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Fermion masses from grand unification with O(14)
Volume lOlB, number 4
PHYSICS LETTERS
14 May 1981
FERMION MASSES FROM GRAND UNIFICATION WITH 0( 14)
Hikaru SAT0 ’ Department
of Physics, University of Tokyo, Hongo, Tokyo I 13, Japan
Received 2 January 1981
A mechanism is presented for generating fermion masses in the O(14) model which unifies color, flavor and generations. The masses of the conjugate generations, having V + A couplings to the ordinary weak currents are predicted to be O(10’) GeV. The mass of the observed generations can be obtained by the radiative corrections to the vanishing tree-level mass.
The 0( 14) is the smallest group with rank 7 which unifies all known quarks and leptons into a single irreducible fundamental (or spinor) representation [ 11. The spinor representation of 0(14) is 128, which is decomposed into 64 Q w under SO(14). The 64 of SO(14) is further decomposed into 16 @16 @% o i6 under SO(10) [2]. There are four 16’s in 128 so that this model predicts the existence of the fourth generation in addition to the known three:
fl =(u,,d,;ve,e;u~,d~;$,eC)L
,
f-2 =(c,,s,;~,,cc;c~,s~;~~,~c)~, f3 =(t,,b,;v,,7;t~,b~;v~,7C)L, f4 = (h, , 0, ; vf, f; h:, 0: ; v;, fCjL ,
(1)
where cr is the color index and c denotes the charge conjugated field *I. There are additional four 16’s in 128. Since a 16 has a decomposition, i6 = (4,1,2) @(3,2,1) into SU(4), QD Sum @SU(2)B, the fermiOnS with 4 Of SU(4), in 16 have a V t A coupling with SU(2), gauge bosons or a V - A coupling with SU(2)B gauge bosons. Thus the mass of the conjugate generations associated with i6 must be heavy. Concerning the mass of the fundamental fermions,
there arise two major--problems: (1) Besides the mass terms of 16-16 and 16-16, there are in general 16-G mass terms. They may acquire super heavy masses of 0( 1015) GeV corresponding to the super strong breaking of SO( 10) [3]. (2) The mass terms for both 16’s and i6’s must break SU(2)L @U(l), so that the 16’s and i6’s may have comparable masses. How can the hierarchical masses between 16 and i6 be understood? In this letter I present the mechanism to evade these difficulties and predict the mass of the conjugate generations to be O(102) GeV. The mass of 16’s can be generated by the radiative correction to the vanishing zeroth-order mass. Consider the strong symmetry breakdown of 0(14). This is achieved by using the Higgs fields of a 104 (traceless symmetric second rank tensor), ati (i,j = 1, ... . 14) and a 128, X of O(14). For a range of parameters in the potential, the vacuum expectation value (VEV) of ~ii can be in the form [4] (~ii> = diag(a ... b ... c ...) ,
(2)
where there are four a’s, six b’s and four c’s along the diagonal, with 4u t 6b t 4c = 0 *2. The VEV, eq. (2), breaks O(l4) down to SU(2)L @ SU(2), 0 SU(4), 0 SU(2)H 0 SU(2);I. Here SU(2)H and SU(2);I are the “horizontal” gauge symmetries which relate different generations [S]. The two generations (fr, f2) are as-
1 Present address: Department of Physics, Hyogo University of Education, Yashiro, Hyogo 673-14, Japan. *r The names of the seventh and the eigth quark are from Greek seven (heptu) and eigth (okt5).
*2 This VEV can be obtained for the potential in ref. [4] with -8hl < X2 < -6ht < 0.
0 03 l-9163/8 I /OOOO-0000/S 02.50 0 North-Holland Publishing Company
233
signed to a doublet of SU(2)I.I) while (f3, f4) are to a doublet of SU(2);I. The symmetry is further broken down to SU(2)L @U(l) 8 W(3), by giving the appropriate VEV’s to 2. The Higgs multiplet
0t
z=
7)
Z has components
1716 =(120,1,1)8(126,3,1)@(126, @(210,2,2)
1,3)
1
2002=2(126,1,1)e(120,3,1)te(120,1,3)
as
@(lo, 1,1)8(210,2,2)8(45,2,2).
