# A neutrinoless double beta decay master formula from effective field theory

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## Abstract

We present a master formula describing the neutrinoless-double-beta decay (0*νββ*) rate induced by lepton-number-violating (LNV) operators up to dimension nine in the Standard Model Effective Field Theory. We provide an end-to-end framework connecting the possibly very high LNV scale to the nuclear scale, through a chain of effective field theories. Starting at the electroweak scale, we integrate out the heavy Standard Model degrees of freedom and we match to an SU(3)_{c} ⊗ U(1)_{em} effective theory. After evolving the resulting effective Lagrangian to the QCD scale, we use chiral perturbation theory to derive the lepton-number-violating chiral Lagrangian. The chiral Lagrangian is used to derive the two-nucleon 0*νββ* transition operators to leading order in the chiral power counting. Based on renormalization arguments we show that in various cases short-range two-nucleon operators need to be enhanced to leading order. We show that all required nuclear matrix elements can be taken from existing calculations. Our final result is a master formula that describes the 0*νββ* rate in terms of phase-space factors, nuclear matrix elements, hadronic low-energy constants, QCD evolution factors, and high-energy LNV Wilson coefficients, including all the interference terms. Our master formula can be easily matched to any model where LNV originates at energy scales above the electroweak scale. As an explicit example, we match our formula to the minimal left-right-symmetric model in which contributions of operators of different dimension compete, and we discuss the resulting phenomenology.

## Keywords

Beyond Standard Model Chiral Lagrangians Effective Field Theories## Notes

### **Open Access**

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## References

- [1]KamLAND-Zen collaboration, A. Gando et al.,
*Limit on neutrinoless ββ decay of*^{136}*Xe from the first phase of KamLAND-Zen and comparison with the positive claim in*^{76}*Ge*,*Phys. Rev. Lett.***110**(2013) 062502 [arXiv:1211.3863] [INSPIRE]. - [2]GERDA collaboration, M. Agostini et al.,
*Results on neutrinoless double-β decay of*^{76}*Ge from Phase I of the GERDA experiment*,*Phys. Rev. Lett.***111**(2013) 122503 [arXiv:1307.4720] [INSPIRE]. - [3]EXO-200 collaboration, J.B. Albert et al.,
*Search for Majorana neutrinos with the first two years of EXO-200 data*,*Nature***510**(2014) 229 [arXiv:1402.6956] [INSPIRE]. - [4]SNO+ collaboration, S. Andringa et al.,
*Current status and future prospects of the SNO+ experiment*,*Adv. High Energy Phys.***2016**(2016) 6194250 [arXiv:1508.05759] [INSPIRE]. - [5]KamLAND-Zen collaboration, A. Gando et al.,
*Search for majorana neutrinos near the inverted mass hierarchy region with KamLAND-Zen*,*Phys. Rev. Lett.***117**(2016) 082503 [arXiv:1605.02889] [INSPIRE]. - [6]S.R. Elliott et al.,
