The Hubble tension and a renormalizable model of gauged neutrino self-interactions
TThe Hubble tension and a renormalizable model ofgauged neutrino self-interactions
Maximilian Berbig, ∗ Sudip Jana, † and Andreas Trautner ‡ Bethe Center for Theoretical Physics und Physikalisches Institut der Universit¨at Bonn,Nussallee 12, 53115 Bonn, Germany Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
We present a simple extension of the Standard Model that leads to renormalizable long-rangevector-mediated neutrino self-interactions. This model can resolve the Hubble tension by delayingthe onset of neutrino free-streaming during recombination, without conflicting with other measure-ments. The extended gauge, scalar and neutrino sectors lead to observable signatures, includinginvisible Higgs and Z decays, thereby relating the Hubble tension to precision measurements at theLHC and future colliders. The model has a new neutrinophilic gauge boson with m Z (cid:48) ∼ O (10 eV)and charged Higgses at a few 100 GeV. It requires hidden neutrinos with active-hidden mixingangles larger than 5 × − and masses in the range 1 ÷
300 eV, which could also play a role forshort baseline neutrino oscillation anomalies.
Introduction.—
There is convincing evidence thatneutrinos played a substantial role during the epoque ofbig bang nucleosynthesis (BBN) at T ∼ MeV, closelymonitored by early element abundances. The lowest tem-perature scale indirectly probed for neutrinos is T ∼ eV,where observations of the cosmic microwave background(CMB) fit well to a history of our universe that doesnot only comply with the cosmological standard model(ΛCDM), but also with the expectation of the StandardModel of particle physics (SM), including exactly threegenerations of neutrinos.However, evidence is accumulating not only for a dis-crepancy between local measurements of today’s Hubblerate H [1–5] and therelike global determinations basedon ΛCDM together with CMB [6], baryonic acoustic os-cillations (BAO) and large scale structure (LSS) datasets[7–16], but also for an increasing tension in other parame-ters, see e.g. [17, 18]. The ultimate resolution of those dis-crepancies might require a modification of ΛCDM, pref-erentially, perhaps, shortly before the era of recombina-tion [19, 20]. Too many new physics (NP) scenarios havebeen discussed to review all of them, see [20–23] and ref-erences therein. Naturally, any consistent modificationof ΛCDM must be in compliance with a consistent mod-ification of the SM.The positive correlation of H and N eff with the am-plitude of the matter power spectrum σ , as observed inCMB data [6], prohibits a resolution of the H tensionsimply by increasing N eff alone (LSS prefers low σ ).However, a delay in the onset of neutrino free stream-ing during recombination could achieve both: breakingthe positive correlation of H and σ , while solving theHubble tension at the cost of increasing ∆ N eff duringrecombination [24–29]. Taking into account an effectivefour-neutrino interaction G ν eff (¯ νν )(¯ νν ) with strength G ν eff ∗ [email protected] † [email protected] ‡ [email protected] a good, bi-modal fit to CMB data is obtained with [28, 29] G ν eff ≡ g m Z (cid:48) ≈ (cid:40) (5 MeV) − (SI), or(100 MeV) − (WI) . (1)The weakly interacting mode (WI) should be interpretedas an upper limit on G ν eff such that the fit of cosmologicalparameters stays close to ΛCDM [24, 25], which then ofcourse does not resolve above tensions. Therefore, wewill focus on the strongly interacting mode (SI), whichconsiderably alters cosmology to resolve the tensions in H and σ while being consistent with local astronomicalobservations [28, 29].While decoupled heavy new physics (NP) certainly issuited to generate G ν eff , we stress that it is not precludedthat also a light mediator with interaction strength sim-ilar to the SI mode might resolve the tensions [30, 31].But of course, light mediators strongly