Finite modular majoron

We point out that the accidental $U(1)_{B-L}$ symmetry can arise from a finite modular symmetry $\Gamma_N$ in the type-I seesaw. The finite modular symmetry is spontaneously broken in such a way that the residual $\mathbb{Z}^T_N$ discrete symmetry, associated with the $T$-transformation which shifts the modulus $\tau \to \tau+ 1$, remains unbroken. This discrete $\mathbb{Z}^T_N$ symmetry mimics $U(1)_{B-L}$, and hence the majoron appears as a pseudo Nambu-Goldstone boson of $U(1)_{B-L}$. Without introducing additional interactions, the modulus $\tau$ can be stabilized by the Coleman-Weinberg (CW) potential given by the Majorana mass terms of the right-handed neutrinos. We study cosmological implications of the majoron, with particular interests in the dark matter and dark radiation, where the latter may alleviate the Hubble tension. We also find that the CW potential can have a wide range of nearly exponential shape which prevents $\tau$ from overshooting, and makes the amount of dark radiation not too large.


Introduction
The type-I see-saw mechanism is one of the most plausible explanations for the tiny neutrino masses, where the active neutrino masses are suppressed by the Majorana mass of the righthanded neutrinos [1][2][3][4].In this mechanism, U (1) B−L symmetry is broken by the Majorana mass term, and hence the breaking scale of U (1) B−L symmetry is related to that of the Majorana mass which should be much smaller than the Planck scale.Although U (1) B−L symmetry can be gauged such as the Pati-Salam model [5], it is also interesting to consider it as a spontaneously broken global symmetry, so that the Nambu-Goldstone (NG) boson, the so-called majoron, appears in the low-energy theory [6,7].As it couples only to the neutrinos, the lifetime of the majoron is so long that it can be the dark matter [8][9][10][11][12][13][14][15][16][17][18][19][20][21] and/or contributes to the dark radiation.The latter can be tested by the measurements of the Cosmic Microwave Background (CMB) [22] and Baryon Acoustic Oscillation (BAO) [23].Such majoron dark matter would be detected in future experiments [24][25][26][27] searching for neutrino flux from the decay of the majoron dark matter.
Theoretically, the origin of the global U (1) B−L symmetry is a fundamental question of the majoron.The global U (1) B−L symmetry guarantees the lightness of the majoron, while it should be broken explicitly so that the majoron is not massless.In other words, U (1) B−L symmetry should be accidental, and the majoron should be a pseudo NG boson.This would be a result of quantum gravity which is believed not to respect any global symmetry [28][29][30][31].However, there is no clear explanation how it is broken, and thus the mass of the majoron can not be predicted from a theory.Also, it is unclear that which interactions are allowed by the accidental U (1) B−L symmetry.For instance, the majoron can not be the dark matter if it sizably couples to lighter Standard Model (SM) fermions by a symmetry breaking interaction.
Our idea in this paper is that the accidental U (1) B−L symmetry is realized from the residual Z n symmetry, and hence the explicit U (1) B−L breaking effects arises in a way such that the Z n symmetry is unbroken.Since the majoron is a pseudo-NG boson protected by the finite modular symmetry, we call it as finite modular majoron.We also note that the finite modular symmetry is flavor universal unlike the other flavor models, so that it realizes the global U (1) B−L symmetry accidentally.
Having a large VEV of the imaginary part of the modulus τ is essential to realize the accidental U (1) B−L symmetry from the residual Z n symmetry and to make the seesaw mechanism work within the perturbative range.Interestingly, it will turn out that the Majorana mass term itself generates the potential of the modulus by the Coleman-Weinberg (CW) mechanism1 .This scenario is similar to the one proposed in Ref. [88], where the CW potential is generated from the mass term for the vector-like quarks in the KSVZ axion model [89,90] 2 .In the vector-like quark case, the accidental U (1) PQ is realized due to the residual Z n symmetry, and the pseudo-NG boson is the QCD axion.We shall show that the right-handed neutrinos play the role of the vector-like quarks, and the modulus is stabilized near the fixed-point where the accidental U (1) B−L symmetry arises.In this model, we can explain the smallness of the Majorana mass term by the FN mechanism from the residual Z n symmetry, in the same way as the explanation for the flavor hierarchies.Furthermore, we can explicitly calculate the masses of the majoron ∼ Re τ and the modulus ∼ Im τ .
We investigate cosmological implications of the majoron in this simple model mainly focusing on the dark matter and dark radiation.The majoron has a typically long lifetime, so its coherent oscillation can be the dark matter.Since the decay constant of the majoron resides at O (10 16 ) GeV, the dark matter abundance is explained when either the majoron mass is O (10 −17 ) eV, or the oscillation starts during a matter-dominated era.In addition, the relativistic majorons produced from the decay of the modulus ∼ Im τ contribute to the effective number of neutrino species ∆N eff , which may alleviate the Hubble tension [22,[95][96][97][98][99][100][101].We study in detail the dynamics of the modulus to estimate ∆N eff and to examine if the modulus can successfully settle down to the potential minimum in the early Universe.
The rest of this paper is organized as follows.In Sec. 2, we selectively review the basic ingredients of finite modular symmetry which are necessary to understand our model introduced in Sec. 3.Then, its cosmological implications are discussed in Sec. 4, and section 5 is devoted to the summary.The details of the dynamics of the modulus are explained both analytically and numerically in Appendix A.

