On behalf of M. Bona, G. Eigen, R. Itoh and E. Kou

α,φ 2 γ,φ 3 β,φ 1 On behalf of M. Bona, G. Eigen, R. Itoh and E. Kou

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: β, α (which decays to consider), γ,, 2β+γ, 2, V ub, V cb, Δm d, A d SL, B(B τν τν),radiative penguins (how to use them) 4p Subsection: Non-B-factories results (briefly on their threatment): ε K, Δm s, A s SL, TD B s J/ J/ψφ, ΔΓ s (with order calculation). 2-3p Rather than having subsubsections we indicate in the table which are inputs for the SM fit and inputs i 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 2

Motivation The CKM matrix is specified by 4 independent parameters, in the Wolfenstein approximation they are λ, A, ρ, and η % 1! 1 2 "2! 1 8 "4 " A" 3 (#! i$) ( ' * V CKM = '!" + A 2 " 5 ( 1 2! #! i$) 1! 1 2 "2! 1 8 "4! 1 2 A2 " 4 A" 2 * + O(" 6 ) ' * ' A" 3 (1! #! i$ )!A" 2 + A" 4 ( 1 2! #! i$) 1! 1 2 A2 " & 4 * ) Unitarity of the CKM matrix specifies relations among the parameters e.g. V ud V * ub + V cd V * cb + V td V * (ρ, η) tb = 0 α,φ 2 Combine measurements from the R u = V V * ud ub R t = V V * td tb * * V B and K systems to overconstrain cd V V cd V cb cb the triangle test if phase of CKM matrix γ,φ 3 1 β,φ 1 is only source of CP violation $!," 2 = arg # V V * ' td tb * % & V ud V ub ( ) $!," 1 = arg # V V * ' cd cb * % & V td V tb ( ) $!," 3 = arg # V V * ' ud ub * % & V cd V3 cb ( )

Motivation In the SM in the absence of errors all measurements of UT properties exactly meet in ρ and η The extraction of ρ, η depends on QCD parameters that have large theory uncertainties We use 3 different fit methods: CKMfitter, UTfit and Scanning method which differ in the treatment of QCD parameters 4

CKMfitter Methodology CKMfitter (Rfit)) is a frequentist-based approach to the global fit of CKM matrix Likelihood function: L[y mod ] = L exp [x exp! x th (y mod )] " L th [y QCD ] First term measures agreement between data, x exp, and prediction, x th Second term expresses our present knowledge on QCD parameters y mod are a set of fundamental and free parameters of theory (m( t, etc) Minimize and determine! 2 (y mod ) " #2 ln(l[y mod ])!" 2 (y mod ) = " 2 2 (y mod ) # " min;ymod where χ 2 min;ymod mod is the absolute minimum value of χ 2 function Separate uncertainties of QCD parameters into statistical (σ)( ) and non-statistical (theory) uncertainties (δ)( statistical uncertainties are treated like experimental errors with a Gaussian likelihood 5

Rfit Methodology Treatment of theory uncertainties (δ): If fitted parameter a lies within the predicted range x 0 ±δx 0 contribution to χ 2 is zero If fitted parameter a lies outside the predicted range x 0 ±δx 0 the likelihood L th [ y QCD ] drops rapidly to zero, define:!2 ln L th [x 0,",#] = 0, $x 0 % ' x 0 ± #&x ) ( 0 * + x 0! x. 0, - "&x 0 / 0 2 +! #. - 0, " / 2, $x 0 1 ' x 0 ± #&x ( 0 ) * 3 different analysis goals Within SM achieve best estimate of y th Within SM set CL that quantifies agreement between data and theory Within extended theory framework search for specific signs of new physics 6

UTfit Methodology UTfit is a Bayesian-based approach to the global fit of CKM matrix For M measurements c j that depend on ρ and η plus other N parameters x i the function f(!, ", x 1,...x N c 1,...c M ) needs to be evaluated by integrating over x i and c j Using Bayes theorem one finds $ f(!, ", x 1,...x N c 1,...c M ) # f j (c j!, ", x 1,..., x N ) f i (x i )f 0 (!, ") j=1,m where f 0 (ρ, η) is the a-priory probability for ρ and η $ i=1,n The output pdf for ρ and η is obtained by integrating over c j and x i 7

