Self-interacting neutrinos: solution to Hubble tension versus experimental constraints

Exotic self-interactions among the Standard-Model neutrinos have been proposed as a potential reason behind the tension in the expansion rate, H0, of the universe inferred from different observations. We constrain this proposal using electroweak precision observables, rare meson decays, and neutrinoless double-\{beta} decay. In contrast to previous works, we emphasize the importance of carrying out this study in a framework with full Standard-Model gauge invariance. We implement this first by working with a relevant set of Standard- Model-Effective-Field-Theory operators and subsequently by considering a UV completion in the inverse See-Saw model. We find that the scenario in which all flavors of neutrinos self-interact universally is strongly constrained, disfavoring a potential solution to the H0 problem in this case. The scenario with self-interactions only among tau neutrinos is the least constrained and can potentially be consistent with a solution to the H0 problem.


I. INTRODUCTION
ere is a tantalizing discrepancy between the value of the Hubble constant (H 0 ) extracted from local measurement versus the one extracted from the Cosmic Microwave Background data [1][2][3][4][5].
Towards this end, the authors of Ref. [6] suggested to give neutrinos a new, extra strong self-coupling in the form of the dimension-six operator We focus on the possibility that the Standard-Model (SM) neutrinos are Majorana, i.e., ν M are four-component Majonara fermion elds. e Dirac case is strongly disfavoured by Big-Bang-Nucleosynthesis (BBN) constraints [7]. is e ective interaction can be induced by the presence of a light scalar mediator -a massive version of the so-called Majoron [8] with an e ective coupling L e Majoron ∼ λφν M ν M .
(3) * klyuaa@connect.ust.hk † emmanuel.stamou@ep .ch ‡ liantaow@uchicago.edu ere have been many studies on the constraints of the neutrino-Majoron coupling. Experimental results like Supernova [14,15], neutrinoless double-β decay [16,17], Meson decays [18][19][20][21][22][23] and Z-pole observables [23,24] all give relevant constraints, see Ref. [7] for a summary of various bounds in the strong self-coupling scenario. However, most of the studies focus on the e ective neutrino-Majoron coupling in Eq. (2), which violates electroweak gauge invariance. is is perfectly ne as long as one focuses on the degrees of freedom well below the weak scale. On the other hand, we have established a very accurate description of the physics around the weak scale, known as the Standard Model (SM). ere are precision measurements that will set relevant constraints on the scenario of self-interacting neutrinos. Indeed, many of the studies did implemented such constraints, e.g., Z decays. It is now mandatory to go beyond the e ective interaction in Eq. (2) and consider weak-scale UV completions. While our analysis here is motivated by the solution to the Hubble tension, the results are general constraints on the neutrino self-coupling, whether it would play a role in interpreting the CMB data or not.
In this work, we take two consecutive steps in this direction. Firstly, we will remain (mostly) agnostic about the speci cs of new physics and assume that, apart from the Majoron itself, it is somewhat heavier than the weak scale. Hence, we will parameterize their e ect by dimension-ve and six e ective operators in the Standard Model E ective eory (SMEFT). One such dimension-six operator contains the Majoron and induces neutrino self-interactions. However, in typical models that modify the neutrino sector and induce neutrinos masses the aforementioned operator is accompanied by additional ones, which do not contain the Majoron, and are typically generated in any UV completion. We will use experimental data to constrain the size of their Wilson coe cients.
Secondly, we will consider the possibility of UV completing this e ective theory into renormalizable models by introducing new degrees of freedom. Neutrinos are embedded in SU(2) doublets, therefore, at the renormalizable level the neutrino-Majoron coupling can only be induced via the (mass) eigenstate mixing a er electroweak symmetry breaking. ere are two paradigms: mixing with a neutrino sector or with a scalar sector. e former is realized in the Type-I seesaw model while the la er in Type-II. In both cases, the mixing angle determines the strength of the neutrino-Majoron coupling. However, we will see that for Type-I, the mixing is proportional to the neutrino mass and is thus too suppressed to provide a su ciently large mixing. Similarly for Type-II, the current bound on the triplet Yukawa coupling and the vev of the triplet scalar implies that it cannot provide a sufciently large mixing either [25,26]. However, we will show that there exist extended seesaw models in which there is no direct connection between the mixing and the neutrino mass. One of them is the so-called inverse seesaw model, which we will consider in detail. We will match the model to the SMEFT operators, and use the constraints derived for them to set limits on the model parameters.
We will nd that within a SM gauge invariant framework, the extent to which neutrino self-interactions may alleviate the H 0 inconsistency depends on the avour structure of the self-couplings. e case in which all avours interact with the same strength (universal) is too constrained from electronsector observables to provide a solution. However, the case in which only tau-avor neutrinos self-interact may still provide a solution due to the weaker constraints from particle-physics observations. e rest of this paper is organized as follows: In section II, we describe the relevant SMEFT framework and match it to seesaw models. In section III, we present the predictions for the observables entering the analysis. In section IV we combine the observables and contrast them to the CMB t and discuss the various regions of the parameter space. We conclude in section V.

