To calculate the lifetime of mesons in hot and dense nuclear matter, it is necessary to compute multidimensional integrals with a complicated integrand function. This work presents an algorithm and methods to calculate such integrals on the basis of the Monte-Carlo method. To optimize the computation, the parallel calculation algorithm is implemented in the C++ programming language using OpenMP and NVIDIA CUDA technologies. The calculations are performed on nodes with multicore CPUs and Intel Xeon Phi coprocessors and the Nvidia Tesla K40 accelerator installed within a heterogeneous cluster of the Meshcheryakov Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna. The code is used to calculate the pion lifetime using all possible pionpion scattering modes.
Proceedings of the 9th International Conference "Distributed Computing and Grid Technologies in Science and
Education" (GRID'2021), Dubna, Russia, July 5-9, 2021
The investigation of particles and their properties, including scattering and decay modes, the lifetime and width of particles in critical conditions of heavy ion collisions, is an important problem in modern physics. The theoretical description of strong-interaction matter under critical conditions is a complicated task, and it is not always possible to find an analytical solution to the problem. Therefore, the use of modern computer technologies plays a key role in theoretical research.
This paper is devoted to the investigation of the pion damping width in a hot pion gas within the NJL model. The main contribution to the width/lifetime comes from pion–pion collisions. To calculate the meson lifetime in hot and dense nuclear matter, it is necessary to compute 5-dimensional integrals with a complicated integrand function. This work presents an algorithm and methods for the calculation of complicated integrals based on the Monte-Carlo method using the parallelization algorithm to optimize the computation time.
The meson lifetime is determined by the particle self-energy [
1 (2 )3 ∫ ∫ ⃗1 1 2 1 8
1 ∫−1 ( )
| ⃗3|2 | ⃗3|2(√ 2+ 1)−| ⃗1| 3 | |2 >(<), (2) where = 1
( 1) 3 ( 3) 4 ( 4), the spectral function is calculated in the BreitWigner form ( ) = 2 ((
1( 3 + 1)( 4 + 1) and − 2)2 + 2 2)−1 with
= 2. The factors > =
< = ( 1 + 1) 3 4 are introduced for brevity, and are
the particle distribution functions = (
statistics [
( √ + ̅2) ± 1)−1 according to the particle | ⃗3|1,2 = | ⃗1| ± √| ⃗1|2 2 2 + ((√ 2 + 1)2 − | ⃗1|2 2((√ 2 + 1)2) − | ⃗1|2 2 2 )( 2 − 4 3(√ 2 + 1)2) (3) with = (√ 2 + 1)2 + 3 − 4 + 1 − 12.
To calculate the meson lifetime, the scattering amplitude | | of the process under investigation should be calculated. The amplitude | | is generally considered as an invariant amplitude and depends on the Mandelstam variables , , , where + + = ∑ .
It is seen from Eq. (2) that the structure of the integrand function is complicated and depends on many variables. The complex structure of the integrand is due to the kinematic conditions of physical processes and is limited by the pole in the denominator and the negative value of the radical expression. To show the 5-dimensional integration domain for the integral (2), we transfer it to the plane ( ⃗1, ⃗3), since by generating the arbitrary point ( 1, 3, 4, ⃗1, cos ) it is possible to determine ⃗3 for the fixed ⃗1 (Fig. 1).
In this work, the Monte Carlo (MC) method [
To obtain the selected point in the integration domain of Eq. (2), it is necessary to generate 5
random numbers. Taking into account the fact that cos(α) changes in [
According to the standard MC method, if the accuracy of the integral calculated over N selected points is insufficient, the number of points should be increased. To optimize calculations, the additional number of the required selected points is equal to the increase step M. The calculated results for M points are added to the previous calculations to get the final answer, and the accuracy is checked for N+M points.
Parallelization is also used to accelerate the machine time for computations. For this purpose,
the number of generated points (N) is evenly distributed among all threads (P). The remainder of the
points (N mod P) is also additionally distributed over the required number of threads (instead of
sending it to one thread). The program is implemented for computations in both single-thread and
multi-thread versions. The multi-thread version uses such parallelization technologies as OpenMP and
NVIDIA CUDA [
Error of S> Error of S< 2 4 6 Points number, 107 8 10 2 4 6 Points number, 107 8 10
The tests are performed at a fixed temperature T = 0.15 GeV, = 0.143 GeV and a fixed amplitude |A| = 1 for a different number of selected points. Fig. 2 (left) shows the results of calculations of Г, Σ>, Σ as a function of the number of selected points. Fig. 2 (right) illustrates the upper error bounds. As seen in Fig. 2 (left), after the limit of 3-4∙107 points, the result becomes insensitive to an increase in the number of calculations, which is caused by a decrease of the upper error bounds.
To speed up calculations, parallelization technologies are used. Table 1 shows the computation time with the generation of 2*107 points for a different number of threads. The calculations are also performed on the HybriLIT heterogeneous platform of the Meshcheryakov Laboratory of Information Technologies, JINR, Dubna [7]. As demonstrated in Table 1, the use of HybriLIT parallelization resources extremely reduces the computation time.
The physical background of this work is the calculation of the pion lifetime in hot nuclear matter. The amplitude of pion scattering is defined within the NJL model with meson-exchange diagrams considered in the pole approximation, which leads to amplitudes similar to the linear sigma model [8] with isospin amplitudes: 0 = 5 4 + 2
2 1 = 2
2 2 = 2 4 + 2 2
3 ( 2 −
1 ( 2 − + −
1 2 −
1 2 − + ) ,
1 2 −
) , 1 ( 2 − +
1 2 − ) , where all parameters, the meson mass and interaction constants , qq, depend on temperature, and 4 is supposed to be a constant of the 4-pion interaction, , , are the Mandelstam variables. The coupling strength is defined as ( , ) = 2 2 Γ ( , ), where Γ ( , ) is the amplitude of the triangle vertex σ → ππ.
The temperature dependence for the masses and constants is chosen within the NJL model and
can be found, for example, in [
The process π0π+⟶π0π+ occurs at the same rate as π0π-⟶π0π-, so we do not calculate them separately; we just put the factor 2. The calculation result is given in Fig. 3. It is clearly seen, that in a hot gas, the pion spectral function significantly broadens (due to an increase in the width) near the critical temperature, which is about 0.17 GeV for this version of the NJL model.
An algorithm for calculating the pion lifetime as a function of temperature is presented. Since the integrals (2) have a complicated integrand function, it is necessary to select a large number of points, which leads to an increase in the computation time.
To minimize the error, tests with a constant amplitude and a fixed temperature are performed (Fig. 2). The goal of the tests is to identify the minimal number of points for the initial step to plot the pion width as a function of temperature (Fig. 3). However, it cannot be argued that the test results obtained at a fixed temperature are relevant for all temperatures and a non-constant amplitude. Therefore, for each subsequent temperature, an additional calculation is carried out. The calculation with an increasing number of points (with the increase step M) is performed until the error obtained in the current iteration becomes worse than in the previous one. Nevertheless, it is difficult to obtain a smooth behavior of the function Γ( ) near critical temperatures (Fig. 3). Fig. 3 shows a comparison of our results with GSL-integration tools (VEGAS) [10].
To optimize the computation time, integral optimization and parallelization technologies (such as OpenMP, CUDA and their combination) are applied to the code. The use of the resources of the HybriLIT cluster reduces the computation time and enhances code efficiency.
The work was supported by the RFBR, grant No. 18-02-40137. 2021.