REVIEW
EDM constraints on flavored CP-violating phases
High Energy Physics - Phenomenology · Submitted 11 Feb 2009 · 49 references
Pith's verdict: The real contribution is Section 2.4 — the CP limit for MFV coefficients is basis-dependent — plus a clean structural result that all spurion phases are flavor off-diagonal; the numerical scan is a window, not a bound.
T0 · AI review
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Referee report
The paper
The paper claims
In the MSSM with minimal flavor violation, all three classes of new CP-violating slepton phases survive the current electron EDM bound, but the flavor-blind one is about to be squeezed.
The reading
What carries the argument
The machinery is the MFV spurion expansion of the slepton soft-breaking terms, made finite by the Cayley–Hamilton identity for $3×3$ matrices and then truncated using the charged-lepton mass hierarchy, $(Y_e^†Y_e)^2 ≈ y_τ^2 Y_e^†Y_e$. Writing the expansion entirely in terms of hermitian operators — $1$, $A$, $B$, $B^2$, $A B + B A$, $B A B$, and the three antisymmetric combinations $i[A,B]$, $i[A,B^2]$, $i(B^2AB - BAB^2)$ — is what makes the classification work: the diagonal entries of hermitian operators are real, so any phase attached to them can only reach the EDM through off-diagonal insertions, i.e. at second order in the mass-insertion approximation. A second piece of machinery is the argument that CP cannot be imposed on the MFV coefficients alone: the Jarlskog invariant $J = i^2 det[Y_ν^†Y_ν, Y_e^†Y_e]$ is nonzero, so the coefficients are always defined up to phases proportional to it, and setting them real is a fine-tuning rather than a symmetry. The third piece is the matching of the general MSSM onto this basis, which converts EDM and $μ→ eγ$ limits into bounds on the expansion coefficients and thereby into a quantitative naturalness test.
Extended reading notes
Core claim
The paper's central result is a hierarchy, not a bound. Once the slepton soft-breaking terms are written in the hermitian MFV operator basis built from $A = Y_e^†Y_e$ and $B = Y_ν^†Y_ν$, the CP-violating phases organize themselves into three classes whose contributions to the electron EDM differ by large factors. The flavor-blind phase, essentially $arg c_1$, enters at leading order in the mass-insertion approximation and dominates; the flavor-diagonal phases $Im c_i$ and $Im d_3$ enter at the same order but are suppressed by charged-lepton or neutrino mass ratios; the flavor off-diagonal phases — the $b_i$, $d_{1,2}$, and every phase living in the neutrino spurion $Y_ν^†Y_ν$ — only start at second order in the mass-insertion approximation and are numerically negligible. Because the operator basis is hermitian, the diagonal entries of these operators are automatically real, which is precisely what shields the EDMs from the Dirac, Majorana and $φ_i$ phases of the neutrino sector. Numerically, for slepton masses around 600 GeV, $|μ| = 400$ GeV and $M_1 = 200$ GeV, and for $M_R$ between $10^9$ and $10^{13}$ GeV, all three classes remain compatible with the current electron EDM bound while $B(μ→ eγ)$ is satisfied, with the flavor-blind phase the only one close enough to the bound to be probed soon.
Load-bearing premise
The numerical verdict rests on keeping only one-loop neutralino and chargino diagrams, with two-loop Barr–Zee contributions and $tan β$-enhanced non-holomorphic Yukawa corrections explicitly dropped, as the authors state in Section 3.1; at the $tan β = 50$ end of their scan those omitted terms can be comparable to or larger than the retained ones, so the quoted EDM values are more permissive than a complete calculation would give.
Editorial extensions
If this is right
- The neutrino-sector CP phases — Dirac, Majorana and the three $φ_i$ — are effectively invisible to lepton EDMs within MFV, because hermiticity of the operator basis forces them to enter only at second order in the mass-insertion approximation.
- For each class of phases there is one dominant operator: $Im c_1$ for the flavor-blind class, $Im c_3$ for the flavor-diagonal class, and the product $b_1 Re c_3$ for the flavor off-diagonal class, so future EDM measurements can be interpreted as constraints on single coefficients.
- The planned order-of-magnitude improvement in the electron EDM reach should cover most of the parameter space generated by the flavor-blind phase of $A_e$, which would either produce a signal or push that phase into the same fine-tuning corner as the phases of $μ$, $M_1$ and $M_2$.
