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Even though radiative corrections have not been taken into account in the calculations, the calculated Kobsayashi-Maskawa matrix seems to be roughly consistent with experiment, as is shown by the following analyses based on the paper of Nir37:

Nir37 gives for sg Å Vcb the following:

where F(x) = 1 - 8x + 8x^3 - x^4 - 12x^2 ln(x) is a phase space factor,

BR(b -> cl`n) is experimentally measured to be 0.121±0.008, and

tb is experimentally measured to be (1.16±0.16) x 10^-12 sec.

Note that tb can be written in GeV^-1 as

(1.16 x 10^-12) / (0.7 x 10-24) Å 1.66 x 10^12 .

If (from the F4 model) GF = 0.97 x 10^-5 x (1/0.93825)^2 =

= 1.1 x 10^-5 in terms of GeV and if (from the F4 model)

mb = 5.63 GeV and mc^2/mb^2 = 2.092/5.632 = 0.1378 so that

F(mc^2/mb^2) = F(0.1378) =

= 1 - 1.1024 + 0.0209 - 0.00036 - (0.228)ln(0.1378) =

= 1 - 1.1024 + 0.0209 - 0.00036 - (0.228)(-1.982) =

= 1 - 1.1024 + 0.0209 - 0.00036 + 0.4519 Å 0.37 , then

sg = Ã((192¹^3/1.21 x 10^-10)(0.121/1.66 x 10^12)(1/(5.63)^5(0.37))) Å

Å Ã((4.92 x 10^13)(7.3 x 10^-14)(4.778 x 10^-4)) Å Ã(17.16 x 10^-4) Å 0.0414

is the value determined from experimental measurements of BR(b -> cl`n) and tb.

It is close to the F4 model value sg = 0.0423484.

Nir37 gives for q = sb / sg Å |Vub| / |Vcb| the following:

q = sb / sg Å |Vub| / |Vcb| =

= [ Ã(G(b -> ul`n) / G(b -> cl`n)) ] [Ã(F(mc^2 / mb^2) / F(mu^2 / mb^2))]

Since in the F4 model mu = 0.3128 Gev,

mu2 / mb2 = (0.3128)^2/(5.63)^2 = 0.0031, and ln(0.0031) = -5.78,

F(mu^2/mb^2) = F(0.0031) Å 1 - 0.025 + 2 x 10^-7 - 10^-10 + 0.0007 Å 0.976, and Ã(F(mc^2/mb^2) / F(mu^2/mb^2)) Å Ã(0.37/0.976) Å 0.616.

Therefore:

q = sb / sg Å |Vub| / |Vcb| = 0.616 [ Ã(G(b -> ul`n) / G(b -> cl`n)) ] .

Even without directly observing the semileptonic decay b -> ul`n , observation of hadronic decays of the type b -> u + d`u have been reported by ARGUS (see Franzini24), with

BR(B± -> p`p¹±) = (3.7±1.3±1.4) x 10^-4 and

BR(B0 -> p`p¹+¹-) = (6.0±2.0±2.2) x 10^-4.

Although the connection between b -> u + d`u and b -> ul`n

may have a lot of theoretical uncertainty, ARGUS indicates that

q = sb / sg Å |Vub| / |Vcb| > 0.07, so that

sb Å |Vub| > (0.07)(0.0423484) Å 0.00296.

The lower bound is consistent with the F4 model value

sb = 0.00460816.

An upper bound is established (see Nir37) by non-observation limits for the semileptonic decay b Æ ul`n ,

G(bÆul`n) / G(bÆcl`n) < 0.08 , so that

sb Å |Vub| < 0.616 sg Ã(0.08) = 0.0074.

The upper bound is also consistent with the F4 model value

sb = 0.00460816.

The F4 model value for q is q = sb / sg Å |Vub| / |Vcb| Å 0.1088.

K-`K parameters Œ and Œ':

K0 = `s d and `K0 = s`d.

Define K01 = (K0+`K0)/Ã2 [CP=+1] and K02 = (K0-`K0)/Ã2 [CP=-1] .

Define K0S = K01 + ŒK02 and K0L = K02 + ŒK01 .

Then define Œ' by h+- = Œ + Œ' and h00 = Œ - 2Œ' .

Nir37 gives for the CP-violation parameter Œ the following:

where

and yc = mc^2/mw^2 and yt = mt^2/mw^2 .

The parameters h1, h2, and h3 are QCD corrections that are taken to be

h1 = 0.7, h2 = 0.6, and h3 = 0.4.

C is a dimensionless constant calculated to be C Å3.8 x 10^4 Å 4 x 10^4 by using GF = 1.166 x 10^(-5) Gev-2, mw = 81 Gev , fK^2 = (0.16 Gev)^2,

MK = 0.498 Gev, and DMK = 3.52 x 10^-15 Gev.

