PHYSICAL REVIEW D 102, 036016 (2020)

Dynamical evidence for a explanation of the ATOMKI nuclear anomalies

† ‡ Jonathan L. Feng ,* Tim M. P. Tait , and Christopher B. Verhaaren Department of Physics and Astronomy, University of California, Irvine, California 92697-4575, USA

(Received 18 June 2020; accepted 30 July 2020; published 17 August 2020)

Recent anomalies in 8Be and 4He nuclear decays can be explained by postulating a fifth force mediated by a new X. The distributions of both transitions are consistent with the same X mass, 17 MeV, providing kinematic evidence for a single new explanation. In this work, we examine whether the new results also provide dynamical evidence for a new particle explanation, that is, whether the observed decay rates of both anomalies can be described by a single hypothesis for the X boson’s interactions. We consider the observed 8Be and 4He excited nuclei, as well as a 12C excited nucleus; together these span the possible JP quantum numbers up to spin 1 for excited nuclei. For each transition, we determine whether scalar, pseudoscalar, vector, or axial vector X can mediate the decay, and we construct the leading operators in a effective field theory that describes them. Assuming parity conservation, the scalar case is excluded and the pseudoscalar case is highly disfavored. Remarkably, however, the protophobic vector , first proposed to explain only the 8Be anomaly, also explains the 4He anomaly within experimental uncertainties. We predict signal rates for other closely related nuclear measurements, which, if confirmed by the ATOMKI group and others, would provide overwhelming evidence that a fifth force has been discovered.

DOI: 10.1103/PhysRevD.102.036016

I. INTRODUCTION them to probe extremely rare processes mediated by particles with very weak interactions. In the of , particles In 2015, researchers working at the ATOMKI pair interact through the electromagnetic, strong, and weak spectrometer experiment reported a 6.8σ anomaly in the forces. Following tradition, if we include gravity, but not decays of an excited beryllium nucleus to its ground state, the force mediated by the , there are four 8 8 − Beð18.15Þ → Beeþe [9]. The observed anomaly is an known forces. The discovery of a fifth force and the particle excess of events at opening angles θeþe− ≈ 140°, suggesting that mediates it would have profound consequences for 8 the production of a new X boson through Beð18.15Þ → fundamental physics. 8BeX, followed by the decay X → eþe−, with best- Nuclear physics provides a fruitful hunting ground for fit parameters m ¼ 16.7 0.35ðstatÞ0.5ðsystÞ MeV new forces. This was obviously true in the past, given the X and Γð8Beð18.15Þ → 8BeXÞ¼1.1 × 10−5 eV, assuming central role of nuclear physics in elucidating the strong and BðX → eþe−Þ¼1. weak forces, but it remains true today. In particular, nuclear In 2016, this possibility was analyzed in detail in experiments are well suited to discovering light and weakly Refs. [10,11]. All possible spin-parity assignments for interacting particles [1–4], which have many particle and the X particle were examined, several explanations were astrophysical motivations [5–8]. These particles may be excluded, and the possibility that the X particle is a vector light enough to be produced in nuclear decays, and the gauge boson mediating a protophobic fifth force was shown extraordinary event rates of nuclear experiments allow to be a viable explanation. Following these studies, other new physics explanations were examined in detail, with a host of implications for nuclear, particle, and astrophysical – *[email protected] observations; see, for example, Refs. [12 25] and Ref. [26] † [email protected] for a brief review. A follow-up study from nuclear theory ‡ [email protected] was also shown to disfavor a nuclear form factor as an explanation [27]. Follow-up measurements of the nuclear Published by the American Physical Society under the terms of transitions have not been reported by other groups, but the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to other experimental collaborations have provided new con- the author(s) and the published article’s title, journal citation, straints on X boson couplings to that exclude part and DOI. Funded by SCOAP3. of the viable parameter space in some models [28,29].

2470-0010=2020=102(3)=036016(19) 036016-1 Published by the American Physical Society FENG, TAIT, and VERHAAREN PHYS. REV. D 102, 036016 (2020)

8 4 Since 2016, the ATOMKI experimentalists have rebuilt anomalous Be and He decay rates have a consistent new their spectrometer and confirmed the 8Beð18.15Þ anomaly physics interpretation. with the new detector. A refined analysis yields best-fit On the experimental side, the width measurement of parameters [30] Eq. (4) requires careful differentiation of signal and back- ground. The leading backgrounds include external pair 3 → 4 γ → þ − mX ¼ 17.01 0.16 MeV; ð1Þ creation (p þ H He þð e e Þ) and internal pair creation from the E0 transition 4Heð20.21Þ → 4Heeþe−.By Γð8Beð18.15Þ → 8BeXÞ¼ð6 1Þ fitting to the expected opening angle distributions for the Γ =Γ 0 10−6Γ 8 18 15 → 8 γ signal and these backgrounds, the ratio X E was × ð Beð . Þ Be Þ determined. The collaboration has noted, however, that ¼ð1.2 0.2Þ × 10−5 eV: ð2Þ an additional background from the E1 transition may also be significant [36]. Including this contribution will modify They have also considered the decays of other excited the best-fit width. With this in mind, we consider models beryllium and carbon nuclei [31,32]. Most remarkably, in that predict widths that differ by orders of magnitude from recent months they have reported the observation of a 7.2σ Eq. (4) to be highly disfavored, whereas models that predict 4 excess at θeþe− ≈ 115° in the transitions of Heð20.49Þ → roughly similar widths are considered viable. These pre- 4Heeþe−, where the beam energy was tuned to dictions will be tested once a refined experimental analysis produce a resonance that sits within the Breit-Wigner peaks including a separate determination of the E1 background is of both the 4Heð20.21Þ and 4Heð21.01Þ excited states. available. Assuming the production of a new particle, the best fit We begin examining the fifth force explanation for parameters are reported to be [33,34] the ATOMKI results in Sec. II, where we consider the production and decay kinematics for the observed 8Be and 4He processes and also for a 12C excited state, which mX ¼ 16.98 0.16ðstatÞ0.20ðsystÞ MeV; ð3Þ completes the possible nuclei JP assignments up to spin 1. Γ 4 20 49 → 4 0 12Γ 4 20 21 → 4 þ − In Sec. III we determine how the observation or exclusion ð Heð . Þ HeXÞ¼ . ð Heð . Þ Hee e ÞE0 of various decays constrains the possible X spin-parity 4 0 1 2 10−5 ¼ð . . Þ × eV; ð4Þ assignments. This is developed further in Sec. IV, where for each of these possibilities we determine the leading where the width uncertainty includes only the uncertainty interactions in an effective field theory (EFT) that describes Γ 4 20 21 → 4 þ − 3 3 1 0 10−4 from ð Heð . Þ Hee e ÞE0 ¼ð . . Þ× eV low-energy nuclear transitions. Using this EFT, the decay [35], the decay width of the standard model E0 transition. rates for the pseudoscalar, axial vector, and vector cases The fact that this excess is at a different angle eliminates are calculated in Secs. V, VI, and VII, respectively. some experimental systematic explanations of both anoma- Remarkably, we find that in the vector case, the proto- lies. At the same time, the fact that the excess is found at the phobic gauge boson, previously advanced as an explan- same X mass provides striking supporting evidence that a ation of the 8Be anomaly, also provides a viable explanation new particle is being produced. for the new 4He observations within experimental uncer- Of course, it would be clarifying if these nuclear decays tainties. In Sec. VIII, we suggest further nuclear measure- were examined by another collaboration. It is important to ments that can provide incisive tests of the new particle note, however, that the observation of a second signal hypothesis, and we summarize our conclusions in Sec. IX. provides an opportunity for another highly nontrivial check of the new physics interpretation beyond simply the II. PRODUCTION AND DECAY KINEMATICS consistency of the best-fit masses. The description in terms of a new particle requires not just kinematic consistency, We consider the production of new X particles in the but also dynamical consistency; that is, the rates for the decays of excited states of 8Be, 12C, and 4He nuclei. anomalous decays must also be consistent with the same set Although the states of interest have different quantum of interaction strengths between the X and . The numbers, leading to different decay dynamics, the kine- best-fit widths are similar numerically, but it is far from matic features of the relevant processes are very similar. 8 4 obvious that this implies consistency: the Be and He In each case, a beam of with kinetic energy P excited states have different J quantum numbers and Ebeam collides with nuclei A at rest to form excited nuclei 8 → different excitation energies, and the Be measurement was through the process p þ A N. The excited nuclei N 4 on resonance, while the He measurement was between two decay to the corresponding ground state nuclei N0 through → → þ − resonances. The decays therefore take place through N N0X, and the X further decay via X e e . operators with different dimensions and different partial In this section we review the experimental results and the waves, and these differences may lead to disparate decay kinematics of these processes. We examine the kinematics widths. A priori, then, it is not at all clear that the observed of resonance production in Sec. II A, and we consider the X

