Gravitational Interactions and Neutrino Masses

PHYSICAL REVIEW D 101, 115024 (2020)

Gravitational interactions and neutrino masses

Hooman Davoudiasl* High Energy Theory Group, Physics Department Brookhaven National Laboratory, Upton, New York 11973, USA

(Received 26 March 2020; accepted 19 May 2020; published 19 June 2020)

We describe a scenario where the smallness of neutrino masses is related to a global symmetry that is only violated by quantum gravitational effects. The coupling of neutrinos to gauge singlet right-handed fermions is attributed to symmetry preserving gravitational operators suppressed by the Planck mass, in this framework. The proposed scenario leads to axion particles that decay into neutrinos, which could be probed through cosmological measurements and may help explain the Hubble parameter tension. Depending on the details of the implementation, the scenario could provide axion dark matter candidates.

DOI: 10.1103/PhysRevD.101.115024

1 I. INTRODUCTION up with tiny mν. However, it is generally expected that gravitational effects lead to violations of global sym- The two outstanding mysteries of particle physics and metries. A macroscopic version of this expectation posits cosmology—the origin of small neutrino masses and the that a black hole destroys global charges and is fully nature of dark matter (DM)—provide the most compelling described by mass, spin, and gauge charges. In this work, phenomenological evidence for new physics. While neu- we will consider a scenario where the smallness of mν is trinos are well-established ingredients of the Standard protected by a global symmetry which is explicitly broken Model (SM), we do not know what type of physics provides only by nonperturbative gravitational effects, as will be the DM content of the Universe—about 25% of its total described below. energy budget [1]. Recent work on implications of nonperturbative gravi- The smallness of neutrino masses, mν ≲ 0.1 eV, could tational processes on low energy effective theories can be be a manifestation of ultraviolet (UV) physics, or it could found in Refs. [4–6]. Reference [7] considers neutrino be explained by a tiny dimensionless coupling ∼Oð10−12Þ. condensation and masses that are gravitationally induced; A popular example of UV physics that could explain why for work along this direction using a different approach see neutrino masses are small is the seesaw mechanism with Ref. [8]. In Ref. [9], a scenario connecting DM and sterile ultraheavy right-handed neutrinos, leading to left-handed neutrinos through gravitational interactions has been exam- Majorana states at low energies [2]. The heavy right- ined. For prior work where generation of heavy right- ∼1014 handed neutrinos could be as heavy as GeV. Larger handed neutrino masses from string theory instanton effects masses would lead to nonperturbative couplings and was considered see Ref. [10]. We will give a more concrete are not generally considered. This mechanism predicts description of our scenario below. However, we will first low energy lepton number violation manifested as rare provide some clarifying comments. neutrinoless double beta decays, suppressed by small Majorana masses ≲0.1 eV. Despite its theoretical appeal, II. CAVEATS the seesaw picture of small mν is quite challenging to verify experimentally and may not yield to direct Before going further, we would like to clarify a few confirmation. points. First, the gravitational effects of interest here can In principle, one could imagine that some global sym- only be fully determined in a consistent theory of quantum metry forbids neutrino masses at the renormalizable level. gravity, which is still under investigation. Nonetheless, Then, if this symmetry is very weakly broken we could end string theory seems to contain all the necessary ingredients for such a framework and many qualitative results can be *[email protected] gleaned from its possible structures. From a general relativistic point of view, semiclassical studies of worm- Published by the American Physical Society under the terms of holes [11,12] and black holes also offer such insights. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1An Abelian gauged symmetry in this spirit was suggested to and DOI. Funded by SCOAP3. describe quark mass hierarchies long ago, in Ref. [3].

