On the investigation of some nonlinear problems in High Energy Particle and Cosmic Ray Physics

1. On the investigation of some nonlinear problems in High Energy Particle and Cosmic Ray Physics L. Alexandrov1, S. Cht. Mavrodiev2 , A. N. Sissakian1 1 BLTP, JINR, Dubna, Russia, 2INRNE, BAS, Sofia, Bulgaria

2. On the investigation of some nonlinear problems in High Energy Particle and Cosmic Ray PhysicsAbstract On the base of new  methods for numerical investigation of nonlinearproblems we find appropriate mathematical models for some phenomenons in High Energy Physics such as Dependence of hadron-hadron total cross sections and the average values of e+e-, pp and app multiplicity on quantum numbers and energy. We also establishe a mathematical model of the Cosmic Ray CerenkovTelescope for precise measuring the energy, mass, charge and direction of  initial Gamma, Proton, Helium, Ferum and other particles in wide energy range. The generalization of above model for all shower components, measured using the modern calorimeter technique, will permit to proof the possibility of creation working in real time World Cosmic Ray Telescopes Set. Finally we propose, after finishing their programs,to discuss the possible using of calorimeters ATLAS, CDF and SMS elements for creating the accelerator simulation target of atmosphere cosmic ray showers, so as to be used for calibration of simulating atmospheric showers computer codes.

3. 1. Introduction • 1.1 Heuristic investigation: experiment-theory or theory-experiment step by step understanding the lows of Nature. • 1.2 Particles physics and Cosmology • 1.3 Accelerators and Cosmic Rays • 2. Non linear inverse problem Dubna solution 2.1 Possibilities • The test of theory and estimation (measurement) the values of physical parameters: energy spectra of states, energy transitions, differential and total cross section, the decay time, the mass spectra of elementary particles, quarks and leptons and so on is based on comparison the experimental data with its theoretical description (or discovering the new, unknown dependences). • One has to solve the problem • YExpti = YThi (X), (1) • where i is a number of arguments in which values is measured YExpti , YThi (X) is a known from the theory function and X is a vector of physical parameters with known, estimated or unknown values. • Some time the experiment is returning the data, which can not be described and explained by theory successfully. In this case, the Dubna nonlinear approach permits, solving the problem (1), to discover the unknown mathematical dependences. • The possibility to compare mathematically the quality of the different functions permits to shoes the better one. The next step is to create the new theory from which one can calculate the same function. The solving again the transformed problem (1) leads to test of the new theory and to estimation the values and errors of X. • One has to stress that the pleasant future of this mathematical construction is that all theorems are constructive. So, the step from mathematical theory to Fortran codes: for example REGN (1972, Dubna), FXY(1997, Dubna), is hard, long, but, clear work. • In the next are presented shortly the formulae of mathematics.

4. R is real axis n-dimentional real CartesianCanonic space is convex unbounded domain has continious second derivativein Maincase

5. Instead of (1) we solve regularized least square problem Double-regularized Gauss-Newton method Linear problem in (3) we solve by - Cholesky Decomposition - Gauss-Jordan Elimination - Singular Value Decomposition (Gene Golub, 1973)

6. Regularizators in the iteration scheme (3) of the problem (2) Tikhonov-Glasko, 1963A. Ramm, 2000 of the process (3)(a) auto-regularizators L. Alexandrov, 1970 The space is normed by Chebyshev norm

7. b) when is normed by Euclidean norm L. Alexandrov, 1980 where is minimal eigenvalue of the matrix

8. Weighting matrix W In the banal case (main case!) where are standard deviations.When errors in experimental data are systematic (not standard) we use both robust weights(Huber, 1981) and LCH-weights

9. Suppose the mathematical model is good but experimental errors are bad or general unknown. In this case we can apply LCH procedure in two steps:1) solving Eq.(3) with we find solution 2) form new weighting matrix and finally solve Eq.(3) with weighting matrix

10. Continuous regularized Gauss-Newton methodis the following Cauchy problem(L. Alexandrov, 1977; A. Ramm, 2000):

11. Local root extraction(L. Alexandrov, 2004) In order to find all solutions of equation (2) in the domain the vector is repeatedly multiplied by the local root extractor in which is the j-th solution of Eq.(2). In the repeated solutions of the transformed problem Process (3) is executed with a new For every solution process (3) is started many times with different and . Each time when j increases the derivatives are automatic computed analytically and the matrices are adaptively scaled by J. More’s method.

12. 2.2 The dependence on quantum numbers of hadrons- hadrons total cross sections. In the framework of quasipotential approach which uses the Lobachevsky space of relative momentum and coordinates (Kadyshevsky formulation of fundamental length) was created a model for total hadrons- hadrons total cross section as a function of quantum numbers of hadrons and energy stot(a1, a2,s), where a1, a2 are the quantum numbers of interacted hadrons and s [Gev] is the interaction energy. 2.4 The average charged multiplicity dependence on quantum numbers. Data Source for average multiplicity as a function of ps for e+e− and app annihilation, and pp and ep collisionare given in http://home.cern.ch/b/biebel/www/RPP02 2.5 The number of quark families.

