New physics via pion capture and simple nuclear reactions
NNew physics via pion capture and simple nuclear reactions
Chien-Yi Chen,
1, 2, 3, ∗ David McKeen, † and Maxim Pospelov
1, 2, ‡ Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2, Canada Perimeter Institute for Theoretical Physics, Waterloo, ON N2J 2W9, Canada Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada (Dated: June 5, 2019)Light, beyond-the-standard-model particles X in the 1-100 MeV mass range can be produced innuclear and hadronic reactions but would have to decay electromagnetically. We show that simpleand well-understood low-energy hadronic processes can be used as a tool to study X productionand decay. In particular, the pion capture process π − p → Xn → e + e − n can be used in a newexperimental setup to search for anomalies in the angular distribution of the electron-positron pair,which could signal the appearance of dark photons, axion-like particles and other exotic states.This process can be used to decisively test the hypothesis of a new particle produced in the Li + p reaction. We also discuss a variety of other theoretically clean hadronic processes, such as p + D(T)fusion, as a promising source of X particles. I. INTRODUCTION
Extensions of the Standard Model (SM) by light andweakly coupled particles such as axions, axion-like par-ticles, dark photons, etc., have been recognized as ageneric possibility. The particle physics community hasevolved towards a very systematic approach to light be-yond the SM (BSM) states, as several new experimentshave been proposed and the results of old experiments re-analyzed [1, 2]. Some of the measurements have delivered“anomalous” results that may signify poorly understoodSM physics, experimental problems, or indeed the exis-tence of light, weakly coupled states. It is important tocheck the origin of such anomalies, and perhaps investi-gate the simplest BSM explanations, as was done in thecase of the muon g − X , the exact conservation of theSM current X couples to (including cancellation of allanomalies) seems to be imperative for the constructionof models that avoid (weak scale) /m X enhancement ofthe production amplitudes (see, e.g., [7, 8]). These the-oretical rules could be “bent” at times (often signalingthat some significant amount of fine-tuning needs to betolerated), when a model can be considered as a candi-date to solve an outstanding anomaly. This is exactlywhat has happened with the recently claimed anomalyin the angular distribution of e + e − pairs produced in the ∗ [email protected] † [email protected] ‡ [email protected] proton capture by Li [9]. The experiment detected anunexpected dependence of pair yield on the relative an-gle θ between the e + e − pairs. The anomaly manifestsitself for the 18.15 MeV intermediate state of Be, and isdifficult to explain from the point of view of conventionalnuclear physics approaches [10]. Instead, a hypothesisof an intermediate particle of mass (cid:39)
17 MeV has beenput forward [9], that would modify the angular distribu-tion in the desired way. The simplest models of vector X coupled to quark currents have been suggested as anexplanation [11–14], which all seem to require some tun-ing of parameters and/or physical amplitudes to avoid m − X -enhanced processes [8]. Nevertheless, this so-called“beryllium anomaly” is intriguing enough to study fur-ther.The goal of the present paper is as follows. We wouldlike to argue that the simplest nucleon-involved processescan be used to search for light weakly coupled states, andcheck for the presence or absence of light resonances inthe sub-20 MeV region. A priori , the dynamics of eightnucleons inside Be can be quite complicated, and despitethe study of [10], one might not be able to currently con-clude with all degree of certainty that the anomaly is notexplained by conventional nuclear physics. On the otherhand, processes that involve fewer nucleons can be wellunderstood, and the loophole of “nuclear physics compli-cation” would not exist