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4.3 Truncated flow equations

Approximate computations of the effective average action can be done in a variety of ways: by perturbation theory, by saddle point approximations of the functional integral, or by looking for approximate solutions of the FRGE. A nonperturbative method of the latter type consists in truncating the underlying functional space. Using an ansatz for Γ k where k-independent local or nonlocal invariants are multiplied by running parameters, the FRGE (4.19View Equation) can eventually be converted into a system of ordinary differential equations for these parameters. In this section we outline how the conversion is done in principle.

A still very general truncated functional space consists of ‘all’ background invariant functionals Γ k[g,¯g, σ,¯σ] which neglect the evolution of the ghost action. The corresponding ansatz reads [179Jump To The Next Citation Point]

¯ ^ Γ k[g,¯g,σ,¯σ ] = Γ k[g] + Γ k[g,¯g] + Sgf[g − ¯g;¯g] + Sgh[g − ¯g,σ, ¯σ;¯g]. (4.29 )
To simplify the notation we wrote gαβ for ⟨gαβ⟩ in the argument of the effective actions as before. In Equation (4.29View Equation) we extracted the classical Sgf and Sgh from Γ k. The remaining functional depends on both gαβ and ¯gαβ. It is further decomposed as ¯ ^ Γ k + Γ k where ¯Γ k is defined in Equation (4.9View Equation) and ^Γ k contains the deviations for ¯g ⁄= g. Hence, by definition, ^Γ k[g,g] = 0, and ^Γ k can be viewed as a quantum correction to the gauge fixing term which vanishes for ¯g = g, too. The ansatz (4.29View Equation) satisfies the initial condition (4.21View Equation) if
¯ ^ Γ k→∞ = S and Γ k→ ∞ = 0. (4.30 )
Inserting the ansatz (4.29View Equation) into the exact form of the RG equation one obtains an evolution equation on the truncated space of Γ [g,¯g]’s:
1 [( )− 1 ] [( ) −1 ] k∂k Γ k[g,¯g] =-Tr κ −2Γ (k2)[g,¯g] + ℛk [¯g] k∂k ℛk [¯g] − Tr − ℳ [g,¯g] + ℛghk [¯g] k ∂kℛghk [¯g(]4.3.1 ) 2
This equation involves
¯ ^ Γ k[g,¯g] := Γ k[g] + Sgf[g − ¯g;¯g] + Γ k[g,¯g] (4.32 )
and Γ (2) k, the Hessian of Γ k[g,g¯] with respect to gαβ at fixed ¯gαβ.

The truncation ansatz (4.29View Equation) is still too general for practical calculations to be easily possible. Computations simplify considerably with the choice ^Γ k = (ZNk − 1)Sgf and a local curvature polynomial for ¯Γ k[g]. The first specialization includes in ^Γ k only the wave function renormalization and gives

Z ∫ √ -- Γ k[g,¯g] = ¯Γ k[g] +-Nk dx ¯g ¯gαβQ αQ β. (4.33 ) 2α
Here Qα = Q α[¯g,f¯] is given by Equation (4.10View Equation) with ¯fαβ := gαβ − ¯gαβ replacing fαβ; it vanishes at g = ¯g.

For ¯Γ k[g] two choices of local curvature polynomials have been considered in detail

∫ ¯Γ [g] = 2κ2Z dx √g-[− R (g) + 2 ¯λ ], (4.34 ) k Nk k ∫ √ -{ } ¯Γ k[g] = dx g 2κ2ZNk [− R (g) + 2¯λk] + ¯νkR2(g) . (4.35 )
The truncation (4.34View Equation) will be called the Einstein–Hilbert truncation. It retains only the terms already present in the classical action, with k-dependent coefficients, though. It is the one where the RG flow has been originally found in [179Jump To The Next Citation Point]. The parameter α in Equation (4.33View Equation) is kept constant (α = 1 specifically; see the discussion below for generalizations) so that in this case the truncation subspace is two-dimensional: The ansatz (4.33View Equation, 4.34View Equation) contains two free functions of the scale, the running cosmological constant ¯λk and ZNk or, equivalently, the running Newton constant gN(k) = (2κ2ZNk )−1. Here κ2 is a scale independent constant related to the fixed Newton constant G in Section 1.5 by −1∕2 κ = (32 πG ).

The truncation (4.35View Equation) will be called the R2-truncation. It likewise keeps the gauge fixing and ghost sector classical as in Equation (4.29View Equation) but includes a local curvature squared term in ¯Γ k. In this case the truncated theory space is three-dimensional. Its natural (dimensionless) coordinates are (g, λ,ν), where

g (k ) gk := kd−2-N----, λk := k−2¯λk, νk := k4− d¯νk. (4.36 ) 16π
Even though Equation (4.35View Equation) contains only one additional invariant, the derivation of the corresponding differential equations is far more complicated than in the Einstein–Hilbert case. We shall summarize the results obtained with Equation (4.35View Equation[133Jump To The Next Citation Point131Jump To The Next Citation Point132Jump To The Next Citation Point] in Section 4.4.

We should mention that apart from familiarity and the retroactive justification through the results described later on, there is no structural reason to single out the truncations (4.34View Equation, 4.35View Equation). Even the truncated coarse graining flow in Equation (4.31View Equation) will generate all sorts of terms in ¯Γ k[g], the only constraint comes from general covariance. Both local and nonlocal terms are induced. The local invariants contain monomials built from curvature tensors and their covariant derivatives, with any number of tensors and derivatives and of all possible index structures. The form of typical nonlocal terms can be motivated from a perturbative computation of Γ k; an example is ∫ -- d4 x√ gR αβγδ ln(− ∇2 )R αβγδ. Since Γ k approaches the ordinary effective action Γ for k → 0 it is clear that such terms must generated by the flow since they are known to be present in Γ. For an investigation of the non-ultraviolet properties of the theory, the inclusion of such terms is very desirable but it is still beyond the calculational state of the art (see however [180Jump To The Next Citation Point]).

The main technical complication comes from evaluating the functional trace on the right-hand-side of the flow equation (4.31View Equation) to the extent that one can match the terms against those occuring on the left-hand-side. We shall now illustrate this procedure and its difficulties in the case of the Einstein–Hilbert truncation (4.34View Equation) in more detail.

Upon inserting the ansatz (4.33View Equation) into the partially truncated flow equation (4.31View Equation) it should eventually give rise to a system of two ordinary differential equations for ZNk and λ¯k. Even in this simple case their derivation is rather technical, so we shall focus on matters of principle here. In order to find k∂ Z k Nk and ¯ k∂kλk it is sufficient to consider (4.31View Equation) for gαβ = ¯gαβ. In this case the left-hand-side of the flow equation becomes ∫ √ -- 2κ2 dx g[− R (g )k∂kZNk + 2k∂k(ZNk ¯λk )]. The right-hand-side contains the functional derivatives of Γ (2); in their evaluation one must keep in mind that the identification gαβ = ¯gαβ can be used only after the differentiation has been performed at fixed ¯gαβ. Upon evaluation of the functional trace the right-hand-side should then admit an expansion in terms of invariants P α[g], among them ∫ √ -- g and ∫ √ -- gR (g). The projected flow equations are obtained by extracting the k-dependent coefficients of these two terms and discarding all others. Equating the result to the left-hand-side and comparing the coefficients of ∫ √g-- and ∫ √gR, the desired pair of coupled differential equations for ZNk and ¯λk is obtained.

In principle the isolation of the relevant coefficients in the functional trace on the right-hand-side can be done without ever considering any specific metric g = ¯g αβ αβ. Known techniques like the derivative expansion and heat kernel asymptotics could be used for this purpose, but their application is extremely tedious usually. For example, because the operators (2) Γ k and gh ℛk, ℛ k are typically of a complicated non-standard type so that no efficient use of the tabulated Seeley–deWitt coefficients can be made. Fortunately all that is needed to extract the desired coefficients is to get an unambiguous signal for the invariants they multiply on a suitable subclass of geometries g = ¯g. The subclass of geometries should be large enough to allow one to disentangle the invariants retained and small enough to really simplify the calculation.

