CKM 2006 CKM 2006 Extracting CKM phase from phase from - PowerPoint PPT Presentation

CKM 2006 CKM 2006 Extracting CKM phase from  phase from Extracting CKM  B ±  K ± and B ±  K - and ±   + +   - B 0 , B 0 K S  +  - B 0 , B  K S  +  0  - hep-ph/0608268 hep-ph/0608268 Ignacio Bediaga, Gabriel Guerrer, Ignacio Bediaga, Gabriel Guerrer, Jussara Miranda Jussara Miranda CBPF – Rio de Janeiro , Brazil CBPF – Rio de Janeiro , Brazil 1 ) Feasibility of |A| 2 +|A'| 2 fit 1 ) Feasibility of |A| 2 +|A'| 2 fit 2 ) Extracting  2 ) Extracting  1 1 Gabriel Guerrer - guerrer@cbpf.br Gabriel Guerrer - guerrer@cbpf.br 15/12/2006 15/12/2006

1 ) Feasibility of |A| Feasibility of |A| 2 +|A'| 2 fit 1 ) 2 +|A'| 2 fit , 2 vs 3 body amplitudes 2 vs 3 body amplitudes a e i i   B 0 a e B 0 K +  - K +  - K*(890) + K*(890) +   - -  j i  a j e i x x BW BW j (E) j a j e j (E) K*(1430) + K*(1430) +   - - B 0 B 0 K s (770) 0 K  (770) 0 s  K s f 0 (980) K f 0 (980) s K s K   c0 s c0 K s  +  - K s  +  - m j m j 2 2 Gabriel Guerrer 15/12/2006 15/12/2006 Gabriel Guerrer

1 ) Feasibility of |A| Feasibility of |A| 2 +|A'| 2 fit 1 ) 2 +|A'| 2 fit , Analysis Formalism* Analysis Formalism* all interm states all interm states A(x) =  a a i e i S i (x)  i i  A(x) =  i i i e S J (x) i J i a i → magnitude , ,  → strong + strong + weak phases a → magnitude  i → weak phases i i x = { s l , s m n } → Dalitz variables x = { s , s m n } → Dalitz variables l m m s s l m = (p l + p m ) 2 = m l + m m + 2 E l E m – 2 p l p m cos  l m = (p + p  ) 2 = m 2 + m 2 + 2 E l E m – 2 p l p m cos  l m    2 2 l m l m l m S i → form factors x BW BW x angular term (spin J) x angular term (spin J) S → form factors x J J i *AKA Isobaric Model *AKA Isobaric Model 3 3 Gabriel Guerrer 15/12/2006 15/12/2006 Gabriel Guerrer

1 ) Feasibility of |A| Feasibility of |A| 2 +|A'| 2 fit 1 ) 2 +|A'| 2 fit , Toy Model Toy Model A = BW 1 + r r e e i BW 2 A = BW 1 + BW i   2 Phase space (Dalitz Plot) points distribution Phase space (Dalitz Plot) points distribution ● ∝ ∝ |A| 2 = |BW 1 | 2 + r r 2 |BW 2 | 2 + |A| 2 = |BW 1 | 2 + 2 |BW 2 | 2 + + 2r r cos cos  Re BW  Re BW 1 BW 2 + 2r r sin sin  Im BW  Im BW 1 BW 2 + 2 1 BW * + 2 1 BW * * * 2 2 DP DP interference interference 1 1 2 2 4 4 Gabriel Guerrer 15/12/2006 15/12/2006 Gabriel Guerrer

1 ) Feasibility of |A| Feasibility of |A| 2 +|A'| 2 fit 1 ) 2 +|A'| 2 fit , Toy Model Toy Model r= = 1,  0 .. 2  r 1,   0 .. 2   surface univocally defined by the set of parameters surface univocally defined by the set of parameters { { r, ,  } ● r  } 5 5 Gabriel Guerrer 15/12/2006 15/12/2006 Gabriel Guerrer

