- Email: [email protected]
Flavor nonconservation and CP violation from quark mixings with singlet quarks
1 January 1998
Physics Letters B 416 Ž1998. 184–191
Flavor nonconservation and CP violation from quark mixings with singlet quarks Isao Kakebe 1, Katsuji Yamamoto
2
Department of Nuclear Engineering, Kyoto UniÕersity, Kyoto 606-01, Japan Received 25 March 1997; revised 12 June 1997 Editor: M. Dine
Abstract Flavor nonconserving and CP violating effects of the quark mixings are investigated in electroweak models incorporating singlet quarks. Especially, the D 0 –D 0 mixing and the neutron electric dipole moment, which are mediated by the up-type quark couplings to the neutral Higgs fields, are examined in detail for reasonable ranges of the quark mixings and the singlet Higgs mass scale. These neutral Higgs contributions are found to be comparable to or smaller than the experimental bounds even for the case where the singlet Higgs mass scale is of the order of the electroweak scale and a significant mixing is present between the top quark and the singlet quarks. q 1998 Elsevier Science B.V.
Some extensions of the minimal standard model may be motivated in various points of view. For instance, in the electroweak baryogenesis, the CP asymmetry induced by the conventional phase in the Cabibbo-Kobayashi-Maskawa ŽCKM. matrix is far too small to account for the observed baryon to entropy ratio, and the electroweak phase transition should be at most of weakly first order consistently with the experimental bound on the Higgs particle mass w1x. In this article, among various possible extensions, we investigate electroweak models incorporating SUŽ2.W = UŽ1. Y singlet quarks with electric charges 2r3 andror y1r3 w2–11x. These sorts of models have some novel features arising from the
1 2
E-mail address: [email protected]. E-mail address: [email protected].
mixings between the ordinary quarks Ž q . and the singlet quarks Ž Q .: The CKM unitarity in the ordinary quark sector is violated, and the flavor changing neutral currents ŽFCNC’s. are present at the treelevel. Furthermore, the q–Q mixings involving new CP violating sources are expected to make important contributions to the electroweak baryogenesis w12x. A singlet Higgs field S introduced to provide the singlet quark mass terms and q–Q mixing terms is even preferable for realizing a strong enough first order electroweak phase transition w1x. It should also be mentioned, as investigated in the earlier literature w6x, that the complex vacuum expectation value Žvev. ² S : of the singlet Higgs field induced by the spontaneous CP violation leads to a non-vanishing CKM phase. The quark mixings with singlet quarks are, on the other hand, subject to the constraints coming from various flavor nonconserving and CP violating pro-
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 3 1 7 - 8
I. Kakebe, K. Yamamotor Physics Letters B 416 (1998) 184–191
cesses w2–6,8,10,11x. In particular, it is claimed in w6x with a simple calculation for the case of d–D mixings that a rather stringent bound, ÕS ' <² S :< R several TeV, on the singlet Higgs mass scale is imposed from the one-loop neutral Higgs contributions to the neutron electric dipole moment ŽNEDM.. ŽThe NEDM in electroweak models with singlet quarks is also considered in w3,4x.. The singlet Higgs field with mass scale in TeV region, however, seems to be unfavorable for the electroweak baryogenesis. Considering this apparently controversial situation, we examine in detail such phenomenological implications of the electroweak models with singlet quarks. We mainly describe the case of singlet U quarks with electric charge 2r3 in the following, keeping in mind the possibility of significant t–U mixing for the electroweak baryogenesis. The analyses are extended readily to the case of singlet D quarks with electric charge y1r3, and similar results are obtained for both cases. In practice, diagonalization of the quark mass matrix is made numerically, in order to calculate precisely the relevant quark couplings to the neutral Higgs fields involving flavor nonconservation and CP violation. Then, for reasonable ranges of the model parameters, systematic analyses are performed on the neutral Higgs contributions to the D 0 –D 0 mixing and the NEDM. In contrast to the earlier expectation w6x, they will turn out to be comparable to or smaller than the experimental bounds even for the case where the singlet Higgs mass scale is of the order of the electroweak scale and a significant t–U mixing is present. The Yukawa couplings relevant for the up-type quark sector are given by L Yukawa s yu 0c l u q0 H y u 0c Ž f U S q f UX S † . U0 y U0c Ž lU S q lXU S † . U0 q h.c.
Ž 1.
with the two-component Weyl fields Žthe generation indices and the factors representing the Lorentz covariance are omitted for simplicity.. Here q0 s Ž u 0 ,V0 d 0 . represents the quark doublets with a unitary matrix V0 , and H s Ž H 0 , Hq . is the electroweak Higgs doublet. A suitable redefinition among the u 0c and U0c fields with the same quantum numbers has been made to eliminate the U0cq0 H couplings without loss of generality. Then, the Yukawa
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coupling matrix l u has been made diagonal by using unitary transformations among the ordinary quark fields. In this basis, by turning off the u–U mixings with f U , f UX ™ 0, u 0 and d 0 are reduced to the mass eigenstates, and V0 is identified with the CKM matrix. The actual CKM matrix is slightly modified due to the u–U mixings Žand possibly the d–D mixings., as shown explicitly later. The Higgs fields develop vev’s, Õ Õ ifS S ² H 0:s ² : Ž 2. '2 , S s e '2 , where ² S : may acquire a nonvanishing phase fS due to either spontaneous or explicit CP violation originating in the Higgs sector. The quark mass matrix is produced with these vev’s as MU s
ž
Du -U , MU
Mu 0
/
Ž 3.
where Mu s
l uÕ
'2
Du -U s Ž f U e i f S q f UX eyi f S .
,
MU s Ž lU e i f S q lXU eyi f S .
ÕS
'2
ÕS
'2
,
.
Ž 4.
The quark mass matrix is diagonalized by unitary transformations V U L and V U R as V U†R MU V U L s diag. Ž m u ,m c ,m t ,mU . .
Ž 5.
ŽWhile the case with one singlet U quark is described hereafter for simplicity of notation, the analyses are readily extended to the case with more than one singlet U quarks, resulting analogous conclusions.. The quark mass eigenstates are determined in terms of the original states by u0 u s V U†L , U U0
ž /
ž /
Ž u c ,U c . s Ž u 0c ,U0c . V U R . Ž 6.
The unitary transformations for the quark mixings may be represented as Vux
V Ux s
e ux
0 X†
ye ux
VUx
w x s L,R x .
Ž 7.
