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Discrete symmetries, Cabibbo universality and flavor mixing angles
Volume 76B, number 1
PHYSICS LETTERS
8 May 1978
DISCRETE SYMMETRIES, CABIBBO UNIVERSALITY AND FLAVOR MIXING ANGLES G.C. BRANCO Physikalisches Institut, Universitiit Bonn, Germany
Received 10 January 1978
A model is proposed where deviations from exact Cabibbo universality are naturally small, and the correct value for the Cabibbo angle is obtained.
One of the outstanding problems in the unified gauge theories of weak and electromagnetic interactions is the understanding of the Cabibbo-like angles [1 ]. Although the occurrence of these angles is natural (in general the weak currents are not flavor diagonal in the basis where the quark mass matrix is diagonal), their value is entirely arbitrary. A related problem is the understanding of Cabibbo universality. In the standard model [2] with only two doublets, Cabibbo universality is a natural property of the theory, i.e. it is satisfied for any value of its free parameters. However, when we extend the theory to more than two doublets, as it seems to be required by the T(9.5) discovery, the experimental validity of universality becomes an accident. In this note, we consider a model with six quarks, where violations of universality are naturally small and the correct value for the Cabibbo angle is obtained. In choosing the gauge model, we were guided by recent experimental results [3 ] : (i) No evidence for righthanded currents in antineutrino scattering experiments, (ii) the absence of parity violation in atomic physics and (iii) the discovery of T(9.5), in hadron-hadron collisions. These results motivate the consideration of a SU(2)L ® SU(2)R ® U(1) gauge model [4], with the following quark multiplets:
where the left- and right-handed multiplets transform as (½,0, 1), (0, 3,x 1), respectively. It has been shown [4] that it is possible to choose the Higgs system in such a way that neutral currents conserve parity and yet the. charged currents are left-handed at low energies Here we will be only concerned with Higgs multiplets transforming as (½, 3,0), 1 since only they can give mass to quarks. We introduce three such multiplets ~i and impose the following two discrete symmetries: RI: Q1L-~ Q3R ,
Q1R-+ Q3L ,
Q2L -~ ein/4O2R, Q2R~ein/4O2L , ~b2 ~ i~b~ 2 ,(2) Q3L-+iQ1R ,
Q3R-~ iQ1L ,
R2: OjL-iQjL,
QjR OjR,
Q1R= 70
L
()
dU R '
s L
()
Q2R- sc R '
Q3L = b L ' Q3R
:(t)
b R,
(1)
~3 --+ei~r/4qS~,
The most general Yukawa interaction which respects this symmetry is: gl(Q1L~IQ1R + Q2L~IQ2R + Q3L~I Q3R)
(3)
+ g2(Q3L~2Q1R) ÷ g3 (Q2L~3 Q1R + Q3L~3Q2R) ÷h.c. After spontaneous symmetry breaking, we obtain the following mass terms:
Earl
[dsb] L Md
+" [uct] L Mu
b R
QIL
~bl ~ b ~ ,
where M d, M u are of the form /3a
.
Eil R
+ h.c.,
(4)
Volume 76B, number 1
PHYSICS LETTERS
The mass matrices for the down and up quarks can be diagonalized by performing biunitary transformations M d._.>(Ad)-lMdA d, M u ~ ( A Lu) -1 M Au R u. I f w e n o w take into account the special form o f M d, M u, the matrices A ~ , A ~ can be evaluated and expressed in terms o f the quark masses. Using the parametrization of Kobayashi and Maskawa*i [5] for the unitary matrices:
Cl
--SlC3
SlC 2
ClC2C3+S2S3
tSlS 2
ClC3S2--c2s 3
-SlS3
I
(5)
ClC2S 3 - s 2 c 3 1 , I c2c3+clS2S3 t
one then obtains for the mixing angles o f A d : md(l+rT),
S2d~
S~d ~ m-~s
md
(rndmsmb)l/3
( */ )
~-~
ms( )
,
(6)
8 May 1978
other quark masses ,2 : md ~ 7.5 MeV, m s ~ 150 MeV, m b ~ 4750 MeV, m u ~ 4 . 2 MeV, m c ~ 1150 MeV, 5000 MeV < m t < oo. One obtains then the following bounds for the Cabibbo angle: 0.18 ~< sin 0 c ~< 0.25 .
