1 Introduction↩︎

In a series of earlier works, we proposed an algebraic framework for the description of three linearly independent fermion generations based on the complex Clifford algebra \(\mathbb{C}\ell(8)\) [1]–[3]¹, and recently its electroweak extension \(\mathbb{C}\ell(10)\) [8]. In that construction, the discrete group \(S_3\), arising from the automorphism structure of the sedenions [9], is identified with a family symmetry acting on algebraic spinors.

Fermionic states are realised as elements of minimal left ideals in \(\mathbb{C}\ell(8)\). Embedding the discrete family symmetry \(S_3\) into \(\mathbb{C}\ell(8)\) yields three linearly independent but gauge-equivalent fermion sectors transforming under the unbroken gauge group \(SU(3)_C\times U(1)_{em}\). Incorporating the electroweak sector \(SU(2)_L\times U(1)_Y\) requires an extension of the underlying algebra from \(\mathbb{C}\ell(8)\) to \(\mathbb{C}\ell(10)\) [8]. The Standard Model gauge generators act via the adjoint (commutator) action \([\,\cdot\,,\cdot\,]\), and are required to commute with the \(S_3\) family symmetry. This ensures that the gauge sector is not replicated, while the fermion sector is organised into three linearly independent but gauge-equivalent families.

What has remained missing, however, is the Higgs sector. Since Yukawa couplings and flavour structure are central to any realistic phenomenology, developing the Higgs sector is the essential next step if one wants to move from an algebraic description of the gauge and fermion sectors toward flavour phenomenology. In the present paper we address precisely this issue. We construct Higgs components as right-action operators mapping weak-doublet sectors into the appropriate weak-singlet sectors, and use a trace pairing to extract the corresponding Yukawa coefficients. This yields two first-generation Higgs doublets with the correct electroweak quantum numbers, transforming as \((1,2,-1)\) and \((1,2,+1)\), and giving a natural Type-II-like separation between down-type and up-type Yukawa channels.

Applying the order-three \(S_3\) generator \(\psi_3\) to these first-generation Higgs doublets triplicates the Higgs sector, producing six Higgs doublets arranged into \(S_3\) orbits. We show that these Higgs states are linearly independent, and then construct the corresponding \(S_3\)-invariant Higgs doublets by averaging over the \(S_3\)-orbits. This exact \(S_3\)-invariant limit provides the simplest Higgs sector compatible with the family symmetry and a useful starting point for flavour analysis. Although the resulting generation-space Yukawa matrices are not diagonal in the algebraic generation basis, the Yukawa sector retains its Type-II structure, and under the usual implementation of electroweak symmetry breaking tree-level flavour-changing neutral currents are not expected.

This construction also makes contact with the broader phenomenology of multi-Higgs and \(S_3\)-based flavour models. In conventional multi-Higgs theories, the Yukawa sector generically leads to tree-level flavour-changing neutral currents unless special conditions are satisfied [10]–[12]. Likewise, realistic \(S_3\) flavour models typically require a nontrivial Higgs-family structure together with some form of symmetry breaking or vacuum alignment [13]–[16]. The present framework differs conceptually from such models in that the family symmetry is algebraically motivated from the outset, and the Higgs-family structure is inherited from that same algebraic origin.

The exact \(S_3\)-invariant Higgs limit studied here should therefore be viewed as a first step in a broader phenomenological programme. Its main role is to establish the Higgs sector of the algebraic \(\mathbb{C}\ell(10)\) three-generation model and to clarify the flavour structure in the symmetric limit. Future work will study the \(S_3\)-doublet Higgs sector, the scalar potential, and the resulting \(S_3\)-breaking Yukawa textures, with the aim of determining whether realistic masses and mixing patterns can emerge naturally from the same algebraic framework.

A number of algebraic approaches have explored how three fermion generations might arise from deeper mathematical structure. These include triality-based proposals, constructions involving larger Clifford algebras, and approaches based on exceptional Jordan structures [17]–[25]. Both triality-based and algebraic considerations have pointed to \(\mathbb{C}\ell(8)\) as a natural setting for three-generation structure [3], [17], [26], [27].

Some recent three-generation algebraic approaches have also touched on the Higgs sector, but in rather different ways. In the triality-based model of [26], the Yukawa sector is discussed through Cartan factorisation and a proposed non-degenerate trilinear-form constraint. In another recent construction based on \(C\ell(8,0)\), a Higgs doublet appears in the mass terms, although the bosonic sector is yet to be fully understood [27]. By contrast, the aim of the present paper is to develop the Higgs sector explicitly within the algebraic \(\mathbb{C}\ell(10)\) three-generation model itself, and to use it as the starting point for a more detailed study of Yukawa structure and flavour phenomenology.

The structure of the paper is as follows. In Section 2 we summarise the algebraic ingredients of the \(\mathbb{C}\ell(10)\) three-generation construction. Section 3 develops the Higgs sector and the associated trace pairing for Yukawa couplings. Section 4 studies the \(S_3\) triplication of the Higgs sector and the resulting \(S_3\)-invariant Higgs doublets. Section 5 analyses the flavour structure in the exact \(S_3\)-invariant limit and discusses the corresponding Glashow–Weinberg picture. We conclude in Section 6 with a discussion of the main results and an outlook on \(S_3\)-breaking, scalar potentials, and realistic flavour phenomenology.

2 Algebraic Setup↩︎

In this section we summarise the algebraic ingredients of the \(\mathbb{C}\ell(10)\) three-generation construction needed in what follows, and refer the reader to [8] for additional details.

We begin by defining the following generating basis for the complex Clifford algebra \(\mathbb{C}\ell(8)\cong \textrm{Mat}(16,\mathbb{C})\): \[\label{Cl8basis95summary} \begin{align} e_1 &= i \sigma_1 \otimes \sigma_1 \otimes \sigma_1 \otimes \sigma_1, &\qquad e_2 &= i \sigma_1 \otimes \sigma_1 \otimes \sigma_3 \otimes \mathbb{I}_2, \\ e_3 &= -i \sigma_1 \otimes \sigma_1 \otimes \sigma_1 \otimes \sigma_3, &\qquad e_4 &= -i \sigma_1 \otimes \sigma_3 \otimes \mathbb{I}_2 \otimes \mathbb{I}_2, \\ e_5 &= -i \sigma_1 \otimes \sigma_1 \otimes \sigma_1 \otimes \sigma_2, &\qquad e_6 &= -i \sigma_1 \otimes \sigma_2 \otimes \mathbb{I}_2 \otimes \mathbb{I}_2, \\ e_7 &= i \sigma_1 \otimes \sigma_1 \otimes \sigma_2 \otimes \mathbb{I}_2, &\qquad e_8 &= i \sigma_2 \otimes \mathbb{I}_2 \otimes \mathbb{I}_2 \otimes \mathbb{I}_2, \end{align}\tag{1}\] with the \(\mathbb{C}\ell(8)\) pseudoscalar defined as \(\omega_8:=e_1e_2e_3e_4e_5e_6e_7e_8\).

Subsequently, define a Witt basis for the algebra as: \[\begin{align} a_j=\frac{1}{2}(-e_j+i e_{j+4}),\qquad a_j^\dagger=\frac{1}{2}(e_j+i e_{j+4}),\qquad j=1,\ldots,4. \end{align}\] These Witt basis generators satisfy the fermionic oscillator relations \[\begin{align} \{a_i,a_j\}=0,\qquad \{a_i^\dagger,a_j^\dagger\}=0,\qquad \{a_i,a_j^\dagger\}=\delta_{ij}. \end{align}\] There are four mutually commuting simple idempotents, defined as \[\begin{align} \pi_j^{(+)}=a_j a_j^\dagger,\quad \pi_j^{(-)}=a_j^\dagger a_j,\quad \pi_j^{(\pm)}\pi_k^{(\pm)}=\pi_k^{(\pm)}\pi_j^{(\pm)}\quad j=1,\ldots,4. \end{align}\] and from these the primitive idempotents are given by \[\begin{align} f_{\varepsilon_1\varepsilon_2\varepsilon_3\varepsilon_4} = \pi_1^{(\varepsilon_1)}\pi_2^{(\varepsilon_2)}\pi_3^{(\varepsilon_3)}\pi_4^{(\varepsilon_4)}, \qquad \varepsilon_i\in\{+,-\}. \end{align}\] These primitive idempotents are mutually orthogonal and complete: \[f_{\varepsilon} f_{\varepsilon'} = \delta_{\varepsilon_1,\varepsilon'_1}\delta_{\varepsilon_2,\varepsilon'_2}\delta_{\varepsilon_3,\varepsilon'_3}\delta_{\varepsilon_4,\varepsilon'_4} f_{\varepsilon}, \qquad \sum_{\varepsilon\in\{+,-\}^4} f_{\varepsilon} = \mathbb{I}_{16},\qquad f_\varepsilon=f_{\varepsilon_1\varepsilon_2\varepsilon_3\varepsilon_4}.\]

In particular, \[\begin{align} f_{++++}=E_{1,1},\qquad f_{---+}=E_{14,14},\qquad f_{----}=E_{6,6},\qquad f_{+++-}=E_{9,9}. \end{align}\] The primitive idempotents are used to decompose the algebra \(\mathbb{C}\ell(8)\) into its minimal left ideals. In the chosen matrix realisation, with \(f_{++++}=E_{1,1}\), the first minimal left ideal is \[\begin{align} I_1=\mathbb{C}\ell(8)f_{++++} =\textrm{Mat}(16,\mathbb{C})\,E_{1,1} =\mathrm{span}_{\mathbb{C}}\{E_{j,1}\}_{j=1}^{16}, \end{align}\] corresponding to the subspace of matrices with support only in the first column.

