AA Standard Model Explanation for the “ATOMKI Anomaly”
A. Aleksejevs, S. Barkanova, Yu.G. Kolomensky,
2, 3 and B. Sheﬀ Grenfell Campus, Memorial University of Newfoundland, Corner Brook, NL, Canada Department of Physics, University of California, Berkeley, CA, USA Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA Department of Physics, University of Michigan, Ann Arbor, MI, USA (Dated: February 3, 2021)Using the e + e − pair spectrometer at the 5 MV Van de Graaﬀ accelerator at the Institute for NuclearResearch, Hungarian Academy of Sciences (ATOMKI), Krasznahorkay et al. have claimed a 6.8 σ excess at high e + e − opening angles in the internal pair creation isoscalar transition Be(18 . → Be e + e − . A hypothetical gauge boson with the mass circa 17 MeV, “X17”, has been proposed asan explanation for the excess. We show that the observed experimental structure can be reproducedwithin the Standard Model by adding the full set of second-order corrections and the interferenceterms to the Born-level decay amplitudes considered by Krasznahorkay et al . We implement adetailed model of the ATOMKI detector, and also show how experimental selection and acceptancebias exacerbate the apparent diﬀerence between the experimental data and the Born-level prediction. INTRODUCTION
The Standard Model (SM) of particle physicshas survived many challenges over the past ﬁftyyears. Occasionally, experimental observations dis-agree with the SM predictions; such deviations areoften reconciled after additional theoretical or exper-imental scrutiny. One such deviation is claimed byan experiment at the Institute for Nuclear Research,Hungarian Academy of Sciences (ATOMKI). In thisexperiment, a 5 MV Van De Graaf-accelerator wasused to produce an 18.15 MeV excited state of Be(18 .
15) (called Be ∗ henceforth) with a subse-quent internal pair conversion (IPC) to the groundstate of Be: Li( p, e + e − ) Be . This channel isof particular interest, as the creation of e + e − pairsprovides a potential avenue for detection of low-massboson candidates that are not present in the Stan-dard Model . A pair spectrometer was constructedto focus on detecting the IPC process with a ﬁne res-olution in the opening angle of e + e − pairs θ + − .Recent results from this experiment indicate anexcess of events at large θ + − over the leading-ordertheoretical predictions, consistent with a new bosonwith a mass of 16 . et al. proposed a new pro-tophobic, ﬁfth force gauge boson, which has sparkeda ﬂurry of additional model-building work and inter-est in popular press. In this paper, we discuss oureﬀort to critically analyze the idea that a non-SM,resonant process is needed to produce the observedexcess in e + e − production at high θ + − . In order tofully understand the interplay between the theoret-ical and experimental eﬀects, we construct a MonteCarlo (MC) model of the detector used at ATOMKI. We use this model to demonstrate that an interfer-ence between the Born-level IPC amplitude and asubleading nonresonant component (an amplitudewith a broad phase-space structure) could produceeﬀects observed by ATOMKI . Similar ideas havebeen explored by other authors [5, 6]. The most ba-sic subleading contribution to the IPC process arisesfrom the second order electromagnetic corrections, i.e. contributions beyond the Born approximation:two-photon box diagrams, vertex corrections, etc.While the calculation of the higher-order correctionsis technically challenging, one would naively expectsuch contributions to be suppressed by a factor of Zα compared to the Born term. However, the in-terference between the box and tree-level diagramsand the nontrivial structure of the box diagrams, in-cluding kinematic singularities and the presence ofexcited nuclear states in the box, can produce un-expected eﬀects. In this paper, we report the fullsecond-order calculation of the Be ∗ → Be e + e − process, and compare the results to the experimen-tal distributions reported by ATOMKI. NLO QED MODEL OF Be ∗ DECAY
