Towards understanding fermion masses and mixings

1. Introduction

The replication of fermion families is one of the main puzzles of particle physics. Three fermion families are in identical representations of the Standard Model (SM) gauge symmetry $\mathrm{\text{SU}}(3)\times \mathrm{\text{SU}}(2)\times U(1)$. The left-handed (LH) quarks $q_{\mathrm{\text{Li}}}={(u_L,d_L)}_i$ and leptons ${\ell }_{\mathrm{\text{Li}}}={({\nu }_L,e_L)}_i$ transform as weak doublets while right-handed (RH) ones $u_{\mathrm{\text{Ri}}},d_{\mathrm{\text{Ri}}},e_{\mathrm{\text{Ri}}}$ are weak singlets, $i=1,2,3$ being the family index. The SM contains the unique order parameter – vacuum expectation value (VEV) of the Higgs doublet $\varphi$, $〈{\varphi }^0〉=v_{\mathrm{\text{w}}}=174$GeV, which spontaneously breaks the electroweak symmetry $\mathrm{\text{SU}}(2)\times U(1)$. It determines the mass scale of the weak bosons $W^{\pm },Z$ as well as fermion masses which emerge via the Yukawa couplings

where $Y_{e,u,d}$ are the Yukawa coupling matrices, and $\tilde{\varphi }=i{\tau }_2{\varphi }^{*}$. Here we use, instead of the RH fermion fields, their complex conjugates as $u_L^c=\mathrm{\text{C}}{\overline{u_R}}^T$ (anti-fields) and omit in the following the subscript L for q, $u^c$, $d^c$ etc., all being the LH Weyl spinors.^a The Yukawa couplings (1), after substituting the Higgs VEV $v_{\mathrm{\text{w}}}$, induce the fermion mass matrices $M_f=Y_f{v}_{\mathrm{\text{w}}}$ ($f=u,d,e$) which are generically non-diagonal. They can be brought to the diagonal form (i.e. to the mass eigenstate basis) via bi-unitary transformations :so that the quark masses $m_u,m_c,m_t$ and $m_d,m_s,m_b$ are the eigenvalues of the mass matrices $M_u$ and $M_d$, as the charged lepton masses $m_e,m_{\mu },m_{\tau }$ are the eigenvalues of $M_e$ (we shall not discuss the neutrino masses in this paper).

In the SM context, the matrices $U_{u,d}$ which rotate the RH fermions have no physical meaning while the “left” ones $V_{u,d}$ give rise to the mixing in the quark charged currents coupled to weak $W^{\pm }$ bosons. This mixing is described by the unitary Cabibbo–Kobayashi–Maskawa (CKM) $V_{\mathrm{\text{CKM}}}$^1,2 :

By rotating away the irrelevant phases, the unitary matrix $V_{\mathrm{\text{CKM}}}$ can be conveniently parameterized in terms of four parameters, three mixing angles and a CP-violating phase.² In the standard parameterization adopted by Particle Data Group (PDG),³ the angles are chosen as ${\theta }_{12}$, ${\theta }_{23}$, ${\theta }_{13}$, and the CKM matrix reads where $s_{ij}=sin{\theta }_{ij}$, $c_{ij}=cos{\theta }_{ij}$ and ${\theta }_{ij}$ are chosen so that $s_{ij},c_{ij}\ge 0$. As a measure of CP violation, the rephasing-invariant quantity $J{\sum }_{m,n}{ϵ}_{ikm}{ϵ}_{jln}=\mathrm{\text{Im}}[V_{ij}{V}_{kl}{V}_{il}^{*}{V}_{kj}^{*}]$ (the Jarlskog invariant)⁴ in the standard parameterization reads :

The SM has a remarkable feature: the natural flavor conservation in neutral currents, namely in the fermion couplings with the Higgs and Z bosons.^5,6,7 However, it contains no theoretical input that could explain the fermion mass spectrum and the weak mixing pattern. In a sense, the SM is technically natural since it can tolerate any pattern of the Yukawa matrices $Y_{u,d,e}$ but it can tell nothing on the origin of the strong hierarchies between their eigenvalues as well as of the nearly aligned structures of the matrices $Y_u$ and $Y_d$. This remains true also in the context of supersymmetric and/or grand unification theories (GUTs). So, the origin of the fermion mass and mixing pattern remains a mystery.

The quark and lepton mass spectrum, schematically shown on Fig. 1, exhibits a strong inter-family hierarchy. The mass hierarchy between the third and first families is $m_b/m_d\sim 10^3$ for the down quarks, and yet stronger for the up quarks, $m_t/m_u\sim 10^5$. Expressed in terms of the small parameters $m_d/m_s={ϵ}_d$, $m_s/m_b={ϵ}_s$ etc., these hierarchies approximately look as :

In a whole, these hierarchies do not exhibit any notable regularities, but only feature some order of magnitude connections. Namely, by comparing the up and down quarks, we see that ${ϵ}_s\sim {ϵ}_d$ while ${ϵ}_u,{ϵ}_c\sim {ϵ}_d^2$. In other words, the up quark masses scale approximately as squares of down quark masses. Comparing with leptons, we see that ${ϵ}_{\mu }\sim {ϵ}_d$ while ${ϵ}_e\sim {ϵ}_c$.

As for the CKM matrix, the quark mixing angles are small (unlike the case of neutrino mixing). Within the experimental uncertainties,³ they exhibit the following pattern

in terms of the small parameter $\lambda \approx \sqrt{1/20}$ (which is incidentally related as ${\lambda }^2\approx {ϵ}_d$ with the mass ratio ${ϵ}_d=m_d/m_s$ in Eq. (6)), and the order one numerical factor $a\approx 0.8$. Then for the Jarlskog invariant one has $J\approx 2a^2{\lambda }^{7}sin\delta \approx 3.5\times 10^{-5}sin\delta$ which means that the smallness of observed CP-violation is originated from the small mixing angles rather than from a small CP-phase $\delta$, and in fact $sin\delta \simeq 1$.

2. Fritzsch Hypothesis

It is tempting to think that the fermion flavour structure is connected to some underlying theory which determines the pattern of the Yukawa matrices with a predictive power, and in particular that the well-known formula for the Cabibbo angle $V_{us}=\sqrt{m_d/m_s}$ is not accidental. Such relations between the fermion masses and mixing angles can be obtained by considering Yukawa matrix textures with reduced number of free parameters, and in particular, by assuming that certain elements in the fermion mass matrices are vanishing by some symmetry reasons. This zero-texture approach was originally thought to calculate the Cabibbo angle in the two-family framework,^8,9,10 in fact before the discovery of b and t quarks. In the frame of six quarks, this picture was extended by Harald Fritzsch^11,12 who suggested the following texture for the mass matrices :

where the non-zero elements are generically complex, with the symmetricity condition $|M_{ij}^{u,d}|=|M_{ji}^{u,d}|$ which is motivated in the context of left–right symmetric models.^b

