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REVIEW 4 major objections 5 minor 85 references

Phenomenological Study of Type II Seesaw with $\Delta(27)$ Symmetry

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that a Delta(27)-symmetric type-II seesaw extension of the Standard Model reproduces the observed neutrino masses, mixing angles, CP phase, baryon asymmetry, and lepton-flavor-violating rates.

desk verdict A systematic Δ(27) type-II seesaw scan whose CP-violating and leptogenesis claims are not derived: δ is set to zero for the diagonalization and then varied to make the plots, and Eq. (49) is identically zero. read the letter →

arxiv 1909.01560 v1 pith:JPPUHKXU submitted 2019-09-04 hep-ph

Itishree Sethi (IIT Hyderabad) , Sudhanwa Patra (IIT Bhilai) This is my paper · ORCID
classification hep-ph PACS 14.60.Pq12.60.Fr
keywords type-IIseesawDelta(27)flavorsymmetryneutrinomassandmixingreactorangleleptogenesisleptonviolationneutrinolessdoublebetadecayscalartriplet
topics Matter-Antimatter Asymmetry
open problems Matter-Antimatter Asymmetry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that a single discrete flavor symmetry, $\Delta(27)$, can organize a type-II seesaw extension of the Standard Model that matches the measured neutrino oscillation data. With two additional Higgs doublets fixing the charged-lepton masses and three scalar triplets generating neutrino masses, the model produces a neutrino mass matrix that is diagonalized by tribimaximal mixing together with one further rotation. A small common perturbation added to the diagonal entries splits the two degenerate light masses, and the resulting parameters then yield the non-zero reactor angle, the Dirac CP phase, a sum of neutrino masses within the cosmological bound, and an effective Majorana mass in the reach of upcoming neutrinoless double $\beta$ decay experiments. The same TeV-scale triplets generate the baryon asymmetry through leptogenesis and give lepton-flavor-violating rates for muon decays near current bounds. If correct, the model would tie several independent neutrino observables to just a few parameters.

What carries the argument

The load-bearing object is the light neutrino mass matrix $M_\nu = \begin{pmatrix} a+\epsilon & c & b \\ c & b+\epsilon & a \\ b & a & c+\epsilon \end{pmatrix}$ in the $\Delta(27)$ basis, together with the two-step diagonalization $U = U_{\text{TBM}}\cdot U_{13}\cdot P$. The tribimaximal matrix $U_{\text{TBM}}$ (the mixing pattern with $\sin^2\theta_{12}=1/3$, $\sin^2\theta_{23}=1/2$, $\theta_{13}=0$) block-diagonalizes the matrix; the perturbation $\epsilon$ splits the degenerate pair, and the $U_{13}$ rotation with angle $\theta$ and phase $\delta$ produces a non-zero $\theta_{13}$, a deviated $\theta_{23}$, and CP violation. All subsequent predictions, from the mixing angles and $J_{CP}$ to Majorana phases, neutrinoless double $\beta$ decay, triplet leptogenesis, and muon lepton-flavor-violating rates, are derived from this same matrix and its diagonalization.

What would settle it

Build the scalar potential for the stated $\Delta(27)$ field content and check whether its vacuum alignment produces a diagonal perturbation $\epsilon$: the model is falsified if the minimum forces $\epsilon=0$ (leaving two degenerate light neutrinos) or gives an off-diagonal $\epsilon$-texture, because then the predicted mixing angles contradict the measured reactor angle.

