Workshop on Pion-Kaon Interactions (PKI2018) Mini-Proceedings. Editors: M. Amaryan, Ulf-G. Meißner, C. Meyer, J. Ritman, and I. Strakovsky
M. Amaryan, M. Baalouch, G. Colangelo, J. R. de Elvira, D. Epifanov, A. Filippi, B. Grube, V. Ivanov, B. Kubis, P. M. Lo, M. Mai, V. Mathieu, S. Maurizio, C. Morningstar, B. Moussallam, F. Niecknig, B. Pal, A. Palano, J. R. Pelaez, A. Pilloni, A. Rodas, A. Rusetsky, A. Szczepaniak, J. Stevens
DDate:
April 19, 2018
Workshop on Pion-Kaon Interactions(PKI2018)Mini-Proceedings
M. Amaryan, M. Baalouch, G. Colangelo, J. R. de Elvira, D. Epifanov, A. Filippi, B. Grube,V. Ivanov, B. Kubis, P. M. Lo, M. Mai, V. Mathieu, S. Maurizio, C. Morningstar, B. Moussallam,F. Niecknig, B. Pal, A. Palano, J. R. Pelaez, A. Pilloni, A. Rodas, A. Rusetsky, A. Szczepaniak,and J. Stevens
Editors : M. Amaryan, Ulf-G. Meißner, C. Meyer, J. Ritman, and I. Strakovsky
Abstract
This volume is a short summary of talks given at the PKI2018 Workshop organized to discusscurrent status and future prospects of π − K interactions. The precise data on π K interaction willhave a strong impact on strange meson spectroscopy and form factors that are important ingredientsin the Dalitz plot analysis of a decays of heavy mesons as well as precision measurement of V us matrix element and therefore on a test of unitarity in the first raw of the CKM matrix. The workshophas combined the efforts of experimentalists, Lattice QCD, and phenomenology communities.Experimental data relevant to the topic of the workshop were presented from the broad range ofdifferent collaborations like CLAS, GlueX, COMPASS, BaBar, BELLE, BESIII, VEPP-2000, andLHC b . One of the main goals of this workshop was to outline a need for a new high intensity andhigh precision secondary K L beam facility at JLab produced with the 12 GeV electron beam ofCEBAF accelerator.This workshop is a successor of the workshops Physics with Neutral Kaon Beam at JLab [1] heldat JLab, February, 2016;
Excited Hyperons in QCD Thermodynamics at Freeze-Out [2] held atJLab, November, 2016;
New Opportunities with High-Intensity Photon Sources a r X i v : . [ h e p - ph ] A p r ontents K L Beam Facility at JLab for Strange Hadron Spectroscopy . . . . . . 5
Moskov Amaryan
Colin Morningstar π Scattering with K L Beam Facility . . . . . . . . . . . . . . . . . . . . . . . . 19
Marouen Baalouch
Antimo Palano
Boris Grube π − K Interactions . . . . . . . . . . . . . . . . . 38
Bilas Pal τ → Kπν
Decay at the B Factories . . . . . . . . . . . . . . . . . . . . . 48
Denis Epifanov πK Amplitudes to πK Form Factors (and Back) . . . . . . . . . . . . . . . 55
Bachir Moussallam
Maxim Mai e + e − → K ¯ Knπ with the CMD-3 Detector at VEPP-2000Collider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Vyacheslav Ivanov
Justin Stevens lessandra Filippi (for the CLAS Collaboration) π − K Scattering Lengths . . . . . . . . . . . . . 95
Jacobo Ruiz de Elvira, Gilberto Colangelo, and Stefano Maurizio
José R. Pelaez, Arkaitz Rodas, and Jacobo Ruiz de Elvira
Vincent Mathieu
Bastian Kubis and Franz Niecknig π K to Understand Heavy Meson Decays . . . . . . . . . . . . . . . . . . . 126
Alessandro Pilloni and Adam Szczepaniak
Akaki Rusetsky
Pok Man Lo
Alessandra Filippi (for the BaBar Collaboration) iii
Preface
1. From February 14-15, 2018, the Thomas Jefferson Laboratory in Newport News, Virginiahosted the PKI2018, an international workshop to explore the physics potential to investigate π -K interactions. This was the fourth of a series of workshops held to establish a neutral kaonbeam facility at JLab Hall D with a neutral kaon flux which will be three orders of magnitudehigher than was available at SLAC. This facility will enable scattering experiments of K L off both proton and neutron (for the first time) targets in order to measure differential crosssection distributions with the GlueX detector.The combination of data from this facility with the self-analyzing power of strange hyperonswill enable precise partial-wave analyses (PWA) in order to determine dozens of predicted Λ ∗ , Σ ∗ , Ξ ∗ , and Ω ∗ resonances up to 2.5 GeV. Furthermore, the KLF will enable strangemeson spectroscopy by studies of the π -K interaction to locate pole positions in the I = 1/2and 3/2 channels. Detailed study of π -K system with PWA will allow to observe and measurequantum numbers of missing kaon states, which in turn will also impact Dalitz plot analysesof heavy meson decays, as well as tau-lepton decay the with π π -K interactions,(c) Results from Chiral Effective Theories(d) Results from Dispersion Relations(e) π -K formfactor and heavy meson and tau decay(f) Hadron Spectroscopy at GlueX, CLAS, CLAS12, BaBar, and COMPASS2. Acknowledgments
The workshop would not have been possible without dedicated work of many people. First,we would like to thank the service group and the staff of JLab for all their efforts. Wewould like to thank JLab management, especially Robert McKeown for their help and en-couragement to organize this workshop. Financial support was provided by the JLab, JülichForschungszentrum, The George Washington and Old Dominion universities.Newport News, March 2018. 1 eferences [1] M. Amaryan, Ulf-G. Meißner, C. Meyer, J. Ritman, and I. Strakovsky, eds., Mini-Proceedings,
Workshop on Physics with Neutral Kaon Beam at JLab (KL2016);arXiv:1604.02141 [hep–ph]. M. Amaryan, E. Chudakov, K. Rajagopal, C. Ratti, J. Rit-man, and I. Strakovsky, eds., Mini-Proceedings,
Workshop on Excited Hyperons in QCDThermodynamics at Freeze-Out (YSTAR2016); arXiv:1701.07346 [hep–ph].[2] M. Amaryan, E. Chudakov, K. Rajagopal, C. Ratti, J. Ritman, and I. Strakovsky, eds., Mini-Proceedings,
Workshop on Excited Hyperons in QCD Thermodynamics at Freeze-Out (YS-TAR2016); arXiv:1701.07346 [hep–ph].[3] T. Horn, C. Keppel, C. Munoz-Camacho, and I. Strakovsky, eds., Mini-Proceedings,
Work-shop on High-Intensity Photon Sources (HIPS2017); arXiv:1704.00816 [nucl–ex].2
Program
Wednesday, February 14, 2018
Welcome and Introductory Remarks – Jianwei Qiu (JLab)9:00am - 9:25am:
KL Facility at JLab – Moskov Amaryan (ODU)9:25am - 9:50am:
Kaon-pion scattering from lattice QCD – Colin Morningstar (CMU)9:50am -10:15am:
Study of k − π interaction with KLF – Marouen Baalouch (ODU)10:15am -10:45am: Coffee breakSession 2: Chair: Eugene Chudakov / Secretary: Chan Kim10:45am -11:15am: Dalitz plot analysis of three-body charmonium decays at BaBar – AntimoPalano (INFN/Bari U.)11:15am -11:45am:
Kaon and light-meson resonances at COMPASS – Boris Grube (TUM)11:45am -12:15pm:
Recent Belle results related to pion-kaon interactions – Bilas Pal (CincinnatiU.)12:15pm - 2:00pm: Conference Photo & Lunch break - on your ownSession 3: Chair: Curtis Meyer / Secretary: Torry Roak2:00pm - 2:25pm:
Study of τ → Kπν decay at the B factories – Denis Epifanov (BINP, NSU)2:25pm - 2:50pm:
From π − K amplitudes to π − K form factors and back – Bachir Moussallam(Paris-Sud U.)2:50pm - 3:15pm: Three-body interactions in isobar formalism
Maxim Mai (GW)3:15pm - 3:40pm:
Study of the processes e + e − → K ¯ Knπ with the CMD-3 detector at VEPP-2000collider – Vyacheslav Ivanov (BINP)3:40pm - 4:10pm: Coffee breakSession 4: Chair: Charles Hyde / Secretary: Tyler Viducic4:10pm - 4:35pm:
The GlueX Meson Program – Justin Stevens (W&M)4:35pm - 5:00pm:
Strange meson spectroscopy at CLAS and CLAS12 – Alessandra Filippi (INFNTorino)5:00pm - 5:25pm:
Non-leptonic charmless three body decays at LHCb – Rafael Silva Coutinho(Zuerich U.) 3:25pm- 5:50pm:
Dispersive determination of the π − K scattering lengths – Jacobo Ruiz de Elvira(Bern U.)5:50pm: Adjourn6:10pm: Networking Reception - CEBAF Center Lobby Thursday, February 15, 2018
Meson-meson scattering from lattice QCD – Jo Dudek (W&M)9:15am - 9:45am:
Dispersive analysis of pion-kaon scattering – Jose R. Pelaez (U. Complutensede Madrid)9:45am -10:15am:
Analyticity Constraints for Exotic Mesons – Vincent Mathieu (JLab)10:15am -10:55am: Coffee breakSession 6: Chair: Jacobo Ruiz / Secretary: Wenliang Li10:55am -11:25am:
Pion-kaon scattering in the final-state interactions of heavy-meson decays –Bastian Kubis (Bonn U.)11:25am -11:55am:
Using πK to understand heavy meson decays – Alessandro Pilloni (JLab)11:55am -12:25pm: Three particle dynamics on the lattice – Akaki Rusetsky (Bonn U.)12:25pm - 2:00pm: Lunch break - on your ownSession 7: Chair: James Ritman / Secretary: Amy Schertz2:00pm - 2:30pm:
S-matrix approach to the thermodynamics of hadrons – Pok Man Lo (WroclawU.)2:30pm - 3:00pm:
Measurement of hadronic cross sections with the BaBar detector – AlessandraFilippi (INFN Torino)3:00pm - 3:30pm:
A determination of the pion-kion scattering length from 2+1 flavor lattice QCD – Daniel Mohler (Helmholtz-Inst. Mainz)3:30pm - 4:10pm: Coffee breakSession 8: Chair: Bachir Moussallam / Secretary: Nilanga Wickramaarachch4:10pm - 4:35pm:
Strangeness-changing scalar form factor from scattering data and CHPT –Michael Döring (GW/JLab)4:35pm - 4:50pm:
Closing Remarks – Bachir Moussallam (Paris-Sud U.)4:50pm: Closing 4
Summaries of Talks K L Beam Facility at JLab for Strange Hadron Spectroscopy
Moskov Amaryan
Department of PhysicsOld Dominion UniversityNorfolk, VA 23529, U.S.A.
Abstract
In this talk, I discuss the photoproduction of a secondary K L beam at JLab to be used withthe GlueX detector in Hall-D for a strange hadron spectroscopy. It is comforting to reflect that the disproportionof things in the world seems to be only arithmetical.
Franz Kafka1.
Introduction
Current status of our knowledge about the strange hyperons and mesons is far from beingsatisfactory. One of the main reasons for this is that first of all dozens of strange hadronstates predicted by Constituent Quark Model (CQM) and more recently by Lattice QCDcalculations are still not observed. The detailed discussions about the missing hyperons perse and in particular their connection to thermodynamics of the Early Universe at freeze-outwere performed respectively in a three preceding workshops [1–3]. Topics discussed in theseworkshops were significant part of the proposal submitted to the JLab PAC45 [4].This is a fourth workshop in this series devoted to the physics program related to the strangemeson states and π − K interactions. As it has been summarized in all four workshopsit is not only the disproportion between the number of currently observed and CQM andLQCD predicted states that makes experimental studies of the strange quark sector to be ofhigh priority. These experiments are crucially important to understand QCD at perturbativedomain and the dynamics of strange hadron production using hadronic beam with the strangequark in the projectile.Many aspects of π − K interactions and their impact on different important problems inparticle physics have been discussed in this workshop.Below we describe conceptually the main steps needed to produce intensive K L beam. Wediscuss momentum resolution of the beam using time-of-flight technique, as well as theratio of K L over neutrons as a function of their momenta simulated based on well knownproduction processes. In some examples the quality of expected experimental data obtainedby using GlueX setup in Hall-D will be demonstrated using results of Monte Carlo studies.5. The K L Beam in Hall D
In this chapter we describe photo-production of secondary K L beam in Hall D. There arefew points that need to be decided. To produce intensive photon beam one needs to increaseradiation length of the radiator up to 10 % radiation length. In a first scenario, E e = 12 GeVelectrons produced at CEBAF will scatter in a radiator in the tagger vault, generating in-tensive beam of bremsstrahlung photons. This may will then require removal of all taggercounters and electronics and very careful design of radiation shielding, which is very hard tooptimize and design.In a second scenario. one may use Compact Photon Source design (for more details seea talk by Degtiarenko in Ref. [1]) installed after the tagger magnet, which will producebremsstrahlung photons and dump electron beam inside the source shielding the radiationinside. At the second stage, bremsstrahlung photons interact with Be target placed on adistance 16 m upstream of liquid hydrogen ( LH ) target of GlueX experiment in Hall Dproducing K L beam. To stop photons a 30 radiation length lead absorber will be installedin the beamline followed by a sweeping magnet to deflect the flow of charged particles.The flux of K L on ( LH ) target of GlueX experiment in Hall D will be measured with pairspectrometer upstream the target. For details of this part of the beamline see a talk by Larinin Ref. [1]. Momenta of K L particles will be measured using the time-of-flight between RFsignal of CEBAF and start counters surrounding LH target. Schematic view of beamlineis presented in Fig. 1. The bremsstrahlung photons, created by electrons at a distance about75 m upstream, hit the Be target and produce K L mesons along with neutrons and chargedparticles. The lead absorber of ∼
30 radiation length is installed to absorb photons exiting Betarget. The sweeping magnet deflects any remaining charged particles (leptons or hadrons)remaining after the absorber. The pair spectrometer will monitor the flux of K L through thedecay rate of kaons at given distance about 10 m from Be target. The beam flux could alsobe monitored by installing nuclear foil in front of pair spectrometer to measure a rate of K S due to regeneration process K L + p → K S + p as it was done at NINA (for a details see atalk my Albrow at this workshop). Figure 14: Schematic view of Hall D beamline on the way e ! ! K L . Electrons first hitthe tungsten radiator, then photons hit the Be target assembly, and finally, neutral kaons hit theLH /LD cryotarget. The main components are CPS, Be target assembly, beam plug, sweep mag-net, and pair spectrometer. See the text for details.and the LH /LD target (located inside Hall D detector) was taken as 16 m in our calculations Itcan be increased up to 20 m. An intense high-energy gamma source is a prerequisite for the production of the K L beam neededfor the new experiments described in this proposal. In 2014, Hall A Collaboration has been dis-cussed a novel concept of a Compact Photon Source (CPS) [116]. It was developed for a Wide-Angle Compton Experiment proposed to PAC43 [117]. Based on these ideas, we suggested (seeRef. [118]) to use the new concept in this experiment. A possible practical implementation ad-justed to the parameters and limitations of the available infrastructure is discussed below. Thevertical cut of the CPS model design, and the horizontal plane view of the present Tagger vaultarea with CPS installed are shown in Fig. 15.The CPS design combines in a single properly shielded assembly all elements necessary for theproduction of the intense photon beam, such that the overall dimensions of the setup are limitedand the operational radiation dose rates around it are acceptable. Compared to the alternative,the proposed CPS solution presents several advantages: much lower radiation levels, both promptand post-operational due to the beam line elements’ radio-activation at the vault. The new de-26
Hall-D beamline and GlueX Setup
Figure 1: Schematic view of Hall D beamline. See a text for explanation.Here we outline experimental conditions and simulated flux of K L based on GEANT4 andknown cross sections of underlying subprocesses [5–7].6he expected flux of K L mesons integrated in the range of momenta P = 0 . − GeV /c will be on the order of ∼ K L /s on the physics target of the GlueX setup under thefollowing conditions: • A thickness of the radiator 10 % . • The distance between Be and LH targets in the range of 24 m. • The Be target with a length L = 40 cm.In addition, the lower repetition rate of electron beam with 64 ns spacing between buncheswill be required to have enough time to measure time-of-flight of the beam momenta and toavoid an overlap of events produced from alternating pulses. Low repetition rate was alreadysuccessfully used by G0 experiment in Hall C at JLab [8].The final flux of K L is presented with 10 % radiator, corresponding to maximal rate .In the production of a beam of neutral kaons, an important factor is the rate of neutrons as abackground. As it is well known, the ratio R = N n /N K L is on the order from primaryproton beams [9], the same ratio with primary electromagnetic interactions is much lower.This is illustrated in Fig. 2, which presents the rate of kaons and neutrons as a function ofthe momentum, which resembles similar behavior as it was measured at SLAC [10]. P (GeV/c) R a t e H z / M e V neutrons PythiaKlongsneutrons DINREGneutrons DINREG, with Pb shieldYields in Be Figure 2: The rate of neutrons (open symbols) and K L (full squares) on LH target of Hall D as afunction of their momenta simulated with different MC generators with K L /sec.Shielding of the low energy neutrons in the collimator cave and flux of neutrons has beenestimated to be affordable, however detailed simulations are under way to show the level ofradiation along the beamline.Th response of GlueX setup, reconstruction efficiency and resolution are presented in a talkby Taylor in Ref. [1].3. Expected Rates
In this section, we discuss expected rates of events for some selected reactions. The pro-duction of Ξ hyperons has been measured only with charged kaons with very low statistical7recision and never with primary K L beam. In Fig. 3, panel (a) shows existing data for theoctet ground state Ξ ’s with theoretical model predictions for W (the reaction center of massenergy) distribution, panel (b) shows the same model prediction [11] presented with expectedexperimental points and statistical error for 10 days of running with our proposed setup witha beam intensity × K L /sec using missing mass of K + in the reaction K L + p → K + Ξ without detection of any of decay products of Ξ (for more details on this topic see a talk byNakayama in Ref. [1]). W (GeV) σ ( µ b ) K − + p → K + + Ξ − (a) W (GeV) σ ( µ b ) K − + p → K + Ξ (b) FIG. 4. (Color online) Total cross section results with individual resonances switched off (a) for K − + p → K + + Ξ − and (b)for K − + p → K + Ξ . The blue lines represent the full result shown in Figs. 2 and 3. The red dashed lines, which almostcoincide with the blue lines represent the result with Λ(1890) switched off. The green dash-dotted lines represent the resultwith Σ(2030) switched off and the magenta dash-dash-dotted lines represent the result with Σ(2250)5 / − switched off. ΛΣ Contact 204060 d σ / d Ω ( µ b / s r) − − − − − cos θ K − + p → K + + Ξ − (a) d σ / d Ω ( µ b / s r) − − − − cos θ K − + p → K + Ξ (b) FIG. 5. (Color online) Kaon angular distributions in the center-of-mass frame (a) for K − + p → K + + Ξ − and (b) for K − + p → K + Ξ . The blue lines represent the full model results. The red dashed lines show the combined Λ hyperonscontribution. The magenta dash-dotted lines show the combined Σ hyperons contribution. The green dash-dash-dotted linecorresponds to the contact term. The numbers in the upper right corners correspond to the centroid total energy of the system W . Note the different scales used. The experimental data (black circles) are the digitized version as quoted in Ref. [50] from theoriginal work of Refs. [31–34, 36, 37] for the K − + p → K + +Ξ − reaction and of Ref. [30, 36, 37, 40] for the K − + p → K +Ξ reaction. p → K + + Ξ − and K − + p → K + Ξ are shown inFigs. 5(a) and 5(b), respectively, in the energy domain upto W = 2 . W = 2 . W = 2 .
33 to 2 .
48 GeV in the charged Ξ − production. Our model underpredicts the yield aroundcos θ = 0. As in the total cross sections, the data for theneutral Ξ production are fewer and less accurate than for the charged Ξ − production. In particular, the Ξ production data at W = 2 .
15 GeV seems incompatiblewith those at nearby lower energies and that the presentmodel is unable to reproduce the observed shape at back-ward angles. It is clear from Figs. 5(a) and 5(b) that thecharged channel shows a backward peaked angular dis-tributions, while the neutral channel shows enhancementfor both backward and forward scattering angles (moresymmetric around cos θ = 0) for all but perhaps the high- Cascade production on proton with K beam
Estimated measurement for 10 days exposition Existing measurements in charged channels (b)
Figure 3: (a) Cross section for existing world data on K − + p → K + Ξ − reaction with modelpredictions from Ref. [11]; (b) expected statistical precision for the reaction K L + p → K + Ξ in10 days of running with a beam intensity × K L /sec overlaid on theoretical prediction [11].The physics of excited hyperons is not well explored, remaining essentially at the pioneeringstages of ’70s-’80s. This is especially true for Ξ ∗ ( S = − and Ω ∗ ( S = − hyperons.For example, the SU (3) flavor symmetry allows as many S = − baryon resonances, asthere are N and ∆ resonances combined ( ≈ ); however, until now only three [ground state Ξ(1382)1 / + , Ξ(1538)3 / + , and Ξ(1820)3 / − ] have their quantum numbers assigned andfew more states have been observed [12]. The status of Ξ baryons is summarized in a tablepresented in Fig. 4 together with the quark model predicted states [13].Historically the Ξ ∗ states were intensively searched for mainly in bubble chamber experi-ments using the K − p reaction in ’60s-’70s. The cross section was estimated to be on theorder of 1-10 µb at the beam momenta up to 10 GeV/c. In ’80s-’90s, the mass or widthof ground and some of excited states were measured with a spectrometer in the CERN hy-peron beam experiment. Few experiments have studied cascade baryons with the missingmass technique. In 1983, the production of Ξ ∗ resonances up to 2.5 GeV were reportedfrom p ( K − , K + ) reaction from the measurement of the missing mass of K + [14]. The ex-perimental situation with Ω −∗ ’s is even worse than the Ξ ∗ case, there are very few data forexcited states. The main reason for such a scarce dataset in multi strange hyperon domainis mainly due to very low cross section in indirect production with pion or in particular-photon beams. Currently only ground state Ω − quantum numbers are identified. Recentlysignificant progress is made in lattice QCD calculations of excited baryon states [15, 16]which poses a challenge to experiments to map out all predicted states (for more details seea talk by Richards at this workshop). The advantage of baryons containing one or more8 and Karl [3]. The 12 excited states were predicted up to 2 GeV/ c , whereas only ⌅(1820) is identified as J P = 3 / state with three stars. FIG. 1. Black bars: Predicted ⌅ spectrum based on the quark model calculation [3]. Colored bars: Observed states. The twoground states and ⌅(1820) are shown in the column of J P = 1 / + , / , respectively. Other unknown J P states are plotted inthe rightest column. The number represents the mass and the size of the box corresponds to the width of each state. Recently it is pointed out that there are two distinct excitation modes when a baryon contains one heavy flavorinside, and the separation of these two modes possibly good enough even at the strange quark mass [4]. Baryonswhich contain single (Qqq) and double (QQq) strange and/or charm flavors might be understood as a “dual” systembased on the spatial parametrization concerning a diquark contribution of (qq) and (QQ). In this sense, it should benoted that cascades and charmed baryons are expected to be closely related.The ⌅ ⇤ states were intensively searched for mainly in bubble chamber experiments using the K p reaction in ’60s ’70s. The cross section was estimated to be an order of 1 µ b at the beam momentum up to ⇠
10 GeV/ c . In ’80s ’90s, the mass or width of ground or some excited states were measured with a spectrometer in the CERN hyperonbeam experiment. There has been a few experiments to study cascade baryons with the missing mass technique. In1983, the production of ⌅ ⇤ resonances up to 2.5 GeV/ c were reported from the missing mass measurement of the p ( K , K + ) reaction, using multi-particle spectrometer at the Brookhaven National Laboratory [5]. Figure 2 showssquared missing mass spectra of p ( K , K + ) reaction. With ten times intense kaon beam combined with 5 p ( K , K + ) reaction. II. THE PHYSICS CASE
The physics case and experimental method are reviewed in the following.
Figure 4: Black bars: Predicted Ξ spectrum based on the quark model calculation [13]. Coloredbars: Observed states. The two ground octet and decuplet states together with Ξ(1820) in thecolumn J P = 3 / − are shown in red color. Other observed states with unidentified spin-parity areplotted in the rightest column.strange quarks for lattice calculations is that then number of open decay channels is in gen-eral smaller than for baryons comprising only the light u and d quarks. Moreover, latticecalculations show that there are many states with strong gluonic content in positive paritysector for all baryons. The reason why hybrid baryons have not attracted the same atten-tion as hybrid mesons is mainly due to the fact that they lack manifest “exotic" character.Although it is difficult to distinguish hybrid baryon states, there is significant theoretical in-sight to be gained from studying spectra of excited baryons, particularly in a framework thatcan simultaneously calculate properties of hybrid mesons. Therefore this program will bevery much complementary to the GlueX physics program of hybrid mesons.The proposed experiment with a beam intensity K L /sec will result in about × Ξ ∗ ’sand × Ω ∗ ’s per month.A similar program for KN scattering is under development at J-PARC with charged kaonbeams. The current maximum momentum of secondary beamline of 2 GeV/c is available atthe K1.8 beamline. The beam momentum of 2 GeV/c corresponds to √ s =2.2 GeV in the K − p reaction which is not enough to generate even the first excited Ξ ∗ state predicted in thequark model. However, there are plans to create high energy beamline in the momentumrange 5-15 GeV/c to be used with the spectrometer commonly used with the J-PARC P50experiment which will lead to expected yield of (3 − × Ξ ∗ ’s and Ω ∗ ’s per month.Statistical power of proposed experiment with K L beam at JLab will be of the same order asthat in J-PARC with charged kaon beam.An experimental program with kaon beams will be much richer and allows to perform acomplete experiment using polarized target and measuring recoil polarization of hyperons.This studies are under way to find an optimal solution for the GlueX setup.9he strange meson spectroscopy is another important subject for K L Facility at JLab. Thehigh intensity K L beam will allow to study final state K − π system. In particluar to performphase shift analysis for different partial-waves, which may have significant imact to all sys-tems having K − π in the final state. This includes heavy D - and B -meson decays as wellas τ → Kπν τ decay.4. Summary
In summary we intend to propose production of high intensity K L beam using photopro-duction processes from a secondary Be target. A flux as high as ∼ K L /sec could beachieved. Momenta of K L beam particles will be measured with time-of-flight. The flux ofkaon beam will be measured through partial detection of π + π − decay products from theirdecay to π + π − π by exploiting similar procedure used by LASS experiment at SLAC [10].Besides using unpolarized liquid hydrogen target currently installed in GlueX experimentthe unpolarized deuteron target may be installed. Additional studies are needed to find anoptimal choice of polarized targets. This proposal will allow to measure KN scatteringwith different final states including production of strange and multi strange baryons with un-precedented statistical precision to test QCD in non perturbative domain. It has a potentialto distinguish between different quark models and test lattice QCD predictions for excitedbaryon states with strong hybrid content. It will also be used to study π − K interactions,which is the topic of the current workshop.5. Acknowledgments
My research is supported by DOE Grant-100388-150.
References [1] M. Amaryan, Ulf-G. Meißner, C. Meyer, J. Ritman, and I. Strakovsky, eds., Mini-Proceedings,
Workshop on Physics with Neutral Kaon Beam at JLab (KL2016);arXiv:1604.02141 [hep–ph].[2] M. Amaryan, E. Chudakov, K. Rajagopal, C. Ratti, J. Ritman, and I. Strakovsky, eds., Mini-Proceedings,
Workshop on Excited Hyperons in QCD Thermodynamics at Freeze-Out (YS-TAR2016); arXiv:1701.07346 [hep–ph].[3] T. Horn, C. Keppel, C. Munoz-Camacho, and I. Strakovsky, eds., Mini-Proceedings,
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Carnegie Mellon UniversityPittsburgh, PA 15213, U.S.A.
Abstract
Recent progress in determining scattering phase shifts, and hence, resonance propertiesfrom lattice QCD in large volumes with nearly realistic pion masses is presented. A crucialingredient in carrying out such calculations in large volumes is estimating quark propagationusing the stochastic LapH method. The elastic I = 1 / , S - and P -wave kaon-pion scatteringamplitudes are calculated using an ensemble of anisotropic lattice QCD gauge field configu-rations with N f = 2 + 1 flavors of dynamical Wilson-clover fermions at m π = 240 MeV.The P -wave amplitude is well described by a Breit-Wigner shape with parameters which areinsensitive to the inclusion of D -wave mixing and variation of the S -wave parametrization.
1. A key goal in lattice QCD is the determination of the spectrum of hadronic resonances fromfirst principles. One of the best methods of computing the masses and other properties ofhadrons from QCD involves estimating the QCD path integrals using Markov-chain MonteCarlo methods, which requires formulating the theory on a space-time lattice. Such calcula-tions are necessarily carried out in finite volume. However, most of the excited hadrons weseek to study are unstable resonances. Fortunately, it is possible to deduce the masses andwidths of resonances from the spectrum determined in finite volume. The method we use isdescribed in Ref. [1] and the references contained therein.To study low-lying resonances in lattice QCD, we first use lattice QCD methods to calcu-late the interacting two-particle lab-frame energies E in a spatial L volume with periodicboundary conditions. Our stationary states have total momentum P = (2 π/L ) d , where d is a vector of integers. We boost the lab-frame energies to the center-of-momentum frameusing E cm = √ E − P , γ = EE cm . (1)Assuming there are N d channels with particle masses m a , m a and spins s a , s a of particle1 and 2 for each decay channel, we can define q ,a = 14 E −
12 ( m a + m a ) + ( m a − m a ) E , (2) u a = L q ,a (2 π ) , s a = (cid:18) m a − m a ) E (cid:19) . (3)The relationship between the finite-volume energies E and the scattering matrix S is givenby det[1 + F ( P ) ( S − , (4)where the F -matrix in the J LSa basis states is given by (cid:104) J (cid:48) m J (cid:48) L (cid:48) S (cid:48) a (cid:48) | F ( P ) | J m J LSa (cid:105) = δ a (cid:48) a δ S (cid:48) S (cid:110) δ J (cid:48) J δ m J (cid:48) m J δ L (cid:48) L + (cid:104) J (cid:48) m J (cid:48) | L (cid:48) m L (cid:48) Sm S (cid:105)(cid:104) Lm L Sm S | J m J (cid:105) W ( P a ) L (cid:48) m L (cid:48) ; Lm L (cid:111) , (5)12nd where J, J (cid:48) indicate total angular momenta,
L, L (cid:48) are orbital angular momenta,
S, S (cid:48) areintrinsic spins, and a, a (cid:48) denote all other identifying information, such as decay channel. The W -matrix is given by − iW ( P a ) L (cid:48) m L (cid:48) ; Lm L = L (cid:48) + L (cid:88) l = | L (cid:48) − L | l (cid:88) m = − l Z lm ( s a , γ, u a ) π / γu l +1 a (cid:115) (2 L (cid:48) + 1)(2 l + 1)(2 L + 1) ×(cid:104) L (cid:48) , l | L (cid:105)(cid:104) L (cid:48) m L (cid:48) , lm | Lm L (cid:105) , (6)and the Rummukainen-Gottlieb-Lüscher (RGL) shifted zeta functions Z lm are evaluated us-ing Z lm ( s , γ, u ) = (cid:88) n ∈ Z Y lm ( z )( z − u ) e − Λ( z − u ) + δ l γπ √ Λ F (Λ u )+ i l γ Λ l +1 / (cid:90) dt (cid:16) πt (cid:17) l +3 / e Λ tu (cid:88) n ∈ Z n (cid:54) =0 e πi n · s Y lm ( w ) e − π w / ( t Λ) (7)where z = n − γ − (cid:2) + ( γ − s − n · s (cid:3) s , (8) w = n − (1 − γ ) s − s · ns , Y lm ( x ) = | x | l Y lm ( (cid:98) x ) (9) F ( x ) = − (cid:90) dt e tx − t / . (10)We choose Λ ≈ for fast convergence of the summation, and the integral in Eq. (7) is doneusing Gauss-Legendre quadrature. F ( x ) given in terms of the Dawson or erf function.The quantization condition in Eq. (4) relates a single energy E to the entire S -matrix. Thisequation cannot be solved for S , except in the case of a single channel and single partialwave. To proceed, we approximate the S -matrix using functions depending on a handfulof fit parameters, then obtain estimates of these parameters using fits involving as manyenergies as possible. It is easier to parametrize a Hermitian matrix than a unitary matrix, sowe use the K -matrix defined as usual by S = (1 + iK )(1 − iK ) − = (1 − iK ) − (1 + iK ) . (11)The Hermiticity of the K -matrix ensures the unitarity of the S -matrix. With time reversalinvariance, the K -matrix must be real and symmetric. The multichannel effective rangeexpansion suggests writing K − L (cid:48) S (cid:48) a (cid:48) ; LSa ( E ) = u − L (cid:48) − a (cid:48) (cid:101) K − L (cid:48) S (cid:48) a (cid:48) ; LSa ( E cm ) u − L − a (12)since (cid:101) K − L (cid:48) S (cid:48) a (cid:48) ; LSa ( E cm ) should behave smoothly with E cm , then the quantization conditioncan be written det(1 − B ( P ) (cid:101) K ) = det(1 − (cid:101) KB ( P ) ) = 0 , (13)13here we define the important box matrix by (cid:104) J (cid:48) m J (cid:48) L (cid:48) S (cid:48) a (cid:48) | B ( P ) | J m J LSa (cid:105) = − iδ a (cid:48) a δ S (cid:48) S u L (cid:48) + L +1 a W ( P a ) L (cid:48) m L (cid:48) ; Lm L ×(cid:104) J (cid:48) m J (cid:48) | L (cid:48) m L (cid:48) , Sm S (cid:105)(cid:104) Lm L , Sm S | J m J (cid:105) . (14)The box matrix is Hermitian for u a real. The quantization condition can also be expressedas det( (cid:101) K − − B ( P ) ) = 0 , (15)and the determinants in Eqs. (13) and (15) are real.The quantization condition involves a determinant of an infinite matrix. To make such deter-minants practical for use, we first transform to a block-diagonal basis, and then truncate inorbital angular momentum. For a symmetry operation G , define theunitary matrix (cid:104) J (cid:48) m J (cid:48) L (cid:48) S (cid:48) a (cid:48) | Q ( G ) | J m J LSa (cid:105) = (cid:26) δ J (cid:48) J δ L (cid:48) L δ S (cid:48) S δ a (cid:48) a D ( J ) m J (cid:48) m J ( R ) , ( G = R ) ,δ J (cid:48) J δ m J (cid:48) m J δ L (cid:48) L δ S (cid:48) S δ a (cid:48) a ( − L , ( G = I s ) , (16)where D ( J ) m (cid:48) m ( R ) are the Wigner rotation matrices, R is an ordinary rotation, and I s is spatialinversion. One can show that the box matrix satisfies B ( G P ) = Q ( G ) B ( P ) Q ( G ) † . (17)If G is in the little group of P , then G P = P , G s a = s a and [ B ( P ) , Q ( G ) ] = 0 , ( G in little group of P ). (18)This means we can use the eigenvectors of Q ( G ) to block diagonalize B ( P ) . The block-diagonal basis can be expressed using | Λ λnJ LSa (cid:105) = (cid:88) m J c J ( − L ; Λ λnm J | J m J LSa (cid:105) (19)for little group irrep Λ , irrep row λ , and occurrence index n . The transformation coefficientsdepend on J and ( − L , but not on S, a . Essentially, the transformation replaces m J by (Λ , λ, n ) . Group theoretical projections with Gram-Schmidt are used to obtain the basisexpansion coefficients. In this block-diagonal basis, the box matrix has the form (cid:104) Λ (cid:48) λ (cid:48) n (cid:48) J (cid:48) L (cid:48) S (cid:48) a (cid:48) | B ( P ) | Λ λnJ LSa (cid:105) = δ Λ (cid:48) Λ δ λ (cid:48) λ δ S (cid:48) S δ a (cid:48) a B ( P Λ B Sa ) J (cid:48) L (cid:48) n (cid:48) ; JLn ( E cm ) , (20)and the (cid:101) K -matrix for ( − L + L (cid:48) = 1 has the form (cid:104) Λ (cid:48) λ (cid:48) n (cid:48) J (cid:48) L (cid:48) S (cid:48) a (cid:48) | (cid:101) K | Λ λnJ LSa (cid:105) = δ Λ (cid:48) Λ δ λ (cid:48) λ δ n (cid:48) n δ J (cid:48) J K ( J ) L (cid:48) S (cid:48) a (cid:48) ; LSa ( E cm ) . (21) Λ is the irrep for the K -matrix, but we need Λ B for the box matrix. When η P a η P a = 1 , then Λ B = Λ , but they differ slightly when η P a η P a = − .Using a large set of single- and two-hadron operators, as described in Ref. [2], for severaldifferent total momenta, we have evaluated a large number of energies in the isodoublet I = × anisotropic lattice with m π ∼ MeV. Each irrep is located in one col-umn, where the energy ratios are shown in the upper panel with a vertical thickness showing thestatistical error. The solid horizontal lines should the two-hadron noninteracting energies, and thegray dashed lines show relevant thresholds. The corresponding columns in the lower panel indicateoverlaps of each interpolating operator onto the finite-volume Hamiltonian eigenstates.15able 1: Best-fit results for various parameters related to the K ∗ (892) calculations.Fit s -wave par. m K ∗ /m π g K ∗ Kπ m π a χ / d . o . f . (1a,1b) LIN . . − . ( . , –)2 LIN . . − . . QUAD . . − . . ERE . . − . . BW . . − . . BW . . − . . strange S = 1 mesonic sector [3]. We used an anisotropic × lattice with a pion mass m π ∼ MeV. The determinations of these energies are possible since we use the stochasticLapH method [4] to estimate all quark propagation. The center-of-momentum energies overthe pion mass are shown in Fig. 1. Horizontal lines indicates the non-interacting two-hadronenergies, and the dashed lines show the Kπ , Kππ , and Kη thresholds. Operator overlapsare shown in the lower panel of the figure.To extract the resonance properties of the K ∗ (892) , we included the L = 0 , , partialwaves. The fit forms used for the P and D partial waves were ( (cid:101) K − ) = 6 πE cm g K ∗ ππ m π (cid:18) m K ∗ m π − E m π (cid:19) , ( (cid:101) K − ) = − m π a , (22)and for the S -wave, several different forms were tried: ( (cid:101) K − ) lin00 = a l + b l E cm , (23) ( (cid:101) K − ) quad00 = a q + b q E , (24) ( ˜ K − ) ERE = − m π a + m π r q m π , (25) ( ˜ K − ) BW = (cid:32) m K ∗ m π − E m π (cid:33) πm π E cm g K ∗ ππ m K ∗ . (26)A summary of our fit results is presented in Table 1. Our determinations of the partial-wave scattering phase shifts are shown in Fig. 2. We found that the qq operators in the A g (0) channel overlap many of the eigenvectors in this channel. Better energy resolution isneeded for a determination of the K ∗ (800) parameters, which will be done in the future, butan NLO effective range parametrization finds m R /m π = 4 . − . i , consistentwith a Breit-Wigner fit. In Ref. [3], our results are compared to the few other recent latticeresults [5–7] that are available. See also Ref. [8].We have also recently determined the decay width of the ρ -meson, including L = 1 , , partial waves [9], as well as the ∆ baryon [10].16 . . . − − E ∗ sub m π (cid:0) q m π (cid:1) c o t δ . . . δ / ◦ . . . E ∗ sub m π (cid:0) q m π (cid:1) c o t δ . . . − δ / ◦ Figure 2: (Left) Our determination of the P -wave I = Kπ scattering phase shift δ and q cot δ using a × anisotropic lattice with m π ∼ MeV. (Right) Our calculations of the S -wavescattering phase shift δ (quadratic fit) and q cm cot δ . Results are from Ref. [3].2. Acknowledgments
This work was done in collaboration with John Bulava (U. Southern Denmark), Ruairi Brett(CMU), Daniel Darvish (CMU), Jake Fallica (U. Kentucky), Andrew Hanlon (U. Mainz),Ben Hoerz (U. Mainz), and Christian W Andersen (U. Southern Denmark). I thank theorganizers, especially Igor Strakovsky, and JLab for the opportunity to participate in thisWorkshop. This work was supported by the U.S. National Science Foundation under awardPHY–1613449. Computing resources were provided by the Extreme Science and Engineer-ing Discovery Environment (XSEDE) under grant number TG-MCA07S017. XSEDE issupported by National Science Foundation grant number ACI-1548562.