*Initial results from the Majorana Demonstrator*,*J. Phys. Conf. Ser.***888**(2017) 012035 [arXiv:1610.01210] [INSPIRE]. - [7]M. Agostini et al.,
*Background-free search for neutrinoless double-β decay of*^{76}*Ge with GERDA*, arXiv:1703.00570 [INSPIRE]. - [8]Majorana collaboration, C.E. Aalseth et al.,
*Search for neutrinoless double-β decay in*^{76}*Ge with the Majorana Demonstrator*,*Phys. Rev. Lett.***120**(2018) 132502 [arXiv:1710.11608] [INSPIRE]. - [9]EXO collaboration, J.B. Albert et al.,
*Search for neutrinoless double-β decay with the upgraded EXO-200 detector*,*Phys. Rev. Lett.***120**(2018) 072701 [arXiv:1707.08707] [INSPIRE]. - [10]CUORE collaboration, C. Alduino et al.,
*First results from CUORE: a search for lepton number violation via*0*νββ decay of*^{130}*Te*,*Phys. Rev. Lett.***120**(2018) 132501 [arXiv:1710.07988] [INSPIRE]. - [11]GERDA collaboration, M. Agostini et al.,
*Improved Limit on neutrinoless double-β decay of*^{76}*Ge from GERDA Phase II*,*Phys. Rev. Lett.***120**(2018) 132503 [arXiv:1803.11100] [INSPIRE]. - [12]CUPID-0 collaboration, O. Azzolini et al.,
*First result on the neutrinoless double-β decay of*^{82}*Se with CUPID-0*,*Phys. Rev. Lett.***120**(2018) 232502 [arXiv:1802.07791] [INSPIRE]. - [13]A. Kobach,
*Baryon number, lepton number and operator dimension in the standard model*,*Phys. Lett.***B 758**(2016) 455 [arXiv:1604.05726] [INSPIRE]. - [14]S. Weinberg,
*Baryon and lepton nonconserving processes*,*Phys. Rev. Lett.***43**(1979) 1566 [INSPIRE].ADSCrossRefGoogle Scholar - [15]A. Zee,
*A theory of lepton number violation and neutrino Majorana masses*,*Phys. Lett.***B 93**(1980) 389 [*Erratum ibid.***B 95**(1980) 461].Google Scholar - [16]
- [17]
- [18]K.S. Babu and E. Ma,
*Natural hierarchy of radiatively induced Majorana neutrino masses*,*Phys. Rev. Lett.***61**(1988) 674 [INSPIRE].ADSCrossRefGoogle Scholar - [19]K.S. Babu, E. Ma and J.T. Pantaleone,
*Model of radiative neutrino masses: mixing and a possible fourth generation*,*Phys. Lett.***B 218**(1989) 233 [INSPIRE]. - [20]K.S. Babu and C.N. Leung,
*Classification of effective neutrino mass operators*,*Nucl. Phys.***B 619**(2001) 667 [hep-ph/0106054] [INSPIRE]. - [21]G. Prezeau, M. Ramsey-Musolf and P. Vogel,
*Neutrinoless double beta decay and effective field theory*,*Phys. Rev.***D 68**(2003) 034016 [hep-ph/0303205] [INSPIRE]. - [22]A. de Gouvêa and J. Jenkins,
*A survey of lepton number violation via effective operators*,*Phys. Rev.***D 77**(2008) 013008 [arXiv:0708.1344] [INSPIRE]. - [23]L. Lehman,
*Extending the standard model effective field theory with the complete set of dimension-7 operators*,*Phys. Rev.***D 90**(2014) 125023 [arXiv:1410.4193] [INSPIRE]. - [24]M.L. Graesser,
*An electroweak basis for neutrinoless double β decay*,*JHEP***08**(2017) 099 [arXiv:1606.04549] [INSPIRE].ADSCrossRefGoogle Scholar - [25]V. Cirigliano et al.,
*Neutrinoless double β decay in chiral effective field theory: lepton number violation at dimension seven*,*JHEP***12**(2017) 082 [arXiv:1708.09390] [INSPIRE].ADSCrossRefGoogle Scholar - [26]V. Cirigliano, W. Dekens, E. Mereghetti and A. Walker-Loud,
