January 01, 1970
We investigate a harmonic interpretation of the hidden flavor coordinates (Q,G,C) previously introduced in a low-rank ternary description of the fermion mass spectrum. The generation coordinate is associated with the odd harmonic mode sequence
\[n_G=(1,3,5),\]
allowing the fermion mass ansatz to be rewritten in a compact harmonic form. The corresponding logarithmic spectrum reveals an additive structure in which the hidden coordinates contribute directly to the observed mass hierarchy.
We argue that the projected quantity
\[L=QG+C,\]
which organizes the fermion masses, does not uniquely characterize a fermion state. This motivates the introduction of the full hidden-coordinate vector
\[X=(Q,G,C),\]
and a geometric interpretation of flavor based on coordinate separations and antisymmetric invariants defined within the hidden-coordinate space.
The harmonic framework further suggests a possible extension beyond the fermion sector. Combining the fermionic harmonic modes generates the bosonic sequence
\[n_B=(2,4,6,8,10),\]
which exhibits a suggestive correspondence with the electroweak and Higgs scales.
Although the resulting framework remains exploratory, it suggests that fermion masses, flavor structure, and bosonic states may reflect different aspects of a common hidden harmonic organization.
The origin of the fermion mass spectrum and flavor structure remains one of the central open questions of particle physics. Within the Standard Model, fermion masses and mixing parameters are described by Yukawa couplings, whose observed values span many orders of magnitude and exhibit a hierarchical pattern that is not explained by the theory itself.
In previous work [1], a low-rank ternary description of the fermion spectrum was proposed in terms of hidden flavor coordinates \((Q,G,C)\), where (Q) labels the fermion family, (G) denotes the generation index, and (C) represents an additional binary coordinate. It was shown that the observed charged-fermion masses can be approximately organized through simple combinations of these coordinates, suggesting the existence of an underlying hidden structure behind the Standard Model flavor sector.
Subsequent work explored a geometric and harmonic interpretation of the hidden coordinates. In particular, the generation coordinate was associated with the odd harmonic sequence
\[n_G = 2G-1 = (1,3,5),\]
while the hidden coordinate space was interpreted as a discrete flavor geometry. This construction naturally led to the appearance of geometric invariants and harmonic structures that organize the fermion spectrum.
The purpose of the present work is to extend this framework from fermion masses to fermion mixing. We investigate whether the same hidden coordinates that describe the mass spectrum can also account for the observed structure of the Cabibbo–Kobayashi–Maskawa (CKM) and Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrices.
To this end, we introduce a hidden flavor metric defined on the (Q,G,C) coordinate space. The metric is constructed from two ingredients: a generation-space distance that measures separation between harmonic shells and a wave-like overlap factor associated with hidden surface coordinates. The resulting framework provides a common geometric description of quark and lepton mixing while preserving the coordinate structure previously used to describe fermion masses.
We further discuss the relation between the hidden flavor metric and the Yukawa sector of the Standard Model. The obtained mixing matrices suggest that the hidden coordinates may provide a useful organizational principle for both fermion masses and flavor mixing, and may serve as a starting point for more complete flavor-symmetry constructions.
The paper is organized as follows. Section 2 reviews the symmetry interpretation of the hidden coordinates and their harmonic realization. Section 3 introduces the hidden flavor metric and its geometric ingredients. Section 4 presents fits to the CKM and PMNS matrices. Section 5 discusses the connection to Yukawa matrices and flavor structure. Finally, possible extensions toward a broader flavor framework and harmonic bosonic sector are briefly outlined.
The experimental fermion masses and mixing parameters used throughout this work are taken from the Particle Data Group review [2].
In the previous work [1], the fermion spectrum was organized using three hidden coordinates,
\[X=(Q,G,C),\]
where (Q) labels the fermion family, (G) denotes the generation index, and (C) is an additional discrete coordinate. The observed fermion masses were found to exhibit a simple dependence on these quantities, suggesting that they may encode aspects of an underlying flavor structure.
The coordinate (Q) distinguishes the four fermion sectors,
\[Q=\{0,1,2,3\},\]
which correspond respectively to neutrinos, charged leptons, down-type quarks, and up-type quarks. The coordinate therefore acts as a flavor-family label and may be viewed as a discrete remnant of a hidden flavor symmetry. In the present work we associate this coordinate with an effective Abelian flavor charge,
\[Q \sim U(1)_F.\]
The generation coordinate (G) takes the values
\[G=(1,2,3).\]
Following the harmonic interpretation proposed previously, we define the corresponding odd harmonic sequence
\[n_G = 2G-1,\]
which yields
\[n_G=(1,3,5).\]
These values may be interpreted as fundamental harmonic modes of the hidden flavor space. In this picture, the three observed generations correspond to successive odd excitations of an underlying discrete harmonic structure.