interacting withneutrinos are highly constrained by the bound on ∆ N eff during BBN, see e.g. [32]. However, while one may feelthat it is just a relatively short time between BBN andrecombination, we recall that it is still six orders of mag-nitude in temperature. This certainly is enough to es-tablish a mass scale, say after a phase transition, andsubsequently integrate it out to obtain a decoupling be-havior of neutrinos during CMB closely resembling (1).In this way, neutrinos re couple by the new interactionsonly after BBN, and fall out of equilibrium shortly beforeor during recombination.In this letter we provide what we think is the simplestrenormalizable and phenomenologically viable extensionof the SM that leads to vector mediated four-neutrinointeractions of above strength. We first outline the pa-rameter space suitable to address the Hubble tension.Subsequently we fully flesh out our model, discuss con-straints on the parameter space and means to test themodel. a r X i v : . [ h e p - ph ] A p r Relevant parameter region.—
The effective four-neutrino interaction strength in our model is G ν eff ≡ g m Z (cid:48) ≡ g X ε m m Z (cid:48) , (2)where g X is the gauge coupling of a new U(1) X symme-try, ε m (cid:28) X charged) neutrinos, and m Z (cid:48) the mass of the new gaugeboson after U(1) X breaking. Equating the resulting ther-mally averaged interaction rate with the Hubble rate H ∼ T /M Pl (for radiation dominated universe – resultsonly change mildly if we switch to matter domination for T (cid:46) . (cid:0) G ν eff (cid:1) T ≈ T /M Pl . (3)Using (1), the decoupling temperature is obtained as T dec . ≈ . T (cid:29) m Z (cid:48) , while ε m is relevant, the new gauge boson willbe effectively massless giving rise to an induced long-range four-neutrino interaction with thermally averagedrate Γ ∼ g T . Requiring this interaction not to ther-malize with neutrinos prior to BBN, but before recombi-nation, results in a narrow parameter window2 × − (cid:46) g X ε m (cid:46) × − . (4)Knowing g eff and G ν eff we can compute m Z (cid:48) to find1 (cid:46) m Z (cid:48) (cid:46)
25 eV (SI) . (5)Furthermore, parametrizing m Z (cid:48) = g X ¯ v we can also con-strain the size of the effective U(1) X breaking vacuumexpectation value (VEV) ¯ v to¯ v := m Z (cid:48) g X ≈ ε m × ξ := ¯ v/v h ≈ ε m × × − (SI) , (7)where v h = 246 GeV is the SM Higgs VEV. The Model.—
Next to the new U(1) X gauge sym-metry we introduce a pair of SM-neutral chiral fermions N , and two new scalars Φ and S with charges as shownin Tab. I [102]. New interaction terms for SM leptons aregiven by L new = − y ¯ L ˜Φ N − M N N + h . c ., (8)where ˜Φ := i σ Φ ∗ , y is a dimensionless Yukawa coupling,and M has mass-dimension one. For brevity, we stick tothe one generation case here, while extensions to three - - - FIG. 1: Thermally averaged four-neutrino interaction raterelative to the Hubble rate as a function of Temperature for m Z (cid:48) = 25 eV and two different values of the Z (cid:48) width.Field Φ N N S X µ Lorentz S RH RH S VSU(2) L × U(1) Y ( , − ) ∅ ∅ ∅ ∅ U(1) X +1 +1 − L − X gauge symmetry as well as under globalLepton number (S=Scalar, RH=right-handed Weyl fermion,V=vector). generations of SM leptons or multiple generations of hid-den fermions are straightforward, and considered below.The most general scalar potential consistent with allsymmetries is V = V H + V Φ + V S + V H Φ + V HS + V Φ S + V , (9)with V Σ := µ Σ † Σ + λ Σ (cid:0) Σ † Σ (cid:1) (Σ = H, Φ , S ) , (10) V H Φ := λ (cid:0) H † H (cid:1) (cid:0) Φ † Φ (cid:1) + λ (cid:0) H † Φ (cid:1) (cid:0) Φ † H (cid:1) , (11) V DS := λ DS (cid:0) D † D (cid:1) ( S ∗ S ) ( D = H, Φ) , (12) V := −√ µ (cid:0) H † Φ (cid:1) S ∗ + h . c . . (13)We decompose the scalars as H = h +1 √ ( h + i a h ) , Φ = φ +1 √ ( φ + i a φ ) , (14)and S = √ ( s + i a s ) . (15)We choose a parameter region such that all neutralscalars obtain VEVs v σ := (cid:104) σ (cid:105) for σ = h, φ, s , and assumeCP conservation in the scalar sector. v h spontaneouslybreaks EW symmetry, v s breaks the new U(1) X , while v φ breaks both. Fixing the Hubble tension requires thehierarchy v h (cid:29) v s , v φ , cf. (7), and we will expand all ofour expressions to leading order in that hierarchy.The photon is exactly the same massless combinationof EW bosons as in the SM, mixed by the electroweakangle c W := m W /m Z [103]. By contrast, the very SM-like Z boson contains a miniscule admixture of the newgauge boson X , Z µ = c X (cid:0) c W W µ − s W B µ (cid:1) + s X X µ , (16)with an angle [104] s X ≈ − c W g X g (cid:18) v φ v h (cid:19) ≪ c X ≈ . (17)The masses of the physical neutral gauge bosons up to O ( ξ ) are m Z ≈ g v h c W and m Z (cid:48) ≈ g X ¯ v := g X (cid:113) v φ + v s . (18)Taking into account v φ , the neutrino mass matrix in thegauge basis (cid:0) ν, N , N (cid:1) is given by M ν = − yv φ / √ − yv φ / √ M M . (19)Upon 13–rotating by an angle ε m withtan ε m := ( yv φ ) / ( √ M ) , (20)this matrix has an exact zero eigenvalue, correspondingto approximately massless active neutrinos, and a Diracneutrino N with mass M N := (cid:113) M + y v φ /
2. Themassless active neutrinos mix with N proportional to s ε m generating the coupling (2). Together withtan γ := v φ /v s , (21)one can show that M = ( y/ √ ε m s γ ( G ν eff ) − / (cid:28) . (22)Owing to constraints discussed below the pa-rameter range one should have in mind is2 × − (cid:46) y (cid:46) × − , ε m (cid:46) .
05 and s γ (cid:46) . M N ≈ M then turns out to be in the range 1 ÷
300 eV.The mass generation for active neutrinos m ν (cid:28) yv φ ismerely a small perturbation to this setting. In particular,our mechanism is compatible with an effective Majoranamass in [ M ν ] , and, therefore, with any type of massgeneration mechanism that gives rise to the Weinbergoperator [33]. Another minimal possibility in the presentmodel would be to populate [ M ν ] like in the inverse see-saw mechanism [34–36]. Also Dirac masses are possiblebut require additional fermions. Ultimately, any of thecommonly considered neutrino mass generation mecha-nisms is compatible with our model. Phenomenology.—
The scalar sector of the modelcorresponds to a 2HDM+scalar singlet. However, both ofthe new scalars are charged under the hidden-neutrino-specific U(1) X which considerably alters phenomenologywith respect to earlier works [37–42]. The masses of thephysical scalars, to leading order in ξ ≡ ¯ v/v h , are givenby [105] m H = 2 λ H v h , m = m A = 2 v h µs γ , (23) m H ± = v h µt γ − λ v h , (24) m h S ≈ ξ v h (cid:18) λ S − λ HS λ H (cid:19) + O ( γµ/v h ) . (25)We diagonalize the neutral scalar mass matrix by threeorthogonal rotations O = R ( θ ) R ( θ ) R ( θ ), such that O T M . s . O = diag (cid:0) m h S , m H , m (cid:1) . (26)The mixing angles, to leading order in ξ , are given by s ≡ s S Φ = s γ , s ≡ s HS = ξ p t γ + q v h λ H , (27) s ≡ s Φ H = ξ s γ µ ( p t γ + q ) − λ H v h p λ H v h ( λ H v h s γ − µ ) , (28)where we use λ := λ + λ and p := λ v H s γ − µ c γ , q := λ HS v H c γ − µ s γ . (29)For the parameter region envisaged to resolve the Hubbletension, there are two new light bosonic fields: next to Z (cid:48) there is a scalar h S with mass in the keV range.To prevent possible reservations about these lightstates straightaway, let us discuss their coupling to theSM. The only way in which h S couples to fermions otherthan neutrinos is via its mixing with the SM Higgs. Oper-ators involving h S linearly, thus, can be written as O h S = c S Φ s HS × O SM H → h S . Hence, couplings to fermions are sup-pressed by their Yukawa couplings and there are no newflavor changing effects. We adopt the bounds on thisscenario of [43]. Besides BBN, which we discuss below,the strongest constraints arise from the burst duration ofSN1987A and requires ( c S Φ s HS ) (cid:46) − in the