High-statistics measurement of the eta->3pi^0 decay at the Mainz Microtron
S. Prakhov, S. Abt, P. Achenbach, P. Adlarson, F. Afzal, P. Aguar-Bartolomé, Z. Ahmed, J. Ahrens, J. R. M. Annand, H. J. Arends, K. Bantawa, M. Bashkanov, R. Beck, M. Biroth, N. S. Borisov, A. Braghieri, W. J. Briscoe, S. Cherepnya, F. Cividini, C. Collicott, S. Costanza, A. Denig, M. Dieterle, E. J. Downie, P. Drexler, M. I. Ferretti Bondy, L. V. Fil'kov, A. Fix, S. Gardner, S. Garni, D. I. Glazier, I. Gorodnov, W. Gradl, G. M. Gurevich, C. B. Hamill, L. Heijkenskjöld, D. Hornidge, G. M. Huber, A. Käser, V. L. Kashevarov, S. Kay, I. Keshelashvili, R. Kondratiev, M. Korolija, B. Krusche, A. Lazarev, V. Lisin, K. Livingston, S. Lutterer, I. J. D. MacGregor, D. M. Manley, P. P. Martel, J. C. McGeorge, D. G. Middleton, R. Miskimen, E. Mornacchi, A. Mushkarenkov, A. Neganov, A. Neiser, M. Oberle, M. Ostrick, P. B. Otte, D. Paudyal, P. Pedroni, A. Polonski, G. Ron, T. Rostomyan, C. Sfienti, V. Sokhoyan, K. Spieker, O. Steffen, I. I. Strakovsky, B. Strandberg, Th. Strub, I. Supek, A. Thiel, M. Thiel, A. Thomas, M. Unverzagt, Yu. A. Usov, S. Wagner, N. K. Walford, D. P. Watts, D. Werthmüller, J. Wettig, L. Witthauer, M. Wolfes, L. A. Zana
aa r X i v : . [ h e p - e x ] J un High-statistics measurement of the η → π decay at the Mainz Microtron S. Prakhov,
1, 2, ∗ S. Abt, P. Achenbach, P. Adlarson, F. Afzal, P. Aguar-Bartolom´e, Z. Ahmed, J. Ahrens, J. R. M. Annand, H. J. Arends, K. Bantawa, M. Bashkanov, R. Beck, M. Biroth, N. S. Borisov, A. Braghieri, W. J. Briscoe, S. Cherepnya, F. Cividini, C. Collicott,
13, 14
S. Costanza,
10, 15
A. Denig, M. Dieterle, E. J. Downie, P. Drexler, M. I. Ferretti Bondy, L. V. Fil’kov, A. Fix, S. Gardner, S. Garni, D. I. Glazier,
6, 8
I. Gorodnov, W. Gradl, G. M. Gurevich, C. B. Hamill, L. Heijkenskj¨old, D. Hornidge, G. M. Huber, A. K¨aser, V. L. Kashevarov,
S. Kay, I. Keshelashvili, R. Kondratiev, M. Korolija, B. Krusche, A. Lazarev, V. Lisin, K. Livingston, S. Lutterer, I. J. D. MacGregor, D. M. Manley, P. P. Martel,
1, 18
J. C. McGeorge, D. G. Middleton,
1, 18
R. Miskimen, E. Mornacchi, A. Mushkarenkov,
10, 20
A. Neganov, A. Neiser, M. Oberle, M. Ostrick, P. B. Otte, D. Paudyal, P. Pedroni, A. Polonski, G. Ron, T. Rostomyan, A. Sarty, C. Sfienti, V. Sokhoyan, K. Spieker, O. Steffen, I. I. Strakovsky, B. Strandberg, Th. Strub, I. Supek, A. Thiel, M. Thiel, A. Thomas, M. Unverzagt, Yu. A. Usov, S. Wagner, N. K. Walford, D. P. Watts, D. Werthm¨uller,
6, 3
J. Wettig, L. Witthauer, M. Wolfes, and L. A. Zana (A2 Collaboration at MAMI) Institut f¨ur Kernphysik, University of Mainz, D-55099 Mainz,Germany University of California Los Angeles, Los Angeles, California 90095-1547, USA Institut f¨ur Physik, University of Basel, CH-4056 Basel, Switzerland Helmholtz-Institut f¨ur Strahlen- und Kernphysik, University of Bonn, D-53115 Bonn, Germany University of Regina, Regina, Saskatchewan S4S 0A2, Canada SUPA School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom Kent State University, Kent, Ohio 44242-0001, USA School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom Joint Institute for Nuclear Research, 141980 Dubna, Russia INFN Sezione di Pavia, I-27100 Pavia, Italy The George Washington University, Washington, DC 20052-0001, USA Lebedev Physical Institute, 119991 Moscow, Russia Dalhousie University, Halifax, Nova Scotia B3H 4R2, Canada Saint Marys University, Halifax, Nova Scotia B3H 3C3, Canada Dipartimento di Fisica, Universit`a di Pavia, I-27100 Pavia, Italy Laboratory of Mathematical Physics, Tomsk Polytechnic University, 634034 Tomsk, Russia Institute for Nuclear Research, 125047 Moscow, Russia Mount Allison University, Sackville, New Brunswick E4L 1E6, Canada Rudjer Boskovic Institute, HR-10000 Zagreb, Croatia University of Massachusetts, Amherst, Massachusetts 01003, USA Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel (Dated: June 13, 2018)The largest, at the moment, statistics of 7 × η → π decays, based on 6 . × η mesonsproduced in the γp → ηp reaction, has been accumulated by the A2 Collaboration at the MainzMicrotron, MAMI. It allowed a detailed study of the η → π dynamics beyond its conventionalparametrization with just the quadratic slope parameter α and enabled, for the first time, a mea-surement of the second-order term and a better understanding of the cusp structure in the neutraldecay. The present data are also compared to recent theoretical calculations that predict a nonlineardependence along the quadratic distance from the Dalitz-plot center. I. INTRODUCTION
For decades, the η → π decay has attracted muchattention from theoretical and experimental studies asit gives access to fundamental physical constants. Thisdecay, which is forbidden by isospin symmetry, mostlyoccurs due to the difference in the mass of the u and d quarks, with Γ( η → π ) ∼ ( m d − m u ) [1]. Therefore,a precision measurement of this decay can be used as asensitive test for the magnitude of isospin breaking in the ∗ Electronic address: [email protected]
Quantum Chromodynamics (QCD) part of the StandardModel (SM) Lagrangian. At the same time, the actual η → π dynamics involve a strong impact from ππ final-state interactions, and the m d − m u magnitude cannot beapproached without a precise experimental measurementof the η → π Dalitz plots, the density of which pro-vides the information needed. Theoretical calculationsof strong-interaction processes at low energy, which couldtypically be performed by using Chiral Perturbation The-ory ( χ PTh) [1–3], were not very successful at describingthe η → π density distributions observed experimen-tally. The main reason was in the final-state rescatteringeffects, the calculation of which turned out to be morereliable with dispersion relations [4, 5], but still insuf-ficient to describe the experimental data. Meanwhile,the experimental progress in both the precise determi-nation of the ππ phase shifts [6–8] and high-statisticsdata on the η → π and η → π + π − π decays [9–14]renewed the interest in theoretical studies of the η → π decay [15–22], which also included the extraction of thequark-mass ratio, Q = ( m s − ¯ m ud ) / ( m d − m u ) with¯ m ud = ( m u + m d ) /