Finite modular symmetry
In this section, we briefly review the finite modular symmetry for the following discussions, see e.g.Refs.[32,102,103] for the details.The modular group Γ = SL(2, Z) is defined as The elements of SL(2, Z) are generated by the three elements, S, T and R, These generators satisfy S 2 = R, R 2 = (ST )3 = 1 and T R = RT .An element of Γ transforms a complex variable τ , with Im τ > 0, as For each generator, we have generated by R. The generators of Γ are S and T satisfying S 2 = (ST ) 3 = 1.We consider the principal congruence group Γ(N ) defined as where N ∈ N is called level.Note that R only exists in Γ(N ) when N = 1 or 2. Thus, we have Γ(N ) := Γ(N )/Z R 2 for N = 1, 2 and Γ(N ) := Γ(N ) for N ≥ 3. The finite modular group is defined as Γ N := Γ/Γ(N ), whose generators S and T satisfy where 1 should be understood as 1 ∈ Γ(N ).There are more relations for N > 5, but we do not use them in this work.Equation (2.6) shows that there are Abelian discrete symmetries Z S 2 , Z ST 3 and Z T N whose fixed points are located at τ = i, e 2πi/3 and i∞, respectively.Here the superscripts represent the corresponding generator.The modular form is a holomorphic function of τ which transforms under γ ∈ Γ N as where 0 ≤ k ∈ Z is the modular weight and ρ(r) is the unitary representation matrix of the representation r.The finite modular group Γ N with N ≤ 5 is isomorphic to the permutation groups as Γ 2n ≃ S 2+n and Γ 2n+1 ≃ A 3+n with n = 1, 2, and hence the representation matrices ρ(r) are the ones in those permutation groups.Hereafter, in this section, we take N = 3 for concreteness.Since Γ N ≃ A 4 , there are three singlet representations 1 t , t = 0, 1, 2 3 , whose representation matrices are given by and a triplet representation 3 with where w := e 2πi/3 .Here, we consider the basis in which ρ T (3) is diagonal.There are k + 1 independent functions for a given weight k.For example, there is only one 3 when k = 2, and there are 1 0 , 1 1 and 3 when k = 4, and so on.In particular, if we consider only k ≤ 10, the non-trivial singlet representation 1 1 exists only at k = 4, 8, 10 and 1 2 exists only at k = 8.These singlet representations will be directly used for our model building in the next section.
The modular form with the modular weight k = 2 is explicitly given by [37], where Here, η(τ ) is the Dedekind eta function, with q := e 2πiτ .The q-expansions of the modular forms for |q| ≪ 1 become (2.13) We choose the normalization of Y 1,2,3 (τ ) such that these are matched to those in Ref. [32].The modular forms with higher weights can be constructed by multiplying those with k = 2 [37].For example, the singlet modular forms for k = 4, 6 are given by The singlet modular forms for k ≥ 8 are easily obtained by scalar multiplications of these.Note that, in the q-expansion, Y 1t ∝ q t/N (1 + O (|q|)) for Im τ ≫ 1 which is close to the fixed point τ = i∞.The value of modular form is suppressed by q t/N which can be understood as the FN mechanism due to the residual Z T N symmetry [72], where a small parameter induced by a flavon VEV is replaced by q 1/N .In our model, this mechanism can be applied to explain the hierarchy between the Majorana mass of right-handed neutrinos and the Planck scale.

