A Dalitz Plot Analysis of 3 Decay · A Dalitz Plot Analysis of !!3ˇDecay Chris Zeoli Florida State...
48
A Dalitz Plot Analysis of ω → 3π Decay Chris Zeoli Florida State University [email protected] June 4, 2015 Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 1 / 48
Transcript of A Dalitz Plot Analysis of 3 Decay · A Dalitz Plot Analysis of !!3ˇDecay Chris Zeoli Florida State...
A Dalitz Plot Analysis of ω → 3π Decay
Chris Zeoli
Florida State University
June 4, 2015
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 1 / 48
Overview
Goals
G12 Data
Fit Function in Brief
Analysis
Next Steps
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 2 / 48
Goals
Main Goal
Fit a modified Khuri-Treiman (KT) Model for the ω → 3π decay.
Fit the decay amplitude via event-based likelihood fits usingAmpTools framework.
Compare fit parameters with known results from other models
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 3 / 48
CLAS G12 Data
Data Total Events
-data 8,200,000-genMC 14,000,000-accMC 1,500,000
Average Acceptance ≈ 0.107 (GeV)ωM0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Eve
nts
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
:[2000-2050]MeVγENevents=65263
Data
Omega Mass, Background Subtracted
G12 data covers incoming photon energy range Eγ :[1150-5400]MeV
Have G12 x-section Eγ :[1150-3800]MeV, extending to 5400MeV,need G12 SDME’s still
We use G11 x-section, SDME’s data; range Eγ :[1107.4-3828.9]MeV
We consider bins in range Eγ :[1150-3800]MeV
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 4 / 48
G12, G11 Cross-Section Comparison
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
1.50 < E < 1.55
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
1
10
1.55 < E < 1.60
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
1.60 < E < 1.65
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
1.65 < E < 1.70
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
1.70 < E < 1.75
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
1.75 < E < 1.80
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
1
10
1.80 < E < 1.85
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
1.85 < E < 1.90
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
1.90 < E < 1.95
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
1.95 < E < 2.00
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.00 < E < 2.05
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
1
10
2.05 < E < 2.10
+ : g11
+ : g12
0
5
100
5
100
5
100
5
10
1 0.5 0 0.5 11 0.5 0 0.5 11 0.5 0 0.5 11 0.5 0 0.5 11 0.5 0 0.5 1
cm ωθcos
b)
µ (
θ
/d
co
s
σd
Eγ:[1500-2010]MeV, Zulkaida Akbar
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 5 / 48
G12, G11 Cross-Section Comparison
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.15 < E < 2.20
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.20 < E < 2.25
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.25 < E < 2.30
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
1
10
2.30 < E < 2.35
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.35 < E < 2.40
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.40 < E < 2.45
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.45 < E < 2.50
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.50 < E < 2.55
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
1
10
2.55 < E < 2.60
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.60 < E < 2.65
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.65 < E < 2.70
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.70 < E < 2.75
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.75 < E < 2.80
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
1
10
2.80 < E < 2.85
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.85 < E < 2.90
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.90 < E < 2.95
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
2.95 < E < 3.00
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
3.00 < E < 3.05
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
1
10
3.05 < E < 3.10
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
3.10 < E < 3.15
+ : g11
+ : g12
0
5
100
5
100
5
100
5
10
1 0.5 0 0.5 11 0.5 0 0.5 11 0.5 0 0.5 11 0.5 0 0.5 11 0.5 0 0.5 1
cm ωθcos
b)
µ (
θ
/d
co
s
σd
Eγ:[2150-3150]MeV, Zulkaida Akbar
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 6 / 48
G12, G11 Cross-Section Comparison
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
3.15 < E < 3.20
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
3.20 < E < 3.25
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
3.25 < E < 3.30
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
1
10
3.30 < E < 3.35
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
3.35 < E < 3.40
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
3.40 < E < 3.45
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
3.45 < E < 3.50
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
3.50 < E < 3.55
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
1
10
3.55 < E < 3.60
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
3.60 < E < 3.65
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
3.65 < E < 3.70
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
3.70 < E < 3.75
+ : g11
+ : g12
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
9
10
3.75 < E < 3.80
+ : g11
+ : g12
0
5
100
5
100
5
100
5
10
1 0.5 0 0.5 11 0.5 0 0.5 11 0.5 0 0.5 11 0.5 0 0.5 11 0.5 0 0.5 1
cm ωθcos
b)
µ (
θ
/d
co
s
σd
Eγ:[3150-3800]MeV, Zulkiada Akbar
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 7 / 48
Paper for Model
JLAB-THY-14-1960
Dispersive Analysis of ω/φ→ 3π, πγ∗
I.V. Danilkin,1, ∗ C. Fernandez-Ramırez,1 P. Guo,2, 3 V. Mathieu,2, 3 D. Schott,4 M. Shi,1, 5 and A. P. Szczepaniak1, 2, 3
(Joint Physics Analysis Center)1Center for Theoretical and Computational Physics,
Thomas Jefferson National Accelerator Facility, Newport News, VA 236062Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403
3Physics Department, Indiana University, Bloomington, IN 474054Department of Physics, The George Washington University, Washington, DC 20052
5Department of Physics, Peking University, Beijing 100871, China(Dated: September 30, 2014)
The decays ω/φ → 3π are considered in the dispersive framework that is based on the isobardecomposition and sub-energy unitarity. The inelastic contributions are parametrized by the powerseries in a suitably chosen conformal variable that properly account for the analytic properties ofthe amplitude. The Dalitz plot distributions and integrated decay widths are presented. Our resultsindicate that the final state interactions may be sizable. As a further application of the formalismwe also compute the electromagnetic transition form factors of ω/φ→ π0γ∗.
PACS numbers: 13.20.Jf, 11.55.Fv, 13.25.Jx, 13.75.Lb
I. INTRODUCTION
Three particle production plays an important role inhadron physics. In the past, analysis of the three-pionspectrum led to the discovery of several prominent mesonresonances [1]. With high precision data already avail-able, for example from the COMPASS Collaboration [2]and expected from Jefferson Lab [3], in the near future itwill be possible to further resolve the three pion spectrumand identify new resonances that do not necessarily fit thequark model template. Indeed, in the charmonium spec-trum several candidates for non-quark model resonanceshave recently been reported [4, 5]. Several of these wereobserved in decays to three-particle final states. Properdescription of interactions in the three-particle system isalso required to advance lattice gauge computations ofscattering amplitudes [6–9].
Because of large production yields, hadron systems arealso an important laboratory for studies of weak interac-tions, symmetry tests and searches for physics beyond theStandard Model [10, 11]. Sensitivity to weak interactionsdemands high precision in determination of hadronic am-plitudes. Near threshold there are first principle con-straints that can help in this process. These low-energyconstraints include, for example, chiral symmetry, par-tial wave and effective range expansions, and unitarity.In general, however, it is impossible to construct a singleanalytical function that describes a reaction amplitudein the entire range of kinematical variables and satisfiesall of the constraints imposed by the relativistic S-matrixtheory. Nevertheless, analyticity is a powerful constraintthat enables to connect different regions of the spectrume.g. constrain resonance parameters by the behavior of
∗Electronic address: [email protected]
the amplitude elsewhere, including both the near thresh-old and the high-mass regions.
In this paper we focus on the analysis of three pionproduction at low energies in particular from decays ofthe light-vector, isoscalar mesons, the ω and the φ. Atlow energies, chiral perturbation theory (χPT) serves asa powerful constraint on amplitudes involving the lightpseudo-scalar mesons [12, 13]. χPT has been applied tothe three pion production from the η decays [14, 15]. Inthe case of ω/φ→ 3π, χPT can be extended by includinglight vector mesons as additional degrees of freedom [16–19]. In a perturbative study, germane to an effective fieldtheory, unitarity is only satisfied order-by-order in theloop expansion. On the other hand, from the perspectiveof the S-matrix theory, unitarity is the key feature thatconstraints singularities of the reaction amplitude andtherefore the amplitude itself. For this reason there hasbeen a lot of interest in application of dispersion relationsto low energy production of pseudoscalar mesons [20–25].
Dispersive methods have been used in description ofrelativistic three body decays in the past [26–29]. Forexample, the decay η → 3π [30–35], is of interest becauseit is sensitive to isospin breaking, which in QCD origi-nates from the mass difference between the up and downquarks. Dispersive analysis of ω decay was performed in[36] and more recently in [37]. It is of interest because itsheds light on the vector mesons dominance and the in-terplay between the QCD dynamics, which is believed tobe responsible for the vector meson formation and its de-cay characteristics restricted by unitarity and long-rangeinteractions.
In relativistic S-matrix theory a function connectingfour external particles describes reaction amplitudes ofall processes related by crossing, i.e. the three 2 → 2scattering channels and, if kinematically allowed, a de-cay channel 1→ 3. Therefore, unitary constraints oughtto be considered in all physical channels connected by thesame analytical function. With the emphasis on unitar-
arX
iv:1
409.
7708
v1 [
hep-
ph]
26
Sep
2014
arXiv:1409.7708v1 [hep-ph] (2014)
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 8 / 48
Illustration2
1
2
3
2
1
3
1
2
3
1
2
3
FIG. 1: Isobar decomposition.
