K-matrix formalism in light-meson spectroscopy

Sebastián Ordóñez

Email: jsordonezs@unal.edu.co

 

 

Supervisor: Diego A. Milanés

Email: damilanesc@unal.edu.co

Phenomenology of High Energy Physics Group

Departamento de Física

Universidad Nacional de Colombia

Outline

  • Introduction

- Dalitz plot and K-matrix formalism (The issue)

  • Analysis

- Implementation of K-matrix formalism in D-decays

- Examples

  • Results and Conclusions

- What we have learnt so far

 

 

 

 

 

Dalitz plots

  • What are Dalitz Plots? Representations of a three-body decay 

  • DP allow us to extract dynamical information from deviations of a "homogeneous event distribution" (phase-space).
\boxed{D^{0}\longrightarrow K_{S}^{0}\pi^{+}\pi^{-}}

D. Milanés, Measurement of D 0 − D̄ 0 mixing in the BABAR experiment. PhD thesis,
Universidad de Valencia, Departmento de Física Teórica, 2010.

... and a question

  • A model for       decay is needed, i.e. 
  • How to deal with the underlying strong dynamics effects?
D^{0}
(K\pi)\pi, K(\pi\pi)

The      S-wave is characterized by broad, overlapping resonances

\pi\pi

Isobar Model

  • Typically for describing resonances is employed a  sum of Breit-Wigner functions (Isobar Model).

The issue

  • Why is this so problematic? Unitarity is not explicitly guaranteed by a simple sum of Breit-Wigner functions (Isobar Model).

The K-matrix

  • What is K-matrix?  It follows from the unitary S-matrix 

\boxed{S = I + 2i\rho^{1/2}T\rho^{1/2}}

We can express any unitary operator in terms of an hermitian operator

\boxed{S=(I-i\rho^{1/2}K\rho^{1/2})^{-1}(I+i\rho^{1/2} K\rho^{1/2})}

In terms of the T-matrix

\boxed{T=(I-iK\rho)^{-1}K}

The advantages of K-matrix approach

  • It heavily simplifies the formalization of any scattering problem since the unitarity of S is automatically respected.
  • For a single-pole problem a K-matrix reduces to the standard BW formula.
  • The K-matrix approach can be extended to production processes.
\boxed{T=(I-iK\rho)^{-1}K}
\boxed{F=(I-iK\rho)^{-1}P}

The advantages of K-matrix approach

  • K-matrix allows for the inclusion of all the knowledge coming from scattering experiments.
\boxed{F=(I-iK\rho)^{-1}P}

Describes coupling of resonances to D

Comes from scattering data

S-wave scattering parameterization

\pi\pi
\boxed{K_{ij}^{00}(s)=\left(\sum_{R}\frac{g_{i}^{R}g_{j}^{R}}{m_{R}^{2}-s}+f_{ij}^{scatt}\frac{1-s_{0}^{scatt}}{s-s_{0}^{scatt}}\right) \left\{\frac{1-s_{A0}}{s-s_{A0}}\left(s-s_{A}\frac{m_{\pi}^{2}}{2}\right)\right\}}

We take the channels j as

\pi\pi,KK,\eta\eta, \eta'\eta', 4\pi

V.V Anisovich and A.V.Sarantsev Eur.Phys.J.A16 (2003) 229

S-wave scattering parameterization

\pi\pi
f_{0}(980)
f_{0}(1500)

S-wave scattering parameterization

\pi\pi

Thank you!

Questions?

p_{1}=p_{2}