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A model of 풏̅ annihilation in experimental searches for 풏 → 풏̅ transformations E. S. Golubeva1, J. L. Barrow2, C. G. Ladd2 1Institute for Nuclear Research, Russian Academy of Sciences, Prospekt 60-letiya Oktyabrya 7a, Moscow, 117312, Russia 2University of Tennessee, Department of Physics, 401 Nielsen Physics Building, 1408 Circle Drive, Knoxville, TN 37996, USA

Searches for number violation, including searches for decay and - transformation (푛 → 푛̅), are expected to play an important role in the evolution of our understanding of beyond Standard Model physics. The 푛 → 푛̅ is a key prediction of certain popular theories of , and the experiments such as the Deep Underground Experiment and the European Spallation Source plan to search for this process with bound- and free-neutron systems. Accurate simulation of this process in Monte Carlo will be important for the proper reconstruction and separation of these rare events from background. This article presents developments towards accurate simulation of the annihilation 12 process for use in a cold, free neutron beam for 푛 → 푛̅ searches from 푛̅퐶 annihilation, as 6퐶 is the target of choice for the European Spallation Source’s NNBar Collaboration. Initial efforts are also made in this 40 paper to perform analogous studies for intra-nuclear transformation searches in 18퐴푟 nuclei.

12 I. INTRODUCTION being allowed to hit a target of carbon ( 6퐶) foil A. Background (with a thickness of ~130 휇푚). This foil would have absorbed , resulting in - As early as 1967, A. D. Sakharov pointed out [1] annihilation which was expected to that for the explanation of the Baryon Asymmetry yield a signal with a star-like topology made of of the (BAU) there should exist several . Particle detectors and calorimeters interactions in which baryonic is violated surrounded the target to record such annihilation besides mere departures from thermal events, and was capable of reconstructing the equilibrium and 퐶푃 symmetry. Thus, vertex of the -star within the central plane of experimental searches for baryon number (퐵) the 12퐶 foil along with the visible . In total, violating processes, and in particular the baryon 6 the target received ~3 × 1018 , with no minus lepton (퐵 − 퐿) number violating process of recorded annihilation events, i.e. with zero neutron—antineutron oscillation (푛 → 푛̅), are of background. This was due to an analysis scheme great importance due to their possible requiring two or more tracks (푛̅-annihilation or connections to the explanations of the observed background-produced , or their decay matter-antimatter asymmetry of the universe—as products) to be reconstructed in the detector as first laid out by V. A. Kuzmin [2] and followed in emanating from the 12퐶 foil. As a result, the developments by many authors, see e.g. recent 6 oscillation limit for free neutrons was established reviews [3-5]. to be Thus, the search for 푛 → 푛̅, along with 8 ( ) decay, remains one of the most important areas of 휏푛→푛̅ ≥ 0.86 × 10 푠. 1 modern physics, hopefully leading to an In the last two decades since obtaining this result, understanding of phenomena related to the BAU. there have been significant technological developments within the field which have The best lower limit on a measurement of the permitted the planning of another transformation oscillation period with free neutrons, 휏 , was 푛→푛̅ experiment, recently proposed at the currently attained at a reactor at the Institut Laue-Langevin under construction European Spallation Source (ILL) [6] in Grenoble, France, with a cold neutron (ESS) [5,7,8]. According to preliminary beam. These neutrons flew through an evacuated, estimates, such an experiment could explore this magnetically shielded pipe of 76 푚 in length process with 2 − 3 orders of magnitude higher (corresponding to a flight time of ~0.1 푠), until sensitivity than in [6], leading next generation

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free neutron experiments to be sensitive to improvement in the separation of signal to 9 10 oscillation time range 휏푛→푛̅~10 − 10 푠. background in new experiments, it will be possible to improve the appearance limit, but Another way to detect 푛 → 푛̅ is through intra- impossible to claim any real discovery. This nuclear searches, and discovery is tantalizing contrasts the tantalizing figure that future possible. Searches for 휏푛→푛̅ can be performed in experiments in large underground detectors could experiments with large underground detectors improve the restrictions on processes where looking for any hints of the instability of matter. Δ퐵 = ±2 up to ~1033 − 1035 푦푟푠 [5] in the Within the nucleus, spontaneous 푛̅ production absence of background. An experiment possibly would lead to annihilation with another capable of such a search for 푛 → 푛̅ within the neighboring nucleon, resulting in the release of 40퐴푟 nucleus is currently under construction, ~2 퐺푒푉 of total energy. However, such intra- 18 using large liquid argon (40퐴푟) time projection nuclear transformations are significantly 18 chamber: the Deep Underground Neutrino suppressed compared to 푛 → 푛̅ in vacuum [5,9- Experiment (DUNE) [20]. 13]. The limit on the 푛 → 푛̅ intra-nuclear transformation time (in matter) 휏푚 is associated Whether or not 푛 → 푛̅ is definitively observed with the square of the free transformation time [5] above background in intra-nuclear experiments through a dimensional suppression factor, 푅: depends critically upon the separability of signal from background and the energy scale at which 2 ( ) 휏푚 = 푅 ⋅ 휏푛→푛̅ 2 the new BSM mechanism will appear. In the case In the nucleus, this suppression is due to of an observation in intra-nuclear experiments the differences between the neutron and antineutron results will be of great importance for the nuclear potentials; however, in high mass understanding of fundamental properties of detectors, this suppression can be compensated matter, along with building a precise theoretical by the large number of neutrons available for model describing these properties. Although in investigation within the large detector volume. A the free neutron search [6] no background was number of nucleon decay search collaborations detected, the question of background separation have been involved in the search for 푛 → 푛̅ in might become essential with the planned increase nuclei, such as Frejus [14] and Soudan-2 [15] in in sensitivity in searches using both free neutrons 56 produced by spallation and bound neutrons in 26퐹푒, and IMB [16], Kamiokande [17], and 16 underground experiments, meriting further study Super-Kamiokande (SK) [18] in 8푂; there has also been a deuteron search performed at SNO beyond this work. [19]. In the Soudan-2 experiment, there is a limit Thus, one requires detailed information about the on the transformation time in iron nuclei of 휏퐹푒 ≥ processes during the annihilation of slow 7.2 × 1031 푦푟푠 [15], which is in line with the antineutrons on nuclei. The purpose of this work limit for the free transformation time of 휏푛→푛̅ ≥ is to create a model describing the annihilation of 8 12 1.3 × 10 푠. In SK, which extracted 24 푛 → 푛̅ a slow antineutron incident upon a 6퐶 nucleus candidate events while expecting a background for the upcoming transformation experiment count of 24.1 atmospheric neutrino events, these using a free neutron beam at ESS. Also, the first 32 limits were 휏푂 ≥ 1.9 × 10 푦푟푠 [18] and steps have also been taken towards a full, realistic 8 휏푛→푛̅ ≥ 2.7 × 10 푠, respectively. simulation of the annihilation resulting from 푛 → 푛̅ within 40퐴푟 nuclei for DUNE. The prevalence of background within SK and 18 other large underground detectors, possibly B. Past simulation for free and bound shrouding a true event, prioritizes the rigorous 풏 → 풏̅ searches modeling of both signal and background within an intra-nuclear context. Without any significant

