REVIEW
mu-tau Interchange symmetry and lepton mixing
High Energy Physics - Phenomenology · Submitted 6 Jul 2012 · 28 references
Pith's verdict: A correct, compact review of mu–tau symmetry from the people who built most of it — solid on the algebra, thin on everyone else's work.
Why Pith says this
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Referee report
The paper
The paper claims
A muon–tau symmetry in the neutrino mass matrix explains maximal atmospheric mixing and ties the small muon mass to a softly broken symmetry.
The reading
What carries the argument
The load-bearing object is the $μ$–$τ$ permutation matrix $S$ in Eq. (7). Imposing $S M_ν S = M_ν$ constrains the neutrino mass matrix to the four-parameter form of Eq. (5), whose antisymmetric eigenvector delivers $θ_{23} = 45^∘$ and $U_{e3} = 0$; imposing instead $S M_ν S = M_ν^{*}$ constrains it to the form of Eq. (18) with real $ee$ and $μμ$–$ττ$ entries, and delivers $|U_{μj}| = |U_{τj}|$ with $sin θ_{13} cos δ = 0$. The second piece of machinery is the split between the Yukawa Lagrangian, where family lepton numbers are exact and the Dirac mass matrix is diagonal, and the right-handed Majorana mass matrix $M_R$, where they are broken softly and where the $μ$–$τ$ symmetry (or its CP version) is preserved; the seesaw formula $M_ν = -M_D^{T} M_R^{-1} M_D$ then transmits the symmetry to the light sector. The third piece is the $K$ symmetry of Eq. (40), which forces $y_4 = -y_5$ and confines the muon–tau mass fine-tuning to the ratio of Higgs vacuum expectation values.
Extended reading notes
Core claim
The central claim is that $μ$–$τ$ interchange symmetry, imposed on the effective light-neutrino Majorana mass matrix $M_ν$ in the basis where the charged-lepton mass matrix is diagonal, is exactly the right amount of structure to produce maximal atmospheric neutrino mixing. The symmetric matrix of the form in Eq. (5) has $(0,1,-1)$ as an eigenvector, so one column of the PMNS matrix is fixed to $(0, e^{-iφ/2}, -e^{-iφ/2})$; identifying it with the third column gives $θ_{23} = 45^∘$ and $U_{e3} = 0$ with no constraint on the neutrino masses or on the solar angle. Because $U_{e3}$ is now known to be non-zero, the authors present the generalized-CP version, defined by $S M_ν S = M_ν^{*}$, whose solutions give $|U_{μj}| = |U_{τj}|$ for all three columns, hence $θ_{23} = 45^∘$ and $sin θ_{13} cos δ = 0$, with Majorana phases restricted to $0$ or $π$. Both effective mass matrices are realized in seesaw models where the family lepton numbers $U(1)_{L_α}$ are exact in the Yukawa sector and broken only by the dimension-three right-handed neutrino mass matrix, so that all lepton mixing originates in one place. The paper then argues that an extra symmetry $K$ reduces the muon and tau masses to $m_μ/m_τ = |(v_2 - v_3)/(v_2 + v_3)|$, and that soft breaking of $K$ by a dimension-two term in the scalar potential yields $m_μ/m_τ ≃ |2μ_{soft}/(λ̃(|v_2|^2+|v_3|^2))|$, making the small ratio a natural consequence of a small soft parameter.
Load-bearing premise
The naturalness claim for $m_μ/m_τ$ rests on the scalar potential's minimum sitting at $v_2 = v_3$ before $K$ is broken, a condition the paper quotes from earlier work rather than deriving, and on the soft parameter $μ_{soft}$ being genuinely small compared with the other dimensionful scales.
Editorial extensions
If this is right
- If $μ$–$τ$ interchange holds in the neutrino mass matrix, atmospheric mixing is exactly maximal, $θ_{23} = 45^∘$, independent of the neutrino mass spectrum and of the solar angle.
- In the generalized-CP version the reactor angle is unconstrained but the Dirac phase is forced to be maximal, $cos δ = 0$, so a precise measurement of $δ$ near $90^∘$ or $270^∘$ would support this class of models.
- All lepton mixing in these models originates in the right-handed neutrino Majorana mass matrix, so the same matrix controls both the PMNS angles and the size of lepton-flavour-violating processes.
- Because flavour violation enters only through soft family-lepton-number breaking, lepton-flavour-changing amplitudes such as $μ → eγ$ are finite at one loop and calculable in terms of the same soft parameters.
- With the $K$ symmetry, the ratio $m_μ/m_τ$ is proportional to the soft breaking parameter $μ_{soft}$, so a small muon mass is technically natural rather than an unexplained cancellation between unrelated Yukawa couplings and vacuum expectation values.
