- Email: [email protected]
The top quark mass and flavour mixing in a seesaw model of quark masses
2 October 1997 PHYSICS
ELSEVIER
LETTERS
B
Physics Letters B 410 (1997) 233-240
The top quark mass and flavour mixing in a seesaw model of quark masses T. Morozumi b CFIF/
a31,T. Satou ay2,M.N. Rebel0 b*3,M. Tanimoto
c,4
a Department of Physics, Hiroshima Uniuersity, l-3-1 Kagamiyama, 739 Higashi Hiroshima, Japan IST and Departamento de Fisica, Instituto Superior Te’cnico, Au. Rouisco Pais, P-1096 Lisboa Codex, Portugal ’ Science Education Laboratory, Ehime Unioersiiy, Bunkyo-cho, 790 Matsuyama, Japan
Received 24 March 1997; revised 24 July 1997 Editor: R. Gatto
Abstract The top quark mass and the flavour mixing are studied in the context of a seesaw model of quark masses based on the gauge group SU(2), X SU(2), X U(1). Six isosinglet quarks are introduced to give rise to the mass hierarchy of ordinary quarks. In this scheme, we reexamine a mechanism for the generation of the top quark mass. It is shown that, in order to prevent the seesaw mechanism to act for the top quark, the mass parametbr of its isosinglet partner must be much smaller than the breaking scale of W(2),. As a result the fourth lightest up quark must have a mass of the order of the breaking scale of SU(2),, and a large mixing between the right-handed top quark and its singlet partner occurs. We also show that this mechanism is compatible with the mass spectrum of light quarks and their flavour mixing. 0 1997 Elsevier Science B.V.
1. Introduction
The seesaw mechanism [1,2] was initially invented to explain the smallness of neutrino masses. Also, in a different framework, the smallness of masses of quarks other than the top quark compared to the scale of the electroweak symmetry breaking can be explained by a seesaw Model of Quark Masses [3-71. In these seesaw models, the Yukawa
mass terms are not invariant under W(2),. 5 The SU(2), symmetry is broken spontaneously by a Higgs field, a W(2), doublet. The representation of the Higgs is chosen so that it does not form any renormalizable Yukawa mass terms for the ordinary quarks. The ordinary quark masses are generated through the exchange of heavy isosinglet quarks. They are proportional to the breaking scales of W(2),, (r)J and X$2),, (~~1 and are inversely proportional to the isosinglet quark mass M, i.e.,
1
E-mail address: [email protected]. ’ E-mail address: t_satou@kakuri2_pc.phys.sci.hiroshimau.a;.jp. E-mail address: [email protected]. 4 E-mail address: [email protected].
ehime-u.ac.jp.
0370-2693/97/%17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)01021-6
7
The r8le of SU(21, can be played by some other symmetry. In the context of GUT, a heavy top quark is naturally obtained by
assigning the top quark to a representation which is not isomorphic to those containing the other light quarks [8,9].
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I: Morozumi ei ai. /Physics
O(y). Conventionally, the isosinglet quark mass M is assumed to be much larger than Q. This assumption leads to an explanation for the smallness of quark masses compared to the scale of the electroweak symmetry breaking. Though the seesaw mechanism explains the smallness of the mass of the five flavours from the up quark to the bottom quark, it has not been shown that the top quark with a mass of O(Q) can be incorporated into the same scheme. In this letter, we study the top quark mass as well as the mass hierarchy of the up and down quark sectors in the context of the seesaw mechanism. We show that the mass formulae for the light quarks proposed before is not valid for the top quark and must be replaced by a new one. The mass hierarchy of the light quarks and flavour mixing are studied in the same context by Refs. [5-71 under the assumption that the isosinglet quark diagonal mass parameter is much larger than the breaking scale of SU(2),. However, under the same assumption the top quark mass would also be much smaller than the breaking scale of SU(2), unless the corresponding Yukawa coupling between isosinglet and isodoublet quarks is chosen to be very large. As we show later, if the diagonal mass parameter for the isosinglet partner of the top quark is much smaller than nR, the seesaw mechanism does not act for the top quark and its mass can be kept at the scale of Q without introducing a large Yukawa coupling. In this case the heavier mass eigenstate is as light as 17~ rather than being given by the singlet mass parameter. Because one of the mass eigenstates is as light as Q~, the flavour mixing and the stability of the light quark masses against the inclusion of flavour off-diagonal Yukawa couplings is a non-trivial problem, we study the light quark mass spectrum and flavour mixing taking into consideration the special r6le played by the top quark. We obtain the approximate mass formulae for quark masses and show that the mass hierarchy is stable against flavour mixing. This paper is organized as follows. In Section 2, we present the results for the diagonalization of the mass matrix for the top quark and its isosinglet partner. The mixing between singlet and doublet quarks is discussed both for left- and right-handed chiralities. In Section 3, the mass spectrum of both light and heavy quarks is obtained by introducing flavour off-diagonal couplings. In Section 4, the
Letters B 410 (1997) 233-240
mixing angles are obtained. rize the results.
In Section 5, we summa-
2. The top quark mass and singlet-doublet mixing in a seesaw model model is SU(21, x quarks, isosinglet quarks and the relevant Higgs scalars are assigned to the following gauge group representations: The
gauge
group
of the
SU(2), X U(1) and the ordinary
:(2,1,1/3),
+; =
L
&!~&,1,~/3),
&:(2,1,1)9
D~,,:(l,L-2,‘3), &:(1,2,1),
(1)
where i = 1,2,3. By introducing an isodoublet Higgs & (c#+) for X$2), (SU(2),), the Yukawa interaction between doublet and singlet quarks and the mass term for the singlet quarks of the model are given by
- ~$&$I,D;
_
,TJ$,,fUi
- y$&b,@~
+ (hc.)
