JIWMLK evolution in Julia with the goal of writing rc-parent-dipole JIMWLK on CUDA.
- Consider a Gaussian field
$\rho$ with the variance (in lattice unites already)
$$ \langle \rho^a( x = a i_x) \rho^b( y =a i_y) \rangle = \frac{1}{N_y a^2} \mu^2 \delta^{ab} \delta_{i_x, i_y} $$
where
- One needs to solve the Posson equation
$\Delta A^a(x) = \rho^a(x)$ In Fourier space, this is simply: $$ A^a(k) = \frac{\rho(k)}{m^2+\frac{4}{a^2} ( \sin^2 (\frac{k_1 a }{ 2 } ) + \sin^2 (\frac{k_2 a }{ 2 } ) )}$$ For periodic boundary conditions$k_{1,2} = \frac{2\pi}{L} i_{1,2}$ , where$i_{1,2}$ are the integers$i_{1,2}=0,...,N$ and$L=a , N$ ; we also accounted for IR regularization with$m^2$ . Thus the procedure involves 1) FFT of$\rho$ to momentum sapce; 2) finding$A(k)$ ; 3) FFT$A$ to coordinate space - Next we need to compute path ordered Wilson line; we start with just a single local Wilson line first.
$$ U(x) = \exp (i t^a A^a(x)), $$ where
$U$ is 3 by 3 matrix;$t^a$ are Gell-Mann matrices.
We use the package https://juliahub.com/ui/Packages/GellMannMatrices/GVbjE/0.1.1 . - We repeat the procedure
$N_y$ times and (matrix) multiply all$U$ together $U = U_1 U_2 U_3 ...$. - The resulting
$U(x)$ serves as the initial condition for JIMWLK evolution
$$ V_{Y+\Delta Y}(x) = \exp \Bigg( - i \frac{\sqrt{\alpha \Delta Y}}{\pi} \int_z K_i (x-z) \cdot (V(z) {\xi_i}(z) V^\dagger(z)) \Bigg) V(x) \exp \Bigg( i \frac{\sqrt{\alpha \Delta Y}}{\pi} \int_z K_i (x-z) \cdot \xi_i(z)\Bigg), $$
where