Is there a phenomenological Lagrangian which can reproduce the long-range potential between quarks?

Question

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?

Answer 1

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...