_ D$,@i, (2)
where i and j are summed over from 1 to 3; yYRIDCuj is the strength of the Yukawa coupling between the down(up) type left-handed {righthanded} isodoublet quark and isosinglet quark; y,(,, are 3 X 3 matrices; M;(ML) are given by ML = M,,M~=M,,M:=M,,M:,=M,,M~=M, and Mz = MB. Without loss of generality, the singlet quark mass matrix can be transformed into a real diagonal matrix through a bi-unitary transformation. The scale of the singlet mass parameter is going to be set later. We first focus on the case in which the flavour mixing is absent and study the top quark sector. The mass eigenstates and eigenvalues for the top quark and its isosinglet partner are obtained by diagonalizing the two by two matrix:
(3)
T. Morozumi et al. /Physics
The mass eigenvalues and the mixing angles for the left-handed chiralities are obtained by diagonalizing MM+:
where 77~~~) are the vacuum expectation values of the neutral Higgs particles,
The vacuum expectation values are related to the masses of the charged SU(2),(,, gauge bosons by M; = &‘I$,
M; = ;g2v;,
(6)
where g is the SU(2),(,, gauge coupling constant. Therefore the mass eigenvalues are written in terms of the gauge boson masses as:
JyLy,I ML
m,zfi (
g2
1
If MT s MR, the lighter mass eigenvalue n, in Eq. I . (7) is reduced to the well known seesaw formulae for light quarks, i.e., 4 WY). H owever, for the top quark, we should take the limit MT -=KMR. Then the suppression factor MJM, is now absent and its mass is given by
Letters B 410 (1997) 233-240
235
times larger than the gauge coupling g. We assume that it is still within the perturbative region. There is a simple reason why the top quark mass is determined by ML and its partner’s mass is proportional to MR. To show this, let us ignore the mass term of the singlet quark and keep only singlet-doublet mixing terms. Then the Yukawa term of the top quark and its singlet partner is _5?= -yLqLtLTR - y;vRTZtR + (h.c.).
(11)
From Eq. (11) the mass term can be diagonalized by the following rotations,
(F),= (A;
(12) (13)
Then the mass for t”’ is 1~~117~ and the mass for T” is JyR)vR. We note that the isosinglet and doublet mixing is maximal for the right-handed sector. The large mixing between the top quark and its singlet partner predicted in this model should be possible to observe in an experiment of t? production. In the limit where flavour mixing is absent we obtain, from the Lagrangian of Eq. (21, mass matrices for the pairs (u,U>, (CC), (d,D), (s,S) and (b,B), generically denoted by (q,Q), similar to the one for the pairs (t,T) in Eq. (3). The eigenvalues are also given by expressions of the form Eq. (7) and Eq. (8) with M, replaced by the corresponding mass parameter M,. In order for the seesaw mechanism to act, so that the masses we obtain for the light quarks are much smaller than ML, we need M, B MR. Then Eq. (9) and Eq. (10) are replaced by
(14) which is at the scale of ML. On the other hand, the heaviest eigenvalue mT in Eq. (8) is
which is at the symmetry breaking scale of SU(2), instead of being given by the singlet mass parameter MT. To reproduce the value of the top quark mass in JZq. (91, the Yukawa coupling yL must be a few
rnQgMQ.
(15)
If MR is fixed, we can estimate the order of magnitude of M, for each case. In these cases the mixing is suppressed by a factor of (7JMJ for the lefthanded components and (vR/MQ) for the righthanded ones [7]. In the next section we extend our analysis to the case where flavour mixing mass terms are present.
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T. Morozumi et al. /Physics
Letters B 410 (I 997) 233-240
3. Mass formulae including flavour mixing in the seesaw model In this section we derive the mass formulae in the case with mixings among different quark flavours. We start with the six by six mass matrix
(16) MDiagis
where matrix,
a three
by three
diagonal
mass
AM* = 77;~;~~ +M&
-M;,
where we assume M,,MC B qRB-M,.As a result the unperturbed part of (r$yi yR + Mkiag) is given by MO2rather than by M,&. Eq. (19) is expressed in the simple form, det[ y,YyL - /\] = 0,
(17)
(21)
where the leading terms of Y, obtained expansion of Eq. (201, are as follows:
by using the
(dYR),3(E'bR)3, and yr and yR are rank 3 matrices. The eigenvalue equation for the quark masses is given by det(
MM+ -A)
= det
I
r]LYLY:. -A 77r
MDiagYL
YLMDiag rl, T’j YAYR + M&g
= I 0.
- A
(YXYR)13( YiYR),,
y12=xlixC (YbR),2[
(hR)33
1
’
y = XUXT(Y~YR),, 13
’
(YbR)33
(18)
The equation which determines the eigenvalues of order of r]z (or smaller than $1 is reduced to a cubic equation:
I ’
(YAYR)33
1
(YLYi?),,( YLYR)32 Y?, = Xc’ (Y; Y,),, [
y=
(YhR)33
’
xCxT(.dyR),3
det[ YL{1 - Mniag(Vi YLYR -A
1
=o,
+ M&g)
_ ’ J4Di.g)
YL
23
(.VbR)33
(19)
where we use the normalized eigenvalue A = A/n,‘. This equation determines the ordinary quark masses. We further use the expansion as follows,
y33
=
l
-
’
(y;t-)33
+
(jZ),,
(22)
’
where Y + = Y and X,, X, and X, are given by
u
4
x,=
xc+
x,;+,
C
TR/(%iji
’
(23) The eigenvalue F(h)
equation
now becomes
= -h3+A’Tr[ylyLY] - -yh + Jdety,l’detY=O.
(24)
T. Morozumi et al. /Physics
The solutions of Eq. (28) are hierarchical, i.e., A = 0(X$), 0(X:), O(1) with X, K X, -=z 1. Taking into account the structure of the Yukawa coupling strengths the solutions are of the form:
The coefficients for each term of Eq. (24) are Tr[ Y~YJ]
= ( Y~YL)~
Y=x,zi(YR2121q2/2(
Y:.YL)33'
detY a X~X~(ia,,12icuR2?
