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Radiative contribution to neutrino masses and mixing in $\mu\nu$SSM

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Pradipta Ghosh, Paramita Dey, Biswarup Mukhopadhyaya, Sourov Roy

High Energy Physics - Phenomenology · Submitted 13 Feb 2010 · 116 references

Pith's verdict: The one-loop machinery is real and reusable; the headline 'accommodation' claim is an existence proof at a single benchmark point, and the abstract's comparative claim outruns it.

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arXiv ID1002.2705 (version 3)
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CitationBiswarup Mukhopadhyaya, Paramita Dey, Pradipta Ghosh, Sourov Roy. "Radiative contribution to neutrino masses and mixing in $\mu\nu$SSM." arXiv:1002.2705 (2010). https://pith.science/paper/LAH5KRHG
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The paper

The paper claims

A supersymmetric neutrino model survives its own one-loop corrections, and normal mass ordering fits most easily.

The reading

The paper asks whether a supersymmetric model that solves the Higgs mass-parameter problem and explains neutrino masses at the same time can survive its own quantum corrections. In the $�mu�nu$SSM, three right-handed neutrino superfields do double duty: their vacuum expectation values generate the effective $�mu$ term, and their mixing with the neutralinos feeds light neutrino masses through a seesaw. Earlier work fitted the three-flavour global neutrino data using only the tree-level mass matrix; here the authors compute the full set of one-loop self-energy diagrams for the ten neutral fermions and ask whether that fit survives. Their answer is yes: the loop-corrected $3��3$ light neutrino mass matrix keeps the same three-term algebraic form as the tree-level one, and scans over the sneutrino vacuum expectation values and neutrino Yukawa couplings still reproduce the measured solar and atmospheric splittings and mixing angles. If correct, the model's neutrino sector is not an artifact of leading-order bookkeeping.

What carries the argument

The load-bearing device is a perturbative block diagonalization of the $10��10$ neutralino-neutrino mass matrix using the expansion matrix $��xi = m_{3��7} M_{7��7}^{-1}$, whose entries are small because the Dirac-type entries $m_{3��7}$ are either left-sneutrino vacuum expectation values or Higgs vacuum expectation values times neutrino Yukawa couplings. This yields an effective light mass matrix $M^{��mathrm{seesaw}}_��nu = -m_{3��7} M_{7��7}^{-1} m_{3��7}^T$ with two competing seesaw structures: an ordinary seesaw scaling as $a_i^2/m_{��nu^c}$ with $m_{��nu^c} = 2��kappa v_c$, and a gaugino-mediated seesaw scaling as $c_i^2/M$ with $1/M = g_1^2/M_1 + g_2^2/M_2$, analogous to a Type-III seesaw. The paper's technical claim is that the one-loop corrections, when rotated back from the mass basis to the flavour basis, reproduce exactly this three-term $a_i a_j$, $c_i c_j$, $a_i c_j$ structure with new coefficients, so the loop analysis can be carried out on the $3��3$ matrix without redoing the full $10��10$ diagonalization.

Extended reading notes

Core claim

The authors compute all one-loop corrections to the $10��10$ neutralino-neutrino mass matrix of the $�mu�nu$SSM in 't Hooft-Feynman gauge and in the $��overline{��mathrm{DR}}$ scheme, including loops with neutral scalars and pseudoscalars, $Z^0$, $W^��$, charged scalars, and quarks and squarks. Rotating the corrections back to the flavour basis and re-applying the seesaw block structure, they argue that the one-loop corrected effective light neutrino mass matrix reduces to $��Delta m_f + M^{��mathrm{seesaw}}_{��nu}$, so that the loop effect can be captured by correcting the $3��3$ light mass matrix alone. That corrected matrix retains the tree-level three-term form $A_1 a_i a_j + A_2 c_i c_j + A_3(a_i c_j + a_j c_i)$, where $a_i = Y^{ii}_��nu v_2$ is the Dirac-type seesaw parameter and $c_i = v'_i$ is the left-sneutrino vacuum expectation value, but with coefficients that now depend on the model parameters and the Passarino-Veltman functions $B_0$ and $B_1$. Numerically, they scan the six parameters of the neutrino sector against the $3��sigma$ ranges of $��Delta m^2_{��mathrm{solar}}$, $|��Delta m^2_{��mathrm{atm}}|$, $��sin^2��theta_{12}$, $��sin^2��theta_{23}$, $��sin^2��theta_{13}$, and the non-oscillation bounds, and report that the data can be accommodated both at tree level and with one-loop corrections included. They further report that the normal hierarchical pattern is easier to fit than the inverted hierarchical or quasi-degenerate patterns once loops are included, with the inverted case in particular failing to find allowed regions for several of the scanned parameters.

