JHEP04(2021)025 Springer April 2, 2021 : August 27, 2020 MeV. To solve : February 22, 2021 November 30, 2020 : 17 : doublet Higgs bosons Published Received Revised Accepted symmetry and identify the SU(2) R symmetric model U(1) R Published for SISSA by https://doi.org/10.1007/JHEP04(2021)025 , , with the mass of about U(1) X c charges of the doublet scalars are determined to R U(1) boson. We also introduce two X . 3 2006.05497 and Takashi Shimomura The Authors. c Beyond Standard Model, Gauge Symmetry

a,b The Atomki collaboration has reported that unexpected excesses have been , seto@particle.sci.hokudai.ac.jp [email protected] Department of Physics, HokkaidoSapporo University, 060-0810, Japan Faculty of Education, Miyazaki University, Miyazaki, 889-2192, Japan E-mail: Institute for the AdvancementHokkaido of University, Higher Sapporo Education, 060-0817, Japan b c a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: the Atomki anomaly, we considernew a gauge model boson with with gauged and the one singlet Higgs boson,scattering. and discuss a It very is stringentevade constraint found the from constraint. neutrino-electron that In the theand end, all we experimental find constraints the can parameter be region simultaneously in satisfied. which the Atomki signal Abstract: observed in the raresuggest the decays existence of of Beryllium a nucleus. new boson, called It is claimed that such excesses can Osamu Seto Atomki anomaly in gauged JHEP04(2021)025 ] ], 3 18 8 7 , 6 (1.1) (1.2) ) pair from the IPC − e ] and Helium (He) [ 5 16 – 1 , , − − e e + + + + 18 e e ) and electron ( 7 + 18 17 e Be + He + 8 4 9 S → 11 → 15 12 14 – 1 – 17 15 MeV) 01 MeV) . . 14 (18 ∗ He(21 Be 4 12 8 boson to fermions 18 X decay branching ratio He, 14 4 4 ∗ boson lifetime 1 Be 19 20 21 8 X and ∗ Be 8 4.2.8 Vacuum expectation value of 4.2.3 Anomalous4.2.4 magnetic moment of charged Effective4.2.5 lepton weak charge electron4.2.6 beam dump experiments Electron-positron4.2.7 collider experiments Parity violating Möller scattering 4.1.1 4.1.2 4.2.1 Rare4.2.2 decay of neutral pion Neutrino-electron scattering 5.1 Model5.2 1 Model 3 4.1 Signal requirement 4.2 Constraints 2.2 Gauge boson masses and mass eigenstates 2.1 Lagrangian respectively. These bumps seem not toeven be if explained parity within violating the standard decays nuclear are physics taken [ into account. The collaboration reported that the The Atomki collaboration has beenin reporting the results Internal that Pair unexpectednuclei. Creation excesses (IPC) In were decay found the of reports,mass Beryllium the and (Be) excesses opening [ appear angledecays as of of an bumps emitted in positron the ( distributions of the invariant 6 Conclusion 1 Introduction 5 Numerical results 3 Couplings of the 4 Signal and experimental constraints Contents 1 Introduction 2 Model JHEP04(2021)025 He 4 boson ’s three and X X ∗ ], neutrino gauge sym- gauge sym- ]. The vec- Be 16 R 8 R the – ]. Experimen- 10 boson to active 1 , 14 25 , 9 – symmetric model, X X U(1) U(1) k )[ 21 L ], we neglected that term, to proton because the 1.1 ]. This hypothesis has − 30 30 X B ], it was shown that there boson. 10 X boson. The ], dark matter [ ] and others [ X 13 – ]. In [ 20 ]. In spite of these advantages, the – 11 29 33 is proportional to – 18 – ] in a gauged MeV is produced through ∗ 26 , measured by the NA48/2 experiment. to quarks can be much smaller to explain 31 ]. Decay widths of these three hypotheses 25 Be 10 . X 8 , γX 0 ], or no interaction of the 34 9 ± 10 pair, respectively. Such a light boson does not in the vector boson hypothesis. Because of this – 2 – → − 68 3 X 0 . e k π - 19 + e is smaller than the mass of the and gauge boson has both vectorial and axial interactions to , in principle, can be a vector, axial-vector, scalar and X ], it is assumed that the neutrino couplings to the k 16 X R . 30 0 ± U(1) . In [ 01 decay into a . boson are also studied in [ X ], the neutrino coupling is assumed to be enough small. In the end, 17 X X 33 term in the partial width. Following the discussion in [ – 3 X ]. 31 k 35 ], lepton anomalous magnetic moments [ ] where the gauge boson is identified with the 17 36 In this work, we pursue the axial-vector hypothesis and consider a An axial-vector boson hypothesis also has been studied in [ The hypothetical boson There is also with the mass of 1 metry [ metry is defined thatnot only charged. right-handed fermions Then, are the charged while left-handed ones are which would be suppressed because neutrino couplings to vanish, and in [ the pseudo-scalar hypothesis wasare studied found in in [ [ momentum dependence, coupling constants of the Atomki anomaly than thatevade in several experimental the constraints vector-boson in hypothesis thewith axial-vector case. boson axial-vector Then, hypothesis. boson it Several is have models constraint possible been from to proposed neutrino-electron in scattering is [ still very stringent and requires to suppress two advantages. Oneevaded is because that the theadvantage decay constraints is receives from that no the the neutral contributionmomentum, partial while pion from decay it decay the is width can proportional axial of to be anomaly. easily The other such a vector bosonneutrino-electron is scattering named measured as by a thethis protophobic TEXONO experiment. constraint boson. and It The neutral is second pion difficultare constraint to constraint introduced comes evade simultaneously. to from Therefore, evade new thisneutrinos leptonic constraint is states in ad [ hoc assumed. tal searches of the are two restrictive constraints tofrom explain the the rare Atomki anomaly.This decay The constraint of first sets neutral constraint a comes decay pion, very branching stringent ratio bound of on the the rare decay coupling is of scaled by the proton coupling. From this fact, pseudo-scalar boson. Amongdue these possibilities, to the the scalar conservationtor boson of boson hypothesis angular hypothesis is momentum wastaking discarded firstly in various experimental studied the constraints in decay intoposed [ eq. account. in contexts ( Then, of many anphysics models extra [ have U(1) been gauge pro- symmetry [ X decays, followed by exist in theconsidered Standard as Model a (SM) signal of of new particle physics beyond physics. the SM. Therefore, the anomaly can be bumps can be well fitted simultaneously under the assumption that a hypothetical boson JHEP04(2021)025 . L 6 and gauge unless as the 4 SU(2) doublet R 1 2 eff , L + N U(1) ∆ to ] that flavour , we assign its SU(3) R SU(2) 1 38 u , H after spontaneous i 37 singlet scalar field is N . Firstly, it is shown S L charge of so that quarks and charged . The new 1 2 1 R SU(2) . In the table, − + symmetry in two Higgs doublet 1 R U(1) current contributions. Therefore, , we give the coupling constants of R U(1) 3 , we introduce our model as a minimal . For the Higgs field 2 U(1) singlet scalar field 1 in the same manner of the SM. Furthermore, L 1 . In the end, we give our conclusion in section – 3 – H 5 should be taken as SU(2) 1 ] that neutrino masses and mixing, dark matter and q must be introduced to cancel the gauge anomalies. charge is assigned to stands for the gauge symmetry of the SM. Under the boson and Majorana masses to 43 – R R X SM gauge symmetry. 39 and a G R 2 as right-handed neutrinos in the following discussions. U(1) U(1) i H N represent left-handed leptons and right-handed charged leptons, U(1) , which are singlet under the SM gauge symmetries. The charge R , where e 3) R , 2 boson and show the allowed region of the gauge coupling constants. , and , which can not be determined from anomaly-free conditions. However, ]. Only with the SM matter content, such a charge assignment generally . Thus its 1 U(1) L R q X 40 = 1 × represent left-handed quarks and right-handed up-type, down-type quarks, i ( R SM U(1) i represent the SM strong, weak and hypercharge gauge groups. The symbols, d G N , Y R doublet scalar field u gauge symmetry, right-handed chiral fermions are charged while left-handed chiral charge to L ] that neutrinos can not be Dirac particles due to the constraints from R R U(1) To explain the Atomki anomaly, we further extend the matter content by adding a This paper is organized as follows. In section 44 and . Therefore, we identify i in [ the coupling constant ofwhen neutrinos right-handed are neutrinos extremely haveintroduced small. Majorana to give masses. This a constraintbreaking mass The can of to be the avoided if requiring the model toleptons be can minimal, form Yukawa interactions with with this charge assignment, left-handedN neutrinos can form the Yukawa interaction with SU(2) respectively, and respectively. Without loss ofoverall generality, normalization. we Then, can the fixanomaly-free gauge the charges conditions of as the other shownU(1) fermions in are the determined from table One of the simplestfermions, solutions for non-vanishingassignment of anomalies the is fermions to inand our add model three is right-handed shownQ in table defined as U(1) ones are singlet [ leads to non-vanishing gauge-anomaliesnew due fermions to charged under our numerical results are shown in section 2 Model We start our discussion by introducing our model. The gauge symmetry of the model is symmetry. Motivated by thesethe previous Atomki works, anomaly we with construct a minimal model tosetup explain to explainfermions the to Atomki the anomaly.Then, In the signal section requirement and experimental constraints are explained in section to satisfy the Atomkicay signal. from vectorial With interactions smallerchanging is couplings, neutral much currents a suppressed. can contribution be Itextension. to suppressed was It neutral due shown was to pion also in de- shownthe [ in muon [ anomalous magnetic moment can be explained in models with the fermions. The existence of the axial interaction allows coupling constants to be smaller JHEP04(2021)025 ), (2.3) (2.1) (2.2c) 2.2d (2.2a) (2.2b) (2.2d) 1 0 S − . c . . In eq. (  2 1 2 + h 2 q H , respectively. In + 0 g SN c 1 2 represents the fermions N is arbitrary, and we will V, and N f 1 1 2 for up, down quarks and . 