# Gentle introduction to rigorous Renormalization Group: a worked fermionic example

Alessandro Giuliani, Vieri Mastropietro, Slava Rychkov

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Alessandro Giuliani, Vieri Mastropietro, Slava Rychkov. Gentle introduction to rigorous Renormalization Group: a worked fermionic example. Journal of High Energy Physics, 2021, 2021 (01), pp.026. 10.1007/JHEP01(2021)026. hal-02933971

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# Gentle introduction to rigorous Renormalization Group: a worked fermionic example

Alessandro Giuliani,<sup>a,b</sup> Vieri Mastropietro<sup>c</sup> and Slava Rychkov<sup>d,e</sup>

E-mail: giuliani@mat.uniroma3.it, vieri.mastropietro@unimi.it

**ABSTRACT:** Much of our understanding of critical phenomena is based on the notion of Renormalization Group (RG), but the actual determination of its fixed points is usually based on approximations and truncations, and predictions of physical quantities are often of limited accuracy. The RG fixed points can be however given a fully rigorous and non-perturbative characterization, and this is what is presented here in a model of symplectic fermions with a nonlocal ("long-range") kinetic term depending on a parameter $\epsilon$ and a quartic interaction. We identify the Banach space of interactions, which the fixed point belongs to, and we determine it via a convergent approximation scheme. The Banach space is not limited to relevant interactions, but it contains all possible irrelevant terms with short-ranged kernels, decaying like a stretched exponential at large distances. As the model shares a number of features in common with $\phi^4$ or Ising models, the result can be used as a benchmark to test the validity of truncations and approximations in RG studies. The analysis is based on results coming from Constructive RG to which we provide a tutorial and self-contained introduction. In addition, we prove that the fixed point is analytic in $\epsilon$, a somewhat surprising fact relying on the fermionic nature of the problem.

**KEYWORDS:** Renormalization Group, Nonperturbative Effects

ARXIV EPRINT: 2008.04361

<footnote><sup>a</sup>Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, L.go S. L. Murialdo 1, 00146 Roma, Italy</footnote>

<footnote><sup>b</sup>Centro Linceo Interdisciplinare Beniamino Segre, Accademia Nazionale dei Lincei, Via della Lungara 10, 00165 Roma, Italy</footnote>

<footnote><sup>c</sup>Dipartimento di Matematica, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy</footnote>

<footnote><sup>d</sup>Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France</footnote>

<footnote><sup>e</sup>Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France</footnote>
# Contents

1 Introduction 1

1.1 The model 4

1.2 Strategy and open questions 5

1.3 Convergence, analyticity and non-perturbative nature of the fixed point 7

1.4 Summary 8

2 Definition of the model and formulation of the problem 9

2.1 Renormalization map 11

3 The fixed point equation at lowest order 13

4 The Banach space of interactions 16

4.1 Representation of interactions by kernels 16

4.1.1 Trimmed representation 17

4.2 Norms 18

4.2.1 The norm of $H_\ell$ 18

4.2.2 The norm of a trimmed sequence 20

5 The renormalization map 21

5.1 Integrating-out map 21

5.2 Trimming 23

5.2.1 Equivalent coupling sequences 23

5.2.2 Trimming map 24

5.3 Dilatation 26

5.4 Renormalization map in the trimmed representation 26

5.5 Fixed point equation 27

5.6 Norm bounds 29

5.6.1 Bounds for $S_\ell^{\ell_1, \dots, \ell_n}$ 30

5.6.2 Bounds for $R_\ell^{\ell_1, \dots, \ell_n}$ 31

6 Construction of the fixed point 32

6.1 Key lemma 32

6.2 Abstract analysis 34

6.2.1 Complex version of the Abstract Lemma 38

6.3 Fixed point theorem 38

6.4 Semigroup property and $\gamma$-independence 40

6.5 Analyticity 42
7 Proof of Key lemma 43

7.1 Case $\ell \ge 8$ 44

7.2 Case $\ell = 6R$ 46

7.3 Case $\ell = 4R$ 47

7.4 Case $\ell = 2R$ 48

7.5 $e_{\nu}^{(0)}$ 48

7.6 $e_{\lambda}^{(0)}$ 49

7.7 Possibility of all choices 49

8 Discussion and open problems 50

8.1 Open questions 52

8.1.1 Extensions to other nonlocal models 52

8.1.2 Further properties of the RG fixed points 53

8.1.3 Conformal invariance 54

8.1.4 Relations with analytic regularization 55

8.1.5 Increasing the range of $\varepsilon$ 55

8.1.6 Extension to non-integer $N$? To non-integer $d$? 57

8.1.7 Connections to Functional Renormalization Group 58

8.1.8 Bosonic fixed points 58

A Gevrey classes and fluctuation propagator bounds 60

A.1 Cutoff function of Gevrey class 60

A.2 Fluctuation propagator bounds 62

A.2.1 k-space 62

A.2.2 x-space 63

B Details about $H_{\text{eff}}$ 64

C Trimming details 65

D Determinant bounds for fermionic expectations 68

D.1 Simple expectations 68

D.2 Connected expectations 70

D.3 Brydges-Battle-Federbush (BBF) formula 72

D.4 Gawedzki-Kupiainen-Lesniewski (GKL) bound 76

D.5 Bound on the number of anchored trees 79

E Proof of $S_l^{\ell_1, \dots, \ell_n}$ norm bound 79

F Estimates of $\Delta_k^{(1)}, \Delta_k^{(2)}$ 81

G One-loop coefficients $I_1$ and $I_2$ 83

H Finite volume and non-perturbative validity of $H_{\text{eff}}$ 86
I Fixed point in a formal power series expansion 92

J Fixed point via the tree expansion 95

J.1 On the flow of the effective couplings 100

K Rigorously constructed bosonic fixed points 101

# 1 Introduction

Renormalization group (RG) is a pillar of theoretical physics, explaining how long-distance collective behavior emerges from microscopic models. Critical phenomena are thus understood in terms of RG fixed points (Wilson [1–3]) and universality is explained in terms of basins of attractions. While this beautiful picture qualitatively works very well, quantitative applications often lead to practical difficulties. To compute critical exponents, one typically does perturbation theory in a small parameter, like $\epsilon = 4-d$ in the $\epsilon$-expansion [4]. The accuracy of this procedure is limited by the proliferation of Feynman diagrams, and by the slow convergence of Borel-resummed series (while without resummation it normally diverges). As a consequence, predictions of critical exponents using perturbative RG [5, 6] are often less accurate than lattice Monte Carlo simulations or the conformal bootstrap [7].

It should be stressed that Wilson did not consider RG limited to situations with a small coupling. Two strongly coupled RG examples can be found [2]. One is his famous solution of the Kondo problem. The other is less known but no less impressive: an RG calculation for the 2D Ising model in a space of 217 couplings, concluding that “one can do precise calculations using pure renormalization group methods with the only approximations being based on locality.”

Other developments in theoretical physics suggest, indirectly, that Wilson ideas are non-perturbatively correct. We can mention here exact results in two-dimensional field theory (see e.g. [8]), obtained by conformal field theory, integrable models, exact S-matrix bootstrap etc., which have provided many examples of exact non-perturbative RG flows, never finding any inconsistency with Wilsonian expectations. In higher dimensions, exact results in supersymmetric theories (see e.g. [9]) as well as the gauge-gravity duality considerations (see e.g. [10, 11]) have always confirmed Wilson ideas.

It is therefore somewhat surprising that the most straightforward interpretation of Wilson’s vision, as a non-perturbative machine which would allow non-perturbatively and with an essentially unlimited precision to compute the properties of any RG fixed point of interest, has not so far been achieved. It is fair to say that this was not for the lack of trying, see e.g. [12] for the early attempts.

Two notable, although not fully successful, attempts have been the Functional Renormalization Group (FRG) [13, 14] and Tensor Network Renormalization (TNR) [15–17]. The FRG calculations include couplings with arbitrary powers of fluctuating field, but
only up to some finite derivative order. Unfortunately, with more derivatives, FRG results tend to become more and more sensitive to the parameters specifying the regulating function [18, 19]. This has been traced to increasing violations of conformal invariance, except at some special parameter values satisfying a “principle of minimal sensitivity” [20]. It remains to be understood why the convergence to the fixed point does not hold more robustly in FRG. As to the TNR, it works well for simple 2D lattice models such as the 2D Ising model but hasn’t been yet as effective in higher dimensions.

So, in spite of these attempts, although it is generally believed that the non-perturbative Wilsonian RG fixed points do exist, at present they often remain Platonic objects, confined to the world of ideas and accessible to us only via approximations of rather limited accuracy. Take e.g. the RG fixed point for the 3D Ising model. If it exists, which Banach space does it belong to? Can we access it via a provably convergent approximation scheme? At the moment these questions are wide open.

Note that Wilson believed in the Ising RG fixed point very concretely: as a fixed-point Hamiltonian invariant under a Kadanoff block-spin transformation. As mentioned above, for 2D Ising, he found an approximate fixed-point Hamiltonian numerically, truncating to a space of 217 lattice spin interactions [2]. But the convergence of his scheme has never been proven, nor has it been implemented in 3D. Incidentally, Wilson did worry about rigorous convergence properties of RG maps; e.g. in [21] a model RG transformation was shown to be convergent for a rescaling parameter larger than $4 \times 10^6$.

Of course, as already mentioned, there are nowadays other methods to get precise values of critical exponents, most notably the conformal bootstrap [7]. However, this does not mean that the RG should be abandoned. First, RG is more general than the conformal bootstrap, since many RG fixed points important for physics do not have conformal invariance, such as any fixed point involving time evolution or relaxation, and having a dynamical critical exponent $z \neq 1$. Second, it is quite possible that there exist much better RG implementations, and we just haven’t found them yet.

Since theoretical physicists have not been able to find a fully successful implementation of non-perturbative RG in spite of many attempts, can mathematical physics give any hint about what we have been doing wrong? In mathematical physics, the rigorous construction of non-trivial RG fixed points has been achieved in different cases using Constructive RG (CRG). It has been obtained in bosonic scalar field theories<fnref n="1" /> [22–26] and interacting fermions [27] with long range interactions, in cases where the system has a scaling dimension differing from marginality by an $\varepsilon$. It has also been achieved in models with marginal interactions of strength $\lambda$ and asymptotically vanishing beta function, such as 1D interacting fermions [28–31] and 2D spin, vertex and dimer models [32–39]. In all these cases, the non-perturbative existence of a non-trivial RG fixed point, close to the Gaussian or free Fermi one, has been proved for $\varepsilon$ or $\lambda$ sufficiently small, and the critical exponents can be computed at an arbitrary precision in terms of resummed perturbative expansions, with rigorous bounds on the remainder. A feature of the CRG is that the fixed point is found, without any approximation, in a Banach space of interactions where all the irrele-

<footnote n="1">¹See section 8.1.8 and appendix K for more details about the rigorously constructed bosonic fixed points.</footnote>
vant terms are nonlocal, even though fast decaying (e.g. like a stretched exponential): this is in striking contrast with the FRG, where the space of interactions is typically spanned by a sequence of *local* functions of the fluctuation field and its derivatives.<fnref n="2" />

One lesson of all this body of rigorous work is that *weakly coupled* non-perturbative RG is possible, both in the bosonic and fermionic case, although it is easier in the fermionic case because in this case convergent perturbation theory captures full non-perturbative physics. We emphasize that, in general, fermionic perturbation theory is expected to be convergent only in the running rather than the bare coupling, see section 1.3. On the other hand, *strongly coupled* non-perturbative RG has so far been out of reach of mathematical physics research. For this reason we will, as a first step, focus in this paper on the weakly coupled fermionic case, well understood by mathematical physicists, and aim to transfer this knowledge into the theoretical physics realm.

