- Email: [email protected]
Multiple hadron production in the quark-gluon string model
UCLEAR PHYSICS
ELSEVIER
Nuclear Physics B (Proc. Suppl.) 52B (1997) 116-119
PROCEEDINGS SUPPLEMENTS
Multiple Hadron Production in the Quark-Gluon String Model Yu.M. ShabelskP apetersburg Nuclear Physics Institute, Gatchina, St.Petersburg, 188350 Russia We give a short description of the Quark-Gluon String Model which allows one to describe quantitatively the Feynman-x and rapidity spectra of different secondaries produced in hadron-nucleon and hadron-nucleus collisions at high energies.
1. I n t r o d u c t i o n
In order to describe the processes of multihadron production at high energies, wide use has been made of models based on the idea of the 1/N expansion of QCD [1,2] in the framework of the scheme of dual topological unitarization (DTU). In Refs. [3-5] only the single-Pomeron contributions were included. In the Quark-Gluon String Model (QGSM) [6-9] the possibility of exchange of several Pomerons is accounted for also. Inelastic processes in the models of this kind are determined [10] by all the possible "cuttings" of one or several Pomeron or by a cutting between them. To calculate the cross section for production of secondary particles, it is necessary to know the distribution functions of the quarks in the colliding hadrons and the functions for fragmentation of quarks and diquarks into secondary hadrons. In the QGSM, it is assumed that all these functions are determined in the regions XF -+ 0 and XF -+ 1 [11] by the corresponding Regge asymptotic behaviors, and in the region of intermediate values of XF by means of interpolation. At the same time, they must satisfy several conservation laws. 2. Inclusive c r o s s s e c t i o n for particle prod u c t i o n in Q G S M We consider the Pomeron intercept greater than unity
Tevatron colliders data included, has led to a somewhat greater value A = 0.12-0.14. The imaginary part of the elastic amplitude is written as a sum of several absorptive parts, corresponding to all possible cuts of the given reggeon diagram, each related with a certain specific group of intermediate states through the schannel unitarity. The one Pomeron exchange gives a multiperipheral hadron ladder with small transverse momenta. The contribution of the double Pomeron exchange to the elastic scattering amplitude is connected with three different types of intermediate states. The cut in between the Pomerons is saturated by elastic scattering or diffraction dissociation. The cut of either one of the two Pomerons produces a multiperipheral ladder of particles with similar properties as in the case of one-Pomeron diagram; it being the first absorption correction to the physical cross section of the production of the single ladder of hadrons. Finally, the cut of both Pomerons yields two hadron ladders and hence, a doubled multiplicity. A straightforward application of the AGK cutting rules [10] yields the cross sections for n cut Pomerons, an, and the diffraction dissociation, a0, [12]. Each value of a (~) should be positive. The inclusive spectrum (i.e. Feynman-x distribution) of a secondary hadron h produced in pp interactions is determined in QGSM by the expression [6,7]
t
ap(t) = l + A + ap . t ,
A>0.
The intercept A was taken equal to 0.07 originally but the later analysis with the S p a S and 0920-5632(97)/S17.00 © 1997 Elsevier Science B.'~ All rights reserved. PII: S0920-5632(96)00855-9
XE
da
~Tinel d X F
Yu.M. Shabelski /Nuclear Physics B (Proc. SuppL ) 52B (1997) 116-119
where XE = E/Ema~,
h fqh (x+,n)fqq(X_,n) +
structure functions obtained from hard processes. The form of the functions u(x, n) is determined in QGSM by the corresponding Regge asymptotic behaviors in the regions x --+ 0 and x ---> 1, and for intermediate values of x by means of a simple interpolation [6,7]. The distribution functions of the quarks and diquarks in the proton which were used in the calculations have the form [7], for example
2(n 1)I~(~+, n)I~(~-, ~),
~(~,n)
Wn
(7n
=
(7n
is the probability of cutting precisely n Pomerons,
¢pPnP'h(xF)
fhq(x+,n)fh(x_,n) +
=
-
=
u~,(x,n)
The quantities fqq, fq and L in (4) correspond to the contributions of the diquark, the valence quark and antiquark, and the sea quarks, while the contributions of the incident particle and the target proton depend on the variables x+ and x_, respectively. The last two terms in (2) take into account the contributions of diffraction dissociation of the incident hadron and the target nucleon. The functions fqhq(x±,n), fqh(x+, n) and fh(x:t:, n) which determine the inclusive spectra of secondary particles, can be expressed in terms of convolutions of the momentum distributions of the diquarks and of the valence and sea quarks in the colliding hadrons, u(x), and the functions for fragmentation of diquarks and quarks into secondary hadrons, G h (z). For example, for the case of pp collisions we have [6,7] fhqq(X:t:, ~t )
:
U,,d(Xl, n)Ghd(x±/Xl )dxl
3
+3
117
u ~ ( x l , n)Ghu(x~:/xl )dxl .