2470-0010=2020=101(11)=115024(7) 115024-1 Published by the American Physical Society HOOMAN DAVOUDIASL PHYS. REV. D 101, 115024 (2020)

While we make no pretense that this work represents an fields that do not share any other type of interactions and ab initio treatment, we will use ideas and results inspired by may not be from the same physical sector. In particular, to the above well-motivated approaches to argue for a generate Dirac masses for neutrinos, “right-handed neu- qualitative picture of how neutrino masses may be low trinos” with no gauge charges are required. We suggest that energy manifestations of Planck scale gravitational proc- it is plausible that such fermions are not part of the SM esses. Obviously, we will not present a definitive model sector, but could couple to the SM fields through gravi- here, but instead we will aim to illustrate the general tational interactions suppressed by powers of MP; if these phenomena that could arise, and their possible signals, in operators do not violate global charges, there is no addi- this picture. tional “instanton” suppression. In what follows, we will assume that new physics, such III. ORGANIZING PRINCIPLES as supersymmetry, which is required to have a consistent UV theory of quantum gravity appears only at or close to In this work, we will entertain the possibility that there is MP. This has the advantage that in minimal implementa- ð1Þ ¼ 0 a global U g symmetry that demands mν . We will tions of our proposal, Planck suppressed operators that set assume, consistent with the above considerations, that the the effective Yukawa couplings for neutrinos provide fairly ð1Þ U g symmetry is only violated by operators that are definite predictions for the required scale of spontaneous suppressed by nonperturbative “gravitational instantons.” global symmetry breaking. ð1Þ Having laid out the organizing principles of our work, Operators that do not violate U g are present in the low energy effective theory, possibly suppressed by powers of we will next provide more specific details for choices of ≈ 1 2 1019 parameters. Planck mass MP . × GeV. This setup gives rise to axions whose mass is generated by gravitational effects that IV. INSTANTON ACTION explicitly break the global symmetry. This is analogous to the well-known Peccei-Quinn mechanism [13] that was We will take the aforementioned gravitational instanton proposed to resolve the strong CP puzzle and yields an effects to correspond to an action S. While the size of this axion whose mass is generated by QCD instantons [14,15]. action depends on the details of the underlying spacetime Our approach has elements in common with the Majoron geometry and the quantum theory of gravity, it has been model [16], in that it involves a global symmetry that leads argued that a typical string theory inspired size for S is to axions. However, the Majoron models address the given by generation of typically large masses for right handed singlet neutrinos that provide the basis for the seesaw mechanism. 2π S ∼ ; ð1Þ These models lead to light Majorana masses for the SM αG neutrinos. In our work, as will be detailed below, we will only consider generating Dirac masses, which require where αG ∼ 1=25 is roughly the grand unified gauge unusually small Yukawa couplings whose explanation coupling [19,20]. Absent a strong motivation for a par- lends itself well to a Planck-suppressed mechanism. ticular value, for the illustrative purposes of our work here Hence, specific models that we will describe later can be we will generally assume that [20] falsified if neutrinoless double beta decay is observed. −S −82 −55 The Majoron model has also been considered in the 1=30 ≲ αG ≲ 1=20 ⇒ e ∼ 10 –10 : ð2Þ context of gravitational global symmetry violation in Ref. [17]. Our approach differs from that of Ref. [17] in As we will show later, the above choice yields numerically that we do not allow gravitational symmetry violations, interesting results that demonstrate the utility of our unless they are mediated by “instanton” effects, making such scenario, while corresponding to UV motivated values. violations exponentially suppressed. We adopt the view that Let us denote SM singlet fermions, often called right- ν these instantons represent tunneling between neighboring handed neutrinos, by R; we postulate that these fermions ð1Þ have charge Q ðν Þ under Uð1Þ , assumed to be respected vacua with different U g charges, in a similar fashion that g R g nonperturbative electroweak processes allow transitions at the classical and renormalizable level. As mentioned among vacua with different b þ l charges, where b and l before, we will take the general view that since these states are baryon and lepton numbers, respectively [18]. are not charged under any SM interactions, they can In our treatment, operators that are only suppressed by reasonably be expected to be from an entirely different powers of MP are presumably generated by perturbative sector and only couple to the SM neutrinos through and nonperturbative gravitational effects, but they do not “gravitational interactions.” ¯ result in global charge violation (a similar approach was If the dim-4 “Dirac mass” term H LνR is forbidden by ð1Þ adopted in Ref. [6], motivated by the results of Ref. [12]). U g, it will lead to zero neutrino masses; here H is the However, since gravity acts universally on all types of Higgs doublet field with vacuum expectation value (vev) particles, we use these gravitational operators to connect hHi ≈ 174 GeV and L is an SM lepton doublet. However,

