New constraint on dark photon at T2K off-axis near detector

The T2K experiment is one of the most powerful long-baseline experiments to investigate neutrino oscillations. The off-axis near detector called ND280 is installed 280 m downstream from the neutrino production target to measure the neutrino energy spectrum. In this paper, we study the capability of the ND280 detector to search for the dark photon produced through the meson rare decay and proton bremsstrahlung processes at the proton beam dump. We find that the ten-year operation of T2K with the ND280 detector excludes the unexplored parameter region for the dark photon mass and kinetic mixing. We also show that a broader parameter region can be searched by the ND280 in the future T2K operation for dark photon as well as U(1)B−L gauge boson.


INTRODUCTION
No direct observation of the dark matter except through gravity leads us to the idea of dark sectors.The dark sector consists of new particles including the dark matter and is sequestered from the Standard Model (SM) sector.It also contains so-called portal particles which weakly connect two sectors.Among such portals, the dark photon is a massive hypothetical vector boson that mixes with the electromagnetic photon.Extensive searches have already placed stringent bounds on its interaction strength and mass.Further searches continue at ongoing and planned experiments with higher statistics and low-background detectors.
The dark photon is in general defined by a gauge boson of a new Abelian symmetry under which the SM particles are uncharged.In such models, there are no direct interactions of the dark photon with the SM matters, except for the kinetic mixing with the electromagnetic photon (or hypercharge gauge boson) [1][2][3][4][5].Due to the mixing, the dark photon can interact with the SM matters through the electromagnetic current.In this sense, it is called "dark photon" [6][7][8].A light dark photon with mass below 1 GeV has been particularly explored in the decays of light mesons (π, η, η ′ ) and bremsstrahlung in electron and proton beam dump experiments.The dark photon can be produced by replacing a photon appearing in these processes.Null results from these experiments excluded the kinetic mixing parameter roughly between 10 −8 and 10 −4 (see refs.[9][10][11] for recent reviews and references).Below the lower bound, the dark photon becomes so feebly interacting and long-lived particle.In order to search for such particles, one needs higher statistics and low-background detectors placed far away from the production point.Several experiments like NA62 [12,13] and FASER [14][15][16][17] are running, and there are also proposals for new experiments: for instance CODEX-b [18,19], FACET [20], MATHUSLA [21,22], SeaQuest [23,24], and SHiP [25,26].
Notably, the NA62 and the FASER collaboration have recently released their first results on the dark photon search [27][28][29], which excluded new parameter space.
The Tokai-to-Kamioka (T2K) experiment [30] can offer a new opportunity to search for the dark photon.The T2K experiment is a long-baseline neutrino oscillation experiment operated since 2010 and will be upgraded by increasing its beam power in the next several years.The proton beams accelerated up to 30 GeV at J-PARC strike graphite targets, and produce a large number of neutral mesons as well as charged ones.The dark photons can be produced through the decays of these neutral mesons and bremsstrahlung of the proton beams.The ND280 off-axis detector is expected to be suitable in searches for the dark photon or long-lived particles decaying into charged particles. 1It is located 280 m away from the neutrino production target and shielded by the ground.Thus, backgrounds originated from mesons, photons, and muons will be reduced significantly.In refs.[31,32] and [33], the searches for heavy neutral leptons and millicharged particles were studied for ND280.In this work, we investigate the capability of dark photon search with the ND280 detector at the T2K experiment, deriving a new exclusion limit on its mass and kinetic mixing.We will show that the ten-year operation of T2K with the ND280 detector excludes the unexplored parameter region for the dark photon mass and kinetic mixing.We also present that a broader parameter region can be searched by the ND280 in the future T2K operation for dark photon as well as U(1) B−L gauge boson.This paper is organized as follows.In section 2, we describe the minimal dark photon model and show their production and decays.In section 3, the simulation for the production of the secondary particles in the target at the T2K proton beam line is described, then the ND280 off-axis detector is explained in section 4. In section 5, we present new constraints and future sensitivity on the mass and kinetic mixing of the dark photon and U(1) B−L gauge boson derived from our analyses.Section 6 is devoted to summary.