2 1 1 1 2 2 1 Y , 1 2 − , H = + + ν ), q H , respectively. The covariant µ 1 + µν ˜ q R , g X ˜ 2.2 as 0 ,Y 2 X N e g 2 1 µν Y yukawa ˜ ixg 1 2 H ˜ H U(1) B L N L +  2 − ν + , so that the stringent constraint from and charge of ˜ Y B and + 1 X d R + Y represent the gauge fields and their field µν 1 2 1 0 R gauge d ˜ ,Y − − iY g ˜ X R charges of each particle. The gauge coupling L charge of X u e U(1) Y U(1) µν − 1 R + R ˜ , µ X – 4 – , are denoted as L and 1 2 2 ˜ 1 4 LH | W 0 L e e U(1) 2 − R ˜ U(1) S scalar B − Y µ ig L gauge boson, + D µν SU(2) | − and + ˜ R U(1) ˜ R W, 1 3 1 2 B ) is given by µ R Y d + d ∂ − − 1 µν 2 | ˜ B U(1) = 2 and 2.2b 1 4 fermion U(1) QH µ H d Y ), and L µ 2 3 1 2 D − Y R D = + | + µν ,N . Matter contents and charge assignment of the model. U(1) ) and ( L + R R ˜ W 2 , u | / 1 Df, L 1 µν 2.2a ˜ 1 6 3 3 3 1 1 1 2 1 1 2 1 1 0 + f Q u H ˜ L, e H W , where we denote the µ Q f Table 1 1 4 represent the u 1 X D | − Y i SU(2) x and L R Y = = = = R , and its magnitude is parameterized by the constant parameter and ), the gauge symmetry of the model allows the gauge kinetic mixing term between SU(3) ˜ U(1) U(1) , u SU(2) X is the scalar potential which is given below. In eqs. ( Y L gauge scalar 2.2c V yukawa fermion L L L L and Q, u ˜ constants of eq. ( B the Dirac Yukawa matricescharged are leptons, denoted neutrinos, respectively. as The Yukawa matrix for right-handed neutrinos is strengths in the interaction basisderivative of in eqs. ( where and ( The Lagrangian of the model takes the form of where each term denotes theare defined fermion, as scalar, gauge and Yukawa sector Lagrangian which neutrino-electron scattering is avoided.discuss The possible charge assignments later.shown The in charge table assignment of the new scalars is2.1 also Lagrangian scalar field is alsoleft-handed introduced. neutrinos and the It plays an important role to reduce the mixing between JHEP04(2021)025 . is 2 Z 1 H (2.6) (2.4) (2.5c) (2.5a) (2.5b) (2.5d) . Such a , 2 H 4 = 1) | S β , | , 2 s | 2 2 λ GeV, for Model = 0 H | + 0 2 2 | 4 λ | 125 S 2 | ]. Therefore we need to = = 0 (tan 0 4 H | λ 0 1 β ., ., 2 46 ., ., 2 , c c + λ c c . . . . 2 45 | + cos 2 1 for each model. + h + h is determined. We classify models 4 + h + h | H 2 1 2 | 2 1 1 2 2 q 2 2 2 2 H | H H , is given by H H H / / / / is the Pauli matrix. Note that flavour , and the other consists of the terms † † | 2 2 1 0 S † † 2 1 1 3 2 q | 1 2 V 2 H H 0 3 − +3 − +5 λ H σ 2 2 λ SH SH S S + 1 2 1 2 + 2 , λ A A κ κ | 2 | 2 . v, S 2 . = = = = | where – 5 – 2 s H 1 2 3 4 ∗ 2 1 , | µ = 1 = 2 V V V V 2 | H s 2 1 − 2 , can be taken real by using phase rotation of 2 2 H , | iσ | 1 2 0 2 Model 1 Model 2 Model 3 Model 4 κ λ H | , and other two are absorbed by the neutral weak boson 2 2 + ± µ + 938 GeV 2 can be divided into two parts. One consists of the terms and | . Then, one massless CP-odd scalar remains in the spectrum, s , W 2 − given in table ˜ 0 charge assignment of 2 Model 1 : ∆ Model 2 : ∆ Model 3 : ∆ Model 4 : ∆ . The charge assignments of X . , 2 V βv H 1 2 | represents † , v 1 R 1 q A 1 = 5 H H ˜ | tan | H 0 4 0 1 1 2 0 3 U(1) λ λ develop vacuum expectation values (VEVs). Two of those are absorbed µ λ Table 2 0 GeV , λ 2 . − + = S √ 19 0 3 . = λ − . Here 0 and N V = = = 246 = 1 2 Y s 1 1 v λ λ A ,H 1 H The charge-dependent scalar potential in each model is given by The scalar potential found as where the parameters, One example of the parameter sets to reproduce the Higgs mass, with different choices of dependent on that. The charge-independent part, and the new gauge boson, which corresponds to themassless broken scalar degree boson causes of seriouswith freedom problems of meson by carrying the decay the phaseintroduce measurement energy rotation of such other of stars as and interaction conflicts ansymmetry terms breaking. axion which In does this give sense, [ the a possible mass choice of to the CP-odd scalar after the denoted as and generation indices are omitted for simplicity. independent of the where we assume thethat mass spontaneous breaking parameters of the as symmetriesappear well successfully occurs, in as the and the no potential. runaway quartic directions Withafter the couplings above to potential, we be obtainby positive five the Nambu-Goldstone so bosons charged weak boson, JHEP04(2021)025 ] ) ' X α 55 992 S . − 1 (2.7) 0 h β to the to the U , 1 ' 1 s GeV [ v h v h 2 cos( for the ]. H 1 ]. Diphoton to top quark 300 49 h + 4 2 GeV 2 ) and charged 2 . 51 ], , h XX a βU ]. H 1 54 50 [ h represent fermions → 58 , βU 1 +cos V = 1343 = 1 57 h 1 3 H β ), and that of h 1 1 h , respectively. Those are and h + cos V ), CP-odd ( tan 1 f βU 3 in H ], however, the mass range of 1 , h 2 h and 2 47 ), the coupling of (sin ] are indeed enhanced, and the H 1 . , h βU 61 where 1 0 2.6 ). The searches for exotic decays of ,, m h 1 ]. Hence this constraint also can be − and sin h 51 1 985 . in ) = H 0 α 1 GeV 7 GeV S model [ . . ' which is much smaller than the bound. For − R β 5 V − – 6 – via a charged Higgs boson loop. It was pointed 10 cos( = 1158 = 1329 U(1) v/m boson itself has not been searched, its decay into an 2 have been done in [ × ± V KX h h g e 3 could be produced at LHC. Once produced, it decays X . 4 in the SM processes, the predictions of our model are 2 1 ), is consistent with the latest bound. Furthermore, the → ≡ GeV in the analysis. Therefore, the gauge boson lighter ]. The decay branching ratio of is the coupling constant for h 1 → GeV for 2.7 h 48 V 15 V κ g , respectively. The mass spectrum and the value of > 490 √ XX πX, B = 1 , m → β , m → and being the components of % C.L. [ 1 being the components of 2 to the SM one. Then, the signal cross section is estimated as at most h K tan 95 H S 996 1 6 GeV 1 . ]. From the parameters given in eq. ( 12 . h h 5 GeV . 0 at . 0 U U ] for 53 , ' MeV mass is about ] that such decays in a 26 56 . 52 f 0 and 17 = 1370 = 125 with 60 , 1 ) bosons are given by 1 a is singlet-like and other scalar bosons are doublet-like. The couplings of 4 dominantly. Although the h H v/m ± GeV is not constrained by the four lepton search. When such an exotic decay is 59 GeV [ − 1 2 m f fb in [ h h boson is restricted as m h Y 1 10 15 U XX fb with converted decay probability given in [ 600 Details of the scalar sector are essentially irrelevant to our study about the Atomki The singlet-like scalar boson X × ≡ ∼ < bosons is much suppressed by a tiny component ( 45 9 f . . out in [ constraints from thesemodes show are a suppressed by tension fine-tunings.tunings with An for extension all the of decay Atomki modes. theand Higgs signal, However, sector left such could unless an for provide extension all our such is beyond future of the work. the scope of In decay this this paper, paper, we focus on the gauge sector which is most present bounds from Higgs measurements andbe electroweak precision evaded measurements due also to can the degenerate masses of theanomaly. heavy One Higgs exceptional bosons issue relevant [ toHiggs the detail scalars, of is the Higgs flavourthe sector, changing especially X charged meson boson decays such as with radiating the longitudinal mode of 0 evaded. The present lowerand bounds charged on Higgs boson the are massesand of the doublet-like heavyin CP our even, above odd, sample point, eq. ( into electron-positron pair could potentially be mis-reconstructedevents as from a photon new [ heavy particlesas have been searched atis LHC suppressed and by the constraints are given the decays and productionsκ of and weak gauge bosons,consistent and with the latest results of combined analyses shown in figure 12 of [ the than identified as invisible decaysis at given LHC, by the presentboson bound with on the invisible branching ratio weak gauge bosons are almost thewith same as those of the SM X 1 the SM Higgs boson Higgs ( where With these parameters, the masses of the CP-even ( JHEP04(2021)025 , (2.8) (2.9) µ (2.15) (2.12) (2.13) (2.14) (2.10) ˜ (2.11c) X (2.11a) (2.11b) . ) µ ˆ ζ ˜ i X 2 s v + . The gauge . 2 2 2 0 s      g g ) + ˆ β 1 2 + ) 2 s correspond to the β 2 1 v + , g 2 ( ˜ matrix as s cos A  2 v 2 2 2 q 1 3 cos q  / 2 √ 2 1 1 µ × q + √ g and ˜ = X . 3 β 0 + 2 ˜ 2 = = Z 2 g v β i i q 2 sin W ,S + S θ h 2 1 , . 2 1 sin q   + 2 µ µ v ( 1 ) , 2 q ˜ ˜ sin µ 2 B B ( v , = ˜ ˆ a ! ˜ Z 2 Z 0 i µ W W 2 2 2 2 g 0 θ θ ˜ , v g F + ) vm ˜ +

