REVIEW
Mass Matrix Models: The Sequel
High Energy Physics - Phenomenology · Submitted 20 Oct 1993 · 22 references
Pith's verdict: A solid, candid entry in the Froggatt–Nielsen program; the new results are real, but the sharpest claim — D–D̄ mixing near the bound — rests on zeros the paper never verifies survive the discrete symmetry.
T0 · AI review
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Referee report
The paper
The paper claims
A single broken U(1) symmetry, with charges assigned to each quark field, can reproduce the entire observed pattern of quark masses and mixing angles as powers of one small parameter λ ≈ 0.2, and — if it commutes with supersymmetry — lineUp
The reading
What carries the argument
The load-bearing device is the selection rule $Y^d_{ij} ∼ λ^{H(Q_i)+H(¯d_j)}$ (and likewise for up-type couplings), read as a statement about holomorphic suppression: because the effective superpotential is analytic in the small breaking parameter, entries with negative total charge vanish rather than being suppressed, so the symmetry can produce genuine zeros in the mass matrices. Those zeros are what make the rest work — they are what allows $|V_{ub}|$ to be pushed below $λ^3$, what allows the down-quark mass matrix to take the aligned form that gives $(V^d_L)_{12} ∼ λ^5$, and what produces the one exact relation. The charge table for the master model, and the analogous tables for the two-$U(1)$ models, are the concrete objects the argument turns on.
Extended reading notes
Core claim
The central claim is that an Abelian horizontal symmetry, explicitly broken by a small parameter $λ ∼ 0.2$, accounts for both the smallness and the hierarchy of the quark sector. The mechanism is a selection rule: a Yukawa entry is of order $λ^{H(Q_i)+H(¯d_j)}$, so the transformation laws of $λ$ under the symmetry fix the order in perturbation theory at which each entry of the mass matrices appears, and hence which parameters are suppressed by more powers of $λ$ than others. With positive charges and the identifications $m_u/m_c ∼ λ^3$, $m_b/m_t ∼ λ^2$, $tan β ∼ 1$, the order-of-magnitude data determine a unique charge assignment; eight discrete choices then explain nine continuous parameters, leaving exactly one order-of-magnitude prediction, $|V_{ub}/V_{cb}| ∼ |V_{us}|$, which the authors check against the then-current data. Extending to $U(1)_{H1} × U(1)_{H2}$ with two breaking parameters lets individual entries of the mass matrices vanish (in the continuous limit), which opens two further results: a class of models in which quark and squark mass matrices are aligned closely enough to suppress supersymmetric contributions to $K$–$Øverline{K}$ mixing without squark degeneracy, and the near-exact relation $m_s^2/m_b^2 = |V_{cb}V_{ub}/V_{us}|$ when six specific entries vanish.
Load-bearing premise
The alignment and the resulting suppression of flavor-changing neutral currents depend on the zeros in the quark mass matrices surviving to high accuracy once the continuous symmetry is replaced by a discrete one, and the authors state in Appendix B that this case needs a more careful analysis.
Editorial extensions
If this is right
- If the framework is right, squarks need not be degenerate for supersymmetry to survive the flavor bounds: the horizontal charges align the quark and squark mass matrices, so the quark–squark–gluino mixing matrices stay near the identity.
- The alignment mechanism makes a sharp, checkable bet: because $(V^d_L)_{12}$ is forced to be tiny, the corresponding up-sector mixing must carry the full Cabibbo angle, so $D$–$Øverline{D}$ mixing sits close to its experimental upper bound, orders of magnitude above the Standard Model.
- The same alignment permits new CP-violating phases of order one, so CP asymmetries in $B^0$ decays can differ from Standard Model expectations by as much as $O(0.3)$, while the corresponding shifts in $B_s$ decays stay near $O(0.01)$.
- A single near-exact relation, $m_s^2/m_b^2 = |V_{cb}V_{ub}/V_{us}|$, follows if six specified entries of the quark mass matrices vanish.
- Embedding the whole picture in a Froggatt–Nielsen construction with heavy mirror quarks, the absence of Landau poles below the Planck scale forces the heavy scale above roughly $10$–$10^2$ TeV in the simplest cases, which is what makes TeV-scale flavor physics unlikely but not excluded.
Reading between the lines
- The paper's single order-of-magnitude prediction, $|V_{ub}/V_{cb}| ∼ |V_{us}|$, is the cheapest place for the framework to fail, and the authors themselves note that the then-new CLEO value was already pulling toward $λ^2$; a confirmed $|V_{ub}/V_{cb}| ≈ 0.04$ would not kill the general idea but would force the more elaborate discrete-symmetry construction of section 3.1.
- The alignment idea inverts the usual logic of the supersymmetric flavor problem: instead of asking for a symmetry that makes squarks heavy and degenerate, it asks for one that makes the misalignment small. That suggests a diagnostic — if non-degenerate squarks are ever found, the pattern of their mixing angles should track the same charge differences that fix the quark mass ratios, which is a stat
- The scale-hierarchy mechanism in section 7, where $⟨S⟩ ∼ M_P(˜m/M_P)^{1/(n-2)}$ follows from a discrete symmetry, is logically independent of the quark-sector story and could be borrowed by any model that needs several separated breaking scales; the price is a set of very flat directions whose cosmological behavior the authors flag as unresolved.
