NMNS/HEPPH-1301
Type-IV Two-Higgs-Doublet Models And Their CP-Violation Chilong Lin∗ National Museum of Natural Science, 1, Guan Chien RD., Taichung, 40453 Taiwan, ROC
arXiv:1308.6039v1 [hep-ph] 28 Aug 2013
Abstract Several new two-Higgs-doublet models which are naturally free of FCNC will be presented in this manuscript. These models solved the FCNC problem in a way quite different to the previous three types of 2HDMs. In these models, the mass matrices corresponding to different Higgs doublets can be diagonalized simultaneously and respectively owing to their specific textures. These textures exhibit obviously hierarchical breaking of SN permutation symmetry among fermion generations. Since we have now multiple textures for the fermion mass matrices, allotting different textures to up- and down-type fermions suitably will bring about complex CKM matrices. However, the CP-violating phases thus derived are not to be referred to as ”spontaneous” since it is the multiplex SN -breaking textures rather than the phase difference θ which gave the complex and distinct transformation matrices.
PACS numbers: 14.80.Fd, 12.60.Fr, 14.60.Pq
∗
Electronic address:
[email protected]
1
I.
INTRODUCTION
In the standard model (SM) of electro-weak interactions, fermions receive their masses from the nonzero vacuum expectation value (VEV) of its unique Higgs doublet through spontaneous symmetry breaking (SSB). The CP-violation in such a model appears explicitly rather than spontaneously since the CP-violating phase potentially generated in its Higgs fields can always be rotated away, or be absorbed into its fermion fields. Attempting to find a new source, a spontaneous one, of CP-violation, Lee [1] extended the Higgs sector of SM with one extra Higgs doublet so that the phases in their VEVs may unlikely be rotated away simultaneously. He expected the phase difference between the Higgs doublets will provide a spontaneous source of CP-violation. Unfortunately, no one has ever reached this goal so far, no matter in the Higgs-potential sector or the Yukawa-coupling sector. Besides the unsolved spontaneous CP-violation (SCPV) problem, such an extension of standard model brought about another problem: the flavor-changing neutral current (FCNC) problem. This problem arises from the fermion mass matrices corresponding to different Higgs doublets may unlikely be diagonalized simultaneously and respectively when transforming from their weak eigenstates to their mass eigenstates. If the respective mass matrices were not diagonalized simultaneously and respectively, non-zero but cancel-eachother off-diagonal elements in each of them will induce flavor-changing neutral interactions at tree level, which is contradict to experimental data. To avoid the FCNC problem appears in a two-Higgs-doublet model (2HDM), Glashow and Weinberg [2] imposed on the model some discrete symmetry which forbids fermions to couple with both Higgs doublets simultaneously. Two such models were proposed which are usually referred to as the 2HDM-I and 2HDM-II, or Type-I and Type-II 2HDMs. A few years after, Branco et al. [3] stated a necessary and sufficient condition for a natural-flavorconserving (NFC), or FCNC-free, 2HDM M1 M2† − M2 M1† = 0,
(1)
where M1 and M2 are fermion mass matrices corresponding to the Higgs doublets Φ1 and Φ2 , respectively. The 2HDM-I and -II can be treated as special cases satisfying this condition through letting some of the mass matrices be vanish. However, it will be better if one revises Eq.(1) as † † Mq1 Mq2 − Mq2 Mq1 = 0,
(2)
where q= u and d are fermion types since the mass matrices of up- and down-type fermions may have different textures. However, in the following discussions, I will neglect the subindex ”q” from time to time if there were no need to indicate which fermion types they correspond to. For 2HDM-I in which only one Higgs doublet, say Φ1 , couples with all fermion types while the other does not, the matrices Mu2 = Md2 = 0 satisfy Eq.(2) obviously. For 2HDMII in which the Higgs doublets couple respectively to only one of the two types of fermions, the matrices will become either Mu1 = Md2 = 0 or Md1 = Mu2 = 0, which also satisfy Eq.(2). Besides the 2HDM-I and -II, there is another type of models usually referred to as the 2HDM-III or type-III 2HDM [4-9]. Such models do not forbid the fermions to couple with 2