Model
Let us consider the type-I see-saw model with supersymmetry (SUSY).We impose the finite modular symmetry Γ N where the modular field τ only couples to the right-handed neutrinos through the Majorana mass term.The Kähler potential and superpotential are given by where h ∈ N. Here, N i 's are the chiral superfields which are identified as the right-handed neutrinos in the type-I seesaw mechanism, and transform under Γ N as such that the Lagrangian is invariant under Γ N .The second term in the superpotential of Eq. (3.1) is the Dirac Yukawa coupling involving the right-handed neutrinos and the SM leptons.K MSSM and W MSSM are respectively the Kähler potential and superpotential of the chiral superfields in the Minimal Supersymmetric Standard Model (MSSM).
For Im τ ≫ 1, the Majorana mass term becomes approximately where τ R := Re τ .This shows how the modular symmetry provides U (1) B−L as an accidental symmetry since it is invariant under with α being a real transformation parameter.The origin of the accidental U (1) B−L symmetry is the Z T 3 symmetry under which τ and The MSSM quarks and leptons transform in the same way as the righthanded neutrinos, given in Eq. (3.3).We assign the Z T 3 charges and the moduar weights of those fields aligned with the B−L number, so that the Yukawa couplings involving the SM fermions, including y ij D , are constant and are independent of the modulus τ .Note that, with this assignment, the discrete anomaly of A 4 ≃ Γ 3 is vanishing since the Z 3 T charge t is flavor universal for the three generations [104,105].The modular weights aligned with the B−L number will also manifest the anomaly cancellation.For simplicity, we assume that the scale of the Majorana mass term is flavor universal, i.e.Λ i =: Λ N .

Radiative modulus stabilization
The modulus field τ can be stabilized at Im τ ≫ 1, where the residual Z T N symmetry is unbroken and the accidental U (1) B−L arises, by the CW potential generated by the Majorana mass term 4 .This mechanism is proposed in Ref. [88] utilizing the vector-like quarks in the KSVZ axion model.We point out that the role of the vector-like quarks can be replaced by the righthanded neutrinos in the type-I seesaw mechanism, and hence we do not need to extend the model.
The CW potential induced by the Majorana mass term is given by [106], with where (−iτ + iτ † ) k is multiplied due to the canonical normalization for the kinetic term in Eq. (3.1) 5 .We introduced the soft SUSY breaking mass m 0 for the right-handed sneutrinos which is assumed to be independent of the modulus τ and flavor universal.To have a nonzero VEV at Im τ > 1 we need m 0 < m N since this minimum does not exist otherwise, as discussed later.The scale parameter µ * is a specific renormalization scale in the MS scheme where the tree-level potential of τ is vanishing.In a general choice of the renormalization scale µ, there must be a tree-level potential associated with soft SUSY breaking to make the effective potential invariant under the choice of µ.Therefore, the scale of µ * can be interpreted as the scale at which such tree-level potential is negligible compared with the CW potential.This might be a similar situation to the radiative breaking of U (1) PQ symmetry [107].
As shown in Ref. [88], the CW potential can have a global minimum at Im τ ≫ 1 for t ̸ = 0.The modular form is expanded by q := e 2πiτ as where the normalization of 1t is chosen so that the coefficient of the leading term is unity.The coefficient β is an integer depending on t and k.At the leading order, the CW potential can be approximated as where τ I := Im τ .This potential has a minimum when is satisfied.We denote τ I0 = Im τ 0 as the solution to this equation, and it can be expressed as where the Lambert function W satisfies W(z)e W(z) = z, which has two values for −e −1 < z < 0, and we take the one which gives τ I0 > 1 6 .As an example, we obtain τ I0 ≃ 4 when k = 8, t = 1 and µ * = Λ N , and τ I0 increases as kN/t increases or µ * /Λ N decreases.The minimization condition can be read as Here, m N corresponds to the canonically normalized mass at the minimum.Thus the mass of the right-handed neutrino is generated at µ * .This simplifies the CW potential in Eq. (3.9) as where we have added a constant 3m 2 0 m 2 N /(16π 2 ) so that the potential energy is zero at the minimum.Note that this minimum does not exist for a large soft mass m 0 > m N .In this case, the leading τ dependence of the potential is proportional to M 2 N (τ ) which is a monotonically decreasing function of τ I for τ I > kN/(4πt) at which the maximum resides.Thus, we restrict our case to m 0 ≪ m N .
The leading potential in Eq. (3.9) does not depend on τ R := Re τ due to the accidental U (1) B−L symmetry at τ I ≫ 1.The leading potential is given by so its size is suppressed by |q| 2 = e −4πτ I ≪ 1 compared with the potential for the τ I direction 7 .This potential has the shift symmetry of τ R → τ R + 1/2 whereas the original periodicity from the finite modular symmetry is τ R → τ R + 1.The smaller shift symmetry τ R → τ R + 1/2 in Eq. (3.14) is an approximate result of the q expansion (3.8), and is broken slightly when higher order corrections are included.

