PoS(NEUTEL2015)039 http://pos.sissa.it/ † ∗ ˘ A Istituto Veneto, Venice, Italy [email protected] Speaker. Currently also at: CERN, Department, Theory Division, Geneva 23 CH-1211, Switzerland. This work is I discuss dynamical generation ofground neutrino space-time geometry in plays unconventional a scenarios crucial where role.Violating and the I (ii) discuss back- Geometries two types with of Torsion.try, backgrounds: In at (i) a the Lorentz scale former M, case, may the beamong violation viewed as of neutrino a Lorentz species, catalyst symme- for which generation surviveamong and the neutrino induced masses. flavour limit oscillations In of the M latterdegrees taken of case, freedom the to correspond (totally infinity, to antisymmetric leading a components pseudoscalarfield to of axion is the) a field assumed torsion in hierarchy to four be space-time mixed,may dimensions. through exist This non-diagonal in kinetic the terms, theory with for ordinary othercouplings. axion reasons fields and The that couple torsion-ordinary-axion-field to mixing neutrinos islous with chirality graphs, responsible, changing for through Yukawa the higher-loop dynamical anoma- realised generation of in Majorana some masses. (compactified) string Theis theory latter played models, scenarios by can where the also the field be (totally strengthexists of antisymmetric) in the the torsion gravitational spin-one string antisymmetric multiplet. tensor (Kalb-Ramond) field, which ∗ † Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

supported in part by theCouncil London via the Centre Advanced for Investigator Terauniverse Grant Studies 267352 and (LCTS), by using STFC funding (UK) from under the the research European grant Research ST/L000326/1. XVI International Workshop on Neutrino Telescopes, 2-6 March 2015 Palazzo Franchetti â Nick E. Mavromatos King’s College London, Physics Department, TheoreticalGroup, Particle Strand, Physics London and WC2R Cosmology 2LS, UK E-mail: Some Novel Ways for Neutrino Mass Generation PoS(NEUTEL2015)039 Nick E. Mavromatos ], in a way consistent 5 . The right-handed Majo- R M , violates Lorentz symmetry spon- 2 ]; it can be associated (but this is only 7 ]. However, up to now, there is no experimental 4 MSM), have been proposed [ ν ]. The second proposal for dynamical generation of neu- 8 . The totally antisymmetric part of the torsion couples, via the 3 ], to be discussed in section 6 ] and string theory [ 3 , 2 ] of the at the CERN Large Hadron Collider (LHC) in 2012 con- 1 is usually considered to be much larger than the lepton or quark masses. The origin ], which necessitates the Majorana nature of the light (active) neutrinos and postu- 2 R M has been the topic of several extensions of the SM in the literature, within the framework ] and is discussed in section 9 R Until therefore such extensions of the SM are discovered, it is legitimate to search for alterna- Motivated by these facts we review in this talk alternative proposals for neutrino mass gen- The discovery [ M gravitational covariant derivative, to all fermionsthat in of a the way Lorentz that and thethe CPT-Violating pseudovector resulting SME. background interaction The with resembles the generation axial ofshall fermion review (right-handed, below, current via sterile) in chiral neutrino anomalous three-loop massesantisymmetric graphs in torsion of that neutrinos quantum interacting case field. with proceeds,an the In totally as axion four we field, space-time whose dimensions, mixing the with latter ordinary is axion represented fields, as that in turn interact with the Majorana trino masses concerns theirsion propagation [ in space-time geometries with quantum-fluctuating tor- of of quantum field theory [ tive mechanisms for neutrino mass generation, thatfor keep the the existence of spectrum right of handed SM neutrinos