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In general, the experiment requires maximum C. This work and its goals efficiency for detection and reconstruction of Our goal is to create an adequately accurate incredibly rare antineutrons to be separated from generator, one which can serve as a platform to background. The development of Monte Carlo be used within all free and intra-nuclear 푛 → 푛̅ (MC) generators for 푛 → 푛̅ searches is not new, experiments. In this article, we present the main and has been an integral part of all past framework and approaches underlying the model, experiments. Sadly, the descriptions of these wherein the annihilation of an antineutron on the MCs, as known, are not always complete or target nucleus is considered to consist of several seemingly consistent, and are not easily sequential and independent stages. We use the accessible. Information about the generator approach originally undertaken in [24,25]. developed for the ILL experiment [6] is few and far between, unavailable [21], and lacking [22] in In the first stage of this approach, one defines the detailed explanation. absorption point of an antineutron by the nucleus in the framework of the optical model. Our Intra-nuclear searches have been completed far modeling was performed for more times than free neutron experiments, and so 10 푚푒푉 12 their accompanying generators are similarly antineutrons incident upon a 6퐶 nucleus [24,25]. 40 abundant. Never-the-less, many of their For 18퐴푟, 푛 → 푛̅ is assumed to occur within the descriptions are scattered throughout a multitude nucleus, where the have some Fermi of dissertations and are poorly defended within motion, and the present paper shows some first published works. Similarly, open access to these steps in this direction; the process of 푛 → 푛̅ 40 simulations is lacking. For instance, SK [18] cites within 18퐴푟 will be the focus of our future work. only three works in reference to their generator, After the point of these quite different initial one of which is a previous work of this paper’s conditions, all of the following stages of the 12 40 lead author, and two of which contain rather process for both 6퐶 and 18퐴푟 do not differ and ancient annihilation data; how exactly are considered within a unified approach. these are implemented within their model is not The second stage in this approach is the actual available. annihilation of the antineutron with one of the The authors are also aware of Hewes’ work in constituent intra-nuclear nucleons. In contrast to relation to 푛 → 푛̅ in DUNE [23]. However, there [24,25], where a statistical model for nucleon- exist similar issues to those seen in [18], among antinucleon annihilation into pions was used, the them the assumption that the annihilation occurs present paper instead uses a combined approach along the density distribution of the nucleus first proposed in [26] and will be described in despite the supposed use of work in [13]. In both Section II D. In this paper we use a version of the [18] and [23], only ~10 of exclusive annihilation annihilation model originating in 1992, utilizing channels are used, whereas our model utilizes corresponding experimental data available at that ~100 derived from experimental and theoretical time. While there do exist more recent findings techniques. No previous studies are known to from later analyses of LEAR data, these are not have been tested on their ability to reproduce many in number and will not greatly affect the antinucleon annihilation data, which is a central conclusions reached for ESS from the model, as feature of our work. Our present model is also the these can only slightly modify the probabilities of first published to incorporate a proper description various annihilation channels within the database of the annihilation’s dependence on the of our simulation; these can be updated at a later interaction radius within carbon. Work is time when we seek even greater precision for 40 underway on a proper implementation of this 18퐴푟.The third stage is the intra-nuclear cascade 40 (INC), initiated by the emergence and nuclear concept within 18퐴푟. transport of mesons from the annihilation; decays

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of short-lived resonances are also handled. In this model, as is usually done for antinucleon paper, we use the original version of the model above several tens of 푀푒푉. Such an interaction which takes into account the nonlinear effect of also cannot be legitimately modeled using decreasing the nuclear density, along with a time antineutron-nucleon cross sections. The approach coordinate [27], which is necessary for the correct used here to describe the interaction between the description of the passage of resonances through nucleus and the incoming slow antineutron the nucleus. resulting from the transformation is based on the integration of optical and cascade models. In the The final stage is the de-excitation of the residual optical-cascade model, the initial conditions for nucleus. the INC are formulated within the optical model. In this paper, we present a general description of This approach was first applied in [28] to describe the model and the first results obtained for 푛̅퐶 in the annihilation of stopping on nuclei preparation for the forthcoming ESS experiment. when the antiproton is absorbed from the bound In future developments, the description of state made by the antiproton orbiting the atom. individual stages of the process can be modified The same approach was used for the antineutron and improved, but the approach remains the by L. A. Kondratyuk [24,25] in the discussion of same. future 푛 → 푛̅ search experiments. The radial (푟) distribution of the absorption probability density The outline of this paper is as follows: Section II 푃푎푏푠(푟) is directly related to the radial nuclear provides a detailed description of the model for density 휌(푟) and the radial wave function 휙(푟), all successive stages of the process under and is derived from the wave equation for a slow consideration. In Section III, a comparison will antineutron: be made between simulation and experiment to 2 2 test the model against existing at-rest 푝̅퐶 푃푎푏푠(푟)~4휋푟 휌(푟) |휙(푟)| . (3) annihilation data. In Section IV, some validation This solution for a slow, plane wave antineutron tests of the 푛̅퐶 annihilation event generator incident on a 12퐶 nucleus was presented in great output data are shown. In Section V, we 6 detail in [24,25]. In order to define the summarize our work, and briefly consider a annihilation point in simulation, it is desirable to future path toward simulation of intra-nuclear 40 use a simple analytic function. Therefore, we transformations in 18퐴푟. approximate the solution 푃푎푏푠(푟) obtained in II. THE MODEL OF THE [24,25] as a Gaussian function, with a maximum ANTINEUTRON ANNIHILATION situated at 푟 = 푐 + 1.2 푓푚, where 푐 is the radius 12 ON THE NUCLEUS of half density (with 푐( 6퐶) = 2.0403 푓푚) with A. Absorption of the slow antineutron by the a width of 휎 = 1 푓푚. This approximated ퟏퟐ ퟔ푪 nucleus function is presented in Fig. 1 as the solid orange curve with arbitrary units to demonstrate the In this work, we simulate the annihilation of a penetration depth of the antineutron inside the 12 cold (~10 푚푒푉) 푛̅ on a 6퐶 nucleus. The nucleus. calculation of the total annihilation cross section 12 The model assumes that the proton density within of an 푛̅ on 6퐶 is a separate problem that is not considered within the scope of this model,, and the nucleus 휌(푟) is described as an electrical instead the annihilation event itself is the starting charge distribution, as obtained in high-energy point. scattering experiments. The function 휌(푟) obeys a Woods-Saxon distribution: The interaction of a slow antineutron with the nucleus cannot be considered within the 휌(푟) 푟−푐 −1 = [1 + 푒 푎 ] , (4) framework of the intra-nuclear cascade (INC) 휌(0)

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where 푎 = 0.5227 푓푚 is the diffuseness For intranuclear 푛 → 푛̅ transformation, the parameter of the nucleus, and 푐 the radius of half conversion of a bound neutron into an associated density [29]. For practical reasons within the antineutron is significantly suppressed within the modeling process, the nucleus is split into seven nuclear environment. The reason for this is the concentric zones, within which the nucleon large difference between the values of the density is considered to be constant. Fig. 1 shows effective nuclear potential for the neutron and 12 the density distribution of the nucleons for 6퐶, antineutron [5,10]. The question of the magnitude calculated by equation (4), along with a step of the suppression of 푛 → 푛̅ occurring within the approximation which divides the nucleus into nucleus has been the subject of in depth nuclear seven zones of constant density. It is seen that theoretical discussions for a number of years [10- although an antineutron penetrates more deeply 13,30-32]. The transformation would be more compared to an antiproton (the dotted line in the probable for neutrons with lower , Fig. 1), the absorption of the antineutron still and the maximum of the antineutron wave- occurs about the periphery of the nucleus. Since function is located beyond the nuclear radius an antineutron would be strongly absorbed even [32]. In contrast to [18,23], for the correct within the diffuse periphery of the nuclear description of the absorption process of the 40 substance, another eighth zone with density antineutron produced by 푛 → 푛̅ within 18퐴푟, it is 휌표푢푡 = 0.001 ⋅ 휌(7) is added which extends far necessary to determine the radial dependence of beyond the nuclear envelope. the probability density of the transformation within the nucleus. This development will be included in our next publication, focused on 40 18퐴푟. However, for a first approximation of the 40 annihilation process within 18퐴푟, the simulation outputs shown in this article are considered for the case when the transformation occurs with equal probability for all neutrons within only the peripheral zone of the nucleus. Thus, we plan to demonstrate the importance of the radial annihilation dependence. With regard to 40 modeling the transformation in the 18퐴푟 nucleus, as discussed in the previous section, the FIG. 1. Left: The radial distribution of the relative difference between the absorption of the slow 12 density of and neutrons throughout the antineutron by the 6퐶 nucleus comes in the first 12 12 6퐶 nucleus (they are identical). The solid black stage only. As with the 6퐶 nucleus, the nucleon 40 line is a Woods-Saxon density distribution, while density distribution of the 18퐴푟 nucleus is the blue step function is an approximation used to described by expression (4) and is approximated divide the nucleus into seven zones of constant as being divided into seven zones of constant density. Right: The radial dependence of the density where it is assumed that the neutrons are 12 absorption probabilities 푃푎푏푠 for the 6퐶 are distributed throughout the nucleus identically to shown for an antineutron (solid orange) and an protons. antiproton (dashed grey) [28]. Note that the C. The nuclear model and nucleon eighth zone extends from the end of zone seven distribution at 푟 = 4.44 푓푚 to 푟 = 10 푓푚. Within the INC model, the nucleus is considered B. Antineutron annihilation within the ퟒퟎ푨풓 ퟏퟖ to be a degenerate, free Fermi gas of nucleons, nucleus enclosed within a spherical potential well with a