relevantregion. Parametrically, ( c S Φ s HS ) ∼ ξ ∼ ε m × − ,implying that we easily avoid this constraint for ε m (cid:46) . Z (cid:48) to SM fermions otherthan neutrinos is by mixing to the Z . Given Eq. (17), Z (cid:48) couples to the SM neutral current with strength2 g X ( v φ /v h ) = 2 g X ξ s γ . For momentum transfer below m Z (cid:48) this gives rise to new four-fermi (and NSI) operatorsof effective strengths (cid:16) G (2 ν )(2 f (cid:54) = ν )eff /G F (cid:17) = − √ ε m s γ , and (30a) (cid:16) G (4 f (cid:54) = ν )eff /G F (cid:17) = 4 √ ξ s γ ≈ ε m s γ × × − . (30b)Such feeble effects are currently not constrained by ex-periment.We note that vector mixing can be modified bygauge-kinetic mixing of the U(1) field strengths L χ = − ( s χ / B µν X µν [44, 45]. This shifts the Z (cid:48) coupling tothe SM neutral current by a negligible amount propor-tional to χ O ( m Z (cid:48) /m Z ) [46, 47] (given m Z (cid:48) (cid:28) m Z , χ (cid:28) Z (cid:48) to the electromagnetic current scaling as c W c X χ . Ex-perimental constraints on this are collected in [48, 49]and our model could, in principle, saturate these limits.Therefore, we stress that χ (cid:54) = 0 would neither affect oursolution to the Hubble tension, nor the H and Z de-cay rates in Eqs. (33) and (34) below (to leading order),which are fixed by Goldstone boson equivalence.We thus shift our attention to effects directly involvingneutrinos. For T (cid:46) v φ , neutrino mixing as in (19) isactive. As required by direct-search bounds [50–52] andPMNS unitarity [53, 54] we are assuming [55, 56] ε ( e ) m ≤ . , ε ( µ ) m ≤ . , ε ( τ ) m ≤ . , (31)for the mixing with e, µ, τ flavors. The couplings of neu-trinos to Z (cid:48) at low T then are given by g X ε m with astrength set by (4). This gives rise to the four-fermionoperators (2,30), but also to the possibility of Z (cid:48) emis-sion in processes involving neutrinos. We stress that(4), together with g X (cid:46)
1, gives rise to a lower bound ε m (cid:38) × − , only two orders of magnitude below thelimits (31). This fuels the intuition that this model istestable.Constraints on neutrinos directly interacting with lightmediators are collected in [21, 57–62]. The strongest lab-oratory constraints arise from meson [57, 63, 64] and nu-clear double-beta decays [65–68]. However, even the moststringent bounds for the least favorable choice of flavorstructure do not exclude couplings g eff (cid:46) − for light m Z (cid:48) , comfortably allowing (4). While most of the labo-ratory constraints are interpreted in terms of light scalar(majoron) emission, the present study makes it worth-while to revisit experimental exclusions in this regionalso for light vectors. The most important constraint isSN1987A neutrino propagation through the cosmic neu-trino background (C ν B) [69]. The exact bound dependson the neutrino masses and rank of y , but even under themost pessimistic assumptions g eff (cid:46) × − cannot beexcluded for m Z (cid:48) <
60 eV.The ¯ νν ↔ ¯ νν scattering cross section via Z (cid:48) exchangeis approximately given by σ (4 ν ) ( s ) = g X ε m π s ( m Z (cid:48) − s ) + m Z (cid:48) Γ Z (cid:48) . (32)To obtain the interaction rate in Fig. 1 we include the t -channel and use Maxwell-Boltzmann thermal averag-ing [70], while noting that a more refined analysis shouldemploy Fermi-Dirac statistics [71, 72]. For m Z (cid:48) > M N , Z (cid:48) decays to N N , N N , ¯ νN ( νN ) and ¯ νν , while for m Z (cid:48) (cid:46) M N only the last channel is accessible. Therespective total widths are Γ Z (cid:48) /m Z (cid:48) ≈ − or 10 − ,see Fig. 1, corresponding to Z (cid:48) lifetimes from micro- totens of picoseconds. For temperatures T (cid:29) v φ a thermalQFT investigation becomes necessary. We show ther-mally averaged rates (dashed) as obtained by dimensionalanalysis for illustration. For temperatures T (cid:28) m Z (cid:48) we reproduce the scaling of the