Hubble Tension and Cosmological Imprints of U (1) X Gauge Symmetry: U (1) B 3 − 3 L i as a case study

The current upper limit on N eff at the time of CMB by Planck 2018 can place stringent constraints in the parameter space of BSM paradigms where their additional interactions may affect neutrino decoupling. Motivated by this fact in this paper we explore the consequences of light gauge boson ( Z ′ ) emerging from local U (1) X symmetry in N eff at the time of CMB. First, we analyze the generic U (1) X models with arbitrary charge assignments for the SM fermions and show that, in the context of N eff the generic U (1) X gauged models can be broadly classified into two categories, depending on the charge assignments of first generation leptons. We then perform a detailed analysis with two specific U (1) X models: U (1) B 3 − 3 L e and U (1) B 3 − 3 L µ and explore the contribution in N eff due to the presence of Z ′ realized in those models. For comparison, we also showcase the constraints from low energy experiments like: Borexino, Xenon 1T, neutrino trident, etc. We show that in a specific parameter space, particularly in the low mass region of Z ′ , the bound from N eff (Planck 2018) is more stringent than the experimental constraints. Additionally, a part of the regions of the same parameter space may also relax the H 0 tension.


INTRODUCTION
The cosmological parameter N eff , associated with the number of relativistic degrees of freedom, is crucial in describing the dynamics of the thermal history of the early universe.At very high temperature of the universe, the photon (along with electrons) and neutrino bath were coupled, whereas at low temperature the interaction rate drops below the Hubble expansion rate, and the two baths decouple [1].N eff is parameterised in terms of the ratio of energy densities of neutrino and photon bath.Within the Standard Model (SM) particle contents, the two aforementioned baths were coupled through weak interactions at high temperature and as temperature drops (T ∼ 2 MeV) they decouple.Assuming such scenario the predicted value of N SM eff turns out to be 3.046 [2,3].This value of N eff deviates from the number of neutrinos (3) in the SM particle content is due to various non-trivial effects like non-instantaneous neutrino decoupling, finite temperature QED corrections and flavour oscillations of neutrinos [2,3].However, in observational cosmology also, measurements of the cosmic microwave background (CMB), baryon acoustic oscillations (BAO), and other cosmological probes provide constraints on N eff .The current Planck 2018 data has precise measurement of N eff at the time of CMB with 95% confidence level, N eff = 2.99 +0. 34 −0.33 [4].Thus the current upper limit from Planck 2018 data shows that there can be additional contribution (apart from SM predicted value) to N eff , indicating the scope for new physics.
It is evident that N eff will change in the presence of any beyond standard model (BSM) particles with sufficient interactions with either of the photon or neutrino bath at temperatures relevant for neutrino decoupling [5][6][7][8] or in presence of any extra radiation [9][10][11][12].Thus the upper limit on N eff from CMB can be used to constrain such BSM paradigms dealing with any extra energy injection.Several studies have been performed to explore the imprints of BSM models in N eff like models with early dark energy [10,13], relativistic decaying dark matter [14] and non-standard neutrino interactions (NSI) [5,6,[15][16][17][18].From the perspective of neutrino physics, the last one is very interesting since the non-standard interactions of light neutrinos may alter the late-time dynamics between photon and neutrino baths, contributing to N eff and the same NSI interactions can be probed from ground based neutrino experiments as well [19,20].
On the other hand, the anomaly-free U (1) X gauge extended BSM models are wellmotivated from several aspects like non zero neutrino masses [21], flavor anomalies [22] etc.The anomaly condition allows the introduction of right-handed neutrinos in the theory and thus it can also explain non-zero neutrino masses via the Type-I seesaw mechanism [21].These scenarios naturally involve a gauge boson (Z ′ ) that originates from the U (1) X abelian gauge symmetry and has neutral current interactions with neutrinos and electrons which may have some nontrivial role in neutrino decoupling and hence in deciding N eff [6,7,15,23].The detectability of gauge boson throughout the mass scale (M Z ′ ) also motivates such scenarios.For TeV-scale Z ′ , constraints arise from collider experiments [24][25][26][27], whereas in the sub-GeV mass region, low energy scattering experiments (neutrino electron scattering [28], neutrino-nucleus scattering [29] etc.) are relevant to constrain the parameter space.However, both types of direct searches become less sensitive in the mass M Z ′ ≲ O(MeV) and lower coupling (g X ≲ 10 −5 ) region.In that case, the CMB observation on N eff plays a crucial role and can impose severe constraints on the M Z ′ − g X plane, which stands as the primary focus of our study.
In cosmology, there exists some discrepancy between the values of expansion rate H 0 obtained from CMB and local measurement [4,[30][31][32][33][34][35].Using direct observations of celestial body distances and velocities, the SHOES collaboration calculated H 0 = 73.04 ± 1.04 Km s −1 Mpc −1 [33].However, the CMB measurements like Planck 2015 TT data predict H 0 = 68.0+2. 6 −3.0 Km s −1 Mpc −1 at 1σ [36] and from recent Planck 2018 collaboration, it turns out to be H 0 = 67.36 ± 0.54 Km s −1 Mpc −1 [4], where both the collaboration analyzes the CMB data under the assumption of ΛCDM cosmology.So there is a disagreement between the local and CMB measurements roughly at the level of 4σ − 6σ [30].Although such discrepancy may arise from systematic error in measurements [37,38], it also provides a hint for BSM scenarios affecting the dynamics of the early universe.One possible way out to relax this so called "H 0 tension" involves increasing N eff (in the approximate range ∼ 3.2 − 3.5) [6,39,40] 1 .It is also worth highlighting that, though BSM radiation with N eff ∼ 3.2 − 3.5 ameliorate the H 0 tension to 3.6σ, it can not resolve the tension completely as revealed by recent studies [39,42].As discussed earlier U (1) X local gauge extension leads to new gauge boson and new interaction with SM leptons which might affect neutrino decoupling and hence contribute to N eff .Thus U (1) X gauge extension can be one possible resolution to the H 0 tension also.Two kinds of U (1) X scenarios have been considered so far in the literature to address the Hubble tension problem and/or the excess in N eff at CMB: U (1) µ−τ [6] and more recently with U (1) B−L [7].However, there exist several other well motivated U (1) X models (different anomaly free combinations of B and L numbers [28,43]) whose consequences in N eff have not been explored till date.Depending on the value of L, each model exhibits distinct signatures in N eff , demanding individual treatment 2 .Therefore an analysis of N eff in the context of light gauge boson in generic U (1) X model is required, which is the main objective of this work.
In this work, we begin with a generic U (1) X extension and explore the aforementioned cosmological phenomena more comprehensively.For simplicity, we assume the light gauge boson was in a thermal bath, and the right-handed neutrinos are heavy enough (> O(10 2 ) MeV) that they hardly play any role in neutrino decoupling.We study the dynamics of neutrino decoupling in the presence of the light gauge boson (Z ′ ) in a generic U (1) X scenario with different U (1) X charges assigned to SM fermions.For some specific values of such charges, the generic U (1) X models can be interpreted as popular U (1) X models as we will discuss in the later part.We evaluate N eff by solving a set of coupled Boltzmann equations that describe the evolution of light particles i.e. electron, SM neutrino, and light gauge boson ( M Z ′ ∼ O(10) MeV).Our analysis shows that depending on the U (1) X charge assignment of the first generation lepton, the generic U (1) X gauged models can be classified in the context of N eff : Z ′ having tree-level coupling with e ± and Z ′ having induced coupling with e ± .This classification leads to two different types of N eff characteristics depending on the U (1) X charges of first generation leptons.Thus we identify the region of parameter space that can be excluded by Planck 2018 observations and indicate the constraints from N eff .We also illustrate the parameter space in which Hubble tension can be relaxed where the value of N eff falls within the range of 3.2 to 3.5 [39].Although N eff for a generic MeV scale particle has been discussed in ref. [44], it rely on model independent formalism in contrast to our case.We also analyze the cosmological observations for specific gauged U (1) X models, U (1) B 3 −3L i (i = e, µ, τ ).Our analysis show that, despite the resemblance with B − L or L µ − L τ model, the bound from N eff significantly differs in B 3 − 3L i models due to the different U (1) X charges of first generation leptons.In context of ground based experiments, these extensions encounter fewer constraints in comparison to others, as these are exclusively linked only to the third generation of quarks and one generation of leptons.Our results show that the upper limit on N eff from Planck 2018 [4] can put stringent bounds on the parameter space of light Z ′ in the mass region ≲ O( 102 ) MeV where the other experimental bounds are comparatively relaxed.
The paper is organized as follows.In sec.2, we begin with a brief model-independent discussion on the generic U (1) X model.Sec.3 is dedicated to a comprehensive analysis and discussion of the dynamics governing neutrino decoupling in terms of the cosmological parameter N eff in the presence of the light Z ′ originated from the generic U (1) X .Subsequently, in sec.4,we present the numerical results in terms of N eff for this generic U (1) X model.In sec.5, we discuss the analysis presented earlier, which applies to the specific U (1) X model, U (1) B 3 −3L i (i = e, µ, τ ).We present a brief discussion on H 0 tension in sec.6.Finally, in sec.7,we summarise and conclude with the outcomes of our analysis.In Appendices A to C, we provide various technical details for the calculation of N eff .

