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Masses and mixing angles in the SU(5) gauge model
Volume 92B, number 1,2
PHYSICS LETTERS
5 May 1980
MASSES AND MIXING ANGLES IN THE SU(5) GAUGE MODEL S. NANDI and K. TANAKA Department o f Physics, The Ohio State University, Columbus, 0H43210, USA Received 31 January 1980
We obtain Georgi and Jarlskog mass relations mu/m e = 9ms/m d, mb = m r above the grand unification mass M = 1015 GeV with two 5's and one 45 Higgs representations of SU(5) and a discrete symmetry. In the lowest order, the KobayashiMaskawa angles are found to be s2 = -(mc/mt) ]/2 and s 3 = -(mu/mt)l/2/Sl, where sl is the sine of the Cabibbo angle. CP violation is considered and the b-quark decays predominantly into c-quarks with a lifetime of r b ~ 10-13 s for mt = 25 GeV.
The SU(5) grand unified model of Georgi and Glashow [ 1] that unifies the color SU(3) and the electroweak SU(2) × U(1) gauge model has generated a great deal of interest. In this model the left-handed fermions for each family are assigned to the 5* and 10 representations in which case the choice of the representation of the Higgs fields (that generate the fermion masses) are 5 and 45. It was suggested [2] that the l e p t o n - q u a r k mass relations above the grand unification mass M = 1015 GeV should be the following * 1: N
mt =mb ,
mu/m e = 9ms/m d ,
(1)
so that the mass renormalization [3,4] leads to m r ~ m b / 3 and the other relation remains unchanged at low energies. Eq. (1) was obtained with the aid of three 5's and one 45 of Higgs. The question of strong CP violation was addressed in a similar model and no violation was found in the tree approximation and the violation was shown to be computable in higher order [5]. We present here an SU(5) gauge model, supplemented by a discrete symmetry, that uses two 5's and one 45 of Higgs to generate the fermion masses. We obtain the mass relations (1) and expressions for the Kobayashi-Maskawa [6] mixing angles s 2 and s 3 in terms of the quark mass ratios and the Cabibbo angle. The weak CP violating phase is taken into account and no strong CP violation is found in the tree approximation. Let the three 10 and 5 multiplets of SU(5) corresponding to the three families of left-handed fermions be denoted by ~abiLan~x/'eR, and the two _5's and one 45,.~of Higgs by ~bd and Eedf, respectively. The indices a, b, c, d, e, and f a r e SU(5) indices and run from 1 to 5, i a n d j refer to the number of families and run from 1 to 3 and k = 1,2. The most general Yukawa couplings of fermions and Higgs bosons in SU(5) are given by f~y
-,.kVab a ~ b - ; a b e ~ab + ~ k - a b T O . c d - e -- -abT O - c d ~ e f + =Jil:~iLXiR@k ~-gij~iLXjR2~c ni/g~iL 7 ~jLqgkeabcde+aijqQL "Y ~ j L ~ e eabde f h.c.
(2)
In eq. (2), tpT70 = ~c where We is the charge conjugate field, the letter T means transpose, eabcde is the completely antisymmetric five index tensor, and a sum over repeated indices is implied. In order to restrict the coupling parameters, the following discrete symmetry is imposed: filL ~ r/2mfflL ,
ff2L ~ r/Zn~bZL ,
X1R - + ~ - ( m - 7 n ) X i R , ~b1 ~ ~(m-5n)~b 1 ,
ff3L -+7?(m+n)~3L ,
X2R "~rT(m+5n)X2R ,
~b2 ~ r/-2(m+n)~b 2 ,
X3R -->r/6nX3R ,
Y~~ r/-(m+3n)z ,
(3)
,1 The actual mass relations given in ref. [2] are mr = rob, mta = 3ms, and me = md/3. The present model also leads to these three mass relations. 107
Volume 92B, number 1,2
PHYSICS LETTERS
5 May 1980
where m and n are arbitrary numbers and [r/i[ = I. The resulting Yukawa coupling is :~r~-ab
a
--ab a
--ab a
b
l :,--ab. c x,ab
£ = ( A WILX2R+A~2LX1R + B~b3LX3R)~bl + ~"~2L,~2R'-'c + l:~-abT
O-cd
abT 0
cd
e
1
abT 0 cd
ef
~ L ~ I L "[ ~g2L+E~3L 3' ~3L)~2eabcde + 15Ft~2L ")' ~3LY~c eabdef + h . c . .
