Covariant spectator theory of $\mathit{np}$ scattering: Effective range expansions and relativistic deuteron wave functions

We present the effective range expansions for the ${}^1{S}_0$ and ${}^3{S}_1$ scattering phase shifts, and the relativistic deuteron wave functions that accompany our recent high precision fits (with ${\chi }^2/N_{\mathrm{data}}\simeq 1$) to the 2007 world $\mathit{np}$ data below 350 MeV. The wave functions are expanded in a series of analytical functions (with the correct asymptotic behavior at both large and small arguments) that can be Fourier-transformed from momentum to coordinate space and are convenient to use in any application. A fortran subroutine to compute these wave functions can be obtained from the authors.

Figure 1

The effective range expansions for the S waves (smooth line) compared to calculated phase shifts (solid dots). The phases are in degrees.

Figure 2

Diagrammatic representation of the vertex function (3.8). The on-shell particle is labeled by the $\times$, so $p_1^2=m^2$, and the relative four-momentum is $p=\frac{1}{2}(p_1-p_2)$. The deuteron is also on-shell, with $P^2=M_d^2$.

Figure 3

Log-log plot of the momentum-space wave functions $z_{\ell }(\mathrm{p})$, defined in Eq. (3.30). (Left) WJC-1 wave functions tabulated in Table , compared to the asymptotic limits given in Eq. (3.41). Solid circles: $u_0$; solid diamonds: $w_2$; open squares: $v_t$; open circles: $v_s$. The solid lines are proportional to $p^{-5}$ and $p^{-4}$. (Right) WJC-2 wave functions tabulated in Table . The solid lines are proportional to $p^{-5}$ and $p^{-4}$, and are the same as those shown in the left panel.

Figure 4

Ratio of the raw wave functions for model WJC-1 (taken from Table  with the probabilities “scaled” to 100% as outlined in Table ) to the scale functions from Eq. (3.42). In each panel the dots are the tabulated wave functions for 60 Gauss points, the solid curve the fit, and the dashed curve the fit without the tail wave function of Eq. (3.47).

Figure 5

Ratio of the raw wave functions for model WJC-2 (taken from Table  with the probabilities “scaled” to 100% as outlined in Table ) to the scale functions from Eq. (3.42). In each panel the dots are the tabulated wave functions for 60 Gauss points, the solid curve the fit, and the dashed curve the fit without the tail wave function of Eq. (3.47). The origin of the many small wiggles in $u$ is not clear, but $u$ is very small in this region, as illustrated in Fig. 3. The first $v_s$ point is probably inaccurate because of numerical cancellations.

Figure 6

Momentum-space wave functions for both models. The largest two wave functions are $u$ (off-scale at small momenta) and $w$ (zero at small momenta), with WJC-1 wave functions solid and WJC-2 dotted. The P-state wave functions are very small, with both $v_t$ wave functions solid ($v_t(1)$ is negative and $v_t(2)$ positive) and both $v_s$ wave functions dotted (with $v_s(1)$ negative and $v_s(2)$ very near zero). Note that the S-state wave function has a zero around 400 MeV and that above this momentum the P-state wave functions for WJC-1 are no longer completely negligible, as illustrated in Fig. 7.

Figure 7

Enlarged view of the momentum-space wave functions for both models. The curves are labeled as in Fig. 6, with a dashed zero reference line. The large positive states are $w$; the large negative ones $u$, with WJC-1 (solid) and WJC-2 (dotted). Both of the $v_t$ wave functions are solid, and both $v_s$ wave functions are dotted, with WJC-1 negative in both cases and WJC-2 both close to zero.

Figure 8

The coordinate space wave functions for both models. The curves are labeled as in Fig. 6. The largest two wave functions are the familiar $u$ and $w$ wave functions, with WJC-1 solid and WJC-2 dotted. The $v_t$ wave functions are both solid, with WJC-1 negative and WJC-2 small and positive. The $v_s$ wave functions are both dotted, with WJC-1 negative and WJC-2 close to zero.

Figure 9

Expanded view of the coordinate space wave functions for both models at small $r$. The curves are labeled as in Fig. 6.

Figure 10

Plot of the mapping function $z(y)$ defined in Eq. (B5). The four curves, each with 90 Gauss points, correspond to $b_0=10$ (small solid triangles), $b_0=1.5$ (open boxes), $b_0=1.2$ (solid boxes), and $b_0=1.1$ (small solid circles).

Figure 11

Plot of the meson propagator $F$ (with $m_b=0.14$, ${\Lambda }_b=4$, $z_0=1$, and $A=1800$) as a function of the mapped variable $y$ defined in Eq. (B5). The four curves, each with 90 Gauss points, correspond to $b_0=10$ (small solid triangles), $b_0=1.5$ (open boxes), $b_0=1.2$ (solid boxes), and $b_0=1.1$ (small solid circles). The right panel covers the region shown in the upper right corner of the left panel.

Figure 12

Plot of the mapping function $z(y)$ defined in Eq. (B8), with $\zeta =4$. The three curves, each with 90 Gauss points, correspond to $z_0=-0.5$, $z_0=0$, and $z_0=0.5$. Each curve flattens out in the region of $z=y=z_0$.

Figure 13

Plot of the meson propagator $F$ (with $m_b=0.14$, ${\Lambda }_b=4$, and $A=1800$) as a function of both the unmapped variable $y$ and the mapped variable $z(y)$, defined in Eq. (B8), for each of the choices $z_0=-0.5$, $z_0=0$, and $z_0=0.5$. The 90 Gauss points distributed between $-1<y<1$ are shown on each curve. All the curves peak at $y=z_0$, with the unmapped functions $F(y)$ having narrower peaks than the mapped function $F[z(y)]$.

Figure 14

The $\mathrm{p}$ dependence (for $\mathrm{k}$ fixed at 816 MeV) of the 16 model WJC-1 deuteron kernels $V_i(\mathrm{p},\mathrm{k})$ defined in Eq. (C6) (but with the nucleon form factors set to unity). The left panel shows clearly that the total kernel (direct plus exchange) approaches a constant at large $\mathrm{p}$, as predicted by (C19) with $\ell =2$. The right panel shows that the asymptotic $\mathrm{p}$ dependence of 5 of the direct kernels behaves like ${\mathrm{p}}^{-4}$, again predicted by (C19) with $n_b=4$ and $\ell =2$. The other direct kernels decrease more rapidly, presumably because they have an $\ell <2$.