# Axions, Majorana neutrino masses and implications for the

## Transcript Of Axions, Majorana neutrino masses and implications for the

PoS(CORFU2017)003

Axions, Majorana neutrino masses and implications for the dark sector of the Universe

Nick E. Mavromatos∗† King’s College London, Physics Department, Theoretical Particle Physics and Cosmology Group, Strand, London WC2R 2LS, UK E-mail: [email protected]

We discuss a novel mechanism for generating masses for right-handed Majorana neutrinos, that goes beyond the conventional seesaw. The mechanism involves quantum ﬂuctuations of a massless Kalb-Ramond (KR) pseudoscalar (axion-like) ﬁeld, which exists in string-inspired extensions of the standard model. We assume a kinetic mixing of the KR ﬁeld with ordinary (massive in general) axions, which also exist in string models, and which are assumed to couple to Majorana (right-handed) neutrinos via non-perturbatively-generated chirality changing Yukawa couplings, breaking the axion-shift symmetry. No vacuum expectation value is assumed for the axion ﬁelds. Majorana masses for the right-handed neutrinos are generated radiatively as a result of anomalous higher-loop axion-neutrino couplings. Implications for the Dark sector of the Universe are discussed. In particular, we explore the possibility of generating masses for the right-handed neutrinos of order of a few tens of keV. Such neutrinos could play an important rôle in the galactic structure, and more generally they could serve as a warm dark matter component in the Universe, providing a potential resolution to the so-called small-scale cosmology “crisis”, that is, discrepancies between observations at galactic scales and numerical simulations based on the ΛCDM model. If such scenarios are realised in nature, they might imply that Dark Matter consists of more than one species (warm and cold), with distinct rôles in the structure and evolution of the Universe: the cold one being still responsible for the large scale structure of the Universe, in accordance to the predictions of the ΛCDM model, which agree with a plethora of cosmological observations, but the warm component (keV sterile neutrino) playing a crucial rôle in the (observed) galactic structure.

Corfu Summer Institute 2017 ’School and Workshops on Elementary Particle Physics and Gravity’ 2-28 September 2017 Corfu, Greece

∗Speaker. †This work is supported in part by STFC (UK) under the research grant ST/P000258/1

c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).

https://pos.sissa.it/

Axions and massive Majorana neutrinos

Nick E. Mavromatos

PoS(CORFU2017)003

1. Introduction and Summary

The discovery [1] of the Higgs boson at the CERN Large Hadron Collider (LHC) in 2012 constitutes an important milestone for the Ultra-Violet (UV) completion of the Standard Model (SM). Although the so-called Higgs mechanism may well explain the generation of most of the particle masses in the SM, the origin of the small neutrino masses still remains an open issue. In particular, the observed smallness of the light neutrino masses may naturally be explained through the see-saw mechanism [2], which necessitates the Majorana nature of the light (active) neutrinos and postulates the presence of heavy right-handed Majorana partners of mass MR. The right-handed Majorana mass MR is usually considered to be much larger than the lepton or quark masses. The origin of MR has been the topic of several extensions of the SM in the literature, within the framework of quantum ﬁeld theory [2, 3] and string theory [4]. However, up to now, there is no experimental evidence for right-handed neutrinos or for any extension of SM, as a matter of fact, although some optimism of discovering supersymmetry in the next round of LHC (operating at 14 TeV energies) exists among particle physicists.

Until therefore such extensions of the SM are discovered, it is legitimate to search for alternative mechanisms for neutrino mass generation, that keep the spectrum of SM intact, except perhaps for the existence of right handed neutrinos that are allowed. Such minimal, non supersymmetric extensions of the Standard Model with three in fact right-handed Majorana neutrinos complementing the three active left-handed neutrinos (termed νMSM), have been proposed [5], in a way consistent with current cosmology. Such models are characterised by relatively light right-handed neutrinos, two of which are almost degenerate, with masses of order GeV, and a much lighter one, almost decoupled, with masses in the keV range, which may play the role of warm dark matter. The keV neutrino warm dark matter has been argued [6] to play an important rôle in the galactic structure, and more general in providing a resolution to the so-called “small-scale cosmology crisis”, that is discrepancies between observations and ΛCDM-model-based simulations for the dark matter distribution in galaxies. The right-handed neutrinos serve the purpose of generating, through seesaw type mechanisms, the active neutrino mass spectrum, consistent with observed ﬂavour oscillations. However, there are no suggestions for microscopic mechanisms for the generation of the righthanded neutrino mass spectrum in such scenarios.

Motivated by these facts we review in the next section 2, an alternative proposal for righthanded Majorana neutrino mass generation [7], through the anomalous interaction of these neutrinos with axion-like ﬁeld that exist in string-inspired extensions of the standard model. These models contain (in their massless gravitational multiplet) a spin-one antisymmetric tensor (KalbRamond) ﬁeld, and there is an abelian gauge invariance that implies the presence of the latter only through its (three-rank) covariant, totally antisymmetry, tensor ﬁeld strength Hµνρ , which in terms of the (lowest order in derivatives - we restrict ourselves to) string effective action appear as a a totally antisymmetric part of a torsion. The latter couples, via the gravitational covariant derivative, to the axial fermion current, summed up over all fermion species in the model, including righthanded neutrinos. The generation of (right-handed, sterile) neutrino masses in that case proceeds, as we shall review below, via chiral anomalous three-loop graphs of neutrinos interacting with the totally antisymmetric torsion quantum KR ﬁeld. In four space-time dimensions, the latter is represented as an axion ﬁeld, whose (kinetic) mixing with ordinary axion ﬁelds, that in turn interact

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with the Majorana right-handed neutrinos via chirality changing Yukawa couplings, is held responsible for the right-handed Majorana neutrino mass generation. We discuss speciﬁcally under which circumstances low (in the keV range) masses for the lightest of the sterile right-handed neutrinos can be generated, which may have important implications for the (warm) dark matter sector of the Universe. These are discussed in section 3, where it is argued that sterile neutrinos with masses of a few tens of keV can play an important rôle in providing agreement between theory and observations for the galactic structure, and more generally for alleviating some of the tensions of the so-called “small-scale” cosmology “crisis” [6]. Our conclusions are presented in section 4.

2. Axions and radiatively generated Majorana neutrino masses

Let us commence our discussion by considering ﬁrst a rather generic discussion concerning

the propagation of Dirac fermions in a torsionful space-time. The extension to the Majorana case is

straightforward. We shall restrict ourselves to the speciﬁc case of interest to us here, in which the

(totally antisymmetric) torsion is provided by the antisymmetric tensor Kalb-Ramond ﬁeld later

on.

The relevant action reads:

i Sψ = 2

d4x√−g ψγ µ D µ ψ − (D µ ψ)γ µ ψ

(2.1)

where D µ = ∇µ + . . . , is the covariant derivative (including gravitational and gauge-ﬁeld con-

nection parts, in case the fermions are charged). The overline above the covariant derivative,

i.e. ∇µ , denotes the presence of torsion, which is introduced through the torsionful spin connec-

tion: ωabµ = ωabµ + Kabµ , where Kabµ is the contorsion tensor. The latter is related to the torsion

two-form

Ta

=

d

ea

+ωa

∧ eb

via

[8,

9]:

Kabc

=

1 2

Tcab − Tabc − Tbcd

. The presence of torsion in

the covariant derivative in the action (2.1) leads, apart from the standard terms in manifolds without

torsion, to an additional term involving the axial current

J5µ ≡ ψγ µ γ5ψ .

(2.2)

The relevant part of the action reads:

Sψ − 34 d4√−g Sµ ψγ µ γ5ψ = − 43 S ∧ J5

(2.3)

where S = T is the dual of T: Sd = 31! εabcdTabc.

