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the search of new gauge interactions has been of great interest . such efforts help us understand whether there is any other new force in nature and how it fits to the grand picture of particle physics . if found , there is also a possibility that the new force carrier provides a portal between the standard model ( sm ) visible sector and a hidden sector involving new dynamics and matter contents .    in the simplest scenario , such a new vector boson may have an origin from some extra @xmath4 gauge symmetry , under which some particles in both visible and hidden sectors are charged . it may even have kinetic or mass mixing with the sm photon or @xmath5 boson field to facilitate the mediation . after its symmetry breaking , as is often assumed , the @xmath4 gauge boson acquires a mass and is commonly called the @xmath6 boson if it is heavier than the electroweak scale or the dark photon if it is lighter . there have been a vast amount of studies on the neutral gauge boson over the years  @xcite . for example , recent direct searches at the lhc have already pushed the lower bound on the sequential @xmath6 mass to about 3  tev  @xcite . further investigations rely on advances in the colliding energy and beam luminosity . on the other hand , probes of the dark photon of mass below sub - gev have been done using nuclear transitions  @xcite or from its effects on the magnetic dipole moment of electron and/or muon  @xcite . the dark photon can be radiated off from some particle by bremsstrahlung and then decay into a pair of leptons in the fixed - target and beam - dump experiments  @xcite . with an appropriate coupling , it can be produced at @xmath7 colliders as well . if the dark photon couples to quarks , one can consider its production in meson decays , if kinematically allowed . recently , there is an elevated interest in the study of dark photon because of an experimental anomaly involved in isoscalar @xmath3be transitions reported by krasznahorkay et . al .  @xcite . in the transition from an excited state to the ground state , the nucleus emitted an electron - positron pair whose open angle and invariant mass were found to deviate from the sm expectation of internal pair creation ( ipc ) by @xmath8 . it was shown that the distributions of open angle and invariant mass could be well fit by introducing a new particle with mass 16.7  mev produced in the transition . in refs . @xcite , the authors claimed that the new particle could be a vector boson @xmath9 and provided the preferred ranges of its couplings with sm particles that were consistent with current dark photon search constraints . there are several analyses on how to further test the model in ref . @xcite using low - energy physical processes  @xcite , as well as proposals of alternative models for the @xmath3be anomaly  @xcite .    in this work , we propose to use rare leptonic decays of kaon and pion , @xmath10 and @xmath11 , as felicitous means to probe the light dark photon in the mass range of about @xmath1  mev , particularly in view of the putative gauge boson @xmath9 with vectorial couplings mentioned above . this is because the final - state electron - positron pair production can be enhanced via the mediation of the @xmath9 boson over the sm background through a virtual photon . we show that the sm background and the signal have very different spectra in the @xmath7 invariant mass : the sm background has a continuous spectrum , whereas the decay through a dark photon features in a sharp resonance peak around the dark photon mass @xmath12 . moreover , the @xmath13 and @xmath14 decays are able to probe most part of the preferred coupling space inferred from the @xmath3be anomaly , assuming specific production rates of kaons / pions and their energy resolutions . we provide the projected experimental limits on the dark photon couplings based on these decay processes . we show that the signal events and sm background events have different behaviours in the @xmath15 angular distribution in the leptonic kaon decay . we also discuss the influence of the kaon / pion structure dependence ( i.e. , the form factors ) , and find their effects almost irrelevant in current considerations . this paper is organized as follows . in section  [ sec : x ] , we review the leptonic meson decays of interest