Masses and decays
Here, we discuss the masses and decays of the modulus τ = τ R + iτ I in our model.After canonically normalizing the kinetic term from the Kähler potential in Eq. (3.1) at the minimum τ I = τ I0 , the physical fields are defined as Since J ∝ Reτ can be considered as a pseudo-NG mode of the U (1) B−L symmetry breaking, we can identify J as the majoron.As it originates from the finite modular symmetry, we call it finite modular majoron.Also we call φ as modulus.From Eqs. (3.14) and (3.15), we define the decay constant of majoron as so that Eq. (3.14) is invariant under J → J + 2πf J .The masses of φ and J are given by The φ mass can be estimated as and the mass ratio of the majoron to the modulus is This exponentially suppressed ratio m J /m φ is a distinctive property of the finite modular majoron.For comparison, the mass originated from an explicit breaking by a higher-order term of the U (1) B−L breaking field, the mass is given by with n being a (large) positive integer depending on the dimension of the explicit breaking operator.In this case, the majoron mass strongly correlates with the decay constant, unlike our model.
The majoron J can be the dark matter as its decay width is suppressed by (m ν /M p ) 2 .It could decay to neutrinos through the Majorana mass term if m J > 2m ν , where the decay width is Γ(J → νν) ≃ m 2 ν m J /M 2 p .This is small, and the corresponding lifetime is much longer than the age of the universe unless m J is greater than 10 10 GeV8 .As will be shown later, the mass of our majoron is smaller than 10 GeV, so the majoron in our model is a good dark matter candidate.
The modulus dominantly decays to majorons through the Kähler potential, and the decay rate is given by [115] Γ (3.20) The corresponding decay temperature is where g * ,ρ (T ) is the effective number of relativistic degrees of freedom for energy density when photon temperature is T .Therefore, the majoron abundance that is produced from φ → JJ may result in a sizable ∆N eff before the last scattering surface.We discuss the cosmology of the majoron and modulus in the next section.The modulus will also decay to the SM fermions by the Kähler potential [115], as we assign non-zero modular weights to the MSSM fields.The branching fraction to a pair of SM fermion is Br(φ → SM) ∼ m 2 t /m 2 φ ∼ O (10 −4 ) for m φ ∼ 10 TeV, where the dominant decay mode is the top quark.Although the decays to quarks are subdominant and T D in Eq. (3.21) is unchanged, such late-time hadronic decays are severely constrained by Big Bang nucleosynthesis (BBN) analysis which becomes especially strong at T D ≲ 10 keV as they can drive a photo-dissociation process of the deuterium abundance [111,[116][117][118].To avoid this, the mass of the modulus should be heavier than O (10 TeV) 9 .

Cosmological implications 4.1 Majoron oscillation as dark matter
As we discussed in the previous section, the majoron in our model is lighter than the GeV scale, and its lifetime is much longer than the age of the Universe.Therefore, its coherent oscillation driven by the misalignment mechanism is a good dark matter candidate.The large f J ∼ O (10 16 ) GeV strongly restricts our scenarios: (i) ultralight dark matter scenario with m J ∼ 10 −17 eV, or (ii) scenario where the oscillation starts during a matter-dominated era.The schematic picture of the two scenarios is shown in Fig. 1.

(i) Ultralight dark matter scenario
If the oscillation starts during the radiation-dominated era, the oscillation energy density per entropy density becomes Figure 1: Schematic picture of the evolution of the masses and the Hubble rate.If the majoron is ultralight (m J ≲ 10 −17 eV), the oscillation can start during radiation domination, which is the case (i), depicted by the green dot-dashed line.Otherwise, the oscillation should start during matter domination, which is the case (ii), depicted by the green dotted line.In any case, to avoid ∆N eff constraint, the modulus φ should start rolling down and settles down to the potential minimum during the matter-dominated era as depicted by the red dashed line.
The mass of φ obeys m φ ∼ H during the rolling along the nearly exponential potential before it starts to oscillate around the minimum.See App.A for the details.
where θ i is the initial misalignment angle and T osc is the temperature when the oscillation starts, i.e.H(T osc ) = m J .Here, g * ,s ≃ 3.91 is the effective number of relativistic degrees of freedom after the electron decoupling for the entropy density.As the observed dark matter relic corresponds to ρ DM /s ≃ 0.44 eV, our majoron can be the dark matter when the mass is around 10 −17 eV [22].Such ultralight dark matter has a de-Broglie wavelength comparable to the galactic size and is often called fuzzy dark matter [119].Note that, in this case, the oscillation starts at T osc ∼ 30 keV after the BBN.

