Neutrino Masses and Hubble Tension Via a Majoron in MFV

Eur. Phys. J. C (2021) 81:28 https://doi.org/10.1140/epjc/s10052-020-08825-8

Regular Article - Theoretical Physics

Neutrino masses and Hubble tension via a Majoron in MFV

Fernando Arias-Aragón1,2,a, Enrique Fernández-Martínez1,2,b, Manuel González-López1,2,c , Luca Merlo1,2,d 1 Instituto de Física Teórica UAM/CSIC, Calle Nicolás Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain 2 Departamento de Física Teórica, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain

Received: 25 September 2020 / Accepted: 28 December 2020 / Published online: 15 January 2021 © The Author(s) 2021

Abstract The recent tension between local and early mea- trum [3] and the one obtained by the SH0ES collaboration [4] surements of the Hubble constant can be explained in a par- from supernovae measurements. ticle physics context. A mechanism is presented where this Although the solution to this discrepancy might be related tension is alleviated due to the presence of a Majoron, aris- to systematics in the measurements (notably the callibration ing from the spontaneous breaking of Lepton Number. The of the supernovae distances) or, more interestingly, point to lightness of the active neutrinos is consistently explained. a modification of the cosmological model, it may instead Moreover, this mechanism is shown to be embeddable in be provided by particle physics, as already discussed in the the minimal (Lepton) flavour violating context, providing literature (see for example [5–18]). In particular, Ref. [14] a correct description of fermion masses and mixings, and suggests that a Majoron that couples to light neutrinos could protecting the flavour sector from large deviations from the reduce the tension in the determination of H0.Itistheninter- Standard Model predictions. A QCD axion is also present esting to investigate whether this setup is compatible with to solve the Strong CP problem. The Lepton Number and possible explanations of other open problems in the Stan- the Peccei–Quinn symmetries naturally arise in the minimal dard Model of particle physics (SM): the focus of this paper (Lepton) flavour violating setup and their spontaneous break- is to study the compatibility with Type-I Seesaw mechanism ing is due to the presence of two extra scalar singlets. The to explain the lightness of the active neutrinos, with specific Majoron phenomenology is also studied in detail. Decays of flavour symmetries to describe the flavour puzzle and with the heavy neutrinos and the invisible Higgs decay provide the presence of an axion to solve the strong CP problem. the strongest constraints in the model parameter space. The Majoron, called ω hereafter, is the Nambu–Goldstone boson (NGB) associated to the spontaneous breaking of lepton number (LN) [19–22], which is only accidental within 1 Introduction the SM and breaks down at the quantum level. It naturally arises in the context of the Type-I Seesaw mechanism, where There is nowadays a considerable tension between late-time, the Majorana mass term, instead of being a simple bilinear of local probes of the present rate of expansion of the Uni- the right-handed (RH) neutrino fields NR, is a Yukawa-like verse, that is the Hubble constant H0, and its value inferred term that couples NR to a scalar field that carries a LN charge, through the standard cosmological model CDM from early labelled as χ in the following. Once this scalar field devel- Universe observations. Local measurements, from type-Ia ops a vacuum expectation value (VEV), then LN is sponta- supernovae and strong lensing, tend to cluster at similar val- neously broken, a Majorana neutrino mass term is generated ues of H0, significantly larger than those preferred by cosmic and the Majoron appears as a physical degree of freedom of microwave background (CMB) and baryon acoustic oscilla- the spectrum. tions probes. The strongest tension, estimated at the level of If a Dirac term that mixes NR and the left-handed (LH) 4 − 6 σ [1,2] depending on the assumptions performed, is lepton doublets L is also present in the Lagrangian, small between the Planck inferred measure from the CMB spec- masses for active neutrinos are generated at low energies, according to the Type-I Seesaw mechanism. At low-energies, the Majoron ω acquires a coupling with a e-mail: [email protected] ν , labelled as λωνν. For Majoron masses b e-mail: [email protected] L c e-mail: [email protected] (corresponding author) d e-mail: [email protected] mω ∈[0.1, 1] eV (1.1) 123 28 Page 2 of 14 Eur. Phys. J. C (2021) 81 :28 and λωνν in the range Table 1 LN assignments. Field U(1)L Charge Fields that are not listed here do −14 −12 not transform under LN λωνν ∈[5 × 10 , 10 ] , (1.2) L , eR 1 NR −L N the tension on the Hubble constant is reduced [14]. Indeed, for χ Lχ such small Majoron-neutrino mixings and Majoron masses, Majorons only partially thermalise after Big Bang Nucle- osynthesis (BBN) or never thermalise [23], enhancing the the phenomenological signatures of this model and Sect. 5 effective number of neutrino species Neff by at least 0.03 and contains the final remarks. at most 0.11, values that may be tested with CMB-S4 experiments [24]. Moreover, a non-vanishing λωνν would reduce neutrino free-streaming, modifying the neutrino anisotropic 2 The Majoron mechanism stress energy tensor [25]. This has an impact in the CMB that results in modifying the posterior for the Hubble constant: as To produce a Majoron and explain the lightness of the active shown in Ref. [14], the inclusion of Majoron-neutrino inter- neutrinos, one can consider a Type-I Seesaw mechanism actions slightly shifts the central value of H0, but largely where the Majorana mass is dynamically generated by the . σ broadens its profile reducing the H0 tension to 2 5 .For spontaneous breaking of LN. The SM spectrum is extended λ > −12 larger couplings, ωνν 10 , these effects are too large by three RH neutrinos1 and a singlet scalar field χ that only and excluded by the same Planck data. transforms under LN. The LN charge assignments can be read Interestingly, Ref. [14] found that the best χ 2 inaMarkov in Table 1, where L , NR and χ have already been defined, Chain Monte Carlo corresponds to Majoron mass and cou- and e refers to the RH charged leptons. Notice that Lχ and = . ± . R pling as in Eqs. (1.1) and (1.2) and Neff 0 52 0 19. The L are integer numbers and are not completely free, but must = N uncertainty in the last observable is very large and Neff 0 obey a series of constraints that will be made explicit in the σ is compatible within 3 . However, values close to the cen- following. tral one can be achieved if a thermal population of Majorons The most general effective Lagrangian in the neutrino sec- is produced in the early Universe and is not diluted during tor invariant under LN is the following2: inflation. This may occur if the reheating temperature is larger + − than the RH neutrino masses [14]. Alternatively, other rel- 1 L N 2L N Lχ χ Lχ χ Lχ −L = ¯ ˜ Y + 1 χ ¯ c Y + , ativistic species, such as axions [26–29], may contribute to ν L H ν NR NR N NR h.c. χ 2 χ N and their presence may justify such a large value. eff (2.2) The existence of both Majorana and Dirac terms, the achievement of the correct scale for the active neutrino ˜ ∗ where H is the SM Higgs doublet, H = iσ H , χ is the cut- masses and at the same time the alleviation of the Hubble 2 off scale up to which the effective operator approach holds, tension via the Majoron strongly depend on the LN charge and Yν is a dimensionless and complex matrix, while YN is assignments of L , NR and χ. In particular, fixing the LN of dimensionless, complex and symmetric. A first condition on L to unity, then the LN of the RH neutrinos needs to have L ,χ arises from requiring that all the terms are local: opposite sign with respect to the LN of the scalar field χ. N This model presents interesting phenomenological features. 1 + L 2L N − Lχ On one hand, the heavy neutrinos are expected to be rela- N , ∈ N . (2.3) tively light, with masses at the MeV or GeV scales, opening Lχ Lχ up the possibility to be studied at colliders. Moreover, the presence of the Majoron may also have other consequences 1 The case with only two RH neutrinos is also viable, and correspond- distinct from the Hubble tension. In particular, its couplings ingly the lightest active neutrino would be massless. to photons and electrons are constrained by CAST and Red 2 Other terms can be added to this Lagrangian inserting χ † instead of χ. Giant observations, respectively, while, due to its coupling to However, once the terms in Eq. (2.2) are local, then their siblings with χ † would not be local and therefore cannot be added to the Lagrangian. the Higgs, the Majoron may contribute to the invisible Higgs The only exception is the term decay, strongly constrained by colliders. + 2L N Lχ Section 2 illustrates the mechanism to produce a Majoron 1 χ Lχ (2.1) that alleviates the H0 tension together with a correct scale 2 χ for the active neutrino masses. In Sect. 3, this mechanism χ † ¯ c Y , NR N NR is introduced in a setup that correctly describes the flavour puzzle of the SM and at the same time produces a QCD that however only provides a suppressed correction with respect to the axion that solves the Strong CP problem. Section 4 gathers Majorana term written in Eq. (2.2) and for this reason it can be neglected. 123 Eur. Phys. J. C (2021) 81 :28 Page 3 of 14 28

