CBPF-NF-41/3 ESTIMATIVE OF INELASTICITY COEFFICIENT FROM EMULSION CHAMBER DATA H M PORTELLA Instituto de F sica, Universidade Federal Fluminense, Av. Litor^anea s/n, Gragoatá, 2421-34, Niterói, RJ, Brazil LCSdeOLIVEIRACECLIMA Centro Brasileiro de Pesquisas F sicas, Rua Dr. Xavier Sigaud 15, Urca, 2229-18, Rio de Janeiro, RJ, Brazil Abstract Atmospheric diffusion of high energy cosmic rays is studied analytically the obtained integral electromagnetic fluxes are compared with the data measured by emulsion chamber detectors at mountain altitudes. We find a good consistency between them when the average nucleon inelasticity coefficient is varying between.5.65. Key-words:Inelasticity; Hadrons; Emulsion Chambers; Electromagnetic Cascades; Inelastic Cross Section; Z-factors. PACS: 13.85.Tp; 13.85.Hd
CBPF-NF-41/3 1 1 Introduction The inelasticity is a global parameter of hadronic interactions it is a crucial attribute in the calculation of the fluxes in cosmic ray cascades. It is defined as the fraction of energy giving up by the leading hadron in a collision induced by an incident hadron on a target nucleon or nucleus. The inelasticity, K N, in a nucleon-induced interaction is related to the spectrumweighted moments evaluated at fl = 1 (where fl is the power index of primary spectrum) as 1 K N = Z NN (fl =1)= f NN ( ) d. Several authors have suggested that the average inelasticity-coefficient is an increasing function of the energy [1, 2], whereas others proposed that it is a decreasing function [3, 4, 5]. The behaviour of high energy cosmic rays, which reflects the nuclear interactions in the energy region 1 ο 1 TeV, indicates a constant value of the mean inelasticity equal to.5 [6]. A decreasing with energy of this parameter points out a strongly suppression of the pion production in this energy region does not agree with cosmic ray observations. The objective of this paper is to estimate the average inelasticity coefficient through the comparison of our calculations with the integral electromagnetic fluxes measured with large Emulsion Chamber Experiments at mountain altitudes. The main purpose is to describe in qualitative detail single events, i.e., shower cores in their early stage of development observed at mountain altitudes. Our calculation is performed by solving analitically the cosmic ray diffusion equations for electromagnetic showers in earth's atmosphere. In these calculations we adopt to the particle production an approximated Feynman scaling in the fragmentation region. We have used the data on proton targets on nuclear targets [7] to obtain the pion spectrum weighted moments. Although exist several functional forms to fit the behaviour of rising cross-section we assume the following one ff = ff (E=ffl) ff. So, beginning with the measured all-particle primary spectrum we diffuse the particle showers down to the detection depths we investigate our numerical results as a function of the adopted average inelasticity coefficient. 2 Hadronic electromagnetic cascades in the Earth's atmosphere The differential hadronic fluxes, H(t; E) at a depth t in the energy range E E + de are obtained integrating the nucleon pion diffusion equations using the semigroup theory. We
CBPF-NF-41/3 2 assume that only the pions are generated in the multiparticle production neglecting the particles with small fractions such askaons, heavy mesons, etc.. In this case the solutions are [8]; 1X ( 1) n t Y n N(t; E) = (1 Z NN (fl;ff; n)) N(;E) (1) n! n= N (E) n i=1 with Z NN (fl; ff) = f NN ( ) fl iff d (2) where is the elasticity coefficient of the nucleons in the atmosphere that is distibuted according to f NN ( ). with The pionic fluxes are Π(t; E) = 1X 1X m= n= ny i=1 Z t dz ( 1)m ( 1) n m! n! (1 Z NN (fl; ff; n)) Z ßß (fl; ff;m;n) = Z Nß (fl;ff; n) = my j=1 t z m z n Z Nß (fl;ff; n) ß (E) N (E) (1 Z ßß (fl; ff; m; n)) N(;E) N (E) (3) f ßß (x) x fl ff(j+n+1) dx (4) f Nß (x) x fl ff(n+1) dx. (5) The f Nß (x) f ßß (x) are the energy spectra of the pions produced in the nucleon-air in the pion-air interactions respectively, x is the Feynman variable (x =2p L = p S). The solutions (1) (3) are subjected to the boundary conditions N(;E)=N E (fl+1) (6) Π(;E)= (7) that represents the nucleon pion fluxes at the top of atmosphere. The above solutions are obtained considering the interaction mean-free paths in the power form, H (E) = H E ff (H = N or ß). Using naive approximations [9] we can write the solutions (1) (3) in the following forms N(t; E) =N(;E) e t! N (E) (8)! Nß (fl;e) Π(t; E) =N(;E)! N (E)! ß (E) e t!ß(e) e t! N (E) The! N (E),! ß (E) the! Nß (fl; E) functions are described in the appendix A.. (9)
