Limiting Fragmentation and the Charge Ratio of Cosmic-Ray Muons

Combining the steeply falling primary cosmic-ray spectrum with the scaling hypothesis for the spectrum of produced secondary particles, we illustrate how to correlate data at machine energies with cosmic-ray phenomena. We find that the p+/p ratio is expected to be energy-independent and greater than 1, regardless of the specific model chosen for multiparticle hadronic reactions. An analysis. of accelerator data gives the value p+/p, = 1. 56, to be compared with the experimental result

The purpose of this paper is to examine some implications for cosmic-ray spectra of recent the- oretical and experimental developments in hadron- ic multiparticle production; in particular, to examine some of the implications of the scaling' or limiting fragmentation hypothesis.
Given the nature of the primary cosmic-ray spectrum, one can use the scaling hypothesis to correlate data at ma- chine energies with cosmic-ray phenomena.
We concentrate primarily on the cosmic-ray g'/p, ratio, in order to see (a) whether scaling and accelerator data can explain this ratio, and (b) whether the observed constancy of the ratio discriminates between models of high-energy reactions.Experimentally, the p+/g ratio at sea level is known to be approximately 1.2-1.3from ener- gies of a few GeV up to 1 TeV (10" eV), as shown in Fig. 1. 3 Although Chou and Yang4 cite this con- stancy as evidence in favor of the class of highenergy theories that give rise to constant topolog- ical cross sections o", we find that it is a much more general result implied by any model consistent with scaling.' To understand the process qualitatively, let us distinguish between two mechanisms for production of hadrons (and consequently of muons), pionization and fragmentation.
Fragmentation products of the projectile (or target) are those particles pro- duced at low energy in the projectile (or target) rest frame.Pionization refers to the copious pro- duction of particles at low energy in the center-ofmass system.In models with a finite correlation length, such as the multiperipheral model, the charge of the incident particles is reflected only in the fragmentation products; the pionization products are independent of the nature of the incident particles.Since in such models the ratio of the number of pionization to fragmentation products increases logarithmically with the incident particle energy, one might expect the g'/p, ratio to decrease asymptotically to unity.The distinctive feature of cosmic-ray phenomena which invalidates this reasoning is the steep decline of the primary flux with increasing energy; experimentally, de/ dE ~E 2' for energies up to 1 TeV.Although pions of energy E can be produced by primaries with any energy E a E" the steeply falling primary spectrum suppresses the contribution for E»E" and enhances the contribution for E=E,.Since pions having E, =E are projectile fragments, we arrive at the conclusion that projectile fragmenta- tion is the dominant process in producing cosmic- ray pions (and muons) of any given energy, despite the existence of pionization." In order to make these considerations quantitative in a way which elucidates the essential phys- ics, let us first consider an oversimplified model, in which the primary spectrum consists exclusive- ly of protons which interact once with the atmospheric nucleii, producing pions which all decay into muons.For simplicity, kaon production is ignored here.The problem is most conveniently formulated in terms of the single-particle distribution for pions' produced in a proton-proton collision, primary proton and secondary pion, respectively, v~~" is the total inelastic proton-proton cross section, and + refers to the charge of the observed pion.If we assume a primary spectrum of the form that the p'/p, ratio is explicitly independent of the muon energy.(2) p, '/p, & 1 because the proton fragments more often into v' than v .
(3) The ratio p+/p, depends on the power y of the primary spectrum, and thus may have an implicit energy dependence through variations in the power law of the primary spectrum.(4) The nature of the target nuclei in the atmosphere is unimportant.This fol- lows from the fact that only fast forward-moving fragments with x&0 contribute to the integral for Z~"and to the extent factorization is valid, f~, (x&0) depends only on the nature of the projec- tile particle, not the target.(The actual propaga- tion of cosmic rays through the atmosphere does depend on the average atomic weight of the atmospheric nuclei, but, as we shall discuss later, such dependence for the muon charge ratio will be expected to be fairly mild.) (5) Pionization products have very little effect on the p. '/g ratio.The power-law behavior of the primary spectrum, appearing as the weighting factor x" in the inte- grand of Z~", suppresses contributions from x=0, the pionization region.For x&O, it is expected that ~+-(Z+y) ) dn w (Ew ) dE~f dEE i"&~f'"(E", Z).
p, (E") x (E,) Z, "' This simple model already embodies the relevant salient features: (1) From EIls. ( 6)-( 8), we see (where y=1.7), then the pion spectrum in this mod- el is as a result of the initial proton charge.This is indeed the case in laboratory experiments performed up to 30 GeV/c, ' '~a nd is expected to hold asymptotically as well.Since the integrand in Z~, never becomes negative, it follows that Z~+"&Z~" and that p, '/y, &1.
After this simplified discussion, we proceed now to consider a more realistic model which takes in- to account the propagation of cosmic rays through the atmosphere, the neutron component of the primaries, and the production and decay of charged kaons in addition to pions.We have followed the general approach described by Barrett et al. " and obtained the muon spectra in terms of the single- particle spectra f», f~, f~", f», f"", and f,", all defined analogously to the previous fp".To il-  lustrate this approach, we present here a brief discussion of the propagation of nucleons through the atmosphere.
These equations are simplified by the following charge-symmetry relations'4: (lib) Inserting these relations into Eqs.(10) and assum- ing a separable dependence" on y and E, we obtain the solutions: lt is known experimentally that A0/N0 = O.V4 +0.01." If f» and f~"are experimentally determined, A/N ls then a known function of y f1'om the above equations.It shouM be interesting to measure A/N as a function of y directly and compare with the results obtained from laboratory data on f» and ft hrough Eq. ( 13).%8 note that many of the characteristic features of our results are similar to those of the isobar model of Ref. 7. However, the explicit use of the scaling property enables us to sharpen the derivations so that the results are less model- dependent, and to relate known or measurable laboratory quantities to cosmic-ray phenomena.For instance, if we take the experimental values" A«=120 g/cm' and" X«=77.6 g/cm', we find ~h~~~A"-=&«[I -(Z»+Z2"}] ' is the attenuation length of nucleons in the atmosphere and A- = X«[1 -(Z»-Z2"}] '.Note that the initial primary spectrum is given by 6'(Z, o)+st, (z, o) =N~-&" », as expected.It is also worth mentioning that both A«/A«and A«/A«are explicitly energy-indepen- dent as a consequence of the power-law behavior of the primary spectrum.From Eqs. ( 12), the charge composition of nucleons at atmospheric depth y is given by 6'(&, y) -&(&, y) t(E, y) +x(z, y) A"'= " =93.2 g/cm'.

