July SLAC-PUB-5051 1989 (T/E)
Phenomenology
of the CKM
Matrix*
YOSEF Stanford Stanford Linear
NIR Center 94309
Accelerator Stanford,
University,
Caliifornia
ABSTRACT
- The way in which an exact determination the Standard Model is demonstrated of the CKM matrix example. elements tests The determiwith an
by a two generation
nation of matrix
elements from meson semi-leptonic
decays is explained,
emphasis on the respective reliability The assumptions
of quark level and meson level calculations. Finally, the
involved in the use of loop processes are described. is presented.
state of the art of our knowledge of the CKM matrix
Invited 1989 International Cornell University,
talk presented at the on Heavy Quark Physics
Symposium
Ithaca, New York, June 13 - 17, 1989
* Work supported by the Department
of Energy, contract DE-AC03-76SF00515
c 1. INTRODUCTION
The free parameters of the quark sector in the Standard Model (SM) are the quark masses and the mixing gauge interactions parameters. In the interaction basis, the charged
are, by definition,
diagonal:
For n generations,
the mass matrices
Mi
and Mi
are general n x n matrices,
while 1 stands for the unit matrix. definition, diagonal:
In the mass basis, the mass matrices are, by
.
The charged gauge interactions, given by the unitary ea.ch mass matrix matrix
V.
however, are no longer diagonal: the mixings are The independent parameters are n eigenvalues of
V.
and (n - 1)2 parameters of the matrix
At present we know
of-three quark generations, in which case V is the Cabibbo - Kobayashi - Maskawa (CKM) mixing matrix of four free parameters: three mixing angles and one phase.
If we ha.ve several independent mea.surements for a given CKM matrix element, or if we find the values of the nine entries, we will have the four mixing parameters
overdetermined.
Therefore, an exact determination a stringent
of the CKM
matrix
elements
provides us with
L --
test of the SM and with
possible clues to physics
beyond it. We explain this by showing what we can tell about the third generation from our present knowledge of the 2 x 2 Cabibbo matrix. We survey the determination meson decays. of different matrix elements from semi-leptonic at either the quark elements:
We explain the shortcomings We concentrate
of calculations
level or the meson level.
PusI, IVcd and IKbi-
on the three above-diagonal
Additional
information
can be derived from loop processes. The assumptions and show the constraints from
made are stronger.
We explain these assumptions
B - B mixing a.nd from the e parameter.
2
c
2.
THE
FIRST
TWO
GENERATIONS
The Cabibbo mixing matrix
for the first two generations is
(3)
The value of lVud 1 is calculated Fermi p transitions:lj2 from the comparison of O+ -+ O+ superallowed
Ivudl = 0.9747f 0.0011
.This is the most accurately determined of all CKM matrix elements.
(4
The calcucorrections.
lation of the above value follows a continuous refinement of radiative The most recent one2 takes into account 0(Z(r2) accurately studied Ft values to agree within corrections,
and brings the eight
less than la.
The value of jV,,l and A --+ r-e+v, :
-
is best determined from the measured rates of li + r + e+y, which give3
p&l
= 0.220 f 0.002.
(5)
I _-
The calculation
cannot be carried out within
the spectator quark model, because:
a. The final spectrum is completely dominated by the single pion state, so that duality is not expected to hold.
h. There are large QCD corrections as the relevant scale for LY,(,v) is p = O(m,), but m, AQCD
(the scale at which, by definition,
LY, - 1).
c. There are large uncertainties
in m S: first, it is a running mass and we do not
know the relevant scale and second, even if we knew the scale, the uncertainty in m, is still about 30%.4 This is significant, depends on (uz,)~.
3
as the phase space for the decay
-
f
Thus, the above value is derived from a phenomenological BR(II -+ rev)
r(K)
model:
= CK p-(0)12
pJ2, only. In general, the major
where CK includes factors with small uncertainties difficulty is in the calculation
of the form factor If+(O)l.
In this case, however, only limit (mu = md = m,)
the three light quarks take part.
In the SU(3) symmetry limit
we have If+(O)1 = 1. D eviations from the symmetry symmetry breaking parameter and calculable.
are second order in the we have a l-1.5% error calculations. by another infit to the
Altogether
from experiment
and about a 2% error in the theoretical of IV,,/
Our confidence in the above calculation
is supported
dependent measurement which gives a consistent value: a simultaneous . rates of A t Consistency
pev, C- + nev and Z- + Aev gives5 IV,,( = 0.220 f 0.001 f 0.003. with the meson decay data was achieved only after recoil corrections
were taken into account. - There are two methods to determine IV&, 1 and IV,, 1. The first one is using data from deep inelastic neutrino - nucleon scattering.
