1 Introduction

The Standard Model of particle physics has endured a multitude of precision tests over the past decades. The discovery of the Higgs Boson in 2012 constitutes a landmark [1, 2]. Albeit, we are far from having a final theory that explains exciting observations, such as dark matter. The LHC (Large Hadron Collider) has reached unprecedented energies, but yet has not been able to detect dark matter particles. The nature of dark matter is unknown, and its nature might be unveiled through the detection of a messenger that can be heavy or light, with no prejudice. As no positive signal has been observed from WIMPs (Weakly Interacting Massive Particles), there is a growing interest in the community for light dark matte particles that belong to a dark sector, which may feature light mediators. Such light force carriers can have different interactions with SM particles and are subject to a multitude of experimental searches by low energy accelerators [3]. Among several possibilities, a vector mediator usually called dark photon or hidden photon has been subject to a multitude of studies. The dark photon is the simplest interaction that can be tested experimentally because it represents a dark Quantum Electrodynamics. In other words, the dark photon interacts with particles proportionally to their electric charges, up to a constant factor. The dark photon interacts with the photon through the gauge-invariant lagrangian [4, 5],

$$\begin{aligned} {\mathcal {L}} = - \frac{\epsilon }{2} F^{\mu \nu } F^{\prime \mu \nu }. \end{aligned}$$
(1)

The presence of such a term is universal to all Abelian extensions of the SM, regardless if the new gauge symmetry is broken at small or high-energy scales. As a result, the dark photon \((A^{\prime })\) inherits an electromagnetic interaction proportional to the kinetic mixing term, \(\epsilon \), which is governed by [6, 7],

$$\begin{aligned} {\mathcal {L}}\supset -e\epsilon J^{\mu }A'_{\mu }, \end{aligned}$$
(2)

where \(J^{\mu }\) is the electromagnetic current, e the electric charge. Notice that \(\epsilon \) links the dark and visible sectors, i.e. the SM spectrum. The mass of the dark photon can be treated as a free parameter [8]. Without loss of generality, we will consider the case in which \(A^{\prime }\) couples to leptons.

Fig. 1
figure 1

A schematic illustration of SeDS experiment devoted to search for Dark Sectors

Full size image

The search for a new force carrier has motivated many theoretical and phenomenological studies [9,10,11,12], stimulated the reanalysis and interpretation of old data [13, 14], and promoted new experimental programs [15,16,17,18]. Our work presents a new proposal devoted to the search for dark photons at the Brazilian Synchrotron Light Laboratory (lnls), a new positron on target experiment with features that makes it a promising dark sector detector.

2 SeDS

The second-generation Brazilian synchrotron light source, UVX, was a 1.37 GeV electron storage ring recently decommissioned in Campinas, Brazil. Its injection system included a 120 MeV linear accelerator and a 500 MeV Booster Synchrotron Injector [19,20,21]. UVX has now been succeeded by Sirius, a fourth-generation store ring [22,23,24]. Several subsystems of UVX could potentially be used to repurpose this old light source into a new \(1{-}3\) GeV positron accelerator to host a new SeDS small-scale fixed target experiment set to look for dark photons via the process \(e^+ e^- \rightarrow \gamma A'\). The mass of the dark photon can be determined using the missing mass technique, which requires the knowledge of the initial parameters of the positron beam, the electron target at rest, and the energy-momentum of the final state photon. Hence, the only assumption of the proposed experimental technique is that the dark photon couples to leptons. From a realistic perspective, we expect to be competitive in the search for dark photons through the annihilation process, since the positron beam at SeDS can reach 1 GeV, and optimistically 3 GeV, which is not achieved by current experiments working with the same technique [16, 25].

3 Experimental strategy and kinematics

The experimental design of the experiment is exhibited in Fig. 1, where accelerated positrons are directed to a diamond target, producing photons and \(A^{\prime }\). We highlight that a Carbon target could be used instead, but we selected diamond mostly because it acts as a solid-state ionization chamber, and has very good thermal conductivity, which is appropriate for high beam intensity, as one can dissipate heat through the side connections. The photons are expected to hit an electromagnetic calorimeter to extract the properties of the final state, and we have a spectrometer to measure charged interactions in a momentum range. The dipole magnet is added to deflect the positron beam and reduce the number of events on the calorimeter.

