Total and differential cross sections of $\eta$-production in proton-deuteron fusion for excess energies between $Q_\eta=13\;\text{MeV}$ and $Q_\eta=81\;\text{MeV}$

New data on both total and differential cross sections of the production of $\eta$ mesons in proton-deuteron fusion to ${}^3\text{He}\,\eta$ in the excess energy region $13.6\;\text{MeV}\leq Q_\eta \leq 80.9\;\text{MeV}$ are presented. These data have been obtained with the WASA-at-COSY detector setup located at the Forschungszentrum J\"ulich, using a proton beam at 15 different beam momenta between $p_p = 1.60\;\text{GeV}/c$ and $p_p = 1.74\;\text{GeV}/c$. While significant structure of the total cross section is observed in the energy region $20\;\text{MeV}\lesssim Q_\eta \lesssim 60\;\text{MeV}$, a previously reported sharp variation around $Q_\eta\approx 50\;\text{MeV}$ cannot be confirmed. Angular distributions show the typical forward-peaking that was reported elsewhere. For the first time, it is possible to study the development of these angular distributions with rising excess energy over a large interval.


Introduction
The production of η mesons off nuclei has been a topic of active research over at least two decades. Inspired by the attractive interaction between η mesons and nuclei, first studied by Bhalerao, Haider and Liu [1,2], extensive experimental effort was put into the study of near-threshold production of η mesons off various nuclei [3,4,5,6,7,8,9,10]. Although the original work suggested studies on heavier nuclei, the reaction pd → 3 He η is one of the most discussed due to its markedly enhanced cross section very close to the production threshold. Here, it was observed that the production cross section σ rises from zero at threshold to around 400 nb within less than 1 MeV of excess energy [11,12,13,14]. This curious behaviour of the production cross section has first been discussed in the context of a strong final state interaction and the presence of a possible (quasi-)bound η 3 He state close to the threshold in [15], which was later followed up on, e.g., in [16,17]. However, while the production cross section of the reaction pd → 3 He η has been studied in great detail close to threshold, at higher excess energies the available database becomes sparse. Measurements by the CELSIUS/WASA [18], COSY-11 [19] and ANKE experiments [20] seem to suggest a cross section plateau away from threshold, whereas a measurement by the GEM collaboration [21] yielded a larger cross section value, albeit with a sizable uncertainty. Recently, in [22], a sharp variation of the total cross section between Q η = 48.8 MeV and Q η = 59.8 MeV has been reported. In order to further investigate the existence and cause of this cross section variation, a new measurement was performed at 15 different beam momenta between p p = 1.60 GeV/c and p p = 1.74 GeV/c, using the experimental apparatus WASA at the COoler SYnchrotron COSY. Apart from determining the total cross section of the proton-deuteron fusion to the 3 He η final state, the focus of the new measurement is on the precise determination of differential cross sections and the study of their development with rising excess energy. Such a comparison between differential distributions at different excess energies has thus far been hindered by large systematic differences between the individual measurements performed in the various experiments mentioned above. For this reason, a coherent measurement over a large range of higher excess energies by a single experiment does for the first time present the possibility for an in-depth study of the dependence of the differen-tial cross section on the excess energy. Here, high quality data are of great importance in order to facilitate theoretical work on the production mechanism of η mesons in proton-deuteron fusion, as has recently been claimed in [23]. Up to now, no model exists that manages to correctly reproduce the total and differential cross sections away from the production threshold. While the two-step model, first studied by [24] in a classical framework and by [25] quantum-mechanically, has some success in describing near-threshold data (see, e.g., [23,25]), at larger excess energies the model no longer describes the available database [26]. In [27], it was reported that the GEM data can be adequately described by a resonance model, in which η mesons are produced from the decay of a N * resonance. Such a model is, however, unlikely to have a large contribution close to threshold due to the large momentum transfer necessary to compensate for the η meson mass. It remains to be resolved if, why, and at which energy the production mechanism of the reaction pd → 3 He η changes. It is for these reasons that in [23] new data at larger excess energies were assessed to be of high importance.