- The paper's own caveat in Appendix B — that replacing continuous $U(1)$'s by discrete symmetries which lift the zeros requires a more careful analysis — marks the boundary of the result: the FCNC suppression is only as good as the accuracy with which the discrete symmetry preserves those zeros.
Original abstract
The smallness of the quark sector parameters and the hierarchy between them could be the result of a horizontal symmetry broken by a small parameter. Such an explicitly broken symmetry can arise from an exact symmetry which is spontaneously broken. Constraints on the scales of new physics arise {}from new flavor changing interactions and {}from Landau poles, but do not exclude the possibility of observable signatures at the TeV scale. Such a horizontal symmetry could also lead to many interesting results: (i) quark -- squark alignment that would suppress, without squark degeneracy, flavor changing neutral currents induced by supersymmetric particles, (ii) exact relations between mass ratios and mixing angles, (iii) a solution of the $\mu$-problem and (iv) a natural mechanism for obtaining hierarchy among various symmetry breaking scales.
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
The smallness of the quark sector parameters and the hierarchy between them could be the result of a horizontal symmetry broken by a small parameter. Such an explicitly broken symmetry can arise from an exact symmetry which is spontaneously broken. Constraints on the scales of new physics arise {}from new flavor changing interactions and {}from Landau poles, but do not exclude the possibility of observable signatures at the TeV scale. Such a horizontal symmetry could also lead to many interesting results: (i) quark -- squark alignment that would suppress, without squark degeneracy, flavor changing neutral currents induced by supersymmetric particles, (ii) exact relations between mass ratios and mixing angles, (iii) a solution of the $\mu$-problem and (iv) a natural mechanism for obtaining hierarchy among various symmetry breaking scales.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The smallness of the quark sector parameters and the hierarchy between them could be the result of a horizontal symmetry broken by a small parameter. | background 0.93 | 0.08 | 0.21 | hedged possibility 1.00 |
| s1 | Such an explicitly broken symmetry can arise from an exact symmetry which is spontaneously broken. | background 0.68 | 0.11 | 0.07 | hedged possibility 0.62 |
| s2 | Constraints on the scales of new physics arise {}from new flavor changing interactions and {}from Landau poles, but do not exclude the possibility of observable signatures at the TeV scale. | result 0.64 | 0.23 | 0.27 | asserted 0.68 |
| s3 | Such a horizontal symmetry could also lead to many interesting results: (i) quark -- squark alignment that would suppress, without squark degeneracy, flavor changing neutral currents induced by supersymmetric particles, (ii) exact relations between mass ratios and mixing angles, (iii) a solution of the $\mu$-problem and (iv) a natural mechanism for obtaining hierarchy among various symmetry breaking scales. | interpretation 0.72 | 0.19 | 0.30 | hedged possibility 1.00 |
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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References
How the literature uses it
73 arXiv papers cite this in 113 sentences
Papers citing it · 62 sentences without a confident label
Compares with it 3 papers
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This illustrates a difference between hierarchical wavefunctions, and Froggatt-Neilson [ref] type models [ref] .
Solving the flavour problem with hierarchical fermion wave functions · Basic setup for the quark sector · 0.86
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Some of these good textures correspond to those identified already in the literature, but, to the best of our knowledge, most of them have not been considered before.[Texture 3 first appeared in the seminal papers [ref] .]
Mapping and Probing Froggatt-Nielsen Solutions to the Quark Flavor Puzzle · Systematic exploration of Froggatt-Nielsen Models · 0.95
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This is reminiscent of the situation in the simplest supersymmetric Froggatt-Nielsen flavor models with a single $U(1)$ flavor symmetry [ref] .
A Supersymmetric Flavor Clockwork · The Supersymmetric Flavor Clockwork · 0.82
Cites it as background 42 papers
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This rather technical but necessary procedure, to which we will refer to as canonical normalisation (CN) in the following, has been discussed in [ref] and more recently in [ref] .
Third Family Corrections to Quark and Lepton Mixing in SUSY Models with non-Abelian Family Symmetry · Introduction · 0.95
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In these models, the flavour structure of the Yukawa matrices is only generated after the breaking of a flavour symmetry [ref] and the flavour structure of the SUSY soft-breaking terms would also originate through the same mechanism [ref] .
Electric dipole moments from flavoured CP violation in SUSY · Introduction · 0.81
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The two most popular solutions to this puzzle are (i) the Froggatt–Nielsen (FN) mechanism and its variations [ref] , and (ii) extra dimensional models where an $𝒪(1)$ difference in the bulk masses of fermions gives rise to an exponential hierarchy between the observed masses in the IR [ref] .
Wrinkles in the Froggatt-Nielsen Mechanism and Flavorful New Physics · Aperitif · 0.97
Discussion
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