ACTA UNIVERSITATIS UPSALIENSIS Uppsala Dissertations from the Faculty of Science and Technology 118

Transcription

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

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3 Measurement of the Dalitz Plot Distribution for η π + π π with KLOE Li Caldeira Balkeståhl

4 Dissertation presented at Uppsala University to be publicly examined in Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 22 January 216 at 9:15 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 Measurement of the Dalitz Plot Distribution for η π + π π with KLOE. Uppsala Dissertations from the Faculty of Science and Technology pp. Uppsala: Acta Universitatis Upsaliensis. ISBN The mechanism of the isospin violating η π + π π 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 24 and 25 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 (19.5 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 η π + π π 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. The distribution is parametrized by determining the coefficients of the third order polynomial in the X and 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 X 2 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 Uppsala, Sweden. Li Caldeira Balkeståhl 215 ISSN ISBN urn:nbn:se:uu:diva (

5 To my parents, for always believing in me

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7 Contents Introduction Motivation Chiral Perturbation Theory Chiral Symmetry Ingredients for the ChPT Lagrangian ChPT Lagrangian Quark Masses from ChPT Dalitz Plot Dalitz Plot Variables Dalitz Plot Parameters More Theory Electromagnetic Corrections Dispersive Calculations Previous Experimental Results Asymmetries Experiment DAΦNE Accelerator KLOE Detector Drift Chamber (DC) Electromagnetic Calorimeter (EMC) Trigger Upgrades DAΦNE KLOE Event Reconstruction and Selection Reconstruction FILFO: Background Filter Event Classification Analysis Selection Simulations Background Rejection Data-MC Comparison... 68

8 4 Results Dalitz Plot and Variable Resolution Fit Description Phase Space Integrals Fit Test on MC Fit Results Systematic Uncertainties Minimum Photon Energy Cut Background Subtraction Choice of Binning Track-Photon Angle Cut Time-of-Flight Cuts Opening Angle Cut Missing Mass Cut Event Classification Procedure Summary of Systematic Effects Final Results for Dalitz Plot Parameters Charge Asymmetries Acceptance Corrected Data Diagonality of the Smearing Matrix Acceptance Correction Comparison with Smearing Matrix Method Results Discussion of Results Dalitz Plot Parameters Acceptance Corrected Data Charge Asymmetries Conclusions Summary in Swedish Acknowledgments References

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 antiquarks 1 [1]. Some of the lightest mesons are the three pions (π +,π,π ) 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].

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,, 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. 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 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 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 This chapter gives the motivation for the η π + π π 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 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 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 γγ or π + μ + ν μ. 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 Λ χ 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 m ρ = 77 MeV Λ χ indicates that ChPT will not work at these energies. 2-flavor ChPT in general converges better than 3- flavor 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 Chiral Symmetry The quark part of the QCD Lagrangian can be written as: L QCD, quarks = q f (i /D m f )q f (1.1) f =u,d,s,c,b,t where q f and q f are the quark fields (with implicit color and spinor indices), m f the mass of quark flavor f and /D the gauge derivative. The gauge derivative includes the gluon-field matrix A μ in /D = γ μ D μ = γ μ μ + igγ μ A μ, where μ =,1,2 or3,γ μ 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 Λ QCD. Concentrating on the light quarks, define a quark flavor vector q = (u,d,s ) and consider the projection operators: P R = 1 2 (I + γ5 )=P R, P L = 1 2 (I γ5 )=P L (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: q R = P R q such that q R = q R γ =(P R q) γ = q P R γ = q γ P L = qp L (1.3) q L = P L q such that q L = qp R (1.4) where the anti-commutation relation for γ 5 is used ({γ 5,γ μ } = ). The light-quark Lagrangian can then be written in terms of the right- and left-handed quarks: L QCD, light quarks = q(i /D M)q = q R i /Dq R + q L i /Dq L q R Mq L q L Mq R (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 Λ QCD, 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: q R U R q R q L U L q L (1.6) where U R,U L are 3 3 special unitary matrices (i.e., U R,U L SU(3)), acting in flavor space. So the Lagrangian L QCD, light quarks = qi /Dq = q Ri /Dq R + q L i /Dq L 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 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: q R Uq R q L Uq L (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: q R Uq R q L U q L (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 1 2-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]: U(x)=e i φ(x) F (1.9) π η 2π + 2K + with φ = 2π π η 2K (1.) 2K 2 K 2 3 η 17

18 where the bosonic fields π, π +, π, K +, K, K, K and η all depend on the space-time coordinate x and have dimension of energy, and F 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 Lagrangian LQCD, light quarks is upgraded to a locally chiral invariant one by introducing external fields v μ, a μ, 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 r μ = v μ + a μ, l μ = v μ a μ, M = s + ip and M = s ip are used. The extended Lagrangian: L ext, light quarks = q R iγ μ D μ q R + q R γ μ r μ q R + q L iγ μ D μ q L + q L γ μ l μ q L q R M q L q L M q R (1.11) is invariant under the local SU(3) R SU(3) L transformation: q R (x) U R (x)q R (x), q R (x) q R (x)u R (x), q L (x) U L (x)q L (x), q L (x) q L (x)u L (x), r μ (x) U R (x)r μ (x)u R (x) +U R (x)i( μ U R (x) ), l μ (x) U L (x)l μ (x)u L (x) +U L (x)i( μ U L (x) ), M U R (x)mu L (x), M U L (x)m U R (x), (1.12) where U R (x),u L (x) SU(3) and depend on the space-time coordinate x. Note that putting v μ = a μ = p = and s = diag(m u,m d,m s )=M one recovers the Lagrangian of equation 1.5. The Lagrangian of the effective field theory, ChPT, will use the same external fields r μ,l μ,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) U R (x)u(x)u L (x). The definition of the chiral gauge covariant derivative of an object A, which transforms as A(x) U R (x)a(x)u L (x), is: D μ A = μ A ir μ A + ial μ (1.13) and transforms as D μ A(x) U R (x)(d μ A(x))U L (x). The field strength tensors: f Rμν = μ r ν ν r μ i[r μ,r ν ], f Lμν = μ l ν ν l μ i[l μ,l ν ] (1.14) 18

19 are also needed, and they transform as: f Rμν (x) U R (x) f Rμν (x)u R (x), f Lμν (x) U L (x) f Lμν (x)u L (x). (1.15) 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 ( U R (x)a(x)u L (x) (U R (x)b(x)u L (x) ) ) = Tr ( U R (x)a(x)u L (x) U L (x)b(x) U R (x) ) = Tr ( U R (x)a(x)b(x) U R (x) ) = Tr ( U R (x) U R (x)a(x)b(x) ) = Tr ( A(x)B(x) ). (1.16) With the fields introduced, examples of entities transforming as A(x) are: U(x),D μ U(x),D ν D μ U(x),M (x), f Rμν (x)u(x) and U(x) f Lμν (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 r μ and l μ are also considered O(q) and thus also the gauge covariant derivative D μ. The field strengths f Rμν and f Lμν are then of O(q 2 ). The boson field matrix is considered O(q ), while M is of O(q 2 ), since the quark masses can be related to the square of the pseudo-goldstone boson masses, see section ChPT Lagrangian The lowest-order Lagrangian in ChPT is of O(q 2 ). At O(q ) 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), orin fact at any O(q n ) where n is odd. The only building block with odd order is D μ, 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(q 2 ). The candidate hermitian structures of the Lagrangian are: Tr ( (D μ U) D μ 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 transformation, M = s + ip s ip = M and U = e i φ F e i φ F = U, so that itr ( U M M U ) itr ( UM MU ) = itr ( U M M U ). At the considered order, charge conjugation invariance does not impose any more 19