2, from the data.The function describing the density of the η → π Dalitz plot follows the standard parametrization forthree-body decay, which is a polynomial expansion of | A ( s , s , s ) | around the center of the Dalitz plot, where s i = ( P η − p i ) , with p i = M i . The parameters areusually normalized to be dimensionless. The standardvariables introduced for the η → π decay are then X = √ T − T ) /Q η = √ s − s ) / (2 m η Q η ) and Y = 3 T /Q η − m η − m π ) − s ) / (2 m η Q η ) − T i is the kinetic energy of pion i in the η restframe, and Q η = m η − m π for the neutral decay and Q η = m η − m π ± − m π for the charged decay. In ad-dition, another dimensionless variable z = X + Y =6 P i =1 ( T i + m π − m η / /Q η = ρ /ρ was introducedto describe the η → π Dalitz-plot density in terms ofthe quadratic distance, ρ , from the plot center. For theneutral η decay, its polynomial expansion A ( s , s , s ) ∼ α ′ X i =1 ( s i − s ) + β ′ X i =1 ( s i − s ) + γ ′ X i =1 ( s i − s ) + ... , (1)with s = m η / − m π , results in [16, 17] | A ( X, Y ) | ∼ αz + 2 β (3 X Y − Y ) + 2 γz + ... , (2)where parameters α ′ , β ′ , and γ ′ are complex in general,and parameters α , β , and γ are real. Representing X = √ z cos( φ ) and Y = √ z sin( φ ) as polar coordinates withrespect to the Dalitz-plot center, Eq. (2) can be rewrittenas | A ( X, Y ) | ∼ αz + 2 βz / sin(3 φ ) + 2 γz + ... , (3)where angle φ = arctan( Y /X ).Due to the low energies of the decay pions, π π rescat-tering in η → π is expected to be dominated by Swaves. Such an assumption leads to the conventionalleading-order parametrization | A ( z ) | ∼ αz [23] ofthe η → π amplitude, with only the quadratic slope pa-rameter α , which was used in all previous measurements.Rather than fitting two-dimensional Dalitz plots, thosemeasurements were based on the deviation of measured z distributions from the corresponding distributions ob-tained from the phase-space simulation of the η → π decay, which is illustrated for both the Dalitz plot and z distribution in Fig. 1.The current value for the η → π quadratic slope pa-rameter, α = − . ± . ) π →η X ( -1 -0.5 0 0.5 1 Y -1-0.500.51 × (a) ) π →η Z ( E n t r i es × (b) FIG. 1: Dalitz plot and its z = X + Y = ρ /ρ distri-bution for the phase-space η → π decay. measurements [9–11, 24–30]. The results of those mea-surements are plotted in Fig. 2 along with values fromvarious calculations [2–4, 16–19, 21, 31]. As shown inFig. 2, all experimental results obtained with compara-bly large statistics are in good agreement within theiruncertainties, and the earlier theoretical calculations con-tradict experimental data more than the most recent.The result with the best accuracy, ( α = − . ± . stat ± . syst , obtained by the A2 Collabora-tion at MAMI, was based on 3 × observed η → π decays [11]. Significant attention in that work was ded-icated to a search for a possible cusp structure in thespectra below the π + π − threshold. Based on the ππ scattering length combination a − a , extracted fromthe analysis of K → π decays [32], and calculationswithin the framework of nonrelativistic effective fieldtheory (NREFT) [15], the cusp effect was expected tobe visible in the m ( π π ) spectrum, reaching ∼
1% atthe 2 π threshold with respect to the spectrum in thecase of the cusp absence. This calculation used the η → π + π − π results from KLOE [12] to describe thecharge-decay amplitude, assuming the isospin limit toconnect it to the neutral decay. In principle, the pre-dicted cusp magnitude should not change much even inthe case of isospin breaking. However, the expected cuspstructure was not confirmed experimentally in Ref. [11].At the same time, the statistical accuracy of data pointsin the measured z distribution made it possible to indi-cate that the conventional leading-order parametrization | A ( z ) | ∼ αz was not sufficient for the proper de-scription of the η → π decay amplitude. This indi-cates that the contributions from the higher-order termsin Eq. (3) need to be checked as well. The cusp structurecannot be described by polynomial expansion but, simi-lar to the NREFT, the cusp range can be parametrized inthe density function as ρ ( s ) = Re q (1 − s/ m π ± ), whichresults in ρ ( s ) = 0 for s ≥ m π ± [33]. Then the densityfunction is given by | A | ∼ αz + 2 βz / sin(3 φ ) + 2 γz + ... +2 δ X i =1 ρ ( s i ) , (4)where the factor 2 in front of the cusp term is added forthe consistency with the other terms. ) π →η ( α -0.05 0 0.05 0.1 0.15 0.2 D a t a & C a l c u l a t i on s Albaladejo et al., K.T.Form., 2017Colangelo et al., Disp.An., 2017JPAC, Disp.An., 2017Kampf et al., Disp.An., 2011Schneider et al., NREFT, 2011Bijnens et al., ChPT NNLO, 2007Borasoy et al., Chir.Unit., 2005Bijnens et al., ChPT NLO, 2002Kambor et al., Disp.An., 1996BESIII, 33k, 2015KLOE, 512k, 2010WASA at COZY, 120k, 2009C.Ball at MAMI-C, 3M, 2009C.Ball at MAMI-B, 1.8M, 2009WASA at CELSIUS, 75k, 2007SND, 12k, 2001C.Ball at AGS, 1M, 2001C.Barrel at LEAR, 98k, 1998GAMS2000, 50k, 1984=-0.0318+/-0.0015) α PDG (
FIG. 2: Comparison of the experimental data [9–11, 24–30] (plotted by black points), which were used in the RPP [23] toobtain the averaged value (shown by the vertical lines) for the η → π quadratic slope parameter α , to each other and tovarious calculations [2–4, 16, 17, 19, 21, 22, 31] (colored points). Ref. [2] gives the magnitude of α for the analysis made inRef. [1], in which its value was not given. A better determination of the η → π decay param-eters, needed for a precise determination of light-quarkmass ratios, was recently the focus of many theoreticalworks. In Ref. [16], a detailed study of the η → π de-cays within the framework of the modified NREFT, inwhich final-state interactions were analyzed beyond oneloop including isospin-breaking corrections, resulted inthe extraction of the Dalitz-plot parameters for both thecharged and neutral decays. The values obtained for theparametrization of the neutral decay with Eq. (3), α = − . β = − . γ = 0 . η → π calculations, involving parameter β , useda unitary dispersive model [18, 19], in which substrac-tion constants were fixed