Neutrino Lines from Majoron Dark Matter

ULB-TH/17-01

Camilo Garcia-Cely1, ∗ and Julian Heeck1, † 1Service de Physique Théorique,UniversitéLibre de Bruxelles, Boulevard du Triomphe, CP225, 1050 Brussels, Belgium Models with spontaneously broken global lepton number can lead to a pseudo-Goldstone boson as a long-lived dark matter candidate. Here we revisit the case of singlet majoron dark matter and discuss multiple constraints. For masses above MeV, this model could lead to a detectable flux of monochromatic mass-eigenstate neutrinos, which have flavor ratios that depend strongly on the neutrino mass hierarchy. We provide a convenient parametrization for the loop-induced majoron couplings to charged fermions that allows us to discuss three-generation effects such as lepton flavor violation. These couplings are independent of the low-energy neutrino parameters but can be constrained by the majoron decays into charged fermions.

I. INTRODUCTION generations of fermions that also allows us to discuss lepton flavor violation (LFV) and perturbativity. Owing to its tree-level coupling, the key feature of ma- The observation of neutrino oscillations has raised the joron DM is arguably its two-body decay into monochro- question why neutrino masses are so much smaller than matic neutrinos, a topic that has received a lot of at- all other known masses. The most-studied solution to tention in recent years in its own right [21–25]. In this puzzle comes in the form of the seesaw mecha- fact, since neutrinos are the least-detectable SM par- nism [1], where heavy right-handed neutrinos suppress ticles, any limit on their flux automatically provides a active-neutrino masses with respect to the electroweak model-independent lower bound on the DM lifetime [26]. scale. As a bonus, these heavy neutrinos can dynami- (The same arguments apply to DM annihilations [27– cally generate a baryon asymmetry in the early Universe 29].) Majorons are a well-motivated DM candidate that via leptogenesis [2], thus solving a further problem of can lead to observable monochromatic neutrino fluxes the Standard Model (SM). An inherent feature of the for energies between MeV and 10 TeV. For energies be- seesaw mechanism is the self-conjugate Majorana nature low the electroweak scale, these neutrino lines do not of neutrinos, which implies that the anomaly-free global receive Bremsstrahlung corrections that could otherwise U(1) symmetry of the SM has to be broken by two B−L lead to observable gamma-ray fluxes [25, 30, 31], so neu- units. Following the success of spontaneous symmetry trino detectors have unique detection possibilities. Ex- breaking in particle physics, one can easily imagine that periments that are sensitive to MeV-scale supernova neu- also this B −L symmetry is broken spontaneously, result- trinos, most prominently Borexino [32], KamLAND [33], ing in a Nambu–Goldstone boson named majoron [3, 4]. and Super-Kamiokande (SK) [34, 35], can thus be used Gravitational or explicit breaking terms then typically as DM detectors as well. generate a mass term, making the majoron a pseudo- Goldstone boson. Since the majoron has couplings that This article is organized as follows: in Sec. II we provide an introduction to the singlet majoron model, its are suppressed by the B − L breaking scale, i.e. the seesaw scale, it can easily be long-lived enough to form the couplings and decay modes. In particular, we introduce dark matter (DM) of our Universe [5–12]. The CP-even a compact parametrization for the one-loop induced ma- partner of the majoron can on the other hand be used joron couplings to charged fermions that is invaluable to to drive inflation [13, 14], although this is not the focus study majorons. In Sec. III we discuss the signatures of here. Since the seesaw scale is far above the electroweak majoron DM, split into neutrino signatures (Sec. III 1) scale for DM stability reasons, leptogenesis will be hardly and visible decay modes (Sec. III 2). Low-energy con- modified by the majoron [15]. straints from e.g. LFV that are also relevant if the majoron is not DM are presented in Sec. IV. Finally, we con- The most salient and well-studied indirect-detection clude in Sec. V. Appendix A is devoted to a discussion signature of majoron DM is its decay into two photons, of neutrino flavor ratios after propagation, highlighting which most prominently arises in cases where one iden- the differences between neutrino production from elec- arXiv:1701.07209v2 [hep-ph] 12 May 2017 tifies the majoron with the axion [16–19], for which new troweak and majoron interactions. particles have to be introduced to create a color anomaly of the U(1), often accompanied by an electromagnetic anomaly as well [20]. In this article we focus on other II. SINGLET MAJORON MODEL possible signatures, most notably from the tree-level decays into neutrinos and from the one-loop decays into charged fermions [7–9]. To this effect we provide a sim- We know from neutrino-oscillation experiments that ple parametrization of the majoron couplings to the three at least two neutrinos are massive, with sub-eV mass splitting. If neutrinos are Majorana particles, this implies that lepton number U(1)L (or U(1)B−L) is broken by two units, and if this breaking is spontaneous we expect a ∗Electronic address: [email protected] Goldstone boson, the majoron J [3, 4]. We will restrict †Electronic address: [email protected] ourselves to the singlet majoron model, where an SM- 2 √ 0 singlet complex scalar σ = (f +σ +iJ)/ 2 with L(σ) = U(1)B−L breaking in the Lagrangian, as required for a −2 couples to three right-handed neutrinos NR, non-zero majoron mass, can lead to severe fine-tuning issues. Some U(1) breaking terms, such as σ3 or even some 1 c L = −LyNRH − 2 N RλNRσ + h.c., (1) Planck-scale suppressed operators, need to be heavily suppressed in order to keep the majoron mass small [5]. with the lepton (scalar) doublet L (H) and the Yukawa Similar issues arise in axion models [40], but can often be matrices y and λ. A generalization to arbitrarily many solved by means of additional particles and symmetries. right-handed neutrinos is straightforward and will not The limit of interest in this article is the seesaw [1] change the discussion [36]. Spontaneous symmetry relation mD MR in Eq. (2), which leads to light neu- breaking at the scale f gives rise to the right-handed 2 √ trino masses of order mD/MR, automatically suppressed Majorana mass matrix MR = fλ/ 2, diagonal with- with respect to the electroweak scale. This allows for a out loss of√ generality. Electroweak symmetry breaking, block-diagonalization and expansion in the small ratio h i = ( 2 0)T , introduces a mixing between the left p H v/ , mD/MR ∼ dl/dh, leading to and right-handed neutrinos via the Dirac mass matrix √  √ q  mD = yv/ 2. The full Majorana mass matrix in the ∗ ∗ † −1 c U −iU dlR dh basis (νL,NR) = V nR is then V '  q √  , (7) −i d−1R d 1 h l 0 mD ∗ †  √ q  T = V diag(m1, . . . , m6)V , (2) T −1 mD MR 1 i dlR dh C '  q √  , (8) −1 ∗ −i d R dl 0 where V is the 6 × 6 mixing matrix to the states nR, h which form the Majorana mass eigenstates n = n + nc . √ q R R B ' i T −1 , (9) The relevant couplings of J, Z, and W − can be rewritten U U dlR dh in terms of these mass eigenstates as [37] where dl = diag(m1, m2, m3) dh = diag(m4, m5, m6), T −1 6 and R = (R ) is a complex orthogonal 3 × 3 ma- iJ X 1 trix that arises in the Casas–Ibarra parametrization of LJ = − ni γ5(mi + mj)( δij − ReCij) √ √ 2f 2 T i,j=1 (3) mD = iU dlR dh and describes the mixing between light and heavy neutrinos [41]. Since the mixing angles +i(m − m )ImC ] n , i j ij j of the Pontecorvo–Maki–Nakagawa–Sakata matrix U and 6 2 2 gw X the mass splittings ∆m21 and |∆m32| are known [42], the LZ = − niZ/ [iImCij − γ5ReCij] nj , (4) 4 cos θ free parameters in the seesaw limit are e.g. m1, dh, R, f, w i,j=1 mJ , and the three CP-violating phases in U. It will prove 6 useful to distinguish three different extreme hierarchies gw X − LW = − √ ìBìjW/ (1 − γ5)nj + h.c. , (5) of light neutrinos: 2 2 i,j=1 Normal Hierarchy (NH): m1 m2 m3 , (10) where Inverted Hierarchy (IH): m3 m2 ' m1 , (11)