I. Kakebe, K. Yamamotor Physics Letters B 416 (1998) 184–191
186
ŽSimilar unitary transformations V D are introduced x if the d–D mixings are also present.. Then, the u–U mixing submatrices are found in the leading orders to be
Ž e u L . i, Ž e uX L . i , Ž Mu Du -U MUy2 . i ; Ž m u irmU . e u -U ,
Ž 8.
Ž e u R . i , Ž e uX R . i , Ž Du -U MUy1 . i ; e u -U .
Ž 9.
Here the parameter e u -U ; Ž< f U < q < f UX <.rŽ< lU < q < lXU <. represents the mean magnitude of the u–U mixings, though there may be some generation dependence more precisely, as seen later. The relation between the masses and Yukawa couplings of the ordinary up-type quarks is slightly modified by the u–U mixings as mui ,
lu iÕ
'2
1 y 12 Ž e u R e u†R . i i .
Ž 10 .
The generalized CKM matrix for the up-type quarks including the U quark is given by
ž
V
V 0 s V U†L 0 0 0
/
ž
Vu†LV0 0 s † 0 e u LV0
/
ž
0 0
/
.
Ž 11 .
ŽThe effect of possible d–D mixings may be included by multiplying V DL from the right.. Here the CKM unitarity within the ordinary quark sector is violated slightly due to the u–U mixings. It should be noticed in Eq. Ž8. that the left-handed u–U mixings are suppressed further by the relevant u–U mass ratios m u irmU . On the other hand, the right-handed u–U mixings given in Eq. Ž9. are actually ineffective by themselves, since u c and U c fields have the same quantum numbers. Hence, in models of this sort with the quark mass matrix of the form given in Eqs. Ž3. and Ž4., the effects of the light ordinary quark mixings with the singlet quarks appear to be rather small with suppression factors of e u -U and m u irmU under the natural relation l u i ; m u irÕ. In fact, the CKM unitarity violation within the ordinary quark sector is found to arise at the order of Ž m u i m u jrmU2 . e u2-U w2,3,5,6,10,11x, which is sufficiently below the experimental bounds w13x. Contrary to this situation, it is in principle possible,
for instance, to take l u1 4 m urÕ by making a finetuning in Eq. Ž10. with a significant mixing between the uŽs u1 . quark and the singlet U quark. However, such a choice will not respect the quark mass hierarchy in a natural sense; the smallness of m u is no longer guaranteed by a chiral symmetry appearing for l u1 ™ 0. The quark couplings to the neutral Higgs fields are extracted from the Yukawa couplings Ž1., which involve flavor nonconservation and CP violation induced by the u–U mixings: neutral L Yukawa sy
U cL a Uf a q h.c. ,
Ý
Ž 12 .
as0,1,2
where U s Ž u,c,t,U . represents the quark mass eigenstates, and f 0 , f 1 , f 2 are the mass eigenstates of the neutral Higgs fields. The original complex Higgs fields are decomposed as H 0 s ² H 0 : q Ž h1 q ih 2 .r '2 and S s ² S : q Ž s1 q is2 .r '2 with real fields. While the Nambu-Goldstone mode h 2 is absorbed by the Z gauge boson, the remaining h1 , s1 , s2 are combined to form the mass eigenstates f a . Then, the coupling matrices in Eq. Ž12. are given by
La s
1
'2
V U†RŽ Oa0 L u q Oa1 LUqq iOa2 LUy . V U L
Ž 13 . with
Lu s
ž
lu 0
0 , 0
/
LU"s
ž
0 0
f U " f UX . lU " lXU
/
Ž 14 .
Here the Higgs mass eigenstates are expressed with a suitable orthogonal matrix O as
f0 h1 f 1 s O s1 . s2 f2
0 0
Ž 15 .
At present the Higgs masses mf a and the mixing matrix O should be regarded as free parameters varying in certain reasonable ranges. The quark mass matrix may be diagonalized perturbatively with respect to the relevant couplings to determine the quark mixing matrices V U L and V U R . Then, it is seen from Eqs. Ž13. and Ž14. that the FCNC’s of the ordinary up-type quarks coupled to
I. Kakebe, K. Yamamotor Physics Letters B 416 (1998) 184–191
the neutral Higgs fields, in particular, have the following specific generation dependence:
Ž L a . i j ; Ž m u jrmU . e u2-U Ž i / j . .
Ž 16 .
This feature is actually confrimed by numerical calculations, and is also valid for the case of d–D mixings. Since these FCNC’s coupled to the neutral Higgs fields are of the first order of the ordinary quark Yukawa couplings, their contributions are expected to be significant in certain flavor nonconserving and CP violating processes such as the NEDM, D 0 –D 0 , K 0 –K 0 , B 0 –B 0 , b ™ sg , and so on w2– 6,8,10,11x. In fact it will be seen below that the neutral Higgs contributions to the D 0 –D 0 mixing and the NEDM become important for reasonable ranges of the model parameters. In contrast to the FCNC’s coupled to the neutral Higgs fields, the q–Q mixing effects on the Z gauge boson couplings appear at the order of Ž m u i m u jrmU2 . e u2-U with the relation m u i ; l u irÕ, which are related to the CKM unitarity violation w2,3,5,10,11x. Since they are of the second order of the u–U mass ratios, the contributions of the FCNC’s in gauge couplings are suppressed much more, being sufficiently below the experimental bounds w13x for the natural choices of the model parameters. Detailed analyses on various flavor nonconserving and CP violating effects coming from the quark mixings with singlet quarks will be presented elsewhere, which are, in particular, mediated by the quark couplings to the neutral Higgs fields. They would serve as signals for the new physics beyond the minimal standard model. The effective Hamiltonian relevant for the D 0 –D 0 mixing is obtained with the quark couplings in Eq. Ž13. mediated by the neutral Higgs fields f a . They are written with the four-component Dirac fields as HfD cs2 s Ý a
1 mf2 a
c Ž GaS . 21 q Ž GaP . 21g 5 4 u
2
,
Ž 17 . where ) Ž GaS . 21 s 12 Ž L a . 12 q Ž L a . 21
,
) Ž GaP . 21 s 12 Ž L a . 12 y Ž L a . 21
.
Ž 18 .
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The D 0 –D 0 transition matrix element is calculated with this effective Hamiltonian as follows w14x: ² D 0 < HfD cs2 < D 0 : 1 sÝ a
S 2 a 21 y
½ŽG
6 mf2 a
.
