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PHYSICAL REVIEW D 104, 023013 (2021)

Modified majoron model for cosmological anomalies

Gabriela Barenboim * Departament de Fisica Teòrica and IFIC, Universitat de Val`encia, Avenida Doctor Moliner, 50, E-46100 Burjassot, Val`encia, Spain † Ulrich Nierste Institut für Theoretische Teilchenphysik (TTP), Karlsruher Institut für Technologie (KIT), 76131 Karlsruhe, Germany

(Received 2 July 2020; revised 7 January 2021; accepted 21 June 2021; published 15 July 2021)

The vacuum expectation value vs of a Higgs triplet field Δ carrying two units of number L induces ∝ vs. The neutral component of Δ gives rise to two Higgs , a pseudoscalar A and a scalar S. The most general renormalizable Higgs potential V for Δ and the Standard-Model Higgs doublet Φ does not permit the possibility that the of either A or S is small, of order vs, while the other mass is heavy enough to forbid the decay Z → AS to comply with LEP 1 data. We present a model with additional dimension-6 terms in V, in which this feature is absent and either A or S can be chosen light. Subsequently we propose the model as a remedy to cosmological anomalies, namely the tension between observed and predicted tensor-to-scalar mode ratios in the cosmic microwave background and the different values of the Hubble constant measured at different cosmological scales. Furthermore, if Δ dominantly couples to the third-generation doublet Lτ ¼ðντ; τÞ, the deficit of ντ events at IceCube can be explained. The singly and doubly charged triplet Higgs are lighter than 280 GeV and 400 GeV, respectively, and could be found at the LHC.

DOI: 10.1103/PhysRevD.104.023013

I. INTRODUCTION its amplitude and the dependence on the scale of such amplitude. However, only a subset of two of these four Although the hot big bang model and general relativity quantities is independent and therefore all our insight of are arguably very robust, they work only if an additional inflation is reduced to two parameters generally chosen to piece is added to the game: inflation. In order to explain the be the spectral index n , i.e., the dependence on the scale of flatness, homogeneity, and isotropy of the Universe and the s the perturbations, and the tensor to scalar (ampli- absence of monopoles and other relics, a period of inflation tude) ratio r. This is the reason why all the inflationary is crucial. models reduce to lines, points or regions in the n − r plane. But inflation is a framework comprising countless s As a consequence to discriminate which region is different inflationary models. Clearly all of them produce favored by experiments is also to select which inflationary a flat, isotropic, homogeneous and relic-free Universe but models remain in the game. The theoretical guidance at this each one leaves some specific imprints (as the inhomoge- stage is crucial. Specific physics models with their neities pattern of the model at hand in the CMB and matter content and interactions should help shed some light structure formation) that can help us find out, which one of on which are the inflationary potentials worth considering, the plethora of models in the market is the correct one. while at the same time making predictions which can be During inflation two types of perturbations are produced: tested elsewhere. scalar or matter perturbations and tensor (metric) pertur- In recent years, tensions of cosmological data with the bations (gravity waves). Each one can be characterized by predictions based on the SM and the ΛCDM model have emerged and a light scalar ϕ interacting with *[email protected] has been considered to alleviate these tensions. † [email protected] Specifically, favored regions for the ratio r of tensor (metric) to scalar (matter) perturbations inferred from the Published by the American Physical Society under the terms of anisotropies of the cosmic microwave background (CMB) the Creative Commons Attribution 4.0 International license. and the spectral index n can be significantly modified and Further distribution of this work must maintain attribution to s the author(s) and the published article’s title, journal citation, therefore the selection rule for successful inflationary and DOI. Funded by SCOAP3. models is vastly affected [1–4]. Furthermore, the Hubble

