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 nonlinear
problems 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 Cerenkov
Telescope 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.
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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.
R is real axis
n-dimentional real Cartesian
Canonic space
is convex unbounded
domain
has
continious second derivative
in
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
solution
2) form new weighting matrix
we find
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
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 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
R( x, y, z, x0, y0, Teta, Fi )
cos( Teta ) ( x1 cos( Fi )y1 sin( Fi ) ) ( x1 sin( Fi )y1 cos( Fi ) )
•
• where,
• . x1xx0z tan( Teta) cos( Fi ) y1yy0z tan( Teta) sin( Fi )
2
2
2
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, r )
R0( E, m, e, A, r )
( E, m, e, A )e
r0( E, m, e, A )e
e
s( E, m, e, A ) e
s( E, m, e, A ) 2 r
2
R0( E, m, e, A, r ) 
( E, m, e, A )
r
( rr0( E, m, e, A ) )

2
( E, m, e, A )
( E, m, e, A ) 2
2
( R0( E, m, e, A, r ) 2 )
C( E, m, e, A )
2
a a ma e 

E 
16 17
18 

 a a ma e( a a ma e ) ln



12
13 14
15
E 
 E0 

 10 11

ln


 E0 

a a ma e 

E 
34 35
36 

 a a ma e( a a ma e ) ln



30
31 32
33
E 
 E0 

 29 29

ln


 E0 

C( E, m, e, A )e
s( E, m, e, A )e
a a ma e 

E 
7 8
9 
 a a ma e( a a ma e ) ln


1
2
3
4
5
6
E 
 E0 



ln


 E0 

a a ma e 

E 
25 26
27 

 a a ma e( a a ma e ) ln



21
22 23
24
E 
 E0 
 19 20

ln

E0




K( E, m, e, A )e
a a ma e 

E 
43 44
45 
 a a ma e( a a ma e ) ln


37
38
39
40
41
42
E0
E 





ln


 E0 

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:
rmax
E2  K( E, m, e, A ) 

0
Q( E, m, e, A, r ) r dr
Q( E, m, e, A, rmax )1
Table of Parameters values and its errors
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
x(i)
1.939024
-0.83362
1.872445
0.159841
0.077143
-0.17309
-2.92089
2.437944
-5.47661
0.300308
-0.85798
1.871813
-0.00531
0.102175
-0.22353
dx(i)
0.01735
0.01583
0.03462
0.00235
0.00208
0.00456
0.03081
0.02929
0.06396
0.02468
0.02198
0.048
0.00331
0.00298
0.00652
i
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
x(i)
-0.430066763
1.304120847
-2.809946841
1.557207613
2.116343322
-4.780808881
-0.667269741
-0.331930847
0.745382538
-2.40401045
-3.617059696
8.298235209
-3.797325259
-0.680576397
1.332398356
dx(i)
0.04511
0.03959
0.08639
0.16891
0.14414
0.31597
0.0252
0.02161
0.04736
0.2492
0.21224
0.46521
0.22304
0.19706
0.43119
i
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
x(i)
-0.055330523
0.096979506
-0.198326287
2.345331288
0.540401991
-0.66694739
5.015475173
-0.799937436
1.62228099
0.116651365
0.167906163
-0.358334689
0.884252585
-1.085482763
2.851134406
dx(i)
0.03126
0.02839
0.06207
0.36225
0.31096
0.68082
0.19678
0.17602
0.38383
0.02861
0.02639
0.05757
0.30612
0.26806
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%
0
10
20
30
40
50
Number of
Restor.
Showers
20235
19845
18626
17013
9668
2500
Persent of
Restor.
Showers
0.96
0.94
0.88
0.80
0.46
0.12
hi2
0.02
0.10
0.28
0.94
4.66
26.33
mean
sdE
Resmean
sRes
0.007
0.024
0.28
0.80
0.010
0.022
0.30
2.50
0.031
0.067
0.35
3.40
0.040
0.400
0.50
4.00
0.097
0.095
0.52
4.34
0.106
0.104
0.51
4.81
Table for comparing the restoration effectiveness for different particle
Error%
0
10
20
30
40
50
Gamma
Number
5292
5265
5024
4609
2616
707
%
100
99.5
94.9
87.1
49.4
13.4
Proton
Number
%
5264
99.5
5187
98
4856
91.8
4438
83.9
2618
49.5
723
13.7
Helium
Number
%
5292
100
5234
98.9
4904
92.7
4481
84.7
2552
48.2
691
13.1
Ferum
Number
4387
4159
3842
3485
1882
379
%
82.9
78.6
72.6
65.9
35.6
7.16
Table for comparison of angle Teta efectiveness
Teta
Error%
0
10
20
30
40
50
TE 000
3375
3306
3114
2826
1626
420
%
95.7
93.7
88.3
80.1
46.1
11.9
TE 050
3374
3304
3109
2853
1626
410
%
95.6
93.7
88.1
80.9
46.1
11.6
TE 100
3374
3310
3111
2834
1607
432
% TE 150
95.6 3374
93.8 3313
88.2 3104
80.3 2832
45.5 1621
12.2 432
%
95.6
93.9
88
80.3
45.9
12.2
TE 200
3374
3311
3105
2831
1598
398
% TE 250
95.6 3363
93.8 3301
88 3083
80.2 2837
45.3 1590
11.3 408
Table for comparison of angle Fi efectiveness
FI
Error%
0
10
20
30
40
50
FI 000
6744
6624
6198
5648
3237
826
%
95.6
93.9
87.8
80
45.9
11.7
FI 800
6745
6595
6217
5682
3175
867
%
95.6
93.5
88.1
80.5
45
12.3
FI 1600
6746
6626
6211
5683
3256
807
%
95.6
93.9
88
80.5
46.1
11.4
%
95.3
93.6
87.4
80.4
45.1
11.6
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!