• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New features!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

123 submissions , 104 unreviewed
3,600 questions , 1,219 unanswered
4,601 answers , 19,547 comments
1,470 users with positive rep
423 active unimported users
More ...

Nuclear physics from perturbative QFT

+ 9 like - 0 dislike

Is there a renormalizable QFT that can produce a reasonably accurate description of nuclear physics in perturbation theory? Obviously the Standard Model cannot since QCD is strongly coupled at nuclear energies. Even the proton mass cannot be computed from perturbation theory, as far as I understand.

The strong force sector should probably be comprised of a nucleon spinor field Yukawa-coupled to a scalar pion field, the later with quartic self-coupling. The electroweak sector should probably be the Glashow-Weinberg-Salam model with hadrons replacing quarks in some manner.

Some key parameters of what a "reasonably accurate description" should be are:

  1. Reproducing the entire spectrum of nuclei and predicting which nuclei are stable
  2. Estimating nuclear masses with precision of ~ 0.1%
  3. Estimating nuclear decay rates with precision ~ 10%

Assuming this is feasible:

How the precision of the results depends on computation loop order?

It is possible to consider the nonrelativistic limit of the QFT, in which nuclei are described by quantum mechanics of nucleons coupled by a (multibody) interaction potential. The potential can be computed in perturbation theory by considering the nonrelativistic limit of nucleon scattering. Obviously this cannot give the right masses, since mass is additive in non-relativistic physics, but does it give reasonably precise stability criteria and binding energies?

What if we remain with the 2-body potential only?

What happens if we extract the potential from the nonrelativistic limit of the full nonperturbative Standard Model (theoretically, I'm not sure it's manageable in practice)?


  1. Let me explain my motivation to think such a renormalizable QFT exists. The (sufficiently) fundamental description of nucleons and pions is QCD, but in low energies they can be described by an effective field theory. Such effective field theories are normally nonrenormalizable but the nonrenormalizable interactions are suppressed by powers of E / Lambda, where E is the energy scale of interest and Lambda is the fundamental energy scale: in this case, the QCD scale. Since nuclear binding energies are significantly lower than the QCD scale (highest binding energy per nucleon is about 9 MeV and the QCD scale is about 200 MeV so the ratio is < 5%), it doesn't seem absurd to use renormalizable interactions only
  2. It is not really possible to compute bound state energies and lifetimes at finite order of perturbation theory. This is because they appear as poles in the S-matrix but these poles don't appear at any finite order. However, maybe it's possible to introduce some meaningul infinite (but not complete) sum which will converge (at least after some resummation technique) and exhibit the required poles? It is certainly possible for bound states in external fields (in this case the infinite sum amounts to including the external field effect in the propagator), maybe there's a way to do it for bound states of dynamic particles as well? Now that I think about it, this issue should have been a separate question... In any case, we can still use perturbation theory to extract the many-body potentials

EDIT: I now realize issue 2 above can be tackled using the Bethe-Salpeter equation. However, I found no good discussion of it so far. Any recommendations? I'd prefer something mathematically-minded

EDIT: Thomas' answer below raised some doubts as to the possibility of describing nuclear physics using a renormalizable QFT. Therefore I wish to expand the question to include nonrenormalizable QFTs. As long as we stick to finite loop order there is a finite number of parameters so the approach makes sense. The question is then: Which QFTs (renormalizable or not) can produce nuclear physics from perturbation theory + Bethe-Salpeter equation? What is the required loop order?

This post has been migrated from (A51.SE)
asked Dec 26, 2011 in Theoretical Physics by Squark (1,705 points) [ no revision ]
Not an expert, but I know that people work on this. E.g., see: http://en.wikipedia.org/wiki/Chiral_perturbation_theory

This post has been migrated from (A51.SE)

2 Answers

+ 7 like - 0 dislike

The theory your asking about is an effective field theory (in this case the nuclear eft developed by Weinberg), so it is not renormalizable. QCD is the only renormalizable field theory that can account for nuclear physics (quantum hadrodynamics, mentioned above, is a renormalizable model field theory, with no real predictive power). Also note that nuclear bound states are non-perturbative, so nuclear eft is a powerful tool to organize calculations, but even leading order calculations are not perturbative (you must solve the Schroedinger equation, or an equivalent Dyson-Schwinger equation). Because of the efimov effect, calculations may require three body forces at leading order, but there is a hierarchy of many body forces. Finally asking for 0.1% accuracy on masses (that is roughly 10% for binding energies) is challenging, but not crazy.

A few more comments: 1) It is not true in general that the leading order terms in an EFT correspond to renormalizable interactions. 2) Solving a Bethe-Salpeter equation that sums two-body ladders with an interaction that is local in time is equivalent to solving the Schroedinger equation (this is used not only in nuclear EFT, but also in NRQED or NRQCD calculations). 3) The issue with the Efimov effect is that according to Weinberg's power counting three-body operators should be suppressed relative to two-body operators, but Weinberg counting can fail non-perturbatively (if there is an Efimov effect).

This post has been migrated from (A51.SE)
answered Dec 27, 2011 by tmschaefer (700 points) [ no revision ]
Most voted comments show all comments
hadro, not hydro

This post has been migrated from (A51.SE)
Sorry, fixed typo.

This post has been migrated from (A51.SE)
@Thomas: welcome and thanks for your answer. You seem to have more than one account, I can merge them if you wish.

This post has been migrated from (A51.SE)
Regarding the comments in the body of the answer:

This post has been migrated from (A51.SE)
1. Well, it might be that no renormalizable interactions are allowed by the symmetries but it doesn't seem to be the case since we have the Nucleon-pion Yukawa coupling and pion-pion quartic self-coupling

This post has been migrated from (A51.SE)
Most recent comments show all comments
1. You may consider linear representations (the linear sigma model), where the leading terms are indeed renormalizable couplings to sigma an pi fields, but as an EFT you should integrate out the sigma, and you end up with a non-linear representation, and a derivatively coupled pion (this is explained in Weinberg's QFT book).

This post has been migrated from (A51.SE)
2. The leading term in the EFT for nucleons is the free Schroedinger lagrangian, but higher order terms contain relativistic corrections (kinetic energy corrections, spin-orbit forces, etc). This is essentially the Foldy-Wouhuysen reduction. The same technology is used in NRQED to do order-by-order to corrections to Coulomb bound states.

This post has been migrated from (A51.SE)
+ 3 like - 0 dislike

You're looking for quantum hadrodynamics. See, e.g. Serot and Walecka.

This post has been migrated from (A51.SE)
answered Dec 26, 2011 by S Huntsman (405 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights