On the four-zero texture of quark mass matrices and its stability

We carry out a new study of quark mass matrices $M^{}_{\rm u}$ (up-type) and $M^{}_{\rm d}$ (down-type) which are Hermitian and have four zero entries, and find a new part of the parameter space which was missed in the previous works. We identify two more specific four-zero patterns of $M^{}_{\rm u}$ and $M^{}_{\rm d}$ with fewer free parameters, and present two toy flavor-symmetry models which can help realize such special and interesting quark flavor structures. We also show that the texture zeros of $M^{}_{\rm u}$ and $M^{}_{\rm d}$ are essentially stable against the evolution of energy scales in an analytical way by using the one-loop renormalization-group equations.


I. INTRODUCTION
The discovery of the Higgs boson [1] signifies the "completion" of the Standard Model (SM) which is not only phenomenologically successful but also theoretically self-consistent.
In particular, it verifies the Brout-Englert-Higgs mechanism and Yukawa interactions which are responsible for the generation of lepton and quark masses. However, the SM is not really "complete" in the sense that it cannot explain the origin of neutrino masses, the structures of lepton and quark flavors, the asymmetry of matter and antimatter in the Universe, the nature of dark matter, etc. Hence one has to go beyond the SM and explore possible new physics behind it in order to solve the afore-mentioned puzzles.
Here let us focus on the flavor puzzles in the SM. The flavor issues mainly refer to the generation of fermion masses, the dynamics of flavor mixing and the origin of CP violation.
Even within the SM in which all the neutrinos are assumed to be massless, there are thirteen free flavor parameters which have to be experimentally determined. On the other hand, one is also puzzled by the observed spectra of lepton and quark masses and the observed patterns of flavor mixing, which must imply a kind of underlying flavor structure [2].
In this paper we restrict ourselves to the flavor issues in the quark sector where there are ten free parameters: six quark masses, three flavor mixing angles and one CP-violating phase.
Thanks to the coexistence of Yukawa interactions and charged-current gauge interactions, the flavor and mass bases of three quark families do not coincide with each other, leading to the phenomenon of flavor mixing and CP violation. The latter is described by a 3 × 3 unitary matrix V , the so-called Cabibbo-Kobayashi-Maskawa (CKM) matrix [3], which can be parameterized in terms of three mixing angles (θ 12 , θ 13 , θ 23 ) and one CPviolating phase (δ) via the definitions of V us = sin θ 12 cos θ 13 , V ub = sin θ 13 e −iδ and V cb = cos θ 13 sin θ 23 and the unitarity of V itself. As V originates from a mismatch between the diagonalizations of the up-type quark mass matrix M u and the down-type one M d , which are equivalent to transforming their flavor bases into their mass bases, an attempt to calculate the flavor mixing parameters should start from the mass matrices in the flavor basis. In view of the experimental results m u m c m t , m d m s m b and θ 13 θ 23 θ 12 ≡ θ C 13 • , where θ C denotes the Cabibbo angle, we believe that the strong hierarchy of three flavor mixing angles must be attributed to the strong hierarchy of quark masses.
Therefore, one is tempted to relate the smallness of three flavor mixing angles with the smallness of four independent mass ratios m u /m c , m c /m t , m d /m s and m s /m b . A famous relation of this kind is the Gatto-Sartori-Tonin (GST) relation sin θ 12 ∼ m d /m s [4]. The Fritzsch ansatz of quark mass matrices [5], can easily lead us to the above GST relation. Note that M F u and M F d possess the parallel structures with the same zero entries. Furthermore, they have been taken to be Hermitian without loss of generality, since a rotation of the right-handed quark fields does not affect any physical results in the SM or its extensions which have no flavor-changing right-handed currents. The Fritzsch ansatz totally involves eight independent parameters, and thus it can predict two relations among six quark masses and four flavor mixing parameters. However, it has been shown that this simple ansatz is in conflict with current experimental data [6].