Experiment
The measurement was performed using the WASA detector setup (which is described in detail in [28]) at the storage ring COSY of the Forschungszentrum Jülich. Utilizing the so-called supercycle mode of the storage ring, the momenta of the beam protons are changed at each injection of a new proton bunch. Eight beam settings can be stored at once and the measurement is composed of two such supercycles (SC), each containing the eight beam momenta (flat-tops) indicated in table 1. In total, data were taken at 15 different beam momenta between p p = 1.60 GeV/c and p p = 1.74 GeV/c with a momentum spread of around ∆p/p = 10 −3 [29] and a stepsize of 10 MeV/c. The measurement at a momentum of p p = 1.70 GeV/c was repeated during both supercycles and in an additional single-energy measurement for systematic checks. Inside the WASA Central Detector the beam protons are steered to collide with a deuterium pellet target. Due to the fixed-target geometry, heavy ejectiles like 3 He are produced near the forward direction and subsequently stopped inside the WASA Forward Detector. Here, using a proportional chamber and various layers of plastic scintillator, both the production angles ϑ and ϕ, and the energy deposit of forward-going particles are reconstructed. Doubly charged Helium ions can be efficiently separated from protons, deuterons and charged pions by their energy deposit. From the deposited energy, the kinetic energy of 3 He nuclei is also reconstructed, thus, in combination with the determined scattering angles, fully reconstructing their four-momenta.

Data Analysis
For a two-particle final state such as 3 He η, the polar angle ϑ3 He and the kinetic energy T3 He of the Helium nuclei are kinematically correlated. Using this relation, the precise measurement of the polar angle ϑ3 He (∆ϑ3 He ≈ 0.2 • ) can be exploited to find a highly accurate calibration of the reconstructed energy. A comparison of the two-dimensional distribution of ϑ3 He versus T3 He between the kinematical expectation for the signal reaction pd → 3 He η and the data obtained at p p = 1.70 GeV/c can be found in Fig. 1. The reaction of interest is identified from the spectra of the final state momentum of 3 He nuclei in the centre-of-mass frame p * 3 He in a missing-mass analysis. Thus, no assumption on the η decay is made. Dividing the cosine of the centre-of-mass scattering angle cos ϑ * η into 100 equally sized bins, the final state momentum spectra are fitted by a background function, excluding the peak region. Here, the background is a sum of Monte Carlo (MC) simulations of twoand three-pion production as well as a third order polynomial, accounting both for other possible background reactions and deviations from simple phase space distributions in the case of the three-pion production. The simulation of double-pion production was performed using a model incorporating the ABC effect and t-channel double-∆(1238) excitation, developed for [30]. An example of such a fit can be found in Fig order to determine the signal yield in a given bin in cos ϑ * η , the background subtracted data are summed over the interval p * η − 3σ ≤ p * 3 He ≤ p * η + 3σ, where p * η and σ are the position and width of the signal peak determined from a fit of an appropriate peak function to the background subtracted data. For most values of cos ϑ * η a simple Gaussian is chosen. However, close to the maximum scattering angle the breakup of 3 He nuclei in the detector leads to asymmetric peaks (see Fig 2) that are fitted by a double-Gaussian. In these cases, peak position and width of the dominant signal contribution are used. Before physically meaningful angular distributions are obtained, the signal yield needs to be corrected for the product of detector acceptance and reconstruction efficiency. This can be derived from MC simulations. In contrast to the earlier work [22], an extension to the GEANT3 software package [31] provided by the authors of [18] was used to simulate nuclear breakup of 3 He nuclei in the scintillator material. Additionally, the possibility that the primary proton-deuteron interaction does not occur with the pellet target but with evaporated target gas was accounted for. First, simulations of the signal reaction pd → 3 He η were performed with cos ϑ * η equally distributed over all values from −1 to +1. From this set of simulations, the product of acceptance times reconstruction efficiency was calculated as the ratio of the number of events reconstructed in a bin of cos ϑ * η divided by the number of events that were generated in that bin. However, only if the detector resolution were perfect, would this ratio directly correspond to the sought-after product of acceptance and reconstruction efficiency. Otherwise, the finite detector resolution, in combination with angular distributions that exhibit a strong angular dependence, causes a bin migration effect in the opposite direction to the gradient of the angular distribution. In addition, the nuclear breakup introduces a tendency to reconstruct the 3 He nuclei at slightly smaller kinetic energies. To account for these effects, the acceptance correction is done in an iterative manner. For this, the angular distributions observed in data after correcting for the acceptance derived from the MC sample equally distributed in cos ϑ * η are fitted by a third order polynomial These polynomials are subsequently used to generate a new set of MC simulations with which the product of acceptance and reconstruction efficiency can again be determined. This procedure is repeated until convergence of all angular distributions is reached. As an example, the angular distribution, along with the product of acceptance and reconstruction efficiency of the sum of the three measurements at p p = 1.70 GeV/c, is displayed in

