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@ARTICLE{Grojean:602012,
      author       = {Grojean, Christophe and Kley, Jonathan and Leflot, Damien
                      and Yao, Chang-Yuan},
      title        = {{T}he flavor invariants of the $\nu${SM}},
      reportid     = {PUBDB-2024-00468, DESY-24-021. arXiv:2406.00094.
                      CERN-TH-2024-076. HU-EP-24/14},
      year         = {2024},
      note         = {27 pages + appendices, 3 figures},
      abstract     = {Sixty years after the experimental discovery of CP
                      violation in the quark sector, the existence of a similar CP
                      violation in the lepton sector is still to be established.
                      Actually, the structure of such a violation depends
                      crucially on the origin of the neutrino masses. In an
                      attempt at categorizing the leptonic sources of CP
                      violation, we studied the $\nu$SM, the Standard Model
                      extended with three generations of sterile neutrinos, that
                      can interpolate continuously between the Dirac and Majorana
                      scenarios of neutrino masses. In particular, we perform a
                      classification of the Jarlskog-like flavor invariants
                      entering CP-violating observables and we study their
                      suppression with the heavy Majorana mass in the seesaw limit
                      of the model. To simplify the construction of the
                      invariants, we introduce a graph-based method. With the
                      guidance of the Hilbert series and plethystic logarithm of
                      the theory, we construct the generating and primary sets of
                      invariants for the $\nu$SM for the first time. Unlike in the
                      Standard Model and some other theories, we find that the
                      numbers of generating invariants and the syzygies among them
                      cannot immediately be read off from the plethystic
                      logarithm, but require a more careful examination. Our
                      analysis reveals that the generating set contains 459
                      invariants, out of which 208 are CP-even and 251 are CP-odd.
                      In the seesaw limit of the $\nu$SM, we show that all
                      parameters of the UV theory can be captured in the effective
                      theory with a certain suppression with the heavy Majorana
                      mass, while these parameters can only appear in a
                      flavor-invariant way with ahigher mass suppression.
                      Furthermore, we discuss how the necessary and sufficient
                      conditions for CP violation can be captured by utilizing
                      these invariants. Along the way, we present useful
                      algorithms to enumerate and build the flavor invariants.},
      cin          = {T},
      cid          = {I:(DE-H253)T-20120731},
      pnm          = {611 - Fundamental Particles and Forces (POF4-611) / DFG
                      project G:(GEPRIS)390833306 - EXC 2121: Quantum Universe
                      (390833306) / ASYMMETRY - Essential Asymmetries of Nature
                      (101086085)},
      pid          = {G:(DE-HGF)POF4-611 / G:(GEPRIS)390833306 /
                      G:(EU-Grant)101086085},
      experiment   = {EXP:(DE-MLZ)NOSPEC-20140101},
      typ          = {PUB:(DE-HGF)25},
      eprint       = {2406.00094},
      howpublished = {arXiv:2406.00094},
      archivePrefix = {arXiv},
      SLACcitation = {$\%\%CITATION$ = $arXiv:2406.00094;\%\%$},
      doi          = {10.3204/PUBDB-2024-00468},
      url          = {https://bib-pubdb1.desy.de/record/602012},
}