20 constraints, and the Lagrangian is: L 2,ChPT = F2 4 Tr( (D μ U) D μ U ) + F2 4 2B Tr ( U M + M U ) (1.18) where F and B are the low-energy constants at this order, F is related to the pion decay and B to the quark condensate. Any process in ChPT O(q 2 ) is calculated by tree level diagrams with vertices from L 2,ChPT. Loop diagrams appear first at O(q 4 ). 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 L 2,ChPT are compensated by terms in L 4,ChPT,bya suitable redefinition of the low-energy constants of L 4,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 L 2n,ChPT each contribute q 2n, e.g. vertices from L 2,ChPT contribute q 2, vertices from L 4,ChPT contribute q 4 ; each pseudo-goldstone boson propagator contributes 1 ; q 2 each independent loop contributes q 4 (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 q D : D = 4N l 2N p + n=1 2nN v,2n (1.19) where N l is the number of independent loops, N p the number of propagators and N v,2n the number of vertices from L 2n,ChPT. In a connected diagram, the number of loops, propagators and vertices are not independent of each other but obey the relation N l N p + n=1 N v,2n = 1 and with this, equation 1.19 can be rewritten as [15]: D = 2N l n=1 2(n 1)N v,2n. (1.2) 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 L 2,ChPT ). As an example of equation 1.2, consider a loop diagram of 2 2 pseudo- Goldstone boson scattering, with two L 2,ChPT vertices connected by two propagators, which has one independent loop, see figure 1.1. According to the 2

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 L 2,ChPT. power counting above, this diagram has chiral dimension D = 4, i.e., it contributes at O(q 4 ). At next to leading order (NLO), i.e.,ato(q 4 ), both one-loop diagrams with an arbitrary number of vertices from L 2,ChPT and tree-level diagrams with one vertex from L 4,ChPT need to be taken into account. The NLO Lagrangian L 4,ChPT has 12 low-energy constants, and can be written as [16]: L 4,ChPT =L 1 [ Tr ( (Dμ U) D μ U )] 2 + L2 Tr ( (D μ U) D ν U ) Tr ( (D μ U) D ν U ) +L 3 Tr ( (D μ U) (D μ U)(D ν U) D ν U ) +L 4 Tr ( (D μ U) D μ U ) Tr ( χ U + χu ) +L 5 Tr ( (D μ U) D μ U(χ U + χu ) ) + L 6 [ Tr ( χ U + χu )] 2 [ ( +L 7 Tr χ U χu )] 2 + L8 Tr ( χ Uχ U + χu χu ) il 9 Tr ( f Rμν (D μ U)(D ν U) + f Lμν (D μ U) D ν U ) +L Tr ( U f Rμν Uf μν ) L +H 1 Tr ( f Rμν f μν R + f Lμν f μν L ) + H2 Tr ( χ χ ), (1.21) where M is now encoded in χ = 2B M. Of the 12 low-energy constants, (L 1,...,L ) have physical significance. The remaining 2 parameters (H 1 and H 2 ) 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(q 6 ), the Lagrangian L 6,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 QCD 1 [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] 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 L 2,ChPT. Expanding U and U in powers of the field matrix φ: U = I + i φ 1 F 2F 2 φ , U = I i φ 1 F 2F 2 φ , (1.22) setting r μ = l μ = p =, s = M (the quark mass matrix), and keeping only terms up to φ 2, the Lagrangian can be written as: L 2,ChPT = L 2,ChPT,kin + L 2,ChPT,mass (1.23) 1 Lattice 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 L 2,ChPT,kin corresponds to the kinetic terms and L 2,ChPT,mass to the mass terms. The kinetic part of the Lagrangian is: ( L 2,ChPT,kin = F2 [ 4 Tr μ I i ( φ ) μ I + i )] φ F F = 1 4 Tr[ μ φ μ φ ] = 1 [ 2 μ π μ π + 2 μ η μ η + 4 μ π + μ π μ K + μ K + 4 μ K μ K ] = 1 2 μπ μ π μη μ η + μ π + μ π + μ K + μ K + μ K μ K. (1.24) These are the usual kinetic terms of scalar hermitian fields (π and η) and scalar non-hermitian fields (π +,π ; K +,K and K, K ). The mass terms are: L 2,ChPT,mass = F2 4 2B Tr(M iφ M φ 2 F 2F 2 M + M + M iφ M φ 2 F 2F 2 ) = F 2 B Tr(M) B 2 Tr(Mφ 2 ) = C B [((π + η ) 3 ) 2 + 2π + π + 2K + K 2 + (2π + π +( π + 3 η ) ) 2 + 2K K m d ] + (2K + K + 2K K + )m 4η2 s. 3 m u (1.25) Dropping the constant term C, of no physical importance, and collecting terms of the same fields gives L 2,ChPT,mass B ( (π ) 2 (m u + m d )+ 2 π η(m u m d ) π + π (m u + m d )+2K + K (m u + m s ) + 2K K (m d + m s )+η 2 ( 1 3 m u m d + 4 ) 3 m s). (1.26) To get the masses of the pseudo-goldstone bosons, the normal form of the mass term of a scalar hermitian field ( 1 2 m2 aa 2 ) and of a scalar non-hermitian 23

24 fields ( m 2 aa a) is used. Neglecting the π η mixing, the masses can be read directly from equation 1.26: m 2 π = m 2 π = m 2 π ± = B (m u + m d ), m 2 K ± = B (m u + m s ), m 2 K = m 2 K = B (m d + m s ), m 2 η = B 1 3 (m u + m d + 4m s ). (1.27) These equations are called the Gell-Man, Oakes, Renner relations [2]. As can be seen, the quark masses are of the order of the pseudo-goldstone boson masses squared, so assigning O(q 2 ) 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. (m 2 K ± m 2 K ) E.M., LO =(m 2 π ± m 2 π ) E.M., LO = Δ E.M. (m 2 K ± m 2 K ) E.M. (m 2 π ± m 2 π ) E.M. = O(e 2 M). (1.28) Using also the fact that the neutral particles do not get any electromagnetic corrections at lowest order and including the unknown Δ E.M., equation 1.27 gives m 2 π = B (m u + m d ), m 2 π ± = B (m u + m d )+Δ E.M., m 2 K ± = B (m u + m s )+Δ E.M., m 2 K = m 2 K = B (m d + m s ). (1.29) With this equation, the quark mass ratios m u m d and m s m d were calculated in [22]: m u m d = 2m2 π m 2 π ± + m 2 K ± m 2 K m 2 K m 2 K ± + m 2 π ± =.56 m s m d = m2 K + m 2 K ± m 2 π ± m 2 K m 2 K ± + m 2 π ± = 2.2. (1.3) 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 K and m 2 K m2 K ± m 2 K m2 π 24, where m 2 K is the isospin averaged kaon mass, i.e., the mass of the m 2 π