FIG. 2: Crossed channel rescattering effects.
ity, the natural starting point for amplitude constructionis the partial wave expansion. At low energies, it is ex-pected that only low partial waves are significant andtherefore the infinite partial waves series can be trun-cated to a finite sum. We refer to such an approximationas the isobar model [38]. The diagrams representing atruncated partial waves series, a.k.a the isobar decompo-sition are shown in Fig.1.
Implementation of unitarity on a truncated set of par-tial waves leads to the so called Khuri-Treiman (KT)equations [26, 27, 39]. In the the KT framework elas-tic unitarity in the three crossed channels is used to de-termine the discontinuity of partial waves which are thenreconstructed using a Cauchy dispersion relation. Conse-quently additional diagrams contribute to the amplitude,see Fig. 2. Since, as discussed above, the model truncatesthe number of partial waves, it is intrinsically restrictedto low energies. In other words the high-energy behaviorin the KT framework is arbitrary. Mathematically, thistranslates into an arbitrariness in choosing the bound-ary condition for the solution of an integral equation,which follows from the dispersion relation. It is thereforemore appropriate to consider the KT framework as a setof constrains on partial wave equations. Furthermore,above threshold of production of inelastic channels theKT amplitudes will couple to other open channels. Anyscheme that tries to reduce the sensitivity of the elasticKT equations to the high-energy contributions in dis-persion integrals should therefore take into account thechange in the analytical properties of the partial waveamplitudes above the inelastic open channels. A novelimplementation of this feature within the KT frameworkis the main new ingredient of the approach presented inthis paper.
In previous works, in order to suppress sensitivity tothe unconstrained high-energy region, subtracted disper-sion relations were used [33, 34, 37]. Moreover, KT equa-tions depend on the elastic 2→ 2 scattering amplitudes.The ππ → ππ amplitudes needed for analysis of ω/φdecays have been studied in Ref. [20]. These studies con-strain the amplitudes only up to certain center of mass
energy (somewhat above KK threshold) and this addsfurther uncertainty into the KT framework. For exam-ple, in previous analyses of the vector meson decays theππ phase shift was extended beyond the elastic regionwith a specific model [37]. In this paper we present analternative to the subtraction procedure, which not onlysuppresses the high-energy contributions to the disper-sive integrals, but also takes into account the change inthe analytical properties induced by the opening of in-elastic channels. Specifically, we split the dispersive in-tegral into elastic and inelastic parts, and parameterizethe latter in terms of an appropriately chosen conformalvariable.
The paper is organized as follows. In the next sectionwe summarize the derivation and main features of the KTframework as applied to the vector meson decays. Thediscontinuity relation and the role that inelastic effectsplay in choosing a suitable solution of the dispersive re-lation are discussed in Sections III and IV. The numericalanalysis of ω/φ→ 3π is presented in Section V A. In Sec-tion V B we consider the electromagnetic (EM) transitionform factors of ω/φ → π0γ∗ as a further application ofour formalism. Summary and outlook are presented inSection VI.
II. PARTIAL WAVE OR ISOBARDECOMPOSITION
The matrix element for the three pion decay of a vectorparticle is given in terms of a helicity amplitude Habc
λ ,
〈πa(p1)πb(p2)πc(p3) |T |V (pV , λ)〉 =
= (2π)4 δ(pV − p1 − p2 − p3)Habcλ . (1)
Here pV and λ are the momentum and helicity of the vec-tor particle, V = ω/φ in our case, p1, p2, p3 are the mo-menta of outgoing pions with a, b, c denoting their Carte-sian isospin indices. The Lorentz-invariant Mandelstamvariables are defined by s = (pV − p3)2, t = (pV − p1)2,u = (pV − p2)2 and satisfy the relation
s+ t+ u = M2 + 3m2π . (2)
The helicity amplitude Habcλ can be expressed in terms
of a single scalar function of the Mandelstam variables,since Lorentz and parity invariance imply that,
Habcλ = i εµναβ ε
µ(pV , λ) pν1 pα2 p
β3
P 1abc√2F (s, t, u) , (3)
Isobar Model
2
1
2
3
2
1
3
1
2
3
1
2
3
FIG. 1: Isobar decomposition.
FIG. 2: Crossed channel rescattering effects.
ity, the natural starting point for amplitude constructionis the partial wave expansion. At low energies, it is ex-pected that only low partial waves are significant andtherefore the infinite partial waves series can be trun-cated to a finite sum. We refer to such an approximationas the isobar model [38]. The diagrams representing atruncated partial waves series, a.k.a the isobar decompo-sition are shown in Fig.1.
Implementation of unitarity on a truncated set of par-tial waves leads to the so called Khuri-Treiman (KT)equations [26, 27, 39]. In the the KT framework elas-tic unitarity in the three crossed channels is used to de-termine the discontinuity of partial waves which are thenreconstructed using a Cauchy dispersion relation. Conse-quently additional diagrams contribute to the amplitude,see Fig. 2. Since, as discussed above, the model truncatesthe number of partial waves, it is intrinsically restrictedto low energies. In other words the high-energy behaviorin the KT framework is arbitrary. Mathematically, thistranslates into an arbitrariness in choosing the bound-ary condition for the solution of an integral equation,which follows from the dispersion relation. It is thereforemore appropriate to consider the KT framework as a setof constrains on partial wave equations. Furthermore,above threshold of production of inelastic channels theKT amplitudes will couple to other open channels. Anyscheme that tries to reduce the sensitivity of the elasticKT equations to the high-energy contributions in dis-persion integrals should therefore take into account thechange in the analytical properties of the partial waveamplitudes above the inelastic open channels. A novelimplementation of this feature within the KT frameworkis the main new ingredient of the approach presented inthis paper.
In previous works, in order to suppress sensitivity tothe unconstrained high-energy region, subtracted disper-sion relations were used [33, 34, 37]. Moreover, KT equa-tions depend on the elastic 2→ 2 scattering amplitudes.The ππ → ππ amplitudes needed for analysis of ω/φdecays have been studied in Ref. [20]. These studies con-strain the amplitudes only up to certain center of mass
energy (somewhat above KK threshold) and this addsfurther uncertainty into the KT framework. For exam-ple, in previous analyses of the vector meson decays theππ phase shift was extended beyond the elastic regionwith a specific model [37]. In this paper we present analternative to the subtraction procedure, which not onlysuppresses the high-energy contributions to the disper-sive integrals, but also takes into account the change inthe analytical properties induced by the opening of in-elastic channels. Specifically, we split the dispersive in-tegral into elastic and inelastic parts, and parameterizethe latter in terms of an appropriately chosen conformalvariable.
The paper is organized as follows. In the next sectionwe summarize the derivation and main features of the KTframework as applied to the vector meson decays. Thediscontinuity relation and the role that inelastic effectsplay in choosing a suitable solution of the dispersive re-lation are discussed in Sections III and IV. The numericalanalysis of ω/φ→ 3π is presented in Section V A. In Sec-tion V B we consider the electromagnetic (EM) transitionform factors of ω/φ → π0γ∗ as a further application ofour formalism. Summary and outlook are presented inSection VI.
II. PARTIAL WAVE OR ISOBARDECOMPOSITION
The matrix element for the three pion decay of a vectorparticle is given in terms of a helicity amplitude Habc
λ ,
〈πa(p1)πb(p2)πc(p3) |T |V (pV , λ)〉 =
= (2π)4 δ(pV − p1 − p2 − p3)Habcλ . (1)
Here pV and λ are the momentum and helicity of the vec-tor particle, V = ω/φ in our case, p1, p2, p3 are the mo-menta of outgoing pions with a, b, c denoting their Carte-sian isospin indices. The Lorentz-invariant Mandelstamvariables are defined by s = (pV − p3)2, t = (pV − p1)2,u = (pV − p2)2 and satisfy the relation
s+ t+ u = M2 + 3m2π . (2)
The helicity amplitude Habcλ can be expressed in terms
of a single scalar function of the Mandelstam variables,since Lorentz and parity invariance imply that,
Habcλ = i εµναβ ε
µ(pV , λ) pν1 pα2 p
β3
P 1abc√2F (s, t, u) , (3)
consequence of elastic unitarityrequirement of model
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 9 / 48
Model
6
FIG. 4: Real and imaginary parts of Ωel(s) in Eq. (32) (Dot-Dashed), Ω′(s) in Eq. (35) (Dashed) and Ω(s) in Eq. (34) (Solid).
The first part is determined entirely by elastic scatteringwhile the second part takes into account inelastic effects.The inelastic contribution is described by an analyticalfunction on the s-plane cut along the real axis aboves = si. It is largely unknown, and often parametrizedthrough an expansion in a conformal variable which mapsthe right-hand cut in the complex s-plane onto the unitdisk. Such a mapping is known as a convenient repre-sentation of functions on a cut plane with the the knownanalytical properties [48],
Σ(s) =∞∑
i=0
ai ωi(s) (29)
The variable
ω(s) =
√si − sE −
√si − s√
si − sE +√si − s
(30)
maps the cut plane onto the unit disk. The parametersi = 1 GeV is identified with the point where inelas-tic contributions are expected to become relevant2 andthe expansion point sE should lie below the cut. Wedefine sE = 0. The conformal mapping technique wassuccessfully applied in other descriptions of two-to-twoamplitudes e.g. in [49–51] it was used to take into ac-count the contributions from the more distant left-handcuts. However, to the best of our knowledge the confor-mal mapping technique has never been used before in thecontext of the KT equations.