effective operator (1) (alsodashed). Before recombination, Γ ν ( T ) differs from theeffective theory. While this should not change conclu-sions based on the (non-)free-streaming nature of neu-trinos [24–29], see also [73], it certainly motivates dedi-cated cosmological analyses to tell if our specific tempera-ture dependence could be discriminated from the effectivemodel.Finally we discuss the coupling of neutrinos to thelight scalar h S . Note that the matrix (19) is diagonal-ized exactly in s ε m , reflecting massless active neutrinosand prevailing lepton number conservation at this stage.This prevents a quadratic coupling of neutrinos to h S .Hence, SM neutrinos couple to h S only in associationwith hidden neutrinos, or suppressed by their tiny mass(e.g. [ M ν ] ∼ m ν produces such a coupling). In bothcases effects are unobservably small, also because of thevastly suppressed coupling of h S to matter targets.Also modification of Z decays to neutrinos are unob-servably small. Even if hidden N sizably mixes with ν ,the invisible Z width is not affected for M N (cid:28) m Z [50].However, the vertex Z ¯ N ν leads to N production fromneutrino upscattering on matter targets. The relevantoperator is suppressed by ε m compared to G F . Whileinteresting per se, N decays invisibly, leaving an unac-companied recoil as only signature.Any consistent model of strong neutrino self-interactions requires a modification of the SM scalar sec-tor and these are amongst the most visible effects of thismodel. The necessary modifications allow for new ex-otic decays of the SM Z and Higgs bosons to invisiblefinal states. To leading order in ξ the rates of the mostprominent decays areΓ H → h S h S = v h π m H (cid:20) λ HS c γ + λ s γ − µ s γ v h (cid:21) , (33)Γ H → Z (cid:48) Z (cid:48) = Γ H → h S h S , Γ Z → Z (cid:48) h S = m Z g s γ π c W , (34)and to leading order in ξ and γ Γ H → ZZ (cid:48) γ (cid:28) = g c W (cid:0) m H − m Z (cid:1) m H m Z ξ s γ π (cid:18) λ HS λ H (cid:19) . (35)Using Γ H → inv . ≤ . new Z → inv . ≤ . Z → Z (cid:48) h S requires γ (cid:46) . H → inv . , in the absence of fine tuning, demands λ HS , λ , ( µs γ /v h ) (cid:46) O (10 − ). In the light of this,Γ H → ZZ (cid:48) is merely a rare Higgs decay with BR( H → ZZ (cid:48) ) ≈ − ε m s γ .A model similar to ours but with S removed is phe-nomenologically excluded by Γ Z → Z (cid:48) h S , which would havea rate as in (34) with γ → π/ H ± couple directly to charged lep-tons and hidden neutrinos via (8), with strengths setby y . Important constraints on y arise from (cid:96) → (cid:96) γ and the measured lepton magnetic moments, both me-diated by a loop of H ± and N . Exact constraints aregiven in [55], while here it suffices to note that certainly y (cid:46) O (1) for all flavors, as we will find much tighterconstraints below. - - - - - - - - FIG. 2: Allowed region in the tan γ –( µ/v h ) plane. Pa-rameters have been chosen as y = 6 × − , ε m = 0 . g X = 2 × − , λ HS = 0 . λ = 0 . λ = 0 . λ Φ = 0 . λ S = 0 . λ Φ S = 0 . The coupling of H ± to quarks is suppressed by s γ ξ such that standard LHC searches [78, 79] do not apply.At LEP, H ± could have been pair-produced via s -channel γ/Z or t -channel hidden neutrinos, or singly-produced inassociation with charged and neutral leptons. H ± dom-inantly decays to final states N α ¯ (cid:96) β with BRs set by y . N further decays to three neutrinos via Z (cid:48) . The finalstate for H ± hence is (cid:96) ± β + MET. We use LEP limitson H ± pair-production [80] as well as a reinterpretedLEP selectron search [81–84] to obtain a lower bound m H ± >
100 GeV [106].Regarding electroweak precision tests, there are no newtree-level contributions to the ρ ≡ α T parameter as weonly introduce EW doublets and singlets. We follow [85]to estimate one-loop corrections. T is always enhancedcompared to the SM one-loop contribution, and stays inthe allowed interval T = 0 . ± .