A. Neutrino self-interactions within the extended SMEFT
We begin with the assumption that, with the exception of the Majoron φ, new physics is heavier than the electroweak scale. In this case, all beyond-the-SM e ects can be parameterized by a set of non-renormalizable operators. In our case, we are interested in a small subset of operators that induce neutrino self-interactions and those that typically accompany them in UV-complete models. More speci cally, the following set su ces to capture the main phenomenological aspects f denotes the neutrino avor with f = e, µ, τ , and Our notation follows closely the ones of Ref. [27]. We ignore avor-changing operators and restrict the discussion to avordiagonal operators.
e SM neutrinos live in the weak doublets L f , thus the Higgs doublet H must be included to form gauge singlets. e dimension-ve Weinberg operator, Q f νν , accounts for neutrino masses.
e operator Q f φ is responsible for generating the self-interaction. e operators Q (1),f HL and Q (3),f HL must also be included, because they are typically also generated at the treelevel in models that induce Q f νν . In particular, the operators Q  4)). e reason behind this tree-level relation is that typical models that generate the Q f νν operator by integrating out some heavy degrees of freedom, also necessarily induce the derivative operator . e presence of these operators lead to important phenomenological consequences, which cannot be captured when one simply works with the e ective coupling in Eq. (2).
To work with dimensionless couplings for the dimensionsix Wilson coe cients we introduce the notationC X = C X v 2 , with v 246 GeV the electroweak vev.
At scales below the electroweak scale the Q f νν operators induce a Majorana mass term for the neutrinos, and the Q f φ couplings of the neutrinos to φ. e resulting Lagrangian reads We note that both the mass and the interaction in Eq. 6 are avor diagonal. We emphasize that this is an assumption, and more general avor structures are certainly possible. However, the aim of this work is to extract main lessons rather than carry out an exhaustive study. Moreover, as we will see in section III D, the e ect of neutrino self-interactions on the CMB has only been studied under a quite (over)simpli ed case. Hence, we will also make simplifying assumptions for the avor structure in our study.

Type-I Seesaw Model
To illustrate how the EFT operators presented in section II A are induced in concrete UV models we start with the simplest Type-I seesaw model. e SM Lagrangian is augmented with an extra heavy right-handed neutrino with N R a four-component chiral eld. A er electroweak symmetry breaking, the mixed Dirac mass is generated m D = y R v/ √ 2. e neutrino mass matrix then reads A er diagonalization, the masses of the light mass eigenstates in the limit m D M R are and the mixing angle between light and heavy eigenstates is Hence, the coupling between the Majoron and light eigenstates reads We see that in this model the coupling to the Majoron is suppressed by the neutrino mass and thus cannot produce strong self-interactions for perturbative values of λ.
To match to the e ective Lagrangian in Eq. (4), we integrate out N R at the tree-level and nd the Wilson coe cients Again, we see that C f νν , which generates the neutrino mass, is correlated to C f φ . Hence, the neutrino-Majoron interaction, proportional to C f φ , is suppressed by the neutrino mass.

