Studies of the Decay η→π+π-π0 with WASA-at-COSY

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ACTA UNIVERSITATIS UPSALIENSIS

Uppsala Dissertations from the Faculty of Science and Technology 100

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Studies of the Decay η→π

+

π

-

π

0

with WASA-at-COSY

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Dissertation presented at Uppsala University to be publicly examined in Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, November 2, 2012 at 10:15 for the degree of Doctor of Philosophy. The examination will be conducted in English.

Abstract

Adlarson, P. 2012. Studies of the Decay η→π+_π-_π0_{with WASA-at-COSY. Acta Universitatis}

Upsaliensis. Uppsala Dissertations from the Faculty of Science and Technology 100. 160 pp. Uppsala. ISBN 978-91-554-8476-7.

In 2008 a large statistics sample of approximately 1·107_{η-decays has been collected with}

the WASA detector at COSY using the pd→3_{Heη reaction at the beam kinetic energy of 1 GeV.}

These data are being used to study the not so rare η decays involving charged pions, like η→π+_π-_π0_{. This decay proceeds mainly via a strong isospin violating contribution, where the}

decay width is proportional to the light quark mass difference squared, (md-mu)2. In addition this

decay can be used to search for C-violating effects. The analysis is presented and the Dalitz plot parameters with statistical and systematical uncertainties are determined from a sample of 1.33·105 η→π+π-π0 events in the Dalitz plot. The asymmetry parameters with statistical uncertainties are presented which show no evidence of C-violation.

Keywords: Hadron physics, Chiral Perturbation Theory, Isospin violation, Quark masses,

Symmetries, Advanced Instrumentation

Patrik Adlarson, Uppsala University, Department of Physics and Astronomy, Nuclear Physics, Box 516, SE-751 20 Uppsala, Sweden.

urn:nbn:se:uu:diva-181236 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-181236) Printed in Sweden by Holmbergs, Malmö, 2012

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1. Introduction

The current understanding of how the world works physically at small scales is summed up in the theoretical framework called the Standard Model of elementary particle physics [CB07, MEP95, AP00]. It describes how the elementary particles, quarks and leptons, interact with each other. There are nineteen free parameters in the Standard Model. That is, these parameter values are not derived from the Standard Model itself, but their numerical values have to be measured by performing high precision experiments. For example the values of the up, down, and strange quark masses are such free parameters which are to be determined. These three quarks are examples of the possible ﬂavors that quarks may have, the others being charm, bottom and top. The fundamental forces between the elementary particles are mediated via spin-1 particles called vector bosons. For the electromagnetic force this particle is the photon, the weak force is mediated by W± and Z0 and the gluons are the mediators of the strong force. The fourth fundamental force in nature, gravitation, is not included in the Standard Model.

The strong force is described by Quantum Chromo Dynamics (QCD) which is a non-abelian gauge field theory where the quarks and gluons are the relevant degrees of freedom. Two important properties in QCD are asymptotic freedom and color confinement. Asymptotic freedom concerns the phenomenological finding that when going to small distances quarks and gluons behave like non-interacting particles. This is equivalent to saying that the coupling of the strong force becomes weaker. For greater distances between particles the strong force increases. Since there is an inverse relation between distance and energy, the strength of the coupling is energy dependent. A simplified analogy of asymptotic freedom and confinement is to imagine two balls attached together with a rubber string, where the balls represent the quarks and the rubber string represents the strong force mediated by the gluons. When the balls are close together, the string does not do any work on the balls and they may move about unaffected, but as the balls are stretched apart the rubber string will pull on the balls.

Quarks and gluons have internal degrees of freedom, denoted color charge and their possible values are red, green and blue. Color and ﬂavor are not meant to be understood as properties which a person sees with his eyes (color) and tastes with his mouth (ﬂavor), but rather as useful

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representations of the quarks. Color confinement (or confinement) describes the phenomenological finding that single quarks can not be isolated but only exist as composites confined in color neutral (white) objects like hadrons. The varying strength of the coupling constant for varying energies also affects the way systematical studies are undertaken. When the strength of the coupling constant is small it is possible to perform systematical improvements by doing perturbative calculations in QCD, but for increasing values of the coupling constant, i.e. for lower energies, this method looses its accuracy. This does not mean that theoretical predictions in lower energy regimes are impossible. One possibility is to construct different models for different types of processes. Another, more preferable method, is to construct an effective field theory for which improvements may be calculated systematically. In an effective field theory the underlying symmetries are the same as in the fundamental theory but the degrees of freedom which are used are those relevant for that energy scale [SW79]. Instead of expanding in powers of the coupling constant as is done in QCD, expansions are made in powers of momentum.

In the study of nature symmetries are often of central importance. Sometimes the symmetries are, to the best of our knowledge, exact and sometimes they are approximate. The mathematician Emmy Noether proved that if there exists a continuous symmetry this leads to a conserved current [EN18]. Symmetry means that the formalism which describes the dynamical system, i.e. the Lagrangian, stays unchanged when the generalized coordinates of the system are transformed in a speciﬁc way. By identifying the symmetries in nature, the description of the system becomes simpler but it has also greater explanatory power for physics since it explains many aspects seen in nature. For instance baryon number and ﬂavor number are such symmetries which to our knowledge never are broken in strong interactions. A symmetry which arises if the up and down quark masses were equal is isospin symmetry. One consequence of this symmetry is the existence of degenerate states, so called isospin multiplets. This is also what is observed in nature- for example the proton and neutron form an isospin doublet and theπ+,π0,π− form an isospin triplet. By also including the strange quark and assuming the masses to be equal even larger multiplets can be formed [GZ64]. Mathematically the up, down and strange quarks form an SU(3)f lavor group. The existence

of particle multiplets have been found quantitatively in nature. Isospin and SU(3)f lavor are only approximate symmetries, because the quark masses are

not equal to each other.

Even though quarks are massive it is helpful to consider symmetry properties if the quark masses are considered massless. The mu and md and, to a

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interactions. Therefore it makes sense to study the chiral limit, i.e. the limit when quarks are massless. In this limit QCD possesses a chiral symmetry SUL(3) × SUR(3), where L and R stands for the left- and right-handedness,

respectively. For massless fermions chirality is conserved, i.e. a left handed (right handed) quark is always left handed (right handed) which means that the QCD Lagrangian is unchanged if the massless left-handed (right-handed) up-, down- and strange-quarks are mixed. Another way to write this symmetry is by introducing vector and axial-vector transformations, denoted by V and A which are linear combinations of the right- and left-handed transformations, so SUL(3) × SUR(3) = SUA(3) × SUV(3). If the axial and

vector symmetries had been realized in nature, there would exist particle pairs with similar properties except for having opposite parity. This is not what is observed in nature. Instead what is found phenomenologically is an approximate SUV(3) symmetry in the form of hadronic multiplets. This is the

ﬂavor symmetry discussed before. The absence of parity partners is explained by spontaneous symmetry breaking. SUL(3) × SUR(3) is reduced to an

SUV(3) symmetry. This generates eight massless pseudoscalar Goldstone

bosons which are identified as theπ, K and η observed in nature. But these particle states are not massless and this is because quarks are massive, due to an explicit symmetry breaking. The masses are small enough compared to typical hadronic mass scales to construct an effective field theory for the Goldstone bosons as the lightest physically realized states. This theory can generate accurate predictions as long as the symmetry breaking is included in the formalism. The effective field theory of QCD at low energies is called Chiral Perturbation Theory and uses these Goldstone bosons as the relevant degrees of freedom [SW79, JG85, SS02, BK07, SL06, SL11].