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radius equal to the nuclear radius. Nucleons fill 1 2 3 3휋 푛푁 all energy levels of the potential well, from the 푝퐹푁 = ℏ ( ) , (9) lowest, when a nucleon can have the largest 푉 negative potential energy and ~0 momentum, to 2 2 2 2 3 the highest echelons of the Fermi level, where the 푝퐹푁 ℏ 3휋 푛푁 푇퐹푁 = = ( ) , (10) nucleon moves with Fermi momentum 푝퐹푁, and 2푚푁 2푚푁 푉 is retained within the nucleus only because of the 4 where 푉 = 휋푅3 is the volume of the nucleus, binding energy 휀 (where 휀 ≈ 7 푀푒푉 per 3 nucleon). and 푚푁 remains the nucleon mass.

In the interval 푝휖[0, 푝퐹푁], the three-momentum If the nucleus is subdivided into concentric of the nucleon can take all permissible values. spherical zones of constant density, the values of The differential probability distribution of the 푝퐹푁 and 푇퐹푁 for each zone are calculated nucleons with respect to the total momentum and similarly to equations (9) and (10), but with an kinetic energy [29] takes the form: 푖-th radius, and the density of the nucleons within 3푝2 this 푖-th zone. Fig. 2 shows the spatial distribution 푊(푝) = 3 , 푝 ≤ 푝퐹푁, (5) of the potential 푉푁 = −(푇퐹푁 + 휀) for protons and 푝퐹푁 12 40 neutrons in both 6퐶 and 18퐴푟 nuclei. 1 3푇2 푊(푇) = 3 , 푇 ≤ 푇퐹푁. (6) 2 2푇퐹푁 Here, 푇 is the kinetic energy of a nucleon within 2 푝퐹푁 the nucleus, and 푇퐹푁 = represents the 2푚푁 boundary Fermi kinetic energy, while 푚푁 is the mass of the nucleon. If the nucleons are distributed evenly throughout the spherical well 1 having a radius 푅 = 푟0퐴3 (and where 푟0 is 1.2­1.4 푓푚), then their Fermi momentum and energy are easily expressed in terms of the radius. FIG. 2: The spatial distribution of the potential Because every cell in phase space 푑3푥 푑3푝 푉푁 = −(푇퐹푁 + 휀), with appropriate partitioning contains a number of states of the nucleus into seven zones for protons (solid histogram) and neutrons (dotted histogram) for 2푠 + 1 3 3 푑 푥 푑 푝 (7) 12 40 12 (2휋ℏ)3 both 6퐶 and 18퐴푟 nuclei (for 6퐶, the solid and dotted histograms lay atop one another). 휀 is the (푠 is the spin of the nucleon) and the total number average nuclear binding energy of 7 푀푒푉 per of protons or neutrons in the nucleus being equal nucleon. to 푛푁, it then follows from the normalization condition that The momentum distribution of the nucleons in individual zones will be the same as for a 3 2푠 + 1 푉푝퐹푁 degenerate Fermi gas, and the probability of a ∫ 푑3푥 푑3푝 = = 푛 , (8) (2휋ℏ)3 3휋2ℏ3 푁 nucleon to have momentum 푝 in the 푖th-zone will continue to be determined by (5), although and one finally gets that corresponding to 푖th-zone’s boundary Fermi momentum value. Fig. 3 shows the momentum 12 40 distributions of nucleons for both 6퐶 and 18퐴푟

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12 nuclei, obtained by summing all the momentum antineutron absorption probability in the 6퐶 distributions for all individual zones. From Figs. nucleus 푃푎푏푠(푟) (also Fig. 1), the point of 2 and 3, we can see that the nucleons located in annihilation is taken randomly by Monte Carlo the central zone of the nucleus have the highest technique. The radius of this point determines the value of 푇퐹푁, and, accordingly, the maximum number of the zone in which the nucleon partner value of the Fermi momentum 푝퐹푁. Therefore, (neutron or proton) is located, and with which the the contribution to the total momentum antineutron annihilates. distribution from the nucleons located in the central (푖 = 1) zone gives the high-momentum The physics underlying the transformation within 40 part which extends up to 250 − 270 푀푒푉/푐. an 18퐴푟 nucleus is different, since the antineutron Conversely, the nucleons located within the generated from 푛 → 푛̅ is not extranuclear in peripheral zone of the nucleus (푖 = 7) have nature, but instead depends entirely on the momenta up to 80 − 100 푀푒푉/푐. Moreover, the magnitude of the intra-nuclear binding energy as contribution to the overall momentum a function of radius. Generally, it is thought that distribution of a particular zone is greater the the bound transformation should occur near the more nucleons within it. Thus, in our model there surface of the nucleus (possibly even outside the is a correlation of the momentum with the density nuclear envelope [32]), we assume that an and, respectively, with the radius. antineutron resulting from the transformation has kinetic energy 푇푛̅ = 푇푛 + 휀, and annihilates on the nearest nucleon neighbor within only the peripheral zone. Further, the simulation is done 12 40 within the same scheme for both 6퐶 and 18퐴푟 nuclei. The annihilation partner has Fermi momentum randomly selected from the momentum distribution for a particular zone; 5 then, the annihilation occurs. D. Annihilation model Unlike papers [24,25], where a statistical model for nucleon-antinucleon annihilation into pions

was used, the present paper uses a combined Fig. 3. The thick histogram shows the momentum approach first proposed in [26]. The phenomena 12 distribution of intra-nuclear nucleons in both 6퐶 of 푁̅푁 annihilation can lead to the creation of 40 and 18퐴푟 nuclei, summed over all zones. The thin many particles through many possible (at times lines show histograms which correspond to ~200) exclusive reaction channels; many neutral contributions from individual zones of the particles may be present, which can make nucleus to the total momentum distribution (only experimental study quite difficult. Experimental odd-numbered zone distributions are shown so information for exclusive channels is known only that the picture is not indecipherable). The zone for a small fraction of possible annihilation number 푖 is shown at the right of each histogram. channels, and therefore a statistical model based Note the logarithmic scale of probability in on 푆푈(3) symmetry [33] has been chosen to arbitrary units. describe the 푁̅푁 annihilation. Work to generalize the unitary-symmetric model for 푁̅푁 Thus, for 12퐶 nuclei in the first stage, the 6 annihilations, along with the development of nucleons are distributed within the nucleus methods for calculating the characteristics of according to the step density function (see Fig. 1). mesons produced from the annihilation, was Next, according to the radial distribution of the performed by I. A. Pshenichnov [34]. According

