15.3 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.=4.58739P[0]=0.151619±(0.0011)P[1]=3.02208±(0.1) {C=(1.12e060.5590.5590.0102)}

00.20.40.60.811.2−9.5−9−8.5−8−7.5−7−6.5−6
$k/m$$\frac{k}{m} \cot \delta $

15.3.1 ML in zeta func and M as normalization

χ2/d.o.f.=4.59698P[0]=0.151542±(0.0011)P[1]=3.02586±(0.1) {C=(1.13e060.5610.5610.0102)}

00.20.40.60.811.2−9.5−9−8.5−8−7.5−7−6.5−6
$k/m$$\frac{k}{m} \cot \delta $

15.3.2 Using M

χ2/d.o.f.=6.7062P[0]=0.164027±(0.0014)P[1]=3.49212±(0.1) {C=(1.87e060.5030.5030.00994)}

00.20.40.60.811.2−9.5−9−8.5−8−7.5−7−6.5−6−5.5
$k/m$$\frac{k}{m} \cot \delta $