Cosmological Constraint on Dark Photon from $N_{\rm eff}$

A new U(1) gauge symmetry is the simplest extension of the Standard Model and has various theoretical and phenomenological motivations. In this paper, we study the cosmological constraint on the MeV scale dark photon. After the neutrino decoupling era at $T = \mathcal{O}(1)\,$MeV, the decay and annihilation of the dark photon heats up the electron and photon plasma and accordingly decreases the effective number of neutrino $N_{\mathrm{eff}}$ in the recombination era. We derive a conservative lower-limit of the dark photon mass around 8.5 MeV from the current Planck data if the mixing between the dark photon and ordinary photon is larger than $\mathcal{O}(10^{-9})$. We also find that the future CMB stage-$\rm I\! V$ experiments can probe up to 17 MeV dark photon.


Introduction
The dark photon which stems from a new U(1) gauge symmetry is one of the simplest extensions of the Standard Model (SM).By assuming no SM fields are charged under the new U(1) gauge symmetry, it couples to the SM sector through the kinetic mixing with the gauge boson of the U(1) Y in the SM at the renormalizable level [1].
The dark photon has various cosmological advantages.For instance, the U(1) symmetry can be the origin of the stability of the dark matter.Moreover, it is discussed that the dark matter selfinteraction via the gauge interaction can solve the small scale structure problems of the collision-less dark matter [2][3][4][5].The dark photon also provides a portal to transfer excessive entropy in the dark sector into the SM sector before the neutrino decoupling [6,7].In light of these features, the dark photon is gathering more and more attention and several new experiments are proposed to probe the sub-GeV dark photon (see Ref. [8] for summary).
In this paper, we study the effective number of neutrino degrees of freedom, N eff , in the presence of the dark photon, which is constrained by the cosmic microwave background (CMB) observations.As the MeV dark photon does not couple to the neutrinos, it would heat up only the electron-photon plasma if it decays or annihilates after the neutrino decoupling.Such late-time energy injection can reduce N eff .In previous analyses, the N eff constraint puts an upper limit on the dark photon lifetime of τ γ < O(1) sec.As we will see, however, the MeV dark photon produced from the photon thermal bath can reduce N eff , thus it is constrained even in the case τ γ < O(1) sec. 1n deriving the constraint, we solve the Boltzmann equation of the dark photon coupling to the photon, the electron, and the neutrino systems.There, we use the full Boltzmann equation of the momentum distribution of the dark photon which includes the Pauli-blocking and the Boseenhancement effects.This treatment is particularly important to derive the constraints on the scenario with freeze-in dark photon.As we will see, the freeze-in dark photon is excluded for ε 10 −9.5 and m γ 8.5 MeV by the latest Planck constraint [11].We also find that the stage-IV CMB experiment [12] is sensitive to the dark photon mass up to about 17 MeV.The constraint on the freeze-in dark photon provides the conservative and initial condition independent constraints, which can be generically applicable as long as we assume that the dark photon exists.We also discuss the constraint in the scenarios where the dark photon decouples from the SM thermal bath in the early universe.
The organization of the paper is as follows.In Sec. 2, we summarize the relevant properties of the dark photon.In Sec. 3, we provides the full Boltzmann equation of the momentum distribution of the dark photon.In Sec. 4, we show the constraints on the dark photon in the freeze-in scenario as well as the scenarios with early decoupled dark photon.The final section is devoted to discussions.