We next remark that the torsion tensor can be decomposed into its irreducible parts [8], of

which

Sd

is

the

pseudoscalar

axial

vector:

Tµ ν ρ

=

1 3

Tν gµρ − Tρ gµν

− 31! εµνρσ Sσ + qµνρ , with

εµνρσ qνρσ = qνρν = 0. This implies that the contorsion tensor undergoes the following decompo-

sition:

Kabc = 1 εabcd Sd + Kabc 2

(2.4)

where K includes the trace vector Tµ and the tensor qµνρ parts of the torsion tensor. The gravitational part of the action can then be written as: SG = 2κ12 d4x√−g R + ∆ +

3 4κ 2

S ∧ S, where ∆ = Kλµν Kνµλ − Kµνν Kµλ λ , with the hatted notation deﬁned in (2.4).

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In a quantum gravity setting, where one integrates over all ﬁelds, the torsion terms appear as non propagating ﬁelds and thus they can be integrated out exactly. The authors of [9] have observed though that the classical equations of motion identify the axial-pseudovector torsion ﬁeld Sµ with the axial current, since the torsion equation yields

Kµab = − 41 ecµ εabcdψγ5γ˜dψ .

(2.5)

From this it follows d S = 0, leading to a conserved “torsion charge” Q = S. To maintain this conservation in quantum theory, they postulated d S = 0 at the quantum level, which can be achieved by the addition of judicious counter terms. This constraint, in a path-integral formulation of quantum gravity, is then implemented via a delta function constraint, δ (d S), and the latter via the well-known trick of introducing a Lagrange multiplier ﬁeld Φ(x) ≡ (3/κ2)1/2b(x). Hence, the relevant torsion part of the quantum-gravity path integral would include a factor

Z∝

DS Db exp i

3 S ∧ S − 3 S ∧ J5 +

3

1/2

bd S

4κ 2

4

2κ 2

= Db exp − i 1 db ∧ db + 1 db ∧ J5 + 1 J5 ∧ J5 ,

2

fb

2

f

2 b

(2.6)

where fb = (3κ2/8)−1/2 = √MP and the non-propagating S ﬁeld has been integrated out. The reader

3π

should notice that, as a result of this integration, the corresponding effective ﬁeld theory contains a non-renormalizable repulsive four-fermion axial current-current interaction, characteristic of any torsionful theory [8].

The torsion term, being geometrical, due to gravity, couples universally to all fermion species, not only neutrinos. Thus, in the context of the SM of particle physics, the axial current (2.2) is expressed as a sum over fermion species

∑ J5µ ≡

ψiγ µ γ5ψi .

i=fermion species

(2.7)

In theories with chiral anomalies, like the quantum electrodynamics part of SM, the axial current is not conserved at the quantum level, due to anomalies, but its divergence is obtained by the one-loop result [10]:

∇ J5µ = e2 F µν F − 1 Rµνρσ R

µ

8π 2

µν 192π2

µνρσ

≡ G(A, ω) .

(2.8)

We may then partially integrate the second term in the exponent on the right-hand-side of (2.6) and

take into account (2.8). The reader should observe that in (2.8) the torsion-free spin connection has

been used. This can be achieved by the addition of proper counter terms in the action [9], which

can convert the anomaly from the initial G(A, ω) to G(A, ω). Using (2.8) in (2.6) one can then

obtain for the effective torsion action in theories with chiral anomalies, such as the QED part of the

SM:

Db exp − i 1 db ∧ db − 1 bG(A, ω) + 1 J5 ∧ J5 .

(2.9)

2

fb

2

f

2 b

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A concrete example of torsion is provided by string-inspired theories, where the totally antisymmetric component Sµ of the torsion is identiﬁed with the ﬁeld strength of the spin-one antisymmetric tensor (Kalb-Ramond (KR) [11]) ﬁeld Hµνρ = ∂[µ Bνρ], where the symbol [. . . ] denotes antisymmetrization of the appropriate indices. The string theory effective action depends only on Hµνρ as a consequence of the “gauge symmetry” Bµν → Bµν + ∂[µ Θν] that characterises all string theories. It can be shown [12] that the terms of the effective action up to and including quadratic order in the Regge slope parameter α , of relevance to the low-energy (ﬁeld-theory) limit of string theory, which involve the H-ﬁeld strength, can be assembled in such a way that only torsionful Christoffel symbols, Γνµρ appear: Γνµρ = Γνµρ + √κ3 Hνµρ = Γρµν , where Γνµρ = Γρµν is the ordinary, torsion-free, symmetric connection, and κ is the gravitational constant. In four space-time dimensions, the dual of the H-ﬁeld is indeed the derivative of an (KR) axion-like ﬁeld, analogous to the ﬁeld b above, Hµνρ ∝ εµνρσ ∂ σ b(x). For completeness we mention at this point that background geometries with (approximately) constant background Hi jk torsion (corresponding to a linear in cosmic time KR axion), where Latin indices denote spatial components of the four-dimensional space-time, may characterise the early universe. In such cases, the H-torsion background constitutes extra source of CP violation, necessary for lepotogenesis, and through Baryon-minus-Lepton-number (B-L) conserving processes, Baryogenesis, and thus the observed matter-antimatter asymmetry in the Universe [13]. Today of course any torsion background should be strongly suppressed, due to the lack of any experimental evidence for it [14]. Scenarios as to how such cosmologies can evolve so as to guarantee the absence of any appreciable traces of torsion today can be found in [13].

In what follows we shall consider the effects of the quantum ﬂuctuations of such a KR Htorsion, which survive the absence of any torsion background. An important aspect of the coupling of the H-torsion (or, equivalently, the KR axion quantum ﬁeld b(x)) to the fermionic matter discussed above is its shift symmetry, characteristic of an axion ﬁeld. Indeed, by shifting the ﬁeld b(x) by a constant: b(x) → b(x) + c, the action (2.9) only changes by total derivative terms, such as c Rµνρσ Rµνρσ and c F µν Fµν . These terms are irrelevant for the equations of motion and the induced quantum dynamics, provided the ﬁelds fall off sufﬁciently fast to zero at space-time inﬁnity. The scenario for the anomalous Majorana mass generation through torsion proposed in [7], and reviewed here, consists of augmenting the effective action (2.9) by terms that break such a shift symmetry. To illustrate this last point, we ﬁrst couple the KR axion b(x) to another pseudoscalar axion ﬁeld a(x). In string-inspired models, such pseudoscalar axion a(x) may be provided by the string moduli [15]. The proposed coupling occurs through a mixing in the kinetic terms of the two ﬁelds. To be speciﬁc, we consider the action (henceforth we restrict ourselves to right-handed Majorana neutrino fermion ﬁelds):

S=

d4x√−g 1 (∂ b)2 + b(x) Rµνρσ R + 1 J5 J5µ + γ(∂ b) (∂ µ a) + 1 (∂ a)2

2µ

192π2 fb

µνρσ

2

f

2 b

µ

µ

2µ

−yaia

ψ

C R

ψR

−

ψ

R

ψRC

+... ,

(2.10)

where the . . . indicate terms in the low-energy string effective action, including SM ones, that are

not of direct relevance to our purposes in the present article. Above, ψRC = (ψR)C is the chargeconjugate right-handed fermion ψR, Jµ5 = ψγµ γ5ψ is the axial current of the four-component Majorana fermion ψ = ψR + (ψR)C, and γ is a real parameter to be constrained later on. Here, we

have ignored gauge ﬁelds, which are not of interest to us, and the possibility of a non-perturbative

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Figure 1: Typical higher-loop Feynman graph giving rise to anomalous fermion mass generation [7]. The black circle denotes the operator a(x) Rµνλρ Rµνλρ induced by KR torsion. The ﬁelds hµν (wavy lines) denote graviton ﬂuctuations. Straight lines with arrows denote right handed neutrino ﬁelds and their conjugates.

mass Ma for the pseudoscalar ﬁeld a(x). Moreover, we remind the reader that the repulsive selfinteraction fermion terms are due to the existence of torsion in the Einstein-Cartan theory. The Yukawa coupling ya of the axion moduli ﬁeld a to right-handed sterile neutrino matter ψR may be due to non perturbative effects. These terms break the shift symmetry: a → a + c.