References [1] C. Morningstar, J. Bulava, B. Singha, R. Brett, J. Fallica, A. Hanlon, and B. Hörz, Nucl. Phys.B , 477 (2017).[2] C. Morningstar, J. Bulava, B. Fahy, J. Foley, Y.C. Jhang, K.J. Juge, D. Lenkner, andC.H. Wong, Phys. Rev. D , 014511 (2013).[3] R. Brett, J. Bulava, J. Fallica, A. Hanlon, B. Hörz, and C. Morningstar, arXiv:1802.03100[hep-lat] (2018). 174] C. Morningstar, J. Bulava, J. Foley, K. J. Juge, D. Lenkner, M. Peardon, and C. H. Wong, Phys.Rev. D , 114505 (2011).[5] S. Prelovsek, L. Leskovec, C. B. Lang, and D. Mohler, Phys. Rev. D88 , 054508 (2013).[6] D. J. Wilson, J. J. Dudek, R. G. Edwards, and C. E. Thomas, Phys. Rev.
D91 , 054008 (2015).[7] G. S. Bali et al. , Phys. Rev.
D93 , 054509 (2016).[8] M. Döring and Ulf-G. Meißner, JHEP , 009 (2012).[9] J. Bulava, B. Fahy, B. Hörz, K. J. Juge, C. Morningstar, and C. H. Wong, Nucl. Phys. B ,842 (2016).[10] C. W. Andersen, J. Bulava, B. Hörz, and C. Morningstar, Phys. Rev. D , 014506 (2018).18 .3 K- π Scattering with K L Beam Facility
Marouen Baalouch
Department of PhysicsOld Dominion UniversityNorfolk, VA 23529, U.S.A. &Thomas Jefferson National Accelerator FacilityNewport News, VA 23606, U.S.A.
Abstract
In this talk, I discuss the importance of the Kπ scattering amplitude analysis, its impacton other physics studies and the possible related elements that need to be improved. Finally Idiscuss the feasibility of performing a Kπ amplitude scattering analysis within KLF. Introduction
The K L Beam Facility can offer a good opportunity to study the kaon-pion interaction exper-imentally, by producing the final state Kπ using the scattering of a neutral kaon off proton orneutron as K L N → [ Kπ ] N (cid:48) . The analysis of the kaon-pion interaction experimentally hasseveral implication on the imperfect phenomenological studies, as the test of the Chiral Per-turbation Theory, Strange Meson Spectroscopy and Physics beyond Standard Model. Thesephenomenological studies require more data to improve the precision on the observable ofinterest. So far, the main experimental data used to study kaon-pion scattering at low energycomes from kaon beam experiments at SLAC in the 1970s and 1980s.2. Chiral Perturbation Theory
In Quantum Chromodynamics (QCD), the strong interaction coupling constant increaseswith decreasing energy. This means that the coupling becomes large at low energies, andone can no longer rely on perturbation theory. Few phenomenological approaches can beused at this energy level, as Lattice QCD or the Chiral Perturbation Theory (ChPT) [1].The purpose of the ChPT is to use an effective Lagrangian where the mesons π , η , and K ,called also Goldstone Bosons, are the fundamental degrees of freedom. The ChPT studieson on the ππ scattering amplitude shows a good agreement with the experimental studies( e.g. , see reference [2]). However, this theory is less successful with the Kπ scatteringamplitude [3–9], and so far no accurate experimental data is available at low energy.3. Strange Meson Spectroscopy
Hadron Spectroscopy plays an important role to understand QCD in the non perturbativedomain by performing a quantitative understanding of quark and gluon confinement, andvalidate Lattice QCD prediction. In the last years, an important number of resonances havebeen identified, especially resonances with heavy flavored quark. However, the sector ofstrange baryons and mesons was not significantly improved and several estimated states bylattice QCD and quark model still not yet observed. Moreover, the identification of thescalar strange light mesons, as κ and K ∗ (1430) , still a long-standing puzzle because of theirlarge decay width that causes an overlap between resonances at low Lorentz-invariant mass.19he indications on the presence of κ resonance have been reported based on the data ofthe E791 [10] and BES [11] Collaborations and several phenomenological studies [12–14]have been made to measure the pole of κ resonance. However, the results from Roy-Steinerdispersive representation [12] not in good agreement with low energy experimental data, andthe confirmation of this pole in the amplitude for elastic Kπ scattering requires more dataat low energy. The K ∗ (1430) is the second scalar strange resonance which is also not wellunderstood. And recently the Kπ S -wave amplitude extracted from η c → KKπ [15] foundto be very different with respect to the amplitude measured by LASS and E791. Fig 1, takenfrom the reference [15], shows the comparison of the amplitude extracted from η c → KKπ , Kp → Kπn and D → KKπ .The light strange scalar mesons can be produced in KN scattering, and more data from thesetype of reactions will certainly improve the understanding of the non well identified strangeresonances.Figure 1: Figure taken from ref [15]: The I=1/2 Kπ S-wave amplitude measurements from η c → KKπ compared to the (a) LASS and (b) E791 results: the corresponding I=1/2 Kπ S-wave phasemeasurements compared to the (c) LASS and (d) E791 measurements. Black dots indicate theresults from the present analysis; square (red) points indicate the LASS or E791 results. TheLASS data are plotted in the region having only one solution.4. Kπ Scattering Amplitude and Physics Beyond Standard Model
The determination of the CKM matrix elements V us is mainly performed using τ or kaondecays. As an example, the V us matrix element is accessible from the K l decays using thebraching ratio function Γ( K → πlν ) ∝ N | V us | | f Kπ + (0) | I lK , (1)where I K = (cid:90) d t m K λ / F ( (cid:101) f + ( t ) , (cid:101) f ( t )) . (2)20n this function f Kπ and f Kπ + represent the form factors of the strangeness changing scalarand vector, respectively. These form factors in the low energy region, can be obtained fromLattice QCD or from the study of the Kπ scattering using dispersion relation analysis [16].The precision on V us depends strongly on the precision of these strangeness changing formfactors. And by improving the precision of V us one can probe the physics beyond the stan-dard model indirectly thanks to the unitarity of the CKM matrix | V ud | + | V us | + | V ub | = 1 . (3)Therefore, any shift from unitarity is a sign of physics beyond Standard Model. Fig 2 showsthe fit to the different CKM elements involved in the unitarity equation.Figure 2: Figure taken from Ref. [17]: Results of fits to | V us | , | V us | , and | V ud /V us | .5. Kaon-Production and GlueX Detector
The hadroproduction of the Kπ system has been intensively studied with charged Kaonbeam [18–22]. However, few studies have been made using a neutral kaon beam. The pro-duction mechanism of the Kπ system with charged kaon beam is proportional to the mech-anism with neutral kaon, the main difference related to the Clebsch-Gordan coefficients. InLASS analysis [18], the Kπ production mechanism is parametrized using a model consistingof exchange degenerate Regge poles together with non-evasive “cut" contributions. Theseparameterization was extrapolated to the neutral kaon beam and used in the simulation of K L p → [ Kπ ] p in KLF where the reconstruction of the events is made by GlueX spectrom-eter. The GlueX spectrometer is built in Hall D at JLab and using photon beam scatteringoff proton to provide critical data needed to address one of the outstanding and fundamen-tal challenges in physics the quantitative understanding of the confinement of quarks andgluons in QCD. The GlueX detector is azimuthally symmetric and nearly hermetic for bothcharged particles and photons, which make it a relevant detector for studying Kπ scatteringamplitude. Fig 3 shows the kinematics region that can be covered by the detector using the21igure 3: The simulated events of the reaction K L p → [ Kπ ] p projected on the plane of theproduction angle (in the Lab frame) θ versus the magnitude of the momentum. The dashed regionrepresents the region where the GlueX detector performance of Particle Identification are low: onthe top left for the proton, on the top right for the kaon and on the bottom for the pion.simulation of K L p → [ Kπ ] p and LASS parameterization. More details about the detectorperformance can be found in reference [23].6. Acknowledgments
My research is supported by Old Dominion University, Old Dominion University ResearchFoundation and Jefferson Laboratory.
References [1] G. Ecker, Prog. Part. Nucl. Phys. (1995) 1.[2] B. Ananthanarayan, G. Colangelo, J. Gasser, and H. Leutwyler, Phys. Rept. (2001) 207.[3] J.P. Ader, C. Meyers, and B. Bonnier Phys. Lett. B (1973) 403.[4] C. B. Lang, Nuovo Cim. A (1977) 73.[5] N. Johannesson and G. Nilsson, Nuovo Cim. A (1978) 376.[6] P. Buttiker, S. Descotes-Genon, and B. Moussallam, talk given at QCD-03 conference, 2-9July 2003, Montpellier; hep-ph/0310045.[7] V. Bernard, N. Kaiser, and Ulf-G. Meißner, Phys. Rev. D (1991) 2757.[8] V. Bernard, N. Kaiser, and Ulf-G. Meißner, Nucl. Phys. B (1991) 129[9] V. Bernard, N. Kaiser, and Ulf-G. Meißner, Nucl. Phys. B (1991) 283.[10] E. M. Aitala et al. (E791 Collaboration), Phys. Rev. Lett. (2002) 121801.[11] J. Z. Bai et al. (BES Collaboration), hep-ex/0304001.[12] S. Descotes-Genon and B. Moussallam, Eur. Phys. J. C (2006) 553.2213] H. Q. Zheng et al. Nucl. Phys. A , 235 (2004).[14] J. R. Pelaez and A. Rodas, Phys. Rev. D , 074025 (2016).[15] A. Palano and M. R. Pennington, arXiv:1701.04881,[16] V. Bernard, M. Oertel, E. Passemar, and J. Stern, Phys. Rev. D (2009) 034034.[17] M. Antonelli et. al. , Eur. Phys. J. C (2010) 399.[18] D. Aston et al. , Nucl. Phys. B , 493 (1988).[19] F. Schweingruber, M. Derrick, T. Fields, D. Griffiths, L. G. Hyman, R.J. Jabbur, J. Lokan, R.Ammar, R.E.P. Davis, W. Kropac, and J. Mott, Phys. Rev. (1968) 1317.[20] G. Ranft et al. [Birmingham-Glasgow-London (I.C.)-Mtinchen-Oxford-Rutherford Labora-tory Collaboration], Nuovo Cimento (1968) 522.[21] J. H. Friedman and R. R. Ross, Phys. Rev. Lett. (1966) 485.[22] Birmingham-Glasgow-Oxford Collaboration, private communication; CERN Topical Conf.on high-energy collisions of hadrons, Vol. II , p. 121 (1968).[23] H. Al Ghoul et al. [GlueX Collaboration], AIP Conf. Proc. , 020001 (2016).23 .4 Dalitz Plot Analysis of Three-body Charmonium Decays at BaBar Antimo Palano (on behalf of the BaBar Collaboration)
I.N.F.N. and University of BariBari 70125, Italy
Abstract
We present results on a Dalitz plot analysis of η c and J/ψ decays to three-body. In particu-lar, we report the first observation of the decay K ∗ (1430) → Kη in the η c decay to K + K − η .We also report on a measurement of the I=1/2 Kπ S -wave through a model independent partialwave analysis of η c decays to K S K ± π ∓ and K + K − π . The η c resonance is produced in two-photon interactions. We perform the first Dalitz plot analysis of the J/ψ decay to K S K ± π ∓ produced in the initial state radiation process. Introduction
Charmonium decays can be used to obtain new information on light meson spectroscopy.In e + e − interactions, samples of charmonium resonances can be obtained using differentprocesses. • In two-photon interactions we select events in which the e + and e − beam particlesare scattered at small angles and remain undetected. Only resonances with J P C =0 ± + , ± + , ++ , ± + .... can be produced. • In the Initial State Radiation (ISR) process, we reconstruct events having a (mostly un-detected) fast forward γ ISR and, in this case, only J P C = 1 −− states can be produced.2. Selection of Two-Photon Production of η c → K + K − η , η c → K + K − π , and η c → K S K ± π ∓ We study the reactions γγ → K + K − η , γγ → K + K − π , γγ → K S K ± π ∓ ,where η → γγ and η → π + π − π [1, 2].Two-photon candidates are reconstructed from the sample of events having the exact numberof charged tracks for that η c decay mode. Since two-photon events balance the transversemomentum, we require p T , the transverse momentum of the system with respect to the beamaxis, to be p T < . GeV /c for η c → K + K − η/π and p T < . GeV /c for η c → K S K ± π ∓ . We also define M rec ≡ ( p e + e − − p rec ) , where p e + e − is the four-momentumof the initial state and p rec is the four-momentum of the three-hadrons system and removeISR events requiring M rec > GeV /c . Figure 1 shows the K + K − η , K + K − π , and K S K ± π ∓ mass spectra where signals of η c can be observed.Selecting events in the η c mass region, the Dalitz plots for the three η c decay modes areshown in Fig. 2. The η c → K ¯ Kπ Dalitz plots are dominated by the presence of horizontaland vertical uniform bands at the position of the K ∗ (1430) resonance. Charge conjugation is implied through all this work. K + K + η , (b) K + K − π , and (c) K S K ± π ∓ mass spectra. The superimposed curvesare from the fit results. In (c) the shaded area evidences definition of the signal and sidebandsregions.Figure 2: (Left) K + K + η , (Center) K + K − π , and (Right) K S K ± π ∓ Dalitz plots. The arrowsindicate the positions of the K ∗ (1430) resonance.25igure 3: η c → K + K − η Dalitz plot analysis. (a) K + K − , and (b) K ± η squared mass projections.Shaded is the background contribution.The η c signal regions contain 1161 events with (76.1 ± η c → K + K − η , 6494events with (55.2 ± η c → K + K − π , and 12849 events with (64.3 ± η c → K S K ± π ∓ . The backgrounds below the η c signals are estimated from thesidebands. We observe asymmetric K ∗ ’s in the background to the η c → K S K ± π ∓ final statedue to interference between I=1 and I=0 contributions.3. Dalitz Plot Analysis of η c → K + K − η and η c → K + K − π We first perform unbinned maximum likelihood fits using the Isobar model [3]. Figure 3shows the η c → K + K − η Dalitz plot projections.The analysis of the η c → K + K − η decay requires significant contributions from f (1500) η (23 . ± . ± . % and K ∗ (1430) + K − (16 . ± . ± . %, where K ∗ (1430) + → K + η :a first observation of this decay mode. It is found that the η c three-body hadronic decaysproceed almost entirely through the intermediate production of scalar meson resonances.A similar analysis performed on the η c → K + K − π allows to obtain the correspondingcontribution from K ∗ (1430) + K − to be (33 . ± . ± . %, where K ∗ (1430) + → K + π .Combining the above information with the measurement of the η c relative branching fraction BR ( η c → K + K − η ) BR ( η c → K + K − π ) = 0 . ± . ± . , (1)we obtain BR ( K ∗ (1430) → ηK ) BR ( K ∗ (1430) → πK ) = 0 . ± . +0 . − . . (2)We perform a Likelihood scan and obtain a measurement of the K ∗ (1430) parameters m ( K ∗ (1430)) = 1438 ± ± M eV /c , Γ( K ∗ (1430)) = 210 ± ± M eV. (3)26igure 4: (Top) η c → K S K ± π ∓ and (Bottom) η c → K + K − π Dalitz plots projections. The super-imposed curves are from the fit results. Shaded is contribution from the interpolated background.4.
Model Independent Partial Wave Analysis of η c → K + K − π and η c → K S K ± π ∓ We perform a Model Independent Partial Wave Analysis (MIPWA) [4] of η c → K + K − π and η c → K S K ± π ∓ . In the MIPWA the Kπ mass spectrum is divided into 30 equally spacedmass intervals 60 MeV/c wide and for each bin we add to the fit two new free parameters,the amplitude and the phase of the Kπ S -wave (constant inside the bin).We also fix the amplitude to 1.0 and its phase to π/ in an arbitrary interval of the massspectrum (bin 11 which corresponds to a mass of 1.45 GeV/c ). The number of additionalfree parameters is therefore 58. Due to isospin conservation in the decays, amplitudes aresymmetrized with respect to the two Kπ decay modes. The K ∗ (1420) , a (980) , a (1400) , a (1310) , ... contributions are modeled as relativistic Breit-Wigner functions multiplied bythe corresponding angular functions. Backgrounds are fitted separately and interpolated intothe η c signal regions. The fits improves when an additional high mass a (1950) → K ¯ K , I=1resonance, is included with free parameters in both η c decay modes. The weighted averageof the two measurement is: m ( a (1950)) = 1931 ± ± MeV/c , Γ( a (1950)) = 271 ± ± MeV. The statistical significances for the a (1950) effect (including systematics) are . σ for η c → K S K ± π ∓ and . σ for η c → K + K − π .The Dalitz plot projections with fit results for η c → K S K ± π ∓ and η c → K + K − π areshown in Fig. 4. We observe a good description of the data.We note that the K ∗ (892) contributions arise entirely from background. The fitted fractionsand phases are given in Table 1. Both η c → K ¯ Kπ decay modes are dominated by the27 Kπ S -wave) ¯ K amplitude, with significant interference effects.Table 1: Results from the η c → K S K ± π ∓ and η c → K + K − π MIPWA. Phases are determinedrelative to the ( Kπ S -wave) ¯ K amplitude which is fixed to π/ at 1.45 GeV/c . η c → K S K ± π ∓ η c → K + K − π Amplitude Fraction (%) Phase (rad) Fraction (%) Phase (rad) ( Kπ S -wave) ¯ K ± ± ± ± a (980) π ± ± ± ± ± ± a (1450) π ± ± ± ± ± ± ± ± a (1950) π ± ± − ± ± ± ± − ± ± a (1320) π ± ± ± ± ± ± ± ± K ∗ (1430) ¯ K ± ± ± ± ± ± ± ± ± ± ± ± χ /N cells We use as figure of merit describing the fit quality the 2-D χ /N cells on the Dalitz plot andobtain a good description of the data with χ /N cells = 1 . and χ /N cells = 1 . for the two η c decay modes.In comparison, the isobar model gives a worse description of the data, with χ /N cells =457 /
254 = 1 . and χ /N cells = 383 /
233 = 1 . , respectively for the two η c decay modes.The resulting Kπ S -wave amplitude and phase for the two η c decay modes is shown inFig. 5. We observe a clear K ∗ (1430) resonance signal with the corresponding expectedphase motion. At high mass we observe the presence of the broad K ∗ (1950) contributionwith good agreement between the two η c decay modes.Comparing with LASS [5] and E791 [4] experiments we note that phases before the Kη (cid:48) threshold are similar, as expected from Watson theorem [6] but amplitudes are very different.A preliminary K-matrix fit which include K − π − → K − π − S -wave data [7], LASS data and η c decays has been performed [8], obtaining a description of the data in terms of three-poles: Pole 1 E P = 659 − i
302 MeV on Sheet II , (4) Pole 2 E P = 1409 − i
128 MeV on Sheet III , (5) Pole 3 E P = 1768 − i
107 MeV on Sheet III . (6)Pole 1 is identified with the κ , the pole position of which was found to be at [(658 ± − i (278 ± MeV, in the dispersive analysis of Ref. [9]. Pole 2 is identified with K ∗ (1430) ,to be compared with [(1438 ± ± − i (105 ± ± MeV using the Breit-Wignerform (Eq. (3)). Pole 3 may be identified with the K ∗ (1950) with a pole mass closer to thatof the reanalysis of the LASS data from Ref. [10] with a pole at E = (1820 ± − i (125 ± MeV. For pole 2, the K ∗ (1430) , a ratio of Kη / Kπ decay rate of 0.05 is obtained,consistent with that reported in the present analysis (Eq. (2)).28igure 5: The I = 1 / Kπ S -wave amplitude (a) and phase (b) from η c → K S K ± π ∓ (solid(black) points) and η c → K + K − π (open (red) points); only statistical uncertainties are shown.The dotted lines indicate the Kη and Kη (cid:48) thresholds.5. Dalitz Plot Analysis of
J/ψ → K S K − π ∓ We study the following reaction: e + e − → γ ISR K S K ± π ∓ where γ ISR indicate the ISR photon [11]. Candidate events for this reaction are selected fromthe sample of events having exactly four charged tracks including the K S candidate. Wecompute M rec ≡ ( p e − + p e + − p K S − p K − p π ) , which peaks near zero for ISR events.We select events in the ISR region by requiring | M rec | < . GeV /c and obtain the K S K ± π ∓ mass spectrum shown in Fig. 6 where a clean J/ψ signal can be observed. Wefit the K S K ± π ∓ mass spectrum using the Monte Carlo resolution functions described by aCrystal Ball+Gaussian function and obtain 3694 ±
64 events with 93.1 ± J/ψ signal region and Fig. 7(Right) shows theDalitz plot projections. We perform the Dalitz plot analysis of
J ψ → K S K | pmπ M P usingthe isobar model and express the amplitudes in terms of Zemach tensors [12,13]. We observethe following features: • The decay is dominated by the K ∗ (892) ¯ K , K ∗ (1430) ¯ K , and ρ (1450) ± π ∓ amplitudeswith a smaller contribution from the K ∗ (1410) ¯ K amplitude. • We obtain a significant improvement of the description of the data by leaving free the K ∗ (892) mass and width parameters and obtain m ( K ∗ (892) + ) = 895 . ± . M eV /c , Γ( K ∗ (892) + ) = 43 . ± . M eV,m ( K ∗ (892) ) = 898 . ± . M eV /c , Γ( K ∗ (892) ) = 52 . ± . M eV. (7)The measured parameters for the charged K ∗ (892) + are in good agreement with those mea-sured in τ lepton decays [14]. 29igure 6: K S K ± π ∓ mass spectrum from ISR events.Figure 7: (Left) J ψ → K S K ± π ∓ Dalitz plot. (Right) Dalitz plot projections with fit results for
J ψ → K S K ± π ∓ . Shaded is the background interpolated from J ψ sidebands.30.
Acknowledgments
This work was supported (in part) by the U.S. Department of Energy, Office of Science,Office of Nuclear Physics under contract DE–AC05–06OR23177.
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Technical University Munich85748 Garching, Germany
Abstract
COMPASS is a multi-purpose fixed-target experiment at the CERN Super Proton Syn-chrotron aimed at studying the structure and spectrum of hadrons. One of the main goals of theexperiment is the study of the light-meson spectrum. In diffractive reactions with a 190 GeVnegative secondary hadron beam consisting mainly of pions and kaons, a rich spectrum ofisovector and strange mesons is produced. The resonances decay typically into multi-bodyfinal states and are extracted from the data using partial-wave analysis techniques.We present selected results of a partial-wave analysis of the K − π − π + final state based ona data set of diffractive dissociation of a 190 GeV K − beam impinging on a proton target. Thisreaction allows us to study the spectrum of strange mesons up to masses of about 2.5 GeV. Wealso discuss a possible future high-intensity kaon-beam experiment at CERN. Introduction
The excitation spectrum of light mesons is studied since many decades but is still not quan-titatively understood. For higher excited meson states experimental information is oftenscarce or non-existent. This is in particular true for the strange-meson sector as illustratedby Fig. 1. The PDG [1] lists only 25 kaon states below 3.1 GeV: 12 states that are consideredwell-known and established and in addition 13 states that need confirmation. Many higherexcited states that are predicted by quark-model calculation (Fig. 1 shows as an example theone from Ref. [2]) have not yet been found by experiments. In addition, for certain combi-nations of spin J and parity P the quark model does not describe the experimental data well.This is most notably the case for the scalar kaon states with J P = 0 + , where the K ∗ (800) seems to be a supernumerous state.In the last 30 years, little progress has been made on the exploration of the kaon spectrum.Since 1990, only four kaon states were added to the PDG and only one of them to the sum-mary table. However, precise knowledge of the kaon spectrum is crucial to understand thelight-meson spectrum. In particular, the identification of supernumerous states that could berelated to new forms of matter beyond conventional quark-antiquark states—like multi-quarkstates, hybrids, or glueballs—requires the observation of complete SU(3) flavor multiplets. Thekaon spectrum also enters in analyses that search for CP violation in multi-body decays of D and B mesons, where kaon resonances appear in the subsystems of various final states.2. Diffractive Production of Kaon Resonances
A suitable reaction to produce excited kaon states is diffractive dissociation of a high-energykaon beam, as it was already measured in the past by the WA3 experiment at CERN (see,e.g., Ref. [3]) and the LASS experiment at SLAC (see, e.g., Refs. [4, 5]). In these peripheralreactions, the beam kaon scatters softly off the target particle and is thereby excited intointermediate states, which decay into the measured n -body hadronic final state.32 . K ∗ K ∗ . K K K ∗ K ∗ K K K ∗ K ∗ M a ss [ G e V / c ] − + − + − + − + − + − Figure 1: Strange-meson spectrum: Comparison of the measured masses of kaon resonances (col-ored points with boxes representing the uncertainty) with the result of a quark-model calculation [2](black lines). States that are considered well established by the PDG [1] are shown in blue, statesthat need confirmation in orange.These reactions were also measured by the COMPASS experiment using a secondary hadronbeam provided by the M2 beam line of the CERN SPS. The beam was tuned to deliver neg-atively charged hadrons of 190 GeV momentum passing through a pair of beam Cherenkovdetectors (CEDARs) for beam particle identification. The beam impinged on a 40 cm longliquid-hydrogen target with an intensity of × particles per SPS spill (10 s extractionwith a repetition time of about 45 s). At the target, the hadronic component of the beamconsisted of 96.8% π − , 2.4% K − , and 0.8% ¯ p . At the beam energy of 190 GeV, the reactionis dominated by Pomeron exchange. Elastic scattering at the target vertex was ensured bymeasuring the slow recoil proton. This leads to a minimum detectable reduced squared four-momentum transfer t (cid:48) of about 0.07 ( GeV ) . By selecting the K − component of the beamusing the CEDAR information, we have studied diffractive production of kaon resonances.Charged kaons that appear in some of the produced forward-going final states were separatedfrom pions by a ring-imaging Cherenkov detector (RICH) in the momentum range between2.5 and 50 GeV.3. Partial-Wave Analysis of K − π − π + Final State
In the 2008 and 2009 data-taking campaigns, COMPASS acquired a large data sample onthe diffractive dissociation reaction K − + p → K − π − π + + p. (1)The measurement is exclusive, i.e., all four final-state particles are measured and energy-momentum conservation constraints are applied in the event selection. In reaction (1), in-termediate kaon resonances are produced that decay into the 3-body K − π − π + final state.33 [GeV] - π + π - Energy(K160 165 170 175 180 185 190 195 200 E ve n t s × p r e li m i n a r y COMPASS 2008 negative hadron beam recoil p - π + π - K → p - K not acceptance corrected ] )[GeV/c - π + π - M (K0.5 1 1.5 2 2.5 ] E v e n t s / [ M e V / c × p r e li m i n a r y COMPASS 2008 negative hadron beam recoil p - π + π - K → p - K not acceptance corrected (1270) K (1400) K (1770) K ] )[GeV/c + π - M (K0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ] E v e n t s / [ M e V / c × p r e li m i n a r y COMPASS 2008 negative hadron beam recoil p - π + π - K → p - K not acceptance corrected K*(892) (1430) K* ] )[GeV/c - π + π M (0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ] E v e n t s / [ M e V / c × p r e li m i n a r y COMPASS 2008 negative hadron beam recoil p - π + π - K → p - K not acceptance corrected (770) ρ (980) f (1270) f Figure 2: Kinematic distributions [7]: (Top left) distribution of the energy sum of the forward-going particles in reaction (1), (Top right) K − π − π + invariant mass distribution, (Bottom left) K − π + invariant mass distribution, and (Bottom right) π − π + invariant mass distribution.A first analysis of this reaction was performed based on a data sample that corresponds toa fraction of the available data and consists of about 270 000 events with K − π − π + massbelow 2.5 GeV and in the range . < t (cid:48) < . GeV ) [6, 7]. This data sample is similarin size to the one of the WA3 experiment. Fig. 2 shows selected kinematic distributions.The distribution of the energy sum of the forward-going particles peaks at the nominal beamenergy. The non-exclusive background below the peak is small. The invariant mass distribu-tion of the K − π − π + system and that of the K − π + and π − π + subsystems exhibit peaks thatcorrespond to known resonances. All three distribution are similar to the ones obtained byWA3 [3].As a first step towards a description of the measured K − π − π + mass spectrum in terms ofkaon resonances, we performed a partial-wave analysis (PWA) using a model similar tothe one used by the ACCMOR collaboration in their analysis of the WA3 data [3]. ThePWA formalism is based on the isobar model and is described in detail in Ref. [8]. ThePWA model takes into account three K − π + isobars [ K ∗ (800) , K ∗ (892) , and K ∗ (1430) ]and three π − π + isobars [ f (500) , ρ (770) , and f (1270) ]. Based on these isobars, a waveset is constructed that consists of 19 waves plus an incoherent isotropic wave, which absorbs34 ) [GeV/c + π - π - Mass of (K1 1.2 1.4 1.6 1.8 2 ] I n t e n s it y / . [ G e V / c × preliminary COMPASS 2008p + π - π - K → p - K /c < t’ < 0.7 GeV /c − π K*(892)[01] + + ] ) [GeV/c + π - π - Mass of (K1 1.2 1.4 1.6 1.8 2 ] I n t e n s it y / . [ G e V / c × preliminary COMPASS 2008p + π - π - K → p - K /c < t’ < 0.7 GeV /c − π K*(892)[21] + + ] ) [GeV/c + π - π - Mass of (K1 1.2 1.4 1.6 1.8 2 2.2 2.4 ] I n t e n s it y / . [ G e V / c × preliminary COMPASS 2008p + π - π - K → p - K /c < t’ < 0.7 GeV /c − π (1430)[02] K + − Figure 3: Intensities of selected waves as a function of the K − π − π + mass [7]: (Top left) + + K ∗ (892) πS wave, (Top right) + + K ∗ (892) πD wave, and (Bottom) − + K ∗ (1430) πS wave.intensity from events with uncorrelated K − π − π + , e.g., non-exclusive background. A partial-wave amplitude is completely defined by the spin J , parity P , spin projection M ε , and thedecay path of the intermediate state. The spin projection is expressed in the reflectivitybasis [9], where M ≥ and ε = ± is the naturality of the exchange particle. Since thereaction is dominated by Pomeron exchange, all waves have ε = +1 . We use the partial-wave notation J P M ε [isobar] πL , where L is the orbital angular momentum between theisobar and the third final-state particle.Fig. 3 shows the intensity distributions of selected waves. The + + K ∗ (892) πS wave in-tensity exhibits two clear peaks at the positions of the K (1270) and K (1400) . We alsosee a clear peak of the K ∗ (1430) in the + + K ∗ (892) πD wave intensity. However, thereis no clear signal from the K ∗ (1980) . The − + K ∗ (1430) πS wave shows a broad bump inthe intensity distribution peaking slightly below 1.8 GeV that could be due to the K (1770) and/or K (1820) . But also contributions from K (1580) and/or K (2250) are not excluded.The analysis is currently work in progress. With an improved beam particle identificationand event selection the full data sample consists of about 800 000 exclusive events, makingit the world’s larges data set of this kind. Also the PWA model will be improved by usingmore realistic isobar parametrizations and parameters and by including the f (980) as an35igure 4: Principle of an RF-separated beam [15]: A momentum-selected beam passes throughtwo RF-cavities that act as a time-of-flight selector. The first RF-cavity deflects the beam. Thisdeflection is cancelled or enhanced by the second RF-cavity, depending on the particle type. Henceparticles can be selected in the plane transverse to the beam by putting absorbers or collimators.additional π − π + isobar. In order to extract kaon resonances and their parameters, we willalso perform a resonance-model fit similar to the one of the π − π − π + final state in Ref. [10].4. Possible Future Measurements with Kaon Beam
The COMPASS collaboration has submitted a proposal for a future fixed-target experimentwithin in the framework of CERN’s “Physics beyond Colliders” initiative. Among otherthings, we propose to perform a high-precision measurement of the kaon spectrum usinga high-energy kaon beam similar to the COMPASS measurements in 2008 and 2009. Thegoal of this experiment would be to acquire a high-precision data set that is at least 10 timeslarger than that of COMPASS. Such a large data set would allow us not only to searchfor small signals but also to apply the novel analysis techniques that we developed for theanalysis of the COMPASS π − π − π + data sample, which consists of × events [8,10,11].In particular, we could study in detail the amplitude of the scalar K − π + subsystem with J P = 0 + in the the K − π − π + final state as a function of the K − π + mass, the K − π − π + mass, and the quantum numbers of the K − π − π + system, similar to the analysis of the scalar π − π + subsystem in the π − π − π + final state in Ref. [8]. With these data, one could learnmore about the scalar kaon states. The PWA could also be performed in bins of the reducedsquared four-momentum t (cid:48) in order to extract the t (cid:48) dependence of the resonant and non-resonant partial-wave components like it was done for the π − π − π + final state in Ref. [10].This gives information about the production processes.To obtain such a high-precision data set for kaon spectroscopy, the rate of beam kaons onthe target must be increased with respect to the COMPASS measurements in 2008 and 2009.This can be achieved by increasing the kaon fraction in the beam using RF-separation tech-niques [12, 13] similar to the ones that have already been used in the past at CERN [14] (seeFig. 4). First preliminary estimates for the M2 beam line at CERN show that a high-energykaon beam with a kaon rate of about . × s − seems to be feasible [15, 16]. This wouldcorrespond to about 10 to × K − π − π + events within a year of running. However,more detailed feasibility studies are still needed.36. Acknowledgments
This work was supported by the BMBF, the Maier-Leibnitz-Laboratorium (MLL), the DFGCluster of Excellence Exc153 “Origin and Structure of the Universe,” and the computingfacilities of the Computational Center for Particle and Astrophysics (C2PAP).