+
The parameter A ,~ 0.2 - 0.3 determines the relative suppression of strange quarks in the sea. The inclusive spectra in ~ - p and Kp collisions can be calculated by the same way. 3. Q u a r k a n d d i q u a r k d i s t r i b u t i o n s in h a d r o n s QGSM considers hadrons consisting from the constituent quarks so we can not use hadron
C~"R--2~B+I(1--
X) -~R+'~-I ,
= C ~ x - a ' ( 1 - x) "a-2~'+'~-I ,
where a n = an(0) ~ 0.5 and ~B = (~B(0) --0.5 are the parameters of the meson and baryon Regge trajectories. The normalization factors Ci are determined by the condition
f ui(x,n)dz = 1 . In addition, as a consequence of the energy conservation law, for each value of n we have
E (x)i = E / u i ( x ' n ) x d x i
= 1.
i
4. Q u a r k a n d d i q u a r k f r a g m e n t a t i o n functions The functions for fragmentation of diquarks and quarks into secondary hadrons are also determined in QGSM by their Regge asymptotic behaviors [11]. For example let us consider the case of a quark fragmentation into a meson. In the limit z --~ 1 the fastest hadron h is separated from all the other hadrons by a large rapidity interval Ay ln[1/(1 - z)]. The probability of this quark fragmentation can he obtained from the corresponding triple-reggeon diagram [11]:
Gh(z) ~ (1
-
z)%Z(°)-2%~(p~') .
Using the relations between the intercepts and slopes of the trajectories of Regge poles containing quarks of various types, as obtained in the planar approximation of the topological expansion [13]
%~(o) + %~(o) = 2a~(0),
118
Yu.M. Shabelski /Nuclear Physics B (Proc. Suppl.) 52B (1997) 116-119
and a(p 2) = a(0) - a ' . p2 we obtain t
a , h ~ (1 - z ) - ~ ( ° ) + ~ , ~
strangeness must hold. For example,
2
p~ .
(1)
B B B [a~(z)a~, (o) + a~B (z)a~Ao) -
fO dz Z
The value ai~(0 ) is the intercept of the secondary trajectory "composed" of a quark and an antiquark of type i ( a ~ = ad-~ = ap = a~ = OR 1/2 , as~ ---- a~o). The behavior of G h ( z ) for z --+ 1 integrated over transverse momentum p~ can be obtained from Eq. 1 by replacing/92 in it by the average
value (p~>:
z ) a u ~ (-~ 0)] a ~ ( z ) a ~~ (0) - a u (-~
= 1.
For the fragmentation function used in the previous studies [6-8], all these conservation law are not exactly satisfied. Therefore the functions were modified appropriately in [9]. However, the Regge asymptotic behaviors at both extremes can be retained.
G h ( z ~ 1) ~ (1 -- z ) - % ~ (°)+~ ,
5. I n c l u s i v e s p e c t r a o n n u c l e a r t a r g e t s 2
A = 2ai-£(pT ) .