to us . we present their decay amplitudes , including both inner bremsstrahlung and structure - dependent parts , and provide the form factors involved in the latter part . in section  [ sec : scen ] , we consider three different dark photon scenarios , depending on how it couples to the sm particles , and three different experimental schemes for the kaon and pion decays . we estimate the projected reach in the dark photon couplings in section  [ sec : proj ] . numerical results are presented in section  [ sec : resu ] , where we also discuss effects coming from the structure - dependent contributions . our findings are summarized in section  [ sec : summary ] . there can be many different ways to realize a light @xmath4 gauge boson and let it couple with sm particles ( directly or via mixing with the photon and/or @xmath5 boson ) . in the following , we will keep the formalism as general as possible , without explicitly referring to any particular new physics model . suppose the gauge coupling @xmath17 of the dark photon @xmath9 has a ratio of @xmath18 to the sm electric coupling , _ i.e. _ , @xmath19 . we denote the coupling charge of electron , muon , @xmath20 quarks , and @xmath21 mesons respectively by @xmath22 , whose values depend on the model .    the dark photon @xmath9 can contribute to the @xmath23 decay , where the variables in the parentheses denote the momenta of the corresponding particles . the radiative kaon decay involves both inner bremsstrahlung ( ib ) and structure - dependent ( sd ) parts , and the total decay amplitude is given by  @xcite @xmath24 ~,\ ] ] where @xmath25 is the fermi decay constant , @xmath26  mev is the kaon decay constant , the cabibbo - kobayashi - maskawa ( ckm ) matrix element @xmath27 , and @xmath28\,,\end{aligned}\ ] ] with the @xmath29 part being due to ib , the @xmath30 part containing the sd form factors , @xmath31 , @xmath32 , and @xmath33 denoting the total width of @xmath9 . the expressions in eqs . ( [ eq1 ] ) and ( [ eq2 ] ) can be readily modified to give those for the sm background by replacing symbols associated with the dark photon by those for photon . we follow the convention in ref . @xcite for the kaon form factors , which is consistent with the one used in ref . , as explicitly shown in eq . ( c.8 ) of ref . we set @xmath34 , and the rest form factors from recent measurements are parametrized as  @xcite @xmath35 with @xmath36 , @xmath37 , @xmath38 , @xmath39  mev , @xmath40  mev , and @xmath41  mev .    in the case of signal events , the distribution in the electron - positron invariant mass is a sharp resonance peak around @xmath12 . the partial width of the dark photon decay into a pair of fermions is @xmath42\,,\ ] ] where @xmath43 , @xmath44 is the color factor , and @xmath45 , @xmath46 are the vector and axial - vector couplings between @xmath9 and @xmath47 . taking @xmath48  mev and assuming vectorial couplings as in refs . @xcite , the dominant decay channel should be @xmath7 , @xmath49 and @xmath50 . for the sm background , one should set @xmath51 , @xmath52 , @xmath53 and replace @xmath54 by @xmath55\,.\ ] ] in this case , the electron - positron invariant mass distribution gives a continuous spectrum . we also consider the pion decay @xmath56 . similar to the kaon decay , the pion decay amplitude is  @xcite @xmath57 where @xmath58 where @xmath59  mev , @xmath60  mev is the pion decay constant , the ckm matrix element @xmath61 , and the form factors  @xcite @xmath62 with @xmath63 , @xmath64 , @xmath65 , and @xmath66 . the ib term in the @xmath67 decay amplitude is proportional to @xmath68 , therefore the ib contribution is more important than that in the @xmath69 decay amplitude . the sd contribution of the @xmath69 becomes more important at the kinematic regime of large @xmath7 invariant mass  @xcite . to explain the @xmath70 anomaly , it is sufficient for the dark photon to have couplings with only fermions in the first family while satisfying gauge anomaly cancellation  @xcite . the preferred ranges of dark photon couplings are found to be  @xcite @xmath71 decays measured at the na48/2 experiment .    if the dark photon also couples to the second family and we require gauge anomaly cancellation within each family , then the @xmath4 charges for muon and strange quark have either @xmath72 or @xmath73 . we will thus consider the following three dark photon scenarios :    * * scenario 1 * : only @xmath74 . * * scenario 2 * : add @xmath75 and @xmath76 to scenario 1 . * * scenario 3 * : add @xmath77 and @xmath78 to scenario 1 .    