CBPF-NF-41/3 3 The production rate of fl-rays originated from the ß decay isgiven by [1] de P fl (z; E fl )=2 2E P ß± N air + P ß± air ß± E fl (1) where P N air ß± P ß± air ß± are the production rate of the charged pions from the nucleon-air pion-air interactions are represented by P H air ß± = H(z; E ) H (E ) f Hß(x) dx x ; H = N or ß. (11) In the expression (1) we have used that the secondary pions are formed with equal multiplicity, so the number of ß is half of the number of charged pions. The differential flux of the electromagnetic component can be written as Z t F fl;e (t; E) = dz de fl P fl (z; E fl )(e + fl)(t z; E fl ;E) (12) E where (e+fl)(t z; E fl ;E) are the photons, electrons positrons produced by a single photon. They are computed using the approximation A [11] are written in the following form (e + fl)(t z; E fl ;E)= 1 2ßi Z C s Efl 1 E E N 1 (s)e 1(s)(t z)=x + N 2 (s)e 2(s)(t z)=x. (13) The functions N i (s), i (s) X are the parameters with stardized definitions in cascade theory [11]. These functions the evaluation of the electromagnetic fluxes are presented in the appendix B. In order to compare our calculations for electromagnetic showers with the emulsion chamber data measured at mountain altitudes we need to obtain the integral fluxes of this component. They are, F fl;e (t; E) = 3 Comparison with experimental data E F fl;e (t; E) de (14) Emulsion Chamber Experiments at mountain altitudes are very simple detectors constituted of multiple swich of material layers (Pb, Fe, etc.) photo-sensitive layers (X-ray films, nuclear emulsions plates, etc.). They can detect electromagnetic showers of high energy (E e;fl 1 TeV) due to its powerful spatial resolution energy determination they can study in details the cosmic-ray interactions. The experimental data are taken from J. R. Ren et al. [19] for Mt. Kanbala; C. M. G. Lattes et al. [13] for Mt. Chacaltaya M. Amenomori et al. [21] for Mt. Fuji. The experimental fluxes at Mt. Chacaltaya are revised according to the energy determination introduced by M.Okamoto T. Shibata [22] where the LPM spacing effects were taken
CBPF-NF-41/3 4 into account. This revision followed the same procedure made by N.M. Amato N. Arata [2] for determination of total hadron fluxes. In order to make a comparison with electromagnetic fluxes measured with these detectors it is necessary to check the main assumptions made in our calculations; a) Primary cosmic-ray radiation The energy spectra obtained by Emulsion Chamber Experiments at mountain altitude are in the region of 3 ο 5 TeV for (e; fl) component 3 ο 4 TeV for hadronic shower. So, we need to include in our calculations the primary spectrum up to knee (ß 1 15 ev). In the present work we assume the primary cosmic-ray radiation in integral form reported by J. Belli Filho et al. [14] from JACEE data [15]. N 1 (; E) =1:12 1 6 (E=5TeV) 1:7 (/cm 2.s.st) (15) where E(TeV/nucleus). Nucleonic intensity att =, equivalent to primary cosmic-ray radiation is given by N(; E) =A 1 fl N 1 (; E) (16) where A is the average mass number of cosmic ray nuclei; we adopta =14: which is the value in low energy region (E/part < 1 14 ev). b) Collision mean free paths We use to fit the behaviour of rising with energy of the p-air inelastic cross-section the following form [7, 14, 16, 17], ff N (E) =ff N (E=TeV) ff (17) with ff N = 3 mb ff =:6 [7, 14]. So, the collision mean free path of the nucleon in the earth's atmosphere is expressed by N (E) = N (E=TeV) ff (18) with N = 8 g/cm 2 [7, 14]. For the pion mean free paths we assume that ß = N ß 1:4 that they have the same energy dependence, like inthenucleon case [14]. Thus, we have ß (E) = ß (E=TeV) ff (19) with ß = 112 g/cm 2. c) Secondary pions:
CBPF-NF-41/3 5 The functions f Nß (x) f ßß (x) are the energy distribution of secondary nucleons pions for nucleon-air pion-air interactions, respectively. We have used in this calculation the data on proton targets on nuclear targets [7] to obtain these functions they are written as f Nß (x) =2:8 (1 x) x e 5x (2) f ßß (x) = 2:6 x 1+ x 3 + :32 :45 x e2(x 1) (21) these functions are for two charged states of pions the contribution of leading pions are taken into account in the last one. Figures 1-3 show the comparison of our solution with the integral electromagnetic fluxes measured at Mt. Kanbala (52 g/cm 2 ), Mt. Chacaltaya (54 g/cm 2 ) Mt. Fuji (65 g/cm 2 ), respectively. Two curves are also drawn in the figures for <K N > =.5.65, taking into account the above mentionated itens. 4 Conclusions We have solved the diffusion equations of cosmic-ray hadrons analytically after this we derived the integral electromagnetic fluxes using the approximation A of the electromagnetic cascade theory. These fluxes at mountain atmospheric depths decrease when we include in our calculation the rising of the cross-section the decreasing of the mean nucleon elasticity. We conclude that the naive assumptions for the secondary production based on approximate scaling in the fragmentation region is possible to explain experimental data on integral electromagnetic fluxes detected in large emulsion chamber experiments. This scenario requires a mean nucleon inelasticity coefficient varying from.5 to.65. At low energy (few TeV) the agreement between experimental data the calculated fluxes shows that <K N >ß :5, that is the adjust indicated by former workers in the cosmic ray literature. For E > 1 TeV we see from the figure that < K N >ß :65. Therefore, with this simple analysis we see that the nucleon inelasticity coefficient appears to show a tendency of rising with energy according to Mini-jet Models [17] Quark-Gluon String Models [2]. These integral fluxes are somewhat affected when the coefficient ff changes in the interval.6 to.1, which is the appropriated values to explain the experimental data for mountain emulsion chambers experiments. In the latter the fit between our calculation the experimental data is obtained for < K N > changing between approximately.45.6. These results are in agreement with that of L. Jones [18] based in an analysis on inclusive reactions data from accelerator, with analytical calculations on nucleon fluxes at sea level [9] with hadron electromagnetic fluxes at mountain altitudes [8, 1].
CBPF-NF-41/3 6 Appendix A Hadronic cascades where The functions! N (E),! ß (E) Z Nß (fl; E) in the hadronic fluxes (8) (9) are [9] 1 1 2 Z NN(fl) 1 2 Z NN(fl; ff)! N (E) =! N E :6 = 1 N (E)! ß (E) =! ß E :6 = 1 ß (E)! Nß (E) = Z Nß(fl;ff) N (E) Z NN (fl; ff) = Z ßß (fl; ff) = 1+ 1 2 Z ßß(fl) 3 2 Z ßß(fl;ff) = 1 N (E) (22) (23) x fl ff f Nß (x) dx (24) fl ff f NN ( ) d (25) x fl ff f ßß (x) dx. (26) The functions f Nß (x) f ßß (x) are the energy distribution of secondary nucleons pions for nucleon-air pion-air interactions, respectively (see item c of section 3). The nucleon elasticity distribution is assumed to be [9] where f NN ( ) =(1+fi)(1 ) fi (27) = E=E fi fulfills a consistency relation between the mean elasticity average inelasticity, so that < N > + < K > = 1, for nucleons. When fi = we have the nucleon uniform distribution of elasticity (< > = :5). Appendix B Electromagnetic cascades The production rate of fl-rays produced from ß decay expression (1) can be written as P fl (z; E fl )= N E (fl+1 ff+z ff! N ) fl e z! N ZNß (fl ff + zff! N ) Z ßß(fl ff + zff! N ) fl +1 ff + zff! N N ß! Nß (E) + N E (fl+1 ff+zff!ß)! N (E)! ß (E) (fl +1 ff + zff! ß ) The Z-factors Z Nß Z ßß is the last expression are! Nß (E)e z!ß Z ßß (fl ff + zff! ß ) (! N (E)! ß (E)) ß. (28) Z Nß (fl ff + zff! N )= x fl ff+zff! N f Nß (x)dx (29)
CBPF-NF-41/3 7 Z ßß (fl ff + zff! i )= The expression (28) was obtained considering the expansion x fl ff+zff! i f ßß (x) dx i = N or ß. (3) 1 H (E) = 1+ffln(E=E ) H ; H = N or ß (31) The differential flux of the electromagnetic component (12) is obtained from this production rate evaluated at the poles s 1 = fl ff + zff! N s 2 = fl ff + zff! ß. The integral flux is put in the form F fl;e (t; E) =F fl 1;e(t; E)+F fl 2;e(t; E) (32) with F fl 1;e(t; E) = F fl 2;e(t; E) = where with Z t Z t dz N E s1 e z! N s 1 (s 1 +1) dz N E s2 e z!ß s 2 (s 2 +1) N 1 (s 1 )e 1(s1)(t z)=x + N 2 (s 1 )e 2(s1)(t z)=x C 1 (s 1 ) N 1 (s 2 )e 1(s2)(t z)=x + N 2 (s 2 )e 2(s2)(t z)=x C 2 (s 2 ) (33) (34) C 1 (s 1 )= Z Nß(s 1 )! Nß (E) Z ßß (s 1 ) (35) N (! N (E)! ß (E)) ß C 2 (s 2 )= Z Nß (s i )= Z ßß (s i )=! Nß (E) Z ßß (s 2 ) (36) (! N (E)! ß (E)) ß x s i f Nß (x) dx (37) x s i f ßß (x) dx. (38) The functions N i i are the usual parameters with definitions in cascade theory can be obtained in the reference [11]. X is the radiation length in the atmosphere (X = 37:1g/cm 2 ).
