Fermion mass relations in one-parameter modular models

Salvador Centelles Chuliá Email: salcen@ific.uv.es Affiliation: Instituto de Física Corpuscular (IFIC), Universidad de Valencia-CSIC,
Paterna (Valencia) E-46980, Spain    Xueqi Li Email: xueqi.li@uci.edu Affiliation: Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575 USA    Xiang-Gan Liu Email: xianggal@uci.edu Affiliation: Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575 USA Affiliation: Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México    Omar Medina Affiliation: Instituto de Física Corpuscular (IFIC), Universidad de Valencia-CSIC,
Paterna (Valencia) E-46980, Spain    J. T. Penedo Email: jpenedo@roma3.infn.it Affiliation: INFN Sezione di Roma Tre, Via della Vasca Navale 84, 00146, Roma, Italy

Abstract

Modular flavour symmetries provide a possible organizing principle for the Standard Model Yukawa sector, by replacing generic couplings with a potentially small number of modular forms controlled by a single complex modulus. We study the extreme limit of this idea: one-parameter modular models, in which each charged-fermion mass matrix is fixed by a single modular invariant contraction. We develop a systematic method to construct such models, showing that the one-parameter modular model requirement is already highly constraining at the level of possible fermion hierarchies. In a concrete realization, the charged-lepton and down-quark sectors are controlled by the common modulus, leading to exact mass relations at the flavour scale,

$$m_{s}^{5}=2\sqrt{2}\,m_{d}^{3}m_{b}^{2},\qquad m_{\mu}^{3}=\sqrt{2}\,m_{e}m_{\tau}^{2},\qquad m_{s}^{2}m_{\tau}=\sqrt{2}\,m_{e}m_{b}^{2}.$$

We show that, once renormalization-group evolution and selective supersymmetric threshold effects are included, these high-scale relations can be made compatible with low-energy charged-fermion data. Our results provide a working proof of principle for one-parameter modular models and point towards a possible route to the flavour puzzle through highly constrained constructions.

I Introduction

The Standard Model (SM) allows for an extraordinarily economical description of fundamental interactions, based on the gauge symmetry principle. However, this economy is lost in its Yukawa sector, which encodes fermion masses, mixing and CP violation (CPV), and where most of the theory’s free parameters reside. Explaining the peculiarities of the observed flavour structures, namely the marked mass hierarchies and contrasting mixing patterns (i.e. large for leptons and small for quarks), constitutes the flavour puzzle and remains an open challenge in theoretical particle physics.

An organizing principle in the flavour sector is desired: it would work orthogonally to the gauge principle, unifying different particle generations into multiplets. Non-Abelian flavour (or horizontal) symmetries provide such a framework, postulating the invariance of the action under the transformations of a symmetry group $G_{f}$, suitably broken by the vacuum expectation values of scalar fields, known as flavons (see e.g. Altarelli:2010gt; Ishimori:2010au; King:2014nza; Petcov:2017ggy; Feruglio:2019ybq; Ding:2024ozt for reviews). The modular invariance generalization of the traditional flavour approach Feruglio:2017spp (see Kobayashi:2023zzc; Ding:2023htn for reviews) focuses instead on a minimal symmetry-breaking sector, described by a single complex field: the modulus $\tau$, with $\im\tau>0$. Within this string-inspired, supersymmetric framework, modular forms $Y(\tau)$ play the role of Yukawa couplings which are significantly constrained. In particular, the holomorphicity of the superpotential restricts the space of available forms, making this setup potentially very predictive. Moreover, model-independent mass relations may emerge from the symmetry structure of the flavour group and of the modular forms in modular-invariant models containing a small number of parameters, see e.g. Chen:2023mwt. Taken to the extreme, one could in principle build a modular model where each Yukawa matrix results from a single invariant contraction. Within such an one-parameter modular model, each fermion mass matrix is thus fully determined by $\tau$ and an overall coefficient.

In this work, we investigate the conditions under which one-parameter modular models may be viable. A first step towards building an one-parameter modular model involves accounting for the hierarchical fermion spectrum, which already places significant constraints on the possible values of the modulus vacuum expectation value. Note that a generic vacuum expectation value of $\tau$ fully breaks the non-linearly realized modular invariance. Instead, at the special values $\tau_{\text{sym}}=i,\,\omega=e^{2\pi i/3},\,i\infty$ some residual $\mathbb{Z}_{n}$ ($n\geq 2$) symmetry is preserved and can be exploited. Indeed, in the vicinity of these fixed points, the residual symmetry is slightly broken (but linearly realized) and Yukawa couplings can be expanded as power series in $\epsilon\sim|\tau-\tau_{\text{sym}}|$, providing a natural origin of the fermion mass hierarchies Feruglio:2021dte; Novichkov:2021evw. This mechanism, which has recently Carducci:2026shs been dubbed modular proximity-induced hierarchies, has been used to derive “golden”-type mass relations in models relying on a limited number of parameters in Ref. Chen:2023mwt. Finally, we stress that the number of possible one-parameter modular models one can construct may be limited, due to the structure of the modular group $\mathrm{SL}(2,\mathbb{Z})$. As a result, obtaining the observed charged-fermion mass hierarchies from an one-parameter modular model is highly non-trivial.

This paper is organized as follows. In Section II we introduce the modular-invariant framework and the conditions under which a single invariant contraction gives rise to an one-parameter modular model Yukawa sector with suitable mass hierarchies. In Section III we use these criteria to search for possible one-parameter modular models. This search first singles out $\Delta(96)$ and the related group $\Delta(384)$, where four inequivalent hierarchy patterns H₂ – H₅ are obtained. Two of these hierarchies, H₂ and H₅, are naturally close to the observed charged-lepton and down-quark spectra. Assigning them to the corresponding sectors in an explicit double one-parameter modular model then fixes the high-scale mass relations studied in the rest of the paper. In Section IV we confront these relations with low-energy data, through renormalization-group running and finite threshold corrections. In Section V we discuss hints on how this framework may provide a solution of the quark flavour puzzle and we conclude in Section VI.

II Framework

We consider a simple setup in flavour modular symmetry. We work in global $\mathcal{N}=1$ supersymmetry, with matter chiral superfields $\psi^{i}$ acquiring their masses from the Yukawa interactions in the superpotential, characterized by cubic terms of the type

$$\mathcal{W}\,\supset\,Y_{ijk}(\tau)\,\psi_{i}\psi_{j}\psi_{k}\,,$$ (1)

where the $Y_{ijk}$ are the components of a vector-valued modular form (or modular form multiplet). These are holomorphic functions that transform under the modular group $\mathrm{SL}(2,\mathbb{Z})$ as

$$Y^{(k_{Y})}(\tau)\,\,\xmapsto{\gamma}\,\,Y^{(k_{Y})}(\gamma\tau)\,=\,(c\,\tau+d)^{k_{Y}}\rho(\gamma)\,Y^{(k_{Y})}(\tau)\,,\qquad\text{with }\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z})\,,$$ (2)

where $\rho$ is a representation of $\mathrm{SL}(2,\mathbb{Z})$, $k_{Y}$ is known as the weight of the modular form, and $\tau$ is the modulus, which transforms as

$$\tau\,\,\xmapsto{\gamma}\,\,\gamma\tau\,=\,\frac{a\,\tau+b}{c\,\tau+d}\,.$$ (3)

Recall that $\mathrm{SL}(2,\mathbb{Z})$ is generated by the matrices $S=\left(\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}\right)$ and $T=\left(\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}\right)$. We focus on the case where the representation $\rho$ has finite image; in such a case, the vector-valued modular forms are those of the finite modular group $\mathrm{SL}(2,\mathbb{Z})/\ker\rho$. A matter field transforms under the modular group as

$$\psi\,\,\xmapsto{\gamma}\,\,(c\,\tau+d)^{-k_{\psi}}\rho_{\psi}(\gamma)\,\psi\,,$$ (4)

where $k_{\psi}$ is the modular weight of the matter field $\psi$ and $\rho_{\psi}$ is a (unitary) representation of the finite modular group. In global supersymmetry, the superpotential is invariant under modular transformations. Then, together with eq. 2, one finds that, in each superpotential term, the sum of the matter field modular weights must equal the weight of the modular form, $k_{Y}=k_{\psi_{i}}+k_{\psi_{j}}+k_{\psi_{k}}$. For relatively low weights, there are only a few, or even a single, modular form(s) available, strongly constraining flavour structures and observables.

In building modular-invariant flavour models, we consider a minimal form for the Kähler potential,

$$\mathcal{K}=-h\,\Lambda_{\tau}^{2}\log(-i\tau+i\bar{\tau})+\sum_{i}(-i\tau+i\bar{\tau})^{-k_{\psi_{i}}}|\psi_{i}|^{2}\,,$$ (5)

where $h>0$ and $\Lambda_{\tau}$ has a mass dimension of 1. In a given charged fermion sector, assuming a single Higgs field $H$ per sector ($H_{u}$ or $H_{d}$), the modular-invariant Yukawa term can be expanded as

$$\mathcal{W}\supset\sum_{i}\alpha_{i}\left(Y_{\mathbf{r}_{i}}^{(k_{i})}(\tau)\,\psi^{c}\psi\right)_{\mathbf{1}}H\;,$$ (6)

where the $\alpha_{i}$ are constant parameters and $(\cdots)_{\mathbf{1}}$ denotes the contraction of flavour indices into a trivial singlet. Each Higgs doublet is taken to transform as a trivial singlet in flavour space with modular weight $k_{H}=0$, without loss of generality. The modular weight $k_{i}$ is then equal to the sum of the weights of the parts of $\psi^{c}$ and $\psi$ participating in the contraction. The sum in eq. 6 runs over all possible modular form representations and weights whose contraction yields a trivial singlet.

We are interested in the extreme one-parameter case, in which there is only one $\alpha_{i}$ for each sector, denoted $\alpha_{f}$ ($f=u,d,e$). In such a scenario, there is exactly one possible contraction for a given $k=k_{\psi}+k_{\psi^{c}}$, i.e. the above sum must reduce to a single term. To satisfy this requirement, two conditions must be met:

1. among the representations $\mathbf{r}_{i}$ under which the vector-valued modular forms $Y_{\mathbf{r}_{i}}^{(k)}$ transform, only one representation, $\mathbf{r}$, can contract with $\psi^{c}$ and $\psi$ to produce a trivial (non-vanishing) singlet;

2. there is only one modular form of weight $k$ furnishing the representation $\mathbf{r}$;

If either of these conditions is not met, more than one contraction will be present.

