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C.2 Flow equation

As anticipated, the scale dependence of Γ Λ,k is governed by a functional differential equation [228Jump To The Next Citation Point229Jump To The Next Citation Point29Jump To The Next Citation Point],
[ ] ∂-- 1- -∂- ( (2) ) −1 k ∂kΓ Λ,k[φ] = 2 Tr k∂k ℛ Λ,k ΓΛ,k[φ] + ℛ Λ,k . (C.11 )
This is the functional renormalization group equation (FRGE) for the effective average action. The ingredients have already been explained in Section 2.3; here we present its derivation. An instructive derivation of Equation (C.11View Equation) starting directly from Equation (C.8View Equation) can be found in [44Jump To The Next Citation Point] (see also [152]). A technically quicker one proceeds as follows [228Jump To The Next Citation Point]:

Taking the k-derivative of Equation (C.6View Equation) with Equations (C.1View Equation, 2.10View Equation) inserted one finds

∫ k-∂-^Γ [φ] = 1- dx dy k-∂-ℛ (x, y)⟨χ(x )χ (y)⟩ , (C.12 ) ∂k Λ,k 2 ∂k Λ,k Λ,k
with the J and k-dependent expectation values defined as in Equation (C.2View Equation). Next it is convenient to introduce the connected 2-point function 2 G Λ,k(x,y ) := δ W Λ,k[J∗]∕δJ∗(x)δJ ∗(y ) and ^Γ (k2)(x,y ) := δ2^Γ Λ,k[φ ]∕δφ (x)δφ(y). Since W Λ,k and ^Γ Λ,k are related by a Legendre transformation, G Λ,k(x, y) and ^(2) ΓΛ,k(x,y) are mutually inverse integral kernels, i.e. ^(2) GΛ,kΓΛ,k = 1 for the associated integral operators. Taking two J-derivatives of Equation (C.1View Equation) one obtains ⟨χ (x )χ(y)⟩ = G (x,y) + φ(x)φ(y) Λ,k Λ,k. Substituting this expression for the two-point function into Equation (C.12View Equation) we arrive at
∂ 1 [ ∂ ] 1 ∫ ∂ k---^Γ Λ,k[φ ] =-Tr k ---ℛk G Λ,k + -- dx dyφ (x )k---ℛ Λ,k(x,y )φ(y). (C.13 ) ∂k 2 ∂k 2 ∂k
In terms of Γ Λ,k this translates into k-∂Γ Λ,k[φ] = 1Tr [k ∂-ℛ Λ,k GΛ,k] ∂k 2 ∂k. The derivation is completed by noting that (2)− 1 (2) −1 [^ΓΛ,k] = (Γ Λ,k + R Λ,k), where the second equality follows by differentiating Equation (C.7View Equation).

We add some comments on the FRGE (C.11View Equation):

  1. The right-hand-side of Equation (C.11View Equation) can be also rewritten in the form of a one-loop expression
    -∂- 1---D--- ( (2) ) k ∂k Γ Λ,k[φ ] = 2 D ln k Tr ln ΓΛ,k[φ ] + ℛ Λ,k . (C.14)
    Here the scale derivative D ∕D ln k acts only on the k-dependence of ℛ Λ,k, not on (2) Γ Λ,k. The Tr ln(...) = lndet (...) expression in Equation (C.11View Equation) differs from a standard one-loop determinant in two ways: It contains the Hessian of the actual effective action rather than that of the bare action SΛ, and it has a built in IR regulator ℛ Λ,k. These modifications make Equation (C.11View Equation) an exact equation.
  2. Observe that the FRGE (C.6View Equation) is independent of the bare action S Λ, which enters only via the initial condition Γ Λ,Λ = SΛ (for large Λ). In the FRGE approach the calculation of the path integral for W Λ,k[J] is replaced with the task of integrating this RG equation from k = Λ, where the initial condition Γ Λ,Λ = S Λ is imposed, down to k = 0, where the effective average action equals the ordinary effective action Γ Λ. The role of the bare action in the removal of cutoff Λ and its relation to the UV renormalization problem has already been decribed in Section 2.3. Here we repeat again that the explicit Λ dependence entering via ℛ Λ,k is harmless, and ℛ Λ,k can essentially be replaced with ℛ ∞,k.

    To see this let us momentarily write G Λ,k(p, q) for the kernel of (2) (ΓΛ,k[φ ] + ℛ Λ,k)−1 in Fourier space. By assumption GΛ,k(p,q) is a family of functions which remains pointwise bounded as Λ → ∞ (but the falloff in p,q may not be strong enough so as to define a bounded operator in the limit). The right-hand-side of Equation (C.11View Equation) then is proportional to ∫ ∂ 2 dp dq δΛ(p + q)k∂kℛk (p )G Λ,k(p,q), which by Equation (C.3View Equation) behaves like ∫ dpk ∂∂kℛk (p2)G∞,k(p,− p) for Λ → ∞. On the other hand by Equation (C.4View Equation) the derivative k-∂ℛk (p2) ∂k has support mostly on a thin shell around p2 ≈ k2, so that the (potentially problematic) large p behavior of G (p,− p) ∞,k is irrelevant.

