ ∗ , φ ∗ ) , x ( γ ) ( θ ∗ , φ ∗ ) = I cos 2 βI c ( θ ∗ , φ ∗ ) ,W y ( γ ) ( θ ∗ , φ ∗ ) = I sin 2 βI s ( θ ∗ , φ ∗ ) , (14.8) W z ( γ ) ( θ ∗ , φ ∗ ) = I I ⊙ ( θ ∗ , φ ∗ ) ,W ix ( γN ) ( θ ∗ , φ ∗ ) = I cos 2 βP ic ( θ ∗ , φ ∗ ) ,W iy ( γN ) ( θ ∗ , φ ∗ ) = I sin 2 βP is ( θ ∗ , φ ∗ ) ,W iz ( γN ) ( θ ∗ , φ ∗ ) = I P i ⊙ ( θ ∗ , φ ∗ ) . The SDME’s of eqn. (14.7) are given by ρ λλ ′ ( S k ) = 14 X λ γ λ N λ H M λλ H ; λ γ λ N M ∗ λ ′ λ H ; λ γ λ N ,ρ i ( γ ) λλ ′ ( S k ) = 14 X λ N λ H λ γ λ ′ γ M λλ H ; λ γ λ N σ iλ γ λ ′ γ M ∗ λ ′ λ H ; λ ′ γ λ N ,ρ i ( N ) λλ ′ ( S k ) = 14 X λ N λ ′ N λ γ λ H M λλ H ; λ γ λ N σ iλ N λ ′ N M ∗ λ ′ λ H ; λ γ λ ′ N ,ρ ij ( γN ) λλ ′ ( S k ) = 14 X λ γ λ ′ γ λ N λ ′ N λ H M λλ H ; λ γ λ N σ iλ γ λ ′ γ σ jλ N λ ′ N M ∗ λ ′ λ H ; λ ′ N λ ′ γ , (14.9)and ξ λλ ′ ( S k , S l ) = 14 X λ γ λ N λ H M λλ H ; λ γ λ N M ∗ λ ′ λ H ; λ γ λ N ,ξ i ( γ ) λλ ′ ( S k , S l ) = 14 X λ N λ H λ γ λ ′ γ M λλ H ; λ γ λ N σ iλ γ λ ′ γ M ∗ λ ′ λ H ; λ ′ γ λ N , (14.10) ξ i ( N ) λλ ′ ( S k , S l ) = 14 X λ N λ ′ N λ γ λ H M λλ H ; λ γ λ N σ iλ N λ ′ N M ∗ λ ′ λ H ; λ γ λ ′ N ,ξ ij ( γN ) λλ ′ ( S k , S l ) = 14 X λ γ λ ′ γ λ N λ ′ N λ H M λλ H ; λ γ λ N σ iλ γ λ ′ γ σ jλ N λ ′ N M ∗ λ ′ λ H ; λ ′ γ λ ′ N . λ H is the helicity of the spectator hadron. This is how some of those expressions look likewhen explicit values of the matrix elements of the Pauli matrices are used, ξ λλ ′ ( S k , S l ) = 14 X λ γ λ N λ H M λλ H ; λ γ λ N M ∗ λ ′ λ H ; λ γ λ N , (14.11) ξ x ( γ ) λλ ′ ( S k , S l ) = 14 X λ γ λ N λ H M λλ H ; λ γ λ N M ∗ λ ′ λ H ; − λ γ λ N ,ξ y ( γ ) λλ ′ ( S k , S l ) = − i X λ γ λ N λ H λ γ M λλ H ; λ γ λ N M ∗ λ ′ λ H ; − λ γ λ N , (14.12) ξ z ( γ ) λλ ′ ( S k , S l ) = 14 X λ γ λ N λ H λ γ M λλ H ; λ γ λ N M ∗ λ ′ λ H ; λ γ λ N .ξ x ( N ) λλ ′ ( S k , S l ) = 14 X λ γ λ N λ H M λλ H ; λ γ λ N M ∗ λ ′ λ H ; λ γ − λ N ,ξ y ( N ) λλ ′ ( S k , S l ) = − i X λ γ λ N λ H λ N M λλ H ; λ γ λ N M ∗ λ ′ λ H ; λ γ − λ N ,ξ z ( N ) λλ ′ ( S k , S l ) = 14 X λ γ λ N λ H λ N M λλ H ; λ γ λ N M ∗ λ ′ λ H ; λ γ λ N . The parity relations of these 16 spin density matrices were also derived. This equation is alsovalid for the non-interference matrices, ξ − λ, − λ ′ = ¯ η ( − S ∗ +1 ζ ( − λ + λ ′ ξ λ,λ ′ ,ξ i ( γ ) − λ, − λ ′ = ¯ η ( − S ∗ +1 ζ ( γ ) i ( − λ + λ ′ ξ i ( γ ) λ,λ ′ ,ξ i ( N ) − λ, − λ ′ = ¯ η ( − S ∗ +1 ζ ( N ) i ( − λ + λ ′ ξ i ( N ) λ,λ ′ ,ξ ij ( γN ) − λ, − λ ′ = ¯ η ( − S ∗ +1 ζ ( γ ) i ζ ( N ) j ( − λ + λ ′ ξ ij ( γN ) λ,λ ′ , ¯ η ≡ ηη ′ . (14.13) η and η ′ are the intrinsic parities of the two interfering hadrons, S ∗ is the spin of either of theinterference hadrons (it makes no difference because both have either integer or half-integer spin),114nd the ζ factors are characteristic of each of the 16 matrices, ζ = − ,ζ i ( γ ) = ( , if i = 2 , − , if i = 1 ,ζ i ( N ) = ( , if i = 1 , − , if i = 2 . . (14.14)There is therefore a factor dependent on the intrinsic parities (¯ η ), one dependent on the statisticsof the hadrons (( − S ∗ +1 ), one dependent on the type of matrix ( ζ i ), and one dependent on thematrix element (( − λ + λ ′ ). These relationships reduce the number of independent matrix elementswhen the reaction is parity conserving.An equivalent way to represent the spin state of a system is to use statistical tensors instead ofSDME’s. The relationships between them are, using matrix notation,ˆ ρ ( S ) = S X L =0 L X M = − L t LM ( S ) ˆ T † LM ( S ) , ˆ ξ ( S i , S j ) = L = S i + S j X L = | S i − S j | L X M = − L τ LM ( S i , S j ) ˆ T † LM ( S j , S i ) , (14.15)where the elements of the polarization operators are[ T LM ( S )] mm ′ ≡ ( − S − m ′ C LMSm ; S − m ′ , [ T LM ( S i , S j )] m,m ′ ≡ ( − S j − m ′ C