In the rest frame of Be ∗ , the doubly-diﬀerentialdecay rate comprises from a phase-space componentand a square of the matrix element, which has aleading-order (LO) contribution | M LO | and an in-terference term between LO and next-to-the-leading(NLO) matrix element, 2 (cid:60) [ M LO M ∗ NLO ]: d Γ dθ + dθ − = (cid:16) | M LO | + 2 (cid:60) [ M LO M ∗ NLO ] (cid:17) Φ , (1) a r X i v : . [ h e p - ph ] F e b where details on phase space element Φ are given insupplemental part of the paper.At the energy scale of the ATOMKI experiment,an eﬀective operator approach can be employed,with Be ∗ and Be as the fundamental degrees offreedom. Speciﬁcally, for the transition Be ∗ → Be + γ , we can write: L (1) = e Λ γ (cid:15) µναβ B ∗ µν F αβ B, (2)where B ∗ µν = ∂ µ B ∗ ν − ∂ ν B ∗ µ and F αβ = ∂ α A β − ∂ β A α are the usual ﬁeld strength tensors for Be ∗ andelectromagnetic ﬁelds, respectively. The constantΛ γ is related to the transition nuclear matrix el-ement and is used as a scaling parameter. How-ever, Eq. (2) is only describing the tree-level tran-sition, which is insuﬃcient to explain a small peakat ∼ . α to the next-to-leading-order (NLO) QED contributions, which re-quires one-loop calculations. The four categories ofFeynman graphs corresponding to α transitions areshown on Fig. 1. Group (a) in Fig. 1 is the vac-uum polarization contributions, which we treat bysplitting the leptonic and hadronic contributions andusing eﬀective masses of light quarks to calculatethe hadronic contributions. Group (b) in Fig. 1)is the electron current vertex corrections. Groups(c) and (d) are represented by boxes with Be and Be ∗ in the loop, respectively. The diagrams (b)-(d)are infrared-divergent. To treat infrared (IR) diver-gences in the vertex correction graph (c), we use thesoft-photon approximation in bremsstrahlung dia-grams and introduce a cut on energy of soft photonsgiven by the threshold conditions in the electron-positron pair production. For the boxes, we takeinto account only the infrared-ﬁnite part. SinceATOMKI does not distinguish between the electronsand positrons, we sum over events in which eitherelectron or positron are observed in the same detec-tor element in the MC simulation. Due to the CPsymmetry of the underlying QED lagrangian, thiscancels out the IR-divergence in the box diagrams.Besides the QED transition Be ∗ → Be + γ , inthe box diagrams, we also use L (2) = − ieZ (cid:2) B ∗ µν B ∗ µ A ν − B ∗ µν B ∗ µ A ν + B ∗ µ B ∗ ν F µν (cid:3) (3)to describe a coupling Be ∗ → Be ∗ + γ , and L (3) = − ieZ [ B∂ µ B − ∂ µ BB ] A µ (4)to introduce coupling Be → Be + γ . In both la-grangians, Z is equal to four. Calculations are done in several stages. Based on the Llagrangian densi-ties in Eqs. (2-4) we develop the model ﬁle, whichwe used in FeynArts  to generate one-loop topolo-gies and produce matrix elements, which we evaluateusing Passarino-Veltman basis in the FormCalc package. Numerical calculations are carried out withthe help of LoopTools  and conﬁrmed by Collier packages.The results of the higher-order QED correctionsfor the cases when one of the angles (either elec-tron or positron) is ﬁxed are given in Fig. 2. Self-energies and vertex corrections are completely sym-metric with respect to electron-positron ﬂip. On thecontrary, boxes show asymmetrical behavior and al-most cancel out each other when we combine theelectron and positron events. The look-up tables forthe MC simulations are available upon request. MONTE CARLO SIMULATION
The momenta and angles of the ﬁnal state par-ticles Be, e + , and e − are generated over the fullthree-body phase space with the probability propor-tional to the diﬀerential decay rate in Eq. (1). Thekinematics are taken in the rest frame of Be ∗ , ne-glecting its boost due to the initial proton momen-tum. e + and e − are assumed to be indistinguish-able in the detector, and we ignore the 4-vector of Be in the subsequent steps. The ﬁnal momentaof e + e − are then fed into the detector simulationand weighted accordingly. We then histogram overthe opening angle and over the invariant mass ofthe e + e − pair and compare against the experimen-tal data.The ATOMKI detector model was constructedbased on description of . We assume each de-tector block to be a rectangle in θφ space withperfect eﬃciency. The center of each rectangle isplaced at φ = 0 , π/ , π/ , π/ , π/ e + e − pairs, and matching the angular accep-tance of the e + e − pairs to the released detector re-sponse curve published , as shown in Fig. 3. Themodel represents the measured detector acceptancevery well for angles larger than 40 ◦ , with compa-rable variance from data to the published expecteddetector response. The ﬁnal block sizes are set to∆ θ = 0 .
59 rad and δφ = 0 .
58 rad, with the uncer-tainty of roughly 0 .