3 3 Quasi-Degenerate (QD): m1 ' m2 ' m3 . (12) X ∗ X ` ∗ Cij ≡ VkiV ,B` j ≡ U V . (6) kj i ìk kj Assuming , the majoron can decay k=1 k=1 m1,2,3 mJ m4,5,6 into the light neutrinos, with partial widths proportional ` 2 Here, U is a unitary mixing matrix from the diagonal- to mj due to the diagonal Jνν couplings in the seesaw ization of the charged-lepton mass matrix which we can limit: assume to be the identity matrix without loss of general- 3 mJ X ity. The neutrino couplings to the CP-even scalars can be Γ(J → νν) ' m2 (13) 16πf 2 j found in Ref. [37] but are of no importance here. We will j=1 0 assume σ to be very heavy, m 0 ∼ f v, and essen- σ 9 2 P 2 ! 1 m 10 GeV mj tially decoupled from the SM to simplify the discussion. ' J j . It could however be used as an inflaton [13, 14], which 3 × 1019 s 1 MeV f 10−3 eV2 has little impact on the discussion in this article. P 2 We will further assume the majoron to be massive, Neutrino oscillations give a lower bound on j mj of i.e. a pseudo-Goldstone boson. The mass could arise be- 2.6×10−3 eV2 (4.9×10−3 eV2) for normal (inverted) mass cause of (quantum-)gravity effects [5, 38], heavy chiral ordering; Cosmology gives a conservative upper limit of fields that render B − L anomalous (which could make J 0.17 eV2 [43], to be used below for the QD regime, al- an axion and identify the seesaw scale with the Peccei– though much stronger limits even below 10−2 eV2 are Quinn scale [16–19]) or simply because of explicit break- possible for certain combinations of datasets [43–45]. We ing in the Lagrangian [9, 39]. The actual mechanism see that a majoron can easily be long-lived enough to be is not important for our analysis, its main impact will DM for typical seesaw scales, assuming J → νν to be the be on the production mechanism for majoron DM and main decay channel. on its decay into two photons, to be discussed below. At the one-loop level one obtains a coupling of J to It must be mentioned, however, that the existence of charged fermions [3, 37], which is crucial for majoron 3

• The oﬀ-diagonal lepton couplings 0 are approxi- a) ℓ, q gJ`` mately chiral due to the hierarchy of charged-lepton ni J Z masses,

im` 0 0 0 ¯ LJ`` ' − 2 K`` J `PL` + h.c., (21) n 8π v j ℓ,¯ q¯ for m` m`0 . b) ℓ • The matrix K is positive semi-deﬁnite if the light- ni est neutrino mass is zero and positive deﬁnite oth- J erwise, with determinant W 6 1 Y n det K = mj ≥ 0 (22) j v3f 3 ℓ¯ j=1

1/3 FIG. 1: Loop-induced majoron couplings to charged fermions and non-negative trace, trK ≥ 3(det K) . All di- with the Majorana neutrino mass eigenstates ni running in agonal entries K`` are real and non-negative. Since the loops. mlightest = 0 is unstable under renormalization group evolution [46], we can take K to be strictly positive definite, which gives Schwarz inequalities phenomenology. The Feynman diagrams are shown in on the off-diagonal entries, Fig. 1 and give rise to the effective on-shell couplings p |K 0 | ≤ K K 0 0 ≤ trK. (23) L = i ¯ ( S + P ) (14) `` `` ` ` J Jf1 gJf1f2 gJf1f2 γ5 f2 , As a result, constraints on tr , e.g. from → ¯ , with flavor-diagonal pseudoscalar quark couplings K J qq constrain all entries of K, courtesy of its positive- P mq q S definite nature. 0 gJqq0 ' 2 δqq T3 trK , gJqq0 = 0 , (15) 8π v † • From the definition K = mDm /(vf) we can esti- and more involved lepton couplings, D mate a simple√ perturbativity condition by demand- ing mD/v < 4π (see also Ref. [47]), P m` + m`0 ` 0 0 gJ``0 ' 2 δ`` T3 trK + K`` , (16) 16π v 9 4πv −6 10 GeV S m`0 − m` |K``0 | < ' 3 × 10 . (24) g 0 ' K``0 , (17) J`` 16π2v f f d,` 1 Typical values for K – without fine-tuned matrix to lowest order in the seesaw limit, where T3 = − 2 = u cancellations, i.e. imaginary R – can on the other −T3 . The dimensionless hermitian 3 × 3 matrix K is defined as hand be estimated as

† dhdl dl −13 mDmD 1 p T ∗p † K ∼ ∼ λ ∼ 2 × 10 λ , (25) K ≡ = U dlR dhR dlU . (18) vf vf fv v