(4)
At the level of SU(3) × SU(2) symmetry, we expect there are three Higgs multiplets (which are SU(3) color singlets and SU(2) weak doublets) that generate masses corresponding to the familiar three generations. In the lagrangian (4) we do have three Higgs fields, two 5's and one 45. Note that the term with the coefficient F does not contribute to the lepton masses. After spontaneous symmetry breaking, the fermion mass matrices M U, M D, and M ~ for charge 2/3 quarks ~0 = (u0, c 0, to), charge - 1 / 3 quarks ~0 = (do, So , b0), and leptons ~0 = (e0, P0, ~'0) take the form, respectively, (t~ el_ MU = 2D ia2
0 M~ =
V l A e ~°q 0
I)2De-i~2 0
0 oF
vF
v2Ee -ic~2
V l A ' e wq
0
-3vC
0
0
1
( ,
MD =
0 VlAe icq
vlA'ei~x
0
vC
0
0
0
v l B e ial
(5,6)
,
(7)
,
V l B e lal
where the vacuum expectation values (vev) are (¢5) = vieiai and ( ~ 5 ) = v(5~ -- 45a46b4), i = 1,2, a, b = 1-4. We note from eqs. (5), (6) and (7) that for M U equal coupling parameters are associated with the same vev, and for M D and M ~ the same situation holds when A = A' (which is approximately the case as we shall see later). The effect is that the mass matrices are independent of the scale, i.e., the size of the vev's. One can eliminate the phases in the mass matrices (5) and (6) by a suitable redefinition of the charge 2/3 and - 1 / 3 quark states so that Arg(Det M U Det M D) = 0 and hence there is no strong CP violation in the tree approximation ,2 Since M U and M D are not hermitian, we diagonalize the square of the mass matrices M U M Ut and M D M D? given by
M U M U~ = D F e i~2
D 2 +F 2
E F e ia2
E F e -ic~2
E2 +F 2
,
M D M Dt =
A ' C e ial
A 2+C 2
0
0
0
B2
,
(8,9)
where the vacuum expectation values Vl, v2, and u are suppressed. The characteristic equation for M U M Uy with 2 m c2 , m t2 ) yields eigenvalues (mu, mu2 + m 2 + m 2 = 2 ( D 2 + F 2 ) + E 2
,
2 t2 + m t2m 2u = 2D2E 2 + (/9 2 + F 2 ) 2 , m u2m 2c + mcrn
mumcm t=DzE,(10)
and for M D M Dt with eigenvalues (m 2, ms, 2 mb) 2 yields rn2_vm2=A,2 +a2 +c2,
mdrn s2 2=A2A,2,
m2=B 2.
(11)
The solution of eq. (10) corresponding to the eigenvalues for the masses (mu, - m c , rnt) of eq. (5) with a 2 = 0 is D = ( r n u m c m t / m t - m e + mu)l/2 ,
E = rn t - m c + m u ,
F = [mum c + mere t - m t m u ~- ( m u m c m l / m t - rn c + mu)] 1/2 .
,2 We thank R.N. Mohapatra for a discussion on this point. 108
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Volume 92B, number 1,2
PHYSICS LETTERS
5 May 1980
There are four unknowns A, A', B, and C in eq. (i 1). We shall find that A ' i s determined by the Cabibbo angle 0c, md, and m s. The mass matrices (8) and (9) are diagonalized, respectively, by the following unitary transformations ¢10 = U I ~ I and ~0 = U2~2 :
- N 2Dim c
N3 D/mt
N 2 e 2ic~2
N 3 e 2ic~2
-(N2F/m c +E ) e i~2
(N 3 Flint - E ) e i~
N1D/mu U 1 = {.
N1 e 2ic~2
I(N1F/rnu - E)e •
ia2
(13)
where N i = [1 + (D/mi)2 + (F/m i - E ) 2 ] -1/2, mi = (mu, _mc ' rot), sin 0
cos 0
0° )
(14)
cos 0 0e -ial
U2 = /-sin 00e-i~l
e-iCq where
(15)
tan20 = (A,2 , md)/(m 2 s2 -- A'2).
