Flavor structure from ‘canonical’ Yukawa interactions and ‘emergent’ kinetic terms

We study the ﬂavor structure of quarks in the standard model from a viewpoint of a canonical type of Yukawa interactions and an emergence of kinetic terms. A realistic structure can be generated based on the emergence proposal that quark kinetic terms appear in the infra-red region, as a result of radiative corrections involving towers of massive states.


Introduction
The origin of the fermion mass hierarchy and flavor mixing in the standard model (SM) has been a big enigma.The reason why it is difficult to uncover the flavor structure is that Yukawa interactions in the SM contain many unobservable parameters, which are eliminated by bi-unitary transformations of global symmetries on fermion kinetic terms, and useful information to determine a physics beyond the SM is not fully obtained from precision measurements of the SM parameters alone.
If the coexistence of matter kinetic terms and Yukawa interactions in the SM and the appearance of flavor symmetries on the kinetic terms complicate an understanding of the flavor structure, it must be better to return to the origin of each term in the SM Lagrangian density.Hence, we go with the idea that the origin of kinetic terms and Yukawa interactions can give a key to solve the enigma.
In the usual case, we assume that kinetic terms (including gauge fields via gauge interactions) exist from the beginning, and then chiral fermion fields are determined, up to some global unitary transformation, by making their kinetic terms the canonical ones.We cast doubt on it by considering a case that kinetic terms are absent and Yukawa interactions such as y i j χ Li ϕη Rj are present in the ultra-violet (UV) region at a fundamental theory level.Here, y i j is a Yukawa coupling matrix, i , j (= 1, 2, 3) are family labels, summation over repeated indices is understood, in most cases, throughout this paper, χ Li and η Rj are chiral fermions and ϕ is a scalar field.In such a case, chiral fermion fields can be defined through Yukawa interactions and the simplest choice of the Yukawa coupling matrix is y i j = δ i j (δ i j equals to 1 for i = j and 0 for i = j ).We refer to this type of Yukawa interactions as 'canonical' Yukawa interactions.We note that there is a freedom to change y i j into δ i j if y i j has non-zero singular values, although fields are not uniquely fixed and couplings to extra fields can become complicated.Then, the problem on the flavor structure is transported into that on the origin of kinetic terms, under the assumption that kinetic terms appear in the infra-red (IR) region at an effective theory level.We refer to this kind of kinetic terms as 'emergent' kinetic terms.In this way, the flavor structure in the SM can be originated from a counterpart in the emergent kinetic terms.
Recently, a generation of kinetic terms with large coefficients has been proposed based on the emergence proposal in the strong version that "In a theory of quantum gravity, all light fields in a perturbative regime have no kinetic terms in the UV.The required kinetic terms appear as an IR effect after integrating out towers of massive states below the quantum gravity cut-off scale."[1,2,3,4,5], and its phenomenological implications including the fermion mass hierarchy and the electro-weak hierarchy problem have been studied [6,7].The emergence proposal has been presented as part of the Swampland program [8].
In this paper, we study the flavor structure of quarks in the SM from a viewpoint of canonical Yukawa interactions and emergent kinetic terms and examine whether a realistic structure can be generated or not based on the above emergence proposal.
The outline of this paper is as follows.In the next section, we review the flavor structure of quarks and study a structure of kinetic terms based on canonical Yukawa interactions in the SM.In Sect.3, we investigate a generation of quark kinetic terms and a formation of the flavor structure, using a simple model.In the last section, we give conclusions and discussions.