with @xmath79 , @xmath80 , @xmath81 , and @xmath82 , the @xmath4 charge of kaon is @xmath83 for scenario  1 , @xmath84 for scenario  2 , and @xmath85 for scenario  3 . the @xmath4 charge of pion is @xmath86 for all three scenarios . here we consider three different experimental schemes for the estimation of projected limits in the determination of dark photon gauge coupling . suppose we produce @xmath87 kaons and pions , respectively , and measure the @xmath7 invariant mass from the decays with an energy resolution of @xmath88 . the schemes are  :    * * scheme 1 * : @xmath89 , @xmath90 , and @xmath91 mev , * * scheme 2 * : @xmath92 , @xmath93 , and @xmath91 mev , * * scheme 3 * : @xmath92 , @xmath93 , and @xmath94 mev . the energy resolution @xmath88 used here is much larger than the dark photon width for @xmath95 , as preferred by the @xmath70 anomaly  @xcite . therefore , we can not determine its width from the resonance peak in the @xmath7 invariant mass distribution . but in contrast , sm background gives a continuous spectrum . .[t1 ] branching ratios of the sm background in the range of @xmath96 , where @xmath48  mev , for different experimental schemes . [ cols="<,^,^,^",options="header " , ]     in this section , we estimate the projected limits for the @xmath4 gauge coupling or , equivalently , the @xmath18 parameter for the above - mentioned three experimental schemes , following the steps outlined in ref . @xcite . first , we use the simple definition of standard deviation @xmath97 where @xmath98 is the number of signal events , and @xmath99 is that number of background events . in the following , we will consider @xmath100 , corresponding to about @xmath101 confidence limit ( c.l . ) . base on the experimental schemes , we write down the ratio of @xmath98 and @xmath99 in terms of the branching ratios @xmath102 where @xmath103 and @xmath104 are the branching ratios of the leptonic kaon ( or pion ) decay through @xmath9 and @xmath105 , respectively , with the requirement that the @xmath7 invariant mass falls within the energy range : @xmath96 . with a fixed @xmath12 , @xmath103 is proportional to @xmath106 due to the breit - wigner description and @xmath107^{-1 } \simeq \frac{\pi}{m_{x}\gamma_{x}}\delta(q'^2-m^2_{x})$ ] . combining eqs . ( [ eq10 ] ) and ( [ eq11 ] ) , we obtain the 2@xmath2 probing limit on the ratio @xmath108 where @xmath109 can be any reasonable value , as long as @xmath110 . the branching ratios of sm background in various schemes are listed in table  [ t1 ] .    in fig . [ f1 ] , we show @xmath111 versus @xmath12 in colored curves for the three experimental schemes from kaon ( left plot ) and pion ( right plot ) decays by fixing @xmath112 and @xmath113 ( _ i.e. _ , scenario 2 ) . we also show in black curves the effect of form factors on @xmath111 by removing the sd part . for the dark photon mass of interest to the @xmath3be anomaly , @xmath114  mev , the effects from kaon or pion form factors are basically negligible .     projected limits on @xmath18 with @xmath115 ( about 95% c.l . ) with @xmath112 , and @xmath113 ( scenario 2 ) . the left panel is for the @xmath116 decay , and the right panel is for the @xmath117 decay . the red , green , and blue curves , including both ib and sd contributions , are for schemes 1 , 2 , and 3 , respectively . for each colored curve , the adjacent black curves include only the ib contribution . , title="fig:",height=211 ]   projected limits on @xmath18 with @xmath115 ( about 95% c.l . ) with @xmath112 , and @xmath113 ( scenario 2 ) . the left panel is for the @xmath116 decay , and the right panel is for the @xmath117 decay . the red , green , and blue curves , including both ib and sd contributions , are for schemes 1 , 2 , and 3 , respectively . for each colored curve , the adjacent black curves include only the ib contribution . , title="fig:",height=211 ]    before concluding this section , we comment on the decay length of the dark photon as produced from the kaon or pion decay . the decay length of @xmath9 produced from @xmath116 is @xmath118 , where @xmath119 is the boost factor , @xmath120 is the lifetime , and @xmath121 is the