Chiral potential renormalized in harmonic-oscillator space

We renormalize the chiral effective field theory potential in harmonic-oscillator (HO) model space. The low energy constants (LECs) are utilized to absorb not just the ultraviolet part of the physics due to the cutoff, but also the infrared part due to the truncation of model space. We use the inverse $J$-matrix method to reproduce the nucleon-nucleon scattering phase shifts in the given model space. We demonstrate that by including the NLO correction, the nucleon-nucleon scattering in the continuum could be well reproduced in the truncated HO trap space up to laboratory energy $T_{\mathrm{lab}}=100\mathrm{MeV}$ with number of HO basis $n_{\max }$ as small as 10. A perturbative power counting starts at subleading order is adopted in this work, and how to extract the perturbative contribution is demonstrated. This work serves as the input to perform ab initio calculations.

Figure 1

${}^{1}S_0$ LO phase shift as a function of laboratory energy $T_{\mathrm{lab}}=0–50\mathrm{MeV}$. Here the black-dot represent the Nijmegen phase shift, and each colored line represents the phase shift with various $n_{\max }$, where $\omega$ is fixed to 20 MeV except for the one with the “+” sign. Here the result is renormalized to give $a_0=-23.7$ fm, and the potential has intrinsic cutoffs ${\mathrm{\Lambda }}_c=800$ MeV, $\mathrm{\Lambda }=600\mathrm{MeV}$.

Figure 2

${}^{1}S_0$ LO phase shift as a function of laboratory energy $T_{\mathrm{lab}}=0–300\mathrm{MeV}$. Here the black circles represent phase shift obtained by the Busch formula, and the red line represents the phase shift obtained by the $J$-matrix method. Here $n_{\max }=10$, $\omega =20\mathrm{MeV}$, and the potential has intrinsic cutoffs ${\mathrm{\Lambda }}_c=800\mathrm{MeV}$, $\mathrm{\Lambda }=600\mathrm{MeV}$.

Figure 3

${}^{1}S_0$ LO phase shift as a function of laboratory energy $T_{\mathrm{lab}}=0–300\mathrm{MeV}$. Here phase shifts obtained by the Busch formula (colored symbol) are compared to those obtained by the $J$-matrix method (colored line). Here $n_{\max }=20,40,60$, and $\omega =20\mathrm{MeV}$. The potential has intrinsic cutoffs ${\mathrm{\Lambda }}_c=1000\mathrm{MeV}$, $\mathrm{\Lambda }=800\mathrm{MeV}$.

Figure 4

${}^{3}S_1$ LO phase shift as a function of laboratory energy $T_{\mathrm{lab}}=0–60\mathrm{MeV}$. Here the black-dot represent the Nijmegen phase shift, and each colored line represents the phase shift with various $n_{\max }$, where $\omega$ is fixed to 20 MeV. The LEC is renormalized to reproduce $a_0=5.4$ fm in the left panel, and is renormalized to reproduce the deuteron binding energy $E_b=-2.225\mathrm{MeV}$ in the right panel. The potential has intrinsic cutoffs ${\mathrm{\Lambda }}_c=800\mathrm{MeV}$, $\mathrm{\Lambda }=600\mathrm{MeV}$.

Figure 5

Leading order phase shifts as a function of laboratory energy $T_{\mathrm{lab}}$. Here the black dot represents the Nijmegen phase shift, and each colored line represents the phase shift with various $n_{\max }$, where $\omega$ is fixed to 20 MeV, and the potential has intrinsic cutoffs ${\mathrm{\Lambda }}_c=800\mathrm{MeV}$, $\mathrm{\Lambda }=600\mathrm{MeV}$.

Figure 6

Leading order phase shifts as a function of laboratory energy $T_{\mathrm{lab}}$. Here the black dot represents the Nijmegen phase shift, and each colored line represents the phase shift with various $n_{\max }$, where $\omega$ is fixed to 120 MeV, and the potential has intrinsic cutoffs ${\mathrm{\Lambda }}_c=1000\mathrm{MeV}$, $\mathrm{\Lambda }=800\mathrm{MeV}$.

Figure 7

Leading order phase shifts as a function of laboratory energy $T_{\mathrm{lab}}$. Here the black dot represents the Nijmegen phase shift, and each colored line represents the phase shift with various $N_{\max }$, where $\omega$ is fixed to 120 MeV, and the potential has intrinsic cutoffs ${\mathrm{\Lambda }}_c=1000\mathrm{MeV}$, $\mathrm{\Lambda }=800\mathrm{MeV}$.

Figure 8

NLO and NNLO coupled-channel phase shifts as a function of laboratory energy $T_{\mathrm{lab}}$. Here the black sdot represent the Nijmegen phase shift, and each colored line represents the phase shift with various $n_{\max }$, where $\omega$ is fixed to 120 MeV and the intrinsic cutoffs are ${\mathrm{\Lambda }}_c=800\mathrm{MeV}$, $\mathrm{\Lambda }=600\mathrm{MeV}$.

Figure 9

NLO and NNLO uncoupled-channel phase shifts as a function of laboratory energy $T_{\mathrm{lab}}$. Here the black dot represents the Nijmegen phase shift, and each colored line represents the phase shift with various $n_{\max }$, where $\omega$ is fixed to 120 MeV and the intrinsic cutoffs are ${\mathrm{\Lambda }}_c=800\mathrm{MeV}$, $\mathrm{\Lambda }=600\mathrm{MeV}$.

Figure 10

NLO and NNLO coupled-channel phase shifts as a function of laboratory energy $T_{\mathrm{lab}}$. Here the intrinsic cutoffs are ${\mathrm{\Lambda }}_c=1000\mathrm{MeV}$, $\mathrm{\Lambda }=800\mathrm{MeV}$, and black dot represents the Nijmegen phase shift. Each colored line represents the phase shift at various order and combination of ($\omega ,n_{\max }$). Label (A) stands for $(\omega ,n_{\max })=(120\left[\mathrm{MeV}\right],8)$ for the ${}^{3}S_1-{}^{3}D_1$ channel and (120 [MeV],6) for the ${}^{3}P_2-{}^{3}F_2$ channel, and (B) stands for $(\omega ,n_{\max })=(60\left[\mathrm{MeV}\right],19)$ for the ${}^{3}S_1-{}^{3}D_1$ channel and (60 [MeV],15) for the ${}^{3}P_2-{}^{3}F_2$ channel.

Figure 11

NLO and NNLO uncoupled-channel phase shifts as a function of laboratory energy $T_{\mathrm{lab}}$. Here the intrinsic cutoffs are ${\mathrm{\Lambda }}_c=1000\mathrm{MeV}$, $\mathrm{\Lambda }=800\mathrm{MeV}$, and the black dot represents the Nijmegen phase shift. Each colored line represents the phase shift at various order and combination of ($\omega ,n_{\max }$). Label (A) stands for $(\omega ,n_{\max })=(120\left[\mathrm{MeV}\right],8)$, and (B) stands for $(\omega ,n_{\max })=(60\left[\mathrm{MeV}\right],17)$.