For a deeper understanding of the Standard Model the quark masses are cru-cial. Therefore it is of interest to determine the masses of the light quarks to a great precision. But, as mentioned before, the quarks are conﬁned inside hadrons and the three lightest quark masses are small compared to typical hadron masses. This makes it a challenge to determine their numerical values [FLAG11]. Therefore processes are studied which are particularly sensitive to the values of the light quark masses. The squares of theπ, K and η-masses are to lowest order directly related to the quark masses which compose these particles [JG85]. But it is possible to obtain even more precise constraints. This can be achieved by studying theη → 3π decay. This is related to the fact that the isospin symmetry is not fully realized in nature but is broken. Had this symmetry not been broken this decay would not occur by the strong interac-tion. If theη → 3π decay is well understood then it is possible to understand the effect of isospin breaking. The isospin breaking occurs because up and down quark masses are not equal and a parameter directly related to the quark mass difference may be measured to high accuracy. To understand the decay both theoretical and experimental input are needed. This is an experimental

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thesis and concerns speciﬁcally the decayη → π+π−π0which was measured using the WASA detector setup at the COSY ring, situated in Jülich, Germany. This work is divided into seven chapters:

1. This introductory chapter.

2. Contains a theoretical motivation as well as a review of previous experi-mental results forη → π+π−π0.

3. Describes the WASA detector.

4. Describes how the events are selected and reconstructed for simulations and in the experiment.

5. Describes the analysis to obtain the ﬁnalη → π+π−π0candidates. 6. Gives the results of the analysis.

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2. Motivation, theory and previous

experimental results

2.1 General properties of

η

Theη meson was ﬁrst discovered in 1961 at the Bevatron of the Lawrence Berkeley National Laboratory as a three-pion resonance in theπ+d-reaction at 1.23 GeV/c [AP61]. The η meson is a pseudoscalar particle which means that it has intrinsic spin 0 and transforms negatively under parity transformations. The η-meson is part of the group of light pseudoscalar mesons, with its partners beingπ0,π±, K0, ¯K0 , K± andη, shown in ﬁgure 2.1. Its quantum numbers are given by IG(JPC) = 0+(0−+) where I,G,J,P and C denote the isospin, G-parity, angular momentum, parity, and charge parity, respectively. It has a mass of 547.853(24) and it has a decay width of 1.30(7) keV which gives it a mean lifetime τ = 5.1(3) · 10−19s [PDG12].

Figure 2.1: The lightest pseudoscalar meson nonet, where Q is the electric charge and S is the strangeness of the particle. Figure obtained from [SW10].

Following the quark model, mesons are combinations of q ¯q states. In the SU(3)f lavor description q= u,d,s are considered to be equal in mass. The

9 possible q ¯q states are structured into an octet and a singlet, 81. The phys-ical particleη is mainly a mixture of the octet and singlet states ψ8andψ1,

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η − decays fraction η → γγ (39.31 ± 0.20) · 10−2 η → π0_π0_π0 _{(32.57 ± 0.23) · 10}−2 η → π+_π−_π0 _{(22.74 ± 0.28) · 10}−2 η → π+_π−_γ _{(4.60 ± 0.16) · 10}−2 η → e+_e−_γ _{(6.9 ± 0.4) · 10}−3 η → μ+_μ−_γ _{(3.1 ± 0.4) · 10}−4 η → π0_γγ _{(2.7 ± 0.5) · 10}−4 η → π+_π−_e+_e−_{(γ) (2.68 ± 0.11) · 10}−4 η → μ+_μ− _{(5.8 ± 0.5) · 10}−6

Table 2.1: Selectedη decays and their branching ratios [PDG12].

The statesψ1andψ8are given by the quark-antiquark combinations

ψ1=√1 3 u ¯u+ d ¯d+ s¯s, (2.2) ψ8= 1 √ 6 u ¯u+ d ¯d− 2s¯s, (2.3) andφ is a mixing angle in the singlet-octet basis which is determined experi-mentally, for example from radiative φ decays: −13.3(3)stat(7)syst(6)th◦ [FA07]. To a much smaller extent theη has a contribution from √1

2(u ¯u − d ¯d)

which will give rise to the isospin breaking decay studied in this thesis. The main decays ofη are either via 3π or radiative, and comprise 99% of all cases. The decays are presented in table 2.1 [PDG12].

2.2 η → π

+

π

−

π

0

- theoretical aspects

In strong reactions, charge conjugation is exact and isospin symmetry is slightly broken. In the following the impact of these symmetries and their possible breaking on theη → 3π decay is discussed. The decay η → 3π is a strong G-parity violating decay. G-parity is deﬁned as charge conjugation following a rotation of 180◦ around the second axis in isospin space. The operator is given as

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The eigenvalue of G-parity, ηG is most easily determined from the neutral

member of an isospin multiplet (i.e.π0for theπ-triplet) and is given as

ηG= (−1)IηC, (2.5)

whereηC is the eigenvalue of the charge conjugation operator. For the pions

ηG(π) = (−1)I=1ηC= (−1)I=11= −1 (2.6)

and forη

ηG(η) = (−1)I=0ηC= (−1)I=01= +1. (2.7)

Therefore neitherη → 3π0norη → π+π−π0conserve G-parity.

Isospin forη is given by |I,I3 = |0,0. If it is not assumed that the decay

η → 3π conserves C, isospin and C have to be discussed separately and not be fused into G-parity. First the decayη → 3π0is discussed. In general, two pions can couple to I= 0,1,2. However, (π0π0)I=1 is excluded because the

Clebsch-Gordan coefﬁcient vanishes. On the other hand, the only way to get I= 0 for the three pion system needed for η would be (π0_π0₎

I=1with which

the thirdπ0 could couple to I= 0. Thus isospin symmetry must be violated forη → π0π0π0.

Next, η → π+π−π0 is discussed. Isospin for π+π−π0 in a |I,I3 = |0,0 state can be written as the combination

|π+_π0_π−_{+ |π}0_π−_π+_{+ |π}−_π+_π0_{− |π}+_π−_π0_{− |π}0_π+_π−_{− |π}−_π0_π+

[BH07]. By exchange of two isospin states with each other, i.e. + ↔ −, + ↔ 0 or − ↔ 0 it is seen that the isospin part of the wave function is antisymmetric. If the charge conjugation is applied to the wave function then

C(π+π−π0)I_I=0₃₌₀= −1 · (π+π−π0)I_I₃=0₌₀, (2.8) which is in contradiction to the assignment of C(η) = +1. On the other hand, C is conserved if the pions are in an isospin I= 1 state

C(π+π−π0)I_I=1₃₌₀= +1 · (π+π−π0)I_I₃=1₌₀, (2.9) but this would mean that isospin I is not preserved in the process. Hence the decay η → π+π−π0 is either C- or I-violating. The possibility that η → 3π proceeds via electromagnetic interaction was ruled out already in the 1960s when the electromagnetic contribution was found to be small [DS66, JC67, HO70]i. Instead it was found that a strong isospin breaking part proportional to the quark mass difference (md− mu) is responsible for the

decay. This can also be seen in the modern language of Chiral Perturbation

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Theory, as follows.