By rotating the phases of upper quarks: $u_k^c\to e^{i{\alpha }_k^{\prime }}{u}_k^c$ and $u_k\to e^{i{\alpha }_k}{u}_k$, and similarly for down quarks, the complex phases in matrices (8) can be removed and the non-zero entries can be rendered real. Namely, the matrices (8) can be parameterized as $M_{u,d}=F_{u,d}^{\prime }{\tilde{M}}_{u,d}{F}_{u,d}^{*}$, where $F_u=\mathrm{\text{diag}}(e^{i{\alpha }_1},e^{i{\alpha }_2},e^{i{\alpha }_2})$, etc. are the phase transformation matrices, and

are real symmetric matrices which can be further diagonalized by orthogonal transformations, $O_{u,d}^T{\tilde{M}}_{u,d}{O}_{u,d}=M_{u,d}^{\mathrm{\text{diag}}}$. The three real parameters $A_d$, $B_d$, $C_d$ can be expressed in terms of the three eigenvalues of ${\tilde{Y}}_d$, i.e. in terms of the down quark masses $m_d,m_s,m_b$. Namely, one gets approximately, up to small corrections, $C_d\approx m_b/v_{\mathrm{\text{w}}}$, $B_d\approx \sqrt{m_s{m}_b}/v_{\mathrm{\text{w}}}$ and $A_d\approx \sqrt{m_d{m}_s}/v_{\mathrm{\text{w}}}$, and similarly for up quarks and charged leptons (provided that the mass matrix of the latter has a structure similar to (9)). Therefore, we have :Hence, the fermion mass hierarchies (6) follow from somewhat milder hierarchies between the input parameters which are the non-zero entries in matrices (9). Notice also the following relation between the down quark and leptons :Later on we shall explore its origin in the GUT context.

The three rotation angles in the orthogonal matrix $O_d$ can be expressed in terms of the mass ratios $m_d/m_s$ and $m_s/m_b$. Analogously, the three angles in $O_u$ can be expressed in terms of the upper quark mass ratios $m_u/m_c$ and $m_c/m_t$. The CKM matrix (3) is obtained as $V_{\mathrm{\text{CKM}}}=O_u^T{F}_u^{*}{F}_d{O}_d$, where the diagonal matrix $F=F_u^{*}{F}_d$ can be parameterized by two phase parameters, $F=\mathrm{\text{diag}}(e^{i\alpha },e^{i\beta },1)$. Then, the four physical elements of the CKM matrix, that is the three mixing angles ${\theta }_{12},{\theta }_{23},{\theta }_{13}$ and the CP-phase $\delta$, can be expressed in terms of the four mass ratios, $m_d/m_s$, $m_s/m_b$, $m_u/m_c$ and $m_c/m_t$, and of two unknown phases $\beta$ and $\gamma =\beta -\alpha$. Namely, in the leading approximation one has :

Besides reproducing the formula for the Cabibbo angle, this texture exhibits a remarkable feature in the light of the interfamily hierarchies $m_d\ll m_s\ll m_b$ and $m_u\ll m_c\ll m_t$. Namely, in the limit $A_{u,d}\to 0$ the light quarks become massless, $m_{u,d}\to 0$, but at the same time the first family decouples in the CKM mixing, $s_{12},s_{13}\to 0$. Next, in the limit $B_{u,d}\to 0$ which renders massless the second family, $m_{c,s}\to 0$, also its mixing with third family disappears, $s_{23}\to 0$.

Generalizing these properties, Fritzsch suggested that in any kind of realistic flavor models the quark mixing pattern should be intimately related to the interfamily hierarchy. The mixing angles ${\theta }_{ij}$ should depend on quark mass ratios $m_d/m_s$, $m_u/m_c$, etc. so that the smallness of the former stems from the smallness of the latter. He hypothesized the following analytic properties¹⁹:

• Decoupling hypothesis: in the limit of massless first family vanish, $m_u,m_d\to 0$, its mixings with the heavier families disappear, i.e. ${\theta }_{12},{\theta }_{13}\to 0$. At the next step, for the massless second family, $m_s,m_c\to 0$, also the 2–3 mixing should disappear, i.e. ${\theta }_{23}\to 0$.

• Scaling hypothesis: in the limit when up and down quark masses become proportional, $m_u:m_c:m_t=m_d:m_s:m_b$, all mixing angles must vanish: ${\theta }_{12},{\theta }_{23},{\theta }_{13}\to 0$.

While the Fritzsch anstaz (9) has the first property (decoupling hypothesis), in general it does not meet the second one (scaling hypothesis), since the Yukawa matrices $Y_u$ and $Y_d$ do not necessarily become aligned in the limit $A_u:B_u:C_u=A_d:B_d:C_d$. In fact, in this case we have $m_u:m_c:m_t=m_d:m_s:m_b$ but the mixing angles do not generally vanish because of arbitrary phases $\beta$ and $\gamma$.

However, the original Fritzsch texture for quarks was excluded when the knowledge of quark masses and the CKM parameters became accurate enough.²⁰ Given the present experimental and lattice results on quark masses and CKM elements, there is no parameter space in which the precision data can be reproduced. More concretely, the small enough value of $|V_{cb}|$ and large enough value of $|V_{ub}/V_{cb}|$ cannot be achieved for any values of the phase parameters $\beta$ and $\gamma$ in (12). A possibility to obtain viable textures is to extend the original Fritzsch texture by replacing one of the zero entries with a non-zero one, e.g. by introducing a non-zero 13 element²¹ or a non-zero 22 element, as e.g. in Refs. 22 and 23 (for a review of different schemes, see Ref. 24). However, the introduction of new parameters reduces the predictive power, and, in addition, these modifications generically do not satisfy the decoupling feature.

On the other hand, instead of decreasing the number of zero entries, one can think to break the symmetricity condition. Namely, an asymmetry in the 23 blocks, $|M_{23}^{u,d}|\ne |M_{32}^{u,d}|$ can be introduced in the mass matrices.^25,26,27,28 In other words, one can consider the Yukawa textures of the form

where the matrices $F,F^{\prime }$ contain only phases and ${\tilde{Y}}_f$ are the real matrices, with $x_{u,d,e}$ being deformation parameters. It is worth noting that such a modification preserves the main properties of the original Fritzsch texture. In particular, it satisfies the decoupling property, and in addition the hierarchy relations (10) still remain valid.

As it was shown in a recent analysis of Ref. 29, such a texture is perfectly compatible with the observed pattern of the CKM matrix. Namely, all defects of the original texture can be corrected at once provided that the upper quark matrix exactly has a ‘symmetric’ Fritzsch texture, that is $x_u=1$, and only the down quark matrix is deformed by a factor $x_d=3$ or so. In fact, such a modification leads to the correct values of the CKM mixing angles as well as CP-violating phase, as the functions of the fermion mass ratios and of the two phase factors.²⁹

In next section we describe how the Fritzsch texture can be obtained within the context of the inter-family gauge group $\mathrm{\text{SU}}{(3)}_H$, and how it can be minimally deformed in the 2–3 blocks by using a scalar field in adjoint (octet) representation of $\mathrm{\text{SU}}{(3)}_H$. In Sec. 4 we analyse predictions of Fritzsch textures in light of the present high precision determinations of quark masses and CKM matrix elements, and show that this flavour structure predicts the mixing angles and the CP-violating phase in perfect agreement with the experimental results.

