1. ,1,1,1,1 ,2,2,2,2 ,3,3,3,3 On behalf of M. Bona, G. Eigen, R. Itoh On behalf of M. Bona, G. Eigen, R. Itoh and E. Kou

2. G. Eigen, Vxb workshop, SLAC, October 31 2009 2 Chapter Outline Section: Introduction and goals 2p Section: Methodology Subsection: CKMfitter 2p Subsection: UTfit 2p Subsection: Scanning method 2p Section: Experimental Inputs Subsection: B-factories results: ,  (which decays to consider), , 2  + ,, V ub, V cb,  m d, A d SL, B(B  ),radiative penguins (how to use them) 4p Subsection: Non-B-factories results (briefly on their threatment):  K,  m s, A s SL, TD B s  J/ ,  s (with order calculation). 2-3p Rather than having subsubsections we indicate in the table which are inputs for the SM fit and inputs for the BSM fits Section Theoretical Inputs Subsection Derivation of hadronic observables 2p Subsection Lattice QCD inputs 4p Benchmark models 5p Section Results from the global fits Section Global fits beyond the Standard Model 4p Subsection New-physics parameterizations 4p Subsection Operator analysis 2p Section Conclusions 1-2p total: 36-38 pages total: 36-38 pages

3. G. Eigen, Vxb workshop, SLAC, October 31 2009 3 Motivation The CKM matrix is specified by 4 independent parameters, The CKM matrix is specified by 4 independent parameters,  in the Wolfenstein approximation they are  in the Wolfenstein approximation they are, A, , and  Unitarity of the CKM matrix specifies relations among the parameters e.g. e.g. Combine measurements from the Combine measurements from the B and K systems to overconstrain B and K systems to overconstrain the triangle the triangle  test if phase of CKM matrix  test if phase of CKM matrix is only source of CP violation is only source of CP violation ,1,1,1,1 ,2,2,2,2 ,3,3,3,3 1 ( ,  )

4. G. Eigen, Vxb workshop, SLAC, October 31 2009 4 Motivation In the SM in the In the SM in the absence of errors absence of errors all measurements of all measurements of UT properties exactly UT properties exactly meet in meet in  and  The extraction of The extraction of ,  depends on QCD depends on QCD parameters that parameters that have large theory have large theory uncertainties uncertainties We use 3 different We use 3 different fit methods: fit methods: CKMfitter, UTfit CKMfitter, UTfit and Scanning method and Scanning method which differ in the which differ in the treatment of QCD parameters treatment of QCD parameters

5. G. Eigen, Vxb workshop, SLAC, October 31 2009 5 CKMfitter Methodology CKMfitter (Rfit) is a frequentist-based approach to the global fit of CKMfitter (Rfit) is a frequentist-based approach to the global fit of CKM matrix CKM matrix Likelihood function: Likelihood function:  First term measures agreement between data, x exp, and prediction, x th  First term measures agreement between data, x exp, and prediction, x th  Second term expresses our present knowledge on QCD parameters  Second term expresses our present knowledge on QCD parameters  y mod are a set of fundamental and free parameters of theory (m t, etc)  y mod are a set of fundamental and free parameters of theory (m t, etc) Minimize Minimize and determine and determine where  2 min;y mod is the absolute minimum value of  2 function where  2 min;y mod is the absolute minimum value of  2 function Separate uncertainties of QCD parameters into statistical (  ) and non-statistical (theory) uncertainties (  ) non-statistical (theory) uncertainties (  )  statistical uncertainties are treated like experimental errors with  statistical uncertainties are treated like experimental errors with a Gaussian likelihood a Gaussian likelihood

6. G. Eigen, Vxb workshop, SLAC, October 31 2009 6 Rfit Methodology Treatment of theory uncertainties (  ): Treatment of theory uncertainties (  ):  If fitted parameter a lies within the predicted range x 0 ±  x 0  If fitted parameter a lies within the predicted range x 0 ±  x 0 contribution to  2 is zero contribution to  2 is zero  If fitted parameter a lies outside the predicted range x 0 ±  x 0  If fitted parameter a lies outside the predicted range x 0 ±  x 0 the likelihood L th [ y QCD ] drops rapidly to zero, define: the likelihood L th [ y QCD ] drops rapidly to zero, define: 3 different analysis goals Within SM achieve best estimate of y th Within SM achieve best estimate of y th Within SM set CL that quantifies Within SM set CL that quantifies agreement between data and theory agreement between data and theory Within extended theory framework search for specific signs of Within extended theory framework search for specific signs of new physics new physics

7. G. Eigen, Vxb workshop, SLAC, October 31 2009 7 UTfit Methodology UTfit is a Bayesian-based approach to the global fit of CKM matrix UTfit is a Bayesian-based approach to the global fit of CKM matrix For M measurements c j that depend on For M measurements c j that depend on  and  plus other N parameters x i the function needs to be evaluated by integrating over x i and c j Using Bayes theorem one finds where f 0 ( ,  ) is the a-priory probability for  and  The output pdf for  and  is obtained by integrating over c j and x i

8. G. Eigen, Vxb workshop, SLAC, October 31 2009 8 UTfit Methodology Measurement inputs and all theory parameters are described by pdfs Errors are typically treated with a Gaussian model, only for B k,  Errors are typically treated with a Gaussian model, only for B k,  and f B √B B a flat distribution representing the theory uncertainty is and f B √B B a flat distribution representing the theory uncertainty is convolved with a Gaussian representing the statistical uncertainty convolved with a Gaussian representing the statistical uncertainty So if available, experimental inputs are represented by likelihoods So if available, experimental inputs are represented by likelihoods The method does not make any distinction between measurement and The method does not make any distinction between measurement and theory parameters theory parameters The allowed regions are well defined in terms of probability  allowed regions at 95% probability means that you expect the  allowed regions at 95% probability means that you expect the “true” value in this range with 95% probability “true” value in this range with 95% probability By changing the integration variables any pdf can be extracted By changing the integration variables any pdf can be extracted  this yields an indirect determination of any interesting quantity  this yields an indirect determination of any interesting quantity

9. G. Eigen, Vxb workshop, SLAC, October 31 2009 9 The Scanning Method The basis is the original approach by M.H. Schune & S. Plaszczynski used for the BABAR physics book The fit method was extended to include over 250 single measurements The four QCD parameters B k, f B, B B,  and V ub, V cb, have significant theory uncertainties, thus they are scanned in the following way We express each parameter in terms of x 0 ±  x ±  x, where  is a statistical uncertainty and  x is the theory uncertainty We select a specific value x *  [x 0 -  x, x 0 +  x ] as a model We consider all models inside the [x 0 -  x, x 0 +  x ] interval In each model the uncertainty  q is treated in a statistical way The uncertainties in the QCD parameters  cc,  ct,  tt, and  B and the quark masses m c (m c ) and m b (m b ) are treated like statistical uncertainties, since these uncertainties are relatively small  however, if necessary, we can scan over any of these parameters

10. G. Eigen, Vxb workshop, SLAC, October 31 2009 10 The Scanning Method We perform maximum likelihood fits using a frequentist approach A model is considered consistent with data if P(  2 M ) min > 5% For consistent models we determine the best estimate and plot a 95% CL ( ,  ) contour  we overlay contours of consistent models  however, though only one of the contours is the correct one, we do not know which and thus show a representative numebr of them For accepted fits we also study the correlations among the theoretical parameters extending their range far beyond the range specified by the theorists We can input ,  2 and ,  3 via a likelihood function or directly using We can input ,  2 and ,  3 via a likelihood function or directly using individual B , , , a 1 , b 1  measurements and individual B , , , a 1 , b 1  measurements and GLW, ADS and Dalitz plot measurements in B  D (*) K (*) & sin(2  +  ), respectively We can further determine PP, PV VV amplitudes and strong phases using Gronau and Rosner parameterizations in powers of Work is in progress to include cos 2 ,  s, A q SL,  s and  s add contours of sin 2 ,  and sin (2  +  ) and improve on display

11. G. Eigen, Vxb workshop, SLAC, October 31 2009 11 Differences among the 3 Methods Fit methodologies differ: 2 frequentist Fit methodologies differ: 2 frequentist approaches vs 1 Bayesian approach approaches vs 1 Bayesian approach Theory uncertainties are treated Theory uncertainties are treated differently in the global fits differently in the global fits Presently, measurement input values differ Presently, measurement input values differ plus some assumptions differ plus some assumptions differ For V ub and V cb there is an issue how to For V ub and V cb there is an issue how to combine inclusive and exclusive results combine inclusive and exclusive results Inclusive/exclusive averages  individual results Resulting errors QCD parameters inputs differ, eg f Bs, B Bs, f Bs /f Bd, B Bs /B Bd  QCD parameters inputs differ, eg f Bs, B Bs, f Bs /f Bd, B Bs /B Bd  f Bd, B Bd,   f Bd  B Bd,  f Bd, B Bd,   f Bd  B Bd,   We need to standardize measurement inputs, QCD parameters (at least numerical values should agree) and assumptions (at least numerical values should agree) and assumptions