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to the model, the 푁̅푁 annihilation allows for the Considering the laws of energy and momentum production of between two and six intermediate conservation for each annihilation, the procedure particles. Given the estimates of the phase space for simulating the characteristics of both the volume at low momenta, the production of a intermediate particles and their various decay larger number of intermediate particles is products consists of the following: first, a single unlikely. Intermediate particles, such as 휋, 휌, 휔 channel from the table is randomly selected via and 휂 mesons, are all possible; the channels with Monte Carlo technique as the initial state, with all strangeness production are not considered within necessary momenta of all annihilation particles this version of the model. This unitary-symmetric determined according to the pertinent phase- statistical model predicts 106 푝̅푝 annihilation space volume. This takes into account the Breit- channels, and 88 푝̅푛 annihilation channels, but Wigner mass distribution for resonances, 퐺푒푉 this differs from experiments, which effectively while all pions have a mass value of 0.14 . 푐2 measure only ~40 channels for 푝̅푝 and ~10 The subsequent disintegration of unstable mesons channels for 푝̅푛 annihilation. However, neither is modeled according to experimentally known the statistical model, nor the experimental data, branching ratios. All major decay modes for can provide a complete and exclusive description meson resonances have been considered, such as of the elementary nucleon-antinucleon in Table III. annihilation processes. For this reason, semi- empirical Tables I and II of annihilation channels All experimental data used for comparison with are employed for use in annihilation modeling. this annihilation model are described in great These are obtained as follows: First, all detail in [26]. However, there do exist more experimentally measured channels were included recent data obtained from LEAR by the Crystal in Tables I and II. Then, by using isotopic Barrel [35] and OBELIX [36] Collaborations on relations, probabilities were found for those some exclusive channels which show somewhat channels that have the same configurations but different branching ratios from those used by us. different particle charges. Finally, the predictions We plan to make a revision of the annihilation of the statistical model with 푆푈(3) symmetry tables in the near future, taking into account all were entered for the remaining intermediate data. Below is a comparison of the simulation channels. Sometimes the probabilities of results and experimental data on 푝̅푝 annihilation intermediate channels measured in different at rest. experiments differ significantly. In this case, the Table IV shows the average multiplicity of data in the semi-empirical tables were corrected mesons formed in 푝̅푝 annihilations at rest. The within experimental accuracies in order to simulation results are within the range of describe the topological cross section for 푝̅푝 and experimental uncertainties. From these 푝̅푛 in a consistent way. In our approach, a simulation results, it follows that more than 35% substantively large collection of experimental of all pions have been formed by the decay of data was used: multi-particle topologies, meson resonances. Fig. 4 shows the pion inclusive spectra, topological pion cross sections, multiplicity distribution generated by 푝̅푝 and branching ratios of various resonance annihilation, while Fig. 5 shows the charged pion channels. The following pages show the semi- momentum distribution. From considering Table empirical tables, with probabilities of various 푝̅푝 IV and Figs. 4 and 5, it follows that the Monte and 푝̅푛 annihilation channels included. In further Carlo and available experimental data are in modeling of 푁̅퐴 interactions, it is considered that general agreement with the main features of channels for 푛̅푛 are identical to 푝̅푝 channels, and 푝̅푝 annihilation. In all, our annihilation model that annihilation channels for 푛̅푝 are charge utilizes a complex series of tables with a much conjugated to 푝̅푛 channels. larger number of predicted and pertinent channels

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than [18,23]. As this approach demonstrates a adequate for an accurate description of 푝̅푛 good description of the experimental data for 푝̅푝 annihilation at rest, and so can be implemented annihilation at rest, we believe that it is also within our 푛̅퐴 annihilation simulation.

10 TABLE I. Probability of intermediate states for 푝̅푝 annihilation at rest (%). Note that 1) indicates a probability attained from experiment; see references used in [26]. Note that 2) indicates that the probabilities are obtained from isotopic relations. The sum of all branching ratios is normalized to 100 percent. Channel Probability (%) Channel Probability (%) Channel Probability (%) 휂휂 0.01 1) 휂휂휋+휋− 0.07 휋+휋+휋+휋−휋−휋− 2.83 휂휔 0.34 1) 휂휂휋0휋0 0.02 휋+휋+휋−휋−휋0휋0 9.76 휔 휔 1.57 1) 휂휔휋+휋− 0.04 휋+휋−휋0휋0휋0휋0 2.68 휋+휋− 0.40 1) 휂휔휋0휋0 0.01 휋0휋0휋0휋0휋0휋0 0.07 휋0휋0 0.02 1) 휋+휋−휋0휂 1.22 휋+휋+휋+휋−휋−휌− 0.02 휋+휌− 1.52 1) 휋0휋0휋0휂 0.17 휋+휋+휋−휋−휋−휌+ 0.02 휋−휌+ 1.52 1) 휋+휋−휋0휔 2.84 휋+휋+휋−휋−휋0휌0 0.06 휋0휌0 1.57 1) 휋0휋0휋0휔 0.40 휋+휋+휋−휋0휋0휌− 0.06 휌−휌+ 3.37 2) 휋+휋−휌0휂 0.06 휋+휋−휋−휋0휋0휌+ 0.06 휌0휌0 0.67 1) 휋+휋0휌−휂 0.06 휋+휋−휋0휋0휋0휌0 0.03 휋0휂 0.06 1) 휋−휋0휌+휂 0.06 휋+휋0휋0휋0휋0휌− 0.01 휋0휔 0.58 1) 휋0휋0휌0휂 0.02 휋−휋0휋0휋0휋0휌+ 0.01 휌0휂 0.90 1) 휋+휋+휋−휋− 2.74 휋+휋+휋−휋−휋0휂 0.31 휌0휔 0.79 1) 휋+휋−휋0휋0 3.89 휋+휋−휋0휋0휋0휂 0.17 휋+휋−휋0 2.34 1) 휋0휋0휋0휋0 0.21 휋0휋0휋0휋0휋0휂 0.01 휋0휋0휋0 1.12 1) 휋+휋+휋−휌− 2.58 1) 휋+휋+휋−휋−휋0휔 0.10 휋+휋−휌0 2.02 1) 휋+휋−휋−휌+ 2.58 1) 휋+휋−휋0휋0휋0휔 0.06 휋+휋0휌− 2.02 2) 휋+휋−휋0휌0 6.29 1) 휂휂휂 0.0036 휋−휋0휌+ 2.02 2) 휋+휋0휋0휌− 5.05 2) 휂휂휌0 0.0002 휋0휋0휌0 1.01 2) 휋−휋0휋0휌+ 5.05 2) 휔휔휋+휋− 0.0002 휋+휌−휌0 1.23 휋0휋0휋0휌0 0.77 2) 휔휌0휋+휋− 0.0005 휋−휌+휌0 1.23 휋+휋+휋−휋−휋0 2.61 휔휌−휋+휋0 0.0005 휋0휌+휌− 1.23 휋+휋−휋0휋0휋0 1.37 휔휌+휋−휋0 0.0005 휋0휌0휌0 0.54 휋0휋0휋0휋0휋0 0.07 휔휌0휋0휋0 0.0002 휋+휋−휂 1.50 1) 휋+휋+휋−휋−휌0 0.08 휌−휌−휋+휋+ 0.0003 휋+휋−휔 3.03 1) 휋+휋+휋−휋0휌− 0.16 휌0휌0휋0휋0 0.0001 휋0휋0휔 0.79 2) 휋+휋−휋−휋0휌+ 0.16 휌+휌−휋+휋− 0.0011 휋+휌−휂 0.84 휋+휋−휋0휋0휌0 0.12 휌0휌0휋+휋− 0.0004 휋−휌+휂 0.84 휋+휋0휋0휋0휌− 0.04 휌−휌0휋+휋0 0.0008 휋0휌0휂 0.44 휋0휋0휋0휋0휌0 0.01 휌+휌+휋−휋− 0.0003 휋+휌−휔 1.10 휋+휋+휋−휋−휂 0.11 1) 휌+휌0휋−휋0 0.0008 휋−휌+휔 1.10 휋+휋−휋0휋0휂 0.22 2) 휌+휌−휋0휋0 0.0004 휋0휌0휔 0.57 휋0휋0휋0휋0휂 0.01 2) 휋+휋−휋0휂휂 0.0055 휂휂휋0 0.11 휋+휋+휋−휋−휔 1.80 1) 휋0휋0휋0휂휂 0.0007 휂휔휋0 0.30 휋+휋−휋0휋0휔 2.58 2) 휔휔휋0 0.37 휋0휋0휋0휋0휔 0.10 2)

