Abstract
In the Standard Model minimally extended to include massive neutrinos, we compute the leading CP-violating zero temperature contributions to the one-loop effective action induced by integration of the leptons. Such contributions start at operators of dimension six and they are P even for Dirac neutrinos and P even or odd for Majorana neutrinos. Dimension four operators are allowed in the mixed Dirac-Majorana case. It is verified by explicit calculation that CP can be violated in two generation settings for Majorana neutrinos. Using different neutrino scenarios we give upper bounds for the couplings of the CP-violating operators. As a rule, we find that lepton-induced couplings are suppressed as compared to quark-induced couplings, whenever the latter are allowed, nevertheless, through virtual lepton-number violating mechanisms, Majorana neutrinos induce new CP-violating operators not present in the quark or Dirac-neutrino cases.
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References
R.N. Mohapatra et al., Theory of neutrinos: A White paper, Rept. Prog. Phys. 70 (2007) 1757 [hep-ph/0510213] [INSPIRE].
J. Bernabeu, A. De Rujula and C. Jarlskog, Neutrinoless Double Electron Capture as a Tool to Measure the ν e Mass, Nucl. Phys. B 223 (1983) 15 [INSPIRE].
S.R. Elliott and P. Vogel, Double beta decay, Ann. Rev. Nucl. Part. Sci. 52 (2002) 115 [hep-ph/0202264] [INSPIRE].
J.G. Morfin, J. Nieves and J.T. Sobczyk, Recent Developments in Neutrino/Antineutrino — Nucleus Interactions, Adv. High Energy Phys. 2012 (2012) 934597 [arXiv:1209.6586] [INSPIRE].
M.C. Gonzalez-Garcia and M. Maltoni, Phenomenology with Massive Neutrinos, Phys. Rept. 460 (2008) 1 [arXiv:0704.1800] [INSPIRE].
G. Bertone, D. Hooper and J. Silk, Particle dark matter: Evidence, candidates and constraints, Phys. Rept. 405 (2005) 279 [hep-ph/0404175] [INSPIRE].
F. Halzen and D. Hooper, High-energy neutrino astronomy: The Cosmic ray connection, Rept. Prog. Phys. 65 (2002) 1025 [astro-ph/0204527] [INSPIRE].
T. Schwetz, M.A. Tortola and J.W.F. Valle, Three-flavour neutrino oscillation update, New J. Phys. 10 (2008) 113011 [arXiv:0808.2016] [INSPIRE].
J. Schechter and J.W.F. Valle, Neutrino Masses in SU(2) × U(1) Theories, Phys. Rev. D 22 (1980) 2227 [INSPIRE].
J. Schechter and J.W.F. Valle, Neutrino Decay and Spontaneous Violation of Lepton Number, Phys. Rev. D 25 (1982) 774.
KAMIOKANDE-II collaboration, K. Hirata et al., Observation of a Neutrino Burst from the Supernova SN 1987a, Phys. Rev. Lett. 58 (1987) 1490 [INSPIRE].
R.M. Bionta et al., Observation of a Neutrino Burst in Coincidence with Supernova SN 1987a in the Large Magellanic Cloud, Phys. Rev. Lett. 58 (1987) 1494 [INSPIRE].
WMAP collaboration, G. Hinshaw et al., Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results, Astrophys. J. Suppl. 208 (2013) 19 [arXiv:1212.5226] [INSPIRE].
D.D. Stancil et al., Demonstration of Communication using Neutrinos, Mod. Phys. Lett. A 27 (2012) 1250077 [arXiv:1203.2847] [INSPIRE].
Z.-z. Xing, Flavor mixing and CP-violation of massive neutrinos, Int. J. Mod. Phys. A 19 (2004) 1 [hep-ph/0307359] [INSPIRE].
A.J. Buras and R. Fleischer, Quark mixing, CP-violation and rare decays after the top quark discovery, Adv. Ser. Direct. High Energy Phys. 15 (1998) 65 [hep-ph/9704376] [INSPIRE].
