Mapping and Probing Froggatt-Nielsen Solutions to the Quark Flavor Puzzle

The possible origin of the wide and varying hierarchies amongst the quark and lepton masses and mixing angles has long invited speculation. While such disparate Lagrangian parameters are technically natural for fermions, the fact that the three matter generations appear identical in all other respects calls for a dynamical explanation of this so-called flavor puzzle.

A common strategy is to describe these patterns in terms of an approximate flavor symmetry, a subgroup of $U(3)^{5}$, whose breaking yields the observed masses and mixing angles. Historically, the first attempt in this direction was made by Froggatt and Nielsen in 1978 [1]. This strategy, expanded in [2, 3], became known as the Froggatt-Nielsen (FN) mechanism (see e.g. Ref. [4] for a modern review). In its simplest form, this mechanism relies on the introduction of a $U(1)_{X}$ symmetry under which fermions of different generations have different charges. This symmetry is broken at some high “flavor scale” $\Lambda_{F}$ by the vacuum expectation value (vev) of a SM-singlet scalar $\phi$, often referred to as the flavon. The SM Yukawa couplings are then generated as effective operators suppressed by factors $(\langle\phi\rangle/\Lambda_{F})^{n}=\epsilon^{n}$, where $\epsilon\lesssim 0.1$ and $n$ is determined by the $U(1)_{X}$ charges of the Higgs and the respective fermions. Assuming that these operators are generated in the UV completion of the model through heavy particle exchange with presumably $\mathcal{O}(1)$ coefficients (see e.g. Refs. [5, 6, 7]), the hierarchies of the Yukawa couplings in the flavor basis are generated by the $U(1)_{X}$ charge assignments of the SM fields and the flavon vev relative to the UV scale, $\epsilon$.

FN models have been extremely well studied over the last decades (see e.g. Refs. [3, 2, 8, 9, 10, 11, 12, 5, 13, 14, 15, 16, 6, 17, 18, 7, 19, 20, 4]), but curiously, the vast literature only considered very few choices of the flavor symmetry charge assignments for the SM fields. The studied scenarios typically identify $\epsilon$ with the Cabibbo angle of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and perform a spurion analysis to find a solution for the required fermion charges. The selection of these charges carries important phenomenological implications, since FN models have a variety of experimental signatures at low energies – most obviously various flavor-changing processes – the details of which depend on the exact charge assignment chosen.

In FN models of the quark sector, flavor-changing neutral current (FCNC) constraints involving the light generations bound the flavor scale to be above $\Lambda_{F}\gtrsim 10-100$ PeV [4]. This makes FN models (like most solutions to the flavor puzzle relying on a single flavor scale $\Lambda_{F}$) notoriously hard to probe experimentally. However, the next decades may see significant advances in our ability to probe flavor violation beyond the SM, whether with new data from on-going experiments (most notably LHCb and Belle II), concrete proposals for future high-energy colliders [21, 22, 23, 24, 25, 26, 27, 28], the hypothetical possibility of future flavor factories, or theoretical advances to improve the precision of SM predictions for experimentally well-measured processes. It is thus pertinent to investigate whether experimental insights can be gained regarding FN models in the foreseeable future, despite their characteristic high energy scale.

In this paper, we perform the first step in such a model-exhaustive program by identifying the most general “theory space” of natural FN models with a single Higgs field that can address the SM flavor problem in the quark sector. As pointed out in Ref. [20], flavor charge assignments beyond the few canonical choices can generate close matches to the SM, seemingly with natural $\mathcal{O}(1)$ choices for the various coefficients. Inspired by this observation, our analysis considers all possible flavor charge assignments up to $|X_{q}|\leq 4$ in a FN setup restricted to the quark sector, then performs numerical scans over natural $\mathcal{O}(1)$ choices of all the Yukawa coefficients, to determine in a maximally agnostic and general fashion which charge assignments and flavon vevs can generate “SM-like” quark masses and mixings. We find that the space of fully natural FN models is much larger than previously understood in the literature, including at least several dozen models depending on the precise interpretation of our results.

