Measurement of the Dalitz Plot Distribution for η→π+π−π0 with KLOE

Full text

(1)ACTA UNIVERSITATIS UPSALIENSIS Uppsala Dissertations from the Faculty of Science and Technology 118.

(2)

(3) Measurement of the Dalitz Plot Distribution for η→π+π−π0 with KLOE Li Caldeira Balkeståhl.

(4) Dissertation presented at Uppsala University to be publicly examined in Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 22 January 2016 at 09:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Simon Eidelman (Budker Institute of Nuclear Physics and Physics Div., Novosibirsk State University). Abstract Caldeira Balkeståhl, L. 2015. Measurement of the Dalitz Plot Distribution for η→π+π−π0 with KLOE. Uppsala Dissertations from the Faculty of Science and Technology 118. 146 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9427-8. The mechanism of the isospin violating η→π+π−π0 decay is studied in a high precision experiment using a Dalitz plot analysis. The process is sensitive to the difference between up and down quark masses. The measurement provides an important input for the determination of the light quark masses and for the theoretical description of the low energy strong interactions. The measurement was carried out between 2004 and 2005 using the KLOE detector at the DAΦNE e+e− collider located in Frascati, Italy. The data was collected at a center of mass energy corresponding to the φ-meson peak (1019.5 MeV) with an integrated luminosity of 1.6 fb−1. The source of the η-mesons is the radiative decay of the φ-meson: e+e−→φ→ηγ, resulting in the world’s largest data sample of about 4.7·106 η→π+π−π0 decay events. In this thesis, the KLOE Monte Carlo simulation and reconstruction programs are used to optimize the background rejection cuts and to evaluate the signal efficiency. The background contamination in the final data sample is below 1%. The data sample is used to construct the Dalitz plot distribution in the normalized dimensionless variables X and Y. The distribution is parametrized by determining the coefficients of the third order polynomial in the X and Y variables (so called Dalitz plot parameters). The statistical accuracy of the extracted parameters is two times better than any of the previous measurements. In particular the contribution of the X2Y term is found to be different from zero with a significance of approximately 3σ. The systematic effects are studied and found to be of the same size as the statistical uncertainty. The contribution of the terms related to charge conjugation violation (odd powers of the X variable) and the measured charge asymmetries are consistent with zero. The background subtracted and acceptance corrected bin contents of the Dalitz plot distribution are provided to facilitate direct comparison with other experiments and with theoretical calculations. Keywords: Hadron physics, Quark masses, Hadronic decays, Light mesons, Meson-meson interactions Li Caldeira Balkeståhl, Department of Physics and Astronomy, Nuclear Physics, Box 516, Uppsala University, SE-751 20 Uppsala, Sweden. © Li Caldeira Balkeståhl 2015 ISSN 1104-2516 ISBN 978-91-554-9427-8 urn:nbn:se:uu:diva-266871 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-266871).

(5) To my parents, for always believing in me.

(6)

(7) Contents. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Chiral Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Ingredients for the ChPT Lagrangian . . . . . . . . . . . . . . . . . . . . . . 1.1.3 ChPT Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Quark Masses from ChPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Dalitz Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Dalitz Plot Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Dalitz Plot Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 More Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Electromagnetic Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Dispersive Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Previous Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14 14 15 16 19 22 28 29 31 31 31 32 34 38. 2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 DAΦNE Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 KLOE Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Drift Chamber (DC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Electromagnetic Calorimeter (EMC) . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Upgrades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 DAΦNE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 KLOE-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40 40 42 44 48 52 53 53 53. 3 Event Reconstruction and Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 FILFO: Background Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Event Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Analysis Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Background Rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Data-MC Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58 58 58 60 61 63 64 68.

(8) 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Dalitz Plot and Variable Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fit Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Phase Space Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Fit Test on MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Fit Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Minimum Photon Energy Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Background Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Choice of Binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Track-Photon Angle Cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.5 Time-of-Flight Cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.6 Opening Angle Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.7 Missing Mass Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.8 Event Classification Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.9 Summary of Systematic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Final Results for Dalitz Plot Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Charge Asymmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73 73 74 75 77 78 87 87 89 92 95 98 103 106 111 118 118 119. 5 Acceptance Corrected Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Diagonality of the Smearing Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Acceptance Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Comparison with Smearing Matrix Method . . . . . . . . . . . . . . . . . . . . . . . 5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 121 121 123 124 127. 6 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Dalitz Plot Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Acceptance Corrected Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Charge Asymmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 130 130 132 135 135. Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.

(9) Introduction. Physics as a science is about understanding the world around us, everything from the big scale of the universe to very small objects like atoms, nucleons and elementary particles, and everything in between. The goal of physics is to describe all these things and to predict how they react, but not necessarily to describe all things with just one equation. Among the many fields in physics dedicated to different aspects of the world around us, this thesis fits into the field of subatomic physics. Subatomic physics aims to describe nuclei, the nucleons that make up the nuclei and also other particles. The description of nuclei is a whole sub-field in itself, but let us focus on things smaller than this, on the particles. The nucleons, the proton and the neutron, are examples of particles called hadrons. In contrast to the electron, hadrons are not elementary particles, that is, hadrons are composed of other particles. These particles are called quarks. Electrons and quarks are, as far as physics has managed to determine, elementary particles. There are two well-established kinds of hadrons: baryons and mesons. Baryons, such as protons and neutrons, are made up of three quarks, while mesons are composed of quarks and antiquarks1 [1]. Some of the lightest mesons are the three pions (π + , π − , π 0 ) and the eta-meson (η), all of which feature prominently in this thesis. The current understanding of the elementary particles and their interactions is expressed in the so called Standard Model (of particle physics) [2, 3]. According to this model, the known particles are grouped depending on how they interact. The interactions correspond to three of the known forces in nature: the strong force, the weak force, the electromagnetic force; and are mediated by particles called gauge bosons. The forth known force in nature, gravity, acts very weakly for the elementary particles and is not included in the Standard Model. An illustration of the particles in the Standard Model and their interactions is shown in figure 1. The quarks are classified according to their flavor (the name of the quantum number used, not at all related to taste) as: up (u), down (d), charm (c), strange (s), top (t) or bottom (b). They possess a type of charge called color charge (also nothing to do with how the quarks look), which means they feel the strong force. The strong interaction is mediated by particles called gluons, and the shaded area in figure 1 surrounding the quarks 1 Antiquarks are the antiparticles of the quarks.. Antiparticles are the same as particles, except for their charge. The most well-known example is the positron, the electron’s antiparticle, having the same mass as the electron but positive instead of negative electric charge.. 9.

(10) Figure 1. The elementary particles of the Standard Model. The interactions felt by different particles are indicated by the lightly shaded areas. Image credit: [4].. 10.

(11) and gluons indicates that these particles interact via the strong force. The particles possessing electric charge: the quarks, the electron, muon, tau and the W bosons all feel the electromagnetic force and interact with and via photons. All quarks and leptons feel the weak force, that is, they interact with and via the W and Z bosons. The Higgs boson is different from the other bosons. It does not mediate any force, but it interacts with the other elementary particles with a strenght depending on their mass. The Higgs boson is part of the mechanism that gives mass to the other elementary particles in the Standard Model [2, 3]. The mathematical formulation of the Standard Model describes how and with which strength all these particles interact. It is formulated as a quantum field theory, where the particles are described as excitations of fields in spacetime. The electromagnetic and weak forces are combined in the electroweak theory while the strong interaction is described separately in the quantum field theory called Quantum Chromodynamics (QCD). In the mathematical formulation of the Standard Model, some free parameters appear [2, 3]. These represent constants that are not predicted by the theory, but must be measured by experiment. There are 19 such parameters in the Standard Model, like the masses of the quarks, the charged leptons and the Higgs, and the coupling strengths of the interactions. To test the Standard Model and to be able to build other theories that address some of its shortcomings, the Standard Model needs to be quantitatively understood very well. This includes knowing the parameters of the Standard Model with high precision. QCD has two features that make it different from the other quantum field theories in physics: confinement and asymptotic freedom. These two properties can be seen as two related extremes of the theory. Asymptotic freedom means that quarks and gluons at high energies, or at small distances, interact weakly [5, 6]. In the limit of the quarks and gluons having infinite relative momenta, they would not interact at all and would behave as free particles. On the other extreme, confinement concerns quarks and gluons at low energies or at large distances. The strength of the strong interaction increases as the distance gets larger, and if it gets large enough, new quarks and antiquarks are created. This implies that the quarks can never be separated and always appear inside hadrons [2]. If one tried to break a hadron, say a proton, by dragging it apart, instead of seeing the separate quarks, one would get more hadrons. These two features illustrate that the strong interaction changes strength depending on the energy of the particles concerned, getting weaker at high energies and stronger at lower energies. This is referred to as the running of the strong coupling constant. Since the quarks are always strongly bound inside hadrons, measuring their mass is much more complicated than for example for the electron. The masses have to be extracted from other quantities or processes which depend on the quark masses. Especially the masses of lightest quarks, the u and d quarks, present a challenge and are still under investigation [7, 8, 9, 10, 11]. 11.

