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Theory of inclusive breakup cross section for Borromean nuclei within a four-body spectator model

We develop a model to treat the inclusive non-elastic break up reactions involving weakly bound three-cluster nuclei. Borromean, two-nucleon, halo nuclei are candidates of unstable three-fragments projectiles. The model is based on the theory of inclusive breakup reactions commonly employed in the treatment of incomplete fusion and surrogate method. The theory was developed in the 80's by Ichimura, Autern and Vincent (IAV) [Phys. Rev. C 32, 431 (1985)] \cite{IAV1985}, Udagawa and Tamura (UT)[Phys. Rev. C 24, 1348 (1981)], \cite{UT1981} and Hussein and McVoy (HM)[Nucl. Phys. A 445, 124 (1985)], \cite{HM1985}. We extend these three-body theories to derive an expression for the fragment yield in the reaction $A\,(a,b)\,X$, where the projectile is $a = x_1 + x_2 + b$. The inclusive breakup cross section is found to be the sum of a generalized four-body form of the elastic breakup cross section plus the inclusive non-elastic breakup cross section which involves the "reaction" cross section of the participant fragments, $x_1$ and $x_2$. The final result is similar to the three-body case reviewed in Austern, et al. (Phys. Rep. \textbf{154}, 125 (1987)), \cite{Austern1987}, but with important genuine four-body effects added, both in the elastic breakup cross section, which now contains the full correlations between the participant fragments, and in the inclusive non- elastic breakup. These developments should encourage experimentalists to seek more information about the $x_{1} + x_{2}$ system in the elastic breakup cross section, and to theorists to further develop and extend the surrogate method, based on the inclusive non-elastic breakup part of the $b$ spectrum.