speed of light . the largest boost factor from @xmath122 is @xmath123 . taking the smallest value of @xmath124 for the @xmath3be anomaly , the largest decay length of @xmath9 with @xmath48 mev is @xmath125 therefore , the dark photon should decay within the detector . on the other hand , a decay length of @xmath9 of @xmath126 1  cm corresponds to @xmath127 . contours of @xmath128 for scenario  1 on the @xmath129-@xmath130 plane . @xmath131 comes from the experimental scheme  1 . the light green area is the region favored by the @xmath3be anomaly  @xcite . the black dotted line is for @xmath132 . the colored lines indicate the contours for various values of @xmath128 ; for example , the red contour is for @xmath133 . , height=336 ]     the 2@xmath2 projected limits from the @xmath134 decay under the experimental scheme 1 ( red ) , scheme 2 ( green ) , and scheme 3 ( blue ) and for scenario 1 ( first row ) , scenario 2 ( second row ) , and scenario 3 ( third row ) . the light green area is the coupling region favored by the @xmath3be anomaly  @xcite . depending on the schemes , the region outside the red , green , or blue areas are experimentally probe - able . the back dotted lines in the left - column plots indicate the case with @xmath135 . the grey regions in the middle - and right - column plots indicate the possibility that the @xmath9 has a decay length larger than 1  cm . , title="fig:",height=144 ]   the 2@xmath2 projected limits from the @xmath134 decay under the experimental scheme 1 ( red ) , scheme 2 ( green ) , and scheme 3 ( blue ) and for scenario 1 ( first row ) , scenario 2 ( second row ) , and scenario 3 ( third row ) . the light green area is the coupling region favored by the @xmath3be anomaly  @xcite . depending on the schemes , the region outside the red , green , or blue areas are experimentally probe - able . the back dotted lines in the left - column plots indicate the case with @xmath135 . the grey regions in the middle - and right - column plots indicate the possibility that the @xmath9 has a decay length larger than 1  cm . , title="fig:",height=144 ]   the 2@xmath2 projected limits from the @xmath134 decay under the experimental scheme 1 ( red ) , scheme 2 ( green ) , and scheme 3 ( blue ) and for scenario 1 ( first row ) , scenario 2 ( second row ) , and scenario 3 ( third row ) . the light green area is the coupling region favored by the @xmath3be anomaly  @xcite . depending on the schemes , the region outside the red , green , or blue areas are experimentally probe - able . the back dotted lines in the left - column plots indicate the case with @xmath135 . the grey regions in the middle - and right - column plots indicate the possibility that the @xmath9 has a decay length larger than 1  cm . , title="fig:",height=144 ]   the 2@xmath2 projected limits from the @xmath134 decay under the experimental scheme 1 ( red ) , scheme 2 ( green ) , and scheme 3 ( blue ) and for scenario 1 ( first row ) , scenario 2 ( second row ) , and scenario 3 ( third row ) . the light green area is the coupling region favored by the @xmath3be anomaly  @xcite . depending on the schemes , the region outside the red , green , or blue areas are experimentally probe - able . the back dotted lines in the left - column plots indicate the case with @xmath135 . the grey regions in the middle - and right - column plots indicate the possibility that the @xmath9 has a decay length larger than 1  cm . , title="fig:",height=144 ]   the 2@xmath2 projected limits from the @xmath134 decay under the experimental scheme 1 ( red ) , scheme 2 ( green ) , and scheme 3 ( blue ) and for scenario 1 ( first row ) , scenario 2 ( second row ) , and scenario 3 ( third row ) . the light green area is the coupling region favored by the @xmath3be anomaly  @xcite . depending on the schemes , the region outside the red , green , or blue areas are experimentally probe - able . the back dotted lines in the left - column plots indicate the case with @xmath135 . the grey regions in the middle - and right - column plots indicate the possibility that the @xmath9 has a decay length larger than 1  cm . , title="fig:",height=144 ]   the 2@xmath2 projected limits from the @xmath134 decay under the experimental scheme 1 ( red ) , scheme 2 ( green ) , and scheme 3 ( blue ) and for scenario 1 ( first row ) , scenario 2 ( second row ) , and