Pos(CORFU2017)003 CDM Λ CDM Model, Which Agree with a Plethora of Cosmolog- Λ † ∗ [email protected] Speaker

PoS(CORFU2017)003 CDM Λ https://pos.sissa.it/ CDM model, which agree with a plethora of cosmolog- Λ † ∗ [email protected] Speaker. This work is supported in part by STFC (UK) under the research grant ST/P000258/1 model. If such scenariosmore are than realised one in species nature, (warm theyUniverse: and might cold), the with imply cold distinct that one rôles Darkaccordance in being Matter to the still consists structure the responsible of and predictions for evolution ofical of the the the observations, large but scale the structure warm(observed) of galactic component the structure. (keV Universe, sterile in neutrino) playing a crucial rôle in the We discuss a novel mechanism forgoes generating beyond masses the for conventional seesaw. right-handed Majorana Theless mechanism neutrinos, Kalb-Ramond involves (KR) that quantum pseudoscalar (axion-like) fluctuations field, of which a existsof in mass- the string-inspired extensions standard model.general) We axions, assume which a also kinetic exist in mixing(right-handed) string of models, neutrinos the and via which KR non-perturbatively-generated are chirality field assumed changingbreaking with to Yukawa the couple ordinary couplings, axion-shift to (massive symmetry. Majorana No in vacuum expectationMajorana value masses is assumed for for the the right-handed axionlous neutrinos fields. higher-loop are axion-neutrino generated couplings. radiatively Implications asdiscussed. for a In the result particular, we Dark of explore sector anoma- the of possibilitytrinos of the of generating Universe masses order are for of the a right-handed fewstructure, neu- and tens more of generally keV. they Such could neutrinos serveproviding could as a a play potential warm an dark resolution important matter to rôle component theancies in in so-called the between the small-scale Universe, observations galactic cosmology at “crisis”, galactic that scales is, discrep- and numerical simulations based on the ∗ † Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Corfu Summer Institute 2017 ’School and2-28 Workshops September on 2017 Elementary Particle Physics andCorfu, Gravity’ Greece Nick E. Mavromatos King’s College London, Physics Department, TheoreticalGroup, Particle Strand, Physics London and WC2R Cosmology 2LS, UK E-mail: Axions, Majorana neutrino masses and implications for the dark sector of the Universe PoS(CORFU2017)003 , which in terms Nick E. Mavromatos µνρ ], in a way consistent H 5 . The right-handed Majo- R M , an alternative proposal for right- 2 ]. However, up to now, there is no experimental 4 1 MSM), have been proposed [ ], through the anomalous interaction of these neu- ] to play an important rôle in the galactic structure, ν 7 6 CDM-model-based simulations for the dark matter dis- Λ ] and string theory [ 3 , 2 ] of the Higgs boson 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 R Motivated by these facts we review in the next section Until therefore such extensions of the SM are discovered, it is legitimate to search for alterna- The discovery [ M of the (lowest order in derivatives -tally we antisymmetric restrict part ourselves of to) a stringto torsion. effective action the The appear latter as axial a couples, fermion a via current,handed to- the neutrinos. summed gravitational The covariant up derivative, generation over of allas (right-handed, we fermion sterile) shall neutrino species review masses below, in via in thetotally chiral that antisymmetric model, anomalous case three-loop torsion proceeds, including graphs quantum right- of KRresented neutrinos field. interacting as with an In the axion four field, space-time whose dimensions, (kinetic) the mixing latter with is ordinary rep- axion fields, that in turn interact and more general in providing adiscrepancies resolution between to observations the and so-called “small-scaletribution in cosmology crisis”, galaxies. that The is right-handedtype neutrinos mechanisms, serve the the active neutrino purpose mass ofHowever, spectrum, generating, consistent there through with are seesaw observed flavour no oscillations. suggestionshanded neutrino for mass microscopic spectrum in mechanisms such for scenarios. the generation of the right- trinos with axion-like field thatmodels contain exist (in in their string-inspired massless extensionsRamond) gravitational field, of multiplet) and the a there standard is spin-one an model.through antisymmetric abelian its tensor gauge These (three-rank) (Kalb- invariance covariant, that totally implies antisymmetry, the tensor presence field of strength the latter only 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 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, with degenerate, masses with in masses the of keVneutrino range, warm order which dark GeV, may and matter play has a the been much role argued lighter of [ warm one, dark almost matter. The keV handed Majorana neutrino mass generation [ lates the presence of heavy right-handed Majorana partners of mass Axions and massive Majorana neutrinos 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(CORFU2017)003 +  ], of (2.3) (2.4) (2.1) (2.2) b ∆ 8 , with + . 4 µνρ R ). q  g 2.4 + − σ S √ x 4 Nick E. Mavromatos d µνρσ R ε 2 1 3! 1 κ 2 5 −  . The presence of torsion in J ? =
 ψ ∧ µ G µν bcd γ S S g ) T ρ Z ψ T − µ 3 4 − D − abc ( µρ T abc parts of the torsion tensor. = g b − K . − ν ψ T ψ + 5 ψ µνρ µ cab d 5 γ 1 3 q S γ T µ D µ  µ = 1 2 2 , with the hatted notation defined in ( ψγ abcd . λ ψγ µ ε ψγ = ]. Our conclusions are presented in section µνρ 2 1  ≡ 6 µλ abc T gS , where it is argued that sterile neutrinos with masses g T b K µ abc 3 5 = d − − J ν K ) leads, apart from the standard terms in manifolds without √ is the contorsion tensor. The latter is related to the torsion √ abc abc µν 4 ]: x ε µ 4 b d 9 K and the tensor 2.1 K 1 3! , d ab µ 8 Z − K T = Z 4 3 λ i d 2 S νµ via [ : b = K b 3 − T e ψ ψ µν , where λ S ∧ S µ b a K , is the covariant derivative (including gravitational and gauge-field con- ab ω = K 0. This implies that the contorsion tensor undergoes the following decompo- + ... b ∆ + a = + µ µ d e ρν ab ν is the dual of q ∇ ω = where T a = , = = ? is the pseudoscalar axial vector: T S µ µ ? includes the trace vector = d ab νρσ ∧ b D , denotes the presence of torsion, which is introduced through the torsionful spin connec- S S K q µ ω S We next remark that the torsion tensor can be decomposed into its irreducible parts [ The gravitational part of the action can then be written as: Let us commence our discussion by considering first a rather generic discussion concerning The relevant action reads: ∇ R 2 3 κ µνρσ 4 torsion, to an additional term involving the axial current where where tion: two-form the propagation of Dirac fermions in astraightforward. torsionful We space-time. shall The restrict extension ourselves to the(totally to Majorana the antisymmetric) case specific is torsion case is of providedon. interest to by us the here, antisymmetric in tensor which the Kalb-Ramond field later 2. Axions and radiatively generated Majorana neutrino masses where nection parts, in casei.e. the fermions are charged). The overline above the covariant derivative, the covariant derivative in the action ( The relevant part of the action reads: which of a few tens ofservations keV for the can galactic play structure, an andso-called important more “small-scale” generally rôle cosmology for “crisis” in alleviating [ providing some agreement of between the tensions theory of and the ob- Axions and massive Majorana neutrinos with the Majorana right-handed neutrinos via chiralitysible changing for Yukawa the couplings, right-handed is Majorana held neutrino respon- masscircumstances generation. low We (in discuss the specifically under keV which can range) be masses generated, for which the may lightest haveUniverse. important of implications the These for sterile are the right-handed discussed (warm) neutrinos dark in matter section sector of the ε sition: PoS(CORFU2017)003 ) is (2.6) with (2.5) (2.7) (2.9) (2.8) ) and 2.2 µ S 2.6 ], which 9 . Hence, the ) x ) one can then . To maintain ( ] have observed S b 9 ? 2 2.6 , and the latter via / R Nick E. Mavromatos ) 1 . ) S field theory contains i ? i 2 = 5 S κ d ) in ( ? J ( Q / , ? 3 δ bd i 2.8 ( ∧ 5 2 / J 5 1 ≡ ? J effective  ) 2 b ∧ 2 x µνρσ f 1 5 ( 3 κ e R 2 J 2 . Φ 2 b .  i . Using ( f 1 ) ) the torsion-free spin connection has 2 ψ ) + 0 at the quantum level, which can be + ψ µνρσ 5 ω 5 d ω γ + , R field has been integrated out. The reader 2.8 = ˜ J , γ ? 5 µ 2 A 5 S J S A γ ( π ∧ ? i ? ( 1 G S ψγ d ∧ ψ 3 4 b bG 192 to d b − 1 abcd f b ) − 1 f ε S 3 ? µ c ω − µν , e + ∧ . e b F 1 4 ∑ b A S ) d ( d ? 2 ? µν − ω G 3 κ , ∧ fermion species F ∧ 4 = = 2 b A b i 2 ( d d π Z e ab i G 8 2 1 ≡ 1 2 µ h µ K 5 Z = ≡ Z J i i exp µ and the non-propagating − 5 − J h π Db h P 0, leading to a conserved “torsion charge” µ 3 S M √ ∇ exp = D exp = S Z ? Db 2 repulsive four-fermion axial current-current interaction, characteristic of any / d ∝ ). The reader should observe that in ( Db 1 Z − ]. Z Z = ) 2.8 8 8 / 2 κ 3 ]: = ( b 10 f The torsion term, being geometrical, due to gravity, couples universally to all fermion species, In a quantum gravity setting, where one integrates over all fields, the torsion terms appear as non-renormalizable relevant torsion part of the quantum-gravity path integral would include a factor the well-known trick of introducing a Lagrange multiplier field 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 take into account ( been used. This can becan achieved convert by the the anomaly addition fromobtain of the for proper the initial counter effective torsion terms action in inSM: theories the with action chiral [ anomalies, such as the QED part of the torsionful theory [ should notice that, as a result of this integration, the corresponding though that the classical equationsthe of axial motion current, identify since the the axial-pseudovector torsion torsion equation field yields not only neutrinos. Thus, in the context of the SM of particle physics, the axial current ( We may then partially integrate the second term in the exponent on the right-hand-side of ( non propagating fields and thus they can be integrated out exactly. The authors of [ Axions and massive Majorana neutrinos where PoS(CORFU2017)003 - as H 2 ) ], and (2.10) µνρ a above, 7 µ H b ∂ ]. ( 1 2 13 is the charge- ) + C a ) denotes antisym- µ R ] Nick E. Mavromatos ∂ ψ ... )( [ b = ( to another pseudoscalar may be provided by the µ C ) ∂ ) R ( x x is the ordinary, torsion-free, ( ψ ( γ b a ) to the fermionic matter dis- + µ ρν ) µ x Γ ( 5 b J ) by terms that break such a shift = 5 µ that characterises all string theories. J ] µ νρ 2.9 2 b ν Γ f 1 Θ 2 µ [ , where the symbol + ∂ ] where νρ + ) only changes by total derivative terms, such , is the axial current of the four-component Ma- B µνρσ µν µ [ 2.9 µ ρν e ψ 4 R B ∂ 5 Γ γ = → µ 6= ]. Scenarios as to how such cosmologies can evolve so torsion (corresponding to a linear in cosmic time KR µνρσ µν ψγ R µ µνρ 14 νρ is a real parameter to be constrained later on. Here, we B b i jk f = H H γ H ..., ) 2 5 3 µ x κ π J ( + √ , the action ( , b c R + i 192 , and ψ C ]) field C ) + µ . These terms are irrelevant for the equations of motion and the in- R νρ ) + x , of relevance to the low-energy (field-theory) limit of string theory, is the gravitational constant. In four space-time dimensions, the dual Γ ψ R ( 11 0 2 µν R ) b e ψ κ F α b = ψ µ µν → ∂ + ( − µ ( νρ ) R R x 2 1 Γ cF . For completeness we mention at this point that background geometries with ( ψ ψ ) h ] that the terms of the effective action up to and including quadratic order in b x C g R ( of the torsion is identified with the field strength of the spin-one antisymmetric = ]. The proposed coupling occurs through a mixing in the kinetic terms of the 12 and b − ψ µ σ ψ . In string-inspired models, such pseudoscalar axion  15 S √ ) ∂ indicate terms in the low-energy string effective action, including SM ones, that are appear: x x ia ]. Today of course any torsion background should be strongly suppressed, due to the 4 ( a µνρσ d a y µ νρ e R ... 13 Γ µνρσ − Z -torsion (or, equivalently, the KR axion quantum field ε H = ∝ µνρσ by a constant: In what follows we shall consider the effects of the quantum fluctuations of such a KR ) S cR x µνρ ( jorana fermion have ignored gauge fields, which are not of interest to us, and the possibility of a non-perturbative two fields. To be specific, weMajorana consider neutrino the fermion action fields): (henceforth we restrict ourselves to right-handed as duced quantum dynamics, provided the fieldsThe fall off scenario sufficiently for fast to the zero anomalous at space-time Majorana infinity. mass generation through torsion proposed in [ symmetric connection, and of the H-field is indeedH the derivative of an (KR) axion-like field, analogous to theaxion), field where Latin indicescharacterise denote the spatial early components universe. of theof In four-dimensional such CP space-time, cases, violation, may the necessary H-torsionconserving for background processes, constitutes lepotogenesis, Baryogenesis, extra and and source Universe through thus [ Baryon-minus-Lepton-number the (B-L) observed matter-antimatterlack of asymmetry any in experimental evidence the foras it to [ guarantee the absence of any appreciable traces of torsiontorsion, today which can survive be the found absence in of [ of any torsion the background. An important aspectcussed of the above coupling is its shiftb symmetry, characteristic of an axion field. Indeed, by shifting the field reviewed here, consists of augmentingsymmetry. the To effective illustrate action this last ( point, we first couple the KR axion where the conjugate right-handed fermion the Regge slope parameter (approximately) constant background not of direct relevance to our purposes in the present article. Above, which involve the H-field strength, cansymbols, be assembled in such a way that only torsionful Christoffel Axions and massive Majorana neutrinos A concrete example of torsion is provided byric string-inspired component theories, where the totally antisymmet- metrization of the appropriate indices.a The consequence string of theory the effective “gauge action symmetry” depends only on axion field string moduli [ tensor (Kalb-Ramond (KR) [ It can be shown [ PoS(CORFU2017)003 , , ) . 2 2 γ self- γ µνρσ (2.12) ]. The (2.11) (2.13) − e 7 R − may be 1 1 , coupled R ) p ψ x p b µνρσ f ( / a 2 R a repulsive . We may now y π 1 .... ) becomes: . (wavy lines) denote c 192 Nick E. Mavromatos + ( µν + . / 2.10 γ i a ) 2 C R γ → µνρσ ψ 6 e − a R R Λ 1 5 ψ ( 4 . − µνρσ π γ κ with the wrong sign of its kinetic i R R a µ 8 2 y 5 ψ γ i.e. √ J C 3 R 5 µ − J √ ψ ], the gravitationally induced Majorana 1 2 ) b  f x 1 2 p 16 ( 2 49152 ) b ia a f a a γ + 2 y µ = π  ∂ ) − ( C the shift symmetry: 2 R  γ 192 ψ 5 2 to right-handed sterile neutrino matter 6 R γ − − a induced by KR torsion. The fields h Λ µνρσ ψ 2 1 4 e − R break ) ( − a b 1 f R µ  torsion, with a modified coupling γ κ µνλρ 2 ∂ ψ a e . Moreover, we remind the reader that the 1 2 ( R µνρσ π C y ) R 1 2 x R + h . This implies that the effective action ( ψ ( à la ) , is estimated to be: 2 ) a g  192 x µνλρ ) x R ( 0 b a ( − R f coupled both to matter fermions and to the operator 2 a b M 2 ) a ) 2 γ √ ) µ x ) corresponds to a canonically normalised axion field γ γ 2 x x ( ∂ π a 4 ( π 1 − ( field has decoupled and can be integrated out in the path integral, − d a ) iy 1 2 1 0 ) + 16 x 2.12 b 192 x h Z ( ( ( p 0 g b b = in ( − ∼ − a a ≡ + R √ S of the axion moduli field ) µ S x M x 5 4 a ( J y 0 d 5 µ b J Z 2 b → f 1 = otherwise the axion field would appear as a ghost, ) 2 , x ( 1 Typical higher-loop Feynman graph giving rise to anomalous fermion mass generation [ + S for the pseudoscalar field b a < to the curvature of space-time, M | . Adopting the effective field-theory framework of [ 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 | . 