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Bilas Pal (for the Belle Collaboration)
Brookhaven National LaboratoryUpton, NY 11973, U.S.A. &University of CincinnatiCincinnati, OH, U.S.A
Abstract
We report the recent results related to π − K interactions based on the data collected by theBelle experiment at the KEKB collider. This includes the branching fraction and CP asym-metry measurements of B + → K + K − π + decay, search for the Λ + c → φpπ , Λ + c → P + s π decays, branching fraction measurement of Λ + c → K − π + pπ , first observation of doublyCabibbo-suppressed decay Λ + c → K + π − p , and the measurement of CKM angle φ ( γ ) with amodel-independent Dalitz plot analysis of B ± → DK ± , D → K S π + π − decay. Introduction
In this report, we present some recent results related to π − K interactions based on thedata, collected by the Belle experiment at the KEKB e + e − asymmetric-energy collider [1].(Throughout this paper charge-conjugate modes are implied.) The experiment took data atcenter-of-mass energies corresponding to several Υ( nS ) resonances; the total data samplerecorded exceeds − .The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon ver-tex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel thresholdCherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation coun-ters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals (ECL) locatedinside a super-conducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented to detect K L mesons and to identify muons(KLM). The detector is described in detail elsewhere [2, 3].2. CP Asymmetry in B + → K + K − π + Decays
In the recent years, an unidentified structure has been observed by BaBar [4] and LHCbexperiments [5, 6] in the low K + K − invariant mass spectrum of the B + → K + K − π + decays. The LHCb reported a nonzero inclusive CP asymmetry of − . ± . ± . ± . and a large unquantified local CP asymmetry in the same mass region. These resultssuggest that final-state interactions may contribute to CP violation [7,8]. In this analysis, weattempt to quantify the CP asymmetry and branching fraction as a function of the K + K − invariant mass, using
711 fb − of data, collected at Υ(4 S ) resonance [9].The signal yield is extracted by performing a two-dimensional unbinned maximum likeli-hood fit to the variables: the beam-energy constrained mass M bc and the energy difference ∆ E . The resulting branching fraction and CP asymmetry are B ( B + → K + K − π + ) = (5 . ± . ± . × − ,A CP = − . ± . ± . , CP asymmetry in the low K + K − invariant mass region, weperform the 2D fit (described above) to extract the signal yield and A CP in bins of M K + K − .The fitted results are shown in Fig. 1 and Table 1. We confirm the excess and local A CP inthe low M K + K − region, as reported by the LHCb, and quantify the differential branchingfraction in each K + K − invariant mass bin. We find a 4.8 σ evidence for a negative CP asymmetry in the region M K + K − < . GeV/ c . To understand the origin of the low-massdynamics, a full Dalitz analysis from experiments with a sizeable data set, such as LHCband Belle II, will be needed in the future.Figure 1: Differential branching fractions (left) and measured A CP (right) as a function of M K + K − . Eachpoint is obtained from a two-dimensional fit with systematic uncertainty included. Red squares with errorbars in the left figure show the expected signal distribution in a three-body phase space MC. Note that thephase space hypothesis is rescaled to the total observed K + K − π + signal yield. Table 1:
Differential branching fraction, and A CP for individual M K + K − bins. The first uncertainties arestatistical and the second systematic. M K + K − d B /dM ( × − ) A CP . ± . ± . − . ± . ± . . ± . ± . − . ± . ± . . ± . ± . − . ± . ± . . ± . ± . − . ± . ± . . ± . ± . − . ± . ± . Search for Λ + c → φpπ and Branching Fraction Measurement of Λ + c → K − π + pπ The story of exotic hadron spectroscopy begins with the discovery of the X (3872) by theBelle collaboration in 2003 [10]. Since then, many exotic XYZ states have been reportedby Belle and other experiments [11]. Recent observations of two hidden-charm pentaquarkstates P + c (4380) and P + c (4450) by the LHCb collaboration in the J/ψp invariant mass spec-trum of the Λ b → J/ψpK − process [12] raises the question of whether a hidden-strangeness39entaquark P + s , where the c ¯ c pair in P + c is replaced by an s ¯ s pair, exists [13–15]. Thestrange-flavor analogue of the P + c discovery channel is the decay Λ + c → φpπ [14, 15],shown in Fig. 2 (a). The detection of a hidden-strangeness pentaquark could be possiblethrough the φp invariant mass spectrum within this channel [see Fig. 2 (b)] if the underlyingmechanism creating the P + c states also holds for P + s , independent of the flavor [15], and onlyif the mass of P + s is less than M Λ + c − M π . In an analogous s ¯ s process of φ photoproduction ( γp → φp ) , a forward-angle bump structure at √ s ≈ . GeV has been observed by theLEPS [16] and CLAS collaborations [17]. However, this structure appears only at the mostforward angles, which is not expected for the decay of a resonance [18].Figure 2:
Feynman diagram for the decay (a) Λ + c → φpπ and (b) Λ + c → P + s π . Previously, the decay Λ + c → φpπ has not been studied by any experiment. Here, we report asearch for this decay, using 915 fb − of data [19]. In addition, we search for the nonresonantdecay Λ + c → K + K − pπ and measure the branching fraction of the Cabibbo-favored decay Λ + c → K − π + pπ .In order to extract the signal yield, we perform a two-dimensional (2D) unbinned extendedmaximum likelihood fit to the variables m ( K + K − pπ ) and m ( K + K − ) . Projections ofthe fit result are shown in Fig. 3. From the fit, we extract . ± . signal events, . ± . nonresonant events, and . ± . combinatorial background events. Thestatistical significances are found to be 2.4 and 1.0 standard deviations for Λ + c → φpπ and nonresonant Λ + c → K + K − pπ decays, respectively. We use the well-established decay Λ + c → pK − π + [11] as the normalization channel for the branching fraction measurements. ) ) (GeV/c π p - K + m(K ) E v e n t s / ( . G e V / c (a) ) ) (GeV/c - K + m(K ) E v e n t s / ( . G e V / c −
10 110 (b) Figure 3:
Projections of the 2D fit: (a) m ( K + K − pπ ) and (b) m ( K + K − ) . The points with the error barsare the data, and the (red) dotted, (green) dashed and (brown) dot-dashed curves represent the combinatorial,signal and nonresonant candidates, respectively, and (blue) solid curves represent the total PDF. The solidcurve in (b) completely overlaps the curve for the combinatorial background. φpπ signal and K + K − pπ nonresonant decays, we set upper limits on their branching fractions at 90% confidence level(CL) using a Bayesian approach. The results are B (Λ + c → φpπ ) < . × − , B (Λ + c → K + K − pπ ) NR < . × − , which are the first limits on these branching fractions.To search for a putative P + s → φp decay, we select Λ + c → K + K − pπ candidates in which m ( K + K − ) is within 0.020 GeV/ c of the φ meson mass [11] and plot the background-subtracted m ( φp ) distribution (Fig. 4). This distribution is obtained by performing 2D fitsas discussed above in bins of m ( φp ) . The data shows no clear evidence for a P + s state. Weset an upper limit on the product branching fraction B (Λ + c → P + s π ) × B ( P + s → φp ) byfitting the distribution of Fig. 4 to the sum of a RBW function and a phase space distributiondetermined from a sample of simulated Λ + c → φpπ decays. We obtain . ± . P + s events from the fit, which gives an upper limit of B (Λ + c → P + s π ) × B ( P + s → φp ) < . × − at 90% CL. for our limit on B (Λ + c → φpπ ) . From the fit, we also obtain, M P + s = (2 . ± . GeV/ c and Γ P + s = (0 . ± . GeV, where the uncertainties are statistical only. ) p) (GeV/c φ m( ) E v e n t s / ( . G e V / c − − Figure 4:
The background-subtracted distribution of m ( φp ) in the φpπ final state. The points with errorbars are data, and the (blue) solid line shows the total PDF. The (red) dotted curve shows the fitted phasespace component (which has fluctuated negative). The high statistics decay Λ + c → K − π + pπ is used to adjust the data-MC differences in the φpπ signal and K + K − pπ nonresonant decays. For the Λ + c → K − π + pπ sample, the massdistribution is plotted in Fig. 5. We fit this distribution to obtain the signal yield. We find
242 039 ± signal candidates and
472 729 ± background candidates. We measurethe ratio of branching fractions, B (Λ + c → K − π + pπ ) B (Λ + c → K − π + p ) = (0 . ± . ± . , B (Λ + c → K − π + p ) = (6 . ± . [20], we obtain B (Λ + c → K − π + pπ ) = (4 . ± . ± . ± . , where the first uncertainty is statistical, the second is systematic, and the third reflects theuncertainty due to the branching fraction of the normalization decay mode. This is the mostprecise measurement of B (Λ + c → K − π + pπ ) to date and is consistent with the recently mea-sured value B (Λ + c → K − π + pπ ) = (4 . ± . ± . by the BESIII collaboration [21]. ) ) (GeV/c π p + π - m(K ) E v e n t s / ( . G e V / c Figure 5:
Fit to the invariant mass distribution of m ( K − π + pπ ) . The points with the error bars are the data,the (red) dotted and (green) dashed curves represent the combinatorial and signal candidates, respectively,and (blue) curve represents the total PDF. of bins) of the fit is 1.43, which indicate that the fit gives a gooddescription of the data. Observation of the Doubly Cabibbo-Suppressed Λ + c Decay
Several doubly Cabibbo-suppressed (DCS) decays of charmed mesons have been observed [11].Their measured branching ratios with respect to the corresponding Cabibbo-favored (CF) de-cays play an important role in constraining models of the decay of charmed hadrons and inthe study of flavor- SU (3) symmetry [22, 23]. On the other hand, because of the smallerproduction cross-sections for charmed baryons, DCS decays of charmed baryons have notyet been observed, and only an upper limit, B (Λ + c → pK + π − ) B (Λ + c → pK − π + ) < . at 90% CL, has beenreported by the FOCUS Collaboration [24]. Here we present the first observation of theDCS decay Λ + c → pK + π − and the measurement of its branching ratio with respect to theCF decay Λ + c → pK − π + , using
980 fb − of data [25].Figure 6 shows the invariant mass distributions of (a) pK − π + (CF) and (b) pK + π − (DCS)combinations. DCS decay events are clearly observed in M ( pK + π − ) . In order to obtainthe signal yield, a binned least- χ fit is performed. From the mass fit, we extract (1 . ± . × Λ + c → pK − π + events and ±
380 Λ + c → pK + π − events. The latterhas a peaking background from the single Cabibbo-suppressed (SCS) decay Λ + c → Λ( → pπ − ) K + , which has the same final-state topology. After subtracting the SCS contribution,we have ± ± DCS events, where the first uncertainty is statistical and the second42s the systematic due to SCS subtraction. The corresponding statistical significance is 9.4standard deviations. We measure the branching ratio, B (Λ + c → pK + π − ) B (Λ + c → pK − π + ) = (2 . ± . ± . × − , where the uncertainties are statistical and systematic, respectively. This measured branchingratio corresponds to (0 . ± .
21) tan θ c , where the uncertainty is the total, which is con-sistent within 1.5 standard deviations with the naïve expectation ( ∼ tan θ c [24]). LHCb’srecent measurement of B (Λ + c → pK + π − ) B (Λ + c → pK − π + ) = (1 . ± . ± . × − [26] is lower thanour ratio at the 2.0 σ level. Multiplying this ratio with the previously measured B (Λ + c → pK − π + ) = (6 . ± . +0 . − . )% by the Belle Collaboration [27], we obtain the the absolutebranching fraction of the DCS decay, B (Λ + c → pK + π − ) = (1 . ± . +0 . − . ) × − , where the first uncertainty is due to the total uncertainty of the branching ratio and the secondis uncertainty due to the branching fraction of the CF decay. After subtracting the contribu-tions of Λ ∗ (1520) and ∆ isobar intermediates, which contribute only to the CF decay, therevised ratio, B (Λ + c → pK + π − ) B (Λ + c → pK − π + ) = (1 . ± .
17) tan θ c is consistent with the naïve expectationwithin 1.0 standard deviation.Figure 6: Distributions of (a) M ( pK − π + ) and (b) M ( pK + π − ) and residuals of data with respect to thefitted combinatorial background. The solid curves indicate the full fit model and the dashed curves thecombinatorial background. φ Measurement with a Model-independent Dalitz Plot Analysis of B ± → DK ± , D → K S π + π − Decay
The CKM angle φ (also denoted as γ ) is one of the least constrained parameters of theCKM Unitary Triangle. Its determination is however theoretically clean due to absenceof loop contributions; φ can be determined using tree-level processes only, exploiting theinterference between b → u ¯ cs and b → c ¯ us transitions that occurs when a process involvesa neutral D meson reconstructed in a final state accessible to both D and ¯ D decays (seeFig. 7). Therefore, the angle φ provides a SM benchmark, and its precise measurement43s crucial in order to disentangle non-SM contributions to other processes, via global CKMfits. The size of the interference also depends on the ratio ( r B ) of the magnitudes of the twotree diagrams involved and δ B , the strong phase difference between them. Those hadronicparameters will be extracted from data together with the angle φ .Figure 7: Feynman diagram for B − → D K − and B − → ¯ D K − decays. The measurement are performed in three different ways: (a) by utilizing decays of D mesonsto CP eigenstates, such as π + π − , K + K − ( CP even) or K S π , φK S ( CP odd), proposedby Gronau, London, and Wyler (and called the GLW method [28, 29]) (b) by making use ofDCS decays of D mesons, e.g. , D → K + π − , proposed by Atwood, Dunietz, and Soni (andcalled the ADS method [30]) and (c) by exploiting the interference pattern in the Dalitz plotof the D decays such as D → K S π + π − , proposed by Giri, Grossman, Soffer, and Zupanc(and called the GGSZ method [31]).Using a model-dependent Dalitz plot method, Belle’s earlier measurement [32] based on adata sample of
605 fb − integrated luminosity yielded φ = (78 . +10 . − . ± . ± . ◦ and r B = 0 . +0 . − . ± . +0 . − . , where the uncertainties are statistical, systematic and Dalitzmodel dependence, respectively. Although with more data one can squeeze on the statisticalpart, the result will still remain limited by the model uncertainty.In a bid to circumvent this problem, Belle has carried out a model-independent analysis [33],using GGSZ method [31], that is further extended in a latter work [34]. The analysis is basedon the
711 fb − of data, collected at the Υ(4 S ) resonance. In contrast to the conventionalDalitz method, where the D → K S π + π − amplitudes are parameterized as a coherent sumof several quasi two-body amplitudes as well as a nonresonant term, the model-independentapproach invokes study of a binned Dalitz plot. In this approach, the expected numberof events in the i th bin of the Dalitz plan for the D mesons from B ± → DK ± is givenby where h B is the overall normalization and K i is the number of events in the i th Dalitzbin of the flavor-tagged (whether D or ¯ D ) D → K S π + π − decays, accessible via thecharge of the slow pion in D ∗± → Dπ ± . The terms c i and s i contain information about thestrong-phase difference between the symmetric Dalitz points [ m ( K S π + ) , m ( K S π − ) ] and[ m ( K S π − ) , m ( K S π + ) ]; they are the external inputs obtained from quantum correlated D ¯ D decays at the ψ (3770) resonance in CLEO [35, 36]. Finally x ± = r B cos( δ B ± φ ) and y ± = r B sin( δ B ± φ ) , where δ B is the strong-phase difference between B ± → ¯ D K ± and B ± → D K ± .We perform a combined likelihood fit to four signal selection variables in all Dalitz bins(16 bins in our case) for the B ± → DK ± signal and Cabibbo-favored B ± → Dπ ± control44able 2: Results of the x, y parameters and their statistical correlation for B ± → DK ± decays. The quoteduncertainties are statistical, systematic, and precision on c i , s i , respectively. Parameter x + +0 . ± . ± . ± . y + +0 . +0 . − . ± . ± . corr( x + , y + ) -0.315 x − − . ± . ± . ± . y − − . +0 . − . ± . ± . corr( x − , y − ) +0.059samples; the free parameters of the fit are x ± , y ± , overall normalization (see Eq. 1) andbackground fraction. Table 2 summarizes the results obtained for B ± → DK ± decays.From these results, we obtain φ = (77 . +15 . − . ± . ± . ◦ and r B = 0 . ± . ± . ± . , where the first error is statistical, the second is systematic, and the last erroris due to limited precision on c i and s i . Although φ has a mirror solution at φ + 180 ◦ , weretain the value consistent with ◦ < φ < ◦ . We report evidence for direct CP violation,the fact that φ is nonzero, at the 2.7 standard deviations level. Compared to results of themodel-dependent Dalitz method, this measurement has somewhat poorer statistical precisiondespite a larger data sample used. There are two factors responsible for lower statisticalsensitivity: 1) the statistical error for the same statistics is inversely proportional to the r B value, and the central value of r B in this analysis is smaller, and 2) the binned approachis expected to have the statistical precision that is, on average, 10–20% poorer than theunbinned one. On the positive side, however, the large model uncertainty for the model-dependent study ( . ◦ ) is now replaced by a purely statistical uncertainty due to limited sizeof the ψ (3770) data sample available at CLEO ( . ◦ ). With the use of BES-III data, thiserror will decrease to ◦ or less.The model-independent approach therefore offers an ideal avenue for Belle II and LHCb intheir pursuits of φ . We expect that the statistical error of the φ measurement using thestatistics of a
50 ab − data sample that will be available at Belle II will reach − ◦ . We alsoexpect that the experimental systematic error can be kept at the level below ◦ , since most ofits sources are limited by the statistics of the control channels.6. Acknowledgments
The author thanks the organizers of PKI2018 for excellent hospitality and for assembling anice scientific program. This work is supported by the U.S. Department of Energy.
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Decay at the B Factories
Denis Epifanov
Budker Institute of Nuclear Physics SB RASNovosibirsk, 630090 Russia
Abstract
Recent results of high-statistics studies of the τ → Kπν decays at B factories are re-viewed. We discuss precision measurements of the branching fractions of the τ − → K S π − ν τ and τ − → K − π ν τ decays, and a study of the K S π − invariant mass spectrum in the τ − → K S π − ν τ decay. Searches for CP symmetry violation in the τ − → K S π − ( ≥ π ) ν τ decaysare also briefly reviewed. We emphasize the necessity of the further studies of the τ → Kπν decays at B and Super Flavour factories. Introduction
The record statistics of τ leptons collected at the e + e − B factories [1] provide unique oppor-tunities of the precision tests of the Standard Model (SM). In the SM, τ decays due to thecharged weak interaction described by the exchange of W boson. Hence, there are two mainclasses of τ decays, leptonic and hadronic τ decays. Leptonic decays provide very clean lab-oratory to probe electroweak couplings [2], while hadronic τ decays offer unique tools forthe precision study of low energy QCD [3]. The hadronic system is produced from the QCDvacuum via decay of the W − boson into ¯ u and d quarks (Cabibbo-allowed decays) or ¯ u and s quarks (Cabibbo-suppressed decays). As a result the decay amplitude can be factorizedinto a purely leptonic part including the τ − and ν τ and a hadronic spectral function.Of particular interest are strangeness changing Cabibbo-suppressed hadronic τ decays. Thedecays τ − → ¯ K π − ν τ and τ − → K − π ν τ (or, shortly, τ → Kπν ) provide the dominantcontribution to the inclusive strange hadronic spectral function, which is used to evaluate s -quark mass and V us element of Cabibbo-Kobayashi-Maskawa (CKM) quark flavor-mixingmatrix [4]. In the τ → Kπν decays the Kπ system is produced in the clean experimentalconditions without disturbance from the final state interactions. Hence, τ → Kπν decaysprovide complementary information about K - π interaction to the experiments with kaonbeams [5, 6]. In the leptonic sector, CP symmetry violation (CPV) is strongly suppressed inthe SM ( A CPSM (cid:46) − ) leaving enough room to search for the effects of New Physics [7]. Ofparticular interest are strangeness changing Cabibbo-suppressed hadronic τ decays, in whichlarge CPV could appear from a charged scalar boson exchange in some Multi-Higgs-Doubletmodels (MHDM) [8].Recently, Belle and BaBar performed an extended study of the τ → Kπν decays andsearches for CPV in these decays [9–14].2.
Study of τ → Kπν
Decays at Belle and BaBar
The first analysis of τ − → K S π − ν τ decay at Belle was done with a
351 fb − data sample thatcontains × τ + τ − pairs [9]. So called lepton-tagged events were selected, in which Unless specified otherwise, charge conjugate decays are implied throughout the paper. + decays to leptons, τ + → (cid:96) + ν (cid:96) ¯ ν τ , (cid:96) = e, µ , while the other one decays to the signal K S π − ν τ final state. Events where both τ ’s decay to leptons were used for the normalization.In the calculation of the τ − → K S π − ν τ branching fraction the detection efficiencies for thesignal and two-lepton events were determined from Monte Carlo (MC) simulation with thecorrections from the experimental data. The obtained branching fraction: B ( τ − → K S π − ν τ ) = (4 . ± . . ) ± . . )) × − is consistent with the other measurements.Figure 1: (a) The K S π mass distribution, points are experimental data, the histogram is the K ∗ (800) − + K ∗ (892) − + K ∗ (1410) − model; (b) The K ∗ (892) − mass measured in differentexperiments.The K S π − invariant mass distribution shown in Fig. 1 (a) is described in terms of the vector( F V ) and the scalar ( F S ) form factors according to Ref. [16]: d Γ d √ s ∼ s (cid:18) − sm τ (cid:19) (cid:18) sm τ (cid:19) P (cid:26) P | F V | + 3( m K − m π ) s (1 + 2 sm τ ) | F S | (cid:27) , (1)where s is squared K S π − invariant mass, P is K S momentum in the K S π − rest frame. Thevector form factor is parametrized by the K ∗ (892) − , K ∗ (1410) − and K ∗ (1680) − meson am-plitudes, while the scalar form factor includes the K ∗ (800) − and K ∗ (1430) − contributions.The K ∗ (892) − alone is not sufficient to describe the K S π − invariant mass spectrum. Todescribe the enhancement near threshold, we introduce a K ∗ (800) − amplitude, while for thedescription of the distribution at higher invariant masses we try to include the K ∗ (1410) − , K ∗ (1680) − vector resonances or the scalar K ∗ (1430) − . The best description is achievedwith the K ∗ (800) − + K ∗ (892) − + K ∗ (1410) − and K ∗ (800) − + K ∗ (892) − + K ∗ (1430) − models. The parameterization of F S suggested by the LASS experiment [5] was also tested: F S = λ √ sP (sin δ B e iδ B + e iδ B BW K ∗ (1430) ( s )) , (2)49here λ is a real constant, P is K S momentum in the K S π − rest frame, and the phase δ B isdetermined from the equation cot δ B = aP + bP , where a , b are the model (fit) parameters.In this parameterization the non-resonant mechanism is given by the effective range term sin δ B e iδ B , while the resonant structure is described by the K ∗ (1430) amplitude.Figure 2: The absolute value of F S from LASS experiment (left) and from the τ − → K S π − ν τ study at Belle (right).The shape of the optimal scalar form factor in the LASS experiment strongly differs (espe-cially, near the threshold of the K S π − production) from that obtained in the fit of the K S π − mass distribution in the study of τ − → K S π − ν τ decay at Belle, see Fig. 2. There is largesystematic uncertainty in the near K S π − production threshold part of the spectrum due tothe large background from the τ − → K S π − K L ν τ decay, whose dynamics is not preciselyknown. In the new study at B factories it will be possible to suppress this background essen-tially applying special kinematical constraints.A fit to the K S π − invariant mass spectrum also provides a high precision measurement ofthe K ∗ (892) − mass and width: M ( K ∗ (892) − ) = (895 . ± . . ) ± . . ) ± . . )) MeV /c , Γ( K ∗ (892) − ) = (46 . ± . . ) ± . . ) ± . . )) MeV .While our determination of the width is compatible with most of the previous measurementswithin experimental errors, our mass value, see Fig. 1 (b), is considerably higher than thosebefore and is consistent with the world average value of the neutral K ∗ (892) mass, whichis (896 . ± . MeV/ c [15].The second analysis of the τ − → K S π − ν τ decay at Belle was based on the data sample withthe luminosity integral of L = 669 fb − which comprises 615 million τ + τ − events [11].One inclusive decay mode τ − → K S X − ν τ and 6 exclusive hadronic τ decay modes with K S ( τ − → K S π − ν τ , τ − → K S K − ν τ , τ − → K S K S π − ν τ , τ − → K S π − π ν τ , τ − → K S K − π ν τ , τ − → K S K S π − π ν τ ) were studied in Ref. [11]. In this study signal events50ere tagged by one-prong tau decays (into eνν , µνν or π/K ( n ≥ π ν final states) withthe branching fraction B − prong = (85 . ± . . In total, events of the τ − → K S π − ν τ decay were selected with the fraction of the non-cross-feed background of (8 . ± . and the detection efficiency ε det = (7 . ± . . The obtained branching fraction: B ( τ − → K S π − ν τ ) = (4 . ± . . ) ± . . ))) × − supersedes the previous Belle result and has the best accuracy.Figure 3: The K − π invariant mass distribution (left) from Ref. [12]. The dots are ex-perimental data, histograms are background MC events with selection and efficiency correc-tions: background from τ τ (dashed line), q ¯ q (dash-dotted line), µ + µ − (dotted line). The K S π − invariant mass distribution (right) from Ref. [13]. The dots are experimental data, his-tograms are signal and background MC: signal events (red), dominant background from the τ − → K S K L π − ν τ decay (yellow), non- τ τ events (magenta).Precision measurement of the branching fraction of the τ − → K − π ν τ decay with a . fb − data sample collected at the Υ(4 S ) resonance has been carried out by BaBar [12]. The result: B ( τ − → K − π ν τ ) = (4 . ± . . ) ± . . ))) × − is consistent with the previous measurements and B ( τ − → K S π − ν τ ) and has the best accu-racy. The K − π invariant mass distribution is shown in Fig. 3.The preliminary result on the B ( τ − → K S π − ν τ ) using a . fb − data sample was alsopublished by BaBar [13]: B ( τ − → K S π − ν τ ) = (4 . ± . . ) ± . . ))) × − . It is consistent with the other measurements. The distribution of the invariant mass of the K S π − system is shown in Fig. 3, experimental data exhibit additional contribution aroundthe invariant mass of 1.4 GeV/ c , which is not included in the signal MC simulation.51. Search for CPV in τ → Kπν
Recent studies of CPV in the τ − → π − K S ( ≥ π ) ν τ decays at BaBar [14] as well as inthe τ − → K S π − ν τ decay at Belle [10] provide complementary information about sourcesof CPV in these hadronic decays.The decay-rate asymmetry A CP = Γ( τ + → π + K S ( ≥ π ) ν τ ) − Γ( τ − → π − K S ( ≥ π ) ν τ )Γ( τ + → π + K S ( ≥ π ) ν τ )+Γ( τ − → π − K S ( ≥ π ) ν τ ) was studied atBaBar with the τ + τ − data sample of (cid:82) Ldt = 476 fb − . The obtained result A CP = ( − . ± . ± . is about . standard deviations from the SM expectation A K CP = (+0 . ± . .At Belle, CPV search was performed as a blinded analysis based on a
699 fb − data sam-ple. Specially constructed asymmetry, which is a difference between the mean values ofthe cos β cos ψ for τ − and τ + events, was measured in bins of K S π − mass squared ( Q = M ( K S π ) ): A CPi ( Q i ) = (cid:82) ∆ Q i cos β cos ψ (cid:16) d Γ τ − dω − d Γ τ + dω (cid:17) dω (cid:82) ∆ Q i (cid:16) d Γ τ − dω + d Γ τ + dω (cid:17) dω (cid:39) (cid:104) cos β cos ψ (cid:105) τ − − (cid:104) cos β cos ψ (cid:105) τ + , where β , θ and ψ are the angles, evaluated from the measured parameters of the final hadrons, dω = dQ d cos θd cos β . In contrary to the decay-rate asymmetry the introduced A CPi ( Q i ) is already sensitive to the CPV effects from the charged scalar boson exchange [17]. NoCP violation was observed and the upper limit on the CPV parameter η S was extracted tobe: | Im( η S ) | < . at 90% CL. Using this limit parameters of the Multi-Higgs-Doubletmodels [18, 19] can be constrained as | Im( XZ ∗ ) | < . M H ± / (1 GeV /c ) , where M H ± is the mass of the lightest charged Higgs boson, the complex constants Z and X describe thecoupling of the Higgs boson to leptons and quarks respectively.4. Further Studies of τ → Kπν
Decays
In the analysis of the τ − → K S π − ν τ decay, it is very desirable to measure separately vec-tor ( F V ), scalar ( F S ) form factors and their interference. The K ∗ (892) − mass and widthare measured in the vector form factor taking into account the effect of the interference ofthe K ∗ (892) − amplitude with the contributions from the radial exitations, K ∗ (1410) − and K ∗ (1680) − . The scalar form factor is important to unveil the problem of the K ∗ (800) − state as well as for the tests of the various fenomenological models and search for CPV. Theinterference between vector and scalar form factors is necessary in the search for CPV in τ − → K S π − ν τ decay.To elucidate the nature of the K ∗ (892) − − K ∗ (892) mass difference it is important to studythe following modes: τ − → K S π − ν τ , τ − → K S π − π ν τ , τ − → K S K − π ν τ . K ∗ (892) − mass and width can be measured in the clean experimental conditions without disturbancefrom the final state interactions in the τ − → K S π − ν τ decay. While a study of the τ − → K S π − π ν τ mode allows one to measure simultaneously in one mode the K ∗ (892) − ( K S π − )and the K ∗ (892) ( K S π ) masses. The effect of the pure hadronic interaction of the K ∗ (892) − ( K ∗ (892) ) and π ( π − ) on the K ∗ (892) − ( K ∗ (892) ) mass can be precisely measured as52ell. It is also important to investigate precisely the effect of the pure hadronic inter-action of the K ∗ (892) − ( K ∗ (892) ) and K S ( K − ) on the K ∗ (892) − ( K ∗ (892) ) mass in the τ − → K S K − π ν τ decay.5. Summary
Belle and BaBar essentially improved the accuracy of the branching fractions of the τ − → ( Kπ ) − ν τ decays. At Belle the K S π invariant mass spectrum was studied and the K ∗ (892) − alone is not sufficient to describe the K S π mass spectrum. The best description is achievedwith the K ∗ (800) − + K ∗ (892) − + K ∗ (1410) − and K ∗ (800) − + K ∗ (892) − + K ∗ (1430) − mod-els. For the first time the the K ∗ (892) − mass and width have been measured in τ decay at B factories. The K ∗ (892) − mass is significantly different from the current world average value,it agrees with the K ∗ (892) mass. Precision study of the τ − → K S π − ν τ , τ − → K S π − π ν τ and τ − → K S K − π ν τ decays at the B factories as well as the e + e − → K S K ± π ∓ reaction atthe VEPP-2000 [20,21] and K − π scattering amplitude at the coming GlueX experiment [6]could provide additional valuable information about the K ∗ (892) − mass, namely unveil animpact of the hadronic and electromagnetic interactions in the final state.6. Acknowledgments
I am grateful to Prof. Igor Strakovsky for inviting me to this very interesting workshop. Thiswork was supported by Russian Foundation for Basic Research (Grant No. 17–02–00897–a).