both Higgs doublets. In fact, they do not really solve the problem by finding a group of mass matrices satisfying Eq.(2). They just evade the problem by assuming the tree-level FCNCs are below the empirical level. However, a third 2HDM naturally free of FCNCs at tree level is not really inachievable. In fact, a group of mass matrices satisfying Eq.(2) were proposed in ref.[10-12] through imposing a S3 permutation symmetry among fermion generations. That model was found to be a special case, the case 1 of totally four to be demonstrated in section II, of a more general model. In section II, I will discuss Branco’s statement and demonstrate the assumptions proposed to simplify the models to a manageable level. There, four independent FCNC-free matrix textures satisfying Eq.(2) will be exhibited. These models will be referred to as 2HDM-IV or Type-IV 2HDMs since they are completely different to the previous three types of models. It is interesting that all of these textures possess a S3 symmetry among fermion generations while some of them even possess S3 +S2 hierarchical breaking textures. It is more interesting that these textures not only solve the FCNC problem, they also derive complex, CP-violating CKM matrices. The key of this achievement is to allot different mass matrix textures to different fermion types. That will evade the problem of V u = V d and VCKM = V u V d† = 13×3 we met in previous works [10-12], in which we have only one texture for all fermion types. However, paradoxically, such derived CP-violations should not be referred to as ”spontaneous” since they are completely independent of the phase difference θ derived in the Higgs fields through SSB. In fact, it is the specific SN -breaking textures and their multiplicity gave us the complex, CP-violating CKM matrices. The details of this matter will be discussed in section III and brief conclusions and discussions are to be given in section IV.
II.
FCNC-FREE 2HDM-IVs
As mentioned in last section, 2HDMs whose fermion mass matrices satisfy Eq.(2) will be free of FCNC naturally. The 2HDM-I and -II proposed by Glashow and Weinberg solved this problem by imposing some discrete symmetry to let some of the mass matrices be vanish. However, people wonder if there were any other models in which Higgs doublets may couple with arbitrary type of fermions without any forbiddance while are still FCNC-free at tree level. In our previous works, one such model has been achieved by imposing a S3 permutation symmetry among fermion generations [10,11]. The matrix textures achieved in that model are the same as what are to be given in Eq.(12) below. The mass matrices M1 and M2 correspond to Higgs doublets Φ1 and Φ2 respectively are both non-vanish and can be diagonalized simultaneously and respectively. However, I would like to study the problem in a different aspect in this manuscript. In this section, I will concentrate on how to solve the FCNC problem by finding groups of matrix textures satisfying Eq.(2). The discussions below are primarily mathematical skills to find FCNC-free matrix textures. It is interesting that some of the derived textures 3
obviously reveal a hierarchical breaking of S3 symmetry down to a residual S2 symmetry and then to nothing. It is also amazing that these textures may lead to several CKM-matrices with complex elements. Looking at Eq.(2), it is obvious that the matrices M1 and M2 are too complicated to find them sets of exact solutions if no assumptions were imposed to simplify their textures. In order to simplify the problem to a manageable level, I will first assume the matrices are Hermitian conjugates of themselves and then Eq.(2) will becomes Mq1 Mq2 − Mq2 Mq1 = 0.
