Chapter 18 GEVP g0.025

with the term $g\phi_0^3\phi_1$ we expect the following spectral decomposition
$$\langle\phi_1(0)\phi_1(t) \rangle \approx |\langle 0|\phi_1|\phi_1\rangle|^2| \left( e^{-M_1t}+e^{-M_1(T-t)}\right) +|\langle 0|\phi_1|\phi_0\rangle|^2 \left( e^{-Mt}+e^{-M(T-t)}\right)\\ +|\langle 0|\phi_1|\phi_0^3\rangle|^2 \left( e^{-E_3t}+e^{-E_3(T-t)}\right) +|\langle \phi_0|\phi_1|\phi_0^2\rangle|^2 e^{-(M)T} \left( e^{-(E_2-M)t}+e^{-(E_2-M)(T-t)}\right)\,.$$

Together with the previous correlator we have $\langle\phi_0(0)\phi_0(t) \rangle$ and $\langle\phi_0^3(0)\phi_0^3(t) \rangle$ which interpolates the same states. However the relevant states should be

$$\langle\phi_0(0)\phi_0(t) \rangle \approx |A_{\phi_0\to\phi_0}| \left( e^{-Mt}+e^{-M(T-t)}\right)$$

and the other should be suppressed. While

$$\langle\phi_0^3(0)\phi_0^3(t) \rangle \approx |A_{\phi_0^3\to\phi_0^3}| \left( e^{-E_3t}+e^{-E_3(T-t)}\right)+ |A_{\phi_0\to\phi_0}| \left( e^{-Mt}+e^{-M(T-t)}\right)\\ +|A_{\phi_0\to\phi_0^2}| e^{-(M)T} \left( e^{-(E_2-M)t}+e^{-(E_2-M)(T-t)}\right)\,.$$

We construct the matrix

$$C(t)=\begin{pmatrix} \langle\phi_0(0)\phi_0(t) \rangle & \langle\phi_0^3(0)\phi_0(t) & \langle\phi_0(0)\phi_1(t) \rangle \rangle \\ & \langle\phi_0^3(0)\phi_0^3(t) \rangle & \langle\phi_0^3(0)\phi_1(t) \rangle \\ & &\langle\phi_1(0)\phi_1(t) \rangle \end{pmatrix}$$ the matrix since $C$ is hermitiian so the lower triangular non-written part can be computed as $C_{ij}=C_{ji}^*$.

We solve the GEVP $$C(t)v_n(t,t_0)=\lambda_n(t,t_0)C(t_0) v_n(t,t_n)$$ We fit $\lambda_n(t,t_0)$ as $$\lambda_n(t,t_0)=C \left(e^{ - E_n (t-t_0)} + e^{ - E_n ((T - t)-t_0)}\right)$$

We also consider the case where we add an other operato to the GEVP

We construct the matrix

$$C(t)=\begin{pmatrix} \langle\phi_0(0)\phi_0(t) \rangle & \langle\phi_0^3(0)\phi_0(t) & \langle\phi_0(0)\phi_1(t) \rangle \rangle &\langle\phi_0(0) O_{A1}(t) \rangle \\ & \langle\phi_0^3(0)\phi_0^3(t) \rangle & \langle\phi_0^3(0)\phi_1(t) \rangle &\langle\phi_0^3(0) O_{A1}(t) \rangle\\ & &\langle\phi_1(0)\phi_1(t) \rangle &\langle\phi_1(0) O_{A1}(t) \rangle \\ && &\langle O_{A1} (0) O_{A1}(t) \rangle \end{pmatrix}$$

18.0.0.1 ../../g0.025/out/G2t_T32_L28_msq0-4.900000_msq1-4.650000_l02.500000_l12.500000_mu5.000000_g0.025000_rep0_output

GEVP_phi0_phi03_phi1_meffl0(L28T32) = 0.127893(91) $\chi^2/dof=$ 0.11154

GEVP_phi0_phi03_phi1_meffl1(L28T32) = 0.36939(57) $\chi^2/dof=$ 315.77

GEVP_phi0_phi03_phi1_meffl2(L28T32) = 0.44696(12) $\chi^2/dof=$ 0.71698

E1_0(L28T32) = 0.127925(91) $\chi^2/dof=$ 0.27266

E1_1(L28T32) = 0.44434(13) $\chi^2/dof=$ 72.605

E3_0(L28T32) = 0.18428(16) $\chi^2/dof=$ 7411.4

18.0.0.2 ../../g0.025/out/G2t_T64_L24_msq0-4.900000_msq1-4.650000_l02.500000_l12.500000_mu5.000000_g0.025000_rep0_output

GEVP_phi0_phi03_phi1_meffl0(L24T64) = 0.128339(68) $\chi^2/dof=$ 0.027406

GEVP_phi0_phi03_phi1_meffl1(L24T64) = 0.40411(43) $\chi^2/dof=$ 3.2019

GEVP_phi0_phi03_phi1_meffl2(L24T64) = 0.446984(70) $\chi^2/dof=$ 2.2663

E1_0(L24T64) = 0.128384(68) $\chi^2/dof=$ 0.22881

E1_1(L24T64) = 0.444533(75) $\chi^2/dof=$ 110.09

E3_0_vev(L24T64) = 0.4096(13) 0.0004845(11) 0.0111(24) 0.000573(11) $\chi^2/dof=$ 0.001885

18.0.0.3 ../../g0.025/out/G2t_T128_L24_msq0-4.900000_msq1-4.650000_l02.500000_l12.500000_mu5.000000_g0.025000_rep0_output

