Unitarity effects in high-energy nuclear abrasion

Nuclear Physics

(1976) 477 - 481;

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UNITARITY

EFFECTS

@

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IN HIGH-ENERGY

NUCLEAR

Co., Amsrerdom from the publisher

ABRASION

D. 1. JULIUS

Weizmann Inslirure of Science, Rehovor, Isruei ’ and Insrirul fiir Theorerische

Kernphysik,

Universitdi

Karlsruhe.

Karlsruhe,

Germany

and A. S. RINAT

Webmann

Insrirute o/’ Science. Rehovor, Israel Received

16 March

1976

Abstract: Using an explicitly unitarity model we have studied an heuristic approach to the inclusion of unitarity effects in high-energy ion collisions introduced by Hiifner, Schlfer and Schiirmann. We infer that their treatment introduces errors of less than a factor of two for the abrasion of not more than five to ten nucleons, depending on the nucleus involved but almost independent of energy.

1. Introduction High-energy reactions on nuclear targets are most expediently described in the eikonal (Glauber) limit I). This description generally employs only simple properties of the nuclei involved and the elusric projectile-nucleon interaction. Complications arise as soon as one wishes to account for the fact that either the projectile or the nucleon change their state in the elementary collision. It is well known that even the inclusion of mere spin-flip or charge-exchange seriously complicates standard Glauber calculations, since the phase a projectile acquires through collisions with more than one scatterer depends on their spatial ordering 2). These difficulties can in principle be overcome if one is willing to replace the simple algebra of a Glauber calculation by a non-commutative one. The situation changes if many channels compete with the elastic projectile-nucleon collision. Below we shall be concerned with the influence of the unitarity defect of elastic amplitudes as for instance displayed in ratios ~‘O’a’/cP’~c’ of order 3-4. We are not aware of previous calculations of this effect except for the heuristic approach used in ref. ‘) +‘. Our interest in this problem was aroused by a description of the inclusive ’ Present address. ” The coupled-channel Glauber approach we shall use is similar to that found in the treatment of meson production processes in nuclei 3). Unlike these calculations, however, our model is solved to UN orders in the inelastic amplitudes. 477

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D. 1. JULIUS

AND A. S. RINAT

process 4*‘) 160+‘Be

(1)

+ %+X,

at an incident momentum of 2.1 GeV/c per nucleon. The description by Hiifner, Schafer and Schtirmann s, adopts the picture of Bowman et al. “) holding that fragments of the projectile are torn off in the collision (“abraded”). The residual projectile further rids itself of excitation energy by emission of one or several nucleons leaving a residue which is finally detected. The above authors performed calculation based on a single-channel Glauber model and reached the following simple result for the inclusive cross section with 1 nucleons removed OA.1

=

OS ;

d’bP(bf’-‘{

1 -P(b)}‘,

with P(b) the probability that a nucleon, hit at impact parameter b, remains in the projectile. The result (2) would have looked different had a single-channel description been consistently followed. Expressions of inclusive processes typically contain IY oN,oN(b)(2, the squared elastic S-matrix element. The authors of ref. ‘) propose that the insufficiency of the single-channel description of the elementary collision a + N be heuristically remedied by the replacement

+ 1 IY&c12= 1. I%,,a,@)12

(3)

It is the prescription (3) which results in the term 1 in the curley bracket of eq. (2). In the following note we check the assumption embodied in eq. (3). Our treatment will consist of lumping all inelasticity effects in UN collisions into a single, effective, second channel a+ N -+ x+ N. For the resulting two-channel system, it is a simple matter to parametrize the 2 x 2 matrix 9’(b) representing the S-matrix at fixed impact parameter. Under certain circumstances, it is possible to arrange that [9(b), Y(b’)] = 0, so that the matrix generalization of Glauber theory is not complicated by the necessity for spatial ordering. This theory can then be solved exactly, in the sense that the matrix Y(b,)Y(b,) . . . for any number of factors can be found. It is important to note that our model is even more artificial than the notion of an “effective” inelastic channel usually implies, in that all inelastic effects in a aN collision appear in our treatment as dissociation of the beam; the target nucleon remains unexcited. To treat both nucleons on an equal footing entails at least a four-channel calculation. In this case, even if a suitable parametrization of the unitary 4 x 4 matrix 9’ could be found, the problem of non-commutativity would render the matrix generalization of Glauber theory intractable. We note also that the use of Glauber theory involves an additional, rather drastic simplification, in that the content of the nucleon excitation x is treated as a single particle state. This