For a model with three generations of massive fermions, all three generations must participate in the contraction so that each generation acquires a mass. Together with the condition in item 1, this implies that the fermions must transform as a triplet of the finite modular group. Otherwise, either some generations remain massless because they do not participate in the contraction, or there is more than one contraction, since each singlet or doublet must be contracted separately. We therefore focus only on the case in which $\psi$ and $\psi^{c}$ transform as triplets, since only in this case can an one-parameter modular model be obtained.

To satisfy the condition in item 2, it is useful to consider only modular forms of the lowest weight. At low modular weight, there is typically only one modular form available. In this work, we restrict our search to weights 1 and 2, which are the lowest weights for modular forms in most finite modular groups.

Type	Max rank at $\tau_{\text{sym}}$	Leading spectrum	Possible asymptotic regions
I	$1$	$c_{1}\,\epsilon^{p}:c_{2}\,\epsilon^{q}:1$	$\tau\simeq i\infty$
II	$2$	$c_{1}\,\epsilon^{p}:c_{2}:1$	$\tau\simeq i\infty,~i$
III	$3$	$c_{1}:c_{2}:1$	$\tau\simeq i\infty,~i,~\omega$

Table 1: Classification of mass spectra in one-parameter modular models, where $p,q\neq 0$ and the constants $c_{i}$ follow from the expansion coefficients of modular forms and Clebsch–Gordan coefficients of the finite modular group.

As discussed in the introduction, we further narrow our focus by considering only models predicting (non-zero) hierarchical masses. Indeed, one can naturally obtain hierarchical fermion spectra by expanding around zeros in modular space Feruglio:2021dte; Novichkov:2021evw. More precisely, the determinant of the mass matrix, which is a one-dimensional modular form (see e.g. Appendix A of Penedo:2024gtb), vanishes at these values of $\tau$. The transformation properties and the zeroes of modular determinants have been discussed in detail in Chen:2025tby. One can show that, within the fundamental domain of $\mathrm{SL}(2,\mathbb{Z})$, the determinant modular forms of weight below 12 can vanish only at three points: $\tau=i\infty$, $i$, and $\omega=e^{{2\pi i}/{3}}$, known as the critical or fixed points. By expanding mass matrix elements around these points in terms of a small deviation parameter $\epsilon\sim|\tau-\tau_{\text{sym}}|$, one can categorize the resulting mass hierarchies into three types, as presented in Table 1.

Note that some spectra are not available in a given asymptotic region. For instance, the spectrum $c_{1}\,\epsilon^{2}:c_{2}\,\epsilon:1$ is not attainable if $\tau\simeq\omega$ in this context. This follows from the stringent requirement that $\psi$ and $\psi^{c}$ are irreducible triplets. Indeed, as noted in Feruglio:2023mii, in this irreducible case one has

• $\psi\,\,\xmapsto{ST}\,\,\diag(1,\omega,\omega^{2})\,\psi$ in an appropriate $ST$-diagonal basis;

• $\psi\,\,\xmapsto{S}\,\,i^{k_{S}}\diag(1,-1,-1)$ in an appropriate $S$-diagonal basis, where $k_{S}$ is an integer in the cases of interest.

This implies that, at the symmetric point $\tau=\omega$ one expects a spectrum of the type $c_{1}:c_{2}:1$, while for $\tau=i$ one expects either $c_{1}:c_{2}:1$ or $0:c_{2}:1$ (or a fully massless spectrum). At nearby values of $\tau$, massless fermions are generically lifted by an appropriate power of $\epsilon$.

For spectra of type I, the hierarchy is naturally given by powers of the small parameter $\epsilon$. For type III, the hierarchy would instead arise purely from the coefficients of the modular forms and group tensor products, an intriguing possibility Chen:2025tby. Type II lies between these two cases. In this work, we focus on spectra of type I, leading us to consider large $\im\tau$. As such, the appropriate expansion parameter $\epsilon$ is an appropriate power of $|q|$, with $q\equiv e^{2\pi i\tau}$ and $q\to 0$ as $\tau\to i\infty$. These criteria define the search performed in the next section.

III One-parameter modular models

An one-parameter modular model is defined, in part, by the finite modular group $\mathrm{SL}(2,\mathbb{Z})/\ker\rho$, which must admit at least one triplet irreducible representation. A list of the non-Abelian finite modular groups with order $<78$ has been given in Ref. Liu:2021gwa. Up to order 100, one finds the following finite modular groups admitting triplet irreducible representations:

$$\begin{array}[]{llll}[24,12]\simeq S_{4}\,\,(2)\,,&[24,13]\simeq A_{4}\times\mathbb{Z}_{2}\,\,(2)\,,&[48,28]\simeq 2O\,\,(2)\,,&[48,29]\simeq GL(2,3)\,\,(2)\,,\\[5.69054pt] [48,30]\simeq S_{4}^{\prime}\,\,(4)\,,&[48,31]\simeq A_{4}\times\mathbb{Z}_{4}\,\,(4)\,,&[48,32]\simeq T^{\prime}\times\mathbb{Z}_{2}\,\,(2)\,,&[48,33]\simeq T^{\prime}\circ\mathbb{Z}_{4}\,\,(2)\,,\\[5.69054pt] \mathbf{[60,5]\simeq A_{5}}\,\,(2)\,,&[72,42]\simeq S_{4}\times\mathbb{Z}_{3}\,\,(6)\,,&[72,44]\simeq A_{4}\times S_{3}\,\,(2)\,,&\mathbf{[96,64]\simeq\Delta(96)}\,\,(6)\,,\\[5.69054pt] [96,66]\simeq T^{\prime}\rtimes\mathbb{Z}_{4}\,\,(4)\,,&[96,69]\simeq T^{\prime}\times\mathbb{Z}_{4}\,\,(4)\,,&\lx@intercol[96,72]\simeq((\mathbb{Z}_{4}\times\mathbb{Z}_{4})\rtimes\mathbb{Z}_{2})\rtimes\mathbb{Z}_{3}\,\,(2)\,,\hfil\lx@intercol\end{array}$$ (7)

where each group is first identified by its GAP Id GAP4 and the number in brackets counts the number of distinct triplet irreducible representations available.

Focusing on hierarchical spectra of type I, i.e. of the type $c_{1}\,\epsilon^{p}:c_{2}\,\epsilon^{q}:1$ with $p\neq q$, we require that the mass matrix obtained from the contraction of the pair of triplets $\psi^{c}\otimes\psi$ has at most rank 1, in the symmetric limit $\tau\to i\infty$. Note that a residual $\mathbb{Z}_{N}^{T}$ symmetry is recovered in that limit, where $N$ is the order of $\rho(T)$. From the decomposition of the triplets $\psi$ and $\psi^{c}$ under $\mathbb{Z}_{N}^{T}$, one can directly infer the rank of the mass matrix and how its zeroes are lifted for $\epsilon\neq 0$ in a $T$-diagonal basis Novichkov:2021evw. This selects only two groups from eq. 7, highlighted in bold, as well as a small number of pairs of triplets as promising one-parameter modular model ingredients.

An $A_{5}$-based one-parameter modular model would represent an attractive possibility. Unfortunately, as already noted in section 3.3.2 of Ref. Novichkov:2021evw, this potential one-parameter modular model predicts a massless fermion in the SUSY limit, since the determinant of the mass matrix vanishes identically for any value of $\tau$. This can be understood from the fact that such a weight-6 determinant, with a zero at $\tau=i\infty$, transforms as a trivial singlet of $\mathrm{SL}(2,\mathbb{Z})$. Following Chen:2025tby, one can see that these properties force it to be constantly zero, even if $\epsilon\neq 0$.

Motivated by the above discussion, one may consider the promising group $\Delta(96)\simeq[96,64]$, which is a member of the $\Delta(6n^{2})$ family of groups, with $n=4$, and is a subgroup of the finite modular group $\Gamma_{8}$ deAdelhartToorop:2011re. In what follows, we mostly concentrate on exploring a larger group, $\Delta(384)\simeq[384,568]\subset\Gamma_{16}$, which contains $\Delta(96)$ as a subgroup and is also a finite modular group. It is, in fact, the first element of the $\Delta(6n^{2})$ family ($n=8$) that contains $\Delta(96)$ as a proper subgroup. The presentations of these groups read:

$$\Delta(96) =\,\langle S,T~\big|~S^{2}=(ST)^{3}=(ST^{-1}ST)^{3}=T^{8}=1\rangle\,,$$ (8)

$$\Delta(384) =\,\langle S,T~\big|~S^{2}=(ST)^{3}=(ST^{-1}ST)^{3}=T^{16}=1\rangle\,,$$

so that $N=8$ for $\Delta(96)$ and $N=16$ for $\Delta(384)$. By explicit construction, we find that one-parameter modular models that can be obtained using $\Delta(96)\simeq\Delta(384)/(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$ also arise in the search based on $\Delta(384)$.

The group $\Delta(384)$ admits as irreducible representations: two singlets $\mathbf{1},\mathbf{1}^{\prime}$, one doublet $\mathbf{2}$, fourteen triplets $\mathbf{3}_{0,1,\dots,13}$, and seven sextets $\mathbf{6}_{0,1,\dots,6}$ (all even representations, i.e. $\rho(S)^{2}=\mathbb{1}$). To obtain one-parameter modular models, as illustrated in Section II, we focus on the lowest non-trivial weight, which is $k=2$ for this group. There are only a few vector-valued modular forms at this weight, namely

$$k=2: Y^{(2)}_{\mathbf{2}}(\tau),~Y^{(2)}_{\mathbf{3}_{0}}(\tau),~Y^{(2)}_{\mathbf{3}_{3}}(\tau),~Y^{(2)}_{\mathbf{3}_{5}}(\tau),~Y^{(2)}_{\mathbf{3}_{7}}(\tau),$$ (9)

$$Y^{(2)}_{\mathbf{6}_{0}}(\tau),~Y^{(2)}_{\mathbf{6}_{2}}(\tau),~Y^{(2)}_{\mathbf{6}_{3}}(\tau),~Y^{(2)}_{\mathbf{6}_{4}}(\tau),~Y^{(2)}_{\mathbf{6}_{5}}(\tau)\,,$$

in our convention. Using these forms and scanning over triplet tensor products $\mathbf{3}_{i}\otimes\mathbf{3}_{j}$, we are able to obtain several one-parameter modular models leading to spectra of the desired type for large $\im\tau$.

