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underground and underwater detectors for neutrino astronomy require simulation tools capable to correctly handle the propagation of high energy muons up to @xmath2 energy region and above
. the new object - oriented ( c++ ) version of the detector simulation tool geant ( geant4 @xcite ) , will probably be suitable for this goal .
+ however many experiments  @xcite still make use of geant3  @xcite to simulate the detector response .
even if such package can in principle perform the simulation of particle propagation above 10 @xmath3 , it has been mainly designed for accelerator experiments , whose tipical energy range does nt exceed the @xmath3 region .
+ the simulation of muonic interactions is actually guaranteed by the authors for muon energy below 10 @xmath3 @xcite .
this is due to the parametrizations of cross sections for radiative processes contained in geant3 , which are reliable only for energy up to 10 @xmath3 .
moreover the description of photonuclear interaction is realized in a frame which significantly disagrees with theoretical calculations , for each muon energy . + our new simulation code ( gmu ) has been carried out to make the geant3 standard library reliable in reproducing uhe muons propagation through matter .
+ gmu replaces the simulation of the radiative muonic interactions performed by geant3 , for each energy and for each material , keeping the same structure of the original program .
the procedure to apply the new code is completely transparent in such a way that no adjustments must be implemented in the programs that make use of the standard geant3 library .
high - energy muons propagating through matter mainly interact by quasi - continuous ( ionization ) and discrete ( bremsstrahlung , direct electron - positron pair production , photonuclear interaction ) energy losses mechanisms .
ionization dominates at energy lower than few hundreds of @xmath1 while above the @xmath3 region energy losses are mainly due to radiative processes .
+ the radiative processes are simulated stocastically by geant3 above a fixed ( user supplied ) transferred energy threshold ; below this threshold they are treated as continuous .
the models which describe the interaction processes are the physical input for the code . in more detail , simulating a given process requires :    * to evaluate the probability of occurrence of the process by sampling the total cross section of the process * to generate the final state after interaction by sampling the differential cross section of the process    the reliability of the simulation is then affected both by the formulas ( and/or parametrizations ) chosen for cross sections and by the alghoritms used for sampling and for numerical integration .
+ standard reference formulas for uhe calculations can be found in the paper by lohmann and voss  @xcite which also tabulate the average muon energy losses for many materials and compounds up to @xmath4 @xmath3 . + direct electron pair production differential cross section has been first calculated by kelner and kotov in the framework of qed theory @xcite .
we have used the well - known parametrization performed by kokoulin and petrukhin @xcite which considers the corrections for atomic and nuclear form factors .
+ for bremsstrahlung differential cross section we have used the formula derived by andreev and bugaev @xcite which takes into account the structure of nuclear target ( elastic and inelastic form factors ) and the exact contributions due to atomic electrons ( screening effect and bremsstrahlung of the muons on electrons ) .
the formula used by lohmann and voss had been carried out just from this one by petrukhin and shestakov @xcite by neglecting bremsstrahlung on electrons and nuclear effects in light materials ( @xmath5 ) .
the discrepancy between the two approaches reaches few percents , in terms of bremsstrahlung energy losses , in the worse case ( for materials with @xmath6 ) and it is almost always negligible in the calculation of the total muon energy losses . + for the photonuclear interaction we have used the differential cross section calculated by bezrukov and bugaev @xcite within the vector meson dominance hypothesis .
we also have considered the recent accelerator data coming from zeus and h1 @xcite @xcite according to the ref .
for this process we have not considered , at present , the angle between the ingoing and outgoing muon , taking the reasonable approximation of completely forward scattering . as a consequence ,
the code is not suitable for studies dedicated to muon - nucleus scattering at large angles .
+ total cross section and average energy loss for each radiative process @xmath7 can be calculated , starting from differential cross sections , as follows : + @xmath8 @xmath9    * @xmath10 and @xmath11 are avogadro s number and the mass number respectively * @xmath12 is the fraction of initial energy @xmath13 lost by muon at the occurrence of the process @xmath7 * @xmath14 and @xmath15 are the kinematical limits for the allowed values of @xmath12 ( @xcite,@xcite ) * @xmath16 is the thickness of throughgone matter , expressed in @xmath17 * @xmath18 is the differential cross section for the process @xmath7 ( explicit formulas are given in the appendix )    the calculation of energy losses performed with the formulas described above agree with the lohmann and voss results at percent level in the energy range they considered ( @xmath19-@xmath20 . )
+ in fig .
