Introduction
The Standard Model (SM) offers no explanation for the broad spread of fermion masses among the three generations. According to the PDG [1 ] , at the scale M Z M_Z M Z m e : m μ : m τ ≈ 1 : 210.7 : 3578 m_e : m_\mu: m_\tau\approx1 : 210.7 : 3578 m e : m μ : m τ ≈ 1 : 210.7 : 3578 1 : 19.9 : 1058 1 : 19.9 : 1058 1 : 19.9 : 1058 1 : 506 : 1.374 × 10 5 1 : 506 : 1.374 \times10^5 1 : 506 : 1.374 × 1 0 5
The classic answer to this puzzle is to impose a flavor symmetry, weakly broken by one or several spurions. Froggatt and Nielsen introduced the first systematic mechanism of this type [2 ] , in which a spontaneously broken Abelian flavor symmetry U ( 1 ) F N \mathrm{U}(1)_{\mathrm{FN}} U ( 1 ) FN ϕ \phi ϕ ⟨ ϕ ⟩ / Λ \langle\phi\rangle/\Lambda ⟨ ϕ ⟩ /Λ A 4 A_4 A 4 S 4 S_4 S 4 Δ ( 27 ) \Delta(27) Δ ( 27 ) [3 , 4 ] , and more recently modular invariance has been made to play the same role [5 ] . Further ideas where the mechanisms that do not rest on a symmetry at all, such as wave-function localisation in extra dimensions [6 ] or partial compositeness [7 ] , generate the hierarchy from exponentially small overlaps or from anomalous dimensions instead.
Over the past decade it has become clear that quantum field theories generically admit symmetries that are not groups at all [8 ] . They are implemented by topological defect operators which fuse with non-group-like rules, so that the defects need not be invertible and the fusion is governed by a fusion category rather than by a group [9 , 10 , 11 , 12 , 13 , 14 ] . Explicit non-invertible duality defects in four dimensions were constructed in refs. [15 , 16 ] . In perturbative string theory, worldsheet non-invertible symmetries govern selection rules on spacetime amplitudes through the loop-order structure of genus-g g g [17 , 18 ] , to control the smallness of neutrino masses [19 ] , and more generally to generate exponential hierarchies [20 ] .
The algebraic structure that forms the backbone of such a symmetry is called a fusion algebra A \mathcal{A} A
The fusion algebra, and its attendant selection rules are presumably the low-energy residue of a symmetry of the high-energy theory. In one realization, loops of the heavy fields generate the hierarchy of entries in the Yukawa matrix; in another, the same entries are generated by insertions of spurion vacuum expectation values. Regardless of the mechanism, after integrating out the heavy fields, we then need to run the Yukawa couplings down to low energies. We don’t want the texture generated at high energies to be wiped out by the RG running. For the up Yukawa matrix, the 1-loop running coupling is
Y u ( E ) = Y u ( Λ ) + ( Y u Y u † − Y d Y d † ) Y u 3 32 π 2 log ( E / Λ ) + ⋯ (1) Y_u(E) = Y_u(\Lambda) + (Y_uY_u^\dagger- Y_dY_d^\dagger)Y_u \frac{3}{32\pi^2}\log(E/\Lambda) + \cdots \tag*{(1)} Y u ( E ) = Y u ( Λ ) + ( Y u Y u † − Y d Y d † ) Y u 32 π 2 3 log ( E /Λ ) + ⋯ (1) where the “⋯ \cdots ⋯
( Y u Y u † − Y d Y d † ) Y u ≲ Y u (2) (Y_uY_u^\dagger- Y_dY_d^\dagger)Y_u \lesssim Y_u \tag*{(2)} ( Y u Y u † − Y d Y d † ) Y u ≲ Y u (2) where by “≲ \lesssim ≲ ϵ \epsilon ϵ ϵ \epsilon ϵ (2) is replaced by
( Y d Y d † − Y u Y u † ) Y d ≲ Y d , Y e Y e † Y e ≲ Y e (3) (Y_dY_d^\dagger- Y_uY_u^\dagger)Y_d \lesssim Y_d,\qquad Y_eY_e^\dagger Y_e \lesssim Y_e \tag*{(3)} ( Y d Y d † − Y u Y u † ) Y d ≲ Y d , Y e Y e † Y e ≲ Y e (3) (1.2) are nontrivial constraints on the form of the textures of the Yukawa matrices. In Appendix § 14 , we show that they follow from the structure of the fusion when the Higgs is assigned an invertible element of the fusion algebra.
A simple way to imbue a Lagrangian with a fusion algebra structure is to use conjugacy classes of a finite group G G G G G G G G G [21 , 22 , 23 ] . In order to effectively explain the texture of masses within the three generations of the SM, we need groups whose conjugacy classes closes at least after two steps. This severely restricts the possible candidate groups. Fusion algebras generalise this construction, and the phenomenology of the resulting Yukawa textures has begun to be explored in refs. [24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 ] .
In this paper, we set up the fusion-algebra selection rule for Yukawa operators and make explicit the one structural concept that organises everything that follows. With the entries of the Yukawa matrices drawn from a geometric hierarchy { 1 , ϵ , ϵ 2 , … , ϵ cl ( A ) } \{1,\epsilon,\epsilon^2,\ldots,\epsilon^{\operatorname{cl}(\mathcal{A})}\} { 1 , ϵ , ϵ 2 , … , ϵ cl ( A ) } ϵ cl ( A ) \epsilon^{\operatorname{cl}(\mathcal{A})} ϵ cl ( A ) O ( 1 ) \mathcal{O}(1) O ( 1 ) cl ( A ) \operatorname{cl}(\mathcal{A}) cl ( A ) A \mathcal{A} A cl ( A ) = 3 \operatorname{cl}(\mathcal{A})=3 cl ( A ) = 3 cl ( A ) = 3 \operatorname{cl}(\mathcal{A})=3 cl ( A ) = 3 ≤ 10 \leq10 ≤ 10 O ( 1 ) \mathcal{O}(1) O ( 1 )
The paper is organized as follows. In Sec. § 2 we review fusion algebras and define the commutator filtration Com ( A ) L \operatorname{Com}(\mathcal{A})^L Com ( A ) L § 3 turns these rules into Yukawa textures for the leptons, quarks and neutrinos, and introduces the reach bound and the scoring metrics. In § 4 we show that algebras with cl ( A ) = 2 \operatorname{cl}(\mathcal{A})=2 cl ( A ) = 2 ≤ 7 \leq7 ≤ 7 § 5 analyses the unique cl = 3 \operatorname{cl}=3 cl = 3 ≤ 9 \leq9 ≤ 9 F R 8 , 6 8 , 6 \mathrm{FR}_{8,6}^{8,6} FR 8 , 6 8 , 6 § 7 with discussion and open questions. The appendices collect the fusion tables in the labelling we have used, the detailed cl ( A ) = 2 \operatorname{cl}(\mathcal{A})=2 cl ( A ) = 2 § 14 ).
The Fusion-Algebra Framework
Let us begin by briefly describing what a fusion algebra is and how it can be used as an organizational tool for effective field theories. A fusion algebra is a finite-dimensional associative algebra A \mathcal{A} A C \mathbb{C} C
a ⋅ b = ∑ c ∈ A N a b c c (4) a \cdot b = \sum_{c\in\mathcal{A}} N_{ab}^{c}c \tag*{(4)} a ⋅ b = c ∈ A ∑ N ab c c (4) with N a b c N_{ab}^{c} N ab c multiplicity-free fusion algebras, for which N a b c ∈ { 0 , 1 } N_{ab}^{c}\in\{0,1\} N ab c ∈ { 0 , 1 } c ≺ a ⋅ b c\prec a\cdot b c ≺ a ⋅ b N a b c = 1 N_{ab}^{c}=1 N ab c = 1 c c c a a a b b b
The multiplicative identity e e e A \mathcal{A} A A \mathcal{A} A N e b a = N b e a = δ b a N_{eb}^{a}=N_{be}^{a}=\delta_{b}^{a} N e b a = N b e a = δ b a
A \mathcal{A} A a ∈ A a\in\mathcal{A} a ∈ A a ˉ ∈ A \bar{a}\in\mathcal{A} a ˉ ∈ A N a a ˉ e = N a ˉ a e = 1 N_{a\bar{a}}^{e}=N_{\bar{a}a}^{e}=1 N a a ˉ e = N a ˉ a e = 1
Frobenius Reciprocity : N a b c = N a c b = N c b a N_{ab}^{c}=N_{ac}^{b}=N_{cb}^{a} N ab c = N a c b = N c b a ∀ a , b , c ∈ A \forall a,b,c\in\mathcal{A} ∀ a , b , c ∈ A
The number of basis elements r = ∣ A ∣ r=|\mathcal{A}| r = ∣ A ∣ rank of the fusion algebra. Fusion algebras of this kind arise as the Grothendieck rings of fusion categories [35 ] , and contain Rep ( G ) \operatorname{Rep}(G) Rep ( G ) Conj ( G ) \operatorname{Conj}(G) Conj ( G ) [36 ] as special cases.
Now suppose we are given a theory with a set of fields ϕ 1 , ϕ 2 , … , ϕ n \phi_1,\phi_2,\ldots,\phi_n ϕ 1 , ϕ 2 , … , ϕ n ϕ [ a ] \phi[a] ϕ [ a ] a ∈ A a \in A a ∈ A ϕ ∗ [ a ˉ ] \phi^*[\bar{a}] ϕ ∗ [ a ˉ ]
Selection rules from fusion algebras: Given an assignment of the basis elements to the respective fields, we allow a tree level vertex O = ϕ 1 ϕ 2 ⋯ ϕ n \mathcal{O} = \phi_1\phi_2\cdots\phi_n O = ϕ 1 ϕ 2 ⋯ ϕ n ϕ i [ a i ] \phi_i[a_i] ϕ i [ a i ]
e ≺ a 1 ⋅ a 2 ⋅ … ⋅ a n (5) e \prec a_1 \cdot a_2 \cdot\ldots\cdot a_n \tag*{(5)} e ≺ a 1 ⋅ a 2 ⋅ … ⋅ a n (5) i.e. the product a 1 ⋅ a 2 ⋅ … ⋅ a n a_1 \cdot a_2 \cdot\ldots\cdot a_n a 1 ⋅ a 2 ⋅ … ⋅ a n e e e
What is new here is that, unlike group representations — whose selection rules are the same at all loop orders — the selection rules of a fusion algebra can differ from one loop order to the next1 Com ( A ) L \operatorname{Com}(A)^L Com ( A ) L L L L
Com ( A ) = { a ∈ A ∣ a ≺ b ⋅ b ˉ for some b ∈ A } . (6) \operatorname{Com}(A) = \{a \in A \mid a \prec b \cdot\bar{b}\text{ for some }b \in A\}. \tag*{(6)} Com ( A ) = { a ∈ A ∣ a ≺ b ⋅ b ˉ for some b ∈ A } . (6) and inductively
Com ( A ) L = { a ∈ A ∣ a ≺ b 1 ⋅ … ⋅ b L for some b i ∈ Com ( A ) } (7) \operatorname{Com}(A)^L = \{a \in A \mid a \prec b_1 \cdot\ldots\cdot b_L\text{ for some }b_i \in\operatorname{Com}(A)\} \tag*{(7)} Com ( A ) L = { a ∈ A ∣ a ≺ b 1 ⋅ … ⋅ b L for some b i ∈ Com ( A )} (7) Note that if a ∈ Com ( A ) L a \in\operatorname{Com}(A)^L a ∈ Com ( A ) L a ˉ ∈ Com ( A ) L \bar{a} \in\operatorname{Com}(A)^L a ˉ ∈ Com ( A ) L a , a ˉ ∈ Com ( A ) L + 1 a,\bar{a} \in\operatorname{Com}(A)^{L+1} a , a ˉ ∈ Com ( A ) L + 1
Loop level selection rules: With these definitions, we can now state the loop level selection rules. A vertex O = ϕ 1 ϕ 2 ⋯ ϕ n \mathcal{O} = \phi_1\phi_2\cdots\phi_n O = ϕ 1 ϕ 2 ⋯ ϕ n L L L b ˉ ∈ Com ( A ) L \bar{b} \in\operatorname{Com}(A)^L b ˉ ∈ Com ( A ) L
e ≺ a 1 ⋅ a 2 ⋅ … ⋅ a n ⋅ b ˉ (8) e \prec a_1 \cdot a_2 \cdot\ldots\cdot a_n \cdot\bar{b} \tag*{(8)} e ≺ a 1 ⋅ a 2 ⋅ … ⋅ a n ⋅ b ˉ (8) This is equivalent to the condition,
∃ b ∈ Com ( A ) L s.t. b ≺ a 1 ⋅ a 2 ⋅ … ⋅ a n . (9) \exists b \in\operatorname{Com}(A)^L \text{ s.t. } b \prec a_1 \cdot a_2 \cdot\ldots\cdot a_n. \tag*{(9)} ∃ b ∈ Com ( A ) L s.t. b ≺ a 1 ⋅ a 2 ⋅ … ⋅ a n . (9) For our purpose, we will only consider fusion algebras that admit new selection rules up to three loops. This is mathematically stated as cl ( A ) = 3 \operatorname{cl}(A) = 3 cl ( A ) = 3 cl ( A ) \operatorname{cl}(A) cl ( A ) L L L Com ( A ) L = Com ( A ) L + 1 \operatorname{Com}(A)^L = \operatorname{Com}(A)^{L+1} Com ( A ) L = Com ( A ) L + 1 N a b c ∈ { 0 , 1 } N^c_{ab} \in\{0,1\} N ab c ∈ { 0 , 1 } [22 ] compiled all possible independent fusion algebras up to rank 7. We first scan over these algebras and then turn to the longer list of fusion algebras up to rank 10 compiled at [37 ] (see [38 ] for an introduction). For cl ( A ) ≤ 3 \operatorname{cl}(A) \le3 cl ( A ) ≤ 3
Yukawa textures from Fusion Algebras
We now outline how one can use a fusion algebra to generate Yukawa textures.