intact,tensions that except of are perhaps the allowed. Standard Such Model minimal, with nonthe supersymmetric three three ex- in active fact left-handed right-handed neutrinos Majorana (termed neutrinos complementing eration, through the interaction oftypes neutrinos of with backgrounds. non-trivial The backgrounds. first [ We examine two such rana mass evidence for right-handed neutrinos or foroptimism any of extension discovering of supersymmetry SM, in as a theexists matter among next of particle round fact, physicists. of although LHC some (operating at 14 TeV energies) with current cosmology. Such modelstwo are characterised of by which relatively are light right-handed almostdecoupled, neutrinos, degenerate, with with masses masses in ofright-handed the order neutrinos keV GeV, serve and range, the a whichactive purpose much neutrino may of lighter mass play generating, one, spectrum, the , almost consistentno role through with suggestions of observed seesaw for flavour warm type microscopic oscillations. dark mechanisms mechanisms,spectrum for matter. However, the in the there such generation scenarios. are The of the right-handed neutrino mass taneously. The background is ofModel a Extension type (SME) existing of in Kostelecky and theone collaborators example) so-called with [ Lorentz-Violating some (LV) string/brane Standard modelssional of the brane Universe, world in which propagates our ininteracting four with a space-time right-handed dimen- bulk neutrinos punctured [ by populations of point-like D-brane defects, lates the presence of heavy right-handed Majorana partners of mass Unconventional Neutrino Mass Generation 1. Introduction and Summary stitutes an important milestone for theAlthough Ultra-Violet the (UV) so-called completion of Higgs the mechanismmasses Standard may in Model well the (SM). explain SM, the the origin generationthe of of observed the smallness most small of of neutrino the masses light the still neutrino particle mechanism masses remains may [ an naturally open be issue. explained In through the particular, see-saw PoS(NEUTEL2015)039 ]. + 8 5 may γ (2.1) (2.3) (2.2) )[ iB µν E + , A µ D = , µ Q D-particles C The relevant case , are dimensionless . Nick E. Mavromatos B c , ,  , 2 A ∆ ) b ] to a quasi-relativistic , 2 6 c Ψ a M τ ] for leptogenesis induced + Ψ ( γ 10 ~ is used both to control the LV 2 · ], where the lightest of them (of keV 1 where 5 ∂ M M ~ , ) corresponds to the specific case b M 0 + i ,  2.1 = Ψ , and the absence of critical coupling MSM [ γ , which leads [ ~  ψ · ν c M µν  , ∆ ∂ M ~ E b i , Q , a + . We shall argue that in this model fermion + = + ! 1 3 2 ∆ µ  m 2 g g 2 a D M ∆ − M . The (tensorial) quantities 0 ] / = 1 3 ∂ ]. The model ( γ ]. i ν g g − 0 7 B γ 11 ∂ . 1

i ψ µ ,  − γ = ) [ = ∂ ~ γ 1 4 τ ~ = ∆ · 2 c Q = ∂ ~ M SME i − µν L − 0 σ γ 0 = ∂ ( ~ C i with  , , ¯ 0 Ψ 0), such that µν ∂ = ∆ > σ 2 2 c a ) is motivated by a gravitational microscopic model, based on the low-energy µν M L i is a fermion flavour doublet, with bare mass zero, and E 2.1 − ! + 1 2 5 0 and = γ ψ ψ 0 > µ

C γ a µ = , contains the LV terms, and can be expanded on a basis of gamma matrices D ∆ Ψ Q + b M When applied to right-handed neutrinos, and in cases, such as the The model to be considered in this part of the talk is defined by the two-flavour fermion The model belongs to an SME of the type The choice ( µ 1 γ = µ mass) plays the role ofwhich may dark play matter, an the important rôle four in fermion galactic interactions structure [ provide an example of self-interacting dark matter, where where A for our study corresponds to non-vanishing coefficients limit of a stringbrane theory observer view on point, a in a three bulk space-time brane punctured universe, with point-like which defects is ( embedded, from an effective three- Lagrangian by the torsion background into geometries lack of of the space. early universe, Theof which totally the we antisymmetric spin-one do torsion antisymmetric not tensor in discuss (Kalb-Ramond) such here, field cases of due corresponds the to string the gravitational field multiplet. 