The charged current coupled to physical quarks are Juc = --0 ~lLT~z if02L = ~ILFTuff2L where the phase transformations I~IL ~ Q1P1~IL and ff2L -+ Q2P2ff2L are introduced to bring P to the form
r=Q e v v2e2o2 '(N1D/mu)cosO-N 1 s i n 0 e -i°
(NlD/mu) sinO+N 1 c o s 0 e -i6
N1F/m u - E
= Q~ -(N2D/mc) cosO e i~ - N 2 sin0
-(N2D/mc) sinO e i6 + N 2 cos0
-(N2F/mc + E ) ei~
(N3D/mt) sin 0 e i8 + N 3 cos 0
(N3F/m t - E ) e i~
(N3D/mt) cos 0 e i~ - N 3 sin 0
Q2 ,
(16)
where P1 = diag(1, e -i~ , e -i6) and P2 = diag(1, 1, e-in), and 6 = ~1 + 2a2,/3 = - cq - a2, and the phase transformations Q1 and Q2 will be'defined shortly. The leading approximation to eq. (16) with the aid of eqs. (1 1) and (12) is I cos 0 - (mu/mc) 1/2 sin 0 e -i6 F=Q~
-sinO-(mu/mc)l/2cos008
sin 0 + (mu/mc)l/2 cos 0 e -i6
(mv/mt) 1/2
cos 0 - (mu/mc)I/2 sin 0 e i~
-(mc/mt)l/2 e i~
(mc/mt)l/2 cos 0
ei6
-sin 0 (mc/mt) 1/2
Q2 •
(17)
Let us define the Cabibbo angle 0 c as cos 0 - (mu/mc)l/2 sin 0 e -i5 = cos 0 c e io ,
sin 0 + (mu/mc) 1/2 cos 0 e -i6 = sin 0 c e i° ,
(18)
where p and o are arbitrary phases. Then the choice of Q1 = diag( eip, e-i°, e - i ° ) and Q2 = diag(1, e i(p-°), e io) yields the Kobayashi-Maskawa form F:
l Cl
Sl
SlS3 I
--s 1
c1
s2ei~c /
S1S2
--c1s 2 -- S3 e i~c
el~C /
I~=
(19)
where 109
Volume 92B, number 1,2 S1 = sin 0c,
PHYSICS LETTERS
5May 1980
s 2 = sin 02 = -(rnc/mt) 1/2 + (mu/mt)l/2 cot 0 c e i~c ,
s 3 = sin 03 = --(mu/mt) 1/2 (sin 0 c ) - l ,
6c = P + o + 6 .
(20)
Note that in this approximation 0 c depends on an u n k n o w n constant A ' as given by eq. (15). If we put the familiar expression tan20 = md/m s in eq. (15) then A ' = A = (mdms) 1/2 with the aid of eqs. (11), while the experimental value tan 0 = 0.23 gives A = A' to within 10%. The lepton mass matrix can be handled in a similar way. The combination of the equations corresponding to eqs. (11) [that follow from (7)], eqs. (11), and the result C 2 >>A 2, A '2 lead to the mass relations (1). The CP violation in the K 0 - K 0 system is proportional to Im P~l ['32 of (19) and is expressible in terms of the mixing angles and 6 c by [7] 2 t2/ m c2) ] . [el = 2 IClC2CgS2S3 sin ~ c I [c2(m2/m 2 -- m 2) ln(m2/m 2) -- c 2 + s2(m
(21)
We obtain from eqs. ( 1 9 ) - ( 2 1 ) lel = 2(mcmsmu/mdm2t) 1/2 [ln(m2/m2c) -- 1 + (mt/mc)][sin 6c1 = 0.5 Isin 6cl,
(22)
with ms/m d = 20, m u = 4 MeV, m c = 1.5 GeV and m t = 25 GeV. The experimental value e = 2 X 10 - 3 is much smaller than (22) unless sin 6 c is very small. We note from (19) and (20) that P(b ~ u + X)/['(b -+ c + X) = mu/m c = 2.7 X 10 -3 so that b-quarks decay predominantly into c-quarks • The b-quark lifetime r b can be obtained from r b 3= r ~ (m u /m b )5/(m c /m t ), where ~-~ = 2.2 × 10 - 6 s is the muon lifetime of mass mu. One obtains r b ~ 2 × 10 -1 s for m t = 25 GeV. One of us (KT) would like to thank J. Prentki for many discussions on the SU(5) gauge model and the hospitality at CERN and C. Quigg for the hospitality at Fermilab. This work was supported in part by the US Department of Energy under Contract No. EY-76-C-02-1545.*000.