0 2 2 F . In the following, we parameterize the µ 2 + g 2 s sin + v. 0 2 2 β, v ˆ h ˜ m 2 v 1 − 2 W 1 ˆ 2 2 g i H T 0 − √ g + cos µ + g → ˜ 3 cos 3 F µ q 2 µ ∓ = = ) v 1 2 v ˜ ˜ 1 µ ) W ( − – 7 – i W ,  2 2 β 0 ˜ W = = 1  W W W 2 g √ ( θ H 2 θ ( m h + 2   cos 1 µ √ mass is given by 2 , = − q β, v L ˜ ! 2 F 2 = = cos = sin + 1 W 0 m v sin µ µ ± β + µ µ ˜ ˜ ˜ 2 Z v Z

2 A ,H W W m 2 2 = 2 sin 1   g √ 1 ) 1 , and symmetries are broken down, the gauge boson masses are gener- 2 q 1 v T (  ˆ a = ) ˜ R Z i 2 i µ i v ˜ 1 + X vm 0 , 2 1 H =1 g U(1) + µ h i X 1 ˆ h ˜ − ˆ Z 1 8 H , + µ 1 00 0 0 0 = ˜ v A      ( 2 , is the charged weak gauge boson of the SM, and is the Weinberg angle of the SM defined by 1 = = ( √ ± mass , ˜ µ 2 F W   W ˜ θ F m = gauge Then, the mass terms of the gauge fields are given by 1 L boson and photon in the SM limit, H Here, where With this parametrization, the charged weak gauge boson mass is given by The mass terms of the neutral gauge bosons can be casted in a boson, Z VEVs as, with and each scalar field is expanded around its VEV as appropriately chosen so thatthe the Atomki new anomaly. gauge boson acquires the2.2 mass required to Gauge explain bosonAfter masses the and EW mass and eigenstates ated via the VEVs of the scalar fields. We denote the VEVs as relevant to the Atomki signal, and assume that the parameters in the scalar potential are JHEP04(2021)025 (2.25) (2.20) (2.21) (2.24) (2.16) (2.17) (2.18) (2.19) (2.23c) (2.22a) (2.23a) (2.22b) (2.23b) F V , ) is given by χ, χ.    ) MeV by the Atomki as 2 2 cos cos . 2.24 δ ) T 17 χ χ ) β , + ) 2 1 ' X 2 1 β sin sin δ δ ˜ 2 1 1 Z cos Z, δ δ + 2 ˜ ˜ Z Z cos q rm 2 s A, , , 2 v − + q 2 basis is given as rm rm    0    2 = ( g β . µ χ + W ( χ 2 W θ 2 − + 2 F F 2 X v. β θ r 1 2 2 sin . 2 χ χ m δ | sin r g cos to sin 2 2 ˜ Z 1 β − cos 1 − sin q + ˜ F, δ r F ( 1 sin ˜ 2 ˜ 1 2 Z r rm Z ˜ ˜ K q cos − 2 Z Z ) sin v ˜ g χ χ 2 Z 0 ( 2 2 m U β g v m m δ q 0 m rm − – 8 – r , the leading term of eq. ( g = 1 2 − + + X = ˜ − + 2 1 F ) sin F, = χ 1 0 0 1 0 0 δ m 2 10 0 cos 0 sin 0 ˜ χ F Z 00 00 0 ˜ 2 q W Z    V    θ +    m  m − 2 s = tan = = ˜ W 1 v Z = sin ) cos θ 2 q 0 2 K F 2 ( F ) can be diagonalized by an orthogonal matrix m K δ g r V v U ( 0 U sin 2 g + ˜  | 2 F r 2.19 2 1 ' − δ m = = + T χ K 1 2 + χ δ δ boson mass defined by U is the mass eigenstates. Their mass eigenvalues are given by 2 2 s v can be expressed as . Then, the mass matrix in the is an input of the model which should be tan T Z = 2 2 0 ) cos / χ g 2 F 1 X ˜ ( 2 Z − , 2 m ) m r 2  = 0 = = A, Z, X − 2 Z 2 A 2 X is an orthogonal matrix given by is the SM m m = ( m ˜ K Z = (1 F U m r The mixing angle Here the mass of experiment. In the situation of where Next, the mass matrix eq. ( with with where To obtain the masseskinetic of term the by changing neutral the gauge basis bosons, of the we fields first diagonalize the gauge boson Here, JHEP04(2021)025 χ is ˜ Z GeV (3.1) m (2.26) (2.27) (2.28) (3.2a) (3.2b) (2.29c) (2.29a) (2.29b) 1876 . and Z . Therefore, m = 91 ) 3 Z , − ), the interaction m χ, 10 NC ,  2 2.29 ' , −  . cos 4 ˜ Z     4 X W − W µ θ m θ and axial-vector couplings m X 2 (10 µ µ χ, V f − 13 boson to quarks and leptons. sin  O X X sin 2 2 U is expressed in terms of other , δ 3 2 r 23 33 cos (100 MeV) X µ + s 2 Z U U v − − W ∼ µ ) with eqs. ( m ) θ + + fX ) 2 χ Z 4 1 χ. ) 2 2 X r µ µ 5  12 v 2.28 cos Z Z boson are modified due to the mixing γ m 2 sin − U 0 cos − A f 22 32 r  − − r )) , g + are roughly X χ U U 2 2 ˜ 2 Z 2 Z µ + =  δ = = A m m cos V f + ) and ( 13 ( 2 23 33       2 – 9 – ( 2 1 ). Then, the difference between 0 W and δ µ g ] and therefore we use θ 0 = ( 2 2.11 , can be written in the following form, 2 g r ) , fγ 62 r 2.25     max( e cos µ µ µ χ, U + NC ' Z = A r  X 2 Z 2 3 1 4 2 1 sin     GeV [ δ int m ( + + U W L ' u, d, ν, N θ 2 X χ χ χ, U boson to fermions = ˜ 2 Z ) should be positive for consistency. 0021 m . sin (= ), the Lagrangian can be written in the mass basis of the gauge     0 m sin cos cos X µ = µ µ f r r r 2.27 ˜ ± − ˜ ˜ W 2 Z s A X R R 2.28 θ v −   2 Z χ, U     1 4 1 4 χ m cos 1876 and its elements are sin . = = r r ) is used. This difference is smaller than the present error of the measured F A u 91 V u ) and ( V   = = cos = K 12 22 32 U 2.11 2.23b U U U = is the proton electric charge. The vector coupling e U In the end, the gauge eigenstates are expressed in terms of the mass eigenstates as are given by boson mass, A f  among the gauge bosons.Lagrangian of fermions, Using eqs. ( where 3 Couplings of the In this section, weThe present gauge the coupling interactions constants of of the the fermions to the From eqs. ( boson. where Right-hand-side of eq. ( Z as an inputparameters value as in the following discussion. Then, is much smaller thanroughly unity given from as, eq. ( where eq. ( In the parameter space of our interest, JHEP04(2021)025 ∗ M β can (3.5) (3.6) (3.7) (3.3) (3.4) L (3.2f) (3.2c) (3.2e) (3.8a) and (3.8b) (3.2d) As we ν cos 2 ν eV. , 1 2 m . 2 0 in the limit , ∼ ! ν NC ), the approxi- where ˜ , 4 Z  4 X m ν m 3.2 m M NC m for  , W , The Majorana mass is 4 p  θ − NC  (1)  . From table W 10 θ O X  cos 2 = m W , θ ˜ sin N 4 Z 4 X 2  ) into eqs. ( Y 1 3 m m . ˜ β. Z sin 3.5 R − χ , m − 2 , 1 4 , Q and neglected higher order terms of χ ]. The left-handed neutrinos − 1 4 cos  .