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Extended reading notes

Core claim

In the paper's own terms, the central result is that an $SU(2)_L$ extension with three $\Delta(27)$-triplet scalar fields plus two additional Higgs doublets realizes the type-II seesaw in a phenomenologically complete way: it reproduces the current $3\sigma$ ranges of the solar, atmospheric, and reactor mixing angles, the two mass-squared differences, and the Dirac CP phase. The key step is the structure of the neutrino mass matrix, which before perturbation is diagonalized by the tribimaximal matrix and has two degenerate eigenvalues; adding a universal diagonal perturbation $\epsilon$ lifts the degeneracy, and a further $U_{13}$ rotation with angle $\theta$ and phase $\delta$ relates the observable mixing to the model parameters. The authors conclude that all mixing angles can be reproduced for a restricted range of the parameter $\alpha_1$, and the model simultaneously gives a total neutrino mass $\sum m_\nu$ consistent with cosmology, an effective Majorana mass $|m_{ee}|$ relevant for neutrinoless double $\beta$ decay, TeV-scale scalar triplet leptogenesis with the observed baryon asymmetry, and lepton-flavor-violating branching ratios near present bounds.

Load-bearing premise

The entire fit rests on a small diagonal perturbation $\epsilon$ that is added by hand to every diagonal entry of the neutrino mass matrix; the paper states that it could come from extra fields but does not construct them, and without $\epsilon$ the two lighter neutrino masses are equal, so the oscillation data cannot be reproduced.

Editorial extensions

If this is right

  • The model reproduces, within $3\sigma$, the observed values of $\Delta m^2_{21}$, $|\Delta m^2_{31}|$, $\sin^2\theta_{12}$, $\sin^2\theta_{23}$, and $\sin^2\theta_{13}$, with the Dirac CP phase $\delta_{CP}$ as a correlated output.
  • The predicted sum of neutrino masses lies in the range $0.12$ to $0.29$ eV, compatible with cosmological bounds, and the effective Majorana mass $|m_{ee}|$ is correlated with the lightest neutrino mass and with $\theta$, giving a concrete target for neutrinoless double beta decay searches.
  • The TeV-scale decay of the lightest scalar triplet can generate the observed baryon asymmetry through flavored leptogenesis, with the CP asymmetry controlled by the same $\theta$ and $\delta$ that govern neutrino mixing.
  • The branching ratios for $\mu\to e\gamma$ and $\mu\to 3e$ are expressed through the same PMNS matrix and light neutrino masses; for TeV-scale triplets they fall near the current experimental upper limits, making the framework testable in upcoming muon experiments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the diagonal $\epsilon$ is ever traced to explicit fields, its flavor-diagonal texture becomes the model's decisive prediction: an $\epsilon$ that is not diagonal would alter the block structure that produces the successful mixing-angle relations.
  • The $U_{\text{TBM}}\cdot U_{13}$ mixing ansatz is not unique to $\Delta(27)$; the same perturbation logic could be transplanted to other discrete flavor groups, so the specific claim of the paper is that $\Delta(27)$ supplies the matrix texture, and that is what a direct symmetry-breaking calculation would have to check.
  • A future measurement of $\delta_{CP}$ combined with a bound on $\mu\to e\gamma$ would select disjoint regions of the $\theta$--$v_\Delta$ parameter space, turning this model into a concrete target that muon experiments can probe.
  • If the $\Delta(27)$ scalar potential fixes the phases that the paper's scan leaves free, the allowed $\alpha_1$ range found here could shrink or disappear, so the claimed restricted range is not yet a prediction from the symmetry alone.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a type-II seesaw extension of the Standard Model with a Δ(27) flavor symmetry, adding two SU(2)_L Higgs doublets and three scalar triplets. The light neutrino mass matrix is assumed to have a form that can be block-diagonalized by the tribimaximal mixing matrix and then fully diagonalized by a U13 rotation. A diagonal perturbation ε is added by hand to break an exact degeneracy of the light neutrino masses. The authors scan their model parameters to reproduce the observed neutrino mass-squared differences and mixing angles, derive the Dirac CP phase, the effective Majorana mass, scalar-triplet leptogenesis, and the branching ratios of μ→eγ and μ→3e. The central claims are that the model reproduces all current neutrino oscillation data, including nonzero θ13 and δ_CP, and gives testable predictions for neutrinoless double beta decay, leptogenesis, and lepton flavor violation.

