11.5 Lattice dispersion relation

given the parametrization

\[ \frac{k}{m} \cot{ \delta}=\frac{1}{am}+\frac{r_0 k^2}{2m}\,, \] with \(P[0]=am\) , \(P[1]=r_0 m\) , \(P[2]=P_2\). We solve the quantization condition finding the value of \(k\) such that \[ \cot{ \delta}=\frac{Z_{00}(1,q^2)}{\pi^{3/2}\gamma q}. \]

From k we compute the two particle energy \(E_2\)

\[ k^2=\frac{E_{CM}^2}{4}-m^2 = \frac{E^2-\vec{P}^2}{4}-m^2. \]

and finally the energy shift \[ \Delta E_2^{predicted} = E_2 - \sqrt{4m^2 + p_1^2}-\sqrt{4m^2 + p_2^2} \]

On the other hand we measure the two particle energy and we compute the energy shift with the lattice dispersion relation \[ \Delta E_2^{latt}=E_2^{measured}-E_2^{free-latt} \] \[\begin{gather} E_2^{free-latt} = \cosh^{-1}{\left( \cosh(m) +\frac{1}{2}\left( \sum_{i=1}^{2}4 \sin\left(\frac{ p_{1i}}{2}\right)^2\right)\right)} \\ + \cosh^{-1}{\left( \cosh(m) +\frac{1}{2}\left( \sum_{i=1}^{2}4 \sin\left(\frac{ p_{2i}}{2}\right)^2\right)\right)} \,. \tag{11.1} \end{gather}\]

Finally we minimise the \(\chi^2\) \[ \chi^2= \sum_i \frac{( \Delta E_2^{predicted} -\Delta E_2^{latt})^2}{\sigma^2} \]

\[\begin{gather} \chi^2/d.o.f.=0.912112 \\ P[0]=-0.120481\pm (0.0015) \\ P[1]=-4.24382\pm (0.13) \\ \end{gather}\] {\[\begin{gather} C=\begin{pmatrix} 2.19e-06& 0.497\\ 0.497& 0.0159\\ \end{pmatrix} \end{gather}\]}

V1	V2	V3	V4	V5	V6	V7
24	2.05526	0.0010315	32	0.0306013	(0,0,0)	0.236697
28	2.03669	0.0015889	32	0.0247557	(0,0,0)	0.192419
30	2.02786	0.0014532	32	0.0215197	(0,0,0)	0.167482
32	2.02312	0.0013810	32	0.0195871	(0,0,0)	0.152493
36	2.01565	0.0015958	32	0.0160638	(0,0,0)	0.125359
24	3.30710	0.0026425	32	0.1072370	(1,0,0)	0.829466
28	3.04846	0.0011070	32	0.0954119	(1,0,0)	0.741610
30	2.94009	0.0020888	32	0.0896310	(1,0,0)	0.697574
32	2.85229	0.0022296	32	0.0853352	(1,0,0)	0.664370
36	2.71088	0.0017537	32	0.0777009	(1,0,0)	0.606361
24	4.07090	0.0039228	32	0.1317890	(1,1,0)	1.019370
28	3.67927	0.0042767	32	0.1173090	(1,1,0)	0.911813
30	3.53246	0.0035146	32	0.1124440	(1,1,0)	0.875121
32	3.40429	0.0024913	32	0.1081130	(1,1,0)	0.841702
36	3.18781	0.0024844	32	0.0991795	(1,1,0)	0.773975
24	4.58073	0.0022710	32	0.2645580	(1-1,0,0)	2.046330
28	4.07458	0.0029542	32	0.2271230	(1-1,0,0)	1.765370
30	3.87570	0.0031294	32	0.2122380	(1-1,0,0)	1.651790
32	3.69684	0.0024715	32	0.1987850	(1-1,0,0)	1.547630
36	3.41622	0.0027037	32	0.1767760	(1-1,0,0)	1.379520
24	4.69010	0.0070158	32	0.1488430	(1,1,1)	1.151280
28	4.21421	0.0078635	32	0.1349640	(1,1,1)	1.049040
30	4.01712	0.0073082	32	0.1282370	(1,1,1)	0.998036
32	3.84782	0.0047845	32	0.1226830	(1,1,1)	0.955142
36	3.57350	0.0035345	32	0.1129190	(1,1,1)	0.881194

png 2

the resulting \(\delta\) determined is