[antognini]

I have idly wondered whether or not there could be a completely different approach to QCD from the usual perturbative techniques. I remember reading in one of Zee's books that back 80s he pointed out to Feynman that the path integral formalism that QFT is based on has no natural way to treat something as simple as a particle in a box. And an object like a proton seems to be more like a particle in a box than a free particle undergoing an interaction.

[morbia]

Yeah as someone who spent 4 years of his life calculating a second order term (Next-to-Next-to-Leading-Order), I have often wondered the same thing! In my original post I grossly simplified how challenging it is to calculate terms in perturbative QCD, even when in the perturbative regime. To name a few:-

* Two loop calculations are extremely challenging on an algebraic level

* You get low energy (called 'infrared red') infinities appearing at low energies. These need to cancel between all your contributing terms, and getting them to cancel is really really challenging.

* The numerical Monte Carlo approaches become extremely computationally intensive because of high dimensional integrals and numerical instability caused by point 2

It was not uncommon for calculations of single terms to involve multiple PhD students over a decade or more.

Throughout my PhD I certainly felt like something was fundamentally 'wrong' with the approach. Alas, I wasn't smart enough to rewrite the field with a whole new way of thinking so bailed instead.

[selectodude]

>It was not uncommon for calculations of single terms to involve multiple PhD students over a decade or more.

Forgive my ignorance, but what does calculating this sort of look like? I am not a mathematician or even math-adjacent.

[morbia]

There are many components to such a calculation that can be split up amongst different research groups. I won't go into detail on all of those but essentially it boils down into two main categories of components to the calculation:-

* Analytical integrals - This is a big algebraic task where you're trying to compute an equation that can be written by hand. For example, if you have 1-loop diagram [1] then the particle in the loop effectively becomes an integral over all possible momentum configurations that particle can have. One-loop is a hard problem but reasonably 'solved', 2-loops is extremely challenging.

* Numerical integrals - This is typically using Monte Carlo techniques to numerically integrate over all possible momentum configurations of the incoming and outgoing particles. Because you can have many particles, it becomes a high dimensional integral pretty quickly. Monte Carlo scales well with dimensionality, but not that well. Therefore you need serious computation power for non-trivial numerical integrals.

Added to this fact is a fun feature of these calculations that infinities spring up all over the place. You have both a numerical and analytical game of getting these guys to cancel (they do, the calculation must be finite) but it is not a straightforward task at all.

[1] https://en.wikipedia.org/wiki/One-loop_Feynman_diagram

[selectodude]

I appreciate the insight. If you happen to see this reply, how does "canceling an infinity" work. Do you just keep trying new numbers until the infinities cancel out? I guess my confusion lies at how something can take years and where the manual aspect of these calculations comes in. I've run calculations that take minutes or even hours, but it's just one of those things where you walk away and do something else and then its done. There seems to be more of a "guess and check" aspect in high energy physics that maybe I'm not grasping.

[morbia]

Ok so to answer your first question, you have to isolate the divergent piece of the terms. So we typically write it as an expansion in epsilson, where you have 1/e as e -> 0 (there are higher order terms, but let's keep it simple). So if we had two contributing terms*, what we should end up with is

X * ((1/e) - (1/e)) + finite terms = finite result.

Easy right? Only there is a problem: the different terms have different dimensional integrals. You can't just do some alebra to cancel these terms, instead what you have to do is construct something called a subtraction scheme which moves the contributions between your different integrals, such that the subtraction terms don't contribute anything to final result and cancel each other, but render the whole calculation finite. This is a by-hand crafted thing, and takes years and years to calculate properly, which is very easy to get wrong.

The complication in particle physics is actually constructing the equations and evaluating the integrals either numerically or alebraically. The algebraic calculations are extremely hard, and checking you are right is really difficult. Typically that involves two independent research groups attempting the same calculation using different approaches and checking you get the same result.

Similarly, for the numerical evaluation of the integrals you really are pushing what a computer can do to the limits. If we consider the term:-

X * (1/e - 1/e) + finite

and I'm trying to integrate this numerically, you typically put in an artifical cutoff term as you get close to the singularity. Problem is you reach the limits of floating point precision pretty quickly: asking for the difference between two massive numbers is the worst case scenario for numerical evaluation. Trying to work around these problems are really, really challenging.

* There are many more than two terms in a real calculation

[davrosthedalek]

Essentially, solving a large number of non-trivial integrals.

[TheOtherHobbes]

Reminds me of:

https://en.wikipedia.org/wiki/Deferent_and_epicycle

[hither_shores]

Lattice methods are probably the most common nonperturbative approach to QCD.