- The muon and tau EDMs remain far out of experimental reach in every scenario, since even the most favorable scaling is set by lepton mass ratios and gives $d_μ$ and $d_τ$ five to six orders of magnitude below their current sensitivities.
- Projecting the general MSSM onto the MFV basis shows that only $a_3/a_1$ and $Im c_1/(a_1 a_7)$ are currently constrained to be of order one to ten, while all other coefficients may take values up to $10^{13}$ without violating data, so present mass-insertion bounds do not yet demonstrate non-natural flavor structure.
Reading between the lines
- Because the argument that CP cannot be defined on MFV coefficients separately from the spurions relies only on the existence of a nonzero Jarlskog invariant, the same reasoning should transfer verbatim to the quark-sector expansions, and the authors say as much; a reader could check whether any quark-sector analysis that assumes real MFV coefficients is thereby making an implicit fine-tuning assum
- The omission of two-loop Barr–Zee diagrams and $tan β$-enhanced non-holomorphic Yukawa corrections, which the authors state explicitly in Section 3.1, likely makes the quoted EDM values conservative in the wrong direction: at $tan β = 50$ those terms are known to be competitive, so the allowed region for $Im c_1$ is probably narrower than the one-loop scan suggests.
- The naturalness table points to $τ → μγ$ as the sharpest future lever on the right-handed slepton sector, which is a testable prediction: if the coefficients of $m_E^2$ are as large as the table permits, an improved $τ → μγ$ bound should bite before any EDM improvement does.
- If the next-generation electron EDM search comes up empty at the $10^{-29}$ e cm level, the flavor-blind phase of the trilinear coupling becomes a fine-tuning problem structurally identical to the familiar MSSM CP problem, which would suggest either very heavy sleptons or a CP-symmetric high-scale boundary condition rather than MFV alone.
Original abstract
The CP-violating phenomenology of the MSSM with Minimal Flavor Violation (MFV) in the lepton sector is revisited. To this end, the most general parametrizations of the slepton soft-breaking terms are constructed assuming a seesaw mechanism of type I. After a critical reassessment of how the CP-symmetry is broken within the MFV framework, all possible CP-violating phases are introduced. From the strong hierarchy of their contributions to the Electric Dipole Moments (EDMs), these phases are split into three classes: flavor-blind, flavor-diagonal and flavor off-diagonal. In particular, the phases from the neutrino sector belong to the last class; they start to contribute only at the second order in the mass-insertion approximation and have thus a negligible effect. It is then shown that to each class of phases corresponds a unique largely dominant term in the MFV expansion. Numerically, for a realistic range of MSSM and neutrino parameters, such that B(mu --> e gamma) does not exceed its experimental bound, the three types of phases are found to be allowed by the current bound on the electron EDM, though the next generation of experiments should constrain tightly the flavor-blind phase. Finally, we relax the MFV hypothesis and show how in the general MSSM, the MFV operator basis can be used to judge of the naturality of the slepton soft-breaking terms.
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 CP-violating phenomenology of the MSSM with Minimal Flavor Violation (MFV) in the lepton sector is revisited. To this end, the most general parametrizations of the slepton soft-breaking terms are constructed assuming a seesaw mechanism of type I. After a critical reassessment of how the CP-symmetry is broken within the MFV framework, all possible CP-violating phases are introduced. From the strong hierarchy of their contributions to the Electric Dipole Moments (EDMs), these phases are split into three classes: flavor-blind, flavor-diagonal and flavor off-diagonal. In particular, the phases from the neutrino sector belong to the last class; they start to contribute only at the second order in the mass-insertion approximation and have thus a negligible effect. It is then shown that to each class of phases corresponds a unique largely dominant term in the MFV expansion. Numerically, for a realistic range of MSSM and neutrino parameters, such that B(mu --> e gamma) does not exceed its experimental bound, the three types of phases are found to be allowed by the current bound on the electron EDM, though the next generation of experiments should constrain tightly the flavor-blind phase. Finally, we relax the MFV hypothesis and show how in the general MSSM, the MFV operator basis can be used to judge of the naturality of the slepton soft-breaking terms.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The CP-violating phenomenology of the MSSM with Minimal Flavor Violation (MFV) in the lepton sector is revisited. | contribution 0.61 | 0.68 | 0.03 | asserted 0.75 |