11 TABLE II. Probability of intermediate states for 푝̅푛 annihilation at rest (%). Similarly note that 1) indicates a probability attained from experiment; see references used in [26]. Note that 2) indicates that the probabilities are obtained from isotopic relations. The sum of all branching ratios is normalized to 100 percent. Channel Probability (%) Channel Probability (%) Channel Probability (%) 휋−휋0 0.49 1) 휋+휋−휋−휔 10.52 1) 휋+휋+휋−휋−휋0휌− 0.07 휋−휔 0.48 1) 휋−휋0휋0휔 7.01 2) 휋+휋−휋−휋−휋0휌+ 0.05 휋−휌0 0.47 1) 휋+휋−휌−휂 0.08 휋+휋−휋−휋0휋0휌0 0.06 휋0휌− 0.47 2) 휋−휋−휌+휂 0.05 휋+휋−휋0휋0휋0휌− 0.03 휌−휌0 3.51 2) 휋−휋0휌0휂 0.06 휋−휋−휋0휋0휋0휌+ 0.02 휋−휂 0.29 1) 휋0휋0휌−휂 0.02 휋−휋0휋0휋0휋0휌0 0.01 휌−휂 2.27 휋+휋−휋−휋0 5.51 휋+휋+휋−휋−휋−휂 0.14 휌−휔 3.51 2) 휋−휋0휋0휋0 1.38 휋+휋−휋−휋0휋0휂 0.30 휋+휋−휋− 2.86 휋+휋−휋−휌0 0.99 휋−휋0휋0휋0휋0휂 0.05 휋−휋0휋0 1.90 휋+휋−휋0휌− 1.97 휋+휋+휋−휋−휋−휔 0.05 휋+휋−휌− 3.62 1) 휋−휋−휋0휌+ 0.99 휋+휋−휋−휋0휋0휔 0.09 휋−휋−휌+ 0.58 1) 휋−휋0휋0휌0 0.75 휋−휋0휋0휋0휋0휔 0.01 휋−휋0휌0 5.61 2) 휋0휋0휋0휌− 0.25 휂휂휌− 0.0003 휋0휋0휌− 3.51 2) 휋+휋+휋−휋−휋− 1.24 휔휔휋−휋0 0.0002 휋+휌−휌− 1.04 휋+휋−휋−휋0휋0 2.72 휔휌−휋+휋− 0.0008 휋−휌+휌− 2.09 휋−휋0휋0휋0휋0 0.37 휔휌+휋−휋− 0.0004 휋−휌0휌0 0.70 휋+휋+휋−휋−휌− 0.12 휔휌0휋−휋0 0.0005 휋0휌−휌0 1.39 휋+휋−휋−휋−휌+ 0.08 휔휌−휋0휋0 0.0003 휋−휋0휂 1.23 휋+휋−휋−휋0휌0 0.16 휌−휌0휋+휋− 0.0011 휋−휋0휔 5.05 휋+휋−휋0휋0휌− 0.16 휌−휌−휋+휋0 0.0005 휋0휌−휂 0.78 휋−휋−휋0휋0휌+ 0.08 휌+휌0휋−휋− 0.0005 휋−휌0휂 0.78 휋−휋0휋0휋0휌0 0.05 휌−휌+휋0휋− 0.0011 휋−휌0휔 1.03 휋0휋0휋0휋0휌− 0.01 휌0휌0휋0휋− 0.0004 휋0휌−휔 1.03 휋+휋−휋−휋0휂 0.37 휌−휌0휋0휋0 0.0004 휂휂휋− 0.21 휋−휋0휋0휋0휂 0.09 휋+휋−휋−휂휂 0.0042 휋−휔휂 0.60 휋+휋−휋−휋0휔 0.40 휋−휋0휋0휂휂 0.0028 휋−휔휔 0.71 휋−휋0휋0휋0휔 0.09 휂휂휋−휋0 0.06 휋+휋+휋−휋−휋−휋0 8.33 휂휔휋−휋0 0.03 휋+휋+휋−휋−휋−휋0 6.67 휋+휋−휋−휂 1.00 휋−휋0휋0휋0휋0휋0 0.56

휋−휋0휋0휂 0.67 휋+휋+휋−휋−휋−휌0 0.02

12

TABLE III. Pertinent decay branching ratios of intermediate resonance particles shown in %.

Channel Probability (%) Channel Probability (%) Channel Probability (%)

휂 → 2훾 39.3 휔 → 휋+휋−휋0 89.0 휌+ → 휋+휋0 100

휂 → 3휋0 32.1 휔 → 휋0훾 8.7 휌− → 휋−휋0 100

휂 → 휋+휋−휋0 23.7 휔 → 휋+휋− 2.3 휌0 → 휋+휋− 100

휂 → 휋+휋−훾 4.9

TABLE IV. Meson multiplicities for simulated and experimental 푝̅푝 annihilations, shown in absolute particle counts. Multiplicity Simulated ̅푝푝 Experimental ̅푝푝 4.98±0.35 [37] 푀(휋) 4.910 4.94±0.14 [38]

3.14±0.28 [37] 푀(±) 3.110 3.05±0.04 [37] 3.04±0.08 [38] 1.83±0.21 [37] 푀(0) 1.800 1.930.12 [37] 1.90±0.12 [38] 0.100.09 [39] 푀() 0.091 0.06980.0079 [37] 0.280.16, [39] 푀() 0.205 0.220.01 [40] 푀(휌+) 0.189 ---

푀(휌−) 0.191 ---

푀(휌0) 0.193 0.260.01 [40] 푓푟표푚 푀(휋)푑푒푐푎푦 1.908 --- + 푓푟표푚 푀( )푑푒푐푎푦 0.606 --- − 푓푟표푚 푀( )푑푒푐푎푦 0.606 --- 0 푓푟표푚 푀( )푑푒푐푎푦 0.695 ---

13

which energy is redistributed between the various degrees of freedom within the nucleus as a finite open system, and 2) the slow equilibrium stage of the decay of the thermalized residual nuclei. The INC model is a phenomenological model describing the out-of-equilibrium stage of inelastic interactions and operates with the notion of the probability of a nuclear system being in a given state. Transitions between different states are caused by two-body interactions, which lead to secondary particles exiting the nucleus, dissipating the excitation energy in the process. FIG. 4. The pion multiplicity distribution for 푝̅푝 However, this phenomenological model is linked annihilation at rest (taking into account the decay to fundamental microscopic theory. It was shown of meson resonances). The solid histogram shows in [42] that it is possible to transform a non- the model, with the points showing experimental stationary Schrödinger equation for a many body data [37]. system into kinetic equations, if large energy (and so short time) wave packet formulations are used. To explain, if the duration of the wave packet’s individual collisions are shorter than the interval of time between consecutive collisions, then the amplitudes of these collisions will not interfere. This condition is essentially analogous to the condition of a free gas approximation: 휏0 < 휏퐹푃, where 휏0 is the duration time of the collision, and 휏퐹푃 is the mean-free-path time. This condition allows for the consideration of a particle’s motion as in a dilute gas with independent particle motion on free path trajectories perturbed by binary collisions. Under these conditions, in a quasi-classical way, one can use the local FIG. 5. The momentum distribution of charged momentum approximation by assigning a particle pions produced in 푝̅푝 annihilation at rest (taking ⃗ into account the decay of meson resonances). The a momentum 푃(푟 ) between consecutive solid histogram shows the model, with the points collisions. In this case, the quantum kinetic showing experimental data [41]. equation is transformed into a kinetic equation of Boltzmann-type describing the transport of E. The Intra-nuclear Cascade (INC) Model particles within nuclear media; this differs from the conventional Boltzmann equation only by Inelastic nuclear interactions are clearly accounting for the Pauli exclusion principle. statistical in nature, as they can be realized in Thus, the INC model is a numerical solution of many possible states. A statistical approach is key the quasi-classical kinetic equation of motion for to describing such systems, and replaces the a multi-particle distribution function using the evolution of a system’s wave function with the Monte Carlo method. description of the evolution of an ensemble of the many possible states of the system. There are two We will now focus our discussion on the scope of dramatically different stages of a deeply inelastic the INC model and the possibility of generalizing interaction: 1) a fast, out-of-equilibrium stage in its use, such as in the event of the absorption of a