For the Einstein–Hilbert truncation the most efficient choice is a family of d-spheres Sd(r), labeled by their radius r. For those geometries ∇ R = 0 ρ αβγδ, so they give a vanishing value on all invariants constructed from g = ¯g containing covariant derivatives acting on curvature tensors. What remains (among the local invariants) are terms of the form ∫ √ -- gP (R ), where P is a polynomial in the Riemann tensor with arbitary index contractions. To linear order in the (contractions of the) Riemann tensor the two invariants relevant for the Einstein–Hilbert truncation are discriminated by the Sd (r) metrics as they scale differently with the radius of the sphere: ∫ √g-∼ rd, ∫ √gR- (g) ∼ rd−2. Thus, in order to compute the beta functions of ¯ λk and ZNk it is sufficient to insert an d S (r) metric with arbitrary r and to compare the coefficients of d r and d−2 r. If one wants to do better and include the three quadratic invariants ∫ R αβγδRαβγδ, ∫ Rα βRαβ, and ∫ R2, the family Sd (r ) is not general enough to separate them; all scale like rd−4 with the radius.

Under the trace we need the operator (2) Γk [g,¯g], the Hessian of Γ k[g,¯g] at fixed ¯g. It is calculated by Taylor expanding the truncation ansatz, Γ k[g¯+ f¯,¯g] = Γ k[¯g,¯g] + O (¯f) + Γ qkuad[f¯;¯g] + O (¯f3), and stripping off the two ¯f’s from the quadratic term, Γ quad= 1∫ f¯Γ (2)f¯ k 2 k. For ¯g αβ a metric on Sd(r) one obtains

∫ { ( ) } Γ quad[f¯;¯g] = 1Z κ2 dx f^ [− ¯∇2 − 2¯λ + C R¯]f^αβ − d-−-2- φ [−∇¯2 − 2¯λ + C ¯R] φ(4,.37 ) k 2 Nk αβ k T 2d k S
CT := d-(d-−-3-) +-4, CS := d-−-4. (4.38 ) d (d − 1) d
In order to partially diagonalize this quadratic form, f¯ αβ has been decomposed into a traceless part ^f αβ and the trace part proportional to φ: −1 f¯αβ = f^αβ + d ¯gαβφ, αβ ¯g ^fαβ = 0. Further, 2 αβ ¯∇ = ¯g ¯∇ α¯∇ β is the Laplace operator of the background geometry, and R¯ = d(d − 1)∕r2 is the numerical value of the curvature scalar on Sd(r).

At this point we can fix the coefficients 𝒵k which appear in the cutoff operators ℛk and gh ℛ k of Equation (4.14View Equation). They should be adjusted in such a way that for every low–momentum mode the cutoff combines with the kinetic term of this mode to − ¯∇2 + k2 times a constant. Looking at Equation (4.34View Equation, 4.35View Equation) we see that the respective kinetic terms for ^fαβ and φ differ by a factor of − (d − 2)∕2d. This suggests the following choice:

[ d − 2 ] 𝒵 αkβγδ= (I − Pφ)αβγδ − -----P αφβγδ ZNk. (4.39 ) 2d
Here (P φ) γδ= d−1¯gαβ¯gγδ αβ is the projector on the trace part of the metric. For the traceless tensor (4.39View Equation) gives 𝒵 = Z k Nk, and for φ the different relative normalization is taken into account. Thus we obtain in the ^ f and the φ-sector, respectively:
( ) −2 (2) [ 2 2 (0) 2 2 ¯ ] κ Γk [g,g] + ℛk ^f^f = ZNk − ∇ + k ℛ (− ∇ ∕k ) − 2λk + CTR , ( (2) ) [ ] (4.40 ) κ −2Γk [g,g] + ℛk = − d−2d2ZNk − ∇2 + k2ℛ (0)(− ∇2 ∕k2) − 2¯λk + CSR . φφ

From now on we may set ¯g = g and for simplicity we have omitted the bars from the metric and the curvature. Since we did not take into account any renormalization effects in the ghost action we set gh Z k ≡ 1 in gh ℛk and obtain similarly, with CV = − 1∕d,

− ℳ + ℛgh = − ∇2 + k2ℛ (0)(− ∇2∕k2 ) + CVR. (4.41 ) k
Looking at Equation (4.37View Equation) we see that for d > 2 the trace φ has a “wrong sign” kinetic term which corresponds to a normalization factor Zφk < 0. The choice (4.39View Equation) complies with the rule (4.25View Equation) motivated earlier. As a result, 𝒵 < 0 k in the φ-sector. The negative Z φ k is a reflection of the notorious conformal factor instability in the Einstein–Hilbert action.