*Neutrinoless double-β decay in effective field theory: The light-Majorana neutrino-exchange mechanism*,*Phys. Rev.***C 97**(2018) 065501 [arXiv:1710.01729] [INSPIRE]. - [27]V. Cirigliano et al.,
*New leading contribution to neutrinoless double-β decay*,*Phys. Rev. Lett.***120**(2018) 202001 [arXiv:1802.10097] [INSPIRE]. - [28]G. Hagen, T. Papenbrock, M. Hjorth-Jensen and D.J. Dean,
*Coupled-cluster computations of atomic nuclei*,*Rept. Prog. Phys.***77**(2014) 096302 [arXiv:1312.7872] [INSPIRE]. - [29]G. Hagen et al.,
*Coupled-cluster calculations of nucleonic matter*,*Phys. Rev.***C 89**(2014) 014319 [arXiv:1311.2925] [INSPIRE]. - [30]G. Hagen, G.R. Jansen, M. Hjorth-Jensen and T. Papenbrock,
*Emergent properties of nuclei from ab initio coupled-cluster calculations*,*Phys. Scripta***91**(2016) 063006 [arXiv:1601.08203] [INSPIRE]. - [31]S. Pastore et al.,
*Neutrinoless double-β decay matrix elements in light nuclei*,*Phys. Rev.***C 97**(2018) 014606 [arXiv:1710.05026] [INSPIRE]. - [32]J. Hyvärinen and J. Suhonen,
*Nuclear matrix elements for*0*νββ decays with light or heavy Majorana-neutrino exchange*,*Phys. Rev.***C 91**(2015) 024613 [INSPIRE]. - [33]J. Menéndez,
*Neutrinoless ββ decay mediated by the exchange of light and heavy neutrinos: The role of nuclear structure correlations*,*J. Phys.***G 45**(2018) 014003 [arXiv:1804.02105] [INSPIRE]. - [34]J. Barea, J. Kotila and F. Iachello, 0
*νββ and*2*νββ nuclear matrix elements in the interacting boson model with isospin restoration*,*Phys. Rev.***C 91**(2015) 034304 [arXiv:1506.08530] [INSPIRE]. - [35]J. Barea, private communication.Google Scholar
- [36]J.C. Pati and A. Salam,
*Lepton number as the fourth color*,*Phys. Rev.***D 10**(1974) 275 [*Erratum ibid.***D 11**(1975) 703] [INSPIRE]. - [37]R.N. Mohapatra and J.C. Pati,
*Left-right gauge symmetry and an isoconjugate model of CP-violation*,*Phys. Rev.***D 11**(1975) 566 [INSPIRE]. - [38]G. Senjanović and R.N. Mohapatra,
*Exact left-right symmetry and spontaneous violation of parity*,*Phys. Rev.***D 12**(1975) 1502 [INSPIRE]. - [39]H. Pas, M. Hirsch, H.V. Klapdor-Kleingrothaus and S.G. Kovalenko,
*A superformula for neutrinoless double beta decay. 2. The Short range part*,*Phys. Lett.***B 498**(2001) 35 [hep-ph/0008182] [INSPIRE]. - [40]H. Pas, M. Hirsch, H.V. Klapdor-Kleingrothaus and S.G. Kovalenko,
*Towards a superformula for neutrinoless double beta decay*,*Phys. Lett.***B 453**(1999) 194 [INSPIRE]. - [41]M.J. Savage,
*Pionic matrix elements in neutrinoless double Beta decay*,*Phys. Rev.***C 59**(1999) 2293 [nucl-th/9811087] [INSPIRE]. - [42]A. Nicholson et al.,
*Neutrinoless double beta decay from lattice QCD*, PoS(LATTICE 2016)017 [arXiv:1608.04793] [INSPIRE]. - [43]V. Cirigliano, W. Dekens, M. Graesser and E. Mereghetti,
*Neutrinoless double beta decay and chiral*SU(3),*Phys. Lett.***B 769**(2017) 460 [arXiv:1701.01443] [INSPIRE]. - [44]A. Nicholson et al.,
