---
title: Residual Lepton Mixing Matrix
url: https://www.emergentmind.com/topics/residual-mixing-matrix
type: topic
---

# Residual Lepton Mixing Matrix

A residual mixing matrix is a lepton (typically neutrino) mixing matrix whose (partial or full) structure is fixed by the residual flavor symmetries that survive the spontaneous breaking of a larger, typically non-Abelian, discrete flavor symmetry group. The concept has proven central for encoding highly predictive relations among the mixing angles and CP phases, and for tightly constraining the parametric freedom in the Standard Model flavor sector. In the context of the PMNS (Pontecorvo–Maki–Nakagawa–Sakata) matrix, the residual symmetry approach captures the scenario where residual symmetries in the neutrino and/or charged-lepton mass matrices lead to mixing matrices of determined or partially determined form. This framework provides the basis for understanding and classifying leading paradigms of lepton flavor mixing, most notably the “trimaximal” (TM), “tri-bimaximal” (TBM), “golden ratio”, and bimaximal patterns, and supplies analytic sum rules for mixing angles and phases derived solely from group-theoretic and modular constraints.

## 1. Residual Symmetry Approach: Core Principles

The residual symmetry formalism emerges from the observation that, after an underlying flavor symmetry group $G$ breaks spontaneously, different residual subgroups may remain unbroken in the charged-lepton and neutrino mass sectors. In the Majorana neutrino case, the effective mass matrix $M_\nu$ admits unitary transformations $G$ such that $G^T M_\nu G = M_\nu$, and analogously for $M_\ell M_\ell^\dagger$ with $T$: $T^\dagger M_\ell M_\ell^\dagger T = M_\ell M_\ell^\dagger$ [1405.3678, 1401.5036]. 

A generic feature is that the charged-lepton and neutrino residual symmetries ($G_\ell$, $G_\nu$) determine the diagonalization basis $U_\ell$, $U_\nu$ up to phases and permutations; the physical mixing matrix $U_{\rm PMNS} = U_\ell^\dagger U_\nu$ is then fully or partially fixed by the group-theoretic structure. For instance, if $G_\nu$ is a Klein group $Z_2\times Z_2$ and $G_\ell$ is a $Z_m$, specific rows or columns of $U_{\rm PMNS}$ are analytically determined [1610.07903]. 

These arguments depend crucially on the requirement that $G$ be a finite group, ensuring all residual generator eigenvalues are roots of unity; trace relations and theorems about vanishing sums of roots of unity provide the classification toolkit for all possible residual mixing patterns [1405.3678, 1407.4722]. 

## 2. Group-Theoretical Construction and Classification

In the general classification [1405.3678], all “residual mixing matrices”—i.e., PMNS matrices determined by residual symmetries—were catalogued under the assumption of Majorana neutrinos and finite underlying $G$. The key steps include expressing the invariant conditions as trace and eigenvalue relations, then reducing the resulting constraints to algebraic—typically trigonometric Diophantine—equations. These equations admit only a small set of solutions compatible with three-neutrino mixing.

The main result is that only 17 distinct “sporadic” mixing patterns plus a unique infinite family (“trimaximal-type series”) satisfy all group-theoretical and phenomenological criteria. The infinite series corresponds to “trimaximal” matrices of the form
$$
|U|^2 = \frac13 
\begin{pmatrix}
1 & 1+\Re\sigma & 1-\Re\sigma \\
1 & 1+\Re(\omega \sigma) & 1-\Re(\omega \sigma) \\
1 & 1+\Re(\omega^2 \sigma) & 1-\Re(\omega^2 \sigma)
\end{pmatrix}
$$
for $\sigma$ a root of unity and $\omega = e^{2\pi i/3}$ [1405.3678]. All 17 sporadic cases (covering TBM, golden ratio, bimaximal, and democratic patterns) are now excluded by the measured value of the reactor angle $\theta_{13}$, except this infinite series, which encompasses the TM family and is consistent with experimental constraints [1405.3678, 1407.4722]. 

The group-theoretic structure is manifest through the residual $Z_2$ and $Z_m$ generators, especially within $\Delta(6N^2)$ and closely related families [1610.07903]. The residual mixing matrix’s columns/rows arise as unique (up to permutation) eigenvectors of the generators, and their moduli can be computed analytically in terms of group parameters.

## 3. Analytic Predictions and Sum Rules

Residual mixing matrices yield sharp analytic predictions for mixing angles and phases, derived algebraically from the group/theoretic constraints without reference to model-specific parameters. For instance, if a column $c_0 = (a, b, c)^T$ is fixed by residual symmetry, then in terms of the standard PMNS parametrization,
\[
U_{\rm PMNS} = 
\begin{pmatrix}
c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i\delta} \\
- s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i\delta} & c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i\delta} & s_{23} c_{13} \\
s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i\delta} & - c_{12} s_{23} - s_{12} c_{23} s_{13} e^{i\delta} & c_{23} c_{13}
\end{pmatrix}
\]
one obtains two real “sum-rule” equations for the identified column, e.g.,
\[
|U_{e2}|^2 = \sin^2\theta_{12} \cos^2\theta_{13},\qquad
|U_{\mu 2}|^2 = |c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i\delta}|^2
\]
for the TM2 column [2410.00565, 1812.11289]. 

