Determining quark and lepton mass matrices by a geometrical interpretation

By designating one eigenvector of the mass matrix, one can reduce the free parameters in the mass matrix effectively. Applying this method to the quark mass matrix and to the lepton mass matrix, we find that this method is consistent with available experimental data. This approach may provide some hints for constructing theoretical models. Especially, in the lepton sector, the Koide's mass relation is connected to the element of the tribimaximal matrix through Foot's geometrical interpretation. In the quark sector, we suggest another mass formula and the same procedure also applies.


I. INTRODUCTION
The understanding of quark masses and mixings has posed a major challenge in particle physics for a long time. Recently, the non-zero neutrino masses and their mixings have been confirmed [1], which implies that the mixing also exists in the lepton sector, just like that in the quark sector.
A key step to understand the masses and mixings of quarks and leptons is to determine the mass matrices of quarks and leptons. One popular method, suggested by Fritzsch [2], is the texture zero structure. For example, the four texture zero structure can survive current experimental tests [3]. In the lepton sector, other matrices, for example, based on the ν µ -ν τ symmetry and the A 4 symmetry, have been suggested [1]. Especially, the nearest-neighbor-interaction form (NNIform), can be implemented in some grand unified theories [4], and is consistent with the current experimental data from quarks and leptons [4,5]. Although the progress have been made in these directions, there is still no a commonly granted standard theoretical model for these problems. Therefore, other phenomenological approaches are necessary and worthy to be explored, which may provide some hints for constructing theoretical models. In this paper, we explore a way that can realize Koide's mass formula [6] through Foot's geometrical interpretation [7].
The Yukawa sector of the standard model has too many free parameters. In order to make definite predictions, we must make efforts to reduce the redundant parameters effectively. As we have emphasized, many papers have been devoted for this purpose. One common character of these papers is to reduce the redundant parameters by virtue of some symmetry [1,4]. In this paper, we explore another way, which is different from these approaches. The main ideas are as

II. THE METHOD
In the standard model, the mass matrices are complex matrices in general, but we can use the freedom of right-hand rotation to make them Hermitian [8]. So without loss of generality, we start our discussion from Hermite mass matrices.
Supposed a general Hermite matrix in which A, B, C, φ D , φ E , φ F are real and D, E, F are nonnegative. The matrix M can be written in another way in which P = diag(1, exp iφF , exp iφD ), and α = φ E − φ D + φ F . Given that M has one eigenvector − → x T = (x 1 , x 2 exp iβ , x 3 exp iγ ) T belonging to its eigenvalue λ, we have the eigenequation in which x 1 , x 2 and x 3 are nonnegative real numbers; β and γ are real numbers. After some simplification, it reduces to We see that Eq. (4) contains complex variables, and it will be difficult to solve them. We notice that it will be simple in a special case, in which we let β = −φ F and γ = −φ D . This implies that we designate a real eigenvector to the matrix M in Eq. (4). By this choice, all of the equations have real variables, and they can be solved with little labor. It is obviously that this is only a conventional choice, which simplifies the equation effectively. Of course, we should consider the possibility that this choice may be not appropriate, hence the equations have no solutions. However, in Sec.III and Sec.IV, we will give the numerical results, which imply that our choice is compatible with experimental data. In this paper, we will restrict our discussions on this simple case. Of course, other cases, in which β + φ F = 0 and γ + φ D = 0, are not excluded if they are needed. Then we have in which I is the identity matrix. In Eq. (5), as A, B, C, E, D, F , λ, x 1 , x 2 and x 3 are real numbers, if E and x 3 are nonzero, α must equals to 0 or π. Therefore, this matrix identity produces three equations. It is well known that two of these equations are independent with each other. We choose the independent equations to be in which α = 0 or π. In addition to Eq. (6) and Eq. (7), we have three eigenequations in which N = cos α = ±1. So far, we have five equations, and we have six free parameters, among which only one parameter is still free. We can let F to be the free parameter. Once we fix the value of F, all other parameters are fixed. These equations can be solved analytically, but the expressions are too complicated. In order to simplify the expressions of the solutions, we give some analysis in Appendix B. When we apply them to the lepton sector in Sec.III and to the quark sector in Sec.IV, we will give the analytical expressions explicitly in Appendix C and in Appendix D. However, when we adjust the left free parameters to fit the experimental data, the numerical approach is needed. In the text, we display the numerical results. Also, it is possible that these equations have no solutions, if the eigenvalue and the eigenvector are not appropriate. However, in Sec.III and Sec.IV, we will show that for the parameters we choose, the solutions always exist, as we will display explicitly.
The key point of our method is to choose the appropriate eigenvalue and the appropriate eigenvector. In the following application, we will choose the eigenvector and eigenvalue according to physical ground.
With the method we suggested above, if we fix the value of the left free parameter, we can fix the matrix M . Now we turn to show how we can use this method to determine the mixing matrix. We take the CKM matrix for example. We let the up-quarks mass matrix to be M u , and the down-quarks mass matrix to be M d . Like M , we can write M u and M d as We designate − → y T = (y 1 , y 2 , y 3 ) T as the eigenvector of M u belonging to its eigenvalue λ u , and − → z T = (z 1 , z 2 , z 3 ) T as the eigenvector of M d belonging to its eigenvalue λ d . Then we have According to our analysis above, all the elements of M u and M d can be determined except two of them. Suppose that we input the values of these two free parameters, then we can determine M u and M d . M u and M d can be diagonalized by orthogonal transformation By Eq. (11), Eq. (13) can be rewritten as The CKM matrix can be defined as By our analysis above, P = P u P † d = diag(1, exp iξ , exp iη ). Therefore, we have four free parameters, i.e., F u and F d respectively in M u and M d , and ξ and η in P . It is well known that the CKM matrix have four free parameters. Hence in principle it is possible that we can adjust the values of our free parameters to make them consistent with the experimental data. The similar procedure also applies to the lepton sector.