χ 2 =− . ± . × −3 2 In the LN broken phase, the ﬁeld can be parametrised matm 2 497 0 028 10 eV for the Inverted Order- as ing (IO) [30].

ω The heavy neutrinos, that mostly coincide with the RH σ + vχ i χ = √ e vχ , (2.4) neutrinos, have a mass matrix that in ﬁrst approximation can 2 be directly read from the Lagrangian in Eq. (2.2), where the angular part ω is the NGB identiﬁed as a Majoron, − 2L N Lχ v σ v Lχ χ is the radial component and χ is its VEV. Notice that the MN εχ √ YN . (2.10) scale appearing in the denominator of the exponent is also 2 vχ in order to obtain canonically normalised kinetic terms The overall scale for the heavy neutrinos must be larger than for the Majoron. A useful notation that will be employed in the overall scale of the active neutrinos, in order for the See- the following is the ratio of the χ VEV and the cut-off scale: saw approximation to hold:

− v 2L N Lχ χ vχ εχ = √ . (2.5) ε Lχ √ Y | 2 | . χ N matm (2.11) 2χ 2

This parameter εχ is expected to be smaller than 1 in order On the other hand, the electroweak and LN breakings give to guarantee a good expansion in terms of 1/χ . Conse- rise to the Majoron Lagrangian that can be written as follows: quently, the χ VEV, which represents the overall scale of the + + 1 L N v LN breaking, is expected to be smaller that the scale χ , 1 μ 1 2 2 1 L N Lχ Lω = ∂μω∂ ω + mωω − i εχ √ ν¯L Yν NRω+ where New Physics should be present and is responsible for 2 2 Lχ 2vχ − 2L N Lχ generating the expression in Eq. (2.2). Lχ L εχ Once the electroweak symmetry is also spontaneously − N √ ¯ c Y ω + , i NR N NR h.c. (2.12) broken, i.e. after that the SM Higgs develops its VEV that in Lχ 2 the unitary gauge reads 2 where the mω term parametrises the Majoron mass intro- h + v duced here as an explicit breaking of its corresponding shift H = √ , (2.6) 2 symmetry. At low energy, a direct coupling of the Majoron to the active neutrinos emerges after performing the same where h is the physical Higgs and v 246 GeV, masses for transformations that give rise to the mass matrix in Eq. (2.7): the active neutrinos are generated according to the Type-I Seesaw mechanism: λ low-energy ωνν c mν low-energy 1 c Lω ⊃ i ων¯ ν + h.c. with λωνν = 2 . Lν = ν¯L mν ν + h.c. with L L v 2 L 2 Lχ χ 2+Lχ (2.13) ε Lχ v2 = χ√ Y Y−1Y T . mν ν N ν (2.7) 3 2vχ From the results in Ref. [14], shown in Eq. (1.2), it is then possible to infer a bound on the product Lχ vχ , once taking In the basis where the charged lepton mass matrix is already 2 diagonal, the neutrino mass matrix can be diagonalised by matm as the overall scale for the neutrino masses: thePMNSmatrixU: | 2 | ˆ ≡ ( , , ) = † ∗ . 2 matm mν diag m1 m2 m3 U mν U (2.8) |Lχ |vχ ∈[0.1, 2] TeV . (2.14) λωνν The overall scale for the active neutrino masses can be written Adopting this relation and substituting it within the expres- in terms of the parameter εχ , the ratio of the VEVs and the Y Y−1Y T sions in Eqs. (2.9) and (2.11), new conditions can be found: product ν N ν : 2+Lχ Lχ − + χ | |ε Y Y 1Y T 2 L Lχ χ ν N ν ε Lχ v2 √ χ −1 T 2 | 2 | √ YνY Y | m | , (2.9) 2 2 matm −13 −12 v N ν atm ∈[1.2 × 10 , 2.4 × 10 ] , (2.15) 2 χ λωνν v2

2 = . +0.028 × −3 2 where matm 2 514−0.027 10 eV for the Nor- 3 This expression coincides with the one in Ref. [14] once identifying mal Ordering (NO) of the neutrino mass spectrum and ω with φ and λωνν with λ. 123 28 Page 4 of 14 Eur. Phys. J. C (2021) 81 :28

2L −Lχ N L > Lχ < ε Lχ λ In the case when N 0 and 0, it is possible χ Y √ωνν . × −14 . to obtain the same results listed above substituting χ by χ † | | N 3 5 10 (2.16) Lχ 2 in the Lagrangian in Eq. (2.2): in this case, the signs in the The following choice of charge assignments leads to a denominators of the exponents would be flipped, compen- completely renormalisable Lagrangian and thus deserves sating the negative sign of Lχ . The opposite case, L N < 0 special mention: and Lχ > 0, is not allowed by the locality conditions. For L N,χ < 0, besides the possibility of CASE R, only CASE R: L N =−1 and Lχ =−2 , (2.17) another choice is allowed by the locality conditions: (L N = −1, Lχ =−1). However, this case would require χ 1, such that the powers of the ratio χ/χ in Eq. (2.2) are not leading to an even more extreme fine-tuning than in CASE present in either the Dirac and the Majorana terms. The rela- R without the appeal of renormalizability. tion in Eq. (2.14) and the two conditions in Eqs. (2.15) and The condition in Eq. (2.14), corresponding to Eq. (1.2), (2.16) further simplify: is only one of the ingredients necessary to lower the H0 tension. A second relevant requirement is Eq. (1.1), regarding | 2 | matm the Majoron mass. For the sake of simplicity, it has been Eq. (2.14) −→ vχ ∈[0.05, 1] TeV . (2.18) introduced directly in the Majoron Lagrangian in Eq. (2.12) λωνν √ as en effective term. Its origin has been widely discussed in | 2 | −1 T 2 matm Eq. (2.15) −→ YνY Y the literature and constitutes in itself an interesting research N ν λ v2 ωνν topic. Any violation of the global LN symmetry would induce − − ∈[1.2 × 10 13, 2.4 × 10 12] , (2.19) a mass for the Majoron. An obvious example are gravi- λ 2 ωνν −13 tational effects, which are expected to break all acciden- Eq. (2.16) −→ YN √ 7 × 10 . (2.20) 2 tal global symmetries. Estimations of their size from non- perturbative arguments via wormhole effects [32] fall too The first expression fixes a range of values for vχ . While the short of their required value. Conversely, their size from third expression represents a lower bound on the overall scale Planck-suppressed effective operators [33] would point to Y Y Y−1Y T of N , the second one implies that the product ν N ν too large a mass, although several possibilities have been should be tuned to a very small value in order to reproduce discussed that would prevent the lower dimension operators the lightness of the active neutrinos. from being generated [34–41]. These options were originally For values of L N,χ different from the previous ones, the introduced as a solution to the axion quality problem. A sim- Lagrangian is necessarily non-renormalisable. An interest- pler possibility, given the singlet nature of the NR,isan ing question is whether the extremely small values of the explicit breaking of LN via a Majorana mass term at the Y Y−1Y T product ν N ν can be avoided exploiting the suppres- Lagrangian level. The Majoron would thus develop a mass sion in χ from the new physics scale χ , similarly to the slightly below this breaking scale from its coupling to the NR Froggatt-Nielsen approach to the flavour puzzle [31]. Con- through self-energy diagrams. In this work we will remain sidering first the case in which L N,χ > 0, then the only agnostic to the origin of the Majoron mass, and a value con- available possibilities are sistent with Eq. (1.1) will be assumed. Besides the Majoron, also the radial component of χ is CASE NR1: L N = 1 and Lχ = 1 present in the spectrum and does play a role modifying the CASE NR2: L N = 1 and Lχ = 2 . (2.21) Higgs scalar potential. Indeed, the most general scalar potential containing H and χ can be written as The corresponding values for vχ , εχ , the overall scale for the 2 † † 2 2 ∗ ∗ 2 † ∗ heavy neutrinos MN and the cut-off scale χ are reported V (H,χ)=−μ H H+λ H H −μχ χ χ+λχ χ χ +gH Hχ χ in Table 2. (2.22) From Eq. (2.15), it can be seen that χ gets smaller for Y Y−1Y T larger values of Lχ (unless tuning the product ν N ν as The minimisation of such potential leads to VEVs for the two in CASE R discussed above, a possibility to be avoided in the fields that read present discussion): although this is not a problem by itself, it λχ μ2 − μ2 λμ2 − μ2 hardens the constraint in Eq. (2.16). It follows that for larger 2 4 2g χ 2 4 χ 2g v = ,vχ = . (2.23) values of L N and Lχ satisfying to the locality conditions in 4λλχ − g2 4λλχ − g2 Eq. (2.3), then the overall scale of the heavy neutrino masses would be as small as the one of active neutrinos and therefore The parameters of this scalar potential need to be such that the expansion in the Type-I Seesaw mechanism would break v takes the electroweak value and vχ acquires the values in down. Table 2. 123 Eur. Phys. J. C (2021) 81 :28 Page 5 of 14 28

