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I learnt here Is a neutron deflected sideways by a laser beam? that a photon beam has no influence on the motion of a free neutron in the first and second approximation. Now I'm interested in what happens when the two particle paths intersect. After all, the photon has a momentum.

Does the photon flow around it? How can I expand my knowledge to include a visual explanation of this process?

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In Quantum Field Theory, each species of particle is a separate field. There can be a photon in a position, and a neutron in the same position, with no problem - they don't need to "cheat" to be in the same position. The theory's Lagrangian determines how the individual particles move when undisturbed, and how they interact when they meet at the same place (e.g., a photon and electron may annihilate and create a new deflected electron - this is how photons interact with electrons).

The photon only interacts with charged particles, which is why it wouldn't interact with a neutron. However, they can still interact through various second-order interactions - e.g., a neutron temporarily converts into a proton and an electron, the photon interacts with the electron, and the electron and the proton turn back into a neutron. These second-order interactions have very low probability of happening ("cross-section") because it is the multiplication of the probabilities of all the separate interactions happening.

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    $\begingroup$ Minor comment (v1): an oscillation like $\rm n\to pe \to n$ requires a flavor-changing weak interaction. A better example is the strong oscillation $\rm n\to p\pi^- \to n$, though it takes some gymnastics to connect that model to the neutron's charge distribution. $\endgroup$
    – rob ♦
    Commented Dec 18, 2023 at 7:08
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    $\begingroup$ Surely the neutron is composed of charged quarks, has a finite size and magnetic moment and therefore a non-zero chance of interaction with a photon? $\endgroup$
    – ProfRob
    Commented Dec 18, 2023 at 7:14
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    $\begingroup$ @ProfRob The wavelength of visible light is so large compared to a neutron that it cannot "see" the individual charges in the neutron. Yes there is probability of scattering but it's so small as to be effectively zero. Use a gamma ray instead and the scattering probability would be larger though still small. $\endgroup$
    – John Rennie
    Commented Dec 18, 2023 at 7:32
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    $\begingroup$ @JohnRennie sure, it's weak but infinitely stronger than zero. I presume then that the second order effects referred to here would correspond to the interaction with that non-zero magnetic moment? $\endgroup$
    – ProfRob
    Commented Dec 18, 2023 at 7:43
  • $\begingroup$ @JohnRennie please write it as an answer $\endgroup$
    – HolgerFiedler
    Commented Dec 18, 2023 at 8:11
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How can I expand my knowledge to include a visual explanation of this process?

It may be useful to compare your question with the following: "How does a photon "cheat" its way past a photon?" So there is practically no interaction between photons. QED predicts scattering of light on light, but it has not been yet demonstrated experimentally, as far as I know.

Let me also note that there is a prediction of non-relativistic neutron spin-flipping by state-of the-art laser beams:

The non-relativistic Pauli equation is used to study the interaction of slow neutrons with a short magnetic pulse. In the extreme limit, the pulse is acting on the magnetic moment of the neutron only at one instant of time. We obtain the scattering amplitude by deriving the junction conditions for the Pauli wave function across the pulse. Explicit expressions are given for a beam of polarized plane wave neutrons subjected to a pulse of spatially constant magnetic field strength. Assuming that the magnetic field is generated by an ultrashort laser pulse, we provide crude numerical estimates.

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