By Bornemann F., Yserentant H.

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The exact flow equation for Vκ is easily obtained by assuming that the field φ is constant in Eq. 132). 141) where G T and G L are, respectively, the transverse and longitudinal components of the propagator: G i j (κ; q) = G T (κ; q) δi j − φi φ j 2ρ + G L (κ; q) φi φ j . 142) By using the LPA effective action, Eq. 143) with V (ρ) = d V /dρ and V (ρ) = d 2 V /dρ2 . g. [50, 58]). In this section, we shall just, for illustrative purposes, solve approximately these equations, by keeping only a few terms in the expansion of the effective potential in powers of ρ (thereby effectively implementing a truncation which ignores the effect of higher n-point functions on the flow).

N1 and N2 are (infinite) normalization constants. The quantity Seff [ψ0 ] is the effective action for ψ0 . Aside from the direct classical field contribution to which we shall return shortly, this effective action receives also contributions which, diagrammatically, correspond to connected diagrams whose external lines are associated to ψ0 , and the internal lines are the propagators of the non-static modes ψn . A few examples are displayed in Fig. 5. Thus, a priori, Seff [ψ0 ] contains operators of arbitrarily high order in ψ0 .

B 347, 80 (1995) 56. : Nucl. Phys. B 509, 662 (1998) 57. : Phys. Lett. B 486, 92 (2000); Phys. Rev. D 64, 105007 (2001); Nucl. Phys. B 631, 128 (2002); Int. J. Mod. Phys. A 16, 2081 (2001) 58. : Phys. Rev. D 67, 065004 (2003) 59. : Nucl. Phys. B 423, 137 (1994) 60. : Nucl. Phys. B 464, 492–511 (1996) 61. : Phys. Lett. B 409, 363–370 (1997) 62. : Phys. Rev. E 67, 066702 (2003) 63. : Phys. Rev. A 69, 061601(R) (2004); Phys. Rev. E. 1 Introduction We give in these notes a short presentation of both the main ideas underlying Wilson’s renormalization group (RG) and their concrete implementation under the form of what is now called the non-perturbative renormalization group (NPRG) or sometimes the functional renormalization group (which can be perturbative).