Table 2 Parameter ranges in the two phenomenologically interesting scenarios

L N Lχ vχ εχ MN χ

CASE NR1 1 1 [0.1, 2] TeV [0.49, 1.4]×10−4 [3.5, 200] MeV [1.4 − 11]×103 TeV CASE NR2 1 2 [0.05, 1] TeV [2.4, 11]×10−7 [35.4, 707] GeV [1.4 − 6.5]×105 TeV

Due to the mixed quartic term, the two physical scalar from flavour observables, and also contains a QCD axion that fields h and σ mix in the broken phase and the mass matrix solves the strong CP problem. describing this mixing is given by λv2 vv M2 = 2 g χ . 3 The Majoron and Axion from a MFV setup 2 (2.24) g vvχ 2 λχ vχ Flavour models aim at describing the heterogeneity of the The two eigenvalues that arise after diagonalising this mass fermion masses and mixings through some underlying argu- matrix are the following ment, such as a symmetry principle. The simplest proposal is known as the Froggatt–Nielsen model [31] and consists 2 = λv2+λ v2 ± λv2 − λ v2 + 2 ϑ, Mh,σ χ χ χ χ 1 tan 2 (2.25) in introducing a global Abelian symmetry to describe the flavour structure in the quark sector. Almost 25 years later, where after the more precise measurements of neutrino oscillations and the introduction of the so-called tri-bimaximal mixing g vvχ ϑ = . texture [43,44], whose main feature is a vanishing reac- tan 2 2 2 (2.26) λχ vχ − λv tor mixing angle in the PMNS matrix, discrete symmetries were considered as an attractive approach to reproduce the The lightest mass in Eq. (2.25) corresponds to the eigenstate observed pattern of all fermion masses and mixings [45– mainly aligned with the SM Higgs, while the heaviest state is 52]. However, after the discovery, in 2011, of a relatively mainly composed of the radial component of χ. Its mass can large reactor mixing angle [53–57], models based on discrete be as small as a few hundred GeV or much larger than the symmetries underwent a deep rethinking and other options TeV. From the relation between the mixed quartic coupling also became popular. A few examples are Froggatt-Nielsen g and the physical parameters, inspired models based on a simple U(1) [58–60] and models based on continuous non-Abelian symmetries. The latter 2 2 Mσ − M g = h sin 2ϑ, (2.27) include Minimal Flavour Violation (MFV) [61] and its lep- 2 vvχ tonic versions [62–64], which are based on a U(3) symmetry, smaller symmetries like U(2) [65,66], or an intermediate it is possible to straightforwardly study the dependence of approach [67]. Mσ with the other model parameters. Notice that the mixing The focus here will be the MFV setup, that will be shown parameter ϑ is constrained by LHC data to be[42] to naturally suggest the presence of the Majoron and of a QCD axion. The key concept behind MFV is to require that 2 sin ϑ 0.11 , (2.28) any flavour and CP violation in physics Beyond the SM has √ the same origin as the one in the SM [68]. This idea has been from a s = 13 TeV analysis of Higgs signal strengths with formulated in terms of the symmetries of the kinetic terms in −1 80 fb of integrated luminosity. Ref. [61]. These symmetries are then broken by a series of As shown in Fig. 1, Mσ can reach very large values, with- fields that are also employed to describe fermion masses and out requiring any fine-tuning on g. On the contrary, very mixings, as well as the suppressions associated to any non- small Mσ can only be achieved for g close to zero. It is then renormalisable flavour violating operator. Considering the natural to focus on the case in which σ is massive enough to SM spectrum augmented by three RH neutrinos, the flavour be safely integrated out. This will be the case in the rest of symmetry of the corresponding kinetic terms is a product of the paper. a U(3) term for each fermion species, The mechanism illustrated in this section allows to soften 6 the H0 tension explaining at the same time the lightness of GF = U(3) . (3.1) the active neutrinos. In the next section, this mechanism will be embedded into a specific flavour model that allows to The non-Abelian terms are responsible for the intergener- account for the flavour puzzle, without violating any bounds ational fermion mass hierarchies and of the mixing matri- 123 28 Page 6 of 14 Eur. Phys. J. C (2021) 81 :28

