The Majoron at Two Loops

UCI-TR-2019-23, arXiv:1909.02029, Phys. Rev. D100 (2019) 095015

The Majoron at two loops

Julian Heeck1, ∗ and Hiren H. Patel2, † 1Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA 2Department of Physics and Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064, USA We present singlet-Majoron couplings to Standard Model particles through two loops at leading order in the seesaw expansion, including couplings to gauge bosons as well as ﬂavor-changing quark interactions. We discuss and compare the relevant phenomenological constraints on Majoron production as well as decaying Majoron dark matter. A comparison with standard seesaw observables in low-scale settings highlights the importance of searches for lepton-ﬂavor-violating two-body decays ` → `0+Majoron in both the muon and tau sectors.

CONTENTS is Weinberg’s dimension-five operator (LH¯ )2/Λ [1]. This operator violates lepton number (∆L = 2) and leads to 2 I. Introduction 1 Majorana neutrino masses of order hHi /Λ after electroweak symmetry breaking, which gives the correct neu- II. Majoron couplings at tree level and one loop 2 trino mass scale for a cutoff Λ ∼ 1014 GeV. Besides ex- A. Neutrino couplings 3 plaining neutrino oscillation data, an EFT scale this high B. Charged fermion couplings 3 has little impact on other observables and thus nicely ac- C. Couplings to gauge bosons 4 commodates the absence of non-SM-like signals in our experiments. III. Majoron couplings at two loops 4 An ever-increasing number of renormalizable realiza- A. Coupling to two gluons 5 tions of the Weinberg operator exist in the literature, B. Coupling to two photons 5 the simplest of which arguably being the famous type-I C. Coupling to Z and photon 6 seesaw mechanism [2] that introduces three heavy right- D. Coupling to two Z bosons 6 handed neutrinos to the SM field content. While the E. Coupling to two W bosons 7 Weinberg operator explicitly breaks lepton number, un- F. Coupling to γ-Higgs and Z-Higgs 7 derlying renormalizable models could have a dynamical G. Flavor changing quark couplings 8 origin for ∆L = 2 via spontaneous breaking of the global U(1)L symmetry. This leads to the same Weinberg oper- IV. Phenomenology 8 ator and thus Majorana neutrino masses, but as a result A. Light Majorons 8 of the spontaneous breaking of a continuous global sym- B. Comparison with seesaw observables 10 metry also a Goldstone boson appears in the spectrum. C. Majoron dark matter 11 This pseudo-scalar Goldstone boson of the lepton number symmetry was proposed a long time ago and was dubbed V. Conclusion 11 the Majoron [3, 4]. Acknowledgements 12 The Majoron is obviously intimately connected and coupled to Majorana neutrinos, but at loop level also References 12 receives couplings to the other SM particles. This makes it a simple renormalizable example of an axion-like particle (ALP), defined essentially as a light pseudo-scalar I. INTRODUCTION with an approximate shift symmetry. Although not our focus here, by coupling the Majoron to quarks it is even The Standard Model (SM) has emerged as an incredi- possible to identify it with the QCD axion [5–9], thus bly accurate description of our world at the particle level. solving the strong CP problem dynamically. The main Even its apparently accidental symmetries, baryon num- appeal of the Majoron ALP is that its couplings are not arXiv:1909.02029v2 [hep-ph] 14 Nov 2019 ber B and lepton number L, are seemingly of high quality free but rather specified by the seesaw parameters, which and have never been observed to be violated. One could, opens up the possibility to reconstruct the seesaw La- however, argue that the established observation of non- grangian by measuring the Majoron couplings [10]. This zero neutrino masses is not only a sign for physics beyond is aided by the fact that the loop-induced effective op- the SM but also for possible lepton number violation by erators that couple the Majoron to the SM are only two units. This argument is based on an interpretation suppressed by one power of the lepton-number break- of the SM as an effective field theory (EFT) and the ob- ing scale Λ, whereas right-handed neutrino-induced op- servation that the leading non-renormalizable operator erators without Majorons are necessarily suppressed by 1/Λ2 [11–17], rendering it difficult to reconstruct the seesaw parameters in that way. In this article we complete the program that was ∗ [email protected] started in the inaugural Majoron article [3] and derive † [email protected] all Majoron couplings to SM particles. The tree-level 2

