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Natural flavour conservation and the fourth generation
Volume 86B, number 3,4
PHYSICS LETTERS
8 October 1979
N A T U R A L F L A V O U R CONSERVATION AND THE FOURTH GENERATION D. GROSSER
Institut fiir TheoretischePhysik, Universitdt~bingen, Germany Received 4 June 1979
In a sequential Weinberg-Salam model with eight quarks and two Higgs doublets we fix the Yukawa interaction by requiring S4 invariance. The model yields naturally: (i) flavour conservation in the neutral currents; (ii) m u = m d = 0; (iii) ms: mb : m~ = m c : m t : mh; (iv) m~ = m s + m b, m h = m c + m t, where h and ~ denote the new quarks. The model provides an explanation of why quark masses are so different. It follows from (iv) that the mass of ~ is not much greater than m T. Perhaps ~ has already been seen.
The sequential Weinberg-Salam model is in remarkable agreement with a wide variety o f experimental data. Nevertheless the Higgs sector o f this model is n o t specified. We do not know the number o f Higgs doublets and their interaction with the fermions, the Yukawa interaction. This interaction provides the fermion masses and the mixing angles. The observed suppression o f the strangeness-changing neutral currents supplies some help as far as the quarks are concerned. It has led to the assumption that all flavours are conserved in neutral currents. Glashow and Weinberg [1 ] proposed the idea that this flavour conservation in the neutral currents should emerge naturally, i.e. as a consequence o f the group structure and the representation content. Discrete symmetries are best suited because their spontaneous breaking will n o t introduce any unwanted particles. These symmetries would restrict if not fix the Yukawa interaction and hence could determine the quark mixing angles and mass ratios. Gatto et al. [2] have shown, however, that since these symmetries have to be chosen such that the quark masses are nondegerate they will not lead to physically acceptable mixing angles in lowest order. In this approach, therefore, mixing angles are a higher order effect. In a previous note [3] we remarked t h a t i f the discrete symmetries in this approach have certain properties then they imply the following relation between quark masses m u : m c : m t : . . . = m d : m s : m b : ....
in zeroth order. This relation makes sense only if mu = md = 0,
(2)
in this order. In this note we give a model w i t h d i s c r e t e symmetries which guarantee flavour conservation in the neutral currents and which have the properties required in ref. [3]. The model therefore yields (1). In addition it gives (2) naturally. We will now give the model and then discuss further properties. We start from the general sequential W e i n b e r g Salam model. One has 2n quarks: Pi (i = 1 ..... n) with charge 2/3 and N i (i = 1 ..... n) with charge - 1 / 3 . The left-handed quarks form doublets (PiL, NiL) and the right-handed quarks (PiR and N/R ) are singlets. The only Higgs bosons which can couple to these quarks are doublets ~ (a = 1 , 2 ; i = 1, ..., m) and their charge conjugates ~c~ = _ % ~ , . The most general SU(2) × U(1) invariant Yukawa interaction is -~y=
(P]L,NI'L) ( (~c2)A}kPkR
+ (P/L' NjL)
q~?
B;kNkR + h.c.,
(3)
where A i and B i are arbitrary matrices. Let ~ denote the total lagrangian. Z?-Z~y is invariant under an arbitrary group G o f discrete symmetries if the quark and Higgs fields transform as
(1) 301
Volume 86B, number 3,4 PL ~ KL(g)PL,
PHYSICS LETTERS
(4)
NR -->KR d (g)NR ,
AI = where K L (g), K~ (g), K d (g), and D ( g ) with g E G are unitary matrices forming a representation ofG. Z?y is invariant under G if the coupling matrices A t and B i satisfy K{ (g) [D+(g)] liA iK~ (g) = A l,
(5)
K{ (g) [D(g)] ilBiK~ (g) = B l .
(6)
Our model is now easily specified. We set n = 4 and m = 2. We choose G = S4, K L = K~ = K d = (3,1) + (4) = the four-dimensional "selfrepresentation" (for notation see Hamermesh [4] ), and D = (22), in particular
(-1/2
x/3/2~
0 ( 2 3 ) =~ x / ~ / 2
1/2
(7)
/ '
21
0
0
-1
0
-1
2
(8)
0
-1
1
0
2 X/-Sa -2v~-a
Together with (9) this implies flavour conservation in the neutral currents. The mass matrices are given by
gp = -(C+~A e, MN = ( ~ ) B " .