Specific patterns, such as the TM2 column (trimaximal mixing), give the solar–reactor sum rule $\sin^2\theta_{12}\cos^2\theta_{13}=1/3$, and analytic relations among the atmospheric angle $\theta_{23}$, reactor angle $\theta_{13}$, and the Dirac phase $\delta$:
\[
\sin^2\theta_{23} = \frac12 + \frac{\sqrt2 \sin\theta_{13} \cos\phi}{2\sqrt{1 - \sin^2\theta_{13}}},\quad
\cos\delta = \frac{\cos 2\theta_{23} \cos 2\theta_{13}}{ \sin 2\theta_{23} \sin\theta_{13} \sqrt{2 - 3 \sin^2\theta_{13}} }
\]
[1812.11289]. These sum rules are independent of the neutrino mass spectrum and thus provide distinguishing tests of residual symmetry models.

## 4. Examples and Representative Patterns

A variety of group choices and residual subgroups instantiate different residual mixing matrices:

- **A$_4$ / Modular case (TM2):** The residual $Z_2^S$ (neutrino sector) with $Z_3^T$ (charged leptons) in $A_4$ or its modular generalization leads to TM2 mixing. This fixes the second column to $(1,1,1)^T/\sqrt{3}$, with angle and phase sum rules as above [1812.11289, 2410.00565].

- **$\Delta(6N^2)$ and general residuals:** For $Z_2\times Z_m$ residuals, one column of $U_{\rm PMNS}$ is given by
  $$
  (\,|1-e^{i\xi}|^2,\,|1-\omega e^{i\xi}|^2,\,|1-\omega^2 e^{i\xi}|^2\,)/6
  $$
  with $\xi=2\pi k/N$, $\omega=e^{2\pi i/3}$. The second column at $\xi=\pi$ is trimaximal; viable third-column predictions emerge for $N=11,17,...$ [1610.07903].

- **A$_5$ and other finite groups:** Some sporadic cases fix a column, e.g., $((3+\sqrt{5})/8,\,1/4,\,(3-\sqrt{5})/8)$ for $A_5$ [1401.5036], but are now strongly disfavored by precision data.

- **Complex scaling symmetry:** A complex extension of the residual scaling symmetry ($Z_2^{\rm scaling}$ plus generalized CP) constrains the mixing matrix so that $\theta_{23} = \arctan(1/k)$ and forces maximal Dirac CP violation ($\cos\delta=0$), with the other two phases restricted to $0$ or $\pi$, $\theta_{13}$ and $\theta_{12}$ unconstrained [1604.06731].

## 5. Modular Invariance and Residual Mixing at Special Moduli

In modular invariant models, the “self-dual” point $\tau=i$ generically yields a residual $Z_2$ (or $Z_2$-antisymmetry) in the neutrino sector, leading to a fixed column in the mixing matrix. Depending on the modular weight, either an ordinary symmetry ($[S_\nu, M_\nu]=0$) or antisymmetry ($\{S_\nu, M_\nu\}=0$) is realized. This can force one neutrino to be massless (antisymmetry) and the corresponding mixing vector to take group-theoretic values, e.g., for $A_4$, $c_\nu = (1,1,1)^T/\sqrt{3}$ [2410.00565]. Such models provide a highly predictive—often unique—residual mixing matrix, with sum rules directly descending from representation and modular properties.

## 6. Phenomenological Implications and Experimental Tests

Residual mixing matrices, given their rigid structure, pronounce clear experimental predictions for PMNS parameters. For instance, the TM2 sum rule $\sin^2\theta_{12}\cos^2\theta_{13}=1/3$ is a discriminant for models with a trimaximal fixed column [1812.11289]. The presence or absence of maximal Dirac CP violation distinguishes complex scaling and certain modular cases from others [1604.06731, 2410.00565].

Neutrinoless double-beta decay matrix elements are also directly correlated with the Majorana phases fixed (or not) by residual symmetries. For the complex-scaling case, only CP-conserving values $(0,\pi)$ for Majorana phases are allowed, leading to distinct $|M_{ee}|$ bands for normal/inverted spectra, providing future experimental exclusion or confirmation [1604.06731].

Constructs with a fully fixed residual mixing matrix (i.e., all entries determined) are now phenomenologically excluded, except for the continuous trimaximal family. Only highly constrained, partial patterns—typically single column or single row fixings—remain within experimental viability [1405.3678, 1401.5036].

## 7. Summary Table: Key Residual Mixing Matrix Patterns

| Group/Construction               | Fixed Structure                    | Viability (current data) |
|----------------------------------|------------------------------------|--------------------------|
| $A_4$, $Z_3^T$ and $Z_2^S$       | TM2 column: $(1,1,1)/\sqrt{3}$     | Allowed                  |
| $\Delta(6N^2)$, $N$ even         | TM2 column                         | Allowed                  |
| $A_5$ ($[60,5]$ extension)       | $((3+\sqrt{5})/8,\,1/4,(3-\sqrt{5})/8)$ | Excluded            |
| Complex-extended scaling         | $\theta_{23} = \arctan (1/k)$, $\delta = \pi/2$ or $3\pi/2$ | Allowed for unconstrained $k$ |
| $S_4$ (Sc. 2, fixed row)         | $(1/4,\,1/4,\,1/2)$ row            | Essentially maximal $\theta_{23}$ |
| TBM, bimaximal, golden ratio     | All entries (fully determined $U$) | Excluded                 |

## References

- Classification of all possible mixing matrices from finite residual symmetries: [1405.3678]
- Trigonometric Diophantine classification and derivation of sum rules: [1407.4722]
- Fixed row/column and analytic trace formulae: [1401.5036]
- Modular invariance and residual symmetry at $\tau=i$: [2410.00565]
- Residual symmetries in $\Delta(6N^2)$ and analytic expressions: [1610.07903]
- Modular $A_4$ models with residual mixing: [1812.11289]
- Complex scaling symmetry, maximal CPV, testable consequences: [1604.06731]
- Global structure and unified flavor/CP residual symmetries: [1911.12043]

Source: https://www.emergentmind.com/topics/residual-mixing-matrix