The partial width for the charged-fermion modes J → ff¯ with the Yukawa coupling λ from Eq. (1). This is is then given by of course nothing but the result one obtains for one fermion generation, as calculated in Ref. [3]. 3 P 2 Γ(J → qq¯ ) ' |gJqq| mJ , (19) † 8π • As shown in Ref. [48], the matrix mDmD (or K in our case) can be used to replace R and d in ¯ 0 1 P 2 S 2 h Γ(J → `` ) ' |g 0 | + |g 0 | mJ , (20) 8π J`` J`` the seesaw parametrization. In other words, the entire seesaw matrix from Eq. (2) can be recon- working again in the limit of small fermion masses. A structed using low-energy neutrino parameters (dl couple of remarks are in order: and U) as well as K and f. This is the parametrization of choice in this article, seeing as K describes • All couplings g are proportional to the corre- Jf1f2 the physical couplings of the majoron to charged sponding fermion masses as required for derivative fermions and furthermore fulﬁlls a number of use- couplings of Goldstone bosons. This in turn implies ful inequalities that would be tedious to translate that the processes J → f f are helicity suppressed 1 2 to e.g. . It is quite remarkable that the seem- as expected for a neutral spin-zero particle decay- R ingly lost high-energy seesaw parameters encoded ing into SM fermions. † in mDmD become available in the form of majoron • The diagonal fermion couplings gJff are of pure couplings, allowing in principle to reconstruct the pseudoscalar nature [37]. seesaw mechanism with low-energy data. 4

¯ 2 Due to the proportionality Γ(J → ff) ∝ mf , the dom- γ inant decay channel of J is typically into the heaviest ni kinematically available fermion, but there are some no- J Z table loopholes: 1) the decay rates into charged fermions f 2 all scale with K and can be made small compared to n √ j γ J → νν in the limit λ ' 2dh/f 1; 2) the diagonal lepton couplings J`` are proportional to trK − 2K``, which could be highly suppressed for up to two leptons FIG. 2: One two-loop contribution to J → γγ via charged despite K being large [37, 49]. For example, the pattern fermions f. Kee = Kµµ Kττ ' 0 turns off the majoron couplings to ee and µµ. Since we are interested in majoron masses in the MeV– has some admixture of a triplet majoron [4]). In the ab- GeV range, the decays J → uu,¯ dd,¯ ss,¯ cc¯ should be re- sence of U(1)B−L-anomaly-inducing heavy fermions, our placed by appropriate decays into hadrons, which in par- singlet-majoron coupling to photons will be generated first at two loops. One contribution comes from the ma- ticular moves the kinematic threshold from mJ ' 2mu joron mixing with the longitudinal component of the Z to mπ, with first allowed channel J → πγγ [50], albeit heavily suppressed. Note that J decays into pairs of boson, which then decays into two photons, see Fig. 2. pseudoscalar mesons are forbidden by CP, so the next Notice that only fermion loops contribute to this piece of the amplitude, because similar diagrams with the W bo- threshold is 3mπ [51]. Seeing as not even the hadronic decay modes of a CP-even Higgs-like scalar with mass son and its Faddeev–Popov ghosts cancel each other [55]. between 0.1–10 GeV have been agreed-on in the litera- The additional diagrams that arise from closing the lep- ture (see e.g. Ref. [52]), we will not attempt to derive tonic lines in the W -boson loop of Fig. 1 b) are much the J → hadron decay rates here, but leave them for more complicated to calculate, but we expect them to future work. Estimates for a pseudoscalar’s decay into be further suppressed by the W mass or even the heavy three mesons can be found in Ref. [53], assuming Higgs- neutrino mass, so we will neglect them for now. Notice like couplings. In the majoron model we have instead a that such a separation of the diagrams is gauge invariant, Higgs-like coupling with additional sign-flip for up- and as the corresponding amplitudes satisfy the Ward iden- down quarks, just like in two-Higgs-doublet models of tities separately. Note also that the Z-boson contribu- type I and X. The only hadronic decay used in the fol- tion depends on different parameters than the W -boson lowing is J → ¯bb, which can be calculated reliably and part (e.g. quark masses), so it is not possible for the ne- > glected diagrams to cancel the entire amplitude; a partial will provide the best constraints on K for mJ ∼ 10 GeV. Let us continue our discussion of majoron decay destructive interference could, of course, be possible. Fo- modes. Still at the one-loop level one has virtual internal cusing only on the gauge-invariant part of the amplitude Bremsstrahlung, J → ffγ¯ , simply by attaching photons induced by J–Z mixing, i.e. Fig. 2, the two-loop rate to the diagrams in Fig. 1. For quarks this merely gives the takes the simple form well-known final-state radiation spectrum, but the addi- 2 2 2 3 2 ! tional diagram with a W boson gives a more interesting α (trK) mJ X f f 2 mJ Γ(J → γγ) ' 7 2 Nc T3 Qf g 2 , result for leptons. The extra photon removes the helic- 4096π v 4mf ity suppression of the amplitude and leads to a photon f spectrum similar to the s-wave Majorana DM annihila- (27) tion into ffγ¯ [54], with characteristic shape for sizable q ` with the color factor Nc = 3, Nc = 1 and the loop func- photon energy tion 1 dΓ(J → ``¯ 0γ) 1 p 2 ' 20x3(1 − x) , (26) g(x) ≡ − log[1 − 2x + 2 x(x − 1)] ¯ 0 4 Γ(J → `` γ) dx x (28) x 8x2 4x3 = 1 + + + + O(x4) . for x = 2Eγ /mJ ∈ [0, 1]. In our case, the helicity sup- 3 45 35 pression of the amplitude A ∝ m is however replaced by ` For m m , the fermion-mass independent contribu- an additional heavy-neutrino propagator, A ∝ 3 2 , J e e mJ /dh tions cancel due to anomaly freedom, leading to a rate so the rate is of higher order in the seesaw expansion and that is dominated by the lightest fermion, hence strongly suppressed. Bremsstrahlung will therefore not give testable signatures and will not be discussed fur- 2 2 7 α (trK) mJ Γ(J → γγ) ' , for mJ me . (29) ther. 15362π7 v2m4 Lastly, let us mention the possible decay mode J → γγ, e which could be the prime discovery channel and has been In particular, the coupling Jγγ vanishes for mJ = 0 discussed extensively in the literature for other models. as expected. Up to a prefactor, the rate of Eq. (27) For a massless majoron, the coupling to photons van- is equivalent to the singlet–triplet majoron case, where ishes because the global U(1)B−L symmetry is anomaly the majoron–Z mixing is induced already at tree level free [9, 37]. The coupling for a pseudo-Goldstone boson by the vacuum√ expectation value of an SU(2)L triplet then depends on the UV completion of the theory, i.e. the ∆ → vT / 2 [8]. The singlet–triplet-majoron rate then 2 2 details of how mJ is generated (and whether the singlet follows from Eq. (27) via trK → 32π vT /(fv). 5

