International Journal of Modern Physics E

Three-body bound, virtual and resonant states in the $dt{\mu }^{-}$ and $dd{\mu }^{-}$ systems were found by solving homogeneous Faddeev equations on the real and complex energy planes. The formalism of searching is conducted for the scattering matrix poles on both physical and nonphysical energy sheets with an analytical continuation of the corresponding Lippmann–Schwinger equations for t-matrix. Some benchmark tests have been done with model and realistic two-body potentials to verify the analytical continuation procedure. Direct calculations revealed five resonant states in the $dt{\mu }^{-}$ and $dd{\mu }^{-}$ systems, as well as two virtual states in the $dt{\mu }^{-}$ system in the strong coupling mode. The key influence of short-range $dt$ and $dd$ interactions was found on the appearance of $dt{\mu }^{-}$ and $dd{\mu }^{-}$ bound states in the energy range of 1–20keV, which disappeared when such interactions were turned off.

In this paper, we determine the location of the neutron drip line for light $\mathrm{\Lambda }$ hypernuclei. Predictions are made for elements up to carbon within the Skyrme–Hartree–Fock approach. It is shown that ${}_{\mathrm{\Lambda }}^{11}$Li, ${}_{\mathrm{\Lambda }}^{17,19}$B and ${}_{\mathrm{\Lambda }}^{22}$C with unstable nuclear cores are bound whereas ${}_{\mathrm{\Lambda }}^{10,11}$He and ${}_{\mathrm{\Lambda }}^{14}$Be are not.

A component of the total cross section of the deuteron-deuteron interaction connected with the vector polarized ($P_y$) one initial deuteron and tensor polarized ($P_{xz}$) another one constitutes a null-test signal of the T-invariance violation but P-parity conservation. The energy dependence of this signal is calculated here at beam kinetic energies per nucleon of 0.1–1GeV within the Glauber spin-dependent theory of multiple scattering in the S-wave approximation for the deuteron wave function.

Accurate simulation of radiative capture reactions (RCRs) requires precise modeling of $\gamma$-ray strength functions (GSF). Various GSF models exist but need normalization based on experimental total radiative width (TRW) data. This study found that current normalization methods fail for neutron-induced RCRs in ${}^{238}$U and ${}^{233}$Th nuclei. In this regard, re-normalization values are proposed in order to improve model consistency at low energies. The standard Lorentzian (SLO) model fits experimental data best with minimal normalization.

Asymptotic normalization coefficients (ANC) are important nuclear characteristics. This paper discusses a method for determining ANC values from experimental data on differential cross-sections of nuclear transfer reactions. The method is based on the use of the analytic continuation of these cross-sections to the pole point of the reaction amplitude with respect to the variable $cos\theta$, where $\theta$ is the center-of-mass scattering angle. The method under consideration is used to determine the ANC for the channel ${}^{17}$O(1/2${}^{+}$; 0.871MeV) $\to {}^{16}$O(0${}^{+}$; 0MeV) $+n$ from the data on the ${}^{16}$O(0${}^{+}$; 0MeV)${(d,p)}^{17}$O (1/2${}^{+}$; 0.871MeV) reaction. When determining the ANC, the corrections caused by the Coulomb interaction in the initial, final and intermediate states of the reaction were taken into account. It is shown that these corrections have a great impact on the extracted ANC value.

We present a novel helix fit, based on an analytical method, on par with FPGA implementations in terms of speed. High-luminosity experiments exhibit event pileup (200 at the upgraded LHC).^, To cope with the data load, the concept of Hardware Track Trigger (in the Event Filter pipeline) emerged. The specific solution at ATLAS deploys a Stratix-10 FPGA chip in conjunction with a pre-fitted track database. The throughput is approximately 1.5 M tracks/s. Comparatively we can fit 7.6 M tracks/s (without pattern recognition, no field non-uniformity) on Xeon-Icelake – which puts our solution in a competitive position vis-á-vis the dedicated hardware one when pattern recognition (say Hough transform for stiff tracks, equivalent to resolution relaxation in Ref. ) is included.