In the limit z --+ 0 the functions G ( z ) are constants ah [11] which do not depend on the type of quark (diquark) i, i.e.
a ~ ( ~ - , 0) = a h . The value of the constant ah is determined by the dynamics of the rupture of the string when a q~ pair is produced from the vacuum. All the particles of the same isotopic multiplet cotrespond to exactly the same constant ah (i.e., a~+ = a~- = a~0 = a t ) , and the constant ah is the same for production of particles and antiparticles (ag+ = a K - = aKo = a-K-O = a g , ap = a F = an ~ a~ :-- a N , etc.). In Refs. [6-8] the simplest interpolation formulae were used for the fragmentation functions at intermediate values of z, for example: G 7ru + (z)
=
a ~ ( 1 - z) -aR(°)+~ ,
Gud(Z)
=
ap(1
Gk+ (z)
=
a k ( 1 - - z) -~*(°)+x .
P
-
z) -aR(°)+x ,
The inclusive spectra of secondaries produced on nuclear targets can be considered also in the framework of DTU approach [14-16], and in particular by QGSM [17]. The multiple scattering theory [18,19] is used successfully for a description of high energy hadron-nucleus collisions. In this theory the interaction with a nucleus is treated as a sum of interactions with different number of nucleons. Such an approach corresponds [20,21] to the Regge space-time picture of the interaction. It turns possible to describe quantitatively elastic and inelastic h A cross sections as well as several important characteristics of the multiple production processes (average multiplicities of secondaries, multiplicity distributions, probabilities of multinucleon interactions, etc.), see, e.g., Ref. [22]. Expressions for the cross sections of hadronnucleus collisions are the most simple in the socalled optical limit of the multiple scattering theory. The secondary production cross section has the well-known forms:
However, the fragmentation functions must obey the energy conservation law
tThAPr°d =
/ d2b[1
_ e-ainT(b)]
where
y~
a)(z)a)(O)d~
:
1.
h
In addition, for each quark-diquark or quarkantiquark string as whole the laws of conservation of the electric and baryon charges and of
T(b) = A ] p(b, z)dz , f
T ( b ) d2b
A,
and p(r = v ~ + z 2) is the nuclear density distri-
bution, a
=
_tot inel O h N ~ f f i n ~--- ~ h N "
Yu.M. Shabelski/Nuclear Physics B (Proc. Suppl.) 52B (1997) 116-119
The cross section of inelastic interaction with a fixed number ~ of nucleons is equal to
and the probabilities of inelastic interaction with a fixed number v = 1, 2, etc. of nucleons, P(A) is equal to ,~(v) /~prod
P~A(v) = vhA,~,hA
.
In calculations of the high-energy hadronnucleus cross sections, it is necessary to take the inelastic screening effects into account, in such a case the considered cross sections have more complicate form (see, e.g., [22,23]). In the case of nuclear target we must consider the possibility of one or several Pomeron cuts in each of the ~ blobs of hadron-nucleon inelastic interactions as well as cuts between Pomerons. For example, for a p A collision one of the cut Pomerons links a diquark and a valence quark of the projectile proton with a valence quark and diquark of one target nucleon. Another Pomeron links the sea quark-antiquark pairs of the projectile proton with diquarks and valence quarks of another target nucleon and with sea quarkantiquark pairs of the target. It is essential to take into account every possible Pomeron configuration and permutation on all diagrams. 6. R e s u l t s a n d conclusion There is no enough place to show the comparison of the calculated results with the experimental data. One can find such comparison in Refs. [7-9, 17, 23-26], in particular in [24] the penetration of cosmic rays throw the atmosphere was considered. As a rule, the agreement of the calculated results with the data is on the level of 20%. The A-dependences of produced light-flavor (u, d, s) secondaries is reproduces quit reasonable also. REFERENCES
1. t'Hooft, Nucl. Phys. B72 (1974) 461. 2. G. Veneziano, Phys. Lett. 52B (1974) 220; Nucl. Phys. Bl17 (1976) 519.