Load-bearing premise

The conclusion that normal hierarchy is easier to fit than inverted or quasi-degenerate rests on holding the soft supersymmetry-breaking parameters and the right-sneutrino vacuum expectation values fixed at one benchmark set while scanning only the six neutrino-sector parameters, and the authors themselves note near the end of Section 4.3 that other choices could yield different orderings.

Editorial extensions

If this is right

  • The $�mu�nu$SSM remains a viable joint solution to the $�mu$-problem and the neutrino mass problem once one-loop corrections are included, not merely at leading order.
  • Because the loop-corrected light mass matrix keeps the same three-term form as the tree-level one, the correlations between neutrino mixing angles and ratios of lightest-neutralino decay branching ratios that were derived at tree level are expected to persist, with shifted coefficients.
  • The allowed ranges of the model parameters $��kappa$ and $��lambda$ shift once loops are included: $��kappa$ moves to lower values and $��lambda$ shrinks toward higher values, so collider studies based on tree-level parameter ranges would need updating.
  • Only a narrow window of $��tan��beta$, roughly 10 to 14, survives all neutrino constraints in the normal-hierarchy scan, which sharpens the model's prediction for the Higgs sector.
  • The authors note that a singlino-dominated lightest supersymmetric particle remains compatible with the neutrino data, which would give the LHC a handle on the right-handed neutrino mass scale through displaced decays.

Reading between the lines

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

  • If the three-term structure really is stable under loops, then the loop corrections act mainly as a renormalization of three effective coefficients rather than as a new source of flavour structure, which suggests the model's neutrino predictions are more rigid, and therefore more falsifiable, than its large parameter count would imply.
  • The reported preference for normal hierarchy is a statement about one benchmark point in the soft supersymmetry-breaking sector; a broader scan would be the natural next step, and the paper's own closing remark in Section 4.3 concedes that other choices could reverse the ordering.
  • The same machinery, applied in reverse, could be used to ask which regions of soft parameter space make the loop contribution dominate over the tree-level seesaw, which is a distinct question from whether the data can be fitted at all.
  • Because the loop corrections are computed for the heavy neutralinos as well as the light neutrinos, the framework already contains the pieces needed for a dedicated radiative-correction analysis of the heavy neutralino masses, which the authors explicitly defer.
Original abstract

In an extension of the minimal supersymmetric standard model (popularly known as the $\mu\nu$SSM), three right handed neutrino superfields are introduced to solve the $\mu$-problem and to accommodate the non-vanishing neutrino masses and mixing. Neutrino masses at the tree level are generated through $R-$parity violation and seesaw mechanism. We have analyzed the full effect of one-loop contributions to the neutrino mass matrix. We show that the current three flavour global neutrino data can be accommodated in the $\mu\nu$SSM, for both the tree level and one-loop corrected analyses. We find that it is relatively easier to accommodate the normal hierarchical mass pattern compared to the inverted hierarchical or quasi-degenerate case, when one-loop corrections are included.

