Shedding New Light on SterileNeutrinos from XENON1TExperiment

Soroush Shakeri, a,b Fazlollah Hajkarim,c She-Sheng Xued

aDepartment of Physics, Isfahan University of Technology, Isfahan 84156-83111, IranbICRANet-Isfahan, Isfahan University of Technology, 84156-83111, IrancInstitut für Theoretische Physik, Goethe Universität, Max von Laue Straße 1,D-60438 Frankfurt am Main, GermanydICRANet Piazzale della Repubblica, 10 -65122, Pescara, Italy,Physics Department, University of Rome La Sapienza,P.le Aldo Moro 5, I–00185 Rome, Italy

E-mail: s.shakeri@iut.ac.ir, hajkarim@th.physik.uni-frankfurt.de, xue@icra.it

Abstract. The XENON1T collaboration recently reported the excess of events from recoilelectrons, possibly giving an insight into new area beyond the Standard Model (SM) of par-ticle physics. We try to explain this excess by considering effective interactions between thesterile neutrinos and the SM particles. In this paper, we present an effective model basedon one-particle-irreducible interaction vertices at low energies that are induced from the SMgauge symmetric four-fermion operators at high energies. The effective interaction strength isconstrained by the SM precision measurements, astrophysical and cosmological observations.We introduce a novel effective electromagnetic interaction between sterile neutrinos and SMneutrinos, which can successfully explain the XENON1T event rate through inelastic scatter-ing of the sterile neutrino dark matter from Xenon electrons. We find that sterile neutrinoswith masses around 90 keV and specific effective coupling can fit well with the XENON1Tdata where the best fit points preserving DM constraints and possibly describe the anomaliesin other experiments.

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Contents

1 Introduction 1

2 Sterile Neutrino and Charged Lepton Couplings 22.1 Sterile Neutrino DM and Coupling Constraints 6

3 Neutrino-Electron Scattering 83.1 SM Neutrino-Electron Scattering 83.2 Sterile Neutrino-Electron Scatterings 9

3.2.1 Nee→ νee 103.2.2 N`e→ N`e 103.2.3 N`e→ ν`e (EM Channel) 103.2.4 νee→ Nee 11

4 Calculation the Electronic Recoil Events 124.1 Incoming Sterile Neutrino 124.2 Outgoing Sterile Neutrino 15

5 Bounds from Stellar Cooling in Astrophysics 15

6 Conclusions and Discussion 16

A Kinematics of the Electronic Recoil and Cross Section 19

B Estimation of the Number of Excess Events 21

1 Introduction

Recently the XENON1T experiment has reported an excess of O(keV) electronic recoil eventsover known Standard Model (SM) background with a statistical significance of ∼ 3σ [1]. Theexcess is peaked around the 2 and 3 keV energy bins in a 1-7 keV recoil energy window. TheXENON collaboration has been unable to exclude the β decay of Tritium as a backgroundresponsible for the surplus events. Even a trace amount of fewer than three 3H atoms perkg of XENON is enough to fit the excess with a 3.2σ significance [2]. Despite viable SMexplanations regarding unresolved backgrounds [3–5], it remains a reasonable possibility thatthis excess is a hint of new physics beyond the SM. The experimental analysis takes the latterinto account and considers the possibility that the excess can be due to a solar axion or asolar neutrino with a sizeable magnetic moment being absorbed by target electrons. However,both of these scenarios are in tension with stellar cooling constraints [6–11].

In the SM, the neutrino-electron scattering as a source of electron recoils, genericallyoriginates from the exchange of the SM gauge bosons W± and Z0 [12–14]. Nonzero mag-netic moments of neutrinos can also provide some additional contributions to the low energyneutrino-electron scattering via electromagnetic (EM) interactions [15–19]. However, the pre-cise value of the neutrino magnetic moment is unknown and it is predicted to be very smallin the SM, proportional to the neutrino masses for Dirac neutrinos [18,20]. The nonstandardinteractions of the SM neutrinos induce large neutrino magnetic moments which can account

– 1 –

for the XENON1T excess events [21–26]. In order to avoid large corrections to neutrinomasses induced by large neutrino magnetic moments, some new symmetries can be intro-duced [22, 27]. Moreover, there are many theoretical attempts to interpret the XENON1Tsignal assuming new physics beyond the SM such as, boosted dark matter [28–33], darkradiation [34, 35], anomalous magnetic moment of muon [24, 36], inelastic DM-electron scat-tering [37–42], axion-like DM [11,43]. Besides, there are some proposals using a gauged U(1)Xextention of the SM, where a gauge boson Z ′ contributes to dark matter (DM) [44–52]. Adetailed study of electron-neutrino scattering in the framework of U(1)X was presented in [53].

In this article, we argue that the sterile neutrino DM can account for the low-energyexcess at XENON1T, while evading cosmological and astrophysical bounds. Right-handedsterile neutrinos are introduced to resolve different theoretical problems in the SM and si-multaneously can be served as a viable DM candidates, see for example Refs. [54–56]. Sterileneutrinos as a warm DM with masses at keV scale may constitute all or a part of galactic DMhalo [57–62]. They can satisfy the bounds from structure formation and the free streaminglength of DM at early epochs [56,58,62], and also explain the deviation of number of effectiveneutrinos measured from cosmic microwave background (CMB) [56, 63]. Moreover, the pres-ence of sterile neutrinos might explain the baryon asymmetry of the Universe [64,65]. As soonas finding any evidences for the sterile neutrino DM and its properties one can gain insightfulinformation about the thermal history of the early Universe and the production mechanismof these particles [56,66].

We present an effective model based on the fundamental symmetries and particle con-tent of the SM, in order to describe the relevant effective interactions of right-handed sterileneutrinos with SM particles at low energies. In this scenario three massive sterile neutrinosare introduced, the SM gauge symmetric four-fermion interactions giving rise to new effectiveinteractions between sterile neutrinos and SM gauge bosons. Using these effective interac-tions, we calculate the scattering cross sections between sterile neutrinos and electrons at theXENON detector. We will show that inelastic scattering of sterile neutrino DM from Xenonelectrons is able to successfully reconstruct the XENON1T event rate. The interaction ofsterile neutrinos with electrons for explaining the XENON1T excess has been recently stud-ied in [67, 68], where an incoming flux of solar neutrinos up scatters to an outgoing state ofsterile neutrinos. In contrast to the case of incoming sterile neutrinos, their results do notrely on the nature of sterile neutrinos as DM particles.

This paper is organized as follows. In the next section we explain the framework forsterile neutrino interactions with other SM particles. Then we calculate the possible interac-tion with electrons (Sec. 3) which might lead to the excess of events for XENON1T electronrecoil at low energy. In Sec. 4 we compute the recoil energy of electrons due to sterile neutrinointeractions either being in the initial state or in the final state. Moreover, we discuss thepossible bounds on the parameters of our effective model. Finally, we discuss our results andconclude in Sec. 6.

2 Sterile Neutrino and Charged Lepton Couplings

Chiral gauge symmetries and spontaneous/explicit breaking of these symmetries are key con-cepts to understand the hierarchy pattern of fermion masses in the SM, and furthermoreplay crucial role in developing possible scenarios beyond the SM for the fundamental particlephysics. As demonstrated by neutrino oscillation phenomenon [69], apart from being Diracor Majorana type, neutrinos are massive particles. This fact can imply the existence of the

– 2 –

right-handed neutrinos and raises the questions of how chiral gauge symmetries have beenbroken to generate neutrino masses.

There are many possibilities with various degree of complexity to capture such a newphysics where more unknown particles with different masses, interaction, spin and strengthmight be included in the model. The seesaw mechanism is one of the most economicalsolution to neutrino mass problem which is implemented in three tree-level ideas so-calledas type-I [70–75], type-II [76] and type-III [77]. In some of the models in order to avoidvery small tree-level Yukawa term, new discrete symmetries or continuous global symmetriesare assumed [78, 79]. Another way is to extend the SM with new gauge symmetries such asU(1)B−L [80–83] , or U(1)R [84], such symmetry extensions in the gauge group may leadto introducing several new beyond SM gauge bosons. Moreover left-right symmetric modelswith direct right-handed neutrino interactions and SM gauge group SU(2)L×SU(2)R×U(1)are among the attractive extensions of the SM [85–87].

On the other hand, the theoretical inconsistency [88–91] between the SM bilinear La-grangian of the chiral gauged fermions and the natural UV cutoff of unknown dynamics orquantum gravity requires that an effective field theory possesses quadrilinear four-fermion in-teractions (operators) of the Nambu-Jona-Lasinio (NJL) type [92] at high-energy scales. Onthe basis of only SM gauge symmetries, four-fermion operators of SM left- and right-handedfermions (ψL , ψR) in the charge sector “Q” and flavor family “f” are introduced [54,55,93–97]

∑f=1,2,3

G[ψfLψfRψfRψfL

]Qi=0,−1,2/3,−1/3

, (2.1)

and three SM gauge-singlet right-handed neutrinos ψfR

= νfR1 are in the sector Q = 0,

playing the role of warm dark matter. There are two fixed points (scaling domains) for theeffective coupling G in (2.1). One is the strong coupling ultraviolet (UV) fixed point atcharacteristic energy TeV, where SM gauge symmetries are preserved by composite particles[94]. A phenomenological study of such composite particles at the LHC is recently presentedin [98]. Another one is the weak-coupling infrared (IR) fixed point at the electroweak scale∼ 246 GeV, where the low-energy SM model is realized. The SM gauge symmetries arespontaneously broken, top quark, W± and Z0 gauge bosons acquire their masses [99]. Thehierarchy massive spectra of SM Dirac fermions are resulted from the explicit symmetrybreaking due to the family mixing, and the seesaw mechanism leads to Majorana neutrinomasses [93].

The relevant feature of this model for the present article is that in the IR low en-ergy domain one-particle-irreducible (1PI) interacting vertices between left- and right-handedfermions are induced by the four-fermion operators (2.1). These lead to additional right-handed currents coupling to SM bosons W± and Z0 in the neutrino sector [54, 100] and thequark sector [95, 97]. These 1PI left-right mixing vertices vanish at low energies, giving riseto the chiral (parity-violated) symmetries of SM. However, they do not vanish at high en-ergies, implying that the parity symmetry could be restored at high energy scale [55, 101].Recently, the effective operators of right-handed currents have been considered for studyingLHC physics [102], the circular polarization of cosmic photons due to right-handed neutrinoDM candidate is also considered in [75].

1It is not excluded that they are all identical or related.

– 3 –

Figure 1: The Feynman diagram of the effective right-handed current couplings in Eq. (2.4).Left: The effective vertex of sterile neutrino and lepton coupling to W− with the mixingmatrix [(U `R)†UνR]. Right: The effective vertex of sterile neutrino and anti-sterile neutrinocoupling to Z0.