MODEL INDEPENDENT DISCUSSION: EFFECTIVE Z ′ MODELS
In this section, we look at effective light Z ′ models.We take a model-independent but minimalist approach and only assume that the Z ′ originates from the breaking of a new local U (1) X abelian gauge symmetry under which the SM particles are charged as shown in Table I.The anomaly cancellation conditions for a typical U (1) X local gauge symmetry requires the presence of additional chiral fermions beyond the SM fermions3 .To keep the minimal scenario we assume that the only BSM fermions needed for anomaly cancellation are the (three) right handed neutrinos ν R which are taken to be charged under the U (1) X symmetry.With the addition of ν R , several different type of gauged U (1) X models including the popular U (1) X models can be constructed4 .We will discuss some of the models later in Sec. 5.
For this purpose within this section, we will not specify the nature of U (1) X symmetry nor the charges of SM particles and ν R under it and will treat them as free parameters and proceed with a generic discussion.We will also not go into details of anomaly cancellation constraints.All these things will be clarified in the following sections during our discussions on some well motivated U (1) X models.In Table I, in addition to ν R (typically required In this section, we will take X Φ = 0 as we are interested in a light Z ′ gauge boson.To simplify our notation later we denote X L i = X ℓ i = X i (i ≡ 1 − 3, signify e, µ, τ ), which are relevant for our discussion on N eff (see the text for detail).
for anomaly cancellation) we also add an SM singlet scalar σ carrying U (1) X charge whose vacuum expectation value (VEV) will break the U (1) X symmetry.Note that apart from the minimal particle content of Table I, most of the U (1) X models available in literature may also contain additional BSM particles 5 .These additional particles are model dependent and we refrain from adding them to proceed with a minimal setup.Now some general model independent simplifications and conclusions can be immediately drawn for the charges of the particles under U (1) X symmetry listed in the aforementioned Table I.
1. Light Z ′ : In the upcoming sections, we delve into an effective resolution of the wellknown Hubble parameter tension [39] by increasing N eff considering a light Z ′ with mass M Z ′ ∼ O(MeV) mass range [6].To avoid any fine-tuning in keeping the M Z ′ very light compared to the SM Z gauge boson mass i.e.M 2 Z ′ ≪ M 2 Z , we consider the corresponding U (1) X charge of the SM Higgs doublet X Φ = 0, so that M 2 Z ′ does not receive any contribution from the SM vev.
2. Mass generation for quarks and charged leptons: For the choice of X Φ = 0, to generate the SM quark and charged lepton masses, we must take and X L i = X ℓ i such that the standard Yukawa term y u ij Qi Φu j involving only SM fields can be written in the canonical form as shown in eq.( 1).It is important to note that although within each generation, the U (1) X charges of quark doublet should be the same as that of the up and down quark singlets, the charges may differ when comparing across different generations i.e.
The same applies to charged leptons.In fact, in later sections, we will indeed consider flavour dependent U (1) X symmetries.To simplify our notation throughout the remaining part of this paper we will denote 3. Quark mixing: As a follow up point note that if the charges of all three generations of quarks are unequal i.e. if we take X and Y d and hence the resulting mass matrices will only have diagonal entries and we will not be able to generate CKM mixing.Thus, the charges of some (but not all) generations should match with each other in order to allow the generation of quark mixing.The same is true for charged lepton mass matrices.Again we will elaborate it further with specific examples in the coming sections.

BSM fermions:
The U (1) X charge of right handed neutrinos is typically fixed by anomaly cancellation conditions as we will discuss in a later section with specific examples.As mentioned earlier, for the sake of simplicity, we will not consider U (1) X symmetries involving chiral fermions beyond the fermion content of Table I.
Based on the aforementioned assumptions one can write the Yukawa and scalar potential for the general U (1) X model as: and It is worth highlighting a couple of salient features of the Yukawa terms and the scalar potential of this scenario.
1.In eq.( 1), L ν refers to the terms needed for light active neutrino mass generation.
These terms depend on the details of the U (1) X symmetry, the charges of leptons under it as well as the nature of neutrinos (Dirac or Majorana) and the mechanism involved for mass generation.Furthermore, this typically requires the presence of additional scalars or fermions or both, beyond the particle content listed in Tab.I.Note that even in the massless limit and absence of any other interactions, the ν R is still interacting with the rest of the particles through its U (1) X gauge interactions and depending on the charges and strength of the U (1) X gauge coupling (g X ), they can be in thermal equilibrium with the rest of the plasma at a given epoch in the evolution phase of the early universe.
2. Since we want a very light Z ′ , therefore the vev of σ field ⟨σ⟩ = v σ which is responsible for Z ′ mass generation should be small.Furthermore, after SSB, if we want the real physical scalar σ R to be light as well, we should have λ Φσ ≪ 1.The condition λ Φσ ≪ 1 will also imply that the 125 GeV scalar (h) is primarily composed of the real part of Φ 0 (a neutral component of Φ) and hence the LHC constraints on it can be trivially satisfied.
Throughout this work we also assumed that the tree level gauge kinetic mixing between U (1) Y and U (1) X symmetries is negligible.Hence for a specific U (1) X model, we have only two BSM free parameters, the gauge coupling (g X ) and mass (M Z ′ ) of the Z ′ gauge boson.With these general assumptions, one can examine the potential range of a Z ′ parameter that may account for the observed excess of N eff in the Planck 2018 data, leaving the charge of the leptons and the gauge coupling as free parameters.