3. Fritzsch-Like Textures from Horizontal Symmetry SU${(3)}_H$

The key for understanding the replication of families, fermion mass hierarchy and mixing pattern may lie in symmetry principles. For example, one can assign to the fermion species different charges of an abelian global flavor symmetry $U(1)$.³⁰ There are also models making use of an anomalous gauge symmetry $U{(1)}_A$ to explain the fermion mass hierarchy while also tackling other naturalness issues.^{31,32,33,34,35} However, it is difficult to obtain the highly predictive quark mass matrices with the texture zeros within this approach.

One can point to a more complete picture by introducing a non-abelian horizontal gauge symmetry $\mathrm{\text{SU}}{(3)}_H$ between three families.^{14,15,16,17,36} This symmetry should have a chiral character, with the LH and RH components of quarks (and leptons) transforming in different representations of the family symmetry. In particular, they can be arranged respectively as the triplet and anti-triplet representations of $\mathrm{\text{SU}}{(3)}_H$ which in our notations means that LH fermions $q,\ell$ as well as anti-fermions $u^c,d^c,e^c$ are $\mathrm{\text{SU}}{(3)}_H$ triplets :

with $i=1,2,3$ being the family $\mathrm{\text{SU}}{(3)}_H$ index. Clearly, in this case the fermion direct Yukawa couplings to the Higgs doublet is forbidden by $\mathrm{\text{SU}}{(3)}_H$, which means that their masses cannot be induced without the breakdown of the horizontal symmetry. On the contrary, a vector-like $\mathrm{\text{SU}}{(3)}_H$, with $q,\ell \sim 3$ and $u^c,d^c,e^c\sim \bar{3}$, would allow the Yukawa couplings inducing a degenerate fermion mass spectrum degenerate between three families, without breaking $\mathrm{\text{SU}}{(3)}_H$.

In addition, the vector-like $\mathrm{\text{SU}}{(3)}_H$ would not allow the grand unified extensions of the SM as $\mathrm{\text{SU}}(5)$³⁷ or $\mathrm{\text{SO}}(10)$³⁸, while the chiral arrangement (14) is compatible with the GUT structures (see, e.g. Ref. 44 for a review on fermion patterns in GUTs). In particular, in $\mathrm{\text{SU}}(5)$ each family is represented by the LH spinors in $\bar{5}$ and 10 representations. Thus, in the context of $\mathrm{\text{SU}}(5)\times \mathrm{\text{SU}}{(3)}_H$ all fermion species in Eq. (14) are embedded in the following representations^15,16,17:^c

As for $\mathrm{\text{SO}}(10)$ GUT, all fermions of one family can be packed into 16-dimensional spinor representation of $\mathrm{\text{SO}}(10)$, $\mathrm{\Psi }=(\bar{F},T,{\nu }^c)$, along with the “right-handed neutrino” ${\nu }_L^c=C{\overline{{\nu }_R}}^T$. Hence, in the context of $\mathrm{\text{SO}}(10)\times \mathrm{\text{SU}}{(3)}_H$ all three families compose the unique multiplet ${\mathrm{\Psi }}_i={(\bar{F},T,{\nu }^c)}_i\sim (16,3)$.^d With this set of fermions, $\mathrm{\text{SU}}{(3)}_H$ would have triangle anomalies. For their cancellation one can introduce additional chiral fermions which are SM singlets but nontrivially transform under $\mathrm{\text{SU}}{(3)}_H$,^14,15,16 and they can be used for the neutrino mass generation.^40,41,42 The easiest way to cancel the anomalies, suggested in Ref. 25, is to share the $\mathrm{\text{SU}}{(3)}_H$ symmetry with a parallel mirror sector of particles having exactly the same physics as ordinary particles (for a review, see e.g. Refs. 43, 44 and 45). In this case $\mathrm{\text{SU}}{(3)}_H$ anomalies will be cancelled between ordinary fermion species and their mirror counterparts of the opposite chiralities.

The chiral character of the fermion representations (14) forbids their direct Yukawa couplings with the Higgs doublets $\varphi$, so that the fermion masses cannot be induced without breaking $\mathrm{\text{SU}}{(3)}_H$. As far as the fermion bilinears $u_i^c{q}_j$, $d_i^c{q}_j$ and $e^c\ell$ transform in representations $3\times 3=6+\bar{3}$, the fermion masses can be induced only via the higher order operators

involving some amount n of horizontal scalars ${\chi }_n$ (coined as flavons) in symmetric (anti-sextets ${\chi }^{\{ij\}}\sim \bar{6}$) or antisymmetric (triplets ${\chi }^{[ij]}={ϵ}^{ijk}{\chi }_k\sim 3$) representations of $\mathrm{\text{SU}}{(3)}_H$ which are gauge singlets of the SM. Here M is some effective scale and the Wilson coefficients $C_n^{(f)}$ ($f=u,d,e$) are generically complex. After inserting the flavon VEVs, operators (16) give rise to the SM Yukawa couplings (1) as $Y_f^{ij}={\sum }_n{C}_n^{(f)}〈{\chi }_n^{ij}〉/M$. In a sense, these operators “project” the VEV pattern of the flavon fields $\chi$ onto the structure of the Yukawa matrices. In the UV-complete pictures such operators can be induced in a seesaw-like manner, via integrating out some extra heavy scalars^16,17 or extra heavy fermions in vector-like representations^14,15 living at the mass scale $\sim M$.

Interestingly, operators (16) being invariant under the local $\mathrm{\text{SU}}{(3)}_H$ symmetry by construction, in fact have a larger global symmetry $U{(3)}_H=\mathrm{\text{SU}}{(3)}_H\times U{(1)}_H$. Namely, they are invariant also under an accidental global chiral $U{(1)}_H$ symmetry, implying the overall phase transformation of fermions $(u_i^c,d_i^c,q_i)\to e^{i\omega }(u_i^c,d_i^c,q_i)$ and flavon scalars ${\chi }_n\to e^{-2i\omega }{\chi }_n$. Hence, all families can become massive only if $U{(3)}_H$ symmetry is fully broken.

This feature allows to relate the fermion mass hierarchy and mixing pattern with the breaking steps of $U{(3)}_H$ symmetry, with a natural realization of the decoupling hypothesis. When $U{(3)}_H$ breaks down to $U{(2)}_H$, the fermions of third fermion family get masses while the first two families remain massless and all mixing angles are vanishing. At the next step, when $U{(2)}_H$ breaks down to $U{(1)}_H$, the second family acquires masses and the CKM mixing angle ${\theta }_{23}$ can be non-zero, but the first family remains massless ($m_u,m_d=0$) and unmixed with the heavier fermions (${\theta }_{12},{\theta }_{13}=0$). Only at the last step, when $U{(1)}_H$ is broken, also the first family can acquire masses and its mixing with heavier families can emerge. In this way, the inter-family mass hierarchy can be related to the hierarchy of flavon VEVs inducing the horizontal symmetry breaking $U{(3)}_H\to U{(2)}_H\to U{(1)}_H\to$nothing.

In the last step of this breaking chain, the chiral global $U{(1)}_H$ symmetry can be associated with the Peccei–Quinn symmetry provided that $U{(1)}_H$ is also respected by the Lagrangian of the flavon fields.^15,36 This can be achieved by forbidding the trilinear terms between the $\chi$-scalars by means of a discrete symmetry. Thus, in this framework, the Peccei–Quinn symmetry can be considered as an accidental symmetry emerging from the local symmetry $SU{(3)}_H$. In this case the axion will have non-diagonal couplings between the fermions of different families.^14,15,36 Phenomenological and cosmological implications of gauge family symmetry with such flavor-changing axion were discussed in Refs. 46–51.

