Effect of deuteron breakup on the deuteron-$\Xi$ correlation function

We adopt the Argonne V4’ parameter WP02 for the $NN$ interaction. The triplet-even ${}^{13}{\rm S}_{1}$ and the singlet-even ${}^{31}{\rm S}_{0}$ states are taken into account. The continua of these states are truncated at $k_{\rm max}=2.0$ fm^-1 ($\sim 400$ MeV/$c$); the size $\Delta_{c}$ of the bin state is set to 0.2 fm^-1 ($\sim 40$ MeV/$c$) and 0.005 fm^-1 ($\sim 1$ MeV/$c$) for the ${}^{13}{\rm S}_{1}$ and ${}^{31}{\rm S}_{0}$ states, respectively¹¹1Because the breakup states are characterized by wave numbers, not momenta, in the CDCC code employed, we represent $k_{\rm max}$ and $\Delta_{c}$ in the unit of fm^-1.. $r_{\rm max}=20$ fm is taken for evaluating the folded potentials.

As for the $N$-$\Xi$ strong interaction, we employ the parameterization by the LQCD work at $a/t=11$ HALQCD-pXi . In the original parameterization, the $N$-$\Xi$ interaction $V_{ts}^{(N\Xi)}$ for each spin ($s$) and isospin ($t$) channel was expressed by the sum of one Yukawa function (with a form factor), one squared Yukawa, and three Gaussians. In this study, we expand each of the former two by 30 Gaussians; the range parameters are chosen in a geometric progression and the minimum and maximum ranges are optimized for each $st$ channel. It is found that $V_{ts}^{(N\Xi)}$ thus obtained gives the $N$-$\Xi$ phase shift that agrees with the result with the original $V_{ts}^{(N\Xi)}$ for six digits. By expressing all the terms of $V_{ts}^{(N\Xi)}$ by Gaussians, one can use the simple analytic form of Eq. (43) for the monopole component of the $N$-$\Xi$ interaction.

The CC equations (20) are integrated up to $R=10$ fm. The Coulomb interaction $V^{\rm C}$ is taken to be

with $R_{0}=1.5$ fm. The dependence of the numerical results shown below on $R_{0}$ is found to be negligibly small (less than 1%).

The source function ${\cal S}$ is assumed to have a Gaussian form

The source size $b$ of the source function is taken to be 1.2 fm; in Fig. 4, results with $b=1.6$ and 3.0 fm are shown for comparison. In the evaluation of the correlation function, the integration over $R$ is carried out up to $R_{\rm max}=10$ fm (15 fm) when $b=1.2$ fm and 1.6 fm (3.0 fm), and the maximum $L$ is taken to be a larger of $K_{0}R_{\rm max}$ and 5.

We show in Fig. 1 $C_{d\Xi^{-}}$ as a function of $q\equiv\hbar cK_{0}$. The inset is an enlarged figure in the region of $30~{}{\rm MeV}/c\leq q\leq 120~{}{\rm MeV}/c$. The solid (red) line represents the result calculated with the present framework of CDCC. The dotted (blue) line is the result of the single-channel calculation, that is, only the ground state of deuteron is considered. If we take only the ${}^{13}{\rm S}_{1}$ channels in $NN$ into account, the dashed (green) line is obtained. The dash-dotted (purple) line is the result obtained with all the strong interactions turned off. For a simple notation, below we designate the ${}^{13}{\rm S}_{1}$ (${}^{31}{\rm S}_{0}$) channel as the $pn$ ($nn$) channel.

The solid line shows a clear enhancement relative to the dash-dotted line for $q\leq 100~{}{\rm MeV}/c$, which indicates that the correlation due to the strong interaction can be deduced from $C_{d\Xi^{-}}$. The difference of the solid line from the dotted line represents an increase in $C_{d\Xi^{-}}$ by the deuteron breakup effect, which is about 6–8 % for $30~{}{\rm MeV}/c\leq q\leq 70~{}{\rm MeV}/c$. At larger $q$, the enhancement due to deuteron breakup decreases monotonically and becomes less than 1% for $q>100~{}{\rm MeV}/c$. We discuss the deuteron breakup effect in more detail in Sec. III.5. The small difference between the dashed and dotted line indicates that the $nn$ breakup states are more significant than the $pn$ breakup states. This can be understood by the behavior of the CC potentials as discussed in Sec. III.4. With a closer look, a shoulder structure is found in the solid line at around 60 MeV$/c$. This corresponds to the strong coupling to low-lying $nn$ breakup states located just below the scattering threshold; the channel energy $E_{c}$ is negative and close to 0. We will return to this point soon below and in Sec. III.5. Compared with the net effect of the strong interaction (difference between the solid and dash-dotted lines), the deuteron breakup effect is found to be not very significant. In other words, including only the deuteron ground state in the calculation of $C_{d\Xi^{-}}$ will be useful except that it will miss a further increase in the correlation function by several percent below about 70 MeV$/c$.