25 kaons if m u = m d. The ratios are: m 2 K m 2 π m 2 K m 2 K ± m 2 K m2 π = m s + ˆm ( 1 + ΔM + O(M 2 ) ) 2ˆm = m d m u m s ˆm ( 1 + ΔM + O(M 2 ) ) (1.31) where the average u,d quark mass ˆm = 1 2 (m u + m d ) is used, and Δ M 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 m u = m d = ˆm for the cases when a charge of the meson is not specified. A new ratio, Q 2, which does not receive a correction at NLO, can be constructed out of the ratios in equation 1.31: m 2 K m2 K ± 1 Q 2 = m2 K m 2 K ± m 2 π m m 2 K m2 π m 2 = 2 K m2 π m K 2 K m 2 π ( 1 + ΔM + O(M 2 ) ) = = m d m u m s ˆm m s + ˆm 2ˆm (1 + Δ M + O(M 2 )) ( 1 + O(M 2 ) ) m d m u m s ˆm m s + ˆm 2ˆm = m d m u m s ˆm 2ˆm ( 1 + O(M 2 ) ) m s + ˆm = m d m u m 2 s ˆm (m u + m d ) ( 1 + O(M 2 ) ) = m2 d m2 u m 2 s ˆm 2 ( 1 + O(M 2 ) ) Q 2 = m2 s ˆm 2 m 2 d m2 u ( 1 + O(M 2 ) ) = m2 K m2 π m 2 K m 2 K ± m 2 K m 2 π. (1.32) With Dashen s theorem, assuming that the mass difference between charged and neutral pions is exclusively due to electromagnetic effects, and inserting m 2 K = 1 2 (m2 K + m 2 K ± ) (only the QCD contribution of the masses), one can calculate Q 2 from the measured meson masses: Q 2 D = (m2 K + m 2 K ± m 2 π ± m 2 π )(m 2 K + m 2 K ± m 2 π ± + m 2 π ) 4(m 2 K m 2 K ± + m 2 π ± m 2 π )m 2 π. (1.33) Inserting the known values of the masses from [7] gives Q D =

26 Knowing Q provides an elliptical constraint on the quark mass ratios, as can be seen by rewriting equation 1.32: 1 Q 2 = m2 d m2 u m 2 s ˆm 2 = m2 d m 2 s 1 m2 u m 2 d 1 ˆm2 m 2 s m2 d m 2 s 1 m2 u m 2 d 1 (1.34) 1 Q 2 m 2 s m 2 d 1 m 2 s Q 2 m 2 + m2 u d m 2 = 1 d = 1 m2 u m 2 d where one has used the fact that the u and d quarks are much lighter than the s quark. The last line is the equation of an ellipse in the m s m vs m u d m plane, d with semi-axes Q and 1. This ellipse is called the Leutwyler ellipse [8] and is shown for the value Q = Q D in figure 1.2, together with the quark mass ratios from equation m s m d 15 5 Q D Weinberg m u m d 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 un- 26

27 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 decay can be used for an alternative determination of Q. The η π + π π decay The amplitude of the η π + π π 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]: A LO (s,t,u)= B (m u m d ) 3 3F 2 ( 1 + 3(s s ) m 2 η m 2 π ) (1.35) where s,t,u are the Mandelstam variables and 3s =(s +t + u)=m 2 η + m 2 π + 2m 2 π ±. The Mandelstam variables are defined similarly as for 2-to-2 scattering: s =(P π + + P π ) 2 =(P η P π ) 2 t =(P π + P π ) 2 =(P η P π +) 2 u =(P π + + P π ) 2 =(P η P π ) 2 (1.36) with P X being the four-momentum of particle X. As can be seen from the definition of s above, the Mandelstam variables are not all independent. A 1- to-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 m u m d, so this decay would not occur if m u = m d. Equation 1.35 can be rewritten in terms of Q. Note that, at LO ChPT, using equations 1.27 and 1.32: B (m u m d )= (m 2 K m 2 K ± )= 1 m 2 K Q 2 m 2 (m 2 K m 2 π) (1.37) π so that the amplitude becomes A LO (s,t,u)= 1 m 2 K m 2 K m2 π Q 2 m 2 π 3 3F 2 ( 1 + 3(s s ) m 2 η m 2 π ). (1.38) Using the value of Q = and integrating over phase space gives the LO result for the decay width Γ LO = 66 ev [23] 2. The NLO result is again more involved, and calculations with the same value of Q give Γ NLO =(16 ± 5) ev [23], later updated to Γ NLO =(168 ± 5) ev [8]. Both results are quite far from the experimental value Γ exp =(3 ± 11) ev [7]. A full NNLO calculation has also been performed [24], and using the same value of Q gives Γ NNLO = 298 ev 3. These results show at best a slow convergence of the SU(3) 2 Q = is the value of Q D from equation 1.33 at the time [23] was written. 3 A value for Γ is not quoted in this reference, but using Q = and m s / ˆm = 27.4 together with their results gives Γ = 298 ev [25]. 27

28 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, 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 ± = m 2 π + m 2 π ± 1 ( (s m 2 2s η + m 2 π )(s + m 2 π ± m 2 π ± ) ) λ 1 2 (s,m 2 η,m 2 π )λ 1 2 (s,m 2 π ±,m 2 π ± ) t ± = m 2 π + m 2 π ± 1 ( ) (s m 2 η + m 2 2s π )s λ 1 2 (s,m 2 η,m 2 π )λ 1 2 (s,m 2 π ±,m 2 π ± ) (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 = x 2 + y 2 + z 2 2(xy + xz + yz), (1.4) m i 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

29 ) 2 t(gev s(gev ) 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 η π + π π Dalitz Plot Variables For the η π + π π decay, historically the X and variables are used to construct the Dalitz plot. These dimensionless variables are defined in the η rest frame as: X = 3 T π + T π Q η (1.41) = 3T π Q η 1 (1.42) with Q η = T π + + T π + T π = m η 2m π ± m π (1.43) and T i 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 (E x ) of the decay particles (x) inthe η rest frame: s =(P η P π ) 2 s = m 2 η + m 2 π 2P η P π s = m 2 η + m 2 π 2m η E π E π = m2 η + m 2 π s 2m η (1.44) 29

30 and similarly E π = m2 η + m 2 π ± u 2m η (1.45) E π + = m2 η + m 2 π ± t 2m η. (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 X = (u t) (1.47) 2m η Q η 3 [ ] = (m η m 2m η Q π ) 2 s 1. (1.48) η Dalitz plot boundary Equation 1.39, together with equations 1.47 and 1.48, allows to calculate the values of the X and variables for all t and s (which also define u) atthe Dalitz plot border, and thus to calculate the border in the variables X and. 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 ( p π + p π + + p π = ), and thus for p π as a function of the other pions momenta (momenta and three-momenta are used interchangeably) p π 2 = p 2 π = p 2 π + + p 2 π + 2 p π + p π p 2 π = p 2 π + + p 2 π + 2p π + p π cos(θ π +,π ) (1.49) where θ π +,π is the angle between the three-momenta of the charged pions and the simplified notation for the modulus of the momenta p π + = p π + is used. The physical region is delimited by 1 cos(θ π +,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 p 2 π p 2 π + p 2 π 2p π + p π. (1.5) Being interested instead in evaluating if a certain point (X, ) is inside the Dalitz plot, one can invert the relations to get the kinetic energies of the pions: 3 T π = Q η ( + 1) 3 (1.51) T π + = Q η 6 (2 + 3X) (1.52) T π = Q η 6 (2 3X). (1.53)