(3)
Obviously, these two matrices are still too complicated. Thus, I will go one step further to make them more simplified. Let us consider a case that one of the mass matrices is purely real, say M1 , and the other is purely imaginary, say M2 . Or, we may realize this activity as decomposing the matrices into a purely real part and a purely imaginary part, respectively. The assumption Mi = Mi† (i=1, 2) states the most general form of these matrices as a b b A B B 1 1 2 1 1 2 M1 = B1 A2 B3 = hΦ1 i b1 a2 b3 , b2 b3 a3 B2 B3 A3 0 c c 0 C C 1 2 1 2 (4) M2 = i −C1 0 C3 = hΦ2 i −c1 0 c3 , −c2 −c3 0 −C2 −C3 0 where Aj , Bj and Cj (j=1, 2, 3) are purely real and the diagonal elements of M2 are all zero. The parameters aj , bj and cj in Yukawa-coupling matrices are still complex. But they will be shown all relatively real in the θ = π2 case which is to be discussed below. At this stage, one still has the freedom to choose phases for the VEVs. Here, I would like to choose a real hΦ1 i to make the following derivations easier. Of course, one may choose a complex hΦ1 i possessing a phase eiθ1 . But, the assumption of a real M1 requires that corresponding Yukawa-coupling matrix must possess a common factor e−iθ1 which will annihilate the phase in hΦ1 i. Thus, all elements in M1 are relatively real. As we choose the phase θ1 = 0, the phase difference between two Higgs doublets will appear only in the VEV of Φ2 as hΦ2 i = eiθ √v22 . As derived in ref.[12], if one imposes some symmetry, no matter continuous or discrete, to distinguish the doublets from each other, the phase difference can only be θ = 0 or π2 . The θ = 0 case was studied in ref.[14] and it was proved that no CP violation could be generated spontaneously in that case. But, the θ = π2 case discussed in ref.[12] induced a purely imaginary VEV, i.e., hΦ2 i = i √v22 . In such a case, all imaginary units in M2 come solely from the VEV of Φ2 and all parameters cj are real. That gives us a hope to generate CP-violation in the 4
Yukawa-coupling sector spontaneously, though the final result is negative. Substituting Eq.(4) into Eq.(2), we will B C + B2 C2 1 1 † M2 M1 = i B2 C3 − A1 C1 −A1 C2 − B1 C3 −B1 C1 − B2 C2 † M1 M2 = i −A2 C1 − B3 C2 −B3 C1 − A3 C2
arrive at A2 C1 + B3 C2 B3 C3 − B1 C1
B3 C1 + A3 C2 A3 C3 − B2 C1
−B1 C2 − A2 C3 −B2 C2 − B3 C3 A1 C1 − B2 C3 A1 C2 + B1 C3 B1 C1 − B3 C3 B1 C2 + A2 C3 . B2 C1 − A3 C3 B2 C2 + B3 C3
(5)
The diagonal elements give the following equation B1 C1 = −B2 C2 = B3 C3
(6)
and the off-diagonal ones give another three (A1 − A2 ) = (B3 C2 + B2 C3 )/C1 , (A3 − A1 ) = (B1 C3 − B3 C1 )/C2 , (A2 − A3 ) = −(B2 C1 + B1 C2 )/C3 .
(7) (8) (9)
But, these equations are dependent since substituting Eq.(6) into the sum of Eq.(7) and Eq.(8) will arrive at Eq.(9). These conditions are still insufficient for finding sets of exact solutions since there are too few conditions and too many unknown parameters. Thus, we have to assume some more correlations among the parameters to reduce the number of them. If we assume that all diagonal elements Aj are the same, i.e., A1 = A2 = A3 = A,
(10)
we will arrive at the following relations B12 = B22 = B32 ,
C12 = C22 = C32 .
(11)
There are four independent cases for the correlations among Bj s and Cj s. They are to be demonstrated and discussed respectively in what follows.
Case 1: B1 = B2 = B3 = B and C1 = −C2 = C3 = C
In this case, the mass matrices can be expressed as 0 iC −iC A B B M1 = B A B , M2 = −iC 0 iC , iC −iC 0 B B A 5
(12)
which are exactly the same as those derived in refs.[10, 11] with a S3 permutation symmetry among three fermion generations. In that model, one of the Higgs doublets was assumed to be totally symmetric and the other anti-symmetric under the transformation of S3 . If one demands the Lagrangian to be S3 invariant, Eq.(12) will be the only exact solution for the mass matrices. The mass eigenvalues thus derived are √ √ (m1 , m2 , m3 ) = (A − B − 3C, A − B + 3C, A + 2B) = (α − β, α + β, γ), (13) √ where α = A − B, β = 3C and γ = A + 2B were redefined to achieve a general form for the fermion mass spectra which is to appear also in other cases. The unitary transformation matrix which may diagonalize M1 and M2 simultaneously and respectively can be derived as √ √ V1 =
−1−i √ 3 2 3 √ −1+i √ 3 2 3 √1 3
−1+i √ 3 2 3 √ −1−i √ 3 2 3 √1 3
√1 3 √1 3 √1 3
,
(14)
where the sub-index k (k=1 to 4) of V indicates to which case it corresponds. The diagonalized mass matrices can be expressed respectively as √ 0 0 − 3C √0 0 A−B † V1 M1 V1† = 0 , V1 M2 V1 = 0 A−B 0 3C 0 . 0 0 A + 2B 0 0 0
(15)
Obviously, no FCNC will appear at tree level in such a model. However, this model only solved the FCNC problem. It is imcompetent to provide a spontaneous source of CP-violation since the CKM matrix thus derived is an identity matrix. Such matrix textures obviously possess a S3 permutation symmetry among the fermion generations. The spontaneous symmetry breaking of Higgs fields also break this symmetry completely. However, in the following three cases, we will see the permutation symmetry may be broken down hierarchically. It is this hierarchical breaking down of permutation symmetry which provides a mechanism to generate CP-violation in thus derived CKM matrices.