References [1] A. J. Bevan et al. [BaBar and Belle Collaborations], Eur. Phys. J. C , 3026 (2014).[2] W. Fetscher and H. J. Gerber, Adv. Ser. Direct. High Energy Phys. , 657 (1995).[3] A. Pich, Adv. Ser. Direct. High Energy Phys. , 453 (1998).[4] E. Gamiz, M. Jamin, A. Pich, J. Prades, and F. Schwab, Phys. Rev. Lett. , 011803 (2005).[5] D. Aston et al. (LASS Collaboration), Nucl. Phys. B , 493 (1988).[6] S. Dobbs [GlueX Collaboration], PoS ICHEP , 1243 (2017).[7] D. Delepine, G. Lopez Castro, and L. -T. Lopez Lozano, Phys. Rev. D , 033009 (2005).[8] J. H. Kuhn and E. Mirkes, Phys. Lett. B , 407 (1997).[9] D. Epifanov et al. [Belle Collaboration], Phys. Lett. B , 65 (2007).[10] M. Bischofberger et al. [Belle Collaboration], Phys. Rev. Lett. , 131801 (2011).[11] S. Ryu et al. [Belle Collaboration], Phys. Rev. D , 072009 (2014).[12] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D , 051104 (2007).5313] B. Aubert et al. [BaBar Collaboration], Nucl. Phys. Proc. Suppl. , 193 (2009).[14] J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D , 031102 (2012); Erratum: [Phys. Rev.D , 099904 (2012)].[15] W.-M. Yao et al. , J. Phys. G , 1 (2006).[16] M. Finkemeier and E. Mirkes, Z. Phys. C , 619 (1996).[17] J. H. Kuhn and E. Mirkes, Phys. Lett. B , 407 (1997).[18] Y. Grossman, Nucl. Phys. B , 355 (1994).[19] S. Y. Choi, K. Hagiwara, and M. Tanabashi, Phys. Rev. D , 1614 (1995).[20] Y. M. Shatunov et al. , Conf. Proc. C (2000) 439.[21] D. Berkaev et al. , Nucl. Phys. Proc. Suppl. (2012) 303.54 .8 From πK Amplitudes to πK Form Factors (and Back)
Bachir Moussallam
IPN, Université Paris-Sud 1191406 Orsay, France
Abstract
The dispersive construction of the scalar ππ and the scalar and vector πK form factors arereviewed. The experimental properties of the ππ and πK scattering J = 0 , amplitudes arerecalled, which allow for an application of final-state interaction theory in a much larger rangethan the exactly elastic energy region. Comparisons are made with recent lattice QCD resultsand with experimental τ decay results. The latter indicate that some corrections to the πKP -wave phase-shifts from LASS may be needed. Introduction
Pions and kaons are the lightest hadrons in QCD. The ππ interaction plays an importantrole in generating stable nuclei. Studying the ππ and the πK interactions allows to probethe chiral symmetry aspects of QCD at low energy and the resonance structure at higherenergy. Sufficiently precise measurements also associated with theoretical, analyticity prop-erties of QCD have allowed to establish the existence of light, exotic, resonances [1] whichhad remained elusive for many decades.In recent years a considerable amount of data on decays of heavy mesons or leptons intolight pseudo-scalar mesons have accumulated. The energy dependencies of these decayamplitudes are essentially controlled by the final-state interactions (FSI) among the lightmesons. FSI theory relies on analyticity and the existence of a right-hand cut, in each energyvariable, which can be associated with unitarity [2].Two-mesons form factors are the simplest functions to which FSI theory can be applied inthe sense that they depend on a single variable and that the unitarity cut is the only one.We review below some aspects of the ππ scalar and the πK scalar and vector form factors.We will recall, in particular, the experimental properties of the interactions which allow foran application of FSI theory in a much larger energy range than the range of purely elasticscattering. We will also consider the possibility, given sufficiently precise determination ofthe form factors, to improve the precision of the determination of the light meson amplitudes.2. The ππ Scalar Form Factors ππ scattering is exactly elastic in QCD in the energy range s ≤ m π . However, it has longbeen known that it can be considered as effectively elastic in a larger range. The two mainassumptions which underlie the dispersive construction of the form factors in Ref. [3] is that,in the S -wave, this effectively elastic region extends up to the K ¯ K threshold and that beyondthis, there exists an energy region in which two-channel unitarity effectively holds. Theseassumptions are supported by the experimental measurements of the ππ phase-shifts andinelasticities. For instance Fig. 5a of Ref. [4] showing the inelasticity η indicates clearlythat the η is driven away from 1 by the f (980) resonance and the cusp-like shape of the55urve suggests that K ¯ K is the leading inelastic channel, while π or π channels play anegligible role up to 1.4 GeV.This can be further probed using experimental measurements of the ππ → K ¯ K amplitudes.Indeed, if two-channel unitarity holds then the modulus of this amplitude is simply related tothe inelasticity. This is illustrated from fig 1 which shows fits to the inelasticity as deducedfrom the ππ → K ¯ K measurements of Refs. [5, 6] compared with the inelasticity measure in ππ . The two experiments are in good agreement in the energy energy region E ≥ . GeVand the figure indicates that the two-channel unitarity picture could hold up to E (cid:39) . GeV. ππ i ne l a s t i c i t y √ s (GeV)HyamsEtkin FitCohen Fit Figure 1:
Inelasticity parameter of the ππ I = 0 S -wave compared to that deduced from ππ → K ¯ K measurements assuming two-channel unitarity. Scalar form factors, for instance Γ π ( t ) = (cid:104) π ( p ) π ( q ) | m u ¯ uu + m d ¯ dd | (cid:105) , with t = ( p + q ) are analytic functions of t in the whole complex plane except for a right-hand cut on the realaxis. Assuming that it has no zeroes, it can be expressed as a phase dispersive representation Γ π ( t ) = Γ π (0) exp (cid:20) tπ (cid:90) ∞ m π dt (cid:48) t (cid:48) ( t (cid:48) − t ) φ π ( t (cid:48) ) . (cid:21) (1)From Watson’s theorem the form factor phase φ π is equal to the ππ scattering phase in theelastic scattering region, that is, according to the preceding discussion φ π ( t ) = δ ( t ) , m π ≤ t ≤ m K . (2)Furthermore, when t → ∞ one must have φ π ( t ) → π which ensures compatibility witha Brodsky-Lepage type behavior [7]. Based on these arguments, a simple interpolatingmodel for the phase in the range [4 m K , ∞ ] was proposed in ref. [8]. As was pointedout in [9] this simple interpolation leads to an overestimate of the pion scalar form factor (cid:104) r (cid:105) πS = ˙Γ π (0) / (6Γ π (0)) because the f (980) resonance effect is not properly accountedfor. A more accurate interpolation can be performed by exploiting the existence of a two-channel unitarity region and generating the form factor from a solution of the corresponding56oupled-channel Muskhelishvili equations (cid:126)F ( t ) = 1 π (cid:90) ∞ m π dt (cid:48) T ( t (cid:48) )Σ( t (cid:48) ) (cid:126)F ∗ ( t (cid:48) ) t (cid:48) − t , (3)where T is the ππ − K ¯ K coupled-channel matrix and Σ( t (cid:48) ) = diag (cid:16)(cid:112) − m π /t (cid:48) , (cid:112) − m K /t (cid:48) θ ( t (cid:48) − m K ) (cid:17) . Beyond the region where two-channel unitarity applies, a proper asymptotic phase interpolation is performed in this modelby imposing asymptotic conditions on the T -matrix. It is also convenient to choose theseconditions such that the so-called Noether index [10] is equal to N = 2 which ensures thatEq. 3 has a unique solution once two conditions are imposed, e.g., Γ π (0) , Γ K (0) .Fig. 2 illustrates the result for the phase of the form factor, φ π , obtained from solving theseequations. The f (980) resonance is predicted to generate a sharp drop in this phase at 1GeV. Nevertheless, it is noteworthy that the asymptotic phase is reached from above in thisapproach as is required in QCD [8] (see right plot in Fig. 2). The result for the pion scalarradius [9]: (cid:104) r (cid:105) πS = (0 . ± . fm is in good agreement with lattice QCD simulations,e.g., [11]. Φ π ( deg r ee s ) √ s (GeV)CohenEtkin Φ π ( deg r ee s ) √ s (GeV) CohenEtkin Figure 2:
Phase of the pion scalar form factor from solving the Muskhelishvili equations 3. The right plotillustrates how the asymptotic value is reached. πK Scalar Form Factor πK scattering in the S -wave is exactly elastic in QCD in the region t ≤ ( m K + 3 m π ) . Byanalogy with ππ one would expect the amplitude to be effectively elastic in a substantiallylarger range. This seems to be confirmed by experiment. Fig. 3 shows the inelasticity pa-rameter η / obtained from a fit to the two most recent amplitudes determinations [12, 13].One sees that η / remains close to 1 up to √ t (cid:39) . GeV and it is driven away from 1very sharply by the K ∗ (1950) resonance. The detailed properties of this resonance are yetunknown. It is often assumed that its main decay channel is η (cid:48) K which would imply theexistence of a two-channel unitarity region for the πK S -wave. A dispersive derivation ofthe πK scalar form factor exactly analogous to that of ππ can then be performed [14].57n important difference with ππ , though, is that the πK scalar form factor is measurable.It can be determined from semi-leptonic decay amplitudes K → πlν l and τ → Kπν τ sincethe matrix element of the vector current involves both the vector and the scalar form factors, f Kπ + , f Kπ √ (cid:104) K + ( p K ) | ¯ uγ µ s | π ( p π ) (cid:105) = f Kπ + ( t ) (cid:18) p K + p π − ∆ Kπ t ( p K − p π ) (cid:19) µ + f Kπ ( t ) ∆ Kπ t ( p K − p π ) µ (4)The prediction from the two-channel dispersive form factor model for the slope parameter λ = m π + ˙ f Kπ (0) /f Kπ (0) is [15] λ = (14 . ± . · − which is in rather good agreementwith the most recent determination by NA48/2, NA62 experiments [16]: λ = (14 . ± . ± . · − . Further verifications of this form factor, in particular in the region of the K ∗ (1430) resonance, would be necessary. This is possible, in principle, from τ → Kπν τ decays if one measures both the energy and the angular distributions in order to disentanglethe contributions from the two form factors. At low energy f Kπ ( t ) displays a strong κ mesoninduced enhancement. As can be seem from fig. 4 (left plot) below, this feature is in accordwith the data already existing on τ decay.4. πK Vector Form Factor and J = 1 Amplitude
For the P -wave amplitude, fits to the experimental data indicate that the quasi-elastic rangeextends up to the K ∗ π threshold (see fig. 3). The inelasticity is driven by the K ∗ (1410) andby the K ∗ (1680) resonance. The decay properties of these two resonances have been studiedin detail in Ref. [17]: they essentially involve the two quasi two-body channels K ∗ π and Kρ .This suggests that a plausible model based on three-channel unitarity can be developed fordescribing the πK P -wave scattering which can be applied to the dispersive constructionof the vector form factor. Models of this type were considered [18, 19]. We present belowslightly updated results from [18]. I n e l a s t i c i t y E π K (GeV)I=1/2 S-wave m K + m η m K + m η ' Fit (Aston) I n e l a s t i c i t y E π K (GeV)I=1/2 P-wave m K * + m π m K + m ρ Fit (Aston)
Figure 3: πK inelasticity from fits to the data of [12, 13]. Left plot: S -wave, right plot: P -wave. At first, a reasonable fit of the experimental data can be achieved (with a χ /d.o.f = 1 . )in the whole energy range . ≤ E ≤ . GeV within a model which implements exact58hree-channel unitarity via a simple standard K -matrix approach, K ij = (cid:88) R g iR g jR m R − s + K backij . (5)The model has four P -wave resonances and contains 15 parameters which are determinedfrom a fit to the πK → πK experimental data. The parameters are also constrained withrespect to the inelastic channels K ∗ π and Kρ by the measured branching fractions of the K ∗ (1410) , K ∗ (1680) resonances.The three vector form factors associated with Kπ , K ∗ π , Kρ can then be determined bysolving the coupled integral dispersive equations analogous to eq. 3. Having chosen N = 3 for the Noether index, it is necessary to provide three constraints in order to uniquely specifythe solution, e.g. the values of the three form factors H i at t = 0 . The value of H (0) = f Kπ + (0) is rather precisely known from chiral symmetry and from lattice QCD simulations(see [20]). Concerning H (0) , H (0) we can only obtain a qualitative order of magnitude inthe limit of exact three-flavor chiral symmetry and assuming exact vector-meson dominance.This gives H (0) = − H (0) = √ N c M V π F V F π (cid:39) . GeV − (6)We can thus parametrize the exact values as H (0) = 1 . a ) , H (0) = 1 . b ) and we expect | a | , | b | < . Fitting the Belle data [21] on τ → Kπν τ with these twoparameters a , b gives a rather poor χ = 8 . . While this could simply be blamed on theinadequacy of the model, we will argue instead that the P -wave phase-shifts of LASS [17]may need some updating. At first, one sees that much of the large χ value originates fromthe K ∗ (892) resonance region and simply reflects a significant difference in the K ∗ (892) resonance width between the LASS data and the Belle data. Indeed, in the recent re-analysisof the LASS data constrained by dispersions relations [22] a value for the width, Γ K ∗ =58 ± MeV is found, which is much larger than the result of Belle: Γ K ∗ = 46 . ± . MeV.In the present approach one can go further and determine how the P -wave results from LASSneed to be modified in order for the vector form factor to be in better agreement with the Belleresults. This is done by refitting the parameters of the three-channel T matrix including bothLASS and Belle data sets. Doing this, a reasonable agreement with the τ data can be achievedwith a χ /N ( Belle ) = 2 . while the χ for the LASS data is χ /N ( LASS ) = 3 . . This isillustrated in fig. 4. The left plot shows the πK energy distribution in the τ → πKν τ modeand the right plot shows the phase of the π + K − → π + K − J = 1 amplitude as a function ofenergy. Beyond the region of the K ∗ (892) a visible modification of the phase in the regionof the K ∗ (1410) seems to be also needed. References [1] I. Caprini, G. Colangelo, and H. Leutwyler, Phys. Rev. Lett. (2006) 132001.[2] R. Omnès, Nuovo Cim. (1958) 316. 59 d N e v en t s / d E π K E π K (GeV) Belle(2007)LASS fitCombined fitScalar Φ S ( deg r ee s ) E π K (GeV) LASSLASS only fitLASS+tau fit Figure 4:
Left plot: energy distribution in the τ → πKν τ mode. Right plot: Phase of the π + K − → π + K − J = 1 amplitude. The dashed blue curves show the results when the T -matrix parameters are fitted to theLASS data only, while the solid red curves when the fit includes both LASS and Belle data. The dash-dotmagenta curve shows the contribution associated with the scalar form factor. [3] J. F. Donoghue, J. Gasser, and H. Leutwyler, Nucl. Phys. B (1990) 341.[4] B. Hyams et al. , Nucl. Phys. B (1973) 134.[5] D. H. Cohen et al. , Phys. Rev. D (1980) 2595.[6] A. Etkin et al. , Phys. Rev. D (1982) 1786.[7] G. P. Lepage and S. J. Brodsky, Phys. Rev. D (1980) 2157.[8] F. J. Yndurain et al. , Phys. Lett. B (2004) 99.[9] B. "Ananthanarayan et al. , Phys. Lett. B (2004) 218.[10] N. Noether, Mathematische Annalen, (1921) 42.[11] V. Gülpers et al. , Phys. Rev. D (2014) 094503.[12] P. Estabrooks et al. , Nucl. Phys. B (1978) 490.[13] D. Aston et al. , Nucl. Phys. B (1988) 493.[14] M. Jamin et al. , Nucl. Phys. B (2002) 279.[15] M. Jamin et al. , Phys. Rev. D (2006) 074009.[16] M. Piccini [for NA48/2 Collaboration], Proceedings, 2017 European Physical Society Con-ference on High Energy Physics (EPS-HEP 2017): Venice, Italy, July 5-12, 2017; PoS EPS-HEP2017 (2018) 235.[17] D. Aston et al. , Nucl. Phys. B (1987) 693.[18] B. Moussallam, Eur. Phys. J. C (2008) 401.6019] V. Bernard, JHEP, (2014) 082.[20] D. Aoki et al. , Eur. Phys. J. C (2017) 112.[21] D. Epifanov et al. [Belle Collaboration], Phys. Lett. B (2007) 65.[22] J. R. Peláez et al. , Eur. Phys. J. C (2017) 91.61 .9 Three-Body Interaction in Unitary Isobar Formalism Maxim Mai
Institute for Nuclear Studies andDepartment of PhysicsThe George Washington UniversityWashington, DC 20052, U.S.A.
Abstract
In this talk, I present our recent results on the three-to-three scattering amplitude con-structed from the compositeness principle of the S -matrix and constrained by two- and three-body unitarity. The resulting amplitude has important applications in the infinite volume, butcan also be used to derive the finite volume quantization condition for the determination ofenergy eigenvalues obtained from ab-initio Lattice QCD calculations of three-body systems. Introduction
Interest in the description of three-hadron systems has re-sparked in recent years due to twoaspects of modern nuclear physics. First, there are substantial advances of experimental fa-cilities such as COMPASS [1], GlueX [2] and CLAS12 [3] experiments, aiming for the studyof meson resonances with mass above 1 GeV including light hybrids. The large branchingratio of such resonances to three pions is expected to generate important features via thefinal state interaction. Furthermore, effects such as the log-like behavior of the irreduciblethree-body interaction, associated with the a (1420) [4] can be studied in detail with a fullthree-body amplitude. Similarly, the study of the XY Z sector [5, 6] currently explored byLHCb, BESIII, Belle and BaBar [7, 8] can be conducted in more detal when the three-bodyinteractions are taken into account. The prominent Roper puzzle can be re-addressed whenthe features of the ππN amplitude are sufficiently under control, see, e.g., Ref. [9]. Finally,the proposed Klong beam experiment at JLab [10–13] can give new insights into propertiesof, e.g., κ -resonance from the Kππ channel. Second, the algorithmic and computationaladvances in ab-intio
Lattice QCD calculations make the analysis of such interesting systemsas the Roper-resonance ( N (1440)1 / + ) or the a (1260) possible. Some first studies in thesesystems have already been conducted, using gauge configurations with unphysically heavyquark masses [14–16], but no 3-body operators have been included there yet. Many othergroups are working on conducting similar studies, such as, e.g., the πρ scattering in the I = 2 sector [17]. Also in these systems the pion mass is very heavy, such that the ρ is stable andthe infinite-volume extrapolation can effectively be carried out using the two-body Lüscherformalism. In the future, it has to be expected that these and similar studies will be carriedout at lower pion masses with an unstable ρ decaying into two pions. For these cases thefull understanding of the infinite volume extrapolations including three-body dynamics isdesired. Important progress has been achieved in the last years [18–30], and first numericalcase studies have now been conducted with three different approaches [19, 30–32]. Whilestill exploratory, they mark an important step in the development of the three-body quanti-zation condition.In this work, we show theoretical developments of one of these approaches [30] as well asnumerical studies based on it. This framework is based on the general formulation of the62 = + T + + + Figure 1: Total scattering amplitude ˆ T consisting of a connected ( ˆ T c ) and a disconnected contribu-tion ( ˆ T d ), represented by the first and second term on the right-hand side of Eq. (1), respectively.Single lines indicate the elementary particle, double lines represent the isobar, empty dots standfor isobar dissociation vertex v , while time runs from right to left. T and ” + ” denote the isobar-spectator scattering amplitude and isobar propagator τ , respectively.infinite volume three-to-three scattering amplitude which respects two- and three-body uni-tarity. At its core, the two-body sub-amplitudes are parametrized by a tower of functions ofinvariant mass with correct right-hand singularities of the corresponding partial-waves. Thetruncation of such a series of functions in a practical calculations is the only approximationof such an approach and takes account of the sparsity of the lattice data. The imaginary partsof such an amplitude are fixed by unitarity, giving rise to a power-law (in M L – a dimension-less product of pion mass and size of cubic lattice volume) finite volume dependence, whenreplacing continuous momenta in such an amplitude by discretized ones due to boundaryconditions imposed in Lattice QCD studies. Therefore, such a framework gives a naturalway to study three-body systems in the infinite and in finite volume simultaneously.2.
Three-Body Scattering in the Infinite Volume
The interaction of three-to-three asymptotic states is described by the scattering amplitude T . We assume here that the particles in question are stable, spinless and identical (of mass M ) for simplicity. The connectedness-structure of matrix elements dictates that the scatter-ing amplitude consists of two parts: the fully connected one and one-time disconnected,denoted by the subscript c and d in the following. As discussed in Refs. [19, 33], the fulltwo-body amplitude can be re-parametrized by a tower of “isobars", which for given quan-tum numbers of the two-body sub-system describe the correct right-hand singularities ofeach partial wave in that system. In this sense the isobar formulation is not an approximationbut a re-parametrization of the full two-body amplitude, see the discussion in the originalwork [34]. In summary, the three-to-three scattering amplitude (depicted in Fig. 1) reads (cid:104) q , q , q | ˆ T | p , p , p (cid:105) = (cid:104) q , q , q | ˆ T c | p , p , p (cid:105) + (cid:104) q , q , q | ˆ T d | p , p , p (cid:105) (1) = 13! (cid:88) n =1 3 (cid:88) m =1 v ( q ¯ n , q ¯¯ n ) v ( p ¯ m , p ¯¯ m ) (cid:32) τ ( σ ( q n )) T ( q n , p m ; s ) τ ( σ ( p m )) − E ( q n ) τ ( σ ( q n ))(2 π ) δ ( q n − p m ) (cid:33) , two particle sub-amplitude are still fully connected, while the third particle takes the role of what we refer to asthe “spectator". P is the total four-momentum of the system, s = W = P and E ( p ) = (cid:112) p + M .All four-momenta p , q , ... are on-mass-shell, and the square of the invariant mass of theisobar reads σ ( q ) := ( P − q ) = s + M − W E ( q ) for the spectator momentum q . Wework in the total center-of-mass frame where P = and denote throughout the manuscriptthree-momenta by bold symbols. The dissociation vertex v ( p, q ) of the isobar decayingin asymptotically stable particles, e.g., ρ ( p + q ) → π ( p ) π ( q ) , is chosen to be cut-free inthe relevant energy region, which is always possible. The notation is such that, e.g., for aspectator momentum q n the isobar decays into two particles with momenta q ¯ n and q ¯¯ n . Finally, T describes the isobar-spectator interaction and is the function of eight kinematic variablesallowed by momentum and energy conservation, such as the full three-to-three scatteringamplitude.The above equation contains three unknown functions τ ( σ ) , T and v . Since real and imag-inary parts of the scattering amplitude are related by unitarity, the latter building blocks ofthe scattering amplitude are related between each other as well. Specifically, including acomplete set of three-particle intermediate states (inclusion of higher-particle states will bestudied in a future work) the three-body unitarity condition can be written as (cid:104) q , q , q | ( ˆ T − ˆ T † ) | p , p , p (cid:105) = (2) i (cid:90) (cid:89) (cid:96) =1 d k (cid:96) (2 π ) (2 π ) δ + ( k (cid:96) − m ) (2 π ) δ (cid:32) P − (cid:88) (cid:96) =1 k (cid:96) (cid:33) (cid:104) q , q , q | ˆ T † | k , k , k (cid:105) (cid:104) k , k , k | ˆ T | p , p , p (cid:105) , where δ + ( k − m ) := θ ( k ) δ ( k − m ) . To reveal the relations between T , τ and v wemake the Bethe-Salpeter Ansatz for the isobar-spectator amplitude T ( q n , p m ; s ) = B ( q n , p m ; s ) + (cid:90) d k (2 π ) B ( q n , k ; s )ˆ τ ( σ ( k )) T ( k, p m ; s ) , (3)which holds for any in/outgoing spectator momenta p m /q n (not necessarily on-shell), andeffectively re-formulates the unknown function T by yet another two unknown functions:the isobar-spectator interaction kernel B and the isobar-spectator Green’s function ˆ τ . Inparticular, we can rewrite the left-hand-side of the unitarity relation (2) into eight differenttopologies, which symbolically read ˆ T − ˆ T † = v ( τ − τ † ) v + v (cid:0) τ − τ † (cid:1) T τ v + vτ † T † (cid:0) τ − τ † (cid:1) v + vτ † ( B − B † ) τ v + vτ † ( B − B † )ˆ τ T τ v + vτ † T † ˆ τ † ( B − B † ) τ v (4) + vτ † T † (ˆ τ − ˆ τ † ) T τ v + vτ † T † ˆ τ † ( B − B † )ˆ τ T τ v . The main goal of such a decomposition is to express the discontinuity of ˆ T as a sum ofdiscontinuities of simpler building blocks in corresponding variables.The same kind of decomposition can be achieved for the right-hand-side of unitarity rela-tion (2), inserting ˆ T = ˆ T d + ˆ T c there, while keeping in mind the permutations of particleindices as in Eq. (1). The exact formulation of this is discussed in detail in the originalpublication [34]. Here we restrict ourselves of mentioning that piecewise comparison of all64tructures to those of Eq. (4) leads to eight independent matching relations. These can befulfilled simultaneously imposing ˆ τ ( σ ( k )) = − (2 π ) δ + ( k − m ) τ ( σ ( k )) , (5) B ( q, p ; s ) − B † ( q, p ; s ) = iv ( Q, q )(2 π ) δ + ( Q − m ) v ( Q, p ) , (6) (cid:16) τ † ( σ ( k )) (cid:17) − − (cid:16) τ ( σ ( k )) (cid:17) − = i (cid:90) d ¯ K π ) δ + (cid:18)(cid:16) ˜ P + ¯ K (cid:17) − m (cid:19) δ + (cid:18)(cid:16) ˜ P − ¯ K (cid:17) − m (cid:19) (7) (cid:16) v (cid:16) ˜ P + ¯ K, ˜ P − ¯ K (cid:17)(cid:17) , where ˜ P := ( P − k ) / and Q := P − p − q . The direct consequence of the above relationsis that the number of unknown in the integral equation defining the three-to-three scatteringamplitude (1) is naturally reduced from three ( ˆ τ , B , τ ) to two ( τ , B ). The remaining two canbe determined using twice subtracted dispersion relation w.r.t the invariant mass of the two-body system ( σ ) as well an un-subtracted dispersion relation in Q , respectively. The exactintegral representation of B and τ is given in the original publication [34] and resemblesthe one-particle exchange as well as a fully dressed isobar propagator. In both cases a real-valued function can be added, without altering the discontinuity relations and, thus, beingallowed by the unitarity constraint discussed here.The above considerations finalize the form of the three-to-three scattering amplitude as de-manded by three-body unitarity and compositeness principle. It is a fully relativistic three-dimensional integral equation, which becomes a coupled-channel equation when more thanone isobar is considered for the parametrization of the two-to-two scattering. The parame-ters of these isobars (subtraction constants) can be fixed from the two-body scattering data,while the real part of B ( q, p ; s ) has to be fixed from the three-body data. The application ofthis approach to systems like the a (1260) with two isobars, i.e., isovector and isoscalar ππ channels, is work in progress.3. Three-Body Scattering in Finite Volume
As discussed in the introduction, one of the main goals for the present investigations on theunitarity constraints on three-body systems is the derivation of the finite-volume spectrumin such systems. To re-iterate, the main idea is that unitarity fixes the imaginary parts of theamplitude, which themselves lead to the power-law dependence of finite-volume corrections.To begin, we note that since v (the isobar dissociation vertex into asymptotically stableparticles) is a cut-free function in momenta, it will not lead to any power-law finite-volumeeffects. Thus, the desired quantization condition will be entirely derived from the part inbrackets of the second line of Eq. (1). The non-trivial part of this expression is the isobar-spectator scattering amplitude T , which in integral form reads T ( q, p ; s ) = B ( q, p ; s ) − (cid:90) d l (2 π ) B ( q, l ; s ) τ ( σ ( l ))2 E ( l ) T ( l, p ; s ) , (8)where the parameters of the isobar-propagator τ can be fixed from two-body scattering data,defining the two-to-two scattering amplitude via T := vτ v .65
00 600 800 1000 12000 W [ MeV ] τ [ a r b i t r a t y un i t s ]
400 600 800 1000 12000 W [ MeV ] [ a r b i t r a r y un i t s ] Figure 2: Left: isobar propagator τ in the finite volume (blue line) and the real part of the infinitevolume one (red dashed line) for a given boost l = (0 , , π/L ) and L = 3 . fm. The graydashed line denotes the onshell-condition of two particles ( σ ( l ) = (2 M ) ), whereas the gray arearepresents the energy range for which σ ( l ) ≤ . Right: Red dashed line (green dotted line)represent real (imaginary) part of the infinite-volume S -wave projection of the isobar-spectatorinteraction kernel B . In comparison, the finite-volume projection B A +1 for the transition from shell1 to shell 1 at L = 6 fm (blue dots).In the finite cubic volume with periodic boundary conditions the momenta are discretized.In particular, in a cube of a size L only the following three-momenta are allowed (organizedby “shells”) q ni = 2 πL r i for { r i ∈ Z | r i = n, i = 1 , . . . , ϑ ( n ) } . (9)where ϑ ( n ) = 1 , , , . . . indicates the multiplicity (number of points in shell n = 0 , , , . . . )that can be calculated as described, e.g., in Refs. [32, 35]. In principle, replacing all mo-menta in Eqs. (1, 8, 12) including the replacement of the appearing integrals over solid angleas (cid:82) d Ω p n → πϑ ( n ) (cid:80) ϑ ( n ) i =1 leads to a generic three-body quantization condition – an equationwhich determines the positions of singularities of such an amplitude in energy. However,several subtleties arise from the breakdown of the spherical symmetry on the lattice, whichwe wish to discuss in the following.First of all, the isobar propagator in the second line of Eq. (12) is evaluated in the isobarcenter-of-mass frame. In the finite volume, however, the allowed momenta given by Eq. (9)are defined in the three-body rest frame at P = . For the calculation of the finite-volumeself-energy one, therefore, has to boost the momenta to the isobar rest frame. In this con-text it is important to recall that the two-body sub-system can become singular when theinvariant mass of the system becomes real, see, e.g., Fig. 2. The tower (for all spectatormomenta in question) of two-body singularities has to cancel such that only genuine three-body singularities remain in the final expression. As it is shown analytically in the originalpublication [30], such cancellation occur in the full quantization condition when all terms(including disconnected topology ˆ T d ) and boosts are taken into account accordingly.Another important observation w.r.t the breakdown of spherical symmetry is that the isobar-spectator interaction kernel is singular for specific combination of momenta and energies.66xpressed differently, when projecting to a partial wave in infinite volume, this term devel-ops an imaginary part below threshold as presented for the S-wave projection of B in Fig. 2.Thus, in finite volume the same term has to have a series of singularities in this region. Thisindeed happens when projecting B to the corresponding irreducible representation ( A +1 forthe depicted case) of the cubic symmetry group O h . Furthermore, the projection to irrepsof O h of the three-body scattering amplitude has two additional advantages. For once, theresults of Lattice QCD calculation are usually projected to these irreps. Additionally, theprojection to different irreps reduces the dimensionality of the three-body scattering ampli-tude and therefore also that of the quantization condition in the finite volume.There are various ways of projecting to definite irreps in the finite volume. In Ref. [32] amethod has been developed, which has a form very similar to the usual partial-wave projec-tion in infinite volume. We refer the reader for more details on the construction techniquesof this method to the original publication [32], and quote here only the corresponding result.A given function f s ( ˆp j ) acting on momenta of the shell s can be expanded as f s ( ˆp j ) = √ π (cid:88) Γ α (cid:88) u f Γ αsu χ Γ αsu ( ˆp j ) for f Γ αsu = √ πϑ ( s ) ϑ ( s ) (cid:88) j =1 f s ( ˆp j ) χ Γ αsu ( ˆp j ) , (10)where Γ , α , and u denote the irrep, basis vector of the irrep and the corresponding index,respectively. In these indices the functions χ Γ αsu build an orthonormal basis of functionsacting on momenta on the shell s .Using the projection method presented above our final result for three-body quantizationcondition for the irrep Γ reads Det (cid:20) B Γ ss (cid:48) uu (cid:48) ( W ) + 2 E s L ϑ ( s ) τ s ( W ) − δ ss (cid:48) δ uu (cid:48) (cid:21) = 0 , (11)where W is the total energy of three-body system, E s = (cid:112) p i + M and τ s ( W ) := τ ( σ ( p i )) for any three-momentum on the shell s . Note that the determinant is taken withrespect to a matrix in shell-indices ( s, s (cid:48) ), as well as basis indices ( u, u (cid:48) ) of the functionalbasis to the irrep Γ , whereas the dependence on α drops off naturally. To demonstrate theusefulness of the derived quantization condition we fix m = 139 MeV and L = 3 . fm.Furthermore, we assume one S-wave isobar with v ( p, q ) := λf (( p − q ) ) with f such thatit is 1 for ( p − q ) = 0 and decreases sufficiently fast for large momentum difference, e.g., f ( Q ) = β / ( β + Q ) to regularize integrals of the scattering equation. Note that one isnot obliged to use form factors but can instead formulate the dispersive amplitude throughmultiple subtractions rendering it automatically convergent, see Eq. (14) in Ref. [34]. Thisleads to B ( q, p ; s ) = − λ f (( P − q − p ) ) f (( P − q − p ) )2 E ( q + p ) ( W − E ( q ) − E ( p ) − E ( q + p ) + i(cid:15) ) + C ( q, p ; s ) , (12) τ ( σ ( l )) = σ ( l ) − M − (cid:90) d k (2 π ) λ ( f (4 k )) E ( k )( σ ( l ) − E ( k ) + i(cid:15) ) ,
200 400 600 800 1000 1200W [ MeV ] Figure 3: The red lines show finite-volume energy eigenvalues for L = 3 . fm and M = 139 MeV.The doted gray vertical lines show the positions of singularities of τ for all used boost momenta,while the dashed green vertical lines show the position of non-interacting energy eigenvalues ofthree particles.where C is a real-valued function of total energy and both spectator momenta, while M isa free parameter that is fixed (together with λ and β ) to reproduce some realistic two-bodyscattering data. In the present case we take the experimental phase-shifts for the ππ scatter-ing in the isovector channel for demonstration, and fix C = 0 and Γ = A +1 for simplicity.The result of the numerical investigation is depicted in Fig. 3. It shows the non-interactinglevels of the three-body system (green, dashed lines) along with the solutions of the quanti-zation condition (11) (full, red lines) representing the interacting energy eigenvalues. Since χ A +1 = 1 / √ π the dimensionality of the matrix is given entirely by the set of consideredshells. We have checked that using more than 8 first shells does not lead to visible changeof the position of the interacting levels. Constraining ourselves to these shells is equivalentto a momentum cutoff of ∼ GeV for the given lattice volume. Note also that the rangeof applicability of the quantization condition is, in principle, also restricted by construction,due to missing intermediate higher-particle states. The later is common to all present studiesof the three-body in finite volume, and has to be eased at some point in future. However, thisstudy demonstrates clearly the usefulness and practical applicability of the derived quantiza-tion condition. Further studies, such as volume dependence and inclusion of multiple isobarsare work in progress.4.
Acknowledgments
The speaker thanks the organizers of the workshop for the invitation and productive dis-cussions during the workshop. Furthermore, he is grateful for the financial support by theGerman Research Foundation (DFG), under the fellowship MA 7156/1-1, as well as for theGeorge Washington University for the hospitality and inspiring environment.
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Vyacheslav Ivanov (on behalf of CMD-3 Collaboration)
Budker Institute of Nuclear PhysicsNovosibirsk, 630090 Russia
Abstract
This paper describes the preliminary results of the study of processes of e + e − annihilationin the final states with kaons and pions with the CMD-3 detector at VEPP-2000 collider. Thecollider allows the c.m. energy scanning in the range from 0.32 to 2.0 GeV, and about pb − of data has been taken by CMD-3 up to now. The results on the K + K − π + π − , K + K − η , K + K − ω (782) and K + K − π final states are considered. Introduction
A high-precision measurement of the inclusive e + e − → hadrons cross section is required fora calculation of the hadronic contribution to the muon anomalous magnetic moment ( g − µ in the frame of the Standard Model. To confirm or deny the observed difference between thecalculated ( g − µ value [1] and the measured one [2], more precise measurements of theexclusive channels of e + e − → hadrons are necessary. The exclusive KK ( n ) π final states areof special interest, since their producltion involves rich intermediate dynamics which allowsthe test of isotopic relations and measurement of intermediate vector mesons parameters.In this paper we describe the current status of the study of e + e − → K + K − π + π − , K + K − η , K + K − ω (782) and K + K − π processes with the CMD-3 detector at VEPP-2000 collider(Novosibirsk, Russia), based on about pb − of data, collected in the runs of 2011-2012years, and about pb − in the runs of 2017 year. The preliminary results for φ (1680) mesonparameters were obtained from K + K − η cross section fitting. We see also the indication onnon trivial behavior of e + e − → K + K − π + π − process cross section at p ¯ p threshold.2. VEPP-2000 collider and CMD-3 detector
The VEPP-2000 e + e − collider [3] at Budker Institute of Nuclear Physics covers the E c . m . range from 0.32 to 2.0 GeV and employs a technique of round beams to reach luminosity upto 10 cm − s − at E c . m . =2.0 GeV. The Cryogenic Magnetic Detector (CMD-3) describedin [4] is installed in one of the two beam interaction regions. The tracking system of theCMD-3 detector consists of a cylindrical drift chamber (DC) and a double-layer cylindri-cal multiwire proportional Z-chamber, installed inside a superconducting solenoid with a1.0–1.3 T magnetic field (see CMD-3 layout in Fig. 1). Amplitude information from theDC wires is used to measure the specific ionization losses ( dE/dx DC ) of charged parti-cles. Bismuth germanate crystals of 13.4 X thickness are used in the endcap calorimeter.The barrel calorimeter, placed outside the solenoid, consists of two parts: internal (basedon liquid Xenon (LXe) of 5.4 X thickness) and external (based on CsI crystals of 8.1 X thickness) [5].The physics program of CMD-3 includes:71igure 1: The CMD-3 detector layout: 1 - beam pipe, 2 - drift chamber, 3 - BGO endcap calorime-ter, 4 - Z-chamber, 5 - superconducting solenoid, 6 - LXe calorimeter, 7 - time-of-flight system, 8- CsI calorimeter, 9 - yoke. • precise measurement of the R = σ ( e + e − → hadrons ) /σ ( e + e − → µ + µ − ) , neccessaryfor clarification of ( g − µ puzzle; • study of the exclusive hadronic channels of e + e − annihilation, test of isotopic relations; • study of the ρ , ω , φ vector mesons and their excitations; • CVC tests: comparison of isovector part of σ ( e + e − → hadrons ) with τ lepton decayspectra; • study of G E /G M of nucleons near threshold; • diphoton physics (e.g. η (cid:48) production).3. Study of e + e − → KK ( n ) π Processes (a)
Charged Kaon/Pion Separation
The starting point of the analysis of the final state with charged kaons and pions isthe kaon/pion separation, and to perform it we use the measurement of the specificionization losses dE/dx of particles in the DC. For the event with n tr DC-tracks thelog-likelihood function (LLF) for the hypothesis that for i = 1 , , ..., n tr the particlewith the momentum p i and energy losses ( dE/dx ) i is the particle of α i type ( α i = K or π ) is defined as L ( α , α , ..., α n tr ) = n tr (cid:88) i =1 ln (cid:32) f α i ( p i , ( dE/dx ) i ) f K ( p i , ( dE/dx ) i ) + f π ( p i , ( dE/dx ) i ) (cid:33) , (1)72here the functions f K/π ( p, dE/dx ) represent the probability density for charged kaon/pionwith the momentum p to produce the energy losses dE/dx in the DC. To perform theparticle identification (PID) we search for the ( α , α , ..., α n tr ) combination (with twooppositely charged kaons and zero net charge) that delivers the maximum to LLF. Inwhat follows we use L K ( n tr − π designation for the LLF maximum value. The cut on L K ( n tr − π value is used to avoid misidentification.Unfortunatelly, the described ( dE/dx ) DC -based separation for single kaons and pionsworks reliably only up to the momenta p < MeV/c, see Fig. 2. For the K + K − , K + K − π , K + K − π π and K S K ± π ∓ final states studies we are developing other tech-nique based on the dE/dx in 14 layers of LXe-calorimeter, see detailed descriptionin [6].(b) Study of the e + e − → K + K − π + π − Process
The study of e + e − → K + K − π + π − process has been performed on the base of pb − of data, collected in 2011-2012. The events with 3 and 4 DC-tracks were consideredwith the kaon/pion separation using the LLF maximization and cuts on L K π and L Kπ , see Fig. 3. For the class of events with 4 tracks the pure sample of signal eventswas selected using energy-momentum conservation law (see Fig. 4). For the 3-tracksclass the signal/background separation was performed by the fitting of energy disbal-ance ∆ E ≡ E K + + E K − + E π + (cid:112) m π + ( (cid:126)p K + + (cid:126)p K − + (cid:126)p π ) − √ s distribution, seeFig. 5. In total we selected about 24000 of signal events.Figure 2: The ( dE/dx ) DC distribution for thesimulated kaons and pions in K + K − π + π − finalstate. Figure 3: The L K π distribution for the 4-tracks events for data (open histogram) and K + K − π + π − simulation (blue histogram).The cut L K π > − . is applied to avoidmisidentification. All c.m. energy points arecombined.The cross section measurement for e + e − → K + K − π + π − process requires the ampli-tude analysis of the final state production. The major intermediate mechanisms werefound to be: • f (500 , φ (1020) ; 73igure 4: The energy disbalance vs. total mo-mentum for the 4-tracks events after kaon/pionseparation (data, E c . m . = 1 .
98 GeV ). The eventsinside the frame are considered to be signal. Figure 5: The signal/background separa-tion by fitting the energy disbalance distri-bution for 3-tracks events (data, E c . m . =1 .
98 GeV ). • ρ (770)( KK ) S − wave ; • ( K (1270 , K ) S − wave → ( K ∗ (892) π ) S − wave K ; • ( K (1400) K ) S − wave → ( ρ (770) K ) S − wave K .The relative amplitudes of these mechanisms at each c.m. energy point were foundusing the unbinned fit of the data, see the Monte-Carlo-data comparison after the fit inFigs. 6a– 6d.The results for the cross section measurement, based on the runs of 2011-2012 (pub-lished in [7]), are shown in Fig. 7. The preliminary results of the analyzis of new dataof 2017 show a drop of about in the visible cross section at p ¯ p threshold (seeFig. 8), similar to that in e + e − → π + π − cross section [9]. Such a drop at p ¯ p thresh-old is firstly observed in the final state with kaons, and, being confirmed, will requiretheoretical explanation.(c) Study of the e + e − → K + K − η, K + K − ω (782) processes The study of e + e − → K + K − η, K + K − ω (782) processes has been performed on thebase of pb − of data, collected in 2011-2012. In these two analyzes the η and ω (782) were treated as the recoil particles. The events with 2, 3 and 4 DC-tracks wereconsidered. The kaon/pion separation was performed using the LLF maximization andcuts on L K , L Kπ and L K π , see Fig. 9. The K + K − π + π − final state dominates thebackground in 3 and 4-tracks classes, but we suppress it’s contribution by the cuts onthe Kπ and K π missing masses, see Fig. 10. Since we observed only φ (1020) η intermediate mechanism of K + K − η production, we applied the cut on the K + K − invariant mass to select the φ (1020) meson region, see Fig. 11. The signal/backgroundseparation in both processes is performed by approximation of the K + K − missingmass distribution, see Fig. 12. The preliminary results for the cross sections are shownin Figs. 13–14. 74igure 6: The distributions of the K ± π ∓ (a), π + π − (b), K + K − (c) invariant masses and theangle between momenta of kaons (d). Data (points) and simulation (open histogram) for E c . m . =1 .