III. THE APPLICATION TO THE LEPTON SECTOR
In recent years, neutrino physics has made great progress. The mixing of neutrinos has been confirmed and the structure of neutrino mixing matrix has been determined to a reasonable precision [1]. It is well known that the neutrino mixing matrix can be expressed with the tribimaximal matrix approximately [11]. It is The important step of our method is to choose the eigenvectors appropriately. In order to choose eigenvectors in the lepton sector, we notice some investigations below.
In the lepton sector, Koide [6] ever suggested an accurate formula Foot [7] gave a geometrical interpretation to it, where( √ m e , √ m µ , √ m τ ) and (1, 1, 1) are interpreted as two vectors, and θ is the angle between them. If we choose θ = π 4 , we get the Koide's mass formula. We have seen that in the tribimaximal matrix, the entry 33 is approximate to be 1 √ 2 . Therefore there is a natural connection between them. The Koide's mass formula and Foot's geometrical interpretation provide hints for choosing the eigenvectors.
We choose ( √ m e , √ m µ , √ m τ ) and (1, 1, 1) as the eigenvectors we want. It is proper that we choose (1, 1, 1) as the eigenvector of neutrino mass matrix belonging to m 3 (the mass of a neutrino) and ( √ m e , √ m µ , √ m τ ) as the eigenvector of the charged lepton matrix belonging to m τ . This implies that we designate the vectors as follows where P = diag(1, exp iξ , exp iη ). As we described above, we need the values of the mass parameters. For the charged leptons, the masses are known accurately. However, for the neutrinos, only the mass squared differences are measured. The results of global analysis read [1,12] If we assume the normal mass hierarchy, i.e., then we have It is obvious that the mixing matrix only depends on the mass ratio [13]. So we only need the ratios of neutrinos masses. It is convenient to normalize the neutrinos masses as follows Note that these values of mass parameters do not stand for the absolute mass, but just stand for the mass ratio. These ratios are consistent with the experimental data we have displayed above. Let x = 1, y = 1 and F = 0.279853 m 3 in the neutrino mass matrix, by Eqs. and Note that in the course of getting the mass matrices M l and M ν above, we have chosen the eigenvectors of M l and M ν to be real, hence the mass matrices are real matrices according to our analysis in Sec.II. Here we emphasize that this just is a conventional choice, which simplifies the equations effectively.
The matrices that diagonalizes M l and M ν are given as Obviously the third column of V l is ( The MNS matrix is given as The phases α 1 and α 2 , known as the Majorana phases, have physical consequences only if neutrinos are Majorana particles. Because there is no clear evidence whether there is CP-violation in the lepton sector, we can not fix the values of ξ and η. Once we can measure the MNS matrix accurately, we can adjust the free parameters F, ξ and η to fit the experimental data. If we let ξ = 0 and η = 0, we obtain The magnitude of the elements are given as which is very close to the tribimaximal matrix.

IV. THE APPLICATION TO THE QUARK SECTOR
The quark mixing matrix has been determined in high precision. The CKM matrix elements can be most precisely determined by a global fit that uses all available measurements and imposes the standard model constraints. The allowed ranges of the magnitudes of all CKM elements are [14]  Just like that in the lepton sector, we also need two vectors. We find that a numerical relation is well satisfied. It reads The reasons that we choose this formula are displayed in Appendix A.
where P = diag(1, exp iξ , exp iη ). It implies that the vector ( mt √ mu+mc+mt ) T is the eigenvector of M u belonging to its eigenvalue m t , and the vector ( ) T is the eigenvector of M d belonging to its eigenvalue m b . As we have emphasized before, if we fix the value of the free parameter, other elements of the mass matrix are determined. The equations can be solved analytically. We display the result in Appendix C.
and  Obviously the third column of V u is ( The CKM matrix is given as Given ξ = 0.76π + π 4.515 and η = 0.76π, the CKM matrix equals to The quantities of rephasing invariance are calculated as and the invariant measure of CP violation is calculated as while the the experimental data [14] are given as They are consistent with each other.

