Resonant production of dark photons in positron beam dump experiments

Positrons beam dump experiments have unique features to search for very narrow resonances coupled superweakly to $e^+ e^-$ pairs. Due to the continue loss of energy from soft photon bremsstrahlung, in the first few radiation lengths of the dump a positron beam can continuously scan for resonant production of new resonances via $e^+$ annihilation off an atomic $e^-$ in the target. In the case of a dark photon $A'$ kinetically mixed with the photon, this production mode is of first order in the electromagnetic coupling $\alpha$, and thus parametrically enhanced with respect to the $O(\alpha^2)$ $e^+e^- \to \gamma A'$ production mode and to the $O(\alpha^3)$ $A'$ bremsstrahlung in $e^--$nucleon scattering so far considered. If the lifetime is sufficiently long to allow the $A'$ to exit the dump, $A' \to e^+e^-$ decays could be easily detected and distinguished from backgrounds. We explore the foreseeable sensitivity of the Frascati PADME experiment in searching with this technique for the $17\,$MeV dark photon invoked to explain the $^8$Be anomaly in nuclear transitions.

The first possibility keeps being actively investigated mainly in collider experiments, with the current high energy frontier set by the LHC experiments. However, the so far unsuccessful search for new heavy states has triggered in recent years an increasing interest in the second possibility, with many proposals and many new ideas to * Corresponding author, email: enrico.nardi@lnf.infn.it hunt for new physics at the intensity frontier (see [1,2] for recent reviews). In particular, the so called dark-photon (DP) or A -boson, that is a massive gauge boson arising from a new U (1) symmetry, can be considered as a natural candidate for a superweakly coupled new state, since its dominant interaction with the SM sector might arise solely from a mixed kinetic term ( /2)F µν F µν coupling the U (1) and QED field strength tensors, with values of naturally falling in a range well below 10 −2 .
From the phenomenological point of view, light weakly coupled new particles have been invoked to account for discrepancies between SM predictions and experimental results, as for example the measured value of the muon anomalous magnetic moment [3], the value of the proton charge radius as measured in muonic atoms [4][5][6][7], or the anomaly observed in excited 8 Be nuclear decays by the Atomki collaboration [8][9][10]. This last anomaly is particularly relevant for the present paper since the new experimental technique that we are going to describe appears remarkably well suited to test, at least in some region of the parameter space, the particle physics explanation involving a new gauge boson with mass m A ∼ 17 MeV kinetically mixed with the photon [11].
The anomaly consists in the observation of a bump in the opening angle and invariant mass distributions of arXiv:1802.04756v2 [hep-ph] 22 Apr 2018 electron-positron pairs produced in the decays of an excited 8 Be nucleus [8], which seems unaccountable by known physics. The anomaly has a high statistical significance of 6.8σ which excludes the possibility that it arises as a statistical fluctuation. The shape of the excess is remarkably consistent with that expected if a new particle with mass m A = 17.0 ± 0.2(stat) ± 0.5(sys) MeV [10] is being produced in these decays. The strength of the A coupling to e + e − pairs, parametrized as = α /α with α the U (1) fine structure constant, is constrained by different experimental considerations. In the Atomki setup, A → e + e − decays must occur in the few cm distance between the target, where the 8 Be excited state is formed, and the detectors. This implies a lower limit / Br(A → e + e − ) > ∼ 1.3 × 10 −5 (we will always quote limits on leaving understood that they apply to its absolute value). In the following we will assume for simplicity Br(A → e + e − ) = 1, if the A decay with a non-negligible rate into invisible "dark" particles χ, with m χ < m A /2, the quoted limits need to be accordingly rescaled. However, in case the invisible decay channel becomes largely dominant, other limits different from the ones discussed in this paper apply. We refer to Ref. [12] for details.
Lower limits on much stronger than what implied by the Atomki experimental setup are obtained from electron beam dump experiments. Old data from KEK [13] and ORSAY [14] have been reanalyzed in Ref. [15] yielding, in the interesting mass range m A ∼ 17 MeV, > ∼ 7 × 10 −5 .
A stronger limit, > ∼ 2 × 10 −4 was obtained in [16] from a reanalysis the E141 experiment at SLAC [17]. However, for a m A ∼ 17 MeV the excluded region is very close to the kinematic limit of the sensitivity (see Fig. 4) and it has been recently pointed out, by direct comparison with exact calculations [18], that the Weizsäker-Williams (WW) approximation [19][20][21] adopted to derive the limits become inaccurate in this kinematic region, tending to overestimate the reach in mass [18,22,23].
More in detail, for primary energies in the the range 10 − 20 GeV, as was the case for the E141 beam [17], and for m A ∼ 20 MeV, the WW approximation yields an A production cross section about 50% larger than the exact calculation (see Fig. 2 in Ref. [23]) and it also overesti-mates the A emission spectrum at large energies (see Fig. 4 in the same reference), in which case the number of expected positrons falling within the 1.1mrad angular acceptance of the experiment would be overestimated both because of the larger boost, and also because of the larger lifetime dilation that would cause the A to decay closer to the detector. Besides this, let us note that an A slightly heavier than the benchmark value of 17 MeV would in any case evade the E141 limit. It is then questionable if, for m A > ∼ 17 MeV, the E141 constraints on the A couplings can be considered as firmly established.
Conservatively, we will assume that the corresponding region is still viable.
Upper bounds on in the relevant A mass range also exist, see Fig. 4. The KLOE-2 experiment has searched for e + e − → γA followed by A → e + e − setting the limit < 2 × 10 −3 [24], while constrains from the anomalous magnetic moment of the electron [25] yield < 1.4 × 10 −3 [26,27]. A comparable limit stems from BaBar searches for A → e + e − decays, but it only applies for m A > 20 MeV [28]. In summary, we will take the interval as the window allowed for a 17 MeV A decaying dominantly into e + e − . This corresponds to a DP width