We summarize our results in Table 2, which lists the corresponding hierarchical spectra, at leading order in an expansion in the small variable $\epsilon$, defined as Novichkov:2021evw

$$\epsilon\equiv|q|^{1/N}=e^{-2\pi\im\tau/N}\ll 1\,.$$ (10)

In particular, we have $\epsilon=|q|^{1/8}$ for $\Delta(96)$ and $\epsilon=|q|^{1/16}$ for $\Delta(384)$. We find a total of 10 triplet pairs that lead to one-parameter modular models based on the $\Delta(96)$ and $\Delta(384)$ finite modular groups. The complete list of triplet pairs and the corresponding hierarchy patterns is collected in Appendix A. These are pairwise physically equivalent, corresponding to each of the 5 spectra in Table 2. Moreover, one sees that the $\Delta(96)$-based H₁ one-parameter modular models are equivalent to the $\Delta(384)$-based H₂ ones, cf. eq. 10. In the case of H₃, the mass matrix vanishes in the symmetric limit and a global $\epsilon$ has been factored out when displaying the corresponding spectrum. Recall that the coefficients in the spectra are fixed by the structure of the modular forms and by the Clebsch–Gordan coefficients of the finite modular group. In short, there are no additional parameters to adjust, and ratios between fermion masses are fully determined in the limit of unbroken SUSY.

Hierarchy label	Leading spectrum $m_{1}:m_{2}:m_{3}$	Finite modular group	$p/q$
H1	$4\sqrt{2}\,\epsilon^{3}:2\,\epsilon:1$	$\Delta(96)$	$3$
H2	$4\sqrt{2}\,\epsilon^{6}:2\,\epsilon^{2}:1$	$\Delta(384)$	$3$
H3	$2\sqrt{2}\,\epsilon^{4}:\sqrt{2}\,\epsilon:1$	$\Delta(384)$	$4$
H4	$8\,\epsilon^{7}:\sqrt{2}\,\epsilon:1$	$\Delta(384)$	$7$
H5	$4\,\epsilon^{5}:2\sqrt{2}\,\epsilon^{3}:1$	$\Delta(384)$	$5/3$

Table 2: Mass hierarchies obtained within one-parameter modular models and the associated flavour groups. Note that H₁ is physically equivalent to H₂.

Curiously, for all the found one-parameter modular models (cf. Table 2), the coefficients $c_{1}$ and $c_{2}$ describing the leading-order spectrum, in the notation $c_{1}\,\epsilon^{p}:c_{2}\,\epsilon^{q}:1$, additionally obey $\sqrt{2}\,c_{1}=c_{2}^{p/q}$, leading to the meta-relation

$$\left(\frac{m_{2}}{m_{3}}\right)^{\frac{p}{q}}\,=\,\sqrt{2}\,\frac{m_{1}}{m_{3}}\,,$$ (11)

which is independent of $\epsilon$ and relates fermion masses within a given sector. It only depends on the ratio $p/q$, which characterizes each hierarchy and is shown explicitly in the last column of Table 2.

In what follows, we present in more detail two representative one-parameter modular models, corresponding to spectra of the type H₂ and H₅. As we will see, these may simultaneously describe, with a common value of $\epsilon$, the charged-lepton and down-quark sectors, respectively. Consider assignments such that

$$k_{e^{c}}+k_{L}=2\,,\qquad{}\,(\rho_{e^{c}},\rho_{L}) \sim(\mathbf{3}_{1},\mathbf{3}_{2})\,,$$ (12)

$$k_{d^{c}}+k_{Q}=2\,,\qquad{}(\rho_{d^{c}},\rho_{Q}) \sim(\mathbf{3}_{7},\mathbf{3}_{8})\,,$$

where $Q$ and $L$ denote the quark and lepton doublet superfields, while $d^{c}$ and $e^{c}$ refer to the down-quark and charged-lepton singlet superfields. Here, the triplet irreducible representations can be uniquely identified by:

$$\frac{1}{2\pi i}\log(\rho_{\mathbf{r}})(T)\,\,\,\,\text{mod}\,\mathbb{Z}\,=\,\begin{cases}\diag\left(\frac{1}{4},\,\frac{1}{2},\,\frac{3}{4}\right)\,,&\text{for }\mathbf{r}=\mathbf{3}_{1}\,,\\ \diag\left(\frac{3}{8},\,\frac{3}{4},\,\frac{7}{8}\right)\,,&\text{for }\mathbf{r}=\mathbf{3}_{2}\,,\\ \diag\left(\frac{1}{16},\,\frac{3}{8},\,\frac{9}{16}\right)\,,&\text{for }\mathbf{r}=\mathbf{3}_{7}\,,\\ \diag\left(\frac{1}{8},\,\frac{5}{8},\,\frac{3}{4}\right)\,,&\text{for }\mathbf{r}=\mathbf{3}_{8}\,.\end{cases}$$ (13)

To check that one obtains a potentially viable one-parameter modular model, one must i) verify that only one contraction is possible at the selected modular weight, and ii) confirm that the resulting Yukawa matrix has a determinant which is not constantly zero. This is the case for the two models presented here, for which the triplet contractions read:

$$\mathbf{3}_{1}\otimes\mathbf{3}_{2}\,=\,\mathbf{3}_{3}\oplus\mathbf{6}_{0}=\overline{\mathbf{3}}_{2}\oplus\overline{\mathbf{6}}_{0}\,,\qquad\quad\mathbf{3}_{7}\otimes\mathbf{3}_{8}\,=\,\mathbf{3}_{5}\oplus\mathbf{6}_{4}=\overline{\mathbf{3}}_{4}\oplus\overline{\mathbf{6}}_{4}\,,$$ (14)

where a bar indicates the conjugate irreducible representation, $\mathbf{r}\otimes\overline{\mathbf{r}}\supset\mathbf{1}$, showing explicitly which representation is required to obtain an invariant. An inspection of eq. 9 indicates that only one contraction (that with a sextet form) is possible for each product. The modular form multiplets of interest, $Y^{(2)}_{\mathbf{6}_{0}}$ and $Y^{(2)}_{\mathbf{6}_{4}}$, can be expanded as

$$Y^{(2)}_{\mathbf{6}_{0}}(\tau)=\begin{pmatrix}-8\,q^{3/8}(1+3\,q+\dots)\\ 2\,q^{1/8}(1+13\,q+\dots)\\ 4\,q^{1/2}(-1+4\,q+\dots)\\ 1-8\,q^{2}+\dots\\ -4\,q^{5/8}(3+7\,q+\dots)\\ -16\,q^{7/8}(1+3\,q+\dots)\end{pmatrix}\,\, \sim\,\,\begin{pmatrix}-8\,\epsilon^{6}\\ 2\,\epsilon^{2}\\ -4\,\epsilon^{8}\\ 1\\ -12\,\epsilon^{10}\\ -16\,\epsilon^{14}\end{pmatrix}\,+\,\mathcal{O}(\epsilon^{18})\,,$$ (15)

$$Y^{(2)}_{\mathbf{6}_{4}}(\tau)=\begin{pmatrix}2\sqrt{2}\,q^{11/16}(5+10\,q+\dots)\\ 2\sqrt{2}\,q^{3/16}(1+9\,q+\dots)\\ -2\,q^{1/2}(1-6\,q+\dots)\\ -1+6\,q+\dots\\ 4\sqrt{2}\,q^{5/16}(1+4\,q+\dots)\\ -4\sqrt{2}\,q^{13/16}(3+7\,q+\dots)\end{pmatrix}\,\, \sim\,\,\begin{pmatrix}10\sqrt{2}\,\epsilon^{11}\\ 2\sqrt{2}\,\epsilon^{3}\\ -2\,\epsilon^{8}\\ -1\\ 4\sqrt{2}\,\epsilon^{5}\\ -12\sqrt{2}\,\epsilon^{13}\end{pmatrix}\,+\,\mathcal{O}(\epsilon^{16})\,,$$ (16)

with $q=e^{2\pi i\tau}$ and $\epsilon=|q|^{1/16}$. Analytical expressions for these forms as well as more complete $q$-expansions are provided in Appendix A. Already at this stage one sees that the approximate $\mathbb{Z}_{N}^{T}$ symmetry strongly suppresses corrections to leading order results, which arise at rather high orders in $\epsilon$.

The modular-invariant Yukawa terms are simply

$$\mathcal{W}_{e}\,=\,\alpha_{e}\left(Y^{(2)}_{\mathbf{6}_{0}}(\tau)\,e^{c}L\right)_{\mathbf{1}}H_{d}\,,\qquad\mathcal{W}_{d}\,=\,\alpha_{d}\left(Y^{(2)}_{\mathbf{6}_{4}}(\tau)\,d^{c}Q\right)_{\mathbf{1}}H_{d}\,,$$ (17)

where $\alpha_{d,e}$ are overall factors that, together with $v_{d}=\langle H_{d}^{0}\rangle$, will set the corresponding mass scales. Indeed, after electroweak symmetry breaking, the charged-lepton and down-quark mass matrices explicitly read

$$M_{e}(\tau) \,=\,\alpha_{e}\,v_{d}\begin{pmatrix}0&\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{0},5}&-\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{0},2}\\ \sqrt{2}\,Y^{(2)}_{\mathbf{6}_{0},4}&Y^{(2)}_{\mathbf{6}_{0},6}&Y^{(2)}_{\mathbf{6}_{0},1}\\ -\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{0},3}&-Y^{(2)}_{\mathbf{6}_{0},1}&-Y^{(2)}_{\mathbf{6}_{0},6}\end{pmatrix}\,\sim\,\begin{pmatrix}0&-12\sqrt{2}\,\epsilon^{10}&-2\sqrt{2}\,\epsilon^{2}\\ \sqrt{2}&-16\,\epsilon^{14}&-8\,\epsilon^{6}\\ 4\sqrt{2}\,\epsilon^{8}&8\,\epsilon^{6}&16\,\epsilon^{14}\end{pmatrix}\,+\,\mathcal{O}(\epsilon^{18})\,,$$ (18)

$$M_{d}(\tau) \,=\,\alpha_{d}\,v_{d}\begin{pmatrix}-\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{4},3}&0&\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{4},4}\\ Y^{(2)}_{\mathbf{6}_{4},5}&-\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{4},1}&-Y^{(2)}_{\mathbf{6}_{4},6}\\ -Y^{(2)}_{\mathbf{6}_{4},6}&-\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{4},2}&Y^{(2)}_{\mathbf{6}_{4},5}\end{pmatrix}\,\sim\,\begin{pmatrix}2\sqrt{2}\,\epsilon^{8}&0&-\sqrt{2}\\ 4\sqrt{2}\,\epsilon^{5}&-20\,\epsilon^{11}&12\sqrt{2}\,\epsilon^{13}\\ 12\sqrt{2}\,\epsilon^{13}&-4\,\epsilon^{3}&4\sqrt{2}\,\epsilon^{5}\end{pmatrix}\,+\,\mathcal{O}(\epsilon^{16})\,,$$ (19)

in a right-left convention. Both determinants are non-vanishing and seen to be $\mathcal{O}(\epsilon^{8})$. As anticipated, taking into account the precise structure of the mass matrices, one obtains

$$m_{e}:m_{\mu}:m_{\tau} =\;4\sqrt{2}\,\epsilon^{6}:2\,\epsilon^{2}:1\,,\quad\qquad(\text{H}_{2})$$ (20)

$$m_{d}:m_{s}:m_{b} =\;4\,\epsilon^{5}:2\sqrt{2}\,\epsilon^{3}:1\,,\quad\qquad(\text{H}_{5})$$

at leading order. Corrections to the spectra, including a possible dependence on $\re\tau$, emerge only at $\mathcal{O}(\epsilon^{12})$ or higher. For a given value of $\epsilon=e^{-\pi\im\tau/8}$, the mass hierarchies are fixed exactly by the modular structure, i.e. without invoking additional $\mathcal{O}(1)$ flavour coefficients beyond the single, overall parameters $\alpha_{d,e}$.

