Solution of the relativistic Schrödinger equation for the ${\delta }^{\prime }$-Function potential in one dimension using cutoff regularization

We study the relativistic version of the Schrödinger equation for a point particle in one dimension with the potential of the first derivative of the delta function. The momentum cutoff regularization is used to study the bound state and scattering states. The initial calculations show that the reciprocal of the bare coupling constant is ultraviolet divergent, and the resultant expression cannot be renormalized in the usual sense, where the divergent terms can just be omitted. Therefore, a general procedure has been developed to derive different physical properties of the system. The procedure is used first in the nonrelativistic case for the purpose of clarification and comparisons. For the relativistic case, the results show that this system behaves exactly like the delta function potential, which means that this system also shares features with quantum filed theories, like being asymptotically free. In addition, in the massless limit, it undergoes dimensional transmutation, and it possesses an infrared conformal fixed point. The comparison of the solution with the relativistic delta function potential solution shows evidence of universality.

Figure 1

The integration contours for obtaining the wave function of the bound state. In the nonrelativistic case, there is a pole inside the contour at $i\sqrt{-2m\mathrm{\Delta }{E}_B}$ but no branch cut (top panel). For relativistic case, there is a branch cut along the positive imaginary axis, starting at $p=im$, and there is a pole at $p=i\sqrt{m^2-E_B^2}$ (bottom panel).

Figure 2

Bound state wave function in coordinate space for the nonrelativistic case. The value of the binding energy is $\mathrm{\Delta }{E}_B=-0.01\mathrm{m}$, and ${\lambda }_1=|{\lambda }_1|$ with different values of $\mathrm{\Lambda }=2\mathrm{m}$, 5 m, 20 m, and $\infty$. The wave function turns to an even function with the increase of $\mathrm{\Lambda }$ (top four figures). For ${\lambda }_1=-|{\lambda }_1|$, the curves are different than the analogs ones for the case ${\lambda }_1=|{\lambda }_1|$; however, the difference diminishes with the increase of $\mathrm{\Lambda }$, and the wave function turns to an even function (bottom four figures).

Figure 3

The integration contours for obtaining the wave function of scattering states. In the nonrelativistic case, there are two poles on the reals axis at $\pm \sqrt{2m\mathrm{\Delta }E}$ but no branch cut (top panel). For the relativistic case, there is a branch cut along the positive imaginary axis, starting at $p=im$, and there are two poles on the real axis at $p=\pm \sqrt{E^2-m^2}$ (bottom panel).

Figure 4

Bound state wave function in coordinate space for the relativistic case with $E_B=m/2$ and different values of $\mathrm{\Lambda }=2\mathrm{m}$, 5 m, 20 m, and $\infty$. The wave function turns to an even function when $\mathrm{\Lambda }\to \infty$ (top four figures). For ${\lambda }_1=-|{\lambda }_1|$, the curves are different than the analogous ones for the case ${\lambda }_1=|{\lambda }_1|$; however, the difference diminishes with the increase of $\mathrm{\Lambda }$, and the wave function turns to an even function (bottom four figures).

Figure 5

The running coupling $\lambda (E,E_B)$ as a function of the scattering energy $E$ in the units of $m$, for $E_B=-1.1\mathrm{m}$, $-2\mathrm{m}$, $-3\mathrm{m}$, and $-4\mathrm{m}$. The graph in the lower right corner was extended to large values of $E$ in order to illustrate the asymptotic freedom of the system when $\lambda (E,E_B)\to 0$ as $E\to \infty$.