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

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, ξ 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 ± σ x ±δ x, where σ is a statistical uncertainty and δ x is the theory uncertainty We select a specific value x * [x 0 - δ x, x 0 + δ x ] as a model We consider all models inside the [x 0 - δ x, x 0 + δ x ] interval In each model the uncertainty σ q is treated in a statistical way The uncertainties in the QCD parameters η cc, η ct, η tt, and η 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 9

The Scanning Method We perform maximum likelihood fits using a frequentist approach A model is considered consistent with data if P(χ 2 M ) min > 5% For consistent models we determine the best estimate and plot a 95% CL (ρ, η) 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 α, φ 2 and γ, φ 3 via a likelihood function or directly using individual B ππb ππ, ρπ, ρρ,, a 1 π,, b 1 π measurements and GLW, ADS and Dalitz plot measurements in B D (*) K (*) & sin(2β+γ), respectively We can further determine PP, PV VV amplitudes and strong phases using Gronau and Rosner parameterizations in powers of λ Work is in progress to include cos 2β, β s, A q SL, ΔΓ s and τ s add contours of sin 2α, γ and sin (2β+γ) and improve on display 10

Differences among the 3 Methods Fit methodologies differ: 2 frequentist approaches vs 1 Bayesian approach Theory uncertainties are treated differently in the global fits Presently, measurement input values differ plus some assumptions differ For V ub and V cb there is an issue how to 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 Bd, ξ f Bd B Bd, ξ Bs /f Bd Bd, B Bs Bs /B Bd We need to standardize measurement inputs, QCD parameters (at least numerical values should agree) and assumptions 11

Input Measurements from B Factories V ub and V cb measured in exclusive and inclusive semileptonic B decays Δm d from B d B d oscillations CP asymmetries a cp (ψk S ) from B cck s decays angle β α from B ππ, B ρρ, & B ρπ CP measurements, add B a 1 π, B b 1 π γ β r u Δm d sin2α GLW, ADS and GGSZ analyses in B D (*) K (*) 12

Input Measurements from B Factories sin(2β+γ) measurement from B D (*) π(ρ) cos 2 β from B J/ψK * and B D 0 π 0 B τν branching fraction sin(2α+ γ) cosβ B(B τν τν) 13

Other Input Measurements ε K from CP violation in K decays ε K Δm d /Δm s from B d B d and B s B s oscillations β s - ΔΓ s from B s measurements at the Tevatron Δm d /Δm s CKM elements V ud, V us, V cd, V cs, V tb 14

Measurement Inputs Observable V us V ub V cb ub [10-3 sin 2β2 α[ ππ, ρπ, ρρ] γ [GGSZ, GLW, ADS] cos 2β 2β+γ -3 ] cb [10-3 -3 ] B(B τν τν) ) [10-4 Δm [ps-1-1 Bd ] Δm [ps-1-1 Bs ] ε K [10-3 ] -4 ] CKMfitter 0.2246±0.0012 3.79±0.09±0.41* 40.59±0.37±0.58* 1.73±0.35 0.507±0.005 17.77±0.12 2.229±0.010 0.671±0.023 1-CL(α) 1-CL(γ) J/ψK* D (*) (*) π(ρ) UTfit 0.2259±0.0009 4.11 +0.27-0.28 (inc) 3.38±0.36 (exc) 41.54±0.73 (inc) 38.6±1.1 (exc( exc) 1.51±0.33 0.507±0.005 17.77±0.12 2.229±0.010 0.671±0.023-2Δln ln(l) -2Δln ln(l) J/ψK*, D 0 π 0 Scanning M 0.2258±0.0021 3.84±0.16±0.29 (ex 40.9±1.0±1.6 (exc( exc) 1.79±0.72 0.508±0.005 17.77±0.12 2.232±0.007 0.68±0.025 B, S, C for ππ & ρρ GGSZ, GLW, ADS To be done -2Δln ln(l) D (*) π(ρ) * use average values 15