Significance. If the derivation were correct, the model would be a useful example of a discrete flavor symmetry producing trimaximal-like mixing with several phenomenological consequences. The paper is clearly organized and uses standard type-II seesaw and LFV formulas, and it does compare with the 3σ ranges of global neutrino oscillation fits. However, the predictive power is severely limited: the neutrino mass spectrum requires an ad hoc perturbation ε with no field-theoretic origin, and the CP-violating predictions rest on an internally inconsistent treatment of the phase δ. Moreover, the leptogenesis asymmetry as written in Eq. (49) is identically zero. With at least ten free inputs used to fit the oscillation observables, the agreement shown is consistency rather than prediction. These issues undermine the central claims of the manuscript in its present form.

major comments (4)
  1. [II.B, Eqs. (14)-(17); III.B; IV] The treatment of the phase δ is internally inconsistent. The manuscript fixes δ = 0 for the diagonalization: Eq. (16) contains a term 2i ε sinδ in the denominator, and the text before Eq. (17) states 'For rest of our analysis we will use δ = 0.' For δ ≠ 0 with ε ≠ 0, the matrix U13^T M_bd U13 is not diagonal; its (1,3) element acquires an imaginary part -2 c s ε sinδ (in addition to real terms). Nevertheless, Sections III.B and IV treat the same δ as a free parameter: sin²θ23 depends on cosδ in Eq. (29), J_CP is proportional to sinδ in Eq. (30), δ_CP is displayed in Fig. 7, and the scalar-triplet CP asymmetry is plotted against δ in Fig. 12. No mechanism is constructed that would generate a nonzero δ while preserving the block-diagonal form, and the scanned phases φ_ba, φ_ca, φ_εa in Eq. (20) are never connected to δ. Consequently, the CP-violating predictions are imposed by hand, not derived from the model.
  2. [II.B, Eq. (10)] The diagonal perturbation ε is introduced without a symmetry origin. The text acknowledges that it 'can also be generated by the inclusion of additional fields,' but no such fields are specified. Without ε, the eigenvalues in Eq. (9) give |m1| = |m3|, so Δm²31 vanishes and the oscillation data cannot be reproduced. The agreement with the neutrino mass splittings is therefore entirely dependent on this unmodeled parameter. The adjacent sentence saying 'two eigenvalues m1 and m2 are degenerate' is also a misstatement: Eq. (9) shows that m1 and m3 have equal magnitude.
  3. [IV.A, Eq. (49)] The simplified CP asymmetry for the lightest scalar triplet is written as ε^{ℓ_i}_{Δ1} ∝ Im[(M_ν^† M_ν)_{ii}]. Since M_ν^†M_ν is Hermitian, its diagonal entries are real, so this expression is identically zero. The flavored asymmetry in Eq. (48) requires products of distinct triplet matrices, Im[(M_{Δα}^† M_{Δβ})_{ii}], which do not reduce to a diagonal element of a single Hermitian matrix. The leptogenesis curves in Figs. 12 and 13 therefore do not follow from the stated equations.
  4. [III, Eq. (20) and Figs. 4-9] The numerical analysis is a scan over at least ten free inputs (|a|, ε, α1, α2, α3, φ_ba, φ_ca, φ_εa, θ, δ) used to fit the two mass-squared differences and the three mixing angles. The 'predictions' for δ_CP, |mee|, and the LFV rates are then evaluated at the same scanned parameter points, so the agreement with oscillation data is a fit rather than a test of the model. No χ², confidence-level, or number-of-degrees-of-freedom statistic is reported, making it impossible to assess the statistical significance of the claimed 'restricted range of parameter space for α1.'
minor comments (5)
  1. [Throughout] There are numerous typos and OCR artifacts, e.g., 'Yukuwa' should be 'Yukawa,' 'corelation' should be 'correlation,' and 'fllowing' should be 'following.' These should be corrected in a revised version.
  2. [III.B, Eq. (31)] The expression for δ_CP in Eq. (31) is not well defined as written and appears to be dimensionally inconsistent; please clarify the intended formula or remove it if it is not used in the numerical analysis.
  3. [IV, Eq. (50)] The relation Y_B = κ·c·ε_{Δ1}/g* is only schematic; the washout factor κ is not evaluated, so the final baryon asymmetry is not actually determined from the model parameters.
  4. [VIII.C, Appendix] The scalar potential and symmetry-breaking discussion mention 'flavon fields,' but Table I does not include any flavon fields. Please clarify how the Δ(27) symmetry is broken to obtain the VEV alignments used in Section II.
  5. [Fig. 12] The text and the figure caption disagree on whether the left panel shows δ or θ; please correct the inconsistency.

Circularity Check

3 steps flagged · score 6.0 of 10

The neutrino-mass 'prediction' is produced by an ad hoc fitted diagonal perturbation ε, and the CP-violating results reuse a phase δ that the mass diagonalization had already fixed to zero.

  1. fitted input called prediction [Section II.B around Eq. (10); scan range in Section III, Eq. (20)]
    "Thus, we add a perturbation term ϵ to all diagonal terms in order to generate non-degenerate neutrino masses. This small perturbation can also be generated by the inclusion of additional fields but here we express all diagonal terms as sum of leading terms plus this perturbation term ϵ ≠ 0."

    Before ε is added, Eq. (9) yields m1 = -m3, so |m1| = |m3| and Δm²31 = 0, which contradicts oscillation data. The text admits that the ε term is not derived from the Δ(27) structure ('can also be generated by the inclusion of additional fields' is not carried out). In Eq. (20) ε is then scanned over [-0.01, 0.01] eV as a free input. The resulting non-zero mass-squared differences are therefore produced by the fitted perturbation rather than predicted by the symmetry; the reported agreement with oscillation data is a restatement of this input.

  2. fitted input called prediction [Section II.B, Eqs. (16)-(17); Section III.B, Eqs. (29)-(31); Section IV, Figs. 12-13]
    "For rest of our analysis we will use δ = 0 and as a result of this, the mixing angle θ is then given by tan 2θ = √3(α1−α2)/(−2+α1+α2). ... The mixing angles prediction are shown as a function of θ in Fig.6. ... sin2θ23 = 1/2(1 + √3 sin 2θ cosδ/(2 + cos 2θ))."

    Eq. (16) has 2i ε sinδ in the denominator of tan2θ, so the U13 rotation in Eq. (14) diagonalizes the real block matrix of Eq. (11) only when δ = 0, which is why the paper fixes δ = 0 at Eq. (17). Nevertheless, Eqs. (29)-(31) and Figs. 7, 12, 13 use the same δ as a free parameter to generate sin²θ23, JCP, δCP and the triplet CP asymmetry. No complex phase or additional field structure is introduced to make δ ≠ 0 consistent with the diagonalization. The CP-violating results are thus scanned inputs presented as predictions, not consequences of the Δ(27) model.

1 more flagged steps
  1. fitted input called prediction [Section III, Eq. (20); Section V, Eqs. (52)-(55); Section VI conclusion]
    "We examine the correlation between model parameters compatible with 3 σ limits of the current oscillation data for which we present a random scan of these model parameters over the following ranges: a∈ [−0.1, 0.1] eV, ϵ∈ [−0.01, 0.01] eV, α1∈ [0, 0.3], α2∈ [0, 1], α3∈ [0, 0.03], φba,ca,ϵa∈ [−π,π]."