2.2.1 η → π

+

π

−

π

0

at tree level

At low energies the strong coupling constant,αs, becomes large and

perturba-tive expansions of QCD fail to converge. Chiral Perturbation Theory (ChPT) is an effective ﬁeld theory describing the strong interactions at low energies [JG84, SS02]. It expands the amplitudes in powers of momenta

LChPT = L2+ L4+ L6+ ... (2.10)

where Ln stands for the n-derivatives or equivalent. Lorentz invariance

restricts the theory so that only even powers of derivatives appear in the series. At each new level more low energy constants are needed in order to ﬁx re-normalizability. AtL2 there are only two constants at 3-ﬂavor structure

(B0 and fπ), 10 (+2 contact terms) atL4 and 90 (+4 contact terms) at L6

[JB05]. The low energy constants include the information from physics at higher energy scales. The values of these are determined from experimental results or can in principle be calculated from QCD.

The lowest order (LO) Lagrangian in Chiral Perturbation Theory is LChPT,LO= f 2 π 4 tr(∂μU †_∂μ_U_{) +} fπ2B0 2 tr(M(U †_+U)), _(2.11)

where f_π is the pion decay constant 92.2 MeV, B0 is a constant that enters

in the masses of the pseudoscalars, U is a unitary matrix which contains the pseudoscalar ﬁelds, U= eiΦ/ fπ_{, Φ =} ⎛ ⎜ ⎜ ⎝ π3+√1₃η8 √ 2π+ √2K+ √ 2π− −π3+√1₃η8 √ 2K0 √ 2K− √2 ¯K0 −√2 3η8 ⎞ ⎟ ⎟ ⎠ (2.12)

and M is the quark mass matrix

M= ⎛ ⎜ ⎝ mu 0 0 0 md 0 0 0 ms ⎞ ⎟ ⎠. (2.13)

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To obtain the decay width at tree levelii theΦ-ﬁelds are expanded to fourth order U= eiΦ/ fπ _{= 1 +}iΦ f_π − Φ2 2 f2 π − iΦ3 6 f_π3 + Φ4 24 f4 π + ..., (2.14) U†= e−iΦ/ fπ _{= 1 −}iΦ f_π − Φ2 2 f2 π + iΦ3 6 f_π3 + Φ4 24 f4 π (2.15) and inserted into equation (2.11). Both the kinetic term and the mass term give contributions to the matrix element. By rotating the contributions ofη8andπ3

to the physical ﬁeldsη and π0

π3= cos(θ)π0− sin(θ)η (2.16)

η8= sin(θ)π0+ cos(θ)η (2.17)

the mass term contribution becomes 1 3√3 B0 f2 π(mu− md) cos2(θ) − sin2(θ)ηπ+π−π0. (2.18) The contribution to the decay from the derivative term is

− 1 3 f2 π[−2sin(θ)cos(θ)[(∂μπ 0_)(∂μ_η)π+_π−_{+ (∂} μπ+)(∂μπ−)π0η] + sin(θ)cos(θ)[(∂μπ0)(∂μπ+)ηπ−+ (∂μη)(∂μπ+)π0π−+ (∂μπ−)(∂μπ0)ηπ++ (∂μπ−)(∂μη)π+π0]],

and this term together with (2.18) constitutes the interaction term,Lint. The

value of the mixing angle is small and is given by θ = √ 3 2 (md− mu) (2ms− md− mu). (2.19) In addition, from equation (2.11) the mass relations

(m2 π+)QCD= B0(mu+ md), (m_π20)QCD= B0(mu+ md) + O(θ2), (m2 η)QCD= B0 3 (mu+ md+ 4ms) + O(θ 2_{), (m}2 K+)QCD= B0(mu+ ms), (m2 K0)QCD= B0(md+ ms), (2.20)

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can be derived. This implies that, (m2 K0− m2_K+) = B0(md− mu), (m2_π0− m2_π+)QCD∼ 0, (m2 η− m2π0)QCD∼2B0 3 (2ms− md− mu), (2.21) from which θ = (m2K0− m2_K+)QCD √ 3(m2 η− m2π)QCD . (2.22)

A numerical estimate may be obtained by describingθ in terms of the physical meson masses. To do this the inﬂuence of electromagnetism on the meson masses must be included. This is done by invoking the Dashen relation [RD69]. It says that the electromagnetic contribution to the mass difference between the neutral and charged states are the same for pions and kaons:

(m2

K0− m2_K+)em= (m_π20− m2_π+)em. (2.23)

If this relation holds then the mass difference, md− mu, can be described

as (m2

K0−m2_K+)QCD+(m2_K0−m2_K+)em−(m2_π0−m2_π+)QCD−(m2_π0−m2_π+)em=

= (m2

K0− m2_K+)QCD− (m2_π0− m2_π+)QCD∼ B0(md− mu). (2.24)

The mixing angle in terms of meson masses is obtained from equations (2.20) and (2.24) which gives [JG85]

θ = m 2 K0− m2_K++ m2_π+− m2_π0 √ 3(m2 η− m2π) = 0.0106, (2.25)

where all small effects in the denominator have been neglected. To obtain the matrix element,Lintis transformed into momentum space

1 3 f2 π B0 √ 3(mu− md) + θ · f cn(pη, pπ+, pπ−, pπ0) , (2.26)

where pxrepresents the four-momentum(E, px, py, pz) for particle x. With the

approximations given above,θ becomes θ =_√B0

3

(md− mu)

(m2

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Using this relation and the relationships,

p_η = p_π++ p_π−+ p_π0, (2.28) m2_η = (p_π++ p_π−+ p_π0)2, (2.29)

and p2_π_±,0= m2_π, the matrix element is simpliﬁed to ALO(s,t,u) =B0(mu− md) 3√3 f2 π 1+3(s − s0) m2 η− m2π , (2.30)

where s,t and u are the Mandelstam variables,

s= (p_π++ p_π−)2= (p_η− p_π0)2= m2_π+_π−, (2.31)

t= (p_π−+ p_π0)2= (pη− p_π+)2= m2_π₋_π0, (2.32) u= (p_π++ p_π0)2= (pη− p_π−)2= m2_π−_π+, (2.33)

and s0= 1₃(m2η+ 2m2π++ m2_π0). This is the tree-level result for the amplitude

[JC67, JB68, JG85]. It is thus seen that in the isospin limit, where mu= md

the tree level amplitude would give no contribution.

The decay width is proportional to the amplitude squared,Γ ∝ |A (s,t,u)|2. This givesΓLO≈ 70 eV when integrating over the available phase space. This

result is equivalent to the Current Algebra result [WB67]. The graph that con-tributes at tree level is shown in ﬁgure 2.2. The tree level result differs by a factor 4.5 from the PDG result 296± 16 eV [PDG12]. To get a better corre-spondence between theory and experiment, higher order terms in ChPT be-come important (section 2.2.5).

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2.2.2 Dalitz plot

To measure how well theoretical predictions compare to experimental results suitable observables must be used. A Dalitz plot [RD51] is a convenient way to present the phase space for a three-body decay using two kinematical variables and it contains the complete information on the dynamical behaviour of the decay particles. For η → π+π−π0 the variables used to describe the Dalitz plot are denoted X and Y ,

X =√3T+− T− Q_η = √ 3 2m_ηQ_η(u −t) (2.34) and Y =3T0 Q_η − 1 = 3 2mηQη (mη− mπ0)2− s − 1, (2.35)

where (s,t,u) are the Mandelstam variables and T_π+, T_π−, T_π0 are the kinetic

energies of the pions in theη-rest frame. Q_η is the excess energy

Q_η= m_η− 2m_π±− m_π0 (2.36)

or

Qη= Tπ++ T_π−+ T_π0. (2.37)

The choice of these variables gives a (nearly) symmetrical Dalitz plot around X= Y = 0. The squared amplitude for η → 3π is expanded around X = Y = 0 and is described by |A (X,Y)|2_{= A}2 0 1+ aY + bY2+ cX + dX2+ eXY + fY3+ gX2Y+ ..., (2.38) where A2₀ is a normalization factor and a,b,...,g,.. are parameters which are used to polynomially ﬁt the Dalitz plot density. Non-vanishing parameter values for odd-powered X terms, like c and e, would imply charge conjugation violation, since exchangingπ+andπ−changes X to−X. Since the Dalitz plot can be obtained via experimental measurements it serves as an important check of the predictive power of ChPT. The Dalitz plot pa-rameters for various theoretical predictions are given in table 2.3.

