Cosmological Constraint on Vector Mediator of Neutrino-Electron Interaction in light of XENON1T Excess

Recently, the XENON1T collaboration reported an excess in the electron recoil energy spectrum. One of the simplest new physics interpretation is a new neutrino-electron interaction mediated by a light vector particle. However, for the parameter region favored by this excess, the constraints from the stellar cooling are severe. Still, there are astrophysical uncertainties on those constraints. In this paper, we discuss the constraint on the light mediator from the effective number of neutrino Neff in the CMB era, which provides an independent constraint. We show that Neff is significantly enhanced and exceeds the current constraint in the parameter region favored for the XENON1T excess. As a result, the interpretation by a light mediator heavier than about 1 eV is excluded by the Neff constraint.


Introduction
The XENON1T experiment has recently reported an excess in the electron recoil energy spectrum above the known expected background spectrum [1].Although the XENON1T experiment has not excluded unknown backgrounds, such as the β-decay of the tritium, the report is intriguing and has prompted many new physics interpretations.
Among various new physics, one of the simplest possibilities is to introduce non-standard neutrinoelectron interactions so that the solar neutrino flux explains the excess [1][2][3][4][5][6][7].For example, the XENON1T has discussed the sizable neutrino magnetic moment as a candidate for such a nonstandard interaction.However, the best fit value is in tension with constraints from the white dwarfs and the globular clusters [8].Refs.[2][3][4][5][6][7] have discussed new neutrino-electron interactions mediated by a light mediator, where the mediator with a mass below O(100) keV and the neutrino-electron coupling, (g e g ν ) 1/2 ∼ 3 × 10 −7 explains the excess.The effects on the stellar cooling severely constrain the coupling constant of the light mediator [9][10][11][12][13][14].However, those astrophysical constraints are still under debate as there are several uncertainties in the case of the mediator heavier than 1 eV [11,15].For example, a new mediator produced inside the astronomical objects is reabsorbed before exiting the objects in the case of a large coupling to electron, which weakens the constraints.
In this paper, we discuss a new constraint on the light mediator from the effective number of neutrino degrees of freedom, N eff , measured by the cosmic microwave background (CMB) observations.In the standard cosmology, it is predicted that N (SM) eff 3.045 [16,17], which is consistent with the current CMB measurement.The introduction of new physics alters the N eff prediction, which provides a constraint independent from the astrophysical constraints.As the light mediator interpretation of the XENON1T excess requires a particle lighter than O(100) keV, the mediator mainly decays into the neutrinos.As we will see, the presence of the mediator significantly enhances N eff even for a very tiny coupling g ∼ 10 −10 .The parameter region favored for the XENON1T excess results in N eff > 5 for m Z ≥ 1 eV, which exceeds the upper limit of the Planck CMB only (joint Planck+BAO) constraint, N eff = 2.92 +0.36  −0.37 (2.99 +0.34 −0.33 ) at 95% C.L. [18].The organization of the paper is as follows.In Sec. 2, we explain the setup of the phenomenological model for the light mediator interpretation of the XENON1T excess.In Sec. 3, we show the full Boltzmann equation of the momentum distribution of the light mediator.In Sec. 4, we obtain the constraints on the neutrino and the electron coupling of the light mediator.The final section is devoted to our conclusions.