M. Neubert, B decays and CP-violation, Int. J. Mod. Phys. A 11 (1996) 4173 [hep-ph/9604412] [INSPIRE].
Y. Grossman, Y. Nir and R. Rattazzi, CP violation beyond the standard model, Adv. Ser. Direct. High Energy Phys. 15 (1998) 755 [hep-ph/9701231] [INSPIRE].
B. Winstein and L. Wolfenstein, The Search for direct CP-violation, Rev. Mod. Phys. 65 (1993) 1113 [INSPIRE].
E.A. Paschos and U. Turke, Quark Mixing and CP-violation, Phys. Rept. 178 (1989) 145 [INSPIRE].
L. Wolfenstein, Present Status of CP-violation, Ann. Rev. Nucl. Part. Sci. 36 (1986) 137 [INSPIRE].
J.F. Donoghue, B.R. Holstein and G. Valencia, Survey of present and future tests of CP violation, Int. J. Mod. Phys. A 2 (1987) 319 [INSPIRE].
CKMfitter Group collaboration, J. Charles et al., CP violation and the CKM matrix: assessing the impact of the asymmetric B factories, Eur. Phys. J. C 41 (2005) 1 [hep-ph/0406184] [INSPIRE].
J.H. Christenson, J.W. Cronin, V.L. Fitch and R. Turlay, Evidence for the 2 pi Decay of the k(2)0 Meson, Phys. Rev. Lett. 13 (1964) 138 [INSPIRE].
NA31 collaboration, H. Burkhardt et al., First Evidence for Direct CP-violation, Phys. Lett. B 206 (1988) 169 [INSPIRE].
BaBar collaboration, B. Aubert et al., Observation of CP-violation in the B 0 meson system, Phys. Rev. Lett. 87 (2001) 091801 [hep-ex/0107013] [INSPIRE].
A.D. Sakharov, Violation of CP Invariance, c Asymmetry and Baryon Asymmetry of the Universe, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32 [INSPIRE].
M. Trodden, Electroweak baryogenesis, Rev. Mod. Phys. 71 (1999) 1463 [hep-ph/9803479] [INSPIRE].
T. Ibrahim and P. Nath, CP Violation From Standard Model to Strings, Rev. Mod. Phys. 80 (2008) 577 [arXiv:0705.2008] [INSPIRE].
M. Pospelov and A. Ritz, Electric dipole moments as probes of new physics, Annals Phys. 318 (2005) 119 [hep-ph/0504231] [INSPIRE].
H.-Y. Cheng, The Strong CP Problem Revisited, Phys. Rept. 158 (1988) 1 [INSPIRE].
M. Kobayashi and T. Maskawa, CP Violation in the Renormalizable Theory of Weak Interaction, Prog. Theor. Phys. 49 (1973) 652 [INSPIRE].
B. Pontecorvo, Mesonium and anti-mesonium, Sov. Phys. JETP 6 (1957) 429 [INSPIRE].
B. Pontecorvo, Neutrino Experiments and the Problem of Conservation of Leptonic Charge, Sov. Phys. JETP 26 (1968) 984 [INSPIRE].
Z. Maki, M. Nakagawa and S. Sakata, Remarks on the unified model of elementary particles, Prog. Theor. Phys. 28 (1962) 870 [INSPIRE].
Particle Data Group collaboration, J. Beringer et al., Review of Particle Physics (RPP), Phys. Rev. D 86 (2012) 010001 [INSPIRE].
RENO collaboration, J.K. Ahn et al., Observation of Reactor Electron Antineutrino Disappearance in the RENO Experiment, Phys. Rev. Lett. 108 (2012) 191802 [arXiv:1204.0626] [INSPIRE].
Double CHOOZ collaboration, Y. Abe et al., Reactor electron antineutrino disappearance in the Double CHOOZ experiment, Phys. Rev. D 86 (2012) 052008 [arXiv:1207.6632] [INSPIRE].
T2K collaboration, K. Abe et al., Evidence of Electron Neutrino Appearance in a Muon Neutrino Beam, Phys. Rev. D 88 (2013) 032002 [arXiv:1304.0841] [INSPIRE].