by fitting recent high-statistics η → π + π − π data from WASA-at-COSY (1 . × de-cays) [13] and KLOE (4 . × decays) [14]. In contrastto Ref. [16], the latter calculations predicted a value of β consistent with zero. Another recent dispersive anal-ysis [21] of the η → π decay amplitudes, in which thelatest η → π + π − π data from KLOE [14] were also fit-ted to determine subtraction constants, predicted a non-linear z dependence for η → π , which turned out tobe in good agreement within the uncertainties with themeasured z dependence from Ref. [11]. However, no nu-merical predictions were provided for the higher-orderterms of Eq. (3). The most recent η → π calculation,which used the extended chiral Khuri-Treiman disper-sive formalism [22], showed that the effect from the twolight resonances f (980) and a (980) in the low energyregion of the η → π decay is not negligible, especiallyfor the neutral mode, and improves the description of the density variation over the Dalitz plot. The η → π parameters obtained in Ref. [22] from their fitted ampli-tude, α = − . β = − . βz / sin(3 φ ) term.Obviously, a better comparison of the experimentaldata with the recent η → π calculations, going beyondthe leading-order parametrization, should now be basedon describing the two-dimensional density distribution ofmeasured Dalitz plots, rather than on one-dimensional z distributions. To obtain reliable experimental results forthe parametrization with Eq. (4), a new measurementof the η → π Dalitz plot, with even higher statisticalaccuracy, is very important.In this paper, we report on a new high-statistics mea-surement of the η → π Dalitz plot, which is basedon 7 × detected decays. The A2 data used in thepresent analysis were taken in 2007 (Run I) and 2009(Run II). Compared to the previous analysis of Run I re-ported in Ref. [11], the present analysis was made with animproved cluster algorithm, which increased the numberof η → π decays reconstructed in Run I from 3 × to3 . × . The γp → ηp → π p → γp data from RunI and Run II used in this work were previously used tomeasure the γp → ηp differential cross sections, the anal-ysis of which was recently reported in Ref. [35]. The new η → π results were obtained with the parametrizationinvolving the higher-order terms of the Dalitz-plot den-sity function and the cusp term. The NREFT frameworkfrom Ref. [34] was also used to check whether the present η → π data can be described together with the KLOE η → π + π − π data [14], assuming the isospin limit. Theexperimental spectra are also compared to recent theo-retical calculations that predict a nonlinear dependencealong the quadratic distance from the Dalitz-plot center. II. EXPERIMENTAL SETUP
An experimental study of the η → π decay wasconducted via measuring the process γp → ηp → π p → γp with the Crystal Ball (CB) [36] as a centralcalorimeter and TAPS [37, 38] as a forward calorime-ter. These detectors were installed in the energy-taggedbremsstrahlung photon beam of the Mainz Microtron(MAMI) [39, 40]. The photon energies were determinedby the Glasgow tagging spectrometer [41–43].The CB detector is a sphere consisting of 672 opticallyisolated NaI(Tl) crystals, shaped as truncated triangularpyramids, which point toward the center of the sphere.The crystals are arranged in two hemispheres that cover93% of 4 π , sitting outside a central spherical cavity witha radius of 25 cm, which holds the target and inner de-tectors. In this experiment, TAPS was initially arrangedin a plane consisting of 384 BaF counters of hexagonalcross section. It was installed 1.5 m downstream of theCB center and covered the full azimuthal range for polarangles from 1 ◦ to 20 ◦ . Later on, 18 BaF crystals, cov-ering polar angles from 1 ◦ to 5 ◦ , were replaced with 72PbWO crystals, allowing for a higher count rate in thecrystals near the photon-beam line. More details on theenergy and angular resolution of the CB and TAPS aregiven in Refs. [11, 44].The present measurement used electron beams withenergies of 1508 and 1557 MeV from the Mainz Mi-crotron, MAMI-C [40]. The data with the 1508-MeVbeam were taken in 2007 (Run I) and those with the1557-MeV beam in 2009 (Run II). Bremsstrahlung pho-tons, produced by the beam electrons in a 10- µ m Curadiator and collimated by a 4-mm-diameter Pb colli-mator, were incident on a liquid hydrogen (LH ) targetlocated in the center of the CB. The LH target was 5cm and 10 cm long in Run I and Run II, respectively.The total amount of material around the LH target, in-cluding the Kapton cell and the 1-mm-thick carbon-fiberbeamline, was equivalent to 0.8% of a radiation length X , which was essential to keep the material budget aslow as possible to minimize the conversion of final-statephotons.The target was surrounded by a Particle IDentification(PID) detector [45] used to distinguish between chargedand neutral particles. The PID consists of 24 scintillatorbars (50 cm long, 4 mm thick) arranged as a cylinderwith the middle radius of 12 cm.In Run I, the energies of the incident photons wereanalyzed up to 1402 MeV by detecting the post-bremsstrahlung electrons in the Glasgow tagged-photonspectrometer (Glasgow tagger) [41–43], and up to1448 MeV in Run II. The uncertainty in the energy ofthe tagged photons is mainly determined by the segmen-tation of the tagger focal-plane detector in combinationwith the energy of the MAMI electron beam used in the experiments. Increasing the MAMI energy increases theenergy range covered by the spectrometer and also hasthe corresponding effect on the uncertainty in E γ . Forboth the MAMI energy settings of 1508 and 1557 MeV,this uncertainty was about ± ∼
320 MeV and thenumber of so-called hardware clusters in the CB (mul-tiplicity trigger) to be two or more. In the trigger, ahardware cluster in the CB was a block of 16 adjacentcrystals in which at least one crystal had an energy de-posit larger than 30 MeV. Depending on the data-takingperiod, events with a cluster multiplicity of two wereprescaled with different rates. TAPS was not includedin the multiplicity trigger for these experiments. In RunII, the trigger threshold on the total energy in the CBwas increased to ∼