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,332 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.32 · human participants 0.01 · controlled comparison 0.13 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.50 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

In an extension of the minimal supersymmetric standard model (popularly known as the $\mu\nu$SSM), three right handed neutrino superfields are introduced to solve the $\mu$-problem and to accommodate the non-vanishing neutrino masses and mixing. Neutrino masses at the tree level are generated through $R-$parity violation and seesaw mechanism. We have analyzed the full effect of one-loop contributions to the neutrino mass matrix. We show that the current three flavour global neutrino data can be accommodated in the $\mu\nu$SSM, for both the tree level and one-loop corrected analyses. We find that it is relatively easier to accommodate the normal hierarchical mass pattern compared to the inverted hierarchical or quasi-degenerate case, when one-loop corrections are included.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In an extension of the minimal supersymmetric standard model (popularly known as the $\mu\nu$SSM), three right handed neutrino superfields are introduced to solve the $\mu$-problem and to accommodate the non-vanishing neutrino masses and mixing. background 0.94 0.17 0.04 asserted 1.00
s1 Neutrino masses at the tree level are generated through $R-$parity violation and seesaw mechanism. background 0.88 0.13 0.04 asserted 1.00
s2 We have analyzed the full effect of one-loop contributions to the neutrino mass matrix. contribution 0.72 0.96 0.05 asserted 0.98
s3 We show that the current three flavour global neutrino data can be accommodated in the $\mu\nu$SSM, for both the tree level and one-loop corrected analyses. result 1.00 0.92 0.60 asserted 0.98
s4 We find that it is relatively easier to accommodate the normal hierarchical mass pattern compared to the inverted hierarchical or quasi-degenerate case, when one-loop corrections are included. result 0.98 0.96 0.29 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.92 · evidence from simulation or computation 0.53.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. One-loop diagrams contributing to the neutralino masses. The various contributions are arising from (clockwise from top left) (a) neutralino-neutralino-neutral scalar loop, (b) neutralino￾neutralino-neutral pseudoscalar loop, (c) neutralino-neutralino-Z 0 µ loop, (d) neutralino-chargino￾charged scalar loop, (e) neutralino-chargino-W± µ loop, (f) neutralino-quark-squark loop. θ13 ≈ 0 ◦ , which is often referred to as the ‘tribimaximal structure’ [103]. Following the above discussions, we can write down the tree level PMNS matrix Uν as, Uν = Ne′ T 3×3 , (3.37) where Ne′ 3×3 is t
Figure 2. Flow chart of the calculational procedure that we have followed in order to calculate the neutrino masses and mixing at the tree level and at one-loop. Here U one−loop PMNS = U ′ ν , U tree PMNS = Uν , L tree f lavour−basis = L mass neutral, Mtree seesaw = Mseesaw ν , L one−loop = L ′ and Mone−loop seesaw = (Mν ′ )ef f in the text. The quasi-degenerate pattern: m1 ≈ m2 ≈ m3 ≫ p |∆m2 atm|, where m1, m2 and m3 are the three light neutrino masses. Here, ∆m2 solar ≡ m2 2 −m2 1 and ∆m2 atm ≡ m2 3 −m2 2 (m2 1 ) for Normal (inverted) Hierarchy. The possibility o
Figure 3. Neutrino mass squared values (m2 i ) vs c 4 i M2 (left panel) and vs a 4 i m2 νc (right panel) plots for the normal hierarchical pattern of light neutrino masses, i = e, µ, τ. In the subsequent plots, we show the variation of the neutrino squared masses (m2 i ) and the atmospheric and solar mass squared differences with the square of the seesaw parameters c 2 i M and a 2 i mνc . Results are shown for the tree level as well as the one-loop corrected neutrino masses. These plots also demonstrate the importance of one-loop corrections to neutrino masses compared to the t
Figure 4. Atmospheric and solar mass squared differences (∆m2 atm, ∆m2 solar) vs c 4 i M2 plots for the normal hierarchical pattern of light neutrino masses, i = e, µ, τ. The full lines are shown for which only the constraints on ∆m2 solar is not within the 3σ limit. The dark coloured portions on these lines are the values of parameters for which all the neutrino constraints are within the 3σ limit. The red (yellow) coloured lines in the plots correspond to the tree (one-loop corrected) regions where all the constraints except ∆m2 solar are within 3σ allowed region. Other parameter choices
Figure 5. Atmospheric and solar mass squared differences (∆m2 atm, ∆m2 solar) vs a 4 i /m2 νc plots for the normal hierarchical pattern of light neutrino masses with i = e, µ, τ. Colour specification is same as described in the context of figure 4. Other parameter choices are discussed in the text. ranges of the parameters a 4 i /m2 ν c , where the neutrino data can be satisfied with both tree and one-loop corrected analysis. We have also considered the variation of light neutrino mass squared differences with the effective bilinear RP violating parameter, εi = Y ijv c j . For this
