Question about feature_alignment align2D

[HCHrobo]

in the paper，the photometric cost is Ic-Ir, and use inverse compositional Lucas-Kanade algorithm, so Ic can be a constant var, the jacobi of uv can be:
J[0] = 0.5 * (it[1] - it[-1]);
J[1] = 0.5 * (it[ref_step] - it[-ref_step]);
but in the source code, the error is:
float res = search_pixel - alpha * (*it_ref) + mean_diff;
search_pixel is in cur_img, it_ref is the patch in ref_img, if use this cost function, the J[0] and J[1] should have a coefficient alpha?
Am i right?
or the gradient expansion at alpha=1 and beta=0，so the J[0] and J[1] is only related to the photometric gradient?