gluon constituent in the wave function, and so the final results. We illustrate the contribution in the above calculations for i = 1 and j = 2. The first term in the above bracket comes from Fig. 1(a), whereas the second term comes from Fig. 1(b). Moreover, at this particular order, quark number is conserved, such that n = n $^{′}$. The gluon polarization tensor is defined as, where v is a light-like vector v · P = 1, ˜ q differs from q in the minus component to take into account the instant propagator contribution [2]. When one solves the Heisenberg eigenvalue problem for light-front QCD as in discretized light-front quantization [3], the light-cone gauge singularity is regulated by the principal value prescription, which corresponds to the antisymmetric boundary condition for the gauge potential A$_{⊥}$ ( x − = + ∞ )+ A$_{⊥}$ ( x − = −∞ ) = 0. However, this prescription will not result into an imaginary part for the light-front wave function from the above interaction kernel. In the following calculation, we will choose the advanced boundary condition: A$_{⊥}$ ( x − = + ∞ ) = 0 whereas A$_{⊥}$ ( x − = −∞ ) = 0 in order to construct the augmented light-front wavefunction. With this boundary condition, the light-cone singularity will be regulated by [7], where the momentum flow of q is toward to the vertex ν . Clearly, this term contains a phase. The imaginary part is simple, and proportional to a Delta function: iπδ ( v · q ). Since we are only interested in the imaginary part of the light-front wave function amplitudes, we simply apply this Delta function to the interaction kernel in Eq. (3). In particular, we find that the dominant contribution comes from the d$_{+}$$_{⊥}$ components of the d$_{µν}$ tensor [7, 15]. All other contributions cancel out between the above two terms or by themselves. Another important consequence is that the helicities are conserved in the interaction kernel: δ$_{λ}$$_{i}$$_{λ}$ ′ $_{i}$δ$_{λ}$$_{j}$$_{λ}$ ′ $_{j}$. After a little algebra, we obtain a rather simple result for the imaginary part of the light-front wave function amplitude generated from lowest order perturbation theory, where [C . F . ] represents the color-factor for the Feynman diagram in Fig. 1 and ψ ( ij ) n = ψ$_{n}$ ( x$_{α}$ ; ℓ$_{i}$$_{⊥}$ = k$_{i}$$_{⊥}$ − q$_{⊥}$, ℓ$_{j}$$_{⊥}$ = k$_{j}$$_{⊥}$ + q$_{⊥}$, ℓ$_{β}$$_{⊥}$ | β = i,j = k$_{β}$$_{⊥}$ ). We emphasize that the wave function at the right hand side only contains real part. 6 d$_{µν}$ = − g$_{µν}$ + v$_{µ}$ ˜ q$_{ν}$ + v$_{ν}$ ˜ q$_{µ}$ [ v · q ] , (4) v$_{µ}$ ˜ q$_{ν}$ + v$_{µ}$ ˜ q$_{ν}$ [ v · q ] | Adv. = v$_{µ}$ ˜ q$_{ν}$ v · q − iϵ + v$_{ν}$ ˜ q$_{µ}$ v · q + iϵ , (5) I [ ψ$_{n}$ ( x$_{α}$, k$_{α}$$_{⊥}$ )] = − α$_{s}$ 2 π [C . F . ] ∫ d $^{2}$q$_{⊥}$ ⃗ q 2 ⊥ ( 1 − P − − ∑ ℓ − P − k − ) ∑ i = j ψ ( ij ) n ( x$_{β}$ ; ℓ$_{β}$$_{⊥}$ ) , (6)