Residual Lepton Mixing Matrix

• Residual mixing matrix is a lepton mixing structure fixed by surviving flavor symmetries after spontaneous symmetry breaking.
• It enables analytic sum rules and constraints on PMNS parameters, highlighting patterns such as trimaximal, tri-bimaximal, and golden ratio mixing.
• This framework tightly restricts parameter freedom in the Standard Model flavor sector, offering testable predictions with current neutrino data.

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 $T$: $$T^\dagger M_\ell M_\ell^\dagger T = M_\ell M_\ell^\dagger$$ (Fonseca et al., 2014, Lavoura et al., 2014).

A generic feature is that the charged-lepton and neutrino residual symmetries ($G_\ell$, $G_\nu$) determine the diagonalization basis $U_\ell$, $M_\nu$0 up to phases and permutations; the physical mixing matrix $M_\nu$1 is then fully or partially fixed by the group-theoretic structure. For instance, if $M_\nu$2 is a Klein group $M_\nu$3 and $M_\nu$4 is a $M_\nu$5, specific rows or columns of $M_\nu$6 are analytically determined (Joshipura et al., 2016).

These arguments depend crucially on the requirement that $M_\nu$7 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 (Fonseca et al., 2014, Hu, 2014).

2. Group-Theoretical Construction and Classification

In the general classification (Fonseca et al., 2014), all “residual mixing matrices”—i.e., PMNS matrices determined by residual symmetries—were catalogued under the assumption of Majorana neutrinos and finite underlying $M_\nu$8. 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

$M_\nu$9

for $G$0 a root of unity and $G$1 (Fonseca et al., 2014). All 17 sporadic cases (covering TBM, golden ratio, bimaximal, and democratic patterns) are now excluded by the measured value of the reactor angle $G$2, except this infinite series, which encompasses the TM family and is consistent with experimental constraints (Fonseca et al., 2014, Hu, 2014).

The group-theoretic structure is manifest through the residual $G$3 and $G$4 generators, especially within $G$5 and closely related families (Joshipura et al., 2016). 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 $G$6 is fixed by residual symmetry, then in terms of the standard PMNS parametrization,

$G$7

one obtains two real “sum-rule” equations for the identified column, e.g.,

$G$8

for the TM2 column (Kashav et al., 2024, Novichkov et al., 2018).

Specific patterns, such as the TM2 column (trimaximal mixing), give the solar–reactor sum rule $G$9, and analytic relations among the atmospheric angle $G^T M_\nu G = M_\nu$0, reactor angle $G^T M_\nu G = M_\nu$1, and the Dirac phase $G^T M_\nu G = M_\nu$2: $G^T M_\nu G = M_\nu$3 (Novichkov et al., 2018). 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$G^T M_\nu G = M_\nu$4 / Modular case (TM2): The residual $G^T M_\nu G = M_\nu$5 (neutrino sector) with $G^T M_\nu G = M_\nu$6 (charged leptons) in $G^T M_\nu G = M_\nu$7 or its modular generalization leads to TM2 mixing. This fixes the second column to $G^T M_\nu G = M_\nu$8, with angle and phase sum rules as above (Novichkov et al., 2018, Kashav et al., 2024).
• $G^T M_\nu G = M_\nu$9 and general residuals: For $M_\ell M_\ell^\dagger$0 residuals, one column of $M_\ell M_\ell^\dagger$1 is given by

$M_\ell M_\ell^\dagger$2

with $M_\ell M_\ell^\dagger$3, $M_\ell M_\ell^\dagger$4. The second column at $M_\ell M_\ell^\dagger$5 is trimaximal; viable third-column predictions emerge for $M_\ell M_\ell^\dagger$6 (Joshipura et al., 2016).