We consider right-handed sterile neutrinos as the DM candidate effectively couple to theSM gauge bosons via the right-handed charged and neutral current interactions as [54,55,97]

L ⊃ GR(gw/√

2) ¯Rγ

µν`RW−µ + FR(gw/

√2) ν`Rγ

µν`RZµ + h.c. (2.2)

where GF /√

2 = g2w/8M

2W and ν`R stands for the right-handed neutrino in the same family

of the charged lepton ¯R, ν`R and `R can be considered as a right-handed doublet (ν`R, `R) in

gauge interacting basis. Summation over three lepton families ` is performed in Eq. (2.2) andno flavour-changing-neutral-current (FCNC) interactions occur. The GR and FR representeffective coupling vertices, in the momentum space, given by (see Fig. 1)

ΓµW = iGR(p1, p2)√

2γµgwPR , ΓµZ = i

FR(p1, p2)√2

γµgwPR . (2.3)

They are complex functions of momenta (p1, p2), and should be suppressed at low energies,as required by SM precision measurements and cosmological constraints. At the electroweakscale v = (GF

√2)−1/2 ≈ 246 GeV, GR ≈ FR ∝ (v/Λcut)

2. The UV cutoff Λcut is the scale atwhich the four-fermion operators (2.1) representing unknown new physics become relevant.

In terms of mass eigenstates (N lR, lR), gauge eigenstates ν`R = (UνR)`l

′N l′R and `R =

(U `R)`l′l′R, where U

νR and U `R are 3× 3 unitary matrices in family flavor space, the 1PI inter-

actions (2.2) take the following form

L ⊃ GR(gw/√

2) [(U `R)†UνR]ll′lRγ

µN l′RW

−µ + FR(gw/

√2) N l

RγµN l

RZµ + h.c. (2.4)

and flavor mixing matrix [(U `R)†UνR] appears in charged current interaction, while neutralcurrent one remains diagonal in lepton family flavor space. The off-diagonal elements ofmixing matrix [(U `R)†UνR] shows interactions between different flavour families through thecharged current channel, for example N e

R−τR. This in fact gives the flavour oscillations of thesterile neutrinos [94]. The mixing matrix [(U `R)†UνR] is not the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [(U `L)†UνL] in the sector of left-handed leptons and neutrinos. Thatis due to the transformations from the mass eigenstates (lL, ν

lL) to the gauge eigenstates

ν`L = (UνL)`l′νl

′L and `L = (U `L)`l

′l′L. Since there is no information about the mixing matrix

– 4 –

W−

``

γ

ν`N`

Figure 2: The effective electromagnetic (EM) interaction vertex of normal and sterile neu-trinos. The PMNS mixing matrix (UνLU

`L) associates to the SM coupling vertex νLγµ`LWµ.

This effective vertex leads to the dominate radiative decay of right-handed sterile neutrinoNl → νl + γ .

[(U `R)†UνR], we assume in this article that off-diagonal mixing elements are very small, namely[(U `R)†UνR]ll

′ ≈ δll′ . It is a reasonable assumption due to the hierarchy structure of charged

lepton masses and thus the diagonal element values of order of unity have been absorbed inthe GR coupling.

The sterile neutrinos via interaction (2.4) at loop level can interact with photons whichis given by Feynman diagram in Fig. 2, where SM gauge boson W and charged lepton arepresent in the loop. In fact, this is an EM channel of SM neutrino and sterile neutrinointeraction. It can be represented as an effective operator

(UνLU`L)ll

′νlLΛµl′N

l′RAµ + h.c., (2.5)

where Aµ is the electromagnetic field and (UνLU`L) is the PMNS mixing matrix associating to

the SM vertex νlγµPL`lWµ in the loop. In the momentum space, the 1PI vertex Λµ is givenby

Λµl′(q) = ieg2wGRml′

16π2

[(C0 + 2C1)pµ1 + (C0 + 2C2)kµ1

]. (2.6)

Here pµ1 and kµ1 denote the four-momenta of incoming sterile neutrino and outgoing SMneutrinos, respectively. The coefficients C0, C1 and C2 are the three-point Passarino-Veltmanfunctions [103],

Ci ≡ Ci(m2N , q

2,m2ν ;mW ,ml,ml)

q2→0−−−→ Ci(m2N , 0,m

2ν ;mW ,ml,ml) ∝ m−2

W , (2.7)

where i = 0, 1, 2 and zero momentum transfer limit q2 = (k1 − p1)2 → 0 (Appendix A)is a good approximation for the evaluation of the loop integral at the low energy regimeleading to Ci ∝ m−2

W . To compute these functions we use the Package-X program [104].The effective operator in Eq. (2.5) represents a novel electromagnetic property of normalneutrino and sterile neutrino coupling to photon, stemming from the effective right-handedcurrent coupling in Eq. (2.4). The same type operator in the quark sector can be obtained,and they associate to the effective Dirac mass operator νLNR (qLqR) of neutrino (quark) bythe Ward-Takahashi identity [93]. This is different from the effective operator νσµννFµν ofneutrino electromagnetic moment, which has been intensively discussed so far in the literature

– 5 –

[15,18,20,85,105,106]. Under the CP transformation of field Aµ → −Aµ and four-momentumpµ → pµ [106], it can be shown that the effective vertex Λµl′ → −Λl′µ and Ci(q

2) are CPinvariant. Therefore the effective operator of Eq. (2.5) is CP invariant, except the CP phases inthe PMNS and [(U `R)†UνR] mixing matrices, both are approximately considered as an identitymatrix in this article.

To this end, we have presented a scenario relevant for studying the electron recoil inXENON1T experiment. There are three massive sterile neutrinos N e

R (meN ), N

µR (mµ

N ) andN τR (mτ

N ). They effectively interact with SM charged leptons eR, µR and τR, respectively,through the SM gauge bosons W± and Z0 of weak interaction. Effective EM interaction ofsterile neutrinos with SM left-handed neutrinos νeL, ν

µL and ντL is through the SM photon γ.

The effective coupling strength GR at low energies is a theoretical parameter to be determinedexperimentally. In such a scenario, four-fermion operators of Eq. (2.1) induce right-handedcurrent couplings [93–95]. We do not need to add any fiducial beyond the SM gauge boson,such as WR,W

′, Z ′ or light particles, except right-hand neutrinos. In the following, we firstconstrain the coupling strength GR from SM precision measurements, cosmological and astro-physical observations, then apply this scenario to compute the relevant observable quantitiesin XENON1T experiment.

2.1 Sterile Neutrino DM and Coupling Constraints

It is essential that the contribution of right-handed sterile neutrinos to the total decay widthof the W gauge boson not to exceed the experimental accuracy of W decay width, 4.2× 10−2

GeV. This gives a weak constraint on the right-handed current coupling as GR . 6×10−3 [75].A more tighter constraint can be obtained from the radiative decay of massive neutrinos[107,108]. It will be shown that the dominate decay channel of massive sterile neutrinos N l

R

is the radiative decay process (Fig. 2) with the corresponding rate as

Γ(N lR → νlL + γ) =

(α g4

w

1024π4

)m2l (m

lN )3G2

R

[(C0 + 2C1)2 + (C0 + 2C2)(C0 + 2C1)

]. (2.8)

For different sterile neutrinos at low energy scale we have

Γ(N eR → νeL + γ) = 1.57× 10−19s−1

(GR

10−4

)2 ( me

511 keV

)2(

M eN

100 keV

)3

, (2.9)

Γ(NµR → νµL + γ) = 6.70× 10−15s−1

(GR

10−4

)2 ( mµ

106MeV

)2(

MµN

100 keV

)3

, (2.10)

Γ(N τR → ντL + γ) = 1.87× 10−12s−1

(GR

10−4

)2 ( mτ

1.77GeV

)2(

M τN

100 keV

)3

, (2.11)

showing that decay rates get larger values for heavier sterile neutrinos and more massiveinternal leptons. Suppose that sterile neutrinos N e

R, NµR and N τ

R are DM candidates requirethat their lifetimes to be at least 4.4×1017 sec of Universe age. This yields further constraintson GR for each type of sterile neutrinos as

GR . 3.8× 10−4

(511 keV

me

)(100 keV

meN

)3/2

, (2.12)

– 6 –

GR . 1.84× 10−6

(106 MeV

mµ

)(100 keV

mµN

)3/2

, (2.13)

GR . 1.1× 10−7

(1.77 GeV

mτ

)(100 keV

mτN

)3/2

. (2.14)

In the seesaw mechanism, there is a transition between active SM neutrinos and sterile neutri-nos which is induced by a "mixing angle". This mixing would impact on the relic abundance ofsterile neutrinos indirectly through the interaction of generated active neutrinos with thermalbath of SM particles [60, 66, 107]. However in the model presented here, there is an effectiveinteraction between right-handed sterile neutrinos and SM gauge bosons (2.4) leading to theeffective scattering of sterile neutrinos from SM particles (Fig. 2). These interactions containthe effective coupling to right-handed currents (see diagrams in Fig. 1), and can be com-pared to those effective interactions in seesaw model, which are induced by the mixing angleand follows by a conversion process between active and sterile neutrinos. If one replaces theright-handed coupling GR in Eq.(2.8) by active-sterile mixing angle θNν , the only deviationfrom the radiative decay rate obtained in the context of seesaw model [62, 105, 107, 109], isthe appearance of m2

`M3N instead of M5

N . This specific property results from the effectivecouplings (2.4) and (2.5) of right-handed sterile neutrinos to SM sector. Although in seesawmodel the radiative decay channel N → ν + γ is a sub-dominant process and the lifetimeof sterile neutrino DM is determined by N → νlνανα [107, 110], the latter is absent in thecontext of our model and the radiative decay of sterile neutrinos turns out to be the dominantdecay channel.

There are various theoretical frames explaining sterile neutrino with keV-scale masseswhich can provide the relic density of DM [111, 112]. We estimate the relic density of sterileneutrinos by using the interaction rate and the Hubble rate during the radiation dominatedera [113–117]

ΓN = nN 〈σv〉 ≈ G2FG2

RT5 , H =

√4π3gρ

45

T 2

MPl≈ 1.66g1/2

ρ

T 2

MPl, (2.15)

where the energy density degrees of freedom is shown by gρ [118] and nN denotes the sterileneutrino number density. When ΓN ≈ H, sterile neutrinos decouple at temperature

TN ≈ 10 GeV(

10−6

GR

)2/3

g1/6ρ , (2.16)

note that SM neutrinos decoupled at Tν ≈ 1 MeV. Since the interaction of sterile neutrinos isweaker than SM neutrinos, they decoupled earlier than SM types in the early Universe. Thenumber density of sterile neutrinos is obtained as

ρN = mNnN = mNYNnγ , (2.17)

where YN = nN/nγ is the fraction of density of sterile neutrinos relative to the densityof photons. Therefore the relic abundance of keV neutrinos produced during the radiationdominated epoch can be found as [117]

ΩNh2 ≈ 76.4

[3gN

4gs(TN )

](mN

keV

), (2.18)

– 7 –

which gN is the sterile neutrino degrees of freedom and gs denotes the entropy density numberof relativistic degrees of freedom of thermal bath particles [118]. DM relic density presentedin Eq. (2.18) obviously demands entropy dilution process after freeze-out, to bring down theabundance consistent with the current DM relic density obtained from CMB observationsΩDMh

2 = 0.120 ± 0.001 [119]. If the coupling limit does not match with the DM producedin the radiation dominated era, a possible matter dominated or kination scenario can explainthe mismatch of the relic abundance [58,120]. Besides, there is another possibility which canbe considered. Then we might consider the oscillation of keV sterile neutrino to heavier sterileneutrinos after its decoupling [93], analogous to the SM neutrino oscillation. Sterile neutrinosdecay to the SM particles, leading to keV DM relic abundance observed today. Moreover,considering sterile neutrinos may influence on the extra number of effective neutrinos ∆Neffdecoupled from the thermal bath in the early Universe [56]. The parameter ∆Neff is con-strained by CMB [119]. Its exact value in our scenario also depends on the cosmic historybefore the big bang nucleosynthesis and after the inflationary era [56, 63] which is unknownthen we will not consider its computation here.