We stress once more that the above diphoton rate was the Higgs resonance [56]. For the production via freeze- obtained by considering only a (gauge-invariant) subset in, Eq. (31) leads to of two-loop diagrams. While we expect the remaining −2 diagrams to be suppressed by mW or dh or even cancel λh mJ ' MeV. (32) completely, a full calculation is beyond the scope of this 2.0 × 10−10 article. Furthermore, the rate can be modified by the details of the scalar (admixture of triplets or CP-violating Freeze-in is thus a viable mechanism to produce majoron mixing with the Higgs) and fermion sector (B−L anoma- DM in the MeV and GeV range. Other production mech- lous fermions that create a Jγγ coupling at one-loop). anisms exist, see e.g. Ref. [9] and references therein. Nev- The reader should therefore be careful when interpreting ertheless, from now on we will remain agnostic about how the diphoton rate used here. DM was produced in the Early Universe and only assume that (cold) majorons constitute all the DM and that its mass lies below the electroweak scale. In any case, the III. DARK MATTER specific indirect detection signatures discussed below do not depend on the details of the majoron mass generation Possible production mechanisms for majoron(-like) or its production mechanism. DM have been extensively discussed in Ref. [9], assuming a restricted set of couplings in order to obtain predictions. 1. Neutrino signatures For example, taking 2 † The only tree-level decay mode of the singlet majoron LL/ = λhσ H H + h.c. J is into neutrinos, Eq. (13), completely specified in ⊃ −λ J 2H†H = − 1 m2 J 2(1 + h/v)2 (30) h 2 J terms of neutrino masses and U(1) breaking scale f. An interesting side effect of the majoron coupling to neutrino to be the only explicit U(1) breaking term in the mass eigenstates is that the emitted neutrinos will not os- scalar potential and neglecting the U(1)-invariant portal cillate, resulting in flavor ratios that can be completely |σ|2H†H, the relic density Ω of J is completely fixed for J different from astrophysical sources [58]; for a detailed a given mass mJ (assuming small Yukawa couplings λ to 2 2 discussion using the density-matrix formalism, see Ap- the heavy neutrinos). For sufficiently large λh = mJ /v , pendix A. The branching ratio of J decaying into νj is a thermal population of majorons is produced in the 2 2 Early Universe from annihilations and the (inverse) de- proportional to mj , and νj contains a fraction |U`j| of cays of the Higgs boson; after the Higgs disappears from flavor `, so the flavor composition of the majoron-decay the thermal plasma, the DM density eventually freezes neutrino flux is given by αe : αµ : ατ with out. The required value for λh in this scenario is typi- P3 2 2 mj |U`j| cally incompatible with constraints from direct detection α ≡ j=1 , (33) ` P3 2 or h → invisible, at least in the mass range of interest j=1 mj here [56]. Another possible situation is to assume that P the number of DM particles was negligible with respect normalized so that ` α` = 1. The self-conjugate Ma- to those of the SM after reheating. If the portal inter- jorana nature ensures that α` = α`¯. Contrary to most other neutrino fluxes, these ratios are the same at the action coupling λh has small values, the population of S majorons never reaches thermal equilibrium; for temper- source, where DM decays, and on Earth, so α` = α` = ⊕ atures much smaller than the Higgs mass – when the α` , up to matter effects inside the Earth. See Fig. 3 for majoron decouples from the SM plasma – its abundance an illustration using a ternary plot with a scan over the approaches a constant value. This leads to [57] 1σ and 3σ ranges of the oscillation parameters obtained in Ref. [42]. 27 2 2.19 × 10 mJ Γ(h → JJ) The mixing angles θ23 ' π/4 θ13 result in an almost ΩJ h ' , (31) sp ρ 2 µ–τ-symmetric mixing matrix, i.e. |Uµj| ' |Uτj|, which g∗ g∗ mh ensures αµ ' ατ independent of the mass ordering. αe where gs and gρ are the number of degrees of freedom on the other hand depends strongly on the neutrino mass ∗ ∗ 2 contributing to the entropy and energy density when hierarchy, with lowest value for NH (αe ' sin θ13) and the majoron decouples. This is the freeze-in mechanism, largest value for IH (αe ' 1/2). Using the best-fit val- which obviously requires mJ < mh/2 and a very small ues from Ref. [42] for the mixing angles, we obtain the decay rate of the Higgs boson into majorons (automati- following benchmark values for the flavor ratios in the cally satisfying LHC constraints on h → invisible). From hierarchical regime, s ρ the observed DM density, and taking g∗ ∼ g∗ ∼ 100, we 2 2 αe : αµ : ατ obtain mJ ' 2.7 MeV for the λh = mJ /v case described above [9]. NH: 0.03 : 0.43 : 0.54 , In a more general case, one can consider separate U(1) IH: 0.48 : 0.22 : 0.30 , (34) breaking terms for the majoron mass and the Higgs por- QD: 1 : 1 : 1 , tal, disentangling relic density and DM mass. For the freeze-out production mechanism, this is just the singlet denoted by stars in Fig. 3. These are the values we will DM scenario, heavily constrained and only viable around use in the following, but most results can be rescaled 6