One may modify the Fritzsch ansatz by reducing the number of its texture zeros. Given a Hermitian or symmetric mass matrix, a pair of its off-diagonal texture zeros are always counted as one zero. Hence the Fritzsch ansatz has six nontrivial texture zeros. It has been shown that adding nonzero (1,1) or (1,3) entries to M F u and M F d does not help much [7], but the following Fritzsch-like ansatz is phenomenologically viable [8,9]: We see that Hermitian M u and M d have the up-down parallelism and four texture zeros. So far a lot of interest has been paid to the phenomenological consequences of Eq. (3) [8][9][10][11]. In particular, the parameter space of this ansatz was numerically explored in Ref. [12], where a mild hierarchy |B|/A ∼B/|B| ∼ 0.24 was found to be favored for both up and down sectors. Here we have omitted the subscript "u" and "d" for the relevant parameters, and we shall do so again when discussing something common to M u and M d throughout this paper. Although there are four complex parameters in Eq. (3), only two linear combinations of the four phases are physical and can simply be denoted as φ 1 = arg(C u ) − arg(C d ) and It has been found that φ 2 is very close to zero or 2π [12], while sin φ 1 is close to ±1 and its sign can be fixed by η u η d sin φ 1 > 0, where the dimensionless coefficients η u and η d will be defined in section II.
In this paper we aim to carry out a new study of the four-zero texture of quark mass matrices and improve the previous works in the following aspects: • We reexplore the parameter space of M u and M d by taking into account the updated values of quark masses and the latest results of the CKM flavor mixing parameters.
The new analysis leads us to a new part of the parameter space, which is interesting but was missed in Ref. [12] and other references.
• We identify two more specific four-zero patterns of M u and M d with fewer free parameters. Namely, there is a kind of parameter correlation in such an ansatz, making the exercise of model building much easier. We present two toy flavor-symmetry models to realize such special and interesting quark flavor structures.
• The running behaviors of M u and M d from a superhigh scale down to the electroweak scale are studied in an analytical way by using the one-loop renormalization-group equations (RGEs), in order to examine whether those texture zeros are stable against the evolution of energy scales. We find that they are essentially stable in the SM.
The remaining parts of this paper are organized as follows. In section II we first explore the complete parameter space of M u and M d and then discuss the relevant phenomenological consequences. Particular attention will be paid to some properties of the four-zero texture that the previous works did not put emphasis on. Section III is devoted to discussions about the special patterns of four-zero quark mass matrices in which some particular relations among the finite matrix elements are possible. Two toy flavor-symmetry models, which can help realize such interesting patterns, will be presented for the sake of illustration. In section IV we derive the one-loop RGE corrections to M u and M d which evolve from a superhigh energy scale down to the electroweak scale. Our analytical results show that those texture zeros are essentially stable against the evolution of energy scales. As a byproduct, the possibility of applying the four-zero texture of quark mass matrices to resolving the strong CP problem is also discussed in a brief way. Finally, we summarize our main results and make some concluding remarks on the quark flavor issues in section V.

II. THE PARAMETER SPACE: RESULTS AND EXPLANATIONS
Before performing an updated and complete numerical analysis of the parameter space of Hermitian M u and M d with four texture zeros, let us briefly reformulate the relations between the parameters of M u,d and the observable quantities [12]. First of all, M u,d can be transformed into a real symmetric matrix M u,d through a phase redefinition: where the subscript "u" or "d" has been omitted, and P = Diag{1, e −iφ C , e −i(φ C +φ B ) } with φ B = arg(B) and φ C = arg(C). Of course, one may diagonalize M as follows: Without loss of generality, we require A and λ 3 to be positive. Then |B|,B and |C| can be expressed in terms of A and the three quark mass eigenvalues λ i (for i = 1, 2, 3, corresponding to m u , m c , m t in the up sector or m d , m s , m b in the down sector): In this case the orthogonal matrix O reads where η = ±1, and the emergence of this coefficient can be understood as follows. Since A and λ 3 have been taken to be positive, λ 1 and λ 2 must have the opposite signs so as to assure a negative value of the determinant of M , When identifying λ 1,2,3 with the physical quark masses, we use η = +1 and −1 to label the cases (λ 1 , λ 2 , λ 3 ) = (−m u , m c , m t ) and (λ 1 , λ 2 , λ 3 ) = (m u , −m c , m t ) in the up sector, respectively. The same labeling is valid for the down sector.
In terms of quark mass eigenstates, the weak charged-current interactions are written as where the CKM matrix V appears in the form The nine elements of V can be explicitly expressed as where In our numerical analysis, we prefer to use |V us |, |V ub |, |V cb | and the CP-violating observable sin 2β as the inputs because their values have been determined to a very good degree of accuracy. Here β stands for one of the inner angles of the CKM unitarity triangle described by the orthogonality relation V * ub V ud + V * cb V cd + V * tb V td = 0 in the complex plane. The three inner angles of this triangle are defined as Obviously, the uncertainty associated with sin 2β is much smaller than those associated with α and γ. The unitarity of V requires α + β + γ = π.
