1. Dalitz Plot Analysis of D Decays Luigi Moroni INFN-Milano

2. Dalitz Analysis of Heavy Flavor Decays Infinite power tool! –It provides a “complete observation” of the decay –Everything could be in principle measured from the dynamical features of the HF decay mechanism, –Relative importance of non-spector processes up to the CP-violating phases, mixing, etc –Just recall  from B o  : probably the only clean way to get it We already learned a lot on charm But, as we know, – strong dynamic effects, if not properly accounted for, would completely hide or at least confuse the underlying fundamental physics.

3. Outline In this talk I will address all the key issues of the HF Dalitz analysis –Formalization problems –Failure of the traditional “isobar” model –Need for the K-matrix approach –Implications for the future Dalitz analyses in the B-sector Will discuss these issues in the context of the recent D +  +  -  + Dalitz analysis we performed in FOCUS –A lot to be learned

4. Formalization Problems The problem is to write the propagator for the resonance r –For a well-defined wave with specific isospin and spin (IJ) characterized by narrow and well-isolated resonances, we know how. r r 3 1 2 | 1 2 3

5. the propagator is of the simple Breit-Wigner type traditional isobar model Spin 0 Spin 1 Spin 2 the decay amplitude is the decay matrix element

6. when the specific IJ–wave is characterized by large and heavily overlapping resonances (just as the scalars!), the problem is not that simple. where K is the matrix for the scattering of particles 1 and 2. In this case, it can be demonstrated on very general grounds that the propagator may be written in the context of the K-matrix approach as Indeed, it is very easy to realize that the propagation is no longer dominated by a single resonance but is the result of a complicated interplay among resonances. i.e., to write down the propagator we need to know the related scattering K-matrix In contrast

7. What is K-matrix? It follows from S-matrix and because of S-matrix unitarity it is real Viceversa, any real K-matrix would generate an unitary S-matrix This is the real advantage of the K-matrix approach: –It drastically simplifies the formalization of any scattering problem since the unitarity of S is automatically preserved

8. From Scattering to Production Thanks to I.J.R. Aitchison (Nucl. Phys. A189 (1972) 514), the K-matrix approach can be extended to production processes In technical language, –From –To The P-vector describes the coupling at the production with each channel involved in the process –In our case the production is the D decay

9. K-Matrix Picture of D +  +  -  + 00 ++ -wave

10. Failure of the Isobar Model At this point, on the basis of a pretty solid theory, it is very easy to understand when we can employ the traditional Isobar Model and when not. It turns out that – for a single pole problem, far away of any threshold, K- matrix amplitude reduces to the standard BW formula. The two descriptions are equivalent –In all the other cases, the BW representation is not any more valid The most severe problem is that it does not respect unitarity

11. Add BW Add K The Unitarity circle Add BW Add K An Explicit Example Adding BWs ala “Isobar Model” –Breaks the Unitarity –And heavily modify the phase motion!

12. The decay amplitude may be written, in general, as a coherent sum of BW terms for waves with well-isolated resonances plus K-matrix terms for waves with overlapping resonances. Can safely say that in general K-matrix formalization is just required by scalars (J=0), whose general form is Summarizing

13. Where can we get a reliable  s-wave scattering parametrization from? In other words, we need to know K to proceed. A global fit to all the available data has been performed! *  p    0 n,  n,  ’n, |t|  0.2 (GeV/c 2 ) GAMS *  p     n, 0.30  |t|  1.0 (GeV/c 2 ) GAMS * BNL *  p -  KKn CERN-Munich :         * Crystal Barrel * * * pp             pp       ,     ,    pp         K + K -  , K s K s  , K +  s   np       -,        K s K -  , K s K s  -  - p    0 n, 0  |t|  1.5 (GeV/c 2 ) E852 * At rest, from liquid At rest, from gaseous At rest, from liquid “ K-matrix analysis of the 00++-wave in the mass region below 1900 MeV’’ V.V Anisovich and A.V.Sarantsev Eur.Phys.J.A16 (2003) 229

14. is the coupling constant of the bare state  to the meson channel describe a smooth part of the K-matrix elements suppresses the false kinematical singularity at s = 0 near the  threshold and is a 5x5 matrix (i,j=1,2,3,4,5) 1=  2= 3=4  4= 5= A&S

15. A&S K-matrix poles, couplings etc.

16. A&S T-matrix poles and couplings A&S fit does not need a  as measured in the isobar fit

17. FOCUS D s +   +  +  - analysis Observe: f 0 (980) f 2 (1270) f 0 (1500) Sideband Signal Yield D s + = 1475  50 S/N D s + = 3.41 PLB 585 (2004) 200

18. First fits to charm Dalitz plots in the K-matrix approach! C.L fit 3 % Low mass projection High mass projection decay channel phase (deg) fit fractions (%)

19. Yield D + = 1527  51 S/N D + = 3.64 FOCUS D +   +  +  - analysis Sideband Signal PLB 585 (2004) 200

20. C.L fit 7.7 % K-matrix fit results Low mass projection High mass projection decay channel phase (deg) fit fractions (%) No new ingredient (resonance) required not present in the scattering!

21. With  Without  C.L. ~ 7.5% Isobar analysis of D +   +  +  would instead require a new scalar meson:  C.L. ~ 10 -6 m = 442.6± 27.0 MeV/c  = 340.4 ± 65.5 MeV/c preliminary

22. What about  -meson then? Can conclude that –Do not need anything more than what is already in the  s-wave phase-shift to explain the main feature of D  3  Dalitz plot Or, if you prefer, –Any  -like object in the D decay should be consistent with the same  -like object measured in the  scattering.

23. Just by a simple insertion of KK -1 in the decay amplitude F We can view the decay as consisting of an initial production of the five virtual states , KK,  ’  and 4  which then scatter via the physical T-matrix into the final state. The Q-vector contains the production amplitude of each virtual channel in the decay Even more: from P to Q-vector

24. Q-vector for D s s-wave dominated by an initial production of  and KK-bar states The two peaks of the ratios correspond to the two dips of the  normalizing modulus, while the two peaks due to the K-matrix singularities, visible in the normalization plot, cancel out in the ratios. The normalizing  modulus Ratio of moduli of Q-vector amplitudes

25. Q-vector for D + The same! –s-wave dominated by an initial production of   and KK-bar states

26. The resulting picture The s-wave decay amplitude primarily arises from a ss-bar contribution –Cabibbo favored for D s –Cabibbo suppressed together with the competing dd-bar contribution for D + The measured fit fractions seems to confirm this picture –s-wave decay fraction, 87% for D s and only 56% for D + –The dd-bar contribution in D + case evidently prefers to couple to a vector state like  (770), that alone accounts for about 30% of the decay.

27. Conclusions Dalitz plot analysis is and will be a crucial tool to extract physics from the HF decays Nevertheless, to fully exploit this unlimited potential a systematic revision of the amplitude formalization is required Thanks to FOCUS, K-matrix approach has been shown to be the real breakthrough Its application has been decisive in clearing up a situation which recently became quite fuzzy and confusing –new “ad hoc” resonances were required to understand data, e.g.  (600) and  (900) Strong dynamics effects in D-decays now seem under control and fully consistent with those measured by light-quark experiments The new scenario is very promising for the future measurements of the CP violating phases in the B sector, where a proper description of the different amplitudes is essential.