2. strength Neutrino Mass Generation due to propagation in Lorentz Violating Backgrounds is the (dimensionless) interaction coupling matrix.scale The and mass scale the strength of themasses and four-fermion flavour interaction oscillations are generated dynamically. dispersion relation, in the sense thatin it an is intermediate relativistic regime, in characterized both byfor the the the infrared mass generation and scale of the dynamical ultraviolet, mass. but not Unconventional Neutrino Mass Generation right-handed neutrinos via chirality changinghanded Yukawa couplings, Majorana is neutrino held mass responsible generation forlatter the through scenario right- the may aforementioned be anomalous motivated by graphs. some This string-inspired proposals [ contain any number of derivatives, includingsymmetry terms which CPT. are The either different odd coefficients orpectation associated even values with under (vev) the these discrete of terms tensors cansymmetry arise of violation, from imposed different by vacuum ranks, experiments ex- and [ should satisfy upper bounds for Lorentz constants ( C PoS(NEUTEL2015)039 ), ), + in 2 Q 2.1 2.1 (2.6) (2.5) (2.7) (2.4) φ 2 4 M = 2 . V )) reads in the ]  6 2 is the frequency, [ , which we shall ) 2.4 . ( φ ∞ ω Nick E. Mavromatos M  , . Ψ / 2 cf. 2 2 φ τ ) → ( is inversely proportional to p 2 ± Ψ τ g h M M 2 + φ φ + − 2 − 1 φ s = 2 g M ) ( h / φ , since 2 3 2 2 M g ∞ M p . 4 / + ( 4 M 2 2 − → p ! ) + ) 3 2 − 2 2 2 M g g + ) M Ψ 2 2 2 M 1 3 are then given by / g  φ / g g equation for the fermion masses 2 s 2 . θ p h ∆ − p

M 2 ( 2 1 + +  g M gap + are the eigenvalues of the coupling matrix. ( ) 1 2 1 1 we are interested in here, 2 3 2  g )( 3 g g )( q ], the coupling is proportional to the density of D-particle 2 2 g 4 γ 8  − ~ ∆ p ], this model provides an elegant construction of · 2 + ± i M 1 6 + 1 p 2 g ~ g + 2 ) g − ) 2 2 ( −  1 2 g 0 ω g +  0 leads to a ( + ) 018 1  − ωγ . 1 γ = ~ 1 0 g · ω s 2 g ( φ d ( = ( ∂  ~ ' + and the mixing angle i≡ ± − q Z 2 φ M λ ± h 1 2 0 g M 3 h γ = dp g 2 0 − ± φ 2 2 φ

009 ∂ 1 ) . p ) ( 2 g i 0 = φ g where 0, which characterises the Lorentz symmetric limit  Z d , ± = = ¯ + 3 ) s / Ψ → 2 m h 1 π − , which is used for the linearisation of the four-fermion interactions in ( ± θ 2 g = λ φ − ( dV g m 0 , from a fundamental theory, which allows the generation of the operator 2 2 1 ( ln tan ∑ L Q =+ + = s . This leads to a mass term in the original Yukawa interaction. This approach ne- + ± ) = λ h φ 2 ( ln φ ) it follows that the dynamically generated mass matrix ( V 2 4 2 ) p , and M 4 π 2.5 | d 2 ( p ~ To study dynamical generation of fermion masses, we introduce a Yukawa coupling of fermions In terms of the microscopic, string-inspired model [ The Lagrangian containing the auxiliary field, equivalent to the original Lagrangian ( Minimization 2 R ≡ | p The mass eigenvalues From ( defects, and hence its vanishingthe is defects consistent density. Hence with the the Lorentz simultaneous symmetric limit limit of that model corresponds to vanishing D-particle-defect density. to an auxiliary field consider in this work andfollows by when means the density of of which D-particles one becomes views vanishingly small. the LV as a catalyst of mass generation, (Lorentz-symmetric) limit of small couplings the LV operator a natural way. In this microscopic context, the Lorentz symmetric limit integrate over the fermions in apotential path integral, and look for a non-trivial minimum for the effective reads Integrating over fermions, we obtain the effective potential for the auxiliary field glects fluctuations of the auxiliary fieldomitted about in its the vacuum limit expectation value (vev), but these can be Unconventional Neutrino Mass Generation Non-trivial interactions of D-particles with stringycarry matter standard occur model only quantum for numbers, such andconstitute matter from that this perfect point does candidates. of not view right-handed As sterile shown neutrinos in [ i tr PoS(NEUTEL2015)039 on i (3.3) (3.1) (3.2) (2.9) (2.8) g Nick E. Mavromatos ]. 