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A n a ly tica l E v a lu a tio n o f th e H a d ro n N u m b e r in E x tensive A ir S h o w ers fo r R is in g I n e la s t ic i tyJiben SidhantaDepartment of Physics, Charuchnadra College, Calcutta-700 029, India

Rajkumar RoychoudhuryElectronics Unit, Indian Statistical Institute, Calcutta-700 035, India

>Z. Naturforsch. 39 a, 1172 - 1174 (1984); received August 8.1984

A purely analytical method is presented in this paper for evaluating the average number of hadrons (in an air shower) with a given energy as the inelasticity of the cross-section rises. Our result is found to be in excellent agreement with numerical solutions and also with experimental data, particularly in the high energy range.

I. Introduction

In view of its importance for multiparticle pro­duction processes in the ultra high energy region the analysis of the hadronic components of extensive air showers (EAS) has become an active field of interest. A crucial point of such an analysis consists in determining the average number of hadrons with a given energy. Unfortunately, most of the studies employ numerical techniques involving a cluttering mass of experimental data and a lot of equipment.

The present paper employs a purely analytical method to determine the average number of hadrons with a given initial energy as the inelasticity of the cross-section rises. In a recent paper [1] a solution for nucleons has been found; in the pion case the diffusion equation itself is somewhat harder but our previous approach works, mutatis mutandis, quite satisfactorily. As in the previous paper [1] we shall follow the rigorous saddle-point method which we developed in a recent work [2] for the inversion of the Mellin transform.

The raison d'etre of the present paper will be apparent when one considers the following facts:

In the first place the coupled integro-differential equations that occur in the present problem have prompted one numerical solution after another [3]-[5] although they are by no means intractable analytically. Perusing some of them which are found to be protracted over sixteen odd years, we

Reprint requests to Dr. Roychoudhury, Electronics Unit, Inidan Statistical Institute, 203 Barrackpore Trunk Road, Calcutta-700 035, India.

have undertaken here to vindicate the feasibility of a purely analytical solution. For the general reader in particular an analytical solution is always more suggestive of novel ideas than any cut and dried numerical sheet. Secondly, the prevalent numerical solutions employ "stronger" assumptions than we need even in our wholly analytical approach. In a standard numerical calculation [3] the nucleon inter­action length is approximated by a constant (see the paragraph preceding Eq. (2.4) of [3]). In another [4] ).n and /.N are accorded identical values throughout. We. however, consider a more general situation in the sense that we do not require kn and AN to be equal. Of course, the present work too needs some assumption. We assume an = aN (see Eqs. (4)-(6) below); but then, this is evidently a more tenable hypothesis than ).n = / N.

II. Analytical Evaluation of the Hadron Number

Consider the relevant integro-differential equa­tions (of diffusion):

8A N' E° N(E', P)_ _ + j ---- - H/nn (£', E )d E ', (1)Cp /-N £ /.N

6/7 1 B ec°N (E \p) dp /.n Ep E An

+ f \ " Wnn{E\ E )d E \E '-n

(2)

where /.m n etc. are written ).n etc. for simplicity, the rest of the symols being as in [5] and [6].

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J. Sidhanta and R. Roychoudhury ■ Analytical Evaluation of the Hadron Number 1173

Assuming B/Ep < 1 and setting 77 = Fne p,/-n and p' = p//.n(E), we reduce (2) into:

aFn t ;.,(£)J - / r , exp {/?//.„(£)) FN

• exp{— /? /(£ " ')} W„N( £ \ £ ) d £ '

k '*(£*) An(E) An(E'),

•Fn Wnn{E ',E )dE ', (3)

where we have written

N(E\ p) = FN (£', />) exp {- p/kN (£')}.

We next express the rise of inelasticity in the standard form:

_o(7in — a I 1 + a ln100

i.e.

n= 11 + fl^N In

-7t,N(4)

As noted before, we do not require butuse the somewhat more generally valid assumption

= aN = a (say).From (4), writing X = E/E', we obtain

and

Also

--------= 1 - a In X,W )

'■n (F) ^ 4 n .---------= -77T (1 ~ a In X )./-n(F')

(5)

(6)

1 1 1— ^ — ------ + — a \n X , (7)

An{E) An(F') 4 4 4

the last equation following from a neglect of second / 1 1 \

order of smallness, viz. a I — .\/.n / N J

Accordingly (3) reduces to (vv = E/E0)

6Fn ' d.v—-f (£,//) ="7T J / , NW - p r ( l - f l l n J r ) F N op ZN »■ ^

• exp {a ln A'I • exp ̂1 - -ry

• exp {ap' In X}, (8)

where we have introduced a naive form of Feynman scaling typified by

i J aU (X ) = EWn^{E, E') = — j E ---- j d/?7.

