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B.1 Standard effective action and its perturbative construction

We begin with a quick reminder on the standard effective action: After coupling χ(x) to a source J (x) one has an (initially formal) functional integral representation for the Euclidean generating functional of the connected Schwinger functions: ∫ ∫ W [J ] = ln 𝒟χ exp{− S[χ] + dx χ(x)J(x )}. Source dependent normalized expectation values of some (smooth) observable 𝒪 (χ) are defined by
( ) ⟨𝒪 ⟩ = e−W [J]𝒪 -δ- eW [J], (B.1 ) J δJ
and it is assumed that (at least for vanishing source) the map 𝒪 ↦→ ⟨𝒪 ⟩ is a positive functional in the sense that positive functions 𝒪 (χ ) have positive expectation values. In order to make the functional integral well-defined a UV cutoff Λ is needed; for example one could replace ℝd by a d-dimensional lattice ℤd with lattice spacing Λ −1. The flat reference functional measure 𝒟 χ would then be proportional to ∏ x∈ℤd dχ (x). In the following we implicitly assume such a UV regularization but leave the details unspecified and use a continuum notation for the fields and their Fourier transforms. The dependence on the cutoff Λ will be specified only when needed. We will also omit overall normalizations in the functional integrals. Whenever 𝒟 χe −S[χ] in addition to being positive is also a normalized measure (on a suitable space of functions χ) it follows from Hölder’s inequality that W [J ] is a convex functional of the source, i.e. W [αJ1 + βJ2] ≤ αW [J1 ] + βW [J2 ], for α + β = 1, α,β ≥ 0. Taking J1 = J, J2 = J + f and expanding in powers of f gives
⟨ ⟩ ∫ δ2W [J] ( ∫ )2 0 ≤ dx dy -----------f(x )f(y) = dx (χ(x ) − ⟨χ(x )⟩J f(x) . (B.2 ) δJ(x)δJ (y) J
This means the second functional derivative (2) W (x, y) is a kernel of positive type; under suitable falloff conditions it defines a positive bounded integral operator on the space of the functions f. Kernels of positive type allow one to (re-)construct a Hilbert space such that schematically W (2)(x,y) is recovered as an inner product (“a two-point function”). A fully fledged reconstruction of the operator picture requires knowledge of all multipoint functions and is roughly the content of the Osterwalder–Schrader reconstruction theorem.

Since W [J] is convex, the effective action can be introduced as the Legendre transform ∫ Γ [φ ] := supJ { φJ − W [J]}, which is a convex functional of φ. Although W [J ] is always convex it may not be differentiable everywhere. In fact, W [J] has ‘cusps’ in the case of spontaneous symmetry breaking. Even on the subspace of homogeneous solutions the supremum in the definition of Γ [φ] may then be reached for several configurations J∗ = J∗[φ] and Γ [φ] is ‘flat’ in these directions. If W [J] admits a series expansion in powers of J, a formal inversion of the series δW [J]∕δJ = φ defines a unique J∗[φ] with the property J∗[φ = 0] = 0 [237Jump To The Next Citation Point]. Often this extra assumption isn’t needed and we shall write J = J [φ] ∗ ∗ for any configuration on which the supremum in the Legendre transform is reached; functional derivatives with respect to J evaluated at J∗[φ] will be denoted by δ∕δJ∗. The defining properties of the Legendre transform then read

∫ Γ [φ] = dxφJ [φ] − W [J [φ]], (B.3 ) ∗ ∗ δW [J ] -----∗- := ⟨χ⟩J∗ = φ, (B.4 ) δJ∗ δΓ [φ] δφ = J∗[φ], (B.5 ) 2 2 δ-Γ [φ]δ-W-[J∗] = 1, (B.6 ) δφ δφ δJ∗δJ∗
where Equation (B.6View Equation) is short for the fact that the φ-dependent integral operators with kernels Γ (2)(x,y) := δ2Γ [φ ]∕δ φ(x)δφ(y) and G (x,y ) := δ2W [J∗]∕δJ∗(x)δJ∗(y) are inverse to each other. Since G (x,y) is a kernel of positive type, so is Γ (2)(x,y).

To switch off the source one selects configurations φ∗ such that δΓ [φ ]∕δφ |φ= φ∗ = J ∗[φ ∗] = 0. As J∗[φ] defined by series inversion one can directly take φ∗ = 0 as the source-free condition. This is typical in the absence of spontaneous symmetry breaking, otherwise one should use the defining relation 0 = δΓ [φ ]∕ δφ|φ=φ∗ for φ∗ to switch off the source. The vertex functions are defined for n ≥ 2 by

(n) δ δ || Γ (x1, ...,xn) :=-------... ------Γ [φ ]| . (B.7 ) δφ (x1) δφ(xn) φ=φ∗
In a situation without spontaneous symmetry breaking the Γ (n) are independent of the choice of φ∗. The original connected Greens functions can be reconstructed from the Γ (n) and [Γ (2)]− 1 = G |φ=φ∗ by purely algebraic means, as can be seen by repeated differentiation of Equation (B.4View Equation). Finally we remark that both the ‘connectedness property’ of the multipoint functions and the ‘irreducibility property’ of the vertex functions can be characterized intrinsically [55], i.e. without going through the above construction.

Inserting Equation (B.3View Equation) into the definition of W [J] one sees that the effective action is characterized by the following functional integro-differential equation:

{ } ∫ ∫ δΓ [φ] exp {− Γ [φ]} = 𝒟χ exp − S[χ] + dx (χ − φ )(x )------ (B.8 ) δφ(x)
(see e.g. [223Jump To The Next Citation Point]). In itself of course Equation (B.8View Equation) is useless because one still has to perform a functional integral. It can be made computationally useful, however, in two ways. First as a tool to generate a recursive algorithm to compute Γ [φ ] perturbatively, and second as a starting point to derive a functional differential equation for Γ [φ]. For the latter we refer to Appendix C.2, here we briefly recap the perturbative construction.

