Quark mass dependence of on-shell and half off-shell scattering amplitudes from Bethe-Salpeter wave function inside the interaction range

We evaluate scattering amplitudes at on-shell and half off-shell for $I=2$ S-wave two-pion system using the Bethe-Salpeter wave function inside the interaction range in the quenched QCD. The scattering length and effective range are extracted from these scattering amplitudes. Quark mass dependence of them is investigated with the pion mass ranged in 0.52–0.86 GeV. We examine consistency between a result by the conventional finite volume method and our estimate, as well as the phenomenological value.

Figure 1

Effective masses of a single pion. Results of a single exponential fit with $1\sigma$ error are denoted by the solid lines.

Figure 2

Effective energies of $I=2$ two pions. Results of a single exponential fit with $1\sigma$ error are denoted by the solid lines.

Figure 3

Ratios of effective BS wave functions of two pions with the fit results represented by solid horizontal lines. The vertical dotted line denotes the Dirichlet boundary position.

Figure 4

Ratio of the reduced wave function $h(\mathbf{x};k)$ over the wave function $\varphi ({\mathbf{x}}_{\mathrm{ref}};k)$ at a reference point ${\mathbf{x}}_{\mathrm{ref}}=(12,7,2)$ using $k=k_t$. The inside panel enlarges data in $7\le x\le 12$.

Figure 5

Momentum dependence of a ratio of the scattering amplitude at half off-shell $H(p;k)$ over the value at on-shell $H(k;k)$. The vertical line expresses the threshold momentum of the two-pion scattering.

Figure 6

$\mathbf{x}$ dependence of the ratio $R(\mathbf{x};k)$ defined in Eq. (34).

Figure 7

$t$ dependence of the scattering amplitude over the wave function $H_L(k;k)/(C_k\varphi ({\mathbf{x}}_{\mathrm{ref}};k))$ with ${\mathbf{x}}_{\mathrm{ref}}=(12,7,2)$. The vertical dotted line denotes the Dirichlet boundary position.

Figure 8

Numerator and denominator of $H_L(k;k)/(C_k\varphi ({\mathbf{x}}_{\mathrm{ref}};k))$ at ${\kappa }_{\mathrm{val}}=0.1340$ multiplied by $e^{E_{k}t}$ are plotted in the left and right panels.

Figure 9

$Y(\mathbf{x};k)$ in Eq. (38) at ${\kappa }_{\mathrm{val}}=0.1340$ as a function of $|\mathbf{x}|$. An arrow expresses three components of each $\mathbf{x}$.

Figure 10

Reference point ${\mathbf{x}}_{\mathrm{ref}}$ dependence of $a_0/m_{\pi }$ obtained from $H(k;k)$. The solid line represents the result with ${\mathbf{x}}_{\mathrm{ref}}=(12,7,2)$ with $1\sigma$ error band.

Figure 11

Quark mass dependence of the ratio of the scattering length over the pion mass $a_0/m_{\pi }$. Open circles are lattice QCD results with the conventional finite volume method [6]. An open square is a phenomenological estimate of the chiral perturbation theory (ChPT) [18]. Open symbols are shifted to some extent for clarification of data.

Figure 12

Quark mass dependence of the effective range $r_{\mathrm{eff}}$. Open circles are $N_f=0$ lattice QCD results with the conventional finite volume method calculated using data in Ref. [6]. An open triangle is the $N_f=2+1$ lattice QCD result with the finite volume method [19]. An open square is a phenomenological estimate of the chiral perturbation theory (ChPT) [18]. Open symbols are shifted to some extent for clarification of data.

Figure 13

Effective masses and two-pion energies with random Z2 and wall sources on a $24^3\times 64$ lattice. The data with random Z2 source on $24^3\times 96$ are also plotted.

Figure 14

Comparison of $h(\mathbf{x};k)$ using $k=k_t$ with random Z2 and wall sources on a $24^3\times 64$ lattice.

Figure 15

$p^2$ dependence of the ratio of $R_{\mathrm{surf}}(k;p)/R_{\mathrm{surf}}(k;k)$ with ${\mathbf{x}}_{\mathrm{ref}}=(12,7,2)$.