(12) Strong CP problem and the light quark masses One of the motivations for measuring the light quark masses precisely is the so called strong CP problem. One proposed solution to this problem is a massless up quark [12]. Although this is not favored by experimental evidence so far, it increases the interest in the precise measurement of the u quark mass. The strong CP problem is related to the CP transformation, a combination of two transformations: one obtained by exchanging particles with their antiparticles (C, charge conjugation) and the other by mirroring the physical system (P, parity). A transformation is said to be a symmetry if the physics description is the same for the process and the transformed process. The process is said to obey the symmetry. C and P were first thought to be symmetries obeyed by all particles and their interactions, until it was shown in an experiment conducted in 1956 that P symmetry was violated in weak interactions [13]. The combined CP symmetry was then proposed as a symmetry that would be conserved also by the weak interaction, but this was also shown not to be true when an experiment measured C and CP violation with K mesons [1]. With CP shown not to be a strict symmetry for the weak interaction, the question arises of why no CP violation has been seen in the strong interaction. In the formulation of QCD, it is possible to include a parameter that, being different from zero, would imply P and CP violation [12]. Since no experimental evidence exists for CP violation in the strong interaction, this suggest that this parameter should be either zero or very small (so small that the experiments so far were not precise enough to see the CP violation). To have a parameter equal to zero or very small just by chance is not intellectually satisfying and is known as a fine-tuning problem. Therefore, other explanations have been proposed for the non-observation of CP violation: for example, that there exists a new type of particle, called axion, that would make CP violation unobservable no matter the size of the parameter; or that the up quark would be massless, which would also make CP violation unobservable [12]. So far no axion has been found [7].. Thesis outline This thesis concerns the experimental measurement of the decay of the etameson to three pions, η → π + π − π 0 . This is a process sensitive to the difference between the u and d quark masses, and thus this thesis contributes to the determination of the quark masses. This contribution is not direct, but the results presented in this thesis, the most precise measurement of the Dalitz plot distribution of the η → π + π − π 0 decay to date, can be used together with theoretical calculations to constrain the masses of the light quarks. The thesis is divided into 6 chapters as follows. Chapter 1 gives a motivation for the η → π + π − π 0 measurement, including a more detailed theoretical 12.

(13) background and previous experimental results. The Dalitz plot and Dalitz plot distribution are also explained in this chapter. Chapter 2 presents an overview of the accelerator facility where the experiment was conducted and of the detector used. In chapter 3, the analysis is described: event selection, reconstruction and background rejection. Chapter 4 concerns the results from this thesis: the Dalitz plot parameters, as well as the charge asymmetries, and their systematic uncertainties. Chapter 5 gives an alternative stating of the results, in the form of the acceptance corrected, arbitrarily normalized Dalitz plot distribution. This alternative stating does not include systematic uncertainties, but it can directly be used for comparison with other experiments or theoretical calculations. In chapter 6, the results are discussed, with emphasis on the comparison with previous experiments, and a conclusion about the work performed in this thesis is presented.. 13.

(14) 1. Motivation for Studying η → π +π −π 0. This chapter gives the motivation for the η → π + π − π 0 Dalitz plot measurement. It starts with a short introduction to Chiral Perturbation Theory (ChPT), and how this theory can be used to relate the decay width of η → π + π − π 0 to the quark masses. The next part introduces the Dalitz plot, the kinematic variables used and how to calculate the boundary. Then come some theory updates for the η → π + π − π 0 that go beyond ChPT and finally a summary of previous experimental results.. 1.1 Chiral Perturbation Theory The introduction on chiral perturbation theory presented here follows Scherer and Schindler’s lecture notes [14], although in a simplified and condensed way. QCD is the quantum field theory of the strong interaction, but due to the running of the strong coupling constant and to the confinement of quarks at low energies, it is impractical for use at low energies. The perturbative methods of calculating QCD processes, which are successful at high energies, cannot be applied since the strong coupling constant cannot be regarded as a small expansion parameter. Instead, one can use Effective Field Theories (EFTs), and the low-energy EFT of QCD is chiral perturbation theory. In general, EFTs approximate a fundamental theory at low energies, and simplify calculations since the full theory need not be used. The fundamental theory needs to have one (or more) energy scales (usually denoted Λ), and the EFT works for energies that are small compared to this scale. The physics of the fundamental theory at higher energies is included in the constants of the EFT, which in principle can be calculated from the full theory. The EFT uses only degrees of freedom relevant for the energy regime in question. The correspondence of the physical observables calculated in the EFT to the ones from the fundamental theory is guaranteed by Weinberg’s conjecture [15]. According to this, for the correspondence to be true, one needs the most general Lagrangian consistent with all symmetries of the fundamental theory. This could mean a Lagrangian with infinitely many terms, which would make predictions impossible. But if one is only interested in a certain accuracy of the EFT, i.e., the results from the EFT need only be the same as the fundamental theory up to a certain numerical digit, then not all the terms in the general Lagrangian need be taken into account (note: the energies and momenta involved 14.

(15) must also be small compared to the scale Λ). Which terms are important is determined by the relevant power counting. Chiral perturbation theory builds on the chiral symmetry of massless quark QCD, and the ChPT Lagrangian also obeys Lorentz invariance, charge conjugation and parity invariance. There are two variants: 2- and 3-flavor ChPT. In the first case, the u and d quarks are considered massless and the s quark as heavy, and the relevant degrees of freedom are the pions. In the second case, the u, d and s quarks are considered massless and the degrees of freedom are the pions, the kaons and the eta-meson (π, K, η). ChPT can thus be used to describe interactions between π’s, K’s and η’s, and including the weak or electromagnetic interaction appropriately (as external fields coupling to the ChPT degrees of freedom) also processes like π 0 → γγ or π + → μ + νμ . In both the 2- and 3-flavor case, the quark masses are actually taken into account as a perturbation, and lead to an explicit chiral symmetry breaking of the Lagrangian. The limit of zero quark masses is referred to as the chiral limit. The power counting in ChPT is done in powers of energy, momenta and quark masses. The scale of chiral symmetry breaking Λχ ∼ 1 GeV determines the region of applicability of ChPT, but also the appearance of other particles not included as degrees of freedom signals the breakdown of ChPT. For example the ρ-meson, with a mass of mρ = 770 MeV ∼ Λχ indicates that ChPT will not work at these energies. 2-flavor ChPT in general converges better than 3flavor ChPT, which is expected since the s quark is significantly heavier than the u and d quarks [7], and thus approximating its mass to zero will require more corrections.. 1.1.1 Chiral Symmetry The quark part of the QCD Lagrangian can be written as: LQCD, quarks =. ∑. f =u,d,s,c,b,t. / − m f )q f q¯ f (iD. (1.1). where q¯ f and q f are the quark fields (with implicit color and spinor indices), / the gauge derivative. The gauge derivative m f the mass of quark flavor f and D / = γ μ Dμ = γ μ ∂μ + igγ μ Aμ , where includes the gluon-field matrix Aμ in D μ μ = 0, 1, 2 or 3, γ are the gamma matrices, g is the strong coupling constant and repeated indices are summed over. The quark flavors can be divided into three light quarks (u, d, s) and three heavy quarks (c, b,t), with the light quarks all having masses smaller than ΛQCD . Concentrating on the light quarks, define a quark flavor vector q† = (u† , d † , s† ) and consider the projection operators: 1 PR = (I + γ 5 ) = PR† , 2. 1 PL = (I − γ 5 ) = PL† 2. (1.2) 15.