[ cross ] we report the total cross sections for the radiative processes as they are tabulated at the initialization procedures by standard geant321 and by geant321 plus gmu . + in the first case the points are calculated with approximated analytical parametrizations whose accuracy is guaranteed ( at least for bremsstrahlung and pair production ) within @xmath21 in the declared energy range ( @xmath22 @xmath3 ) . + in the second case the points are calculated by performing numerical integration of the formulas given in the appendix .
( this operation requires a small amount of cpu time at the initialization , depending also on the number of materials considered . ) + the photonuclear cross sections differ by more than one order of magnitude . this discrepancy has already been observed in previous works @xcite and it is expected to affect the total energy loss especially in the case of light materials .
the total cross sections for bremsstrahlung and pair production below 100 @xmath3 seem to be in excellent agreement but for energy above 100 @xmath3 the geant321 cross sections strongly deviate from expectation .
in order to evaluate the correctness of the simulation code we have performed some checks for different materials . for each test
run we have activated the full stochastical regime and we have evaluated the average energy loss resulting from the simulation .
the values achieved with standard geant321 and with geant321 plus gmu have been compared with numerical calculations of the same quantities [ see eq.([losses ] ) ] .
+ in fig .
[ water ] and [ std ] we present the results for water and for standard rock respectively . the simulation performed by geant321 plus
gmu reproduces very well the numerical calculation at any energy , both for total energy losses and for the individual processes .
+ for such materials , the geant321 simulation is in good agreement up to 100 @xmath3 , but it produces much greater average energy loss for energy exceeding 100 @xmath3 . + in table 1 we report the total and the photonuclear energy losses in hydrogen as they result from simulation performed by the two codes ( @xmath23 geant321 ; @xmath24 geant321+gmu ) .
the discrepancy is not negligible , even at low energy , being around @xmath25 at 10 @xmath3 .
the main source of this disagreement is clearly due to a different estimation of the photonuclear process , whose relative importance grows for light materials .
+ in order to further verify the reliability of the simulation code we performed some specific runs by recording the fraction @xmath12 of energy lost by muon at the occurrence of each process .
( indeed the average energy loss is not especially sensitive to very big or very small energy transfer ) .
the distributions of such variable have been compared with analytical expressions for differential cross sections .
the results are shown in fig .
[ sampling1 ] , [ sampling2 ] and [ sampling3 ] for a fixed muon energy ( @xmath26 @xmath3 ) : they confirm the correctness of the sampling performed by gmu .
the simulation of muon propagation carried out by standard geant3 seems to be out of control for energy above @xmath27 @xmath1 .
+ moreover geant3 slightly underestimates the energy losses for very light materials ( at lower energies too ) , due to a strong discrepancy between expected and simulated values of the muon photonuclear cross section .
+ geant321 in addition to the new code gmu performs complete agreement with expectation in the tested energy range ( up to @xmath0 @xmath1 ) .
+ further improvements will consist in a more accurate simulation of the direction of outgoing particle produced in photonuclear interaction and in taking into account the influence of the medium ( lpm effect ) wich becomes important in the process of muon bremsstrahlung at extreme high energy . in this region direct muon pair production by muons
should be taken into account too .
+   + the gmu code can be requested by conctacting the authors .
e - mail :    * bottai`@`fi.infn.it * lorenzo`@`le.infn.it
we would like to thank the members of the lecce macro group and prof .
v.a.naumov for useful discussions and concrete help .
* @xmath10 is avogadro s number * @xmath28 and @xmath11 are the atomic number and the atomic weight of the material * @xmath12 is the fraction of initial energy @xmath13 lost in the interaction ( @xmath29 ) * @xmath30 is the fine structure constant * @xmath31 is the classical electron radius * @xmath32 is the electron compton wavelength * @xmath33 , @xmath34 , @xmath35 and @xmath36 are the electron , muon , pion and proton rest masses ( @xmath37 ) respectively    the formulae for radiative differential cross sections are taken from ref .
@xcite for @xmath38 pair - production , from ref .
@xcite for bremsstrahlung and from ref .
