NEAP53, April 1997 
hepph/9705203 
Flavor nonconservation and violation
from quark mixings with singlet quarks
Isao Kakebe , Katsuji Yamamoto
Department of Nuclear Engineering, Kyoto University, Kyoto 60601, Japan
abstract
Flavor nonconserving and violating effects of the quark mixings are investigated in electroweak models incorporating singlet quarks. Especially, the  mixing and the neutron electric dipole moment, which are mediated by the uptype 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.
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Some extensions of the minimal standard model may be motivated in various points of view. For instance, in the electroweak baryogenesis, the asymmetry induced by the conventional phase in the CabibboKobayashiMaskawa (CKM) 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 [1]. In this article, among various possible extensions, we investigate electroweak models incorporating singlet quarks with electric charges and/or [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. These sorts of models have some novel features arising from the mixings between the ordinary quarks () and the singlet quarks (): The CKM unitarity in the ordinary quark sector is violated, and the flavor changing neutral currents (FCNC’s) are present at the treelevel. Furthermore, the  mixings involving new violating sources are expected to make important contributions to the electroweak baryogenesis [12]. A singlet Higgs field introduced to provide the singlet quark mass terms and  mixing terms is even preferable for realizing a strong enough first order electroweak phase transition [1]. It should also be mentioned, as investigated in the earlier literature [6], that the complex vacuum expectation value (vev) of the singlet Higgs field induced by the spontaneous violation leads to a nonvanishing CKM phase.
The quark mixings with singlet quarks are, on the other hand, subject to the constraints coming from various flavor nonconserving and violating processes [2, 3, 4, 5, 6, 8, 10, 11]. In particular, it is claimed in [6] with a simple calculation for the case of  mixings that a rather stringent bound, , on the singlet Higgs mass scale is imposed from the oneloop neutral Higgs contributions to the neutron electric dipole moment (NEDM). (The NEDM in electroweak models with singlet quarks is also considered in [3, 4].) 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 quarks with electric charge 2/3 in the following, keeping in mind the possibility of significant  mixing for the electroweak baryogenesis. The analyses are extended readily to the case of singlet quarks with electric charge , 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 violation. Then, for reasonable ranges of the model parameters, systematic analyses are performed on the neutral Higgs contributions to the  mixing and the NEDM. In contrast to the earlier expectation [6], 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  mixing is present.
The Yukawa couplings relevant for the uptype quark sector are given by
(1) 
with the twocomponent Weyl fields (the generation indices and the factors representing the Lorentz covariance are omitted for simplicity). Here represents the quark doublets with a unitary matrix , and is the electroweak Higgs doublet. A suitable redefinition among the and fields with the same quantum numbers has been made to eliminate the couplings without loss of generality. Then, the Yukawa coupling matrix has been made diagonal by using unitary transformations among the ordinary quark fields. In this basis, by turning off the  mixings with , and are reduced to the mass eigenstates, and is identified with the CKM matrix. The actual CKM matrix is slightly modified due to the  mixings (and possibly the  mixings), as shown explicitly later. The Higgs fields develop vev’s,
(2) 
where may acquire a nonvanishing phase due to either spontaneous or explicit violation originating in the Higgs sector. The quark mass matrix is produced with these vev’s as
(3) 
where
(4) 
The quark mass matrix is diagonalized by unitary transformations and as
(5) 
(While the case with one singlet quark is described hereafter for simplicity of notation, the analyses are readily extended to the case with more than one singlet quarks, resulting analogous conclusions.) The quark mass eigenstates are determined in terms of the original states by
(6) 
The unitary transformations for the quark mixings may be represented as
(7) 
(Similar unitary transformations are introduced if the  mixings are also present.) Then, the  mixing submatrices are found in the leading orders to be
(8) 
(9) 
Here the parameter represents the mean magnitude of the  mixings, though there may be some generation dependence more precisely, as seen later. The relation between the masses and Yukawa couplings of the ordinary uptype quarks is slightly modified by the  mixings as
(10) 
The generalized CKM matrix for the uptype quarks including the quark is given by
(11) 
(The effect of possible  mixings may be included by multiplying from the right.) Here the CKM unitarity within the ordinary quark sector is violated slightly due to the  mixings.