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, Figure 1: Axions and massive Majorana neutrinos black circle denotes the operator a interaction fermion terms areYukawa due coupling to the existence of torsion in the Einstein-Cartan theory. The follows: graviton fluctuations. Straight lines with arrows denote right handed neutrino fields and their conjugates. mass due to non perturbative effects. These terms Evidently, the action PoS(CORFU2017)003 at to a R y (2.14) M 3, the (2.16) (2.17) (2.15) > , of which n n ,  ], in addition ,..., are related. 1 2 1. In this case, 3, in which the , + P 15 1 i ∼ a a ≥ M i , so as to obtain a a a Nick E. Mavromatos n 1 1 , which gives rise to M + i n + / i ) , 2 i a M 2 a i as free parameters to be M M M ] the existence of Yukawa M a δ δ / 7 1 . 1 2 , the anomalously generated a M has a Yukawa coupling − < = 2 5 ∑ a i n δ to be , M -independent) result for axion fields, n 1 of the axion fields and not on .  M 1 2 a δ R 3 + Λ a n a i 2 a 3 ( δ , P 2 − i M − , − M with next-to-neighbour mixing as n M M n ) ) M = . Assuming, for simplicity, that all M δ n δ ) 3 1 δ 2 a n , ) 1 2 δ a a 2 2 ) 2 , a  + M µ 2  1 i E M , γ γ 2 i ∂ δ M a ( a / 2 a ( and only δ 1 )( M γ − y M ( will depend on higher powers of the energy b b ) 3 δ n 1 µ 2 ∼ − R 3 2 and the Planck mass scale ( have a next-to-neighbour mixing pattern. In − ∂ γ ) , described by the propagator matrix element 4 n 1 − ( M n 2 n a ) 6 Λ π γ 10 -axion-mixing models with − and a 2 Λ and the torsion-axion kinetic mixing coefficient M 8 n and mass-mixings ,..., of quantum gravity. In order to get an estimate p 1 2 + a 5 3 6 ( δ ( , a ∼ a √ →  ( 4 2 M / y Λ 2 i 5 a 1 γ κ M are of the same order, i.e. π 1 6 5 a P M a = ) 8 a y a ∝ γ κ 2 i M − √ M 3. Instead, for axion-mixing scenarios with a 3 ) 49152 M a 2 y M p ) δ √ i ≤ M ( 3 are constrained to be ( a n a n 1 µ √ ∼ γ 1 with the KR axion ∂ + 49152 a 2 i a ( R , 2 i γ γ y ∆  1, these higher-order effects are expected to be subdominant. M ∼ 3 1 M − n − = R ∑ i δ 1  M 10 1 2 Λ  Kalb-Ramond field. One can then assume [ . The other axions κ ∼ g R , originating from flux fields that exist in the spectrum [ and their masses − µν a R n ψ a B √ M M x M 4 ,... -axion-mixing scenarios, we note that the anomalously generated Majorana δ , but if d 2 n 6 , , we treat the axion masses Z > 1 ), we may estimate the Majorana neutrino mass R n M = Λ = i 2.15 , kin a ) , i.e. x S , changes drastically, i.e. 3. It is then not difficult to see that three axions ( has a kinetic mixing term 3, and ) i Λ 1 p a . To make this point explicit, let us consider a scenario with > ≤ a ( In the multi-axion models outlined above, the anomalously generated Majorana neutrino mass In the above In particular, as already mentioned, in string theory there are several axion-like (pseudoscalar) 1 n will still depend on the Yukawa coupling n n a 1 R a , besides the assumed UV completion scale employing ( axion mass mixing M discussed above would be sufficient to obtain a UV finite (cut-off- their masses themselves, as long as γ of the size of constrained by phenomenology. Let us assume a mass term will only depend on the mass-mixing parameters cut-off for the two-loop level. Of course, beyond the two loops, induced Majorana neutrino masses arean proportional additional to suppression the for factor heavy axions with masses fields couplings with right-handed neutrinos,sponsible provided for some a non-perturbative breakingeach instanton of other. effects the Such are shift a mixing symmetry. re- Fig. can These give rise string-theory to axion reducedonly fields UV sensitivity could of mix the with right-handed two-loop neutrinos graph shown in Majorana mass may be estimated to be for Axions and massive Majorana neutrinos In a UV complete theory such as strings, the cutoff to the aforementioned such a model, the kinetic terms of the effective action are given by stable positive mass spectrum forthe all UV axions. behaviour of As the a off-shell consequence transition of the next-to-neighbour mixing, axion masses and mixings are equal, i.e. ∆ where the mixing mass terms PoS(CORFU2017)003 , R 1. 83 . M 0 − (3.1) GeV, = 10 16 γ ), which < ∼ 10 a . This is in CDM-based M CDM model and 2.12 3 = / Λ Λ 3 − R Λ − in ( 10 M 1, the size of 10 π Nick E. Mavromatos = 4 .  = γ 6 √ γ a  y > ∼ = 2 a to qualify as a warm dark γ compensates for the mass y GeV R 2 − γ ψ 18 γ 1 − Λ 1 10 p / × lies in the range a 4 . y m 2 47 (3.2) ≡ . . An obvious caveat to this result would  0 R 1 eff a − M ≤ y γ 10 1 ) above. Assuming that is at the TeV scale, for m − / R g 7   16 keV, for the choice 2 M 2.13 3 a − SIDM ∼ , at small radii. This is, e.g., evidenced in the standard y cm σ 1 R 10 − M  r ≤ close to the GUT scale and/or fine-tune the torsion-axion ) ]. In addition, such self-interacting neutrino WDM models a 1 ∝ . given in ( 19 0 M ρ GeV R ], the introduction of appropriate self-interactions among the is highly model dependent. Taking, for example, the quantum 11 6 (or, as is also known, the cuspy-halo problem), refers to a dis- M ). The latter possibility, however, might result in an unnaturally R 10 ]. M × GeV, we find that to 1, in a way such that the factor 2.17 ) to be 20 1 . γ 17 3 in ( of a dark matter particle with mass ( 1 TeV considered in the literature thus far, we find that 2.13 10 5 ∼ ]. ) ≤ P = R a 18 SIDM M M Λ , σ M / a 17 M δ ( The Core-Cusp problem For completeness, we mention that discrepancies between observations at galactic scales(i) and It is interesting from a cosmological viewpoint to provide a numerical estimate of the anoma- Moreover, as discussed in [ CDM-numerical simulations, appear in three areas: the preferred ballpark region formatter the (WDM) (right-handed) [ sterile neutrino However, if we take thewe quantum obtain gravity a scale right-handed to neutrino be mass close to the GUT scale, i.e. gravity scale to be be to have ultra-heavy axion masses kinetic mixing parameter srerile right-handed neutrino darktion matter of (DM), the observed can core-halo serve galacticthe structure, as Ruffini-Argüelles-Rueda upon profile a assuming [ specific means core of profiles,can specifically, also providing alleviate some an of explana- the discrepancies between numerical simulations based on Λ crepancy between the observed darkprofiles matter predicted density by profiles of cosmological low-mass N-body(DM galaxies only) simulations. and simulations the form Characteristically, dark density all matterthe the halos density increasing which steeply, have i.e. “cuspy" as dark matter distributions, with lously generated Majorana mass which implies extra-ordinary small Majorana masses suppression large (non-perturbative) “effective” Yukawa coupling will bring us outsidedetailed the phenomonological perturbative and astrophysical framework analysis thatbeyond of the we all scope possible have of axion-mixing been the scenarios present considering falls talk. here. A more 3. Implications for the Dark Sector of the Universe and observations at galactic scalestotal (“small-scale cross cosmology section crisis”), provided the self interaction where the upper limit has beentechniques inferred and observables by [ recent studies of several merging galaxies, using novel Axions and massive Majorana neutrinos For axion masses may be estimated from ( Obviously, the generation of PoS(CORFU2017)003 (3.3) ], or, 25 CDM sim- Λ Nick E. Mavromatos ] 6 ]. 22 CDM simulations predict that Λ in the range [ , χ m ]) is to make the core more compact than ]. 6 24 345 keV ] develop always an extended plateau on halo . 6 2 , 8 c 19 χ ]. On the contrary, the rotation curves of most of the m 21 . (or, as is also known, the dwarf galaxy problem), arises CDM simulations and the observed dynamics of the brightest , that is a discrepancy arising between the most massive sub- Λ ], whilst at the same time the fermi gas of fermionic DM may 47 keV 19 CDM-based numerical cosmological simulations that predict the ]. ]. For sterile neutrino masses Λ ], for resolving the Core-Cusp problem, because the density profiles 23 6 19 the luminosity of the Sun [ 5 ], have been argued to play a rôle, either in the too-big-to-fail problem [ 5 The too big to fail problem The “missing satellite problem” In this latter respect, WDM sterile neutrinos, with masses in the range of tens of keV, as in the All three problems have their root in the fact that the cold DM particles, which the (iii) (ii) MSM model [ provide an alternative to the blackthe hole lower in bound the depends centre crucially of on(and the the its galaxy. details detailed of It properties) the must of galactic beless a profile, stressed robust supermassive as though than black well that the hole as upper in bound, of and the the can galactic existence be centre. further relaxed. Hence it is much scales (starting immediately after the quantuminteractions core). due The to rôle massive of vector fields self in interactionsin the (attractive the model vector non-interacting of case [ [ where the (robust) upper boundhalo corresponds properties to the of limit galaxies of (inthe gravitational agreement galaxy collapse, with profile the observations) correct proposed are DM in guaranteed [ within the context of based on fermionic phase-space distributions [ ν ulations rely upon, have too short freetherefore streaming they length form during too the epochs clumped ofsome and galaxy of too formation, these and many problems structures could comparedbaryonic be to feedback, and partly those may alleviated observed. have by their Although root morelite in precise problem astrophysical observations reasons may (for and/or be instance including attributed themay missing partly have satel- to been the stripped fact apartit that gravitationally would dwarf by be galaxies larger desirable have galaxies if aand and a tendency hence this unified to invisible), is explanation merge nonetheless is the or found pointparticles. within of the view context taken of in fundamental the physics, approach of including self interactions among thein DM the self-interacting case [ haloes predicted in (dissipationless) dwarf spheroidal galaxies in the Milky way. In other words, the Axions and massive Majorana neutrinos Navarro-Frenk-White (NFW) DM profile [ observed dwarf galaxies indicate flat central density profiles ("cores") [ from a discrepancy between evolution of the distribution of matter inDM the (where smaller universe - halos merge pointing to towards form aenough larger hierarchical halos) observed clustering - normal-sized of and galaxies observations. to Although account there fordwarf seem such galaxies to is a be orders numerical of distribution, the magnitude number lowerexample, than