0 + - decays in a factorization approach Analysis done in - PowerPoint PPT Presentation

Dalitz plot analysis of D 0  K S 0  +  - decays in a factorization approach Analysis done in collaboration with Robert Kamiński (Institute of Nuclear Physics PAS, Kraków, Poland), Jean-Pierre Dedonder and Benoit Loiseau (LPNHE, Paris, France) published recently in Physical Review D 89, 094018 (2014), arXiv: 1403.2971 [hep-ph].

Motivation Studies of the D 0  K S 0  +  - reaction are useful in: 1. measurements of the D 0 -  D 0 mixing parameters, 2. determination of the Cabibbo- Kobayashi- Maskawa angle  in the decay amplitude B   D K  , D  K S 0  +  - , 3. description of the final state interactions between mesons, in particular in the S-waves, 4. testing theoretical models of meson form factors , 5. understanding properties of the meson resonances and their interference effects on the Dalitz plot .

Isobar model and its problems 1. Amplitudes in the isobar model are not unitary neither in three-body decay channels nor in two-body subchannels. 2. It is difficult to distinguish the S - wave amplitude from the background terms. Their interference is often very strong. 3. Some branching fractions extracted in such analyses could be unreliable. 4. The isobar model has many free parameters (at least two fitted parameters for each amplitude component). Recently Belle used 49 fitted parameters and BaBar 43 parameters.

Why unitarity is important? Unitary model allows for: 1. proper construction of the D-decay amplitudes, 2. partial wave analyses of final states, 3. explanation of structures seen in Dalitz plots, 4. adequate determination of branching fractions and CP asymmetries for different quasi-two-body decays, 5. extraction of standard model parameters (weak amplitudes), 6. application not only in analyses of D decays but also in studies of other reactions.

Towards a unitary approach 1. Construction of unitary three-body strong interaction amplitudes in a wide range of effective masses is difficult. 2. As a first step we attempt to incorpotate in our model two-body unitarity into the D-decay amplitudes with final state interactions in the following subchannels: a) K 0  S-wave amplitude, b)   S-wave amplitude, c)   P-wave amplitude.

Allowed and suppressed tree transitions c  G Transition su d  * F ( ) ( ) O V V s c u d   1 cs ud V A V A 2 u  +     C = Cabibbo angle cos V V W +  cs ud C d c s  K 0  d allowed D 0 d  -  u G  * c  F ( ) ( ) O V V d c u s Transition du s   2 cd us V A V A 2  + u  d          , , sin 0 . 225 V V W +  s d cd us C K 0 d c doubly Cabibbo suppressed  - D 0  u

Tree diagrams with internal W lines allowed doubly Cabibbo suppressed

Annihilation decay amplitudes c s 0 K d d c  D 0 W +  - Transition su d u u  + allowed d u c d  - u c  u Transition D 0 W + du s  + d d K 0 doubly Cabibbo suppressed u s

Factorization approach     ( ) , ( ) , ' ( ) , ' ( ) j s c j u d j u c j s d Quark currents :     1 V A 2 V A 1 V A 2 V A   * / 2 main part of the effective Hamiltonian: H G V V j j 1 2 F cs ud                0 0 0 0 Factorization: | | | | | | 0 K j j D K j D j 1 2 1 2         0 0 | ' | | ' | 0 j D K j 1 2         0 0 0 | ' | | ' | 0 j D K j 1 2        f  - pion decay constant | | 0 j if p   2      0 | ' | 0 K j if K p f K - kaon decay constant 2 K       0 f D - D decay constant 0 | 1 | ' j D if D p D

Types of decay amplitudes 27 amplitudes for the D 0  K S 0  +  - decay: a) 7 allowed tree amplitudes, b) 6 doubly Cabibbo suppressed tree amplitudes, c) 14 annihilation (W-exchange) amplitudes (7 allowed and 7 doubly Cabibbo suppressed). Seven partial wave amplitudes: 1. S-, P- and D- wave amplitudes in the K  subsystem, 2. S-, P- and D- wave amplitudes in the  +  - subsystem, including in addition the ω   +  - P- wave transition .