IKdl = 0.21 f 0.03 (7) /&s/v,dI > 3.3
One gets:6
The bound on the ratio is derived with a mild assumption on the ratio of strange : .sea to anti-quark sea in the nucleon, 2s 5 u + D. is from D semi-leptonic decays. A reliable qua,rk level is ques-
The second method calculation
is still impossible due to the lightness of the c quark: Duality
tionable and QCD corrections may be large. However, the uncertainty given scale is small, so the question here is that of the relevant scale. At the meson level we have:
in m, at a
BR(Do + xi;
T(DO >
e+v)
where CD includes factors with small uncertainties
4
only. The uncertainty
from the
Do lifetime
is common to both determinations,7 comes from the calculation
~(0 )
= .422 f
.008 f
.OlO psec. The charm Various
A large uncertainty
of the form factor.
quark is too heavy to make an SU(4) sy mmetry useful for the calculation. calculations
of the form factors, using quark models and QCD sum rules, give:
(9)
The main difference between the determination from the experimental measurements.
of ]vcd] and that of IV,, 1 comes
For c --+ s there are two measurements:
(3.8 f 0.5 f 0.6) x 1O-2 4R(D t K-esv) = (3.4 f 0.5 f 0.4) x 1o-2
[ll] [12] (10)
With enough confidence in the models for the form factor one may give a value for ]I/ ,,], e.g. ]I/,,] = 1.1 f 0.2 for If+(O)] = .7 f .l. However, for c t one measurement and with large uncertainties:12 d there is only
BR(DO
-+ H-e+v)
= (3.9 ;:;
f 0.4) x 10-3.
(11)
The ratio ]Vcd/xs] the ratio
lffi/ff+
1s f ree of the uncertainties
). in ~(0
Moreover, it depends on which is expected to
I: this ratio is 1 in the SU(3) limit,
hold within
10%. Thus, we get:
Ivcd/vcsl
= 0.25 f 0.06.
(12)
With present experimental
errors and theoretical uncertainties, scattering,
the more restrictive of D semi-
bounds come from deep inelastic
but the measurements
leptonic decays give further confidence in these results.
5
-
To conclude,
f
different elements:
direct measurements give the following
range for the
Cabibbo matrix
vc =
.21 f .OOll .9747 f .03
2.60 .220 f .002 > .
(13)
would imply
Now, suppose we knew about two generations only. Then unitarity that the above matrix depends on one parameter only:
vc =
Cl2
s12 Cl2
(
-s12
>.
(14)
the Cabibbo angle.
With the above measurements we have certainly overdetermined The test to the two generation SM is the following:
Can we find a range for the
Cabibbo angle which is consistent with all measurements? The answer is positive: for .219 < 512 < .222 we get the following ranges for the matrix elements:
vc =
.219 - .222 .9750 - .9758
.9750 - .9758 > .219 - .222
(15)
which is consistent with the measurements (13). Thus, the two generation picture is still consistent and we could not tell that there is a third generation if not for its direct observation (or from CP violation). l&l From our knowledge about lVcbl and
we know that the third generation mixings would be probed only if we reached of lVuil or lop3 in the determination
an accuracy level of 10m4 in the determination of II&l
(i = d,s);
th is is well beyond the present level of accuracy. At present, the mild bounds on the possible mixings of a
values in (13) imply only the following third generation:
I: .07
v=
1. .78
(16)
Additional
information
on the parameters of the first two generations can be
derived from indirect measurements, namely SM loop processes. To extract useful
6
-
c -.
information,
we need to know all the significant
contributions
to such a process.
Thus, we make two major assumptions: a. There are no additional generations. This assumption is unnecessary in the
case of direct measurements.
b. There are no significant
beyond standard contributions.
For direct meaprocesses which
surements we assume that there are no beyond standard
compete with the tree level SM processes, which is indeed the case for most reasonable models (with charged Higgs). the possible exception of models with a light
no
For indirect
measurements we assume that
there are
processes which compete with SM loop processes (which are suppressed by .the high order in the weak interaction coupling and by the GIM mechanism).