340 MeV, and the multiplicity triggerrequired three or more hardware clusters in the CB.
III. DATA ANALYSIS
The η → π decays were measured via the process γp → ηp → π p → γp from events having six or sevenclusters reconstructed by a software analysis in the CBand TAPS together. Seven-cluster events were analyzedby assuming that all final-state particles were detected,and six-cluster events by assuming that only the six pho-tons were detected, with the recoil proton going unde-tected. The offline cluster algorithm [46] was optimizedfor finding a group of adjacent crystals in which the en-ergy was deposited by a single-photon electromagnetic(e/m) shower. This algorithm also works well for recoilprotons. The software threshold for the cluster energywas chosen to be 12 MeV. Compared to the previous η → π analysis of Run I [11], the cluster algorithm wasimproved for a better separation of e/m showers partiallyoverlapping in the calorimeters, which is especially im-portant for processes with large photon multiplicity inthe final state and for conditions of the forward energyboost of the outgoing photons in the laboratory system.At the same time, the cluster algorithm has also to beefficient for reconstructing one photon splitting into twonearby e/m showers. The new optimization of the clus-ter algorithm was needed to improve its efficiency forhigher energies of MAMI-C. Particularly for the process γp → ηp → π p → γp , its reconstruction efficiencywas improved by ∼ γp → ηp → π p → γp hypothesis, 15combinations are possible to pair six photons into threeneutral pions. To reduce the number of combinationstested with the kinematic fit, invariant masses of clusterpairs for each combination were tested prior to fitting.For seven-cluster events, where seven combinations arepossible to select the proton cluster, this number was re-duced by a cut on the cluster polar angle, the value ofwhich is limited by the recoil-proton kinematics in thelaboratory system. The events for which at least onepairing combination satisfied the tested hypothesis at the1% confidence level, CL, (i.e., with a probability greaterthan 1%) were selected for further analysis. The pairingcombination with the largest CL was used to reconstructthe reaction kinematics. The combinatorial backgroundfrom mispairing six photons into three pions was found tobe quite small and could be further reduced by tighteninga selection criterion on the kinematic-fit CL. Misidentifi-cation of the proton cluster with the photons was foundto be negligibly small for seven-cluster events. The six-cluster sample, which includes ∼
20% from all detected η → π decays, had a small contamination from eventsin which one of the photons, instead of the proton, wasundetected. Because such misidentification mostly oc-curred for clusters in TAPS, those events were success-fully removed, based on the cluster’s time-of-flight infor-mation, which provides good separation of the γp → ηp recoil protons from photons in the present energy range.To minimize systematic uncertainties in the determi-nation of experimental acceptance, Monte Carlo (MC)simulations of the production reaction γp → ηp werebased on the actual spectra measured with the same datasets [35]. The η → π decay was generated according tophase space (i.e., with the slope parameter α = 0). Thesimulated events were propagated through a GEANT (version 3.21) simulation of the experimental setup. Toreproduce the resolutions observed in the experimentaldata, the
GEANT output (energy and timing) was sub-ject to additional smearing, thus allowing both the simu-lated and experimental data to be analyzed in the sameway. Matching the energy resolution between the ex-perimental and MC events was achieved by adjustingthe invariant-mass resolutions, the kinematic-fit stretchfunctions (or pulls), and probability distributions. Suchan adjustment was based on the analysis of the samedata sets for reactions that could be selected with thekinematic fit practically without background from otherreactions (namely, γp → π p , γp → ηp → γγp , and γp → ηp → π p were used). The simulated events werealso tested to check whether they passed the trigger re-quirements.For η → π decays, physical background can onlycome from the γp → π p events that are not producedfrom η decays. As shown in Ref. [47], those 3 π events aremostly produced via baryon decay chains, with a smallerfraction from γp → K S Σ + → π p . For selected γp → ηp → π p events, this background is negligibly smallnear the η production threshold, and reaches ∼
4% nearbeam energy E γ = 1 . Integral 4442112 ] ) [GeV/c π m(3 E ve n t s
10 Integral 4442112
Run I
Integral 4460397 ] ) [GeV/c π m(3
10 Integral 4460397
Run II
FIG. 3: m (3 π ) invariant-mass distributions for events se-lected at CL >
1% by testing the γp → π p → γp hypothesisfor the data of Run I (left) and Run II (right). selection criteria have to be applied instead to reduce theremaining background to a level ≤ π background under the η → π peak canbe seen in the m (3 π ) invariant-mass distributions forevents selected at CL >
1% by testing the γp → π p → γp hypothesis, which has no constraint on the η mass.These distributions are shown in Fig. 3. It was checkedthat the level of the direct 3 π background ≤
1% in the η → π data sample could be reached by requiring CL > .
5% for the γp → ηp → π p → γp hypothesis alongwith rejecting events having E γ > . γp → ηp → π p → γp events and whichcould directly be subtracted from the experimental spec-tra. The first background is due to interactions of thebremsstrahlung photons in the windows of the target cell.The evaluation of this background is based on the anal-ysis of data samples that were taken with the target cellemptied of liquid hydrogen. The weight for the subtrac-tion of empty-target spectra is usually taken as a ratioof the photon-beam fluxes for the data samples with thefull and the empty target. Because, in the present ex-periments, the amount of empty-target data were muchsmaller than with the full target, the subtraction of thisbackground would cause larger statistical uncertainties.It was checked that, for the selection criteria used, thefraction of the empty-target background is ≤ η → π decaysthat were just produced in interactions with the target-cell material. Thus, the subtraction of the empty-targetbackground was neglected in the present analysis.The second background was caused by random coinci-dences of the tagger counts with the experimental trig-ger. It mostly includes γp → ηp → π p → γp eventsreconstructed with random E γ , resulting in poorer χ and resolution after kinematic fitting. The subtraction ofthis background was carried out by using event samplesfor which all coincidences were random (see Ref. [11] formore details). The fraction of random background was6.7% for Run I, and 6.9% for Run II. The actual back-ground samples included much more events to diminishthe impact from statistical fluctuations in the distribu-tions used for the subtraction. ) π →η X ( -1 -0.5 0 0.5 1 Y -1-0.500.51 0.80.850.90.951 A2 (a) ) π →η X ( A2 (b) ) π - π + π→η X ( -1 -0.5 0 0.5 1-1-0.500.51 00.511.52
KLOE (c)
FIG. 4: Comparison of the experimental η → π Dalitz plots: (a) the full η → π plot (six entries per decay) from the presentanalysis of ∼ × decays; (b) one sextant of the η → π plot (one entry per decay) for the angle range 30 ◦ < φ < ◦ inEq. (3); (c) the η → π + π − π plot (without boundary bins) from the KLOE analysis of ∼ . × decays [14]. IV. RESULTS AND DISCUSSION
The full Dalitz plot obtained from ∼ × η → π decays of Run I and Run II is shown in Fig. 4(a). Becausethere are three identical particles in the final state, vari-ables X and Y can be determined in six different ways,with the same value for variable z and different angle φ from Eq. (3). Each of these six combinations in X and Y goes into six different sextants, repeating the densitystructure every 60 degrees. The difference between thosesextants is only in their different orientation with respectto each other and to the plot binning. Also, this Dalitzplot is symmetric with respect to the Y axis. In principle,one sextant is sufficient to analyze the Dalitz-plot shapeand to obtain the corresponding results with proper sta-tistical uncertainties. Such a sextant plot, obtained forthe angle range 30 ◦ < φ < ◦ , is shown in Fig. 4(b).As seen, this sextant plot has bins with limited physicalcoverage not only along the external edge but also alongangle φ = 30 ◦ . To avoid any dependence of the results onsuch an effect and on the sextant orientation with respectto the plot binning, one half of the Dalitz plot ( X <
X >
0) can be used to analyze its shape. Because halfof the plot has three entries per event, the parameter er-rors from fitting to such a plot must be multiplied by thefactor of √ η → π plots shown in Figs. 4(a) and4(b), the plots with the measured decays from Runs I andII were divided by the corresponding plots obtained fromthe analysis of the γp → ηp → π p MC simulations forthose data sets. Because η → π decays were generatedas phase space, the ratio of the experimental and the MCplots provides both the acceptance correction for the fullarea and the cancellation of the phase-space factor com-ing from the limited physical coverage, which is typicalfor boundary bins. Then those boundary bins can betreated in the same way as the inner bins while fittingthe acceptance-corrected Dalitz plots with density func-tions. The only difference from the inner bins is in using X and Y coordinates averaged inside the boundary binsover the available phase space, instead of taking the bincenters. To combine the acceptance-corrected plots fromdifferent data sets (namely from Runs I and II), theirnormalization should be done in the same way. In thepresent analysis, an identical normalization was made bytaking the weight of the MC Dalitz plot as the ratio ofthe event numbers in the experimental and the MC plots.As shown in Fig. 4(a), the largest density of events isaccumulated in the center of the η → π Dalitz plot,with a smooth decrease of a few percent toward the plotedge. To compare such a structure with the charged de-cay, the acceptance-corrected η → π + π − π Dalitz plotfrom KLOE [14] (with excluded boundary bins) is illus-trated in