Covariant spectator theory of $\mathit{np}$ scattering: Phase shifts obtained from precision fits to data below 350 MeV

Using the covariant spectator theory (CST), we present two one-boson-exchange kernels that have been successfully adjusted to fit the 2007 world $\mathit{np}$ data (containing 3788 data) below 350 MeV. One model (which we designate WJC-1) has 27 parameters and fits with a ${\chi }^2/N_{\mathrm{data}}=1.06$. The other model (designated WJC-2) has only 15 parameters and fits with a ${\chi }^2/N_{\mathrm{data}}=1.12$. Both of these models also reproduce the experimental triton binding energy without introducing additional irreducible three-nucleon forces. One result of this work is a new phase-shift analysis, updated for all data until 2006, which is useful even if one does not work within the CST. In carrying out these fits we have reviewed the entire database, adding new data not previously used in other high precision fits and restoring some data omitted in previous fits. A full discussion and evaluation of the 2007 database is presented.

Figure 1

(Top line) diagramatic representation of the covariant spectator equation (2.2) with particle 1 on-shell (the on-shell particle is labeled with a ×). (Second line) Diagrammatic representation of the definition of the antisymmetrized kernel (2.6).

Figure 2

Diagramatic representation of the covariant spectator equation for the three-body bound-state vertex function Γ with particles 1 and 2 on-shell (labeled with a ×). Here particle 1 is the spectator to the last two-body interaction between particles 2 and 3, described by the scattering amplitude $M$ with particle 3 off-shell. The spectator has three-momentum $q$ after the two-body interaction and $q^{\text{'}}$ before.

Figure 3

Log-log scatter plot showing the value of $z={\chi }^2/n$ for each data set with $n$ data. The sets that are retained are represented by small circles; those rejected by larger boxes. The $z_{\max }$ (solid line) and $z_{\min }$ (dashed line) limits given by Eq. (4.3) are shown. The five labeled data sets are discussed in Figs. 5, 7, and 8.

Figure 4

Distribution in ${\chi }^2/n$ of data sets with $n>50$. The continuous curves are the theoretical distributions (normalized to this number of sets) with $n=100$ (solid curve) and $n=60$ (dashed curve). [Both curves are given to show the dependence of the theoretical distributions (4.1) on $n$.]

Figure 5

Total cross-section as a function of laboratory energy. The lower two panels compare the three rejected cross-section data sets, PO82 [149] (shown in the lower left panel on an expanded energy scale and also in the lower right panel as three indistinguishable points near zero), ME66 [153], and DE73 [161] with theory. The very small ${\chi }^2$ for ME66 is due to an overestimate of the errors (not consistent with the expected statistical fluctuations), whereas the other data sets disagree strongly with the fit (theory), and all three sets lie well outside the boundaries (see their locations on Fig. 4). The solid line is the fit (referred to as the theory).

Figure 6

Example of the quality of the fit to some recent 194 MeV differential cross-section data [126].

Figure 7

Differential cross-section measurements at 162 MeV [122, 155]. The upper two panels show both data sets, the lower-left panel shows only the data set BO78 [122] that is kept, the bottom-right panel only the data set RA98 [155] that is excluded. RA98 seems to have some unexplained systematic error, particularly at the backward angles, giving it the wrong shape and a large ${\chi }^2/n$ of 6.9.

Figure 8

Differential cross-section measurements at 319 MeV [130] and 320.1 MeV [156]. The data set FR00 [156] shown in the bottom-right panel is excluded; do to some large unexplained systematic error it has the wrong shape and a large ${\chi }^2/n$ of 8.4. The lower-left panel shows the effect of scaling on the KE82 data [130], as discussed in the text. In this panel the triangles are unscaled data not shown in the upper panels; the small circles are scaled data shown also in the upper panels.

Figure 9

Phase shifts and mixing parameters for all states with $J⩽1$. Models WJC-1 (solid line) and WJC-2 (dotted line) are compared to the Nijmegen phases (dashed line).