Case 2: B1 = B2 = −B3 = B and C1 = −C2 = −C3 = C
In this case, the mass matrices A B M1 = B A B −B
can be expressed as 0 iC −iC B −B , M2 = −iC 0 −iC . iC iC 0 A 6
(16)
The mass eigenvalues can be derived as √ √ (m1 , m2 , m3 ) = (A + B − 3C, A + B + 3C, A − 2B) = (α − β, α + β, γ), (17) √ where α = A + B, β = 3C and γ = A − 2B were redefined to have the same texture of mass spectra as in Eq.(13). The corresponding transformation matrix thus derived can be expressed as √ √ V2 =
However, if A B M1 = B A B B
1−i √ 3 2 3 √ 1+i √ 3 2 3 −1 √ 3
we reformulate Eq.(17) as 0 0 0 B B + 0 0 −2B , 0 −2B 0 A
−1−i √ 3 2 3 √ −1+i √ 3 2 3 √1 3
√1 3 √1 3 1 √ 3
.
(18)
0 0 0 0 iC −iC . M2 = −iC 0 iC + 0 0 −2iC (19) 0 2iC 0 iC −iC 0
It is obvious that the first parts of both matrices are exactly the same as those in Eq.(12) which possess a S3 symmetry, while the second parts of them obviously reveal a residual S2 symmetry among the second and third generations. As to be shown in next section, these residual S2 -symmetric parts are the key to generate CP-violations in the CKM matrices. Case 3: B1 = −B2 = B3 = B and C1 = C2 = C3 = C
In this case, the mass matrices can be expressed as 0 iC iC A B −B M1 = B A B , M2 = −iC 0 iC . −iC −iC 0 −B B A
(20)
The mass eigenvalues can be given as √ √ (m1 , m2 , m3 ) = (A + B − 3C, A + B + 3C, A − 2B) = (α − β, α + β, γ), (21) √ where the elements of its mass eigenvalues were redefined as α = A + B, β = 3C and γ = A − 2B to has it the same texture as in Eq.(13). The corresponding transformation matrix can be derived as √ √ V3 =
−1+i √ 3 2 3 √ −1−i √ 3 2 3 √1 3
7
1+i √ 3 2 3 √ 1−i √ 3 2 3 −1 √ 3
√1 3 1 √ 3 1 √ 3
.
(22)
In this case, the mass matrices can also be reformulated as two S3 -symmetric and S2 symmetric parts respectively, just like what happened in case 2. 0 0 2iC 0 iC −iC 0 0 −2B A B B . M1 = B A B + 0 0 0 , M2 = −iC 0 iC + 0 0 0 (23) −2iC 0 0 iC −iC 0 −2B 0 0 B B A This case also satisfy Eq.(2) and V3 is parameter independent, too. The residual S2 symmetry appears in such a hierarchical breaking of permutation symmetry is between the first and third generations.
Case 4: B1 = −B2 = −B3 = −B and C1 = C2 = −C3 = −C
In this case, the mass matrices can be expressed as: 0 −iC −iC A −B B M1 = −B A B , M2 = iC 0 iC . iC −iC 0 B B A
(24)
The mass eigenvalues thus derived are √ √ (m1 , m2 , m3 ) = (A + B − 3C, A + B + 3C, A − 2B) = (α − β, α + β, γ), (25) √ where the mass eigenvalues were redefined as α = A + B, β = 3C and γ = A − 2B to has it the same form as in Eq.(13). The corresponding transformation matrix can be derived as √ √ V4 =
1−i √ 3 2 3 √ 1+i √ 3 2 3 √1 3
1+i √ 3 2 3 √ 1−i √ 3 2 3 −1 √ 3
√1 3 1 √ 3 √1 3
.