It is helpful to restore the implicit dependence on the UV cutoff Λ and we write Γ Λ[φ] from now on. In outline, the perturbative algorithm based on Equation (B.8View Equation) involves the following steps. One introduces the loop counting parameter λ as follows: Γ Λ ↦→ 1λΓ Λ, S Λ ↦→ 1λSΛ, √ -- χ − φ = λf, where S Λ is the bare action depending on Λ. After the rescaling one expands the exponent on the right-hand-side in powers of √ -- λ. Schematically the result is

√ -- √ -- i SΛ [φ + λf ] − λf ∂iΓ Λ[φ] = ( ) λ i j √ -- 1 i j k i 2 SΛ[φ] + -f f ∂i∂jSΛ[φ] + λ λ --∂i∂j∂kS Λ[φ]f f f − f ∂iΓ Λ,1[φ ] + O (λ ), (B.9 ) 2 3!
where we momentarily use DeWitt’s ‘condensed index notation’ [68Jump To The Next Citation Point], that is, functional differentiation δ∕δ φ(x) is denoted by ∂i and the index contraction is short for a x-integration. Further we used an ansatz for Γ Λ[φ] of the form
∑ ℓ Γ Λ[φ] = SΛ[φ] + Γ Λ,ℓ[φ]λ . (B.10 ) ℓ≥1
Note that the term linear in f drops out without assuming that φ is an extremizing configuration. Equation (B.9View Equation) is now re-inserted into Equation (B.8View Equation) and the exponentials involving positive powers of √ -- λ are expanded. This reduces the evaluation of the functional integral on the right-hand-side to the evaluation of correlators ⟨fi1f i2 ...fin ⟩Λ with respect to the Gaussian measure with covariance ∂i∂jSΛ [φ ]. By the source-free condition the ones with an odd number of f’s vanish, so that also the right-hand-side gives an expansion in integer powers of λ. Matching both sides of Equation (B.9View Equation) then gives a recursive algorithm for the computation of the Γ Λ,ℓ[φ], ℓ ≥ 1. The first two equations are
1 i Γ Λ,1[φ] =--TrΛ ln∂ ∂iSΛ [φ ], 2 -1- i1 i2 i3 i4 Γ Λ,2[φ] = 24 ∂i1∂i2∂i3∂i4S Λ[φ]⟨f f f f ⟩Λ (B.11 ) − -1-∂i1∂i2∂i3S Λ[φ]∂j1∂j2∂j3S Λ[φ ]⟨fi1f i2f i3fj1fj2fj3⟩Λ 72 1 i i i j 1 i j + -∂i1∂i2∂i3S Λ[φ]∂jΓ Λ,1[φ]⟨f 1f2f 3f ⟩Λ − --∂iΓ Λ,1[φ]∂jΓ Λ,1[φ]⟨f f ⟩Λ. 6 2
The expression for Γ Λ,1[φ ] is the familiar regularized one-loop determinant. Inserting this into the second equation the reducible parts cancel and one can verify the equivalence to the two-loop result in [237] (Equation (A6.12)). The presence of the UV cutoff in SΛ renders the expressions for Γ Λ,ℓ[φ], ℓ ≥ 1, well-defined.

The removal of the cutoff is done by a recursive procedure which is based on the following crucial fact: In a perturbatively (quasi) renormalizable QFT the divergent (as Λ → ∞) part of the ℓ loop contribution to the effective action Γ Λ,ℓ[φ ] is local, i.e. it equals a single dx integral over a local function in the fields φ and their derivatives. Moreover this divergent part has the same structure as the bare action (2.2View Equation) with specific parameter functions uα,ℓ(u(μ),Λ, μ). This can be used to compute the parameter functions in the bare action (2.2View Equation) recursively in the number of loops.

Schematically one proceeds as follows. Let SΛ,≤L[χΛ] denote the bare action (2.2View Equation) at L loop order, with the parameter functions u (u(μ),Λ,μ ) α,ℓ, ℓ ≤ L, known. In particular the limits lim Λ→ ∞ S Λ,≤L [χ Λ] = S μ,≤L [χ μ] is finite and defines the L-loop renormalized action. Further the parameter functions are such that ∑ ℓ≤L Γ Λ,ℓ[φ ]λ ℓ has a finite limit as Λ → ∞. Using this bare action S Λ,≤L one computes the effective action Γ Λ,L+1 at the next order. According to the above result it can be decomposed as

c.t. Γ Λ,L+1 [φ ] = Γ ∞,ℓ[φ] − SΛ,L+1[φ], (B.12 )
where c.t. ∑ SΛ,L+1[φ] = αu α,L+1(u (μ ),Λ,μ )P α[φ], determines the parameter functions at the next order. The “counter term” action L+1 c.t. λ S Λ,L+1 is then added to the bare action SΛ,≤L to produce S Λ,≤L+1. Re-computing Γ Λ,L+1 with this new bare action, the limit Λ → ∞ turns out to be finite, that is subdivergencies cancel as well. The traditional proof of this result (e.g. for scalar field theories) involves the analysis of Feynman diagrams and their (sub-)divergencies. More elegant is the use of flow equations (see [123] for a recent review). Once the renormalized ∑ ℓ Γ ∞[φ] = ℓ≥0Γ ∞,ℓ[φ ]λ is known (to exist), the vertex functions (B.7View Equation) and the S-matrix elements are in principle likewise known to all loop orders. The latter can then be shown to be invariant under local redefinitions of the fields.
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