Atomki anomaly in gauged $U(1)_R$ symmetric model

The Atomki collaboration has reported that unexpected excesses have been observed in the rare decays of Beryllium nucleus. It is claimed that such excesses can suggest the existence of a new boson, called $X$, with the mass of about $17$ MeV. To solve the Atomki anomaly, we consider a model with gauged $U(1)_R$ symmetry and identify the new gauge boson with the $X$ boson. We also introduce two $SU(2)$ doublet Higgs bosons and one singlet Higgs boson, and discuss a very stringent constraint from neutrino-electron scattering. It is found that the $U(1)_R$ charges of the doublet scalars are determined to evade the constraint. In the end, we find the parameter region in which the Atomki signal and all experimental constraints can be simultaneously satisfied.


I. INTRODUCTION
The Atomki collaboration has been reporting results that unexpected excesses were found in the Internal Pair Creation (IPC) decay of Beryllium (Be) [1][2][3][4][5] and Helium (He) [6,7] nuclei. In the reports, the excesses appear as bumps in the distributions of the invariant mass and opening angle of an emitted positron (e + ) and electron (e − ) pair from the IPC decays of 8 Be * and 4 He, 8 Be * (18.15 MeV) → 8 Be + e + + e − , (1) 4 He(21.01 MeV) → 4 He + e + + e − , respectively. These bumps seem not to be explained within the standard nuclear physics [8], even if parity violating decays are taken into account. The collaboration reported that the bumps can be well fitted simultaneously under the assumption that a hypothetical boson X with the mass of 17.01 ± 0.16 and 19.68 ± 0.25 MeV is produced through 8 Be * and 4 He decays, followed by X decay into a e + -e − pair, respectively. Such a light boson does not exist in the Standard Model (SM) of particle physics. Therefore, the anomaly can be considered as a signal of new physics beyond the SM. The hypothetical boson X, in principle, can be a vector, axial-vector, scalar and pseudoscalar boson. Among these possibilities, the scalar boson hypothesis is discarded due to the conservation of angular momentum in the decay Eq. (1) [9,10]. Vector boson hypothesis was firstly studied in [9,10] in a gauged B − L symmetric model, taking various experimental constraints into account. Then, many models have been proposed in contexts of an extra U(1) gauge symmetry [11][12][13], dark matter [14][15][16], neutrino physics [17], lepton anomalous magnetic moments [18][19][20] and others [21][22][23][24][25]. Experimental searches of the X boson are also studied in [26][27][28][29]. In [10], it was shown that there are two restrictive constraints to explain the Atomki anomaly. The first constraint comes from the a rare decay of neutral pion, π 0 → γX, measured by the NA48/2 experiment. This constraint sets a very stringent bound on the coupling of X to proton because the decay branching ratio of the rare decay is scaled by the proton coupling. From this fact, such a vector boson is named as a protophobic boson. The second constraint comes from neutrino-electron scattering measured by the TEXONO experiment. It is difficult to evade this constraint and neutral pion constraint simultaneously. Therefore, new leptonic states are introduced to evade this constraint in [10], or no interaction of the X boson to active neutrinos is ad hoc assumed.
An axial-vector boson hypothesis also has been studied in [30]. This hypothesis has two advantages. One is that the constraints from neutral pion decay can be easily evaded because the decay receives no contribution from the axial anomaly. The other advantage is that the partial decay width of 8 Be * is proportional to k X , 1 the X's three momentum, 1 There is also k 3 X term in the partial width. Following the discussion in [30], we neglected that term, which would be suppressed because k X is smaller than mass of the X boson.
while it is proportional to k 3 X in a vector boson hypothesis. Because of this momentum dependence, coupling constants of X to quarks can be much weaker to explain the Atomki anomaly than that in the vector-boson hypothesis case. Then, it is possible to evade several experimental constraints in the axial-vector boson hypothesis. Several models with axialvector boson have been proposed in [31][32][33]. In spite of these advantages, the constraint from neutrino-electron scattering is still very stringent and requires to suppress neutrino couplings to X. In [30], it is assumed that neutrino couplings to the X boson vanish, and in [31][32][33], many fermions are introduced to cancel the neutrino couplings. In the end, pseudoscalar hypothesis was studied in [34]. Decay widths of these three hypotheses are found in [35].
In this work, we pursue the axial-vector hypothesis and consider a U (1) R gauge symmetry [36] where the gauge boson is identified with the X boson. The U (1) R gauge symmetry is defined that only right-handed fermions are charged while left-handed ones are not charged. Then, the U (1) R gauge boson has both vectorial and axial interactions to fermions. The existence of the axial interaction allows coupling constants to be weaker to satisfy the Atomki signal. With weaker couplings, a contribution to neutral pion decay from vectorial interactions is much suppressed. It was shown in [37,38] that flavour changing neutral currents can be suppressed due to U (1) R symmetry in two Higgs doublet extension. It was also shown in [39][40][41][42] that neutrino masses and mixing, dark matter and the muon anomalous magnetic moment can be explained in models with the U (1) R gauge symmetry. Motivated by these previous works, we construct a minimal model to explain the Atomki anomaly with U (1) R gauge symmetry. This paper is organized as follows. In Sec. II, we introduce our model as a minimal setup to explain the Atomki anomaly. In Sec. III, we give the coupling constants of fermions to the X boson and show the allowed region of gauge coupling constant. Then, the signal requirement and experimental constraints are explained in Sec. IV and our numerical results are shown in Sec. V. In the end, we give our conclusion in Sec. VI.