At this point the operator under the first trace on the right-hand-side of Equation (4.31View Equation) has become block diagonal, with the ^f ^f and φφ blocks given by Equation (4.40View Equation). Both block operators are expressible in terms of the Laplacian ∇2, in the former case acting on traceless symmetric tensor fields, in the latter on scalars. The second trace in Equation (4.31View Equation) stems from the ghosts; it contains (4.41View Equation) with 2 ∇ acting on vector fields.

It is now a matter of straightforward algebra to compute the first two terms in the derivative expansion of those traces, proportional to ∫ dx √g--∼ rd and ∫ dx √gR- (g) ∼ rd−2. Considering the trace of an arbitrary function of the Laplacian, W (− ∇2), the expansion up to second order derivatives of the metric is given by

{ ∫ √ -- 1 ∫ √ -- } Tr[W (− ∇2 )] = (4π )−d∕2tr(I ) Qd∕2[W ] ddx g + --Qd∕2−1[W ] dx gR(g) + O (R2) .(4.42 ) 6
The Qn’s are defined as
∫ --1-- ∞ n− 1 Qn [W ] = Γ (n ) dz z W (z) (4.43 ) 0
for n > 0, and Q0 [W ] = W (0) for n = 0. The trace tr(I) counts the number of independent field components. It equals 1, d, and (d − 1)(d + 2)∕2, for scalars, vectors, and traceless tensors, respectively. The expansion (4.42View Equation) is derived using the heat kernel expansion
( )d∕2 ∫ -{ } Tr[exp(− is∇2 )] = -i-- tr(I) dx √ g 1 − 1-isR (g) + O (R2 ) , (4.44 ) 4πs 6
and Mellin transform techniques [179Jump To The Next Citation Point]. Using Equation (4.42View Equation) it is easy to calculate the traces in Equation (4.31View Equation) and to obtain the RG equations in the form ∂tZNk = ... and ¯ ∂t(ZNk λk) = .... We shall not display them here since it is more convenient to rewrite them in terms of the dimensionless parameters (4.36View Equation).

In terms of the dimensionless couplings g and λ the RG equations become a system of autonomous differential equations

k∂kgk = [d − 2 + ηN ]gk =: βg(gk,λk ), (4.45 ) k∂kλk = βλ(gk,λk).
Here ηN := − k∂k ln ZNk, the anomalous dimension of the operator √-- gR (g), is explicitly given by
ηN = --gkB1-(λk-)--, (4.46 ) 1 − gkB2 (λk )
with the following functions of λ k:
1- 1−d∕2[ 1 2 1 2 ] B1 (λk) := 3(4π ) d(d + 1)Φ d∕2−1(− 2λk)− 6d(d − 1)Φ d∕2(− 2λk)− 4d Φd∕2−1(0)− 24Φ d∕2(0) , (4.47 ) [ ] B (λ ) := − 1(4π)1−d∕2 d(d + 1)^Φ1 (− 2λ ) − 6d(d − 1)^Φ2 (− 2λ ) . 2 k 6 d∕2−1 k d∕2 k

The beta function for λ is given by

1 [ ] βλ = − (2 − ηN)λk + -gk (4 π)1−d∕2 2d(d + 1)Φ1d∕2(− 2λk) − 8dΦ1d∕2(0) − d(d + 1)ηN^Φ1d∕2(− 2λk)(4..48 ) 2
The Φ’s and ^Φ’s appearing in Equations (4.47View Equation, 4.48View Equation) are certain integrals involving the normalized cutoff function ℛ (0),
∫ ∞ (0) (0) Φp (w) := --1-- dz zn−1ℛ---(z)-−-z∂zℛ---(z), n Γ (n) 0 [z + ℛ (0)(z) + w]p ∫ ∞ (0) (4.49 ) ^Φpn(w) := --1-- dz zn−1-----ℛ---(z)------, Γ (n) 0 [z + ℛ(0)(z) + w ]p
for positive integers p, and n > 0.

With the derivation of the system (4.45View Equation) we managed to find an approximation to a two-dimensional projection of the FRGE flow. Its properties and the domain of applicability or reliability of the Einstein–Hilbert truncation will be discussed in Section 4.4. It will turn out that there are important qualitative features of the truncated coupling flow (4.45View Equation) which are independent of the cutoff scheme, i.e. independent of the function ℛ (0). The details of the flow pattern on the other hand depend on the choice of the function (0) ℛ and hence have no intrinsic significance.