Daya Bay collaboration, F.P. An et al., Improved Measurement of Electron Antineutrino Disappearance at Daya Bay, Chin. Phys. C 37 (2013) 011001 [arXiv:1210.6327] [INSPIRE].
M.E. Shaposhnikov, Structure of the High Temperature Gauge Ground State and Electroweak Production of the Baryon Asymmetry, Nucl. Phys. B 299 (1988) 797 [INSPIRE].
M. Dine, P. Huet, J. Singleton, Robert L. and L. Susskind, Creating the baryon asymmetry at the electroweak phase transition, Phys. Lett. B 257 (1991) 351 [INSPIRE].
X. Zhang and B.L. Young, Effective Lagrangian approach to electroweak baryogenesis: Higgs mass limit and electric dipole moments of fermion, Phys. Rev. D 49 (1994) 563 [hep-ph/9309269] [INSPIRE].
A. Lue, K. Rajagopal and M. Trodden, Semianalytical approaches to local electroweak baryogenesis, Phys. Rev. D 56 (1997) 1250 [hep-ph/9612282] [INSPIRE].
J. García-Bellido, D.Y. Grigoriev, A. Kusenko and M.E. Shaposhnikov, Nonequilibrium electroweak baryogenesis from preheating after inflation, Phys. Rev. D 60 (1999) 123504 [hep-ph/9902449] [INSPIRE].
A. Tranberg and J. Smit, Baryon asymmetry from electroweak tachyonic preheating, JHEP 11 (2003) 016 [hep-ph/0310342] [INSPIRE].
J. Smit, Effective CP-violation in the standard model, JHEP 09 (2004) 067 [hep-ph/0407161] [INSPIRE].
C. Jarlskog, Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP-violation, Phys. Rev. Lett. 55 (1985) 1039 [INSPIRE].
G.R. Farrar and M.E. Shaposhnikov, Baryon asymmetry of the universe in the minimal Standard Model, Phys. Rev. Lett. 70 (1993) 2833 [Erratum ibid. 71 (1993) 210] [hep-ph/9305274] [INSPIRE].
A. Hernandez, T. Konstandin and M.G. Schmidt, Sizable CP-violation in the Bosonized Standard Model, Nucl. Phys. B 812 (2009) 290 [arXiv:0810.4092] [INSPIRE].
C. Garcia-Recio and L.L. Salcedo, CP violation in the effective action of the Standard Model, JHEP 07 (2009) 015 [arXiv:0903.5494] [INSPIRE].
T. Brauner, O. Taanila, A. Tranberg and A. Vuorinen, Temperature Dependence of Standard Model CP-violation, Phys. Rev. Lett. 108 (2012) 041601 [arXiv:1110.6818] [INSPIRE].
L.L. Salcedo, Leading order one-loop CP and P violating effective action in the Standard Model, Phys. Lett. B 700 (2011) 331 [arXiv:1102.2400] [INSPIRE].
T. Brauner, O. Taanila, A. Tranberg and A. Vuorinen, Computing the temperature dependence of effective CP-violation in the standard model, JHEP 11 (2012) 076 [arXiv:1208.5609] [INSPIRE].
A. Tranberg, A. Hernandez, T. Konstandin and M.G. Schmidt, Cold electroweak baryogenesis with Standard Model CP-violation, Phys. Lett. B 690 (2010) 207 [arXiv:0909.4199] [INSPIRE].
T. Konstandin and G. Servant, Natural Cold Baryogenesis from Strongly Interacting Electroweak Symmetry Breaking, JCAP 07 (2011) 024 [arXiv:1104.4793] [INSPIRE].
T. Yanagida, Horizontal Symmetry and Masses of Neutrinos, Prog. Theor. Phys. 64 (1980) 1103 [INSPIRE].
M. Gell-Mann, P. Ramond and R. Slansky, Complex Spinors and Unified Theories, Conf. Proc. C 790927 (1979) 315 [arXiv:1306.4669] [INSPIRE].