of we that mention expected that from there the were simulations.and observed As only to a around be concrete 11 around 38 orbiting dwarfMilky the galaxies Way Milky dwarf in Way, satellites the yet [ Local one Group, dark matter simulation predicted around 500 the most massive subhaloes of theluminosity Milky higher way than are 10 too dense to host any of its bright satellites, with PoS(CORFU2017)003 ] 26 (3.4) ): MSM 2 ν ], when 27 ], may affect the 6 Nick E. Mavromatos ] observations of the 27 ]). The recent analysis 26 2 MSM WDM sterile neutrino ], employing observations 0 ν 1 26 ], leaves a very small allowed [ s b 18 ]. In that analysis, any mixing of o χ , 6 Ω PreviousX-ray constraints , 17 et al. ] profiles in their study). An analysis > s ν χ 28 Ω ] ], sterile neutrinos can acquire radiatively

7 V 16 keV e ], employed in the work of [ 1 k

. 0 [ 19 1

2 χ ] that the lightest of the sterile neutrinos provides c 9 5 m χ

s m ] and Einasto [ b o χ . 21 Ω

M

S MSM [ <

M NuSTAR GC 2016 ν s ν MSM. These latest studies exclude almost half of the previously and ν χ ν Ω phaseconstraints space 10 keV

MW satellite counts 0 0 5 0 1 2 3 4 7 8 9 )), within the model of [ 1 - - - 1 1 1 1 1 1 ------0 0 0 0 0 0 0 0 0

1 1 1

3.1

1 1 1 1 1 1

( 2 n i s θ 2 cf. MSM mixing-angle-mass parameters (thick (red) line and hatched region in the figure, ], employs NuSTAR (Nuclear Spectroscopic Telescope Array) [ ν 26 ], the mixing of the lightest of the sterile right-handed neutrinos which, notably, is 5 ], the sterile neutrino might constitute only one component of the DM in the Universe, Up-to-date Cosmological and Astrophysical Constraints for the 6 However, the above region is quite sensitive to the galactic profile used (the authors of [ In this respect, we mention a crucial feature of the model of [ framework [ used versions of the Navarro-Frenk-White [ making use of the Ruffini-Argüelles-Rueda profile [ window in the mixing-angle-sterile-neutrino-mass parameter plane (see white region in fig. masses in this range for quite natural values of the parameters. with other equally dominant DMrôles in species the structure in formation the and energy underlying budget microscopic of the model Cosmos. playing On different theassumed other to hand, play in the the rôle ofage, the can dominant be DM constrained species, mainly havingto by a the its life-time large decay longer into than scale the X-rays structure. Universe and the The DM recent production constraints and of contribution Perez the sterile neutrinos with the Standardspirit Model of sector [ had been ignored, assumed very small. In the of the Galactic Centrecombined with with the the basic assumption Nuclear of Spectroscopicthe dominant Telescope component of Array DM, thus (NuSTAR) leadingin [ to additional the constraints early for Universe sufficient and DM production efficient rôle in structure formation [ As we discussed above ( Figure 2: Axions and massive Majorana neutrinos of Perez et al.Galactic Centre, [ together withstructure additional formation constraints in associated the withof Universe, DM the which production lightest stem and of from the itsallowed the sterile region rôle requirement of neutrinos in that of DM the comprises predominantly the white region being the currently allowed one). mass versus mixing angle with the Standard Model sector (picture from ref. [ PoS(CORFU2017)003 cf. )), once 3.1 ( cf. ], due to a massive Nick E. Mavromatos 6 ], there are additional 7 ). Such a study is still pending. Moreover, 3.4 10 ) is concerned. From this general point of view, therefore, 3.4 ] in the near future. 6 will be modified. For instance, the lower bound on the mixing can only be interpreted as providing upper limits for the mixing sterile neutrinos can he produced in the early Universe by decays 2 2 ] a massless pseudoscalar (axion-like) field (KR axion), which exists 7 e.g. ] and/or [ 7 , which, to lowest order in a derivative expansion of the effective low-energy ], in particular the upper limit in ( µνρ 26 H is dual to the aforementioned KR axion. In the model discussed in the talk, the KR )). Hence, if the sterile neutrino does not constitute the whole of DM, or its production MSM model, ν µνρ 2.14 H ). This induces a radiative generation of the sterile (right-handed) neutrino mass, which can ),( 1 As a result of such couplings, as well as the torsion-like features of the KR axion field, there is a In this talk we reviewed a rather novel mechanism, of semi-geometrical origin, for the genera- 2.12 axion has a kinetic mixing with otherturn (massive) couple axions that with exist Yukawa-type terms in to string-inspired the models,generated right which by in handed non-perturbative neutrinos. string effects Such that Yukawa couplings breakaxion may the models. be shift symmetry that usually characterises coupling of quadratic curvature terms to the sterilegin neutrinos of (hence their the mass” term “semi-geometrical used ori- above),(fig. through anomalous higher loop graphs, involving axion exchanges Lagrangian, including gravity, appears as asions, totally antisymmetric part of a torsion. In four dimen- be determined in terms of thethe effective pertinent ultraviolet cut-off Yukawa couplings of and the kinetic low-energycomplete theory mixing theories, ( parameters like of strings, the the cut-off model scaleUpon are is a specified