ν L DECOUPLING IN PRESENCE OF LIGHT Z ′ AND N eff
It is a well known fact that one of the most precisely measured quantities from cosmology is the effective neutrino degrees of freedom (N eff ), which may get altered by the presence of light BSM particles.This change in N eff from the SM value can in principle provide one of the solutions to relax the Hubble tension [39].In this work, we focus on a scenario with light Z ′ interacting with SM neutrinos (ν i , (i ≡ e, µ, τ )) that may lead to change in N eff .After having a brief description of generic features of light Z ′ models emerging from U (1) X symmetries in the previous section, we now move to the most crucial part of our paper which is the cosmological implications of those scenarios.Before delving into the analysis of N eff in the presence of such light Z ′ , we would like to mention some key aspects of N eff within the SM framework.In a standard cosmological scenario at temperature T ∼ 20 MeV6 , only e ± , ν i are particles coupled to thermal (photon) bath as the energy densities of other heavier SM particles are already Boltzmann suppressed.As the universe cools, once the weak interactions involving e ± and ν i drop below the Hubble expansion rate H(T ), neutrinos decouple from the photon bath.Considering only the SM weak interactions, the neutrino decoupling7 temperature turns out to be 2 MeV [1,46].After that, there exist two separate baths of photon (+e ± ) and ν i , each with different temperatures; T γ and T ν respectively.Approximately, at a temperature below T γ ≲ 0.5 MeV, e ± annihilate, and the entropy is transferred entirely to photon bath leading to a increment in T γ compared to neutrino bath.This difference in temperature is parameterised in terms of N eff which is given by [1], At the time of CMB formation, the predicted value of N eff within the SM particle content is N CMB eff = 3.046 [2], whereas the recent Planck 2018 data [4] estimates it to be N CMB eff = 2.99 +0. 33 −0.34 at 95% confidence level (C.L.).In this scenario, the Z ′ arising from the aforementioned U (1) X models introduces new interactions with both ν i and e ± , which can potentially impact the neutrino decoupling consequently altering N CMB eff .As pointed out previously, the SM particles relevant for ν L decoupling are only e ± and ν i , whereas the energy densities of heavy SM particles are negligible due to Boltzmann suppression.The light quarks also do not take part due to the QCD confinement at a much higher temperature around ∼ 150 MeV.Hence, neutrino decoupling in presence of light Z ′ is independent of the generation of baryons (B number) gauged under the new U (1) symmetry.Following the same argument used above, in BSM U (1) X scenario Z ′ must be light enough (M Z ′ ≲ O(1) MeV) to affect ν L decoupling which will be shown in the later part of this section.There are two other BSM particles present in our model: the BSM scalar σ and the RHN ν R .As we elaborated in sec.2, the BSM scalar σ can be taken as heavy enough so that they are irrelevant for phenomenology at the MeV scale temperature.Hence, we integrate out σ to perceive the sole effect of light Z ′ in late time cosmology.In the Majorana type mass models, ν R are also too heavy to affect ν L decoupling [47,48] and we can neglect them also.However, in Dirac-type mass models, ν R are relativistic at MeV temperature and can significantly alter N eff [9].In this work, we only consider heavy Majorana RHN and ignore their contribution to temperature evaluation.In Fig. 1(a) we present the relevant mass scales by a schematic diagram.
So, in our proposed scenario, we have to trace the interactions between only three baths i.e. e, ν L and Z ′ to evaluate T ν or N eff (eq.( 3)).We describe the scenario using a cartoon diagram in Fig. 1(b).It is essential to take care of various energy transfers among these 3 particles as they will play a key role in computing temperature evolution equations (see Appendix A).The energy transfer rates are dictated by the various collision processes and the distribution functions of respective particles [5,49,50].Here we enlist the relevant processes to consider for the successful evaluation of N eff .
1. SM contributions: The SM weak interactions are active at temperature T ∼ MeV.
At this point, active neutrino annihilations (ν i νi ↔ e + e − ) as well as elastic scatterings (ν i e ± ↔ ν i e ± ) mediated by SM Z or W ± take place to maintain the required thermal equilibrium of the early universe.
2. BSM contributions to γ bath: For a light Z ′ with mass M Z ′ ∼ O(MeV) sufficient energy density can be pumped into the thermal bath via the decay and inverse decay between Z ′ and electrons (Z ′ ↔ e + e − ) at temperature around O(MeV).Additional contributions to the thermal bath may in principle come from scattering processes like Z ′ Z ′ ↔ e + e − , Z ′ γ ↔ e + e − .However, it turns out that for a very light Z ′ , with mass M Z ′ ∼ O MeV, its coupling g X with SM fermions is highly constrained from various experimental data, g X ≲ (10 −3 − 10 −5 ).For such a small coupling, the aforementioned decay process of Z ′ significantly dominates over the scattering processes.Thus we ignore the scattering contributions in our numerical calculations.
3. BSM contributions to ν bath: Similarly, Z ′ can transfer energy to ν i bath through decay and inverse decay (Z ′ ↔ ν i νi ) processes.Moreover, (ν i νi ↔ e + e − ) scattering process can play an important role through one loop coupling of Z ′ with electron.This one-loop coupling of Z ′ with electrons is responsible for connecting two separate thermal baths containing ν i and electrons.This feature can be seen in certain scenarios of the U (1) X models and we will elaborate on this issue in great detail in a later section.
4. Within ν bath: In this case, if we assume that different ν i flavours have different temperatures then ν i νi ↔ ν j νj , (i ̸ = j) mediated by both Z and Z ′ will have significant impact on the overall ν thermal bath.We will address this point in detail in the last part of this section.
To construct the temperature equations and compute the energy transfer rates we adopt the formalism already developed in ref. [5,6,51,52].It is worth highlighting the approximations made in the formalism prescribed in ref. [5] before the description of temperature equations.Firstly, Maxwell Boltzmann distributions were considered to characterize the phase space distribution of all particles in equilibrium, aiding in simplifying the collision term integral.The use of the Fermi Dirac distribution in the collision term does not alter the energy transfer rates substantially [16,53].Additionally, the electron's mass was ignored to simplify the collision terms as non-zero electron mass would have resulted in a minimal modification of the energy transfer rate, typically less than a few percent [54].It is to be noted that the ν L masses can be easily neglected as the relevant temperatures for neutrino decoupling is sufficiently higher than ν L -mass.
After successfully demonstrating all relevant processes and stating the assumptions, we are now set to construct the temperature evolution equations.The temperature equations are derived from the Liouville equation for phase space distribution of particles in a thermal bath (eq.(A1)) and the collision terms take care of the energy transfers among involved particles through the processes discussed before [55].Following the detailed calculations of the temperature evolution for the SM and BSM scenarios as displayed in appendices A and B, here we quote the final results of aforementioned temperature evolution equations [5,6]: where, ρ r , P r and T r signify the energy density, pressure density, and temperature of species r.Terms like δρ a→b δt indicate the energy transfer rate from bath a to b and is determined by integrating the collision terms (see eq.( A2)).The energy transfer rates are discussed in great detail in Appendix B8 .Here we assume all three ν i generations share the same temperature T ν L 9 and ρ ν L is the summation over the energy densities of three generations of ν i .
Note that, the equations in eq.(4-6) are dependent on the thermal history of heavy Z ′ when it remains in equilibrium with thermal bath in the early universe (T Z ′ ≳ M Z ′ ) through its interaction with fermions (f ).Z ′ preserves its thermal equilibrium via decay, inverse decay (Z ′ ↔ f f ) and also through scattering (f f ↔ Z ′ Z ′ (γ)) process.Thanks to processes like f f ↔ Z ′ Z ′ (γ) the chemical potentials (µ i (T )) are suppressed and Z ′ remain in chemical equilibrium with the SM bath.The condition for the thermal equilibrium of Z ′ incorporates a lower bound on g X (≳ 10 −9 ) and the lower bound may shift slightly depending on the specific choice of U (1) X charge and the value of M Z ′ .Alternatively, it is highly plausible that Z ′ was initially not in thermal equilibrium in the early universe, but produced from other SM particles via freeze in process.In these scenarios, it is necessary to solve the coupled equations for µ i (T ) to compute N eff [7].This alternate scenario requires a detailed complementary study which will be reported elsewhere.
As mentioned above we affix to the simplest BSM scenario assuming Z ′ in thermal equilibrium and set the initial condition for the set of equations eq.(4-6) as for the reasons already discussed before.In such case, one can further simplify the scenario assuming T Z ′ = T ν L as Z ′ remains coupled to ν bath for a longer time than with γ bath [6].This makes the term δρ Z ′ →ν L /δt = 0 and reduces the three equations in eq.( 4)-( 6) into two.However, the equations described in the above format are useful for generic scenarios.
So far, we have assumed that all three ν L share common temperature T ν as mentioned earlier.The approximation is valid as neutrino oscillations are active around MeV temperature leading to all three generations of neutrinos equilibriating with each other [46,[56][57][58].However one can also evaluate the temperatures assuming different temperatures T ν i , (i = e, µ, τ ) for all three generations, and the relevant equation reads as (see eq.(B27)), Note that earlier eq.( 4) differs from the one described in eq.( 7) as the later one also contains the energy transfer rate between different generations of ν i .However the value of N eff does not change significantly (≲ 10%) if one solves the temperature equation without considering different T ν i (see Appendix C) [6].For the same reason we stick to the simpler scenario assuming a common temperature of ν L bath and solve eq.( 4)-( 6) for numerical estimation of N eff throughout this paper. .For this plot, we treat the U (1) X charges of leptons (X i ) as free parameters and for some specific values of such charges, it will lead to the popular U (1) X models.
After a detailed discussion of the basic framework, we are now set to perform an exhaustive numerical analysis.In Fig. 2 we show case the evolution of temperature ratio (T γ /T ν ) as well as ∆N eff (≡ N CMB eff − 3.046) with T γ for different U (1) X charge combinations.Here we denote T ν ≡ T ν i with the assumption all three neutrinos share a common temperature.Following the discussion in the previous paragraphs, we stress that the relevant charges for ν L decoupling are X L i = X ℓ i ≡ X i or more precisely their modulus values as their squared value will enter in the collision terms (see Appendix B).In the aforementioned plot, we present the simplest scenario where all 3 ν L share the same temperature.We consider benchmark parameter (BP) values M Z ′ = 10 MeV with g X = 10 −7 (Fig. 2(a) & 2(c)) and g X = 10 −8 (Fig. 2(b) &2(d)).We will justify the importance of such a light Z ′ in this process at the end of this section.For such a light M Z ′ and g X = 10 −7 , we show the variation of T γ /T ν with T γ and ∆N eff with T γ in Fig. 2(a) and Fig. 2(c) respectively.From the above figure, it is very clear that around T γ ≳ 10MeV, T γ /T ν = 1 as both ν L and Z ′ was coupled to photon bath at that time.However, the ratio starts to increase after T γ ∼ 0.5 MeV and then saturates at low temperature (T γ ∼ 10 −2 MeV) as almost all the processes mentioned earlier gradually become inefficient at low T γ .In Fig. 2(c) we portray the corresponding variation in ∆N eff for all the U (1) X charge combinations in Fig. 2(a).As around T γ ∼ 10 −2 MeV the values of ∆N eff become saturated we can surmise that it will remain unchanged till recombination epoch (T γ ∼ 0.1 eV) and say ∆N eff (T γ ∼ 10 −2 MeV) ≡ ∆N CMB eff .In a similar way, for g X = 10 −8 also we display the evolution of T γ /T ν and ∆N eff in Fig. 2(b) and Fig. 2(d) respectively.In Fig. 2(c) and Fig. 2(d) we also showcase the 2σ exclusion limit ∆N CMB eff = 0.28 from Planck 2018 [4] and shown in black dotted line.Note that this limit is only valid at CMB and it excludes any values of ∆N eff at that time above the black dotted line.It is easy to infer that in the presence of the light Z ′ , the values differ from SM prediction.Before spelling out the physical implications of the U (1) X scenario we tabulate our findings from Fig. 2 for the ease of understanding in TableII.for different U (1) X charges inferred from Fig. 2. As mentioned earlier, here we treat the U (1) X charges of leptons (X i ) as free parameters and for some specific values of such charges it will lead to the popular U (1) X models.
Both from Fig. 2 and TableII it is evident that in our proposed U (1) X scenario the value of temperature ratio or N CMB eff differs from the values predicted by SM only.One can interpret this feature from the new processes involved in ν L decoupling apart from the SM weak interactions (see Fig. 1).Thus the light Z ′ acts as the bridge between photon and ν L bath and tries to balance their energy densities (through decays and scatterings) and hence reduces the temperature ratio from the value predicted by the SM only.From both Fig. 2(a) and Fig. 2(b) we notice that for a fixed value of g X and X L 1 ̸ = 0 the ratio T γ /T ν (at very late time) and ∆N CMB eff grow with an increase in |X 1 |.For X 1 ̸ = 0 the following BSM processes affect ν L decoupling: (i) Z ′ decaying to both e + e − and ν i νi and (ii) scattering process ν i νi → e + e − mediated by Z ′ .Thus with an increase in |X 1 |, the effective coupling (X 1 g X ) governing these BSM processes increases and hence boosts the BSM contribution (see eq.(B18) and eq.(B21)).As a result, picking higher values of X 1 leads to a higher interaction rate between ν bath and photon bath leading to an enhancement in ∆N CMB eff or, more precisely a diminution in T γ /T ν .
On the contrary when X 1 = 0 the only BSM process relevant for ν L decoupling is Z ′ ↔ ν µ,τ ν µ,τ as there is no tree level coupling of Z ′ with electrons.At T ν < M Z ′ eventually all Z ′ decay to ν L transferring all their energy density to ν bath only.As all the equilibrium number density of Z ′ finally gets diluted to ν L bath (with 100% branching ratio) it does not depend on the coupling strength (X 2/3 g X ).There is no change in N eff with the change in charge assignments (X 2 , X 3 ) for X 1 = 0.For the same reason described above, we infer that for X 1 ̸ = 0, N eff increases with an increase in g X whereas with X 1 = 0 it does not change at all with change in g X (comparing Fig. 2(c) and Fig. 2(d) ).Due to the fact that for X 1 = 0, Z ′ has only decay mode to ν µ,τ , T ν starts to increase before e ± decouples (T γ ∼ 0.5 MeV).This causes a slight dip in the T γ /T ν evolution line at higher temperature for X 1 = 0 in Fig. 2(a) and Fig. 2(b).
The approximate behaviour of N eff for the case with X 1 = 0 can also be understood by analytic estimates.As mentioned above, to compute the BSM effect in ν L decoupling, we have to consider only the Z ′ → ν ν decay.Using the co-moving energy conservation rule in two different epochs.The first epoch ("1") is just after ν L decoupling from SM bath and the second ("2") is at CMB formation when Z ′ has completely decayed to ν L .For these two epochs, one can write, where a 1 (T 1 ) and a 2 (T 2 ) signify the scale factors at the two epochs.We are using instantaneous ν L decoupling at T 1 ≈ 2 MeV which leads to slightly less value, similar to what happens the SM case as well (discrepancy of 0.046 [5]).At T 2 = 2 MeV ν L decouples from photon bath.However, for simplicity we assume Z ′ to follow its initial equilibrium distribution.Thus we can use their respective equilibrium energy densities 10 .Now in SM scenario 10 The equilibrium energy densities are given as (absence of Z ′ ) ρ SM ν L only red-shifts after decoupling and hence one can write, Combining eq.( 8) and eq.( 9) one can write Using this eq.(10) we can rewrite the earlier definition of N eff in eq.( 3) as [11], From the above eq.(11) on notices that the reason behind the enhanced N eff in U (1) X models is the additional Z ′ energy density at an earlier epoch.As hinted before, for X 1 = 0 scenario, only one decay mode of Z ′ exists and thus eventually all the Z ′ density gets transferred to ν L bath irrespective of g X X 2,3 .Also, it is easier to understand that for X 1 = 0 scenario M Z ′ plays the most crucial role to decide its energy density at the decoupling epoch and hence also N eff .Now for M Z ′ = 10 MeV, we get from eq.( 11), N eff = 3.28 in presence of Z ′ which is close to the one obtained numerically in Table II.Even though the Z ′ density is suppressed (exp(−M Z ′ /T 1 )) at T 1 = 2 MeV in the vicinity of ν L decoupling, it injects all the energy (via decays) to ν L sector enhancing N eff .Following the same argument, decreasing M Z ′ will lead to more contribution to N eff , as will be explored in Sec.4 11 .Before concluding the discussion about Fig. 2, we point out the fact that we compute N eff with a basic assumption that all ν L share same temperature and hence all ν L equilibrate with each other even if Z ′ decays (transfers energy) to any one of them.For this reason we observe similar behaviour of T γ /T ν for all U (1) X charge combinations with same value of X 1 and the same phenomenology elaborated in the earlier paragraph accounts for that.One should note that the U (1) X charge combination used in Fig. 2 is not an exhaustive one, yet one can easily deduce the outcome of other combinations from the reasoning we made above.For example all charge combinations with |X 1 | = 1 and |X 1 | = 0 will lead to N CMB eff = 3.38 and 3.34 respectively for g X = 10 −7 ,M Z ′ = 10 MeV.This is the most pivotal point of our analysis and we will revisit this in the next section.At this juncture, it is worth pointing out that the dependence of N CMB eff on g X for | X i |= 0, indicating the absence of tree-level Z ′ e + e − coupling.However, some effective coupling between Z ′ and e ± can be generated 11 However, it is worth highlighting that these analytic estimates are drawn based on two assumptions: (1) Z ′ decays to ν L only and not to e ± ; and (2) Z ′ is in equilibrium.Thus, these estimates do not work for the case when X 1 ̸ = 0.They also fail if the coupling (g X ≪ 10 −9 ) is very small and mass (M Z ′ ≪ 1) MeV where equilibrium formalism cannot be used [6,15].depending on the specific U (1) X model [6].In the subsequent section, we will extensively discuss such a scenario where the induced Z ′ e + e − coupling plays a crucial role in deciding N CMB eff .For the ease of our notation from now on whenever we say N eff , we refer to N CMB eff only.Having discussed the dependence of N eff on the two most important parameters of the BSM model: the Z ′ universal gauge coupling g X and U (1) X charge combinations, now we turn our attention to investigate its dependence on the light M Z ′ .In Fig. 3 we show variation of ∆N CMB eff 12 with M Z ′ for a fixed coupling g X = 10 −7 with different U (1) X charge combinations.Rather than showing all the U (1) X charge combinations used in Fig. 2, we just portray only three distinct combinations of them in Fig. 3 increases and eventually it becomes almost zero, reproducing the SM value of N eff when M Z ′ ≳ 30 MeV.This feature can be interpreted from our previous discussion in the context of Fig. 1.The energy density of heavier Z ′ gets Boltzmann suppressed at ν L decoupling temperature (T γ ∼ 2 MeV) making the BSM contribution less significant in deciding ν L decoupling.On the other hand Z ′ mediated scattering processes between ν and e, also propagator suppressed for higher M Z ′ .At very high M Z ′ (≳ 30 MeV), the BSM contribution hardly plays any role in deciding ν L decoupling resulting ∆N CMB eff = 0. Following the same argument we infer that a lower value of M Z ′ will enhance the BSM contribution and hence 12 To amplify the change in N eff with M Z ′ and portray more lucidly, for this particular plot we switch to ∆N CMB eff ≡ N eff − 3.046.The variation in N eff will immediately follow from it.
∆N CMB eff .For a fixed value of M Z ′ , the difference in ∆N CMB eff for different U (1) X charge combinations can be easily apprehended from the discussion in the context of Fig. 2. So far we explored the dependence of ∆N CMB eff on various model parameters, and the dependence of N eff also can be easily understood from that.Keeping in mind the key findings from U (1) X Z ′ models in the context of N eff , in the following section we will explore their contribution to alleviating the Hubble tension.