(16) where I is the identity matrix and γ 5 the fifth gamma matrix. These operators project the quark fields into right- and left-handed fields: qR = PR q. such that. q¯R = q†R γ 0 = (PR q)† γ 0 = q† PR γ 0 = q† γ 0 PL = qP ¯ L (1.3). qL = PL q. such that. q¯L = qP ¯ R. (1.4). where the anti-commutation relation for γ 5 is used ({γ 5 , γ μ } = 0). The light-quark Lagrangian can then be written in terms of the right- and left-handed quarks: / − M)q LQCD, light quarks = q(i ¯ D / R + q¯L iDq / L − q¯R MqL − q¯L MqR = q¯R iDq. (1.5). where M is a 3 × 3 diagonal matrix with the quark masses. As can be seen, the right- and left-handed quarks are only coupled by the mass part of the Lagrangian. Since the quark masses are light compared to ΛQCD , they can be approximated to zero (chiral limit). In this case, the Lagrangian is invariant under transformations of the right- and left-handed quarks separately, according to: qR → UR qR qL → UL qL. (1.6). where UR ,UL are 3×3 special unitary matrices (i.e., UR ,UL ∈ SU(3)), acting in 0 / = q¯R iDq / R + q¯L iDq / L ¯ Dq flavor space. So the Lagrangian LQCD, light quarks = qi is invariant under transformations of the group SU(3)R × SU(3)L . This invariance is called chiral symmetry. The 2-flavor case of ChPT corresponds to considering only the u and d quarks as light, i.e., in equation 1.5 only the u and d quarks are included. In this case, in the limit where both these quark masses go to zero, the Lagrangian is invariant under SU(2)R × SU(2)L .. 1.1.2 Ingredients for the ChPT Lagrangian Before introducing the ChPT Lagrangian, the concepts of spontaneous symmetry breaking and Goldstone bosons are needed. Spontaneous symmetry breaking is when the ground state of a theory is not symmetric under the full symmetry group of the Lagrangian. According to the Goldstone theorem [2], a broken continuous symmetry, i.e., a continuous symmetry of the Lagrangian that is not a symmetry of the ground state, gives rise to massless, spin-less bosons called Goldstone bosons. There is one Goldstone boson for each generator of the broken symmetry, and these bosons have the same quantum numbers as the generators. 16.

(17) Even in the case of a spontaneously broken approximate symmetry of the Lagrangian, spin-less bosons appear, but in this case they are not massless (but usually light) and are instead called pseudo-Goldstone bosons. An approximate symmetry of the Lagrangian implies a symmetry which is explicitly broken in the Lagrangian, but only by a small parameter. For example, the Lagrangian of equation 1.5 has an approximate chiral symmetry, since it would have full chiral symmetry if the quark masses were zero, but these are nonzero and small, i.e., the quark masses are the small parameters explicitly breaking the chiral symmetry. In the case of QCD, the broken symmetry is suggested by the low lying hadron spectrum to be SU(3)A [3]. The symmetry group SU(3)R × SU(3)L is equivalent to SU(3)V × SU(3)A , where transformations according to SU(3)V imply: qR → UqR qL → UqL. (1.7). where U ∈ SU(3) (i.e., the left- and right-handed quarks are transformed in the same way); and transformations according to SU(3)A imply: qR → UqR qL → U † qL. (1.8). where U ∈ SU(3). From the symmetry of the spectrum one can infer the symmetry of the ground state. In the hadron spectrum, one can identify octets (for mesons and baryons) and decuplets (for baryons) consistent with SU(3)V flavor symmetry and the assumption that mesons consist of quark and anti-quark while baryons consist of three quarks. If the full SU(3)R × SU(3)L symmetry was realized in the spectrum, one would expect degenerate octets (or decuplets) with opposite parity. The fact that this is not realized in the spectrum, e.g. there is no low-lying octet of negative parity 12 -spin baryons, implies a breaking of the full symmetry, in fact, a breaking of the SU(3)A symmetry. The broken SU(3)A symmetry implies 8 pseudo-Goldstone bosons, which are spin-less, nearly degenerate low-mass states. These can be identified with the octet of light pseudo-scalar mesons: the three π’s, the four K’s and the η. The pseudo-Goldstone bosons are the degrees of freedom used in ChPT, and they appear in the Lagrangian in the SU(3) matrix [14]: φ (x). i F 0. U(x) = e. √ + √ +⎞ π 0 + √13 η 2π 2K √ 0⎟ ⎜ √2π − 1 0 −π + √3 η 2K ⎠ with φ = ⎝ √ − √ 0 2K 2K¯ − √23 η. (1.9). ⎛. (1.10). 17.

(18) where the bosonic fields π 0 , π + , π − , K + , K − , K 0 , K¯ 0 and η all depend on the space-time coordinate x and have dimension of energy, and F0 is the pion decay constant in the chiral limit (which makes the exponential dimensionless). In order to get the most general Lagrangian, the globally chiral invariant 0 Lagrangian LQCD, light quarks is upgraded to a locally chiral invariant one by introducing external fields vμ , aμ , s and p. These fields transform under Lorentz transformation as vector, axial-vector, scalar and pseudo-scalar respectively. In fact, instead of these fields, the combinations rμ = vμ + aμ , lμ = vμ − aμ , M = s + ip and M † = s − ip are used. The extended Lagrangian: Lext, light quarks = q¯R iγ μ Dμ qR + q¯R γ μ rμ qR +q¯L iγ μ Dμ qL + q¯L γ μ lμ qL. (1.11). −q¯R M qL − q¯L M qR †. is invariant under the local SU(3)R × SU(3)L transformation: qR (x) → UR (x)qR (x), q¯R (x) → q¯R (x)UR (x)† , qL (x) → UL (x)qL (x), q¯L (x) → q¯L (x)UL (x)† , rμ (x) → UR (x)rμ (x)UR (x)† +UR (x)i(∂μ UR (x)† ),. (1.12). lμ (x) → UL (x)lμ (x)UL (x)† +UL (x)i(∂μ UL (x)† ), M → UR (x)M UL (x)† , M † → UL (x)M †UR (x)† , where UR (x),UL (x) ∈ SU(3) and depend on the space-time coordinate x. Note that putting vμ = aμ = p = 0 and s = diag(mu , md , ms ) = M one recovers the Lagrangian of equation 1.5. The Lagrangian of the effective field theory, ChPT, will use the same external fields rμ , lμ , M and M † , with the same transformation properties under the local SU(3)R × SU(3)L transformation, as well as the Goldstone boson field matrix U(x), which transforms as U(x) → UR (x)U(x)UL (x)† . The definition of the chiral gauge covariant derivative of an object A, which transforms as A(x) → UR (x)A(x)UL (x)† , is: Dμ A = ∂μ A − irμ A + iAlμ. (1.13). and transforms as Dμ A(x) → UR (x)(Dμ A(x))UL (x)† . The field strength tensors: fRμν = ∂μ rν − ∂ν rμ − i[rμ , rν ], fLμν = ∂μ lν − ∂ν lμ − i[lμ , lν ] 18. (1.14).