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TABLE I. Production and decay kinematic parameters. Beams of protons with kinetic energy Ebeam collide with → nuclei A at rest to form excited nuclei N, which then decay to the ground state nucleus N0 through N N0X.We m 17 E m m v N fix X ¼ MeV, and for each of the relevant processes, we give the values of beam, A, N , N (the velocity θmin þ − 4 20 49 in the lab frame), vX (the X velocity in the N rest frame), and eþe− (the minimum e e opening angle). Heð . Þ indicates the resonance energy probed in Ref. [33], which sits between the 4Heð21.01Þ and 4Heð20.21Þ states.

min p þ A → N E [MeV] m [MeV] m [MeV] v =c v =c θ − beam A N N X eþe p þ 7Li → 8Beð18.15Þ 1.03 6533.83 7473.01 0.0059 0.350 139° p þ 7Li → 8Beð17.64Þ 0.45 6533.83 7472.50 0.0039 0.267 149° p þ 11B → 12Cð17.23Þ 1.40 10252.54 11192.09 0.0046 0.163 161° p þ 3H → 4Heð21.01Þ 1.59 2808.92 3748.39 0.0146 0.587 108° p þ 3H → 4Heð20.49Þ 0.90 2808.92 3747.87 0.0110 0.557 112° p þ 3H → 4Heð20.21Þ 0.52 2808.92 3747.59 0.0084 0.540 115°

4 boson decay and the - opening angle θeþe− For the He anomaly [33], the experiment ran between in Sec. II B. the broad 0þ 4Heð20.21Þ and 0− 4Heð21.01Þ resonances. The proton beam energy was tuned to produce 4He nuclei A. Resonance production with an excitation energy of E ¼ 20.49 MeV, which is well within the widths of both excited states; see Fig. 1. For resonance production, the required proton energy is B. X boson decay 2 2 2 m − m − mp N A Once produced, the excited nucleus decays through Ep ¼ 2 ; ð5Þ mA → N N0X. In the N rest frame, the X boson is produced with energy where m and m are the nuclear (not atomic) masses. A N 2 2 2 The proton beam energy, or proton kinetic energy, is m þ m − m E N X N0 ≈ m − m ; 6 − ’ X ¼ N N0 ð Þ Ebeam ¼ Ep mp, the proton s momentum is pp ¼ 2m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 2 − 2 Ep mp, and the N nucleus is created with velocity where the last expression exploits the fact that m , m − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX N vN ¼ pp=ðEp þ mAÞ in the lab frame. The values of Ebeam, ≪ 1 − 2 mN0 mN , mN0 . The X velocity is vX ¼ ðmX=EXÞ m m v A, N , and N for the considered processes are given in Table I, and the properties of the relevant excited nuclear Ebeam (MeV) states are summarized in Table II. 0 0.5 1 2 8 The anomaly observed in the Be system is unambig- + uously associated with the 8Beð18.15Þ state [9]. The 0 0 8Beð18.15Þ and 8Beð17.64Þ resonances do not overlap significantly, and the anomaly was found to appear and disappear as the proton beam energy was varied to scan through the 8Beð18.15Þ resonance.

P TABLE II. Nuclear excited states N and their spin-parity J , T Γ isospin , total decay width N , and branching fraction [37–42] [or, in the case of 4Heð20.21Þ, the branching fraction for þ − the E0 decay into e e [35] ]. The N states are labeled by their 8 energy above the nuclear ground state N0 in MeV. The Be 19.5 20.0 20.5 21.0 21.5 22.0 excited states mix; we list the dominant isospin component. E (MeV) P N T Γ [keV] BðN → N0γÞ J N FIG. 1. The Breit-Wigner resonance curves of the 4 − 4 8Beð18.15Þ 1þ 0 138 1.4 × 10−5 0þ Heð20.21Þ and 0 Heð21.01Þ excited states as a function 8Beð17.64Þ 1þ 1 10.7 1.4 × 10−3 of the excitation energy E above the ground state (bottom) 12 − −5 Cð17.23Þ 1 1 1150 3.8 × 10 and the proton beam energy Ebeam (top). The solid and dashed − 4Heð21.01Þ 0 0 840 0 vertical lines at beam energies of 0.90 MeV and 0.52 MeV 4Heð20.21Þ 0þ 0 500 6.6 × 10−10 (E0) correspond to the beam energy used in Ref. [33] and the beam energy required to maximize the 0þ resonance, respectively.

036016-3 FENG, TAIT, and VERHAAREN PHYS. REV. D 102, 036016 (2020) and is shown in Table I for the various processes, assuming 180 17 → þ − mX ¼ MeV. The X boson decays through X e e .In C 17.23 the X rest frame, the electron and positron velocities 160 are ve ≈ 0.998.

All of the processes of interest have the hierarchies deg Be 18.15 140 ≲ 0 01 ≪ ≈ 1 e v v

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⊕ typically remains strongly peaked at the minimum value J ¼ L JX; ð16Þ due to phase space. −1 L Thus, any X bosons produced in the processes under P ¼ð Þ PX: ð17Þ θmin consideration lead to an excess at eþe− . In Fig. 3 we plot min contours of m in the θ − ;m − m plane, using the We analyze the implications of these conservation laws for X ð eþe N N0 Þ PX simple relation of Eq. (13). The blue data points indicate the JX , considering spin-1 nuclear excited states and spin-0 parameters where the 7σ excesses have been found in 8Be excited states in turn. and 4He nuclear decays. The observations are at different opening angles, but both are consistent with the production A. Spin-1 excited states: 8Be and 12C of a boson with mass m ≈ 17 MeV, a striking consistency X In the case of 8Be, the excited states of interest are both check of the new particle hypothesis. 1þ states: one with energy 18.15 MeV and the other with III. SPIN-PARITY ANALYSIS energy 17.64 MeV. The conservation laws for production from the 8Beð18.15Þ state require A first layer of particle dynamics, added to the kinematic foundation laid in the previous section, is to consider 1 ¼ L ⊕ JX; ð18Þ the allowed spin and parity assignments of an X boson L capable of accounting for the ATOMKI observations. The þ1 ¼ð−1Þ PX: ð19Þ assumption that parity is conserved warrants some dis- cussion. Although parity is a very good symmetry with If JX ¼ 0, then L ¼ 1, and PX ¼ −1: X cannot be a scalar, respect to the internal nuclear dynamics and the electro- but it can be a pseudoscalar produced in the P wave. If magnetic interaction, the fundamental description of the JX ¼ 1, then L ¼ 0; 1; 2, and PX ¼þ1; −1; þ1, respec- X boson interactions must be organized with respect to tively: X can be either a vector produced in a P wave or an chirality and would require engineering to end up parity axial vector produced in an S or D wave. symmetric. At the same time, nuclear processes for the In the case of 12C, the excited state is a 1− state with same spin, but opposite parity, typically proceed via energy 17.23 MeV above the stable ground state. Relative different partial waves, and since the X is produced at to the 8Be case, the parity is reversed, and so all the 8Be most semirelativistically (see Table I), the decay of a mixed results apply with the substitutions scalar ↔ pseudoscalar state is likely to be highly dominated by a single parity and vector ↔ axial vector. component. With this in mind, we organize our discussion by treating the X boson as a state of definite parity, and B. Spin-0 excited states: 4He consider the more general case of a vector particle with 4 − 4 mixed vector and axial vector interactions in Sec. VIII. The He excited states of interest are the 0 Heð21.01Þ 4 We continue to label each nuclear decay process as state and the 0þ Heð20.21Þ state. The anomaly is seen at an N → N0X, where N is an excited nucleus and N0 is the intermediate energy chosen to excite both resonances. For − ground state nucleus. The cases of primary interest are the 0 excited state, the conservation laws require those where anomalies have been observed: N ¼ 8Be [9], 4 0 ⊕ where the excited states are 1þ states, and He [33], where ¼ L JX; ð20Þ the excited states are 0þ and 0− states. However, to −1 −1 L complete the analysis of all JP combinations up to spin ¼ð Þ PX: ð21Þ 1, we also consider the case N ¼ 12C, where 12Cð17.23Þ is 0 0 −1 an excited 1− state that can be used to search for the X If JX ¼ , then L ¼ , and PX ¼ : X cannot be a scalar, boson (see, e.g., Ref. [43]). but it can be a pseudoscalar produced in the S wave. If P P0 JX ¼ 1, then L ¼ 1, and PX ¼þ1: X cannot be a vector, The spin-parity of N, N0, and X are denoted by J , J0 , P but it can be an axial vector produced in a P-wave state. and J X , respectively. Parity and angular momentum X Note that X cannot be a vector gauge boson, and so conservation imply symmetries also forbid decays to single for this transition. J ¼ L ⊕ J0 ⊕ J ; ð14Þ X For the 0þ excited state, the parity is reversed, and so all 0− ↔ L the results apply with the substitutions scalar P ¼ð−1Þ P0P ; ð15Þ X pseudoscalar and vector ↔ axial vector. where L is the final state orbital angular momentum and ⊕ denotes the addition of angular momentum. In all the cases C. Spin-parity summary considered, the ground states have quantum numbers The spin-parity analysis results are summarized in P0 þ P J0 ¼ 0 , leading to Table III. All four J possibilities up to spin 1 have been

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P TABLE III. Nuclear excited states N, their spin-parity J , and the possibilities for X (scalar, pseudoscalar, vector, axial vector) allowed by angular momentum and parity conservation, along with the operators that mediate the decay and references to the equation numbers where these operators are defined. The operator subscripts label ’ Oð0Þ the operator s dimension and the partial wave of the decay, and the superscript labels the X spin. For example, 4P is a dimension-four operator that mediates a P-wave decay to a spin-0 X boson.