in such systems. To that end,we would like to study the simplest hadronic reactions,such as pion capture on protons as a possible source of X particles. We would also like to point out other promis-ing reactions, such as proton fusion with deuterium ortritium as a source of exotic 17 MeV boson. In a lessspeculative vein, we forecast sensitivity to dark photonsand other exotic bosons in a quasi-realistic pion captureset-up with attainable pion intensities.Previously, the π − p → e + e − n reaction has been stud-ied by the SINDRUM collaboration [15], along with thepair production from π produced in the charge-exchangereaction. The resulting constraints on dark photon pa-rameter space were derived in Ref. [16]. Unfortunately, a r X i v : . [ h e p - ph ] J un the experimental setup in [15] did not allow probing pairswith invariant mass less than 20 MeV, and therefore thispast experiment cannot place constraints on a hypothet-ical 17 MeV particle.Pion capture on a proton is a powerful, robust way tosearch for the production of new, light, weakly coupledbosons. In the SM, capture of a negatively charged pionon a proton leads to the production of neutron which,to conserve momentum, recoils against a neutral bosonwith a mass less than about m π ± = 139 . π . This should be contrasted with capture onnuclei. In that case, the capture process typically ejectsa nucleon from the nucleus. The nucleon can then recoilagainst the nuclear remnant, thereby conserving momen-tum. As a result, fewer light, neutral bosons are releasedper event in π − capture on nuclei as compared to cap-ture on a proton. In addition, capture on a proton hasan additional advantage compared to that on a nucleus:less theoretical uncertainty in the calculation of rates be-cause capture on a proton can be well understood in thecontext of a chiral effective Lagrangian. (Pion capture ondeuterium can also be useful: while the yield of photonsand new physics states are only a factor of ∼ π in the finalstate may prove to be advantageous in reducing radiativebackgrounds.)New, weakly coupled bosons, X , with masses m X (cid:46) m π ± can also be produced in pion capture on a proton, π − p → Xn , if they couple to hadrons (or quarks). Pro-duction of these exotic bosons is relatively enhanced in π − capture on a proton as compared to nuclei for thesame reason as π and γ production. Estimation of therates of X production are also theoretically cleaner in thenucleon system.This paper is organized as follows. In the next sectionwe review the radiative and pair-production pion capturein the SM. In Sec. 3 we provide relevant formulae for cal-culating the rate of X production in the pion capture,where X is either a dark photon or “protophobic” vec-tor [11]. In the same section, we estimate the sensitivitythat can be achieved with modern sources of negativepions. In Sec. 4, we discuss additional nuclear physicschannels (such as deuterium-proton fusion) that can beused in a new experiment looking for X . We reach ourconclusion in Sec. 5, and generalize our results to axion-like particles in the Appendix. II. RADIATIVE PION CAPTURE AND PAIRPRODUCTION IN THE STANDARD MODEL
Historically, the study of pion capture, the exother-mic π − p reaction, was very important for understandingthe simplest hadronic processes [17, 18]. To begin, wecalculate the cross section for a π − bound to a protonto produce a photon and a neutron. We will use theprediction of this rate (and corresponding experimentalmeasurement) to normalize the production rates of exotic π − p γn pp π − π − γn nγ FIG. 1. Diagrams responsible for the reaction π − p → γn fromthe interactions in Eq. (1). states later on.This process is described well in the low-energy chiralLagrangian involving nucleons and pions. The relevantterms for this process read L ⊃ (cid:12)(cid:12) ( ∂ µ + ieA µ ) π − (cid:12)(cid:12) − m π π + π − + ¯ p ( i (cid:54) ∂ + e (cid:54) A − m N ) p + ¯ n ( i (cid:54) ∂ − m N ) n − g A √ F π ¯ nγ µ γ p ( ∂ µ + ieA µ ) π − , (1)where g A = 1 .