@xcite and  @xcite for photonuclear interaction .
@xmath67 \\ \\ & + \ ; { \displaystyle}{\frac{2\xi m_{\mu}^2}{t}}\left[g\left(z\right){\displaystyle}{\frac{2m_1 ^ 2}{m_1 ^ 2+t } } + { \displaystyle}{\frac{m_2 ^ 2}{t}}\ln \left(1+{\displaystyle}{\frac{t}{m_2 ^ 2}}\right)\right ] \big\ } \\
\label{photonuclear } \end{split}\ ] ]    @xmath68 is the energy of virtual photon @xmath69\right\},\\ \\
z=0.00282 a^{\frac{1}{3}}{\sigma}_{{\gamma}p}(\nu ) , \;\;\;\ ; t={\displaystyle}{\frac{m_{\mu}^2 v^2 } { 1-v } } ,    \\ \\
m_1 ^ 2=0.54 \rm{gev}^2 , \;\ ; m_2 ^ 2=1.80 \rm{gev}^2 , \;\ ; \xi=0.25 \;. \end{array}\ ] ]    the differential cross section is proportional to the total cross section @xmath70 , for absorption of a real photon of energy @xmath71 by a nucleon . in this calculation
we have used the regge - type parametrization for @xmath70 , according ref .
@xcite this model performs the best fit to accelerator data  ( @xcite,@xcite ) .
@xmath72 \;\mu b\ ] ] the integration limits are : @xmath73    figure 1 : total cross sections for radiative interaction processes vs muon energy ( pp  @xmath74 @xmath38 pair - production , bs  @xmath74  bremsstrahlung , ph  @xmath74  photonuclear interaction ) .
the plot shows the values tabulated by geant321 ( dashed lines ) and by geant321 plus gmu ( full lines ) .
the material is standard rock ( @xmath75 , @xmath76 ) .
+ figure 2 : average total muon energy loss in water ( @xmath77 ) vs the @xmath78 ( @xmath13 is the muon energy ) .
the plot shows the results from geant321 ( empty triangles ) and from geant321 plus gmu ( full stars ) ; the contributions from individual radiative processes ( performed by geant321 plus gmu ) and results from numerical calculations ( lines ) are also shown .
statistical error bars are comparable with dots size in case of photonuclear interactions ( ph ) and bremsstrahlung ( br ) .
+ figure 3 : average total muon energy loss in standard rock ( @xmath75 , @xmath76 ) vs the @xmath78 ( @xmath13 is the muon energy ) .
the plot shows the results from geant321 ( empty triangles ) and from geant321 plus gmu ( full stars ) ; the contributions from individual radiative processes ( performed by geant321 plus gmu ) and results from numerical calculations ( lines ) are also shown .
statistical error bars are comparable with dots size in case of photonuclear interactions ( ph ) and bremsstrahlung ( br ) .
+ figure 4 : differential cross section for muon bremsstrahlung vs @xmath79 ( @xmath12 is the fraction of energy @xmath13 lost by muon in the interaction ) . the numerical calculation ( dashed line ) is compared with the result obtained from the simulation performed with the code geant321 plus gmu .
the material is standard rock ( @xmath26 @xmath3 ) . + figure 5 : differential cross section for pair production vs @xmath79 ( @xmath12 is the fraction of energy @xmath13 lost by muon in the interaction ) .
the numerical calculation ( dashed line ) is compared with the result obtained from the simulation performed with the code geant321 plus gmu .
the material is standard rock ( @xmath26 @xmath3 ) .
+ figure 6 :    differential cross section for photonuclear interaction vs @xmath79 ( @xmath12 is the fraction of energy @xmath13 lost by muon in the interaction ) . the numerical calculation ( dashed line )
is compared with the result obtained from the simulation performed with the code geant321 plus gmu .
the material is standard rock ( @xmath26 @xmath3 ) .
a simulation package for the transport of high energy muons has been developed . 
 it has been conceived to replace the muon propagation software modules implemented in the detector simulation program geant3 . here 
 we discuss the results achieved with our package and we check the agreement with numerical calculations up to @xmath0 @xmath1 . 
 +   + _ keywords _ : muon transport , monte carlo simulation . 
 + _ pacs _ numbers : 25.30.mr , 02.50.ng