Normalization
For the measurement presented here, normalization consists of two steps. The luminosity of the sum of the three measurements at p p = 1.70 GeV/c (Q η = 61.7 MeV) is determined by comparison of the integral over the fit to the 3 He η angular distribution displayed in Fig. 3 and the total cross section value of σ = (388.1 ± 7.2 stat. ) nb (with an additional 15% normalization uncertainty), as measured by the ANKE collaboration at Q η = 60 MeV [20]. Then, the measurements at the 14 remaining beam momenta are normalized relative to the luminosity derived for p p = 1.70 GeV/c. Whereas this relative°/ normalization was performed using the single pion production pd → 3 He π 0 in [22], a different ansatz was used in this work. Here, the proton-deuteron elastic scattering is used for two rea-sons. On the one hand, data from [32,33,34,35,36] suggest that, within the experimental uncertainties, the pd elastic differential cross section dσ/dt does not vary with p p , thus allowing one to cancel the literature cross section and its uncertainty in a relative normalization. On the other hand, as one of the objectives of this new measurement is to examine the cross section variation observed in [22], it is desirable to use an independent normalization method. The elastic pd scattering can be identified by demanding coincident charged particles in the forward and central detector. As the forward-going protons are minimum ionizing, the measurement of their energy deposit does not aid in determining their kinetic energy. Instead, the analysis is performed using only the measured scattering angles of the charged particles in the forward and central detector. First, a cut on the coplanarity of the two tracks is set at 120 • < |ϕ FD − ϕ CD | < 240 • . Afterwards, the polar angles of the two particles are compared. In the case of a two-particle final state, the polar angles of both particles are directly related and can each be expressed as a function of the other angle. In Fig. 4a, this relation is displayed for data in comparison to the kinematical expectation. Under the assumption of an event corresponding to proton-deuteron elastic scattering, the momentum transfer t is calculated as a function of the polar angle of the forward-going proton. In addition, the minimum distance d to the kinematic expectation for pd elastic scattering is calculated for each pair of measured polar angles ϑ FD and ϑ CD . As can be seen from Fig. 4b, the distance d exhibits a narrow peak close to d = 0 for momentum transfers in the region 0.140 (GeV/c) 2 ≤ −t ≤ 0.215 (GeV/c) 2 on top of a strong background contribution caused by quasi-elastic proton-proton scattering pd → ppn spec . For 15 bins in the momentum transfer range indicated above, the distribution of the distance d is fitted by a fourth order polynomial for the background, again excluding the signal region. In this way, the acceptance-corrected event yield for proton-deuteron elastic scattering as a function of −t is determined for each beam momentum (see Fig. 4c). A fit to the combined database from [32,33,34,35,36], given by f (−t) = exp(12.45−27.24 (GeV/c) −2 ·|t|+26.31 (GeV/c) −4 ·|t| 2 ) originally performed by the ANKE collaboration and already applied for the luminosity determination in [37], is scaled to the observed distribution dN/dt in a one-parameter fit. The data used for the luminosity determination in [37] is currently being prepared for publication [38]. Under the assumption, discussed later, that the cross section dσ/dt is independent of the beam momentum, the relative luminosity of a measurement at p ′ p compared to the sum of the three measurements at p p = 1.70 GeV/c is directly given by the ratio of the two scaling factors.