LMS i ,m ; S j , − m ′ . (14.16)Using index notation, the expression is ρ mm ′ ( S ) = S X L =0 L X M = − L ( − S − m t LM ( S ) C LMSm ′ ; S − m ,ξ mm ′ ( S i , S j ) = S i + S j X L = −| S i − S j | M = L X M = − L τ LM ( S i , S j )( − S i − m C LMS j m ′ ; S i − m . (14.17)The inverse relations are t LM ( S ) = h ˆ T LM ( S ) i = Tr[ˆ ρ ( S ) ˆ T LM ( S )]= S X m,m ′ = − S ( − S − m ρ mm ′ ( S ) C LMsm ′ ; s − m ,τ LM ( S i , S j ) = D ˆ ξ ( S i , S j ) ˆ T LM ( S j , S i ) E = Tr[ ˆ ξ ( S i , S j ) ˆ T LM ( S j , S i )]= S i X m = − S i S j X m ′ = − S j ( − S i − m ξ m,m ′ ( S i , S j ) C LMS j ,m ′ ; S i , − m . (14.18)115he last two equations show that the SDME’s and the statistical tensors are linear combinationsof each other. It is therefore easy to change from using one representation to the other.The expressions for the general decay distributions and observables were also derived in termsof the statistical tensors. The decay distribution is W ( θ ∗ , φ ∗ ; s ) = X i S i X L =0 L X M = − L κ L ( S i ; s )( − M t LM ( S i ) Y L, − M ( θ ∗ , φ ∗ )+ X i>j S i + S j X L = | S i − S j | L X M = − L κ L ( S i , S j ; s ) ℜ n ( − M τ LM ( S i , S j ) Y L, − M ( θ ∗ , φ ∗ ) o , (14.19)where κ L ( S i ; s ) ≡ r π L + 1 s X m = − s ( − S i − m C L S i ,m ; S i , − m = r π L + 1 s X m = − s [ ˆ T L ( S i )] mm ,κ L ( S i , S j ; s ) ≡ r π L + 1 s X m = − s ( − S i − m C L S j ,m ; S i , − m = r π L + 1 s X m = − s [ ˆ T L ( S i , S j )] mm , . (14.20)We have also shown how for the non-interference statistical tensors only the terms with even L appear in the distributions because κ L ( S i ; s ) = 0 for L odd . (14.21)If written in terms of cosine and sine functions of φ ∗ , W ( θ ∗ , φ ∗ ; s ) = X i S i X L =0 L X M ≥ e κ L ( S i ; s ) P ML (cos θ ∗ ) (cid:18) ℜ h t LM ( S i ) i cos( M φ ∗ )+ ℑ h t LM ( S i ) i sin( M φ ∗ ) (cid:19) (14.22)+ X i>j S i + S j X L = | S i − S j | L X M = − L e κ L ( S i , S j ; s ) P ML (cos θ ∗ ) (cid:18) ℜ h τ LM ( S i , S j ) i cos( M φ ∗ )+ ℑ h τ LM ( S i , S j ) i sin( M φ ∗ ) (cid:19) , P ML (cos θ ∗ ) are the associated Legendre polynomials and e κ LM ( S i ; s ) ≡ s ( L − M )!( L + M )! s X m = − s ( − S i − m C L S j m ; S i − m , (14.23) e κ LM ( S i , S j ; s ) ≡ s ( L − M )!( L + M )! s X m = − s ( − S i − m C L S j m ; S i − m . Just like the spin density matrices, the statistical tensors can also be decomposed into 16 othertensors, t LM = t LM + X j Λ j ( γ ) t j ( γ ) LM + X i Λ jN t jNLM + X jk Λ j ( γ ) Λ kN t jk ( γN ) LM ,τ LM = τ LM + X j Λ j ( γ ) τ j ( γ ) LM + X i Λ jN τ jNLM + X jk Λ j ( γ ) Λ kN τ jk ( γN ) LM . (14.24)The observables that involve only the beam and/or target in terms of statistical tensors can alsobe written, and they have the same mathematical form as the decay distribution shown in eqn.(14.22) but with the appropriate superscript, W i ( θ ∗ , φ ∗ ; s ) = X i S i X L =0 L X M ≥ e κ L ( S i ; s )( − M P ML (cos θ ∗ ) (cid:18) ℜ h t iLM ( S i ) i cos( M φ ∗ )+ ℑ h t iLM ( S i ) i sin( M φ ∗ ) (cid:19) + X i>j S i + S j X L = | S i − S j | L X M = − L e κ L ( S i , S j ; s ) P ML (cos θ ∗ ) (cid:18) ℜ h τ iLM ( S i , S j ) i cos( M φ ∗ )+ ℑ h τ iLM ( S i , S j ) i sin( M φ ∗ ) (cid:19) . (14.25)where the relationship between the ˆ ρ i ’s and ˆ ξ i ’s to the t iLM ’s and τ iLM ’s has the same form as thoseshown in eqn. (14.17), ρ kmm ′ ( S ) = S X L =0 L X M = − L ( − S − m t kLM ( S ) C LMSm ′ ; S − m ,ξ kmm ′ ( S i , S j ) = S i + S j X L = −| S i − S j | M = L X M = − L τ kLM ( S i , S j )( − S i − m C LMS j m ′ ; S i − m . (14.26)The parity relations for the statistical tensors were also derived, t kLM ( S i ) = − ηζ k ( − L + M t kL − M ( S i ) ,τ kLM ( S i , S j ) = − ηζ k ( − L + M + S i − S j τ kL − M ( S i , S j ) . (14.27)117hese relationships reduce the number of independent statistical tensors when the reaction is parityconserving. These can also be used to prove which observables are even or odd in φ ∗ when thereaction is parity conserving. Table 12.1 shows which non-recoil observables are even and whichones are odd in the variable φ ∗ .Finally, we also derived general expression for the observables that involve the recoil baryon forthe case of a decaying baryon resonance, O ix ( θ ∗ , φ ∗ ) = X k S k X L =0 L X M = − L κ L ( S k ; s ) (cid:18) d LM − ( θ ∗ ) − d LM ( θ ∗ ) (cid:19) × (cid:18) ℜ h t iLM ( S k ) i cos( M φ ∗ ) + ℑ h t iLM ( S k ) i sin( M φ ∗ ) (cid:19) + X k>l S k + S l X L = | S k − S l | L X M = − L κ L ( S k , S l ; s ) (cid:18) χ L ( S k , S l ) d LM − ( θ ∗ ) − d LM ( θ ∗ ) (cid:19) × (cid:18) ℜ h τ iLM ( S k , S l ) i cos( M φ ∗ ) + ℑ h τ iLM ( S k , S l ) i sin( M φ ∗ ) (cid:19) , (14.28) O iy ( θ ∗ , φ ∗ ) = X k S k X L =0 L X M = − L κ L ( S k ; s ) (cid:18) d LM − ( θ ∗ ) + d LM ( θ ∗ ) (cid:19) × (cid:18) ℑ h t iLM ( S k ) i cos( M φ ∗ ) − ℜ h t iLM ( S k ) i sin( M φ ∗ ) (cid:19) + X k>l S k + S l X L = | S k − S l | L X M = − L κ L ( S k , S l ; s ) (cid:18) χ L ( S k , S l ) d LM − ( θ ∗ ) + d LM ( θ ∗ ) (cid:19) × (cid:18) ℑ h τ iLM ( S k , S l ) i cos( M φ ∗ ) − ℜ h τ iLM ( S k , S l ) i sin( M φ ∗ ) (cid:19) , (14.29)and O iz ( θ ∗ , φ ∗ ) = X k S k X L =0 L X M = − L κ L ( S k ; s ) d LM (cos θ ∗ ) × (cid:18) ℜ h t iLM ( S k ) i cos( M φ ∗ ) + ℑ h t iLM ( S k ) i sin( M φ ∗ ) (cid:19) + X k>l S k + S l X L = | S k − S l | L X M = − L κ L ( S k , S l ; s ) d LM (cos θ ∗ ) × (cid:18) ℜ h τ iLM ( S k , S l ) i cos( M φ ∗ ) + ℑ h τ iLM ( S k , S l ) i sin( M φ ∗ ) (cid:19) , (14.30)118here we have defined κ L ( S k , S l ; s ) ≡ r s ( s + 1)(2 s + 1)3 s X m,m ′ = − s ( − S k − s C L S l m ′ ; S k − m C sm ′ ; s − m ,κ L ( S k , S l ; s ) ≡ r s ( s + 1)(2 s + 1)3 s X m,m ′ = − s ( − S k − s C L S l m ′ ; S k − m C sm ′ ; s − m , (14.31) κ L ( S k ; s ) ≡ κ L ( S k , S k ; s ) ,κ L ( S k ; s ) ≡ κ L ( S k , S k ; s ) , and χ L ( S k , S l ) ≡ ( − S k + S l − L = ± . (14.32)The second subscript in O ij refers to the axis along which the recoil baryon’s polarization is mea-sured. The first one refers to the superscript of the statistical tensor in the expression (e.g., x ( γ ) , z ( N ) , zy ( γN ), etc.). The observable P ⊙ x ′ , for example, is equal to O zx . The parity relationsin eqn. (14.27) can also be used to show which ones of these observables are even or odd in φ ∗ when the reaction is parity conserving. Tables 13.1, 13.2 and 13.3 shows which recoil observablesare even and which ones are odd in the variable φ ∗ . These expressions are not valid for the caseof a decaying meson resonance, since those cannot be expressed in terms of the statistical tensorsof said meson. Rather, they would necessarily have to be expressed in terms of statistical tensorsdescribing the entire meson-baryon quasi-two-body state.119 HAPTER 15CONCLUSIONS
As stated at the beginning of this thesis, this work was initiated due to the current interest inphotoproduction reactions in hadron spectroscopy. Because of the importance of polarization mea-surements in these experiments, we consider that it is important to understand as much as possiblethe relationship between polarization observables and the resonances that contribute to these re-actions.To this end, we have investigated the role that spin plays in two pseudoscalar (and scalar)photoproduction, γN → M M B , where M and M are mesons and B is a baryon. But morespecifically, for reaction channels involving the photoproduction of a