119
3. A. Capella et al. Phys. Lett 81B (1979) 68; Z. Phys. C3 (1980) 329. 4. H. Minakata, Phys. Rev. D20 (1979) 1656. 5. G. Cohen-Tannoudji et al., Phys. Rev. D19 (1979) 3397; D21 (1980) 2699. 6. A.B. Kaidaiov, Phys. Lett. l16B (1982) 459. 7. A. B. Kaidalov and K. A. Ter-Martirosyan, Phys. Lett. l17B (1982) 247; Yad. Fiz. 39 (1984) 1545; 40 (1984) 211. 8. A.B. Kaidalov and O. I. Piskunova, Yad. Fiz. 41 (1985) 1278; Yad. Fiz. 43 (1986) 1545. 9. Yu. M. Shabelski, Yad. Fiz. 44 (1986) 186. 10. V. A. Abramovsky, V. N. Gribov and O. V. Kancheli, Yad. Fiz. 18 (1973) 595. 11. A. B. Kaldalov, Yad. Fiz. 33 (1981) 1369; Yad. Fiz. 45 (1987) 1452. 12. K. A. Ter-Martirosyan, Phys. Lett. 44B (1973) 377. 13. A. B. Kaidalov, Z. Phys. C12 (1982) 63. 14. A. Capella and J. Tran Thanh Van, Phys. Lett. B93 (1980) 146. 15. W. Q. Chao, C. B. Chiu, Z. He and D. M. Tow, Phys. Rev. Lett. 44 (1980) 518. 16. J. Ranft and S. Z. Ritter, Z. Phys. C20 (1983) 347. 17. A. B. Kaldalov, K. A. Ter-Martirosyan and Yu. M. Shabelski, Yad. Fiz. 43 (1986) 1282. 18. R. J. Glanber, Phys. Rev. 100 (1955) 242; High Energy Physics and Nuclear Structure (Amsterdam 1967). 19. A. G. Sitenko, Ukr. Fiz. Journal 4 (1959) 152. 20. V. N. Gribov JETP 56 (1969) 892; 57 (1969) 1306. 21. O. V. Kancheli and S. G. Matinyan, Yad. Fiz. 11 (1970) 1305. 22. Yu. M. Shabelski, Nucl.Phys B132 (1978) 491. 23. Yu. M. Shabelski, Z. Phys. C57 (1993) 409. 24. A. D. Erlykin, N. P. Krutikova and Yu. M. Shabelski, Yad. Fiz. 45 (1987) 1075. 25. Yu. M. Shabelski, Z. Phys. C38 (1988) 569. 26. Yu. M. Shabelski Yad. Fiz. 49 (1989) 1081.
ELSEVIER
Nuclear Physics B (Proc. Suppl.) 52B (1997) 116-119
PROCEEDINGS SUPPLEMENTS
Multiple Hadron Production in the Quark-Gluon String Model Yu.M. ShabelskP apetersburg Nuclear Physics Institute, Gatchina, St.Petersburg, 188350 Russia We give a short description of the Quark-Gluon String Model which allows one to describe quantitatively the Feynman-x and rapidity spectra of different secondaries produced in hadron-nucleon and hadron-nucleus collisions at high energies.
1. I n t r o d u c t i o n
In order to describe the processes of multihadron production at high energies, wide use has been made of models based on the idea of the 1/N expansion of QCD [1,2] in the framework of the scheme of dual topological unitarization (DTU). In Refs. [3-5] only the single-Pomeron contributions were included. In the Quark-Gluon String Model (QGSM) [6-9] the possibility of exchange of several Pomerons is accounted for also. Inelastic processes in the models of this kind are determined [10] by all the possible "cuttings" of one or several Pomeron or by a cutting between them. To calculate the cross section for production of secondary particles, it is necessary to know the distribution functions of the quarks in the colliding hadrons and the functions for fragmentation of quarks and diquarks into secondary hadrons. In the QGSM, it is assumed that all these functions are determined in the regions XF -+ 0 and XF -+ 1 [11] by the corresponding Regge asymptotic behaviors, and in the region of intermediate values of XF by means of interpolation. At the same time, they must satisfy several conservation laws. 2. Inclusive c r o s s s e c t i o n for particle prod u c t i o n in Q G S M We consider the Pomeron intercept greater than unity
Tevatron colliders data included, has led to a somewhat greater value A = 0.12-0.14. The imaginary part of the elastic amplitude is written as a sum of several absorptive parts, corresponding to all possible cuts of the given reggeon diagram, each related with a certain specific group of intermediate states through the schannel unitarity. The one Pomeron exchange gives a multiperipheral hadron ladder with small transverse momenta. The contribution of the double Pomeron exchange to the elastic scattering amplitude is connected with three different types of intermediate states. The cut in between the Pomerons is saturated by elastic scattering or diffraction dissociation. The cut of either one of the two Pomerons produces a multiperipheral ladder of particles with similar properties as in the case of one-Pomeron diagram; it being the first absorption correction to the physical cross section of the production of the single ladder of hadrons. Finally, the cut of both Pomerons yields two hadron ladders and hence, a doubled multiplicity. A straightforward application of the AGK cutting rules [10] yields the cross sections for n cut Pomerons, an, and the diffraction dissociation, a0, [12]. Each value of a (~) should be positive. The inclusive spectrum (i.e. Feynman-x distribution) of a secondary hadron h produced in pp interactions is determined in QGSM by the expression [6,7]
t
ap(t) = l + A + ap . t ,
A>0.