It should be noticed in eq.(8) that the lefthanded  mixings are suppressed further by the relevant  mass ratios . On the other hand, the righthanded  mixings given in eq.(9) are actually ineffective by themselves, since and fields have the same quantum numbers. Hence, in models of this sort with the quark mass matrix of the form given in eqs.(3) and (4), the effects of the light ordinary quark mixings with the singlet quarks appear to be rather small with suppression factors of and under the natural relation . In fact, the CKM unitarity violation within the ordinary quark sector is found to arise at the order of [2, 3, 5, 6, 10, 11], which is sufficiently below the experimental bounds [13]. Contrary to this situation, it is in principle possible, for instance, to take by making a finetuning in eq.(10) with a significant mixing between the quark and the singlet quark. However, such a choice will not respect the quark mass hierarchy in a natural sense; the smallness of is no longer guaranteed by a chiral symmetry appearing for .
The quark couplings to the neutral Higgs fields are extracted from the Yukawa couplings (1), which involve flavor nonconservation and violation induced by the  mixings:
(12) 
where represents the quark mass eigenstates, and , , are the mass eigenstates of the neutral Higgs fields. The original complex Higgs fields are decomposed as and with real fields. While the NambuGoldstone mode is absorbed by the gauge boson, the remaining , , are combined to form the mass eigenstates . Then, the coupling matrices in eq.(12) are given by
(13) 
with
(14) 
Here the Higgs mass eigenstates are expressed with a suitable orthogonal matrix O as
(15) 
At present the Higgs masses 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 and . Then, it is seen from eqs.(13) and (14) that the FCNC’s of the ordinary uptype quarks coupled to the neutral Higgs fields, in particular, have the following specific generation dependence:
(16) 
This feature is actually confrimed by numerical calculations, and is also valid for the case of  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 violating processes such as the NEDM, , , , , and so on [2, 3, 4, 5, 6, 8, 10, 11]. In fact it will be seen below that the neutral Higgs contributions to the  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  mixing effects on the gauge boson couplings appear at the order of with the relation , which are related to the CKM unitarity violation [2, 3, 5, 10, 11]. Since they are of the second order of the  mass ratios, the contributions of the FCNC’s in gauge couplings are suppressed much more, being sufficiently below the experimental bounds [13] for the natural choices of the model parameters. Detailed analyses on various flavor nonconserving and 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  mixing is obtained with the quark couplings in eq. (13) mediated by the neutral Higgs fields . They are written with the fourcomponent Dirac fields as
(17) 
where
(18) 
The  transition matrix element is calculated with this effective Hamiltonian as follows [14]:
(19)  
(20) 
(In 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 meson mass difference is given by
(21) 
It is seen from eq.(16) that is, in particular, proportional to rather than , providing a significant contribution to the  mixing. The boson contribution to the  mixing is investigated in ref. [8] by considering a possible significant FCNC between the and quarks. In contrast to that analysis, the boson FCNC of and arises at the order of with the quark mass matrix (3) respecting the relation . Hence its contribution to the  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 quark EDM, which is one of the main components of the NEDM, is induced by the  mixings in the oneloop diagrams involving the quark and neutral Higgs intermediate states. (The gauge couplings, on the other hand, do not contribute to the NEDM at the oneloop level, which is due to the same situations as in the minimal standard model [15, 10].) The total contribution of the neutral Higgs fields to the quark EDM is calculated by a formula
(22) 
where
(23) 
and represent the quark mass eigenvalues. (This form of may be modified for the intermediate state of . However, the contribution with the quark intermediate state is in any case negligible due to the very small mass .) These contributions to the quark EDM in fact arise at the first oder of the quark mass (). This feature is understood by considering the limit , where the quark EDM is vanishing due to a relevant chiral symmetry. It is also noticed from eq.(16) that the top quark contributions with the couplings factors become important together with those of the singlet quark intermediate states.