2 Note the following inelastic thresholds: mπ + mω = 0.92 GeV(only for the ω decay), 2mK = 0.99 GeV and 8mπ = 1.1 GeV(we omit the 4π and 6π multi-particle thresholds since they areknown to be weak).
With the inelastic contributions parametrized by thefunction, Σ(s), the integral equation for the KT ampli-tude takes the form of
F (s) = Ω(s)
(1
π
∫ si
sπ
ds′ρ(s′) t∗(s′)
Ω∗(s′)F (s′)s′ − s + Σ(s)
).
(31)This is an alternative to the standard way which employssubtractions to reduce sensitivity of the dispersive inte-gral to the high-energy region [37]. The problem withsubtractions is twofold. First, the dispersive integrals,including computation of Ω(s) run over inelastic regions,while the dispersion relation contains only the elastic con-tributions. Furthermore, subtracting an analytical func-tion of s does not account for the change in the analyti-cal behavior of the amplitudes due to opening of inelasticchannels.
In Eq. (31) there is no need for subtractions in the dis-persive integral since it is restricted to the elastic-region,which the only part of the right hand cut controlled byelastic unitarity. The unknown, inelastic contributionsare parametrized by Σ(s), and are to be determined bycomparing with the experimental data, or other theoret-ical approaches that treat inelastic channels explicitly.Moreover, with the dispersive integral restricted to a fi-nite interval over s there are uncontrollably large con-tributions from higher-partial waves, which otherwise re-quire more and more subtractions.
Besides F (s), the problem with the determination ofinelastic contributions also affects computation of Ω(s).The unitarity condition in Eq. (25) does not determine itabove the inelastic threshold s = si. Therefore, we seekits solution given in terms of the Omnes function [52, 53]taken only over the elastic region
Ωel(s) ≡ exp
(s
π
∫ si
sπ
ds′
s′δ(s′)s′ − s
), (32)
The Decay Amplitude
6
FIG. 4: Real and imaginary parts of Ωel(s) in Eq. (32) (Dot-Dashed), Ω′(s) in Eq. (35) (Dashed) and Ω(s) in Eq. (34) (Solid).
The first part is determined entirely by elastic scatteringwhile the second part takes into account inelastic effects.The inelastic contribution is described by an analyticalfunction on the s-plane cut along the real axis aboves = si. It is largely unknown, and often parametrizedthrough an expansion in a conformal variable which mapsthe right-hand cut in the complex s-plane onto the unitdisk. Such a mapping is known as a convenient repre-sentation of functions on a cut plane with the the knownanalytical properties [48],
Σ(s) =∞∑
i=0
ai ωi(s) (29)
The variable
ω(s) =
√si − sE −
√si − s√
si − sE +√si − s
(30)
maps the cut plane onto the unit disk. The parametersi = 1 GeV is identified with the point where inelas-tic contributions are expected to become relevant2 andthe expansion point sE should lie below the cut. Wedefine sE = 0. The conformal mapping technique wassuccessfully applied in other descriptions of two-to-twoamplitudes e.g. in [49–51] it was used to take into ac-count the contributions from the more distant left-handcuts. However, to the best of our knowledge the confor-mal mapping technique has never been used before in thecontext of the KT equations.
2 Note the following inelastic thresholds: mπ + mω = 0.92 GeV(only for the ω decay), 2mK = 0.99 GeV and 8mπ = 1.1 GeV(we omit the 4π and 6π multi-particle thresholds since they areknown to be weak).
With the inelastic contributions parametrized by thefunction, Σ(s), the integral equation for the KT ampli-tude takes the form of
F (s) = Ω(s)
(1
π
∫ si
sπ
ds′ρ(s′) t∗(s′)
Ω∗(s′)F (s′)s′ − s + Σ(s)
).
(31)This is an alternative to the standard way which employssubtractions to reduce sensitivity of the dispersive inte-gral to the high-energy region [37]. The problem withsubtractions is twofold. First, the dispersive integrals,including computation of Ω(s) run over inelastic regions,while the dispersion relation contains only the elastic con-tributions. Furthermore, subtracting an analytical func-tion of s does not account for the change in the analyti-cal behavior of the amplitudes due to opening of inelasticchannels.
In Eq. (31) there is no need for subtractions in the dis-persive integral since it is restricted to the elastic-region,which the only part of the right hand cut controlled byelastic unitarity. The unknown, inelastic contributionsare parametrized by Σ(s), and are to be determined bycomparing with the experimental data, or other theoret-ical approaches that treat inelastic channels explicitly.Moreover, with the dispersive integral restricted to a fi-nite interval over s there are uncontrollably large con-tributions from higher-partial waves, which otherwise re-quire more and more subtractions.
Besides F (s), the problem with the determination ofinelastic contributions also affects computation of Ω(s).The unitarity condition in Eq. (25) does not determine itabove the inelastic threshold s = si. Therefore, we seekits solution given in terms of the Omnes function [52, 53]taken only over the elastic region
Ωel(s) ≡ exp
(s
π
∫ si
sπ
ds′
s′δ(s′)s′ − s
), (32)
inelastic contribution(a0 = IgorParameter)
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 10 / 48
Dalitz Analysis
9
FIG. 6: The Dalitz plots for ω → 3π (left-hand panel) and φ → 3π (right-hand panel) decays. The distributions are dividedby the p-wave phase space P and normalized to 1 at x = y = 0. This is a parameter free result, because we kept only one termin the conformal expansion (29) which is responsible for the overall normalization. See main text for details.
TABLE I: Dalitz Plot parameters and√χ2 of the polynomial parametrization (40) for ω → 3π. In addition to our results
we also show the selected results from Niecknig et al. [37] (dispersive study with incorporated crossed-channel effects) andTerschlusen et al. [19] (Lagrangian based study with the pion-pion rescattering effects).
α× 103 β × 103 γ × 103 δ × 103√χ2 × 103
This paper (F = 0) 136 - - - 3.5
This paper (full) 94 - - - 3.2
Niecknig et al. [37] 84...96 - - - 0.9...1.1
Terschlusen et al. [19] 202 - - - 6.6
This paper (F = 0) 125 30 - - 0.74
This paper (full) 84 28 - - 0.35
Niecknig et al. [37] 74...84 24...28 - - 0.052...0.078
Terschlusen et al. [19] 190 54 - - 2.1
This paper (F = 0) 113 27 24 - 0.1
This paper (full) 80 27 8 - 0.24
Niecknig et al. [37] 73...81 24...28 3...6 - 0.038...0.047
Terschlusen et al. [19] 172 43 50 - 0.4
This paper (F = 0) 114 24 20 6 0.005
This paper (full) 83 22 1 14 0.079
Niecknig et al. [37] 74...83 21...24 0...2 7...8 0.012...0.011
Terschlusen et al. [19] 174 35 43 20 0.1
postpone the comprehensive data analysis to the futureand for now only consider the application to electromag-netic (EM) transition form factors of ω/φ. In partic-ularly, the transition ω → πγ∗ is of interest since theexisting data in the time-like region seems to be incom-patible with the vector meson dominance model (VMD)[60, 61].
B. ω/φ→ πγ∗
In this section we discuss the EM transition form fac-tors of the ω and φ mesons. The Dalitz decay of thevector mesons into pion and a lepton pair
〈π0(p0) l+(p+) l−(p−) |T |V (pV , λ)〉 =
(2π)4 δ(pV − p0 − p+ − p−)HV π , (42)
Expected Dalitz Plot
8
V. NUMERICAL RESULTS
A. ω/φ→ 3π
We solve the integral equation in Eq.(31) by numericaliteration 3. The convergence is fast, typically after threeto four iterations, no significant variations in the solutionare observed. From the amplitude, it is straightforwardto compute the Dalitz plot distribution, the partial decayand the total, 3π decay widths, [1]
d2Γ
ds dt=
1
(2π)3
1
32M3
1
3P (s, t) |F (s, t, u)|2 , (37)
where P (s, t) = φ(s, t)/4 is the kinematic factor discussedin Sec. II. In the computations of the Dalitz plot that fol-low, the conformal expansion in Eq. (29) is truncated at
0th order i.e. only a constant term is kept and this isthe only free parameter of the model. It is fixed to repro-duce the measured 3π decay width for ω and φ, whichare Γexpω→3π = 7.57 MeV and Γexpφ→3π = 0.65 MeV, respec-
tively [1]. Since the integral equation is linear in F (s),the one parameter that is fitted is responsible for theoverall normalization, while the Dalitz plot distributionis only affected by higher order terms in Σ(s).
In Fig. 5 we show the solution of the integral equa-tion (31) together with the invariant mass distribution.The significance of the three-body effects, given by thecross-channel terms, is accessed by keeping or eliminat-ing F from the discontinuity relation. In either case Σ(s)is represented by a constant which is fitted to reproducethe decay width. As can be seen in Fig. 5 the effect ofthe crossed-channels for ω → 3π is less significant thanfor φ → 3π. In both cases, the invariant mass distribu-tion are quite similar. The three body effects are morepronounced for the Dalitz plot distributions to which weturn next.