2.2.3 Q and R

Using the assumption of Dashen’s theorem (2.20) together with the approxi-mation that m2_π+= m2_π0= m2π it is possible to write the amplitude (eqn 2.30)

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in terms of meson masses where the quark masses appear in one term, Qiii, as A (s,t,u) = 1 Q2 m2 K m2 π(m 2 K− m2π) M (s,t,u) 3√3 f2 π (2.39) where M (s,t,u) = 3s− 4m2π m2 η− m2π, (2.40) m2_K =1 2(m 2 K0+ m2_K+), m2π= 1 2(m 2 π0+ m2_π+) (2.41) and Q2= m 2 s− ˆm2 m2 d− m2u , ˆm = 1 2(mu+ md). (2.42) Furthermore, it is possible to describe Q itself as a function of meson masses which gives Q2= (ms+ ˆm)(ms− ˆm) (md+ mu)(md− mu)∼ Q 2 D= (m2 K+12(m2π0− m2_π+))(m2_K− m2π) m2_π0(m2_K0− m2_K+) = (24.2)2_. (2.43) If ˆm is neglected in the numerator of equation (2.42), it may be rewritten

m2 u m2 d + 1 Q2 m2 s m2 d = 1. (2.44)

Equation (2.44) describes an ellipse in the mu

md,

ms

md plane where Q deﬁnes

the major semi-axis and gives a constraint on the allowed quark mass ratios [HL96]. An example is shown in ﬁgure 2.3 for Q= 22.7 ± 0.8 [AA96] with the allowed region for the mass ratios of the three lightest quarks from PDG [PDG12].

The numerical value of(m_K0− m_K+)_em, assuming that Dashen’s relation in

equation (2.20) holds, is 1.3 MeV [GC10]. Higher order corrections and cal-culations using different models modify the numerical value of Dashen’s the-orem, which leads to a modiﬁed Q. Modiﬁcations have been determined by ChPT together with a 1/Nc approach [JB93], incorporating chiral symmetry

with Vector Meson Dominance models [JD93], inclusion of ChPT contribu-tions arising from resonances within a photon loop [RB96b] or estimates from Lattice QCD calculations [FLAG11]. The value of Q ranges from 20.6 Q 24.2 [CD09], also seen in ﬁgure 2.4. Studying η → 3π is a precise way to determine the value of Q, since electromagnetic contributions are negligible

iii_{Not to be confused with Q}

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Figure 2.3: Constraints for the light quark mass ratios. The light gray boxed area are the constraints as given by the Particle Data Group, [PDG12], and the light gray ellipse represents equation (2.44) with Q= 22.7 ± 0.8 [AA96]. The darker shaded gray area shows the overlap of these constraints and thus the allowed region for quark mass ratios.

(see section 2.2.4). WithΓ ∝ |A (s,t,u)|2the relation Γ = QD Q 4 ¯ Γ (2.45)

is obtained, where ¯Γ is the η → 3π decay width calculated from ChPT in the Dashen limit, QDis the Q calculated in the Dashen limit (eqn. 2.43) andΓ is

the experimentally measured decay width. As

Q QD

₋₁

comes with a power of four, a precision value of Q can be obtained from a precision measurement ofΓ, if ¯Γ is known. The theoretical calculation has to reproduce the decay dynamics and since the complete dynamical information is contained in the Dalitz plot distribution it is a suitable testing ground [AK09].

Another way to parametrize the up and down quark mass differences is in terms of R [HL96],

R= ms− ˆm

md− mu, (2.46)

where the relation to Q is

Q2=1 2(1 +

ms

ˆ

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Figure 2.4: Q as deﬁned in equation (2.42) as a function of the electromagnetic mass splitting,Δmem_K in the kaon system. The Dashen value is at Q= QD= 24.2 The other

analyses, from left to right are from [AD96], [JB97], [JD97] and [BA04]. Picture from [SLaTh11].

The constraint that this gives on the ms

md versus

mu

md plane is shown in ﬁgure 2.5.

Determinations for both Q and R fromη → 3π for various different theoretical approaches are shown in table 2.2. Good agreement is seen when comparing the different theoretical estimates.

2.2.4 Electromagnetic corrections to

η → π

+

π

−

π

0

Electromagnetic corrections can contribute both indirectly or directly. Indi-rectly via affecting parameters such as the pion decay constant f_πor directly, speciﬁcally for theη → 3π-decay [RB96]. At tree level the electromagnetic interaction can be described by an effective Lagrangian of the form

L_{e f f}(2),em= C· < QchUQchU†>, (2.48)

where U is the unitary 3× 3 matrix with the pseudoscalar meson ﬁelds, Qch is the charge matrix of the three light quarks and C is a low energy

constant which determines the electromagnetic contribution to the masses of the charged mesons in the chiral limit [GE89]. The tree level Lagrangian (eqn. 2.48) gives only an equal contribution to the charged pion and kaon masses which corresponds to Dashen’s theorem [RD69]. There is no contribution at leading order to the decay η → π+π−π0, in agreement with Sutherland’s theorem [DS66]. Hence both Suther-land’s theorem and Dashen’s theorem are derived within the ChPT framework.

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Reference Q R

LO [JB07] 15.6 19.1

NLO [JB07] 20.1 31.8

NNLO [JB07] 23.2 42.2

dispersive calc. [AA96] 22.7 ± 0.8

-dispersive calc. [JK96] 22.4 ± 0.9

-dispersive calc. (PLM) [KK11] 23.1 ± 0.7 37.7 ± 2.2

dispersive calc. (BLV)1[SLa11] 22.7 ± 0.7

-dispersive calc. (BLV)2[SLa11] 21.3 ± 0.6

-Lattice QCD av. [FLAG11] 22.8 ± 1.2 36.3 ± 3.8

Table 2.2: Values of Q (eqn. 2.42) and R (eqn. 2.46) obtained from theη → 3π decay.

In addition a lattice QCD estimate of Q and R is shown for comparison. The subscript i=1,2 for (BLV)_idenotes the dispersive method when matching to the one-loop calcu-lation (i=1) [JG85] and when matching to the KLOE data (i=2) [FA08]. PLM stands for Prague-Lund-Marseille and BLM stands for Bern-Lund-Valencia.

Higher order electromagnetic contributions ofO(e2p2) were considered up to L_{e f f}(4),embut the contributions were found to be at most of a few percent com-pared to the NLO result [JG85] and do not therefore change the theoretical result signiﬁcantly [RB96]. Recently also terms of the orderO(e2(mu− md))

were studied [CD09]. The inclusion of these terms compared to the NLO re-sult also give corrections on the percent-level . This means that theη → 3π decay can be used as a probe for the isospin breaking effect, via the measure-ment of Q .

