1. The theory of nucleus-nucleus scattering has recently acquired a modern impetus from the large number of the currently available experimental data (see e.g. [1–4]). The theoretical predictions for the scattering at the comparatively high energies more than several hundreds MeV per nucleon are standardly provided by the Glauber theory [5, 6]. It proves 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. [8–10]). Apart from these models there are only numerical calculations based on Monte-Carlo method or on its modifications [11, 12].

In the present paper we propose a novel approach based on the analytically derived generating function the Glauber amplitudes for nucleus-nucleus scattering. It allows to reach the same accuracy as the numerical Monte-Carlo calculations but in much more simple way.

2. The amplitude of the elastic scattering of the incident nucleus \(A\) on the fixed target nucleus \(B\) in the Glauber theory reads [14, 15]

$$F_{{AB}}^{{el}}(q) = \frac{{ik}}{{2\pi }}\int {{d}^{{{\kern 1pt} 2}}}b{\kern 1pt} {{e}^{{iqb}}}[1 - {{S}_{{AB}}}(b)],$$
(1)

where \(q\) is the transferred momentum and \(k\) is the mean momentum carrying by a nucleon in nucleus \(A\). The two-dimensional impact momentum \(b\) lies in the transverse plain to the vector \(k\). The main assumption underlying the Glauber theory is that the radius of 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, \({{\chi }_{{NN}}}(b)\), the function \({{S}_{{AB}}}(b)\) takes the form

$$\begin{gathered} {{S}_{{AB}}}(b) = \int \prod\limits_{i = 1}^A {{d}^{{{\kern 1pt} 2}}}{{x}_{i}}\int \prod\limits_{j = 1}^B {{d}^{2}}{{y}_{j}}\rho _{A}^{ \bot }({{x}_{1}} - b, \ldots ,{{x}_{A}} - b){\kern 1pt} \\ \times \,\,\rho _{B}^{ \bot }({{y}_{1}}, \ldots ,{\kern 1pt} {{y}_{B}})\left\{ {\prod\limits_{i{\kern 1pt} j} \text{[}1 - {{\Gamma }_{{NN}}}({{x}_{i}} - {{y}_{j}})]} \right\}, \\ \end{gathered} $$
(2)

with

$${{\Gamma }_{{NN}}}(b) = 1 - {{e}^{{i{{\chi }_{{NN}}}(b)}}} = \frac{1}{{2\pi ik}}\int {{d}^{2}}q{\kern 1pt} {{e}^{{ - iqb}}}f_{{NN}}^{{{\text{el}}}}(q),$$
(3)

where \(f_{{NN}}^{{{\text{el}}}}(q)\) is the nucleon-nucleon scattering amplitude. The product in (2) comprises all pairwise interactions between the nucleons from the projectile and target nuclei \(A\) and \(B\), \({{x}_{i}}\) and \({{y}_{j}}\) being the nucleons’ positions in the transverse plain. The nucleon densities in the transverse plain, \(\rho _{{A,B}}^{ \bot }\), are determined through three-dimensional ones integrated over longitudinal coordinates,

$$\begin{gathered} \rho _{N}^{ \bot }({{x}_{1}}, \ldots ,{{x}_{N}}) = \int \prod\limits_{i = 1}^N d{{z}_{i}}{\kern 1pt} {{\rho }_{N}}({{z}_{1}},{{x}_{1}}, \ldots ,{{z}_{N}},{{x}_{N}}), \\ \int \prod\limits_{i = 1}^N {{d}^{{{\kern 1pt} 3}}}{{r}_{i}}{\kern 1pt} {{\rho }_{N}}({{r}_{1}}, \ldots ,{{r}_{N}}) = 1. \\ \end{gathered} $$

