Determination of Unitarity Triangle parameters Achille Stocchi LAL-Orsay Phenomenology Workshop on Heavy Flavours Ringberg Schloss 28 April – 2 May 2003

-Introduction (Unitarity Triangle) - Statistical method – Comparison (very brief) -The measurements/theoretical inputs - Results. Determination of the Unitarity Triangle parameters

All these topics extensively discussed at

~ 46 theorists ~ 52 experimentalists 98 authors Organised as a coherent document 330 pages THE CKM MATRIX AND THE UNITARITY TRIANGLE (work during 12 months between the 2 Workshops) hep-ph/ will be submitted as CERN-Yellow Book

(b  u)/(b  c) 2 + 2, λ 1,F(1),… KK [ ( 1– ) + P] BKBK mdmd (1– ) f B B B  m d /  m s (1– ) ξ A CP (J/ψ,K S ) sin (2β) - ρη η ρ ρ η ρη 2 Λ theories which give the link from quarks to hadrons OPE /HQET/Lattice QCD …. Need to be tested,m t Standard Model+

Short review on the inputs

b c l V cb b sl Br Fclb  )(    f(  2   m b,, m c  s,  D (or 1/m b 3  mbmb (  Fermi movement)( also named  ) 22 cb V 2 Based on OPE  sl = (0.434  (1 ± 0.018)) MeV V cb - Inclusive Method  2% Moments of distributions HADRONIC mass, LEPTON Momentum, Photon energy b  s  M b,kin (1GeV) = 4.59 ± 0.08 ± 0.01GeV m c,kin. (1GeV) = 1.13 ± 0.13 ± 0.03GeV   2 = 0.31 ± 0.07 ± 0.02GeV 2  D 2 = 0.05 ± 0.04 ± 0.01GeV 2  terms 1/m b 3 (under control?)/small !  4.23(mb(mb)) V cb (inclusive)= ( 41.4 ± 0.6 ± 0.7(theo.) ) Exp  2   m b,,  D ….absorbed !) Pert. QCD.  s, terms 1/m b 4 hep-ph/ C.Bauer,Z.Ligeti,M.Luke,A.Manohar hep-ph/ , M.Battaglia et al. (P.Gambino,N.Uraltsev) hep-ph/ D. Benson,I.Bigi,T.Mannel,N.Uralstev

Based on HQET At zero recoil (w=1), as M Q  F(1)  1 V cb -Exclusive Method 38.1 ± 1.0F(1) |V cb |= F(1) |V cb 22 V cb (exclusive)= ( 42.1± 1.1 ± 1.9 ) F(1) ~ 0.91 ± 0.04

V ub Inclusive methods B  X u l + (End Point) b  u b  c b  u \\\\ b  c Backgr. BABAR Backg. substructed b  c b  u b  c b  u DELPHI CLEO  D Fit  q2 and M X E l

Conservative approach syst. fully correlated V ub = ( 4.09 ± 0.46 ± 0.36) 10 -3

Exclusive methods B  (  l … Error : dominated by form factor errors as F(1) in V cb V ub = ( 3.30 ± 0.24 ± 0.46) Common to all analyses

Oscillations in B 0 d system :  m d d, sb b t,c,u WW WW B 0 d,s t,c,u V ts V td ))1((     cbBB d VBf m dd tdcbBB VVBf dd   m d = ± ps -1 LEP/SLD/CDF/B-factories ( today dominated by Belle-BaBar ) Precise measurement (1.2%)

 m s > 14.4 ps -1 at 95% CL Sensitivity at 19.3 ps -1 LEP/SLD/CDF-I Oscillations in B 0 s system :  m s cbBBtdBBs VBfVBfm ssss   

1.02±0.02 Calculation partially unquenched (N f =2 or 2+1) in agreement Chiral extrapolation : light quarks simulated typically in a range [m s /2 - m s ]

(syst not correlated ~m b ) Calculation partially unquenched (N f =2 or 2+1) in agreement 1.09± ±0.15 unquenching factor 1.05±0.05 SU(3) effects factor

CP violation comes from interference between decays with and without mixing mixing decay

Determination of V ud, V us Attributing it to an understimate of syst. error (theo/exp) or (an unlikely stat. fluctuaction)  inflate the error V us = ± Neutron  decay  transition of J P =0 + nuclei K l3 decays : K   l V ud = ± V us = ± Using unitarity V 2 ud +V 2 us +V 2 ub =1,  (V us ) = 1/  (V ud ) V us = ±  discrepancy 1%

Rfit Bayesian p.d.f. from convolution (sum in quadrature) Likelihood Delta Likelihood Likelihood summing linearly the two errors Delta Likelihood [ ] [ ]At 68% CL Scan Treatment of the inputs Ex : B K = 0.86 ± 0.06 (Gaus.) ± 0.14 (theo.)