13 [86] for | m H ± − m Φ | (cid:46)
120 GeV. For λ > A are heavierthan H ± . We assume them to be heavier than m H toavoid a small parameter window with mixed heavy-lightdecays. BBN.—
With Z (cid:48) , N , and h S there are three new lightspecies which could potentially distort BBN. Ultimately,a thermal QFT analysis seems worthwhile to fully explorethe early universe cosmology of this model for T (cid:29) v φ .The full set of coupled Boltzmann equations then shouldbe solved to track abundances precisely, but this is be-yond the scope of this letter. Nontheless, a simple orderof magnitude estimate suffices to clarify that there is aparameter region in which BBN can proceed as usual. The tight bound on ∆ N eff during BBN [32] does not allowany of the new particles to be in thermal equilibrium withthe SM. While m (cid:48) Z is fixed by (5), the mass of N is limitedby (22), and m h S ≈ ξv h √ λ S ≈ ε m √ λ S MeV. Hence,given the allowed parameter space, neither of these statescan simply be pushed beyond the MeV scale in order toavoid BBN constraints. Instead, we discuss the possibil-ity that all of the new states are sufficiently weakly cou-pled to the SM at the relevant temperatures such that athermal abundance is not retained.While all of the new fields thermalize with the SMat EW temperatures, this connection is lost once theheavy scalars freeze out and decay. The initial abun-dance of new states subsequently is depleted by reheat-ing in the SM, for example, at the QCD phase transi-tion. We thus focus on the temperature region aroundBBN. The coupling of Z (cid:48) to the SM, as well as active-hidden neutrino mixing ε m , is only effective after theU(1) X breaking phase transition. This warrants that Z (cid:48) and N do not thermalize with the SM between EWand BBN, realizing a generic mechanism [87] to recon-cile short baseline neutrino oscillation anomalies withcosmology. Nontheless, Z (cid:48) exchange would thermal-ize an abundance of N ’s, which thus has to be ab-sent. The leading process thermalizing N ’s with theSM is e + e − ( ν ¯ ν ) ↔ N ¯ N via t -channel H ± (Φ , A ) ex-change, which scales as Γ ∼ ( y/m H ± (Φ) ) T . Re-quiring this to be absent after QCD (EW) epoquerequires y (cid:46) × − ( m H ± (Φ) /
100 GeV). Togetherwith above bounds on ε m and γ (cid:46) . M (cid:46)
300 eV (QCD), or M (cid:46) m ν (cid:28) yv φ (cid:28) M , implying a lower bound y (cid:29) × − ( m ν / .