Lagrangian
We extend the SM by adding a new Abelian gauge symmetry U(1) ′ under which all the SM particles are neutral.In this case, the SM Lagrangian is augmented by only two kinetic terms where F µν and F ′ µν are the field strength tensors of the SM U(1) hypercharge symmetry and U(1) ′ , respectively.The second term is the so-called kinetic mixing term, and it causes interaction between the U(1) ′ gauge boson and the SM particles.
In the diagonal basis of the gauge kinetic terms, we consider the following Lagrangian: where e is the electric charge, and J µ EM is the electromagnetic current of the SM.The new gauge boson associated with U(1) ′ is denoted as A ′ µ , that is to say, A ′ µ stands for the dark photon.Note that we assume U(1) ′ is broken, and thus A ′ µ has a non-zero mass m A ′ .In the first term, we define ε ≡ ε ′ cos θ w , where θ w is the weak mixing angle, and will refer to ε as the kinetic mixing parameter in what follows.
In addition to the meson decays, dark photons can be also produced from proton bremsstrahlung.In particular, for 0.5 GeV < m A ′ < 1 GeV, the production rate could be resonantly enhanced via the mixing with ρ and ω mesons, and hence a large amount of A ′ is expected in that mass range.The exact calculation of the bremsstrahlung process, however, is difficult due to nonperturbative QCD effects.In order to estimate the number of events, we rely on the Weizsäcker-Williams (WW) approximation [36][37][38][39][40][41] and follow the calculation procedure outlined in ref. [14].We will discuss in more detail in section 5.

Decay of dark photon
Through the kinetic mixing, dark photons can decay into the SM particles if kinematically allowed.For decays into leptons, the partial decay widths are given by where ℓ = e, µ, τ .The decay widths for hadronic final states can be calculated by using the hadronic production cross section measured at e + e − colliders, such as where R(s) = σ(e + e − → hadrons)/σ(e + e − → µ + µ − ), and s is the center of mass energy.
In our numerical calculations, nevertheless, we make use of the date files included in the FORESEE package [42].

T2K BEAM SIMULATION
GEANT4 package [43]  where r is a radius from the z-axis.
Injecting 10 5 protons on the target, the momentum and angle with respect to the beam axis distributions are obtained for π 0 , η, and η ′ , that are generated from interactions of the protons in the target.They are used as the initial condition to simulate production of the dark photons from the decays of these particles.

T2K NEAR DETECTORS
The ND280 detector is designed to measure the neutrino energy spectrum with neutral and charged current interactions, and is placed at 280 m from the neutrino production target and 2.5 • away from the beam axis.It consists of the Pi-zero Detector (PØD) [45], Fine Grained Detectors (FGDs) [46], Time Projection Chambers (TPCs) [47], Electromagnetic CALorimeter (ECAL) [48] and Side Muon Range Detector (SMRD) [49], that are installed in the previous UA1 magnet operated at 0. In T2K-II starting from 2023, PØD is replaced by three new detectors, Super-Fine Grained Detector (SuperFGD) [50], High Angle (HA) TPC [51] and Time Of Flight (ToF) [52].SuperFGD consists of 2.1 million scintillator cubes with 1 × 1 × 1 cm 3 , that are traversed by three Wave Length Shifting (WLS) fibers.The total size of SuperFGD is 192 × 182 × 56 cm 3 .HA-TPC is placed bellow and above the SuperFGD.It consists of two TPC layers with resistive Micromegas modules, and its total size is 181 × 223 × 85 cm 3 .The ToF system surrounds the SuperFGD and high angle TPCs, and consists of six panels with 250 × 230 cm 2 which is comprised of 20 scintillator bars.
In this study, we assume the region of TPCs in the ND280 at T2K-II as the effective area, which is 82% of the total volume.For the signal selection, we require ∆Φ < 90 • and cos θ > 0.992, where ∆Φ is angle between two charged tracks in the final states and θ is polar angle of the reconstructed dark photon.These requirements follow those used in search for heavy neutrinos in the final states of µ ± π ∓ or e ± π ∓ with the ND280 at T2K [32], except for ignoring the selection cut of the invariant mass less than 700 MeV.We assume 0.25 of the signal selection efficiency by referring to Fig. 4 in ref. [32] while it will be improved with ND280 at T2K-II due to the better track reconstruction efficiency.

