Constraining the CKM matrix

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Constraining the CKM matrix from experiments
International School on CP violation and Heavy
Quark physics Prerow, Germany, September 21 27,
2003 Heiko Lacker, TU Dresden
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OUTLINE
PART I 1. Introduction a) CKM-matrix
b) Unitarity Triangle 2. Determination of the
CKM-matrix a) Constraints on the CKM-matrix
b) Theoretical uncertainties c)
Statistical approaches 3. The Standard CKM fit
a) Determination of Vud Vus gt ? b)
Determination of Vcb gt A c)
Determination of Vub d) ?md ?ms e)
eK f) sin2ß g) The global CKM fit
PART II 4) Constraints on a B?pp 5)
Constraints on ? sin(2ß?) 8) New Physics
in B mixing? 9) Conclusion Outlook
1) There is much more to say than this
Many topics have been skipped due to time
constraints. 2) Several exercises/problems
will be posed during these lectures. You
are free to discuss them during the
evening seminars at the beach.
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Modified Couplings in charged Currents
Wolfenstein-Parametrisation (Expansion in ?
0.22)
1) Appearance of the CP- violating phase
depends on the parametrisation! 2) For
proper use Higher order terms have to
be taken into account !
Max J 1/6v3 O(0.1)
CP-Violation in SM, if
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The Unitarity Triangle
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What is our Goal ?
Within the Standard Model I) Test
consistency between data SM. Can all data
be described by one parameter set (?, A, ?,
?) ? II) If data SM consistent
Constrain (?, A, ?, ?) gt Quantify CP
violation in the SM J Do we then
understand CP violation? No! Outside the SM
I) Data consistent with physics beyond the SM ?
II) Constrain New Physics parameters.
Attention Outside the framework of the SM
certain constraints can not be visualized
any longer e.g. in the ?-? plane !
Attention NP does not necessarily enter in
in the quark sector!
J/2
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Statistical Issues ...
Probability Theory Probability Density Function
(PDF) characterised by parameter p f(xp) gt
Probability to measure a value x in x-dx/2,
xdx/2 f(xp)
dx Statistics N measurements of x1,... xN p
unknown gt Likelihood L(px1,... xN) f(x1p)
... f(xNp) (A-posteriori probability)

Function of p not a
density Theoretical parameter Frequentist
Not a random variable
Has a
fixed (but a-priori unknown) value
Bayesian Random
variable Why does it matter? Theory calculation
gives p ?p(estimated) Statistically, no value
in the range p ?p is prefered Frequentist
Const. Likelihood L(p) 1 , p
?p L(p)0 outside Bayesian
Const. PDF PDF(p) 1/(2 ?p), p
?p PDF(p)0 outside P p1 p2 L(P)
1 gt Adding theoretical
errors linearly PDF(P) ?
const. gt Adding them in quadrature
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Xtheo(ymodel
, yQCD)
yQCD(BK,fB,BBd, )
 Guesstimates 
Frequentist
Bayesian
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Statistical approaches
1) First ?-? fit M. Schmidler, K. R. Schubert
(1992) (Likelihood fit Schubert Nogowski,
CKM WS 2003) 2) Based on Bayesian Statistics
F. Parodi, P. Roudeau, A. Stocchi et al. (1997
- ) 3) Alternative Method based on Frequentist
Statistics Scan Method (see e.g. BABAR
Physics Book, 1998) 4) Gaussian Method (P.
Faccioli et al., 2000) 5) Rfit method (2001)
Based on Frequentist statistics
Definition of a CL ...
Final Remark From the statistical point of view
the problem is ill-posed. The meaning of a
theoretical error is not at all a clear
concept. This can not be solved by any method.
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http//ckmfitter.in2p3.fr

• If p-value(SM) good
• Obtain limits on
• New Physics parameters
• If p-value(SM) bad
• Hint for New Physics ?!

