$K\pi$ interaction in finite volume and the $K^{*}$ resonance

We evaluate energy levels of the $K\pi$ system in the $K^{*}$ channel in finite volume using chiral unitary theory. We use these energy levels to obtain $K\pi$ phase shifts and then obtain the $K^{*}$ mass and its decay width. We investigate their dependence on the pion mass and compare this with lattice QCD calculations. We also compare our method with the standard Lüscher approach and solve the inverse problem to obtain the $K\pi$ phase shifts from these “synthetic” lattice data.

Figure 1

The solid curve shows $K\pi$ scattering $P$-wave phase shifts obtained using Eqs. (1) and (9), and the dotdashed curve shows the results from Ref. [21]. The experimental data are taken from Refs. [28] and [29], shown using solid circles and triangles, respectively.

Figure 2

The real part of Eq. (11). Here, we choose $L=2.5m_{\pi }^{-1}$ and $E=800\mathrm{MeV}$.

Figure 3

Energy levels as functions of the cubic box size $L$, derived using $\tilde{G}(s,L)$ from Eq. (11). We perform an average for different $q_{\max }$ values between 1200 and 2000 MeV.

Figure 4

Solid curves are $K\pi$ scattering energy levels evaluated using Eqs. (4) and (11), and dashed curves are energy levels evaluated using Eqs. (14) and (11), when $q_{\max }^{\prime }=724.70\mathrm{MeV}$ [21].

Figure 5

On the left-hand side, the curves ending with $|\text{−}|$, $\parallel \text{−}\parallel$, and $|\parallel \text{−}|\parallel$ correspond to the phase shifts extracted from the first (lowest), the second, and the third energy levels of Fig. 3, respectively. On the right-hand side, the solid curve is the phase shift extracted from the lowest energy level, the dashed curve is the phase shifts calculated in the infinite volume, and the experimental data are denoted as solid circles [28] and triangles [29].

Figure 6

Uncertainties of energy levels and phase shifts.

Figure 7

Energy levels as functions of the cubic box size $L$. The left, middle, and right figures correspond to $m_{\pi }=1.5m_{\pi }^0$, $m_{\pi }=2.0m_{\pi }^0$, and $m_{\pi }=2.5m_{\pi }^0$, respectively. The solid curves are obtained using the physical kaon mass $m_K=496\mathrm{MeV}$, and the dotted curves are obtained using the nonphysical kaon mass evaluated using Eq. (21).

Figure 8

$K\pi$ phase shifts with different pion masses. The left, middle, and right figures correspond to $m_{\pi }=1.5m_{\pi }^0$, $m_{\pi }=2.0m_{\pi }^0$, and $m_{\pi }=2.5m_{\pi }^0$, respectively. The solid curves are obtained using the physical kaon mass $m_K=496\mathrm{MeV}$, and the dotted curves are obtained using the nonphysical kaon mass evaluated using Eq. (21).

Figure 9

The $K\pi$ energy levels obtained using the chiral unitary theory. In the left panel, we use $m_{\pi }=240\mathrm{MeV}$ and $m_K=548\mathrm{MeV}$ to compare to the results of Ref. [16], and in the right panel, we use $m_{\pi }=266\mathrm{MeV}$ and $m_K=552\mathrm{MeV}$ to compare to the results of Ref. [17]. We also show the lattice results for comparisons [16, 17]. Here, we do not evaluate their uncertainties but note that they are similar to Fig. 6.

Figure 10

The $K^{*}$ mass (left), the coupling constant $g_{K^{*}\pi K}$ (middle), and the decay width ${\mathrm{\Gamma }}_{K^{*}}$ (right) as functions of $m_{\pi }$. The solid curves are obtained using the physical kaon mass $m_K=496\mathrm{MeV}$, and the dashed curves are obtained using the nonphysical kaon mass evaluated using Eq. (21).

Figure 11

The $K^{*}$ mass (left), the coupling constant $g_{K^{*}\pi K}$ (middle), and the decay width ${\mathrm{\Gamma }}_{K^{*}}$ (right) as functions of $m_{\pi }$. The solid curves are obtained using the lowest $K\pi$ energy level, the dotted-dashed curves are obtained using both the first and the second energy levels, and the dashed curves are obtained using the three lowest energy levels. We find that these results are almost the same.

Figure 12

The comparison of our results with results obtained from the standard Lüscher approach. The dashed curves are the phase shift evaluated using the standard Lüscher approach, while the solid curves are our results. The left panel is obtained using the lowest $K\pi$ energy level shown in Fig. 3, in which $m_{\pi }=138\mathrm{MeV}$ and $m_K=496\mathrm{MeV}$, while the right panel is obtained using the lowest $K\pi$ energy level shown in the right panel of Fig. 9, in which $m_{\pi }=266\mathrm{MeV}$ and $m_K=552\mathrm{MeV}$ [17].

Figure 13

Contributions of the second, the third, and the fourth terms of Eq. (22), in which physical pion and kaon mass are used, $m_{\pi }=138\mathrm{MeV}$, and $m_K=496\mathrm{MeV}$. The solid curves are our results; the dashed curves are the phase shift evaluated using the standard Lüscher approach, i.e., the first term of Eq. (22); the short-dashed curves are obtained using the first and the second terms of Eq. (22); the middle-dashed curves are obtained using the first and the third terms of Eq. (22); and the long-dashed curves are obtained using the first and the fourth terms of Eq. (22). We note that sometimes the middle-dashed and long-dashed curves overlap with each other. In the left panel, we show phase shifts extracted from the first energy level of Fig. 3, and in the right panel, we show phase shifts obtained using the second energy level of Fig. 3.

Figure 14

Contributions of the second, the third, and the fourth terms of Eq. (22), in which nonphysical pion and kaon masses are used, $m_{\pi }=266\mathrm{MeV}$, and $m_K=552\mathrm{MeV}$ [17]. We note that sometimes the middle-dashed and long-dashed curves overlap with each other.

Figure 15

The solid curves are the best-fit results ${\chi }_{\min }^2=0.064$, and the bands are fits that fulfill the ${\chi }^2<{\chi }_{\min }^2+1$ criterion. We note that these bands are not error bars, so it is ok that the rightmost point in the right panel is within them. The discrete points on the right figure are the results of the direct determination from each “data” points on the left figure.