(11)
We can still choose the phases of the quark fields in such a way that the elements ofMp andM N are real and non-negative. Assuming T invariance we find from (9) and (11) (12)
0
2pl IPl - P2~4~1
'
0
(9)
pl*p2~l'
(13) This and (1) has been found as an approximate result in a different model by Pakvasa and Sugawara [5]. In addition (13) implies m h = m e + m t. Then because of (12) one has m~ = m s + mb. As has been pointed out in ref. [3] (1) implies m t to be approximately 20 GeV or 35 GeV depending on the value O f m c / m s. (13) explains why Pl and P2 generate very different quark masses even if they are of the same order of magnitude. The relation m~ = m s + m b states that m~ is not much greater than mb. Let us consider the possibility that £~has already been seen. Herb et al. [6] measuring the invariant mass dis-
where Oi = I ( ~ ) 1 . Thus we have m u
where b is an arbitrary constant. This means in particular that the Yukawa interaction has been fixed up to two constants, the most one can achieve by discrete symmetries. Since A 1 and A 2 commute, the quark fields can be 302
(o
M2= 21al ( 0
where a is an arbitrary constant. Since (5) and (6) are identical in our model we have B i = bA i,
A2=
where e is a constant. This implies (1). From (10) and (11) we obtain
/:° i1 -1
/_lll
(1 O)
-
-1
0
-2a
M N = eMp,
where (if) denotes the transposition i ~ j. The solution of (5) is
=aV~
4a
-2a
(°i ~ Dik (g)$k,
A2
(0 1
chosen such that the Ai are diagonal
NL ~ KL(g)NL ,
PR ~ K ~ ( g ) P R ,
8 October 1979
= m d = 0.
Volume 86B, number 3,4
PHYSICS LETTERS
tribution o f ~t+/~- pairs found two peaks. The one with lighter mass is interpreted as arising from the decay of b b - T. The other peak has been interpreted as the T ' However, when Herb et al. [7] compared improved data with the DESY data [8] on the T ' then they concluded that the second peak arises n o t only from the T ' . They proposed that it is partly generated by the decay of the T " . Our model suggests that this second peak contains the @- instead o f or in addition to the t ~ tt
8 October 1979
References [1] S.L. Glashow and S. Weinberg, Phys. Rev. D15 (1977) 1958. [2] R. Gatto, G. Morchio and F. Strocchi, Phys. Lett. 83B (1979) 348. [3] D. Grosser, Phys. Lett. 83B (1979) 355. [4] M. Hamermesh, Group theory and its application to physical problems (Addison-Wesley, Reading, 1962). [5] S. Pakvasa and H. Sugawara, Phys. Lett. 82B (1979) 105. [6] S.W. Herb et al., Phys. Rev. Lett. 39 (1977) 252. [7] S.W. Herb et al., Phys. Rev. Lett. 42 (1979) 486. [8] J.K. Bienlein et al., Phys. Lett. 78B (1978) 360; C.W. Darden et al., Phys. Lett. 78B (1978) 364.
303
PHYSICS LETTERS
8 October 1979
N A T U R A L F L A V O U R CONSERVATION AND THE FOURTH GENERATION D. GROSSER
Institut fiir TheoretischePhysik, Universitdt~bingen, Germany Received 4 June 1979
In a sequential Weinberg-Salam model with eight quarks and two Higgs doublets we fix the Yukawa interaction by requiring S4 invariance. The model yields naturally: (i) flavour conservation in the neutral currents; (ii) m u = m d = 0; (iii) ms: mb : m~ = m c : m t : mh; (iv) m~ = m s + m b, m h = m c + m t, where h and ~ denote the new quarks. The model provides an explanation of why quark masses are so different. It follows from (iv) that the mass of ~ is not much greater than m T. Perhaps ~ has already been seen.
The sequential Weinberg-Salam model is in remarkable agreement with a wide variety o f experimental data. Nevertheless the Higgs sector o f this model is n o t specified. We do not know the number o f Higgs doublets and their interaction with the fermions, the Yukawa interaction. This interaction provides the fermion masses and the mixing angles. The observed suppression o f the strangeness-changing neutral currents supplies some help as far as the quarks are concerned. It has led to the assumption that all flavours are conserved in neutral currents. Glashow and Weinberg [1 ] proposed the idea that this flavour conservation in the neutral currents should emerge naturally, i.e. as a consequence o f the group structure and the representation content. Discrete symmetries are best suited because their spontaneous breaking will n o t introduce any unwanted particles. These symmetries would restrict if not fix the Yukawa interaction and hence could determine the quark mixing angles and mass ratios. Gatto et al. [2] have shown, however, that since these symmetries have to be chosen such that the quark masses are nondegerate they will not lead to physically acceptable mixing angles in lowest order. In this approach, therefore, mixing angles are a higher order effect. In a previous note [3] we remarked t h a t i f the discrete symmetries in this approach have certain properties then they imply the following relation between quark masses m u : m c : m t : . . . = m d : m s : m b : ....
in zeroth order. This relation makes sense only if mu = md = 0,
(2)
in this order. In this note we give a model w i t h d i s c r e t e symmetries which guarantee flavour conservation in the neutral currents and which have the properties required in ref. [3]. The model therefore yields (1). In addition it gives (2) naturally. We will now give the model and then discuss further properties. We start from the general sequential W e i n b e r g Salam model. One has 2n quarks: Pi (i = 1 ..... n) with charge 2/3 and N i (i = 1 ..... n) with charge - 1 / 3 . The left-handed quarks form doublets (PiL, NiL) and the right-handed quarks (PiR and N/R ) are singlets. The only Higgs bosons which can couple to these quarks are doublets ~ (a = 1 , 2 ; i = 1, ..., m) and their charge conjugates ~c~ = _ % ~ , . The most general SU(2) × U(1) invariant Yukawa interaction is -~y=
(P]L,NI'L) ( (~c2)A}kPkR
+ (P/L' NjL)
q~?
