Z c (3900) AS A MOLECULE FROM POLE COUNTING RULE Yu-fei Wang in collaboration with Qin-Rong Gong, Zhi-Hui Guo, Ce Meng, Guang-Yi Tang and Han-Qing Zheng School of Physics, Peking University 2016/11/1 Yu-fei Wang School of Physics, Peking University 1 / 34
CONTENTS 1 Introduction 2 Theoretical framework 3 Phenomenological analyses 4 iscussions and Conclusions Yu-fei Wang School of Physics, Peking University 2 / 34
Introduction Experimental Backgrounds ISCOVERY OF Z c (3900), J/Ψ SPECTRUM The hadron exotic state Z c (3900) was first observed at BESIII in e + e X(4260) J/Ψ + process, J/Ψ spectrum Ablikim et al. (BESIII), PRL 110,252001(2013) Mass: 3899.0±3.6±4.9 MeV. Width: 46±10±20 MeV Yu-fei Wang School of Physics, Peking University 3 / 34
Introduction Experimental Backgrounds ISCOVERY OF Z c (3900), SPECTRUM Ablikim et al. (BESIII), PRL 112,022001(2014) Quantum numbers: I(J P ) = 1(1 + ) Yu-fei Wang School of Physics, Peking University 4 / 34
Introduction Experimental Backgrounds CONFIRM OF Z c BY BELLE AN CLEO Yu-fei Wang School of Physics, Peking University 5 / 34
Introduction Possible Theoretical Interpretations POSSIBLE THEORETICAL INTERPRETATIONS Z c can not be embedded into conventional c c quark model, while it s close to threshold. molecular state Phenomenological Lagrangian: [ong et al, PR 88,014030(2013)] Heavy quark flavor symmetry: [Guo et al, PR 88,054007(2013)] Electromagnetic structure: [Wilbring, Hammer, and Meissner, PLB 726,326(2013)] Yu-fei Wang School of Physics, Peking University 6 / 34
Introduction Possible Theoretical Interpretations POSSIBLE THEORETICAL INTERPRETATIONS Elementary tetraquark state QC sum rule: [Wang and Huang, PR 89,054019(2014)] tetraquark model: [Maiani et al., PR 87,111102(2013)] QC sum rule: [ias et al, Phys. Rev. 88,016004(2013)] Non-resonance cusp effect: [Chen, Liu, and Matsuki, PR 88,036008(2013)] triangle diagram singularity: [Liu, Oka, and Zhao, PLB 753,297(2016)] Pang s talk. Yu-fei Wang School of Physics, Peking University 7 / 34
Introduction Summary SUMMARY Z c (3900): M 3900 MeV, width tens of MeV, close to threshold. Significantly coupled to and J/Ψ channels (h c less marked) I(J P ) = 1(1 + ) Possible theoretical interpretations: molecule compact tetra-quark state (i.e. elementary state) non-resonance (cusp effect, ATS, etc. ) Yu-fei Wang School of Physics, Peking University 8 / 34
Theoretical framework Strategies Strategies: effective Lagrangian under C, P, χ and isospin symmetries calculation of the X(4260) Z c, J/Ψ, h c amplitudes combined fit to J/Ψ channel J/Ψ, data, data and h c data (as background) find pole position and use pole counting rule Pole counting rule [Morgan, NPA 543,632(1992)] : based on potential scattering and effective range expansion only one pole near threshold the pole corresponds to a molecular state two poles near threshold the poles indicate an elementary state Yu-fei Wang School of Physics, Peking University 9 / 34
Theoretical framework Strategies EFFECTIVE RANGE EXPANSION In non-relativistic k plane, for s wave phase shift k: k cotδ = 1/a+r e /2k 2 + o(k 4 ) a: scattering length. r e : effective range. Partial wave S matrix: S(k) = r e 2 k 2 + ik 1 a r e2 k 2 ik 1 a Constraint on the two poles: k 1 + k 2 k 1 + k 2 = 2/ r e Yu-fei Wang School of Physics, Peking University 10 / 34