- I(Y,&. (YR,\*),
(25)
where we are keeping the leading term for each coefficient. Here cy and 0 are complex vectors which are related to the three complex vectors (y,,y,,y,) of the matrix yL and yR. To show the relation, let us first write y, and y, as follows. y, =
(YLIvYLZ’YL3)’
YR =
(YRdR2’yR3)’
(26)
Then (Y and /? are defined by (YLI = YLI
237
Letters B 410 (1997) 233-240
mf z
1yLu31*qf?
m2, %;;I
mf
PRU1121
=
x~\aRu2?~aLu2~2~~~
PLu112r1,2.
(29)
The heavier eigenvalues are also obtained as follows, rnt=Mi, rns=Mj, m+2iyRu3/2q~. (30) In the case of the down type quarks the masses are obtained by assuming M, z=MS z=M, * qR:
YL
‘YLI 2 YL39 IYL31
--
YjP3 ‘YRI -
aRf=yRl
7YR39 bR3l
pLI
=
aL,
-
2,
m,=
’ aL1 aL2, laL212
a’2
where I = 1,2. Geometrically a,(Z = 1,2) are the projections of y, onto the plane perpendicular to the vector y,, and PI is the projection of cr, onto the line perpendicular to (Ye and yS. To find the solutions of Eq. (241, we first set the order of magnitude of X, and X, as follows. If we neglect flavour mixing, in the seesaw model, the up quark mass is given by MLMR/MD. with yttR) = g. Therefore the seesaw suppression factor X, = M,/M, is of the order of mu/M, = 10-5. With a similar argument, we have Xc = M,/M, = lo-’ by assuming that the strength of the Yukawa coupling does not depend significantly on the flavour. By setting the magnitude of X, and Xc in this way, we obtain an approximate mass formula for the case of flavour mixing. The point is that in the approximation that the strengths of all Yukawa couplings do not depend significantly on the flavour, the eigenvalue equation Eq. (24) has the following form,
F(h)= -h3+O(l)P-O(X~)A +O(x;x;)=o.
(28)
M&
m2,EA4,2,
rni=Mi.
(31)
with X, = qR/M,,X, = qR/Ms and X, = qR/M,. We would like to stress the following points. - The mass formulae for the five light quarks are of the seesaw type. They are proportional to ?,I~:77R and inversely proportional to the singlet quark mass. They are also proportional to the appropriate projection of the Yukawa couplings y,,,, y,,,( I = 1,2) for the up and the charm quarks and yLD,,yRD,(I = 1,2,3) for the down type quarks. * The top quark mass Cm,) is proportional to vL and the length of the vector y,,,. The other mass eigenvalue (m,) is proportional to qR and the length of yRU3. - The masses of the remaining five heavier quarks are given to a good approximation by the mass parameters M,,(i= 1,2) and MDi(i= 1,2,3).
4. Flavour mixing In order to find the mixing angles (CKM matrix) among left (right) chiralities of quarks, it is convenient to perform a unitary transformation among ordinary quarks such that the singlet-doublet Yukawa
238
T. Morozumi et al./ Physics Letters B 410 (1997) 233-240
x0 0 1
couplings, ces,
i.e., yL and y, become triangular
YL(R)U(D)= %(D)L(RJ x I x
x x
0 x
matri-
’
Table I Singlet and Doublet mixing angles
(32)
where the U’s are unitary matrices. This decomposition can always be done as proved in the Appendix. Further we introduce a new basis denoted by u’,d’ which is related to the original basis by the unitary matrices, u;. = U&,
u’R= U&uR,
d’, = U&d,,
u; = U&dR.
(33)
In this new basis, the 6 X 6 matrix formed into
M is trans-
(34) The explicit form of M’ for the up type quarks is given by
,+f'=
0
0
0
PLUll?lL
0
0
0
PLU211)L
I"LL'ZI')L
0
0
0
=LoilvL
~LUIZ'IL
PR*"lPlR
1
/(IRUZ'IR I
0
0
0
case,
IYLOhL
MU
0
0
~';U32%
0
MC
0
IYWhR
0
0
MT
PI;o2,1IX an'u,Pln
0
0
II
sine,e’+q
(35)
where
I/3Lu,,l=I~Lu,I
and
lPRuI,I=
l&,_,,I;
their
phases can be obtained from the Appendix; cyLDZ, and yLU, are defined by Eq. (26) and YLD3' aLU2 Eq. (27). In this form, u’ only couples to U. Therefore the up quark mass is determined by the mass of the heaviest singlet quark lJ and is not affected by the presence of the other lighter singlet quarks. The c’ couples to both U and C. However its mass is mainly determined by the mass of C and the effect of U is tiny because 44, >> n/l,. As a result the matrix M’ can be approximated by the following block diagonal matrix M”,
M'" =
0
0
0
0
0
0
0
0
0
laL":hL
0
0
0
0
0
0
0
0
MU
0
0
l~.Q"&lR
0
0
MC
0
0
0
0
Pdu,,%
I
BLL'IIVL
IYRUhR
The off-diagonal elements of M’ which do not appear in M’* can be treated as a perturbation. In this approximation, each doublet quark only couples to one of the singlet quarks, So the eigenvalues and eigenstates are obtained by diagonalizing two by two matrices. It is interesting to see that the eigenvalues of M” agree with those obtained in Eq. (29) and Eq. (301 with the same assumption for the singlet quark mass parameters. In addition the mass eigenstates are defined by the following rotation between the doublet quark q’ and the singlet quark Q,
where the CKM matrix is written, in terms of the unitary matrices defined before, as:
IYL"h. 0
v, = UJLIJDL.
