A bottom-up approach to fermion mass hierarchy: a case with vector-like fermions

We propose a bottom-up approach that a structure of a high-energy physics is explored by accumulating existence proofs and/or no-go theorems in the standard model or its extension. As an illustration, we study fermion mass hierarchies based on an extension of the standard model with vector-like fermions. It is shown that a magnitude of elements of Yukawa coupling matrices can become $O(1)$ and a Yukawa coupling unification can be realized in a theory beyond the extended model, if vector-like fermions mix with three families. In this case, small Yukawa couplings in the standard model can be highly sensitive to a small variation of matrix elements, and it seems that the mass hierarchy occurs as a result of a fine tuning.


Introduction
One of the most fascinating riddles in particle physics is the origin of the fermion mass hierarchy and flavor mixing in the standard model (SM). Various intriguing attempts have been performed to solve it. Most of them are based on the top-down approach [1,2,3,4,5,6]. Starting from high-energy theories (HETs) such as grand unified theories and superstring theories or extensions of the SM with some flavor symmetries, Yukawa coupling matrices are constructed or ansatzes called texture zeros are proposed, to explain the flavor structure in the SM. In spite of endless efforts, we have not arrived at satisfactory answers, because a theory beyond the SM has not yet been confirmed and there is no powerful guiding principle to determine it. Although flavor symmetries are possible candidates 1 , any solid evidence has not yet been discovered.
In the exploration of the flavor physics, the bottom-up approach has also been carried out [16,17,18,19,20,21]. Observed values of fermion masses and mixing angles become a springboard for study on the origin of flavor structure. In some cases, the analyses are made based on specific models. In other cases, the form of Yukawa coupling matrices can be specified to some extent by adopting a guiding principle and/or taking reasonable assumptions, independent of models in HETs, in the framework of the SM or its extension. This approach has also a limitation, because global U(3) symmetries exist in the fermion kinetic terms, Yukawa coupling matrices contain unphysical parameters, and the structure of HETs cannot be completely identified from experimental data alone. However, it can offer useful information on HETs, depending on how it is used. In concrete, we make plausible conjectures in the extension with extra particles and/or under additional assumptions, and then we can give some statements as theorems, by examining whether they are correct or not.
In this paper, based on the bottom-up approach, we make conjectures on Yukawa couplings and pursue whether they are realized or not in an extension of the SM including heavy vector-like fermions, without specifying HETs beyond the extension. We consider two conjectures. One is that a magnitude of Yukawa couplings can become O(1) in a HET. It comes from Dirac's naturalness. Here, Dirac's naturalness means that the magnitude of dimensionless parameters on terms allowed by symmetries should be O(1) in a fundamental theory. If it is true, the hierarchical structure of Yukawa couplings occurs after a transition from a HET to the extension of the SM or through a mechanism in some lower-energy physics. The other is that Yukawa couplings can be unified in a HET. It stems from a symmetry principle. It is deeply related to a grand unification based on SO(10) [22] and E 6 [23]. One of our goals is to present our strategy, its availability, and its limitation, and hence we focus on the quark sector, in the following.
The outline of this paper is as follows. In the next section, we introduce our strategy based on the SM and explain our setup on an extension of the SM. In Sect. 3, we examine whether above-mentioned conjectures hold or not. In the last section, we give conclusions and discussions.