FIG. 1(a) shows the allowed region of A u and A d , which are rescaled as r u = A u /m t and Since the results of r u and r d in the other three cases are not quite different from that illustrated in FIG. 1(a), here we just concentrate on the (η u , η d ) = (+1, +1) case for the sake of simplicity. Now that r u r d is a quite good approximation as shown in FIG. 1(a), we simply use r to denote both r u and r d when their difference needs not to be mentioned. We find that the region of r can be roughly divided into two parts: (1) r is close to 1 and mainly lies in the range of 0.8 to 0.9; (2) r is around 0.5 and mainly ranges from 0.4 to 0.6. These two parts will be referred to as the r ∼ 1 and r ∼ 0.5 regions, respectively, in the following discussions. The reasonableness of case for example, we find that φ 2 is very close to 2π and thus its allowed range can also be denoted as φ 2 0. In comparison, the allowed range of φ 1 is much wider but it can also be divided into two parts in a reasonable approximation: φ 1 ∼ 0.5π and φ 1 ∼ 1.5π. There is actually a correlation between r and φ 1 : in the r ∼ 1 region φ 1 ∼ 0.5π holds, and in the r ∼ 0.5 region φ 1 ∼ 1.5π holds. After examining all the four (η u , η d ) = (±1, ±1) cases, we obtain the more general correlation between r and φ 1,2 as follows: Note that only the constraint η d sin φ 2 < 0 is numerically exact, and the other three constraints serve for good approximations in which most scattered points are satisfied. Such correlative constraints can also be given in a more explicit way, as listed in TABLE I. Finally let us point out that the r ∼ 1 region and its corresponding parameter correlation found here are consistent with the results presented in Ref. [12], but the r ∼ 0.5 region and its parameter correlation are our new findings which were missed in the previous works (mainly because φ 1 takes totally different values in this region from our expectation based on its values in the r ∼ 1 region).
All the correlative constraints listed in TABLE I can find an explanation once the analytical expression of the CKM matrix V is explicitly presented. No matter whether the region r ∼ 1 or r ∼ 0.5 is concerned, one can easily check that A is close to the mass of the third-family quark and thus it is much larger than the masses of the first-and second-family quarks. As a result, the orthogonal matrices O u and O d can approximate to Because Although the factor m s /(r d m b ) is actually much smaller than 1, it is kept in the (3,1) and (3,3) entries of O d since it will play a crucial role in explaining the correlation η d sin φ 2 < 0.
Given the approximate results of O u and O d in Eq. (16), it is straightforward to calculate all the CKM matrix elements by using Eq. (10). We are particularly interested in Among them |V cb | deserves special attention and can be decomposed as follows: where V cb = e −iφ 1 V cb has been defined. Clearly, neither |Re (V cb )| nor |Im (V cb )| is allowed to be larger than the experimental result |V cb | 0.04. That is why r u is always nearly equal to r d and φ 2 is so close to 0 or 2π. For either r u ∼ r d ∼ 1 or r u ∼ r d ∼ 0.5, the fact of φ 2 ∼ 0 (or 2π) allows us to simplify the expression of |V us | to It is known that the term m d /m s itself can fit the experimental value of |V us | to a good degree of accuracy (i.e., the GST relation), and hence one has to control the contribution from the smaller term m u /m c by adjusting the CP-violating phase φ 1 . This observation immediately leads to cos φ 1 ∼ 0, or equivalently φ 1 ∼ 0.5π or 1.5π. As first pointed out in Ref. [15], the relation in Eq. (19) is essentially compatible with the orthogonality relation V * ub V ud +V * cb V cd +V * tb V td = 0 after the latter is rescaled by V * cb , leading to the striking prediction α φ 1 ∼ 0.5π for the corresponding CKM unitarity triangle. Needless to say, this prediction is consistent with current experimental data shown in Eq. (14).
In order to understand the correlation between the signs of sin φ 1,2 and those of η u,d , one needs to consider the impact of the CP-violating observable sin 2β on the parameter space of M u and M d . Eqs. (10) and (16) allow us to obtain Then the definition of β in Eq. (13) leads us to Given the experimental value of sin 2β in Eq. (14), we arrive at tan β = 0.394 ± 0.015. In the r ∼ 1 region the first term of Eq. (21) is dominant, and thus η u η d sin φ 1 is required to be positive. Note that this term is at most 0.322, if the values of quark masses in Eq. (12) are input. Hence the second term of Eq. (21) has to be positive too. In other words, η d sin φ 2 should be negative because Im(V cb ) is proportional to sin φ 2 . Furthermore, η u η d cos φ 1 is likely to be negative to enhance the contribution of the second term of Eq. (21) to tan β.