5  5 J . The presence of torsion in ? ψ  ∧ µ , , γ S bcd ) ) ) T θ θ Z ψ 2 2 µ − 3 4 ( ( , , D − ) ( abc ) ) cos cos in the form 3 θ as T , = ) ) i − − − 2 . i 2 ( g − − µ µ , − to be taken, in such a way that we are left g ψ ψ µ µ 1 5 ψ µ + + sin 5 γ ∞ cab − − γ ) = + + µ D T µ i + + ). − µ µ µ →  ( ( µ µ , µ ψγ 1 2 ( + α α ψγ 2.8 µ ψγ M + µ + ( − i  = M p p ≡ + a gS g − − − 2 2 µ √ µ can be described by the Lorentz-symmetric limit of our 5 − µ µ − ( abc − J ) leads, apart from the standard terms in manifolds without θ µ α = K √ √ + + 4 i x + + 4 g d ]: p 3.1 and are fixed by the “experimental” (in case of realistic mod- µ µ d 2 and 13 Z is the contorsion tensor. The latter is related to the torsion two- Z , ± MSM and warm dark matter studies [ 4 3 = = = µ i ν 2 m finite 1 2 3 12 ab g g g = K 3 − ψ ψ via [ S S . This expression shows the explicit dependence of the couplings b e θ , is the covariant derivative (including gravitational and gauge-field con- are kept i ∧ , where a a ... µ and ω + ab Therefore one can write the couplings ± + . µ K m a ± ∇ M + m , for the Lorentz symmetric limit µ d e = = ab M µ = ω ± a D µ , denotes the presence of torsion, which is introduced through the torsionful spin connection: = T µ Let us for concreteness consider Dirac fermions in a torsionful space-time. The extension to The model can be straightforwardly extended to Majorana fermions, as well as three genera- µ ∇ ab the Majorana case is straightforward. The relevant action reads: 3. Neutrino Mass Generation due to Interactions with Quantum Torsion where torsion, to an additional term involving the axial current ω nection parts, in casei.e. the fermions are charged). The overline above the covariant derivative, with two relativistic free fermions, for whichTherefore flavour any oscillations set have been of generated values dynamically. for model, by considering the coupling constants ( form the covariant derivative in the action ( The relevant part of the action reads: tions of fermions, including seesawpriate type couplings Lagrangians, one where can by generate,hierarchies a of in judicious relevance, the choice e.g. of Lorentz-symmetric to the limit, appro- (right-handed) neutrino mass Unconventional Neutrino Mass Generation From this we can express the dimensionless couplings where where the constants els) values of the scale PoS(NEUTEL2015)039 + ) is  (3.6) ], of (3.4) (3.5) (3.7) (3.8) b ∆ , with ] have 3.2 12 + 13 µνρ R ). q  . Hence, the g 3.4 ) + x − . To maintain σ ( S S b √ ? 2 x , and the latter via / 4 R Nick E. Mavromatos ) 1 d ) S field theory contains i ? 2 µνρσ = R S ε κ d ? 2 ( Q 1 / 3! , 1 κ 3 δ 2 bd i ( 5 − 2 / J =  1 ≡ ? effective  ) G µν ∧ 2 x µνρσ S g 5 ( 3 κ e R ρ J 2 . Φ T 2 b .  i f 1 − 2 ψ 0 at the quantum level, which can be + ψ µνρσ 5 5 d γ + R field has been integrated out. The reader µρ = ˜ J abc γ parts of the torsion tensor. ? 