Charged-Lepton Yukawa Textures
First, consider the lepton sector of the Standard Model. The term that is responsible for the lepton masses is the Yukawa coupling term
L ⊃ Y i j e H ˉ l i e j c + h . c . (10) \mathcal{L} \supset Y^e_{ij}\bar{H}l_i e^c_j + \mathrm{h.c.} \tag*{(10)} L ⊃ Y ij e H ˉ l i e j c + h.c. (10) where H H H l i l_i l i e j c e^c_j e j c Y i j e Y^e_{ij} Y ij e [39 ] . The masses of the leptons are proportional to the singular values of the Yukawa matrix Y e Y^e Y e ∣ ⟨ H ⟩ ∣ |\langle H\rangle| ∣ ⟨ H ⟩ ∣ m e : m μ : m τ ≈ 1 : 210.7 : 3578 m_e : m_\mu: m_\tau\approx1 : 210.7 : 3578 m e : m μ : m τ ≈ 1 : 210.7 : 3578
H ∗ [ a ] , l i [ b i ] , e j c [ c j ] (11) H^*[a], \quad l_i[b_i], \quad e^c_j[c_j] \tag*{(11)} H ∗ [ a ] , l i [ b i ] , e j c [ c j ] (11) where a , b i , c j ∈ A a,b_i,c_j \in\mathcal{A} a , b i , c j ∈ A
The entry Y i j e Y^e_{ij} Y ij e
If e ≺ a b i c j e \prec ab_i c_j e ≺ a b i c j Y i j e Y^e_{ij} Y ij e Y i j e ∼ 1 Y^e_{ij} \sim1 Y ij e ∼ 1
If it’s not allowed at tree level but ∃ w ∈ Com ( A ) \exists w \in\operatorname{Com}(\mathcal{A}) ∃ w ∈ Com ( A ) w ≺ a b i c j w \prec ab_i c_j w ≺ a b i c j Y i j e Y^e_{ij} Y ij e Y i j e ∼ ε Y^e_{ij} \sim\varepsilon Y ij e ∼ ε
If it’s not allowed at tree level or one-loop level but ∃ w ∈ Com ( A ) 2 \exists w \in\operatorname{Com}(\mathcal{A})^2 ∃ w ∈ Com ( A ) 2 w ≺ a b i c j w \prec ab_i c_j w ≺ a b i c j Y i j e ∼ ε 2 Y^e_{ij} \sim\varepsilon^2 Y ij e ∼ ε 2
If it’s not allowed at tree level, one-loop or two-loop levels but ∃ w ∈ Com ( A ) 3 \exists w \in\operatorname{Com}(\mathcal{A})^3 ∃ w ∈ Com ( A ) 3 w ≺ a b i c j w \prec ab_i c_j w ≺ a b i c j Y i j e ∼ ε 3 Y^e_{ij} \sim\varepsilon^3 Y ij e ∼ ε 3
Since all the algebras we consider are such that cl ( A ) ≤ 3 \operatorname{cl}(\mathcal{A}) \le3 cl ( A ) ≤ 3
Benchmark value of ε \varepsilon ε In order to compare the singular values to the observed mass ratios, we need to assign a value for ε \varepsilon ε
ε = m e m τ ≈ 0.017 (12) \varepsilon= \sqrt{\frac{m_e}{m_\tau}} \approx0.017 \tag*{(12)} ε = m τ m e ≈ 0.017 (12) In the remainder of the paper we quote every predicted mass ratio and mixing element in two ways: as the decimal obtained by substituting the benchmark ε = 0.017 \varepsilon= 0.017 ε = 0.017 leading power of ε \varepsilon ε . The decimals allow a direct comparison with the measured values, while the powers of ε \varepsilon ε
An example calculation
To illustrate the procedure, consider the algebra C.9 which has a commutator length cl ( A ) = 2 \operatorname{cl}(A)=2 cl ( A ) = 2 { 1 , 2 , 3 , 4 , 5 , 6 } \{1,2,3,4,5,6\} { 1 , 2 , 3 , 4 , 5 , 6 }
Table 11.
Let us make the following assignment:
H ∗ [ 1 ] , l 1 [ 4 ] , l 2 [ 1 ] , l 3 [ 3 ] . e 1 c [ 6 ] , e 2 c [ 2 ] , e 3 c [ 3 ] (13) H^{*}[1],\quad l_{1}[4],l_{2}[1],l_{3}[3].\quad e^{c}_{1}[6],e^{c}_{2}[2],e^{c}_{3}[3] \tag*{(13)} H ∗ [ 1 ] , l 1 [ 4 ] , l 2 [ 1 ] , l 3 [ 3 ] . e 1 c [ 6 ] , e 2 c [ 2 ] , e 3 c [ 3 ] (13) Since H H H l i e j c l_i e^{c}_j l i e j c Com ( C .9 ) \operatorname{Com}(\mathrm{C.9}) Com ( C.9 )
Com ( C .9 ) = { 1 , 4 , 5 } (14) \operatorname{Com}(\mathrm{C.9})=\{1,4,5\} \tag*{(14)} Com ( C.9 ) = { 1 , 4 , 5 } (14) We then list the different products and the order in which it appears:
Table 12.
Therefore the resulting Yukawa matrix has the texture
Y e ∼ ( ε ε ε 2 ε 2 ε 2 ε 2 ε 1 ε 2 ) (15) Y^{e}\sim\begin{pmatrix}
\varepsilon& \varepsilon& \varepsilon^{2}\\
\varepsilon^{2} & \varepsilon^{2} & \varepsilon^{2}\\
\varepsilon& 1 & \varepsilon^{2}
\end{pmatrix} \tag*{(15)} Y e ∼ ε ε 2 ε ε ε 2 1 ε 2 ε 2 ε 2 (15) The singular values of this matrix are in the ratio
m e : m μ : m τ ≈ 1 : 58.8 : 3522 ( ∼ 1 : ε − 1 : ε − 2 ) (16) m_{e}:m_{\mu}:m_{\tau}\approx1:58.8:3522\qquad\left(\sim1:\varepsilon^{-1}:\varepsilon^{-2}\right) \tag*{(16)} m e : m μ : m τ ≈ 1 : 58.8 : 3522 ( ∼ 1 : ε − 1 : ε − 2 ) (16) for the benchmark value of ε \varepsilon ε O ( 1 ) \mathcal{O}(1) O ( 1 )
The quark sector
The lepton sector alone is not sufficient to test the applicability of this framework. Applying it in the quark sector is a stronger test for three reasons. First, there are two independent quark textures, for the up-type and down-type quarks. Second, the up quarks show a wider spread compared to the leptons and the down quarks. Third, quark mixing provides additional observables (in the form of the CKM matrix) that can be compared with predictions in addition to masses. The terms that are responsible for the quark masses are the Yukawa coupling terms
L ⊃ Y i j u H q i u j c + Y i j d H ˉ q i d j c + h . c . (17) \mathcal{L} \supset Y^{u}_{ij}Hq_i u^{c}_j + Y^{d}_{ij}\bar{H}q_i d^{c}_j + \mathrm{h.c.} \tag*{(17)} L ⊃ Y ij u H q i u j c + Y ij d H ˉ q i d j c + h.c. (17) where q i q_i q i u j c , d j c u^{c}_j,d^{c}_j u j c , d j c Y i j u , Y i j d Y^{u}_{ij},Y^{d}_{ij} Y ij u , Y ij d ∣ ⟨ H ⟩ ∣ \lvert\langle H\rangle\rvert ∣⟨ H ⟩∣ m u : m c : m t ≈ 1 : 506 : 137400 m_u:m_c:m_t \approx1:506:137400 m u : m c : m t ≈ 1 : 506 : 137400 m d : m s : m b ≈ 1 : 19.9 : 1058 m_d:m_s:m_b \approx1:19.9:1058 m d : m s : m b ≈ 1 : 19.9 : 1058
The neutrino sector
Finally, one can apply the fusion algebra approach to generate neutrino mass hierarchies. We assume that the neutrino masses are generated by the Weinberg operator
O W ( i j ) ∼ Y ν i j Λ ( l i H ) ( l j H ) . (18) \mathcal{O}^{(ij)}_{W} \sim\frac{Y^{ij}_{\nu}}{\Lambda}(l_iH)(l_jH). \tag*{(18)} O W ( ij ) ∼ Λ Y ν ij ( l i H ) ( l j H ) . (18) The neutrino sector is worth including since it is fixed by the lepton and Higgs labels that we have already chosen for the lepton sector. On the other hand, this is one place where the framework is applied to a higher dimensional operator. Given a representation for the leptons l i l_i l i H H H M ν i j = v 2 Λ Y ν i j M^{ij}_{\nu} = \frac{v^2}{\Lambda}Y^{ij}_{\nu} M ν ij = Λ v 2 Y ν ij
The Weinberg operator is a four-point operator, built from two lepton doublets l i [ b i ] l_i[b_i] l i [ b i ] H [ a ˉ ] H[\bar{a}] H [ a ˉ ] H H H H ∗ H^* H ∗ [ a ] [a] [ a ] H ∗ H^* H ∗ [ a ˉ ] [\bar{a}] [ a ˉ ] H H H a ˉ ⋅ a ˉ ⋅ b i ⋅ b j \bar{a}\cdot\bar{a}\cdot b_i\cdot b_j a ˉ ⋅ a ˉ ⋅ b i ⋅ b j L L L w ∈ Com ( A ) L w\in\operatorname{Com}(\mathcal{A})^L w ∈ Com ( A ) L w ≺ a ˉ ⋅ a ˉ ⋅ b i ⋅ b j w\prec\bar{a}\cdot\bar{a}\cdot b_i\cdot b_j w ≺ a ˉ ⋅ a ˉ ⋅ b i ⋅ b j
The reach of a texture
The loop-order dictionary has an immediate consequence. Let Y Y Y 3 × 3 3\times3 3 × 3 Y i j = c i j ε n i j Y_{ij}=c_{ij}\varepsilon^{n_{ij}} Y ij = c ij ε n ij 0 ≤ n i j ≤ cl ( A ) 0\le n_{ij}\le\operatorname{cl}(\mathcal{A}) 0 ≤ n ij ≤ cl ( A ) O ( 1 ) \mathcal{O}(1) O ( 1 ) c i j c_{ij} c ij O ( 1 ) \mathcal{O}(1) O ( 1 ) O ( ε cl ( A ) ) \mathcal{O}(\varepsilon^{\operatorname{cl}(\mathcal{A})}) O ( ε cl ( A ) )
m 3 m 1 ≲ ε − cl ( A ) . (19) \frac{m_3}{m_1} \lesssim\varepsilon^{-\operatorname{cl}(\mathcal{A})}. \tag*{(19)} m 1 m 3 ≲ ε − cl ( A ) . (19) We refer to ε − cl ( A ) \varepsilon^{-\operatorname{cl}(\mathcal{A})} ε − cl ( A ) reach of the algebra. At the benchmark ε = 0.017 \varepsilon= 0.017 ε = 0.017 3.5 × 10 3 3.5 \times10^3 3.5 × 1 0 3 cl ( A ) = 2 \operatorname{cl}(\mathcal{A}) = 2 cl ( A ) = 2 2.0 × 10 5 2.0 \times10^5 2.0 × 1 0 5 cl ( A ) = 3 \operatorname{cl}(\mathcal{A}) = 3 cl ( A ) = 3
m τ m e ≈ 3.6 × 10 3 , m b m d ≈ 1.1 × 10 3 , m t m u ≈ 1.4 × 10 5 , (20) \frac{m_\tau}{m_e} \approx3.6 \times10^3,\qquad
\frac{m_b}{m_d} \approx1.1 \times10^3,\qquad
\frac{m_t}{m_u} \approx1.4 \times10^5, \tag*{(20)} m e m τ ≈ 3.6 × 1 0 3 , m d m b ≈ 1.1 × 1 0 3 , m u m t ≈ 1.4 × 1 0 5 , (20) we see that the charged-lepton and down-type sectors sit comfortably inside the cl ( A ) = 2 \operatorname{cl}(\mathcal{A}) = 2 cl ( A ) = 2 ε − 3 \varepsilon^{-3} ε − 3 (19) can of course be violated at non-generic coefficients, and § 4 shows that this is exactly what happens for C.9 — which is why the bound must be tested against an ensemble of coefficients rather than at c i j = 1 c_{ij} = 1 c ij = 1
Scoring metrics
To compare algebras and assignments we use the following measures throughout. For a sector whose predicted singular values are in the ratio 1 : α : β 1 : \alpha: \beta 1 : α : β 1 : α o b s : β o b s 1 : \alpha_{\mathrm{obs}} : \beta_{\mathrm{obs}} 1 : α obs : β obs
ρ = ∣ log 10 α α o b s ∣ + ∣ log 10 β β o b s ∣ , (21) \rho= \left|\log_{10}\frac{\alpha}{\alpha_{\mathrm{obs}}}\right| + \left|\log_{10}\frac{\beta}{\beta_{\mathrm{obs}}}\right|, \tag*{(21)} ρ = log 10 α obs α + log 10 β obs β , (21) so that ρ = 0 \rho= 0 ρ = 0 ρ \rho ρ ρ ℓ \rho_\ell ρ ℓ ρ u \rho_u ρ u ρ d \rho_d ρ d
ρ C K M = 1 9 ∑ i , j ∣ log 10 ∣ V i j a l g ∣ ∣ V i j o b s ∣ ∣ , (22) \rho_{\mathrm{CKM}} = \frac{1}{9}\sum_{i,j}\left|\log_{10}\frac{\left|V_{ij}^{\mathrm{alg}}\right|}{\left|V_{ij}^{\mathrm{obs}}\right|}\right|, \tag*{(22)} ρ CKM = 9 1 i , j ∑ log 10 V ij obs V ij alg , (22) and we combine the two quark-mass scores into
ρ m a s s = ρ u 2 + ρ d 2 2 . (23) \rho_{\mathrm{mass}} = \sqrt{\frac{\rho_u^2 + \rho_d^2}{2}}. \tag*{(23)} ρ mass = 2 ρ u 2 + ρ d 2 . (23) Finally, the combined figure of merit used to select a single assignment across all sectors is
ρ t o t = 1 4 ( ρ ℓ 2 + ρ u 2 + ρ d 2 + ρ C K M 2 ) . (24) \rho_{\mathrm{tot}} = \sqrt{\frac{1}{4}\left(\rho_\ell^2 + \rho_u^2 + \rho_d^2 + \rho_{\mathrm{CKM}}^2\right)}. \tag*{(24)} ρ tot = 4 1 ( ρ ℓ 2 + ρ u 2 + ρ d 2 + ρ CKM 2 ) . (24) Beyond unit Wilson coefficients
Every ratio quoted from a texture alone assumes coefficients of the form 1 × ε n 1 \times\varepsilon^n 1 × ε n O ( 1 ) \mathcal{O}(1) O ( 1 )
Y i j = s i j ∣ c i j ∣ ε n i j , log ∣ c i j ∣ ∼ U [ − log 2.5 , + log 2.5 ] , s i j = ± 1 (25) Y_{ij} = s_{ij}|c_{ij}|\varepsilon^{n_{ij}},\qquad\log|c_{ij}| \sim U[-\log2.5,+\log2.5],\qquad s_{ij} = \pm1 \tag*{(25)} Y ij = s ij ∣ c ij ∣ ε n ij , log ∣ c ij ∣ ∼ U [ − log 2.5 , + log 2.5 ] , s ij = ± 1 (25) and quote the median and the 68% band of every observable over the ensemble. The difference between the median and the value at c i j = 1 c_{ij}=1 c ij = 1 § 4.2 it is what disqualifies the cl ( A ) = 2 \operatorname{cl}(\mathcal{A})=2 cl ( A ) = 2
cl(A)=2 Selection Rules and Their Limits
Every multiplicity-free fusion algebra of rank ≤ 7 \le7 ≤ 7 [22 ] has cl ( A ) ≤ 2 \operatorname{cl}(\mathcal{A})\le2 cl ( A ) ≤ 2 § 11 , and the fusion tables in Appendix § 8 .