In what follows the three-dimensional nuclear densities are assumed to be the product of one-nucleon densities,

$${{\rho }_{N}}({{r}_{1}}, \ldots ,{{r}_{N}}){\kern 1pt} \, = \,\,{\kern 1pt} \prod\limits_{i = 1}^N {{\rho }_{N}}({{r}_{i}}), \int {{d}^{3}}r{{\rho }_{N}}(r) = 1,$$

and consequently

$$\rho _{N}^{ \bot }({{x}_{1}}, \ldots ,{{x}_{N}}) = \prod\limits_{i = 1}^N \rho _{N}^{ \bot }({{x}_{i}}),\,\,\,\,\int {{d}^{{{\kern 1pt} 2}}}x\rho _{N}^{ \bot }(x) = 1.$$
(4)

Each pair \(\{ i,j\} \) enters the product (2) only once, meaning that each nucleon from the projectile nucleus can scatter on each nucleon on the target one but no more than once.

The total interaction cross section is

$$\sigma _{{AB}}^{{{\text{tot}}}} = \frac{{4\pi }}{k}\operatorname{Im} F_{{AB}}^{{{\text{el}}}}(q = 0) = \int {{d}^{{{\kern 1pt} 2}}}b[1 - {{S}_{{AB}}}(b)],$$
(5)

while the integrated elastic cross section evaluates to

$$\sigma _{{AB}}^{{{\text{el}}}} = \int {{d}^{{{\kern 1pt} 2}}}b{\kern 1pt} {{[1 - {{S}_{{AB}}}(b)]}^{2}}.$$
(6)

The difference of these two values determines the total inelastic, or reaction, cross section,

$$\sigma _{{AB}}^{r} = \sigma _{{AB}}^{{{\text{tot}}}} - \sigma _{{AB}}^{{{\text{el}}}} = \int {{d}^{{{\kern 1pt} 2}}}b{\kern 1pt} [1 - {{\left| {{{S}_{{AB}}}(b)} \right|}^{2}}].$$
(7)

3. A main obstacle to deal with the Glauber amplitude (2) is its complicated combinatorial structure. To treat it analytically we present (2) as a functional integral. Let us consider the identity

$$\begin{gathered} {{C}_{0}}\int D\Phi D\Phi {\text{*}}\exp \{ - \int {{d}^{{{\kern 1pt} 2}}}x{{d}^{{{\kern 1pt} 2}}}y\Phi (x){{\Delta }^{{ - 1}}}(x - y)\Phi {\text{*}}(y) \\ + \,\,\sum\limits_i \Phi ({{x}_{i}}) + \sum\limits_j \Phi {\text{*}}({{y}_{j}})\} \\ = \exp \left\{ {\sum\limits_{i,j} \Delta ({{x}_{i}} - {{y}_{j}})} \right\} = \prod\limits_{i,j} {{e}^{{{\kern 1pt} \Delta ({{x}_{i}} - {{y}_{j}})}}}, \\ \end{gathered} $$
(8)

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 variables \(\Phi (x)\) at each space point \(x\), the inverse of the propagator, \({{\Delta }^{{ - 1}}}(x - y)\), being understood as \(\int {{d}^{{{\kern 1pt} 2}}}z{{\Delta }^{{ - 1}}}(x - z)\Delta (z - y) = {{\delta }^{{(2)}}}(x - y)\). If this function is chosen to obey the equation

$${{e}^{{{\kern 1pt} \Delta (x - y)}}}{\kern 1pt} {\kern 1pt} 1 = - {{\Gamma }_{{NN}}}(x - y),$$
(9)

the right hand side of (8) recovers the product in (2). The function \(\Delta (x - y)\) plays a role similar to that of Mayer propagator (function) in statistical mechanics, the analogy between Glauber theory and statistical mechanics has been remarked earlier (see, e.g. [16]). Then we get