2
2
Ž GaP . 21 5 MA0 2
q y Ž GaS . 21 q 11 Ž GaP . 21 MP0 ,
½
MA0 s f D2 m2D ,
MP0 s
5
f D2 m 4D
Ž mu q mc .
2
.
Ž 19 . Ž 20 .
ŽIn the present analysis, it is enough to use the vacuum insertion approximation, by considering various ambiguities in choosing the model parameter values.. Then, the neutral Higgs contribution to the neutral D meson mass difference is given by
DmD Ž f . s
1 mD
¦
;
D 0 < HfD cs2 < D 0 .
Ž 21 .
It is seen from Eq. Ž16. that Ž L a .12 is, in particular, proportional to m crmU rather than m urmU , providing a significant contribution to the D 0 –D 0 mixing. The Z boson contribution to the D 0 –D 0 mixing is investigated in Ref. w8x by considering a possible significant FCNC between the c and u quarks. In contrast to that analysis, the Z boson FCNC of u and c arises at the order of Ž m u m crmU2 . e u2-U with the quark mass matrix Ž3. respecting the relation l u i ; m u irÕ. Hence its contribution to the D 0 –D 0 mixing becomes rather small in the present case. Important contributions are also provided to the NEDM from the quark mixings with singlet quarks. Here, the u quark EDM, which is one of the main components of the NEDM, is induced by the u–U mixings in the one-loop diagrams involving the quark and neutral Higgs intermediate states. ŽThe gauge couplings, on the other hand, do not contribute to the NEDM at the one-loop level, which is due to the same situations as in the minimal standard model w15,10x.. The total contribution of the neutral Higgs
I. Kakebe, K. Yamamotor Physics Letters B 416 (1998) 184–191
188
fields f a to the u quark EDM is calculated by a formula du Ž f . s y
Ž f U . i s k i e u -U e i a i Ž < MU
2e 3 Ž 4p .
=Ý
X
2
Ž f UX . i s k iX e u -U e i a i Ž < MU
Ý
a UKsu i ,U
=
mUK mf2 a
½
Im Ž L a . 1 K Ž L a . K 1
5
Ž1yX .
3 2
y
1 q
2
X
s '2 Ž MU rÕS . .
I Ž m2U K rmf2 a . ,
1
Ž 24 .
lU e i f S q lXU eyi f S' < lU
Ž 22 .
where
IŽ X . s
singlet Higgs field may be parametrized by considering Eqs. Ž4. and Ž9. as
2
Xy
1 1yX
ln X ,
Ž 23 .
Ž 25 .
The generation dependence of the u–U couplings is taken into account with the factors k i and k iX . The lU and lXU couplings are taken under the condition in Eq. Ž4. so as to reproduce the given value of < MU < with varying ÕS , where < MU < is approximately equal to the U quark mass mU up to the corrections due to the u–U mixings. More specifically these parameters are taken in the ranges of
e u -U ; 0.1 , 0 F k i , k iX F 1 , 0 F a i , a iX - 2p ,
Ž 26 . and m U K s m u i ,mU represent the quark mass eigenvalues. ŽThis form of I Ž X . may be modified for the intermediate state of UK s u1. However, the contribution with the u quark intermediate state is in any case negligible due to the very small mass m u .. These contributions to the u quark EDM in fact arise at the first oder of the u quark mass m u Ž l u1 ; m urÕ .. This feature is understood by considering the limit l u1 ™ 0, where the u quark EDM is vanishing due to a relevant chiral symmetry. It is also noticed from Eq. Ž16. that the top quark contributions with th e c o u p lin g s fa c to rs Ž L a . 1 3 Ž L a . 3 1 ; Ž m u m trmU2 . e u4-U become important together with those of the singlet U quark intermediate states. We now make a detailed analysis of the u–U mixing effects on D 0 –D 0 mixing and the u quark EDM. Numerical calculations are performed systematically in the following way: First, by taking the model parameters in certain reasonable ranges, the quark mass matrix MU is diagonalized numerically to obtain the quark mixing matrices V U L and V U R . Then, the quark couplings L a to the neutral Higgs fields are determined with Eqs. Ž13. and Ž14.. Finally, D m D Ž f . and d uŽ f . are calculated by using the formulae Ž17. – Ž23.. Practically, the relevant model parameters are taken as follows: The Yukawa couplings to the
< MU < , mU ; several= 100 GeV ,
0 F b , b X - 2p . Ž 27 .
Here the U quark with mU ; m t is favored, since significant contributions to the electroweak baryogenesis are then expected to be obtained through the CP violating t–U mixing w12x. ŽIt is, however, necessary to take at least mU ) m t , since the U quark has a dominant decay mode U ™ b q W similarly to the top quark.. The diagonal coupling matrix l u for the ordinary up-type quarks is, on the other hand, chosen suitably with the relation Ž10. so that the quark masses m u , m c and m t are reproduced within the experimentally determined ranges w13x. ŽThe corrections due to the u–U mixings in Eq. Ž10. are actually at most of 10 % for e u -U ; 0.3.. The decay constant of the D meson is taken to be f D s 0.3 GeV w13x. The parameters concerning the Higgs fields are taken as mf 0 ; 100 GeV ,
mf 1 ,mf 2 ; ÕS R 100 GeV ,
Ž 28 .
O 01 , O 02 ; ÕrÕS ,
O 12 ; 1 , 0 F fS - 2p .
Ž 29 .
Here the mass of f 0 is fixed to be a somewhat smaller value of 100 GeV or so, as suggested from the requirement that the first order electroweak phase
I. Kakebe, K. Yamamotor Physics Letters B 416 (1998) 184–191
Fig. 1. The neutral Higgs contributions of < d u Ž f .< versus D m D Ž f . are plotted for the case of ÕS s 400 GeV together with the experimental upper bounds on the NEDM Ždashed line. and on D m D Ždotted line.. The relevant parameters are taken to be e u -U s 0.3, MU s 300 GeV, mf 0 s100 GeV, and mf 1s mf 2 s 0.5ÕS .
transition be strong enough. The mixings between h1 and s1 , s2 are supposed to scale as ÕrÕS in a viewpoint of naturalness, since f 0 f h1 Žthe standard neutral Higgs. for the extreme case of ÕS 4 Õ , 246 GeV. The resultant < d uŽ f .< versus D m D Ž f . are shown in Figs. 1 and 2 together with the experimental upper bounds on the NEDM Ždashed line. and on D m D ' < m D 0 y m D 0 < Ždotted line. w13x, where the relevant 1 2 model parameters are taken typically as e u -U s 0.3, MU s 300 GeV Ž, mU ., mf 0 s 100 GeV, and mf 1 s
189
mf 2 s 0.5ÕS with ÕS s 400 GeV ŽFig. 1., and 4 TeV ŽFig. 2.. In these scatter plots, each dot corresponds to a random choice for the set of parameters such as: the complex phases a i , a iX , b , b X and fS ; the generation dependent factors k i and k iX s 2r3 – 1; the coupling ratio < lXU
ž
Vu†LV0 y V0
e u†LV0
/
10y4 10y4 ; 10y5 10y3
10y4 10y4 10y4 10y2
10y5 10y4 . 10y2 10y1
0
Ž 30 .