31 From the kinetic energies of the pions one can calculate the modulus of their three-momenta by p i = T i (T i + 2m i ) and use equation 1.5 to check if the point is inside the Dalitz plot. The shape of the boundary of the Dalitz plot in the X variables can be seen in figure 4.3 on page 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, )=(,): A(X, ) 2 N(1+a +b 2 +cx +dx 2 +ex + f 3 +gx 2 +hx 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 Theory To better understand the η π + π π decay, one can go beyond pure pseudo- Goldstone 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 Electromagnetic Corrections to η π + π π The decay η π + π π 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: Q ch = e 2 1 (1.55)

32 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(Q ch UQ ch U ) [3]. 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 mass terms 2Ce2 (π + π + K + K ). As can be seen, the electromagnetic contribution to the charged pions and charged kaons mass is the same at this F 2 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, at the leading order of the electromagnetic expansion (O(e 2 q )), the decay is forbidden. Calculations at O(e 2 q 2 ) in the isospin limit, i.e, with m u = m d [29], show only small differences from the pure ChPT O(q 4 ) result, both for the decay width but also for the shape in the Dalitz plot. Calculations at order O(e 2 q 2 ) including the effects of O(e 2 (m d m u )) [31] show that the O(e 2 (m d m u )) effects are comparable in size to other O(e 2 (m q )) effects (where m q is a typical light quark mass), in contradiction to the assumption in the previous O(e 2 q 2 ) calculation [29]. The total effect of the electromagnetic corrections, however, remains very small, and the conclusion is that the η π + π π decay is very sensitive to the strong isospin breaking due to the quark mass difference m u m d. It is worth noting that electromagnetic corrections can also enter indirectly in the constants used for the ChPT calculations. For example, F can be identified with the pion decay constant F π. The value of F π changes by 1% when including radiative corrections [32], which also changes the η π + π π amplitude through equation 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

33 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 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 Prague- Lund-Marseille method [] 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 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

34 1.4 Previous Experimental Results Several experiments have measured the η π + π π 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 is reconstructed through missing mass techniques. For more information on the experimental setup, see [37]. The final Dalitz plot contains 3 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(7) c in the table. The authors also performed the fit for the Dalitz plot parameters by folding the distribution around X =, labeled Gormley(7) 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 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 Dalitz plot distribution from 323 events [39]. The experiment was carried out at the LEAR accelerator, using the reaction pp ηπ π. 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 138 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 invariant mass. This analysis only considered the Dalitz plot distribution s dependence on, assuming c = 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 events in the Dalitz plot, is by the KLOE collaboration [4]. 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

35 Table 1.1. Summary of Dalitz plot parameter results, both from experiments and theoretical calculations. Row Gormley(7)c includes also a result for the c parameter, c =.5(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 [4]. 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 b d f Gormley c (7)[36] 1.18(2).2(3).4(4) - Gormley(7)[36] 1.17(2).21(3).6(4) - Layter(73)[35] 1.8(14).3(3).5(3) - CBarrel(98)[39] 1.22(7).22(11).6(fixed) - KLOE(8)[4] 1.9(5)( ).124(6)().57(6)(+7 16 ).14(1)(2) WASA (14)[41] 1.144(18).219(19)(47).86(18)(15).115(37) BESIII(15)[42] 1.128(15)(8).153(17)(4).85(16)(9).173(28)(21) Calculations a b d f ChPT LO[24] ChPT NLO[24] ChPT NNLO[24] 1.271(75).394(2).55(57).25(16) dispersive[33] BLV ChPT[43] 1.266(42).516(65).47(11).52(31) g=.5(7) BLV KLOE[43] 1.77(25).126(15).62(8).7(17) g=.37(8) disp WASA[11] 1.116(32).188(12).63(4).91(3) g=.42(9) 35

36 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(8) (figure from [4]). The distribution is binned with a bin width of.125 in X and, with a total of 154 bins used. The two most recent measurements come from the WASA-at-COS collaboration [41] and the BESIII collaboration [42]. The WASA experiment was carried out at the COS 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 12 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 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 η event candidates and is binned in.2 wide bins in X and. The shape of the Dalitz plot is shown in figure 1.5, normalized to the bin with center at X = =. 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

37 X Figure 1.5. The acceptance corrected Dalitz plot distribution from WASA-at-COS, normalized to the bin at X = = (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

38 J/ψ ηγ (π + π π )γ hypothesis. The Dalitz plot contains 8 events, with a background contamination of.1.2%. 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 Asymmetries To test C-invariance in the η π + π π 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 (A LR ) tests overall C-invariance [44, 45]. The quadrant asymmetry (A Q ) is sensitive to C-violating transitions with isospin change ΔI = 2 and the sextant asymmetry (A S ) to transitions with ΔI = [46, 47]. The asymmetries are defined as follows: A LR = N + N N + + N (1.56) A Q = N I N II + N III N IV N I + N II + N III + N IV (1.57) A S = N 1 N 2 + N 3 N 4 + N 5 N 6 N 1 + N 2 + N 3 + N 4 + N 5 + N 6 (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-COS are from a PhD thesis [49] and have not been published. All results are consistent with zero except for A LR from [5], which most likely was due to a systematic bias (unmeasured effects in the spark chamber due to the electric and magnetic fields [7]). 38

39 X X 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 decay. Systematic errors are only explicitly quoted for the KLOE(8) results. Experiment ALR 2 AQ 2 AS 2 Gormley(68)[5] Layter(72)[51] Jane(74)[48] KLOE(8)[4] WASA(14)[49] 1.5(5).5(22).28(26).9()(+9 14 ).9(33).7(22).3(25).5()(+3 5 ).22(33).5(5).(22).2(25).8()(+8 13 ).6(33) 39

40 2. Experiment This chapter gives an overview of the DAΦNE accelerator and the KLOE detector. In the last part, the crabbed waist upgrade of the DAΦNE accelerator is presented, as well as the recent upgrades to the KLOE detector, now named KLOE DAΦNE Accelerator The DAΦNE accelerator [52], Double Annular φ-factory for Nice Experiments, is an e + e collider located in Frascati, Italy. The accelerator is optimized for a center of mass energy of s M φ = ±.19 MeV, the φ-meson mass [7]. Figure 2.1. Schematic view of the DAΦNE accelerator [53]. Figure 2.1 shows a schematic view of the DAΦNE accelerator. The linear accelerator (LINAC) can accelerate electrons up to 8 MeV, and positrons up 4

