Zc(4430), Zc(4200), Z 1 (4050), and Z 2 (4250) as triangle - PowerPoint PPT Presentation

Zc(4430), Zc(4200), Z 1 (4050), and Z 2 (4250) as triangle singularities arXiv:1901.07385 (to appear in PRD, Rapid Comm.) PRD 100 011504(R) (2019) Satoshi Nakamura University of Science and Technology of China Collaborator : Kazuo Tsushima (Univ. Cruzeiro do Sul, Brazil)

Introduction

Discoveries of Z c and Z b ± ± + + + + Zc(4430) Z 1 (4050), Z 2 (4250) Zc(4200) Belle (2008) Belle (2008) Belle (2014) B 0 → χ c 1 π + K − B 0 → ψ (2 S ) π + K − B 0 → J / ψ π + K − Beyond the convenXonal quark model ? If they are charged quarkonium-like states Z + Z + c : ccud b : bbud Minimally 4-quark states and not à clear signature of exoXcs qq

Z c (4430) has been outstanding exoXc candidate h[ps://home.cern/news/news/experiments/lhcb-confirms-existence-exoXc-hadrons

Incomplete list of previous interpretaXons of Zc Zc(4430) diquark-anXdiquark (tetraquark) : Ebert et al. EPJC 58 (2008); Maiani et al. PRD 89 (2014); Deng et al. PRD 92 (2015) hadron molecule : Liu et al, PRD 77 (2008); Ding et al, PRD 79 (2009); Lee et al, PLB 661 (2008); Zhang et al. PRD 80 (2009); Ma et al, PRD 90 (2014) kinemaXcal cusp : Rosner, PRD 76 (2007); Bugg, JPG 35 (2008) ß already ruled out Z 1 (4050) tetraquark : Patel et al. EPJA 50 (2014); Deng et al, PRD 92 (2015) hadron molecule : disfavored by meson exchange models of Liu et al. EPJC 61 (2009); Liu et al, PRC 80 (2009); Ding et al, PRD 79 (2009) All surviving theoreXcal works interpret Zc as tetraquark (including hadron molecule)

AlternaXve interpretaXon: Triangle Singularity 1 c ∫ A ~ dq 1 E − E 1 − E 2 + i Γ H 2 3 b q 2 2( ! ∫ d Ω q q ) × E − E 2 − E 3 − E c + i ε a logarithmic singularity 0 Γ 0 200 400 600 800 1000 q (MeV)

AlternaXve interpretaXon: Triangle Singularity 1 c ∫ A ~ dq 1 E − E 1 − E 2 + i Γ H 2 3 b q 2 2( ! ∫ d Ω q q ) × E − E 2 − E 3 − E c + i ε a logarithmic singularity 0 0 Γ Γ 0 200 400 600 800 1000 0 200 400 600 800 1000 q (MeV) q (MeV)

AlternaXve interpretaXon: Triangle Singularity 1 c ∫ A ~ dq 1 E − E 1 − E 2 + i Γ H 2 3 b q 2 2( ! ∫ d Ω q q ) × E − E 2 − E 3 − E c + i ε a logarithmic singularity Singularity is relaxed because of finite Γ In small kinemaXcal window where the process is kinemaXcally allowed to occur at classical level • intermediate states are on-shell • momenta are collinear 0 0 Γ Γ Amplitude is significantly enhanced ! 0 200 400 600 800 1000 0 200 400 600 800 1000 q (MeV) q (MeV)

Triangle singulariXes for Zc(4430), Zc(4200) Κ – Κ – K * (892) * (1430) K 2 B 0 B 0 π + π + π + π + Y (4260) ψ (3770) ψ (2 S ) J / ψ + + Zc(4430) Zc(4200) At zero-width limit, diagrams exactly hit triangle singulariXes at (using PDG averaged masses) m ψ (2 S ) π + = 4420 MeV m J / ψπ + = 4187 MeV For finite (realisXc) widths, triangle singulariXes are somewhat relaxed à Spectrum peak posiXon associated with TS can be a bit different from above à will be examined by numerical calculaXon

