Backward nucleon production by heavy baryonic resonances in proton-nucleus collisions

TL;DR

This paper investigates backward nucleon production in proton-nucleus collisions, attributing it to secondary reactions of heavy baryonic resonances within the nucleus, supported by simulations consistent with experimental data.

Contribution

It introduces a detailed mechanism involving baryonic resonances and their secondary reactions as the source of backward nucleons, supported by Ultra-relativistic Quantum Molecular Dynamics simulations.

Findings

Backward nucleons outside p+N kinematic limits are due to resonance secondary reactions.

Simulations support the resonance mechanisms and match experimental data.

Two reaction pathways for backward nucleon production are analyzed.

Abstract

The production of backward nucleons, $N(180^\circ)$, at $180^\circ$ in the nuclear target rest frame in proton-nucleus ($\mathrm{p}+A$) collisions is studied. The backward nucleons appearing outside of the kinematically allowed range of proton-nucleon ($\mathrm{p}+N$) reactions are shown to be due to secondary reactions of heavy baryonic resonances produced inside the nucleus. Baryonic resonances $R$ created in primary $\mathrm{p}+N$ reactions can change their masses and momenta due to successive collisions $R+N\rightarrow R +N $ with other nuclear nucleons. Two distinct mechanisms and kinematic restrictions are studied: the reaction $R+N\rightarrow N(180^\circ)+N$ and the resonance decay $R\rightarrow N(180^\circ)+\pi$. Simulations of $\mathrm{p}+A$ collisions using the Ultra-relativistic Quantum Molecular Dynamics model support these mechanisms and are consistent with available data on proton backward production.

Equations

$$\displaystyle\sqrt{p^{2}+m^{2}}+m+m=\sqrt{k_{2}^{2}+m^{2}}+\sqrt{p_{1}^{2}+m^{2}}+\sqrt{p_{2}^{2}+m^{2}}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\$$

$$E_2^{\ast }=m+\sqrt{p^2+m^2}-2\text{leavevmode}\sqrt{m^2+{\left(\frac{p+k_2^{\ast }}{2}\right)}^2}\text{leavevmode}.$$

$$R+N\text{leavevmode}\to \text{leavevmode}N(180°)\text{leavevmode}+\text{leavevmode}N\text{leavevmode}.$$

$$M_1^2={\left[\sqrt{p^2+m^2}-\sqrt{{\left(\frac{p+k_2^{\ast }}{2}\right)}^2+m^2}+m\right]}^2-{\left[p-\left(\frac{p+k_2^{\ast }}{2}\right)\right]}^2\text{leavevmode}.$$

$$\sqrt{p^2+m^2}+nm=\sqrt{k_n^2+m^2}+\sum _{i=1}^n\sqrt{p_i^2+m^2}\text{leavevmode},\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}p=\sum _{i=1}^n{p}_i-k_n\text{leavevmode}.$$

$$E_n^{\ast }=(n-1)m+\sqrt{p^2+m^2}-\sqrt{n^2{m}^2+{\left(p+k_n^{\ast }\right)}^2}\text{leavevmode}.$$

$$M_{n-1}^2$$

$$\text{leavevmode}-\text{leavevmode}{\left[p-(n-1)\left(\frac{p+k_n^{\ast }}{n}\right)\right]}^2\text{leavevmode}.$$

$$P_{n-1}=p-(n-1)p_n=p-(n-1)\text{leavevmode}\frac{p+k_n^{\ast }}{n}\text{leavevmode}.$$

$$\sqrt{p^2+m^2}+nm=\sqrt{k_n^2+m^2}+\sum _{i=1}^n\sqrt{p_i^2+m^2}+\sqrt{p_{\pi }^2+m_{\pi }^2}\text{leavevmode},$$

$$p=\sum _{i=1}^n{p}_i+p_{\pi }-k_n\text{leavevmode},$$

$$R\to N(180°)+\pi \text{leavevmode},$$

$$E_n^{\ast }$$

$$M_n^2$$

$$P_n=p-n{p}_n=p-n\text{leavevmode}\frac{p+k_n^{\ast }}{n+(m_{\pi }/m)}\text{leavevmode}.$$

Full text

Key words: proton-nucleus reactions, backward proton production, UrQMD simulations.