(ii) Early matter domination scenario
As shown in Eq. (4.1), the majoron oscillation can easily end up with the overclosure problem if m J > 10 −17 GeV.To avoid this, we consider a matter-dominated era during which majoron starts oscillation.The matter domination is driven by an additional field χ which we call reheaton.In this case, the majoron abundance is given by [120] ρ so we need a mild O (0.1) tuning of the initial misalignment angle θ i .T R is the reheating temperature after χ domination, which must be greater than 5 MeV to avoid the BBN bound as studied in Refs.[121][122][123][124][125]. Since the majoron should start oscillation before reheating, m J must be greater than H(T R ) ≃ 4 × 10 −14 eV(T R /10 MeV) 2 for consistency.

Modulus dynamics and dark radiation
As the modulus φ decays to majorons around T D ∼ O (100 keV) and has the mass around the TeV scale, the energy density of relativistic majorons from φ → JJ provides an additional relativistic degree of freedom and increases the Hubble rate at T ≲ T D .In the following, we estimate it in terms of the effective number of neutrino species N eff , where the estimation of the initial abundance of the modulus field is important.
In this section, we assume that φ evolves during the matter domination driven by χ.Otherwise, if the modulus starts its motion and ends up with an oscillation during the radiationdominated era, the corresponding energy density is so large that it easily dominates the energy of the Universe.In this case, the modulus domination will be finished with its decay into the majorons, which does not reheat the SM plasma, and therefore it cannot be compatible with the current Universe.To avoid this, we have to restrict our scenario to the case where the modulus dynamics and oscillation starts during the matter-dominated era.
The modulus field τ I (or equivalently φ) has the nearly exponential potential ∝ τ k I e −(4πt/N )τ I which is shown in the left panel of Fig. 2, where we depict the shape of the potential given by Eq. (3.13).In this figure, we set the parameters as k = 4, h = t = 1 and N = 3, and the location of the minimum is at τ I0 = 5, 7.5 and 10 on the blue, green, and red lines, respectively.The potential is normalized by V 0 := 3m 2 0 m 2 N /(16π 2 ) and its fourth root is plotted in the log scale.The potential has its maximum at τ I,max = kN /(4πt) ∼ 1.0, and there appears a range of τ I between τ I,max and τ I0 where the potential is nearly exponential in τ I .
We solve the equation of motion numerically and show the evolution of τ I as functions of log(a/a i ) with the scale factor a in the right panel of Fig. 2, see App.A.1 for the details of numerical calculation.The colors of the lines correspond to the shape of the potential on the left panel, i.e. the location of the minimum.The thick red, green, and blue lines show the evolution starting from τ I = 2.5, whereas it starts from τ I = 7.5 (5) for the thin red (dashed) line for the potential with τ I0 = 10.The modulus evolves like τ I ∝ log(a) and does not scale like a power of a due to the Hubble friction when it rolls down along the exponential slope.Consequently, the modulus does not overshoot from the minimum even though its initial point is far away from the potential minimum unless the starting point is close to the maximum of the potential where its shape is no longer exponential.
As shown numerically by the red line on the right panel of Fig. 2, and analytically in App.A.2, the remaining oscillation after rolling down is almost independent of the initial position.This implies an important fact that the energy density ratio between the coherent oscillation of the modulus and the SM plasma ρ rad at the reheating temperature is given by insensitively to its initial condition and also to the reheating temperature T R .We confirm this feature numerically as shown in Fig. 3 which shows the values of ξ with changing the initial value τ Ii := Im τ i .The left panel is in the case of the potential shown in Fig. 2 with the same color coding for the location of the minimum.The right panel is the same figure as the left one, but k = 8.The colored bands are the values from our analytical estimation in App.A.2.We  see that the analytical estimation nicely agrees with the numerical evaluation particularly for larger τ I0 in which the time of slow rolling on the exponential slope is longer.The value of ξ does not change as long as the initial value is on the exponential slope.For smaller τ Ii where it is closer to the maximum, where τ I ∼ 1.0 (1.9) for k = 4 (8), the modulus overshoots from the minimum and does not oscillate around the minimum.For τ Ii ∼ τ I0 , it is simply the oscillation around the minimum, and the ratio ξ is not changed from the initial value, which is 10 −4 in the plot.In both panels, the region around the maximum, where the modulus overshoots, overlaps with that around the minimum, where the modulus oscillates, and hence there is no exponential slope for τ I0 = 5.Now, let us estimate the modification of the Hubble rate in terms of the effective number of neutrino species.Since, at T < T R , we have the scaling behavior of ρ φ,osc ∝ a −3 ∝ T 3 and ρ rad ∝ T 4 , their ratio scales as which is valid until φ decays.Assuming that φ decays instantaneously at T = T D and that T D < T R , we have ρ J,dec /ρ rad = ξ(T R /T D ), where ρ J,dec is the energy density of the majoron produced from the modulus decay.Altogether, ∆N eff is given by  ) GeV so that m N < 10 14 GeV for the perturbativity of the Dirac Yukawa coupling y D , and m 0 < m N for the existence of the minimum.However, this lower bound of ∆N eff is negligibly small unless T R ≫ O (MeV) and it seems challenging to test our model completely.
The produced majorons remains relativistic until today if where p 0 J is the momentum of the majoron today and T 0 = 0.234 meV is the temperature of the current Universe measured from the CMB.Hence, the produced majoron is relativistic if m J ≲ keV and m φ ≲ 10 TeV.For m J ≫ p J0 , it initially contributes to the dark radiation, but to the dark matter at a later time.Its contribution to the dark matter relic today can be estimated as This amount is much smaller than the whole dark matter abundance 0.4 eV for m J ≪ m φ .The CMB analysis gives the upper bound on ∆N eff < 0.3 at 95% confidence level [22], which provides an upper bound on the modulus mass m φ in our model, see Eq. (4.5).However, The right-handed neutrino mass is heavier than 10 14 GeV on the light gray region, and it is lighter than the soft mass m 0 in the dark gray region.The allowed window is in between these gray regions.The left panel represents case (ii) where the oscillation of heavy majoron starts during early matter domination and becomes the dark matter.The right panel corresponds to case (i) where the oscillation of ultralight majoron becomes the dark matter during radiation domination.
the boundary of this constraint is not robust in the presence of the Hubble tension [22,95], see also Refs.[96][97][98][99][100][101] for reviews.For instance, ∆N eff ∼ 0.4 can alleviate the tension as analyzed in Refs.[126,127].Therefore, ∆N eff ∼ O (0.1) may be preferable in the context of the Hubble tension, which can be alleviated by the majoron in our model.