Unlike previous such analyses, we select the “SM-like” choices in a Bayesian-inspired fashion without explicitly fitting to the SM. We believe this more accurately reflects the intended spirit of the FN solution, that the SM hierarchies should emerge “naturally.”¹¹1Our approach to quantifying flavour naturalness is very similar to [29], though that study focused on lepton CP violation in the MSSM for a handful of natural charge assignments. For an example of applying more strictly Bayesian analyses to select FN models of the lepton sector, see [30, 31, 32]. Reassuringly, we do also find that the SM-like FN setups easily yield many stable exact fits to the SM, and that our numerical results can be understood in the context of an analytical spurion analysis that assumes small rotations between the flavor- and mass-bases.

With this natural theory space of viable FN models now defined, we study its phenomenology by estimating the size of various flavor-violating observables, most notably meson-antimeson mixing, in each model and comparing the range of predictions to current constraints. This analysis shows that different charge assignments or “textures” yield distinct predictions for flavor-violating observables, and highlights the specific observables that could be most effective in distinguishing between different FN solutions to the flavor puzzle in the future.

We hope that this kind of analysis can be helpful for future studies of flavor physics, including SMEFT global fits, by providing a “theoretical lamp post” that can direct attention on the most well-motivated parts of the vast landscape of FN models and flavor observables.

This paper is structured as follows. In Section II we briefly review the Froggatt-Nielsen mechanism. The numerical analysis of all possible FN models with $|X_{q}|\leq 4$ is presented in Section III. The results of this analysis are corroborated by an analytical spurion analysis in Section IV. he phenomenological implications are explored in Section V, and we conclude in Section VI. Details on SM fits, a custom tuning measure appropriate for FN models, and a variety of necessary statistical checks are collected in the Appendices.

In this Section we briefly review the Froggatt-Nielsen mechanism in its minimal form. Our treatment and notation parallels closely that of Ref. [20]. As stated in the introduction, the FN mechanism consists of adding a new $U(1)_{X}$ global flavor symmetry to the SM, along with a scalar field $\phi$ whose vev spontaneously breaks this symmetry. Without loss of generality we take $X_{\phi}=1,X_{H}=0$ and the vev of $\phi$ to be in the positive real direction, $\braket{\phi}=\braket{\phi}^{\dagger}=\epsilon\Lambda_{F}$, where $\Lambda_{F}$ is the characteristic scale of spontaneous symmetry breaking. SM matter fields are then assigned $U(1)_{X}$ charges, forcing the scalar to appear in the SM Yukawa couplings in order to preserve the symmetry. We denote the left-handed quark doublets as $Q_{i}$ and the right-handed up- and down-quark singlets $u_{i}$ and $d_{i}$, where $i=1,2,3$ is an index running over the fermion families. The quark mass terms in the low-energy SM effective field theory then take the form:

where $H$ is the SM Higgs doublet, and $c^{u,d}$ are arbitrary $3\times 3$ complex matrices with (presumably) $\mathcal{O}(1)$ entries. Comparing Eq. 1 to the usual SM Yukawa Lagrangian

we can read off the Yukawa matrices in terms of the FN parameters:

where

From this expression it is clear that, if $\epsilon\ll 1$, the SM Yukawa couplings can exhibit large hierarchies even if all entries of the coefficient matrices $c^{u,d}$ are $\mathcal{O}(1)$. These can be related to observable quantities – the quark masses and mixing angles – by performing unitary flavor rotations of the quark fields.

We consider all possible charge assignments with $|X|\leq 4$ for all SM quarks. Since $y_{t}\approx 1$, baryon number is conserved, and permutations of the quark fields do not constitute a physical difference between FN models, we can set $X_{q_{3}}=X_{u_{3}}=0$ and adopt the ordering convention that $|X_{Q_{i}}|\geq|X_{Q_{j}}|$ for $i<j$ when all $X_{Q}$ have the same sign, otherwise $X_{Q_{i}}\geq X_{Q_{j}}$. (Similarly for $X_{u}$, or $X_{d}$.) In addition, following [20] we remove “mirror” charges which are related by multiplying all of the charges $X$ by $-1$ (and then enforcing the above ordering convention). This gives a total of 167,125 charge assignments that are physically inequivalent in the IR.