In particular, for the charged gauge interactions of the t quark and the U quark, < Vt b < s 0.95 y 1 and < VU b < s 0 y 0.2 are obtained. This amount of slight modification in the electroweak gauge interactions of the quarks provide contributions smaller than about 0.2 to the oblique parameters w7x, which are still consistent with the experimental bounds Žsee for instance p. 104 in Ref. w13x.. The modification in the neutral currents U †s m VZ U coupled to the Z boson has also been estimated as 10y1 1 10y9 VZ y VZŽ0. ; 10y7 10y6
10y9 10y7 10y4 10y4
10y7 10y4 10y2 10y1
10y6 10y4 , 10y1 10y2
0
Ž 31 . Fig. 2. The neutral Higgs contributions of < d u Ž f .< versus D m D Ž f . are plotted for the case of ÕS s 4 TeV. The same values are taken for the relevant parameters as in Fig. 1.
where VZŽ0. represents the usual neutral currents in the absence of u–U mixings. ŽThe neutral currents of U c is not modified.. This should be compared to
I. Kakebe, K. Yamamotor Physics Letters B 416 (1998) 184–191
190
the quark couplings to the neutral Higgs fields, which have been estimated as 10y5 10y6 La ; 10y6 10y6
10y4 10y3 10y4 10y4
10y2 10y2 10y1 10y1
10y1 10y1 . 10y1 10y1
0
Ž 32 .
This indicates that the FCNC’s in neutral Higgs couplings are more important than those in Z boson couplings in these sorts of models with the quark mass matrix of the form given in Eq. Ž3. respecting the quark mass hierarchy. As for the case of d–D mixings, the neutral Higgs contributions to the K 0 –K 0 mixing and the NEDM should be investigated as well. It has actually been checked by numerical calculations that the contributions to the d quark EDM are analogous to those obtained for the u quark EDM in the presence of u–U mixings. It should also be mentioned that the CP violation parameter e for the K 0 –K 0 mixing, in particular, can be as large as 10y3 for e d - D ; 0.1 and ÕS ; Õ. Detailed analyses will be presented elsewhere. Finally, some comments are presented for possible variants of the present model. Ži. The complex Higgs field S may be replaced by a real field with f UX s lXU s 0. Then, the coupling matrix lU can be made real and diagonal by a redefinition among the U0 and U0c fields. Even in this case, similar contributions are obtained to D m D and d u with more than one U quarks and the complex Yukawa couplings f U . ŽIf only one U quark is present, the complex phases in f U are absorbed by the u 0c fields, resulting in the vanishing of d uŽ f ... It should, however, be mentioned that with only one real singlet Higgs field the t–U mixing is ineffective for electroweak baryogenesis. This is because the complex phases in the t–U couplings to the real singlet Higgs field can be eliminated by rephasing the U0 and U0c fields. Žii. It may be considered that the singlet Higg field S is absent, regarding Du -U and MU as explicit mass terms in Eq. Ž3.. Even in this case, the significant FCNC’s are still present in the quark couplings to the standard neutral Higgs field f 0 s h1. It should, however, be mentioned that the one-loop neutral Higgs contribution to d u vanishes, just as does the
one-loop Z boson contribution. This is because the standard neutral Higgs field h1 and the NambuGoldstone mode h 2 couple to the quarks in the same way. In conclusion, flavor nonconserving and CP violating effects of the quark mixings have been investigated in electroweak models incorporating singlet quarks. It is found especially that the neutral Higgs contributions to the neutral D meson mass difference and the NEDM are still consistent with the present experimental bounds even for the case where the singlet Higgs mass scale is comparable to the electroweak scale and a significant t–U mixing is present. This, in particular, implies that there is a good chance for these types of models with singlet quarks to generate a sufficient amount of baryon number asymmetry in the electroweak phase transition. A sufficiently strong electroweak phase transition can be realized with the singlet Higgs field with ÕS ; Õ, and the asymmetries of certain quantum numbers contributing to the baryon number chemical potential can be produced by the CP violating t–U mixing through the bubble wall. We would like to thank R.N. Mohapatra and also F. del Aguila and J.A. Aguilar-Saavedra for informing us of their papers, which are relevant for the present article.
References w1x For a review see for instance, A.G. Cohen, D.B. Kaplan, A.E. Nelson, Ann. Rev. Nucl. Part. Sci. 43 Ž1993. 27, and references therein; K. Funakubo, Prog. Theor. Phys. 46 Ž1996. 652, and refereces therein. w2x G.C. Branco, L. Lavoura, Nucl. Phys. B 278 Ž1986. 738. w3x F. del Aguila, J. Cortes, ´ Phys. Lett. B 156 Ž1985. 243. w4x K.S. Babu, R.N. Mohapatra, Phys. Rev. Lett. 62 Ž1989. 1079. w5x P. Langacker, D. London, Phys. Rev. D 38 Ž1988. 886; Y. Nir, D. Silverman, Phys. Rev. D 42 Ž1990. 1477; L. Lavoura, J.P. Silva, Phys. Rev. D 47 Ž1993. 1117; G.C. Branco, T. Morozumi, P.A. Parada, M.N. Rebelo, Phys. Rev. D 48 Ž1993. 1167; V. Barger, M.S. Berger, R.J.N. Phillips, Phys. Rev. D 52 Ž1995. 1663. w6x L. Bento, G.C. Branco, Phys. Lett. B 245 Ž1990. 599. w7x L. Lavoura, J.P. Silva, Phys. Rev. D 47 Ž1993. 2046.
