Increasing p-p and p-nucleus cross sections

Volume 46B, number 1

PHYSICS LETTERS

3 September 1973

I N C R E A S I N G p-p A N D p - N U C L E U S C R O S S S E C T I O N S * U. MAOR and S. NUSSINOV Department of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Israel Receded 26 June 1973 p-Nucleus cross sections are calculated assuming that total p-p cross sections increase with energy. The dependence on the nucleon structure in b-space is examined and it is found that p-nucleus cross sections g~ow very rapidly with A if the increasing p-p cross section is attributed to an expanding nucleon halo.

Recent ISR experiments [1] indicate an increase of OTPP,the total p-p cross section, from 38.5 mb at s ~ 100 GeV 2 to 43.5 ± 1 mb at s -~ 2800 GeV 2. The parametrization opp = [38.4 + 0.49 log2(s/122)] mb

(1)

was suggested by Leader and Maor [2] (LM) to fit this increase. An earlier analysis of inelastic p-air cross sections at cosmic-ray energies ( 1 0 3 - 5 × 104 GeV) by Yodh et al., [3] (YPT) suggested also a rising p-p cross section o pp = [38.8 + 0.4 log 2 (s/130)] m b .

(2)

If the cosmic-ray and ISR results are indeed reliable, the agreement of (1) and (2) suggests that these parametrizations hold up to very high energies. At such energies eTPPreaches values larger by 50% then the "constant" pre ISR value of ~ 4 0 rob, In view of the importance of p-nucleus cross sections, we discuss its estimates using multiple scattering theory [4, 5]. In general, o(p-N) depends on the p-p interaction and the nuclear parameters (e.g. A the total nucleon number and (R 2) the r.m.s. radius). It is commonly believed [3, 5] that whereas o(p-N) depends strongly on a pp it depends only weakly on other parameters describing the p-p interaction. It is purpose of this note to re.examine this point. We find that when the increasing p-p cross sections are attributed [2] to a rapidly expanding thin nucleon halo, the resulting o(p-N) grows rather rapidly with A. This can be visualized as the result of diminished shadowing [6]. Such an effect becomes important for p-air collision once the nucleon halo * Research supported in part by the Israeli Academy of Science.

exceeds the radius of the nucleus. On the other hand, if the increasing p-p cross sections are confirmed within a conventional interaction radius (about 1 fermi) we re-obtain the standard result [3, 5], namely,-weak dependance of o(p-N) on the detailed p-p parameters. Indeed, the rather large difference between these two types of calculations, over the energy range considered by YPT, puts non trivial constraints on models of the type suggested by LM, provided one has good estimates of oin(P-N). It was suggested that the constant and increasing parts of o~p represent two distinct "components" generated by different mechanisms so as to correspond to different types of multiparticle final states [7] and/or different ranges of initial p-p impact parameters

Lbt =b. Most specific with respect to the last point- the only. one relevant in the present context - were LM. For the component corresponding to the constant cross section (component 1) they suggest:

pl(b) = pl(0) exp(-F/b

)

(3)

where Pl(0) = 0.73, bl2 = 0.82 fm 2. These parameters were chosen so as to obtain a t = 0 logarithmic slope of [2]

Ifdo

dt \dt/t=O/\dt/t=O

=

10.8 GeV -2

(4)

and o pp = 38.5 mb. These are the experimental values at s ~, 120 GeV 2. An expression similar to (3) is proposed for the second component (component 2) which is responsible for the logarithmically increasing term i n ( l ) . One gets p2(0 ) = 0.021, b~ = 0.38 log2(s/122) fm 2, which reproduce LM second component. Let us recall that the scattering amplitude (assumed to be purely imaginary) is presented by

99

Volume 46B, number 1

PHYSICS LETTERS

= i f db p(b) exp(-iA "b)

(5)

°r = 2 f d b p (b)

t2pC )- P2~)I

~bA(~'~) = (l/rrr i ) exp(--y2/r 2 )

(11)

where rA 2 is two thirds of the mean square nuclear radius. Eq. (8) yields then for Pz(~,) = Pl(O)exp(-b2/b2):

where A is the momentum transfer ( - A 2 = t) and

om=f@

3 September 1973

_p/0)b,

(6)

f/(~) r2 + b2

•

exp [-b2/(rl + b/2)]

.

(12)

The complete differential cross section is given by Thus we finally obtain for the LM model:

ill(A) +f2(A)12 dt ~ ~

do =

(7)

and it was agreed by LM that the interference term accounts for the break observed [ 1] in the p-p loga rithimic slope at t ~- 0.1 GeV 2. Alternatively, one may assume that P2(b) has roughly the same shape as Pl(b), at least up to energies of ~<50 X 103 GeV. This was implicitly assumed by YPT, and is also consistent with the analysis of ref. [7]. As we will now show, considerably different oT(P-N) are obtained for these two alternatives. Our calculation procedure closely follows that of ref. [5]. Let us assume that a single nucleon bound inside a nucleus, with atomic number A, is described by a wave function ~kA (r). The scattering amplitude, at impact parameter b, of a fast proton incident on this bound nucleon is given by the convolution

[~) =f p(b.-.L,) iaA~)d.L,

(8)

where

~A~) = f dzlXI'A(r=Y~+ze)12

(9)

p~M(b) = 1-

Pl(0)bl2rl +b~-

{1

P2(0) b 2

exp

[-b2l(r 2~

exp [ - b 2 / ( r l +b2)]}a .