Fig. 1 Mσ vs. g parameter space, ﬁxing v = 246 GeV, Mh = 125 GeV and taking three values for LN breaking scale, vχ ={50 GeV, 100 GeV, 2TeV}.Thewhiteareaisthe one allowed by the bound in Eq. (2.28), while the green region is the excluded one. The dashed line correspond to the bound in Eq. (4.13), being the area above such line allowed

ces, while the Abelian terms are associated to the hierarchies The Lagrangian invariant under the aforementioned sym- among the masses of the different third family fermions. The metries is the following: latter can be written as follows 3 3 −L =¯q H˜ Y u + q¯ HY d + ¯ HY e G A = ( ) × ( ) × ( ) × ( ) × ( ) × ( ) , Y L u R L d R L e R F U 1 qL U 1 u R U 1 dR U 1 L U 1 eR U 1 NR + (3.2) 1 xN χ xχ ¯ ˜ + L HYν NR χ − 2xN xχ where qL refers to the LH quark doublets, u R to the RH up- χ xχ + 1 χ ¯ c Y + , type quarks and dR to the RH down-type quarks, while L , eR NR N NR h.c. (3.5) 2 χ and NR to the leptons as already introduced in the previous section. It is possible to rearrange this product, provided that the new combinations are still linearly independent, identify- where stands for the cut-off scale associated to the scalar ing among them baryon number, LN, weak hypercharge and the Peccei–Quinn (PQ) [69] symmetry: field . According to the MFV approach, Yu,d,e,ν,N are not simple matrices, but are promoted to be spurion fields that do G A = ( ) × ( ) × ( ) × ( ) × ( ) × ( ) . F U 1 B U 1 L U 1 Y U 1 PQ U 1 eR U 1 NR transform under the non-Abelian part of the flavour symme- (3.3) try group GF . These fields can be thought of as dimensionless scalar fields that do not have kinetic terms, but acquire background values that play the role of VEVs. In the MFV In the model described in this section, fermion charges under approach, to reproduce correctly quark masses and mixings baryon number and hypercharge are assigned as in the SM, and charged lepton masses, the background values of Yu,d,e while the LN charges are given in Table 1. Moreover, PQ should read charges are chosen so as to explain the suppression of the bottom and τ masses with respect to the top mass: only dR † mu mc and eR transform under PQ, with a charge equal to 3. In an Yu = ct V diag , , 1 , analogous way to LN, the PQ symmetry is formally exact at mt mt the Lagrangian level after introducing a second scalar field md ms Y = c diag , , , d b 1 that transforms non-trivially only under PQ with a charge mb mb − 1: me mμ Ye = cτ diag , , 1 , (3.6) mτ mτ ρ + v i a = √ e v , (3.4) 2 where V is the CKM mixing matrix and ci are order 1 parameters. The possible origin of these values is under study [71– where ρ is the radial component, a is the angular one and 74]. v is its VEV. Once this scalar field develops a VEV and While these matrices present hierarchies among their electroweak symmetry gets broken, masses for bottom and entries, the ratios mb/mt and mτ /mt are still not described τ are generated, suppressed with respect to the top one. As a within this approach, but can be explained by the sponta- byproduct of this mechanism, the axion a is originated [70]. neous breaking of the PQ symmetry. Indeed, the overall scale Finally, the last two symmetries in Eq. (3.3) do not play any of down-type quarks and charged leptons is multiplied by the role in this model and are explicitly broken. cubic power of the ratio among the VEV and the cut-off 123 Eur. Phys. J. C (2021) 81 :28 Page 7 of 14 28