The charged-lepton and down-quark mass matrices are controlled by only three continuous parameters: two overall normalisations, which can be fixed e.g. by $m_{\tau}$ and $m_{b}$, and the common expansion parameter $\epsilon$. Therefore, the resulting hierarchies lead to three exact mass relations at the flavour-breaking scale $M_{F}\gg\Lambda_{\rm EW}$. It is convenient to start from the two intra-sector relations, which are found separately, within each sector:

$$R_{q}:\quad 1 =\frac{1}{2\sqrt{2}}\,\frac{m_{s}^{5}}{m_{b}^{2}\,m_{d}^{3}}\,,$$ (21)

$$R_{\ell}:\quad 1 =\frac{1}{\sqrt{2}}\,\frac{m_{\mu}^{3}}{m_{\tau}^{2}\,m_{e}}\,.$$ (22)

These relations can be rewritten as $m_{s}^{5}=2\sqrt{2}\,m_{d}^{3}m_{b}^{2}$ and $m_{\mu}^{3}=\sqrt{2}\,m_{e}m_{\tau}^{2}$, and follow directly from eq. 20. One may additionally form inter-sector relations. Indeed, the fact that the same $\epsilon$ controls both sectors implies

$$R_{\ell q}:\quad 1 =\frac{1}{\sqrt{2}}\,\frac{m_{s}^{2}}{m_{b}^{2}}\,\frac{m_{\tau}}{m_{e}}\,,$$ (23)

that can be rewritten as $m_{s}^{2}m_{\tau}=\sqrt{2}\,m_{e}m_{b}^{2}$, while other such relations can be obtained algebraically. In particular, combining eqs. 22 and 21 yields a relation involving all six masses,

$$\frac{m_{e}\,m_{\tau}^{2}}{m_{\mu}^{3}}\;=\;2\,\frac{m_{d}^{3}\,m_{b}^{2}}{m_{s}^{5}}\,,$$ (24)

and one can also find

$$\frac{m_{\mu}}{m_{\tau}}\;=\;\sqrt{2}\,\frac{m_{d}}{m_{s}}\,.$$ (25)

IV Running and threshold corrections

The mass relations derived above are exact at the flavour scale $M_{F}$, where the modular symmetry is imposed in the supersymmetric theory. At this scale, the Yukawa structures are fixed in terms of a single parameter $\epsilon$. To compare these predictions with low-energy data, one must account for the renormalization group evolution between $M_{F}$ and the electroweak scale. We take $M_{F}=2\times 10^{16}\,\mathrm{GeV}$ and consider a supersymmetric completion in which the renormalization group evolution is governed by MSSM beta functions above the intermediate scale $M_{\rm SUSY}$, and by SM beta functions below it. We assume that possible direct SUSY-breaking corrections to the high-scale modular Yukawa operators are negligible Criado:2018thu, so that the one-parameter modular model relations define the boundary conditions for the dimensionless Yukawa couplings. We do not specify the mediation mechanism of SUSY breaking. Instead, we first isolate the effect of renormalization group running, and later include finite SUSY threshold corrections at $M_{\rm SUSY}$, treating the soft spectrum phenomenologically.

IV.1 Running without finite threshold corrections

We first neglect finite SUSY threshold corrections. This provides a clean diagnostic of which part of the flavour pattern is already captured by the one-parameter structure before invoking matching effects.

We begin with a purely pedagogical exercise. Starting from the experimentally extracted Yukawa ratios at $M_{Z}$, we evolve them upwards to $M_{F}=2\times 10^{16}\,\mathrm{GeV}$, using SM renormalization group equations below $M_{\rm SUSY}$ and MSSM renormalization group equations above $M_{\rm SUSY}$. We use REAP Antusch:2005gp and SUSYTC Antusch:2015nwi; Antusch:2020ztu to compute the running. The result is shown in Figure 1 for different choices of $M_{\rm SUSY}$ and $\tan\beta$. The running of the ratios is clearly sector-dependent: the charged-lepton ratios are almost scale-independent in the SM regime, but can receive visible MSSM running above $M_{\rm SUSY}$, especially at large $\tan\beta$; the down-quark ratios also show a significant dependence on both $M_{\rm SUSY}$ and $\tan\beta$. Thus the comparison between the high-scale modular relations and low-energy data is not SUSY-scale-independent. The parameters $M_{\rm SUSY}$ and $\tan\beta$ affect the image of the low-energy data at $M_{F}$, and hence the value of $\epsilon$ selected by the observed hierarchies.

(a)

(b)

(c)

(d)

Figure 1: renormalization group evolution of the experimentally extracted Yukawa ratios from $M_{Z}$ to $M_{F}=2\times 10^{16}\,\mathrm{GeV}$, for different choices of $M_{\rm SUSY}$ and $\tan\beta$. The evolution is performed with SM renormalization group equations below $M_{\rm SUSY}$ and MSSM renormalization group equations above $M_{\rm SUSY}$, without finite SUSY threshold corrections.

We then run the mass relations of eqs. 21, 22 and 23 from $M_{Z}$ to $M_{F}$, for a particular choice of $M_{\rm SUSY}=10^{9}$ GeV and $\tan\beta=50$. The result is shown in Figure 2. In this plot, no model prediction is imposed yet as a boundary condition at $M_{F}$. Instead, it asks whether the low-energy data, when lifted to high scales, can approach the relations predicted at the flavour scale. The result already captures the central pattern. The down-sector relation of eq. 21 is compatible with unity within the propagated uncertainty over a broad range of SUSY scales and values of $\tan\beta$, while the inter-sector relation of eq. 23 is also close to unity for $M_{\rm SUSY}\sim 10^{9}$ GeV and high $\tan\beta$. By contrast, the charged-lepton relation of eq. 22 remains far from unity and is not rescued by running. This identifies the muon mass prediction as the main discrepancy: the model tends to predict a muon mass around $20\%$ higher than the experimentally measured one.

Figure 2: Running of the mass relations in eqs. 21, 23 and 22. Recall that the one-parameter modular model predicts $R_{q}=R_{\ell q}=R_{\ell}=1$ at the flavour scale $M_{F}$. The experimental Yukawa ratios and their uncertainties (shaded bands) are extracted at $M_{Z}$ and evolved upwards to $M_{F}=2\times 10^{16}\,{\rm GeV}$, using SM renormalization group equations below $M_{\rm SUSY}$ and MSSM renormalization group equations above $M_{\rm SUSY}$, without finite threshold corrections. The plot is shown for a representative choice of $M_{\rm SUSY}$ and $\tan\beta$. The down-sector and inter-sector relations are close to unity within the running uncertainties, while the charged-lepton relation, controlled by the muon-to-tau hierarchy, remains clearly displaced.

We now impose the high-scale one-parameter predictions explicitly. At $M_{F}$, the relevant ratios are

$$\left.\frac{y_{d}}{y_{b}}\right|_{M_{F}}=4\,\epsilon^{5}\,,\qquad\left.\frac{y_{s}}{y_{b}}\right|_{M_{F}}=2\sqrt{2}\,\epsilon^{3}\,,$$ (26)

$$\left.\frac{y_{e}}{y_{\tau}}\right|_{M_{F}}=4\sqrt{2}\,\epsilon^{6}\,,\qquad\left.\frac{y_{\mu}}{y_{\tau}}\right|_{M_{F}}=2\,\epsilon^{2}\,.$$ (27)

For each pair $(\epsilon,M_{\rm SUSY})$, we impose these relations at $M_{F}$, run the Yukawa matrices down to $M_{Z}$, and compare with the experimental values at $M_{Z}$. The behaviour of the mass relations in Figure 2 motivates a first fit in which the muon ratio is not included. We therefore fit only

$$\frac{y_{d}}{y_{b}}\,,\qquad\frac{y_{s}}{y_{b}}\,,\qquad\frac{y_{e}}{y_{\tau}}\,,$$ (28)

and leave $y_{\mu}/y_{\tau}$ as an independent diagnosis of the threshold corrections that will be needed. Since charged-lepton SUSY thresholds are $\tan\beta$-enhanced, we use $\tan\beta=50$ as a benchmark for the threshold analysis. The best-fit point is

$$\epsilon_{\rm best}=0.1892\,,\qquad M_{\rm SUSY}^{\rm best}\simeq 5.8\times 10^{9}~\mathrm{GeV}\,.$$ (29)

The result of the fit is shown in Figure 3. With only two continuous parameters, $\epsilon$ and $M_{\rm SUSY}$, the model brings the two down-quark ratios and $y_{e}/y_{\tau}$ within their corresponding $1\sigma$ experimental regions. This is already a non-trivial alignment between the down-quark and charged-lepton sectors. The same point, however, predicts $y_{\mu}/y_{\tau}$ significantly above the experimental value. Thus the no-threshold analysis isolates a single problem: the model captures the down-quark hierarchy and the electron-to-tau hierarchy, while the muon-to-tau ratio requires an additional effect that may arise from matching.