, ) cos 2 W R rc  2 2 2 63  = 0 = θ + W . q q q O R θ NC  ∗ ∗ + χ due to  − + − + 1 4 NC β β cos takes a specific value of 1 4 χ   1 1 and 1 sin ˜ 2 Z q q 1 2 2 X q cos = β W  − ( r θ cos m m − W ≡ A N χ −  θ + W  ∗ – 10 – = . Inserting eq. ( ) sin θ 2 β ˜ χ = 2 Z cos 2 X L W q cos ν ) θ r   V N cos + sin 2 X R m /m . Thus, the mixing is smaller than r  3 cos 2 1 1 3 2 X m cos r − 1 4 q + = ,  m − 10 − − − + W Model 1 : cos 2 Model 2 : cos 2 NC = ( ˜ NC θ 2 Z < χ χ  R =  0 4 1 ) vanishes when Q m g as A e ( Q represents the neutral current contribution defined by cos cos cos −   3.5 W r r R NC θ = = R R NC −  as,    Q A ν A u 4 1 4 1 R ˜   1 2 Z cos GeV for boson through the weak neutral current. Thus, the coupling constant − − − − m Q 10 for convenience as = = = = |  X Q Q A d V e V ν V d , where ' − ' − |     /e 0 NC ), we neglect the mixing between left and right handed neutrinos. g  and = 3.2 1 R  is proportional to and larger than |  The first term of eq. ( To obtain approximate formulae of the coupling constants, we expand ) L The mixing between the left and right-handed neutrinos is roughly given by s χ 2 | ν v ( with are the active neutrinoO mass and Majorana mass, respectively. Taking Then, the remaining term isis much given smaller in than each Model by In the expansion, we keptthese the leading couplings term in of eachmate power of expression of thesignal coupling requirement constants and can constraints be as obtained, we which will is explain useful later. to understand where we define of explained above, one of thetering most of stringent reactor constraints neutrinos comesinteract measured from with at neutrino-electron the TEXONO scat- [ of In eqs. ( JHEP04(2021)025 . 4 − (3.9) 10 (3.8c) (3.8d) should . Thus, 1 β in Model and 4 ≥ . In figure, we − tan = 0 Q Q 10 Q × , the constraint can = 5 10  9 01 . Q=0 , which we will explain 0 8 Q=1 4 Be . However, nuclear matrix Q=0.1 Q=0.01 8 − ≤ 7 10 Q 6 , and the constraints from various , . 5 and 4  1 2 3 2 %, which is calculated from while it does not for 4 . Red, blue, green and brown lines Q − − − 1 R 1 scattering while Model 2 and 4, with 2 3 4 .  4 0 = = 10 e  - plane is almost independent of ∗ ∗ ν ≤ × Q β β x10 3 Be decay NC 8 Q R  ' - ϵ . Red, blue, green and brown lines correspond to = 5 R ∗  R β   ] and we need further study to reduce the uncertainty. – 11 – , while there are no solutions for ) more than 3 35 ∗ 2 , respectively. Solid and dashed ones correspond tan , the dashed and solid curves are almost the same. β 1 3.7 − He [ and as a function of 4 = 0 tan β | 1 and Q Model 3 : cos 2 Model 4 : cos 2 NC 1 tan  . | of our interest in Model 1 and 3. In this case, 0 R , as a function of 1  . Gray filled regions with solid and dashed edges are exclusion 01 4 . -2 -3 -4 -5 -6 -7 -8 -9 NC -10 -11 -12 0 − 

He anomaly in our analysis.

10 10 10 10 10 10 10 10

4

10 10 10 , 10 Except for NC | ϵ | , respectively. Solid and dashed ones correspond to 1 3 × . is a plot of = 0 5 and 1 can evade the constraint from the Q 1 . 3 4.2.2 ], which can be consistently explained by a light particle for and 0 6 4 , . − . The coupling and can be vanished in Model 01 He [ . 4 . Figure 0 1 = 0 4 Q , are excluded for the choice of the parameters. When , 1 Q = 10 It should be noted that the exclusion regionAs in we explained in the Introduction, the Atomki collaboration also reported that a peak like excess was  3 4 ≥ = 0 and fixed found in elements have significant uncertaintyThus for we will not indicate for Model 1 (3). 4 Signal and experimentalWe summarize constraints the signal requirement fromexperiments. the Q be evaded in thebe region tuned of to the value obtained from ( to region by neutrino-electron scatteringin for subsection One can see thatModel the allowed region exists for Thus, 2 correspond to Figure 1 Q Gray filled region are exclusion region by the TEXONO results. JHEP04(2021)025 ]. X 30 and (4.4) (4.1) (4.2) (4.3) ∗ (4.5a) (4.5b) Be 8 , 2 01(16) 6(1) . i| Average 1 . 17 || decay of ∗ p σ γ Be || 8 0 h σ p , a ) 90 17(7) . 7(21) . Exp2 . − + 4 4 e 17 i ]. The collaboration measured + 1 e 5 , || 6 – n particle, followed by the decay of boson mass have been constrained 1 − σ → || X X 10 σ 0 . X h , we employ the results given in [ , , × 37 ∗ Be [ 86(6) n . 8(10) . 8 1 1 Exp1 . into an electron-positron pair. From the a 7 7 Br( | 6 a a Be 16 ) ) 8 of ∼ < ! X 2 2 − + γ 0 MeV X . ∗ 2 k 2 X 0 0 X ) and the a a Be E Be 17 m B Be 8 ] 8 are the three momentum and energy of the – 12 – 8 1 = = 4.2 ' ∼ < → p → E n particle and branching ratio of 7 2 + X a eV is the partial width of the ∗ a ∗ −

X m MeV for the lower and upper bounds, respectively. 4) into the assumed Be 10 Be . σ π = ∆ 7 8 8 8 . 0 . 8 ∗ k . 5 × ]). These values have a relatively large uncertainty, which 70(51) k 18 6 Γ( . Γ( 16 5 ± Be 4 E 8 16 9 . ≡ , are defined as ) = n with a mass of and (1 X a X and Previous Result [ MeV being the difference of the energy level. The proton and 6 B X . ' Be 2 X . The mass of 15 MeV excited state ) . 8 17 and m γ p 15 → ) . a − Be 6 ∗ (taken from [ 8 2 = 18 − 18 3 E is the decay branching ratio of Table 3 Be 10 decay branching ratio E → 8 ) ∆ (MeV) × ∆ ( ∗ − , is defined by ∗ Γ( q e X − is taken to X + Be e m Significance Be = 8 e B X + 8 e k Γ( m → To calculate the decay branching ratio of X into boson, with neutron couplings, where The partial decay width from the axial part of the gauge interaction is expressed as Therefore, we employ a rather conservative range for our numerical calculation where where Br( Atomki experiment, the branching ratio ( as given in table may originate from systematic uncertainty of unstable beam position in the experiment. The signal branching ratioX of angular correlations, and found aresult significant is peak-like mostly enhancement well-fitted atan under larger intermediate the angles. particle assumption This of the creation and subsequent decay of 4.1 Signal requirement 4.1.1 The Atomki collaboration hastransition of reported the an anomalous internal pair creation for the M1 JHEP04(2021)025 as 2 ) (4.9) (4.7c) (4.13) (4.10) (4.11) (4.12) (4.6a) (4.7a) (4.8a) (4.6b) (4.7b) (4.8b) A e can be  ( A e  , 1 |  , , A d Q  | ), we have neglected A u ) , ]  . p , ) ( ˜ ), is expressed by 2 Z ) 2 R 2 X ˜ s 64 . 2 Z 2 X p  4.10 ( 2 and m is proportional to m  ) m d m + ∗ , 1 A e 4.10 W  ∆ 2   θ L ) ( = 2∆ 2 ν Be , , W  −  W 8 θ , ! 2 R A d 3 2  ) θ W 2 cos    , . |  p 2 k ) θ 2 X 2 ( A e χ p + −  cos | u ( E Q cos m 2 s ), we obtain the branching ratio of 4 ). In eq. ( R + + A e 897(27) 367(27)  026(4) + . .  . ) 2 R 0 4.2 3.4 0 2 + R V e 047(29) 132(33) Q + + 4 cos . .  )

− − Q 0 0 2 R Q ) + 2∆ ) = 2(∆ W π  V e − − A d A d θ  ( k    = = 0   = 2 18 1 4 3 2 = = ) ) ) ) and ( ] + − W n n – 13 – n ∝ ( θ is given by + − ( ( ) = A u A u 30 d u s ) 2 3.2 4 sin 1 −   kSi kSi R − e p  n e X )( )( − ) ' sin + σ ) ) σ + p p e k = ∆ = ∆ = ∆ k Q ) e ( ( Be − + 8 ) ) ) + d d − − p p p 0 e 0 → 1 4 ( ( ( → h ∆ h s + d u → , the partial decay width of  (1 e X (1 + (1 + ∆ − terms, the branching ratio, eq. ( A e X ∗ ∆ ∆ ∆ 1 4  − 1 4 ) ) ) ). The quark coefficients take values [ p p → ˜ Be ( ( Br( 2 Z = 8 u u 3.2 ' − X A u ' − Γ( /m  A e V e 2 X  − Br(  are given in eqs. ( = (∆ = (∆ m = ( 0 1 L a a ) with these numbers into eq. ( ν A d O   4.4 ) is used. Similarly, the vector coupling of electron is approximated as and 3.5 is given in eqs. ( e V,A  decay. A u,d  Before closing this subsection, we show the parameter dependence of the signal branch- ∗ Be and the nuclear matrix elements takes [ Thus, neglecting where eq. ( where the masses ofapproximated electron as and neutrino. In the case of The decay branching ratio of ing ratio. Using Inserting eq. ( 8 and where JHEP04(2021)025 4 − 10 (4.14) (4.15) × (green), 9 = 5 . 0  . We fixed as R  boson propagates 10 X . . 9 2 . # ) (blue) and ) Q=0 8 2 R R −   5 Q=0.9 increases. This is because e . ( 7 0 + + . Red, blue and green curves Q e ), the signal branching ratio 2 R 6 )  Q=0.5  → 2 R 4.13 W  5 (red), θ , becomes large for non-vanishing X 2 as a function of MeV, the most stringent constraint Q NC 4 Br( 4  − 17 = 0 4 e q ) and ( ' + 5 Q + 4 cos e − L ], we require that the X x10 R 3 2 ν 4.11   10 R m 10 → ( 3 2 as a function of 2 ϵ × 3 2 A e − +  X – 14 – 3 e . − 2 1 + + A e 1 2 e  2 " ≥ V e 2 R + 2 →   ) 2 2 boson with A e ) X V e   (dashed) and as Q X 2 , respectively. Solid and dashed ones correspond to ) 4 + ( 9 − . e A − 2 0  ) ( (1 10 V e 1  ∝ ∝ and (

.