PP + Pn
The charge composition of nucleons in air is then based On the experiment of I al et al. , as quoted in their Ref. 14. ] We take this to be an encourag- ing sign for the validity of the general approach we have adopted.
We have treated the pion, kaon, and muon fluxes by similar diffusion equations, and obtained the resulting muon spectrum.The results are particularly simple in the limit of large muon energies (E0cos8»100 GeV; 8 is the zenith angle of the muon).Ignoring the generation of nucleons by v or X fragmentation and the conversion of m to X, or vice versa, we obtain: (18} where A; =Z,'"(I+A,/N, )+Z"'"(I -A,/N, ), )n(A /ll") with similar definitions for A~a nd R~.Here, Z~" A", and A'"are the same as before, g'""is the corresponding quantity for neutron fragmentlllg into F A, =~"(I-Z~, + -Z"+, -)-', A, '=X, (I-Z, +,++Z, +, ) ', and A. " is the pion-interaction mean free path in air, Similar quantities are introduced for the kaons and appear in A~a nd R~i n exactly the same way.Finally, (y) ~e-0.00242 This is in reasonable agreement with a cosmic-ray measurement of A/N at mountain altitude, referred to in Ref. V. [Our Eq. ( 13) corresponds to their Eq. (III.2), for which they gave the numerical expression (7 " is the pion lifetime; ~r is the kaon lifetime; b~" is the branching ratio of K-LU.decay; and [-(I+ y)] is the power of the primary energy spec- trum.} If we ignore the kaons, and assume that the pion charge-exchange fragmentation is small (i.e. , Z, +, -=0), then Eq. ( 18) simplifies to To obtain Eq. (22}, we have used the charge-sym- metry relations'~: f",(x) =f",-(x),   (24a)   f"(x)=f".+(x).(24b)   It is important to note that Eq. ( 22) depends only on the ratio X,/X", and not on X, and X"separately.
While A, , or A."depends significantly on the average atomic weight A of the nuclei targets in the atmo- sphere, A. , /A. "has only a mild dependence (see Ref. 18), and the laboratory results at 40 GeV/c on carbon should be close to those for the atmo- sphere.This is also true for the full relation ( 18), which depends also on X"/A."; consequently, the muon charge ratio p'/p, can also be regarded as independent of the nature of the atmospheric nuclei, as we have mentioned earlier.
It is interesting to note that in Eq. (22) Z~"/Z~" & 1 and S'"/S, &1 always imply p'/p, &1.One can show that S+/S, &1 as long as A"/A"'&1, which is always the case.
We have attempted to give a numerical estimate of p, '/p.as given by Eqs. ( 22) and (18), using presently available accelerator data for the various single-particle distributions.' "'"'" We find Z, +,+=Z, -, -=0.242,X, =111g/cm' (from Ref. 18)  and the ratio Z~"/Z~, ranges from -1.8 to -2. 25, depending on the data used and the method of estimation.While the suppression of contributions to the integrals Z~~f rom the x=0 region is sufficiently effective to minimize the effects of pioniza- tion, the actual behavior of the single-particle distributions f~, at x = 0 may still be important for accurate determination of the ratio Z~, /Z~".In this connection, we note that accelerator data at E 630 GeV are probably not yet scaling for x ~0.1, as is evidenced by the inequality f~;(0)&f~,(0) from these data "" .Taking the (unweighted) average value Z~, /Z~, =2.02, we obtain the muon ratio from Eq. ( 22) as v'/I =1.58.This is somewhat higher than the experimental values (see Fig. 1), although the latter fluctuate between 1.2 and 1.45 for E & 100 GeV " We can readily identify two possible contribu- tions to this discrepancy: the nonscaling behavior of present accelerator data at x = 0, and the pos- sible inaccuracy of our factorization assumption.
Finally, our estimate using existing accelerator data in Eq. (18}shows that contributions from kaons and from the pion charge-exchange fragmentation tend to cancel each other, so that the value of p'/p is only slightly modified from Eq. ( 25).
In conclusion, we wish to emphasize the role which the scaling behavior of the single-particle distributions can play in the analysis of cosmicray phenomena.Although we have concentrated our attention on the muon charge ratio, the formu- lation can have much wider applications.As a general result, it allows one to correlate informa- tion obtained from particle production distribu- tions at accelerator energies with the spectra of secondary cosmic rays at much higher energies.
FIG, 1. Compilation of the muon charge ratio at sea level as a function of muon energy.