CBPF-NF-41/3 8 References [1] J. D. de Deus, Phys. Rev. D32 (1985) 2334. [2] A. B. Kaidalov K. A. Ter-Martyrosian, Phys. Lett. B117 (1982) 247; A. B. Kaidalov K. A. Ter-Martyrosian, Sov. J, Nucl. Phys. 4 (1984) 135. [3] G. N. Fowler et al., Phys. Rev. D35 (1987) 87; G. N. Fowler et al., Phys. Rev. C4 (1989) 1219. [4] A. Ohsawa K. Sawayanagi, Phys. Rev. D49 (1992) 3128. [5] Z. Wlodarczyk et al., J. Phys. G 21 (1995) 281. [6] C. R. A. Augusto et al., Phys. Rev. D61 (1999) 123; A. Ohsawa, Prog. Theor. Phys., 92 (1994) 15. [7] H. M. Portella, PhD Thesis (CBPF: Centro Brasileiro de Pesquisas F sicas, Brazil), (1989). [8] H. M. Portella, N. Amato, R. H. C. Maldonado A. S. Gomes, J. Phys. A: Math. Gen. 31 (1998) 6861. [9] J. Belli Filho et al., Il Nuovo Cimento A11 (1989) 897. [1] H. M. Portella et al., ICRR-Report 454-99-12 Institute for Cosmic Ray Research, University oftokyo (1999) 31. [11] J. Nishimura, Hbuch Der Physik (Reading: Springer-Verlag) (1967) 1. [12] M. Akashi et al., Il Nuovo Cimento A65 (1981) 355. [13] C. M. G. Lattes et al., Proc. 13 th Int. Symp. on Very High Energy Cosmic Ray Interactions (Denver) 13 (1973) 2219. [14] J. Belli Filho et al., Prog. Theor. Phys. 83 (199) 58. [15] T. H. Burnett et al. (JACEE Collaboration), Proc. of the Int. Symp. on Cosmic Ray particle Physics (ICRR, Univeristy oftokyo) (1984) 468. [16] B. Z. Kopeliovich et al., Phys.Rev. D39 (1989) 769. [17] T. K. Gaisser, Cosmic Ray Particle Physics (Cambridge: Cambridge University Press), (199). [18] L. W. Jones, Proc. of 2 th Int. Symp. on Very High Energy Cosmic Ray Interactions (La Paz-Rio de Janeiro) (1982) 1. [19] J. R. Ren et al., Nuovo Cimento 1C (1987) 43. [2] N. M. Amato N. Arata, Proc. 5 th Int. Symp. on Very High Energy Cosmic Ray Interactions (Lodz) (1988) 49. [21] M. Amenomori et al., Proc. 18 th Int. Cosmic Ray Conf. (Bangalore) 11 (1983) 57. [22] M. Okamoto T. Shibata, Nucl. Inst. Meth. A257 (1987) 155.
CBPF-NF-41/3 9 Figure Captions Figure 1 - Integral electromagnetic spectrum at Mt. Kanbala (52 g/cm 2 ). The full lines represent the calculated flux for <K N >=:65 the dashed lines for <K N >=:5. All lines are for ff =:6. The experimental data are taken from J. R. Ren et al. [19]. Figure 2 - Integral electromagnetic spectrum at Mt. Chacaltaya (54 g/cm 2 ). The full lines represent the calculated flux for <K N >=:65 the dashed lines for <K N >=:5. All lines are for ff =:6. The experimental data are taken from C. M. G. Lattes et al.(black dots) [13]. Figure 3 -Integral electromagnetic spectrum at Mt. Fuji (65 g/cm 2 ). The full lines represent the calculated flux for <K N >=:65 the dashed lines for <K N >=:5. All lines are for ff =:6. The experimental data are taken from M Amenomori et al. [21].
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