P N J Scalar X Pseudoscalar X Vector X Axial Vector X 8Beð18.15Þ 1þ … Oð0Þ Oð1Þ Oð1Þ Oð1Þ 4P (27) 5P (37) 3S (29), 5D (34) − 12Cð17.23Þ 1 Oð0Þ … Oð1Þ Oð1Þ Oð1Þ 4P (27) 3S (29), 5D (34) 5P (37) − 4Heð21.01Þ 0 … Oð0Þ … Oð1Þ 3S (39) 4P (40) 4Heð20.21Þ 0þ Oð0Þ … Oð1Þ … 3S (39) 4P (40)

X considered, and so these results are broadly applicable to LI ¼ ciOi: ð22Þ many other nuclear decays. i By comparing results from more than one excited nucleus, O one can in principle determine whether two signals can be The i are operators built out of combinations of fields attributed to a single X boson, and constrain the possible with powers of the nuclear scale Λ included to make them spin-parity assignments of the X boson. The current situation dimension four, and the dimensionless Wilson coefficients is somewhat clouded by the experimental decision to target a ci encode the nuclear physics. Corrections due to finite size nonresonant 4He state, where the anomaly can be attributed effects are represented by (even) higher dimensional non- to transition through either the 0þ or the 0− intermediate renormalizable interactions. The coefficients describing the state (or both, if parity is not conserved). interactions can in principle be extracted from theoretical That said, it is worth noting that if the X particle is being nuclear physics computations or matched to experimental produced in both 0− 4He and 8Be decays, X must be either a observations of nuclear transitions. In cases where neither pseudoscalar or an axial vector; it cannot be a vector (in is available, they must be estimated using dimensional contrast to the conclusion of Ref. [33]). In addition, this analysis. As above, we take the spin-1 and spin-0 cases in 8 4 0þ case implies that the Be decay and the He decay are either turn. In each case we denote the ground state by N0 and ðμÞ S wave and P wave or vice versa. On the other hand, if the the excited state by N , where a vector index is included X particle is produced in both 0þ 4He and 8Be decays, the when it is a spin-1 state. only possibility is that the X boson is a vector and that both The EFT consists of all possible interaction terms decays are P-wave. We construct an effective field theory consistent with the symmetries, including Lorentz invari- for these nuclear decays in Sec. IV, and then use it to ance, gauge invariance (for the photon), number, examine the pseudoscalar, axial vector, and vector pos- and (to good approximation) parity and strong isospin. We sibilities in more detail in Secs. V, VI, and VII, respectively. review its construction in detail, though in some cases the material is well-known. Rather than break the operators IV. EFT FOR NUCLEAR TRANSITIONS into components to analyze parity, we recognize that the contraction of a vector Vμ (JP ¼ 1−) with a derivative is The next refinement in exploring the dynamical con- parity even, sistency of the 8Be and 4He anomalies is to explore whether a single set of X interactions can consistently account for μ∂ Parity⟶ μ∂ both of them. This is complicated by the fact that a V μ V μ; ð23Þ fundamental description of the X boson specifies its μ coupling to quarks and , whereas the states partici- while the contraction of a derivative with an axial vector A P 1þ pating in the reaction are complicated nuclei. It is useful to (J ¼ ) is parity odd, employ the language of an effective field theory, which μ Parity μ exploits the fact that the typical momentum transfer for the A ∂μ⟶ − A ∂μ: ð24Þ transitions of interest are far smaller than the sizes of the participating nuclei, allowing for an expansion in p × rN, Similarly, the contraction of two vector or two axial vectors where p denotes the typical energy/momentum of the is parity even, while the contraction of a vector with an −1 transition, and rN ∼ ð200 MeVÞ characterizes the size axial vector is odd. These observations account for nearly of the 4He, 8Be, and 12C nuclei. all of the needed operators. The remaining cases employ the In this limit, each energy level of the nucleus can be Levi-Civita tensor εμναβ. If an operator that includes four represented as a separate pointlike quantum field, with contracted vector indices has definite parity, the operator interactions described by terms in an effective Lagrangian, composed of the same fields but with the vector indices

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μ contracted into the Levi-Civita tensor has the opposite vector), X must be a scalar (pseudoscalar). To see what N parity. decay partial waves result from this operator, we consider We also make use of the equations of motion for a Oð0Þ μ the matrix element hN0Xj 4P jNi in the N rest frame, massive vector: which takes the form 2 μ μ μ ð□ þ m ÞV ¼ J ; ∂μV ¼ 0; ð25Þ M ∝ ϵ 0⃗i V V ð Þ pXi: ð28Þ μ μ ⃗ where mV and JV are the mass and current of the V Because of the single factor of pX, this operator mediates a particle, respectively. Unlike in the massless case, the P-wave decay, in agreement with the spin-parity analysis of second condition is unrelated to a choice of gauge and the previous section. ϵμ ⃗ 0 ϵμ ⃗ implies that, in general, pμ VðpÞ¼ , where VðpÞ is the The leading operator for a massive spin-1 X particle is μ polarizationpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vector of a V state with momentum μ 2 2 ð1Þ † μ ⃗ ⃗ O ΛN N Xμ: 29 p ¼ð mV þ p ; pÞ. This can be rewritten as 3S ¼ 0 ð Þ μ i μ p If N is a vector (axial vector), then parity demands that X ϵ0 ⃗ V ϵi ⃗ μ VðpVÞ¼ VðpV Þ: ð26Þ must also be a vector (axial vector). The N decay EV amplitude is proportional to Note that for a massive vector at rest, ϵ0 ð0⃗Þ¼0 for all three V M ∝ Λϵ 0⃗iϵ ⃗ polarization states. ð Þ XiðpXÞð30Þ Finally, it is well-known that simply enumerating all operators consistent with a given set of symmetries tends to and is clearly S wave, after summing over the X polari- introduce ambiguities in how the dynamics is described, zation states. although the physical outputs are the same. This is because At higher order, there are three dimension-five operators operators are often related by integration by parts or to consider. The first is through the equations of motion for a given field. In what † 1 N follows we choose to focus on the operators that make Oð Þ 0 ∂ ν ∂μ 5S ¼ ð μNÞ Xν: ð31Þ agreement with the partial wave analysis in the decay Λ process most transparent. In doing so, derivatives are The contraction of derivatives is parity even, requiring the μ μ typically moved from N0 to X, but the final results do contraction of X with N to be parity even as well. Thus, not depend on this choice. μ μ X and N must either both be vectors or both be axial vectors. By integrating by parts 3 times, this operator can be 8 12 A. Spin-1 excited states: Be and C rewritten as

In the spin-1 cases of beryllium and carbon, the excited † † μ μ Parity μ N0 ν μ Nν † μ ν N0 μ ð∂μN Þ∂ Xν ¼ − ð∂μN0Þ∂ X − N □Xμ state N transforms as N∂μ→ð−ÞN∂μ when the state is Λ Λ Λ a vector (axial vector). The lowest dimension operator ν † X N μ describing the decay into a spinless X particle is the ∂μ ∂ † − 0 □ ¼ Λ ð NνÞ μN0 Λ N Xμ dimension-four operator μ □ † þ NXμ N0 ð0Þ † μ Λ O ¼ N N ∂μX: ð27Þ 4P 0 † † N0 ν μ N0 μ − ∂μN ∂ Xν − N □Xμ Here and below, the operator’s subscripts denote the ¼ Λ ð Þ Λ operator’s dimension and partial wave of the decay it □ † μ † − N0 □ μ mediates, and the superscript denotes the spin of the X þ NXμ Λ N0 Λ Xμ N: ð32Þ Oð0Þ boson. Operator 4P is one of a family of operators that † μ † μ Using the leading order equations of motion of the three includes ð∂μN0Þ N X and N0ð∂μN ÞX. All three of these Oð1Þ operators are related to one another by integration by parts, fields, the final three terms are proportional to 3S , μ implying that this operator is not independent and so only two are linearly independent. However, ∂μN vanishes for on-shell processes by the equations of motion, † 2 2 2 N m − m þ m μ ð0Þ 0 ν μ N N0 X † O ð∂μN Þ∂ Xν ¼ − N XμN so we may choose 4P as the single independent operator Λ 2Λ 0 describing processes mediated by this interaction for which μ m E † μ − N X N is on-shell. ¼ Λ N0XμN; ð33Þ Since N0 is parity even, we find that for the entire μ operator to be parity even when N is a vector (axial where we have used Eq. (6) in the last equality.