The processes involving dark photon production by \(({\mathcal {O}})\) GeV positrons impinging on a thin target are \(e^+ e^- \rightarrow A'\gamma \) and \(e^+ Z \rightarrow e^+ Z A'\), the so-called annihilation and \(A'\)-strahlung production. Annihilation processes occur when dark photons interact with electrons in the material of the target, shield, and detector, thereby producing real photons in a process similar to Compton scattering. The \(A'\)-strahlung, instead, represents the radiative \(A'\) emission by an impinging \(e^+\) in the electromagnetic field of a target nucleus. Both processes are similar to the ones for ordinary photons, as shown in Fig. 2, and their cross-section scale with \(\epsilon ^2\).

Fig. 2
figure 2

Dark photon production mechanisms by high energy positrons on a fixed target. Left: \(A'\)-production in \(e^+e^-\) annihilation. Right: \(A'\)-production via Bremsstrahlung

Full size image

The main goal of the SeDS experiment is to search for a dark photon produced in the annihilation of positrons of the beam and electrons at rest in a diamond target. This material has a low atomic number \((Z=6)\) which allows limiting the bremsstrahlung interactions (cross-section proportional to \(Z^2\)), which is the main background in annihilation searches [26, 27]. Choosing a graphite target (which has the same atomic number \(Z = 6\)), the experiment would have to run for a longer period to access the same sensitivity found with the diamond target because of its lower density. Furthermore, a diamond target produced by chemical vapor deposition process is not much more expensive. A dark photon signal is assessed by missing-mass events. The ordinary photon in the final state can be observed and its four-momentum measured; the invisible \(A'\) will appear as a bump in the missing mass spectrum,

$$\begin{aligned} M^2_{\text {miss}}=(p_{e^-} + p_{\text {beam}}-p_{\gamma })^2, \end{aligned}$$
(3)

and no assumptions about the \(A'\) decay mode are indeed necessary. Two-photon annihilation is the dominant process of high-energy photon production. The emission angle of the final photon \(\theta _{\gamma }\) with respect to the direction of the positron beam defines the value of the photon energy \(E_{\gamma }\). In the case of two-photon production \(E_{\gamma \gamma }^{\text {lab}} \approx E_{\text {beam}}(1-\cos \theta _{\gamma }^{\text {CM}})\), whereas for \(A'\)-boson production: \(E_{A'\gamma }=E_{\gamma \gamma }^{\text {lab}}(1-M_{A'}^2/s)\). The maximum positron energy of 1 and 3 GeV allows the production of \(A'\) bosons through annihilation up to a center of mass (CM) energy \(\sqrt{s}=\sqrt{2E_{\text{ beam }} m_e}=31.9\) and 55.3 MeV, respectively, where \(m_e\) is the electron mass. The experiment luminosity can be computed using the relation,

$$\begin{aligned} L_{\text {inst}} = \frac{\text {P.O.T}}{\text {s}} N_{A} \frac{Z\rho d}{A}. \end{aligned}$$
(4)

SeDS would be a fixed target experiment with a diamond target of \(d=100{-}500\,\upmu \)m. Its instantaneous and integrated luminosities can be calculated using \(Z=6\), density of diamond, \(\rho =3,51\) g/cm\(^{3}\), and \(A = 12.01\) g diamond’s gram-molecular weight. The experiment can generate 10 bunches per second with \(10^9\) positrons in each bunch, which corresponds to \(10^{10}\) positrons on target (P.O.T) per second.

4 Background

Several processes can be identified as potential background sources. The cross-sections for the major background contributions at \(E_{\text {beam}} = 1~\text {GeV}\) are shown in Table 1. The value of 30 MeV cut on the gamma energy was chosen as 1% of the maximal beam energy, assuming a calorimeter is able to achieve such resolution on the energy measurement.

Table 1 Cross-sections of the dominant background contributions to the search for \(e^+ e^-\rightarrow \gamma A'\). \(E_{\gamma } > 30~\text {MeV}\) for the infrared divergent processes
Full size table
Fig. 3
figure 3

Cross-section for emission of a high energy photon for as a function of the beam energy

Full size image

The dominant contribution is the radiation of a hard photon \(\gamma \) from the beam positrons, either due to bremsstrahlung in the field of the diamond nuclei Z, or due to radiative Bhabha scattering. The bremsstrahlung cross-section increases logarithmically with the beam energy (Fig. 3 [28]), but the opening angle \(\langle \theta _{\gamma }\rangle \) of the emitted photon decreases as 1/E and the emission is highly forward. For the \(e^+e^- \rightarrow \gamma A'\) process, the invariant mass scales as \(\sqrt{E}\), and the opening angle \(\langle \theta _{\gamma }\rangle \) for the recoil photon shrinks with \(1/\sqrt{E}\).