2470-0010=2021=104(2)=023013(6) 023013-1 Published by the American Physical Society GABRIELA BARENBOIM and ULRICH NIERSTE PHYS. REV. D 104, 023013 (2021) constant determined from local measurements disagrees there are two extra neutral Higgs bosons, a scalar S and a with the value inferred from CMB data and new neutrino pseudoscalar A. These approximately coincide with hs and interactions might remedy this as well [5,6]. as, respectively. By assigning two units of lepton number L Δ ð1Þ Historically, the first interest into light scalars interacting to and choosing an U L-invariant Higgs potential one ð1Þ with neutrinos was driven by the attempt to build majoron arrives at a model which breaks U L spontaneously. Then models breaking lepton number spontaneously, first real- A is a massless Goldstone mode, the majoron [7–9]. ized through an SU(2) singlet [7,8] or triplet [9] field, or Postulating an effective interaction of neutrinos with a with several fields in both representations [10]. The triplet light scalar ϕ, models (and those employing doublet fields) did not comply with LEP 1 data on invisible Z decays [11] shifting L ¼ gαβν¯ανβϕ; ð2Þ the focus entirely to SU(2) singlet fields ϕ [11–13].But eff couplings of singlets to active neutrinos are tiny, because ν SU(2) forbids such couplings at dimension-4 level and the where α generically denotes any of the light left-handed effective coupling is necessarily suppressed by small neutrino fields or its charge-conjugate, one can alleviate the mixing angles (e.g., between active and sterile neutrinos). tensions in cosmological data mentioned in the introduction − E.g., Ref. [11] finds coupling constants below 10−3.Aswe while simultaneously modifying the ns r region to include well motivated inflationary models which were will see below, such small couplings do not permit – solutions to the cosmological problems in the most inter- previously ruled out [1 5]. In this paper we only consider couplings of ϕ to τ-neutrinos ντ, for which terrestrial and esting region with mϕ > 30 keV. Thus to date there is no astrophysical data do not imply strict bounds and it is not viable model supporting the idea of Refs. [1–6] in this relevant for us whether L is identified with the total lepton mass range. In the following we will present a model of neutrinos that number or with Lτ. We aim at the formulation of a minimal − triplet majoron model addressing the cosmological anoma- can not only modify the allowed region in the ns r plane ν ν changing this way the inflationary models that survive the lies and the majoron couplings to e and μ are neither experimental scrutiny but also provides a viable modifi- conceptually nor phenomenologically relevant for this. cation of the SU(2) triplet majoron idea [9]. Since triplet Leff is not gauge invariant, a meaningful interaction must ϕ fields have renormalizable, dimension-4 couplings to be formulated in terms of lepton doublets. Identifying , even Oð1Þ couplings of the majoron to active with the majoron amounts to completing Leff to neutrinos are possible a priori. Thus we reopen a large Δ portion of the parameter space with consequences for other Δ yτ ¯c L ¼ L ΔL3 þ H:c: applications of a light scalar coupling to neutrinos. y 2 3 Δ Furthermore, our model makes predictions beyond cos- mν ⊃− τ ðν¯ νc þ ν¯cν Þ mology and neutrino physics and can be tested in forth- 2 τ τ τ τ coming experiments. Δ yτ This paper is organized as follows: In the following pffiffiffi ¯c c − ½ðhs þ iasÞντ ντ þðhs − iasÞν¯τντ Þ; ð3Þ section we present a class of majoron models in which 2 2 either S or A is light, while the other boson is heavy enough Z → AS ¼ðν τÞT c ¼ðτc −νcÞT Δ ¼ Δ to forbid . Next we discuss phenomenological where L3 τ; , L3 ; τ , and mντ yτ vs is “ ” consequences and smoking gun features. Finally we the contribution of Δ to the Majorana mass of ντ. However, conclude. all known Higgs potentials predict that S and A are either both light, with masses around vs, or both heavy, with II. THE MODEL masses of order v or larger. This finding holds for the most An SU(2) triplet Higgs field general renormalizable Higgs potential, irrespective of whether L is broken spontaneously or explicitly. For this þ ! pδ ffiffi δþþ reason the original triplet majoron models were discarded Δ ¼ 2 ð Þ with the advent of LEP 1 data on the invisible Z width þ δþ 1 þ hspffiffiias − pffiffi → vs 2 2 which left no room for the decay Z AS, whose decay rate is entirely fixed by the value of the SU(2) gauge coupling.1 ϕ developing a vacuum expectation value (vev) vs in the Finally, in models in which in Eq. (2) is a singlet field neutral component generates Majorana masses of light mixing with a heavy S or A the coupling g ≡ gττ is neutrinos in a natural way via its coupling to the lepton suppressed by a tiny mixing angle and is far too small doublets. Electroweak precision data imply vs ≪ v, where to solve the cosmological tensions in the interesting region v ¼ 174 GeV is the vev of the doublet Higgs field Φ of the (SM). The physical Higgs fields are 1The decays Z → SS and Z → AA are forbidden by CP mixtures of the components of Δ and Φ; in particular invariance of the electroweak gauge interaction.