+b2)]

(13)

Eq. (13) should be compared with ~P'r(,b) =

(14)

which is the p-N "proNe" resulting from the convert. tional assumption that the additional cross section is spread over a similar impact parameter range as the constant cross section part. Eq. (13) and (14) were substituted in eq. (6) with the parameters defined above. The calculation of o(p-N) was carried out forA values of 4, 12, 15, 27 and 64 at lab energies ranging from 500 GeV to

--no

~rmb

ez is a unit vector along the z axis. PA~)

is the normalized two dimensional distribution of the nucleon inside the nucleus, and p ~ ) is the N-N amplitude in impact parameter-b space. Consider next the joint effect of A nucleons moving independently with the same wave function xI'4 (r). The total transmission coefficient SA (b) = (S(b)) a and we have the standard relation S = 1 - f . The resulting p-N amplitude PA ~ ) is thus given by ,oa (b) = 1 - ( 1 _ f ~ ) y l

(112)

The total and inelastic p-N cross sections are obtained by substituting (10) in (6). Following ref. [5] we conveniently take 100

•

,

'

,

....

i

55O 5O0 450

o~..u

400

O-VPT

350 300 250

io'

I0:

10 4

s GeV =

Fig. 1. Calculated inelastic cross sections for p-air. The data taken from ref. [3] does not contain the upper error bar.

Volume 46B, number 1

PHYSICS LETTERS

3 September 1973

Table 1 SummarT,..9~ calculated inelastic p-N cross-sections in rob. Each entry contains two numbers. The upper one is o ~~x and the lower one is oil'*"-. PL(10aGeV)

A =4

A = 12

A = 15

A = 27

0.5

108 107

253 252

300 299

472 471

915 914

1.0

112 110

262 257

311 305

486 479

937 929

5.0

131 120

303 275

358 325

553 507

1042 975

10.0

143 125

331 284

391 335

600 521

1114 999

20.0

156 130

365 293

431 346

652 537

1209 1026

30.0

165 133

388 299

458 353

700 546

1275 1041

40.0

172 136

406 303

479 358

732 553

1327 1053

50.0

177 138

420 305

496 361

758 558

1369 1061

100.0

256 144

468 317

553 373

846 575

1518 1090

100 000 GeV. rA 2 was approximated by r 2 = 2/3 A z/3 which is consistent with the r.m.s, valves quoted in ref. [5]. The results are summarized in table 1. Partic, ular attention should be given to the inelastic cross sections which are the only ones available experimentally at very high energies. The conventional calculation (eq. 14) is found to be consistent with YPT calculations. When carried out at accelerator energies our calculations reproduce the measured cross sections. Our main f i ~ is that ¢rbmM(p-N)is considerable larger than o.~"~ (p-N) for sufficiently large energies and A values. This general characteristic is demonstrated in fig. 1 where we have presented oin(P-air) as obtained from our calculations. Drawn on the same figure are also the experimental estimates obtained from cosmic ray data [3]. The experimental points given are the most probable ones and the l o w e r b o u n d corresponding to 95% confidence level. We have not presented the complete error bars as the upper b o u n d is not given in ref. [3]. To conclude, we find that the cosmic ray data, crude as it may be, does constrain models like LM with quickly expanding p-p profiles. Our calculations suggest that Oin(P-air) becomes suasitive to the

A = 64

particular features of p-p interaction once b 2 I> 5 - 10 fin2. A more careful and comprehensive cosmic ray work could thus narrow even further the possible leeway in speculations on the impact parameter structure o f the rising component in N-N interactions. We would like to thank E. computer work.

Komay

for his help with

References [1] C E R N - Rome Collaboration, Phys. Lett. 44B (1973) 112;

[2] [3] [4] [5] [6] [7]

Pisa-Stony Brook Collaboration, Phys. Lett. 44B (1973) 119; These results and other relevant ISR data are summarized in U. Amaldi, CERN NP Internal Report 73-5. E. Leader and U. Maor, Phys. Lett. 43B (1973) 505. G.B. Yodh, Yash Pal and J.S. Treffl, Phys. Rev. Lett. 28 (1972) 1005. R.J. Glanber, in High energy physics ,and nuclear structure, ed. S. Devons (Plenum, New York, 1970). J. Newmeyer and J.S. Treffl, Nucl. Phys. B23 (1970) 315. M. Froissart, Theoretical physics (Trieste, 1962) p. 397. A. Casber, S. Nussinov and L. Susskind, Phys. Lett. 44B (1973) 511. 101