’
where E (77)transforms as 64 (64) of SO(14). A component of each t and g with the respective quantum numbers, (1, 1,2,2,1) and (1, 1,2, 1,2) under SU(3)c 8 SU(2)L @ SU(2)R 8 SU(2)H @ SU(2); can acquire VEV for suitable conditions on the Higgs potential. This completes the strong symmetry breaking of 0(14) down to SU(2), o U( 1) o SU(3)c. I should emphasize here that the Higgs bosons with 104 or 128 of 0(14) cannot couple to the fermions with 128, hence the fermions can escape to get super heavy masses associated with the super strong breaking of 0(14) *3. The Higgs multiplets which can give masses to the fundamental spinors are those in 128 o 128. The Yukawa couplings are given by
sc’=f(~T,
14 May 1981
PHYSICS LETTERS
Volume lOlB, number 4
G*~)(;:: ;;:)(;) +h.c.
(6)
Since the Higgs multiplets which transform as 10, 126 [120] of SO(10) contribute to the mass matrices symmetric [antisymmetric] under the flavor indices [61, @(lo, 1, 1), $(126,1,1), @(120,3,1) and $(120, 1, 3) have no contribution to masses. The Higgs fields #(*, 2,2) generate the mixing between 16 and 16 of SO(10) in F or G*, therefore the VEV of $(*, 2,2) must be small. Thus the mass and the mixing angles are produced by the Higgs bosons in 364 and/or 1716
[71. For the neutrinos to acquire tiny Majorana masses, the Higgs bosons with 126 of SO(10) are necessary [8] The only choice left is then $I I = $I I (17 16) and 4~~ = @,,(1716). In 648~64 = 1@91@ 1001 e3003, the 9 1, which I use as @I2 = & , is decomposed into SG( 10) @SU(2)R @ SU(2);I as 91 =(45,1,1)fB(1,3,l)e(l,
1,3)
e(10,2,2). =f
\kT+P t h.c. ,
(4)
where the 128 spinors are put into
G*T=(&g;,f~,f4)~
(5)
Here F [G*] transforms as 64 [64] of SO(14) and fi [g;] transforms as 16 [i6] of SO(10). Note that (g!, g;) and (g;, &) are doublets of SU(2)H and SU(2)k, respectively. The Higgs bosons @II [&] transform as 64~64 [64@64] and@12 [&I] as64@64ofS0(14). The product decomposition of 64 o 64 is 64 @64 = 14 $364 8 1716 @2002, which have the SO(10) @SU(2), @ SU(2)h decompositions as
364 =(120,1,1)@(10,3, e(l,
l)e(i&
1,3)
292) >
*3 The radiative -correction
to the 16-16 mass will be of the order of OHM(16) = 1 GeV. See the discussion given later.
234
The Higgs fields @(lo, 2,2) generate the mixing between 16 in F and 16 in G* so that (u, d; c, s) and (t, b; h, o) can couple. Other Higgs bosons in 91 cause the mixing between 16 (i6) in F and i6 (16) in G*, therefore these VEV’s must be small. The mass m(16)of the conjugate generations is produced by the (126,1,3) and (126,3,1) components in 4I I and $22. Since these VEV’s violate SU(2)L 8 U(l), the mass of weak gauge bosons must be M,
=&11(126,
1,3)) =:8(@22(126, 331)) ,
with the gauge coupling constant g. The mass of m’s is then m(16)*:f($II(126,
1,3))=cf/g)Mw.
(8)
m(16)= O(102) GeV. The mass of the observed generations, m (16)must be much lighter than m(16)*4.This is due to ($II(126,3, I)), ($,2(126, 1,3))9@,,(126, 1,3)),etc. Forfeg,
14=(10,1,1)8(1,2,2),
@(45,2,2)
(7)
*’ I do not consider here the heavy Majorana They can be treated tail elsewhere.