II. MODEL
We start our discussion by introducing our model. The gauge symmetry of the model is defined as G SM × U (1) R , where G SM stands for the gauge symmetry of the SM. Under the U (1) R gauge symmetry, right-handed chiral fermions are charged while left-handed chiral ones are singlet [40]. Only with the SM matter content, such a charge assignment generally leads to non-vanishing gauge-anomalies due to U (1) R current contributions. Therefore, new fermions charged under U (1) R must be introduced to cancel the gauge anomalies. One of the simplest solutions for non-vanishing anomalies is to add three right-handed fermions, N i (i = 1, 2, 3), which are singlet under the SM gauge symmetries. The charge assignment of the fermions in our model is shown in Table I. In the Table,  the SM strong, weak and hypercharge gauge groups. The symbols, Q and u R , d R represent left-handed quarks and right-handed up-type, down-type quarks, respectively, and L and e R represent left-handed leptons and right-handed charged leptons, respectively. Without loss of generality, we can fix the U (1) R charge of u R to + 1 2 as the overall normalization. Then, the gauge charges of the other fermions are determined from anomaly-free conditions as shown in the Table I. For the Higgs field H 1 , we assign its U (1) R charge to q 1 , which can not be determined from anomaly-free conditions. However, if requiring for the model to be minimal, q 1 should be taken as + 1 2 so that quarks and charged leptons can form Yukawa interactions with H 1 in the same manner of the SM. Furthermore, with this charge assignment, left-handed neutrinos can form the Yukawa interaction with N i . Therefore, we identify N i as right-handed neutrinos in the following discussions.
To explain the Atomki anomaly, we further extend the matter content by adding a SU (2) L doublet scalar field H 2 and a SU (2) L singlet scalar field S. Firstly, it is shown in [43] that neutrinos can not be Dirac particle due to the constraints from ∆N eff unless the coupling constant of neutrinos are extremely small. This constraint can be avoided when right-handed neutrinos have Majorana masses. The SU (2) L singlet scalar field is introduced to give a mass to the X boson and Majorana masses to N i after spontaneous breaking of U (1) R . Thus its U (1) R charge is assigned to −1. The new SU (2) L doublet scalar field is also introduced. It plays an important role to reduce the mixing between left-handed neutrinos and the U (1) R gauge boson, X, so that the stringent constraint from neutrino-electron scattering is avoided. The U (1) R charge of H 2 is arbitrary, and we will discuss possible charge assignments later. The charge assignment of the new scalars is also shown in Table I, where we denote the U (1) R charge of H 2 as q 2 .