The \(S_3\) symmetry, responsible for generating three families of fermions, is generated by the order-three map \(\psi_3\) and the order-two map \(\epsilon\). The action of \(\psi_3\) on the Clifford generators is given by \[\begin{align} \psi_3: e_i&\mapsto& \frac{1}{4}e_i+\frac{\sqrt{3}}{4}g_i-\frac{\sqrt{3}}{4}e_i e_8-\frac{3}{4}g_i e_8,\qquad i=1,\ldots,7,\\ \psi_3: e_8&\mapsto& e_8, \end{align}\] where \[\begin{align} \nonumber g_1 &=& \tfrac{1}{2}e_1e_8(-e_{2345}+e_{2367}+e_{4567}-1) ,\quad g_{2} = \tfrac{1}{2}e_2e_8(e_{1346}+e_{1357}+e_{4567}-1), \\ g_{3} &=& \tfrac{1}{2}e_3e_8(e_{1256}-e_{1247}+e_{4567}-1), \quad g_{4} =\tfrac{1}{2}e_4e_8(e_{1256}+e_{1357}+e_{2367}-1), \\ \nonumber g_{5} &=& \tfrac{1}{2}e_5e_8(-e_{1247}+e_{1346}+e_{2367}-1), \quad g_{6} = \tfrac{1}{2}e_6e_8(-e_{1247}+e_{1357}-e_{2345}-1), \\ \nonumber g_{7} &=& \tfrac{1}{2}e_7e_8(e_{1256}+e_{1346}-e_{2345}-1). \end{align}\]

The order-two generator meanwhile is defined as \[\begin{align} \epsilon:e_i\mapsto -e_i,\qquad i=1,\ldots,8. \end{align}\] Together, \(\psi_3\) and \(\epsilon\) satisfy \(\psi_3^3=\mathrm{Id},\quad \epsilon^2=\mathrm{Id},\quad \epsilon\psi_3=\psi_3^2\epsilon\), and generate \(S_3\).

2.1 \(\mathbb{C}\ell(10)\) extension and \(S_3\) action↩︎

Subsequently, the algebra \(\mathbb{C}\ell(8)\) is enlarged to \(\mathbb{C}\ell(10)\) via the graded tensor product embedding \[\begin{align} \iota:\mathbb{C}\ell(8)\hookrightarrow \mathbb{C}\ell(10),\qquad \iota(e_i)=\mathbb{I}_2\otimes e_i,\qquad i=1,\ldots,8, \end{align}\] with the remaining two basis elements defined by \[\begin{align} e_9:=-i\bar{\sigma}_1\otimes \omega_8,\qquad e_{10}:=i\bar{\sigma}_2\otimes \omega_8, \end{align}\] where \(\bar{\sigma}_i\) are Pauli matrices, and the bar is used to avoid confusion with the Pauli matrices used to define our \(\mathbb{C}\ell(8)\) basis. The fifth Witt pair is then \[\begin{align} a_5=\frac{1}{2}(-e_9+i e_{10}),\qquad a_5^\dagger=\frac{1}{2}(e_9+i e_{10}), \end{align}\] with associated simple idempotents \[\begin{align} \pi_5^{(+)}=a_5 a_5^\dagger,\qquad \pi_5^{(-)}=a_5^\dagger a_5. \end{align}\] Accordingly, the primitive idempotents of \(\mathbb{C}\ell(10)\) are \[\begin{align} f_{\varepsilon_1\varepsilon_2\varepsilon_3\varepsilon_4\varepsilon_5} = \pi_1^{(\varepsilon_1)}\pi_2^{(\varepsilon_2)}\pi_3^{(\varepsilon_3)}\pi_4^{(\varepsilon_4)}\pi_5^{(\varepsilon_5)}. \end{align}\] In particular, \[\begin{align} f_{+++++}=E_{1,1},\qquad f_{++++-}=E_{17,17},\qquad f_{+++-+}=E_{9,9},\qquad f_{+++--}=E_{25,25}. \end{align}\]

The action of the \(S_3\) generators on \(\mathbb{C}\ell(8)\) extends naturally to \(\mathbb{C}\ell(10)\) through the graded tensor product decomposition \[\begin{align} \mathbb{C}\ell(10)\cong \mathbb{C}\ell(2)\,\hat{\otimes}\,\mathbb{C}\ell(8). \end{align}\] The action of \(S_3\) on the \(\mathbb{C}\ell(2)\) factor is taken to be trivial, so in particular \(\psi_3(\bar{\sigma}_i)=\bar{\sigma}_i\) and \(\epsilon(\bar{\sigma}_i)=\bar{\sigma}_i\) for \(i=1,2,3\). In this extension, \(\psi_3\) and \(\epsilon\) act block-diagonally on the two \(\mathbb{C}\ell(8)\) blocks, so that \[\begin{align} \iota:\psi_3\mapsto \psi_3\oplus\psi_3,\qquad \iota:\epsilon\mapsto \epsilon\oplus\epsilon. \end{align}\] Since the additional generators \(e_9\) and \(e_{10}\) are defined using the \(\mathbb{C}\ell(8)\) pseudoscalar \(\omega_8\), their transformation is induced from the action of \(S_3\) on \(\omega_8\). Thus \(\psi_3\) acts nontrivially on \(e_9\) and \(e_{10}\), whereas \(\epsilon(\omega_8)=\omega_8\) and hence fixes the fifth Witt pair: \[\begin{align} \epsilon(a_5)=a_5,\qquad \epsilon(a_5^\dagger)=a_5^\dagger. \end{align}\] It follows that the three Witt bases \[\begin{align} \{a_i,a_i^\dagger\},\qquad \{\psi_3(a_i),\psi_3(a_i^\dagger)\},\qquad \{\psi_3^2(a_i),\psi_3^2(a_i^\dagger)\},\qquad i=1,\ldots,5, \end{align}\] again satisfy the fermionic anticommutation relations. In this way the \(S_3\) family symmetry of \(\mathbb{C}\ell(8)\) lifts consistently to \(\mathbb{C}\ell(10)\), and continues to permute the relevant primitive idempotents and fermion sectors without replicating the gauge generators.

2.2 Fermion sectors and three generations↩︎

The first-generation isospin-up doublet sector is \[V_1^+:=\Bigl(\mathrm{span}_{\mathbb{C}}\bigl\{ a_1^\dagger a_2^\dagger a_3^\dagger a_4^\dagger,\; a_i^\dagger a_4^\dagger,\; a_i^\dagger a_j^\dagger,\; a_4^\dagger a_5^\dagger \bigr\}\Bigr)f_{+++++}\subset I_1, \label{eq:V1plus95full95again}\tag{2}\] with \(i,j\in\{1,2,3\}\) and \(i<j\). The corresponding isospin-down doublet sector is \[V_1^-:=\Bigl(\mathrm{span}_{\mathbb{C}}\bigl\{ a_1^\dagger a_2^\dagger a_3^\dagger a_4^\dagger a_5,\; a_i^\dagger a_4^\dagger a_5,\; a_i^\dagger a_j^\dagger a_5,\; a_4^\dagger \bigr\}\Bigr)f_{++++-}\subset I_{17}. \label{eq:V1minus95full95again}\tag{3}\]

The weak singlet sectors are \[U_1^-:=\Bigl(\mathrm{span}_{\mathbb{C}}\bigl\{ a_1^\dagger a_2^\dagger a_3^\dagger,\; a_i^\dagger,\; a_i^\dagger a_j^\dagger a_4,\; a_5^\dagger \bigr\}\Bigr)f_{+++-+}\subset I_9, \label{eq:U1minus95again}\tag{4}\] and \[U_1^+:=\Bigl(\mathrm{span}_{\mathbb{C}}\bigl\{ a_1^\dagger a_2^\dagger a_3^\dagger a_5,\; a_i^\dagger a_5,\; a_i^\dagger a_j^\dagger a_4a_5,\; 1 \bigr\}\Bigr)f_{+++--}\subset I_{25}, \label{eq:U1plus95again}\tag{5}\]

These fermion subspaces are related by right multiplication: \[\begin{align} V_1^-=V_1^+a_5,\qquad U_1^-=V_1^+a_4,\qquad U_1^+=V_1^+a_4a_5. \end{align}\] Equivalently, \[\begin{align} U_1^-=V_1^-a_4a_5^{\dagger},\qquad U_1^+=V_1^-a_4, \end{align}\] where these equalities are understood at the level of subspaces, so any overall minus signs arising from the fermionic anticommutation relations are immaterial.