Number of signal events
As we mentioned in section 2, we consider two types of production modes of dark photons, i.e., via the meson decays and proton bremsstrahlung.Thus, the number of signal events is divided into two parts and given by In the parentheses, the first term, dN meson DP , stands for the number of dark photons produced from the meson decays, and it is written by where N pot is protons on target (POT).The momentum of a meson X, its angle with respect to the beam axis, and the differential flux are denoted by p X , θ X and d2 N X /d|p X |dθ X , respectively, and they are numerically generated as explained in section 3. We incorporate productions from X = π 0 , η mesons with Y = γ, and X = η ′ with Y = γ and ρ.The overall factor P det (|p A ′ |, θ A ′ ) in eq.(5.1) is a probability that a dark photon having a momentum p A ′ and an angle θ A ′ decays inside the detector, and it is given in eq.(A.7).Note that p A ′ is determined by |p X |, θ X and the angles of the dark photon momentum; Those angles are generated and integrated out by running Monte Carlo simulations.For the dark photon branching ratio BR(A ′ → f f ), in this work, we take into account f = e ± , µ ± , π ± with the charged track reconstruction efficiency of EFCY = 0.25.Lastly, in order to take into account the signal events only in TPCs, the factor 0.82 (82%) is multiplied.
As for the bremsstrahlung production, we follow the calculation procedure summarized in ref. [14] 2 and define the number of produced dark photons as where z = p A ′ ,z /P beam with P beam being the beam momentum, p A ′ ,t is the transverse momentum of a dark photon, w(z, p 2 A ′ ,t ) is the splitting function derived in ref. [41] 3 , and Θ(Λ 2 QCD − q 2 min ) is a Heaviside step function to ensure the validity of the WW approximation.As for the proton-proton inelastic cross section, σ pp , we read off the data provided by the particle data group [35].Note that the probability P det (|p A ′ |, θ A ′ ) is integrated over z and p 2 A ′ ,t for the case of N brmss DP .The timelike form factor F (m 2 A ′ ) is multiplied to take into account the mixing with ρ and ω mesons.We adopt the parametrization proposed in ref. [54], which is based on the extended vector meson dominance approach and written as where V = ρ, ρ ′ , ρ ′′ , ω, ω ′ , ω ′′ .The masses of ρ and ω mesons are assumed to be the same for each family and set as m ρ = m ω = 0.77 GeV, m ′ ρ = m ′ ω = 1.25 GeV, and m ′′ ρ = m ′′ ω = 1.45 GeV.As for the decay widths, Γ ρ = 0.15 GeV, Γ ω = 0.0085 GeV, Γ ′ ρ = Γ ′ ω = 0.3 GeV, and Γ ′′ ρ = Γ ′′ ω = 0.5 GeV are assumed.We do not include the uncertainties of the masses and decay widths in our calculations.With these values, it is shown in refs.[55,56] that the parametrization can fit the experimental data for f ρ = 0.616, 339, and f ′′ ω = 0.369.It should be noted, however, that the experimental data in the timelike region are limited, in particular, near the resonance; The validity of eq.(5.4) is still under discussion at present.