(9)
In order to obtain a further restriction on the Cabibbo angle, we need to estimate the value of the top quark mass. For that, we use our mixing scheme and the known value o f the K L - K S mass difference. Most of the contribution to the X + fi ~ 7t + n effective lagrangian [6] comes from box diagrams where two vector gauge bosons are exchanged , 3 . Since in the present model m~v R~.. >> m 2 L , the dominant contribution comes from WL exchange and therefore the K 0 - K 0 transition amplitude essentially coincides with that o f the KM model. We can thus use the result o f Ellis et al. [7] who obtained: M K ° - K ° ~ (K01 - ,.~eff[ K 0) ~ ( G F / ~ / 2 ) f 2 m 2 (o~/4rr) e,
where Sid refer to down quarks and ~ = [(ms/mb)/ (rod~ms)] 1/3. Identical expressions hold for the angles of A ~ , with m u, m c, m t instead of rod, ms, m b, respectively. The left-handed charged current can be written
with
as
The K L - K s mass difference is given by m L - m S ~.MK°-K°/m K. From the experimental value (m L - m s ) / m K ~ 7.1 × 10 -15 , it follows that e 0.7 × 1 0 - 4 , and using eqs. (7), (8), one obtains ,4 m t ~ 10 GeV. For this value o f rot, one gets for the Cabibbo angle sin 0 c -~ 0.2, which is to be compared with the experimental value [8] sin 0 c = 0.230 -+0.003. The other mixing angles are calculated using eqs. (6), (8) and one obtains s 2 ~ 0.3, s 3 ~ 0.1. This leads to rather small deviations from exact Cabibbo universality, since s2s 2 ~ 4 × 10 - 4 . Experimentally [91 one has cos20 c + sin20 c = 1.001 + 0.004, which implies the upper bound s2s 2 < 3 × 10 - 3 .
Lb'j L tl - 1 d where U L = ( A L ) A L and the primed quarks are eigenstates of the mass matrix. Using eq. (6), one can then evaluate UL, and the Cabibbo angle is given by: sin 0 c ~. s 1 .~ Sld -- Slu
[
I
~m d (I +,7) 1/2 _
mc (1+n'
,
(7)
(10) 2 2 2
2 2 2 ~c4m2+s2m2+ 2s2C2mc
--,lClC3 t 2
c 2 t d
2
, [ m t ~']
Jnt
J}il)
where r/'= [(mc/mt)/(mu/mc) ] 113. Similarly, one gets for the other mixing angles o f U L :
SlS 2 ~ S3u(Sld - Slu ) + Sld(S2d - S2u),
(8) SlS 3 ~ S3d(Sld -- Slu ) + Slu(S2d-- S2u ) . In order to evaluate 0c, we assume that the T is a bb bound state and use the conventional values for the
4:1 For simplicity, we neglect CP violation.
,2 We took here the values quoted by Weinberg in the first paper of ref. [1 ]. ,3 We assume that the physical Higgs particles which remain after spontaneous symmetry breaking are sufficiently heavy for their contribution to the AS = 2 effective lagrangian to be negligible. ,4 It is clear that due to the uncertainty in the value of the "known" quark masses, this estimation of m t (within the proposed mixing scheme) should be taken at most as an order of magnitude. 71
Volume 76B, number 1
PHYSICS LETTERS
References [1] S. Weinberg, Harvard preprint HUTP-77/A057; H. Fritzsch, Phys. Lett. 70B (1977) 436; F. Wilczek and A. Zee, Phys. Lett. 70B (1977) 418; A. De Rujula, H. Georgi and S.L. Glashow, Ann. Phys. 109 (1977) 258. [2J S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: Proc. 8th Nobel Syrup., ed. N. Svartholm (Stockholm, 1968) p. 367. [3] Proc. Intern. Symp. on Lepton and photon interactions (Hamburg, 1977). [4] R.N. Mohapatra and D. Sidhu, Phys. Rev. Lett. 38 (1977) 667;
72
8 May 1978
A. De Rujula, H. Georgi and S.L. Glashow, Ann. Phys. 109 (1977) 242; H. Fritzsch and P. Minkowski, Nucl. Phys. B103 (1976) 61. [5] M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49 (1973) 652. [6J M.K. Galliard and B.W. Lee, Phys. Rev. DI0 (1974) 897. [7] J. Ellis, M.K. Galliard, D.V. Nanopoulos and S. Rudaz, CERN preprint TH 2346. [8] K. Kleinknecht, Proc. 17th Intern. Conf. on High energy physics (London, 1974). [9] C. Jarlskog et al., Nucl. Phys. B109 (1976) 1.