1 ) Feasibility of |A| Feasibility of |A| 2 +|A'| 2 fit 1 ) 2 +|A'| 2 fit , Fitting |A| 2 2 Fitting |A| given a set of points modelled by A : given a set of points modelled by A : ● = 1 ex: r= 1.2, = 1 ex: r= 1.2, BW 1 + r r e e i BW 2 | 2 2 | BW 1 + BW 2 | i  PDF = = 1 1 |  PDF M M M = ∫ ds 13 ds 23 |A| 2 M = ∫ ds 13 ds 23 |A| 2 Likelihood Fit : r, r,  extracted extracted : Likelihood Fit    Fit of Isobaric amplitudes in 3 body decays has been ● Fit of Isobaric amplitudes in 3 body decays has been successfully used in many analysis from BaBar, Belle, Focus, etc successfully used in many analysis from BaBar, Belle, Focus, etc ex: ex:  (500) discovery, Fermilab E791, Phys Rev Lett 86 , 770 ●  (500) discovery, Fermilab E791, Phys Rev Lett 86 , 770 6 6 Gabriel Guerrer 15/12/2006 15/12/2006 Gabriel Guerrer

1 ) Feasibility of |A| Feasibility of |A| 2 +|A'| 2 fit , 1 ) 2 +|A'| 2 fit B B 0 K s  +  - resonances  K s  +  resonances 0  - - + + - - + + - K s K s K*(890) K*(890) K*(890) K*(890)     - + + - 1 - + 1 + - K*(1430) K*(1430)  K*(1430)  K*(1430)   B 0 B 0 B 0 B  + 0  + K s (770) 0 K s (770) 0 K  (770) K  (770) 0 0   s s 2 2 f 0 f 0 K s f (980) K s f (980) K 0 (980) K 0 (980)  - -  s s K s K s K  c0 K  c0   c0 c0 s s 3 3 |A'| 2 2 |A'| |A| 2 2 |A| s 13 s s 13 s 13 13 s 12 s s 12 s 7 12 7 Gabriel Guerrer 15/12/2006 15/12/2006 Gabriel Guerrer 12

1 ) Feasibility of |A| Feasibility of |A| 2 +|A'| 2 fit 1 ) 2 +|A'| 2 fit , B B 0 K s  +  - amplitudes  K s  +  amplitudes 0  - time independent time independent  K  K B 0 K s  +  - : A, B B 0 K s  -  + : A' 0  0  B s  +  : A, s  -  : A' - + B 0 / B 0 mixing   B 0 / B 0 mixing time dependence time dependence Prob. (t= Prob. (t= 0) B 0  (t= t) K S  +  - 0) B 0  (t= t) K S  +  - M (t)= M (t)= e -( [A cos(  m t/2) – i q/p A' sen(  m t/2)] e /2-iM)t [A cos(  m t/2) – i q/p A' sen(  m t/2)] -(   /2-iM)t Prob. (t= 0) B 0  (t= t) K S  +  - Prob. (t= 0) B 0  (t= t) K S  +  - M' (t)= M' (t)= e -( [A' cos(  m t/2) – i p/q A sen(  m t/2)] e /2-iM)t [A' cos(  m t/2) – i p/q A sen(  m t/2)] -(   /2-iM)t 8 8 Gabriel Guerrer 15/12/2006 15/12/2006 Gabriel Guerrer

1 ) Feasibility of |A| Feasibility of |A| 2 +|A'| 2 fit 1 ) 2 +|A'| 2 fit , Good news Good news G. Burdman et al , Phys Rev 45 45 , 187 , 187 if | p/q| = 1 G. Burdman et al , Phys Rev if | p/q| = 1 or B Prob. ( B 0 0 or B 0 K s  +  - ) ∝ Prob. ( B  K s  +  - ) ∝ 0  |M(t)| 2 +|M'(t)| 2 = e e - t ( |A| 2 +|A'| 2 ) |M(t)| 2 +|M'(t)| = ( |A| 2 +|A'| 2 ) 2 - t   mixing effects cancels out! mixing effects cancels out!  time independent amp  NO TAGGING time independent amp NO TAGGING 9 9 Gabriel Guerrer 15/12/2006 15/12/2006 Gabriel Guerrer

1 ) Feasibility of |A| Feasibility of |A| 2 +|A'| 2 fit 1 ) 2 +|A'| 2 fit , In practice In practice DATA DATA { r 1 ,  , ...} r |A| 2 2 { 1 ,  1 , ...} |A| 1 B 0 0 + + B 0 0 B B ? ? |A'| 2 2 { r 2 ,  , ...} |A'| { r ,  2 , ...} 2 2 Is possible to extract parameters Is possible to extract parameters separately from A and A' in a joint sample? separately from A and A' in a joint sample? 10 10 Gabriel Guerrer 15/12/2006 15/12/2006 Gabriel Guerrer

