The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies

L.G. DedenkoM.V. Lomonosov Moscow State University,119992 Moscow, Russia

Content

Introduction Hybrid multilevel scheme The 5-level scheme for the atmosphere Examples Conclusion

GOALS

Simulations of cascades at ultra-high energies

Acoustical (radio) signals production Transport of acoustical (radio) signals in

the real matter Detections of signals

ENERGY SCALE

SPACE SCALE

Transport equations for hadrons:

here k=1,2,....m – number of hadron types; - number of hadrons k in bin E÷E+dE and depth bin x÷x+dx; λk(E) – interaction length; Bk – decay constant; Wik(E′,E) – energy spectra of hadrons of type k produced by hadrons of type i.

),(/),(),(

)/(),()(/),(),(

1

EEEWxEPEd

xExEPBExEPxxEP

iik

m

ii

kkkkk

dEdxxEPk ),(

The integral form:

here

E0 – energy of the primary particle; Pb (E,xb) – boundary condition; xb – point of interaction of the primary particle.

),,())/ln()/()(/)(exp(

))/ln()/()(/)(exp(),(),(

EfxEBExd

xxEBExxxEPxEPx

x

bbbbk

b

0

)(/),(),(),(1

E

Eiiki

m

i

EEEWEPEdEf

The decay products of neutral pions are regarded as a source function Sγ(E,x) of gamma quanta which give origins of electron-photon cascades in the atmosphere:

Here – a number of neutral pions decayed at depth x+ dx with energies E΄+dE΄

.0),(

/)),(2),(0

0

xES

EEdExEPxES

e

E

E

EdxEP ),(0

The basic cascade equations for electrons and photons can be written as follows:

where Г(E,t), P(E,t) – the energy spectra of photons and electrons at the depth t; β – the ionization losses; μe, μγ – the absorption coefficients; Wb, Wp – the bremsstrahlung and the pair production cross-sections; Se, Sγ – the source terms for electrons and photons.

EdГWEdPWSEPPtP pbee //

'/ dEPWStГ b

The integral form:

where

At last the solution of equations can be found by the method of subsequent approximations. It is possible to take into account the Compton effect and other physical processes.

)])((exp(),(),( 00 ttEtEГtEГ

,)],(),(),()][)(([exp0

EEWEPEdEStEd b

t

t

,)](,[),( EdtEEWEPA be

t

te dtEtttEPtEP

0

))]([exp(]),([),( 00

t

t

t

eeee BAtEStdttEd0

]]),([[)]([exp(

EdtEEWEГB pe )](,[),(

Source functions for low energy electrons and gamma quanta

x=min(E0;E/ε)

)),(),(),(),((),(

),(),(),(

0

EEWtEEEWtEPEdtES

EEWtEPEdtES

p

E

Ebe

x

Eb

For the various energies Emin≤ Ei ≤ Eth (Emin=1 MeV, Eth=10 GeV)

and starting points of cascades0≤Xk≤X0 (X0=1020 g∙cm-2)

simulations of ~ 2·108 cascades in the atmosphere with help of CORSIKA code and responses (signals) of the scintillator detectors using GEANT 4 code

SIGNγ(Rj,Ei,Xk)SIGNγ(Rj,Ei,Xk)10m≤Rj≤2000m

have been calculated

SIGNAL ESTIMATION

s

th

E

E

x

e rxxEQGSrxxEQExESdxdEQ0

000 ,,,,,

Responses of scintillator detectors at distance Rj from the shower core (signals S(Rj))

Eth=10 GeV

Emin=1 MeV

)),,(),(),,(),(()(min

0

ERSIGNESERSIGNESdEdRS jee

E

Ej

x

xj

th

b

ENERGY DEPOSITION

POSITIVE CHARGE (GEANT4)

NEGATIVE CHARGE (GEANT4)

FOR HADRON CASCADESFLUCTUATIONS ARE OF IMPORTANCE

CHARGE EXCESS (GEANT4)