Reading between the lines
- The two variants make sharply different bets that current data can already begin to separate: the plain interchange version is now excluded by $U_{e3} ≠ 0$, while the generalized-CP version survives and converts the reactor angle measurement into a prediction for $δ$, so the two are distinguishable by the same oscillation data.
- If the mechanism is right, the same $S$ that fixes $θ_{23}$ also restricts the Majorana phases to $0$ or $π$, which sharpens the predicted rate for neutrinoless double beta decay relative to generic Majorana models.
- The naturalness argument for $m_μ/m_τ$ is a statement about the scalar potential, not about the neutrino sector, so it could be tested independently by the Higgs spectrum: the same $φ_2$–$φ_3$ exchange symmetry that ties the mass ratio to $μ_{soft}$ also organizes the physical scalar masses and their couplings.
- The construction suggests a broader template: any family symmetry that is exact in the Yukawa sector and broken only in the dimension-three right-handed neutrino mass matrix gives mixing with a single source, which is a cheaper route to predictivity than building a full discrete flavour group.
Original abstract
We focus on the use of a mu-tau interchange symmetry to explain features of lepton mixing, especially maximal atmospheric neutrino mixing. We review two models which achieve this goal and are based on the seesaw mechanism and on the soft breaking of the family-lepton-number symmetries. We also note that that symmetry may be embedded in a generalized CP symmetry. We show that, in the context of some of our models, arguments of naturalness may be used for explaining the smallness of the mass ratio m_mu / m_tau.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We focus on the use of a mu-tau interchange symmetry to explain features of lepton mixing, especially maximal atmospheric neutrino mixing. We review two models which achieve this goal and are based on the seesaw mechanism and on the soft breaking of the family-lepton-number symmetries. We also note that that symmetry may be embedded in a generalized CP symmetry. We show that, in the context of some of our models, arguments of naturalness may be used for explaining the smallness of the mass ratio m_mu / m_tau.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We focus on the use of a mu-tau interchange symmetry to explain features of lepton mixing, especially maximal atmospheric neutrino mixing. | contribution 0.86 | 0.57 | 0.19 | asserted 0.81 |
| s1 | We review two models which achieve this goal and are based on the seesaw mechanism and on the soft breaking of the family-lepton-number symmetries. | contribution 0.79 | 0.51 | 0.10 | asserted 0.99 |
| s2 | We also note that that symmetry may be embedded in a generalized CP symmetry. | interpretation 0.52 | 0.27 | 0.07 | hedged possibility 1.00 |
| s3 | We show that, in the context of some of our models, arguments of naturalness may be used for explaining the smallness of the mass ratio m_mu / m_tau. | result 0.63 | 0.90 | 0.14 | asserted 0.94 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
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- R cross-paper relations
References
How the literature uses it
34 arXiv papers cite this in 36 sentences
Papers citing it · 22 sentences without a confident label
Cites it as background 14 papers
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A notable example is the $-$ reflection symmetry [ref] that predicts the maximal 2-3 mixing $þ_23 = π/4$, the maximal Dirac phase $=̣ ±π/2$ and Majolana phases 0 or $π/2$ [ref] .
Almost general analysis of $\mu$-$\tau$ reflection symmetry perturbed by charged leptons and its testability by DUNE and T2HK · Introduction · 0.85
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The invariance of the light neutrino mass matrix under $μ-τ$ reflection in the charged lepton diagonal basis leads to maximal atmospheric mixing angle $θ_23$ and maximal Dirac CP phase $δ_CP$ with $δ_CP=±π/2$ [ref] .
Deviation from Bimaximal Mixing and Leptonic CP Phases in $S_4$ Family Symmetry and Generalized CP · Introduction · 0.89
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This symmetry[For a review and original references, see Ref. [ref] ] postulated only in the neutrino sector, stipulates an invariance of all neutrino terms in the Lagarangian under the interchange of the flavor indices $2↔3$.
Maximal zero textures of the inverse seesaw with broken $\mu\tau$ symmetry · Inverse seesaw, maximal zeros and $μτ$ symmetry · 0.95
What it cites · 5 sentences without a confident label
Cites as background 2 papers
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[ref] and [ref] for the model-building and group-theoretical, respectively, aspects of that problem.
Discrete Flavor Symmetries and Models of Neutrino Mixing · Introduction · 1.00
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[ref] and [ref] for the model-building and group-theoretical, respectively, aspects of that problem.
Non-Abelian Discrete Symmetries in Particle Physics · Introduction · 1.00
Discussion
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