6 and one-loop couplings were obtained a long time ago; gw X here we go to the two-loop level in order to calculate the LZ = niZ/ [CijPL − CjiPR] nj , (3) 4 cos θW remaining couplings, which include the phenomenologi- i,j=1 cally important couplings to photons as well as to quarks 6 3 gw X X − ∗ + of different generations. Armed with this complete set LW = √ `αBαjW/ PLnj + njB W/ PL`α , 2 αj of couplings we then discuss various phenomenological j=1 α=1 consequences and constraints on the parameters. This 6 h X includes a discussion of Majorons as dark matter (DM). L = − n [C (m P + m P ) h 2v i ij i L j R The rest of this article is structured as follows: in i,j=1 Sec. II we introduce the singlet Majoron model and re- +Cji(mjPL + miPR)] nj , produce the known tree-level and one-loop couplings. In Sec. III we present results of our novel two-loop cal- where gw = e/ sin θW with Weinberg angle θW and culations necessary for the Majoron couplings to gauge 3 3 bosons and to quarks of different generations. The phe- X ∗ X ` ∗ ∗ nomenological aspects of all these couplings are discussed Cij ≡ VkiVkj ,Bαj ≡ VαkVkj = Vαj . (4) in Sec. IV. Finally, we conclude in Sec. V. k=1 k=1 ` In the last equation we used Vαk = 11αk since we work in the basis where the charged-lepton mass matrix is diag- II. MAJORON COUPLINGS AT TREE LEVEL onal. AND ONE LOOP The 6 × 6 matrix C and the 3 × 6 matrix B satisfy a number of identities [20, 21] that are particularly impor- In this article we consider the minimal singlet Majoron tant in order to establish ultraviolet (UV) finiteness of model [3], amplitudes involving neutrino loops: c L = −LyN H − 1 N λN σ + h.c. − V(H, σ), (1) R 2 R R C = C† = CC, which introduces three right-handed neutrinos NR cou- BB† = 11 ,CM CT = 0 , pled to the SM lepton doublets L and Higgs doublet n † T H, and one SM singlet complex scalar σ carrying lep- B B = C,BMnC = 0 , (5) ton number L = −2, minimally coupled to the right- T BC = B,BMnB = 0 . handed neutrinos proportional to the Yukawa matrices y and λ. We do not specify the scalar potential√ V(H, σ) So far we have not made any assumption about the but simply assume that σ = (f + σ0 + iJ)/ 2 obtains scale of MR. In the following we will work in the a vacuum expectation value f, which then gives rise√ to seesaw limit MD MR, resulting in a split neutrino the right-handed Majorana mass matrix MR = fλ/ 2. spectrum with three heavy neutrinos with mass matrix J is the Majoron, σ0 is a massive CP-even scalar with MR and three light neutrinos with seesaw mass matrix −1 T mass around f, assumed to be inaccessibly heavy in the Mν ' −MDMR MD, naturally suppressed compared to following. Both MR and the charged-lepton mass ma- the electroweak scale v. This hierarchy permits an ex- trix are chosen to be diagonal without loss of generality, pansion of all relevant matrices in terms of the small 3 × 3 matrix A ≡ U †M M −1, where U is the unitary effectively shifting all mixing parameters√ into y. Elec- D R T 3 × 3 Pontecorvo–Maki–Nakagawa–Sakata (PMNS) ma- troweak symmetry breaking via hHi√= (v/ 2, 0) yields trix. Parametrically this corresponds to an expansion in the Dirac mass matrix MD = yv/ 2. The full 6 × 6 c the scale hierarchy v f which we refer to as the seesaw neutrino mass matrix in the basis (νL,NR) = V nR is then expansion. To leading order, the matrices take the form † 1 c T 0 MD 11 − AA A L = − n¯RV T V nR + h.c. C ' , 2 M MR † † D (2) A A A 1 c B ' U(11 − 1 AA†) UA , (6) ≡ − n¯RMnnR + h.c. , 2 2 T −AMRA 0 where V is the unitary 6 × 6 mixing matrix to the states Mn ' . 0 MR nR, which form the Majorana mass eigenstates n = nR + nc . The diagonal mass matrix M = diag(m , . . . , m ) T R n 1 6 Note that AMRA is diagonal, which imposes constraints consists of the physical neutrino masses arranged in as- on A and provides an implicit definition of U. These con- cending order. Throughout this article, we denote mass straints may be automatically satisfied using the Casas– matrices with capital letters Mx and individual mass Ibarra parametrization [22]; however, more useful for eigenvalues with small letters mi. In the mass eigenstate our purpose is the Davidson–Ibarra parametrization [23], − basis, the tree-level neutrino couplings to J, Z, W , and T † which uses Mν = −AMRA and MDMD as the indepen- h, take the form [18, 19] dent matrices containing all seesaw parameters. Since 6 M is essentially already fixed by neutrino oscillation iJ X ν LJ = − ni [Cij(miPL − mjPR) experiments (modulo the phases, hierarchy, and overall 2f i,j=1 mass scale), the next step is to experimentally deter- † +Cji(mjPL − miPR) + δijγ5mi] nj , mine MDMD. As we will see, this could in principle be 3 achieved by measuring Majoron couplings without ever the phenomenology despite the additional loop suppres- observing the heavy right-handed neutrinos. sion [10]. Therefore, a thorough discussion of the Ma- To this effect let us point out some interesting prop- joron requires knowledge of all loop-induced couplings † erties of the hermitian matrix MDMD [10]: its determi- that are leading order in the seesaw expansion. Using † Q6 the tree-level couplings of Eq. (3) we calculate the loop- nant is simply det MDMD = det Mn = j=1 mj, which is strictly positive in the model at hand even if one of induced Majoron couplings to the rest of the SM particles the light neutrinos were massless at tree level [24]. Thus, and provide them below. † MDMD is positive definite, which yields a chain of in- † equalities for the off-diagonal entries (MDMD)ij, i 6= j B. Charged fermion couplings (see e.g. Ref. [25]):

† q † † The leading order couplings to charged fermions are |(MDMD)ij| < (MDMD)ii(MDMD)jj obtained from the one-loop diagrams in Fig. 1. These † † were calculated long ago, both in the one-generation (MDM )ii + (MDM )jj ≤ D D (7) case [3] and in the three-generation case, which leads to 2 1 oﬀ-diagonal Majoron couplings to leptons [18]. At lead- ≤ tr(M M † ) . ing order in the seesaw expansion, these couplings take a 2 D D simple form [10], with (diagonal) quark couplings This provides a useful way to constrain magnitudes of the elements of M M † since its trace appears in many iJ † ¯ D D LJqq = tr(MDM ) dMdγ5d − uM¯ uγ5u , (10) couplings of the Majoron. 16π2v2f D From Eq. (3) all loop-induced Majoron couplings are necessarily proportional to 1/f. But many couplings con- and charged lepton couplings tain additional powers of M −1 ∝ 1/f, which make them R iJ ¯ † higher order in the seesaw expansion. We will neglect LJ`` = 2 2 ` M` tr(MDMD)γ5 these suppressed couplings and focus on those that are 16π v f (11) † † down by only one power of 1/f. For the sake of gen- +2M`MDMDPL − 2MDMDM`PR ` , erality, we determine the Majoron couplings assuming an explicit shift-symmetry-breaking Majoron mass term where M`,u,d denote the diagonal mass matrices of the 1 2 2 − 2 mJ J , making J a pseudo-Goldstone boson. This appropriate SM fermions. In addition to exhibiting de- mass could be explicit [26, 27] or arise from quantum- coupling in the seesaw limit MR ∼ f → ∞, these cou- gravity eﬀects [28–30]. plings vanish in the electroweak symmetric limit v → 0 as expected since J is an electroweak singlet. The quark couplings can be used to derive the Majoron couplings to A. Neutrino couplings nucleons N = (p, n)T , using the values from Ref. [37]:

By inserting Eq. (6) into Eq. (3), the tree-level Ma- † iJ tr(MDMD) ¯ −1.30mp 0 joron coupling to the light active Majorana neutrinos in LJNN ' N γ5N. 16π2v2f 0 1.24mn the seesaw limit is (12) 3 iJ X L = m n γ n . (8) At this point let us make some remarks about CP vi- J 2f j j 5 j j=1 olation. Already in the one-loop processes above one en- counters loop-induced Majoron mixing with the Brout– These diagonal Majoron couplings to neutrinos are Englert–Higgs boson h, which would result in Majoron formally second order in the seesaw expansion since couplings to the scalar bilinear ff¯ as opposed to the 2 2 ¯ m1,2,3/f ∼ MD/(MRf) ∼ (v/f) . The omitted oﬀ- pseudo-scalar fiγ5f. It was noted in Ref. [18] that the diagonal Jninj couplings are determined by the ma- relevant J–h mixing diagrams vanish for mJ = 0. For † T 3 2 trix AA AMRA /f ∼ (v/f) , which are further sup- mJ 6= 0 the J–h amplitude is of order (v/f) in see- pressed, and lead to irrelevantly slow active-neutrino de- saw and hence negligible. This can be understood by cays ni → njJ [4]. noting that CP-violating phases in the Davidson–Ibarra Assuming for simplicity mJ m1,2,3, the Majoron’s parametrization reside both in the active-neutrino mass partial decay rate into light neutrinos is

3 mJ X Γ(J → νν) = m2 . (9) 16πf 2 j j=1 For suﬃciently large f the Majoron becomes a long-lived DM candidate [27, 29, 31–36], discussed in Sec. IV C. As mentioned earlier, the Majoron couplings to all FIG. 1. Loop-induced Majoron couplings to charged other SM particles are leading order in the seesaw expan- fermions with the Majorana neutrino mass eigenstates ni run- sion, i.e. proportional to 1/f, and may easily dominate ning in the loops. 4

symmetry beyond mJ due to the absence of electroweak instantons without B + L violation [41–43]. Therefore, electroweak instantons cannot generate Majoron mass.

FIG. 2. Loop-induced Majoron couplings to WW and ZZ III. MAJORON COUPLINGS AT TWO LOOPS at one loop. These diagrams generate couplings that are sub- dominant in the seesaw expansion. In this section we present Majoron couplings for which the leading seesaw behavior arises at two loops. To auto- mate the evaluation of some O(100) Feynman diagrams T matrix Mν = −AMRA and in the off-diagonal entries contributing to each effective coupling we implemented † of the hermitian matrix MDMD, each containing three this model in Feynman gauge including all Goldstone complex phases [23]. CP-violating Majoron couplings bosons [19] in FeynRules [44], and generated the nec- 2 via Mν are unavoidably suppressed by Mν /f ∼ (v/f) , essary amplitudes with FeynArts [45]. We validated our † leaving only MDMD as a potential source. However, implementation by reproducing the tree-level and one- closing lepton loops implies an amplitude dependence loop couplings above. † on tr(MDMDg(M`)), with some function g(M`) of the The Feynman diagrams naturally divide into two sets charged-lepton mass matrix. Since the latter is diago- (see Fig. 3). Set I diagrams contain the one-particle ir- nal, the trace depends only on the real diagonal entries reducible (1PI) two-loop diagrams. Set II diagrams con- † tain the reducible diagrams that are dominated by J–Z of MDMD, resulting in an effectively CP-conserving am- † mixing. We used an in-house Mathematica implementa- plitude. The CP phases of MDMD thus only appear in the off-diagonal Majoron couplings to leptons, at least to tion of expansion by regions as described in Ref. [46] to lowest order in the seesaw expansion. carry out a double asymptotic expansion m4,5,6 → ∞, m1,2,3 → 0 of the two-loop vertex integrals, and to alge- braically reduce the one-loop [47] and two-loop [48] ten- C. Couplings to gauge bosons sor integrals in dimensional regularization. We treated γ5 naively, such that it anti-commutes with all other Dirac matrices while also preserving the cyclic property At one-loop order, the only non-vanishing couplings to of traces. Finally, after expanding around four space- gauge bosons are to WW and ZZ, with typical diagrams time dimensions, we summed over fermion generations shown in Fig. 2. However, they are higher order in the to extract the leading seesaw behavior of each Majoron seesaw expansion, which can be understood as follows: coupling. We found the couplings to be expressible as the amplitudes come with a factor of M M † /f in order D D simple sums of one-loop functions and rational terms. In to achieve the necessary NR–νL mixing to close the loop; principle, two-loop self-energy and vacuum integrals may on dimensional grounds there is an additional M −2 sup- R be present, but they cancel away in the course of reduc- pression since this is the only high mass scale in the loop. tion, leaving behind rational terms. Explicit one-loop formulae can be found in Refs. [38, 39]. We have checked our results by confirming that all am- The leading seesaw behavior for coupling to gauge bosons plitudes are proportional to the expected tensor struc- without the M −2 suppression starts at two-loop order. R tures, are UV finite upon using the relations of Eq. (5), At this point it is appropriate to discuss the connec- and have the expected limiting low-energy/small-mass tion to anomalies in the minimal Majoron model. There behavior. Additionally, we confirmed that our results are is some confusion in the literature regarding the question insensitive to the treatment of γ5 by reevaluating them of whether the Majoron is the Goldstone boson of the in several different ways, including projecting the inte- anomaly-free U(1)B−L or the anomalous U(1)L. Both grals onto form factors and also starting from cyclically choices seem equally valid because baryon number re- reordered Dirac traces, and finding the same answer upon mains unaltered by the Lagrangian in Eq. (1). Since expanding around four spacetime dimensions. according to common lore Goldstone couplings to gauge We pause to comment on how we quote our bosons are determined by the anomaly structure of the results for couplings to gauge bosons {VV 0} = theory, this leads to a paradox when attempting to guess {gg,γγ,Zγ,ZZ,W +W −}. We phrase our results in the form of Majoron couplings to W and Z. The reso- terms of on-shell decay amplitudes lution was recently presented in Ref. [40], where it was explained that Goldstone couplings to gauge bosons are 0 M(J → V (k1)V (k2)) = driven entirely by non-anomalous processes. Anomalies µνρσ ∗ ∗ still serve as a useful bookkeeping device for the couplings − gJVV 0 ε εµ(k1)εν (k2)k1,ρk2,σ . (13) to vector-like gauge bosons such as gluons and photons, It is commonplace to see these amplitudes interpreted as but fail for chiral gauge bosons. Disregarding anomalies effective couplings as they appear to match onto EFT it is then necessary to calculate Goldstone couplings to operators of the form [49] gauge bosons in perturbation theory, the results of which gJVV 0 µν ˜ 0 we present in the next section. Additionally, we em- L = − JV Vµν , (14) phasize that the non-vanishing Majoron coupling to elec- 4 troweak gauge bosons (Eqs. (24), (28), and (35) below) where V µν is the appropriate field-strength tensor and does not lead to nonperturbative violation of the shift V˜ µν its dual. However, we caution the reader that the 5

solve the strong CP problem even at higher loop order [50, 51]. B. Coupling to two photons