PHYSICS LETTERS
8 May 1978
DISCRETE SYMMETRIES, CABIBBO UNIVERSALITY AND FLAVOR MIXING ANGLES G.C. BRANCO Physikalisches Institut, Universitiit Bonn, Germany
Received 10 January 1978
A model is proposed where deviations from exact Cabibbo universality are naturally small, and the correct value for the Cabibbo angle is obtained.
One of the outstanding problems in the unified gauge theories of weak and electromagnetic interactions is the understanding of the Cabibbo-like angles [1 ]. Although the occurrence of these angles is natural (in general the weak currents are not flavor diagonal in the basis where the quark mass matrix is diagonal), their value is entirely arbitrary. A related problem is the understanding of Cabibbo universality. In the standard model [2] with only two doublets, Cabibbo universality is a natural property of the theory, i.e. it is satisfied for any value of its free parameters. However, when we extend the theory to more than two doublets, as it seems to be required by the T(9.5) discovery, the experimental validity of universality becomes an accident. In this note, we consider a model with six quarks, where violations of universality are naturally small and the correct value for the Cabibbo angle is obtained. In choosing the gauge model, we were guided by recent experimental results [3 ] : (i) No evidence for righthanded currents in antineutrino scattering experiments, (ii) the absence of parity violation in atomic physics and (iii) the discovery of T(9.5), in hadron-hadron collisions. These results motivate the consideration of a SU(2)L ® SU(2)R ® U(1) gauge model [4], with the following quark multiplets:
where the left- and right-handed multiplets transform as (½,0, 1), (0, 3,x 1), respectively. It has been shown [4] that it is possible to choose the Higgs system in such a way that neutral currents conserve parity and yet the. charged currents are left-handed at low energies Here we will be only concerned with Higgs multiplets transforming as (½, 3,0), 1 since only they can give mass to quarks. We introduce three such multiplets ~i and impose the following two discrete symmetries: RI: Q1L-~ Q3R ,
Q1R-+ Q3L ,
Q2L -~ ein/4O2R, Q2R~ein/4O2L , ~b2 ~ i~b~ 2 ,(2) Q3L-+iQ1R ,
Q3R-~ iQ1L ,
R2: OjL-iQjL,
QjR OjR,
Q1R= 70
L
()
dU R '
s L
()
Q2R- sc R '
Q3L = b L ' Q3R
:(t)
b R,
(1)
~3 --+ei~r/4qS~,
The most general Yukawa interaction which respects this symmetry is: gl(Q1L~IQ1R + Q2L~IQ2R + Q3L~I Q3R)
(3)
+ g2(Q3L~2Q1R) ÷ g3 (Q2L~3 Q1R + Q3L~3Q2R) ÷h.c. After spontaneous symmetry breaking, we obtain the following mass terms:
Earl
[dsb] L Md
+" [uct] L Mu
b R
QIL
~bl ~ b ~ ,
where M d, M u are of the form /3a
.
Eil R
+ h.c.,
(4)
Volume 76B, number 1
PHYSICS LETTERS
The mass matrices for the down and up quarks can be diagonalized by performing biunitary transformations M d._.>(Ad)-lMdA d, M u ~ ( A Lu) -1 M Au R u. I f w e n o w take into account the special form o f M d, M u, the matrices A ~ , A ~ can be evaluated and expressed in terms o f the quark masses. Using the parametrization of Kobayashi and Maskawa*i [5] for the unitary matrices:
Cl
--SlC3
SlC 2
ClC2C3+S2S3
tSlS 2
ClC3S2--c2s 3
-SlS3
I
(5)
ClC2S 3 - s 2 c 3 1 , I c2c3+clS2S3 t
one then obtains for the mixing angles o f A d : md(l+rT),
S2d~
S~d ~ m-~s
md
(rndmsmb)l/3
( */ )
~-~
ms( )
,
(6)
8 May 1978
other quark masses ,2 : md ~ 7.5 MeV, m s ~ 150 MeV, m b ~ 4750 MeV, m u ~ 4 . 2 MeV, m c ~ 1150 MeV, 5000 MeV < m t < oo. One obtains then the following bounds for the Cabibbo angle: 0.18 ~< sin 0 c ~< 0.25 .
