Low energy theorem

The fermionic determinant Detiow averaged over instanton anti-instanton positions, orientations and sizes leads to a partition function of light quarks Z. Then the properties of the hadrons and their interactions are concentrated in the QCD effective action written in terms of the quasiparticles. This approach leads to the Diakonov-Petrov(DP) effective action (D.I. Diakonov et.al., 1996). It was shown that DP effective action is a good tool in the chiral limit but fails beyond this limit, checked by the calculations of the axial-anomaly low energy theorems (M.M. Musakhanov et.al., 1997 E. Di Salvo et.al., 1998). 

Within this approach it was proposed so called improved effective action which is more properly takes into account current quark masses and satisfies axial-anomaly low energy theorems also beyond the chiral limit (M.M. Musakhanov, 1999) at least at 0(m). 

In order to check the gauging method, applied here, by axial-anomaly low-energy theorem, GG —> 2 photons correlator (M.M. Musakhanov et.al., 2003) was calculated. It was found that this gauged QCD low-energy effective action perfectly satisfies the theorem (M.M. Musakhanov et.al., 2003). 

A relevant feature of this theory is the so-called low-energy theorem, analogous to corresponding theorems which hold in ferromagnetism and in particle physics. The invariance of the phenomenological field equations under the transformations (3.53)-(3.55) requires a corresponding invariance of the 5 matrix. In the case of an infinite volume [namely, f x) = 1], this requirement is shown to imply... 

Let us now look at the form the low-energy theorem assumes for rj O. The field equation becomes... 

Furlan.G., Paver,N., Verzegnassi.C, Low Energy Theorems and Photo- and Electroproduction Near Threshold by Current Algebra (Vol. 62)... 

If U0 and U1 were the functions of a sufficient number of identically distributed random variables, then AU would be Gaussian distributed, which is a consequence of the central limit theorem. In practice, the probability distribution Pq (AU) deviates somewhat from the ideal Gaussian case, but still has a Gaussian-like shape. The integrand in (2.12), which is obtained by multiplying this probability distribution by the Boltzmann factor exp (-[3AU), is shifted to the left, as shown in Fig. 2.1. This indicates that the value of the integral in (2.12) depends on the low-energy tail of the distribution - see Fig. 2.1. 

Much current theoretical work is devoted to finding a more fundamental and general underlying theory from which the standard model of particle physics could be deduced as the low energy limit. String theory and the inclusion of the gravitational interaction are central viewpoints, and in such theories particles may have structure and the CPT theorem may be violated. 

This effective Hamiltonian proposed by Gadea et for the cations of conjugated molecules is able to give many more eigenvectors of the positive ion than Koopman s theorem. It provides a direct estimate of the spectrum of the positive ion, involving the non-Koopmans states, which appear to occur at quite low energy and are described as two-hole one-particle states in the delocalized MO-CI language. 

Fig. 3.8. Calibration of the laboratory energy of an ion beam using the retarding field (upper curve) and the time of flight method. Over a wide range the mean kinetic (E ) is proportional to the voltage U applied to the ion guide. At/ is an usually small additive correction accounting for space or surface charges. At low energies, the energy spread of the ion beam plays a role (Liouville theorem). For a detailed description of the technique see Ref. 7.

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