UNITARITY EFFECTS

479

factdone would be sufficient to preclude a comparison of the model with experiment; its more limited purpose is to clarify the role played by unitarity in a simplified

framework, in the hope that the results remain relevant to more refined theories. The restriction to two channels means that we cannot apply our model to ion collisions of the type (l), but only to particle induced reactions. It is for such reactions that we shall compare the results of our unitary model calculation with the heuristic treatment of unitarity described by eq. (3). We discuss in the last section what conclusion germane to the ion collision can be drawn. Sect. 2 gives a description of our model ; the parameters of the S-matrix to be used are then fixed by a comparison with the pN data as discussed in sect. 3. Results for selected targets are found in sect. 4 and we present some concluding remarks in sect. 5. 2. Inclusive cross sections in B two-channeImodel Consider the process a+A -+ (A-I)+&,

(4)

where a projectile a incident on a target A produces a residual nucfeus A - I (assumed to be in its ground state} and further unobserved fragments X in some state n. We wish to determine the inclusive cross section in terms of the wave functions of target and residual nucleus and the aff amplitudes. We shall consider only two channels, namety the elastic ~h~nel a f N -+ a+ N and a reactive channel a+ N -+ x+N, which is supposed to make up the unitary defect of the former. The elementary unitary S-matrix at fixed impact parameters b can then be parametrized as follows Y(b) =

v(b) exp 2iJ Ab) ( i( 1 -q’(b))* exp i@,(b)+ d,(b))

i(l - q2(b))* exp i@,(b) -t-6,(b))

q(b) exp 23,(b)

>’

(5)

Callingf,.($ II; A -+ A - 1, X,,) the amplitude for the process where r stands for the channel index fr = 1 denotes the elastic channel), we thus wish to calculate (9 = k’ -k, d2fZ * dz~/~‘~

As an expedient tool we shall use Glauber theory, which, however, becomes unwieldy if internal degrees of freedom prevent individual S-matrices from commuting at different b. We thus demand

i?w,+wqj= 4

(71

which is achieved by setting 6, = 6, in (5). As a result we are feft with (a, = (f) i)) 9’(b) = exp &S(b)[rljb) I + $1 - q2(b))fcJ.

(81

480

D. I. JULIUS

AND A. S. RINAT

If eq. (7) is fulfilled, Glauber theory for our two-channel model looks exactly like the single-channel case, namely S(b>{rj})

=

fi

Y(b-bj),

(9)

j=l

where {rj> are the coordinates of the nuclear scatterers and {bj) are their projections perpendicular to the beam direction. Next we need wave functions for target and residual nucleus in thier ground state ‘. We assume an independent particle model JiA = fi Gus);

$A_1 = ‘ii

j=l

~(~-*)(rj),

(10)

j=l

and #(“) and #@-” will differ substantially if I is not small compared with A. The amplitudes for the process (4) then read

f,,tE,

4;

A -+ A- I, X,)

01)

Here s is the projection of r normal to the beam direction. In (11) we employ a mixed density PI(r)(- pA [(r)) = i#+A)‘(r)f#P-t)(r)

(12)

and use X, as a notation for the state of the fragments X. Next we integrate over angles, sum over final states t of the beam particle and sum over all fragment states. In our independent particle model of the nucleus, this last step means summing over all states except the ground state for each particle contained in the fragment X. Using the unitarity of Y we obtain d2b(r = ll[l - un(a)ub(6)1’[uXb)uT(bf3A- ‘lr =

1>,

(13)

where u@) =

p,(r)Yl(b -s)dr. s

(14)

The combinatorial factor (f) in eq. (13) results from treating the target nucleons in the final state as distinguishable. It is not difficult to show that complete antisymrnetrization of the target final state leads to the same result, provided the overlap t See note added in proof.

UNfTAR1T-Y

EFFECTS

481

integral

is negligible. In the appendix we evaluate the matrix products in eq. (1J), take the (1, 1) element and find

Eqs, (15), (18) aBd (19j obvious$ a&w a numerical ~~rn~ar~soncmor:the densities and the matrix Y is specified.