Setup
As a phenomenological setup, we consider a light vector boson which couples to the neutrinos ν i and the charged leptons ψ i , r D g 9 a 5 g l 7 W h a s I r 8 l x J c q 2 4 s r + F K p n L 4 9 j I 2 v 7 9 O 2 L e O + + f 5 m n X 2 P s A F 8 I 8 + L 3 o G r s d j v m Z u f p 2 2 5 7 6 0 X x r W i Q e + Q + 2 U B F n p E t 8 g q 7 2 S G O 9 k 3 7 p f 3 W / i z 9 W P r X W G g s 5 q 5 z W q G 5 S 8 5 d j e X / c e v a m Q = = < / l a t e x i t > (a) electron annihilation e ± e ± Z < l a t e x i t s h a 1 _ b a s e 6 4 = " K L / q Q J G i Y 3 0 5 g B w J + 5 E r 3 I c n w Y W e f P U m q P v w w X W v w J d 9 4 c r D a 3 J S 8 7 K r d N V 8 C j W z s / 0 4 y N o 5 e / v C 3 7 s f X 5 f p t w Q n w B f C u v w 9 q A v 7 v a 6 1 3 X 3 6 p t f Z e V F + K 3 T y g D w k W + j I M 7 J D X u E 0 e 8 T V v m u / t T / a 3 5 V f u q b r e q s w X d J K z j q 5 c O n r / w E c x N v c < / l a t e x i t > (b) electron scattering where i, j, = e, µ, τ .The new vector mediator Z has a mass m Z .Hereafter, we assume that the mediator is lighter than an electron-positron pair, m Z < 2m e , which is favored by the XENON1T excess [2].Accordingly, the decay into the e ± is kinematically forbidden.
As the left-handed charged leptons and the neutrinos are in the same multiplets in the Standard Model, simple introduction of a new U (1) gauge interaction tends to predict g ν = g .It is, however, possible to achieve g ν g in, for example, the U (1) Lµ−Lτ gauge symmetry [19,20].In this case, g ν corresponds to the U (1) Lµ−Lτ gauge coupling, while g e is provided through the gauge kinetic mixing between the U (1) Lµ−Lτ gauge boson and the photon.In the following, we take g ν and g e to be independent free parameters.We also assume that the mediator couples to one flavor of the neutrinos.Since the neutrino oscillation is fast enough for T < O(1) MeV, the choice of the neutrino flavor is irrelevant for the following arguments.Now, let us discuss cosmology of the mediator at the temperature below O(10) MeV, which is crucial for the determination of N eff .In this setup, the mediators are produced from the thermal bath through, e − + e + ↔ γ + Z , e ± + γ ↔ e ± + Z , ν + ν ↔ Z , and ν + ν ↔ Z + Z (see Fig. 1).These production processes are relatively enhanced compared to the Hubble expansion rate as the temperature decreases.Hence, the mediator is produced at the lower temperature even if it has the zero initial abundance after inflation.In our analysis, we adopt this "freeze-in" scenario, which leads to the most conservative constraint.
In order to comprehend the overview of the cosmology, let us estimate the temperature at which the production processes become effective.For the electron annihilation or scattering production, if that is, these processes are effective.Here, M Pl is the reduced Planck scale.Thus, the production from the γ-e thermal bath is effective before the e ± annihilation for In this region, Z and e ± are thermalized together and share the same temperature.
For the production via the neutrino annihilation, it becomes efficient for Thus, Z and ν are thermalized by the temperature, T ∼ m Z , for As we consider m Z < 2m e , the mediator mainly decays into a pair of neutrinos.The decay rate at the temperature T m Z is given by where we assume that the mediator couples to one flavor of the three neutrinos.We also treat the neutrinos massless throughout this paper.The light mediator exhibits the in-equilibrium decay at the neutrino temperature Note that if g ν is between Eqs. ( 6) and ( 8), only the inverse decay is effective.In this case, the neutrino thermal bath may have a non-zero chemical potential due to the number conservation of Z and ν [21,22].When Z thermalization occurs by the neutrino decoupling era, the energy injected into the ν sector changes the ratio of the neutrino and the photon energy densities, ρ ν /ρ γ , from the one in the standard cosmology.In this case, N eff is changed drastically, which conflicts with the CMB observations.