The Majoron coupling to photons at two-loop level receives contributions from both sets of Feynman diagrams in Fig. 3,

I II gJγγ = gJγγ + gJγγ , (17) and yields the partial decay rate into two photons, FIG. 3. Representative two-loop diagrams contributing to |g |2m3 loop-induced Majoron couplings to vector bosons {VV 0} = Γ(J → γγ) = Jγγ J . (18) {gg,γγ,Zγ,ZZ,W +W −}. Set I contains the two-loop 1PI 64π diagrams and Set II contains reducible diagrams dominated The contributions from Set II were calculated in by J–Z mixing. Here, ni and nj are Majorana neutrino mass Ref. [10] with the result eigenstates and f SM fermions (not necessarily all identical). α X m2 gII = tr(M M † ) N f Q2 T f h J , (19) Jγγ 8π3v2f D D c f 3 4m2 identification with effective couplings in this way is some- f f what clumsy for the following reasons. First, matching already simplified with the help of the electroweak onto local operators should be carried out for off-shell P f 2 f anomaly cancellation condition f Nc Qf T3 = 0. Here, Green functions which have been expanded in the ex- u,d ` u Nc = 3 = 3Nc is the number of colors, T3 = 1/2 = ternal momenta. Second, our effective couplings gJVV 0 d,` cannot be viewed in a Wilsonian sense, since degrees of −T3 the isospin, and {Q`,Qd,Qu}√= {−1, −1/3, +2/3} freedom lighter than the Majoron contribute in certain the electric charge in units of e = 4πα. We complete mass ranges, nor can it be viewed in the 1PI sense since the evaluation of gJγγ here by computing the additional the couplings include Set II diagrams that are not one- contributions arising from Set I diagrams, which give particle irreducible. Therefore, interpreting our results α X m2 as coefficients of effective operators should be done with gI = (M M † ) h J . (20) Jγγ 8π3v2f D D `` 4m2 care. ` ` Just as for the gluon coupling, the amplitude vanishes as 2 gJγγ ∼ mJ for small Majoron masses, implying that the A. Coupling to two gluons leading effective operator this amplitude matches onto in 2 µν the derivative expansion is (∂ J)Fµν F˜ rather than the Assuming a sufficiently heavy Majoron m > Λ , µν J ∼ QCD typically occurring JFµν F˜ . the coupling to free gluons comes entirely from J–Z mix- For mJ mf we can relate the total Majoron–photon ing diagrams of Set II in Fig. 3. A straightforward eval- coupling gJγγ to the dimensionless diagonal fermion cou- uation of the decay amplitude J → gg at leading order ¯ plings of Eqs. (10) and (11), gJff Jfiγ5f, as in the seesaw expansion yields the simple expression 2 αmJ X f 2 gJff 2 gJγγ ' − Nc Qf 3 , (21) αS † X q mJ 12π m g = tr(M M ) T h , (15) f f Jgg 16π3v2f D D 3 4m2 q=u,d q which agrees with the EFT result of Ref. [52]. Since u d the gJff couplings can have different signs and magni- with T3 = −T3 = 1/2 and the loop function tudes, the Jγγ coupling for mJ < me could be heav- 1 2 ily suppressed. The key point and crucial result of this h(x) ≡ − log[1 − 2x + 2px(x − 1)] − 1 4x full two-loop calculation is that the Jγγ coupling has ( x 8x2 4x3 4 (16) a richer structure than anticipated in Ref. [10] based 3 + 45 + 35 + O(x ) , x → 0 , on the evaluation of gII alone. This is illustrated in = 2 Jγγ (π+i log(4x)) −2 −1 + 4x + O(x ) , x → ∞ . Fig. 4 where we show |gJγγ | × f for a variety of hi- † erarchies of the diagonal entries (MDMD)``. The SM- 2 For small mJ , the amplitude vanishes as gJgg ∼ mJ and fermion mass thresholds together with the different signs indicates that at leading order in the derivative expansion in gJff potentially suppress gJγγ by orders of magni- 2 aµν ã the amplitude matches onto an operator (∂ J)G Gµν tude. The typical size of the coupling for mJ > MeV aµν ã −5 −1 † 2 instead of JG Gµν as in Eq. (14). This implies that is |gJγγ | ∼ 10 f (MDMD)/(100 GeV) , simply due to the Majoron does not solve the strong CP problem, as the unavoidable suppression factor α/(8π3). In Fig. 4 this operator is insensitive to a constant shift J → J + c we used the current-quark masses to evaluate gJγγ ; for < that could otherwise be used to cancel the strong CP θ mJ ∼ ΛQCD they should be replaced by hadronic loops. term [40]. Furthermore, contrary to the claim in Ref. [38], We have not attempted this, but we refer the interested Majorons without tree-level couplings to quarks cannot reader to standard axion literature on the topic [53–55]. 6

10-5

-6 † 2 10 (MDMD)jj=(100 GeV) † f ⨯ | (MDMD)ττ=0 J γγ † | g (MDMD)ee=0 -7 10 † † (MDMD)ee=(MDMD)ττ=0

10-8 10-4 10-3 10-2 10-1 100 101 102 103

mJ [GeV]

† † FIG. 4. Jγγ coupling for various hierarchies in the diagonal elements of MDMD. The black solid line shows (MDMD)ee = † † 2 † † 2 † (MDMD)µµ = (MDMD)ττ = (100 GeV) ; the blue dashed line shows (MDMD)ee = (MDMD)µµ = (100 GeV) ,(MDMD)ττ = 0 † † 2 † (which corresponds to gJee = gJµµ = 0); the red dotted line shows (MDMD)µµ = (MDMD)ττ = (100 GeV) ,(MDMD)ee = 0; † 2 † † and the green dot-dashed line shows (MDMD)µµ = (100 GeV) (MDMD)ee,(MDMD)ττ .