An extension of the standard model 2.1 Strategy
Before we explain our setup, we introduce our strategy using the SM. Let us start with the quark sector in the SM, described by the Lagrangian density: where q Li are left-handed quark doublets, u Ri and d Ri are right-handed up-and downtype quark singlets, i , j (= 1, 2, 3) are family labels, summation over repeated indices is understood with few exceptions throughout this paper, y (u) i j and y (d) i j are Yukawa coupling matrices, φ is the Higgs doublet,φ = i τ 2 φ * and h.c. stands for hermitian conjugation of former terms.
Here, we assume that there exists a set of privileged field variables (q ′ L , u ′ R , d ′ R ) relating to the flavor structure. A candidate is a unitary basis of flavor symmetries [20,21]. By the change of field variables as the above Lagrangian density is rewritten by where N q , N u , and N d are 3×3 complex matrices, k (q) i j , k (u) i j , and k (d) i j are quark kinetic coefficients, and y 1 i j and y 2 i j are Yukawa coupling matrices for privileged fields. They yield the relations: where R are unitary matrices and, using them, the Yukawa coupling matrices are diagonalized as V KM is the Kobayashi-Maskawa matrix defined by [24] Using experimental values of quark masses, y (u) diag , y (d) diag , and V KM are roughly estimated at the weak scale as [25] In the final expressions, λ n means O λ n with λ = sin θ C ∼ = 0.225 (θ C is the Cabibbo angle [26]). 2 Physical parameters, in general, receive radiative corrections, and the above values should be evaluated by considering renormalization effects to match with their counterparts of HET at a high-energy scale.
L ′ quark SM is supposed to be obtained as a result of a transition from a theory beyond the SM and can possess useful information on what is behind the flavor structure. Note that non-canonical matter kinetic terms appear in L ′ quark SM . 3 2 Although the magnitude of (V KM ) 13 is 0.00365 ± 0.00012 and is regarded as O λ 4 , we treat it as O λ 3 with respect for the Wolfenstein parametrization [27]. 3 Several works on the flavor physics have been carried out based on non-canonical matter kinetic terms [28,29,30,31,32,33,34,35].
Because the kinetic coefficients are hermitian and positive definite, they are written by where U q , U u , and U d are 3×3 unitary matrices, and J q , J u , and J d are real 3×3 diagonal matrices. Then, N q , N u , and N d are expressed by where V q , V u , and V d are 3 × 3 unitary matrices. As a magnitude of elements in unitary matrices is not beyond 1, we obtain the inequalities: from the requirement that the magnitude of matter kinetic coefficients should be at most O(1) according to Dirac's naturalness and due to the appearance of suppression factors in terms containing higher-dimensional operators. Then, from Eqs. (6) and (10) Next, we examine whether the Yukawa couplings can be unified or not. Using L ′ quark SM , we obtain the following no-go theorem.
[Theorem] Under the kinetic unification such as k i j , the exact Yukawa coupling unification such as y 1 i j = y 2 i j does not occur in the framework of SM.
It is understood from the observation that we obtain unrealistic features such that Yukawa coupling matrices have same eigenvalues, degenerate masses are derived between up-and down-type quarks, and the Kobayashi-Maskawa matrix becomes a unit matrix, if the kinetic coefficients are common and the up-type Yukawa coupling matrix is identical to the down-type one.
In the case that the unifications are required from a symmetry, symmetry breaking terms appear and ruin some unification relations in the broken phase. Here, we consider the case that the kinetic unification is destroyed. In this case, from Eqs. (5) and (6), the following relation should hold to realize y 1 i j = y 2 i j . Using Eqs. (9), (10), (11), and (15), we obtain the relation: If a fine tuning is absent in Eq.
and then every element becomes small, i.e., y 1 i j = y 2 i j ≤ O λ 3 .
In this way, we have obtained no-go theorems on the Yukawa couplings using experimental data in the framework of SM, without specifying HETs. Through such a downto-earth approach, knowledge and information on the flavor structure are expected to be accumulated, and some clues to the origin of flavor and hints on HETs are provided.
In the following, we apply this strategy in an extension of the SM.