It is convenient to diagonalize the axion kinetic terms by redeﬁning the KR axion ﬁeld as follows: b(x) → b (x) ≡ b(x) + γa(x). This implies that the effective action (2.10) becomes:

S = d4x√−g 12 (∂µ b )2 + 12 1 − γ2 (∂µ a)2

+ 1 J5 J5µ + b (x) − γa(x) Rµνρσ R − yaia

2

f

2 b

µ

192π2 fb

µνρσ

ψ

C R

ψR

−

ψ

R

ψRC

+... .

(2.11)

Thus we observe that the b ﬁeld has decoupled and can be integrated out in the path integral, leaving behind an axion ﬁeld a(x) coupled both to matter fermions and to the operator Rµνρσ Rµνρσ , thereby playing now the rôle of the torsion ﬁeld. We observe though that the approach is only valid for |γ| < 1 , otherwise the axion ﬁeld would appear as a ghost, i.e. with the wrong sign of its kinetic terms, which would indicate an instability of the model. This is the only restriction of the parameter γ. In this case we may redeﬁne the axion ﬁeld so as to appear with a canonical normalised kinetic term, implying the effective action:

Sa = −

d4x√−g 1 (∂ a)2 −

γ a(x)

Rµνρσ R

2µ

192π2 fb 1 − γ2

µνρσ

iya

a

ψ

C R

ψR

−

ψ

R

ψRC

+

1 J5J5µ

.

1−γ2

2

f

2 b

µ

(2.12)

Evidently, the action Sa in (2.12) corresponds to a canonically normalised axion ﬁeld a(x), coupled both to the curvature of space-time, à la torsion, with a modiﬁed coupling γ/(192π2 fb 1 − γ2),

and to fermionic matter with chirality-changing Yukawa-like couplings of the form ya/ 1 − γ2.

The mechanism for the anomalous Majorana mass generation is shown in Fig. 1. We may now

estimate the two-loop Majorana neutrino mass in quantum gravity with an effective UV energy cut-

off Λ. Adopting the effective ﬁeld-theory framework of [16], the gravitationally induced Majorana

mass for right-handed neutrinos, MR, is estimated to be:

√

1

ya γ κ4Λ6

3 ya γ κ5Λ6

MR ∼

(16π 2 )2

192π2 fb(1 − γ2)

=

√ 49152 8 π4(1 − γ2)

.

(2.13)

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In a UV complete theory such as strings, the cutoff Λ and the Planck mass scale MP are related. In particular, as already mentioned, in string theory there are several axion-like (pseudoscalar)

ﬁelds ai(x), i = 1, 2, . . . n, originating from ﬂux ﬁelds that exist in the spectrum [15], in addition to the aforementioned Bµν Kalb-Ramond ﬁeld. One can then assume [7] the existence of Yukawa couplings with right-handed neutrinos, provided some non-perturbative instanton effects are responsible for a breaking of the shift symmetry. These string-theory axion ﬁelds could mix with each other. Such a mixing can give rise to reduced UV sensitivity of the two-loop graph shown in Fig. 1. To make this point explicit, let us consider a scenario with n axion ﬁelds, a1,2,...,n, of which only a1 has a kinetic mixing term γ with the KR axion b and only an has a Yukawa coupling ya to right-handed neutrinos ψR. The other axions a2,3,...,n have a next-to-neighbour mixing pattern. In such a model, the kinetic terms of the effective action are given by

Sakin =

∑ ∑ d4x √−g 1 n (∂ a )2 − M2 + γ(∂ b)(∂ µ a ) − 1 n−1 δ M2 a a ,

2 i=1 µ i i

µ 1 2 i=1 i,i+1 i i+1

(2.14)

where the mixing mass terms δ Mi2,i+1 are constrained to be δ Mi2,i+1 < MiMi+1, so as to obtain a stable positive mass spectrum for all axions. As a consequence of the next-to-neighbour mixing,

the UV behaviour of the off-shell transition a1 → an, described by the propagator matrix element

∆a1an(p), changes drastically, i.e. ∆a1an(p) ∝ 1/(p2)n ∼ 1/E2n. Assuming, for simplicity, that all

axion masses and mixings are equal, i.e. Mi2 = Ma2 and δ Mi2,i+1 = δ Ma2, the anomalously generated

Majorana mass may be estimated to be

√

M∼

3

ya

γ

κ5 √

Λ6−2n

(δ

Ma2

)n

,

R 49152 8 π4(1 − γ2)

(2.15)

for n ≤ 3, and

√ M ∼ 3 ya√γ κ5(δ Ma2)3 (δ Ma2)n−3 ,

R 49152 8 π4(1 − γ2) (Ma2)n−3

(2.16)

for n > 3. It is then not difﬁcult to see that three axions a1,2,3 with next-to-neighbour mixing as

discussed above would be sufﬁcient to obtain a UV ﬁnite (cut-off-Λ-independent) result for MR at

the two-loop level. Of course, beyond the two loops, MR will depend on higher powers of the energy cut-off Λ, i.e. Λn>6, but if κΛ 1, these higher-order effects are expected to be subdominant.

In the above n-axion-mixing scenarios, we note that the anomalously generated Majorana mass term will only depend on the mass-mixing parameters δ Ma2 of the axion ﬁelds and not on their masses themselves, as long as n ≤ 3. Instead, for axion-mixing scenarios with n > 3, the induced Majorana neutrino masses are proportional to the factor (δ Ma2/Ma2)n, which gives rise to an additional suppression for heavy axions with masses Ma δ Ma.

In the multi-axion models outlined above, the anomalously generated Majorana neutrino mass

MR will still depend on the Yukawa coupling ya and the torsion-axion kinetic mixing coefﬁcient

γ, besides the assumed UV completion scale Λ of quantum gravity. In order to get an estimate

of the size of MR, we treat the axion masses Ma and mass-mixings δ Ma as free parameters to be

constrained by phenomenology. Let us assume a n-axion-mixing models with n ≥ 3, in which the

axion mass mixing δ Ma and their masses Ma are of the same order, i.e. δ Ma/Ma ∼ 1. In this case,

employing (2.15), we may estimate the Majorana neutrino mass MR to be

MR 10−3 ya γ (δ Ma)6 10−3 ya γ δ Ma 5

∼

∼

.

Ma 1 − γ2 Ma MP5

1 − γ2 MP

(2.17)

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For

axion

masses

Ma

≤

1

TeV

considered

in

the

literature

thus

far,

we

ﬁnd

that

MR/Ma

< ∼

10−83,

which implies extra-ordinary small Majorana masses MR. An obvious caveat to this result would

be to have ultra-heavy axion masses Ma close to the GUT scale and/or ﬁne-tune the torsion-axion

kinetic

mixing

parameter

γ

to

1,

in

a

way

such

that

the

factor

γ 1−γ 2

compensates

for

the

mass

suppression (δ Ma/MP)5 in (2.17). The latter possibility, however, might re√sult in an unnaturally

large (non-perturbative) “effective” Yukawa coupling yeaff ≡ ya/

1

−

γ2

> ∼

4π in (2.12), which

will bring us outside the perturbative framework that we have been considering here. A more

detailed phenomonological and astrophysical analysis of all possible axion-mixing scenarios falls

beyond the scope of the present talk.

3. Implications for the Dark Sector of the Universe

It is interesting from a cosmological viewpoint to provide a numerical estimate of the anomalously generated Majorana mass MR given in (2.13) above. Assuming that γ 1, the size of MR may be estimated from (2.13) to be

MR ∼ (3.1 × 1011 GeV) ya 10−3

γ 10−1

Λ6 2.4 × 1018 GeV .