What a cl(A)=2 rule achieves
Scanning all assignments of ( H , ℓ i , e j c , q i , u j c , d j c ) (H,\ell_i,e^c_j,q_i,u^c_j,d^c_j) ( H , ℓ i , e j c , q i , u j c , d j c ) (21) , the outcome is easy to state.
A two-stage rule produces a two-stage hierarchy. Entries in { 1 , ε , ε 2 } \{1,\varepsilon,\varepsilon^2\} { 1 , ε , ε 2 } 1 : ε − 1 : ε − 2 1:\varepsilon^{-1}:\varepsilon^{-2} 1 : ε − 1 : ε − 2
One algebra stands out: the rank-6, non-categorifiable algebra C.9. With H ∗ [ 1 ] H^*[1] H ∗ [ 1 ] ℓ i = ( 4 , 1 , 3 ) \ell_i=(4,1,3) ℓ i = ( 4 , 1 , 3 ) e j c = ( 6 , 2 , 3 ) e^c_j=(6,2,3) e j c = ( 6 , 2 , 3 ) m e : m μ : m τ ≈ 1 : 58.8 : 3522 m_e:m_\mu:m_\tau\approx1:58.8:3522 m e : m μ : m τ ≈ 1 : 58.8 : 3522 1 : 210.7 : 3578 1:210.7:3578 1 : 210.7 : 3578 ρ ℓ = 0.561 \rho_\ell=0.561 ρ ℓ = 0.561 q i = ℓ i q_i=\ell_i q i = ℓ i d j c = e j c d^c_j=e^c_j d j c = e j c m d : m s : m b ≈ 1 : 58.8 : 3522 m_d:m_s:m_b\approx1:58.8:3522 m d : m s : m b ≈ 1 : 58.8 : 3522 1 : 19.9 : 1058 1:19.9:1058 1 : 19.9 : 1058 ρ d = 0.994 \rho_d=0.994 ρ d = 0.994 ≤ 7 \le7 ≤ 7 ε − 3 \varepsilon^{-3} ε − 3 § 11 .§ 1 and Sec. B.3.
The neutrino sector prefers the same algebra. Through the Weinberg operator of § 3.4 , C.9 gives a normal hierarchy 1 : ε : ε 2 1:\varepsilon:\varepsilon^2 1 : ε : ε 2 Δ m 31 2 ≈ 2.5 × 10 − 3 e V 2 \Delta m^2_{31}\approx2.5\times10^{-3}\,\mathrm{eV}^2 Δ m 31 2 ≈ 2.5 × 1 0 − 3 eV 2 Λ ≈ 6 × 10 14 G e V \Lambda\approx6\times10^{14}\,\mathrm{GeV} Λ ≈ 6 × 1 0 14 GeV 1 : 1 : 1 1:1:1 1 : 1 : 1 Δ m 21 2 \Delta m^2_{21} Δ m 21 2 § 11 .§ 2 ).
There is no good fit to the CKM matrix. The mass-optimal C.9 assignment gives a near-maximal 1–2 rotation, ∣ V u s ∣ ∼ 0.70 |V_{us}|\sim0.70 ∣ V u s ∣ ∼ 0.70 O ( 1 ) O(1) O ( 1 ) ρ C K M = 0.316 \rho_{\mathrm{CKM}}=0.316 ρ CKM = 0.316 ρ m a s s \rho_{\mathrm{mass}} ρ mass
Why the up-type sector fails
The C.9 up-type fit with unit Wilson coefficients, m u : m c : m t ≈ 1 : 7162 : 2.98 × 10 5 m_u:m_c:m_t\approx1:7162:2.98\times10^5 m u : m c : m t ≈ 1 : 7162 : 2.98 × 1 0 5 (19) for cl ( A ) = 2 \operatorname{cl}(\mathcal{A})=2 cl ( A ) = 2
Let us express the up type Yukawa matrix as Y i j u ≡ c i j ε n i j Y^{u}_{ij} \equiv c_{ij}\varepsilon^{n_{ij}} Y ij u ≡ c ij ε n ij
Y u ∼ ( c 11 ε c 12 ε 2 c 13 ε c 21 ε c 22 ε 2 c 23 ε 2 c 31 ε 2 c 32 ε 2 c 33 ε 0 ) (26) Y^{u} \sim\begin{pmatrix} c_{11}\varepsilon& c_{12}\varepsilon^{2} & c_{13}\varepsilon\\ c_{21}\varepsilon& c_{22}\varepsilon^{2} & c_{23}\varepsilon^{2} \\ c_{31}\varepsilon^{2} & c_{32}\varepsilon^{2} & c_{33}\varepsilon^{0} \end{pmatrix} \tag*{(26)} Y u ∼ c 11 ε c 21 ε c 31 ε 2 c 12 ε 2 c 22 ε 2 c 32 ε 2 c 13 ε c 23 ε 2 c 33 ε 0 (26) We see that
det Y u = c 33 ( c 11 c 22 − c 12 c 21 ) ε 3 + O ( ε 4 ) . (27) \det Y^{u} = c_{33}(c_{11}c_{22} - c_{12}c_{21})\varepsilon^{3} + O(\varepsilon^{4}). \tag*{(27)} det Y u = c 33 ( c 11 c 22 − c 12 c 21 ) ε 3 + O ( ε 4 ) . (27) At c i j = 1 c_{ij}=1 c ij = 1 O ( ε 4 ) O(\varepsilon^{4}) O ( ε 4 )
The results for the C.9 texture are summarized in Table 1 .
Table 1. The C.9 assignment of Secs. 3.1 and B.3 under the ensemble of Eq. (25) . Note that the median among the MC scan is O ( ε ) O(\varepsilon) O ( ε )
The charged-lepton and down-type fits are therefore genuine predictions of the selection rule. The up-type fit on the other hand is not: its median m t / m u m_{t}/m_{u} m t / m u 3.85 × 10 3 3.85 \times10^{3} 3.85 × 1 0 3 1.37 × 10 5 1.37 \times10^{5} 1.37 × 1 0 5 cl ( A ) = 2 \operatorname{cl}(\mathcal{A})=2 cl ( A ) = 2 { 1 , ε , ε 2 } \{1,\varepsilon,\varepsilon^{2}\} { 1 , ε , ε 2 } ε − cl ( A ) \varepsilon^{-\operatorname{cl}(\mathcal{A})} ε − cl ( A ) m t / m u ≈ 1.37 × 10 5 m_{t}/m_{u} \approx1.37 \times10^{5} m t / m u ≈ 1.37 × 1 0 5 ε − 3 \varepsilon^{-3} ε − 3 ε = 0.017 \varepsilon=0.017 ε = 0.017
This is the obstruction that organises the rest of the paper. It is not a statement about C.9 in particular, nor about rank: it is the reach bound (19) applied to cl ( A ) = 2 \operatorname{cl}(\mathcal{A})=2 cl ( A ) = 2 O ( 1 ) O(1) O ( 1 ) ≤ 7 \le7 ≤ 7
The unique algebra of rank≤ 9 with cl(A) = 3
In order to explain the up-quark hierarchy, we need a cl ( A ) = 3 \operatorname{cl}(\mathcal{A})=3 cl ( A ) = 3 cl ( A ) \operatorname{cl}(\mathcal{A}) cl ( A ) [40 ] . Computing the commutator length for all fusion algebras of ranks up to 9, we find that there is a unique algebra2 cl ( A ) = 3 \operatorname{cl}(\mathcal{A})=3 cl ( A ) = 3 cl ( A ) > 3 \operatorname{cl}(\mathcal{A})>3 cl ( A ) > 3 F R 8 8 , 6 \mathrm{FR}_{8}^{8,6} FR 8 8 , 6
Its invertible objects { 1 , 2 , 3 , 4 } \{1,2,3,4\} { 1 , 2 , 3 , 4 } Z 4 \mathbb{Z}_4 Z 4 3 ˉ = 4 \bar{3}=4 3 ˉ = 4 5 ˉ = 6 \bar{5}=6 5 ˉ = 6 7 ˉ = 8 \bar{7}=8 7 ˉ = 8 1 , 2 1,2 1 , 2
Com ( A ) = { 1 , 7 , 8 } ⊊ Com ( A ) 2 = { 1 , 3 , 4 , 5 , 6 , 7 , 8 } ⊊ Com ( A ) 3 = A , cl ( A ) = 3. (28) \operatorname{Com}(\mathcal{A})=\{1,7,8\}\subsetneq\operatorname{Com}(\mathcal{A})^2=\{1,3,4,5,6,7,8\}\subsetneq\operatorname{Com}(\mathcal{A})^3=\mathcal{A},\qquad\operatorname{cl}(\mathcal{A})=3. \tag*{(28)} Com ( A ) = { 1 , 7 , 8 } ⊊ Com ( A ) 2 = { 1 , 3 , 4 , 5 , 6 , 7 , 8 } ⊊ Com ( A ) 3 = A , cl ( A ) = 3. (28) The best assignment
We can now ask for whether there is an assignment that fits all the textures at once. For this, we define the combined metric,
ρ t o t = 1 4 ( ρ ℓ 2 + ρ u 2 + ρ d 2 + ρ C K M 2 ) , (29) \rho_{\mathrm{tot}}=\sqrt{\frac{1}{4}\left(\rho_{\ell}^{2}+\rho_{u}^{2}+\rho_{d}^{2}+\rho_{\mathrm{CKM}}^{2}\right)}, \tag*{(29)} ρ tot = 4 1 ( ρ ℓ 2 + ρ u 2 + ρ d 2 + ρ CKM 2 ) , (29) Then, within one draw of assignments, we build all three Yukawa matrices with a Monte-Carlo scan of the O ( 1 ) \mathcal{O}(1) O ( 1 ) (21) , form V C K M V_{\mathrm{CKM}} V CKM Y u Y^{u} Y u Y d Y^{d} Y d ρ t o t \rho_{\mathrm{tot}} ρ tot c i j = 1 c_{ij}=1 c ij = 1 F R 8 8 , 6 \mathrm{FR}_{8}^{8,6} FR 8 8 , 6
H ∗ [ 4 ] , ℓ i = ( 2 , 4 , 8 ) , e j c = ( 2 , 3 , 7 ) , q i = ( 2 , 4 , 6 ) , u j c = ( 4 , 7 , 8 ) , d j c = ( 1 , 5 , 6 ) , (30) H^{*}[4],\qquad\ell_i=(2,4,8),\qquad e^c_j=(2,3,7),\qquad q_i=(2,4,6),\qquad u^c_j=(4,7,8),\qquad d^c_j=(1,5,6), \tag*{(30)} H ∗ [ 4 ] , ℓ i = ( 2 , 4 , 8 ) , e j c = ( 2 , 3 , 7 ) , q i = ( 2 , 4 , 6 ) , u j c = ( 4 , 7 , 8 ) , d j c = ( 1 , 5 , 6 ) , (30) giving the textures
Y e ∼ ( ε 2 ε 3 ε 2 1 ε 2 ε 2 ε ε ε ) , Y u ∼ ( ε 3 ε ε 2 ε 2 ε ε ε 2 1 ε ) , Y d ∼ ( ε 2 ε 2 ε ε 3 ε ε ε 2 ε 1 ) , (31) Y^{e}\sim
\begin{pmatrix}
\varepsilon^2 & \varepsilon^3 & \varepsilon^2\\
1 & \varepsilon^2 & \varepsilon^2\\
\varepsilon& \varepsilon& \varepsilon
\end{pmatrix},
\qquad
Y^{u}\sim
\begin{pmatrix}
\varepsilon^3 & \varepsilon& \varepsilon^2\\
\varepsilon^2 & \varepsilon& \varepsilon\\
\varepsilon^2 & 1 & \varepsilon
\end{pmatrix},
\qquad
Y^{d}\sim
\begin{pmatrix}
\varepsilon^2 & \varepsilon^2 & \varepsilon\\
\varepsilon^3 & \varepsilon& \varepsilon\\
\varepsilon^2 & \varepsilon& 1
\end{pmatrix}, \tag*{(31)} Y e ∼ ε 2 1 ε ε 3 ε 2 ε ε 2 ε 2 ε , Y u ∼ ε 3 ε 2 ε 2 ε ε 1 ε 2 ε ε , Y d ∼ ε 2 ε 3 ε 2 ε 2 ε ε ε ε 1 , (31) and the scores of Table 2 .
Table 2. The assignment of Eq. (30) under the ensemble of Eq. (25) .
The generic reaches of the three textures of Eq. (31) are ε − 2 \varepsilon^{-2} ε − 2 ε − 3 \varepsilon^{-3} ε − 3 ε − 2 \varepsilon^{-2} ε − 2 Y e Y^{e} Y e Y u Y^{u} Y u Y d Y^{d} Y d Table 2 agrees with what that reach predicts. Therefore, nothing here is carried by an accidental cancellation. The top-quark hierarchy comes out at 1.05 × 10 5 1.05 \times10^5 1.05 × 1 0 5 1.37 × 10 5 1.37 \times10^5 1.37 × 1 0 5 ε − 3 \varepsilon^{-3} ε − 3
The CKM matrix for F R 8 8 , 6 \mathrm{FR}_{8}^{8,6} FR 8 8 , 6
In this section, we will discuss the CKM structure that arises from the Yukawa matrices with the textures we get from the fusion algebra structure. The CKM matrix is given by V C K M = V L u V L d † V_{\mathrm{CKM}} = V_L^u V_L^{d\dagger} V CKM = V L u V L d † V L u V_L^u V L u V L d V_L^d V L d
Y u = V L u Σ u V R u † , Y d = V L d Σ d V R d † (32) Y^u = V_L^u \Sigma^u V_R^{u\dagger}, \qquad Y^d = V_L^d \Sigma^d V_R^{d\dagger} \tag*{(32)} Y u = V L u Σ u V R u † , Y d = V L d Σ d V R d † (32) with Σ u \Sigma^u Σ u Σ d \Sigma^d Σ d Y u Y^u Y u Y d Y^d Y d V L u V_L^u V L u V L d V_L^d V L d u , c , t u,c,t u , c , t d , s , b d,s,b d , s , b
Diagonalising the two quark textures of Eq. (31) across the ensemble gives
∣ V C K M ∣ = ( 0.998 5 − 0.0046 + 0.0012 0.04 0 − 0.029 + 0.058 0.02 6 − 0.019 + 0.029 0.04 0 − 0.029 + 0.058 0.998 4 − 0.0044 + 0.0013 0.02 6 − 0.019 + 0.029 0.02 6 − 0.019 + 0.029 0.02 6 − 0.019 + 0.029 0.999 1 − 0.0020 + 0.0007 ) , ρ C K M = 0.408 , (33) \left|V_{\mathrm{CKM}}\right| =
\begin{pmatrix}
0.9985^{+0.0012}_{-0.0046} & 0.040^{+0.058}_{-0.029} & 0.026^{+0.029}_{-0.019} \\
0.040^{+0.058}_{-0.029} & 0.9984^{+0.0013}_{-0.0044} & 0.026^{+0.029}_{-0.019} \\
0.026^{+0.029}_{-0.019} & 0.026^{+0.029}_{-0.019} & 0.9991^{+0.0007}_{-0.0020}
\end{pmatrix}, \qquad\rho_{\mathrm{CKM}} = 0.408, \tag*{(33)} ∣ V CKM ∣ = 0.998 5 − 0.0046 + 0.0012 0.04 0 − 0.029 + 0.058 0.02 6 − 0.019 + 0.029 0.04 0 − 0.029 + 0.058 0.998 4 − 0.0044 + 0.0013 0.02 6 − 0.019 + 0.029 0.02 6 − 0.019 + 0.029 0.02 6 − 0.019 + 0.029 0.999 1 − 0.0020 + 0.0007 , ρ CKM = 0.408 , (33) where the error bars correspond to the 68% band as shown in Table 2 .