References [1] [2] [3] [4] [5] [6] [7]
110
H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. H. Georgi and C. Jarlskog, Phys. Lett. 86B (1979) 297. H. Georgi, H.R. Quinn and S. Weinberg, Phys. Rev. Lett. 33 (1974) 451. A. Buras, J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nuel. Phys. B135 (1978) 66. R.N. Mohapatra and D. Wyler, SLAC-PUB-2382. M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49 (1973) 652. J, Ellis, M.K. GaiUardand D.V. Nanopoulos, Nucl. Phys, B109 (1976) 213.
PHYSICS LETTERS
5 May 1980
MASSES AND MIXING ANGLES IN THE SU(5) GAUGE MODEL S. NANDI and K. TANAKA Department o f Physics, The Ohio State University, Columbus, 0H43210, USA Received 31 January 1980
We obtain Georgi and Jarlskog mass relations mu/m e = 9ms/m d, mb = m r above the grand unification mass M = 1015 GeV with two 5's and one 45 Higgs representations of SU(5) and a discrete symmetry. In the lowest order, the KobayashiMaskawa angles are found to be s2 = -(mc/mt) ]/2 and s 3 = -(mu/mt)l/2/Sl, where sl is the sine of the Cabibbo angle. CP violation is considered and the b-quark decays predominantly into c-quarks with a lifetime of r b ~ 10-13 s for mt = 25 GeV.
The SU(5) grand unified model of Georgi and Glashow [ 1] that unifies the color SU(3) and the electroweak SU(2) × U(1) gauge model has generated a great deal of interest. In this model the left-handed fermions for each family are assigned to the 5* and 10 representations in which case the choice of the representation of the Higgs fields (that generate the fermion masses) are 5 and 45. It was suggested [2] that the l e p t o n - q u a r k mass relations above the grand unification mass M = 1015 GeV should be the following * 1: N
mt =mb ,
mu/m e = 9ms/m d ,
(1)
so that the mass renormalization [3,4] leads to m r ~ m b / 3 and the other relation remains unchanged at low energies. Eq. (1) was obtained with the aid of three 5's and one 45 of Higgs. The question of strong CP violation was addressed in a similar model and no violation was found in the tree approximation and the violation was shown to be computable in higher order [5]. We present here an SU(5) gauge model, supplemented by a discrete symmetry, that uses two 5's and one 45 of Higgs to generate the fermion masses. We obtain the mass relations (1) and expressions for the Kobayashi-Maskawa [6] mixing angles s 2 and s 3 in terms of the quark mass ratios and the Cabibbo angle. The weak CP violating phase is taken into account and no strong CP violation is found in the tree approximation. Let the three 10 and 5 multiplets of SU(5) corresponding to the three families of left-handed fermions be denoted by ~abiLan~x/'eR, and the two _5's and one 45,.~of Higgs by ~bd and Eedf, respectively. The indices a, b, c, d, e, and f a r e SU(5) indices and run from 1 to 5, i a n d j refer to the number of families and run from 1 to 3 and k = 1,2. The most general Yukawa couplings of fermions and Higgs bosons in SU(5) are given by f~y
-,.kVab a ~ b - ; a b e ~ab + ~ k - a b T O . c d - e -- -abT O - c d ~ e f + =Jil:~iLXiR@k ~-gij~iLXjR2~c ni/g~iL 7 ~jLqgkeabcde+aijqQL "Y ~ j L ~ e eabde f h.c.
(2)
In eq. (2), tpT70 = ~c where We is the charge conjugate field, the letter T means transpose, eabcde is the completely antisymmetric five index tensor, and a sum over repeated indices is implied. In order to restrict the coupling parameters, the following discrete symmetry is imposed: filL ~ r/2mfflL ,
ff2L ~ r/Zn~bZL ,
X1R - + ~ - ( m - 7 n ) X i R , ~b1 ~ ~(m-5n)~b 1 ,
ff3L -+7?(m+n)~3L ,
X2R "~rT(m+5n)X2R ,
~b2 ~ r/-2(m+n)~b 2 ,
X3R -->r/6nX3R ,
Y~~ r/-(m+3n)z ,
(3)
,1 The actual mass relations given in ref. [2] are mr = rob, mta = 3ms, and me = md/3. The present model also leads to these three mass relations. 107
Volume 92B, number 1,2
PHYSICS LETTERS
5 May 1980
where m and n are arbitrary numbers and [r/i[ = I. The resulting Yukawa coupling is :~r~-ab
a
--ab a
--ab a
b
l :,--ab. c x,ab
£ = ( A WILX2R+A~2LX1R + B~b3LX3R)~bl + ~"~2L,~2R'-'c + l:~-abT
O-cd
abT 0
cd
e
1
abT 0 cd
ef
~ L ~ I L "[ ~g2L+E~3L 3' ~3L)~2eabcde + 15Ft~2L ")' ~3LY~c eabdef + h . c . .