'Canonical' Yukawa interactions
First, we review the quark sector in the SM, based on the usual 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, 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.The Yukawa coupling matrices are diagonalized as diag by bi-unitary transformations and the quark masses are obtained as where are unitary matrices, v / 2 is the vacuum expectation value of neutral component in the Higgs doublet, family labels are omitted, and m u , m c , m t , m d , m s and m b are masses of up, charm, top, down, strange and bottom quarks, respectively.
As seen from eqs. ( 2) and (3), the quark Yukawa coupling matrices are expressed by using diag and the Cabibbo-Kobayashi-Maskawa matrix defined by [9,10] The 3 × 3 matrices V (u) L , V (u) R and V (d) R are completely unknown in the SM, because they can be eliminated by the global U(3) × U(3) × U(3)/U(1) symmetry that the quark kinetic terms possess.Here, a global U (1) phase is constrained by a global U (1) invariance in Yukawa interactions.We have a situation that information on a physics beyond the SM is not fully obtained by observable parameters such as y (u)  diag , y (d)  diag and V CKM alone.In Table 1, the number of independent parameters relating to y (u)  i j and y (d) i j is listed.As y (u)  i j and y (d) i j are 3 × 3 complex matrices, they totally have 36 parameters.The V CKM Yukawa couplings and related ones Number of parameters Table 1: The number of parameters relating to quark Yukawa couplings contains three mixing angles and a CP violating phase.The Next, we consider an unusual case with 'canonical' Yukawa interactions given by in preparation for the study on the emergence of kinetic terms in the next section.Using the field variables with a prime, the Lagrangian density in the quark sector is written by where k (q) i j , k (u) i j and k (d) i j are kinetic coefficient matrices denoted by Here, W (u) is a 3×3 complex matrix.Note that non-canonical quark kinetic terms appear in L ′ quark SM . 1 Because W (u) is an arbitrary matrix, the expressions ( 8) -( 10) are not unique.For instance, we obtain the relations: u) .As another choice, we have the relations: where u) .We find that a seed of the mass hierarchy and flavor mixing can be hidden in various places.
In Table 2, the number of independent parameters concerning k i j and k (d) i j totally have 27 independent parameters because they are Kinetic coefficients and related ones Number of parameters Table 2: The number of parameters relating to quark kinetic coefficients hermitian matrices.Note that a global U (1) phase in W (u) is canceled out and does not appear in eqs.( 8) -( 10) and then the total number of independent parameters in W (u) is 17.
Let us show that L ′ quark SM is equivalent to L quark SM .The k (q) i j , k (u) i j and k (d)  i j are also written by where 3 × 3 complex matrices X q , X u and X d parametrized by using eqs.( 8) - (10).Note that unitary matrices V (u) L , V (u) R and V (d) R made of unobservable parameters appear.Using X q , X u and X d , the quarks q L , u R and d R in Using eqs.( 18) -( 21) and ( 4), the canonical Yukawa interactions are rewritten as φd Rj + h.c.
= −y (u) i j q Li φu Rj − y (d) i j q Li φd Rj + h.c., (22) and then Yukawa interactions in L quark SM are obtained.In this way, we can set a goal to obtain the quark kinetic coefficients given in eqs.( 8) - (10) or its equivalent ones, under the assumption that L ′ quark SM effectively describes a relic from emergent kinetic terms as a physics beyond the SM.Then, we need kinetic coefficients with huge values, because the eigenvalues of y (u)−1 diag 2 and y (d)−1 diag 2 are roughly estimated at the weak scale as [29] y (u)−1 diag 2 diag 6.9 × 10 9 , 1.9 × 10 4 , 1.0 , ( 23) We look into how the flavor structure can be induced in the next section.