14

slow antineutron. The principles underlying the the main criterion for the applicability of the model are altogether justified if the following model. conditions are met [29,42,43]: The standard INC model is based on a numerical a) The wavelengths, 휆, of the majority of solution of the kinetic equation using a linearized moving particles should be less than the mean approximation, which implies that the density of distance between nucleons within the the media does not change in the development of nucleus, i.e. 휆 < Δ, where the cascade, i.e. 푁푐 ≪ 퐴푡 (where 푁푐 is the 1 number of cascading particles, and 퐴 is the 4휋푅3 3 푡 Δ ≈ [ ] ≈ 푟0 ≈ 1.3 푓푚. number of nucleons making up the target 3퐴 nucleus). Such an approximation is violated in the In this case, the system acquires quasi- case of multi-pion production in 푝퐴 and 휋퐴 classical characteristics, and one can speak of interactions at 퐸푝,휋 ≥ 3 − 5 퐺푒푉, and also in the the trajectories of particles and two-body case of annihilation, especially when considering interactions within the nucleus. For light nuclei such as 12퐶. A version of the model, individual nucleons, this corresponds to an 6 which takes into account the effect of a local energy of more than tens of 푀푒푉. Of course, reduction in nuclear density, was first proposed in this condition cannot be met in the case of a [43]. This version of the model considers the slow antineutron, and therefore, its nucleus as consisting of separate nucleons, the absorption is described in the framework of position of their centers computed by Monte the optical model. Carlo method according to the prescribed density b) The interaction time should be less than the distribution 휌(푅) such that the distance between time between successive interactions 휏푖푛푡 ≤ their centers is no less than 2푟푐, where 푟푐 = 휏퐹푃, where 푟 0.2 푓푚 is the nucleon core radius. A cascading 휏 ≈ 푁 ≈ 10−23푠, and 푟 is the nucleon 푖푛푡 푐 푁 particle may interact with any intra-nuclear radius. The mean-free-path-length time is nucleon which lies inside the cylinder of diameter 3 −22 푙 1 4휋푅 3 × 10 2푟푖푛푡 + 휆 extending along the particle’s velocity 휏퐹푃 = = ≈ ≈ 푠 푐 휌휎푐 3퐴휎푐 휎 vector (here, 푟푖푛푡 is the interaction radius, while 휆 (휎 is the cross section in 푚퐵). This is the deBroglie wavelength of the particle). The requirement is equivalent to the condition of 푟푖푛푡 is a parameter of the model and is chosen for requiring sufficiently small cross sections of better agreement with the experimental data. The elementary interactions and proves key point to understand here is the ability to problematic for pions produced from the determine the probability of the cascading annihilation and lying within the energy particle interacting with another constituent range of the Δ-resonance, where 휎 > nucleon. We now consider this process in more 100 푚퐵. However, it should be kept in mind detail. that the effective mean-free-path-length Within the standard cascade model, the randomly within the nucleus is increased by the Pauli chosen interaction point is computed from a exclusion principle; secondarily, because the Poisson distribution for the mean-free-path- uptake of the antineutron is predominately on length. In this case, the probability 휔(푘) of the the periphery of the nucleus, where the particle experiencing 푘 collisions along the path- nuclear density is low and the distance length 퐿 in media with density 휌, where the between the nucleons large, one can expect particle has a total cross section 휎, is defined as: that the INC model would work in this case. Never-the-less, the comparison of the (휌휎퐿)푘 휔(푘) = 푒−휌휎퐿 . (11) simulation results with experimental data is 푘!

15

If on the path-length 퐿 there lie 푛 individual • The nuclear target is a degenerate Fermi particle centers, each has an equal collision gas of protons and neutrons within a probability 푝 for the particle to collide on 푘 of 푛 spherical potential well with a diffuse centers and 푞 = 1 − 푝. This probability is nuclear boundary. The real nuclear described by a binomial distribution: potentials for nucleons (푉푁), antinucleons (푉 ), and mesons 푛! 푁⃐⃗⃗ 휔(푘, 푛, 푝) = 푝푘푞푛−푘. (12) (푉 , 푉 , 푉 ) effectively takes into account 푘! (푛 − 푘)! 휋 휂 휔 the influence on the particle of all intra- From the Poisson distribution (11), it follows nuclear nucleons. The depth of the directly that the probability of a particle potential well for the antinucleon and experiencing no collisions along 퐿 is simply: mesons within the nucleus remains a free-parameter of the model. −휌휎퐿 휔(0) = 푒 . (13) Recognizing that the annihilation process The same probability for this process can be usually occurs on the periphery of the obtained from the binomial distribution in (20): nucleus, a good approximation for this is considered to be 푉⃐⃗⃗ ≈ 0 and 푉휋,휂,휔 ≈ 0. 푛 푛 푁 휔(0, 푛, 푝) = (1 − 푝) = 푞 . (14) In the future, a detailed study is planned If one takes 휔(0) = 휔(0, 푛, 푝), and when to focus on the influence of these 2 potentials on the simulation output. considering that 푛 = 휌휋퐿(푟푖푛푡 + 휆/2) , then: • Hadrons involved in collisions are 휌휎퐿 푞 = 1 − 푝 = exp [− ] treated as classical particles. A hadron 푛 can initiate a cascade of consecutive, 2 = exp[−휎/휋(푟푖푛푡 + 휆/2) ] . (15) independent collisions upon nucleons within the target nucleus. The An essential feature of present version of the INC interactions between cascading particles model is the fact that after interactions occur are not taken into account. inside the nucleus, the nucleon is considered to be cascade particle and not a constituent part of the • The cross sections of hadron-nucleon nuclear system. Thus, a reduction in nuclear interactions are considered within the density takes place during the cascade nucleus to be identical to those in development. In order to describe the evolution vacuum, except that Pauli’s exclusion of the cascade and the decays of unstable meson principle explicitly prohibits transitions resonances over time, an explicit time-coordinate of cascade nucleons into states already has been incorporated into the model. occupied by other nucleons. So, one can summarize the physical considerations that underlie the INC model as follows:

푁푁 → 푁푁 푁푁 → 휋푁푁 푁푁 → 푖휋 푁푁 (푖 ≥ 2) (16) 휋푁 → 휋푁 휋 + (푁푁) → 푁푁 휋푁 → 휋휋푁

Elementary processes, such as those seen in the experimental data on 푁푁 and 휋푁 interactions at channels (16) shown above, are described by kinetic energies 푇 < 20퐺푒푉 [29,43] empirical approximations from analysis of

16

Now consider some of the features of the INC factor. The mean lifetime of the 휂-meson is large model related to the introduction of unstable enough for the particle to be considered stable meson resonances into the model. Modeling within the nucleus, which can then decay upon annihilation with meson resonances (i.e. 푁̅푁 → exit. The model uses the experimentally 푖휋 + 푗휌 + 푛휂 + 푛휔) was described in the measured decay modes of the meson resonances preceding section. It is assumed within the model described above. When the annihilation products that 휌-mesons produced by annihilation decay are allowed to disintegrate, their three-body quickly enough to avoid interacting with any decay is simulated by evaluation of the intra-nuclear nucleons. In contrast, 휔-mesons permissible phase-space volume. produced by annihilation can both interact with To accommodate the passage of 휂-and-휔-mesons other intra-nuclear nucleons and decay within or through nuclear material, in addition to channels outside the nucleus. The competition between the listed in (16), the following interactions are also decay of the 휔-meson and its interaction with considered: intra-nuclear nucleons is determined by the expression for the mean-free-path: 1 1 1 = + , (17) 휆 휆푑푒푐 휆푖푛푡 where 휆 = (휌 휎푡표푡)−1, 휆 = 훾훽(ℎ훤 )−1, 푖푛푡 푛 휔푁 푑푒푐 휔 휌푛 is the nuclear density, and 훾 is the Lorentz 휂푁 → 휂푁 휂푁 → 휋푁 휂푁 → 휋휋푁 휂 + (푁푁) → 푁푁 휂 + (푁푁) → 휋푁푁 휔푁 → 휔푁 휔푁 → 휋푁 휔푁 → 휋휋푁 휔 + (푁푁) → 푁푁 휔 + (푁푁) → 휋푁푁 (18)

Along with the creation of 휂- and 휔-mesons by stage begins (휏푒푣 ≫ 휏0) involving the annihilation, the model also accounts for the disintegration of the highly excited residual −22 creation of mesons through interactions between nucleus (note that 휏0 ≤ 10 푠, which is the annihilation pions and nucleons, such as average time required for a particle to pass completely through the nucleus). The INC model 휋푁 → 휂푁 휋푁 → 휔푁 (19) is able to describe the dissipation of energy For cross sections of reactions in (18), estimates throughout the nucleus. At the end of the cascade given in [26] were employed. For those few stage, the nuclear degenerate Fermi gas contains reactions shown in (19), experimental cross a number of “holes” 푁ℎ, which is equal to the sections were taken from compilation [44]. As number of collisions of cascade particles with these interactions are considered at relatively low nucleons within the nucleus. Also, there exists energy, the angular distributions for reactions some number of excited particles 푁푝, which is shown in (18) and (19) are assumed to be equal to the number of slow cascade nucleons isotropic in the center of mass of the system. trapped by the nuclear potential well. The Reactions with three particles in the final state are excitation energy of the residual nucleus 퐸∗, is the simulated via their pertinent phase-space volume. sum of the energy of all such quasiparticles calculated from the Fermi energies 휀 : F. De-excitation of the residual nucleus 푖