Resonances in decay amplitudes Channel: wave: name:       * * 0 S ( 800 ) or , ( 1430 ) K K K 0 0    * * * P ( 892 ) , ( 1410 ) , ( 1680 ) K K K  * ( 1430 ) K D 2   0 K same list as above but with pion charge +     σ ( 500 ) or , ( 980 ), ( 1400 ) S f f f 0 0 0    P ( 770 ), ( 1450 ), ( 782 ) ( 1270 ) f D 2 Very rich resonance spectrum  complexity of final state interactions

Selected formulae of decay amplitudes 1        0 0 0    | (| | ) 0 0 K S K K D K 2 S    1  0 V * K cs V Allowed transitions with final state interactions S ud       0  m eff. masses of , eff. mass a 1 - effective Wilson coefficient K m  S 0 G       * π 2 2 DK 2 K 2 F ( )) ( ) ( ) S-wave: A a f m m F m F m 0 0  π π π 1 1 1 0 0 S D 2  * DK 2 * transition scalar form factor - D to K 0 ( ) F m 0 π 0  m    scalar form factor 2 2 π ( ) K 2 - K 0 m p p ( ) F 0    0 π 0 K P-wave:   2 2 2 2 ( )( ) G f m m m m   *          DK 2 2 2 2 K F [ D K ] ( ) ( ) A a m m A m F m 0    1 1 1 0 0 1 P 2 2 f m      * 2 2 * transition vector form factor ( ) DK 2 - D to K 0 m p p ( ) A m 0   0 π π 0 K     m 2 2 ( )  vector form factor m p p π K 2 - K 0 ( )  F 0 0 π π  1

kaon-pion scalar form factor

pion scalar form factor

 2 joint fit    2 2 2 χ χ χ χ 2 τ 0 Br D Data for: 1. D 0  K S 0  +  - decays, A. Poluektov et al. (Belle Coll.), Phys. Rev. D 81, 112002 (2010), 2.  -  K S 0  -   decays, D. Epifanov et al. (Belle Coll.), Phys. Lett. B 654, 65 (2008), 3. total branching fraction Br exp = (2.82  0.19) %. Number of degrees of freedom: ndf= 6321 + 89 + 1 – 33 free model param.= 6378. Result:  2 = 9451 which gives  2 / ndf = 1.48.

Dalitz plot density distribution for the D 0  K S 0  +  - decay   2 ( ) s p p   0 π K S   2 ( ) s p p   0 π K S

0  - effective mass Comparison of the K S squared distributions with the Belle data

0  + and  +  - effective Comparison of the K S mass squared distributions with the Belle data 0  +  +  - K S

Comparison with the Belle data on the  -  K S 0  -   decay m K  (GeV )

Branching fractions Channel Br (%) Br (tree) Annihil. low. limit 0  - ] S  + 25.0  3.6 8.2  0.1 7.9  0.1 [K S 0 [  -  + ] S 16.9  1.3 14.7  0.2 2.9  0.1 K S 0  - ] P  + 62.7  4.5 24.7  5.7 8.7  3.0 [K S 0 [  -  + ] P 22.0  1.6 4.4  0.1 6.7  0.04 K S 0  ) = (21.2  0.5) % Br (K S Exp. Br (K*(892) +  - ) = (62.9  0.8) %

Summary 1. The D 0  K S 0  +  - decays are analysed using the factorization approximation. 2. The annihilation (via W-echange) amplitudes are added to the weak-decay tree amplitudes. 1. The strong interactions between kaon-pion and pion-pion pairs in the S-, P- states are described in terms of the corresponding form factors . For D-waves we use relativistic Breit-Wigner formulae. 2. The kaon-pion and pion-pion scalar form factors are constrained using unitarity, analyticity and chiral symmetry and by the present Dalitz plot analysis. 5. A good agreement with the Belle and BABAR Dalitz plot density distributions and with the  -  K S 0  -   decay data is achieved. 6. The lower-limit values of the branching fractions of the annihilation amplitudes are significant.

Lower limit of annihilation amplitudes 10     T i – tree ampl. A i – annihilation ampl. ; T A M M M i i i i  1 i 2 d Br    2 ρ 2 - i i | | | ; fitted ampl. e c M c M M M i i i i ds ds   . tree ann 2 2 d Br d Br   2 2 i i | | ; | | c T c A i i ds ds ds ds       ρ i A e M T  = phase of the K S 0  amplitude i i i Lower limit of the annihilation branching fraction:     . ann low tree 2 | | | | Br Br Br ds ds M T   i i i i i