275 [19] is the nucleon axial vector couplingand F π = 92 MeV [20] is the pion decay constant. We donot differentiate between proton and neutron mass, ne-glect particle form-factors and anomalous magnetic mo-ment contributions. There are three separate amplitudes,shown in Fig. 1, that contribute to the process π − p → γn .A straightforward calculation of the cross section forthis process gives( σv ) π − p → γn = αg A F π m π / (2 m N )1 + m π /m N = 5 . × − cm × c. (2)The branching fraction for radiative pion capture canbe computed using the measured value of the Panofskyratio [21], P = ( σv ) π − p → π n ( σv ) π − p → γn = 1 . ± . , (3)which relates the strength of this mode to the other finalstate, π n . From this one finds a radiative branchingBr π − p → γn = 11 + P = 0 . ± . . (4)Calculation of the rate can be contrasted with the mea-surement of the 1 s width [22], which can be converted tothe capture rate,( σv ) meas π − p → γn = Γ meas1 s Br π − p → γn × | ψ s (0) | − (cid:39) . × − cm × c, (5)with ∼
3% experimental error. The accuracy of our the-oretical result in Eq. (2) is within ∼
10% of this measure-ment, which is rather good given crude nature of some ofthe approximations. This accuracy is entirely sufficientfor our purposes of studying new physics.We also calculate the rate for the π − p → e + e − n process, given the Lagrangian of Eq. (1). The re-sult, in terms of the invariant mass of a pair, m ee , for π − p Vn pp π − π − VV nn
FIG. 2. Diagrams responsible for the reaction π − p → γn fromthe interactions in Eq. (1). m e /m π (cid:28) d ( σv ) π − p → e + e − n ( σv ) π − p → γn = 2 α π dm ee m ee × f (cid:18) m ee m N , m π m N (cid:19) (6)with f ( x, y ) = (cid:115) − x y (cid:0) − z (cid:1) / (1 − xz ) (1 − xz/y ) × (cid:26) − z y (cid:20) − y − x y (cid:0) − y − y + x (cid:1)(cid:21)(cid:27) (7)and z = x/ (2+ y ). This function is defined so that f → m ee (cid:28) m π . The logarithmic term, ∝ dm ee /m ee , isperfectly consistent with the classic paper by Kroll andWada [23]. The remaining integral can be performed tofind the full probability of emitting a pair relative to emit-ting a photon, which is dominated by small m ee and isabout 7 . × − . In practice, the relative angle betweenfinal state charged particles, rather than their invariantmass, represent a more convenient variable. III. PRODUCTION OF NEW BOSONS IN PIONCAPTURE
Having calculated the rate for radiative π − captureon a proton, we would now like to estimate the rates toemit new, weakly coupled bosons, X , in the same process.We begin with the well-studied “dark photon,” X = A (cid:48) .The A (cid:48) is a vector boson with mass m A (cid:48) that kineticallymixes with the photon with strength (cid:15) . Its couplings toSM particles are then the same as the photon’s timesthe mixing strength. We can therefore use Eq. (1 withthe replacement A µ → (cid:15)A (cid:48) µ to describe the interactionsof the dark photon with hadrons. (We do not considerthe m A (cid:48) → π − p → A (cid:48) n ,relative to that of standard photons is then( σv ) π − p → A (cid:48) n ( σv ) π − p → γn = (cid:15) f (cid:18) m A (cid:48) m N , m π m N (cid:19) , (8)with f from Eq. (7) describing the phase space depen-dence of the process.As another benchmark, we take the recently proposed“protophobic” vector boson [11] that can explain the ex-cess e + e − events seen in nuclear transitions of Be [9]. � �� �� �� �� ��� ��� ��������������������������������� � [ ��� ] � � � � � � � �� � � �� � [ % ] ���� ����������������� ������ ϵ = �� - � FIG. 3. The branching ratios for π − p → A (cid:48) n (dark photon)and π − p → V n (protophobic vector) as functions of the vectormasses. We have taken (cid:15) = 10 − in both cases. In this setup, a new vector boson X = V , with a mass m V (cid:39)