Figure 2 displays the convergence of $C_{d\Xi^{-}}$ regarding $k_{\rm max}$. In all the calculations, we take the size $\Delta_{c}$ of the bin state to be 0.2 fm^-1 (0.005 fm^-1) for the $pn$ ($nn$) continuum. The solid (red), dashed (green), dotted (blue), and dash-dotted (purple) lines correspond to $k_{\rm max}=0.2$, 0.5, 1.0, and 2.0 fm^-1, respectively. The dash-dotted line is the same as the solid line in Fig. 1. The result with $k_{\rm max}=2.5$ fm^-1 is found to agree with the dash-dotted line within the width of the line (not shown). It should be noted that almost all of the $NN$ states included in the converged CDCC calculation serve as a closed channel. For instance, at $q=100$ MeV$/c$ ($E_{0}\sim 4.22$ MeV), $NN$ states having $k\mathrel{\mathchoice{\lower 3.6pt\vbox{ \halign{$\mathsurround=0pt\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower 3.6pt\vbox{ \halign{$\mathsurround=0pt\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower 3.6pt\vbox{ \halign{$\mathsurround=0pt\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower 3.6pt\vbox{ \halign{$\mathsurround=0pt\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}0.32$ fm^-1 are all closed, whereas we need the $NN$ states up to 2.0 fm^-1 ($\varepsilon\sim 166$ MeV) to achieve a convergence of $C_{d\Xi^{-}}$.

The convergence of the CDCC result regarding $\Delta_{c}$ for the $nn$ continuum is shown in Fig. 3; $\Delta_{c}=0.2$ fm^-1 is used for the $pn$ continuum and $k_{\rm max}$ is set to 2.0 fm^-1 for both $pn$ and $nn$ continua. The solid (red), dashed (green), dotted (blue), and dash-dotted (purple) lines correspond to $\Delta_{c}=1.0$, 0.5, 0.2, and 0.005 fm^-1, respectively. The dash-dotted line is the same as the solid-line in Fig. 1. The dashed line turns out to have a rather sharp peak around 93 MeV$/c$. This happens when the lowest (pseudo) $nn$ state is located just below the threshold energy; note that the eigenenergy of a discretized continuum state is defined by Eq. (4) and depends on $\Delta_{c}$. When $\Delta_{c}=0.2$ fm^-1, the eigenenergy of the lowest $nn$ state becomes 0.55 MeV and the peak appears at $q\sim 65$ MeV$/c$. At the same time, another peak is found around 97 MeV$/c$, which corresponds to the second-lowest $nn$ state. As $\Delta_{c}$ becomes smaller, a larger number of peaks appear and the characteristics of each peak become less emphasized. It is found that with $\Delta_{c}=0.005$ fm^-1, a reasonably smooth $C_{d\Xi^{-}}$ is obtained. The shoulder structure of the dash-dotted line around 60 MeV$/c$ is due to many tiny peaks corresponding to low-lying $nn$ breakup states. It should be noted that for breakup states that do not strongly couple to the deuteron ground state, the above-mentioned threshold effect is negligibly small. This is why we can use a rather large bin-size, $\Delta_{c}=0.2$ fm^-1, for the $pn$ breakup states. The properties of the CC potentials for the $pn$ and $nn$ breakup states are discussed in Sec. III.4.

We show in Fig. 4 the dependence of $C_{d\Xi^{-}}$ on the source size $b$ of the source function; Fig. 4(a) and Fig. 4(b) correspond to $b=1.6$ and 3.0 fm, respectively. The meaning of each line is the same as in Fig. 1. As $b$ increases, the correlation due to the strong interaction becomes weak, as well as the deuteron breakup effect. This is simply because the non $s$-wave contribution of the $d$-$\Xi$ scattering wave function is large in the outer region of $R$. Notwithstanding, the effect of the strong interaction on $C_{d\Xi^{-}}$ will remain at small $q$.

