REVIEW
Tri-Bimaximal Neutrino Mixing, A4 and the Modular Symmetry
High Energy Physics - Phenomenology · Submitted 8 Dec 2005 · 12 references
Pith's verdict: A careful, honest 4D SUSY A4 model with a genuinely useful modular-group appendix — but eq. (40) for |U_e3| looks dimensionally wrong as printed and should be checked.
T0 · AI review
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Referee report
The paper
The paper claims
A four-dimensional supersymmetric model in which a broken A4 flavour symmetry produces tri-bimaximal neutrino mixing, with controlled corrections and a quark-sector mismatch.
The reading
What carries the argument
The load-bearing object is the pair of $A_4$ triplet flavons and their vacuum alignment. $A_4$ is presented through two generators $S$ and $T$ obeying $S^2 = (ST)^3 = T^3 = 1$, with $T$ diagonal and $S$ the symmetric matrix with $-1$ on the diagonal and $2$ off it (divided by three). A triplet vacuum along $(1,0,0)$ preserves the $Z_3$ generated by $T$; a triplet vacuum along $(1,1,1)$ preserves the reflection generated by $S$. The driving superpotential, linear in the driving fields $ϕ_T^0$, $ϕ_S^0$ and $ξ^0$ and written with all contractions allowed by $A_4$, $Z_3$ and $U(1)_R$, has its minimum exactly at those two directions, with an extra singlet $ξ$ required so that the $ϕ_S$ alignment has a non-trivial solution at all. The same $S$–$T$ presentation is what connects the construction to the modular group, whose defining relations are $s^2 = (st)^3 = 1$ with $t$ of infinite order; $A_4$ is the finite group obtained by truncating $t$ to order three, and the paper classifies the three-dimensional unitary representations of the modular group to show where the $A_4$ representation sits among them.
Extended reading notes
Core claim
The central claim is that a four-dimensional supersymmetric $A_4$ model with a driving-field sector, a $Z_3$ symmetry and a continuous $U(1)_R$ symmetry reproduces the tri-bimaximal mixing matrix at leading order without extra dimensions, and that it is natural in the technical sense: no parameter must be tuned to obtain the mixing angles. The mechanism is vacuum alignment. The driving superpotential forces the flavon $S$ to $(v_S,v_S,v_S)$ and the flavon $T$ to $(v_T,0,0)$, up to equivalent $A_4$ images. The first direction leaves unbroken the subgroup generated by $S$, which is the low-energy symmetry of the neutrino sector; the second leaves unbroken the subgroup generated by $T$, which is the low-energy symmetry of the charged-lepton sector. The resulting charged-lepton mass matrix is diagonal and the neutrino mass matrix has eigenvalues $a+b$, $a$, $-a+b$ with the tri-bimaximal matrix as its diagonalizer. The authors then classify the next-to-leading operators in three groups — corrections to the charged-lepton masses, to the neutrino masses, and to the vacuum alignment — and derive the induced shifts in $|U_{e3}|$, $tan^2 θ_{12}$ and $tan^2 θ_{23}$. All are suppressed by ratios of flavon vacuum expectation values to the cutoff, so the leading-order picture survives provided those ratios lie in the stated interval. The paper also shows that the same field assignment applied to quarks yields a diagonal CKM matrix at leading order and, because the first-order vacuum corrections are nearly identical in the up and down sectors, they cancel in the mixing matrix; the departures from the identity are therefore far too small to accommodate the observed quark mixing angles.
Load-bearing premise
The naturalness claim depends on the singlet vacuum expectation value $u$, which the leading-order potential leaves undetermined, actually being stabilized at a large scale by one-loop radiative corrections and by unspecified additional fields — the paper states this as an assumption rather than deriving it, so if that stabilization fails the ratio window that keeps corrections small is not fixed.