Fig. 4(c), showing a sharp decrease in its densityfrom the smallest Y to the largest. In the present work,this η → π + π − π plot was used to check whether it couldbe described together with the η → π data within theNREFT framework [34], assuming the isospin limit.The advantage of analyzing the η → π + π − π decayis the fact that the X and Y variables can be defineduniquely. Then the experimental raw (i.e., uncorrectedfor the acceptance) Dalitz plot can be fitted with the cor-responding plots of the phase-space MC events weightedwith the density-function terms. Because the weightsare calculated from the generated variables, but fillingthe MC plots is done according to the reconstructed vari-ables, such a fit takes into account both the experimentalacceptance and resolution. For the η → π decay, the X and Y generated in one sextant could be reconstructedin another sextant, which allows proper fitting a sextantof the raw Dalitz plot with the density function depen-dent only on z (which is the same for all pairs of X and Y ) but not on φ . Therefore, all fits with the higher-orderterms were made only for the acceptance-corrected Dalitzplots. The sensitivity of the results to the experimentalresolution, which could be determined by comparing tothe fits to the raw Dalitz plots, was only checked for theleading-order parametrization.The traditional z distributions, which were used in all ) π → η z( E x p ./ P h . S p . (a) This Work: Run IThis Work: Run IIA2, PRC79(2009)035204NREFT Bonn ,KLOE) π - π + π→η DA Bern(Khuri-Treiman form. ) π → η z( (b) This Work: A2z α fit with 1+2), A2 β , α Dn( ), A2 δ , β , α Dn( ), KLOE+A2 π →η NREFT( ), A2 π →η NREFT(
FIG. 5: Experimental z distributions obtained (a) individually from Run I (blue circles) and Run II (red triangles), and(b) the combined results (black triangles). The earlier A2 data from Ref. [11] are depicted in (a) by green open squares.The NREFT calculation by the Bonn group [16] is shown in (a) by the black long-dash-dotted line. The prediction from thedispersive analysis by the Bern group [21] is shown in (a) by the magenta long-dashed line with an error band. The predictionbased on the extended chiral Khuri-Treiman formalism [22] is shown in (a) by the black dotted line. The fit of the combined z distribution with the leading-order term (fit no. 2 in Table I) is shown in (b) by the cyan long-dashed line. The fits of theDalitz-plot sextant with Eq. (4), namely nos. 4 and 6 from Table I, are shown in (b) by the yellow solid and the green dashedlines, respectively. The isospin-limit results from fitting both the present η → π and KLOE’s η → π + π − π [14] data withinthe NREFT framework from Ref. [34] are shown by the blue dash-dotted line. The isospin-breaking results from fitting solelythe η → π data within the same NREFT framework are shown by the red dotted line. previous measurements of the slope parameters α , wereobtained individually for Run I and Run II. Similar to theindividual Dalitz plots, their normalization was based onthe ratio of the total number of events in the experimen-tal and the MC distributions, which allows the propercombination of the two independent measurements. Theindividual z distributions from Run I and Run II are com-pared in Fig. 5(a) with each other and with the earlierA2 data from Ref. [11], demonstrating good agreementwithin their statistical uncertainties. The combined z distribution, shown in Fig. 5(b), has a statistical accu-racy in its 30 data points that appears to be sufficient toreveal the deviation from a linear dependence.The ratios of the experimental m ( π π ) invariant-massdistributions to phase space, in which a cusp structureis expected to be seen, were obtained in the same wayas the z distributions. The agreement of the individual m ( π π ) distributions from Run I, Run II, and the ear-lier A2 data from Ref. [11], can be seen in Fig. 6(a). Thecombined m ( π π ) distribution is shown in Fig. 6(b), sig-nificantly improving the statistical accuracy in the cuspregion, compared to the previous measurement [11].In addition to fitting the present η → π data withthe density function from Eq. (4), the NREFT frameworkfrom Ref. [34] was used to check whether the neutral-decay data can be fitted well together with the KLOE η → π + π − π data [14] by assuming the isospin limit.Next, the solely η → π data were fitted in the sameframework by assuming isospin breaking. In Ref. [34],the decay amplitude is decomposed into up to two loops, A ( η → π ) = A tree + A − loop + A − loop , with the tree am-plitude complemented by final-state interactions of one and two loops. The tree amplitudes are parametrized as A tree ( η → π ) = K + K ( T + T + T ) and A tree ( η → π + π − π ) = L + L T + L T + L ( T − T ) , where T i = E i − m π is the kinetic energy of pion i in the η restframe. For the conventional Dalitz plot variables, the treeamplitudes can be rewritten as A tree ( η → π ) = u + u z and A tree ( η → π + π − π ) = v + v Y + v Y + v X ,where, at the tree level, the quadratic slope parameter is α = u /u , and the coefficients u i and v i are strictly con-nected to K i and L i , respectively. Note that the shape ofthe actual η → π Dalitz plot is determined by the to-tal amplitude; therefore, a measured α could be differentfrom the ratio u /u of the tree-amplitude coefficients.The coefficients K i and L i (or u i and v i ) are also in-volved in the calculation of A − loop and A − loop for boththe neutral and charged decays. The cusp structure be-low 2 m π ± appears in A ( η → π ) − loop , and the cuspsign and magnitude is mostly determined by the scatter-ing length combination a − a [32] and the η → π + π − π tree-amplitude coefficients L i . In the isospin limit, thecoefficients of the tree amplitude for the neutral decaycan be rewritten via the coefficients of the charged decay: K = − (3 L + L Q η − L Q η ) and K = − ( L +3 L ) [15],with Q η = m η − m π . The isospin-limit fit to both the η → π and η → π + π − π Dalitz plots has only five freeparameters ( L i =1 , , and two normalization parameters),with fixed L = 1. The η → π data can also be fittedindependently of the η → π + π − π decay by assumingisospin breaking, which requires the addition of K and K as free parameters, but leaves just one normalizationparameter.Consistency of the present results for z and m ( π π ) ] ) [GeV/c π π m( E x p ./ P h . S p . (a) This Work: Run IThis Work: Run IIA2, PRC79(2009)035204NREFT Bonn ,KLOE) π - π + π→η DA Bern(Khuri-Treiman form. ] ) [GeV/c π π m( (b) This Work: A2), A2 α Dn( ), A2 β , α Dn( ), A2 δ , β , α Dn( ), KLOE+A2 π →η NREFT( ), A2 π →η NREFT(
FIG. 6: Ratios of the experimental m ( π π ) invariant-mass distributions to phase space obtained (a) individually from RunI (blue circles) and Run II (red triangles), and (b) the combined results (black triangles). The earlier A2 data from Ref. [11]are depicted in (a) by green open squares. The NREFT calculation by the Bonn group [16] is shown in (a) by the blacklong-dash-dotted line. The prediction from the dispersive analysis by the Bern group [21, 33] is shown in (a) by the magentalong-dashed line. The prediction based on the extended chiral Khuri-Treiman formalism [22] is shown in (a) by the blackdotted line. The combined m ( π π ) distribution is compared in (b) to the results of fitting a sextant (30 ◦ < φ < ◦ ) ofthe acceptance-corrected η → π Dalitz plot with the density function of Eq. (4): fits no. 1 (cyan long-dashed line), no. 4(yellow solid line), and no. 6 (green dashed line) in Table I. The isospin-limit results from fitting both the present η → π and KLOE’s η → π + π − π [14] data within the NREFT framework from Ref. [34] are shown by the blue dash-dotted line. Theisospin-breaking results from fitting solely the η → π data within the same NREFT framework are shown by the red dottedline. with theoretical calculations that predict a nonlinear z dependence [16, 21, 22] is illustrated in Figs. 5(a)and 6(a). The results of fits to the present data withvarious density functions, including the NREFT fits, aredepicted in Figs. 5(b) and 6(b). The fit results with thedensity function from Eq. (4) are also listed in Table Ifor different combinations of the density-function termsinvolved in a particular fit.Fit no. 1 in Table I was made to a sextant (30 ◦ < φ < ◦ ) of the acceptance-corrected Dalitz plot with the den-sity function including only the leading-order term. Fitno. 2 was similar, but to the acceptance-corrected z dis-tribution as in all previous measurements. As shown, thevalues obtained there for α are practically the same andare in agreement within the fit errors with the RPP value α = − . ± . χ /ndf values indicate that the use of the leading-orderterm only may be insufficient for a good description ofthe η → π decay. Fit no. 2 is shown in Fig. 5(b) and fitno. 1 in Fig. 6(b) by the cyan long-dashed lines, confirm-ing that it is not sufficient to use only the leading-orderterm. Fit no. 3 in Table I was made to the same sextantof the raw Dalitz plot with the technique taking both theacceptance and the experimental resolution into account(see the text above). This fit results in a slightly bet-ter χ /ndf value and a slightly larger quadratic slope,which was expected because of some smearing of theacceptance-corrected distributions by the experimentalresolution. In the end, the difference between the α re-sults for the acceptance-corrected and the raw distribu-tions can be considered as the magnitude of its systematicuncertainty due to the limited experimental resolution. Fit no. 4 in Table I, which also involves the nextdensity-function term 2 βz / sin(3 φ ), does improve the χ /ndf value, whereas including the 2 γz term in fit no. 5practically does not. In addition, the parameters α and γ in fit no. 5 become strongly correlated, which resultsin large fit errors for them. Fit no. 4, shown in Figs. 5(b)and 6(b) by the yellow solid line, demonstrates a quitedecent description of the z and m ( π π ) distributions,except in the region where the cusp is expected. Asshown in the m ( π π ) distribution, the 2 βz / sin(3 φ )term curves the spectrum up at the lowest masses, whichis opposite to the effect expected from the cusp. Inthe z distribution, the same term causes a kink up at z ≈ .