2.2.5 η → π

+

π

−

π

0

at order p

4

and higher

To evaluate the next-to-leading order amplitude, ANLO(s,t,u), one-loop

diagrams from L2 and tree diagrams from L4 have to be calculated

[JG85]. The types of interactions needed at order p4 are shown in ﬁgure 2.6 [JB02]. η → π+π−π0 _{calculated at the one-loop level gives}

Γη→π_NLO+π−π0 = 160 ± 50 eViv_{. This is an enhancement of the decay rate}

from tree-level with a factor 2.4, where half of the additional contribution comes fromππ re-scattering between ﬁnal state pions [CR81, JG85]. Vertex corrections and tree graphs are the other contributions. The next-to-leading

iv_{Two historical remarks regarding this result. In 1985 when this result was published}_Γ

EXP=

197±29 eV. The theoretical prediction was thus compatible with the experimental decay width at the time. Second, in [JG85] the pion decay constant value 93.3 MeV was used, slightly larger than today’s value of 92.2 MeV [PDG12].

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Figure 2.5: The plane of quark mass ratios ms

md and

mu

md with the elliptical constraint

Q= 22.7(8), Δm which accounts for the breaking of SU(3) and the interval of R =

[38,44] as deﬁned by equation (2.46). Constraints on R and Δm discussed in [HL96],

from where the ﬁgure was obtained. The dot is an estimate from Steven Weinberg [SW77] and the cross is an estimate in [JG82].

order (NLO) corrections are large but still not sufﬁcient to be in agreement with the experimentally measured decay width,Γη→π_EXP+π−π0 = 296 ± 16 eV [PDG12].

The next-to-next-to-leading (NNLO) order calculations have also been performed [JB07]. The ﬁrst motivation for the calculation was to study if the full NNLO contributions are large compared to the NLO calculation. The second motivation was to study the convergence properties of SU(3)f lavor

which was expected to be less convergent compared to SU(2)f lavordue to the

relatively large strange quark mass. The change on the Dalitz plot parameter values from NLO to NNLO is not very large, especially not for the charged parameters d and f , but the overall normalization, A2₀, increases with 70%. There is no direct value for the NNLO decay width, but can be extracted together with R= 38.5 [FLAG11] which gives Γη→π_NNLO+π−π0= 339 eV [JB12]. The Dalitz plot parameter values for the ChPT NLO and NNLO calculations are given in table 2.3. The uncertainty of the NNLO result is estimated to be one half of the NNLO contribution. The higher order calculations contain many low energy constants which have to be determined from experiment or from lattice calculations. Uncertainties in determining these values also affect the predictivity of ChPT [KK11]. What is

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seen in section 2.3 is that the Dalitz plot parameter values from Chiral Perturbation Theory are not in good agreement, or at least ambiguous, with respect to the experimentally measured values. This is one of the moti-vations for why other approaches have been developed, to which we turn next.

Figure 2.6: The Feynman diagrams at order p4in NLO ChPT. Figure obtained from [JB07].

2.2.6 Dispersive treatments

By studying the NLO result [JG85] it is seen that a sizable contribution comes fromππ-rescattering. If it is assumed that the dominant contribution to the amplitude comes from pion-rescattering, other approaches may be employed [KK11]. One approach is to study theππ ﬁnal state re-scattering using disper-sive methods which connects the imaginary part of the decay amplitude to the amplitude itself. The amplitude toO(p6) can be split up into the rescatterings of two of the ﬁnal state pions with given isospin,

A (s,t,u) = A0(s) + (s − u)A1(t) + (s −t)A1(u) + A2(t) + A2(u) −2

3A2(s), (2.49) where AI represents the amplitude contributions with isospin I = 0,1,2

and s,t,u are the Mandelstam variables (eqns. 2.31-2.33) [AA96]. If the amplitude is described as an analytic function it is uniquely characterized by its singularities and can be represented as integrals over the discontinuities. With unitarity and analyticity it is possible to determine the amplitude via the ππ phase shifts up to some subtraction constants. These subtraction constants can be determined with a ﬁt to the ChPT NLO calculations [AA96, AK09]. Two theoretical results from 1996 exist using dispersive methods [AA96, JK96]. Their difference lies in the formalism and choice of subtraction constants but they use the same assumptions and approximations [JB02]. Both calculations give a small total enhancement with a 14% increase of the total decay rate in [JK96] and a 5% enhancement in the centre of the Dalitz plot in [AA96] with respect to the NLO result [JG85]. A simpliﬁed dispersive method was also performed in [JB02] called tree dispersive and absolute dispersive method.

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In recent years new dispersive calculations have emerged. The Bern-Lund-Valencia dispersive treatment [SLa09, SLa11, SLaTh11] follow closely the method proposed in [AA96]. The motivations for re-doing the calculations from 1996 are among other things more precisely determinedππ phase shifts, several experimental measurements onη → 3π (mostly for the neutral decay) and the NNLO ChPT calculations. Another recent dispersive treatment is the Prague-Lund-Marseille approach [KK11]. Their approach is to parametrize theη → 3π amplitude into a polynomial part, Px, and a unitary part, Ux,

Mx(s,t,u) = Px(s,t,u) +Ux(s,t,u). (2.50)

Px(s,t,u) contains six free parameters corresponding to the symmetric

expansion around the Dalitz plot center and Ux(s,t,u) can be expressed in

terms of these six parameters as well as with a few additional analytical parameters which describe the ππ-scattering. Both of the newer dispersive approaches may either try to estimate the Dalitz plot parameters by matching to the ChPT calculations (NLO for [SLa11], NNLO for [KK11]), but they may also estimate the Dalitz plot parameters by using experimental data as input. This has to be supplemented with some points or regions in ChPT where the amplitude is expected to be reliable. The region where the real part of the amplitude is small is called the Adler region ([SA65a, SA65b]) and [SLa11] uses the region around the Adler zero for matchingv, while [KK11] rely less on this region and use other parts of the amplitude for matching. The estimate of R and Q for the dispersive calculations are in table 2.2 and show fairly good agreement for Q and R compared to the other theoretical estimates. A difference compared to the estimates of Q and R from lattice QCD (row 9, table 2.2) is that the uncertainties of the predicted values have smaller uncertainties.

2.2.7 Non-Relativistic Effective Field Theory

The modiﬁed Non-Relativistic Effective Field Theory (NREFT) is a framework which is used to study low-energy scattering and decay processes, which have been used in studies of cusp effects in K → 3π [GC06, MB08, MB09],η → 3π0 [CG08] andη→ ηππ [BK09]. The term non-relativistic refers to the fact that inelastic thresholds outside the physical region are summed up into effective coupling constants [SS10].

