000602012 001__ 602012 000602012 005__ 20250801212027.0 000602012 0247_ $$2INSPIRETeX$$aGrojean:2024qdm 000602012 0247_ $$2inspire$$ainspire:2793720 000602012 0247_ $$2arXiv$$aarXiv:2406.00094 000602012 0247_ $$2datacite_doi$$a10.3204/PUBDB-2024-00468 000602012 037__ $$aPUBDB-2024-00468 000602012 041__ $$aEnglish 000602012 088__ $$2DESY$$aDESY-24-021 000602012 088__ $$2arXiv$$aarXiv:2406.00094 000602012 088__ $$2CERN$$aCERN-TH-2024-076 000602012 088__ $$2Other$$aHU-EP-24/14 000602012 1001_ $$0P:(DE-H253)PIP1023796$$aGrojean, Christophe$$b0$$udesy 000602012 245__ $$aThe flavor invariants of the $\nu$SM 000602012 260__ $$c2024 000602012 3367_ $$0PUB:(DE-HGF)25$$2PUB:(DE-HGF)$$aPreprint$$bpreprint$$mpreprint$$s1754056986_3349141 000602012 3367_ $$2ORCID$$aWORKING_PAPER 000602012 3367_ $$028$$2EndNote$$aElectronic Article 000602012 3367_ $$2DRIVER$$apreprint 000602012 3367_ $$2BibTeX$$aARTICLE 000602012 3367_ $$2DataCite$$aOutput Types/Working Paper 000602012 500__ $$a27 pages + appendices, 3 figures 000602012 520__ $$aSixty years after the experimental discovery of CP violation in the quark sector, the existence of a similar CP violation in the lepton sector is still to be established. Actually, the structure of such a violation depends crucially on the origin of the neutrino masses. In an attempt at categorizing the leptonic sources of CP violation, we studied the $\nu$SM, the Standard Model extended with three generations of sterile neutrinos, that can interpolate continuously between the Dirac and Majorana scenarios of neutrino masses. In particular, we perform a classification of the Jarlskog-like flavor invariants entering CP-violating observables and we study their suppression with the heavy Majorana mass in the seesaw limit of the model. To simplify the construction of the invariants, we introduce a graph-based method. With the guidance of the Hilbert series and plethystic logarithm of the theory, we construct the generating and primary sets of invariants for the $\nu$SM for the first time. Unlike in the Standard Model and some other theories, we find that the numbers of generating invariants and the syzygies among them cannot immediately be read off from the plethystic logarithm, but require a more careful examination. Our analysis reveals that the generating set contains 459 invariants, out of which 208 are CP-even and 251 are CP-odd. In the seesaw limit of the $\nu$SM, we show that all parameters of the UV theory can be captured in the effective theory with a certain suppression with the heavy Majorana mass, while these parameters can only appear in a flavor-invariant way with ahigher mass suppression. Furthermore, we discuss how the necessary and sufficient conditions for CP violation can be captured by utilizing these invariants. Along the way, we present useful algorithms to enumerate and build the flavor invariants. 000602012 536__ $$0G:(DE-HGF)POF4-611$$a611 - Fundamental Particles and Forces (POF4-611)$$cPOF4-611$$fPOF IV$$x0 000602012 536__ $$0G:(GEPRIS)390833306$$aDFG project G:(GEPRIS)390833306 - EXC 2121: Quantum Universe (390833306)$$c390833306$$x1 000602012 536__ $$0G:(EU-Grant)101086085$$aASYMMETRY - Essential Asymmetries of Nature (101086085)$$c101086085$$fHORIZON-MSCA-2021-SE-01$$x2 000602012 588__ $$aDataset connected to INSPIRE 000602012 693__ $$0EXP:(DE-MLZ)NOSPEC-20140101$$5EXP:(DE-MLZ)NOSPEC-20140101$$eNo specific instrument$$x0 000602012 7001_ $$0P:(DE-H253)PIP1098464$$aKley, Jonathan$$b1$$udesy 000602012 7001_ $$0P:(DE-H253)PIP1103590$$aLeflot, Damien$$b2 000602012 7001_ $$0P:(DE-H253)PIP1100776$$aYao, Chang-Yuan$$b3$$eCorresponding author 000602012 8564_ $$uhttps://bib-pubdb1.desy.de/record/602012/files/HTML-Approval_of_scientific_publication.html 000602012 8564_ $$uhttps://bib-pubdb1.desy.de/record/602012/files/PDF-Approval_of_scientific_publication.pdf 000602012 8564_ $$uhttps://bib-pubdb1.desy.de/record/602012/files/2406.00094v1.pdf$$yRestricted$$zStatID:(DE-HGF)0599 000602012 8564_ $$uhttps://bib-pubdb1.desy.de/record/602012/files/nuSM_invariants.pdf$$yOpenAccess$$zStatID:(DE-HGF)0510 000602012 8564_ $$uhttps://bib-pubdb1.desy.de/record/602012/files/2406.00094v1.pdf?subformat=pdfa$$xpdfa$$yRestricted$$zStatID:(DE-HGF)0599 000602012 8564_ $$uhttps://bib-pubdb1.desy.de/record/602012/files/nuSM_invariants.pdf?subformat=pdfa$$xpdfa$$yOpenAccess$$zStatID:(DE-HGF)0510 000602012 909CO $$ooai:bib-pubdb1.desy.de:602012$$popen_access$$popenaire$$pdnbdelivery$$pec_fundedresources$$pVDB$$pdriver 000602012 9101_ $$0I:(DE-588b)2008985-5$$6P:(DE-H253)PIP1023796$$aDeutsches Elektronen-Synchrotron$$b0$$kDESY 000602012 9101_ $$0I:(DE-588b)2008985-5$$6P:(DE-H253)PIP1098464$$aDeutsches Elektronen-Synchrotron$$b1$$kDESY 000602012 9101_ $$0I:(DE-HGF)0$$6P:(DE-H253)PIP1098464$$aExternal Institute$$b1$$kExtern 000602012 9101_ $$0I:(DE-HGF)0$$6P:(DE-H253)PIP1103590$$aExternal Institute$$b2$$kExtern 000602012 9101_ $$0I:(DE-588b)2008985-5$$6P:(DE-H253)PIP1100776$$aDeutsches Elektronen-Synchrotron$$b3$$kDESY 000602012 9131_ $$0G:(DE-HGF)POF4-611$$1G:(DE-HGF)POF4-610$$2G:(DE-HGF)POF4-600$$3G:(DE-HGF)POF4$$4G:(DE-HGF)POF$$aDE-HGF$$bForschungsbereich Materie$$lMatter and the Universe$$vFundamental Particles and Forces$$x0 000602012 9141_ $$y2024 000602012 915__ $$0StatID:(DE-HGF)0510$$2StatID$$aOpenAccess 000602012 915__ $$0LIC:(DE-HGF)CCBY4$$2HGFVOC$$aCreative Commons Attribution CC BY 4.0 000602012 915__ $$0StatID:(DE-HGF)0580$$2StatID$$aPublished 000602012 9201_ $$0I:(DE-H253)T-20120731$$kT$$lTheorie-Gruppe$$x0 000602012 980__ $$apreprint 000602012 980__ $$aVDB 000602012 980__ $$aI:(DE-H253)T-20120731 000602012 980__ $$aUNRESTRICTED 000602012 9801_ $$aFullTexts