With this in mind, we will present here the rigorous construction of a non-Gaussian fixed point for a fermionic model with weakly relevant quartic interaction. This is possibly the simplest model of this kind where to test field-theoretical RG methods, and a perfect example to provide an introduction to CRG accessible to a wider audience. We will explain how these methods allow one to characterize a non-trivial fixed point without any ad-hoc assumption or any uncontrolled approximation. The above mentioned crucial role played by the space of *mildly nonlocal* interactions, as opposed to expanding all interactions in local functions of the field, will be evident from our presentation.

A complementary goal of our work will be to prove a new result, which is the *analyticity* of the $\varepsilon$-expansion for our non-Gaussian fermionic fixed point. This is in contrast with bosonic $\varepsilon$-expansions, which are, at best, Borel summable. Although we will focus on $\varepsilon > 0$ in much of the paper, eventually we will show analyticity in a complex disk around $\varepsilon = 0$. Although the sign of $\varepsilon$ is correlated with the sign of the fixed point quartic interaction, Dyson's argument against analyticity does not apply for our model because fermions are allowed to have quartic interaction of either sign. Moreover, analyticity in $\varepsilon$ of the fixed point is not in contradiction with the divergence of perturbation theory in the bare couplings, see section 1.3 for further comments.

Note that in this paper we only construct RG fixed points, and we do not discuss in detail the RG flow between the microscopic model and the constructed fixed points. In any case, the result about analyticity is only valid for the fixed point and does not extend to the full RG flow, whose very structure changes discontinuously with the sign of $\varepsilon$. For positive $\varepsilon$ we will have the gaussian model at short distances, perturbed by the relevant quadratic and quartic couplings and flowing at long distances to the nontrivial RG fixed point. For negative $\varepsilon$ it will be the other way around: starting from the nontrivial RG fixed point we may flow to the gaussian model at long distances, with the quartic coupling then describing the leading irrelevant interaction at long distances. The former situation is referred to as ‘IR fixed point’, while the latter as ‘UV fixed point’. To avoid any ambiguity we stress that no reverse RG flow is implied: all RG flows are from short to long distances.

<footnote n="2">²For a fair comparison it should be noted that FRG calculations are often performed in terms of the 1PI effective action, not the Wilsonian effective action used here. Also, some FRG schemes do attempt to go beyond the local derivative expansion. See section 8.1.7.</footnote>
The convergence of our CRG scheme lets us hope that it will be used as a starting point for developing systematic and stable truncation schemes for the fermionic FRG. Although we have been able to prove our theorems only for small $\epsilon$, it might be that the range of practical applicability of our scheme is order one and includes strongly coupled fixed points — in the future one should try to see if this is the case (see section 8.1.5).

Rigorous non-perturbative constructions of bosonic fixed points present several complications compared to the fermionic case discussed here (see section 8.1.8). Extracting lessons from those constructions for practical RG calculations remains another important open problem for the future.

## 1.1 The model

The model we consider is schematically described by the following action:

$$a\text{MFT}(\psi) + \nu_0 \int d^d x \psi^2 + \lambda_0 \int d^d x \psi^4, \qquad (1.1)$$

where aMFT is an ‘anticommuting Mean Field Theory’ of the fermionic<fnref n="3" /> field $\psi$ with $N$ components, and the two additional terms are quadratic and quartic interactions preserving $Sp(N)$ global symmetry. Our fermions will be scalars under rotation, rather than spinors. So the model is not reflection-positive, but reflection positivity will play no role in the RG analysis.

Local models of this kind were considered in [40] under the name ‘symplectic fermions’, and are relevant for the description of polymers and loop-erased random walks [41, 42]. In 3D, interacting symplectic fermion models have been recently considered in the context of dS/CFT correspondence [43]. See also [44, 45] for theoretical studies of related models, and [46–48] for other speculative appearances of symplectic fermions in physics. Our model is a variant of those, with a nonlocal (“long-range”) kinetic term of the schematic form $\psi (\partial^2)^\# \psi$ where $(\partial^2)^\#$ is a non-integer power of the Laplacian. Similar long-range models in a large $N$ limit were recently considered in [49] in relation to the SYK model.

More precisely, our model is defined as follows: take an even number $N$ of real Grassmann fields $\psi = (\psi_a)_{a=1}^N$ in $\mathbb{R}^d,^4$ with $d = 1, 2, 3$. The reference Gaussian theory (the aMFT mentioned above) is characterized by the following two-point function:

$$\langle \psi_a(x) \psi_b(y) \rangle = \Omega_{ab} P(x-y) \equiv G_{ab}(x,y), \qquad (1.2)$$

where $P(x) \propto \frac{1}{|x|^{d/2-\epsilon}}$ at large distances (see the next section for the explicit expression), with $\epsilon$ small and positive, and $\Omega_{ab}$ is the symplectic $N \times N$ matrix:

$$\Omega_{ab} = \begin{bmatrix} 1 & & & & \\ & -1 & & & \\ & & \ddots & & \\ & & & 1 & \\ & & & & -1 \end{bmatrix} \qquad (1.3)$$

<footnote n="3">³Fermionic=anticommuting=Grassmann in this paper.</footnote>

<footnote n="4">⁴We will not make a distinction between the lower and upper $Sp(N)$ and Euclidean indices whose position is determined only by typographic convenience: $\psi^a \equiv \psi_a$ and $\partial_\mu \psi \equiv \partial^\mu \psi$.</footnote>
The quadratic and quartic monomials $\psi^2$ and $\psi^4$ in (1.1) must be interpreted as $\Omega_{ab}\psi_a\psi_b$ and $(\Omega_{ab}\psi_a\psi_b)^2$. Given the form of $P(x)$, the fields $\psi_a$ are assigned the scaling dimension $[\psi] = d/4 - \epsilon/2$, so that the quadratic and quartic terms in (1.1) are both relevant, the quartic one being barely so for $\epsilon$ small and positive. The parameter $\epsilon$ plays a role similar to the deviation of spatial dimension $d$ from 4, $\epsilon = 4 - d$ in the Wilson-Fisher $\epsilon$-expansion, which, contrary to ours, is not at present suitable for a rigorous non-perturbative RG analysis because the space of $4 - \epsilon$ dimensions has not been rigorously defined so far.<fnref n="5" /> For $d = 1, 2, 3$, there are no other local relevant or marginal terms in addition to those included in (1.1). In perturbative RG, the lowest order RG equations for the fixed point are, letting $\gamma$ be the scaling parameter:

$$ \nu = \gamma^{d/2+\epsilon}(\nu + I_1\lambda + \dots), \quad \lambda = \gamma^{2\epsilon}(\lambda + I_2\lambda^2 + \dots), \quad (1.4) $$

where $I_1$ and $I_2$ are positive constants given by the one-loop Feynman diagrams. Neglecting the higher-order terms, we get a nontrivial fixed point $\lambda_* = (1 - \gamma^{2\epsilon})/I_2$, $\nu_* = I_1\lambda_*/(1 - \gamma^{d/2+\epsilon})$, which is $O(\epsilon)$, close to the Gaussian one. This is just an approximation, and we want to be sure that the existence of the fixed point is not spoiled by non-perturbative effects caused by higher orders or irrelevant terms. Moreover, we want to define a scheme whose truncations provide arbitrarily good approximations of the actual fixed point, with apriori bounds on the error made.

## 1.2 Strategy and open questions

Our rigorous construction of the fixed point goes as follows. First, we identify a space which is left invariant by the RG iteration. We cannot restrict to the (finite) space of relevant couplings, since the RG transformation generates the irrelevant interactions, whatever the input action is. Similarly, we cannot restrict to the space of local irrelevant interactions, because it too is not left invariant by the RG map. The right choice turns out to be the span of all possible monomials in $\psi$ and $\partial\psi$ with nonlocal, but sufficiently fast decaying, kernels: this space is left invariant. We stress that this mild nonlocality is unrelated to the long-range character of our reference Gaussian theory; it has to do with the fact that the IR-cutoff propagator is not fully local although short-range; it would be present also for the local kinetic term. Note that we couldn't find an invariant space of nonlocal monomials involving $\psi$ only: in our construction the presence of derivative fields $\partial\psi$ is generated by what we call the trimming operation, which consists in extracting from a nonlocal monomial of order 2 or 4 its local part, and in re-expressing the nonlocal remainder in terms of irrelevant monomials of the form $\partial\psi\partial\psi$ or $\psi^3\partial\psi$. Note also that our construction does not exclude the existence of other invariant spaces, with different (non)locality properties of the kernels; in particular, it remains to be seen whether there exists an invariant Banach space consisting of local monomials in $\psi$ and its derivatives (of arbitrary order), but we are not aware of any rigorous result in this sense. The construction of an invariant Banach space of interactions comes, in particular, with a non-perturbative definition of the RG

<footnote n="5">⁵See however [50] for a non-perturbative analysis in non-integer $d$ using the conformal bootstrap.</footnote>
map: this is achieved via combinatorial cancellations due to the $\pm$ signs in the series expansion, ultimately due to the fermionic nature of the fields. In order to take advantage of these cancellations, we need to organize the perturbative expansion in the form of series of determinants, rather than in the more standard form of series of Feynman diagrams. This may be thought of as a smart rearrangement and partial resummation of the perturbative series: while the Feynman diagrams expansion is *not* absolutely convergent, the determinant expansion is. Once the invariant space has been identified and the RG map defined at a non-perturbative level, we prove that the RG map is *contractive* in a suitable neighborhood of the approximate lowest-order fixed point: this implies existence and uniqueness of the actual fixed point in such a neighborhood. (More precisely, the RG map is contractive near the fixed point along all directions but $\psi^2$, but this complication is easily taken care of.) Remarkably, such fixed point is analytic in $\varepsilon$.

Therefore, the problem of obtaining the correct Banach space to which the fixed point belongs, and of computing the fixed point via a provably convergent approximation scheme, while still open for 3D Ising, is completely solved in our fermionic case, at least when $\varepsilon$ is sufficiently small.

Our results have similarities with those of Gawedzki and Kupiainen (GK) [27], with some differences. GK had fermions transforming as spinors and the model (long-range Gross-Neveu) was reflection positive. This is a minor difference and we could have considered their model, the only complication being an extra spinorial index. Their quartic interaction was weakly irrelevant rather than weakly relevant, and so they have obtained an ultraviolet fixed point,<fnref n="6" /> while our fixed point for $\varepsilon > 0$ is in the infrared. Our proof establishes estimates on the irrelevant fixed point interactions which are of natural size suggested by perturbation theory. Finally, we establish fixed point analyticity that, as far as we know, has not been previously pointed out. Let us mention that the fixed point we construct can also be obtained by using a different, rigorous, CRG scheme, based on a tree expansion [51], which bypasses the use of the contraction mapping theorem, as well as the apriori definition of an invariant Banach space of irrelevant interactions (see appendix J).

Open questions, to be addressed in future work, include: the computation of critical exponents and their independence from the cutoff, rigorous derivation of conformal invariance and the operator product expansion (OPE) the connection between our mildly nonlocal representation of the fixed point with the local operators used in conformal field theory, the relation with analytic regularization and Wilson-Fisher $\epsilon$-expansion, computer-assisted computation of the radius of convergence, crossover to the local symplectic fermion fixed point for $\varepsilon = \varepsilon_* = O(1)$ in $d=3$ (Do critical exponents coincide with the Wilson-Fisher $\epsilon$-expansion for the local symplectic fermions in such a limiting case?), etc. See section 8 for a complete list of open problems (9 pages!).

<footnote n="6">⁶One of the purposes of [27] was to construct rigorously a healthy theory at short distances from a non-renormalizable effective theory at long distances, hence their title. This was made possible by the small parameter (weakly irrelevant interaction). Unfortunately, their paper is often misunderstood as a license to search for the UV theory in terms of IR degrees of freedom even when there is no weak coupling in sight (as e.g. in the asymptotic safety program for gravity).</footnote>
## 1.3 Convergence, analyticity and non-perturbative nature of the fixed point

Some readers may feel that our result about the analyticity of the $\epsilon$-expansion contradicts quantum field theory lore, and here we wish to explain why this is not the case.