'Emergent' kinetic terms
To produce the quark mass hierarchy, it is needed that kinetic coefficients can possess a hierarchy with huge values when they are diagonalized.It can be realized based on the emergence proposal that fermion kinetic terms in the SM can be generated radiatively by loop corrections involving towers of massive states [6,7].First, we give some basic assumptions.(a) The SM fermions have no kinetic terms in the UV region.(b) Yukawa interactions among the SM fields exist, and the SM fermion fields are defined by making Yukawa interactions the canonical types.(c) Towers of massive states exist with canonical kinetic terms.(d) The SM fermions strongly couple to towers of massive states.(e) The SM fermion kinetic terms (including gauge bosons via gauge interactions) appear emergently.
Let us study a simple model to grasp a feature of our proposal and see if a realistic flavor structure can be generated or not.The model has Yukawa interactions among each q ′ L , u ′ R and d ′ R and massive particles such that where Li are massive fermions and Φ (n) q , Φ (n) u and Φ (n) d are massive scalar fields.We take quantized couplings where n Qi , n Ui and n Di are integers and no summation is done for repeated indices. 2ere and hereafter, we treat f Q i j , f U i j and f D i j as complex matrices with elements of O(1), although one of them can become a diagonal form after performing a suitable biunitary transformation, keeping both quark Yukawa interactions in the SM and kinetic terms of massive particles the canonical ones.Then, the kinetic coefficients of q ′ L at the one-loop level are calculated based on the diagram in Figure 1.After summing over towers of states up to the UV cutoff scale Λ, we estimate the contribution of k (q) i j as Here, we assume that the one-loop contributions are dominant to generate the SM fermion kinetic terms in the IR region, and renormalized Kaluza-Klein propagators and coupling constants are used.It is based on the fact that higher loop diagrams connected by the SM fermion propagators do not contribute in the absent of the SM fermion kinetic terms and a conjecture that the dynamics in the UV region can be well-controlled by a topological nature of a fundamental theory, even in a strong coupling regime.
In the same way, we obtain the following type of kinetic coefficients: where ξ q , ξ u and ξ d are positive diagonal matrices given by respectively.Then, the kinetic coefficients are rewritten as respectively.Here, V q , V u and V d are 3 × 3 unitary matrices.In terms of Now, the kinetic terms take a canonical form with the global U(3) × U(3) × U(3)/U(1) symmetry, and hence we have Yukawa coupling matrices such as by replacing V q q L , V u u R and V d d R with q L , u R and d R , respectively.Let us consider the case that complex matrices ( 1) and assume that there exist hierarchies such that ξ because of a large mass difference in the up-type quark sector and a small flavor mixing.Then, y (u)  i j and y (d) i j are approximated by the formulas:3 and, after performing suitable bi-unitary transformations, they can be diagonalized as where ξ−1 d1 , ξ−1 d2 and ξ−1 d3 are some positive numbers.Note that ξ−1 di (i = 1, 2.3) do not necessarily agree with ξ −1 di because a hierarchy such as is not assumed.The unitary matrices are given by The expression of V (d) R is not determined without specifying a magnitude relationship among ξ −1 d1 , ξ −1 d2 and ξ −1 d3 or giving explicit values, either.From y (u) diag = diag(y u , y c , y t ), y (d)  diag = diag(y d , y s , y b ) and eqs.( 38) and (39), we obtain the relations: From eqs. ( 5) and (40), we obtain the relations: and derive the relations (V CKM ) i j ≃ (V CKM ) j i and (V CKM ) 13 ≃ (V CKM ) 12 (V CKM ) 23 .

Conclusions and discussions
We have studied the flavor structure of quarks in the SM from a viewpoint of a canonical type of Yukawa interactions and an emergence of kinetic terms.We have found that a realistic structure can be generated based on the emergence proposal that quark kinetic terms appear in the IR region, as a result of radiative corrections involving towers of massive states, and the quark mass hierarchy and flavor mixing can originate from a milder mass hierarchy on massive fermions.A similar analysis can be applied to find out the origin of the lepton flavor structure [7].
There are several problems that remains to be solved.Why do the SM fields have no kinetic terms in the UV region?It might be caused by some topological nature of a fundamental theory including a quantum gravity [31].In contrast, why do towers of massive fields have kinetic terms in the UV region?And what is the origin of such kinetic terms?What is the origin of gauge fields and gauge interactions?What is the origin of Yukawa interactions?In the first place, what is the origin of quantum fields including the SM particles and towers of massive states?A fundamental theory such as superstring theory and/or M theory is expected to answer the above questions.
where we use the relation Λ ≃ N Qk m Qk (k = 1, 2, 3, no summation over k) and A ≃ B means a = O(b) (a and b are values of A and B, respectively).