Setup
We adopt several assumptions. (a) A theory beyond the SM, which is referred to as HET, has higher gauge symmetries. It owns a seed of the flavor structure, and the form of Yukawa coupling matrices is determined by HET. Fermion kinetic terms do not necessarily take a canonical form, where the origin of flavor structure is unveiled [20,21].
(b) At a high-energy scale, the theory turns out to be an extension of the SM, i.e., a model with the SM gauge group G SM (= SU(3) C × SU(2) L × U(1) Y ) and extra particles. We refer to it as "SM + α". One should be careful not to confuse SM + α with HET. 4 (c) The extra particles have large masses, compared with the weak boson mass, and the SM particles survive after the decoupling of heavy ones. There are 4th generation fermions and their mirror particles, as extra particles. Here, mirror particles are particles with opposite quantum numbers under G SM . A fermion and its mirror one obey a vector representation in pairs, and hence they are often referred to as a vector-like fermion. We consider the Lagrangian density of quarks in SM + α, described by where q ′ LI are counterparts of left-handed quark doublets, u ′ RI and d ′ RI are those of righthanded up-and down-type quark singlets, and I and J run from 1 to 4.  Table 1. For reference, we list the gauge quantum numbers and the chirality of mirror quarks in Table  2. In Tables 1 and 2, Y is the weak hypercharge. Table 1: The gauge quantum numbers and the chirality of quarks in SM + α.
The Yukawa interactions and mass terms in L ′ quark SM+α are compactly written by where A and B run from 1 to 5, and U ′ LA , U ′ RA , D ′ LA , and D ′ RA consist of 5 components such that and M (U ) AB and M (D) AB are 5 × 5 complex matrices given by respectively. The ellipsis in Eq. (18) stands for terms containing a charged component of the Higgs doublet.
To apply the bottom-up approach to our model, we need to know the relationship between the privileged fields in L ′ quark SM+α and mass eigenstates in the SM. In the following, we will see that the kinetic terms change into the canonical form and mass terms including Yukawa interactions are diagonalized by redefining field variables.
First, we pay attention to the fact that the quark kinetic coefficients are hermitian and are expressed by , N u L I 5 = 0, N u L 5J = 0 and so on. Then, after the change of variables: we obtain the canonical type of quark kinetic terms: where q LI and q L(m) c are SU(2) L doublets given by respectively.
Here, we give comments. The transformation matrices N u arbitrary unitary matrices, and using this arbitrariness, M (U ) and M (D) are diagonalized, as will be described below. Then, V KM appears in q LI i D q LI on the mass eigenstates. The quark kinetic terms in L ′ quark SM+α cannot be compactly written by using U ′ L , U ′ R , D ′ L , and D ′ R , because these variables contain fields with different quantum numbers under SU(2) L × U(1) Y and do not treat c as SU(2) L doublets.
Next, by the change of variables such as Eqs. (22) and (23), including suitable unitary where, for simplicity,Ṽ u 2 are large masses of extra quarks. The experimental bounds on 4th generation quark masses require that an extra up-type quark is heavier than 1160 GeV from neutral-current decays and an extra down-type quark is heavier than 880 GeV from charged-current decays [25].

Examination on conjectures
We carry out order estimations using λ( ∼ = 0.225), and examine whether the conjectures on the Yukawa couplings hold or not, based on the extension of the SM described by L ′ quark SM+α . The analyses on the case with partial multiplets are carried out in Appendix A.

Seeking transformation matrices
Using λ, the magnitude of the right-hand sides in Eqs. (26) and (27) is parametrized as where n 1 , n 2 , n 3 , and n 4 are positive integers, as seen from the lower mass bounds 1160 GeV and 880 GeV, and v / 2 is the vacuum expectation value of a neutral component of the Higgs doublet. Note that v is used for the sake of convenience, although m (U ) 1 , m (U ) 2 , m (D) 1 , and m (D) 2 must be irrelevant to the breakdown of electroweak symmetry. Using Eqs. (20), (26), (27), and (28), we obtain the relations: According to Dirac's naturalness, we impose the following conditions on the kinetic coefficients, The conjectures on the Yukawa couplings are written by and respectively. The examination on conjectures is carried out by studying whether N u L , N d L , N u R , and N d R exist or not, to satisfy Eqs. (32) and (33), based on Eqs. (21), (29), (30), and (31), and specifying the form of those matrices.
Let us take the ansatzes: 5 First, the relations (21) yield the conditions: These conditions are satisfied with suitable components, if the rank of 4×4 sub-matrices N u L I J , N d L I J , N u R I J , and N d R I J is 4. In our case, they are automatically satisfied, because the transformation matrices are given in the form asṼ N whereṼ is an arbitrary unitary matrix and N is a block-diagonal matrix with N I 5 = 0 and N 5J = 0. 5 Although a generality is lost by adopting them, it is enough if N u L , N d L , N u R , and N d R are found with this choice, from the viewpoint of a possible existence.
Next, from the relations (29) and (30), the following relations for the Yukawa couplings are obtained, = O λ n 3 and/or N d Hereafter, we consider the case with "and" in Eqs. (43) and (44). Then, from N d = O λ n 4 , respectively. Then, the transformation matrices take the form: From K 1, 2, 3). Then, we obtain the transformation matrices such as In this way, we find that an existence of a large mixing between d ′ Then, we obtain the transformation matrices such as For large masses concerning extra quarks, we derive the relations: From the fact that the 4th terms in the right hand side of Eqs. , we obtain n 1 = n 3 , and then the following relation holds When taken together, we have arrived at the transformation matrices to realize the conjecture (32), such as under the parametrization (28)  Finally, we study whether the Yukawa couplings can be unified or not. As in the SM, we have the following no-go theorem.
[Theorem] Under the kinetic unification such as k SM+α . Here, we consider the case that the kinetic unification and the mass unification are spoiled. In this case, from Eqs. (39) and (41), we find that y (U ) I J = y (D) I J is realized with the relation: where mass parameters are replaced with values at the unification scale. Here, we use quark masses defined by m u = y u v / 2, m c = y c v / 2, m t = y t v / 2, m d = y d v / 2, m s = y s v / 2, and m b = y b v / 2. In general, there appear breaking terms that contribute to the Yukawa coupling matrices, on the breakdown of a higher gauge symmetry. Then, the relation (52) is modified and their effects should be considered in a model-building. Our results are summarized as follows. There is a possibility that a magnitude of elements of Yukawa coupling matrices is O(1) and the Yukawa couplings are unified at some high-energy scale, if transformation matrices take a particular form such as Eq. (51).

