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 Problem Statement

Hi there,
I'm trying to reproduce the results of one article and writing a program that performs calculation of twopoint Green's function G(x'x) in 4D using Quantum MonteCarlo approach.
I'm cosidering the theory of a massive scalar field with a quartic potential (lambdapsi^4).
Here I will denote scalar field as 'psi' just not to mix it up with an azimuthal coordinate 'phi'.
I use spherical coordinates and want to calculate the Green's function along azimuthal coordinate and also to get its dependency on rcoordinate. I consider two points that have similar t, r and theta coordinates but phicoordinates differ on pi. So that G(x'x) = G(pi).
For different rcoordinates I will get different G(pi).
The problem is that I don't understand how to write down the exact computational formula to obtain this rdependency.
 Relevant Equations

After all the standard steps of Quantum MonteCarlo simulation I get the configurations of the scalar fields psi(t, r, theta, phi).
Then G(pi) = <psi(t, r, theta, phi)*psi(t, r, theta, phi+pi)>
Introducing the spacetime spherical symmetric lattice, I use the following notifications in my program.
i  index enumerating the nodes along tcoordinate,
j  along the rcoordinate,
k  along the thetacoordinate,
l  along the phicoordinate.
N_t  the number of nodes along tcoordinate.
N_r  along the rcoordinate,
N_theta  along the thetacoordinate,
N_phi  along the phicoordinate.
I tried to get the rdependency as following. I fixed the coordinate r by fixing the jindex, and then calculated G(pi) using the formula:
C++ code
Here (l+N_phi/2)%N_phi ensures that x'x = π.
However, this algorithm seems to be incorrect. Comparing the results with an article, I obtain incorrect graphics for G(pi) dependence on r.
Would be glad if someone gives an advice on it. Thanks in advance!
i  index enumerating the nodes along tcoordinate,
j  along the rcoordinate,
k  along the thetacoordinate,
l  along the phicoordinate.
N_t  the number of nodes along tcoordinate.
N_r  along the rcoordinate,
N_theta  along the thetacoordinate,
N_phi  along the phicoordinate.
I tried to get the rdependency as following. I fixed the coordinate r by fixing the jindex, and then calculated G(pi) using the formula:
C++ code
C++:
for(int j=0; j<N_r; j++)
{
G(j) = 0;
for(int i =0: i<N_t; i++)
{
for(int k=0; k<N_theta; k++)
{
for(int l = 0; l<N_phi; l++)
{
G(j)=G(j) + psi(i, j, k, l)*psi(i, j, k, (l+N_phi/2)%N_phi);
}
}
}
G(j) = G(j)/(N_t*N_theta*N_phi) //average along t, theta, phi directions
}
Here (l+N_phi/2)%N_phi ensures that x'x = π.
However, this algorithm seems to be incorrect. Comparing the results with an article, I obtain incorrect graphics for G(pi) dependence on r.
Would be glad if someone gives an advice on it. Thanks in advance!