NUMERICAL RESULTS
In the previous section, we pinned down the key aspects of light Z ′ from generic U (1) X extension and showed that the U (1) X charge assignments play a key role in deciding N eff as well as its dependence on coupling g X .In this section, we will take a closer look at the model parameters and explore their cosmological implication through exhaustive numerical scans.Though one can have numerous U (1) X charge assignments as suggested in the Table I., in the context of N eff the arbitrary charge assignments can be categorised into only two classes when we assume all ν L share the same temperature (through oscillation).Needless to say, it is the coupling of Z ′ with electron (apart from ν i ) that affects the N eff and not the couplings with τ, µ or quarks, for the reasons already discussed earlier.Thus the light Z ′ models can be broadly classified into two pictures: X 1 ̸ = 0 and X 1 = 0, more precisely, whether Z ′ has coupling with electron or not (see Fig. 4).In our discussions on the light Z ′ phenomenology so far we have mainly focused on its tree level couplings with fermions.Yet, it is important to highlight that even in the absence of tree level Z ′ e + e − interaction (X 1 = 0), Z ′ can develop induced coupling13 with e ± [59].As a result, for | X 1 |= 0, while computing N eff , we need to consider the following effective Z ′ e + e − interaction Lagrangian: where ϵ is the induced effective coupling.If, for instance, the induced coupling is generated from the γ − Z ′ kinetic mixing at one loop level, its expression is given as (see Appendix A of ref. [59]).
where, ∆ ℓ = m 2 ℓ − x(1 − x)q 2 , Λ denotes an arbitrary mass scale and the summation includes all U (1) X charged fermions with mass m ℓ .This induced coupling will have a significant impact in scenarios where X 1 = 0, as we'll shortly discover.While we aim to maintain a model-independent discussion, the effective coupling described in eq.( 12) relies on the characteristics of particular U (1) X models.Therefore, to compare our findings with existing literature, we opt for the benchmark value of the effective coupling, setting ϵ = − g X 70 , which is very commonly used for L µ − L τ models (with X 1 = 0) [6,45].Henceforth, in all our numerical results, we will use this particular value of ϵ.(a) In Fig. 5, we aim to investigate the role of g X on ∆N CMB eff while maintaining a constant M Z ′ = 10 MeV and as stated earlier the variation of N eff also follows from it.This scrutiny involves two distinct scenarios for the Z ′ e + e − coupling: (a) solely with tree-level couplings (ϵ = 0) depicted in Fig. 5 that is excluded by the 2σ upper limit obtained from the Planck 2018 measurement [4].Note that the dependence of ∆N CMB eff (and hence N eff ) on g X for the cases with X 1 ̸ = 0 remains the same even after including mixing.This feature is easy to realize as the Z ′ e + e − induced coupling (ϵ) is suppressed by an order of magnitude compared to the corresponding Ze + e − tree level interaction.Hence, in the presence of the tree-level Z ′ e + e − interaction (X 1 ̸ = 0), one can easily ignore the contribution of Z ′ arising due to the induced coupling ϵ.For the reasons already discussed in sec.3, the BSM contribution increases as the values of g X rise, leading to an increase in ∆N CMB eff (also N eff ) also, as shown in the aforementioned figure.
However, comparing Fig. 5(a) and Fig. 5(b) one notices that the dependence ∆N CMB eff on g X for the case with X 1 = 0 changes drastically when the induced coupling (ϵ ̸ = 0) is present.In Fig. 5(a), we discern that ∆N CMB eff remains unchanged with variations in g X for X 1 = 0 in the absence of induced coupling (ϵ = 0).In this scenario, regardless of the associated coupling, the Z ′ decay is limited to ν L exclusively, transferring its entire energy density, as reasoned in sec.3 14 .The numerically obtained value of N eff is also close to the analytical estimates presented in the previous section.This observation upholds our earlier analysis discussed in the preceding section.
On the other hand, when |X 1 | = 0, in the presence of the induced coupling (ϵ ̸ = 0), Z ′ can couple to e ± .Thus for a nonzero ϵ the decay Z ′ → e + e − and ν i νi → e + e − scattering processes mediated by Z ′ continue during ν L decoupling temperature (T γ ∼ 1 MeV).Among these two processes, the scattering process tries to balance e and ν L bath by increasing T ν or increasing N eff .Therefore, as g X increases, the contribution of BSM scenarios also increases, resulting in an overall rise in ∆N CMB eff .This feature is reflected in Fig. 5(b) (red line) for g X ≳ 4 × 10 −8 .On the other extreme, for g X ≲ 4 × 10 −8 , the scattering process (∝ ϵ 2 g 2 X ) is unable to compete with the tree level decay Z ′ → ν L νL (∝ g 2 X ) process.Hence, for lower values of g X (≲ 4 × 10 −8 ), Z ′ promptly decays to ν bath transferring all its energy to ν sector and the scattering processes become inefficient to dilute this extra energy density to e bath.So, when g X (≲ 4 × 10 −8 ) we see a distinct rise in ∆N CMB eff for X 1 = 0 in Fig. 5(b).When g X (≲ 7 × 10 −8 ) is very small, the BSM contribution that affects the evolution of the energy density of ν L is primarily dominated by Z ′ decay processes.Irrespective of the coupling, at this level, the Z ′ particle transfers all its energy density to ν L .It is essential to note that at this point, the value of ∆N CMB eff for X 1 = 0 turns out to be identical for both scenarios with and without induced coupling when comparing Fig. 5(a) and Fig. 5(b).This outcome serves as validation for our previous argument.At this point, it is worth pointing out another aspect of this tree-level vs induced Z ′ coupling in generating non-zero contributions of BSM physics to ∆N CMB eff .When g X exceeds 10 −8 and X 1 = 0, the other two U (1) X charge combinations with non-zero | X 1 | result in the tree-level Z ′ e + e − interaction yielding greater contributions to ∆N CMB eff compared to the aforementioned induced Z ′ coupling.This distinction is particularly noticeable in Fig. 5(b).Now, we put forward the main thrust of this paper and split the analysis of light Z ′ realized in different U (1) X models in the context of N eff into two categories as shown in Fig. 4. The consequences of all other U (1) X charge combinations can be easily anticipated from the broad classifications.Following this line of thought we will now perform numerical scans to explore the imprints in N eff in the presence of light Z ′ emerging from two broad classes of generic U (1) X models in the following two subsections.where the H 0 tension can be relaxed as pointed out in ref. [30].
Here we consider the charge assignment |X 1,2,3 | = 1 and show the contours of constant N eff 15 in M Z ′ vs. g X plane in Fig. 6.In the same plot, we also showcase the 2σ bound from Planck 2018 data [4] shown by the grey dashed line.The parameter space to the left of the grey dashed line, as shown by the grey region, is excluded by Planck 2028 data at 2σ [4].
From the figure we observe that for a fixed g X , N eff decreases with an increase in M Z ′ as we explained in the context of Fig. 3.For very high M Z ′ the contribution to N eff becomes negligible.We consider the lowest value of g X = 10 −9 as below that, Z ′ fails to thermalize in the early universe.Following the discussion made earlier in this section, it is evident that increasing the value of |X 1 | will lead to a gradual shift of the contour for Planck 2018 upper limit (grey dashed line) towards right i.e. towards higher values of M Z ′ .Thus this bound applies to all U (1) X models with |X 1 | = 1 irrespective of other U (1) X charges.Note that for |X 1 | = n (n ̸ = 1) these bounds do not apply, rather the bounds for such models should be evaluated explicitly as shown in sec.5.Here we consider the following charge assignment |X 1 | = 0, |X 2,3 | = 1 and show the contours of constant N eff in M Z ′ vs. g X plane in Fig. 7. Similar to Fig. 6 we portray the contour lines for N eff = 3.2, N eff = 3.5 and 2σ upper limit from Planck 2018 [4] depicted by blue, red and grey dashed lines.The grey region to the left of the grey dashed line in each figure is excluded by the 2σ upper bound from Planck 2018 data [4].We show the numerical results for the case |X 1 | = 0 without induced coupling (ϵ = 0) in Fig. 7(a).The dependence of N eff on the model parameter g X is pretty straight forward as we noticed in earlier Fig. 5(a).For |X 1 | = 0, in the absence of induced coupling, N eff stays unchanged despite the variation in g X , resulting in distinct vertical lines of N eff contours in M Z ′ − g X plane in Fig. 7(a).Also, from the same figure we note a decrease in N eff with an increase in M Z ′ for a fixed g X , as elaborated earlier.Note that, in the absence of induced coupling (ϵ = 0)the contour for Planck 2018 upper limit (grey dashed line) will not change with increasing X 2,3 for such kind (X 1 = 0) of U (1) X models as argued before in the context of Fig. 5(a).Following the discussions in context of eq.( 11) we can analytically estimate that around M Z ′ = 8 and M Z ′ = 12 MeV the predicted values will be N eff = 3.48 and N eff = 3.17 respectively, which are very close to the numbers obtained numerically in Fig. 7(a) and are independent of g X .This justifies our argument that for X 1 = 0, N eff is only dependent on M Z ′ and not on g X for our chosen parameter space 16 .
However, the results are quite different after including the induced coupling (ϵ ̸ = 0) of Z ′ with electrons as shown in Fig. 7(b).Due to the induced coupling of Z ′ with electrons, the N eff contours replicate the feature of the |X 1 | = 1 case (as shown in Fig. 6) for higher values of g X (≳ 10 −8 ) with a knee like pattern around g X ∼ 10 −8 .Such non-trivial dependence is the consequence of the interplay between tree-level decay ( Z ′ → ν L νL ) and the induced Z ′ mediated scattering as explained earlier in detail.We notice the bend in the contour lines in Fig. 7(b) since the collision term accounting for the BSM contribution in computing N eff is decay dominated in lower coupling region (g X ≲ 4 × 10 −8 ).It is worth mentioning that for ϵ ̸ = 0 the contour for Planck 2018 upper limit (grey dashed line) will shift towards the right with increasing X 2,3 for g X ≳ 4 × 10 −8 and will remain unchanged for g X ≲ 4 × 10 −8 .Again note that for all the models with |X 1 | = 0 and |X 2 |, |X 3 | ̸ = 0 this bounds apply when one considers ϵ = 0.However, for ϵ ̸ = 0 the bounds for such models with different |X 2 |, |X 3 | should also be evaluated explicitly as shown in sec.5.
Thus from this section, we propound that the cosmological imprints of light Z ′ models due to different charge assignments leading to different U (1) X models can be put under the same roof following our prescription.However, in the model independent analysis, we ignored the experimental constraints which are inevitably relevant for our parameter space.For completeness, we will show the numerical results for some specific U (1) X models along with experimental constraints in the next section.
We conclude this section with a comment on futuristic limits on N eff by several proposed experiments.We list a few such experiments and their expected limits on N eff in Table III.There also exists BBN bounds on N eff which rule out N eff > 3.16 at 1σ [60].However, applying BBN constraints requires dedicated analysis which is beyond the scope of this work.For our chosen parameter space the Z ′ remains nonrelativistic at the onset of BBN and we show only CMB constraints.