5 µ 2 g b 5 K S J S γ ν π ∧ ? i ? T 1 + S ψγ d ∧ ψ µνρ d 3 4 1 3 b q 192 S d − abcd b − = 1 f ε S , with the hatted notation defined in ( ? abcd µ c . λ ε µν e + ∧ . e F µνρ 2 1 1 4 ∑ b S ) µλ abc T d 2 T b ? µν − K ω = d 3 κ , fermion species ν F ∧ 4 = = 2 A abc b i abc 2 µν ( ε d π Z e ab b K and the tensor i K G 8 2 1 ≡ 1 3! µ h µ µ K − 5 Z T ≡ = = J i λ exp µ d and the non-propagating − 5 S νµ J h π : b Db P 0, leading to a conserved “torsion charge” K µ 3 T S M √ ∇ µν exp = D λ = b S K Z ? Db 2 repulsive four-fermion axial current-current interaction, characteristic of any / d ∝ = 1 0. This implies that the contorsion tensor undergoes the following decompo- Z ]. − b ∆ Z = ) = 12 8 / ρν ν 2 is the dual of q κ where 3 T = , ? is the pseudoscalar axial vector: S ]: ? includes the trace vector = ( = d b νρσ ∧ b 14 S K S f q S The gravitational part of the action can then be written as: The torsion term, being geometrical, due to , couples universally to all fermion species, We next remark that the torsion tensor can be decomposed into its irreducible parts [ In a quantum gravity setting, where one integrates over all fields, the torsion terms appear R with the axial current, since the torsion equation yields 2 non-renormalizable µ 3 κ µνρσ 4 relevant torsion part of the quantum-gravity path integral would include a factor the well-known trick of introducing a Lagrange multiplier field where From this it follows this conservation in quantum theory,achieved by they the postulated addition of judiciousof counter quantum terms. gravity, is This then constraint, implemented in a via path-integral a formulation delta function constraint, a expressed as a sum over fermion species In theories with chiral anomalies, like thenot conserved quantum at electrodynamics the part quantum of level, SM, due theresult to axial [ anomalies, current but is its divergence is obtained by the one-loop torsionful theory [ not only neutrinos. Thus, in the context of the SM of , the axial current ( should notice that, as a result of this integration, the corresponding which observed though that the classical equationsS of motion identify the axial-pseudovector torsion field as non propagating fields and thus they can be integrated out exactly. The authors of [ Unconventional Neutrino Mass Generation where ε sition: where PoS(NEUTEL2015)039 , c µνρ (3.9) ) and H ) + x . These ( 3.6 ], which b µν e 13 F is the ordi- → ) µν µ ρν x ) one can then ( Γ cF b denotes antisym- 3.6 = ] Nick E. Mavromatos . and i µ νρ ... [ 5 Γ ) in ( J ? ]. The proposed coupling 3.8 µνρσ ∧ . In string-inspired models, e R 17 5 ) x J where ( 2 b , a that characterises all string the- f by a constant: 1 µνρσ ] 2 ) ν µ ρν x ( cR . Using ( Γ Θ ) ) the torsion-free spin connection has b µ ) + [ ω ∂ ω 6= , 3.8 , A + A ( , where the symbol µ ( ] νρ G µν H ], and reviewed here, consists of augmenting νρ B bG 3 9 B to κ b √ 1 f µ is the gravitational constant. In four space-time ) [ → ]. Today of course any torsion background should ∂ + ω κ − , 10 µν = b µ νρ A B d ( ? Γ G µνρ ∧ = b H d , of relevance to the low-energy (field-theory) limit of string 0 1 2 µ νρ α torsion, where Latin indices denote spatial components of the to the fermionic matter discussed above is its shift symmetry, Γ Z ) i i jk x ]) field ( to another pseudoscalar axion field H − may be provided by the string moduli [ b 15 ) h ) x ] that the terms of the effective action up to and including quadratic x ( ( appear: b ]. exp 16 a ) by terms that break such a shift symmetry. To illustrate this last point, we 10 µ νρ ). The reader should observe that in ( Db Γ 3.9 Z of the torsion is identified with the field strength of the spin-one antisymmetric 3.8 µ S ) only changes by total derivative terms, such as 3.9 above. For completeness we mention at this point that background geometries with (approx- b In what follows we shall consider the effects of the quantum fluctuations of torsion, which terms are irrelevant for thefields equations fall of off motion sufficiently and fast the torana induced zero mass quantum at generation dynamics, space-time through provided infinity. torsion the The proposed scenario in for the [ anomalous Majo- occurs through a mixing in the kinetic terms of the two fields. To be specific, we consider the action nary, torsion-free, symmetric