115024-2 GRAVITATIONAL INTERACTIONS AND NEUTRINO MASSES PHYS. REV. D 101, 115024 (2020) as mentioned earlier, it is generally expected that non- which has ΔQg ¼ 0 and hence can be generated by ð1Þ perturbative gravitational effects would not respect U g. gravitational effects unsuppressed byp instantonffiffiffi effects. Yet such violations of the associated charge would be If the vev of Φ is nonzero, hΦi¼ϕ0= 2, we will then −ΔQgS exponentially suppressed by e , where ΔQg is the net get neutrino Yukawa couplings to the Higgs yν of the size magnitude of the charge of the operator. hΦi As a first attempt, it seems natural to assume that we only ∼ ð Þ ðν Þ¼1 yν : 4 need to have Qg R , with all SM fields uncharged MP ð1Þ under U g (we will later show why this minimal setup would not yield acceptable values of mν). Therefore, we To get the correct mass for the neutrinos, we need −S ¯ h i ∼ 0 1 ∼ 10−12 hΦi ∼ could have a Dirac mass term e H LνR in the low energy yν H . eV. This requires yν and hence effective theory. However, this interaction will lead to 107 GeV. negligibly tiny masses ≲10−44 eV for neutrinos, given To generate hΦi ≠ 0, we consider the potential the reference values in Eq. (2). Note that a “Majorana” ν −2S ν ν ðΦÞ¼− 2Φ†Φ þ λðΦ†ΦÞ2 ð Þ mass term for R of the form e MP R R could be V m ; 5 generated through gravitational effects, but it would be extremely small ≲10−91 eV. For comparison, the inverse where m is the mass parameter of Φ and λ is its Oð1Þ self- size of the visible Universe, given by the present day coupling constant. Given the above considerations, we −33 ϕ ∼ hΦi Hubble parameter H0 ∼ 10 eV, is enormously larger. expect a heavy scalar of similar mass, mϕ , upon ð1Þ ð1Þ Hence, for all intents and purposes both types of neutrino spontaneous breaking of U g. However, if the U g is masses are zero at the renormalizable level. fully respected we would also end up with a massless Next, we will consider a minimal model, dubbed “Model “Goldstone” boson or axion a in the low energy effective I,” that accommodates viable values of mν and leads to theory. The vev hΦi is then identified with the decay ð1Þ potentially observable cosmological signals. constant of a. In passing, we note that if the U g breaking entails a first order phase transition, it could lead to V. MODEL I primordial gravitational waves. However, the above scale is a factor of ∼102 beyond the sensitivity of future ground- Since the gravitationally generated dim-4 interactions do based gravitational wave observatories [21]. Assuming a not yield the inferred values of mν ∼ 0.1 eV, we need to lower gravitational scale than MP, perhaps corresponding consider other contributions from higher dimension oper- to a more fundamental description, could possibly bring the ð Þ2 ators. Note that operators of the form HL =MP would requisite symmetry breaking scale hΦi within the reach of generate Majorana masses for neutrinos that are about 5 those experiments. orders of magnitude too small. Therefore, we are led to As discussed earlier, we expect gravitational effects to consider additional fields that allow forming Uð1Þ neutral ð1Þ g violate U g through the action of nonperturbative instan- operators, to avoid severe suppressions from instanton tons and thus to generate a potential for a, given by [19] effects. Φ ð1Þ ðΦÞ a Let us introduce a scalar with U g charge Qg . ∼− −S 4 ð Þ Va e MP cos : 6 We will attempt to generate acceptable “Dirac masses” mν. ϕ0 Hence, to avoid further suppressions, we need to make sure that any induced Majorana masses for νR satisfy mR ≪ mν. The above yields a mass for the axion Generally speaking, we then need to ensure that operators of the type Φnν ν , with n ≥ 1, are sufficiently suppressed. M4 R R 2 ∼ −S P ð Þ We will hence choose a set of charges that will make this ma e 2 ; 7 ϕ0 possible and also lead to operators that can provide the right size of mν. Our choice for the purposes of illustration here which for our choice of parameters yields will be ðQgðΦÞ;QgðLÞ;QgðνRÞÞ¼ð1; −2; −3Þ, with all other fields uncharged under Uð1Þ . Note that this charge −10 3 g 10 GeV ≲ ma ≲ 3 × 10 GeV ðModel IÞ: ð8Þ assignment is presumably not unique, but we will show that it could lead to interesting results. In what follows, we will Let us parametrize Φ as refer to this choice as “Model I.” With the above charges, we can write down the following ϕ þ ϕ0 Φ ¼ pffiffiffi eia=ϕ0 : ð9Þ dim-5 operator 2 ¯ ΦH LνR O5 ∼ ; ð3Þ Using Eq. (3), we find that the coupling of a to Dirac MP neutrinos ν is given by