95 GeV . 75igure 7: The e + e − → K + K − π + π − cross section, measured on the base of the runs of 2011-2012years (black circles), along with the BaBar results (open bars) [8].Since the φ (1020) η production is dominated by φ (1680) meson decay, the approxima-tion of the e + e − → φ (1020) η cross section allows us to measure the φ (1680) parameters.We perform the approximation using the following formula: F ( s ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A non − φ (cid:48) ( s ) e i Ψ + (cid:115) (Γ φ (cid:48) ee B ( φ (cid:48) → φη ))Γ φ (cid:48) m φ (cid:48) | (cid:126)p φ ( m φ (cid:48) ) | D φ (cid:48) ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2)In these formulae D φ (cid:48) ( s ) = 1 / ( s − m φ (cid:48) + i √ s Γ φ (cid:48) ( s )) and D φ ( p φ ) = 1 / ( p φ − m φ + i (cid:113) p φ Γ φ ( p φ )) are the inverse denominators of the φ and φ (cid:48) propagators, | (cid:126)p φ ( √ s ) | is themomentum of the φ in the φ (cid:48) → φη decay in φ (cid:48) rest frame, | (cid:126)p K ( (cid:113) p φ ) | is the momentumof the kaon in the φ → K + K − decay in φ rest frame, θ normal is the polar angle of thenormal to the plane, formed by the (cid:126)p K + and (cid:126)p K − vectors, d Φ K + K − η is the element ofthree-body phase space, the function A non − φ (cid:48) ( s ) = a/s is introduced to describe thepossible contribution of the resonances apart from φ (cid:48) ( a is a constant), Ψ is the relativephase between two amplitudes. The results of the fit, shown in the Table 1, are in goodagreement with those in BaBar study [10].For the σ ( e + e − → K + K − ω (782)) approximation we used the shape, obtained from theintegration of the squared matrix element of e + e − → φ (cid:48) → K + K − ω (782) , ω (782) → ρ ± , π ∓ , → π + π − π decay chain over the 5-body phase space.Table 1: The φ (cid:48) parameters obtained from the fit.Parameter Value χ / n . d . f 46 . / ≈ . φ (cid:48) ee B ( φ (cid:48) → φη ) , (cid:48) eV 163 ± stat ± mod m φ (cid:48) , MeV 1690 ± stat ± mod Γ φ (cid:48) , MeV 327 ± stat ± mod e + e − → K + K − π + π − process with the drop at p ¯ p threshold(on the base of 2017 year runs).(d) Study of the e + e − → K + K − π Process
The study of e + e − → K + K − π process has been performed on the base of pb − ofdata, collected in 2011-2012. The events with two oppositely charged DC-tracks and noless than 2 photons with the E γ >
40 MeV were considered. Then, assimung energy-momentum conservation, the 4C-kinematic fit was performed with the χ C < cut,see Fig. 15. The major background sources were found to be K + K − γ , K + K − π , K S,L K ± π ∓ , π + π − π , π + π − π . The suppression of these backgrounds was done us-ing the training of BDT classifiers (see Fig. 16) with the following input variables: 1) ( dE/dx ) DC ; 2) momenta and angles of charged particles and photons; 3) missing massof K + K − system. The preliminary results for the e + e − → K + K − π cross section areshown in Fig. 17.4. Conclusion
The current status of the study of processes of e + e − → KK ( n ) π with the CMD-3 detectorwas considered. The CMD-3 has alredy collected about pb − of data and now is contin-uing datataking to collect about 1 fb − in the next few years. We are in good disposition toprovide the best precision for the φ (1680) vector meson parameters and to perform the studyof KK ( n ) π final states in all charge modes to test the isotopic relations between them. Thedrop in the e + e − → K + K − π + π − cross section, seen in the preliminary analysis of the dataof 2017 year runs, is firstly observed in the final state with kaons, and, being confirmed, willrequire theoretical explanation.5. Acknowledgments
We thank the VEPP-2000 personnel for the excellent machine operation. The work wassupported by the Russian Fund for Basic Research grants RFBR 15–02–05674–a, RFBR77igure 9: The distribution of L Kπ parameterin the experiment and simulation of signal andmajor background processes. All c.m. energypoints are combined. Figure 10: The distribution of the K π miss-ing mass in the experiment and simulation ofsignal and major background processes. Allc.m. energy points are combined.Figure 11: The distribution of K + K − invariantmass in the experiment and simulation of sig-nal and major background processes. All c.m.energy points are combined. Figure 12: The distribution of K + K − missingmass in the experiment and simulation of sig-nal and major background processes. All c.m.energy points are combined.78igure 13: The results for the e + e − → K + K − η cross section (red - CMD-3, preliminary; black- BaBar [10]). Figure 14: The results for the e + e − → K + K − ω (782) cross section (red -CMD-3, preliminary; black - BaBar [11]).Figure 15: The distribution of the χ of the 4C-kinematic fit of the events (red - experiment,green - simulation of signal process). The en-ergy points in range .
54 GeV < E c . m . < .
65 GeV are combined. Figure 16: The distribution of the BDT re-sponse of the events (red - experiment, green- simulation of signal process). The energypoints in range . < E c . m . < .
58 GeV are combined.79igure 17: The results for the e + e − → K + K − π cross section (blue and red - CMD-3 2011-2012,preliminary; black - BaBar [10]).14–02–00580–a, RFBR 16–02–00160–a, RFBR 17–02–00897–a. Part of this work relatedto the photon reconstruction algorithm in the electromagnetic calorimeter is supported bythe Russian Science Foundation (project No. 14–50–00080). References [1] K. Hagiwara, R. Liao, A.D. Martin, D. Nomura, and T. Teubner, J. Phys. G , 085003(2011).[2] G.W. Bennett et al. (Muon g-2 Collaboration), Phys. Rev. D , 072003 (2006).[3] D.E. Berkaev et al. , JETP , 213 (2011).[4] B.I. Khazin et al. (CMD-3 Collaboration), Nucl. Phys. B (Proc. Suppl.) , 376 (2008).[5] V.M. Aulchenko et al. , JINST , P10006 (2015).[6] V. L. Ivanov et al. , JINST , no.09, C09005 (2017).[7] D.N. Shemyakin et al. , Phys. Lett. B , 153 (2016).[8] J.P. Lees et al. [BaBar Collaboration], Phys. Rev. D , (2012) 012008.[9] R.R. Akhmetshin et. al. , Phys. Lett. B , 82 (2013).[10] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D , (2008) 092002.[11] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D , (2008) 092005.80 .11 The GlueX Meson Program Justin Stevens (for the GlueX Collaboration
Department of PhysicsCollege of William & MaryWilliamsburg, VA 23187, U.S.A.
Abstract
The GlueX experiment is located in Jefferson Lab’s Hall D, and provides a unique capabil-ity to study high-energy photoproduction, utilizing a 9 GeV linearly polarized photon beam.Commissioning of the Hall D beamline and GlueX detector was recently completed and thedata collected in 2017 officially began the GlueX physics program. Meson Photoproduction
The GlueX experiment, shown schematically in Fig. 1, utilizes a tagged photon beam de-rived from Jefferson Lab’s 12 GeV electron beam. Coherent bremsstrahlung radiation froma thin diamond wafer, yields a linearly polarized photon beam with a maximum intensitynear 9 GeV. The primary goal of the experiment is to search for and ultimately study an un-conventional class of mesons, known as exotic hybrid mesons which are predicted by LatticeQCD calculations [1]. barrelcalorimeter time-of-flight forward calorimeter photon beamelectronbeamelectronbeam superconductingmagnet targettagger magnet tagger to detector distanceis not to scale diamondwafer G lue X central driftchamberforward driftchambersstartcounter DIRC
Figure 1: A schematic of the Hall D beamline and GlueX detector at Jefferson Laboratory. TheDIRC detector upgrade will be installed directly upstream of the time-of-flight detector in theforward region.To pursue the search for exotic hybrid mesons in photoproduction the production of conven-tional states, such as pseudoscalar and vector mesons, must first be understood. Previous81 ) c (GeV/ -t Σ <9.0 GeV γ GlueX 8.4 As described above, an initial physics program to search for and study hybrid mesons whichdecay to non-strange final state particles is well underway. However, an upgrade to theparticle identification capabilities of the GlueX experiment is needed to fully exploit itsdiscovery potential, by studying the quark flavor content of the potential hybrid states.This particle identification upgrade for GlueX will utilize fused silica radiators from theBaBar DIRC (Detection of Internally Reflected Cherenkov light) detector [3], with new,compact expansion volumes to detect the produced Cherenkov light. The GlueX DIRC willprovide π/K separation for momenta up to 4 GeV, significantly extending the discovery po-tential of the GlueX program [4]. The charged kaon identification provided by the DIRC mayalso yield useful identification of charged kaons for the K L beam facility (KLF) proposedfor Hall D [5]. This may be particularly relevant in the production of strange mesons fromthe high momentum component of the K L beam, where the charged kaons are produced atforward angles.3. Acknowledgments This work is supported by the Department of Energy Early Career Award contract DE–SC0018224. References [1] J. J. Dudek et al. , Phys. Rev. D , 094505 (2013).822] H. Al Ghoul et al. [GlueX Collaboration], Phys. Rev. C , 042201 (2017).[3] I. Adam et al. [BaBar DIRC Collaboration], Nucl. Instrum. Meth. A , 281 (2005).[4] J. Stevens et al. , JINST , C07010 (2016).[5] S. Adhikari et al. (GlueX Collaboration), arXiv:1707.05284 [hep-ex].83 .12 Strange Meson Spectroscopy at CLAS and CLAS12 Alessandra Filippi I.N.F.N. Sezione di Torino10125 Torino, Italy Abstract The CLAS Experiment, that had been operating at JLAB for about one decade, recentlyobtained the first high statistics results in meson spectroscopy exploiting photon-induced reac-tions. Some selected results involving production of strangeness are reported, together with adescription of the potentialities of the new CLAS12 apparatus for studies of reactions inducedby quasi-virtual photons at higher energies. Introduction The identification of states containing open and hidden strangeness is still an open issuein light meson spectroscopy investigations. Apart form a handful of confirmed states, stilllittle is known, for instance, about the radial excitations of the φ (1020) meson, and evenless about strangeonia with quantum numbers other than −− . Below 2.1 GeV just abouthalf the open strangeness kaonia states (composed by a strange and a light quark) expectedby the Constituent Quark Model (CQM) [1] have been observed so far, while less than tenstrangeonia states, mesons of ¯ ss structure, have been steadily observed out of a total of atleast 20 expected states.Nonetheless, the knowledge of the properties of such states can provide important inputs forhadron spectroscopy. Strangeonia, in fact, feature an intermediate mass between the heaviersystems where the quark model is approximately valid and the lighter meson sector. Unfor-tunately, their experimental signatures are less clear as compared to charmonium states, asthey are broader and lie in a mass range where the overlap probability with other light quarkstates is very strong, and moreover most of the decay channels modes are shared among allof them.To further complicate the problem, in the same mass region some other structures of exoticcomposition ( ¯ qqg states, known as hybrids, ggg states, the so-called glueballs, or even molec-ular states of multi-quark composition) are expected by QCD. Recent QCD calculations onthe lattice are able to predict most of the conventional meson spectrum in good agreementwith experimental findings [2]; these confirmations of course strengthen the confidence intheir predictive power for the searches of new states. According to these calculations, thelightest hybrids and glueballs are predicted in the 1.4–3 GeV mass range: namely, at 2 GeVfor the lightest J P C = 0 + − state, and at 1.6 GeV for the − + one. This is actually the massregion where signatures of still unobserved strangeonia or kaonia are expected to show up,and indeed a number of candidates for exotics has been suggested over the years, still allawaiting for confirmation, fitting the same slots where strangeonia would be expected.The use of photons as probes to study strangeness was not used extensively in the past dueto the small production rates, and the lack of beams of suitable intensity and momentumresolution. In fact, the production of strange quarks in a non-strange environment involves84isconnected quark diagrams whose occurrence is suppressed as a consequence of the OZIrule. However, a good step forward is expected from the results obtained with the pho-ton beam produced by brehmsstrahlung from the continuous electron beam at the CEBAFmachine at JLAB.Smoking guns for the existence of open and hidden strangeness states would by their obser-vations in the φη or φπ invariant mass systems [3]; especially in the first case, due to thestrange content of the η meson, the production of strangeness should be eased [4]. A typicaldiagram for the photoproduction of the φη final state is shown in Fig. 1.Figure 1: Quark line diagram for the photoproduction of the φη final state.Very few events in these channels have been observed so far; however, there is a good pos-sibility, as will be shown in Sec. 5, that in the upcoming meson spectroscopy experiment atCLAS12 good samples of such reactions could be collected, opening therefore new oppor-tunities to widen the knowledge of this sector of the meson spectrum.2. Meson Spectroscopy Studies in Photon Induced Reactions Several reactions and beams have been exploited so far for meson spectroscopy searches:among most important, high-energy meson (mainly pion) and proton beams, based on theperipheral and central production mechanisms, antinucleon annihilation at rest and in flight,which convey the formation of a gluon-rich environment suitable for glueball production,and e + e − annihilation. The latter reaction has been studied extensively since the LEP era,and is an environment where also γγ collisions can be measured, which provide quite usefulinformation as they are a natural anti-glueball filter. The e + e − annihilation differs fromhadronic reactions for the fact that only −− systems can be formed, so these reactionsprovide naturally a powerful quantum number selection.On the other hand, the use of electromagnetic probes, and in particular of photons, in fixedtarget reactions was not used very extensively for meson spectroscopy purposes, as mean-tioned earlier. Nevertheless, electromagnetic reactions could deliver important complemen-tary information. In fact, first of all, electromagnetic processes can be exactly reproduced toa high level of precision through QED diagrams thanks to the smallness of the electromag-netic coupling, which is prevented in the case of strong interactions. Moreover, photons can85xcite with larger probability the production of spin-1 mesons as compared to pion or kaoninduced reactions, since in the latter case a spin-flip is required. Therefore, the productionof vector hybrid mesons could occur in photoproduction reactions with a rate comparable tothose of conventional vector mesons [3, 4]. This feature applies not only to hybrids, but alsoto spin-1 ¯ ss excitations.3. Selected Results from CLAS The CLAS apparatus, which had operated up to 2010, is described in detail elsewhere [5].In the following some selected results from recent meson spectroscopy papers involvingstrangeness production will be summarized, in connection with some still open issues.(a) The Scalar Glueball Search Case In spite of the efforts by the experiments in the early Nineties, in particular those study-ing antinucleon annihilations, like Crystal Barrel and OBELIX, the full composition ofthe scalar meson sector is not completely clear yet. Observations have been madeof several mesons whose existence was not foreseen by the CQM; one of them, the f (1500) , seemed to have the right features as lightest scalar glueball candidate [6].Among these, the most important is the fact that it was observed to decay in severalchannels, a clear hint to its flavor-blindness. However, more data are still desirable toconfirm these properties as several other structures tentatively identified as scalars aswell. All the existing observations make the interpretation of the scalar sector diffi-cult, since it is populated more than expected by broad and overlapping states. In thismass region one should also recall the existence of the σ state (also known as f (600) ),corresponding to a very broad ππ non-resonant iso-scalar S -wave interaction, whosenature and properties are still unclear. Also scarcely known are the features of the κ ,the analogous of σ observed in the KK channel.A search was carried on in CLAS exploiting the γp → pK S K S reaction [7], with realphotons of energy in the ranges (2 . − and (3 . − . GeV. The K S were fullyreconstructed through their decay in two pions, while the proton was identified fromthe event missing mass. A strong correlation between the two K S was found, whichallowed to collect a clean sample of good statistics, enough to perform selections inmomentum transfer t (= Q ) . The selection in t is useful to understand the produc-tion mechanism of the intermediate states: small values of momentum transfers arecorrelated to a dominant production from the t -channel, while s -channel production ischaracterized by a wider range of transferred momenta. Fig. 2 shows the distributionsof the ( K S K S ) invariant mass system, after proper background subtraction, in two mo-mentum transfer ranges: for | t | < GeV on the left side, and for | t | > GeV onthe right. A clean peak at about 1500 MeV appears in the first case, while at largermomentum transfer no evidence for it can be observed. This means that the structure at1500 MeV, a possible indication for the f (1500) observed in its K S K S decay mode,is predominantly produced via t -channel. This observation could support its possibleinterpretation as glueball.In order to fully characterize the features of this state a complete spin-parity analysis ofthe sample is required. Unfortunately, the task is not straightforward due to the limited86igure 2: Invariant mass of the ( K S K S ) system for events selected in the γp → pK S K S reaction,in two momentum transfer ranges: left, for | t | < GeV , right for | t | > GeV .apparatus acceptance at small forward and backwards angles. Nonetheless, an angularanalysis was attempted to test which of the spin-parity hypotheses for the events in thepeak region would provide a better fit to the Gottried-Jackson angular distributions.We recall that a ( K S K S ) bound system may just have J P C = ( even ) ++ quantumnumbers, so the spin 0 and 2 hypotheses need to be tested. An example of angulardistribution is reported in Fig. 3, in a ( K S K S ) mass window centered at 1525 MeV. Thecurves superimposed to the experimental data show the contributions of S and D waves(blue and red, respectively) to the total fit (green). The D -wave contribution plays amarginal rôle, larger for masses higher than 1550 MeV, from which one can deducethat the scalar hypothesis for the observed resonance is mostly supported; therefore, itcan be more easily identified as the f (1500) .Figure 3: Gottfried-Jackson angular distribution for K S K S events selected in the 1525 MeV invari-ant mass slice, with superimposed the expected trend, on the basis of Monte Carlo simulations, forproduction from S wave (blue), D wave (red), and the global fit with the two components (green).(b) The Axial/Pseudoscalar Sector at 1.4 GeV Case Kaonia radial excitations were widely studied in the past together with η excitations inthe same mass range, to search for possible exotic states. Many observation of η ’s and f mesons have been reported since the Sixties, when the issue of the overlap of manyaxial and pseudoscalar states and the difficulty of their identification posed the so-called87 /ι puzzle [8]. While annihilation experiments, and in particular OBELIX, couldprovide a solution to this puzzle addressing the production of several pseudoscalarand axial states, high statistics photoproduction reactions are expected to deliver newcomplementary information which will be able to improve the knowledge in this sector.A systematic study was performed by CLAS to study the photoproduction of states,decaying into ηπ + π − and K K ± π ∓ and recoiling against a proton, in the γp reactionwith photons in the energy range (3 − . GeV [9]. Fig. 4 shows the missing massplot of the system recoiling against a proton for γp → pηπ + π − selected events, wherea clean signal due to the η (cid:48) (958) appears, together with a structure at about 1280 MeV,that can tentatively be identified as the f (1285) .Figure 4: Missing mass recoling against a proton for events selected in the γp → pηπ + π − reaction.Concerning the reaction witk kaons, the K ± were identified by CLAS through time offlight techniques, while the K via the missing mass information. The missing massplots for the system recoiling against the proton for events selected in the two channels γp → pK K + π − and γp → pK K − π + are shown, respectively, in Fig. 5(a) and (b).Figure 5: Missing mass recoling against a proton for events selected in the a) γp → pK K + π − and b) γp → pK K − π + reactions.In both of them a clear peak appears at about 1.3 GeV, but no further evidence forhigher mass states that could be addressed to additional pseudoscalar states, like the η (1405) and η (1470) , or to axial states, as the f (1420) of f (1510) , is present.88rom the richest ηπ + π − sample, one can get information for the identification of theobserved state. There are a few hints that support the identification as axial f (1285) against the pseudoscalar η (1295) . The first one is given by the values obtained for themass and width of the signal, M = (1281 . ± . MeV and Γ = (18 . ± . MeV,which are closer to those already observed for f (1285) [10]. The ratio itself of thedecay rates in ηπ + π − versus KKπ , that amounts to about five, is consistent with thevalue quoted by PDG for the f (1285) decays, while no such ratio was ever measuredfor η (1295) [10].The second hint is given by the trend of the differential cross section, for events selectedin the band of η (cid:48) (958) and around 1280 MeV: as shown in Fig. 6, the two are remarkablydifferent as a function of the center-of-mass angle, indicating a different productionmechanism and possibly a different spin configuration of the produced state.Figure 6: Differential cross section of the γp → ηπ + π − p reaction for events selected in the η (cid:48) (958) mass band (red open points) and the f (1285) one (blue points). The energy in the center of massis fixed at 2.55 GeV.The comparison of the cross sections for events in the 1280 MeV band with the ex-pectations from t -channel based models [11–13], shown in Fig. 7, indicates clearly apoor match with this production hypothesis: a substantial contribution from s -wavecould be needed, or a mechanism different from meson exchange involving N ∗ excita-tions or KK ∗ molecular interactions. In both cases the identification of the structure as f (1285) would get larger support.4. Prospects for CLAS12 The CLAS12 spectrometer, an upgraded version of CLAS, is a multipurpose facility dedi-cated to hadron physics studies, from nuclear properties and structure to meson spectroscopyinvestigations. A full description of CLAS12 is given elsewhere [14]. A second experimentinstalled on the CEBAF machine, GlueX, is operating in parallel with a program fully com-mitted to meson spectroscopy [15].Due to the increased beam energy, as compared to the previous installation, the CLAS12spectrometer has a more compact structure, which limits its geometric acceptance, although89igure 7: Differential cross sections of the γp → ηπ + π − p reaction for events selected in the f (1285) mass band in two center-of-mass energy bins (left: 2.45 GeV, right: 2.85 GeV), comparedwith a few t -channel exchange based models (none of which matches the experimental points):solid red line from Ref. [11], blue dashed line from Ref. [12], black dotted line from Ref. [13].allowing an acceptable hermeticity. However, its detectors feature better momentum resolu-tion and particle identification capabilities as compared to GlueX, therefore the two experi-mental setups offer complementary qualities.While GlueX will use a real photon bremsstrahlung beam, this is not be possible for CLAS12as the bending dipole magnet available in Hall B will not be powerful enough to steer the11 GeV electron beam into the tagger beam dump. A new technique was therefore con-ceived to produce a photon beam for the study of photoproduction reactions, based on elec-tron scattering at very low transferred momentum ( Q < − GeV ). In this situation theelectrons are scattered at very small polar angles, and the reaction on protons is induced byquasi-real photons. A dedicated part of the CLAS12 apparatus, the Forward Tagger , wasdevised to detect these forward scattered electrons. For momentum transfers in the interval . < Q < . GeV , the electrons scattered between . ◦ and . ◦ in polar angle inthe lab have an energy in the 0.5–4.5 GeV range, being produced by the interaction of 6.5–10.5 GeV quasi-real photons on protons. Measuring the electron momentum, each virtualphoton can be tagged. The use of the CLAS12 spectrometer to measure in coincidence theparticles produced in the photoproduction reaction will allow to perform a complete eventrecontruction, necessary for meson spectroscopy studies. With this tagging technique onecan measure also the virtual photon polarization, that is linear and can be deduced, event byevent, from the energy and the angle of the scattered electron. The systematic uncertaintyaffecting the polarization depends only on the electron momentum resolution. High electroncurrents may be used, therefore a good luminosity can be obtained even with thin targets,that are not operable with real photon bremsstrahlung beams. For instance, using a 5 cmlong LH target, the resulting hadronic rate will be equivalent to that achievable by a realphoton flux of about × γ /s.The Forward Tagger equipment, described in detail in Ref. [16], is located about 190 cmaway from the target and fits within a 5 ◦ cone around the beam axis. It is made up of: • an electromagnetic calorimeter (FT-Cal): composed by 332 PbWO crystals, 20 cmlong and with square × mm cross-section. It is used to identify the scatteredelectron and measure its energy, from which the photon energy and its polarization can90e deduced (the polarization being given by (cid:15) − ∼ ν / ( EE (cid:48) ) , where ν = E − E (cid:48) is the photon energy, and E and E (cid:48) the energies of the incident and of the scatteredelectron, respectively). It is also used to provide a fast trigger signal. Its expecteddesign resolution is σ E /E ∼ (2% / (cid:112) E (GeV) ⊕ ; • a scintillator hodoscope made of plastic scintillator tiles: located in front of the calorime-ter, is used to veto photons; its spatial and timing resolution is required to be compara-ble with FT-Cal’s; • a tracker: located in front of the hodoscope and composed by Micromegas detectors, isused to measure the angle of the scattered electron and the photon polarization plane.A sketch of the Forward Tagger region is shown in Fig. 8.Figure 8: Schematic view of the Forward Tagger equipment, to be hosted in CLAS12.5. Strange Meson Spectroscopy with CLAS12 The Meson-EX experiment at CLAS12 (Exp-11-005 [17]) was proposed to study of themeson spectrum in the 1–3 GeV mass range through quasi-real photon induced reactions,for the identification of gluonic excitations of mesons and other exotic quark configurationsbeyond CQM. The use of the Forward Tagger will allow to identify the photoproductionreaction on protons through the tagging of the forward scattered electron, while the fullCLAS12 apparatus will perform a complete reconstruction and identification of the chargedand neutral particles produced in the interaction. Some golden channels have been selectedas particularly suitable for the search of exotic or still unknown particles, whose production,as mentioned before, will be favored in photoproduction especially in the case of spin 1particles. In particular, for the search of hybrids with open strangeness and strangeonia,the following channels are expected to provide interesting new indications: γp → φπ p , γp → φηp and γp → KKπp . For the first two, extensive simulations were carried over to91est the feasibility of such measurements and the expected collectable statistics, in a 80-dayslong data taking at the full CEBAF luminosity ( ∼ s − cm − ) with a total expected triggerrate for photoproduction reactions less than 10 kHz. Assuming an apparatus acceptance forfour track events of the order of 15%, it will be possible to collect as many as 3000 events per10 MeV mass bin, for reactions with a cross section as small as 10 nb, expected for instancefor strangeonia production. This statistics is considered to be enough to perform detailedpartial-wave analysis studies.More in detail, concerning the φη channel and the possible production of the φ (1850) exci-tation, simulations showed that an overall acceptance of about 10% can be expected, due tothe coverage at small angles provided by the Forward Tagger electromagnetic calorimeter inwhich the photons from the η decay can be detected. The full event would be reconstructedidentifying, in addition, at least one of the kaons from the φ decay and the recoiling proton,the K − being identifiable through the reaction missing mass. A cross section on the order of10 nb is tentatively expected, given an existing estimation by the CERN Omega Collabora-tion [18] of about 6 nb at higher energies, for the production of φ (1850) and its decay in the K + K − mode, whose branching ratio is expected to be about twice as large as compared to φη .On the other hand, concerning the φπ intermediate state which could be a possible sourceof exotic systems, again a resonable detection efficiency could be achieved requiring a fullidentification of the positive kaon from the φ and the reconstruction of the π through twophotons in the Forward Tagged calorimeter. The negative kaon acceptance, largely reducedas it bends inwards at small angles, therefore in a region of scarce detector coverage, can berecovered identifying the particle via the missing mass information; this of course demandsagain a good resolution on the photon energy. In these conditions, an acceptance similar tothose of more “conventional” intermediate states (composed for instance by pions only) canbe expected, with basically no impact by the chosen intensity of the magnetic field.6. Conclusion Nowadays many communities are involved all over the world in hadron spectroscopy stud-ies, both from experimental and from the theoretical point of view. Experimental facilitiesare presently delivering, and will deliver in the near future, large amounts of good qualitydata that will allow to extract for the first time, from different reactions and experimentalenvironments, new information on many still unsolved issues.Photoproduction experiments at JLab will play a big role in the coming future. At JLab,GlueX and CLAS12 will provide data samples of unprecedented quality and richness. Theircontribution is expected to grow to sizeable relevance and have a big impact in the presentand future meson spectroscopy experimental scenario, complementing the information thatwill be provided by e + e − collisions (by BESIII and BelleII), by pion induced interactions(by COMPASS), by proton-proton interactions (at fixed target by LHCb, and in high-energy pp collisions by ATLAS and CMS) and by antiproton-proton annihilations (by PANDA atFAIR). 92 eferences [1] S. Godfrey and J. Napolitano, Rev. Mod. Phys. , 1411 (1999).[2] J. Dudek et al. , Phys. 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Dickson et al. , Phys. Rev. C , 065202 (2016).[10] C. Patrignani et al. [Particle Data Group], Chin. Phys. C , 100001 (2016).[11] N. I. Kochelev et al. , Phys. Rev. C, , 025201 (2009).[12] S. K. Domokos et al. , Phys. Rev. D, , 115018 (2009).[13] Y. Huang et al. , Int. Mod. Phys. E , 1460002.[14] CLAS12 Technical Design [CLAS Collaboration], CLAS12 Technical Design Report , Tech.Rep. Jefferson Laboratory, 2008.[15] [GlueX Collaboration], Mapping the Spectrum of Light Mesons and Gluonic Excitations withLinearly Polarized Photons , Proposal to Jefferson Laboratory PAC30, 2006;url: http://argus.phys.uregina.ca/cgibin/public/DocDB/ShowDocument?docid=1226[16] CLAS Collaboration, CLAS12 Forward Tagger (FT) Techni-cal Design Report, Tech. Rep. Jefferson Lab, 2012; url:http://clasweb.jlab.org/wiki/index.php/CLAS12_Technical_Design_Report9317] CLAS Collaboration, Meson Spectroscopy with low Q electron scattering in CLAS12 , Pro-posal to Jefferson Laboratory PAC37, 2011[18] D. Aston t al. , Phys. Lett. , 231 (1981).94 .13 Dispersive Determination of the π − K Scattering Lengths Jacobo Ruiz de Elvira, Gilberto Colangelo, and Stefano Maurizio Albert Einstein Center for Fundamental PhysicsInstitute for Theoretical Physics, University of Bern3012 Bern, Switzerland Abstract The pion-kaon scattering lengths are one of the most relevant quantities to study the dy-namical constraints imposed by chiral symmetry in the strange-quark sector and hence, theyare a key quantity for understanding the interaction of hadrons at low energies. In this talk wereview the current status of their determination. After discussing the predictions expected fromchiral symmetry at different orders in the chiral expansion, we review current experimental andlattice determinations. We then focus on the dispersive determination, based on a Roy-Steinerequation analysis of pion-kaon scattering, and discuss in detail the current tension between thechiral symmetry and dispersive solutions. We finish this talk providing an explanation of thisdisagreement. Introduction Pion-kaon scattering is one of the simplest processes to test our understanding of the chiralsymmetry-breaking pattern in the presence of the strange quark. In particular, its low-energyparameters, most notably the scattering lengths, encode relevant information about the spon-taneous and explicit chiral symmetry breaking in this sector. Being low-energy observables,their properties can be efficiently studied using the effective field theory of Quantum Chro-modynamics (QCD) at low energies, Chiral Perturbation Theory (ChPT) [1–3], constructedas a systematic expansion around the chiral limit of QCD in terms of momenta and quarkmasses.Pion-kaon scattering can be expressed in terms of two independent invariant amplitudes withwell defined isospin I = 1 / and I = 3 / in the πK → πK channel, namely T / and T / .Nevertheless, for convenience, it is useful to combine them in terms of isospin-even and -oddamplitudes I = ± , which are defined as T ab = δ ab T + + 12 [ τ a , τ b ] T − , (1)where a and b denote pion isospin indices and τ a stand for the Pauli matrices. Both basis arerelated by simple isospin transformations: T / = T + + 2 T − , T / = T + − T − . (2)At leading order (LO) in the chiral expansion, i.e., in the expansion in pion and kaon massesand momenta, the scattering amplitude is given by the Feynman diagram shown in Fig. 1a,resulting in the well-known low-energy theorems for the S -wave scattering lengths, the am-plitudes evaluated at threshold [3, 4]: a − = m π m K π ( m π + m K ) f π + O (cid:0) m i (cid:1) , a +0 = O (cid:0) m i (cid:1) , (3)95 K π π /f π ( a ) KK π πL i ( b ) Figure 1: (a) Leading-order diagrams for πK scattering in chiral perturbation theory. Kaon aredenoted by full, pions by dashed lines. (b) Next-to-leading-order diagrams depending on low-energy constants L − .where m i denotes the pion ( m π ) or kaon ( m K ) mass. The isospin-odd scattering lengthis hence predicted solely in terms of the pion and kaon masses as well as the pion decayconstant f π , while the isospin-even one is suppressed at low energies. The LO ChPT valuefor the pion-kaon scattering lengths is denoted by a star in Fig. 2, where the pion-kaonscattering length plane is plotted in the I = 1 / and I = 3 / basis.The pion-kaon scattering amplitude at next-to-leading order (NLO) was derived and studiedfirst in [5, 6]. It involves one-loop diagrams, which might generate large contributions. Nev-ertheless, they are suppressed at threshold and hence their role for the pion-kaon scatteringlengths is relatively small. In addition, the πK scattering amplitude depends at NLO on alist of low-energy constants (LECs), conventionally denoted by L − , which, encoding in-formation about heavier degrees of freedom, can not be constrained from chiral symmetrysolely, Fig. 1b. Once determined in one process, these LECs can subsequently be used topredict others. The NLO pion-kaon scattering lengths were analyzed in detail in [5]. Thecontribution from the LECs reads [5]: a − (cid:12)(cid:12) LECs = m K m π π ( m π + m K ) f π L + O (cid:0) m i (cid:1) , (4) a +0 (cid:12)(cid:12) LECs = m K m π π ( m π + m K ) f π (4( L + L − L ) + 2 L − L + 2(2 L + 2 L )) + O (cid:0) m i (cid:1) . On the one hand, whereas the NLO LECs contribution to the isospin-odd scattering lengthis suppressed by a m π factor, the isospin-even scattering length is proportional to m K . Thus,one should expect a small higher-order correction in the chiral expansion for a − but a muchlarger shift for a +0 . On the other hand, while the isovector scattering length only dependson one LEC, L , which is indeed well constrained from the pion and kaon mass and decayconstant values, the isoscalar scattering length involve seven LECs and hence sizable uncer-tainties for this quantity associated with LECs errors are expected. The constraint imposedby the isovector scattering length at NLO in ChPT is depicted by a dark-green band in Fig. 2,where the central value and error for L is taken from the FLAG group estimate [7]. Notethat the small width of this band is completely determined by the L uncertainty. Further-more, using the LECs collected in Table 1 in Ref. [8] (first column), one can also includethe NLO prediction for the isoscalar scattering length, which leads to the solid-red ellipse in96 .14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a M π −0.080−0.075−0.070−0.065−0.060−0.055−0.050−0.045−0.040 a M π CA FLAG 16BE14 p C i = 0BE14 p RχPT p Dirac 17Flynn et al. 07NPLQCD 06Fu 12PACS 14B ¨uettiker et al. 04Universal Band Figure 2: Different determinations of the pion-kaon scattering lengths in the I = 1 / , I = 3 / basis. The LO ChPT value, just a result of current algebra, is denoted by a star. The NLO predictionof the isospin-odd scattering length is given by the dark-green band labelled as FLAG16. Theinclusion of the isospin-even scattering length, using the LECs provided in Ref. [8], leads to thesolid-red ellipse BE14 p . The red dot-dashed ellipse, BE14 p C i = 0 , corresponds to the NNLOchiral prediction when the O (cid:0) p (cid:1) LECs are set to zero. The full NNLO result is represented bythe dashed-red ellipse, albeit, as explained in the text, this result is biased to reproduce the RSdispersive values given in Ref. [10], solid-green ellipse. The NNLO results with O (cid:0) p (cid:1) LECsestimated by resonance saturation is denoted by the dashed-blue ellipse, RχP T p . The universalband obtained from the RS analysis performed in this work is given by the violet band. Theremaining experimental and lattice results are explained in the main text.Fig. 2. As we have already anticipated, this ellipse is stretched out in the isoscalar directionbut shrunk in the isovector one.One might wonder how stable are the NLO predictions against higher-order corrections. Apion-kaon low-energy theorem [5] imposes higher O (cid:0) m ni (cid:1) contributions to the isospin-odd97cattering length arising from contact terms to be at most: a − (cid:12)(cid:12) N n LO ∝ a − (cid:12)(cid:12) LO (cid:18) m π πf π (cid:19) (cid:18) m K πf π (cid:19) n with n ≥ . (5)Whereas the factor ( m K / πf π ) ∼ . is relatively large, the prefactor m π / (4 πf π ) ∼ . suppresses higher order corrections by roughly two orders of magnitude, i.e., the isospin-odd scattering length is protected from higher-order correction and hence one should expectsmall deviations from the NLO ChPT prediction for a − .The pion-kaon scattering amplitude at next-to-next-to-leading order (NNLO) in the chiralexpansion was derived in [9]. It involves a set of 32 new O (cid:0) p (cid:1) LECs, the so called C − ,which unfortunately are still not well constrained from experiment. As a first step, one couldestimate the size of the NNLO chiral corrections by setting all the C i to zero. Using for the O (cid:0) p (cid:1) L i the corresponding fit in [8], the outcome is the red dot-dashed ellipse plotted inFig. 