(26)
In this case, the mass matrices can also be reformulated as two S3 -symmetric and S2 symmetric parts respectively, just like what happened in case 2. 0 −2iC 0 0 iC −iC 0 −2B 0 A B B . M1 = B A B + −2B 0 0 , M2 = −iC 0 0 0 (27) iC + 2iC 0 0 0 iC −iC 0 0 0 0 B B A This case also satisfy Eq.(2) and V4 is parameter independent, too. The residual S2 symmetry appears in such a hierarchical breaking of permutation symmetry is between the 8
first and second generations. It is interesting that the matrix textures in cases 2, 3 and 4 not only possess a S3 permutation symmetry among three fermion generations. They also reveal residual S2 symmetries between two of the three generations. In fact, they all have a S3 +S2 hierarchy except for between which two generations the residual S2 textures apply. Besides, since the matrices M1 and M2 in these four cases can be diagonalized simultaneously, arbitrary hybrids of them will also be diagonalized by the same corresponding Vi . Thus, the mass matrices can be complex and the assumption of a purely real M1 and a purely imaginary M2 at the beginning of this section is not indeed necessary. However, it is still very useful during the derivation of the mass matrix textures.
III.
CP-VIOLATION IN THE CKM-MATRICES
In a 2HDM, the CP-violations induced through SSB may appear in two sectors of its Lagrangian, one is the Higgs potential sector and the other is the Yukawa-coupling sector. In ref.[12], it was proved that no CP-violation can be induced spontaneously in the Higgs potential of any 2HDM if one imposes some symmetry, no matter continues or discrete, to distinguish the Higgs doublets from each other. That still holds true in this manuscript. At the other hand, the new textures derived in last section induced several complex, CP-violating CKM matrices in the Yukawa-coupling sector. However, paradoxically, these CP-violations should not be referred to as ”spontaneous” due to the reasons to be discussed below. If one reviews the previous S3 -model in refs.[10, 11], he may find the major cause of its failure to give a CP-violating CKM-matrix is: There was only one texture for all fermion types, which gives V u = V d and then VCKM = V u V d† = 13×3 an identity matrix. Thinking contrarily, V u 6= V d will be a necessary but not sufficient condition for deriving a CP-violating CKM matrix. Besides, at least one of the transformation matrices must be complex, or the CKM matrix will be purely real and CP-conserving. As derived in last section, we have now totally four different textures for the mass matrices. Up- and down-type fermions may be allotted to different textures and thus satisfy the first condition mentioned in last paragraph. The CKM-matrices thus derived are tabulated as in TABLE I. The diagonal elements of TABLE I are all the same and 13×3 indicates a 3 × 3 identity matrix. The hyper-index ”*” means the complex conjugate. The details of the matrices
9
V1d†
VCKM
V2d† V3d† V4d†
V1u
13×3 D
D∗
V2u
D∗
13×3 G
V3u
D
G
13×3 E ∗
V4u
F
E
E∗
F E
13×3
TABLE I: Various assembles of CKM matrix.
D, E, F and G are given as what follows √ √ 1−i 3 3
D =
F =
1 3 √ 1+i 3 3
1 3 √ 1+i 3 3 √ 1−i 3 3
2 3
−1 3
2 3
−1 3
2 3
2 3
2 3
2 3
−1 3
1+i 3 3 √ 1−i 3 3 1 3
, E =
, G =
1 3 √ 1+i 3 3 √ 1−i 3 3
−1 3
2 3
2 3
2 3
−1 3
2 3
2 3
2 3
−1 3
√ 1−i 3 3 1 3 √ 1+i 3 3
√ 1+i 3 3 √ 1−i 3 3 1 3
,
(28)
.
