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Is there a direct way to see the failure of semiclassical approximation in infrared Yang-Mills from the bare lattice theory?

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Usually the reasoning comes from the behavior of the running of the coupling: since it gets very strong at long distance, the Boltzmann weight becomes evened out for all paths, classical or non-classical, which means semiclassical approach ceases to be a good approximation. I'm wondering if there is any other simple (I consider running coupling argument simple enough) way to see that the non-classical paths get more and more important at long distances, other than the running coupling picture.

asked Mar 22, 2016 in Theoretical Physics by Jia Yiyang (2,465 points) [ revision history ]

One of the reasons for IR divegences is the presence of bound states. The concept of bound states makes sense only on an infinite lattice, hence all problems associated with it arise only in that limit.

@ArnoldNeumaier, I'm not sure if the kind of IR divergence you mentioned is what I'm thinking about, QED has bound states and IR divergences, but it doesn't have the property of "non-classical paths get more and more important at long distances" because it runs to a trivial theory at long distance.

QED has no bound states. Positronium is only a resonance.

@ArnoldNeumaier, How about QED with an external potential? The point is that it's hard to see why the existence of bound states is tied to an infrared-slaved type of running.

1 Answer

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I am not sure whether this addresses yuor question, but one of the reasons for IR divergences in YM theory is the presence of bound states. The concept of bound states makes sense only on an infinite lattice, hence all problems associated with it arise only in that limit.

Bound states completely alter the renormalized action, since to get the S-matrix correct one has to add for each bound state a field with unrenormalized coefficent Z=0. (Cf. the somewhat cryptic remark in the middle of p.110 of Weinberg's Vol. 1.) Failure to do so results in severe divergences. Already for nonrelativistic scattering problems of a single particle in an external field, the Born series diverges.

In case of QED, one has therefore to treat the Coloumb external field problem in a completely different way than the standard case. I haven't seen anywhere a sensible treatment of the quantized field case. But see Chapter 6.2 of Derezinski's lecture notes for the case of a Dirac Fermion in an external electromagnetic field (i.e., QED with external field but without radiative corrections).

I don't know how this would show up in a lattice version of the theory.

I just found the following articles; haven't read them yet: 

Baldicchi, M., and G. M. Prosperi. "Infrared behavior of the running coupling constant and bound states in QCD." Physical Review D 66.7 (2002): 074008.

Ganbold, Gurjav. "Hadron spectrum and infrared-finite behavior of QCD running coupling." Physics of Particles and Nuclei 43.1 (2012): 79-105.

answered Mar 27, 2016 by Arnold Neumaier (12,355 points) [ revision history ]
edited Mar 27, 2016 by Arnold Neumaier

I don't think this addresses my question. Really I'm asking for an alternative picture for the specific running behavior of Yang-Mills, equivalently formulated as that non-classical path becomes more and more important. The latter picture looks more intuitive to me, but is normally justified by invoking the running behavior, which is (arguably) more abstract. My question was asked in the spirit that "intuitive result normally has an intuitive justification.", but I guess there lies some potential wishful thinking in my hope.

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