푁ℎ 푁푝 For inelastic nuclear reactions, after the rapid ∗ ℎ 푝 퐸 = ∑ 휀푖 + ∑ 휀 . (20) stage of the intra-nuclear cascade (휏푐푎푠 ≃ 휏0) and 푗 푖=1 푗=1 once statistical equilibrium (휏푒푞 ≅ (5 − 10)휏0) is established inside the residual nucleus, a slow

17

The resulting residual nuclei have a broad on the surface of the nucleus, most of the mesons distribution on the excitation energies 퐸∗, produced fly out of the nucleus without any momenta, masses, and charges. The INC model interaction. In the case of a light nucleus such as 12 correctly accounts for the fluctuations of the 6퐶, the effect of absorption of annihilation cascade particles, and reliably defines the entire mesons is not large and the average multiplicity set of characteristics for residual nuclei. of pions emitted appear to be quite similar to the multiplicity of pions in 푝̅푝 annihilation (4.910). The de-excitation mechanism for a residual For comparison, Table V also shows results nucleus is determined from the accumulated which simulate the annihilation of a slow excitation energy of the nucleus [45]. Under low 12 푀푒푉 antineutron on a 6퐶 nucleus. The comparison excitation energies (where 퐸∗ ≤ 2 − 3 ), 푛푢푐푙푒표푛 shows that the average pion multiplicity for an 푛̅퐶 the primary de-excitation mechanism is the annihilation is somewhat lower, and that the consecutive emission (evaporation) of particles average multiplicity for exiting nucleons slightly from the compound nucleus [46]. When the higher than the case of a stopped antiproton. This excitation energy of the nucleus is approaching is due to the fact that the antineutron penetrates 푀푒푉 the total binding energy (where 퐸∗ ≥ 5 ), the more deeply into the nucleus (seen in the solid 푛푢푐푙. prevalent mechanism is explosive decay [47]. For line shown in Fig. 1) compared to an antiproton intermediate energies, both mechanisms coexist. (seen in the dashed line shown in Fig. 1), and so there are more intra-nuclear interactions between III. COMPARISON WITH annihilation mesons and constituent nucleons. EXPERIMENT Thus, the number of mesons emitted from the nucleus and their total energy 퐸 are reduced, The optical-cascade model described throughout 푡표푡 while instead the number of nucleons that were this work has been used to analyze experimental kicked from the original nucleus during the fast data taken from antiproton annihilation at rest on cascading stage (and then emitted from the 12퐶 target nuclei. Table V shows the average 6 nucleus during the de-excitation process) is multiplicity of emitted pions and protons. increased. In the case of peripheral annihilation Experimental data on average pion multiplicities of an antineutron on 40퐴푟, the pions are almost (values of which are shown at the bottom of the 18 entirely free to leave the nucleus, increasing the 푝̅퐶 row) are taken from [38]. The final column of value of 퐸 , and so the number of emitted the table indicates the average energy of pions 푡표푡 nucleons is significantly lower than 푛̅퐶 and (resulting from the decay of 휂- and annihilation. Note here that the calculation 휔-mesons) emitted from the nucleus. Calculated completed for 40퐴푟 is made in a very rough values for the average multiplicities of pions 18 approximation with respect to the annihilation (values of which are shown at the top of the 푝̅퐶 radius; this property requires further detailed row) are within accuracies of the experimental investigation. data. Since the antiproton primarily annihilates TABLE V. The average outgoing particle multiplicities emitted after 푝̅퐶, 푛̅퐶 and 푛̅퐴푟 annihilation and all decays. Experimental data taken from [38] is used as a comparison for 푝̅퐶. In the first row (calculation for 푝̅퐶), an option of the model with the antineutron potential was used.

Type 푴흅 푴흅+ 푴흅− 푴흅ퟎ 푴풑 푴풏 푬풕풐풕 (푴풆푽) calculation 4.557 1.208 1.634 1.715 1.138 1.209 1736 풑̅푪 experiment 4.57 ± 0.15 1.25 ± 0.06 1.59 ± 0.09 1.73 + 0.10 ------1758 ± 59 풏̅푪 calculation 4.451 1.558 1.182 1.712 1.543 1.317 1679 풏̅푨풓 calculation 4.599 1.651 1.267 1.680 0.727 0.804 1751

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dashed histogram). As was expected, the differences in these distributions, as with the mean number of emitted pions, are not significant, although there appears to be some bias towards a smaller number of pions for 푛̅퐶 and a larger number for 푛̅퐴푟.

Fig. 7 shows the distribution by number of events with the charge 푄 carried out by pions. For the 푝̅퐶 annihilation the maxima of the distribution are 푄 = −1 and 푄 = 0, which practically corresponds to mesons exiting the nucleus without any interaction with nucleons. The FIG. 6. The probability (%) of formation of a optical-cascade model demonstrates good given number of charged pions in antinucleon- agreement with the experimental data. In the case of an annihilation with an 푛̅, the distribution has nuclei annihilation. The solid histogram shows a maximum 푄 that is shifted to 푄 = 0 and 푄 = 푝̅퐶. Experimental data: circles-[48], squares- +1, respectively. In the case of a peripheral [49]. The dotted histogram shows an 푛̅퐶 annihilation for 푛̅퐴푟, the distribution has a simulation; the dashed histogram shows an 푛̅퐴푟 narrower maximum than 푛̅퐶. simulation. Fig. 8 shows the distribution of the number of emitted protons. The analysis of experimental data and simulation results show that a significant number of events (from ~40% for 푛̅퐶, to ~60% for 푛̅퐴푟) do not have any exiting protons.

FIG. 7. The probability (%) of particular values of total charge 푄 carried out by pions emitted from the nucleus. The solid histogram shows a 푝̅퐶 calculation. Experimental data: open squares- [50], circles-[51]. The dotted histogram shows an 푛̅퐶 simulation; the dashed histogram shows an FIG. 8. The probability (%) of the events with a 푛̅퐴푟 simulation. given number of exiting protons. The solid Now, consider and compare the Monte Carlo histogram shows a 푝̅퐶 calculation. Experimental calculation to other available experimental data data: solid squares-[49], open squares-[50]. The and features for 푝̅퐶 annihilation at rest. Fig. 6 dotted histogram shows an 푛̅퐶 simulation. The shows the charged pion multiplicity distribution dashed histogram shows an 푛̅퐴푟 simulation. emitted from the nucleus due to 푝̅퐶 annihilation (shown as the solid histogram with points), 푛̅퐶 Fig. 9 shows the momentum distribution for 휋+ (shown as the dotted histogram), and 푛̅퐴푟 (the exiting the nucleus, which is rather similar to the

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momentum distribution of pions created by 푝̅푝 shows good agreement with the available annihilation (as seen in Fig. 5). To understand the experimental data. uncertainty of the model, calculations were done 1) without any nuclear potential for the antineutron, and, as an option, 2) with a model where the antineutron nuclear potential is introduced similarly to [26]. For mesons propagating inside the nucleus, we have not assumed any nuclear potentials. Both model calculations are presented in Fig. 9, and show rather good agreement with experimental data, although there is some exaggerated absorption behavior corresponding to the Δ-resonance 푀푒푉 region (~260 ). The difference between 푐 experimental measurements appears to be of the same order as the uncertainty in the calculation. FIG. 10. The exiting proton kinetic energy Never-the-less, in the near future, we plan to spectrum due to antiproton annihilation at rest on 12 study this question in more detail. 6퐶 nuclei. The solid histogram shows the simulation result. The dotted histogram shows the contribution which evaporative protons impart to the whole distribution. The points show the experimental data taken in [53]. From the comparisons made above between the simulation results of the optical-cascade model and experimental data of 푝̅퐶 annihilation at rest, it follows that the model as a whole describes experiments well, thus accurately reflecting the dynamics of the annihilation process and the propagation of annihilation mesons throughout the nucleus.