*Heavy physics contributions to neutrinoless double beta decay from QCD*,*Phys. Rev. Lett.***121**(2018) 172501 [arXiv:1805.02634] [INSPIRE].ADSCrossRefGoogle Scholar - [45]A. Manohar and H. Georgi,
*Chiral quarks and the nonrelativistic quark model*,*Nucl. Phys.***B 234**(1984) 189 [INSPIRE]. - [46]A.J. Buras, M. Misiak and J. Urban,
*Two loop QCD anomalous dimensions of flavor changing four quark operators within and beyond the standard model*,*Nucl. Phys.***B 586**(2000) 397 [hep-ph/0005183] [INSPIRE]. - [47]A.J. Buras, S. Jager and J. Urban,
*Master formulae for*Δ*F*= 2*NLO QCD factors in the standard model and beyond*,*Nucl. Phys.***B 605**(2001) 600 [hep-ph/0102316] [INSPIRE]. - [48]M. González, M. Hirsch and S.G. Kovalenko,
*QCD running in neutrinoless double beta decay: short-range mechanisms*,*Phys. Rev.***D 93**(2016) 013017 [arXiv:1511.03945] [INSPIRE]. - [49]J. Gasser and H. Leutwyler,
*Chiral perturbation theory to one loop*,*Annals Phys.***158**(1984) 142 [INSPIRE]. - [50]V. Bernard, N. Kaiser and U.-G. Meissner,
*Chiral dynamics in nucleons and nuclei*,*Int. J. Mod. Phys.***E 4**(1995) 193 [hep-ph/9501384] [INSPIRE]. - [51]D.B. Kaplan, M.J. Savage and M.B. Wise,
*Nucleon-nucleon scattering from effective field theory*,*Nucl. Phys.***B 478**(1996) 629 [nucl-th/9605002] [INSPIRE]. - [52]A. Nogga, R.G.E. Timmermans and U. van Kolck,
*Renormalization of one-pion exchange and power counting*,*Phys. Rev.***C 72**(2005) 054006 [nucl-th/0506005] [INSPIRE]. - [53]M. Pavón Valderrama and D.R. Phillips,
*Power counting of contact-range currents in effective field theory*,*Phys. Rev. Lett.***114**(2015) 082502 [arXiv:1407.0437] [INSPIRE]. - [54]B. Pontecorvo,
*Superweak interactions and double beta decay*,*Phys. Lett.***B 26**(1968) 630.Google Scholar - [55]J.D. Vergados,
*Pion double charge exchange contribution to neutrinoless double β decay*,*Phys. Rev.***D 25**(1982) 914 [INSPIRE]. - [56]A. Faessler, S. Kovalenko, F. Šimkovic and J. Schwieger,
*Dominance of pion exchange in R-parity violating supersymmetry contributions to neutrinoless double beta decay*,*Phys. Rev. Lett.***78**(1997) 183 [hep-ph/9612357] [INSPIRE]. - [57]M.E. Peskin and D.V. Schroeder,
*An Introduction to quantum field theory*, Addison-Wesley, Reading U.S.A. (1995).Google Scholar - [58]Particle Data Group collaboration, C. Patrignani et al.,
*Review of particle physics*,*Chin. Phys.***C 40**(2016) 100001 [INSPIRE]. - [59]T. Bhattacharya et al.,
*Axial, scalar and tensor charges of the nucleon from*2 + 1 + 1*-flavor lattice QCD*,*Phys. Rev.***D 94**(2016) 054508 [arXiv:1606.07049] [INSPIRE]. - [60]M. Horoi and A. Neacsu,
*Towards an effective field theory approach to the neutrinoless double-β decay*, arXiv:1706.05391 [INSPIRE]. - [61]F. Simkovic et al.,
*Anatomy of nuclear matrix elements for neutrinoless double-beta decay*,*Phys. Rev.***C 77**(2008) 045503 [arXiv:0710.2055] [INSPIRE]. - [62]M. Doi, T. Kotani and E. Takasugi,
*Double β decay and Majorana neutrino*,*Prog. Theor. Phys. Suppl.***83**(1985) 1 [INSPIRE].ADSCrossRefGoogle Scholar - [63]S.M. Bilenky and C. Giunti,