C. Coupling to Z and photon

I II Next we present the Z-photon coupling gJZγ = gJZγ + gJZγ , which receives contributions from Set I and Set II diagrams in Fig. 3. The results are

2 2  2 mJ 2 mZ  mJ h 2 − mZ h 2 I α † 2 X † 4m` 4m` gJZγ = − 3 2 tr(MDMD) − (1 − 4sW ) (MDMD)`` 2 2  , (22) 16π cW sW v f m − m ` J Z 2 2 2 mJ 2 mZ mJ h 4m2 − mZ h 4m2 II α † X f f f 2 f f f gJZγ = − 3 2 tr(MDMD) 2Nc Qf T3 (2Q sW − T3 ) 2 2 , (23) 16π cW sW v f m − m f J Z

P f 2 f P f f 2 with cW ≡ cos θW and sW ≡ sin θW . We have used f Nc Qf T3 = 0 = f Nc Qf (T3 ) to simplify the formula. In the limit mJ , mZ → 0, the amplitude is non-vanishing,

† α tr(MDMD) gJZγ ∼ − 3 2 , (24) 16π cW sW v f

µν and matches onto the eﬀective operator JZ F˜µν , as in Eq. (14).

D. Coupling to two Z bosons

I II The Majoron coupling gJZZ = gJZZ + gJZZ to two Z bosons receives contributions from Sets I and II:

α h 2 2 2 2 I 2 † X † 2 mJ mZ gJZZ = − 3 2 2 2 1 − 2sW tr(MDMD) − 2 2 (MDMD)``m` g( 4m2 ) − g( 4m2 ) 32π c s v f m − 4m ` ` W W J Z ` 2 2 2 2 2 2 2 2 2 i + 2sW (1 − 2sW )mJ + (1 − 4sW ) mZ C0(mJ , mZ , mZ , m`, m`, m`) , (25)

α 1 h 2 2 2 II † X f f 2 f mJ mZ gJZZ = 3 2 2 2 tr(MDMD) 2 2 Nc T3 mf T3 g( 4m2 ) − g( 4m2 ) 4π c s v f m − 4m f f W W J Z f 2 f 2 2 f 2 2 2 2 2 2 i + sW Qf T3 − Qf sW mJ + T3 − 2Qf sW mZ C0(mJ , mZ , mZ , mf , mf , mf ) , (26) 7 where

q 1 p g(x) ≡ 1 − x log[1 − 2x + 2 x(x − 1)] (27) 2 2 −1 and C0 is the scalar three-point Passarino–Veltman function. Despite the appearance of (mJ −4mZ ) , the amplitude is regular at threshold mJ → 2mZ . The coupling gJZZ is nonvanishing in the limit mJ , mZ → 0, † 2 α tr(MDMD) gJZZ ∼ −(1 − 3sW ) 3 2 2 2 , (28) 48π cW sW v f µν and matches onto JZ Z˜µν .

E. Coupling to two W bosons

+ − Finally, we present results for the two-loop amplitude for J → W W in order to extract the coupling gJWW , which receives contributions from diagrams in Set I and Set II I II,` II,d II,u gJWW = gJWW + gJWW + gJWW + gJWW , (29) where we have separated the Set II J–Z mixing contributions based on the type of SM fermions running in the loop. The Set I diagrams give † 2 α 1 X (MDM )`` m gI = D 4m4 − m2 m2 + 4m2m2 g J JWW 64π3s2 v2f m2 − 4m2 m2 W J W ` W 4m2 W J W ` W ` (30) 2 2 2 2 m` 2 2 2 2 +4m` (mW − m` ) − log 2 2 + mW C0(mJ , mW , mW , m`, m`, 0) , m` − mW the Set II J–Z mixing diagrams with two charged leptons in the loop give † 2 α tr(MDM ) 1 X m gII,` = − D m2 m2 g J JWW 32π3s2 v2f m2 m2 − 4m2 ` W 4m2 W W J W ` ` (31) 2 2 2 m` 2 2 2 2 +(mW − m` ) − log 2 2 + mW C0(mJ , mW , mW , m`, m`, 0) , m` − mW the Set II J–Z mixing diagrams with two down quarks in the loop give

† ( " 2 2 2 # II,d 3 α tr(MDMD) 1 X 2 2 2 mJ mW mW g = − |(Vq)ji| m m g − g , JWW 32π3s2 v2f m2 m2 − 4m2 di W 4m2 m2 m2 W W J W i,j di di uj (32) m +(m2 − m2 + m2 ) log uj + m2 C (m2 , m2 , m2 , m , m , m ) , W di uj W 0 J W W di di uj mdi and the Set II J–Z mixing diagrams with two up quarks in the loop give

† ( " 2 2 2 # II,u 3 α tr(MDMD) 1 X 2 2 2 mJ mW mW g = |(Vq)ij| m m g − g , JWW 32π3s2 v2f m2 m2 − 4m2 ui W 4m2 m2 m2 W W J W i,j ui ui dj (33) m +(m2 − m2 + m2 ) log dj + m2 C (m2 , m2 , m2 , m , m , m ) . W ui dj W 0 J W W ui ui dj mui

Here, Vq is the unitary Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix. In the last two formulae, the two- argument loop function is s ! p x + y − 1 (x + y − 1)2 g(x, y) = (x + y − 1)2 − 4xy log √ + − 1 . (34) 2 xy 4xy

The coupling is regular at threshold mJ → 2mW , and nonvanishing in the limit mJ , mW → 0,   † m4 − m4 + 2m2 m2 log(m2 /m2 )! 3α tr(MDM ) X ui dj dj ui dj ui g ∼ − D 1 − |(V ) |2 , (35) JWW 128π3s2 v2f  q ij (m2 − m2 )2  W i,j ui dj

+ µν − and matches onto JW Wµν .

F. Coupling to γ-Higgs and Z-Higgs allows couplings of J to hZ and hγ. The former arises

Besides the usually considered pseudo-scalar couplings to two gauge bosons discussed above, CP-invariance also 8

For sub-TeV Majoron mass and large-enough right- > handed neutrino masses, log (MR/mW ) ∼ 1 as is assumed in our seesaw expansion, the leading logarithmic sub contribution LJdd0 dominates over LJdd0 , which we neglect in the following for simplicity. † 2 The MFV matrix MdVq MuVq relevant for LJdd0 makes it clear that oﬀ-diagonal terms would vanish if all up quarks were degenerate, so these terms are necessarily proportional to up-quark mass diﬀerences. Numerically this matrix evaluates to