  FIT COMPARISON-same inputs Quantitative differences in the selected (  ) regions between Bayesian and frequentist are small Ratio between sizes of intervals corresponding to a given CL

Both methods use the same likelihood Conclusion of the CERN Workshop: “The main origin of the difference on the output quantities between the Bayesian and the Rfit method comes from the likelihood associated to the input quantities” “ If same (and any) likelihood are used the output results are very similar ”

ParameterValueError(Gaussian)Error(Flat) V cb (  ) (excl.) V cb (  ) (incl.) V ub (  ) (excl.) V ub (  ) (incl.)  m d (ps -1 )  m s (ps -1 ) > 14.4 ps -1 at 95% CL m t (GeV)1675 m c (GeV) (MeV)  BKBK sin2 

If the theoretical/statistical errors are - Convoluted (Bayesian) - Linearly (frequentist-Rfit) V cb know at ~2% Differences if : (difference of 95% C.L.) V cb = ( 41.5 ± 0.8) No correlation between the incl/excl measurements Combination of V cb and V ub incl/excl V ub know at ~10% V ub = ( 35.7 ± 3.1) Precision driven by incl. method

Results on Unitarity Triangle parameters Buras,Ciuchini,Franco,Lubicz,Martinelli,Parodi,Roudeau,Silvestrini,Stocchi

sin2  ± ( ) B  J/  K 0 s Coherent picture of CP Violation in SM from sides-only Crucial Test of the SM in the fermion sector

Fit of the Unitarity Triangle in SM

Indirect determination of the UT angles : sin2α, sin2β and γ  > 90° Prob~ ° Without  m s   > 90° Prob~0.005

RED: WITH ALL CONSTRAINTS / BLUE: WITHOUT Δ m s By removing the constraint from Δm s : γ = (65 ± 7)° → γ = (60 ± 9)°  > 90° Prob~0.005

Prediction for  m s Without limit on  m s With limit on  m s

Indirect determination of the non-perturbative QCD parameters 

V ub / V cb  m d /  m s A CP (J/ψ,K S ) Using : So in particular knowing  V ub / V cb A CP (J/ψ,K S ) Using : You can take this examples to show how the system is starting to be overconsrained

Looking for “ New physics ” by measuring Δm s Δm s > 26ps -1 New Physics at 3  >30.5ps -1 New Physics at 5  Almost independently of the precision on the measurement of Δm s If the value measured for Δm s will fall in the SM region [(12-26) ps -1 ] important theoretical improvements have to be forseen to test the SM Red lines: σ(Δm s ) = 1.0, 0.5, 0.2, 0.1 ps -1 Blue line: all errors divided by 2 33 55 ΔmsΔms indirectdirect Delta(Δm s )

Looking for “ New physics ” by measuring  Red lines: σ = 20, 15, 10, 5 degrees Blue line: all errors divided by 2 Suppose  can be measured with an error of 10 o   >100 o New Physics at 3  Importance of reducing some theo. errors ( ,B K …) to perform a more powerful test of the SM for ex: if all theo. errors/2   >90 o New Physics at 3    >80 o New Physics at 3  with an error of 5 o

SM prediction without A(J/ψK s ) sin2  = ± Red lines: σ(sin2  ) = 0.2, 0.1, 0.05, 0.02 Blue line: all errors divided by 2 ° [ ] 3  SM region if  (sin2  )=0.02 [ ] 3  SM region if  (sin2  )=0.05 (today) Improving the precision on sin2  using A(J/ψK s ) ? Obviously difficult to find any discrepancy with SM

SM prediction with A(J/ψK s ) sin2  = ± Measuring sin2  using a different channel B →φK s ? A( φK s ) = –0.39   If the experimental error goes down by a factor 2 (  0.2) A( φK s ) < 0 4  Red lines: σ(sin2  ) = 0.4, 0.2, 0.1,  A( φK s ) ~ 

Important progress in the last years Next (in a ~year-time scale) hopes :  m s, A( φK s ) V cb ~2% V ub ~10%  m d ~ 1.2%  m s > 14.4 ps -1 at 95% CL sin2   -1 ~ 20-25% B K ~ 15% ~ 15% m t ~ 3% Success of SM + LQCD/OPE/HQET Standardissssssimo

ADDITIONAL MATERIAL

f(,, x|c 1,...,c m ) ~ ∏ f j (c|,,x) ∏ f i (x i ) f o (, ) ηρρηρη j=1,mi=1,N The Bayes Theorem: f(, |c) ~ L (c|, ) f o (, ) ρρρηηη x  x 1,...,x n = m t, B K, F B …. c  c 1,...,c m =  K,  m d /  m s, A CP (J/ψ,K S )