G. Eigen, U Bergen

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. B0dB0d and B0sB0s mixing CP violation in the B system, CP asymmetry of BJ/Ks Rare B decays BXs

G. Eigen, U Bergen

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

=sinc=0.22 is best-measured parameter (1.5%), A.8 (~5%) while - are poorly knownwithand

G. Eigen, U Bergen

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.Hcker 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,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(bu,cl), BK, fBBB,

G. Eigen, U Bergen

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

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(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

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 withtruncated at 0For An=12.4667, l=0.055 ps and reproduces 95% CL limit of14.4 ps-1

G. Eigen, U Bergen

The 2 Function in Standard Model

G. Eigen, U Bergen

Observables Presently eight different observables provide useful information:

Vcbaffected by non-GaussianuncertaintiesVubphase space corrected rate inB D*l extrapolated for w1excl:incl:excl:incl:branching fractionat (4S) & Z0branching fractionat (4S)branching fractionat (4S) & Z0

G. Eigen, U Bergen

Observables

In the future additional measurements will be added, such as sin 2, and the K+ + & KL 0 branching fractionsmBdmBsKfrom CP asymmetriesin bccd modestheoretical parameters with large non-Gaussian errorsQCD parameters that have smallnon Gaussian errors (except 1) account for correlation of mc in 1 & S0(xc)

G. Eigen, U Bergen

Measurements B(bul) = (2.030.22exp0.31th)10-3 (4S) B(bul) = (1.710.48exp0.21th)10-3 LEP B(bcl) = 0.10700.0028 (4S) B(bcl) = 0.10420.0026 LEP B(Bl) = (2.680.43exp0.5th)10-3 CLEO/BABAR B0-l+ |Vcb |F(1)=0.03880.0050.009 LEP/CLEO/Belle mBd= (0.5030.006) ps-1 world average mBs > 14.4 ps-1 @95%CL LEP |K |= (2.2710.017)10-3 CKM workshop Durham sin 2 = 0.7310.055 BABAR/Belle aCP(KS) average =0.22410.0033 world average For other masses and lifetimes use PDG 2002 values

G. Eigen, U Bergen

Theoretical Parameters Theoretical parameters that affect Vub and Vcb Loop parameters QCD parameters

G. Eigen, U Bergen

Scan of Theoretical Uncertainties in Vub Check effect of scanning theoretical uncertainties for individual parameters between 2thVub inclusive Vub exclusive measured in B0 -l+ and Vub inclusive measured in B Xul+ are barely consistent effect of quark-hadron duality or experiments?Vub exclusive1th-2th to -1th 1th to 2th

G. Eigen, U Bergen

Study Correlations among Theoretical Parameters Perform global fits with either exclusive Vub/Vcb or inclusive Vub/Vcb and plot Vubvs fBBB vs 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 inclusiveVub/Vcb exclusive

G. Eigen, U Bergen

Present Status of the Unitarity Trianglecentral valuesfrom individualfits to models Range of - values resulting from fits to different modelsContour ofindividual fitOverlay of 95% CLcontours, each represents a model

G. Eigen, U Bergen

Comparison of Results in the 3 Methods0.0670.0200.0240.12.616.63.3

ParameterScan Method0.103-0.337, =0.0260.280-0.409, =0.034A0.80-0.85, =0.027mc1.27-1.32, =0.1(20.7-26.9)0, =6.2 (83.1-116.3)0, =5.4 (40.5-74.5)0, =8.2

G. Eigen, U Bergen

Comparison of different Fit Methods Bayesian approachadd theoretical & experimental errors in quadratureoverlay of 95% CL contoursupper limitof 95%CLupper limitsRFitScan method

G. Eigen, U Bergen

Comparison of Results in the 3 Methods Listed are 95% CL ranges Use inputs specified after CKM workshop at CERN

PrmScan MethodRfit Bayesian-0.143-0.394 0.091-0.3170.137-0.2950.230-0.4870.273-0.4080.295-0.409 (51.0-126.9)0(32.7-109.6)0 42.1-75.747-70.0

G. Eigen, U Bergen

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

Include CP asymmetry in BKS BKs is an bsss 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

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.:

Model-independent analysis of UT

G. Eigen, U Bergen

Model independent Analysis: rd-d 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 Using present measurementsrdd

G. Eigen, U Bergen

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 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 aCP(KS) from aCP(KS) is interesting but not yet significant Model-independent parameterizations will become important in the future

G. Eigen, U Bergen

Future Scenario Use precision of measured quantities and theoretical uncertainties specified in previous table expected by 2011

In addition, branching ration for Bl 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)rdd

G. Eigen, U Bergen