There are two main reasons for the divergence of perturbation theory in quantum field theory: classical solutions (instantons) and renormalons. Since our theory is fermionic, it does not contain instantons. A related difference of fermions vs bosons is that bosons only make sense for positive quartic while fermions are defined for quartic of any sign, and indeed our fixed point coupling $\lambda_*$ will be positive or negative depending on the sign of $\epsilon$.

As for the renormalons and associated divergences (see e.g. reviews [52, 53]), they exist both for fermions and bosons, but only if there is running over a long range of scales. Also in our model, the full RG flow from UV to IR would not be analytic, for reasons similar to renormalons in asymptotically free theories like QCD. However, in this paper we focus exclusively on the fixed point physics, so there is no running, and we are immune to renormalons.

Let us illustrate this point by a short computation, considering for definiteness the weakly relevant quartic case (positive $\epsilon$). Note that the infrared fixed point can be constructed in two equivalent ways. The first, which is the one we use in the rest of this paper, is to construct it as the fixed point of the single step Wilsonian RG transformation. The second, which we briefly discuss here and in appendix J.1, is to construct it dynamically, as the infrared limit of the flow of the running couplings. We will not consider the full flow from the gaussian fixed point, but a “half-flow” which starts at an intermediate scale and flows to the IR fixed point. Even such a “half-flow” is already non-analytic, as we will see.

In our model, the beta function flow equation for $\lambda(t)$ (the running quartic coupling at scale $t$, where $t \le 0$ is the logarithm of the infrared cutoff scale) at lowest order has the following form:

$$ \frac{d\lambda(t)}{dt} = -2\epsilon\lambda(t) + c_2\lambda^2(t), \qquad (1.5) $$

for a suitable positive constant $c_2$. The solution to (1.5) with initial condition $\lambda(0) = \lambda_0$, which we assume to be positive and smaller than $2\epsilon/c_2$, is:

$$ \lambda(t) = \frac{\lambda_0}{e^{2\epsilon t} + \frac{c_2\lambda_0}{2\epsilon}(1 - e^{2\epsilon t})}. \qquad (1.6) $$

The infrared fixed point is $\lambda_* = \lim_{t \to -\infty} \lambda(t) = 2\epsilon/c_2$, which is obviously analytic in $\epsilon$. At any finite $t$, $\lambda(t)$ is analytic in $\lambda_0$, but non-uniformly in $t$, as $|t|$ grows. This effect is clearly due to the running and to the nontrivial structure of the RG flow: small positive $\lambda_0$ eventually flow to $\lambda_*$, while small negative $\lambda_0$ flow away to large negative values of the coupling. However, $\lambda(t)$ is Borel-summable in $\lambda_0 > 0$ uniformly in $t$. The complete flow is more complicated than the toy model (1.5), but it retains the same qualitative features as the above illustration. In our fermionic setting, fixed point observables, such as critical exponents, are expected to be convergent power series in $\lambda_*$ and, therefore, analytic in $\epsilon$. Observables (e.g. correlation functions) at intermediate distance scales can be expressed as convergent power series in the whole sequence $\{\lambda(t)\}_{t \le 0}$ (see appendix J.1 for further
details on this point). Due to the non-analytic dependence of $\lambda(t)$ in the bare coupling $\lambda_0$, such observables are expected to be “just” Borel-summable in $\lambda_0$.

Finally, let us comment on the setup of massless perturbation theory, i.e. when the gaussian fixed point is perturbed by only the (weakly relevant) quartic coupling, setting mass to zero and working directly in the continuum limit. Such a setup, under the name of “conformal perturbation theory” [54–56], is often considered when perturbing non-gaussian fixed points (see e.g. [57, 58] for recent applications), but it could be used in our problem as well. It is a form of perturbative expansion in the bare coupling. At a small but fixed $\epsilon$, the first $n$ terms of the resulting perturbative expansion will be finite, where $n \sim 1/\epsilon$, while subsequent terms have infinite coefficients (because the corresponding integrals diverge at long distances). Thus, the perturbative expansion itself is ill-defined beyond the first few terms in this framework.

Some authors, e.g. ref. [56], argued that this pathology is a possible signal of the appearance of non-analytic terms in the infrared fixed point observables (although, as [56] admits, “their actual presence is unclear at the moment”). It has to be emphasized that we are talking here about the situation when the RG flow leads to a fixed point, and only about the infrared fixed point observables, such as the critical exponents. We are not concerned with the situation when the flow leads to a massive phase, in which case the mass of the particles is indeed generically non-analytic in the bare couplings. While non-analytic terms may affect bosonic relatives of our model (see section 8.1.8), in our fermionic model we rigorously exclude them, see remark 5.1 and appendix H. Our analytic-in-$\epsilon$ fixed point defines the infrared theory in a fully non-perturbative way. Thereby, results based on convergent perturbation theory lead in our case to a fully non-perturbative description. The key point allowing this to happen is that in finite volume Grassmann integrals are finite dimensional. Therefore, in presence of any finite-volume cutoff, there is no room for non-analytic terms. Furthermore, uniformity in the volume of our estimates, in combination with uniqueness theorems for the limit of uniformly convergent analytic functions, imply that the absence of non-analytic terms carries over to the infinite-volume limit, see appendix H for details.

## 1.4 Summary

The paper is structured as follows. In section 2 we present the model and we state informally our main results. In section 3 we identify an approximate nontrivial fixed point by truncating the RG map at lowest order (explicit perturbative computations are in appendix G). The rest of the paper will be devoted to a non-perturbative proof of its existence: in section 4 we introduce the Banach space of interactions consisting of monomials in the fields with mildly nonlocal interactions, and we equip it with a suitable norm, tailored for our smooth slicing cutoffs (whose properties are in appendix A). In section 5 we show that the assumed form of the interaction is left invariant by the RG map, a fact made apparent rearranging the output via a trimming operation (more details are in appendices B and C). We show also there that the action of the RG map can be expressed as a series which is absolutely convergent in norm; this follows from a number of results described in appendix D, such as determinant bounds for simple fermionic expectations and a suitable
representation of connected expectations. Absolute convergence allows to rigorously estimate the action of the RG map, and this allows us in section 6 to prove, see theorem 6.1, the existence of the fixed point, together with its independence of the slicing parameter and its analyticity in $\epsilon$. This result relies on the crucial Key Lemma 6.1 and its variants, which ensures that the Banach space is invariant and the RG map is contractive. The key lemma is in a sense optimal, as it predicts a dependence on $\epsilon$ of the effective interactions which is exactly the one suggested by perturbation theory; this is obtained by a careful choice of constants done in the proof, presented in section 7 and appendix F. section 8 is devoted to conclusions and open problems. The fact that our convergent analysis fully reconstruct the theory and provides non-perturbative information is proved in appendix H. In appendix I we show that the fixed point can be obtained via a formal series expansion; perturbation theory is similar for boson or fermionic models but for fermions the series converges, a fact offering, see appendix J, a way to construct the RG fixed point alternative to the path via Banach space and contraction method, using instead the direct tree expansion technique. Finally in appendix K a review and comparison with previous results in bosonic theories is presented.

## 2 Definition of the model and formulation of the problem

Let us now discuss the model more in detail. The propagator (1.2) is defined in terms of $P(x)$, which is chosen in the form

$$P(x) = \int \frac{d^d k}{(2\pi)^d} \hat{P}(k) e^{ikx}, \qquad \hat{P}(k) = \frac{\chi(k)}{|k|^{\frac{d}{2}+\epsilon}}. \qquad (2.1)$$

The function $\chi(k)$ here is a “UV cutoff”, a short-distance regulator of the model. We will choose it satisfying the following conditions (see figure 1):

$$\chi \text{ is a radial } C^\infty \text{ function, } 0 \le \chi(k) \le 1, \quad \chi(k) = \begin{cases} 1, & (|k| \le 1/2) \\ 0, & (|k| \ge 1), \end{cases} \qquad (2.2)$$

In fact we will require something a bit stronger than $\chi \in C^\infty$, namely:

$$\chi \text{ belongs to the Gevrey class } G^s \text{ for some } s > 1. \qquad (2.3)$$

This “Gevrey condition” will be defined in section 4.2, see eq. (4.14), and is not used until then. As explained there, it is needed so that the fluctuation propagator $g(x)$ (see section 2.1) decays at infinity as a stretched exponential. There are many cutoff functions satisfying both conditions (2.2) and (2.3); an explicit example is given in appendix A.1.

As a consequence of (2.1) and (2.2), $P(x)$ is uniformly bounded, and its large-$x$ asymptotics is proportional to $1/|x|^{d/2-\epsilon}$, as stated after (1.2).

We denote by $d\mu_P(\psi)$ the Gaussian Grassmann integration with propagator (1.2), which can be formally written as:

$$d\mu_P(\psi) = D\psi e^{S_2(\psi)}, \\ S_2(\psi) = \frac{1}{2} \int \frac{d^d k}{(2\pi)^d} \hat{P}(k)^{-1} \Omega_{ab} \psi_a(k) \psi_b(-k). \qquad (2.4)$$
<figure><img src="image_2.png" /><figcaption>**Figure 1.** The function $\chi(k)$ (red curve), and the resulting function $\chi(k) - \chi(\gamma k)$, eq. (2.10) (blue dots).</figcaption></figure>

[Since $\hat{P}(k)^{-1}$ is non-analytic in $k^2$ near $k=0$, such an action is called “long-range”.]

More precisely, $d\mu_P(\psi)$ is characterized by the expectations of an even number $2s$ of fields, via:

$$ \begin{aligned} \langle \psi_{a_1}(x_1) \dots \psi_{a_{2s}}(x_{2s}) \rangle &\equiv \int d\mu_P(\psi) \psi_{a_1}(x_1) \dots \psi_{a_{2s}}(x_{2s}) \\ &= \sum_{\pi} (-)^{\pi} \prod_{i=1}^{s} G_{\pi(a_{2i-1})\pi(a_{2i})}(x_{2i-1}, x_{2i}), \end{aligned} \quad (2.5) $$

where the sum is over all pairings of $2s$ fields and $(-)^{\pi}$ is the sign of the corresponding permutation.