Comment on Pakhlov et al.’s model Phys. Le[. B 702, 139 (2011) Κ – Phys. Le[. B 748, 183 (2015) D ' S B 0 D * D ' S : hypotheXcal charmed-strange hadron π + D ψ (2 S ) CLAIM: The above triangle diagram generates Zc(4430)-like bump Comments • The process is kinemaXcally forbidden at the classical level à no triangle singularity (Coleman-Norton theorem) • Argand plot is clockwise à ruled out by LHCb data (counter-clockwise Argand plot) • Our calculaXon does not find Zc(4430)-like bump from above triangle diagram à expected from Coleman-Norton theorem

Model

Triangle amplitudes for Zc(4430), Zc(4200) 1 c d 3 q ∫ T B 0 → abc = v 23 → ab 1 E − E 2 − E 3 − E c + i ε B 0 3 b 2( ! q ) 1 Γ B 0 → 12 × Γ a E − E 1 − E 2 + i Γ 1 → 3 c 2 v 23 → ab ( ! p a , ! p b ; ! p 2 , ! p 3 ) = f ( p ab ) f ( p 23 ) ! ε a ⋅ ! ε 2 : s-wave interacXon; consistent with J P =1 + of Zc(4430), Zc(4200) Γ R → ij ( ! p R ; ! p i , ! j | SS z )( LMSS z | S R S z ∑ f ( p ij ) ( s i s i z s j s z R ) Y LM ( ˆ p j ) = p ij ) LS f ( p ij ) : dipole form factor with cutoff 1 GeV (data not available to fix form factors) Spectrum shape is mostly determined by kinemaXcal effect à insensiXve to cutoff è to be checked All parXcle masses and widths are taken from PDG average

Results for Z c (4430) and Z c (4200)

Invariant mass spectrum + + Zc(4430) Zc(4200) Breit-Wigner model Κ – Κ – Κ – * (892) K * (1430) K 2 0 B 0 0 B B π + π + π + π + π + + Z c ψ Y (4260) ψ (3770) ψ (2 S ) J / ψ (a) (b) Triangle diagram 4 d Γ / dm ψ f π (a.u.) 3 Breit-Wigner fit 2 Phase-space 1 Specta are normalized to 0 give unity when integrated 4 4.2 4.4 4.6 3.8 4 4.2 4.4 4.6 m ψ (2S) π (GeV) m J/ ψ π (GeV) Clear resonance-like peaks are generated by triangle diagrams Absolute magnitude is unknown à experimental inputs needed

Invariant mass spectrum + + Zc(4430) Zc(4200) (a) (b) 4 Triangle diagram d Γ / dm ψ f π (a.u.) 3 Breit-Wigner fit 2 Phase-space 1 0 4 4.2 4.4 4.6 3.8 4 4.2 4.4 4.6 m ψ (2S) π (GeV) m J/ ψ π (GeV) Breit-Wigner parameters + + Zc(4430) Zc(4200) (a) Belle (2013) LHCb (2014) (b) Belle (2014) Mass (MeV) Width (MeV) Remarkable agreement with data Ranges of the parameters from the model are cutoff-dependence ( Λ =0.5 - 2 GeV )

Zc(4430) Argand plot Angle-independent part of amplitude + constant background A ( m 2 ∫ c bg , c norm : fi[ed to data ( ˆ ψ (2 S ) π + ) = c bg + c norm d Ω K − Y p K − ) M B 0 → ψ (2 S ) π + K − 1 Resonance-like counter-clockwise moXon 0.2 is reproduced by triangle diagram, not a resonance 0 Im A (a.u.) Curved segment and point of same -0.2 color belong to same bin m 2 ψ (2 S ) π + -0.4 Data: LHCb, PRL 112 222002 (2014) -0.4 -0.2 0 0.2 Re A (a.u.)