For simplicity, we shall use Gaussian single particle densities

482

D. I. JULIUS AND

A. S. RINAT

pi(r) = rc-tR,,:exp

(- r’/R:

(12) then read

R,:

,),

(21)

= +(R, 2 + Ri!,).

3.2. THE S-MATRIX The

elements in eq. (8) were parametrized

as follows:

26(b) = I exp [ - W&J21, q(b) = 1 -q. exp [ -

@/bo)21.

(22)

The three parameters 1, ‘lo and b, may, for instance, be related to o“‘“l/rP’~el and cP*, and the slope of the elastic cross section in the forward direction B = ~(d/dt)(da”/dt)~(do”L/dt)J,=, (all quantities here refer to pp scattering). The first ratio actually comes close to a definition of the energy. The following relations can easily be established: 0 *‘tar= 27rbi[qo sin A/A+ Cin A], 0 ~~9~’= 27rnbi[rj,(sinA/A- l)+$$j + Cin A],

B=Lb2Re 2 0

where Si ‘z, Cin z are the sine integral and a function relation to the cosine integral [see ref. ‘) p. 2311. The experimental values of R = ~~“~~/o““*~and 4 = cP”/B [ref. *)I are gr‘venas functions of beam momentum in table 1. The data change only insubstantially above 10 GeV/c. We have chosen a smoothed set of parameters corresponding to the experimental value of R at each momentum, but TABLE I Parameters of the nucleon-nucleon interaction

0.9 1.4 2.0 2.8 4.0 5.8 10.5

1.1 1.5 2.0 2.5 3.0 3.5 4.0

60 29 24 20 17 14 12

0.10 0.40 0.60 0.80 0.90 0.95 0.95

2.10 2.10 1.80 1.58 1.27 0.84 0.50

UNITARITY

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a constant value of 4 = 12, as is obtained at high energy. The large values of this quantity at lower energy are clearly incompatible with our small angle kinematics. The parameters ‘to, I are fixed in this way at the values shown in table 1. The quantity b, is now determined from the experimental value of crcota’.We found b, = 0.79 fm, constant to within 2 % for the entire kinematical range shown in table 1.

4. Results In figs. l-3 we display results for protons incident on “C, 60Ni and 2oePb. The ratios 4”/0, and a$“/a, [eqs. (15), (18) and (19)3 of inclusive cross sections are displayed using two versions of a single channel theory and our two-channel approach. The ratios are given as functions of the number of removed (and unobserved) nucleons and r~~~~~‘/8”*~‘, i.e. the energy. Inspection shows that the Hiifner-Schafer-Schtirmann (HSS) prescription indeed accounts for some unitarity defect, in that it comes substantially closer to the full two-channel result than does a consistent one-channel calculation containing no reference to unitarity whatever. As expected, the HSS treatment comes closest to the two-channel result for near-elastic reactions (I a A). Somewhat paradoxically, however, the departure of e/a, from unity for fixed I decreases with increasing

Fig. I. Left hand : Ratio of inclusive cross section p + ’ rC + (A -I) + X computed with a single-channel model to that for a two-channel model. The former has heuristically applied unitary-defect corrections. The ratios are given as functions of R = u~~‘/u~~~~’(essentially the energy, cf. table I) and the number of removed particles I. The right hand graph shows the same without the heuristic correction.

484

D. I. JULIUS

AND

A. S. RINAT

2

468lOLl4l6l620

R Fig. 2. Same as fig. 1 but for 60Ni.

/a’“”e’(increasing energy). This is especially striking for light nuclei, as is seen from fig. 1. There, for R = 4, Gss/oA deviates from unity by < 35 % for all 1, which is an enormous improvement over the strict one-channel result of fig. 1.

ototal

5. Conclusion As noted in the introduction, the two-channel description of projectile-nucleon interactions limits our scope to particle-induced reactions; we cannot treat the 160 + ‘Be + “C+X reaction described by the authors of ref. 6). However, it is natural and appealing to equate the number 1of nucleons removed from the target nucleus in the particle-induced reaction with the number of nucleons abraded from the detected nucleus in the ion collision. Of course, the states describing the debris of the second collision partner are more complicated for the ion collision than for the particle-induced reaction, but in both cases the only property required for calculation is the completeness of the states. We may then conclude from our calculations that the heuristic treatment of unitarity by HSS in ion collisions will introduce errors of less than a factor two in the cross section, provided the detected

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Fig. 3. Same as fig. 1 but for “*Pb.