(3.1)

Obviously, the generation of MR is highly model dependent. Taking, for example, the quantum gravity scale to be Λ = 1017 GeV, we ﬁnd that MR is at the TeV scale, for ya = 10−3 and γ = 0.1. However, if we take the quantum gravity scale to be close to the GUT scale, i.e. Λ = 1016 GeV, we obtain a right-handed neutrino mass MR ∼ 16 keV, for the choice ya = γ = 10−3. This is in the preferred ballpark region for the (right-handed) sterile neutrino ψR to qualify as a warm dark matter (WDM) [17, 18].

Moreover, as discussed in [6], the introduction of appropriate self-interactions among the srerile right-handed neutrino dark matter (DM), can serve as a means of providing an explanation of the observed core-halo galactic structure, upon assuming speciﬁc core proﬁles, speciﬁcally, the Rufﬁni-Argüelles-Rueda proﬁle [19]. In addition, such self-interacting neutrino WDM models can also alleviate some of the discrepancies between numerical simulations based on ΛCDM model and observations at galactic scales (“small-scale cosmology crisis”), provided the self interaction total cross section σSIDM of a dark matter particle with mass m lies in the range

σSIDM/m 0.1 ≤ cm2 g−1 ≤ 0.47

(3.2)

where the upper limit has been inferred by recent studies of several merging galaxies, using novel techniques and observables [20].

For completeness, we mention that discrepancies between observations at galactic scales and ΛCDM-numerical simulations, appear in three areas:

(i) The Core-Cusp problem (or, as is also known, the cuspy-halo problem), refers to a discrepancy between the observed dark matter density proﬁles of low-mass galaxies and the density proﬁles predicted by cosmological N-body simulations. Characteristically, all the ΛCDM-based (DM only) simulations form dark matter halos which have “cuspy" dark matter distributions, with the density increasing steeply, i.e. as ρ ∝ r−1, at small radii. This is, e.g., evidenced in the standard

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Navarro-Frenk-White (NFW) DM proﬁle [21]. On the contrary, the rotation curves of most of the observed dwarf galaxies indicate ﬂat central density proﬁles ("cores") [22].

(ii) The “missing satellite problem” (or, as is also known, the dwarf galaxy problem), arises from a discrepancy between ΛCDM-based numerical cosmological simulations that predict the evolution of the distribution of matter in the universe - pointing towards a hierarchical clustering of DM (where smaller halos merge to form larger halos) - and observations. Although there seem to be enough observed normal-sized galaxies to account for such a numerical distribution, the number of dwarf galaxies is orders of magnitude lower than that expected from the simulations. As a concrete example, we mention that there were observed to be around 38 dwarf galaxies in the Local Group, and only around 11 orbiting the Milky Way, yet one dark matter simulation predicted around 500 Milky Way dwarf satellites [23].

(iii) The too big to fail problem, that is a discrepancy arising between the most massive subhaloes predicted in (dissipationless) ΛCDM simulations and the observed dynamics of the brightest dwarf spheroidal galaxies in the Milky way. In other words, the ΛCDM simulations predict that the most massive subhaloes of the Milky way are too dense to host any of its bright satellites, with luminosity higher than 105 the luminosity of the Sun [24].

All three problems have their root in the fact that the cold DM particles, which the ΛCDM simulations rely upon, have too short free streaming length during the epochs of galaxy formation, and therefore they form too clumped and too many structures compared to those observed. Although some of these problems could be partly alleviated by more precise observations and/or including baryonic feedback, and may have their root in astrophysical reasons (for instance the missing satellite problem may be attributed partly to the fact that dwarf galaxies have a tendency to merge or may have been stripped apart gravitationally by larger galaxies and hence invisible), nonetheless it would be desirable if a uniﬁed explanation is found within the context of fundamental physics, and this is the point of view taken in the approach of including self interactions among the DM particles.

In this latter respect, WDM sterile neutrinos, with masses in the range of tens of keV, as in the νMSM model [5], have been argued to play a rôle, either in the too-big-to-fail problem [25], or, in the self-interacting case [6], for resolving the Core-Cusp problem, because the density proﬁles based on fermionic phase-space distributions [19, 6] develop always an extended plateau on halo scales (starting immediately after the quantum core). The rôle of self interactions (attractive vector interactions due to massive vector ﬁelds in the model of [6]) is to make the core more compact than in the non-interacting case [19]. For sterile neutrino masses mχ in the range [6]

47 keV mχ c2 345 keV ,

(3.3)

where the (robust) upper bound corresponds to the limit of gravitational collapse, the correct DM halo properties of galaxies (in agreement with observations) are guaranteed within the context of the galaxy proﬁle proposed in [19], whilst at the same time the fermi gas of fermionic DM may provide an alternative to the black hole in the centre of the galaxy. It must be stressed though that the lower bound depends crucially on the details of the galactic proﬁle, as well as of the existence (and its detailed properties) of a supermassive black hole in the galactic centre. Hence it is much less robust than the upper bound, and can be further relaxed.

8

Axions and massive Majorana neutrinos

Nick E. Mavromatos

PoS(CORFU2017)003

sin22θ

10-7

10-8

Ωνχs > Ωoχbs

10-9 10-10 10-11 10-12

MW sataellite pcohnasstenrdaspaccoeunts ints νMSM

10-13 10-14 10-11500

Ωνχs < Ωoχbs

NuSTAR GC 2016 101

mχ [ keV ]

cPXor-nreavyious straints

102

Figure 2: Up-to-date Cosmological and Astrophysical Constraints for the νMSM WDM sterile neutrino mass versus mixing angle with the Standard Model sector (picture from ref. [26]). The recent analysis of Perez et al. [26], employs NuSTAR (Nuclear Spectroscopic Telescope Array) [27] observations of the Galactic Centre, together with additional constraints associated with DM production and its rôle in the structure formation in the Universe, which stem from the requirement that DM comprises predominantly of the lightest of the sterile neutrinos of νMSM. These latest studies exclude almost half of the previously allowed region of νMSM mixing-angle-mass parameters (thick (red) line and hatched region in the ﬁgure, the white region being the currently allowed one).

In this respect, we mention a crucial feature of the model of [6]. In that analysis, any mixing of the sterile neutrinos with the Standard Model sector had been ignored, assumed very small. In the spirit of [6], the sterile neutrino might constitute only one component of the DM in the Universe, with other equally dominant DM species in the underlying microscopic model playing different rôles in the structure formation and energy budget of the Cosmos. On the other hand, in the νMSM framework [5], the mixing of the lightest of the sterile right-handed neutrinos which, notably, is assumed to play the rôle of the dominant DM species, having a life-time longer than the Universe age, can be constrained mainly by its decay into X-rays and the DM production and contribution to the large scale structure. The recent constraints of Perez et al. [26], employing observations of the Galactic Centre with the Nuclear Spectroscopic Telescope Array (NuSTAR) [27], when combined with the basic assumption of νMSM [5] that the lightest of the sterile neutrinos provides the dominant component of DM, thus leading to additional constraints for sufﬁcient DM production in the early Universe and efﬁcient rôle in structure formation [17, 18], leaves a very small allowed window in the mixing-angle-sterile-neutrino-mass parameter plane (see white region in ﬁg. 2):

10 keV mχ c2 16 keV ,

(3.4)

As we discussed above (cf. (3.1)), within the model of [7], sterile neutrinos can acquire radiatively masses in this range for quite natural values of the parameters.