PACS numbers: 25.40.Ep, 25.75.-q, 25.75.Dw, 25.90+k.

I Introduction

A large fraction of the particles produced in inelastic nucleon-nucleon ($N+N$) collisions appears from the decays of meson and baryon resonances, e.g., from $\Delta$ and $N^{*}$ isobars, which decay subsequently into a stable baryon and one or more mesons, dileptons or photons. The large number of resonance states and their large decay widths have led Rolf Hagedorn to postulate that the resonance mass spectrum behaves as a continuous exponentially increasing function Hagedorn (1965). This is experimentally indeed found up to masses of approximately 3 GeV Tanabashi et al. (2018), larger masses are not easily identified. An exponentially increasing mass spectrum $\rho(m)\sim\exp(m/T_{\rm H})$ of the hadronic states at $m\rightarrow\infty$ leads to the limiting temperature $T=T_{\rm H}$ for strongly interacting matter. Later theoretical suggestions Hagedorn and Rafelski (1980); Gorenstein et al. (1981); Stoecker et al. (1981) transformed the concept of a limiting temperature to the concept of a temperature of a phase transition or a crossover to a new high-temperature state – the quark-gluon plasma. The crossover model with Hagedorn states does indeed describe the lattice QCD data Vovchenko et al. (2019). Microscopic transport models of high energy collisions Bass et al. (1998); Bleicher et al. (1999); Cassing and Bratkovskaya (2008, 2009); Belkacem et al. (1998) model the Hagedorn states by string excitations Andersson et al. (1983) or include them directly Beitel et al. (2014).

Experimentally produced nucleons emitted in proton-nucleus (p + $A$) collisions have been observed in backward direction, at $180\degree$, in the nuclear target rest frame (these nucleons will be further denoted as $N(180\degree)$ and named as backward nucleons). Note, that the backward nucleons appear in a kinematic region forbidden in binary proton-proton (p + p) reactions. The production of backward nucleons and mesons was observed experimentally Baldin et al. (1973, 1975) and is referred to as a ’cumulative effect’ since two or more nuclear nucleons should be involved in this process. Several models were proposed to explain the data. One group of models suggests that an extension of the kinematic limit of nucleon-nucleon ($N+N$) collision is possible in p + $A$ reactions due to the short range proton-neutron correlations in the nuclear target Amado and Woloshyn (1976); Frankel (1977); Frankfurt and Strikman (1977, 1981, 1988); Frankfurt et al. (2008). These correlations lead to the presence of long tails in the nucleon momentum distributions inside a nucleus and thus extend the kinematic region of the emitted particles in p + $A$ reactions. It was also suggested that the cumulative effect is a result of the presence of multi-nucleon targets (“grains”) with masses $2m$, $3m$, $\ldots$ (where $m$ is the nucleon mass) inside the nucleus (see Refs. Baldin (1977); Burov et al. (1977); Efremov (1976); Efremov et al. (1988); Braun and Vechernin (2001)). In both these explanations, all necessary requisites for the cumulative hadron production are present inside the nucleus before the p + $A$ reaction.

Our present study is based on an alternative model scenario Gorenstein and Zinovev (1977), see also Refs. Gorenstein et al. (1977); Bogatskaya et al. (1978, 1980); Anchishkin et al. (1982); Golubyatnikova et al. (1984); Kalinkin and Shmonin (1990); Motornenko and Gorenstein (2017a, b). The kinematically forbidden regions in $N+N$ collisions can be explored in p + $A$ reactions due to creation of heavy hadronic states and their successive collisions with nuclear nucleons. We assume that a heavy baryonic resonance $R$, created in a primary proton-nucleon (p + $N$) reaction, can propagate further through the nucleus, and it has a chance to interact with another nuclear nucleon earlier than its decay to a stable hadron occurs. Therefore, several nuclear nucleons are involved in the backward nucleon production. Two mechanisms for the production of a backward nucleon $N(180\degree)$ will be considered: the reaction $R+N\rightarrow N(180\degree)+N$ and the resonance decay to nucleon and pion $R\rightarrow N(180\degree)+\pi$.