Mass spectrum and Hubble tension
In this section, we present the mass spectrum of various particles focusing on the parameter range where ∆N eff ≃ 0.3 is satisfied.One can consider our choice as the 2σ boundary of the Planck constraint, or the preferred parameters that can alleviate the Hubble tension.For example, if we take m 0 lower than the values presented in the following, m φ decreases, and hence ∆N eff may become too large.So, one can interpret the following values of m 0 and m φ as lower bounds.We also take the scale for the Majorana mass term as the Planck scale, i.e.Λ N = M p .The physical mass of the right-handed neutrino is given by a function of τ I0 , and exponentially suppressed as τ I0 increases, as shown in Eq. (3.12).Similarly, m φ is also expressed as a function of τ I0 and also m 0 .We fix m 0 such that ∆N eff ≃ 0.  (4.9) The modulus mass m φ can be greater than the TeV scale depending on T R .However, for the case (ii) in which the majoron oscillation starts during matter domination, T R should not be too far from the 10 MeV scale to avoid tuning of θ i , and hence m φ should be around TeV scale.Lastly, the majoron mass can be obtained by using Eq.(3.19), where the majoron mass is exponentially lighter than the modulus mass roughly by e −2πτ I0 .Figure 4 shows the masses of the particles with varying the location of the minimum τ I0 .The modular weight is k = 4 (k = 24) on the left (right) panel, with the other parameters are set to Λ N = M p , h = t = 1, N = 3, T R = 10 MeV and ∆N eff = 0.30.The initial value of τ I is set at τ Ii = 2.5 (τ Ii = 10) for k = 4 (k = 24) 10 .In the light gray region, the right-handed neutrino mass is heavier than 10 14 GeV, and the Dirac Yukawa coupling constant for the SM neutrino y D is non-perturbatively large.Whereas, m N < m 0 in the dark gray region where the approximation for the potential in Eq. (3.9) is broken down and the minimum disappears.Thus the window in between the gray regions is the viable parameter space.The majoron 10 Regarding the modular form for k = 24, we choose the one Y (24) 11 4 which gives β = 448.In general, the value β is not unique for a large k, where there are more than one modular form for a given representation.decay constant f J , the majoron momentum p 0 J and the temperatures T R and T D are also shown in the plot.
On the left panel where k = 4 and t = 1, the case (ii) can be realized, i.e. the majoron oscillation starts during the matter domination.The decay constant is always O (10 16 GeV) with slight change due to f J ∝ τ −1 I0 , and hence O (0.1) tuning of the initial amplitude is necessary to explain the DM density, see Eq. (4.2).T D /T R ∼ O (0.1 ∼ 0.01) is required to have ∆N eff = 0.30, which can be achieved when m φ is at O (TeV) by increasing m 0 .The majoron momentum is shown by the orange dot-dashed lines, and it is equal to the majoron mass on the vertical line at τ I0 ∼ 4.9.The majoron is relativistic only for the right to this line and contributes to ∆N eff .
The right panel is for the case of k = 24.Such a large modular weight is chosen so that the majoron mass is so light that it becomes the ultralight dark matter in case (i).Note that the window for m 0 < m N < 10 14 GeV shifts to larger τ I0 region since m N ∝ Λ N τ k/2 I0 e −2πtτ I0 /N .The shift of the window changes the majoron mass ∝ e −2πτ I0 drastically, whereas the other parameters are not changed significantly.The majorons produced from the modulus decay are always relativistic as they are ultralight.
The case of k = 8 is also shown in Fig. 5.The parameters are the same as those in Fig. 4, but k = 8 and t = 1 (t = 2) on the left (right) panel.The initial value is τ Ii = 3.5 (1.6) for t = 1 (t = 2).We see that the mass spectrum is quite similar in the window except for the majoron mass, and m N ∼ 10 10∼14 GeV, m 0 ∼ 10 8∼10 GeV and m φ ∼ 10 0∼1 TeV.Interestingly, the right-handed neutrino mass of the usual typical type-I seesaw is favored and lighter masses are not allowed to stabilize the modulus by the CW potential.The soft mass for the right-handed sneutrino is heavier than the observable scale, and hence the sparticles in the MSSM would be too heavy to be detected at the LHC, although it is possible that some of the sparticles are much lighter, such as in the split-SUSY scenario [128,129].We note that the modulus in this scenario avoids the usual modulus problems due to the suppression of its energy density due to the exponential slow roll.In the left panel (t = 1), the majorons produced from the modulus decay are relativistic on the right to the vertical line as in Fig. 4, while it is always non-relativistic in the window in the right panel (t = 2).Since the majoron mass is O (GeV) in this case, the majoron produced from the φ decay also contributes to the DM with a sizable fraction, see Eqs. (4.2) and (4.7).The existence of such exotic dark matter would give an interesting signature, but it is beyond the scope of this paper.
Before closing, we summarize the range of the majoron mass, which exponentially depends on the value of τ I0 , as e −2πτ I0 .The value of τ I0 is determined to satisfy m 0 < m N < 10 14 GeV where m N ∝ τ k/2 I0 e 2πtτ I0 /N .From the figures, the majoron mass is typically MeV, 10 −17 eV, keV and GeV, for (k, t) = (4, 1), (24, 1), (8, 1) and (8, 2), respectively.The majoron oscillation always contributes to the dark matter, while those produced from the modulus decay contribute to the dark matter for m J ≳ keV or the dark radiation for m J ≲ keV.Those two components of the majoron contribution to the dark components would be discriminated by the measurements of the CMB.If the majoron is heavy enough, the majoron dark matter would be probed by the searches for neutrino flux from the majoron decay, where the current limits exist for m J ≳ O (10 GeV) and f J ≳ O (10 15 GeV) [18,[130][131][132][133].