3 Neutrino-Electron Scattering

3.1 SM Neutrino-Electron Scattering

In the low energy regime of the SM, when the momentum carried by intermediate vectorbosons W± and Z0 are much less than their masses, the effective Lagrangian describing theneutrino-electron scattering via the charged current (CC) or neutral current (NC) are givenby

LCCeff = −2√

2GF

([νeγ

µPLue][ueγµPLνe]), (3.1)

LNCeff = −√

2GF∑

α=e,µτ

(ueγβ(gαLPL + gαRPR)ue

)[ναγ

βPLνα] , (3.2)

where PR,L = 12(1 ± γ5), and gαR = 2 sin2 θW ± 1 with ” + ” for electron neutrino flavour

α = e, and ” − ” for α = τ, µ and gαL = 2 sin2 θW for all the neutrino flavours α = e, µ, τ .Note that sin2 θW = 0.23 represents the weak mixing angle. The corresponding cross sectionof the neutrino-electron scattering induced by these low-energy current interactions is givenby [12,14,121,122]

dσeνα→eναSM

dEr=

2G2Fme

π

[(gαL)2 + (gαR)2

(1− Er

Eν

)2

− gαRgαLmeErE2ν

], (3.3)

where Er is the recoil energy of electron in the detector and Eν is the energy of incomingneutrinos2.

2Note that this cross section can be also re-written in terms of gV and gA where gV,A = 12(gR ± gL) as

dσSMdEr

=G2Fme

2π

[(gV + gA)

2 + (gV − gA)2(1− Er

Eν)2 + (g2A − g2V )

2meErE2ν

]. (3.4)

– 8 –

3.2 Sterile Neutrino-Electron Scatterings

We compute the contribution of sterile neutrinos to the electron recoil event. In the modelpresented in Sec. 2, sterile neutrino interactions with electrons give rise to additional contri-butions to the recoil process. The effective Lagrangian describing the low energy scatteringbetween sterile neutrinos and electrons regarding effective vertices presented in Fig. 1, aregiven by

LSCCeff = −2√

2GF

([NγµGRPRue][ueγµPLνe] + [NγµGRPRue][ueγµGRPRN ]

)(3.5)

= −2√

2GF

([NγµGRPRue][ueγµPLνe]

)+GFO(G2

R)

and

LSNCeff = −√

2GF

(ueγβ(gLPL + gRPR)ue

)[FRNγβPRN ] . (3.6)

In principle there are two possibilities in which sterile neutrinos can participate in the electronrecoil event;

I) Sterile neutrinos may appear as incoming particle contributing to the galactic DM halowhere initial flux of sterile neutrinos can be inferred from the local density of DM, inthis case our final states are recoiled electrons and SM neutrinos.

II) Sterile neutrinos may appear as outgoing particle whether or not dark matter is madeof sterile neutrinos, in this case initial flux is provided by solar neutrinos leading toemission of sterile neutrinos after recoil events.

In general, scattering processes can be divided into two types; elastic and inelastic. It is cru-cially important to clarify, in which types of scattering processes and also scattering channels,sterile neutrinos can produce recoil signal in the desired energy range detected by XENON1Texperiment. It has been recently shown that an elastic scattering process between a DMparticle and electron, only if DM satisfies velocity vDM ∼ 0.1 and mass mDM & 0.1 MeV, canproduce the electron recoil excess in the energy range of O(keV ) [30,123,124]. While such ahigh velocity is far above the local escape velocity of the Milky Way ∼ 10−3, it is argued thata long range attractive force might accelerate DM towards the Earth and boosted to highvelocities near the Earth surface [31]. In contrast to the elastic scattering where the kineticenergy of a non-relativistic light DM is not sufficient to produce recoil electron at keV scale,an inelastic DM-electron scattering thanks to the momentum transfer in the order of ∼ mDM

leads to the electron kinetic energy Te ≈ m2DM/2me = 2.45 keV(mDM/50 keV)2 at the final

state [37, 123–127].The DM inelastic scattering from atomic electrons may lead to electron ionization or

electronic excitation, and DM-nuclear scattering may cause molecular dissociation [128–130].For electron ionization, the total energy EN of an incoming sterile neutrino must be largerthan the energy of a bound electron in level i, this defines a threshold condition for the processto occur in the various bound states (see Eq. 4.2). It has been shown that the scatteringfrom bound electrons leads to larger recoil than the free electron case [125, 129]. In fact theeffect of bound electrons in atomic shells is important at low-energy electronic recoil arounda few keV and causes some deviations from the approximation of free electron [129–134].

In the following we consider inelastic scattering of the sterile neutrinos with electronstaking into account effective interactions given by Eqs. (3.5) and (3.6).

– 9 –

W−

eRNe

νeeL

Z

NlNl

ee

Figure 3: Feynman diagrams corresponding to sterile neutrino-electron scattering throughboth the charged current and neutral current. In these diagrams time flows from left to right.

3.2.1 Nee→ νee

The differential cross section of Ne → νee corresponding to the first diagram in Fig. (3), isobtained as follows

dσ(Nee→ eνe)

dEr=

2G2F

π

G2Rme

|~pN |2EN

[EN +

m2N

2me

], (3.7)

where |~pN | = (E2N −m2

N )1/2 is the momentum of sterile neutrinos. We assume that the sterileneutrinos as DM have the same velocity as in the standard halo model vN ≈ 220km/s andEN ∼ mN , for sterile neutrinos with masses in O(keV) range

dσ(Nee→ eνe)

dEr≈ 1.94× 10−53

(GR

10−6

)2(10−3

vN

)2

cm2[keV−1

], (3.8)

obviously right hand side of Eq. (3.7) is independent of the recoil energy Er and only providea flat recoil spectrum which can not be used to explain XENON1T excess at low energy.

3.2.2 N`e→ N`e

If we consider the elastic scattering between sterile neutrinos and electrons (second diagramin Fig. (3)), the differential cross section is given by

dσ(N`e→ N`e)

dEr=G2F

2π

G2Rme

|~pN |2[(geL)2(EN − Er)2 + (geR)2E2

N − geLgeR(m2N −meEr)

]. (3.9)

Since in such an elastic scattering, there is not enough momentum transfer provided by non-relativistic keV mass sterile neutrinos, it is not possible to produce observable signal from theelectron recoil in N e→ N e process. Note that an extra contribution to N e→ N e originatesfrom W exchange which is at higher order O(G4

R) and we ignore it here.

3.2.3 N`e→ ν`e (EM Channel)

Here we consider electron-neutrino scattering via electromagnetic (EM) channel [15, 106].The neutrino magnetic-moment µν contribution to the electron-neutrino scattering process is

– 10 –

given by [20]

dσ(eνe → eνe)

dEr=πα2

m2e

(1

Er− 1

Eν

)(µνµB

)2

, (3.10)

where µB is the Bohr Magneton. For neutrino-electron scattering the magnetic momentcontribution is dominant over the standard electroweak contribution at low recoil ener-gies. However, a strict astrophysical bound on the neutrino magnetic moment is providedby the observation of properties of globular cluster stars and stellar cooling process µν .10−12µB [9, 135, 136]. The upper bound for the neutrino magnetic moment gives dσ/dEr ≈10−45cm2keV−1(keV/Er), it is claimed in Ref. [1] that this 1/Er enhancement at low en-ergy could provide an explanation for XENON1T excess but with less statistical significancecompared to other alternative possibilities.

In the following, we consider the sterile neutrino EM scattering from electrons throughlepton loop correction (see Fig. 4), namely through the effective EM vertex of Eq. (2.5) orFig. 2. To find the cross section of Ne→ νle regarding leptons l = e, µ, τ at loop level of thesame family of SM and sterile neutrinos, we use Eq. (A.7) and the kinematics used for itsderivation. Then the differential cross section is given by

dσ(N`e→ νle)

dEr=

(m2l

8πme q4 p2N

) (α g2

w GR4π

)2

(2mNmν + 2meEr +M2N )× (3.11)

(2C1 + C0)2 [me(2meEN +M2N )(EN − Er))]+

(2C2 + C0)2 [me(2MNEr +M2N )(EN − Er)]+

(2C1 + C0)(2C2 + C0) [2m2e(E

2N + E2

r − 2ENEr) +mem2N (EN − Er)]

.

The recoil energy transfers to an electron through the inelastic scattering of sterile neutrinoq2 = (k1 − p1)2 = 2meEr (Appendix A). Here we assume vanishing neutrino mass mν ≈ 0in favour of the sterile neutrino and electron masses. To give more insight at the low energyscale, Eq. (3.11) can be approximated as

dσN`e→νledEr

∝

(αGFGRm`

4π3/2vNErm1/2e

)2

(M2N + 2meEr) . (3.12)

3.2.4 νee→ Nee

The cross section of neutrino-electron scattering when a sterile neutrino contribution as anoutgoing particle is

dσ(νee→ Nee)

dEr=

(G2FG2

R|~pN |2

2πEνE2N

) [m2N + 2meEr

](2meEν −m2

N )

me(Er + |~pN |) + Eν(|~pN | − EN ) +m2N

, (3.13)

here EN = Eν − Er is the energy of outgoing sterile neutrinos in terms of the energy ofstandard model neutrinos Eν and the recoil energy of electrons Er, and |~pN | =

√E2N −m2

N .Our estimation shows this process will not provide a significant contribution to the XENON1Texcess (see Appendix B for the outgoing process). We will not compute the cross section forthe process ν`e→ N`e at one loop level like Sec. 3.2.3 when the mediator is photon. Since itwill not give a large enough cross section to explain the excess.

– 11 –

W−

``

γ

ν`N`

ee

Figure 4: Sterile neutrino-electron inelastic (EM channel) scattering.

4 Calculation the Electronic Recoil Events

We consider two different possibilities for the sterile neutrino interacting with electrons in theXENON detector which might explain the reported anomaly.

4.1 Incoming Sterile Neutrino

The differential number of events in reconstructed electron recoil energy Erec, for the inelas-tic scattering of sterile neutrino DM with electrons per unit detector mass as observed byXENON1T is

dR

dErec= NT

ρDMmN

∫dEr

∫ vesc

vmin

dv v f(v)dσtotdEr

ε(Er)G(Erec, Er) . (4.1)

Here ρDM is the local DM density, mN is the mass of the sterile neutrino, f(v) is the DM halovelocity distribution in the lab frame, ε(Er) is the detector response efficiency and dσtot/dEris the differential cross section (see Appendix B for the estimation of event rate). We use astep-function approximation for the energy of bound electrons in XENON atoms [1,131,132]

dσtotdEr

=54∑i=1

Θ(Er −Bi)dσ

dEr, (4.2)

where the binding energy for an electron in shell i is Bi using the data of Ref. [137]. Infact, one would expect a sharp enhancement in the differential cross section when reachingto a new atomic shell, the stepping approximation in Eq. (4.2) is used to set an upperbound here [131, 132]. This approximation is consistent with more robust calculations in-volving the internal structure of the XENON nucleus [138]. Meanwhile, we use the datafrom Ref. [1] for the detector efficiency ε(Er) as a function of recoil energy Er. The num-ber of events are then smeared with a Gaussian distribution G(Erec, Er) with σ(Er)/Er =(0.3171/

√Er[keV ]) + 0.0015 to account for the response of the photo-multiplier tubes as a

function of the reconstructed recoil energy in the XENON1T detector [139]. For integratingthe halo velocity distribution we use closed-form analytic expressions [140–142] and assumea standard DM halo with ρDM ' 0.4 GeV cm−3 [143], the local circular velocity v0 = 220

– 12 –

km/s, the most probable velocity ve = 232 km/s and the escape velocity vesc = 544 km/s.The minimum energy that includes the relevant minimum velocity is described in AppendixA.

