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Modular Invariance Faces Precision Neutrino Data

by Juan Carlos Criado, Ferruccio Feruglio

This is not the latest submitted version.

This Submission thread is now published as SciPost Phys. 5, 042 (2018)

Submission summary

As Contributors: Juan Carlos Criado · Ferruccio Feruglio
Arxiv Link: (pdf)
Date submitted: 2018-07-17 02:00
Submitted by: Feruglio, Ferruccio
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
  • High-Energy Physics - Phenomenology
Approach: Theoretical


We analyze a modular invariant model of lepton masses, with neutrino masses originating either from the Weinberg operator or from the seesaw. The constraint provided by modular invariance is so strong that neutrino mass ratios, lepton mixing angles and Dirac/Majorana phases do not depend on any Lagrangian parameter. They only depend on the vacuum of the theory, parametrized in terms of a complex modulus and a real field. Thus eight measurable quantities are described by the three vacuum parameters, whose optimization provides an excellent fit to data for the Weinberg operator and a good fit for the seesaw case. Neutrino masses from the Weinberg operator (seesaw) have inverted (normal) ordering. Several sources of potential corrections, such as higher dimensional operators, renormalization group evolution and supersymmetry breaking effects, are carefully discussed and shown not to affect the predictions under reasonable conditions.

Ontology / Topics

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Modular invariance Weinberg operator
Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2018-9-23 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1807.01125v2, delivered 2018-09-23, doi: 10.21468/SciPost.Report.585


1- Original aproach to lepton masses and mixings patterns
2- The comparison with the data is excellent
3- A relatively small number of parameters used, leading to several predictions
4- Clarity of the arguments, in a subject that is rather technical


It would be interesting to find the vacuum structure by a dynamical minimization starting from a microscopic model for the flavons and the modulus field.


Excellent paper using an original approach, using techniques from field theory, string theory and neutrino physics. Impressive amount of work with an excellent output in comparing with data.
I am very happy to propose the paper for publication in the current form.
Just a question for my curiosity: in eq. 3, Kahler transformations require that the superpotential transforms with $e^{-f}$. This term becomes one in the global SUSY limit $M_P \to \infty$, but is necessary in supergravity. Moreover, in the simplest examples of compactifications, superpotential Yukawas do not have zero modular weight. It seems to me that the invariance of the superpotential under modular transformations was imposed by hand. Unless the invariance under the discrete flavor symmetry $\Gamma_3$ is the key ingredient. Am I wrong ?

Requested changes


  • validity: top
  • significance: top
  • originality: top
  • clarity: top
  • formatting: perfect
  • grammar: excellent

Author:  Ferruccio Feruglio  on 2018-09-24  [id 321]

(in reply to Report 2 on 2018-09-23)
answer to question

First of all, thank you for your comments.

Concerning the case of local supersymmetry you are perfectly
right. The superpotential is not expected to be modular invariant. For instance, with the `minimal'
choice of Kahler potential done in our paper, namely $K=-h log(-i \tau+i\bar\tau)+...$,
the superpotential should transform with weight $-h$, that is $w\to (c \tau+d)^{-h} w$.
However this fact can be easily incorporated in our construction by suitably shifting the weight
assigned to the matter supermultiplets, our eq. (2). In the bottom-up approach we are considering,
the choice of the weights for the matter supermultiplets is part of the freedom of the model.
Also in a sugra context, with an appropriate choice of the weights, we can reproduce the same Yukawa
couplings of the rigid case, while allowing the superpotential $w$ to transform non trivially
under modular transformations as requested.

Please do not hesitate to ask further questions if this explanation is not sufficiently clear.

Report 1 by George Leontaris on 2018-9-15 (Invited Report)

  • Cite as: George Leontaris, Report on arXiv:1807.01125v2, delivered 2018-09-15, doi: 10.21468/SciPost.Report.581


1)The use of Modular invariance is an interesting alternative approach to mass textures
2) Good predictions on the neutrino sector


1) Arguments about charged lepton mass matrix are rather weak
2) Implications of modular invariance on the quark sector not discussed.


In this work the authors pursue an approach based on modular invariance
to determine the lepton mass matrices and give predictions for neutrino data.
In their analysis the neutrino masses are generated either from a Weinberg operator or from a see-saw mechanism. In both cases, modular symmetry puts strong constraints and reduces the number of arbitrary parameters. RGEs and SUSY effects are also taken into account.

This work is an interesting improvement and continuation of past effords in
this particular subject. It is also well motivated since modular invariance
appears also in string theories.

In general the paper is well written and the results agree with the existing data. However, I find the discussion of section 3 (above equation (18)), regarding the issue of the reactor angle $\theta_{13}$ and the charged lepton
mass matrix, somewhat confusing and not very convincing.
At least in the way it is presented, it is also not clear what exactly the authors are using from ref[20].
Also, some expectations for the quark sector should be discussed.
In section 5 the author refer to "approximate symmetry" for SUSY. I think that they mean broken supersymmetry.

There are also a few minor flaws in the text. For example "by a phase transformations" (page 2), Kahler instead of K\"ahler, a missing fullstop, "loosing generality" instead of the math-jargon "without loss of generality"
( page 8) .
I would recommend publication after the authors consider the above corrections and suggestions.

Requested changes

1) Improvement and clarification of discussion in section 3, above eq (18)
2) Short discussion for the quark sector
3) Correction on language mistakes

  • validity: good
  • significance: good
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: good

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