Excitations of heavy baryonic states and their subsequent re-scattering can be also probed in the sub-threshold production Aichelin and Ko (1985); Mosel (1991); Hartnack et al. (1994) of massive hadrons in $A+A$ collisions. Recently this approach was used to describe the data on strange and charm particle production Steinheimer and Bleicher (2016); Steinheimer et al. (2017); Gallmeister et al. (2018). The two phenomena in p + $A$ reactions – the production of hadrons outside the kinematic region of $N+N$ collisions and the sub-threshold production of strange and charmed heavy hadrons – have probably the same origin, namely, the creation of heavy resonances and their further interactions with nuclear nucleons.

This paper is organized as follows. In Sec. II we calculate the maximal energy for the backward production of nucleons in p + $A$ reactions if $n=2,3,\ldots$ nuclear nucleons are involved in this reaction. The mass of the baryonic resonance needed for this production is also calculated. These results are obtained as a consequence of energy-momentum conservation. In Sec. III the results of the Ultra-relativistic Quantum Molecular Dynamics simulations are presented. Section IV summarizes the paper.

II Kinematic restrictions for the backward nucleons

In this section we consider general restrictions on the energy of the backward nucleon emitted at 180$\degree$ relative to the direction of the projectile proton in p + $A$ reactions in the rest frame of the target nucleus. The restrictions obtained are a consequence of the energy-momentum conservation laws. We are interested in the maximal value of the backward nucleons energy. The production of any additional particles and/or a presence of a non-zero transverse particle momentum in the final state would require extra energy and would cause a reduction of the final energy of the backward nucleon. Thus, to find the maximal value of its energy we assume that no new hadrons are created, and all nucleons move longitudinally. Therefore, our kinematic analysis is reduced to the one-dimensional (longitudinal) problem. Besides, we assume $m=0.94$ GeV for the nucleon mass and neglect the small difference between proton and neutron masses.

II.1 $R+N\rightarrow N+N(180\degree)$

The energy-momentum conservation in the reaction p + $N\rightarrow N+\ldots$ does not permit the backward nucleon production in the target nucleon rest frame. If two nuclear nucleons are involved in the p + $A$ reaction the conservation laws for energy and momentum are:

[TABLE]

where $p$ is the momentum of the projectile proton, $k_{2}>0$ is the momentum of the backward nucleon $N(180\degree)$, the lower index in $k_{2}$ denotes the number of nuclear nucleons involved, whereas $p_{1}$ and $p_{2}$ are the final longitudinal momenta of the two nucleons.

Let us denote the maximal value of $k_{2}$ as $k_{2}^{*}$. From Eq. (1) and the maxima criteria $\partial k_{2}/\partial p_{1}=\partial k_{2}/\partial p_{2}=0$ one finds nucleon momenta $p_{1}=p_{2}=(p+k_{2})/2$ which maximize the backward nucleon momentum $k_{2}=k_{2}^{*}$. This leads to the following algebraic equation for the maximal kinetic energy, $E_{2}^{*}\equiv\sqrt{(k^{*}_{2})^{2}+m^{2}}-m$, of the backward nucleon:

[TABLE]

The solution of Eq. (2) for the kinetic energy, $E^{*}_{2}$, is presented in Fig. 1 (a) by the lower solid (red) line.

The value of $E_{2}^{*}$ increases with the projectile proton momentum $p$, and the upper limit $E_{2}^{*}\cong 0.24$ GeV is reached at $p\rightarrow\infty$.