The advantages of NREFT compared to ChPT is its transparency of the analytic representation as well as its efﬁciency in including isospin breaking which are more involved in ChPT and unexplored in dispersive relations [BK10, SS10]. A disadvantage compared to dispersive calculations is that

v_{The Adler zero is for the value os s where}_A

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while NREFT is only calculated up to two loops, dispersion relations resum to higher orders also. NREFT and dispersive calculations are not predictive in the sense that they can not independently give information on the size of the isospin breaking. Both need to match to ChPT for proper normalization. A disadvantage with respect to dispersion relations is that NREFT only works well inside the physical Dalitz plot, whereas dispersive relations also should work in regions beyond the Dalitz plot region [KK10]. In [AA96] and [SLa11] this region is used for matching, but this is not feasible to do for NREFT since it is too far from the physical region. That may be seen as a disadvantage depending on how reliable this region is expected to be. In the NREFT approach matching is done to ChPT NLO at the center of the Dalitz plot. Just as for dispersive calculations physically measured scattering parameters are used in calculations. To judge which diagrams are expected to be more important a power-counting scheme is used in terms of a non-relativistic parameter ε and the effective range parameter a_ππ based on ππ-scattering characteristics. The calculations in [SS10] include orders up to O(ε4), O(a_ππε4), O(a_ππ2 ε4) and partial inclusion of O(a2_ππε6) and O(a2

ππε8). A result for the understanding of charged Dalitz plot parameters

relates to the sizable isospin breaking shifts in the Dalitz plot parameters. The center of the experimental Dalitz plot X= Y = 0 does not coincide with s= t = u. In fact the corresponding point gives Y = −0.052 for s = t = u [JB07]. The central values of the theoretical predictions come somewhat closer to the experimentally obtained parameters compared to the NNLO calculation. The Dalitz plot parameter values for NREFT are shown in table 2.3.

2.2.8 Resummed ChPT

Chiral perturbation theory is a perturbative scheme. The basic assumption is that calculations converge when including higher order diagrams into the cal-culations. Resummed ChPT investigates whether or not this assumption is valid or if ChPT actually converges slowly or irregularly. Therefore resummed ChPT is not a new alternative method to ChPT as the dispersive methods or NREFT, but rather an approach to study the convergence properties and as-sumptions of ChPT [SDG04, SDG07, MK08, MK11]. This can be done by expressing low-energy constants in terms of free parameters and studying the effect on the Dalitz plot parameters when varying the free parameters. Some work has been done onη → 3π, on the charged Dalitz plot parameter b and the neutral parameterα (section 2.5) but more comprehensive work is in prepara-tion [MKXX].

2.2.8.1 Relativistic coupled-channels approach

A more model dependent analysis that also looks at s- and p-wave re-scattering with a U(3) framework in combination with a relativistic

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couple-channels approach ﬁnds reasonable agreement with the experimental data (called BSE in table 2.3) [BB05]. This is based on a ﬁtting procedure which uses the information of several hadronic η and η decay channels, including η → 3π. The Dalitz plot parameter values obtained with this method is therefore not considered to be an unbiased prediction [SS10].

2.2.9 Other considerations: Decay width

As was shown in eqn. (2.45), precise knowledge of theoretical calculations for the decay width under the Dashen assumption can give an accurate estimate of Q, provided that the experimental decay width is precisely determined. How-ever, experimental measurements mainly measure branching ratios. Therefore Γtot also plays a role. The main contribution to the total width besidesη → 3π

isη → γγ. Therefore a more precise determination of the decay width also plays a role in determining Q [KK10].

2.3 η → π

+

π

−

π

0

- previous experiments

There have been some experimental measurements made. The measurements with higher statistics which will be described are [MG70], [JL72], [JL73], [MJ74], [CA95], [AA98] and [FA08], where [FA08] is the most important. In the late sixties and early seventies some other measurements were also made, but the statistics in those experiments were rather poor, ranging from approximately 500 to 7000 events in the Dalitz plot. In addition, the ﬁts to obtain the parameter values were only performed with a constant and a linear term in Y . The discussion of these results have been omitted. The experimental values and their references can be found in [JL73].

The ﬁrst three experimental results mentioned, [MG70], [JL72], [JL73], were measured in the Princeton- Pennsylvania detector with η from the reaction π−_p_{→ ηn where the neutron was detected in a time of ﬂight detector and}

the charged pions were detected in sonic chambers. The first experimental result, [MG70], contained 30 000 events in the Dalitz plot. For each bin a flat background subtraction was performed and divided by the average detection efficiency determined from Monte Carlo. They found a result with a slight charge asymmetry, or a linear term in X.

A compatible result was obtained when folding the plot about X= 0 and in-stead ﬁtting the parameters to|M(|X|,Y)|2= 1+aY +bY2+dX2. Both results are presented in table 2.3. After improving the experiment a new measure-ment was carried out with signiﬁcantly higher statistics [JL72]. The

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right-left-Prediction -a b d f g LO 1.039 0.27 0.000 0.000 -NLO 1.371 0.452 0.053 0.027 -NNLO 1.271(75) 0.394(102) 0.055(57) 0.025(160) -dispersive [JK96] 1.16 0.26 0.10 - -tree disp [JB02] 1.10 0.31 0.001 - -abs disp [JB02] 1.21 0.33 0.04 - -dis. BLV1, [SLaTh11] 1.266(42) 0.516(65) 0.047(11) -0.052(31) -0.050(7) dis. BLV2, [SLaTh11] 1.077(25) 0.126(15) 0.062(8) 0.107(17) -0.037(8) NREFT [SS10] 1.213(14) 0.308(23) 0.050(3) 0.083(19) -0.039(2) BSE [BB05] 1.054 0.185 0.079 0.064 -Experiment -a b d f g Gormley1[MG70] 1.18(2) 0.20(3) 0.04(4) - -Gormley2[MG70] 1.17(2) 0.21(3) 0.06(4) - -Layter et al [JL72] -0.54(ﬁxed) - - - -Layter et al [JL73] 1.080(14) 0.03(3) 0.05(3) - -CBarrel-95 [CA95] 0.94(15) 0.11(27) - - -CBarrel-98 [AA98] 1.22(7) 0.22(11) 0.06 (ﬁxed) - -KLOE [FA08] 1.090(5)(+19₋₈) 0.124(6)(10) 0.057(6)(+7₋₁₆) 0.14(1)(2) ∼ 0

Table 2.3: Dalitz plot parameters from theoretical predictions and experimental

re-sults. Results at LO, NLO and NNLO ChPT are taken from [JB07]. The value inside the parentheses denote the uncertainty of the obtained parameter. BLV1 and BLV2

refers to the Bern-Lund-Valencia dispersive result when ﬁtting to NLO [JG85] (BLV1)

and to a KLOE-like [FA08] experimental distribution (BLV2). Gormley1refers to the

result of the Dalitz plot ﬁt with c= 0.05(2). Gormley2is the result when folding the

Dalitz plot about X= 0. The ﬁrst parentheses in the KLOE data give the statistical uncertainty and the second parentheses give the systematical uncertainty.

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Experiment ALR· 10−2 AQ· 10−2 AS· 10−2

Layter et al, [JL72] −0.05(22) −0.07(22) 0.10(22)

Jane et al, [MJ74] −0.28(26) −0.20(25) −0.30(25) KLOE, [FA08] 0.09(10)(+9)₍₋₁₄₎ −0.05(10)(+3)₍₋₅₎ 0.08(10)(+8)₍₋₁₃₎

Table 2.4: Left-right asymmetry, quadrant asymmetry and sextant asymmetry from the

Dalitz plot forη → π+π−π0_{. All values are consistent with 0. The uncertainties of A}

Q

and ASfor [MJ74] are statistical. For KLOE the ﬁrst and second parentheses denote

the statistical and systematical uncertainties, respectively.

asymmetry deﬁned as

ALR=

N₊− N₋

N₊+ N₋, (2.51)

where N± is the number of events whereπ± has a larger energy thanπ∓ in theη rest frame, was found to be ALR= −0.05(22)·10−2, i.e. consistent with

0 (see ﬁgure 2.7). The quadrant asymmetry, AQ, and the sextant asymmetry,

AS, deﬁned as AQ= NI+ NIII− NII− NIV NI+ NIII+ NII+ NIV (2.52) and AS= NI+ NIII+ NV− NII− NIV− NV I NI+ NIII+ NV+ NII+ NIV+ NV I, (2.53)

were also measured. The sextant asymmetry regions are shown in ﬁgure 2.7. The quadrant asymmetry is deﬁned analogously, with the upper right quadrant numbered I, with increasing numbers in the clock-wise direction. The AQis a test of C-violation with I= 2 of the 3π system and ASis a test of

C-violation with I= 1 [CJ02]. No violation of these symmetries were found as shown in table 2.4. In addition, a ﬁt with 63 degrees of freedom was done to c and e, where a was ﬁxed. Both c and e were found to be consistent with 0 (see table 2.3). The a parameter value was obtained from an unpublished Ph.D. thesis and strongly deviates from other experimental a-values.