0
MT
In Table 1, we show the mixing angles between singlet and doublet quarks for both chiralities. It can be seen that the singlet to doublet mixing for the five light quarks is strongly suppressed being at least of the order qR(vLL)/M for right (left) -handed quarks. For the top quark, there is a suppression of the left-handed mixing angles of the order qLMT/$ while the right-handed mixing angle is not suppressed. In the following analysis, we only keep the right-handed mixing angle for the top quark and its singlet partner and set the other singlet to doublet mixing angles to zero. In this approximation, the left-handed charged currents and the CKM matrix are given by ._I&/_ = cy&VLij d:i 3 (38)
I (36)
Within this approximation, for the left-handed neutral
(39) we do not have FCNC isospin current. On the
T. Morozumi et ai. /Physics
other hand, the right-handed charged currents are given by
+ COSO,,$~,V,,~~,“~
+
where VR = ud,if,,,, and 8,, = tan- ‘(1y,&?jR/Mr) as shown in Table 1. The right-handed neutral isospin currents are =$
i&,u,”
+~y,cR”
+
(d,~)~i=,&t~
(
+(sin8,R)2F,rypT[ +(h.c.)
-FRj’$dgj),
239
is a characteristic of our model and may be checked in the top quark production experiments.
6. Note added
sinO,,Ty,V,,jd$j, (40)
.3 JMR
Letters B 410 (1997) 233-240
+ sin8,,cosB,,Tm,y,t; (41)
in both currents a sum over j running from 1 to 3 is implied. Because @,a+ ; as M, + 0, there is a large mixing for T/ in the right-handed charged current jJrR and the right-handed neutral current j&.
5. Summary We study a mechanism for the generation of the top quark mass and flavour mixing in the context of a seesaw model for quark masses. When the mass parameter Mr for the isosinglet quark is smaller than the symmetry breaking scale of X!(2),, the lightest of the two eigenstates is at the breaking scale of SU(2), and can be identified with the top quark, the heaviest one will be found at the symmetry breaking scale of S(I(2),. The smallness of M, compared to the breaking scale of SU(2), may be originated from the zero texture of the singlet quark mass matrix at a GUT scale. Much work remains to be done along these lines. We also study the mass hierarchy of ordinary quarks by including the flavour mixing. The stability of the light quark masses against flavour mixing is explicitly shown. The dependence on the strength of the Yukawa couplings is nontrivial and we have given a simple geometrical interpretation for it. The singlet-doublet mixing is suppressed except in the case of the mixing between the right-handed top quark and its singlet partner. This large mixing angle
After completing our work, we have received a paper by Y. Koide and H. Fusaoka 1101in which the enhancement of the top quark mass in a seesaw model is studied. They found that the top quark mass is enhanced to the electroweak symmetry breaking scale at the singular point (bf = - l/3) of their singlet quark mass matrix (democratic matrix + bf X unit matrix). Their singular point would correspond to MT = 0 in our case. Ignoring flavour mixing, the mass scale we obtained for the fourth lightest up type quark together with the mass formula for the top quark coincides with their result at the singular point. Still, our approach differs considerably from theirs and the mass formulae and the flavour mixing we obtain for the light quarks are quite different. We attribute the mass hierarchy of the five light quarks to the mass hierarchy of the five heavy singlet quark mass parameters and assume that the Yukawa couplings among the singlet and doublet quarks do not depend significantly on the flavour whilst they start from a specific type of the singlet quark mass matrix and introduce flavour dependent Yukawa couplings. As a result our formulae for the five light quarks cannot be directly translated into their framework. Furthermore their model gives constraints on the CKM matrix unlike ours where we cannot make definite predictions.
Acknowledgements We would like to thank T. Muta and G.C. Branco for helpful discussions and A. Sugamoto for a suggestion. We would like to thank Y. Koide for conversations about his work and results. This work is supported by the Grant-in-Aid for Joint International Scientific Research (#08044089) from the Ministry of Education, Science and Culture, Japan, and JSPS which made exchange programs of M.N.R. and T.M. possible. The work of T.M. is supported in part by Grant-in-Aid for Scientific Research on Priority Areas (Physics of CP violation).
T. Morozumi et al/Physics
240
Appendix
A
By multiplying triangular form,
We prove the decomposition of y,(,, into a unitary matrix and a triangular matrix. Let us write a rank 3 matrix y with three complex vectors in C3, y= (YPY,,Y3)9
(A.11
where y is the singlet-doublet Yukawa coupling yLcRj in our case. Choose three orthonormal vectors U= (ut,u2,~3) with ~3 = &. Multiplying Ut on the left hand side of y, we obtain
lJ+y=
where r?
Letters B 410 (19971233-240
aI1
aI2
0
a21
ff22
0
I a31
o32
,
(A4
IY3II
aij = ui .yj. Then we define two vectors in
(A.31 We also define two orthonormal vectors Y, and v2 with v2 = a2/l~y2( in C 2. With these two vectors, we can form another unitary matrix V,
v=[‘;;“0: ;I.
(A.41
v+u+y=
Vt
to Uty,
PI,
0
0
p2,
Ia21
0
a31
a32
we finally
)
obtain
a
(A.9
b3l
I I where pii = v! . aj. Then y is written as the product of a unitary matrix and a triangular matrix.
References [l] T. Yanagida, in: Proceedings of the Workshop on The Unified Theory and the Baryon Number of the Universe, Eds. Osamu Sawada, Akio Sugamoto, KEK 13-14 (Feb. 1979), (KEK-79- 18). [2] M. Gell-Mann, P. Ramond, R. Slansky in Sanibel Talk, CALT-68-709 (Feb. 1979). and in: Supergravity (North-Holland, Amsterdam, 1979). [3] Z.G. Berezhiani. Phys. Lett. B 129 (1983) 99; B 150 (1985) 177. [4] D. Chang, R.N. Mohapatra, Phys. Rev. Lett. 58 (1987) 1600. [s] J. Rajpoot. Phys. Lett. B 191 (1987) 122. [6] A. Davidson, K.C. Wab, Phys. Rev. Lett. 59 (1987) 393. [7] KS. Babu. R.N. Mohapatra, Phys. Rev. Lett. 62 (1989) 1079. [8] R. Barbieri, G. Dvali, A. Sttumia, 2. Berezhiani, L. Hall, Nucl. Phys. B 432 (1994) 49. [9] G. Dvali, S. Pokorski, Phys. Lett. B 379 (1996) 126. [lo] Y. Koide. H. Fusaoka, Z. Phys. C 71 (1996) 459.