We assume that a backward nucleon with kinetic energy $E_{2}^{*}$ is created through a two-step process. First, the reaction p + $N\rightarrow R+N$ takes place, and a resonance $R$ with the mass $M_{1}$ is created. The backward nucleon production takes then place at the second step in the following reaction111 Note that the reactions $\Delta+N\rightarrow N+N$ were discussed in Ref. Gazdzicki et al. (1998) where these reactions were proposed as a mechanism of the pion suppression in nucleus-nucleus reactions at small collision energies (see also Ref. Weyer (1990) ). :

[TABLE]

To reach the maximal energy $E_{2}^{*}$ (2) of the backward nucleon the baryonic resonance mass $M_{1}$ after the first p + $N$ collision should be equal to:

[TABLE]

The resonance mass $M_{1}$ after the p + $N$ collision is shown in Fig. 1 (b) as a function of the projectile momentum $p$ by the lower solid (red) line.

If $n\geq 2$ nucleons are involved in the reaction p + $A\rightarrow N(180\degree)+\ldots$ with a backward nucleon production, the energy and momentum conservation equations are:

[TABLE]

Similar to the case of $n=2$, one finds the nucleon momenta $p_{1}=p_{2}=\ldots=p_{n}=(p+k_{n})/n$ which maximize the backward nucleon momentum $k_{n}=k_{n}^{*}$. This leads to the following equation for the maximal kinetic energy of the backward nucleon:

[TABLE]

A solution of this equation for $n=3$ is presented in Fig. 1 (a) by an upper solid (blue) line. The value of $E_{3}^{*}$ increases with the projectile proton momentum $p$, and the upper limit $E_{3}^{*}\cong 0.63$ GeV is reached at $p\rightarrow\infty$. Note also that the resulting conservation laws and final expression  (6) for $E_{n}^{*}$ are the same as for a collision of the projectile proton with $n$-nucleon ”grain” in a nucleus.

To produce a backward nucleon with energy $E_{n}^{*}$ (6) in reaction (3) a baryonic resonance with mass $M_{n-1}$ needs to be formed in $n-1$ preceding collisions with nuclear nucleons. For example, to reach the energy $E_{3}^{*}$ within the reaction (3) the following two preceding reactions have to take place: p + $N\rightarrow R_{1}+N$ and then $R_{1}+N\rightarrow R_{2}+N$.

The straightforward calculations give:

[TABLE]

For $n=2$, Eqs. (6) and (7) are reduced to Eqs. (2) and (4), respectively. The resonance mass $M_{2}$ after two successive collisions with nuclear nucleons is shown in Fig. 1 (b) as a function of the projectile proton momentum $p$ by the upper solid (blue) line.

From Fig. 1 one observes that $E_{n}^{*}$ increases strongly with the projectile proton momentum $p$ up to $p\sim 10$ GeV/$c$. This corresponds to a mass region of the baryonic resonances not larger than $(3-4)$ GeV. A further strong increase of the baryonic resonance masses at $p>10$ GeV/$c$ seen in Fig. 1 (b) leads to only a slight increase of $E_{n}^{*}$ shown in Fig. 1 (a).

The momentum of a baryonic resonance with mass $M_{n-1}$ after $n-1$ collisions with nuclear nucleons should be equal to:

[TABLE]

The solutions of Eq. (8) for $n=2$ and $3$ are presented in Fig. 2 as functions of the projectile proton momentum by the solid upper red and lower blue line, respectively.