THE PADME EXPERIMENT AT LNF
Collider searches for dark photons have been carried out in electron beam dump experiments (see [15] for a review) assuming A -strahlung as the leading production mechanism in electron-nucleon scattering. Parametrically, this process is of order α 3 , see Fig. 1

(a). As regards
A searches with positron beams, there are only few facilities which, in the next future, will be able to provide beams suitable for fixed target experiments, and correspondingly only a few experimental proposals have been put forth [29][30][31]. The production mechanism considered so far is analogous to the usual QED process of positron annihilation off an atomic target electron with two final state photons, where one photon is replaced by one A  see Fig. 1(b), corresponding to a process of O(α 2 ). This is the specific production process envisaged for the Frascati PADME experiment [31] that we will now describe briefly.
The PADME experiment [31,32]   detector is thus able to detect photons and charged particles and it will be sensitive to invisible (A → χχ) as well as to visible (A → e + e − ) DP decays. PADME will start taking data already during May 2018.
In this Letter we point out that for A masses > ∼ 1 MeV, the process of resonant e + e − annihilation into on-shell A depicted in Fig. 1(c), represents another production mechanism which, being of O(α), is parametrically enhanced with respect to the previous two production channels. Besides this, A production via resonant e + e − annihilation has several other advantages that we will illustrate below, which altogether suggest that it might be particularly convenient to operate the PADME (as well as other) positron beam fixed target experiment in a dedicated mode in order to search for A via resonant production. Besides experiments with positron beams, resonant e + e − → A annihilation must also be accounted for in a correct analysis of electron beam dump experi-ments since, as is remarked in [35], positrons are abundantly produced in the electromagnetic (EM) showers inside the dump. This feature was recently exploited in [35] in reanalysing old results from the SLAC E137 experiment [36] by including A production via resonant annihilation (and, but less importantly, also A -strahlung in annihilation). As a result, it was found that due to the contribution of resonant A production, the E137 data exclude a parameter space region larger than it was previously though [15,16]. The extended excluded region corresponds to the area in light grey color towards the bottom of the plot in Fig. 4. Hence, in analysing electron beam dump data, A production from annihilation of secondary positrons via the diagrams in Fig. 1

