6 ms·

Actually there's a bit more to it than that. Coupling constants change as a function of their energy, they're called 'running coupling constants'. As a result o

by morbia 4y ago

Actually there's a bit more to it than that. Coupling constants change as a function of their energy, they're called 'running coupling constants'. As a result of this phenomenon, there are domains where alpha_s is small and therefore a perturbative expansion of terms is possible. This happens at very high energies, so at the LHC we can happily calculate the higher order terms that you talk about and each successive term is a smaller contribution than the last. Unfortunately, alpha_s is large at low energies, and by low I mean at the atomic and nuclear scale. There you are well and truly in the domain that perturbative QCD is impossible. The only option at that point is something called lattice QCD at the quark/gluon level. Edit: Typo

antognini 4y ago
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 4y ago
  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.
  - TheOtherHobbes 4y ago
    Reminds me of: https://en.wikipedia.org/wiki/Deferent_and_epicycle https://en.wikipedia.org/wiki/Deferent_and_epicycle
  - selectodude 4y ago
    >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.
    - davrosthedalek 4y ago
      Essentially, solving a large number of non-trivial integrals.
    - morbia 4y ago
      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 https://en.wikipedia.org/wiki/One-loop_Feynman_diagram
      - 4y ago
- hither_shores 4y ago
  Lattice methods are probably the most common nonperturbative approach to QCD.

by morbia 4y ago

antognini 4y ago

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 4y ago
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.
- TheOtherHobbes 4y ago
  Reminds me of: https://en.wikipedia.org/wiki/Deferent_and_epicycle https://en.wikipedia.org/wiki/Deferent_and_epicycle
- selectodude 4y ago
  >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.
  - davrosthedalek 4y ago
    Essentially, solving a large number of non-trivial integrals.
  - morbia 4y ago
    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 https://en.wikipedia.org/wiki/One-loop_Feynman_diagram
    - 4y ago
hither_shores 4y ago
Lattice methods are probably the most common nonperturbative approach to QCD.