nucleus resulted from an abrasion of not more than v(A) nucleons, with v(A) = 5, 7, 7, 9, 10 for A = 12, 40, 60, 108, 208 respectively. A final remark is in order l3XWXn@thea~ofEketwoWrfiodeh It has already been stressed in the introduction that all inelastic effects in the particlenucleon collision have been cast into the form of a change of state of the projectile. This has two related implications. The first, is that the model presumably becomes less reliable with increasing 1, the number of knocked-out nucleons. The reason is that if a struck nucleon remains bound, its excitational degrees of freedom, which in our model have been “transferred” to the projectile, are completely suppressed. Secondly since the observed in&r&&y is asc&ed entire@ to the ~BR$&&, this particle is given too much excitational freedom. Furthermore, whatever the content of the projectile’s second state, it will not propagate coherently from one ,zat&ng to another, as we have assumed, except perhaps at extremely high energy 4. An obvious, if crude, compensation for this shortcoming would be to adopt a value of 6tOtd/p”l somewhat smaller than the physical one at a given energy. However,

486

D. I. JULIUS

AND A. S. RINAT

from figs. l-3 that our numerical results are quite insensitive to the value of this parameter over its entire range. we see

Appendix

We evaluate here the matrix products in eq. (13) in order to obtain the inclusive cross section (15). We first write (0, = (y A)) n,(b) = a,(b) I+ B&r,,

The products urul in eq. (13) have a similar form which can be exponentiated follows : MM(b)

as

= /dbl a+ v,(bb, = ew (_M) ~+~l@b,).

(A-2)

One easily checks that

l%l2 + lA12, .A= 9ln@f--v3, PI =

vI = 91 =

2 ReMY),

coth- l h/d.

(A.3)

The expression (A.2) for uIul is inserted into eq. (13), which is then expanded using the binomial theorem. The required (1, 1) matrix element is picked out after writing each term of the form exp (1a.J as cash (12)+ gX sinh (A). The result is (lI[l -u,u;]r[u,uf]“-‘Il)

i

= i m=O

(_ l)memfo+(d-Ofl cash [mg, + (A - 0gr]- (A.4)

0

The hyperbolic cosine is now written out in exponentials which allows the sum over m to be performed. The result involves exponentials oFthe form exp t-&+-g& Using the identity c&h-‘(x) = 3 In [(x + 1)/(x - l)] the quantities go and g, may be written as logariis leading to the simple rest&s exptf;&-gJ = ++& We note finally from (A.3) that CL, f vt = Ial+ j?J2. Our final formula (15) for the inchrsive cross section then follows immediately. Note added in procfi The abrasion process is fast compared with the ablation of nucleons responsible for the transition of the observed target fragment to its ground state. It would therefore be more appropriate to place the residual target nucleons

UNITARLTY EFFECTS

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in the ground state of the initial target nucleus, rather than the ground state of the observed fragment as we have done. We owe this observation to Hiifner. We have examined nume~~lly the effect of the above change on the ratios of cross sections displayed in figs. 1-3. The ratios are changed by less than 10 % for the number of removed nucleons 1 < Z,(A) with 1, = 7, 10, 16 for A = 12,60,208 respectively.

1) R. J. Glauber, int. Conf. on high-energy physics and nuclear structure, Rehovot, Israel (NorthHolland, Amsterdam, 1967); High energy physics and nuclear structure (Plenum Press, NY. 1970) 2) R. J. Glauber and V. Franco, Phys. Rev. 1% (3967) 1685; D. Harrington, Nucl. Phys. B59 (1973) 305; B. Schiirmann and W. E. Frahn, ibid. B62 (1973) 365 3) K. S. Kiilbig and B. Margohs, Nucl. Phys. B6 (1968) 85 4) P. f. Lindstrom et at., Report LBL-3650 (19X5), unpublished 5) J. Hnfner, K. Sehgfer and B. Schihmann, Phys. Rev. CI2 (19‘S) 1888 6) J. D. Bowman, W. J. Swiatecki and C. F. Tsang, Abrasion and ablation of heavy ions, unpublished UCRL report, July 1973 7) M. Abramovitz and I. !&gun, Handbook of mathematical functions 8) Particle data group, UCRL 20,000 NN (1970) NN, ND interactions 9) D. R. Warrington and D. I. Julius, Nucl. Phys. B88 (1975) 411