We now wish to assess these textures based on how “typical” it is for them to reproduce the flavor hierarchies of the SM. Given a certain charge assignment $\bm{X}=\left\{X_{Q},X_{u},X_{d}\right\}$, we randomly draw some large number of $\mathcal{O}(1)$ complex coefficient matrices $c^{u,d}$ (details discussed below) and calculate the SM parameters for each instance of $c^{u,d}$ as a function of $\epsilon$. For a given choice of $\epsilon$, we can then calculate the observable with the maximum fractional deviation from its SM value:

where $\mu_{i}$ ranges over the six quark masses, the three independent CKM entries $|V_{12}|,|V_{13}|$, and $|V_{23}|$, and the absolute value of the Jarlskog invariant $|J|$. The precise numerical values we use can be found in Table 2 in Appendix B. Since $\epsilon$ plays the central role of generating the overall hierarchy, we minimize $\Xi_{\mathrm{SM}}$ with respect to $\epsilon$ for each instance of the $c^{u,d}$ coefficient matrices to match as closely as possible the SM values.

After repeating this process for each charge assignment $\bm{X}$, we define $\mathcal{F}_{2}$ ($\mathcal{F}_{5}$) as the fraction of random coefficient choices that yields $\Xi_{\mathrm{SM}}\leq 2$ $(5)$ for each model. This allows us to define “global naturalness criteria” (i.e. independent of a particular fit to the SM) for a given texture to be a good candidate for reproducing the SM hierarchies, namely requiring that $\mathcal{F}_{2}$ or $\mathcal{F}_{5}$ be above some lower bound.

Out of the total 167,125 possible charge assignments, we only find about 10 for which $\mathcal{F}_{2}\gtrsim 1\%$. We adopt the latter criterion as the most stringent measure of naturalness for a texture to solve the SM quark flavor problem, and show the top-20 possible charge assignemnts ranked by $\mathcal{F}_{2}$ in Table 1.We also show the average value of $\epsilon$ for each texture. The distribution of preferred $\epsilon$ values is very narrow, indicating that each texture has a uniquely suited value of $\langle\phi\rangle/\Lambda_{F}$ to reproduce the SM. Some of these good textures correspond to those identified already in the literature, but, to the best of our knowledge, most of them have not been considered before.²²2Texture 3 first appeared in the seminal papers [3, 2].

This is quite remarkable, given how similarly they all naturally generate SM-like mass and mixing hierarchies. Textures where all quark masses and CKM entries lie within a factor of 5 of their measured SM values are even more common, with about 430 textures satisfying $\mathcal{F}_{5}\gtrsim 10\%$. This constitutes a large collection of textures that are well-motivated in this model-independent way. The complete ranking of these textures is included in the file charges.csv provided in the auxiliary material.³³3Note that because the auxiliary file is rank ordered by $\mathcal{F}_{5}$, it does not match the order given in Table 1, and the precise ordering of textures with very similar values of $\mathcal{F}_{5}$ (or $\mathcal{F}_{2}$) is subject to small numerical fluctuations. Several textures in this larger list have been studied in the past, for example [33] examines a texture with $\mathcal{F}_{5(2)}=33(0.1)\%$.

As mentioned above, to understand how close a texture generically is to the SM, we may look at the distribution of $\Xi_{\mathrm{SM}}$ values for a given texture over many different random choices of the $c^{u,d}$ matrices. In Fig. 1, we show such a distribution for the top texture from Table 1. As we can see, the distribution is clustered around $\Xi_{\mathrm{SM}}\sim 4$ with broad tails on either side, indicating that this texture generically results in physical parameters with the proper SM-like hierarchies regardless of the $\mathcal{O}(1)$ coefficients $c^{u,d}$. Of course getting the precise SM values requires a precise choice of these coefficients, but the hierarchies themselves are robust.