separately
neutrinos and will be discussed
in 16. in de-
Volume lOlB, number 4
PHYSICS LETTERS
(16, 2,1 1
tl6,2,1)
(126,1,31 Fig. 1. The divergent selfenergy diagram, which vanishes due to the mismatch of the quantum numbers. The quantum numbers indicated are of the SO(lO)@ Su(&@ SU(2)fI. Since m (16) = am (16), it is attractive to consider that at the tree level only the conjugate generations acquire nonvanishing masses of 0(102) GeV, while the 16’sremain massless. The higher-order radiative corrections will generate the mass of the 16’s of the order of am(g). This scenario can actually happen in this model as outlined below. (i) It may be possible to arrange the Higgs potential for @ij such that only #I1(126,1,3) and $22(126, 3,l) acquire nonzero VEV’s. Thus the fermions of the conjugate generations become massive and others remain massless. (ii) There is no divergent self-energy diagram (fig. 1). Consider the correction to the fermions with (16,2, 1) of SO(10) QKI SU(2)R @ SU(2); in 64 of SG(l4). Since the gauge bosons connect two fermions in 64 or a, the intermediate heavy fermions are in (i6, 1,2) C 64, whose mass is generated by ($r L(126, 1,3)). There is no common channel between (126,1,3) and (16,2,1) 8 (16,2, I), hence fig. 1 vanishes. The correction to the (16, 1,2) fermions in 64 is also absent to this order.
14 May 1981
(iii) There is no contribution from the diagrams with a mixing among the gauge bosons as in fig. 2. Suppose the mixing of the (10,2,2) gauge bosons is induced by the VEV, (a>, and consider the correction to the (16,2, 1) fermions. Then it is easy to see that (a) o(~t)@($~~(126, 1,3)) cannot make (120, 1, 1) or (126,3, l)fora=&iiiZZ, $11(126,1,3) and d92(126, 3, 1). Note in particular that for a = Z, (16, 1,2) 8 (16, 1,2)~@(126,1,3)seemstomake(120,1,1).However, 1 of SU(2)R in (16,2,1) QX(16,2, 1) must be antisymmetric with respect to exchange of two 16’s in a doublet of SU(2), , which cannot be made by (0 a(,@) @(@LL).Although these diagrams give the finite mass to the 16’s, they may cause the divergent wave function renormalization for ~ii(12O, 1, l), $1 I (126, 3, l), r$22(126, 1,3) if the two end fermion lines are closed [9]. Absence of these diagrams is necessary to guarantee the existence of the solution for (ail) stated in (i). (iv) The first nonvanishing contribution to the mass comes from fig. 3. The (10,2,2) gauge bosons acquire super heavy masses due to the breaking of O(14) by (104). If the breaking by (128) is comparable to (lW, g(Z) =:M(lO, 2,2) =i&. Then the contribution from fig. 3 reads
= (g2/4n) m
Fig. 2. The mixing among the (10,2,2) gauge bosons occurs through nonvanishing VEV, (a) witha = @ii, E, @rr, @22.The diagrams for the (16,2,1) and (16, 1,2) fermions vanish.
(9)
The Cabibbo mixing between f, and f2 or f3 and f4 is generated by the mass matrices, hhich have the transformation property of (120, 1, l), (126,3, 1) and (126, 1,3)underS0(10)@SU(2)H@SU(2);I [7]. (v) The mixing between (fl , f2) and (f3, f4) is induced by Q12(91) whose VEV is chosen to be zero in (i), so that there is no mixing at the tree level. This
(16, 2,1) (126,1,31
(ii?) .
(i-,1,21 II
(16,2,11
1 (126,1,3)
Fig. 3. The diagram leading to the nonvanishing mass corrections. 235
<1>
Thus the of which radiative relatively
<$> ,
:
(gt)___iL’.---
--.\ ;:
__
+‘-I
(l&Z,))
!’
14 May 1981
PHYSICS LETTERS
Volume lOlB, number 4
\
(1%,1,2)
‘:
(16,1,2)
mass and mixing of the four generations, three are observed ones, have been generated by the corrections. This will explain naturally the small m (16) compared to m(G).
