Chapter 10 report fit phase shift $\delta$

The phase shif 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 toal momentum $\vec{P}$ via $$E_{CM}^2=E^2-\vec{P}^2\,.$$ For the $Z$ function we use the rzeta package. The fit function for the phase shift is $$k \cot{ \delta}=\frac{1}{a}+\frac{r_0 k^2}{2}-Pr_0^3 k^4$$

$T$	$L$	$k \cot(\delta)$	err	$k$	$q^2$
32	24	-1.08593	1.84135e-02	0.0306013	0.013662888
32	28	-1.03311	4.09708e-02	0.0247557	0.012170453
32	30	-1.09773	5.20706e-02	0.0215197	0.010557357
32	32	-1.08661	5.88078e-02	0.0195871	0.009951316
32	36	-1.12134	1.08256e-01	0.0160638	0.008471140
32	24	-1.80346	1.00101e-01	0.1072370	0.167784585
32	28	-1.50407	5.11882e-02	0.0954119	0.180784592
32	30	-1.77873	1.24245e-01	0.0896310	0.183146767
32	32	-1.59680	1.24582e-01	0.0853352	0.188884639
32	36	-1.38922	9.23453e-02	0.0777009	0.198197131
32	24	-4.60620	1.55925e+00	0.1317890	0.253408438
32	28	-112.19000	4.90134e+03	0.1173090	0.273287011
32	30	-5.52459	3.15745e+00	0.1124440	0.288240728
32	32	-2.45020	5.07289e-01	0.1081130	0.303176865
32	36	-2.15729	5.14408e-01	0.0991795	0.322915650
32	24	-3.76511	1.47172e-01	0.2645580	1.021185276
32	28	-2.79984	1.37815e-01	0.2271230	1.024421201
32	30	-2.37259	8.48897e-02	0.2122380	1.026902147
32	32	-2.39690	1.03216e-01	0.1987850	1.024961235
32	36	-2.05591	1.07701e-01	0.1767760	1.025868914
32	24	-56.57610	6.31984e+02	0.1488430	0.323235890
32	28	-6.99573	2.15198e+01	0.1349640	0.361736397
32	30	35.66020	3.57882e+02	0.1282370	0.374894848
32	32	15.36720	6.20218e+01	0.1226830	0.390399268
32	36	6.61338	8.66737e+00	0.1129190	0.418580808

$$\begin{gather} \chi^2/d.o.f.=313.097 \\ P[0]=8.01648e+11\pm (3.7e+10) \\ P[1]=4.81083\pm (0.0093) \\ P[2]=9.60444\pm (0.056) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 1.36e+21& -0.333& -0.333\\ -0.333& 8.59e-05& 1\\ -0.333& 1& 0.00308\\ \end{pmatrix} \end{gather}$$ }

10.0.1 Lattice dispersion relation

We correct the energy with

$$E_2=E_2^{measured}-E_2^{free-latt}+E_2^{free-cont}$$ with $E_2^{free-latt}$ defined in (??)

$T$	$L$	$k \cot(\delta)$	err	$k$	$q^2$
32	24	-1.085930	0.0184135	0.0306013	0.013662888
32	28	-1.033110	0.0409708	0.0247557	0.012170453
32	30	-1.097730	0.0520706	0.0215197	0.010557357
32	32	-1.086610	0.0588078	0.0195871	0.009951316
32	36	-1.121340	0.1082560	0.0160638	0.008471140
32	24	-1.352870	0.0545292	0.1072370	0.167784585
32	28	-1.188310	0.0311852	0.0954119	0.180784592
32	30	-1.351540	0.0697130	0.0896310	0.183146767
32	32	-1.247710	0.0743168	0.0853352	0.188884639
32	36	-1.120780	0.0588649	0.0777009	0.198197131
32	24	-1.372560	0.1374780	0.1317890	0.253408438
32	28	-2.000900	0.3896670	0.1173090	0.273287011
32	30	-1.504980	0.2263330	0.1124440	0.288240728
32	32	-1.123700	0.1067700	0.1081130	0.303176865
32	36	-1.063960	0.1293490	0.0991795	0.322915650
32	24	-2.252280	0.0532213	0.2645580	1.021185276
32	28	-1.919640	0.0650388	0.2271230	1.024421201
32	30	-1.723870	0.0449533	0.2122380	1.026902147
32	32	-1.750520	0.0552633	0.1987850	1.024961235
32	36	-1.577160	0.0634087	0.1767760	1.025868914
32	24	-1.037000	0.1695280	0.1488430	0.323235890
32	28	-0.945481	0.2374770	0.1349640	0.361736397
32	30	-1.168270	0.4128580	0.1282370	0.374894848
32	32	-1.247460	0.4082810	0.1226830	0.390399268
32	36	-1.464220	0.4252110	0.1129190	0.418580808

$$\begin{gather} \chi^2/d.o.f.=407.394 \\ P[0]=9.31691e+11\pm (3e+10) \\ P[1]=4.05769\pm (0.0065) \\ P[2]=9.48081\pm (0.045) \\ \end{gather}$$ { $$\begin{gather} C=\begin{pmatrix} 8.92e+20& -0.272& -0.272\\ -0.272& 4.19e-05& 1\\ -0.272& 1& 0.00206\\ \end{pmatrix} \end{gather}$$ }