The undergrad physics curriculum includes the standard college sequence of calculus and differential equations. Many physics students take more math than this, or today, probably some computer science. We took a "physics" course called "theoretical physics" that filled some gaps in the math sequence, most notably multivariate calculus in cylindrical and spherical coordinates, path integrals, and some stuff like that. Specific upper level physics courses also introduced math topics as needed.

Graduate coursework re-iterated the same "theoretical physics" course, but it was called "mathematical methods." We used a textbook by Arfken, that covered the same multivariate calculus plus vector and tensor math. Finding out what the current textbook du jour is for this course, and perusing the table of contents, would give you a pretty good idea of what to expect.

I didn't go into scattering theory as a specialty, but we covered it in some of the other courses, and I remember doing difficult integrals and differential equations in spherical polar coordinates. The coursework quickly runs you out of problems that can be solved in closed form. I'd be shocked if the real work isn't done today with mostly numerical techniques.

You need to work up to (and through) Quantum Field Theory to understand this.

Thankfully, you don't need to be as good or as thorough as Gerard 't Hooft as you learn all those things!

QFT itself has many prerequisites; you need to be well versed in special relativity, electromagnetism, classical field theory and quantum mechanics to get a hold on it. Each of these areas have their own mathematical prerequisites, which include algebra and calculus, multivariable calculus, pdes, complex analysis and fourier analysis.

General relativity and string theory are also comparative to QFT in scope, prerequisite knowledge and utility in other areas of mathematical physics. You would need to learn these if you wanted to start reading a bit more broadly in scattering amplitudes or in another part of math phys, but you can go quite a long way in scattering amplitudes without too much of them.

As someone who has taught myself significantly more than formal education could, I know how valuable it is to have this sort of knowledge trail accessible. Much appreciated.

Do you have any guesses for what might be the connection between these two different scattering amplitudes in seemingly disconnected physical phenomenon?

Dixon et al. find a duality linking form factors and scattering amplitudes. Scattering amplitudes are written as a 'loop expansion', which you could think of as a bit like a Taylor expansion, and I think a remarkable feature of their result is that it appears to hold for all orders of this expansion. I've seen some results relating form factors and scattering amplitudes at only the first order of this expansion before.

Their result is based on a mathematical observation that the functions that the scattering amplitudes are built from at each loop order (called polylogarithms) have a natural operation on them called the 'antipodal map', and in applying this operation to every polylogarithm in the amplitude they recover the form factor. It's intuitive to me that if the mathematical building blocks of your amplitude behave nicely under some operation, you might expect the amplitude to have nice properties under that operation also.

So it's clear mathematically why this new relationship exists, and I guess the question is, 'what does it mean physically?'. Dixon et al. comment that they don't know, and I don't really have any thoughts on this either. What I can say is that, over the last 20 or 30 years there have been many different discoveries of duality between seemingly disparate areas of mathematical physics. The first and most famous was the AdS-CFT relationship which relates gravity to gauge theory, and there is lots of work in amplitudes currently on a different relationship which writes gravitational physics as the square of physics in gauge theory. There are many others.

So, my experience working in the field has taught me that there are many unexpected dualities and relationships between different physical theories, and that nature is more deeply connected at a fundamental level than is indicated by the initial (apparently disparate) mathematical formulations of those theories.