(19) are also needed, and they transform as: fRμν (x) → UR (x) fRμν (x)UR (x)† ,. (1.15). fLμν (x) → UL (x) fLμν (x)UL (x)† .. Locally chiral invariant Lagrangians can be built out of flavor traces of products of the form AB† , where A and B transform as A(x) above. This is easily seen using the cyclicity of traces:. Tr(AB† ) →Tr UR (x)A(x)UL (x)† (UR (x)B(x)UL (x)† )† = . Tr UR (x)A(x)UL (x)†UL (x)B(x)†UR (x)† = (1.16) . Tr UR (x)A(x)B(x)†UR (x)† = . . Tr UR (x)†UR (x)A(x)B(x)† = Tr A(x)B(x)† . With the fields introduced, examples of entities transforming as A(x) are: U(x), Dμ U(x), Dν Dμ U(x), M (x), fRμν (x)U(x) and U(x) fLμν (x). There is an infinite amount of these entities, and thus an infinite amount of different invariant traces that one could construct. To decide which terms are needed, a power counting is introduced. Let q be a small energy or momentum, of the order of the masses of the pseudo-Goldstone bosons. Derivatives are of order O(q), so to be consistent, the fields rμ and lμ are also considered O(q) and thus also the gauge covariant derivative Dμ . The field strengths fRμν and fLμν are then of O(q2 ). The boson field matrix is considered O(q0 ), while M is of O(q2 ), since the quark masses can be related to the square of the pseudo-Goldstone boson masses, see section 1.1.4.. 1.1.3 ChPT Lagrangian The lowest-order Lagrangian in ChPT is of O(q2 ). At O(q0 ) only constant terms can contribute to the Lagrangian, e.g. Tr(UU † ) = 3, and these have no information on the dynamics of the fields. There is no term at O(q), or in fact at any O(qn ) where n is odd. The only building block with odd order is Dμ , but since Lorentz invariance requires Lorentz indices to be contracted, the derivatives will always appear in pairs and thus give terms of even order. The lowest non-trivial Lagrangian is thus of O(q2 ). The candidate hermitian structures of the Lagrangian are: . . . Tr (Dμ U)† Dμ U , Tr U † M + M †U and iTr U † M − M †U . (1.17) The last structure is forbidden by parity conservation: under the parity transiφ. −i φ. formation, M † and U = e F0 → e F0 = U † , so that ip → s− ip = † M =† s + iTr U M − M U → iTr UM † − M U † = −iTr U † M − M †U . At the considered order, charge conjugation invariance does not impose any more 19.

(20) constraints, and the Lagrangian is: L2,ChPT =. F2 . F02 Tr (Dμ U)† Dμ U + 0 · 2B0 Tr U † M + M †U 4 4. (1.18). where F0 and B0 are the low-energy constants at this order, F0 is related to the pion decay and B0 to the quark condensate. Any process in ChPT O(q2 ) is calculated by tree level diagrams with vertices from L2,ChPT . Loop diagrams appear first at O(q4 ). One complication that appears with loop diagrams is the fact that these diverge. In renormalizable theories, the infinities arising from the loops are compensated with counter terms. ChPT is in general not renormalizable, but it is renormalizable order by order, as the higher order Lagrangians contain the counter terms for the loops of the lower order Lagrangians. For example, one-loop diagrams from L2,ChPT are compensated by terms in L4,ChPT , by a suitable redefinition of the low-energy constants of L4,ChPT . With a suitable renormalization in place, the order at which a loop diagram contributes can be understood using the following contributions to the power counting: • vertices from L2n,ChPT each contribute q2n , e.g. vertices from L2,ChPT contribute q2 , vertices from L4,ChPT contribute q4 ; • each pseudo-Goldstone boson propagator contributes q12 ; • each independent loop contributes q4 (because it introduces a momentum integration in four dimensions). This can be summarized in a formula for the “chiral dimension” D of an arbitrary diagram which contributes at order qD : ∞. D = 4Nl − 2N p + ∑ 2nNv,2n. (1.19). n=1. where Nl is the number of independent loops, N p the number of propagators and Nv,2n the number of vertices from L2n,ChPT . In a connected diagram, the number of loops, propagators and vertices are not independent of each other but obey the relation Nl − N p + ∑∞ n=1 Nv,2n = 1 and with this, equation 1.19 can be rewritten as [15]: ∞. D = 2Nl + 2 + ∑ 2(n − 1)Nv,2n .. (1.20). n=1. From this equation it is easy to see that the lowest value for D is 2, when there are no loops, no vertices with n > 1 and an arbitrary number of vertices with n = 1 (i.e., from L2,ChPT ). As an example of equation 1.20, consider a loop diagram of 2 → 2 pseudoGoldstone boson scattering, with two L2,ChPT vertices connected by two propagators, which has one independent loop, see figure 1.1. According to the 20.

(21) Figure 1.1. Feynman diagram of 2 → 2 pseudo-Goldstone boson scattering with one independent loop. The vertices indicated by a dot are from L2,ChPT .. power counting above, this diagram has chiral dimension D = 4, i.e., it contributes at O(q4 ). At next to leading order (NLO), i.e., at O(q4 ), both one-loop diagrams with an arbitrary number of vertices from L2,ChPT and tree-level diagrams with one vertex from L4,ChPT need to be taken into account. The NLO Lagrangian L4,ChPT has 12 low-energy constants, and can be written as [16]: 2 . L4,ChPT =L1 Tr (Dμ U)† Dμ U + L2 Tr (Dμ U)† Dν U Tr (Dμ U)† Dν U . +L3 Tr (Dμ U)† (Dμ U)(Dν U)† Dν U. +L4 Tr (Dμ U)† Dμ U Tr χ †U + χU †. 2 . +L5 Tr (Dμ U)† Dμ U(χ †U + χU † ) + L6 Tr χ †U + χU †. 2. +L7 Tr χ †U − χU † + L8 Tr χ †U χ †U + χU † χU † . −iL9 Tr fRμν (Dμ U)(Dν U)† + fLμν (Dμ U)† Dν U μν +L10 Tr U † fRμν U fL . μν μν +H1 Tr fRμν fR + fLμν fL + H2 Tr χ † χ , (1.21) where M is now encoded in χ = 2B0 M . Of the 12 low-energy constants, 10 (L1 , . . . , L10 ) have physical significance. The remaining 2 parameters (H1 and H2 ) relate to terms including only external fields, so they have no physical significance, although they are needed for the renormalization of the one-loop diagrams. At next to next to leading order (NNLO), O(q6 ), the Lagrangian L6,ChPT is needed. This Lagrangian has 94 low-energy constants, of which 4 concern only external fields and have no physical significance [17]. The low-energy constants of ChPT, as for any EFT, contain the physics of the original theory at energies not covered by the EFT. In principle, these could be calculated from QCD, but our inability to solve QCD at low energies is one 21.

(22) of the things prompting the use of an EFT like ChPT in the first place. Nevertheless, lattice QCD1 [18] can be used to calculate the low-energy constants. These constants can also be fixed from experimental data, i.e., some data is used to calculate these constants, and once they are fixed, ChPT has predictive power for other processes. At present, the accuracy of lattice QCD is for most low-energy constants not competitive with determinations from experimetnal data, but it can be used as a cross-check or to determine low-energy constants that are not easily extracted from experiment. For a recent determination of low-energy constants using both experimental data and lattice results see [19].. 1.1.4 Quark Masses from ChPT The quark masses m f in the QCD Lagrangian (equation 1.1) are free parameters of the theory. Since the quarks are confined in hadrons, their masses cannot be measured directly. For the light quarks, the quark mass term appearing in the ChPT Lagrangians (equation 1.18, equation 1.21, etc.) enables the calculation of quark mass ratios from the pseudo-Goldstone bosons’ masses and interactions. More information, for example from lattice QCD, is needed to get the absolute value of the quark masses. At leading order, the masses of the pseudo-Goldstone bosons can be directly related to the quark masses, by looking at the mass terms of L2,ChPT . Expanding U and U † in powers of the field matrix φ : 1 i φ − 2 φ2 +..., F0 2F0 i 1 U† = I − φ − 2 φ2 + ..., F0 2F0 U =I+. (1.22). setting rμ = lμ = p = 0, s = M (the quark mass matrix), and keeping only terms up to φ 2 , the Lagrangian can be written as: L2,ChPT = L2,ChPT,kin + L2,ChPT,mass. 1 Lattice. (1.23). QCD is a numerical method based on the discretization of QCD on a space-time grid, using Monte Carlo simulations to sample from possible configurations in QCD.. 22.