*Neutrinoless double-β decay: a probe of physics beyond the standard model*,*Int. J. Mod. Phys.***A 30**(2015) 1530001 [arXiv:1411.4791] [INSPIRE]. - [64]S. Stoica and M. Mirea,
*New calculations for phase space factors involved in double-β decay*,*Phys. Rev.***C 88**(2013) 037303 [arXiv:1307.0290] [INSPIRE]. - [65]G. Bambhaniya, P.S.B. Dev, S. Goswami and M. Mitra,
*The scalar triplet contribution to lepton flavour violation and neutrinoless double β decay in left-right symmetric model*,*JHEP***04**(2016) 046 [arXiv:1512.00440] [INSPIRE].ADSGoogle Scholar - [66]P.S. Bhupal Dev, S. Goswami and M. Mitra,
*TeV scale left-right symmetry and large mixing effects in neutrinoless double β decay*,*Phys. Rev.***D 91**(2015) 113004 [arXiv:1405.1399] [INSPIRE]. - [67]V. Tello et al.,
*Left-right symmetry: from LHC to neutrinoless double β decay*,*Phys. Rev. Lett.***106**(2011) 151801 [arXiv:1011.3522] [INSPIRE].ADSCrossRefGoogle Scholar - [68]M. Nemevšek, F. Nesti, G. Senjanović and V. Tello,
*Neutrinoless double β decay: low left-right symmetry scale?*, arXiv:1112.3061 [INSPIRE]. - [69]J. Barry and W. Rodejohann,
*Lepton number and flavour violation in TeV-scale left-right symmetric theories with large left-right mixing*,*JHEP***09**(2013) 153 [arXiv:1303.6324] [INSPIRE].ADSCrossRefGoogle Scholar - [70]G. Senjanović,
*Spontaneous breakdown of parity in a class of gauge theories*,*Nucl. Phys.***B 153**(1979) 334 [INSPIRE]. - [71]R.N. Mohapatra,
*Unification and supersymmetry. The frontiers of quark-lepton physics*, Springer, Berlin Germany (1986).Google Scholar - [72]A. Maiezza, M. Nemevšek, F. Nesti and G. Senjanović,
*Left-right symmetry at LHC*,*Phys. Rev.***D 82**(2010) 055022 [arXiv:1005.5160] [INSPIRE]. - [73]G. Senjanović and V. Tello,
*Restoration of parity and the right-handed analog of the CKM matrix*,*Phys. Rev.***D 94**(2016) 095023 [arXiv:1502.05704] [INSPIRE]. - [74]V. Cirigliano, M. Gonzalez-Alonso and M.L. Graesser,
*Non-standard charged current interactions: β decays versus the LHC*,*JHEP***02**(2013) 046 [arXiv:1210.4553] [INSPIRE]. - [75]M. Nemevšek, G. Senjanović and V. Tello,
*Connecting Dirac and Majorana neutrino mass matrices in the minimal left-right symmetric model*,*Phys. Rev. Lett.***110**(2013) 151802 [arXiv:1211.2837] [INSPIRE]. - [76]CMS collaboration,
*Searches for dijet resonances in pp collisions at*\( \sqrt{s}=13 \)*TeV using data collected in 2016*, CMS-PAS-EXO-16-056 (2016). - [77]ATLAS collaboration,
*Search for new phenomena in dijet events using*37*fb*^{−1}*of pp collision data collected at*\( \sqrt{s}=13 \)*TeV with the ATLAS detector*,*Phys. Rev.***D 96**(2017) 052004 [arXiv:1703.09127] [INSPIRE]. - [78]ATLAS collaboration,
*Search for heavy Majorana neutrinos with the ATLAS detector in pp collisions at*\( \sqrt{s}=8 \)*TeV*,*JHEP***07**(2015) 162 [arXiv:1506.06020] [INSPIRE]. - [79]CMS collaboration,
*Search for a heavy right-handed W boson and a heavy neutrino in events with two same-flavor leptons and two jets at*\( \sqrt{s}=13 \)*TeV*,*JHEP***05**(2018) 148 [arXiv:1803.11116] [INSPIRE]. - [80]CMS collaboration,