The coordinate (C) takes the values
\[C=\{1,-1,2,0\},\]
for neutrinos, charged leptons, down-type quarks, and up-type quarks, respectively. Although originally introduced as an empirical coordinate required to organize the fermion spectrum, it admits a possible interpretation in terms of discrete symmetry labels.
Introducing a binary sector charge
\[S= \begin{cases} 0, & \text{quarks},\\ 1, & \text{leptons}, \end{cases}\]
together with parity and time-reversal eigenvalues,
\[P=\pm1, \qquad T=\pm1,\]
the coordinate (C) may be represented as
\[C= \frac{1}{2}\left(1+P-T-PT\right)-ST.\]
For the assignments
\[(u,d,e,\nu): \quad (P,T,S)= (+1,+1,0), (+1,-1,0), (+1,+1,1), (-1,-1,1),\]
one obtains
\[C_u=0, \qquad C_d=2, \qquad C_e=-1, \qquad C_\nu=1.\]
Although this construction remains phenomenological, it provides a possible symmetry-based interpretation of the third hidden coordinate.
Taken together, the coordinates (Q,G,C) define a discrete hidden flavor space. The mass structure studied previously depended primarily on the projected combination
\[L = QG + C.\]
While this quantity successfully organizes the fermion spectrum, it does not uniquely specify a point in the hidden-coordinate space. Different fermion states can possess similar values of (L) while corresponding to distinct coordinate triples (Q,G,C). This observation motivates the use of the full hidden-coordinate vector
\[X=(Q,G,C)\]
as the fundamental object.
The present work adopts this viewpoint and treats the hidden coordinates as elements of a discrete geometric space. Rather than focusing only on the scalar combination (L), we investigate distances and overlaps between points in the full coordinate space. This leads naturally to the construction of a hidden flavor metric capable of describing not only fermion masses but also the observed pattern of flavor mixing.
Having introduced the hidden-coordinate space
\[X=(Q,G,C),\]
we now construct a metric that quantifies the overlap between different fermion states. The central hypothesis of the present work is that flavor mixing originates from geometric separations within the hidden-coordinate space rather than from independent parameters assigned to each mixing angle.
Following the harmonic interpretation of the generation coordinate, we associate to each fermion state a hidden radial coordinate
\[R_i = r_i (2G_i-1)\pi,\]
where (\(r_i\)) denotes a sector-dependent radius parameter obtained from the fermion mass analysis and
\[n_G = 2G-1=(1,3,5)\]
represents the odd harmonic modes associated with the three generations.
The quantity
\[R_i\]
may be interpreted as the radius of a hidden harmonic shell. The corresponding separation between two fermion states is
\[\Delta R_{ij}=R_i-R_j.\]
In addition to the radial separation, we introduce a generation-space distance (\(D_G(i,j)\)). This quantity measures the effective separation between generations through adjacent harmonic shells. For three generations we define
\[D_G(1,1)=D_G(2,2)=D_G(3,3)=0,\]
\[D_G(1,2)=a_{12},\]
\[D_G(2,3)=a_{23},\]
and
\[D_G(1,3)=a_{12}+a_{23}.\]
The distance therefore corresponds to the length of the shortest path connecting two generations through neighboring shells.
We then define the hidden flavor metric
\[M_{ij} = e^{-D_G(i,j)} \left| \cos\left(k\Delta R_{ij}\right) \right|^{1/4}.\]
The exponential factor suppresses transitions between distant generations and may be interpreted as a tunneling probability in generation space. The cosine factor represents a wave-like overlap of hidden harmonic shells.
By construction,
\[\Delta R_{ii}=0,\]
which implies
\[M_{ii}=1.\]
The diagonal elements therefore correspond to maximal self-overlap, while off-diagonal elements describe the overlap between distinct fermion states.
The metric may also be written in wave-function form,
\[M_{ij}=|\psi_{ij}|^2,\]
with
\[\psi_{ij} = e^{-D_G(i,j)/2} \left| \cos\left(k\Delta R_{ij}\right) \right|^{1/8}.\]
Within this interpretation, flavor mixing emerges from the combination of generation-space tunneling and hidden-shell interference.
