One-loop approximation

Fig. 13,4. The ratio TCc / (/ = 1) as function of the parameter 9o, evaluated in one-loop approximation...

In general we can calculate only low order (one loop) approximations and thus have to live with these ambiguities. 

We have calculated the energy of an /V-rod bundle explicitly, using the one-loop approximation (the same approximation that we used for two rods) [30,46]. Because the repulsion weakens as the bundle size grows, we find that the repulsion does not limit the bundle size. Instead, the free energy decreases linearly with N, the number of rods in the bundle. Thus the counterion-mediated attraction is no different from more pedestrian attractions such as the dispersion attraction. This calculation does not include the translational... 

Osmotic pressure of dilute and semi-dilute solutions in moderately good solvent one-loop approximation (b 0, c = 0)... 

We verify that the one-loop approximation is valid only for small values of z (poor solvent). 

In the present chapter, we ignored the three-body interactions, but we shall come back to them in chapter 14, where we deal with poor solvents. The quantities calculated here by perturbation are expanded to first-orders in powers of z. For a chain ensemble, the other dimensionless parameter is C S 2. An expression of the osmotic pressure corresponding to one-loop approximation and valid for small values of z is given above. 

One-loop approximation of the osmotic pressure in the Kuhnian limit... 

At large stiffness, the distribution can be calculated from the one-loop approximation to the path integral which leads to the Fourier integral (Kleinert 2006 ... 

In the so-called one-loop approximation, corresponding to the first order of perturbation... 

The effective Hamiltonian (29) is the starting point for a study of the polymer limit m 0 of the weakly diluted Stanley model with long-range-correlated disorder. Imposing the renormalization conditions of the massive scheme in the one-loop approximation leads to the following expressions for the RG functions [72] ... 

Then, with FqY we recover the standard mass insertion while the one loop approximation to Fj f is the term considered by MH. The generalized two loop contribution to the bare vertex function may be written as... 

The adduced relationships are valid in the one-loop approximation. In the zeroth order has been obtained as well. 

Fig. 3.1 Graphs which contribute to the (3 function in the one loop approximation (a) fermion loop, (b) gluon loop...

In the above, the propagator , vertex and box corrections are given in the one-loop approximation by... 

Steinberger (1949) calculated the rate for the decay tt 27 in the proton-antiproton or one-loop approximation as shown in Fig. 9.2. 

It is appropriate to mention at this juncture that the Leibler theory used the one-wave-number approximation for the calculation of the order parameter, while the M-O theory used higher order harmonics by extending the Leibler theory. Barrat and Fredrickson [52] also extended the Leibler theory within the spirit of a self-consistent one-loop approximation without making an assumption about the wave vector dependence of the correlation functions. This allowed them to calculate the shift in the peak of the fluctuation spectrum observed experimentally in smaU-angle neutron scattering experiments. 

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