41 to 55 MeV. The positrons are created in an intermediate stage of the LINAC: a high intensity beam of 25 MeV electrons impinges on a tungsten target, producing photons, electrons and positrons in electromagnetic showers; the positrons are separated from the electrons by a chicane of dipoles that brings positrons to the beam axis and electrons to a collimator, where they are stopped [54]. The rest of the process is the same for electrons and positrons, although not at the same time. From the LINAC, the electrons or positrons are transferred to the accumulator ring, where the longitudinal and transverse beam emittance is damped. From the accumulator, they are then injected into one bunch in the storage rings (DAΦNE), at an energy of 5 MeV. Due to the short lifetimes of the beams in the storage rings, the accelerator is topped up several times per hour [55]. An example of this topping up is shown in figure 2.2. Figure 2.2. Example of the DAΦNE currents under 1 hour of operation. The blue line is for electrons, the red one for positrons. The peaks correspond to the top up. Image from the KLOE display software. There are two separate storage rings, one for electrons and one for positrons, and they intersect in two points with a crossing angle of θ cross = mrad. DAΦNE is operated with collisions at only one interaction region at a time, in the other interaction region the beams are kept vertically separated. The peak current in the storage rings is 2.4 A for electrons and 1.5 A for positrons. The number of bunches is and the bunch spacing is 2.7 ns. The bunch size at the interaction point is σ x = 2 mm, σ y =.2 mm and σ z = 3. cm, where x is the horizontal coordinate transverse to the beam trajectory, y is the vertical coordinate and z is the horizontal coordinate along the beam trajectory. During the best period of operation (25-27), the DAΦNE collider has reached peak luminosities of L peak cm 2 s 1, and while running with the KLOE detector, an integrated daily luminosity of 8.5 pb 1. The accelerator has later been upgraded and reached even higher luminosities see section

42 2.2 KLOE Detector The KLOE detector has been operating at one of the interaction regions of the DAΦNE accelerator, from 1999 to 26. In fact, after commissioning of the accelerator and detector, the KLOE data taking occurred at two periods: from 21 to 22, with about 45 pb 1 of integrated luminosity, and with about 2 pb 1 of integrated luminosity [55]. Most of these data were taken at a center of mass energy of the φ-meson mass, s 19.5 MeV, but 25 pb 1 of the integrated luminosity was taken at s = 1 GeV and there was also a scan in s from MeV to 3 MeV, comprising four points in s and integrating pb 1 of luminosity. One of the aims of the KLOE experiment is to measure with great precision decays of the φ-meson. The main decays and their branching ratios are shown in table 2.1. Special attention is given to the decays to neutral kaons, and the subsequent decays of the kaons. This task requires a high acceptance and efficiency, as well as a good resolution and the ability to detect both neutral and charged particles. Table 2.1. The decays of the φ meson with a branching ratio bigger than 1% [7]. Decay channel Branching ratio (%) φ K + K 48.9 ±.5 φ KL K S 34.2 ±.4 φ ρπ and φ π + π π ±.32 φ ηγ 1.39 ±.24 If the φ-meson is produced at rest, then the neutral kaons from its decay have momentum p K = 1 MeV/c. For the long lived KL, with lifetime τ K = ±.21 8 s [7], it corresponds to a mean path in the laboratory frame of λ LF = γτβc = 3.4 m (here γ is the Lorentz factor and β =.22 L is the velocity of the KL in the lab frame, in units of c). This implies that the detector must have a large volume in order to measure the KL decays, e.g. radius of 3.4 m to be able to detect 1 1/e = 63% of the decays. Due to the crossing angle of the beams, φ-mesons produced in KLOE have a small horizontal momentum towards the center of the accelerator of p φ = 13 MeV/c. The neutral kaons are thus not monochromatic in the lab frame, and their momentum varies from 4 MeV/c to 117 MeV/c [55]. A compromise between the size and the complexity of the detector leads to the radius of 2 m, meaning 4% of the KL decay within this region and can be detected. KLOE, depicted in figure 2.3, consists mainly of two detectors: a drift chamber (DC) to measure the momentum of charged particles, and an electromagnetic calorimeter (EMC) to mainly measure energy, time and impact position of photons. As can be seen, surrounding both these detectors there is a superconducting coil and an iron yoke, giving rise to an axial magnetic field of.52 T. At the interaction point, the beam pipe is an Al-Be spherical shell 42

43 with a radius of cm and thickness.5 mm [55]. The low atomic number (Z), low density material minimizes the energy loss of charged particles passing the beam pipe. The spherical shape with a radius of cm ensures almost all KS decay in the vacuum inside the beam pipe. This minimizes K S K L regeneration on the beam pipe. With lifetime τ K = 8.954±.4 11 s [7] S and considering the momentum p K = 1 MeV/c from the φ decay at rest, the mean path of the KS in the lab frame is λ K S = 5.96 mm, and the radius of cm corresponds to 16λ KS, thus ensuring most KS have already decayed in the vacuum. There is also a tile calorimeter surrounding the beam pipe around the interaction region quadrupoles, whose main purpose is to measure photons from the KL decay which would otherwise be lost in the beam pipe. This detector is not used in the current analysis. Figure 2.3. Vertical cross-section of the KLOE detector, showing the DC, EMC and superconducting coil. Figure from [56]. 43

44 2.2.1 Drift Chamber (DC) The tracking detector of KLOE is a gas filled drift chamber, cylindrical in shape, with length 3.3 m, inner radius 25 cm and outer radius 2 m [57]. In a drift chamber, wires shape the electric field in the gas filled space. When a charged particle traverses the chamber, it ionizes the gas, creating electronion pairs along its trajectory. The created electrons drift towards the positive voltage wires, and when close to the wires, the high electric field causes an avalanche of electrons and ions to be created out of the gas. The drifting of the ions away from the wire induces a signal on it that can be measured at the wire s end [58]. In KLOE, to minimize multiple Coulomb scattering, K L K S regeneration and absorption of photons before reaching the calorimeter, the materials used for the walls of the chamber and as the filling gas have low Z and low density [59]. Carbon fiber is used for the mechanical support, i.e., the drift chamber walls. The gas used is a mixture of helium and isobutane (9% He - % ic 4 H ). This gas mixture has a radiation length X 13 m, but this is lowered to an effective radiation lenght of X 9 m if the tungsten wires are taken into account. There are sense wires in the drift chamber. They are made of goldplated tungsten and are 25 μm in diameter. To shape the electric field, silver-plated aluminium field wires, with diameter 8 μm are used, of which 168 form an inner guard and 768 an outer guard to the sensitive volume. There are in total wires, and the voltage difference between field and sense wires is 18 2 V [57]. The wires are arranged in cells of almost square transverse cross-section, consisting of one sense wire surrounded by 8 field wires. Figure 2.4 shows one example cell. The cells are arranged in cylindrical layers around the beam pipe. To account for the higher track density close to the interaction region, due to the usual small momenta of the charged particles coming from the φ decays close to rest, the first 12 layers have a cell size in the transverse plane of 2 2cm 2 and the remaining 46 layers of 3 3cm 2 [57]. In order to reconstruct the tracks in three dimensions, some wires need to be at an angle to the drift chamber axis. Together with the requirement of uniform efficiency, this consideration led to an all stereo geometry (i.e., all wires have an angle to the drift chamber axis), where consecutive radial layers have opposite signs of the stereo angle. The definition of the stereo angle is shown in figure 2.5. As can also be seen in figure 2.5, the stereo angle implies that the distance of the wire from the chamber axis is not constant, with the minimum distance R at the middle of the chamber (z = ) and the maximum R p at the end plates. To ensure that the wires fill the chamber uniformly, the stereo drop is kept constant at R p R = 1.5 cm. This implies that the stereo angle changes with the radius, increasing in absolute value from 6 mrad to 15 mrad. 44