17 MeV is introduced that couples with strength (cid:15) × e (choosing this normalization to make contact withthe dark photon case) to neutrons and negative pions butnot to protons. Note that the coupling of V to n and π − has the same sign. In addition, V has a separate couplingto electrons that mediates its prompt decay to e + e − .To calculate the rate of V production in pion capture,we need its couplings to hadrons. These can be easilylifted from the Lagrangian of Eq. (1) with suitable ad-justments of the covariant derivatives. The resulting in-teractions are described by L ⊃ (cid:12)(cid:12) ( ∂ µ − i(cid:15)eV µ ) π − (cid:12)(cid:12) − m π π + π − + ¯ p ( i (cid:54) ∂ − m N ) p + ¯ n ( i (cid:54) ∂ + (cid:15)e (cid:54) V − m N ) n − g A √ F π ¯ nγ µ γ p ( ∂ µ − i(cid:15)eV µ ) π − (9)As shown in Fig. 2, the amplitude to produce a pho-tophobic vector, π − p → V n , contains three terms, as inthe (dark) photon case. However, in this case, insteadof emission off of a proton, one term involves emissionfrom the neutron. Adding these contributions together,the capture rate with the photophobic vector productionis found to be( σv ) π − p → V n ( σv ) π − p → γn = (cid:15) (1 + m π /m N ) g (cid:18) m V m N , m π m N (cid:19) , (10)where the phase space function is given by g ( x, y ) = (cid:115) − x y (cid:34) − (cid:18) x y (cid:19) (cid:35) / × (cid:40) (cid:20) xy (1 + y )(2 + y ) y − x (cid:21) (cid:41) . (11)The branching ratios for π − p → A (cid:48) n (dark photon)and π − p → V n (protophobic vector) can be obtainedby multiplying Br π − p → γn in Eq. 4 by Eqs. 8 and 10,respectively. We show the branching ratios as functionsof the vector masses in Fig. 3, where we have taken (cid:15) =10 − in both cases. IV. EXPERIMENTAL CONCEPT
The signature of a new boson that decays to e + e − produced in π − capture (both the dark photon and pro-tophobic vector considered above) is the production of an e + e − pair with an invariant mass peaked at m ee = m X .Since the sum of the electron and positron energies isfixed to be close to m π , a peak in the invariant mass alsotranslates into a sharp feature in the opening angle of thepair. A hypothetical particle invoked as an explanationof the anomaly in Be ∗ decay would be seen in pion cap-ture as a sudden increase of lepton pairs with openingangles larger than ∼
14 degrees.This signal sits on top of Standard Model processesleading to a pair-creation. In particular, Dalitz decays offinal state π , π → γe + e − would constitute a significantsource of background, which can however be removed byrequiring E e + + E e − > m π /
2. Another source of back-ground, the photon conversion in material, following theradiative capture can also be controlled using experimen-tal means and requiring a symmetric distribution of e + and e − energies. There is, however, one source of an irre-ducible SM background from pion capture that producesan off-shell photon, π − p → γ ∗ n → e + e − n . Because ofthe photon pole in the process, the SM invariant mass dis-tribution is a monotonically decreasing function of m ee ,given in Eq. (6).Ignoring the electron mass, m ee (cid:39) E + E − (1 − cos θ )with E + and E − the energies of the e + and e − respec-tively and θ the opening angle between them. Therefore,the precision reconstructing m ee is affected by the finiteenergy and angular resolution of any experimental setup.We can estimate the reach of a “bump hunt” in m ee byconsidering the number of signal events in a bin of width δ ee centered on m ee = m X which is roughly N sig ∼ (cid:15) N cap Br π − p → γn , (12)with N cap the number of π − captures and (cid:15) the X cou-pling in units of the positron charge. The size of the bin δ ee is determined by the experimental resolution that canbe achieved for E ± and θ . The number of backgroundevents in the bin centered around m X can be found usingEq. (6) N bkg ∼ α π δ ee m X N cap Br π − p → γn , (13)where we have ignored terms of O ( δ ee /m X ).Requiring that the number of signal events in this bin islarger than a 3 σ statistical fluctuation of the background,one can estimate the values of (cid:15) that can be probed, which scales as ( N cap ) − / , (cid:15) (cid:38) . (cid:18) δ ee /m X (cid:19) / × (cid:18) N cap (cid:19) / ,N sig (cid:38) N / (14)The size of the bin near m ee = m X , δ ee , is importantin controlling the sensitivity to X production. This isdetermined by the experimental resolution on m ee whichdepends on the resolution measuring the e ± energies andopening angle. A well known difficulty arises from the fol-lowing: E + and E − are unlikely to be measured very pre-cisely using calorimetric tools. But if the measurement ofmomentum via tracking of charged particles in the mag-netic field is employed, then the rescattering of leptonsin the tracking layers lead to the significant broadeningof θ . Large values of δ ee would necessarily bring largenumber of the background events in the correspondingbin.In practice, obtaining large rates while keeping the en-ergy resolution small enough to have good precision on m ee is a challenge. (For example, a ∼