    Eight continuous parameters are scanned to select points whose mass-squared differences and mixing angles lie inside the 3σ oscillation windows. The same UPMNS elements and Δm² values then enter the LFV formulas, e.g. Eq. (54) for |(f†f)eμ| and Eq. (55) for Br(µ→3e), and the same inputs feed the |mee| and leptogenesis figures. The agreement with oscillation data is therefore imposed by the scan; the subsequent rates are evaluations of those fitted parameters rather than independent predictions. The concluding statement that the model 'can reproduce all the mixing angles' is a consistency of the scan, not a derivation of those angles from the symmetry.

full rationale

The paper does not rely on a load-bearing self-citation chain: the citations to previous Δ(27) models and to the UPMNS parameterization are contextual or standard, and the mixing-matrix identity U = UTBM·U13·P is an ansatz, not a uniqueness theorem. The circularity is internal. The most concrete reduction is the ε perturbation: the model without ε has |m1| = |m3| and hence cannot produce the observed Δm²31, so ε is introduced by hand, acknowledged as not derived from the field content, and then scanned as a free parameter. The mass spectrum that is later showcased as fitting oscillation data is therefore constructed from that fitted input. A second, sharper problem is the treatment of δ: the mass diagonalization explicitly requires δ = 0, but the mixing-angle, JCP, δCP and leptogenesis analyses all vary δ as a free CP phase. No mechanism is supplied to reconcile δ ≠ 0 with the earlier diagonalization condition, so the CP-violating 'predictions' reduce to inputs. Finally, the parameter scan has enough freedom (|a|, ε, α1, α2, α3 and three phases, plus θ and δ in the mixing section) that the reported 3σ agreement is a selected outcome of the scan rather than a sharp test. For these reasons the central claims are partially circular, meriting a score of 6 rather than a higher score, since the correlations among mixing angles and the type-II seesaw relations do provide some non-trivial structure independent of the fitted parameters.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

The central fit depends on at least eight free inputs: |a|, α1, α2, α3, three relative phases, θ, and δ, plus the chosen triplet masses and VEVs. The least derived input is the ad hoc diagonal perturbation ε, which is not generated by any field in the model. The model contains no genuinely new particle species beyond the well-known scalar triplets and extra doublets of the cited type-II seesaw and Δ(27) literature.

free parameters (8)
  • |a| (overall mass scale) = 0.025-0.1 eV (scanned)
    Scanned in [-0.1, 0.1] eV in Eq. (20); sets the absolute neutrino mass scale and feeds directly into the sum of masses and the effective Majorana mass.
  • α1 = |b/a| = 0-0.3 (scanned)
    Ratio of triplet VEVs, scanned in Eq. (20) and constrained by the solar-to-atmospheric mass-squared ratio r.
  • α2 = |c/a| = 0-1 (scanned), α2=1 used in Figs. 1-5
    Second ratio of triplet VEVs; the special value α2=1 is adopted for the correlation plots in Section III.A.
  • α3 = |ε/a| = 0-0.03 (scanned)
    Size of the ad hoc diagonal perturbation relative to a; scanned in Eq. (20) to split the degenerate masses.
  • Relative phases φba, φca, φεa = [-π, π] (scanned)
    CP phase differences among the complex VEVs; scanned in Eq. (20), and set to zero in the special case of Section III.A.
  • Internal rotation angle θ = 0-180 degrees (varied)
    Angle in U13 that generates the reactor angle through sin²θ13 = (2/3) sin²θ; effectively fitted to the measured θ13.
  • Internal phase δ = 0-180 degrees (varied)
    Used in the PMNS matrix and CP-asymmetry plots, despite the text setting δ=0 for the mass diagonalization in Eq. (17).
  • Scalar triplet masses mΔ and VEV vΔ = mΔ ~ 1 TeV, vΔ ≤ 1 eV (chosen)
    Chosen by hand to realize TeV-scale leptogenesis and to keep LFV rates near current bounds.
assumptions (7)
  • domain assumption Type-II seesaw formula Mν = f vΔ with vΔ ≈ μΔ v² / (2 mΔ²)
    Invoked from the standard seesaw literature, refs. [17-25, 33], in Eq. (38); it is the assumed origin of small neutrino masses.
  • standard math The tri-bimaximal mixing matrix UTBM diagonalizes the leading-order mass matrix
    Used in Eq. (6)-(8) as a basis choice from the Harrison-Perkins-Scott TBM ansatz, ref. [45].
  • ad hoc to paper Neutrino Yukawa couplings fα, f'α, f''α are set equal
    Section II.B states 'Considering, fα=f′α=f′′α' before Eq. (4); this equality is not forced by Δ(27) and is chosen to obtain the symmetric texture.
  • ad hoc to paper The diagonal perturbation ε can be added without changing the model's symmetry content or VEV alignment
    Equation (10): 'we add a perturbation term ε to all diagonal terms' with the admission that it could be generated by additional fields, but those fields are not constructed.
  • ad hoc to paper δ=0 for mass diagonalization while δ is later varied for CP observables
    Equation (17) follows from 'For rest of our analysis we will use δ = 0', yet Section III.B and Figs. 6-16 vary δ in degrees and use it to produce δ_CP.
  • domain assumption Scalar vacuum alignment preserves Δ(27)
    The vacuum alignments <φ_i> = v_i/√2 (0,1)^T and triplet VEVs are stated in Section VIII.C without a stability analysis of the full scalar potential.
  • domain assumption Flavored triplet leptogenesis formulas from ref. [83] apply
    The CP-asymmetry expressions in Eqs. (46)-(49) are taken from Aristizabal Sierra, Dhen and Hambye without independent derivation or a complete Boltzmann calculation.