However, the above region is quite sensitive to the galactic proﬁle used (the authors of [26] used versions of the Navarro-Frenk-White [21] and Einasto [28] proﬁles in their study). An analysis making use of the Rufﬁni-Argüelles-Rueda proﬁle [19], employed in the work of [6], may affect the

9

Axions, Majorana neutrino masses and implications for the dark sector of the Universe

Nick E. Mavromatos∗† King’s College London, Physics Department, Theoretical Particle Physics and Cosmology Group, Strand, London WC2R 2LS, UK E-mail: [email protected]

We discuss a novel mechanism for generating masses for right-handed Majorana neutrinos, that goes beyond the conventional seesaw. The mechanism involves quantum ﬂuctuations of a massless Kalb-Ramond (KR) pseudoscalar (axion-like) ﬁeld, which exists in string-inspired extensions of the standard model. We assume a kinetic mixing of the KR ﬁeld with ordinary (massive in general) axions, which also exist in string models, and which are assumed to couple to Majorana (right-handed) neutrinos via non-perturbatively-generated chirality changing Yukawa couplings, breaking the axion-shift symmetry. No vacuum expectation value is assumed for the axion ﬁelds. Majorana masses for the right-handed neutrinos are generated radiatively as a result of anomalous higher-loop axion-neutrino couplings. Implications for the Dark sector of the Universe are discussed. In particular, we explore the possibility of generating masses for the right-handed neutrinos of order of a few tens of keV. Such neutrinos could play an important rôle in the galactic structure, and more generally they could serve as a warm dark matter component in the Universe, providing a potential resolution to the so-called small-scale cosmology “crisis”, that is, discrepancies between observations at galactic scales and numerical simulations based on the ΛCDM model. If such scenarios are realised in nature, they might imply that Dark Matter consists of more than one species (warm and cold), with distinct rôles in the structure and evolution of the Universe: the cold one being still responsible for the large scale structure of the Universe, in accordance to the predictions of the ΛCDM model, which agree with a plethora of cosmological observations, but the warm component (keV sterile neutrino) playing a crucial rôle in the (observed) galactic structure.

Corfu Summer Institute 2017 ’School and Workshops on Elementary Particle Physics and Gravity’ 2-28 September 2017 Corfu, Greece

∗Speaker. †This work is supported in part by STFC (UK) under the research grant ST/P000258/1

c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).

https://pos.sissa.it/

Axions and massive Majorana neutrinos

Nick E. Mavromatos

PoS(CORFU2017)003

1. Introduction and Summary

The discovery [1] of the Higgs boson at the CERN Large Hadron Collider (LHC) in 2012 constitutes an important milestone for the Ultra-Violet (UV) completion of the Standard Model (SM). Although the so-called Higgs mechanism may well explain the generation of most of the particle masses in the SM, the origin of the small neutrino masses still remains an open issue. In particular, the observed smallness of the light neutrino masses may naturally be explained through the see-saw mechanism [2], which necessitates the Majorana nature of the light (active) neutrinos and postulates the presence of heavy right-handed Majorana partners of mass MR. The right-handed Majorana mass MR is usually considered to be much larger than the lepton or quark masses. The origin of MR has been the topic of several extensions of the SM in the literature, within the framework of quantum ﬁeld theory [2, 3] and string theory [4]. However, up to now, there is no experimental evidence for right-handed neutrinos or for any extension of SM, as a matter of fact, although some optimism of discovering supersymmetry in the next round of LHC (operating at 14 TeV energies) exists among particle physicists.

Until therefore such extensions of the SM are discovered, it is legitimate to search for alternative mechanisms for neutrino mass generation, that keep the spectrum of SM intact, except perhaps for the existence of right handed neutrinos that are allowed. Such minimal, non supersymmetric extensions of the Standard Model with three in fact right-handed Majorana neutrinos complementing the three active left-handed neutrinos (termed νMSM), have been proposed [5], in a way consistent with current cosmology. Such models are characterised by relatively light right-handed neutrinos, two of which are almost degenerate, with masses of order GeV, and a much lighter one, almost decoupled, with masses in the keV range, which may play the role of warm dark matter. The keV neutrino warm dark matter has been argued [6] to play an important rôle in the galactic structure, and more general in providing a resolution to the so-called “small-scale cosmology crisis”, that is discrepancies between observations and ΛCDM-model-based simulations for the dark matter distribution in galaxies. The right-handed neutrinos serve the purpose of generating, through seesaw type mechanisms, the active neutrino mass spectrum, consistent with observed ﬂavour oscillations. However, there are no suggestions for microscopic mechanisms for the generation of the righthanded neutrino mass spectrum in such scenarios.

Motivated by these facts we review in the next section 2, an alternative proposal for righthanded Majorana neutrino mass generation [7], through the anomalous interaction of these neutrinos with axion-like ﬁeld that exist in string-inspired extensions of the standard model. These models contain (in their massless gravitational multiplet) a spin-one antisymmetric tensor (KalbRamond) ﬁeld, and there is an abelian gauge invariance that implies the presence of the latter only through its (three-rank) covariant, totally antisymmetry, tensor ﬁeld strength Hµνρ , which in terms of the (lowest order in derivatives - we restrict ourselves to) string effective action appear as a a totally antisymmetric part of a torsion. The latter couples, via the gravitational covariant derivative, to the axial fermion current, summed up over all fermion species in the model, including righthanded neutrinos. The generation of (right-handed, sterile) neutrino masses in that case proceeds, as we shall review below, via chiral anomalous three-loop graphs of neutrinos interacting with the totally antisymmetric torsion quantum KR ﬁeld. In four space-time dimensions, the latter is represented as an axion ﬁeld, whose (kinetic) mixing with ordinary axion ﬁelds, that in turn interact

1

Axions and massive Majorana neutrinos

Nick E. Mavromatos

PoS(CORFU2017)003

with the Majorana right-handed neutrinos via chirality changing Yukawa couplings, is held responsible for the right-handed Majorana neutrino mass generation. We discuss speciﬁcally under which circumstances low (in the keV range) masses for the lightest of the sterile right-handed neutrinos can be generated, which may have important implications for the (warm) dark matter sector of the Universe. These are discussed in section 3, where it is argued that sterile neutrinos with masses of a few tens of keV can play an important rôle in providing agreement between theory and observations for the galactic structure, and more generally for alleviating some of the tensions of the so-called “small-scale” cosmology “crisis” [6]. Our conclusions are presented in section 4.

2. Axions and radiatively generated Majorana neutrino masses

Let us commence our discussion by considering ﬁrst a rather generic discussion concerning

the propagation of Dirac fermions in a torsionful space-time. The extension to the Majorana case is

straightforward. We shall restrict ourselves to the speciﬁc case of interest to us here, in which the

(totally antisymmetric) torsion is provided by the antisymmetric tensor Kalb-Ramond ﬁeld later

on.

The relevant action reads:

i Sψ = 2

d4x√−g ψγ µ D µ ψ − (D µ ψ)γ µ ψ

(2.1)

where D µ = ∇µ + . . . , is the covariant derivative (including gravitational and gauge-ﬁeld con-

nection parts, in case the fermions are charged). The overline above the covariant derivative,

i.e. ∇µ , denotes the presence of torsion, which is introduced through the torsionful spin connec-

tion: ωabµ = ωabµ + Kabµ , where Kabµ is the contorsion tensor. The latter is related to the torsion

two-form

Ta

=

d

ea

+ωa

∧ eb

via

[8,

9]:

Kabc

=

1 2

Tcab − Tabc − Tbcd

. The presence of torsion in

the covariant derivative in the action (2.1) leads, apart from the standard terms in manifolds without

torsion, to an additional term involving the axial current

J5µ ≡ ψγ µ γ5ψ .

(2.2)

The relevant part of the action reads:

Sψ − 34 d4√−g Sµ ψγ µ γ5ψ = − 43 S ∧ J5

(2.3)

where S = T is the dual of T: Sd = 31! εabcdTabc.

We next remark that the torsion tensor can be decomposed into its irreducible parts [8], of

which

Sd

is

the

pseudoscalar

axial

vector:

Tµ ν ρ

=

1 3

Tν gµρ − Tρ gµν

− 31! εµνρσ Sσ + qµνρ , with

εµνρσ qνρσ = qνρν = 0. This implies that the contorsion tensor undergoes the following decompo-

sition:

Kabc = 1 εabcd Sd + Kabc 2

(2.4)

where K includes the trace vector Tµ and the tensor qµνρ parts of the torsion tensor. The gravitational part of the action can then be written as: SG = 2κ12 d4x√−g R + ∆ +

3 4κ 2

S ∧ S, where ∆ = Kλµν Kνµλ − Kµνν Kµλ λ , with the hatted notation deﬁned in (2.4).

