Baryon-baryon scattering in the Gross-Neveu model: The large $N$ solution

The scattering of two Dashen-Hasslacher-Neveu baryons in the large $N$ Gross-Neveu model is solved exactly using the relativistic, time-dependent Hartree-Fock approach. Unlike the special case of kink-antikink scattering, the scattering of Dashen-Hasslacher-Neveu baryons is sensitive to the backreaction of the fermions bound inside the baryons. Correspondingly the solution is much more complicated than the kink-antikink scattering solutions, which can be expressed in terms of sinh-Gordon solitons. Nevertheless, we present a simple ansatz form that gives closed analytic expressions for both the space–time-dependent mean fields and the Dirac spinors for all continuum and bound states. The solution can also be applied to the scattering of polarons and solitons in conducting polymers.

Figure 1

A sketch of the anatomy of a DHN baryon, here shown for parameter $y=0.999999$. The scalar field $S_{\mathrm{DHN}}(x)$ is shown as the solid blue line going asymptotically to 1 (in units of the dynamically generated fermion mass in the GN model). The fermion density, $\rho ={\psi }^{†}\psi$, for the bound valence mode is shown as a black dotted line; note that the density is localized at the kink and antikink of the baryon scalar field. The pseudoscalar condensate, $\pi (x)=\overline{\psi }i{\gamma }_5\psi$, for the valence state is plotted as a red dashed curve, while the scalar condensate, $\sigma (x)=\overline{\psi }\psi$, for the valence state is plotted as a purple dash-dotted curve ($\sigma$ has been multiplied by a factor of 100, for visualization purposes). The scalar condensate is nonzero inside the DHN baryon, while the pseudoscalar condensate is localized at the edges, on the kink and antikink, like the density $\rho$ but with a change of sign.

Figure 2

The upper figure shows the scattering of a CCGZ-kink baryon and a CCGZ-antikink baryon, as was already discussed in 15, 17. In contrast, the lower figure shows the problem treated in this current paper: the scattering of two DHN baryons. The solid (blue) lines show the scalar potential, while the dotted (black) lines show the density of the bound valence fermions. Note that each asymptotic DHN baryon can have different internal structure (i.e., a different $y$ parameter), which affects the baryon size and shape as well as the internal fermion densities; this will be important for understanding the scattering processes. By varying the structural parameters, and also the relative velocity parameter, we can probe various interesting physical regimes, spanning continuously from relativistic to nonrelativistic.

Figure 3

Scattering of a small ($y_1=0.8$) and a large ($1-y_2={10}^{-7}$) DHN baryon with velocities $\pm 0.4$. The self-consistent scalar potential $S$ is shown for a range of ($x$, $t$) values in the vicinity of the collision.

Figure 4

Same as Fig. 3, but total fermion density $\rho$ is shown. In the large baryon, the density has two peaks near the kink and antikink boundaries of the baryon. A negative time delay occurs during the collision.

Figure 5

Scattering of a small DHN baryon ($y=0.8$) on a CCGZ kink ($y=1$), at velocities $\pm 0.2$. Notice the chiral reflection which the small baryon undergoes during the collision.