 −8 † 2 0 0 8 × 10 MdV M Vq q u 7 × 10−8 3 × 10−7 8 × 10−6 FIG. 5. Representative two-loop diagrams for oﬀ-diagonal 3 '   , (39) v −5 −4 −3 Majoron couplings to quarks. 7 × 10 3 × 10 8 × 10

keeping only the largest entries. The biggest amplitude already at one-loop level but is seesaw suppressed; the is therefore b → sJ, which is, however, experimentally dominant contributions to both couplings then arise at less clean than s → dJ, further discussed in Sec. IV A. two-loop level. Because of the large number of diagrams From the dominant ﬂavor-changing down-quark cou- and low phenomenological relevance of decays such as plings in LJdd0 we immediately obtain the corresponding h → γJ compared to the processes derived above we will ﬂavor-changing up-quark couplings as not, however, present the results here. 1 MR † LJuu0 = − tr MD log M 128π4v4f mW D G. Flavor changing quark couplings 2 † × (iJ u¯RMuVqMd Vq uL + h.c.) , (40)

At the two-loop level we find off-diagonal Majoron cou- with the markedly smaller MFV coupling matrix plings to quarks, which can lead, for example, to s → dJ   or K → πJ at hadron level. Such flavor-changing cou- 2 † 0 0 0 MuVqMd Vq plings have long been advocated to search for light bosons ' 3×10−10 3 × 10−9 6 × 10−8 . (41) v3   and axions [56, 57] and have enjoyed increased atten- 7 × 10−7 8 × 10−6 2 × 10−4 tion in recent years [58–61], partly because of an improved reach at existing and upcoming experiments such Taken together with the weaker experimental limits on as NA62 and Belle II. u → u0J we can ignore these couplings in practice. In the Majoron model the relevant flavor-changing From the point of view of the seesaw expansion, quark-level couplings arise at two loops and involve a the quark-flavor changing couplings are actually the large number of diagrams; see Fig. 5. The leading loga- dominant Majoron couplings. They only decouple as rithmic contribution to coupling to down quarks is log(MR)/f, whereas all other couplings decouple at least as 1/f. It was noted before that Goldstone bosons 1 with effective diagonal couplings Jmqq¯iγ5q yield flavor- MR † LJdd0 = − tr MD log M changing quark couplings at one loop that depend log- 128π4v4f mW D arithmically on the UV scale [63, 64], whereas an ini- ¯ † 2 × (iJ dRMdVq MuVq dL + h.c.) , (36) µν tial coupling JWµν W˜ does not have such a dependence [58]. In our case JW W˜ µν gives only a seesaw- arising from the Set I diagrams in Fig. 5. The couplings µν suppressed contribution to Jqq0, and the log(M ) terms are of minimal-flavor-violating (MFV) type [62], as ex- R originate from an effective coupling to Goldstone bosons, pected from the fact that J is quark-flavor blind. The JG+G−. subleading contribution from the remaining diagrams is given by

† IV. PHENOMENOLOGY sub −tr(MDMD) † L 0 = (iJ d¯ M V F V d +h.c.) , (37) Jdd 512π4v4f R d q u q L Having obtained all Majoron couplings to leading or- where the diagonal matrix Fu has entries der in the seesaw expansion we can discuss existing constraints and signatures. 4 2 2 4 2 7mu + 3mumW − 8mW 2 mJ Fu = 2 2 + 4mu g 2 mu − mW 4mu 2 4 2 2 4 2 A. Light Majorons 2mu(mu − 2mumW + 2mW ) mW (38) + 2 2 2 log 2 (mu − mW ) mu 2 2 2 We start with the simplest case of a massless Majoron, − 4mumW C0(0, 0, mJ , mu, mW , mu) . which most importantly gives a vanishing coupling to 9 photons. It proves convenient to phrase our discussion in tr(K) (B→KJ) terms of the dimensionless parameters 1 tr(K) Belle II (K→πJ) (M M † ) D αβ ) v f /( K ≡ D , (42) |Kτℓ | αβ αβ NA62 (τ→ℓJ) vf ) † D 0.01 M

D Belle as they capture the Majoron couplings in most cases [10]. tr(K) gJγγ, SN1987

† ( M = The off-diagonal entries of MDMD are directly constrained by the lepton-flavor-violating (LFV) decays ` → αβ 10-4 0 tr(K) ` J [18, 65, 66]. For m`0 m`, the partial widths read gJNN, SN1987 |Kμe| (μ→eJ) 0 2 K on Limit Γ(` → ` J) 3 v 2 -6 0 10 0 ' 2 2 |K`` | (43) Mu3e Γ(` → ` ν`ν¯`0 ) 16π m` and involve a left-handed final-state lepton, leading to an 0.001 0.010 0.100 1 10 anisotropic decay [10]. The constraints in the tau sector Majoron massm J in GeV are Br(τ → `J) < O(10−3) [67] and lead to [10]

FIG. 6. Upper limits on combinations of Kαβ = −3 −3 |Kτe| < 6 × 10 , |Kτµ| < 9 × 10 , (44) † (MDMD)αβ /(vf) for Majoron masses above MeV. The shaded regions exclude |Kαβ | or tr(K) by non-observed rare which can be improved by Belle and Belle-II [68–70]. In decays, the dashed lines show the potential future reach; see the muon sector, the best constraints on a Majoron with text for details. K → πJ and B → KJ further scale with anisotropic emission come from µ → eJ [71] (to be im- log(MR/mW ), which has been set to 11here. The black re- proved with Mu3e [72]) and µ → eJγ [73]. The latter is gion is a very naive estimate of SN1987 constraints on the also sensitive to mJ = 0 and provides a limit di-photon coupling, setting for simplicity K`` = tr(K)/3. The yellow region is the SN1987 constraint on the JNN cou- −5 pling [78]. The oﬀ-diagonal entries have to satisfy |K | < |Kµe| < 10 . (45) αβ tr(K)/2; see Eq. (7).