(23) where L2,ChPT,kin corresponds to the kinetic terms and L2,ChPT,mass to the mass terms. The kinetic part of the Lagrangian is:

(24) F02 i i μ I+ φ Tr ∂μ I − φ ∂ L2,ChPT,kin = 4 F0 F0. 1. = Tr ∂μ φ ∂ μ φ 4 1. = 2∂μ π 0 ∂ μ π 0 + 2∂μ η∂ μ η + 4∂μ π + ∂ μ π − + 4. 4∂μ K + ∂ μ K − + 4∂μ K 0 ∂ μ K¯ 0 1 1 = ∂μ π 0 ∂ μ π 0 + ∂μ η∂ μ η + ∂μ π + ∂ μ π − + ∂μ K + ∂ μ K − + 2 2 ∂μ K 0 ∂ μ K¯ 0 . (1.24) These are the usual kinetic terms of scalar hermitian fields (π 0 and η) and scalar non-hermitian fields (π + , π − ; K + , K − and K 0 , K¯ 0 ). The mass terms are: F02 iφ φ2 iφ φ2 · 2B0 Tr(M − M − 2 M + M + M − M 2 ) 4 F0 F0 2F0 2F0 B 0 = F02 B0 Tr(M) − Tr(Mφ 2 ) 2

(25) B0 η 2 0 + − + − =C− mu (π + √ ) + 2π π + 2K K 2 3. η 2 + − 0 0 ¯0 √ ) + 2K K md + 2π π + (−π + 3. 4η 2 + 2K + K − + 2K 0 K¯ 0 + ms . 3 (1.25). L2,ChPT,mass =. Dropping the constant term C, of no physical importance, and collecting terms of the same fields gives. B0 2 (π 0 )2 (mu + md ) + √ π 0 η(mu − md ) L2,ChPT,mass ≈ − 2 3 + − + 2π π (mu + md ) + 2K + K − (mu + ms ) (1.26) 1 1 4 + 2K 0 K¯ 0 (md + ms ) + η 2 ( mu + md + ms ) . 3 3 3 To get the masses of the pseudo-Goldstone bosons, the normal form of the mass term of a scalar hermitian field (− 12 m2a a2 ) and of a scalar non-hermitian 23.

(26) fields (−m2a a† a) is used. Neglecting the π 0 − η mixing, the masses can be read directly from equation 1.26: m2π = m2π 0 = m2π ± = B0 (mu + md ), m2K ± = B0 (mu + ms ), (1.27). m2K 0 = m2K¯ 0 = B0 (md + ms ), 1 m2η = B0 (mu + md + 4ms ). 3. These equations are called the Gell-Man, Oakes, Renner relations [20]. As can be seen, the quark masses are of the order of the pseudo-Goldstone boson masses squared, so assigning O(q2 ) to M is consistent. To be able to use the physical meson masses, the electromagnetic interaction and its effect on the masses also has to be taken into account. According to Dashen’s theorem [21], the electromagnetic contribution to the mass difference of pions and kaons is the same at leading order, i.e. (m2K ± − m2K 0 )E.M., LO = (m2π ± − m2π 0 )E.M., LO = ΔE.M. ⇔ (m2K ± − m2K 0 )E.M. − (m2π ± − m2π 0 )E.M. = O(e2 M).. (1.28). Using also the fact that the neutral particles do not get any electromagnetic corrections at lowest order and including the unknown ΔE.M. , equation 1.27 gives m2π 0 = B0 (mu + md ), m2π ± = B0 (mu + md ) + ΔE.M.,. (1.29). m2K ± = B0 (mu + ms ) + ΔE.M. , m2K 0 = m2K¯ 0 = B0 (md + ms ). With this equation, the quark mass ratios. mu md. and. ms md. were calculated in [22]:. 2m2π 0 − m2π ± + m2K ± − m2K 0 mu = = 0.56 md m2K 0 − m2K ± + m2π ± m2 0 + m2K ± − m2π ± ms = K2 = 20.2. md mK 0 − m2K ± + m2π ±. (1.30). Including the next order in chiral perturbation theory is of course more complicated. Gasser and Leutwyler [16] noted that the ChPT NLO corrections are the same for the two pseudo-Goldstone bosons’ squared mass ratios m2 0 −m2 ± K K m2K −m2π. 24. m2K m2π. and. , where m2K is the isospin averaged kaon mass, i.e., the mass of the.

(27) kaons if mu = md . The ratios are:. m2K ms + mˆ = 1 + ΔM + O(M 2 ) 2 mπ 2mˆ 2 2. mK 0 − mK ± md − mu = 1 + ΔM + O(M 2 ) 2 2 ms − mˆ mK − mπ. (1.31). where the average u, d quark mass mˆ = 12 (mu + md ) is used, and ΔM is the same NLO correction (for the exact formula see [16]). The leading order part of this result is easily seen with equation 1.27 and setting mu = md = mˆ for the cases when a charge of the meson is not specified. A new ratio, Q2 , which does not receive a correction at NLO, can be constructed out of the ratios in equation 1.31: m2 0 −m2 ± K. K. m2K 0 − m2K ± m2π 1 m2K −m2π = = 2 2 2 m2K Q mK − m2π mK m2π. md −mu 2) 1 + Δ + O(M M m −mˆ = mss +mˆ 2 2mˆ (1 + ΔM + O(M )) = = = = ⇔ Q2 =. md −mu ms −mˆ ms +mˆ 2mˆ. . 1 + O(M 2 ). (1.32). md − mu 2mˆ 1 + O(M 2 ) ms − mˆ ms + mˆ . md − mu 1 2 (mu + md ) 1 + O(M 2 ) 2 2 ms − mˆ 2. m2d − m2u 2 1 + O(M ) m2s − mˆ 2. m2s − mˆ 2 m2K − m2π m2K 2 1 + O(M ) = . m2d − m2u m2K 0 − m2K ± m2π. With Dashen’s theorem, assuming that the mass difference between charged and neutral pions is exclusively due to electromagnetic effects, and inserting m2K = 12 (m2K 0 + m2K ± ) (only the QCD contribution of the masses), one can calculate Q2 from the measured meson masses: Q2D =. (m2K 0 + m2K ± − m2π ± − m2π 0 )(m2K 0 + m2K ± − m2π ± + m2π 0 ) 4(m2K 0 − m2K ± + m2π ± − m2π 0 )m2π 0. .. (1.33). Inserting the known values of the masses from [7] gives QD = 24.3. 25.

(28) Knowing Q provides an elliptical constraint on the quark mass ratios, as can be seen by rewriting equation 1.32: m2d − m2u 1 = Q2 m2s − mˆ 2 ⎛ ⎞ m2u 1 − 2 2 m ⎜ md ⎟ = d2 ⎝ ⎠ ms 1 − mˆ 22 m s. m2d 1 − m2s 1 m2 1 m2 ⇔ 2 2s = 1 − 2u Q md md ≈. ⇔. m2u m2d. (1.34). 1 m2s m2u + =1 Q2 m2d m2d. where one has used the fact that the u and d quarks are much lighter than mu plane, the s quark. The last line is the equation of an ellipse in the mms vs m d d with semi-axes Q and 1. This ellipse is called the Leutwyler ellipse [8] and is shown for the value Q = QD in figure 1.2, together with the quark mass ratios from equation 1.30. 25. 20. 15. ms md. 10. 5. QD Weinberg 77. 0 0. 0.2. 0.4. 0.6. mu md. 0.8. 1. Figure 1.2. The Leutwyler ellipse [8] for Q = 24.3 and the values of the quark mass ratios from Weinberg [22].. Weinberg’s result [22], supplemented by Leutwyler’s ellipse [8], means that the u quark mass is non-zero, but to what accuracy? To further test the un26.