To compare with, we also give the observed CKM matrix, which is given by [1 ]
∣ V C K M o b s ∣ ≈ ( 0.9744 0.2250 0.0037 0.2249 0.9735 0.0418 0.0086 0.0411 0.9991 ) , (34) \left|V_{\mathrm{CKM}}^{\mathrm{obs}}\right| \approx
\begin{pmatrix}
0.9744 & 0.2250 & 0.0037 \\
0.2249 & 0.9735 & 0.0418 \\
0.0086 & 0.0411 & 0.9991
\end{pmatrix}, \tag*{(34)} V CKM obs ≈ 0.9744 0.2249 0.0086 0.2250 0.9735 0.0411 0.0037 0.0418 0.9991 , (34) We see that some of the qualitative features of the CKM matrix are captured by the fusion algebra structure, such as the fact that the third generation is mostly decoupled from the first two generations. Comparing with Eq. (34) , the near-unit diagonal and the third-generation mixings are reproduced at the few-percent level. The 1–2 mixing remains the weak point: the ensemble gives ∣ V u s ∣ = 0.04 0 − 0.029 + 0.058 \left|V_{us}\right| = 0.040^{+0.058}_{-0.029} ∣ V u s ∣ = 0.04 0 − 0.029 + 0.058 cl ( A ) = 2 \operatorname{cl}(\mathcal{A}) = 2 cl ( A ) = 2
It would be interesting to ask if there are directions on the coefficient space that takes us closer to the observed CKM matrix. To see, we look at the top 1% of draws, which gives a median ρ C K M = 0.134 \rho_{\mathrm{CKM}} = 0.134 ρ CKM = 0.134 ρ C K M = 0.026 \rho_{\mathrm{CKM}} = 0.026 ρ CKM = 0.026 ρ C K M \rho_{\mathrm{CKM}} ρ CKM ρ C K M = 0.026 \rho_{\mathrm{CKM}} = 0.026 ρ CKM = 0.026
Neutrinos from F R 8 8 , 6 \mathrm{FR}_{8}^{8,6} FR 8 8 , 6
We now look at what this assignment predicts for the neutrino sector. The Weinberg-operator selection rule of § 3.4 applies unchanged: with ℓ i [ b i ] \ell_i[b_i] ℓ i [ b i ] H [ a ˉ ] H[\bar{a}] H [ a ˉ ] O W ( i j ) \mathcal{O}_{W}^{(ij)} O W ( ij ) L L L w ∈ Com ( A ) L w \in\operatorname{Com}(\mathcal{A})^L w ∈ Com ( A ) L w ≺ a ˉ a ˉ b i b j w \prec\bar{a}\,\bar{a}\,b_i\,b_j w ≺ a ˉ a ˉ b i b j
Unlike every other sector, the neutrino sector involves no choice: the assignment of Eq. (30) fixes H ∗ [ 4 ] H^{*}[4] H ∗ [ 4 ] ℓ i = ( 2 , 4 , 8 ) \ell_i=(2,4,8) ℓ i = ( 2 , 4 , 8 )
The predicted texture: Since 4 ˉ = 3 \bar{4}=3 4 ˉ = 3 F R 8 8 , 6 \mathrm{FR}_{8}^{8,6} FR 8 8 , 6 O W \mathcal{O}_{W} O W 3 3 3 3 ⋅ 3 ⋅ b i ⋅ b j = 2 ⋅ b i ⋅ b j 3\cdot3\cdot b_i\cdot b_j=2\cdot b_i\cdot b_j 3 ⋅ 3 ⋅ b i ⋅ b j = 2 ⋅ b i ⋅ b j ℓ i = ( 2 , 4 , 8 ) \ell_i=(2,4,8) ℓ i = ( 2 , 4 , 8 )
M ν ∼ v 2 Λ ( ε 3 ε 2 ε ε 2 1 ε ε ε ε ) (35) M_{\nu}\sim\frac{v^2}{\Lambda}\begin{pmatrix}\varepsilon^3 & \varepsilon^2 & \varepsilon\\ \varepsilon^2 & 1 & \varepsilon\\ \varepsilon& \varepsilon& \varepsilon\end{pmatrix} \tag*{(35)} M ν ∼ Λ v 2 ε 3 ε 2 ε ε 2 1 ε ε ε ε (35) symmetric, as the commutativity of the fusion algebra guarantees. Since this texture predicts a split between the highest eigenvalues, inverted ordering for the neutrinos is disfavored. Henceforth, we will assume normal ordering.
The Weinberg operator is often taken to be generated by integrating out some heavy SM singlet fermions, N a N_a N a
L = ⋯ − y i a ( ℓ i H ) N a − Λ γ a b N a N b + … (36) \mathcal{L}=\cdots-y^{ia}(\ell_iH)N_a-\Lambda\gamma^{ab}N_aN_b+\ldots \tag*{(36)} L = ⋯ − y ia ( ℓ i H ) N a − Λ γ ab N a N b + … (36) so that
Y ν i j = y i a y j b ( γ − 1 ) a b (37) Y_{\nu}^{ij}=y^{ia}y^{jb}(\gamma^{-1})_{ab} \tag*{(37)} Y ν ij = y ia y j b ( γ − 1 ) ab (37) In the present context, we can for example assign the labels N 1 [ 1 ] N_1[1] N 1 [ 1 ] N 2 [ 7 ] N_2[7] N 2 [ 7 ] N 3 [ 8 ] N_3[8] N 3 [ 8 ]
γ a b ∼ ( 1 ε ε ε ε 1 ε 1 ε ) (38) \gamma^{ab}\sim\begin{pmatrix}1 & \varepsilon& \varepsilon\\ \varepsilon& \varepsilon& 1 \\ \varepsilon& 1 & \varepsilon\end{pmatrix} \tag*{(38)} γ ab ∼ 1 ε ε ε ε 1 ε 1 ε (38) and Yukawa matrix with texture
y i a ∼ ( ε 2 ε ε 2 1 ε ε ε ε 1 ) (39) y^{ia}\sim\begin{pmatrix}\varepsilon^2 & \varepsilon& \varepsilon^2 \\ 1 & \varepsilon& \varepsilon\\ \varepsilon& \varepsilon& 1\end{pmatrix} \tag*{(39)} y ia ∼ ε 2 1 ε ε ε ε ε 2 ε 1 (39) Putting these together, we reproduce
M ν i j = v 2 Λ ( y ⋅ γ − 1 ⋅ y t ) i j ∼ v 2 Λ ( ε 3 ε 2 ε ε 2 1 ε ε ε ε ) , (40) \begin{aligned}
M_{\nu}^{ij}=\frac{v^2}{\Lambda}(y\cdot\gamma^{-1}\cdot y^t)^{ij} \\
&\sim\frac{v^2}{\Lambda}\begin{pmatrix}\varepsilon^3 & \varepsilon^2 & \varepsilon\\ \varepsilon^2 & 1 & \varepsilon\\ \varepsilon& \varepsilon& \varepsilon\end{pmatrix}, \tag*{(40)}
\end{aligned} M ν ij = Λ v 2 ( y ⋅ γ − 1 ⋅ y t ) ij ∼ Λ v 2 ε 3 ε 2 ε ε 2 1 ε ε ε ε , (40) Table 3 gives the spectrum under the ensemble of Eq. (25) , with one modification required by the physics: the coefficients are drawn symmetrically, c i j = c j i c_{ij}=c_{ji} c ij = c j i M ν M_{\nu} M ν
Table 3. Table 3: The neutrino sector predicted by the assignment of Eq. (30) , from 4 × 10 5 4 \times10^5 4 × 1 0 5 (25) with symmetric coefficients. The scale Λ \Lambda Λ m ν 3 = 0.05 m_{\nu_3}=0.05 m ν 3 = 0.05 [41 ] .
What we find is Δ m 21 2 / Δ m 31 2 = 7.0 × 10 − 4 \Delta m^2_{21}/\Delta m^2_{31}=7.0\times10^{-4} Δ m 21 2 /Δ m 31 2 = 7.0 × 1 0 − 4 3.0 × 10 − 2 3.0\times10^{-2} 3.0 × 1 0 − 2 2.9 × 10 − 4 2.9 \times10^{-4} 2.9 × 1 0 − 4 Λ ≈ 6 × 10 14 \Lambda\approx6 \times10^{14} Λ ≈ 6 × 1 0 14
The PMNS matrix
Diagonalising Y e Y^e Y e (31) and M ν M_\nu M ν (35) across the ensemble, with independent coefficients for the two operators, gives
∣ U P M N S ∣ = ( 0.85 1 − 0.070 + 0.082 0.52 5 − 0.166 + 0.100 0.00 1 − 0.000 + 0.001 0.52 5 − 0.166 + 0.099 0.85 0 − 0.070 + 0.082 0.02 6 − 0.019 + 0.029 0.01 2 − 0.009 + 0.016 0.02 2 − 0.016 + 0.025 1.00 0 − 0.001 + 0.000 ) (41) \left|U_{\mathrm{PMNS}}\right| =
\begin{pmatrix}
0.851^{+0.082}_{-0.070} & 0.525^{+0.100}_{-0.166} & 0.001^{+0.001}_{-0.000} \\
0.525^{+0.099}_{-0.166} & 0.850^{+0.082}_{-0.070} & 0.026^{+0.029}_{-0.019} \\
0.012^{+0.016}_{-0.009} & 0.022^{+0.025}_{-0.016} & 1.000^{+0.000}_{-0.001}
\end{pmatrix} \tag*{(41)} ∣ U PMNS ∣ = 0.85 1 − 0.070 + 0.082 0.52 5 − 0.166 + 0.099 0.01 2 − 0.009 + 0.016 0.52 5 − 0.166 + 0.100 0.85 0 − 0.070 + 0.082 0.02 2 − 0.016 + 0.025 0.00 1 − 0.000 + 0.001 0.02 6 − 0.019 + 0.029 1.00 0 − 0.001 + 0.000 (41) quoting the median with its 68% band, against the observed magnitudes [41 ]
∣ U P M N S o b s ∣ ≈ ( 0.823 0.548 0.148 0.275 0.613 0.741 0.497 0.569 0.655 ) (42) \left|U_{\mathrm{PMNS}}^{\mathrm{obs}}\right| \approx
\begin{pmatrix}
0.823 & 0.548 & 0.148 \\
0.275 & 0.613 & 0.741 \\
0.497 & 0.569 & 0.655
\end{pmatrix} \tag*{(42)} U PMNS obs ≈ 0.823 0.275 0.497 0.548 0.613 0.569 0.148 0.741 0.655 (42) We see that ∣ U e 1 ∣ = 0.851 \lvert U_{e1}\rvert=0.851 ∣ U e 1 ∣ = 0.851 ∣ U e 2 ∣ = 0.5 \lvert U_{e2}\rvert=0.5 ∣ U e 2 ∣ = 0.5 ∣ U e 3 ∣ \lvert U_{e3}\rvert ∣ U e 3 ∣
Unlike the case of CKM matrix, here the MC scan does identify a direction that takes us closer to the observed magnitudes. Again, looking at the top 1% of draws, we find that c 22 c_{22} c 22 c 23 c_{23} c 23
Discussion
The fusion-algebra framework explains the fermion mass hierarchy in a way that differs structurally from the familiar flavour mechanisms. In Froggatt–Nielsen models [2 , 42 , 43 ] and in non-Abelian or modular flavour models [3 , 4 , 5 ] , the small parameter is the vacuum expectation value of a new field, and the charges or representations are chosen to place its powers in the right entries. Here, we find that the suppression is equally well applicable as possibly from a flavon or as a consequence of UV loop effects. The hierarchy is fixed entirely by the multiplication table of A \mathcal{A} A ε \varepsilon ε ε n \varepsilon^n ε n n ∈ { 0 , … , cl ( A ) } n \in\{0,\ldots,\operatorname{cl}(\mathcal{A})\} n ∈ { 0 , … , cl ( A )} Com ( A ) cl ( A ) = A \operatorname{Com}(\mathcal{A})^{\operatorname{cl}(\mathcal{A})}=\mathcal{A} Com ( A ) cl ( A ) = A
The commutator length cl ( A ) \operatorname{cl}(\mathcal{A}) cl ( A ) { 0 , … , cl ( A ) } \{0,\ldots,\operatorname{cl}(\mathcal{A})\} { 0 , … , cl ( A )} ε − cl ( A ) \varepsilon^{-\operatorname{cl}(\mathcal{A})} ε − cl ( A ) O ( 1 ) \mathcal{O}(1) O ( 1 ) (19) ). This turns the up-quark spectrum into a direct probe of the depth of the symmetry. With m t / m u ≃ 1.4 × 10 5 m_t/m_u \simeq1.4 \times10^5 m t / m u ≃ 1.4 × 1 0 5 ε − 2 ≃ 3.5 × 10 3 \varepsilon^{-2} \simeq3.5 \times10^3 ε − 2 ≃ 3.5 × 1 0 3 cl ( A ) ≥ 3 \operatorname{cl}(\mathcal{A}) \ge3 cl ( A ) ≥ 3 cl ( A ) = 2 \operatorname{cl}(\mathcal{A})=2 cl ( A ) = 2 m t / m u ≃ 3.9 × 10 3 m_t/m_u \simeq3.9 \times10^3 m t / m u ≃ 3.9 × 1 0 3 § 4.2 ).