In Fig. 6 we show the Dalitz plot distribution in termsof Lorentz invariant, dimensionless parameters
x =
√3
Q(T1 − T2) =
√3(t− u)
2M(M − 3mπ),
y =3T3
Q− 1 =
3(sc − s)2M(M − 3mπ)
. (38)
Here Ti is the kinetic energy of the i-th pion in the three-particle rest frame and, using the isospin-averaged pion
3 Note, that the double integral in Eq. (31) (F (s) is given by acontour integral over t as shown in Eq. (19)) can be invertedusing the Pasquier method [28, 55]. In this method the order ofthe s and t integration is reversed with the latter deformed ontoa real axis that needs can be calculated analytically or numeri-cally only once. This leads to a single-variable integral equationfor F (s) with a kernel that depends on the input two-body scat-tering amplitude. This is an equivalent method to solve the KTequation which has its advantages and disadvantages [56].
mass, Q = M − 3m2π and sc = 1
3 (M2 + 3m2π) represents
the location of the center of the Mandelstam triangle.Dalitz plot distribution is symmetric under the x ↔ −xreflection as a consequence of the t ↔ u symmetry. Forω decays it is convenient to parametrize the Dalitz plotdistribution in terms of a polynomial expansion in x andy around the center of the plot. We follow the procedureoutlined in [37] and introduce polar variables
x =√z cosϑ , y =
√z sinϑ , (39)
and fit the following polynomial expansion
|Fpar(z, ϑ)|2 = |N |2(1 + 2α z + 2β z3/2 sin(3ϑ) + 2 γ z2
+2 δ z5/2 sin(3ϑ) +O(z3))
(40)
to our matrix element. In (40) N is the overall nor-malization constant. To find Dalitz plot parameters weminimize
χ2 =
∫
D
dz dϑ
ND
(P (z, ϑ)2(|Fpar(z, ϑ)|2 − |F (z, ϑ)|2)
P (0, 0)2|N |2)2
ND =
∫
D
dz dϑ , (41)
where the integration range (D) is limited by the Dalitzplot. The results are summarized in Table I. There is anon negligible deviation between the Dalitz plot param-eters with and without three body effects. In particular,the three-body effects result in the decrease of intensityby approximately 5% at the boundary of the Dalitz plotand an increase by approximately 2% in the center. Asimilar, but even more significant effect is observed forφ → 3π, where the Dalitz plot intensity decreases atthe boundary by 42% and increases by 6% in the centralarea. In Table I we also compare our results with othertheoretical calculations from [19] and [37]. We find ourDalitz plot parameters to be quite similar to [37] (whichis not surprising since our formalisms are similar) andin general smaller than the ones given in [19]. The lat-ter calculation is based on a chiral Lagrangian modifiedby explicit inclusion of light vector mesons [18]. In [18]the unknown coupling constants of the Lagrangian wereobtained from the decay properties of the vector mesons[57]. To this extent, the result of [19] provides a goodestimate for the decay width, while in the present anal-ysis the decay width was used to fix the normalization.The shortcoming of the approach in [19] is that it doesnot fully comply with unitarity. Though the two-bodypartial waves were unitarized, the crossed-channel effectswere not included.
On the experimental side, the situation is the follow-ing: The measurements of φ → 3π were performed byKLOE [58] and CMD-2 [59] collaborations. As for ω de-cay we expect new data from CLAS12, WASA at COSYand KLOE collaborations. Since the main purpose of thepresent paper is to outline a novel theoretical scheme, we
Lorentz Invariant Variables
x =√z cos θ
y =√z sin θ
8
V. NUMERICAL RESULTS
A. ω/φ→ 3π
We solve the integral equation in Eq.(31) by numericaliteration 3. The convergence is fast, typically after threeto four iterations, no significant variations in the solutionare observed. From the amplitude, it is straightforwardto compute the Dalitz plot distribution, the partial decayand the total, 3π decay widths, [1]
d2Γ
ds dt=
1
(2π)3
1
32M3
1
3P (s, t) |F (s, t, u)|2 , (37)
where P (s, t) = φ(s, t)/4 is the kinematic factor discussedin Sec. II. In the computations of the Dalitz plot that fol-low, the conformal expansion in Eq. (29) is truncated at
0th order i.e. only a constant term is kept and this isthe only free parameter of the model. It is fixed to repro-duce the measured 3π decay width for ω and φ, whichare Γexpω→3π = 7.57 MeV and Γexpφ→3π = 0.65 MeV, respec-
tively [1]. Since the integral equation is linear in F (s),the one parameter that is fitted is responsible for theoverall normalization, while the Dalitz plot distributionis only affected by higher order terms in Σ(s).
In Fig. 5 we show the solution of the integral equa-tion (31) together with the invariant mass distribution.The significance of the three-body effects, given by thecross-channel terms, is accessed by keeping or eliminat-ing F from the discontinuity relation. In either case Σ(s)is represented by a constant which is fitted to reproducethe decay width. As can be seen in Fig. 5 the effect ofthe crossed-channels for ω → 3π is less significant thanfor φ → 3π. In both cases, the invariant mass distribu-tion are quite similar. The three body effects are morepronounced for the Dalitz plot distributions to which weturn next.
In Fig. 6 we show the Dalitz plot distribution in termsof Lorentz invariant, dimensionless parameters
x =
√3
Q(T1 − T2) =
√3(t− u)
2M(M − 3mπ),
y =3T3
Q− 1 =
3(sc − s)2M(M − 3mπ)
. (38)
Here Ti is the kinetic energy of the i-th pion in the three-particle rest frame and, using the isospin-averaged pion
3 Note, that the double integral in Eq. (31) (F (s) is given by acontour integral over t as shown in Eq. (19)) can be invertedusing the Pasquier method [28, 55]. In this method the order ofthe s and t integration is reversed with the latter deformed ontoa real axis that needs can be calculated analytically or numeri-cally only once. This leads to a single-variable integral equationfor F (s) with a kernel that depends on the input two-body scat-tering amplitude. This is an equivalent method to solve the KTequation which has its advantages and disadvantages [56].
mass, Q = M − 3m2π and sc = 1
3 (M2 + 3m2π) represents
the location of the center of the Mandelstam triangle.Dalitz plot distribution is symmetric under the x ↔ −xreflection as a consequence of the t ↔ u symmetry. Forω decays it is convenient to parametrize the Dalitz plotdistribution in terms of a polynomial expansion in x andy around the center of the plot. We follow the procedureoutlined in [37] and introduce polar variables
x =√z cosϑ , y =
√z sinϑ , (39)
and fit the following polynomial expansion
|Fpar(z, ϑ)|2 = |N |2(1 + 2α z + 2β z3/2 sin(3ϑ) + 2 γ z2
+2 δ z5/2 sin(3ϑ) +O(z3))
(40)
to our matrix element. In (40) N is the overall nor-malization constant. To find Dalitz plot parameters weminimize
χ2 =
∫
D
dz dϑ
ND
(P (z, ϑ)2(|Fpar(z, ϑ)|2 − |F (z, ϑ)|2)
P (0, 0)2|N |2)2
ND =
∫
D
dz dϑ , (41)
where the integration range (D) is limited by the Dalitzplot. The results are summarized in Table I. There is anon negligible deviation between the Dalitz plot param-eters with and without three body effects. In particular,the three-body effects result in the decrease of intensityby approximately 5% at the boundary of the Dalitz plotand an increase by approximately 2% in the center. Asimilar, but even more significant effect is observed forφ → 3π, where the Dalitz plot intensity decreases atthe boundary by 42% and increases by 6% in the centralarea. In Table I we also compare our results with othertheoretical calculations from [19] and [37]. We find ourDalitz plot parameters to be quite similar to [37] (whichis not surprising since our formalisms are similar) andin general smaller than the ones given in [19]. The lat-ter calculation is based on a chiral Lagrangian modifiedby explicit inclusion of light vector mesons [18]. In [18]the unknown coupling constants of the Lagrangian wereobtained from the decay properties of the vector mesons[57]. To this extent, the result of [19] provides a goodestimate for the decay width, while in the present anal-ysis the decay width was used to fix the normalization.The shortcoming of the approach in [19] is that it doesnot fully comply with unitarity. Though the two-bodypartial waves were unitarized, the crossed-channel effectswere not included.
On the experimental side, the situation is the follow-ing: The measurements of φ → 3π were performed byKLOE [58] and CMD-2 [59] collaborations. As for ω de-cay we expect new data from CLAS12, WASA at COSYand KLOE collaborations. Since the main purpose of thepresent paper is to outline a novel theoretical scheme, we
Dalitz Plot Amplitude Expansion
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 11 / 48
Dalitz Plot Parameter Comparison
8
V. NUMERICAL RESULTS
A. ω/φ→ 3π
We solve the integral equation in Eq.(31) by numericaliteration 3. The convergence is fast, typically after threeto four iterations, no significant variations in the solutionare observed. From the amplitude, it is straightforwardto compute the Dalitz plot distribution, the partial decayand the total, 3π decay widths, [1]
d2Γ
ds dt=
1
(2π)3
1
32M3
1
3P (s, t) |F (s, t, u)|2 , (37)
where P (s, t) = φ(s, t)/4 is the kinematic factor discussedin Sec. II. In the computations of the Dalitz plot that fol-low, the conformal expansion in Eq. (29) is truncated at
0th order i.e. only a constant term is kept and this isthe only free parameter of the model. It is fixed to repro-duce the measured 3π decay width for ω and φ, whichare Γexpω→3π = 7.57 MeV and Γexpφ→3π = 0.65 MeV, respec-
tively [1]. Since the integral equation is linear in F (s),the one parameter that is fitted is responsible for theoverall normalization, while the Dalitz plot distributionis only affected by higher order terms in Σ(s).