30% resolution inenergy, would result in δ ee /m X ∼
15% even with a per-fect measurement of θ .) In contrast, the angular resolu-tion can typically be kept under control; for this reason θ can be a useful discriminant for distinguishing signal fromirreducible background, as taken advantage of in [9].For e + e − production through an on-shell boson X → e + e − produced in pion capture, there is a lower boundon the opening angle at θ min = cos − (cid:0) − m X /m π (cid:1) = 2 m X m π + O (cid:18) m X m π (cid:19) . (15)The background through an off-shell photon has no suchcutoff and is peaked toward θ = 0. The intrinsic broad-ening of θ may come from various sources. The uncer-tain position of pion capture within a target leads to ageometric uncertainty in θ . The thickness of the target(and in case of a cryogenic H target, the thickness ofwalls around the target) contributes to multiple Coulombscattering. We estimate that an electron and positronpair with individual energies ∼ m π / θ by∆ θ ∼ . (such targetdimensions are sufficient to stop π − with p π ≤
70 MeV).If a detector registering leptons is placed 1 m away fromthe target, the geometric broadening is small. Thus, webelieve that the resolution of 1 ◦ on θ can be achieved.To understand the potential sensitivity of a pion cap-ture experiment more qualitatively, we simulate bothbackground and signal from a protophobic vector in (9)decaying to e + e − with m V = 17 MeV. We show theresulting m ee distribution with (cid:15) = 0 . e + and e − energies and a momentumdirection resolution of 1 ◦ in Fig. 4. We impose a cut E + + E − > m π / π ’s. We also show the opening angle distribu-tion of both signal and background in Fig. 4, requiring m ee [ MeV ] d B r / d m ee [ - M e V - ] θ d B r / d θ [ - ] FIG. 4. The distribution of e + e − invariant mass (top) andopening angle (bottom) in π − capture on a proton due toboth background through an off-shell photon (gray) and back-ground plus the signal of a protophobic vector (black), cf.Eq. (9). We take m V = 17 MeV as suggested by the Beanomaly and (cid:15) = 0 .
1. We assume a 1 ◦ resolution on the e ± direction and a 50% resolution on their energies. that − . ≤ E + − E − E + + E − ≤ . π − capture on a protonfor 10 captures (corresponding to ∼ m V that the e + e − opening angle satisfies θ − θ min ( m V ) ∈ [ − ◦ , +4 ◦ ].We further enforce E + + E − > m π / − . ≤ ( E + − E − ) / ( E + + E − ) ≤ . ◦ energy andangular resolutions, respectively. The results of this pro-cedure are shown in Fig. (5). This minimum value of (cid:15) that can be probed is determined by the requirement thatthe number of signal events with this selection is larger ★★ - - m V [ MeV ] ϵ %× π %× π %× π FIG. 5. Minimum values of the coupling of a protophobicboson that can be probed through production in π − captureon a proton for 10 captures, an e ± energy resolution of 50%and 1 ◦ resolution on the direction of the e ± momenta. Foreach mass point we require the reconstructed e + e − openingangle is within [ − ◦ , +4 ◦ ] of θ min ( m V ) and − . ≤ ( E + − E − ) / ( E + + E − ) ≤ .
5. The reach is determined by requiringthe number of signal events is larger than a 3 σ statisticalfluctuation of the background. The solid, dashed, and dottedblue lines show the reach for detectors that have 10%, 20%,and 100% coverage of the full 4 π solid angle, respectively.The green dot shows the coupling and mass able to explainthe Be anomaly [11]. than a 3 σ statistical fluctuation of the background. Weshow the reach for detectors that have 10%, 20%, and100% coverage of the full 4 π solid angle along with theparameters that explain the Be anomaly [11]. (A re-quired value of (cid:15) in the protophobic case can be esti-mated via v − V × (Γ V / Γ γ ), where v V (cid:39) .