Fitting with the XENON1T Excess

We fit the XENON1T data with our model for the incoming sterile neutrino DM scattersfrom electrons to the SM neutrinos using Eq. (4.1) to obtain the required number of excessevents. Moreover, we consider the χ2 test to check the reliability of our fit. As a consequence,we use [25,31,144]

χ2 =∑j

1

σ2j

[dRexpdErec

− dRthdErec

]2

, (4.3)

where σi is the variance of experimental data by XENON1T. The statistical significance ofour fit can be identified by δχ2 = χ2

min − χ2bg where χ2

bg ' 46.5 for the background estimatedby the XENON Collaboration [31, 144]. The parameter χ2

min is the minimum value of χ2

in Eq. (4.3) that can be found by scanning over a range of values for the coupling GR andthe sterile neutrino mass m`

N . The statistical importance of background is shown by χ2bg

calculated from the background estimation and the measured data points by the XENON1Texperiment. To match the presence of sterile neutrino DM with the data of XENON1T excessone requires δχ ' 3.2 which is reported by the experiment [1]. The scattering processes ofincoming sterile neutrinos given by the cross sections in Eqs. (3.7) and (3.9), do not have 1/Erenhancement factor to reproduce event excess at low energy recoil spectrum. However, theymay have a spectrum at higher energies assuming a large enough coupling that is constrainedby DM lifetime in Eq. (2.12). Then, in practice we can not have any observational effect fromthese processes in XENON1T experiment.

However, the sterile neutrino loop interaction described in Fig. 4 and computed inEq. (3.11), besides giving enhancement recoil energy factors at low energy, due to the presenceof the internal loop leptons masses enhance the cross section. Consequently, it can providethe sufficient event rate for specific choices of coupling and mass.

The value of chi-squared for the fitting parameters used in Figs. 5a, 5b and 5c whenwe include all measured data points are χ2|e ' 38.3, χ2|µ ' 38.3 and χ2|τ ' 38.4 thatimplies δχ|e ' 2.9, δχ|µ ' 2.9, and δχ|τ ' 2.8, respectively. These results are obtained byconsidering the first data point around 1 keV reported by the experiment. If one assumessome deviation from that data point due to the low resolution of detector at low energies,then it is possible to increase the GR values to get a better fit by considering the minimum ofchi-squared. These values respect the bound from lifetime of sterile neutrinos in Eqs. (2.12)- (2.14) as DM candidate. The value of relic abundance does not put a strong bound on thesterile neutrino since to satisfy the relic density constraint one can consider a nonstandardcosmology scenario which was dominated prior to big bang nucleosynthesis [56,66].

The best fit results presented in Figs. 5a - 5c weakly depend on the sterile neutrino massme,µ,τN ∼ 90 keV of each family, but GR values are rather different and respecting to the DM

constrains obtained from Eqs. (2.12) - (2.14). On the basis of these results, we present thefollowing physical interpretations in terms of (GR,M l

N ) values, for massive sterile neutrinosN eR (me

N ), NµR (mµ

N ) and N τR (mτ

N ):

– 13 –

0 5 10 15 20 25 300

20

40

60

80

100

120

Er[keV]

Events/(tonne.y.keV

)

(a)

0 5 10 15 20 25 300

20

40

60

80

100

120

Er[keV]

Events/(tonne.y.keV

)

(b)

0 5 10 15 20 25 300

20

40

60

80

100

120

Er[keV]

Events/(tonne.y.keV

)

(c)

Figure 5: In the left panels events versus recoil energy including the error bars are shown in bluecolor. The solid red line is the background model computed by XENON Collaboration [1]. Theadditional recoil due to the sterile neutrino DM interaction with the Xenon electrons are shown bygreen solid curve (the best fit points in the right panels are shown by light green points). To find theminimum χ2 we scanned over the coupling GR and the sterile neutrino mass ml

N considering differentcharged leptons in the loop (right panels). In case electron (upper panels), muon (middle panels) andtau (lower panels) are in the loop of diagram of Fig. 4 the value of coupling and mass of best fit pointsare [GR,ml

N ] ' a) [1.4× 10−4, 90.6 keV], b) [6.4× 10−7, 90.6 keV] and c) [4.0× 10−8, 90.6 keV]. Theorange lines in right panels are lifetime constraint for sterile neutrinos to be the DM candidate.

– 14 –

(i) If GR ∼ O(10−4), only N eR is present today as DM component and its mass me

N ∼ 90keV, and Nµ

R and N τR have already decayed to SM particles, e.g., photons and SM

neutrinos;

(ii) If GR ∼ O(10−6), sterile neutrinos N eR and Nµ

R are present today as DM particles. N τR

has already decayed to the SM particles. The main contribution to the XENON1Telectron recoil comes from the µ channel. This indicates mµ

N ∼ 90 keV;

(iii) If GR ∼ O(10−7), all sterile neutrinosN eR, N

µR, andN

τR are present today as DM particles.

The main contribution to the XENON1T electron recoil comes from the τ channel. Thisindicates mτ

N ∼ 90 keV.

These cases are realized by assuming the mass hierarchy meN . mµ

N . mτN and also neglect-

ing family mixing phenomenon. Regarding the XENON1T electron recoil data, there is adegeneracy between three scenarios. In order to break the degeneracy and figure out whichsituation was selected by the nature, we need other independent experiments/observations tofurther constrain/determine the values of GR and ml

N .

4.2 Outgoing Sterile Neutrino

When the SM neutrino inelastically scattered off the electron producing the sterile neutrinos,the number of events per electron recoil energy Erec, per unit detector mass as observed byXENON1T reads

dR

dErec= NT

∫dEr

∫Eν |min

dEνdΦν

dEνPν

dσtotdEr

ε(Er)G(Erec, Er) . (4.4)

Here NT is the number of target electrons in the detector per unit mass, dΦν/dEν is the low-energy solar neutrino differential flux on Earth per the incoming neutrino energy Eν takenfrom [145, 146]. The minimum energy of neutrinos to produce the recoil energy is denotedby Eν |min. The parameter Pν is the survival probability of one family of neutrinos (e.g.electron neutrino) arriving at Earth including matter effects and conversion from the twoother neutrino families [147, 148], dσtot/dEr is the analogous form of Eq. (4.2) for neutrino-electron scattering (see Appendix B for the rough estimation of number of events). UsingEq. (4.4) and differential cross sections νe → Ne at tree level in Eq. (3.13) and its EMchannel obtained from the Feynman diagram similar to Fig. 4 and Eq. (3.11) with exchangingsterile and SM neutrinos. Suppose that GR value preserves SM precision measurements andastrophysical constraints (which will be discussed in the following section), one can not getenough excess to explain XENON1T results.

5 Bounds from Stellar Cooling in Astrophysics

It is important to check the consistency of beyond SM couplings (2.4), and (2.5) with ob-served astrophysical data and to clarify the available parameter space (GR,ml

N ) required tofit XENON1T excess data points. The right-handed current couplings (2.4), and their in-duced neutrino and sterile neutrino EM coupling (2.5) have the potential to affect on thestellar energy loss through a plasmon decay channel γ∗ → Nlνl. In order to prevent fromanomalous cooling during stellar evolution of the sun, red giants (RGs), horizontal branchstars, supernova, white dwarf, and etc, [136,149,150], we need to put some constraints on themodel parameters.

– 15 –

Sterile neutrino can be easily produced only if sterile neutrino mass mN are smallerthan the plasma frequency ωp, instead, when mN > ωp the stellar cooling driven by sterileneutrinos is highly suppressed due to kinematical reasons. The most strict astrophysicalcooling constraint can be obtained from the RGs where the core temperature of a typical RGjust before helium ignition was estimated as 18 keV [136,150]. Obviously, this temperature iswell below our best fit mass in the range 50 keV. mN . 100 keV, causes less important effectson the cooling process and therefore weaker constraint on GR. However, we can estimate anupper bound on GR using the similar approach of [151] by defining an effective transitionfactor from Eq. (2.5) and (2.6) [see Fig. 2], effective transition dipole moments in active-sterile neutrino mixing have been previously studied in [67,152–155]. The effective transitionmagnetic moment operator µeff νLσµνNRF

µν has been considered more recently in [25, 151,156,157] in order to explain XENON1T excess. It was shown that while the magnetic momentof active neutrino required to explain the XENON1T excess is µν ' 2× 10−11µB, the active-sterile neutrino transition moment needed for the XENON1T excess is µeff ' 10−10µB [151].

By qualitatively comparing the neutrino and sterile neutrino EM coupling (2.5) withthe effective transition magnetic moment µeff, and the decay rate (2.8) with the decay rateΓµeff = µ2

effm3N/(16π) [158], we arrive at an estimation

µeffµB∼ GFme

4√

2π2GRm` ≈ 1.87× 10−12

(GR

10−2

)( m`

1.77GeV

), (5.1)

in order to obtain the upper bounds on GR from astrophysical data that have imposed theconstrain on µeff. By setting the upper bound µeff using more stringent constraint on neutrinomagnetic moment µν . 2.2 × 10−12µB inferring from the RGs cooling observations [9, 151],we can specify allowed parameter region of GR for each types of the sterile neutrinos in orderto be consistent with cooling data can be defined as (N τ

R,GR . 1.17×10−2), (NµR,GR . 0.19)

and (N eR,GR . 40.83) from Eq. (5.1). These bounds are much weaker than the DM bounds

(2.12), (2.13) and (2.14) and also the range of GR values used to fit XENON1T electron recoilexcess in Figs. 5a - 5c. This shows that our results are consistent with stellar cooling data.There are various types of radiative interactions giving contributions to the stellar coolingprocess [159]. While some of the astrophysical observations are subject to large uncertaintiesin both the measurement and the model [160], a large region in the parameter space (GR,mN )preserving DM constraints (2.12) - (2.14) is offered to be consistent with both the astrophysicalobservation and XENON1T new results.

6 Conclusions and Discussion

In the original XENON1T paper [1], it is argued that solar axions can provide more satisfac-tory interpretation while the other explanation such as neutrino magnetic moment or evensome possible systematic errors such as the energy spectrum of tritium decay obtained lessstatistical significant [5]. In spite of providing more satisfactory statistical significant by solaraxions, the axion couplings required to fit the data are ruled out by several astrophysicalbounds on stellar cooling [9–11, 161–163]. As an important example it was recently claimedthat the solar axion explanation of the XENON1T excess is in strong tension with the RGcooling bounds [6].