  3. By repeated differentiation of Equation (C.11View Equation) and evaluation at an extremizing configuration φ∗ one obtains a coupled infinite system of flow equations for the n-point functions (B.7View Equation). For example for a translation invariant theory [34Jump To The Next Citation Point],
    ∫ { k ∂kΓ (2)(p,− p) = --dq--k∂k ℛ Λ,k(q,− q) GΛ,k(q2)Γ (3)(p,q, − p − q) Λ,k (2π )d Λ,k 2 (3) 2 ×G Λ,k((p + q))Γ Λ,k(− p,p + q,− q)GΛ,}k(q ) 1 2 (4) 2 − 2G Λ,k(q )Γ Λ,k(p,− p,q,− q)G Λ,k(q ) . (C.15)
    Here we defined Γ (Λn,k)(x1, ...,xn) in analogy to Equation (B.7View Equation) and wrote (n) d ΓΛ,k(p1,...,pn)(2π) δ(p1 + ⋅⋅⋅ + pd) for its Fourier transform. Similarly for G −1(q2) := Γ (2)(q,− q) + ℛ Λ,k(q,− q) Λ,k Λ,k. One sees that lower multipoint functions couple to higher order ones in a way so that only the infinite system closes. However if all external momenta are small, 2 2 pi < k, the k ∂kℛ Λ,k insertion will ensure that also the internal momentum is small. This is the rationale for the derivative (small momentum) expansion. For an approximation suited for uniformly large momenta p2i ≥ k2 see [34].
  4. The above FRGE is in the spirit of Wilson–Kadanoff renormalization ideas, but with the iterated coarse graining procedure replaced by a direct mode cutoff. Since the kernel in Equation (A.2View Equation) cuts off momenta with p2 < k2, the right-hand-side of Equation (A.2View Equation) corresponds to Equation (C.2View Equation) evaluated at J = 0. One could also have derived a flow equation for (a suitable variant of) W [0] Λ,k.

    However, there are conceptual differences between the effective average action and a genuine Wilsonian action S = SW l l, as discussed in Appendix A.

    First, in the literature the running scale l on which a Wilsonian action depends is frequently referred to as an ultraviolet cutoff and is denoted by Λ W. This is due to a difference in perspective: If all modes of the original system with momenta between infinity (or Λ) are integrated out, l = ΛW is an infrared cutoff for the original model, but it plays the role of an UV cutoff for the “residual theory” of the modes below this scale, which are to be integrated out still. For them SWΛW has the status of a bare action.

    Second, the Wilsonian action SWΛW describes a set of different actions, parameterized by ΛW and subject to a flow equation like Equation (A.4View Equation), for one and the same system; the Greens functions are independent of Λ W and have to be computed from SW ΛW by further functional integration. In contrast the effective average action Γ Λ,k can be thought of as the standard effective action for a family of different systems; for any value of k it equals the standard effective action (generating functional for the vertex- or 1-PI Green’s functions) for a model with bare action SΛ + C Λ,k. The latter is of course not subject to a Wilsonian type flow equation like Equation (A.4View Equation). In particular the multi-point functions do depend on k. This is a desired property, however, as these k-dependent Green’s functions are supposed to provide an effective field theory description of the physics at scale k, without further functional integration. See [29Jump To The Next Citation Point] for a detailed discussion.

    There exists a variety of different functional renormalization group equations. We refer to [21Jump To The Next Citation Point16629Jump To The Next Citation Point146229Jump To The Next Citation Point] for reviews. To a certain extent they contain the same information but encoded in different ways (see e.g. [147Jump To The Next Citation Point]); the differences become important in approximations (the ‘truncations’ described below) where simple truncations adapted to a certain application in one FRGE might correspond to more complicated and less adapted truncations in another. We use the effective average action [22822929Jump To The Next Citation Point] here because of its effective field theory properties and because via the background field method it has been extended to gravity (see [179]). FRGEs invariant under field reparameterizations have been developed in [16544] but have not yet been applied in computations.

  5. To solve the functional flow equation (C.11View Equation) approximations are indispensible. One which does not rely on a perturbative expansion is by truncation of the space of candidate continuum functionals Γ tkrunc[φ] to one where the initial value problem for the flow equation (2.12View Equation) can be solved in reasonably closed form. In this case one can then by ‘direct inspection’ determine the initial data for which a global solution exists. The existence of a nontrivial unstable manifold for Γ trunc k can then be taken as witnessing the renormalizability of an implicitly defined ‘hierarchical’ dynamics (see Section 2.3).