13. 3. The world set Cosmic ray telescopes, particle physics (standard model, Higgs boson, fundamental length) and cosmology (Big bang, Star’s evolution, Universe evolution, dark matter and energy) After creating the standard model on the basis of symmetries ideas and experiments on huge particle accelerators the next step of experimental particle physics, probably, will be combination of accelerators and cosmic ray high energy atmospheric showers telescopes experiments. A classic example of Cosmic ray telescope technique are the Cerenkov telescopes which measure the number of photons. But they can work only in Moonless and cloudless night. The next telescope step was realized in The Pierre Auger Observatory, combining fluorescence telescopes with an extensive air shower array of water Cerenkov detectors. The used form of lateral distribution function is S(r) = S(r0)[r(1+r/rs)/rs]-b, where r0, rs and bare fitting parameters. Using the Corsica simulating data for Cerenkov photons distribution in atmospheric showers and Dubna approach for searching the unknown dependences we found 45 parameters lateral distribution function, which permits to estimate the energy, mass, charge and axis parameters of initial particle in real time after the master condition of the telescope is realized.

14. 3.1 The inverse problem for the lateral distribution function • Q(Energy, mass, charge, R(x,y,z,x0,y0,Teta,Fi), x1,…, xn) • - where x, y, z are the coordinates of detectors, x0, y0, Teta and Fi are the shower axis coordinates and angles correspondingly, and • X=[ x1,…, xn] the values of fit parameters and its errors. • Corsica simulation data for the number distribution of Cerenkov photons at attitude 650 g/cm3 was calculated for Gamma, Proton Helium and Ferum initial cosmic ray particles for energies from 1012 to 1016 eV till distances rmax from the shower axis, where the number of photons is approximately one per squared meter. The value of rmaxdepends on energy, the shower starting attitude and from the kind of particle. • The distance between the detector with coordinate x,y,z and the shower axis with coordinates x0,y0 and angles Teta, Fi is • where, • .

15. In the next figure the spiral set of detectors is presented as well as the coordinate of shower axis, which was used for recovering tests, performed to demonstrate the real time work of telescope The mathematical model of Lateral distribution function for Cerenkov photons and parameters values was discovered using the nonlinear inverse methods, described in 2. The result is nonlinear composition of Bright-Wigner and Gauss functions. The dependence on quantum numbers is exponential and on energy is logarithmic.

16. In the next formulae the function Q is presented in coordinate system, where the shower axis is the vertical z- axis. So, the distances from axis to the detector is r =R(x,y,z,x0=0,y0=0,Teta=0,Fi=0). The integral calculation of the energy from the zero to rmax, is important supplementary condition, with clear physical sense for the velocity distribution of the shower components:

17. Table of Parameters values and its errors

18. The next 4 figures demonstrate the description of simulated data from the model and the energy behavior of functions Amp = exp(C(E,m,e,A)), r0, g, s, K.

22. 3.2 The estimation of mass, charge and axis parameters of initial cosmic ray particle using the quasiexperimental data with fluctuation errors We tested the possibility for estimation the mass, charge and axis parameters of initial cosmic ray particle solving the inverse problem for 64 telescope detectors data for different particles, energies, and axis parameters, using the random generated fluctuation of detector response in the interval 0-50%. Such test can be considered as a model of real time working telescope. One has to stress that all computer code executions was automatically performed as it has to be in a real telescope. The results for 21168 showers restoration test at different initial particles, energies, and axis parameters are presented in the next tables

24. The next two figures illustrate the energy and shower coordinate restoration for different fluctuations (0,10,20,30,40%)

26. The next two figures illustrate the restoration of energy of initial particle and the shower coordinates. We remind that the telescope radius is 0.2 km. The different colors indicate the values of fluctuations

28. In the next figure the restored mass distributions of initial particle for different fluctuations are presented

29. In the next figure the restored charge distribution of initial particle for different fluctuations are presented

30. In the next 4 figure the Hi2 distributions at different fluctuations are presented

31. In the next 4 figure the ResidualR = Rin-Rout distribution at different fluctuations as function of Hi2 is presented

32. In the next 4 figure the ResidualR = Rin-Rout distribution at different fluctuations as function of dE = Ein-Eout is presented

33. 3.2 The proposal for future using of accelerators for calibration of Intensive Atmospheric Showers simulating codes Today calorimeter simulating codes are well calibrated in the process of accelerator and calorimeter building. We propose to use this technique for calibrating the atmospheric shower simulating code by creating an accelerator model of atmosphere as a target in different proton machine. Because of logarithmic dependences on energy we hope that the lateral distribution function will work at EeV and higher energies. It seems that the energy interval from 1012 to 1014 eV will be enough for calibration the simulating code and to test the accuracy of electromagnetic, hadrons and leptons lateral distribution functions to recover the energy, mass and charge composition and space distribution of cosmic rays in wider energy interval, including EeV and higher energies.

34. 3.3 The world set Cosmic ray telescopes, particle physics (standard model, Higgs boson, fundamental length) and cosmology (Big bang, Star’s evolution, Universe evolution, dark matter and energy) Because the telescope model test demonstrates accuracy restoration for Teta to 25 degrees, we can hope that the world set of 8 telescopes from North to Sough Earth poles will give not only the energy, mass and charge composition but space distribution of Cosmic rays for one, two years in wide energy interval as well. Thank you for attention!