In this paper, we made an attempt to understand the low-energy excess of electronicrecoils in XENON1T experiment regarding the sterile neutrino DM in the keV mass rangewhich can preserve different bounds. Our scenario utilizes an effective theory that couples

– 16 –

the right-handed sterile neutrino to the SM particles. We investigated two different caseswhere sterile neutrinos can interact with Xenon electrons either as incoming or as outgoingparticles. We showed that the excess of events can be explained by inelastic scattering ofsterile neutrinos from electrons through EM channel of Eq. (3.11) in Fig. 4, assuming thatincoming flux of sterile neutrinos is provided by the local DM density. During this processthe mass of the initial sterile neutrino can be converted into kinetic energy of outgoing SMneutrinos and recoil electrons in final-state. It is shown in Figs. 5a - 5c that the event rateobserved by XENON1T can be reconstructed for the three different families of sterile neutrinoswith mass me,µ,τ

N ∼ 90 keV and different values of GR where best-fit points preserving DMconstraints (2.12) - (2.14). In the case of appearing the sterile neutrino at final state, thesolar neutrino flux arrives at detector on the Earth and scatters from Xenon electrons, theprocess can not produce enough recoil event at low energy. This is due to small values ofGR limited by SM precision measurements and astrophysical tests in our model, and also lessflux of solar neutrinos compare to local density of sterile neutrino DM.

The rich phenomenology of the scenario presented in this article by introducing threemassive sterile neutrinos couple to the SM gauge bosons W± and Z0 through effective right-handed current interaction (2.4), and its new effective EM interaction (2.5) of SM neutrinos νlLand sterile neutrinos N l

R might explain a wide range of anomalous observational evidences [62,164, 165] besides satisfying essential astrophysical constraints [136]. The main advantages ofour model compare to large growing numbers of beyond SM models on neutrino hidden sector,is that besides having minimal numbers of free parameters, it relies only on the fundamentalsymmetries and particle content of the SM. It is noteworthy to study this model to explainthe significant excess of electron-like events detected in LSND and MiniBooNE experiments[159,166–168]. The scattering between solar neutrinos and electrons taking place in Borexinoexperiment [169] may also provide further restrictions on (GR,ml

N ) via SM neutrino-sterileneutrino interaction [170–172]. Moreover, in astrophysics the observed 3.5 keV line from thecenter of Galaxy [173,174], can be considered in the scenario presented in this article. We leavemore detailed consideration of different anomalies detected in experiments and observationsfor further investigations in near future.

Different scenarios to explain XENON1T excess are required to be confirmed by furtherinvestigations and more precise instruments. In addition to explain XENON1T anomaly,the scenario presented in this article has some distinctive features which can be used todistinguish between our scenario and other beyond SM proposals. The new effective EMvertex [Fig. 2] should give non-vanishing corrections to lepton Anomalous Magnetic Moment(AMM), and CP violating phases in PMNS and [(U `R)†UνR] mixing matrices provide some newcontributions to charged lepton Electric Dipole Moments (EDMs). These new contributionscould potentially change EDM and AMM lower bounds to be within future experimentalsensitivities and more studies are needed in these directions.

In spite of the sterile neutrino-electron scattering through EM channel (Fig. 4), thescattering via charged lepton coupling (left in Fig. 3) leads to asymmetry between left- andright-handed recoil electrons. It can be shown that polarized scattering processes such asN + eL/R → ν+ eL/R and N + eR → ν+ eL are suppressed with respect to N + eL → ν+ eR.In the limit of all particles being massless due to the conservation of angular momentum, onecan easily verify that N mainly scatters from eL giving rise to recoil eR. In order to probe thisfeature experimentally, we may define the asymmetry quantity A = (σR − σL)/(σR + σL),where σR(σL) is the cross section of sterile neutrinos with electrons in right-handed (left-handed) helicity state [55, 93, 101, 175]. Non-vanishing asymmetry (A 6= 0) induced by the

– 17 –

charged current can be served as a peculiar signature of our model. However, we knowthat the measurement of the asymmetry would not be easily achieved in practice, especiallyfor small numbers of events. In upgrade phase of XENON experiments or other upcomingDM experiments, if an instrument indirectly detecting the polarization of recoil electrons isimplemented then the measurement of the asymmetry could possibly verify or falsify thisfeature.

In addition to EM channel of the sterile neutrino-electron scattering (3.2.3), our modelpredicts other scattering channels (3.2.1), (3.2.2) and (3.2.4) which may potentially pro-duce observable signals but not in the sensitivity range of current XENON1T experiment.Consequently, the next generation of XENON detectors such as XENONnT [176], LZ [177],PandaX-II [178] and DARWIN [179] can further probe the parameters space (GR,M l

N ) in ourmodel, which will be discussed in our future papers. Thanks to the multiton-year exposuretime of upcoming precise experiments, it may shed light on the dark matter and low-energyneutrino physics beyond SM.

Acknowledgments

The authors are very thankful to Rahul Mehra for insightful discussions on various aspectsof direct detection experiments. FH thanks Raghuveer Garani for useful discussions. SS isgrateful to Razie Pakravan for her patience and support during the COVID-19 pandemic issuewhen this work was in progress. The work of FH is supported by the Deutsche Forschungs-gemeinschaft (DFG) through the project number 315477589-TRR 211.

– 18 –

Figure 6: Scattering at a 4-vertex.

A Kinematics of the Electronic Recoil and Cross Section

In this appendix we represent the framework for the computation of neutrino-electron scat-tering using our proposed effective interaction taking into account the contribution of ster-ile neutrinos. The differential cross section can be evaluated from a general formula givenby [13,180]

dσ =(2π)4δ4(p1 + p2 − k1 − k2)

4[(p1 · p2)2 −m2

p1m2p2

]1/2

d3~k1

(2π)32Ek1

d3~k2

(2π)32Ek2|M |2 , (A.1)

where p1 and p2 are four momenta of incoming particle and target electron, respectively, andk1 and k2 are outgoing scattered particle and the recoil electron, respectively. The absolutevalue of Feynman amplitude denoted by |M |.

Incoming Sterile Neutrino

When modeling direct detection experiments similar to XENON1T, it is optimal to workwithin the Lab reference frame, where ~p2 = 0. In case Ne → νe the energy conservationimplies EN +me = Ek1 + Ek2 , p1 + p2 = k1 + k2, Ek1 = EN − Er and Ek2 = me + Er. HereEr is the recoil energy. Then assuming mν ' 0 we get

Ek1 =2meEN +m2

N

2me + 2(EN −

√E2N −M2

N cos Θ) , (A.2)

for the kinetic energy of the outgoing SM neutrino which implies the minimum energy ofnonrelativistic sterile neutrino and the relevant recoil energy to be

EN |min 'Er2

+1

2

√E2r + 4meEr + 2M2

N , (A.3)

and

Er '2E2

N −M2N

2(me + EN ). (A.4)

– 19 –

The energy of scattered electron is derived as

Ek2 =

√m2e + E2

k1+ (E2

N −m2N )− 2Ek1

√E2N −m2

N cos Θ . (A.5)

The parameter Θ is the angle between the incoming sterile neutrino p1 and the outgoingneutrino k1, which can be determined by

cos Θ =1√

E2N −m2

N

(EN −

m2N + 2Erme

2(EN − Er)

). (A.6)

Using Eqs. (A.1), (A.2), (A.5) and (A.6) leads to

dσ

dEr=

|M |2

32πmep2N

. (A.7)

Outgoing Sterile Neutrinos

In the case of νee→ Ne we have Eν+me = EN+Ek2 and ~pν = ~k1+~k2 from energy momentum

conservation. Then by using Ek2 = me + Er, Ek1 = EN = Eν − Er and pN =√E2N −m2

N

the following equations for the energy of outgoing electron and the incoming SM neutrino canbe obtained

Ek2 =

√m2e + E2

ν + (E2N −m2

N )− 2Eν

√E2N −m2

N cos Θ , (A.8)

and

Ep1 = Eν =2meEN −m2

N

2me − 2(EN −

√E2N −M2

N cos Θ) . (A.9)

The angle between p1 and k1 reads

cos Θ = 1−(m2N + 2meEr

2Eν(Eν − Er)

). (A.10)

The minimum energy of incoming neutrinos that causes the recoil of electron fromEq. (A.9)

Eν |min =1

2

(1 +

m2N

2meEr

)(Er +

√E2r + 2meEr

), (A.11)

and Eν |max = 420 keV from the maximum of flux of neutrinos. Then we can obtain electronrecoil energy from (A.11) in terms of the energy of the incoming SM neutrinos as

Er = 12me(me+2Eν)

(me[2E

2ν −m2

N ]−mem2N + Eν

√4me(E2

N −m2N )−m2

N (2meEν −m2N )). (A.12)

In the limit of mN → 0 Eqs. (A.9) and (A.10) correspond to the Compton scattering ofphotons from electrons in the Lab frame [180,180]. Therefore the differential cross section is

dσ

dEr=

1

πme

(pN

8EνEN

)2[m2N + 2meEr

]|M |2

me(Er + pN ) + Eν(pN − EN ) +m2N

. (A.13)

– 20 –

B Estimation of the Number of Excess Events

Here we present an order of magnitude estimation of the rate of recoil electrons in the detectorfor incoming and outgoing sterile neutrinos. The efficiency factor ε(Er) and the Gaussiansmearing function G(Er, Erec) are at the order O(1) and do not affect the estimation much.

Incoming Sterile Neutrino

Assuming that the sterile neutrino DM inelastically scattered from the Xenon electrons leadingto the SM neutrinos and recoil electrons at the final state, we estimate electronic recoil eventrate by using Eq. (4.1) as

dRthdEr

≈ NTρDMmN

⟨f(v)

dσNe→νeedEr

⟩≈ NT

ρDMmN

∫dvf(v)v

(dσNedEr

), (B.1)

taking the scattering cross section in EM channel from Eqs. (3.11) and suppose that mN ≈100 keV, EN ' mN , vN ≈ 10−3 and pN ' mNvN . One can find the following approximatevalue of the differential cross section for recoil energy around a few keV

dσ

dEr≈ 5.6× 10−37G2

R

[1

cm−2keV

]. (B.2)

Meanwhile, we assume that the density of sterile neutrinos is obtained from the local densityof DM halo as nDM = (ρDM/mN ) ≈ 1014cm−2s−1. Then our theoretical estimation for theevent rate excess relate to recoil electrons generated via the interaction with sterile neutrinosin a sample of 1 tonne liquid XENON is

dRthdEr

≈ NTρDMmN

dσ

dEr≈ 8.4× 1015G2

R

[1

tonne. year. keV

], (B.3)

regarding the fact that the total number of electrons in XENON target is NT ' 4×1027/tonneand exposure time of data taking is around 1 year. For the integral over halo velocity distri-bution we use the result of Refs. [140–142]. On the other hand the event excess that XENONcollaboration reported in [1] at low energy is roughly

dRexpdEr

≈ 50

[1

tonne. year. keV

]. (B.4)

This shows that, for example in the case of incoming tau sterile neutrinos where tau leptonsare turning inside the loop in Fig. 4, by assuming GR ≈ 7.23× 10−8, one can simply explainthis amount of event excess. Note that this value of GR is consistent with SM precisionmeasurements such as W− decay width and the sterile neutrino lifetime as a DM candidateshowing in Eq. (2.8).