Speculations
We examine features on Yukawa coupling matrices and transformation matrices inferred from our conjectures.
First, we point out that our matrices possess different features from ordinary ones in the following points. One is that a small mixing of quark flavors between the weak and the mass eigenstates can occur as a result of a cancellation between a large mixing in N u L and N d L . In our setup, the flavor mixing is defined by the matrix: and N mix contains V KM as the sub-matrix (N mix ) i j (i , j = 1, 2, 3). For reference, we explain the difference between consequences from a small and a large mixing in transformation matrices, based on the SM. In an ordinary case, from Eqs. (8) and (11), transformation matrices are also assumed to be the same form with a small mixing as V KM , In this case, using Eqs. (8), (54), and the relations: we find that the Yukawa coupling matrices take the forms: They are often used in the Froggatt-Nielsen mechanism [6]. Using y u , y c , y t , y d , y s , and y b , y (u) and y (d) are also expressed as On the other hand, in the case with a large mixing among up-type quarks given by the up-type Yukawa coupling matrix takes the form: where we take ⋆ = λ in (59). In this case, a small mixing in V KM originates from a cancellation between a large mixing in V (u) L and V (d)

L
The other is that small Yukawa couplings in the SM have a high-sensibility for a small variation of matrix elements, and it suggests that the quark mass hierarchy occurs as a consequence of a fine tuning among large parameters. This feature is, in general, different from that achieved from the top-down approach, and lies the other end of that obtained by a 'stability' principle [18,19]. Here, the stability principle means that a tiny dimensionless parameter should not be sensitive to the change of matrix elements including it. In our model, the Yukawa coupling matrices are expressed by where y u i = (y u , y c , y t ), a are 4 × 4 complex matrices given by . (63) The conditions that y u i are stable under a change of the (I , J ) element are given by To fulfill the conditions (68) for y u and y d , the following inequalities are needed, for the (I , J ) element with T (u) (1). In this way, we conjecture that the Yukawa couplings associated with a large mixing hardly satisfy the stability condition.
Let us illustrate this feature in the SM. Using Eq. (60), for the entries (i , j ) = (1, 1), (1, 3), (3, 1), (3,3), the sensibility of y u in y (u) i j is roughly evaluated as where no summations on i and j are done. Eq. (71) shows that y u is highly sensitive to the change of other parameters. In contrast, in the ordinary case, using Eq. (57), for the entry (i , j ) = (1, 1), the sensibility of y u in y (u) i j is roughly evaluated as δy u /y u δy (u) 11 /y (u) and then the sensibility becomes much milder than that in Eq. (71). For the sake of completeness, we give an example to satisfy the condition ∆ y u i j ≤ O(1). If we take the transformation matrices: y (u) becomes as Then, we find that ∆ y u 11 = O(1) for p ≥ 2 and r ≥ 4. Finally, we give a speculation on the Yukawa coupling unification. After and this is realized by transformation matrices which satisfy and this is realized by transformation matrices which satisfy They show that d ′ R(m) c and d ′ R4 in the down-type quark sector play a role of u ′ L3 and u ′

R3
in the up-type one, respectively.