quasi-two-body state, such as γN → M ∗ B → M M B , γN → M B ∗ → M M B and γN → M B ∗ → M M B , where M ∗ and B ∗ is an unstable meson and baryon, respectively, that undergo a two-body decay.The interest was in examining the connection between the spin state of the unstable hadron, M ∗ or B ∗ , and polarization measurements, and also its connection with the decay angular distributionsof the decaying hadron in a model-independent way. Another way of stating this goal is as follows:How much information can we gain about the spin state of the unstable hadron from experimentalobservables without having to rely on a specific model?Note how we have used the generic labels M , M and B in order to emphasize that our conclu-sions are independent of the type of hadrons involved in the reaction. Therefore, our conclusionscan also be applied to reactions such as γN → ππN,γN → πηN,γN → ηηN,γN → K ¯ KN,γN → πKY,γN → KK Ξ , (15.1)120ust to name a few. The only property of relevance in this research is the spin of the hadronsinvolved in the reaction.This led us to deriving general expressions for the decay distributions of the contributing res-onances and of the polarization observables in terms of quantities that describe the spin of theresonances: either SDME’s or statistical tensors. Eqns. (14.1) and (14.2), which have been pre-viously derived in the literature, are special cases of the general expressions. These two equationshad been used to aid in the interpretation of the data gathered from experiments, but can onlybe applied when vector mesons and spin-3/2 baryons contribute to the reaction. By contrast, thegeneral expression derived in this work can be applied to contributions for hadrons of any spin,and it also takes into account the interference effects among the different reaction channels. Asmentioned in the introduction, this is relevant because in the energy region where the “missing”resonances are expected to be found there are many broad and overlapping resonances, so it isuseful to have an expression that can be applied to more general situations.The general expression for the decay distribution of the resonance in terms of SDME’s is shownin eqn. (14.3). It has a very simple structure: it consists of a linear sum of the real and imaginaryparts of all of the independent spin density matrix elements (the hermiticity condition shown ineqn. (4.8) reduces the number of elements that are independent, which is the reason one of thesums only has the terms with λ > λ ′ ). The terms with the real parts are proportional to cosinefunctions of φ ∗ , while the imaginary parts are proportional to sine functions of this angle, of theform cos[ nφ ∗ ] and sin[ nφ ∗ ], where n is an integer. The value of n is completely determined by theSDME it has as its factor: it does not depend on the spin of the decaying hadron, but it is thedifference between the two helicity indices of the matrix element, n = λ − λ ′ . Each of the termswill also be proportional to a trigonometric function of θ ∗ , which is also a function of the spin andof the decaying hadron (or hadrons, for the interference matrix elements). The explicit form ofthese functions, e d S i λλ ′ ( θ ∗ ) for the diagonal submatrices and e d S i S j λλ ′ ( θ ∗ ) for the off-diagonal interferencesubmatrices, shown in eqn. (14.4), are bilinear combinations of Wigner d -functions. As such, whiletheir explicit forms may be long and complicated, they are related to well known functions in aneasy way.With this simple general expression, the decay distribution in terms of the SDME’s can begenerated for any number of contributing hadrons, all of arbitrary spin, and including the effects121f interference. It can even be shown