Boltzmann equations
We solve the Boltzmann equation of the phase space distribution of Z , f Z , the energy densities of the γ-e thermal bath, ρ γe , the neutrino thermal bath, ρ ν and the number density of the neutrino thermal bath, n ν .Here, we need to treat ρ ν and n ν independently, since the neutrino thermal bath obtains a non-vanishing chemical potential from the Z interaction as we will see later.Hereafter, we assume that γ has the Bose-Einstein distribution, and e ± and ν have the Fermi-Dirac distributions, which gives a good approximation after the neutrino decoupling [21][22][23].We treat the three flavor neutrinos as a fluid with a single temperature/chemical potential to mimic the effect of the neutrino oscillations as in Refs.[21][22][23].
The evolution of the momentum distribution for the mediator Z is determined by the Boltzmann equation, where H is the Hubble expansion rate and C[f Z ] is a sum of collision terms.In this work, we include the decay, the scattering and the annihilation processes for the calculation of the collision terms.
With the aid of the formalism in Ref. [24], all the Z collision terms are written in the approximated form Here, f BE Z is the Bose-Einstein distribution function for a single degree of freedom of Z , µ ν is a chemical potential of the neutrino, and T γ , T ν are the temperatures of the γ-e and the neutrino thermal bathes, respectively.Each term for the process including neutrinos is written as Here, we define where p 0 ν = E 0 ν = m Z /2 are the momentum and energy of the neutrino at the rest frame of the dark photon, Z .For collision terms of the νν annihilation processes, we use the integrated amplitude For the scattering process with the γ-e thermal bath, we use the same formulae in the appendix of Ref. [24] with parameters replaced as εg → g e , m γ → m Z , In the above, we define | M| 2 as the amplitude squared averaged over spins of all the initial and final states.Thus, we multiply factors of spin degrees of freedom when we integrate | M| 2 over phase space volumes.We determine the thermal evolution of the SM particles by solving the zeroth and first moment of the Boltzmann equations, Z →νe νe (T ν , µ ν ) + C (1) eZ ↔eγ (T γ ) + C (1) where is the n-th energy moment of a collision term.Here, n ν = n νe + n νµ + n ντ and ρ ν = ρ νe + ρ νµ + ρ ντ are the number and energy density of the neutrinos, and ρ γe and p γe are the density and pressure of the γ-e thermal bath, which include thermal corrections [25][26][27].According to Ref. [22], we have included the effect of spin-statistics and the electron mass in the collision terms between e ± and the neutrinos, C e↔ν and C (1) e↔ν .The Appendix A describes the detail of these collision terms.The new interactions in L Z also induces a new channel of ν-e scattering via the off-shell Z exchange.As shown in Ref. [2], however, rather small couplings √ g ν g e ∼ 3 × 10 −7 are preferred to explain the XENON1T excess.In such a small coupling region, the ν-e scattering mediated by the off-shell Z is negligible compared to the weak interactions at around the neutrino decoupling temperature, T ν-dec = O(1) MeV.Thus, we ignore the Z mediated scattering process.We set the following initial conditions of the Boltzmann equations at T init = T γe = T ν = 20 MeV, where ρ QED γe is the QED loop correction to the electromagnetic energy density.