05 eV). Noteworthy, this allows M N rightin the correct ballpark to resolve short baseline neutrinooscillation anomalies; not only in the well-known waywith eV-scale states, see e.g. [88, 89], but also providinga definite model realization to the idea of decaying sterileneutrino solutions [90, 91].The most relevant processes for thermalization of h S are e + e − ↔ h S h S , e − γ ↔ e − h S , and ν ¯ ν ↔ h S h S . Nonenone of them ever reaches thermal equilibrium due to thehighly suppressed couplings of h S .Finally, we note that despite bearing some dangerfor successful BBN, the new states N and h S can alsobe a virtue: In order to explain the Hubble tensionwith self-interacting neutrinos, N eff must be enhanced to∆ N eff ≈ m h S ∼ O (10 keV) implies that an h S abundance could be present in a non-thermal state dur-ing BBN, and subsequently decay to reheat the neutrinobackground. h S decays to N ν and N N , but in ab-sence of L -violation not to ¯ νN , N N or ¯ νν (orall processes barred), with a total width proportionalto Γ h S ∝ y s γ . τ h S , therefore, is extremely depen-dent on the exact parameters ranging somewhere frommilli- to picoseconds. Also two-body decays N → Z (cid:48) ν from a supposed non-thermal population of hidden neu-trinos could contribute to ∆ N eff during CMB, provided M N > m Z (cid:48) + m ν . For M N < m Z (cid:48) + ν , on the other hand,only three body decays N → (2 ν )¯ ν are possible with life-time scaling as τ N ∼ (8 π ) − M N G ν eff 2 ε − m . Depending onthe exact parameters, a population of N , thus, could butbut doesn’t have have to decay before recombination. Discussion.—
In summary, we have presented aconsistent (renormalizable and phenomenologically vi-able) model that leads to vector-mediated neutrino self-interactions. In a narrow region of parameter space theseinteractions have the right strength to resolve the ten-sions between local and global determinations of H and σ [108]. To consistently implement such interactions inthe SM, we had to introduce a second Higgs doublet anda hidden Dirac neutrino, both charged under a new gaugesymmetry U(1) X . Phenomenological consistency (invis-ible Z decays) furthermore required the introduction ofthe U(1) X charged SM singlet scalar S .The scalar spectrum then consists of several newstates, all with very lepton-specific couplings: h S withmass of O (10 keV), as well as Φ, pseudo-scalar A and thecharged scalars H ± all with masses of O (100 GeV). Thenew, naturally neutrinophilic fore carrier has a mass of m Z (cid:48) ∼ O (10 eV) and the new hidden neutrinos massesin the range M N ∼ ÷
300 eV. A preferred region ofparameter space has charged Higgses at a few 100 GeV,sizable BR(Higgs → inv . ) as well as eV-scale hidden neu-trinos. Other, perhaps testable signatures would thenbe non-standard neutrino matter interactions of strength G (2 ν )(2 f (cid:54) = ν )eff ∼ O (10 − ) G F , cf. Eq. (30a), as well as active-hidden neutrino mixing with an angle ε m > × − .That our model works without specifying the mecha-nism of neutrino mass generation may feel like a draw-back to some. However, we think it is a virtue, as it renders this scenario compatible with all standard neu-trino mass generation mechanisms.The least appealing feature of our model, perhaps, isthe introduction of several new scales ( v φ , v s , µ, M ), andsome hierarchies among them. We have nothing to sayhere about this or any other hierarchy problem but sim-ply accepted this fact for the reason that we are con-vinced that this is the simplest renormalizable model inwhich active neutrinos pick up gauged self-interactions.Stabilizing these hierarchies against radiative correctionsmight require smaller scalar quartic cross-couplings thanthe direct constraints discussed above. Suchlike wouldnot contradict any of our findings.Finally, our analysis also shows that “model indepen-dent” considerations, which previously seemingly ruledout this model, are actually not always valid in con-crete models. On the contrary, it is only in completeand consistent models that early universe cosmology likethe Hubble tension can, and in fact must be, directlyrelated to physics testable in laboratories. Acknowledgments
We would like to thank V. Brdar, L. Graf, andM. E. Krauss for useful conversations as well as O. Fis-cher, A. Ghoshal, M. Ratz, and S. Vogl for commentson the manuscript. This work benefited from using thecomputer codes
SARAH [92],
FeynArts/FormCalc [93, 94],and
PackageX [95].
Note added.—
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