The model of dark photon
The massive dark photon has the kinetic mixing interaction with the QED photon, Here, F µν (F µν ) represents the field strength of the QED (dark) photon, A µ (A µ ) is the SM (dark) photon field, and J QED is the QED current.The gauge coupling constant of QED is given as e, while m γ and ε are the dark photon mass and the mixing parameter, respectively.Throughout this paper, we assume that the kinetic mixing is tiny, ε 1. 2 The redefinition of the QED photon field eliminates the kinetic mixing term, which induces the dark photon interaction, εeA µ J µ QED .Accordingly, the partial decay rate of the dark photon into a pair of the electron and positron is given by where α = e 2 /4π is the QED fine structure constant and m e = 0.511 MeV is the electron mass.
Since the dark photon coupling to the neutrinos are suppressed, it heats up only the electron-photon plasma if it decays or annihilates after the neutrino decoupling.This effect reduces N eff , which can be constrained by the CMB observations.Let us discuss the production of the dark photon in the cosmological history.There are various production mechanisms of the dark photon in the early Universe, depending on the details of the underlying dark sector.In this work, we focus on two mechanisms: production from the SM thermal plasma and that from the dark sector. 3he first one is the thermal freeze-in mechanism, in which the dark photons are produced from the SM plasma via the interaction in Eq. (1).In particular, the inverse decay from e + + e − pairs is a dominant process for m γ > 2m e .In this scenario, the produced dark photons may reach to the thermal equilibrium with the electron-photon plasma depending on the value of the mixing parameter ε.It should be emphasized that this production mechanism scarcely depends on the reheating temperature T R of the Universe after the primordial inflation as far as T R > m e and m γ .This is because the dark photon production from the electron-photon plasma is dominated at where T γe is the temperature of the electron-photon plasma.This mechanism guarantees the minimum amount of the dark photon in the early Universe, regardless of the initial condition of the Universe and details of the dark sector.Therefore we can obtain the most conservative constraint if we consider only the freeze-in contribution.Hereafter, we refer this conservative case as the "freeze-in scenario." The second scenario is the case that the dark photon used to be thermalized with the SM sector through the dark sector in the very early Universe.For instance, if the dark sector contains heavy dark Higgs particles which have sizable couplings to the SM Higgs, the dark photon can be thermalized and have the same temperature as the SM sector.As the temperature gets lower, the dark photon is no longer in equilibrium with the SM sector at the decoupling temperature T D .If we assume the case of sudden decoupling and the Hubble rate much greater than Γ γ , the momentum distribution of the dark photon in max [m e , m γ ] T γe < T D is given by, where g * S is given by the entropy density of the SM thermal of temperature T : s = 2π 2 g * S (T )T 3 /45.Due to the large effective massless degrees of freedom of the SM, the number density of the dark photon is diluted by g * S (T γe )/g * S (T D ) compared to the thermalized case.We refer this case as the "decoupling scenario", which can be characterized by the decoupling temperature T D .Before closing this section, let us comment on the dark photon decay for m γ < 2m e .In this regime, the main mode of the dark photon decay is either the one into the three photons or the one into the neutrinos through the mixing with the SM Z-boson.The decay rate into the three-photon is given by, where the prefactor corresponds to the decay rate in the Euler-Heisenberg limit [15], while the enhancement factor F(x) is given in Ref. [16]. 4 The decay into the neutrino is induced by the kinetic mixing to the SM Z-boson, where Z µν denotes the field strength of the SM Z-boson, and θ W is the Weinberg angle.After eliminating the kinetic mixing term and diagonalizing the mass term, the dark photon has the coupling to the SM neutral current, where J µ Z is the neutral current in the SM.Accordingly, the dark photon decay rate into a pair of the neutrinos is given by, where m Z is the mass of the SM Z-boson and g is the gauge coupling constant of SU(2) L gauge interaction of the SM.Thus, we find that the three photon mode is dominant for m γ > O(10) keV, while the neutrino mode is dominant for a lighter dark photon.0 e + e < l a t e x i t s h a 1 _ b a s e 6 4 = " q h d h P U j X a D s B L 4 G y h Z J Z d P 7 3