The intercept A was taken equal to 0.07 originally but the later analysis with the S p a S and 0920-5632(97)/S17.00 © 1997 Elsevier Science B.'~ All rights reserved. PII: S0920-5632(96)00855-9
XE
da
~Tinel d X F
Yu.M. Shabelski /Nuclear Physics B (Proc. SuppL ) 52B (1997) 116-119
where XE = E/Ema~,
h fqh (x+,n)fqq(X_,n) +
structure functions obtained from hard processes. The form of the functions u(x, n) is determined in QGSM by the corresponding Regge asymptotic behaviors in the regions x --+ 0 and x ---> 1, and for intermediate values of x by means of a simple interpolation [6,7]. The distribution functions of the quarks and diquarks in the proton which were used in the calculations have the form [7], for example
2(n 1)I~(~+, n)I~(~-, ~),
~(~,n)
Wn
(7n
=
(7n
is the probability of cutting precisely n Pomerons,
¢pPnP'h(xF)
fhq(x+,n)fh(x_,n) +
=
-
=
u~,(x,n)
The quantities fqq, fq and L in (4) correspond to the contributions of the diquark, the valence quark and antiquark, and the sea quarks, while the contributions of the incident particle and the target proton depend on the variables x+ and x_, respectively. The last two terms in (2) take into account the contributions of diffraction dissociation of the incident hadron and the target nucleon. The functions fqhq(x±,n), fqh(x+, n) and fh(x:t:, n) which determine the inclusive spectra of secondary particles, can be expressed in terms of convolutions of the momentum distributions of the diquarks and of the valence and sea quarks in the colliding hadrons, u(x), and the functions for fragmentation of diquarks and quarks into secondary hadrons, G h (z). For example, for the case of pp collisions we have [6,7] fhqq(X:t:, ~t )
:
U,,d(Xl, n)Ghd(x±/Xl )dxl
3
+3
117
u ~ ( x l , n)Ghu(x~:/xl )dxl .
+
The parameter A ,~ 0.2 - 0.3 determines the relative suppression of strange quarks in the sea. The inclusive spectra in ~ - p and Kp collisions can be calculated by the same way. 3. Q u a r k a n d d i q u a r k d i s t r i b u t i o n s in h a d r o n s QGSM considers hadrons consisting from the constituent quarks so we can not use hadron
C~"R--2~B+I(1--
X) -~R+'~-I ,
= C ~ x - a ' ( 1 - x) "a-2~'+'~-I ,
where a n = an(0) ~ 0.5 and ~B = (~B(0) --0.5 are the parameters of the meson and baryon Regge trajectories. The normalization factors Ci are determined by the condition
f ui(x,n)dz = 1 . In addition, as a consequence of the energy conservation law, for each value of n we have
E (x)i = E / u i ( x ' n ) x d x i
= 1.
i
4. Q u a r k a n d d i q u a r k f r a g m e n t a t i o n functions The functions for fragmentation of diquarks and quarks into secondary hadrons are also determined in QGSM by their Regge asymptotic behaviors [11]. For example let us consider the case of a quark fragmentation into a meson. In the limit z --~ 1 the fastest hadron h is separated from all the other hadrons by a large rapidity interval Ay ln[1/(1 - z)]. The probability of this quark fragmentation can he obtained from the corresponding triple-reggeon diagram [11]:
Gh(z) ~ (1
-
z)%Z(°)-2%~(p~') .