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The next operator is To preserve parity, either both N and X are scalars or both are pseudoscalars, and it is clear that the decay mediated by † ð1Þ N0 ν μ this operator is S wave. O ∂μN ∂νX ; μ 5D ¼ Λ ð Þ ð34Þ If X is spin 1, the leading independent operator is

μ μ which again requires X and N to have equal parity. In the ð1Þ † μ μ μ O ¼ N X ∂μN : ð40Þ N p m ; 0; 0; 0 4P 0 rest frame, where ¼ð N Þ, the decay matrix element takes the form A similar operator with the derivative acting on N0 is m μ m obtained by integrating by parts and recalling that N 0 ⃗i N ⃗i j j M ∝ δμϵ ð0Þ p ϵ ðp⃗ Þ¼ ϵ ð0Þ p p ϵ ðp⃗ Þ; ∂ μ 0 Λ Xi X X ΛE Xi X X X μX ¼ . The parity analysis is simple: if N is parity X μ∂ μ even, then so is X μ,soX is a vector. If N is a ð35Þ μ pseudoscalar, then X must be an axial vector. The amplitude for N decay in its own rest frame is where we have used Eq. (26). Since this depends on two factors of p⃗ , it describes a D-wave decay. The factor of X μ m 0 N i i m in the numerator of this decay might cause one to M ∝ ϵ ðp⃗ Þm δμ ¼ p ϵ ðp⃗ Þ; ð41Þ N X X N E X X X wonder if it is really suppressed relative to the S-wave X decay. However, comparing Eq. (33) and Eq. (35), we see that the ratio of the D-wave operator to the S-wave operator where we have used the identity in Eq. (26). This is a is parametrically P-wave decay, and again we find complete agreement between the spin-parity and operator analyses of the helium p2 decays. These results are summarized in Table III. X ¼ v2 ; ð36Þ E2 X X V. PSEUDOSCALAR X which is exactly what we expect when we compare a When X is a pseudoscalar, its couplings to protons and D-wave amplitude with an S-wave amplitude. can be written as The final dimension-five operator contracts the four vector indices of the previous two operators with the L ⊃ X½ε p¯ γ5p þ ε n¯γ5n Levi-Civita tensor: X p n  1 5 1 5 † ¼ X ðεp þ εnÞJ0 þ ðεp − εnÞJ1 ; ð42Þ ð1Þ N0 μναβ 2 2 O ε ∂μN ν ∂αXβ: 5P ¼ Λ ð Þ ð37Þ where As mentioned in the introduction to this section, this μ implies that Xμ and N have opposite parity, in contrast 5 ¯ γ ¯γ 5 ¯ γ − ¯γ to the two previous operators, where they had equal parity. J0 ¼ p 5p þ n 5n and J1 ¼ p 5p n 5n ð43Þ In the rest frame of the excited state, the amplitude for this decay is proportional to are the isosinglet and isotriplet currents, respectively. Nuclear transitions between states of the same isospin m 5 M ∝ N ε0ijkϵ ð0⃗Þp ϵ ðp⃗ Þ; ð38Þ couple to J0, while transitions that change isospin by one Λ i Xj Xk X 5 unit couple to J1. For the 8Be case, the decay of the 8Beð18.15Þ excited which is P wave. Thus, we have explicitly confirmed nucleus to the ground state transitions from (predomi- that the operator-based analysis conforms with the spin- nantly) isosinglet to isosinglet, so X must couple through parity analysis for both the beryllium and the carbon decays J5 in every particular. These results are summarized in the 0 field. In terms of currents, then, the decay Table III. amplitude is 1 B. Spin-0 excited states: 4He M ε ε 5 ¼hN0Xj 2 ð p þ nÞXJ0jNi For helium the operator analysis is nearly identical for 1 ε ε 8 5 8 18 15 both the scalar and the pseudoscalar excited states. If X is ¼ 2 ð p þ nÞh BejJ0j Beð . Þi: ð44Þ spin 0, the leading operator that mediates the decay is In the nuclear EFT, the P-wave decay amplitude, deter- ð0Þ † O Λ ð0Þ 3S ¼ N0NX: ð39Þ O mined by 4P in Eq. (27), can be written as

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1 M ε ε Oð0Þ and the decay width is ¼hN0Xj 2 ð p þ nÞCP;Be 4P jNi 1 ε ε 2 2 Λ2 ε ε ϵμ 4He ð p þ nÞ CP;He ¼ ð p þ nÞCP;Be pXμ; ð45Þ Γ ðN → N0XÞ¼ p : ð51Þ 2 P 32πm2 X;He N where C is the Wilson coefficient of the operator. This P;Be Therefore, for pseudoscalar X bosons, the ratio of the coefficient is expected to be Oð1Þ in the nuclear EFT [44]. From matching Eqs. (44) and (45), it is determined by beryllium and helium decay widths of Eqs. (47) and (51) is predicted to be ϵμ 8 5 8 18 15 CP;Be pXμ ¼hBejJ0j Beð . Þi; ð46Þ 8 Γ Be 1 C2 m2 p2 P P;Be He pX;Be X;Be 4 ¼ He 3 2 2 Λ2 where the nuclear matrix element has been estimated in Γ CP;He mBe pX;He Ref. [15] using a shell model. Note that, since the excited P   C2 2 beryllium state is spin-1, the matrix element must depend ≈ 1 7 10−6 P;Be GeV μ . × 2 Λ ; ð52Þ on ϵ. Because the matrix element is a scalar, this CP;He polarization vector must be contracted into some combi- nation of the particle momenta. Conservation of momen- where we have assumed m ¼ 17 MeV. The coefficients μ X tum allows us to write this as a linear combination of p C and C are expected to be Oð1Þ in the EFT, but μ μX P;Be P;He and p, but the latter vanishes when contracted with ϵ, they are not both readily available in the literature, leading leaving only pXμ in Eq. (46). Note also that the combina- to significant uncertainty in their ratio. On the experimental ε ϵμ ν α β side, the fact that the ATOMKI 4He data were collected off- tion μναβ pp0pX vanishes because the momenta are not independent, but could not appear in any case because it is resonance requires some care (see Sec. VIII). Nevertheless, odd under parity. even given these theoretical and experimental uncertainties, Γ8Be Γ4He ∼ 10−6 The decay width is then the EFT prediction P = P is clearly difficult to reconcile with the observations of Eqs. (2) and (4), which ε ε 2 2 8 4 8Be ð p þ nÞ CP;Be 3 Γ Be ∼ Γ He X Γ ðN → N0XÞ¼ p ; ð47Þ imply P P , and a pseudoscalar explanation for P 96πm2 X;Be N both anomalies is strongly disfavored. Note also that the extreme hierarchy in expected decay where pX;Be is the magnitude of the 3-momentum of the X widths also implies that X is unlikely to be a scalar and particle produced in the beryllium decay. pseudoscalar mixture, since a significant pseudoscalar 12 12 − For the C case, as shown in Sec. III, the Cð17.23Þ 1 component is required to explain the beryllium anomaly state cannot decay into the 0þ ground state and a pseudo- (since the beryllium state cannot decay to a scalar), but even scalar. If an X boson were observed in 12Cð17.23Þ decay, it a small pseudoscalar component would result in a very would exclude the possibility that X is a pure pseudoscalar. large helium decay width, which has not been observed. In the 4He case, the decay is from the 4Heð21.01Þ 0− isosinglet state. In terms of nucleon currents, the decay VI. AXIAL VECTOR X amplitude is therefore Similar to the pseudoscalar case, an axial vector X 1 M ε ε 5 couples to the nucleon currents ¼hN0Xj 2 ð p þ nÞXJ0jNi 1 μ5 ε ¯ γμγ ε ¯γμγ ε ε 4 5 4 21 01 JX ¼ pp 5p þ nn 5n ¼ 2 ð p þ nÞh HejJ0j Heð . Þi: ð48Þ 1 μ5 1 μ5 ¼ ðεp þ εnÞJ0 þ ðεp − εnÞJ1 ; ð53Þ In the nuclear EFT, the decay is mediated by the operator 2 2 0 Oð Þ S 4P in Eq. (27), leading to the -wave decay amplitude where 1 ð0Þ μ5 μ5 M ¼hN0Xj ðε þ ε ÞC O jN i ¯ γμγ ¯γμγ ¯ γμγ − ¯γμγ 2 p n P;He 4P J0 ¼ p 5p þ n 5n and J1 ¼ p 5p n 5n 1 ε ε Λ ð54Þ ¼ 2 ð p þ nÞCP;He : ð49Þ are the isospin-preserving and isospin-changing currents, O 1 The ð Þ Wilson coefficient CP;He is determined simply by respectively. For the isospin-preserving 8Be decay, the nucleon-level Λ 4 5 4 21 01 CP;He ¼hHejJ0j Heð . Þi; ð50Þ amplitude is