*A search for doubly-charged Higgs boson production in three and four lepton final states at*\( \sqrt{s}=13 \)*TeV*, CMS-PAS-HIG-16-036 (2016). - [81]ATLAS collaboration,
*Search for doubly charged Higgs boson production in multi-lepton final states with the ATLAS detector using proton-proton collisions at*\( \sqrt{s}=13 \)*TeV*,*Eur. Phys. J.***C 78**(2018) 199 [arXiv:1710.09748] [INSPIRE]. - [82]O. Naviliat-Cuncic and M. González-Alonso,
*Prospects for precision measurements in nuclear β decay at the LHC era*,*Annalen Phys.***525**(2013) 600 [arXiv:1304.1759] [INSPIRE]. - [83]M. Gonzalez-Alonso, O. Naviliat-Cuncic and N. Severijns,
*New physics searches in nuclear and neutron β decay*,*Prog. Part. Nucl. Phys.***104**(2019) 165 [arXiv:1803.08732] [INSPIRE]. - [84]CMS collaboration,
*Search for high-mass resonances in final states with a lepton and missing transverse momentum at*\( \sqrt{s}=13 \)*TeV*,*JHEP***06**(2018) 128 [arXiv:1803.11133] [INSPIRE]. - [85]ATLAS collaboration,
*Search for a new heavy gauge boson resonance decaying into a lepton and missing transverse momentum in*79*.*8*fb*^{−1}*of pp collisions at*\( \sqrt{s}=13 \)*TeV with the ATLAS experiment*, ATLAS-CONF-2018-017 (2018). - [86]M.L. Graesser,
*Broadening the Higgs boson with right-handed neutrinos and a higher dimension operator at the electroweak scale*,*Phys. Rev.***D 76**(2007) 075006 [arXiv:0704.0438] [INSPIRE]. - [87]M.L. Graesser,
*Experimental constraints on Higgs boson decays to TeV-scale right-handed neutrinos*, arXiv:0705.2190 [INSPIRE]. - [88]CMS collaboration,
*Search for displaced supersymmetry in events with an electron and a muon with large impact parameters*,*Phys. Rev. Lett.***114**(2015) 061801 [arXiv:1409.4789] [INSPIRE]. - [89]J.A. Evans and J. Shelton,
*Long-lived staus and displaced leptons at the LHC*,*JHEP***04**(2016) 056 [arXiv:1601.01326] [INSPIRE]. - [90]CMS collaboration,
*Search for new long-lived particles at*\( \sqrt{s}=13 \)*TeV*,*Phys. Lett.***B 780**(2018) 432 [arXiv:1711.09120] [INSPIRE]. - [91]CMS collaboration,
*earch for long-lived particles with displaced vertices in multijet events in proton-proton collisions at*\( \sqrt{s}=13 \)*TeV*, CMS-PAS-EXO-17-018 (2017). - [92]A. de Gouvêa,
*See-saw energy scale and the LSND anomaly*,*Phys. Rev.***D 72**(2005) 033005 [hep-ph/0501039] [INSPIRE]. - [93]C.F. Jiao, J. Engel and J.D. Holt,
*Neutrinoless double-β decay matrix elements in large shell-model spaces with the generator-coordinate method*,*Phys. Rev.***C 96**(2017) 054310 [arXiv:1707.03940] [INSPIRE]. - [94]Y. Iwata et al.,
*Large-scale shell-model analysis of the neutrinoless ββ decay of*^{48}*Ca*,*Phys. Rev. Lett.***116**(2016) 112502 [*Erratum ibid.***117**(2016) 179902] [arXiv:1602.07822] [INSPIRE]. - [95]N. López Vaquero, T.R. Rodr´ıguez and J.L. Egido,
*Shape and pairing fluctuations effects on neutrinoless double beta decay nuclear matrix elements*,*Phys. Rev. Lett.***111**(2013) 142501 [arXiv:1401.0650] [INSPIRE]. - [96]J.M. Yao et al.,