I. Kakebe, K. Yamamotor Physics Letters B 416 (1998) 184–191 w8x G.C. Branco, P.A. Parada, M.N. Rebelo, Phys. Rev. D 52 Ž1995. 4217. w9x W.-S. Hou, H.-c. Kao, Phys. Lett. B 387 Ž1996. 544; G. Bhattacharyya, G.C. Branco, W.-S. Hou, Phys. Rev. D 54 Ž1996. 2114. w10x Y. Takeda, I. Umemura, K. Yamamoto, D. Yamazaki, Phys. Lett. B 386 Ž1996. 167. w11x F. del Aguila, J.A. Aguilar-Saavedra, G.C. Branco, UG-FT69r97, hep-phr9703410, March 1997.
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w12x J. McDonald, Phys. Rev. D 53 Ž1996. 645; T. Uesugi, A. Sugamoto, A. Yamaguchi, Phys. Lett. B 392 Ž1997. 389. w13x Particle Data Group, Phys. Rev. D 54 Ž1996. 1. w14x B. McWilliams, O. Shanker, Phys. Rev. D 22 Ž1980. 2853. w15x For a review see for instance, S.M. Barr, W.J. Marciano, CP Violation, ed. C. Jarlskog, World Scientific, Singapore, 1989, p.455.
Physics Letters B 416 Ž1998. 184–191
Flavor nonconservation and CP violation from quark mixings with singlet quarks Isao Kakebe 1, Katsuji Yamamoto
2
Department of Nuclear Engineering, Kyoto UniÕersity, Kyoto 606-01, Japan Received 25 March 1997; revised 12 June 1997 Editor: M. Dine
Abstract Flavor nonconserving and CP violating effects of the quark mixings are investigated in electroweak models incorporating singlet quarks. Especially, the D 0 –D 0 mixing and the neutron electric dipole moment, which are mediated by the up-type quark couplings to the neutral Higgs fields, are examined in detail for reasonable ranges of the quark mixings and the singlet Higgs mass scale. These neutral Higgs contributions are found to be comparable to or smaller than the experimental bounds even for the case where the singlet Higgs mass scale is of the order of the electroweak scale and a significant mixing is present between the top quark and the singlet quarks. q 1998 Elsevier Science B.V.
Some extensions of the minimal standard model may be motivated in various points of view. For instance, in the electroweak baryogenesis, the CP asymmetry induced by the conventional phase in the Cabibbo-Kobayashi-Maskawa ŽCKM. matrix is far too small to account for the observed baryon to entropy ratio, and the electroweak phase transition should be at most of weakly first order consistently with the experimental bound on the Higgs particle mass w1x. In this article, among various possible extensions, we investigate electroweak models incorporating SUŽ2.W = UŽ1. Y singlet quarks with electric charges 2r3 andror y1r3 w2–11x. These sorts of models have some novel features arising from the
1 2
E-mail address: [email protected]. E-mail address: [email protected].
mixings between the ordinary quarks Ž q . and the singlet quarks Ž Q .: The CKM unitarity in the ordinary quark sector is violated, and the flavor changing neutral currents ŽFCNC’s. are present at the treelevel. Furthermore, the q–Q mixings involving new CP violating sources are expected to make important contributions to the electroweak baryogenesis w12x. A singlet Higgs field S introduced to provide the singlet quark mass terms and q–Q mixing terms is even preferable for realizing a strong enough first order electroweak phase transition w1x. It should also be mentioned, as investigated in the earlier literature w6x, that the complex vacuum expectation value Žvev. ² S : of the singlet Higgs field induced by the spontaneous CP violation leads to a non-vanishing CKM phase. The quark mixings with singlet quarks are, on the other hand, subject to the constraints coming from various flavor nonconserving and CP violating pro-
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 3 1 7 - 8
I. Kakebe, K. Yamamotor Physics Letters B 416 (1998) 184–191
cesses w2–6,8,10,11x. In particular, it is claimed in w6x with a simple calculation for the case of d–D mixings that a rather stringent bound, ÕS ' <² S :< R several TeV, on the singlet Higgs mass scale is imposed from the one-loop neutral Higgs contributions to the neutron electric dipole moment ŽNEDM.. ŽThe NEDM in electroweak models with singlet quarks is also considered in w3,4x.. The singlet Higgs field with mass scale in TeV region, however, seems to be unfavorable for the electroweak baryogenesis. Considering this apparently controversial situation, we examine in detail such phenomenological implications of the electroweak models with singlet quarks. We mainly describe the case of singlet U quarks with electric charge 2r3 in the following, keeping in mind the possibility of significant t–U mixing for the electroweak baryogenesis. The analyses are extended readily to the case of singlet D quarks with electric charge y1r3, and similar results are obtained for both cases. In practice, diagonalization of the quark mass matrix is made numerically, in order to calculate precisely the relevant quark couplings to the neutral Higgs fields involving flavor nonconservation and CP violation. Then, for reasonable ranges of the model parameters, systematic analyses are performed on the neutral Higgs contributions to the D 0 –D 0 mixing and the NEDM. In contrast to the earlier expectation w6x, they will turn out to be comparable to or smaller than the experimental bounds even for the case where the singlet Higgs mass scale is of the order of the electroweak scale and a significant t–U mixing is present. The Yukawa couplings relevant for the up-type quark sector are given by L Yukawa s yu 0c l u q0 H y u 0c Ž f U S q f UX S † . U0 y U0c Ž lU S q lXU S † . U0 q h.c.
Ž 1.
with the two-component Weyl fields Žthe generation indices and the factors representing the Lorentz covariance are omitted for simplicity.. Here q0 s Ž u 0 ,V0 d 0 . represents the quark doublets with a unitary matrix V0 , and H s Ž H 0 , Hq . is the electroweak Higgs doublet. A suitable redefinition among the u 0c and U0c fields with the same quantum numbers has been made to eliminate the U0cq0 H couplings without loss of generality. Then, the Yukawa
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coupling matrix l u has been made diagonal by using unitary transformations among the ordinary quark fields. In this basis, by turning off the u–U mixings with f U , f UX ™ 0, u 0 and d 0 are reduced to the mass eigenstates, and V0 is identified with the CKM matrix. The actual CKM matrix is slightly modified due to the u–U mixings Žand possibly the d–D mixings., as shown explicitly later. The Higgs fields develop vev’s, Õ Õ ifS S ² H 0:s ² : Ž 2. '2 , S s e '2 , where ² S : may acquire a nonvanishing phase fS due to either spontaneous or explicit CP violation originating in the Higgs sector. The quark mass matrix is produced with these vev’s as MU s
ž
Du -U , MU
Mu 0
/
Ž 3.
where Mu s
l uÕ
'2
Du -U s Ž f U e i f S q f UX eyi f S .