Theoretical framework Strategies POLE COUNTING RULE Potential scattering r e L (the range of potential interaction) AT MOST one pole lie in k < 1/L. Two poles near threshold poles are generated by hidden channel other than the explicit potential there exists an elementary state. In multi-channel scattering, only one pole (bound state or virtual state) coupled to the inelastic threshold may split to a pair of resonance, which also represents a potential generated molecular. Yu-fei Wang School of Physics, Peking University 11 / 34
Theoretical framework Strategies r e L Single channel case: r e = 2 0 dr[ϕ 2 0 (r) ψ2 0 (r)] ψ0 2 (r): solution of radial Schrödinger equation at k = 0. ϕ 2 0 (r): asymptotic form of ψ2 0 (r) when r. For short-ranged and weak potential r e = 2 L 0, and the integrand takes cos 2 (r/2l) form so r e L. Multi-channel case and relativistic partial wave dispersion relations give the same result. Two poles near threshold C poles in dispersion relation: k cotδ = (ν p m 2 k 2 )/(m 2 g 2 ) +, poles k = ±m ν p. Yu-fei Wang School of Physics, Peking University 12 / 34
Theoretical framework Strategies UNCERTAINTIES The condition r e L dose not hold for too strong binding case/ potentials with strange mathematical structure. In multi-channel case, pole counting rule is satisfied only when the pertinent poles are dominated by one channel. Pole counting rule is a qualitative result rather than a quantitative theorem. Yu-fei Wang School of Physics, Peking University 13 / 34
Theoretical framework Effective Lagrangian EFFECTIVE LAGRANGIANS For X(4260) J/ψ(h c ) channel: L XJ/ψ = g 1 X µ ψ ν u µ u ν +g 2 X µ ψ µ u ν u ν +g 3 X µ ψ µ χ + + L XZc = g 4 ν X µ Z c µ u ν + L ZcJ/Ψ = g 7 ν ψ µ Z c µ u ν + L Xhc = f 8 λ ρ X µ H ν u λ u σ ǫ µνρσ + L Zch c = f 9 α H ν Z cµ u β + Field operators: X : X(4260); ψ : J/ψ; Z c : Z c (3900); H : h c µ X = µ X +[Γ µ, X] Γ µ = 1 2 [u+ ( µ ir µ )u+u( µ il µ )u + ] χ ± = u + χu + ± uχ + u, u µ = i{u + µ u u µ u + } χ = 2B(s+ip), u = exp( iφ 2F ) Yu-fei Wang School of Physics, Peking University 14 / 34
Theoretical framework Effective Lagrangian For X(4260) channel: L X = f 1 ν X µ µ u ν +f 2 X µ ν µ u ν +f 3 ν X µ ν u µ +f 4 X µ µ ν u ν + L Zc = f 7 [( 0 µ + + + µ 0 )Zc µ + h.c.] L = λ 1( +µ 0 µ 0 + + 0µ 0 µ +h.c.)+ λ 2 ( +µ 0 0 µ + + 0µ µ 0 + h.c.) L J/ψ = λ 3 ν ψ µ µ u ν +λ 4 ψ µ ν µ u ν +λ 5 ν ψ µ ν u µ +λ 6 ψ µ µ ν u ν + L h c = λ 9 α H ν µ u β +λ 10 H ν α µ u β + Heavy quark spin symmetry λ 1 = λ 2 Yu-fei Wang School of Physics, Peking University 15 / 34
Theoretical framework Feynman diagrams FEYNMAN IAGRAMS = 1 + 2 + 2 + J/ψ J/ψ 5 4 J/ψ 5 J/ψ = 2 + 2 + 2 + 4 4 4 = 3 + 2 + 2 + hc hc 6 4 hc 6 hc Yu-fei Wang School of Physics, Peking University 16 / 34
Theoretical framework Feynman diagrams COMPOUN 3 VERTICES X/γ 1 = + Zc J/ψ J/ψ J/ψ X/γ 2 = + Zc X/γ 3 = + Zc hc hc hc Yu-fei Wang School of Physics, Peking University 17 / 34