$$\begin{gathered} {{S}_{{AB}}}(b) = {{C}_{0}}\int D\Phi D\Phi {\text{*}} \\ \times \,\,\exp \left\{ { - \int {{d}^{{{\kern 1pt} 2}}}x{{d}^{{{\kern 1pt} 2}}}y{\kern 1pt} \Phi (x){{\Delta }^{{ - 1}}}(x - y)\Phi {\text{*}}(y)} \right\} \\ \times \,\,{{\left[ {\int {{d}^{{{\kern 1pt} 2}}}x{\kern 1pt} \rho _{A}^{ \bot }(x - b){{e}^{{\Phi (x)}}}} \right]}^{A}}{{\left[ {\int {{d}^{{{\kern 1pt} 2}}}y{\kern 1pt} \rho _{B}^{ \bot }(y){{e}^{{\Phi *(y)}}}} \right]}^{B}}. \\ \end{gathered} $$
(10)

This form suggests that it is natural to introduce the generating function,

$$\begin{gathered} Z(u,v) = \int D\Phi D\Phi {\text{*}}\\ \times \,\,\exp \left\{ { - \int {{d}^{2}}x{{d}^{2}}y\Phi (x){{\Delta }^{{ - 1}}}(x - y)\Phi {\text{*}}(y)} \right. \\ {\kern 1pt} \left. { + \,\,u\int {{d}^{{{\kern 1pt} 2}}}x{{\rho }_{ \bot }}(x - b){{e}^{{\Phi (x)}}} + v\int {{d}^{{{\kern 1pt} 2}}}x{\kern 1pt} {{\rho }_{ \bot }}(x){{e}^{{\Phi{\text{*}}(x)}}}} \right\}, \\ \end{gathered} $$
(11)

so that

$${{S}_{{AB}}}(b) = \frac{1}{{Z(0,0)}}\frac{{{{\partial }^{A}}}}{{\partial {{u}^{A}}}}\frac{{{{\partial }^{B}}}}{{\partial {{v}^{B}}}}{\kern 1pt} {{\left. {Z(u,v)} \right|}_{{u = v = 0}}}.$$
(12)

The small interaction range, the property Glauber theory is based on, makes the complex functional integral (11) feasible. The standard parametrization of the elastic nucleon-nucleon scattering amplitude,

$$f_{{NN}}^{{{\text{el}}}}(q) = k\frac{{\sigma _{{NN}}^{{{\text{tot}}}}}}{{4\pi }}{{e}^{{ - \frac{1}{2}\beta {{q}^{2}}}}},$$
(13)

where \(\sigma _{{NN}}^{{{\text{tot}}}}\) is the total nucleon-nucleon cross section. It gives according to (3)

$${{\Gamma }_{{NN}}}(x) = \frac{{\sigma _{{NN}}^{{{\text{tot}}}}}}{{4\pi \beta }}{{e}^{{ - \frac{{{{x}^{2}}}}{{2\beta }}}}},$$
(14)

the value \(a = \sqrt {2\pi \beta } \) 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,

$${{\Gamma }_{{NN}}}(x) \simeq \frac{1}{2}\sigma _{{NN}}^{{{\text{tot}}}}{{\delta }^{{(2)}}}(x).$$
(15)

If \(\Delta (x - y)\) is point-like the integrals over \(\Phi (x)\) fields in (11) are independent for different coordinate values. It turns the functional integral into the infinite product of finite dimension integrals, that can be separately evaluated for each \(x\). It results into the generating function (see [7] for details)

$$\begin{gathered} Z(u,v) = C{{e}^{{{{W}_{y}}(u,v)}}}, \\ {{W}_{y}}(u,v) = \frac{1}{{{{a}^{2}}}}\int {{d}^{2}}x \\ \times \,\,\ln \left( {\sum\limits_{M \leqslant A,N \leqslant B} \frac{{z_{y}^{{MN}}}}{{M!N!}}{{{[{{a}^{2}}u{{\rho }_{A}}(x - b)]}}^{M}}{{{[{{a}^{2}}v{{\rho }_{B}}(x)]}}^{N}}} \right), \\ \end{gathered} $$
(16)

with \(u\)- and \(v\)-independent constant \(C\) irrelevant in (12) and

$${{z}_{y}} = 1 - \frac{1}{2}\frac{{\sigma _{{NN}}^{{{\text{tot}}}}}}{{{{a}^{2}}}}.$$

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 (12). Put differently, the number of contributions to the generating function does not exceed the number of various brackets in the initial product (2).