0 Z c (4430) and Z c (4200) in Λ b à J / ψ π - p LHCb PRL 117 082003 (2016) : • Z c (4200) contribuXon significantly improves the descripXon of data • Z c (4430) contribuXon hardly improves No other Z c (4430) and Z c (4200) as triangle singulariXes give a consistent explanaXon explanaXon yet Z c (4200)-like peaks from triangle diagrams p possibly a coherent sum in reality 4 d Γ / dM ψ f π (a.u.) 0 Ν * 3 Λ b N * = N (1440)1/ 2 + π - 2 π - N * = N (1520)3/ 2 − ψ (3770) 1 J / ψ N * = N (1680)5 / 2 + 0 3.8 4 4.2 4.4 4.6 M J/ ψ π (GeV) No triangle diagram is available to create Z c (4430)-like peak

Results for Z 1 (4050) and Z 2 (4250)

Triangle singulariXes for Z 1 (4050), Z 2 (4250) Κ – Κ – K * (892) * (1430) K 2 B 0 B 0 π + π + π + π + X (3872) ψ (3770) χ c 1 χ c 1 + + Z 1 (4050) Z 2 (4250) similar Κ – * (1430) K 2 0 B π + π + ψ (3770) J / ψ Zc(4200)

Invariant mass spectrum Κ – Κ – K * (892) * (1430) K 2 B 0 B 0 π + π + π + π + X (3872) ψ (3770) χ c 1 χ c 1 + + + Z 1 (4050) 1 - Z 2 (4250) 1 + Z 2 (4250) 1 - 6 (a) (b) [x2] (c) [x2] 5 d Γ / dm χ c1 π (a.u.) 4 3 2 1 0 3.8 4 4.2 4.4 4.6 4 4.2 4.4 4.6 4.8 4 4.2 4.4 4.6 4.8 m χ c1 π (GeV) m χ c1 π (GeV) m χ c1 π (GeV) Triangle diagram Clear resonance-like peaks are generated by triangle diagrams Breit-Wigner fit Absolute magnitude is unknown à experimental inputs needed Phase-space

Invariant mass spectrum + + + Z 1 (4050) 1 - Z 2 (4250) 1 + Z 2 (4250) 1 - 6 (a) (b) [x2] (c) [x2] 5 d Γ / dm χ c1 π (a.u.) 4 3 2 1 0 3.8 4 4.2 4.4 4.6 4 4.2 4.4 4.6 4.8 4 4.2 4.4 4.6 4.8 m χ c1 π (GeV) m χ c1 π (GeV) m χ c1 π (GeV) Very good agreement with data Breit-Wigner parameters (range ß cutoff dependence: Λ =1 - 2 GeV) (a) Belle (2008) (b) (c) Belle (2008) J P Mass (MeV) Width (MeV)

Comparison with spectrum from Belle Spectra from triangle diagrams are scaled and 40 incoherent background is added to fit data 35 30 Triangle diagrams similar to Events/0.024 GeV ê 25 + Z 1 (4050) 1 - 20 + Z 2 (4250) 1 + 15 10 + Z 2 (4250) 1 - 5 Data: Belle, PRD 78 072004 (2008) 0 3.6 3.8 4 4.2 4.4 4.6 4.8 m χ c1 π (GeV) • Spectra from triangle diagrams capture the Belle data feature • highly asymmetric Z 1 (4050) peak is well reproduced Note : QualitaXve comparison; no interference with other mechanisms considered

Origin of asymmetric shape of Z 1 (4050) peak Κ – 6 K * (892) B 0 5 π + d Γ / dm χ c1 π (a.u.) π + 4 X (3872) χ c 1 3 2 Spectrum has abrupt bent at m χ c 1 π + ~ 4.01 GeV 1 where channel opens X (3872) π + 0 3.9 4 4.1 4.2 m χ c1 π (GeV) : original triangle diagram

Origin of asymmetric shape of Z 1 (4050) peak Κ – 6 K * (892) B 0 5 π + d Γ / dm χ c1 π (a.u.) π + 4 X (3872) χ c 1 3 2 1 0 3.9 4 4.1 4.2 m χ c1 π (GeV) : original triangle diagram : on-shell contribuXon turned off X (3872) π + • on-shell makes the spectrum significantly asymmetric X (3872) π + • threshold energy is close to the peak posiXon à effect of proximity ? X (3872) π +