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The long-range potential between quarks in a confining gauge theory increases linearly with the potential: $$ V(r)=\sigma r \tag{1} $$ where $\sigma$ is the string tension.

In QFT, one can calculate the form of the potential between particles by taking the Fourier transform of the $2\to2$ scattering amplitude -- see http://www.damtp.cam.ac.uk/user/tong/qft.html section 3.5.2 and 6.6.1 where it is shown, in this way, that the potential in Yukawa theory and quantum electrodynamics is $e^{-mr}/r$ and $1/r$ respectively.

My question is whether there is a phenomenological Lagrangian in gauge theory which can give us the linearly rising potential (1), assuming, of course, that it is calculated by taking the Fourier transform of some scattering amplitude (the amplitude itself being calculated from the sought-after Lagrangian) as David Tong does in the document I linked.

My own feeling is that such a Lagrangian will depend explicitly on the space(-time) coordinates to give us the desired behaviour of the potential. Is that a reasonable hypothesis?

dennis
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    Your $2\to2$ scattering experiment assumes asymptotically FREE incoming/outgoing quarks. Are you sure that is feasible? – MadMax Oct 11 '23 at 14:04
  • " assuming, of course, that it is calculated by taking the Fourier transform of some scattering amplitude (the amplitude itself being calculated from the sought-after Lagrangian) as David Tong does in the document I linked." Where did this absurd "of course" come from? It has been long appreciated that infrared slavery upends this expectation by invalidating perturbative approaches such as for the Yukawa potential. There have been heuristic lattice estimates/arguments to connect Richardson-type potentials to gauge theory but your insistence on barking up the wrong tree cannot lead anywhere. – Cosmas Zachos Oct 11 '23 at 14:50
  • @CosmasZachos The whole point of a phenomenological Lagrangian is to imitate exact effects at tree level though – dennis Oct 11 '23 at 17:30
  • Obviously; but exact effects are not always calculable in the manner you suggest, out of a tree diagram. The non-perturbative estimate of the Richardson phenomenological lagrangian arises out of lattice setups, as crudely outlined in 26.7.2 of Schwartz, suggested. The linear potential is but a cartoon of it. – Cosmas Zachos Oct 11 '23 at 17:58
  • @CosmasZachos I never suggested that I was calculating the phenomenological Lagrangian from first principles...I was simply asking if there is a Lagrangian which reproduces the correct long-range potential – dennis Oct 11 '23 at 18:37
  • I recommend reading Section 26.7.2 of the Schwartz textbook: QUNTUM FIELD THEORY and the STANDARD MODEL. – Navid Oct 11 '23 at 11:00
  • I should add, for future readers, that my David Tong comment is to specify exactly what I mean by "reproduces" in the comment above: i.e. how does one go from a Lagrangian to a potential? I'm proposing to use David Tong's route. I am NOT proposing to use that route on the original Yang-Mills Lagrangian (which is what Cosmos, mistakenly, thinks I'm doing -- if we did, we would find a Coulomb potential as in QED) but rather on the phenomenological Lagrangian. – dennis Oct 13 '23 at 11:35
  • (just a minor point: the linearly rising behaviour doesn't continue forever, you do at some point observe 'string breaking' where the ground state becomes two mesons, instead of a long string connecting the quarks. At this point, the potential flattens out, see e.g. fig 13 https://arxiv.org/pdf/hep-lat/0505012.pdf) – QCD_IS_GOOD Oct 13 '23 at 17:11
  • I should also add that I am not concerned with exact phenomenological details of the confining potential. I'm more concerned with what kind of effective Lagrangians can give us potentials which look even remotely like confining potentials in gauge theory. – dennis Oct 14 '23 at 18:00

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I never suggested that I was calculating the phenomenological Lagrangian from first principles...I was simply asking if there is a Lagrangian which reproduces the correct long-range potential .

The phenomenological cottage industry of long-range potentials in QCD was pioneered by Richardson 1978 and kept going, cf., here.

By study of the nonperturbative behavior of lattice QCD, the basic long-distance linear behavior of its potential has been confirmed, cf the above cite to M Schwartz's book, §26.7.2, and a spate of efforts such as this; but there is no derivation along the perturbative lines of your question's references.

Given the derived/modeled potentials, there is no point in summarizing their structure in a phenomenological lagrangian; the long-distance effective lagrangians of QCD are chiral model constructions of hadrons, not quarks. Chiral-bag models deal with constituent quarks and χSB Goldstone peudoscalars ("pions"), but they don't lead to these potential, they assume them.

PS (Geeky). If you are asking an abstract question of principle on dimensional analysis of the Born approximation of such an untenable effective theory, indeed, the propagator for the "spring particle" pulling the quarks together would need an additional "nonlocal" form factor of $f(\vec k)\sim 1/\vec k^2$, from elementary dimensional analysis. But don't go there, really...

Cosmas Zachos
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  • I find your PS more interesting than your answer. Can you say why you think $f(\vec k)=1/\vec k^2$ is "non-local"? – dennis Oct 12 '23 at 13:24
  • Standard QFT. Corresponds to a lagrangian with $\partial^4$, so a propagator like $1/\vec k^4$. You are not talking about the strong interactions, though, as detailed; You are trying to go down the paths of the early 1970s, all of them pathological.... – Cosmas Zachos Oct 12 '23 at 14:41
  • Why is $\partial^4$ "non-local"? It's less local than $\partial^2$ but it's still local – dennis Oct 12 '23 at 14:50
  • This is a formal question, possibly represented someplace on this site. Higher derivatives in the lagrangian are generically nonlocal, as they correspond to expansions of translation operators, $e^{\delta x ~ \partial}$... – Cosmas Zachos Oct 12 '23 at 14:58
  • Here are some refs; and; and; and.... – Cosmas Zachos Oct 12 '23 at 15:09