Because the selection rule is determined by a fusion ring, the space of models is not a continuum of charges but a finite list that can be searched exhaustively. Every multiplicity-free algebra of rank ≤ 7 \le7 ≤ 7 [22 ] has cl ( A ) ≤ 2 \operatorname{cl}(\mathcal{A}) \le2 cl ( A ) ≤ 2 ≤ 10 \le10 ≤ 10 [37 , 40 ] , exactly four have cl ( A ) = 3 \operatorname{cl}(\mathcal{A})=3 cl ( A ) = 3 cl ( A ) > 3 \operatorname{cl}(\mathcal{A})>3 cl ( A ) > 3 cl ( A ) ≥ 3 \operatorname{cl}(\mathcal{A}) \ge3 cl ( A ) ≥ 3 F R 8 8 , 6 \mathrm{FR}_{8}^{8,6} FR 8 8 , 6 cl ( A ) = 3 \operatorname{cl}(\mathcal{A})=3 cl ( A ) = 3
Once the lepton doublet and Higgs labels are fixed by the charged-lepton fit, the Weinberg operator’s texture is fully determined. For F R 8 8 , 6 \mathrm{FR}_{8}^{8,6} FR 8 8 , 6 Λ ∼ 6 × 10 14 G e V \Lambda\sim6 \times10^{14}\ \mathrm{GeV} Λ ∼ 6 × 1 0 14 GeV ∣ U e 1 ∣ ≃ 0.85 |U_{e1}| \simeq0.85 ∣ U e 1 ∣ ≃ 0.85 ∣ U e 2 ∣ ≃ 0.53 |U_{e2}| \simeq0.53 ∣ U e 2 ∣ ≃ 0.53 Δ m 21 2 / Δ m 31 2 \Delta m_{21}^2/\Delta m_{31}^2 Δ m 21 2 /Δ m 31 2
A fusion-algebra texture is generated at the scale Λ \Lambda Λ Λ \Lambda Λ β \beta β Y β Y β † Y α Y^{\beta}Y^{\beta\dagger}Y^{\alpha} Y β Y β † Y α h h ˉ = 1 h\bar{h}=1 h h ˉ = 1 § 14 that the texture is stable under 1-loop RGE running and survives from the scale Λ \Lambda Λ
Several questions remain open at this point. F R 8 8 , 6 \mathrm{FR}_{8}^{8,6} FR 8 8 , 6 cl ( A ) = 3 \operatorname{cl}(\mathcal{A})=3 cl ( A ) = 3
This is particularly pressing if one wants to go beyond the crude analysis of this paper and fix the O ( 1 ) O(1) O ( 1 ) ε \varepsilon ε O ( 1 ) O(1) O ( 1 )
Quick Reference: All Fusion Algebra Tables
This Appendix tabulates the fusion rules for the commutative algebras3 cl ( A ) = 2 \operatorname{cl}(\mathcal{A})=2 cl ( A ) = 2 [22 ] . For each algebra we give its fusion table, whether it is categorifiable and list its Frobenius-Perron dimension (an algebraic number which is a rough measure of its complexity). We also provide its name in the conventions of link text . Finally, we provide the same data for F R 8 8 , 6 \mathrm{FR}_{8}^{8,6} FR 8 8 , 6 ≤ 9 \le9 ≤ 9 cl ( A ) ≥ 3 \operatorname{cl}(\mathcal{A})\ge3 cl ( A ) ≥ 3
Rank-4 Algebras
C.1a (F R 4 4 , 1 , 2 \mathrm{FR}_{4}^{4,1,2} FR 4 4 , 1 , 2 D F P 2 = 8 + 4 2 D_{\mathrm{FP}}^{2}=8+4\sqrt{2} D FP 2 = 8 + 4 2
Table 13.
C.1b (F R 4 4 , 1 , 0 \mathrm{FR}_{4}^{4,1,0} FR 4 4 , 1 , 0 Z 2 \mathbb{Z}_2 Z 2 6 _6 6 D F P 2 = 8 + 4 2 D_{\mathrm{FP}}^{2}=8+4\sqrt{2} D FP 2 = 8 + 4 2
Table 14.
Rank-5 Algebra
C.2 (F R 5 5 , 1 , 0 \mathrm{FR}_{5}^{5,1,0} FR 5 5 , 1 , 0 D F P 2 = 13 + 13 D_{\mathrm{FP}}^{2}=13+\sqrt{13} D FP 2 = 13 + 13
Table 15.
Rank-6 Algebras
C.3 (F R 6 6 , 1 , 0 \mathrm{FR}_{6}^{6,1,0} FR 6 6 , 1 , 0 10 _{10} 10 D F P 2 = 24 + 12 3 D_{\mathrm{FP}}^{2}=24+12\sqrt{3} D FP 2 = 24 + 12 3
Table C.3.
C.5 (F R 6 6 , 1 , 4 \mathrm{FR}_{6}^{6,1,4} FR 6 6 , 1 , 4 D F P 2 = 12 + 4 3 D_{\mathrm{FP}}^{2}=12+4\sqrt{3} D FP 2 = 12 + 4 3
Table C.5.
C.6 (F R 5 6 , 1 , 2 \mathrm{FR}_{5}^{6,1,2} FR 5 6 , 1 , 2 D F P 2 = 12 + 4 3 D_{\mathrm{FP}}^{2}=12+4\sqrt{3} D FP 2 = 12 + 4 3
Table C.6.
C.7 (F R 6 6 , 1 , 2 \mathrm{FR}_{6}^{6,1,2} FR 6 6 , 1 , 2 D F P 2 = 12 + 4 3 D_{\mathrm{FP}}^{2}=12+4\sqrt{3} D FP 2 = 12 + 4 3
Table C.7.
C.8 (F R 3 6 , 1 , 0 \mathrm{FR}_{3}^{6,1,0} FR 3 6 , 1 , 0 D F P 2 = 12 + 4 3 D_{\mathrm{FP}}^{2}=12+4\sqrt{3} D FP 2 = 12 + 4 3
Table C.8.
C.9
C.9 (F R 7 6 , 1 , 4 \mathrm{FR}_{7}^{6,1,4} FR 7 6 , 1 , 4 D F P 2 = 12 + 6 2 D_{\mathrm{FP}}^{2}=12+6\sqrt{2} D FP 2 = 12 + 6 2
Table 16.
C.10
C.10 (F R 10 6 , 1 , 0 \mathrm{FR}_{10}^{6,1,0} FR 10 6 , 1 , 0 D F P 2 = 12 + 6 2 D_{\mathrm{FP}}^{2}=12+6\sqrt{2} D FP 2 = 12 + 6 2
Table 17.
C.11
C.11 (F R 8 6 , 1 , 4 \mathrm{FR}_{8}^{6,1,4} FR 8 6 , 1 , 4 D F P 2 = 39 2 + 9 2 13 D_{\mathrm{FP}}^{2}=\frac{39}{2}+\frac{9}{2}\sqrt{13} D FP 2 = 2 39 + 2 9 13
Table 18.
C.12
C.12 (F R 10 6 , 1 , 2 \mathrm{FR}_{10}^{6,1,2} FR 10 6 , 1 , 2 D 2 ≈ 36.78 D^{2}\approx36.78 D 2 ≈ 36.78
Table 19.
C.13
C.13 (F R 15 6 , 1 , 2 \mathrm{FR}_{15}^{6,1,2} FR 15 6 , 1 , 2 D 2 ≈ 36.78 D^{2}\approx36.78 D 2 ≈ 36.78
Table 20.
Quick Reference: All Fusion Algebra Tables
Rank-7 Algebras
C.14 (F R 5 7 , 1 , 2 \mathrm{FR}_{5}^{7,1,2} FR 5 7 , 1 , 2 Non-categorifiable , D F P 2 = 17 + 17 D_{\mathrm{FP}}^{2}=17+\sqrt{17} D FP 2 = 17 + 17
Table C.14.
C.15 (F R 2 7 , 1 , 0 \mathrm{FR}_{2}^{7,1,0} FR 2 7 , 1 , 0 Non-categorifiable , D F P 2 = 17 + 17 D_{\mathrm{FP}}^{2}=17+\sqrt{17} D FP 2 = 17 + 17
Table C.15.
C.16 (F R 11 7 , 1 , 0 \mathrm{FR}_{11}^{7,1,0} FR 11 7 , 1 , 0 Non-categorifiable , D F P 2 = 20 + 12 2 D_{\mathrm{FP}}^{2}=20+12\sqrt{2} D FP 2 = 20 + 12 2
Table C.16.
C.17 (F R 12 7 , 1 , 0 \mathrm{FR}_{12}^{7,1,0} FR 12 7 , 1 , 0 Non-categorifiable , D F P 2 = 42 D_{\mathrm{FP}}^{2}=42 D FP 2 = 42
Table C.17.
C.18 (F R 9 7 , 1 , 0 \mathrm{FR}_{9}^{7,1,0} FR 9 7 , 1 , 0 Non-categorifiable , D F P 2 = 29 + 29 D_{\mathrm{FP}}^{2}=29+\sqrt{29} D FP 2 = 29 + 29
Table C.18.
Rank-8 Algebra
F R 8 8 , 6 \mathrm{FR}_{8}^{8,6} FR 8 8 , 6 Non-categorifiable, D F P 2 = 16 + 8 2 D_{\mathrm{FP}}^{2}=16+8\sqrt{2} D FP 2 = 16 + 8 2 cl ( A ) = 3 \operatorname{cl}(\mathcal{A})=3 cl ( A ) = 3
Table 21.
The cl ( A ) = 2 \operatorname{cl}(\mathcal{A})=2 cl ( A ) = 2
This appendix collects the full analysis of the multiplicity-free fusion algebras of rank ≤ 7 \le7 ≤ 7 cl ( A ) = 2 \operatorname{cl}(\mathcal{A})=2 cl ( A ) = 2 § 4 ; the fusion tables are in Appendix § 8 . Throughout, scores are computed with the metrics of § 3.6 and assignments are ranked by ensemble medians. The loop-order dictionary is that of § 3 truncated at cl ( A ) = 2 \operatorname{cl}(\mathcal{A})=2 cl ( A ) = 2
Throughout, the assignments are ranked by the median of the relevant score over the ensemble of Eq. (25) .
Charged leptons: scan over all algebras
In order to find the best possible hierarchy in each algebra listed in Appendix § 8 , we assign the basis elements to the fields in all possible ways, compute the resulting mass ratios, and select the assignment minimising the metric ρ \rho ρ (21) .
Table 22.
We see that there are multiple algebras that give a pretty good fit to the observed mass ratios. However, when we also consider the quark sector and the neutrino sector, we will see that the available algebras get limited.
Neutrinos
The C.9 neutrino texture
With the specific example we considered in the previous section, we have H ∗ [ 1 ] H^{*}[1] H ∗ [ 1 ] l 1 [ 4 ] , l 2 [ 1 ] , l 3 [ 3 ] l_{1}[4], l_{2}[1], l_{3}[3] l 1 [ 4 ] , l 2 [ 1 ] , l 3 [ 3 ]
M ν ∼ v 2 Λ ( ε ε ε 2 ε 1 ε 2 ε 2 ε 2 ε 2 ) (43) M_{\nu} \sim\frac{v^{2}}{\Lambda}
\begin{pmatrix}
\varepsilon& \varepsilon& \varepsilon^{2} \\
\varepsilon& 1 & \varepsilon^{2} \\
\varepsilon^{2} & \varepsilon^{2} & \varepsilon^{2}
\end{pmatrix} \tag*{(43)} M ν ∼ Λ v 2 ε ε ε 2 ε 1 ε 2 ε 2 ε 2 ε 2 (43) The singular values of this matrix are in the ratio
m ν 1 : m ν 2 : m ν 3 ≈ ε 2 : ε : 1 (44) m_{\nu_{1}} : m_{\nu_{2}} : m_{\nu_{3}} \approx\varepsilon^{2} : \varepsilon: 1 \tag*{(44)} m ν 1 : m ν 2 : m ν 3 ≈ ε 2 : ε : 1 (44) which gives a hierarchy of neutrino masses.
In order to check with observations, we need to fix the scale Λ \Lambda Λ m ν 3 = 0.05 m_{\nu_{3}} = 0.05 m ν 3 = 0.05 Λ \Lambda Λ Λ ≈ 6.17 × 10 14 \Lambda\approx6.17 \times10^{14} Λ ≈ 6.17 × 1 0 14
Δ m 31 2 ≈ m 3 2 ≈ 2.5 × 10 − 3 e V 2 (45) \Delta m_{31}^{2} \approx m_{3}^{2} \approx2.5 \times10^{-3}\ \mathrm{eV}^{2} \tag*{(45)} Δ m 31 2 ≈ m 3 2 ≈ 2.5 × 1 0 − 3 eV 2 (45) which is consistent with the observed value. But, note that this will be the case by construction as long as there is a hierarchy in the neutrino masses in our setup since we are fixing the largest neutrino mass to be 0.05 eV.
Scan over all algebras
As before we scan over all the algebras, in each case evaluating the Weinberg texture at that algebra’s best charged-lepton assignment and quoting the median of the ensemble of Eq. (25) with symmetric coefficients. The charged-lepton score is repeated in the second column.
Table 4. Table 4: Neutrino spectra for the algebras that survive the charged-lepton fit, each evaluated at its best-fit left-handed lepton assignment. For every algebra the scale is fixed by hand through m ν 3 = 0.05 m_{\nu_3}=0.05 m ν 3 = 0.05 Λ ≈ 6 × 10 14 \Lambda\approx6 \times10^{14} Λ ≈ 6 × 1 0 14
A comment on the PMNS matrix: Unlike the CKM matrix, the observed PMNS matrix has no hierarchical structure, so any assignment that predicts a hierarchical PMNS matrix is disfavored. Hierarchical textures for Y e Y^e Y e M ν M_\nu M ν
Quarks
Down-type Yukawa texture
Going forward with the algebra C.9, we assign for the down-type quarks d j c d^c_j d j c e i c e^c_i e i c q i q_i q i l i l_i l i
H ∗ [ 1 ] , q 1 [ 4 ] , q 2 [ 1 ] , q 3 [ 3 ] , d 1 c [ 6 ] , d 2 c [ 2 ] , d 3 c [ 3 ] (46) H^*[1],\quad q_1[4],q_2[1],q_3[3],\quad d^c_1[6],d^c_2[2],d^c_3[3] \tag*{(46)} H ∗ [ 1 ] , q 1 [ 4 ] , q 2 [ 1 ] , q 3 [ 3 ] , d 1 c [ 6 ] , d 2 c [ 2 ] , d 3 c [ 3 ] (46) Therefore, the resulting down-type Yukawa matrix will have the same texture as the lepton Yukawa matrix.