The full model is defined by an “interacting” Grassmann measure

$$ Z^{-1} d\mu_P(\psi) e^{H(\psi)}, \quad (2.6) $$

where the “interaction” $H(\psi)$ is a bosonic function of Grassmann fields, and $Z = \int d\mu_P(\psi) e^{H(\psi)}$ is the partition function. The simplest interaction includes only the local quadratic and quartic terms:<fnref n="7" />

$$ H_L(\psi) = \nu \int d^d x \Omega_{ab} \psi_a \psi_b + \lambda \int d^d x (\Omega_{ab} \psi_a \psi_b)^2. \quad (2.7) $$

This interaction has $Sp(N)$ global symmetry rotating the fermion indices, as well as $O(d)$ spatial invariance. We will assume $N \ge 4$ so that the quartic interaction does not vanish identically.<fnref n="8" /> We will furthermore assume

$$ d \in \{1, 2, 3\}, \quad 0 < \varepsilon < \min(2 - d/2, d/6) = d/6, \quad (2.8) $$

where the second condition guarantees that the two terms in $H_L(\psi)$ are the only $O(d) \times Sp(N)$-invariant interactions which are relevant, see section 5.3.<fnref n="9" />

<footnote n="7">⁷Here and below we denote the local couplings by $\nu, \lambda$, rather than by $\nu_0, \lambda_0$, as in (1.1). The change of notation is meant to highlight the difference between the bare couplings $\nu_0, \lambda_0$, and the running ones, which will be their meaning from now on. In fact, in the following, we shall construct the interaction $H$ corresponding to the infrared fixed point, whose local quadratic and quartic couplings correspond to the fixed point values $\nu_*, \lambda_*$ computed at lowest order in section 3. The notation $\nu, \lambda$ is used for generic values of the parameter entering the RG equations.</footnote>

<footnote n="8">⁸For $N=2$ the quartic interaction vanishes, while for $N=4$ it is proportional to $\psi_1\psi_2\psi_3\psi_4$.</footnote>

<footnote n="9">⁹The notion of relevance in our setup involving mildly nonlocal kernels will be made precise below in eq. (5.37), and it will agree with the usual rule that the interaction containing $l$ fields and $p$ derivatives is relevant if $l[\psi] + p < d$.</footnote>
RG transformation of the model will be acting in a more general space of interactions

$$H(\psi) = H_L(\psi) + H_{\text{IRR}}(\psi), \qquad (2.9)$$

where $H_{\text{IRR}}(\psi)$ stands for an infinite number of generally nonlocal (although mildly so) terms corresponding to irrelevant interactions. Like $H_L(\psi)$, interactions in $H_{\text{IRR}}(\psi)$ will respect $Sp(N) \times O(d)$ invariance.<fnref n="10" />

**Remark 2.1** Eq. (2.6) as written is not immediately meaningful in infinite volume, because partition function is infinite: $Z = \infty$. To give it a rigorous meaning, we should e.g. put the model in finite volume and pass to the limit. To speed up this introductory part of the paper, let us work directly in infinite volume and consider the interacting measure (2.6) in the sense of formal perturbative expansion in $H(\psi)$. In perturbation theory, the normalization factor $Z^{-1}$ in (2.6) means that diagrams with disconnected interaction vertices should be excluded when computing expectations. In the main text we will show that infinite-volume perturbation theory is convergent (this is a general feature of fermionic models at weak coupling). The rigorous definition as a limit from finite volume is postponed to appendix H. Taking this limit will be easy once the infinite volume behavior is understood. See also remark 5.1 below.

## 2.1 Renormalization map

Let us fix a “rescaling parameter”<fnref n="11" /> $\gamma \ge 2$. We will define the “renormalization map” which maps $H(\psi)$ to another interaction $H'(\psi)$. It will be a composition of integrating-out and dilatation.

Integrating-out consists in splitting the field $\psi$ as $\psi = \psi_\gamma + \phi$ where $\psi_\gamma$ is the “low-momentum component” of $\psi$, and defining the effective interaction $e^{H_{\text{eff}}(\psi_\gamma)}$ by eliminating $\phi$. Concretely, we split the Grassmann propagator as (see figure 1)

$$P(x) = P_{\gamma}(x) + g(x), \quad \hat{P}_{\gamma}(k) = \frac{\chi(\gamma k)}{|k|^{\frac{d}{2}+\epsilon}}, \quad \hat{g}(k) = \frac{\chi(k) - \chi(\gamma k)}{|k|^{\frac{d}{2}+\epsilon}}. \qquad (2.10)$$

Note that $P_\gamma$ is just a rescaled version of $P$ (see eq. (2.14)), while $g(x)$ is called “fluctuation propagator”. This decomposition implies factorization of the integration measure $d\mu_P(\psi)$ as

$$d\mu_P(\psi) = d\mu_{P_\gamma}(\psi_\gamma)d\mu_g(\phi), \quad \psi = \psi_\gamma + \phi, \qquad (2.11)$$

where $\psi_\gamma$ and $\phi$ are two independent Grassmann fields with propagators $P_\gamma$ and $g$. As mentioned, eq. (2.3) will guarantee that $g(x)$ decays at infinity as a stretched exponential.

Correlation functions of $\psi_\gamma$ with respect to the interacting measure (2.6) can equivalently be computed with respect to the measure

$$d\mu_{P_\gamma}(\psi_\gamma)e^{H_{\text{eff}}(\psi_\gamma)} \qquad (2.12)$$

<footnote n="10">¹⁰In the trimmed representation of section 4.1.1, $H_{\text{IRR}}(\psi)$ will consist of $H_{2R}$, $H_{4R}$, $H_{6SL}$, $H_{6R}$ and $H_\ell$ for $\ell \ge 8$.</footnote>

<footnote n="11">¹¹Although at this point any $\gamma > 1$ would do, we assume $\gamma \ge 2$ from the start, as some estimates below, specifically eq. (5.43), will require that $\gamma$ is separated from 1. The fixed point construction will require raising $\gamma$ even further.</footnote>
(normalization understood) where $e^{H_{\text{eff}}(\psi_{\gamma})}$ is defined by “integrating out the fluctuation field” $\phi$:<fnref n="12" />

$$e^{H_{\text{eff}}(\psi_{\gamma})} = \int d\mu_g(\phi)e^{H(\psi_{\gamma}+\phi)}. \qquad (2.13)$$

Note that the propagator $P_{\gamma}$ is related to $P$ via

$$P_{\gamma}(x) = \gamma^{-2[\psi]} P(x/\gamma) \qquad (2.14)$$

with $[\psi] = d/4 - \varepsilon/2$ as above. This motivates to consider the dilatation transformation:

$$\psi_{\gamma}(x) \mapsto \gamma^{-[\psi]} \psi(x/\gamma), \qquad (2.15)$$

which maps the measure (2.12) to the measure $d\mu_P(\psi)e^{H'(\psi)}$ with the same gaussian factor as in (2.6) but with a different interaction:

$$H'(\psi) = H_{\text{eff}}[\gamma^{-[\psi]} \psi(\cdot/\gamma)]. \qquad (2.16)$$

This formula defines the renormalization map $R = R(\varepsilon, \gamma) : H \mapsto H'$ (also called “RG transformation”). Note that $R$ also depends on $d, N, \chi$ but this dependence will be left implicit. As a function of $\gamma$ for a fixed $\varepsilon$, the renormalization map satisfies the semigroup property:

$$R(\varepsilon, \gamma_1)R(\varepsilon, \gamma_2) = R(\varepsilon, \gamma_1\gamma_2). \qquad (2.17)$$

Our main goal will be to construct the fixed point of the RG transformation. We would like to remind the reader that although our RG transformation is obtained by integrating out the degrees of freedom with momenta between $\Lambda \sim 1$ and $\Lambda_{\text{IR}} \sim \Lambda/\gamma$, one should not think of $\Lambda_{\text{IR}}$ as some sort of mass which breaks criticality of our fixed point. The correct interpretation is that we have only one RG scale, $\Lambda$, while $\Lambda_{\text{IR}}$ entered the game because we find it technically convenient to consider the discrete RG transformation rather than the continuous one, such as Polchinski’s equation [59]. At an intuitive level discrete RG transformation can be obtained by integrating the continuous one, and they are expected to have the same fixed points (although to make rigorous sense of the continuous RG may be nontrivial, see remark 5.4 below). In particular, it would be wrong to think that some sort of ‘IR cutoff removal’ has to be performed with our result to extract the fixed point physics. On the contrary, all of this physics is already contained in the fixed point $H_*$. E.g., the critical exponents can be obtained by linearizing the RG transformation (the same one which leads to the fixed point), near the fixed point, and computing the eigenvalues.

After this warning, the informal formulation of our main result goes as follows: Fix $\chi, d \in \{1, 2, 3\}$, and $N \ge 4, N \ne 8$. For $\gamma$ large enough and $\varepsilon > 0$ small enough, there exists a nontrivial interaction $H_*(\varepsilon)$ which is a fixed point of $R(\gamma, \varepsilon)$ for all $\gamma$:

$$R(\varepsilon, \gamma)[H_*(\varepsilon)] = H_*(\varepsilon). \qquad (2.18)$$

<footnote n="12">¹²We will drop the $\psi_{\gamma}$-independent term in $H_{\text{eff}}(\psi_{\gamma})$, since this constant drops out when normalizing and does not affect the expectations. As we will discuss in appendix H, this constant is finite in finite volume although it becomes infinite in the infinite-volume limit.</footnote>
Moreover, $H_*(\varepsilon)$ can be extended to an analytic function of $\varepsilon$ in a small neighborhood of the origin.

A precise statement is the content of Theorems 6.1 and 6.2, which rely on Key Lemma 6.1 and Abstract Lemma 6.2. The condition $N \neq 8$ comes from requiring a non-vanishing one-loop beta-function. The condition that $\gamma$ is sufficiently large arises for the following technical reason: the fixed point $H_*(\varepsilon)$ will live in a Banach space, and only for sufficiently large $\gamma$ will we be able to show that $R(\varepsilon, \gamma)$ is a bounded operator on this Banach space, so that eq. (2.18) makes sense.

The definition of the Banach space requires a suitable representation of the interactions, called trimmed representation, and discussed in section 4 below. In order to define it, we will distinguish, quite naturally, the local (relevant) terms from the nonlocal (irrelevant) ones. Moreover, we will rewrite the nonlocal quadratic or quartic interactions in terms of derivative fields, via the trimming operation, defined in section 5.2 below: the usefulness of a representation in terms of derivative fields is to make the irrelevance of these interactions apparent, already at the level of the linearized RG map. An additional feature of the trimmed representation is that it distinguishes a so-called “semilocal” sextic term from the fully nonlocal sextic interaction. This splitting may look strange at first sight. In our setup with a smooth cutoff in momentum space, it is needed to obtain the correct lowest-order approximation to the fixed point, which is in turn important for defining a neighborhood in the Banach space where the RG map (or, better, a suitable rewriting thereof) is contractive.

In order to better motivate it, let us explain more explicitly the structure of the splitting of the sextic term and the intuitive reason behind its definition: we'll do it in the next section, before getting to the formal definition of the trimmed representation.

### 3 The fixed point equation at lowest order

Let us go back to the lowest order fixed point equation (FPE), whose structure was anticipated in eq. (1.4), and let us discuss its derivation more carefully, in view of our choice of a smooth cutoff function. The most naive approximation one can do is to compute the FPE by neglecting all couplings but the relevant ones, $\nu$ and $\lambda$, and, assuming these couplings to be of order $\varepsilon$, to retain only the dominant contributions to the beta functions for $\nu$ and $\lambda$, which are of order $\varepsilon$ and $\varepsilon^2$, respectively. While very natural, we would like to convince the reader that such a naive approximation leads to a wrong lowest order FPE, whose solution differs from the correct one by $O(\varepsilon)$ rather than $O(\varepsilon^2)$: the important contribution missed by this scheme is the $O(\varepsilon^2)$ contribution to the beta function for the quartic coupling $\lambda$, due to the self contraction of the “semilocal” sextic term (the tree graph contribution to the sextic interaction, of order $O(\varepsilon^2)$), see below for details.

Let us start by describing the most naive approximation (the wrong one). Consider a local interaction, $H(\psi) = H_L(\psi)$, see (2.7), and integrate the fluctuation field via (2.13). After this integrating-out step the local couplings are modified as follows:

$$ \nu \rightarrow \nu + \Delta\nu \equiv \nu_{\text{eff}}, \quad \lambda \rightarrow \lambda + \Delta\lambda \equiv \lambda_{\text{eff}}, \qquad (3.1) $$
where the leading contributions to $\Delta\nu$, $\Delta\lambda$ are given by the diagrams

$$ \begin{aligned} \Delta\nu &= \underbrace{\mathcal{O}}_{\text{loc}} + \dots \\ \Delta\lambda &= \left(\underset{\text{loc}}{\mathcal{X}}\right) + \dots \end{aligned} \qquad (3.2) $$

Here $(\cdot)_{\text{loc}}$ stands for the local part of the nonlocal term generated by the second diagram.<fnref n="13" /> Note that the diagram $\mathcal{X} = O(\varepsilon^2)$ does not contribute to $\Delta\lambda$ because its local part vanishes. Indeed, the propagator $g$ vanishes in momentum space at $k=0$, or equivalently $\int d^dxg(x) = 0$. For this reason, $\nu$ insertions on external legs never give rise to local terms: $(\mathcal{X})_{\text{loc}} = 0$. One can easily check by inspection that, starting from $H = H_L$, there are no other contributions of $O(\varepsilon)$ to $\Delta\nu$ and of $O(\varepsilon^2)$ to $\Delta\lambda$, beyond those shown in (3.2).