ELSEVIER
LETTERS
B
Physics Letters B 410 (1997) 233-240
The top quark mass and flavour mixing in a seesaw model of quark masses T. Morozumi b CFIF/
a31,T. Satou ay2,M.N. Rebel0 b*3,M. Tanimoto
c,4
a Department of Physics, Hiroshima Uniuersity, l-3-1 Kagamiyama, 739 Higashi Hiroshima, Japan IST and Departamento de Fisica, Instituto Superior Te’cnico, Au. Rouisco Pais, P-1096 Lisboa Codex, Portugal ’ Science Education Laboratory, Ehime Unioersiiy, Bunkyo-cho, 790 Matsuyama, Japan
Received 24 March 1997; revised 24 July 1997 Editor: R. Gatto
Abstract The top quark mass and the flavour mixing are studied in the context of a seesaw model of quark masses based on the gauge group SU(2), X SU(2), X U(1). Six isosinglet quarks are introduced to give rise to the mass hierarchy of ordinary quarks. In this scheme, we reexamine a mechanism for the generation of the top quark mass. It is shown that, in order to prevent the seesaw mechanism to act for the top quark, the mass parametbr of its isosinglet partner must be much smaller than the breaking scale of W(2),. As a result the fourth lightest up quark must have a mass of the order of the breaking scale of SU(2),, and a large mixing between the right-handed top quark and its singlet partner occurs. We also show that this mechanism is compatible with the mass spectrum of light quarks and their flavour mixing. 0 1997 Elsevier Science B.V.
1. Introduction
The seesaw mechanism [1,2] was initially invented to explain the smallness of neutrino masses. Also, in a different framework, the smallness of masses of quarks other than the top quark compared to the scale of the electroweak symmetry breaking can be explained by a seesaw Model of Quark Masses [3-71. In these seesaw models, the Yukawa
mass terms are not invariant under W(2),. 5 The SU(2), symmetry is broken spontaneously by a Higgs field, a W(2), doublet. The representation of the Higgs is chosen so that it does not form any renormalizable Yukawa mass terms for the ordinary quarks. The ordinary quark masses are generated through the exchange of heavy isosinglet quarks. They are proportional to the breaking scales of W(2),, (r)J and X$2),, (~~1 and are inversely proportional to the isosinglet quark mass M, i.e.,
1
E-mail address: [email protected]. ’ E-mail address: t_satou@kakuri2_pc.phys.sci.hiroshimau.a;.jp. E-mail address: [email protected]. 4 E-mail address: [email protected].
ehime-u.ac.jp.
0370-2693/97/%17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)01021-6
7
The r8le of SU(21, can be played by some other symmetry. In the context of GUT, a heavy top quark is naturally obtained by
assigning the top quark to a representation which is not isomorphic to those containing the other light quarks [8,9].
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I: Morozumi ei ai. /Physics
O(y). Conventionally, the isosinglet quark mass M is assumed to be much larger than Q. This assumption leads to an explanation for the smallness of quark masses compared to the scale of the electroweak symmetry breaking. Though the seesaw mechanism explains the smallness of the mass of the five flavours from the up quark to the bottom quark, it has not been shown that the top quark with a mass of O(Q) can be incorporated into the same scheme. In this letter, we study the top quark mass as well as the mass hierarchy of the up and down quark sectors in the context of the seesaw mechanism. We show that the mass formulae for the light quarks proposed before is not valid for the top quark and must be replaced by a new one. The mass hierarchy of the light quarks and flavour mixing are studied in the same context by Refs. [5-71 under the assumption that the isosinglet quark diagonal mass parameter is much larger than the breaking scale of SU(2),. However, under the same assumption the top quark mass would also be much smaller than the breaking scale of SU(2), unless the corresponding Yukawa coupling between isosinglet and isodoublet quarks is chosen to be very large. As we show later, if the diagonal mass parameter for the isosinglet partner of the top quark is much smaller than nR, the seesaw mechanism does not act for the top quark and its mass can be kept at the scale of Q without introducing a large Yukawa coupling. In this case the heavier mass eigenstate is as light as 17~ rather than being given by the singlet mass parameter. Because one of the mass eigenstates is as light as Q~, the flavour mixing and the stability of the light quark masses against the inclusion of flavour off-diagonal Yukawa couplings is a non-trivial problem, we study the light quark mass spectrum and flavour mixing taking into consideration the special r6le played by the top quark. We obtain the approximate mass formulae for quark masses and show that the mass hierarchy is stable against flavour mixing. This paper is organized as follows. In Section 2, we present the results for the diagonalization of the mass matrix for the top quark and its isosinglet partner. The mixing between singlet and doublet quarks is discussed both for left- and right-handed chiralities. In Section 3, the mass spectrum of both light and heavy quarks is obtained by introducing flavour off-diagonal couplings. In Section 4, the
Letters B 410 (1997) 233-240
mixing angles are obtained. rize the results.
In Section 5, we summa-
2. The top quark mass and singlet-doublet mixing in a seesaw model model is SU(21, x quarks, isosinglet quarks and the relevant Higgs scalars are assigned to the following gauge group representations: The
gauge
group
of the
SU(2), X U(1) and the ordinary
:(2,1,1/3),
+; =
L
&!~&,1,~/3),
&:(2,1,1)9
D~,,:(l,L-2,‘3), &:(1,2,1),
(1)
where i = 1,2,3. By introducing an isodoublet Higgs & (c#+) for X$2), (SU(2),), the Yukawa interaction between doublet and singlet quarks and the mass term for the singlet quarks of the model are given by
- ~$&$I,D;
_
,TJ$,,fUi
- y$&b,@~
+ (hc.)