A. Lagrangian
The Lagrangian of the model takes the form of where each term denotes the fermion, scalar, gauge and Yukawa sector Lagrangian which are defined as and V is the scalar potential which is given below. In Eqs. (4), f represents the fermions (Q, u L , u R and L, e R , N ), andW ,B andX represent the gauge fields and their field strengths in the interaction basis of SU (2) L , U (1) Y and U (1) R , respectively. The covariant derivative in Eqs. (4a) and (4b) is given by where Y and x represent the U (1) Y and U (1) R charges of each particle. The gauge coupling constants of SU (2) L . U (1) Y and U (1) R are denoted as g 2 , g 1 and g , respectively. In Eq. (4c), the gauge symmetry of the model allows the gauge kinetic mixing term betweenB andX, and its magnitude is parameterized by the constant parameter . In Eq. (4d), the Dirac Yukawa matrices are denoted as Y u , Y d and Y e , Y ν for up, down quarks and charged leptons, neutrinos, respectively. The Yukawa matrix for right-handed neutrinos is denoted as Y N . HereH 1 represents iσ 2 H * 1 where σ 2 is the Pauli matrix. Note that flavour and generation indices are omitted for simplicity.
The scalar potential V can be divided into two parts. One consists of the terms independent of the U (1) R charge assignment of H 2 , and the other consists of the terms dependent on that. The charge-independent part, V 0 , is given by where we assume the mass parameters as well as the quartic couplings to be positive so that spontaneous breaking of the symmetries successfully occurs, and no runaway directions appear in the potential. With the above potential, we obtain five Nambu-Goldstone bosons after H 1 , H 2 and S develop vacuum expectation values (VEVs). Two of those are absorbed by the charged weak boson, W ± , and other two are absorbed by the neutral weak boson Z and the new gauge boson,X. Then, one massless CP-odd scalar remains in the spectrum, which corresponds to the broken degree of freedom of the phase rotation of H 2 . Such a massless scalar boson causes serious problems by carrying the energy of stars and conflicts with meson decay measurement such as an axion does [44,45]. Therefore we need to introduce other interaction terms which give the mass to the CP-odd scalars after the symmetry breaking. In this sense, a possible choice of q 2 is determined. We classify models with different choices of q 2 given in Table II.
The charge-dependent scalar potential in each model is given by where the parameters, A 1,2 and κ 3,4 , can be taken real by using phase rotation of H 2 . One example of the parameter sets to reproduce the Higgs mass, 125 GeV, for Model 1 is found as v = 246.0 GeV, v s = v, cos 2β = 0.1 (tan β = 0.9045), With these parameters, the Higgs couplings to the weak gauge bosons are the same as those of the SM, and the coupling to the X boson vanishes. The messes of other extra Higgs scalars are also large enough. However, details of the scalar sector is essentially irrelevant for our study about the Atomki anomaly. Therefore, in the following discussions, we assume that the parameters in the scalar potential are appropriately chosen so that the new gauge boson acquires the mass required to explain the Atomki anomaly.

B. Gauge boson Masses and Mass Eigenstates
After the EW and U (1) R symmetries are broken down, the gauge boson masses are generated via the VEVs of the scalar fields. We denote the VEVs as and each scalar field is expanded around its VEV as Then, the mass terms of the gauge fields are given by Here, θ W is the Weinberg angle of the SM defined by sin θ W = g 1 / g 2 1 + g 2 2 . The gauge boson, W ± , is the charged weak gauge boson of the SM, andZ andÃ correspond to the Z boson and photon in the SM limit, (g , ) → 0. In the following, we parameterize the VEVs as, 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 3 × 3 matrix as whereF µ = (Ã µ ,Z µ ,X µ ) T , and m 2F is given by Here, mZ is the SM Z boson mass defined by To obtain the masses of the neutral gauge bosons, we first diagonalize the gauge boson kinetic term by changing the basis of the fieldsF to F = (A, Z, X) T as where U K is an orthogonal matrix given by with r = (1 − 2 ) −1/2 . Then, the mass matrix in F µ basis is given as with Next, the mass matrix Eq. (20) can be diagonalized by an orthogonal matrix V F where F = (A, Z, X) T is the mass eigenstates. Their mass eigenvalues are given by The mixing angle χ can be expressed as Here the mass of X is an input of the model which should be 17 MeV by the Atomki experiment. In the situation of mZ m X , the leading term of Eq. (25) is given by In the parameter space of our interest, g and are roughly O(10 −4 − 10 −3 ). Therefore, χ is much smaller than unity from Eq. (26). Then, the difference between m Z and mZ is roughly given as, where Eq. (24b) is used. This difference is smaller than the present error of the measured Z boson mass, 91.1876 ± 0.0021 GeV [46] and therefore we use mZ m Z = 91.1876 GeV as an input value in the following discussion. Then, v s is expressed in terms of other parameters as Right-hand-side of Eq. (28) should be positive for consistency. In the end, the gauge eigenstates are expressed in terms of the mass eigenstates as where U = U K V F and its elements are From Eqs. (12) and (29), the Lagrangian can be written in the mass basis of the gauge boson.