12. G. Eigen, Vxb workshop, SLAC, October 31 2009 12 Input Measurements from B Factories V ub and V cb measured in exclusive and inclusive semileptonic B decays  m d from B d B d oscillations CP asymmetries a cp (  K S ) from B  ccK s decays  angle   from B , B , & B  CP measurements, add B  a 1 , B  b 1  rurururu  mdmdmdmd sin2   GGSZ GLW, ADS and GGSZ analyses in B  D (*) K (*)

13. G. Eigen, Vxb workshop, SLAC, October 31 2009 13 Input Measurements from B Factories sin(2  +  ) measurement from B  D (*)  (  ) cos 2  from B  J/  K * and B  D 0  0 B  branching fraction sin(2  +  ) cos  B(B  )

14. G. Eigen, Vxb workshop, SLAC, October 31 2009 14 Other Input Measurements |  K | from CP violation in K decays  m d /  m s from B d B d and B s B s oscillations  s -  s from B s measurements at the Tevatron CKM elements V ud, V us, V cd, V cs, V tb |K||K||K||K| md/msmd/msmd/msmd/ms

15. G. Eigen, Vxb workshop, SLAC, October 31 2009 15 Measurement Inputs * use average values * use average values ObservableCKMfitterUTfit Scanning M |V us | 0.2246±0.00120.2259±0.00090.2258±0.0021 |V ub | [10 -3 ] 3.79±0.09±0.41* 4.11 +0.27 -0.28 (inc) 3.38±0.36 (exc) 3.84±0.16±0.29 (ex |V cb | [10 -3 ] 40.59±0.37±0.58* 41.54±0.73 (inc) 38.6±1.1 (exc) 40.9±1.0±1.6 (exc) B(B  ) [10 -4 ] 1.73±0.351.51±0.331.79±0.72  m B d [ps -1 ] 0.507±0.0050.507±0.0050.508±0.005  m B s [ps -1 ] 17.77±0.1217.77±0.1217.77±0.12 |  K | [10 -3 ] 2.229±0.0102.229±0.0102.232±0.007 sin 2  0.671±0.0230.671±0.0230.68±0.025  [ , ,  ] 1-CL(  ) -2  ln( L ) B, S, C for  &   [GGSZ, GLW, ADS] 1-CL(  ) -2  ln( L ) GGSZ, GLW, ADS cos 2  J/  K* J/  K*, D 0  0 To be done 2+2+2+2+ D (*)  (  ) -2  ln( L ) D (*)  (  )

16. G. Eigen, Vxb workshop, SLAC, October 31 2009 16 Lattice QCD Inputs Parameter CKMfitter CKMfitter UTfit UTfitScanning Mean  stat  theo Mean  Mean  stat  theo f B s [f Bd ] 228 ±3 ±17 245 ±25 [216 ±10 ± 20] f B s /f B d 1.199 ±0.008 ±0.023 1.21 ±0.04 B B s [B Bd ] 1.23 ±0.03 ±0.05 1.22 ±0.12 [1.29 ±.05 ±.08] B B s /B B d [  ] 1.05 ±0.02 ±0.05 1.00 ±0.03 [1.2 ±.028 ±.05] B K [2 GeV] 0.525 ±0.0036 ±0.052 BKBKBKBK 0.721 ±0.005 ±0.040 0.75 ±0.07 0.79 ±0.04 ±0.09 m c (m c ) [GeV] 1.286 ±0.013 ±0.040 1.3 ±0.1 1.27±0.11 m t (m t ) [GeV] 165.02 ±1.16 ±0.11 161.2 ±1.7 163.3 ±2.1  cc Calculated from m c (m c ) &  s 1.38 ±0.53 1.46±0.22  ct 0.47±0.04 0.47 ±0.04 0.47±0.04  tt 0.5765±0.0065 0.574 ±0.004 0.5765±0.0065  B (MS) 0.551±0.007 0.55 ±0.01 0.551±0.007 ssss0.1176±0.0020 0.119 ±0.03 0.118

17. G. Eigen, Vxb workshop, SLAC, October 31 2009 17 Global Fit Results Inputs: |V ud |, |V us | |V cb |, |V ub | |V cb |, |V ub | B (B  ) B (B  ) |  K | |  K |  m B d,  m B s  m B d,  m B s sin 2 , 1-CL(  ), 1-CL(  ) sin 2 , 1-CL(  ), 1-CL(  ) Note scales are not the same! Note scales are not the same! 2008inputs