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Nick E. Mavromatos

PoS(CORFU2017)003

In a quantum gravity setting, where one integrates over all ﬁelds, the torsion terms appear as non propagating ﬁelds and thus they can be integrated out exactly. The authors of [9] have observed though that the classical equations of motion identify the axial-pseudovector torsion ﬁeld Sµ with the axial current, since the torsion equation yields

Kµab = − 41 ecµ εabcdψγ5γ˜dψ .

(2.5)

From this it follows d S = 0, leading to a conserved “torsion charge” Q = S. To maintain this conservation in quantum theory, they postulated d S = 0 at the quantum level, which can be achieved by the addition of judicious counter terms. This constraint, in a path-integral formulation of quantum gravity, is then implemented via a delta function constraint, δ (d S), and the latter via the well-known trick of introducing a Lagrange multiplier ﬁeld Φ(x) ≡ (3/κ2)1/2b(x). Hence, the relevant torsion part of the quantum-gravity path integral would include a factor

Z∝

DS Db exp i

3 S ∧ S − 3 S ∧ J5 +

3

1/2

bd S

4κ 2

4

2κ 2

= Db exp − i 1 db ∧ db + 1 db ∧ J5 + 1 J5 ∧ J5 ,

2

fb

2

f

2 b

(2.6)

where fb = (3κ2/8)−1/2 = √MP and the non-propagating S ﬁeld has been integrated out. The reader

3π

should notice that, as a result of this integration, the corresponding effective ﬁeld theory contains a non-renormalizable repulsive four-fermion axial current-current interaction, characteristic of any torsionful theory [8].

The torsion term, being geometrical, due to gravity, couples universally to all fermion species, not only neutrinos. Thus, in the context of the SM of particle physics, the axial current (2.2) is expressed as a sum over fermion species

∑ J5µ ≡

ψiγ µ γ5ψi .

i=fermion species

(2.7)

In theories with chiral anomalies, like the quantum electrodynamics part of SM, the axial current is not conserved at the quantum level, due to anomalies, but its divergence is obtained by the one-loop result [10]:

∇ J5µ = e2 F µν F − 1 Rµνρσ R

µ

8π 2

µν 192π2

µνρσ

≡ G(A, ω) .

(2.8)

We may then partially integrate the second term in the exponent on the right-hand-side of (2.6) and

take into account (2.8). The reader should observe that in (2.8) the torsion-free spin connection has

been used. This can be achieved by the addition of proper counter terms in the action [9], which

can convert the anomaly from the initial G(A, ω) to G(A, ω). Using (2.8) in (2.6) one can then

obtain for the effective torsion action in theories with chiral anomalies, such as the QED part of the

SM:

Db exp − i 1 db ∧ db − 1 bG(A, ω) + 1 J5 ∧ J5 .

(2.9)

2

fb

2

f

2 b

3

Axions and massive Majorana neutrinos

Nick E. Mavromatos

PoS(CORFU2017)003

A concrete example of torsion is provided by string-inspired theories, where the totally antisymmetric component Sµ of the torsion is identiﬁed with the ﬁeld strength of the spin-one antisymmetric tensor (Kalb-Ramond (KR) [11]) ﬁeld Hµνρ = ∂[µ Bνρ], where the symbol [. . . ] denotes antisymmetrization of the appropriate indices. The string theory effective action depends only on Hµνρ as a consequence of the “gauge symmetry” Bµν → Bµν + ∂[µ Θν] that characterises all string theories. It can be shown [12] that the terms of the effective action up to and including quadratic order in the Regge slope parameter α , of relevance to the low-energy (ﬁeld-theory) limit of string theory, which involve the H-ﬁeld strength, can be assembled in such a way that only torsionful Christoffel symbols, Γνµρ appear: Γνµρ = Γνµρ + √κ3 Hνµρ = Γρµν , where Γνµρ = Γρµν is the ordinary, torsion-free, symmetric connection, and κ is the gravitational constant. In four space-time dimensions, the dual of the H-ﬁeld is indeed the derivative of an (KR) axion-like ﬁeld, analogous to the ﬁeld b above, Hµνρ ∝ εµνρσ ∂ σ b(x). For completeness we mention at this point that background geometries with (approximately) constant background Hi jk torsion (corresponding to a linear in cosmic time KR axion), where Latin indices denote spatial components of the four-dimensional space-time, may characterise the early universe. In such cases, the H-torsion background constitutes extra source of CP violation, necessary for lepotogenesis, and through Baryon-minus-Lepton-number (B-L) conserving processes, Baryogenesis, and thus the observed matter-antimatter asymmetry in the Universe [13]. Today of course any torsion background should be strongly suppressed, due to the lack of any experimental evidence for it [14]. Scenarios as to how such cosmologies can evolve so as to guarantee the absence of any appreciable traces of torsion today can be found in [13].

In what follows we shall consider the effects of the quantum ﬂuctuations of such a KR Htorsion, which survive the absence of any torsion background. An important aspect of the coupling of the H-torsion (or, equivalently, the KR axion quantum ﬁeld b(x)) to the fermionic matter discussed above is its shift symmetry, characteristic of an axion ﬁeld. Indeed, by shifting the ﬁeld b(x) by a constant: b(x) → b(x) + c, the action (2.9) only changes by total derivative terms, such as c Rµνρσ Rµνρσ and c F µν Fµν . These terms are irrelevant for the equations of motion and the induced quantum dynamics, provided the ﬁelds fall off sufﬁciently fast to zero at space-time inﬁnity. The scenario for the anomalous Majorana mass generation through torsion proposed in [7], and reviewed here, consists of augmenting the effective action (2.9) by terms that break such a shift symmetry. To illustrate this last point, we ﬁrst couple the KR axion b(x) to another pseudoscalar axion ﬁeld a(x). In string-inspired models, such pseudoscalar axion a(x) may be provided by the string moduli [15]. The proposed coupling occurs through a mixing in the kinetic terms of the two ﬁelds. To be speciﬁc, we consider the action (henceforth we restrict ourselves to right-handed Majorana neutrino fermion ﬁelds):

S=

d4x√−g 1 (∂ b)2 + b(x) Rµνρσ R + 1 J5 J5µ + γ(∂ b) (∂ µ a) + 1 (∂ a)2

2µ

192π2 fb

µνρσ

2

f

2 b

µ

µ

2µ

−yaia

ψ

C R

ψR

−

ψ

R

ψRC

+... ,

(2.10)

where the . . . indicate terms in the low-energy string effective action, including SM ones, that are

not of direct relevance to our purposes in the present article. Above, ψRC = (ψR)C is the chargeconjugate right-handed fermion ψR, Jµ5 = ψγµ γ5ψ is the axial current of the four-component Majorana fermion ψ = ψR + (ψR)C, and γ is a real parameter to be constrained later on. Here, we

have ignored gauge ﬁelds, which are not of interest to us, and the possibility of a non-perturbative

4

Axions and massive Majorana neutrinos

Nick E. Mavromatos

PoS(CORFU2017)003

Figure 1: Typical higher-loop Feynman graph giving rise to anomalous fermion mass generation [7]. The black circle denotes the operator a(x) Rµνλρ Rµνλρ induced by KR torsion. The ﬁelds hµν (wavy lines) denote graviton ﬂuctuations. Straight lines with arrows denote right handed neutrino ﬁelds and their conjugates.

mass Ma for the pseudoscalar ﬁeld a(x). Moreover, we remind the reader that the repulsive selfinteraction fermion terms are due to the existence of torsion in the Einstein-Cartan theory. The Yukawa coupling ya of the axion moduli ﬁeld a to right-handed sterile neutrino matter ψR may be due to non perturbative effects. These terms break the shift symmetry: a → a + c.