CMB-HD [63]
N eff < 3.07 Let's now look at the type of U (1) X symmetry where the symmetry is flavour dependent in both quarks as well as the lepton sector namely the U (1) B 3 −3L j gauge symmetries [22].In this case, we will consider the following three gauged U (1) symmetries: U (1) B 3 −3Le , U (1) B 3 −3Lµ and U (1) B 3 −3Lτ .The charges of three of such symmetries are listed in Table IV.The charges of the other U (1) B 3 −3L j symmetries are analogous and can be similarly written without difficulty.
3 ) (0, 0, 1) (0, 0, 1) (0, 0, 1) 3 ) (0, 0, 1) (0, 0, 1) (0, 0, 1) For all three cases, the charges of ν R i can be fixed by the anomaly cancellation conditions.A convenient anomaly free charge assignment for ν R i is ν R i = −3, ν R j = 0; j ̸ = i for the U (1) B i −3L j symmetry e.g. for say U (1) B 3 −3L 2 gauge symmetry the charges can be ν R ∼ (0 − 3, 0) [22].The charges of the σ field depend on the details of the model and we will not go into details of the model building unless needed 17 .And as we mentioned earlier in sec.3,we assume this ν R and σ to be heavy enough that they are irrelevant for the analysis of N eff .As elaborated in the previous sections, the value of N eff in the presence of a light Z ′ depends only on its leptonic couplings (L number) and not on the couplings with quarks.Thus U (1) B 2 −3L 1 and U (1) B 3 −3L 1 models will exhibit similar imprints on N eff .However, the other experimental constraints indeed depend on the Z ′ quark couplings and change significantly with the B number.
Before exploring the phenomenology of the specific U (1) X , we would like to outline an overview of the exclusion bounds on the mass of the light-gauge boson (M Z ′ ) and corresponding gauge coupling (g X ) within the mass range M Z ′ ∼ O(10) MeV, which is our point of interest.Here, we will briefly review various types of low-energy experimental observations that can be used to constrain the scenarios involving any U (1) X .In the following subsections, we will present the exclusion bound identified by each low-energy experiment for a specific U (1) B 3 −3L i scenario (i = e, µ, τ ), along with our cosmological findings.

