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Global FITS of the CKM Matrix

Global FITS of the CKM Matrix. G. Eigen, University of Bergen. In collaboration with G. Dubois-Felsmann, D. Hitlin, F. Porter. EPS conference, Aachen 17-07-2003. Motivation. In the hunt for New Physics (e.g. Supersymmetry ) the Standard

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Global FITS of the CKM Matrix

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  1. Global FITS of the CKM Matrix G. Eigen, University of Bergen In collaboration with G. Dubois-Felsmann, D. Hitlin, F. Porter EPS conference, Aachen 17-07-2003

  2. Motivation • In the hunt for New Physics (e.g. Supersymmetry) the Standard Model (SM) needs to be scrutinized in various ways • One interesting area is the study of the Cabbibo-Kobayashi- Maskawa (CKM) matrix, since • Absolute values of the CKM elements affect hadronic decays • Its single phase predicts CP violation in the SM • Especially interesting are processes that involve b quarks, as effects of New Physics may become visible • e.g. B0dB0dand B0sB0s mixing • CP violation in the B system,  CP asymmetry of BJ/Ks • Rare B decays BXs G. Eigen, U Bergen

  3. The Cabbibo-Kobayashi-Maskawa Matrix • A convenient representation of the CKM matrix is the small-angle Wolfenstein approximation to order O(6) with and • The unitarity relation that represents a triangle (called Unitarity Triangle) in the - planeinvolvesall 4 independent CKM parameters , A, , and  • =sinc=0.22 is best-measured parameter (1.5%), A.8 (~5%) while -arepoorly known G. Eigen, U Bergen

  4. Global Fit Methods • Present inputs for determinig the UT are based on measurements of B semi-leptonic decays,md,ms, & |K| that specify the sides and the CP asymmetry acp(KS) that specifies angle  • Though many measurements are rather precise already the precision of the UT is limited by non gaussian errors in theoretical quantities th(bu,cl), BK, fBBB,  • Different approaches exist: • The scanning method a frequentist approach first developed for the BABAR physics book (M. Schune, S. Plaszynski), extended by Dubois-Felsmann et al • RFIT, a frequentist approach that maps out the theoretical parameter space in a single fit A.Höcker et al, Eur.Phys.J. C21, 225 (2001) • The Bayesian approach that adds experimental & theoretical errors in quadrature M. Ciuchini et al, JHEP 0107, 013 (2001) • A frequentist approach by Dresden group K. Schubert and R. Nogowski • The PDG approachF. Gilman, K. Kleinknecht and D. Renker G. Eigen, U Bergen

  5. The Scanning Method • An unbiased, conservative approach to extract from the observables is the so-called scanning method • We have extended the method adopted for the BABAR physics book (M.H. Schune and S. Plaszczynski) to deal with the problem of non-Gassian theoretical uncertainties in a consistent way • We factorize quantities affected by non-Gaussian uncertainties (th) from the measurements • We select specific values for the theoretical parameters & perform a maximum likelihood fit using a frequentist approach • We perform many individual fits scanning over the allowed theoretical parameter space in each of these parameters • We either plot - 95% CL contours or use the central values to explore correlations among the theoretical parameters G. Eigen, U Bergen

  6. The Scanning Method • For a particular set of theoretical parameters (called a model M) we perform a 2 minimization to determine • Here <Y> denotes an observable & Y accounts for statistical and systematic error added in quadrature while F(x) represents the theoretical parameters affected by non-Gaussian errors • For Gaussian error contributions of the theoretical parameters we include specific terms in the 2 • Minimization has 3 aspects:  Model is consistent with data if P(2M)min > 5%  For these obtain best estimates for plot 95%CL contour The contours of various fits are overlayed  For accepted fits we study the correlations among the theoretical parameters extending their range far beyond the range specified by the theorists G. Eigen, U Bergen

  7. Treatment of ms • Ciuchini et al use 2 term (A-1)2/A2-A2/A2 to include limit on ms in the global fits • We found that for individual fits that the 2 is not well behaved • Therefore, we have derived a 2 term based on the significamce of measuring ms with For An=12.4667, l=0.055 ps and <ms> reproduces 95% CL limit of 14.4 ps-1 truncated at 0 G. Eigen, U Bergen

  8. The 2 Function in Standard Model G. Eigen, U Bergen

  9. Observables • Presentlyeight different observables provide useful information: Vcb phase space corrected rate in B D*l extrapolated for w1 excl: branching fraction at (4S) & Z0 incl: affected by non-Gaussian uncertainties Vub branching fraction at (4S) excl: branching fraction at (4S) & Z0 incl: G. Eigen, U Bergen

  10. Observables theoretical parameters with large non-Gaussian errors mBd • In the future additional measurements will be added, such as sin 2,  and the K+ + & KL 0 branching fractions QCD parameters that have small non Gaussian errors (except 1) K account for correlation of mc in 1 & S0(xc) mBs  from CP asymmetries in bccd modes G. Eigen, U Bergen