_ D$,@i, (2)
where i and j are summed over from 1 to 3; yYRIDCuj is the strength of the Yukawa coupling between the down(up) type left-handed {righthanded} isodoublet quark and isosinglet quark; y,(,, are 3 X 3 matrices; M;(ML) are given by ML = M,,M~=M,,M:=M,,M:,=M,,M~=M, and Mz = MB. Without loss of generality, the singlet quark mass matrix can be transformed into a real diagonal matrix through a bi-unitary transformation. The scale of the singlet mass parameter is going to be set later. We first focus on the case in which the flavour mixing is absent and study the top quark sector. The mass eigenstates and eigenvalues for the top quark and its isosinglet partner are obtained by diagonalizing the two by two matrix:
(3)
T. Morozumi et al. /Physics
The mass eigenvalues and the mixing angles for the left-handed chiralities are obtained by diagonalizing MM+:
where 77~~~) are the vacuum expectation values of the neutral Higgs particles,
The vacuum expectation values are related to the masses of the charged SU(2),(,, gauge bosons by M; = &‘I$,
M; = ;g2v;,
(6)
where g is the SU(2),(,, gauge coupling constant. Therefore the mass eigenvalues are written in terms of the gauge boson masses as:
JyLy,I ML
m,zfi (
g2
1
If MT s MR, the lighter mass eigenvalue n, in Eq. I . (7) is reduced to the well known seesaw formulae for light quarks, i.e., 4 WY). H owever, for the top quark, we should take the limit MT -=KMR. Then the suppression factor MJM, is now absent and its mass is given by
Letters B 410 (1997) 233-240
235
times larger than the gauge coupling g. We assume that it is still within the perturbative region. There is a simple reason why the top quark mass is determined by ML and its partner’s mass is proportional to MR. To show this, let us ignore the mass term of the singlet quark and keep only singlet-doublet mixing terms. Then the Yukawa term of the top quark and its singlet partner is _5?= -yLqLtLTR - y;vRTZtR + (h.c.).
(11)
From Eq. (11) the mass term can be diagonalized by the following rotations,
(F),= (A;
(12) (13)
Then the mass for t”’ is 1~~117~ and the mass for T” is JyR)vR. We note that the isosinglet and doublet mixing is maximal for the right-handed sector. The large mixing between the top quark and its singlet partner predicted in this model should be possible to observe in an experiment of t? production. In the limit where flavour mixing is absent we obtain, from the Lagrangian of Eq. (21, mass matrices for the pairs (u,U>, (CC), (d,D), (s,S) and (b,B), generically denoted by (q,Q), similar to the one for the pairs (t,T) in Eq. (3). The eigenvalues are also given by expressions of the form Eq. (7) and Eq. (8) with M, replaced by the corresponding mass parameter M,. In order for the seesaw mechanism to act, so that the masses we obtain for the light quarks are much smaller than ML, we need M, B MR. Then Eq. (9) and Eq. (10) are replaced by
(14) which is at the scale of ML. On the other hand, the heaviest eigenvalue mT in Eq. (8) is
which is at the symmetry breaking scale of SU(2), instead of being given by the singlet mass parameter MT. To reproduce the value of the top quark mass in JZq. (91, the Yukawa coupling yL must be a few
rnQgMQ.
(15)
If MR is fixed, we can estimate the order of magnitude of M, for each case. In these cases the mixing is suppressed by a factor of (7JMJ for the lefthanded components and (vR/MQ) for the righthanded ones [7]. In the next section we extend our analysis to the case where flavour mixing mass terms are present.
236
T. Morozumi et al. /Physics
Letters B 410 (I 997) 233-240
3. Mass formulae including flavour mixing in the seesaw model In this section we derive the mass formulae in the case with mixings among different quark flavours. We start with the six by six mass matrix
(16) MDiagis
where matrix,
a three
by three
diagonal
mass
AM* = 77;~;~~ +M&
-M;,
where we assume M,,MC B qRB-M,.As a result the unperturbed part of (r$yi yR + Mkiag) is given by MO2rather than by M,&. Eq. (19) is expressed in the simple form, det[ y,YyL - /\] = 0,
(17)
(21)
where the leading terms of Y, obtained expansion of Eq. (201, are as follows:
by using the
(dYR),3(E'bR)3, and yr and yR are rank 3 matrices. The eigenvalue equation for the quark masses is given by det(
MM+ -A)
= det
I
r]LYLY:. -A 77r
MDiagYL
YLMDiag rl, T’j YAYR + M&g
= I 0.
- A
(YXYR)13( YiYR),,
y12=xlixC (YbR),2[
(hR)33
1
’
y = XUXT(Y~YR),, 13
’
(YbR)33
(18)
The equation which determines the eigenvalues of order of r]z (or smaller than $1 is reduced to a cubic equation:
I ’
(YAYR)33
1
(YLYi?),,( YLYR)32 Y?, = Xc’ (Y; Y,),, [
y=
(YhR)33
’
xCxT(.dyR),3
det[ YL{1 - Mniag(Vi YLYR -A
1
=o,
+ M&g)
_ ’ J4Di.g)
YL
23
(.VbR)33
(19)
where we use the normalized eigenvalue A = A/n,‘. This equation determines the ordinary quark masses. We further use the expansion as follows,
y33
=
l
-
’
(y;t-)33
+
(jZ),,
(22)
’
where Y + = Y and X,, X, and X, are given by
u
4
x,=
xc+
x,;+,
C
TR/(%iji
’
(23) The eigenvalue F(h)
equation
now becomes
= -h3+A’Tr[ylyLY] - -yh + Jdety,l’detY=O.
(24)
T. Morozumi et al. /Physics
The solutions of Eq. (28) are hierarchical, i.e., A = 0(X$), 0(X:), O(1) with X, K X, -=z 1. Taking into account the structure of the Yukawa coupling strengths the solutions are of the form:
The coefficients for each term of Eq. (24) are Tr[ Y~YJ]
= ( Y~YL)~
Y=x,zi(YR2121q2/2(
Y:.YL)33'
detY a X~X~(ia,,12icuR2?