We may also look at such a distribution for a charge assignment that does not perform well by this metric. This category includes many textures previously mentioned in the literature, including for example most of the textures listed in Tables 1 and 2 of Ref. [20]. The third texture in Table 2 of [20] results in the distribution for $\Xi_{\mathrm{SM}}$ shown in Fig. 2. It is quite reasonable to ask why these textures show up as good in that analysis but not ours, and the answer is simple: Ref. [20] searches for textures for which there is at least one technically natural choice of $c^{u,d}$ coefficients that approximately reproduces the SM, whereas we look for textures which generically produce SM-like hierarchies. With random $\mathcal{O}(1)$ coefficient matrices, a texture like that of Fig. 2 results in a typical deviation from SM parameters of more than an order of magnitude, but there does exist a very special choice of $c^{u,d}\sim\mathcal{O}(1)$ that results in the SM. This choice of coefficients may be technically natural but it is still exceedingly rare, as Fig. 2 shows. The difference thus stems from our “global” choice of metric for naturalness, which prioritizes textures that can lead to SM-like parameters without need for additional dynamics or precise restrictions on the coefficients.

To verify that our global naturalness criterion also produces viable and “locally natural” solutions to the SM flavor problem, we also have to check that each of the textures in Table 1 yields exact and untuned numerical fits to the SM, including uncertainty on the SM parameters. This necessitates the definition of a custom tuning measure that is more appropriate for FN models than standard choices like the Barbieri-Giudice measure [34, 35], taking into account that any change in the UV theory would likely perturb all coefficients in Eq. 1 at once, rather than one at a time. The details are in Appendix A, but the upshot is that each of the textures in Table 1 readily yields many good SM fits that are completely untuned with respect to simultaneous random uncorrelated perturbations of all coupling coefficients. This confirms that our global naturalness criterion guarantees particular technically natural solutions to the SM flavor problem as well.

We perform several statistical checks to make sure our conclusions are robust. The results obtained here depend on the details of the statistical distributions from which we draw the coefficients in the $c^{u,d}$ matrices. Taking a Bayesian-inspired point of view, these distributions can be thought of as prior probability distributions for the $c^{u,d}$ coefficients. For the analysis shown here, we use a “log-normal” prior in which the logarithm of the magnitude of each $c^{u,d}$ entry is drawn from a Gaussian distribution. We have repeated the same analysis for a wider log-normal distribution, and a third “uniform” prior, in which the real and imaginary parts of each $c^{u,d}$ are drawn uniformly from the range $[-3,3]$. Up to modest reorderings in our ranked list of top textures, our results are robust with respect to changing the prior. We also confirm that the SM-like hierarchies for our good textures are robustly determined by the flavor charges and the small $\epsilon$ rather than anomalous hierarchies amongst the randomly drawn $c^{u,d}$ coefficients, and that individual SM observables are roughly uncorrelated and drawn from a distribution roughly centered on the correct SM value for each good texture. For details, see Appendix B.

Our numerical study identified many new plausible FN textures beyond those that have been studied in the literature. It would be useful to understand analytically why the textures in Table 1 reliably yield SM-like hierarchies. If viable FN models involve no large rotations between the flavor- and the mass-basis, then it is possible to estimate masses and mixings via an analytical spurion analysis. We can easily test this assumption.