In ultraperipheral collisions (UPC) of relativistic nuclei, the nuclear density distributions do not overlap, and the collision partners interact through Lorentz-contracted Coulomb fields. This leads to electromagnetic dissociation (EMD) of the nuclei induced by equivalent Weizsäcker–Williams photons. The RELDIS model was used to simulate the EMD of ${}^{209}$Bi and ${}^{124}$Xe beam nuclei in the MPD experiment at NICA, focusing on neutron and proton emissions in ${}^{209}$Bi–${}^{209}$Bi and ${}^{124}$Xe–${}^{184}$W collisions in collider and fixed target experiments, respectively. These simulations are crucial for understanding the potential contamination of detected hadronic events by EMD events. In addition, EMD of ${}^{124}$Xe in UPC with a CsI target at the BM@N experiment was modeled. A monoenergetic neutron beam resulting from EMD at the BM@N setup can be used for detector calibration and radiobiological research.

The results of application of the Cluster Channels Orthogonal Functions Method in combination with the R-matrix theory for calculating the cross-sections of the reactions induced by ⁶Li + ²H collisions are presented. As part of this work, first, very high lying resonances that have a decisive influence on the cross-sections of these reactions were identified. Second, for the first time in the framework of a theoretical study of nuclear reactions, to the cross-sections of all binary reactions pass through one and the same compound nucleus were computed using a consistent unified theoretical approach. The yield of sequential tripartition reactions is also a subject of this study.

Baryonic Matter at Nuclotron (BM@N) is a fixed-target experiment designed to probe the properties of the strongly interacting matter in the region of high baryon densities. In the beginning of 2023, BM@N has conducted the first physical experiment collecting several hundred millions of $\mathrm{\text{Xe}}+\mathrm{\text{CsI}}$ collisions at the beam energy of 3.8AGeV ($\sqrt{s_{NN}}=3.28$GeV). We report the preliminary results for the directed flow $v_1$ of protons with respect to the spectator symmetry plane from the first physics run at the BM@N facility. The results were compared with the existing data from previous measurements of directed flow in heavy-ion collisions at the beam energies of several GeV.

The SArov Tritium Neutrino Experiment (SATURNE), will study neutrino–atom collisions at unprecedentedly low energies for neutrino scattering experiments. A high-intensity tritium source of electron antineutrinos will be employed, with a total tritium mass of at least 1 kg (about 10 MCi) and possibly up to 4kg (about 40 MCi). The goal of SATURNE is to provide the first experimental evidence of coherent elastic neutrino-atom scattering and to search for the neutrino magnetic moment. It is expected that the detection and measurement of the elastic neutrino–atom interaction channel will be achieved using a 1–m³ liquid He-4 detector in a superfluid state. With five-year data from this detector, one will be able to probe the neutrino magnetic moment at a level of ${\mu }_{\nu }\sim 10^{-13}{\mu }_B$, which is an order of magnitude better than the world-leading upper limits. SATURNE will also study the ionization channel of neutrino-atom scattering. To this end, a cryogenic 4-kg Si crystal detector and a low-temperature 14-kg SrI₂(Eu) scintillation detector are being developed. Both detectors will have record low energy thresholds for their respective detector types. They are expected to achieve a sensitivity to the electron-antineutrino ${\mu }_{\nu }$ value of the order of $\sim 10^{-12}{\mu }_B$ already after one year of taking data.

Based on the microscopic model of proton–nucleus optical potential in which the amplitude of scattering of an incident proton on a bound nuclear nucleon is explicitly revealed, the experimental data of the proton elastic and inelastic scattering on ${}^{12}$C, ${}^{40}$Ca, ${}^{58}$Ni and ${}^{208}$Pb target nuclei at the energy range between 200MeV and 1000MeV are analyzed. Three parameters of the proton amplitude characterizing the scattering of protons on the bound nuclear nucleons are adjusted to the proton–nucleus scattering data and compared to the same parameters in case of proton scattering on free unbound nucleons to estimate “in-medium” effect on the proton–nucleus scattering process.

In this paper, we present a novel vertexing algorithm for determining the point of primary interaction, based on the linear extrapolation of tracks in the vicinity of the beamline. The algorithm considers tracks as infinitely elongated ellipsoids, thereby reducing the track vertexing problem to a vertex-to-points proximity problem. We have implemented the concept in C++ with the aid of our previously reported NXV4 package three for vectors and matrices, which gives very good CPU performance.

Accurate estimates of (anti)neutrino spectra and luminosities are needed to determine the possibility of pre-supernova neutrino detection. Using the recently proposed thermal quasiparticle random-phase approximation (TQRPA) method, we have studied thermal effects on the spectra and luminosities of (anti)neutrinos emitted in nuclear weak-interaction processes under pre-supernova conditions. We show that a thermodynamically consistent consideration of Gamow–Teller transitions in hot nuclei increases the energy luminosity and results in larger average (anti)neutrino energies.

The scissors mode structure is studied in the actinides within the Wigner Function Moments method. The theory predicts a splitting of the scissors mode into three branches due to spin degrees of freedom. The excitation energies and $B(M1)$ transition probabilities for the scissors states are calculated for even–even axial nuclei of the actinide mass region. The origin of the double-humped structure of the scissors spectrum observed in the lightest actinides is discussed. The energy centroids and integrated $M1$ strengths for the transuranium nuclides up to ${}^{256}$No are predicted. The results of calculations for ${}^{232}$Th, ${}^{238}$U and ${}^{254}$No are compared with available experimental data.

In this paper, we model the fission process by assuming that at certain elongation, after crossing the fission barrier, a fissile nucleus can be treated as a superposition of dinuclear systems (DNS). The distribution of primary fission fragments is described as a result of competition between evolution of initially formed DNS and its decay in relative distance. The level densities required for the calculations were microscopically derived accounting for deformation and excitation energy effects. The calculations performed for even ${}^{244-260}$Fm isotopes give overall good description of mass and neutron multiplicity distributions. To describe sudden onset of symmetric fission in ${}^{258}$Fm, the fissile nucleus is treated as superposition of DNS at smaller elongations than for lighter Fm isotopes, which is in line with significant reduction of half-life for ${}^{258}$Fm. Our results indicate the presence of bimodality due to coexistence of spherical and deformed mass symmetric fission modes.

The process of pion production in AA collisions at central rapidities with a large transverse momentum in the region kinematically forbidden for reactions with free nucleons (cumulative production) is analyzed. It is shown that in this region the dominant contribution is from the flucton–flucton interaction, leading to the emission of a quark, which then fragments into a pion. The asymptotic behavior of the inclusive cross-section at high initial energies near the kinematic boundary of the process is calculated and quark counting rules for the inclusive cross-section in this region are formulated. The found patterns can be tested in the MPD and SPD experiments at the NICA collider.

Rapidity-odd directed flow ($v_1$) of multi-strange baryons ($\mathrm{\Xi }$ and $\mathrm{\Omega }$) at mid-rapidity is reported for Au+Au collisions as recorded by the STAR detector at the Relativistic Heavy Ion Collider (RHIC). We focus on particle species where all constituent quarks are produced, such as $K^{-}(ūs)$, $\bar{p}(ūū\bar{d})$, $\bar{\mathrm{\Lambda }}(ū\bar{d}\bar{s})$, $\varphi (s\bar{s})$, ${\overline{\mathrm{\Xi }}}^{+}(\bar{d}\bar{s}\bar{s})$, ${\mathrm{\Omega }}^{-}(sss)$ and ${\overline{\mathrm{\Omega }}}^{+}(\bar{s}\bar{s}\bar{s})$, and demonstrate using a novel analysis method that the coalescence sum rule holds in a common kinematic region for hadron combinations with identical quark content. We examine the coalescence sum rule as a function of rapidity for nonidentical quark content having the same quark-level mass but different strangeness ($\mathrm{\Delta }S$) and electric charge ($\mathrm{\Delta }q$). The difference in the directed flow of different quark and anti-quark combinations, e.g., $v_1({\mathrm{\Omega }}^{-}(sss))-v_1({\bar{\mathrm{\Omega }}}^{+}(\bar{s}\bar{s}\bar{s}))$, is a measure of coalescence sum rule violation, and we call it directed flow splitting ($\mathrm{\Delta }{v}_1$). We measure $\mathrm{\Delta }S$ and $\mathrm{\Delta }q$ dependence of the $\mathrm{\Delta }{v}_1$ slope ($d\mathrm{\Delta }{v}_1∕dy$) between produced quarks and anti-quarks in Au+Au collisions at $\sqrt{s_{NN}}=27$GeV and 200GeV. Measurements have been compared with A Multi-Phase Transport (AMPT) and Parton-Hadron String Dynamics (PHSD) models with electromagnetic field calculations.