Results
The resulting total cross sections at all 15 excess energies are given in table 2 and displayed in Fig. 5, where they are compared to the database available in the literature. As the cross section at Q η = 61.7 MeV is fixed to the value of σ = 388.1 nb, the statistical uncertainty of the new measurement at that excess energy needs to be considered as a collective uncertainty of the whole chain of points relative to the fixed value. Additionally, asymmetric systematic uncertainties were found. Generally, using the supercycle mode of the accelerator, systematic effects due to changes to the experiment or environmental conditions can be ruled out. Also, by individual analysis and comparison of the three measurements at p p = 1.70 GeV/c, no systematic changes between the data-taking periods were found. Performing a relative normalization, systematic effects due to inefficiencies are largely canceled out. Two main sources of systematic uncertainty remain. The distribution and density of evaporated target gas in the scattering chamber is not known to high precision. As a shift of the vertex location along the beam axis leads to a loss of information for large polar angles ϑ, variation of density and distribution in Monte Carlo simulations has implications on the geometrical acceptance that are larger for higher excess energies, when the maximum scattering angle of the 3 He nuclei is larger. In addition, while the assumption that the differential cross section of pd elastic scattering dσ/dt is constant as a function of the beam momentum is in accordance with the precision of the available data, calculations by [39,40] have shown that the integral over the cross section dσ/dt in the interval 0.140 (GeV/c) 2 ≤ −t ≤ 0.215 (GeV/c) 2 changes slightly but linearly with the beam momentum. Keeping the measured value at p p = 1.70 GeV/c at a fixed position, the linear change in dσ/dt introduces a change in normalization by roughly 4% at p p = 1.60 GeV/c and −2% at p p = 1.74 GeV/c. In addition, the overall normalization factor from the comparison of the Q η = 61.7 MeV data with the total cross section published in [20] comes with an uncertainty of 16.3%. Of this, 15% is the normalization uncertainty of the literature cross section and an additional 6.3% uncertainty was found when different subparts of the differential cross section were used for normalization instead of the total cross section. These 16.3% are, however, irrelevant when the energy dependence of the total cross section is studied. From Fig. 5, it is apparent that the sharp variation of the total cross section that was previously reported in [22] is not confirmed by the present measurement. However, repeating the normalization procedure used in [22], it could be shown that the behaviour observed in [22] is indeed reproduced. In [41], it is shown in detail that the effect is caused by an incorrect assumption regarding the differential cross section of the single pion production. In the excess energy interval 20 MeV Q η 60 MeV, the increase and subsequent leveling of the total cross section of the reaction pd → 3 He η that was observed in [18,20] is also observed in the present work. It can, however, be studied in a lot more detail than was previously possible. The differential cross sections derived in the present work are displayed in Fig. 6. Generally, the distributions at all energies exhibit the forward-peaking that was previously observed in other experiments. At all energies, the differential cross sections can be described by a third order polynomial, with no need for a quartic term. Due to the large amount of data gathered, the angular distributions as well as their energy dependence can be studied in unprecedented detail. The values of the fit parameters Cyan stars are from [11], blue boxes from [12], green open triangles from [21], orange open diamonds from [18], purple filled circles from [13], gray upward filled triangles from [19], black downward filled triangles from [14,20], brown open circles from [22] and red filled diamonds from the present work. Here, the error bars indicate the statistical point-to-point uncertainty, red boxes indicate the statistical chain-to-point uncertainty relative to the fixed cross section at Q η = 61.7 MeV and gray boxes indicate the systematic uncertainty. In addition, a normalization uncertainty of 16.3% is not displayed here. Similarly, the normalization uncertainties of the literature data are not displayed. the systematic uncertainty due to the aforementioned evaporated gas, an additional element arises from minor imprecisions in the determination of the polar angle. In relative normalization, this effect cancels and thus does not influence the determination of the total cross section. The asymmetry parameter α is of special importance, as it is regularly used to study an inter-ference between s-and p-waves in the near-threshold data (see, e.g., [14,16]) in the search for indications of η-mesic states below threshold. In Fig. 7, the three lowest energy values of the present work are compared to the values of the asymmetry parameter extracted in [14,20] and [13]. Slightly better agreement with the higher values from [13] is found. The ANKE value at Q = 19.5 MeV is in strong conflict to the findings reported here, however, as is already argued in [20], the inclusion of this point into a combined fit with the data from [14] yields an unsatisfactory result.    Fig. 5. The data from [21] were omitted due to the large uncertainties.

Summary
In the course of this work, total and differential cross sections of the η meson production in proton-deuteron fusion were extracted. The differential distributions exhibit the same forwardpeaking behaviour as previously observed away from the reaction threshold. Due to the amount and quality of the data, it is possible for the first time to study changes in the shape of the angular distributions with rising excess energy in a large interval between 13.6 MeV and 80.9 MeV. In this way, the contributions of higher partial waves might be studied which will greatly aid in the investigation of the production process that remains largely unknown. A previously reported sharp variation of the total cross section around Q η ≈ 50 MeV is not confirmed. However, the fluctuating structure of the production cross section between Q η ≈ 10 MeV and Q η ≈ 60 MeV that had already been observed by both the WASA/PROMICE and ANKE experiments [18,20], albeit in much less detail, is nicely reproduced. Close to the production threshold, effects of a strong final state inter-    action are thought to be a dominating contribution to the total cross section. The observed structure reported here might indicate the energy region in which the final state interaction loses its importance. With none of the available theoretical models being able to reproduce the forward-peaking in the angular distributions as well as the observed total cross section, further theoretical effort is clearly needed in order to fully understand the production of η mesons off 3 He nuclei. : Asymmetry parameter α of the angular distributions of the reaction pd → 3 He η, comparing the presented data (red diamonds) to values extracted by the ANKE [14,20] and COSY11 [13] experiments (black downward and gray upward triangles, respectively). Systematic uncertainties of the present work are shown as gray boxes. In the case of the data from [14,20] and [13], thick lines represent statistical uncertainties, thin lines systematic ones.