In general, the flavor-basis Yukawa matrix $Y^{u}$ (identically for $Y^{d}$) can be related to the diagonal mass-basis Yukawa $\hat{Y}^{u}$ via the usual singular value decomposition $Y^{u}=U_{u}\hat{Y}^{u}W_{u}^{\dagger}$. Under the near-aligned assumption, and adopting the strict ordering of charges defined at the beginning of Section III, the mass-basis Yukawa couplings are simply given by

where the exponent matrices $n^{u,d}$ are defined terms of the $X_{Q,u,d}$ charges in Eq. 4. Under the same strict assumptions,⁴⁴4Eq. LABEL:e.UW does not apply for flavor charge assignments where the heaviest up and down quarks are not in the same generation, but those are irrelevant for our analysis. the magnitudes of the elements of the rotation matrices can be written as

This makes it straightforward to obtain magnitude estimates for the elements of $V_{\mathrm{CKM}}=U_{u}^{\dagger}U_{d}$.⁵⁵5 For about half of the textures in Table 1, this yields $V_{\mathrm{CKM}}$ estimates that follow the Wolfenstein parameterization (with $\lambda\rightarrow\epsilon$), while the other half show slight deviations from this pattern.

For a given FN charge assignment and choice of $\epsilon$, we can now compute the spurion estimate $\mu_{i}^{\text{spurion}}$ for $i=$ each of the six quark masses and $|V_{\mathrm{CKM}}|_{12,23,13}$, using Eqns. 6, LABEL:e.UW verbatim, working to lowest order in $\epsilon$, and setting all coefficients to 1. To assess how SM-like a given FN charge assignment is, we consider the logs of the ratios by which these estimates deviate from the SM, added in quadrature,

with $\epsilon$ chosen to minimize $d$. This measure always takes all SM observables into account and is more appropriate for the approximate nature of the spurion analysis than Eq. 5. We would expect that FN textures that naturally give a good fit to the SM would satisfy $d\lesssim\mathcal{O}(1)$.

Indeed, we find that all the textures in Table 1 satisfy $d<2.4$. Of all 167,125 possible charge assignments, there are only 36 additional possibilities that also satisfy this criterion, and all of them are in the top 160 of our ranked list of textures obtained in the numerical analysis of the previous section. This analytical spurion estimate is therefore fully consistent with our full numerical study, which gives us additional confidence that the global naturalness criterion derived in our numerical study is theoretically plausible. Furthermore, this convergent result confirms the expectation that natural FN models always feature a high degree of alignment between the flavor and the mass bases.⁶⁶6This was confirmed in the numerical analysis, where coefficient choices with small values of $\Xi_{\mathrm{SM}}$ for textures in Table 1 always feature small mixing angles in $U_{u,d},W_{u,d}$.

Obviously, the specific $d<2.4$ criterion was derived a posteriori from the results of the numerical study, but if one were to guess at a reasonable upper bound for $d$ that natural SM-like FN models must satisfy, a number like $d<3$ might plausibly come to mind, which yields a still very modest total of 149 textures (including the top 20). As with any such definition, the cutoff for what constitutes a natural model is somewhat arbitrary, and if one wants to make quantitative statements the fully numerical approach of the previous section is necessary. However, it is interesting to note that the numerical study was not necessary to make the much more important qualitative observation that there are many different FN charge assignments, of the order of several dozen at least, that very naturally give SM-like hierarchies. The crucial ingredient is merely to formulate a well-defined and general criterion and check it across all possible flavor charges.

Flavor-violating effects are the most relevant experimental signatures of FN models. At energies well below $\Lambda_{F}$ these can be effectively parameterized within the framework of Standard Model Effective Field Theory (SMEFT). In the “FN flavor basis” of Eq. 3, where flavor charges are well-defined for each quark generation, the couplings of flavor-changing operators are suppressed by powers of $\epsilon^{n}$, where the value of $n$ is set by the charges of the fields appearing in the operator. For example, for a generic 4-quark operator in the SMEFT ⁷⁷7For the sake of simplicity, Dirac structures as well as Lorentz and gauge group indices are left implicit. Upper-case roman indices $A$ are taken to label the different possible dimension-6 flavor-violating quark operators. Flavor indices are indicated explicitly.

where $i,j,k,l$ denote different quark flavors, and $q$ can represent either $Q,d$ or $u$ type quarks, we expect the associated effective scale to be given by

where $\Lambda_{F}$ is the UV scale of the model, and $c^{A}_{ijkl}$ is an unknown dimensionless complex coefficient.