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hHi mν g aνγ¯ 5ν ¼ pffiffiffi aνγ¯ 5ν ¼ aνγ¯ 5ν; ð10Þ a 2 ϕ MP 0 which is—as expected for an axion—proportional to neutrino masses and suppressed by the axion decay −17 constant. We then find ga ∼ 10 for the above model. The lifetime τa of a from decay into neutrinos is given by 8π 20 10−17 2 τ ¼ ∼ 1013 MeV ð Þ 2 s : 11 gama ma ga

17 For comparison, the age of the Universe is tU ∼ 4 × 10 s ∼ −42 −1 ð2 × 10 GeVÞ and the cosmic microwave background FIG. 1. Unstable fraction f of dark matter versus ma in Model I ∼ 1013 from Eq. (14). Values above the dashed line are excluded, (CMB) era roughly corresponds to tCMB s. Given the possibility of a cosmologically long lifetime for the axion, it corresponding to the 95% C.L. limits from Ref. [22] in the t ≲ τ ≲ t regime (see the text for details). is interesting to consider it as a possible signature of the CMB U above model, as we will discuss next. ϕ The decay of DM could leave an observable imprint on which a starts its oscillation is large compared to 0. Φ the evolution of the Universe. In Ref. [22], this general Assuming that the field is initially in thermal equilib- possibility was considered and its effects on CMB and rium, one would expect that it gets a vev after a phase ∼ ϕ matter power spectra were examined. These authors find transition at T 0. Hence, for sufficiently large ma, the ≈ 3H that in the long lifetime regime, which roughly corresponds condition ma would correspond to temperatures ð1Þ where the symmetry U g is typically unbroken and there to τ ≳ tU, the fraction f of DM that decays at a rate Γ is is no axion. As a representative range of parameters, we bounded by [22] −3 consider ma ∈ ½10 ; 20 MeV; this range would corre- fΓ < 15.9 × 10−3 Gyr−1 ð95% CLÞ: ð12Þ spond to the onset of axion oscillation temperatures T ≲ 108 GeV, with the axion typically expected to be Using Eq. (11), we can recast the above bound as present (due to spontaneously broken symmetry). We note that potential electroweak gauge boson couplings propor- 8.9 × 10−42 tional to anomalies are not required and could in principle fm < GeV: ð13Þ a g2 be set to zero; this may require further assignment of a charges. Hence, thermal production of a does not pose an In the intermediate regime, corresponding roughly to impediment to implementation of our scenario. ∈ ½10−3 20 t ≲ τ ≲ t , based on the analysis of Ref. [22],we For ma ; MeV, assuming the maximum CMB U ¼ ϕ ∼ 100 use the typical bound f ≲ 0.038. amplitude ai 0 and g , we present the values 2 2 2 of f from Eq. (14) versus ma in Fig. 1, shown as the solid The initial energy density stored in a is of order maai = , where a is the initial amplitude of the axion oscillations, line. The horizontal dashed line is the intermediate i ≲ τ ≲ ≲ 0 038 1=2 2 lifetime (tCMB tU) 95% C.L. bound f . , from commencing when ma ≈ 3H, where H ¼ðcgÞ T = 3 Ref. [22]. Values of f above this line are excluded. M —with T the temperature, c ≡ ð2πÞ =90, and g the P Constraints beyond the intermediate lifetime bound, cor- relativistic degrees of freedom—is the Hubble parameter responding to m ≳ 20 MeV (τ ≲ t ) are weaker [22]. during the radiation dominated era. The oscillating modu- a CMB Also, the long lifetime constraints (τ ≳ t ) are not very lus energy density drops with the expansion of the Universe U constraining and are not shown. For m ≳ 20 MeV, as long like that of matter, that is like T3. We interpret f to be the a as the reheat temperature is much larger than ϕ0,we ratio of the axion energy density to that of cosmic DM at ∼ ¼ 1 generally expect that the axion starts its oscillation only matter-radiation equality marked by Teq eV; for f upon spontaneous symmetry breaking and hence f grows the axion is assumed to constitute all DM. Thus, demand- with m2 in this mass range. For m ∼ 2 GeV, correspond- ∼ 4 ∼ a a ing the axion energy density redshift to fTeq at T Teq, τ ∼ 0 01 ∼ 1 ing to . tCMB, we roughly get f , which suggests we find above this mass one perturbs standard cosmology, since the pffiffiffiffiffiffi new unstable component of DM starts to be significant and 2 9 3=4 m ≈ ai cg a ð Þ the parameters are likely not viable [22]. f 2 2 : 14 MP Teq Our mechanism could potentially provide a resolution of a persistent tension between values of present time The above formula, however, should be used with care, Hubble parameter H0 determined from the CMB [23] and since at sufficiently large values of ma the temperature at local [24,25] measurements, with the latter yielding a result