Summary
In this paper, we have proposed a simple model of finite modular majoron where the accidental B − L symmetry arises from the finite modular symmetry.The modulus field τ only couples to the right-handed neutrinos in the type-I seesaw via the Majorana mass term being a modular form in terms of τ .If it is given by one of the non-trivial singlets under Γ N , the right-handed neutrino mass is exponentially suppressed compared to the Planck scale as in Eq. (3.12).
The accidental U (1) B−L symmetry is realized by the finite modular symmetry Γ N at the leading order of q-expansion, and hence the Re τ mode can be identified as the pseudo-NG boson of U (1) B−L symmetry breaking, which we have called majoron and denoted by J.The accidental B−L symmetry could arise because of the residual Z T 3 symmetry at Im τ ≫ 1, where the modulus transforms as τ → τ + 1 under Z T 3 .The majoron acquires its mass because U (1) B−L is an accidental symmetry in the leading power of q := e 2πiτ , and is explicitly broken in the higher order corrections at |q| ≪ 1.So, the majoron mass is also exponentially suppressed compared to the modulus mass despite the large decay constant at O (10 16 ) GeV.Since the majoron has a long lifetime, its coherent oscillation can be the dark matter.Due to the large decay constant, there are two viable scenarios without having an overabundance; (i) m J ∼ 10 −17 eV and becomes the ultralight dark matter, or (ii) the oscillation starts during an early matter-dominated era.
In addition, relativistic majorons that are produced by the decay of the modulus φ ∝ Im τ , can also contribute to the dark radiation.The dynamics of φ should start during an early matter-dominated era regardless of the dark matter scenarios, so that the amount of dark radiation is not too large.For the estimation of ∆N eff , we have further studied the dynamics of φ both numerically and analytically with different initial conditions and found that if φ is initially located along the exponential slope of the CW potential, it slowly rolls down to its minimum due to the Hubble friction, which prevents the modulus from overshooting beyond the potential barrier.Another important consequence of the slow rolling is that the energy density of the final φ oscillation is insensitive to the initial condition, and is given by O (10 −4 ) of the total energy density.This suppression allows modulus mass heavier than O (TeV) with ∆N eff = O (0.1), which may alleviate the Hubble tension, as shown in Figs. 4 and 5. Assuming the Majorana mass scale in the superpotential at the Planck scale, the right-handed neutrino mass m N , soft right-handed sneutrino mass m 0 and the modulus mass m φ are expected to be 10 10 ≲ m N ≲ 10 14 GeV, 10 8 ≲ m 0 ≲ 10 10 GeV and m φ ∼ O (TeV) in the phenomenologically viable parameter space.Whereas, the majoron mass can be in the range of keV ≲ m J ≲ GeV for the modular weight k ≤ 8.For a larger k, the majoron mass can be lighter, e.g.m J ∼ 10 −17 eV for k = 24, so that it can be the ultralight dark matter.
In our case, we assume that V φ /V is slowly changing and the form of the tracker solution is hold.The solution is .16)where F is defined in Eq. (A.6).The potential is nearly exponential where F ≫ 1 and is far from the minimum, and hence the last factor of Eq. (A. 16) is almost constant.Thus the change of the ratio V φ /V is only from the shift of τ I0 in the second factor which is assumed to be slow.For X 2 , Y 2 ≪ 1 i.e. energy domination by the reheaton, value of ξ is estimated as e F (F − 1) + 1 F e F 2 , (A.17) while the modulus dynamics follows the quasi-tracker solution.The value of τ I in terms of F is given by Then, ξ does not follow the solution as it closes to the minimum and starts to oscillation around the minimum.Since the ratio ξ = ρ φ /ρ χ is a constant during the oscillation for T > T R , the value of ξ is frozen at this time.We assume that the oscillation starts when F = 1 correspond to the time when the potential value is same as the potential barrier.Finally, we estimate the ξ factor after the oscillation starts as (A. 19) Here, we neglect the difference between τ I0 and τ I at F = 1.Since the value of ξ is determined by the value of τ I when it starts to oscillate, it is almost independent of the initial value as long as it follows the quasi-tracker solution.