It is convenient to diagonalize the axion kinetic terms by redeﬁning the KR axion ﬁeld as follows: b(x) → b (x) ≡ b(x) + γa(x). This implies that the effective action (2.10) becomes:

S = d4x√−g 12 (∂µ b )2 + 12 1 − γ2 (∂µ a)2

+ 1 J5 J5µ + b (x) − γa(x) Rµνρσ R − yaia

2

f

2 b

µ

192π2 fb

µνρσ

ψ

C R

ψR

−

ψ

R

ψRC

+... .

(2.11)

Thus we observe that the b ﬁeld has decoupled and can be integrated out in the path integral, leaving behind an axion ﬁeld a(x) coupled both to matter fermions and to the operator Rµνρσ Rµνρσ , thereby playing now the rôle of the torsion ﬁeld. We observe though that the approach is only valid for |γ| < 1 , otherwise the axion ﬁeld would appear as a ghost, i.e. with the wrong sign of its kinetic terms, which would indicate an instability of the model. This is the only restriction of the parameter γ. In this case we may redeﬁne the axion ﬁeld so as to appear with a canonical normalised kinetic term, implying the effective action:

Sa = −

d4x√−g 1 (∂ a)2 −

γ a(x)

Rµνρσ R

2µ

192π2 fb 1 − γ2

µνρσ

iya

a

ψ

C R

ψR

−

ψ

R

ψRC

+

1 J5J5µ

.

1−γ2

2

f

2 b

µ

(2.12)

Evidently, the action Sa in (2.12) corresponds to a canonically normalised axion ﬁeld a(x), coupled both to the curvature of space-time, à la torsion, with a modiﬁed coupling γ/(192π2 fb 1 − γ2),

and to fermionic matter with chirality-changing Yukawa-like couplings of the form ya/ 1 − γ2.

The mechanism for the anomalous Majorana mass generation is shown in Fig. 1. We may now

estimate the two-loop Majorana neutrino mass in quantum gravity with an effective UV energy cut-

off Λ. Adopting the effective ﬁeld-theory framework of [16], the gravitationally induced Majorana

mass for right-handed neutrinos, MR, is estimated to be:

√

1

ya γ κ4Λ6

3 ya γ κ5Λ6

MR ∼

(16π 2 )2

192π2 fb(1 − γ2)

=

√ 49152 8 π4(1 − γ2)

.

(2.13)

5

Axions and massive Majorana neutrinos

Nick E. Mavromatos

PoS(CORFU2017)003

In a UV complete theory such as strings, the cutoff Λ and the Planck mass scale MP are related. In particular, as already mentioned, in string theory there are several axion-like (pseudoscalar)

ﬁelds ai(x), i = 1, 2, . . . n, originating from ﬂux ﬁelds that exist in the spectrum [15], in addition to the aforementioned Bµν Kalb-Ramond ﬁeld. One can then assume [7] the existence of Yukawa couplings with right-handed neutrinos, provided some non-perturbative instanton effects are responsible for a breaking of the shift symmetry. These string-theory axion ﬁelds could mix with each other. Such a mixing can give rise to reduced UV sensitivity of the two-loop graph shown in Fig. 1. To make this point explicit, let us consider a scenario with n axion ﬁelds, a1,2,...,n, of which only a1 has a kinetic mixing term γ with the KR axion b and only an has a Yukawa coupling ya to right-handed neutrinos ψR. The other axions a2,3,...,n have a next-to-neighbour mixing pattern. In such a model, the kinetic terms of the effective action are given by

Sakin =

∑ ∑ d4x √−g 1 n (∂ a )2 − M2 + γ(∂ b)(∂ µ a ) − 1 n−1 δ M2 a a ,

2 i=1 µ i i

µ 1 2 i=1 i,i+1 i i+1

(2.14)

where the mixing mass terms δ Mi2,i+1 are constrained to be δ Mi2,i+1 < MiMi+1, so as to obtain a stable positive mass spectrum for all axions. As a consequence of the next-to-neighbour mixing,

the UV behaviour of the off-shell transition a1 → an, described by the propagator matrix element

∆a1an(p), changes drastically, i.e. ∆a1an(p) ∝ 1/(p2)n ∼ 1/E2n. Assuming, for simplicity, that all

axion masses and mixings are equal, i.e. Mi2 = Ma2 and δ Mi2,i+1 = δ Ma2, the anomalously generated

Majorana mass may be estimated to be

√

M∼

3

ya

γ

κ5 √

Λ6−2n

(δ

Ma2

)n

,

R 49152 8 π4(1 − γ2)

(2.15)

for n ≤ 3, and

√ M ∼ 3 ya√γ κ5(δ Ma2)3 (δ Ma2)n−3 ,

R 49152 8 π4(1 − γ2) (Ma2)n−3

(2.16)

for n > 3. It is then not difﬁcult to see that three axions a1,2,3 with next-to-neighbour mixing as

discussed above would be sufﬁcient to obtain a UV ﬁnite (cut-off-Λ-independent) result for MR at

the two-loop level. Of course, beyond the two loops, MR will depend on higher powers of the energy cut-off Λ, i.e. Λn>6, but if κΛ 1, these higher-order effects are expected to be subdominant.

In the above n-axion-mixing scenarios, we note that the anomalously generated Majorana mass term will only depend on the mass-mixing parameters δ Ma2 of the axion ﬁelds and not on their masses themselves, as long as n ≤ 3. Instead, for axion-mixing scenarios with n > 3, the induced Majorana neutrino masses are proportional to the factor (δ Ma2/Ma2)n, which gives rise to an additional suppression for heavy axions with masses Ma δ Ma.

In the multi-axion models outlined above, the anomalously generated Majorana neutrino mass

MR will still depend on the Yukawa coupling ya and the torsion-axion kinetic mixing coefﬁcient

γ, besides the assumed UV completion scale Λ of quantum gravity. In order to get an estimate

of the size of MR, we treat the axion masses Ma and mass-mixings δ Ma as free parameters to be

constrained by phenomenology. Let us assume a n-axion-mixing models with n ≥ 3, in which the

axion mass mixing δ Ma and their masses Ma are of the same order, i.e. δ Ma/Ma ∼ 1. In this case,

employing (2.15), we may estimate the Majorana neutrino mass MR to be

MR 10−3 ya γ (δ Ma)6 10−3 ya γ δ Ma 5

∼

∼

.

Ma 1 − γ2 Ma MP5

1 − γ2 MP

(2.17)

6

Axions and massive Majorana neutrinos

Nick E. Mavromatos

PoS(CORFU2017)003

For

axion

masses

Ma

≤

1

TeV

considered

in

the

literature

thus

far,

we

ﬁnd

that

MR/Ma

< ∼

10−83,

which implies extra-ordinary small Majorana masses MR. An obvious caveat to this result would

be to have ultra-heavy axion masses Ma close to the GUT scale and/or ﬁne-tune the torsion-axion

kinetic

mixing

parameter

γ

to

1,

in

a

way

such

that

the

factor

γ 1−γ 2

compensates

for

the

mass

suppression (δ Ma/MP)5 in (2.17). The latter possibility, however, might re√sult in an unnaturally

large (non-perturbative) “effective” Yukawa coupling yeaff ≡ ya/

1

−

γ2

> ∼

4π in (2.12), which

will bring us outside the perturbative framework that we have been considering here. A more

detailed phenomonological and astrophysical analysis of all possible axion-mixing scenarios falls

beyond the scope of the present talk.

3. Implications for the Dark Sector of the Universe

It is interesting from a cosmological viewpoint to provide a numerical estimate of the anomalously generated Majorana mass MR given in (2.13) above. Assuming that γ 1, the size of MR may be estimated from (2.13) to be

MR ∼ (3.1 × 1011 GeV) ya 10−3

γ 10−1

Λ6 2.4 × 1018 GeV .