IV. 풏̅푪 ANNIHILATION GENERATOR FIG. 9. The 휋+ momentum distribution is shown 12 VALIDATION for antiproton annihilation at rest on 6퐶 nuclei. The points show experimental data from [38,52]. Colleagues within the ESS NNBar Collaboration The histograms show calculations, where the have tested the model through its corresponding solid line shows an option of the model with a event generator comprehensively. The event nuclear potential for the antineutron, and the generator outputs an annihilation point within 12 dashed line is the calculation done without this 6퐶, annihilation product particle identities, potential. The nuclear potential for annihilation energies, momenta, etc. These particles and mesons inside the nucleus is assumed to be zero. variables are tracked as outputs and saved to file in three successive stages: 1) after the primary Fig. 10 shows the energy spectrum of protons annihilation, 2) after all cascading exiting the nucleus from 푝̅퐶 annihilation at rest. (푛, 푝, 휋, 휌, 휂, 휔) and evaporation particles have In the low energy regime (up to 50 푀푒푉), left the nucleus, and 3) after all decays of the evaporative protons provide a significant meson resonances emitted from the contribution to the spectrum. The model again nucleus (휌, 휂, 휔) are modeled.

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As discussed in the previous sections, Similarly, for Fig. 12, we see that the momentum multiplicity, charge, momentum and energy distribution reconstructed from initial distributions of particles show good agreement annihilation mesons is perturbed and expanded with antiproton annihilation experimental data, by transport through the nuclear environment. and all simulated variables quantitatively satisfy The structure shown in the dotted histogram the fundamental tenets of the required physics. shows a similar distribution as seen in Fig. 3, Specifically, the generator has been shown to though implicitly convolved with Fig. 1, and conserve charge, energy, momentum, baryon considerate of different scales. After transport, number, etc., through all three stages of this distribution distorts as particles cascade simulation. The output file type is .txt, and through the nucleus, shifting values up to as far 퐺푒푉 formatted in such a way as to easily separate the as ~0.8 . particle content and their respective physical 푐 variables through the stages. Analysis of the output has been completed by ESS colleagues using C++ and the CERN ROOT 5.34 scientific software framework [54]. 100,000 simulated 푛̅퐶 annihilation events are available upon request from the authors. The currently available event files and the following plots are created from the completed simulation file data without either antineutron or meson potentials. An important characteristic for any relativistic many particle system is the . One may analyze the invariant mass distribution for FIG 11. The distribution of invariant mass of 푛̅퐶 annihilation mesons at the annihilation point, and annihilation products. The dotted histogram then see how it distorts due to interactions shows the distribution of invariant mass due only throughout the nucleus. Detector performance to original annihilation mesons. The solid might affect the invariant mass further, but this histogram shows the invariant mass of pions and study in not the focus of this paper. Fig. 11 shows photons emanating from the nucleus after how the distribution of invariant mass changes transport. for all outgoing pions and photons generated by 푛̅퐶 annihilation products (solid), a result of interactions with nuclear media. The dotted line shows the original distribution of invariant mass of the initial 푛̅푁 annihilation products. Fig. 11 shows that the intra-nuclear interactions of annihilation mesons with nucleons have resulted in a significant redistribution of energy between mesons and other nuclear constituents, shifting and smearing the initial distribution of 푀푖푛푣 down to values of ~1.2 퐺푒푉⁄푐2. Note that the higher the initial value of 푀 , or the deeper the 푖푛푣 penetration of the antineutron into the nucleus, the larger the number of mesons which will FIG. 12. The distribution of total momentum of interact with the nuclear environment, quickly 푛̅퐶 annihilation products. The dotted histogram devouring this particular part of the distribution. shows the distribution of total momentum of all

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original annihilation mesons. The solid histogram initial stage of the simulation. A combination of shows the distribution of total momentum of experimental data with the results of a statistical pions and photons emanating from the nucleus model employing 푆푈(3) symmetry is used to after transport. describe the annihilation process. The propagation of annihilation-produced pions and Following discussion of Fig. 2 within [18], a heavier meson resonances within the nucleus is highly relevant plot of total momentum versus described by the intra-nuclear cascade model, invariant mass output variables is shown below in which takes into account the nonlinear effect of Fig. 13 for outgoing pions and photons. The decreasing nuclear density. The process of de- projection of the 푥-axis is precisely Fig. 11, while excitation of residual nuclei is described by a the 푦-axis is Fig. 12. Across Figs. 11-13, all bin 2 combination of the evaporation model and the widths are identical (10 푀푒푉⁄푐 or 푀푒푉⁄푐), Fermi model of explosive disintegration. This and all counting scales are logarithmic. Note the combined approach shows good agreement with 퐺푒푉 bright spot at ~1.9 in invariant mass and 푐2 experimental data in the modeling of antiproton 퐺푒푉 12 ~0.1 in total momentum; this shape curves annihilation at rest on 6퐶 nuclei, and provides a 푐 reliable predictor of the characteristics of slow slightly upward and to the right, and contains the antineutron annihilation on 12퐶. This model can ~35% of all exiting pions and photons which go 6 thus be used as an event generator in the design through the nucleus without interaction. of detector systems for planned experimental searches for neutron—antineutron transformation at the European Spallation Source, employing a free beam of cold neutrons. This approach is universal, and can be used for simulating antineutron annihilation on many different nuclei. However, to search for the transformation occurring within nuclei (for 40 example, within 18퐴푟, with no external source of antineutrons), a valid model can only be created when a proper definition of radial annihilation FIG. 13: Stage 3 total momentum vs. invariant probability density is incorporated, allowing for mass of pions and photons. Note the double lobe the derivation of intra-nuclear 푛 → 푛̅ structure; this is due to the absorption of a single transformation constraints. The model proposed pion during transport. Also recognize the thin, in this work can be thought of as a first step in the sickle-like shape in the lower right-hand corner of preliminary modeling of this full process. the figure; this is due to invariant mass of the Expressing an aside into future developments, it initial Stage 1 mesons which did not interact with is planned that more precise modeling will be the nucleus. rendered for the intra-nuclear 푛 → 푛̅ V. CONCLUSION transformation thought possibly to take place 40 with 18퐴푟. Proper simulation of such a signal will This work has endeavored to demonstrate the be important to assess the feasibility of detail of the optical-cascade model for describing significantly improved transformation searches in 12 antineutron annihilation on 6퐶 nuclei. It is quite the DUNE experiment. The goal of accurate and important that the absorption of a slow precise simulation is sought for the absolute antineutron is described within the framework of suppression of the atmospheric neutrino the optical model and that radial dependence of background in the DUNE experiment, which will the annihilation probability is used within the allow for an improvement in the search limits by

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several orders of magnitude. Inevitably, this will EG is grateful to the INR, Moscow for supporting also require careful study and accurate simulation her travel. of atmospheric neutrino events in DUNE. The basic physics, model, file types, and analysis techniques presented in this article will continue to be employed, though special care must be taken in the integration of proper intra-nuclear transformation and radial annihilation probability distributions into the simulation. One other 푛 → 푛̅ generator, already developed internally to the DUNE collaboration [23], is currently being studied by multiple colleagues in both the DUNE and ESS collaborations; complex techniques have been developed using neural networks and multivariate boosted decision trees to study the separability of supposed signal from atmospheric neutrino background with promising results. However, when considering the subtleties of the simulation assumptions and techniques, some room for improvements within the generator are thought to be possible. Thus, the independent generator development described in this work, 40 along with its future 18퐴푟 extension, will help in understanding the potential and limits of exploration of rare processes like 푛 → 푛̅ where separation from background plays a major role. Altogether, this will hopefully lead to a fruitful collaboration and collective meta-analysis between generators and groups, which could reputably assess the probability of separating signal from background sources in such a large, underground experiment. A. Acknowledgements The authors would like to thank Jean-Marc Richard and Eduard Paryev for fruitful discussions of these nuclear processes, and to Yuri Kamyshkov for his permanent interest in this work. We wish to thank our colleagues in the NNBar ESS collaboration for the encouragement of this work. The work of JB was supported through DOE Grant DE-SC0014558 and the University of Tennessee Department of Physics.

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