PHYSICS OF HIGH ENERGY COSMIC RAY MUONS

                                  E.V.Bugaev
        Institute for Nuclear Research, Academy of Sciences of Russia,
                               Moscow, Russia
   
       
    The first part of the present talk is dedicated to the review of 
author's works on photonuclear and electromagnetic interactions of high 
energy muons.
    The main feature of muon-nucleon (nucleus) interaction studies in 
cosmic ray experiments is that usually the average (over total 
statistics of experimental events) square of 4-momentum transfer is very 
small and, in opposite, the energy, transferred from muon to hadronic 
target (in laboratory system) is very large. So, cosmic ray experiments 
study the inelastic muon scattering in diffractive region. For this 
region there is no quantitative QCD model of the process (because in 
this case we have the soft interaction and perturbation theory does not 
work), there are Regge-type fits  only.
    The well known approach for an description of the inelastic lepton 
scattering in the diffractive region is an use of Generalized Vector 
Dominance Models (GVDM) in which total cross sections of virtual 
photon-nucleon interaction are expressed through imaginary parts of 
vector meson-nucleon scattering amplitudes. This approach is based on an 
existence of hadronic vacuum fluctuations of the photon (these 
fluctuations are clearly sufficient, i.e. the photon is an structured 
object when we study the lepton-nucleon interaction in the region where 
the square of transferred 4-momentum is small).
    The main problem of GVDM is that the sums over vector mesons 
entering the formulas for nucleon structure functions are, in general, 
divergent. It is assumed, in nondiagonal variants of GVDM [1-3], that 
the convergence of these sums is provided by the destructive 
interference of diagonal and nondiagonal amplitudes. In these models the 
relative contributions of nondiagonal amplitudes are model parameters 
adjusted to agree with photoproduction data. It is shown in the present 
talk that from quantitative point of view the model elaborated in [3] is 
good enough (because it agrees with data on small-x nucleon structure 
functions and with data on photon-nucleus scattering for real as well as 
virtual photons).
    However, as is well known, in diffractive scattering of hadrons the 
nondiagonal transitions are usually rather small. So, the assumption 
about the destructive interference of diagonal and nondiagonal 
scattering amplitudes must be checked by straightforward calculation 
using some model of vector meson-nucleon scattering. This check was 
performed in the recent work of author and Mangazeev [4]. The model used 
in [4] for a calculation of the scattering amplitudes contains several 
assumptions: 1) the nondiagonal transitions between members of the 
vector meson family (consisting of a ground state meson and its radial 
excitations) are of diffraction dissociation type; 2)constituent quark 
model of hadrons is valid, at least approximately; 3) meson-nucleon 
scattering can be described by two gluon exchange approximation of QCD; 
4) vector mesons are bound states of quark and antiquark described by 
Bethe-Salpeter equation with free propagators. The interaction kernel of 
this equation describes the long range quark-antiquark interaction of 
harmonic oscillator type (leading to the confinement). The solution of 
BS equation gives simultaneously the two particle wave functions of 
vector mesons and the mass spectrum of the mesons belonging to a family. 
For the parameterization of scattering amplitudes we use the two-pole 
Regge formula [5] (the second pole term is added for the correct 
description of an high energy behavior of the amplitudes).
    The result of the calculation [4] is the following: the nondiagonal 
amplitudes are rather small  ( about 10% of nearest diagonal ones) and, 
therefore, the destructive interference effects are relatively 
inessential.
    In [4] the QCD-motivated GVDM which is alternative to the 
nondiagonal approaches is elaborated. The main assumption is that not 
entire phase space of the quark-antiquark pair produced by the (virtual) 
photon is used for transforming into vector meson before striking the 
target. By other words, the cut-off factors must be introduced in the 
sums over vector mesons in expressions of GVDM taking into account that 
the process is not always mediated by photon-vector meson transitions. 
The idea goes back to Bjorken's aligned jet model. The parameter of the 
cut-off is qualitatively given by the evident physical considerations 
(connecting with the quark confinement), the accurate adjusting is done 
by the comparison with the data on the photoabsorbtion cross section. 
The introduction of the cut-off leads to the convergence of the GVDM 
sums and to appearing of additional terms in GVDM expressions (which can 
be calculated using the perturbation theory of QCD). It can be shown 
that these terms are negligibly small if transferred energy (in 
laboratory system) does not exceed several TeV.
    In the end of this part of the talk the process of muon 
bremsstrahlung on a heavy atom is briefly considered. It is shown that 
in the case of muon bremsstrahlung we have in contrast with the electron 
case, two non-Born corrections[6]: the first one is the usual correction 
of Davis, Bethe and Maximon arising from the region of large impact 
parameters (and,correspondingly, small momentum transfers). The second 
correction is due to small impact parameters (and large momentum 
transfers) and is not negligibly small only in the case of muon 
bremsstrahlung (because in this case the characteristic impact parameter 
is about a size of the target nucleus). The quantitative calculation 
shows that these two corrections have the same order of magnitude and 
different signs so they almost compensate each other[6]. 
So,surprisingly, there is no need to take into account the non-Born 
corrections to muon bremsstrahlung.
    In second part of the talk the questions connected with the prompt 
muon problem in cosmic ray physics are considered. At the beginning the 
review of hadronic charm production is presented. All main models of 
charm production in hadronic collisions are briefly considered 
(perturbative QCD model, intrinsic charm approach, recombination models, 
string models of Lund and Regge types). It is stressed that purely 
perturbative QCD model has serious difficulties (absence of the leading 
effects, too steep inclusive spectra). It is probable that charm 
particles are produced in soft collisions, i.e. that nonperturbative 
effects are important. The steepness of inclusive spectra depend on the 
model: if,e.g. the charm is intrinsic, the spectra are relatively flat. 
Experiments with cosmic rays, as is well known, are sensitive just to 
the region of large Feynman' x (in contrast with the accelerator 
experiments). It is clear therefore that the detection of prompt muons 
in cosmic ray experiments would give very interesting information about 
the process of hadronic charm production (which cannot be obtained from 
present day accelerators). Hadronic charm production may be one of 
examples of "forward physics", especially if charm in a nucleon is 
intrinsic.
    Prompt muon component if it is noticeable leads to a flattening of 
sea level spectrum of high energy muons and to a change of the zenith 
angle distribution. In last part of the talk the following problem is 
considered: which physical information can be obtained from underground 
muon data and what can be said, using this information, about prompt 
muon component in sea level muon spectrum.
    Authors of underground experimennt give, as a final output (i.e. 
after recalculation) vertical differential or integral muon spectra at 
sea level and depth-intensity relation (absorbtion curve). Authors of 
recent theoretical work[7] calculated sea level muon spectrum and 
absorbtion curves for a number of experiments using several models of 
hadronic charm production. The idea was the following: if some change of 
spectrum slope really takes place at high muon energies (as is claimed 
by some experimental groups) it must be visible also in absorbtion curve 
behavior. We suppose that the only source of possible flattening of the 
spectrum is prompt muon contribution. The final conclusion (see point 
3.2 of transparencies) is : the question about existence of large charm 
component, visible in vertical spectrum at several tens of TeV is still 
open in spite of the flattening observed in four underground 
experiments.
     
References

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4.E.Bugaev and B.Mangazeev,talk on "School on Microphysics",   
  Irkutsk, Russia, 1998 (unpublished)
5.B.Kopeliovich et al, JJINR E2-86-125 (1986)
6.Yu.Andreev and E.Bugaev, Phys.Rev.D57,N3 (1997)
7.E.Bugaev et al, Phys.Rev.D58 (1998)