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1 μ5 a vector, while Xμ is an axial vector, the inner product of M ¼hN0Xj ðε þ ε ÞXμJ jN i 2 p n 0 their polarization vectors is parity odd. Therefore, the 1 ε ε 8 μ5 8 18 15 ϵ matrix element must be proportional to the axial vector ¼ ð p þ nÞh BejJ0 j Beð . Þi Xμ: ð55Þ εμναβ ϵ 2 pν αpXβ. The resulting decay width is

At the nucleus level, the decay is mediated by the operator 2 12C 2 CA;C 3 ð1Þ Γ → ε − ε O A ðN N0XÞ¼ð p nÞ 2 pX;C: ð62Þ 3S in Eq. (29), which leads to the S-wave decay 48πΛ amplitude Combining Eqs. (58) and (62), the ratio of the beryllium 1 ð1Þ and carbon decay widths is M N0 ε ε O ¼h Xj 2 ð p þ nÞCA;Be 3S jNi   8Be 2 2 4 2 1 μ Γ 3 ε ε C Λ ε ε Λϵ ϵ A ð p þ nÞ A;Be pX;Be pX;Be ¼ ð p þ nÞCA;Be Xμ; ð56Þ 12 ¼ 1 þ 2 C 2 ε − ε 2 2 2 3 3 2 Γ ð p nÞ CA;C mBe pX;C mX A   O 1 2 2 4 where CA;Be is the ð Þ Wilson coefficient of the operator. ðε þ ε Þ C Λ ≈ 8 1 103 p n A;Be : Matching these amplitudes, we find . × ε − ε 2 2 ð63Þ ð p nÞ CA;C GeV Λϵμ 8 μ5 8 18 15 CA;Be ¼hBejJ0 j Beð . Þi; ð57Þ Although this result is quantitatively limited by our knowl- edge of CA;Be and CA;C, these coefficients are expected to be where the nuclear matrix element has been obtained via 8Be 12C Oð1Þ in the nuclear EFT, and so we expect Γ ≫ Γ . many-body techniques in Ref. [19]. Note that the nuclear A A μ Given that the carbon total decay width and photon decay matrix element must be proportional to ϵ. The quantity εμναβϵ width are both about an order of magnitude larger than their νpαpXβ cannot appear because it would make beryllium counterparts, for pseudoscalar X bosons, the the matrix element odd under parity. The resulting decay carbon decay rate will be greatly suppressed relative to the width is beryllium decay rate and very difficult to observe.   The 4He decay must proceed from the 4Heð21.01Þ ε ε 2 2 Λ2 2 8Be ð p þ nÞ CA;Be pX;Be state. It is isospin preserving, and so the nucleon-level Γ ðN → N0XÞ¼ p 1 þ : A 32πm2 X;Be 3m2 N X amplitude is ð58Þ 1 M ε ε μ5 ¼hN0Xj ð p þ nÞXμJ0 jNi 12 2 For C, the decay of the excited nucleus requires 1 changing isospin by one unit, and so the nucleon-level 4 μ5 4 ¼ ðεp þ εnÞh HejJ0 j Heð21.01ÞiϵXμ: ð64Þ amplitude is 2 1 1 At the nucleus level, the P-wave decay is mediated by Oð Þ M ε − ε μ5 4P ¼hN0Xj 2 ð p nÞXμJ1 jNi in Eq. (40), with amplitude 1 ε − ε 12 μ5 12 17 23 ϵ ¼ ð p nÞh CjJ1 j Cð . Þi Xμ: ð59Þ 1 ð1Þ 2 M N0 ε ε O ¼h Xj 2 ð p þ nÞCA;He 4P jNi ð1Þ 1 μ In the nuclear EFT, the operator O5 of Eq. (37) gives rise ε ε ϵ P ¼ ð p þ nÞCA;Hep Xμ; ð65Þ to the amplitude 2

1 1 where M N X ε − ε C Oð Þ N ¼h 0 j 2 ð p nÞ A;C 5P j i μ μ 4 5ν 4 21 01 1 1 CA;Hep ¼ PXνh HejJ0 j Heð . Þi ð66Þ ε − ε εμναβ ϵ ϵ ¼ 2 ð p nÞCA;C Λ pμ νpXα Xβ; ð60Þ and where the Wilson coefficient CA;C is determined by μ μ μ pXpXν P ¼ δν − ð67Þ 1 Xν p2 εμναβ ϵ 12 μ5 12 17 23 X CA;C pν αpXβ ¼h CjJ1 j Cð . Þi: ð61Þ Λ μ is the projection matrix into the subspace orthogonal to pX. The excited nuclear state has spin 1, so the nuclear matrix The matrix element has a vector Lorentz index, but is μ μ element must be proportional to ϵ. However, because N is composed of scalar fields. This, along with momentum

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4 μ5 4 M 8 μ 8 18 15 conservation, implies that h HejJ0 j Heð21.01Þi is propor- ¼hBeXjXμJXj Beð . Þi μ μ tional to a linear combination p and p . However, the 1 X ε ε 8 μ 8 18 15 ϵ latter vanishes when contracted with the X polarization ¼ 2 ð p þ nÞeh BejJ0j Beð . Þi Xμ: ð72Þ vector. The decay width is ð1Þ At the nucleus level, the P-wave decay is mediated by O5 ε ε 2 2 P 4He ð p þ nÞ CA;He 3 in Eq. (37), leading to the decay amplitude Γ ðN → N0XÞ¼ p : ð68Þ A 32πm2 X;He X 1 M ε ε Oð1Þ ¼hN0Xj ð p þ nÞeCV;Be 5 jN i Thus, combining Eqs. (58) and (68), for an axial vector X 2 P 1 1 boson, we find ε ε εμναβ ϵ ϵ ¼ 2 ð p þ nÞeCV;Be Λ pμ νpXα Xβ: ð73Þ   Γ8Be 2 2 2 A CA;Be 2 mX pX;Be pX;Be 4 ¼ Λ 1 þ Matching the amplitudes, we find that the Wilson coef- He 2 2 3 3 2 Γ CA;He mBe pX;He mX ficient C is determined by A   V;Be C2 Λ 2 ≈ 1 8 10−2 A;Be 1 . × 2 ; ð69Þ μναβ 8 μ 8 C ε p νϵ αp β Be J Be 18.15 : 74 CA;He GeV V;Be Λ X ¼h j 0j ð Þi ð Þ where we have assumed mX ¼ 17 MeV. In contrast to the Because the excited nuclear state is spin-1, the nuclear μ pseudoscalar case, the decay widths are not expected to be matrix element must contain a factor of ϵ. However, since μ μ separated by 2 orders of magnitude, not 4. Within the N is an axial vector, while X is a vector, the matrix uncertainties of the measurements and the nuclear matrix εμναβ ϵ element must be proportional to pν αpXβ to preserve elements, it may be possible that an axial vector X boson 8 4 parity. could explain both the Be and He anomalies. The resulting decay width is

α 2 VII. VECTOR X 8Be 2 CV;Be 3 Γ N → N0X ε ε p ; X ð Þ¼ð p þ nÞ 12Λ2 X;Be ð75Þ In contrast to the pseudoscalar and axial vector cases, the necessary nuclear matrix elements for decays to a pure where p is the magnitude of the 3-momentum of the X P 1− X;Be J ¼ vector X boson are related to standard model boson. For the photon, the width is similarly computed. decays to photons and E0 transitions and cancel out in Note that the mediating operator is “accidentally” gauge appropriately defined ratios. Oð1Þ 1 2Λ †εμναβ ∂ invariant, 5P ¼ =ð ÞN0 ð μNνÞXαβ, where Xαβ is the X boson’s field strength, and so the results have a A. Beryllium decays to vector X bosons smooth limit mX → 0 and are immediately applicable to We begin by reviewing the analysis for beryllium decays photons. The resulting photon width is [10,11]. The vector X boson couples to the nucleon current α 2 8Be CV;Be 3 Γγ N → N0γ p ; 76 μ ε ¯ γμ ε ¯γμ ð Þ¼ 12Λ2 γ;Be ð Þ JX ¼ pep p þ nen n 1 1 ε ε μ ε − ε μ and the ratio of widths is ¼ 2 ð p þ nÞeJ0 þ 2 ð p nÞeJ1; ð70Þ 8   Γ Be p3 2 3=2 where X 2 X;Be 2 mX 8 ¼ðε þ ε Þ ≈ ðε þ ε Þ 1 − : Be p n p3 p n m − m 2 Γγ γ;Be ð N N0 Þ μ ¯ γμ ¯γμ μ ¯ γμ − ¯γμ J0 ¼ p p þ n n and J1 ¼ p p n n ð71Þ ð77Þ are the isospin-preserving and isospin-changing vector Because the same nuclear matrix element occurs in both the currents, respectively. In contrast to the previous two vector X boson and the photon decays, the ratio of the two sections, we have included a factor of the photon coupling decay widths is particularly simple and independent of μ e to simplify comparisons between X and photon decays. nuclear matrix elements. 8Beð18.15Þ and the 8Be ground state have equal isospin. As shown in Ref. [11], the presence of the nearby spin-1, If we assume isospin is conserved and neglect isospin isospin-triplet 8Beð17.64Þ state induces some isospin mix- μ mixing in the nuclear states, only the J0 current contributes, ing corrections to this result. Including the isospin mixing, and the decay amplitude is the ratio of widths is modified to

