Model independent determination of γ from B ± →D(K 0 S π + π − )K ±

1. Model independent determination of γfrom B±→D(K0Sπ+π−)K± Jim Libby (University of Oxford) LHCb Tuesday Meeting

2. Outline • Current e+e−B-factory and LHCb status • Why use a model independent method? • Explanation of the model independent method • Importance of CLEOc • Implementation to LHCb environment and results • Background • Acceptance • Systematic uncertainties • Outlook LHCb Tuesday Meeting

3. B±→D(K0Sπ+π−)K± Vub e-iγ • Decays of D0or D0 to common final state gives sensitivity to γ • For B+→D(K0π+π−)K+ • Assume isobar model (sum of Breit-Wigners) • Fit D-Dalitz plots from B-decay to extractγ, rB and δB (770) K*(892) LHCb Tuesday Meeting

4. Current results • Current best direct constraints on γ: • These measurements are theoretically clean • No penguin CKM standard candle • D-mixing ignored in formalism but < 1° correction • However, large error from isobar model assumptions • BABAR and Belle use large samples of flavour tagged D*+D0π+events to find parameters of the isobar model • Excellent knowledge of |f|2 but phases less well known • Model uncertainties from assumptions about the resonance structures in the model LHCb Tuesday Meeting

5. K*0(1430) Isobar model uncertainty =rB sin(δ+γ) BABAR CKM06 • Most challenging aspects of the model uncertainty come from Kπand ππS-wave BABAR (PRL 95 121802,2005) Fit to flavour tag sample LHCb Tuesday Meeting

6. Aside: K-matrix • Breit Wigner description of broad overlapping resonances violates unitarity and requires non-physical σ • K-matrix description preserves unitarity • First studies (Lauren Martin/JL) of LHCb γfit with one K-matrix parameterisation of the ππS-wave • Difference between assuming K-matrix and BW model consistent with B-factory observations • Note with EB – draft available from CPWG webpage • Explore different physical K-matrix parameterisation to evaluate systematic rather than introduce σ will reduce model uncertainty 5 pole K-matrix LHCb Tuesday Meeting

7. Model uncertainty impact on LHCb • Recent studies for LHCb (LHCb-048-2007) showed that the model-dependent fit would yield an uncertainty onγbetween 7-12° for an rB=0.1 • Range represents differing assumptions about the background • However, the best current model uncertainty is 10-15° with an rB=0.1 • Uncertainties 1/rB • Without improvements LHCb sensitivity will be dominatedby model assumptions within 1 year of data taking • Motivates a model-independent method that relies on a binned analysis of the Dalitz plot • Disadvantage is that information is lost via binning • Rest of this talk will discuss this method and its implementation at LHCb LHCb Tuesday Meeting

8. Binned method • Proposed in the original paper by Giri, Grossman, Soffer and Zupan and since been extended significantly by Bondar and Poluektov • GGSZ, PRD 68, 054018 (2003) • BP, Eur. Phys. J. C47, 347 (2006)+hep-ph/0703267 • Bin the Dalitz plot symmetrically about m−2= m+2 then number of entries in B− decay given by:  # events in bin of flavour tagged D0 decays Average cosine and sine of strong phase difference between D0 and D0 decay amplitudes (ΔδD) in this bin LHCb Tuesday Meeting

9. Binned method continued • Can determine si and ci at the same time as extracting γ, rB and δB from B data • 3 + Nbins free parameters (ci =c-i and si =−s-i) and 2 Nbins • Huge loss in γsensitivity not practical until you have O(106) events (2500/fb-1 @ LHCb) • However, CP-correlated e+e−→ ψ″→D0D0datawhere onedecay is to KSππand the other decays to a CP eigenstate or KSππallowssi and ci to be determined LHCb Tuesday Meeting

10. CLEO-c: double tagged ψ(3770) events CLEO-c has collected ~ 800 fb-1 at the ψ (3770) DDbar produced in quantum entangled state: Reconstruct one D in decay of interest for γ analysis (eg. Kππ), & other in CP eigenstate (eg. KK, Ksπ0 …) then CP of other is fixed. Almost background free Can use KL From talk by E. White at Charm 07 LHCb Tuesday Meeting

11. CLEO-c measurement status 1/3 of total data Studies not complete but projected uncertainties on c and s will lead to 3-4 degree uncertainty on γ LHCb Tuesday Meeting

12. Inkblot test Absolute value of strong phase diff. (BABAR model used in LHCb-48-2007) • Bondar and Poluektov show that the rectangular binning is far from optimal for both CLEOc and γanalyses • 16 uniform bins has only 60% of the B statistical sensitivity • c and s errors would be 3 times larger from the ψ″ • Best B-data sensitivity when cos(ΔδD) and sin(ΔδD) are as uniform as possible within a bin Good approximation and the binning that yields smallest s and c errors is equal ΔδD bins-80% of the unbinned precision LHCb Tuesday Meeting