. Assuming that this ratio is given by Diagonalising mν corresponds to diagonalising the product Yν Yν T and, given the fact that the the lepton mix- v √ 0.23 , (3.7) ing angles are relatively large, then no hierarchies should 2 be expected among the entries of Yν Yν T , contrary to what happens in the quark case. Note that some setups, the m /m and mτ /m ratios are also correctly described. b t t such as the so-called sequential dominance scenarios, Any non-renormalisable operator that describes flavour- obtain large mixing angles even if there exists a strong violating processes should be invariant under the flavour hierarchy among the Yukawa couplings [87]. However, symmetry. This is accomplished by inserting proper pow- this possibility is disfavoured by the general philosophy ers of the spurions: once they acquire their background val- of MLFV.In the same spirit, the overall scale of this prod- ues, the non-renormalisable operator under consideration uct is of O(1), as any explanation of the neutrino masses gets suppressed. The main consequence is that the scale that should reside in the model itself, and not be due to any can be studied considering flavour observables is at the level fine-tuning. of 1−10 TeV [61,75–85], instead of 100 TeV in a generic II): G NA → SU(3) + × SU(3) [64]. case [86], opening up the possibility of discovering New L L NR eR Assuming that the three RH neutrinos transform as a Physics (NP) at colliders. triplet under the same symmetry group of the lepton dou- For the neutrino sector, the discussion is slightly more blets, complicated. Indeed, there are two spurions in the neutrino sector, Yν and YN , and both enter in the definition of the L , NR ∼ (3, 1)G NA eR ∼ (1, 3)G NA , (3.9) active neutrino masses, see Eq. (2.7). It follows that it is not L L possible to identify univocally either Yν or YN in terms of neutrino masses and the PMNS matrix entries. Thus, the sup- then Schur’s Lemma guarantees that Yν transforms as pression in the non-renormalisable flavour violating opera- a singlet of the symmetry group. Then, Yν is a unitary tors cannot be directly linked to neutrino masses and mixings, matrix [88,89], which can always be rotated to the iden- losing the predictivity that characterizes the MFV approach tity matrix by a suitable unitary transformation acting in the quark sector. The solutions that have been proposed only on the RH neutrinos. The only meaningful quanti- Y ∝ 1 Y are to assume N [62,63] or to consider ν as a unitary ties in this context are Ye and YN , so neutrino masses matrix [64]. and lepton mixings are encoded uniquely into YN ,

G NA → ( ) × ( ) × ( ) × 2+Lχ I): L SU 3 L SU 3 eR SO 3 NR CP[62,63]. ε Lχ v2 Under the assumption that the three RH neutrinos are = χ√ Y−1 . Y ∝ 1 mν N (3.10) degenerate in mass, i.e. N , then the non-Abelian 2vχ ( ) symmetry associated to the RH neutrinos, SU 3 NR ,is ( ) broken down to SO 3 NR . The additional assumption of Moreover, all flavour changing effects involving leptons no CP violation in the lepton sector is meant to force Yν can be written only in terms of Ye and YN , and therefore 4 to be real . With these simplifications, the expression for any flavour changing process can be predicted in terms the active neutrino mass in Eq. (2.7) simplifies to of lepton masses and mixings.

2+Lχ As for the previous case, the diagonalisation of the Lχ εχ v2 active neutrino mass coincides with the diagonalisation of = √ Y Y T , −1 mν ν ν (3.8) YN , that therefore does not present any strong hierar- 2vχ chy among its entries. Moreover, its overall scale should O( ) and all flavour changing effects involving leptons can be be 1 according to the MLFV construction approach. T written in terms of YνYν and Ye. These are the only relevant combinations entering any d = 6 operator involving In both cases, the constraints on NP considering the lepton fields and describing flavour violating effects. It present available data on flavour changing processes in the follows that any flavour changing process can be pre- lepton sector are as low as a few TeV [62–64,90–93]. dicted in terms of lepton masses and mixings. Once the PQ symmetry is spontaneously broken, the axion arises as its NGB and its Lagrangian can be written as 4 Strictly speaking, the condition of CP conservation in the leptonic sector forces the Dirac CP phase to be equal to δ ={0,π} and the CP 1 μ 3ia 3ia α ={ ,π, π} L = ∂ ∂ − v Y − v Y Majorana CP phases to be 21,31 0 2 .However,Yν is real only a μa a e q L H d dR e L H eeR α ={ , π} α = π 2 if 21,31 0 2 , and therefore 21,31 needs to be disregarded in α order to guarantee predictivity. The CP conservation condition assumed s aμν a + θQCDG Gμν , (3.11) in Refs. [62,63] is then stronger than the strict definition. 8π 123 28 Page 8 of 14 Eur. Phys. J. C (2021) 81 :28

a ≡ 1 aρσ with Gμν 2 μνρσ G and μνρσ the totally antisymmet- in Eq. (1.1), the strongest constraints on the effective coupling ric tensor such that 1230 = 1. The last term is the well-known to photons are set by CAST [102], which establishes the QCD θ-term, allowed by the QCD Lagrangian, which con- upper bound stitutes a source of CP violation. The θQCD parameter con- −10 −1 tributes to the neutron electric dipole moment [94,95] and λωγ γ 10 GeV , (4.1) can thus be experimentally constrained [96] where λωγ γ is defined as −10 θQCD 10 . (3.12) low-energy 1 μν ˜ θ Lω ⊃ λωγ γ ω F Fμν (4.2) The presence of the axion naturally explains why QCD is 4 so small, providing a solution to the Strong CP problem: couplings of the axion to the gauge fields, and in particu- ≡ 1 ρσ with Fμν 2 μνρσ F . lar to gluons, are generated at the quantum level and the As the Majoron does not couple at tree-level to charged θ QCD parameter can be reabsorbed by an axion field redefini- particles, then the process ω → γγ occurs only at two loops. tion [69,97,98]. The effective axion potential (see for exam- Ref. [106] provides an estimate for its decay width: under the ple Ref. [99]) predicts a vanishing VEV for the axion, that assumption mω me, finally solves the Strong CP problem. Moreover, the same potential provides the axion with a mass, 2 α2 7 † mω m Dm D ω→γγ = Tr (4.3) 2π 7 v2 4 vv 1012 GeV 1536 me χ ma ∼ 6 μeV , (3.13) fa 2 where α ≡ e /4π and with m D the Dirac neutrino mass being fa the axion scale, that is connected to the VEV matrix. in this model by the relation fa = v/9. For QCD axions, Computing the same process by means of the effective the most stringent constraint on fa comes from the axion couplings in Eq. (4.2), couplings to photons [100–102] and to electrons [103–105], . × 7 2 3 which push its value to be larger than fa 1 2 10 GeV λωγ γ mω and f 8 × 108 GeV, respectively. ω→γγ = (4.4) a 32π Summarising, the Majoron together with axion constitute the natural Abelian completion of MFV scenarios. The it is then possible to match the two expressions for the Majoron does not affect (at tree level) the physics associ- ω → γγ decay width providing the expression for the λωγ γ ated to the axion and the quark and charged lepton flavour coupling: physics. Thus, this model, besides describing fermion masses + and mixings and solving the Strong CP problem, is able to α 2 2 2L N mω Lχ † alleviate the Hubble tension with the only inclusion of three λωγ γ = √ χ Tr YνYν . (4.5) 384 2π 3m2vχ RH neutrinos and two extra singlet scalars. In particular, as e no fine-tuning is allowed within this approach on Yν or Table 3 shows the numerical estimations for the predicted Y , then only CASE NR1 and CASE NR2 are viable in N values of the Majoron coupling to photons, assuming that the MFV framework. In the following section, the analysis the trace gives an O(1) number, as already discussed: the of this model will be completed with the study of its phe- experimental bound is still far from the theoretical prediction. nomenological signatures.