5. To elaborate Eq. (16), further we expand \({{W}_{y}}(u,v)\) into the series built of the densities overlaps,

$${{t}_{{m,n}}}(b) = \frac{1}{{{{a}^{2}}}}\int {{d}^{{{\kern 1pt} 2}}}x{\kern 1pt} {{[{{a}^{2}}\rho _{A}^{ \bot }(x - b)]}^{m}}{\kern 1pt} {{[{{a}^{2}}\rho _{B}^{ \bot }(x)]}^{n}}.$$
(17)

Since \({{t}_{{0,1}}}(b) = {{t}_{{1,0}}}(b) = 1\) we have \({{W}_{y}}(u,v) = \) \(u + v + F(u,v)\) and the amplitude reads

$$\begin{gathered} {{S}_{{AB}}}(b) = \sum\limits_{k,j \leqslant A,B} \frac{{A!B!}}{{(A - k)!(B - j)!}} \\ \times \,\,\frac{1}{{k!}}\frac{{{{\partial }^{{{\kern 1pt} k}}}}}{{\partial {{u}^{k}}}}\frac{1}{{j!}}\frac{{{{\partial }^{j}}}}{{\partial {{v}^{j}}}}{{\left. {{{e}^{{{\kern 1pt} F(u,v)}}}} \right|}_{{u = v = 0}}}. \\ \end{gathered} $$
(18)

For \(A,B \gg 1\) one may assume that \(k,j \ll A,B\) and \({{A!} \mathord{\left/ {\vphantom {{A!} {(A - k)!}}} \right. \kern-0em} {(A - k)!}}{{B!} \mathord{\left/ {\vphantom {{B!} {(B - j)!}}} \right. \kern-0em} {(B - j)!}} \approx {{A}^{k}}{{B}^{j}}\), that gives

$${{S}_{{AB}}}(b) \approx {{e}^{{F(A,B)}}}.$$
(19)

Really the functions \({{t}_{{m,n}}}(b)\) decrease as the indices \(m\), \(n\) grow. Keeping only the lowest, \(m = n = 1\), we arrive at the well known optical approximation [8]

$$\begin{gathered} F(A,B) = - \frac{1}{2}{{\sigma _{{NN}}^{{{\text{tot}}}}} \mathord{\left/ {\vphantom {{\sigma _{{NN}}^{{{\text{tot}}}}} {{{a}^{2}}}}} \right. \kern-0em} {{{a}^{2}}}}{\kern 1pt} {{T}_{{AB}}}(b), \\ {{T}_{{AB}}}(b) = AB{{t}_{{1,1}}}(b). \\ \end{gathered} $$
(20)

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 [9, 10]. 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 fairly good works even for the equal atomic weights [15]. It requires one density, say, \(\rho _{A}^{ \bot }(x)\), to be kept in Eq. (16) only in the linear order, permitting at the same time any powers of \(\rho _{B}^{ \bot }(x)\). It gives

$$\begin{gathered} {{W}_{y}}(u,v) = \frac{1}{{{{a}^{2}}}}\int {{d}^{2}}x{\kern 1pt} \ln \left( {\sum\limits_N \frac{1}{{N!}}{{{[{{a}^{2}}v\rho _{B}^{ \bot }(x)]}}^{N}}} \right. \\ + \,\,{\kern 1pt} [{{a}^{2}}\rho _{A}^{ \bot }(x - b)]\left. {\sum\limits_N \frac{{z_{y}^{N}}}{{N!}}{{{[{{a}^{2}}v\rho _{B}^{ \bot }(x)]}}^{N}}} \right) \\ = \,\,{\kern 1pt} v + u\int {{d}^{2}}x{\kern 1pt} \rho _{A}^{ \bot }(x - b){{e}^{{{{a}^{2}}{\kern 1pt} v{\kern 1pt} ({{z}_{y}} - 1)\rho _{B}^{ \bot }(x)}}}, \\ \end{gathered} $$