,
MU s Ž lU e i f S q lXU eyi f S .
ÕS
'2
ÕS
'2
,
.
Ž 4.
The quark mass matrix is diagonalized by unitary transformations V U L and V U R as V U†R MU V U L s diag. Ž m u ,m c ,m t ,mU . .
Ž 5.
ŽWhile the case with one singlet U quark is described hereafter for simplicity of notation, the analyses are readily extended to the case with more than one singlet U quarks, resulting analogous conclusions.. The quark mass eigenstates are determined in terms of the original states by u0 u s V U†L , U U0
ž /
ž /
Ž u c ,U c . s Ž u 0c ,U0c . V U R . Ž 6.
The unitary transformations for the quark mixings may be represented as Vux
V Ux s
e ux
0 X†
ye ux
VUx
w x s L,R x .
Ž 7.
I. Kakebe, K. Yamamotor Physics Letters B 416 (1998) 184–191
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ŽSimilar unitary transformations V D are introduced x if the d–D mixings are also present.. Then, the u–U mixing submatrices are found in the leading orders to be
Ž e u L . i, Ž e uX L . i , Ž Mu Du -U MUy2 . i ; Ž m u irmU . e u -U ,
Ž 8.
Ž e u R . i , Ž e uX R . i , Ž Du -U MUy1 . i ; e u -U .
Ž 9.
Here the parameter e u -U ; Ž< f U < q < f UX <.rŽ< lU < q < lXU <. represents the mean magnitude of the u–U mixings, though there may be some generation dependence more precisely, as seen later. The relation between the masses and Yukawa couplings of the ordinary up-type quarks is slightly modified by the u–U mixings as mui ,
lu iÕ
'2
1 y 12 Ž e u R e u†R . i i .
Ž 10 .
The generalized CKM matrix for the up-type quarks including the U quark is given by
ž
V
V 0 s V U†L 0 0 0
/
ž
Vu†LV0 0 s † 0 e u LV0
/
ž
0 0
/
.
Ž 11 .
ŽThe effect of possible d–D mixings may be included by multiplying V DL from the right.. Here the CKM unitarity within the ordinary quark sector is violated slightly due to the u–U mixings. It should be noticed in Eq. Ž8. that the left-handed u–U mixings are suppressed further by the relevant u–U mass ratios m u irmU . On the other hand, the right-handed u–U mixings given in Eq. Ž9. are actually ineffective by themselves, since u c and U c fields have the same quantum numbers. Hence, in models of this sort with the quark mass matrix of the form given in Eqs. Ž3. and Ž4., the effects of the light ordinary quark mixings with the singlet quarks appear to be rather small with suppression factors of e u -U and m u irmU under the natural relation l u i ; m u irÕ. In fact, the CKM unitarity violation within the ordinary quark sector is found to arise at the order of Ž m u i m u jrmU2 . e u2-U w2,3,5,6,10,11x, which is sufficiently below the experimental bounds w13x. Contrary to this situation, it is in principle possible,
for instance, to take l u1 4 m urÕ by making a finetuning in Eq. Ž10. with a significant mixing between the uŽs u1 . quark and the singlet U quark. However, such a choice will not respect the quark mass hierarchy in a natural sense; the smallness of m u is no longer guaranteed by a chiral symmetry appearing for l u1 ™ 0. The quark couplings to the neutral Higgs fields are extracted from the Yukawa couplings Ž1., which involve flavor nonconservation and CP violation induced by the u–U mixings: neutral L Yukawa sy
U cL a Uf a q h.c. ,
Ý
Ž 12 .
as0,1,2
where U s Ž u,c,t,U . represents the quark mass eigenstates, and f 0 , f 1 , f 2 are the mass eigenstates of the neutral Higgs fields. The original complex Higgs fields are decomposed as H 0 s ² H 0 : q Ž h1 q ih 2 .r '2 and S s ² S : q Ž s1 q is2 .r '2 with real fields. While the Nambu-Goldstone mode h 2 is absorbed by the Z gauge boson, the remaining h1 , s1 , s2 are combined to form the mass eigenstates f a . Then, the coupling matrices in Eq. Ž12. are given by
La s
1
'2
V U†RŽ Oa0 L u q Oa1 LUqq iOa2 LUy . V U L
Ž 13 . with
Lu s
ž
lu 0
0 , 0
/
LU"s
ž
0 0
f U " f UX . lU " lXU
/
Ž 14 .
Here the Higgs mass eigenstates are expressed with a suitable orthogonal matrix O as
f0 h1 f 1 s O s1 . s2 f2
0 0
Ž 15 .
At present the Higgs masses mf a and the mixing matrix O should be regarded as free parameters varying in certain reasonable ranges. The quark mass matrix may be diagonalized perturbatively with respect to the relevant couplings to determine the quark mixing matrices V U L and V U R . Then, it is seen from Eqs. Ž13. and Ž14. that the FCNC’s of the ordinary up-type quarks coupled to
I. Kakebe, K. Yamamotor Physics Letters B 416 (1998) 184–191
the neutral Higgs fields, in particular, have the following specific generation dependence:
Ž L a . i j ; Ž m u jrmU . e u2-U Ž i / j . .
Ž 16 .
This feature is actually confrimed by numerical calculations, and is also valid for the case of d–D mixings. Since these FCNC’s coupled to the neutral Higgs fields are of the first order of the ordinary quark Yukawa couplings, their contributions are expected to be significant in certain flavor nonconserving and CP violating processes such as the NEDM, D 0 –D 0 , K 0 –K 0 , B 0 –B 0 , b ™ sg , and so on w2– 6,8,10,11x. In fact it will be seen below that the neutral Higgs contributions to the D 0 –D 0 mixing and the NEDM become important for reasonable ranges of the model parameters. In contrast to the FCNC’s coupled to the neutral Higgs fields, the q–Q mixing effects on the Z gauge boson couplings appear at the order of Ž m u i m u jrmU2 . e u2-U with the relation m u i ; l u irÕ, which are related to the CKM unitarity violation w2,3,5,10,11x. Since they are of the second order of the u–U mass ratios, the contributions of the FCNC’s in gauge couplings are suppressed much more, being sufficiently below the experimental bounds w13x for the natural choices of the model parameters. Detailed analyses on various flavor nonconserving and CP violating effects coming from the quark mixings with singlet quarks will be presented elsewhere, which are, in particular, mediated by the quark couplings to the neutral Higgs fields. They would serve as signals for the new physics beyond the minimal standard model. The effective Hamiltonian relevant for the D 0 –D 0 mixing is obtained with the quark couplings in Eq. Ž13. mediated by the neutral Higgs fields f a . They are written with the four-component Dirac fields as HfD cs2 s Ý a
1 mf2 a
c Ž GaS . 21 q Ž GaP . 21g 5 4 u
2
,
Ž 17 . where ) Ž GaS . 21 s 12 Ž L a . 12 q Ž L a . 21
,
) Ž GaP . 21 s 12 Ž L a . 12 y Ž L a . 21
.