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8 Γ Be p3 As with beryllium, we can relate the X decay rate to the X − 0 09 ε ε 1 09 ε − ε 2 X;Be 8 ¼j . ð p þ nÞþ . ð p nÞj 3 photon decay rate and form a ratio that is independent of Γ Be p γ γ;Be nuclear matrix elements. Unfortunately, the procedure is 2 ≈ 0.043j − 0.09ðεp þ εnÞþ1.09ðεp − εnÞj ; ð78Þ less straightforward than in the beryllium case, because, as is clear from the mX → 0 limit of Eq. (83), the analysis where we have set mX ¼ 17 MeV in deriving the final above is not appropriate for the photon decay. This is numerical result. The coefficients of the isosinglet and simply because, when the photon is substituted for X in isotriplet components are related to the degree of mixing ð1Þ ð1Þ O3 , the result is not gauge invariant. Instead of using O3 and relative strengths of the respective M1 transitions in the S S 8 immediately, we must first determine the gauge-invariant Be system. A further refinement to fit experimental data by operator that mediates photon decay and then relate it to introducing isospin breaking in the electromagnetic tran- 1 Oð Þ. The decay amplitude is sition operators was introduced in Ref. [11]. With both 3S isospin mixing and isospin breaking, the width ratio is 1 modified to M O ¼hN0Xj 2 eCγ;C γjNi; ð84Þ 8 Γ Be p3 X 2 X;Be where Oγ is the gauge-invariant operator and Cγ; is its 8 ¼j0.05ðε þ ε Þþ0.95ðε − ε Þj C Be p n p n 3 Γγ pγ;Be Wilson coefficient. 2 At leading order, there appear to be two possible ≈ 0.043j0.05ðε þ ε Þþ0.95ðε − ε Þj : ð79Þ p n p n dimension-five gauge-invariant operators:

We use both Eqs. (78) and (79) in presenting our 1 1 results below. O μν †∂ O0 μν ∂ † γ ¼ Λ F N0 νNμ and γ ¼ Λ F Nμ νN0; ð85Þ B. Carbon decays to vector X bosons where Fμν is the usual electromagnetic field strength. For 12C, it is straightforward to calculate the X boson However, integrating by parts, we find decay rate. The excited state is a vector with isospin 1 1 T ¼ . Consequently, the transition to the ground state O −O0 − †∂ μν þ μ γ ¼ γ NμN0 νF : ð86Þ 0 is through the J1 current of Eq. (71). As with the case of Λ 8 Be decaying to axial vector X bosons, this carbon decay is μν ν 1 ∂ Oð Þ Given the photon equation of motion μF ¼ JEM, the last mediated by the operator 3S of Eq. (29), but now with μ μ term is suppressed by the small coupling in the photon both N and X vectors instead of both axial vectors. The current. To leading order, then, the two operators are decay amplitudes are interchangeable, and the difference involving the current ν 1 J can be incorporated systematically through Feynman M ε − ε μ EM ¼hN0Xj 2 ð p nÞeXμJ1jNi diagrams. We can therefore simply consider Oγ. O Oð1Þ μν → μν O 1 To relate γ to 3S , we replace F X in γ and use ε − ε 12 μ 12 17 23 ϵ ¼ 2 ð p nÞeh CjJ1j Cð . Þi Xμ ð80Þ 1 μν † 1 μ ν † 1 ν μ † X N0∂νN μ ¼ ð∂ X ÞN0∂νN μ − ð∂ X ÞN0∂νN μ and Λ Λ Λ 1 † m E † μ ∂μ ν ∂ N X 1 ¼ ð X ÞN0 νNμ þ N0NXμ; M ε − ε Oð1Þ Λ Λ ¼hN0Xj 2 ð p nÞeCV;C 3S jNi ð87Þ 1 ε − ε Λ ϵμϵ ¼ ð p nÞeCV;C Xμ; ð81Þ 2 where, in the second line, we have applied the identity given in Eq. (33). The second term on the right has the same where Oð1Þ form as 3S in Eq. (29). The first mediates the D-wave Λ ϵμ 12 μ 12 17 23 decay and is smaller than the dimension-three term by a CV;C ¼h CjJ1j Cð . Þi; ð82Þ 2 ∼ 0 03 factor of vX . [see Eq. (36) and Table I], so we neglect and the resulting decay width is [see Eq. (58)] Oð1Þ O it. We find, then, that the Wilson coefficients of 3S and γ   α 2 Λ2 2 are related by 12C 2 CV;C pX;C Γ ðN → N0XÞ¼ðε − ε Þ p 1 þ : X p n 8m2 X;C 3m2 Λ2 N X Cγ ¼ C : ð88Þ ;C V;C m E ð83Þ N X

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With this identification, we can calculate the decay ampli- where we have set mX ¼ 17 MeV in deriving the final tude for the photon decay and determine an expression for numerical result. The ratio is independent of nuclear matrix the ratio of X boson and photon widths that is independent elements, as desired, and the singularity as mX → 0 is now of nuclear matrix elements. readily understood. It is instructive to determine the decay amplitude in a general way that encompasses both the massless and C. Helium decays to vector X bosons massive vector cases. Specifically, we determine the For 4He, it is also straightforward to calculate the decay amplitude rate into the X boson. The 0− state cannot decay to vector 1 bosons, but the 0þ state can. The 0þ state has the same M ε O ¼hN0Xj 2 eCγ;C VjNi; ð89Þ isospin as the ground state, and so the amplitudes are 1 where M ε ε μ ¼hN0Xj 2 ð p þ nÞeXμJ0jNi κ μ ν † 1 ν μ † O ∂ ∂ − ∂ ∂ 1 4 μ 4 V ¼ ð V ÞN0 νNμ ð V ÞN0 νNμ: ð90Þ ε ε 20 21 ϵ Λ Λ ¼ 2 ð p þ nÞeh HejJ0j Heð . Þi Xμ ð96Þ

When κ ¼ 1, OV is Oγ, the gauge-invariant operator of and Eq. (85) that mediates decays to a massless photon. But 1 1 1 κ 0 O Oð Þ M N X ε ε eC Oð Þ N when ¼ , V is proportional to 3S , the operator that ¼h 0 j 2 ð p þ nÞ V;He 4P j i mediates decays to a massive X, and we expect to recover 1 the decay width calculated in Eq. (83). The amplitude is ε ε μϵ ¼ 2 ð p þ nÞeCV;Hep Xμ: ð97Þ 1 1 M ε ϵ ϵ κ μ ν − ημν Matching these two amplitudes, the Wilson coefficient is ¼ eCγ;C μ Vν½ pVp ðpV · pÞ ; ð91Þ 2 Λ determined by which leads to the decay width μ μ 4 ν 4 20 21 CV;Hep ¼ PXνh HejJ0j Heð . Þi; ð98Þ Γ12C → V ðN N0VÞ μ where P ν, defined in Eq. (67), projects into the subspace 2  4  X μ αCγ p orthogonal to p . The nuclear states are scalars, so the ε2 ;C p 3m2 2 2 − κ p2 1 − κ 2 V;C : X ¼ 24Λ2 V;C V þ ð Þ V;C þð Þ 2 Lorentz index in the matrix element must be attached to a mV linear combination of the particle momenta. By momentum ð92Þ μ conservation, this can be chosen to be a combination of p μ and p , but the latter vanishes when contracted with the X For the photon, we take κ ¼ 1, mV → 0, and ε ¼ 1 to X obtain polarization vector. The resulting decay rate is α 2 2 4He 2 CV;He 3 12 αCγ Γ → ε ε C ;C 3 X ðN N0XÞ¼ð p þ nÞ 2 pX; : ð99Þ Γγ ðN → N0γÞ¼ p : ð93Þ 8 He 12Λ2 γ;C mX Following the 8Be and 12C examples above, we hope to For the massive vector X boson, we take κ ¼ 0 and ε ¼ normalize the X decay rate to the photon decay rate. ε − ε to find 4 p n Unfortunately, the relevant He excited states do not decay   0− α 2 2 to photons. For the state, the spin-parity analysis of 12C 2 Cγ;C 2 pX;C Γ N → N0X ε − ε p E 1 : Sec. III showed that decays to vector bosons, either X ð Þ¼ð p nÞ 8Λ2 X;C X;C þ 3 2 mX massless or massive, are forbidden. This is also easily ð94Þ seen from the operator point of view. The only gauge- invariant operator that could mediate a photon decay is