Inverse Seesaw Model
In order to break the correlation between C f νν and C f φ we consider an inverse seesaw model [28][29][30][31][32][33] augmented with an additional real scalar, φ, that couples to one species of the heavy neutrinos: with F = F L + F R . e fermion elds F L and F R above are four-component chiral elds, i.e., only two components are non-zero. By choosing to couple the Majoron only to F L and not to F R we break the correlation between neutrino mass and Majoron coupling. e subscript, f , stands for the avor. For simplicity we consider the heavy-neutrino se ing for each avor separately and do not consider their mixing.
We match to the e ective Lagrangian in Eq. (4) by integrating out the heavy elds F R and F L at the tree-level. For the case δ L , δ R M and up to dimension-six the Wilson coe cients we obtain are Contrary to the Type-I model, we see that the neutrino mass and the neutrino-Majoron coupling are controlled by independent parameters, δ L and λ, respectively. It is thus possible to induce a sizable neutrino-Majoron coupling without it being suppressed by the neutrino mass. At the same time, we see that C f φ and C f ew are correlated to some extent, which has important phenomenological consequences.
To include constraints from electroweak-precision observables, we also compute the Wilson coe cient of the operator that contributes to the T -parameter at tree level. In the Warsaw basis this operator is Q HD ≡ |H † D µ H| 2 . e one-loop matching at a scale µ M gives where we only kept terms of O(y 4 R,f ). We include the leading terms of O(y 2 R,f g 2 1 , y 2 R,f y 2 e ) by solving the renormalization group (RG) within SMEFT (see section III B).

III. OBSERVABLES
In this section we discuss the most relevant observables in our analysis and their predictions within the SMEFT framework. We choose as the numerical input for the electroweak parameters G F , α, and m Z . As extensively discussed in the literature, e.g., Ref. [34] and references within, the presence of dimension-six operators a ects the determination of the electroweak-parameter input. In the case at hand, only the operators in Eq. (5) are induced at the tree-level and in fact out of them only the operators Q unchanged. e G F shi a ects all electroweak observables. To take it into account, one substitutes, e.g., Ref. [34], where G F is still the experimental input value.
A. Z decays A er electroweak-symmetry breaking the operator combination Q HL does not (directly) a ect the charged lepton sector, but it does induce an anomalous Z coupling to the neutrino species f , i.e., Together with the shi in G F this modi es the partial width to the neutrinos We also include the three-body partial width Z →ν f ν f φ, which is, however, formally higher order in the EFT, i.e., it is proportional to (C f φ ) 2 . For the region of interest m φ m Z , we nd for a neutrino species coupled to φ via the operator Q f φ the width Notice that this rate diverges for m φ → 0. For small m φ it is thus necessary to resum the logarithms. However, for the masses that we are considering this is not necessary. Also due to the double EFT suppression this rate is numerically small.
Similarly we evaluate the e ect of the shi in G F in the partial width to charged leptons and to hadrons. In the SM the partial width to a fermion f with charge Q f is

B. T -parameter
Heavy sterile neutrinos can a ect electroweak-precision observables, i.e., the T -parameter. Within the SMEFT framework the new-physics contributions to the T -parameter are controlled by the Wilson coe cient of the Q HD operator evaluated at the electroweak scale, µ ew ∼ m Z , via αT = T = − v 2 2 C HD (µ ew ) [35]. In our setup we integrate out the heavy degrees of freedom at a scale M µ ew and obtain C HD (µ ew ) via the RG evolution to µ ew (see Refs. [36,37] for the corresponding anomalous dimensions). Operators that have been induced at the tree-level at M can mix into Q HD . In our case we nd that at leading-log accuracy e singlet operators Q (1),f HL mix into Q HD , introducing the dependence on C (1),f HL = C f ew .

