, EPS03G. Eigen, U Bergen2 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. B 0 d B 0 d and B 0 s B 0 s mixing   CP violation in the B system,  CP asymmetry of B  J/  K s   Rare B decays B  X s 

, EPS03G. Eigen, U Bergen3 The Cabbibo-Kobayashi-Maskawa Matrix   A convenient representation of the CKM matrix is the small-angle Wolfenstein approximation to order O( 6 )   The unitarity relation that represents a triangle (called Unitarity Triangle) in the  -  plane involves all 4 independent CKM parameters, A, , and    =sin  c =0.22 is best-measured parameter (1.5%), A .8 (~5%) while  -  are poorly known with and

, EPS03G. Eigen, U Bergen4 Global Fit Methods   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 approach F. Gilman, K. Kleinknecht and D. Renker   Present inputs for determinig the UT are based on measurements of B semi-leptonic decays,  m d,  m s, & |  K | that specify the sides and the CP asymmetry a cp (  K S ) 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 ), B K, f B  B B, 

, EPS03G. Eigen, U Bergen5 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

, EPS03G. Eigen, U Bergen6 The Scanning Method   For a particular set of theoretical parameters (called a model M) we perform a  2 minimization to determine   Here 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(  2 M ) 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

, EPS03G. Eigen, U Bergen7 Treatment of  m s   Ciuchini et al use  2 term (A-1) 2 /  A 2 -A 2 /  A 2 to include limit on  m s 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  m s with truncated at 0 For A n = ,  l =0.055 ps and reproduces 95% CL limit of reproduces 95% CL limit of 14.4 ps -1

, EPS03G. Eigen, U Bergen8 The  2 Function in Standard Model

, EPS03G. Eigen, U Bergen9 Observables   Presently eight different observables provide useful information:     V cb affected by non-Gaussian non-Gaussianuncertainties V ub phase space corrected rate in B  D * l extrapolated for w  1 excl: incl: excl: incl: branching fraction at  (4S) & Z 0 branching fraction at  (4S) branching fraction at  (4S) & Z 0

, EPS03G. Eigen, U Bergen10 Observables     In the future additional measurements will be added, such as sin 2 ,  and the K +   + & K L   0 branching fractions  m Bd  m Bs KKKK  from CP asymmetries in b  ccd modes theoretical parameters with large non-Gaussian errors QCD parameters that have small non Gaussian errors (except  1 ) account for correlation of m c in  1 & S 0 (x c )

, EPS03G. Eigen, U Bergen11 Measurements   B(b  ul ) = (2.03±0.22 exp ±0.31 th )   (4S)   B(b  ul ) = (1.71±0.48 exp ±0.21 th )  LEP   B(b  cl ) = ±  (4S)   B(b  cl ) = ± LEP   B(B  l ) = (2.68±0.43 exp ±0.5 th )  CLEO/BABAR B 0  - l +   |V cb |F(1)=0.0388±0.005±0.009 LEP/CLEO/Belle    m B d = (0.503±0.006) ps -1 world average    m B s > 14.4 ps LEP   |  K |= (2.271±0.017)  CKM workshop Durham   sin 2  = 0.731±0.055 BABAR/Belle a CP (  K S ) average   =0.2241± world average   For other masses and lifetimes use PDG 2002 values

, EPS03G. Eigen, U Bergen12 Theoretical Parameters   Theoretical parameters that affect V ub and V cb   Loop parameters   QCD parameters

, EPS03G. Eigen, U Bergen13 Scan of Theoretical Uncertainties in V ub  Check effect of scanning theoretical uncertainties for individual parameters between ±2  th individual parameters between ±2  th V ub inclusive  V ub exclusive measured in B 0   - l + and V ub inclusive measured in B  X u l + are barely consistent in B  X u l + are barely consistent effect of quark-hadron duality or experiments? effect of quark-hadron duality or experiments? V ub exclusive ±1  th -2  th to -1  th 1  th to 2  th

, EPS03G. Eigen, U Bergen14 Study Correlations among Theoretical Parameters   Perform global fits with either exclusive V ub /V cb or inclusive V ub /V cb and plot V ub vs f B  B B vs B K 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 V ub /V cb inclusive V ub /V cb exclusive

, EPS03G. Eigen, U Bergen15 Present Status of the Unitarity Triangle central values from individual fits to models  Range of  -  values resulting from fits to different models Contour of individual fit Overlay of 95% CL contours, each represents a “model”

, EPS03G. Eigen, U Bergen16 Comparison of Results in the 3 Methods ParameterScan Method  ,  =±  ,  =± A ,  =± mcmc ,  =± 0.1  ( ) 0,  =± 6.2  ( ) 0,  =± 5.4  ( ) 0,  =±

, EPS03G. Eigen, U Bergen17 Comparison of different Fit Methods Bayesian approach add theoretical & experimental errors in quadrature overlay of 95% CL contours upper limit of 95%CL upper limits RFit Scan method

, EPS03G. Eigen, U Bergen18 Comparison of Results in the 3 Methods PrmScan MethodRfitBayesian    ( ) 0  ( )  Listed are 95% CL ranges  Use inputs specified after CKM workshop at CERN

, EPS03G. Eigen, U Bergen19 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

, EPS03G. Eigen, U Bergen20 Include CP asymmetry in B  K S  B  K s is an b  sss penguin decay  In SM a CP (  K s )=a CP (  K s ), but it may be different in other models different in other models  Present BABAR/Belle measurements yield for sine term in CP asymmetry in CP asymmetry  Parameterize a CP (  K s ) by including additional phase  s in global fit  ssss  For present measurements the  -  plane is basically unaffected  The discrepancy between a(  K s ) and a(  K s ) is absorbed by  s  With present statistics  s is consistent with 0

, EPS03G. Eigen, U Bergen21   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 r r Under these circumstances new physics effects can be described by two parameters: Then, e.g.: Model-independent analysis of UT

, EPS03G. Eigen, U Bergen22 Model independent Analysis: r d -  d  The introduction of phase  d weakens effect of sin 2  measurement  contours exceed both upper & lower bounds measurement  contours exceed both upper & lower bounds  The introduction of scale factor weakens  m d &  m s bounds letting the fits extend into new region letting the fits extend into new region  Using present measurements rdrdrdrd dddd

, EPS03G. Eigen, U Bergen23 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  significant  This is important for believing that any observed significant discrepancy discrepancy is 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 a CP (  K S ) from a CP (  K S ) is interesting but not yet significant   Model-independent parameterizations will become important in the future

, EPS03G. Eigen, U Bergen24 Future Scenario  Use precision of measured quantities and theoretical quantities and theoretical uncertainties specified in uncertainties specified in previous table expected previous table expected by 2011 by 2011  In addition, branching ration for B  l ration for B  l was increased by ~25% was increased by ~25% and |V cb |F(1) was and |V cb |F(1) was decreased by ~4% decreased by ~4% since otherwise no fits since otherwise no fits with P(  2 )>5% are found! with P(  2 )>5% are found!  In this example, preferred  -  region is disjoint with  -  region is disjoint with sin 2  band from a(  K s ) sin 2  band from a(  K s ) rdrdrdrd dddd