(9)
In order to obtain a further restriction on the Cabibbo angle, we need to estimate the value of the top quark mass. For that, we use our mixing scheme and the known value o f the K L - K S mass difference. Most of the contribution to the X + fi ~ 7t + n effective lagrangian [6] comes from box diagrams where two vector gauge bosons are exchanged , 3 . Since in the present model m~v R~.. >> m 2 L , the dominant contribution comes from WL exchange and therefore the K 0 - K 0 transition amplitude essentially coincides with that o f the KM model. We can thus use the result o f Ellis et al. [7] who obtained: M K ° - K ° ~ (K01 - ,.~eff[ K 0) ~ ( G F / ~ / 2 ) f 2 m 2 (o~/4rr) e,
where Sid refer to down quarks and ~ = [(ms/mb)/ (rod~ms)] 1/3. Identical expressions hold for the angles of A ~ , with m u, m c, m t instead of rod, ms, m b, respectively. The left-handed charged current can be written
with
as
The K L - K s mass difference is given by m L - m S ~.MK°-K°/m K. From the experimental value (m L - m s ) / m K ~ 7.1 × 10 -15 , it follows that e 0.7 × 1 0 - 4 , and using eqs. (7), (8), one obtains ,4 m t ~ 10 GeV. For this value o f rot, one gets for the Cabibbo angle sin 0 c -~ 0.2, which is to be compared with the experimental value [8] sin 0 c = 0.230 -+0.003. The other mixing angles are calculated using eqs. (6), (8) and one obtains s 2 ~ 0.3, s 3 ~ 0.1. This leads to rather small deviations from exact Cabibbo universality, since s2s 2 ~ 4 × 10 - 4 . Experimentally [91 one has cos20 c + sin20 c = 1.001 + 0.004, which implies the upper bound s2s 2 < 3 × 10 - 3 .
Lb'j L tl - 1 d where U L = ( A L ) A L and the primed quarks are eigenstates of the mass matrix. Using eq. (6), one can then evaluate UL, and the Cabibbo angle is given by: sin 0 c ~. s 1 .~ Sld -- Slu
[
I
~m d (I +,7) 1/2 _
mc (1+n'
,
(7)
(10) 2 2 2
2 2 2 ~c4m2+s2m2+ 2s2C2mc
--,lClC3 t 2
c 2 t d
2
, [ m t ~']
Jnt
J}il)
where r/'= [(mc/mt)/(mu/mc) ] 113. Similarly, one gets for the other mixing angles o f U L :
SlS 2 ~ S3u(Sld - Slu ) + Sld(S2d - S2u),
(8) SlS 3 ~ S3d(Sld -- Slu ) + Slu(S2d-- S2u ) . In order to evaluate 0c, we assume that the T is a bb bound state and use the conventional values for the
4:1 For simplicity, we neglect CP violation.
,2 We took here the values quoted by Weinberg in the first paper of ref. [1 ]. ,3 We assume that the physical Higgs particles which remain after spontaneous symmetry breaking are sufficiently heavy for their contribution to the AS = 2 effective lagrangian to be negligible. ,4 It is clear that due to the uncertainty in the value of the "known" quark masses, this estimation of m t (within the proposed mixing scheme) should be taken at most as an order of magnitude. 71
Volume 76B, number 1
PHYSICS LETTERS
References [1] S. Weinberg, Harvard preprint HUTP-77/A057; H. Fritzsch, Phys. Lett. 70B (1977) 436; F. Wilczek and A. Zee, Phys. Lett. 70B (1977) 418; A. De Rujula, H. Georgi and S.L. Glashow, Ann. Phys. 109 (1977) 258. [2J S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: Proc. 8th Nobel Syrup., ed. N. Svartholm (Stockholm, 1968) p. 367. [3] Proc. Intern. Symp. on Lepton and photon interactions (Hamburg, 1977). [4] R.N. Mohapatra and D. Sidhu, Phys. Rev. Lett. 38 (1977) 667;
72
8 May 1978
A. De Rujula, H. Georgi and S.L. Glashow, Ann. Phys. 109 (1977) 242; H. Fritzsch and P. Minkowski, Nucl. Phys. B103 (1976) 61. [5] M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49 (1973) 652. [6J M.K. Galliard and B.W. Lee, Phys. Rev. DI0 (1974) 897. [7] J. Ellis, M.K. Galliard, D.V. Nanopoulos and S. Rudaz, CERN preprint TH 2346. [8] K. Kleinknecht, Proc. 17th Intern. Conf. on High energy physics (London, 1974). [9] C. Jarlskog et al., Nucl. Phys. B109 (1976) 1.