Generating function for nucleus-nucleus scattering amplitudes in Glauber theory

A new approach to deal with the scattering amplitudes in Glauber theory is proposed. It relies on the use of generating function, that has been explicitly found. The method is applied to the analytical calculation of the nucleus-nucleus elastic scattering amplitudes in the all interaction orders of the Glauber theory.


I. INTRODUCTION
The theory of nucleus-nucleus interaction is an important aspect of a long-standing multiple scattering problem. It has acquired a modern impetus from the large number of the currently available experimental data (see e.g. Refs. [1][2][3][4]). The theoretical calculations provide, in particular, a way to get information on scattering of both stable and unstable nuclei at the comparatively high energies of more than several hundreds of MeV per nucleon. The calculations are standardly carried out in the Glauber approach [5,6]. It has proven to be highly efficient for the hadron-nucleus collision, supplying rather simple analytical expressions for the scattering amplitudes. The case of the nucleus-nucleus scattering is much more involved. Additional simplifying approximations are commonly used to obtain an analytical expression such as the optical model or the rigid target model (see e.g.Refs. [7][8][9]). Apart from these models there are only numerical calculations based on the Monte-Carlo method or on its modifications [10][11][12][13].
In the present paper we propose a novel approach. Assuming the range of nucleon-nucleon interaction to be small compared to the typical nucleus size we have derived the analytical expression for the generating function giving the Glauber amplitudes for nucleus-nucleus scattering. Its relatively simple form allows one to reach the same accuracy as that provided with the numerical Monte-Carlo calculations without being as time-consuming as they are. * shabelsk@thd.pnpi.spb.ru † shuvaev@thd.pnpi.spb.ru

II. SCATTERING AMPLITUDES IN THE GLAUBER THEORY
To begin, we briefly outline the basics of the Glauber theory. The amplitude of the elastic scattering of the incident nucleus A on the fixed target nucleus B reads [14,15] where q is the transferred momentum and k is the mean momentum carryed by a nucleon in nucleus A. The two-dimensional impact momentum b lies in the transverse plane to the vector k. The main assumption underlying the Glauber theory is that the radius of the nucleon-nucleon interaction is much smaller than the typical nucleus size. Then assuming the phase shifts of the nuclear scattering to be the sum of those for each nucleon-nucleon scattering, χ N N (b), the function S AB (b) takes the form with where f el N N (q) is the nucleon-nucleon scattering amplitude. The product in Eq. (2) comprises all pairwise interactions between the nucleons from the projectile and target nuclei A and B, with symbol A, B | · · · | A, B standing for an average over the nucleons' positions x i and y j in the transverse plane. Each pair {i, j} enters the product only once, meaning that each nucleon from the projectile nucleus can scatter on each nucleon on the target once but no more than once.

The total interaction cross section is
while the integrated elastic cross section is The difference between these two values determines the total inelastic, or reaction, cross section,

III. GENERATING FUNCTION
A main obstacle to dealing with the Glauber amplitude (2) is its complicated combinatorial structure. To treat it analytically we firstly rewrite it more explicitly through nucleon distributions in the colliding nuclei, Here the nucleon densities in the transverse plane, ρ ⊥ A,B , are determined through threedimensional ones integrated over longitudinal coordinates, For later convenience an extra parameter g counting the number of interactions is introduced in Eq. (7), really g = −1. We also assume in what follows that the three-dimensional nuclear densities are reduced to the product of one-nucleon densities, This assumption means the nucleon-nucleon correlations are neglected.
The next step is to represent Eq. (7) as a functional integral. To this end let us consider the identity where C 0 is the normalization constant and the functional integral can be thought of as an infinite product of two dimensional integrals over the auxiliary fields Φ(x) at each space point x. The inverse of the propagator, ∆ −1 (x − y), is understood in a functional sense, . If this function is chosen to obey the equation the right-hand side of Eq. (9) recovers the product in Eq. (7). The function ∆(x − y) plays a role similar to that of the Mayer propagator (function) in statistical mechanics, the analogy between Glauber theory and statistical mechanics has been remarked on earlier (see, e.g., Ref. [16]). Then we get This form suggests that it is natural to introduce the generating function, The generating function (12) is the focus of the present paper. Being analytically evaluated it comprises all interaction orders of the Glauber theory for nucleus-nucleus collision.

IV. EXPLICIT EVALUATION OF THE GENERATING FUNCTION
As it has been mentioned above the Glauber theory essentially relies on the short-distance nature of the nucleon-nucleon interaction. The same property, the small interaction range, makes the complex functional integral (12) feasible. The standard parametrization of the elastic nucleon-nucleon scattering amplitude is where σ tot N N is the total nucleon-nucleon cross section. It gives according to Eq. (3) the value a = √ 2πβ being of the order of the interaction radius. Assuming a to be small at the nuclear scale the nucleon-nucleon amplitude can be treated as a point-like function, If ∆(x − y) is point-like the integrals over the Φ(x) fields in Eq. (12) are independent for different coordinate values. This turns the functional integral into the infinite product of finite-dimension integrals, which can be separately evaluated for each x.
To do this accurately we replace the continuous integrals in the exponent by discretized sums. The discrete version of the identity (9) reads where δ x i ,y j is Kronecker symbol for the discrete nucleons' coordinates, δ the right hand side of the identity (17) yields whereas Eq. (10) translates into The generating function becomes the product of independent integrals at the points x i , or, after evaluating Φ(x) integrals in (19), The densities are slowly varying at the size a, which allows to replace the sum over x i with the integral, with u-and v-independent constant C being irrelevant in Eq. (13) and The sums over M and N can always be truncated up to A and B because the higher terms obviously do not contribute to the derivatives in Eq. (13). Put differently, the number of contributions to the generating function does not exceed the number of various brackets in the initial product (2).

V. RELATION TO THE KNOWN APPROXIMATIONS
The formulas (20) and (21) are the net result for the generating function. To elaborate it further we expand W y (u, v) into the series built of the densities overlaps, Since and the amplitude reads For A, B ≫ 1 one may assume that k, j ≪ A, B and A!/ Really the functions t m,n (b) decrease as the indices m and n grow. Keeping only the lowest, m = n = 1, we arrive at the well-known optical approximation [7] F (A, The optical approximation is equivalent to the requirement that each nucleon from one nucleus interacts with another nucleus no more than once. Another known approximation easily reproduced here is the rigid target (or projectile) approximation [8,9]. It allows any nucleon from the projectile to interact with several nucleons from the target, whereas any target nucleon can interact no more than once. Though it seems to be rather natural when the atomic weight of the projectile is significantly smaller than that of the target nucleus, this approximation works fairly good even for the equal atomic weights [15]. It requires one density, say, ρ ⊥ A (x), to be kept in the formula (20) only in the linear order, permitting at the same time any powers of ρ ⊥ B (x). It gives yielding the generating function which produce for B ≫ 1 VI. RESULTS FOR 12 C -12 C SCATTERING Below we present the results obtained with the full generating function (20) for the 12 C -12 C scattering in the energy interval 800 -1000 MeV per projectile nucleon, where the experimental data exist [17]. The total cross section σ tot N N = 43 mb has been taken from averaging over pp and pn values, and the slope value has been chosen to be β = 0.2 fm 2 .
The nucleon density is parametrized by harmonic oscillator distribution well suited for light nuclei with the atomic weight A ≤ 20, with ρ 0 being the normalization, and the factor λ being adjusted to match the nuclear mean square radius, R rms = r 2 A , where r 2 A = d 3 r r 2 ρ A (r). Upon evaluating W y (u, v) through all t mn (b) functions (22) for m, n ≤ A = 12 in the parametrization (27) with R rms = 2.49 fm fitted for this parametrization in Ref. [13] from Monte-Carlo simulation of a 12 C -12 C collision, we have calculated the reaction cross section (6) and the total cross section (4). Table I compares their values obtained in the optical approximation (25), in the rigid target approximation (26) and with the full generating function for two cases, first assuming A ≫ 1 and using the approximate formula (24) and second by exact differentiating the generating function. The last two numbers in the upper row of the table are in reasonable agreement with the experimental value 853 ± 6 mb [17]. One should bear in mind that the experimentally measured value actually refers to the so-called interaction cross section rather than to the reaction one. The difference between them can be at the several percents level [18]. At the same time the obtained values are close to those of the Monte Carlo calculations with the same parameters and the density parametrization [13].
A word of caution with respect to the formula (24)  Representing the amplitude (7) as a power series in the parameter g enables one to pick up the individual contribution of n-fold interaction as a coefficient in front of the g n term.
They are large when taken separately, but due to the opposite signs they almost cancel each other returning a final sum much smaller than any of them. Thus the treatment of the amplitude in terms of the number of interactions seems to be quite unreasonable.

VII. HALO NUCLEI SCATTERING
An interesting topic to apply our method to is the scattering of halo nuclei. The distinguishing feature of these nuclei is their large size exceeding that of the stable nuclei. They are assumed to be a composite systems of a core and a halo [19,20], the density distribution being the sum of the two components [21][22][23][24], The first and the second terms stand here for the core and for the halo densities, N c is the number of the nucleons in the core, and N v is the number of the valence neutrons in the halo [25]. Both the densities are taken in the Gaussian form, and the core density reads where R c is the core mean square radius, while the second part, ρ v , usually admits three different parametrizations depending on the shell state the halo neutrons are supposed to occupy [25]: where R v is the halo mean square radius. There exists also a "non halo" distribution that neglects the halo and uses the density (27) fitted to the experimental matter radius [23].
In Table II we present the reaction cross sections for 11 Li -12 C, 11 Be -12 C and 14 Be -12 C scattering at the energy 790 MeV per nucleon obtained through exactly differentiated full generating function. The calculations have been carried out with the density (28) (normalized to unity) for all three parameterizations with [25] R c = 2.50 fm, R v = 5.04 fm, N c = 9 , N v = 2 for 11 Li, R c = 2.30 fm, R v = 5.39 fm, N c = 10 , N v = 1 for 11 Be, R c = 2.59 fm, R v = 5.45 fm, N c = 12 , N v = 2 for 14 Be. Besides we add the results for the "non halo" distributions with R rms = 3.12 fm for 11 Li, R rms = 2.73 fm for 11 Be and R rms = 3.16 fm for 14 Be [17]. The parametrization of 12 C is the same as in Table I.   Table II. The reaction cross sections in mb for 11 Li -12 C, 11 Be -12 C and 14 Be -12 C collisions at the energy 790 MeV per nucleon. The first three columns are for the three density parametrizations (29), and the fourth column is for the "non halo" distributions.
The experimental points in the fifth column are taken from Ref. [17].  The results for 11 Li and 14 Be are in a quite good agreement with the experimental data especially regarding the relatively large error bars. The evaluated cross sections for the 11 Be beam are systematically smaller although the deviation is not large. One might doubt whether the parameters are determined for this case with the proper accuracy. On the other hand the reason for the discrepancy could be in the nucleons' correlations.
It is worth pointing out that it is the reaction cross sections that have been calculated here, whereas the experimentally measured value is the interaction cross section. It is smaller than the reaction one, but their difference does not exceed 1 -2% [18].

VIII. CONCLUSION
In this article we set up a novel approach to deal with Glauber amplitudes for nucleusnucleus scattering at energies higher than several hundreds of MeV per nucleon. It is based on the closed expression obtained for the generating function. The main advantage it has is in a relatively simple analytical form that allows one to carry out calculations avoiding the complexities encountered in the Monte Carlo technique. As an example we apply our method to 12 C -12 C scattering at the energy 950 MeV per nucleon for which there exist the experimental data [17]. We have calculated the reaction and the total cross sections for the mean square nuclear radius R rms = 2.49 fm, the value taken from the Monte Carlo analysis in the harmonic oscillator parametrization in Ref. [13]. Our results are in good agreement with those obtained in that paper. As another example we have calculated the cross section of several halo nuclei scattering on a 12 C target at the energy 790 Mev per nucleon.
The proposed generating function (20) is appropriate for any pairs of colliding nuclei regardless their atomic weight. Apart from the above-considered integrated cross sections it can provide a consistent evaluation of the differential elastic cross section (1) as well. In this case, however, one has to account for the Coulomb corrections at small scattering angles for heavy nuclei