45 Figure 2.4. A cell of the drift chamber, showing the definition of the angles β and φ. The filled circles correspond to sense wires, the open circles to field wires (adapted from [55]). Figure 2.5. Definition of the stereo angle ε (adapted from [55]). 45

46 The almost square shape of the cells comes from the stereo geometry. Layer k of wires is defined as all the wires sharing the radius with either the sense wire at radius R k or the field wire just below, at R k. All wires at R k have the same stereo angle, and it is almost the same as the stereo angle for the wires at R k, i.e. ε k ε k. The field wires of the upper part of the cell, on the other hand, belong to the next layer, and thus have a stereo angle with the opposite sign. This results in a cell shape varying periodically along the axis of the chamber, and varying also with the radius and the azimuthal angle. Tracking The signals from the wires of the drift chamber are the drift times. To reconstruct a track, the drift times first have to be translated into drift distances of the electrons from the ionization to the wire. For this, 232 space-time relations are used, depending on cell shape and track impact parameter (see calibration on page 47). Then the track reconstruction program works in three steps: pattern recognition, track fit and vertex fit. A hit is a measured signal from a wire. First, space-time relations averaged over cell shape and track impact parameter are used (since these two variables depend on the track, they can only be calculated after the track fitting) to get the drift distances from the drift times, the signal. For each of the two stereo views, the pattern recognition starts at the outermost layer and works inward, associating hits close in space to track candidates. After the association is done, the track candidates in each view are fit and the track parameters extracted. Then tracks from both views with the same curvature and compatible geometry are combined into a three dimensional track. The three dimensional track is fit again, providing also information about the z coordinate. The pattern recognition program outputs tracks together with a first estimation of their parameters. The next step, the track fitting, refines the track parameters through a χ 2 minimization, with: χ 2 = i ( d track i ) 2 d i (2.1) σ(d i ) where the sum goes over the hits in the track, di track are the drift distances calculated from the fitted track parameters, d i the drift distances as measured from the drift time, dependent on the space-time relations, and σ(d i ) is the drift distance resolution. This is done in an iterative procedure, where the space-time relations are first used to calculate the drift distances d i. In the first iteration, the space-time relations are calculated from the output of the pattern recognition. The drift distances are then fit to a track by the χ 2 minimization, and new track parameters are obtained. The procedure is repeated with these new track parameters until a sufficiently good track is obtained. Since the space-time relations depend on the track parameters, each time new track parameters are found, new space-time relations are used, and thus new drift 46

47 distances d i. After the first iteration, procedures to improve the quality of the track fit are employed: adding hits missed by the pattern recognition, rejecting wrong hits, identifying split tracks and joining them to one track. The vertex fit then associates track pairs to a vertex, by extrapolating the tracks and checking their point of closest approach. Primary vertices are found by extrapolating the tracks to the beam crossing point, after this a search for secondary vertices is done, ignoring tracks already associated to a vertex. Charged tracks with polar angle larger than 45 are reconstructed with a momentum resolution of σ p /p.4%, and the spatial resolution is σ xy 2 μm in the transverse plane and σ z = 2 mm in the axial direction. The resolution of the vertex position is σ V 1 mm. Calibration The calibration of the DC is performed with cosmic ray muons, selected by requiring two calorimeter clusters separated in time and a track in the DC. The time signal measured in the drift chamber has contributions from the drift time, T drift, the propagation time along the wire, T wire, and a time offset, T. Using the time from the calorimeter cluster measurement, the time of flight between the calorimeter and the wire, T tof, also needs to be taken into account, but the calorimeter information allows the determination of T wire and T tof event by event. The distribution of T drift + T for each wire can then be fitted to extract the time offset for each wire. This procedure is performed once per run period, but it is repeated after interventions in the front-end electronics. To relate the drift time to drift distances, 232 space-time relations are used. These describe the dependence of the drift distance on the drift time, the cell shape and the track impact parameter. The variable β is used to classify the cell shape, and φ for the track impact parameter. Both these variables are shown for the example cell in figure 2.4. To parametrize the space-time relations as a function of only drift time, six different values of β characterize the cell shape, and 36 evenly spaced intervals of φ characterize the track impact, giving a set of 116 space-time relations for the big cells ( 3 3cm 2 ) and the same number for the small cells ( 2 2cm 2 ). Each of the space-time relations is parametrized by a fifth-order polynomial of the drift time, resulting in calibration coefficients (C k ). At the start of each run, an online filter selects 8 [55] cosmic ray events, fits the hits to tracks and checks the residuals of the space-time relations. The residuals are defined as the difference between the drift distance calculated from the track parameters and the drift distance calculated from the drift time using the space-time relations (di track d i ). If these are too big, the calibration procedure for the space-time relations is started, which collects 3 cosmic ray events and finds new calibration coefficients for the 232 space-time relations. The starting values for the coefficients C k are taken from the previous calibration run. Before the track is fit, there is no information on the β and φ parameters, so at first coefficients are averaged over all cell shapes 47

48 and track impact angles, as for the regular track fitting procedure. After the track fitting iterations are done, a fit for new C k coefficients for each of the 232 space-time relations is performed, this time by minimizing the absolute value of the residuals ( di track d i, i.e., only changing d i ). With these new C k, the track fitting and C k fitting is performed in an iterative procedure, until the residuals are small enough Electromagnetic Calorimeter (EMC) The electromagnetic calorimeter of KLOE [6] consists of a cylindrical barrel surrounding the drift chamber, and two end-caps perpendicular to the beam axis, see figure 2.3. It is a sampling calorimeter composed of lead and scintillating fibers. A photon with energy larger than a few MeV interacts with matter mainly via pair-production. The created electrons and positrons in turn radiate photons which, if their energy is high enough, create more electronpositron pairs and so on, in what is called an electromagnetic shower. In a material with high density and high Z, these interactions are more probable and if the material is thick enough, the original photon will deposit all its energy in the material, in the form of electron-positron pairs and photons. Measuring the energy of these created photons and e + e gives the energy of the original photon. In the KLOE calorimeter, the lead serves as a passive material that due to its high density and Z accelerates the showering process, while the scintillating fibers are the active part and convert the deposited energy into light, which is measured by photomultiplier tubes (PMTs). The detector is built in layers of.5 mm thick lead foils, with grooves to accommodate the fibers, and 1 mm diameter, clad scintillating fibers. The fibers are glued to the lead foils with epoxy. The final material has a volume of 42% lead, 48% scintillating fibers and % epoxy, and is shaped into 23 cm thick modules. Its radiation length X is 1.5 cm, so the module thickness corresponds to 15X, corresponding to % of absorbed energy. For the barrel, 24 modules, 4.3 m long, with trapezoidal cross-section are used, while each endcap consists of 32 C-shaped modules, m long of rectangular cross-section with variable width. The calorimeter covers 98 % of the full solid angle, see figure 2.3. For the read-out, each module is subdivided into 4 4cm 2 cells, which are matched to the circular area of the PMTs by light guides. Each module is read-out at both ends. Cells at the same depth (same value of r in the barrel or z in the endcap) form a so called calorimeter plane, for a total of 5 planes in depth over the whole calorimeter. The energy resolution of the calorimeter, σ E /E = 5.7%/ E(GeV), is determined with radiative Bhabha events, i.e., e + e e + e γ events, where the electron and positron are measured in the DC. The time resolution is 57 ps σ t (E) = 14 ps [55], where the second term is added in quadrature and includes calorimeter miscalibrations and trigger jitter. The time E(GeV) reso- 48