When the r ∼ 0.5 region is concerned, we find that the second term of Eq. (21) becomes important, so η d sin φ 2 is still required to be negative. Since this term has a chance to saturate the experimental value of tan β, the first term of Eq. (21) is possible to be negative in such a case. In fact, η u η d sin φ 1 must be negative in the r ∼ 0.5 region if we take into account the constraint from |V ub |. With the help of Eqs. (17) and (18), we have Taking r d = 0.5 for example, we find that the first term of Eq. (22) is about 0.0044, larger than the experimental value |V ub | 0.0036. In this case the second term of Eq. (22) should be negative so as to partially offset the contribution from the first term. Therefore, we are left with η u η d sin φ 1 < 0. However, when the first term is large enough, the second term will fail to offset its extra contribution, bringing about a lower bound about 0.3 for r d as shown in FIG. 1(a).
To summarize, we have performed a new numerical analysis of the four-zero ansatz of quark mass matrices by using the updated values of quark masses and CKM parameters.
We find a new part of the parameter space of this ansatz -the r ∼ 0.5 region together with the relevant correlation between φ 1,2 and η u,d . We have also explained the salient features of the whole parameter space of M u and M d in some analytical approximations. As a byproduct, FIG. 3 shows the numerical outputs of |V ub |/|V cb | versus sin 2β and α versus γ in the (η u , η d ) = (+1, +1) case. One can see that the uncertainties associated with the CP-violating quantities α and γ remain quite significant, and they mainly originate from the uncertainties of m s , r u and r d . In the next section, we shall go back to the quark mass matrices themselves to look at their structures and to see whether they assume some special patterns with fewer free parameters.

III. SPECIAL FOUR-ZERO PATTERNS AND MODEL BUILDING
In the Fritzsch ansatz of quark mass matrices, the first term of Eq. (22) should be replaced with (m s /m b ) m d /m b , whose size is about 0.00078. This result can easily be understood from the trace of M F u,d (i.e., A = λ 1 + λ 2 + λ 3 ), which gives rise to r d = 1 − O(m s /m b ) in the down sector. Since the upper limit that the second term of Eq. (22) can reach is about 0.00236, the experimental value of |V ub | has no way to be saturated by these two terms in the Fritzsch ansatz. When the four-zero texture of M u,d is concerned, the existence of nonzero (2,2) entries modifies the trace of M u,d to the form A +B = λ 1 + λ 2 + λ 3 . The constraint on r d is consequently relaxed, and it actually becomes a free parameter. Given a typical value r d = 0.9 in the r ∼ 1 region, for example, the first term of Eq. (22) contributes a value 0.0014 to |V ub | such that the experimental result of |V ub | can be well fitted. In this case the magnitude ofB is about 0.1λ 3 . To avoid a relatively large λ 2 from such a largeB, the three parameters A,B and |B| must satisfy an approximate geometrical relation [9] up to a correction of O(m 2 /m 3 ): . In short, the (2,3) sectors of M u and M d have the same structure which can be parameterized as where (1 − r) /r , and its value is about 0.3 in the r ∼ 1 region of the parameter space. Eq. (24) hints at a common origin of the (2,3) sectors of M u and M d , and thus it can be taken as a guideline for model building. However, a numerical analysis shows that such an up-down parallelism is slightly broken by the (2,2) entries of quark mass matrices. In addition, their (1,2) entries do not share this kind of parallelism, as one can see in FIG. 1(d).
With the help of Eq. (6), we typically take r u r d 0.9 and illustrate the finite matrix elements of M u and M d as follows: It is worth reiterating that the mild hierarchy in the (2,3) sectors of quark mass matrices is crucial to fit current experimental data.
In the r ∼ 0.5 region of the parameter space of quark mass matrices M u and M d , there is a particularly interesting case, which deserves special attention. We have verified that these exact equalities are really allowed in our numerical calculations. The corresponding parameter space is certainly a part of the parameter space restricted by r ∼ 0.5. In this special case, the (2,3) sectors of M u and M d have a neat form: A typical numerical illustration of the structures of M u and M d turns out to be One can see that the (2,3) sectors of quark mass matrices are suggestive of an underlying flavor symmetry which controls the second and third quark families.