5 X

0.2 0.0 1.0 0.8 0.6 0.4

q

. → ( ) e X Br

e 1 0

B

- + , , the branching ratio is simply determined by 1 = 0 (solid) and Q |  4 cm from its production point, which gives the condition as boson lifetime − Q | shows the branching ratio of 1 . The branching ratio of 10 . X 2 4 × − Using the approximate expressions, eqs. ( 10 as shown in figure = 5 Figure 4.2 Constraints 4.2.1 Rare decayThe of coupling neutral constants pion ofdecay the experiments. light The gauge gauge boson bosonkinematically can allowed. to be For produced quarks the in can rare be meson decays constrained when those by are meson electro-positron pair can beless detected. than As in [ Thus, for 4.1.2 To explain the Atomki anomaly, the new vector should decay inside the detector so that the is scaled by the parameters as  respectively. One can see thatthe the coupling branching ratio to decrease neutrinos, as Q which is proportional to Figure 2 correspond to and JHEP04(2021)025 ,
(4.16) (4.17) (4.18) T (4.20c) (4.19a) (4.20a) (4.19b) (4.20b) e m ) 0 L g boson is given R with a photon, gauge boson to g is the energy of X L X + ν ] puts the following − E 0 R , and other coupling g . Therefore, our result is B 65 F L Q g G ( , ) interaction to quarks can − A e ,  2 ) W . X + for small T θ ) V e 2 5 , − fit is performed. As we have shown in  −
− e ( 2 . ν T 3 e + 3 χ 10 ) is simplified as e e E − − ( = sin = , m 0 10 R 10 → V e R 0 R R 4.16 g 5  g model, and constrain the parameters by × L ≤ L X − g

3 R g .  = 0 − Br( W + 2 , eq. ( θ V d 2 1 2 p ν U(1)  ) e – 15 – E T + 0 ], the contributions from the L |  g − V u | ≤ + cos 67  Q R ν | , g V d 2 g R  E , g 2  ( W ) ]. The most stringent constraint for the 1 4 θ 2 L + ) A e