49 lution is determined using cosmic rays, e + e γγ, radiative φ decays (i.e. φ γx) and φ π + π π events. Excluding the trigger jitter contribution to the resolution results in an intrinsic calorimeter time resolution of 57 ps σ t (E) = ps. The cluster position is reconstructed with a resolution of σ rφ σ xz 1.3 cm transverse to the fibers, and along the fibers, E(GeV) 1.2 cm from the time measurement, with a resolution of σ z = σ y = (for the E(GeV) barrel the coordinate along the fibers is z, for the endcaps y). Reconstruction For each cell, time (T ) and energy (S) is measured at both ends (A and B) ina time to digital converter (TDC) and an analog to digital converter (ADC), and expressed in counts. The energy of each side (E A and E B ), for each cell (i), is corrected for the pedestal, calibrated relative to the response to minimum ionizing particles (mip) and multiplied by a factor accounting for the absolute energy scale [6]: E A,B i = SA,B i S A,B,i K (2.2) S mip,i where S,i A is the pedestal of the amplitude scale of side A of cell i, S mip,i is the response of cell i to mip passing through its center, both in ADC counts, and K is the absolute energy scale calibration constant. The time at each side of the cell is converted to nanoseconds using calibration constants: t A,B i = c A,B i T A,B i, where t is the time in nanosecons and c the calibration constants. The particle arrival time at the cell (t i ) and the its position along the fiber (s i ) is determined from the measured times: t i (ns)= ta i +t B i 2 ta,i +tb,i 2 L i 2v, (2.3) s i (cm)= v 2 ( t A i t A,i (t B i t B,i) ), (2.4) where t,i A is the time offset for side A of cell i, L i is the lenght of the cell (in cm) and v is the velocity of light in the fibers (in cm/ns, 17 cm/ns). The definition of s i assumes s i = cm in the middle of the cell. The total energy deposited in the cell (E i ) is taken as the mean value of the determination at both ends, corrected by a factor A A,B i (s i ) accounting for the attenuation of light along the fiber. Note that the attenuation factor depends on the position of the particle along the fiber: E i = EA i AA i (s i)+ei BAB i (s i). (2.5) 2 To reconstruct energy and time of incidence of a particle in the calorimeter, the information from the different cells is joined by a clustering algorithm. First, cells adjacent to each other in r φ (barrel) or x z (endcaps) 49

50 are grouped together if they have both energy and time information from both sides. Then, the algorithm uses the longitudinal coordinates (s i above, corresponding to z for barrel and y for endcaps) and the incidence times to further join and/or split cells to form a cluster. At this stage, cells missing time or amplitude information are recovered if their φ (barrel) or x (endcaps) is close enough to the clusters corresponding variable. The energy of the whole cluster is evaluated as the sum of the energies of the cells: E clu = E i (2.6) i while the time of arrival of the particle and its position are calculated as energy weighted averages: t clu = i t i E i i E i, (2.7) R clu = i r i E i i E i (2.8) where i stands for the ith cell included in the cluster and r or R clu for the position vector relative to the KLOE reference frame (as (r,φ,z) or (x,y,z)). The time of flight of the particle from the interaction point to the calorimeter, t tof, can be related to the calorimeter time t clu by t clu = t tof + δ C N bc T RF [55], where δ C is a number accounting for the electronic offset and delay due to cable length, N bc is the number of bunch crossings needed to start the TDCs and T RF = ns is the machine radio frequency period. The last term is needed because the particles can have a big spread of arrival times at the calorimeter. The KLOE trigger (see section 2.2.3) cannot identify the bunch crossing related to each event. Instead, the KLOE fast trigger is used as the start signal for the TDCs, and this trigger is phase-locked with a replica of the machine RF, in a 4 T RF period clock, giving rise to the N bc T RF term. The correct bunch crossing, and correspondingly the quantity N bc, are offline determined event by event. The quantities δ C and T RF are determined for each run with e + e γγ events. For these events, the calorimeter is detecting photons, and their time of flight should be t tof = R clu /c. The distribution of t clu R clu /c, seen in figure 2.6, shows well separated peaks, corresponding to diffrent values of N bc. The distance between peaks gives T RF, and δ C is chosen as the mean value of the peak with largest statistics. Note that although the choice of δ C is arbitrary, the same definition has to later be used in determining N bc event by event. Calibration The energy calibration consists of finding the pedestals S A,B,i for each side A and B and the response to mip, S mip,i, for each cell i; as well as finding the absolute energy calibration constant K. The pedestals are determined with cosmic 5

51 Figure 2.6. Distribution of t clu R clu /c for e + e γγ events (adapted from [6]). ray runs without circulating beams, and cross-checked with pulser triggered runs [6]. S mip,i, which serves as a relative energy calibration between cells, is done with cosmic ray minimum ionizing particles which cross the center of the cells. For each cell, the peak of the energy distribution defines S mip,i. The same data is used for determining the light attenuation in the fibers, A A,B i (s i ). This dedicated cosmic ray run is performed before the start of each long data taking period [55]. The absolute energy scale factor K is determined with e + e γγ events. The monochromatic 5 MeV photons are identified and used to set the energy scale (in MeV/S mip ). The K calibration is repeated every 4 nb 1 of collected luminosity. The time calibration determines the time offsets t A,B,i, the light velocity in the fibers v and the time calibration constants c A,B i. The time offsets and the light velocity v are determined with high momentum cosmic rays, which can be collected in parallell with data taking, every few days of data taking. The tracks are identified in the DC, and an iterative procedure minimizes the time residuals between the tracks and the calorimeter. The quantities t A tb, ta +tb and v are determined from time distributions, and from these t A and tb are calculated. The values of t A tb and v are also checked using the s coordinate as determined from the extrapolation of the track to the calorimeter. Equation 2.4, if rewritten to ti A ti B as a function of s, is a straight line with slope 2/v 51