It is a plasure to acknowledge with Dr. K. Higashijima. (126,1,3)
References
Fig. 4. The correction due to Higgs bosons, which generates the (u, d; c, s) and (t, b; h, o) mixing.
mixing is produced by the corrections due to the Higgs bosons. Since no gauge bosons connect 64 to 64 in 128, they cannot.generate the mixing between 64 and 64. The first nonvanishing corrections come from fig. 4. To give the finite corrections, the mixing of $I 1(1716) and $1~(91) is necessary. This occurs by a quartic term in the Higgs potential: V=xz:t~~~a=
hS.~~,,(i?i&)~~,,(91)~
t . .. . (10)
Assuming the massMH of the Higgs bosons in fig. 4 to be of the order of (G/g)&, the off-diagonal masses responsible for the mixing between (u, d; c, s) and (t, b; h, o) read
= (g2/4n) m
236
(3) .
useful conversations
(11)
111 F. Wilczek and A. Zee, Spinors and families, preprint (1979); H. Sato, 0 (14) super unification of color, flavor and generations, Univ. of Tokyo preprint UT-338 (1980); M. Ida, Y. Kayama and T. Kitazoe, Kobe Univ. preprint (1980). PI J.C. Pati and A. Salam, Phys. Rev. D10 (1974) 275; H. Fritzsch and P. Minkowski, Ann. Phys. (NY) 93 (1975) 193;Nucl. Phys. B103 (1976) 61; H. Georgi, Particles and fields (1974) (APS/DPF Williamsburg) ed. C.E. Carlson (AIP, New York, 1975). [31 H. Georgi, Nucl. Phys. B156 (1979) 126. [41 L.-F. Li, Phys. Rev. D9 (1974) 1723. iSI F. Wilczek and A. Zee, Phys. Rev. Lett. 42 (1979) 421. 161 M.S. Chanowitz, J. Ellis and M.K. Gaillard, Nucl. Phys. B128 (1977) 506; H. Georgi and D.V. Nanopoulos, Nucl. Phys. B155 (1979) 52. f71 ?I. Sato, Phys. Rev. Lett., to be published. [81 M. Gell-Mann, P. Ramond and R. Slansky, unpublished; E. Witten, Phys. Lett. 91B (1980) 81; R.N. Mohapatra and G. Senjanovid, Phys. Rev. Lett. 44 (1980) 912. PI H. Georgi and S.L. Glashow, Phys. Rev. D7 (1973) 2457.
PHYSICS LETTERS
14 May 1981
FERMION MASSES FROM GRAND UNIFICATION WITH 0( 14)
Hikaru SAT0 ’ Department
of Physics, University of Tokyo, Hongo, Tokyo I 13, Japan
Received 2 January 1981
A mechanism is presented for generating fermion masses in the O(14) model which unifies color, flavor and generations. The masses of the conjugate generations, having V + A couplings to the ordinary weak currents are predicted to be O(10’) GeV. The mass of the observed generations can be obtained by the radiative corrections to the vanishing tree-level mass.
The 0( 14) is the smallest group with rank 7 which unifies all known quarks and leptons into a single irreducible fundamental (or spinor) representation [ 11. The spinor representation of 0(14) is 128, which is decomposed into 64 Q w under SO(14). The 64 of SO(14) is further decomposed into 16 @16 @% o i6 under SO(10) [2]. There are four 16’s in 128 so that this model predicts the existence of the fourth generation in addition to the known three:
fl =(u,,d,;ve,e;u~,d~;$,eC)L
,
f-2 =(c,,s,;~,,cc;c~,s~;~~,~c)~, f3 =(t,,b,;v,,7;t~,b~;v~,7C)L, f4 = (h, , 0, ; vf, f; h:, 0: ; v;, fCjL ,
(1)
where cr is the color index and c denotes the charge conjugated field *I. There are additional four 16’s in 128. Since a 16 has a decomposition, i6 = (4,1,2) @(3,2,1) into SU(4), QD Sum @SU(2)B, the fermiOnS with 4 Of SU(4), in 16 have a V t A coupling with SU(2), gauge bosons or a V - A coupling with SU(2)B gauge bosons. Thus the mass of the conjugate generations associated with i6 must be heavy. Concerning the mass of the fundamental fermions,