related rather to (for natural the UV string choice and oftrino compactification the mass scales). parameters can of be the made model, to the fit radiatively interesting generated phenomenological sterile models neu- of sterile neutrinos, where the in the massless stringabelian multiplet of gauge low-energy symmetry string-inspired of effectiveric strings, field three this theories. form field Due appears to only an in the form of a totally antisymmet- tion of sterile right-handed neutrino Majorana masses,The which mechanism goes beyond involves the [ conventional seesaw. vector meson field, will modifywith significantly the the production of the sterile neutrinos asYukawa compared interactions of the sterilepart neutrinos of a with dark (massive) sector axion-like of the( fields, theory, and which there may is kinetic well mixing be of these axions with the KR axion ( diagrams like the oneparameter in of fig. the sterile neutrinosonly with robust the constraints Standard are Model the sector. onesBag associated It Nucleosynthesis with (BBN) goes the mechanism. without maintainance We saying of hopewithin that to the the the come predictions models back of of to the [ a Big detailed study of such issues 4. Conclusions and Outlook is modified by additionalthe interactions gray in shaded a areas hidden in sector fig. of the model, then the lines bounding the existence of self interactions among the sterile neutrinos in the model of [ of the massive vector field. In the model discussed in the present work [ angle may be significantly reduced, orproduction even alleviated, exists. if a different This mechanismparticular would for as sterile imply far neutrino as partial the upper or bound of total ( relaxation of the associated constraints, in Axions and massive Majorana neutrinos conclusions of [ PoS(CORFU2017)003 ap- , a y ´ c, Phys. , Rept. et al. Nick E. Mavromatos )) may be determined 2.16 ] or other models, to be almost ) and ( 18 , , 113 (2007) [hep-th/0609191]; 5 ], but in such multi-axion cases, as discussed 2.15 7 , 30 (2012) [arXiv:1207.7235 [hep-ex]]. 771 ), ( . In fact, in such scenarios, constant (in 716 2 , 1 (2012) [arXiv:1207.7214 [hep-ex]]; MSM [ , 086002 (2007) [hep-th/0703028]. 2.13 ν 76 716 , 2227 (1980); G. Lazarides, Q. Shafi and 11 (spatial components), which violate spontaneously 22 i jk ], which, for instance, allows two of the heavier right- H 29 , 287 (1981); For recent reviews see: R. N. Mohapatra , 421 (1977); M. Gell-Mann, P. Ramond and R. Slansky, in 67 181 Workshop on the Unified Theory and the Baryon Number in the Universe [CMS Collaboration], Phys. Lett. B CDM model and observations at galactic scales. Λ et al. , 1757 (2007) [hep-ph/0510213] and references therein. [ATLAS Collaboration], Phys. Lett. B )), the axion field has to be superheavy in order for the mass of the sterile neutrino to be at least in , 912 (1980). , eds. D.Z. Freedman and P. van Nieuwenhuizen (North-Holland, Amsterdam, 1979); T. ], which by the way the KR axion field has also an important rôle to play in, as ] and mentioned previously, in section 70 44 30 13 2.17 ( et al. cf. MSM, where the sterile neutrinos are considered as the dominant DM species. We have ν . 1 Tsukuba, Japan, 1979, eds. O. SawadaRev. and Lett. A. Sugamoto; R. N. Mohapatra and G. Senjanovi M. Cvetic, R. Richter and T. Weigand, Phys. Rev. D Supergravity Yanagida, in Proc. of the C. Wetterich, Nucl. Phys. B Prog. Phys. S. Chatrchyan It is important to notice that for exactly three axion species, there is no dependence of the induced right handed In the talk we discussed the pertinent constraints of such models, within our context. We I would like to close the talk with the comment that the values of the Yukawa couplings 1 [2] P. Minkowski, Phys. Lett. B [3] J. Schechter and J. W. F. Valle, Phys. Rev. D [1] G. Aad [4] R. Blumenhagen, M. Cvetic and T. Weigand, Nucl. Phys. B neutrino Majorana mass on the effective field theory cut-off in our model [ the keV range. previously ( by some underlying discrete symmetry [ degenerate in mass. This latterleptogenesis feature [ is also requireddiscussed for in enhanced [ CP violation of relevance to Axions and massive Majorana neutrinos lightest of the latter play the rôlerange of a warm dark matter component, with mass in a few-tens-of-keV stress once more that ourstring model includes axions more and than one sterilean potential neutrinos, embedding species along of of dark the with matter: model otherfeature into massive implies potential fully-fledged, a phenomenologically DM significant realistic, candidates, relaxation, string inlike originating theories. our the from case, This of the constraints characterising other models, pearing in the radiative mass for the sterile neutrinos (( handed neutrinos out of the three that characterise the cosmic time) backgrounds of the KRLorentz form and CPT symmetries, mayand induce CPT-Violating lepton decays asymmetries of in (sufficiently) the heavydiscussed early sterile here Universe neutrinos which from (distinct play the from the CP further the rôle exploration of lightest and DM ones we components). hope we These shall are be interesting able issues to tackle that some deserve of themReferences in the near future. also discussed how (self-interacting) sterile neutrinosrôle with in the masses galactic of structure, a as fewsimulations well tens as based of offer on keV a resolution can of play some a discrepancies between numerical PoS(CORFU2017)003 , 81 , 008 (2008) , 393 (1976); The Quantum Nick E. Mavromatos , 191 (2009) 0808 48 59 , no. 1, 622 (2015) , no. 04, 038 (2016) 451 1604 , 6229, 1462 (2015) and , no. 03, 1730007 (2016). , no. 10, 514 (2015) 26 347 , 493 (1997) doi:10.1086/304888 75 , 401 (2006). , 151 (2005) , 215 (1992). 