(29) derstanding of QCD and the standard model at low energies, it is useful to determine these quantities in alternative ways. The η → π + π − π 0 decay can be used for an alternative determination of Q. The η → π + π − π 0 decay The amplitude of the η → π + π − π 0 decay can be calculated in ChPT, at leading order using equation 1.18, by expanding the matrix U up to order φ 4 . The simplified result is [23]: 3(s − s0 ) B0 (mu − md ) √ 1+ 2 (1.35) ALO (s,t, u) = mη − m2π 3 3F02 where s,t, u are the Mandelstam variables and 3s0 = (s + t + u) = m2η + m2π 0 + 2m2π ± . The Mandelstam variables are defined similarly as for 2-to-2 scattering: s = (Pπ + + Pπ − )2 = (Pη − Pπ 0 )2 t = (Pπ 0 + Pπ − )2 = (Pη − Pπ + )2. (1.36). u = (Pπ + + Pπ 0 )2 = (Pη − Pπ − )2 with PX being the four-momentum of particle X. As can be seen from the definition of s0 above, the Mandelstam variables are not all independent. A 1to-3 decay of spin-less particles has only two independent variables, and it is enough to use two of the Mandelstam variables. The amplitude is proportional to the quark mass difference mu − md , so this decay would not occur if mu = md . Equation 1.35 can be rewritten in terms of Q. Note that, at LO ChPT, using equations 1.27 and 1.32: B0 (mu − md ) = −(m2K 0 − m2K ± ) = −. 1 m2K 2 (m − m2π ) Q2 m2π K. so that the amplitude becomes 1 m2 m2 − m2 ALO (s,t, u) = − 2 K2 K√ 2π Q mπ 3 3F0. . 3(s − s0 ) 1+ 2 mη − m2π. (1.37). .. (1.38). Using the value of Q = 24.15 and integrating over phase space gives the LO result for the decay width ΓLO = 66 eV [23]2 . The NLO result is again more involved, and calculations with the same value of Q give ΓNLO = (160 ± 50) eV [23], later updated to ΓNLO = (168 ± 50) eV [8]. Both results are quite far from the experimental value Γexp = (300 ± 11) eV [7]. A full NNLO calculation has also been performed [24], and using the same value of Q gives ΓNNLO = 298 eV 3 . These results show at best a slow convergence of the SU(3) Q = 24.15 is the value of QD from equation 1.33 at the time [23] was written. value for Γ is not quoted in this reference, but using Q = 24.15 and ms /mˆ = 27.4 together with their results gives Γ = 298 eV [25].. 2. 3A. 27.

(30) chiral expansion and that the theoretical uncertainty estimate is not under control (cf. the error in the NLO result with the NNLO result). It turns out that the biggest part of the corrections at NLO comes from final state interactions between the pions [23]. But Dashen’s theorem is also known to be a leading order result, and the corresponding corrections should be taken into account when calculating the value of Q. Instead of relying on Dashen’s theorem and its corrections to predict the decay width of η → π + π − π 0 , the experimental decay width can be used to extract the value of Q. For this approach, one needs a good theoretical description of the decay dynamics together with accurate experimental knowledge of the decay width. The theoretical description of the decay should be checked with accurate experimental measurements of the Dalitz plot distribution.. 1.2 Dalitz Plot The physical region in a 1-to-3 body decay is called Dalitz plot [26] and is usually defined using two of the Mandelstam variables from equation 1.36, but it can also be defined using variables linearly related to these. The Dalitz plot distribution is the decay amplitude squared in the Dalitz plot, and can be written as a function of the same variables. Since there are only two independent variables in a 1-to-3 decay of spin-less particles, this distribution contains all the information on the dynamics of the decay. Considering four-momentum conservation, the boundary of the Dalitz plot can be calculated. The equation for the boundary of the Dalitz plot in the s − t plane can be written for t in terms of s as [26]: t ± = m2π 0 + m2π ± − 1. 1 (s − m2η + m2π 0 )(s + m2π ± − m2π ± ) 2s 1. ∓λ 2 (s, m2η , m2π 0 )λ 2 (s, m2π ± , m2π ± ) ⇔ 1 1 1 t ± = m2π 0 + m2π ± − (s − m2η + m2π 0 )s ∓ λ 2 (s, m2η , m2π 0 )λ 2 (s, m2π ± , m2π ± ) 2s (1.39) where the Källén function λ is given by √ √ √ √ λ (x, y, z) = x − ( y + z)2 x − ( y − z)2 = (x − y − z)2 − 4yz. (1.40). = x2 + y2 + z2 − 2(xy + xz + yz), mi is the mass of particle i and the ± in the superscript of t stands for which of the equations to use (the one with − or + before the Källén functions, respectively). The boundary is shown in figure 1.3, with different line types for t + and t − . 28.

(31) 2. t(GeV ). 0.18 0.16. 0.14 0.12 0.1 0.08 0.08. 0.1. 0.12. 0.14. 0.16. 0.18 s(GeV2). Figure 1.3. The Dalitz plot boundary in the s − t plane, where the dashed line corresponds to t + and the full line to t − .. 1.2.1 η → π + π − π 0 Dalitz Plot Variables For the η → π + π − π 0 decay, historically the X and Y variables are used to construct the Dalitz plot. These dimensionless variables are defined in the η rest frame as: √ Tπ + − Tπ − 3 Qη 3T 0 Y = π −1 Qη with Qη = Tπ + + Tπ − + Tπ 0 = mη − 2mπ ± − mπ 0 X=. (1.41) (1.42) (1.43). and Ti the kinetic energy of particle i (in the η rest frame). These variables are related to the Mandelstam variables defined in the previous section by calculating the energies (Ex ) of the decay particles (x) in the η rest frame: s = (Pη − Pπ 0 )2 ⇔ s = m2η + m2π 0 − 2Pη Pπ 0 ⇔ s = m2η + m2π 0 − 2mη Eπ 0 ⇔ Eπ 0 =. (1.44). m2η + m2π 0 − s 2mη 29.

(32) and similarly Eπ − =. m2η + m2π ± − u 2mη. (1.45). Eπ + =. m2η + m2π ± − t . 2mη. (1.46). Since the kinetic energy is defined as T = E − m, this can be substituted in equations 1.41 and 1.42 for √ 3 (u − t) (1.47) X= 2mη Qη 3 (mη − mπ 0 )2 − s − 1. (1.48) Y= 2mη Qη Dalitz plot boundary Equation 1.39, together with equations 1.47 and 1.48, allows to calculate the values of the X and Y variables for all t and s (which also define u) at the Dalitz plot border, and thus to calculate the border in the variables X and Y . For these variables though, a more intuitive way can be used to calculate the boundary of the Dalitz plot. In the η rest frame, the pions’ three-momenta sum to zero (pπ 0 +pπ + +pπ − = 0), and thus for pπ 0 as a function of the other pions’ momenta (momenta and three-momenta are used interchangeably) |pπ 0 |2 = p2π 0 = p2π + + p2π − + 2pπ + ·pπ − p2π 0 = p2π + + p2π − + 2pπ + pπ − cos(θπ + ,π − ). (1.49). where θπ + ,π − is the angle between the three-momenta of the charged pions and the simplified notation for the modulus of the momenta |pπ + | = pπ + is used. The physical region is delimited by −1 ≤ cos(θπ + ,π − ) ≤ 1, and the border corresponds to the extreme cases, the equalities. For any values of the modulus of the three-momenta of the three pions, it is easy to check if this momentum configuration is inside the Dalitz plot or not, by checking if |p2π 0 − p2π + − p2π − | ≤ 2pπ + pπ − .. (1.50). Being interested instead in evaluating if a certain point (X,Y ) is inside the Dalitz plot, one can invert the relations 1.41-1.43 to get the kinetic energies of the pions: (1.51). Tπ +. (1.52). Tπ − 30. Qη (Y + 1) 3 √ Qη = (2 −Y + 3X) 6 √ Qη = (2 −Y − 3X). 6. Tπ 0 =. (1.53).

(33) From the kinetic energiesof the pions one can calculate the modulus of their three-momenta by pi = Ti (Ti + 2mi ) and use equation 1.50 to check if the point is inside the Dalitz plot. The shape of the boundary of the Dalitz plot in the X −Y variables can be seen in figure 4.3 on page 76.. 1.2.2 Dalitz Plot Parameters To allow for a direct comparison of the Dalitz plot distribution between theory and experiment, the amplitude squared of the decay is usually parametrized by a polynomial expansion around (X,Y ) = (0, 0): |A(X,Y )|2 N(1 + aY + bY 2 + cX + dX 2 + eXY + fY 3 + gX 2Y + hXY 2 + lX 3 ) (1.54) The experimental or theoretical distribution can then be fit to this formula to extract the parameters a, b, . . ., called the Dalitz plot parameters. Note that c, e, h and l must be zero assuming charge conjugation symmetry which implies that the decay probability should not change if π + and π − are interchanged. This interchange will, however, change the sign of X according to equation 1.41 and therefore all Dalitz plot parameters in terms containing odd powers of X’s must vanish.. 1.3 More η → π + π − π 0 Theory To better understand the η → π + π − π 0 decay, one can go beyond pure pseudoGoldstone boson ChPT. One important part is the calculation of electromagnetic contributions to the decay. Another extension is the use of dispersion relations to calculate the pion rescattering in the final state to all orders in ChPT.. 1.3.1 Electromagnetic Corrections to η → π + π − π 0 The decay η → π + π − π 0 can also occur via the electromagnetic interaction. In fact, this was the initial hypothesis considered for this decay, but it was shown that this electromagnetic transition is forbidden [27, 28], which obviously contradicted the comparatively large experimental decay width. Later on, the framework of ChPT has been used, including the photons as additional degrees of freedom, to calculate the electromagnetic corrections at higher order in ChPT. The photons are included as fields in the covariant derivative, and the photon field appears multiplied by the quark electric charge matrix: ⎛ ⎞ 2 0 0 e (1.55) Qch = ⎝0 −1 0 ⎠ 3 0 0 −1 31.