2. This result is consistent with our previous statement, i.e., whereas the shift betweenthe NLO and NNLO ellipsis is small in the isovector direction, it is much larger in theisoscalar one.The C i entering in pion-kaon scattering were also estimated in [8] by performing a global fitto different ππ and πK observables. Nevertheless, among them, the dispersive determinationof the πK scattering lengths in [10] was used as constraint. Consequently, the full O (cid:0) p (cid:1) results in [8] are not a genuine ChPT prediction but they are biased to satisfy the resultsgiven in [10]. The scattering length results in [8] and [10] are denoted in Fig. 2 by thedashed-red and dashed-green circle-filled ellipse, respectively. As we will discuss in detailbelow, the large difference one finds between the NLO and NNLO chiral estimates in theisovector direction is just a consequence of the large discrepancy between the dispersiveresult in [10] and chiral expectations. Alternatively, one can estimate the value of the C i by using resonance saturation. The contribution from vector and scalar resonances to thesaturation of the O (cid:0) p (cid:1) LECs was also studied in [9]. Using the vector and scalar resonanceparameter values extracted in [11] from a global ππ and πK fit, one obtains the dashed-blueellipse in Fig. 2, which is now consistent with the NLO prediction for the isovector scatteringlength.The only direct experimental information about the pion-kaon scattering lengths comes fromthe DIRAC experiment at CERN [12], where the lifetime of hydrogen-like πK atoms wasmeasured. They are a electromagnetically bound state of charged pions and kaons, π + K − and π − K + , which decay predominantly by strong interactions to the neutral pairs π ¯ K and π K . The πK atom lifetime and the scattering length are related through the so-calledmodified Deser formula [13–15], namely Γ S = 8 α µ p a − (1 + δ K ) , (6)where α is the fine structure constants, µ is the reduced mass of the π ± K ∓ system, p isthe outgoing momentum in the centre-of-mass frame and δ K accounts for isospin breakingcorrections [14–16]. The experimental determination of Γ S obtained at CERN yields [12] a − = (cid:0) . +0 . − . (cid:1) m − π , (7)98hich is denoted in Fig. 2 by a light-blue squared-filled band. Unfortunately, the experi-mental errors are still too large to provide useful information about the pion-kaon scatteringlengths. Nevertheless, there is still room for improvement, the statistical precision is ex-pected to improve by a factor 20 if the DIRAC Collaboration manages to run its experimentusing the LHC 450 GeV proton beam. The proposed kaon beam experiment at JLabcould certainly help to improve the current experimental information about the pion-kaon scattering lengths. On the lattice side, there is a plethora of results and we will only consider unquenchedanalyses. From a lattice analysis of the πK scalar form factor in semileptonic K l decays, thevalue a / = 0 . 179 (17) (14) m − π was reported in [17] for the pion-kaon scattering length inthe I = 1 / channel. This value corresponds to the gray squared-filled band in Fig. 2. Thefirst fully dynamical calculation with N f = 2 + 1 flavors was performed by the NPLQCDcollaboration, leading to [18] a / = (cid:0) . +0 . − . (cid:1) m − π , a / = (cid:0) − . +0 . − . (cid:1) m − π , (8)which is denoted in Fig. 2 by a dotted-blue ellipse. Further dynamical results for N f = 2 + 1 flavors were reported in [19] using a staggered-fermion formulation, a / = 0 . 182 (4) m − π ,a / = − . 051 (2) m − π , and by the PACS collaboration considering an improved Wilsonaction [20], a / = 0 . 183 (18) (35) m − π , a / = − . 060 (3) (3) m − π . These results aredepicted in Fig. 2 by a solid-yellow and a dotted-brown ellipse, respectively.As we have seen, all the previous results are consistent with chiral predictions, i.e., all ofthem are consistent within one standard deviation for the isospin-odd direction, whereasmuch larger differences are found in the isospin-even component. Nevertheless, the mostprecise up-to-date result was reported in [10] by solving a complete system of Roy-Steinerequations, corresponding to a / = 0 . m − π , a / = − . m − π . (9)This result is denoted in Fig. 2 by a light-green circle-filled ellipse and it lies more than3.5 standard deviations away from the NLO ChPT result. This disagreement is particularlypuzzling in the isospin-odd direction, where the ChPT prediction is protected by the low-energy theorem given in (5) and one should expect NLO ChPT to provide a reasonablyprecise value for the pion-kaon scattering lengths. In fact, previous dispersive analyses for ππ scattering provided results for the scattering lengths only within a universal band [22,23].High accuracy values were reached only after constraining dispersive results with chiralsymmetry. Thus, one might wonder why should things be different in pion-kaon scattering.In the remaining part of this talk we will try to answer that question.2. Roy–Steiner Equations for πK Scattering Dispersion relations have repeatedly proven to be a powerful tool for studying processes atlow energies with high precision. They are built upon very general principles such as Lorentzinvariance, unitarity, crossing symmetry, and analyticity.For ππ scattering, Roy equations (RE) [21] are obtained from a twice-subtracted fixed- t dis-persion relation, where the t -dependent subtraction constants are determined by means of99 ↔ t crossing symmetry, and performing a partial-wave expansion. This leads to a cou-pled system of partial-wave dispersion relations (PWDRs) for the ππ partial waves wherethe scattering lengths—the only free parameters—appear as subtraction constants. The useof RE for ππ scattering has led to a determination of the low-energy ππ scattering ampli-tude with unprecedented accuracy [22–24], which, for the first time, allowed for a precisedetermination of the f (500) pole parameters [25, 26].In the case of πK scattering, a full system of PWDRs has to include dispersion relations fortwo distinct physical processes, πK → πK ( s -channel) and ππ → ¯ KK ( t -channel), andthe use of s ↔ t crossing symmetry will intertwine s - and t -channel equations. Roy-Steiner(RS) equations [27] are a set of PWDR that combine the s - and t - channel physical regionby means of hyperbolic dispersion relations. The construction and solution of a completesystem of RS equations for πK scattering has been presented in [10].In more detail, the starting point for the work in [10] is a set of fixed- t dispersion for the pion-kaon isospin-even and -odd scattering amplitudes, where the t -dependent subtraction con-stants are expressed in terms of hyperbolic dispersion relations passing through the thresh-old, i.e., where the internal and external Mandelstam variables s and u satisfy the condition s · u = m K − m π . A twice- and once-subtracted version was considered for the isospin-even-odd amplitude, respectively, where the subtraction constants are the a ± scattering lengthsand the slope of the hyperbola in the t -direction for the isospin-even amplitude, b + , which,in the end, is written in terms of a sum rule involving the a − scattering length. Finally, thesolution of the RS equations is achieved by minimizing the χ -like function χ = (cid:88) l,I s N (cid:88) j =1 (cid:0) Re f I s l ( s j ) − F [ f I s l ]( s j ) (cid:1) , (10)where f I s l denotes pion-kaon partial-waves with angular momentum l and isospin I s , F [ f I s l ] stands for the functional form of the RS equations for the f I s l partial wave, and the mini-mizing parameters are the partial waves and the pion-kaon scattering lengths. In this way,the minimum of (10) provides as an output the pion-kaon scattering length values givenin (9). A relevant question is whether this solution is unique. In principle, the subtractedversion built in [10] is constructed in such a way that it matches the conditions ensuring aunique RS equation solution investigated in [28]. In the ππ RE case studied in Ref. [22], itwas observed that the ππ scattering lengths were determined only within a universal band.Something similar was observed in the RS solution for πN presented in Ref. [29], whereprecise results were obtained once the πN scattering lengths were imposed as constraints.More precisely, the problem is connected with the number of no-cusp conditions required inorder to ensure a smooth matching in the three partial waves between the dynamical solutionof the RS equations and the input considered at higher energies. In [10], no-cusp condi-tions for the f / and f / partial waves were imposed, matching precisely the number offree subtraction constants, the two pion-kaon scattering lengths a ± . However, in Ref. [22] itwas found for ππ scattering that only one no-cusp condition was enough to ensure a smoothmatching, leading to a ππ scattering length universal band.In order to analyze whether something similar might happen in the πK case, we have studiedfurther possible cusp-free RS solutions in a grid of points in the pion-kaon scattering length100igure 3: Value of the πK RS χ -like function defined in (2.1) for a grid of points on the I = 1 / , I = 3 / scattering length plane. This result suggests that one can achieved an exact solution of thepion-kaon RS equations on the universal band.plane. The results are plotted in Fig. 3, where one can see that RS equation solutions for πK scattering can be achieved within a universal band. Although the solution presented in [10]lies perfectly within this universal band, it is clearly not enough to fully constrain the valuesof the pion-kaon scattering lengths. As we can see in Fig. 2, this universal band is indeedconsistent with both chiral predictions and the different lattice results studied above. Thenext step of this project will be to study whether the combination of RS equations with sumrules for subtraction constants allows one to obtain a unique and consistent solution of the πK scattering lengths.3. Acknowledgments We would like to thank the organizers for a wonderful workshop, and for the invitation to101alk about our work on pion–kaon scattering. We are grateful to B. Moussallam for helfuldicussions and to J. Bijnens for providing us the code for the NNLO pion-kaon scatteringamplitudes. We also acknowledge useful discussions with M. Hoferichter, B. Kubis andU.-G. Meißner. Financial support by the Swiss National Science Foundation is gratefullyacknowledged. References [1] S. Weinberg, Physica A (1979) 327.[2] J. Gasser and H. Leutwyler, Annals Phys. (1984) 142.[3] J. Gasser and H. Leutwyler, Nucl. Phys. B (1985) 465.[4] S. 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Pelaez and Arkaitz Rodas Departamento de Física TeóricaUniversidad Complutense28040, Madrid, Spain Jacobo Ruiz de Elvira Albert Einstein Center for Fundamental PhysicsInstitute for Theoretical Physics, University of Bern3012 Bern, Switzerland Abstract After briefly motivating the interest of πK scattering and light strange resonances, wediscuss the relevance of dispersive methods to constrain the amplitude analysis and for thedetermination of resonances parameters. Then we review our recent results on a precise de-termination of πK amplitudes constrained with Forward Dispersion Relations, which are laterused together with model-independent methods based on analyticity to extract the parametersof the lightest strange resonances. In particular we comment on our most recent determina-tions of the κ/K ∗ (800) pole using dispersive and/or techniques based on analytic propertiesof amplitudes. We also comment on the relevance that a new kaon beam at JLab may have fora precise knowledge of these amplitudes and the light strange resonances. Motivation to Study πK Scattering Pion and kaons are the Goldstone bosons of the spontaneous SU (3) chiral symmetry break-ing in SU(3) and their masses are due to the small explicit breaking due to non-vanishingquark masses. Thus, by studying their interactions we are testing our understanding of thisspontaneous symmetry breaking, which is rigorously formulated in terms of the low-energyeffective theory of QCD, namely Chiral Perturbation Theory, as well as the role of quarkmasses and the breaking of the flavor SU(3) symmetry. In addition, pion and kaons ap-pear in the final state of almost all hadronic interactions involving strangeness and a preciseunderstanding of πK scattering is therefore of relevance to describe the strong final-stateinteractions of many hadronic processes, including those presently under an intense experi-mental and theoretical study: B decays, D decays, CP violation, etc...Finally, most of our knowledge of strange resonances below 2 GeV comes from πK exper-iments. Strange resonances are very helpful in order to determine how many flavor multi-plets exist, which in turn helps to determine how many non-strange resonances are neededto complete these multiplets. Any additional flavorless state would then clearly suggest theexistence of glueball states. Moreover, the mass hierarchy between the strange and non-strange members of a multiplet can also reveal the internal nature of meson resonances (or-dinary q ¯ q states, tetraquarks, molecules, etc...). In particular, as we will see below, the strongtheoretical constraints on πK scattering provide the most reliable method to determine theexistence of the lightest strange scalar meson, the controversial κ or K ∗ (800) meson, whichstill “Needs Confirmation” according to the Review of Particle Properties (RPP) [1].104. Analyticity, Dispersion Relations, and Resonance Poles Analyticity constraints in the s, t, u Mandelstam variables are the mathematical expression ofcausality. For example, as illustrated on the left side of Figure 1, causality implies that two-body scattering amplitudes T ( s, t ) have no singularities in the first Riemann sheet of complex s -plane for fixed t , except for cuts in the real axis. In particular, there is one “physical” cuton the real axis that reflects the existence of a threshold, which extends from that thresholdto + ∞ . In addition, due to crossing symmetry, the physical cuts in the u -channel are seenas cuts in the complex s -plane from −∞ to − t . By using Cauchy Theorem one can nowcalculate the value of the amplitude anywhere in the complex plane as an integral of theamplitude over the contour C . If the amplitude decreases sufficiently fast when s → ∞ then, by sending the curved part of contour C to infinity, one is left only with integralsover the right and left cuts in the real axis. Since the amplitude in the upper half plane isconjugate to the amplitude in the lower plane, these are integrals over the imaginary part ofthe amplitude. If the amplitude does not decrease sufficiently fast at infinity, one repeats theargument but for the amplitude divided by a polynomial of sufficiently high degree. Thenthe amplitude is determined up to a polynomial of that degree, whose coefficients are called“subtraction constants”. The resulting relation is called a subtracted Dispersion Relation.Figure 1: Left: Analytic structure in the s -plane for fixed- t . Right: Two-sheet structure for elasticpartial waves. Poles appear in the second sheet. In the same location a zero appears in the S -matrix.Dispersion relations are useful for calculating the amplitude where there is no data, to con-strain data analyses or to look for resonance poles in the complex plane. Actually, the rig-orous mathematical definition of a resonance involves the existence of a pair of conjugatedpoles in the second Riemann sheet of a partial wave amplitude. This sheet is reached bycrossing continuously the physical cut, as seen on the right side of Figure 1. The positionof the pole in the lower half plane is related to the mass M and with Γ of the resonance as √ s pole = M − i Γ / .So far we have discussed the physical cut and its sheet structure, but in Figure 2 we show thecomplicated analytic structure of πK scattering partial waves, which includes the "unphysi-cal" left cut and an additional circular cut which appears due to the different pion and kaonmasses. Now, when the pole of a resonance is far from other singularities and close to thereal axis, namely Γ << M , as it is the case of the K ∗ (892) , whose width is ∼ MeV, it105s seen in the real axis, i.e., experimentally, as a peak in the amplitude. Simple parameteri-zations of data just around its nominal mass, like a Breit-Wigner formula, can describe suchresonances rather well. However, for very wide resonances like the κ/K ∗ (800) , the energyregion around its nominal mass is more distant than other energy regions and particularlythe threshold region. This threshold will distort dramatically the peak naively expected fora resonance. In addition, chiral symmetry implies the existence of the so-called Adler zerobelow threshold, which once again is close to the pole ans distorts its simple shape whenseen at physical energies. Finally, the nominal mass region is also at a comparable distanceto the pole as the circular cut and the left cut and their contributions should not be discardeda priori but should taken into account for precise pole determinations.Therefore, the κ/K ∗ (800) resonance cannot be described with precision just by the knowl-edge of data around its nominal mass, and the pole cannot be extracted with simple formulasthat do not posses the correct analytic structures or that do not have the correct low-energyconstraints of the QCD spontaneous chiral symmetry breaking. This is the reason why dis-persion relations and analyticity properties are relevant tools to analyze data and the useof simple models can lead to considerable confusion or artifacts. In addition, data in thenear threshold region impose strong constraints to the pole position, even if the resonance isnominally at a higher mass. This partly explains the situation of the κ/K ∗ (800) in the RPPwhere, as we will see in the final section, the poles obtained with Breit-Wigner formulas arequite spread and do not coincide with those obtained using more rigorous formalisms.Figure 2: Analytic structure of the πK partial waves in the complex s -plane (in MeV ). Note theexistence of a left cut up to ( m K − m π ) and a circular cut. We have shown the position of thepoles associated to the K ∗ (892) and the κ/K ∗ (800) .Note that in order to write Cauchy’s Theorem as we described above we need to consider theamplitude as a one-variable function. For this, there are two approaches:(a) Fix one variable and obtain a dispersion relation in terms of the other. This leads tothe popular fixed- t dispersion relations, although one could also obtain fixed- u or evenfixed- s dispersion relations. But one can also fix one variable in terms of the otherone by means of a constrain. For instance, this is the case of Hiperbolic DispersionRelations, where one imposes a constraint ( s − a )( u − a ) = b . Complicated relationslike the latter are used to maximize the applicability region of the Dispersion Relationbut if that is not needed for a particular study a simple choice could be more convenient.106f particular interest are Forward Dispersion Relations (FDRs), t = 0 , since theyyield rather simple expressions and because the higher part of the integrals can beexpressed in terms of total cross sections thanks to the Optical Theorem. These FDRshave also the advantage that they can be applied in principle up to arbitrarily highenergies and that their analytic structure is very simple. This makes it easier to usecrossing symmetry in order to rewrite the left-cut contributions in terms of physicalamplitudes over the physical cut. Their expressions are rather simple for πK scatteringand as we will see below provide stringent constraints on the description of existingdata. Unfortunately it is not possible to use the integral FDRs to access the secondRiemann sheet in search for poles.(b) Partial-wave Dispersion Relations. Now the scattering angle dependence is eliminatedby projecting the amplitudes in partial waves, t J ( s ) . The main advantage is that thesecond Riemann sheet, where resonance poles can appear, is easily accessible. Thereason is that, due to unitarity, for elastic partial waves the second Riemann sheet ofthe S matrix is just the inverse of the first, namely S IIJ ( s ) = 1 /S IJ ( s ) . Recalling that S J ( s ) = 1 − iσ ( s ) t J ( s ) with σ ( s ) = k/ √ s , where k is the CM-momentum, thenthe Second Riemann Sheet partial wave amplitude can be obtained from the first asfollows: t IIJ ( s ) = t IJ ( s )1 − iσ ( s ) t IJ ( s ) (1)Unfortunately, when the scattering particles do not have equal masses, the partial-waveprojection leads to a more complicated analytic structure. For πK partial waves thisstructure was already shown in Fig. 2 and includes a new circular cut and a longer leftcut. The most difficult part of using Dispersion Relation is to calculate the contribu-tions along these unphysical cuts, and two main approaches appear once again in theliterature:i. Do not use crossing and just approximate the left cut. This leads to simple disper-sion relations (see for instance [2]) and has become very popular when combinedwith Chiral Perturbation Theory to approximate the left cut and the subtractionconstants. Within this approach, the dispersion relations are written for the in-verse partial-wave, since unitarity fixes its imaginary part in the elastic region.This approach is known as the Inverse Amplitude Method [3] that is one instanceof Unitarized Chiral Perturbation Theory. It provides a fairly good description ofdata and yields the poles of all resonances that appear in the two-body scatteringof pions and kaons below 1 GeV. In particular, a pole for the kappa is found at [4] (753 ± − i (235 ± MeV. Despite not being good for precision, it is veryconvenient to connect the meson scattering phenomenology with QCD parameterslike N c or the quark masses [5] and thus study the nature of resonances.ii. Use crossing to rewrite the contributions from unphysical cuts in terms of Usecrossing to rewrite the contributions from unphysical cuts in terms of partial waveson the physical cut. Unfortunately, when projecting into a partial wave of the s -angle, one does not extract a single partial wave in the u -channel angle, so thatthe whole tower of partial waves is still present when rewriting the unphysical cutsinto the physical region using crossing. This leads to an infinite set of coupled dis-107ersion relations. These are called Roy-Steiner dispersion relations [6] (differentversions can be obtained starting from fixed- t or hyperbolic dispersion relations).They are more cumbersome to use, and usually they are applied only to the lowestpartial waves, taking the rest as fixed input. However, they yield the most rigorousdetermination of the κ/K ∗ (800) pole so far [7]: (658 ± − i (278 . ± MeV.Note that in these works Roy-Steiner equations were solved in the elastic regionfor the S and P waves, using data on higher energies and higher waves as input,but no data on the elastic region below ∼ . It was also shown that the poleappears within the Lehman-ellipse that ensures the convergence of the partial-waveexpansion (actually, even inside a more restrictive region that also ensures the useof hyperbolic dispersion relations). Nevertheless the RPP still considers that theexistence of the κ/K ∗ (800) “ Needs Confirmation”.(c) Amplitudes Constrained with Forward Dispersion Relations Hence, in order to provide the required confirmation of the κ/K ∗ (800) existence andof the values of its parameters, we have recently performed a Forward Dispersion Re-lation study of πK scattering data [8]. The explicit expressions of the FDRs for thesymmetric amplitude T + and the antisymmetric one T − and all other details can befound in [8]. We took particular care to use simple parameterizations of data on partialwaves that could be easily used later in further theoretical or experimental studies. Thefirst relevant result, as shown in the left column of Fig. 3 is that theUnconstrained Fits to Data (UFD) do not satisfy FDRs well, particularly at high en-ergies and in the threshold region. For this reason we imposed the FDRs to obtainConstrained Fits to Data (CFD). Once this is done the agreement is remarkable up to1.6 GeV for the symmetric FDR and up to 1.74 GeV for the antisymmetric one, asseen in the central column of Fig. 3. In the right column of that figure, we show thatthe difference between the UFD and CFD is significant above 1.4 GeV and also in thethreshold region for the phase, where there is very little experimental information. Aswe have already commented when discussing the κ pole and Fig.2, this area is partic-ularly relevant for the determination of the κ/K ∗ (800) resonance from experimentaldata. It would be very interesting to have new data to confirm our findings and thiscould certainly be achieved with the proposed kaon beam at JLab .Moreover, it can be seen in that last column of Fig.2, that the most reliable data source,which is coming from the LASS/SLAC spectrometer [9], only provides a combinationof isospins, but not the separated I = 1 / and I = 3 / partial waves. Actually,the information on I = 3 / is rather scarce and sometimes contradictory [9]. Thiswill be one of the main sources of uncertainty in our calculation of the κ pole. As ithas already been explained in other talks of this conference that a clear isospinseparation could be achieved with the proposed kaon beam facility .(d) Analytic Methods to Extract Resonances Our parameterizations of the data are obtained from piece-wise analytic functionswhich are carefully matched to impose continuity. Therefore, none of the pieces byitself has all the relevant information to determine the resonance poles with precision.Nevertheless, the lowest energy piece was constructed by means of a conformal expan-sion, which exploits the analyticity of the partial wave on the full complex plane and108igure 3: Left Column: The input from Unconstrained Fits to Data (UFD) do not agree wellwith the Dispersive output of FDRs. Central Column: By imposing FDRs as constraints on datafits (CFD) they can be well satisfied up to 1.6 GeV without spoiling the data description. RightColumn: The difference between UFD and CFD data descriptions. Data comes from Ref. [9].has the correct analytic structure while satisfying the elastic unitarity constraint thatallows for a continuation to the second Riemann sheet as explained in Eq. 1. We haveactually checked that this simple parameterization yields a pole for the κ/K ∗ (800) at: (680 ± − i (334 ± . . However, this still makes use of a specific parameteri-zation and, no matter how good it might be, is still model-dependent and the smalluncertainties can only be understood within that model.However, there is a procedure to extract poles [10], which is model independent inthe sense that it does not rely on a specific parameterization choice. It is based on apowerful theorem of Complex Analysis, which ensures that in a given domain wherea function is analytic except for a pole, one can construct a sequence of Padé approx-imants that contains a pole that converges to the actual pole in the function. Thissequence is built in terms of the values of the partial wave and the derivatives at agiven point within that domain. Thus, using our amplitudes constrained with FDRs(CFD), we have searched [11] for a point in the real axis in which we can define a do-main that includes the expected κ/K ∗ (800) pole and built a sequence of Padés. In thisway we have confirmed the existence of the κ/K ∗ (800) pole [11] and its location at: (670 ± − i (295 ± . This time the errors are larger because the determination doesnot depend on a specific parameterization. Similar results have been obtained for allthe resonances that appear in πK scattering below 1.8 GeV (except for the K ∗ (1680) ),thus providing further support for their existence and a determination of their parame-ters that do not depend on approximations or specific choices of parameterizations like109reit-Wigner parameterizations. Further details can be found in [11].Of course, the best would be a new analysis of the Roy-Steiner type, independent fromthat of [7]. Actually our Madrid group already has a preliminary result of this kind ofanalysis which finds a pole at (662 ± − i (289 ± MeV [12]. For this we usepartial-wave Roy-Steiner-like equations obtained from hyperbolic dispersion relationusing as input the amplitudes constrained to satisfy FDRs [8], and therefore using databelow 1 GeV instead of solving the equations in the elastic region as in Ref [7]. Weare showing this preliminary result in red in Fig. 4, together with all the other polesmentioned in this mini-review.Note however that there is considerable room for improvement by reducing the un-certainties. For this it would be of the uttermost importance not only to have moreprecise data, but also data in the regions where there is none nowadays, and particu-larly with a clear separation of partial waves with different isospin. As we have seen inthis workshop, all this could be achievable with the proposed kaon beam at JLab that therefore could provide a sound experimental basis for the understanding of thespectroscopy of strange mesons, which nowadays relies on strong model-dependentassumptions and data sets with strong inconsistencies among themselves and with thefundamental dispersive constraints.Figure 4: Compilation of results for the κ/K ∗ (800) resonance pole positions. We list as emptysquares all the determinations using some for of Breit-Wigner shape listed in the RPP [ ? ]. Therest of references correspond to: Zhou et al. [2], Pelaez [4], Bugg [13], Bomvicini [14], Descotes-Genon et al. [7], Padé result [11], Conformal CFD [8] and our preliminary HDR result usingRoy-Steiner equations [12] 110. Acknowledgments JRP and AR are supported by the Spanish Project FPA2016-75654-C2-2-P. The work of JREwas supported by the Swiss National Science Foundation. AR would also like to acknowl-edge financial support of the Universidad Complutense de Madrid through a predoctoralscholarship. References [1] C. Patrignani et al. (Particle Data Group), Chin. Phys. C , 100001 (2016) and 2017 update.[2] H. Q. Zheng, et al. Nucl. Phys. A , 235 (2004); Z. Y. Zhou and H. Q. Zheng, Nucl. Phys.A , 212 (2006).[3] T. N. Truong, Phys. Rev. Lett. , 2526 (1988); A. Dobado, M. J. Herrero, and T. N. Truong,Phys. Lett. B (1990) 134; A. Dobado and J. R. Pelaez, Phys. Rev. D , 3057 (1997);A. Dobado and J. R. Pelaez, Phys. Rev. D , 4883 (1993); A. Gomez Nicola, J. R. Pelaezand G. Rios, Phys. Rev. D , 056006 (2008).[4] J. R. Pelaez, Mod. Phys. Lett. A , 2879 (2004).[5] J. R. Pelaez, Phys. Rev. Lett. , 102001 (2004); C. Hanhart, J. R. Pelaez, and G. Rios, Phys.Rev. Lett. (2008) 152001; J. Nebreda and J. R. Pelaez., Phys. Rev. D , 054035 (2010).[6] S. M. Roy, Phys. Lett. , 353 (1971); F. Steiner, Fortsch. Phys. (1971) 115; G. E. Hiteand F. Steiner, Nuovo Cim. A , 237 (1973).[7] P. Buettiker, S. Descotes-Genon, and B. Moussallam, Eur. Phys. J. C (2004) 409;S. Descotes-Genon and B. Moussallam, Eur. Phys. J. C , 553 (2006).[8] J. R. Pelaez and A. Rodas, Phys. Rev. D , no. 7, 074025 (2016).[9] P. Estabrooks, R. K. Carnegie, A. D. Martin, W. M. Dunwoodie, T. A. Lasinski, andD. W. G. S. Leith, Nucl. Phys. B (1978) 490; D. Aston et al. , Nucl. Phys. B , 493(1988).[10] P. Masjuan and J. J. Sanz-Cillero, Eur. Phys. J. C (2013) 2594; P. Masjuan, J. Ruiz deElvira and J. J. Sanz-Cillero, Phys. Rev. D (2014) no.9, 097901; I. Caprini, P. Masjuan,J. Ruiz de Elvira, and J. J. Sanz-Cillero, Phys. Rev. D , no. 7, 076004 (2016).[11] J. R. Pel?z, A. Rodas, and J. Ruiz de Elvira, Eur. Phys. J. C , no. 2, 91 (2017).[12] J. R. Pelaez and A. Rodas, in preparation.[13] D. V. Bugg, Phys. Lett. B , 1 (2003) Erratum: [Phys. Lett. B , 556 (2004)]; D. V. Bugg,Phys. Rev. D , 014002 (2010).[14] G. Bonvicini et al. [CLEO Collaboration], Phys. Rev. D , 052001 (2008).111 .15 Analyticity Constraints for Exotic Mesons Vincent Mathieu Thomas Jefferson National Accelerator FacilityNewport News, VA 23606, U.S.A. Abstract Dispersive techniques have drastically improved the extraction of the pole position of thelowest hadronic resonances, the σ and κ mesons. I explain how dispersion relations can beused in the search of the lowest exotic meson, the π meson. It is shown that a combinationof the forward and backward elastic πη finite energy sum rules constrains the production ofexotic mesons. Introduction Despite the knowledge of the Quantum ChromoDynamics (QCD) Lagrangian for about 50years and the abundance of data, the pole position of the lowest QCD resonances, the σ and κ mesons also called f (500) and K ∗ (800) respectively, have remained inaccurate for decades.The situation of the σ has, however, changed recently. The combined use of dispersionrelations, crossing symmetry and experimental data have led independent teams to a preciselocation of the complex pole [1–3]. The interested readers can find a historical and technicalreview in Ref. [4].The situation of the κ is going in the same direction. A recent analysis [5] using dispersionrelation and crossing symmetry have determined its pole location accurately. Nevertheless,the lowest resonance with strangeness is currently not reported in the summary tables in theReview of Particle Properties [6]. This situation might change when the results of Ref. [5]will be confirmed independently. Studies in this direction are in progress [7].With these recent confirmation of existence and precise location of the σ and κ poles, thenext challenge in meson spectroscopy is the existence of exotic mesons. The theoreti-cal and experimental works tend to support the existence of an isovector with the quantumnumbers J P C = 1 − + , denoted π , around 1400-1600 MeV. The experimental status is stillnevertheless controversial [8]. As in the case of the σ and the κ meson, the (possible) largewidth of the π prevents the use of standard partial-wave parametrizations, such as the Breit-Wigner formula, to extract the pole location. The parametrization of the exotic wave can,nevertheless, be constrained thanks to dispersive techniques.Dispersion relations, in meson-meson scatterings, are typically used in their subtracted formto reduce the influence of the, mostly unknown, high energy part. One can alternatively writedispersion relations for moments of the amplitudes. Using a Regge form for the high energypart, they lead to the finite energy sum rules (FESR) [9, 10]. FESR were applied recentlyin hadro- and photo-production on a nucleon target [11–13] in which data in the low andhigh energy mass region are available. With the advent of high statistic generation of hadronspectroscopy meson, one now has access to data in a wide energy range in meson-meson By exotic, I mean in the quark model sense. A quark-antiquark pair cannot couple to the π quantum number I G J P C = 1 − − + . A review of exotic quantum numbers and notation is presented in Ref. [8]. πη and πη (cid:48) (acceptance corrected)partial waves from threshold to 3 GeV and has recorded data up to 5 GeV [14]. CLAS12and GlueX are currently taking data on two mesons photo- and electro-production with anexpected coverage up to at least 3 GeV. These data sets allow then to constrain the resonanceregion with the Regge region via FESR.Schwimmer [15] has demonstrated that duality and the non-existence of odd wave in elastic πη scattering lead to a degeneracy relation between Regge exchange residues in the equalmass case. In order words, in the equal mass case, the equality between the forward andbackward region at high energy is equivalent to the non-existence of odd waves. This results,showing the self-consistency of the quark model, was actually already included in the generalresults of Mandula, Weyers and Zweig [16] also in the SU (3) symmetry limit.Here I demonstrate how FESR in elastic πη scattering can be used to constrain the exotic P -wave. I examine Schwimmer argument with exact kinematics and derive the forward andbackward FESR for πη scattering. It is shown how the deviation from forward/backwardequality and the mass difference is related to the amount of exotic meson in πη scattering.The kinematics are reviewed in Section 2 and the FESR are written in Section 3. The mainresults are derived in Section 4.2. Kinematics Figure 1: The s -channel πη → πη and the t -channel ππ → ηη .The s − channel reaction πη → πη . The u − channel is identical to the s − channel and the t − channel is ππ → ηη , as depicted in Fig 1. All channels are described by a single function A ( s, t, u ) . Let m π and m η be the masses of the π and η mesons, and s + t + u = Σ =2 m π + 2 m η . The center-of-mass momentum and scattering angle in the s -channel are q = 14 s (cid:0) s − ( m π + m η ) (cid:1) (cid:0) s − ( m π − m η ) (cid:1) , (1a) z s = 1 + t q = − − u − u q , u = ( m η − m π ) s (1b)The unitarity threshold of the s - and u -channels is ( m π + m η ) . In the t -channel the thresholdis m π . I will do fixed- t and fixed- u dispersion relations and finite energy sum rules. Fixed- s sum rules are identical to fixed- u sum rules. The crossing variables are ν = ( s − u ) / and113 (cid:48) = ( s − t + 4 m π − ( m η + m π ) ) / . ν (cid:48) is designed to symmetrize the two cuts at fixed u , i.e., ν (cid:48) ( s = ( m η + m π ) ) = − ν (cid:48) ( t = 4 m π ) . That will allow to place both cuts under thesame integral in the fixed- u sum rules. I will need to trade the Mandelstam variables for thepair ( ν, t ) and ( ν (cid:48) , u ) via s ( ν, t ) = + ν − ( t − Σ) / , s ( ν (cid:48) , u ) = + ν (cid:48) − ( u − m η + ( m η − m π ) ) / , (2a) u ( ν, t ) = − ν − ( t − Σ) / , t ( ν (cid:48) , u ) = − ν (cid:48) − ( u − m π − ( m η − m π ) ) / , (2b)so that s ( − ν, t ) = u ( ν, t ) .3. Finite Energy Sum Rules Figure 2: Exchanges in forward (left) and backward (right) πη elastic scattering.I consider small t = x , i.e., small angles with the pion going forward. The two exchangesat high energy are the Pomeron and the f Regge pole. At high energy, the Regge form is Im A ( s, t, u ) = β P ( t ) ν α P + β f ( t ) ν α f for ν > Λ . The threshold is ν ( x ) = 2 m η m π + x/ .The FESR at fixed t reads (cid:90) Λ ν ( t ) Im (cid:8) A [ s ( ν, x ) , x, u ( ν, x )] + ( − ) k A [ s ( − ν, x ) , x, u ( − ν, x )] (cid:9) ν k dν = (cid:88) τ (cid:2) τ ( − ) k (cid:3) β τ ( t ) Λ α τ ( t )+ k +1 α τ ( t ) + k + 1 (3)The s - and u -channel are identical, so A ( s, x, u ) = A ( u, x, s ) and there can only be evenmoments. This is consistent with the exchanges that can only have positive signature τ ( P ) = τ ( f ) = +1 . I obtain, with n a positive integer (cid:90) Λ ν ( x ) Im A [ s ( ν, x ) , x, u ( ν, x )] ν n dν = β P ( x ) Λ α P ( x )+2 n +1 α P ( x ) + 2 n + 1 + β f ( x ) Λ α f ( x )+2 n +1 α f ( x ) + 2 n + 1 (4)The factorization of Regge pole allow write β P ( x ) = β ππ P ( x ) β ηη P ( x ) and β f ( x ) ≡ β ππf ( x ) β ηηf ( x ) ,as represented in Fig. 2.I now consider small u = x , i.e., small angles with the eta meson going forward. Thethreshold is ν (cid:48) ( x ) = (4 m π − ( m η − m π ) + x ) / . Similarly to the fixed- t case, the FESR at114xed u read (cid:90) ¯Λ ν (cid:48) ( x ) Im (cid:8) A [ s ( ν (cid:48) , x ) , t ( ν (cid:48) , x ) , x ] + ( − ) k A [ s ( − ν (cid:48) , x ) , t ( − ν (cid:48) , x ) , x ] (cid:9) ν (cid:48) k dν (cid:48) = (cid:88) τ (cid:2) τ ( − ) k (cid:3) β τ ( x ) ¯Λ α τ ( x )+ k +1 α τ ( x ) + k + 1 (5)There are two kinds of exchanges. The positive signature a pole and the negative signature π pole. So for k even there is the a pole and k odd the π (exotic) pole. I thus obtain twobackward FESR: (cid:90) ¯Λ ν (cid:48) ( x ) Im { A [ s ( ν (cid:48) , x ) , t ( ν (cid:48) , x ) , x ] + A [ s ( − ν (cid:48) , x ) , t ( − ν (cid:48) , x ) , x ] } ν (cid:48) n dν (cid:48) = β a ( x ) ¯Λ α a ( x )+2 n +1 α a ( x ) + 2 n + 112 (cid:90) ¯Λ ν (cid:48) ( x ) Im { A [ s ( ν (cid:48) , x ) , t ( ν (cid:48) , x ) , x ] − A [ s ( − ν (cid:48) , x ) , t ( − ν (cid:48) , x ) , x ] } ν (cid:48) n +1 dν (cid:48) = β π ( x ) ¯Λ α π ( x )+2 n +2 α π ( x ) + 2 n + 2 (6)Again, the factorization of Regge pole leads to β a ( x ) = (cid:2) β πηa ( x ) (cid:3) and β π ( x ) = (cid:2) β πηπ ( x ) (cid:3) as displayed in Fig. 2. For small x , the amplitude A [ s ( ν (cid:48) , x ) , t ( ν (cid:48) , x ) , x ] represents the re-action πη → πη in the backward direction and A [ s ( − ν (cid:48) , x ) , t ( − ν (cid:48) , x ) , x ] represents the t -channel reaction ππ → ηη .4. Constraint on Exotica Production Several models are available for the elastic πη S -wave [17–20]. For higher waves, little infor-mation is known beside the resonance content. The elastic πη D -wave is certainly dominatedby the a (1320) meson. At higher mass, the excitation a (cid:48) (1700) has been recently confirmby a joint publication by the JPAC and COMPASS collaborations [21], but its branchingfraction to πη is not known. The branching ratio of the a (2040) is not known either butquark model calculations yields the width Γ( a → πη ) ∼ (1 . − . MeV [22, 23]. Experimental information on ππ → ηη are available above the ηη threshold [ ? ]. Belowthe physical threshold, one can use Watson’s theorem and the ππ elastic phase shift fromthe unitarity threshold up to the K ¯ K threshold. Unfortunately between the K ¯ K and the ηη thresholds, little information is known about the phase shift and inelasticities of the ππ → ηη amplitude. Although some couple channels calculations exist, see for instance [25].The a , f are lying on the trajectory degenerate with the ρ and ω one with α a ( t ) = α f ( t ) ≡ α N ( t ) ∼ . t + 0 . . The exotic π is lying on a trajectory below the leading natural exchangetrajectory. A mass around 1.6 GeV yields the intercept α π (0) ∼ α N ( t ) − , which makesboth right-hand-side of Eq. (6) for the same n of the same order up to the residues β a and β π . The existence of an exotic would make β π not zero. However since its pole would befar away from the physical region, there is a possibility of β π ( x ) (with x < ) being small.In this case, one could neglect the right-hand-side of the second sum rule in Eq. (6) for any n , which would lead to Im A [ s ( ν (cid:48) , x ) , t ( ν (cid:48) , x ) , x ] ≈ Im A [ s ( − ν (cid:48) , x ) , t ( − ν (cid:48) , x ) , x ] (7) The a was denoted δ in these publications. t = x and backward u = x sum rulesto obtain the constraint (cid:90) Λ ν ( x ) Im { A [ s ( ν, x ) , x, u ( ν, x )] − A [ s ( ν, x ) , t ( ν, x ) , x ] } ν n d ν = β P ( x ) Λ α P ( x )+2 n +1 α P ( x ) + 2 n + 1 + [ β f ( x ) − β a ( x )] Λ α N ( x )+2 n +1 α N ( x ) + 2 n + 1 (8)In term of partial waves, the forward and backward amplitudes read A ( s, x, u ) = (cid:88) (cid:96) (2 (cid:96) + 1) t (cid:96) ( s ) P (cid:96) ( z ) , z = +1 + x q , (9a) A ( s, t, x ) = (cid:88) (cid:96) (2 (cid:96) + 1) t (cid:96) ( s ) P (cid:96) ( z ) , z = − − x − u q , (9b)with q and u evaluated at s ( ν, x ) . If the forward and backward direction were exactly equal, i.e., u = 0 the even waves would exactly cancel in Eq. (8). The reader can then see that thetwo components duality and the absence of odd wave would then imply degeneracy betweenthe f and a Regge exchanges as demonstrated by Schwimmer [15]. In the presence of onlyone odd wave (cid:96) = 1 , its absorptive part is constrained by (cid:90) Λ ν ( x ) Im t [ s ( ν, x )] [ P ( z ) − P ( z )] ν n dν = β P ( x ) Λ α P ( x )+2 n +1 α P ( x ) + 2 n + 1+ [ β f ( x ) − β a ( x )] Λ α N ( x )+2 n +1 α N ( x ) + 2 n + 1 − (cid:88) (cid:96) ∈ even (cid:90) Λ ν ( x ) Im t (cid:96) ( s ν )∆ (cid:96) ( ν, x ) ν n dν (10)The even waves enter via the quantity ∆ (cid:96) ( ν, x ) = (2 (cid:96) + 1) [ P (cid:96) ( z ) − P (cid:96) ( z )] . Note that ∆ (cid:96) ∝ u (for (cid:96) even) vanishes in the limit m η → m π . Of course the S -wave cancels ∆ = 0 .We also have [ P ( z ) − P ( z )] = 1 + x/ q − u / q . The individual contribution of theright-hand-side of Eq. (10) can be evaluated.5. Acknowledgments I thank M. Albaladejo, A. Jackura, and J. Ruiz de Elvira for providing their elastic πη so-lutions. This work was supported by the U.S. Department of Energy under grants No. DE-AC05-06OR23177 References [1] I. Caprini, G. Colangelo and H. Leutwyler, Phys. Rev. Lett. , 132001 (2006). In the two component duality hypothesis, one separate the background dual to the Pomeron and the resonancedual to the Regge exchange [26]. ,072001 (2011),[3] M. Albaladejo and J. A. Oller, Phys. Rev. D , 034003 (2012).