there arise two major--problems: (1) Besides the mass terms of 16-16 and 16-16, there are in general 16-G mass terms. They may acquire super heavy masses of 0( 1015) GeV corresponding to the super strong breaking of SO( 10) [3]. (2) The mass terms for both 16’s and i6’s must break SU(2)L @U(l), so that the 16’s and i6’s may have comparable masses. How can the hierarchical masses between 16 and i6 be understood? In this letter I present the mechanism to evade these difficulties and predict the mass of the conjugate generations to be O(102) GeV. The mass of 16’s can be generated by the radiative correction to the vanishing zeroth-order mass. Consider the strong symmetry breakdown of 0(14). This is achieved by using the Higgs fields of a 104 (traceless symmetric second rank tensor), ati (i,j = 1, ... . 14) and a 128, X of O(14). For a range of parameters in the potential, the vacuum expectation value (VEV) of ~ii can be in the form [4] (~ii> = diag(a ... b ... c ...) ,
(2)
where there are four a’s, six b’s and four c’s along the diagonal, with 4u t 6b t 4c = 0 *2. The VEV, eq. (2), breaks O(l4) down to SU(2)L @ SU(2), 0 SU(4), 0 SU(2)H 0 SU(2);I. Here SU(2)H and SU(2);I are the “horizontal” gauge symmetries which relate different generations [S]. The two generations (fr, f2) are as-
1 Present address: Department of Physics, Hyogo University of Education, Yashiro, Hyogo 673-14, Japan. *r The names of the seventh and the eigth quark are from Greek seven (heptu) and eigth (okt5).
*2 This VEV can be obtained for the potential in ref. [4] with -8hl < X2 < -6ht < 0.
0 03 l-9163/8 I /OOOO-0000/S 02.50 0 North-Holland Publishing Company
233
signed to a doublet of SU(2)I.I) while (f3, f4) are to a doublet of SU(2);I. The symmetry is further broken down to SU(2)L @U(l) 8 W(3), by giving the appropriate VEV’s to 2. The Higgs multiplet
0t
z=
7)
Z has components
1716 =(120,1,1)8(126,3,1)@(126, @(210,2,2)
1,3)
1
2002=2(126,1,1)e(120,3,1)te(120,1,3)
as
@(lo, 1,1)8(210,2,2)8(45,2,2).
’
where E (77)transforms as 64 (64) of SO(14). A component of each t and g with the respective quantum numbers, (1, 1,2,2,1) and (1, 1,2, 1,2) under SU(3)c 8 SU(2)L @ SU(2)R 8 SU(2)H @ SU(2); can acquire VEV for suitable conditions on the Higgs potential. This completes the strong symmetry breaking of 0(14) down to SU(2), o U( 1) o SU(3)c. I should emphasize here that the Higgs bosons with 104 or 128 of 0(14) cannot couple to the fermions with 128, hence the fermions can escape to get super heavy masses associated with the super strong breaking of 0(14) *3. The Higgs multiplets which can give masses to the fundamental spinors are those in 128 o 128. The Yukawa couplings are given by
sc’=f(~T,
14 May 1981
PHYSICS LETTERS
Volume lOlB, number 4
G*~)(;:: ;;:)(;) +h.c.
(6)
Since the Higgs multiplets which transform as 10, 126 [120] of SO(10) contribute to the mass matrices symmetric [antisymmetric] under the flavor indices [61, @(lo, 1, 1), $(126,1,1), @(120,3,1) and $(120, 1, 3) have no contribution to masses. The Higgs fields #(*, 2,2) generate the mixing between 16 and 16 of SO(10) in F or G*, therefore the VEV of $(*, 2,2) must be small. Thus the mass and the mixing angles are produced by the Higgs bosons in 364 and/or 1716
[71. For the neutrinos to acquire tiny Majorana masses, the Higgs bosons with 126 of SO(10) are necessary [8] The only choice left is then $I I = $I I (17 16) and 4~~ = @,,(1716). In 648~64 = 1@91@ 1001 e3003, the 9 1, which I use as @I2 = & , is decomposed into SG( 10) @SU(2)R @ SU(2);I as 91 =(45,1,1)fB(1,3,l)e(l,
1,3)
e(10,2,2). =f
\kT+P t h.c. ,
(4)
where the 128 spinors are put into
G*T=(&g;,f~,f4)~
(5)
Here F [G*] transforms as 64 [64] of SO(14) and fi [g;] transforms as 16 [i6] of SO(10). Note that (g!, g;) and (g;, &) are doublets of SU(2)H and SU(2)k, respectively. The Higgs bosons @II [&] transform as 64~64 [64@64] and@12 [&I] as64@64ofS0(14). The product decomposition of 64 o 64 is 64 @64 = 14 $364 8 1716 @2002, which have the SO(10) @SU(2), @ SU(2)h decompositions as
364 =(120,1,1)@(10,3, e(l,
l)e(i&
1,3)
292) >
*3 The radiative -correction
to the 16-16 mass will be of the order of OHM(16) = 1 GeV. See the discussion given later.