We now make a detailed analysis of the  mixing effects on  mixing and the 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 is diagonalized numerically to obtain the quark mixing matrices and . Then, the quark couplings to the neutral Higgs fields are determined with eqs. (13) and (14). Finally, and are calculated by using the formulae (17) – (23).
Practically, the relevant model parameters are taken as follows: The Yukawa couplings to the singlet Higgs field may be parametrized by considering eqs.(4) and (9) as
(24) 
(25) 
The generation dependence of the  couplings is taken into account with the factors and . The and couplings are taken under the condition in eq. (4) so as to reproduce the given value of with varying , where is approximately equal to the quark mass up to the corrections due to the  mixings. More specifically these parameters are taken in the ranges of
(26) 
(27) 
Here the quark with is favored, since significant contributions to the electroweak baryogenesis are then expected to be obtained through the violating  mixing [12]. (It is, however, necessary to take at least , since the quark has a dominant decay mode similarly to the top quark.) The diagonal coupling matrix for the ordinary uptype quarks is, on the other hand, chosen suitably with the relation (10) so that the quark masses , and are reproduced within the experimentally determined ranges [13]. (The corrections due to the  mixings in eq. (10) are actually at most of 10 % for .) The decay constant of the meson is taken to be [13]. The parameters concerning the Higgs fields are taken as
(28) 
(29) 
Here the mass of is fixed to be a somewhat smaller value of 100 GeV or so, as suggested from the requirement that the first order electroweak phase transition be strong enough. The mixings between and , are supposed to scale as in a viewpoint of naturalness, since (the standard neutral Higgs) for the extreme case of .
The resultant versus are shown in figs. 1 and 2 together with the experimental upper bounds on the NEDM (dashed line) and on (dotted line) [13], where the relevant model parameters are taken typically as , (), , and with (fig. 1), and (fig. 2). In these scatter plots, each dot corresponds to a random choice for the set of parameters such as: the complex phases , , , and ; the generation dependent factors and = 2/3 – 1; the coupling ratio = 0.1 – 10; and the Higgs mixing matrix O. It is clearly found in fig. 1 that the neutral Higgs contributions are comparable to or smaller than the experimental bounds even for the case of with . On the other hand, if or larger, they become sufficiently below the experimental bounds, as seen in fig. 2. It should also be remarked that if while , namely, that only the top quark has a significant coupling to the singlet quark, then these flavor nonconserving and violating effects for the light quarks become much smaller.
The modification in the CKM matrix (11) induced by the  mixings has been estimated by the numerical calculations for the parameter choices taken in figs. 1 and 2. It is roughly given as
(30) 
In particular, for the charged gauge interactions of the quark and the quark, and 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 [7], which are still consistent with the experimental bounds (see for instance p. 104 in ref. [13]).
The modification in the neutral currents coupled to the boson has also been estimated as
(31) 
where represents the usual neutral currents in the absence of  mixings. (The neutral currents of is not modified.) This should be compared to the quark couplings to the neutral Higgs fields, which have been estimated as
(32) 
This indicates that the FCNC’s in neutral Higgs couplings are more important than those in boson couplings in these sorts of models with the quark mass matrix of the form given in eq.(3) respecting the quark mass hierarchy.
As for the case of  mixings, the neutral Higgs contributions to the  mixing and the NEDM should be investigated as well. It has actually been checked by numerical calculations that the contributions to the quark EDM are analogous to those obtained for the quark EDM in the presence of  mixings. It should also be mentioned that the violation parameter for the  mixing, in particular, can be as large as for and . Detailed analyses will be presented elsewhere.