[4] J. R. Pelaez, Phys. Rept. , 1 (2016).[5] S. Descotes-Genon and B. Moussallam, Eur. Phys. J. C , 553 (2006).[6] C. Patrignani et al. [Particle Data Group], Chin. Phys. C , no. 10, 100001 (2016).[7] J. R. Pelaz, A. Rodas ,and J. Ruiz de Elvira, Eur. Phys. J. C , no. 2, 91 (2017).[8] C. A. Meyer and Y. Van Haarlem, Phys. Rev. C , 025208 (2010).[9] R. Dolen, D. Horn and C. Schmid, Phys. Rev. , 1768 (1968).[10] P. D. B. Collins, Cambridge Monograph 1977[11] V. Mathieu et al. [JPAC Collaboration], Phys. Rev. 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A ,313 (1983).[25] M. Albaladejo and J. A. Oller, Phys. Rev. Lett. , 252002 (2008).[26] H. Harari, Phys. Rev. Lett. , 1395 (1968).117 .16 Pion–Kaon Final-State Interactions in Heavy-Meson Decays Bastian Kubis and Franz Niecknig Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) andBethe Center for Theoretical PhysicsUniversität Bonn53115 Bonn, Germany Abstract We discuss the description of final-state interactions in three-body hadronic decays basedon Khuri–Treiman equations, in particular their application to the charmed-meson decays D + → ¯ Kππ + . We point out that the knowledge of pion–pion and pion–kaon scattering phaseshifts is of prime importance in this context, and that there is no straightforward application ofWatson’s theorem in the context of three-hadron final states. What’s not to Like about the Isobar Model? In experimental analyses, the Dalitz-plot distributions of hadronic three-body decays arestill conventionally described in terms of the isobar model : amplitudes are constructed interms of subsequent two-body decays, and the resonant intermediate states mostly modeledin terms of Breit–Wigner line shapes. This is problematic in several respects. First of all,many resonances cannot be described by Breit–Wigner functions at all, be it due to the closeproximity of thresholds (e.g., the f (980) or the a (980) among the light mesons, and manyof the newly found, potentially exotic states in the charmonium and bottomonium sectors),or because their poles lie too far in the complex plane: prime examples include the lightestscalar resonances, the f (500) or σ (see the comprehensive review [1] and references therein)as well as, in the strange sector, the K ∗ (800) [2, 3]. These problems can be avoided by usingknown phase shifts as input, potentially including coupled channels; see, e.g., Refs. [4, 5]for recent applications in the context of heavy-meson decays. On the other hand, models ofsubsequent two-body decays ignore rescattering effects between all three strongly interactingparticles in the final state, which will in general affect the phase motion of the amplitudes inquestion. Close control over the various amplitude phases is, e.g., important in the contextof CP-violation studies: rapidly varying, resonant strong phases may significantly enhancesmall weak phases (originating in the Cabibbo–Kobayashi–Maskawa matrix) locally in theDalitz plot, which may give the search for CP violation in three-body charmed-meson decaysthe edge over two-body decays, which occur at fixed energy.A tool to consistently treat full iterated two-body final-state interactions in three-body decaysare the so-called Khuri–Treiman equations [6], originally derived for the description of K → π , and recently increasingly popular for the description of various low-energy decays suchas η → π [7–12], η (cid:48) → ηππ [13], or ω/φ → π [14, 15]. We will illustrate the formalismwith the last example in the following, before turning to the application in the more complexDalitz plot studies of D + → ¯ Kππ + [16, 17].2. Dispersion Relations for Three-Body Decays (1): V → π Particularly simple examples of three-body decays are those of isoscalar vector mesons intothree pions, V → π , V = ω, φ . Pairs of pions can only be in odd relative partial waves;118eglecting discontinuities in F - and higher partial waves, the decay amplitude can be de-composed into single-variable functions according to M ( s, t, u ) = i(cid:15) µναβ (cid:15) µV p νπ + p απ − p βπ (cid:8) F ( s ) + F ( t ) + F ( u ) (cid:9) , (1)where (cid:15) µV denotes the vector meson’s polarization vector. The function F ( s ) obeys a discon-tinuity equation [14, 18] disc F ( s ) = 2 i (cid:2) F ( s ) + ˆ F ( s ) (cid:3) θ (cid:0) s − M π (cid:1) sin δ ( s ) e − δ ( s ) , ˆ F ( s ) = 32 (cid:90) − d z (cid:0) − z (cid:1) F (cid:0) t ( s, z ) (cid:1) , (2)where z = cos θ s denotes the cosine of the s -channel scattering angle, and δ ( s ) ≡ δ ( s ) isthe isospin I = 1 P -wave ππ scattering phase shift. Without the inhomogeneity ˆ F ( s ) inthe discontinuity equation, the latter’s solution could be given in terms of an Omnès func-tion [19] like a simple form factor, and the phase of F ( s ) would immediately coincide with δ ( s ) according to Watson’s theorem [20].The presence of the partial-wave-projected crossed-channel contributions ˆ F ( s ) renders thesolution of Eq. (2) slightly more complicated; it is given as F ( s ) = Ω( s ) (cid:26) a + sπ (cid:90) ∞ M π d xx sin δ ( x ) ˆ F ( x ) | Ω( x ) | ( x − s ) (cid:27) , Ω( s ) = exp (cid:26) sπ (cid:90) ∞ M π d xx δ ( x ) x − s (cid:27) , (3)where Ω( s ) is the Omnès function, and a a subtraction constant that has to be fixed phe-nomenologically. Care has to be taken in performing the angular integral for ˆ F ( s ) in Eq. (2)as to avoid crossing the right-hand cut: in decay kinematics, ˆ F ( s ) itself becomes complex, 750 800 850 900 950 1000 1050 1100 1150 1200 1250 bin number Figure 1: Selected slices through the φ → π Dalitz plot measured by the KLOE Collabora-tion [21], compared to the theoretical description of Ref. [14]. The fit with Omnès functionsonly, neglecting crossed-channel rescattering, is shown in red, once-subtracted Khuri–Treimanamplitudes in blue, and twice-subtracted Khuri–Treiman solutions in black. The widths of theuncertainty bands are due to variation of the phase shift input as well as fit uncertainties.119 .0 0.2 0.4 0.6 0.8 1.0 s [GeV ] -4-2024 Omnes ω(782)φ(1020) s [GeV ] Omnes ω(782)φ(1020) Figure 2: Real (left) and imaginary (right) parts of single-variable amplitudes F ( s ) for varyingdecay masses, compared to the Omnès function.signaling the appearance of three-particle cuts, and a simple phase relation between F ( s ) and the elastic scattering phase shift is lost.The solution to Eq. (3) is compared to high-statistics Dalitz plot data for φ → π by theKLOE collaboration [21] in Fig. 1. Different ππ phases have been employed, derived fromRoy (and similar) equations [22, 23]. Non-trivial, crossed-channel rescattering effects im-prove the data description significantly; introducing a second subtraction constant in orderto better suppress imperfectly known high-energy behavior of the dispersion integral leadsto a perfect fit [14]. A first modern experimental investigation of the ω → π Dalitz plotby WASA-at-COSY [24] is not yet accurate enough to discriminate these subtle rescatteringeffects.It is also interesting to investigate the dependence of F ( s ) on the decay mass: it differs for ω and φ decays [14, 15], but even has a well-defined high-energy limit, as shown in Fig. 2.For high decay masses, F ( s ) approaches the input Omnès function, i.e., crossed-channelrescattering effects vanish [25]. This conforms with physical intuition: if the third pion hasa very large momentum relative to the other pair, its influence on the pairwise, resonantinteraction should be small. Note that this limit is only formal, as inelastic effects as well ashigher partial waves are not taken into account. Such a partial wave for variable and higherdecay masses has been employed both for the description of the reaction e + e − → π , whichserves as an input for a dispersive analysis of the π transition form factor [26], and in astudy of the J/ψ → π γ ∗ transition form factor [27], in analogy to preceding studies of theconversion decays of the light vector mesons [28].3. Dispersion Relations for Three-Body Decays (2): D + → ¯ Kππ + In the first attempt to employ Khuri–Treiman equations to describe D -meson decays, wehave chosen to consider the Cabibbo-favored processes D + → K − π + π + and D + → ¯ K π π + . The data situation for these is rather good, with high-statistics Dalitz plot mea-surements available by the E791 [29], CLEO [30], and FOCUS [31, 32] Collaborations forthe fully charged final state, and more recently by BESIII for the partially neutral one [33].120heoretical analyses of D + → K − π + π + have typically concentrated on improved descrip-tions of the πK S -wave [34–38], however neglecting rescattering with the third decay par-ticle. Ref. [39] is the only previous combined study of both decay channels, using Faddeevequations to generate three-body rescattering effects.An interesting aspect of these two decay channels is that they are coupled by a simple charge-exchange reaction, and can be related to each other by isospin; however, this relation islargely lost as long as only two-body rescattering is considered: e.g., the isospin I = 1 ππ P -wave only features indirectly in the fully charged final state, where the ρ (770) resonance canobviously not be observed. Otherwise, we truncate the partial-wave expansion consistentlybeyond D -waves, but among the exotic, non-resonant ones only retain the S -waves (the I = 2 ππ and the I = 3 / πK S -waves). This way, beyond the aforementioned ρ (770) ,the following πK resonances are included in the Dalitz plot description as the dominantstructures: K ∗ (800) , K ∗ (1430) , K ∗ (892) , [ K ∗ (1410) ,] and K ∗ (1430) .The decomposition of the two decay amplitudes up-to-and-including D -waves has been de-rived [16] and proven in the sense of the reconstruction theorem [25]. With the exceptionof the D -wave, the number of subtractions has been determined by imposing the Froissartbound [40]; note that some of them can be eliminated consistently due to the constraint s + t + u = const.. The full system then reads F ( u ) = Ω ( u ) u π (cid:90) ∞ u th d u (cid:48) u (cid:48) ˆ F ( u (cid:48) ) sin δ ( u (cid:48) ) | Ω ( u (cid:48) ) | ( u (cid:48) − u ) , F ( u ) = Ω ( u ) (cid:40) c + c u + u π (cid:90) ∞ u th d u (cid:48) u (cid:48) ˆ F ( u (cid:48) ) sin δ ( u (cid:48) ) (cid:12)(cid:12) Ω ( u (cid:48) ) (cid:12)(cid:12) ( u (cid:48) − u ) (cid:41) , F / ( s ) = Ω / ( s ) (cid:40) c + c s + c s + c s + s π (cid:90) ∞ s th d s (cid:48) s (cid:48) ˆ F / ( s (cid:48) ) sin δ / ( s (cid:48) ) (cid:12)(cid:12) Ω / ( s (cid:48) ) (cid:12)(cid:12) ( s (cid:48) − s ) (cid:41) , F / ( s ) = Ω / ( s ) (cid:40) s π (cid:90) ∞ s th d s (cid:48) s (cid:48) ˆ F / ( s (cid:48) ) sin δ / ( s (cid:48) ) (cid:12)(cid:12) Ω / ( s (cid:48) ) (cid:12)(cid:12) ( s (cid:48) − s ) (cid:41) , F / ( s ) = Ω / ( s ) (cid:40) c + sπ (cid:90) ∞ s th d s (cid:48) s (cid:48) ˆ F / ( s (cid:48) ) sin δ / ( s (cid:48) ) (cid:12)(cid:12) Ω / ( s (cid:48) ) (cid:12)(cid:12) ( s (cid:48) − s ) (cid:41) , F / ( s ) = Ω / ( s ) (cid:40) c + sπ (cid:90) ∞ s th d s (cid:48) s (cid:48) ˆ F / ( s (cid:48) ) sin δ / ( s (cid:48) ) (cid:12)(cid:12) Ω / ( s (cid:48) ) (cid:12)(cid:12) ( s (cid:48) − s ) (cid:41) , (4)where s (and t ) are the Mandelstam variables describing πK invariant masses, while u refersto the ππ system; the lower limits of the dispersion integrals are given by s th = ( M K + M π ) and u th = 4 M π . Explicit formulae for the various ˆ F IL are given in Ref. [16]. We haveused the πK phase shifts of Ref. [41] as input (note also other new [42] and ongoing [43]analyses of πK scattering). Eq. (4) therefore contains 8 complex subtractions constants,which (subtracting one overall normalization and one overall phase) need to be fitted to data.In particular, the subtraction in F / ( s ) was introduced a posteriori , as it turned out to benecessary to describe both Dalitz plots consistently [17]. As we work in the approximation of elastic unitarity, we refrain from describing the parts of the Dalitz plot for which √ s, √ t ≥ combined CLEO/FOCUS combined FF (1 . ± . ± FF (23 ± —FF / (36 ± ± FF / (8 . ± . . ± . FF / (6 ± . ± . FF / (0 . ± . . ± . Table 1: Fit fractions of the various partial waves for the best combined fit. The errors are eval-uated by varying the basis functions within their uncertainty bands. The fit fractions of the πK amplitudes in the s - and t -channel are summed together. M η (cid:48) + M K ≈ . GeV, which is taken as the typical onset of significant inelasticities inparticular in the S -wave. Note that no isoscalar ππ S -wave can contribute in these decays,with its sharp onset of inelasticities around the f (980) .Moduli and phases of the resulting single-variable amplitudes F IL are shown in Fig. 3. Obvi-ously, the different data sets constrain the various amplitudes differently: the ππ P -wave isonly well constrained when including the BESIII data on the partially neutral final state (inwhich it features directly), while in contrast, the ππ I = 2 S -wave is determined with muchhigher precision in the K − π + π + final state.Given the relatively large number of subtraction constants compared to the φ → π anal-ysis described in the previous section, the necessity to include Khuri–Treiman amplitudesas opposed to simple polynomial-times-Omnès functions is demonstrated less by the best χ / d.o.f., but rather by the resulting fit fractions, which become implausibly large in par-ticular for the nonresonant waves in the case of Omnès fits [16]. Similar observationsare made when neglecting the D -wave. The fit fractions for the best combined fit to theCLEO, FOCUS, and BESIII data are shown in Table 1. The total fit quality characterizedby χ / d.o.f. ≈ . is at best satisfactory, but of similar quality as isobar fits performed bythe experimental collaborations [31–33]. We point out that three-body rescattering effectschange the phases of the amplitudes quite significantly in comparison to the input phaseshifts, see Fig. 3: in particular the I = 1 / πK S -wave phase is seen to rise much morequickly around √ s ≈ GeV.4. Summary / Outlook We have demonstrated that dispersion relations constitute an ideal tool to analyze the final-state interactions of pions and kaons systematically. While two-body form factors obey uni-versal phase relations, in three-body decays non-trivial rescattering effects can affect phasemotions and line shapes significantly. This was demonstrated succinctly for the ideal demon-stration case φ → π that can be described in terms of one single partial wave only; ananalysis of two coupled D + → ¯ Kππ + decay modes proved to be far more involved due tothe proliferation of subtraction constants.122 F F / F / F / F / √ s [GeV] 864444432222 11 11 1 11 1 11 1 111 1 110 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3: Left: Moduli of single-variable amplitudes, in arbitrary units: CLEO/FOCUS fits (blue),BESIII (red), combined fit with improved D -wave (turquoise). Right: Phases of the single-variableamplitudes and input scattering phases (black) in radiant. The dashed lines visualize the fittedarea: from threshold to the η (cid:48) K threshold for the πK amplitudes, the full phase space for the ππ amplitudes. Figure taken from Ref. [17]. 123uture work could improve on the treatment of inelastic rescattering effects by using coupledchannels, not the least in order to extend the amplitude description to the full D + → ¯ Kππ + Dalitz plots. The consistent inclusion of higher partial waves in Khuri–Treiman equationsstill requires further investigations, as does the role of three-body unitarity in constraining thephases of the subtraction constants. Finally, it should be attempted to match the subtractionconstants to short-distance information on the weak transitions involved.5. Acknowledgments B.K. thanks the organizers for the invitation to and support at this most enjoyable workshop.Financial support by DFG and NSFC through funds provided to the Sino–German CRC 110“Symmetries and the Emergence of Structure in QCD” is gratefully acknowledged. References [1] J. R. Peláez, Phys. Rept. , 1 (2016).[2] S. Descotes-Genon and B. Moussallam, Eur. Phys. J. C , 553 (2006).[3] J. R. Peláez, A. Rodas, and J. 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Adam Szczepaniak Theory CenterThomas Jefferson National Accelerator FacilityNewport News, VA 23606, U.S.A. &Center for Exploration of Energy and MatterIndiana UniversityBloomington, IN 47403, U.S.A. &Physics DepartmentIndiana UniversityBloomington, IN 47405, U.S.A. Abstract Several of the mysterious XY Z resonances have been observed in 3-body B → πK ( c ¯ c ) decays. A better description of the πK dynamics is required to improve the understanding ofthese decays, and eventually to confirm the existence of the exotic states. As an example, wediscuss the B → J/ψπK decay, where the Z (4430) has been observed. We critically reviewthe formalisms to build amplitudes available in the literature. Introduction The last decade witnessed the observation of many unexpected XY Z resonances in theheavy quarkonium sector. Their spectrum and production and decay rates are not compatiblewith a standard charmonium interpretation [1–3]. Some of these states have been observedin 3-body B → πK ( c ¯ c ) decays, with ( c ¯ c ) = J/ψ, ψ (cid:48) , χ c . Understanding the three-bodydynamics, and specifically the effect of the πK interaction, is mandatory to confirm theexistence of these exotic states, and to better establish their properties.We focus here on the so-called Z (4430) . The state was claimed in 2007 by Belle, as a peakin the B → K + ( ψ (cid:48) π − ) channel [4], and it was the first observation of a charged charmo-niumlike state. The rich structure of the πK resonances has led to opposite claims aboutthe need of an exotic state to describe data [5–7]. The high-statistic analysis 4D analysis byLHCb provided further evidence for the existence of such state [8]. In particular, the analy-sis of the Legendre moments in [9] suggests that the πK waves with J ≤ are not able todescribe the data, calling either for an unexpected contribution of higher spin K ∗ , or for anexotic resonance in the crossed channel. This state is extremely interesting, because is farfrom any reasonable open-charm threshold with the correct quantum numbers [10, 11]. Theaveraged mass and width are M = (4478 ± M eV and Γ = (180 ± M eV , whereasthe favored signature is J P C = 1 + − (see Fig. 1).126 [GeV - p ' y m16 18 20 22 ) C a nd i d a t e s / ( . G e V LHCb ] [MeV/c p (2S) y m ) Y i e l d / ( M e V / c LHCb Figure 1: Left panel: Invariant mass distributions in ψ (cid:48) π − channel, from Ref. [8]. The red solid(brown dashed) curve shows the fit with (without) the additional Z (4430) resonance. Right panel:the Legendre moments populated by πK waves with J ≤ are not able to describe data (fromRef. [9]). Although the agreement in the figure may look good, the significance of higher momentais σ .2. Analyticity constraints for B → ψ (cid:48) πK In the modern literature, there seems to be a lot of confusion regarding properties of thereaction amplitudes employed in analyses of such processes. This is often stated in thecontext of a potentially nonrelativistic character of certain approaches [12–17]. As discussedin Ref. [18], rather than arising from relativistic kinematics, the differences between thevarious formalisms have a dynamical origin.We consider the scattering process ψ (cid:48) B → πK , related to the decay we are interested invia crossing symmetry. The spinless particles B , π , K are stable against the strong inter-action, and the ψ (cid:48) is narrow enough to completely factorize its decay dynamics. We use p i , i = 1 . . . to label the momenta of ψ (cid:48) , B , π , and K respectively. We denote the helicity am-plitude by A λ ( s, t ) , λ being the helicity of ψ (cid:48) . The amplitude depends on the standard Man-delstam variables s = ( p + p ) , t = ( p − p ) , and u = ( p − p ) with s + t + u = (cid:80) i m i .We discuss the parity violating amplitude. We call p ( q ) to the magnitude of the incoming(outgoing) three momentum in the s -channel center of mass, and θ s the scattering angle. Thequantities depend on the Mandelstam invariants through z s ≡ cos θ s = s ( t − u ) + ( m − m )( m − m ) λ / λ / , p = λ / √ s , q = λ / √ s , (1)with λ ik = ( s − ( m i + m k ) ) ( s − ( m i − m k ) ) . We assume the isobar model, and incorpo-rate the πK resonances up to spin J max via A λ ( s, t, u ) = 14 π J max (cid:88) j = | λ | (2 j + 1) A jλ ( s ) d jλ ( z s ) , (2)127here A jλ ( s ) are the helicity partial wave amplitudes in the s -channel. In Eq. (2) the entire t dependence enters though the d functions. The d functions have singularities in z s which leadto kinematical singularities in t of the helicity amplitudes A λ . An extensive discussion andthe full characterization of the kinematical singularities can be found in Refs. [19–24]. Werecall that d jλ ( z s ) = ˆ d jλ ( z s ) ξ λ ( z s ) , where ξ λ ( z s ) = (cid:16)(cid:112) − z s (cid:17) | λ | is the so-called half an-gle factor that contains all the kinematical singularities in t . The reduced rotational function ˆ d jλ ( z s ) is a polynomial in s and t of order j −| λ | divided by the factor λ ( j −| λ | ) / λ ( j −| λ | ) / . Thehelicity partial waves A jλ ( s ) have singularities in s . These have both dynamical and kine-matical origin. The former arise, for example, from s -channel resonances. The kinematicalsingularities, just like the t -dependent kinematical singularities, arise because of externalparticle spin. We explicitly isolate the kinematic factors in s , and denote the kinematicalsingularity-free helicity partial wave amplitudes by ˆ A jλ ( s ) , A j ( s ) = K ( pq ) j ˆ A j ( s ) , (3a) A j ± ( s ) = K ± ( pq ) j − ˆ A j ± ( s ) , (3b)with K = 2 m /λ / , and K ± = λ / / √ s . The j = 0 amplitude is exceptional, A ( s ) = ˆ A ( s ) /K . The ˆ A jλ ( s ) are left as the dynamical functions we are after, usuallyparameterized in terms of a sum of Breit-Wigner amplitudes with Blatt-Weisskopf barrierfactors. We now seek a representation of A λ ( s, t ) in terms of the scalar functions, A λ ( s, t ) = (cid:15) µ ( λ, p ) (cid:20) ( p − p ) µ − m − m s ( p + p ) µ (cid:21) C ( s, t )+ (cid:15) µ ( λ, p )( p + p ) µ B ( s, t ) , (4)where the functions B ( s, t ) and C ( s, t ) are the kinematical singularity free scalar amplitudes.We can match Eqs. (2) and (4), and express the scalar functions as a sum over kinematicalsingularity free helicity partial waves. √ C ( s, t ) = 14 π (cid:88) j> (2 j + 1)( pq ) j − ˆ A j ± ( s ) ˆ d j ( z s ) , (5) πB ( s, t ) = ˆ A ( s ) + 4 m λ (cid:88) j> (2 j + 1)( pq ) j × (cid:20) ˆ A j ( s ) ˆ d j ( z s ) + s + m − m √ m ˆ A j + ( s ) z s ˆ d j ( z s ) (cid:21) . (6)Neither B ( s, t ) nor C ( s, t ) can have kinematical singularities in s or t . In Eqs. (5)-(6), ˆ d j ( z s ) is regular in t , and the s singularities at (pseudo)thresholds are canceled by the factor ( pq ) j − . For the same reason the sum in Eq. (6) has no kinematical singularities in s and t ,however the /λ factor in front of the sum generates two poles at s ± = ( m ± m ) , unlessthe expression in brackets vanishes at those points. This means that the ˆ A jλ ( s ) with different λ cannot be independent functions at the (pseudo)threshold. Explicitly, using the expansionof the Wigner d -function for z s → ∞ , we get ˆ A j ( s ) ( z s ) j (cid:104) j − , 0; 1 , | j, (cid:105) − s + m − m √ m ˆ A j + ( s ) ( z s ) j √ (cid:104) j − , 0; 1 , | j, (cid:105) . (7)128his combination has to vanish to cancel the /λ , thus one finds (for j > ) ˆ A j + ( s ) = (cid:104) j − , 0; 1 , | j, (cid:105) g j ( s ) + λ f j ( s ) , (8a) ˆ A j ( s ) = (cid:104) j − , 0; 1 , | j, (cid:105) s + m − m m g (cid:48) j ( s ) + λ f (cid:48) j ( s ) , (8b)where g j ( s ) , f j ( s ) , g (cid:48) j ( s ) , and f (cid:48) j ( s ) are regular functions at s = s ± , and g j ( s ± ) = g (cid:48) j ( s ± ) .Together with Eq. (8), the expressions in Eqs. (4), (5) and (6) provide the most generalparameterization of the amplitude that incorporates the minimal kinematic dependence thatgenerates the correct kinematical singularities as required by analyticity.Upon restoration of the kinematic factors, the original helicity partial wave amplitudes read A j + ( s ) = p j − q j (cid:20) (cid:104) j − , 0; 1 , | j, (cid:105) g j ( s ) + λ f j ( s ) (cid:21) , (9a) A j ( s ) = p j − q j (cid:20) (cid:104) j − , 0; 1 , | j, (cid:105) s + m − m m √ s g (cid:48) j ( s ) + m √ s λ f (cid:48) j ( s ) (cid:21) , (9b)and A ( s ) = λ / / (2 m ) ˆ A ( s ) , where ˆ A ( s ) is regular at (pseudo)threshold. A particularchoice of the functions g j ( s ) , g (cid:48) j ( s ) , f j ( s ) and f (cid:48) j ( s ) constitutes a given hadronic model.We now compare the general expression for the helicity partial waves with the spin-orbitLS partial waves, ˆ G jL ( s ) . These match the general form in Eq. (8) when g j ( s ) = (cid:113) j − j +1 ˆ G jj − ( s ) , f j ( s ) = 14 s (cid:113) j +32 j +1 (cid:104) j + 1 , 0; 1 , | j, (cid:105) ˆ G jj +1 ( s ) , (10a) g (cid:48) j ( s ) = 2 m √ ss + m − m (cid:113) j − j +1 ˆ G jj − ( s ) , f (cid:48) j ( s ) = 14 m √ s (cid:113) j +32 j +1 (cid:104) j + 1 , 0; 1 , | j, (cid:105) ˆ G jj +1 ( s ) . (10b)The common lore is that the LS formalism is intrinsically nonrelativistic. However, thematching in Eq. (10) proves that the formalism is fully relativistic, but care should be takenwhen choosing a parameterization of the LS amplitude so that the expressions in Eqs. (10)are free from kinematical singularities. For example, if one takes the functions ˆ G jj − ( s ) and ˆ G jj +1 ( s ) to be proportional to Breit-Wigner functions with constant couplings, the amplitudes g (cid:48) j ( s ) and f (cid:48) j ( s ) would end up having a pole at s = m − m . It is clear that using Breit-Wigner parameterizations, or any other model for helicity amplitudes, i.e., the left-hand sidesof Eq. (10), instead of the LS amplitudes helps prevent unwanted singularities.We also consider the Covariant Projection Method (CPM) approach of [12–15], based on theconstruction of explicitly covariant expressions. We limit ourselves to the special case of anintermediate K ∗ with j = 1 . We start with the tensor amplitude for the scattering process ψB → K ∗ → πK , A λ ( s, t ) = (cid:15) µ ( λ, p ) (cid:18) − g µν + P µ P ν s (cid:19) X ν ( q, P ) g S ( s )+ (cid:15) ρ ( λ, p ) X ρµ ( p, P ) (cid:18) − g µν + P µ P ν s (cid:19) X ν ( q, P ) g D ( s ) , (11)129here P is the K ∗ momentum. The final P-wave orbital tensor is X ν ( q, P ) = q ⊥ ν = q ν − P ν P · q/s . The D-wave orbital tensor X ρµ ( p, P ) = 3 p ρ ⊥ p µ ⊥ / − g ρµ ⊥ p ⊥ / , with p µ ⊥ = p µ − P µ P · p/s , and g ρµ ⊥ = g ρµ − P ρ P µ /s . Explicitly, A + ( s, θ s ) = − q sin θ s √ (cid:20) g S ( s ) + p g D ( s ) (cid:21) , A ( s, θ s ) = q E m cos θ s (cid:2) g S ( s ) − p g D ( s ) (cid:3) , (12)and matching with Eq. (8) gives g ( s ) = g (cid:48) ( s ) = 4 π g S ( s ) , f ( s ) = 2 π s g D ( s ) , f (cid:48) ( s ) = − π s s + m − m m g D ( s ) . (13) M π K (GeV )012345 / Γ d Γ / d M π K ( G e V − ) Without Blatt-Weisskopf factors JPACCPM scatteringCPM decayLS scatteringLS decay 0.5 1.0 1.5 2.0 2.5 M π K (GeV )0.00.51.01.52.0 / Γ d Γ / d M π K ( G e V − ) With Blatt-Weisskopf factors Figure 2: Comparison of the lineshape of K ∗ (892) and K ∗ (1410) in the πK -invariant mass dis-tribution, constructed with the different formalisms. In the left panel we show the result with nobarrier factors. In the right panel, we include the customary Blatt-Weisskopf factors. From [18].The threshold conditions g ( s ± ) = g (cid:48) ( s ± ) are satisfied, and the functions f ( s ) and f (cid:48) ( s ) are regular at the thresholds. Finally, we show the relation between the CPM and the LSamplitudes: π G ( s ) = g S ( s ) q (cid:114) (cid:18) E m + 2 (cid:19) − g D ( s ) q p (cid:114) (cid:18) E m − (cid:19) , (14a) π G ( s ) = g D ( s ) q p (cid:114) (cid:18) E m + 1 (cid:19) − g S ( s ) q (cid:114) (cid:18) E m − (cid:19) . (14b)Although the g S ( s ) and g D ( s ) of the CPM formalism, see Eq. (11), are typically interpretedas the S and D partial wave amplitudes, we see that this is the case only at (pseudo)threshold s = s ± , where the factor E /m − vanishes. An extensive discussion about the samecalculation performed in the decay kinematics, which turns out into an explicit violation ofcrossing symmetry, can be found in [18].130o explore the differences between the various approaches, we consider the example oftwo intermediate vectors in the πK channel: the K ∗ (892) , with mass and width M K ∗ =892 MeV, Γ K ∗ = 50 MeV, and the K ∗ (1410) , with M K ∗ = 1414 MeV, Γ K ∗ = 232 MeV.In Fig. 2, we show the results for the differential decay width in five different scenarios. Weconsider the CPM formalism (for the scattering and decay kinematics, respectively), setting g S ( s ) = 0 and g D ( s ) = T K ∗ ( s ) , with T K ∗ ( s ) being the sum of Breit-Wigners for the tworesonances. For the LS formalism, we choose the couplings to be ˆ G ( s ) = 0 , ˆ G ( s ) = T K ∗ ( s ) . The LS amplitude in the decay kinematics differs from the one in the scatteringkinematics because of the breakup momentum of B → ψK ∗ , calculated in the B rest frameor in the K ∗ rest frame, respectively. In Ref. [18] we also propose an alternative model.We see in Fig. 2 that the Kπ invariant mass squared distribution is significantly distorted inall models. This is important to extract the physical couplings, and may enhance the highmass contributions of the higher spin K ∗ , thus affecting the extraction of the properties ofthe Z (4430) .3. Conclusions (cid:72) Ψ ' Π (cid:76) (cid:72) MeV (cid:76) E v e n t s Figure 3: Fit to the ψ (cid:48) π invariant mass, in the GeV ≤ m ( πK ) ≤ . GeV slice, where thesignal of the Z (4430) is more prominent. The red curve includes the K ∗ resonances with J ≤ ,whereas the blue curve includes the tails of the K ∗ (2045) and the K ∗ (2380) . Data from [9].The rich structure of the πK system affects the extraction of several exotic candidates, asthe Z (4200) in B → J/ψπK [25], the Z (4050) and Z (4250) in B → χ c πK [26, 27],and in particular the Z (4430) in B → ψ (cid:48) πK we have discussed here [4–9]. As we showin a temptative fit in Fig. 3, the inclusion of higher spin K ∗ resonances can improve thedescription of the ψ (cid:48) π invariant mass, even without adding an exotic state. Although thismay not be enough to challenge the existence of the Z (4430) , this can dramatically affectthe estimate of the resonance parameters and of its quantum numbers. This also calls formore refined analysis, which take into account unitarity in the isobar model [28–35], orinclude the effect of higher spin resonances [36].131. Acknowledgments I wish to thank Igor Strakovsky for his kind invitation to this workshop, and Tomasz Skwar-nicki for the fruitful collaboration. 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J. C , no. 8, 508 (2017).[35] M. Albaladejo et al. [JPAC Collaboration], arXiv:1803.06027 [hep-ph].