scenario 3 ( third row ) . the light green area is the coupling region favored by the @xmath3be anomaly  @xcite . depending on the schemes , the region outside the red , green , or blue areas are experimentally probe - able . the back dotted lines in the left - column plots indicate the case with @xmath135 . the grey regions in the middle - and right - column plots indicate the possibility that the @xmath9 has a decay length larger than 1  cm . , title="fig:",height=144 ]   the 2@xmath2 projected limits from the @xmath134 decay under the experimental scheme 1 ( red ) , scheme 2 ( green ) , and scheme 3 ( blue ) and for scenario 1 ( first row ) , scenario 2 ( second row ) , and scenario 3 ( third row ) . the light green area is the coupling region favored by the @xmath3be anomaly  @xcite . depending on the schemes , the region outside the red , green , or blue areas are experimentally probe - able . the back dotted lines in the left - column plots indicate the case with @xmath135 . the grey regions in the middle - and right - column plots indicate the possibility that the @xmath9 has a decay length larger than 1  cm . , title="fig:",height=144 ]   the 2@xmath2 projected limits from the @xmath134 decay under the experimental scheme 1 ( red ) , scheme 2 ( green ) , and scheme 3 ( blue ) and for scenario 1 ( first row ) , scenario 2 ( second row ) , and scenario 3 ( third row ) . the light green area is the coupling region favored by the @xmath3be anomaly  @xcite . depending on the schemes , the region outside the red , green , or blue areas are experimentally probe - able . the back dotted lines in the left - column plots indicate the case with @xmath135 . the grey regions in the middle - and right - column plots indicate the possibility that the @xmath9 has a decay length larger than 1  cm . , title="fig:",height=144 ]   the 2@xmath2 projected limits from the @xmath134 decay under the experimental scheme 1 ( red ) , scheme 2 ( green ) , and scheme 3 ( blue ) and for scenario 1 ( first row ) , scenario 2 ( second row ) , and scenario 3 ( third row ) . the light green area is the coupling region favored by the @xmath3be anomaly  @xcite . depending on the schemes , the region outside the red , green , or blue areas are experimentally probe - able . the back dotted lines in the left - column plots indicate the case with @xmath135 . the grey regions in the middle - and right - column plots indicate the possibility that the @xmath9 has a decay length larger than 1  cm . , title="fig:",height=144 ]     same as fig . [ f4 ] , but with only the ib contribution included . , title="fig:",height=144 ]   same as fig . [ f4 ] , but with only the ib contribution included . , title="fig:",height=144 ]   same as fig . [ f4 ] , but with only the ib contribution included . , title="fig:",height=144 ]   same as fig . [ f4 ] , but with only the ib contribution included . , title="fig:",height=144 ]   same as fig . [ f4 ] , but with only the ib contribution included . , title="fig:",height=144 ]   same as fig . [ f4 ] , but with only the ib contribution included . , title="fig:",height=144 ]   same as fig . [ f4 ] , but with only the ib contribution included . , title="fig:",height=144 ]   same as fig . [ f4 ] , but with only the ib contribution included . , title="fig:",height=144 ]   same as fig . [ f4 ] , but with only the ib contribution included . , title="fig:",height=144 ]     same as fig . [ f3 ] , but for the @xmath136 decay . the back dotted lines of the left - column plots indicate the case with @xmath137 . , title="fig:",height=144 ]   same as fig . [ f3 ] , but for the @xmath136 decay . the back dotted lines of the left - column plots indicate the case with @xmath137 . , title="fig:",height=144 ]   same as fig . [ f3 ] , but for the @xmath136 decay . the back dotted lines of the left - column plots indicate the case with @xmath137 . , title="fig:",height=144 ]   same as fig . [ f3 ] , but for the @xmath136 decay . the back dotted lines of the left - column plots indicate the case with @xmath137 . , title="fig:",height=144 ]   same as fig . [ f3 ] , but for the @xmath136 decay . the back dotted lines of the left - column plots indicate the case with @xmath137 . , title="fig:",height=144 ]   same as fig . [ f3 ] , but for the @xmath136 decay . the back dotted lines of the left - column plots indicate the case with @xmath137 . , title="fig:",height=144 ]   same as fig . [ f3 ] , but for the @xmath136 decay . the back dotted lines of the left - column plots indicate the case with @xmath137 . , title="fig:",height=144 ]   same as fig . [ f3 ] , but for the @xmath136 decay . the back dotted lines of the left - column plots indicate the case with @xmath137 . , title="fig:",height=144 ]   same as fig . [ f3 ] , but for the @xmath136 decay . the back dotted lines of the left - column plots indicate the case with @xmath137 . , title="fig:",height=144 ]    since the best - fit mass of dark photon for the @xmath3be anomaly is 16.7  mev  @xcite , we will fix @xmath48  mev in the following discussions . refs . @xcite worked out the ranges of dark photon couplings with neutron , proton , and electron that could explain the @xmath3be anomaly and evade other experimental constraints . we will show how the leptonic kaon and pion decays can probe this region . first , we take the experimental scheme 1 and consider the kaon decay in scenario 1 as an example . the sm background branching ratio is @xmath138 . then the signal branching ratio corresponding to @xmath139 upper limit of @xmath140 is @xmath141 using @xmath131 as a basic unit , we draw the contours of @xmath128 on the @xmath142 parameter plane in fig .  [ f2 ] . the black dotted line in the plot indicates @xmath132 and thus @xmath143 . the region with @xmath144 is between the two red lines . this is the region where the dark photon in scenario 1 can not be checked using the leptonic kaon decay under the experimental scheme 1 . however , all the region outside the red contour , including most of the light green region , can be probed . we perform a scan over the dark photon couplings , @xmath145 . in fig . [ f3 ] , we show the @xmath139 projected limits for scenarios 1 , 2 , and 3 in experimental schemes 1 , 2 , and 3 . most part of the light green area , outside the red , green , or blue regions , can be probed by the @xmath134 decay . in the left column , the black dotted lines indicate the case with @xmath135 . in this case , the leptonic kaon decay loses sensitivity to probing the dark photon . in the right column , the leptonic kaon decay also has less sensitivity to the dark photon along the semi - major axis of the ellipses . this is because when @xmath146 , a cancellation occurs in the ib amplitude and , therefore , @xmath147 is suppressed . [ f4 ] is the same as fig . [ f3 ] , except that we turn off the sd contribution in the amplitude of leptonic kaon decay . there is almost no noticeable numerical difference between fig . [ f3 ] and fig . this demonstrates that uncertainties from the kaon form factors have little effects on the analysis , especially for @xmath148 mev .    in fig . [ f5 ] , we show the @xmath139 projected limits from the @xmath149 decay as another possible probe . comparing to fig . [ f3 ] , the projected limits from the leptonic pion decay are not as good . but a good portion of the preferred parameter region for the @xmath3be anomaly ( light green area ) can still be probed . angular distribution of the @xmath150 decays , where @xmath151 denotes the angle of @xmath152 with respect to the 3-momentum of @xmath9 , but measured in the @xmath7 center - of - mass frame . the @xmath153 is the ratio between the number of events within each bin to the total number of events . virtually no difference can be seen among the four scenarios . , height=336 ]     the @xmath154 distribution of the @xmath150 decays , where @xmath155 denotes the angle of @xmath156 with respect to the @xmath15 3-momentum sum , but measured in the @xmath15 center - of - mass frame . both ib and sd contributions are included in the calculations . , title="fig:",height=211 ]   the @xmath154 distribution of the @xmath150 decays , where @xmath155 denotes the angle of @xmath156 with respect to the @xmath15 3-momentum sum , but measured in the @xmath15 center - of - mass frame . both ib and sd contributions are included in the calculations . , title="fig:",height=211 ]    we now turn to the discussions of angular distributions of different subsystems in the @xmath157 final state . first , we show in fig . [ f6 ] the distribution of @xmath158 from the leptonic kaon decay , where @xmath151 denotes the angle of @xmath152 with respect to the boost direction of @xmath9 , but measured in the @xmath7 center - of - mass frame . the parabolic shape of the angular distribution signifies the feature of vectorial couplings of the @xmath9 boson and the photon , and it is almost independent of @xmath4 charges and the sd part .    