g ]. In [ 2  66 T V u e  ). For + 67 − . 2 | m 2 ν V ν V e F + sin  E 4.12 e ( G 2 R + 2 2 1 e e g gauge boson was shown. In this work, we derive the interference 2 X m = = = 2 ]. 0 ν 2 F m L 0 ν g 0 L L 2 ν 20 ( G g g g e π − are the recoil energy and the mass of electron, and 2 πE m ], this constraint is computed using data and B e 2 √ 20 . Theoretically, only vectorial parts of the m = = γX int SM and   → T , the coupling constant of neutrinos is much smaller than dσ dT 0 dσ dT 1 scattering have been studied. The authors analyzed the differential cross section with The differential cross section in the SM and the interference term between the SM and π  Recently, in [  boson contributions are given by e 5 - e figure consistent with that of [ incident neutrino, respectively. Theconstants Fermi are constant given is by denoted as where term of the differential crosscomparing section the in differential the cross section in the SM. X by the TEXONO experiment [ ν respect to the recoilsizable energy contributions. of scattered Basedcoupling electron on of and the the showed analyses, the the interference allowed term region gives of the mass and gauge 4.2.2 Neutrino-electron scattering The interaction of the gauge bosonby to neutrino-electron leptons, scattering especially [ to neutrinos, is tightly constrained The left-hand-side of the constraint is rewritten by and approximated as eq. ( i.e. contribute to the decay. Thebound, latest result of the NA48/2 experiment [ among such meson decays comes from the rare decay of neutral pion into JHEP04(2021)025 ] = 68 T as, ], the (4.26) (4.23) (4.21) (4.22) 1 (4.24a) (4.25a) 16(sys) (4.24b) (4.25b) 70 . 0 |  ± Q | MeV and 0 . ]. From [ and 21(stat) 74 1 . – = 3 0 , , 2 6 ν 71 ± − |  − is always negative while E . , χ l 9 08 | l , ) can be done numerically 10 10 . y − ) 1  . δa × . ) × x l 10 R 4.24 l y −  ( y /Q × ) 419 8 484 A . . 3) x I , − . )(4 2 8 79) . 2 ) − . 10 , 0 x A l 0 + 1 l  ) = 9 ) = 3 ∼ > y ( < − e µ is important for this constraint. Unless ± ) Q ) y y MeV, respectively. + (1 7 R x x − ( ( x 7 Q 2 61 int ) .  A A SM . 10 − l x −   + ( y ( × 3 dσ dT (1 dσ dT V are given by 0 x ,I ,I 2 = (2 .  I  + (1 7 4 2 – 16 – = 105 ) x 2 l − − 2 (5 ) µ y < SM µ x dx ( V l 10 10 a  m 1 A ( ] and theoretical predictions [ 0 64 dx I ' − × × . −  Z 1 0 l 0 2 70 1 2 y , Z − int π SM exp µ e 881 894 and  4 . .  a 69 ) l dσ dT ) = ) = dσ = dT = y l l MeV and  ( l  y y µ ( ( ) = 7 ) = 6 V range. We use our numerical analysis a ) can be approximated for the case of e δa µ I A V y y I ∆ I σ ( ( 5110 3 . V V 4.21 I I , and = 0 l 2 e m /m X 2 m = ], the event rate relative to that of the SM is given by l y 63 The muon anomalous magnetic moment has exhibited a long-standing discrepancy The ratio in eq. ( MeV, respectively. is very close to zero, the constraint excludes 0 . between experimental results [ discrepancy is given by for electron and muon as where we used It should be noticedthe that vector contribution the is axial positive. coupling The contribution integration to of eqs. ( where The anomalous magnetic moments ofby experiments the and charged also leptons predicted precisely haveto in the been charged the measured leptons, SM. can accurately The shiftvia new the quantum anomalous vector boson, magnetic loop moments that corrections. from couples the One SM loop predictions contribution of the vector boson is given by [ It can be understoodQ from above equation that 4.2.3 Anomalous magnetic moment of charged lepton 3 in the TEXONO experiment. Thus we require which corresponds to From [ JHEP04(2021)025 , of γX W (4.29) (4.30) (4.31) (4.27) (4.28) Q . → A 0 I π , . and ], uncertainty of ], we impose a 71 . 71 V . 0 30 σ 30 I 0 2 ≤ ]. Although recent ≤  80 can be positive when , also has been mea- ), results in too large  2 – 2 2 µ / 77 e δa 4.14 2) 8 ], via bremsstrahlung from . 8 . 0 − 88 0 MeV) 0 . . , 0 MeV) g . ] employed in [ 13 2 R is the axial coupling term and  − (17 ) 81 (17 µ = ( µ  10 . y   e ( δa 9  × a − A V d  I 8  2 10 ) ) W ∼ < θ N ] and NA64 [ × Q ), one finds that e due to the smaller value of boson should be less than 87 − 58 δa cos + 2 . µ MeV mass, one obtains the following constraint 1 X discrepancy between the measurement and SM (1 Z ∼ < – 17 – Z 4.25b δa 2 ∼ < 17 σ 2 π 13 5 | + 4 e + ( . − µ 64 2 R V u  10 ], Orsay [  δa ) | = ) . further worsens the discrepancy. Thus a special care to | × is required. Following the discussion in [ R N N 86 µ set by the signal requirement, ( µ  , µ + 26 + a δa A e δa 85 |  − boson with Z ∆ Z are the number of proton and neutrino in Cs nucleus, respec- (  (2 X  R 16  is given by also exhibits A e ], and predicted precisely within the SM [  2 = 78 e 2 e W a 76 F e represent the anomalous magnetic moment by the measurements and 2 , F Q 2 N G √ G 75 ∆ √ 2 SM µ 2 , a ]. For the 1 and − 84 ' − . However, such parameter region is excluded by the constraint from | = and ]. The constraint is given by the measurement of the effective weak charge |  A e W  W = 58 | Q ), Q 83 | exp µ , Q 10 Z a ∆ ∆ 82 4.26 ∼ > For The anomalous magnetic moment of electron, in this situation. Then, the dominant part in | V e V e  4.2.5 electron beamAnother dump constraint experiments is obtained byiments, searches such for as gauge SLAC bosonelectron E141 at and electron [ nuclei beam scatterings. dump(1) exper- The the null gauge results boson of can not these be searches produced are due interpreted to as very small either coupling, or (2) the gauge boson tively. the Cs atom [ where This constraint is weaker than that from 4.2.4 Effective weakThe charge axial-vector coupling of electron can(Cs) be [ restricted by atomic parity violation in Cesium sured accurately [ results claimed that predictions, we impose a rather conservative constraint [ This above constraint is scaled by the parameter as thus it is determined mostly by implement the constraint of constraint that the contributioneq. from ( the SM predictions, respectively. From| eqs. ( because the lower bound on that negative contribution to where JHEP04(2021)025 is 0 ], one (4.35) (4.36) (4.37) (4.38) (4.32) (4.33) (4.34) < 2 that the 89 q , β 88 boson has been set X . ) . 0 − e MeV is given as > + . . , the requirement turns out , e ) 6 17 2 X − 4 X − − , e e 0 → m m + 10 + 3 e e − > X −  , the exclusion region for × . 2 2 collider experiment such as KLOE- 2 R 2 → R ˜ 4 δ 2 Z → . 10  .  Br( − 2 Z 2 4 7 /e m X e q × 8 − X - q m 8 | ≥ − 2 4 2 2 + 10 r − q 10 e 2 R boson to electrons induce an extra parity | Br( , the constraint is given by 10  × − A e 10 × p p ×  1 X + 0 . )) × . 1 2 3 2 ≤ 2 S ) – 18 – 8 δ boson to satisfy the Atomki signal requirement, .  2 and > ) 6 + 3 | ≤ followed by + W | A e 2 X A e V e θ 2  ≥ 1  R   boson with the mass of δ  7 2 | V e ( . γX )  2 + ( | X e A r 2  . Thus, the constraint excludes the parameter space when is the same order of → ) 2 +  + 4 cos V e / −  + ( 29 + 6 2 Z e ( R 2  + m ) ( = 1 q e ( e V ) must be positive. Since 1 2 X  q ( m q 2.27 ] experiments. The most stringent limit on the and 91 = 0 Q ]. The constraint for the 92 ] and BaBar [ is negative. Assuming that 90 2 constraints listed in theconstant of previous left-handed section. neutrinoconstraint in As from Model we 2 neutrino-electron explained and scatteringsignal 4 in requirement can is and section not so the III, be largeTherefore, constraints, and we for the evaded. show found any coupling our no value We numerical allowed of results have region on analyzed in Model these the 1 two and models. 3. 5 Numerical results In this section, we show our numerical results of the signal requirement and experimental where we set q given by 4.2.8 Vacuum expectationFor value consistency, of eq. ( to be This constraint is also approximated as Vector and axial-vector interactionsviolation of in the Möller scattering.ment The [ cross section was measured at the SLAC E158 experi- This constraint is weaker than that of4.2.7 electron beam dump experiment. Parity violating Möller scattering 4.2.6 Electron-positron colliderThe experiments coupling to electron2 is [ also constrained by by KLOE-2, searching for the latter one restricts theobtain electron the couplings. constraint, From the latest result of NA64 [ Using the approximate expression of decays rapidly in the dump. For the JHEP04(2021)025 MeV (5.1) (dark 7 . µ 2) plane. In 16  − - (brown). g R ( and S  6 . in for comparison as 1 17 05 . 0 MeV for the constraints 01 . 17 and also scattering (purple), 0 e - ν to β β, is fixed to be , while it is taken to be cos 2 X cos 2 X m − – 19 – = decay (yellow), ) and exclusion regions for Model Q 0 π 4.3 . We took = 0 β cos 2 , which determines the constraints from neutrino-electron scattering Q vanishes for Q . Allowed parameter region and signal in Model 1. The red band is the signal region shows the signal region eq. ( 3 and hence shown in the top of eachbecause panels. the The constraints mass are of less sensitive to 5.1 Model 1 Figure this model, the factor and signal, is given by, Figure 3 for the Atomki results.excluded by White the region constraintsblue), is from effective allowed weak rare by charge (light experiments, blue), while electron other beam colored dump (green), regions and are VEV of JHEP04(2021)025 , 0 2 ≤ . νν < 3 . (5.4) 12 (5.2) (5.3) → R 2 chosen , which ∼ <  X R 4 β  10 cos 2 | ×  | is not suppressed . The qualitative 0 , respectively. For ∼ < 0 , the consistent re- scattering (purple), . NC 5 ) with > .  e 05 8 12 - . is given by R 0 ν 4.22  ∼ < Q − exclude the region in large 4 constraint excludes the re- and S = 10 4 . 4 β γX | × ∼ <  (yellow), → | 4 cos 2 0 ∼ < π , , 10 γX β, 4 4 , the signal requirement shows slight 4 ). The last one is the constraints from . This is because . 1 − − . 8 and VEV of | × 0 → 10 10 . In this model, . For R µ = 0 ± 4.38 0  4 . We found a narrow consistent region in | 3 . × × 2 cos 2 2) π β and 1 = 4 5 ), the exclusion region from this constraint is ), ( ∼ 12 < − . . − 4 . In this model, the constraint from the VEV ), we found wider parameter region consistent . 4 2 g 9 4 1 – 20 – Q 4 . ( ∼ . is needed to satisfy the signal by enhancing the < cos 2 2 − > > 4.38 4.33 . The coupling constant for this region is 4 = 0 11 ∼ < R | | 5  . = as figure 4 10 R R Q 0 ≤   ( 3 | | Q 10 − 4 (outside), respectively. Color filled regions are exclusion . The general behavior of the signal requirement and ). For 6 | × 5 ) and (  . 