Line-shape analysis of the ψ(3770)

Transcription

1 2017 International Summer Workshop on Reaction Theory June 21, 2017, Bloomington, Indiana, USA Line-shape analysis of the ψ(3770) Susana Coito Collaborator: Francesco Giacosa Jan Kochanowski University, Kielce, Poland

2 Introduction Mysteries in the data The vector meson ψ(3770), reported in PDG with average parameters M = MeV and Γ = 27.2 ± 1.0 MeV, has a deformed, i.e., non Breit-Wigner line-shape Figure: σ = σ(e) in [nb, GeV]. PRL101,102004(2008) BES, e + e hadrons

3 Other data: PRL93,051803(2004) Belle B + ψ(3770)k + PRD76,111105(2007) BaBar ISR

4 The Novosibirsk group: KEDR Collaboration, PLB711,292(2012). Figure: e + e D D Analysis: interference between resonant and nonresonant part, where the nonresonant part is related to the nonresonant part of the form factor (use of vector dominance model).

5 Phenomenological Studies In PRD88,014010(2013) contribution of off-shell PV channels PLB718,1369(2013) production e + e D D, including interference with nonresonant background due to ψ(2s), and rescattering of final states: D 0 D0 D + D

6 PRD80,074001(2009) E. van Beveren, G. Rupp Interference with background due to D D threshold enhancement

7 An Effective Lagrangian Model Defining the Lagrangian We consider the following decay of a vector to two pseudovectors: ψ(3770) D + D We define a Lagrangian density as L = L 0 + L I L 0 = 1 4 ΨµνΨµν m2 ψψ µψ µ ( µd + µ D m 2 DD + D ) Ψ µν = µψ ν νψ µ L I = igψ µ ( µ D + D µ D D +)

8 The 3-level decay width is given by dγ(e) dω = d 3 p 1 (2π) 3 2E 1 d 3 p 2 (2π) 3 2E 2 δ 4 (P p 1 p 2) (2π)4 2 s Which turns, for the spherically symmetric case, Γ(E) = 1 8π p(e) M 2. s The amplitude M 2 is computed from the Lagrangian. It comes where f (E) is a cutoff function M 2 = g 2 3 p2 (E)f (E), f (E) = e 2p2 (E)/Λ 2. Free parameters: g coupling for the vertex ψ(3770) to D + D, Λ cutoff parameter, Λ 1 <r 2 >

9 Propagator The propagator for an unstable particle is given by (E) = where the loop-function, or self-energy, is 1 E 2 m 2 ψ + Σ(E) Σ(E) = Ω(E) + ieγ(e) Dispersion relation, cauchy principal value Ω(E) = 1 π s th s Γ(s ) s s ds The introduction of Ω(E) in the propagator leads to a normalized spectral function.

10 Spectral function with D + D loop but no rescattering Considering the once-subtracted dispersion relation Ω 1S (E) = Ω(E) Ω(m ψ ) Σ(E) = Ω 1S (E) + ieγ(e) (E) = Cross section (line-shape) 1 E 2 m 2 ψ + Ω 1S(E) + ieγ(e) σ(e) = g 2 ψe + e Im (E)

11 Estimating the cutoff through the wave-function of a system c c (D wave) D + D (P wave) 6 4 R(r) 2 r 4 c c (l = 2) 2 D + D (l = 1) r(gev 1 ) < r 2 > = 4.74 GeV fm.

12 1 Cutoff parameter: Λ = 211 MeV 4.74GeV σ(nb) BES BaBar BES E(GeV) Two poles are found: 3744 i11 MeV and 3775 i6 MeV!

13 Dependence of the spectral function on the coupling g 6 σ 4 g = 0.7g 2 g = g g = 1.3g E(GeV) 3741 i20 & 3774 i i11 & 3775 i i4 & 3778 i9

14 Spectral function with D + D loop and rescattering of final states Redefining the loop function Σ (E) = Σ + ΣλΣ + = Σ n=0(λσ) n = Σ(E) 1 λσ(e) Σ 1(E) = Σ (E), Σ 2(E) = Σ (E) Ω(m ψ ), Σ 3(E) = Σ (E) Ω (m ψ ) a new free parameter λ is introduced (rescattering coupling)

15 Influence of the rescattering over the spectral function, for Λ = 211 MeV 3 σ E(GeV)

16 Width dependence on the cutoff Γ(MeV) Λ = Λ = 506 MeV Λ = 211 MeV E(GeV)

17 Influence of the rescattering over the spectral function, for Λ = 506 MeV σ E(GeV)

18 Fit with rescattering: channel D + D σ(nb) E(GeV) Λ = 211 MeV Λ = 506 MeV 3775 i6, 3744 i i3, 3742 i13

19 Summary and Perspectives Given the increasing number of XYZ states, and the existence of many thresholds in the charmonium energy region, correct analysis of data are very important to understand the signals. We are employing an effective Lagrangian approach to study the line-shape of the ψ(3770). Independently of the final state rescattering, we find two poles associated with the cross section fitted to data.