234
The Higgs fields @(lo, 2,2) generate the mixing between 16 in F and 16 in G* so that (u, d; c, s) and (t, b; h, o) can couple. Other Higgs bosons in 91 cause the mixing between 16 (i6) in F and i6 (16) in G*, therefore these VEV’s must be small. The mass m(16)of the conjugate generations is produced by the (126,1,3) and (126,3,1) components in 4I I and $22. Since these VEV’s violate SU(2)L 8 U(l), the mass of weak gauge bosons must be M,
=&11(126,
1,3)) =:8(@22(126, 331)) ,
with the gauge coupling constant g. The mass of m’s is then m(16)*:f($II(126,
1,3))=cf/g)Mw.
(8)
m(16)= O(102) GeV. The mass of the observed generations, m (16)must be much lighter than m(16)*4.This is due to ($II(126,3, I)), ($,2(126, 1,3))9@,,(126, 1,3)),etc. Forfeg,
14=(10,1,1)8(1,2,2),
@(45,2,2)
(7)
*’ I do not consider here the heavy Majorana They can be treated tail elsewhere.
separately
neutrinos and will be discussed
in 16. in de-
Volume lOlB, number 4
PHYSICS LETTERS
(16, 2,1 1
tl6,2,1)
(126,1,31 Fig. 1. The divergent selfenergy diagram, which vanishes due to the mismatch of the quantum numbers. The quantum numbers indicated are of the SO(lO)@ Su(&@ SU(2)fI. Since m (16) = am (16), it is attractive to consider that at the tree level only the conjugate generations acquire nonvanishing masses of 0(102) GeV, while the 16’sremain massless. The higher-order radiative corrections will generate the mass of the 16’s of the order of am(g). This scenario can actually happen in this model as outlined below. (i) It may be possible to arrange the Higgs potential for @ij such that only #I1(126,1,3) and $22(126, 3,l) acquire nonzero VEV’s. Thus the fermions of the conjugate generations become massive and others remain massless. (ii) There is no divergent self-energy diagram (fig. 1). Consider the correction to the fermions with (16,2, 1) of SO(10) QKI SU(2)R @ SU(2); in 64 of SG(l4). Since the gauge bosons connect two fermions in 64 or a, the intermediate heavy fermions are in (i6, 1,2) C 64, whose mass is generated by ($r L(126, 1,3)). There is no common channel between (126,1,3) and (16,2,1) 8 (16,2, I), hence fig. 1 vanishes. The correction to the (16, 1,2) fermions in 64 is also absent to this order.
14 May 1981
(iii) There is no contribution from the diagrams with a mixing among the gauge bosons as in fig. 2. Suppose the mixing of the (10,2,2) gauge bosons is induced by the VEV, (a>, and consider the correction to the (16,2, 1) fermions. Then it is easy to see that (a) o(~t)@($~~(126, 1,3)) cannot make (120, 1, 1) or (126,3, l)fora=&iiiZZ, $11(126,1,3) and d92(126, 3, 1). Note in particular that for a = Z, (16, 1,2) 8 (16, 1,2)~@(126,1,3)seemstomake(120,1,1).However, 1 of SU(2)R in (16,2,1) QX(16,2, 1) must be antisymmetric with respect to exchange of two 16’s in a doublet of SU(2), , which cannot be made by (0 a(,@) @(@LL).Although these diagrams give the finite mass to the 16’s, they may cause the divergent wave function renormalization for ~ii(12O, 1, l), $1 I (126, 3, l), r$22(126, 1,3) if the two end fermion lines are closed [9]. Absence of these diagrams is necessary to guarantee the existence of the solution for (ail) stated in (i). (iv) The first nonvanishing contribution to the mass comes from fig. 3. The (10,2,2) gauge bosons acquire super heavy masses due to the breaking of O(14) by (104). If the breaking by (128) is comparable to (lW, g(Z) =:M(lO, 2,2) =i&. Then the contribution from fig. 3 reads
= (g2/4n) m
Fig. 2. The mixing among the (10,2,2) gauge bosons occurs through nonvanishing VEV, (a) witha = @ii, E, @rr, @22.The diagrams for the (16,2,1) and (16, 1,2) fermions vanish.