115024-4 GRAVITATIONAL INTERACTIONS AND NEUTRINO MASSES PHYS. REV. D 101, 115024 (2020) that is a few standard deviations larger than that obtained call this extension “Model II.” Here, we propose to expand ð1Þ0 from the former. We note that this tension could be a result the model by another global symmetry U g.FromaUV of underestimated or unknown systematic uncertainties, (string theory) point of view, one expects a multitude of such ð1Þ ð1Þ0 however it has persisted for some time and its significance symmetries, see e.g., Ref. [19]. In principle, U g and U g has been at an interesting level (recent work in Ref. [26] symmetries would be violated by separate instantons of suggests that it now stands at 4.4σ). Hence, it is reasonable to action S and S0, respectively. We will assume that S ¼ S0,as entertain the possibility that it could be due to new physics. distinct numerical values are not necessary for our illustrative One of the proposed resolutions of the above H0 tension examples, given the broad range considered in Eq. (2). postulates late time DM decay into dark particles or We will also assume that there is an additional scalar Φ0. – ð1Þ radiation [27 31]; see also Ref. [22]. Such resolutions of The U g charge assignments of the fields are as follows: the H0 tension could require that only a subdominant 0 ðQgðΦÞ;QgðΦ Þ;QgðLÞ;QgðνRÞÞ¼ð1; 0;qþ 1;qÞ and the component of DM decay by the present epoch; see, for corresponding Uð1Þ0 assignments are ð0; 1; 0; −1Þ. example, Ref. [27]. We would then need a cosmologically g Therefore, we can write down the following gravitationally stable component, that we will not specify here, to account mediated dim-6 operator for the DM observed today. Axion decays in Model I would lead to a population of relativistic neutrinos that behave ΦΦ0 ¯ ν ∼ H L R ð Þ like dark radiation. A more detailed study is required to O6 2 ; 17 examine whether our scenario could plausibly alleviate the MP Hubble parameter tension. Nonetheless, given that the which, in order to generate the correct size for mν requires general features of a resolution are present in our proposal, let us elaborate on this possibility some more. hΦihΦ0i ∼ 10−12 ð Þ If axions make up a fraction f of DM energy density, 2 : 18 today’s flux could be of order MP