The first generation of fermions is then \[\begin{align} S_1:=V_1^+\oplus V_1^-\oplus U_1^-\oplus U_1^+. \end{align}\] with \[\begin{align} V_1^+ &\leftrightarrow& \left(e_L^+,\;\bar{d}_L^{(\bar{3})},\;u_L^{(3)},\;\nu_L\right),\quad V_1^- \leftrightarrow \left(\bar{\nu}_L,\;\bar{u}_L^{(\bar{3})},\;d_L^{(3)},\;e_L^-\right),\\ U_1^+ &\leftrightarrow& \left(e_R^+,\;\bar{d}_R^{(\bar{3})},\;u_R^{(3)},\;\nu_R\right),\quad U_1^- \leftrightarrow \left(\bar{\nu}_R,\;\bar{u}_R^{(\bar{3})},\;d_R^{(3)},\;e_R^-\right). \end{align}\] Here an overbar denotes the corresponding antiparticle state, so for example \(\bar{u}\) and \(\bar{d}\) denote antiup and antidown quarks, respectively. The explicit algebraic representation of the first generation of fermion states is given in Appendix 7.

The remaining two generations are obtained via the \(S_3\) action, \[\begin{align} S_2:=\psi_3(S_1),\qquad S_3:=\psi_3^2(S_1). \end{align}\] Thus the order-three map \(\psi_3\) cyclically permutes the three generations.

2.3 Gauge generators and their action↩︎

The \(SU(3)\) gauge symmetry generators in \(\mathbb{C}\ell(10)\) are given by \[\begin{align} \nonumber\Lambda_1 &=& -a_2^\dagger{a_1}-a_1^\dagger{a_2}, \quad \Lambda_2 = ia_2^\dagger{a_1}-ia_1^\dagger{a_2}, \quad \Lambda_3 = a_2^\dagger{a_2}-a_1^\dagger{a_1}, \\ \Lambda_4 &=& -a_1^\dagger{a_3}-a_3^\dagger{a_1}, \quad \Lambda_5 = -ia_1^\dagger{a_3}+ia_3^\dagger{a_1}, \quad \Lambda_6 = -a_3^\dagger{a_2}-a_2^\dagger{a_3}, \\ \nonumber\Lambda_7 &=& ia_3^\dagger{a_2}-ia_2^\dagger{a_3}, \quad \Lambda_8 = -\frac{1}{\sqrt{3}}(a_1^\dagger{a_1}+a_2^\dagger{a_2}-2a_3^\dagger{a_3}), \end{align}\]

The \(SU(2)_L\) weak gauge generators are defined as \[\begin{align} \label{eq:weakgenerators} T_1&=&\frac{1}{2}(-i a_5+i a_5^\dagger)\omega_8 P,\quad T_2=\frac{1}{2}(a_5+a_5^\dagger)\omega_8 P,\quad T_3=\frac{1}{2}(a_5^\dagger a_5-a_5 a_5^\dagger)P, \end{align}\tag{6}\] where \(P\) is an \(S_3\)-invariant projector defined as \[\begin{align} P:=f_{+++++}+f_{---++}+f_{++++-}+f_{---+-}. \end{align}\]

The first-generation \(SU(2)_L\) doublets are then \[\begin{align} \left(\begin{array}{c} a_1^\dagger a_2^\dagger a_3^\dagger a_4^\dagger f_{+++++}\\ a_1^\dagger a_2^\dagger a_3^\dagger a_4^\dagger a_5 f_{++++-} \end{array} \right)&,&\; \left(\begin{array}{c} a_i^\dagger a_j^\dagger f_{+++++}\\ a_i^\dagger a_j^\dagger a_5 f_{++++-} \end{array} \right),\;\\ \left(\begin{array}{c} a_i^\dagger a_4^\dagger f_{+++++}\\ a_i^\dagger a_4^\dagger a_5 f_{++++-} \end{array} \right)&,&\; \left(\begin{array}{c} a_4^\dagger a_5^\dagger f_{+++++}\\ a_4^\dagger f_{++++-} \end{array} \right). \end{align}\]

The \(U(1)_{em}\) electromagnetic charge generator is defined as \[\begin{align} \label{eq:Q39def} Q'&:=&Q+(2P-\mathbb{I})a_5^\dagger a_5, \end{align}\tag{7}\] where \[\begin{align} \label{eq:Qdef} Q&:=&\frac{1}{3}\left(Q_1+\psi_3(Q_1)+\psi_3^2(Q_1)\right), \end{align}\tag{8}\] is the \(\psi_3\)-invariant charge operator obtained by averaging the first-generation generator \[\begin{align} \label{eq:Q1def} Q_1&:=&\frac{1}{3}\left(a_1a_1^\dagger+a_2a_2^\dagger+a_3a_3^\dagger-3a_4a_4^\dagger\right), \end{align}\tag{9}\] over its \(\psi_3\) orbit.

Finally, the \(U(1)_Y\) hypercharge generator is given by \[\begin{align} Y&:=&2(Q'-T_3). \end{align}\]

The gauge generators act on the fermion states via the adjoint (commutator) action \([\,\cdot\,,\cdot\,]\). On the physically relevant fermion sectors this action simplifies: the colour generators \(\Lambda_i\) reduce to a left action, while the weak generators \(T_i\) reduce to a right action. This is precisely the structural feature that makes it natural to introduce the Higgs below as a right-action operator from the weak-doublet sectors \(V_1^\pm\) to the weak-singlet sectors \(U_1^\pm\).

A crucial property for the present construction is that the Standard Model gauge generators are \(S_3\)-invariant. In particular, the action of the family symmetry permutes the fermion sectors while leaving the gauge algebra fixed, so the gauge sector is generation-independent and is not triplicated.

3 The Higgs Sector↩︎

In this section we develop the Higgs sector in \(\mathbb{C}\ell(10)\). We will treat the Higgs components as right-action operators that take the weak doublet sectors into the appropriate weak singlet sectors. A numerical Yukawa coefficient can then be extracted using a suitable bilinear pairing. For the latter we use the Hilbert–Schmidt inner product in the matrix realisation \(\mathbb{C}\ell(10)\cong\text{Mat}(32,\mathbb{C})\).

3.1 A naive algebraic Yukawa ansatz↩︎

At first glance one might try to proceed exactly as in ordinary field theory and simply replace the spacetime fields in a term such as \[\begin{align} \overline{\psi}_L\, H\, \psi_R \end{align}\] by the corresponding algebraic representatives from the \(\mathbb{C}l(10)\) model². One would then attempt to determine the Higgs components by demanding that the resulting algebraic product be nonzero. This idea is natural, but it is not the correct algebraic notion of a Yukawa coupling. For example, consider a colour-diagonal down-type channel. Using the algebraic representatives of the first-generation fermion states listed in Appendix 7, take \[\begin{align} d_L = a_2^\dagger a_3^\dagger f_{+++++},\qquad d_R = a_1^\dagger f_{+++-+}. \end{align}\] One then considers the product \[\begin{align} a_2^\dagger a_3^\dagger \bigl(f_{+++++} H f_{-++-+}\bigr)a_1^\dagger \neq 0, \end{align}\] where we have used \(a_1^\dagger f_{+++-+}=f_{-++-+}a_1^\dagger\). Demanding that the term in parentheses be nonzero forces the Higgs component to contain a colour-carrying factor such as \[\begin{align} H \sim a_1 a_4 f_{-++-+}, \end{align}\] which is immediately unacceptable as a Higgs candidate.

The reason this simplistic approach fails is that the algebraic product of three state representatives is not the correct analogue of a spacetime Yukawa invariant. In ordinary field theory, the barred spinor \(\bar{\psi}\) is not simply another particle state but instead a Dirac adjoint spinor that belongs to the dual space. Therefore one should not expect the raw algebraic product of three state representatives to be the final physical invariant.

3.2 Higgs operators as right-action maps↩︎

Within the \(\mathbb{C}\ell(10)\) framework, the doublet states live in the minimal ideals \(V^+\) and \(V^-\), whereas the singlet states live in the minimal ideals \(U^+\) and \(U^-\). The Yukawa terms therefore connect states belonging to distinct minimal ideals. This implies that the Higgs should be treated as a right-action operator mapping weak doublet sectors into weak singlet sectors. A suitable bilinear pairing can subsequently be used to extract a numerical Yukawa coefficient.

Let us first identify Higgs operators \(H\) for which the map \[R_H : X_L \mapsto X_L H \label{eq:right95action95intertwiner95preview}\tag{10}\] takes weak doublet states into weak singlet states. Here \(X_L\in \mathbb{C}\ell(10)\) is an internal algebraic state representing a left-handed fermion.

Already at the algebraic level, the two singlet sectors suggest the relevant Higgs maps. The maps \[\begin{align} V_1^+ \xrightarrow{\;a_4\;} U_1^-, \quad V_1^- \xrightarrow{\;a_4 a_5^\dagger\;} U_1^-, \quad V_1^+ \xrightarrow{\;a_4 a_5\;} U_1^+, \qquad V_1^- \xrightarrow{\;a_4\;} U_1^+, \end{align}\] are precisely the ones needed to reproduce the Yukawa patterns. This suggests that the Higgs sector should consist of two weak doublets: one down-type doublet mapping to \(U_1^-\), and one up-type doublet mapping to \(U_1^+\).