The diagonal couplings K`` of a massless Majoron are constrained by astrophysics, stellar cooling in particular, and imply decays, especially K → πJ and B → KJ (limits and future sensitivity taken from Ref. [61], based on −5 |Kee − Kµµ − Kττ | < 2 × 10 , (46) Refs. [58, 79, 80]). These decays probe the quantity −6 † tr(K) < 5 × 10 , (47) tr(MD log(MR/mW )MD), but to simplify comparison with the other limits we set log(MR/mW ) = 11to obtain from the electron [74] and nucleon coupling [75], respec- a limit on tr(K). It should be kept in mind, however, that tively. The bound on the trace tr(K) is particularly a larger log enhancement can make these rare processes powerful since it provides upper bounds on all entries of even more relevant. Also potentially relevant are the two- the positive-definite K [10] by means of the inequality of loop Majoron couplings to photons via the effective cou- Eq. (7). This then puts an upper bound on the Majoron pling gJγγ from Sec. III B. Astrophysical limits on this < coupling to muons and taus, which is far better than any coupling are extremely strong for mJ ∼ 100 MeV [81]; 2 < direct bound on these couplings. It also ensures that rare since gJγγ is mJ suppressed for mJ ∼ MeV, the region decays such as K → πJ and Z → γJ [54, 55, 76, 77] where the photon coupling is important is between MeV are unobservably suppressed for a massless Majoron. and 100 MeV. This is unfortunately precisely the mass Two-loop couplings are hence irrelevant for massless Ma- region where the light quarks that run in the J-γ-γ loops jorons. should be replaced by hadrons; as a very naive way to Overall we see that a massless Majoron gives seesaw- incorporate this we simply set mu = md = mπ and † < −5 m = m in Eq. (19). The g limit in Fig. 6 should parameter constraints of order MDMD/(vf) ∼ 10 – s K Jγγ 10−6. While this is far off the “natural” value therefore not be taken too seriously. In light of these un- † −13 MDMD/(vf) ∼ Mν /v ∼ 10 , it can be realized by certainties we do not discuss how the gJγγ coupling de- assuming certain matrix structures in MD that suppress pends on the various diagonal K``, but rather set them −1 T † all equal to tr(K)/3 to allow a comparison to the other Mν ' −MDMR MD but not MDMD, to be discussed in more detail in Sec. IV B. As we have seen, the relevant limits. It is clearly possible to suppress gJγγ significantly couplings of a massless Majoron are those to nucleons in the region of interest by choosing hierarchical K``, as and electrons, but even µ → eJ could be observable. shown in Fig. 4. The phenomenology becomes more interesting for non- Also illustrated in Fig. 6 are SN1987 constraints on zero Majoron mass, specifically values above ∼ 10 keV the Majoron–nucleon coupling, Eq. (12), adopted from in order to kinematically evade the stellar cooling con- Ref. [78], which reach up to mJ ∼ 250 MeV and constrain straints. This is shown in Fig. 6 for Majoron masses tr(K) between 5 × 10−5 and 0.06. above 1 MeV. In addition to the one-loop lepton-flavor- As can be appreciated from Fig. 6, even the strong violating decays that probe Kαβ we now also have rele- astrophysical constraints on Majorons do not rule out vant constraints from the two-loop quark-flavor-violating flavor-violating rare decays, with significant experimen- 10 tal progress expected in the near future. Even the two- with complex z and real ξα without loss of general- 0 1 loop suppressed d → d J decays provide meaningful con- ity [17]. This product structure of MD implies straints. † µ → eJ is well constrained already, and we expect it (MD b(MR)MD)αβ ∝ ξαξβ , (52) to eventually become the most sensitive probe of the K for any function b. The off-diagonal entries are then real matrix entries for m < m , even beating out stellar J µ and entirely determined by the diagonal ones: cooling limits. τ → `J, on the other hand, is mainly † relevant for mJ between ∼ 100 MeV and mτ . For smaller (MD b(MR)MD)αβ (53) mJ the flavor-conserving constraints on K from gJγγ and q † † gJNN become stronger, which suppresses the LFV modes = (MD b(MR)MD)αα(MD b(MR)MD)ββ . via the inequality of Eq. (7). For mJ > mτ , the main rare decays are B → KJ and Z → Jγ [49], the former is Notice that this violates the strict inequality derived ear- shown in Fig. 6, and the latter gives irrelevant constraints lier in Eq. (7), which assumed non-vanishing neutrino on the K entries of order 104. masses. Equation (53) drastically simplifies the discussion of the seesaw parameter space, seeing as all observables B. Comparison with seesaw observables now only depend on the three real diagonal entries of † MD b(MR)MD instead of the nine parameters it could So far we have discussed the interactions of the Ma- contain in general. Furthermore, all Majoron and non- joron, but, of course, the right-handed neutrinos NR Majoron parameter matrices have the same flavor struc- also mediate non-Majoron processes, discussed at length ture and only differ in their absolute magnitude. Equa- in the literature [11–17]. Assuming again that the NR tion (53) also shows that any low-scale seesaw texture are heavy enough to be integrated out, the relevant automatically maximizes the off-diagonal flavor-violating dimension-six operators involving SM fields all depend entries of the relevant coupling matrices (Eq. (48) or on the matrices Eq. (49)). We compare Majoron and non-Majoron limits in −2 † −2 † MDMR MD and MDMR log (MR/mW ) MD , (48) Fig. 7, the latter adopted from Ref. [17]. We set MR/TeV = 11 as well as f = 1 TeV and stress again which drive LFV processes such as ` → `0γ as well that the Majoron limits can be suppressed arbitrarily as lepton-universality violating effects such as Γ(Z → † by increasing f. Figure 7 (left) shows the (MDM )ee ``¯)/Γ(Z → `0`¯0), recently discussed thoroughly in D vs. (M M † ) parameter space, setting (M M † ) = Ref. [17]. In comparison, we have seen above that all D D ττ D D µµ 0. As already noted in Ref. [17], the non-Majoron limits Majoron operators depend on the matrices are completely dominated by flavor-conserving observ- + − M M † M log (M /m ) M † ables such as Z → e e and other electroweak preci- D D and D R W D . (49) sion data. LFV such as τ → eγ and τ → eee reside f f far in the already excluded region and even future im- provements, e.g. in Belle-II, do not reach the allowed pa- For f ∼ MR, this makes the Majoron operators poten- rameter space. In comparison, Majorons with masses tially dominant, while f MR suppresses them to an arbitrary degree [10]. To properly compare Majoron and between ∼ 100 MeV and mτ do give relevant constraints non-Majoron processes it is necessary to pick a structure from τ → eJ and will probe significantly more parameter space with upcoming Belle and Belle-II analyses [70]. for MD, which is guided by our experimental reach. As we have seen above, even future limits on Ma- Standard LFV in the τe (and τµ in complete analogy) joron production cannot reach the natural seesaw scale sector are hence doomed to be unobservable in the see- 14 saw model, but the Majoron LFV channels τ → `J could MR ∼ 10 GeV, and the same is true for other NR- mediated processes [17]. By no means does this preclude be observable and deserve more experimental attention. † † observable effects, since it is possible to use the ma- Figure 7 (right) shows the (MDMD)ee vs. (MDMD)µµ −1 T † trix structure of MD to suppress Mν ' −MDMR MD parameter space, setting (MDMD)ττ = 0. Standard † LFV, currently dominated by µ conversion in nuclei [90], while keeping MDMD large [82, 83], potentially realiz- ing a lepton-number symmetry [84] as in the inverse see- provides important constraints on the parameter space, saw [85–87]. Following Refs. [17, 88] we can solve Mν = 0 and all future µe LFV will probe uncharted terrain. For for MD, which then requires only tiny perturbations δMD mJ < mµ, the Majoron channel µ → eJ already sets to produce the observable neutrino masses via better limits than µN → eN and can continue to dominate over µ → eγ and µ → 3e in the future. Ultimately, −1 T −1 T Mν ' −δMDMR MD − MDMR (δMD) . (50) µN → eN conversion in Mu2e [91] and COMET [92] has the best future reach. † The key observation is that δMD is negligible in MDMD. Mν = 0 requires the low-scale seesaw structure