Theoretical framework Feynman diagrams COMPOUN 4 VERTICES 4 = + Zc 5 = + Zc J/ψ J/ψ J/ψ 6 = + Zc hc hc hc Yu-fei Wang School of Physics, Peking University 18 / 34
Theoretical framework Feynman diagrams IFFERENT MECHANISMS TO GENERATE Z c Z c : dynamically generated particle / basic d.o.f ifferent mechanisms: pure dynamical (Fit I,, bubble chain) g 4, g 5, f 5, f 7 = 0 pure explicit Z c field (Fit II, Breit-Wigner Z c propagator) λ 1 = 0 a mixture λ 1 0, f 7 0 Yu-fei Wang School of Physics, Peking University 19 / 34
Phenomenological analyses Fit to J/Ψ channel, spectrum FIT TO J/Ψ CHANNEL, SPECTRUM Events 35 30 25 20 15 10 5 BaBar ata FIt I Fit II 0 0.2 0.4 0.6 0.8 1.0 1.2-5 M /Gev -10 Events 45 40 35 30 25 20 15 10 5 Belle ata Fit I FIt II 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-5 M /GeV Yu-fei Wang School of Physics, Peking University 20 / 34
Phenomenological analyses Fit to J/Ψ channel, spectrum 120 100 BESIII ata Fit I Fit II 80 Events 60 40 20 0 0.2 0.4 0.6 0.8 1.0 1.2 M /GeV Yu-fei Wang School of Physics, Peking University 21 / 34
Phenomenological analyses Fit to J/Ψ, h c, and spectrums FIT TO J/Ψ, h c, AN SPECTRUMS Events 100 90 80 70 60 50 40 30 20 10 BESIII ata Fit I Fit II 0 3.6 3.7 3.8 3.9 4.0 4.1 M max (J/ )/GeV Events 45 40 35 30 25 20 15 10 5 BESIII ata Fit I Fit II 0 3.80 3.82 3.84 3.86 3.88 3.90 3.92 3.94 M(h c )/GeV Yu-fei Wang School of Physics, Peking University 22 / 34
Phenomenological analyses Fit to J/Ψ, h c, and spectrums Events 100 90 80 70 60 50 40 30 20 10 BEIII ata Fit I Fit II 0 3.85 3.90 3.95 4.00 4.05 4.10 4.15 M *-/GeV 0 Events 120 100 80 60 40 20 BESIII ata Fit I Fit II 0 3.85 3.90 3.95 4.00 4.05 4.10 4.15 M *0/GeV + Yu-fei Wang School of Physics, Peking University 23 / 34
Phenomenological analyses Fit quality FIT QUALITY Fit quality: Fit method Pure bubble (Fit I) Pure Breit-Wigner (Fit II) χ 2 d.o.f 454 291 28 = 1.726 497 291 26 = 1.875 A mixed fit does not obviously improve the total χ 2. The overall quality of Fit I and Fit II are quite similar! Yu-fei Wang School of Physics, Peking University 24 / 34
Phenomenological analyses Pole searching POLE SEARCHING Ignorance of h c channel a two channel system I IV Riemann sheet: (ρ J/Ψ,ρ ) (+,+)(,+)(, )(+, ) Pole position (only near-threshold poles) Sheet I Sheet II Sheet III Sheet IV Fit I 3.87988 ± 0.00390i GeV Fit II 3.87909 ± 0.00143i GeV Only one pole near threshold Z c is molecular state! (Fit II also contains another far away pole. ) Yu-fei Wang School of Physics, Peking University 25 / 34
iscussions and Conclusions Summary SUMMARY Main propose: study the nature of Z c (3900) state. Effective Lagrangians with C,P,χ and isospin symmetry are employed. Near threshold singularities are built via resummation. Three fits to experimental data are taken: one for pure dynamically generated Z c, another for pure explicit Z c field, and the last for mixture case. ifferent fits give similar χ 2, while only one pole is found in s plane. According to pole counting rule, Z c is a molecular. Yu-fei Wang School of Physics, Peking University 26 / 34