II.2 $R\rightarrow N(180\degree)+\pi$

Let us assume now a resonance decay into the backward nucleon and pion. If $n$ nuclear nucleons are involved, the conservation laws for the energy and momentum are:

[TABLE]

[TABLE]

where $m_{\pi}$ and $p_{\pi}>0$ are the pion mass and longitudinal momentum, respectively. At a given value of the projectile proton momentum $p$, the maximal value of the backward nucleon $k_{n}$ is reached at the conditions: $p_{1}=p_{2}=...=p_{n},$ and p_{\pi}/m_{\pi}=p_{n}/m\leavevmode\nobreak\, i.e., the $n$ nuclear nucleons and created pion should move with the same velocity. The maximal kinetic energy $E_{n}^{*}$ of the backward nucleon $N(180\degree)$ and the mass $M_{n}$ of a resonance before its decay into the backward nucleon and pion,

[TABLE]

are calculated as:

[TABLE]

The solutions of Eqs. (12) and (13) for $n=2$ and $n=3$ as functions of the projectile proton momentum are presented in Figs. 1 (a) and (b) by lower (red) and upper (blue) dashed lines, respectively.

The momentum of the baryonic resonance before its decay to the backward nucleons and pions is:

[TABLE]

The solutions of Eq. (14) for $n=2$ and $n=3$ are presented in Fig. 2 by upper red and lower blue dashed lines, respectively. One can see that the resonance should move backwards at small $p$ to produce a backward nucleon with maximal energy.

III UrQMD simulations of ${\rm p}+A$ reactions

In this section an analysis of the backward production of protons within the Ultrarelativistic Quantum Molecular Dynamics (UrQMD) transport model Bass et al. (1998); Bleicher et al. (1999) is performed. The mechanisms of the backward nucleon production by heavy baryonic resonances suggested in previous sections are probed by microscopical simulations of ${\rm p}+A$ collisions. The UrQMD model performs simulations by the Monte-Carlo calculations of stochastic two-particle collisions and resonance decays, it includes nucleon $N^{*}$ and $\Delta$ resonances with pole masses in the region $1.232-2.250$ GeV, higher masses are possible due to the finite widths in Breit-Wigner distribution for unstable particles. The hadron-like states with masses higher than $2.250$ GeV are effectively modeled by the string excitation, this mechanism is dominating for inelastic hadron collisions at the center of mass energy of nucleon pair $\sqrt{s_{\rm NN}}>3$ GeV. Contrary to hadronic degrees of freedom, the string degrees of freedom do not interact with other objects, they are only subject to fragmentation 222Note a novel string model for high energy p+A and A+A collisions where the string-string interactions are considered Bierlich et al. (2018).. Therefore, in UrQMD simulations the strings are not able to contribute to suggested mechanism of successive collisions with nuclear nucleons. Within UrQMD the Fermi motion of nucleons is modeled with the random distribution of nuclear nucleons momenta in the range of 0 - 300 MeV in a nucleus rest frame. The Fermi motion allows to widen the available kinematic region for backward proton production. An another theorized source of cumulative particle production, nuclear short range correlations Frankfurt and Strikman (1988), cannot be studied within UrQMD because of a lack of implementation of the phenomenon. The implementation of the nuclear short range correlations in a transport model is still a conceptual problem and a subject for future studies.

For the simulations of ${\rm p}+A$ energetic collisions the standard setup of UrQMD-3.4 is used. In this setup the mean fields and the hydrodynamic stage of collision are not considered. The backward nucleons are defined as nucleons emitted in a narrow cone with respect to beam axis, $180^{\degree}\pm 6^{\degree}$. The UrQMD simulations are performed in ${\rm p+A}$ collisions with different nuclei ${\rm{}^{4}He}$, ${\rm{}^{12}C}$, and ${\rm{}^{208}Pb}$ at the two values of the projectile proton momenta $p=6.9$ GeV/$c$ and $p=158$ GeV/$c$. These two momenta correspond to available energies at Dubna Synchrophasotron and the top CERN SPS energy, respectively. The backward nucleon production is a rare phenomenon and requires a large sample of events to study energetic cumulative particles. For each of p + $A$ reaction a number of $N_{\rm ev}=10^{8}$ collision events was simulated. To enlarge a production of the backward nucleons the central p + $A$ collisions with zero impact parameter $b=0$ are mostly considered. The calculated spectra are presented in Fig. 3. To avoid experimental problems with neutrons, only the backward protons are considered in the analysis. Therefore, a straightforward comparison with experimental data can be done.