10 − = | | × = 0 0 R 10 4.14 β − ∼ < β  | × 4.28 region. In such region, the decay branching ratio of |  × 4 | . From eq. ( . = 7  ∼ < 8 | ), ( cos 2 ≤ | cos 2 β 1 , . = 0 4 5 2 . . is 4 4.18 8 , Q 1 cos 2 . ∼ < 05 . − 4 in small e , while it disappears for and 1 +  while the constraints from 10 . 5 (inside) and = 0 e . | , there are no consistent region found in our analysis.  7 2 | β 1 . − = 0 → | × (dark blue), effective weak charge (light blue), electron beam dump experiment 0 ≤ | R 2 10  β X 4 | is the same plot for model ∼ > cos 2 × − (brown). | 10 4 g ∼ < β S excludes most of the signal region, even if the neutrino-electron scattering constraint cos 2 2 About the constraints, one can also see that the We first explain the general behavior of the signal requirement and constraints. In = 4 . . The central region is excluded by the constraint from electron beam dump experiment. | | × | S 2 X R R cos 2   which is in good agreement with our numerical results. obtained as while that in Model constraints is almost thethe same parameter as space in for | Model of is avoided by taking 5.2 Model 3 Figure and hence we took with the Atomki signalis and all of thegion constraints. in The the coupling parameterand space constants for for is this found in region | the constraints, eqs. ( neutrino-electron scattering, which ishere. well It approximated is by seen eq.for that ( the constraint excludes largein region the of former the cases. parameter space For in gion in large | The constraint from effective weakbehavior charge of excludes these the exclusion region regions of can be understood by the approximated expressions of each panel, one can seeis that the well signal approximated requirement is bydependence almost eq. determined on only ( by is not negligible, anddecay therefore of larger B region by the experimentalmuon constraints from (green) and also by thefield theoretical constraints from the positive VEV squared of the scalar for the signals, respectively. The red transparent band represents the signal region with JHEP04(2021)025 + , 2 / 3 ± , 2 / and in Model He anomalies 1 4 − 05 . 0 = 2 < q | Be and β 8 symmetric model. As a R cos 2 | charges of two doublet Higgs R ] appeared, in which the axial- U(1) in Model 3. and also other models have been 35 3 β U(1) cos respectively, we found that consistent regions doublet and one SM singlet Higgs scalar parti- 3 – 21 – and SU(2) 1 . Other values of 5 . The same figure as figure . 0 − = β Figure 4 , the consistent region can be found for , called Model 1 2 / cos 2 3 − While we were finishing this work, ref. [ and 2 boson. The Atomki signal requirement and other experimental constraints have / 1 17 − We first classified the models depending on the X . Two of them leads to large neutrino coupling to electron, and hence such cases are = 2 , it is found for / 2 excluded by experimental and theoretical constraints. Comment. vector hypothesis wascould examined. be explained The by significant authors uncertainty concluded in nuclear that matrix elements. q with the signal and constraints exist.scattering In is such suppressed regions, the duedoublets. constraint to from In neutrino-electron the Model cancellation3 of the gauge charges between two Higgs fields, by requiring allthat of the CP-odd possible as choices of well the5 as gauge the charges are CP-even limited scalars toexcluded to four by cases be the massive. constraint We from found neutrino-electron scattering. Then, for other models with cancellation of gauge anomalies. Two cles are also introduced to evadeand the relativistic stringent degree constraints at from neutrino-electron thethe scattering early Universe. Then, thebeen new studied gauge analytically boson and is numerically identified in with this model. 6 Conclusion We have discussed theminimal model Atomki to anomaly solve in the the anomaly, three gauged right-handed neutrinos are introduced for the JHEP04(2021)025 , ]. ]. ] , Be Be 8 8 (2018) Be 8 SPIRE SPIRE → 232 IN IN [ ∗ Be nuclear ][ 8 1056 Be Be: a possible 8 8 ]. ]. , arXiv:1504.01527 SPIRE ]. [ IN -boson coupled to ]. ][ EPJ Web Conf. X SPIRE , IN SPIRE (2017) 159 ][ MeV particle created in IN SPIRE arXiv:1704.02491 ]. ]. IN [ ][ J. Phys. Conf. Ser. PoS(BORMIO2017)036 17 773 , , ][ (2016) 042501 SPIRE SPIRE IN IN MeV X boson in the 17.6 MeV M1 116 [ ][ MeV anomaly in beryllium nuclear decays The protophobic (2020) 909 17 arXiv:1604.07411 17 Be anomaly Be anomaly 51 [ 8 8 Phys. Lett. B arXiv:1606.05171 , ]. [ arXiv:1710.03906 – 22 – [ arXiv:1609.01605 model for excited beryllium decays (2017) 01019 [ 0 ]. SPIRE Phys. Rev. Lett. IN 142 , [ keV line: cascade annihilating dark matter with the U(1) (2016) 071803 arXiv:1608.03591 Protophobic light vector boson as a mediator to the dark (2017) 209 [ SPIRE (2018) 112 ]. ), which permits any use, distribution and reproduction in 511 IN ]. 117 [ 919 Can nuclear physics explain the anomaly observed in the internal On the creation of the New results on the New results on the New evidence supporting the existence of the hypothetic X17 New experimental results for the Observation of anomalous internal pair creation in 78 Acta Phys. Polon. B , (2017) 015008 Realistic model for a fifth force explaining anomaly in SPIRE SPIRE ]. IN 95 (2017) 08010 IN [ ][ EPJ Web Conf. CC-BY 4.0 Confirmation of the existence of the X17 particle (2017) 035017 , Protophobic fifth-force interpretation of the observed anomaly in Particle physics models for the ]. 137 SPIRE This article is distributed under the terms of the Creative Commons IN Be 95 8 [ Nucl. Phys. B Phys. Rev. Lett. , , A family-nonuniversal Explanation of the SPIRE Eur. Phys. J. C ]. arXiv:1910.10459 , IN , Phys. Rev. D [ decay , − e SPIRE + arXiv:1703.04588 IN arXiv:1911.10482 sector anomaly e quantum electrodynamics pair production in the Beryllium-8[ nucleus? transitions Phys. Rev. D particle (2020) 04005 012028 transition of indication of a light,[ neutral boson EPJ Web Conf. T. Kitahara and Y. Yamamoto, L.-B. Jia, P.-H. Gu and X.-G. He, M.J. Neves, E.M.C. Abreu and J.A. Helayël-Neto, B. Puliçe, J.L. Feng et al., J.L. Feng et al., A.J. Krasznahorkay et al., D.S. Firak et al., X. Zhang and G.A. Miller, A.J. Krasznahorkay et al., A.J. Krasznahorkay et al., A.J. Krasznahorkay et al., A.J. Krasznahorkay et al., A.J. Krasznahorkay et al., [9] [6] [7] [8] [3] [4] [5] [1] [2] [14] [15] [11] [12] [13] [10] Attribution License ( any medium, provided the original author(s) and source are credited. References This work is19K03865, and MEXT supported, KAKENHI Grant No. 19H05091Nos. in (O.S.), 18K03651, and 18H01210 JSPS part, KAKENHI and Grant MEXT by KAKENHI Grant JSPS No.Open 18H05543 KAKENHI (T.S.). Access. Grant Nos. 19K03860 and Acknowledgments JHEP04(2021)025 ] ]. 97 ] Phys. , JHEP ] 17 , 2 , ]. SPIRE ]. 101 − IN g 1008 (2019) Eur. Phys. ][ , SPIRE colliders SPIRE 34 ]. Phys. Rev. A IN model and the IN − [ , e ][ (2019) 055022 α arXiv:1704.03436 + 3 e [ 99 SPIRE arXiv:1809.00177 Phys. Rev. A [ , IN ]. arXiv:1802.10449 Nucl. Phys. A Atomki anomaly in ][ [ , Be nuclear transitions and 8 ]. (2020) 235 (2016) 039 MeV Atomki anomaly in a SPIRE arXiv:1610.08112 07 11 [ IN 17 Implication of hidden sub-GeV (2018) 956 (2017) 115024 SPIRE process? ][ Phys. Rev. D Classical electrodynamics and gauge IN ]. Int. J. Mod. Phys. A γ arXiv:2006.01018 [ , arXiv:1612.01525 78 96 , , [ New physics suggested by Atomki + JHEP JHEP , , γ (2018) 1850148 Anomalies in ]. SPIRE Light axial vector bosons, nuclear transitions, 33 IN arXiv:2002.07496 [ (2017) 095032 ][ ]. ]. ]. ]. SPIRE 95 IN Phys. Rev. D – 23 – Explanation of the (2017) 115024 ][ , Accommodating three low-scale anomalies ( ]. arXiv:1812.05497 SPIRE SPIRE SPIRE SPIRE Eur. Phys. J. C [ Could the 21-cm absorption be explained by the dark IN IN IN IN 95 , arXiv:2001.08995 ][ ][ ][ ][ , Atomic physics constraints on the X boson SPIRE Searching for new light gauge bosons at (2020) 095024 IN ]. ]. ]. He anomaly, proton charge radius, EDM of fermions, and dark Possible explanation of the electron positron anomaly at 4 ][ Atomki anomaly and dark matter in a radiative seesaw model Phys. Rev. D (2019) 73 102 , arXiv:1710.10131 Int. J. Mod. Phys. A 7 Be- [ , 8 extension of the standard model SPIRE SPIRE SPIRE , Phys. Rev. D 0 IN IN IN µ Be transitions? , arXiv:1907.09819 ][ ][ ][ 2) 8 [ U(1) Fifth force and hyperfine splitting in bound systems symmetry − Evidence of quantum phase transition in carbon-12 in a g arXiv:1804.03096 arXiv:2003.05722 arXiv:2003.07207 arXiv:2001.04864 L ( MeV Atomki anomaly from short-distance structure of spacetime - [ [ [ [ Phys. Rev. D Open string QED meson description of the X17 particle and dark matter Is the X17 composed of four bare quarks? B , 17 (2018) 115004 Front. in Phys. X17: A new force, or evidence for a hard : towards a minimal combined explanation arXiv:1906.09229 Be transitions through a light pseudoscalar Be anomaly ]. ]. ]. (2020) 231 , 8 8 [ 97 e,µ 80 -extended two Higgs doublet model 2) 0 (2020) 165 − SPIRE SPIRE SPIRE g arXiv:2005.00028 IN IN arXiv:1811.07953 arXiv:1609.01669 IN [ anomaly MeV in [ U(1) [ family-dependent (2021) 122143 (2020) 062503 and the Rev. D (2018) 042502 J. C 08 problem of hypothetical X17 boson ( [ symmetry of the X-boson [ 1950140 bosons for the axion portal matter suggested by [ with gauged Lamb shift, and Atomki) in the framed standard model L. Delle Rose, S. Khalil, S.J.D. King and S.U. Moretti, Ellwanger and S. Moretti, L. Delle Rose, S. Khalil and S. Moretti, L. Delle Rose, S. Khalil, S.J.D. King, S. Moretti and A.M. Thabt, B. Koch, U.D. Jentschura, J. Kozaczuk, D.E. Morrissey and S.R. Stroberg, H.-X. Chen, I. Alikhanov and E.A. Paschos, U.D. Jentschura and I. Nándori, C.-Y. Wong, E.M. Tursunov, M.J. Neves, L. Labre, L.S. Miranda and E.M.C. Abreu, C.H. Nam, D.V. Kirpichnikov, V.E. Lyubovitskij and A.S. Zhevlakov, C. Hati, J. Kriewald, J. Orloff and A.M. Teixeira, O. Seto and T. Shimomura, J. Bordes, H.-M. Chan and S.T. Tsou, L.-B. Jia, X.-J. Deng and C.-F. Liu, [33] [34] [31] [32] [28] [29] [30] [25] [26] [27] [23] [24] [21] [22] [19] [20] [17] [18] [16] JHEP04(2021)025 , ] ]. 