Figure 2 :
Figure 2: The left panel shows the shape of the potential for t = 1, k = 4 and τ I0 = 5, 7.5 and 10 for the blue, green and red lines, respectively.On the right panel, evolutions of Im τ from τ Ii = 2.5 are shown.The colors of the lines are the same as the left panel.The evolutions from τ Ii = 5 (dashed) and 7.5 (thin solid) on the potential with τ I0 = 10 are also shown by the red lines.

Figure 3 : 2 φ
Figure 3: Values of ξ with respect to the initial value τ Ii for the potential with τ I0 = 5 (blue dashed), 7.5 (green dotdashed) and 10 (red).The left panel shows the case of k = 4 and t = 1 whose potential is shown in Fig. 2. The right panels is the case of k = 8 and t = 1.The horizontal bands are the values from the analytical expression Eq. (A.17) with 0.7 < F < 1.0.

Figure 4 :
Figure 4: Masses, momentum of J and temperatures are depicted for k = 4 (left) and k = 24 (right) with h = t = 1 and Λ N = M p .m 0 is chosen to have ∆N eff = 0.30 when T R = 10 MeV.The right-handed neutrino mass is heavier than 10 14 GeV on the light gray region, and it is lighter than the soft mass m 0 in the dark gray region.The allowed window is in between these gray regions.The left panel represents case (ii) where the oscillation of heavy majoron starts during early matter domination and becomes the dark matter.The right panel corresponds to case (i) where the oscillation of ultralight majoron becomes the dark matter during radiation domination.