Using Eq. (88) to write Cγ;C in terms of CV;C, we recover 1 μναβ † ε Fμν∂αN ∂βN : exactly the result of Eq. (83), confirming the Wilson Λ2 0 ð100Þ coefficient relation of Eq. (88). Combining Eqs. (93) and (94) yields Upon integration by parts, we find this operator vanishes,

12   either because of the symmetry of the derivative operator or Γ C 3 2 εμναβ∂ 0 X 2 pX;C 2 pX;C 2 by the Bianchi equation, βFμν ¼ . 12 ¼ ðε − ε Þ E 1 þ ≈ 0.25ðε − ε Þ ; C 2 p n 3 X;C 3 2 p n 0þ Γγ pγ;C mX The state can decay to a massive , but unfortunately, it cannot also decay to a photon. When N is ð95Þ the 0þ state, we can write the gauge-invariant operator

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Λ2 CE0 ¼ 2 CV;He: ð106Þ mX Now that we have related the Wilson coefficients, we can μ calculate the E0 decay width. In Lorentz gauge, ∂μA ¼ 0, FIG. 4. Feynman diagrams for the E0 transition (left) and the and so the E0 operator of Eq. (101) is decay into Xμ (right). 1 μ ν ν μ † O 0 ∂ A − ∂ A ∂μN ∂νN E ¼ Λ2 ð Þð Þ 0 1 † μν 1 μ ν 2 μ † O 0 ¼ F ∂μN0∂νN : ð101Þ − ∂ ∂ − ∂ ∂ E Λ2 ¼ Λ2 ½ νA A N0 μN

1 † 2 μ However, if we use this operator to calculate the decay ¼ N0ð∂μN Þ∂ A ; ð107Þ Γ → γ Λ2 width ðN N0 Þ, we find the amplitude vanishes. This O occurs because we can integrate E0 by parts and use the μν ν which makes clear that the interaction vanishes when the photon equation of motion ∂μF ¼ 0 þ J . It is this EM photon is on-shell. We use this vertex to calculate the N leading zero that is captured by the amplitude, reflecting the decay shown in the left panel of Fig. 4. Summing over final 0þ fact that the excited state cannot decay into the ground state spins, we find the squared amplitude state and an on-shell photon. The piece that is higher order in the electromagnetic coupling, X 4 2 2 e CE0 jMj ¼ ½2ðp · p0Þðp− · p0Þ 1 Λ4 þ ν † spin J N ∂νN ; Λ2 EM 0 ð102Þ − 2 − 2 2 mN0 ðpþ · p−Þ memN0 ; ð108Þ is nonvanishing and mediates the E0 transition 0þ → 4 þ − Hee e shown in Fig. 4. This is an electromagnetic where p is the e momentum, and p0 is the N0 transition, but one in which the photon is never on-shell. momentum. It is possible, however, to normalize the X boson decay to The integration over the three-body phase space is most the 0þ E0 transition. Following the procedure of Sec. VII B, conveniently expressed as integrals over the electron and ð1Þ O O positron energies, E and E−, we relate E0 to 4P . The E0 decay amplitude is þ

1 2 2 Z M O α C 0m ¼hN0Xj eCE0 E0jNi; ð103Þ Γ E N 2 − 2 2 E0 ¼ 4 dE dE−½m m 8πΛ þ N N0 where C 0 is the Wilson coefficient of the O 0 operator. − 2 4 E E mN ðEþ þ E−Þþ EþE−; ð109Þ Similar decays to eþe− from the 0− state can only appear though operators of higher dimension and are consequently with the limits of integration chosen so that the outer (e.g., neglected. E ) limits of integration are O þ We can relate the E0 operator to the massive vector Oð1Þ m − m 2 − m2 m2 operator 4P in Eq. (40) by using the X equations of ð N eÞ N0 þ e EþMin ¼ me;EþMax ¼ ; motion. If we begin with the same operator as in Eq. (101), 2ðmN − meÞ but replace Fμν with the X field strength Xμν,wehave ð110Þ

1 μν † X ∂μN ∂νN : Λ2 0 ð104Þ while the inner limits are

Integrating by parts and using the X equations of motion E−ðÞ ∂ μν 2 ν ν μX þ mXX ¼ JX, we find this operator includes 1 ¼ 2 2½mN ðmN − 2E Þþm m2 þ e X μ †∂ 2 2 2 X N0 μN; ð105Þ × fðm − E Þ½m ðm − 2E Þ − m þ 2m Λ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN þ N N þ N0 e 2 2 2 2 2 2 2 ð1Þ ðm − E Þ½ðm − 2m E − m Þ − 4m m g; O N þ N N þ N0 N0 e which is exactly the form of 4P in Eq. (40). The remaining μ term that includes the X current is subleading, because it ð111Þ contains additional factors of the small X coupling, and its effects can be included diagrammatically. The Wilson where the positive and negative signs correspond to the Oð1Þ O coefficients of 4P and E0 are therefore related by maximum and minimum, respectively.

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4 Γ He ε ε 2 π m2 p3 In general, the integral does not have a simple solution. X ≈ ð p þ nÞ X X;He ≈ 360 ε ε 2 −13 5 ð p þ nÞ ; ð113Þ However, neglecting the electron mass results in Γ 0 6.0 × 10 α m E N Γ 3 3 1 10−4 2 2 5   2 6 8  where E0 ¼ð . Þ × eV [35] and we have set α C 0m 1 m m m E N N0 N0 N0 m ¼ 17 MeV in deriving the final numerical result. The Γ 0 ≈ 1 − 8 þ 8 − X E 8πΛ4 96 m2 m6 m8 ratio is independent of nuclear matrix elements and is larger N N N 4  than ratios of X widths to photon widths derived here, since m m N0 N the E0 transition is a 3-body decay. þ 4 4 ln mN mN0   2 2 5 5 α C 0m 2 m − m D. Vector X summary ≈ E N N N0 8πΛ4 15 mN0 For the case of a vector boson, the nuclear decay widths 2 2 5 are functions of mX and the couplings εp and εn.The α C 0m ≈ E N 6 03 10−13 : constraints on these parameters from all other experiments 8πΛ4 ð . × Þ ð112Þ have been determined in Refs. [10,11].AboundfromNA48/ −3 2 on the process π → Xγ [45] requires jεpj < 1.2 × 10 , Evaluating the integral numerically with the electron mass that is, that the X boson be protophobic. Once this constraint included, we find the final numerical coefficient changes by is satisfied, no other constraints further restrict the values of 5 97 10−13 a tiny amount to . × . mX and εn favored by the beryllium anomaly. Combining Eqs. (99) and (112), and using Eq. (106) to In Figs. 5 and 6, we plot the regions of the ðmX; εnÞ plane relate CE0 and CV;He, we find that the ratio of the X width to that can explain the beryllium anomaly, with mX ¼ the E0 width is 17.01 0.16 MeV and [30]

FIG. 5. Predictions for decay widths into a protophobic X gauge boson in the ðεn;mXÞ plane for the fixed values of εp indicated, where 8 εn is the X boson’s coupling to neutrons, and mX is its mass. The blue shaded regions are favored by the Be anomaly, with isospin Γ Γ 4 20 21 Γ Γ mixing included [see Eq. (78)]. Also shown are solid black contours of X= E0 for Heð . Þ decays and dashed red contours of X= γ for 12Cð17.23Þ decays.

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FIG. 6. As in Fig. 5, but with both isospin mixing and isospin breaking included (see Eq. (79)).