We now rescale the fields as in (2.16) and find $\nu' = \gamma^{d-2[\psi]}\nu_{\text{eff}}$ and $\lambda' = \gamma^{d-4[\psi]}\lambda_{\text{eff}}$, that is, recalling (3.1),

$$ \nu' = \gamma^{\frac{d}{2}+\varepsilon}(\nu + I_1\lambda + \dots), \qquad (3.3) $$

$$ \lambda' = \gamma^{2\varepsilon}(\lambda + I_2^{\mathcal{X}}\lambda^2 + \dots), \qquad (3.4) $$

where $I_1$ and $I_2^{\mathcal{X}}$ are the one-loop diagrams in the two lines of (3.2), respectively. Performing $\Omega$-tensor contractions, one finds $I_2^{\mathcal{X}} \propto N-8 \neq 0$, since we are assuming $N \neq 8$.<fnref n="14" /> It is now extremely tempting to conclude that the fixed point equation for $\lambda$ is

$$ \lambda = \gamma^{2\varepsilon}(\lambda + I_2^{\mathcal{X}}\lambda^2 + \dots), \qquad (3.5) $$

up to terms of $O(\varepsilon^3)$, so that the fixed point is $\lambda_* = (1 - \gamma^{2\varepsilon})/I_2^{\mathcal{X}}$ up to an error $O(\varepsilon^2)$; plugging this into the fixed point equation for $\nu$, one would find $\nu_* = I_1\lambda_*/(1 - \gamma^{d/2+\varepsilon})$ up to an error $O(\varepsilon^2)$. Even if extremely tempting, **this conclusion is wrong!**

Where is the problem? The point is that neglecting the irrelevant terms, and in particular the sextic one, leads to an error of $O(\varepsilon^2)$ in the FPE for $\lambda$; such an error is comparable in size with the term $I_2^{\mathcal{X}}\lambda^2$ that we included above: therefore, dropping blindly the irrelevant terms is not consistent even at the lowest order. To see this, notice that, by starting with a local interaction, $H = H_L$, after having integrated out the fluctuation field, we obtain an effective interaction $H_{\text{eff}}$, whose sextic term contains the tree diagram $\mathcal{X} = O(\varepsilon^2)$. Therefore, in order to find an interaction $H$ solving the fixed point equation $H' = H$ at $O(\varepsilon^2)$, we cannot avoid assuming that $H$ contains a sextic irrelevant term with the same structure as $\mathcal{X}$. Let us then take $H = H_L + H_{\text{IRR}}$, with $H_{\text{IRR}}$ containing the following sextic 'x-term' interaction:

$$ \mathcal{X} = \Omega_{ab}\Omega_{a'b'}\Omega_{cc'} \int d^dx d^dy (\psi_a\psi_b\psi_c)(x) \mathcal{X}(x-y)(\psi_{a'}\psi_{b'}\psi_{c'})(y), \qquad (3.6) $$

<footnote n="13">¹³This operation is done in momentum space by evaluating the diagram with all external momenta set to zero. Alternatively, in position space one replaces the kernel of the nonlocal operator by its integral.</footnote>

<footnote n="14">¹⁴See appendix G for the computations of these coefficients, where we also comment that vanishing of $I_2^{\mathcal{X}}$ is an accident which does not reproduce at higher loops.</footnote>
with the particular shown contraction of $\psi$ indices. This term might be called 'semilocal': there are two $\psi^3$ groups interacting via one nonlocal kernel. The gothic $\mathcal{X}$ is meant to remind about the shape of this diagram. Upon integrating out, the unique new contribution to $\mathcal{X}$ comes from the tree-level diagram contracting two quartic vertices:

$$ \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}]
\draw (-0.5,0) -- (0.5,0);
\node[above] at (-0.5,0) {$\Rightarrow$};
\node[below] at (0.5,0) {$\Rightarrow$};
\node[above] at (-0.5,0.5) {$\Delta\mathcal{X}(x) = -8\lambda^2 g(x)$};
\node[below] at (0.5,0.5) {$\Delta\mathcal{X}(x) = -8\lambda^2 g(x)$};
\end{tikzpicture} \qquad (3.7) $$

At the fixed point, we thus expect $\mathcal{X} = O(\lambda_*^2) = O(\varepsilon^2)$. On the other hand, $\mathcal{X}$ gives a direct contribution to $\Delta\lambda$, which therefore has to be included explicitly: the equation for $\Delta\lambda$ thus has to be corrected as follows:

$$ \Delta\lambda = \left( \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}]
\draw (-0.5,0) -- (0.5,0);
\node[above] at (-0.5,0) {$\Delta\lambda$};
\node[below] at (0.5,0) {$\Delta\lambda$};
\node[above] at (-0.5,0.5) {$+ \varkappa$};
\node[below] at (0.5,0.5) {$+ \varkappa$};
\node[above] at (-0.5,0.5) {$+ \varkappa$};
\node[below] at (0.5,0.5) {$+ \varkappa$};
\node[above] at (-0.5,0.5) {$+ \varkappa$};
\node[below] at (0.5,0.5) {$+ \varkappa$};
\end{tikzpicture} \right)_{\text{loc}} + \dots \qquad (3.8) $$

Assuming that $\mathcal{X}$ is $O(\varepsilon^2)$ and that all the other irrelevant interactions of order 6 or more in the fields are $O(\varepsilon^3)$ or smaller (while the nonlocal, irrelevant, contributions of order 2 or 4 are $O(\varepsilon^2)$), one can check by inspection that the dots in (3.8) are $O(\varepsilon^3)$. At this point some readers may be thrown out of balance: who has ever seen this second diagram? In fact Wilson and Kogut discuss it, [1], eq. (5.23) and below. They do observe that it is $O(\varepsilon^2)$ and thus would need to be included. Only if one uses a sharp cutoff, then this diagram drops out because its local part then vanishes (momenta along the wavy and curved lines do not overlap). Since we use a smooth cutoff, we have to include both diagrams. The corrected leading approximation to the FPE thus involves $\nu$, $\lambda$ and the $\mathcal{X}$-term parametrized by $\mathcal{X}(x)$; it takes the form

$$ \nu = \gamma^{\frac{d}{2}+\varepsilon}[\nu + I_1\lambda + O(\varepsilon^2)], \qquad (3.9) $$

$$ \lambda = \gamma^{2\varepsilon}[\lambda + I_2^{\mathcal{X}}\lambda^2 + (N-8)\int d^dx \mathcal{X}(x)g(x) + O(\varepsilon^3)], \qquad (3.10) $$

$$ \mathcal{X}(x) = \gamma^{2d-6[\psi]}[\mathcal{X}(x\gamma) - 8\lambda^2 g(x\gamma)], \qquad (3.11) $$

the factor $(N-8)$ coming from $\Omega$-tensor contractions. This allows us to compute the fixed point couplings $\nu_*$ and $\lambda_*$ at order $\varepsilon$, while $\mathcal{X}_*$ will be order $\varepsilon^2$. The approximation is consistent: all the irrelevant terms not explicitly shown contribute to the error terms only. Note that (3.11) allows us to express the fixed point $\mathcal{X}_*$ in terms of $\lambda$ as a geometric series:

$$ \mathcal{X}_*(x) = -8\lambda^2 \sum_{n=1}^{\infty} \gamma^{(2d-6[\psi])n} g(x\gamma^n). \qquad (3.12) $$

Plugging $\mathcal{X} = \mathcal{X}_*$ in the right side of (3.10) gives the FPE for the quartic coupling $\lambda = \gamma^{2\varepsilon}(\lambda + I_2\lambda^2)$ up to an error $O(\varepsilon^3)$, with $I_2 = I_2^{\mathcal{X}} + I_2^{\mathcal{X}}$ and $I_2^{\mathcal{X}}$ the constant coming from the term $(N-8) \int d^dx \mathcal{X}(x)g(x)$. We are thus led to the FPE (1.4), which we now expect to be correct at dominant order, contrary to (3.5). Interestingly, in the $\varepsilon \to 0$ limit the sum of the two diagrams $I_2 = I_2^{\mathcal{X}} + I_2^{\mathcal{X}} \approx \bar{I}_2 \log \gamma$, where $\bar{I}_2$ is independent of the choice of the cutoff function (appendix G). Therefore the fixed point coupling $\lambda_*$ is universal at order $\varepsilon$. This is similar to the well-known scheme independence of the first two beta-function coefficients.
In conclusion, the inclusion of the irrelevant sextic terms is crucial for computing the correct coefficients in the lowest order FPE. One expects that the inclusion of more and more irrelevant terms will produce better and better approximation to the FPE, unless non-perturbative effects come into play. Therefore, in order to compute the exact FPE, we'll assume that $H$ belongs to a space of interactions including all possible irrelevant terms, of arbitrary high order in the fields, as discussed in the next section.

# 4 The Banach space of interactions

## 4.1 Representation of interactions by kernels

In order to conveniently represent the interaction, we use the following condensed notation for fields, their first derivatives, and products thereof:

$$ \Psi_A = \begin{cases} \psi_a, & A = a, \\ \partial_\mu \psi_a, & A = (a, \mu), \end{cases} \quad \Psi(\mathbf{A}, \mathbf{x}) = \prod_{i=1}^{l} \Psi_{A_i}(x_i), \quad (4.1) $$

where $\mathbf{A} = (A_1, \dots, A_l)$ and $\mathbf{x} = (x_1, \dots, x_l)$ are finite sequences. $|\mathbf{A}|$ will denote the length of $\mathbf{A}$, and $d(\mathbf{A})$ the number of derivative fields in $\Psi(\mathbf{A}, \mathbf{x})$. See section 5.2.2 for why we allow fields with zero or one (but not more) derivatives.

An interaction $H(\psi)$ is a sum of terms with some kernels $H(\mathbf{A}, \mathbf{x})$:

$$ H(\psi) = \sum_{\mathbf{A}} \int d^d\mathbf{x} H(\mathbf{A}, \mathbf{x}) \Psi(\mathbf{A}, \mathbf{x}), \quad (4.2) $$

where $d^d\mathbf{x} = d^d x_1 \dots d^d x_l$. The kernels satisfy various obvious constraints following from the Grassmann nature of the fields and from the $Sp(N) \times O(d)$ symmetry of the model. E.g., the kernels are assumed antisymmetric.<fnref n="15" /> The individual interaction terms being bosonic, “the number of legs” $l = |\mathbf{A}|$ must be even.

The local quadratic and quartic interactions in (2.7) correspond to $\delta$-function kernels:

$$ \nu \int d^d x \Omega_{ab} \psi_a \psi_b \leftrightarrow \nu \Omega_{ab} \delta(x_1 - x_2), \quad (4.3) $$

$$ \lambda \int d^d x (\Omega_{ab} \psi_a \psi_b)^2 \leftrightarrow \frac{1}{3} \lambda q_{abce} \delta(x_1 - x_2) \delta(x_1 - x_3) \delta(x_1 - x_4), \quad (4.4) $$

where $q_{abce} = \Omega_{ab}\Omega_{ce} - \Omega_{ac}\Omega_{be} + \Omega_{ae}\Omega_{bc}$ is totally antisymmetric. We will represent interactions of $H_{IRR}$ by nonlocal kernels rather than expanding them in local interactions.