02 rad in each dimension. Thesevalues are then applied to the MC simulation by ap-plying a 0 weight to any e + e − pair that includes atleast one particle not at an angle that crosses one of γγ γγ γγ γ γγγ γ γBe ∗ Beee ( a ) ( b ) ( c ) ( c )( d ) ( d ) FIG. 1: Higher order QED contributions in decay of Be ∗
10 12 14 16 18 - - - ω [ MeV ] R e | M L O M N L O * θ - = o
10 12 14 16 18 - - - ω [ MeV ] R e | M L O M N L O * θ + = o FIG. 2: The results for the interference term in Eq. (1)(2 (cid:60) [ M LO M ∗ NLO ]) for electron (top) and positron (bot-tom) angles ﬁxed to θ − ( θ + ) = 40 ◦ . Dashed line (yel-low) represents vacuum polarization graphs (a), dot-dashed line (green) is the result of vertex correctiongraph (b) with soft-photon bremsstrahlung treatment.Dotted graph (blue) is the IR-ﬁnite part of the boxes(c)-(d). the detectors.Based on , an additional cut is applied tothe detected particle pair, removing particularlyasymmetric e + e − pairs. The cut was appliedto only allow | y | < .
5, where disparity y = FIG. 3: Comparison of simulated and measured detectorresponses to uniformly distributed particle pairs, alongwith the published expected response. χ calculatedwith all standard errors assumed to be 1 for compari-son of relative variance of new model and ATOMKI’smodel ( E e − − E e + ) / ( E e − + E e + )We note that while e + and e − are indistinguish-able in the ATOMKI detector, the response of theplastic scintillator to electrons and positrons is dif-ferent . It is not clear if this diﬀerence was incor-porated in the original calibration of the ATOMKIapparatus . We approximate the diﬀerence in re-sponse by adding an additional 1 MeV of visible en-ergy to the positron signal before applying the dis-parity cut. This introduces a small asymmetry inthe detector response. ANALYSIS
We ﬁt the simulated distributions of e + e − open-ing angles to the ATOMKI data using a binned χ ﬁt by varying two parameters: the overall normal-ization of the Born contribution, and the coeﬃcientof the interference term between the Born and thesecond-order diagrams. The sign and the magnitudeof the interference term are allowed to ﬂoat, allow-ing for a possible phase shift between the two terms.The best ﬁt reproduces the ATOMKI spectrum witha χ = 38 for 7 degrees of freedom. We note that thelargest χ contribution comes from the lowest θ + − bin, where our assumption of the uniform detectoreﬃciency may be incorrect. The best-ﬁt value forthe interference coeﬃcient is (cid:15) = 57 . ± .
06, indi-cating that the box diagram propagator may havecontributions from more than one J P = 1 + state( e.g. FIG. 4: Results of a Monte Carlo simulation usingunderlying particle distributions from the calculationsabove and applying detector acceptance and disparitylimiting cuts. Shown are the (orange dashed) tree levelcalculation, (green dot-dashed) loop-level, (blue solid)their ﬁtted sum, as compared to (purple dotted) the pub-lished result  and (red points) the data collected in thesame source.
With a reasonable agreement between the SM ﬁtand the experimental data in the X17 ”signal” regionat the large values of θ + − , we compute the e + e − invariant mass distribution. First, for each of thetwo angular distributions used in the ﬁt, a histogramof the e + e − invariant mass is produced. We usethe X17 contribution reported in  as the detectormass resolution at ¯ m = 16 . m ( m ) = f (cid:18) m − m m ¯ m + ¯ m (cid:19) (5) where m is the true e + e − mass and m is the mea-sured mass. The theoretically predicted mass distri-bution is then smeared with this resolution function.The resulting mass distribution is shown in Fig. 5.We do not attempt to make a quantitative compar-ison with the ATOMKI mass distribution , sinceit depends critically on the detector energy resolu-tion function, detection eﬃciency as a function of e − and e + energies, and details of the detector calibra-tion. Such details are not provided in the experi-mental publications [1, 3]. However, we note a strik-ing qualitative feature: a bump-like structure in thepredicted invariant mass distribution that does notrequire a new boson. This shows how non-resonantSM processes can combine to produce a false bumpin the detector. FIG. 5: Distribution of electron positron pair cre-ation over the invariant mass. Signiﬁcant peak structureshown arising from interference between loop and treelevel eﬀects. Relative magnitudes of the components aretaken from the ﬁt in Fig. 4.