The UrQMD results presented in Fig. 3 suggest that the backward proton spectra increase strongly with the atomic number $A$ of the target nucleus. Both the number of the backward protons and their largest kinetic energy increase with $A$. This behavior goes in line with the results from the analysis in Sec. II as the possible number of primary and successive collisions with nuclear nucleons increases strongly with $A$. The maximal energy of the backward protons does not show a noticeable increase with $p$ from $p$ = 6.9 GeV/$c$ to $p$ = 158 GeV/$c$. This observation just reflects an absence of baryonic resonances with a mass larger than $M>3$ GeV in the present version of the UrQMD model. An inclusion of resonances with masses larger than $M>3$ GeV or an implementation of string-hadron interactions in the UrQMD simulatioms would allow to widen the kinematic range for cumulative particles. This is a point for future studies.

In Fig. 4 (a) the UrQMD values of the backward proton spectra at the kinetic energy $E=E_{2}^{*}=0.17$ GeV are presented in p + $A$ collisions at $p=6.9$ GeV/$c$ as a function of $A$. Approximating these values by $\sim A^{\alpha}$ dependence, one finds $\alpha\cong 2.46$ for light nuclei (from He to C) and $\alpha\cong 0.67$ for heavy nuclei (from C to Pb).

The UrQMD results demonstrate also a strong dependence of the backward proton spectra on the centrality in p + $A$ reactions. In Fig. 4 (b) we present the backward proton spectra for UrQMD simulations in p + Cu reactions at the projectile momentum $p=9.5$ GeV/$c$. The lower (red) histogram presents the UrQMD results for the peripheral collisions with impact parameter $b=7$ fm, the upper (yellow) histogram for the central collisions with $b=0$, while the intermediate (green) histogram for the minimum bias p + Cu reactions. These minimum bias UrQMD results are in a good agreement with the data presented in Ref. Frankel et al. (1979) and shown in Fig. 4 as full circles. Note, however, that in our calculations a backward proton is defined as observed at angles of $180\degree\pm 6\degree$. We have to use an additional normalization factor for the measured data to compare them with our results.

Figures 5 (a) and (c) show the contributions of different sources to the final spectrum of the backward protons as functions of the proton kinetic energy at the projectile proton momentum 158 GeV/$c$ in p + C and p + Pb reactions, respectively. To enlarge the event statistics we define now backward protons as being emitted within the cone of backward angles $180^{\degree}\pm 15^{\degree}$.

The Fermi motion of nucleons inside nuclei with momenta up to $\sim 300$ MeV is implemented in the UrQMD model. This allows to produce backward nucleons even in a primary p + $N$ collision with the nuclear nucleons. On the other hand, the strong nucleon motion inside the nucleus due to the short range correlation effects are absent in the present version of the UrQMD model. The multi-nucleon target ‘grains’ inside nucleus are also absent.

From Figs. 5 (a) and (c) one observes that there are three sources of the backward protons: (multiple) elastic $N+N$ (re)scatterings which take into account the Fermi motion of nuclear nucleons, the resonance decays $R\rightarrow N+\ldots$, and the reactions $R+N\rightarrow N+N$. At low kinetic energies of the backward proton, elastic (re)scattering effects dominate. On the other hand, at larger $E$-values the backward proton production due to reactions $R+N\rightarrow N+N$ becomes the dominant source.

Figures  5 (b) and (d) show the ratios of the spectra of the backward protons created in reaction $R+N\rightarrow N+N$ by resonances with heavy masses $M>1.5$ GeV to the backward protons produced in this type of reactions by all resonances $R$ with any mass. A partial contribution from heavy resonances with $m>1.5$ GeV appears to be rather small, but it strongly increases with $E$.