(a)

(b)

Figure 3: Low-energy predictions of the one-parameter modular model before finite SUSY threshold corrections. The curves show the image at $M_{Z}$ of the high-scale relations as $\epsilon$ and $M_{\rm SUSY}$ are varied. The best-fit point is obtained by fitting $y_{d}/y_{b}$, $y_{s}/y_{b}$, and $y_{e}/y_{\tau}$, leaving $y_{\mu}/y_{\tau}$ as a prediction. The result is $\epsilon\simeq 0.189$ and $M_{\rm SUSY}\simeq 10^{9.77}$ GeV. The fitted ratios are brought into agreement with data, while the muon-to-tau ratio remains significantly displaced, identifying the required threshold correction.

IV.2 Targeted charged-lepton threshold correction

The no-threshold result shows that we do not need arbitrary threshold effects to rescue the full mass pattern: the double one-parameter modular model structure already captures the two down-quark ratios and the electron-to-tau ratio with a common $\epsilon$. The remaining discrepancy is isolated in the muon-to-tau ratio. In the following we adopt a bottom-up treatment of the soft supersymmetry-breaking sector. The soft terms are taken to be generic matching parameters at $M_{\rm SUSY}$, and are not assumed to follow from a modular-covariant supersymmetry-breaking sector. They therefore act as additional sources of modular-flavour breaking. A more constrained possibility would be to impose modular covariance also on the soft terms, see e.g. Kikuchi:2022pkd; Ding:2022nzn. We leave this direction for future work.

We parametrize the finite matching correction to charged-lepton Yukawa ratios at $M_{\rm SUSY}$ as

$$\left.\frac{y_{\ell_{i}}}{y_{\tau}}\right|_{\mathrm{SM}}=\kappa_{i}\left.\frac{y_{\ell_{i}}}{y_{\tau}}\right|_{\mathrm{MSSM}}\,,\qquad\kappa_{i}\equiv\frac{1+\Delta_{i}}{1+\Delta_{\tau}}\qquad\quad(i=e,\mu)\,.$$ (30)

The best-fit point described above requires

$$\kappa_{e}=1\,,\qquad\kappa_{\mu}=0.764\,,$$ (31)

up to the quoted numerical precision. Thus the required correction preserves $y_{e}/y_{\tau}$ and modifies only $y_{\mu}/y_{\tau}$ relative to $y_{\tau}$. In the quark sector we assume flavour-universal threshold corrections, so that the down-quark ratios $y_{d}/y_{b}$ and $y_{s}/y_{b}$ are also preserved.

We now show that a correction of this form can be generated by standard one-loop electroweak SUSY threshold corrections. Following Ref. Antusch:2008tf, and working in the large-$\tan\beta$ regime while neglecting trilinear terms, the charged-lepton correction can be written as

$$\Delta_{i}=\frac{\tan\beta}{16\pi^{2}}\left[\tau_{i}^{W}+\tau_{i}^{B}\right]\,,$$ (32)

with

$$\tau_{i}^{W}=\frac{3}{2}g_{2}^{2}\frac{M_{2}}{\mu_{H}}h_{2}\!\left(\frac{M_{2}^{2}}{\mu_{H}^{2}},\frac{m_{L_{i}}^{2}}{\mu_{H}^{2}}\right)\,,$$ (33)

and

$$\tau_{i}^{B}=\frac{3}{5}g_{1}^{2}\bigg[-\frac{\mu_{H}}{M_{1}}h_{2}\!\left(\frac{m_{e_{i}}^{2}}{M_{1}^{2}},\frac{m_{L_{i}}^{2}}{M_{1}^{2}}\right)+\frac{M_{1}}{\mu_{H}}h_{2}\!\left(\frac{m_{e_{i}}^{2}}{\mu_{H}^{2}},\frac{M_{1}^{2}}{\mu_{H}^{2}}\right)-\frac{1}{2}\frac{M_{1}}{\mu_{H}}h_{2}\!\left(\frac{M_{1}^{2}}{\mu_{H}^{2}},\frac{m_{L_{i}}^{2}}{\mu_{H}^{2}}\right)\bigg]\,.$$ (34)

Here $M_{1}$ and $M_{2}$ are the bino and wino masses, $\mu_{H}$ is the higgsino mass parameter, and $m_{L_{i}}$, $m_{e_{i}}$ are the left- and right-handed charged-slepton soft masses. The loop function is

$$h_{2}(x,y)\equiv\frac{x\log x}{(1-x)(x-y)}+\frac{y\log y}{(1-y)(y-x)}\,,$$ (35)

with the degenerate limits understood by continuity.

For our present purpose it is sufficient to consider a simple analytic ansatz. We take

$$M_{1}=M_{2}=\mu_{H}\equiv M\,,$$ (36)

and

$$m_{e_{1}}=m_{L_{1}}=m_{e_{3}}=m_{L_{3}}\equiv x_{e\tau}M\,,\qquad m_{e_{2}}=m_{L_{2}}\equiv x_{\mu}M\,.$$ (37)

This ansatz makes the first and third charged-lepton generations identical from the point of view of threshold corrections. Therefore $\Delta_{e}=\Delta_{\tau}$ and $\kappa_{e}=1$ as required. The muon correction is instead controlled by the single ratio $x_{\mu}$, while the common electron–tau threshold is controlled by $x_{e\tau}$.

Under eqs. 36 and 37, the correction becomes a function only of the dimensionless ratio $x_{i}=m_{i}/M$,

$$\Delta(x_{i})\,=\,\frac{\tan\beta}{16\pi^{2}}\bigg\{\frac{3}{2}g_{2}^{2}h_{2}(1,x_{i}^{2})+\frac{3}{5}g_{1}^{2}\left[-h_{2}(x_{i}^{2},x_{i}^{2})+h_{2}(x_{i}^{2},1)-\frac{1}{2}h_{2}(1,x_{i}^{2})\right]\bigg\}\,.$$ (38)

The relevant matching factors are then

$$\kappa_{e}=1\,,\qquad\kappa_{\mu}=\frac{1+\Delta(x_{\mu})}{1+\Delta(x_{e\tau})}\,.$$ (39)

Using the gauge couplings evaluated at the best-fit matching scale,

$$g_{1}(M_{\rm SUSY}^{\rm best})=0.515\,,\qquad g_{2}(M_{\rm SUSY}^{\rm best})=0.5703\,,\qquad\tan\beta=50\,,$$ (40)

we find that the simple choice

$$x_{\mu}=0.5\,,\qquad x_{e\tau}=0.012137\,,$$ (41)

gives

$$\frac{1+\Delta_{\mu}}{1+\Delta_{\tau}}\simeq 0.764\,,\qquad\frac{1+\Delta_{e}}{1+\Delta_{\tau}}=1\,.$$ (42)

This benchmark should be viewed as an explicit proof of existence. A fully specified split spectrum would require the corresponding multi-scale matching treatment, which lies beyond the scope of the present work.

In the down sector we take the squark soft masses to be flavour universal, $m_{Q_{i}}=m_{Q}$ and $m_{d_{i}}=m_{d}$. The corresponding finite threshold corrections are then generation independent, $\Delta_{d}=\Delta_{s}=\Delta_{b}$, so that they cancel in the ratios $y_{d}/y_{b}$ and $y_{s}/y_{b}$. The down-quark ratios are protected by construction, while the only non-universal correction relevant for our purpose is the muonic one in eq. 42. This provides a proof of existence: a selective muon threshold correction of the required size can be generated at one loop, while preserving the successful electron-to-tau and down-quark ratios.

The complete running of the flavour prediction from $M_{F}$ to $M_{Z}$ including this effective threshold correction is shown in Figure 4. At $M_{F}$, the four ratios are fixed by the one-parameter modular relations. They are then evolved down to $M_{\rm SUSY}^{\rm best}$, where the matching factor in eq. 42 is applied to the muon-to-tau ratio. After subsequent SM running down to $M_{Z}$, all four ratios lie in the corresponding experimental bands. The need for sizeable threshold corrections shifts the prediction of the model from a sharp flavour prediction to a constraint on the SUSY spectrum.

Figure 4: renormalization group evolution of the rescaled Yukawa ratios from $M_{F}=2\times 10^{16}\,\mathrm{GeV}$ to $M_{Z}$ at $\epsilon_{\rm best}$ and $M_{\rm SUSY}^{\rm best}$. The green bands denote the experimental $1\sigma$ intervals at $M_{Z}$. The high-scale relations for $y_{d}/y_{b}$, $y_{s}/y_{b}$, and $y_{e}/y_{\tau}$ evolve into the corresponding experimental bands, while the muon ratio requires a selective threshold correction. The displayed benchmark applies $(1+\Delta_{\mu})/(1+\Delta_{\tau})=0.764$ at $M_{\rm SUSY}^{\rm best}$.

V Hints towards a solution of the quark flavour puzzle

The results of the previous sections show that one-parameter modular models can lead to viable charged-fermion mass relations once renormalization-group evolution and selective threshold corrections are taken into account. We now take a broader perspective and ask whether the same framework may contain hints towards a more complete solution of the quark flavour puzzle.

A first important point is that the one-parameter predictions are genuinely restrictive. At the flavour scale $M_{F}$, each one-parameter modular model hierarchy defines a one-dimensional curve in the hierarchy plane

$$\left(\frac{m_{1}}{m_{3}},\frac{m_{2}}{m_{3}}\right).$$

The comparison with data is not completely direct, since the flavour relations are imposed at $M_{F}$, whereas the charged-fermion masses are measured at low energies. In a supersymmetric setup, this introduces several additional ingredients: the flavour scale $M_{F}$, the SUSY scale $M_{\rm SUSY}$, $\tan\beta$, and possible finite threshold corrections at $M_{\rm SUSY}$. One may therefore worry that these effects could wash out the one-parameter predictions.

Figure 5: High-scale hierarchy plane illustrating the non-triviality of the one-parameter predictions and the possible up-quark hint. The coloured regions are obtained by lifting the measured low-energy up-quark, down-quark and charged-lepton hierarchies taken from ParticleDataGroup:2024cfk to the high-scale plane, scanning over $M_{F}$, $M_{\rm SUSY}$, and $\tan\beta$, and applying threshold-like deformations to the individual Yukawa eigenvalues. We use $\Delta_{i}\in[-0.15,0.15]$ for the down-quark and charged-lepton sectors, and $\Delta_{i}^{u}\in[-0.05,0.05]$ for the up-quark sector. Even with this enlarged freedom, the allowed regions remain small compared with the full hierarchy plane. The known one-parameter modular model curves show that H₅ naturally matches the down-quark hierarchy, while H₂ lies close to the charged-lepton region once sizeable muon-threshold effects are allowed. The hypothetical H₆ curve, with $p/q=2$, passes close to the up-quark region for the same order of $\epsilon$, namely $\epsilon\sim\theta_{C}$.

