20.3 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

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

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

$$\begin{gather} \chi^2/d.o.f.=0.852098 \\ P[0]=-0.147936\pm (0.002) \\ P[1]=-4.14553\pm (0.18) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 1& 0.00157\\ 0.00157& 1\\ \end{pmatrix} \end{gather}$$ }

20.3.1 $M_L$ in zeta func and $M_\infty$ as normalization

$$\begin{gather} \chi^2/d.o.f.=0.852883 \\ P[0]=-0.147924\pm (0.002) \\ P[1]=-4.1478\pm (0.18) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 1& 0.00157\\ 0.00157& 1\\ \end{pmatrix} \end{gather}$$ }

20.3.2 Using $M_\infty$

$$\begin{gather} \chi^2/d.o.f.=1.33012 \\ P[0]=-0.150333\pm (0.0032) \\ P[1]=-4.18722\pm (0.23) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 1& 0.00286\\ 0.00286& 1\\ \end{pmatrix} \end{gather}$$ }

20.3.3 no correction

$$\begin{gather} \chi^2/d.o.f.=7.45428 \\ P[0]=-0.152828\pm (0.0026) \\ P[1]=-48.4841\pm (1.4) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 1& -0.000143\\ -0.000143& 1\\ \end{pmatrix} \end{gather}$$ }