Figure 6. Plots showing the variations of ∆m2 atm, ∆m2 solar with model parameters λ and κ for normal hierarchy. Values of all other parameters are given by table 1 and 2. Colour specification is same as described in the context of figure 4. Other parameter choices are discussed in the text. Finally, we will discuss the tanβ dependence of ∆m2 atm and ∆m2 solar. These plots are shown in figure 7. The quantity ∆m2 atm decreases with the increasing values of tanβ and nearly saturates for larger values of tanβ. However, the one-loop corrected result for ∆m2 atm is not much different from that a
Figure 7. ∆m2 atm, ∆m2 solar vs tanβ plots for the normal hierarchical pattern of light neutrino masses. The allowed values of tan β are shown by bold points. Other parameter choices are shown in table 1 and 2. on the mixing angles. It was shown in ref. [73] that for the normal hierarchical pattern of neutrino masses, when the parameter bi ∼ ai , the neutrino mixing angles θ23 and θ13 can be written as (with the tree level analysis), sin2 θ23 ≈ b 2 µ b 2 µ + b 2 τ , (4.6) and sin2 θ13 ≈ b 2 e b 2 µ + b 2 τ . (4.7) On the other hand, the mixing angle θ12 is a much more co
Figure 8. Variation of sin2 θ23 with c 2 µ (c 2 µ+c 2 τ ) , sin2 θ12 with c 2 e c 2 µ , sin2 θ13 with c 2 e (c 2 µ+c 2 τ ) for normal hierarchy of light neutrino masses. Other parameter choices are discussed in the text. If one looks at the plot of sin2 θ13 in figure 8, then it is evident that the amount νe flavour in the heaviest state (ν3) decreases a little bit with the inclusion of one-loop corrections for a fixed value of the quantity c 2 e (c 2 µ+c 2 τ ) . Very small sin2 θ13 demands c 2 e ≪ c 2 µ , c2 τ . This feature is also consistent with the
Figure 9. Neutrino mass squared values (m2 i ) vs c 4 i M2 (left panel) and vs a 4 i m2 νc (right panel) plots for the inverted hierarchical pattern of light neutrino masses, i = e, µ, τ. (particularly for ε 2 1 and ε 2 2 ) and then becomes flat. In the one-loop corrected results we do not find any range of values for parameters where the neutrino data are satisfied. The variation of mass squared differences with λ and κ have also been analyzed. The variation of |∆m2 atm| and ∆m2 solar with λ and κ are found to be opposite to those of normal hierarchical scenario. The one-loop c
Figure 10. Atmospheric and solar mass squared differences (|∆m2 atm|, ∆m2 solar) vs c 4 i M2 plots for the inverted hierarchical pattern of light neutrino masses with i = e, µ, τ. Colour specification is same as described in the context of figure 4. Choices of other parameters are discussed in the text. result for ∆m2 solar is lower than the corresponding tree level result for tan β > 10. For the chosen values of other parameters we see that the one-loop corrected analysis does not provide any value of tan β where the neutrino data can be satisfied. We conclude the discussion on inverted hie
Figure 11. Atmospheric and solar mass squared differences (|∆m2 atm|, ∆m2 solar) vs a 4 i /m2 νc plots for the inverted hierarchical pattern of light neutrino masses with i = e, µ, τ. Colour specification is same as described in the context of figure 4. Choices of other parameters are discussed in the text. hierarchy. A similar feature is obtained for the variation with a 2 µ (a 2 µ+a 2 τ ) . On the other hand, the correlations of sin2 θ12 with c 2 e c 2 µ and a 2 e a2 µ and the correlations of sin2 θ13 with c 2 e (c 2 µ+c 2 τ ) and a 2 e (a 2 µ+a 2 τ )
Figure 12. ∆m2 atm|, ∆m2 solar vs tanβ plots for the inverted hierarchical pattern of light neutrino masses. Choices of other parameters are discussed in the text. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 c 2 µ / (c 2 µ + c 2 τ ) 0.3 0.4 0.5 0.6 0.7 Sin 2 θ23 INVERTED HIERARCHY TREE + ONE LOOP (all constraints within 3σ limit) TREE LEVEL (all constraints within 3σ limit) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 a 2 µ / (a 2 µ + a 2 τ ) 0.3 0.4 0.5 0.6 0.7 Sin 2 θ23 INVERTED HIERARCHY TREE LEVEL (all constraints within 3σ limit) TREE + ONE LOOP (all constraints within 3σ limit)
Figure 13. Variation of sin2 θ23 with c 2 µ (c 2 µ+c 2 τ ) and a 2 µ (a2 µ+a2 τ ) for inverted hierarchy of light neutrino masses. Choices of other parameters are discussed in the text. 4.3 Quasi-degenerate spectra The discussion on the light neutrino mass spectrum remains incomplete without a note on the so-called “quasi-degenerate” scenario. A truly degenerate scenario of three light neutrino masses is, however, inconsistent with the oscillation data (see eq.(4.1)). Hence, the quasi-degenerate scenario of light neutrino masses is defined in such a way that in this case all the
Figure 14. Neutrino mass squared values (m2 i ) vs c 4 e M2 (left panel) and vs a 4 e m2 νc (right panel) plots for the quasi-degenerate pattern of light neutrino masses. As mentioned earlier, one can play with the model parameters and obtain a spectrum with a different ordering of masses termed as “Quasi-degenerate-II” in table 3. However, for such an ordering of masses, we found that it was rather impossible to find any region of parameter space where the one-loop corrected result satisfies all the constraints on neutrino masses and mixing. Nevertheless, we must emphasize here that it