iscussions and Conclusions Some remarks SOME REMARKS A near-threshold state does not trivially correspond to a molecular state, e.g. X(3872). An explicit Z c field does not trivially leads to an elementary Z c the Breit-Wigner propagator always produces two poles, but in molecular case one of them would be far away from threshold. As for non-resonance case, anomalous triangle singularity (ATS) is investigated (Pang s talk). Alternative criterion to distinguish molecular states and elementary states besides pole counting rule? Yu-fei Wang School of Physics, Peking University 27 / 34
iscussions and Conclusions Some remarks SPECTRAL ENSITY FUNCTION SUM RULE Spectral density function sum rule [Baru et. al. PLB 586,53(2003)] : based on a coupled channel non-relativistic QM: C(E) E 0 + d 3 k χ k (E) E (2) 3 k, one elementary state E 0 plus infinite continuous states. For resonance states with energy E and momentum k hidden in continuous state, w(e) = µk C k (E) 2 (E = 2 2 k 2 /2µ). µ: two particle reduced mass. Sum rule: C( B) 2 + w(e)de = 1, with Z = C( B) 2 0 the possibility for a bound state to be elementary. The possibility for a resonance to be elementary: Z = Emax E min w(e)de. The integral interval is where the resonance locate. Yu-fei Wang School of Physics, Peking University 28 / 34
iscussions and Conclusions Some remarks SPECTRAL ENSITY FUNCTION SUM RULE For Z c, two channel Flatté formula (E = 0: threshold) w(e) = 1 2 g 1 2µEθ(E)+Γ0 E E 0 + i g 2 1 2µE+ i Γ 2 0 2 Parameters from fit and matching: E 0 = 0.02800 GeV, Γ 0 = 0.01333 GeV, g 1 = 0.4910. Integral intervals: [E c nγ, E c + nγ], for E c = 0.005000 GeV and Γ = 0.02984 GeV; n = 1/2, 1, 2, 3. Yu-fei Wang School of Physics, Peking University 29 / 34
iscussions and Conclusions Some remarks SPECTRAL ENSITY FUNCTION SUM RULE Results n 1/2 1 2 3 Z 12.58% 20.62% 31.97% 39.67% This method contains some ambiguities and uncertainties since the integral interval is arbitrary to some extent. However, the possibility is still smaller than 50% even if the interval become as large as 6Γ. So in quality Z c is a molecular state. Yu-fei Wang School of Physics, Peking University 30 / 34
iscussions and Conclusions Some remarks Thank you! Yu-fei Wang School of Physics, Peking University 31 / 34
iscussions and Conclusions Appendix Appendix Yu-fei Wang School of Physics, Peking University 32 / 34
iscussions and Conclusions Appendix ECOMPOSITION OF AMP.S, propagators: Π µν = kµkν g µν d k m 2 (2) (k 2 m 2 )[(p k) 2 m 2 ] = P TµνΠ T + P Lµν Π L P Tµν = g µν pµpν p 2,P Lµν = pµpν p 2 Transverse and longitudinal components (p: the momentum of Z c ): transverse part 1 i(λ 1 + f 7 2 )Π p 2 m 2 T pertinent poles Z longitudinal part 1 i(λ 1 f 7 2 )Π m 2 L far away poles Z Yu-fei Wang School of Physics, Peking University 33 / 34
iscussions and Conclusions Appendix PARTIAL WAVE ANALYSIS & FSI Extract S-wave part for contact tree vertex Γ Zc 40MeV ignorable final state interaction (FSI) in the second picture For X(4260) J/ψ(h c ) FSI: A = A tree X J/Ψ α 1(s)T +A tree X J/ΨK K α 2(s)T K K +A α i = ci 0 s s A + c i 1 + ci 2 s+ (s A : Adler zero. A : amplitude w/o contact tree. T: amplitudes given by unitarized χ PT. ) Yu-fei Wang School of Physics, Peking University 34 / 34