    Concretely the truncation is usually done by assuming an ansatz of the form

    trunc ∑N Γk [φ ] = uα (k )Pα[φ], (C.16) α=1
    where the u α’s are scale dependent (‘running’) parameters as in the previous general discussion, and the k-independent functionals Pα [φ] span the subspace selected. The ‘art’ of course consists in choosing a set of Pα [φ ] small enough to be computationally manageable and yet such that the projected flow encapsulates the essential physics features of the exact flow. The projected RG flow then is described by a set of ordinary differential equations for the parameters u (k) α. Schematically those equations arise as follows. Let us assume the finite set P α[φ ], α = 1,...,N, can be extended to a ‘basis’ (in the sense discussed in Section 2.1) Pα, α ∈ ℕ, of the full space of functionals. Expanding the φ dependence of Tr[...] on the right-hand-side of the FRGE (C.11View Equation) in this basis an expression of the form
    1 ∑∞ --Tr[...] = bα(u;k) Pα[φ] (C.17) 2 α=1
    arises. Here the bα(u;k) play the role of generalized beta functions for the parameters uα, α ∈ ℕ. Neglecting, (i) the terms with α > N in Equation (C.17View Equation) and (ii) the dependence of b1,...,bN on the parameters u α, α > N, a closed system of ordinary differential equations arises: k ∂-uα = bα (u1, ...,uN;k) ∂k, α = 1, ...,N. As in Equation (A.3View Equation) one can reparameterize the Γ [φ] k in Equation (C.14View Equation) in terms of inessential parameters and couplings g i, i = 1,...,n, with the latter ones made dimensionless. The corresponding bi functions then become the beta functions proper and the system of differential equations reads
    k∂ g (k) = β (g ,...,g ), i = 1,...,n. (C.18) k i i 1 n
    The βi’s have no explicit k dependence and define a ‘time independent’ vector field on the space of couplings {g1,...,gn }.

    Another approximation procedure for the solution of Equation (2.12View Equation) is the local potential approximation [21111147]. Here the functionals Γ k’s are constrained to contain only a standard kinetic term plus arbitrary non-derivative terms

    ∫ { 1 } Γ trkunc[φ] := dx --(∂ φ(x))2 + Uk(φ(x )) . (C.19) 2
    Since the potential function φ ↦→ U (φ) could be (Taylor-) expanded one can view (C.19View Equation) as a simple infinite parametric version of Equation (C.16View Equation). The truncated flow equations for Equation (C.19View Equation) now amount to a partial differential equation (in two variables) for the running potential Uk(φ). It is obtained by inserting Equation (C.19View Equation) into the FRGE and projecting the trace onto functionals of the form (C.19View Equation). This is most easily done by inserting a constant field φ = ϕ = const into both sides of the equation since it gives a nonvanishing value precisely to the non-derivative Pα’s. As trunc(2) 2 ′′ [Γ k ] (ϕ ) = − ∂ + U k(ϕ ), ′′ 2 2 U (ϕ) := d Uk ∕dϕ has no explicit x dependence the trace is easily evaluated in momentum space. This leads to the following partial differential equation:
    ∫ 2 k∂ U (ϕ ) = 1- -dp--------k∂kℛk-(p-)-----. (C.20) k k 2 (2 π)d p2 + ℛk (p2) + Uk′′(ϕ )
    This equation describes how the classical (or microscopic) potential U∞ = Vclass evolves into the standard effective potential U0 = Veff. Remarkably, for an appropriate choice of ℛk the limit limk →0 Uk is automatically a convex function of φ, a feature the full effective action must have but which is usually destroyed in perturbation theory. For a detailed discussion of this point we refer to [29]. Generally convexity can be used as guideline to identify good truncations.

    A slight extension of the local potential approximation is to allow for a (φ-independent) wave function renormalization, i.e. a running prefactor of the kinetic term: trunc ∫ 1 2 Γk [φ] = dx { 2Zk(∂φ ) + Uk (φ)}. Using truncations of this type one should employ a slightly different normalization of ℛk (p2), namely ℛk (p2) ≈ Zkk2 for p2 ≪ k2. Then ℛk combines with Γ (2) k to the inverse propagator trunc(2) 2 2 [Γ k ] [φ] + ℛk = Zk(p + k ) + ..., as required if the IR cutoff is to give rise to a 2 (mass) of size 2 k rather than 2 k ∕Zk. In particular in theories with more than one field it is important that all fields are cut off at the same k2. This is achieved by a cutoff function of the form

    2 2 (0) 2 2 ℛk (p ) = k 𝒵k ℛ (p ∕k ), (C.21)
    with ℛ (0) : ℝ → ℝ + + as in Equation (4.15View Equation). Here 𝒵 k is in general a matrix in field space. In the sector of modes with inverse propagator (i) 2 Z k p + ... the matrix 𝒵k is chosen diagonal with entries 𝒵k = Z (ki). In a scalar field these 𝒵k factors are automatically positive and the flow equations in the various truncations are well-defined.

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