In scenarios with the horizontal $\mathit{\text{SU}}{(3)}_H$ symmetry the Fritzsch textures can be naturally obtained by a suitable choice of the representations and VEV configurations of $\chi$-flavons. As the simplest set, one can take two triplets ${\chi }_1$, ${\chi }_2$, and one anti-sextet ${\chi }_3$, with their VEVs in the following form¹⁵ :

i.e. the VEV of ${\chi }_3$ is given by a symmetric rank-1 matrix which for a convenience can be directed towards the 3rd axis in the $\mathrm{\text{SU}}{(3)}_H$ space. As for the VEVs of ${\chi }_1$ and ${\chi }_2$, they are respectively parallel and orthogonal to $〈{\chi }_3〉$ (so that $〈{\chi }_3〉$ can be oriented towards 3rd axis without losing generality; the detailed analysis of the flavon scalar potential is given in Ref. 16). Thus, the total matrix of flavon VEVs has the Fritzsch texture :After $\mathrm{\text{SU}}{(3)}_H$ breaking, the theory reduces to the SM containing one standard Higgs doublet $\varphi$, and the ‘projective’ operators (16) transfer the Fritzsch-like pattern of the VEV matrix (18) to the Yukawa matrices $Y_f={\sum }_n{C}_n^{(f)}〈{\chi }_n〉/M$ ($f=u,d,e$), modulo different constants $C_n^{(u)}$, $C_n^{(d)}$ and $C_n^{(e)}$ in the expansion (16).^e ($-$ signs between off-diagonal terms are irrelevant since they can be eliminated by quark phase transformations). Namely, after rotating away the complex phases, the Yukawa matrices acquire the ‘symmetric’ Fritzsch forms $\widetilde{Y_f}$ as in (9). The inter-family mass hierarchies can be related to a hierarchy in the horizontal symmetry breaking chain $U{(3)}_H\to U{(2)}_H\to U{(1)}_H\to$nothing, that is $V_3\gg V_2\gg V_1$. In particular, the hierarchies (10) between the Yukawa entries can be originated from the hierarchy of the flavon VEVs which can be estimated as

In the context of $\mathrm{\text{SU}}(5)\times \mathrm{\text{SU}}{(3)}_H$ theory,^15,16,17 with the quarks and leptons in representations (15), one can consider $\mathrm{\text{SU}}(5)$-invariant effective operators

where H is the scalar 5-plet which contains the SM Higgs doublet $\varphi$. In fact, after $\mathrm{\text{SU}}(5)$ breaking down to the SM, the last two terms in (16) emerge from the decomposition of the last term in (20). Then, if all flavons ${\chi }_n$ are $\mathrm{\text{SU}}(5)$ singlets, the Yukawa constants of down quarks and charged leptons would be identical, $Y_e=Y_d^T$. Though $b-\tau$ unification is a successful prediction of $\mathrm{\text{SU}}(5)$, for the first two families this would imply the wrong relations $m_b:m_s:m_d=m_{\tau }:m_{\mu }:m_e$, in obvious contradiction with the observed pattern given in Eq. (6). In addition, in the first term of (20) the antisymmetric flavons cannot contribute by symmetry. Hence, one has $A_u,B_u=0$ and so u and c quarks remain massless. However, this shortcoming can be avoided by assuming that some (at least one) of the flavon triplets transforms as 24-plet of $\mathrm{\text{SU}}(5)$.^15 f This can avoid the wrong relation $Y_e=Y_d^T$ between the down quark and lepton Yukawas, and also induce the non-zero $A_u$ and $B_u$ in $Y_u$.

Let us turn back to the general case of the SM, with the quark and lepton species in $\mathrm{\text{SU}}{(3)}_H$ representations (14). In the UV-complete pictures the operators (16) can be induced via renormalizable interactions after integrating out some extra heavy fields, scalars,^16,17 or verctor-like fermions^14,15 (see also Ref. 36). The latter possibility is more economical and natural, and we shall exploit it in our further considerations.^g Namely, one can introduce the following vector-like set of chiral LH fermions of the up- and down-quark type in weak doublet and singlet representations

Then the matrices of Yukawa couplings $Y_{u,d}$ in Eq. (1) are induced after integrating out these fermion species with large mass terms. Analogously, the charged lepton Yukawa couplings can be induced by using the vector like lepton states, weak singlets $E_i\sim 3$, $E^{ci}\sim \bar{3}$ and weak doublets $L^i\sim \bar{3}$, $L_i^c\sim 3$. In particular, for the upper quarks this seesaw-like mechanism is illustrated by two diagrams in Fig. 2 while the analogous diagrams will work for the down quarks and leptons. For the heavy vector-like species having $\mathrm{\text{SU}}{(3)}_H$ invariant masses, this mechanism will induce operators (16) which ‘project’ the Fritzsch-texture (18) of the flavon VEVs on the quark and lepton Yukawa matrices.

However, by using the diagrams of Fig. 2, one can also achieve a realistic deformation of the Fritzsch ansatz in the following way. Let us assume that the masses of vector-like fermions (21) are induced by spontaneous symmetry breaking, from the VEVs of some scalars. Their mass terms transform as $\bar{3}\times 3=1+8$ and so one can introduce the Yukawa couplings of $D,D^c$ fermions

and analogous couplings of $U,U^c$ and $Q^c,Q$ species. with the scalars in singlet and octet representations of $\mathrm{\text{SU}}{(3)}_H$, $S\sim 1$ and $\mathrm{\Phi }\sim 8$. In fact, one can introduce an adjoint scalar $\mathrm{\Phi }$ of $\mathrm{\text{SU}}{(3)}_H$ in analogy to the adjoint scalar of $\mathrm{\text{SU}}(5)$, the 24-plet $\mathrm{\Sigma }$, which possibility was discussed in Refs. 25 and 27. We also assume that the cross-interaction term of $\mathrm{\Phi }$ with $\chi$-flavons in the scalar potential aligns the VEV of $\mathrm{\Phi }$ towards the largest VEV $V_3$ in Eq. (18), i.e. it is proportional to ${\lambda }_8$ generator of $\mathrm{\text{SU}}{(3)}_H$: $〈\mathrm{\Phi }〉=V\mathrm{\text{diag}}(1,1,-2)$, and it breaks $\mathrm{\text{SU}}{(3)}_H$ symmetry down to $\mathrm{\text{SU}}{(2)}_H\times U{(1)}_H$. Then masses of heavy species are contributed by singlet and octet VEVs and their mass matrices have diagonal forms, namely ${\mathrm{\text{M}}}_D=g_D〈S〉+f_D〈\mathrm{\Phi }〉=M_D\mathrm{\text{diag}}(X_D,X_D,1)$, and analogously for U and Q species, where generically $X_{U,D,Q}\ne 1$ are complex numbers. In this way, one obtains the Yukawa matrices with the minimally deformed Fitzsch textures Eq. (13), where the ‘asymmetries’ $x_{u,d}$ are introduced between 23 and 32 elements of ${\tilde{Y}}_{u,d}$, without changing the amount of zero elements.