Elastic electron-neutrino scattering (EνES):
The elastic scatterings of neutrinos with electrons (ν α e → ν α e, α = e, µ, τ ) in laboratory experiments serve as one of the probes for non-standard interaction of neutrino and electron with the light-gauged boson (Z ′ ).In SM, the e − ν e scattering involves both charged-current (CC) and neutral-current (NC) weak interactions, whereas e − ν µ,τ scattering is solely governed CC interaction [28].The elastic e − ν e,µ,τ scattering can be altered in the presence of an additional NC interaction mediated by the U (1) X gauge boson Z ′ , which can be probed in the low-energy scattering experiments.Note that the interference terms in the matrix amplitude between the SM (W and Z-mediated ) and BSM (Z ′ -mediated), play a critical role in altering the scattering rate.The interference term in differential cross section is given by [59], where E R and E ν denote electron recoil energy and incoming neutrino energy respectively.g L,R is defined in appendix A. For the cases where X 1 = 0 the term (g X X 1 ) will be replaced by ϵe in eq.( 14).Thus observing the electron recoil rate imposes constraints on the M Z ′ − g X plane for a model-specific scenario.The coupling strength of the light gauge boson with leptons varies over gauge extensions.As a result, the constraints will vary from model to model.Borexino [19] and dark matter experiments like XENON 1T [64] dedicated to measuring the electron recoil rate, can be relevant in the context of e − ν α elastic scattering [20].The Borexino experiment is designed to study the low-energy solar neutrinos (ν e ) 18 produced via decay of 7 Be by observing the electron recoil rate through the neutrino-electron elastic scattering process [20].These solar neutrinos undergo flavour change as they travel from the sun to the detector.This flavor changing phenomena ν e → ν α (α = e, µ, τ ) can be accounted using the transition probability P eα [65] .Therefore the number of events for solar neutrinos (ν e ) interacting with the electrons in the Borexino will be N νe ∝ α=e,µ,τ P eα σ e−να [28,65].Note that in Solar neutrino flux P ee (∼ 50%) > P eµ = P eτ and the dominant contribution in interference term is due to ν e as it has both CC and NC interaction with electron [28].
Hence for X 1 ̸ = 0 one can approximate the differential recoil rate [28], For this reason the event rate for EνES in the presence of Z ′ depends only on the Z ′ e + e − and Z ′ ν e νe coupling (X 1 g X ).Thus the constraint on the parameter space of Z ′ in B 3 − 3L e model will be analogous to B − L model [20], except an overall scaling of g X due to the charge X 1 = 3 in the former one.Note that when X 1 = 0 i.e. in the absence of tree-level coupling of Z ′ with electron one has to consider the elastic scattering (via loop induced coupling with Z ′ ) of an electron with ν µ/τ with specific transition probabilities.Thus the EνES constraints for B 3 −3L µ model can be drawn from L µ −L τ model [20], with the overall scaling as mentioned before.
Coherent elastic neutrino-nucleus scattering (CEνNS): The COHERENT experiment investigates coherent elastic scattering between neutrinos and nucleus in CsI material (ν N → ν N ) [29,45,66].This mode of interaction opens up new opportunities for studying neutrino interactions, including the introduction of a new light gauge boson in this case.In SM, the elastic neutrino-nucleus scattering (ν − N ) takes place via the NC interactions between neutrinos (ν e,µ,τ ) and nucleons or more precisely with the first generation of quarks (q = {u, d}).The additional NC interaction introduced by the light gauge boson Z ′ of U (1) X can contribute to the CEνNS process.The modification to SM CEνNS, due to the Z ′ can be utilized to constrain the parameter space of M Z ′ vs g X plane.Similar to EνES, the exclusion limit is also dependent on the specific gauged scenarios.This is because the coupling strength of quarks and neutrinos with the light-gauged boson is influenced by the specific gauge choice.However, in contrast to EνES experiment, CEνNS is lepton flavor independent and one has to consider interaction with all 3 ν.For a detailed discussion see ref. [20].For the B 3 − 3L j model there is no tree level coupling of Z ′ with u, d, and hence the coupling with the nucleus will be an induced coupling (as discussed later).The CEνNS constraints for B 3 − 3L e and B 3 − 3L µ model can be deduced from B − L and L µ − L τ model [20] respectively, with an overall scaling to take into account the induced coupling of Z ′ with first generation quarks.
Supernova 1987A (SN1987A): Non-standard neutrino interactions with a light gauge boson can also affect the cooling of core-collapse supernovae (SN) which is a powerful source of neutrinos (ν e ) [67,68].The non-standard neutrino interaction gives rise to the initial production of light-gauged bosons in the core of a supernova (SN), ν α ν α → Z ′ , resulting in energy loss in the SN core [6].After production the late decay of these Z ′ into neutrinos (Z ′ → ν i ν i ) has the potential to modify the neutrino flux emitted by the SN [68].For a detailed study on supernova constraints on such light mediators see ref. [69][70][71].As there is no nucleon coupling of Z ′ in B 3 − 3L j model, for simple order estimation the supernova cooling bound can be adopted from L µ − L τ model [6].However, there can be additional processes relevant for SN 1987A cooling in the presence of light Z ′ models [72] and explicit derivation of that is beyond the scope of this work.
Neutrino trident: The neutrino trident production mode, in which neutrinos interact with a target nucleus to produce a pair of charged leptons without changing the neutrino flavor (ν N → ν N µ + µ − ), is a powerful tool for probing new physics [73].The gauge extension introduces new interactions between leptons and light gauge bosons, increasing the rate of neutrino trident production, as predicted by the SM.Therefore any gauge extended scenarios associated with muon (X 2 ̸ = 0) face constraints from existing neutrino trident production experimental results [22].
After having a generic discussion on the constraints relevant to our analysis, in the following two subsections we present our numerical results for specific U (1) X gauge choice.We portray the bound from N eff on these models along with the constraints discussed above.In the same plane, we also indicate the parameter space that can relax the H 0 tension.

Gauged U (1) B 3 −3Le symmetry
In this subsection we show the values of N eff for light Z ′ in U (1) B 3 −3Le gauge extension.We show our numerical results in M Z ′ vs. g X plane in Fig. 8.In the same plane, we also portray other relevant astrophysical and experimental constraints.The grey dashed line in the same plot signify the 2σ upper limit on N eff from Planck 2018 data [4].It excludes the parameter space to the left of that line as shown by the grey shaded region.One can easily infer from the figure that for a fixed g X with increasing M Z ′ the value of N eff decreases which is explained in detail in the previous section.Note that for U (1) B 3 −3Le , Z ′ has tree level coupling with e ± (X 1 = −3) and hence it belongs to the group of U (1) X models with |X 1 | ̸ = 0 following our discussion in previous section 4.1.For this, we notice that the behavior of N eff with g X is similar to what we observed in the context of Fig. 6.However, for U (1) B 3 −3Le model |X 1 | = 3 and hence for fixed g X and M Z ′ the value of N eff will be higher compared to the case considered in Fig. 6 for the reason already elaborated before.As a result we notice the N eff lines in Fig. 8 has moved rightwards compared to Fig. 6.
In the same plane in Fig. 8 we showcase the relevant phenomenological constraints that we mentioned earlier.Since Z ′ has tree level coupling with electron in U (1) B 3 −3Le model, it attracts strong constraint from EνES [20].The exclusion region from EνES with XENON1T experiment is shown by the region shaded by dark cyan diagonal lines.Similarly, Borexino also constrains the parameter space shown by the magenta shaded region [20].In U (1) B 3 −3Le model the light Z ′ has no tree-level coupling with the first two generations of quarks (u, d), (c, s) which is important for CEνNS.However, Z ′ can have induced coupling with first-generation quarks via CKM mixing [22] and can contribute to CEνNS.In SM the FIG.8: Constraints from cosmology and low energy experiments in M Z ′ vs. g X plane for a generic U (1) B 3 −3Le gauge extension.The 2σ upper bound from Planck 2018 [4] is shown by the grey dashed line and it excludes the parameter space to the left side of that line as shown by the grey shaded region.The region between the N eff = 3.2 (blue dashed) and N eff = 3.5 (red dashed) lines helps to relax the H 0 tension, as mentioned in the ref.[30].
CKM matrix V CKM is constructed from charge current interaction betweeen quarks [74], where, q u ≡ (u c t) † and q d ≡ (d s b) † .U L and D L are rotation matrices that diagonalize mass matrices for q u and q d respectively.The CKM matrix can be constructed as . For our estimation we take U L = I and hence D L = V † CKM [22].In our scenario as Z ′ couples to only third generation of quarks (with L int ⊃ g X J µ Z ′ Z ′ µ ) it may induce flavor changing neutral current (FCNC) and the Noether current (J µ Z ′ ) is given by, After inserting the values of the elements of D L matrix [75], in eq.( 17) the Z ′ effective coupling with d quark takes the following form: Using this we translate the bound from CEνNS experiment in our scenario and obtain that it excludes the parameter space for g X ≳ 10 −2 .This limit is weaker than EνES and hence not shown in Fig. 8.As Z ′ has coupling with b quark it generates B s − B s mixing and hence it attracts constraint indicated by orange shaded region [22].Finally, the bound from supernova cooling is shown by the region shaded with grey vertical lines [6].where the H 0 tension can be relaxed as mentioned in ref. [30].
In this subsection, we present our analysis with N eff for a light Z ′ in the gauged extension of U (1) B 3 −3Lµ .Analogous to the previous subsection we show our numerical results in M Z ′ vs. g X plane in Fig. 9 along with all relevant astrophysical and experimental constraints.The grey dashed line in the same plot signify the 2σ upper limit on N eff from Planck 2018 data [4] and likewise before.The grey shaded region depicts the exclusion of the parameter space to the left of the grey dashed line.
However, unlike previous scenario, for U (1) B 3 −3Lµ , Z ′ has no tree level coupling with e ± (X 1 = 0) and hence it belongs to other class of U (1) X models with |X 1 | = 0 as discussed in previous sub-section 4.2.We present our numerical results with the two values of effective coupling, ϵ = 0 and ϵ = −g X /70 in Fig. 9(a) and Fig. 9(b) respectively.For ϵ = 0 we observe no change in N eff with g X in Fig. 9(a) as argued in the previous sections in the context of Fig. 7(a).On the other hand, for ϵ = −g X /70 we observe the non-trivial dependence of N eff with g X in Fig. 9(a) due to the interplay between scattering and decays as elaborated in detail in the context of Fig. 7(b).
In the same plane for both the plots Fig. 9(a) and Fig. 9(b) we portray the constraints arising from EνES experiment with XENON1T [20] shown by the regions shaded by dark cyan diagonal lines.Similarly, we showcase the exclusion region from Borexino depicted by magenta shaded region [20].Note that these bounds are weaker for U (1) B 3 −3Lµ compared to Fig. 8 as in this case Z ′ has no tree level coupling with e ± and the recoil rate gets smaller here (see eq.( 14)).Because, in this scenario Z ′ couples directly with µ, the Neutrino trident experiment significantly constrains the parameter space shown by the dark blue dashed dot line [22].The parameter space above that line is excluded.In this model also Z ′ can generate B s − B s mixing and attracts constraint depicted by orange shaded region [22].Similar to the previous subsection the bound from CEνNS experiment is weaker than EνES and hence not shown in Fig. 9.The bound from supernova cooling is shown by the region shaded with grey vertical lines [6].We do not show the results for U (1) B 3 −3Lτ explicitly, though the imprint in N eff for this model will be similar like the one in U (1) B 3 −3Lτ as both these models have For both of the aforementioned models, one can observe that for M Z ′ ≲ 1 MeV the parameter space is almost ruled out from astrophysical and laboratory searches.However, for M Z ′ ≳ 10 MeV parameter space remains unconstrained.Thus the MeV scale Z ′ with M Z ′ ≳ 1 MeV should be explored through alternate search strategies which is the main goal of this work.Since the ν L decoupling generally happens at T ∼ MeV, the bound from N eff lies in the mass region M Z ′ ∼ MeV.