The two photon annihilation can be kept under control due to the symmetric kinematics of the outgoing photons. The three photon annihilation, despite the small cross-section, is one of the dominant backgrounds due to the loss of symmetry in the event and the non-peaking distribution of \(M_{miss}^2\). The employed photon vetoing system should be based on detectors with high efficiency and hermeticity to control this background.

The background suppression will govern the design of the detector system. A high granularity electromagnetic calorimeter will be necessary to precisely measure the \(\theta _{\gamma }\) of the impinging photons and provide sufficient separation and identification of possibly overlapping showers. A full simulation of the experimental setup is necessary to reliably assess the background contribution, together with a simulation of the major characteristics of the chosen readout system. Since this is outside the scope of our work, an indication of the physics potential of the proposed experiment is presented by the single event sensitivity, under the assumption that we can identify and/or veto the background events.

5 Production at lnls

In the experimental setup proposed, the dark photon is produced with a total cross-section at tree-level that reads,

$$\begin{aligned} \sigma _{\text {ann.}}= & {} \frac{8\pi \alpha ^2\epsilon ^2}{s}\left[ \left( \frac{s-m_{A'}^2}{2s} + \frac{m_{A'}^2}{s-m_{A'}^2}\right) \log \frac{s}{m_e^2} \right. \nonumber \\{} & {} \left. - \frac{s-m_{A'}^2}{2s}\right] . \end{aligned}$$
(5)
Fig. 4
figure 4

\(A'\) boson production cross-section as a function of its mass for different beam energies, \(E=250\) MeV, 750 MeV and 3 GeV. We adopted \(\epsilon =10^{-3}\)

Full size image
Fig. 5
figure 5

In the left panel we display SeDS sensitivity on \(\varepsilon ^2\) with 95% CL as a function of dark photon mass in the case where the background is negligible for a 1GeV positron beam impinging on a diamond target of \(100~\upmu \)m for \({\mathcal {L}} = 820~\text{ pb}^{-1}\) and \({\mathcal {L}} = 3330~\text{ pb}^{-1}\). In the right panel, we adopted a 3GeV positron beam with a diamond target of \(500~\upmu \)m, for \({\mathcal {L}} = 4105~\text{ pb}^{-1}\) and \({\mathcal {L}} = 16{,}650~\text{ pb}^{-1}\)

Full size image

The expected annihilation cross-section as a function of the mass of the dark photon for different beam energies is displayed in Fig. 4. Notice that when we increase the energy beam, the cross-section decreases, and this feature translates into a smaller sensitivity on the kinetic mixing for a given dark photon mass. On the other hand, the larger the beam energy, the larger the dark photon mass is kinematically accessible in this process. This trade-off between energy beam and cross-section drives our sensitivity. For instance, from Fig. 4, for low dark photon masses the production cross-section starts below 1000 pb using a positron beam of 3 GeV, whereas for a positron beam of 250 MeV the mass and cross-section ramps up by over a factor of five. Having that in mind, we derive the projected sensitivity to dark forces next.

Fig. 6
figure 6

SeDS sensitivity on \(\varepsilon ^2\) with 95% CL as a function of dark photon mass for the case where the background can be reduced to produce \(N_{sig} \sim N_{back}\). In the left (right) panel, we display the sensitivity of the experiment for a 1 GeV (3 GeV) positron beam impinging on a diamond target of \(100~\upmu \)m (\(500~\upmu \)m) with for \({\mathcal {L}} = 820~\text{ pb}^{-1}\) and \({\mathcal {L}} = 3330~\text{ pb}^{-1}\) \(({\mathcal {L}} = 4105~\text{ pb}^{-1}\) and \({\mathcal {L}} = 16{,}650~\text{ pb}^{-1}\))