1   To ensure Mν = 0 to all orders in the seesaw expansion and at ξe q √ q m5 2 m6 loop level one has to further impose either m5 = m4 and z = ±i M = v ξµ 1 z ±i 1 + z , (51) D   m4 m4 or m6 = m4 and z = 0, both of which correspond to a conserved ξτ lepton number [84, 89]. 11

† † † 1/2 MR =f= 1 TeV,(M DMD)eτ = [(MDMD)ee (MDMD)ττ] † † † 1/2 MR =f= 1 TeV,(M DMD)eμ = [(MDMD)ee (MDMD)μμ] 1

Excluded by flavor-conserving observables(G F ,Z→ee,…) 0.010

0.100 ↑ excluded byμN→eN ↑ excluded by 0.001 2 2 τ→eJ

/ v ↑ excluded by μ→eJ / v μ→eγ ττ μμ ) ) -4 † † D 0.010 τ→eJ(Belle) D 10 M M D D μ→3e ( M ( M

10-5 μ→eJ(Mu3e) 0.001

μN→eN 10-6

10-4 10-7 10-4 0.001 0.010 0.100 1 10-7 10-6 10-5 10-4 0.001 0.010 † 2 † 2 (MDMD)ee/v (MDMD)ee/v

† 2 FIG. 7. Limits on (MDMD)αβ /v from flavor-conserving non-Majoron (gray region) and flavor-violating observables (colored lines), dashed lines denoting future reach, adopted from Ref. [17]. We set MR/TeV = 11, f = 1 TeV, and assume a matrix † q † † texture that gives (MDMD)αβ = (MDMD)αα(MDMD)ββ ; see text for details. The Majoron mass is assumed to be between 100 MeV and GeV in the left figure and between MeV and 100 MeV in the right figure.

C. Majoron dark matter illuminating, but stress that this diphoton suppression makes the neutrino modes even more dominant. Returning to the “standard” high-scale seesaw scenario with huge hierarchy v f it is clear that a massive Majoron can be long lived even on cosmological scales, V. CONCLUSION e.g. from Eq. (9), Majorons, the Goldstone bosons of spontaneously bro- 1 m 109 GeV 2 ken lepton number, were proposed in the early 1980s in Γ(J → νν) ∼ J . (54) 400 Gyr MeV f models for Majorana neutrino masses. Since then experiments have indeed found evidence for non-zero neutrino In this region of parameter space Majorons can form masses, although it is not yet clear whether they are of DM [27, 29, 31–36], with a production mechanism that Majorana type. With the motivation for Majorons as can be unrelated to the small decay couplings [10, 27, 93, strong as ever, we have set out in this article to com- 94]. plete the program that was started almost 40 years ago The defining signature of Majoron DM is a flux of neu- and calculate all Majoron couplings to SM particles. The trinos from DM decay with Eν ' mJ /2 and a known couplings to neutrinos (tree level) as well as charged lep- > tons and diagonal quarks (one loop) were known pre- flavor composition [10]. For mJ ∼ MeV these neutrino lines could potentially be observable via charged- viously. Here we presented the two-loop couplings to current processes in detectors such as Borexino or Super- gauge bosons (Jγγ, JγZ, JZZ, JWW , Jgg) and flavor- Kamiokande [10], while lower masses are more difficult to changing quarks (Jdd0, Juu0). Phenomenologically rel- probe [95]. evant of these are currently only the Majoron coupling 0 The loop-level couplings generate the much more con- to photons as well as the Jdd couplings behind the rare strained decays J → ff¯0, γγ, which, however, depend on decays K → πJ and B → KJ. the matrix K and are hence complementary to the neu- Standard seesaw effects in an EFT approach are en- −2 † trino signature, as discussed in detail in Ref. [10]. This coded in the matrix MDMR MD, which drives, for ex- analysis used only the Set II of diagrams to calculate ample, ` → `0γ. Majoron couplings, on the other hand, † J → γγ, namely the expression proportional to tr(K), depend on the matrix MDMD/f, which is parametrically Eq. (19). The new full expression presented in Sec. III B larger in the seesaw limit and can indeed give better con- leads in general to a suppression of the diphoton rate straints in parts of the parameter space. For example, and a more involved dependence on the K matrix entries while τ → `γ and other τ LFV are unlikely to be ob- (Fig. 4). We omit a full recasting of existing DM→ γγ servable in the seesaw model, τ → `J can be observably limits onto our J → γγ expression since it is not very large and deserves more experimental attention. 12

The singlet Majoron model together with the coupling ACKNOWLEDGEMENTS texture of Eq. (51) implied by low-scale seesaw is a very minimal UV-complete realization of an axion-like parti- JH would like to thank Arvind Rajaraman, Raghuveer cle and thus a well-deﬁned benchmark model. The dom- Garani, and C´edricWeiland for useful discussions. HHP inant theoretical challenge not addressed here is the re- would like to thank Wolfgang Altmannshofer, Michael placement of quarks by hadrons in loops, which we leave Dine, and Stefano Profumo for useful discussions. JH for future work. We expect future studies to elucidate is supported, in part, by the National Science Founda- additional aspects of this model, in particular when the tion under Grant No. PHY-1620638, and by a Feodor Majoron is used as a portal to dark matter. Lynen Research Fellowship of the Alexander von Hum- boldt Foundation. The work of JH was performed in part at the Aspen Center for Physics, which is supported by the National Science Foundation under Grant No. PHY- 1607611. HHP is partly supported by U.S. Department of Energy Grant No. de-sc0010107.

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