• Define
• ymod a µ
• ?, ?, A,?,yQCD,...
• Set Confidence Levels in
• a space, irrespective of
• the µ values
• Fit with respect to µ
• ?²min µ (a) minµ ?²(a, µ)
• ??²(a)?²min µ(a)?²minymod
• CL(a) Prob(??²(a), Ndof)

• Evaluate global minimum
• ?²minymod(ymod-opt)
• Fake perfect agreement
• xexp-opt xtheo(ymod-opt)
• generate xexp using Lexp
• Perform many toy fits ?²min-toy(ymod-opt)
  ? F(?²min-toy)

p-value(SM)
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Significance Level

Caveat This is not the probability that the SM
is correct!
p-value(SM) P(?² gt ?²minSM)57
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(?) Observables may also depend on ? and A - not
always explicitly noted
NA48
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(?) Observables may also depend on ? and A - not
always explicitly noted
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Determination of ?Wolfenstein
At present the most stringent constraints on ?
from Vus and Vud Vud 1) Super-allowed
nuclear ß-decays 2) Neutron ß-decays
3) Pionic ß-decays Vus 1) Semileptonic
Kaon decays 2) Semileptonic Hyperon
decays 3) t-decays Vcd Dimuon
production from Not discussed in
these lectures neutrinos on nuclei
Vcs semileptonic D-decays
Promising prospects to determine
Vcd and Vcs with high precision in

semileptonic D-decays at
CLEO-c
(data
taking starting this fall)
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Vud from nuclear ß-decays
Super-allowed nuclear ?-decays
Fermi-transitions 0?0 within same isospin
multiplet pure
vector-current
Radiative Correction (nucleus-independent)
1) PS Integral ( E05) 2) Radiative Correction
(nucleus-dependent) 3) Isospin-symmetry
breaking
?R (2.40 ? 0.08)
Vud 0.9740 ? 0.0001 ft,exp
? 0.0004 ? ?
0.0003 Ft/ft
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Vud from neutron ß--decays
Neutron ?--decays n ? p e- ?e Vector
transition GV gV GF Vud (CVC ltgt
Isospin Cons. gV1) Axial-V. transition GA
gA GF Vud (PCAC gA/gV ? ? 1)
gt Measure tn and ?
Gamov-Teller-transition gt gA
Fermi-transition gt gV
? can be measured from polarisation
observables, e.g. from the ß-asymmetry
coefficient A
The Wu-experiment with neutrons!
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Vud from neutron ß--decays
P(?2gt15.5,NDOF 4) 0.004 Likely, at least one
experi- ment has underestimated its errors. PDG
recipe If not clear which one force ?2 NDOF
by scaling all errors by a factor S.
P A/Araw-1 0.989 0.02 0.97
- 0.98 0.15 0.70 0.30 0.98 0.13
PDG 2003 ? -1.2695 0.0029 (error
scaled) tn (885.7 0.8) s
Exercise Derive the scaling factor S! What is
the scaling factor for the error on the average
value?
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Vud from pion ß-decay
Br 1.025(34) .10-8 (PDG 2002)
Exercise Knowing the pion and neutron lifetimes
as well as the energy releases in neutron and
pionic ß-decay Can you estimate BR(p?p0e?e)
O(10-8)?
Preliminary result BF ( ? ? ?0e?e )
(1.044 ? 0.007 ? 0.009) 10-8 Final error may
decrease by 2.
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Vus from semileptonic kaon decays (Kl3)
K l3 decays K ? ?0 l ?e and KL ? ?- l ?e ,
0- ? 0- (pure Vector transitions)
Phase Space Integral II(f, (ml /mK )2 f0) gt
Ke3 prefered
Normalisation Kl3 C 1/v2 K0l3 C 1
Chiral limit
Corrections SU(3)-breaking ² Ademollo-Gatto
SU(2)-breaking
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Vus from semileptonic kaon decays (Kl3)
BNL-E865
Primary goal lepton number violating decay K
?pµe- One week running (1998) for K ?p0e?e
, p0 ? ee-? Normalisation using K ?pp0, pp0p0

BNL-E865 BF(K ?p0e?e) (5.13 0.02 0.09
0.04) PDG 2002 BF(K ?p0e?e) (4.87
0.06), 2.6 s
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Vus from semileptonic kaon decays (Kl3)
Vus Kl3,old 0.2201 ? 0.0016exp ?
0.0018theo
Vus E865 0.2285 ? 0.0023exp
? 0.0019theo
Vus Resc. Av. 0.2228 ? 0.0039exp
? 0.0018theo
KLOE at DAPHNE ee-? ?(1020) ? KSKL ,
currently 500 pb-1 ? 5 .108 KSKL K0l3 results
for 78 pb-1 (prelim.) Normalisation tag with KS
?pp-
Exercise Do you understand the
difference BR(F(1020) ? KSKL) 33.7 BR(F(1020)
? KK-) 49.2 ?
Exercise A pair of neutral K-mesons
emerging from f(1020)-decays can never be KSKS
or KLKL at the same time. Why?
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Summary Vud Vus
Unitarity in the first row with 3 families ?