T. Morii, C. Lim and S. Mukherjee, The physics of the standard model and beyond, World Scientific, Singapore (2002).
C. Giunti and C.W. Kim, Fundamentals of Neutrino Physics and Astrophysics, Oxford University Press, Oxford U.K. (2007).
L.L. Salcedo, The Invariant factor of the chiral determinant, Eur. Phys. J. C 58 (2008) 423 [arXiv:0807.1696] [INSPIRE].
N.G. Pletnev and A.T. Banin, Covariant technique of derivative expansion of one loop effective action. 1., Phys. Rev. D 60 (1999) 105017 [hep-th/9811031] [INSPIRE].
L.L. Salcedo, The method of covariant symbols in curved space-time, Eur. Phys. J. C 49 (2007) 831 [hep-th/0606071] [INSPIRE].
L. Álvarez-Gaumé and E. Witten, Gravitational Anomalies, Nucl. Phys. B 234 (1984) 269 [INSPIRE].
S.L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177 (1969) 2426 [INSPIRE].
J.S. Bell and R. Jackiw, A PCAC puzzle: π 0 → γγ in the σ-model, Nuovo Cim. A 60 (1969) 47 [INSPIRE].
W.A. Bardeen, Anomalous Ward identities in spinor field theories, Phys. Rev. 184 (1969) 1848 [INSPIRE].
J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95 [INSPIRE].
E. Witten, Global Aspects of Current Algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE].
L.L. Salcedo, Derivative expansion for the effective action of chiral gauge fermions: The Normal parity component, Eur. Phys. J. C 20 (2001) 147 [hep-th/0012166] [INSPIRE].
L.L. Salcedo, Derivative expansion for the effective action of chiral gauge fermions. the abnormal parity component, Eur. Phys. J. C 20 (2001) 161 [hep-th/0012174] [INSPIRE].
L.L. Salcedo, Direct construction of the effective action of chiral gauge fermions in the anomalous sector, Eur. Phys. J. C 60 (2009) 387 [arXiv:0804.2118] [INSPIRE].
J.W. Negele and H. Orland, Frontiers in physics. Vol. 68: Quantum many particle systems, Addison-Wesley, Redwood City U.S.A. (1988).
J.S. Dowker and R. Critchley, Effective Lagrangian and Energy Momentum Tensor in de Sitter Space, Phys. Rev. D 13 (1976) 3224 [INSPIRE].
S.W. Hawking, Zeta Function Regularization of Path Integrals in Curved Space-Time, Commun. Math. Phys. 55 (1977) 133 [INSPIRE].
E. Elizalde, S. Odintsov, A. Romeo, A. Bytsenko and S. Zerbini, Zeta regularization techniques with applications, World Scientific, Singapore (1994).
R.D. Ball, Chiral Gauge Theory, Phys. Rept. 182 (1989) 1 [INSPIRE].
K. Huang, Quarks, leptons & gauge fields, World Scientific, Singapore (1992).
F.J. Moral-Gamez and L.L. Salcedo, Derivative expansion of the heat kernel at finite temperature, Phys. Rev. D 85 (2012) 045019 [arXiv:1110.6300] [INSPIRE].
R.I. Nepomechie, Calculating heat kernels, Phys. Rev. D 31 (1985) 3291 [INSPIRE].
L.L. Salcedo and E. Ruiz Arriola, Wigner transformation for the determinant of Dirac operators, Annals Phys. 250 (1996) 1 [hep-th/9412140] [INSPIRE].
K.N. Abazajian et al., Cosmological and Astrophysical Neutrino Mass Measurements, Astropart. Phys. 35 (2011) 177 [arXiv:1103.5083] [INSPIRE].
Planck collaboration, P.A.R. Ade et al., Planck 2013 results. XVI. Cosmological parameters, Astron. Astrophys. (2014) [arXiv:1303.5076] [INSPIRE].
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García-Recio, C., Salcedo, L.L. Leptonic CP violating effective action for Dirac and Majorana neutrinos. J. High Energ. Phys. 2014, 156 (2014). https://doi.org/10.1007/JHEP08(2014)156
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DOI: https://doi.org/10.1007/JHEP08(2014)156