Figures reproduced from arXiv:1002.2705 by the authors.

References
How the literature uses it

42 arXiv papers cite this in 69 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 51 sentences without a confident label

Uses its method or tool 1 paper
Cites it as support 2 papers
  • The smallness of the left sneutrino VEVs for a correct description of the neutrino sector, compatible with current data, has been shown in Refs. [ref] .

    $U(1)'$ extensions of the $\mu \nu$SSM · The scalar potential · 0.99

  • In our numerical studies we fitted the experimentally well measured quantities [equation] We restricted ourselves in the neutrino sector to a tree-level analysis, because the one-loop corrections turn out to be moderate in size (in the normal hierarchy pattern) [ref] and can always be compensated by a small shift in the neutrino Yukawa couplings $Y^ν_ij$ without affecting the conclusions drawn in the scalar sector, in particular for the observables related to the SM-like Higgs boson.

    Precise prediction for the Higgs-Boson Masses in the $\mu\nu$SSM with three right-handed neutrino superfields · Neutral fermion sector · 0.97

Cites it as background 14 papers

What it cites · 36 sentences without a confident label

Compares with 4 papers
  • Note that the one-loop diagrams in figure[ref], contributing to the neutrino mass matrix are very similar to those obtained in bilinear R-parity violating scenario [ref] .

    Neutrino masses in the R-parity violating MSSM · Radiative corrections to neutrino mass terms · 0.99

  • Note that the one-loop diagrams in figure[ref], contributing to the neutrino mass matrix are very similar to those obtained in bilinear R-parity violating scenario [ref] .

    Basis independent neutrino masses in the $R_p$ violating MSSM · Radiative corrections to neutrino mass terms · 0.99

  • Note that the one-loop diagrams in figure[ref], contributing to the neutrino mass matrix are very similar to those obtained in bilinear R-parity violating scenario [ref] .

    Neutrino masses in R-parity violating supersymmetric models · Radiative corrections to neutrino mass terms · 0.99

Cites as background 19 papers

Discussion

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