that the two equations from the literature, eqn. (14.1) andeqn. (14.2), can be obtained from this general equation by including only one hadron with S i = 1or S i = 3 / θ ∗ dependence of the distributions is a function of the spins of the hadrons thatcontribute, the fitting procedure can also be used to determine the values of spin that are notinvolved in the reaction: after doing the fit procedure, if the coefficient in a particular value ofone of the e d S i is equal to zero, this indicates that hadrons of spin S i are not contributing to thereaction.The general expression for the decay distribution using statistical tensors is shown in eqn.(14.19) in terms of spherical harmonics and in eqn. (14.22) in terms of the associated Legendrepolynomials. These equations show that the expression for the decay distribution acquires an evensimpler form when written in terms of the tensors. One of the reasons is that the θ ∗ and φ ∗ dependence is entirely contained in the spherical harmonics, which are very well known functions.Another is that, unlike in the expression in terms of the SDME’s, this expression is manifestlyrotationally invariant because contractions between a contravariant and a covariant tensor, such as t LM and ( − M Y L − M , are rotationally invariant. In fact, the constraint of rotational invariance isenough to conclude that each term in the distribution has to be proportional to this factor. Whenthe distribution is instead expressed in terms of the Legendre polynomials, the θ ∗ dependence incontained in them, which are also well known functions. In this expression the φ ∗ dependence isentirely contained in cosine and sine functions, cos[ M φ ∗ ] and sin[ M φ ∗ ]. Since the prefactor in theirarguments is M , it is easy to know which trigonometric functions will be accompanying each of thetensors in the sum. The values of the prefactors, κ L and e κ LM , are proportional to a factor verysimilar to the trace of the polarization operator ˆ T L (it will in general be a sum of only some of itsdiagonal elements). 122nother reason for its simplicity is that each of the cosine and sine functions of φ ∗ are multipliedby a single tensor. In contrast, when the distribution is expressed in terms of SDME’s, each of thecosine and sine functions of φ ∗ are multiplied by a linear combination of them. This means thatif these expressions for the distributions were to be used to perform fits to a sum of trigonometricfunctions of the form cos[ nφ ∗ ] and sin[ nφ ∗ ], with n being an integer, the fit parameters will end upbeing sums of SDME’s. See for example the decay distribution for vector mesons in eqn. (14.1) and(14.2). Therefore, when the distribution is expressed in terms of the statistical tensors, it becomesmanifest the fact that the number of independent parameters describing the spin of the resonance(the SDME’s or the statistical tensors) that can be extracted from such fits is actually less thantheir total number. This is seen from eqn. (14.21), which shows that for the non-interferencetensors only those with even L appear in the distribution.Expressions for the polarization observables were also derived and are shown in eqns. (14.7)and (14.22), where the W i ’s relation to the observables are shown in eqn. (14.8). They have thesame mathematical form as the decay distributions shown in eqns. (14.1) and (14.2) but with theSDME’s or statistical tensors substituted with the appropriate one of the 16 expansion SDME’sand tensors in eqns. (14.5) and (14.24). Note that W i ’s arise from the decomposition of the decaydistribution W , as shown in eqn. (14.6).The 16 SDME’s or statistical tensors in these expressions for the observables can be extracted