connection, and four-dimensional space-time, may characterisebackground the constitutes early extra universe. sourceBaryon-minus-Lepton-number In of (B-L) such CP conserving cases, processes, violation, the Baryogenesis,matter-antimatter necessary asymmetry H-torsion and in for thus the Universe lepotogenesis, the [ observed and through the effective action ( first couple the KR axion such pseudoscalar axion dimensions, the dual of the H-fieldfield is indeed the derivative ofimately) an constant axion-like background field, analogous to the be strongly suppressed, due tosuch the cosmologies lack can of evolve any sotoday experimental as can evidence be to for found guarantee it. in the [ Scenarios absence as of to any how appreciable traces ofsurvive torsion the absence of any torsion(or background. KR An axion) important quantum aspect of field characteristic the of coupling an of axion the field. torsion Indeed, by shifting the field order in the Regge slope parameter take into account ( been used. This can becan achieved by convert the the addition anomaly of fromobtain proper the for counter the initial terms effective torsion in action the inSM: action theories [ with chiral anomalies, such as the QED part of the ories. It can be shown [ theory, which involve the H-fieldChristoffel strength, symbols, can be assembled in such a way that only torsionful Unconventional Neutrino Mass Generation We may then partially integrate the second term in the exponent on the right-hand-side of ( A concrete example of torsion is provided byric string-inspired component theories, where the totally antisymmet- metrization of the appropriate indices.as a The consequence string of theory the “gauge effective symmetry” action depends only on the action ( tensor (Kalb-Ramond (KR) [ PoS(NEUTEL2015)039 , , ) . 2 2 γ 2 γ ) µνρσ (3.12) (3.11) (3.13) (3.10) − a e R − µ 1 1 ∂ , coupled ( ) p x is the axial p b 1 2 µνρσ f ( / a 2 R a are related. ψ . We may now y π 5 ) + 1 P γ . a ) becomes: µ M µ 192 Nick E. Mavromatos ∂ the shift symmetry: i ( ψγ . C / R 3.10 )( to right-handed sterile γ ) b = ψ 2 a . Moreover, we remind is a real parameter to be µ R 5 γ µ ) ∂ µνρσ break J γ x 6 ψ ( e ( − R , γ Λ a 1 R − 5 ( + R ψ 4 . , and µ µνρσ ψ π γ κ with the wrong sign of its kinetic i 5 C R C a µ 8 ) R J 2 y 5 R 5 µ γ i.e. ψ √ J 3 J ψ 5 µ  − 2 b J √ 2 f ], the gravitationally induced Majorana 1 1 2 ) ia ) b + ( f 2 a x a 1 p 18 ( R y 2 µ 49152 b a + f ∂ ψ γ − + ( 2 and the Planck mass scale = π  =  Λ ) 2 C 2 R µνρσ γ ψ γ of the axion moduli field 192 µνρσ ψ e R 6 e R − a R − − y Λ 1 ψ 2 1 4  ) ( − µνρσ a b for the pseudoscalar field 1 2 f µνρσ R µ R torsion, with a modified coupling γ κ a 2 ∂ R ψ b + a ( f ) π C y M 2 R 1 2 ) x 2 ) ( x 0 h b . This implies that the effective action ( ψ à la π , ( f , is estimated to be: a ) b g  192 b x 2 R γ µ a ( − coupled both to matter fermions and to the operator 2 ∂ π i 192 M 2 a − ) ( √ ) C ) corresponds to a canonically normalised axion field γ γ 2 ) R x 1 2 x + x a 4 ( π 1 field has decoupled and can be integrated out in the path integral, 192 − h ψ ( 2 d a 0 iy R 1 ) 0 g self-interaction fermion terms are due to the existence of torsion in the ) + 16 3.12 b b b x Z ( ψ − ( p µ + b = ∂ in ( √ − − ∼ ( µ a x a R ≡ 5 R 4 2 1 S is the charge-conjugate right-handed fermion J ψ ) d h S M 5 µ x may be due to non perturbative effects. These terms C g R C ( J repulsive Z 0 ) R 2 − b ψ b R f ψ 1 =  ψ √ 2 x → ia 4 otherwise the axion field would appear as a ghost, S + a ) , d = ( y x ( 1 . C − Z b c R < ψ + = to the curvature of space-time, | . Adopting the effective field-theory framework of [ a The mechanism for the anomalous Majorana mass generation is shown in Fig. It is convenient to diagonalize the axion kinetic terms by redefining the KR axion field as γ Λ | S → . In this case we may