By construction the normalized cutoff function ℛ (0)(u ), u = p2∕k2, in Equation (C.21View Equation) describes the “shape” of ℛk (p2) in the transition region where it interpolates between the prescribed behavior for 2 2 p ≪ k and 2 2 k ≫ p, respectively. It is referred to as the shape function therefore.

In the literature various forms of ℛ (0)’s have been employed. Easy to handle, but disadvantageous for high precision calculations is the sharp cutoff  [181Jump To The Next Citation Point] defined by ℛk (p2) ∼ lim ˆR→ ∞ ˆR θ(1 − p2∕k2), where the limit is to be taken after the 2 p integration. This cutoff allows for an evaluation of the Φ and ^ Φ integrals in closed form. Taking d = 4 as an example, Equations (4.45View Equation) boil down to the following simple system then

gk[ 5 ] ∂tλk = − (2 − ηN)λk − ---5 ln(1 − 2λk) − 2ζ(3) + -ηN , π 2 ∂tgk = (2 + ηN )gk, (4.50 ) 2gk [ 18 ] ηN = − --------- --------+ 5 ln (1 − 2λk ) − ζ (2 ) + 6 . 6 π + 5gk 1 − 2λk
(For orientation, Equation (4.50View Equation) corresponds to the sharp cutoff with s = 1 in [181Jump To The Next Citation Point]). The flow described by Equation (4.50View Equation) is restricted to the halfplane {(λ, g),− ∞ < λ < 1∕2, − ∞ < g < ∞ } since the beta functions are singular along the boundary line λ = 1∕2. When a trajectory hits this line it cannot reach the infrared (k = 0) but rather terminates at a nonzero kterm. Within the Einstein–Hilbert truncation this happens for all trajectories which approach a positive ¯λk at low k.

Also the cutoff with ℛ (0)(u) = (1 − u)θ(1 − u) allows for an analytic evaluation of the integrals; it has been used in the Einstein–Hilbert truncation in [136Jump To The Next Citation Point]. In order to check the scheme (in)dependence of certain results it is desirable to perform the calculation, in one stroke, for a whole class of ℛ (0)’s. For this purpose the following one parameter family of exponential cutoffs has been used [205Jump To The Next Citation Point133Jump To The Next Citation Point131Jump To The Next Citation Point]:

ℛ (0)(u) = ---su--. (4.51 ) s esu − 1
The precise form of the cutoff is controlled by the “shape parameter” s. For s = 1, Equation (4.51View Equation) coincides with the standard exponential cutoff (4.15View Equation). The exponential cutoffs are suitable for precision calculations, but the price to be paid is that their Φ and ^ Φ integrals can be evaluated only numerically. The same is true for a one-parameter family of shape functions with compact support which was used in [133Jump To The Next Citation Point131Jump To The Next Citation Point].

The form of the expression (4.46View Equation) for the anomalous dimension illustrates the nonperturbative character of the method. For g B (λ ) < 1 k 2 k Equation (4.46View Equation) can be expanded as

∑ ηN = gkB1 (λk ) gnkB2(λk )n, (4.52 ) n≥0
which illustrates that even a simple truncation can sum up arbitrarily high powers of the couplings. It is instructive to consider the approximation where only the lowest order is retained in Equation (4.52View Equation). In terms of the dimensionful Gk := gk∕k2 one has in d = 4 and for λk ≈ 0, ηN = B1(0)G0k2 + O(G20k4), so that integrating k∂kGk = ηNGk yields
[ 1 ] Gk = G0 1 + --B1(0)G0 k2 + O (G20 k4) . (4.53 ) 2
The constant B1(0) := [Φ1(0) − 24Φ2 (0)]∕(3π ) 1 2 is ℛ (0)-dependent via the threshold functions defined in Equation (4.49View Equation). However, for all cutoffs one finds that B1 (0) < 0. An acceptable shape function (0) ℛ (z) must interpolate between (0) ℛ (0) = 1 and (0) ℛ (∞ ) = 0 in a monotonic way, the ‘drop’ occuring near z = 1. This implies that Φ11(0) and Φ22(0) are both positive. Since typically they are of order unity this suggests that B1(0) should be negative. This has been confirmed for one-parameter families of cutoffs such as Equation (4.51View Equation) or those of [133Jump To The Next Citation Point131Jump To The Next Citation Point], for the family of sharp cutoffs, and for the optimized cutoff. Using also numerical methods it seems impossible to find an acceptable ℛ (0) which would yield a positive B1 (0). The approximation (4.53View Equation) is valid for k2 ≪ G −1= M 2 0 Pl. One sees that at least in this regime G k is a decreasing function of k. This corresponds to the antiscreening behavior discussed in Sections 1.1 and 1.5.