Figure 10

Singlet an uncoupled triplet phase shifts for all states with $2⩽J⩽4$. Curves are drawn as in Fig. 9.

Figure 11

Triplet coupled phase shifts and mixing parameters for states with $2⩽J⩽4$. Curves are drawn as in Fig. 9.

Figure 12

The family of WJC-1 models with ${\nu }_{{\sigma }_0}$ constrained to various fixed values. The left-hand axis shows the best ${\chi }^2/N_{\mathrm{data}}$ that can be found for each value of ${\nu }_{{\sigma }_0}$ (the data shows some scatter with respect to the solid line, which is a cubic fit to the 7 cases shown) and the right-hand axis shows the triton binding energy ($E_t$ in MeV) for each member of the family. Note that the correct binding energy (shown by the dashed horizontal line) is obtained for the value of ${\nu }_{{\sigma }_0}$ that also gives the best fit to the data.

Figure 13

The family of WJC-2 models with ${\nu }_{{\sigma }_0}$ constrained to various fixed values. Here the fit to ${\chi }^2/N_{\mathrm{data}}$ is a quadratic, and the binding energy for the best case, 8.50, is shifted slightly from the experimental value. However, shifting ${\nu }_{{\sigma }_0}$ by about 0.1 should give exact agreement with the binding energy, and this shift changes ${\chi }^2/N_{\mathrm{data}}$ by less that 0.003 (1 standard deviation for about 3500 data, shown as a shaded band on the figure). (See the caption to Fig. 12 for further explanation.)

Figure 14

Diagrams showing the collapse of off-shell interactions into an effective contact interaction, as derived in Eq. (6.1).

Figure 15

Examples, from the two-body sector, of one and two-loop diagrams generated by iteration of off-shell OBE couplings. The off-shell couplings automatically generate these diagrams, but the same result could be obtained from a theory without off-shell couplings if these diagrams (and an infinite number of others) were added explicitly to the two-body force. The lines marked with an × are on-shell particles and in each case a mechanism similar to that shown in Fig. 14 collapses the off-shell propagation to a point.

Figure 16

Examples, from the three-body sector, of no-loop and one-loop diagrams generated by iteration of off-shell OBE couplings. The off-shell couplings automatically generate these diagrams, but the same result could be obtained from a theory without off-shell couplings if these diagrams (and an infinite number of others) were added explicitly to the three-body force. The lines marked with an × are on-shell particles and in each case a mechanism similar to that shown in Fig. 14 collapses the off-shell propagation to a point.

Figure 17

Diagrammatic representation of the symmetrized OBE kernel given in Eq. (B9) (with the factor of 1/2 suppressed). Particle 1 in the final state is on-shell (denoted by the ×) and the particles in the initial state, with relative four-momentum $p^{*}$, are both on-shell. With our convention, Feynman diagrams are written so that the end of each external line is always labeled with the same four-momentum and Dirac index ($\alpha ,{\alpha }^{\text{'}},\beta ,{\beta }^{\text{'}}$); this requires crossing the initial nucleon lines in the exchange term (b). Diagrammatically, the symmetry relation (B11) corresponds to crossing the initial nucleon lines and interchanging the ${\alpha }^{\text{'}}$ and ${\beta }^{\text{'}}$ indices, giving the same two diagrams multiplied by the factor ${\eta }_I$.

Figure 18

Diagrammatic representation of the interchange rule for the symmetrized kernel, Eq. (B15). Here the solid box represents the full kernel, which is the sum of the two diagrams as shown in Fig. 17. The left-hand diagram is $V^2$ and the right-hand diagram is $V^1$ (with indices and momenta exchanged).

Figure 19

Drawing showing how the interchange rule of Fig. 18 [and Eq. (B15)] can be derived diagrammatically. All Feynman diagrams are equal to their twisted versions. In this example we start with $V^2$, equal to the two boson exchange diagrams shown in the first row. Each diagram is then twisted [obtained, in this example, by exchanging the vertices $a\leftrightarrow b$ and $c\leftrightarrow d$ while leaving the labeling of all external particles unchanged], as shown in the second row, and then the two twisted diagrams are collected into ${\eta }_I{V}^1$ (with a change in the sign of the final state three-momentum and the final Dirac indices exchanged, as suggested by the labeling of the diagram).