In the case of $\mathrm{\text{SU}}(5)\times \mathrm{\text{SU}}{(3)}_H$,¹⁵ with ordinary quarks and leptons arranged as in Eq. (15), the vector-like fermions can be introduced in the representations

which include all vector-like quarks (21) together with lepton states $L,L^c$ and $E^c,E$.

In principle, all Yukawa couplings $Y_{u,d,e}$ can be induced by diagrams similar to that of Fig. 2 involving ${𝒯},\overline{{𝒯}}$ states, while $Y_{d,e}$ can be contributed also by exchange of ${ℱ},\overline{{ℱ}}$ states.¹⁵ Using these tools, in Ref. 29 the model was built which gives the Yukawa matrices (13) in a particularly constrained form. Namely, the (de-phased) up quark Yukawa is predicted to have a ‘symmetric’ Fritzsch texture, $x_u=1$ :

while for the down quarks and leptons one has deformed patterns with $x_{d,e}\ne 1$ :We shall not describe the model of Ref. 29 in details here, but only briefly mention its key ingredients:

(i)	flavon fields ${\chi }_1$ and ${\chi }_3$ are taken as $\mathrm{\text{SU}}(5)$ singlets, i.e. ${\chi }_1\sim (1,3)$ and ${\chi }_3\sim (1,\bar{6})$, while ${\chi }_2$ is taken in a mixed representation ${\chi }_2\sim (24,3)$, with the horizontal directions of VEVs as in Eq. (17) and hierarchy (19) between their values. This ‘mixed’ choice of ${\chi }_2$ induces non-zero $B_u$ in Eq. (24). It also introduces $\mathrm{\text{SU}}(5)$ breaking Clebsch factors in 23 blocks of Eq. (25), so that $B_d\ne B_e$, thus avoiding the wrong scaling relation $m_b:m_s:m_d=m_{\tau }:m_{\mu }:m_e$ between down quark and lepton masses. But $\mathit{\text{SU}}(5)$ invariance for the other two entries is kept, i.e. $C_d=C_e$ and $A_d=A_e$, which in turn implies the successful relation (11).
(ii)	heavy 10-plets ${𝒯},\overline{{𝒯}}$ get the dominant mass from the Yukawa coupling with a field S which is singlet of $\mathrm{\text{SU}}(5)\times \mathrm{\text{SU}}{(3)}_H$, with a subdominant contribution of the $\mathrm{\text{SU}}(5)$ adjoint Higgs $\mathrm{\Sigma }$. This explains the smallness of $A_u/C_u$ with respect to $A_d/C_d$ (see Eq. (10); in fact, without the latter contribution of $\mathrm{\Sigma }$, we would have $A_u=0$;
(iii)	masses of heavy 5-plets ${ℱ},\overline{{ℱ}}$ are strongly contributed by the Yukawa couplings with the $\mathrm{\text{SU}}{(3)}_H$ adjoint scalar $\mathrm{\Phi }$ which induces the deformation factors $x_{d,e}\ne 1$.

In the next section we mainly concentrate on the implications of textures (24) and (25) for the quark sector, following the analysis performed in Ref. 29, but having in mind that in the context of grand unification analogous considerations can be extended to leptons.

4. Deformed Fritzsch Textures: CKM Matrix vs. Quark Mass Ratios

Let us consider the Yukawa matrices ${\mathrm{\text{Y}}}_{u,d}$ having the form (13). They can be brought to the diagonal forms

via biunitary transformations (2) with the unitary matrices $V_u=F_u{O}_u$ and $V_d=F_d{O}_d$, where $F_{u,d}$ are the diagonal matrices containing phases, and $O_{u,d}$ are the real orthogonal matrices. In other terms, by phase transformations $F_{u,d}^{\prime }{Y}_{u,d}{F}_{u,d}$ with $F_{u,d}=\mathrm{\text{diag}}(e^{i{\alpha }_{u,d}},e^{i{\beta }_{u,d}},e^{i{\gamma }_{u,d}})$ (and similarly $F_{u,d}^{\prime }$), the Yukawa matrices $Y_{u,d}$ are brought to the real forms ${\tilde{Y}}_{u,d}$ which can be further diagonalized by bi-orthogonal transformations $O_{u,d}^{\prime }{\tilde{Y}}_{u,d}{O}_{u,d}=Y_{u,d}^{\mathrm{\text{diag}}}$.

Thus, for the CKM matrix of quark mixing we obtain

where the matrix $F=F_u^{†}{F}_d$ without loss of generality can be parameterized by the two phases $\tilde{\beta }$ and $\tilde{\delta }$ while the orthogonal matrices $O_{u,d}$ can be parametrized as with $c_{ij}^d=cos{\theta }_{ij}^d$ and $s_{ij}^d=sin{\theta }_{ij}^d$. Analogously, for up-quarks we have $O_u=O_{u23}{O}_{u13}{O}_{u12}$, as well as for the rotations of right-handed states $O_{u,d}^{\prime }$.

Hence, ${\tilde{Y}}_d$ contains four parameters, $A_d,B_d,C_d$ and $x_d$, which determine the three Yukawa eigenvalues $y_{d,s,b}$ and the three rotation angles in $O_d$. Analogously, the four parameters in ${\tilde{Y}}_u$ determine the Yukawa eigenvalues $y_{u,c,t}$ and the three angles in $O_u$. Therefore, we have 10 real parameters $A_{u,d},B_{u,d},C_{u,d},x_{u,d}$ and two phases $\tilde{\beta },\tilde{\delta }$ which have to match 10 observables, the 6 Yukawa eigenvalues and 4 independent parameters of the CKM matrix (see Eq. (4)).

The Yukawa eigenvalues and rotation matrices O and $O^{\prime }$ can be found by considering the “symmetric” squares respectively of the Yukawa matrices ${\tilde{Y}}_f^T{\tilde{Y}}_f$ and ${\tilde{Y}}_f{\tilde{Y}}_f^T$, $f=u,d$. In doing so, we obtain the following relations

where we omit the indices $f=u,d$ and imply $Y_{1,2,3}=y_{u,c,t}$ for the Yukawa eigenvalues of upper quarks and $Y_{1,2,3}=y_{d,s,b}$ for down quarks.