IV Summary

The production of backward nucleons $N(180\degree)$ in p + $A$ collisions in the nuclear target rest frame is studied. This kinematic region is forbidden in p + $N$ reactions. It is suggested that the backward nucleons can be created by heavy baryonic resonances produced in several successive collisions with nuclear nucleons. A baryonic resonance formed in a primary p + $N$ collision propagates through the nucleus and can interact with other nuclear nucleons earlier than it decays. Thus, several nuclear nucleons, $n=2,3,\ldots$, are involved in the backward nucleon production. To find the largest possible energy of the backward nucleons in such a scenario the two competitive mechanisms are considered. The first one assumes that the baryonic resonance $R$ is formed in the $n-1$ preceding collisions, and then it creates the backward nucleon in the $n$th collision via the reaction $R+N\rightarrow N(180^{\degree})+N$. The second mechanism assumes the baryonic resonance formation in $n$ successive collisions with nuclear nucleons, and then its decay into the backward nucleons and pion, $R\rightarrow N(180^{\degree})+\pi$. In both considered mechanisms, the largest possible energy of the backward nucleon increases with the number of nuclear nucleons involved in the p + $A\rightarrow N(180^{\degree})+\ldots$ reaction. It also noticeably increases with the projectile proton momentum $p$ up to $p\sim 10$ GeV/$c$ and then goes gradually to its limiting value at $p\rightarrow\infty$. The largest energies of the backward nucleon appear to be close to each other in both discussed mechanisms and coincide at $p\rightarrow\infty$. However, the masses and longitudinal velocities of baryonic resonances in these two mechanisms are rather different.

Different aspects of the backward proton production in p + $A$ reactions are also studied within the UrQMD simulations. The reactions p + 4He, p + 12C, and p + 208Pb at the projectile momenta $p=6.9$ GeV/$c$ and $p=158$ GeV/$c$, energy at Dubna Synchrophasotron and top CERN SPS energy, respectively, are considered. The energy spectrum of the backward protons behaves as $\sim A^{\alpha}$, where $\alpha\cong 2.46$ for light nuclei and $\alpha\cong 0.67$ for heavy nuclei. We have also found that the maximal energy of the backward proton increases strongly with $A$.

The UrQMD results show that the decays $R\rightarrow N(180^{\degree})+\pi$ and the reactions $R+N\rightarrow N(180^{\degree})$ dominate in the UrQMD production of the backward nucleons with large kinetic energy. The second reaction appears to be the main source of the backward nucleons in p + Pb reactions. This is qualitatively different from the pion backward production considered in Ref. Motornenko and Gorenstein (2017a) where only resonance decays $R\rightarrow\pi(180^{\degree})+N$ are permitted.

Short range $p$-$n$ correlations inside the nuclei are not implemented in the present version of the UrQMD model. Their implementation would result in a strong increase of particle production in the kinematic regions forbidden in p + p reactions due to the quasi-elastic collisions of the projectile with nuclear target nucleons. Considered simultaneously both the short range correlations effects and inelastic re-scatterings of heavy hadron-like states in nuclei are expected to be the competitive mechanisms for a production of final hadrons with momenta forbidden in p + p reactions and for the sub-threshold hadron production in low energy p + $A$ and $A+A$ collisions. We hope that further experimental studies of p + $A$ reactions allow to search for the new heavy hadron-like states and an extension of the hadron mass spectrum to higher mass values. This research can be done by the NA61/SHINE collaboration at the SPS CERN as well as in GSI-HADES, and future FAIR-CBM and NICA-MPD experiments.

Acknowledgements.

The authors are thankful to M. Gazdzicki, M. Strikman, and G.M. Zinovjev for fruitful discussions. The work of M.I.G. is supported by the Goal-Oriented Program of Cooperation between CERN and National Academy of Science of Ukraine “Nuclear Matter under Extreme Conditions” (agreement CC/1-2019, No.0118U005343). H.St. acknowledges the support through the Judah M. Eisenberg Laureatus Chair at Goethe University, and the Walter Greiner Gesellschaft, Frankfurt.