C. Leptonic meson decays
Non-standard neutrino interactions a ect the decays of pseudoscalar mesons. e most stringent constraints originate from the semileptonic decays of charged pseudoscalars. e modi cation with respect to the SM originates both from the shi in G F and the anomalous coupling W f ν f proportional to e √ 2swC (3),f HL . e two-body partial width of a pseudoscalar, P , to a neutrino and a charged lepton then reads with f P the decay constant of the meson and V CKM = |V uidj | 2 the corresponding CKM elements. As in the SM the two-body widths are helicity suppressed and thus proportional to the charged lepton mass. e helicity suppression is li ed in the three-body decay P → f ν f φ. Expanding in the mass of the charged lepton we nd where x φ = m 2 φ /m 2 P .
D. Neutrino self-scattering (Gν ) e presence of new, neutrino self-interactions can modify the neutrino standard free-streaming behavior during the radiation-dominated era. 2 → 2 sca ering among neutrinos modi es the momentum dependence of the neutrino distribution functions and can thus a ect cosmological observables such as the CMB. e cosmological t of Ref. [6] is performed for the case in which the φ mass is much larger than the typical energy scale of the sca ering event. In this case we can to an excellent approximation integrate out φ and describe the neutrino self-interactions via four-fermion contact interactions.
Starting from the (per assumption) avor-diagonal Lagrangian for the four-component Majorana fermions ν M,i in Eq. (6) with Here, the indices i, j = 1, 2, 3 indicate the avors e, µ, τ , respectively. Note that when φ couples avour-diagonally to more that one avour a mixed four-fermion operator is necessarily generated. In order to make contact with the results of the CMB t of Ref. [6] we present here the corresponding collision terms for neutrino sca ering. In the general, avour-diagonal case there are three independent processes: the self-sca ering of one species (ν i + ν i → ν i + ν i ), s-channel annihilation (ν i + ν i → ν j + ν j with i = j), and t-channel sca ering (ν i + ν j → ν i + ν j with i = j).
eir respective squared matrix-elements summed over initial-and nal-state spins are: with s, t, u the usual Mandelstam variables. No symmetry factors for identical particles have been included above. What enters the evolution of the neutrino distributions are the collision integrals for each process. Adapting the generic expression from Ref. [38] we nd that for a speci c neutrino species i and j = i the collision integrals for the three processes above are: where dΠ i = d 3 pi (2π) 3 2Ei and F [. . . ] de ned as in Ref. [6]. e factor 1/g, with g = 2 the spin degrees of freedom, has been omi ed in Ref. [6]. No additional symmetry factors for identical particles in initial and nal state need to be included in Eq. (32) (see Ref. [39]). e factor 1/2 in Eq. (33) is due to the identical particles j = i in the nal or initial state.
We see that the collision integrals in Eqs. (33) and (34) couple the evolution of the distribution function of the three species. is cross-talk has been been neglected in Ref. [6]. Instead each neutrino avour was assumed to self-interact independently with the same strength and the t to the CMB provided the best-t value for the parameter G ν de ned via the collision integral for each species [6] C Ref. [6] νi(p1)νi(p2)↔νi(p3)νi(p4) = We emphasize again that this is an over-simplifying assumption that does not follow from avor universality. Nevertheless, we will use it since it provides a direct comparison between the explicit t performed in Ref. [6] and the constraints obtained in this paper. Under this simplifying assumption, i.e., neglecting cross-talk, we nd by comparing Eqs. (32) and (35) that for the "universal case" (C 1 νν = C 2 νν = C 3 νν = C 12 νν = C 13 νν = C 23 νν ≡ C νν ) Additionally to the "universal" case, we also consider " avor speci c" cases (one C i νν = 0 and all other couplings zero) in which the self-interactions take place only among a single species instead of among all three. ese cases are governed by the evolution of the thermal bath of one neutrino with the collision integral in Eq. (32). e t of Ref. [6] does not cover these cases, a dedicated re-analysis is required, which is beyond the scope of the present work. Roughly, the total strength of self-interactions are weaker if a single species self-interacts than in the "universal case". To at least partially take this into account we interpret the t of Ref. [6] for the " avor speci c" case via the rescaling e results of a future CMB t for these cases could then be obtained by a simple rescaling of Eq. (37). We caution that while we expect this scaling to partially take into account the di erence between the e ective coupling strength, the factor of √ 6 is just an educated guess. More complete numerical study is needed to obtain the precise factor.

IV. NUMERICAL ANALYSIS
As we have discussed above, we focus on two distinct limits in both of which the self-interactions via φ are assumed to be aligned to the neutrino mass-eigenstates.
"Flavor-speci c" cases: Self-interactions are present only for one, the f -th species of neutrinos with f = e, µ, τ . In "Universal" case: All three neutrinos species interact with equal strength such thatC φ ≡C e φ =C µ φ =C τ φ and C ew ≡C e ew =C µ ew =C τ ew .

A. Experimental input / Constraints
To illustrate the relative importance of various particlephysics and cosmological observables in the " avor-speci c" and "universal" cases we perform χ 2 ts combining information from multiple observables. Below we summarize the experimental input relevant for the ts. Any additional, unspeci ed numerical input is taken from Ref. [40].