Figure 5 shows that this is not the case. Instead of running each model curve down to $M_{Z}$, we lift the measured low-energy hierarchies up to the high-scale hierarchy plane. For each charged-fermion sector ($f=u,d,e$) we start from

$$\left(\frac{m_{1}^{f}}{m_{3}^{f}},\frac{m_{2}^{f}}{m_{3}^{f}}\right)_{M_{Z}}\,,$$

and evolve the ratios upward, scanning over representative values of $M_{F}$, $M_{\rm SUSY}$, and $\tan\beta$. We also include a deliberately aggressive threshold broadening. For the down-quark and charged-lepton sectors, we allow

$$\Delta_{i}^{e},\,\Delta_{i}^{d}\,\in\,[-0.15,0.15]\,,$$

while for the up-quark sector we use the smaller range

$$\Delta_{i}^{u}\,\in\,[-0.05,0.05]\,,$$

since these corrections are not $\tan\beta$-enhanced. At the level of ratios, these deformations act as

$$\frac{m_{1}}{m_{3}}\,\longrightarrow\,\frac{m_{1}}{m_{3}}\frac{1+\Delta_{3}}{1+\Delta_{1}}\,,\qquad\frac{m_{2}}{m_{3}}\,\longrightarrow\,\frac{m_{2}}{m_{3}}\frac{1+\Delta_{3}}{1+\Delta_{2}}\,.$$

This prescription is not meant to represent a detailed threshold calculation, but as a stress test: we intentionally enlarge the regions selected by the data in order to see whether a generic one-parameter curve could be made viable by running and threshold freedom alone.

The allowed regions remain small even after this deliberately generous broadening. In other words: running effects, the choice of $M_{F}$, $M_{\rm SUSY}$, $\tan\beta$, and sizeable threshold-like deformations do not turn the hierarchy plane into a free fit. Most one-parameter curves still miss the data. The cases that do work therefore represent non-trivial alignments between the modular prediction and the charged-fermion hierarchies.

The hierarchy patterns found in Section III provide a further clue. As discussed around eq. 11, the one-parameter modular model hierarchies obtained so far are not arbitrary curves in the hierarchy plane, but follow a common meta-relation given by eq. 11. In the hierarchy plane, the ratio $p/q$ controls the slope of the corresponding one-parameter curve. The down-quark hierarchy is naturally matched by H₅, while the charged-lepton hierarchy is close to H₂ once a muon threshold correction is included. Interestingly, the up-quark hierarchy lies close to the direction selected by

$$\frac{p}{q}=2\,.$$

This motivates the following hypothetical hierarchy, obtained by taking $p/q=2$ and imposing the relation in eq. 11,

$$\text{H}_{6}:\qquad m_{u}:m_{c}:m_{t}=4\sqrt{2}\,\epsilon^{8}:2\sqrt{2}\,\epsilon^{4}:1\,.$$ (43)

This pattern is not one of the one-parameter modular model hierarchies explicitly constructed in Section III. Nevertheless, it is suggestive for two reasons. First, it follows the simple (integer) $p/q=2$ scaling selected by the up-quark hierarchy. Second, it intersects the up-quark region for the same value of $\epsilon$ as the one selected by the charged-lepton and down-quark sectors. Numerically, this common value is of the order of the Cabibbo angle,

$$\epsilon\sim\theta_{C}\,.$$

Taken together, these observations suggest a possible route towards the quark flavour puzzle. One may imagine a modular construction in which the down-quark sector is controlled by the H₅ hierarchy, the up-quark sector by an H₆-type hierarchy, and the CKM structure arises from mixing effects controlled by the same small parameter $\epsilon\sim\theta_{C}$. In such a scenario, the observed quark mass hierarchies and the size of quark mixing would have a common modular origin. In addition to the down-quark mass relation of eq. 21 we would have

$$m_{c}^{2}=\sqrt{2}m_{u}\,m_{t}\,,\qquad\frac{m_{s}^{4}}{m_{b}^{4}}=2\sqrt{2}\,\frac{m_{c}^{3}}{m_{t}^{3}}\,.$$ (44)

Moreover, if the charged leptons are also assigned to the H₂ hierarchy, the $5$ mass relations of eqs. 21, 22, 23 and 44 would be satisfied at the same flavour scale. Alternatively, by algebraically combining those we could build a mass relation involving all the $9$ charged fermions,

$$\frac{1}{2\sqrt{2}}\left(\frac{m_{e}m_{\mu}}{m_{\tau}^{2}}\right)\left(\frac{m_{d}m_{s}}{m_{b}^{2}}\right)\left(\frac{m_{t}^{3}}{m_{u}m_{c}^{2}}\right)=1\,.$$ (45)

Using the PDG values and $1\sigma$ uncertainties at the $M_{Z}$ scale ParticleDataGroup:2024cfk, we find

$$\frac{1}{2\sqrt{2}}\left.\left(\frac{m_{e}m_{\mu}}{m_{\tau}^{2}}\right)\left(\frac{m_{d}m_{s}}{m_{b}^{2}}\right)\left(\frac{m_{t}^{3}}{m_{u}m_{c}^{2}}\right)\right|_{M_{Z}}=1.05\pm 0.05\,.$$ (46)

However, at present this remains speculative. We have not constructed an explicit one-parameter modular model realizing the H₆ hierarchy, nor have we derived the CKM matrix from a complete pair of up- and down-quark modular mass matrices. Finally, while eq. 11 holds for the models based on the finite modular groups $\Delta(96)$ and $\Delta(384)$, it is not guaranteed to apply to all conceivable one-parameter modular models.

VI Conclusions

In this work we have investigated the extreme predictive limit of one-parameter modular models, in which each charged-fermion mass matrix is generated by a single modular-invariant contraction and is therefore controlled, up to an overall normalization, only by the modulus $\tau$. We formulated the conditions under which this can happen: the matter fields must be assigned to triplets, the modulus $\tau$ must be close to the cusp (large $\im\tau$) and the corresponding mass matrix has rank at most one in the symmetric point, that is hierarchically lifted to 3 in its vicinity. We then performed a systematic search of small finite modular groups that could contain one-parameter modular models.

The possible groups and representations are highly restricted by the one-parameter modular model requirement, before any phenomenological input is imposed. We found $5$ hierarchy patterns (four inequivalent) allowed by the modular versions of the groups $\Delta(96)$ and $\Delta(384)$, summarized in Table 2. one-parameter modular models do not provide a flexible parametrization of fermion masses, but rather a discrete set of predictive curves in the hierarchy plane, see Figure 5.

Within the congruence finite-image framework considered here, $\mathrm{SL}(2,\mathbb{Z})$ is known to have only finitely many irreducible three-dimensional representations. Consequently, in one-parameter modular models, the total number of possible assignments of representations for left-handed and right-handed fermion fields is finite. Moreover, the dimension of vector-valued modular form spaces is also known to be finite Liu:2021gwa. In particular, when constructing one-parameter modular models, we typically have the option to choose only the lowest-weight vector-valued modular form as the Yukawa coupling. Thus, in practice, one-parameter modular models based on $\mathrm{SL}(2,\mathbb{Z})$ modular symmetry are quite limited, yielding only a finite number of distinct prediction sets. As a result, finding a pattern that matches quark or lepton masses among these finitely many one-parameter modular models is highly non-trivial.

As an explicit proof of principle, we have constructed a double one-parameter modular model based on $\Delta(384)$, in which the charged-lepton and down-quark sectors are controlled by a common modulus. Up to corrections of very high-order in $\epsilon=e^{-\pi\im\tau/8}$, the charged-lepton sector realizes the H₂ hierarchy,

$$m_{e}:m_{\mu}:m_{\tau}=4\sqrt{2}\,\epsilon^{6}:2\epsilon^{2}:1\,,$$

while the down-quark sector realizes the H₅ hierarchy,

$$m_{d}:m_{s}:m_{b}=4\epsilon^{5}:2\sqrt{2}\,\epsilon^{3}:1\,.$$

Since the two sectors share the same expansion parameter $\epsilon$, the model predicts three exact high-scale mass relations,

$$m_{s}^{5}=2\sqrt{2}\,m_{d}^{3}m_{b}^{2}\,,\qquad m_{\mu}^{3}=\sqrt{2}\,m_{e}m_{\tau}^{2}\,,\qquad m_{s}^{2}m_{\tau}=\sqrt{2}\,m_{e}m_{b}^{2}\,.$$

These relations are fixed by the modular construction and do not rely on additional order-one flavour coefficients.

We then compared these high-scale relations with low-energy charged-fermion data by including the renormalization group evolution between the flavour and electroweak scales. Running effects alone already reveal a non-trivial alignment: both down-quark hierarchies and the electron-to-tau hierarchy can be brought into agreement with data for a common value of $\epsilon$, while the main remaining discrepancy is isolated in the muon-to-tau ratio. We showed that this discrepancy can be corrected by a selective charged-lepton SUSY threshold effect. In this sense, the model provides a controlled proof of existence: the high-scale one-parameter modular model structure captures most of the charged-fermion hierarchy, while the remaining adjustment can be associated with a specific pattern of threshold effects.

The construction also suggests a possible broader direction. The known one-parameter modular model curves remain sparse in the hierarchy plane even after allowing for running and deliberately generous threshold-like deformations. The agreement of H₅ with the down-quark sector, and the proximity of H₂ to the charged-lepton sector after the muon correction, are therefore non-trivial alignments rather than the result of a dense set of available curves. Moreover, each of the 4 found curves can be characterized by a specific ratio of integers $p/q$, see eq. 11 and Table 2. Interestingly, the observed up-quark hierarchy points towards a potential curve of the same kind, described by $p/q=2$. The corresponding speculative pattern,

$$m_{u}:m_{c}:m_{t}=4\sqrt{2}\,\epsilon^{8}:2\sqrt{2}\,\epsilon^{4}:1\,,$$

would pass close to the up-quark region for $\epsilon$ of the order of the Cabibbo angle. This observation hints at a possible modular origin of both quark mass hierarchies and CKM mixing, controlled by a common small parameter $\epsilon\sim\theta_{C}$. Intriguingly, the mass relation involving all nine charged fermions that stems from this hypothetical one-parameter modular model construction is perfectly satisfied at the $M_{Z}$ scale, see eq. 46.