(9)
The Cabibbo mixing between f, and f2 or f3 and f4 is generated by the mass matrices, hhich have the transformation property of (120, 1, l), (126,3, 1) and (126, 1,3)underS0(10)@SU(2)H@SU(2);I [7]. (v) The mixing between (fl , f2) and (f3, f4) is induced by Q12(91) whose VEV is chosen to be zero in (i), so that there is no mixing at the tree level. This
(16, 2,1) (126,1,31
(ii?) .
(i-,1,21 II
(16,2,11
1 (126,1,3)
Fig. 3. The diagram leading to the nonvanishing mass corrections. 235
<1>
Thus the of which radiative relatively
<$> ,
:
(gt)___iL’.---
--.\ ;:
__
+‘-I
(l&Z,))
!’
14 May 1981
PHYSICS LETTERS
Volume lOlB, number 4
\
(1%,1,2)
‘:
(16,1,2)
mass and mixing of the four generations, three are observed ones, have been generated by the corrections. This will explain naturally the small m (16) compared to m(G).
It is a plasure to acknowledge with Dr. K. Higashijima. (126,1,3)
References
Fig. 4. The correction due to Higgs bosons, which generates the (u, d; c, s) and (t, b; h, o) mixing.
mixing is produced by the corrections due to the Higgs bosons. Since no gauge bosons connect 64 to 64 in 128, they cannot.generate the mixing between 64 and 64. The first nonvanishing corrections come from fig. 4. To give the finite corrections, the mixing of $I 1(1716) and $1~(91) is necessary. This occurs by a quartic term in the Higgs potential: V=xz:t~~~a=
hS.~~,,(i?i&)~~,,(91)~
t . .. . (10)
Assuming the massMH of the Higgs bosons in fig. 4 to be of the order of (G/g)&, the off-diagonal masses responsible for the mixing between (u, d; c, s) and (t, b; h, o) read
= (g2/4n) m
236
(3) .
useful conversations
(11)
111 F. Wilczek and A. Zee, Spinors and families, preprint (1979); H. Sato, 0 (14) super unification of color, flavor and generations, Univ. of Tokyo preprint UT-338 (1980); M. Ida, Y. Kayama and T. Kitazoe, Kobe Univ. preprint (1980). PI J.C. Pati and A. Salam, Phys. Rev. D10 (1974) 275; H. Fritzsch and P. Minkowski, Ann. Phys. (NY) 93 (1975) 193;Nucl. Phys. B103 (1976) 61; H. Georgi, Particles and fields (1974) (APS/DPF Williamsburg) ed. C.E. Carlson (AIP, New York, 1975). [31 H. Georgi, Nucl. Phys. B156 (1979) 126. [41 L.-F. Li, Phys. Rev. D9 (1974) 1723. iSI F. Wilczek and A. Zee, Phys. Rev. Lett. 42 (1979) 421. 161 M.S. Chanowitz, J. Ellis and M.K. Gaillard, Nucl. Phys. B128 (1977) 506; H. Georgi and D.V. Nanopoulos, Nucl. Phys. B155 (1979) 52. f71 ?I. Sato, Phys. Rev. Lett., to be published. [81 M. Gell-Mann, P. Ramond and R. Slansky, unpublished; E. Witten, Phys. Lett. 91B (1980) 81; R.N. Mohapatra and G. Senjanovid, Phys. Rev. Lett. 44 (1980) 912. PI H. Georgi and S.L. Glashow, Phys. Rev. D7 (1973) 2457.