We divide the kernels into groups (“couplings”) $H_l$ according to their number of legs $l$ ($l \ge 2$ even):

$$ H_l = \{H(\mathbf{A}, \mathbf{x})\}_{|\mathbf{A}|=l}. \quad (4.5) $$

<footnote n="15">¹⁵This means $H(\pi\mathbf{A}, \pi\mathbf{x}) = (-)^\pi H(\mathbf{A}, \mathbf{x})$ for any permutation acting simultaneously on $\mathbf{A}$ and $\mathbf{x}$. If not already antisymmetric, the antisymmetrized kernels $H^A(\mathbf{A}, \mathbf{x}) = \frac{1}{l!} \sum_{\pi \in S_l} (-)^\pi H(\pi\mathbf{A}, \pi\mathbf{x})$ define the same interaction. The kernel dependence on $Sp(N)$ indices will be made out of $\Omega_{ab}$ tensors. Their dependence on $x_i$ will be an $SO(d)$-invariant tensor built out of $(x_i - x_j)_\mu$ where $\mu$ are spatial indices contained in $\mathbf{A}$ (if any), times a function of pairwise distances $|x_i - x_j|$. Finally, $O(d)$ also contains spatial parity $\psi_a(x) \rightarrow \psi_a(-x)$, $\Psi(\mathbf{A}, \mathbf{x}) \rightarrow (-1)^{d(\mathbf{A})} \Psi(\mathbf{A}, -\mathbf{x})$. Therefore, the kernels will satisfy $H(\mathbf{A}, -\mathbf{x}) = (-1)^{d(\mathbf{A})} H(\mathbf{A}, \mathbf{x})$.</footnote>
The interaction is thus represented by a coupling sequence $(H_l)_{l \ge 2}$. This is a general representation. It is useful to introduce a notation also for the restriction of $H_l$ to the kernels with a specified number of derivatives:

$$H_{l,p} = \{H(\mathbf{A}, \mathbf{x})\}_{|\mathbf{A}|=l, d(\mathbf{A})=p}. \qquad (4.6)$$

We emphasize that $\mathbf{A}$'s are sequences: the ordering is important and terms with different orderings appear separately in (4.2). This convention leads to somewhat simpler combinatorics.

### 4.1.1 Trimmed representation

General representation has too much redundancy in the couplings with a small number of legs. In view of the discussion of section 3, it is convenient to assume that the interaction has a more specific structure. In particular, we want that: $H_2$ consists of a local term like (4.3) plus an irrelevant term schematically of the form $(\partial\psi)^2$; $H_4$ consists of a local term like (4.4) plus an irrelevant term schematically of the form $\psi^3\partial\psi$; $H_6$ consists of a semilocal term like (3.6) plus higher order terms. More precisely, we will assume that the interaction $H$, to be used as the input for the RG map, is written in the trimmed representation, which imposes the following extra requirements on kernels with $l \le 6$:

1. For $H_2$ we require:

i. $H_{2,0}$ should be purely local, i.e. be the $\delta$-function kernel reproducing the local quadratic interaction $\nu \int \psi^2$ given in (4.3),

ii. $H_{2,1} = 0$ (no kernels with one derivative).

We will denote the nonzero parts of trimmed $H_2$ as

$$H_{2L} = H_{2,0}, \quad H_{2R} = H_{2,2} \qquad (4.7)$$

With some abuse of notation, we will identify $H_{2L}$ with the prefactor $\nu$ in front of the delta function, and similarly below for $H_{4L}$ with $\lambda$ and for $H_{6SL}$ with $\mathcal{X}$.

2. For $H_4$ we require that $H_{4,0}$ should be purely local, i.e. be the $\delta$-function kernel reproducing the local quartic interaction $\lambda \int \psi^4$ given in (4.4). We denote the parts of trimmed $H_4$ as

$$H_{4L} = H_{4,0}, \quad H_{4R} = \{H_{4,p}\}_{p \ge 1}. \qquad (4.8)$$

3. For $H_6$ we demand that it comes split into two pieces:

$$H_6 = H_{6SL} + H_{6R}, \qquad (4.9)$$

where $H_{6SL}$ contains only a 'semi-local' interaction with no derivatives, of the form<fnref n="16" /> (3.6), parametrized by a function $\mathcal{X}(x)$. Thus we have

$$H_{6,0} = H_{6SL} + H_{6R,0}, \qquad (4.10)$$

$$H_{6,p} = H_{6R,p} \quad (p \ge 1).$$

<footnote n="16">¹⁶Its kernel is the antisymmetrization of $\Omega_{a_1 a_2} \Omega_{a_4 a_5} \Omega_{a_3 a_6} \delta(x_1 - x_2) \delta(x_1 - x_3) \mathcal{X}(x_1 - x_4) \delta(x_4 - x_5) \delta(x_4 - x_6)$.</footnote>
For the moment we do not make any specific requirement on the no-derivative part of $H_{6R}$.<fnref n="17" />

Mnemonically, L stands for local, SL for semi-local, R for the rest. The trimmed representation thus corresponds to a coupling sequence ($H_ℓ$) where the index $ℓ$ takes values from the ‘trimmed list’

$$ℓ \in \text{TL} = \{2L, 2R, 4L, 4R, 6SL, 6R, 8, 10, 12...\}. \qquad (4.11)$$

The corresponding number of legs, an integer, will be denoted by $|ℓ|$. As for the general representation, we let $H_{ℓ,p}$ be the restriction of $H_ℓ$ to the terms with $p$ derivatives. We will always use $ℓ$ for labels from the trimmed list (4.11) and $l$ for integer labels: $l \ge 2$ even. When $ℓ$ and $l$ appear in the same equation, they are related by $l = |ℓ|$.

**Remark 4.1** The reason for requirements 1,2 is as follows. The $H_{2R}$ and $H_{4R}$ interactions are irrelevant due to the presence of derivatives, while $H_{2,0}$, $H_{2,1}$, $H_{4,0}$ would be relevant by the same criterion, see section 5.3. However, all these “would-be relevant” interactions with arbitrary kernels can be equivalently written as local quadratic and quartic couplings plus an irrelevant $H_{2R}$ and $H_{4R}$ (section 5.2). Therefore, requirements 1,2 make manifest the fact that our model has only two relevant couplings: $ν$ and $λ$.

Requirement 3 originates from the fact that, as discussed in section 3, isolating the semilocal sextic term is important for deriving the correct lowest order FPE; this, in turn, will be crucial for defining the correct neighborhood on which the FPE (or, better, a suitable rewriting thereof) is contractive, see section 6.

**Remark 4.2** Even if the input interaction $H$ is in the trimmed representation, in general the interaction $H_{\text{eff}}$, obtained via the integrating-out step (2.13), won’t be trimmed. In order to put it in trimmed form, we will need to suitably manipulate the kernels of $H_{\text{eff}}$, via an operation called trimming, discussed in section 5.2 below. This is one of the operations needed for proving that the space of interactions is left invariant by the action of the RG map.

## 4.2 Norms

The interactions in the trimmed representation form a vector space. In order to promote it to a Banach space, we need to equip it with a norm: for this purpose, we will first specify the norm in the subspace associated with $ℓ \in \text{TL}$, see (4.11), and then the norm of a trimmed sequence.

### 4.2.1 The norm of $H_ℓ$

We will be measuring the size of interaction kernels by means of the weighted $L_1$ norm

$$\|H(\mathbf{A})\|_w = \int_{x_1=0} d^d\mathbf{x} |H(\mathbf{A}, \mathbf{x})| w(\mathbf{x}), \qquad (4.12)$$

<footnote n="17">¹⁷Eventually, we shall impose a norm condition which will make the no-derivative part of $H_{6R}$ smaller by an extra $ε$ factor, compared with $H_{6SL}$.</footnote>
where $w(\mathbf{x})$ is a translationally invariant weight function. In view of translation invariance we perform the integral fixing one of the $x$ coordinates to zero. We also let

$$ \|H_l\|_w = \max_{|\mathbf{A}|=l} \|H(\mathbf{A})\|_w, \qquad (4.13) $$

and similarly $\|H_\ell\|_w$ and $\|H_{\ell,p}\|_w$ are defined as the maximum of weighted norms of all kernels belonging to the corresponding trimmed coupling.<fnref n="18" />

By choosing $w(\mathbf{x})$ growing at infinity appropriately, we will incorporate the information about the decay of the kernels $H(\mathbf{A}, \mathbf{x})$, induced by the decay of the fluctuation propagator $g(x)$. Recall that we are requiring the Gevrey condition (2.3): $\chi \in G^s$, $s > 1$. Concretely, this means that derivatives of $\chi$ of arbitrary order $\alpha$ are uniformly bounded by

$$ \max_{k \in \mathbb{R}^d} |\partial^\alpha \chi(k)| \le C^{|\alpha|} |\alpha|^{|\alpha|s} \qquad (4.14) $$

for some constant $C = C(\chi) > 0$. The Gevrey condition is stronger than $C^\infty$ but weaker than real analyticity. Importantly for us, Gevrey class contains compactly supported functions. An explicit example of a cutoff functions satisfying both condition (2.2) and (2.3) is given in appendix A.1.

As usual, decay of $g(x)$ is related to the smoothness of its Fourier transform $\hat{g}(k)$, i.e. to the smoothness of $\chi(k)$. It turns out that the Gevrey condition (4.14) implies a stretched exponential decay. Namely, we have the following bound for the fluctuation propagator, as well as its first and second derivatives needed below:

$$ |g(x)|, |\partial_\mu g(x)|, |\partial_\mu \partial_\nu g(x)| \le M(x) \equiv C_{\chi1} e^{-C_{\chi2} |x/\gamma|^\sigma} \quad (x \in \mathbb{R}^d), \qquad (4.15) $$

where $\sigma = 1/s < 1$. The constants $C_{\chi1}, C_{\chi2}$ depend on $\chi$ but are independent of $\gamma$. See appendix A.2 for a detailed proof, while here we only give two simple remarks. First, the decay scale $x \sim \gamma$ in (4.15) is as expected from the IR momentum cutoff $\sim \gamma^{-1}$. Second, stretched exponential is the best we could hope for: exponential decay ($\sigma = 1$) would require analyticity of $\chi$, incompatible with the compact support.

Kernels $H(\mathbf{A}, \mathbf{x})$ are expected to decay at large separation with the same rate as (4.15). We choose $w(\mathbf{x})$ growing with a similar rate. A convenient choice turns out to be

$$ w(\mathbf{x}) = e^{\bar{C}(\mathrm{St}(\mathbf{x})/\gamma)^\sigma}, \qquad (4.16) $$

where $\mathrm{St}(\mathbf{x})$ is the Steiner diameter of the set $\mathbf{x}$, defined [60] as the length of the shortest tree $\tau$ connecting the points in $\mathbf{x}$ (the tree may contain extra vertices as in figure 2).<fnref n="19" /> We will fix $\bar{C} = \frac{1}{2}C_{\chi2}$ so that $M(x)$ has a finite weighted norm:

$$ \|M\|_w = \int d^d x M(x) e^{\bar{C}(|x|/\gamma)^\sigma} = \mathrm{Const.} \gamma^d < \infty. \qquad (4.17) $$

<footnote n="18">¹⁸To avoid any misunderstanding, we stress that $\|H_{6SL}\|_w$ and $\|H_{6R}\|_w$ are two independently defined quantities.</footnote>

<footnote n="19">¹⁹To be precise, $\mathrm{St}(\mathbf{x}) = \min_{\mathbf{x}'} \min_{\tau(\mathbf{x} \cup \mathbf{x}')'} L(\tau)$, the minimum taken over all possible trees $\tau$ with vertices $\mathbf{x} \cup \mathbf{x}'$, with the tree length $L(\tau)$ defined as the sum of the edge lengths. $\mathrm{St}(\mathbf{x})$ coincides with the usual diameter if all points lie on a line (e.g. for sets of 2 points). See appendix E for an explanation of why we use the Steiner diameter.</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 2.** The optimal Steiner tree $\tau$ for this configuration of 4 points contains two extra vertices.</figcaption></figure>

