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 $i_x$, $i_y$, $x$ and $y$ are two-dimensional vectors

• 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 $\xi_i(x) = t^a \xi_i^a(x)$ and $\xi_i^a(x)$ is a random Gaussian noise with unit variance:

$$ \langle \xi^a_i (x) \xi^b_j (x') \rangle = \frac{1}{a^2} \delta_{ij}\delta_{ab} \delta_{i_x, i_y} ,. $$