Having in mind the relations (6), we can use them in the synthetic form $y_t:y_c:y_u\sim 1:{ϵ}_u:{ϵ}_u^2$ and $y_b:y_s:y_d\sim 1:{ϵ}_d:{ϵ}_d^2$, noting that phenomenologically ${ϵ}_u\sim {ϵ}_d^2$. Then, in leading approximation (up to relative corrections of order $ϵ\sim Y_2/Y_3\sim Y_1/Y_2$) we have^26,29 :

so that $C_f:B_f:A_f\sim 1:{ϵ}_f^{1/2}:{ϵ}_f^{3/2}$. Since the ratios $C_d:B_d:A_d\sim V_3:V_2:V_1$ in fact reflect the hierarchy in the horizontal symmetry breaking (19) (and also taking into account that off-diagonal elements in $Y_u$ have additional suppression in the context of $\mathrm{\text{SU}}(5)\times \mathrm{\text{SU}}{(3)}_H$ model²⁹), this means that the inter-family mass hierarchy can actually be induced by a milder hierarchy between the flavon VEVs. The matrix entries $A_f$, $B_f$ and $C_f$ depend on the deformation $x_f$ only at higher orders in ${ϵ}_f$. As regards the rotation matrices, the angles in Eq. (28), up to small corrections of order ${ϵ}_d$, are given by The expressions for $s_{d23}^{\prime }$, $s_{d12}^{\prime }$ and $s_{d13}^{\prime }$ are the same with the replacement $x_d\to x_d^{-1}$, Analogously, the up quark rotation angles in $O_u$ can be expressed in terms of Yukawa ratios $y_u/y_c$ and $y_c/y_t$. Then, at leading order, for the elements of the CKM matrix we obtain :It can be noticed that for fixed values of the asymmetries $x_d$, $x_u$, $V_{us}$ depends on the phase $\tilde{\delta }$ while $V_{cb}$ only on the phase $\tilde{\beta }$. It is also worth noting that for $x_d=1$, the contribution of $s_{13}^d$ in $|V_{ub}|$ is negligible and the Fritzsch texture implies the prediction $|V_{ub}/V_{cb}|\approx \sqrt{y_u/y_c}$. Similar considerations can be inferred for the other off-diagonal elements, with the prediction $|V_{td}/V_{ts}|\approx \sqrt{y_d/y_s}$ for $x_d=1$. As regards the complex part of $V_{\mathrm{\text{CKM}}}$, we can consider the rephasing-invariant quantity $J=-\mathrm{\text{Im}}(V_{us}^{*}{V}_{cb}^{*}{V}_{ub}{V}_{cs})$, the Jarlskog invariant. In our scenario we have

In the following we are going to test the viability of asymmetric Fritzsch textures given present precision of experimental data on moduli and phases of the mixing elements. The values of the CKM matrix parameters and their uncertainties are given in Table 1.

The input values in our analysis are the ratios of the Yukawa eigenvalues. More specifically, we want to reproduce the CKM elements as functions of the Yukawa couplings assuming that the $Y_{u,d}$ acquire the Fritzsch form at some high energy scales of new physics. Thus, we need to consider the evolution of Yukawa matrices according to the renormalization group (RG) equations. For that purpose, first of all we bring the running masses of all quarks to the scale $\mu =m_t\approx v_{\mathrm{\text{w}}}$, using the QCD renormalization factors (see, e.g. in Eq. 55), and determine their ratios (i.e. the Yukawa constant ratios) at this scale. In doing so we obtain :

In particular, here we used the PDG data on top quark mass and the precision results of the lattice QCD computations for the ratios $m_b/m_c=4.579(9)$, $m_b/m_s=53.94(12)$, $m_c/m_s=11.768(34)$, $m_u/m_d=0.477(19)$ and $m_s/m_{ud}=27.31(10)$, where $m_{ud}=(m_u+m_d)/2$.³ In particular, from the two latter results we obtain the mass ratios of light quarks, $m_s/m_d=m_s/m_{ud}(m_u/m_d+1)/2$ — see Fig. 3.

The obtained pattern of the quark mass ratios (34) can be compared to that of the lepton masses known with much greater precision: $m_e/m_{\mu }=206.76852^{-1}$ and $m_{\mu }/m_{\tau }=16.817^{-1}$. Then we observe that the following combinations of the down quark and lepton masses coincide with extremely good precision :

In the context of the Fritzsch-like textures (25) this relation implies $A_d/C_d=A_e/C_e$ which in turn stems from the $\mathit{\text{SU}}(5)$ symmetry in the context of $\mathit{\text{SU}}(5)\times \mathit{\text{SU}}{(3)}_H$ model in Ref. [29].

Now we are going to test the viability of textures (13) when $x_u=1$ but $x_d\ne 1$, i.e. when ${\tilde{Y}}_u$ has a ‘symmetric’ form (24) while ${\tilde{Y}}_d$ is deformed as in (25). In this case we have 9 real parameters $A_{d,u}$, $B_{d,u}$, $C_{d,u}$, $\tilde{\beta }$, $\tilde{\delta }$ and $x_d$, which have to match 10 observables, the 6 Yukawa eigenvalues and the 4 independent parameters of the CKM matrix.

In our numerical analysis the ratios of quark running masses at $\mu =m_t$ will be fixed as in Eq. (6). However, we consider that the ‘starting’ scale (i.e. the mass scale of heavy fermions (21)) at which the Yukawa matrices acquire the Fritzsch form (13) can be much larger, say from $10^3$GeV to $10^{16}$GeV.^h For energy scales $\mu \gg m_t$, the quark mass ratios are no more given by Eq. (6) since their RG evolution will be influenced by additional contributions from the top Yukawa constant $\sim 1$. However, we anticipate that these effects are small and the obtained results are practically independent of the choice of the ‘starting’ scale $\mu$.²⁹

The complete analysis was done in Ref. 29. In Fig. 4 we show its results for a choice $x_d=3.3$ which demonstrates that all CKM parameters can be obtained within their $1\sigma$ uncertainties for a proper choice of phases $\tilde{\delta }$ and $\tilde{\beta }$.

For having an insight of how it works, we can use the approximate relations (32) and (31) (though in Ref. 29 the complete analysis was done without any approximation). As it is apparent from Eqs. (32), the element $|V_{us}|=s_{12}$ (the Cabibbo angle) is fixed by the phase $\tilde{\delta }$ and has no dependence on $\tilde{\beta }$ in leading approximation. In other words, for the ratios $y_d/y_s$ and $y_u/u_c$ fixed as in Eq. (6), $\tilde{\delta }\approx \pm 2\pi /3$ is fixed by the value of $s_{12}$ as shown in Fig. 4(a). Conversely, $|V_{cb}|=s_{23}$ depends only on $\tilde{\beta }$, and thus, for the given $x_d$, the latter phase is fixed as $\tilde{\beta }\approx \pm \pi /3$ by the value of $|V_{cb}|$ — see Fig. 4(b). Thus, in fact we have two possible signs for both $\tilde{\delta }$ and $\tilde{\beta }$, but as one can see from other plots on Fig. 4, the correct values of $|V_{ub}/V_{cb}|$ and J can be obtained only for both phases being on negative side, which in Fig. 4 are indicated by the green and yellow vertical bands respectively for $\tilde{\delta }$ and $\tilde{\beta }$. In fact the effect of the asymmetry parameter $x_d>1$ is that the rotation angle $s_{23}^d$ decreases while $s_{13}^d$ increases. This causes the prediction of $|V_{cb}|$ to shift towards lower values and in addition, large $s_{13}^d$ originates a dependence of $|V_{ub}|$ on both $\tilde{\delta }$ and $\tilde{\beta }$, and for the ‘negative’ choices of these parameters the value of $|V_{ub}/V_{cb}|$ intercepts the experimental determination. We also show in Fig. 5 how well the expectations on $V_{td}$ and $V_{ts}$ correspond to these phase parameters.

