14.3 Luescher analysis

The phase shift can be computed from the formula

cotδ=Z00(1,q2)π3/2γq where γ=E/ECM, q=kL/2π with k the scattering momentum k2=E2CM4m2=E2P24m2. The Energy in the center of mass is related to the one in a generic frame with total momentum P via E2CM=E2impP2. Eimp is the energy measured in the lattice E=EmeasuredEfreelatt+Efreecont Efreelatt=cosh1(cosh(m)+12(3i=14sin(p1i2)2))+cosh1(cosh(m)+12(3i=14sin(p2i2)2)).

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

kmcotδ=1a0m+r0m2k2m2

P[0]=am , P[1]=r0m

χ2/d.o.f.=4.15496P[0]=0.141378±(0.0003)P[1]=2.97907±(0.043) {C=(9.03e080.5340.5340.00183)}

00.511.52−13−12−11−10−9−8−7
$k/m$$\frac{k}{m} \cot \delta $

14.3.1 ML in zeta func and M as normalization

χ2/d.o.f.=4.50788P[0]=0.140408±(0.0003)P[1]=2.99413±(0.043) {C=(8.89e080.5340.5340.00188)}

00.511.52−13−12−11−10−9−8−7
$k/m$$\frac{k}{m} \cot \delta $

14.3.2 Using M

χ2/d.o.f.=35.7968P[0]=0.184981±(0.001)P[1]=3.44905±(0.035) {C=(1.09e060.4720.4720.00119)}

00.511.52−13−12−11−10−9−8−7−6−5
$k/m$$\frac{k}{m} \cot \delta $