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, 2 INRNE, BAS, Sofia, Bulgaria

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

Abstract

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.

• 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 = YTh

i (X), (1)• where i is a number of arguments in which values is measured YExpt

i , 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.

R is real axis

n-dimentional real CartesianCanonic space

is convex unbounded domain has continious second derivativein

Maincase

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)

Regularizators in the iteration scheme (3)of the problem (2)

Tikhonov-Glasko, 1963

A. Ramm, 2000

of the process (3)(a) auto-regularizators

L. Alexandrov, 1970

The space is normed by Chebyshev norm

b) when is normed by Euclidean norm

L. Alexandrov, 1980

where is minimal eigenvalue of the matrix

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

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

Continuous regularized Gauss-Newton method

is the following Cauchy problem(L. Alexandrov, 1977; A. Ramm, 2000):

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.

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

tot(1,2,s), where 1,2 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 collision are given in http://home.cern.ch/b/biebel/www/RPP02

2.5 The number of quark families.

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 b are 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.

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 rmax depends 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,• .

( )R , , , , , ,x y z x0 y0 Teta Fi

( )cos Teta 2 ( )x1 ( )cos Fi y1 ( )sin Fi 2 ( )x1 ( )sin Fi y1 ( )cos Fi 2

x1 x x0 z ( )tan Teta ( )cos Fi y1 y y0 z ( )tan Teta ( )sin Fi

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.

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).

( )Q , , , ,E m e A re

( )C , , ,E m e A( ) , , ,E m e A e

( ) ( )R0 , , , ,E m e A r 2

( )R0 , , , ,E m e A r 2 ( ) , , ,E m e A 2 r( ) , , ,E m e A

( )R0 , , , ,E m e A r r2

( ) , , ,E m e A 2

( )r ( )r0 , , ,E m e A 2

( ) , , ,E m e A 2 ( )C , , ,E m e A e

a1a

2m a

3e ( ) a

4a

5m a

6e

ln

EE0

a7a

8m a

9e

ln

EE0

( ) , , ,E m e A e

a10

a11m a

12e ( ) a

13a

14m a

15e

ln

EE0

a16

a17m a

18e

ln

EE0 ( ) , , ,E m e A e

a19

a20m a

21e ( ) a

22a

23m a

24e

ln

EE0

a25

a26m a

27e

ln

EE0

( )r0 , , ,E m e A e

a29

a29m a

30e ( ) a

31a

32m a

33e

ln

EE0

a34

a35m a

36e

ln

EE0 ( )K , , ,E m e A e

a37

a38m a

39e ( ) a

40a

41m a

42e

ln

EE0

a43

a44m a

45e

ln

EE0

E 2 ( )K , , ,E m e A d0

rmax

( )Q , , , ,E m e A r r r ( )Q , , , ,E m e A rmax 1

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:

Table of Parameters values and its errors

i x(i) dx(i) i x(i) dx(i) i x(i) dx(i) 1 1.939024 0.01735 16 -0.430066763 0.04511 31 -0.055330523 0.03126 2 -0.83362 0.01583 17 1.304120847 0.03959 32 0.096979506 0.02839 3 1.872445 0.03462 18 -2.809946841 0.08639 33 -0.198326287 0.06207 4 0.159841 0.00235 19 1.557207613 0.16891 34 2.345331288 0.36225 5 0.077143 0.00208 20 2.116343322 0.14414 35 0.540401991 0.31096 6 -0.17309 0.00456 21 -4.780808881 0.31597 36 -0.66694739 0.68082 7 -2.92089 0.03081 22 -0.667269741 0.0252 37 5.015475173 0.19678 8 2.437944 0.02929 23 -0.331930847 0.02161 38 -0.799937436 0.17602 9 -5.47661 0.06396 24 0.745382538 0.04736 39 1.62228099 0.38383

10 0.300308 0.02468 25 -2.40401045 0.2492 40 0.116651365 0.02861 11 -0.85798 0.02198 26 -3.617059696 0.21224 41 0.167906163 0.02639 12 1.871813 0.048 27 8.298235209 0.46521 42 -0.358334689 0.05757 13 -0.00531 0.00331 28 -3.797325259 0.22304 43 0.884252585 0.30612 14 0.102175 0.00298 29 -0.680576397 0.19706 44 -1.085482763 0.26806 15 -0.22353 0.00652 30 1.332398356 0.43119 45 2.851134406 0.58273

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.

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

Table of restored shower number as function of fluctuation errors

Error%

Number ofRestor.

Showers

Persent ofRestor.

Showers hi2 mean sdEResmean

Res

0 20235 0.96 0.02 0.007 0.024 0.28 0.80

10 19845 0.94 0.10 0.010 0.022 0.30 2.50

20 18626 0.88 0.28 0.031 0.067 0.35 3.40

30 17013 0.80 0.94 0.040 0.400 0.50 4.00

40 9668 0.46 4.66 0.097 0.095 0.52 4.34

50 2500 0.12 26.33 0.106 0.104 0.51 4.81

Table for comparing the restoration effectiveness for different particleGamma Proton Helium Ferum

Error% Number % Number % Number % Number %

0 5292 100 5264 99.5 5292 100 4387 82.9

10 5265 99.5 5187 98 5234 98.9 4159 78.6

20 5024 94.9 4856 91.8 4904 92.7 3842 72.6

30 4609 87.1 4438 83.9 4481 84.7 3485 65.9

40 2616 49.4 2618 49.5 2552 48.2 1882 35.650 707 13.4 723 13.7 691 13.1 379 7.16

Table for comparison of angle Teta efectivenessTeta

Error% TE 000 % TE 050 % TE 100 % TE 150 % TE 200 % TE 250 %

0 3375 95.7 3374 95.6 3374 95.6 3374 95.6 3374 95.6 3363 95.3

10 3306 93.7 3304 93.7 3310 93.8 3313 93.9 3311 93.8 3301 93.6

20 3114 88.3 3109 88.1 3111 88.2 3104 88 3105 88 3083 87.4

30 2826 80.1 2853 80.9 2834 80.3 2832 80.3 2831 80.2 2837 80.4

40 1626 46.1 1626 46.1 1607 45.5 1621 45.9 1598 45.3 1590 45.1

50 420 11.9 410 11.6 432 12.2 432 12.2 398 11.3 408 11.6

Table for comparison of angle Fi efectivenessFI

Error% FI 000 % FI 800 % FI 1600 %

0 6744 95.6 6745 95.6 6746 95.6

10 6624 93.9 6595 93.5 6626 93.9

20 6198 87.8 6217 88.1 6211 88

30 5648 80 5682 80.5 5683 80.5

40 3237 45.9 3175 45 3256 46.1

50 826 11.7 867 12.3 807 11.4

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

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

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

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

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

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

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

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.

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!