(29)
The matrices 13×3 , F and G are purely real and obviously CP-conserving. While those matrices D, E and their complex conjugates are complex, which means they could be CP-violating. However, such derived CP-violations should not be referred to as spontaneously generated. They were originated in the specific SN -breaking textures of mass matrices rather than the non-zero phase θ = π2 of hΦ2 i. The phase difference θ between Higgs doublets appear neither in the transformations V u,d nor in the CKM matrices. Besides the problem of spontaneous CP-violation, the derived CKM matrices have still another problem, the magnitudes of their elements do not fit the experimental values well. The matrices D and E can be expressed as 1 2 −i π6 1 2 i π6 2 −i π6 2 i π6 e e e e 3 3 3 3 3 3 1 π π π π 2 2 1 2 2 i −i i −i (30) D= 3 e 6 3e 6 , E = 3e 6 e 6 . 3 3 3 π π π π 2 −i 6 1 2 i6 1 2 i6 2 −i 6 e e e e 3 3 3 3 3 3 They both differ greatly from the experimental values given in ref.[13]. That indicates the need of more complicated textures for the mass matrices which may induce parameterdependent transformation matrices. If such textures were achieved, the CP-violating phases in the CKM matrices may be directly related to the phase θ and thus be spontaneously generated. Besides, additional parameters will give us a space to tune the parameters to fit them with the experimental CKM elements.
10
IV.
CONCLUSIONS
In this manuscript, several new 2HDMs, the type-IV models, were presented. These models solved the FCNC problem in a way very different to the previous three types of 2HDMs. The Higgs doublets in the models couple with fermions arbitrarily without any forbiddance and their corresponding mass matrices can be diagonalized simultaneously and respectively. The key of this achievement is the specific textures of their mass matrices which reveal obviously hierarchical breaking of the SN permutation symmetry among fermion generations. Besides solving the FCNC problem, these models also derived several complex CKM matrices. The key of this achievement is the multiplicity of the SN -breaking mass matrix textures, totally four were derived in this manuscript, which enable us to allot various textures to different fermion types so that the CKM matrix derived are not always a 13×3 identity matrix just like what happened in ref.[10, 11]. But, though the CP-violations thus derived were originated in specific textures which were originated in the spontaneous symmetry breaking of Higgs fields, they are not to be referred to as ”spontaneous” since it is the SN -breaking textures rather than the SSB phase θ gave us the complex transformation matrices. Besides, the magnitudes of the elements of CKM matrices thus derived do not fit the empirical data well. That also hints a need of textures more complicated than those derived in this manuscript. After solving the FCNC problem of 2HDMs, one may expect to go one step further to derive a CP-violating CKM matrix texture with satisfactory element magnitudes. It is obvious that the textures derived here are still too simple to generate CP-violation spontaneously and to derive a satisfactory CKM matrix. One needs more complicated matrix textures which may bring about at least one parameter-dependent transformation matrix so that the phase difference θ can be embedded in the CKM matrix explicitly. That may also give us a space to tune the parameters to fit the empirical values of CKM matrix elements.
[1] T. D. Lee, Phys. Rev. D 8, 1226 (1973). [2] S. L. Glashow and S. Weinberg, Phys. Rev. D 15, 1958 (1977). [3] G. C. Branco, A.J.Buras and J.M. Gerard, Nucl. Phys. B 259, 306 (1985). [4] W.S. Hou, Phys. Lett. B 296, 179 (1992). [5] D. Chang, W. S. Hou and W. Y. Keung, Phys. Rev. D 48, 217 (1993). [6] S. Nie and M. Sher, Phys. Rev. D 58, 097701 (1998). [7] M. Sher and Y. Yuan, Phys. Rev. D 44, 1461 (1991). [8] D. Atwood, L. Reina and A. Soni, Phys. Rev. D 55, 3156 (1997). [9] D. Atwood, L. Reina and A. Soni, Phys. Rev. Lett. 75, 3800 (1995). [10] C.E. Lee, C. Lin and Y.W. Yang, Chin. J. Phys. 26, 180 (1988). [11] C.E. Lee, C. Lin and Y.W. Yang, Phys. Rev. D 42, 2355 (1990). [12] C. Lin, C.E. Lee and Y.W. Yang, Chin. J. Phys. 30, 41 (1994). [13] K. Nakamura et al., J. Phys. G 37, 75021 (2010).
11
[14] S. Bertolini, Nucl. Phys. B 272, 77 (1986)
12