Boltzmann Equations
In this section, we summarize the Boltzmann equations relevant for the calculation of N eff .The equation for the momentum distribution of the dark photon, f γ (p γ ), is written as Here, f eq γ (p γ , T ) is the Bose-Einstein (BE) distribution:.f eq γ (p γ , T ) = 1/(exp(E γ /T ) − 1).The function G γ ↔e represents the collision term for the decay of γ and its inverse process.In deriving the collision term of f γ , we use the BE distribution for the photon distribution, and the Fermi-Dirac (FD) distribution for the electron and the positron distributions.In this case, the function ϕ is given by where p 0 e = m 2 γ − 4m 2 e /2 and E 0 e = m γ /2 are the momentum and energy of the electron at the rest frame of the dark photon, γ .
In addition to the decay and the inverse decay processes, we also take into account γ +γ ↔ e − +e + and γ + e ± ↔ γ + e ± .Such processes are subdominant for m γ > 2m e as they are suppressed by an additional power of α compared with Eq. ( 9).For m γ < 2m e , on the other hand, they are the main production/annihilation processes of the dark photon, where the decay and the inverse decay are ineffective.We show the Boltzmann equation for those processes in the Appendix B.
When we calculate the collision terms of these processes, we encounter two types of the infrared (IR) divergences for m γ > 2m e (see the Appendix B).One of which stems from the Bose enhancement of the scattered photon, and the other is from the soft photon emission/absorption. 5 In order to take care of these IR divergences appropriately, we have to add up 1-loop diagrams of the dark photon decay and the tree-level diagrams of the soft photon emission/absorption with finite temperature fermion propagators [17].However, as we noted before, contributions from γ + γ ↔ e − + e + and γ + e ± ↔ γ + e ± are subdominant for m γ > 2m e which is the region of our main interest.Therefore, in the calculation of the collision terms of these two processes, we simply introduce a thermal mass effects as a soft photon mass cut-off to avoid the IR divergences.For m γ < 2m e where these processes are dominant, on the other hand, we do not have IR singularities from the tree-level contributions.
To estimate N eff , we need to solve the Boltzmann equations for the SM sector simultaneously.In our analysis, we follow Ref. [18], which allows an efficient and precise estimation of N eff .There, the Boltzmann equations for SM part are given by Here, ρ γe = ρ γ + ρ e + δρ and p γe = p γ + p e + δP each represents the electron-photon plasma density and pressure, and ρ νµ,τ is the sum of the densities of ν µ and ν τ . 6hermodynamical quantities ρ i and p i are calculated from the relation where g i is the degrees of freedom and the sign of denominator depends on the statistics of the particle.δρ and δP are the QED loop corrections to the energy density and the pressure of the electron-photon plasma calculated as [18][19][20][21], With these quantities, the Hubble expansion rate is defined as where ρ heavy (T γe ) represents the energy density from heavier SM particles (e.g., muon and hadrons) other than the electron, neutrino and photon.We assume they have the same temperature as the electron and photon sector and adopt the result of Ref. [22] for the numerical estimation.
After solving the Boltzmann equation, N eff is given by 11 7 4/3 ρ νe (T νe ) + ρ νµ (T νµ ) + ρ νµ (T ντ ) which is evaluated at the temperature much below the dark photon decay temperature and the electron mass.In our numerical analysis, we stop solving the Boltzmann equation at T γe = 0.5 keV below which the double Compton scattering becomes ineffective [23].Below this temperature, interactions with neutrinos are already decoupled, thus the further evolution does not affect the value of N eff , in the case dark photons decay above 0.5 keV.The dark photon which decays (or annihilates) below T γe 0.5 keV is constrained by the CMB spectrum distortion [24,25], which has been applied to the dark photon in Refs.[26,27].Several comments are in order.First, although we assume the BE or FD statistics for the photon and the electron/neutrinos in the calculation of thermodynamical quantities, we use the approximation that electrons and neutrinos obey the Maxwell-Boltzmann distribution and the Pauli blocking effects are negligible in the calculation of C e↔ν i and C ν i ↔ν j .We also ignore the masses of the electrons and the neutrinos in the calculations of the collision terms.Those approximations are validated in [18], which affect N eff less than 1%.The explicit forms of these collision terms are given in the Appendix A. In the following analysis, we shift the value of N eff obtained in our analysis by where N SM eff = 3.045 is the most precise evaluation in the SM [28].