in fig . [ f7 ] , we show the distribution of @xmath154 , where @xmath155 is the angle of @xmath156 with respect to the boost direction of the @xmath15 system , but measured in the @xmath15 center - of - mass frame . apparently , the sm background and signal from dark photon have different behaviours . from the sm background , the @xmath156 tends to fly toward the direction of @xmath9 in the @xmath116 rest frame . for the signal events , however , it has an isotropic distribution . in the left plot , when @xmath159 and @xmath160 , we are left with only the first term in the curly brackets of @xmath29 in eq . ( [ eq2 ] ) that has no angular dependence . when @xmath132 and @xmath161 , on the other hand , the second term in the curly brackets survives and has some minor angular dependence . when @xmath162 , the angle - independent parts cancel , rendering a more dramatic angular dependence , as shown by the blue - dotted histogram . the same cancellation also happens for the sm background , as the electric charges of @xmath116 and @xmath163 have opposite signs . the result is shown by the black - dotted histogram in the right plot . in this work , we propose to use the rare leptonic kaon and pion decays , @xmath164 to probe a light dark photon of @xmath1  mev mass . this is particularly suitable for probing the putative @xmath9 gauge boson hinted at by the recent @xmath3be anomaly . we consider three scenarios for the standard model particle couplings with the dark photon . the first scenario assumes that the dark photon only couples to fermions in the first family , as directly relevant for the @xmath3be anomaly . in the second and third scenarios , the dark photon is allowed to also couple with fermions in the second family , with equal or opposite signs to the couplings in the first family . we also consider three schemes for the estimation of experimental reach . we estimate the projected limits by calculating the numbers of events from both dark photon and sm background . we perform a scan of dark photon couplings , @xmath145 , and compare the results with the region favored by the @xmath3be anomaly . moreover , we show the angular distributions of final - state @xmath7 and @xmath15 systems in their own center mass frames .    in general , scheme  1 has the best sensitivity to probe the dark photon couplings due to a larger number of events as well as a better energy resolution . most of the parameter space preferred by the @xmath3be anomaly can be probed by both @xmath10 and @xmath11 decays . the effects from kaon and pion form factors are found to be negligible in the projected limits for @xmath48  mev . in each specific experimental scheme , the projected limits from the kaon decay are stronger than those from the pion decay . except for some special case where @xmath116 and @xmath156 have opposite couplings with the dark photon , the signal events and the sm background events have different behaviours in the @xmath15 angular distributions . the @xmath7 angular distribution can directly reveal the vectorial nature of the dark photon interaction . c.w.c would like to thank the hospitality of kavli ipmu during his visit in 2016 summer , where this project was initiated . this research was supported in part by the ministry of science and technology of taiwan under grant no . nsc 104 - 2628-m-002 - 014-my4 , and in part by world premier international research center initiative ( wpi ) , mext , japan .
rare leptonic kaon and pion decays @xmath0 can be used to probe a dark photon of mass @xmath1  mev , with the background coming from the mediation of a virtual photon . 
 this is most relevant for the 16.7-mev dark photon proposed to explain a 6.8@xmath2 anomaly recently observed in @xmath3be transitions by the atomki collaboration . 
 we evaluate the reach of future experiments for different scenarios of how the dark photon couples with the standard model particles , and show that a great portion of the preferred 16.7-mev dark photon parameter space can be decisively probed . 
 we also show the use of angular distributions to further distinguish the signal from the background .