The first-generation fermion states of the \(\mathbb{C}\ell(10)\) model are organised into the weak doublet sectors \[V_1^+ \subset I_1 = \mathbb{C}\ell(10)f_{+++++}, \qquad V_1^- \subset I_{17} = \mathbb{C}\ell(10)f_{++++-}, \label{eq:V1pm95again}\tag{11}\] and the weak singlet sectors \[U_1^- \subset I_9 = \mathbb{C}\ell(10)f_{+++-+}, \qquad U_1^+ \subset I_{25} = \mathbb{C}\ell(10)f_{+++--}. \label{eq:U1pm95again}\tag{12}\] The target primitive idempotents are therefore \[f_{+++-+}=E_{9,9}, \qquad f_{+++--}=E_{25,25}, \label{eq:target95idempotents95again}\tag{13}\] whereas the source primitive idempotents for the weak doublets are \[f_{+++++}=E_{1,1}, \qquad f_{++++-}=E_{17,17}. \label{eq:source95idempotents95again}\tag{14}\]

For primitive idempotents \(f_a\) and \(f_b\), define \[H_{a\to b} := f_a\,\mathbb{C}\ell(10)\,f_b. \label{eq:intertwiner95space95def}\tag{15}\]

For the physical Higgs maps of interest there are four relevant operator spaces: \[H_{1\to 9} := f_{+++++}\mathbb{C}\ell(10)f_{+++-+}, \qquad H_{17\to 9} := f_{++++-}\mathbb{C}\ell(10)f_{+++-+}, \label{eq:Hd95intertwiner95spaces}\tag{16}\] \[H_{1\to 25} := f_{+++++}\mathbb{C}\ell(10)f_{+++--}, \qquad H_{17\to 25} := f_{++++-}\mathbb{C}\ell(10)f_{+++--}. \label{eq:Hu95intertwiner95spaces}\tag{17}\]

Each such space is one-dimensional, so once the source and target primitive idempotents are fixed, the corresponding Higgs operator is unique up to an overall scalar. Choosing a nonzero generator, one finds for example \[\begin{align} H_{1\to 9} &=& f_{+++++}\mathbb{C}\ell(10)f_{+++-+},\\ &=& f_{+++++}a_4f_{+++-+},\\ &=& a_4f_{+++-+}f_{+++-+}, \qquad (\text{since } f_{+++++}a_4=a_4f_{+++-+}),\\ &=& a_4f_{+++-+}, \qquad (\text{since } f_{+++-+}f_{+++-+}=f_{+++-+}). \end{align}\] Likewise, \[\begin{align} H_{17\to 9} = a_4a_5^{\dagger}f_{+++-+}, \qquad H_{1\to 25} = a_4a_5f_{+++--}, \qquad H_{17\to 25} = a_4f_{+++--}. \end{align}\] In the chosen matrix realisation, each space \(H_{a\to b}\) is spanned by the corresponding matrix unit \(E_{a,b}\). By a slight abuse of notation, we use the same symbol \(H_{a\to b}\) for a chosen nonzero generator of the one-dimensional space \(f_a\mathbb{C}\ell(10)f_b\).

If \(X_L\in V_1^+\), then right multiplication by \(H_{1\to 9}=a_4f_{+++-+}\) gives \[R_{H_{1\to 9}}:X_L \mapsto X_L\,H_{1\to 9} =X_L\,a_4 f_{+++-+}\in U_1^-. \label{eq:right95map95Vplus95to95Uminus}\tag{18}\] Likewise, if \(Y_L=X_Li a_5\in V_1^-\) with \(X_L\in V_1^+\), then right multiplication by \(H_{17\to 9}=a_4a_5^\dagger f_{+++-+}\) gives \[R_{H_{17\to 9}}:Y_L \mapsto Y_L\,H_{17\to 9} =X_L a_5 a_4a_5^\dagger f_{+++-+} =X_L a_4 f_{+++-+}\in U_1^-. \label{eq:right95map95Vminus95to95Uminus}\tag{19}\]

Similarly, right multiplication by \(H_{1\to 25}\) and \(H_{17\to 25}\) maps \(V_1^+\) and \(V_1^-\) into \(U_1^+\), respectively. Thus the Higgs sector should be thought of as two pairs of right-action maps.

3.3 Candidate Higgs doublets↩︎

It is now natural to assemble the four right-action operators into two candidate Higgs doublets. \[H_d^{(1)} := \begin{pmatrix} H_{17\to 9} \\ H_{1\to 9} \end{pmatrix},\quad H_u^{(1)} := \begin{pmatrix} H_{17\to 25} \\ H_{1\to 25} \end{pmatrix}.\] We therefore identify the four right-action operators with the Higgs components as \[H_{17\to 9}=H_d^0,\quad H_{1\to 9}=H_d^-,\quad H_{17\to 25}=H_u^+,\quad H_{1\to 25}=H_u^0, \label{eq:H95component95identification}\tag{20}\] so that \[H_d^{(1)} =\begin{pmatrix} H_d^0 \\ H_d^- \end{pmatrix} = \begin{pmatrix} a_4 a_5^\dagger f_{+++-+} \\ a_4 f_{+++-+} \end{pmatrix},\quad H_u^{(1)} =\begin{pmatrix} H_u^+ \\ H_u^0 \end{pmatrix} =\begin{pmatrix} a_4 f_{+++--} \\ a_4 a_5 f_{+++--} \end{pmatrix}.\]

In the explicit \(32\times 32\) matrix representation of \(\mathbb{C}\ell(10)\), these Higgs components are \[H_d^0 = -E_{17,9},\qquad H_d^- = iE_{1,9},\qquad H_u^+ = iE_{17,25},\qquad H_u^0 = E_{1,25}. \label{eq:Higgs95matrix95components}\tag{21}\]

3.4 Trace pairing and spacetime Yukawa terms↩︎

In the chosen matrix realisation \(\mathbb{C}\ell(10)\cong \mathrm{Mat}(32,\mathbb{C)}\), we equip \(\mathbb{C}\ell(10)\) with the Hermitian pairing induced from the matrix algebra, namely the Hilbert–Schmidt inner product \[\langle A,B\rangle := \mathrm{Tr}(A^\dagger B), \qquad A,B\in \mathbb{C}\ell(10). \label{eq:trace95inner95product}\tag{22}\] In the matrix-unit basis this satisfies \[\mathrm{Tr}(E_{ij}^\dagger E_{kl})= \mathrm{Tr}(E_{ji}E_{kl})=\delta_{ik}\delta_{jl}. \label{eq:matrix95unit95trace95pairing}\tag{23}\] So distinct matrix units are orthonormal.

This pairing allows one to extract a complex number from the image \(X_LH\) by comparing it with a chosen right-handed algebraic state \(X_R\). We therefore define the Yukawa coefficient by \[\mathcal{Y}(X_L,H,X_R):= \mathrm{Tr}\bigl((X_LH)^\dagger X_R\bigr). \label{eq:Yukawa95pairing95def}\tag{24}\]

Comparing 24 with 23 , we see that \(\mathcal{Y}(X_L,H,X_R)\) simply extracts the coefficient of \(X_R\) in the expansion of \(X_LH\) in the matrix-unit basis. In particular, \(\mathcal{Y}(X_L,H,X_R)=0\) whenever \(X_LH\) has no component along \(X_R\), while \(\mathcal{Y}(X_L,H,X_R)\neq 0\) whenever the \(X_R\) component is nonzero. For normalized basis states, \(\mathcal{Y}(X_L,H,X_R)=\pm 1\) when \(X_LH=\pm X_R\). Thus the algebra determines the selection rules and the structure of the Yukawa sector, while the overall coupling constants remain free parameters, just as in ordinary field theory.

It is important to distinguish the internal algebraic states from the physical spacetime spinor fields. An algebraic state \(X\in\mathbb{C}\ell(10)\) is not itself a Dirac or Weyl spinor in spacetime. Rather, a physical fermion field is taken to be of the form \[\Psi_{L,R}(x)=\chi_{L,R}(x)\otimes X_{L,R},\] where \(\chi_{L,R}(x)\) is the usual spacetime spinor field and \(X_{L,R}\in\mathbb{C}\ell(10)\) is the corresponding algebraic representative.

The barred spacetime field is then \[\overline{\Psi}_{L,R}(x)=\overline{\chi}_{L,R}(x)\otimes X_{L,R}^\dagger,\] where the bar on \(\chi\) denotes the usual Dirac adjoint in spacetime, while the dagger on \(X\) denotes Hermitian conjugation in the matrix algebra.

Once the algebraic Yukawa coefficient has been extracted, the corresponding spacetime Yukawa term takes the form \[\mathcal{L}_Y \supset y\,\mathcal{Y}(X_L,H,X_R)\,\overline{\chi}_L(x)\,\varphi_H(x)\,\chi_R(x)+\mathrm{h.c.},\] where \(y\) is an overall coupling constant, \(\varphi_H(x)\) is the spacetime Higgs field associated with the algebraic right-action operator \(H\), and \(\mathcal{Y}(X_L,H,X_R)\) is the algebraically computed coefficient.

