Volume 52B, number 2 PHYSICS LETTERS 30 September 1974

F A C T O R I Z A T I O N IN R E L A T I V I S T I C H E A V Y - I O N S C A T T E R I N G ~

W.L. WANG Lawrence Berkeley Laboratory, University of Cali[ornia. Berkeley, California 94720, USA

Received 2 July 1974

We have calculated the total cross sections for relativistic nucleus-nucleus scattering in the Glauber theory and conclude that there will be.no factorization, due to the short-range nature of nucleon-nucleon interaction as com- pared to the sizes of the colliding nuclei.

In this letter, we shall discuss the problems of ap- plying the concept o f factorization in the heavy-ion collisions at the Bevatron energies [1 ]. We base our discussions on the results obtained in a standard Glau- ber approximation [2], which has been extended to the nucleus-nucleus scattering by Czy~ and Maximon [3]. In the optical limit, the elastic scattering ampli- tude FAB(q 2) is given as [3],

FAB(qa)=2~fd~b exp [iq'b]

X (1 - exp [ixfd2b'TA(b') TB(b - b')] } (1)

where q is the momentum transfer and k is the inci- dent momentum. The two-dimensional densities T(b) are related to the nuclear density distribution p(r) by

TA(b ) = : PA(r =-- b + z)dz, (2) _oo

where PA(r) is normalized to unity. The interaction parameter x is related to the nucleon-nucleon total cross section Onn by

X = 21-(i + oOJJB tInn , ( 3 )

where A and B are the mass numbers of the colliding nuclei, a is the ratio of real to imaginary parts of the nucleon-nucleon elastic scattering amplitude. In our calculation we use Onn = 44.5 mb and a = -0 .2 . The nuclear density distribution p(r) is taken to have the spherical Woods-Saxon form. The parameters we

* Work performed under the auspices of the U.S. Atomic Energy Commission.

need in the calculations are listed in ref. [4]. To simplify our discussions, we first define the

"factorizabili ty" PAB for A-B scattering as

["aB = (OAB)2/OAA O'BB ' (4)

where OAB is the total cross section for A-B scattering, and OAA that for A-A scattering, etc. We may obtain the total cross sections from FAB(q ) of eq. (1) through the optical theorem. A strict factorization means I'AB = 1 .

We now show our results of the total cross sections. In fig. 1, we have plotted the total cross section as a function of the effective radius

Reff = ro(A1/3 + B 1/3) (5)

I I i i I i i |

• G l o u b e r c o r c u l o t i o n

o-Ae :Zy r ro z (A I / z+B ' /3 )z ( re= 1 . 2 5 f m )

~ 2°''b ~

I o

I 0 ~'S I I I I I I " I 4 5 6 7 8 9 I 0 I I 12

Ref f / r o

Fig. 1. The total cross section (in fm 2) of the nucleus A and B, plotted against the effective radius Reff in units of r0. The points represent the results of the Glauber theory; the values for the total cross sections are from table 1. The solid line is a fit to the results of the calculation, by choosing re = 1.25 fm in eq. (5).

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Volume 52B, number 2 PHYSICS LETTERS 30 September 1974

Table 1 Total cross section and factorizability. Nucleus-nucleus total cross sections (fm 2) calculated in the optical model. The averaged nucleon-nucleon cross section O--nn = 44.5 mb and a = -0.2 (see eq. (3)). The nuclear density is of a Fermi distribution. The fac- torizability (the lower left table) is defined by eq. (4). The values in the parentheses are from eqs. (7) and (8).

B 4 12 16 27

4 I 35.2 89 104 164 i

12 1.52 148 165 260 (1.29) I

16 1.63 0.97 I 189 298 (1.41) (1.01) I I

27 2.34 1.40 1.44 I 326 (1.80) (1.08) (1.04) t

32 2.78 1.70 1.76 1.01 (2.03) (1.14) (1.08) (1.01)

40 2.93 1.66 1.69 1.15 (2.21) (1.18) (1.11) (1.02)

88 3.64 1.71 1.66 1.08 (2.98) (1.40) (1.28) (1.10)

208 5.18 2.12 1.98 1.26 (4.44) (1.81) (1.61) (1.29)

with r 0 = 1.25 fm. From these results, it is clear that the Glauber theory gives qualitatively the same results as predicted by a simple black-sphere model, in which the total cross section is

°AB = 2 (RA + RB)2 (6)

where R a and R B are the black-sphere equivalent radii of the nuclei. In this simple model, the factoriza- bility becomes

lAB = (1 +TAB)4/167A21~ (7)

where 7A B is the ratio of the radii o f the two nuclei

"YAB = RA/R B" (8)