13. Implementation at LHCb (γ=60°, rB=0.1 and δB=130°) • Generate samples of B±→D(K0Sππ)K±with a mean of 5000 events split between the charges • Bin according to strong phase difference, ΔδD • Minimise χ2 • Ki, ci and si amplitudes calculated from model • In reality from flavour tagged samples and CLEO-c LHCb Tuesday Meeting

14. No background with predicted 2 fb-1 yield 5000 experiments Input parameters γ=60°, rB=0.1 and δB=130° The four Cartesian coordinates and normalization are free parameters All pulls are normal therefore calculate γ, rBandδB with propagated Cartesian uncertainties LHCb Tuesday Meeting

15. No background with predicted 2 fb-1 yield Model independent average uncertainty 7.7° (c.f. Model dependent 5.9°) LHCb Tuesday Meeting

16. Acceptance • Acceptance in each bin calculated as a weighted average of the acceptance function used for model dependent studies • 15% relative difference amongst bins • Modifies the fit function: • Average γuncertainty increases to 8.1° Can be calculated from Dπ LHCb Tuesday Meeting

17. Background • 3 types of background to consider • B→D(KSππ)π(DC04 B/S = 0.24) • rB(Dπ) O(10-3) so Dalitz plots are likeD0andD0forB−andB+, respectively • Combinatoric (DC04 B/S<0.7) • Admixtures of two types considered • DKcomb: real D→ D(KSππ) combined with a bachelor K • Dalitz plot a even sum of D0andD0decays • PScomb:combinatoric D with a bachelor K • Follows phase space • Integrate background PDFs used in model-dependent analysis over each bin, then scaled to background level assumed: fractional area of Dalitz space covered by bin LHCb Tuesday Meeting

18. γ uncertainties with 5000 toy experiments LHCb Tuesday Meeting

19. Systematic related to acceptance • The acceptance varies over the Dalitz plane • The relative acceptance in each bin can be measured using the B→Dπcontrol sample with DK selection applied without bachelor K PID • With the DC04 selection expect 60k events/2 fb-1 • Relative relative-efficiency uncertainty 1-4%/ΔδD bin with 2 fb-1 • Increased statistics reduces error • Toy MC study smearing bin efficiencies in event generation by this amount leads to an additional 1° uncertainty without background and 3.2° uncertainty with DKcomb B/S=0.7 • Small effect compared to statistical uncertainty • NB: the efficiency related to the PID of the bachelor π/K can be factored out and will be determined from the D*→D(Kπ)π data to better than one percent-ignore at present LHCb Tuesday Meeting

20. Asymmetry in efficiency in Dalitz space • Considered charge asymmetries in the efficiency across the Dalitz plane • ε(m2+, m2 −)≠ε(m2−, m2 +) • Generated with the efficiency biased relative to one another depending on whether the event had m2+>m2 − or m2+<m2 − • Maximum bias onγinduced was <1° for 10% relative effect and full background • 10% effects would be evident in the Dπ sample LHCb Tuesday Meeting

21. Resolution • ΔδD binning has some narrow regions in Dalitz space • Investigation of how resolution on the Dalitz variables might affected the extraction of γ • 10 MeV2/c4 resolution (DC04) on Dalitz variables and generated toy experiments with this smearing • Found that this led to a few bins with largest (red) and smallest (dark blue) phase difference having a 2-3% relative changes in expected yields due to resolution induced migration • Fit results on toy experiments where resolution included in generation but ignored in fit found no significant bias (<0.5°) on γ LHCb Tuesday Meeting

22. Background fractions • Combinatoric background rate will be determined from B and D mass sidebands which will cover at least 2-3 times the area of the signal region • Use 10× in DC04 background studies but this will probably be unrealistic with data • If background distributions relatively flat in masses one can estimate that this leads to B/S will be determined absolutely to around 0.01 or better • Toy studies suggest that there is no impact on γprecision with this kind of uncertainty • Maybe complications depending on Dalitz space distribution of the PS background but can only speculate until we have the data in hand LHCb Tuesday Meeting

23. Conclusion Model independent Model dependent σ(model)=10° • Implemented model independent fit with binning that yields smallest error from exploiting CLEO-c data • Binning depends on model - only consequence of incorrect model is non-optimal binning • Such a measurement will be better than model dependent method if we cannot improve the model error • 10 fb-1 statistical uncertainty 4-6°depending on background • Experimental systematic uncertainties considered • can be controlled from data • acceptance determined from B→Dπ sample contributes ~3° with 2 fb-1 • Liaison required with CLEO-c to get common binning and to best understand all uncertainties related to the ψ″data • Note with EB since beginning of October • Available from CPWG webpage σ(model)=5° LHCb Tuesday Meeting

24. Additional slides LHCb Tuesday Meeting

25. Toy experiment results: γ (2 fb−1) LHCb Tuesday Meeting

26. Toy experiment results: γ (10 fb−1) LHCb Tuesday Meeting

27. Toy experiment results: rB (2 fb−1) LHCb Tuesday Meeting

28. Toy experiment results: rB (10 fb−1) LHCb Tuesday Meeting