- I(Y,&. (YR,\*),
(25)
where we are keeping the leading term for each coefficient. Here cy and 0 are complex vectors which are related to the three complex vectors (y,,y,,y,) of the matrix yL and yR. To show the relation, let us first write y, and y, as follows. y, =
(YLIvYLZ’YL3)’
YR =
(YRdR2’yR3)’
(26)
Then (Y and /? are defined by (YLI = YLI
237
Letters B 410 (1997) 233-240
mf z
1yLu31*qf?
m2, %;;I
mf
PRU1121
=
x~\aRu2?~aLu2~2~~~
PLu112r1,2.
(29)
The heavier eigenvalues are also obtained as follows, rnt=Mi, rns=Mj, m+2iyRu3/2q~. (30) In the case of the down type quarks the masses are obtained by assuming M, z=MS z=M, * qR:
YL
‘YLI 2 YL39 IYL31
--
YjP3 ‘YRI -
aRf=yRl
7YR39 bR3l
pLI
=
aL,
-
2,
m,=
’ aL1 aL2, laL212
a’2
where I = 1,2. Geometrically a,(Z = 1,2) are the projections of y, onto the plane perpendicular to the vector y,, and PI is the projection of cr, onto the line perpendicular to (Ye and yS. To find the solutions of Eq. (241, we first set the order of magnitude of X, and X, as follows. If we neglect flavour mixing, in the seesaw model, the up quark mass is given by MLMR/MD. with yttR) = g. Therefore the seesaw suppression factor X, = M,/M, is of the order of mu/M, = 10-5. With a similar argument, we have Xc = M,/M, = lo-’ by assuming that the strength of the Yukawa coupling does not depend significantly on the flavour. By setting the magnitude of X, and Xc in this way, we obtain an approximate mass formula for the case of flavour mixing. The point is that in the approximation that the strengths of all Yukawa couplings do not depend significantly on the flavour, the eigenvalue equation Eq. (24) has the following form,
F(h)= -h3+O(l)P-O(X~)A +O(x;x;)=o.
(28)
M&
m2,EA4,2,
rni=Mi.
(31)
with X, = qR/M,,X, = qR/Ms and X, = qR/M,. We would like to stress the following points. - The mass formulae for the five light quarks are of the seesaw type. They are proportional to ?,I~:77R and inversely proportional to the singlet quark mass. They are also proportional to the appropriate projection of the Yukawa couplings y,,,, y,,,( I = 1,2) for the up and the charm quarks and yLD,,yRD,(I = 1,2,3) for the down type quarks. * The top quark mass Cm,) is proportional to vL and the length of the vector y,,,. The other mass eigenvalue (m,) is proportional to qR and the length of yRU3. - The masses of the remaining five heavier quarks are given to a good approximation by the mass parameters M,,(i= 1,2) and MDi(i= 1,2,3).
4. Flavour mixing In order to find the mixing angles (CKM matrix) among left (right) chiralities of quarks, it is convenient to perform a unitary transformation among ordinary quarks such that the singlet-doublet Yukawa
238
T. Morozumi et al./ Physics Letters B 410 (1997) 233-240
x0 0 1
couplings, ces,
i.e., yL and y, become triangular
YL(R)U(D)= %(D)L(RJ x I x
x x
0 x
matri-
’
Table I Singlet and Doublet mixing angles
(32)
where the U’s are unitary matrices. This decomposition can always be done as proved in the Appendix. Further we introduce a new basis denoted by u’,d’ which is related to the original basis by the unitary matrices, u;. = U&,
u’R= U&uR,
d’, = U&d,,
u; = U&dR.
(33)
In this new basis, the 6 X 6 matrix formed into
M is trans-
(34) The explicit form of M’ for the up type quarks is given by
,+f'=
0
0
0
PLUll?lL
0
0
0
PLU211)L
I"LL'ZI')L
0
0
0
=LoilvL
~LUIZ'IL
PR*"lPlR
1
/(IRUZ'IR I
0
0
0
case,
IYLOhL
MU
0
0
~';U32%
0
MC
0
IYWhR
0
0
MT
PI;o2,1IX an'u,Pln
0
0
II
sine,e’+q
(35)
where
I/3Lu,,l=I~Lu,I
and
lPRuI,I=
l&,_,,I;
their
phases can be obtained from the Appendix; cyLDZ, and yLU, are defined by Eq. (26) and YLD3' aLU2 Eq. (27). In this form, u’ only couples to U. Therefore the up quark mass is determined by the mass of the heaviest singlet quark lJ and is not affected by the presence of the other lighter singlet quarks. The c’ couples to both U and C. However its mass is mainly determined by the mass of C and the effect of U is tiny because 44, >> n/l,. As a result the matrix M’ can be approximated by the following block diagonal matrix M”,
M'" =
0
0
0
0
0
0
0
0
0
laL":hL
0
0
0
0
0
0
0
0
MU
0
0
l~.Q"&lR
0
0
MC
0
0
0
0
Pdu,,%
I
BLL'IIVL
IYRUhR
The off-diagonal elements of M’ which do not appear in M’* can be treated as a perturbation. In this approximation, each doublet quark only couples to one of the singlet quarks, So the eigenvalues and eigenstates are obtained by diagonalizing two by two matrices. It is interesting to see that the eigenvalues of M” agree with those obtained in Eq. (29) and Eq. (301 with the same assumption for the singlet quark mass parameters. In addition the mass eigenstates are defined by the following rotation between the doublet quark q’ and the singlet quark Q,
where the CKM matrix is written, in terms of the unitary matrices defined before, as:
IYL"h. 0
v, = UJLIJDL.