18. G. Eigen, Vxb workshop, SLAC, October 31 2009 18 Tension from B (B  ) B (B  ) is proportional to |V ub | 2 and f 2 B d |V ub | 2 and f 2 B d from global fit from global fit B (B  )=(0.79 +0.016 )  10 -4 B (B  )=(0.79 +0.016 )  10 -4 WA: WA: B (B  )=(1.73±0.35)  10 -4 B (B  )=(1.73±0.35)  10 -4 2.4  discrepancy 2.4  discrepancy If B (B  ) or sin 2  are removed  2 min in global fit drops by 2.4  2 min in global fit drops by 2.4 |V ub |, |V cb | remain unaffected |V ub |, |V cb | remain unaffected direct measurement allowed region by fit -0.010

19. G. Eigen, Vxb workshop, SLAC, October 31 2009 19 Constraints in the m H -tan  Plane From BABAR/Belle average From BABAR/Belle average we extract we extract We can use the 95% CL to present exclusions at to present exclusions at 95% CL in the m H+ -tan  plane 95% CL in the m H+ -tan  plane B  1.2< 10 9  a  <4.6 BXsBXsBXsBXs Dark matter mHmHmHmH 95% C.L. exclusions Tevatron K  tan  LEP rHrHrHrH tan  /m H 95%CL H+H+H+H+ ATLAS 5  discovery curve

20. G. Eigen, Vxb workshop, SLAC, October 31 2009 20 Model-Independent Analysis of UT Assume that new physics only affects short-distance part of  B=2 We use model-independent parameterization for B d and B s We use model-independent parameterization for B d and B s where H full = H SM + H NP where H full = H SM + H NP Several observables are Several observables are modified by the magnitude modified by the magnitude or phase of  q NP or phase of  q NP In B d system we compare In B d system we compare R u &  with sin 2 , sin2  R u &  with sin 2 , sin2  and  m d that may be modified and  m d that may be modified by NP parameters (  d,  d ) by NP parameters (  d,  d ) In B s system we compare R u &  with  m s,  s, and  s that may be In B s system we compare R u &  with  m s,  s, and  s that may be modified by NP parameters (  d,  d ) modified by NP parameters (  d,  d )

21. G. Eigen, Vxb workshop, SLAC, October 31 2009 21 Model-Independent Analysis of UT for |  d |-  d Inputs:  m d,  m s, sin 2 , ,  d, A SL Bd, A SL Bs, w/o B (B  ) Dominant constraints come from  and  m d Dominant constraints come from  and  m d Semileptonic asymmetries A SL exclude symmetric solution with  <0 Semileptonic asymmetries A SL exclude symmetric solution with  <0  d =1 (SM) is disfavored by 2.1  (discrepancy B (B  ) and sin 2  )  d =1 (SM) is disfavored by 2.1  (discrepancy B (B  ) and sin 2  )   d NP =(-12 +9 -6 ) 0 @ 95% CL (  discrepancy is 0.6  w/o B (B  ) )   d NP =(-12 +9 -6 ) 0 @ 95% CL (  discrepancy is 0.6  w/o B (B  ) ) |d||d||d||d|

22. G. Eigen, Vxb workshop, SLAC, October 31 2009 22 Model-Independent Analysis of UT for |  s |-  s Inputs:  s,  m d,  m s, A SL Bd, A SL Bs,  s,  s, B (B  ) Dominant constraints come from direct measurements of  s,  s Dominant constraints come from direct measurements of  s,  s in B  J/  and  m s from the Tevatron in B  J/  and  m s from the Tevatron  s is 2.2  away from the SM prediction  s is 2.2  away from the SM prediction  s =1 is disfavored at 1.9  level independent of B (B  )  s =1 is disfavored at 1.9  level independent of B (B  ) |s||s||s||s|

23. G. Eigen, Vxb workshop, SLAC, October 31 2009 23 Final Remarks Among the 3 global CKM fitting methods we need to standardize All measurement inputs What QCD parameters to use, their central values, their statistical errors and their theory errors The notation for quantities in the text, on plots and in equations We need to specify which input parameters are used in the fits and standardize on the assumptions  This is important for comparing results We will present results in form of plots with values listed in tables  we accompany the results with a few remarks  in particular in cases of discrepancies we need to discuss them Most of the writing probably has to be done by the co-conveners

24. G. Eigen, Vxb workshop, SLAC, October 31 2009 24 More Recent Publications CKMfitter publications J. Charles et al., Eur. Phys. J. C41, 1-135, 2005. UTfit publications: M. Bona et al., JHEP 0610:081, 2006. M. Bona et al., Phys.Rev.D76:014015, 2007. M. Bona et al., Phys.Rev.Lett.97:151803, 2006. M Bona et al., Phys.Lett.B687:61-69, 2010. Scanning method Scanning method G. Eigen et al., Eur.Phys.J.C33:S644-S646,2004. G.P. Dubois-Felsmann et al., hep-ph/0308262.