Z decays:
We implement the constraints from the partial width measurements of the Z boson by centering the corresponding χ 2 's around the SM predictions and using the experimental uncertainties [40] ∆Γ + − = 0.086 MeV , ∆Γ had = 2.0 MeV , ∆Γ inv = 1.5 MeV .
2. T -parameter: When discussing the inverse seesaw model we also include the constraint from the Tparameter as it can be a ected by the presence of heavy neutrinos. We use the current best t value of T = 0.06 ± 0.06 [40].
3. Meson decays: Analogously to Z decays also for meson decays we assume that the experimental measurements of their branching ratios and their lifetimes are centered around their SM predictions and add the corresponding experimental uncertainties in their χ 2 . We neglect subleading theory uncertainties associated to form-factors. We consider constraints from branchings fractions of two-body leptonic decays of π + , K + , D + s , as well as their lifetimes [40]: Note that o en measurement of ratios of branching fractions are more constraining than those from the branching ratios above. However, using such ratios can leave certain directions unconstrained when more than one neutrino species self-interact, i.e., in the "universal" case. e combination of constraints are, however, similar when folded with the lifetimes measurements and Z decays. erefore, to enable a be er comparison between di erent cases we do not include ratios of branching ratios in the ts. 4. Neutrinoless double β-decay: As discussed in Ref. [41], current neutrinoless double β-decay experiments like NEMO-3 [42] and KamLAND-Zen [16] can stringently constrain light e-avor Majorons. is will be illustrated by mapping the results of Figure 4 of Ref. [41] into our corresponding exclusion plots. φ . Each panel corresponds to one of the four cases (νe, νµ, ντ , and universal). In purple and green the constraints from Z and leptonic meson decays, respectively. In black (gray) the combined allowed 68.27% (95.45%) CL region. e red regions in the second and third plot correspond to the 1σ preferred region for MIν in Ref. [6], cf., Eq. (38). e best-t regions for the SIν case and the MIν case not appearing in the rst and last plot lie outside the ranges. All coloured regions correspond to m φ = 10 MeV. For the combined constraints we show the allowed region for m φ = 1 MeV and m φ = 100 MeV in dashed and do ed lines, respectively.

B. SMEFT t
We rst investigate the constraints on the SMEFT Wilson coe cients for the four di erent cases (e, µ, τ , universal) without specifying a UV model. In each case there are three independent parametersC (i) ew ,C (i) φ , and m φ . In gure 1, we consider the four cases and show the allowed 68.27% and 95.45% CL regions for the two Wilson coe cients. e purple and green regions are the allowed regions from Z and mesons decays, respectively, for the case m φ = 10 MeV. e black and gray regions are the combined allowed regions. e dashed lines enclose the allowed region for m φ = 1 MeV and the do ed ones the region for m φ = 100 MeV. We also show the best-t regions for the strength of neutrino self-interactions from Ref. [6], cf., Eq. (38) when they lie within the plot ranges.
Inspecting gure 1 we observe that: • e constraints from Z decays (blue) and meson decays (green) are o en complementary, e.g., in the ν e case.
• e main di erence between the three " avor-speci c" cases are the constraints from meson decays. ey are strongest for the ν e case (top-le plot) and rather weak for the ν τ case (bo om-le plot). e reason is the di erent helicity suppression of the two-body meson decays, phase-space, and the fact that the ν τ case is only constrained by D + s → τ + ν. In contrast, the ν e and ν µ cases receive strong constraints from π + and K + decays to e + ν e and µ + ν µ .
• e "universal" case (bo om-right plot) is to a large extent controlled by the its ν e component and is thus similarly stringently constrained as the ν e case.
• e particle physics constraints on the ν e and "universal" cases cannot be accommodated in neither the SIν nor the MIν best-t regions of Ref. [6] for m φ > 1 MeV.
• e MIν best-t regions (red) are compatible with particle-physics constraints for the ν µ and ν τ cases. Note, however, that the corresponding values forC φ are O(1) thus close to the validity region of the EFT.