(3.1)

Obviously, the generation of MR is highly model dependent. Taking, for example, the quantum gravity scale to be Λ = 1017 GeV, we ﬁnd that MR is at the TeV scale, for ya = 10−3 and γ = 0.1. However, if we take the quantum gravity scale to be close to the GUT scale, i.e. Λ = 1016 GeV, we obtain a right-handed neutrino mass MR ∼ 16 keV, for the choice ya = γ = 10−3. This is in the preferred ballpark region for the (right-handed) sterile neutrino ψR to qualify as a warm dark matter (WDM) [17, 18].

Moreover, as discussed in [6], the introduction of appropriate self-interactions among the srerile right-handed neutrino dark matter (DM), can serve as a means of providing an explanation of the observed core-halo galactic structure, upon assuming speciﬁc core proﬁles, speciﬁcally, the Rufﬁni-Argüelles-Rueda proﬁle [19]. In addition, such self-interacting neutrino WDM models can also alleviate some of the discrepancies between numerical simulations based on ΛCDM model and observations at galactic scales (“small-scale cosmology crisis”), provided the self interaction total cross section σSIDM of a dark matter particle with mass m lies in the range

σSIDM/m 0.1 ≤ cm2 g−1 ≤ 0.47

(3.2)

where the upper limit has been inferred by recent studies of several merging galaxies, using novel techniques and observables [20].

For completeness, we mention that discrepancies between observations at galactic scales and ΛCDM-numerical simulations, appear in three areas:

(i) The Core-Cusp problem (or, as is also known, the cuspy-halo problem), refers to a discrepancy between the observed dark matter density proﬁles of low-mass galaxies and the density proﬁles predicted by cosmological N-body simulations. Characteristically, all the ΛCDM-based (DM only) simulations form dark matter halos which have “cuspy" dark matter distributions, with the density increasing steeply, i.e. as ρ ∝ r−1, at small radii. This is, e.g., evidenced in the standard

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Axions and massive Majorana neutrinos

Nick E. Mavromatos

PoS(CORFU2017)003

Navarro-Frenk-White (NFW) DM proﬁle [21]. On the contrary, the rotation curves of most of the observed dwarf galaxies indicate ﬂat central density proﬁles ("cores") [22].

(ii) The “missing satellite problem” (or, as is also known, the dwarf galaxy problem), arises from a discrepancy between ΛCDM-based numerical cosmological simulations that predict the evolution of the distribution of matter in the universe - pointing towards a hierarchical clustering of DM (where smaller halos merge to form larger halos) - and observations. Although there seem to be enough observed normal-sized galaxies to account for such a numerical distribution, the number of dwarf galaxies is orders of magnitude lower than that expected from the simulations. As a concrete example, we mention that there were observed to be around 38 dwarf galaxies in the Local Group, and only around 11 orbiting the Milky Way, yet one dark matter simulation predicted around 500 Milky Way dwarf satellites [23].

(iii) The too big to fail problem, that is a discrepancy arising between the most massive subhaloes predicted in (dissipationless) ΛCDM simulations and the observed dynamics of the brightest dwarf spheroidal galaxies in the Milky way. In other words, the ΛCDM simulations predict that the most massive subhaloes of the Milky way are too dense to host any of its bright satellites, with luminosity higher than 105 the luminosity of the Sun [24].

All three problems have their root in the fact that the cold DM particles, which the ΛCDM simulations rely upon, have too short free streaming length during the epochs of galaxy formation, and therefore they form too clumped and too many structures compared to those observed. Although some of these problems could be partly alleviated by more precise observations and/or including baryonic feedback, and may have their root in astrophysical reasons (for instance the missing satellite problem may be attributed partly to the fact that dwarf galaxies have a tendency to merge or may have been stripped apart gravitationally by larger galaxies and hence invisible), nonetheless it would be desirable if a uniﬁed explanation is found within the context of fundamental physics, and this is the point of view taken in the approach of including self interactions among the DM particles.

In this latter respect, WDM sterile neutrinos, with masses in the range of tens of keV, as in the νMSM model [5], have been argued to play a rôle, either in the too-big-to-fail problem [25], or, in the self-interacting case [6], for resolving the Core-Cusp problem, because the density proﬁles based on fermionic phase-space distributions [19, 6] develop always an extended plateau on halo scales (starting immediately after the quantum core). The rôle of self interactions (attractive vector interactions due to massive vector ﬁelds in the model of [6]) is to make the core more compact than in the non-interacting case [19]. For sterile neutrino masses mχ in the range [6]

47 keV mχ c2 345 keV ,

(3.3)

where the (robust) upper bound corresponds to the limit of gravitational collapse, the correct DM halo properties of galaxies (in agreement with observations) are guaranteed within the context of the galaxy proﬁle proposed in [19], whilst at the same time the fermi gas of fermionic DM may provide an alternative to the black hole in the centre of the galaxy. It must be stressed though that the lower bound depends crucially on the details of the galactic proﬁle, as well as of the existence (and its detailed properties) of a supermassive black hole in the galactic centre. Hence it is much less robust than the upper bound, and can be further relaxed.

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Axions and massive Majorana neutrinos

Nick E. Mavromatos

PoS(CORFU2017)003

sin22θ

10-7

10-8

Ωνχs > Ωoχbs

10-9 10-10 10-11 10-12

MW sataellite pcohnasstenrdaspaccoeunts ints νMSM

10-13 10-14 10-11500

Ωνχs < Ωoχbs

NuSTAR GC 2016 101

mχ [ keV ]

cPXor-nreavyious straints

102

Figure 2: Up-to-date Cosmological and Astrophysical Constraints for the νMSM WDM sterile neutrino mass versus mixing angle with the Standard Model sector (picture from ref. [26]). The recent analysis of Perez et al. [26], employs NuSTAR (Nuclear Spectroscopic Telescope Array) [27] observations of the Galactic Centre, together with additional constraints associated with DM production and its rôle in the structure formation in the Universe, which stem from the requirement that DM comprises predominantly of the lightest of the sterile neutrinos of νMSM. These latest studies exclude almost half of the previously allowed region of νMSM mixing-angle-mass parameters (thick (red) line and hatched region in the ﬁgure, the white region being the currently allowed one).

In this respect, we mention a crucial feature of the model of [6]. In that analysis, any mixing of the sterile neutrinos with the Standard Model sector had been ignored, assumed very small. In the spirit of [6], the sterile neutrino might constitute only one component of the DM in the Universe, with other equally dominant DM species in the underlying microscopic model playing different rôles in the structure formation and energy budget of the Cosmos. On the other hand, in the νMSM framework [5], the mixing of the lightest of the sterile right-handed neutrinos which, notably, is assumed to play the rôle of the dominant DM species, having a life-time longer than the Universe age, can be constrained mainly by its decay into X-rays and the DM production and contribution to the large scale structure. The recent constraints of Perez et al. [26], employing observations of the Galactic Centre with the Nuclear Spectroscopic Telescope Array (NuSTAR) [27], when combined with the basic assumption of νMSM [5] that the lightest of the sterile neutrinos provides the dominant component of DM, thus leading to additional constraints for sufﬁcient DM production in the early Universe and efﬁcient rôle in structure formation [17, 18], leaves a very small allowed window in the mixing-angle-sterile-neutrino-mass parameter plane (see white region in ﬁg. 2):

10 keV mχ c2 16 keV ,

(3.4)

As we discussed above (cf. (3.1)), within the model of [7], sterile neutrinos can acquire radiatively masses in this range for quite natural values of the parameters.

However, the above region is quite sensitive to the galactic proﬁle used (the authors of [26] used versions of the Navarro-Frenk-White [21] and Einasto [28] proﬁles in their study). An analysis making use of the Rufﬁni-Argüelles-Rueda proﬁle [19], employed in the work of [6], may affect the

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