Above we illustrated the general ideas and constructions underlying truncated gravitational RG flows by means of the simplest example, the Einstein–Hilbert truncation (4.34View Equation). The flow equations for the R2 truncation are likewise known in closed form but are too complicated to be displayed here. These ordinary differential equations can now be analyzed with analytical and numerical methods. Their solution reveals important evidence for asymptotic safety. Before discussing these results in Section 4.4 we comment here on two types of possible generalizations.

Concerning generalizations of the ghost sector truncation, beyond Equation (4.29View Equation) no results are available yet, but there is a partial result concerning the gauge fixing term. Even if one makes the ansatz (4.33View Equation) for Γ k[g, ¯g] in which the gauge fixing term has the classical (or more appropriately, bare) structure one should treat its prefactor as a running coupling: α = αk. After all, the actual “theory space” of functionals Γ [g,¯g,σ, ¯σ] contains “¯ Γ-type” and “gauge-fixing-type” actions on a completely symmetric footing. The beta function of α has not been determined yet from the FRGE, but there is a simple argument which allows us to bypass this calculation.

In nonperturbative Yang–Mills theory and in perturbative quantum gravity α = αk = 0 is known to be a fixed point for the α evolution. The following heuristic argument suggests that the same should be true beyond perturbation theory for the functional integral defining the effective average action for gravity. In the standard functional integral the limit α → 0 corresponds to a sharp implementation of the gauge fixing condition, i.e. exp(− Sgf) becomes proportional to δ[Qα ]. The domain of the 𝒟f αβ integration consists of those fαβ’s which satisfy the gauge fixing condition exactly, Q α = 0. Adding the infrared cutoff at k amounts to suppressing some of the fαβ modes while retaining the others. But since all of them satisfy Q α = 0, a variation of k cannot change the domain of the f αβ integration. The delta functional δ[Qα ] continues to be present for any value of k if it was there originally. As a consequence, α vanishes for all k, i.e. α = 0 is a fixed point of the α evolution [137].

In other words we can mimic the dynamical treatment of a running α by setting the gauge fixing parameter to the constant value α = 0. The calculation for α = 0 is more complicated than at α = 1, but for the Einstein–Hilbert truncation the α-dependence of βg and β λ, for arbitrary constant α, has been found in [133Jump To The Next Citation Point]. The R2 truncations could be analyzed only in the simple α = 1 gauge, but the results from the Einstein–Hilbert truncation suggest the UV quantities of interest do not change much between α = 0 and α = 1 [133Jump To The Next Citation Point131Jump To The Next Citation Point].

Up to now we considered pure gravity. As far as the general formalism is concerned, the inclusion of matter fields is straightforward. The structure of the flow equation remains unaltered, except that now (2) Γ k and ℛk are operators on the larger space of both gravity and matter fluctuations. In practice the derivation of the projected FRG equations can be quite formidable, the main difficulty being the decoupling of the various modes (diagonalization of (2) Γk) which in most calculational schemes is necessary for the computation of the functional traces.

Various matter systems, both interacting and non-interacting (apart from their interaction with gravity) have been studied in the literature. A rather detailed analysis of the fixed point has been performed by Percacci et al. In [72171170Jump To The Next Citation Point] arbitrary multiplets of free (massless) fields with spin 0,1∕2, 1 and 3∕2 were included. In [170] a fully interacting scalar theory coupled to gravity in the Einstein–Hilbert approximation was analyzed, with a local potential approximation for the scalar self-interaction. A remarkable finding is that in a linearized stability analysis the marginality of the quartic self-coupling is lifted by the quantum gravitational corrections. The coupling becomes marginally irrelevant, which may offer a new perspective on the triviality issue and the ensued bounds on the Higgs mass. Making the number of matter fields large O (N ), the matter interactions dominate at all scales and the nontrivial fixed point of the 1∕N expansion [216217203] is recovered [169].

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