7.3 Luescher analysis

The phase shift can be computed from the formula

$$\cot{ \delta}=\frac{Z_{00}(1,q^2)}{\pi^{3/2}\gamma q}$$ where $\gamma=E/E_{CM}$, $q=kL/2\pi$ with $k$ the scattering momentum $$k^2=\frac{E_{CM}^2}{4}-m^2 = \frac{E^2-\vec{P}^2}{4}-m^2.$$ The Energy in the center of mass is related to the one in a generic frame with total momentum $\vec{P}$ via $$E_{CM}^2=E^{2}_{imp}-\vec{P}^2\,.$$ $E_{imp}$ is the energy measured in the lattice $$E=E^{measured}-E^{free-latt}+E^{free-cont}$$ $$E^{free-latt} = \cosh^{-1}{\left( \cosh(m) +\frac{1}{2}\left( \sum_{i=1}^{3}4 \sin\left(\frac{ p_{1i}}{2}\right)^2\right)\right)} \\ + \cosh^{-1}{\left( \cosh(m) +\frac{1}{2}\left( \sum_{i=1}^{3}4 \sin\left(\frac{ p_{2i}}{2}\right)^2\right)\right)} \,.$$

For the $Z$ function we use the rzeta package. The fit function for the phase shift is

$$\frac{k}{m} \cot{ \delta}=\frac{1}{a_0m}+\frac{r_0 m }{2}\frac{k^2}{m^2}$$

$P[0]=am$ , $P[1]=r_0m$

$$\begin{gather} \chi^2/d.o.f.=0.887249 \\ P[0]=-0.151165\pm (0.0015) \\ P[1]=-3.19192\pm (0.13) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 1& 0.00107\\ 0.00107& 1\\ \end{pmatrix} \end{gather}$$ }

7.3.1 $M_L$ in zeta func and $M_\infty$ as normalization

$$\begin{gather} \chi^2/d.o.f.=0.879456 \\ P[0]=-0.151075\pm (0.0015) \\ P[1]=-3.19427\pm (0.13) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 1& 0.00106\\ 0.00106& 1\\ \end{pmatrix} \end{gather}$$ }

7.3.2 Using $M_\infty$

$$\begin{gather} \chi^2/d.o.f.=1.41617 \\ P[0]=-0.160813\pm (0.0031) \\ P[1]=-3.50648\pm (0.16) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 1& 0.00237\\ 0.00237& 1\\ \end{pmatrix} \end{gather}$$ }

7.3.3 no correction

$$\begin{gather} \chi^2/d.o.f.=17.0811 \\ P[0]=-0.153688\pm (0.0019) \\ P[1]=-22.6356\pm (0.58) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 1& 0.000378\\ 0.000378& 1\\ \end{pmatrix} \end{gather}$$ }