Several research directions remain open. The most immediate one is to search for an explicit one-parameter modular model realizing the $p/q=2$ up-quark hierarchy and to embed it, together with the down-quark sector, in a complete construction leading to a viable CKM matrix. A second direction is to study the soft supersymmetry-breaking sector in a more top-down way, including modular-covariant soft terms and a systematic treatment of finite threshold corrections, and to understand how the sharp mass relations derived here are modified by non-minimal Kähler effects. Finally, it would be important to investigate whether the one-parameter modular models constructed here can be realized in string theory. Finite modular groups such as $S_{3}$, $T^{\prime}$, and $2D_{3}\subset S^{\prime}_{4}$ have been shown to arise naturally in $T^{2}/\mathbb{Z}_{N}$ heterotic orbifolds Baur:2024qzo. Extending this picture to larger groups such as $\Delta(96)$ and $\Delta(384)$ would provide a compelling ultraviolet origin for the present construction. The fact that lowest-weight vector-valued modular forms often appear in leading trilinear couplings in string constructions is suggestive in this respect. Clarifying these issues will determine whether one-parameter modular models can be promoted from a predictive proof of principle to a complete framework for the charged-fermion flavour puzzle.

Acknowledgements

X. L. and X.-G. L. would like to thank Michael Ratz and Mu-Chun Chen for useful discussions and support. X.-G. L. was also supported by the Universidad Nacional Autónoma de México Postdoctoral Program (POSDOC). S. C. Ch. acknowledges support from the Spanish grants PID2023-147306NB-I00, CNS2024-154524 and CEX2023-001292-S (MICIU/AEI/10.13039/501100011033).

Appendix A vector-valued modular forms and one-parameter modular models

We collect here the complete list of one-parameter modular model triplet pairs found, as described in Section III. Throughout this appendix we consider $k_{\psi^{c}}+k_{\psi}=2$, as well as a trivial and weightless Higgs representation, $\rho_{H}\sim\mathbf{1}$ and $k_{H}=0$. For each one-parameter modular model, the superpotential takes the form

$$\mathcal{W}_{f}=\alpha_{f}\left(Y^{(2)}_{\mathbf{r}_{Y}}(\tau)\,\psi^{c}\psi\right)_{\mathbf{1}}H\,,$$ (47)

where the modular-form representation $\mathbf{r}_{Y}$ is specified below.

The corresponding hierarchical spectra arise in the vicinity of the cusp, with $\epsilon=|q|^{1/N}$, where $N=8$ for $\Delta(96)$ and $N=16$ for $\Delta(384)$. For $\Delta(96)$, the two triplet pairs leading to an one-parameter modular model are shown in Table 3. They both realize the same hierarchy pattern, denoted H₁ in Table 2. For $\Delta(384)$, the eight triplet pairs found in the search are given in Table 4. They come in four pairs, each leading to one of the four leading-order hierarchy patterns $\text{H}_{2},\dots,\text{H}_{5}$, see Table 2. The two $\Delta(384)$ models realizing H₂ reproduce the same hierarchy as the $\Delta(96)$ models realizing H₁. The pairwise degeneracy of the OPMs observed in Table 3 and  Table 4 is a direct manifestation of the outer automorphisms of the finite modular groups. Specifically, there exists a non-trivial outer automorphism that permutes the triplet irreducible representations, such as $(\mathbf{3}_{0},\mathbf{3}_{9})\leftrightarrow(\mathbf{3}_{1},\mathbf{3}_{2})$, while leaving the sextet multiplets $\mathbf{6}_{0}$ invariant. Since the paired models share the exact same sextet vector-valued modular form $Y^{(2)}_{\mathbf{6}_{0}}$ for their Yukawa couplings, this algebraic symmetry ensures that their respective Clebsch–Gordan contractions are structurally isomorphic. Consequently, the resulting mass matrices and the predicted mass hierarchies remain physically equivalent for each pair.

Model	$(\rho_{\psi^{c}},\rho_{\psi})$	$\mathbf{r}_{Y}$	$m_{1}:m_{2}:m_{3}$	Hierarchy
$\Delta(96)$-1	$(\mathbf{3}_{0},\mathbf{3}_{5})$	$\mathbf{6}$	$4\sqrt{2}\,\epsilon^{3}:2\epsilon:1$	H1
$\Delta(96)$-2	$(\mathbf{3}_{1},\mathbf{3}_{2})$	$\mathbf{6}$	$4\sqrt{2}\,\epsilon^{3}:2\epsilon:1$	H1

Table 3: Triplet pairs leading to one-parameter models based on $\Delta(96)$. Here $\epsilon=|q|^{1/8}$.

Model	$(\rho_{\psi^{c}},\rho_{\psi})$	$\mathbf{r}_{Y}$	$m_{1}:m_{2}:m_{3}$	Hierarchy
$\Delta(384)$-1	$(\mathbf{3}_{0},\mathbf{3}_{9})$	$\mathbf{6}_{0}$	$4\sqrt{2}\,\epsilon^{6}:2\epsilon^{2}:1$	H2
$\Delta(384)$-2	$(\mathbf{3}_{1},\mathbf{3}_{2})$	$\mathbf{6}_{0}$	$4\sqrt{2}\,\epsilon^{6}:2\epsilon^{2}:1$	H2
$\Delta(384)$-3	$(\mathbf{3}_{0},\mathbf{3}_{13})$	$\mathbf{6}_{5}$	$2\sqrt{2}\,\epsilon^{4}:\sqrt{2}\epsilon:1$	H3
$\Delta(384)$-4	$(\mathbf{3}_{1},\mathbf{3}_{6})$	$\mathbf{6}_{5}$	$2\sqrt{2}\,\epsilon^{4}:\sqrt{2}\epsilon:1$	H3
$\Delta(384)$-5	$(\mathbf{3}_{2},\mathbf{3}_{10})$	$\mathbf{6}_{3}$	$8\epsilon^{7}:\sqrt{2}\epsilon:1$	H4
$\Delta(384)$-6	$(\mathbf{3}_{5},\mathbf{3}_{9})$	$\mathbf{6}_{3}$	$8\epsilon^{7}:\sqrt{2}\epsilon:1$	H4
$\Delta(384)$-7	$(\mathbf{3}_{3},\mathbf{3}_{12})$	$\mathbf{6}_{4}$	$4\epsilon^{5}:2\sqrt{2}\epsilon^{3}:1$	H5
$\Delta(384)$-8	$(\mathbf{3}_{7},\mathbf{3}_{8})$	$\mathbf{6}_{4}$	$4\epsilon^{5}:2\sqrt{2}\epsilon^{3}:1$	H5

Table 4: Triplet pairs leading to one-parameter models based on $\Delta(384)$. Here $\epsilon=|q|^{1/16}$. For the H₃ entries, a common overall factor of $\epsilon$ has been factored out.

The leading behaviour of the relevant $\Delta(384)$ sextet vector-valued modular forms close to the cusp is

$$Y^{(2)}_{\mathbf{6}_{0}}(\tau)\sim\begin{pmatrix}-8\,\epsilon^{6}\\ 2\,\epsilon^{2}\\ -4\,\epsilon^{8}\\ 1\\ -12\,\epsilon^{10}\\ -16\,\epsilon^{14}\end{pmatrix}\,,\quad Y^{(2)}_{\mathbf{6}_{5}}(\tau)\sim\begin{pmatrix}-\sqrt{2}\,\epsilon^{2}\\ 3\,\epsilon^{9}\\ \epsilon\\ -4\sqrt{2}\,\epsilon^{10}\\ 4\,\epsilon^{5}\\ -4\,\epsilon^{13}\end{pmatrix}\,,\quad Y^{(2)}_{\mathbf{6}_{3}}(\tau)\sim\begin{pmatrix}-8\sqrt{2}\,\epsilon^{7}\\ 8\sqrt{2}\,\epsilon^{15}\\ -6\,\epsilon^{8}\\ 1\\ -7\sqrt{2}\,\epsilon^{9}\\ \sqrt{2}\,\epsilon\end{pmatrix}\,,\quad Y^{(2)}_{\mathbf{6}_{4}}(\tau)\sim\begin{pmatrix}10\sqrt{2}\,\epsilon^{11}\\ 2\sqrt{2}\,\epsilon^{3}\\ -2\,\epsilon^{8}\\ -1\\ 4\sqrt{2}\,\epsilon^{5}\\ -12\sqrt{2}\,\epsilon^{13}\end{pmatrix},$$ (48)

up to ($\re\tau$)-dependent phases and higher-order corrections in $\epsilon$. These leading powers, combined with the Clebsch–Gordan contractions for the triplet pairs in Table 4, up to transposition and permutations and sign flips of rows and columns, result in the Yukawa matrices:

$$\mathcal{Y}_{2}(\tau) \,\propto\,\begin{pmatrix}0&\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{0},5}&-\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{0},2}\\ \sqrt{2}\,Y^{(2)}_{\mathbf{6}_{0},4}&Y^{(2)}_{\mathbf{6}_{0},6}&Y^{(2)}_{\mathbf{6}_{0},1}\\ -\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{0},3}&-Y^{(2)}_{\mathbf{6}_{0},1}&-Y^{(2)}_{\mathbf{6}_{0},6}\end{pmatrix}\,\,\,\sim\,\,\,\begin{pmatrix}0&-12\sqrt{2}\,\epsilon^{10}&-2\sqrt{2}\,\epsilon^{2}\\ \sqrt{2}&-16\,\epsilon^{14}&-8\,\epsilon^{6}\\ 4\sqrt{2}\,\epsilon^{8}&8\,\epsilon^{6}&16\,\epsilon^{14}\end{pmatrix}\,,$$ (49a)

$$\mathcal{Y}_{3}(\tau) \,\propto\,\begin{pmatrix}\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{5},3}&-\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{5},2}&0\\ Y^{(2)}_{\mathbf{6}_{5},5}&-Y^{(2)}_{\mathbf{6}_{5},6}&-\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{5},1}\\ -Y^{(2)}_{\mathbf{6}_{5},6}&Y^{(2)}_{\mathbf{6}_{5},5}&-\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{5},4}\end{pmatrix}\,\,\,\sim\,\,\,\begin{pmatrix}\sqrt{2}\,\epsilon&-3\sqrt{2}\,\epsilon^{9}&0\\ 4\,\epsilon^{5}&4\,\epsilon^{13}&2\,\epsilon^{2}\\ 4\,\epsilon^{13}&4\,\epsilon^{5}&8\,\epsilon^{10}\end{pmatrix}\,,$$ (49b)

