Effective theory of bound states from QCD?

2019 Award

Main Question or Discussion Point

Effective theory of bound states from QCD??

Correct, it is not possible to construct few-particle states from vacuum, quarks and gluons and to study their scattering below Lambda b/c quarks and gluons are "the wrong d.o.f." in this regime. But of course we should try something like Bogoljubov trf., dressing, integrating out d.of.s, ... in order to derive an effective theory.
Do you know any work that actually succeeds in producing the action of an effective
field theory for nucleons and mesons, starting from the QCD action?

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tom.stoer

Afaik there are no strict derivations of effective degrees of freedom. There are several effective theories "inspired" by QCD, e.g. Chiral Perturbation Theory, Skyrme models and extensions (including vector mesons, ...), Heavy Quark Effective Theories, (old-fashioned) Non-Relativistic Quark Models, ... but I guess this is not what you are asking for.

2019 Award

Afaik there are no strict derivations of effective degrees of freedom. There are several effective theories "inspired" by QCD, e.g. Chiral Perturbation Theory, Skyrme models and extensions (including vector mesons, ...), Heavy Quark Effective Theories, (old-fashioned) Non-Relativistic Quark Models, ... but I guess this is not what you are asking for.
It is not clear to me what you mean by ''strict''. I don't ask for a rigorous derivation, but for a derivation that starts with the QCD action, then makes certain transformations and approximations, and from that deduces (heuristically) the effective action of hadrons.

This is different from merely ''inspired'' theories that borrow from QCD some ideas but then build an effective theory from scratch rather than showing why it comes from the QCD action by means of some approximation scheme.

So what I am asking for is whether any of the inspired effective theories have a heuristic derivation in which the basic fields of the effective theory are clearly related to the basic fields of QCD.

I think that's what tom meant --- he said "strict" when he meant "any derivation".

tom.stoer

You derive the chiral / flavor structure of chiral effective theories from QCD. That means upfront you have no idea regarding relevant / irrelevant operators (as you turn a renormalizable theory into a non-renormalizable one), you have no ideas regarding coupling constants, ... but you can derive the chiral / flavor structure of certain operators and translate them into terms of an effective Lagrangian.

I don't know if there is a something like a spacetime renormalization program a la Kadanoff.

I know that in 1+1 dim. QCD one can use a kind of bosonization but I have never seen something like that in the 3+1 dim. theory.

2019 Award

You derive the chiral / flavor structure of chiral effective theories from QCD. That means upfront you have no idea regarding relevant / irrelevant operators (as you turn a renormalizable theory into a non-renormalizable one), you have no ideas regarding coupling constants, ... but you can derive the chiral / flavor structure of certain operators and translate them into terms of an effective Lagrangian.
The step I am inquiring about is

-- whether the field operators of the effective Lagrangian are equipped by hand with the properties derived from QCD and the Lagrangian itself is then built just by using these properties but without further reference to the QCD Lagrangian (which I'd regard as an inspired theory only)

-- or whether there exists, say, a renormalization group calculation that relates the effective Lagrangian to the full QCD action via variation of the energy scale (which I'd regard as a derivation, even if nothing quantitative could be said about the numerical relationships).

In the second case, I'd appreciate references.

tom.stoer

-- or whether there exists, say, a renormalization group calculation that relates the effective Lagrangian to the full QCD action via variation of the energy scale (which I'd regard as a derivation, even if nothing quantitative could be said about the numerical relationships).
I haven't seen those calculations, but I am out of the business since some years.

The step I am inquiring about is

-- whether the field operators of the effective Lagrangian are equipped by hand with the properties derived from QCD and the Lagrangian itself is then built just by using these properties but without further reference to the QCD Lagrangian (which I'd regard as an inspired theory only)
Disclaimer: I am currently studying this topic so this is all new to me at this point.

As far as I am aware the derivation follows from assuming the up and down quarks to be massless and the heavier quarks to be non-existent. The Lagrangian is one of two massless fermions with the usual SU(3) gauge bosons. Then follows an ad-hoc* introduction of a fermion condensate which breaks chiral symmetry.

Then a state |U> is assumed to exist for which the expectation value of said anti-fermion fermion operator only slowly rotates in flavor space, giving a 2x2 unitary matrix. This is taken to be an effective field. It is written as the exponential of three real scalar fields multiplying the Pauli matrices, these are identified with the pions.

* (i haven't seen a justification for this, although the existence of symmetry breaking is justified by the lack of a parity partner for the proton) see [2]

A mass term is added which contains the quark masses. Because of the fermion condensate this term becomes a mass term for the pions.

This can be expanded to an SU(3) which then gives 8 pseudo-Goldstone bosons.

[1] I learnt it from here: Mark Srednicki, Quantum Field Theory, http://www.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf page 502
[2] http://arxiv.org/abs/hep-ph/0108111

Nonrelativistic bound states are studied using an effective field theory. Large logarithms in the effective theory can be summed using the velocity renormalization group. For QED, one can determine the structure of the leading and next-to-leading order series for the energy, and compute corrections up to order α8 ln3 α, which are relevant for the present comparison between theory and experiment. For QCD, one can compute the velocity renormalization group improved quark potentials. Using these to compute the renormalization group improved Image t production cross-section near threshold gives a result with scale uncertainties of 2%, a factor of 10 smaller than existing fixed order calculations.

nrqed
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The step I am inquiring about is

-- whether the field operators of the effective Lagrangian are equipped by hand with the properties derived from QCD and the Lagrangian itself is then built just by using these properties but without further reference to the QCD Lagrangian (which I'd regard as an inspired theory only)

-- or whether there exists, say, a renormalization group calculation that relates the effective Lagrangian to the full QCD action via variation of the energy scale (which I'd regard as a derivation, even if nothing quantitative could be said about the numerical relationships).

In the second case, I'd appreciate references.

One example of the second type of effective field theory is NRQCD, for non-relativistic quarks. But the matching of the coefficients in the eft with QCD is made using lattice calculations.

Most low-energy effective lagrangians are of the first type.

2019 Award

One example of the second type of effective field theory is NRQCD, for non-relativistic quarks. But the matching of the coefficients in the eft with QCD is made using lattice calculations.
Could you please give a reference where this is actually done?