(4)
At the level of SU(3) × SU(2) symmetry, we expect there are three Higgs multiplets (which are SU(3) color singlets and SU(2) weak doublets) that generate masses corresponding to the familiar three generations. In the lagrangian (4) we do have three Higgs fields, two 5's and one 45. Note that the term with the coefficient F does not contribute to the lepton masses. After spontaneous symmetry breaking, the fermion mass matrices M U, M D, and M ~ for charge 2/3 quarks ~0 = (u0, c 0, to), charge - 1 / 3 quarks ~0 = (do, So , b0), and leptons ~0 = (e0, P0, ~'0) take the form, respectively, (t~ el_ MU = 2D ia2
0 M~ =
V l A e ~°q 0
I)2De-i~2 0
0 oF
vF
v2Ee -ic~2
V l A ' e wq
0
-3vC
0
0
1
( ,
MD =
0 VlAe icq
vlA'ei~x
0
vC
0
0
0
v l B e ial
(5,6)
,
(7)
,
V l B e lal
where the vacuum expectation values (vev) are (¢5) = vieiai and ( ~ 5 ) = v(5~ -- 45a46b4), i = 1,2, a, b = 1-4. We note from eqs. (5), (6) and (7) that for M U equal coupling parameters are associated with the same vev, and for M D and M ~ the same situation holds when A = A' (which is approximately the case as we shall see later). The effect is that the mass matrices are independent of the scale, i.e., the size of the vev's. One can eliminate the phases in the mass matrices (5) and (6) by a suitable redefinition of the charge 2/3 and - 1 / 3 quark states so that Arg(Det M U Det M D) = 0 and hence there is no strong CP violation in the tree approximation ,2 Since M U and M D are not hermitian, we diagonalize the square of the mass matrices M U M Ut and M D M D? given by
M U M U~ = D F e i~2
D 2 +F 2
E F e ia2
E F e -ic~2
E2 +F 2
,
M D M Dt =
A ' C e ial
A 2+C 2
0
0
0
B2
,
(8,9)
where the vacuum expectation values Vl, v2, and u are suppressed. The characteristic equation for M U M Uy with 2 m c2 , m t2 ) yields eigenvalues (mu, mu2 + m 2 + m 2 = 2 ( D 2 + F 2 ) + E 2
,
2 t2 + m t2m 2u = 2D2E 2 + (/9 2 + F 2 ) 2 , m u2m 2c + mcrn
mumcm t=DzE,(10)
and for M D M Dt with eigenvalues (m 2, ms, 2 mb) 2 yields rn2_vm2=A,2 +a2 +c2,
mdrn s2 2=A2A,2,
m2=B 2.
(11)
The solution of eq. (10) corresponding to the eigenvalues for the masses (mu, - m c , rnt) of eq. (5) with a 2 = 0 is D = ( r n u m c m t / m t - m e + mu)l/2 ,
E = rn t - m c + m u ,
F = [mum c + mere t - m t m u ~- ( m u m c m l / m t - rn c + mu)] 1/2 .
,2 We thank R.N. Mohapatra for a discussion on this point. 108
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Volume 92B, number 1,2
PHYSICS LETTERS
5 May 1980
There are four unknowns A, A', B, and C in eq. (i 1). We shall find that A ' i s determined by the Cabibbo angle 0c, md, and m s. The mass matrices (8) and (9) are diagonalized, respectively, by the following unitary transformations ¢10 = U I ~ I and ~0 = U2~2 :
- N 2Dim c
N3 D/mt
N 2 e 2ic~2
N 3 e 2ic~2
-(N2F/m c +E ) e i~2
(N 3 Flint - E ) e i~
N1D/mu U 1 = {.
N1 e 2ic~2
I(N1F/rnu - E)e •
ia2
(13)
where N i = [1 + (D/mi)2 + (F/m i - E ) 2 ] -1/2, mi = (mu, _mc ' rot), sin 0
cos 0
0° )
(14)
cos 0 0e -ial
U2 = /-sin 00e-i~l
e-iCq where
(15)
tan20 = (A,2 , md)/(m 2 s2 -- A'2).