Finally, it will be convenient to use definition (4.13) also for the $\delta$-function kernels of the local quadratic and quartic interactions. Since $w = 1$ when all points coincide, we have (see (4.3), (4.4))

$$ \|H_{2L}\|_w = |\nu|, \quad \|H_{4L}\|_w = \frac{1}{3}|\lambda|. \qquad (4.18) $$

### 4.2.2 The norm of a trimmed sequence

The norm of an interaction $H$ associated with the trimmed sequence $(H_\ell)$, $\ell \in \text{TL}$, will have the form $\|H\| = \sup_{\ell \in \text{TL}} \|H_\ell\|_w / \delta_\ell$, for a sequence $\delta_\ell$ to be fixed conveniently. In order to decide how to let $\delta_\ell$ scale with $\ell$, let us first develop an intuition about the expected size of $H_\ell$ at the fixed point. If we parametrize $H$ by $(\nu, \lambda, \mathcal{X}, u)$, with $u = (H_\ell)_{\ell \neq 2L, 4L, 6SL} \equiv (u_\ell)_{\ell \in \{2R, 4R, 6R, 8, 10, \dots\}}$, we expect that at the fixed point $\nu$ and $\lambda$ are of order $\varepsilon$, $\mathcal{X}$ is equal to the kernel $\mathcal{X}_*$ in eq. (3.12), which is of order $\varepsilon^2$, and $u_\ell$ is of the order of the corresponding tree graph (i.e., the leading Feynman diagram with vertices all of type $\lambda$ contributing to the interaction labelled $\ell$), namely: of order $\varepsilon^2$ if $\ell = 2R, 4R$; of order $\varepsilon^3$ if $\ell = 6R$; of order $\varepsilon^{l/2-1}$ if $\ell = l \ge 8$.

In the following, in order to determine the fixed point, we will fix $\mathcal{X} = \mathcal{X}_*$, thought of as a function of $\lambda$, see (3.12), and parametrize the fixed point interaction by the remaining coordinates, $y = (\nu, \lambda, u)$. On this subspace, we will use the following norm (depending on the parameters $\delta, A_0, A_0^R, A_1^R, A_2^R, A$):

$$ \|y\|_Y = \max \left\{ \frac{|\nu|}{A_0 \delta}, \frac{|\lambda|}{A_0 \delta}, \frac{\|u_{2R}\|_w}{A_0^R \delta^2}, \frac{\|u_{4R}\|_w}{A_1^R \delta^2}, \frac{\|u_{6R}\|_w}{A_2^R \delta^3}, \sup_{l \ge 8} \frac{\|u_l\|_w}{A \delta^{l/2-1}} \right\}, \quad (4.19) $$

where, motivated by the intuitive discussion above, the parameter $\delta$ will be chosen to be proportional to $\varepsilon$. Eq. (4.19) defines the Banach space of interest.<fnref n="20" /> Eventually, the constants $A_0, A_0^R, A_1^R, A_2^R, A$ will be fixed in such a way that the action of the RG map (or better, of a suitable equivalent rewriting thereof, called $F$ in the following, see section 6.2) on the sequence $y$ returns a new sequence $y'$ in the same Banach space $Y$. Even more, we will show that there is a neighborhood $Y_0$ in $Y$ on which the fixed point map $F$ is a contraction and, therefore, $F$ admits a unique fixed point in $Y_0$. All this will be proved in sections 6 and 7 below. As a preparation to these proofs, we need to specify how the RG map explicitly acts on the space of trimmed sequences. This will be discussed in section 5.

<footnote n="20">²⁰It is easy to see in particular that the space is complete with respect to the introduced norm (because weighted $L_1$ spaces are complete).</footnote>
In particular, the result of the RG map on the trimmed sequence ($H_ℓ$) will be expressed in the form of a series, see eq. (5.24) below, which is absolutely convergent in the norm of interest, thanks to the bounds discussed in section 5.6 below.

# 5 The renormalization map

In this section we detail the structure of the RG map, thought of as a map from the vector space of trimmed sequences into itself. We proceed in steps: we first describe the integrating-out map (2.13), assuming the input to be a trimmed sequence (section 5.1). In general, the output of the integrating-out map is not trimmed: therefore, we explain how to make it so, via the trimming operation (section 5.2). Next, we perform the rescaling (2.16) (section 5.3). In section 5.4 we combine these three steps and derive the representation eq. (5.24), which expresses the image of the RG map as a series in the multi-indices ($ℓ_i$)$_{i=1}^n$, $ℓ_i \in$ TL. Remarkably, this series turns out to be absolutely convergent in the relevant norms, i.e., those introduced in section 4.2 above, thanks to the norm bounds discussed in section 5.6.

## 5.1 Integrating-out map

Let $H$ be an interaction associated with the trimmed sequence ($H_ℓ$), $ℓ \in$ TL, and consider the integrating-out map (2.13). For the effective interaction $H_{\text{eff}}$ we have a well-known perturbative formula in terms of connected expectations (see appendix D.2):<fnref n="21" />

$$H_{\text{eff}}(\psi) = \sum_{n=1}^{\infty} \frac{1}{n!} \underbrace{\langle H(\psi + \phi); H(\psi + \phi); \dots; H(\psi + \phi) \rangle_c}_{n \text{ times}} \quad (5.1)$$

We write the interaction in the trimmed representation using the same eq. (4.2) as in the general representation. Kernels $H(\mathbf{A}, \mathbf{x})$ now come from couplings $H_ℓ$. The $H_{2,1}$ kernels are absent due to trimming requirements. The $H_{6,0}$ kernels are understood as a sums of $H_{6SL}$ and $H_{6R,0}$ kernels, see (4.10). All the other kernels $H(\mathbf{A}, \mathbf{x})$ are associated with a unique coupling $H_ℓ$.

Replacing $\psi \to \psi + \phi$ in (4.2), each term gives rise to ‘interaction vertices’ with ‘external’ $\psi$ legs and ‘internal’ $\phi$ legs. Parametrizing the external legs by a subsequence $\mathbf{B} \subset \mathbf{A}$,<fnref n="22" /> and the internal ones by $\bar{\mathbf{B}} = \mathbf{A} \setminus \mathbf{B}$, we write:

$$H(\psi + \phi) = \sum_{\mathbf{A}} \sum_{\mathbf{B} \subset \mathbf{A}} (-)^{\#} \int d^d\mathbf{x} H(\mathbf{A}, \mathbf{x}) \Psi(\mathbf{B}, \mathbf{x}_{\mathbf{B}}) \Phi(\bar{\mathbf{B}}, \mathbf{x}_{\bar{\mathbf{B}}}), \quad (5.2)$$

where $(-)^{\#}$ is the sign, which we won't need to track, produced by reordering the fields to put all $\psi$'s first. The $\mathbf{x}_{\mathbf{B}}$ and $\mathbf{x}_{\bar{\mathbf{B}}}$ are the corresponding restrictions of the coordinate vector $\mathbf{x}$. Substituting (5.2) into (5.1), we obtain the following formula for the kernels of

<footnote n="21">²¹In eq. (2.13) the argument of $H_{\text{eff}}$ was $\psi_\gamma$, the low-momentum component of $\psi$. The momentum-range restriction turns out unimportant for working out (5.1), so we replaced $\psi_\gamma$ by a generic $\psi$.</footnote>

<footnote n="22">²²Sequences being ordered sets, a subsequence inherits ordering from the parent sequence.</footnote>
the effective interaction (see appendix B for more details):

$$ H_{\text{eff}}(\mathbf{B}, \mathbf{x}_{\mathbf{B}}) = \mathcal{A} \sum_{n=1}^{\infty} \frac{1}{n!} \sum_{\substack{\mathbf{B}_1, \dots, \mathbf{B}_n \\ \sum \mathbf{B}_i = \mathbf{B}}} \sum_{\substack{\mathbf{A}_1, \dots, \mathbf{A}_n \\ \sum \mathbf{A}_i \supset \mathbf{B}_i}} (-)^n \int d^d \mathbf{x}_{\overline{\mathbf{B}}} \mathcal{C}(\mathbf{x}_{\overline{\mathbf{B}}}) \prod_{i=1}^{n} H(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}), \quad (5.3) $$

where the sum is over all ways to represent $\mathbf{B}$ as a concatenation $\mathbf{B}_1 + \dots + \mathbf{B}_n$, and then over all ways to extend $\mathbf{B}_i$'s to $\mathbf{A}_i \supset \mathbf{B}_i$. The integration is over points $\mathbf{x}_{\overline{\mathbf{B}}}, \overline{\mathbf{B}} = \mathbf{B}_1 + \dots + \mathbf{B}_n, \overline{\mathbf{B}}_i = \mathbf{A}_i \setminus \mathbf{B}_i$, while the unintegrated parts of the vectors $\mathbf{x}_{\mathbf{A}_i}$ form $\mathbf{x}_{\mathbf{B}} = \mathbf{x}_{\mathbf{B}_1} + \dots + \mathbf{x}_{\mathbf{B}_n}$, and the integration kernel $\mathcal{C}(\mathbf{x}_{\overline{\mathbf{B}}})$ is the connected expectation:

$$ \mathcal{C}(\mathbf{x}_{\overline{\mathbf{B}}}) = \langle \Phi(\overline{\mathbf{B}}_1, \mathbf{x}_{\overline{\mathbf{B}}_1}); \dots; \Phi(\overline{\mathbf{B}}_n, \mathbf{x}_{\overline{\mathbf{B}}_n}) \rangle_c. \quad (5.4) $$

Finally, $\mathcal{A}$ in eq. (5.3) denotes the antisymmetrization operation, see footnote 15. Note that the set of kernels $H_{\text{eff}}(\mathbf{B}, \mathbf{x}_{\mathbf{B}})$ produced by this formula will not in general satisfy the trimming requirements, even if the kernel $H(\mathbf{A}, \mathbf{x})$ did. This will be dealt with in the next section.

We write eq. (5.3) more compactly and abstractly as

$$ (H_{\text{eff}})_l = \sum_{(\ell_i)_1^n} S_l^{\ell_1, \dots, \ell_n}(H), \quad H = (H_\ell)_{\ell \in \text{TL}}. \quad (5.5) $$

Each term in (5.5) is numbered by a sequence $(\ell_i)_1^n = (\ell_1, \dots, \ell_n)$, $n \ge 1$, $\ell_i \in \text{TL}$, and by an even $l \ge 2$. The map $S_l^{\ell_1, \dots, \ell_n}(H)$ in (5.5) is the sum of all terms in (5.3) which have $|\mathbf{B}| = l$ and $H(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}) \in H_{\ell_i}$, while $\mathbf{B}_i$ can be arbitrary, subject to the requirements $\mathbf{B}_i \subset \mathbf{A}_i, \sum \mathbf{B}_i = \mathbf{B}$.<fnref n="23" />

Let $B_l$ and $B_\ell$ be the vector spaces of couplings $H_l$ and trimmed couplings $H_\ell$, respectively, and let $B_{\text{trim}} = \otimes_{\ell \in \text{TL}} B_\ell$ be the vector space of trimmed coupling sequences $H = (H_\ell)_{\ell \in \text{TL}}$. The map $S_l^{\ell_1, \dots, \ell_n}$ then acts from $B_{\text{trim}}$ to $B_l$ and is homogeneous of degree $n$. Of course, this map only depends on the couplings $H_{\ell_i}$ whose index $\ell_i$ occurs in the sequence $(\ell_i)_1^n$. If index $\ell_i$ occurs $n_i$ times, this map has homogeneity degree $n_i$ in $H_{\ell_i}$.