N eff constraints
Here, we show the results of the Boltzmann equations.For the freeze-in scenario with the empty dark photon in the early universe, the dark photon is mainly produced at the low temperature of Here we adopt m γ = 5 MeV and ε = 10 −8 (red), 10 −9 (blue) and 10 −10 (green), respectively.For comparison we show the energy density of the dark photon in the thermal equilibrium (black).We solve the Boltzmann equation from temperature 300 MeV which is well above the neutrino decoupling temperature.The evolution of T γe /T νe for ε 10 −8 is almost identical to the completely thermalized case.figure shows that the dark photon is thermalized for ε = 10 −8 , while it deviates from the thermal equilibrium and exhibits the out-of-equilibrium decay for ε = 10 −10 .In each choice of ε, we find sizable amount of the dark photon energy density is released to the electron-photon plasma below the neutrino decoupling temperature, T ν 2 MeV [18].
In Fig. 3, we show the value of N eff for the dark photon which completely freezes-in and is in the thermal equilibrium with the photon thermal bath.Such a scenario is achieved for ε 10 −8 .As the dark photon energy density follows the value in the thermal equilibrium (i.e. the black line in Fig. 2), the predicted N eff does not depend on ε.The red line in the figure shows the lower limit of the present Planck constraint, N eff = 2.99 +0. 34 −0.33 at the 95%CL [11].The blue line shows the prospected sensitivity at the 2σ of the stage-IV CMB, δN eff = 0.06 [12].As a result, we find the dark photon mass m γ < 8.5 MeV has been excluded by the current Planck data for the completely freezed-in dark photon.The stage-IV CMB observation will be also sensitive to the dark photon mass, m γ 17 MeV.In Fig. 4, we show the contour plot of N eff on the (m γ , ε) plane (left panel).As we have mentioned above, the dark photon freezes-in completely for ε 10 −8 , and hence, the predicted N eff does not depend on ε.For a smaller ε, on the other hand, the dark photon is not completely freezed-in, and the dark photon effect on N eff becomes small for ε 10 −9 .In the figure, we shade the present Planck constraint (95%CL) by red, while the stage-IV CMB sensitivity (2σ) is shaded by blue.The figure shows the freeze-in dark photon is excluded for ε 10 −10 and m γ 8.5 MeV by the latest Planck constraint.For m γ < 2m e , the decay temperature Here we consider the case that the mixing ε is large enough for the dark photon to be thermalized with the electron and photon.In this case, the predicted N eff does not depend on ε.We shift the value of N eff according to Eq. ( 24).
It should be noted that the dark photon effect on N eff is enhanced at around ε = O(10 −10 ) where the dark photon decays in an out-of-equilibrium way.As the dark photon is a massive particle, the relative energy density of the dark photon is enhanced by the cosmic expansion by the time of the decay, which enlarges the dark photon effect on N eff .As a result, we find that the stage-IV CMB is sensitive to the dark photon mass of m γ 30 MeV for ε 10 −10 .Light particles produced in a supernova explosion can alter the neutrino burst spectrum.Thus, dark photons with m γ 100 MeV are constrained from the observation of neutrino burst of SN1987A [33,[55][56][57][58]. Recently, however, it is pointed out that there are uncertainties in a model of the neutrino burst and there is a possibility that the constraints from SN1987A are discarded [59].In view of such astrophysical uncertainties, we do not show the supernova constraints.
Next, we consider the decoupling scenario where the dark and the SM sector thermal bath are in the equilibrium in the very early universe and then decouple at a certain low temperature.In the right panel of Fig. 4, we show N eff for the decoupling scenario of T D = 1 TeV.In this case, the initial condition of f γ is given by Eq. −0.33 (95%) [11].The blue region shows that the sensitivity of the stage IV CMB experiment δN eff = 0.06 [12].The yellow shaded region is excluded by the constraint on the CMB distortion (µ distortion) [29] and the effects on the reionization history [30].The green line corresponds to the parameters where the lifetime of the dark photon is τ γ = 1 sec.The gray shaded region is the compilation of the constraints from the beam dump and the collider experiments in Ref. [8], NA64 [50] and the electron g − 2 constraint [51][52][53][54].
abundance well before the freeze-in production, the parameter region with τ γ = O(1) sec has been excluded by the Planck constraint.For the parameter region with τ γ O(1) sec, the constraint is identical with that in the freeze-in scenario.
Finally, let us comment on the X(16.7 MeV) boson, which is reported in the 8 Be * nucleus decay, i.e., the so-called Berillium anomaly [60,61].To explain this anomaly, the hidden vector boson is required to have a sizable coupling to the electrons [62,63] and thus we can apply the current N eff constraint.For X(16.7 MeV), the predicted ∆N eff 0.6 and is still consistent with the current Planck data but can be probed with the future CMB stage-IV experiment.

Discussions
In this paper, we studied the N eff constraint on the dark photon in detail.As the MeV dark photon coupling to the neutrinos is suppressed, the dark photon heats up only the electron-photon plasma and reduces N eff , if it decays or annihilates after the neutrino decoupling.For the dark photon mass above the electron-positron threshold, which is the main interest in this paper, we solve the Boltzmann equations of the dark photon with the Pauli-blocking and the Bose-enhancement fully included.We estimated the effects of the energy injection after the neutrino decoupling caused by the decay or annihilation of the MeV-scale dark photon.As a result, we found that this effect leads to the decrease of the N eff , and the CMB stage-IV experiments can test a wide range of the MeV-scale dark photon.
Although we considered the Boltzmann equations which include the Pauli-blocking and the Boseenhancement for m γ > 2m e , we have not taken into account the thermal effects on the kinetic mixing [26,27].Such effects could slightly enhance the production rate of the dark photon.We also adopt an approximated treatment of the thermal mass effects to take care of the IR singularity for the scattering processes for m γ > 2m e .The analysis with the full Boltzmann equations of the dark photon momentum distribution with those effects requires consistent treatment of the thermal effects including the higher-order corrections, though we expect such effects are insignificant since the dark photon production is dominated by the decay and the inverse decay process for m γ > 2m e .