35 is the finalstate velocity of the suggested 17 MeV particle, and ra-tio of exotic-to-gamma emission Γ V / Γ γ in the decay ofthe 18.15 MeV state is the claimed experimental value of5 . × − [9].) Clearly, larger boson masses require a de-tector that can cover larger solid angles since they movewith smaller velocity in the lab frame so that the e + e − pair is less boosted and emitted closer to back-to-back. V. EXOTIC BOSONS IN NUCLEARREACTIONS
Pion capture represents perhaps the simplest hadronicreaction where a new boson can be explored or con-strained without much complication of nuclear physics.At the same time, powerful sources of π − (with sub-100MeV momentum so that they can be stopped) exist onlyin a handful of laboratories around the world. It is thenimportant to consider other very simple hadronic reac-tions, that involve up to 4 nucleons, and where nuclearphysics may also be considered “under control.”One can discuss the following as candidate radiativereactions, p + n → D + γ ( X ) , Q = 2 . , (17)D + p → He + γ ( X ) , Q = 5 . , (18)D + n → T + γ ( X ) , Q = 8 . , (19)T + p → He + γ ( X ) , Q = 19 . , (20) He + n → He + γ ( X ) , Q = 20 . , (21)where instead of a γ one could emit an exotic boson X .The last two reactions have enough energy to search fora 17 MeV state even for a small energy of incoming par-ticles, while the first three reactions would require some-what energetic beam of protons or neutrons.On the other hand, if a beam of energetic protons upto a few tens of MeV is available, the D + p → He + γ ( X ) reaction is an ideal testing ground for the search ofexotic bosons. The protons provide a powerful tool forsuch a search due to the potentially much larger statisticsrelative to pion capture. In addition, this process is inprinciple calculable to a great degree of accuracy [24], sothat the complications of nuclear theory are not an issue.Moreover, pair production via D + p → He + e + e − hasbeen successfully studied experimentally in the past [25].It is easy to relate the energy of a proton beam, E p ,in the lab frame to the maximum available energy in thecenter-of-mass frame for the emission of X , E max = E p m D m D + m p + Q = 23 E p + Q. (22)For example, a proton of energy 20 MeV capturing onD corresponds to E max = 18 . m X = 17 MeVmass.Assuming the dominance of the E X to the emissionof γ in the D( p, γ ) He reaction. We start with the darkphoton, X = A (cid:48) . In the assumption that the matrixelements of longitudinal current are the same as for thetransverse current (see, e.g., [23] for the details of suchseparation, and recent calculations of e + e − emission fromnuclear reactions in [26]), the rate is rather simply relatedto the photon rate, σ D+ p → He+ A (cid:48) = (cid:15) σ D+ p → He+ γ × v A (cid:48) (3 − v A (cid:48) )2 . (23)Here v A (cid:48) = (1 − m A (cid:48) /E max ) / is the velocity of theoutgoing A (cid:48) particle in the center-of-mass frame. In thelimit of the small mass, the rate into dark photons issimply (cid:15) from the rate to the regular photons. Goingthrough the calculation of the relative dipole for the D − p system, one can see that in the protophobic case the sizeof the dipole is the same as in the standard D + p → He + γ reaction, multiplied by (cid:15) . Therefore, the verysame formula (23) would apply to the protophobic caseas well. We note that the rate of dark photon emissioncan be related to the SM D + p → He + e + e − ratebypassing any theoretical assumptions, as for m ee = m A (cid:48) the emission of the pair and the emission of dark photonhas the very same kinematic dependence on all relevantenergies, cf. Eqs. (6) and (8), σ D+ p → He+ A (cid:48) = (cid:15) dσ D+ p → He+ e + e − /dm ee α/ (3 πm ee ) (cid:12)(cid:12)(cid:12)(cid:12) m ee = m A (cid:48) . (24)It is easy to see that for realistic proton beams ( E p ∼ −