3.5 Illustrative Yukawa channels↩︎

Before proceeding, it is useful to work through two explicit examples.

Take the first-generation up-type doublet and right-handed down-quark states \[u_L^{(i)} = a_i^\dagger a_4^\dagger f_{+++++},\quad d_R^{(j)} = a_j^\dagger f_{+++-+}. \label{eq:uL95example}\tag{25}\] Here \(i,j=1,2,3\) denote the colour labels of \(u_L\) and \(d_R\). Acting on \(u_L^{(i)}\) with the right-action operator \(H_d^-=a_4f_{+++-+}\), we obtain \[R_{H_d^-}:u_L^{(i)} \mapsto u_L^{(i)} H_d^-=a_i^\dagger a_4^\dagger f_{+++++} a_4 f_{+++-+}.\] Using \(f_{+++++}a_4 = a_4f_{+++-+}\), this becomes \[u_L^{(i)} H_d^-=a_i^\dagger a_4^\dagger a_4 f_{+++-+}.\] Since \(a_4^\dagger a_4 = \pi_4^{(-)}\) and \(f_{+++-+}\) already contains \(\pi_4^{(-)}\), we obtain \[u_L^{(i)} H_d^-=a_i^\dagger f_{+++-+}=d_R^{(i)}.\] Therefore \[\mathcal{Y}\bigl(u_L^{(i)},H_d^-,d_R^{(j)}\bigr)=\mathrm{Tr}\bigl((d_R^{(i)})^\dagger d_R^{(j)}\bigr)= \delta_{ij},\] so colour diagonality is enforced automatically by the trace pairing.

As a second example, consider the neutrino channel in the up-type family. Take \[\nu_L = a_4^\dagger a_5^\dagger f_{+++++},\qquad\nu_R = f_{+++--}.\] Acting with \(H_u^0=a_4 a_5 f_{+++--}\), we obtain \[\begin{align} R_{H_u^0}:\nu_L \mapsto\nu_L H_u^0&=&a_4^\dagger a_5^\dagger f_{+++++} a_4 a_5 f_{+++--}\nonumber\\ &=&a_4^\dagger a_5^\dagger a_4 a_5 f_{+++--}\qquad\text{using }f_{+++++}a_4a_5=a_4a_5f_{+++--}\nonumber\\ &=&-\,a_5^\dagger a_4^\dagger a_4 a_5 f_{+++--}\nonumber\\ &=&-\,\pi_5^{(-)} \pi_4^{(-)} f_{+++--}\nonumber\\ &=&-\,f_{+++--}\nonumber\\ &=&-\,\nu_R. \end{align}\] Thus the map again lands on the desired right-handed singlet, up to a sign. Subsequently \[\mathcal{Y}\bigl(\nu_L,H_u^0,\nu_R\bigr)=\mathrm{Tr}\bigl(-\nu_R^\dagger \nu_R\bigr)= -1.\]

Likewise, for the lower weak partner \(e_L^- = a_4^\dagger f_{++++-}\) one finds \[\begin{align} R_{H_u^+}:e_L^- \mapsto e_L^- H_u^+&=a_4^\dagger f_{++++-} a_4 f_{+++--}\nonumber\\ &=\nu_R, \end{align}\] and therefore, \[\mathcal{Y}\bigl(e_L^-,H_u^+,\nu_R\bigr)=\mathrm{Tr}\bigl(\nu_R^\dagger \nu_R\bigr)= +1.\] So the two members of the weak doublet map into the same right-handed singlet sector, as required.

3.6 Gauge quantum numbers and Type-II structure↩︎

The Higgs components are first seen to be \(SU(3)_c\) singlets. Indeed, the colour generators \(\Lambda_i\) act only on the first three Witt modes \(a_1,a_2,a_3\), whereas the nontrivial Higgs factors are built from \(a_4\) and \(a_5\), with the remaining factor \(\pi_1^{(+)}\pi_2^{(+)}\pi_3^{(+)}\) in the primitive idempotents being colour invariant. Hence \[[\Lambda_i,H]=0,\qquad i=1,\ldots,8,\] for \(H\in\{H_d^0,H_d^-,H_u^+,H_u^0\}\).

Using the \(SU(2)_L\) generators given in 6 , one readily verifies that the two Higgs pairs \(H_d\) and \(H_u\) are genuine weak doublets. In particular we find that \[\begin{align} [T_3,H_d^0]=+\frac{1}{2} H_d^0,\quad [T_3,H_d^-]=-\frac{1}{2} H_d^-,\quad [T_3,H_u^+]=+\frac{1}{2} H_u^+,\quad [T_3,H_u^0]=-\frac{1}{2} H_u^0. \end{align}\]

Unlike for fermions, for which the commutator action of \(SU(2)_L\) reduces to a right action on both \(V^+\) and \(V^-\), \[[T_i,V^+]=-V^+T_i,\qquad [T_i,V^-]=-V^-T_i,\qquad i=1,2,3,\] the Higgs sector behaves differently under the commutator action. For \(H\in\{H_d^0,H_d^-,H_u^+,H_u^0\}\), the right action of the weak generators vanishes because \[f_{+++-+}P=0,\qquad f_{+++--}P=0,\] while the left action survives since \[PH=H.\] Therefore the commutator action reduces to a left action, \[[T_i,H]=T_iH.\]

Likewise we can determine the commutator action of \(Q'\) and \(Y\) on the Higgs components. The eigenvalues are summarised in Table 1

The commutator relations above, together with Table 1, show that the two first-generation Higgs doublets transform under \(SU(3)_c\times SU(2)_L\times U(1)_Y\) as \[H_d=\begin{pmatrix} H_d^0\\ H_d^- \end{pmatrix} \sim (1,2,-1), \qquad H_u=\begin{pmatrix} H_u^+\\ H_u^0 \end{pmatrix} \sim (1,2,+1).\]

This is the same opposite-hypercharge \(H_u/H_d\) basis familiar from supersymmetric model building, where one works directly with two Higgs doublets carrying hypercharges \(+1\) and \(-1\), rather than starting from a single doublet and its conjugate.

The two doublets are algebraically distinct because \(H_d\subset I_9\) and \(H_u\subset I_{25}\) take values in different minimal-ideal sectors. They are not related by the usual electroweak conjugation \(i\sigma_2 H^*\), because neither complex conjugation nor left multiplication changes the primitive idempotents (and thus the minimal left ideal). At the level of the algebraic construction, the model yields a genuinely Type-II-like two-Higgs-doublet structure.

Although the two doublets are related by the purely internal algebraic operation of right multiplication, \[\begin{align} H_u = H_d a_5,\quad H_d = H_u a_5^{\dagger} \end{align}\] this does not identify them as the same Higgs field. Rather, it is merely an internal relation between two distinct Higgs doublets.

4 \(S_3\) Triplication of the Higgs Sector↩︎

We now turn to one of the central structural consequences of the present construction: the Higgs sector as constructed is not immediately invariant under the action of the \(S_3\) family symmetry. Instead, the two first-generation Higgs doublets are mapped via \(\psi_3\) into two further pairs of Higgs doublets. The result is a triplication of the Higgs sector.

Let us first rewrite the first generation Higgs doublets as \[H_d^{(1)}:= \begin{pmatrix} a_4 a_5^\dagger\,f_{+++-+} \\ a_4\,f_{+++-+} \end{pmatrix},\qquad H_u^{(1)}:= \begin{pmatrix} a_4\,f_{+++--}\\ a_4 a_5\,f_{+++--} \end{pmatrix}.\] Since \(\psi_3\) is an algebra automorphism, it acts on these doublets component-wise: \[H_d^{(2)}:=\psi_3\!\left(H_d^{(1)}\right)= \begin{pmatrix} \psi_3(a_4 a_5^\dagger)\,\psi_3(f_{+++-+}) \\ \psi_3(a_4)\,\psi_3(f_{+++-+}) \end{pmatrix},\] \[H_u^{(2)}:=\psi_3\!\left(H_u^{(1)}\right)= \begin{pmatrix} \psi_3(a_4)\,\psi_3(f_{+++--}) \\ \psi_3(a_4 a_5)\,\psi_3(f_{+++--}) \end{pmatrix}.\] Likewise, \[H_d^{(3)}:=\psi_3^2\!\left(H_d^{(1)}\right)= \begin{pmatrix} \psi_3^2(a_4 a_5^\dagger)\,\psi_3^2(f_{+++-+}) \\ \psi_3^2(a_4)\,\psi_3^2(f_{+++-+}) \end{pmatrix},\] \[H_u^{(3)}:=\psi_3^2\!\left(H_u^{(1)}\right)= \begin{pmatrix} \psi_3^2(a_4)\,\psi_3^2(f_{+++--}) \\ \psi_3^2(a_4 a_5)\,\psi_3^2(f_{+++--}) \end{pmatrix}.\]

The full Higgs sector therefore consists of six doublets \[\{H_d^{(1)},H_d^{(2)},H_d^{(3)}\},\qquad\{H_u^{(1)},H_u^{(2)},H_u^{(3)}\},\] that are permuted cyclically by \(\psi_3\): \[H_d^{(1)} \xrightarrow{\;\psi_3\;} H_d^{(2)}\xrightarrow{\;\psi_3\;} H_d^{(3)} \xrightarrow{\;\psi_3\;} H_d^{(1)},\] \[H_u^{(1)} \xrightarrow{\;\psi_3\;} H_u^{(2)}\xrightarrow{\;\psi_3\;} H_u^{(3)} \xrightarrow{\;\psi_3\;} H_u^{(1)}.\] The \(\psi_3\)-images are distinct algebraic objects.