This is not the case in many extensions of the SM. Finally, we note that as the GIM mechanism is in operation, quarks. the results have
strong dependence on the masses of intermediate
The only loop process which does not a priori necessitate the existence of a third generation is AMI{-, the mass difference between the two neutral K-mesons:
i!bkf,NK (1 - D) BI C
(17)
NK E Gz,f;iyiM = 2.1 x 10-l GeV. parameter gives
I
--
The NI, parameter is a known quantity, The long distance contributions the ratio insertion
are given by D . An/l,-.
The BK
between the short distance contribution approximation.
and its value in the vacuum 71 = 0.7.
The ~1 parameter gives the QCD corrections, for two generations by putting
In the above we used unitarity
(18)
We note the strong dependence on m,. K mixing13 was performed,
When the original
study of the I< discovered.
the c-quark was not yet experimentally
7
-
c
Thus one could use eq. (17) to predict the mass of the c-quark. calculation, long-distance the vacuum saturation contributions approximation
In the original
was used (Bk* = l), and neither were taken into account (D to the correct prediction:
=
nor QCD corrections
0, 71 = 1). This led, somewhat coincidentally, 1.5 GeV. With the full range of uncertainties
m, =
in BK and D one gets:
8 x lo-6
<
Ak=
(l
-
O>
I
5 x
1o-5
-
NK
BK
(19)
the
which gives 1.3 GeV
2 m, < 3.2 GeV.
As we now know that m, c 1.4 GeV, even when information
two generation picture is still self-consistent, process AMI(- is taken into account. generation, . matrix.
3.
THE ABOVE-DIAGONAL ELEMENTS
from the loop
Due to the very small mixings of the third in the Cabbibo
at present we could not find it from inconsistencies
In this section we concentrate on the determination elements:
of the three above diagonal
~
(20)
from semi-leptonic previous section:
meson decays. The determination
of IV,,/
was explained in the but
the s quark is too light to allow a quark level calculation,
light enough to allow a reliable calculation
of the form factor at the meson level.
B decays: B -+ X,ev,.
The value of 1Vcb1is best determined semi-leptonic from At the quark level the process is b t (1. The dominant semi-leptonic cev,. In this case:
modes are those with X, about 10%.
= D,
D*.
Duality
should hold for the decay rate within
b. The relevant scale for QCD corrections
is of order mb. As oS(mb) - 0.2, a 4% or so.
first-order
calculation
should be fine to within 8
-
c
c. The mass of the b quark at a certain energy scale is known at the 2% accuracy level. Consequently, the crucial question is that of the relevant energy scales. We will argue that there is no ambiguity accurately, in the ratio m,/mb. of energy scales for m, or, more
However, the question of energy scale for mb in mb
in the (mb)5 factor is still open and remains the main source of uncertainty the calculation. One possible way to overcome this difficulty The fit is model-dependent, is by fitting
to the leptonic spectrum.
but if we use several
models and let their parameters vary in a reasonable range, we may learn what is the uncertainty involved t . Within the spectator quark model:
BR(b -+ cev) Tb = m
.on two scales:
G2F [ 1
m~F~s(Pc)FQCD(pc)l~b12.
(21)
The experimantal
quantities on the left hand side are known with about 15% error, The phase-space factor Fps and the QCD As mentioned,
mainly from the b lifetime determination.
correction factor FQCD both depend on the mass ratio pc = m:/rni. a priori there is an ambiguity,
because quark masses are rumring, so that p depends
The question is what are the relevant scales p, and pb. The answer is14 that to every choice of two scales, there corresponds a specific QCD correction factor. The modification of FQCD is such that the product Fps(p) +FQCD(P) is independent of
the choice of scales:
&&~)FQcD(&) = 0.46 f 0.04. (23)
Various arguments suggest that the value of mb should be taken as
mb = 4.9 f 0.3 GeV. (24)
As the deca,y width
depends on (mb)5, this gives a 30% uncertainty.
for discussions on this point. 9
With
the
t We thank K. Schubert and G. Altarelli
-
above values we get:
c
It&l
= 0.046 f 0.008.