The metric introduced above is formulated in terms of dimensionless hidden-shell coordinates. A physical length scale may be introduced by defining
\[R_i^{\rm phys}=\ell_0 R_i,\]
where \(\ell_0\) is a fundamental length scale.
If the outer shell associated with the first-generation quark sector is identified with the onset of hadronization,
\[R_{\rm conf}\sim 1~{\rm fm},\]
then the corresponding energy scale is of order
\[\Lambda_{\rm QCD}\sim 200~{\rm MeV}.\]
Within this interpretation, the hidden-shell radii may be viewed as dimensionless coordinates measured relative to a confinement-scale length. The flavor metric itself remains unchanged, while the parameter \(k\) may be interpreted as an effective surface-wave number defined on the hidden shells.
The possible relation between hidden-shell overlap and QCD confinement remains speculative and will not be pursued further in the present work.
In the following section we determine the parameters of the metric by fitting the observed CKM and PMNS mixing matrices.
We now test whether the hidden flavor metric introduced in the previous section can reproduce the observed quark and lepton mixing patterns.
The metric
\[M_{ij} = e^{-D_G(i,j)} \left| \cos\left(k\Delta R_{ij}\right) \right|^{1/4}\]
depends on the generation-space distances (\(a_{12}\)) and (\(a_{23}\)) together with a wave parameter (k). The sector-dependent radii (\(r_i\)) are fixed from the fermion-mass analysis and are not adjusted in the mixing fit.
For quarks the best fit is obtained with
\[a_{12}^{q}=1.375,\]
\[a_{23}^{q}=3.092,\]
\[k_q=-10.275.\]
The resulting matrix is
\[M_{\rm CKM}^{\rm fit} = \begin{pmatrix} 0.9736 & 0.2281 & 0.0037 \\ 0.2121 & 0.9763 & 0.0423 \\ 0.0091 & 0.0370 & 0.9993 \end{pmatrix}.\]
This should be compared with the observed CKM matrix
\[M_{\rm CKM}^{\rm obs} = \begin{pmatrix} 0.9740 & 0.2250 & 0.0037 \\ 0.2250 & 0.9730 & 0.0410 \\ 0.0087 & 0.0400 & 0.9990 \end{pmatrix}.\]
The largest deviation is approximately (7.5%), occurring in the (3,2) element.
An interesting feature of the fit is the hierarchy
\[\frac{a_{23}^{q}}{a_{12}^{q}} = 2.25,\]
indicating that the effective separation between the second and third generations is larger than that between the first and second generations.
For leptons the best fit is obtained with
\[a_{12}^{\ell}=0.425,\]
\[a_{23}^{\ell}=0.092,\]
\[k_{\ell}=-25.023.\]
The fitted matrix is
\[M_{\rm PMNS}^{\rm fit} = \begin{pmatrix} 0.8439 & 0.5137 & 0.1547 \\ 0.3524 & 0.6951 & 0.6267 \\ 0.4509 & 0.4190 & 0.7881 \end{pmatrix}.\]
This may be compared with the observed PMNS matrix
\[M_{\rm PMNS}^{\rm obs} = \begin{pmatrix} 0.8200 & 0.5500 & 0.1500 \\ 0.3500 & 0.7000 & 0.6200 \\ 0.4400 & 0.4500 & 0.7700 \end{pmatrix}.\]
The largest deviation is approximately (6.9%).
In contrast to the CKM fit, the lepton sector exhibits
\[\frac{a_{23}^{\ell}}{a_{12}^{\ell}} = 0.216,\]
indicating a substantially smaller effective separation between the second and third generations.
The same hidden-coordinate framework is therefore capable of reproducing both quark and lepton mixing patterns. The principal difference between the two sectors is encoded in the generation-space distances.
The quark sector favors
\[a_{23}^{q}>a_{12}^{q},\]
while the lepton sector favors
\[a_{23}^{\ell}<a_{12}^{\ell}.\]
This suggests that quarks and leptons probe the same hidden flavor geometry but experience different effective localization properties within the generation space.
The results support the interpretation of flavor mixing as a combination of generation-space tunneling and wave-like overlap between hidden harmonic shells.