Y d ∼ ( ε ε ε 2 ε 2 ε ε 2 ε 1 ε 2 ) , m d : m s : m b ≈ 1 : 58.8 : 3522 ( ∼ 1 : ε − 1 : ε − 2 ) , (47) Y^d \sim\begin{pmatrix} \varepsilon& \varepsilon& \varepsilon^2 \\ \varepsilon^2 & \varepsilon& \varepsilon^2 \\ \varepsilon& 1 & \varepsilon^2 \end{pmatrix},\qquad m_d:m_s:m_b \approx1:58.8:3522\quad\left(\sim1:\varepsilon^{-1}:\varepsilon^{-2}\right), \tag*{(47)} Y d ∼ ε ε 2 ε ε ε 1 ε 2 ε 2 ε 2 , m d : m s : m b ≈ 1 : 58.8 : 3522 ( ∼ 1 : ε − 1 : ε − 2 ) , (47) against the observed 1 : 19.9 : 1058 1:19.9:1058 1 : 19.9 : 1058 O ( 1 ) \mathcal{O}(1) O ( 1 )
Up-type Yukawa texture
The up-type quarks have the largest hierarchy in the Standard Model. Therefore, we need to find an assignment that gives a large hierarchy. For the specific algebra C.9, with the choice of assignments for q i q_i q i H H H
u 1 c [ 4 ] , u 2 c [ 3 ] , u 3 c [ 2 ] (48) u_1^c[4],u_2^c[3],u_3^c[2] \tag*{(48)} u 1 c [ 4 ] , u 2 c [ 3 ] , u 3 c [ 2 ] (48) giving the up-type Yukawa matrix the texture
Y u ∼ ( ϵ ϵ 2 ϵ ϵ ϵ 2 ϵ 2 ϵ 2 ϵ 2 1 ) (49) Y^u \sim\begin{pmatrix} \epsilon& \epsilon^2 & \epsilon\\ \epsilon& \epsilon^2 & \epsilon^2 \\ \epsilon^2 & \epsilon^2 & 1 \end{pmatrix} \tag*{(49)} Y u ∼ ϵ ϵ ϵ 2 ϵ 2 ϵ 2 ϵ 2 ϵ ϵ 2 1 (49) giving the singular values in the ratio
m u : m c : m t ≈ 1 : 7162 : 297956 ( ∼ 1 : ϵ − 2 : ϵ − 3 ) (50) m_u:m_c:m_t \approx1:7162:297956 \quad(\sim1:\epsilon^{-2}:\epsilon^{-3}) \tag*{(50)} m u : m c : m t ≈ 1 : 7162 : 297956 ( ∼ 1 : ϵ − 2 : ϵ − 3 ) (50) against the observed 1 : 506 : 137400 1:506:137400 1 : 506 : 137400 ϵ − 3 \epsilon^{-3} ϵ − 3 c i j = 1 c_{ij}=1 c ij = 1 (25) the median ratios collapse to 1 : 100.7 : 3854 1:100.7:3854 1 : 100.7 : 3854 § 4.2 . The leading term of det Y u \det Y^u det Y u
Scan over all algebras
Repeating the same procedure as in the lepton sector, we scan over all the algebras and find the best fit for the up-type and down-type quark mass ratios. We find following best fits for both the up-type and down-type quarks
Table 5. Best up-type fit per algebra with the corresponding score.
Table 6. Best down-type fit per algebra with the corresponding score.
The CKM matrix
The CKM matrix is constructed from Y u Y^u Y u Y d Y^d Y d § 5.2 .
The C.9 example
Let us consider the algebra C.9 with the assignments we have chosen for the quarks, namely H ∗ [ 1 ] H^*[1] H ∗ [ 1 ] q 1 [ 4 ] q_1[4] q 1 [ 4 ] q 2 [ 1 ] q_2[1] q 2 [ 1 ] q 3 [ 3 ] q_3[3] q 3 [ 3 ] u 1 c [ 4 ] u^c_1[4] u 1 c [ 4 ] u 2 c [ 3 ] u^c_2[3] u 2 c [ 3 ] u 3 c [ 2 ] u^c_3[2] u 3 c [ 2 ] d 1 c [ 6 ] d^c_1[6] d 1 c [ 6 ] d 2 c [ 2 ] d^c_2[2] d 2 c [ 2 ] d 3 c [ 3 ] d^c_3[3] d 3 c [ 3 ]
Y u ∼ ( ε ε 2 ε ε ε 2 ε 2 ε 2 ε 2 1 ) , Y d ∼ ( ε ε ε 2 ε 2 ε 2 ε 2 ε 1 ε 2 ) . (51) Y^u \sim\begin{pmatrix} \varepsilon& \varepsilon^2 & \varepsilon\\ \varepsilon& \varepsilon^2 & \varepsilon^2 \\ \varepsilon^2 & \varepsilon^2 & 1 \end{pmatrix}, \qquad Y^d \sim\begin{pmatrix} \varepsilon& \varepsilon& \varepsilon^2 \\ \varepsilon^2 & \varepsilon^2 & \varepsilon^2 \\ \varepsilon& 1 & \varepsilon^2 \end{pmatrix}. \tag*{(51)} Y u ∼ ε ε ε 2 ε 2 ε 2 ε 2 ε ε 2 1 , Y d ∼ ε ε 2 ε ε ε 2 1 ε 2 ε 2 ε 2 . (51) Diagonalizing these matrices, and using the best fit value of ε = 0.017 \varepsilon= 0.017 ε = 0.017
∣ V C K M C .9 ∣ ≈ ( 0.719 0.695 0.0002 0.695 0.719 0.0002 0.00001 0.0003 1.000 ) . (52) \lvert V_{\mathrm{CKM}}^{\mathrm{C.9}} \rvert\approx\begin{pmatrix} 0.719 & 0.695 & 0.0002 \\ 0.695 & 0.719 & 0.0002 \\ 0.00001 & 0.0003 & 1.000 \end{pmatrix}. \tag*{(52)} ∣ V CKM C.9 ∣ ≈ 0.719 0.695 0.00001 0.695 0.719 0.0003 0.0002 0.0002 1.000 . (52) Expanding the same diagonalization to leading order in ε \varepsilon ε
∣ V C K M C .9 ∣ ∼ ( O ( 1 ) O ( 1 ) O ( ε 2 ) O ( 1 ) O ( 1 ) O ( ε 2 ) O ( ε 3 ) O ( ε 2 ) 1 ) . (53) \lvert V_{\mathrm{CKM}}^{\mathrm{C.9}} \rvert\sim\begin{pmatrix} \mathcal{O}(1) & \mathcal{O}(1) & \mathcal{O}(\varepsilon^2) \\ \mathcal{O}(1) & \mathcal{O}(1) & \mathcal{O}(\varepsilon^2) \\ \mathcal{O}(\varepsilon^3) & \mathcal{O}(\varepsilon^2) & 1 \end{pmatrix}. \tag*{(53)} ∣ V CKM C.9 ∣ ∼ O ( 1 ) O ( 1 ) O ( ε 3 ) O ( 1 ) O ( 1 ) O ( ε 2 ) O ( ε 2 ) O ( ε 2 ) 1 . (53) which carries information the decimals hide. The upper-left O ( 1 ) \mathcal{O}(1) O ( 1 ) ∣ V u s ∣ ∼ sin θ 12 \lvert V_{us}\rvert\sim\sin\theta_{12} ∣ V u s ∣ ∼ sin θ 12 θ 12 ≈ 43 ∘ \theta_{12} \approx43^\circ θ 12 ≈ 4 3 ∘ O ( 1 ) \mathcal{O}(1) O ( 1 ) ∣ V t d ∣ ≈ 10 − 5 \lvert V_{td}\rvert\approx10^{-5} ∣ V t d ∣ ≈ 1 0 − 5 O ( ε 3 ) \mathcal{O}(\varepsilon^3) O ( ε 3 )
To compare with, recall the observed CKM matrix of Eq. (34) . We see that some of the qualitative features of the CKM matrix are captured by the fusion algebra structure, such as the fact that the third generation is mostly decoupled from the first two generations. Furthermore, the first two generations are strongly mixed with each other. However, the mixing between the first and second generation is too large compared to the observed value. These discrepancies could be fixed by adding O ( 1 ) \mathcal{O}(1) O ( 1 )
Scan over algebras
Using ρ C K M \rho_{\mathrm{CKM}} ρ CKM (22) and the combined mass score ρ m a s s \rho_{\mathrm{mass}} ρ mass (23) , we scan over all the algebras and find the following. First, the assignment that best fits the quark mass ratios as given in Sec. B.3 is not the best fit for the CKM matrix. The mass-optimal assignment gives median scores ρ C K M = 0.444 \rho_{\mathrm{CKM}} = 0.444 ρ CKM = 0.444 ρ m a s s = 1.79 \rho_{\mathrm{mass}} = 1.79 ρ mass = 1.79 ρ C K M = 0.316 \rho_{\mathrm{CKM}} = 0.316 ρ CKM = 0.316 ρ m a s s = 2.19 \rho_{\mathrm{mass}} = 2.19 ρ mass = 2.19 H ∗ [ 1 ] H^{*}[1] H ∗ [ 1 ] q i = ( 1 , 2 , 5 ) q_i = (1,2,5) q i = ( 1 , 2 , 5 ) u i c = ( 1 , 2 , 5 ) u_i^c = (1,2,5) u i c = ( 1 , 2 , 5 ) d i c = ( 1 , 5 , 6 ) d_i^c = (1,5,6) d i c = ( 1 , 5 , 6 ) ρ C K M \rho_{\mathrm{CKM}} ρ CKM
H [ 4 ] , q i = [ 4 , 1 , 2 ] , u i c = [ 1 , 6 , 5 ] , d i c = [ 1 , 2 , 6 ] , (54) H[4], \quad q_i = [4,1,2], \quad u_i^c = [1,6,5], \quad d_i^c = [1,2,6], \tag*{(54)} H [ 4 ] , q i = [ 4 , 1 , 2 ] , u i c = [ 1 , 6 , 5 ] , d i c = [ 1 , 2 , 6 ] , (54) giving
∣ V C K M ∣ ≈ ( 0.800 0.600 0.0001 0.600 0.800 0.0257 0.016 0.021 0.9997 ) , ρ C K M = 0.36 , (55) \lvert V_{\mathrm{CKM}}\rvert\approx
\begin{pmatrix}
0.800 & 0.600 & 0.0001 \\
0.600 & 0.800 & 0.0257 \\
0.016 & 0.021 & 0.9997
\end{pmatrix},
\qquad\rho_{\mathrm{CKM}} = 0.36, \tag*{(55)} ∣ V CKM ∣ ≈ 0.800 0.600 0.016 0.600 0.800 0.021 0.0001 0.0257 0.9997 , ρ CKM = 0.36 , (55) or, to leading order in ε \varepsilon ε
∣ V C K M ∣ ∼ ( O ( 1 ) O ( 1 ) O ( ε 2 ) O ( 1 ) O ( 1 ) O ( ε ) O ( ε ) O ( ε ) 1 ) . (56) \lvert V_{\mathrm{CKM}}\rvert\sim
\begin{pmatrix}
\mathcal{O}(1) & \mathcal{O}(1) & \mathcal{O}(\varepsilon^2) \\
\mathcal{O}(1) & \mathcal{O}(1) & \mathcal{O}(\varepsilon) \\
\mathcal{O}(\varepsilon) & \mathcal{O}(\varepsilon) & 1
\end{pmatrix}. \tag*{(56)} ∣ V CKM ∣ ∼ O ( 1 ) O ( 1 ) O ( ε ) O ( 1 ) O ( 1 ) O ( ε ) O ( ε 2 ) O ( ε ) 1 . (56) Comparing with the mass-optimal texture of Eq. B.11 shows exactly what the CKM-optimal assignment is trading away. The third-generation mixings are enhanced by one power of ε \varepsilon ε ∣ V c b ∣ \lvert V_{cb}\rvert ∣ V c b ∣ ∣ V t d ∣ \lvert V_{td}\rvert ∣ V t d ∣ ∣ V t s ∣ ∼ ε \lvert V_{ts}\rvert\sim\varepsilon ∣ V t s ∣ ∼ ε ε 2 \varepsilon^2 ε 2 ε 3 \varepsilon^3 ε 3 ∣ V c b ∣ ≈ 0.042 \lvert V_{cb}\rvert\approx0.042 ∣ V c b ∣ ≈ 0.042 ∣ V t s ∣ ≈ 0.041 \lvert V_{ts}\rvert\approx0.041 ∣ V t s ∣ ≈ 0.041 ρ m a s s \rho_{\mathrm{mass}} ρ mass
CKM Best-Fit Parameters for Each Algebra
Table 7. Best CKM fit per algebra under the ensemble of Eq. (25) . For each algebra we list the assignment ( H , q i , u i c , d i c ) (H, q_i, u_i^c, d_i^c) ( H , q i , u i c , d i c ) median of ρ C K M \rho_{\mathrm{CKM}} ρ CKM ρ m a s s \rho_{\mathrm{mass}} ρ mass
The spread across algebras is small — twelve of the nineteen lie between 0.31 and 0.34 — so the CKM matrix on its own does very little to discriminate between them. What discriminates is the mass sector, and every CKM-optimal assignment has a poor ρ m a s s \rho_{\mathrm{mass}} ρ mass
Feynman Diagrams for the C.9 lepton texture
In this appendix we present the Feynman diagrams corresponding to the C.9 lepton texture. Note that the fusion rules can be easily interpreted in a diagrammatic way as follows.
A point to note is that there are multiple ways to fuse the lines together. Here we show one possible way to do so, but there are other ways to do it as well. The assignments for which the diagrams below contribute is
H ˉ = 1 , l i = { 4 , 1 , 3 } , e i c = { 6 , 2 , 3 } (57) \bar{H}=1 \qquad, \qquad l_i=\{4,1,3\} \qquad, \qquad e_i^c=\{6,2,3\} \tag*{(57)} H ˉ = 1 , l i = { 4 , 1 , 3 } , e i c = { 6 , 2 , 3 } (57) In drawing the figures, we have shown one particular example where a choice on the Lorentz spin of the fields running in the loops has been made. Dashed lines represent scalars and bold lines represent fermions.
Tree-level contribution
The coupling H ˉ l 3 e 2 c \bar{H}l_3e_2^c H ˉ l 3 e 2 c
Figure 1. Tree-level realization of the C.9 charged-lepton texture.
One-loop contributions
Figure 2. Representative one-loop realizations of the C.9 charged-lepton texture.
Two-loop contributions
!Five representative two-loop Feynman diagrams for the C.9 charged-lepton texture, panels (a)–(e), with labels H ˉ [ 1 ] \bar{H}[1] H ˉ [ 1 ] l 1 [ 4 ] l_1[4] l 1 [ 4 ] l 2 [ 1 ] l_2[1] l 2 [ 1 ] l 3 [ 3 ] l_3[3] l 3 [ 3 ] e 1 c [ 6 ] e_1^c[6] e 1 c [ 6 ] e 2 c [ 2 ] e_2^c[2] e 2 c [ 2 ] bbox=0.14,0.13,0.86,0.77
Figure 3. Representative two-loop realizations of the C.9 charged-lepton texture. The diagrams are grouped by loop order. Other fusion channels may give alternative realizations of the same effective operators. And, for each fusion channel, we have made one choice for how to route the fermion number through the diagram.
Rank-10 algebras with cl ( A ) = 3 \operatorname{cl}(\mathcal{A}) = 3 cl ( A ) = 3
So far we have considered fusion algebras up to rank 9 and were able to find only one with cl ( A ) = 3 \operatorname{cl}(\mathcal{A}) = 3 cl ( A ) = 3 [37 , 40 ] has provided a catalogue of multiplicity free fusion rings up to rank 10. Extending our analysis to rank 10, we find that, among the 444 multiplicity free algebras, there are three with a cl ( A ) = 3 \operatorname{cl}(\mathcal{A}) = 3 cl ( A ) = 3 cl ( A ) > 3 \operatorname{cl}(\mathcal{A}) > 3 cl ( A ) > 3 5 F R 13 10 , 0 \mathrm{FR}_{13}^{10,0} FR 13 10 , 0 F R 22 10 , 4 \mathrm{FR}_{22}^{10,4} FR 22 10 , 4 F R 11 10 , 8 \mathrm{FR}_{11}^{10,8} FR 11 10 , 8
The three rank 10 algebras with cl ( A ) = 3 \operatorname{cl}(\mathcal{A}) = 3 cl ( A ) = 3
The fusion tables for each of the algebra are:
F R 13 10 , 0 \mathrm{FR}_{13}^{10,0} FR 13 10 , 0
Table FR10,0.
