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