Results
The standard neutrino interactions with matter in the TPC become the background in this analysis.Requiring existence of two charged particles and applying the selection cuts (∆Φ < 90 • and cos θ > 0.992), most of the neutrino events like ν µ n → µ − p charged current quasi elastic events can be removed.The remaining background is dominated by µ ± π ∓ final states in the charged current coherent pion production due to existence of two charged particles and difficulty to separate µ ± and π ± with the ND280, of that we take into account in this study.The number of events with the µ ± π ∓ final states after the selection cuts which is shown in TABLE II of ref. [32] is scaled to N pot and the volume of TPCs which is 60% larger in T2K-II than T2K.GENIE v3.0.6 [58] is used to simulate the final state objects in the coherent pion production and create the invariant mass distributions with the µ ± π ∓ final states.Figure 1 (left) shows the invariant mass distribution of µ ± π ∓ events before the event selection (blue), with the event selection of ∆Φ < 90 • (red), and ∆Φ < 90 3.8x10 21   1.0x10 22  3.7x10 22  The gray shaded areas are excluded by previous experiments and extracted from refs.[42,57].Also, as a reference, the recent results of 90% C.L. exclusion regions by NA62 [27,28] and FASER [29] are shown as purple and blue shaded areas, respectively.
As described in Sec. 4, the selection cut of m µπ < 700 MeV, that is used in ref. [32], is not applied in this study.Figure 1 (left) shows that the amount of the µ ± π ∓ events is not under estimated without any events at m µπ > 700 MeV.Then, based on this distribution, the invariant mass cut with the 2 σ resolution 14% is additionally applied, considering the momentum resolution for the charged particles with ND280. Figure 2 shows the number of µ ± π ∓ events that contaminate the signal region after the mass cut as a function of m µπ for N pot = 3.8 × 10 21 (blue), 1.0 × 10 22 (red) and 3.7 × 10 22 (green).
We calculate 95% C.L. exclusion regions by assuming the null observation of dark photons and rounding up the expected number of backgrounds.The dimensions of ND280 assumed in this study are summarized in table 1 in appendix A. The T2K experiment started physics In this figure, we also depict constraints by previous experiments as shaded areas.As can be seen, the new exclusion regions obtained in this work place a new constraint in the parameter region around m A ′ ≃ 0.7 -0.9 GeV and ε ≃ 10 −8 -10 −7 .Also, there is a marginal exclusion region around m A ′ ≃ 0.3 -0.4 GeV and ε ≃ 10 −7 .Compared with the past beam dump experiments, the ND280 detector is sensitive to a small kinetic mixing region.This is because the proton beam energy of the T2K experiment is lower and hence the produced dark photon is not boosted very much.As a result, the kinetic mixing must be small so that the dark photon can reach the detector.Although the production cross section of the dark photon is suppressed in such a situation, the high intensity of the proton beam can produce a sufficient amount of the dark photon.In order to see the difference of the production modes, in figure 4, we show regions predicting more than three (left panel) and ten (right panel) signal events for each production process.It is found that the new constraints are conduced from the η meson decay and bremsstrahlung production.π meson is produced more than η ′ however it is too light to give a new constraint while η ′ is less produced due to its mass.Also, in figure 5, we show the 95% C.L. exclusion regions for each final state: e + e − (top left), µ + µ − (top right) and π + π − (bottom).The π + π − final state mainly excludes the the expected sensitivity reaches of Belle II (green) [59], DUNE (red) [60], FACET (purple) [20], FASER2 (blue) [61] and SeaQuest (orange) [23].The shaded areas are the excluded regions by other experiments shown in figure 3.
mass region around 0.8 GeV.This is due to the mixing with ρ and ω mesons.
The ND280 detector is upgraded in 2023 as described in section 4 and will accumulate N pot = 1.0 × 10 22 data by the end of 2027.Furthermore, after 2027, the experiment with Hyper-Kamiokande (HK) will start, and N pot = 3.7 × 10 22 is expected in ten years.In figure 6, we show the expected sensitivity regions for these future projects and compare them with the projections of other future experiments.As can be seen from the figure, the sensitivity regions of T2K are very similar to that of the DUNE experiment [60], and they are sensitive for smaller ε compared with those of collider and beam dump experiments.experiments: Belle II (green) and SeaQuest (orange) are taken from ref. [9], DUNE (red) from ref.

It is interesting that
where g B−L stands for the new gauge coupling constant.The sum runs over all the SM fermions with the U(1) B−L charge Q f = 1/3 for all the quarks, while Q f = −1 for leptons.In accordance with refs.[55,57,64], we multiply g 2 B−L /(εe) 2 and g 2 B−L /(2εe) 2 by BR(π 0 → A ′ γ) and BR(η → A ′ γ) in eq.(2.3), respectively4 , while we ignore production from η ′ mesons since it is suppressed in comparison with π 0 and η for the case of U(1) B−L .Similarly, we multiply g 2 B−L /(εe) 2 by both the partial decay width in eq.(2.5) and the number of events from the bremsstrahlung production in eq.(5.3).Here, it should be stressed that U(1) B−L gauge bosons can decay into neutrinos; We assume three generations of massless Majorana neutrinos 5 with Γ νν = 3 × g 2 B−L m A ′ /(24π).Note also that U(1) B−L gauge bosons do not mix with ρ mesons because of the universal charge assignment for quarks.As a result, the decay branching ratio of A ′ → π + π − is suppressed to be negligible level.Thus, we ignore A ′ → π + π − and, also, turn off the resonant mixing with ρ, ρ ′ , ρ ′′ mesons in the timelike form factor included in eq.(5.3).By incorporating only e + e − and µ + µ − final states, in figure 7, we derive the 95% C.L. exclusion region and expected sensitivity regions of T2K for U(1) B−L .In comparison with the case of dark photons, sensitivity regions disappear around 0.2 GeV -0.6 GeV because of the lack of the resonant mixing with ρ mesons.