All three copies of each Higgs type carry the same gauge quantum numbers. All the gauge generators of the \(\mathbb{C}\ell(10)\) model are \(S_3\)-invariant. Thus for any such gauge generator \(G\) and any algebra element \(X\), \[\psi_3([G,X])=[\psi_3(G),\psi_3(X)]=[G,\psi_3(X)],\] and similarly for \(\psi_3^2\). It follows that if \(X\) is an eigenvector of the adjoint action of \(G\) with eigenvalue \(\lambda\), then \(\psi_3(X)\) is also an eigenvector with the same eigenvalue. Therefore all gauge quantum numbers are preserved along the \(\psi_3\)-orbit.

4.1 Triplicated Higgs doublets↩︎

It is instructive to display one explicit transformed component explicitly. In the \(32\times32\) matrix representation, the first-generation down-type neutral component is \[H_d^{0(1)} =a_4 a_5^\dagger f_{+++-+}= -\,E_{17,9}.\] Applying \(\psi_3\) gives \[\begin{align} \nonumber H_d^{0(2)}&:=&\psi_3\left(H_d^{0(1)}\right),\\ \nonumber &=&\left(\frac{1}{4}\,a_4 a_5^\dagger-\frac{\sqrt{3}}{4}\,a_1^\dagger a_2^\dagger a_3^\dagger a_4 a_5^\dagger\right) f_{+++-+} +\left(-\frac{\sqrt{3}}{4}\,a_1 a_2 a_3 a_4 a_5^\dagger-\frac{3}{4}\,a_4 a_5^\dagger\right) f_{----+},\\ &=&-\frac{1}{4} E_{17,9}-\frac{i\sqrt{3}}{4} E_{30,9}-\frac{i\sqrt{3}}{4} E_{17,6}+\frac{3}{4} E_{30,6}, \end{align}\] while \[\begin{align} \nonumber H_d^{0(3)}&:=&\psi_3^2\left(H_d^{0(1)}\right)\\ \nonumber &=&\left(\frac{1}{4}\,a_4 a_5^\dagger+\frac{\sqrt{3}}{4}\,a_1^\dagger a_2^\dagger a_3^\dagger a_4a_5^\dagger\right) f_{+++-+} +\left(\frac{\sqrt{3}}{4}\,a_1 a_2 a_3 a_4 a_5^\dagger-\frac{3}{4}\,a_4a_5^\dagger\right) f_{----+},\\ &=&-\frac{1}{4} E_{17,9}+\frac{i\sqrt{3}}{4} E_{30,9}+\frac{i\sqrt{3}}{4} E_{17,6} +\frac{3}{4} E_{30,6}. \end{align}\] We see that \(H_d^{0(2)}\) and \(H_d^{0(3)}\) has support in two columns (minimal left ideals), and that \[\operatorname{span}_{\mathbb{C}}\!\left\{H_d^{0(1)},\,H_d^{0(2)},\,H_d^{0(3)}\right\}\subset I_{6}\oplus I_{9}.\] where \(I_{6}=\mathbb{C}\ell(10)f_{----+}\) and \(I_{9}=\mathbb{C}\ell(10)f_{+++-+}\) are the minimal left ideals corresponding to the 6th and 9th column in the \(\text{Mat}(32,\mathbb{C})\) representation of \(\mathbb{C}\ell(10)\).

An analogous computation for the up-type Higgs gives components supported in \(I_{22}\oplus I_{25}\), so that \[\operatorname{span}_{\mathbb{C}}\left\{H_u^{+(1)},\,H_u^{+(2)},\,H_u^{+(3)}\right\}\subset I_{22}\oplus I_{25}.\] A full list of all three generations of Higgs terms in the Witt basis is given in Appendix 9.

4.2 Linear independence of the triplicated Higgs Sector↩︎

The fermion sector is \(96\)-dimensional [3], with \[\dim\!\left(S_1\oplus S_2\oplus S_3\right)=96,\] where \[S_1=V_1^+\oplus V_1^+a_5\oplus V_1^+a_4\oplus V_1^+a_4a_5,\qquad S_2=\psi_3(S_1),\qquad S_3=\psi_3^2(S_1).\]

The triplicated Higgs sector contributes twelve further linearly independent states, \[\dim\;\operatorname{span}_{\mathbb{C}}\!\left(\{H_d^{0(r)},H_d^{-(r)},H_u^{+(r)},H_u^{0(r)}\}_{r=1,2,3}\right)=12.\] Furthermore, one verifies³ that these twelve Higgs states are linearly independent from the \(96\) fermion states. Hence the combined fermion-Higgs sector has dimension \(108\).

4.3 Symmetric and orthogonal Higgs combinations↩︎

For each Higgs component, the three family copies \[\begin{align} H^{(1)},\qquad H^{(2)}=\psi_3(H^{(1)}),\qquad H^{(3)}=\psi_3^2(H^{(1)}), \end{align}\] are cyclically permuted by the action of \(\psi_3\). It is therefore natural to form their orbit average, in direct analogy with the construction of the \(S_3\)-invariant electric charge generator \(Q'\) in 7 from the first-generation operator 9 . We therefore define \[\begin{align} \bar{H}_d^0&:=&\frac{1}{3}\left(H_d^{0(1)}+H_d^{0(2)}+H_d^{0(3)}\right),\\ \bar{H}_d^-&:=&\frac{1}{3}\left(H_d^{-(1)}+H_d^{-(2)}+H_d^{-(3)}\right),\\ \bar{H}_u^+&:=&\frac{1}{3}\left(H_u^{+(1)}+H_u^{+(2)}+H_u^{+(3)}\right),\\ \bar{H}_u^0&:=&\frac{1}{3}\left(H_u^{0(1)}+H_u^{0(2)}+H_u^{0(3)}\right). \end{align}\]

Algebraically, the resulting symmetric Higgs doublets are \[\begin{align} \bar{H}_d=\begin{pmatrix} \bar{H}_d^0 \\ \bar{H}_d^- \end{pmatrix} = \begin{pmatrix} \frac{1}{2}a_4a_5^{\dagger}(f_{+++-+}-f_{----+}) \\ \frac{1}{2}a_4(f_{+++-+}+f_{----+}) \end{pmatrix}, \end{align}\] and \[\begin{align} \bar{H}_u=\begin{pmatrix} \bar{H}_u^+ \\ \bar{H}_u^0 \end{pmatrix} = \begin{pmatrix} \frac{1}{2}a_4(f_{+++--}+f_{-----}) \\ \frac{1}{2}a_4a_5(f_{+++--}-f_{-----}) \end{pmatrix}. \end{align}\]

Rather than working with the full family-resolved Higgs sector, we now restrict attention to these symmetric combinations. The remaining two independent directions are still present algebraically, but will not be needed in the exact \(S_3\)-symmetric limit considered here. They become important once \(S_3\) breaking and flavour phenomenology are introduced, and a detailed analysis of that sector lies beyond the scope of the present paper.

The three \(\psi_3\)-related Higgs copies may also be decomposed into two orthogonal directions. For each Higgs component \(H\in\{H_d^0,H_d^-,H_u^+,H_u^0\}\), define \[\begin{align} H_\alpha &:=& H^{(1)} - \frac{1}{2}\bigl(H^{(2)} + H^{(3)}\bigr),\\ H_\beta &:=& \frac{\sqrt{3}}{2}\bigl(H^{(2)} - H^{(3)}\bigr). \end{align}\] Under \(\psi_3\), the pair \((H_\alpha,H_\beta)\) transforms as a rotation by \(2\pi/3\). In the present paper we restrict attention to the symmetric Higgs pair \(\bar H_d,\bar H_u\). The remaining two directions will become relevant once \(S_3\) breaking and realistic flavour structure are studied.

4.4 Effect of passing to the symmetric Higgs limit↩︎

When the fermions are kept in the first-generation sectors, replacing the family-resolved Higgs operators by the \(S_3\)-invariant combinations does not change the set of allowed Yukawa channels. The reason is not only the linearity of \[\begin{align} \mathcal{Y}(X_L,H,X_R)=\mathrm{Tr}\bigl((X_LH)^\dagger X_R\bigr), \end{align}\] but also the ideal support of the \(\psi_3\)-images.