0
MT
In Table 1, we show the mixing angles between singlet and doublet quarks for both chiralities. It can be seen that the singlet to doublet mixing for the five light quarks is strongly suppressed being at least of the order qR(vLL)/M for right (left) -handed quarks. For the top quark, there is a suppression of the left-handed mixing angles of the order qLMT/$ while the right-handed mixing angle is not suppressed. In the following analysis, we only keep the right-handed mixing angle for the top quark and its singlet partner and set the other singlet to doublet mixing angles to zero. In this approximation, the left-handed charged currents and the CKM matrix are given by ._I&/_ = cy&VLij d:i 3 (38)
I (36)
Within this approximation, for the left-handed neutral
(39) we do not have FCNC isospin current. On the
T. Morozumi et ai. /Physics
other hand, the right-handed charged currents are given by
+ COSO,,$~,V,,~~,“~
+
where VR = ud,if,,,, and 8,, = tan- ‘(1y,&?jR/Mr) as shown in Table 1. The right-handed neutral isospin currents are =$
i&,u,”
+~y,cR”
+
(d,~)~i=,&t~
(
+(sin8,R)2F,rypT[ +(h.c.)
-FRj’$dgj),
239
is a characteristic of our model and may be checked in the top quark production experiments.
6. Note added
sinO,,Ty,V,,jd$j, (40)
.3 JMR
Letters B 410 (1997) 233-240
+ sin8,,cosB,,Tm,y,t; (41)
in both currents a sum over j running from 1 to 3 is implied. Because @,a+ ; as M, + 0, there is a large mixing for T/ in the right-handed charged current jJrR and the right-handed neutral current j&.
5. Summary We study a mechanism for the generation of the top quark mass and flavour mixing in the context of a seesaw model for quark masses. When the mass parameter Mr for the isosinglet quark is smaller than the symmetry breaking scale of X!(2),, the lightest of the two eigenstates is at the breaking scale of SU(2), and can be identified with the top quark, the heaviest one will be found at the symmetry breaking scale of S(I(2),. The smallness of M, compared to the breaking scale of SU(2), may be originated from the zero texture of the singlet quark mass matrix at a GUT scale. Much work remains to be done along these lines. We also study the mass hierarchy of ordinary quarks by including the flavour mixing. The stability of the light quark masses against flavour mixing is explicitly shown. The dependence on the strength of the Yukawa couplings is nontrivial and we have given a simple geometrical interpretation for it. The singlet-doublet mixing is suppressed except in the case of the mixing between the right-handed top quark and its singlet partner. This large mixing angle
After completing our work, we have received a paper by Y. Koide and H. Fusaoka 1101in which the enhancement of the top quark mass in a seesaw model is studied. They found that the top quark mass is enhanced to the electroweak symmetry breaking scale at the singular point (bf = - l/3) of their singlet quark mass matrix (democratic matrix + bf X unit matrix). Their singular point would correspond to MT = 0 in our case. Ignoring flavour mixing, the mass scale we obtained for the fourth lightest up type quark together with the mass formula for the top quark coincides with their result at the singular point. Still, our approach differs considerably from theirs and the mass formulae and the flavour mixing we obtain for the light quarks are quite different. We attribute the mass hierarchy of the five light quarks to the mass hierarchy of the five heavy singlet quark mass parameters and assume that the Yukawa couplings among the singlet and doublet quarks do not depend significantly on the flavour whilst they start from a specific type of the singlet quark mass matrix and introduce flavour dependent Yukawa couplings. As a result our formulae for the five light quarks cannot be directly translated into their framework. Furthermore their model gives constraints on the CKM matrix unlike ours where we cannot make definite predictions.
Acknowledgements We would like to thank T. Muta and G.C. Branco for helpful discussions and A. Sugamoto for a suggestion. We would like to thank Y. Koide for conversations about his work and results. This work is supported by the Grant-in-Aid for Joint International Scientific Research (#08044089) from the Ministry of Education, Science and Culture, Japan, and JSPS which made exchange programs of M.N.R. and T.M. possible. The work of T.M. is supported in part by Grant-in-Aid for Scientific Research on Priority Areas (Physics of CP violation).
T. Morozumi et al/Physics
240
Appendix
A
By multiplying triangular form,
We prove the decomposition of y,(,, into a unitary matrix and a triangular matrix. Let us write a rank 3 matrix y with three complex vectors in C3, y= (YPY,,Y3)9
(A.11
where y is the singlet-doublet Yukawa coupling yLcRj in our case. Choose three orthonormal vectors U= (ut,u2,~3) with ~3 = &. Multiplying Ut on the left hand side of y, we obtain
lJ+y=
where r?
Letters B 410 (19971233-240
aI1
aI2
0
a21
ff22
0
I a31
o32
,
(A4
IY3II
aij = ui .yj. Then we define two vectors in
(A.31 We also define two orthonormal vectors Y, and v2 with v2 = a2/l~y2( in C 2. With these two vectors, we can form another unitary matrix V,
v=[‘;;“0: ;I.
(A.41
v+u+y=
Vt
to Uty,
PI,
0
0
p2,
Ia21
0
a31
a32
we finally
)
obtain
a
(A.9
b3l
I I where pii = v! . aj. Then y is written as the product of a unitary matrix and a triangular matrix.
References [l] T. Yanagida, in: Proceedings of the Workshop on The Unified Theory and the Baryon Number of the Universe, Eds. Osamu Sawada, Akio Sugamoto, KEK 13-14 (Feb. 1979), (KEK-79- 18). [2] M. Gell-Mann, P. Ramond, R. Slansky in Sanibel Talk, CALT-68-709 (Feb. 1979). and in: Supergravity (North-Holland, Amsterdam, 1979). [3] Z.G. Berezhiani. Phys. Lett. B 129 (1983) 99; B 150 (1985) 177. [4] D. Chang, R.N. Mohapatra, Phys. Rev. Lett. 58 (1987) 1600. [s] J. Rajpoot. Phys. Lett. B 191 (1987) 122. [6] A. Davidson, K.C. Wab, Phys. Rev. Lett. 59 (1987) 393. [7] KS. Babu. R.N. Mohapatra, Phys. Rev. Lett. 62 (1989) 1079. [8] R. Barbieri, G. Dvali, A. Sttumia, 2. Berezhiani, L. Hall, Nucl. Phys. B 432 (1994) 49. [9] G. Dvali, S. Pokorski, Phys. Lett. B 379 (1996) 126. [lo] Y. Koide. H. Fusaoka, Z. Phys. C 71 (1996) 459.