Angular Distribution Bias
Of interest in this investigation are the origins ofthe bump-like structure in the angular and mass dis-tributions in Fig. 4-5. Much of this eﬀect appearsto arise from non-uniformities in the angular accep-tance of the ATOMKI spectrometer. Running thesimulation with complete 4 π acceptance, and settingthe interference coeﬃcient to be the same as in theabove ﬁt, we arrive at the results shown in Fig. 6-7.As shown, the bump structure is highly suppressed,implying that it is a result of an interplay betweensystematic biases in the detector and the eﬀects ofloop diagrams. FIG. 6: Angular separation distribution with detectorset to full 4 π acceptance. Energy disparity cuts still in-cluded. Shown are the (orange dashed) tree level calcula-tion, (green dot-dashed) loop-level, and (blue solid) theirsum. Relative magnitude of each component is takenfrom the ﬁt found in Fig. 4.FIG. 7: Mass Distribution with the detector set tofull 4 π acceptance. Energy disparity cuts still included.Shown are the (orange dashed) tree level calculation,(green dot-dashed) loop-level, and (blue solid) their sum.Relative magnitude of each component is taken from theﬁt found in Fig. 4 CONCLUSIONS
We show that the ATOMKI data  can be rea-sonably described by taking into account the second-order contributions to the IPC process, and account-ing for detector and analysis bias. As these are non-resonant, Standard Model processes, they do not re-quire the addition of a new boson. Clearly, an ele-ment of caution must be taken before conﬁdence can be placed in an entirely new particle having beendiscovered.An independent measurement of the IPC spectrain the Be and other isoscalar nuclear systems isclearly called for. Such measurements should aimto measure the IPC distributions with large accep-tance, minimal detector bias, and ideally better in-variant mass resolution, in order to be able to dis-criminate between a resonant and a non-resonantcontributions. Tagging the initial nuclear state inorder to eliminate initial-state interference eﬀects would also be of value.
We would like to thank the members of UC Berke-ley and Lawrence Berkeley National Lab (LBNL)Weak Interactions Group for their support in thiswork, and Gerald Miller and Xilin Zhou for stim-ulating discussions. This work was supported inpart by the Natural Sciences and Engineering Re-search Council of Canada (NSERC), by the NationalScience Foundation (NSF), and by the US Depart-ment of Energy (DOE) Oﬃce of Science. B. Sheﬀthanks UC Berkeley Institute for International Stud-ies Merit Scholarship Program and UC BerkeleyPhysics department for their support. A. Alekse-jevs and S. Barkanova would also like to thank UCBerkeley and LBNL for hospitality and support.  A.J. Krasznahorkay et al. , Phys. Rev. Lett., ,042501 (2016). J.L. Feng et al. , Phys. Rev. Lett. , 071803(2016). J. Guly´as et al. , Nucl. Instrum. Methods Phys. Res.A , 21 (2016). B. Sheﬀ, Honors Thesis, UC Berkeley, 2017 (unpub-lished). P. K´alm´an and T. Keszthelyi, Eur. Phys. J. A ,205 (2020). X. Zhang and G. A. Miller, Phys. Lett. B , 159(2017). T. Hahn, Comput. Phys. Commun. , 418 (2001). T. Hahn and M. Perez-Victoria, Comput. Phys.Commun. , 153 (1999). A. Denner, S. Dittmaier and L. Hofer, Comput.Phys. Commun. , 220 (2017). W.J. Meiring, J. van Klinken, and V.A. Wichers,Phys. Rev. A , 2960 (1991). APPENDIX
The phase space for doubly diﬀerential decay rate in Eq. (1) can be written as follows:Φ = 1(2 π ) J m Be ∗ E − E Be sin θ − sin θ + and J = sin ( θ + + θ − )2 R sin θ − (cid:20) m Be ∗ (cos ( θ + + θ − ) − (cid:18) m Be ∗ sin ( θ + + θ − )tan θ − − R (cid:19) − m Be (cid:21) ,R = (cid:26) m Be + sin θ + sin θ − (cid:18) m Be ∗ + m Be sin θ + sin θ − (cid:19) + (cid:0) m Be ∗ − m Be (cid:1) sin (2 θ + + θ − )sin θ − (cid:27) / , where subscripts + / − correspond to the positron and the electron, respectively. ���� (cid:1) �� Φ [ Ev ] FIG. 8: Phase space of three body decay for Be ∗ → Be e + e − . The phase space of Be and the electron-positron pair in the ﬁnal state is shown on Fig. 8, demonstratingthe characteristic three-body decay behavior. The correlation between the invariant masses in the Dalitzplot plane and the e + e − opening angle is shown in Fig. 9. The matrix elements projected on the 3-bodyphase space Dalitz plot are shown in Fig. 10. FIG. 9: Separation angle between electron and positron as a function of invariant mass of the e + e − system and the e + 8 Be systemFIG. 10: α order matrix element contribution as a function of invariant mass of the e + e − system and the e + 8+ 8