bymeasuring the polarization observables and performing fits with these expressions with the SDME’sand tensors as fit parameters. And since these expressions are general, they can be used for anyresonance that contributes with any spin, and also for situations in which more than one resonancecontributes to the reaction.This is of great importance because in order to accomplish a complete experiment, the helicityamplitudes of the process must be found. But these amplitudes are related to the SMDE’s byeqns. (14.9) and (14.10) (or to the statistical tensors, since they are related to these SDME’s byeqns. (14.17) and (14.18)). Once the helicity amplitudes are known, they can be used to performpartial wave analyses, which are of great help in the search of resonances. As has been mentioned,one of the reasons why these searches are important is because there is an interest in findingthe “missing” resonances that have been predicted by quark models but not found. The generalexpressions derived in this thesis could therefore be of help in these searches.123or parity conserving reactions, we found the general parity properties of the 16 SDME’s andstatistical tensors in eqns. (14.13), (14.14), and (14.27). These parity relations can be used to findwhether a polarization observable is even or on in the variable φ ∗ . It turns out that when theseparity relations are applied to the expressions for the observables, the factor ζ i is the only one thatdoes not vanish. Therefore, whether an observable is even or odd in φ ∗ is only dependent on thetype of observable, not on the properties of the hadrons involved in the reaction. Being even in φ ∗ means that every term in the expression for the observables will be proportional to a cosinefunction of φ ∗ , while being odd means the terms will be proportional to sine function of φ ∗ . Asummary of which observables are even or odd in φ ∗ can be found in table 12.1.For the case of a decaying baryon resonance, the general expressions for polarization observablesthat involve a measurement of the recoil baryon’s polarization where also derived and are shownin eqns. (14.28), (14.29), and (14.30). It was shown that, for the non-interference tensors, onlyterms with odd L show up in the expressions. By contrast, in the expressions for the observablesthat only involve the beam and/or target, only terms with even L show up in the sum for thenon-interference tensors. This shows how not all of the independent elements that describe thespin of the resonance (in this case, the statistical tensors) can be extracted using only polarizedbeams and targets. Experiments in which the polarization of the recoil baryon can be measuredmust also be performed in order to achieve a complete experiment.Future work could aim to derive the general expressions for the recoil observables for the caseof a decaying meson resonance. These would necessarily have to be expressed in terms of thestatistical tensors of the full meson-baryon quasi-two-body state, rather than the statistical tensorsof the meson subsystem.It is important to note that for each resonance that contributes to these expressions, the anglesused to describe its decays will be different because they can decay into three different particlepairs: M M , M B , and M B . This means that there are three different sets of θ ∗ and φ ∗ thatwill appear in the general expressions, even though only one of the sets is independent. AppendixA describes how to relate these different sets of angles, shown in eqns. (A.6), (A.8) and (A.9).However, they are not related in a simple manner. If the general expressions are used for doing fits,instead of rewriting the non-independent sets of angles in terms of the dependent ones, all three setscould be measured in experiments and used in the general expressions to perform the fits. Future124esearch could be devoted to finding a more compact way of expressing these relations between thedifferent sets of angles, perhaps in terms of well known functions such as Wigner D -functions orspherical harmonics.It is important to remember the limitations of these general expression. They are only validfor reactions that only have contributions from quasi-two-body states. Also, as shown from eqns.