C. Inverse seesaw model
In the previous section we considered the particle-physics constraints in conjunction with the preferred region from the CMB t within the mostly model-independent framework of SMEFT. In concrete models, the SMEFT Wilson coe cients can be correlated, reducing the number of free parameters and leading to correlated signals. To illustrate this, we now study the phenomenology of the inverse-seesaw model from section II B 2. Similarly to before we consider separately the three " avor-speci c" cases and the "universal" one. In each case, we vary the φ mass and the e ective Majoron coupling to neutrinos,C (f ) φ , while keeping the UV coupling λ xed.
In gure 2, we show the resulting constraints in thē C (f ) φ − m φ plane. First-, second-, and third-row plots correspond to the avor-speci c e-, µ-, and τ -case, respectively. Plots of the fourth row correspond to the "universal" case. Plots of each column present the case of di erent values of λ. e colored regions are excluded at 90% CL: in purple the combined constraints from Z decays, in green the combined constraints from meson decays, and in grey the constraints from neutrinoless double-β decay [41]. Dashed lines indicated in the legend show the constraints from each meson sector separately, i.e., from π + , K + , and D + s decays. e red-do ed regions are the preferred 1σ regions of the CMB t. e horizontal, dashed lines show the constraint from the T -parameter when the heavy-neutrino scale is M = 500 GeV and M = 1 TeV. By inspecting gure 2 we recover some of the conclusions from the SMEFT analysis of the previous section.
• e best-t regions of the CMB t cannot be accommodated in the " avor-speci c" ν e and "universal" cases.
• While the SIν scenario is strongly disfavoured, the particle-physics constraints are compatible with the MIν scenario in the " avor-speci c" ν µ and ν τ cases, but only for masses m φ 10 MeV and large values of λ, i.e., λ 1, close to its perturbativity limit. is in turn implies that this scenario must have a cut-o close to the mass scale of exotic fermions.
• e non-trivial structure of the π + (dashed-do ed lines) and K + (dashed lines) constraints in the ν e and "universal" case is due to the interplay between the twobody decays, which suppresses the branching ratio BR(M → ν(φ)), and the three-body decay, which enhances it.
• e scenario is being further tested at colliders by searches for the heavy neutrinos. e analyses, for example Refs. [44,45], typically search for the heavyneutrino decays to W s and either electrons or muons, thus placing limits on the mass of the heavy neutrino for the avor speci c e and µ cases, and not the τ case.
In both e and µ case, the present limits are rather weak, i.e., M 100 GeV [44,45] for a mixing of the order 10 −2 − 10 −3 between light and heavy neutrinos.
alitatively the results of this section are similar to [7,43], but there are important di erences. In particular, the constraints from Z decays, which are dictated by gauge invariance, provide powerful constraints. ey restrict the allowed parameter-space of the ν τ " avor-speci c" case more than meson decays. e allowed region corresponds to large couplings, close to their perturbativity bound.

V. CONCLUSIONS
Motivated by the approach of using neutrino selfinteractions to address the tension in the H 0 measurement, we investigated the experimental constraints on this scenario. In contrast to previous studies on this setup, we began with an e ective-eld-theory framework that respects the full Standard Model gauge symmetry. is is important as many of the constraints are from experiments performed around the electroweak scale, where the e ect of electroweak symmetry is essential. In addition to the SMEFT framework, we have also considered a UV completion within an inverse-seesaw type model. We performed an careful derivation of the constraints from Z decay, T -parameter, and meson decays. We also took into account the limits from the search of neutrinoless double-β decay and BBN. e constraints depends on the avor structure of the couplings. To illustrate this, we considered two scenarios. In one of them, the self-interaction act in a " avor universal" way to all avors of neutrinos. In the other one, there is only interaction between one speci c avor species. We showed that, in the " avor universal" case, the neutrino self-interaction as a solution to the H 0 problem is strongly disfavored. Only the " avor-speci c" ν µ and ν τ cases in the MIν scenario may be provide a solution. However, the scalar mass must be low and the scalar-neutrino couplings large, close to their perturbativity limits. e SIν scenario is strongly disfavoured.
Future experimental searches are promising in further testing these scenarios. e experimental measurements considered in this paper will be improved signi cantly at on-going and future facilities. e scenarios under consideration also point to new particles, for example the new heavy neutrinos,  not far away from the weak scale. ey can be searched for directly in the upcoming LHC runs and at potential higherenergy colliders.