V. CONCLUSION
We have illustrated the method in Sec.II, and apply it to the lepton sector in Sec.III and to the quark sector in Sec.IV respectively. The character of this method is to designate the eigenvector and the eigenvalue for the mass matrix appropriately. In the lepton sector, we use the Koide's formula, and in the quark sector, we use a similar formula that is well satisfied. Now we give some comments about the mass formula we used.
(1) In the lepton sector, the Koide's mass formula, is satisfied in high precision. It is energy scale insensitive [15], and its other characters were also discussed [16]. Several explanations that can realize this formula have been given [17]. Among these explanations, Foot's geometrical interpretation seems fascinating phenomenologically. However, there is still no a theoretical model that can realize it. It seems that our method can implement this interpretation. If there is no CP-violation and the MNS matrix is tribimaximal, the righthand of the mass formula is connected to the element of the MNS matrix by Foot's geometrical interpretation. This is an approach that can lead to the Koide's mass formula.
(2) In the quark sector, the mass formula like Koide's is unsuccessful [18]. However, because the CKM matrix is very different from the MNS matrix, such a mass formula is useless for us. Alternately, we find another well satisfied mass formula, This mass formula is connected to the element of the CKM matrix.
(3) There are two reasons that we choose the mass formula as the vectors. First, the vectors are expressed by the mass parameters, so we do not need to introduce extra parameters. Obviously that this is an economic choice. Second, there exists such mass formula. They are excellent and are well satisfied in high precision, like the Koide's mass formula, but we can not realize them in a concise and convincing way. Our method provides an approach that can realize them, nevertheless in the special situation if there is no CP-violation. Of course other choices of the vectors are also permitted.
Finally we give some comments about the texture zero structure and our method. The differences between our method and the texture zero structure are: in our present approach, at first we choose one eigenvector of the mass matrix, and then we can determine other elements of the mass matrix; in the texture zero structure, some elements of the mass matrix are supposed to zero, which equals to designate the eigenvectors, but we do not know the eigenvector at first. Therefore, in our approach, we have the freedom to choose the eigenvectors to satisfy other request. For example, the Koide's mass formula can be realized in our approach through Foot's geometrical interpretation.
As the texture zero structure, our approach is also consistent with current experimental data, and this approach has the merit that it can realize some well satisfied mass formula, for example, the Koide's mass formula and the mass formula suggested by us. However, just as many texture zero structures, our approach is purely phenomenological, and there is still no a theoretical model to realize it. Therefore it is worthy to investigate whether our approach can be realized in some theoretical models. If this is true, it will provide a new theoretical and phenomenological approach to deal with the mass and mixing problems. Besides, we point out that in our paper we just restrict our discussions in a simple case, in which we choose the eigenvectors to be real, hence we get the real mass matrices. This is just a conventional choice that simplifies the equations. Other cases, in which the eigenvectors are complex, are permitted, and they should be considered if they are needed by some underlying theories.  In order to express the other elements of the mass matrix with the quarks masses and the free parameter F , we have to solve the equations displayed below It is well known that in which s = λ 1 + λ 2 + λ 3 .
If λ 1 = λ 2 , by Eqs. (3), (4) and (6), we obtain in which r = λ 1 λ 3 + λ 1 λ 2 + λ 2 λ 3 . By Eqs. (1) and (2), we can express D and E in terms of A, B and F in which we have let x = x1 x3 and y = x2 x3 , supposing that x 3 = 0. We obtain five new equations AB + BC + AC = r + (D 2 + E 2 + F 2 ), Let N = cos α = ±1, by these equations, we can express A, B, C, D and E in terms of F . In the following we will deduce another formula, which will simplify the expressions effectively. By Eq. (6), we obtain We have argued that N = cos α = ±1 in Sec.II. So by Eq. (9), we have In Eq. (7), we express C, D and E in terms of A, B and F by Eqs. (8), (10) and (11). After some simplification, we obtain Then B can be solved as Hence if we know the values of A and F , the value of B is determined by the Eq. (13).

APPENDIX C
In this appendix, we give the analytical expressions of the solutions for the quark sector.
In the lepton sector, the analysis in Appendix B and C applies similarly. Let λ = λ 3 and α = 0. We find that the solutions we need can be analytically expressed as follows in which b ′ , a ′ , c ′ are expressed as the same as that in Appendix C. Eqs. (8), (9), (10) and (13) in Appendix C apply similarly.