In fact, the 2 ↔ 3 permutation symmetry of quark mass matrices, which is quite similar to the striking µ ↔ τ permutation symmetry in the lepton sector [16], has been conjectured long before [17]. Under this simple flavor symmetry the mass matrix takes the form But such a scenario has been ruled out by the present experimental data, as pointed out in Ref. [18]. This situation can be easily understood by taking a look at the expression of |V ub | in Eq. (17), where the two terms originate from O u and O d in the following way: If there were an exact 2 ↔ 3 permutation symmetry, both θ u 13 and θ d 13 would have to be vanishing. However, the second term alone is unable to fit the experimental value of |V ub |, as already discussed above. Hence we conclude that quark mass matrices might possess a partial 2 ↔ 3 permutation symmetry such that Since there is a large hierarchy between (1,2) and ( Under the parity symmetry, a flavor theory should be invariant when a left-handed fermion field is replaced by its right-handed counterpart (i.e., ψ L → ψ R ), or vice versa.
As for the Yukawa interactions of quark fields, the parity transformation is where i and j are the quark flavor indices, and H stands for the vacuum expectation value (VEV) of the Higgs field. The invariance of Yukawa interactions under parity transformation requires the Yukawa coupling matrix elements to satisfy the condition y ij = y * ji , and hence the corresponding quark mass matrix must be Hermitian in the flavor space. We are therefore motivated to consider Hermitian quark mass matrices in the framework of the Left-Right (LR) symmetric model with an explicit parity symmetry [21].
The LR model extends the SM gauge groups to SU (2) L × SU (2) R × U (1) B−L , where SU (2) R is the opposite of SU (2) L and acts only on the iso-doublets constituted by the righthanded fields, and B−L stands for the baryon number minus the lepton number. All the fermion fields are grouped into iso-doublets as follows: In the present work we concentrate on the quark sector and leave out the lepton fields L i L and L i R . At the scale Λ R which is higher than the electroweak scale, SU (2) R × U (1) B−L is broken to U (1) Y . The residual SU (2) L and U (1) Y are exactly the SM gauge groups which are subsequently broken by a bi-doublet field Φ under SU (2) L × SU (2) R : The six quarks acquire their masses via their Yukawa interactions with Φ: In the non-supersymmetric LR model κ has a relative phase as compared with κ, and this may violate the Hermiticity of quark mass matrices. Hence we prefer to work in the framework of the supersymmetric (SUSY) LR model [22]. Note that the y ij term in Eq. (35) will be forbidden by the holography requirement of the superpotential in this framework.
Now that the issue of Hermiticity has been settled, let us continue to build quark mass models under certain flavor symmetries in a usual way. We begin with a model that can lead to a four-zero texture of M u and M d in the r ∼ 1 region. It is easy to derive the special pattern of M u,d in Eq. (24) with the help of the Froggatt-Nielsen (FN) mechanism [19]. The point is to introduce a global U (1) FN symmetry to structure the quark mass matrices. All the fields relevant for quark masses and their charges under U (1) FN are listed in TABLE II.
According to the convention in SUSY, Q i R is represented by its corresponding left-handed chiral superfield Q ic R . In the SUSY LR models, at least two bi-doublets are needed to avoid the exact parallelism between M u and M d . In our model four bi-doublets are introduced, and their VEVs are written as In addition, two gauge singlets S 1 and S 2 are introduced to spontaneously break the U (1) FN flavor symmetry.
For clarity, let us explore the phenomenological consequences of this model step by step.
The contribution from Φ 1 can be expressed as where y 22 and y 33 are real, but y 23 is complex. Λ is the scale where all the fields associated with the FN mechanism reside. The non-renormalizable operators arise from integrating out the heavy fields which are not explicitly given in which can reproduce the flavor structure in Eq. (24). There is the exact parallelism between up and down quark sectors, because they have the same origin (i.e., from Φ 1 here). However, this situation also brings about two phenomenological problems. One of them is that the (2,2) entries of M u and M d actually do not respect this exact parallelism, as we have seen in Eq. (25). The other problem is that the (2,3) entries of M u and M d should have a phase difference, so as to assure φ 2 = 0 or 2π.