SUMMARY
We have studied the constraints on the dark photon and U(1) B−L gauge boson from T2K off-axis ND280 detector.We have simulated the dark photon production from light meson decays and proton bremsstrahlung assuming 3.8 × 10 21 POT, corresponding to ten-years operation of T2K.It is found that the unexplored parameter region is excluded due to no observation of two electron, muon and pion tracks as shown in figures 3 and 7, for dark photon and U(1) B−L gauge boson, respectively.We also analyzed the expected sensitivity for the future upgrade of the ND280 detector and proton beam line.The results are shown in figures 6 and 7.One can see that the upgraded T2K experiment will complement the searches by other future experiments such as FASER2, SeaQuest, and FACET.
The analyses given in this paper can be applied to other new particles beyond the SM, for instance, axion-like particles and light scalar bosons.These particles also can be produced from the meson decays and proton bremsstrahlung.When one considers a light scalar boson as the origin of the dark photon mass, the decay of such a scalar boson will give significant contributions to the production of the light gauge boson (see, for instance, ref. [65]).We leave these topics for our future works.In our numerical calculations, we assume that the ND280 detector is a rectangular with dimensions of 2.4 m × 2.4 m × 5.8 m.We also assume that the target is point like, the detector is located 280.1 m away from the target, positioned in the direction of 2 •6 from the Nevertheless, in our numerical calculations, we drop the z dependence in P (θ A ′ , z) by fixing r at the center of the detector, i.e., r = 283 m × tan θ A ′ .This can be validated by the fact that the detector depth is much shorter than the distance between the target and detector.
In this case, the z integral in eq.(A.1) can be analytically done, only case (ii) applies, and the probability is simplifyed to be where θ upr and θ lwr are given in eq.(A.3).
version 4.11.0.3 with physics package of QGSP BERT is used to generate the primary protons and simulate interactions of particles in the materials.The protons with 30 GeV are injected to the graphite target with 13 mm in radius and 90 cm in length.The origin of the coordinate is defined to be at the center of the graphite target in the x-y plane and at the upstream end of the graphite target in the z-coordinate which is parallel to the proton beam direction.Three Electro-Magnetic Horns (EMHorns) with two layers of a simplified shape are placed right after the target to focus secondary charged particles.The radii of the inner (outer) layers are set to 27 (200), 40 (500), and 70 (700) mm for the first, second, and third EMHorns, respectively, with 3 mm thickness of the aluminum.Their lengths and central positions are set to (1.5 m, 0.75 m), (2.0 m, 2.81 m), and (2.5 m, 9.98 m), respectively.The magnetic field is applied with a formula of B[T] = 0.063/r [m] to reproduce that shown in Fig. 7 in ref. [44], 2 T. The PØD is a scintillator based tracking calorimeter optimized to measure π 0 produced in the neutral current interactions of neutrinos.The FGDs consist of two layers of scintillator bars and are placed between three TPC layers.The charged particles created in the charged current interactions are measured with FGDs and TPCs.The ECAL is placed to surround PØD, FGDs and TPCs, and the SMRD is integrated inside the return yoke of the magnet.

Figure 2 .
Figure 2. The number of µ ± π ∓ events that contaminate into the signal region after the mass cut with 2 σ resolution as a function of m µπ .The blue, red and green points correspond to N pot = 3.8 × 10 21 , 1.0 × 10 22 and 3.7 × 10 22 , respectively.

Figure 3 .
Figure 3. Exclusion regions for the dark photon model.The regions surrounded by the black solid curves are the 95% C.L. exclusion regions with N pot = 3.8 × 10 21 , that are obtained in this work.

Figure 4 .
Figure 4. Left (Right): the regions predicting more than three (ten) events for each production process with N pot = 3.8 × 10 21 .The black solid, blue dashed, green dotted, and orange dasheddotted curves correspond to productions from the bremsstrahlung process, π 0 decay, η decay, and η ′ decay, respectively.The shaded areas are the excluded regions by other experiments shown in figure 3.

Figure 7 .
Figure 7.The 95% C.L. exclusion region and expected sensitivity regions of T2K for the U(1) B−L model.The black solid, dashed and dotted curves correspond to N pot = 3.8 × 10 21 , 1.0 × 10 22 , and 3.7×10 22 , respectively.The gray shaded areas are excluded by previous experiments and extracted from refs.[42, 57].The colored dashed curves represent the expected sensitivity reaches of other photon model.Before closing this study, we demonstrate the applicability of our calculation to other light gauge bosons.As an example, we here consider a U(1) B−L gauge boson having the following Lagrangian:

Figure 8 .Figure 9 .
Figure 8.A schematic view of the location of ND280.

Table 1 .
The dimensions and location of the ND280 detector.