(b) and
(c) should be also accounted for.
In this section we consider the sensitivity of the PADME experiment to the production process e + e − → A → e + e − . In order to exploit the resonant production mechanism, however, an experimental setup slightly different from the one originally conceived is more convenient. The thin diamond target should be replaced by a tungsten target of several cms of length, and this for two main reasons. The first one is that of absorbing most of the incoming positron beam and of the related EM showers, and in any case to degrade sufficiently the energy of the residual emerging particles, so that the charged particles background can be easily deflected and disposed of.
The A produced in e + e − annihilation, if sufficiently long lived, will escape the dump without interacting, and will decay inside the downstream vacuum vessel, producing an e + e − pair of well defined energy. The thick tungsten target thus allows to take advantage of the full beam intensity of 10 18 pot/yr, with a gain of five orders of magnitude with respect to the thin target running mode, see The probability that a positron with initial energy E will have an energy E e after traversing t = ρ·z/X 0 radiation lengths (with ρ the density of the material in g/cm −3 and X 0 = 6.76 g/cm −2 the unit radiation length in tungsten), is given by [37,38] where b = 4/3 and Γ is the gamma function. Eq. (2) neglects secondary positrons from EM showers, as well as the loss of primary positrons from e + e − → γγ annihilation, but is still sufficiently accurate for our purposes.
The e + energy distribution after t radiation length is given by: Integrating  terfere with the analogous QED process with an off-shell γ, nor with t-channel amplitudes that can then be neglected. Using the narrow width approximation σ res can be written as: with s 2m e E e , σ peak 12π/m 2 A and Γ A 2 αm A /3.
In the numerical computation we take into account m e effects both in the cross section and in the width, and we also account for the emission of soft photons from the initial state (see e.g. [39]) up to energies ∆E/E b ≈ 1%, which can radiatively enhance the resonance width, and thus the production rate. With respect to other DP production mechanisms, resonant production has some peculiarities and advantages:    however, the annihilation probability is suppressed below 10 −5 . Accordingly, we find that a good fit to the experimental and calculated distributions [40] can be obtained with the sum of just three terms: where v e = p e − /m e , N ∼ 12 is a normalization factor, and the first term in parenthesis accounts for 5d electrons,     [15] and from above by the (g − 2)e orange line [26,27]. For reasons explained in the text we do not consider as firmly excluded the region around m A ≈ 17 MeV delimited by the black-dashed curve of the E141 SLAC experiment [15,16]. The region that could be excluded by PADME running in thin target mode is hatched in black, while the three trapezoidal-shaped areas give the PADME reach in thick target mode, respectively for a 10, 5 and 2 cm tungsten dump, assuming zero background. These regions extend to A masses lower than the mass corresponding to the minimum beam energy (m A ∼ 16 MeV for E min b = 250 MeV depicted with the thin brown vertical line) because of positron energy losses in propagating trough the material. The lower region in light gray extending the E137 exclusion limits is from the reanalysis in Ref. [35].
scintillator veto few mm thick, or a silicon detector of a few hundreds of µm, placed right at the end of the dump, to ensure that the e + e − pairs originate from decays in the vacuum vessel outside the dump. Additionally, if the experiment could be equipped with a suitable tracker, able to provide an accurate e + e − invariant mass reconstruction, many sources of backgrounds could be further reduced. In particular, given that the invariant mass of the e + e − originating from photon conversion m 2 e + e − = 0 is very far from m 2 e + e − ∼(17 MeV) 2 expected from resonant annihilation, the punch-through photon background could be efficiently eliminated.
In Fig. 4 we show the status of the current limits for DP searches assuming visible A decays into e + e − pairs with unit branching fraction and suppressed couplings to the proton. As is discussed in Ref. [42] the last assumption is required in order to evade the tight constraints from π 0 → γA obtained by the NA48/2 experiment [43], and to render thus viable an explanation of For this reason we have not included in in Fig. 4 the limits from the NA48/2 experiment [43] nor those from the ν-Cal I experiment at the U70 accelerator at IHEP Serpukhov [44,45] which also do not apply for protophobic A . In the figure, the vertical black line gives the location of the DP resonance at m A = 17 MeV. Leav-ing aside the limits from the SLAC E141 experiment for which, as explained in the introduction, the reach in A mass might be overestimated, a viable window remains between the Orsay/KEK lines ( > ∼ 7 · 10 −5 ) and the (g − 2) e line ( < ∼ 1.4 · 10 −3 ). The black hatched region depicts the forecasted sensitivity of PADME in thin target mode, that will search for DP via the e + e − → A γ process. The limits assume 10 13 pot/yr. The light cyan trapezoidal regions represent instead the constraints that PADME could set by running in thick target mode with  Fig. 4 could be explored in less than one year of running. In particular, the allowed window for the 8 Be DP could be sizeably reduced, or its existence could be unambiguously established.

CONCLUSIONS
In this letter we have suggested a new way to search for narrow resonances, and specifically DP, coupled to e + e − pairs, via resonant production in e + e − annihilation. There are only a few facilities around the world where positron beams in the 100 MeV -few GeV range will be available. The Frascati BTF is one of those and it can provide beams with energy between 250 − 550 MeV.
Coincidentally, this range covers precisely the c.m. energy needed to produce via resonant e + e − annihilation the m A ∼ 17 MeV DP invoked to explain the anomaly observed in 8 Be nuclear transitions [8][9][10]. By exploiting this production process, the Frascati PADME experiment, presently under commissioning, will be able to reach well inside the interesting parameter space region. Fig. 4 shows that a gap will remain between the large region that can be bounded by searching for A produced via e + e − → γA , and the small region that can be efficiently explored via resonant e + e − → A production.
The reason for this gap is that the first process, being of O(α 2 2 ), looses quickly sensitivity when the value of is decreased too much, while A production via resonant annihilation becomes inefficient when becomes too large, so that most A → e + e − decays occur inside the dump. Resonant e + e − → A production is not relevant for PADME running in thin target mode, because the large beam energy E b ∼ 550 MeV implies that positrons will always have energies far from any narrow resonance with mass < ∼ 23.7 MeV, given that positron energy losses in the 100 µm diamond target are negligible. However, it is conceivable that by reducing the beam energy down to ∼ 282 MeV, by increasing the size of the target to several 100 µm to enhance A resonant production, and keeping the beam intensity well below 10 18 pot/yr to keep counting rates inside the detector under control, at least part of the remaining region for the 17 MeV DP could be explored, and maybe the whole gap could be closed. We are presently exploring this possibility. Before concluding, we stress again that resonant e + e − → A production can be relevant also for electron beam dump experiments, since secondary positrons that could trigger the annihi-lation process are abundantly produced in EM showers. This feature has been recently exploited in reanalysing the SLAC E137 data [35], with the result of extending the previously excluded region [15,16] towards smaller values, as is shown by the light gray area in Fig. 4.