We now show the results of oAB and VAB from our numerical calculations (using eqs. (1) and (4) in table 1, where we also list the factorizabilities as obtained from the black-sphere model, eqs. (7) and (8). To be consistent in our comparison, we have determined the equivalent radius from the values of OAn when A = B, i.e.,

R 2 = aAa/8rr (9)

with oAa obtained from the Glauber calculation. The

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A 32 40 88 208

200 221 315 502

321 341 443 659

369 389 493 720

405 424 523 754

409 428 530 762

0.94 474 577 819 (1.00)

0.88 0.91 775 1051

(1.05) (1.03) I 1.03 1.02 1.03 1383

(1.20) (1.15) (1.04)

reason we use eq. (9) for the radius is dictated by the fact that there is no simple prescription for the strong interaction radius for the nucleus [4]. We have used the nuclear densities directly from electron scattering without modification for the strong interactions [4]. The actual values for OAB may be much larger for low- mass nuclei.

As concluding remarks, we would like to note that eq. (1) is obtained with an explicit assumption that the range of nucleon-nucleon interaction (the inter- action radius) is much smaller than the nuclear radius [2, 3]. This is the reason that the relative sizes of the colliding objects remain important and the scattering process is still determined by the geometrical proper- ties of the objects [5]. It is therefore clear that there will be no strict factorization in the nucleus-nucleus scattering as long as the sizes o f the colliding objects remain much larger than the range of nucleon-nucleon interaction. This remark is based on the Glauber the- ory; however, it would be unlikely for the Glauber theory to be grossly invalid at the Bevatron energies. The conclusions made in this note will also not de- pend much on the model.

The essential features of the short-range interaction

Volume 52B, number 2 PHYSICS LETTERS 30 September 1974

and the large sizes of the colliding objects are still true even at the present extreme high energies. I t is important also to point out that , although the fac- torizabil i ty is always close to unity, the "approximate" factorization proper ty in nucleus-nucleus scattering does not indicate dominance by a single pole exchange [5] , and therefore should not be interpreted in the original ideas o f factorization hypothesis [7] . The most convincing example of this point is perhaps the fact that the nucleus-nucleus scattering achieves the black-sphere limit at a rather low energy, say, at 10 MeV/nucleon [e.g., 6] .

Finally, if we accept the Glauber model for any composite particles scattering, the strict factorization could imply that the interaction between the consti- tuents has a range larger than the sizes o f the particles [7].

The author would like to thank H. Gr~f for his suggestions on the overlap integration in eq. (1).

References

[1] General interest in the high energy behavior, including Regge factorization, of nucleus-nucleus scattering has

been reviewed by H. Steiner, Physics research with high energy heavy ions, in the Adriatic Meeting on Particle Physics, Rovinj, Yugoslavia, September-October, 1973; Lawrence Berkeley Laboratory Report #2144.

[2] R.J. Glauber, in: Lectures in Theoretical Physics, eds. W.E. Brittin et al. (Interscience, New York, 1959), Vol. I.

[3] W. Czyz and L.C. Maximon, Ann. Phys. (N.Y.) 52 (1969) 59.

[4] L.R.B. Elton, Nuclear sizes (Oxford University Press, 1961, Oxford, England) p. 31.

[5 ] See recent preprints of P.M. Fishbane and J.S. Trefil and of S. Barshay, C.B. Dover and J.P. Vary. We used the same values of the parameters c~ and Onn as in Fishbane and Trefil, so that the results may be directly compared. Barshay et al. show the approximate factorization as a purely geometric property of the composite particles (consistent with our picture); however, Fishbane and Trefil also discuss single Regge pole exchange.

[6] J.S. Blair, Phys. Rev. 108 (1957) 827. [7] At extremely high energies, the factorization properties

were discussed by V.N. Gribov, Yad. Fiz. 9 (1969) 64 [Soviet J. of Nucl. Phys. 9 (1969) 369], and by B.M. Udgaonkar and M, GeU-Mann, Phys. Rev. Lett. 8 (1962) 346. A study of high energy factorization was also dis- cussed for 4He-4He scattering by V. France, Phys. Rev. Lett. 32 (1974) 911. These authors used a nucleon- nucleon interaction radius which depends on energy, and discussed the asymptotic region where nucleus-nucleus scattering may obey the strict (or Regge) factorization (i.e., FAB = 1) as derived from the pomeranchuk trajec- tory exchange in Regge theory for elementary particle interactions.

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