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C.1 Definition and basic properties

The construction of Γ Λ,k[φ] starts out from a modified form, W Λ,k[J], of the standard generating functional W Λ[J]:
∫ { ∫ } exp{W Λ,k[J]} = 𝒟 χ exp − SΛ[χ] − CΛ,k[χ] + dx χ(x)J (x) . (C.1 )
The extra factor exp{ − C Λ,k[χ ]} suppresses the “IR modes” with 2 2 p < k. The modified W Λ,k[J ] is easily seen to be still a convex functional of the source. The corresponding Λ, k, and J-dependent expectations of some (smooth) observable 𝒪 (χ) are defined as in Equation (B.1View Equation)
∫ { ∫ } ⟨𝒪 ⟩Λ,k = 𝒟 χ 𝒪 (χ) exp − SΛ [χ ] − C Λ,k[χ] + dx χ(x )J (x) . (C.2 )
The cutoff functional CΛ,k is a quadratic form in the fields and has already been displayed in Equation (2.10View Equation). In the literature it is often denoted by ΔkS to indicate that it should be thought of as modifying the bare action.

The kernel ℛ Λ,k defining CΛ,k is conveniently chosen such that both ℛ Λ,k and k∂kℛ Λ,k define a trace-class operator [157] (see Equation (2.10View Equation)). Once the trace-class condition is satisfied one can adjust the other features of the kernel to account for the mode suppression. These features are arbitrary to some extent; what matters is the limiting behavior for p2,q2 ≫ k2 and (with foresight) Λ → ∞. In the simplest case we require that ℛ Λ,k(p,q) is smooth in all variables and of the factorized form

ℛ Λ,k(p,q ) = ℛk (p2)δΛ(p + q), lim δΛ(p) = δ(p), (C.3 ) Λ→∞ { k2 for p2 ≪ k2, ℛk (p2) ≈ 2 2 (C.4 ) 0 for k ≪ p .
In the first condition δΛ (p) is a smooth approximation to the delta distribution, normalized such that ∫ --dp- δ (p) = 1 (2π)d Λ. In Fourier space the finiteness of the trace then amounts to ∫ Tr [ℛ Λ,k] = δΛ (0) (d2πp)d ℛk (p2) < ∞. The condition (C.4View Equation) guarantees that the large momentum modes are integrated out in the usual way, while the ℛ (p2) ≈ k2 k behavior for small p2 leads to a suppression of the small momentum modes by a soft mass-like IR cut-off. Indeed, if the bare action has the structure ∫ 1 2 1 2 2 S Λ[χ ] = dx {2(∂μχ ) + 2m Λχ + interactions}, the addition of a C Λ,k[χ] term as in Equation (2.10View Equation) leads to
∫ 1- -dp--- 2 2 2 2 SΛ[χ] + CΛ,k[χ] = 2 (2 π)d [p + m + ℛk (p )]|^χ(p)| + ..., (C.5 )
where m is the renormalized mass and the dots indicate the interaction terms and terms which vanish for Λ → ∞. Obviously the cutoff function ℛ (p2) k has the interpretation of a momentum dependent mass square which vanishes for 2 2 p ≫ k and assumes the constant value 2 k for 2 2 p ≪ k. How 2 ℛk (p ) is assumed to interpolate between these two regimes is a matter of calculational convenience. In practical calculations one often uses the exponential cutoff ℛk (p2) = p2[exp (p2 ∕k2) − 1 ]−1, but many other choices are possible [29Jump To The Next Citation Point146Jump To The Next Citation Point].

Next one introduces the Legendre transform of W Λ,k,

{ ∫ } ^Γ Λ,k[φ] := sup dxJ (x)φ(x) − W Λ,k[J ] , (C.6 ) {J}
which is a convex functional of φ. Making the usual simplifying assumption that W [J] Λ,k admits a series expansion in powers of J, a formal inversion of the series δW [J]∕δJ = φ defines a unique configuration J = J∗[φ ] with the property J ∗[φ = 0] = 0 and φ(x) = ⟨χ(x )⟩ = δW Λ,k[J∗]∕ δJ∗(x). The actual effective average action is defined by
Γ Λ,k[φ ] := ^Γ Λ,k[φ ] − C Λ,k[φ]. (C.7 )
The subtraction of the mode suppression term is essential for the properties listed below. The main properies of the effective average action are:
  1. If the bare action S Λ is quadratic (free field theory) the action Γ Λ,k (but not ^Γ Λ,k) is independent of k and equals the bare one: Γ Λ,k = SΛ, for 0 ≤ k < Λ.
  2. It satisfies the functional integro-differential equation for the standard effective action with SΛ + C Λ,k playing the role of the bare action, i.e. 
    ∫ { ∫ } δΓ Λ,k[φ-] exp{ − Γ Λ,k[φ]} = 𝒟 χ exp − SΛ [χ ] − C Λ,k[χ − φ] + dx (χ − φ )(x ) δφ(x) . (C.8)
    Equation (C.8View Equation) readily follows by converting Equation (C.1View Equation) via the definitions using the relation J∗[φ] = δ^Γ Λ,k[φ ]∕δφ = − ℛ Λ,kφ + δΓ Λ,k[φ]∕δφ, which is ‘dual’ to δW Λ,k[J∗]∕ δJ∗ = φ.
  3. It interpolates between the SΛ [φ ] and the UV regularized standard effective action Γ Λ[φ], according to
    k→ Λ k→0 S Λ[φ] ← − Γ Λ,k[φ] − → Γ Λ [φ ]. (C.9)
    The first relation, limk →0 Γ Λ,k = Γ Λ, follows trivially from Equation (B.8View Equation) and the fact that ℛk (p2) vanishes for all p2 > 0 when k → 0. The k → Λ limit of Equation (2.8View Equation) is more subtle. A formal argument for limk →Λ Γ k ≈ S Λ is as follows. Since ℛk (p2) approaches k2 for k ≈ Λ, and Λ large, the second exponential on the right-hand-side of Equation (2.8View Equation) becomes
    { ∫ } exp − k2 --dp----dq--δ (p + q)(χ − φ)(p)(χ − φ)(q) . (C.10) (2π )d (2π )d Λ
    For k ≈ Λ this approaches a delta-functional δ[χ − φ ], up to an irrelevant normalization. The χ integration can be performed trivially then and one ends up with limk→ Λ Γ k[φ] ≈ SΛ[φ], for Λ large. In a more careful treatment [29Jump To The Next Citation Point] one shows that the saddle point approximation of the functional integral in Equation (2.8View Equation) about the point χ = φ becomes exact in the limit k ≈ Λ → ∞. As a result lim Γ k≈ Λ→∞ Λ,k and S Λ differ at most by the infinite mass limit of a one-loop determinant, which is ignored in Equation (2.7View Equation) since it plays no role in typical applications (see [188] for a more careful discussion).

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