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}
K-matrix formalism
By Sebastian Ordoñez