Fν ∼ fρ =m ; ð15Þ As before, any dim-4 Dirac masses for neutrinos would 0 DM a be exponentially suppressed by instantons. We will assume −6 −3 the same instanton processes would lead to violations of where ρ ∼ 1.3 × 10 GeV cm is the cosmic value of DM Uð1Þ and Uð1Þ0 symmetries. Also, Majorana masses for DM energy density. We are interested in DM decay after the g g ν ∼ 1013 R would dominantly originate from operators of the from CMB era, corresponding to tCMB s. From Eq. (11), τ ≳ ≲ 20 tCMB requires ma MeV. Using Eq. (14), we then 2q 0 2 ν −2 −1 hΦi hΦ i νRνR have F0 ≳ 100 cm s . To see if this flux is detectable we ð Þ 2qþ1 : 19 need to know its typical energy at the present time. MP During the matter dominated era, corresponding to ¼ 2 t ≳ t , we have t ∼ R3=2, with R the cosmic expansion For q , using the scalar vacuum expectation values CMB ∼4 10−17 scale factor. If the decay takes place at time t ∼ τ below, we find a Majorana mass × eV which is d ≪ (instantaneous approximation), then the energy of the mν and hence we could take mν to be a Dirac mass ν generated from the O6 operator in Eq. (17), to excellent neutrinos at present time E0 is roughly given by accuracy. pffiffiffi pffiffiffi 2 3 hΦi¼ϕ 2 hΦ0i¼ϕ0 2 m τ = Let 0= and 0= ;wewilldenotethe ν ∼ a ð Þ 0 E0 2 ; 16 axions associated with these fields by a and a , respectively. tU 9 0 17 Let us choose ϕ0 ¼ 10 GeV and ϕ0 ¼ 10 GeV, as where the energy of decay final state neutrinos is assumed illustrative examples. By analogy with the discussion of to be ∼ma=2, i.e., of order the cosmologically unstable Model I and Eq. (7), we find DM mass, which we have identified with ma. Hence, for ν 10−12 ≲ ≲ 30 ð ÞðÞ ma ≲ 20 MeV, we find E0 ≲ 10 keV. This energy is small GeV ma GeV Model II 20 enough that it presents a challenge to detection, which typically requires OðMeVÞ energies. and

−20 −7 10 GeV ≲ m 0 ≲ 3 × 10 GeV ðModel IIÞ: ð21Þ VI. MODEL II a 0 To explore further possibilities of the gravitational neu- One could show that the couplings of the a and a axions to −19 trino mass generation scenario proposed here, let us consider neutrinos in Model II are, as expected, ga ¼ mν=ϕ0 ∼ 10 0 −27 a simple extension of the above setup. Though this comes and ga0 ¼ mν=ϕ0 ∼ 10 , respectively. at the expense of minimality, it would lead to potentially The bound in Eq. (12), relevant to the regime τ ≳ tU,is interesting and broader options for phenomenology. We will equivalent to