The charged current coupled to physical quarks are Juc = --0 ~lLT~z if02L = ~ILFTuff2L where the phase transformations I~IL ~ Q1P1~IL and ff2L -+ Q2P2ff2L are introduced to bring P to the form
r=Q e v v2e2o2 '(N1D/mu)cosO-N 1 s i n 0 e -i°
(NlD/mu) sinO+N 1 c o s 0 e -i6
N1F/m u - E
= Q~ -(N2D/mc) cosO e i~ - N 2 sin0
-(N2D/mc) sinO e i6 + N 2 cos0
-(N2F/mc + E ) ei~
(N3D/mt) sin 0 e i8 + N 3 cos 0
(N3F/m t - E ) e i~
(N3D/mt) cos 0 e i~ - N 3 sin 0
Q2 ,
(16)
where P1 = diag(1, e -i~ , e -i6) and P2 = diag(1, 1, e-in), and 6 = ~1 + 2a2,/3 = - cq - a2, and the phase transformations Q1 and Q2 will be'defined shortly. The leading approximation to eq. (16) with the aid of eqs. (1 1) and (12) is I cos 0 - (mu/mc) 1/2 sin 0 e -i6 F=Q~
-sinO-(mu/mc)l/2cos008
sin 0 + (mu/mc)l/2 cos 0 e -i6
(mv/mt) 1/2
cos 0 - (mu/mc)I/2 sin 0 e i~
-(mc/mt)l/2 e i~
(mc/mt)l/2 cos 0
ei6
-sin 0 (mc/mt) 1/2
Q2 •
(17)
Let us define the Cabibbo angle 0 c as cos 0 - (mu/mc)l/2 sin 0 e -i5 = cos 0 c e io ,
sin 0 + (mu/mc) 1/2 cos 0 e -i6 = sin 0 c e i° ,
(18)
where p and o are arbitrary phases. Then the choice of Q1 = diag( eip, e-i°, e - i ° ) and Q2 = diag(1, e i(p-°), e io) yields the Kobayashi-Maskawa form F:
l Cl
Sl
SlS3 I
--s 1
c1
s2ei~c /
S1S2
--c1s 2 -- S3 e i~c
el~C /
I~=
(19)
where 109
Volume 92B, number 1,2 S1 = sin 0c,
PHYSICS LETTERS
5May 1980
s 2 = sin 02 = -(rnc/mt) 1/2 + (mu/mt)l/2 cot 0 c e i~c ,
s 3 = sin 03 = --(mu/mt) 1/2 (sin 0 c ) - l ,
6c = P + o + 6 .
(20)
Note that in this approximation 0 c depends on an u n k n o w n constant A ' as given by eq. (15). If we put the familiar expression tan20 = md/m s in eq. (15) then A ' = A = (mdms) 1/2 with the aid of eqs. (11), while the experimental value tan 0 = 0.23 gives A = A' to within 10%. The lepton mass matrix can be handled in a similar way. The combination of the equations corresponding to eqs. (11) [that follow from (7)], eqs. (11), and the result C 2 >>A 2, A '2 lead to the mass relations (1). The CP violation in the K 0 - K 0 system is proportional to Im P~l ['32 of (19) and is expressible in terms of the mixing angles and 6 c by [7] 2 t2/ m c2) ] . [el = 2 IClC2CgS2S3 sin ~ c I [c2(m2/m 2 -- m 2) ln(m2/m 2) -- c 2 + s2(m
(21)
We obtain from eqs. ( 1 9 ) - ( 2 1 ) lel = 2(mcmsmu/mdm2t) 1/2 [ln(m2/m2c) -- 1 + (mt/mc)][sin 6c1 = 0.5 Isin 6cl,
(22)
with ms/m d = 20, m u = 4 MeV, m c = 1.5 GeV and m t = 25 GeV. The experimental value e = 2 X 10 - 3 is much smaller than (22) unless sin 6 c is very small. We note from (19) and (20) that P(b ~ u + X)/['(b -+ c + X) = mu/m c = 2.7 X 10 -3 so that b-quarks decay predominantly into c-quarks • The b-quark lifetime r b can be obtained from r b 3= r ~ (m u /m b )5/(m c /m t ), where ~-~ = 2.2 × 10 - 6 s is the muon lifetime of mass mu. One obtains r b ~ 2 × 10 -1 s for m t = 25 GeV. One of us (KT) would like to thank J. Prentki for many discussions on the SU(5) gauge model and the hospitality at CERN and C. Quigg for the hospitality at Fermilab. This work was supported in part by the US Department of Energy under Contract No. EY-76-C-02-1545.*000.
References [1] [2] [3] [4] [5] [6] [7]
110
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