The invertible elements form a Z 2 × Z 2 \mathbb{Z}_2 \times\mathbb{Z}_2 Z 2 × Z 2
Com ( A ) = { 1 , 3 , 5 , 9 , 10 } ⊊ Com ( A ) 2 = A ∖ { 2 } ⊊ Com ( A ) 3 = A (58) \operatorname{Com}(\mathcal{A}) = \{1, 3, 5, 9, 10\} \subsetneq\operatorname{Com}(\mathcal{A})^2 = \mathcal{A} \setminus\{2\} \subsetneq\operatorname{Com}(\mathcal{A})^3 = \mathcal{A} \tag*{(58)} Com ( A ) = { 1 , 3 , 5 , 9 , 10 } ⊊ Com ( A ) 2 = A ∖ { 2 } ⊊ Com ( A ) 3 = A (58) F R 22 10 , 4 \mathrm{FR}_{22}^{10,4} FR 22 10 , 4
Table FR10,4.
Similar to the previous algebra, the invertible elements here also form a Z 2 × Z 2 \mathbb{Z}_2 \times\mathbb{Z}_2 Z 2 × Z 2
Com ( A ) = { 1 , 3 , 5 , 7 , 8 } ⊊ Com ( A ) 2 = A ∖ { 2 } ⊊ Com ( A ) 3 = A (59) \operatorname{Com}(\mathcal{A}) = \{1, 3, 5, 7, 8\} \subsetneq\operatorname{Com}(\mathcal{A})^2 = \mathcal{A} \setminus\{2\} \subsetneq\operatorname{Com}(\mathcal{A})^3 = \mathcal{A} \tag*{(59)} Com ( A ) = { 1 , 3 , 5 , 7 , 8 } ⊊ Com ( A ) 2 = A ∖ { 2 } ⊊ Com ( A ) 3 = A (59) × 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 1 5 3 8 6 10 7 9 3 3 1 5 2 4 7 9 6 10 8 4 4 5 2 3 1 10 8 9 6 7 5 5 3 4 1 2 9 10 7 8 6 6 6 8 7 10 9 1 + 7 + 8 3 + 6 + 9 2 + 6 + 10 5 + 7 + 10 4 + 8 + 9 7 7 6 9 8 10 3 + 6 + 9 5 + 7 + 10 1 + 7 + 8 4 + 8 + 9 2 + 6 + 10 8 8 10 6 9 7 2 + 6 + 10 1 + 7 + 8 4 + 8 + 9 3 + 6 + 9 5 + 7 + 10 9 9 7 10 6 8 5 + 7 + 10 4 + 8 + 9 3 + 6 + 9 2 + 6 + 10 1 + 7 + 8 10 10 9 8 7 6 4 + 8 + 9 2 + 6 + 10 5 + 7 + 10 1 + 7 + 8 3 + 6 + 9 \begin{array}{c|cccccccccc}
\times& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
2 & 2 & 4 & 1 & 5 & 3 & 8 & 6 & 10 & 7 & 9 \\
3 & 3 & 1 & 5 & 2 & 4 & 7 & 9 & 6 & 10 & 8 \\
4 & 4 & 5 & 2 & 3 & 1 & 10 & 8 & 9 & 6 & 7 \\
5 & 5 & 3 & 4 & 1 & 2 & 9 & 10 & 7 & 8 & 6 \\
6 & 6 & 8 & 7 & 10 & 9 & 1+7+8 & 3+6+9 & 2+6+10 & 5+7+10 & 4+8+9 \\
7 & 7 & 6 & 9 & 8 & 10 & 3+6+9 & 5+7+10 & 1+7+8 & 4+8+9 & 2+6+10 \\
8 & 8 & 10 & 6 & 9 & 7 & 2+6+10 & 1+7+8 & 4+8+9 & 3+6+9 & 5+7+10 \\
9 & 9 & 7 & 10 & 6 & 8 & 5+7+10 & 4+8+9 & 3+6+9 & 2+6+10 & 1+7+8 \\
10 & 10 & 9 & 8 & 7 & 6 & 4+8+9 & 2+6+10 & 5+7+10 & 1+7+8 & 3+6+9
\end{array} × 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 1 5 3 8 6 10 7 9 3 3 1 5 2 4 7 9 6 10 8 4 4 5 2 3 1 10 8 9 6 7 5 5 3 4 1 2 9 10 7 8 6 6 6 8 7 10 9 1 + 7 + 8 3 + 6 + 9 2 + 6 + 10 5 + 7 + 10 4 + 8 + 9 7 7 6 9 8 10 3 + 6 + 9 5 + 7 + 10 1 + 7 + 8 4 + 8 + 9 2 + 6 + 10 8 8 10 6 9 7 2 + 6 + 10 1 + 7 + 8 4 + 8 + 9 3 + 6 + 9 5 + 7 + 10 9 9 7 10 6 8 5 + 7 + 10 4 + 8 + 9 3 + 6 + 9 2 + 6 + 10 1 + 7 + 8 10 10 9 8 7 6 4 + 8 + 9 2 + 6 + 10 5 + 7 + 10 1 + 7 + 8 3 + 6 + 9 Here, the invertible elements form a Z 5 \mathbb{Z}_5 Z 5
Com ( A ) = { 1 , 7 , 8 } ⊆ Com ( A ) 2 = { 1 , 4 , 5 , 7 , 8 , 9 , 10 } ⊊ Com ( A ) 3 = A (60) \operatorname{Com}(\mathcal{A}) = \{1,7,8\} \subseteq\operatorname{Com}(\mathcal{A})^2 = \{1,4,5,7,8,9,10\} \subsetneq\operatorname{Com}(\mathcal{A})^3 = \mathcal{A} \tag*{(60)} Com ( A ) = { 1 , 7 , 8 } ⊆ Com ( A ) 2 = { 1 , 4 , 5 , 7 , 8 , 9 , 10 } ⊊ Com ( A ) 3 = A (60) Best assignments and textures:
For each algebra, we scan every assignment of ( H , l i , e j c , q i , u j c , d j c ) (H,l_i,e^c_j,q_i,u^c_j,d^c_j) ( H , l i , e j c , q i , u j c , d j c ) ρ t o t \rho_{\mathrm{tot}} ρ tot
F R 13 10 , 0 : H ∗ [ 3 ] , l i = ( 1 , 4 , 7 ) , e j c = ( 1 , 3 , 9 ) , q i = ( 1 , 5 , 9 ) , u j c = ( 2 , 4 , 5 ) , d j c = ( 5 , 9 , 10 ) \mathrm{FR}_{13}^{10,0}: \quad H^*[3],\ l_i=(1,4,7),\ e^c_j=(1,3,9),\ q_i=(1,5,9),\ u^c_j=(2,4,5),\ d^c_j=(5,9,10) FR 13 10 , 0 : H ∗ [ 3 ] , l i = ( 1 , 4 , 7 ) , e j c = ( 1 , 3 , 9 ) , q i = ( 1 , 5 , 9 ) , u j c = ( 2 , 4 , 5 ) , d j c = ( 5 , 9 , 10 ) F R 22 10 , 4 : H ∗ [ 3 ] , l i = ( 1 , 4 , 9 ) , e j c = ( 1 , 3 , 8 ) , q i = ( 1 , 5 , 7 ) , u j c = ( 2 , 4 , 5 ) , d j c = ( 5 , 7 , 8 ) \mathrm{FR}_{22}^{10,4}: \quad H^*[3],\ l_i=(1,4,9),\ e^c_j=(1,3,8),\ q_i=(1,5,7),\ u^c_j=(2,4,5),\ d^c_j=(5,7,8) FR 22 10 , 4 : H ∗ [ 3 ] , l i = ( 1 , 4 , 9 ) , e j c = ( 1 , 3 , 8 ) , q i = ( 1 , 5 , 7 ) , u j c = ( 2 , 4 , 5 ) , d j c = ( 5 , 7 , 8 ) F R 11 10 , 8 : H ∗ [ 4 ] , l i = ( 1 , 4 , 6 ) , e j c = ( 2 , 4 , 8 ) , q i = ( 1 , 4 , 7 ) , u j c = ( 1 , 5 , 7 ) , d j c = ( 3 , 6 , 9 ) (61) \mathrm{FR}_{11}^{10,8}: \quad H^*[4],\ l_i=(1,4,6),\ e^c_j=(2,4,8),\ q_i=(1,4,7),\ u^c_j=(1,5,7),\ d^c_j=(3,6,9) \tag*{(61)} FR 11 10 , 8 : H ∗ [ 4 ] , l i = ( 1 , 4 , 6 ) , e j c = ( 2 , 4 , 8 ) , q i = ( 1 , 4 , 7 ) , u j c = ( 1 , 5 , 7 ) , d j c = ( 3 , 6 , 9 ) (61) with the resulting textures
F R 13 10 , 0 : Y e ∼ ( ε 1 ε 2 ε 3 ε 2 ε 2 ε ε 2 ε ) , Y u ∼ ( ε 2 ε 3 ε ε 2 ε 2 1 ε ε 2 ε ) , Y d ∼ ( ε ε 2 ε 2 1 ε ε ε ε ε 2 ) . (62) \mathrm{FR}_{13}^{10,0}: \quad Y^e \sim\begin{pmatrix} \varepsilon& 1 & \varepsilon^2 \\ \varepsilon^3 & \varepsilon^2 & \varepsilon^2 \\ \varepsilon& \varepsilon^2 & \varepsilon\end{pmatrix}, \quad Y^u \sim\begin{pmatrix} \varepsilon^2 & \varepsilon^3 & \varepsilon\\ \varepsilon^2 & \varepsilon^2 & 1 \\ \varepsilon& \varepsilon^2 & \varepsilon\end{pmatrix}, \quad Y^d \sim\begin{pmatrix} \varepsilon& \varepsilon^2 & \varepsilon^2 \\ 1 & \varepsilon& \varepsilon\\ \varepsilon& \varepsilon& \varepsilon^2 \end{pmatrix}. \tag*{(62)} FR 13 10 , 0 : Y e ∼ ε ε 3 ε 1 ε 2 ε 2 ε 2 ε 2 ε , Y u ∼ ε 2 ε 2 ε ε 3 ε 2 ε 2 ε 1 ε , Y d ∼ ε 1 ε ε 2 ε ε ε 2 ε ε 2 . (62) F R 22 10 , 4 : Y e ∼ ( ε 1 ε 2 ε 3 ε 2 ε 2 ε ε 2 ε ) , Y u ∼ ( ε 2 ε 3 ε ε 2 ε 2 1 ε ε 2 ε ) , Y d ∼ ( ε ε 2 ε 2 1 ε ε ε ε 2 ε ) . (63) \mathrm{FR}_{22}^{10,4}: \quad Y^e \sim\begin{pmatrix} \varepsilon& 1 & \varepsilon^2 \\ \varepsilon^3 & \varepsilon^2 & \varepsilon^2 \\ \varepsilon& \varepsilon^2 & \varepsilon\end{pmatrix}, \quad Y^u \sim\begin{pmatrix} \varepsilon^2 & \varepsilon^3 & \varepsilon\\ \varepsilon^2 & \varepsilon^2 & 1 \\ \varepsilon& \varepsilon^2 & \varepsilon\end{pmatrix}, \quad Y^d \sim\begin{pmatrix} \varepsilon& \varepsilon^2 & \varepsilon^2 \\ 1 & \varepsilon& \varepsilon\\ \varepsilon& \varepsilon^2 & \varepsilon\end{pmatrix}. \tag*{(63)} FR 22 10 , 4 : Y e ∼ ε ε 3 ε 1 ε 2 ε 2 ε 2 ε 2 ε , Y u ∼ ε 2 ε 2 ε ε 3 ε 2 ε 2 ε 1 ε , Y d ∼ ε 1 ε ε 2 ε ε 2 ε 2 ε ε . (63) F R 11 10 , 8 : Y e ∼ ( ε 2 ε 3 ε 2 1 ε 3 ε 3 ε 2 ε ε ) , Y u ∼ ( ε 2 ε 3 ε 2 1 ε 2 ε ε 2 ε 3 ε 2 ) , Y d ∼ ( ε 3 ε 2 ε 3 ε 2 ε ε 2 ε 3 ε 2 ε 2 ) . (64) \mathrm{FR}_{11}^{10,8}: \quad Y^e \sim\begin{pmatrix} \varepsilon^2 & \varepsilon^3 & \varepsilon^2 \\ 1 & \varepsilon^3 & \varepsilon^3 \\ \varepsilon^2 & \varepsilon& \varepsilon\end{pmatrix}, \quad Y^u \sim\begin{pmatrix} \varepsilon^2 & \varepsilon^3 & \varepsilon^2 \\ 1 & \varepsilon^2 & \varepsilon\\ \varepsilon^2 & \varepsilon^3 & \varepsilon^2 \end{pmatrix}, \quad Y^d \sim\begin{pmatrix} \varepsilon^3 & \varepsilon^2 & \varepsilon^3 \\ \varepsilon^2 & \varepsilon& \varepsilon^2 \\ \varepsilon^3 & \varepsilon^2 & \varepsilon^2 \end{pmatrix}. \tag*{(64)} FR 11 10 , 8 : Y e ∼ ε 2 1 ε 2 ε 3 ε 3 ε ε 2 ε 3 ε , Y u ∼ ε 2 1 ε 2 ε 3 ε 2 ε 3 ε 2 ε ε 2 , Y d ∼ ε 3 ε 2 ε 3 ε 2 ε ε 2 ε 3 ε 2 ε 2 . (64) Scores
The scores for the three algebras are shown in Table 8 We note that the scores do not change appreciably from the rank 8 algebra scores. The corresponding spectra are show in Table 9
Table 8. Median score components for the selected rank-10 fusion-ring assignments, evaluated over the ensemble of Eq. (25) . The total score ρ t o t \rho_{\mathrm{tot}} ρ tot
Table 9. Monte Carlo medians of the fermion mass ratios for the selected rank-10 fusion-ring assignments of Eq. (D.4), together with the observed ratios for comparison. The main variation among the assignments appears in the up-type quark spectrum, particularly for F R 11 10 , 8 \mathrm{FR}_{11}^{10,8} FR 11 10 , 8
The CKM matrix
The CKM matrix across the ensemble gives the following intervals for the entries with the corresponding score,
∣ V C K M ∣ F R 13 10 , 0 = ( 0.998 8 − 0.0030 + 0.0010 0.032 3 − 0.0228 + 0.0441 0.025 7 − 0.0188 + 0.0292 0.032 3 − 0.0229 + 0.0441 0.998 7 − 0.0030 + 0.0010 0.025 8 − 0.0188 + 0.0293 0.025 7 − 0.0188 + 0.0291 0.025 8 − 0.0187 + 0.0294 0.999 1 − 0.0020 + 0.0007 ) , ρ C K M = 0.428. (65) \left|V_{\mathrm{CKM}}\right|_{\mathrm{FR}_{13}^{10,0}} =
\begin{pmatrix}
0.9988^{+0.0010}_{-0.0030} & 0.0323^{+0.0441}_{-0.0228} & 0.0257^{+0.0292}_{-0.0188} \\
0.0323^{+0.0441}_{-0.0229} & 0.9987^{+0.0010}_{-0.0030} & 0.0258^{+0.0293}_{-0.0188} \\
