Scattering from an external field in quantum chromodynamics at high energies: from foundations to interdisciplinary connections

We thank the Referees for their positive feedback and valuable suggestions.

AUTHORS' RESPONSE TO REPORT #2

1) We are happy to take the Referee's suggestion into account by adding a few sentences to the paragraph 'Eikonal scattering'.

2) The term 'fundamental representation' is used in the physics literature with various meanings, not always aligning with the mathematical definition. Probably in most cases, it is used as a synonym for the 'defining' or 'standard' representation. For clarity, we will adopt this terminology in the revised version, allowing us to omit the term 'defining representation' (except for a new footnote), and refer solely to the 'fundamental representation.' We will use the term 'anti-fundamental representation' to denote the complex conjugate of the defining representation.

3) We will refine the definition of the branching random walk model and, in particular, clarify that the constraint on the parameters is necessary to ensure that the elementary probabilities are well-defined. Regarding the notation in Eq. (240), the zeros below the 'location bucket graphs' were intended to indicate the lattice site labeled '0,' where the particle starts. Since the transition probabilities in Eq. (236) are constant and independent of the lattice site, these labels are actually unnecessary. Therefore, we have decided to remove them from all equations where they appeared, namely Eqs. (240) and (253).

AUTHORS' RESPONSE TO REPORT #1

1) It will indeed be better to provide a formula for the 'inverse derivative' operator, especially since its definition requires an explicit choice of boundary conditions.

2) Equation (232) is just the first in an infinite hierarchy of equations. However, as the Referee points out, alternative formulations for the rapidity evolution of dipole-nucleus S-matrix elements (and even for any correlator of Wilson lines at a finite number of colors) do exist, and they are expressed in terms of closed equations. We agree that it will be worth mentioning this in Sec. 4.3.3. (NB: The complexity of these formulations lies in the fact that they appear either as functional equations or, equivalently, as stochastic differential equations.)

3) The initial idea was to discuss random walks and Brownian motion separately from pure branching (i.e., zero-dimensional) models in the first stage, before combining the two processes to address branching Brownian motion. Exact solutions can indeed be derived for the relevant observables in the former processes, and we found it useful to work through these.

Zero-dimensional branching models in general, and Eqs. (247) and (249) in particular, were discussed as toy models for QCD evolution. In the revised version, we emphasize this fact and include the references suggested by the Referee.

- In Sec. 2.2 (paragraph "Eikonal scattering"), we added a physical explanation of why the boost does not affect the external potential.

- In Sec. 2.3.1, we included a paragraph to clarify the group theory terminology used throughout our discussion.

- Further down in Sec. 2.3.1, we provided the full definition of the inverse derivative operator and its "square." We slightly modified a statement made in Sec. 2.3.3, right after the definition of the 4-polarization vector (and Dirac spinors), in order to refer to this newly-introduced definition of the inverse derivative operator.

- In Sec. 4.3.3, we added a paragraph introducing the JIMWLK equation.

- In Sec. 5.1.1, we clarified the definition of the branching random walk model we consider.

- Further down in Sec. 5.1.1, just before the paragraph on "Branching Brownian motion," we included a paragraph referencing relevant works on zero-dimensional models.

- Various inconsequential phrasing improvements have been made, along with corrections to a few typographical errors.