Γð8Beð18.15Þ → 8BeXÞ E1 background discussed in Sec. I, and the fact that the ¼ð6 1Þ × 10−6: ð114Þ Γð8Beð18.15Þ → 8BeγÞ prediction of Eq. (115) is for on-resonance decay, while the measurement of Eq. (4) is for off-resonance decay. But with The three panels are for the purely protophobic case those uncertainties in mind, the protophobic vector boson currently provides an amazingly consistent explanation of εp ¼ 0 and the minimal and maximal allowed values −3 both the beryllium and helium anomalies. We strongly ε ¼1.2 × 10 . In Fig. 5, we use the theoretical pre- p recommend that the experimental measurement be updated diction of Eq. (78), which includes the isospin mixing of to include the E1 background and that a future measure- the beryllium excited states. In Fig. 6, we use the theoretical ment at the 4Heð20.21Þ resonance be made to provide an prediction of Eq. (79), which includes both isospin mixing unambiguous test of the prediction of Eq. (115). and isospin breaking. The rate of the ATOMKI beryllium Finally, in Figs. 5 and 6, we also overlay red contours for anomaly requires ε ≈ 10−2. n Γ 12C 17.23 → 12CX normalized to the photon decay In Figs. 5 and 6, we also overlay black contours for ð ð Þ Þ width, using Eq. (95). For parameters that explain the Γð4Heð20.21Þ → 4HeXÞ normalized to the E0 decay width, beryllium anomaly, the carbon width is expected to be in using Eq. (113). We see that for vector X bosons that can the range explain the beryllium anomaly, the predicted value of the helium decay width is Γð12Cð17.23Þ → 12CXÞ 1 − 5 10−5Γ 12 17 23 → 12 γ Γ 4 20 21 → 4 1 − 11 10−2Γ ¼ð Þ × ð Cð . Þ C Þ ð Heð . Þ HeXÞ¼ð Þ × E0 −3 −5 ¼ð0.4 − 2.2Þ × 10 eV: ð116Þ ¼ð0.3 − 3.6Þ × 10 eV: ð115Þ The expected branching ratios for decays to X are similar This theoretical prediction overlaps with the experimen- for the beryllium and carbon cases, and an observation tally measured range of Γð4Heð20.49Þ → 4HeXÞ¼ð4.0 of these carbon decays, with the predicted branching −5 1.2Þ × 10 eV given in Eq. (4). The comparison at present ratio and consistent with mX ≈ 17 MeV, would provide is clouded by the experimental uncertainties regarding the overwhelming evidence for both the existing beryllium

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3 and helium anomalies and the protophobic vector boson treating N as a of p and H with mass M and explanation. spin J. However, for off-shell running, one should broaden the δ function into the resonance peak. This can be VIII. GENERAL SPIN-1 COUPLINGS approximated by taking the narrow width approximation in reverse, The 4He experimental setup, with its ability to run at off-resonance energies, has the potential to provide addi- 2 2 M Γ =π δ E − M → ; ð CM Þ 2 − 2 2 2Γ2 ð122Þ tional discriminating power among possible X parities. We ðECM MÞ þ M illustrate how this works by assuming a spin-1 X with both vector and axial vector couplings to quarks. Similar leading to the relation remarks apply to the case of a spin-0 X boson with both 4πð2J þ 1ÞΓ σ p 3 → N scalar and pseudoscalar interactions, although, as noted at ð þ H Þ¼ 2 − 2 2 2Γ2 the end of Sec. V, spin-0 states with mixed parity are now ðECM MÞ þ M 8 4 Γ → 3 strongly disfavored as explanations of the Be and He × ðN p þ HÞ: ð123Þ anomalies. Γ If the X boson is a mixture of vector and axial vector, it In this result, is the full width of the bound state, could generically be produced by both the 0− and 0þ 4He whereas the particular production mode may only be 0þ excited states. For the purposes of this analysis, we related to a particular partial width. The nearly always decays to protons, but the 0− decays to proton final states consider the true excited state N to be a linear combination of 0− and 0þ. The full production cross section is then split only 76% of the time [46]. We therefore obtain σ σ up into − and þ with σ 0 76Γ2 2 − 2 2 2 Γ2 − . − ðECM MþÞ þ Mþ þ ¼ 2 2 2 2 2 2 ; ð124Þ σ ≡ σ 3 → 0 σ Γ ðE − M−Þ þ M−Γ− ðp þ H Þ: ð117Þ þ þ CM 0 Since X is assumed to be produced through both states, where M is the nuclear mass of the excited state. the complete X production cross section is Putting everything together, we find

− σ Γ 0þ → Γð0 → XÞ Γð0þ → XÞ X ð XÞ σ ¼ σ− þ σ ; ð118Þ ¼ X Γ þ Γ σ 0 Γ 0 − þ E E 0 76Γ 2 − 2 2 2 Γ2 Γ 0− → . − ðECM MþÞ þ Mþ þ ð XÞ where Γ is the total width of the 0 excited state. Because þ 2 2 2 2 2 ; Γ ðE − M−Þ þ M−Γ− Γ 0 the experimental signal is normalized to the E0 electro- þ CM E magnetic transition, which occurs only through the 0þ ð125Þ σ state, we define E0 in a similar way as which relates the total number of X events observed at a Γ given E to the individual partial widths Γ 0þ → X σ σ E0 CM ð Þ E0 ¼ þ Γ : ð119Þ and Γð0− → XÞ. þ As can be seen in Fig. 1, by varying the proton beam energy 0þ Consequently, the ratio of the production cross sections is Ebeam, one can scan over the peak. By measuring the σ σ variation of X= E0 in such a scan, which requires the proper σ Γ 0þ → σ Γ Γ 0− → X ð XÞ − þ ð XÞ simulation of all relevant backgrounds, the two widths σ ¼ Γ þ σ Γ Γ : ð120Þ Γ 0 → X E0 E0 þ − E0 ð Þ can be extracted. In particular, if the ratio remains constant, it would suggest that Γð0− → XÞ¼0, implyingthat This ratio is proportional to the number of X decay events the X boson is a pure vector. Such a result would also exclude divided by the number of E0 events recorded by the the possibility of X being a pseudoscalar. experiment. The goal is to extract the X decay widths of the two individual states, with the E0 width and the two IX. CONCLUSIONS Γ total widths known experimentally. The more subtle quantity is σ−=σ . The transitions between energy levels of light nuclei are þ ∼ One might hope to obtain these cross sections from the a natural laboratory to explore MeV mass particles with widths by using the relation ultraweak interactions with the standard model. These nuclear transitions are able to probe models that are 4π2ð2J þ 1Þ interesting from the point of view of theories of dark σðp þ 3H → N Þ¼ ΓðN → p þ 3HÞ M matter and dark forces, and they provide information complementary to searches for such particles from accel- δ 2 − 2 × ðECM MÞ; ð121Þ erator and astrophysical sources.

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Several years ago, the ATOMKI group discovered a 33 34 ∶ Γ 4 20 21 → 4 resonance structure in the of eþe− pairs ATOMKI Experiment ½ ; ð Heð . Þ HeXÞ produced in the transition of an excited state of 8Be to its ¼ð2.8 − 5.2Þ × 10−5 eV: ground state. The results have so far defied plausible ð127Þ explanation through prosaic nuclear physics and are sug- gestive of a new particle with a mass around 17 MeV. Since The reported 7σ anomalies reported in 8Be and 4He then, the new particle interpretation has been bolstered by nuclear decays are both kinematically and dynamically the discovery of viable new particle explanations that are consistent with the production of a 17 MeV protophobic consistent with all other experimental constraints. More gauge boson. recently, the ATOMKI group has claimed to have con- What is the path forward? Clearly, now is the time for firmed their original results, both in 8Be, using new detector 4 other collaborations to perform the same nuclear measure- elements, and by new observations in a transition of He ments to check the ATOMKI results. But in this work, we whose kinematics point to the same 17 MeV mass required 8 also propose simple modifications of the ATOMKI setup to explain the Be results. that could provide incisive tests of the new particle In this work, we provide a comprehensive theoretical interpretation. The comparison between theory and experi- analysis of these anomalies by examining the dynamical ment will be sharpened considerably by including the E1 consistency of the size of the two observed excesses. We background in the experimental analysis and running on construct an effective field theory to describe the nuclear the 4Heð20.21Þ 0þ resonance. In addition, scanning through transitions of interest, and we examine the possibility that the 4Heð20.21Þ 0þ resonance can provide important infor- the new particle X is a scalar, pseudoscalar, vector, or axial mation to disentangle vector and axial vector X bosons vector boson coupling to and electrons. We find and quantify the properties of particles with mixed cou- that it is very difficult to simultaneously explain the 8Be and 4 plings. Last, we find that the protophobic vector boson He results with a spin-0 particle of either parity (or a could also be observable in the decays of the 12Cð17.23Þ 1− mixture). For the axial vector case, the EFT predicts that the excited state, and we have provided precise predictions for 8 4 ∼102 Be and He decay widths differ by a factor of ,in this rate. contrast to the observations, which find that they are The results of this study therefore lay out a roadmap for similar, but this discrepancy could conceivably be recon- possible future measurements that will further shed light on ciled by significant uncertainty in nuclear matrix elements. the ATOMKI anomalies. If the predictions are confirmed, Inthevector case, however, precisepredictions can bemade these measurements will provide overwhelming evidence that are independent of nuclear physics matrix elements by that a fifth force has been discovered. normalizing them to known electromagnetic transitions. The result is that if the X boson is a vector particle, the protophobic 8 4 ACKNOWLEDGMENTS couplings required by the Be results predict rates for the He transitions with no free parameters. The results are We thank Bart Fornal, Iftah Galon, Susan Gardner, and

4 4 Attila Krasznahorkay for helpful correspondence. This Protophobic vector boson∶ Γð Heð20.21Þ → HeXÞ work is supported in part by U. S. National Science ¼ð0.3 − 3.6Þ × 10−5 eV; Foundation Grant No. PHY-1915005. The work of J. L. F. and C. B. V. is supported in part by Simons Investigator ð126Þ Award No. 376204.

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