Editorial extensions
If this is right
- The tri-bimaximal mixing pattern follows from a broken discrete symmetry plus supersymmetric vacuum alignment alone, with no extra spatial dimensions required, so the earlier extra-dimensional version and this version are two realizations of the same low-energy mixing.
- Next-to-leading corrections to $|U_{e3}|$, $tan^2 θ_{12}$ and $tan^2 θ_{23}$ are all of the same parametric size, roughly the ratio of flavon vacuum expectation values to the cutoff, so a measurement of any one of them bounds the others.
- The window $0.0022 < v_S/Λ ≈ v_T/Λ ≈ u/Λ < 0.05$ is fixed from two sides: the lower end by keeping the tau Yukawa coupling perturbative, the upper end by not spoiling the leading-order agreement with $Δm^2_{sol}/Δm^2_{atm}$ and $tan^2 θ_{12}$.
- The construction carries over to a see-saw realization with right-handed neutrino triplets, where both normal and inverted hierarchies are attainable depending on the relative phase between the two mass parameters.
- Extending the leptonic $A_4$ assignment literally to quarks gives an identity CKM matrix at leading order and first-order corrections that cancel between up and down sectors, so quark mixing must come from new symmetry-breaking sources specific to quarks.
Reading between the lines
- Because the leading-order prediction is $|U_{e3}| = 0$ with corrections of order the flavon-to-cutoff ratio, a measured reactor angle near the current experimental bound would push the cutoff-suppression window to its upper edge or beyond, giving a direct handle on the scale of flavour breaking.
- The same $S$–$T$ presentation that makes $A_4$ a truncation of the modular group suggests the symmetry could arise from a compactification modulus rather than being imposed by hand; the paper stops at noting the coincidence and does not build such a model.
- The quark-sector failure is informative in the reverse direction: it says that any unified flavour scheme must break the symmetry differently in the up and down sectors, which is a constraint on grand-unified embeddings of this construction.
- The classification of three-dimensional unitary representations of the modular group places the $A_4$ representation at a symmetric interior point of the allowed region, which invites the question of whether a dynamical principle selects that point.
Original abstract
We formulate and discuss a 4-dimensional SUSY version of an A4 model for tri-bimaximal neutrino mixing which is completely natural. We also study the next-to-the-leading corrections and show that they are small, once the ratios of A4 breaking VEVs to the cutoff are fixed in a specified interval. We also point out an interesting way of presenting the A4 group starting from the modular group. In this approach, which could be interesting in itself as an indication on a possible origin of A4, the lagrangian basis where the symmetry is formulated coincides with the basis where the charged leptons are diagonal. If the same classification structure in A4 is extended from leptons to quarks, the CKM matrix coincides with the unit matrix in leading order and a study of non leading corrections shows that the departures from unity of the CKM matrix are far too small to accomodate the observed mixing angles.
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 formulate and discuss a 4-dimensional SUSY version of an A4 model for tri-bimaximal neutrino mixing which is completely natural. We also study the next-to-the-leading corrections and show that they are small, once the ratios of A4 breaking VEVs to the cutoff are fixed in a specified interval. We also point out an interesting way of presenting the A4 group starting from the modular group. In this approach, which could be interesting in itself as an indication on a possible origin of A4, the lagrangian basis where the symmetry is formulated coincides with the basis where the charged leptons are diagonal. If the same classification structure in A4 is extended from leptons to quarks, the CKM matrix coincides with the unit matrix in leading order and a study of non leading corrections shows that the departures from unity of the CKM matrix are far too small to accomodate the observed mixing angles.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We formulate and discuss a 4-dimensional SUSY version of an A4 model for tri-bimaximal neutrino mixing which is completely natural. | contribution 1.00 | 0.93 | 0.43 | asserted 1.00 |
| s1 | We also study the next-to-the-leading corrections and show that they are small, once the ratios of A4 breaking VEVs to the cutoff are fixed in a specified interval. | result 0.77 | 0.94 | 0.14 | asserted 1.00 |
| s2 | We also point out an interesting way of presenting the A4 group starting from the modular group. | contribution 0.99 | 0.87 | 0.06 | asserted 0.97 |
| s3 | In this approach, which could be interesting in itself as an indication on a possible origin of A4, the lagrangian basis where the symmetry is formulated coincides with the basis where the charged leptons are diagonal. | result 0.41 | 0.80 | 0.07 | asserted 0.69 |
| s4 | If the same classification structure in A4 is extended from leptons to quarks, the CKM matrix coincides with the unit matrix in leading order and a study of non leading corrections shows that the departures from unity of the CKM matrix are far too small to accomodate the observed mixing angles. | result 0.67 | 0.88 | 0.22 | asserted 0.93 |
What was examined, and what was not
- arXiv listing abstract v2
- 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
Figures
Figures reproduced from arXiv:hep-ph/0512103 by the authors.