| s1 | To this end, the most general parametrizations of the slepton soft-breaking terms are constructed assuming a seesaw mechanism of type I. | method 0.92 | 0.86 | 0.02 | asserted 0.95 |
| s2 | After a critical reassessment of how the CP-symmetry is broken within the MFV framework, all possible CP-violating phases are introduced. | method 0.64 | 0.84 | 0.03 | asserted 0.97 |
| s3 | From the strong hierarchy of their contributions to the Electric Dipole Moments (EDMs), these phases are split into three classes: flavor-blind, flavor-diagonal and flavor off-diagonal. | method 0.41 | 0.85 | 0.09 | asserted 1.00 |
| s4 | In particular, the phases from the neutrino sector belong to the last class; they start to contribute only at the second order in the mass-insertion approximation and have thus a negligible effect. | result 0.95 | 0.76 | 0.10 | asserted 1.00 |
| s5 | It is then shown that to each class of phases corresponds a unique largely dominant term in the MFV expansion. | result 0.89 | 0.88 | 0.17 | asserted 1.00 |
| s6 | Numerically, for a realistic range of MSSM and neutrino parameters, such that B(mu --> e gamma) does not exceed its experimental bound, the three types of phases are found to be allowed by the current bound on the electron EDM, though the next generation of experiments should constrain tightly the flavor-blind phase. | result 0.99 | 0.94 | 0.62 | asserted 0.99 |
| s7 | Finally, we relax the MFV hypothesis and show how in the general MSSM, the MFV operator basis can be used to judge of the naturality of the slepton soft-breaking terms. | contribution 0.97 | 0.91 | 0.12 | asserted 1.00 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:0902.1949 by the authors.
References
How the literature uses it
33 arXiv papers cite this in 68 sentences
Papers citing it · 55 sentences without a confident label
Builds on it 1 paper
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Actually, this reparametrization provides a unique way to specify fully generic RPV couplings independently of the flavor basis chosen for the (s)quark fields, and thus extends to the RPV sector the procedure proposed in Refs. [ref] .
Baryonic R-parity violation and its running · Introduction · 0.98
Uses its method or tool 3 papers
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With the help of the Cayley-Hamilton identity it is then possible to reduce these sums to a finite number of terms [ref] and the final result reads [equation]
Revisiting CP-violation in Minimal Flavour Violation · MFV in the MSSM · 0.89
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Using the Cayley-Hamilton identity [equation] for an invertible 3$×$3 matrix $X$, one can resum the infinite series into a finite number of terms [ref] [equation] where $$ denotes a 3$×$3 unit matrix.
Fermion EDMs with Minimal Flavor Violation · Minimal flavor violation framework · 0.84
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Taking $f(, )$ as an example, its explicit summed expression can be written as [ref] [equation] During the resummation, higher-order polynomials of $$ and $$ contribute to the coefficients $ϵ_i$, and make them complex generally.
Deciphering the Belle II data on $B\to K \nu \bar\nu$ decay in the (dark) SMEFT with minimal flavour violation · Quark sector · 0.81
Compares with it 1 paper
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These three models are then compared in the last two steps with a flavour model with left-handed (CKM-like) currents only [ref] , with a flavour blind MSSM (FBMSSM) [ref] and with the MFV MSSM with additional CP phases [ref] .
Anatomy and Phenomenology of FCNC and CPV Effects in SUSY Theories · Introduction and Motivation · 0.98
Cites it as support 1 paper
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In fact, the $G_f$ symmetry does not protect the MFV coefficients from being complex and it is not even possible to define a CP-limit independently for the spurions and for the coefficients [ref] .
Revisiting CP-violation in Minimal Flavour Violation · MFV in the MSSM · 0.99
Cites it as background 7 papers
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A particularly interesting set-up is obtained introducing flavour-blind CPV phases compatible with the MFV symmetry principle [ref] .
EDMs vs. CPV in B_{s,d} mixing in two Higgs doublet models with MFV · 1. Introduction · 0.82
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Namely, we consider the generic framework where the Yukawa matrices are the only sources of breaking of the $SU(3)_q^3$ flavour group, but they are not the only allowed sources of CP-violation (a general discussion about this framework can be found in Ref. [ref] ).
Higgs-mediated FCNCs: Natural Flavour Conservation vs. Minimal Flavour Violation · Introducing flavour-blind phases · 0.91
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The explorations of the MSSM CP-violating phases within various constructs can be found in the literature such as in Refs. [ref] .
MFV Reductions of MSSM Parameter Space · The MSSM-30 fit · 1.00
Discussion
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