Zhi-Yong Zhou Southeast university Zhangjiajie 周智勇 东南大学

How to precisely model the final state strong interaction is important to understand the weak interactions in shorter distance. The biggest uncertainties in determining the CKM angle,  =(65  7) o, from the difference of and decays is due to our inability to model the final state interactions.

In calculation of Dyson-Schwinger equation, the propagator of the ρ-meson expressed in terms of quark line graphs. At lowest order it is assumed to be a meson, which decays at higher order by coupling to pion pairs.

The analytic structure of the ρ-propagator in the complex s- plane. At lowest order, the propagator is real with a pole on the real axis corresponding to a bare meson. The corrections at higher orders, dominated by pion loops, give the full propagator with a pole on the nearby unphysical sheet.

Start by considering a simple model at the hadron level, in which the inverse meson propagator could be represented as Π n (s) is the self-energy function for the n-th decay channel. Here, the sum is over all the opened channels or including nearby virtual channels. Π n (s) is an analytic function with only a right-hand cut starting from the n-th threshold, and so one can write its real part and imaginary part through a dispersion relation

Based on Cutkosky rule, the imaginary part of the self- energy function could be represented pictorially as

1, Most of states below 2.0 GeV could be described in a consistent and unified picture. Z.Zhou and Z.Xiao, Phys.Rev.D83,014010,2011

1.The masses of charmed and charmed-strange mesons and their decays could be described simultaneously. 2.The low mass puzzle of is solved naturally in this scheme. 3.In a prilliminary work, we obtained good results about charmonium spectra and their decays, which is consistent to the observed values in experiment. Z.Zhou and Z.Xiao, Phys.Rev.D84,034023,2011

isobar picture

Unitarity for P  (c) Or see Aitchson 1977, Caprini 2006, Pennington 2006

UNITARITY : decays in spectator picture T = K 1 - i  K F = P 1 - i  K =  T coupling function If c is not a spectator?

Brian Meadows

m 2 (K -  + low ) (GeV/c 2 ) 2 m 2 (K -  + high ) (GeV/c 2 ) 2 Events/0.04(GeV/c 2 ) non-resonant dominates

Brian Meadows m 2 (K -  + low ) (GeV/c 2 ) 2 m 2 (K -  + high ) (GeV/c 2 ) 2 Events/0.04(GeV/c 2 ) 2 

Brian Meadows

E791 vs elastic scattering (LASS) phases (degrees) M (K  ) GeV E791 LASS

Rescattering

phases simply related if no rescattering Watson’s theorem elastic Rescattering : Unitarity

Including rescattering effect Rescattering : Unitarity

Discontinuity relation of decay amplitude: After making a partial wave projection, Write it in short,

Elastic region Inelastic region Unitarity requires four points on Argond diagram, t*, a + h, (0, 1) and (0, Im[a]), stay on a circle. Pictorially represented as

Reproduced K\pi scattering phase by E791 result

Q:Whether there is the phase ambiguity of ? A: Perhaps yes.

How to obtain a better Dalitz analysis for the processes with strong final state interaction? Building the following relations into analyses may help.