In Fig. 5 we show the solution of the integral equa-tion (31) together with the invariant mass distribution.The significance of the three-body effects, given by thecross-channel terms, is accessed by keeping or eliminat-ing F from the discontinuity relation. In either case Σ(s)is represented by a constant which is fitted to reproducethe decay width. As can be seen in Fig. 5 the effect ofthe crossed-channels for ω → 3π is less significant thanfor φ → 3π. In both cases, the invariant mass distribu-tion are quite similar. The three body effects are morepronounced for the Dalitz plot distributions to which weturn next.
In Fig. 6 we show the Dalitz plot distribution in termsof Lorentz invariant, dimensionless parameters
x =
√3
Q(T1 − T2) =
√3(t− u)
2M(M − 3mπ),
y =3T3
Q− 1 =
3(sc − s)2M(M − 3mπ)
. (38)
Here Ti is the kinetic energy of the i-th pion in the three-particle rest frame and, using the isospin-averaged pion
3 Note, that the double integral in Eq. (31) (F (s) is given by acontour integral over t as shown in Eq. (19)) can be invertedusing the Pasquier method [28, 55]. In this method the order ofthe s and t integration is reversed with the latter deformed ontoa real axis that needs can be calculated analytically or numeri-cally only once. This leads to a single-variable integral equationfor F (s) with a kernel that depends on the input two-body scat-tering amplitude. This is an equivalent method to solve the KTequation which has its advantages and disadvantages [56].
mass, Q = M − 3m2π and sc = 1
3 (M2 + 3m2π) represents
the location of the center of the Mandelstam triangle.Dalitz plot distribution is symmetric under the x ↔ −xreflection as a consequence of the t ↔ u symmetry. Forω decays it is convenient to parametrize the Dalitz plotdistribution in terms of a polynomial expansion in x andy around the center of the plot. We follow the procedureoutlined in [37] and introduce polar variables
x =√z cosϑ , y =
√z sinϑ , (39)
and fit the following polynomial expansion
|Fpar(z, ϑ)|2 = |N |2(1 + 2α z + 2β z3/2 sin(3ϑ) + 2 γ z2
+2 δ z5/2 sin(3ϑ) +O(z3))
(40)
to our matrix element. In (40) N is the overall nor-malization constant. To find Dalitz plot parameters weminimize
χ2 =
∫
D
dz dϑ
ND
(P (z, ϑ)2(|Fpar(z, ϑ)|2 − |F (z, ϑ)|2)
P (0, 0)2|N |2)2
ND =
∫
D
dz dϑ , (41)
where the integration range (D) is limited by the Dalitzplot. The results are summarized in Table I. There is anon negligible deviation between the Dalitz plot param-eters with and without three body effects. In particular,the three-body effects result in the decrease of intensityby approximately 5% at the boundary of the Dalitz plotand an increase by approximately 2% in the center. Asimilar, but even more significant effect is observed forφ → 3π, where the Dalitz plot intensity decreases atthe boundary by 42% and increases by 6% in the centralarea. In Table I we also compare our results with othertheoretical calculations from [19] and [37]. We find ourDalitz plot parameters to be quite similar to [37] (whichis not surprising since our formalisms are similar) andin general smaller than the ones given in [19]. The lat-ter calculation is based on a chiral Lagrangian modifiedby explicit inclusion of light vector mesons [18]. In [18]the unknown coupling constants of the Lagrangian wereobtained from the decay properties of the vector mesons[57]. To this extent, the result of [19] provides a goodestimate for the decay width, while in the present anal-ysis the decay width was used to fix the normalization.The shortcoming of the approach in [19] is that it doesnot fully comply with unitarity. Though the two-bodypartial waves were unitarized, the crossed-channel effectswere not included.
On the experimental side, the situation is the follow-ing: The measurements of φ → 3π were performed byKLOE [58] and CMD-2 [59] collaborations. As for ω de-cay we expect new data from CLAS12, WASA at COSYand KLOE collaborations. Since the main purpose of thepresent paper is to outline a novel theoretical scheme, we
9
FIG. 6: The Dalitz plots for ω → 3π (left-hand panel) and φ → 3π (right-hand panel) decays. The distributions are dividedby the p-wave phase space P and normalized to 1 at x = y = 0. This is a parameter free result, because we kept only one termin the conformal expansion (29) which is responsible for the overall normalization. See main text for details.
TABLE I: Dalitz Plot parameters and√χ2 of the polynomial parametrization (40) for ω → 3π. In addition to our results
we also show the selected results from Niecknig et al. [37] (dispersive study with incorporated crossed-channel effects) andTerschlusen et al. [19] (Lagrangian based study with the pion-pion rescattering effects).
α× 103 β × 103 γ × 103 δ × 103√χ2 × 103
This paper (F = 0) 136 - - - 3.5
This paper (full) 94 - - - 3.2
Niecknig et al. [37] 84...96 - - - 0.9...1.1
Terschlusen et al. [19] 202 - - - 6.6
This paper (F = 0) 125 30 - - 0.74
This paper (full) 84 28 - - 0.35
Niecknig et al. [37] 74...84 24...28 - - 0.052...0.078
Terschlusen et al. [19] 190 54 - - 2.1
This paper (F = 0) 113 27 24 - 0.1
This paper (full) 80 27 8 - 0.24
Niecknig et al. [37] 73...81 24...28 3...6 - 0.038...0.047
Terschlusen et al. [19] 172 43 50 - 0.4
This paper (F = 0) 114 24 20 6 0.005
This paper (full) 83 22 1 14 0.079
Niecknig et al. [37] 74...83 21...24 0...2 7...8 0.012...0.011
Terschlusen et al. [19] 174 35 43 20 0.1
postpone the comprehensive data analysis to the futureand for now only consider the application to electromag-netic (EM) transition form factors of ω/φ. In partic-ularly, the transition ω → πγ∗ is of interest since theexisting data in the time-like region seems to be incom-patible with the vector meson dominance model (VMD)[60, 61].
B. ω/φ→ πγ∗
In this section we discuss the EM transition form fac-tors of the ω and φ mesons. The Dalitz decay of thevector mesons into pion and a lepton pair
〈π0(p0) l+(p+) l−(p−) |T |V (pV , λ)〉 =
(2π)4 δ(pV − p0 − p+ − p−)HV π , (42)
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 12 / 48
Analysis: Kinematics
W, GeV1.6 1.8 2 2.2 2.4 2.6 2.8 3
Eve
nts
0
200
400
600
800
1000
1200
1400
1600
1800
2000W
dataacc (fit)
e_gamma, GeV1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4
Eve
nts
0
500
1000
1500
2000
2500
3000e_gamma
dataacc (fit)
2mt, GeV0.5 1 1.5 2 2.5 3 3.5 4
Eve
nts
0
0.2
0.4
0.6
0.8
1mt
dataacc (fit)
cos_cm-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
100
200
300
400
500
600
700
800
900 cos_cmdataacc (fit)
m_omega, GeV0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Eve
nts
0
100
200
300
400
500
600
700
800
900
m_omegadataacc (fit)
Eγ:[1500-1600]MeV and t:[0.100-0.200]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 13 / 48
Analysis: Kinematics
W, GeV1.6 1.8 2 2.2 2.4 2.6 2.8 3
Eve
nts
0
1000
2000
3000
4000
5000
Wdataacc (fit)
e_gamma, GeV1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4
Eve
nts
0
500
1000
1500
2000
2500
3000
3500e_gamma
dataacc (fit)
2mt, GeV0.5 1 1.5 2 2.5 3 3.5 4
Eve
nts
0
0.2
0.4
0.6
0.8
1mt
dataacc (fit)
cos_cm-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
200
400
600
800
1000cos_cm
dataacc (fit)
m_omega, GeV0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Eve
nts
0
500
1000
1500
2000
2500m_omega
dataacc (fit)
Eγ:[1500-1600]MeV and t:[0.300-0.400]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 14 / 48
Analysis: Kinematics
W, GeV1.6 1.8 2 2.2 2.4 2.6 2.8 3
Eve
nts
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Wdataacc (fit)
e_gamma, GeV1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4
Eve
nts
0
500
1000
1500
2000
2500
3000e_gamma
dataacc (fit)
2mt, GeV0.5 1 1.5 2 2.5 3 3.5 4
Eve
nts
0
500
1000
1500
2000
2500
3000
3500
4000
4500mt
dataacc (fit)
cos_cm-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
100
200
300
400
500
600
700 cos_cmdataacc (fit)
m_omega, GeV0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Eve
nts
0
200
400
600
800
1000
1200
1400
1600
1800m_omega
dataacc (fit)
Eγ:[1500-1600]MeV and t:[0.500-0.600]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 15 / 48
Analysis: Kinematics
W, GeV1.6 1.8 2 2.2 2.4 2.6 2.8 3
Eve
nts
0
500
1000
1500
2000
2500
3000
3500
4000
4500W
dataacc (fit)
e_gamma, GeV1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4
Eve
nts
0
500
1000
1500
2000
2500
3000 e_gammadataacc (fit)
2mt, GeV0.5 1 1.5 2 2.5 3 3.5 4
Eve
nts
0
500
1000
1500
2000
2500
3000
3500
4000
4500mt
dataacc (fit)
cos_cm-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
100
200