the flavor ratios of the secondary neutrinos to fall into Ν Pure Μ the red contour of Fig. 3, because they are created as 0. 1. flavor eigenstates (see Appendix A). Let us also mention 0.1 0.9 that in models with a larger dark sector it is possible to obtain, for example, boosted majorons that decay into a 0.2 0.8 continuous neutrino spectrum, for which the flavor ratios 0.3 0.7 could again be more important. 0.4 0.6 As mentioned above, the spectral feature of J → νν Α should serve as a sufficient discriminant from the contin- Å Å Τ 0.5 0.5 Μ Α NH ø uous background. As shown in Ref. [62], this two-body 1 Σ 0.4 0.6 QD decay mode is suppressed compared to J → ννh(h) for 3 Σ ø > 0.7 0.3 mJ ∼ 10 TeV, which induces a continuous spectrum. We IH ø will further restrict ourselves to masses mJ < 100 GeV 0.8 0.2 astrophysical in this analysis, in order to avoid discussing effects from neutrinos 0.9 0.1 e.g. J → W W, ZZ that could be induced in some UV- completions of our model. We stress, however, that 1. 0. 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1. J → νν could still be an important discovery channel Å Pure ΝΤ Αe Pure Νe for majoron masses up to 10 TeV, for which IceCube becomes the ideal observatory [63, 64]. The neutrino (plus antineutrino) flux per flavor ` from the J → νν decay in FIG. 3: Majoron DM decay J → νν yields neutrino fla- our galaxy is given by [26, 65] vor ratios αe : αµ : ατ that depend on the neutrino mass hierarchy. The 1σ (3σ) ranges of the neutrino-oscillation pa- dΦ J α Γ(J → νν) dN rameters from Ref. [42] correspond to the green (blue) lines; ` = ` , (35) lighter colors correspond to a larger lightest-neutrino mass, dEν 4π mJ dEν converging to 1 : 1 : 1 for the QD spectrum. The three stars R ∞ denote the benchmark values of Eq. (34). The expected flavor where J = 0 ρHalods is the astrophysical factor associ- ratios from realistic astrophysical processes (e.g. pion decay ated to the DM density ρHalo in the Milky Way halo. For followed by averaged-out neutrino oscillations) fall in the red simplicity we write here the flux associated to the full sky, contour, taking into account the uncertainties in the mixing the general case for an angular signal is a straightforward parameters (95% C.L.) [59]. generalization of this case. The J -factor introduces uncertainties in the determination of the flux because the precise shape of ρHalo is unknown in the center of the without much effort. The NH composition with its tiny Galaxy. Nevertheless, in contrast to DM annihilations 2 νe fraction αe ' sin θ13 is particularly interesting, be- for which the J factor scales quadratically with ρHalo cause there is no astrophysical mechanism that would and thus varies by many orders of magnitude depending suppress νe to such a degree without physics beyond the on the assumptions on the DM halo, the uncertainty for SM [58]. This is illustrated in Fig. 3, where we also show DM decays is of less than one order of magnitude [26] and the expected flavor ratios from astrophysical processes the determination of neutrino fluxes or limits on them is (red contour) under the assumption that the neutrino more robust. Notice that here we are neglecting the neu- oscillations have been averaged out when the flux arrives trino flux arising from DM decays outside our Galaxy, at Earth [58, 59], see Appendix A for details. As can whose spectrum is in any case red-shifted and not neces- be seen, the NH region lies outside of the typical as- sarily line-like [66, 67]. trophysical expectation, making flavor ratios a potential For the two-body decay J → νν we have dN/dEν = discriminatory tool for DM detection. 2δ(Eν − mJ /2), which is smeared out by the velocity Seeing as the majoron itself forms cold DM in our distribution and detector resolution. Low-threshold neu- model, the neutrino spectrum with its line-like feature trino detectors such as Borexino [32], KamLAND [33], will be a much better discovery tool than the flavor ratios and SK [34, 35] give limits on the (monochromatic) flux of Fig. 3. Let us mention, however, that the monochro- ofν ¯e from searches for the diffuse supernova neutrino matic signature becomes less important as soon as we background. Due to the large cross section and tag- consider mJ above the electroweak scale; since the cou- ging possibilities, the detection channel of choice here + pling to neutrinos of Eq. (3) also induces a coupling to the is inverse beta decayν ¯ep → ne , which has a kinematic 2 SM Higgs of the form Jνjνj(mj/f)(1 + h/v) , the decay threshold of Eν > 1.8 MeV. This makes it difficult to ob- < modes J → ννh(h) open up for mJ > (2)mh, and in fact tain limits for mJ ∼ 4 MeV, seeing as the background > dominate over J → νν for mJ ∼ 10 TeV [62]. The neu- from reactor neutrinos also increases dramatically for < trino spectrum from J → ννh(h) is then obviously no such low energies. For 5 MeV ∼ mJ < O(100) MeV on longer monochromatic, but the flavor ratios of the pri- the other hand, searches for supernovaν ¯e neutrinos give mary neutrinos illustrated in Fig. 3 continue to be valid. useful constraints on DM-induced neutrino fluxes, as can In addition, there will be secondary neutrinos with a dif- be seen in Fig. 4. Note that in our notation this is a 1 ferent spectrum and flavor ratio from the h decay and limit on the flux Φν¯e = 2 Φe, because only half of our electroweak Bremsstrahlung. A thorough discussion of electron neutrinos are antineutrinos. A near-future im- these effects will be discussed elsewhere, but we expect provement of these limits is realistic, especially with the 7

1015

SK QD , 14 +ΝΜ anisotropy 10 NH ΝΜ atm. Νe ΝΜ+ΝΜ 1013 L GeV H KamLAND

12 SK f 10 e e Ν Ν , on Cosmology

Borexino ® invisible

limit J 11 e

10 Ν from Lower 1010

109

108 10-3 10-2 10-1 100 101 102

mJ GeV

FIG. 4: Lower bound on the U(1)B−L breaking scale f for majoronH JLDM, assuming QD (solid lines) or NH (dashed), IH lying in between. The purple exclusion comes from cosmological constraints such as the CMB [60]. Adopted limits from supernova ν¯e searches come from Borexino [32] (green), KamLAND [33] (red), SK [34, 35] (blue), and reinterpreted SK data (orange) [26]. The black lines for mJ > 0.1 GeV come from a comparison with atmospheric νµ spectra [26], while pink shows the preliminary limit from a designated DM search using angular-anisotropy SK data [61]. proposed Gadolinium-extension of SK [68], which should using tri-bimaximal mixing values as an approximation. reduce background and potentially reach the diffuse su- For the sake of clarity, it is therefore sufficient to discuss pernova regime. Even ton-scale liquid-xenon detectors the limits for the regimes QD and NH in Fig. 4, as those build for the direct detection of DM, such as XENONnT, associated to IH happen to fall in between. LZ or DARWIN, could be sensitive to O(10 MeV) neutrino lines and might give useful information about the flavor ratios [69]. In any case, dedicated DM searches by Less direct limits on the J → νν decay come from the experimental collaborations are desirable, especially cosmology. The most conservative bound is surely to considering the apparent gap of official limits between demand J to have a lifetime that exceeds the age of mJ = 60 MeV and GeV. Above GeV, we have prelimi- our Universe, τ ' 4 × 1017 s. Better limits can be ob- nary SK limits on DM decay into muon neutrinos [61]. tained by studying the effect that the decay of a non- In the gap 60 MeV < mJ < GeV, we adapt the limits relativistic DM particle into relativistic daughter par- of Ref. [26], based on a reinterpretation of SKν ¯e data ticles has on e.g. the matter power spectrum. A re- as well as a comparison to the well-understood atmo- cent analysis provides a 95% C.L. constraint of order spheric muon neutrino flux (see also Ref. [67]). Here, we τ > 5 × 1018 s [70]. Future measurements of the cos- strongly urge the SK collaboration to check for neutrino mic microwave background (CMB), e.g. by CORE, could lines, both in electron and muon neutrinos. Hyper-K is improve the bound on τ by a factor of 2 [71]. This is cur- expected to further improve on the higher-energy region. rently the only constraint on the J → νν mode for ma- Depending on the neutrino mass hierarchy, these flux joron masses below 4 MeV and will be hard to improve limits can be translated into a lower bound on the with line searches due to the huge neutrino background U(1)B−L breaking scale f, see Fig. 4. The latter is natu- below 10 MeV from e.g. reactor neutrinos [29]. P 2 2 rally strongest for QD, seeing as Γ(J → νν) ∝ j mj /f scales with the neutrino masses. In contrast, the weak- est bounds arise for NH, which is quite obvious for limits The limits on f from Fig. 4 can be translated into up- that come from bounds on the total lifetime or from the per bounds on |Kαβ| with the help of the perturbativity α ' sin2 θ suppressed ν flux; surprisingly, limits from e 13 e constraint of Eq. (24). For mJ = 1 MeV (100 GeV) this Φµ lead to roughly the same bounds on f for NH and IH, implies |K| < 5 × 10−6 (3 × 10−11) for NH and about because of the accidental numerical relation an order of magnitude stronger for QD. These limits are 2 ¯ 0 ( m3 1 2 much weaker then the direct constraints from J → ff X 2 2 2 ' 2 |∆m32| for NH, Φµ ∝ mj |Uµj| ' 2 2 derived below (Fig. 5), but are valid even if the J decay m1 m2 1 2 j 6 + 3 ' 2 |∆m32| for IH, is kinematically forbidden. 8