For FN models that satisfy our global naturalness criterion, it is plausible to expect the different coefficients $c^{A}_{ijkl}$ to be relatively uncorrelated and have $\mathcal{O}(1)$ sizes.⁸⁸8Support for this assumption can be derived from the near-uncorrelated near-log-normal distributions of the individual SM observables in our numerical scans, see Fig. 5 in Appendix B. It is plausible to expect other observables to behave similarly. We can therefore extend our Bayesian-inspired numerical approach to study the phenomenology of the most SM-like FN quark sector models. ⁹⁹9For a similar approach, applied to find Yukawa-coupling modifications generated by select FN models due to SMEFT operators involving two fermions and additional Higgs fields, see [37].

By construction, the most interesting observables in this context are quark flavor-changing observables. The implementation proceeds as follows: For each set of flavor charges in Table 1, we make random choices for the Yukawa coefficients $c^{u,d}$ in Eq. (3) as outlined in Section III. If a given coefficient choice gives a SM-like quark spectrum ($\Xi_{\mathrm{SM}}<2$), we use it as the starting point to find an exact (and fully natural) SM fit, as detailed in Appendix A, thereby determining the corresponding $\epsilon$ and rotation matrices $U_{u,d},W_{u,d}$. We then randomly select coefficients for all possible SMEFT operators in the flavor basis, using the same prior distribution as for $c^{u,d}$. At this point, the theory is fully specified up to the unknown flavor scale $\Lambda_{F}$, and for each particular set of Yukawa and dimension-6 coefficient choices, we can perform the rotation to the quark mass basis and the matching to the Low-Energy Effective Theory (LEFT), which is ultimately used to obtain predictions for the observables. These predictions, as well as the corresponding SM predictions and experimental bounds, are taken from the package flavio [38], in the context of a new version of smelli [39, 40]. Repeating this many times for each texture in Table 1 gives a picture of the “theoretically preferred” distribution of observables for each texture.

Unsurprisingly, the most stringent constraint on $\Lambda_{F}$ – typically around $\sim 10^{5}~\mathrm{TeV}$ – is set by the absorptive part of $K-\bar{K}$ mixing, followed by the absorptive part of $D-\bar{D}$ mixing (encoded in the observables $\epsilon_{K}$ and $x_{12}\sin\phi_{12}$, respectively). This is illustrated in the top row of Figure 3, where we show the distribution of the flavor scale saturating the bounds on these observables for four representative textures from Table 1. For all such textures, the overall flavor scale is relatively close to the flavor-anarchic expectation. This is because large right-handed rotations between the flavor and mass bases significantly undermine the suppression of flavor-changing interactions present in the flavor basis. Nevertheless, different textures clearly exhibit distinct distributions, especially in the $D-\bar{D}$ direction.

While it may be possible to observe a signal in $D-\bar{D}$ mixing in the future if the flavor scale is as low as the $\epsilon_{K}$ bounds permit, distinguishing between different textures would be more plausible if deviations in other flavor observables, such as $B_{d}-\bar{B}_{d}$ and $B_{s}-\bar{B}_{s}$ mixing, were also observed. To further illustrate this, we set the flavor scale to saturate the bound on $\epsilon_{K}$, and obtain predictions for $x_{12}\sin\phi_{12}$ and for the new physics contribution to the $B_{d,s}-\bar{B}_{d,s}$ mixing amplitudes, $\Delta_{\rm NP}m_{B_{d,s}}$. We plot the distributions for $x_{12}\sin\phi_{12}$ and $\Delta_{\rm NP}m_{B_{d}}$ in the bottom row of Figure 3. (The distribution and experimental reach for $\Delta_{\rm NP}m_{B_{s}}$ is very similar.) The gray band is excluded by current bounds on $D-\bar{D}$ mixing [41]. The analogous exclusion from $B_{d}-\bar{B}_{d}$ mixing is not visible in the plot, the main limit being the current precision in the SM prediction. However, we show the current measurement precision on $\Delta m_{B_{d}}$ as a dashed light blue line [36], to indicate what sensitivity is experimentally plausible if significant theoretical advances were made in computing the SM prediction.