Using the relations between the intercepts and slopes of the trajectories of Regge poles containing quarks of various types, as obtained in the planar approximation of the topological expansion [13]
%~(o) + %~(o) = 2a~(0),
118
Yu.M. Shabelski /Nuclear Physics B (Proc. Suppl.) 52B (1997) 116-119
and a(p 2) = a(0) - a ' . p2 we obtain t
a , h ~ (1 - z ) - ~ ( ° ) + ~ , ~
strangeness must hold. For example,
2
p~ .
(1)
B B B [a~(z)a~, (o) + a~B (z)a~Ao) -
fO dz Z
The value ai~(0 ) is the intercept of the secondary trajectory "composed" of a quark and an antiquark of type i ( a ~ = ad-~ = ap = a~ = OR 1/2 , as~ ---- a~o). The behavior of G h ( z ) for z --+ 1 integrated over transverse momentum p~ can be obtained from Eq. 1 by replacing/92 in it by the average
value (p~>:
z ) a u ~ (-~ 0)] a ~ ( z ) a ~~ (0) - a u (-~
= 1.
For the fragmentation function used in the previous studies [6-8], all these conservation law are not exactly satisfied. Therefore the functions were modified appropriately in [9]. However, the Regge asymptotic behaviors at both extremes can be retained.
G h ( z ~ 1) ~ (1 -- z ) - % ~ (°)+~ ,
5. I n c l u s i v e s p e c t r a o n n u c l e a r t a r g e t s 2
A = 2ai-£(pT ) .
In the limit z --+ 0 the functions G ( z ) are constants ah [11] which do not depend on the type of quark (diquark) i, i.e.
a ~ ( ~ - , 0) = a h . The value of the constant ah is determined by the dynamics of the rupture of the string when a q~ pair is produced from the vacuum. All the particles of the same isotopic multiplet cotrespond to exactly the same constant ah (i.e., a~+ = a~- = a~0 = a t ) , and the constant ah is the same for production of particles and antiparticles (ag+ = a K - = aKo = a-K-O = a g , ap = a F = an ~ a~ :-- a N , etc.). In Refs. [6-8] the simplest interpolation formulae were used for the fragmentation functions at intermediate values of z, for example: G 7ru + (z)
=
a ~ ( 1 - z) -aR(°)+~ ,
Gud(Z)
=
ap(1
Gk+ (z)
=
a k ( 1 - - z) -~*(°)+x .
P
-
z) -aR(°)+x ,
The inclusive spectra of secondaries produced on nuclear targets can be considered also in the framework of DTU approach [14-16], and in particular by QGSM [17]. The multiple scattering theory [18,19] is used successfully for a description of high energy hadron-nucleus collisions. In this theory the interaction with a nucleus is treated as a sum of interactions with different number of nucleons. Such an approach corresponds [20,21] to the Regge space-time picture of the interaction. It turns possible to describe quantitatively elastic and inelastic h A cross sections as well as several important characteristics of the multiple production processes (average multiplicities of secondaries, multiplicity distributions, probabilities of multinucleon interactions, etc.), see, e.g., Ref. [22]. Expressions for the cross sections of hadronnucleus collisions are the most simple in the socalled optical limit of the multiple scattering theory. The secondary production cross section has the well-known forms:
However, the fragmentation functions must obey the energy conservation law
tThAPr°d =
/ d2b[1
_ e-ainT(b)]
where
y~
a)(z)a)(O)d~
:
1.
h
In addition, for each quark-diquark or quarkantiquark string as whole the laws of conservation of the electric and baryon charges and of
T(b) = A ] p(b, z)dz , f
T ( b ) d2b
A,
and p(r = v ~ + z 2) is the nuclear density distri-
bution, a
=
_tot inel O h N ~ f f i n ~--- ~ h N "
Yu.M. Shabelski/Nuclear Physics B (Proc. Suppl.) 52B (1997) 116-119
The cross section of inelastic interaction with a fixed number ~ of nucleons is equal to
and the probabilities of inelastic interaction with a fixed number v = 1, 2, etc. of nucleons, P(A) is equal to ,~(v) /~prod
P~A(v) = vhA,~,hA
.