10-12 + + -13 Kee KΜΜ KΤΤ 10 from J®ΓΓ 10-14 È È 10-15 '

{ -16 { 10 -K +KΜΜ+KΤΤ K ee + -

on from J®e e 10-17

limits È È 10-18 Kee-KΜΜ+KΤΤ , KeΜ -19 + - Upper 10 from J®Μ Μ

K +KΜΜ-KΤΤ , K Τ , KΜΤ 10-20 È È Èee È e from J®Τ+Τ- - CMB 10 21 Γ- ray telescopes Kee+KΜΜ+KΤΤ AMS-02 p pÈ È È È È È -22 from J®bb 10 AMS-02 e+ 10-23 È È 10-3 10-2 10-1 100 101 102

mJ GeV

FIG. 5: Upper bounds on the matrix elements K``0 or combinations of them from CMB measurements [72] and indirect DM searches with AMS-02 [65, 73]; γ-ray telescope limits on J → γγH andL J → ¯bb refer to INTEGRAL [74] for mJ < 7 MeV, to COMPTEL/EGRET [75] for 7 MeV ≤ mJ ≤ 400 MeV, and to Fermi-LAT [76, 77] for mJ > 400 MeV. For indirect DM searches, we only show the most constraining limits in a given channel. We remind the reader that K is expected to have an order of magnitude of 10−13λ, where λ is the Yukawa in Eq. (1).

2. Signatures from visible decay channels vations of the sky [75, 77, 79–83], but these are typically less stringent than those of positrons for DM masses be- > < < low 100 GeV. In addition, for mJ ∼ 10 GeV, the majoron Having identified MeV ∼ mJ ∼ 100 GeV as the region of interest where majoron DM can lead to a par- decays dominantly into bottom quarks, which subse- ticularly clean observable flux of monochromatic neutri- quently decay and fragment producing antiprotons. The nos, let us discuss the constraints from the visible decay AMS-02 experiment has also measured the corresponding flux [84], which, within astrophysical uncertainties, can channels, i.e. J → ff¯ at one loop and J → γγ at two loop. There are stringent constraints on DM decays into be interpreted as originating from only cosmic ray colli- charged fermions from a wide range of indirect searches, sions with the interstellar material [73]. Slightly stronger see e.g. Ref. [78] for a review. In our model, the majoron bounds can be obtained with Fermi [77]. This allows to decay modes into charged fermions all depend on the ma- set a strong upper bound on the decay rate into bottom quarks, trK < 10−22 at 95% C.L. for m > 10 GeV, as trix = † ( ) introduced in Eq. (18). A crucial ∼ J K mDmD/ vf shown in Fig. 5. observation here is that K does not depend on the low- energy neutrino parameters, but is a completely free pa- The strongest of the indirect detection bounds is the rameter matrix in the seesaw limit, up to the inequalities one on trK by J → ¯bb. As a matter of fact, this bound given below Eq. (18). Typical values can be estimated also applies to all entries of K due to the inequality dhdl −13 of Eq. (23). Thus, majorons with masses greater than as K ∼ fv ' 2 × 10 λ, but it is entirely possible to have values orders of magnitude larger or smaller. While ∼ 10 GeV are severely constrained, because such a small −9 the J → νν modes discussed above gave a direct limit on K would require tiny Yukawa couplings λ ∼ 10 . We the seesaw scale f, the charged-fermion decay modes will expect constraints on trK from the hadronic decay modes give limits on the remaining parameters of our model, even below 10 GeV, but the corresponding decays into which are encoded in the elements of K. The decays mesons are difficult to calculate reliably. Notice that be- J → νν and J → ff¯ thus provide completely orthogonal cause of these constraints, it is hopeless to observe Ma- information about majoron DM. joron DM in direct detection experiments looking for nu- clear recoils. Majoron decays into charged leptons are in particular constrained by the AMS-02 measurements of the positron Below few GeVs, indirect detection bounds become flux in cosmic rays [65]. The corresponding 95% C.L. up- very weak compared to CMB bounds. If DM decays per bounds on the K matrix elements are shown in Fig. 5. into photons or charged particles during the time between For masses above a few GeVs, other limits on the leptonic recombination and reionization, when the Universe was decay channels arise from the diffuse-gamma-ray obser- transparent and no large-scale structures were formed, 9 it injects energy into the photon–baryon fluid and poten- IV. OTHER CONSTRAINTS tially modifies the anisotropies of the CMB and its black- body shape. Consequently, the precise measurements of For mJ > mf + mf , the best constraints on the ma- Planck [85] set stringent constraints on majoron decays 1 2 joron couplings gJf1f2 come from the decay J → f1f2 or into charged fermions. We calculate the corresponding J → γγ, as we have seen in Fig. 5. Let us briefly dis- 1 constraints on 0 following Ref. [72], and show them K`` cuss limits from the production of J, e.g. from f1 → f2J, in Fig. 5. These bounds are very important for two rea- which gives limits on gJf1f2 for mJ < mf1 − mf2 . For sons. On the one hand, they constrain majoron decays at m > MeV, all these constraints turn out to be weaker + − + − J the MeV scale, where J → e e and J → µ µ are the than the perturbativity bounds of Eq. (24) in connection dominant decay channels. On the other hand, they do with the limits on f from Fig. 4, which imply that |K| not suffer from astrophysical uncertainties such as those −6 can be at most 5 × 10 for mJ ' MeV. Even stronger associated to halo DM densities or cosmic-ray propaga- bounds apply when considering the limits from J → γγ tion parameters. (Fig. 5). We nevertheless list the direct constraints below for completeness, and stress that they can be relevant Finally, let us discuss constraints from J → γγ, ar- for mJ < MeV or if J makes up only a subcomponent of guably the most popular decay channel for majorons [8, DM. 10, 11]. Using our estimate for this two-loop decay The off-diagonal majoron couplings are directly con- of Eq. (27), we can translate γ-line limits from IN- strained by the lepton flavor violating (LFV) decays TEGRAL [74], COMPTEL/EGRET [75], and Fermi- ` → `0J [37, 88, 89], with strongest bound in the muon LAT [76, 77] into upper bounds on trK (Fig. 5). These sector, Br(µ → eJ) < 2.6 × 10−6 [90], and Br(τ → `J) < γ-ray telescope limits are stronger than the correspond- O(10−3) [91]. The strong µ → eJ bound of Ref. [90] rests ing CMB limits on J → γγ [72], so we will not show on the assumption of isotropic electron emission; in our them here. Due to the suppression by α2 and an addi- case, however, the emission is maximally anisotropic, see tional loop compared to J → ff¯ , the limits from J → γγ Eq. (21), with dominant emission of the left-handed elec- are for the most part weaker than those from charged tron in the direction opposite to the muon polarization. fermions. Nevertheless, the diphoton decays probe trK, This also happens to be the region where the background which in turn limits all entries of K via Eq. (23), whereas from µ → eνν is largest, diminishing the limit by an order 0 < −5 the J → `` decays only probe specific linear combi- of magnitude [92] to |Kµe| ∼ 1 × 10 for mJ mµ. An nation of K elements. This makes the J → γγ (and almost identical limit has been obtained long ago by con- J → ¯bb) constraints particularly interesting. Future sidering µ → eJγ with a massless J, which does not de- prospects for this channel are good, with considerable pend on the chirality properties of the Jµe coupling, but current effort to improve limits in the MeV gap between is of course further suppressed by α and phase space [93]. < < 7 MeV ∼ mJ ∼ 400 MeV, for example by AdEPT [86] We checked explicitly that the rate for µ → eJγ in our and e-ASTROGAM [87]. An improvement by several or- model is well described by the effective off-diagonal cou- ders of magnitude seems feasible, which could open the pling of Eq. (21) followed by Bremsstrahlung, leading to door to a double-line observation in the MeV range, both the same differential distributions given in Refs. [89, 93].2 in neutrinos and γ-rays. For mJ < MeV, the diphoton Since the Bremsstrahlung rate formally diverges for small decay is the only feasible DM detection channel, seeing photon energies and small electron–photon opening an- as sub-MeV neutrinos are extremely difficult to detect, gle, the number of events crucially depends on the de- especially when it comes to their spectral shape and fla- tector resolution. It would be interesting to see how cur- vor. rent and future experiments such as MEG and Mu3e can improve on these 30-year-old limits with their modern In summary, the constraints on majoron DM from its detectors [89], but this will be discussed elsewhere. visible decay channels provide information on the model For m`0 , mJ m`, the partial widths are simply that is complementary to the main decay mode J → νν. 0 † 2 2 < < Γ(` → ` J) 3 |(mDmD)``0 | 3 v 2 In the region of interest for neutrino lines, MeV mJ 0 ∼ ∼ 0 ' 2 2 2 = 2 2 |K`` | , 100 GeV, the constraints on the elements of K range from Γ(` → ` ν`ν¯`0 ) 16π m` f 16π m` 10−13 to 10−23, which translates into typical values for (36) the Yukawa coupling λ of 1 to 10−10 via Eq. (25). This should not be taken too literally in the three-generation which then translate to the bounds framework, but can give some idea about the level of < −5 tuning necessary to evade the bounds. In particular, the |Kµe| ∼ 1 × 10 , for mJ mµ , > < −3 region mJ ∼ 10 GeV could be regarded as less motivated, |Kτe| ∼ 6 × 10 , for mJ mτ , (37) which is however a highly subjective statement. < −3 |Kτµ| ∼ 9 × 10 , for mJ mτ ,