Figure 20

Diagrammatic representation of (incorrect) uncoupled integral equations for the half on-shell scattering amplitudes $M^1$ and $M^2$. The proposed equations are given on the first and third lines. The second and forth lines give the scattering amplitudes iterated to fourth order. These equations satisfy the reflection property (B15), but are unsatisfactory because when both final-state particles are on-shell diagrams (a) and (b) are not equal to diagrams (a′) and (b′).

Figure 21

(Top row) diagrams (a) and (b) from the fourth order expansion of $M^1$; (bottom row) diagrams (a${}^{\text{'}}$) and (b${}^{\text{'}}$) from the fourth order expansion of $M^2$ (in both cases the overall factor of $\frac{1}{2}$ has been suppressed). The right shows the twisted versions of each diagram on the left. Comparison of the two top-left diagrams with the two bottom-right ones shows that the symmetry (B15) is indeed satisfied (and similarly for the top-right and bottom-left). However, when both final-state particles are on shell, these amplitudes are not equal; for example, the leftmost figures (a) and (á) differ by which particle is on-shell inside the box.

Figure 22

Diagrammatic representation of coupled integral equations for the half on-shell scattering amplitudes $M^1$ and $M^2$. The equations are given on the first and third lines. The second and forth lines give the scattering amplitudes iterated to fourth order. These equations satisfy the reflection property (B15), and also give identical results when both final-state particles are on-shell, provided that, for this on-shell point, $V^{22}=V^{12}$ and $V^{11}=V^{21}$.

Figure 23

Here $p$ and $k$ are in units of $m$, with $W=2.1\hspace{1em}m$. The dark shaded area is the region where $q_{\mathrm{ex}}^2(1)>m_{\mathrm{ex}}^2$, with $m_{\mathrm{ex}}$ equal to the pion mass. The light shaded area is the region where $m_{\mathrm{ex}}^2>q_{\mathrm{ex}}^2(1)>0$, and the white area has $0>q_{\mathrm{ex}}^2(1)$ [except for the tiny triangular region near the origin where $q_{\mathrm{ex}}^2(1)⩾q_{\mathrm{ex}}^2(-1)>0$]. The singularities occur along the boundary between the dark and light shaded regions. The three horizontal lines mark $k=0.2$, 0.6, and 1.2.

Figure 24

Plots of the six dimensionless functions $F(p)=V_{\mathrm{ex}}(p,k_i)$ (solid lines) and $F(p)=V_{\mathrm{direct}}(p,k_i)$ (dashed lines) for the dimensionless values of $k_i=0.2,0.6$, and 1.2 as shown in Fig. 23. The curves can be distinguished by looking at their behavior at small $p$, which is roughly proportional to $1{k/}_i$.

Figure 25

Plots of the 6 dimensionless functions $F(p)=V_{\mathrm{ex}}(p,k_i)$ (solid lines) and $F(p)=V_{\mathrm{ex}}^C(p,k_i)$ (dotted lines) for the dimensionless values of $k_i=0.2,0.6$, and 1.2 as shown in Fig. 23. The curves can be distinguished by looking at their behavior at small $p$, which is roughly proportional to $1{k/}_i$.

Figure 26

The four poles in the complex $p_0$ plane arising from the term $[(m^2-p_1^2)(m^2-p_2^2)]{}^{-1}$. The CST (with particle 1 on-shell) keeps only the pole at $p_0=E(p)-\frac{1}{2}W$ (#1). When $W\to 0$ the pole at $p_0=E(p)+\frac{1}{2}W$ (#3) cannot be neglected. The full description in this case requires the two-channel spectator equation.

Figure 27

Diagramatic representation of the transformation of the exchange term, with relative energy $p_0$ into a direct term with relative energy $-p_0$ (called an “alternating” term in Ref. [5]). The phase η arises from the exchange operation derived in Eq. (E26).