*Systematic study of nuclear matrix elements in neutrinoless double-β decay with a beyond-mean-field covariant density functional theory*,*Phys. Rev.***C 91**(2015) 024316 [arXiv:1410.6326] [INSPIRE]. - [97]J.D. Holt and J. Engel,
*Effective double-β-decay operator for*^{76}*Ge and*^{82}*Se*,*Phys. Rev.***C 87**(2013) 064315 [arXiv:1304.4202] [INSPIRE]. - [98]F. Šimkovic, V. Rodin, A. Faessler and P. Vogel, 0
*νββ and*2*νββ nuclear matrix elements, quasiparticle random-phase approximation and isospin symmetry restoration*,*Phys. Rev.***C 87**(2013) 045501 [arXiv:1302.1509] [INSPIRE]. - [99]D.-L. Fang, A. Faessler and F. Simkovic, 0
*νββ-decay nuclear matrix element for light and heavy neutrino mass mechanisms from deformed quasiparticle random-phase approximation calculations for*^{76}*Ge,*^{82}*Se,*^{130}*Te,*^{136}*Xe and*^{150}*Nd with isospin restoration*,*Phys. Rev.***C 97**(2018) 045503 [arXiv:1803.09195] [INSPIRE]. - [100]L.-J. Wang, J. Engel and J.M. Yao,
*Quenching of nuclear matrix elements for*0*νββ decay by chiral two-body currents*,*Phys. Rev.***C 98**(2018) 031301 [arXiv:1805.10276] [INSPIRE]. - [101]J. Engel and J. Menéndez,
*Status and future of nuclear matrix elements for neutrinoless double-β decay: a review*,*Rept. Prog. Phys.***80**(2017) 046301 [arXiv:1610.06548] [INSPIRE]. - [102]F.F. Deppisch, L. Graf, J. Harz and W.-C. Huang,
*Neutrinoless double beta decay and the baryon asymmetry of the universe*,*Phys. Rev.***D 98**(2018) 055029 [arXiv:1711.10432] [INSPIRE]. - [103]D. Stefanik, R. Dvornicky, F. Simkovic and P. Vogel,
*Reexamining the light neutrino exchange mechanism of the*0*νββ decay with left- and right-handed leptonic and hadronic currents*,*Phys. Rev.***C 92**(2015) 055502 [arXiv:1506.07145] [INSPIRE]. - [104]J. Kotila and F. Iachello,
*Phase space factors for double-β decay*,*Phys. Rev.***C 85**(2012) 034316 [arXiv:1209.5722] [INSPIRE]. - [105]P. Duka, J. Gluza and M. Zralek,
*Quantization and renormalization of the manifest left-right symmetric model of electroweak interactions*,*Annals Phys.***280**(2000) 336 [hep-ph/9910279] [INSPIRE]. - [106]K. Kiers, M. Assis and A.A. Petrov,
*Higgs sector of the left-right model with explicit CP-violation*,*Phys. Rev.***D 71**(2005) 115015 [hep-ph/0503115] [INSPIRE]. - [107]Y. Zhang, H. An, X. Ji and R.N. Mohapatra,
*General CP-violation in minimal left-right symmetric model and constraints on the right-handed scale*,*Nucl. Phys.***B 802**(2008) 247 [arXiv:0712.4218] [INSPIRE]. - [108]W. Dekens and D. Boer,
*Viability of minimal left-right models with discrete symmetries*,*Nucl. Phys.***B 889**(2014) 727 [arXiv:1409.4052] [INSPIRE]. - [109]L. Graf, F.F. Deppisch, F. Iachello and J. Kotila,
*Short-range neutrinoless double beta decay mechanisms*,*Phys. Rev.***D 98**(2018) 095023 [arXiv:1806.06058] [INSPIRE]. - [110]S.L. Adler et al.,
*Renormalization constants for scalar, pseudoscalar and tensor currents*,*Phys. Rev.***D 11**(1975) 3309 [INSPIRE].