Outgoing Sterile Neutrino

When the sterile neutrino appears in the outgoing state, using Eq. (4.4), one can estimatethe event rate as

dRthdEr

≈ NT

⟨φνdσνe→NedEr

⟩≈ NT

∫dEν

dφνdEν

(dσ

dEr

), (B.5)

– 21 –

the incoming neutrino flux is estimated by the neutrino flux from the proton-proton fusioninside the sun that is around φν,pp ≈ 6 × 1010 cm−2·s−1 and the maximum energy of neu-trinos is 420 keV [145–147]. Then assuming mN = 100 keV and using Eqs. (3.13) and(4.2) for the tree level interaction we can obtain dσ/dEr ≈ 2 × 10−47G2

R [cm2/keV]. Forthe outgoing sterile neutrino case at one loop level (the Feynman diagram similar to Fig. 4and Eq. (3.11) with exchanging sterile and SM neutrinos) the estimated cross section isdσ/dEr ≈ 5.6 × 10−37G2

R [cm2/keV]. Taking into account the fact that the total numberof electrons is NT ' 4 × 1027/tonne in addition to the exposure time of the XENON1Texperiment [1]. As a consequence, one can provide an explanation for the observed excessdRexp/dEr ≈ 50 [tonne. year. keV]−1 at a few keV recoil energy, based on the scaled physicalparameters which gives

dRthdEr

≈ 8.4× 10−1 G2R

[1

tonne. year. keV

], (B.6)

taking the tree level cross section νe→ Ne, and

dRthdEr

≈ 4.2× 10−4 G2R

[1

tonne. year. keV

], (B.7)

applying the EM channel scattering cross section νe→ Ne. These estimations show that toget an excess at the order of Eq. (B.4), one requires large values for the coupling GR becauseof the small flux of solar neutrinos compare to the DM local density around the earth. Theselarge values of GR are in conflict with SM precision measurements and astrophysical tests.We note that, it is not essential for the outgoing sterile neutrinos to respect the DM lifetimeconstraints in Eqs. (2.9) - (2.11). Consequently, in our model the outgoing sterile neutrinoscan not explain the XENON1T excess.

References

[1] XENON collaboration, Observation of Excess Electronic Recoil Events in XENON1T,2006.09721.

[2] XENON1T collaboration, E. Shockley, Search for New Physics with Electronic-Recoil Eventsin XENON1T, 2020.

[3] M. Szydagis, C. Levy, G. Blockinger, A. Kamaha, N. Parveen and G. Rischbieter,Investigating the XENON1T Low-Energy Electronic Recoil Excess Using NEST, 2007.00528.

[4] B. Bhattacherjee and R. Sengupta, XENON1T Excess: Some Possible Backgrounds,2006.16172.

[5] A. E. Robinson, XENON1T observes tritium, 2006.13278.

[6] L. Di Luzio, M. Fedele, M. Giannotti, F. Mescia and E. Nardi, Solar axions cannot explain theXENON1T excess, 2006.12487.

[7] C. Gao, J. Liu, L.-T. Wang, X.-P. Wang, W. Xue and Y.-M. Zhong, Re-examining the SolarAxion Explanation for the XENON1T Excess, 2006.14598.

[8] J. B. Dent, B. Dutta, J. L. Newstead and A. Thompson, Inverse Primakoff Scattering as aProbe of Solar Axions at Liquid Xenon Direct Detection Experiments, 2006.15118.

– 22 –

[9] S. Arceo Díaz, K.-P. Schröder, K. Zuber, D. Jack and E. E. Bricio Barrios, Constraint on theaxion-electron coupling constant and the neutrino magnetic dipole moment by using thetip-RGB luminosity of fifty globular clusters, arXiv e-prints (2019) arXiv:1910.10568[1910.10568].

[10] N. Viaux, M. Catelan, P. B. Stetson, G. Raffelt, J. Redondo, A. A. R. Valcarce et al.,Neutrino and axion bounds from the globular cluster M5 (NGC 5904), Phys. Rev. Lett. 111(2013) 231301 [1311.1669].

[11] P. Athron et al., Global fits of axion-like particles to XENON1T and astrophysical data,2007.05517.

[12] R. Mohapatra and P. Pal, Massive neutrinos in physics and astrophysics. Second edition,vol. 60. World scientific, 1998.

[13] C. Itzykson and J. Zuber, Quantum Field Theory, International Series In Pure and AppliedPhysics. McGraw-Hill, New York, 1980.

[14] O. Tomalak and R. J. Hill, Theory of elastic neutrino-electron scattering, Phys. Rev. D 101(2020) 033006 [1907.03379].

[15] C. Broggini, C. Giunti and A. Studenikin, Electromagnetic Properties of Neutrinos, Adv. HighEnergy Phys. 2012 (2012) 459526 [1207.3980].

[16] J. F. Carlson and J. R. Oppenheimer, The impacts of fast electrons and magnetic neutrons,Phys. Rev. 41 (1932) 763.

[17] A. de Gouvea and J. Jenkins, What can we learn from neutrino electron scattering?, Phys.Rev. D 74 (2006) 033004 [hep-ph/0603036].

[18] K. Fujikawa and R. Shrock, The Magnetic Moment of a Massive Neutrino and Neutrino SpinRotation, Phys. Rev. Lett. 45 (1980) 963.

[19] K. Bhattacharya and P. B. Pal, Neutrinos and magnetic fields: A Short review, Proc. IndianNatl. Sci. Acad. A 70 (2004) 145 [hep-ph/0212118].

[20] P. Vogel and J. Engel, Neutrino Electromagnetic Form-Factors, Phys. Rev. D 39 (1989) 3378.

[21] A. N. Khan, Can nonstandard neutrino interactions explain the XENON1T spectral excess?,2006.12887.

[22] K. Babu, S. Jana and M. Lindner, Large Neutrino Magnetic Moments in the Light of RecentExperiments, 2007.04291.

[23] M. Chala and A. Titov, One-loop running of dimension-six Higgs-neutrino operators andimplications of a large neutrino dipole moment, 2006.14596.

[24] d. Amaral, Dorian Warren Praia, D. G. Cerdeno, P. Foldenauer and E. Reid, Solar neutrinoprobes of the muon anomalous magnetic moment in the gauged U(1)Lµ−Lτ

, 2006.11225.

[25] O. Miranda, D. Papoulias, M. Tórtola and J. Valle, XENON1T signal from transitionneutrino magnetic moments, Phys. Lett. B 808 (2020) 135685 [2007.01765].

[26] G. Arcadi, A. Bally, F. Goertz, K. Tame-Narvaez, V. Tenorth and S. Vogl, EFT Interpretationof XENON1T Electron Recoil Excess: Neutrinos and Dark Matter, 2007.08500.

[27] M. Lindner, B. Radovˇ cić and J. Welter, Revisiting Large Neutrino Magnetic Moments,JHEP 07 (2017) 139 [1706.02555].

– 23 –

[28] B. Fornal, P. Sandick, J. Shu, M. Su and Y. Zhao, Boosted Dark Matter Interpretation of theXENON1T Excess, 2006.11264.

[29] Y. Jho, J.-C. Park, S. C. Park and P.-Y. Tseng, Gauged Lepton Number and Cosmic-rayBoosted Dark Matter for the XENON1T Excess, 2006.13910.

[30] H. Alhazmi, D. Kim, K. Kong, G. Mohlabeng, J.-C. Park and S. Shin, Implications of theXENON1T Excess on the Dark Matter Interpretation, 2006.16252.

[31] H. Davoudiasl, P. B. Denton and J. Gehrlein, An Attractive Scenario for Light Dark MatterDirect Detection, 2007.04989.

[32] U. K. Dey, T. N. Maity and T. S. Ray, Prospects of Migdal Effect in the Explanation ofXENON1T Electron Recoil Excess, 2006.12529.

[33] D. McKeen, M. Pospelov and N. Raj, Hydrogen portal to exotic radioactivity, 2006.15140.

[34] G. Alonso-Álvarez, F. Ertas, J. Jaeckel, F. Kahlhoefer and L. Thormaehlen, Hidden PhotonDark Matter in the Light of XENON1T and Stellar Cooling, 2006.11243.

[35] C.-W. Chiang and B.-Q. Lu, Evidence of A Simple Dark Sector from XENON1T Anomaly,2007.06401.

[36] D. Borah, S. Mahapatra, D. Nanda and N. Sahu, Inelastic Fermion Dark Matter Origin ofXENON1T Excess with Muon (g − 2) and Light Neutrino Mass, 2007.10754.

[37] K. Harigaya, Y. Nakai and M. Suzuki, Inelastic Dark Matter Electron Scattering and theXENON1T Excess, 2006.11938.

[38] N. F. Bell, J. B. Dent, B. Dutta, S. Ghosh, J. Kumar and J. L. Newstead, Explaining theXENON1T excess with Luminous Dark Matter, 2006.12461.

[39] H. M. Lee, Exothermic Dark Matter for XENON1T Excess, 2006.13183.

[40] J. Bramante and N. Song, Electric But Not Eclectic: Thermal Relic Dark Matter for theXENON1T Excess, 2006.14089.

[41] J. Smirnov and J. F. Beacom, Co-SIMP Miracle, 2002.04038.

[42] D. Choudhury, S. Maharana, D. Sachdeva and V. Sahdev, Dark Matter, Muon AnomalousMagnetic Moment and the XENON1T Excess, 2007.08205.

[43] F. Takahashi, M. Yamada and W. Yin, XENON1T anomaly from anomaly-free ALP darkmatter, 2006.10035.

[44] A. Bally, S. Jana and A. Trautner, Neutrino self-interactions and XENON1T electron recoilexcess, 2006.11919.

[45] N. Okada, S. Okada, D. Raut and Q. Shafi, Dark Matter Z ′ and XENON1T Excess fromU(1)X Extended Standard Model, 2007.02898.

[46] M. Lindner, Y. Mambrini, T. B. de Melo and F. S. Queiroz, XENON1T Anomaly: A LightZ ′, 2006.14590.

[47] C. Boehm, D. G. Cerdeno, M. Fairbairn, P. A. Machado and A. C. Vincent, Light new physicsin XENON1T, 2006.11250.

– 24 –

[48] D. Aristizabal Sierra, V. De Romeri, L. Flores and D. Papoulias, Light vector mediatorsfacing XENON1T data, 2006.12457.

[49] G. Choi, M. Suzuki and T. T. Yanagida, XENON1T Anomaly and its Implication forDecaying Warm Dark Matter, 2006.12348.

[50] G. Choi, T. T. Yanagida and N. Yokozaki, Feebly Interacting U(1)B−L Gauge Boson WarmDark Matter and XENON1T Anomaly, 2007.04278.

[51] J. Kim, T. Nomura and H. Okada, A radiative seesaw model linking to XENON1T anomaly,2007.09894.

[52] S. Baek, J. Kim and P. Ko, XENON1T excess in local Z2 DM models with light dark sector,2006.16876.

[53] M. Lindner, F. S. Queiroz, W. Rodejohann and X.-J. Xu, Neutrino-electron scattering:general constraints on Z′ and dark photon models, JHEP 05 (2018) 098 [1803.00060].

[54] S.-S. Xue, Neutrino masses and mixings, Mod. Phys. Lett. A 14 (1999) 2701[hep-ph/9706301].

[55] S.-S. Xue, On the standard model and parity conservation, Journal of Physics G: Nuclear andParticle Physics 29 (2003) 2381 [hep-ph/0106117].

[56] M. Drewes, The Phenomenology of Right Handed Neutrinos, Int. J. Mod. Phys. E 22 (2013)1330019 [1303.6912].