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0 ϕ0 18 FIG. 3. Initial amplitude of oscillations ai in units of 0, versus FIG. 2. Lifetime, in units of τ ¼ f2.0 × 10 s, for the 0 0 m in Model II, assuming that the axion a constitutes all DM (see unstable DM fraction f of axion a in Model II, assuming a ¼ a i the text for more details). ϕ0 (see the text for details). Values below the dashed line, corresponding to the 95% C.L. limit (for τ ≳ tU) from Ref. [22], are excluded. accurate geological models, could probe parts of the parameter space of Model II. (For constraints on neutrino flux τ >f2.0 × 1018 s: ð22Þ from DM Majoron decays, using extraterrestrial antineutrino searches by Boexino [37] and KamLAND [38],seeRef.[39]. These constraints would allow the rough sample reference Using Eqs. (11) and (14), and assuming ai ¼ ϕ0,in Fig. 2 we have plotted the values of the axion a lifetime values adopted above, for nondegenerate neutrinos.) 18 In Fig. 3, we have presented the initial misalignment, in in units of τ ≡ f2.0 × 10 s, as a function of f.The units of ϕ0 , needed for the axion a0 to constitute all DM, model parameters are allowed by the cosmological con- 0 that is for f ¼ 1. Note that the lifetime of a0 is much larger straints above the horizontal dashed line. Hence, Model II Oð1020Þ can lead to a non-negligible fraction of unstable DM that than tU,by or more, over the entire range of masses in Eq. (21). Since over the reference range of ma0 decays on timescales comparable to or longer than tU, and thus a potentially detectable flux of neutrinos from the misalignment required is less than unity, all values can → νν¯ represent potentially viable DM candidates. For the lower a . Let us estimate the flux of these neutrinos. −11 0 ≳ 10 We will assume that there is a spherical distribution end of the range, corresponding to ma eV, such of DM particles centered around the Earth, with radius D. DM may be produced copiously by spinning solar mass One can then show that the flux F of neutrinos from the black holes [40,41], which can be probed by gravitational decay of DM arriving at Earth is given by wave measurements [42,43].

Dfρ VII. SUMMARY F ≈ ; ð23Þ mτ To summarize, we have proposed that the small masses ρ of neutrinos may be a hint for a global symmetry that where is the DM energy density. In the Galactic neighbor- requires them to vanish. Such a symmetry is expected to ρ ≈ 0 3 −3 hood of the Solar System, we have S . GeV cm [32], be violated by exponentially suppressed nonperturbative ∼ 0 5 ∼ 5 ¼ ϕ say, for D . kpc. For ma MeV and ai 0, gravitational “instanton” effects. Our approach allows for ∼ 0 3 τ ∼ 4 1017 using Eq. (14), we find f . and × s. Based symmetry preserving gravitational interactions among on this set of possible parameters, we see that the flux various fields, suppressed by powers of Planck scale ∼ 2 5 of neutrinos with energy Eν . MeV will be given by but without instanton suppression. In this view, “right- 5 −2 −1 F ∼ 10 cm s , including a mixture of flavors of both handed” neutrinos which may not be part of the SM neutrinos and antineutrinos. Interestingly, this is not far sector could couple to it and generate Dirac masses for from the level of “geo-neutrino” flux that has been observed neutrinos. Since we also require that the symmetry be by both KamLAND and Boerxino collaborations [33,34]. spontaneously broken, our scenario leads to appearance ν¯ For example, the KamLAND result for the flux of e is of light axions, which generically couple to neutrinos and 3 4þ0.8 106 −2 −1 . −0.8 × cm s . We see that current the uncertainty can decay into them. As the axions could constitute a in this measurement [35] is an order of magnitude above the fraction or all of dark matter, we could expect interesting neutrino flux from axion decays; similar conclusions apply imprints of this scenario on cosmological evolution, as to the Borexino results [36]. In principle, more precise discussed in this work. Simple extensions of the basic measurements of geo-neutrino flux, together with more model can accommodate cosmologically stable dark

115024-6 GRAVITATIONAL INTERACTIONS AND NEUTRINO MASSES PHYS. REV. D 101, 115024 (2020) matter axions, as well as an unstable axion population ACKNOWLEDGMENTS that could potentially lead to an observable neutrino flux. We thank Djuna Croon, Peter Denton, Julian Heeck, and We pointed out that the decay of this subdominant component into neutrinos could help alleviate the current Jure Zupan for comments and discussions. This work is tension between local and cosmological determinations of supported by the United States Department of Energy the Hubble parameter. under Grant Contract No. DE-SC0012704.

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