These proceedings discuss recent jet measurements by the STAR experiment at RHIC to study jet substructure in $p+p$ and jet quenching in $\mathrm{\text{Au}}+\mathrm{\text{Au}}$ collisions at $\sqrt{s_{\mathrm{\text{NN}}}}=200$GeV. Furthermore, STAR’s future plans for precision jet measurements with the upcoming data-taking periods in 2023–2025 are presented.

Jets are collimated sprays of hadrons created by the fragmentation of high energy partons, and can serve as an experimental tool for studying quantum chromodynamics. In particular, we can explore the properties of parton shower and evolution by measuring jet sub-structure. One of the techniques that allows experimental access to the parton shower is the jet grooming technique called SoftDrop. This analysis extends recent measurements of the jet sub-structure observables based on the SoftDrop algorithm, including groomed radius ($R_g$) and shared momentum fraction ($z_g$), in $p+p$ collisions at $\sqrt{s}$ = 200GeV in the STAR experiment. We present fully corrected multi-differential measurements of jet sub-structure observables at the first split and their correlations for jets of different transverse momenta and radii. To further explore the jet sub-structure, we present the first measurement of the jet sub-structure observables at the first, second and third splits via the iterative SoftDrop procedure. We show a strong dependence of the $z_g$ on the $R_g$ and the split number with the observation that selecting on narrower opening angles is similar to progressing further along the jet clustering tree. We compare our measurements to the state-of-the-art Monte Carlo models.

In this paper, we review the calculation of suppression of Chiral Separational and Chiral Vortical Effects for strange quarks (which allegedly yield spin polarization of $\mathrm{\Lambda }∕\bar{\mathrm{\Lambda }}$-hyperons in peripheral Heavy-Ion Collisions) by the nonperturbative interactions in hot deconfined QCD with the Field Correlator Method. The parameter range in the temperature-baryon density plane is expected to cover LHC-ALICE and RHIC-STAR data.

This paper deals with a microscopic approach for the description of the pygmy dipole resonance (PDR) as well as of the giant dipole resonance (GDR). The formalism is based on a Hamiltonian which is summing a mean field term, the pairing for alike nucleons and the dipole–dipole interaction. Two features are specific for our description: (a) The use of a projected spherical single particle basis and (b) the involved dipole operator is a Schif-like operator including a corrective term which is cubic in the radial coordinate. The dipole strength distribution with energy is plotted for the PDR region, i.e., [0,10] MeV. The dominant transitions have an isovector character for three isotopes and isoscalar for another three. The PDR states carry only 0.4–1.8$\%$ of the total EWSR. The dependence of the dipole strength on nuclear deformation is evidenced. For the whole interval of [0,20] MeV the dipole strength was first folded by Lorentzian of 1MeV width and then plotted as function of the QRPA energies. The peaks belonging to each of the two ranges are analyzed in detail and their nature, isovector/isoscalar, was pointed out. The r-cubic term and the nuclear deformation have opposite effects on the dipole strength. The famous Thomas–Reiche–Kuhn sum rule formula is generalized to the case of the Schiff dipole momentum. The new EWSR is very well satisfied. The photoabsorbtion cross-section is calculated and compared with experimental data in a figure showing its dependence on energy. The total cross-section, ${\sigma }_0$, the moments of the integrated cross-sections, ${\sigma }_{-1}$, ${\sigma }_{-2}$, the dipole polarizability and the thickness of the neutron crust are also calculated. The main features of PDR and GDR are realistically described.

The macroscopic model is formulated for a neutron star (NS) as a perfect liquid drop at the equilibrium. We use the leptodermic approximation $a∕R\ll 1$, where a is the crust thickness of the effective surface (ES) of NS, and R is the mean radius of the ES curvature. Within the approximate Schwarzschild metric solution to the general relativity theory equations for the spherically symmetric systems, the macroscopic gravitation is taken into account in terms of the total separation particle energy and incompressibility. Density distribution $\rho$ across the ES in the normal direction to the ES was obtained analytically for a general form of the energy density $\mathcal{ℰ}(\rho )$. For the typical crust thickness, and effective radius, one finds the leading expression for the density $\rho$. NS masses are analytically calculated as a sum of the volume and surface terms, taking into account the radial curvature of the metric space, in reasonable agreement with the recently measured masses for several NSs. We derive the simple macroscopic equation of state (EoS) with the surface correction. The analytical and numerical solutions to Tolman–Oppenheimer–Volkoff equations for the pressure are in good agreement with the volume part of our EoS.