yielding the generating function

$$Z(u,v) = {{e}^{{v + u{{T}_{{rg}}}(v,b)}}}, \,\,\,\,{{T}_{{rg}}}(v,b) = \sum\limits_{n = 0}^\infty \frac{1}{{n!}}{\kern 1pt} {{t}_{{1,n}}}(b){\kern 1pt} {{v}^{n}},$$

that produce for \(B \gg 1\)

$$\begin{gathered} {{S}_{{AB}}}(b) = {{[{{T}_{{rg}}}(b)]}^{A}}, \\ {{T}_{{rg}}}(b) = \int {{d}^{{{\kern 1pt} 2}}}x{\kern 1pt} \rho _{A}^{ \bot }(x - b){\kern 1pt} {{e}^{{ - \frac{1}{2}\sigma _{{NN}}^{{{\text{tot}}}}\rho _{B}^{ \bot }(x)}}}. \\ \end{gathered} $$
(21)

6. Below we present the results obtained with the full generating function (16) for the \(^{{12}}{\text{C}}\)–\(^{{12}}{\text{C}}\) scattering in the energy interval 800–1000 MeV per projectile nucleon, where the experimental data exist [17]. The total cross section \(\sigma _{{NN}}^{{{\text{tot}}}}C\) mb has been taken from averaging over \(pp\) and \(pn\) values, the slope value has been chosen to be \(\beta = 0.2\) fm\(^{2}\). The nucleon density is parameterized by harmonic oscillator distribution well suited for light nuclei with the atomic weight \(A \leqslant 20\),

$${{\rho }_{A}}(r) = {{\rho }_{0}}\left[ {1 + \frac{1}{6}(A - 4)\frac{{{{r}^{2}}}}{{{{\lambda }^{2}}}}} \right]{{e}^{{ - \frac{{{{r}^{2}}}}{{{{\lambda }^{2}}}}}}},$$
(22)

\({{\rho }_{0}}\) being the normalization, and the factor \(\lambda \) adjusted to match the nuclear mean square radius, \({{R}_{{{\text{rms}}}}} = \sqrt {r_{A}^{2}} \), \(r_{A}^{2} = \int {{d}^{3}}r{\kern 1pt} {{r}^{2}}{{\rho }_{A}}(r)\).

Upon evaluating \({{W}_{y}}(u,v)\) through all \({{t}_{{mn}}}(b)\) functions (17) for \(m,n \leqslant A = 12\) in the parametrization (22) with \({{R}_{{{\text{rms}}}}} = 2.49\) fm fitted for this parametrization in [12] from Monte-Carlo simulation of \(^{{12}}{\text{C}}\)–\(^{{12}}{\text{C}}\) collision, we have calculated the reaction cross section (7) and the total cross section (5). The table compares their values obtained in the optical approximation (20), in the rigid target approximation (21) and with the full generating function for two cases, first assuming \(A \gg 1\) and using approximate formula (19) 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 \(853 \pm 6\) 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 [12].

7. Concluding, the proposed generating function (16) is appropriate for any pairs of colliding nucleus 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.

Taking the nucleon density as the product of single particle ones (4) we thereby neglect the nucleon-nucleon correlations. The particular correlations can be in principle accounted for in our approach, provided an appropriate wavefunction is known.

Table 1.   The reaction and the total cross sections of   \(^{{12}}{\text{C}}\)–\(^{{12}}{\text{C}}\) collision at the energy 950 MeV per nucleon and \({{R}_{{{\text{rms}}}}} = 2.49\) fm. The first two columns present the results of the optical and rigid target approximations, the second two columns are for the results obtained with the full generating function, assuming \(A \gg 1\) (third column) and exactly differentiating it (fourth column)
Full size table