[57] S. Dodelson and L. M. Widrow, Sterile-neutrinos as dark matter, Phys. Rev. Lett. 72 (1994)17 [hep-ph/9303287].

[58] A. Boyarsky, M. Drewes, T. Lasserre, S. Mertens and O. Ruchayskiy, Sterile Neutrino DarkMatter, Prog. Part. Nucl. Phys. 104 (2019) 1 [1807.07938].

[59] A. Boyarsky, D. Iakubovskyi and O. Ruchayskiy, Next decade of sterile neutrino studies, Phys.Dark Univ. 1 (2012) 136 [1306.4954].

[60] S. Dodelson and L. M. Widrow, Sterile neutrinos as dark matter, Phys. Rev. Lett. 72 (1994)17.

[61] A. Kusenko, Sterile neutrinos: The Dark side of the light fermions, Phys. Rept. 481 (2009) 1[0906.2968].

[62] M. Drewes et al., A White Paper on keV Sterile Neutrino Dark Matter, JCAP 01 (2017) 025[1602.04816].

[63] G. B. Gelmini, P. Lu and V. Takhistov, Visible Sterile Neutrinos as the Earliest Relic Probesof Cosmology, Phys. Lett. B 800 (2020) 135113 [1909.04168].

[64] T. Asaka and M. Shaposhnikov, The νMSM, dark matter and baryon asymmetry of theuniverse, Phys. Lett. B 620 (2005) 17 [hep-ph/0505013].

[65] M. Shaposhnikov, The nuMSM, leptonic asymmetries, and properties of singlet fermions,JHEP 08 (2008) 008 [0804.4542].

[66] G. Gelmini, S. Palomares-Ruiz and S. Pascoli, Low reheating temperature and the visiblesterile neutrino, Phys. Rev. Lett. 93 (2004) 081302 [astro-ph/0403323].

– 25 –

[67] I. M. Shoemaker and J. Wyenberg, Direct Detection Experiments at the Neutrino DipolePortal Frontier, Phys. Rev. D 99 (2019) 075010 [1811.12435].

[68] S.-F. Ge, P. Pasquini and J. Sheng, Solar Neutrino Scattering with Electron into MassiveSterile Neutrino, 2006.16069.

[69] Super-Kamiokande collaboration, Evidence for oscillation of atmospheric neutrinos, Phys.Rev. Lett. 81 (1998) 1562 [hep-ex/9807003].

[70] P. Minkowski, µ→ eγ at a Rate of One Out of 109 Muon Decays?, Phys. Lett. B 67 (1977)421.

[71] S. Glashow, The Future of Elementary Particle Physics, NATO Sci. Ser. B 61 (1980) 687.

[72] M. Gell-Mann, P. Ramond and R. Slansky, Complex Spinors and Unified Theories, Conf.Proc. C 790927 (1979) 315 [1306.4669].

[73] J. Schechter and J. Valle, Neutrino Masses in SU(2) x U(1) Theories, Phys. Rev. D 22 (1980)2227.

[74] R. N. Mohapatra and G. Senjanović, Neutrino mass and spontaneous parity nonconservation,Phys. Rev. Lett. 44 (1980) 912.

[75] M. Haghighat, S. Mahmoudi, R. Mohammadi, S. Tizchang and S.-S. Xue, Circularpolarization of cosmic photons due to their interactions with Sterile neutrino dark matter,Phys. Rev. D 101 (2020) 123016 [1909.03883].

[76] T. Cheng and L.-F. Li, Neutrino Masses, Mixings and Oscillations in SU(2) x U(1) Models ofElectroweak Interactions, Phys. Rev. D 22 (1980) 2860.

[77] R. Foot, H. Lew, X. He and G. C. Joshi, Seesaw Neutrino Masses Induced by a Triplet ofLeptons, Z. Phys. C 44 (1989) 441.

[78] E. Ma and R. Srivastava, Dirac or inverse seesaw neutrino masses with B − L gaugesymmetry and S3 flavor symmetry, Phys. Lett. B 741 (2015) 217 [1411.5042].

[79] E. Ma, N. Pollard, R. Srivastava and M. Zakeri, Gauge B − L Model with Residual Z3

Symmetry, Phys. Lett. B 750 (2015) 135 [1507.03943].

[80] A. Davidson, b− l as the fourth color within an SU(2)L ×U(1)R ×U(1) model, Phys. Rev. D20 (1979) 776.

[81] R. N. Mohapatra and R. E. Marshak, Local b− l symmetry of electroweak interactions,majorana neutrinos, and neutron oscillations, Phys. Rev. Lett. 44 (1980) 1316.

[82] R. Marshak and R. N. Mohapatra, Quark - Lepton Symmetry and B-L as the U(1) Generatorof the Electroweak Symmetry Group, Phys. Lett. B 91 (1980) 222.

[83] C. Wetterich, Neutrino Masses and the Scale of B-L Violation, Nucl. Phys. B 187 (1981) 343.

[84] P.-H. Gu and U. Sarkar, Radiative Neutrino Mass, Dark Matter and Leptogenesis, Phys. Rev.D 77 (2008) 105031 [0712.2933].

[85] W. Marciano and A. Sanda, Exotic Decays of the Muon and Heavy Leptons in GaugeTheories, Phys. Lett. B 67 (1977) 303.

[86] R. N. Mohapatra and J. C. Pati, "natural" left-right symmetry, Phys. Rev. D 11 (1975) 2558.

– 26 –

[87] G. Senjanovic and R. N. Mohapatra, Exact left-right symmetry and spontaneous violation ofparity, Phys. Rev. D 12 (1975) 1502.

[88] H. B. Nielsen and M. Ninomiya, Absence of Neutrinos on a Lattice. 1. Proof by HomotopyTheory, .

[89] H. B. Nielsen and M. Ninomiya, Absence of Neutrinos on a Lattice. 2. Intuitive TopologicalProof, Nucl. Phys. B 193 (1981) 173.

[90] H. B. Nielsen and M. Ninomiya, A no-go theorem for regularizing chiral fermions, PhysicsLetters B 105 (1981) 219.

[91] H. B. Nielsen and M. Ninomiya, Intuitive understanding of anomalies: A Paradox withregularization, Int. J. Mod. Phys. A 6 (1991) 2913.

[92] Y. Nambu and G. Jona-Lasinio, DYNAMICAL MODEL OF ELEMENTARY PARTICLESBASED ON AN ANALOGY WITH SUPERCONDUCTIVITY. II, Phys. Rev. 124 (1961)246.

[93] S.-S. Xue, Hierarchy spectrum of SM fermions: from top quark to electron neutrino, JHEP 11(2016) 072 [1605.01266].

[94] S.-S. Xue, An effective strong-coupling theory of composite particles in UV-domain, JHEP 05(2017) 146 [1601.06845].

[95] S.-S. Xue, Vectorlike W±-boson coupling at TeV and third family fermion masses, Phys. Rev.D 93 (2016) 073001 [1506.05994].

[96] S.-S. Xue, Ultraviolet fixed point and massive composite particles in TeV scales, Phys. Lett. B737 (2014) 172 [1405.1867].

[97] S.-S. Xue, Quark masses and mixing angles, Phys. Lett. B 398 (1997) 177 [hep-ph/9610508].

[98] R. Leonardi, O. Panella, F. Romeo, A. Gurrola, H. Sun and S.-S. Xue, Phenomenology at theLHC of composite particles from strongly interacting Standard Model fermions viafour-fermion operators of NJL type, Eur. Phys. J. C 80 (2020) 309 [1810.11420].

[99] W. A. Bardeen, C. T. Hill and M. Lindner, Minimal Dynamical Symmetry Breaking of theStandard Model, Phys. Rev. D 41 (1990) 1647.

[100] S.-S. Xue, A Further study of the possible scaling region of lattice chiral fermions, Phys. Rev.D 61 (2000) 054502 [hep-lat/9910013].

[101] S.-S. Xue, Higgs Boson and Top-Quark Masses and Parity-Symmetry Restoration, Phys. Lett.B 727 (2013) 308 [1308.6486].

[102] S. Alioli, V. Cirigliano, W. Dekens, J. de Vries and E. Mereghetti, Right-handed chargedcurrents in the era of the Large Hadron Collider, JHEP 05 (2017) 086 [1703.04751].

[103] G. Passarino and M. Veltman, One Loop Corrections for e+ e- Annihilation Into mu+ mu- inthe Weinberg Model, Nucl. Phys. B 160 (1979) 151.

[104] H. H. Patel, Package-X: A Mathematica package for the analytic calculation of one-loopintegrals, Comput. Phys. Commun. 197 (2015) 276 [1503.01469].

[105] B. W. Lee and R. E. Shrock, Natural Suppression of Symmetry Violation in Gauge Theories:Muon - Lepton and Electron Lepton Number Nonconservation, Phys. Rev. D 16 (1977) 1444.

– 27 –

[106] C. Giunti and A. Studenikin, Neutrino electromagnetic interactions: a window to new physics,Rev. Mod. Phys. 87 (2015) 531 [1403.6344].

[107] K. Abazajian, G. M. Fuller and M. Patel, Sterile neutrino hot, warm, and cold dark matter,Phys. Rev. D 64 (2001) 023501 [astro-ph/0101524].

[108] A. Gvozdev, N. Mikheev and L. Vasilevskaya, The Radiative decay of the massive neutrino inthe external electromagnetic fields, Phys. Rev. D 54 (1996) 5674 [hep-ph/9610219].

[109] P. B. Pal and L. Wolfenstein, Radiative decays of massive neutrinos, Phys. Rev. D 25 (1982)766.

[110] V. D. Barger, R. Phillips and S. Sarkar, Remarks on the KARMEN anomaly, Phys. Lett. B352 (1995) 365 [hep-ph/9503295].

[111] T. Asaka, S. Blanchet and M. Shaposhnikov, The nuMSM, dark matter and neutrino masses,2005. 10.1016/j.physletb.2005.09.070.

[112] R. Mohapatra and J. Valle, Neutrino Mass and Baryon Number Nonconservation inSuperstring Models, Phys. Rev. D 34 (1986) 1642.

[113] K. A. Olive, TASI lectures on dark matter, in Theoretical Advanced Study Institute inElementary Particle Physics (TASI 2002): Particle Physics and Cosmology: The Quest forPhysics Beyond the Standard Model(s), pp. 797–851, 1, 2003, astro-ph/0301505.

[114] S. H. Hansen, J. Lesgourgues, S. Pastor and J. Silk, Constraining the window on sterileneutrinos as warm dark matter, Mon. Not. Roy. Astron. Soc. 333 (2002) 544[astro-ph/0106108].

[115] T. Lin, Dark matter models and direct detection, PoS 333 (2019) 009 [1904.07915].

[116] E. W. Kolb and M. S. Turner, The Early Universe, vol. 69. 1990.

[117] A. Biswas, D. Borah and D. Nanda, keV Neutrino Dark Matter in a Fast Expanding Universe,Phys. Lett. B 786 (2018) 364 [1809.03519].

[118] M. Drees, F. Hajkarim and E. R. Schmitz, The Effects of QCD Equation of State on the RelicDensity of WIMP Dark Matter, JCAP 06 (2015) 025 [1503.03513].

[119] Planck collaboration, Planck 2018 results. VI. Cosmological parameters, 1807.06209.