Full size image

6 Projected sensitivity for \(A'\)

Assuming that we can control the background events as discussed in the previous sections, the statistical significance can be computed under the assumption that the signal events, after the cuts, are much larger than the background events [29]. With this procedure, we find Fig. 5, which displays the projected exclusion regions in the \(\{M_{A'}, \varepsilon ^2\}\) plane at 95% CL. In the left-panel we plot the projected sensitivity of SeDS (green shaded regions) operating with a positron beam of 1 GeV with a diamond target of \(100~\upmu \)m, for 90 days of data-taking \(({\mathcal {L}}= 825~\text {pb}^{-1}),\) and 1 year of data-taking \(({\mathcal {L}}= 3300~\text {pb}^{-1}).\) A portion of the parameter space has been excluded by previous experiments represented by gray contours, but there is a region for \(\epsilon ^2\) between \(10^{-6}{-}10^{-9}\) and \(M_{A^{\prime }} \sim \) 10 MeV–30 MeV that SeDS could potentially discover dark photons. It is clear that when we increase the luminosity, the projected exclusion region improves and that reflects in a shift towards lower values of \(\epsilon \). The sharp increase in sensitivity for \(M_{A^{\prime }} =30\) MeV is the result of the denominator in Eq. 5 that has a \(s-m_{A'}^2\) term, leading to a large cross-section when the dark photon mass approaches the CM. A similar feature occurs in the right-panel, where we repeat our exercise for a positron beam of 3 GeV, but there we adopt a diamond target of \(500~\upmu \)m, which boosts our luminosity by a factor of five see (Eq. 4). As aforementioned the beam energy dictates the largest dark photon mass probed in the annihilation process. Hence, with a 3 GeV positron beam, \(\sqrt{s}=55.3\) MeV, the sensitivity region towards larger dark photon masses. However, as we increase the CM, the production cross-section decreases, see Eq. 5, weakening our sensitivity on \(\epsilon \), except for the resonance peak. Notoriously, with a 3 GeV positron beam, we can probe a sizeable unexplored region of parameter space with the potential to discover the presence of a dark photon, i.e. a new force carrier in nature. If we are unable to utilize all available positrons on target due to pile-up effects, we would have a smaller luminosity. Consequently, a shift in our sensitivity toward larger values of \(\epsilon \) would occur. Although, from Fig. 5, we still expect to probe an unexplored region of parameter, currently inaccessible by other techniques.

In the figures, we overlay the expected exclusion bounds from planned experiments based on the missing mass technique, namely PADME which uses a 550 MeV positron beam, and consequently can probe a dark photon up to masses of 23.7 MeV and \(\epsilon ^2\sim 10^{-8}\) [30]; VEPP3 which aims to hit a positron beam of 500 MeV on a gas of hydrogen, that could probe \(\epsilon ^2=3\times 10^{-8}\) for \(M_{A^{\prime }}=5{-}20\) MeV [16, 31]; MMAPS that features a much more energetic positron beam of 6 GeV incident on thick beryllium target, aimed to reach \(\epsilon \sim 10^{-8}\) for \(M_{A^{\prime }}=20{-}78\) MeV [17]. Unfortunately, PADME is the only experiment currently taking data [32], and as shown in Fig. 5 the projected limit falls into a region that could be fully covered by SeDS.

However, as aforementioned, the sensitivity shown in Fig. 5 is derived by considering the optimistic case where it is possible to reduce the background with cuts in a way that the signal is much larger than the background. In order to present a more realistic scenario where the number of background events is similar to the signal events, \(N_{sig} \sim N_{back}\), we find derive Fig. 6. In this case, SeDs sensitivity slightly weakens, going toward larger values of the kinetic mixing, but our overall conclusion does not change. Therefore, we can solidly conclude that SeDs represents a great laboratory for dark sectors.

7 Discussions

It is clear from Fig. 5 that SeDS represents a promising dark sector detector. We emphasize that a full GEANT simulation would be needed to grasp all technical details of the detector, including energy resolution, and background rejection. These factors would certainly affect our sensitivity and move our projected limits toward larger values of the kinetic mixing, \(\epsilon \), similarly to what happened in Fig. 6. Be that as it may, worsening our sensitivity on \(\epsilon \) would not change our overall conclusions, because we would still be able to cover an explored region of parameter, and discover a dark force in nature.

8 Conclusions

We have proposed the first search for dark sector at the Brazilian Light Source Laboratory using UVX, a second-generation store ring, which is currently decommissioned and could potentially be repurposed to search for dark photons, reaching unprecedented sensitivity. It has the potential to probe an unexplored region of parameter space, for \(\epsilon \sim 10^{-6}{-}10^{-9}\) and \(M_{A^{\prime }}=10{-}30\) MeV using 1 GeV positron impinging on a diamond target of either \(100~\upmu \)m. We have shown that this sensitivity could be greatly improved by either using a thicker target or, more costly, with an increase in the energy of the positron beam. Our conclusions are valid even in the regime where the number of background events are large, compared to the number of signal events. In summary, SeDS represents a great opportunity to search for dark sectors.