75, which again is opposite to the effect expectedfrom the cusp [11, 15]. As shown in Figs. 5(a) and 6(a),the calculation within the framework of the modifiedNREFT [16] predicts a behavior that is very similar tofit no. 4, but with a smaller general slope. This can beexplained by a smaller quadratic slope, α = − . γ = 0 . α , it isstill in agreement with the corresponding value from fitno. 4. In contrast to the calculation from Ref. [16], theprediction based on the extended chiral Khuri-Treimanformalism [22] lies below the experimental data points,which is mostly determined by the larger quadratic slope, α = − . βz / sin(3 φ ) term, β = − . β = − . γ cannot be determined reliably in order to becompared with the prediction from Ref. [16]. TABLE I: Results from fitting to the acceptance-corrected sextant (ACS), 30 ◦ < φ < ◦ , of the η → π Dalitz plot withthe density function of Eq. (4) are considered as the main results, and from the other fits as their cross checks. The results forthe leading-order parametrization were also obtained for the acceptance-corrected z (ACZ) distribution and the same sextantof the raw (RawS) Dalitz plot. The result errors from fitting to the acceptance-corrected half (ACH), − ◦ < φ < ◦ , of theDalitz plot are multiplied by the factor of √
3, correcting for three entries per event. The results from fitting to the independentdata of Run I and Run II are added to illustrate systematic effects due to different experimental conditions. For convenience,calculations involving the higher-order terms are listed as well.Fit no. Data used χ /ndf α β γ δ − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . α β γ —1 [16] — − . − . γ = 0 . − . . − . − . As seen from fit no. 6 in Table I, further improve-ment in the description of the η → π data wasreached by adding the 2 δ P i =1 ρ ( s i ) term, which allows acusp parametrization to be included in the density func-tion. Such a fit results in a slightly smaller quadraticslope, compared to fit no. 4, but also in a stronger2 βz / sin(3 φ ) term. In Figs. 5(b) and 6(b), fit no. 6,which is shown by the green dashed line, demonstratesgood agreement with both z and m ( π π ) distributions.Based on the results of fit no. 6, the contributions fromthe 2 βz / sin(3 φ ) and the cusp terms partially canceleach other in the z and especially in the m ( π π ) dis-tribution. Though, according to the result of fit no. 6for the cusp term, the magnitude of the cusp effect at m ( π π ) = 2 m π is almost 1%, its visibility here isstrongly diminished by the 2 βz / sin(3 φ ) term. The un-derstanding of such a feature became possible due to fit-ting the η → π Dalitz plot based on high experimentalstatistics.The isospin-limit NREFT fit to the present η → π data together with KLOE’s η → π + π − π Dalitz plot [14]is shown in Figs. 5(b) and 6(b) by the blue dash-dottedline. As shown in the m ( π π ) distribution, the majordeviation of this fit from the data is in the cusp region,which is much more prominent in the fit curve. The de-scription of the z distribution deviates from the data as well. The cusp magnitude obtained at m ( π π ) = 2 m π is close to 1%, which is similar to the corresponding re-sult of fit no. 6 in Table I. The discrepancy seems to comefrom inability of the isospin-limit fit to describe prop-erly the 2 βz / sin(3 φ ) term. Though the isospin-limitNREFT fit results in a good description of the chargeddecay, with χ /ndf=1.072, it gives χ /ndf=1.290 for theneutral decay. The numerical results for L i were ob-tained as L = 1(0), L = − . L = − . L = 5 . K i recalculated from L i as K = − . K = 25 . η → π data, which is shown in Figs. 5(b) and 6(b)by the red dotted line, resulted in a much better descrip-tion of the neutral decay, χ /ndf=1.112, with the numer-ical results for K i and L i as K = − . K =25 . L = 1(0), L = − . L = − . . L = − . . η → π decays, unless the NREFT frame-work in Ref. [34] could be improved for a better simul-taneous description of both decay modes. As illustrated0in Figs. 5(a) and 6(a), a recent dispersive analysis by theBern group [21, 33], in which the η → π + π − π data [14]were used to determine subtraction constants, did pro-vide predictions that described the η → π data well.The results of this work provide a strong indicationthat the parametrization of the η → π decay withonly the leading-order term is insufficient, and the RPPvalue α = − . ± . α are also closer to recent calculations reportedin Refs. [16, 18, 19] (see also Fig. 2).The exact systematic uncertainties in the results for α and for the other parameters are difficult to estimate reli-ably because the results themselves depend on the num-ber of density-function terms included in the fit. Thesystematic effect due to the limited experimental resolu-tion was discussed above for a fit with the leading-orderterm only (no. 3 in Table I). The sensitivity of the resultsto the sextant orientation with respect to the plot bin-ning and to additional boundary bins was checked withfits to other sextants and to half of the Dalitz plot. Allthose tests demonstrated practically identical results, af-ter multiplying the half-plot errors by the factor of √ γ and δ ; however, allresults obtained from the different data sets are in agree-ment within the fit errors. The magnitude for parameter γ cannot be determined reliably from the experimentaldata because of the large correlation with parameter α .Therefore, the value obtained for α with the 2 γz termomitted actually reflects the combined effect from thosetwo terms.According to the present analysis, the density functionof Eq. (4) with only three parameters is sufficient fora good description of the experimental η → π Dalitzplot. The