(25)
Various phenomenological against the experimental
models are, at present, in the stage of being tested data. However, they all give l&b/ values which are somequark model value. To account for the model
what higher than the spectator dependence of the calculation
we take:
l&l = 0.048 f 0.009. (26)
The value of J&b I can be determined from semi-leptonic
B --+ X,eu,.
charmless B decays: is subject
At the quark level the process is b t
uev,. The calculation
to uncertainties 4 = II&b/v&l
similar to those of /I& I. It is advantageous to consider the ratio
rather than II&,/ itself:
BR(b BR(b -+ uev) -+ cev) = F~&u) Fp&c) FQCDh) FQCD(Pc) 2 ' *
(27)
The ratio is free of the uncertainties
in (mb)5 and 3-b. Moreover, the ratio between
the QCD correct8ion factors does not depend (to O(cr,)) on the choice of scale for o, and, due to the lightness of the u quark, Fps(pu) get:
&&)FQcD(Pu) = O-85. (28)
= 1 with no uncertainty.
We
The only theoretical
uncertainty
is then in Fps(pc).
BR(b BR(b t
We get:
uev) 112
q = (0.74 f 0.03)
---f cev) I
(29)
from the measured decay rate, the large
Experiment observation semi-leptonic
does not provide us, at present, with BR(b of charmless B decays. rate the theoretically
+ uev) as there is no direct
If one tried to subtract
calculated
charmed semi-leptonic
one would be left with zero and the b --+ u contribution error bars.
10
buried within
-
c -.
Instead, [V&l IS d et ermined from the electron energy spectrum. '
The spectator
quark model is not appropriate for this analysisr5, while various phenomenological models give very different weakest theoretical results. The strongest experimental results with the
constraints
give16:
q 5 0.16.
(30)
The CLEO
collaboration
recently reportedI
a measurement of BR(b --+ uev) # by other
0, but as the errors are still large and the result is not yet confirmed experiments we do not use it here. The above diagonal elements in the CKM matrix
To summarize:
are best de-
termined from semi-leptonic light quarks, quark-meson nated by one final state. would have practical and large uncertainties
meson decays. For light mesons, or correspondingly duality does not hold because the spectrum is domi-
Moreover, even if the spectator in the calculation
quark model held, we
difficulties
due to large QCD corrections
in the light quark masses. On the other hand, we are able models, due to the approxheavier quarks, of the inclusive
to calculate rather accurately within phenomenological imate flavor symmetry. the spectator
For heavier mesons, or correspondingly
quark model should give a reasonable description
decay rate. QCD corrections are small and heavy quark masses are known rather
: --
well, though they remain the major source of uncertainty. quarks, phenomenological the hadronic matrix experimental models have no approximate
In the case of heavy to help control
symmetry
elements, and at this stage they should be tested against the elements.
results rather than used to estimate the CKM matrix
Direct measurements give:
IV,,/
= 0.220 f 0.002,
II&l
= 0.048 f 0.009,
q s /
cb
5 0.16.
(31)
11
2,
4.
INDIRECT
MEASUREMENTS
We now proceed in the same manner as in the two generation case. We assume that there are only three generations. for the CKM matrix elements: .220 f .002
2 .60
Unitarity
implies that the following
values
.9747 f .OOll
V CI(M =
5 .009 .048 f .009 5 .9992
.21 f .03 5 .14
(32)
5 .77
should be consistent with a parametrization
.Cl2
of four free parameters only:
-i6
s12
s13e
i6
V CKM
=
-sl2c23 sl2s23 -
-
C12s23sl3e c12s23s13e
i6
i6
Cl2C23 --cl2s23 -
s12s23sl3e sl2C23s13e
s23 c23
i6
The above parametrization,
recently
adopted by the Particle
cl3
Data Group,l*
is
given here with the only approximation than any of the experimental
= 1, which is good to 0(10w4), better
determinations. parameters consistent with all data.
Indeed, there is a range for the mixing
It is simple to find it, as the values of the three mixing
angles are equal to the
absolute values of the above diagonal elements, which were derived in the previous section. Thus, the allowed ranges for the parameters is:
~12
= .220 f .002, ~23 = .048 f .009, q = s13 5 .16.
s23
(34)
Direct measurements do not constrain S: 0 5 S 5 360. Additional information on the matrix elements is derived from indirect meaon
surements, namely loop processes. At present, we have no direct information
12
-
-. z -.
the mixings of the top quark:
v=
Vtd vts Vtb
(35)
from unitarity, but Vtd is still poorly
The values of Vi, and Vtb are determined determined:
I&b1 = 1, Iv,,/
= I&bl,
lvtdl 5 -022.
(36)
The GIM m...