023013-2 MODIFIED MAJORON MODEL FOR COSMOLOGICAL ANOMALIES PHYS. REV. D 104, 023013 (2021) with mϕ > 30 keV. The same remark applies, if a singlet ϕ 0.100 couples to heavy sterile neutrinos which mix with ντ. For the phenomenological analysis of the relevant collider bounds and astrophysical constraints discussed 0.001 below it does not matter whether S or A is the light

g − scalar corresponding to ϕ in Eq. (2) and for definiteness 10 5 ϕ ¼ ≃ we considerp theffiffiffi case S hs. Then Eq. (3) entails Δ ¼ − ð2 2Þ − g yτ = . 10 7 Global fits to cosmological data constrain the combination [4] −9 10 − − − 10 6 10 5 10 4 0.001 0.010 0.100 1 2 Δ 2 g yτ G ¼ ¼ ; ð Þ ms/MeV eff 2 8 2 4 mϕ mS FIG. 1. The contour of the extra radiation ΔNeff ≡Neff −3¼0.6 while successful big bang nucleosythesis (BBN) is sensi- at a temperature of 1 MeVin the mS-g plane. The (light-blue) region tive to the mass and coupling of the scalar particle in a above the blue line is forbidden by primordial helium and different combination. More specifically, BBN is very deuterium abundances, thus mS is in the sub- keV range or ≥ 30 10−2 2 sensitive to the amount of extra radiation. The observed mS keV. The red region corresponds to =MeV < 10−1 2 2σ primordial abundances of deuterium and helium set strong Geff < =MeV , which is the CMB favored region for 10−4 2 constraints on the coupling and the mass of our scalar Geff > =MeV [4]. Unlike SU(2) singlet models our triplet model covers the whole allowed parameter region, up to particle. These bounds are reflected in Fig. 1, where it can −7 ms ∼ 1 MeV. For the displayed region with g ≥ 10 the bound be seen that two regions are consistent with BBN and at the ≤ 1 ≤ 3 5 mντ eV implies vs . MeV. same time result in a Geff able to change the CMB temperature and polarization spectra.2 A heavy or MeV region with 0.03 MeV ≤ mS ≤ 1 MeV and a light or keV the mass of either S or A above MZ. All parameters are region with mS ≤ 0.1 keV. chosen real, so that V is invariant under charge conjugation Bounds from decays are irrelevant in our case C, and we only consider solutions with real vevs. L is a (and therefore not shown) as our new interaction concerns good symmetry of V for β ¼ δ1 − δ2 ¼ 0. Any other dim-6 only neutrinos. We further stress that none of the operator can be expressed as a linear combination of the τ Δ ¼ 0 bounds derived on Leff in Eq. (2) from or tauonic Z terms in Eq. (5) and a L operator, which does not decays (see, e.g., [14] for a recent study) applies to a contribute to the A-S mass splitting. The minimization complete model like ours: The proper cancellation of conditions ∂V=∂v ¼ 0 ¼ ∂V=∂vs read: → 0 infrared singularities (for mϕ ) requires the inclusion   4 of virtual corrections, which in turn involve also heavy 2 2 vs Δ μ ¼ 2λv þ O ; fields (like the charged components of ) to cancel ultra- v2   violet divergences. 2 2 þ þ The Higgs potential of Δ in Eq. (1), Φ¼ðϕ ;vþhpffiffiiaÞT, β ¼ m þ 4δ 2 þ 2λ vs ð Þ 2 vs 2 1v Δ 2 6 Φc ¼ð þ hp−ffiffiia −ϕ−Þ v v and v 2 ; reads:

2 † 2 † † 2 2 2 2 2 V ¼ −μ Φ Φ − μΔTrðΔ ΔÞþλðΦ ΦÞ with m ≡ −μΔ þ α2v þ α3v ð7Þ † 2 † † † † þ λΔ½TrðΔ ΔÞ þ α1Φ Δ ΔΦ þ α2Φ ΔΔ Φ m2 þ α Φ†Φ ðΔ†ΔÞ − βðΦc†Δ†Φ þ Φ†ΔΦcÞ The parameter will govern the mass of the desired light 3 Tr state. Avoiding fine-tuning between different terms in c† † † c 2 þ δ1ðΦ Δ Φ þ Φ ΔΦ Þ Eq. (7) we must have − δ ðΦc†Δ†Φ − Φ†ΔΦcÞ2 ð Þ 2 : 5 2 2 2 2 μΔ; α2v ; α3v ≤ Oðm Þ; ð8Þ V is complete up to terms of dimension 43 and the dimension-6 terms involving δ1;2 are instrumental to lift Using the minimization conditions in Eq. (6) we trade μ and β for v and vs in the formulas below for fields and masses. 2 2 3 5 10−2 −2 Oðvs Þ As was shown in [4] Geff in the ballpark of - × MeV Neglecting v2 terms the physical states are opens up the allowed window in the nS − r plane, significantly affecting the selection criteria for acceptable models of inflation. 3 λ0 ðΔ†Δ†Þ ðΔΔÞ vs vs The term ΔTr Tr is phenomenologically G ¼ a þ 2 as;A¼ as − 2 a; ð9Þ irrelevant. v v

023013-3 GABRIELA BARENBOIM and ULRICH NIERSTE PHYS. REV. D 104, 023013 (2021)

2μ2 þ 8δ 4 ¼ − δ 1v vs H h 2 2 4 hs; m − 4λv þ 4δ1v v 2μ2 þ 8δ 4 ¼ þ δ 1v vs ð Þ S hs 2 2 4 h; 10 m − 4λv þ 4δ1v v

pffiffiffi v pffiffiffi v Gþ ¼ ϕþ þ 2 s δþ;Hþ ¼ δþ − 2 s ϕþ; ð11Þ v v

Hþþ ¼ δþþ: ð12Þ

FIG. 2. UV completion with heavy real scalars χ1 2: V in Here G and Gþ are the massless Goldstone bosons eaten by ; s þ Eq. (18) contains the renormalizable couplings entering the left Z and W , respectively. Neglecting subdominant terms the diagrams. Integrating out χ1;2 in these diagrams generates the desired (squared) Higgs masses read depicted dim-6 operators Φc†Δ†ΦΦ†ΔΦc (top) and ðΦc†Δ†ΦÞ2 (bottom). By flipping all arrows in the bottom row one generates 2 ¼ 4δ 4 þ 2 þ 2λ 2 ð Þ mA 2v m Δvs 13 the Hermitian conjugate of the second operator.

2 ¼ 4δ 4 þ 2 þ 6λ 2 ð Þ mS 1v m Δvs 14 It is easy to find a UV completion generating the dim-6 terms in Eq. (5): Consider heavy real scalar singlets χ1;2 2 ¼ 4λ 2 ¼ð125 Þ2 ð Þ mH v GeV ; 15 coupling as