values obtained for these three parameters are α = − . stat )(9 syst ), β = − . stat )(9 syst ),and δ = − . stat )(7 syst ), where the main numberscome from fit no. 6 in Table I, and the systematic un-certainties are taken as half of the differences betweenthe results of fits nos. 13 and 14. The new result for thequadratic slope parameter α strongly indicates that itsabsolute value is smaller by ≈ βz / sin(3 φ ) term is found to bedifferent from zero by ∼ . m ( π π ) = 2 m π from the2 δ P i =1 ρ ( s i ) term is close to 1%, but with an uncer- tainty greater than 50%. This result is consistent withthe prediction for the η → π cusp magnitude madewithin the NREFT model [15].The data presented in this work are expected to serveas a valuable input for new refined analyses by theoreticalgroups, which are interested in a better understanding of η → π decays and extracting the quark-mass ratios fromsuch data. V. SUMMARY AND CONCLUSIONS
The largest, at the moment, statistics of 7 × η → π decays, based on 6 . × η mesons produced inthe γp → ηp reaction, has been accumulated by the A2Collaboration at the Mainz Microtron, MAMI. The re-sults of this work provide a strong indication that theparametrization of the η → π decay with only theleading-order term is insufficient, and the RPP value for α reflects the combined effect from higher-order termsand the cusp structure, whereas the actual quadraticslope is smaller by ≈ η → π Dalitz plot, the cusp magnitude at m ( π π ) = 2 m π is about 1%, but its visibility is stronglydiminished by the second-order term of the density func-tion, the magnitude of which is found to be differentfrom zero by ∼ . η → π and KLOE’s η → π + π − π data withinthe NREFT framework indicate a strong isospin break-ing between the charged and the neutral decay modes.At the same time, the predictions based on the most re-cent dispersive analysis by the Bern group, in which the η → π + π − π data were used to determine subtractionconstants, were found to be in good agreement with thepresent η → π data. The data points from the exper-imental Dalitz plot and the ratios of the z and m ( π π )distributions to phase space are provided as supplementalmaterial to the paperl [48]. Acknowledgments
The authors acknowledge the excellent support of theaccelerator group and operators of MAMI. We thankH. Leutwyler, G. Colangelo, and B. Kubis for fruit-ful discussions and constant interest in our work. Thiswork was supported by the Deutsche Forschungsgemein-schaft (SFB443, SFB/TR16, and SFB1044), DFG-RFBR(Grant No. 09-02-91330), the European Community-Research Infrastructure Activity under the FP6 “Struc-turing the European Research Area” program (HadronPhysics, Contract No. RII3-CT-2004-506078), Schweiz-erischer Nationalfonds (Contracts No. 200020-156983,No. 132799, No. 121781, No. 117601, No. 113511), theU.K. Science and Technology Facilities Council (STFC57071/1, 50727/1), the U.S. Department of Energy (Of-fices of Science and Nuclear Physics, Awards No. DE-FG02-99-ER41110, No. DE-FG02-88ER40415, No. DE-FG02-01-ER41194) and National Science Foundation1(Grants No. PHY-1039130, No. IIA-1358175), INFN(Italy), and NSERC of Canada (Grant No. FRN-SAPPJ-2015-00023). A. Fix acknowledges additional supportfrom the Tomsk Polytechnic University competitiveness enhancement program. We thank the undergraduate stu-dents from Department of Physics of Mount Allison Uni-versity and from Institute for Nuclear Studies of TheGeorge Washington University for their assistance. [1] J. Gasser and H. Leutwyler, Nucl. Phys. B , 539(1985).[2] J. Bijnens and J. Gasser, Phys. Scripta T , 034 (2002).[3] J. Bijnens and K. Ghorbani, JHEP , 030 (2007).[4] J. Kambor et al. , Nucl. Phys. B , 215 (1996).[5] A. Anisovich and H. Leutwyler, Phys. Lett. B , 335(1996).[6] G. Colangelo, J. Gasser, and H. Leutwyler, Nucl. Phys.B , 125 (2001).[7] R. Kami´nski, J. R. Pel´aez, and F. J. Yndur´ain, Phys.Rev. D , 054015 (2008),[8] R. Garc´ıa-Mart´ın, R. Kami´nski, J. R. Pel´aez, J. Ruiz deElvira, and F. J. Yndur´ain, Phys. Rev. D , 074004(2011),[9] W. B. Tippens et al. , Phys. Rev. Lett. , 192001 (2001).[10] M. Unverzagt et al. , Eur. Phys. J. A , 169 (2009)[11] S. Prakhov et al. , Phys. Rev. C , 035204 (2009).[12] F. Ambrosino et al. , JHEP , 006 (2008),[13] P. Adlarson et al. , Phys. Rev. C , 045207 (2014)[14] A. Anastasi et al. , JHEP , 019 (2016).[15] C. O. Gullstr¨om, A. Kup´s´c, and A. Rusetsky, Phys. Rev.C , 028201 (2009).[16] S. P. Schneider, B. Kubis, and C. Ditsche, JHEP , 028(2011).[17] K. Kampf, M. Knecht, J. Novotn´y, and M. Zdr´ahal,Phys. Rev. D , 114015 (2011).[18] P. Guo, I. V. Danilkin, D. Schott, C. Fern´andez-Ram´ırez,V. Mathieu, and A. P. Szczepaniak, Phys. Rev. D ,054016 (2015).[19] P. Guo, I. V. Danilkin, D. Schott, C. Fern´andez-Ram´ırez,V. Mathieu, and A. P. Szczepaniak, Phys. Lett. B ,497 (2017).[20] M. Koles´ar and J. Novotn´y, Eur. Phys. J. C , 41(2017).[21] G. Colangelo, S. Lanz, H. Leutwyler, and E. Passemar,Phys. Rev. Lett. , 022001 (2017).[22] M. Albaladejo and B. Moussallam, Eur. Phys. J. C ,508 (2017).[23] C. Patrignani et al. , (Particle Data Group), Chin. Phys.C , 100001 (2016). [24] D. Alde et al. , Z. Phys. C , 225 (1984).[25] A. Abele et al. , Phys. Lett. B , 193 (1998).[26] M. N. Achasov et al. , JETP Lett. , 451 (2001).[27] M. Bashkanov et al. , Phys. Rev. C , 048201 (2007).[28] C. Adolph et al. , Phys. Lett. B , 24 (2009).[29] F. Ambrosino et al. , Phys. Lett. B , 16 (2011).[30] M. Ablikim et al. , Phys. Rev. D , 012014 (2015).[31] B. Borasoy and R. Nissler, Eur. Phys. J. A , 383(2005).[32] J. R. Batley et al. , Eur. Phys. J. C , 589 (2009).[33] Heinrich Leutwyler, private communication.