For example, consider the first-generation charged lepton channel \[e_L^- = a_4^\dagger f_{++++-}\in V_1^-, \qquad e_R^- = a_5^\dagger f_{+++-+}\in U_1^-.\] For the first-generation Higgs component one finds \[\begin{align} e_L^- H_d^{0(1)}&=&a_5^\dagger f_{+++-+}=e_R^-. \end{align}\] For the second Higgs copy one obtains \[\begin{align} \nonumber e_L^- H_d^{0(2)} &=& a_4^\dagger f_{++++-} \Bigl[ \Bigl(\frac{1}{4} a_4a_5^\dagger-\frac{\sqrt3}{4}a_1^\dagger a_2^\dagger a_3^\dagger a_4a_5^\dagger\Bigr)f_{+++-+}\\ \nonumber &&+ \Bigl(-\frac{\sqrt3}{4}a_1a_2a_3a_4a_5^\dagger-\frac{3}{4} a_4a_5^\dagger\Bigr)f_{----+} \Bigr] \\ \nonumber &=& \frac{1}{4}\,a_5^\dagger f_{+++-+} +\frac{\sqrt3}{4}\,a_1a_2a_3a_5^\dagger f_{----+},\\ &=& \frac{1}{4}\,e_R^-+\frac{\sqrt3}{4}\,a_1a_2a_3a_5^\dagger f_{----+}. \end{align}\] Similarly, \[\begin{align} e_L^- H_d^{0(3)}=\frac{1}{4}\,e_R^- -\frac{\sqrt3}{4}\,a_1a_2a_3a_5^\dagger f_{----+}. \end{align}\] Hence \[\begin{align} e_L^- \bar H_d^0 &=& \frac{1}{3}\Bigl(e_L^- H_d^{0(1)}+e_L^- H_d^{0(2)}+e_L^- H_d^{0(3)}\Bigr)= \frac{1}{2}\,e_R^-, \end{align}\] and therefore \[\begin{align} \mathcal{Y}(e_L^-,\bar H_d^0,e_R^-)=\frac{1}{2}, \end{align}\]

Thus passing from \(H_d^{0(1)}\) to the \(S_3\)-invariant combination \(\bar H_d^0\) rescales the Yukawa coefficient from \(1\) to \(\frac{1}{2}\), but does not change the Yukawa channel itself. The same reasoning applies to the other Higgs components. In particular, \(\bar H_d\) still couples only to the \(U^-\)-type singlet sector, while \(\bar H_u\) still couples only to the \(U^+\)-type singlet sector. Hence replacing the family-resolved Higgs sector by the \(S_3\)-invariant Higgs doublets does not introduce any new wrong-type Yukawa couplings and does not change the allowed-versus-forbidden pattern of Yukawa interactions.

What does change is only the numerical value of the Yukawa coefficient: the \(S_3\)-invariant Higgs is an average over the \(\psi_3\)-orbit, so the corresponding coefficient is modified by the surviving projection onto the relevant source and target ideals. At the present stage this is not yet of physical importance, since our goal is only to determine which Yukawa channels are allowed and which are forbidden, rather than to extract fermion masses or mixing angles.

5 Flavour Structure in the Exact \(S_3\)-Invariant Limit↩︎

5.1 Type-II structure in the symmetric limit↩︎

In generic multi-Higgs models, tree-level flavour-changing neutral currents (FCNCs) are generally present unless the Yukawa sector has special structure. A standard sufficient criterion for natural flavour conservation is the Glashow–Weinberg condition: fermions of a given charge sector should couple to only one Higgs doublet. In the present construction, the relevant comparison is with a Type II pattern.

At the level of the two first-generation Higgs doublets, the algebra clearly separates the Yukawa channels: \[\begin{align} H_d^{(1)} \text{ maps to the singlet family } U_1^- \subset I_9,\\ H_u^{(1)} \text{ maps to the singlet family } U_1^+ \subset I_{25}. \end{align}\] Thus down-type quarks and charged leptons couple through \(H_d^{(1)}\), whereas up-type quarks and neutrinos couple through \(H_u^{(1)}\). In this sense, the first-generation Higgs sector is naturally Type-II-like.

Once the Higgs sector is triplicated by \(\psi_3\), however, this simple Type-II-like picture is obscured: in the algebraic generation basis \((S_1,S_2,S_3)\), with \(S_2=\psi_3(S_1)\) and \(S_3=\psi_3^2(S_1)\), the Higgs triplication is not generation-diagonal, so the full Higgs sector is not simply three decoupled copies of a two-Higgs-doublet model.

For the \(S_3\)-invariant Higgs doublets, by contrast, the relevant right-action maps satisfy \[\begin{align} R_{\bar H_d^-}\bigl(\psi_3^i(V_1^+)\bigr)=\psi_3^i(U_1^-),\qquad R_{\bar H_d^0}\bigl(\psi_3^i(V_1^-)\bigr)=\psi_3^i(U_1^-),\\ R_{\bar H_u^0}\bigl(\psi_3^i(V_1^+)\bigr)=\psi_3^i(U_1^+),\qquad R_{\bar H_u^+}\bigl(\psi_3^i(V_1^-)\bigr)=\psi_3^i(U_1^+), \end{align}\] where \(i=0,1,2\). At the level of these right-action maps, the invariant Higgs doublets preserve the Type-II structure already identified for the first-generation Higgs sector: \(\bar H_d\) couples only to the down-type singlet family, whereas \(\bar H_u\) couples only to the up-type singlet family. Thus, in the exact \(S_3\)-invariant limit, the algebraic Yukawa structure is again Type-II-like, even though, as we now discuss, the resulting generation-space Yukawa matrices are not diagonal in the algebraic generation basis.

5.2 Generation-space Yukawa matrices↩︎

The Yukawa matrices are not diagonal in the algebraic generation basis. This is because, although the states in the three sectors \(S_1\), \(S_2\), and \(S_3\) are linearly independent, they are not mutually orthogonal under the Hilbert–Schmidt pairing.

This can be seen explicitly in the down-type quark sector. One finds \[\begin{align} \left\langle d_{R,1}^{(i)},\,d_{R,2}^{(j)}\right\rangle&:=&\mathrm{Tr}\left(\left(d_{R,1}^{(i)}\right)^\dagger d_{R,2}^{(j)}\right)=\frac{1}{4}\,\delta_{ij}. \end{align}\] On the other hand, for the invariant down-type Higgs one has \[\begin{align} u_{L,1}^{(i)}\,\bar H_d^-=\frac{1}{2}\,d_{R,1}^{(i)}. \end{align}\] Therefore \[\begin{align} \nonumber\mathcal{Y}\!\left(u_{L,1}^{(i)},\,\bar H_d^{-},\,d_{R,2}^{(j)}\right) &=& \mathrm{Tr}\left(\left(u_{L,1}^{(i)}\bar H_d^-\right)^\dagger d_{R,2}^{(j)}\right)\\ &=& \nonumber\mathrm{Tr}\left(\left(\frac{1}{2}\,d_{R,1}^{(i)}\right)^\dagger d_{R,2}^{(j)}\right)\\ &=& \frac{1}{8}\,\delta_{ij}. \end{align}\] Thus the invariant Higgs does not mix generations at the level of the right-action map, but the resulting state can still have nonzero overlap with fermion states in the other \(\psi_3\)-related sectors.

For definiteness, consider again the down-type quark sector. Define the generation-space Yukawa matrices by \[\begin{align} \mathcal{Y}\!\left(u_{L,r}^{(i)},\,H_d^{-(a)},\,d_{R,s}^{(j)}\right) =(M_a)_{rs}\,\delta_{ij}, \qquad a,r,s=1,2,3. \end{align}\] Evaluating the pairings gives \[\begin{align} M_1&=& \begin{pmatrix} 1 & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & -\frac{1}{8} \\ \frac{1}{4} & -\frac{1}{8} & \frac{1}{4} \end{pmatrix},\quad M_2= \begin{pmatrix} \frac{1}{4} & \frac{1}{4} & -\frac{1}{8} \\ \frac{1}{4} & 1 & \frac{1}{4} \\ -\frac{1}{8} & \frac{1}{4} & \frac{1}{4} \end{pmatrix},\quad M_3= \begin{pmatrix} \frac{1}{4} & -\frac{1}{8} & \frac{1}{4} \\ -\frac{1}{8} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & 1 \end{pmatrix}. \end{align}\] Linearity of the Yukawa pairing then gives \[\begin{align} (\bar M)_{rs}:=\mathcal{Y}\!\left(u_{L,r}^{(i)},\,\bar H_d^{-},\,d_{R,s}^{(j)}\right) =\frac{1}{3}\Bigl((M_1)_{rs}+(M_2)_{rs}+(M_3)_{rs}\Bigr), \end{align}\] and hence \[\begin{align} \bar M=\frac{1}{3}(M_1+M_2+M_3)= \begin{pmatrix} \frac{1}{2} & \frac{1}{8} & \frac{1}{8} \\ \frac{1}{8} & \frac{1}{2} & \frac{1}{8} \\ \frac{1}{8} & \frac{1}{8} & \frac{1}{2} \end{pmatrix}. \end{align}\]

Thus, in the algebraic generation basis \((S_1,S_2,S_3)\), the averaged Yukawa matrix is not diagonal. The explicit nonzero entry \[\begin{align} \mathcal{Y}\!\left(u_{L,1}^{(i)},\,\bar H_d^{-},\,d_{R,2}^{(j)}\right)=\frac{1}{8}\,\delta_{ij} \end{align}\] is precisely one example of this off-diagonal structure.