Based on the same sample as [JL72], [JL73] re-measured the Dalitz plot parameter, excluding the linear term in X. This analysis included more stringent cuts, reducing the statistical sample from 220 000 to approximately 81 000 events. The parameters are given in table 2.3 where also a appears with a negative sign. Reference [MJ74] performed an independent measurement of the charge asymmetry by using the Proton Synchrotron NIMROD at 8 GeV/c. Their conclusion was that no charge asymmetry is

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found (see table 2.4). Experiment Events Gormley1, [MG70] 3.0 · 104 Gormley2, [MG70] 3.0 · 104 Layter et al., [JL72] 2.2 · 105 Layter et al., [JL73] 8.1 · 104 Jane et al., [MJ74] 1.7 · 105 CBarrel-95, [CA95] 1.1 · 103 CBarrel-98, [AA98] 3.2 · 103 KLOE, [FA08] 1.3 · 106

Table 2.5: Number of events in Dalitz plot for the experimental data.

Figure 2.7: Asymmetry of the Dalitz plot for the measurement done in [JL72]. a) The asymmetry is plotted as a function of the Dalitz plot coordinate X. In b) the asymme-try is plotted as function of Dalitz plot coordinate Y. c) Shows the number of events in the different sextants of the Dalitz plot. ’This experiment’ refers to [JL72] and ’pre-vious experiment’ from [MG70] where the charge asymmetry was measured. Figure obtained from [JL72].

The next two analyses, [CA95] and [AA98], reported measurements using the Crystal Barrel detector where 200 MeV/c anti-protons collided with a liquid hydrogen target. The charged pions were detected in drift chambers while the photons from the π0 decay were detected in an electromagnetic calorimeter. In [CA95] the chosen channel to detect η → π+_π−_π0 _{was p ¯}_p_{→ π}+_π−_{η because of high branching ratio combined}

with a low combinatorial background. A Gaussian ﬁt to the η signal and a four-degree Legendre polynomial to the background was made in the invariant mass spectrum ofπ+π−π0. The events were selected by imposing energy-momentum conservation using kinematical ﬁt and keeping events with a probability greater than 10%. Here only a projection on Y was used to

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Figure 2.8: Y-axis Dalitz plot projection normalized to phase space. The data points with errors from [CA95] are shown with the ﬁt (1+ aY + bY2) (solid line), current algebra prediction (dotted line) and ChPT with one loop prediction (dashed line) ([JG85]). Figure obtained from [CA95]. Borders have been enhanced.

fit the parameter since their data showed almost no dependence on X. Their fit|A |2(Y) = A₀2(1 + aY + bY2) is performed over ten bins in Y, as shown in figure 2.8. Unfortunately the statistical sample was only 1100 events. When studying the compatibility of the total projection of the Dalitz plot Y -parameter a in figure 2.8 between ChPT and their result, the tree-level prediction is found to be more compatible with their data than the one loop correction calculated by [JG85]vi. The comment on this discrepancy was that the one-loop correction is within the errors of the measurement.

In [AA98] one instead uses the channel p ¯p → π0π0η. Here the statistics were three times larger and the Y -region was divided up into nineteen bins as is shown in figure 2.9. Again there is no fit over X. Instead a fixed value of the X2 dependence was set to

vi_{Since the Crystal Barrel did not ﬁnd any dependence in X, the X-value was not further}

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0.06 obtained from theory and other experiments. The parameters are given in table 2.3 and show good agreement with the NNLO parameter values.

Figure 2.9: Y-axis Dalitz plot projection normalized to phase space. The data points from [AA98] is shown with the ﬁt (solid line). Figure obtained from [AA98]. The borders have been enhanced.

The most recent experimental result comes from the KLOE-collaboration [FA08] and is the most precise result both in terms of statistical and systematical uncertainties. It has statistics more than an order of magnitude larger than the other experimental results combined, as may be seen from table 2.5. Data from KLOE is collected at the e+e−-collider DAΦNE situated in Frascati, Italy. The reaction where η is produced is e+e− → φ → ηγ at a center of mass energy 1020 MeV/c2. The detector itself consists of a cylindrical drift chamber surrounded by an electromagnetic calorimeter with a 0.52 T magnetic ﬁeld. Selection ofη is straightforward since the photon coming from the decay ofΦ is monochromatic with an energy of 363 MeV. The main cuts of the analysis were imposed to remove various background channels connected toΦ decays [KLOEn215]. The total Dalitz plot contains 1.34 million events split up into 154 different Dalitz plot bins with a bin size of 0.125 for both X and Y. The Dalitz plot is shown in ﬁgure 2.10. The different asymmetries, ALR, AQ, AS, were also measured and found to be

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and [MJ74].

Figure 2.10: Dalitz plot distribution from KLOE containing 1.34 million events in 256 bins. Figure from [FA08].

Figure 2.11: Comparison between KLOE result (gray area with black solid line boundaries) [FA08] and ChPT predictions (blue solid LO; green solid line-NLO; red solid line- NNLO), as given by [JB07]. For the left frame, LO is constant at

1.

The KLOE result is also the ﬁrst to include degrees to the third order for ﬁt-ting the Dalitz plot distribution. While it is found that the cubic terms gX3+ hX2Y+lXY2are consistent with zero, fY3is found to be non-zero. Compared to the other experiments, the b-parameter was found to disagree with the ex-perimental results from the seventies, while agreeing with the two most recent experiments within the uncertainties. With respect to the theoretical calcula-tion there is a tension between the KLOE result and the NNLO prediccalcula-tion, mainly concerning the b parameter value. The comparison between the pa-rameter values with uncertainties obtained from KLOE and the central values

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of the chiral perturbation theory predictions are shown in ﬁgure 2.11 for slices in X and Y . From the ﬁgure it can be seen that while the distribution in X agrees well between higher order predictions and KLOE, the distribution as function of Y disagrees.

2.4 Comparison experiment and theory

η → π

+

π

−

π

0

When comparing the KLOE result with the theoretical predictions (table 2.3) the most signiﬁcant deviations are found for the b and f parameter values. ChPT at LO is closest to the KLOE a parameter, but deviations become larger at NLO and NNLO. To quantify the discrepancy between theoretical predic-tions and the KLOE result,

χ2

KLOE.c f .theory= (pKLOE− ptheory)

2

σ2

pKLOE+ σp2theory

(2.54)

can be deﬁned, where pKLOE and ptheory denotes the central values of the

charged Dalitz plot parameter values from KLOE and theoretical predictions, respectively, and σp_KLOE and σp_theory give the combined uncertainty of the

statistical and systematical uncertainties for the experimental measurement and the uncertainties from the theoretical calculations [KK11]. Three examples of theoretical predictions have been compared to the KLOE measurement: the NNLO ChPT calculation [JB07], the NREFT calculation [SS10] and the Berne-Lund-Valencia dispersive calculation [SLaTh11] (BLV1 in table 2.3). All these theoretical predictions quote uncertainties for

the Dalitz plot parameters.