[34] M. Bissegger, A. Fuhrer, J. Gasser, B. Kubis andA. Rusetsky, Phys. Lett. B , 576 (2008).[35] V. L. Kashevarov et al. , Phys. Rev. Lett. , 212001(2017).[36] A. Starostin et al. , Phys. Rev. C , 055205 (2001).[37] R. Novotny, IEEE Trans. Nucl. Sci. , 379 (1991).[38] A. R. Gabler et al. , Nucl. Instrum. Methods Phys. Res.A , 168 (1994).[39] H. Herminghaus et al. , IEEE Trans. Nucl. Sci. , 3274(1983).[40] K.-H. Kaiser et al. , Nucl. Instrum. Methods Phys. Res.A , 159 (2008).[41] I. Anthony et al. , Nucl. Instrum. Methods Phys. Res. A , 230 (1991).[42] S. J. Hall et al. , Nucl. Instrum. Methods Phys. Res. A , 698 (1996).[43] J. C. McGeorge et al. , Eur. Phys. J. A , 129 (2008).[44] E. F. McNicoll et al. , Phys. Rev. C , 035208 (2010).[45] D. Watts, Proceedings of the 11th International Confer-ence on Calorimetry in Particle Physics , Perugia, Italy,2004 (World Scientific, Singapore, 2005), p. 560.[46] S. Prakhov et al. , Phys. Rev. C , 025204 (2009).[47] P. Aguar-Bartolom´e et al. , Phys. Rev. C , 044601(2013).[48] See Supplemental Material for the data points from theexperimental Dalitz plot and the ratios of the z and m ( π0
The authors acknowledge the excellent support of theaccelerator group and operators of MAMI. We thankH. Leutwyler, G. Colangelo, and B. Kubis for fruit-ful discussions and constant interest in our work. Thiswork was supported by the Deutsche Forschungsgemein-schaft (SFB443, SFB/TR16, and SFB1044), DFG-RFBR(Grant No. 09-02-91330), the European Community-Research Infrastructure Activity under the FP6 “Struc-turing the European Research Area” program (HadronPhysics, Contract No. RII3-CT-2004-506078), Schweiz-erischer Nationalfonds (Contracts No. 200020-156983,No. 132799, No. 121781, No. 117601, No. 113511), theU.K. Science and Technology Facilities Council (STFC57071/1, 50727/1), the U.S. Department of Energy (Of-fices of Science and Nuclear Physics, Awards No. DE-FG02-99-ER41110, No. DE-FG02-88ER40415, No. DE-FG02-01-ER41194) and National Science Foundation1(Grants No. PHY-1039130, No. IIA-1358175), INFN(Italy), and NSERC of Canada (Grant No. FRN-SAPPJ-2015-00023). A. Fix acknowledges additional supportfrom the Tomsk Polytechnic University competitiveness enhancement program. We thank the undergraduate stu-dents from Department of Physics of Mount Allison Uni-versity and from Institute for Nuclear Studies of TheGeorge Washington University for their assistance. [1] J. Gasser and H. Leutwyler, Nucl. Phys. B , 539(1985).[2] J. Bijnens and J. Gasser, Phys. Scripta T , 034 (2002).[3] J. Bijnens and K. Ghorbani, JHEP , 030 (2007).[4] J. Kambor et al. , Nucl. Phys. B , 215 (1996).[5] A. Anisovich and H. Leutwyler, Phys. Lett. B , 335(1996).[6] G. Colangelo, J. Gasser, and H. Leutwyler, Nucl. Phys.B , 125 (2001).[7] R. Kami´nski, J. R. Pel´aez, and F. J. Yndur´ain, Phys.Rev. D , 054015 (2008),[8] R. Garc´ıa-Mart´ın, R. Kami´nski, J. R. Pel´aez, J. Ruiz deElvira, and F. J. Yndur´ain, Phys. Rev. D , 074004(2011),[9] W. B. Tippens et al. , Phys. Rev. Lett. , 192001 (2001).[10] M. Unverzagt et al. , Eur. Phys. J. A , 169 (2009)[11] S. Prakhov et al. , Phys. Rev. C , 035204 (2009).[12] F. Ambrosino et al. , JHEP , 006 (2008),[13] P. Adlarson et al. , Phys. Rev. C , 045207 (2014)[14] A. Anastasi et al. , JHEP , 019 (2016).[15] C. O. Gullstr¨om, A. Kup´s´c, and A. Rusetsky, Phys. Rev.C , 028201 (2009).[16] S. P. Schneider, B. Kubis, and C. Ditsche, JHEP , 028(2011).[17] K. Kampf, M. Knecht, J. Novotn´y, and M. Zdr´ahal,Phys. Rev. D , 114015 (2011).[18] P. Guo, I. V. Danilkin, D. Schott, C. Fern´andez-Ram´ırez,V. Mathieu, and A. P. Szczepaniak, Phys. Rev. D ,054016 (2015).[19] P. Guo, I. V. Danilkin, D. Schott, C. Fern´andez-Ram´ırez,V. Mathieu, and A. P. Szczepaniak, Phys. Lett. B ,497 (2017).[20] M. Koles´ar and J. Novotn´y, Eur. Phys. J. C , 41(2017).[21] G. Colangelo, S. Lanz, H. Leutwyler, and E. Passemar,Phys. Rev. Lett. , 022001 (2017).[22] M. Albaladejo and B. Moussallam, Eur. Phys. J. C ,508 (2017).[23] C. Patrignani et al. , (Particle Data Group), Chin. Phys.C , 100001 (2016). [24] D. Alde et al. , Z. Phys. C , 225 (1984).[25] A. Abele et al. , Phys. Lett. B , 193 (1998).[26] M. N. Achasov et al. , JETP Lett. , 451 (2001).[27] M. Bashkanov et al. , Phys. Rev. C , 048201 (2007).[28] C. Adolph et al. , Phys. Lett. B , 24 (2009).[29] F. Ambrosino et al. , Phys. Lett. B , 16 (2011).[30] M. Ablikim et al. , Phys. Rev. D , 012014 (2015).[31] B. Borasoy and R. Nissler, Eur. Phys. J. A , 383(2005).[32] J. R. Batley et al. , Eur. Phys. J. C , 589 (2009).[33] Heinrich Leutwyler, private communication.[34] M. Bissegger, A. Fuhrer, J. Gasser, B. Kubis andA. Rusetsky, Phys. Lett. B , 576 (2008).[35] V. L. Kashevarov et al. , Phys. Rev. Lett. , 212001(2017).[36] A. Starostin et al. , Phys. Rev. C , 055205 (2001).[37] R. Novotny, IEEE Trans. Nucl. Sci. , 379 (1991).[38] A. R. Gabler et al. , Nucl. Instrum. Methods Phys. Res.A , 168 (1994).[39] H. Herminghaus et al. , IEEE Trans. Nucl. Sci. , 3274(1983).[40] K.-H. Kaiser et al. , Nucl. Instrum. Methods Phys. Res.A , 159 (2008).[41] I. Anthony et al. , Nucl. Instrum. Methods Phys. Res. A , 230 (1991).[42] S. J. Hall et al. , Nucl. Instrum. Methods Phys. Res. A , 698 (1996).[43] J. C. McGeorge et al. , Eur. Phys. J. A , 129 (2008).[44] E. F. McNicoll et al. , Phys. Rev. C , 035208 (2010).[45] D. Watts, Proceedings of the 11th International Confer-ence on Calorimetry in Particle Physics , Perugia, Italy,2004 (World Scientific, Singapore, 2005), p. 560.[46] S. Prakhov et al. , Phys. Rev. C , 025204 (2009).[47] P. Aguar-Bartolom´e et al. , Phys. Rev. C , 044601(2013).[48] See Supplemental Material for the data points from theexperimental Dalitz plot and the ratios of the z and m ( π0 π0