N eff constraint
Here, we show the results for the freeze-in scenario of Z .Fig. 2 shows the contour plots of N eff on the (g e , g ν ) plane for m Z = 1 eV, 10 eV, 100 eV, 1 keV, 10 keV, and 100 keV.Here, N eff is defined by, at T γ = 0.26 eV.In the figure, we show the contours of N eff ≤ 10.In each plot, the red (orange) shaded region shows the consistent region with the Planck CMB only (joint Planck+BAO) constraint, N eff = 2.92 +0.36 −0.37 (2.99 +0.34 −0.33 ) at 95% C.L. [18].The each blue band is the parameter region favored for the light vector mediator interpretation of the XENON1T excess [2], i.e., √ g e g ν ∼ 3 × 10 −7 . 1 The figure shows that N eff in the favored region exceeds N eff = 5 for m Z ≥ 1 eV.
It should be noted that the massive mediator becomes long-lived enough to behave like "dark matter" at the recombination time in the region below the purple dashed lines.In this case, the above N eff constraint cannot be applied.However, since such a region is far off from the favored region for the XENON1T excess, our conclusions are not affected.
As we discussed in Sec. 2, the mediator production through e ± +γ ↔ e ± +Z and e + +e − ↔ γ +Z become less effective for g e 10 −10 .The mediator production through ν +ν ↔ Z +Z also becomes less effective for g ν 10 −5 .On the other hand, the mediator production via the inverse decay remains effective as long as g ν satisfies the inequality in Eq. ( 8).Let us summarize the expected value of N eff when Z is thermalized for various parameter regions.
(i) Simultaneous thermalization with both e ± and ν For g e 10 −8 and g ν 10 −5 , the on-shell productions of the mediator from the γ-e thermal bath and the neutrino thermal bath are both effective, which delay the neutrino decoupling from the γ-e thermal bath until T m e .As a result, T ν T γ is kept until Z annihilates away.Then, the in-equilibrium decay of the mediator before the recombination heats up the neutrino temperature relative to the photon temperature by a factor of where N F = 3 is the number of flavors of the neutrinos and d Z = 3 is a spin degrees of freedom of Z .Here, we have used the conservation of the entropy per comoving volume in the ν-Z thermal bath.As a result, the expected N eff for g e 10 −8 and g ν 10 −5 is given by, The first factor is due to the delay of the neutrino decoupling, i.e.T ν T γ , while the second factor is due to the in-equilibrium decay of Z .In Fig. 3a, we show the time evolution of the energy densities of ν and Z for g e = 10 −7 , g ν = 10 −4 and m Z = 100 eV.For m Z 1 eV, Z behaves as the dark radiation without heating up the ν temperature.In such a case, the expected N eff is given by, where the second term in the last parenthesis denotes the Z energy density.
(ii) Thermalization with e ± followed by ν-inverse decay For g e 10 −8 but for g ν 10 −5 while satisfying the condition in Eq. ( 8), the mediator remains in equilibrium with γ-e, even after the neutrino decouples as in the standard cosmology.In this case, the temperature of the γ-Z thermal bath after the electron annihilation is given by, After e ± have annihilated away, the (inverse) decay of Z into the neutrinos becomes effective.The (inverse) decay changes the temperature and the chemical potential of ν-Z thermal bath as for µ ν-Z < 0. For µ ν-Z → 0, on the other hand, the conditions are given by Here, T ν denotes the neutrino temperature in the absence of the inverse decay of Z .The mediator distribution is approximated by the Bose-Einstein distribution of the massless particle with the temperature T and the chemical potential µ.The first and the second arguments of the energy/number densities are the temperature and the chemical potential, respectively.The zero momentum contribution to the number density is denoted by n Z , of which the energy density is neglected in the massless approximation.The conditions in Eqs.(32) and (34) are due to the conservation of n ν +2n Z in the (inverse) decay process, which also imposes µ Z = 2µ ν .The above conditions lead to The result in Eq. ( 35) shows that Z exhibits a dilute Bose-Einstein condensation (BEC).In the presence of the BEC, the zero momentum contribution should be treated properly in the Boltzmann equation (see, e.g., Refs.[28,29]).In our numerical analysis, however, we use the Boltzmann equation neglecting the zero momentum contribution.Since n Z is subdominant, our approximation fairly reproduces the expected N eff discussed below (see also Fig. 3b).
The in-equilibrium decay of Z before the CMB era heats up the neutrino temperature as The resultant temperature and the chemical potential of ν are given by, where T ν is again the neutrino temperature in the absence of the inverse decay nor the decay of Z .As a result, we find 2 In Fig. 3b, we show the time evolution of the energy densities of ν and Z for g e = 10 −7 , g ν = 10 −10 and m Z = 100 eV.For m Z 1 eV, the decay of Z takes place after the recombination, and hence, it contributes to N eff as dark radiation as in the previous case.The resultant N eff is given by, N eff 12 . (40) 2 For d Z = 1, for example, we find N eff = 12 for m Z ≥ 1 eV and N eff = 11 for m Z 1 eV.
(v) Thermalization with ν after neutrino decoupling For g e 10 −9 and g ν 10 −5 , the thermalization with neutrino takes place after the neutrino decoupling.Although such parameter region is far off from that favored by the XENON1T excess (see Fig. 2), let us briefly comment on the behavior of N eff in this region.
The resultant temperature and the chemical potential of ν are given by, Here, T ν is again the neutrino temperature in the absence of the (inverse) decay of Z .Then, the in-equilibrium decay of Z before the recombination heats up the neutrino temperature as, n ν (T ν-Z , µ ν-Z ) + 2n Z (T ν-Z , 2µ ν-Z ) = n ν (T ν , µ ν ) .
The resultant temperature and the chemical potential of ν are given by,

A Neutrino-Electron Collision Terms
Here, we present the explicit form of the collision terms for neutrino-electron scatterings used in this work.In the text, we define the collision terms of these processes for the zeroth and first moment i=e,µ,τ C (j) e↔ν i , j = 0, 1 (58) 3 (C e + e − ↔ν i νi + C e ± ν i ↔e ± ν i + C e ± νi ↔e ± νi ) , (59) Using the results of Appendix A.3 of Ref. [22], these are written in the form of T 1 e µ 2 where the numerical factors f FD a = 0.884 and f FD s = 0.829 represent the Pauli blocking effect from the Fermi-Dirac distribution in the annihilation and scattering processes, respectively.Note that we assume that each neutrino flavor has the same temperature and chemical potential.

Figure 1 :
Figure 1: The Feynman diagrams relevant for the Z production.