$$\mathcal{Y}_{4}(\tau) \,\propto\,\begin{pmatrix}\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{3},5}&\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{3},6}&0\\ Y^{(2)}_{\mathbf{6}_{3},1}&Y^{(2)}_{\mathbf{6}_{3},2}&-\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{3},3}\\ Y^{(2)}_{\mathbf{6}_{3},2}&Y^{(2)}_{\mathbf{6}_{3},1}&\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{3},4}\end{pmatrix}\,\,\,\sim\,\,\,\begin{pmatrix}-14\,\epsilon^{9}&2\,\epsilon&0\\ -8\sqrt{2}\,\epsilon^{7}&8\sqrt{2}\,\epsilon^{15}&6\sqrt{2}\,\epsilon^{8}\\ 8\sqrt{2}\,\epsilon^{15}&-8\sqrt{2}\,\epsilon^{7}&\sqrt{2}\end{pmatrix}\,,$$ (49c)

$$\mathcal{Y}_{5}(\tau) \,\propto\,\begin{pmatrix}-\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{4},3}&0&\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{4},4}\\ Y^{(2)}_{\mathbf{6}_{4},5}&-\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{4},1}&-Y^{(2)}_{\mathbf{6}_{4},6}\\ -Y^{(2)}_{\mathbf{6}_{4},6}&-\sqrt{2}\,Y^{(2)}_{\mathbf{6}_{4},2}&Y^{(2)}_{\mathbf{6}_{4},5}\end{pmatrix}\,\,\,\sim\,\,\,\begin{pmatrix}2\sqrt{2}\,\epsilon^{8}&0&-\sqrt{2}\\ 4\sqrt{2}\,\epsilon^{5}&-20\,\epsilon^{11}&12\sqrt{2}\,\epsilon^{13}\\ 12\sqrt{2}\,\epsilon^{13}&-4\,\epsilon^{3}&4\sqrt{2}\,\epsilon^{5}\end{pmatrix}\,,$$ (49d)

where we have indicated the $\epsilon$ power structure for each matrix, which produce the hierarchy patterns $\text{H}_{2},\dots,\text{H}_{5}$.

In the phenomenological construction discussed in the main text, only the H₂ and H₅ models are used explicitly. We therefore give below the analytical expressions in terms of theta constants and the corresponding $q$-expansions for the two sextet forms entering that construction, $Y^{(2)}_{\mathbf{6}_{0}}$ and $Y^{(2)}_{\mathbf{6}_{4}}$.

The relevant $\Delta(384)$ vector-valued modular forms can be constructed from weight-$1/2$ modular forms on $\Gamma(8)$ and $\Gamma(16)$. In the convention used here, these building blocks are the theta constants

$$\theta_{i}(\tau) \equiv\sum_{n=-\infty}^{\infty}e^{4\pi i\tau(n+\frac{i-1}{4})^{2}}\,,\qquad i=1,\dots,4\,,$$ (50a)

$$\vartheta_{i}(\tau) \equiv\sum_{n=-\infty}^{\infty}e^{8\pi i\tau(n+\frac{i-1}{8})^{2}}\,,\qquad i=1,\dots,8\,.$$ (50b)

In terms of these theta constants, the two sextet modular forms used in the explicit $\text{H}_{2}+\text{H}_{5}$ construction are

$$Y^{(2)}_{\mathbf{6}_{0}} =\left(\begin{array}[]{c}-4\left(\vartheta_{1}+\vartheta_{5}\right)\left(\vartheta_{3}+\vartheta_{7}\right)\left(\vartheta_{4}\vartheta_{6}+\vartheta_{2}\vartheta_{8}\right)\\ \left(\vartheta_{2}^{2}+\vartheta_{4}^{2}+\vartheta_{6}^{2}+\vartheta_{8}^{2}\right)\left(\vartheta_{1}+\vartheta_{5}\right){}^{2}+2\left(\vartheta_{3}+\vartheta_{7}\right){}^{2}\left(\vartheta_{2}\vartheta_{6}+\vartheta_{4}\vartheta_{8}\right)\\ 4\left(\vartheta_{3}\vartheta_{5}-\vartheta_{1}\vartheta_{7}\right)\left(\vartheta_{1}\vartheta_{3}-\vartheta_{5}\vartheta_{7}\right)\\ \left(\vartheta_{1}^{2}-\vartheta_{5}^{2}\right){}^{2}-\left(\vartheta_{3}^{2}-\vartheta_{7}^{2}\right){}^{2}\\ -2\left(\vartheta_{2}\vartheta_{6}+\vartheta_{4}\vartheta_{8}\right)\left(\vartheta_{1}+\vartheta_{5}\right){}^{2}-\left(\vartheta_{3}+\vartheta_{7}\right){}^{2}\left(\vartheta_{2}^{2}+\vartheta_{4}^{2}+\vartheta_{6}^{2}+\vartheta_{8}^{2}\right)\\ -4\left(\vartheta_{1}+\vartheta_{5}\right)\left(\vartheta_{3}+\vartheta_{7}\right)\left(\vartheta_{2}\vartheta_{4}+\vartheta_{6}\vartheta_{8}\right)\end{array}\right)\,,$$

$$Y^{(2)}_{\mathbf{6}_{4}} =\left(\begin{array}[]{c}\sqrt{2}\left(\theta_{1}^{2}\left(\theta_{2}\vartheta_{4}+\theta_{4}\vartheta_{6}\right)+\theta_{3}\left(\theta_{2}+\theta_{4}\right)\theta_{1}\left(\vartheta_{2}+\vartheta_{8}\right)+\theta_{3}^{2}\left(\theta_{4}\vartheta_{4}+\theta_{2}\vartheta_{6}\right)\right)\\ \sqrt{2}\left(\theta_{2}\left(\theta_{1}^{2}\vartheta_{8}+\theta_{3}\theta_{1}\left(\vartheta_{4}+\vartheta_{6}\right)+\theta_{3}^{2}\vartheta_{2}\right)+\theta_{4}\left(\theta_{1}^{2}\vartheta_{2}+\theta_{3}\theta_{1}\left(\vartheta_{4}+\vartheta_{6}\right)+\theta_{3}^{2}\vartheta_{8}\right)\right)\\ \left(\theta_{3}^{3}-\theta_{1}^{2}\theta_{3}\right)\left(\vartheta_{1}-\vartheta_{5}\right)-\theta_{1}\left(\theta_{2}^{2}-\theta_{4}^{2}\right)\left(\vartheta_{3}-\vartheta_{7}\right)\\ \left(\theta_{1}\theta_{3}^{2}-\theta_{1}^{3}\right)\left(\vartheta_{1}-\vartheta_{5}\right)-\theta_{3}\left(\theta_{2}^{2}-\theta_{4}^{2}\right)\left(\vartheta_{3}-\vartheta_{7}\right)\\ \sqrt{2}\left(\theta_{2}+\theta_{4}\right)\left(\theta_{3}\left(\theta_{2}\vartheta_{4}+\theta_{4}\vartheta_{6}\right)+\theta_{1}\left(\theta_{2}\vartheta_{2}+\theta_{4}\vartheta_{8}\right)\right)\\ -\sqrt{2}\left(\theta_{2}+\theta_{4}\right)\left(\theta_{1}\left(\theta_{4}\vartheta_{4}+\theta_{2}\vartheta_{6}\right)+\theta_{3}\left(\theta_{4}\vartheta_{2}+\theta_{2}\vartheta_{8}\right)\right)\end{array}\right)\,.$$

Expanding around the cusp, with $q\equiv e^{2\pi i\tau}$, one obtains

$$Y^{(2)}_{\mathbf{6}_{0}}(\tau)=\begin{pmatrix}-8q^{3/8}(1+3q+5q^{2}+10q^{3}+12q^{4}+11q^{5}+18q^{6}+15q^{7}+17q^{8}+31q^{9}+21q^{10}+\dots)\\ 2q^{1/8}(1+13q+18q^{2}+31q^{3}+48q^{4}+42q^{5}+57q^{6}+80q^{7}+84q^{8}+74q^{9}+121q^{10}+\dots)\\ 4q^{1/2}(-1+4q-6q^{2}+8q^{3}-13q^{4}+12q^{5}-14q^{6}+24q^{7}-18q^{8}+20q^{9}-32q^{10}+\dots)\\ 1-8q^{2}+24q^{4}-32q^{6}+24q^{8}-48q^{10}+\dots\\ -4q^{5/8}(3+7q+16q^{2}+15q^{3}+19q^{4}+39q^{5}+27q^{6}+31q^{7}+48q^{8}+48q^{9}+54q^{10}+\dots)\\ -16q^{7/8}(1+3q+3q^{2}+4q^{3}+7q^{4}+6q^{5}+9q^{6}+13q^{7}+9q^{8}+10q^{9}+15q^{10}+\dots)\end{pmatrix}\,,$$ (52)

and

$$Y^{(2)}_{\mathbf{6}_{4}}(\tau)=\begin{pmatrix}2\sqrt{2}q^{11/16}(5+10q+21q^{2}+29q^{3}+21q^{4}+48q^{5}+53q^{6}+42q^{7}+69q^{8}+64q^{9}+63q^{10}+\dots)\\ 2\sqrt{2}q^{3/16}(1+9q+16q^{2}+18q^{3}+33q^{4}+41q^{5}+35q^{6}+48q^{7}+65q^{8}+57q^{9}+81q^{10}+\dots)\\ -2q^{1/2}(1-6q+12q^{2}-8q^{3}+7q^{4}-30q^{5}+36q^{6}-8q^{7}+18q^{8}-54q^{9}+48q^{10}+\dots)\\ -1+6q-14q^{2}+20q^{3}-30q^{4}+40q^{5}-36q^{6}+48q^{7}-62q^{8}+42q^{9}-72q^{10}+\dots\\ 4\sqrt{2}q^{5/16}(1+4q+9q^{2}+13q^{3}+12q^{4}+18q^{5}+25q^{6}+21q^{7}+36q^{8}+37q^{9}+20q^{10}+\dots)\\ -4\sqrt{2}q^{13/16}(3+7q+7q^{2}+15q^{3}+20q^{4}+16q^{5}+27q^{6}+26q^{7}+24q^{8}+39q^{9}+43q^{10}+\dots)\end{pmatrix}\,.$$ (53)