1 Phys. ]. , ] SPIRE (2018) 166 LHEP (2010) 557 IN Higgs gauge Phys. Rev. , SPIRE gauge , ][ 06 IN H 82 R gauge Eur. Phys. J. C ][ ]. , R U(1) ]. U(1) JHEP (2018) 092001 Ann. Rev. Nucl. Part. , arXiv:2006.01151 U(1) , SPIRE [ (2019) 916 98 IN (2012) 202 arXiv:1408.6845 SPIRE TeV [ ][ 79 IN gauge symmetry ][ 717 ]. ]. R ]. ]. arXiv:0907.4112 Rev. Mod. Phys. = 13 [ , s Diphoton excess through dark U(1) arXiv:1904.05105 √ TeV collected with the ATLAS SPIRE SPIRE TeV with the ATLAS detector [ SPIRE SPIRE (2014) 256 (2020) 036016 IN IN Phys. Rev. D IN ]. IN , ][ ][ ][ = 13 ][ ]. = 13 739 102 Top quark forward-backward asymmetry s s TeV Phys. Lett. B Eur. Phys. J. C √ SPIRE , √ , ]. arXiv:1909.02845 (2010) 015004 IN [ ]. Dynamical evidence for a fifth force explanation SPIRE arXiv:1704.03382 ][ = 13 IN 81 [ (2019) 231801 s – 24 – ][ SPIRE ]. √ IN SPIRE 122 ][ IN Phys. Rev. D Phys. Lett. B Faking ordinary photons by displaced dark photon decays , , ][ arXiv:0807.3125 SPIRE arXiv:1603.01256 arXiv:1309.7156 Axions: theory and cosmological role arXiv:1603.00024 IN [ [ [ (2020) 012002 ]. ]. ]. ][ Minimal Dirac neutrino mass models from A resolution of the flavor problem of two Higgs doublet models Higgs phenomenology in Type-I 2HDM with ]. Phys. Rev. D (2017) 015016 Loop suppressed light fermion masses with Minimal realization of right-handed gauge symmetry An inverse seesaw model with , 101 Axions and the strong CP problem symmetry 96 arXiv:1707.04147 SPIRE SPIRE SPIRE Search for new phenomena in high-mass diphoton final states using Combination of searches for invisible Higgs boson decays withCombined the measurements of Higgs boson production and decay using Search for Higgs boson decays to beyond-the-Standard-Model light [ L SPIRE - IN IN IN (2016) 063 (2014) 016 arXiv:1707.00929 symmetry for Higgs flavor IN Phys. Rev. Lett. Search for physics beyond the standard model in high-mass diphoton B [ ][ ][ ][ , arXiv:1301.1123 ][ H (2019) 049902] [ 07 01 (2017) 015027 [ arXiv:1707.07858 91 of proton-proton collision data at 95 U(1) [ (2017) 105 1 Phys. Rev. D JHEP JHEP Phys. Rev. D , − arXiv:1806.01714 Phenomenology of the gauge symmetry for right-handed fermions Unbroken , , , [ fb collaboration, collaboration, collaboration, collaboration, ]. ]. of proton-proton collisions collected at 775 (2013) 69 1 collaboration, (2018) 015015 80 − 63 (2018) 103 fb 97 SPIRE SPIRE arXiv:1809.00327 arXiv:1802.03388 Erratum ibid. arXiv:1904.07407 IN arXiv:1204.4588 IN ATLAS 37 Lett. B CMS events from proton-proton collisions at [ experiment Phys. Rev. D mediators bosons in four-lepton events[ with the ATLAS detector at ATLAS ATLAS experiment ATLAS up to [ Sci. ATLAS symmetry and left-right asymmetry[ at colliders [ D (2018) 10 78 [ symmetry symmetry of the ATOMKI nuclear anomalies [ from new t-channel physics with an extra Y. Tsai, L.-T. Wang and Y. Zhao, C.-Y. Chen, M. Lefebvre, M. Pospelov and Y.-M. Zhong, J.E. Kim and G. Carosi, M. Kawasaki and K. Nakayama, S. Jana, V.P.K. and S. Saad, J. Heeck, T. Nomura and H. Okada, T. Nomura and H. Okada, W. Chao, P. Ko, Y. Omura and C. Yu, T. Nomura and H. Okada, S. Jung, H. Murayama, A. Pierce and J.D. Wells, P. Ko, Y. Omura and C. Yu, J.L. Feng, T.M.P. Tait and C.B. Verhaaren, [52] [53] [50] [51] [48] [49] [45] [46] [47] [43] [44] [40] [41] [42] [38] [39] [36] [37] [35] JHEP04(2021)025 ] , ) ]. ]. 2 Z Phys. ]. m , ( = 13 ] (2018) α SPIRE s ]. SPIRE IN √ 98 SPIRE (2018) 675 IN and ][ IN ][ bosons in the 2 78 ][ SPIRE Z − (2020) 051801 (2016) 132 IN g arXiv:1803.00060 ][ (2012) 042044 [ (2015) 033009 38 125 Update of the global Constraints on dark 92 375 Phys. Rev. D Prog. Theor. Exp. Phys. , , arXiv:1808.03599 [ decays, and light dark matter (2017) 827 ]. (2018) 098 Eur. Phys. J. C arXiv:1912.01431 ]. ]. Υ arXiv:1609.09072 , 77 [ Light weakly coupled axial forces: 05 [ arXiv:1705.06726 Neutrino-electron scattering: [ ]. SPIRE hep-ex/0602035 Nuovo Cim. C and Phys. Rev. D [ , IN SPIRE Phys. Rev. Lett. SPIRE (2018) 085 , ψ , IN ][ Reevaluation of the hadronic vacuum IN of proton-proton collisions at [ JHEP Type-I 2HDM under the Higgs and 11 , ][ SPIRE 1 Chiral effective theory of dark matter direct (2020) 131 − TeV IN (2017) 002 J. Phys. Conf. Ser. fb , ][ 08 05 JHEP = 13 (2017) 141803 139 Eur. Phys. J. C , New constraints on light vectors coupled to Dark forces coupled to nonconserved currents s – 25 – , (2006) 072003 Review of particle physics Review of particle physics √ annihilations, JHEP 119 JHEP 73 , − , e + hep-ph/0702176 arXiv:1611.00368 e [ arXiv:2009.14791 [ ]. ]. ]. ]. ]. , ]. arXiv:1707.01503 collisions at Final results of nu-e-bar electron scattering cross-section Final report of the muon E821 anomalous magnetic moment and dark photon models collaboration, collaboration, [ final states using 0 NA48/2 studies of rare decays SPIRE SPIRE SPIRE SPIRE SPIRE Search for charged Higgs bosons decaying into top and bottom quarks Search for heavy resonances decaying into a pair of Search for heavy Higgs bosons decaying into two tau leptons with the pp Z Phys. Rev. D ¯ SPIRE ν IN IN IN IN IN . , ν Phys. Rev. Lett. (2017) 009 IN ][ ][ ][ ][ ][ − , ` ][ (2007) 115017 02 + ]. ` 75 TeV with the ATLAS detector and collaboration, collaboration, (2017) 075036 JCAP SPIRE U-boson production in collaboration, collaboration, collaboration, collaboration, 0− , ]. ]. IN ` 96 [ + = 13 0 1 (2020) 083C01 ` s − √ SPIRE SPIRE ` + arXiv:1502.07763 arXiv:1508.01307 IN arXiv:1201.4675 IN arXiv:1803.01853 arXiv:2002.12223 arXiv:1706.09436 Particle Data Group 2020 polarisation contributions to theusing Standard Model newest predictions hadronic of cross-section the data [ muon [ Phys. Rev. D Muon g-2 measurement at BNL [ general constraints on [ photon from neutrino-electron scattering experiments measurements and constraints on new[ physics detection NA48/2 models, constraints, and projections Particle Data Group 030001 TEXONO Electroweak Precision Measurements anomalous currents Rev. D at [ electroweak fit and constraints on[ two-Higgs-doublet models ` TeV with the ATLAS detector ATLAS ATLAS detector using [ ATLAS ATLAS M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, P. Fayet, M. Lindner, F.S. Queiroz, W. Rodejohann and X.-J. Xu, S. Bilmis, I. Turan, T.M. Aliev, M. Deniz, L. Singh and H.T. Wong, F. Bishara, J. Brod, B. Grinstein and J. Zupan, Y. Kahn, G. Krnjaic, S. Mishra-Sharma and T.M.P. Tait, N. Chen, T. Han, S. Li, S. Su, W.J.A. Su Dror, and R. Y. Lasenby Wu, and M. Pospelov, J.A. Dror, R. Lasenby and M. Pospelov, J. Haller, A. Hoecker, R. Kogler, K. Mönig, T. Peiffer and J. Stelzer, [70] [71] [68] [69] [66] [67] [64] [65] [61] [62] [63] [58] [59] [60] [57] [55] [56] [54] JHEP04(2021)025 , ] , 91 JHEP ]. , in ] ] 109 2 Eur. 2 102 , − ) Phys. Lett. − g 2 Z SPIRE , g ]. m IN ( ]. ][ α Phys. Rev. D , SPIRE arXiv:1712.06060 ]. IN SPIRE (2011) 052122 [ ][ IN : a new data-based Phys. Rev. Lett. ) ][ 83 arXiv:1203.2947 ]. , Phys. Rev. Lett. 2 Z arXiv:1803.07748 [ SPIRE , (2018) 022003 [ M IN ( (2008) 120801 ][ α SPIRE 121 IN 100 arXiv:1908.00921 and ]. (2018) 036001 ][ implications for parity violation, rare 2 Tenth-order QED contribution to the Tenth-order electron anomalous Z Phys. Rev. A 97 − , A new evaluation of the hadronic vacuum (2012) 115019 arXiv:0906.0580 MeV gauge boson and dark photons in the g (2018) 231802 SPIRE [ 7 Cavity control of a single-electron quantum New fixed-target experiments to search for arXiv:1412.8284 . IN 85 16 ][ ‘Dark’ 120 (2020) 410] [ Muon ]. ]. ]. Phys. Rev. Lett. Testing new physics with the electron Precision determination of electroweak coupling New measurement of the electron magnetic arXiv:1802.02995 , 80 Phys. Rev. Lett. [ – 26 – ]. , Phys. Rev. D Revised and improved value of the QED tenth-order , arXiv:1704.06996 SPIRE SPIRE SPIRE (2009) 075018 [ IN IN IN Phys. Rev. D SPIRE ][ ][ ][ An unambiguous search for a light Higgs boson (2017) 019901] [ , 80 IN Calculation of the hadronic vacuum polarization contribution [ ]. 96 hep-ph/0410260 Phys. Rev. Lett. ]. ]. [ , ]. Erratum ibid. (2018) 114025 Constraints on the parity-violating couplings of a new gauge boson [ (2017) 232 SPIRE 97 SPIRE SPIRE IN Search for a hypothetical SPIRE A search for short lived axions in an electron beam dump experiment IN IN 772 (1987) 755 ][ IN (2005) 87 ][ ][ Phys. Rev. D [ , collaboration, 59 arXiv:0902.0335 arXiv:1205.5368 Erratum ibid. arXiv:1208.6583 (2020) 241 and an improved value of the fine structure constant 608 [ [ [ [ 2 High-precision calculation of the 4-loop contribution to the electron 80 − Phys. Rev. D ]. ]. ]. g , (1989) 150 collaboration, Phys. Lett. B , (2012) 113 229 SPIRE SPIRE SPIRE IN IN arXiv:1009.4831 arXiv:0801.1134 arXiv:1801.07224 IN B NA64 NA64 experiment at CERN [ from atomic parity violation(2009) and 181601 implications for particle physics Phys. Rev. Lett. dark gauge forces Phys. Lett. B meson decays, and Higgs[ physics electron anomalous magnetic moment [ 11 magnetic moment — Contribution(2015) of 033006 diagrams without closed lepton loops QED cyclotron: measuring the electron[ magnetic moment electron (2012) 111807 Phys. J. C moment and the fine[ structure constant analysis RBC, UKQCD to the muon anomalous[ magnetic moment polarisation contributions to the muon anomalous magnetic moment and to M. Davier and H. Nguyen Ngoc, E.M. Riordan et al., J.D. Bjorken, R. Essig, P. Schuster and N. Toro, H. Davoudiasl, H.-S. Lee and W.J. Marciano, S.G. Porsev, K. Beloy and A. Derevianko, G.F. Giudice, P. Paradisi and M. Passera, C. Bouchiat and P. Fayet, T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, S. Laporta, T. Aoyama, T. Kinoshita and M. Nio, T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, D. Hanneke, S. Fogwell and G. Gabrielse, D. Hanneke, S.F. Hoogerheide and G. Gabrielse, M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, A. Keshavarzi, D. Nomura and T. Teubner, [87] [88] [85] [86] [83] [84] [81] [82] [78] [79] [80] [77] [75] [76] [73] [74] [72] JHEP04(2021)025 ] ]. , SPIRE Uγ IN → ][ Phys. Rev. − , e + e ]. arXiv:1509.00740 [ SPIRE IN ][ arXiv:1912.11389 [ (2015) 633 collisions at BaBar − e 750 ]. + e SPIRE hep-ex/0504049 IN [ (2020) 071101 ][ – 27 – 101 Phys. Lett. B , (2005) 081601 Precision measurement of the weak mixing angle in Möller 95 Phys. Rev. D arXiv:1406.2980 , [ Search for a dark photon in Improved limits on a hypothetical X(16.7) boson and a dark photon pairs Limit on the production of a low-mass vector boson in − e + e collaboration, with the KLOE experiment Phys. Rev. Lett. (2014) 201801 , − ]. collaboration, e collaboration, + e 113 SPIRE → IN SLAC E158 scattering decaying into U [ BaBar Lett. NA64 A. Anastasi et al., [92] [90] [91] [89]