3.2 Weight shift
Here we compute the two particle energy with a different procedure. Starting from the correlator
\[ \langle O_{2p_x} O_{2p_x}^\dagger(t)\rangle = |A_{0\to2}|^2 \left( e^{-E_{2p} t}+e^{-E_{2p} (T-t)}\right)+ |A_{\phi(0)\to\phi(p)}|^2 \left(e^{-\omega_{p} T-m t+\omega_{p}t}+e^{-m T-\omega_{p} t+mt} \right)\\ =|A_{0\to2}|^2 e^{-E_{2p} \frac{T}{2}} \cosh\left(E_{2p}(t-\frac{T}{2}) \right)+ |A_{\phi(0)\to\phi(p)}|^2 e^{-(\omega_p+m ) \frac{T}{2}} \cosh\left((\omega_p-m )(t-\frac{T}{2}) \right) \] We divide the correlator by the exponential of the termal pollution term \[ c_w(t)=\frac{c(t)}{e^{-(\omega_p+m ) \frac{T}{2}} \cosh\left((\omega_p-m )(t-\frac{T}{2}) \right)} \] then we do a shift of the correlator \[ c_{ws}(t)=c_w(t+1)-c_w(t) \] in this way we eliminate the dependence over \(|A_{\phi(0)\to\phi(p)}|^2\) and we fit the two parameters \(E_2\) and \(A_{0\to2}\) \[ c_{ws}(t)=|A_{0\to2}|^2 \left\{\frac{e^{-E_{2p} \frac{T}{2}} \cosh\left(E_{2p}(t+1-\frac{T}{2}) \right)} {e^{-(\omega_p+m ) \frac{T}{2}} \cosh\left((\omega_p-m )(t+1-\frac{T}{2}) \right)}- \frac{e^{-E_{2p} \frac{T}{2}} \cosh\left(E_{2p}(t-\frac{T}{2}) \right)} {e^{-(\omega_p+m ) \frac{T}{2}} \cosh\left((\omega_p-m )(t-\frac{T}{2}) \right)} \right\} \]
3.2.0.1 ../out/G2t_T48_L30_msq0-4.900000_msq1-4.650000_l02.500000_l12.500000_mu5.000000_g0.000000_rep0_output
E2_0(L30T48) = 0.26043(14) \(\chi^2/dof=\) 0.25211
E2_0_p1_ws(L30T48) = 0.37667(26) 0.00143477(33) \(\chi^2/dof=\) 0.29432
E2_0_p11_ws(L30T48) = 0.45288(44) 0.00125112(40) \(\chi^2/dof=\) 0.76591
E2_0_p111_ws(L30T48) = 0.51327(70) 0.00114417(42) \(\chi^2/dof=\) 0.72888
E2_0_A1(L30T48) = 0.49564(16) \(\chi^2/dof=\) 2.5133
3.2.0.2 ../out/G2t_T96_L30_msq0-4.900000_msq1-4.650000_l02.500000_l12.500000_mu5.000000_g0.000000_rep0_output
E2_0(L30T96) = 0.26036(10) \(\chi^2/dof=\) 0.12577
E2_0_p1_ws(L30T96) = 0.37628(26) 0.00143292(32) \(\chi^2/dof=\) 0.30754
E2_0_p11_ws(L30T96) = 0.45140(46) 0.00124898(44) \(\chi^2/dof=\) 0.78333
E2_0_p111_ws(L30T96) = 0.51210(75) -0.00114271(39) \(\chi^2/dof=\) 0.19456
E2_0_A1(L30T96) = 0.49594(11) \(\chi^2/dof=\) 2.8021