0.0257^{+0.0291}_{-0.0188} & 0.0258^{+0.0294}_{-0.0187} & 0.9991^{+0.0007}_{-0.0020}
\end{pmatrix},
\qquad\rho_{\mathrm{CKM}} = 0.428. \tag*{(65)} ∣ V CKM ∣ FR 13 10 , 0 = 0.998 8 − 0.0030 + 0.0010 0.032 3 − 0.0229 + 0.0441 0.025 7 − 0.0188 + 0.0291 0.032 3 − 0.0228 + 0.0441 0.998 7 − 0.0030 + 0.0010 0.025 8 − 0.0187 + 0.0294 0.025 7 − 0.0188 + 0.0292 0.025 8 − 0.0188 + 0.0293 0.999 1 − 0.0020 + 0.0007 , ρ CKM = 0.428. (65) ∣ V C K M ∣ F R 22 10 , 4 = ( 0.998 8 − 0.0030 + 0.0010 0.032 3 − 0.0228 + 0.0441 0.025 9 − 0.0188 + 0.0293 0.032 3 − 0.0229 + 0.0441 0.998 7 − 0.0030 + 0.0010 0.025 8 − 0.0188 + 0.0293 0.025 9 − 0.0188 + 0.0292 0.025 8 − 0.0188 + 0.0294 0.999 1 − 0.0020 + 0.0007 ) , ρ C K M = 0.429. (66) \left|V_{\mathrm{CKM}}\right|_{\mathrm{FR}_{22}^{10,4}} =
\begin{pmatrix}
0.9988^{+0.0010}_{-0.0030} & 0.0323^{+0.0441}_{-0.0228} & 0.0259^{+0.0293}_{-0.0188} \\
0.0323^{+0.0441}_{-0.0229} & 0.9987^{+0.0010}_{-0.0030} & 0.0258^{+0.0293}_{-0.0188} \\
0.0259^{+0.0292}_{-0.0188} & 0.0258^{+0.0294}_{-0.0188} & 0.9991^{+0.0007}_{-0.0020}
\end{pmatrix},
\qquad\rho_{\mathrm{CKM}} = 0.429. \tag*{(66)} ∣ V CKM ∣ FR 22 10 , 4 = 0.998 8 − 0.0030 + 0.0010 0.032 3 − 0.0229 + 0.0441 0.025 9 − 0.0188 + 0.0292 0.032 3 − 0.0228 + 0.0441 0.998 7 − 0.0030 + 0.0010 0.025 8 − 0.0188 + 0.0294 0.025 9 − 0.0188 + 0.0293 0.025 8 − 0.0188 + 0.0293 0.999 1 − 0.0020 + 0.0007 , ρ CKM = 0.429. (66) ∣ V C K M ∣ F R 11 10 , 8 = ( 0.706 7 − 0.2958 + 0.2051 0.707 0 − 0.2972 + 0.2043 0.015 3 − 0.0110 + 0.0225 0.707 0 − 0.2972 + 0.2043 0.706 7 − 0.2958 + 0.2051 0.015 3 − 0.0110 + 0.0224 0.016 9 − 0.0093 + 0.0205 0.016 9 − 0.0094 + 0.0208 0.999 6 − 0.0011 + 0.0003 ) , ρ C K M = 0.370. (67) \left|V_{\mathrm{CKM}}\right|_{\mathrm{FR}_{11}^{10,8}} =
\begin{pmatrix}
0.7067^{+0.2051}_{-0.2958} & 0.7070^{+0.2043}_{-0.2972} & 0.0153^{+0.0225}_{-0.0110} \\
0.7070^{+0.2043}_{-0.2972} & 0.7067^{+0.2051}_{-0.2958} & 0.0153^{+0.0224}_{-0.0110} \\
0.0169^{+0.0205}_{-0.0093} & 0.0169^{+0.0208}_{-0.0094} & 0.9996^{+0.0003}_{-0.0011}
\end{pmatrix},
\qquad\rho_{\mathrm{CKM}} = 0.370. \tag*{(67)} ∣ V CKM ∣ FR 11 10 , 8 = 0.706 7 − 0.2958 + 0.2051 0.707 0 − 0.2972 + 0.2043 0.016 9 − 0.0093 + 0.0205 0.707 0 − 0.2972 + 0.2043 0.706 7 − 0.2958 + 0.2051 0.016 9 − 0.0094 + 0.0208 0.015 3 − 0.0110 + 0.0225 0.015 3 − 0.0110 + 0.0224 0.999 6 − 0.0011 + 0.0003 , ρ CKM = 0.370. (67) We note that the two Z 2 × Z 2 \mathbb{Z}_2 \times\mathbb{Z}_2 Z 2 × Z 2 F R 8 8 , 6 \mathrm{FR}_{8}^{8,6} FR 8 8 , 6 F R 11 10 , 8 \mathrm{FR}_{11}^{10,8} FR 11 10 , 8
Neutrinos
As for the neutrinos, we find that the rank 10 algebras perform worse than the rank 8 algebra. We show the results in Table 10 .
Table 10. Neutrino-sector predictions for the selected rank-10 fusion-ring assignments of Eq. (D.4). The scale Λ \Lambda Λ m ν 3 = 0.05 m_{\nu_3}=0.05 m ν 3 = 0.05 [41 ] .
Renormalization Group stability of the textures
In this Appendix, we show that the Yukawa textures are not ruined by renormalization group flows. Once the heavy fields are integrated out, the running of the couplings comes from loops of Standard Model fields. At one loop, as mentioned in the introduction, we get the following contributions
d Y u d t = 1 16 π 2 [ 3 2 ( Y u Y u † − Y d Y d † ) + … ] Y u d Y d d t = 1 16 π 2 [ 3 2 ( Y d Y d † − Y u Y u † ) + … ] Y d d Y e d t = 1 16 π 2 [ 3 2 Y e Y e † + … ] Y e (68) \begin{aligned}
\frac{dY^u}{dt} &= \frac{1}{16\pi^2}\left[\frac{3}{2}\left(Y^uY^{u\dagger}-Y^dY^{d\dagger}\right)+\ldots\right]Y^u \\
\frac{dY^d}{dt} &= \frac{1}{16\pi^2}\left[\frac{3}{2}\left(Y^dY^{d\dagger}-Y^uY^{u\dagger}\right)+\ldots\right]Y^d \\
\frac{dY^e}{dt} &= \frac{1}{16\pi^2}\left[\frac{3}{2}Y^eY^{e\dagger}+\ldots\right]Y^e \tag*{(68)}
\end{aligned} d t d Y u d t d Y d d t d Y e = 16 π 2 1 [ 2 3 ( Y u Y u † − Y d Y d † ) + … ] Y u = 16 π 2 1 [ 2 3 ( Y d Y d † − Y u Y u † ) + … ] Y d = 16 π 2 1 [ 2 3 Y e Y e † + … ] Y e (68) where the … \ldots …
( δ Y α ) i j ∝ ( Y β Y β † Y α ) i j (69) (\delta Y^\alpha)_{ij} \propto(Y^\beta Y^{\beta\dagger}Y^\alpha)_{ij} \tag*{(69)} ( δ Y α ) ij ∝ ( Y β Y β † Y α ) ij (69) For the textures to be stable under RG running, we need to show that
Y β Y β † Y α ≤ Y α Y^\beta Y^{\beta\dagger}Y^\alpha\leq Y^\alpha Y β Y β † Y α ≤ Y α where, as in the introduction, “≤ \leq ≤ ε \varepsilon ε ε \varepsilon ε
We make two restrictions on the assignment of simple objects of the fusion algebra to Standard Model fields.
To each element of the triples q i q_i q i u i c u_i^c u i c d i c d_i^c d i c l i l_i l i e i c e_i^c e i c i = 1 , 2 , 3 i = 1, 2, 3 i = 1 , 2 , 3 A A A
The terms of interest are chains of three Yukawa couplings,
( Y α Y α † Y β ) i j = ∑ k , l Y i k α Y l k α ∗ Y l j β , x , T ∈ { u , d } , (70) \left(Y^\alpha Y^{\alpha\dagger}Y^\beta\right)_{ij}=\sum_{k,l}Y^\alpha_{ik}Y^{\alpha*}_{lk}Y^\beta_{lj},\qquad x,T\in\{u,d\}, \tag*{(70)} ( Y α Y α † Y β ) ij = k , l ∑ Y ik α Y l k α ∗ Y l j β , x , T ∈ { u , d } , (70) where β \beta β α \alpha α ε n i k α + n l k α + n l j β \varepsilon^{n^\alpha_{ik}+n^\alpha_{lk}+n^\beta_{lj}} ε n ik α + n l k α + n l j β ε s i j \varepsilon^{s_{ij}} ε s ij
s i j = min α , k , l ( n i k α + n l k α + n l j β ) . (71) s_{ij}=\min_{\alpha,k,l}\left(n^\alpha_{ik}+n^\alpha_{lk}+n^\beta_{lj}\right). \tag*{(71)} s ij = α , k , l min ( n ik α + n l k α + n l j β ) . (71) We can then write the one-loop correction as,
δ Y i j Y i j ∼ ε s i j − n i j × 1 16 π 2 ln Λ μ (72) \frac{\delta Y_{ij}}{Y_{ij}}\sim\varepsilon^{s_{ij}-n_{ij}}\times\frac{1}{16\pi^2}\ln\frac{\Lambda}{\mu} \tag*{(72)} Y ij δ Y ij ∼ ε s ij − n ij × 16 π 2 1 ln μ Λ (72) where we have suppressed possible order one factors (for example, coming from the top quark coupling). Since, we assume the high scale is sub-Planckian and thus 1 16 π 2 ln Λ μ < 1 \frac{1}{16\pi^2}\ln\frac{\Lambda}{\mu}<1 16 π 2 1 ln μ Λ < 1
n i j ≤ s i j ∀ i , j (73) n_{ij}\leq s_{ij}\ \forall\ i,j \tag*{(73)} n ij ≤ s ij ∀ i , j (73) The proof is as follows. Consider first, the case α = β = u \alpha=\beta=u α = β = u
Figure 4. The two shapes of a one-loop flavour-changing diagram. Each dot is a matched Yukawa coupling drawn as an allowed vertex with a spurion leg.
In the spurion form, we know from the vertices that
\begin{aligned}
Y^u_{ik}&:\quad1\prec hq_i u_k \bar{w}_1,\qquad w_1\in\operatorname{Com}n^\mathord{u}_{ik},\\
Y^{u*}_{lk}&:\quad1\prec h\bar{q}_l\bar{u}_k w_2,\qquad w_2\in\operatorname{Com}n^\mathord{u}_{lk},\\
Y^u_{lj}&:\quad1\prec hq_l u_j\bar{w}_3,\qquad w_3\in\operatorname{Com}n^\mathord{u}_{lj}. \tag*{(74)}
\end{aligned}
But, we know that
1 ≺ ( h q i u k w ˉ 1 ) ( h q ˉ l u ˉ k w 2 ) ( h q l u j w ˉ 3 ) (75) 1\prec(hq_i u_k\bar{w}_1)(h\bar{q}_l\bar{u}_k w_2)(hq_l u_j\bar{w}_3) \tag*{(75)} 1 ≺ ( h q i u k w ˉ 1 ) ( h q ˉ l u ˉ k w 2 ) ( h q l u j w ˉ 3 ) (75) Let us now use two results to simplify this relation.
If X , Y X,Y X , Y r r r 1 ≺ X ⋅ r 1 \prec X \cdot r 1 ≺ X ⋅ r 1 ≺ r ˉ ⋅ Y 1 \prec\bar{r} \cdot Y 1 ≺ r ˉ ⋅ Y 1 ≺ X ⋅ Y 1 \prec X \cdot Y 1 ≺ X ⋅ Y 1 ≺ X ⋅ r 1 \prec X \cdot r 1 ≺ X ⋅ r r ˉ ≺ X \bar{r} \prec X r ˉ ≺ X r ≺ Y r \prec Y r ≺ Y
1 ≺ r ⋅ r ˉ ≺ X ⋅ Y (76) 1 \prec r \cdot\bar{r} \prec X \cdot Y \tag*{(76)} 1 ≺ r ⋅ r ˉ ≺ X ⋅ Y (76) If X X X 1 ≺ X ⋅ h ⋅ h ˉ 1 \prec X \cdot h \cdot\bar{h} 1 ≺ X ⋅ h ⋅ h ˉ r ≺ h ⋅ h ˉ r \prec h \cdot\bar{h} r ≺ h ⋅ h ˉ 1 ≺ X ⋅ r 1 \prec X \cdot r 1 ≺ X ⋅ r h h h r = 1 r=1 r = 1 1 ≺ X 1 \prec X 1 ≺ X
Using the first of these, we first obtain,
1 ≺ ( h q i w ˉ 1 ) ( h w ˉ 2 ) ( h u j w ˉ 3 ) (77) 1 \prec(h q_i \bar{w}_1)(h \bar{w}_2)(h u_j \bar{w}_3) \tag*{(77)} 1 ≺ ( h q i w ˉ 1 ) ( h w ˉ 2 ) ( h u j w ˉ 3 ) (77) Then, using the second and that h h h
( h q i u j ‾ ) ≺ ( w ˉ 1 w 2 w ˉ 3 ) (78) (\overline{h q_i u_j}) \prec(\bar{w}_1 w_2 \bar{w}_3) \tag*{(78)} ( h q i u j ) ≺ ( w ˉ 1 w 2 w ˉ 3 ) (78) but note that the first paranthesis is precisely the operator we defined to be an element of Com n i j u \operatorname{Com}^{n^u_{ij}} Com n ij u ( w ˉ 1 w 2 w ˉ 3 ) ∈ Com n i k u + n l k u + n l j u (\bar{w}_1w_2\bar{w}_3) \in\operatorname{Com}^{n^u_{ik}+n^u_{lk}+n^u_{lj}} ( w ˉ 1 w 2 w ˉ 3 ) ∈ Com n ik u + n l k u + n l j u
n i j ≤ n i k u + n l k u + n l j u (79) n_{ij} \leq n^u_{ik} + n^u_{lk} + n^u_{lj} \tag*{(79)} n ij ≤ n ik u + n l k u + n l j u (79) 