B;kNkR + h.c.,
(3)
where A i and B i are arbitrary matrices. Let ~ denote the total lagrangian. Z?-Z~y is invariant under an arbitrary group G o f discrete symmetries if the quark and Higgs fields transform as
(1) 301
Volume 86B, number 3,4 PL ~ KL(g)PL,
PHYSICS LETTERS
(4)
NR -->KR d (g)NR ,
AI = where K L (g), K~ (g), K d (g), and D ( g ) with g E G are unitary matrices forming a representation ofG. Z?y is invariant under G if the coupling matrices A t and B i satisfy K{ (g) [D+(g)] liA iK~ (g) = A l,
(5)
K{ (g) [D(g)] ilBiK~ (g) = B l .
(6)
Our model is now easily specified. We set n = 4 and m = 2. We choose G = S4, K L = K~ = K d = (3,1) + (4) = the four-dimensional "selfrepresentation" (for notation see Hamermesh [4] ), and D = (22), in particular
(-1/2
x/3/2~
0 ( 2 3 ) =~ x / ~ / 2
1/2
(7)
/ '
21
0
0
-1
0
-1
2
(8)
0
-1
1
0
2 X/-Sa -2v~-a
Together with (9) this implies flavour conservation in the neutral currents. The mass matrices are given by
gp = -(C+~A e, MN = ( ~ ) B " .
(11)
We can still choose the phases of the quark fields in such a way that the elements ofMp andM N are real and non-negative. Assuming T invariance we find from (9) and (11) (12)
0
2pl IPl - P2~4~1
'
0
(9)
pl*p2~l'
(13) This and (1) has been found as an approximate result in a different model by Pakvasa and Sugawara [5]. In addition (13) implies m h = m e + m t. Then because of (12) one has m~ = m s + mb. As has been pointed out in ref. [3] (1) implies m t to be approximately 20 GeV or 35 GeV depending on the value O f m c / m s. (13) explains why Pl and P2 generate very different quark masses even if they are of the same order of magnitude. The relation m~ = m s + m b states that m~ is not much greater than mb. Let us consider the possibility that £~has already been seen. Herb et al. [6] measuring the invariant mass dis-
where Oi = I ( ~ ) 1 . Thus we have m u
where b is an arbitrary constant. This means in particular that the Yukawa interaction has been fixed up to two constants, the most one can achieve by discrete symmetries. Since A 1 and A 2 commute, the quark fields can be 302
(o
M2= 21al ( 0
where a is an arbitrary constant. Since (5) and (6) are identical in our model we have B i = bA i,
A2=
where e is a constant. This implies (1). From (10) and (11) we obtain
/:° i1 -1
/_lll
(1 O)
-
-1
0
-2a
M N = eMp,
where (if) denotes the transposition i ~ j. The solution of (5) is
=aV~
4a
-2a
(°i ~ Dik (g)$k,
A2
(0 1
chosen such that the Ai are diagonal
NL ~ KL(g)NL ,
PR ~ K ~ ( g ) P R ,
8 October 1979
= m d = 0.
Volume 86B, number 3,4
PHYSICS LETTERS
tribution o f ~t+/~- pairs found two peaks. The one with lighter mass is interpreted as arising from the decay of b b - T. The other peak has been interpreted as the T ' However, when Herb et al. [7] compared improved data with the DESY data [8] on the T ' then they concluded that the second peak arises n o t only from the T ' . They proposed that it is partly generated by the decay of the T " . Our model suggests that this second peak contains the @- instead o f or in addition to the t ~ tt
8 October 1979
References [1] S.L. Glashow and S. Weinberg, Phys. Rev. D15 (1977) 1958. [2] R. Gatto, G. Morchio and F. Strocchi, Phys. Lett. 83B (1979) 348. [3] D. Grosser, Phys. Lett. 83B (1979) 355. [4] M. Hamermesh, Group theory and its application to physical problems (Addison-Wesley, Reading, 1962). [5] S. Pakvasa and H. Sugawara, Phys. Lett. 82B (1979) 105. [6] S.W. Herb et al., Phys. Rev. Lett. 39 (1977) 252. [7] S.W. Herb et al., Phys. Rev. Lett. 42 (1979) 486. [8] J.K. Bienlein et al., Phys. Lett. 78B (1978) 360; C.W. Darden et al., Phys. Lett. 78B (1978) 364.
303