In the F4 model, yc = (2.09)^2/(81)^2 = 0.000666 and

yt = (130)^2/(81)^2 = 2.58, so that

f2(yt) = 1- 0.75 x 2.58 x 3.58 x [1+(5.16 x 0.948/(-5.6564)]/2.50 =

= 1 - 6.93 x 0.1352/2.50 = 0.625 and

f3(yt) = 8.26 - 0.75 x 2.58 x [1+2.58 x 0.948/(-1.58)]/(-1.58) =

= 8.26 - 1.94 x (-0.548)/(-1.58) = 8.26 - 0.67 = 7.59.

Therefore, from the F4 theoretical values of sa, sb, sg, and e,

|Œ| Å BK(4 x 10^4 x 0.042349^2 x 0.1088 x {[0.4 x 7.59-0.7] x 0.000666 x 0.2222+ +0.6 x 2.58 x 0.625 x 0.042349^2 x 0.2222}) Å

Å BK (7.805 x {0.000352+0.000386}) Å 0.00576 BK

As experiment shows that |Œ| = 2.3 x 10^(-3) = 0.0023, BK Å 0.399.

The value of BK Å 0.399 is consistent with the range 0.3 < BK < 1 that is usually accepted. BK is a hadronic parameter with much theoretical and experimental uncertainty. According to Franzini38,

QCD sum rule calculations give a value of BK = 0.33 ± 0.09 and SU(3)+chiral invariance+experimental results give BK = 0.33 ± 0.2, while lattice QCD calculations give BK = 0.85 ± 0.20.

Nir37 gives for the CP-violation parameter ratio Œ'/Œ the following:

Œ'/Œ Å 6.0 C" (sg sb sine / sa)

where C" = [ Im˜C6 / (-0.1) ] [< ¹¹|Q6|K0 > / (1.0 Gev)^3 ]

and where Q6 is the penguin operator and ˜C6 is the Wilson coefficient with the KM factor taken out. Both Q6 and < ¹¹|Q6|K0 > are very uncertain theoretically, but it may be reasonable to estimate C" Å 1.

Since Œ'/Œ Å C" x 6.0 x 0.042349 x 0.00460816 / 0.2222 Å 5.27 x 10^-6 C",

then if C" Å1 is assumed, then Œ'/Œ Å 5.27 x 10^-3. That is a little higher than the measurement of the real part of Œ'/Œ by NA31 at CERN to be Re(Œ'/Œ) = (3.3 ±1.1) x 10^-3 by observing 10^6 K0L Æ ¹¹ decays and 10^7 K0S Æ ¹¹ decays to determine h00 / h+- = 0.980 ±0.004 ±0.005.

Nir37 gives for the Bd-`Bd mixing parameter xd the following:

where the parameter h is a QCD correction taken to be h Å 0.85,

MB is taken to be MB Å 5.28 Gev, and BB is taken to be BB Å 1.

The parameter fB^2 is very uncertain. The other parameters have been discussed above. Using those values, including F4 model values,

xd Å 1.66 x 10^12 x (1.1 x 10^-5)^2 x 0.85 x 5.28 x BBfB^2 x (81)^2 x 2.58 x 0.625 x

x |0.00940048+0.00448916i|^2 x (0.999092)^2/6¹^2 Å

Å 18.939 BBfB™

The parameter xd is defined as xd = DM / G .

It is related to the mixing parameter rd by rd = xd^2/(2+xd^2) .

ARGUS has observed rd Å 0.21, giving a value for xd of xd Å 0.73. Therefore BBfB2 Å (0.196 Gev)^2 , a value that is at the upper limit of the range BBfB2 Å (0.15±0.05 Gev)^2 used by Nir37.

Nir37 gives for the Bs-`Bs mixing parameter xs the following:

xs/xd = |Vts|^2 / |Vtd|^2

For the F4 model values and xd Å 0.73,

xs Å

Å 0.73 x (-0.0412923-0.00102199i)^2 / (0.00940048-0.00448916i)^2 Å

Å 0.73 x 0.0017 / 0.00011 Å 11.28, and rs Å xs^2 / 2+xs^2 Å 0.9845 ,

a value that is near to the value rs Å 0.985 expected by Nir37.

Another way to parameterize B-`B mixing is by cs and cd, where

c Å (1/2) (DM)^2 / [(DM)^2 + G^2 ]

(Section 10.1.3 of Barger and Phillips33)

so that, since x = DM / G, c Å (1/2) x^2 / (1 + x^2) and the F4 model gives cs Å 0.496 for cd Å 0.174 .

Experimental measurements of c at CDF39 in 1991 are consistent with the F4 model.

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