redefine the axion field so as to appear with a canonical normalised kinetic Thus we observe thatleaving behind the an axion field thereby playing now the rôle offor the torsion field. We observeterms, though which that would the indicate approach an is instabilityγ only of valid the model. This is theterm, only implying restriction the of effective the action: parameter both and to fermionic matter with chirality-changing Yukawa-like couplings of the form estimate the two-loop Majorana neutrino mass inoff quantum gravity with an effective UV energymass cut- for right-handed neutrinos, In a UV complete theory such as strings, the cutoff Unconventional Neutrino Mass Generation (henceforth we restrict ourselves to right-handed Majorana neutrino fermion fields) where current of the four-component Majorana fermion a follows: Einstein-Cartan theory. The Yukawa coupling constrained later on. Here,possibility we have of ignored a gauge non-perturbative fields, mass the which reader are that not the of interest to us, and the neutrino matter Evidently, the action PoS(NEUTEL2015)039 ]. ]. 1. . 5 , 0 . In 17 GeV, 1 (3.14) = , but if 16 6 γ > ]. MSM [ n 10 ν Λ ], which for 19 and = 3 . This is in the 20 Λ 3 − − , i.e. ]. The black circle 10 9 Λ 3. Nick E. Mavromatos 10 . = 6 > = a  n y γ = for ) to be GeV a 3 y 3 18 − of the axion fields and not on − denote graviton fluctuations. n Λ n ) 3.13 ) 2 10 a 2 a 2 a µν M at the two-loop level. Of course, × M M δ ( ( 4 δ R . ) 2 , 30 (2012) [arXiv:1207.7235 [hep-ex]]. 3 2 M ) γ 2 a −  M 1 716 , 1 (2012) [arXiv:1207.7214 [hep-ex]]; ( δ 1 4 ( with next-to-neighbour mixing as discussed ], or in general characterises the − 5 π γ 3 8 716 , 10 to qualify as a warm dark matter [ γ κ 10 2 √ is at the TeV scale, for , a 1 y R R a 3 ψ   √ M 49152 3 16 keV, for the choice a − y ∼ ∼ 10 R R may be estimated from (  3. As a final comment we mention that the values of the ) M M R ≤ M n GeV induced by torsion. The fields h is highly model dependent. Taking, for example, the quantum 11 3, and , 421 (1977); M. Gell-Mann, P. Ramond and R. Slansky, in R 10 67 ≤ µνλρ M e R × n GeV, we find that Workshop on the Unified Theory and the Baryon Number in the Universe 1 . will depend on higher powers of the energy cut-off 17 3 for ( R µνλρ 1, the size of 10 [CMS Collaboration], Phys. Lett. B R n M ∼ ) ) may be determined by some underlying discrete symmetry [ ) 2 a  x = R 2 ( M a γ -axion-mixing scenarios, we note that the anomalously generated Majorana γ M y δ Λ − n ( et al. 1 n [ATLAS Collaboration], Phys. Lett. B ( 2 , eds. D.Z. Freedman and P. van Nieuwenhuizen (North-Holland, Amsterdam, 1979); T. 4 − π 6 8 Λ 5 √ et al. γ κ a Typical Feynman graph giving rise to anomalous fermion mass generation [ y 49152 3 1, these higher-order effects are expected to be subdominant. √ Supergravity Yanagida, in Proc. of the S. Chatrchyan In a string-theoretic framework, many axions might exist that could mix with each other [ In the above It is then not difficult to see that three axions It is interesting to provide a numerical estimate of the anomalously generated Majorana mass  ∼ . Assuming that R R Λ [2] P. Minkowski, Phys. Lett. B [1] G. Aad above would be sufficient to obtain a UV finite result for mass term will only depend on the mass-mixing parameters preferred ballpark region for the sterile neutrino Such a mixing can givesuch rise cases, to the reduced anomalously UV generated Majorana sensitivityM mass of may the be two-loop estimated graph to be: shown in Fig. beyond the two loops, κ their masses themselves, as longYukawa as couplings instance allows two of the right-handedenhanced neutrinos CP violation to of be relevance almost to degenerate leptogenesisThese in [ are mass, interesting as issues required that for deserve further exploration. 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