Second-order vertex function

The second order perturbation theory term with two one-loop self-energy operators does not generate any logarithm squared contribution for the state with nonzero angular momentum since the respective nonrelativistic wave function vanishes at the origin. Only the two-loop vertex in Fig. 3.24 produces a logarithm squared term in this case. The respective perturbation potential determined by the second term in the low-momentum expansion of the two-loop Dirac form factor [111] has the form... 

Physically the polaron mass enhancement is brought about by the virtual excitation of phonons. In the H (g a Holstein model no restriction is imposed on exciting multiple phonons, implying that all the terms in Fig. lb for the vertex function contribute, while in the g e JT model, there is a severe restriction due to the existence of the conservation law intimately related to the 50(2) rotational symmetry in the pseudospin space. Actually, among the first- and second-order terms for the vertex function, only the term T2/ contributes, leading to the smaller polaron mass enhancement factor m jm than that in the Holstein model in which the correction r 1 is known to enhances m /m very much. In this way, the applicable range of the Migdal s approximation [48] becomes much wider in the g e JT system [63]. 

Fig. 23. The self-energy-like Feynman diagrams for the hyperfine-structure splitting in second order in e and first order in the external vector field, (a) and (b) are again wave-function corrections and (c) is the modification due to the perturbation of the propagator, the exact vertex correction.

Figure 5.42. Order go diagrams contributing to the vertex function To the right is shown separation of the diagrana to reveal it.s explicit dependence on the number of components n of the field vector. In polymer theory (n = 0), the second term of the right-hand side (a loop) is absent (Duplantier, 1982) [Reprinted with permission from B.Diiplantier. J. dc Phys. 43 (1982) 991 1019. Copyright 1982 by EDP Sciences]...

To calculate the higher-order terms, vertex parts and vertex runclions (see section 2.6) must be introduced. I he same functions are used to calculate the second virial coelTicienl which characterizes the intermolecular interactions. 

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