The result of the comparisons is given in table 2.6 and shows that d agrees well between the theoretical and experimental estimates. What is interesting to note is the discrepancy between theoretical and experimental estimates for the a, b and f parameters when using the combined uncertainties of the exper-imental and theoretical predictions. This is seen for a, b and to some degree also for f when including the uncertainties of the theoretical estimates. The NNLO theoretical uncertainties are substantial, but still give a large discrep-ancy for a and b. Graphically, the KLOE result parameters compared to the ChPT predictions are shown in ﬁgure 2.11 when Y = 0 (left frame) and when X= 0 (right frame).

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Comp. exp. vs theory,χ2 χ_a2 χ_b2 χ_d2 χ2_f

KLOE [FA08] vs NNLO [JB07] 5.6 6.9 0.001 0.5

KLOE [FA08] vs NREFT [SS10] 37.5 50.9 0.3 3.8

KLOE [FA08] vs BLV1[SLaTh11] 15.7 35.2 0.4 25.2

Table 2.6: Comparison between the KLOE result and the theoretical estimates for

the different charged Dalitz plot parameters (eqn. 2.54), where the uncertainties from theory and experiments are combined in quadrature.

2.5 η → π

0

π

0

π

0

The neutral decay η → π0π0π0 is also of interest and has bearing on the chargedη → 3π decay. The amplitude to the lowest order is given by

¯

A (s,t,u) =B0(m_√d− mu)

3 f2

π

. (2.55)

The neutral amplitudeA (s,t,u) and the charged amplitude A (s,t,u) is re-¯ lated via

¯

A (s,t,u) = A (s,t,u) + A (t,u,s) + A (u,s,t) [CZ64, JG85], (2.56) which holds at leading order in the isospin breaking for O(e2) but is bro-ken at O(e2(md− mu)) [SS10]. The Dalitz plot for the neutral pion decay

is parametrized by using polar coordinates instead of X- and Y-coordinates, where

X =√Zsinφ, Y =√Zcosφ. (2.57)

The Dalitz plot has a sextant symmetry and therefore it is possible to limit the range toφ = 0 − 60◦, and fold the sextants onto each other. This gives

Z= X2+Y2=2 3 3

∑

i=1 Ti− < T > < T > 2 , (2.58)

where 3< T >= T1+ T2+ T3, Ti is the kinetic energy of a neutral pion in the

η rest frame and Z ranges from 0 to 1 [AK09]. The squared amplitude in the expansion around the Dalitz plot center is given as

|A (Z,φ)|2_{= |N |}2_{(1 + 2αZ + 2βZ}3/2_{sin(3φ) + 2γZ}2_{) + ...,} _(2.59)

where|N | is the normalization and α, β are the Dalitz plot parameters for the neutral decay. The expansion is only valid for Z< 0.598 due to ﬁnal state rescattering of the form π+π− → π0π0 which produces a cusp [SS10]. So far many experimental analyses have conﬁrmed the value of the parameterα

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Figure 2.12: Comparison of slope parameter,α between experimental results (above solid black line) and theoretical predictions (below the black line) with PDG average −3.15(15) · 10−2 _{marked in blue. The blue band corresponds to the PDG average.}

The experimental references from tob to bottom are [DA84, AA98a, ST00, MNA01, WBT01, MB07, CA09, SP09, MU09, FA10, PDG12]. The theoretical references from top to bottom are [WB67, JG85, JK96, JB02, NB03, BB05b, JB07, SS10, KK11, SLaTh11]. Figure obtained from [MW12].

[DA84, AA98a, MNA01, WBT01, MB07, CA09, SP09, MU09, FA10]. Most of the experimental results of the slope parameter are consistent with each other. The theoretical predictions have had a difﬁculty to obtain a negative value of theα parameter as is shown in ﬁgure 2.12.

There are also some relationships between the charged and neutral decay. One is the ratio of the decay rates with respect to each other:

r= Γη→π0π0π0 Γ_η→π+_π−_π0.

(2.60)

This value changes from r= 1.54 to r = 1.47 from LO to NNLO ChPT cal-culations [JB07] in comparison to the PDG average result r= 1.48 ± 0.05 [PDG12]. This ratio is however not expected to deviate drastically from 1.5 which means that it is not a very sensitive observable [JB02]. Another relation given in the literature is the inequality between the Dalitz plot parameters for

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the neutral and the charged decay. These relations are based on ChPT but have no dependence on the values for the NNLO low-energy constants:

α ≤1

4(d + b − 1 4a

2_{) [JB07, KK11].} _(2.61)

For the experimental result of KLOE this inequality matches quite well. An-other relation found in the literature is

rel= a3− 4ab + 4ad + 8 f − 8g = 0[KK11]. (2.62) The second relation is consistent with 0 for both the KLOE parameter values (rel= 0.12(21)) and NNLO calculations (rel = −0.03(72)).

2.6 Motivation for a new measurement

Estimating Q (eqn. 2.42) and R (eqn. 2.46) gives constraints in the plane of the quark mass ratios ms/md and mu/md. These may be estimated from

lattice QCD and other methods, but η → 3π can give a particularly narrow constraint since electromagnetic contributions are suppressed. A precise estimate of Q can serve as important input to lattice QCD since simulations of QCD+QED on the lattice are still exploratory [GC10]. Q determined from ChPT serves also as an independent cross check of lattice QCD results [GC10b].

To rely on estimates of R and Q fromη → 3π requires that experimental and theoretical observables are in reasonable agreement with each other for both the neutral and chargedη → 3π decay. Since the determination of R and Q withη → 3π is determined within the framework of ChPT, this also serves as an important test of the convergence properties of ChPT [MK11]. For the neutral decay the slope parameterα has a negative value which is difﬁcult to reproduce from theoretical estimates (with some exceptions, e.g. [SS10]). Experimentally the neutral decay parameter has been determined and many independent measurements exist.

For the charged decay the experimental situation is less clear. There is a strong experimental indication that charge conjugation is preserved as measured from the asymmetry parameters, but apart from this, when comparing the results for the charged Dalitz plot parameters, different experiments obtain different results. Thus far the KLOE result [FA08] is the experiment to date which has measured the Dalitz plot parameters to the highest precision and is the only experiment which has measured f . A re-analysis of the Dalitz plot parameters with more statistics and with the aim of decreasing the systematical effects is underway [LCB11]. It

Studies of the Decay η→π+π-π0 with WASA-at-COSY

Studies of the Decay η→π

+

π

-

π

0

with WASA-at-COSY

Contents

1. Introduction

2. Motivation, theory and previous

experimental results

2.1

General properties of

η

2.2

η → π

π

π

- theoretical aspects

2.2.1

η → π

π

π

at tree level

2.2.2

Dalitz plot

2.2.3

Q and R

2.2.4

Electromagnetic corrections to

η → π

π

π

2.2.5

η → π

π

π

at order p

and higher

2.2.6

Dispersive treatments

2.2.7

Non-Relativistic Effective Field Theory

2.2.8

Resummed ChPT

2.2.9

Other considerations: Decay width

2.3

η → π

π

π

- previous experiments

2.4

Comparison experiment and theory

η → π

π

π

2.5

η → π

π

π

∑

2.6

Motivation for a new measurement