A Neutrino-Electron Collision Terms
Here, we present the definition of the collision terms and the explicit forms of the electron-neutrino and the neutrino-neutrino collision terms.
First, the general form of the Boltzmann equation for the process ψ + i ↔ f is given by where X and Y can be multi-particle states and g is a spin degrees of freedom of each particle.M 2 is a amplitude squared averaged over spin degrees of freedom of all particles in ψ + X and Y .The factor S is a symmetrization factor which gives 1/2! for each pair of identical particles in X and Y .8 In the text, we define According to Ref [18], the e-ν collision terms are written in the form of where G F is the Fermi constant.Using the same F function, we can write the ν e -ν µ,τ term as

B Dark Photon Production via Scattering
In this appendix, we summarize the collision terms of the dark photon through the electron scattering, Here, | M| 2 denotes the squared amplitude with all the spins averaged.We put on the distribution for a later use, although f = f .In the following, we use the Maxwell-Boltzmann (MB) approximation in the following way.For ee ↔ γγ , we approximate X = ψ, S simply gives factor 1. Also, note that if there is a set n identical particles, S gives a factor 1/n! for each set.
where f e and f ē are the MB distribution while f γ is taken to be the BE distribution.For eγ ↔ eγ , on the other hand, we take where f e and f γ are the MB distribution, while f e is the FD distribution.With these approximations, the distribution of γ converges to the BE distribution in the equilibrium limit.Therefore, we obtain the collision term as, where Following Ref. [64], the above integration can be further reduced to respectively.
The t-integration of the spin averaged squared matrices are given by These results are consistent with those of Ref. [27].It should be noted that the factor (s − m 2 γ ) −2 appearing in Ge + e − ↔γγ causes a linear IR divergence for m γ > 2m e . 9 Finally, let us comment on our treatment on the decay of dark photon for m γ < 2m e .As our main interest in the present paper is on m γ > 2m e , we adopt an approximate treatment of the three-body for m γ < 2m e , where Γ γ →3γ is in given Eq. ( 4).
a 9 e u L 9 + 4 u b J 6 a 0 8 k P e 7 R X S 9 h C d 9 3 H U F Z G N N d G U p G 9 1 N O n c h l t O l 2 3 y p 9 8 4 R y E S b x R 9 l P 6 W H k d O L Q D z 1 H A j p e r R y 1 X N o J 4 8 y P f D 9 k d J D 5 t B 9 3 u J M G g 5 p h j J U a e j J 4 Z

Figure 1 :
Figure 1: The Feynman diagrams relevant for the dark photon decay (left), annihilation (middle) and the Compton-scattering like process (right).

Fig. 2 ,T 4 γeFigure 2 :
Figure2: The time evolution of (a): the energy density of the dark photon ρ γ and (b): the ratio of photon and neutrino temperatures for the freeze-in scenario.Here we adopt m γ = 5 MeV and ε = 10 −8 (red), 10 −9 (blue) and 10 −10 (green), respectively.For comparison we show the energy density of the dark photon in the thermal equilibrium (black).We solve the Boltzmann equation from temperature 300 MeV which is well above the neutrino decoupling temperature.The evolution of T γe /T νe for ε 10 −8 is almost identical to the completely thermalized case.

Figure 3 :
Figure3: The N eff as a function of the dark photon mass m γ .Here we consider the case that the mixing ε is large enough for the dark photon to be thermalized with the electron and photon.In this case, the predicted N eff does not depend on ε.We shift the value of N eff according to Eq. (24).

T D = 1 Figure 4 :
Figure 4: The contour plots of N eff on the m γ -ε plane.The left panel corresponds to the freeze-in scenario, while the right panel to the early decoupled scenario.The red region shows the present Planck constraint N eff 2.99 +0.34−0.33 (95%)[11].The blue region shows that the sensitivity of the stage IV CMB experiment δN eff = 0.06[12].The yellow shaded region is excluded by the constraint on the CMB distortion (µ distortion)[29] and the effects on the reionization history[30].The green line corresponds to the parameters where the lifetime of the dark photon is τ γ = 1 sec.The gray shaded region is the compilation of the constraints from the beam dump and the collider experiments in Ref.[8], NA64[50] and the electron g − 2 constraint[51][52][53][54].