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Cite this review

Pith. "Pith review of Phenomenological Study of Type II Seesaw with $\Delta(27)$ Symmetry." pith.science (2026). https://pith.science/paper/JPPUHKXU

@misc{pith2026190901560,
  author       = {Pith},
  title        = {Pith review of: Phenomenological Study of Type II Seesaw with $\Delta(27)$ Symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JPPUHKXU}},
  note         = {Machine review of arXiv:1909.01560}
}
abstract

We discuss the phenomenology of type-II seesaw by extending the Standard Model with additional Higgs doublets and scalar triplets additionally invoked with $\Delta(27)$ flavor symmetry for the explanation of non-zero neutrino masses and mixings, matter-antimatter asymmetry and lepton flavor violation. The non-zero neutrino masses can be realized via type-II seesaw mechanism by introducing scalar triplets transforming as triplets under $\Delta(27)$ while we add additional $SU(2)_L$ scalar doublets to have correct charge lepton masses. We further demonstrate with detailed numerical analysis in agreement with neutrino oscillation data like non-zero reactor mixing angle, $\delta_{CP}$, the sum of the light neutrino masses, two mass squared differences and its implication to neutrinoless double beta decay. We also discuss on the matter-antimatter asymmetry of the universe through leptogenesis with the decay of TeV scale scalar triplets and variation of CP-asymmetry with input model parameters. Finally, we comment on implication to lepton flavor violating decays like $\mu \to e \gamma$, $\mu \to 3 e$ processes.

Figures

Figures reproduced from arXiv: 1909.01560 by the authors.

Figure 1
Figure 1. FIG. 1. This plot shows the variation between [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. This plot represents the variation between (a) [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. This plot represents the variation between (a) a and [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: FIG. 4. This plot shows the correlation between (a) total neutrino mass with [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. This plot shows the corelation between the atmospheric squared neutrino mass with r. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Variation of solar, reactor and atmospheric mixing angles with internal mixing angle [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Variation of solar, reactor and atmospheric mixing angles along with Dirac CP-phase with internal [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Contour plot for ee-element of neutrino mass matrix in the plane of input model parameters. [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Variation of ee-element of neutrino mass matrix and its square value with input model parameters. [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Variation of CP-asymmetry via decay of scalar triplets with input model parameters like [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Contour plot for CP-asymmetry in the plane of [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Contour plot for [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Variation of [PITH_FULL_IMAGE:figures/full_fig_p023_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Contour plot for branching ratio [PITH_FULL_IMAGE:figures/full_fig_p024_16.png]

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