To address these two problems, let us take account of the contribution from Φ 2 as follows: This treatment modifies Eq. (38) If the ratios κ 1 /κ 2 and κ 1 /κ 2 are close but not exactly equal to each other, the difference between the ( Finally, Φ 3 and Φ 4 can offer finite masses for the first quark family through the terms from which we obtain M u12 = M * u21 = y 12 κ 3 + y 12 κ 4 , M d12 = M * d21 = y 12 κ 3 + y 12 κ 4 .
The reason that we arrange Φ 3 and Φ 4 to have the same quantum number is rather simple: in this case the phases of M u12 and M d12 can be different, such that we are left with a nonzero φ 1 . In a complete flavor-symmetry model the smallness of M u12 and M d12 should also be explained via the FN mechanism as we have done for the (2,3) sectors of M u and M d . Instead of repeating a similar exercise, here we simply assume that κ 3 , κ 3 , κ 4 and κ 4 are much smaller than their counterparts κ 1 , κ 1 , κ 2 and κ 2 . Of course, the elements M 11 and M 13 are vanishing as limited by the relevant flavor quantum numbers.
When it comes to the particular case M 22 = M 33 , a non-Abelian flavor symmetry is needed to realize this equality. The simplest candidate of this kind is the S(3) group which has three irreducible representations 1, 1 and 2. The tensor products of these representations can be decomposed as follows [23]: All the relevant fields are organized to be the representations of S(3) in the following way: where Φ 1 , Φ 2 and Φ 3 are bi-doublets, and their VEVs are specified to be In this model the equality of M 22 and M 33 results from the operator and their values are given by In comparison, the elements M 23 and M 32 are generated by the operator which leads us to A problem is that φ 2 always equals zero, because y 2 is a real number. This is a consequence of the Hermiticity of quark mass matrices as required by the LR models and of the fact that (Q 2c R , Q 3c R ) T and (Q 2 L , Q 3 L ) T are in the same S (3) representation. Hence one has to go beyond the present framework to resolve this problem. But such an attempt is beyond the scope of this paper and will be made elsewhere. Another noteworthy point is that the operators in Eqs. (46) and (48) are completely independent of each other, so it is difficult to understand why M 23 is so close to M 22 and M 33 . We conjecture that these two operators are possible to come from the same tensor product in a larger group, so that M 22 = M 33 = M 23 can be obtained as the leading-order approximation.
Finally, let us consider the operators They lead us to the nonzero (1,2) entries of quark mass matrices: Note that it is Φ 3 = 0 that ensures the vanishing of M 13 and M 31 . There is also a problem that φ 1 equals zero, but it can be overcome by introducing the column vector (Φ 4 , Φ 5 ) T .
Similar to (Φ 2 , Φ 3 ) T , Φ 4 acquires its VEV but Φ 5 does not. In this case Eq. (51) is modified to the form We just need κ 4 /κ 2 = κ 4 /κ 2 to make φ 1 nonzero. The last remarkable issue is that one needs to impose the FN quantum numbers on Q 1 L and Q 1c R , in order to explain why the magnitude of M 12 is suppressed by a power of as compared with those of M 22 and M 23 .
Such a treatment can also help avoid a large M 11 arising from the operator y 5 Q 1c R Φ 1 Q 1 L . If we assign a FN quantum number n to both Q 1 L and Q 1c R , for instance, the contribution of this operator will be suppressed by 2n and thus negligibly small.
In short, we have identified two special four-zero patterns of quark mass matrices and discussed two toy models for realizing them. We admit that some of our discussions about model building remain quite preliminary. Hence a further study is desirable in this respect.
As shown in section III, the four-zero texture of quark mass matrices may result from an underlying flavor symmetry. But the failure in discovering any new physics of this kind indicates that it is likely to reside in a superhigh energy scale, such as the grand unification theory (GUT) scale. This means that a flavor-symmetry model should be built somewhere far above the electroweak scale and the RGE running effects have to be taken into account when its phenomenological consequences are confronted with the experimental data at low energies [24]. One may follow two equivalent ways to consider the evolution of energy scales, provided there is no new physics between the flavor symmetry scale Λ FS and the electroweak scale M Z [25]: (a) the first step is to figure out quark masses and flavor mixing parameters from M u and M d at Λ FS , and the second step is to run these physical quantities down to M Z via their RGEs; (b) the first step is to evolve M u and M d from Λ FS down to M Z via their RGEs, and the second step is to calculate quark masses and flavor mixing parameters from the corresponding quark mass matrices at M Z . Here we take advantage of way (b) to examine the stability of texture zeros of M u and M d against the evolution of energy scales in an analytical way. The RGE effect on the Fritzsch texture of quark mass matrices has been studied in a similar way [25,26]. At the one-loop level, the RGEs of the quark Yukawa coupling matrices in the SM can be written as where t = ln(µ/M Z ), and the subscript "q" stands for "u" and "d". The contributions of the charged leptons and neutrinos to Eq. (53) have been omitted, because they are negligibly small in the SM. Denoting the VEV of the Higgs field as v, we can express the four-zero texture of Y u and Y d at Λ FS as follows: Without loss of generality for CP violation, we have chosen b u and c u to be real in Eq. (54).