2 2 2 m þþ ¼ 2m þ ¼ α1v : ð16Þ H H c† † c† † Vs ¼ðρ1 þ iσ1Þχ1Φ Δ Φ þðρ2 þ iσ2Þχ2Φ Δ Φ ð1Þ We start our discussion with the role of U L symmetry þ H:c: ð18Þ 2 2 in V. For this it is helpful to use Eq. (6) to trade m þ 2λΔvs β 2 for in mA in Eq. (13) to find with real ρ1;2, σ1;2. Under charge conjugation we have β 2 Δ ↔ Δ Φ ↔ Φ χ ↔ χ χ ↔ −χ 2 ¼ 4ðδ − δ Þ 4 þ v ð Þ , . Choosing further 1 1, 2 2 mA 2 1 v : 17 vs under C and demanding C invariance of Vs implies σ1 ¼ ρ2 ¼ 0. Integrating out the heavy singlet fields as ð1Þ β ¼ 0 δ ¼ δ δ ¼ ρ2 ð2 2 Þ δ ¼ σ2 ð2 2 Þ For exact U L symmetry with and 1 2 we shown in Fig. 2 gives 1 1= mχ1 and 2 2= mχ2 , verify that A is the massless of this broken ≫ and e.g., mχ1 mχ2 produces the scenario with heavy A symmetry. This solution corresponds to the vanishing of and light S. Thus the UV completion in Eq. (18) generates ¼ 0 the bracket in Eq. (6). (For the second solution with vs the desired dim-6 terms in Eq. (18) without producing any the symmetry is unbroken and Eq. (17) does not hold.) other dim-6 operators. β ¼ δ ¼ 0 δ ≠ 0 ð1Þ An interesting case is 1 with 2 :U L is Instead of invoking C symmetry one can also work explicitly broken by a higher-dimensional term only: The with L, by assigning L ¼ 2 to χ ¼ χ1 þ iχ2 enforcing minimization equation (6) is not sensitive to this term and ρ2 þ iσ2 ¼ iðρ1 þ iσ1Þ. Compared to the minimal model ð1Þ features spontaneous U L as in the original, renormaliz- described above the UV sector must be richer and break able majoron models. The phenomenological effect of LðτÞ in a way to produce the desired mass hierarchy. δ ≠ 0 2 is to render mA massive, with the possibility of Loop effects and/or a small vev of χ1 may render β ≠ 0, mA >MZ, and we may view this case as spontaneous which is a welcome feature as discussed in the paragraph symmetry breaking without Goldstone. above Eq. (18). Note that χ does not need to acquire a vev. Δ ≲ 2 For fixed Geff the perturbativity limit yτ entails an If one chooses to consider scenarios with a nonzero χ vev, upper limit on mS through Eq. (4). In the scenario with this vev must be small to avoid large corrections to β ð1Þ β ¼ 0 spontaneous U L breaking (where ) one necessarily spoiling jβj ≲ Oðv Þ impliedbyEq.(6). Note that all ≲ Δ ¼ Δ ≲ 1 s has m vs. As a consequence, mντ yτ vs eV pushes other dim ≲4 terms in V conserve L and therefore do Δ yτ and mS far below their otherwise theoretically allowed not receive contributions linear in the L-breaking vev of χ. ≲ 10 ≥ upper bounds. Specifically, mS keV for Geff We do not aim at an exhaustive discussion of all attractive 10−4 MeV−2 and the BBN constraint of Fig. 1 tightens UV completions here. Instead, we consider it as an ≲ 0 3 ð1Þ this to mS . keV. In the scenario, with explicit U L advantage that V in Eq. (5) allows us to fully study the breaking, however, one easily infers from Eq. (6) that one low-energy phenomenology (including loop effects where can choose m (and thereby mS) and vs independently needed) without specifying the underlying fundamental thanks to the free parameter β. theory.

