1. Fermilab, June 26 th 2001 Thoughts on the fitting procedure for the  c + lifetime with the        channel Gianluigi Boca

2. Outline Explanation of the fitting procedure Some check s to show it works Try to answer to the 3 major issues of the lifetime Committee Discussion about Eduardo’s method of treating the double solutions

3. Peculiarity of the       channel : existence of two solutions for the kink (   ) Selection cuts     linked, at least 3 chmbrs, no  no REME L/  > 7 Primary in target region and Z  c < 2.3 ISO2< 0.001  t < 70 fs For the : Wobs(K) < Wobs  –2 For the    n   channel only : from kink not belenging to a Vee with 0.485 < M(K)<0.515 For the    p  0 channel only : proton from not linked and Wobs(p)<Wobs  – 5 Double solution treatment Two solutions for the kink  two  c + effective mass ( = m1, m2) If two solutions survive the physics cuts accept them both with weight 1 unless  m) = | m1 – m2 | < 30 MeV in which case accept them both with weight 0.5

4. Peculiarity of the       channel … cntd Long tails of  c + signal, due to kink double solutions, ‘contaminate’ the sidebands. Extraction of background reduced proper time (t’) plot, used for lifetime fit, from Sidebands, is slightly trickier than usual MC plots from 6,000,000    n   and 6,000,000    p  0 events

5. Two approaches to solve the problem a) Modify the fitting procedure such that it can work with any double solution treatment (my method) b) try to eliminate tails in sidebands by proper choices of one of the two double solutions (Eduardo’s method)

6. Description of my method Based on two assumptions 1)MC reproduces correctly mass shape and t’ evolution of  c events 2)The t’ evolution of  c signal in the sideband region is the same as the one in the peak region and it is proportional to

7. Description of my method, contd Combined fit of a)Diplot of the t’ vs       mass  in peak region (from 2.445 to 2.485) b)       mass  in the OUTER region Peak region Each bin content is predicted Each mass bin content in the outer region is predicted Outer region

8. Likelihood function

9. Likelihood, cntd Outer region mass plot predicted entries g(i) taken from MC and normalized A, C 0, C 1 = fit parameters m i = mass in the center of i th bin =+

10. Diplot t’ vs mass in peak region predicted entries =Ag(j)  +(C 0 +C 1 )  BkBk =+  )

11. Extraction of the B k from sidebands Sidebands definition : from 2.39 to 2.44 and from 2.49 to 2.54 + + (1-Y)  = Y  BkBk

12. Checking the method with  c  c +   + (p  0 )  +   sample (courtesy of Cristina) Selections No stubs l/ > 5 CLD > 0.03 Picon > 6 p from kink not lnkd  ps    dof  PDG average =  ps

13. Results for the  c +        all the sample  ps    dof 

14. A consistency check :  c + lifetime varying the doubles weighting scheme Standard : if two solutions survive the physics cuts accept them both with weight 1 Unless  m) = | m1 – m2 | < 30 MeV in which case accept them both with weight 0.5 Variation 1 : if two solutions survive the discard the event Variation 2 : if two solutions survive accept both with w =1 Variation 3 : if two solutions survive accept both with w = 0.5 Variation 4 : if two solutions survive accept only solution 1 Variation 5 : if two solutions survive accept only solution 2 Variation 6 (the mad man’s scheme) : if two solutions survive accept only solution 2 with w=5. Variation 7 (similar to Eduardo’s scheme) : if two solutions survive accept solution 1 if solution 2 is not in the 2.45 – 2.5 range, else discard the event

15. A consistency check :  c + lifetime varying the doubles weighting scheme, cntd Method is robust against different doubles treatment

16. The  c + lifetime Committee 3 major issues on this fitting method 1)“Is it true the  c + signal present in the sideband region have the same t’ distribution as the  c + signal in the peak ?”  LOOK AT THE MC, ALL SAMPLE ratio

17. 3 major issues, cntd 2) “Is it true that B (t’ k ) is independent of mass and the values of A, C 0, C 1 are independent on t’ k ?” IN OTHER WORDS : IS TRUE ? Yes, B (t’ k ) is independent of mass since it is extracted from sidebands and in all E687-E831 lifetime analysis we always have assumed that the sidebands represent well the background under the peak region (if no nasty reflections are present like in the D0 case). If it represents the bckgrnd under the peak, there are no reasons to doubt that it represent the bckgrnd also in any particular mass bin of the mass region!

18. 3 major issues, cntd 3) “Can the  c +       and  c +        samples be merged in one lifetime fit ?“ Yes, just multiply the likelihood for the two samples together and make A fit with 8 parameters ( , the 4 parameters for the backgrounds of the  c +       channel (type1, type2, kink, MultiVee) and A, C0, C1 for the  c +        channel.

19. Eduardo’s method Reduces long tails in sidebands caused by ‘wrong’ kink solution His double treatment (I hope this summary is accurate) : apply all the physics cuts and, for the events with both solutions surviving the cuts if (  m < 30 MeV ) then choose solution 1 else if ( m 2 NOT in the 2.45 to 2.5 range ) then choose solution 1 else discard the event end if

20. Eduardo’s method, cntd works well with kink4 signal shape, but … MC, kink2, doubles selected solution MC, kink4, doubles selected solution

21. … but, I have 2 questions Consider the background (non  c + events) having both solutions passing the physical cuts Toy model montecarlo, 10,000,000 entries and apply Eduardo’s algorithm artificial bump created  in lifetime analysis one needs to determine the amount of background actually present under the peak 1)

22. The artificial peak stems from the  m < 30 MeV case. One can remove it and use condition : if ( NOT 2.45< m 2 < 2.5 ) then choose solution 1 else discard the event Price to pay :some loss of  c + events beforeafter

23. Second question : What about the events for which only one solution passed the physical cuts ? Option 1 : ACCEPT IT Option 2 : REMEMBER (or recalculate) OTHER SOLUTION AND APPLY AGAIN EDUARDO’S CRITERION Advantages : for the kink4 fixes the tail problem Disadvantages : a) loose more statistics b) discarded solutions reenters from backdoor after being discarded previously by physical cuts 2) Tails remain !

24. Conclusions I demonstrated (I hope) my fitting method works with any kind of double soulution scheme for the kink I addressed the three issues of the Committee of the lifetime paper I think it is the preferable method to be usde in the  c + lifetime calculation for       channel