The terms G q (t) and T (t) read which arise from quantum corrections to the quark and Higgs field strengths, respectively.
They are flavor-blind, and thus proportional to the identity matrix in the flavor space. Since their effects are simply to rescale quark mass matrices as a whole at a lower energy scale, they will be dropped for the moment. Namely, we are mainly concerned about the first term in Eq. (53): q , let us rewrite Eq. (53) by dropping its G q (t) and T (t) terms: In a good approximation S q can be expressed as where ∆ 1 = a 2 u + b 2 u , ∆ 2 = b 2 u +b 2 u and ∆ 3 = b u (a u +b u ). Then we solve the differential equations in Eq. (56) and obtain where the elements denoted as "· · · " can be directly read off by considering the Hermiticity of H u , and ρ describes the RGE running effects from Λ FS to M Z : Here t FS = ln (Λ FS /M Z ), and y(t ) is the Yukawa coupling eigenvalue of the top quark which evolves according to For illustration, ρ ∼ 0.9 when Λ FS ∼ 10 15 GeV, as shown in FIG. 4. On the other hand, The RGE-corrected quark mass matrices can then be extracted from Eqs. (58) and (61): where is the overall rescaling factor of quark mass matrices brought back from the G q (t) and T (t) RGE correction. Because of the parallelism between (a u , b u ,b u ) and (a d , So the real part of the second term of M d23 at M Z is suppressed by a factor proportional to (1 − ρ) m c /m t 10 −3 as compared with the real part of its first term. In other words, the real part of M d23 is approximately equal to ρ −1 Re(b d ) at M Z . According to our phase assignment in Eq. (54), φ 2 = arg(b u ) − arg(b d ) = − arg(b d ) holds. Hence the phase of b d is equal to −φ 2 and must be close to 0 or −2π. In the r ∼ 1 region where ∆ 2 is much smaller than ∆ 1 , it is easy to see that the imaginary part of the (2,3) and the value of sin 2β is almost unchanged from M Z to Λ FS (or vice versa) within the accuracy that we need. The choice of this specific scale is for two simple reasons: on the one hand, it is expected to be around the canonical seesaw [27] and leptogenesis [28] scales; on the other hand, it is close to the energy scale relevant for the possible vacuum stability issue of the SM [29]. Therefore, In comparison, the corresponding quark mass matrices at the electroweak scale are where ε = ρ −1 − 1 is a small value of O(0.1) or much smaller. This numerical exercise confirms our qualitative analysis made above. In particular, the imaginary part of the (2,3) entry of M d (M Z ) is really independent of ρ, and its real part is proportional to ρ −1 . Now let us turn to the running behaviors of quark masses and flavor mixing parameters.
Since c u is negligibly small in magnitude as compared with a u , b u andb u , the invariants of the (2,3) submatrix of M u (Λ FS ) and M u (M Z ) lead us to These relations indicate that m c and m t change with the energy scale in the following way: When c u is concerned, a similar trick yields m u (Λ FS ) σ −1 u m u (M Z ). It is easy to verify that the similar relations hold in the down sector: and These results clearly show that the mass ratios m u /m c and m d /m s are essentially free from the RGE corrections.
To see how the flavor mixing parameters evolve from Λ FS down to M Z , we take a new look at Eq. (17). Above all, the dimensionless parameters r u and r d are independent of the energy scale to a good degree of accuracy. The reason is simply that m t (or m b ) and A u (or A d ) have nearly the same running behaviors, as one can see from Eqs. (62), (70) and (71). It is also straightforward to conclude that |V us | is stable against the evolution of energy scales.
In view of r u r d and φ 2 0, we arrive at the approximation Given Eqs. (65) and (71), the running behavior of |V cb | turns out to be With the help of this result and Eq. (17), we immediately obtain In addition, Eq. (21) tells us that β is nearly scale-independent. It is easy to check that α and γ, the other two inner angles of the CKM unitarity triangle, are also free from the RGE corrections at the one-loop level [30].