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¼ 70 06þ2.21 III. PHENOMENOLOGY constant H0, i.e., at 95% C.L. H0 . −2.31 km=s=Mpc as compared with the ΛCDM value of H0 ¼ Studies of perturbativity for the SM [15] and 2HDM 68 35þ1.94 = = [16,17] have shown that self-couplings should be smaller . −1.84 km s Mpc. This is specially welcome, as the CMB provides a 2-3σ than ≈5. Applying this bound to α1 in Eq. (16) implies that [18] lower value of the Hubble constant than local mHþþ ≲ 400 GeV and mHþ ≲ 280 GeV. Current collider bounds are much weaker, because the production of measurements do [19]. In a simple 1-parameter extension, G these heavy charged Higgs bosons is an electroweak gauge by means of eff this tension is only weakenedP but not resolved. However, including also N and mν as free process (e.g., fusion at the LHC). Favorable eff þþ þ þ þ þ Δ parameters to the analysis completely relaxes the tension in decays are H → τ τ and H → τ ντ, unless yτ is too the Hubble parameter. Such an extension is however small. In the latter case one must resort to gauge-coupling beyond the scope of the present work. driven decays like Hþ → WþS; WþA. For a cutoff scale of Λ ∼ 0.5 TeV (and Oð1Þ couplings in the UV completion) 2 IV. CONCLUSIONS we have δ2v ∼ 0.1 and Eq. (14) gives mA ∼ 120 GeV. A is Δ produced through gauge interactions, and now a small yτ is The possibility of a light scalar particle interacting with welcome to suppress the decay into neutrinos. Detection neutrinos receives a lot of attention to alleviate several tensions in cosmological data. The com- through A → ZS will fail if MA − MZ is smaller than the trigger threshold for missing transverse momentum. It is munity is interested in this kind of interaction as a means therefore advisable to focus on the searches for the charged to break lepton number L spontaneously, providing a bosons. natural framework for neutrino Majorana masses. So far The model can be tested by its astrophysical signatures all cosmological analyses have employed the effective as well. Depending whether our scalar field is in the MeVor interaction of Eq. (2), which violates electroweak SU(2) keV range different signals can be expected. As mentioned symmetry. In this paper we have presented a viable realization of a model of a scalar interacting with neutrinos, before, CMB expriments are sensitive only to Geff and therefore both ranges give exactly the same phenomenol- by complementing the original SU(2) triplet models (fea- ogy cosmology-wise. This is not the case regarding turing spontaneous or explicit L violation) with higher- astrophysical experiments. For scalars in the MeV range, dimensional terms and devising possible UV completions the interaction introduced will make high energy (∼TeV) generating these terms. With our lagrangian one can now tau neutrinos from astrophysical sources scatter resonantly consistently calculate constraints from Z, charged lepton, with the CMB tau neutrinos and therefore a deficit of tau and meson decays, which was not possible with Eq. (2).To neutrinos can be expected. More precisely a dip in the tau bypass these constraints we have discussed the cosmologi- neutrino spectrum corresponding to the resonant energy [4] cal and astrophysical implications for the case that the light scalar couples dominantly to tau neutrinos. Depending on m2 the mass range of the scalar particle, characteristic signals ≃ ϕ ð Þ are possible at the IceCube or T2HK experiments. The LHC Eresonant 2 19 mντ can search for the charged members of the SU(2) triplet,

4 whose masses must be below 400 GeV. For appropriate is to be expected in experiments like IceCube and KM3Net. choices of the parameters our Higgs potential conserves L Remarkably, IceCube seems to be seeing a deficit in tau at the level of the renormalizable terms and shares essential neutrinos although the effect is not significant yet. features of the original triplet majoron model [9]. For scalars in the keV range the resonant energies involved make it ideal to detect such interactions in ACKNOWLEDGMENTS experiments sensitive to the diffuse supernova neutrino background, like T2HK. A detailed analysis of both signals G. B. acknowledges support from the MEC and FEDER will be given elsewhere. (EC) Grant No. FPA2017-845438 and the Generalitat As mentioned before, another very attractive property of Valenciana under Grant No. PROMETEOII/2017/033, also this model is that it provides a higher value of the Hubble partial support from the European Union FP10 ITN ELUSIVES (No. H2020-MSCAITN-2015-674896) and

4 INVISIBLES-PLUS (No. H2020- MSCARISE- 2015- Note that when the mass drops below the 690575). The work of U. N. is supported by project C3b neutrino CMB temperature the resonance energy becomes independent of the neutrino mass. In this case the resonance of the DFG-funded Collaborative Research Center TRR falls into the IceCube sensitivity range, unless the mass of the 257, “Particle Physics Phenomenology after the Higgs neutrino is close to the current cosmological limit. Discovery”.

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