Ž 18 .
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The D 0 –D 0 transition matrix element is calculated with this effective Hamiltonian as follows w14x: ² D 0 < HfD cs2 < D 0 : 1 sÝ a
S 2 a 21 y
½ŽG
6 mf2 a
.
2
2
Ž GaP . 21 5 MA0 2
q y Ž GaS . 21 q 11 Ž GaP . 21 MP0 ,
½
MA0 s f D2 m2D ,
MP0 s
5
f D2 m 4D
Ž mu q mc .
2
.
Ž 19 . Ž 20 .
ŽIn the present analysis, it is enough to use the vacuum insertion approximation, by considering various ambiguities in choosing the model parameter values.. Then, the neutral Higgs contribution to the neutral D meson mass difference is given by
DmD Ž f . s
1 mD
¦
;
D 0 < HfD cs2 < D 0 .
Ž 21 .
It is seen from Eq. Ž16. that Ž L a .12 is, in particular, proportional to m crmU rather than m urmU , providing a significant contribution to the D 0 –D 0 mixing. The Z boson contribution to the D 0 –D 0 mixing is investigated in Ref. w8x by considering a possible significant FCNC between the c and u quarks. In contrast to that analysis, the Z boson FCNC of u and c arises at the order of Ž m u m crmU2 . e u2-U with the quark mass matrix Ž3. respecting the relation l u i ; m u irÕ. Hence its contribution to the D 0 –D 0 mixing becomes rather small in the present case. Important contributions are also provided to the NEDM from the quark mixings with singlet quarks. Here, the u quark EDM, which is one of the main components of the NEDM, is induced by the u–U mixings in the one-loop diagrams involving the quark and neutral Higgs intermediate states. ŽThe gauge couplings, on the other hand, do not contribute to the NEDM at the one-loop level, which is due to the same situations as in the minimal standard model w15,10x.. The total contribution of the neutral Higgs
I. Kakebe, K. Yamamotor Physics Letters B 416 (1998) 184–191
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fields f a to the u quark EDM is calculated by a formula du Ž f . s y
Ž f U . i s k i e u -U e i a i Ž < MU
2e 3 Ž 4p .
=Ý
X
2
Ž f UX . i s k iX e u -U e i a i Ž < MU
Ý
a UKsu i ,U
=
mUK mf2 a
½
Im Ž L a . 1 K Ž L a . K 1
5
Ž1yX .
3 2
y
1 q
2
X
s '2 Ž MU rÕS . .
I Ž m2U K rmf2 a . ,
1
Ž 24 .
lU e i f S q lXU eyi f S' < lU
Ž 22 .
where
IŽ X . s
singlet Higgs field may be parametrized by considering Eqs. Ž4. and Ž9. as
2
Xy
1 1yX
ln X ,
Ž 23 .
Ž 25 .
The generation dependence of the u–U couplings is taken into account with the factors k i and k iX . The lU and lXU couplings are taken under the condition in Eq. Ž4. so as to reproduce the given value of < MU < with varying ÕS , where < MU < is approximately equal to the U quark mass mU up to the corrections due to the u–U mixings. More specifically these parameters are taken in the ranges of
e u -U ; 0.1 , 0 F k i , k iX F 1 , 0 F a i , a iX - 2p ,
Ž 26 . and m U K s m u i ,mU represent the quark mass eigenvalues. ŽThis form of I Ž X . may be modified for the intermediate state of UK s u1. However, the contribution with the u quark intermediate state is in any case negligible due to the very small mass m u .. These contributions to the u quark EDM in fact arise at the first oder of the u quark mass m u Ž l u1 ; m urÕ .. This feature is understood by considering the limit l u1 ™ 0, where the u quark EDM is vanishing due to a relevant chiral symmetry. It is also noticed from Eq. Ž16. that the top quark contributions with th e c o u p lin g s fa c to rs Ž L a . 1 3 Ž L a . 3 1 ; Ž m u m trmU2 . e u4-U become important together with those of the singlet U quark intermediate states. We now make a detailed analysis of the u–U mixing effects on D 0 –D 0 mixing and the u quark EDM. Numerical calculations are performed systematically in the following way: First, by taking the model parameters in certain reasonable ranges, the quark mass matrix MU is diagonalized numerically to obtain the quark mixing matrices V U L and V U R . Then, the quark couplings L a to the neutral Higgs fields are determined with Eqs. Ž13. and Ž14.. Finally, D m D Ž f . and d uŽ f . are calculated by using the formulae Ž17. – Ž23.. Practically, the relevant model parameters are taken as follows: The Yukawa couplings to the
< MU < , mU ; several= 100 GeV ,
0 F b , b X - 2p . Ž 27 .
Here the U quark with mU ; m t is favored, since significant contributions to the electroweak baryogenesis are then expected to be obtained through the CP violating t–U mixing w12x. ŽIt is, however, necessary to take at least mU ) m t , since the U quark has a dominant decay mode U ™ b q W similarly to the top quark.. The diagonal coupling matrix l u for the ordinary up-type quarks is, on the other hand, chosen suitably with the relation Ž10. so that the quark masses m u , m c and m t are reproduced within the experimentally determined ranges w13x. ŽThe corrections due to the u–U mixings in Eq. Ž10. are actually at most of 10 % for e u -U ; 0.3.. The decay constant of the D meson is taken to be f D s 0.3 GeV w13x. The parameters concerning the Higgs fields are taken as mf 0 ; 100 GeV ,
mf 1 ,mf 2 ; ÕS R 100 GeV ,
Ž 28 .
O 01 , O 02 ; ÕrÕS ,
O 12 ; 1 , 0 F fS - 2p .
Ž 29 .