In this subsection, we discuss the properties of the $NN$ states included in the current study. For transparent discussion, the results below are evaluated with $\Delta_{c}=0.2$ fm^-1 ($\sim 40$ MeV/$c$) for both the $pn$ and $nn$ channels. As shown in Fig. 3, apart from the threshold effect of the low-lying $nn$ breakup states, $C_{d\Xi^{-}}$ calculated with $\Delta_{c}=0.2$ fm^-1 well reproduces that with $\Delta_{c}=0.005$ fm^-1 ($\sim 1$ MeV/$c$). Therefore, discussion on $\varphi_{c}$ generated with $\Delta_{c}=0.2$ fm^-1 will be meaningful to understand the role of the $NN$ continuum in this study.

The $s$-wave phase shift $\delta^{(NN)}_{TS}$ of the $NN$ system is shown in Fig. 5(a) as a function of the $NN$ c.m. energy $\varepsilon$. The solid (red) and dashed (green) lines represent $\delta^{(NN)}_{01}$ ($pn$ channel) and $\delta^{(NN)}_{10}$ ($nn$ channel), respectively. As is well known, $\delta^{(NN)}_{10}$ shows a rapid increase near $\varepsilon=0$, which is due to the virtual state (pole) of the $nn$ system. Although it is different from a resonance, the $nn$ wave function near the zero energy has a compact form as shown below. In Fig. 5(b) and Fig. 5(c), respectively, we show $\varphi_{i01}$ and $\varphi_{i10}$; in each panel, the dashed (green), dotted (blue), and dash-dotted (purple) lines correspond to the first bin ($k=0.0$–0.2 fm^-1, $\varepsilon_{c}=0.55$ MeV), the third bin ($k=0.4$–0.6 fm^-1, $\varepsilon_{c}=10.5$ MeV), and the sixth bin ($k=1.0$–1.2 fm^-1, $\varepsilon_{c}=50.3$ MeV) states, respectively. For comparison, the deuteron wave function is shown by the solid (red) line in Fig. 5(b). As mentioned, the first bin state of the $nn$ channel behaves like a bound state. On the other hand, for the $pn$ channel, the amplitude of the first bin state in the inner region is very small, which makes this state almost decoupled from the deuteron ground state and other $NN$ states. As for the third bin state, the $pn$ wave function is slightly more shrunk than the $nn$ one, reflecting the difference in the phase shift shown in Fig. 5(a). The dependence of the sixth bin state on the spin-isospin is found to be very small, which is the case also for higher bin states.

The $N$-$\Xi$ interactions in individual spin-isospin channels as a function of the $N$-$\Xi$ distance are shown in Fig. 6(a) and the corresponding folded potentials for the ground-ground channel, $f^{(ts)}_{00}$, are shown in Fig. 6(b) as functions of $R$. In each panel, the potentials for ${}^{11}{\rm S}_{0}$, ${}^{31}{\rm S}_{0}$, ${}^{13}{\rm S}_{1}$, and ${}^{33}{\rm S}_{1}$ are represented by the solid (red), dashed (green), dotted (blue), and dash-dotted (purple) lines, respectively. Through the folding procedure, the characteristics of the potential for each channel becomes very clear. The potential in the ${}^{11}{\rm S}_{0}$ channel is attractive, while that in ${}^{31}{\rm S}_{0}$ repulsive. The feature of the potential in the ${}^{13}{\rm S}_{1}$ (${}^{33}{\rm S}_{1}$) channel is similar to that in the ${}^{11}{\rm S}_{0}$ (${}^{31}{\rm S}_{0}$) channel but with the absolute value weaken considerably. Note, however, that the attractive nature of the $N$-$\Xi$ potential in the ${}^{33}{\rm S}_{1}$ channel is found to remain when folded by the deuteron density; the $d$-$\Xi^{-}$ scattering length evaluated by taking only the ${}^{33}{\rm S}_{1}$ channel in the single-channel calculation is negative. Here, we use the nuclear physics convention for the scattering length, that is,

where $\kappa$ is the relative wave number of the two particles, $\delta$ is the $s$-wave scattering phase shift, $a_{s}$ is the scattering length, and $r_{s}$ is the effective range. The negative scattering length thus means that there is no bound state. The qualitative features of $f^{(ts)}_{00}$ mentioned above are found to remain for other components of the folded potential.

From now on, we discuss the properties of the CC potentials $U_{cc^{\prime}}^{\left(\sigma\right)}$. For simplicity, we take only four states of the $NN$ system, that is, the deuteron ground state ($d$), the third bin state in the $pn$ channel ($pn$), the first bin state in the $nn$ channel (${}^{2}n$), and the third bin state in the $nn$ channel ($nn$); we abbreviate these four state as denoted in the parentheses. Here, as in Sec. III.3, we take $\Delta_{c}=0.2$ fm^-1 ($\sim 40$ MeV/$c$) for both channels.