In the Standard Model, fermion masses and flavor mixing originate from the Yukawa sector,
\[\mathcal{L}_{Y} = \bar Q_L,Y_d,H,d_R \bar Q_L,Y_u,\widetilde{H},u_R \bar L_L,Y_e,H,e_R +\mathrm{h.c.},\]
where (\(Y_u\)), (\(Y_d\)), and (\(Y_e\)) are the Yukawa matrices. Following electroweak symmetry breaking,
\[\langle H\rangle = \frac{v}{\sqrt2},\]
the fermion mass matrices become
\[M_f = \frac{v}{\sqrt2}Y_f.\]
Within the Standard Model, the entries of the Yukawa matrices are free parameters. The hidden-coordinate framework developed in the present work suggests a possible geometric origin for these quantities.
The flavor metric introduced in the previous section naturally defines an overlap between two fermion states,
\[M_{ij} = e^{-D_G(i,j)} \left| \cos\left(k\Delta R_{ij}\right) \right|^{1/4}.\]
This quantity may be interpreted as a geometric overlap in hidden-coordinate space. Motivated by this observation, we propose that the Yukawa matrices inherit their hierarchical structure from hidden-coordinate overlaps,
\[Y_{ij} \propto M_{ij}.\]
The Yukawa couplings therefore become functions of hidden-coordinate distances rather than independent parameters.
The hidden-coordinate description separates two distinct effects.
The fermion mass spectrum is primarily governed by the projected coordinate
\[L=QG+C,\]
which determines the hierarchy of masses.
The flavor metric
\[M_{ij}\]
describes overlaps between different hidden-coordinate states and determines the mixing structure.
In this picture, masses and mixing arise from the same hidden-coordinate geometry but probe different aspects of the underlying space.
The hidden coordinates may also be incorporated into effective flavor operators analogous to Froggatt–Nielsen constructions [3],
\[\mathcal{O}_{ij} = \left( \frac{\Phi}{\Lambda} \right)^{n_{ij}} \bar f_{Li} H f_{Rj},\]
where (\(n_{ij}\)) depends on hidden-coordinate differences.
A natural possibility is that the exponent is determined by the hidden-coordinate projection,
\[n_{ij} = |L_i-L_j|,\]
or more generally by the geometric distance between hidden-coordinate vectors,
\[n_{ij} = D(X_i,X_j).\]
The resulting Yukawa matrices would then be generated dynamically from hidden-coordinate separations.
Although the present work does not attempt a complete ultraviolet completion, the observed success of the hidden flavor metric suggests that the coordinates (Q,G,C) may provide a useful organizing principle for both the fermion mass spectrum and the Yukawa sector.
In the next section we discuss possible extensions of this framework and its relation to hidden harmonic excitations beyond the Standard Model.
The hidden-coordinate framework suggests that both fermion masses and flavor mixing may originate from a common geometric structure. This motivates the search for an effective flavor symmetry underlying the coordinates
\[X=(Q,G,C).\]
In the present work, the coordinate (Q) is interpreted as an effective flavor charge associated with a hidden Abelian symmetry,
\[U(1)_F.\]
The generation coordinate (G) is associated with the odd harmonic sequence
\[n_G=(1,3,5),\]
while the coordinate (C) provides an additional discrete flavor label.
The observed fermion masses depend primarily on the projected quantity
\[L = QG + C,\]
suggesting that the Yukawa hierarchy may arise from powers of a flavor-breaking field. In analogy with Froggatt–Nielsen constructions, one may consider effective operators of the form
\[\mathcal{L}_{\rm eff} = \sum_{ij} y_{ij} \left( \frac{\Phi}{\Lambda} \right)^{L_{ij}} \bar f_{Li} H f_{Rj} + {\rm h.c.},\]
where (\(\Phi\)) denotes a flavor-breaking scalar field and (\(\Lambda\)) is the corresponding flavor scale.
Within such a framework, the hidden coordinates determine the suppression of the effective Yukawa couplings, while the flavor metric introduced in the previous sections controls the overlap between different hidden-coordinate states.
The resulting picture separates two complementary effects:
the projected coordinate (L=QG+C) governs the mass hierarchy,
the hidden flavor metric governs the mixing structure.
Although the present construction does not provide a complete ultraviolet completion, it suggests that the observed pattern of fermion masses and mixing may emerge from a common hidden-coordinate geometry.
This observation motivates the exploration of additional harmonic excitations associated with the hidden-coordinate space, which may provide a possible route toward an extended flavor sector.
