Chapter 4 E2 connected

We define the generator of the correlation function as $$Z[J_0,J_1]=N\int D\phi_0D\phi_1 e^{-S+\int J_0\phi_0+\int J_1\phi_1} \,.$$ with $N=1/Z[0,0]$ such that $Z[0,0]=1$. the correlation function acting with derivativesrespect to $J$ and then setting it to zero, for example $$\langle \phi_0(x)^2 \phi_0(y)^2\rangle= \frac{\partial^2}{\partial^2 J_0(x)}\frac{\partial^2}{\partial^2 J_0(y)}Z[J_0,J_1]\bigg|_{J_0=J_1=0}\,.$$ We can define the connected correlation function deriving logarithm of $Z$ i.e. $$\langle \phi_0(x)^2 \phi_0(y)^2\rangle_c= \frac{\partial^2}{\partial^2 J_0(x)}\frac{\partial^2}{\partial^2 J_0(y)}\log Z[J_0,J_1]\bigg|_{J_0=J_1=0}\\ =\langle \phi_0(x)^2 \phi_0(y)^2\rangle-\langle \phi_0(x) \phi_0(y)\rangle^2 -\langle \phi_0(x)^2\rangle\langle \phi_0(y)^2\rangle$$ we use the fact that $\langle \phi (x)\rangle=0$.

Below we plot the effective mass computed from

• $\langle \phi_0(x)^2 \phi_0(y)^2\rangle$
• $\langle \phi_0(x)^2 \phi_0(y)^2\rangle-\langle \phi_0(x) \phi_0(y)\rangle^2$
• $\langle \phi_0(x) \phi_0(y)\rangle^2$

In all cases we remove the constant $\langle \phi_0(x)^2\rangle$ term shifting the correlator.

4.0.0.1 ../out/G2t_T32_L24_msq0-4.900000_msq1-4.650000_l02.500000_l12.500000_mu5.000000_g0.000000_rep0_output

E2_0(L24T32) = 0.26571(17) $\chi^2/dof=$ 0.45542

E2_0_con(L24T32) = 0.27326(28) $\chi^2/dof=$ 0.17525

E1_0_sq(L24T32) = 0.25850(10) $\chi^2/dof=$ 1.0445

E1_0(L24T32) = 0.129285(59) $\chi^2/dof=$ 0.039038

4.0.1 FT $\phi^2$

Below we add to the plot the FT of the operator $\phi(x)^2$ difined as

$$\widetilde{\phi(t,p)^2}=\sum_{\vec x}\phi(t,\vec x)^2 e^{i\vec{p}\cdot\vec{x}}$$ we call its effective mass E2_0_p0

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

E2_0(L28T64) = 0.26027(12) $\chi^2/dof=$ 1.1137

E2_0_con(L28T64) = 0.26479(21) $\chi^2/dof=$ 0.45586

E1_0_sq(L28T64) = 0.255803(69) $\chi^2/dof=$ 2.0787

E2_0_p0(L28T64) = 1.3590(23) $\chi^2/dof=$ 824.81

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

4.0.2 Test ensemble