References
How later papers used it
192 later papers cite it
2 later papers by other authors are listed below: 2 compared their own results with it.
Compares with it 2 of 10 papers with this label
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Stated that the main difference between its models and the original models reported in this paper and another work is the mass scale of the right-handed neutrino fields.
Majorana Phases and Leptogenesis in See-Saw Models with A_4 Symmetry (2009) · 1.00 · paraphrased: the licence does not allow quoting
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Counted degrees of freedom and found its model has the same number as the minimal model without a vacuum alignment mechanism in this paper.
Natural Vacuum Alignment from Group Theory: The Minimal Case (2011) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
164 arXiv papers cite this in 396 sentences
Papers citing it · 264 sentences without a confident label
Contests it 4 papers
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We conclude that the operator $((φ_Tφ_S)_3A(lh_u l h_u)_3S)$ is missing from the basis discussed in [ref] .
Discrete symmetries and Efficient Counting of Operators · Case studies · contest 0.89
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(74) of [ref] : one $ϕ_S$ should be relabeled to $ϕ_T$ in $I_2,3^X$ (e.g. $(ϕ_S ϕ_S) →(ϕ_S ϕ_T)$, otherwise these operators are not invariant under the AF $ℤ_3$ shaping symmetry.] carry exclusively $S$ or $T$ labels.
Residual Symmetries and Scalar Multiplet Vacuum Alignment in Non-Abelian Flavour Models · Higher-Order Operators and RFS Violation · contest 0.80
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Early attempts of extending models based on $A_4$ to quarks [ref] and to construct grand unified versions [ref] so far have not been completely satisfactory, e.g. do not offer natural mechanisms for mass hierarchies and for the vacuum alignment.
A SUSY SU(5) Grand Unified Model of Tri-Bimaximal Mixing from A4 · Introduction · contest 0.77
Builds on it 4 papers
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In the present work, we investigate the generation of non-zero $U_e3$ through see saw mechanism by considering a small perturbation in $m_D$, the Dirac neutrino mass matrix, keeping the same texture of the right-handed Majorana neutrino mass matrix as proposed in Ref. [ref] .
Nonzero U_{e3}, CP violation and leptogenesis in a see-saw type softly broken A_4 symmetric model · Introduction · 0.92
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We have used two real scalar copies $ϕ_1$ and $ϕ_2$ of the faithful representation $41$, where the triplet contraction $(ϕ_1 ϕ_2)_31$ couples to neutrinos and thus plays the role of $ϕ_S$ in the model of Altarelli and Feruglio [ref] .
Natural Vacuum Alignment from Group Theory: The Minimal Case · Conclusions · 0.80
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For instance, the $A_4$ model from Ref. [ref] can be modified [ref] by adding right-handed neutrinos transforming as $3$, leading to an “inverted” sum-rule [equation] Again, by replacing the singlet $1$ by two singlets $1'$, $1”$ and assuming the same tuning between their VEVs and couplings as above, the sum-rule [ref] [equation] is found.