490 638 , 407 (2013) [arXiv:1304.5433 [gr-qc]]. 631 387 Science 725 , 11 (2011) [arXiv:0801.0287 [hep-ph]]. , 385 (1987); , 124038 (2012) [arXiv:1209.6387 [hep-ph]]. , no. 3, 2359 (2013) [arXiv:1211.0968 [hep-ph]]. (Cambridge University Press 2001) ISBN 83 86 293 73 12 , 41 (1987). , 381 (1972); see, also: S. Weinberg, 291 40 , 2273 (1974). 9 , 3874 (1994) [gr-qc/9405057]. , no. 2, 113 (2018) doi:10.1140/epjc/s10052-018-5587-5 , no. 7, 656 (2015) doi:10.1134/S1063772915070124 50 78 , 113 (2002) [hep-th/0103093] and references therein 59 357 , no. 6, 801 (2014) doi:10.3938/jkps.65.801 [arXiv:1402.0700 [astro-ph.GA]]; 65 doi:10.1146/annurev.nucl.010909.083654 [arXiv:0901.0011 [hep-ph]]. doi:10.1093/mnras/stv1016 [arXiv:1409.7365 [astro-ph.GA]]. See also: I.and Siutsou, R. C. Ruffini, R. Astron. Argüelles Rep. references therein. [astro-ph/9611107]. [arXiv:1402.0695 [astro-ph.GA]]; C. R. Argüelles, R.Phys. Ruffini, Soc. I. Siutsou and B. Fraga, J. Korean J. Ellis, N. E. Mavromatos andM. S. de Sarkar, Cesare, Phys. N. Lett. E. B Mavromatosdoi:10.1140/epjc/s10052-015-3731-z and [arXiv:1412.7077 S. [hep-ph]]. Sarkar, T. Eur. Bossingham, Phys. N. J.and E. C S. Mavromatos Sarkar, Eur. Phys. J.[arXiv:1712.03312 C [hep-ph]]. 123530 (2010) [arXiv:0905.4720 [hep-th]]; M. Cicoli,arXiv:1206.0819 M. [hep-th]. Goodsell and A. Ringwald, Theory of Fields. Volume II: Modern0-521-55002-5. Applications. D. J. Gross and J. H. Sloan, Nucl. Phys. B I. L. Shapiro, Phys. Rept. [arXiv:0804.4542 [hep-ph]] and references therein. doi:10.1088/1475-7516/2016/04/038 [arXiv:1502.00136 [astro-ph.GA]]; N. E. Mavromatos, C. R. Argüelles, R. Ruffini anddoi:10.1142/S0218271817300075 J. A. Rueda, Int. J. Mod. Phys. D doi:10.1016/j.physletb.2005.09.070 [hep-ph/0503065]; M. Shaposhnikov, JHEP [9] M. J. Duncan, N. Kaloper and K. A. Olive, Nucl. Phys. B [6] C. R. Argüelles, N. E. Mavromatos, J. A. Rueda and R. Ruffini, JCAP [5] T. Asaka, S. Blanchet and M. Shaposhnikov, Phys. Lett. B [8] F. W. Hehl, P. Von Der Heyde, G. D. Kerlick and J. M. Nester, Rev. Mod. Phys. [7] N. E. Mavromatos and A. Pilaftsis, Phys. Rev. D [19] R. Ruffini, C. R. Argüelles and J. A. Rueda, Mon. Not. Roy. Astron. Soc. [21] J. F. Navarro, C. S. Frenk and S. D. M. White, Astrophys. J. [18] A. Boyarsky, O. Ruchayskiy and M. Shaposhnikov, Ann. Rev. Nucl. Part. Sci. [20] D. Harvey, R. Massey, T. Kitching, A. Taylor, E. Tittley [14] V. A. Kostelecky and N. Russell, Rev. Mod. Phys. [16] J. F. Donoghue, Phys. Rev. D [13] N. E. Mavromatos and S. Sarkar, Eur. Phys. J. C [17] T. Asaka, M. Shaposhnikov and A. Kusenko, Phys. Lett. B [15] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper and J. March-Russell, Phys. Rev. D [11] M. Kalb and P. Ramond, Phys. Rev. D [12] R. R. Metsaev and A. A. Tseytlin, Nucl. Phys. B [10] R. Delbourgo and A. Salam, Phys. Lett. B Axions and massive Majorana neutrinos PoS(CORFU2017)003 et , 524 The , no. 12, , 735 95 454 , L40 (2011) , 87 (1965); 5 415 Nick E. Mavromatos , 0179 (2014) 29 , 012017 (2009) 171 , 89 (1989). 13 223 , 103 (2013) doi:10.1088/0004-637X/770/2/103 770 , no. 3, 2836 (2017) doi:10.1093/mnras/stx621 [arXiv:1611.00005 468 , 1203 (2012) [arXiv: 1111.2048]. , 043506 (2011) doi:10.1103/PhysRevD.83.043506 [arXiv:1004.1459 , 82 (1999) doi:10.1086/307643 [astro-ph/9901240]; E. Polisensky and 83 422 522 (6), 96 (2015). , Advances in Astronomy 2010 789293 (2010) [arXiv:0910.3538]; Se-Heon Oh ibid. , Astrophys. J. 149 370, 629 - 631 (25 August 1994); doi:10.1038/370629a0; W. J. G. de Blok, et al. Nature , Astrophys. J. Mon. Not. Roy. Astron. Soc. Core-Cusp Problem al. L19 (1999) doi:10.1086/312287 [astro-ph/9907411]; A. A.F. Prada, Klypin, Astrophys. A. J. V. Kravtsov, O. ValenzuelaM. and Ricotti, Phys. Rev. D [astro-ph.CO]]. [arXiv:1103.0007]; [astro-ph.GA]]. doi:10.1088/1742-6596/171/1/012017 [arXiv:0904.1182 [hep-ph]], and references therein. 123002 (2017) doi:10.1103/PhysRevD.95.123002 [arXiv:1609.00667 [astro-ph.HE]]. (2008) doi:10.1038/nature07153 [arXiv:0805.1244 [astro-ph]]; This isoriginal a profile shallow suggested version in: of J. the Einasto,U. Trudy Haud Astrofizicheskogo and Instituta J. Alma-Ata Einasto, Astron. Astrophys. [arXiv:1309.5785 [hep-ph]], but with their resultsreferences applied therein. to the right-handed neutrino sector, and [arXiv:1301.7307 [astro-ph.IM]]. Axions and massive Majorana neutrinos [22] B. Moore, [25] M. R. Lovell, V. Gonzalez-Perez, S. Bose, A. Boyarsky, S. Cole, C. S. Frenk and O. Ruchayskiy, [24] M. Boylan-Kolchin, J. S. Bullock and M. Kaplinghat, Mon. Not. Roy. Astron. Soc. [26] K. Perez, K. C. Y. Ng, J. F. Beacom, C. Hersh, S. Horiuchi and R. Krivonos, Phys. Rev. D [23] B. Moore, S. Ghigna, F. Governato, G. Lake, T. R. Quinn, J. Stadel and P. Tozzi, Astrophys. J. [29] See for instance: G. K. Leontaris and N. D. Vlachos, Mod. Phys. Lett. A [27] F. A. Harrison [30] For a review see, e.g. A. Pilaftsis, J. Phys. Conf. Ser. [28] J. Diemand, M. Kuhlen, P. Madau, M. Zemp, B. Moore, D. Potter and J. Stadel, Nature