The hidden-coordinate framework developed in this work suggests a possible extension in terms of a flavor symmetry associated with the harmonic generation sequence
\[n_G=(1,3,5).\]
A natural candidate is a flavor charge
\[Q_F=(2G-1)(B-L),\]
where
\[B-L \equiv \text{baryon number} - \text{lepton number}\]
is the standard baryon-minus-lepton quantum number of the Standard Model. This promotes the harmonic structure to an effective \(U(1)_F\) flavor symmetry.
Within such a framework, the Yukawa hierarchy may arise from flavor-breaking scalar fields carrying harmonic charges. In particular, the bosonic modes
\[n_B=(8,10)\]
may be associated with scalar fields
\[\chi_8, \qquad \phi_{10},\]
whose vacuum expectation values generate effective flavor operators.
The resulting effective Lagrangian takes the schematic form
\[\mathcal{L} = \mathcal{L}_{\rm SM} + \mathcal{L}_{\rm flavor} + \mathcal{L}_{\rm Yukawa} - V(H,\chi_8,\phi_{10}).\]
The detailed operator structure, anomaly cancellation conditions, and phenomenological consequences remain subjects for future study. Nevertheless, the construction demonstrates that the hidden-coordinate framework can be embedded naturally into a conventional flavor-symmetry setting.
The hidden-coordinate framework developed in this work extends the previous mass construction based on the coordinates \((Q,G,C)\) to the description of flavor mixing.
The central observation is that the projected quantity
\[L=QG+C\]
appears sufficient to organize the fermion mass hierarchy, while the full coordinate vector
\[X=(Q,G,C)\]
is required to describe flavor mixing. This suggests that masses and mixing probe different aspects of the same hidden flavor space.
The hidden flavor metric introduced in this work combines two geometric ingredients. The first is a generation-space distance \(D_G(i,j)\) describing separations between neighboring harmonic shells. The second is a wave-like overlap term depending on the hidden-shell coordinates \(R_i\). Together these contributions reproduce both the CKM and PMNS matrices with a relatively small number of effective parameters.
An intriguing feature of the fits is the different hierarchy of generation distances in the quark and lepton sectors. The quark fit prefers
\[a_{23}^{q}>a_{12}^{q},\]
whereas the lepton fit favors
\[a_{23}^{\ell}<a_{12}^{\ell}.\]
This may indicate different localization properties of quarks and leptons within the same hidden-coordinate geometry.
The harmonic interpretation of the generation coordinate,
\[n_G=(1,3,5),\]
further suggests a possible connection between flavor structure and an underlying flavor symmetry. In particular, the charge
\[Q_F=(2G-1)(B-L),\]
provides a natural way to promote the harmonic sequence into an effective flavor quantum number. Here \(B-L\) denotes the conventional baryon-minus-lepton charge of the Standard Model and should not be confused with the hidden-coordinate quantity
\[L=QG+C.\]
Although the present framework remains phenomenological, it suggests that fermion masses, flavor mixing, Yukawa hierarchies, and possible flavor-breaking sectors may originate from a common hidden geometric structure.
In this work we extended the hidden-coordinate framework previously developed for the fermion mass spectrum to the description of flavor mixing.
The central idea is that the hidden coordinates
\[X=(Q,G,C)\]
define a discrete flavor space whose geometry influences both fermion masses and mixing. While the projected quantity
\[L=QG+C\]
organizes the mass hierarchy, the full coordinate vector is required to describe flavor transitions between different states.
To quantify these transitions, we introduced a hidden flavor metric combining generation-space distances with wave-like overlaps of hidden harmonic shells. The resulting framework provides a unified geometric description of quark and lepton mixing.
Using a small number of effective parameters, the proposed metric reproduces the observed CKM and PMNS matrices with good accuracy. The fits suggest that quarks and leptons may probe the same hidden flavor geometry while exhibiting different localization properties within generation space.
The harmonic interpretation of the generation coordinate,
\[n_G=(1,3,5),\]
further motivates a possible connection between flavor geometry and effective flavor symmetries. In this context, Yukawa hierarchies, mixing matrices, and flavor-breaking operators may emerge as different manifestations of a common hidden-coordinate structure.
The framework developed here remains phenomenological. Nevertheless, the results indicate that hidden-coordinate geometry provides a useful organizational principle for flavor physics and may offer a starting point for future studies of flavor symmetries, Yukawa structures, and possible extensions beyond the Standard Model.