We employ a theoretical framework motivated by a microscopic phenomenological approach involving Skyrme force model to account for the interaction of $\alpha$-particle with the nucleons of $\alpha$-radioactive nucleus. The realistic calculations for $\alpha$-particle preformation probabilities ($P_{\alpha }$) of even–even nuclei whose atomic numbers lie in the range $84\le Z\le 118$ are performed. Through the calculations of assault frequency possessed by $\alpha$-particle, shell effects are naturally incorporated in this methodology. The obtained results are then subjected to fitting procedures, resulting in a comprehensive fit within $N_p{N}_{n}I$-scheme. These fitted $\alpha$-particle preformation probabilities are utilized to predict the half-lives of unknown superheavy nuclei (SHN). This work not only refines our understanding about radioactive properties of these nuclei but also provides valuable insights into their stability against $\alpha$-decay and spontaneous fission (SF). Twelve $\alpha$-decay chains originating from unknown SHN have been identified in this work. The outcomes of this research would contribute to the advancement of predictive models to study the fundamental properties of SHN, offering an experimental verification as well.

The microscopic variant of the Grodzins relation and the Quasiparticle Phonon Model are applied to predict the excitation energies of the $2_1^{+}$ states of neutron-deficient U and Pu isotopes. The P-factor systematics is used to determine the quadrupole deformation of nuclei under consideration. The excitation energies of the $4_1^{+}$ states are predicted based on the simple universal anharmonic vibrator-type relation.

In this work, we study the r-mode instability of neutron stars and present critical angular velocity ${\mathrm{\Omega }}_c$ associated with bulk and shear viscosity for rotational stellar models. For this aim, we employ the lowest order constrained variational method and utilize realistic nucleon-nucleon interactions to determine the equation of state of neutron stars. We work within the framework of the slow rotation approximation and use various equations of state to calculate the angular velocity, fiducial gravitational radiation timescale and the rates of frequency change for both the maximum mass configuration and 1.4$M_{\odot }$ stars. Additionally, we compute the equation of state of magnetized neutron matter and investigate the effects of strong magnetic field on the r-mode instability window. In addition, we examine the influence of neuron star crust on a star’s angular velocity.

The positive and negative parity rotational band structures of odd–odd Nb isotopes with atomic masses 90, 92 and 94 are investigated using the theoretical framework of the projected shell model. These odd–odd nuclei are found to be ideal for investigating the significant structural changes occurring in the region situated far from the valley of stability. This is the first time that such comprehensive theoretical data for both positive and negative parities together for the odd–odd ${}^{90-94}$Nb isotopes have been disclosed. The g-factor and electromagnetic transition probabilities are among the various nuclear structure quantities that are examined for the first time. Using a broad configuration space for both protons and neutrons, the computed data have been anticipated up to high spin values.

In this paper, we obtained the mean mass density parameter (MMDP) ${\rho }_m(Z,N)$ based on the nuclear mass AME2020 and nuclear charge radius CR2013 databases. On this basis, we study the dependence relationship between neutron number N and the two neighboring isotopes MMDP difference $\delta \rho (Z,N)$, and then obtained the empirical formula for $\delta \rho (Z,N)$. By combining this empirical formula with the AME2020 and CR2013 databases, the root-mean-square deviation (RMSD) for 606 nuclear charge radii with $Z,N\ge$ 21 between the calculated and experimental values is 0.0046fm. Considering the odd–even staggering and isospin term correction, the study found that the RMSD is reduced to 0.0040fm. This work also uses Levenberg–Marquart (L-M) neural network approach to study the nuclear charge radius. The RMSD of nuclear charge radius is only 0.0030 fm, where the accuracy is increased by about 25%. Furthermore, the predicted values of nuclear charge radius obtained using this modified empirical formula and Neural Network model are competitive with the results of other models. These results indicate that nuclear charge radius relation and Neural Network model based on $\delta \rho (Z,N)$ proposed in this work have a certain accuracy, simplicity and reliability.