crin a

Subjecting (8) to Mellin transform (defined by

Mn = \F nEsdE\ A/n = |F NF 5dF), we have for

large F0,

8A4 =ÖF' "

where

and

zng (s) e'-P' A/n (5, />') + Mn (.9, //) 0 (s), (9)

G (s) = j xs+ap'~1 /(A) (1 — a In A') d.x 0

£= 1 ;0 ' / N

Equation (9) may be considered a linear differ­ential equation for Mn whose solution (ignoring the p' dependence of g (5) which is inconsequential in view of the smallness of a) is given by

Mn = C ep'9 {s) + ep'd{s)- j - / N

■jg(s) e(£'# (s»p' Mn (s,p') dp', (10)

where C is an arbitrary integration constant. But

CCA/n (j, p') = j Fn Es dE ^ A/n (s, 0) ep'^ s),

0

where g (s) is related to the process NN -> p + X. Since Mn (s, 0) = 0, C in (10) can be known.

Also, for g(s) + e g (5), we have

" ' ~ ' e'Mn ^ — Mn (.s, 0) [ep'g(s) -/N

Again,

A/n ^ A/n (s, 0) ep'g(s)

Hence

M7I + -^-M n = ^ A / n (5, 0)e /.N zN

P'SU)

(11)

(12)

(13)

What is still left is just the inversion of this Mellin-transformed relation. This is found from our previous work of inversion [2] by a comparison. We

1174 J. Sidhanta and R. Roychoudhury • Analytical Evaluation of the Hadron Number

accordingly find

/-NI r v

Aft E V P'

beta function as follows:

(14)•exp J p' [ (̂.s'o) - l]s0 In H'J .

Also, from another previous work of ours [1],

CnN = l t V y (15)

where p" = and s0 is the saddle-point ob-

tained by solving (1). Note that the elegance of the solution (14) is that the only free parameter here is C0; for details of the inversion procedure we allude to our previous works [1,2]. From (14) and (15), then, the total hadron spectrum 77 + N can easily be obtained. If, however, we use /.n ~ (13) alone would yield the total hadron spectrum:

Crn + N = exp {p' [g (s0) — 1 ] — J0 ln If} ,E ' P (16)

dg{s)is

ln Wandwhere s0 is given by

= - £>P' + 1 • Note that in particular, for the process NN —►

n ± + X, f(X) can be simulated by the formula [7] f (X )~ A 0(a -X ) \

Thengr (j) = J A' es+aP'~1 f(X ) (1 - a In X) dX

o= A5 B (5 + ap ', s) (17)

where B is the usual beta function. In that event S* admits of an approximate representation in terms of

— B{s +ap ', 5) d^

ln Ws=so A0 p

(18)

The last expression can be solved by successive approximation.

For comparing the result of our calculation with a standard work [8] we use a = 0.03 and obtain excel­lent agreement as evinced by Fig. 1, which more­over compares our result with the numerical calcu­lations [5] especially in the higher reaches of the energy range.

The authors thank Dr. D. P. Bhattacharya who made some references on experimental data avail­able.

10* I o 5€ t G e v )

Fig. 1. Energy spectra of hadrons in EAS for £0= 106GeVand a = 0.03 at p = 720 gm/cm2. ------Numerical resultsof [5]: ---- presented work; o Dubovoi and Nestero(1975).

[1] J. Sidhanta, R. K. Roychoudhury, Pratibha Pal, and D. P. Bhattacharya, 18th International Cosmic Ray Conf. Bangalore, FIE4-40, 5,368 (1983).

[2] J. Sidhanta, R. K. Roychoudhury, and D. P. Bhatta­charya, Phys. Rev. D28, 1113 (1983).

[3] P. M. Fishbane. T. K. Gaisser, R. H. Maurer, and J. S. Trill. Phys. Rev. D9, 3038 (1974).

[4] E. V. Denisov et al.. Proc. Int. Conf. Cosmic Rays 1965, HE3. 824 (1965).

[5] M. Boyadzhyan et al., Sov. J. Nucl. Phys. 34 (1), 67 (1981).

[6] A. D. Erlykin, L. K. Ng, and A. W. Wolfendale, J. Phys. A7,2059 (1974).

[7] J. Nishimura, Handbuch der Physik, S. Flügge (ed.), Springer Verlag. Berlin 1967; Vol. 46, p. 1.

[8] A. Odulovoi and N. N. Nesterov, Brief. Commun. on Phys. No. 8, Phys. Inst. Acad. Sei. USSR 1978, p. 18.