1 For second and third generation fermions, these limits were re- ported only for DM masses above 10 GeV. Following the proce- 2 Note an unfortunate typo in Ref. [93], where the double- dure described in Ref. [72], we rederive the limits and extend differential distributions are given as a function of x = 2Ee/mµ, them to lower masses. when it is actually 2Eγ /mµ. 10 neglecting the dependence on the majoron mass for sim- from the electron [97] and nucleon coupling [98], respec- plicity. Perturbativity plus J → νν limits give stronger tively. For mJ up to 100 keV one has (slightly weaker) P limits, unless mJ < MeV; J → γγ even requires mJ < direct-detection bounds on gJee from EDELWEISS [99], 10 keV for LFV to be observable, at least if J makes up XENON [100], XMASS [101], and MAJORANA [102], < 100% of DM. Since such low-mass DM is typically not assuming J to be DM; this gives |Kee − Kµµ − Kττ | ∼ −4 cold, a dedicated analysis is necessary to evaluate its va- 10 [101] for mJ = 100 keV, roughly ten orders of lidity. magnitude weaker than the bound at mJ = O(1) MeV Additional LFV in the form of ` → `0γ arises from the (Fig. 5). The couplings to quarks are much less con- right-handed neutrinos, which is independent of the ma- strained for mJ > 10 keV; since there are no flavor- joron or breaking scale, with the strongest bound com- changing processes in the quark sector mediated by the ing from Br(µ → eγ) < 4.2 × 10−13 [94]. In the seesaw majoron at the one-loop level, quark-flavor constraints limit, m1,2,3 mW m4,5,6, the partial widths take are suppressed. Going to the next loop level we can es- the form [36, 95] timate constraints from K → πJ etc. following Ref. [53], which give constraints tr < 2×10−2 for 100 MeV, 0 2 K ∼ mJ < Γ(` → ` γ) 3α −2 † much weaker for larger mJ . In the region of interest in 0 ' mDdh mD , (38) Γ( → ¯ 0 ) 8 ``0 ` ` ν`ν` π this article, MeV ≤ mJ ≤ 100 GeV, majoron production which has a different matrix structure than K, making it gives weaker limits on K than perturbativity in combi- difficult to directly compare limits. In principle one can nation with the neutrino limits on f, and much weaker than the J → γγ bounds. calculate the above for a given dl, U, K and f [48], but the expression will be far from illuminating. Large rates Lastly, let us mention another signature of our model: typically require some fine-tuning, e.g. large Im(R) in the neutrinoless double beta decay 0νββ [103]. In the see- Casas–Ibarra parametrization, or a symmetry-motivated saw limit, the amplitude for this ∆L = 2 process is dominated by light-neutrino exchange, proportional to structure in mD [96]. Let us focus on the case of degen- ( T ) = P3 2 . This is in particular sen- erate heavy neutrinos, dh = M × 1, for which the above UdlU ee j=1 Uejmj 2 sitive to the Majorana CP phases in , which can- is proportional to |K``0 | , allowing us to directly compare U the two LFV rates, not be measured via neutrino oscillations. Current experiments probe the QD regime, with limits of order Γ(` → `0γ) m2 f 2 |( T ) | 0 2 eV [104]. Future experiments are ' 2πα ` . (39) UdlU ee < . Γ(` → `0J) M 2 M 2 expected to ultimately reach the IH regime, while NH leads to discouragingly small rates. The observation of The ratio is heavily suppressed for M ∼ f m , making ` 0νββ would be an incredible discovery and prove beyond the majoron final state the prime LFV channel despite its doubt that neutrinos are Majorana particles, leading fur- more difficult signature; the photon rate can dominate for √ ther credence to the seesaw mechanism. This would of small Yukawa coupling, = 2 1, implying not- λ M/f course be good news for our majoron model at hand, as too-heavy right-handed neutrinos. Both channels should it would in particular fix the rather strong dependence of hence be searched for experimentally. e.g. J → νν on the neutrino hierarchy. It should be men- The diagonal majoron couplings, i.e. the diagonal K tioned, however, that our (sub-MeV) majoron DM gives entries, are constrained via the J coupling to electrons completely negligible rates for the associated 0νββJ pro- and quarks. At low energies, we typically require the cou- cess (A, Z) → (A, Z + 2) + 2e− + J [103], seeing as the plings to nucleons N = (p, n)T instead of quarks, which majoron couplings to neutrinos m /f are minuscule. The can be estimated naively as i tr (16 2 ). ν JN γ5σ3N mN K/ π v discovery of such a mode would therefore strongly hint The coupling to quarks and nucleons is of particular in- at a more complicated majoron realization. Due to the terest, because it depends on trK, which automatically small Jνν coupling, supernova constraints are also easily limits all entries in due to Eq. (23), even the LFV K evaded [105]. couplings. Limits on (light) pseudoscalars can be readily found in the literature, usually assuming an effective coupling to fermions that is then used to calculate scattering V. CONCLUSION processes etc.; this will be at best an approximately ac- curate procedure in our model, because our effective Jff couplings from Eq. (14) are by construction only valid for In this work, we have revisited the singlet majoron on-shell particles. As such, scattering processes – which model, in which lepton number is a nearly exact symme- necessarily involve off-shell particles – would have to be try that is spontaneously broken at the seesaw scale. The calculated from scratch using the loop diagrams to ob- corresponding pseudo-Goldstone boson, the majoron, is tain the correct dependence of the cross sections on our stable on cosmological scales due to its highly suppressed parameters. couplings and can act as DM. One of the most remarkable A full calculation of all the required scattering rates features of this model is the prediction of monochromatic being beyond the scope of this work, let us assume that neutrinos arising from DM decays, practically testable at energies between MeV (e.g. Borexino) and 100 GeV the Jff couplings provide a reasonable estimate for ma- (e.g. Super-K), potentially up to 10 TeV (IceCube). We joron scattering. For mJ < 10 keV, the best limits then come from astrophysics and imply urge the experimental collaborations to perform designated searches for such low-energy DM-induced neutrino −5 −5 |Kee − Kµµ − Kττ | < 2 × 10 , trK < 10 , (40) lines. Since the majoron couples directly to the neutrino 11 mass eigenstates, the decay neutrinos do not oscillate and Appendix A: Neutrino Oscillations have flavor ratios on Earth that depend strongly on the neutrino mass hierarchy, see Figs. 3 and 4. In particular, An astrophysical source producing neutrinos with an the electron-neutrino flux is suppressed compared to the S energy E and flavor ratios α` leads to the density matrix other flavors for the normal mass hierarchy. S S ρ``0 = α` δ``0 . Neutrinos oscillate during their travel from Other constraints on the model arise from the majoron the source to Earth, as can be seen from the fact that couplings to charged fermions, induced at the one-loop ρS does not commute with the propagation Hamiltonian, level, and the decay into two photons, induced by two- m2 given in the neutrino-mass basis by ' ( + i ) . In loop diagrams. We have provided a convenient and com- Hij E 2E δij fact, the density matrix describing the flux of neutrinos pact three-generation parametrization of these couplings after a distance L at Earth reads ρ⊕ = e−iHLρSeiHL, or in terms of the matrix † , which contains precisely mDmD more precisely, those seesaw parameters that are usually unobservable at low energies. A measurement of the majoron couplings 2 m2L X mi L j ⊕ −i 2E ∗ S i 2E ∗ could then in principle complete our knowledge of the ρ``0 ' Uì e U`00i α`00 U`00j e Ul0j . (A1) seesaw mechanism. In the DM context, majoron decays i,j,`00 into charged fermions and diphotons are constrained by CMB observations and indirect DM searches, which put For a sufficiently large oscillation length L, neutrino os- † m2−m2 strong limits on mDm , especially for mJ > 10 GeV, i j D cillations average out and exp{−i 2E L} → δij, which as illustrated in Fig. 5. Our parametrization also allows ⊕ P ∗ S ∗ leads to ρ 0 ' 00 UìU 00 α 00 U`00i U 0 . The flavor us to study constraints from lepton flavor violation; the `` i ` ` i ` l i 0 composition at Earth, given by the diagonal elements of rates for anisotropic ` → ` J turn out to be small for the density matrix, is thus > 0 mJ ∼ MeV if J makes up all of DM, but ` → ` γ can be observable for not-too-heavy right-handed neutrinos. ⊕ S X 2 2 α ' P α with P``0 = |Uì| |U`0i| . (A2) i By varying the oscillation angles within the ranges al- Acknowledgements lowed by neutrino experiments and assuming an arbitrary composition of flavors at the source, we obtain the red We would like to thank Anna Lamperstorfer, Thomas contour of Fig. 3. Hambye, Sergio Palomares-Ruiz and Hiren Patel for use- The situation is different for neutrinos arising from ma- ful discussions. CGC is supported by the IISN and the joron decay. In this case, the branching ratios associ- Belgian Federal Science Policy through the Interuniver- 2 ated to J → νiνj are proportional to mj δij, at least in sity Attraction Pole P7/37 “Fundamental Interactions”. the lowest seesaw order we consider. Accordingly, the JH is a postdoctoral researcher of the F.R.S.-FNRS. density matrix at the source is diagonal in the mass ba- We acknowledge the use of Package-X [106, 107] and sis and commutes with the Hamiltonian. As a result, JaxoDraw [108]. ⊕ −iHL S iHL S ⊕ S ρ = e ρ e = ρ and therefore α` = α` .

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