Chiral Perturbation Theory

Fettes N., Meissner U. G., Steininger S. Pion-nucleon scattering in chiral perturbation theory (I) Isospin-symmetric case. Nucl. Phys.A 640, 199-234 (1998)... 

Recently the density dependence of the symmetry energy has been computed in chiral perturbation effective field theory, described by pions plus one cutoff parameter, A, to simulate the short distance behavior [23]. The nuclear matter calculations have been performed up to three-loop order the density dependence comes from the replacement of the free nucleon propagator by the in-medium one, specified by the Fermi momentum ItF... 

We want now to see how this state of affairs is affected by the chiral perturbation of our reaction-diffusion equations [term in M in equation (29)]. To this end we follow the lines of imperfection theory (Section II.C) and expand the variables and parameters in series around X = X,. We also set the frequency fl of the solution to be identical to the external frequency w, and assume that o> is close to the linearized intrinsic frequency, ft, in the absence of the field ... 

The contribution of the hadronic l-l scattering effect (32) is relatively small but is potentially a source of serious problem because it is difficult to express in terms of experimentally accessible observables it must be evaluated by theoretical consideration. It has been estimated by two groups, within the framework of chiral perturbation theory and the 1 /Nc expansion [64,65], Recently the theory dependence of these calculations has been reduced [31] by improving a part of the calculation incorporating the measurements of the P77 form factors [66], where P stands for 7r°, 77, and rj. The value quoted in (32) is the result of this work. Evaluation of these effects in lattice QCD will be particularly timely and interesting. 

These parameters can be evaluated using effective Lagrangian and Chiral Perturbation Theory [18,19,20,21,22] which is mathematically equivalent to QCD [19,21]. At present time, the value ao — a2 has been determined within 2% [23], The QCD Lagrangian and effective Lagrangians are determined by Lorentz invariance, P and C-invariance and by the chiral symmetry. For this reason, the measurement of do — < 21 provides an opportunity to check our understanding of the chiral symmetry breaking of QCD. 

The pion-nucleon interaction has been subject both to experimental and theoretical studies since the very beginning of the development of particle physics. On the theoretical side the description of the pion-nucleon system with QCD is considered to be a fundamental issue in the development of this theory. The understanding of strong interaction in the confinement regime has advanced recently, as chiral perturbation theory was developed to perform calculations at low energies [1,2]. 

Its extension to heavy baryon chiral perturbation theory (HBCHPT) [3] allows to calculate many of the experimentally accessible processes in the meson nucleon sector. The check of the soundness of this approach requires high precision experiments. This resembles the situation in the development of QED during the last 50 years, where the measurement of the Lamb shift contributed much to the development of QED. In a comparable way the measurement of strong interaction shift and width in pionic hydrogen may be a key experiment in strong interaction physics at low energies. 

The interaction between the matter and the light beam is weak and I compute the state TOt using perturbation theory based on the complete set of exact states /> , with energies ha> of the chiral medium in the absence of the light beam, noting that the information yielded by the experiment can then be related to the optically active medium alone. The density matrix, , for the medium in the absence of the light beam can be given a spectral representation in terms of this complete set of states, by virtue of the spectral theorem,... 

On the lowest level the carbonyl chromophore is viewed as the prototype of an inherently symmetric, but chirally perturbed chromophore This means that in a chiral ketone the particular carbonyl chromophore retains (largely) its (achiral) symmetry and optical activity will be induced by the chiral environment (i.e. the chiral arrangement of other groups in the whole molecule). In terms of perturbation theory this can be expressed by the statement that in zeroth order the chromophore is achiral and chirality is induced by perturbations in first (or higher) order. On the other hand, if, for a given electronic excitation, optical activity exists in zeroth order (without perturbation by the chiral field of the remaining groups of the molecule not directly involved in the electronic transition) the chromophore is termed inherently chiral ... 

The term in parentheses vanishes for any molecule thatpossesses reflection planes, a center of inversion, or rotation-reflection axes, and jS is thus only non-zero for a chiral molecule. It is of opposite sign for the enantiomers of a chiral molecule. In Rayleigh-Schrodinger perturbation theory the isotropic component of the first electric-dipolar hypeipolarizability at Wj = Wj q- may be written as... 

The importance of this to the development of a CD model arises in the following way. Consider the complex to comprise two chromophores, A and B, such that A is achiral, and B is the chiral perturber. The CD at the transitions of A arises through simple coupling of A and B as described by perturbation theory, yielding expressions containing the transition moments and energies of the unperturbed chromophore. The utility of the perturbation theory therefore depends critically on the definition of the unperturbed chromophores. The dominant terms in the perturbation expansion may then be extracted unambiguously. 

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