Coupling to electrons 4 Phenomenological signatures Astrophysical measurements can also constrain Majoron The only tree-level coupling of the Majoron is to neutrinos. couplings. Ref. [103] provides an upper bound on the However, at quantum level, couplings to gauge bosons, other Majoron effective coupling to electrons SM fermions and the Higgs are originated. low-energy Lω ⊃ i λωee ω ee¯ , (4.6) Coupling to photons based on observations on Red Giants: The searches for very light pseudoscalars, usually addressed −13 to axions, can also apply to Majorons. In the range of masses λωee < 4.3 × 10 . (4.7) 123 Eur. Phys. J. C (2021) 81 :28 Page 9 of 14 28

Table 3 Predictions for the λωγ γ λω λωνν Majoron effective couplings to ee electrons, photons and − − − − − − − CASE NR1 [10 39, 10 36] GeV 1 [10 25, 10 24][10 14, 10 12] neutrinos, for the window of the −34 −32 −1 −20 parameter space where Hubble CASE NR2 [10 , 10 ] GeV 10 tension is alleviated. In the last Exp. upper bounds 10−10 GeV−1 10−13 10−5 line, the corresponding experimental upper bounds

The decay width of the Majoron to two electrons reads [106]

1 2 ωee |λωee| mω , (4.8) 8π where † † 1 me m Dm D 1 m Dm D λωee − Tr , (4.9) 8π 2 v vvχ 2 vvχ 11 with [...]11 standing for the (1, 1) entry of the matrix in the brackets. Substituting explicitly the expression for m D,the coupling becomes

+ 2 2L N 1 me Lχ † † λωee = χ YνYν − Tr YνYν . 16π 2 vχ 11 (4.10)

† Assuming as before that the elements of the product YνYν Fig. 2 Feynman Diagram for the neutrinoless-double-beta decay with are O(1) numbers, also the Majoron-electron coupling is pre- the emission of a Majoron dicted to be much smaller than the corresponding experimental bound, as shown in Table 3. Coupling with the SM Higgs: Higgs invisible decay Coupling to neutrinos: Majoron emission in 0νββ decays The Majoron–Higgs coupling follows due to the mixing The tree level couplings of the Majoron to neutrinos does between the radial component of χ and the physical Higgs, not have an impact only on cosmology, but may be relevant as described at the end of Sect. 2. Indeed, expanding the for low-energy terrestrial experiments. In particular, searches kinetic term of the ﬁeld χ,aσωωcoupling arises that induces for neutrinoless-double-beta decay could also be sensitive an effective coupling hωω, via the mixing ϑ. This coupling to processes in which Majorons are produced, such as in opens up a new decay channel for the Higgs that contributes KamLAND-Zen [107] and NEMO-3 [108] experiments. to the Higgs invisible decay observable. The width of this Current measurements set a lower bound on the half-life of process is given by the neutrinoless-double-beta decay of the order of 1024 years. In the particular case discussed here, where the Majoron 2 3 sϑ Mh could only be produced by the annihilation of two neutrinos h→ωω = 0.8MeV, (4.12) 32πv2 (see Fig. 2), this bound can be translated into a constraint on χ the Majoron-neutrino coupling [109], such that where the last inequality has been obtained considering that −5 λωνν < 10 , (4.11) the invisible Higgs decay width may constitute at most the 19% of its total width [110]. where λωνν is deﬁned in Eq. (2.13). This result can be translated into a strong bound on vχ , The predicted value of the Majoron-neutrino coupling that reads can be read out in Table 3 and it is much smaller than the corresponding experimental value and the bound from vχ λ < O( −12) 5TeV. (4.13) Planck [14]( ωνν 10 ). |sϑ | 123 28 Page 10 of 14 Eur. Phys. J. C (2021) 81 :28

