Comment by morbia

3 years ago

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.