Concluding this discussion, we have shown that minimally deformed Fritzsch texture with $x_u=1$ and $x_d\approx 3$, successfully predicts all observables of the CKM matrix. The choice of $x_u=1$ can be justified in the context of grand unified picture SU$(5)\times \text{SU}{(3)}_H$, but $x_d$ remains a free parameter.²⁹ It is tempting to think that $x_d=3$ could also be originated from some more constrained scheme which is perhaps based on a larger GUT. Notice also that the results imply $\tilde{\delta }+\tilde{\beta }\approx -\pi$ which relation can be originated in the case of spontaneous CP-violation.

5. Discussion: Variations on the Theme

The hierarchy between the fermion masses and their weak mixing pattern remains a mystery. One can hope that some new physics beyond the SM can shed a light on these problems. However, a complete theory which could be coined as quantum flavordynamics does not exist yet, and one can discuss some pieces of the puzzle. The approach of zero-textures put forward by Fritzsch^11,12 is oriented towards the predictivity which in particular can relate the CKM mixing angles to the fermion mass ratios. Such predictive textures can be obtained within the models with the inter-family ‘horizontal’ symmetries as we described in this paper, but they do not address the origin of mass hierarchies: in fact, the hierarchies between the input parameters as (10) can be related to the hierarchy of the horizontal symmetry breaking as (19), but the latter is in fact introduced at hands. The predictive power of the textures can be enhanced by a clever use of the grand unification theories as $\mathrm{\text{SU}}(5)$ or $\mathrm{\text{SO}}(10)$ unifying the fermions of one family. This can provide some successful relations between the quark and lepton masses, such as Eq. (11), but yet the explanation of the inter-family mass hierarchies remains beyond their reach.

The fermion mass spectrum schematically given in Fig. 1 indeed looks very special. The only fermion with the mass of the order of electroweak scale $v_{\mathrm{\text{w}}}=174$GeV is top quark, implying the Yukawa constant $y_t\simeq 1$. The other Yukawa constants of the 3rd family are two orders of magnitude less, $y_b,y_{\tau }\sim 10^{-2}$ which indeed looks puzzling. Certainly, the coincidence of running constants $y_b(\mu )\simeq y_{\tau }(\mu )$ at the GUT scale $\mu \simeq 10^{16}$ GeV, so-called $b-\tau$ unification, is a clear success of $\mathrm{\text{SU}}(5)$, especially in its supersymmetric version, but the origin of $y_{b,\tau }\ll y_t$ still remains open. The question becomes more insistent in $\mathrm{\text{SO}}(10)$ theory where $t,b$ and $\tau$ are unified in one multiplet, implying the Yukawa unification $y_{\tau }=y_b=y_t$ at the GUT scale, and the only way to obtain the ‘vertical’ splitting in the 3rd family masses, $m_{b,\tau }\sim 10^{-2}{m}_t$, is to introduce the large $tan\beta$ at hands. The question is exacerbated by the fact that in the 2nd family the vertical splitting is much less, $m_{s,\mu }\sim 10^{-1}{m}_c$, Paradoxically, under the assumption of large $tan\beta$, this would imply the Yukawa constant $y_c$ an order of magnitude smaller than $y_{s,\mu }$ that of strange quark, namely $y_s\sim y_{\mu }\sim 10y_c$. This in turn implies $y_c\sim 10^{-2}{y}_{b,\tau }$, in obvious contrast with the mass values of c-quark (2nd family) and b-quark (3rd family) which in fact are of the same order, $m_c\sim m_{b,\tau }$ (see Fig. 1). Moreover, at the normalization scale $\mu$ taken as GUT scale, the masses of the first family become quasi-degenerate: $m_d/m_u\simeq 2$ while $m_e/m_u\simeq 1/2$, an additional difficulty for large $tan\beta$ scenario.

The overall pattern of fermion mass spectrum can be more naturally understood in the inverse hierarchy approach¹⁴ which implies that the fermion mass hierarchies are inversely proportional to the hierarchy of the inter-family symmetry breaking (19). In fact, Fig. 1 shows the following scaling laws for inverse masses :

and similarly for leptons, with the near degeneracy $m_d\simeq m_u\simeq m_e$ points that the Yukawa unification at the GUT scale may take place in the first rather in third family, i.e. $y_d=y_u=y_e$ instead of $y_b=y_t=y_{\tau }$. This in turn suggests the following pattern for the inverse Yukawa matrices^65,66:ⁱwhere ${ϵ}_f={ϵ}_u,{ϵ}_d,{ϵ}_e$ are small parameters and $P_{1,2,3}$ are symmetric rank-1 matrices with generically complex $O(1)$ elements. Without losing generality, one can take $P_1={(1,0,0)}^T•(1,0,0)$, $P_2={(a,b,0)}^T•(a,b,0)$ and $P_3={(x,y,z)}^T•(x,y,z)$. Therefore, the inverse Yukawa matrices have onion-like structures :Such a structure of the Yukawa matrices in frames of $\mathrm{\text{SO}}(10)$ symmetry was discussed in Refs. 65 and 66. Three fermion families in representations $16^i$, acquire masses via their seesaw-like mixing with extra vector-like fermions $16_i^{\prime }+{\overline{16}}_i^{\prime }$ while the latter get masses via couplings $16_i^{\prime }[{\chi }_1^{ij}+(45/M){\chi }_2^{ij}+{(45/M)}^2{\chi }_3^{ij}]{\overline{16}}_j^{\prime }$. In the above 45 is a scalar in adjoint representation of $\mathrm{\text{SO}}(10)$ having the VEV $M_G\sim 10^{16}$GeV (the GUT scale), $M>M_G$ is some large (string?) scale, and ${\chi }_{1,2,3}^{ij}\sim$ are flavon fields in symmetric (sextet) representations of $\text{SU}{(3)}_H$, having the rank-1 VEV configurations in family space (as $〈{\chi }_3〉$ in Eq. (17)) disoriented by large angles.⁶⁶ In this way, after integrating out the heavy fermions, one obtains the quark and lepton Yukawa matrices of the form (37), with rank-1 projectors originated from the flavon VEVs: $P_n=〈{\chi }_n〉/M$, and with small expansion parameters ${ϵ}_f\sim M_G/M$. The VEV $〈45〉$ produces different Clebsch factors for different fermion species so that generically ${ϵ}_u$, ${ϵ}_d$ and ${ϵ}_e$ in Eq. (37) have different values. However, $\mathrm{\text{SO}}(10)$ symmetry implies remarkable relation between the expansion parameters :Eigenvalues of (38) reproduce the scaling pattern (36) implying that ${ϵ}_u\ll {ϵ}_d\ll 1$. However, the term $a^2{ϵ}_d$ in $Y_d$ cannot be neglected. Namely, for its eigenvalues we have $y_d\approx y/|1+a^2{ϵ}_d|$, $y_s\approx y|1+a^2{ϵ}_d|/|b^2{ϵ}_d|$, $y_b\approx y/|z{ϵ}_d{|}^2$ and similarly for leptons. However, Eq. (39) implies that $y_d{y}_s\approx y_e{y}_{\mu }$ and $y_b\approx y_{\tau }$, and thus relation (35). As for up quarks, $a^2{ϵ}_u$ is negligible, and we have $y_u\approx y$, $y_c\approx y/|b^2{ϵ}_d|$, $y_t\approx y/|z{ϵ}_u{|}^2$. In addition, the CKM mixing is dominated by the diagonalization of $Y_d$. Then, taking into account that ${ϵ}_e\approx -{ϵ}_d$, we get :Thus, the last relation for the Cabibbo angle implies that $|a^2{ϵ}_d|\simeq 1$ which allows to achieve mass splitting in first family, $m_u/m_d\approx 1/2$ and $m_u/m_e\approx 2$, by properly choosing the phase of $a^2{ϵ}_d$. As for other mixing angles, they are naturally $s_{23}\sim {ϵ}_d$ and $s_{13}\sim {ϵ}_d^2$ in a fine correspondence with Eq. (7).

