18.3 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

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

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

$$\begin{gather} \chi^2/d.o.f.=2.50407 \\ P[0]=-0.154299\pm (0.0023) \\ P[1]=-3.48222\pm (0.21) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 5.51e-06& 0.793\\ 0.793& 0.0442\\ \end{pmatrix} \end{gather}$$ }

18.3.1 $M_L$ in zeta func and $M_\infty$ as normalization

$$\begin{gather} \chi^2/d.o.f.=2.48851 \\ P[0]=-0.15426\pm (0.0024) \\ P[1]=-3.48283\pm (0.21) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 5.55e-06& 0.792\\ 0.792& 0.0441\\ \end{pmatrix} \end{gather}$$ }

18.3.2 Using $M_\infty$

$$\begin{gather} \chi^2/d.o.f.=4.07909 \\ P[0]=-0.15747\pm (0.0029) \\ P[1]=-3.50149\pm (0.21) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 8.35e-06& 0.734\\ 0.734& 0.0435\\ \end{pmatrix} \end{gather}$$ }

18.3.3 no correction

$$\begin{gather} \chi^2/d.o.f.=13.5714 \\ P[0]=-0.160133\pm (0.0032) \\ P[1]=-38.1117\pm (1.2) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 1.03e-05& -0.133\\ -0.133& 1.4\\ \end{pmatrix} \end{gather}$$ }