[120] R. Allahverdi et al., The First Three Seconds: a Review of Possible Expansion Histories of theEarly Universe, 2006.16182.

[121] G. Radel and R. Beyer, Neutrino electron scattering, Mod. Phys. Lett. A 8 (1993) 1067.

[122] J. Formaggio and G. Zeller, From eV to EeV: Neutrino Cross Sections Across Energy Scales,Rev. Mod. Phys. 84 (2012) 1307 [1305.7513].

[123] H.-J. He, Y.-C. Wang and J. Zheng, EFT Analysis of Inelastic Dark Matter for XenonElectron Recoil Detection, 2007.04963.

[124] K. Kannike, M. Raidal, H. Veermäe, A. Strumia and D. Teresi, Dark Matter and theXENON1T electron recoil excess, 2006.10735.

[125] M. D. Campos and W. Rodejohann, Testing keV sterile neutrino dark matter in future directdetection experiments, Phys. Rev. D 94 (2016) 095010 [1605.02918].

– 28 –

[126] M. Baryakhtar, A. Berlin, H. Liu and N. Weiner, Electromagnetic Signals of Inelastic DarkMatter Scattering, 2006.13918.

[127] S. Ando and A. Kusenko, Interactions of keV sterile neutrinos with matter, Phys. Rev. D 81(2010) 113006 [1001.5273].

[128] R. Essig, J. Mardon and T. Volansky, Direct Detection of Sub-GeV Dark Matter, Phys. Rev.D 85 (2012) 076007 [1108.5383].

[129] W. Liao, X.-H. Wu and H. Zhou, Electron events from the scattering with solar neutrinos inthe search of keV scale sterile neutrino dark matter, Phys. Rev. D 89 (2014) 093017[1311.6075].

[130] G. Gounaris, E. Paschos and P. Porfyriadis, Electron spectra in the ionization of atoms byneutrinos, Phys. Rev. D 70 (2004) 113008 [hep-ph/0409053].

[131] J.-W. Chen, H.-C. Chi, C. P. Liu and C.-P. Wu, Low-energy electronic recoil in xenondetectors by solar neutrinos, Phys. Lett. B 774 (2017) 656 [1610.04177].

[132] J.-W. Chen, H.-C. Chi, K.-N. Huang, C. P. Liu, H.-T. Shiao, L. Singh et al., Atomicionization of germanium by neutrinos from an ab initio approach, Phys. Lett. B 731 (2014)159 [1311.5294].

[133] B. Roberts and V. Flambaum, Electron-interacting dark matter: Implications fromDAMA/LIBRA-phase2 and prospects for liquid xenon detectors and NaI detectors, Phys. Rev.D 100 (2019) 063017 [1904.07127].

[134] J.-W. Chen, H.-C. Chi, K.-N. Huang, H.-B. Li, C.-P. Liu, L. Singh et al., Constrainingneutrino electromagnetic properties by germanium detectors, Phys. Rev. D 91 (2015) 013005[1411.0574].

[135] Particle Data Group collaboration, Review of Particle Physics, Phys. Rev. D 98 (2018)030001.

[136] G. Raffelt, Stars as laboratories for fundamental physics: The astrophysics of neutrinos,axions, and other weakly interacting particles. 5, 1996.

[137] A. Kramida, Yu. Ralchenko, J. Reader and and NIST ASD Team. NIST Atomic SpectraDatabase (ver. 5.7.1), [Online]. Available: https://physics.nist.gov/asd [2020, June 25].National Institute of Standards and Technology, Gaithersburg, MD., 2019.

[138] C.-C. Hsieh, L. Singh, C.-P. Wu, J.-W. Chen, H.-C. Chi, C.-P. Liu et al., Discovery potentialof multiton xenon detectors in neutrino electromagnetic properties, Phys. Rev. D 100 (2019)073001 [1903.06085].

[139] XENON collaboration, Energy resolution and linearity in the keV to MeV range measured inXENON1T, 2003.03825.

[140] M. Drees and R. Mehra, Neutron EDM constrains direct dark matter detection prospects,Phys. Lett. B 799 (2019) 135039 [1907.10075].

[141] V. Barger, W.-Y. Keung and D. Marfatia, Electromagnetic properties of dark matter: Dipolemoments and charge form factor, Phys. Lett. B 696 (2011) 74 [1007.4345].

[142] M. Cirelli, E. Del Nobile and P. Panci, Tools for model-independent bounds in direct darkmatter searches, JCAP 10 (2013) 019 [1307.5955].

– 29 –

[143] M. Weber and W. de Boer, Determination of the Local Dark Matter Density in our Galaxy,Astron. Astrophys. 509 (2010) A25 [0910.4272].

[144] C. Dessert, J. W. Foster, Y. Kahn and B. R. Safdi, Systematics in the XENON1T data: the15-keV anti-axion, 2006.16220.

[145] J. N. Bahcall and C. Pena-Garay, Solar models and solar neutrino oscillations, New J. Phys.6 (2004) 63 [hep-ph/0404061].

[146] J. Billard, L. Strigari and E. Figueroa-Feliciano, Implication of neutrino backgrounds on thereach of next generation dark matter direct detection experiments, Phys. Rev. D 89 (2014)023524 [1307.5458].

[147] I. Lopes and S. Turck-Chièze, Solar neutrino physics oscillations: Sensitivity to the electronicdensity in the Sun’s core, Astrophys. J. 765 (2013) 14 [1302.2791].

[148] W. Haxton, R. Hamish Robertson and A. M. Serenelli, Solar Neutrinos: Status and Prospects,Ann. Rev. Astron. Astrophys. 51 (2013) 21 [1208.5723].

[149] G. Raffelt and A. Weiss, Red giant bound on the axion - electron coupling revisited, Phys. Rev.D 51 (1995) 1495 [hep-ph/9410205].

[150] S. Arceo-Díaz, K.-P. Schröder, K. Zuber and D. Jack, Constraint on the magnetic dipolemoment of neutrinos by the tip-RGB luminosity in ω -Centauri, Astropart. Phys. 70 (2015) 1.

[151] I. M. Shoemaker, Y.-D. Tsai and J. Wyenberg, An Active-to-Sterile Neutrino TransitionDipole Moment and the XENON1T Excess, 2007.05513.

[152] S. Gninenko, The MiniBooNE anomaly and heavy neutrino decay, Phys. Rev. Lett. 103 (2009)241802 [0902.3802].

[153] S. N. Gninenko, A resolution of puzzles from the LSND, KARMEN, and MiniBooNEexperiments, Phys. Rev. D 83 (2011) 015015 [1009.5536].

[154] D. McKeen and M. Pospelov, Muon Capture Constraints on Sterile Neutrino Properties,Phys. Rev. D 82 (2010) 113018 [1011.3046].

[155] M. Masip and P. Masjuan, Heavy-neutrino decays at neutrino telescopes, Phys. Rev. D 83(2011) 091301 [1103.0689].

[156] S. Karmakar and S. Pandey, XENON1T constraints on neutrino non-standard interactions,2007.11892.

[157] V. Brdar, A. Greljo, J. Kopp and T. Opferkuch, The Neutrino Magnetic Moment Portal:Cosmology, Astrophysics, and Direct Detection, 2007.15563.

[158] P. Coloma, P. A. Machado, I. Martinez-Soler and I. M. Shoemaker, Double-cascade eventsfrom new physics in icecube, Physical Review Letters 119 (2017) .

[159] G. Magill, R. Plestid, M. Pospelov and Y.-D. Tsai, Dipole Portal to Heavy Neutral Leptons,Phys. Rev. D 98 (2018) 115015 [1803.03262].

[160] W. DeRocco, P. W. Graham and S. Rajendran, Exploring the robustness of stellar coolingconstraints on light particles, 2006.15112.

[161] M. M. Miller Bertolami, B. E. Melendez, L. G. Althaus and J. Isern, Revisiting the axionbounds from the Galactic white dwarf luminosity function, JCAP 10 (2014) 069 [1406.7712].

– 30 –

[162] A. Ayala, I. Domínguez, M. Giannotti, A. Mirizzi and O. Straniero, Revisiting the bound onaxion-photon coupling from Globular Clusters, Phys. Rev. Lett. 113 (2014) 191302[1406.6053].

[163] M. Giannotti, I. G. Irastorza, J. Redondo, A. Ringwald and K. Saikawa, Stellar Recipes forAxion Hunters, JCAP 10 (2017) 010 [1708.02111].

[164] A. Diaz, C. Argüelles, G. Collin, J. Conrad and M. Shaevitz, Where Are We With LightSterile Neutrinos?, 1906.00045.

[165] K. Abazajian et al., Light Sterile Neutrinos: A White Paper, 1204.5379.

[166] LSND collaboration, Evidence for neutrino oscillations from the observation of νe appearancein a νµ beam, Phys. Rev. D 64 (2001) 112007 [hep-ex/0104049].

[167] MiniBooNE collaboration, Significant Excess of ElectronLike Events in the MiniBooNEShort-Baseline Neutrino Experiment, Phys. Rev. Lett. 121 (2018) 221801 [1805.12028].

[168] E. Bertuzzo, S. Jana, P. A. Machado and R. Zukanovich Funchal, Dark Neutrino Portal toExplain MiniBooNE excess, Phys. Rev. Lett. 121 (2018) 241801 [1807.09877].

[169] BOREXINO collaboration, Comprehensive measurement of pp-chain solar neutrinos, Nature562 (2018) 505.

[170] P. Coloma, P. A. Machado, I. Martinez-Soler and I. M. Shoemaker, Double-Cascade Eventsfrom New Physics in Icecube, Phys. Rev. Lett. 119 (2017) 201804 [1707.08573].

[171] T. Bringmann and M. Pospelov, Novel direct detection constraints on light dark matter, Phys.Rev. Lett. 122 (2019) 171801 [1810.10543].

[172] O. Miranda, D. Papoulias, M. Tórtola and J. Valle, Probing neutrino transition magneticmoments with coherent elastic neutrino-nucleus scattering, JHEP 07 (2019) 103 [1905.03750].

[173] E. Bulbul, M. Markevitch, A. Foster, R. K. Smith, M. Loewenstein and S. W. Randall,Detection of An Unidentified Emission Line in the Stacked X-ray spectrum of GalaxyClusters, Astrophys. J. 789 (2014) 13 [1402.2301].

[174] A. Boyarsky, J. Franse, D. Iakubovskyi and O. Ruchayskiy, Checking the Dark Matter Originof a 3.53 keV Line with the Milky Way Center, Phys. Rev. Lett. 115 (2015) 161301[1408.2503].

[175] PVDIS collaboration, Measurement of parity violation in electron–quark scattering, Nature506 (2014) 67.

[176] XENON collaboration, Physics reach of the XENON1T dark matter experiment, JCAP 04(2016) 027 [1512.07501].

[177] LZ collaboration, LUX-ZEPLIN (LZ) Conceptual Design Report, 1509.02910.

[178] PandaX-II collaboration, Dark Matter Results From 54-Ton-Day Exposure of PandaX-IIExperiment, Phys. Rev. Lett. 119 (2017) 181302 [1708.06917].

[179] DARWIN collaboration, DARWIN: towards the ultimate dark matter detector, JCAP 11(2016) 017 [1606.07001].

[180] M. E. Peskin and D. V. Schroeder, An Introduction to quantum field theory. Addison-Wesley,Reading, USA, 1995.

– 31 –