(11.11) to (11.16) and from (11.19) to (11.24), which are specific cases of the general expressions inwhich three resonances contribute to the reaction, the expressions can become very long, especiallywhen the spin of the resonances is high and when many of them contribute. However, we emphasizethat the general expressions are very simple, and shows that, while expressions for specific casesare long, it is straightforward to generate them from the general expressions.Another thing to consider is the fact that the only property of the decaying resonance thatplays a role in the form of it’s distribution as a function of its SDME’s or statistical tensors isits total spin. Therefore, for cases where you have more than one resonance with the same spin,the factors of the trigonometric functions in the expressions will be the sum of the SDME’s orstatistical tensors of all of the resonances of the same spin. As such, the general expressions alongwith measurements of the angles θ ∗ and φ ∗ will not be able to show on their own if more than oneresonance of a particular spin is contributing to the reaction.125 PPENDIX ACHANGING PHASE SPACE COORDINATES
The expression for the angular distribution γN → M M N is given in terms of Wigner D -functions.The arguments to these functions are the decay angles of the products of the unstable resonance.However, as noted in section 3, each of the tree pathways A, B, and C, defined in eqns. (3.3),(3.4) and (3.5), are described in terms of different angles, each defined in different reference frames.When the expression of the distribution involves more than one decaying resonance and interferenceterms, each of the arguments of the Wigner D -functions will be different. It would be convenientto be able to relate these three different set of angles to each other, so that the expression for thedecay distribution can be written in terms of only one set of angles. Refer to section 3 for thedefinition of the kinematic variables.Since we are assuming an unstable resonance will decay into two of the hadrons in the three-bodyfinal state, its four-momentum will have three possible values, q = q + q (A.1) q = p ′ + q (A.2) q = q + p ′ , (A.3)where these four-momenta are defined in the overall center of mass frame. Decaying mesons willalways have four-momentum equal to q , while decaying baryons can have either four-momentum q or q . As has been shown, W will be a function of Wigner D -functions, and these contain theangular dependence on the decay angles. Since there are three different channels, each Wigner D -function that appears in the expression will be a function of one of three sets of angles: { θ ∗ , φ ∗ } , { θ ∗ , φ ∗ } , { θ ∗ , φ ∗ } , which are the decay angles defined in the rest frame of the decaying reso-nance. These angles and the coordinate axes of these frames are defined analogously to the primedcoordinate system defined in eqn. (3.17), ˆ z ij = q ij | q ij | , ˆ y ij = k × q ij | k × q ij | , ˆ x ij = ˆ y ij × ˆ z ij , θ ∗ ij ) = ˆ π ij · ˆ z ij , cos( φ ∗ ij ) = ˆ y ij · ( ˆ z ij × ˆ π ij ) | ˆ z ij × ˆ π ij | , sin( φ ∗ ij ) = − ˆ x ij · ( ˆ z ij × ˆ π ij ) | ˆ z ij × ˆ π ij | , ˆ π ij = q ∗ i ( ij ) | q ∗ i ( ij ) | , (A.4) i, j ∈ { , , } , where i and j label the three possible rest frames, q ij is the spatial part of the four-momenta definedin eqn. (A.1), and q ∗ i ( ij ) is the spatial part of the four-vector of the i -th final state particle in thecenter of mass frame of the i -th and j -th final state particle, i.e., the frame in which q ∗ ij = 0 (Thenumbers in parentheses in q ∗ i ( ij ) are used to label the coordinate frame). The primed coordinatesystem that we defined in eqn. (3.17) is the case when i = 1 and j = 2. We will assume the set { θ ∗ , φ ∗ } is the one chosen to describe the scattering reaction so we will derive expressions for theother two sets in terms of this one.The first step is to find the four-momenta of the two final state mesons q and q in the overallcenter of mass frame from their four-momenta in their center of mass frame q ∗ and q ∗ . Eqns. (3.18)and (3.19) show the dependence of these two four-vectors on the phase space coordinates s M M , θ ∗ and φ ∗ . We find the four-momenta in the overall center of mass frame by applying this boostand rotation, q µi = Λ µν q ∗ νi , ˆΛ = θ ) 0 sin( θ )0 0 1 00 − sin( θ ) 0 cos( θ ) E q √ s M M | q |√ s M M | q |√ s M M E q √ s M M , (A.5)where θ is the scattering angle and E q , | q | and s M ,M can be found in eqn. (3.14). Theexpression for q and q is therefore q i = √ s M ,M ( E q E ∗ q i + η i | q || q ∗ | cos( θ ∗ )) η i | q ∗ | cos( θ ) sin( θ ∗ ) cos( φ ∗ ) + sin( θ ) √ s M ,M ( | q | E ∗ q i + η i E q | q ∗ | ) η i | q ∗ | sin( θ ∗ ) sin( φ ∗ ) − η i | q ∗ | sin( θ ) sin( θ ∗ ) cos( φ ∗ ) + cos( θ ) √ s M ,M ( | q | E ∗ q i + η i E q | q ∗ | ) , (A.6) η i = ( , i = 1 − , i = 2 , E ∗ q i and | q | are defined in eqn. (3.19). This leads us to the expression for the unit vector ˆ n normal to the plane that contains the three-momenta of the three reaction products, q × q | q × q | = ˆ n = − cos( θ ) sin( φ ∗ )cos( φ ∗ )sin( θ ) sin( φ ∗ ) , (A.7)where q and q are the spatial parts of the four-vectors q and q .Eqn. (A.6), along with eqn. (3.13), gives the expressions for the four-momenta of the final stateparticles, q , q , and p ′ , as functions of the 5 independent phase space variables and the massesof the particles involved, m B , m M , and m M . These four-vectors will be used to find the otherset of angles that we are interested in, either { θ ∗ , φ ∗ } or { θ ∗ , φ ∗ } . In order to accomplish this,we must find the expression for the four-vector q ∗ if we are interested in { θ ∗ , φ ∗ } or for thefour-vector q ∗ if we are interested in { θ ∗ , φ ∗ } . For generality, will refer to the four-vector ofinterest as q ∗ i ( ij ) for ( i, j ) ∈ { (3 , , (2 , } , in what follows.To find q ∗ i ( ij ) , we first rotate the { ˆ x , ˆ y , ˆ z } system into the { ˆ x ij , ˆ y ij , ˆ z ij } system by applying arotation around the ˆ z axis followed by a rotation around the ˆ y . We then apply a boost in the ˆ z ij direction to reach the frame in which q ∗ ij = 0. q ∗ i ( ij ) is therefore related to q i by q ∗ µi ( ij ) = Λ µν q νi , ˆΛ = P ij ( q ij ) | q ij | ( q ij ) | q ij | ( q ij ) P ij ( q ij ) θ ij ) 0 sin( θ ij )0 0 1 00 − sin( θ ij ) 0 cos( θ ij ) × φ ij ) sin( φ ij ) 00 − sin( φ ij ) cos( φ ij ) 00 0 0 1 , (A.8)where ( q ij ) = q µ q µ and θ ij and φ ij are the polar and azimuthal angles of q ij . θ ij and φ ij are givenby tan( θ ∗ ij ) = q ( q ij ) + ( q ij ) q ij , tan( φ ∗ ij ) = q ij q ij , (A.9)128here the expressions for q ij , q ij , and q ij can be found from eqns. (3.13), (3.14), (3.19), (A.1) and(A.6) and the correct solution for φ ij is chosen based on the signs of q ij and q ij . 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