Finally, some comments are presented for possible variants of the present model. (i) The complex Higgs field may be replaced by a real field with . Then, the coupling matrix can be made real and diagonal by a redefinition among the and fields. Even in this case, similar contributions are obtained to and with more than one quarks and the complex Yukawa couplings . (If only one quark is present, the complex phases in are absorbed by the fields, resulting in the vanishing of .) It should, however, be mentioned that with only one real singlet Higgs field the  mixing is ineffective for electroweak baryogenesis. This is because the complex phases in the  couplings to the real singlet Higgs field can be eliminated by rephasing the and fields. (ii) It may be considered that the singlet Higg field is absent, regarding and as explicit mass terms in eq. (3). Even in this case, the significant FCNC’s are still present in the quark couplings to the standard neutral Higgs field . It should, however, be mentioned that the oneloop neutral Higgs contribution to vanishes, just as does the oneloop boson contribution. This is because the standard neutral Higgs field and the NambuGoldstone mode couple to the quarks in the same way.
In conclusion, flavor nonconserving and 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 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  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 , and the asymmetries of certain quantum numbers contributing to the baryon number chemical potential can be produced by the violating  mixing through the bubble wall.
We would like to thank R. N. Mohapatra and also F. del Aguila and J. A. AguilarSaavedra for informing us of their papers, which are relevant for the present article.
References
 [1] For a review see for instance, A. G. Cohen, D. B. Kaplan and A. E. Nelson, Ann. Rev. Nucl. Part. Sci. 43 (1993) 27, and references therein; K. Funakubo, Prog. Theor. Phys. 46 (1996) 652, and refereces therein.
 [2] G. C. Branco and L. Lavoura, Nucl. Phys. B 278 (1986) 738.
 [3] F. del Aguila and J. Cortés, Phys. Lett. B 156 (1985) 243.
 [4] K. S. Babu and R. N. Mohapatra, Phys. Rev. Lett. 62 (1989) 1079.
 [5] P. Langacker and D. London, Phys. Rev. D 38 (1988) 886; Y. Nir and D. Silverman, Phys. Rev. D 42 (1990) 1477; L. Lavoura and J. P. Silva, Phys. Rev. D 47 (1993) 1117; G. C. Branco, T. Morozumi, P. A. Parada and M. N. Rebelo, Phys. Rev. D 48 (1993) 1167; V. Barger, M. S. Berger and R. J. N. Phillips, Phys. Rev. D 52 (1995) 1663.
 [6] L. Bento and G. C. Branco, Phys. Lett. B 245 (1990) 599.
 [7] L. Lavoura and J. P. Silva, Phys. Rev. D 47 (1993) 2046.
 [8] G. C. Branco, P.A. Parada and M.N. Rebelo, Phys. Rev. D 52 (1995) 4217.
 [9] W.S. Hou and H.c. Kao, Phys. Lett. B 387 (1996) 544; G. Bhattacharyya, G. C. Branco and W.S. Hou, Phys. Rev. D 54 (1996) 2114.
 [10] Y. Takeda, I. Umemura, K. Yamamoto and D. Yamazaki, Phys. Lett. B 386 (1996) 167.
 [11] F. del Aguila, J. A. AguilarSaavedra and G. C. Branco, UGFT69/97, hepph/9703410, March 1997.
 [12] J. McDonald, Phys. Rev. D 53 (1996) 645; T. Uesugi, A. Sugamoto and A. Yamaguchi, Phys. Lett. B 392 (1997) 389.
 [13] Particle Data Group, Phys. Rev. D 54 (1996) 1.
 [14] B. McWilliams and O. Shanker, Phys. Rev. D 22 (1980) 2853.
 [15] For a review see for instance, S. M. Barr and W. J. Marciano, CP Violation, ed. C. Jarlskog, World Scientific, Singapore, 1989, p.455.
Figure Captions
 Fig. 1

The neutral Higgs contributions of versus are plotted for the case of together with the experimental upper bounds on the NEDM (dashed line) and on (dotted line). The relevant parameters are taken to be , , , and .
 Fig. 2

The neutral Higgs contributions of versus are plotted for the case of . The same values are taken for the relevant parameters as in fig. 1.
Fig. 1
Fig. 2