Neutrino Mass Sum-rules in Flavor Symmetry Models · Sum-rules in flavor symmetry models · 0.97
Uses its method or tool 10 papers
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[ref] and leads to the result anticipated in the previous section: [equation] with $u$ undetermined and $g_3≡3g̃_3^2$ , $g_4≡-g̃_4^2$.
A SUSY SU(5) Grand Unified Model of Tri-Bimaximal Mixing from A4 · Vacuum Alignment · 0.93
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The vacuum alignment problem can be solved by the supersymmetric driving field method introduced in Ref. [ref] .
A Model for Tri-bimaximal Mixing from a Completely Broken $A_4$ · Vacuum alignment · 1.00
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Indeed, the VEV pattern of the flavons is determined dynamically, in which the vacuum alignment problem can be solved by the supersymmetric driving field method [ref] [There is another generic way for the vacuum alignment problem by extending the model with a spacial extra dimension [ref] .].
Flavored Peccei-Quinn symmetry · Vacuum configuration · 0.96
Uses its data or values 1 paper
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We only display leading matrix elements which correspond to the neutrino mass matrix in Ref. [ref] :
Non-Abelian Discrete Symmetries in Particle Physics · $A_4$ flavor model · 0.82
Compares with it 9 papers
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Thus, we would expect that three of the five operators in the first expression of [ref] should reproduce the Yukawa terms when we plug in vevs, and the remaining two operators correspond to the other two non-trivial operators identified in [ref], in order to reproduce the statements made in [ref] .
Discrete symmetries and Efficient Counting of Operators · Case studies · 0.87
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The main difference between the models we are discussing and the original models reported in [ref] and [ref] is in the mass scale of the RH neutrino fields.
Majorana Phases and Leptogenesis in See-Saw Models with A_4 Symmetry · Neutrino Masses and CP Violating Phases in the $A_4$ Models · 1.00
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For the VEVs of the flavons we confirm the result given in [ref] , which we list in the Appendix for completeness.
Vacuum Alignment in SUSY A4 Models · Generic soft SUSY breaking terms · 0.84
Cites it as support 4 papers
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But the corrections to it turn out to be strongly constrained by the leptonic sector, because lepton mixings are nearly TB, and, in the simplest models, are proven to be too small to accommodate the observed quark mixing angles [ref] .
Theoretical Models of Neutrino Mixing: Recent Developments · $A_4$, quarks and GUT's · 0.97
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Actually in the models we consider relatively small values of $tanβ$ are preferable [ref] .
Majorana Phases and Leptogenesis in See-Saw Models with A_4 Symmetry · Introduction · 0.96
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This seems to be well motivated from a model-building perspective as quite often there are a multitude of higher-dimensional operators contributing at NLO order.[Since in typical models (e.g. [ref] ) these higher dimensional operators do not respect any remnant symmetries this agnostic approach seems warranted.
Quark and Leptonic Mixing Patterns from the Breakdown of a Common Discrete Flavor Symmetry · Towards Quantifying the Predictive Power Of Discrete Groups · 0.97
Cites it as background 71 papers
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The $A_4$ models [ref] we discuss in this paper share several features: the left-handed lepton doublets $l$ and the three RH neutrinos $ν^c$ transform as triplets under $A_4$.
Charged Lepton Flavour Violating Radiative Decays $\ell_i \to \ell_j + \gamma$ in See-Saw Models with $A_4$ Symmetry · $A_4$ Models · 0.80
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In this section we recapitulate the main features of the AF [ref] and the AM model [ref] .
Majorana Phases and Leptogenesis in See-Saw Models with A_4 Symmetry · Variants of the Two $A_4$ Models · 0.99
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[ref] there is a brief analysis of the possible hierarchies allowed by the seesaw realization of the AF model.
Fine tuning in an A4-based Tri-Bimaximal neutrino-mixing model · Seesaw realization · 0.82
Discussion
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