THE NEUTRINO PORTAL SOLUTION TO THE HUBBLE TENSION
The change in N eff due to the presence of light Z ′ gauge bosons can also lead to an interesting consequence namely a simple and elegant resolution to the cosmological Hubble constant problem.Recall that the Hubble constant problem is the discrepancy between the obtained value of the Hubble constant using early and late time probes [30,33].As mentioned in the introduction the discrepancy in H 0 value can be ameliorated with increasing N eff [30].Several different resolutions have been proposed to address this discrepancy.One of the simplest possible resolution is the presence of light degrees of freedom changing the N eff [6,9,17,18].There are already a few model specific works successfully addressing this problem for particular U (1) X extensions such as U (1) µ−τ [6] and U (1) B−L [51].
The general formalism developed in this work also can be used to show that the U (1) X symmetries with light Z ′ gauge bosons, in general, can address the Hubble constant problem reducing the tension.According to the Planck 2015 TT data [30,39], the H 0 tension issue can be relaxed upto 1.8σ with the N eff value between 3.2 to 3.5.However, the Planck 2018 polarization measurements provide a more stringent bound on N eff [4] and it is difficult to reach H 0 > 70 Km s −1 Mpc −1 [42] within the ΛCDM.Hence there still exists a disagreement of 3.6σ in the H 0 value predicted from CMB and local measurement.In our discussion, we only highlight the parameter space where N eff lies between 3.2 to 3.5 in M Z ′ − g X plane, that may relax H 0 tension followed by the analysis in ref. [30,39].In sec.4 we portray the parameter space for generic U (1) X models in Fig. 6 (for |X 1 | ̸ = 0)and Fig. 7 (|X 1 | = 0) where the blue and red dashed lines indicate the values of N eff = 3.2 and 3.5 respectively.Similarly, in sec.5 we also showcase the same lines with constant N eff = 3.2 (blue dashed line) and 3.5 (red dashed line) contours for the specific models: U B 3 −3Le (in Fig. 8) and U B 3 −3Lµ (in Fig. 9).The region between these two contour lines may ameliorate H 0 tension.However, the quoted values of N eff can be excluded by future experiment CMB-S4 [62].Also, recent studies show that even with increasing N eff can not resolve the H 0 tension completely [42,76,77].Hence, claiming any possible resolution of H 0 problem with quantitative effectiveness, itself requires a dedicated analysis which is beyond the scope of this work [30].

SUMMARY AND CONCLUSION
The Planck experiment has a very accurate measurement of the CMB which has established stringent limits on ∆N eff , representing the number of effective relativistic degrees of freedom during the early epoch of the universe [4].This sensitivity makes ∆N eff a useful probe for investigating various BSM scenarios that affect the neutrino decoupling at the time of the CMB.In this work, we have studied the impact of light Z ′ particle, arising from generic U (1) X model, on N eff .In the presence of this light Z ′ , N eff receives two-fold contributions: the decay of Z ′ → ν L νL , e + e − and scattering of light SM leptons (ν L , e) via this Z ′ .At first, we considered a generic U (1) X model with arbitrary charge assignments.Apart from the light Z ′ gauge boson, the model also contains a BSM scalar σ and RHN ν R which are necessary for the model construction.To understand the sole effect of the light Z ′ in early universe temperature evolution, we assume the other BSM particles are sufficiently heavy (≳ 100 MeV) that they decouple at the time of neutrino decoupling.In this work, we only consider the scenario where Z ′ was initially (T ≳ M Z ′ ) in a thermal bath.We adopt the formalism developed in ref. [6] and solve the temperature equations to evaluate N eff in sec.3.We enumerate our findings below.
• Firstly, we noticed that the U (1) X charges for e, ν e,µ,τ play a crucial role in N eff for our proposed generic U (1) X model in sec.2 as these are the only SM particles relevant for ν decoupling.In a similar vein, we found that the mass of the BSM gauge boson Z ′ should be around ≲ 30 MeV to affect neutrino decoupling.
• After a careful analysis in sec.3 we observed that the value of N eff depends differently on the coupling g X for two distinct scenarios those are |X 1 | = 0 and |X 1 | ̸ = 0. Based on this fact, in sec.4 we categorise U (1) X models into two classes: (a) electrons having tree level coupling with Z ′ (|X 1 | = 0) and (b) without tree level coupling of an electron with Z ′ (|X 1 | ̸ = 0).However, in the second scenario electron may have some induced coupling with Z ′ even if |X 1 | = 0 and we explored that scenario too.For both the two classes we present the contours from the upper limit on N eff from Planck 2018 [4] in M Z ′ vs. g X plane as shown in Fig. 6 and Fig. 7.In the same plane, we also highlight the parameter space favoured to relax H 0 tension shown in earlier analyses [6,39].
• For comparison with existing constraints from ground based experiments, in sec.5 we considered specific U (1) X models as examples and discuss the cosmological implications in the context of N eff .We present the numerical results for U (1) B 3 −3Le (|X 1 | = 3) and U (1) B 3 −3Lµ (|X 1 | = 0) models in sub-sec.5.1 and sub-sec.5.2 respectively which have not been explored in the existing literature.Depending on the coupling of an electron with Z ′ these two models also lead to distinguishable N eff contours in M Z ′ vs. g X plane as portrayed in Fig. 8 and Fig. 9. Our detailed analysis shows that the bounds from N eff on the M Z ′ vs. g X plane for some of these models are stronger than B − L or L µ − L τ model due to higher U (1) X charge of an electron.A priori these bounds can not be scaled from the existing N eff bound on other models with different U (1) X lepton charges from the existing literature (B − L or L µ − L τ ).
• For both U (1) B 3 −3Le and U (1) B 3 −3Lµ models we also showcase the relevant astrophysical and laboratory constraints in Fig. 8 and Fig. 9 respectively.We displayed the constraints from EνES with XENON1T [20] and Borexino [65], neutrino trident [22], B s − B s mixing [22] as well as from SN1987A [6].We checked the constraints from CEνNS [20] is weaker than the other constraints as our chosen U (1) X models do not have Z ′ coupling with first generation quarks.For U (1) B 3 −3Lµ model electron has no BSM gauge coupling and hence, the bounds from EνES for this model are weaker than the other one.In the same plane, we also indicate the parameter space that can relax the H 0 tension [6,39].
For both U (1) B 3 −3Le and U (1) B 3 −3Lµ models we have shown that the bounds on N eff from Planck 2018 data [4] can provide more stringent bound on the parameter space than the laboratory searches for M Z ′ ≲ O(1) MeV.The future generation experiments like CMB-S4 ( ∆N eff = 0.06 at 2σ) [62] can even probe more parameter space for such models.We also show that there is a certain parameter space still left to relax H 0 tension [6,39] allowed from all kinds of constraints.On the other hand, as U (1) X models are well motivated BSM scenarios and widely studied in several aspects, this analysis may enhance insights to explore their connection with the H 0 problem too.Besides the explicit results of B 3 − 3L i , our discussion for the generic U (1) X models shows how the the L number (X i ) plays a crucial role in deciding the constraint on the parameter space from N eff .From the classification of generic U (1) X models depending on the L number, one can anticipate the consequences in N eff for various other exotic U (1) X models that have not been explored so far [28,43].Thus our generalized prescription for N eff is extremely helpful to put stringent constraints on the light Z ′ parameter space from cosmology complementary to the bounds obtained from ground based experiments.is the energy density transfer rate from χ → j(or k).P χ is the pressure density.So, we can rewrite energy density equation eq.(A3) for χ, Now for SM ν L decoupling we consider the energy transfers between ν i (i = e, µ, τ ) to e ± , γ photon bath.For photon bath containing γ, e ± we consider a common temperature T γ and add their energy density equations and rearrange accordingly, where, in the last step we have used Similarly, following eq.(A3)we can write the respective temperature (T ν i ) equations for each type of ν, Once we have the temperature equations the remaining step is to evaluate the energy transfer rates i.e. the collision terms.The relevant processes in the SM scenario are given in Table V.All the processes are either elastic scatterings between ν i and e or annihilations and all are mediated by W ± , Z.As we are focusing on the MeV scale we can integrate out the mediators and write in terms of Fermi constant G F .In Table V, p 1,2 and p 3,4 denote the momentum of incoming and outgoing particles.Here, W and s W = sin θ W where θ W is Weinberg angle.The reason behind the difference of 1 between g Le and g Rµ, τ is the additional charged current process available for ν e .
We have not written the corresponding interactions with quark sectors since around ∼ 2 MeV quarks are already confined [78].The µ, τ particles also do not take part as they have suppressed energy density around MeV temperature.In the presence of these Z ′ the following things will affect neutrino decoupling (see Fig. 1).
1. Depending on charge assignments (X 1 ̸ = 0) Z ′ can decay to e + e − and the inverse decay can also happen upto T γ = M Z ′ .So there will be energy transfer from Z ′ to e ± sector(γ bath) and vice versa.Similarly, Z ′ can transfer energy to ν bath with the decays and inverse decays to ν i νi .
We will discuss the above points in the following subsections.As discussed in Sec.3, in this work we will only focus on the scenarios where Z ′ was in thermal equilibrium with e, ν i at T > T ν dec .Throughout this section we will simplify the collision terms in the effective operator limit assuming √ s ∼ T ≪ M Z ′ , for the purpose of ν L decoupling calculations.where, k, p 1 , p 2 denote the four momenta of respective particles.Following eq.(A2) and eq.(A3) we calculate the energy transfer rate from Z ′ to e bath, Here the degrees of freedom and energy of the corresponding particle are denoted as g ℓ &E ℓ , (ℓ ≡ Z ′ , 1, 2) respectively.f eq i , (i ≡ Z ′ , e) signify the equilibrium distribution function of i particle The decay width of Z ′ to e + e − is given by, where m e is the electron mass.
Plugging this in eq.(A2) we will get the total energy transfer rate (ν ↔ e) Three sources contribute to the total energy transfer rate; pure SM, SM-BSM interference, and pure BSM respectively.For simplicity, we calculate the energy transfer rates for each of them separately.The amplitudes for pure SM contributions can be found in sec.A.And the energy transfer rates for pure SM contributions are given in eq.(A8-A10) [5].
The amplitude for BSM processes (with proper momentum assignment) will be (s ≪ M 2 Z ′ ) The interference amplitudes in eq.(B14) are tabulated in Tab.-VII.Following the formalism developed in ref. [52] we write the collision terms for the interference terms: The first one in the above two collision terms accounts for elastic scattering of ν i with both e ± .Here we have multiplied with an additional factor 2 to count the effect of νi .The second collision term in eq.(B17) is due to annihilations.So, the total energy transfer rate from ν to e bath due to the interference with U (1) X gauge boson, In a similar fashion, we now move to calculate the collision term due to pure BSM amplitudes (third term in eq.(B14)).The pure BSM amplitudes in eq.(B14) are tabulated in Tab.-VIII.