The same \(S_3\)-invariant texture \(\bar M\) arises in the other allowed Yukawa sectors as well, up to overall sign conventions determined by the algebraic representatives. Since the algebraic construction fixes the matrix texture but not the overall sector-dependent Yukawa constants, these signs do not affect the flavour structure. In the exact \(S_3\)-invariant limit, it is therefore natural to parameterise the allowed Yukawa sectors by separate overall couplings \(y_d\) and \(y_u\) for the down-type and up-type quark sectors, together with the corresponding Higgs vacuum expectation values \(v_d\) and \(v_u\). Analogous overall constants appear in the charged-lepton and neutrino sectors. Thus the nontrivial result of the present construction is the algebraic determination of the matrix texture itself, rather than the absolute mass scales.

5.3 Eigenvalues and comparison with conventional \(S_3\) textures↩︎

The averaged Yukawa matrix \(\bar{M}\) can be written as \[\bar{M}=\tfrac{3}{8}\mathbf{I}_3+\tfrac{1}{8}\mathbf{J}_3,\] where \(\mathbf{I}_3\) is the \(3\times 3\) identity matrix and \(\mathbf{J}_3\) is the \(3\times 3\) all-ones matrix. This is readily diagonalised. The eigenvalues are \(\lambda_1=\tfrac{3}{4}\) with eigenvector \(\tfrac{1}{\sqrt{3}}(1,1,1)\), and \(\lambda_2=\lambda_3=\tfrac{3}{8}\) (doubly degenerate) in the orthogonal complement. Thus, if electroweak symmetry breaking is implemented in the usual way so that the corresponding mass matrix is proportional to \(\bar M\), the exact \(S_3\)-invariant limit yields a mass pattern with ratios \(2:1:1\) up to an overall sector-dependent scale. The eigenvector corresponding to \(\tfrac34\) is the family-symmetric direction \((1,1,1)/\sqrt3\), while the doubly degenerate eigenvalue \(\tfrac38\) spans the orthogonal two-dimensional subspace. In this sense, the exact \(S_3\)-invariant limit distinguishes one family-symmetric combination and leaves a degenerate pair. This indicates that additional \(S_3\)-breaking effects, or vacuum expectation values in the orthogonal Higgs directions, are needed to lift the degeneracy.

It is instructive to compare \(\bar{M}\) with the democratic mass matrix \(M_{\mathrm{dem}}=c\,\mathbf{J}_3\) familiar from the \(S_3\) flavour literature [13], [16]. The democratic matrix has eigenvalues \((3c,0,0)\), giving two exactly massless generations in the symmetric limit. By contrast, \(\bar{M}=\tfrac{3}{8}\mathbf{I}_3+\tfrac{1}{8}\mathbf{J}_3\) has eigenvalues \((\tfrac{3}{4},\tfrac{3}{8},\tfrac{3}{8})\), so no generation is massless even before \(S_3\) breaking. The crucial difference is the identity component \(\tfrac{3}{8}\mathbf{I}_3\), which arises from the intra-generation trace pairings. Moreover, the ratio of diagonal to off-diagonal entries is fixed at \(4:1\) by the algebraic construction. In many conventional \(S_3\) flavour models, by contrast, such a ratio is not fixed by the family symmetry alone, but depends on the Yukawa parameters, Higgs assignments, and vacuum expectation values [13]–[15], [28].

It is also worth noting that \(\bar M\) is not of the fully democratic form, for which all entries are equal and the eigenvalues are proportional to \((3,0,0)\). Instead, the diagonal and off-diagonal entries take the specific values \(\frac{1}{2}\) and \(\frac{1}{8}\), fixed by the algebraic construction. In this sense the present framework differs from conventional \(S_3\) flavour models, in which the texture is typically parametrised phenomenologically and the matrix entries remain free up to symmetry constraints. Here the relative pattern of entries is not chosen by hand but follows directly from the underlying algebra.

5.4 FCNCs in the exact symmetric limit↩︎

The nonzero off-diagonal entries in the algebraic generation basis do not by themselves imply tree-level FCNCs. In the exact \(S_3\)-invariant limit there is only one neutral Yukawa matrix for each fermion charge sector. If electroweak symmetry breaking is implemented in the usual way, so that the neutral component of the corresponding Higgs doublet acquires a vacuum expectation value, then the fermion mass matrix for that sector is proportional to this same Yukawa matrix. The neutral Higgs couplings are therefore aligned with the mass matrix, and are diagonalised by the same change of fermion basis. Thus, in the exact \(S_3\)-invariant limit, no tree-level FCNCs are expected to arise. At the same time, this exact \(S_3\)-invariant limit is likely too restrictive to account for realistic flavour, thereby motivating the need for \(S_3\)-breaking.

A more complete understanding of this limit will require an analysis of the scalar potential and vacuum structure in the symmetric and orthogonal Higgs directions. We leave this to future work.

6 Discussion and Outlook↩︎

The present paper establishes the Higgs sector of the algebraic \(\mathbb{C}\ell(10)\) three-generation model. Starting from the weak-doublet and weak-singlet fermion sectors, we identified Higgs components as right-action operators and used a trace pairing to extract the corresponding Yukawa coefficients. This yields two first-generation Higgs doublets with the correct electroweak quantum numbers and a natural Type-II-like separation between down-type and up-type Yukawa channels. Acting with the \(S_3\) family symmetry then triplicates this Higgs sector, producing six Higgs doublets organised into \(S_3\)-orbits. We showed that these Higgs states are linearly independent, and analysed the exact \(S_3\)-invariant limit.

These results point to a potentially important advantage of the algebraic approach. In conventional flavour model building, the family symmetry, Higgs-family structure, and Yukawa textures are typically specified at the model-building level, subject to phenomenological and symmetry constraints. Here, by contrast, the same algebraic framework that organises the three fermion generations also constrains the Higgs sector and its Yukawa couplings. In particular, the relative pattern of the generation-space Yukawa matrices is fixed by the algebraic construction rather than chosen by hand. The exact \(S_3\)-invariant limit therefore provides a controlled and nontrivial starting point for flavour phenomenology. If electroweak symmetry breaking is implemented in the usual way, so that the corresponding fermion mass matrix is proportional to the \(S_3\)-invariant Yukawa matrix \(\bar M\), then this limit already yields a nontrivial mass pattern with three nonzero eigenvalues, although two remain degenerate.

In the exact \(S_3\)-symmetric limit discussed in this paper, the Higgs sector may be written in terms of the two \(S_3\)-invariant Higgs doublets \(\bar H_d\) and \(\bar H_u\). This limit is useful because it preserves the Type-II structure of the Yukawa sector and, under the usual implementation of electroweak symmetry breaking, does not lead to tree-level FCNCs. At the same time, it is likely too restrictive to account for realistic flavour. A realistic flavour sector is therefore expected to arise only once one goes beyond the exact \(S_3\)-invariant limit and studies the effects of \(S_3\) breaking.

This situation is closely analogous to the existing \(S_3\) flavour literature. In known \(S_3\) extensions of the Standard Model, realistic fermion masses and mixing angles also require nontrivial Higgs-family structure, vacuum alignment, or explicit or soft breaking of the family symmetry [14], [29]–[31]. The important difference in the present algebraic model is that the underlying \(S_3\) family structure, and with it the associated Higgs-family structure, is not imposed ad hoc, but is motivated directly by the underlying algebraic construction. In this sense, the present framework has the potential to provide a more constrained and less ad hoc starting point for flavour model building.

A particularly interesting direction for future work is therefore to develop the present Higgs construction further in close contact with existing \(S_3\) flavour models. Since the present algebraic model already contains a family-resolved Higgs structure before one passes to the \(S_3\)-invariant combination, it is natural to investigate whether known phenomenological mechanisms can be implemented here in a less ad hoc way. A further direction, beyond the immediate Higgs-family analysis pursued here, is to compare the present framework with tri-hypercharge constructions [32], [33]. In such models, family-dependent weak hypercharges and staged symmetry breaking can generate nontrivial flavour textures. Since the present algebraic framework already contains a family-resolved hypercharge structure before one passes to the \(S_3\)-invariant combination, it is natural to ask whether similar flavour-generating mechanisms can emerge here from the algebraic construction itself, rather than being imposed phenomenologically.

7 Algebraic Identification of First Generation of Fermions↩︎

8 Summary Table of Allowed and Forbidden Yukawa Channels↩︎

For the first-generation Higgs components, the allowed and forbidden Yukawa channels are summarised in Tables 3 and 4. The same pattern is inherited by the \(\psi_3\)-images and by the \(S_3\)-invariant Higgs doublets, with only the nonzero coefficients being rescaled in the latter case.

9 \(\psi_3\) Images of Higgs States↩︎

10 \(\psi_3\) Images of Fermion States↩︎

References↩︎