52 and intercept t A tb, which allows the determination of these two values from the distribution of ti A ti B vs s [6]. The time calibration constants c A,B i have been determined in a laboratory test stand, and are good up to an overall scaling factor. To get the absolute time calibration, the value of T RF is used. The scale factor needed is the ratio between the T RF measured with e + e γγ and the value of the period obtained from the accelerator RF signal T RF,DAΦNE (i.e. T RF /T RF,DAΦNE ) Trigger The KLOE trigger uses information from both the EMC and the DC [61]. The high events rates at DAΦNE, mostly due to background, make it desirable for the trigger to accept all φ decays and at the same time to reject the main backgrounds: e + e (Bhabha) scattering at small angles and machine related background due to lost particles from the beams. The trigger should also accept Bhabha events and e + e γγ events at large angles (for detector monitoring and calibration) and reject cosmic rays. Both cosmic ray events and Bhabha scattering at small angles should, however, be accepted in a downscaled sample for monitoring puposes. To allow determination of the bunch crossing corresponding to the event, the trigger should also be fast. These considerations have led to the choice of a two level trigger: the level 1 trigger (T1, fast trigger) has minimal delay, is synchronised to the accelerator radio frequency and starts the data aquisition at the front-end electronics; the level 2 trigger (T2, validation trigger) uses more information from the detector, validates the T1 and starts the whole data aquisition. The T1 trigger is generated by either the EMC or the DC. For this purpose, the EMC is divided into sectors of 3 calorimeter cells (3 cells in the barrel and 2,25 or 3 in the endcaps). The EMC level 1 trigger requires at least two fired trigger sectors, with energy deposit greater than 5 MeV in the barrel and 15 MeV in the endcaps [55]. If there are only two fired sectors, and these are in the same endcap, the event is rejected, since this topology is mainly from machine background. The DC level 1 trigger requires at least 15 hits in the DC within 25 ns. The Bhabha rejection is done at this level with EMC information: as in the normal EMC trigger, two fired sectors are required, but here with a minimum energy deposit of 35 MeV, and only for topologies where both sectors are in the barrel or in different endcaps. If this condition is met, the T1 is vetoed, except for a downscaled amount of the events. After each T1, there is a fixed dead-time of 2.6 μs where no new T1 can occur. In the case where T1 is due to the EMC, the T2 occurs automatically after afixedtimeof 1.5 μs. If the T1 is due to the DC, a validation trigger from the DC is needed: this requires 12 hits within 1.2 μs. As with the Bhabha trigger, the T2 signal can be vetoed by the cosmic ray trigger: two EMC sectors with signals from the outer cells of the detector, with energy 52

53 deposit above 3 MeV, identify the cosmic rays. If no T2 signal arrives within the 2.6 μs of fixed dead-time from T1, all read-out is reset. The background events from Bhabha scattering, cosmic rays and machine background that survive the trigger are later rejected at the beginning of the offline reconstruction by the background filter FILFO, see section Upgrades The upgrades presented in this section have no direct impact on the results of this thesis, since the data used is from before the upgrades. Nonetheless, it is interesting to know what the present and near future hold for the KLOE detector, and the DAΦNE upgrade has a direct impact on this DAΦNE In 27, the DAΦNE accelerator was upgraded in order to increase its luminosity. To achieve this, the beam horizontal size σ x is reduced and the crossing angle increased (this reduces the overlap region of the beams), which allows to decrease the vertical beam size, while the crabbed waist scheme is used to suppress resonances [62]. Figure 2.7 shows the differences in the crossing angle, which in the upgrade is doubled to θ cross = 2 25 mrad, and in the horizontal beam size. The crabbed waist uses sextupoles to rotate the minimum of the vertical beam size of a beam, such that the position of the minimum is aligned along the central trajectory of the other beam. The upgrade also included other hardware improvements [63]. It has been used for the SIDDHARTA experiment [64] and since 2 for the KLOE-2 experiment. With the first experiment, the peak luminosity reached was of cm 2 s 1, with a maximum daily integrated luminosity exceeding 15 pb 1. The more complex setup of the KLOE-2 experiment, specially the magnetic field, makes it hard to reach the same performance. The peak luminosity of cm 2 s 1 has been reached, but with higher backgrounds than during the KLOE run KLOE-2 The higher values of luminosity possible at DAΦNE after the upgrade has prompted a new physics program [65] and upgraded detector, called KLOE- 2. The KLOE-2 detector consists of the DC and EMC as in KLOE, but with several new detectors added: a high and a low energy tagger (HET and LET) for each lepton ring, two new calorimeters CCALT and QCALT and an new inner tracker (IT) placed between the interaction region and the DC. 53

54 Figure 2.7. Representation of the DAΦNE interaction region in the old interaction scheme (upper) and the upgrade scheme (lower). 54

55 The purpose of the LET and HET is to detect the electrons and positrons from e + e e + e γ γ e + e X types of processes, where X are hadrons [66]. The HET [67] is designed to detect electrons and positrons with small energy transfer, and thus are close to the nominal orbit in the accelerator. There is one such detector in each of the electron and positron arms, and they are located after the first dipole following the interaction region, 11 m from the beam crossing. The detector consists of 28 plastic scintillators, and measures the position of the lepton passing through from which of the scintillators was fired. The detector is located after the bending dipole which serves as a spectrometer and the position at the detector is related to the momentum of the measured lepton, see figure 2.8. The HET can detect leptons in the energy range 4 5 MeV. Figure 2.8. The HET detector, located after the first bending dipole. The light guides which transfer the light from the scintillators to PMTs are shown in orange. Figure adapted from [67]. The LET [68] is an energy sensitive detector comprised of 2 LSO crystals (Lutetium ttrium Orthosilicate), read out by silicon photomultipliers (SiPM). There are two LETs, one for electrons and one for positrons, located inside the KLOE detector, 1 m from the interaction region. The LETs cover a limited angular region, and they are replacing the corresponding part of the QCALT. Figure 2.9 shows the LET casing mounted with the QCALT, and the window in the QCALT to allow the leptons to reach the LET. The LETs detect leptons in the energy range MeV. The crystal calorimeter with timing, CCALT [7], is a small angle calorimeter with the purpose of extending the KLOE angular acceptance for photons from the interaction region down to θ = (the EMC only goes down to θ = 21 ). There are two CCALT detectors, one on each side of the interaction region. They each contain 48 crystals, read out by SiPMs. Figure 2. shows one of four wedges of one CCALT, with four crystals visible. 55

56 Figure 2.9. The LET (gray casing with black label) mounted on the QCALT (with black casing) [69]. Figure 2.. Part of the CCALT, with 4 crystals showing. Figure from [71]. 56

57 The QCALT [72] (tile quadrupole calorimeter) is a new calorimeter surrounding the beam pipe and covering the quadrupoles inside KLOE. There are two QCALTs, one on each side of the interaction region, see figure These new detectors were needed because of the changes done to the beam crossing region for the DAΦNE upgrade. The main purpose of the QCALTs is to improve detection efficiency of photons coming from KL decays, which might otherwise be lost in interactions with the beam pipe and the quadrupoles. The QCALT is composed of alternating layers of scintillator plates and tungsten plates, for a total depth of 4.75 cm ( 5.5X )and1minlenght. The scintillating plates are divided in tiles, 2 per plane, each with a fiber embedded to transmit the light to SiPMs for read-out. Figure The detectors around the new KLOE-2 interaction region: the IT (in the middle with the Italian flag) and the two QCALTs (to the left and right of the IT). One can also see the space left in the QCALTs for the LET. The CCALT is inside the IT, not visible. Figure from [56]. The IT (inner tracker) is a tracker for the inner region, and fits between the beam pipe and the DC. It is composed of four layers of cylindrical triple gas electron multipliers (GEM) [73] with the purpose of improving acceptance of low transverse momentum tracks and improving the vertex resolution. One layer of a cylindrical tripple GEM consists of concentrical electrodes: a cathode, three GEM foils (for multiplication) and an anode, which also functions as the read-out. The read-out is done in two coordinates, but instead of the usual X read-out for planar GEMs, it uses a XV view, where the V strips are at an angle of 4. The IT mounted on the interaction region can be seen in figure All these new detectors have been installed and the KLOE-2 detector is operating since November