(34) where e is the proton charge. To keep a consistent chiral counting scheme, i.e., the covariant derivative as O(q), the photon fields are considered as O(1) while e is considered O(q) [29]. The leading order Lagrangian including electromagnetic effects, in addition to the photon terms included in the covariant derivative and in a photon field strength tensor, also gets a term with the quark charge matrix and the pseudo-Goldstone boson fields C ·Tr(QchUQchU † ) [30]. This term, for example, is responsible for the electromagnetic part of the pseudo-Goldstone bosons’ masses: expanding U and U † up to φ 2 and looking only at terms quadratic in pseudo-Goldstone bosons gives the electromagnetic 2 mass terms −2Ce (π + π − + K + K − ). As can be seen, the electromagnetic con2 F 0. tribution to the charged pions’ and charged kaons’ mass is the same at this order, in agreement with Dashen’s theorem, and the contribution to the neutral pseudo-Goldstone bosons’ mass is zero. For the η → π + π − π 0 , at the leading order of the electromagnetic expansion (O(e2 q0 )), the decay is forbidden. Calculations at O(e2 q2 ) in the isospin limit, i.e, with mu = md [29], show only small differences from the pure ChPT O(q4 ) result, both for the decay width but also for the shape in the Dalitz plot. Calculations at order O(e2 q2 ) including the effects of O(e2 (md − mu )) [31] show that the O(e2 (md − mu )) effects are comparable in size to other O(e2 (mq )) effects (where mq is a typical light quark mass), in contradiction to the assumption in the previous O(e2 q2 ) calculation [29]. The total effect of the electromagnetic corrections, however, remains very small, and the conclusion is that the η → π + π − π 0 decay is very sensitive to the strong isospin breaking due to the quark mass difference mu − md . It is worth noting that electromagnetic corrections can also enter indirectly in the constants used for the ChPT calculations. For example, F0 can be identified with the pion decay constant Fπ . The value of Fπ changes by ∼ 1% when including radiative corrections [32], which also changes the η → π + π − π 0 amplitude through equation 1.38.. 1.3.2 Dispersive Calculations The NLO ChPT result showed that the biggest part of the corrections relative to the LO result arise from the rescattering of pions in the final state, specifically the 2-to-2 pion rescattering [23]. These corrections are expected to be considerable even at higher orders and therefore it is useful to have an exact method to calculate them. The dispersive calculations use the decay amplitude’s unitarity, analyticity and crossing symmetry to calculate ππ rescattering to all orders. Assuming that these corrections are separable from other corrections in ChPT and then matching to ChPT yields ChPT predictions corrected for ππ scattering at all orders. 32.

(35) Scattering (or decay) amplitudes, if extended to the complex plane, are analytic functions, except for where they have singularities and discontinuities. Without going into details, dispersion relations build on the following. Using Cauchy’s integral formula, the value of the amplitude is related to a closed integral of the amplitude in the complex plane, where the integral contour avoids discontinuities. The countour of the integral extends to infinity, and assuming the integrand vanishes quickly enough there, this contribution disappears, leaving only the integral along the discontinuities. Thus, the amplitude is related to an integral along its discontinuities in the complex plane. Crossing symmetry takes care of the fact that, if the amplitude is decomposed in terms of amplitudes in the s,t and u channels, where s,t and u are the Mandelstam variables, these amplitudes need to be related, and in fact, singularities in one channel appear as discontinuities in the other channels. In general, the discontinuities are non-linearly related to the scattering amplitudes themselves via the optical theorem [2]. Therefore, a set of integral equations is obtained, which must be solved self-consistently. If the integrand is not vanishing quickly enough at infinity, then so called subtraction is used, giving rise to the subtraction constants of this method. These constants are free parameters and need to be determined from elsewhere, e.g. by comparison to ChPT, by ensuring that the final amplitude matches that of ChPT in some region of the complex energy plane where ChPT converges well. At present, the choice of this region differs for different research groups. Dispersion relations were first used in 1996 for the η → π + π − π 0 decay [33, 34]. Both calculations match the amplitude to the ChPT NLO result and find a small enhancement of the partial decay width compared to this. Newer dispersive calculations have appeared recently, making use of the precise values for the ππ phase shifts which became available (the ππ phase shifts enter in the integrand). The Bern-Lund-Valencia method [9, 25] and the PragueLund-Marseille method [10] differ both in the construction of the amplitude and the determination of the subtraction constants. Both calculations have been matched to NLO ChPT to give predictions of the η → π + π − π 0 decay width and Daltiz plot distribution. However, both methods can instead use as input the experimental Dalitz plot distribution data to extract some of the subtraction constants, and calculate a value for Q. Since the quantity Q appears in the ChPT amplitude and not naturally in the dispersive amplitude, and the experimental Dalitz plot distributions cannot easily provide the absolute normalization, the dispersive treatments still need to match to ChPT for the rest of the subtraction constants to determine Q. This method of matching to data and ChPT has also been used by a third dispersive method [11].. 33.

(36) 1.4 Previous Experimental Results Several experiments have measured the η → π + π − π 0 decay. Here, only the high statistics experiments which measured the Dalitz plot distribution and which extracted at least the b parameter (see equation 1.54) will be mentioned. For references on earlier experiments see [35]. The experiment reported in [36] was performed at the Brookhaven National Laboratory Alternating Gradient Synchrotron (AGS). The protons from the AGS produce a beam of π − used in the experiment in the reaction π − p → nη. The neutron is detected in a forward counter and its momentum is determined by time-of-flight. The π + and π − from the η decay are measured in sonic spark chambers inside a magnetic field. The π 0 is reconstructed through missing mass techniques. For more information on the experimental setup, see [37]. The final Dalitz plot contains 30 000 events, and the results for the Dalitz plot parameters are seen in table 1.1. This experiment found a small charge asymmetry and a corresponding non-zero value for c, labelled as “Gormley(70)c ” in the table. The authors also performed the fit for the Dalitz plot parameters by folding the distribution around X = 0, labeled “Gormley(70)” in the table. The experiment reported in [35] used a similar setup. It was performed at the Princeton-Pennsylvania Accelerator, with a beam of π − produced from accelerated protons. The studied reaction was again π − p → nη, with the neutron’s time-of-flight measured in scintillation detectors and the π + and π − detected in sonic spark chambers [38]. The Dalitz plot contains 80 884 events and the results for the Dalitz plot parameterns are seen in table 1.1, labeled “Layter(73)”. The charge conjugation violating parameter c was assumed to be zero. The value for b is found consistent with zero, unlike the previous experiment. The Crystal Barrel collaboration has measured the η → π + π − π 0 Dalitz plot distribution from 3230 events [39]. The experiment was carried out at the LEAR accelerator, using the reaction pp ¯ → ηπ 0 π 0 . The Crystal Barrel detector consists of two multiwire proportional chambers and a jet drift chamber in a magnetic field, to measure charged particles, surrounded by an electromagnetic calorimeter comprised of 1380 CsI(Tl) crystals, to detect photons. It covers almost the 4π solid angle. The analysis required two tracks measured in the jet drift chamber and six photons in the calorimeter. The η was identified from the π + π − π 0 invariant mass. This analysis only considered the Dalitz plot distribution’s dependence on Y , assuming c = 0 and different values of d. The values of a and b were not sensitive to the assumed values of d. One such fit is reported in table 1.1. The previous measurement with the highest statistics, of 1.34 · 106 events in the Dalitz plot, is by the KLOE collaboration [40]. The detector and setup is the same as for the present analysis (see chapter 2), but a different data set was used. The η originates from the φ → ηγ decay, and all final state particles are 34.