300
400
500
600
700cos_cm
dataacc (fit)
m_omega, GeV0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Eve
nts
0
200
400
600
800
1000
1200
1400
1600
1800 m_omegadataacc (fit)
Eγ:[1500-1600]MeV and t:[0.700-0.800]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 16 / 48
Analysis: Kinematics
W, GeV1.6 1.8 2 2.2 2.4 2.6 2.8 3
Eve
nts
0
500
1000
1500
2000
2500
3000 Wdataacc (fit)
e_gamma, GeV1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4
Eve
nts
0
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
e_gammadataacc (fit)
2mt, GeV0.5 1 1.5 2 2.5 3 3.5 4
Eve
nts
0
500
1000
1500
2000
2500
3000 mtdataacc (fit)
cos_cm-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
50
100
150
200
250
300
350
400 cos_cmdataacc (fit)
m_omega, GeV0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Eve
nts
0
200
400
600
800
1000
1200 m_omegadataacc (fit)
Eγ:[1500-1600]MeV and t:[0.900-1.000]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 17 / 48
Analysis: Kinematics
lambda, GeV^6
0 0.0020.0040.0060.0080.010.0120.0140.0160.0180.02
Eve
nts
0
50
100
150
200
250
300
350
lambdadataacc (fit)
theta_adair
0 0.5 1 1.5 2 2.5 3
Eve
nts
0
20
40
60
80
100theta_adair
dataacc (fit)
phi_adair
-3 -2 -1 0 1 2 3
Eve
nts
0
10
20
30
40
50
60
70
80phi_adair
dataacc (fit)
Eγ:[1500-1600]MeV and t:[0.100-0.200]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 18 / 48
Analysis: Kinematics
lambda, GeV^6
0 0.0020.0040.0060.0080.010.0120.0140.0160.0180.02
Eve
nts
0
200
400
600
800
1000
lambdadataacc (fit)
theta_adair
0 0.5 1 1.5 2 2.5 3
Eve
nts
0
50
100
150
200
250theta_adair
dataacc (fit)
phi_adair
-3 -2 -1 0 1 2 3
Eve
nts
0
20
40
60
80
100
120
140
160
180
200
220
phi_adairdataacc (fit)
Eγ:[1500-1600]MeV and t:[0.300-0.400]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 19 / 48
Analysis: Kinematics
lambda, GeV^6
0 0.0020.0040.0060.0080.010.0120.0140.0160.0180.02
Eve
nts
0
100
200
300
400
500
600
700 lambdadataacc (fit)
theta_adair
0 0.5 1 1.5 2 2.5 3
Eve
nts
0
20
40
60
80
100
120
140
160
180
200
theta_adairdataacc (fit)
phi_adair
-3 -2 -1 0 1 2 3
Eve
nts
0
20
40
60
80
100
120
140
160
phi_adairdataacc (fit)
Eγ:[1500-1600]MeV and t:[0.500-0.600]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 20 / 48
Analysis: Kinematics
lambda, GeV^6
0 0.0020.0040.0060.0080.010.0120.0140.0160.0180.02
Eve
nts
0
100
200
300
400
500
600
700
800
lambdadataacc (fit)
theta_adair
0 0.5 1 1.5 2 2.5 3
Eve
nts
0
50
100
150
200
250
theta_adairdataacc (fit)
phi_adair
-3 -2 -1 0 1 2 3
Eve
nts
0
20
40
60
80
100
120
140
160
phi_adairdataacc (fit)
Eγ:[1500-1600]MeV and t:[0.700-0.800]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 21 / 48
Analysis: Kinematics
lambda, GeV^6
0 0.0020.0040.0060.0080.010.0120.0140.0160.0180.02
Eve
nts
0
100
200
300
400
500 lambdadataacc (fit)
theta_adair
0 0.5 1 1.5 2 2.5 3
Eve
nts
0
20
40
60
80
100
120
140 theta_adairdataacc (fit)
phi_adair
-3 -2 -1 0 1 2 3
Eve
nts
0
20
40
60
80
100
120
phi_adairdataacc (fit)
Eγ:[1500-1600]MeV and t:[0.900-1.000]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 22 / 48
Analysis: Kinematics
0
1
2
3
4
5
6
7
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
DalitzLI_dat
0
10
20
30
40
50
60
70
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
DalitzLI_acc
2DalitzLI_pX, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
20
40
60
80
100DalitzLI_pX
dataacc (fit)
2DalitzLI_pY, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
20
40
60
80
100
120
140 DalitzLI_pYdataacc (fit)
Eγ:[1500-1600]MeV and t:[0.100-0.200]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 23 / 48
Analysis: Kinematics
0
2
4
6
8
10
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
DalitzLI_dat
0
10
20
30
40
50
60
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
DalitzLI_acc
2DalitzLI_pX, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
50
100
150
200
250DalitzLI_pX
dataacc (fit)
2DalitzLI_pY, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
50
100
150
200
250
300
350
DalitzLI_pYdataacc (fit)
Eγ:[1500-1600]MeV and t:[0.300-0.400]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 24 / 48
Analysis: Kinematics
0
2
4
6
8
10
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
DalitzLI_dat
0
5
10
15
20
25
30
35
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
DalitzLI_acc
2DalitzLI_pX, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
50
100
150
200
250DalitzLI_pX
dataacc (fit)
2DalitzLI_pY, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
50
100
150
200
250 DalitzLI_pYdataacc (fit)
Eγ:[1500-1600]MeV and t:[0.500-0.600]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 25 / 48
Analysis: Kinematics
0
1
2
3
4
5
6
7
8
9
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
DalitzLI_dat
0
5
10
15
20
25
30
35
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
DalitzLI_acc
2DalitzLI_pX, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
50
100
150
200
250DalitzLI_pX
dataacc (fit)
2DalitzLI_pY, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
50
100
150
200
250 DalitzLI_pYdataacc (fit)
Eγ:[1500-1600]MeV and t:[0.700-0.800]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 26 / 48
Analysis: Kinematics
0
1
2
3
4
5
6
7
8
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
DalitzLI_dat
0
5
10
15
20
25
30
35
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
DalitzLI_acc
2DalitzLI_pX, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
20
40
60
80
100
120
140
160 DalitzLI_pXdataacc (fit)
2DalitzLI_pY, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
20
40
60
80
100
120
140
160
180
DalitzLI_pYdataacc (fit)
Eγ:[1500-1600]MeV and t:[0.900-1.000]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 27 / 48
Analysis: Kinematics
0
20
40
60
80
100
310×
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
F_LI_dat
0
10
20
30
40
50
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
F_LI_acc
2F_LI_pX, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
20
40
60
80
100
120
140
160
180
310×
F_LI_pXdataacc (fit)
2F_LI_pY, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
20
40
60
80
100
120
140
160
310×
Eγ:[1500-1600]MeV and t:[0.100-0.200]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 28 / 48
Analysis: Kinematics
0
200
400
600
800
1000
310×
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
F_LI_dat
0
5
10
15
20
25
30
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
F_LI_acc
2F_LI_pX, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
200
400
600
800
1000
310×
F_LI_pXdataacc (fit)
2F_LI_pY, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
200
400
600
800
1000
12003
10×
Eγ:[1500-1600]MeV and t:[0.300-0.400]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 29 / 48
Analysis: Kinematics
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
22000
24000
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
F_LI_dat
0
5
10
15
20
25
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
F_LI_acc
2F_LI_pX, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
10000
20000
30000
40000
50000
60000
70000
80000
90000 F_LI_pXdataacc (fit)
2F_LI_pY, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
20
40
60
80
100
120
310×
Eγ:[1500-1600]MeV and t:[0.500-0.600]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 30 / 48
Analysis: Kinematics
0
20
40
60
80
100
120
310×
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
F_LI_dat
0
5
10
15
20
25
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
F_LI_acc
2F_LI_pX, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
20
40
60
80
100
120
140
160
180
200
310×
F_LI_pXdataacc (fit)
2F_LI_pY, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
20
40
60
80
100
120
140
160
180
200
2203
10×
Eγ:[1500-1600]MeV and t:[0.700-0.800]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 31 / 48
Analysis: Kinematics
0
10000
20000
30000
40000
50000
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
F_LI_dat
0
5
10
15
20
25
2x, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
2y,
GeV
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
F_LI_acc
2F_LI_pX, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
10000
20000
30000
40000
50000
60000
70000
80000 F_LI_pXdataacc (fit)
2F_LI_pY, GeV-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Eve
nts
0
20
40
60
80
100
120
310×
Eγ:[1500-1600]MeV and t:[0.900-1.000]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 32 / 48
Analysis: Decay Amplitude Parameter (IgorParameter)
)2t (GeV200 300 400 500 600 700 800 900 1000
Igo
rPar
amet
er
-10
-9
-8
-7
-6
-5
-4
-3
-2
Fit Parameter: IgorParameter