In calculations of the high-energy hadronnucleus cross sections, it is necessary to take the inelastic screening effects into account, in such a case the considered cross sections have more complicate form (see, e.g., [22,23]). In the case of nuclear target we must consider the possibility of one or several Pomeron cuts in each of the ~ blobs of hadron-nucleon inelastic interactions as well as cuts between Pomerons. For example, for a p A collision one of the cut Pomerons links a diquark and a valence quark of the projectile proton with a valence quark and diquark of one target nucleon. Another Pomeron links the sea quark-antiquark pairs of the projectile proton with diquarks and valence quarks of another target nucleon and with sea quarkantiquark pairs of the target. It is essential to take into account every possible Pomeron configuration and permutation on all diagrams. 6. R e s u l t s a n d conclusion There is no enough place to show the comparison of the calculated results with the experimental data. One can find such comparison in Refs. [7-9, 17, 23-26], in particular in [24] the penetration of cosmic rays throw the atmosphere was considered. As a rule, the agreement of the calculated results with the data is on the level of 20%. The A-dependences of produced light-flavor (u, d, s) secondaries is reproduces quit reasonable also. REFERENCES
1. t'Hooft, Nucl. Phys. B72 (1974) 461. 2. G. Veneziano, Phys. Lett. 52B (1974) 220; Nucl. Phys. Bl17 (1976) 519.
119
3. A. Capella et al. Phys. Lett 81B (1979) 68; Z. Phys. C3 (1980) 329. 4. H. Minakata, Phys. Rev. D20 (1979) 1656. 5. G. Cohen-Tannoudji et al., Phys. Rev. D19 (1979) 3397; D21 (1980) 2699. 6. A.B. Kaidaiov, Phys. Lett. l16B (1982) 459. 7. A. B. Kaidalov and K. A. Ter-Martirosyan, Phys. Lett. l17B (1982) 247; Yad. Fiz. 39 (1984) 1545; 40 (1984) 211. 8. A.B. Kaidalov and O. I. Piskunova, Yad. Fiz. 41 (1985) 1278; Yad. Fiz. 43 (1986) 1545. 9. Yu. M. Shabelski, Yad. Fiz. 44 (1986) 186. 10. V. A. Abramovsky, V. N. Gribov and O. V. Kancheli, Yad. Fiz. 18 (1973) 595. 11. A. B. Kaldalov, Yad. Fiz. 33 (1981) 1369; Yad. Fiz. 45 (1987) 1452. 12. K. A. Ter-Martirosyan, Phys. Lett. 44B (1973) 377. 13. A. B. Kaidalov, Z. Phys. C12 (1982) 63. 14. A. Capella and J. Tran Thanh Van, Phys. Lett. B93 (1980) 146. 15. W. Q. Chao, C. B. Chiu, Z. He and D. M. Tow, Phys. Rev. Lett. 44 (1980) 518. 16. J. Ranft and S. Z. Ritter, Z. Phys. C20 (1983) 347. 17. A. B. Kaldalov, K. A. Ter-Martirosyan and Yu. M. Shabelski, Yad. Fiz. 43 (1986) 1282. 18. R. J. Glanber, Phys. Rev. 100 (1955) 242; High Energy Physics and Nuclear Structure (Amsterdam 1967). 19. A. G. Sitenko, Ukr. Fiz. Journal 4 (1959) 152. 20. V. N. Gribov JETP 56 (1969) 892; 57 (1969) 1306. 21. O. V. Kancheli and S. G. Matinyan, Yad. Fiz. 11 (1970) 1305. 22. Yu. M. Shabelski, Nucl.Phys B132 (1978) 491. 23. Yu. M. Shabelski, Z. Phys. C57 (1993) 409. 24. A. D. Erlykin, N. P. Krutikova and Yu. M. Shabelski, Yad. Fiz. 45 (1987) 1075. 25. Yu. M. Shabelski, Z. Phys. C38 (1988) 569. 26. Yu. M. Shabelski Yad. Fiz. 49 (1989) 1081.