(37) Table 1.1. Summary of Dalitz plot parameter results, both from experiments and theoretical calculations. Row “Gormley(70)c” includes also a result for the c parameter, c = 0.05(2). The rows BLV correspond to the Bern-Lund-Valencia dispersive calculations (both with a value for g), in the row labeled ChPT the dispersive calculation is matched to the ChPT NLO result, while the row labeled KLOE is instead fit to the experimental data from [40]. The row labeled “disp WASA” correspond to the dispersive calculations in [11], where the amplitude has been fit to the WASA data [41]. Experiment. −a. Gormleyc (70)[36] 1.18(2) Gormley(70)[36] 1.17(2) Layter(73)[35] 1.080(14) CBarrel(98)[39] 1.22(7) KLOE(08)[40] 1.090(5)(+19 −8 ) WASA (14)[41] 1.144(18) BESIII(15)[42] 1.128(15)(8). b. d. f. 0.20(3) 0.21(3) 0.03(3) 0.22(11) 0.124(6)(10) 0.219(19)(47) 0.153(17)(4). 0.04(4) 0.06(4) 0.05(3) 0.06(fixed) 0.057(6)(+7 −16 ) 0.086(18)(15) 0.085(16)(9). 0.14(1)(2) 0.115(37) 0.173(28)(21). Calculations. −a. b. d. f. ChPT LO[24] ChPT NLO[24] ChPT NNLO[24] dispersive[33] BLV ChPT[43]. 1.039 1.371 1.271(75) 1.16 1.266(42) g=−0.050(7) 1.077(25) g=−0.037(8) 1.116(32) g=−0.042(9). 0.27 0.452 0.394(102) 0.26 0.516(65). 0 0.053 0.055(57) 0.10 0.047(11). 0 0.027 0.025(160) −0.052(31). 0.126(15). 0.062(8). 0.107(17). 0.188(12). 0.063(4). 0.091(3). BLV KLOE[43] disp WASA[11]. 35.

(38) measured. A kinematic fit is used to improve the resolution, mostly affecting the photon energies. This experiment was the first to report a value for the f parameter, and the results for the parameters are shown in table 1.1. The Dalitz plot distribution is shown in figure 1.4.. Figure 1.4. The Dalitz plot distribution from KLOE(08) (figure from [40]). The distribution is binned with a bin width of 0.125 in X and Y , with a total of 154 bins used.. The two most recent measurements come from the WASA-at-COSY collaboration [41] and the BESIII collaboration [42]. The WASA experiment was carried out at the COSY accelerator, using a proton beam on a deuterium pellet target, with the reaction pd →3 Heη. The WASA detector consists of a forward and a central part. The forward part is comprised of plastic scintillators and a straw tube tracker, and provide energy, time and tracking information for the forward going particles, in this case the 3 He. The central part contains a small drift chamber in a magnetic field, to detect momentum of charged particles, a plastic scintillator and an electromagnetic calorimeter with 1012 CsI(Na) crystals, to measure photon energy. The central detector is used for the decay particles of the mesons, in this case, for the π + , π − and the photons from the π 0 decay. The analysis requires the detection of all final state particles and a kinematic fit is performed with the pd →3 Heπ + π − γγ hypothesis to improve the resolution. The Dalitz plot is constructed out of 1.74 · 105 η event candidates and is binned in 0.2 wide bins in X and Y . The shape of the Dalitz plot is shown in figure 1.5, normalized to the bin with center at X = Y = 0. The results for the Dalitz plot parameters are shown in table 1.1. The BESIII experiment is situated at the BEPCII e+ e− collider in Beijing. For this analysis, the radiative decay of the J/ψ is used as the source of the η (J/ψ → ηγ). The BESIII detector consists of a drift chamber, plastic scintillators (for time-of-flight measurements), an electromagnetic calorimeter of CsI(Tl) crystals and a counter system, all in a magnetic field. All final state particles are detected and a kinematic fit is performed with the 36.

(39) 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 X. 0.5 0 −0.5. −1. 1 Y. 0.5. 0. −0.5. −1. Figure 1.5. The acceptance corrected Dalitz plot distribution from WASA-at-COSY, normalized to the bin at X = Y = 0 (obtained from table IV in [41]). In total 59 bins are used.. Figure 1.6. The Dalitz plot distribution from BESIII, figure from [42].. 37.

(40) J/ψ → ηγ → (π + π − π 0 )γ hypothesis. The Dalitz plot contains ∼80 000 events, with a background contamination of 0.1 − 0.2%. The Dalitz plot distribution is shown in figure 1.6. An unbinned maximum likelihood fit is used to extract the Dalitz plot parameters seen in table 1.1. In addition to the Dalitz plot parameters for the mentioned experiments, table 1.1 includes also some theoretical calculations. As can be seen there is some disagreement between the experiments, specially for the b but also for the a parameters. Looking at the theory, both the b and the f parameters are hard to get in agreement with experiment.. 1.4.1 Asymmetries To test C-invariance in the η → π + π − π 0 decay one can also look at asymmetries, which might be more sensitive to C-violation than the c, e, h and l Dalitz plot parameters. The left-right asymmetry (ALR ) tests overall C-invariance [44, 45]. The quadrant asymmetry (AQ ) is sensitive to C-violating transitions with isospin change ΔI = 2 and the sextant asymmetry (AS ) to transitions with ΔI = 0 [46, 47]. The asymmetries are defined as follows: N+ − N− N+ + N− NI − NII + NIII − NIV AQ = NI + NII + NIII + NIV N1 − N2 + N3 − N4 + N5 − N6 AS = N1 + N2 + N3 + N4 + N5 + N6. ALR =. (1.56) (1.57) (1.58). where N is the number of acceptance corrected events in the regions defined in figure 1.7. Some of the experiments described above have also measured the charge asymmetries, and one additional experiment at the Rutherford Laboratory reported only the asymmetries [48]. This experiment also used the reaction π − p → nη to produce the η, and an axially symmetric setup. Table 1.2 summarizes the results. The values quoted for WASA-at-COSY are from a PhD thesis [49] and have not been published. All results are consistent with zero except for ALR from [50], which most likely was due to a systematic bias (unmeasured effects in the spark chamber due to the electric and magnetic fields [7]).. 38.

(41) 1. Y. Y. 1 0.8. 0.8. 0.6. 0.6. 0.4. . . 0.2. 0. 0. -0.2. -0.2. -0.4. -0.4. -0.6. -0.6. -0.8. . . -0.8 -0.8. -0.6. -0.4. -0.2. 0. 0.2. Y. -1 -1. . 0.4. . 0.2. 0.4. 0.6. 0.8. -1 -1. 1 X. -0.8. -0.6. -0.4. -0.2. 0. 0.2. 0.4. 0.6. 0.8. 1 X. 1 0.8. . 0.6 0.4. . 0.2. . . 0 -0.2. . . -0.4 -0.6 -0.8 -1 -1. -0.8. -0.6. -0.4. -0.2. 0. 0.2. 0.4. 0.6. 0.8. 1 X. Figure 1.7. Definition of the kinematic regions used for the asymmetries ALR , AQ and AS .. Table 1.2. Summary of charge asymmetry results in the η → π + π − π 0 decay. Systematic errors are only explicitly quoted for the KLOE(08) results. Experiment. ALR · 102. AQ · 102. AS · 102. Gormley(68)[50] Layter(72)[51] Jane(74)[48] KLOE(08)[40] WASA(14)[49]. 1.5(5) −0.05(22) 0.28(26) 0.09(10)(+9 −14 ) 0.09(33). −0.07(22) −0.30(25) −0.05(10)(+3 −5 ) −0.22(33). 0.5(5) 0.10(22) 0.20(25) 0.08(10)(+8 −13 ) −0.06(33). 39.

No results found