Here the mass of f 0 is fixed to be a somewhat smaller value of 100 GeV or so, as suggested from the requirement that the first order electroweak phase
I. Kakebe, K. Yamamotor Physics Letters B 416 (1998) 184–191
Fig. 1. The neutral Higgs contributions of < d u Ž f .< versus D m D Ž f . are plotted for the case of ÕS s 400 GeV together with the experimental upper bounds on the NEDM Ždashed line. and on D m D Ždotted line.. The relevant parameters are taken to be e u -U s 0.3, MU s 300 GeV, mf 0 s100 GeV, and mf 1s mf 2 s 0.5ÕS .
transition be strong enough. The mixings between h1 and s1 , s2 are supposed to scale as ÕrÕS in a viewpoint of naturalness, since f 0 f h1 Žthe standard neutral Higgs. for the extreme case of ÕS 4 Õ , 246 GeV. The resultant < d uŽ f .< versus D m D Ž f . are shown in Figs. 1 and 2 together with the experimental upper bounds on the NEDM Ždashed line. and on D m D ' < m D 0 y m D 0 < Ždotted line. w13x, where the relevant 1 2 model parameters are taken typically as e u -U s 0.3, MU s 300 GeV Ž, mU ., mf 0 s 100 GeV, and mf 1 s
189
mf 2 s 0.5ÕS with ÕS s 400 GeV ŽFig. 1., and 4 TeV ŽFig. 2.. In these scatter plots, each dot corresponds to a random choice for the set of parameters such as: the complex phases a i , a iX , b , b X and fS ; the generation dependent factors k i and k iX s 2r3 – 1; the coupling ratio < lXU
ž
Vu†LV0 y V0
e u†LV0
/
10y4 10y4 ; 10y5 10y3
10y4 10y4 10y4 10y2
10y5 10y4 . 10y2 10y1
0
Ž 30 .
In particular, for the charged gauge interactions of the t quark and the U quark, < Vt b < s 0.95 y 1 and < VU b < s 0 y 0.2 are obtained. This amount of slight modification in the electroweak gauge interactions of the quarks provide contributions smaller than about 0.2 to the oblique parameters w7x, which are still consistent with the experimental bounds Žsee for instance p. 104 in Ref. w13x.. The modification in the neutral currents U †s m VZ U coupled to the Z boson has also been estimated as 10y1 1 10y9 VZ y VZŽ0. ; 10y7 10y6
10y9 10y7 10y4 10y4
10y7 10y4 10y2 10y1
10y6 10y4 , 10y1 10y2
0
Ž 31 . Fig. 2. The neutral Higgs contributions of < d u Ž f .< versus D m D Ž f . are plotted for the case of ÕS s 4 TeV. The same values are taken for the relevant parameters as in Fig. 1.
where VZŽ0. represents the usual neutral currents in the absence of u–U mixings. ŽThe neutral currents of U c is not modified.. This should be compared to
I. Kakebe, K. Yamamotor Physics Letters B 416 (1998) 184–191
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the quark couplings to the neutral Higgs fields, which have been estimated as 10y5 10y6 La ; 10y6 10y6
10y4 10y3 10y4 10y4
10y2 10y2 10y1 10y1
10y1 10y1 . 10y1 10y1
0
Ž 32 .
This indicates that the FCNC’s in neutral Higgs couplings are more important than those in Z boson couplings in these sorts of models with the quark mass matrix of the form given in Eq. Ž3. respecting the quark mass hierarchy. As for the case of d–D mixings, the neutral Higgs contributions to the K 0 –K 0 mixing and the NEDM should be investigated as well. It has actually been checked by numerical calculations that the contributions to the d quark EDM are analogous to those obtained for the u quark EDM in the presence of u–U mixings. It should also be mentioned that the CP violation parameter e for the K 0 –K 0 mixing, in particular, can be as large as 10y3 for e d - D ; 0.1 and ÕS ; Õ. Detailed analyses will be presented elsewhere. Finally, some comments are presented for possible variants of the present model. Ži. The complex Higgs field S may be replaced by a real field with f UX s lXU s 0. Then, the coupling matrix lU can be made real and diagonal by a redefinition among the U0 and U0c fields. Even in this case, similar contributions are obtained to D m D and d u with more than one U quarks and the complex Yukawa couplings f U . ŽIf only one U quark is present, the complex phases in f U are absorbed by the u 0c fields, resulting in the vanishing of d uŽ f ... It should, however, be mentioned that with only one real singlet Higgs field the t–U mixing is ineffective for electroweak baryogenesis. This is because the complex phases in the t–U couplings to the real singlet Higgs field can be eliminated by rephasing the U0 and U0c fields. Žii. It may be considered that the singlet Higg field S is absent, regarding Du -U and MU as explicit mass terms in Eq. Ž3.. Even in this case, the significant FCNC’s are still present in the quark couplings to the standard neutral Higgs field f 0 s h1. It should, however, be mentioned that the one-loop neutral Higgs contribution to d u vanishes, just as does the
one-loop Z boson contribution. This is because the standard neutral Higgs field h1 and the NambuGoldstone mode h 2 couple to the quarks in the same way. In conclusion, flavor nonconserving and CP violating effects of the quark mixings have been investigated in electroweak models incorporating singlet quarks. It is found especially that the neutral Higgs contributions to the neutral D meson mass difference and the NEDM are still consistent with the present experimental bounds even for the case where the singlet Higgs mass scale is comparable to the electroweak scale and a significant t–U mixing is present. This, in particular, implies that there is a good chance for these types of models with singlet quarks to generate a sufficient amount of baryon number asymmetry in the electroweak phase transition. A sufficiently strong electroweak phase transition can be realized with the singlet Higgs field with ÕS ; Õ, and the asymmetries of certain quantum numbers contributing to the baryon number chemical potential can be produced by the CP violating t–U mixing through the bubble wall. We would like to thank R.N. Mohapatra and also F. del Aguila and J.A. Aguilar-Saavedra for informing us of their papers, which are relevant for the present article.
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w12x J. McDonald, Phys. Rev. D 53 Ž1996. 645; T. Uesugi, A. Sugamoto, A. Yamaguchi, Phys. Lett. B 392 Ž1997. 389. w13x Particle Data Group, Phys. Rev. D 54 Ž1996. 1. w14x B. McWilliams, O. Shanker, Phys. Rev. D 22 Ž1980. 2853. w15x For a review see for instance, S.M. Barr, W.J. Marciano, CP Violation, ed. C. Jarlskog, World Scientific, Singapore, 1989, p.455.