40 MeV, up to 1 mA currents) and gas D targets,one can achieve production of V -bosons at rates as highas 10 s − for (cid:15) = 10 − . Therefore, the main difficulty insuch an experiment would not be low statistics but strongbackgrounds from the elastic and inelastic scattering of p on D. This could be circumvented if the final state He isdetected in coincidence with electrons and positrons (see,e.g., [27]). Therefore, we believe that the simplest nuclearreactions, including the ones that require energetic pro-tons, can also be used as a tool for studying X -bosons inrealistic experiments, without much complication comingfrom nuclear physics as in the case of Be ∗ . VI. CONCLUSIONS
Simple low-energy hadronic processes may provide anefficient way of probing light new physics. The simplestprocess, π − capture on protons, can be used to searchfor light degrees of freedom coupled to quarks (i.e. topions and nucleons at low energy), and decaying electro-magnetically. Our analysis shows that with a powerfulsource of sub-100 MeV π − , one can probe the dark pho-ton parameter space down to (cid:15) ∼ − in the mass rangeof a few MeV to m π . We note that only a very few ex-periments are sensitive in the 10 −
30 MeV mass range,where higher energy probes (e.g., B -factories) becomeless efficient.We have also considered the production of the so-called“protophobic” dark vector V that was suggested [11] asa candidate explanation for the unusual angular corre-lation observed in the decay of a highly excited state of Be [9]. In a hypothesis that the angular anomaly stemsfrom a new X particle, pion capture should exhibit sim-ilar unexplained variation of distribution in the relativeelectron-positron angle at around 14 degrees. The “pro-tophobic” dark vector hypothesis can be then decisivelytested this way, free of any possible nuclear physics com-plications.Another possible avenue for searches of X is given bynuclear reactions in few-nucleon systems. In particular,the previously studied D + p → He + e + e − process maybe used for X searches. With proton beams in excess of20 MeV, and with the same coupling size to explain theberyllium anomaly, copious production of X bosons canbe achieved, potentially opening a new avenue for theirstudy. Acknowledgements
We thank Drs. D. Bryman, J. Feng and A. Papa forhelpful discussions and useful communications. MP isgrateful to Drs. B. Bastin, A. Coc and his colleaguesat CSNSM Orsay, for productive discussions of possi-ble X -boson search strategies. Research at PerimeterInstitute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontariothrough the Ministry of Economic Development & Inno-vation. C.-Y.C is supported by the DOE grant DE-FG02-91ER40684 and the NSF grant NSF-1740142. DM is sup-ported by a Discovery Grant from the Natural Sciencesand Engineering Research Council of Canada (NSERC)and TRIUMF which receives funding through a contri-bution agreement with the National Research Council ofCanada (NRC). Appendix A: New physics models
In this appendix, we will discuss some potential modelswhere the new boson is a pseudoscalar or an axial vector.Each of them can be in the flavor SU(2) singlet or tripletrepresentation. As with the vector bosons we discussed inthe text, such states can be produced in simple hadronicreactions, such as pion capture on a proton. We giveexpressions for this capture rate to facilitate the easycomputation of sensitivities.
1. Singlet and triplet pseudoscalars
The Lagrangian for a singlet pseudoscalar ( a ) couplingto the axial current j µ ≡ ¯ N γ µ γ N can be written as L ⊃ g Sa F a ∂ µ a ¯ N γ µ γ N + g S (cid:48) a F a F π π b a ¯ N τ b N. (A1)where τ b for b = 1 to 3, are the Pauli matrices and N = ( p n ) T is a SU (2) doublet with p and n the protonand neutron, respectively. The second term is analogousto the effective operator 1 / (2 F π ) π ¯ N N σ , in the chiralperturbation theory, where σ = m u + m d (cid:104) N | ¯ uu + ¯ dd | N (cid:105) and g Sa = ∆ u + ∆ d (cid:39) .
521 [28, 29] with 2 s µ ∆ u = (cid:104) p | ¯ uγ µ γ u | p (cid:105) and 2 s µ ∆ d = (cid:104) p | ¯ dγ µ γ d | p (cid:105) , where s µ is theproton spin vector. From the fact that [30] m u − m d (cid:104) N | ¯ uu − ¯ dd | N (cid:105) = − .