IgorParameterIgorParameter
Eγ:[1500-1600]MeV and t:[0.400-1.100]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 33 / 48
Analysis: Decay Amplitude Parameter (IgorParameter)
)2t (GeV200 400 600 800 1000 1200
Igo
rPar
amet
er
-12
-10
-8
-6
-4
-2
0
Fit Parameter: IgorParameter
IgorParameterIgorParameter
Eγ:[2000-2100]MeV and t:[0.400-1.300]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 34 / 48
Analysis: Decay Amplitude Parameter (IgorParameter)
)2t (GeV500 1000 1500 2000 2500 3000
Igo
rPar
amet
er
-16
-14
-12
-10
-8
-6
-4
-2
0
Fit Parameter: IgorParameter
IgorParameterIgorParameter
Eγ:[2500-2600]MeV and t:[0.400-3.100]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 35 / 48
Analysis: Decay Amplitude Acceptance (IgorParameter)
)2t (GeV
200 300 400 500 600 700 800 900 1000
Nd
at
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
22000
24000
Integrated data Event-Count Per Bin
Integrated data, Ndat
Ndat
)2t (GeV
200 300 400 500 600 700 800 900 1000
Nac
c ak
a N
sim
MC
0
5000
10000
15000
20000
25000
Integrated simMC Event-Count Per Bin
Integrated simMC, Nacc (fit)
Nacc
)2t (GeV
200 300 400 500 600 700 800 900 1000
eps
aka
Acc
epta
nce
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
Integrated Acceptance Per Bin
Integrated Acceptance, eps
eps
Eγ:[1500-1600]MeV and t:[0.400-1.100]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 36 / 48
Analysis: Decay Amplitude Acceptance (IgorParameter)
)2t (GeV
200 400 600 800 1000 1200
Nd
at
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
22000
Integrated data Event-Count Per Bin
Integrated data, Ndat
Ndat
)2t (GeV
200 400 600 800 1000 1200
Nac
c ak
a N
sim
MC
0
5000
10000
15000
20000
25000
Integrated simMC Event-Count Per Bin
Integrated simMC, Nacc (fit)
Nacc
)2t (GeV
200 400 600 800 1000 1200
eps
aka
Acc
epta
nce
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Integrated Acceptance Per Bin
Integrated Acceptance, eps
eps
Eγ:[2000-2100]MeV and t:[0.400-1.300]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 37 / 48
Analysis: Decay Amplitude Acceptance (IgorParameter)
)2t (GeV
500 1000 1500 2000 2500 3000
Nd
at
0
1000
2000
3000
4000
5000
6000
7000
8000
Integrated data Event-Count Per Bin
Integrated data, Ndat
Ndat
)2t (GeV
500 1000 1500 2000 2500 3000
Nac
c ak
a N
sim
MC
0
2000
4000
6000
8000
10000
Integrated simMC Event-Count Per Bin
Integrated simMC, Nacc (fit)
Nacc
)2t (GeV
500 1000 1500 2000 2500 3000
eps
aka
Acc
epta
nce
0
0.05
0.1
0.15
0.2
0.25
Integrated Acceptance Per Bin
Integrated Acceptance, eps
eps
Eγ:[2500-2600]MeV and t:[0.400-3.100]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 38 / 48
Next Steps
discussed
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 39 / 48
Analysis: Dalitz Plot Parameter (α)
)2t (GeV-3 -2.5 -2 -1.5 -1
om
ega1
_re
33000
34000
35000
36000
37000
38000
39000
40000
Fit Parameter: omega1_re
omega1_re
omega1_re
)2t (GeV-3 -2.5 -2 -1.5 -1
alp
ha
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
Fit Parameter: alpha
alpha
alpha
Eγ:[2500-3000]MeV and t:[0.400-3.400]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 40 / 48
Analysis: Dalitz Plot Parameters (α,β)
)2t (GeV
-3 -2.5 -2 -1.5 -1
om
ega1
_re
33000
34000
35000
36000
37000
38000
39000
40000
Fit Parameter: omega1_re
omega1_re
omega1_re
)2t (GeV
-3 -2.5 -2 -1.5 -1
alp
ha
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
Fit Parameter: alpha
alpha
alpha
)2t (GeV
-3 -2.5 -2 -1.5 -1
bet
a
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
Fit Parameter: beta
beta
beta
Eγ:[2500-3000]MeV and t:[0.400-3.400]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 41 / 48
Analysis: Dalitz Plot Parameters (α,β,γ)
)2t (GeV
-3 -2.5 -2 -1.5 -1
om
ega1
_re
1250
1300
1350
1400
1450
1500
1550
Fit Parameter: omega1_re
omega1_re
omega1_re
)2t (GeV
-3 -2.5 -2 -1.5 -1
alp
ha
4000
5000
6000
7000
8000
Fit Parameter: alpha
alpha
alpha
)2t (GeV
-3 -2.5 -2 -1.5 -1
bet
a
700
750
800
850
900
950
1000
Fit Parameter: beta
beta
beta
)2t (GeV
-3 -2.5 -2 -1.5 -1
gam
ma
-12000
-11000
-10000
-9000
-8000
-7000
-6000
-5000
Fit Parameter: gamma
gamma
gamma
Eγ:[2500-3000]MeV and t:[0.400-3.400]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 42 / 48
Analysis: Dalitz Plot Parameters (α,β,γ,δ)
)2t (GeV
-3 -2.5 -2 -1.5 -1
om
ega1
_re
0
5000
10000
15000
20000
25000
30000
35000
Fit Parameter: omega1_re
omega1_re
omega1_re
)2t (GeV
-3 -2.5 -2 -1.5 -1
alp
ha
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
Fit Parameter: alpha
alpha
alpha
)2t (GeV
-3 -2.5 -2 -1.5 -1
bet
a
-4000
-3000
-2000
-1000
0
1000
Fit Parameter: beta
beta
beta
)2t (GeV
-3 -2.5 -2 -1.5 -1g
amm
a
-14000
-12000
-10000
-8000
-6000
-4000
-2000
0
Fit Parameter: gamma
gamma
gamma
)2t (GeV
-3 -2.5 -2 -1.5 -1
del
ta
0
2000
4000
6000
8000
Fit Parameter: delta
delta
delta
Eγ:[2500-3000]MeV and t:[0.400-3.400]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 43 / 48
Analysis: Integrated Populations and Acceptance
)2t (GeV-3 -2.5 -2 -1.5 -1
Nd
at
0
10000
20000
30000
40000
50000
60000
70000
80000
Integrated data Event-Count Per Bin
Integrated data, NdatNdat
)2t (GeV-3 -2.5 -2 -1.5 -1
Ng
en a
ka N
raw
MC
0
100
200
300
400
500
310×
Integrated rawMC Event-Count Per Bin
Integrated rawMC, NgenNgen
)2t (GeV-3 -2.5 -2 -1.5 -1
Nac
c ak
a N
sim
MC
0
10000
20000
30000
40000
50000
60000
70000
80000
Integrated simMC Event-Count Per Bin
Integrated simMC, NaccNacc
)2t (GeV-3 -2.5 -2 -1.5 -1
eps
aka
Acc
epta
nce
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Integrated Acceptance Per Bin
Integrated Acceptance, epseps
Eγ:[2500-3000]MeV and t:[0.400-3.400]GeV2
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 44 / 48
The Decay Amplitude
6
FIG. 4: Real and imaginary parts of Ωel(s) in Eq. (32) (Dot-Dashed), Ω′(s) in Eq. (35) (Dashed) and Ω(s) in Eq. (34) (Solid).
The first part is determined entirely by elastic scatteringwhile the second part takes into account inelastic effects.The inelastic contribution is described by an analyticalfunction on the s-plane cut along the real axis aboves = si. It is largely unknown, and often parametrizedthrough an expansion in a conformal variable which mapsthe right-hand cut in the complex s-plane onto the unitdisk. Such a mapping is known as a convenient repre-sentation of functions on a cut plane with the the knownanalytical properties [48],
Σ(s) =∞∑
i=0
ai ωi(s) (29)
The variable
ω(s) =
√si − sE −
√si − s√
si − sE +√si − s
(30)
maps the cut plane onto the unit disk. The parametersi = 1 GeV is identified with the point where inelas-tic contributions are expected to become relevant2 andthe expansion point sE should lie below the cut. Wedefine sE = 0. The conformal mapping technique wassuccessfully applied in other descriptions of two-to-twoamplitudes e.g. in [49–51] it was used to take into ac-count the contributions from the more distant left-handcuts. However, to the best of our knowledge the confor-mal mapping technique has never been used before in thecontext of the KT equations.
2 Note the following inelastic thresholds: mπ + mω = 0.92 GeV(only for the ω decay), 2mK = 0.99 GeV and 8mπ = 1.1 GeV(we omit the 4π and 6π multi-particle thresholds since they areknown to be weak).
With the inelastic contributions parametrized by thefunction, Σ(s), the integral equation for the KT ampli-tude takes the form of
F (s) = Ω(s)
(1
π
∫ si
sπ
ds′ρ(s′) t∗(s′)
Ω∗(s′)F (s′)s′ − s + Σ(s)
).
(31)This is an alternative to the standard way which employssubtractions to reduce sensitivity of the dispersive inte-gral to the high-energy region [37]. The problem withsubtractions is twofold. First, the dispersive integrals,including computation of Ω(s) run over inelastic regions,while the dispersion relation contains only the elastic con-tributions. Furthermore, subtracting an analytical func-tion of s does not account for the change in the analyti-cal behavior of the amplitudes due to opening of inelasticchannels.
In Eq. (31) there is no need for subtractions in the dis-persive integral since it is restricted to the elastic-region,which the only part of the right hand cut controlled byelastic unitarity. The unknown, inelastic contributionsare parametrized by Σ(s), and are to be determined bycomparing with the experimental data, or other theoret-ical approaches that treat inelastic channels explicitly.Moreover, with the dispersive integral restricted to a fi-nite interval over s there are uncontrollably large con-tributions from higher-partial waves, which otherwise re-quire more and more subtractions.