THIS FUNCTIONS SHOULD BE ESTIMATED WITH THE GEANT4 CODE WITH STATISTICS OF 10**6

FOR E=10**12 GEV NEARLY10**12 PARTICLES SHOULD BETAKEN INTO ACCOUNT

FOR ELECTRON-PHOTON CASCADES FLUCTUATIONS ARE VERY IMPORTANT DUE TO THE LPM-EFFECT

EXAMPLES

tQ

Ctp

cp

p

2

2

2

1

tqQ

2

2

2

2

2

1tq

Ctp

cp

p

pqCc

p

2

or

012

2

2 tp

cp

ctRS

dSRq

rtzyxp

4,,,

The Poisson formulae

00,,,`00,,,

zyxpzyxp

0t zyxqzyxp ,,0,,, ss hd

58 107107

I.C.:

It is possible at time

because

Energy deposition Q=dE/dV in water

Energy deposition in water

Energy deposition in water

Energy deposition in water

ENERGY DEPOSITION IN WATER

ENERGY DEPOSITION IN WATER

ENERGY DEPOSITION IN WATER

ENERGY DEPOSITION IN WATER

ENERGY DEPOSITION IN WATER

Charge excess

Lateral distributions of gammas, electrons and positrons

ENERGY DEPOSITION in detector

Energy distributions of gammas, electrons, positrons

Ratio of a signal to a charge particle density

el_ed.jpg

ga_ed.jpg

pos_ed.jpg

Conclusion

The hybrid multilevel scheme has been suggested to estimate acoustical (radio) signals produced by eγ and eh cascades in dense medium.

Acknowledgements

We thank G.T. Zatsepin for useful discussions, the RFFI (grant 03-02-16290), INTAS (grant 03-51-5112) and LSS-1782.2003.2 for financial support.

Number of muons in a group with hk(xk) and Ei :

here P(E,x) from equations for hadrons; D(E,Eμ) – decay function; limits Emin(Eμ), Emax(Eμ); W(Eμ,Ethr,x,x0) – probability to survive.

1 1 max

min

)(

)(0 ),,(),(),,,(

k

k

i

i

x

x

E

E

EE

EEthr xEPEED

EdExxEEWdE

xdxN

here p0=0.2 ГэВ/с.

,/)/exp()( 200 pdppppdppf

Transverse impulse distribution:

here hk= hk(xk) – production height for hadrons.

,// Ecphrtg kjj

The angle α:

Direction of muon velocity is defined by directional cosines:

All muons are defined in groups with bins of energy Ei÷Ei+ΔE; angles αj÷αj+Δαj,

δm÷ δm+Δ δm and height production hk÷ hk +Δhk. The average values have been used: , , and . Number of muons and were regarded as some weights.

cossinsincoscos;sinsincoscossinsincoscossinsin;sinsinsincossincoscoscoscossin

E jm kh

N N

The relativistic equation:

here mμ – muon mass; e – charge; γ – lorentz factor; t – time; – geomagnetic field.

,BVedtVdm

B

The explicit 2-d order scheme:

here ; Ethr , E – threshold energy and muon energy.

);5.0()(2/1ty

nzz

ny

nnx

nx hBVBVCHEVV )5.0(2/1

tnx

nn hVxx

);5.0()(2/1tz

nxx

nz

nny

ny hBVBVCHEVV

);5.0()(2/1tx

nyy

nx

nnz

nz hBVBVCHEVV

;)( 2/12/12/11ty

nzz

ny

nnx

nx hBVBVCHEVV

;)( 2/12/12/11tz

nxx

nz

nny

ny hBVBVCHEVV

;)( 2/12/12/11tx

nyy

nx

nnz

nz hBVBVCHEVV

)5.0(2/1t

ny

nn hVyy

)5.0(2/1t

nz

nn hVzz

tnx

nn hVxx 2/11

tny

nn hVyy 2/11

,2/11t

nz

nn hVzz

)/( EEeCHE thr

Ratio with to without magnetic field