In a completely different approach, the fermion mass pattern can be explained by exceptionally clever choice of the underlying GUT, even without introducing any type of flavor symmetry. Namely, supersymmetric grand unification based on $\mathrm{\text{SU}}(6)$⁷⁰ solves the fundamental problems of grand hierarchy and doublet–triplet splitting in elegant way, since the Higgs doublets emerge as pseudo-Goldstone modes of the accidental global symmetry $\mathit{SU}(6)\times \mathit{SU}(6)$ in the Higgs superpotential.^70,71,72 The mass parameters at low energies (soft supersymmetry breaking masses as well as supersymmetric $\mu$-term) are all originated from the SUSY breaking terms. The pseudo-Goldstone nature of the Higgs has important consequences: it can have the renormalizable coupling in the superpotential, with the Yukawa constant $y_t\sim 1$, only to one of the three up-type quarks that can be identified as the top quark.^73,74 For the rest of fermion species, the Yukawa couplings can emerge only from the higher order operators and thus are suppressed. Interestingly, the structure of these operators also leads to $y_c\sim y_{b,\tau }\ll y_t$, with a perfect match to small values of $tan\beta$. As for the mass hierarchy between the 1st and 2nd families, it cannot be explained without help of flavor symmetries, but some discrete symmetries can be sufficient.⁷⁴

Postscriptum from Zurab Berezhiani

I was a Ph.D. student when I encountered the works of Harald Fritzsch^11,12 which deeply influenced my further career. I started to think on the role of inter-family symmetry $\mathrm{\text{SU}}{(3)}_H$ for understanding the fermion masses and mixings. The first time I met Harald personally was in 1990 in the last days of my one month visit to the Max-Planck Institute in Munich. During our conversations I discussed with him my papers^14,15 in which I succeeded to realize the Fritzsch mass textures in a theoretically appealing way by relating them to the pattern of $\mathrm{\text{SU}}{(3)}_H$ symmetry breaking. As a consequence, Harald invited me for a Humboldt fellowship to Ludwig-Maximilian University where I spent more than one year, and we never lost contacts after: he regularly invited me for seminars in Munich and for conferences that he organized worldwide, and many times he visited me in L’Aquila. Benedetta and I had discussions with Harald on the viable modifications of Fritzsch zero-texture during his last visit which work was completed later as Ref. 29.

Harald often asked me about Georgia and he was eager to visit Tbilisi. After COVID era, we decided to organize a conference in Tbilisi on “Recent Advances in Fundamental Physics”, planned for the fall 2022. But in August 2022 I received a sad news from Bigitte that Harald passed away.

I feel privileged to have had close scientific and personal ties with Harald Fritzsch. I remember him as a renowned physicist who authored many breakthrough works in different areas of particle physics but also as a man with a pleasant and exceptionally humble personality.

Acknowledgments

The work of Z.B. was partially supported by the research grant No. 2022E2J4RK “PANTHEON: Perspectives in Astroparticle and Neutrino THEory with Old and New messengers” under the program PRIN 2022 funded by the Italian Ministero dell’Università e della Ricerca (MUR) and by the European Union – Next Generation EU.

The work of B.B. was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant 396021762 – TRR 257.

ORCID

Zurab Berezhiani  https://orcid.org/0000-0002-4156-1686

Benedetta Belfatto  https://orcid.org/0000-0002-6191-8494

Notes

^a With these notations, the description can be conveniently extended for a supersymmetric version of the SM, in which case the terms (1) correspond to the Yukawa superpotential, with $\varphi ={\varphi }_u$ and $\tilde{\varphi }={\varphi }_d$ being two Higgs doublets taken as left-chiral superfields.

^b In the original works,^11,12 the ‘zeros’ in these matrices were obtained at the price of introducing several Higgs bi-doublets differently transforming under some discrete flavor symmetry. At present, this underlying theoretical construction looks rather obsolete. Namely, the need for several Higgs bi-doublets spoils the natural flavor conservation^5,6,7 and unavoidably leads to severe flavor-changing effects.¹³ In a more natural way, the Fritzsch texture for the Yukawa matrices $Y_{u,d,e}$ (rather than for the mass matrices) was obtained in Refs. 14 and 15 in the context of models with horizontal SU${(3)}_H$ gauge symmetry between the three fermion families.^16,17,18

^c $\mathrm{\text{SU}}(5)\times \mathrm{\text{SU}}{(3)}_H$ can be embedded into $\mathrm{\text{SU}}(8)$ group¹⁸ but this possibility introduces in the particle spectrum some extra fermions in exotic representations.

^d More generically, in the SM context the maximal chiral symmetry which can be achieved in the limit of vanishing Yukawa couplings (1) is $U{(3)}^5$ independently transforming the different species $q,u^c,d^c,\ell ,e^c$. In the context of $\mathrm{\text{SU}}(5)$, the maximal flavor symmetry is reduced to $U{(3)}^2=U{(3)}_{\bar{F}}\times U{(3)}_T$ independently transforming fermionic $\bar{5}$- and 10-plets, while the $\mathrm{\text{SO}}(10)$ structure allows a unique flavor symmetry $U{(3)}_H$ from which we gauge a non-abelian $\mathrm{\text{SU}}{(3)}_H$ part, with chiral species arranged as in Eq. (14).

^e Thus, in this construction the flavor is naturally conserved in neutral currents, in difference from the Fritzsch’s original model^11,12 involved several Higgs doublets.

^f More generally, the effective coefficients $C_n$ in Eq. (20) can be functions of a scalar 24-plet $\mathrm{\Sigma }$ which breaks $\mathrm{\text{SU}}(5)$ down to the SM $\mathrm{\text{SU}}(3)\times \mathrm{\text{SU}}(2)\times U(1)$.³⁹

^g In particular, in the context of supersymmetric theories with $\mathrm{\text{SU}}{(3)}_H$ family symmetry this mechanism can lead to interesting relations between the fermion Yukawa couplings and the soft SUSY breaking terms which naturally realize the minimal flavor violation scenarios.^25,52,53,54

^h The latter choice is natural in the context of grand unified model $\mathrm{\text{SU}}(5)\times \mathrm{\text{SU}}{(3)}_H$, whereas the former is interesting since the quark mixing with the vector-like quarks (21) in the TeV mass range can be at the origin of the recently observed Cabibbo angle anomalies⁵⁷ and can also induce various flavor-changing phenomena accessible in the SM precision tests^{58,59,60,61,62,63} (see also Ref. 64 for a review).

ⁱ For quarks ($f=u,d$) such a pattern was first obtained in Refs. 67,68,69 in the context of left–right symmetric models.