A recent observational study has set a constraint on the maximum mass of neutron stars (NSs), specifically focusing on PSR J0952-0607 and the compact star remnant HESS J1731-347, particularly within the low-mass regime. In our recent study, Ref. , we developed an energy density functional named NITR, which successfully produced the mass limit of the aforementioned pulsar but did not fully meet other observational constraints, such as those from NICER+XMM and GW170817. In this study, we introduce a new EDF named “NITR-I”, which not only reproduces the mass limit of PSR J0952-0607 but also aligns its canonical radius with NICER+XMM data, and its canonical dimensionless tidal deformability is consistent with the GW170817 event, thereby demonstrating the robustness of our model. The low-mass constraint associated with HESS J1731-347 suggests various possible compositions for the NS. The NITR-I model alone does not satisfy the HESS J1731-347 constraint; thus, we explore the possibility of incorporating dark matter (DM) inside the NS to meet this constraint. This approach proves successful when a specific value of Fermi momentum is considered. We also examine the impact of DM with varying Fermi momentum on different NS properties, such as tidal deformability and nonradial f-mode oscillation, using various relativistic mean-field (RMF) models. For the NITR-I EOS, the f-mode frequency is about 2.15kHz at 1.4 $M_{\odot }$ when $k_f^{\mathrm{\text{DM}}}=0$GeV, and it slightly increases to around 2.32kHz with $k_f^{\mathrm{\text{DM}}}=0.03$GeV. This increase in frequency due to DM suggests a possible reduction in tidal deformability, indicating that neutron stars with higher DM content are less susceptible to deformation by tidal forces which could be detectable in gravitational wave signals from neutron star mergers. Finally, we explore various universal relations (URs) for DM-admixed NSs, such as the relation between compactness and tidal deformability, the f-mode frequency and tidal deformability, and estimate the canonical values corresponding to both compactness and f-mode frequency using the GW170817 data.

The nucleon–nucleon (NN) potential is the residual interaction of the strong interaction in the low-energy region and is also the fundamental input to the study of atomic nuclei. Based on the nonperturbative properties of the quantum chromodynamics (QCD), NN potential is not yet directly accessible from QCD theory. Therefore, various models of NN interactions have been constructed based on Yukawa’s meson exchange pictures since the 1930s, including one-boson-exchange models, coordinate operator models and chiral effective field models. Analysis of extensive NN scattering data has shown that the two-body nuclear force exhibits a short-range repulsion and intermediate-range attraction, and decays rapidly with increasing distance. A series of charge-dependent high-precision NN interactions have been further developed in the past 30 years such as the AV18 potential, CD-Bonn potential, pvCD-Bonn potentials and the chiral effective nuclear potentials with momentum expansion up to the fifth order. In this work, the phase shifts at different channels, the cross-sections, the entanglement entropy in spin space, and the equations of state of symmetric nuclear matter and pure neutron matter from these high-precision NN interactions are calculated and systematically compared. It can be found that they have significant differences in the cases with high-angular momentum, high-laboratory energy and high-density regions.

The elastic cross-section is calculated for the elements ${}^{12}$C, ${}^{40}$Ca and ⁴He at high energies using the radial Klein–Gordon equation. These calculations were made for the case of pions using the Yukawa potential. To this end, the coupling constant of the Yukawa potential for the elements ${}^{12}$C, ${}^{40}$Ca and ⁴He was calculated first by fitting the numerical solutions to the experimental data. Our results are in agreement with the experimental data at low angular distribution. Finally, we calculate the bounds for the parameter of the model.

Recently, with the upgradation of LHC, it is realized that study of heavy-flavored hadrons, namely ${\mathrm{\Lambda }}_c^{+}$ and ${\mathrm{\text{D}}}^0$ in hadronic collisions, could reveal the possibility of thermalization of charm quarks. With this motivation, we study the production dynamics of these hadrons using a pQCD-inspired Monte Carlo event generator called PYTHIA8 in proton+proton collisions at $\sqrt{s}=13$TeV. The production dynamics of these hadrons are analyzed through charged-particle multiplicity, transverse momentum and pseudorapidity. With the help of the established PYTHIA8 tunes to mimic the behavior of flow-like features, we investigated the variation of effective temperature and degree of nonextensivity using thermodynamically consistent nonextensive Tsallis statistics. We further attempted to establish a connection between the initial state and final state by estimating the correlation between the number of multi-partonic interactions (${\mathrm{\text{n}}}_{\mathrm{\text{MPI}}}$) with the Knudsen number.