III. COUPLINGS OF THE X BOSON TO FERMIONS
In this section, we present the coupling constants of the X boson to quarks and leptons. The gauge interactions of the fermions to the X boson are modified due to the mixing among the gauge bosons. Using Eqs. (12) and (29) with Eqs. (30), the interaction Lagrangian of fermions, f (= u, d, ν, N ), can be written in the following form, where e is the proton electric charge. The vector coupling V f and axial-vector couplings A f are given by with R = g /e, where NC represents the neutral current contribution defined by In Eqs. (32), we neglect the mixing between left and right handed neutrinos. 2 As we explained above, one of the most stringent constraints comes from neutrino-electron scattering of reactor neutrinos measured at TEXONO [47]. The left-handed neutrinos ν L can interact with the X boson through the weak neutral current. Thus, the coupling constant of ν L is proportional to NC as To obtain approximate formulae of the coupling constants, we expand NC in the limit of |χ| 1 and |Q| 1 as, where we define Q for convenience as In the expansion, we kept the leading term of and R and neglected higher order terms of these couplings in each power of m 2 X /m 2 Z . Inserting Eq. (35) into Eqs. (32), the approximate expression of the coupling constants can be obtained, which is useful to understand signal requirement and constraints as we will explain later.
The first term of Eq. (35) vanishes when β takes a specific value of Then, the remaing term is much smaller than due to mZ m X . From Table II, cos 2β * is given in each Model by Model 1 : cos 2β * = 0, (38a) Thus, Q can be vanished in Model 1 and 3, while there are no solutions for Q = 0 in Model 2 and 4. Figure 1 is a plot of | NC | as a function of R . Red, blue, green and brown lines correspond to Q = 0, 0.01, 0.1 and 1, respectively. Solid and dashed ones correspond to = 10 −4 and 5 × 10 −4 . Gray filled regions with solid and dashed edges are exclusion region by neutrino-electron scattering for = 5×10 −4 and 10 −4 , which we will explain in subsection IV B 2. 3 Except for Q = 0, the dashed and solid curves are almost the same. One can see 3 It should be noted that the exclusion region in R -NC plane is almost independent of Q. In figure, we fixed Q = 0.

IV. SIGNAL AND EXPERIMENTAL CONSTRAINTS
We summarize the signal requirement from the 8 Be decay 4 and the constraints from various experiments.
The signal branching ratio of 8 Be * into the assumed X particle, followed by the decay of X into e + e − , is defined by where Γ( 8 Be * → 8 Beγ) (1.9 ± 0.4) eV is the partial width of the γ decay of 8 Be * and Br(X → e + e − ) is the decay branching ratio of X into an electron-positron pair. From the 4 As we explained in the Introduction, the Atomki collaboration also reported that a peak like excess was found in 4 He [6], which can be consistently explained by a light particle for 8 Be . However, nuclear matrix elements have significant uncertainty for 4 He [35] and we need further study to reduce the uncertainty. Thus we will not indicate 4 He anomaly in our analysis.
Atomki experiment, the branching ratio (40) and the X boson mass have been constrained as given in Table III (taken from [5]). These values have a relatively large uncertainties, which may originate from systematic uncertainty of unstable beam position in the experiment. Therefore, we employ rather conservative range for our numerical calculation where m X is taken to 17.6 and 16.7 MeV for the lower and upper bounds, respectively. To calculate the decay branching ratio of 8 Be * , we employ the results given in [30]. The partial decay width from the axial part of the gauge interaction is expressed as where k = ∆E 2 − m 2 X and E k = ∆E are the three momentum and energy of the X boson, with ∆E = 18.15 MeV being the difference of the energy level. The proton and neutron couplings, a p and a n , are defined as where and A u,d is given in Eqs. (32). The quark coefficients take values [48] ∆u (p) = ∆d (n) = 0.897 (27), and the nuclear matrix elements takes [30] 0 + σ p S = −0.047 (29), (46a) 0 + σ n S = −0.132 (33).
Before closing this subsection, we show the parameter dependence of the signal branching ratio. Using A d = − A u = A e , the partial decay width of 8 Be * is proportional to ( A e ) 2 as The decay branching ratio of X → e + e − is given by where V,A e and ν L are given in Eqs. (32) and (34). In Eq. (48), we have neglected the masses of electron and neutrino. In the case of |χ| 1 and |Q| 1, A e can be approximated as where Eq. (35) is used. Similarly, the vector coupling of electron is approximated as Thus, neglecting O(m 2 X /m 2 Z ) terms, the branching ratio, Eq. (48), is expressed by (51) Figure 2 shows the branching ratio of X → e + e − as a function of R . We fixed as = 5×10 −4 (solid) and 10 −4 (dashed) and as Q = 0 (red), 0.5 (blue) and 0.9 (green), respectively. One can see that the branching ratio decrease as Q increases. This is because the coupling to neutrinos, which is proportional to NC , becomes large for non-vanishing Q as shown in Fig. 1.
Using the approximate expressions, Eqs. (49) and (51), the signal branching ratio is scaled by the parameters as Thus, for |Q| 1, the branching ratio is simply determined by ( R ) 2 .

X boson lifetime
To explain the Atomki anomaly, the new vector should decay inside the detector so that the electro-positron pair can be detected. As in [10], we require that X boson propagates less than 1 cm from its production point, which gives the condition as B. Constraints

Rare decay of neutral pion
The coupling constants of the light gauge boson to quarks can be constrained by meson decay experiments. The gauge boson can be produced in rare meson decays when those are kinematically allowed. For the X boson with m X 17 MeV, the most stringent constraint among such meson decays comes from the rare decay of neutral pion into X with a photon, i.e. π 0 → γX. Theoretically, only vectorial parts of the X interaction to quarks can contribute to the decay. The latest result of the NA48/2 experiment [49] puts the following bound, The left-hand-side of the constraint is rewritten by and approximated as Eq. (50). For |Q| 1, Eq. (54) is simplified as

Neutrino-electron scattering
The interaction of the gauge boson to leptons, especially to neutrinos, is tightly constrained by neutrino-electron scattering [50]. The most stringent constraint for the X boson is given by the TEXONO experiment [51]. In [51], the contributions from the B − L gauge boson to ν e -e scattering have been studied. The authors analyzed the differential cross section with respect to the recoil energy of scattered electron and showed the interference term gives sizable contributions. Based on the analyses, the allowed region of the mass and gauge coupling of the B − L gauge boson was shown. In this work, we derive the interference term of the differential cross section in the U (1) R model, and constrain the parameters by comparing the differential cross section in the SM 5 .
The differential cross section in the SM and the interference term between the SM and X boson contributions are given by where T and m e are the recoil energy and the mass of electron, and E ν is the energy of incident neutrino, respectively. The Fermi constant is denoted as G F , and other coupling constants are given by From [47], the event rate relative to that of the SM is given by 1.08 ± 0.21(stat) ± 0.16(sys) in the TEXONO experiment. Thus we require which corresponds to 3σ range. We use our numerical analysis E ν = 3.0 MeV and T = 3.0 MeV, respectively. The ratio in Eq. (59) can be approximated for the case of |χ| 1 and |Q| 1 as, It can be understood from above equation that Q is important for this constraint. Unless Q is very close to zero, the constraint excludes R > ∼ 10 −8 /Q. 5 Recently, in [20], this constraint is computed using data and χ 2 fit is performed. As we have shown in Fig. 1, the coupling constant of neutrinos is much smaller than 10 −5 for small Q. Therefore, our result is consistent with that of [20].

Anomalous magnetic moment of charged lepton
The anomalous magnetic moment of the charged leptons have been measured accurately by experiments and also predicted precisely in the SM. The new vector boson, that couples to the charged leptons, can shift the anomalous magnetic moments from the SM predictions via quantum loop corrections. One loop contribution of the vector boson is given by [52] where y l = m 2 X /m 2 l , and I V (y l ) and I A (y l ) are given by It should be noticed that the axial coupling contribution to δa l is always negative while the vector contribution is positive. The integration of Eqs. (62) can be done numerically for electron and muon as I V (y e ) = 6.894 × 10 −7 , I A (y e ) = 3.484 × 10 −6 , (63a) where we used m e = 0.5110 MeV and m µ = 105.7 MeV, respectively. The muon anomalous magnetic moment has exhibited a long-standing discrepancy between experimental results [53,54] and theoretical predictions [55][56][57][58]. From [54], the discrepancy is given by where a exp µ and a SM µ represent the anomalous magnetic moment by the measurements and SM predictions, respectively. From Eqs. (63b), one finds that δa µ can be positive when However, such parameter region is excluded by the constraint from π 0 → γX, because the lower bound on A e set by the signal requirement, (52), results in too large V e in this situation. Then, the dominant part in δa µ is the axial coupling term and that negative contribution to δa µ further worsens the discrepancy. Thus a special care to implement the constraint of ∆a µ is required. Following the discussion in [30], we impose a constraint that the contribution from the X boson should be less than 2σ uncertainty of Eq. (64), This above constraint is scaled by the parameter as thus it is determined mostly by R . The anomalous magnetic moment of electron, a e = (g − 2) e /2, also has been measured accurately [59,60], and predicted precisely within the SM [61][62][63][64]. Although recent results claimed that a e also exhibits 2.5σ discrepancy between the measurement and SM predictions, we impose a rather conservative constraint [65] employed in [30], This constraint is weaker than that from δa µ due to the smaller value of I V and I A .

Effective weak charge
The axial-vector coupling of electron can be restricted by atomic parity violation in Caesium (Cs) [66,67]. The constraint is given by the measurement of the effective weak charge Q W of the Cs atom [68]. For the X boson with 17 MeV mass, one obtains the following constraint where Z = 58 and N = 78 are the number of proton and neutrino in Cs nucleus, respectively. For |Q| 1, ∆Q W is given by

electron beam dump experiments
Another constraint is obtained by searches for gauge boson at electron beam dump experiments, such as SLAC E141 [69,70], Orsay [71] and NA64 [72], via bremsstrahlung from electron and nuclei scatterings. The null results of these searches are interpreted as either (1) the gauge boson can not be produced due to very small coupling, or (2) the gauge boson decays rapidly in the dump. For the X boson to satisfy the Atomki signal requirement, the latter one restricts the electron couplings. From the latest result of NA64 [72], one obtain the constraint, Using the approximate expression of V e and A e , the constraint is given by

electron-positron collider experiments
The coupling to electron is also constrained by e + -e − collider experiment such as KLOE-2 [73] and BaBar [74] experiments. The most stringent limit on the X boson has been set by KLOE-2, searching for e + e − → γX followed by X → e + e − , This constraint is weaker than that of electron beam dump experiment.

Parity violating Møller scattering
Vector and axial-vector interactions of the X boson to electrons induce an extra parity violation in Møller scattering. The cross section was measured at the SLAC E158 experiment [75]. The constraint for the X boson with the mass of 17 MeV is given as

vacuum expectation value of S
For consistency, Eq. (28) must be positive. Since m 2 Z m 2 X , the requirement turns out to be This constraint is also approximated as 29 + 6.7 2 + 3.0 × 10 8 q 2 where we set Q = 0 and q 1 = 1/2. Thus, the constraint excludes the parameter space when q 2 is negative. Assuming that is the same order of R , the exclusion region for q 2 < 0 is given by

V. NUMERICAL RESULTS
In this section, we show our numerical results of the signal requirement and experimental constraints listed in the previous section. As we explained in section III, the coupling constant of left-handed neutrino in Model 2 and 4 is so large for any value of β that the constraint from neutrino-electron scattering can not be evaded. We have analyzed the signal requirement and the constraints, and found no allowed region in these two models. Therefore, we show our numerical results on Model 1 and 3. See text for detail.
A. Model 1 Figure 3 shows the signal region Eq. (41) and exclusion regions for Model 1 in Rplane. In this model, the factor Q, which determines the constraints from neutrino-electron scattering and signal, is given by, and hence Q vanishes for cos 2β = 0. We took cos 2β to 0 and also 0.1 for comparison as shown in the top of each panels. The mass of X is fixed to be 17.01 MeV for the constraints because the constraints are less sensitive to m X , while it is taken to be 17.6 and 16.7 MeV for the signals, respectively. Red transparent band represents the signal region with B X = 4 × 10 −7 (inside) and 7 × 10 −6 (outside), respectively. Color filled regions are exclusion region by the experimental constraints from π 0 → γX (yellow), ν-e scattering (purple), muon g − 2 (dark blue), effective weak charge (light blue), electron beam dump experiment (green) and also by the theoretical constraints from the positive VEV squared of the scalar field S (brown). We first explain the general behavior of the signal requirement and constraints. In each panel, one can see that the signal requirement is almost determined only by R , which is well approximated by Eq. (52). For Q = ±0.1, the signal requirement shows slight dependence on in small | | region. In such region, the decay branching ratio of X → νν is not negligible, and therefore larger R is needed to satisfy the signal by enhancing the decay of X → e + e − .
About the constraints, one can also see that the π 0 → γX constraint excludes the region in large | | while the constraints from (g − 2) µ and VEV of S exclude the region in large | R |. The central region is excluded by the constraint from electron beam dump experiment. The constraint from effective weak charge excludes the region of R > 0. The qualitative behavior of these exclusion regions can be understood by the approximated expressions of the constraints, Eqs. (56), (66) and (71), (76). The last one is the constraints from neutrinoelectron scattering, which is well approximated by Eq. (60) with cos 2β chosen here. It is seen that the constraint excludes large region of the parameter space in R < 0 for | cos 2β| = 0.1, while it disappears for cos 2β = 0. This is because NC is not suppressed in the former cases.

B. Model 3
and hence we took cos 2β = −0.5. The general behavior of the signal requirement and constraints is almost the same as in Model 1. We found a narrow consistent region in the parameter space for cos 2β = −0.5. The coupling constant for this region is 2.3 ≤ | R |×10 4 ≤ 2.5 and 5.1 ≤ | | × 10 4 ≤ 11.9. In this model, the constraint from the VEV of S excludes most of the signal region, even if the neutrino-electron scattering constraint is avoided by taking Q = 0. From Eq. (76), the exclusion region from this constraint is obtained as while that in Model 1 is which is in good agreement with our numerical results.

VI. CONCLUSION
We have discussed the Atomki anomaly in the gauged U (1) R symmetric model. As a minimal model to solve the anomaly, three right-handed neutrinos are introduced for the cancellation of gauge anomalies. Two SU (2) doublet and one SM singlet Higgs scalar particles are also introduced to evade the stringent constraints from neutrino-electron scattering and relativistic degree at the early Universe. Then, the new gauge boson is identified with the X(17) boson. The Atomki signal requirement and other experimental constraints have been studied analytically and numerically in this model.
We first classified models depending on the U (1) R charges of two doublet Higgs fields, by requiring all of CP-odd as well as CP-even scalars to be massive. We found that the possible choices of the gauge charges are limited to four cases q 2 = −1/2, ± 3/2, + 5/2. Two of them leads to large neutrino coupling to electron, and hence such cases are excluded by the constraint from neutrino-electron scattering. Then, for other models with q 2 = −1/2 and −3/2, called Model 1 and 3 respectively, we found that consistent regions with the signal and constraints exist. In such regions, the constraint from neutrino-electron scattering is suppressed due to the cancellation of the gauge charges between two Higgs doublets. In Model 1, the consistent region can be found for | cos 2β| < 0.1 and in Model 3, it is found for cos 2β = −0.5. Other values of cos β and also other models have been excluded by experimental and theoretical constraints.
Comment: While we were finishing this work, ref. [35] appeared, in which the axialvector hypothesis was examined. The authors concluded that 8 Be and 4 He anomalies could be explained by significant uncertainty in nuclear matrix elements.