Conclusions and discussions
We have studied an origin of fermion mass hierarchy based on an extension of the SM with vector-like fermions, using a bottom-up approach. It is shown that the magnitude of elements of Yukawa coupling matrices can be sizable of O (1)  Our results are obtained, independent of a theory beyond our setup, and hence they could hold in various models. In fact, large masses of extra fermions are free parameters, and it is difficult to determine their magnitude from theoretical considerations alone. Vector-like fermions including down-type ones can exist at a terascale as remnants of unification and supersymmetry [36]. Conversely, if vector-like fermions are discovered and their masses are precisely measured, they would provide useful information about our setup and take hints for HETs.
We explain preceding works on the fermion masses based on models with vectorlike fermions. Vector-like fermions are used to generate small quark masses through a see-saw type mechanism [37,38,39,40]. The flavor structure has been studied in unified theories with vector-like fermions [41,42,43,44,45,46,47,48]. It would be meaningful to reexamine various models with our conclusions in mind. For a realistic model-building based on a grand unification, we must consider leptons as well as quarks. Our method is also applicable to the lepton sector and models with several Higgs doublets.
In our case with a large mixing in transformation matrices, Yukawa coupling matrices contain tiny parameters such as y u and y d possessing a high-sensibility for a small variation of matrix elements, and hence the quark mass hierarchy can occur as a consequence of accidental cancellations among large parameters. Because this feature is different from that of the top-down approach, and lies the other end of that by the stability principle, it seems to be unnatural. However, if it becomes evident that a small flavor mixing in V KM stems from a cancellation of large mixing angles among various fermions, it must surge one to reconsider implications of fine tuning or naturalness, in connection with a naturalness problem relating to the Higgs boson mass.
Finally, we point out a limitation of our approach and problems left behind. From the bottom up, it is possible to show a possible existence (if not an existence proof) and to offer some suggestions for HETs, but it is difficult to specify the structure of HETs from our findings alone. It would be a next task to answer the following questions by exploiting to various methods. Are transformation matrices with a large mixing realistic? Is there any circumstantial evidence to support them? Or is there a model, theory or mechanism to realize them?
At present, a theory beyond the SM has not been yet known, and hence it is worth pursuing every possibility including a large mixing or a fine tuning. Then, our approach would be useful as a complimentary one to solve riddles in the SM.
where N u L and N u R are 5×5 complex matrices, and N d L and N d R are 4×4 complex matrices.
Relations of mass matrices are written by where the magnitude of m (U ) 1 , m (U ) 2 , and m (D) is given by O λ −n v / 2 . Then, we obtain the following transformation matrices which realize y (U ) using Eqs. (80), (81), and (82). The relation y (U ) I j = y (D) I j is realized if the following relation holds, where we neglect tiny contributions including m u , m c , m d , m s , and m b .
In the absence of u ′ R4 and u ′ R(m) , the quark kinetic coefficients are written by where N u L and N u R are 4×4 complex matrices, and N d L and N d R are 5×5 complex matrices.
Relations of mass matrices are written by where the magnitude of m (U ) , m (D) 1 , and m (D) 2 is given by O λ −n v / 2 . Then, we obtain the following transformation matrices which realize y (U ) I j , y (D) using Eqs. (86), (87), and (88). The relation y (U ) I j = y (D) I j is realized if the following relation holds, where we neglect tiny contributions including m u , m c , m d , m s , and m b .
In the absence of q ′ L4 and q ′ L(m) , the quark kinetic coefficients are written by where N u L , N d L , N u R , and N d R are 4 × 4 complex matrices.
Relations of mass matrices are written by where the magnitude of m (U ) and m (D) is given by O λ −n v / 2 . Then, we obtain the following transformation matrices which realize y (U ) using (92), (93), and (94). The relation y (U ) i J = y (D) i J is realized if the following relation holds, where the magnitude of m (U ) and m (D) is given by O λ −n v / 2 . Then, we obtain the following transformation matrices which realize y (U ) I j , y (D) I j = O (1), using (97), (98), and (99). The relation y (U ) I j = y (D) I j is realized if the following relation holds, where we neglect tiny contributions including m u , m c , m d , m s , and m b .
A.5 Case with u ′ R4 and u ′

R(m)
In the absence of q ′ L4 , q ′ L(m) , d ′ R4 , and d ′ R(m) , the quark kinetic coefficients are written by where N u L and N u R are 4×4 complex matrices, and N d L and N d R are 3×3 complex matrices. Relations of mass matrices are written by where N u L and N u R are 3×3 complex matrices, and N d L and N d R are 4×4 complex matrices. Relations of mass matrices are written by where the magnitude of m (D) is given by O λ −n v / 2 . Then, we obtain the following transformation matrices which realize y (U ) i j , y (D) i J = O (1), using (105), (106), and (107). The relation y (U ) i j = y (D) i j is realized if the following relation holds, where we neglect tiny contributions including m u , m c , m d , m s , and m b . As a simple case, the above relation holds with In this case, d ′ R(m) c and d ′ R4 in the down-type quark sector play a role of u ′ L3 and u ′ R3 in the up-type one, respectively.