  11. Measurements • B(bul) = (2.03±0.22exp±0.31th)10-3 (4S) • B(bul) = (1.71±0.48exp±0.21th)10-3LEP • B(bcl) = 0.1070±0.0028 (4S) • B(bcl) = 0.1042±0.0026 LEP • B(Bl) = (2.68±0.43exp±0.5th)10-3CLEO/BABAR B0-l+ • |Vcb |F(1)=0.0388±0.005±0.009 LEP/CLEO/Belle • mBd= (0.503±0.006) ps-1 world average • mBs > 14.4 ps-1 @95%CL LEP • |K |= (2.271±0.017)10-3CKM workshop Durham • sin 2 = 0.731±0.055 BABAR/Belle aCP(KS) average • =0.2241±0.0033 world average • For other masses and lifetimes use PDG 2002 values G. Eigen, U Bergen

  12. Theoretical Parameters • Theoretical parameters that affect Vub and Vcb • Loop parameters • QCD parameters G. Eigen, U Bergen

  13. Scan of Theoretical Uncertainties in Vub • Check effect of scanning theoretical uncertainties for individual parameters between ±2th Vub exclusive Vub inclusive 1th to 2th -2th to -1th ±1th • Vub exclusive measured in B0 -l+ and Vub inclusive measured in B Xul+ are barely consistent effect of quark-hadron duality or experiments? G. Eigen, U Bergen

  14. Study Correlations among Theoretical Parameters • Perform global fits with either exclusive Vub/Vcb or inclusive Vub/Vcband plot Vubvs fBBBvs BK for different conditions • 1. solid black outermost contour: fit probability > 32 % • 2. next solid black contour: restrict the other undisplayed theoretical parameters to their allowed range • 3. colored solid line: fix parameter orthogonal to plane to allowed range • 4. colored dashed line: fix latter parameter to central value • 5. dashed black line: fix all undisplayed parameters to central values Vub/Vcb exclusive Vub/Vcb inclusive G. Eigen, U Bergen

  15. Present Status of the Unitarity Triangle Contour of individual fit central values from individual fits to models Overlay of 95% CL contours, each represents a “model” • Range of - values resulting from fits to different models G. Eigen, U Bergen

  16. Comparison of Results in the 3 Methods 0.067 0.020 0.024 0.1 2.6 16.6 3.3 G. Eigen, U Bergen

  17. Comparison of different Fit Methods RFit Bayesian approach add theoretical & experimental errors in quadrature Scan method upper limit of 95%CL upper limits overlay of 95% CL contours G. Eigen, U Bergen

  18. Comparison of Results in the 3 Methods • Use inputs specified after CKM workshop at CERN • Listed are 95% CL ranges G. Eigen, U Bergen

  19. Differnces in the 3 Methods • There are big differences wrt the “Bayesian Method”, both conceptually and quantitatively (treatment of non-Gaussian errors).  We assume no statistical distribution for non-Gaussian theoretical uncertainties  The region covered by contours of the scanning method in space is considerably larger than the region of Bayesian approach • Rfit “scans” finding a solution in the theoretical parameter space  In Rfit the central range has equal likelihood, but no probability statements can be made for individual points  In the scanning method the individual contours have statistical meaning: a center point exist which has the highest probability • The mapping of to is not one-to-one In the scan method one can track which values of are preferred by the theoretical parameters G. Eigen, U Bergen

  20. Include CP asymmetry in BKS • BKs is an bsss penguin decay • In SM aCP(Ks)=aCP(Ks), but it may be different in other models • Present BABAR/Belle measurements yield for sine term in CP asymmetry • Parameterize aCP(Ks) by including additional phase s in global fit s  • For present measurements the - plane is basically unaffected • The discrepancy between a(Ks) and a(Ks) is absorbed by s • With present statistics s is consistent with 0 G. Eigen, U Bergen

  21. Model-independent analysis of UT • In the presence of new physics: • i) • remain primarily tree level • ii) there would be a new contribution to K-K mixing • constraint: small • iii) unitarity of the 3 generation CKM matrix is maintained • if there are no new quark generations • Under these circumstances new physics effects can be described by two parameters: • Then, e.g.: G. Eigen, U Bergen

  22. Model independent Analysis: rd-d • Using present measurements d rd • The introduction of phase d weakens effect of sin 2 measurement  contours exceed both upper & lower bounds • The introduction of scale factor weakens md & ms bounds letting the fits extend into new region G. Eigen, U Bergen

  23. Conclusions • The scanning method provides a conservative, robust method with a reasonable treatment of non-gaussian theoretical uncertainties This allows to to avoid fake conflicts or fluctuations • This is important for believing that any observed significant discrepancyis real indicating the presence of New Physics • In future fits another error representing discrepacies in the quark-hadron duality may need to be included • The scan methods yields significantly larger ranges for  &  than the Bayesian approach • Due to the large theoretical uncertainties all measurements are consistent with the SM expectation • The deviation of aCP(KS) from aCP(KS) is interesting but not yet significant • Model-independent parameterizations will become important in the future G. Eigen, U Bergen

  24. Future Scenario • Use precision of measured quantities and theoretical uncertainties specified in previous table expected by 2011 • In addition, branching ration for Bl was increased by ~25% and |Vcb|F(1) was decreased by ~4% since otherwise no fits with P(2)>5% are found! • In this example, preferred - region is disjoint with sin 2 band from a(Ks) d rd G. Eigen, U Bergen

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