• A$M_\ell M_\ell^\dagger$7 and other finite groups: Some sporadic cases fix a column, e.g., $M_\ell M_\ell^\dagger$8 for $M_\ell M_\ell^\dagger$9 (Lavoura et al., 2014), but are now strongly disfavored by precision data.
• Complex scaling symmetry: A complex extension of the residual scaling symmetry ($T$0 plus generalized CP) constrains the mixing matrix so that $T$1 and forces maximal Dirac CP violation ($T$2), with the other two phases restricted to $T$3 or $T$4, $T$5 and $T$6 unconstrained (Samanta et al., 2016).

5. Modular Invariance and Residual Mixing at Special Moduli

In modular invariant models, the “self-dual” point $T$7 generically yields a residual $T$8 (or $T$9-antisymmetry) in the neutrino sector, leading to a fixed column in the mixing matrix. Depending on the modular weight, either an ordinary symmetry ($$T^\dagger M_\ell M_\ell^\dagger T = M_\ell M_\ell^\dagger$$0) or antisymmetry ($$T^\dagger M_\ell M_\ell^\dagger T = M_\ell M_\ell^\dagger$$1) is realized. This can force one neutrino to be massless (antisymmetry) and the corresponding mixing vector to take group-theoretic values, e.g., for $$T^\dagger M_\ell M_\ell^\dagger T = M_\ell M_\ell^\dagger$$2, $$T^\dagger M_\ell M_\ell^\dagger T = M_\ell M_\ell^\dagger$$3 (Kashav et al., 2024). 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 $$T^\dagger M_\ell M_\ell^\dagger T = M_\ell M_\ell^\dagger$$4 is a discriminant for models with a trimaximal fixed column (Novichkov et al., 2018). The presence or absence of maximal Dirac CP violation distinguishes complex scaling and certain modular cases from others (Samanta et al., 2016, Kashav et al., 2024).

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 $$T^\dagger M_\ell M_\ell^\dagger T = M_\ell M_\ell^\dagger$$5 for Majorana phases are allowed, leading to distinct $$T^\dagger M_\ell M_\ell^\dagger T = M_\ell M_\ell^\dagger$$6 bands for normal/inverted spectra, providing future experimental exclusion or confirmation (Samanta et al., 2016).

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 (Fonseca et al., 2014, Lavoura et al., 2014).

7. Summary Table: Key Residual Mixing Matrix Patterns

Group/Construction	Fixed Structure	Viability (current data)
$$T^\dagger M_\ell M_\ell^\dagger T = M_\ell M_\ell^\dagger$$7, $$T^\dagger M_\ell M_\ell^\dagger T = M_\ell M_\ell^\dagger$$8 and $$T^\dagger M_\ell M_\ell^\dagger T = M_\ell M_\ell^\dagger$$9	TM2 column: $G_\ell$0	Allowed
$G_\ell$1, $G_\ell$2 even	TM2 column	Allowed
$G_\ell$3 ($G_\ell$4 extension)	$G_\ell$5	Excluded
Complex-extended scaling	$G_\ell$6, $G_\ell$7 or $G_\ell$8	Allowed for unconstrained $G_\ell$9
$G_\nu$0 (Sc. 2, fixed row)	$G_\nu$1 row	Essentially maximal $G_\nu$2
TBM, bimaximal, golden ratio	All entries (fully determined $G_\nu$3)	Excluded

• Classification of all possible mixing matrices from finite residual symmetries: (Fonseca et al., 2014)
• Trigonometric Diophantine classification and derivation of sum rules: (Hu, 2014)
• Fixed row/column and analytic trace formulae: (Lavoura et al., 2014)
• Modular invariance and residual symmetry at $G_\nu$4: (Kashav et al., 2024)
• Residual symmetries in $G_\nu$5 and analytic expressions: (Joshipura et al., 2016)
• Modular $G_\nu$6 models with residual mixing: (Novichkov et al., 2018)
• Complex scaling symmetry, maximal CPV, testable consequences: (Samanta et al., 2016)
• Global structure and unified flavor/CP residual symmetries: (Chuliá et al., 2019)