The above results can simply be translated into the ones for three flavor mixing angles and one CP-violating phase in the standard parametrization of the CKM matrix: Of course, α, β and γ are all the functions of δ in this parametrization. As for the Jarlskog invariant J = cos θ 12 sin θ 12 cos 2 θ 13 sin θ 13 cos θ 23 sin θ 23 sin δ [31], it is easy to arrive at Finally, let us briefly comment on a possible implication of the loss of Hermiticity of M d running from Λ FS down to M Z . We conjecture that it might have something to do with the strong CP problem [32], which is put forward due to the unnatural smallness of the parameter θ = θ QCD + θ QFD . Here θ QCD is the coefficient of the CP-violating term in the QCD Lagrangian [33], and θ QFD comes from the quark flavor sector, The experimental upper bound of θ is at the 10 −11 level [34], in sharp contrast with a natural value of O(1) from a theoretical point of view. The demand for explaining why θ is so tiny poses the strong CP problem. An attractive solution for this problem is the Peccei-Quinn mechanism [35] in which an anomalous U (1) symmetry is introduced to ensure a complete cancellation between θ QCD and θ QFD . Another competitive strategy is to remove θ QCD by imposing a spontaneously broken P or CP symmetry (e.g., in the LR symmetric model), and to keep the second term vanishing in the meantime [36,37].
Being Hermitian, the four-zero texture of quark mass matrices automatically satisfies the Although φ 2 is very small, it cannot be exactly zero as shown in our numerical analysis.
Given r = 0.9 and Λ FS = 1 TeV, for instance, Eq. (78) leads us to a value of O(10 −5 ), much larger than the upper bound of θ. One way out of this problem is to fine-tune the value of r. But the possibility of r 1 has phenomenologically been ruled out, as discussed at the beginning of section III. If the parallelism between the forms of M u and M d is given up, the situation will change. For example, in a flavor basis with M u being diagonal, the value of θ QFD was estimated to be of O(10 −16 ) in Ref. [38]. In short, it seems difficult to directly employ the four-zero texture of quark mass matrices to solve the strong CP problem in the scenario of spontaneous CP violation. But a more detailed study of this issue is needed before a firm conclusion can be achieved.

V. SUMMARY
We have carried out a new study of the four-zero texture of Hermitian quark mass matrices and improved the previous works in several aspects. In our numerical analysis what really matters is that we have found a new part of the parameter space, corresponding to A ∼ |B| ∼B (or r ∼ 0.5), and confirmed the known part corresponding to A > |B| >B (or r ∼ 1). In particular, the exact equality between A andB is allowed, and this opens an interesting window for model building.
We have identified two special four-zero patterns of quark mass matrices and constructed two toy flavor-symmetry models to realize them. One of the patterns possesses a mild hierarchy A ∼ |B| ∼ 2B with being about 0.3, and it can be obtained with the help of the FN mechanism. The other pattern assumes A =B, which can be realized by means of the S(3) flavor symmetry. Both of them show a similarity between the (2,3) sectors of M u and M d , indicating that the latter could have the same origin. We have done two model-building exercises in the SUSY LR framework with an explicit parity symmetry, which ensures the Hermiticity of quark mass matrices at the flavor symmetry scale Λ FS .
We have also studied the RGE effects on the four-zero texture of quark mass matrices in an analytical way, from Λ FS down to the electroweak scale M Z . Our results show that the texture zeros of M u and M d are essentially stable against the evolution of energy scales, but their finite entries are rescaled due to the RGE-induced corrections. An interesting consequence of the RGE running is the loss of the Hermiticity of M d at M Z in the SM.
As a byproduct, the possibility of applying the four-zero texture of quark mass matrices to resolving the strong CP problem has been discussed in a very brief way.
Although the predictive power of texture zeros has recently been questioned in the lepton sector [39], they remain useful in the quark sector to understand the correlation between the hierarchy of quark masses and that of flavor mixing angles. We remark that possible flavor symmetries are behind possible texture zeros, and they are phenomenologically important to probe the underlying flavor structure before a complete flavor theory is developed.
Note added. While our paper was being finished, we noticed a new preprint [40] in which a systematic survey of possible texture zeros of quark mass matrices was done but the four-zero texture of Hermitian quark mass matrices with the up-down parallelism was not explicitly discussed.