GEVP_phi0_phi03_phi1_meffl0(L24T128) = 0.128475(49) $\chi^2/dof=$ 0.14153

GEVP_phi0_phi03_phi1_meffl1(L24T128) = 0.40571(24) $\chi^2/dof=$ 2.3905

GEVP_phi0_phi03_phi1_meffl2(L24T128) = 0.447005(89) $\chi^2/dof=$ 0.55427

E1_0(L24T128) = 0.128525(48) $\chi^2/dof=$ 0.0040655

E1_1(L24T128) = 0.444192(83) $\chi^2/dof=$ 107.73

E3_0_vev(L24T128) = 0.4073(21) 0.0004844(21) 0.73(25) 0.000566(21) $\chi^2/dof=$ 0.0010442

18.0.0.4 ../../g0.025/out/G2t_T64_L28_msq0-4.900000_msq1-4.650000_l02.500000_l12.500000_mu5.000000_g0.025000_rep0_output

GEVP_phi0_phi03_phi1_meffl0(L28T64) = 0.127768(50) $\chi^2/dof=$ 0.3018

GEVP_phi0_phi03_phi1_meffl1(L28T64) = 0.39594(39) $\chi^2/dof=$ 0.49502

GEVP_phi0_phi03_phi1_meffl2(L28T64) = 0.447000(97) $\chi^2/dof=$ 0.2441

E1_0(L28T64) = 0.127807(50) $\chi^2/dof=$ 0.81852

E1_1(L28T64) = 0.444506(98) $\chi^2/dof=$ 80.579

E3_0_vev(L28T64) = 0.39752(97) 0.00024751(14) 0.0(0.1)e-2 0.00030418(81) $\chi^2/dof=$ 0.0013618

18.0.0.5 ../../g0.025/out/G2t_T64_L32_msq0-4.900000_msq1-4.650000_l02.500000_l12.500000_mu5.000000_g0.025000_rep0_output

GEVP_phi0_phi03_phi1_meffl0(L32T64) = 0.127450(66) $\chi^2/dof=$ 0.44402

GEVP_phi0_phi03_phi1_meffl1(L32T64) = 0.39086(42) $\chi^2/dof=$ 0.45843

GEVP_phi0_phi03_phi1_meffl2(L32T64) = 0.446926(75) $\chi^2/dof=$ 1.3097

E1_0(L32T64) = 0.127484(66) $\chi^2/dof=$ 0.8008

E1_1(L32T64) = 0.444649(75) $\chi^2/dof=$ 73.043

E3_0_vev(L32T64) = 0.39275(71) -0.00013691(13) 0.0(0.6)e-3 -0.00016823(36) $\chi^2/dof=$ 0.0013822

18.0.0.6 ../../g0.025/out/G2t_T64_L34_msq0-4.900000_msq1-4.650000_l02.500000_l12.500000_mu5.000000_g0.025000_rep0_output

GEVP_phi0_phi03_phi1_meffl0(L34T64) = 0.127577(74) $\chi^2/dof=$ 0.10803

GEVP_phi0_phi03_phi1_meffl1(L34T64) = 0.38950(42) $\chi^2/dof=$ 0.72253

GEVP_phi0_phi03_phi1_meffl2(L34T64) = 0.447041(61) $\chi^2/dof=$ 0.12482

E1_0(L34T64) = 0.127611(74) $\chi^2/dof=$ 0.014865

E1_1(L34T64) = 0.444823(66) $\chi^2/dof=$ 61.492

E3_0_vev(L34T64) = 0.39047(65) 0.000104471(92) -0.0002(15) 0.0001282(22) $\chi^2/dof=$ 0.0010512

18.0.0.7 ../../g0.025/out/G2t_T64_L36_msq0-4.900000_msq1-4.650000_l02.500000_l12.500000_mu5.000000_g0.025000_rep0_output

GEVP_phi0_phi03_phi1_meffl0(L36T64) = 0.127426(55) $\chi^2/dof=$ 0.35818

GEVP_phi0_phi03_phi1_meffl1(L36T64) = 0.38744(48) $\chi^2/dof=$ 0.16284

GEVP_phi0_phi03_phi1_meffl2(L36T64) = 0.446835(77) $\chi^2/dof=$ 0.32183

E1_0(L36T64) = 0.127460(55) $\chi^2/dof=$ 0.7874

E1_1(L36T64) = 0.444504(84) $\chi^2/dof=$ 63.021

E3_0_vev(L36T64) = 0.39011(76) -809.8(0.7)e-7 0.1(0.3)e-4 99.5(0.1)e-6 $\chi^2/dof=$ 0.014451

18.0.0.8 ../../g0.025/out/G2t_T64_L40_msq0-4.900000_msq1-4.650000_l02.500000_l12.500000_mu5.000000_g0.025000_rep0_output

GEVP_phi0_phi03_phi1_meffl0(L40T64) = 0.127361(68) $\chi^2/dof=$ 0.1417

GEVP_phi0_phi03_phi1_meffl1(L40T64) = 0.38688(77) $\chi^2/dof=$ 0.69793

GEVP_phi0_phi03_phi1_meffl2(L40T64) = 0.447166(60) $\chi^2/dof=$ 0.75002

E1_0(L40T64) = 0.127400(68) $\chi^2/dof=$ 0.32872

E1_1(L40T64) = 0.445157(66) $\chi^2/dof=$ 42.03

E3_0_vev(L40T64) = 0.3903(15) 50.4(0.1)e-6 -0.00175(57) -5.5(0.5)e-5 $\chi^2/dof=$ 0.0019332