1 ) Feasibility of |A| Feasibility of |A| 2 +|A'| 2 fit 1 ) 2 +|A'| 2 fit , In practice In practice Yes! Yes! DATA DATA 2 | A| 2 N 1 | A| N B 0 0 + + B 0 0 B B ? 1 ? reason: different, non-overlap : different, non-overlap reason interferences interferences | A'| 2 2 | A'|   different surfaces different surfaces N 2 N also possible to extract the number of also possible to extract the number of 2 events N 1 , N 2 of each set, measuring events N 1 , N 2 of each set, measuring direct CPV direct CPV 11 11 Gabriel Guerrer 15/12/2006 15/12/2006 Gabriel Guerrer

1 ) Feasibility of |A| Feasibility of |A| 2 +|A'| 2 fit 1 ) 2 +|A'| 2 fit , Fit |A| Fit |A| 2 2 + + |A'| |A'| 2 2 how to how to PDF = PDF =  S 1 { | S { | S 1 + r r 1 e i S 2 +...| | 2 2 + +| | r r S S 1 + r r 2 e i S 3 +...| | 2 2 } } 1 1 + 1 e 2 +... 1 + 2 e S 3 +... i 1  1 i 2  2  M 1 + M 2 M 1 + M 2 M 1 = ∫ds 13 ds 23 |A| 2 , M 2 = ∫ds 13 ds 23 |A'| 2 M 1 = ∫ds 13 ds 23 |A| 2 , M 2 = ∫ds 13 ds 23 |A'| 2 Likelihood Fit Likelihood Fit r univocally extracted r 1 ,  , r r 2 ,  , r r , ... , ... → univocally extracted 1 ,  1 , 2 ,  2 , → 2 1 r r scale, fundamental for measuring scale, fundamental for measuring N N 1 and N 2 → 1 and N → 2 N N 1 , N N 2 N 1 / N N 2 = M M 1 / M 2 → N 1 / 2 = 1 / M 1 , 2 → 2 12 12 Gabriel Guerrer 15/12/2006 15/12/2006 Gabriel Guerrer

1 ) Feasibility of |A| Feasibility of |A| 2 +|A'| 2 fit 1 ) 2 +|A'| 2 fit , Feasibility test test Feasibility We generated and fitted 100 samples of We generated and fitted 100 samples of 100K 100K ● 0 + 0 events each, using the method discussed B 0 + B 0 events each, using the method discussed B B previously previously The input resonances magnitudes and phases The input resonances magnitudes and phases ● hep-ex:0507004 were inspired in BaBar results hep-ex:0507004 were inspired in BaBar results The errors that will be shown are The errors that will be shown are RMS RMS from the from the ● 100 sample distributions 100 sample distributions 13 13 Gabriel Guerrer 15/12/2006 15/12/2006 Gabriel Guerrer

1 ) Feasibility of |A| Feasibility of |A| 2 +|A'| 2 fit 1 ) 2 +|A'| 2 fit , Feasibility test test Feasibility decay B0 / B0 input fitted  c0 Ks a0 / b0 Fixed / ( 0.3 ± ± 0.03 ) 0.3 / 0.3  0 / 0 3.78 / 3.78 fixed / fixed  K*(890)  a1 / b1 ± 0.06) / (1.30 ± ± 0.01) 1.17 / 1.3 (1.17 ±  1 / 1 ± 0.08) / (5.99 ± ± 0.07) 0.4 / 5.98 (0.41 ±  K*(1430)  a2 / b2 ± 0.11) / (2.72 ± ± 0.13) 2.45 / 2.72 (2.45 ±  2 / 2 ± 0.08) / (6.00 ± ± 0.06) (0.39 ±  0.375 / 6.0  0 Ks a3 / b3 ± 0.04) / (0.60 ± ± 0.04) 0.6 / 0.6 (0.60 ±  3 / 3 ± 0.09) / (1.20 ± ± 0.07) (1.22 ±  1.2 / 1.2 f 0 Ks a4 / b4 ± 0.06) / (1.04 ± ± 0.05) 1.03 / 1.03 (1.02 ±  4 / 4 ± 0.07) / (2.30 ± ± 0.08) 2.3 / 2.3 (2.30 ±  N 1 / N 2 = 0.84 ± ± 0.12 0.12 N 1 / N 2 = 0.84 ● small errors small errors ● 14 14 Gabriel Guerrer 15/12/2006 15/12/2006 Gabriel Guerrer