[36] A. P. Szczepaniak and M. R. Pennington, Phys. Lett. B , 283 (2014).133 .18 Three Particle Dynamics on the Lattice Akaki Rusetsky Helmoltz-Institut für Strahlen- und KernphysikUniversität Bonn &Bethe Center for Theoretical PhysicsUniversität BonnD-53115 Bonn, Germany Abstract In my talk, I review the recent progress in understanding the three-particle quantizationcondition, which can be used for the extraction of physical observables from the finite-volumespectrum of the three-particle system, measured on the lattice. It is demonstrated that the finite-volume energy levels allow for a transparent interpretation in terms of the three-body boundstates, as well as the three-particle and particle-dimer scattering states. The material, coveredby this talk, is mostly contained in the recent publications [1–3]. Introduction The Lüscher approach [4] has become a standard tool for the analysis of lattice data inthe two-body scattering sector. It has been further generalized to study the scattering ofparticles with arbitrary spin, the scattering in the moving frames, as well as the scatteringin coupled two-body channels. Moreover, in their seminal paper [5], Lellouch and Lüscherhave discussed the application of the finite-volume formalism to the calculation of the two-body decay matrix elements. This approach has been also generalized to include multipledecay channels and applied to study of various formfactors in the timelike region and matrixelements with resonances (in the latter case, the analytic continuation in the complex planeto the resonance pole has to be considered) [6–9].The formulation of the finite-volume approach for the three-body problem (in analogy withthe Lüscher approach) has however proven to be a challenging task. Despite the significanteffort during the last few years [1–3, 10–20], the progress has been rather slow. Recently,the tree-body quantization condition, which is a counterpart of the Lüscher equation in thetwo-body case, has been derived in different settings [1, 2, 11, 12, 17] (it has been shown [3]that all these different settings are essentially equivalent to each other, so in practice all boilsdown to the choice of merely the most convenient one). On the basis of the quantizationcondition, the finite-volume energy levels have been calculated in some simple cases. Suchstudies are extremely interesting because, at this stage, one does not yet have enough insightin the problem and lacks the intuition to predict the volume dependence of the three-bodyspectrum, which would emerge in lattice simulations. Moreover, such studies may facilitatethe interpretation of this volume dependence in terms of the observable characteristics of thethree-particle systems in a finite volume, in analogy with the two-particle case where, e.g.,the avoided level crossing in the spectrum is often related to a nearby narrow resonance.The study of the three-particle (and, in general, many-particle) systems on the lattice isinteresting at least from two different perspectives. First, we would refer to the potentialapplications in study of the nuclear physics problems in lattice QCD. Second, it would be134nteresting to study the systems, whose decay into the final states with three and more par-ticles cannot be neglected. In the meson sector, the simplest example could be the decay K → π (a counterpart of the process K → π , which was considered in the original paperby Lellouch and Lüscher [5]), but physically more interesting processes like the decays of a (1260) or a (1420) as well. In the baryon sector, the most obvious candidate is the Roperresonance. In order to obtain an analog of the Lellouch-Lüscher formula for such systems,however, one has first to understand the final-state interactions in a finite volume, and it iswhere the study of the solutions of the three-body quantization condition might help.In my talk, I shall briefly cover the formalism of Refs. [1, 2], which is based on the use ofthe effective field theory in a finite volume, and will further discuss the solution of theseequations, which includes the projection of the quantization condition onto the differentirreducible representations (irreps) of the octahedral group [3]. It will be shown that thesesolutions allow for a nice interpretation in terms of the three-particle bound states, as well asthree-particle and particle-dimer scattering states.2. The Formalism In Refs. [1, 2], the three-particle quantization condition was obtained by using effective fieldtheory approach in a finite volume. Moreover, it has been shown that it is very convenientto use the so-called particle-dimer picture. It should be stressed that this approach is notan approximation, but an equivalent description of the three-particle interactions. This ismost easily seen in the path integral formalism, where the introduction of a dimer amountsto using an additional dummy integration variable, without changing the value of the pathintegral. Moreover, using the particle-dimer picture does not necessarily imply the existenceof a shallow bound state or a narrow resonance in the two-body sector.Below we consider the case of three identical spinless bosons with the S-wave pair interac-tions and use, for simplicity, the non-relativistic kinematics. In order to derive the quantiza-tion condition, we consider the finite-volume Bethe-Salpeter equation for the particle-dimeramplitude M L M L ( p , q ; E ) = Z ( p , q ; E ) + 1 L (cid:88) k Z ( p , k ; E ) τ L ( k ; E ) M L ( k , q ; E ) . (1)Here, p and q are three-momenta of the incoming and outgoing spectators and E is the totalenergy of three particles. All momenta are discretized, e.g., p = 2 π n /L, n ∈ Z and,similarly, for other momenta. Further, L is the spatial size of the cubic box and Λ denotesthe ultraviolet cutoff. The quantity τ L corresponds to the two-body scattering amplitude (thedimer propagator, in the particle-dimer language), and is given by πτ − L ( k ; E ) = k ∗ cot δ ( k ∗ ) + S ( k , ( k ∗ ) ) ,S ( k , ( k ∗ ) ) = − πL (cid:88) l k + kl + l − mE , (2)where k ∗ is the magnitude of the relative momentum of the pair in the rest frame, k ∗ = (cid:114) k − mE . (3)135n Eq. (2), unlike Eq. (1), the momentum sum is implicitly regularized by using dimen-sional regularization and δ ( k ∗ ) is the S-wave phase shift in the two-particle subsystem. Theeffective range expansion for this quantity reads k ∗ cot δ ( k ∗ ) = − a + 12 r ( k ∗ ) + O (( k ∗ ) ) , (4)where a, r are the two-body scattering length and the effective range, respectively. In the nu-merical calculations, for illustrative purpose, we shall use a simplified model, assuming thatthe effective range r and higher-order shape parameters are all equal to zero, correspondingto the leading order of the effective field theory for short range-interactions. The equation (1)is valid, however, beyond this approximation.Finally, the quantity Z denotes the kernel of the Bethe-Salpeter equation. It contains theone-particle exchange diagram, as well as the local term, corresponding to the particle-dimerinteraction (three-particle force). In general, the latter consists of a string of monomials inthe 3-momenta p and q . In the numerical calculations, we shall again restrict ourselves tothe model, where only the non-derivative coupling, which is described by a single constant H (Λ) , is non-vanishing. The kernel then takes the form Z ( p , q ; E ) = 1 − mE + p + q + pq + H (Λ)Λ . (5)The dependence of H (Λ) on the cutoff is such that the infinite-volume scattering ampli-tude is cutoff-independent. In a finite volume, this ensures the cutoff-independence of thespectrum.At the next step, the three-particle Green function can be expressed in terms of the particle-dimer scattering amplitude M L by using the LSZ formalism, and hence the poles of thelatter can be mapped onto the finite-volume energy spectrum of the three particle system.It can be shown [3] that the poles arise at the energies, where the determinant of the linearequation (1) vanishes. This finally gives the quantization condition we are looking for det( τ − L − Z ) = 0 . (6)The l.h.s. of the above equation defines a function of the total energy E , which, for a fixed Λ and L , depends both on the two-body input (the scattering phase δ both above and belowthe two-body threshold) as well as the three-body input (the non-derivative coupling H (Λ) ,higher-order couplings). The former input can be independently determined from the sim-ulations in the two-particle sector and extrapolation below threshold. Hence, measuring thethree-particle energy levels, one will be able to fit the parameters of the three-body force.Finally, using the same equations in the infinite volume with the parameters, determined onthe lattice, one is able to predict the physical observables in the infinite volume.3. The Projection of the Quantization Condition The three-body quantization condition, Eq. (6), determines the entire finite-volume spectrumof the system. However, the eigenvalue problem in a cubic box has the octahedral symmetry,which is a remnant of the rotational symmetry in the infinite volume. This means that all136nergy levels can be assigned to one of the ten irreps A ± , A ± , E ± , T ± , T ± of the octahedralgroup G and the spectrum in each irrep can be measured separately with a proper choice ofthe source/sink operators. It is possible to use this symmetry and to project the quantiza-tion condition onto the different irreps – the obtained equations will determine the energyspectrum in each irrep separately [3].In order to do this, we shall act in a close analogy with the partial-wave expansion in theinfinite volume. A substitute for radial integration will be the sum over shells, defined as setsof momenta with equal magnitude, which can be obtained from any vector (referred hereafteras the reference vector), belonging to the same shell, by applying all transformations of theoctahedral group. Note that the vectors with the same magnitude, which are not connected bya group transformation belong, by definition, to different shells. Furthermore, the integrationover the solid angle in the infinite volume is replaced by the sum over all G = 48 elementsof the octahedral group.On the cubic lattice, the analog of the partial-wave expansion is given by f ( p ) = f ( g p ) = (cid:88) Γ (cid:88) ρσ T Γ σρ ( g ) f Γ ρσ ( s ) , Γ = A ± , A ± , E ± , T ± , T ± , (7)where T Γ σρ ( g ) are the matrices of the irreducible representations, p denotes the referencemomentum, and s is the shell to which both p and p belong. Nothing depends on thechoice of p .Using the orthogonality of the matrices of the irreducible representations, it is possible toproject out the quantity f Γ ρσ ( s ) : (cid:88) g ∈G ( T Γ λδ ( g )) ∗ f ( g p ) = Gs Γ f Γ δλ ( s ) , (8)where s Γ is the dimension of the irrep Γ , and the indices λ, δ, . . . run from 1 to s Γ .Next, we note that both the kernel of the Bethe-Salpeter equation and the dimer propagatorare invariant with respect to the group G : Z ( g p , g q ; E ) = Z ( p , q ; E ) , τ L ( g k ; E ) = τ L ( k ; E ) , for all g ∈ G . (9)Using this property, it can be shown that the projection of the quantization condition ontothe irrep Γ takes the form det (cid:18) τ L ( s ) − δ rs δ σρ − ϑ ( s ) GL Z Γ σρ ( r, s ) (cid:19) = 0 , (10)where ϑ ( s ) is the multiplicity of the shell s (the number of independent vectors in this shell),and Z Γ λρ ( r, s ) = (cid:88) g ∈G ( T Γ ρλ ( g )) ∗ Z ( g p , k ) . (11)Here, the reference vectors p and k belong to the shells r and s , respectively. In the nextsection, we shall solve this equation and obtain the spectrum of both the bound and scatteringstates. 137. The Energy Spectrum (a) The Choice of the Model In order to gain insight into the volume-dependence of the three-particle spectrum, weshall solve the equation (10) in the irrep Γ = A +1 in a simple model, described insection 2. The parameters of the model are fixed as follows. First, we take m = a = 1 .This means that there exists a bound dimer with the binding energy B = ma =1 . Further, we fix the ultraviolet cutoff Λ = 225 – large enough so that all cutoffartifacts can be safely neglected. The last remaining parameter H (Λ) is fixed from therequirement that there exists a three-body bound state with the binding energy B = 10 .This gives H (Λ) = 0 . . Of course, for a different choice of Λ , one may adjust H (Λ) , so that all low-energy spectrum remains the same.(b) Bound States Except of the already mentioned deeply bound three-body state with B = 10 , themodel contains an extremely shallow bound state with B = 1 . – just below theparticle-dimer threshold. Since the characteristic size of such system is much largerthan that of the dimer, it is conceivable that it should behave like a two-body boundstate of a particle and a dimer. For the deep bound state, two-particle and three-particlebound-state scales are comparable in magnitude and hence one might expect a behaviorthat interpolates between the extreme cases of a three-particle shallow bound state anda particle-dimer bound state.It is quite intriguing that the study of the finite-volume spectrum allows one to make achoice among the above alternatives. It is, in particular, well known that the Lüscherequation leads to the finite-volume correction ∝ exp( − ∆ L ) /L to the infinite-volumebinding energy (here, ∆ is some mass scale determined by the kinematics). The func-tional dependence of the three-body shallow bound state is different, ∝ exp( − ∆ L ) /L / , see, e.g., Refs. [18]. So, the volume dependence of the bindingenergy contains information about the nature of the three-body bound states.Among the solutions of the quantization condition (10) one can readily identify thelevels that tend to the infinite-volume binding energies. It is seen that the volume-dependence of the shallow bound state can be approximated by the function ∝ exp( − ∆ L ) /L very well. For the deeper state, the situation is different and one needsa linear combination of the above two functions to get a decent fit. All this of courseperfectly matches our expectations. More details can be found in Ref. [3].(c) Scattering States There are two types of the non-interacting energy levels, corresponding to the differentasymptotic states in the three-body problem. In particular, we have the particle-dimerscattering states with different back-to-back momenta. The threshold for such states isgiven by the dimer binding energy and lies at E th = − . In addition to this, we havefree 3-particle states with the threshold E th = 0 . Of course, in the interacting theory,which we are considering, the energy levels are displaced from their “free” values,but the displacement is relatively small almost everywhere. For this reason, one canidentify the levels of the interacting theory with the different free levels.138 ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ ♡ - E Figure 1: The three lowest-lying states above threshold. The result obtained using Eq. (12) isgiven by the black solid curve. The red dashed curve shows the free particle-dimer states withback-to-back momenta (0 , , , (0 , , and (1 , , , whereas the blue dotted lines denote thefree three-particle states (the lowest level at E = 0 corresponds to the threshold, where all threeparticles are at rest).In Fig. 1, the volume-dependence of the three lowest scattering states, obtained fromthe solution of Eq. (10), is shown. For comparison, in the same figure, we plot the freeparticle-dimer and three-particle energy levels.Last but not least, we plot the displaced ground-state energy of three particles which,up to and including order L − , is given by the formula (see, e.g., Refs. [13, 21]) E = 12 πaL − a L I + 12 a πL ( I + J ) + O ( L − ) , (12)where I (cid:39) − . and J (cid:39) . are numerical constants. It is seen that, at thisaccuracy, the energy shift from the unperturbed value E = 0 contains only the singleparameter a which is fixed from the beginning. Thus, Eq. (12) constitutes a predictionthat allows to identify the three-particle ground state.The interpretation of the levels, shown in Fig. 1 is crystal clear. The lowest level isa displaced particle-dimer ground state. Its relatively large displacement is caused bythe existence of a very shallow three-body bound state, which pushes all other particle-dimer levels up. At small values of L , the next level is the displaced three-body groundstate. Around L (cid:39) , free particle-dimer and three-particle levels cross each other. Thisfact leads to the avoided level crossing in the interacting spectrum, the second level goesdown to the particle-dimer threshold and the third level becomes the displaced ground-state three-particle level. This pattern repeats itself each time, when we have a crossingof two levels corresponding to the different asymptotic states. For more details, see139ef. [3].5. Conclusions i) We use the effective field theory approach in a finite volume to obtain the three-particlequantization condition. The fit of the solutions of this equation to the three-particlefinite-volume spectrum determines the particle-dimer coupling constant(s), which canbe further used in the Bethe-Salpeter equation to reconstruct the infinite-volume scatter-ing amplitudes in the three-particle sector. This is the essence of the approach proposedin Refs. [1, 2].ii) Using the octahedral symmetry of the problem in a cubic box, the quantization condi-tion was projected onto the different irreps of this symmetry group.iii) The equation was numerically solved in the A +1 irrep, both for the bound states andscattering states. It was shown that a nice interpretation of the energy levels is possiblein terms of the different asymptotic states of the three-particle system. In particular, itwas shown that the avoided level crossing occurs at those values of L , when the freelevels, corresponding to these states, have the same energy.6. Acknowledgments The author acknowledges the support from the CRC 110 “Symmetries and the Emergenceof Structure in QCD” (DFG grant no. TRR 110 and NSFC grant No. 11621131001). Thisresearch was also supported in part by Volkswagenstiftung under contract no. 93562 and byShota Rustaveli National Science Foundation (SRNSF), grant No. DI–2016–26. References [1] H. W. Hammer, J. Y. Pang, and A. Rusetsky, JHEP (2017) 109.[2] H.-W. Hammer, J.-Y. Pang, and A. Rusetsky, JHEP (2017) 115.[3] M. Döring, H.-W. Hammer, M. Mai, J.-Y. Pang, A. Rusetsky, and J. Wu, arXiv:1802.03362[hep-lat].[4] M. Lüscher, Nucl. Phys. B (1991) 531.[5] L. Lellouch and M. Lüscher, Commun. Math. Phys. (2001) 31.[6] V. Bernard, D. Hoja, U.-G. Meißner, and A. Rusetsky, JHEP (2012) 023; A. Agadjanov,V. Bernard, U.-G. Meißner, and A. Rusetsky, Nucl. Phys. 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D (2007) 074507.141 .19 S-matrix Approach to Hadron Gas: a Brief Review Pok Man Lo Institute of Theoretical PhysicsUniversity of WroclawPL-50204 Wrocław, Poland &Extreme Matter Institute (EMMI), GSID-64291 Darmstadt, Germany Abstract I briefly review how the S-matrix formalism can be applied to analyze a gas of interactinghadrons. Introduction The S-matrix formulation of statistical mechanics by Dashen, Ma, and Bernstein [1] ex-presses the grand canonical potential in terms of the scattering matrix elements. When ap-plied to describe the system of interacting hadrons [2], the logarithm of the partition functioncan be written as a sum of two pieces : ln Z = ln Z + ∆ ln Z, (1)where ln Z = V × (cid:88) i ∈ gs d i (cid:90) d k (2 π ) e − β √ k + m i (2)is the grand potential for an uncorrelated gas of particles that do not decay under the stronginteraction (i.e., ground-state particles), such as pions, kaons, and nucleons. The interactingpart of the grand potential, ∆ ln Z , can be written in the form ∆ ln Z = V × (cid:90) d √ s d P (2 π ) e − β √ P + s ρ eff ( √ s ) , (3)where √ s is the invariant mass of the relevant scattering system. The quantity ρ eff ( √ s ) can be understood as an effective level density due to the interaction. A key step of the S-matrix approach to study thermodynamics is to identify such effective level density, in thelow density limit where only binary collisions are important, with ρ eff ( √ s ) → ρ smat ( √ s ) = (cid:88) int d IJ × dd √ s (cid:18) π δ IJ ( √ s ) (cid:19) . (4)Here the sum is over all interaction channels, d g is the relevant degeneracy factor, and δ IJ ( √ s ) is the scattering phase shift. Note that the standard Hadron Resonance Gas (HRG)model [3] can also be expressed in this form, with the replacement ρ eff ( √ s ) → ρ HRG ( √ s ) = (cid:88) res d IJ × dd √ s (cid:0) θ ( √ s − m res ) (cid:1) , (5)142 δ (I=1/2J=0)(degree) M (GeV) Estabrooks et al.Aston et al.Baker et al.this work a S = 0 . m δ (I=1/2J=1)(degree) M (GeV) Estabrooks et al.Aston et al.Mercer et al.this work a P = 0 . m ⇡ Estabrooks et al.Jongejans et al.Cho et al.Baker et al.this work -30-20-100 0.6 0.8 1 1.2 1.4 1.6 1.8 a S = . m ⇡ In Figs. 1 and 2, we present results for the interaction contribution to the thermodynamicpressure, computed within the S-matrix approach for πK and ππ scatterings respectively. For simplicity, we present the formulae for the case of vanishing chemical potentials and Boltzmann statistics. Inpractical calculations, suitable fugacity factors and the correct quantum statistics have to be implemented. ( ) ( ) + f ( ) ��� � � ����� I = j = I = j = + I = j = free gasS-matrix I=1I=0,2 a S = . m -20-15-10-500.2 0.4 0.6 0.8 1 1.2 � ���������������� ������ Froggatt & Petersen ���� ���� -20-15-10-500.2 0.4 0.6 0.8 1 1.2 a S = 0 . m � ���������������� ������ Estabrooks & MartinFroggatt & PetersenGrayer et al.Rosselet et al.Belkov et al.this work a S = 0 . m � ���������������� ������ Estabrooks & MartinFroggatt & PetersenProtopopescu et al.this work Figure 2: ππ scattering phase shifts for I = 0 , , and their contributions to thermodynamicpressure.Also shown are the corresponding results from the HRG model.The key input here is the scattering phase shifts. Extensive experimental [15–21] and the-oretical [22–25] efforts are devoted to study these quantities. An efficient way to com-pute the effective level density in Eq. (4) is to perform a phenomenological fit to the phaseshift data [22, 26]. This offers some insights into the relative importance of resonant andnon-resonant contributions in an interaction channel. Nevertheless, more fundamental ap-proaches such as chiral perturbation theory [23] and LQCD [24, 25] are in better position toimplement known theoretical constraints and predict phase shifts for channels that are notyet measured.Let us now discuss several key features of the results in Figs. 1 and 2.(a) S-wave Channel In the I = 1 / ( I = 0 ) channel of πK ( ππ ) scattering we observe a slowly rising phaseshift in the low-energy region. This portion of the phase shift can reflect the physicalproperties of the unconfirmed κ - (confirmed σ -) meson. In particular, the phase shiftdoes not reach ◦ before K ∗ (1430) ( f (980) ) emerges. In the same energy range wealso observe a major cancellation effect from the I = 3 / ( I = 2 ) channel after themultiplication of an appropriate degeneracy factor. The two effects combined severely144uppress the overall pressure from these channels, down to the value of a free gasof K ∗ (1430) ( f (980) ). This underpins the proposed prescription that κ - ( σ -) mesonshould not be included in the HRG particle sum [26, 27].(b) P-wave Channel For the πK system, the S-matrix approach yields a similar result on the thermodynamicpressure as the HRG model. The latter is based on a point-like treatment K ∗ (892) . Onthe other hand, the approach gives an enhanced effect from the two-body scatteringbeyond the free gas result for the ππ system. dn/p T dp T (GeV -2 fm -3 ) p T (GeV) S-matrixzero width ⇡ + Baryon Sector The S-matrix approach has also been applied to study the pion-nucleon system [33]. Usingthe empirical phase shifts from the SAID PWA database [34], it is demonstrated that the nat-ural implementation of the repulsive forces and the consistent treatment of broad resonancescan improve our understanding of fluctuation observables computed in lattice QCD, such asthe baryon electric charge correlation.Lattice study of thermal QCD also indicates a larger interaction strength in the strange-baryon channel than that predicted by the HRG model using only the list of confirmed res-145nances [35]. A simple extension of the HRG model [36], which uses the lattice resultsto guide the incorporation of some extra hyperon states, suggests that the “missing” statesrequired is roughly consistent with the trend of the observed but unconfirmed resonances (1and 2 stars) in the PDG. The corresponding S-matrix based analysis is currently pursued. Itis clear that a detailed experimental knowledge of the hyperon spectrum is critical for thistask.4. Summary The S-matrix approach offers a consistent way to incorporate attractive and repulsive forcesbetween hadrons. Using the input of empirical phase shifts from hadron scattering experi-ments, the important physics of resonance widths and the contribution from purely repulsivechannels are naturally included. Further research in extending the scheme [31] to includeinelastic effects and three-body scatterings [37–39] has begun.The proposed new K L -beam facility can potentially strengthen our understanding of thehadron spectroscopy, such as the investigation of the “missing" hyperon states. This is crucialfor a reliable thermal description of the hadronic medium within the S-matrix approach.We expect interesting results when applying the approach to explore various phenomena ofheavy ion collisions.5. Acknowledgments I would like to thank Igor I. Strakovsky for the invitation to the Workshop. I am also grate-ful to Ted Barnes, Michael Döring, Jose R. Pelaez and James Ritman for the encouragingremarks and the fruitful discussion. This research benefits greatly from the discussion withHans Feldmeier in GSI. I would also like to thank Bengt Friman, Krzysztof Redlich, andChihiro Sasaki for the productive collaboration. Support from the Extreme Matter InstituteEMMI at the GSI is gratefully acknowledged. This work was partially supported by thePolish National Science Center (NCN), under Maestro grant DEC–2013/10/A/ST2/00106. References [1] R. Dashen, S. K. Ma, and H. J. Bernstein, Phys. Rev. , 345 (1969).[2] R. Venugopalan and M. Prakash, Nucl. Phys. A , 718 (1992).[3] P. Braun-Munzinger, K. Redlich and J. Stachel, in Quark-Gluon Plasma 3 , edited byR. C. Hwa et al. (World Scientific, Singapore, 2004), pp. 491-599.[4] C. Patrignani et al. [Particle Data Group], Chin. Phys. C , no. 10, 100001 (2016).[5] J. Gasser and H. Leutwyler, Annals Phys. , 142 (1984).[6] J. A. Oller, E. Oset, and J. R. Pelaez, Phys. Rev. Lett. , 3452 (1998).[7] J. 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Sezione di Torino10125 Torino, Italy Abstract An overview of the measurements performed by BaBar of e + e − annihilation cross sectionsin exclusive channels with kaons and pions in the final state is given. The Initial State Radia-tion technique, which allows to perform cross section measurements in a continuous range ofenergies, was employed. Introduction The BaBar experiment [1] at the SLAC National Accelerator Laboratory has performed,over the last decade, a complete set of measurements of e + e − annihilation cross sectionsof exclusive hadronic channels at low energies, exploiting the Initial State Radiation (ISR)technique. The main purpose of this study is to provide the most precise and complete inputfor the calculation of the muon anomalous magnetic moment a µ = ( g µ − / and the runningelectromagnetic constant α EW . However, also several important and useful indications onthe composition of the light and charmed meson spectrum could be obtained as by-products,by the inspection of the trends of the cross sections and the composition of the intermediatestates of the studied reactions, as well as their decay branching fractions.The knowledge of the total hadronic cross section is fundamental for the determination of thehadronic vacuum polarization contribution, at leading order, to a µ [2]. It is well known thata discrepancy at the level of 3.5 σ exists between the value of the muon anomolous magneticmoment expected from the Standard Model and what is experimentally observed, and thehadronic term of the sum adding up to a µ is the one that bears the largest uncertainty. Thedominant contribution to this term is played by the hadronic cross sections below 2 GeV, thatuntil BaBar were measured mainly inclusively. Only recently measurements of exclusivechannels could be performed accurately enough. The goal of BaBar, almost completelyaccomplished, was to provide measurements for all the exclusive channels below 2 GeV,comparing them with previous inclusive measurements and pQCD predictions.In the following a description of the measurements performed for all channels containingkaons in final states composed by a total of three or four particles will be described.2. The Initial State Radiation (ISR) Technique and Kaon Detection with BaBar BaBar operated at the PEP-II machine at SLAC and had been taking data for about 10 years,up to 2008, integrating a luminosity of about 500 fb − of e + e − annihilations taken at a fewfixed energies values corresponding mainly to the excitation of Υ(4 S ) , with smallest samplesat the peaks of Υ(2 S ) and Υ(3 S ) .In spite of the fixed energy in the center of mass, resorting to the ISR technique it washowever possible to perform cross section measurements in a continuous range of energies,potentially up to at least 8 GeV. Given a hadronic system f , the differential cross section of149he process e + e − → γf in which a photon is radiated from the electron or positron beforetheir interaction can be related to that of the non-radiative process by means of the equation: dσ e + e − → γf ( s, m f ) dm f d cos θ ∗ γ = 2 m f s W ( s, x, θ ∗ γ ) σ e + e − → f ( m f ) (1)where √ s is the e + e − center-of-mass energy, E γ and θ ∗ γ are the energy and the center-of-masspolar angle of the emitted ISR photon, and x = 2 E γ / √ s . W ( s, x, θ ∗ γ ) is a QED radiationfunction known with an accuracy better than 0.5%. Therefore, the measurement of reactionsin which an additional photon, of variable energy, is detected together the hadronic systemmay allow to infer the non radiative cross-section in a continuous range of energies. The ISRphoton is generally emitted along the e + e − collision axis, and the hadronic system is mainlyproduced back-to-back with respect to it. Due to the limited detector acceptance, the massregion below 2 GeV can only be studied if the ISR photon is detected: to perform exclusivecross section measurements the events are required to feature a photon with a center-of-massenergy larger than 3 GeV, and a fully reconstructed and identified recoiling hadronic system.In order for the latter to be fully contained in the detector fiducial volume, the ISR photonmust be emitted at large angles.Among the main advantages of the large-angle ISR method over conventional e + e − mea-surements one can count on a weak dependence of the detection efficiency on the dynamicsof the hadronic system and on its invariant mass; therefore, measurements in wide energyranges can be performed applying the same selection criteria. The exclusivity of the finalstates moreover allows the application of stringent kinematic fits, which can largely improvethe mass resolution and provide effective background suppression.The identification of charged kaons in BaBar was performed through standard techniquesbased on specific energy loss, time of flight and Cerenkov radiation, combined to provide alikelihood value for each particle identification hypothesis. The identification efficiency ofcharged kaons was as large as 89%, with a Kπ misidentification rate not larger than 2%.Neutral K S ’s were identified through their π + π − decay reconstucting a displaced vertexformed by two oppositely charged tracks, while neutral K L ’s were identified following theirnuclear interaction with the material of the electromagnetic calorimeter, which producedan energy cluster with a shape not consistent with that typical of a photon. The minimumrequired energy deposition per cluster was typically 200 MeV. The K L detection efficiencywas measured through the e + e − → φγ events, selected without the detection of the K L [3].By means of charged and neutral kaon identification the measurement of cross sections ofbasically all exclusive channels with kaons and pions (except those with two K L ’s) could beperformed:; in this way the use of isospin relationships to assess the value of the total crosssection could be avoided.The measurement of cross sections in a given channel proceeds from counting of the num-ber of events in intervals of total center-of-mass energy, after proper background subtrac-tion (usually based on Monte Carlo simulations, normalized to the existing data in rate andshape); the number of events is then normalized to the detection efficiency, and to the inte-grated luminosity in the selected energy range.150. Three Particles Final States: KKπ (a) Measurement of e + e − → K S K ± K ∓ Cross Section The purpose of this analysis was to look for possible signatures of the Y (4260) state [3].As can be seen by the trend of the cross section reported in Fig. 1(a) a clean signal dueto the J/ψ can be observed but, as shown in the inset, just an excess of events is presentat about 4.2 GeV, with a significance of 3.5 σ only. This prevents to assess the existenceof any new state, but just allows to quote an upper limit, at 90% C.L., for the elec-tronic width of the possible resonance times its branching ratio in the studied channel: U.L. (Γ Y (4260) ee B Y (4260) K S K ± π ∓ ) = 0.5 eV. The cross section reaches a maximum of about 4 nbat ∼ φ (1680) intermediate statedecaying in K ∗ K .Figure 1: a) The e + e − → K S K ± π ∓ cross section measured by BaBar. b) Dalitz plot of the e + e − → K S K ± K ∓ reaction: squared invariant masses of the ( K S π ± ) versus the ( K ± π ∓ ) systems.From the inspection of the Dalitz plot shown in Fig. 1(b), which is fairly asymmetric, itis evident that the main contribution to the intermediate state is given by K ∗ (892) ± K ∓ and K ∗ (892) K S and, at masses above 2 GeV, by K ∗ (1430) ± K ∓ and K ∗ (1430) K S .A Dalitz plot analysis was performed [3] to extract the isovector and isoscalar contri-butions to the cross section: interference effects model differently the production ofcharged and neutral K ∗ , which is mirrored in the asymmetry of the Dalitz plot. Thedominant component was found to be the isoscalar one, with a clear resonant behaviorto be identified with the mentioned φ (1680) , while a broad contribution of marginalimportance, that could be identified as the ρ (1450) , emerged from the fit.(b) Measurement of e + e − → K + K − π Cross Section As the previous analysis, also in this case the main target was the possible identificationof the Y(4260) state [3]. The trend of the measured cross section, reported in Fig. 2(a),shows even less evidence than in the previous case for the existence of an event excess,and again only un upper limit could be quoted for the decay branching ratio to thischannel times the electronic width of the tentative signal, 0.6 eV at 90% C.L. A clear151igure 2: a) The e + e − → K + K − π cross section measured by BaBar. b) Dalitz plot of the e + e − → K + K − π reaction: squared invariant masses of the ( K − π ) versus the ( K + π ) systems.indication for J/ψ production emerges from the cross section, whose maximum, ∼ nb, is reached at about 1.7 GeV.The reaction Daliz plot, shown in Fig. 2, differently from the previous analysis is sym-metric and features a dominant production of the final state through K ∗ (892) ± K ∓ and K ∗ (1430) ± K ∓ . Selecting the K + K − invariant mass system in the φ (1020) mass win-dow, the φπ system could be studied: even if the available statistics was limited, as canbe seen by the green points in Fig. 3, some indications for the existence of stuctures like ρ (1700) /ρ (1900) , observed as a dip, and a possible enhancement at about 1480 MeV(a possible hint for the long-sought isospin 1 C (1480) ? [4]) are present.Figure 3: ( φπ ) cross section for events selected from the e + e − → K + K − π reaction (greenpoints) and from the e + e − → K S K L π reaction (black points).(c) Measurement of e + e − → K S K L π Cross Section Fig. 4 shows the trend of the e + e − → K S K L π cross section for this channel [5],whose maximum is about 3 nb corresponding to the excitation energy of the φ (1680) .152he systematic uncertainty of the measurement is 10% at the peak, and increases up to30% at 3 GeV. For the first time a clear signal of J/ψ is observed in the K S K L π decaychannel. The dominant intermediate state is K ∗ (892) K , which almost saturates thechannel intensity: a clear signal is obtained for K ∗ (892) for the combinations withboth K S and K L . Some hints for the production of the K ∗ (1430) K are also observed.Figure 4: The e + e − → K S K L π cross section measured by BaBar.The invariant mass of the φ (1020) π system, with the φ (1020) selected through its K S K L decay, is shown by the black points in Fig. 3: even in this case some hints for aresonant activity at about 1.6 GeV can be observed, which could be a signature for anisospin 1 exotic structure.4. Four Particles Final States (a) Measurement of e + e − → K S K L π + π − sross section In the e + e − → K S K L π + π − channel the contribution of the background, coming bothfrom ISR and non-ISR multihadronic events, is rather sizeable. Also the systematicuncertainty of the cross section, shown in Fig. 5(a), is larger, rising from about 10% atthe peak, about 1 nb at ∼ J/ψ can be observed.From the reaction scatter plot, shown in Fig. 5(b), clear bands of K ∗ (892) ± can beobserved with a weak indication for K ∗ (1430) ± . A strong correlated production of K ∗ (892) + K ∗ (892) − and K ∗ (892) ± K ∗ (1430) ∓ appears.(b) Measurement of e + e − → K S K S π + π − Cross Section The cross section for the e + e − → K S K S π + π − reaction features a maximum of about0.5 nb, affected by a 5% error, at about 2 GeV [6] (see Fig. 6(a)). From the inspectionof the scatter plot shown in Fig. 6(b) one can note a clear correlated production of K ∗ (892) , dominant below 2.5 GeV, and a mild indication for K ∗ (1430) ± . There isbasically no correlated production of K ∗ (892) ± K ∗ (1430) ∓ , and just a small strenghtfor the K ∗ (1430) ± K S π ∓ intermediate state can be observed.153igure 5: a) The e + e − → K S K L π + π − cross section measured by BaBar. b) Scatter plot of the e + e − → K S K L π + π − reaction: invariant masses of the ( K L π ∓ ) versus the ( K S π + ) systems.Figure 6: a) The e + e − → K S K S π + π − cross section measured by BaBar. b) Scatter plot of the e + e − → K S K S π + π − reaction: invariant masses of the ( K S π − ) versus the ( K S π ∓ ) systems.(c) Measurement of e + e − → K + K − π + π − Cross Section The cross section for this channel was measured several years ago by the DM1 Ex-periment [7], up to an energy of about 2.5 GeV: the comparison with the most recentmeasurements by BaBar [8], reported in Fig. 7(a), show that the early results, repre-sented by the red open points, were most probably overestimated and affected by asizeable systematic error. In the present case, the systematic uncertainty was evaluatedas about 20% below 1.6 GeV, lowering to as little as 2% in the region around 2 GeVto rise again up to about 10% above 3 GeV. Narrow peaks from the formation of char-monium ( J/ψ and ψ (2 S ) ) and possibly other structures which may be produced uponthe opening of reaction thresholds are visible in the cross section, whose maximum is ∼ K ∗ (892) and K ∗ (1430) , and the evidence, seen as horizontal bands inFig. 7(c), of contributions from the K (1270) and K (1400) axial excitations decay-154igure 7: a) The e + e − → K + K − π + π − cross section measured by BaBar and DM1 (red openpoints [7]). b)-c) Scatter plots of the e + e − → K + K − π + π − reaction: b) invariant masses ofthe ( K − π + ) versus the ( K + π − ) systems, c) invariant masses of the ( K ∗ π ∓ ) versus the ( K ∗ K ± ) systems.Figure 8: a) The e + e − → K + K − π π cross section measured by BaBar. b) Scatter plot of the e + e − → K + K − π π reaction: invariant masses of the ( K − π ) versus the ( K + π ) systems.ing in K ∗ (892) π ± . The ( π + π − ) invariant mass spectrum (not shown) displays a clearenhancement corresponding to the ρ (770) signal, which can be an evidence for the pos-sible decay of the two axial mesons in Kρ . No other signal is observed in the ( π + π − ) system.(d) Measurement of e + e − → K + K − π π Cross Section The cross section for the e + e − → K + K − π π reaction is shown in Fig. 8: it doesnot exceed 1 nb and the maximum is located around 2 GeV [8]. A clear charmoniumsignal is present. The systematic uncertainty of the cross section is about 7% at lowenergies, rising to 18% above 3 GeV. The dominant contribution to the intermediatestates is played by K ∗ (892) .The corresponding scatter plot of the invariant masses of the ( K − π ) versus the ( K + π ) systems is shown in Fig. 8(b), and displays a clear correlated production of K ∗ (892) e + e − → K S K L π π cross section measured by BaBar.pairs as well as K ∗ (892) + K ∗ (1430) − ; however, no evidence of resonant production inthe three particles systems ( K + K − π ) or ( K ± π π ) can be observed.(e) Measurement of e + e − → K S K L π π Cross Section The e + e − → K S K L π π cross section was found to be relatively small, reaching atmost 0.6 nb at its maximum at ∼ J/φ was observed inthis decay channel [9]. The cross section is shown in Fig. 9.While the dominant contribution to the intermediate states is played, as usual, by K ∗ (892) , there is no significant contribution to this channel of the correlated produc-tion of two K ∗ ’s.(f) Measurement of e + e − → K S K ± π ∓ π Cross Section The K S K ± π ∓ π channel features a rich intermediate state composition [10]. Apartfrom the clean J/ψ signal, whose strength exceeds the maximum of the cross section(about 2 nb, measured with a systematic uncertainty of ∼ 7% below 3 GeV), as shownin Fig. 10(a), several intermediate states can be observed from the inspection of thescatter plots reported in Fig. 10(b).The dominant ones are K ∗ (892) Kπ and K S K ± ρ ∓ . This last channel is partly fedby the decay of the axial excitations K (1270) , K (1400) and K (1650) . The corre-lated K ∗ K production is relatively small (less than 15%), and almost saturated by the K ∗ + K ∗− charged mode.5. Summary and Conclusions The total cross sections for e + e − annihilation in KKπ and KK π , thanks to the new BaBarmeasurements described above, can now be evaluated basically without any model assump-tions, nor resorting to isospin symmetry relationships [11]. In the comparison of all theexclusive hadronic cross section measured by BaBar, the dominant contribution is played byfull pionic channels: in particular between 1 and 2 GeV the four pion production dominates,while above 2 GeV the six pion production gives the largest contribution.Adding up all channels with kaons and a total of three particles in the final state, the crosssection amounts to about 12% of the total hadronic cross section, at the maximum located at156igure 10: a) The e + e − → K S K ± π ∓ π cross section measured by BaBar. b)-c) Scatter plots ofthe e + e − → K S K ± π ∓ π reaction: b) invariant masses of the ( K S π ) versus the ( Kπ ) systems, c)invariant masses of the ( Kπ ) versus the ( K S π ) systems.about 1.6–1.7 GeV.Concerning the four particle final state, the largest contribution is played by the K + K − π + π − and K S K + π − π channels; the total cross section at 2 GeV is about 1/4 of the total hadroniccross section.The precise knowledge of the hadronic cross sections, for which BaBar contributed providingan unprecedented bulk of new results in more than 20 channels (including those with pionsonly and η ’s not described in this paper), could improve the accuracy of new evaluations ofthe a had,LOµ term of the muon anomalous magnetic moment by 21% as compared to previousassessments [12].Moreover, the program of ISR measurements with BaBar could provide new interestinginformation about hadronization at low energies and the properties of the light meson spec-trum.New more precise results are awaited soon from the application of this technique at the newgeneration of charm and B-Factories. 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C (2017), 827.158 List of Participants of PKI2018 Workshop • Miguel Albaladejo Serrano, U. Murcia Pok Man Lo, University of Wroclaw