Pseudo-majoron as light mediator of singlet scalar dark matter

In the singlet-triplet majoron model of neutrino mass, lepton number is spontaneously broken. If it is also softly broken, then a naturally light pseudoscalar particle ηI exists. It may then act as a light mediator for a real singlet scalar χ with odd dark parity. It is itself unstable, but decays dominantly to two neutrinos through its triplet scalar component, thereby not disturbing the cosmic microwave background (CMB). It also mixes with the standard-model Higgs boson only in one loop, thereby not contributing significantly to the elastic scattering of χ off nuclei in dark-matter direct-search experiments.


Introduction
In the singlet-triplet majoron model of neutrino mass [1], small Majorana neutrino masses are obtained through a scalar Higgs triplet ∆ = (∆ ++ , ∆ + , ∆ 0 ) with lepton number L = −2, through the Yukawa interactions where f ij = f ji , resulting in m ν ij = 2f ij ∆ 0 . In the Higgs potential with the usual Φ = (φ + , φ 0 ) doublet of the standard model (SM), the trilinear coupling Φ † ∆Φ * is forbidden by L conservation. However, if a scalar singlet σ with L = 2 is added, then the quadrilinear term λ σΦ † ∆Φ * + H.c. (1.2) is allowed. The spontaneous breaking of SU(2) L × U(1) Y and L through the vacuum expectation values v 1,2,3 as defined by (1.3) results in four massless Goldstone bosons. The linear combinations become the longitudinal component of the W ± and Z bosons, and the linear combination (1.5)

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is the majoron. If σ is absent, this becomes the triplet majoron model [2] which is ruled out by Z decay because ∆ 0 would contribute too much to its invisible width. Here, assuming that v 1 v 3 , this effect can be suppressed. We now break L also explicitly but softly with the σ 2 term, then the majoron η I becomes massive. It may be assumed naturally light because it is protected by a would-be symmetry. To accommodate dark matter, we add two complex singlet scalars χ 1,2 which have L = 1. The trilinear scalar terms χ i χ j σ * are now allowed. As a result, χ 1,2 have self-interactions through the light mediator η I , and the enhanced elastic scattering cross section [3] is a possible resolution of the cusp-core anomaly in the density profile of dwarf galaxies [4]. As shown below, our model has one very important feature, namely the decay of the majoron η I is dominantly to two neutrinos. Its lifetime will be very short, and does not disturb the standard big bang nucleosynthesis (BBN). It mixes with the SM Higgs boson only in one loop, and this mixing may be arbitrarily small because it is not needed for it to decay as in any other model of a light scalar mediator [5]. This avoids the problem [6] of too large a cross section in direct-search experiments. Further, since η I decays dominantly to two neutrinos, it avoids the problem [7,8] of too much disruption to the cosmic microwave background (CMB) if it decays to electrons or photons as in all other models of a light scalar mediator.

Scalar sector
In our version of the singlet-triplet Majoron model of neutrino mass, the Higgs potential is given by where σ is a complex neutral singlet and and the m 2 4 term has been added to break L softly. Note that m 2 4 has been chosen real by rotating the phase of σ, and λ as well by rotating the relative phase of ∆ and Φ. Now the minimum of V is determined by

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As a result, the 3 × 3 mass-squared matrix spanning (σ I , φ I , ∆ I ) is given by If m 2 4 = 0, this matrix would have two zero eigenvalues, corresponding to the longitudinal component of the Z boson and the majoron of eq. (1.5). Removing the former, the reduced 2 × 2 matrix spanning σ I and (v 2 ∆ I − 2v 3 φ I )/ v 2 2 + 4v 2 3 becomes We know that v 3 has to be small compared to v 2 from precision electroweak data. We know also that v 3 has to be small compared to v 1 because the triplet majoron is ruled out from the measurement of the Z invisible width. Hence the mixing between the two states is small, i.e. v 3 /v 1 , with the two physical states having the squares of their masses equal to 2m 2 4 for the light pseudo-majoron η I and λ v 1 v 2 2 /v 3 for the other scalar which is heavy. (2.8) This means that the observed 125 GeV scalar at the Large Hadron Collider (LHC) may have a small singlet component which does not couple to quarks or leptons.

Dark sector
Consider now the addition of two complex neutral singlet scalars χ 1,2 with L = 1. We add to V the following scalar potential: The sum V + V is invariant under U(1) L in all its dimension-four and dimension-three terms. The only term which breaks U(1) L explicitly is the dimension-two m 2 4 , so that the symmetry of V + V becomes Z 4 , under which the charges of Φ, σ, ∆, χ 1,2 are 1, −1, −1, i, i respectively. Once the spontaneous breaking of V + V occurs with v 1,2,3 , then the residual symmetry becomes Z 2 , under which Φ, σ, ∆ are even and χ 1,2 odd. Hence the lightest χ is JHEP07(2017)140 a dark-matter candidate. Note that if only one copy of χ is used, V + V would have only real parameters, and there would not be a trilinear coupling linking χ to the light would-be pseudoscalar majoron. Hence the dark matter in this case would not have any enhanced self-interactions. Note also that this is another example [9] of the derivation of dark parity from lepton parity, i.e. (−1) L+2j from (−1) L .

Dark matter interactions
In our model, χ 0 is dark matter and the pseudo-majoron η I (mostly σ I ) is its light mediator. The elastic scattering cross section of χ 0 is dominated through the t-and u-channel exchanges of η I as shown in figure 1. For a center-of-mass energy squared s, it is given by In case of the exchange of a massless η I the infrared singularity for zero momentum transfer would have to be cured by the usual procedure of adding multiple soft emissions, as in QED.
However, in the model proposed here η I is massive. For the benchmark value of σ/m χ 0 ∼ 1 cm 2 /g for self-interacting dark matter, (4.2) is satisfied for example with m χ 0 = 100 GeV, m η I = 10 MeV, Of course, the above lowest-order approximation could be changed including the muliple exchange of η I . We have thus computed the non-perturbative Sommerfeld enhancement [11,12] but find only an enhancement of about 0.4%. Note that we are considering here the elastic scattering of χ 0 and not its annihilation which is indeed very much enhanced [8] at late times. Other contributions to σ(χ 0 χ 0 → χ 0 χ 0 ), such as those of the quartic χ 4 0 coupling or the s-channel exchanges of σ R , φ R , ∆ R , are not enhanced by the appearance of m 4 η I in the denominator, and are thus ignored. We note also that if η I were a true massless Goldstone boson, then it has necessarily derivative couplings to the leptonic current. For example, the kinetic term ∂ µ χ∂ µ χ * becomes

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The tree-level elastic χχ scattering amplitude through η I exchange is then zero because the χ 0 χ 0 η I coupling is then proportional to (ac + bd)(p 1 + p 2 ) µ (p 1 − p 2 ) µ and the elastic cross sections (4.1) and (4.2) are no longer valid. The relic abundance of χ 0 is determined by its annihilation to particles of even dark parity, which includes all SM particles (through the s-channel exchange of the Higgs boson h which is mostly φ R ) as well as η I , σ R . We assume that this cross section is dominated by the annihilation of χ 0 χ 0 → η I η I and χ 0 χ 0 → σ R σ R . The former may proceed through the t− and u−channel exchanges of χ 0 through µ I 0 as in χ 0 elastic scattering. However, for µ I 0 = 7 GeV, the resulting cross section is much smaller than the canonical value σ 0 × v rel = 3 × 10 −26 cm 3 /s for the correct dark matter relic abundance of the Universe. On the other hand, χ 0 annihilation to η I and σ R may also proceed through the quartic λ σχ coupling of eq. (3.7) as well as through s-channel exchanges of σ R , φ R , ∆ R , as shown in figure 2.
In analogy to eq. (3.7), let the φ R χ 0 χ 0 coupling be √ 2v 2 λ φχ , where (4.4) and the ∆ R χ 0 χ 0 coupling be √ 2v 3 λ ∆χ , where The effective quartic coupling of χ 0 χ 0 → η I η I is then whereas that of χ 0 χ 0 → σ R σ R is Note that the ∆ R contributions in both cases are negligible because v 3 is very small. The σ R contributions are also expected to be subdominant. We obtain thus which works for λ η λ σ = 0.05. As noted already, φ R is mostly the SM Higgs boson h which couples to quarks and leptons. Hence direct χ 0 annihilation to SM particles is JHEP07(2017)140 also possible. They have been neglected here for simplicity. If they are non-negligible, they could be important for the indirect detection of χ 0 in space. As for η I , although it does not mix with h at tree level, there is an allowed η I η I h coupling which will keep it in thermal equilibrium with the SM particles until it decays away.

Decay of the pseudo-majoron
As shown in ref. [5], η I mixes with h through χ 1,2 in one loop. This phenomenon of radiative Higgs mixing has only been discovered recently [10]. If this were the dominant decay mode of η I , then its decay product, i.e. e − e + , would disturb the CMB, and because of the large Sommerfeld enhancement [11] for late-time decays, this effect would rule out [8] any selfinteracting dark matter with s-wave annihilation which is strong enough to address the small-scale problems of structure formation. There is also an important constraint [6] from direct-search experiments. For m χ 0 ∼ 100 GeV, the non-observation of dark matter so far places a bound on the η I − h mixing, which makes the η I lifetime too long to accommodate the success of the standard BBN.
Both problems are solved here because η I decays to two neutrinos at tree level through its ∆ I component. Using eqs. (1.1) and (1.5) The decay rate of η I to ν i ν j andν iνj is then (5.1) Setting m η I = 10 MeV, ij |m ν ij | 2 = 10 −2 eV 2 , and v 1 = 1 GeV, we find Γ −1 = 0.7 s which is less than the benchmark of 1 s for the η I lifetime not to be a problem for the standard BBN. The η I − h mixing is loop-suppressed and may well be negligible, so the direct-search bound is not applicable, and since η I decays dominantly to neutrinos, the strong constraints of the CMB are also avoided.

Phenomenological consequences
As shown in a previous section, all the components of the scalar triplet (∆ ++ , ∆ + , ∆ 0 ) are heavy with mass squared roughly λ v 1 v 2 2 /v 3 . Three scalar particles remain: the SM Higgs h which is mostly φ R , the pseudo-majoron η I which is mostly σ I , and the orthogonal scalar to h which is mostly σ R . For v 1 ∼ 1 GeV, m σ R ∼ 1 GeV is expected, in which case h → σ R σ R is possible through λ 12 of eq. (2.1) with a decay width of The subsequent decay of σ R through its mixing with h to SM particles may be searched for [13] at the LHC. In fact, the final state of four muons is expected.
The dark sector has two complex scalars χ 1,2 where all four components mix as shown in eq. (3.2). The lightest mass eigenstate χ 0 is dark matter. It couples to the pseudomajoron η I through the trilinear term µ 0 I η I χ 2 0 and the quartic interaction λ σχ η 2 I χ 2 0 . For

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m η I = 10 MeV, m χ 0 = 100 GeV, and µ I 0 = 7 GeV, the elastic scattering cross section of χ 0 is large enough to explain the cusp-core discrepancy of the density profile of dwarf galaxies. The annihilation cross sections of χ 0 to η I and σ R are also just right for it to account for the observed relic abundance of dark matter. The decay of η I is dominantly to two neutrinos. It does not disturb the standard BBN or the CMB.
The mixing of η I with h is small. For a given m η I , it is constrained by direct-search experiments. However, since its value is unknown, it may still be large enough for χ 0 to be detected in underground experiments through η I exchange in the future. The occasional annihilation of χ 0 in space produces η I , but since the latter decays dominantly to neutrinos, it will be difficult to observe in satellite or ground-based experiments.

Concluding remarks
In the singlet-triplet majoron model of neutrino mass, a light pseudo-majoron η I is natural and can be chosen as the light mediator of self-interacting dark matter χ 0 based on the conservation of lepton parity extended to dark parity. The important property of η I is that it decays dominantly to neutrinos, thus avoiding strong constraints from the CMB, as well as the potential conflict between direct-search bounds and the standard BBN. We also predict a singlet scalar σ R of about 1 GeV, which mixes with the standard Higgs boson h. From h → σ R σ R → µ − µ + µ − µ + decay, it may be discovered at the LHC. The η I − h mixing may also allow underground experiments to discover χ 0 , but the annihilation of χ 0 in space to η I would not be easy to detect. However, it is possible that the χ 0 χ 0 cross section to SM particles through the Higgs portal is not so small and will become observable.