Similar differences across textures can be observed in other flavor-violating processes, for example rare $B$ and $K$ decays. However, in these cases, the absolute size of the deviations is several orders of magnitude below current bounds, making them experimentally inaccessible in the foreseeable future. For instance, the new physics contribution to $B_{s}\to\mu^{+}\mu^{-}$ is $7-8$ orders of magnitude below the SM one, well beyond experimental reach. Thus, $\Delta F=2$ observables remain the only viable experimental probes of FN models in the quark sector. These conclusions also hold if the Higgs has a non-zero FN charge, as SMEFT operators involving the Higgs affect these observables only at the loop level.

Our analysis demonstrates how the phenomenology of different FN textures can be understood and potentially used to distinguish between UV models. In practice, the high flavor scale due to near-anarchic mixing in the right-handed quark sector, and challenges in improving purely hadronic observables both theoretically and experimentally, may make an actual detection of deviations difficult. However, we anticipate that extending our analysis to FN models of the lepton sector would yield more optimistic experimental predictions for observables involving lepton-flavor violation and possibly simultaneous lepton- and quark-flavor violation. We leave this for future work.

The flavor puzzle remains one of the most profound mysteries in the Standard Model, with the peculiar structure of fermion mass matrices hinting at an underlying principle or symmetry yet to be uncovered. Directly probing the mechanisms that generate this structure is difficult, as in most scenarios, tight constraints on flavor violation push the relevant scale of flavor physics to very high values far beyond our ability to probe directly in the intermediate future. Nevertheless, future experiments may offer the prospect of directly or indirectly probing this elusive flavor structure, making its detection and diagnosis a compelling goal.

Our work provides a systematic framework for exploring the vast landscape of Froggatt-Nielsen solutions to the quark flavor problem. By significantly expanding the range of viable charge assignments beyond those commonly studied, we present a much more comprehensive picture of FN models. Even so, our global criterion – requiring a natural generation of the SM mass and mixing hierarchies across the entire parameter space of a model – remains stringent enough to yield a manageable set of FN models, as presented in Table 1.

We demonstrated that this most natural subset of FN benchmark models can generate a range of flavor-violating effects, with varying magnitudes and correlations influenced by the specific flavor charge assignments. In particular, we found that $\Delta F=2$ observables like $K-\bar{K}$, $D-\bar{D}$, and $B_{d,s}-\bar{B}_{d,s}$ mixing could potentially be used to differentiate between these models if deviations from the Standard Model are observed. This explicitly shows how, in principle, identifying the correct solution to the quark flavor problem could become feasible once flavor-violating signals beyond the SM are unambiguously detected.

Global SMEFT fits that incorporate a specific FN model might be more constraining [42, 43, 44, 45], and it would be interesting to perform such fits for each of the natural FN models identified in Table 1, using well-defined priors for the unknown $\mathcal{O}(1)$ coefficients. This approach could further refine our understanding of how different FN textures manifest in low-energy observables and help narrow down viable model candidates.

Our approach to identifying natural FN models can be extended to other sectors, such as FN models for the lepton sector, or more complex frameworks involving multiple or non-abelian flavor symmetries [46, 47, 48, 49, 50]. These extensions could uncover additional flavor-violating observables that offer the best prospects for probing and understanding the underlying physics of the flavor puzzle. It may also be interesting to study the impact of our work on dark sectors that are related to the SM via a discrete symmetry [51], and to revisit our phenomenological analysis if the broken $U(1)_{X}$ is an approximate global symmetry, leading to a flavon degree of freedom in the spectrum (see e.g. [14, 13, 52]). We leave such investigations for future work.