Temperature evolution
Now with all the collision terms, we are set to formulate the energy evaluation equations.Following the prescription in appendix-A we can write the following energy density evaluation equations: where the i indicates the summation of energy density transfer rates over all 3 generatation of ν (e, µ, τ ).And δρν i →ν j δt tot bears a similar kind of form as in the SM case.However, in the presence of the light Z ′ , it will get some additional contribution from Z ′ mediation.So the G 2 F in eq.(A10) will be simply replaced by G F → (G F + (X i g X )(X j g X )/M 2 Z ′ ).Similar to the previous section we can write the temperature evaluation equations using the partial derivatives and we get,

(B29)
Hence solving the aforementioned five Boltzmann equations on can track the evolution of ν L decoupling.By exploiting the neutrino oscillations which are active around ∼ MeV temperature [46,[56][57][58], one can further simplify the scenario by assuming all 3 ν i equilibrate with each other and acquire a common temperature i.e.T νe = T νµ = T ντ ≡ T ν .The benefit of such an assumption is that we no longer have to keep track of the energy transfer rates within different ν sectors i.e.Throughout the paper we evaluated N eff assuming all three ν i , (i ≡ e, µ, τ ) share the same temperature.However, the values of N eff change if we allow all ν i to develop different temperatures as prescribed in context of eq.(B27-B29).We show our results for g X = 10 −8 , M Z ′ = 10 MeV for three different U (1) X charge combinations in Fig. 10 assuming only tree level couplings (i.e.ϵ = 0).only ν e has BSM interactions which lead to enhancing T νe , whereas T ν µ/τ reproduces the same value as predicted in SM scenario (where, (T γ /T ν i ) SM ∼ 1.4 [5]) as portrayed in Fig. 10(a).It is interesting to note that this value of N eff = 3.09 is less (∼ 8%) than the one predicted for the same charge configuration in sec.3.This is due to the fact we overestimated the neutrino energy increment in presence of Z ′ while assuming all 3 ν i share the same temperature.
For Fig. 10(b) and Fig. 10(c) also we notice the calculated values of N eff is slightly less than what was predicted in sec.3.For |X 1 | = 0, |X 2,3 | = 1, Z ′ promptly decays to ν µ and ν τ and not in electron.This causes a sudden dip (before T γ ≈ 0.5 MeV ) in the respective temperature ratio curves shown by the blue and green line in Fig. 10(b).For the same reason we notice similar dip in the temperature ratio curve (green line) for ν τ in Fig. 10(c).The value of N eff is slightly higher for |X 1 | = 0, |X 2,3 | = 1 in Fig. 10(b) compared to the one obtained for Fig. 10(c) as the former one contains two type of ν i having BSM interactions.

FIG. 1 :
FIG. 1: Cartoon diagram of (a) particle spectrum and (b) interactions between three sectors.The vertical axis in (a) denotes the mass scale (or, temperature) and the dotted line indicates the temperature where BBN started.The blue lines in (a) signify the interactions between particles that affect N eff .Here ν in the figure signifies SM neutrinos (ν i ) only.

FIG. 2 :
FIG.2: Evolution of T γ /T ν with photon bath temperature T γ assuming all 3 ν L share same temperature for different U (1) X charge combinations.We chose a benchmark parameter value M Z ′ = 10 MeV for both the plots.The couplings are taken g X = 10 −7 for (a) and g X = 10 −8 for (b).The lines corresponding to different charge combinations are indicated by the plot legends in the figure.The legends named N eff in the plots refer to N CMB eff

FIG. 3 :
FIG. 3: Variation of ∆N CMB effwith M Z ′ assuming all three ν L share same temperature for different U (1) X charge combinations.We choose a fixed value of g X = 10 −7 .

FIG. 5 :
FIG. 5: Variation of ∆N CMB eff with g X in (a) with ϵ = 0 and in (b) with ϵ = − g X 70 assuming all 3 ν L share same temperature for different U (1) X charge combinations.We choose a fixed value of M Z ′ = 10 MeV for both (a) and (b).

4. 1 .FIG. 6 :
FIG.6:The parameter space for N eff is depicted in the M Z ′ vs. g X plane for a generic U (1) X gauge extension with the charge assignment |X 1,2,3 | = 1.The 2σ upper bound from Planck 2018[4] is depicted by the grey dashed line, excluding the parameter space to the left of that line shown by the grey region.The blue and red dashed lines indicate the values of N CMB eff = 3.2 and 3.5, respectively, where the H 0 tension can be relaxed as pointed out in ref.[30].

4. 2 .X 1 FIG. 7 :
FIG.7: Parameter space space for N eff in M Z ′ vs. g X plane for a generic U (1) X gauge extension with the charge assignment |X 1 | = 0, |X 2,3 | = 1.We show the results without kinetic mixing (ϵ = 0) in (a) and with kinetic mixing (ϵ ̸ = 0) in (b).The grey dashed line in each figure represents the 2σ upper bound from Planck 2018[4], excluding the parameter space to its left, as depicted by the grey region.The blue and red dashed lines in each figure denote N CMB eff values at 3.2 and 3.5, respectively, indicating the intermediate regions where the H 0 tension can be relaxed[30].

5. 2 .UFIG. 9 :
FIG.9: Constraints from cosmology and low energy experiments in M Z ′ vs. g X plane for a generic U (1) B 3 −3Lµ gauge extension.We present the results without loop induced mixing (ϵ = 0) in (a) and with loop induced mixing (ϵ ̸ = 0) in (b).The grey dashed line represents the 2σ upper bound from Planck 2018[4], excluding the parameter space on the left side of the line as shown by the grey shaded region.The region between the blue dashed line (N eff = 3.2) and the red dashed line (N eff = 3.5) indicates the parameter space where the H 0 tension can be relaxed as mentioned in ref.[30].

TABLE I :
Particle content and U (1) X gauge charge assignments of Standard Model and new particles.

TABLE II :
Values of N CMB eff

TABLE III :
2σ upper limit on N eff at CMB from different experiments.Note that, all the bounds in the table indicate N eff at CMB. 5. SPECIFIC U (1) X SYMMETRIES:U (1) B 3 −3L j

TABLE IV :
Particle content and U (1) B 3 −3L j gauge charge assignments of Standard Model and new particles.The charges of ν R i can be fixed by anomaly cancellation condition while the charge of σ depends on the details of the nature and model for neutrino mass generation.

TABLE V :
[57]red amplitudes for interactions relevant for ν decoupling in SM.The i in subscripts stand for different generations of ν L[57].