52 MeV ¯
N τ N, (A2)we know that g S (cid:48) a ¯ N τ N = ( m u + m d ) (cid:104) N | ¯ uu − ¯ dd | N (cid:105) = − R + 1 R − .
52 MeV ¯
N τ N (A3) where R = m u /m d = 0 . ± . , is the mass ratio ofthe up quark to the down quark. This implies that g S (cid:48) a (cid:39) .
98 MeV.The Lagrangian for a triplet pseudoscalar, a b for b =1 −
3, can be written as,
L ⊃ g Ta F a ∂ µ a b ¯ N γ µ γ τ b N + g T (cid:48) a F a F π π b a ¯ N τ b N, (A4)where g Ta = ∆ u − ∆ d (cid:39) .
264 [29] and g T (cid:48) a ¯ N τ N = m u − m d (cid:104) N | ¯ uu − ¯ dd | N (cid:105) . According to Eq. A2, this implies g T (cid:48) a (cid:39) .
52 MeV.Using the Lagrangians introduced above one can ob-tain the expressions for the ratios of the pseudoscalarproduction cross sections to that of a photon. The ex-pression for the singlet pseudoscalar reads( σv ) π − p → an ( σv ) π − p → γn = (cid:18) g Sa eF a (cid:19) m N x × (A5) (cid:18) − g S ( x − y − y + 2)4 m N x (1 + y ) (cid:19) f Sa ( x, y ) , with x = m a /m N , y = m π /m N , and f Sa ( x, y ) = (cid:34) − (cid:18) x y (cid:19) (cid:35) (cid:20) − x y (cid:21) − × (cid:32) x − x (cid:0) y + y (cid:1) y (2 + y ) (cid:33) / . (A6)Here g S = 2 g S (cid:48) a / ( g Sa g A ).The expression for the triplet pseudoscalar is as fol-lows,( σv ) π − p → an ( σv ) π − p → γn = (cid:18) g Ta eF a (cid:19) m N y y ) × (A7) (cid:18) g V ( x − y − y + 2) m N [ y (2 + y ) − x (2 + 2 y + y )] (cid:19) f Ta ( x, y ) , with x and y as above and f Ta ( x, y ) = (cid:115) − x y (cid:34) − (cid:18) x y (cid:19) (cid:35) / × (cid:18) − x (1 + y ) y (2 − x + y ) (cid:19) . (A8)Here g V = 2 g T (cid:48) a / ( g Ta g A ).
2. Singlet and triplet axial vectors
Similarly, the Lagrangian for a singlet axial vector ( A µ )coupling to the axial current has the following form, L ⊃ g SA Q AN A µ ¯ N γ µ γ N, (A9)where Q AN is the axial charge of the nucleon doublet N .The Lagrangian for a triplet axial vector, A bµ , can bewritten as L ⊃ g TA A bµ ¯ N γ µ γ τ b N (A10)The cross section ratio for the singlet axial vector is( σv ) π − p → An ( σv ) π − p → γn = (cid:18) Q AN g SA e (cid:19) y (1 + y ) f SA ( x, y ) , (A11) with x = m A /m N , y = m π /m N , and f SA ( x, y ) = (cid:20) − x y (cid:21) (cid:34) − (cid:18) x y (cid:19) (cid:35) (cid:20) − x y (cid:21) − × (cid:32) x − x (cid:0) y + y (cid:1) y (2 + y ) (cid:33) / . (A12)The ratio for the triplet axial vector can be written as,( σv ) π − p → An ( σv ) π − p → γn = (cid:18) g TA e (cid:19) y x (1 + y ) f TA ( x, y ) , (A13)with x and y given above and f TA ( x, y ) = (cid:20) − x y (cid:21) (cid:20) − x y (cid:21) − (cid:32) x − x (cid:0) y + y (cid:1) y (2 + y ) (cid:33) / × (cid:32) x (1 + y ) − x (2 + y ) ( y − y − y −
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