Besides F (s), the problem with the determination ofinelastic contributions also affects computation of Ω(s).The unitarity condition in Eq. (25) does not determine itabove the inelastic threshold s = si. Therefore, we seekits solution given in terms of the Omnes function [52, 53]taken only over the elastic region
Ωel(s) ≡ exp
(s
π
∫ si
sπ
ds′
s′δ(s′)s′ − s
), (32)
KT Amplitude
6
FIG. 4: Real and imaginary parts of Ωel(s) in Eq. (32) (Dot-Dashed), Ω′(s) in Eq. (35) (Dashed) and Ω(s) in Eq. (34) (Solid).
The first part is determined entirely by elastic scatteringwhile the second part takes into account inelastic effects.The inelastic contribution is described by an analyticalfunction on the s-plane cut along the real axis aboves = si. It is largely unknown, and often parametrizedthrough an expansion in a conformal variable which mapsthe right-hand cut in the complex s-plane onto the unitdisk. Such a mapping is known as a convenient repre-sentation of functions on a cut plane with the the knownanalytical properties [48],
Σ(s) =∞∑
i=0
ai ωi(s) (29)
The variable
ω(s) =
√si − sE −
√si − s√
si − sE +√si − s
(30)
maps the cut plane onto the unit disk. The parametersi = 1 GeV is identified with the point where inelas-tic contributions are expected to become relevant2 andthe expansion point sE should lie below the cut. Wedefine sE = 0. The conformal mapping technique wassuccessfully applied in other descriptions of two-to-twoamplitudes e.g. in [49–51] it was used to take into ac-count the contributions from the more distant left-handcuts. However, to the best of our knowledge the confor-mal mapping technique has never been used before in thecontext of the KT equations.
2 Note the following inelastic thresholds: mπ + mω = 0.92 GeV(only for the ω decay), 2mK = 0.99 GeV and 8mπ = 1.1 GeV(we omit the 4π and 6π multi-particle thresholds since they areknown to be weak).
With the inelastic contributions parametrized by thefunction, Σ(s), the integral equation for the KT ampli-tude takes the form of
F (s) = Ω(s)
(1
π
∫ si
sπ
ds′ρ(s′) t∗(s′)
Ω∗(s′)F (s′)s′ − s + Σ(s)
).
(31)This is an alternative to the standard way which employssubtractions to reduce sensitivity of the dispersive inte-gral to the high-energy region [37]. The problem withsubtractions is twofold. First, the dispersive integrals,including computation of Ω(s) run over inelastic regions,while the dispersion relation contains only the elastic con-tributions. Furthermore, subtracting an analytical func-tion of s does not account for the change in the analyti-cal behavior of the amplitudes due to opening of inelasticchannels.
In Eq. (31) there is no need for subtractions in the dis-persive integral since it is restricted to the elastic-region,which the only part of the right hand cut controlled byelastic unitarity. The unknown, inelastic contributionsare parametrized by Σ(s), and are to be determined bycomparing with the experimental data, or other theoret-ical approaches that treat inelastic channels explicitly.Moreover, with the dispersive integral restricted to a fi-nite interval over s there are uncontrollably large con-tributions from higher-partial waves, which otherwise re-quire more and more subtractions.
Besides F (s), the problem with the determination ofinelastic contributions also affects computation of Ω(s).The unitarity condition in Eq. (25) does not determine itabove the inelastic threshold s = si. Therefore, we seekits solution given in terms of the Omnes function [52, 53]taken only over the elastic region
Ωel(s) ≡ exp
(s
π
∫ si
sπ
ds′
s′δ(s′)s′ − s
), (32)
inelastic contribution
6
FIG. 4: Real and imaginary parts of Ωel(s) in Eq. (32) (Dot-Dashed), Ω′(s) in Eq. (35) (Dashed) and Ω(s) in Eq. (34) (Solid).
The first part is determined entirely by elastic scatteringwhile the second part takes into account inelastic effects.The inelastic contribution is described by an analyticalfunction on the s-plane cut along the real axis aboves = si. It is largely unknown, and often parametrizedthrough an expansion in a conformal variable which mapsthe right-hand cut in the complex s-plane onto the unitdisk. Such a mapping is known as a convenient repre-sentation of functions on a cut plane with the the knownanalytical properties [48],
Σ(s) =∞∑
i=0
ai ωi(s) (29)
The variable
ω(s) =
√si − sE −
√si − s√
si − sE +√si − s
(30)
maps the cut plane onto the unit disk. The parametersi = 1 GeV is identified with the point where inelas-tic contributions are expected to become relevant2 andthe expansion point sE should lie below the cut. Wedefine sE = 0. The conformal mapping technique wassuccessfully applied in other descriptions of two-to-twoamplitudes e.g. in [49–51] it was used to take into ac-count the contributions from the more distant left-handcuts. However, to the best of our knowledge the confor-mal mapping technique has never been used before in thecontext of the KT equations.
2 Note the following inelastic thresholds: mπ + mω = 0.92 GeV(only for the ω decay), 2mK = 0.99 GeV and 8mπ = 1.1 GeV(we omit the 4π and 6π multi-particle thresholds since they areknown to be weak).
With the inelastic contributions parametrized by thefunction, Σ(s), the integral equation for the KT ampli-tude takes the form of
F (s) = Ω(s)
(1
π
∫ si
sπ
ds′ρ(s′) t∗(s′)
Ω∗(s′)F (s′)s′ − s + Σ(s)
).
(31)This is an alternative to the standard way which employssubtractions to reduce sensitivity of the dispersive inte-gral to the high-energy region [37]. The problem withsubtractions is twofold. First, the dispersive integrals,including computation of Ω(s) run over inelastic regions,while the dispersion relation contains only the elastic con-tributions. Furthermore, subtracting an analytical func-tion of s does not account for the change in the analyti-cal behavior of the amplitudes due to opening of inelasticchannels.
In Eq. (31) there is no need for subtractions in the dis-persive integral since it is restricted to the elastic-region,which the only part of the right hand cut controlled byelastic unitarity. The unknown, inelastic contributionsare parametrized by Σ(s), and are to be determined bycomparing with the experimental data, or other theoret-ical approaches that treat inelastic channels explicitly.Moreover, with the dispersive integral restricted to a fi-nite interval over s there are uncontrollably large con-tributions from higher-partial waves, which otherwise re-quire more and more subtractions.
Besides F (s), the problem with the determination ofinelastic contributions also affects computation of Ω(s).The unitarity condition in Eq. (25) does not determine itabove the inelastic threshold s = si. Therefore, we seekits solution given in terms of the Omnes function [52, 53]taken only over the elastic region
Ωel(s) ≡ exp
(s
π
∫ si
sπ
ds′
s′δ(s′)s′ − s
), (32)
dispersion relation
a0 = “IgorParameter”
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 45 / 48
Previous Parameter Values and Errors,Eγ:[2500-3500]MeV
2.6 2.8 3 3.2 3.4 3.60
10
20
30
40
50
60
omega_s1_omega1_reomega_s1_omega1_re
2.6 2.8 3 3.2 3.4 3.60
200
400
600
800
1000
1200
1400
1600IgorParameter
IgorParameter
2.6 2.8 3 3.2 3.4 3.6
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44rho00
rho00
2.6 2.8 3 3.2 3.4 3.6-0.06
-0.04
-0.02
0
0.02
0.04
0.06
rho10rho10
2.6 2.8 3 3.2 3.4 3.6-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
rho1m1rho1m1
Note: Broken paddle, Eγ:[3000-3100]MeVChris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 46 / 48
Previous Comments by Mike P., Adam S., Igor D. andCarlos S.
Fitted values are unstable. Have 8M data and 1.5M MC accepted,Need at lest the same amount of of acc-MC and data.
Could you bin in t? Your rho parameters are t dependent and thattrend will reflect in your fits and in the Igor’s parameter that you willfinally want to extract.
Adam suggested that you may have too many parameters in the fit,as rho00 is also part of the overall normalization.
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 47 / 48
Comments by Mike P., Adam S., Igor D. and Carlos S.
4) It will be good to start with a fit using F = 1 (w/o Igor correction),and extract the rho parameters in that way to compare with g11results, and then include the new Igor parameter.
5) Igor’s parameter should be independent of production, i.e. onEbeam and t.
Chris Zeoli (FSU) Dalitz Analysis of ω → 3π June 4, 2015 48 / 48
![Least Squares Optimization and Gradient Descent Algorithm · 2019. 11. 21. · SCATTER PLOT Plot all (X i, Y i) pairs, and plot your learned model !4 0 20 40 60 0 20 40 60 X Y [WF]](https://static.fdocument.org/doc/165x107/6124df642da9ad37a74372ef/least-squares-optimization-and-gradient-descent-algorithm-2019-11-21-scatter.jpg)
![Biological Research in India - · PDF fileRamachandran plot or a [φ,ψ] plot developed in 1963 by G. N. Ramachandran, C. ... biotechnology, bioinformatics, and the various ‘omics’](https://static.fdocument.org/doc/165x107/5ab3b6597f8b9a1d168ea056/biological-research-in-india-plot-or-a-plot-developed-in-1963-by-g-n.jpg)