In Fig. 7(a), we show the diagonal part of the CC potentials for the four states; the total channel spin $\sigma$ is taken to be $1/2$. The solid (red), dashed (green), dotted (blue), and dash-dotted (purple) lines correspond to the $d$, $pn$, ${}^{2}n$, and $nn$ states, respectively. The former two are repulsive in the interior region ($R\mathrel{\mathchoice{\lower 3.6pt\vbox{ \halign{$\mathsurround=0pt\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower 3.6pt\vbox{ \halign{$\mathsurround=0pt\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower 3.6pt\vbox{ \halign{$\mathsurround=0pt\scriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower 3.6pt\vbox{ \halign{$\mathsurround=0pt\scriptscriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}}1$ fm) and weakly attractive at larger $R$, whereas the latter two are attractive in the entire region. This is due to the spin-isospin selection given by Eqs. (26) and (28) combined with the spin-isospin dependence of the folded potential shown in Fig. 6(b). The result shown in Fig. 7(a) indicates that an $nn$ pair can be closer to the $\Xi$ particle than a $pn$ pair including deuteron. This is one of the main reasons for the significant breakup effect from the $nn$ channel.

Figure 7(b) represents the coupling potential between the deuteron channel and each of the other three; the solid (red), dashed (green), and dotted (blue) lines correspond to the $d$-$pn$, $d$-${}^{2}n$, and $d$-$nn$ couplings, respectively. It should be noted that the sign of non-diagonal coupling potentials has no meaning. As mentioned above, the folded potential $f^{(ts)}_{cc^{\prime}}$ does not strongly depend on the combination of the channels, $c$ and $c^{\prime}$. Consequently, the qualitative feature of the $d$-$pn$ coupling potential is similar to that of the $d$-$d$ diagonal potential. The behavior of the $d$-${}^{2}n$ and $d$-$nn$ couplings can be understood by Eq. (27) and Fig. 6(b). An important remark is that the magnitude of the $d$-${}^{2}n$ coupling potential is comparable to the $d$-$nn$ and $d$-$pn$ ones because of the compactness of the ${}^{2}n$ wave function as shown in Fig. 5(c) (the dashed line). This feature is also crucial for making the breakup effect of the $nn$ channel important. Note that the coupling between the deuteron ground state and a low-lying $pn$ state is significantly weaker than the results shown in Fig. 7(b).

To complete the discussion on the breakup effect, we need to consider the scattering threshold effect as well. When the $d$-$\Xi^{-}$ c.m. scattering energy $E_{0}$ is low, the channel energy $E_{c}$ for the ${}^{2}n$ channel becomes negative. In this case, even though the $d$-${}^{2}n$ coupling is strong and the ${}^{2}n$-${}^{2}n$ diagonal potential is attractive, the scattering wave $\chi_{c}^{(\sigma)}$ has to be considerably quenched because of the damping boundary condition of Eq. (12). An exception occurs when $E_{c}$ is very close to the threshold, that is, $E_{c}\sim 0$. This is how the shoulder structure of $C_{d\Xi^{-}}$ is developed (see also Sec. III.5).

We show in Fig. 8 the coupling potentials with $\sigma=3/2$, for which the $nn$ states are not allowed. The solid (red), dashed (green), and dotted (blue) lines show the $d$-$d$ diagonal, $d$-$pn$ coupling, and $pn$-$pn$ diagonal potentials, respectively. The features of the results can be understood by Eq. (29) and Fig. 6(b). It is found that the absence of the $nn$ channel makes the breakup effect negligibly small when $\sigma=3/2$, as shown in Sec. III.5.

Figure 9 displays the nuclear scattering phase shift $\delta^{(NN\mbox{-}\Xi)}_{\tau\sigma}$ of the $NN$-$\Xi$ system as a function of the c.m. scattering energy. The solid (red) and dashed (green) lines show $\delta^{(NN\mbox{-}\Xi)}_{1/2,1/2}$ and $\delta^{(NN\mbox{-}\Xi)}_{1/2,3/2}$, respectively. In the calculation, a single-channel scattering problem with $U_{00}^{\left(\sigma\right)}\left(R\right)$ is solved for $\sigma=1/2$ and 3/2. As shown in Fig. 9, the net effect of $U_{00}^{\left(\sigma\right)}\left(R\right)$ is found to be attractive, and the attraction of $U_{00}^{\left(3/2\right)}\left(R\right)$ is stronger than that of $U_{00}^{\left(1/2\right)}\left(R\right)$. In Table 1, we show the $s$-wave scattering length $a_{s}$ and the effective range $r_{s}$ for the $NN$-$\Xi$ scattering.