Parameter f Bs [f Bd f Bs /f B K α s /f Bd Bd ] B [B Bs Bd ] B Bs /B [ξ] Bd B K [2 GeV] m c (m c ) [GeV[ GeV] Mean Lattice QCD Inputs CKMfitter σ stat δ theo 228 ±3 ±17 1.199 ±0.008 ±0.023 1.23 ±0.03 ±0.05 1.05 ±0.02 ±0.05 0.525 ±0.0036 ±0.052 0.721 ±0.005 ±0.040 1.286 ±0.013 ±0.040 0.1176±0.0020 UTfit Mean σ 245 ±25 1.21 ±0.04 1.22 ±0.12 1.00 ±0.03 0.75 ±0.07 1.3 ±0.1 0.119 ±0.03 Scanning Mean σ stat [216 ±10 ± 20] δ theo [1.29 ±.05 ±.08] [1.2 ±.028 ±.05] 0.79 ±0.04 ±0.09 1.27±0.11 m t (m t ) [GeV[ GeV] 165.02 ±1.16 ±0.11 161.2 ±1.7 163.3 ±2.1 η cc Calculated from m c (m c ) & α s 1.38 ±0.53 1.46±0.22 η ct 0.47±0.04 0.47 ±0.04 0.47±0.04 η tt 0.5765±0.0065 0.574 ±0.004 0.5765±0.0065 η B (MS) 0.551±0.007 0.55 ±0.01 0.551±0.007 0.118 16

Inputs: V ud, V us V cb, V ub B(B τν) ε K us ub Global Fit Results Δm Bd, Δm Bs sin 2β, 2, 1-CL(α), 1-CL(γ) 2008 inputs Note scales are not the same! 17

Tension from B(B τν) B(B τν) is proportional to Vub 2 and f2bd from global fit B(B τν)=(0.79+0.016 ) 10-4 -0.010 WA: B(B τν)=(1.73±0.35) 10-4 2.4σ discrepancy direct measurement If B(B τν) or sin 2β are removed χ2min in global fit drops by 2.4 Vub, Vcb remain unaffected allowed region by fit 18

Constraints in the m H -tanβ Plane From BABAR/Belle average we extract r H = 1.67 ± 0.34 exp ± 0.36 fb,v ub % r H = ' 1! & ' 2 m ( B 2 (1 + " 0 # tan $) tan2 $ * ) * m H + 95% C.L. exclusions 2 H + We can use the 95% CL to present exclusions at 95% CL in the m H+ -tanβ plane m H Dark matter 1.2< 10 9 Δa <4.6 µ r H 95%CL B X s γ B τν K µν µν Tevatron tan β/m H ATLAS 5σ 5 discovery curve LEP tan β 19

Model-Independent Analysis of UT Assume that new physics only affects short-distance part of ΔB=2 We use model-independent parameterization for B d and B s B 0 full q H!B =2 B 0 SM q H!B =2 where H full = H SM + H NP B q 0 B q 0 =! NP q =! NP NP q exp { 2i" q } Several observables are modified by the magnitude or phase of Δ NP q In B d system we compare R u & γ with sin 2β, 2, sin2α and Δm d that may be modified by NP parameters (Δ d, φ d ) In B s system we compare R u & γ with Δm s, β s, and ΔΓ s that may be modified by NP parameters (Δ( d, φ d ) 20

Model-Independent Analysis of UT for Δ d - φ d Inputs: Δm d, Δm s, sin 2β, 2 α, ΔΓ d, A Bd SL, A Bs SL, w/o B(B τν) Dominant constraints come from β and Δm d Semileptonic asymmetries A SL exclude symmetric solution with η<0 Δ d =1 (SM) is disfavored by 2.1σ (discrepancy B(B τν) ) and sin 2β) 2 φ NP d =(-12 +9-6 )0 @ 95% CL ( discrepancy( is 0.6σ w/o B(B τν) ) ) +9-6 Δ d 21

Model-Independent Analysis of UT for Δ s - φ s Inputs: φ s, Δm d, Δm s, A Bd SL, A Bs SL, ΔΓ s, τ s, B(B τν) Δ s Dominant constraints come from direct measurements of φ s, ΔΓ s in B J/ B J/ψφ and Δm s from the Tevatron φ s is 2.2σ away from the SM prediction Δ s =1 is disfavored at 1.9σ level independent of B(B τν) 22

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 23

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 Lett.97:151803, 2006. M Bona et al., Phys.Lett Lett.B687:61-69, 2010. Scanning method G. Eigen et al., Eur.Phys.J.C33:S644-S646,2004. G.P. Dubois-Felsmann et al., hep-ph/0308262. 24