Table 4 Expectation for the M sin2 θ Z ω heavy neutrino mass and mixing N s N→3ν N→3ν between heavy and active − − − − CASE NR1 [3.5, 200] MeV [2.5 × 10 10, 1.4 × 10 8] O(10 38) O(10 68) neutrinos CASE NR2 [35.4, 707] GeV [7.1 × 10−14, 1.4 × 10−12] O(10−27) O(10−66)

For a light σ , the mixing would be close to its current upper the bound on the mixing is [130]: bound, Eq. (2.28), and vχ should be larger than ∼ 1.5TeV, which would exclude CASE NR2 and part of the parameter 2 mν −15 −17 sin θs ≡ 10 − 10 , (4.14) space of CASE NR1, see the dashed line in Fig. 1. However, MN as stated at the end of Sect. 2, the assumption made here is that σ is sufﬁciently heavy to be integrated out and this much smaller than the expected value that can be read in corresponds to much smaller values of the mixing angle ϑ, Table 4. relaxing in this way the bound on vχ .

5 Conclusions

The Hubble tension may well be due to systematic uncertain- 4.1 Heavy neutrinos ties but it might point to a deviation from the CDM model or an extension of the Standard Model of particle physics. In In both cases discussed in Sect. 2, the heavy neutrino masses particular, the presence of a Majoron with couplings to the lie in ranges that may lead to detection in various experi- active neutrinos represents an interesting avenue to alleviate mental facilities. Neutrinos with masses ranging from tens this tension. This paper is focussed on a mechanism where to hundreds of MeV can be probed for and be potentially neutrino masses and the presence of a Majoron are explained detected at beam dump or even near detectors of neutrino by the spontaneous breaking of Lepton Number. Besides the oscillation experiments [111–118], such as DUNE or SHiP, introduction of three RH neutrinos and a new complex scalar whereas those with masses in the range of tens to hundreds singlet, three sets of Lepton Number charge assignments are of GeV have interesting prospects of being produced at the identified that correctly explain the lightness of the active LHC or future colliders [112,119–125]. neutrinos and provide the Majoron with couplings that lower On the other hand, given their extremely small couplings, the Hubble tension from 4.4σ down to 2.5σ . the heavy neutrinos produced in the early Universe would This mechanism is completely general and could be not be Boltzmann suppressed when decoupled from the ther- embedded in different types of flavour models. A compelling mal bath, leading to an unacceptably large contribution to the case presented here is the embedding in the Minimal (Lep- relativistic degrees of freedom of the Universe after their sub- ton) Flavour Violating setup: Lepton Number is part of the sequent decay [126–130]. If their decay takes place before the complete flavour symmetry and therefore naturally arises in onset of BBN, the decay products would quickly thermalise this context. The non-Abelian factors are mainly responsi- and BBN would then proceed as in the standard CDM sce- ble of the intergenerational mass hierarchies and of the mix- nario. However, if the decay of the heavy neutrinos happens ings, while Lepton Number, besides being associated to the after BBN and neutrino decoupling, their contribution to the Majoron, is involved in the explanation of the smallness of effective number of neutrinos would be too high and ruled the active neutrino masses with respect to the top mass. The out. If the decay takes place during BBN, the decay prod- ratios between the bottom and τ masses can be explained ucts could also alter the production of primordial helium and via the spontaneous breaking of the PQ symmetry, which strong constraints also apply [131–134]. This would be the completes the non-Abelian flavour symmetry together with situation of CASE NR1, for which the neutrino masses and Lepton Number. A QCD axion arises as a byproduct of the PQ mixings predict decay rates comparable to or larger than the symmetry breaking, allowing to solve the Strong CP problem onset of BBN. Conversely, the larger masses that character- within the same framework. ize CASE NR2 lead to decays faster than BBN, eluding these The Majoron only couples at tree level to neutrinos. The cosmological constraints. Hence, BBN and CMB observa- neutrinoless-double-beta decay represents a natural observ- tions disfavor CASE NR1 unless the heavy neutrino decay able where to look for it, but the model prediction turns out is faster than BBN in some part of the parameter space or to be much smaller than the current experimental sensitivity some other modification of the standard CDM scenario is and much stronger constraints can be derived from its impact considered. Indeed, if the heavy neutrinos decay after BBN, in neutrino free streaming and the effective number of neu- for heavy neutrino masses in the range [3.5, 200] MeV, then trino species from CMB observations. Effective couplings to 123 Eur. Phys. J. C (2021) 81 :28 Page 11 of 14 28 charged fermions and photons arise at one and two loops, vide a link to the Creative Commons licence, and indicate if changes respectively. For this reason, the model is not significantly were made. The images or other third party material in this article constraint from CAST and Red Giant observations that oth- are included in the article’s Creative Commons licence, unless indi- cated otherwise in a credit line to the material. If material is not erwise provide stringent bounds for light pseudoscalars. included in the article’s Creative Commons licence and your intended Due to the mixing between the physical Higgs and the use is not permitted by statutory regulation or exceeds the permit- radial degree of freedom of the scalar field producing the ted use, you will need to obtain permission directly from the copy- Majoron, the latter does couple to the Higgs. This has an right holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. impact on the invisible Higgs decay, that indeed provides a Funded by SCOAP3. strong bound on the VEV of the new scalar, that is, on the Lepton Number breaking scale. 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