It will be also useful to define a closely related map $S_l^{\ell_1, \dots, \ell_n}$, obtained by replacing $\prod_{i=1}^n H(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i}) \to \prod_{i=1}^n h_i(\mathbf{A}_i, \mathbf{x}_{\mathbf{A}_i})$ in (5.3) with independent $h_i \in B_{\ell_i}$. This gives a multilinear map:

$$ S_l^{\ell_1, \dots, \ell_n} : B_{\ell_1} \times \dots \times B_{\ell_n} \to B_l. \quad (5.6) $$

Note that this map is symmetric, i.e. invariant under the interchanges of indices $\ell_i$ accompanied by the simultaneous interchange of arguments. By identifying the arguments of $S_l^{\ell_1, \dots, \ell_n}$, we get back the map $S_l^{\ell_1, \dots, \ell_n}$:

$$ S_l^{\ell_1, \dots, \ell_n}(H) = S_l^{\ell_1, \dots, \ell_n}(h_1, \dots, h_n), \quad h_i = H_{\ell_i} \quad (5.7) $$

It will be very important that the maps $S_l^{\ell_1, \dots, \ell_n}$ and $S_l^{\ell_1, \dots, \ell_n}$ vanish unless $\sum |\ell_i| \ge l + 2(n-1)$. Otherwise, the number of fields in the connected expectation (5.4), which is $\sum |\ell_i| - l$, is not enough to get a connected Wick contraction.

<footnote n="23">²³As noted above, $H_{6,0}$ kernels are a sum of $H_{6SL}$ and $H_{6R,0}$. Picking one or the other part of the sum is understood when defining the map $S_l^{\ell_1, \dots, \ell_n}$ with $\ell_i = 6SL$ or $\ell_i = 6R$.</footnote>
We will see that the defined maps are continuous with respect to the norms from section 4.2.1, see section 5.6.1 below.

While generally $S_l^{\ell_1, \dots, \ell_n}$ acts into $B_l$, in two cases it can be considered to act into $B_\ell$:

* For $\ell \ge 8$ since then $B_\ell = B_l$.

* For $n=1$ and $l = |\ell_1|$, because in this case $S_l^{\ell_1}$ is the identity map: $S_l^\ell(H_\ell) = H_\ell$.

**Remark 5.1** Eq. (5.3) is going to be the basis for all further considerations. Although derived so far by perturbation theory, in our model this equation will be non-perturbatively true. Let us discuss first why this may be expected on physical grounds. Non-perturbative validity of eq. (5.3) means that in our model the full effective action is captured by perturbation theory, with no extra contributions. Extra “instantonic” contributions are common in models involving bosonic fields, but these are absent in our model since we only have fermions. Perturbation theory may also break down if fermions form a bosonic bound state, but this typically requires a coupling that becomes large when iterating the RG, and in our model all couplings will stay weak. In the main text we will show in particular that the series in the r.h.s. is convergent provided that $H(\psi)$ is sufficiently small, and so $H_{\text{eff}}(\psi)$ is well defined.<fnref n="24" /> In appendix H we will give a rigorous justification of eq. (5.3), by first deriving this equation in finite volume and then passing to the limit, in line with remark 2.1.

## 5.2 Trimming

We wish to realize the renormalization map in the space of trimmed couplings. Unfortunately, as mentioned, the kernels $(H_{\text{eff}})_l$ provided by eq. (5.5) are not in general trimmed. To correct this, we need an extra “trimming” step, which will find an equivalent trimmed representation of the same interaction. This step corresponds, in different notation, to the rewriting of $H_{\text{eff}}$ in the equivalent form $\mathcal{L}H_{\text{eff}} + \mathcal{R}H_{\text{eff}}$ used in many papers in CRG, see in particular [51] and [71]. In the CRG literature, $\mathcal{L}H_{\text{eff}}$ and $\mathcal{R}H_{\text{eff}}$ are usually called the “local” and “regularized” parts of the effective interaction, respectively. Before describing this step in detail, let us first discuss what it means for two representations to be equivalent.

### 5.2.1 Equivalent coupling sequences

A coupling sequence $(N_l)$ is called null if the corresponding interaction vanishes as a function of classical Grassmann fields $\psi_a(x)$ (an explicit example of what we mean is discussed

<footnote n="24">²⁴We wish to draw here a parallel with fermionic models of condensed-matter physics, which have convergent perturbation theory at finite temperatures (a notable exception being fermions at finite density in 3d continuous space, unstable with respect to collapse to a point for attractive interaction). One often studied example is the Fermi-Hubbard model, whose perturbative series in the onsite repulsion $U$ has a finite, $T$-dependent, radius of convergence. A simple extension of the GKL bound for connected expectations discussed in appendix D.4 (see section 3 of [28]; see also [61] and section 6 of [62]) easily implies convergence of the series, but with a far-from-optimal temperature dependence ($U \propto T^{d+1}$, with $d$ the spatial dimension of the lattice). For realistic estimates on the convergence radius, see e.g. [63–68]. The perturbation series for Fermi-Hubbard can be evaluated to high order, and convergence checked, by Diagrammatic Monte Carlo (DiagMC) method [69, 70]. We thank Kris Van Houcke and Felix Werner for discussions about DiagMC. See also footnote 55.</footnote>
below). Two coupling sequences with a null difference are equivalent (represent the same interaction). Interactions are thus identified with coupling sequences modulo this equivalence relation.

The basic mechanism to produce equivalent couplings, referred to as “interpolation”, starts with the Newton-Leibniz formula:

$$ \psi_a(x) = \psi_a(y) + \int_0^1 ds \partial_s \psi_a(y+s(x-y)) = \psi_a(y) + \int_0^1 ds (x-y)^\mu \partial_\mu \psi_a(y+s(x-y)). \quad (5.8) $$

We can also integrate the r.h.s. in y against some function $f(x, y)$ of unit total integral: $\int dy f(x, y) = 1$. We get a family of “interpolation identities” expressing $\psi^a(x)$ as a weighted linear combination of $\psi^a(y)$ and $\partial_\mu \psi_a(y)$:

$$ \psi_a(x) = \int dy [f(x, y)\psi_a(y) + f^\mu(x, y)\partial_\mu \psi_a(y)], \quad (5.9) $$

where $f^\mu(x, y)$ can be expressed in terms of $f$.<fnref n="25" />

Now take a single interaction term $\int dx H(\mathbf{A}, \mathbf{x}) \Psi(\mathbf{A}, \mathbf{x})$ corresponding to some $\mathbf{A}$ with at least one field not differentiated, e.g. the first one: $A_1 = a_1$. Pick a function $f(x, y)$ and replace $\psi_{a_1}(x_1)$ inside this interaction term via the identity (5.9). This generates an equivalent representation of the same interaction of the form

$$ \sum_{\mathbf{B}} \int d^d \mathbf{x} \tilde{H}(\mathbf{B}, \mathbf{x}) \Psi(\mathbf{B}, \mathbf{x}), \quad (5.10) $$

where the sum has $d+1$ terms: either $\mathbf{B} = \mathbf{A}$ or it is obtained from $\mathbf{A}$ replacing $A_1 \to (a_1, \mu)$, $\mu = 1, \dots, d$. The corresponding kernels $\tilde{H}(\mathbf{B}, \mathbf{x})$ are obtained integrating $H(\mathbf{A}, \mathbf{x})$ against $f$ and $f^\mu$. We can also apply this procedure to multiple interaction terms in the original interaction $H(\psi)$, summing up the new kernels $\tilde{H}(\mathbf{B}, \mathbf{x})$ to the kernels $H(\mathbf{B}, \mathbf{x})$ to which the transformation has not been applied. The resulting total interaction, which we call $\tilde{H}(\psi)$, is equivalent to $H(\psi)$. The difference of coupling sequences ($H_l$) and ($\tilde{H}_l$) is null.

We stress that the Newton-Leibniz formula and the interpolation identities will be applied only to those $\psi$'s in the interaction terms which do not carry any derivatives. Then, all the produced terms contain $\psi$'s with at most one derivative. This explains why in (4.1) we allowed fields with zero or one (but not more) derivatives.

### 5.2.2 Trimming map

We now explain the trimming map, which maps the sequence $(H_{\text{eff}})_l$ in (5.5) to an equivalent sequence of trimmed couplings. Of course, the restriction of the sequence $(H_{\text{eff}})_l$ to $l \ge 8$ is already trimmed (see the end of section 5.1), so we only need to do something for $l \le 6$.

For $l=6$, we define

$$ (H_{\text{eff}})_{6\text{SL}} = H_{6\text{SL}} + S_6^{4\text{L},4\text{L}}(H_{4\text{L}}, H_{4\text{L}}), \quad (5.11) $$

<footnote n="25">²⁵Explicitly $f^\mu(x, y) = \int_0^1 \frac{ds}{(1-s)^d} [(x-z)^\mu f(x, z)]_{z=(y-sx)/(1-s)}$. This is finite if $f(x, y)$ decreases sufficiently fast at large y.</footnote>
i.e. the two terms in the $(H_{\text{eff}})_6$ series which manifestly have the form (3.6). We define $(H_{\text{eff}})_{6R}$ as the sum of all the other terms in the $(H_{\text{eff}})_6$ series.

For $l = 2, 4$ trimming will involve “localization” and “interpolation”. “Localization” extracts the local parts of $(H_{\text{eff}})_{2,0}$ and $(H_{\text{eff}})_{4,0}$ (see the example below):

$$ (H_{\text{eff}})_{2L} = T_{2L}^{2,0}(H_{\text{eff}})_{2,0}, \quad (H_{\text{eff}})_{4L} = T_{4L}^{4,0}(H_{\text{eff}})_{4,0}. \qquad (5.12) $$

“Interpolation” rearranges the other components setting to zero the parts of $(H_{\text{eff}})_{2R,p}$ and $(H_{\text{eff}})_{4R,p}$ in agreement with the trimming requirements, and making sure that the resulting coupling sequence is equivalent as in section 5.2.1. This operation will have the following structure:

$$ (H_{\text{eff}})_{4R,p} = \begin{cases} 0 & \text{if } p = 0 \\ (H_{\text{eff}})_{4,1} + T_{4R}^{4,0}(H_{\text{eff}})_{4,0} & \text{if } p = 1 \\ (H_{\text{eff}})_{4,p} & \text{if } p > 1, \end{cases} \qquad (5.13) $$

$$ (H_{\text{eff}})_{2R,p} = \begin{cases} 0 & \text{if } p = 0, 1 \\ (H_{\text{eff}})_{2,2} + T_{2R}^{2,1}(H_{\text{eff}})_{2,1} + T_{2R}^{2,0}(H_{\text{eff}})_{2,0} & \text{if } p = 2. \end{cases} \qquad (5.14) $$

This can be also written succinctly as

$$ (H_{\text{eff}})_{4R} = T_{4R}^{4}(H_{\text{eff}})_{4}, \quad (H_{\text{eff}})_{2R} = T_{2R}^{2}(H_{\text{eff}})_{2}, \qquad (5.15) $$

where (5.13), (5.14) define components of $T_{4R}^4$, $T_{2R}^2$ in subspaces with a definite number of derivatives.

Consider $l = 4$ as an example. The coupling $(H_{\text{eff}})_{4,0}$ corresponds to an interaction the form

$$ \Omega_{ab}\Omega_{ce} \int d^d\mathbf{x}F(\mathbf{x})\psi_a(x_1)\psi_b(x_2)\psi_c(x_3)\psi_e(x_4) \qquad (5.16) $$

(plus two other terms with $\Omega_{ac}\Omega_{be}$ and $\Omega_{ae}\Omega_{bc}$). Substitute into this an interpolation identity

$$ \psi_a(x_1)\psi_b(x_2)\psi_c(x_3)\psi_e(x_4) = (\psi_a\psi_b\psi_c\psi_e)(x_1) + \psi_a(x_1) \int_0^1 dt \partial_t [\psi_b(x_2^t)\psi_c(x_3^t)\psi_e(x_4^t)], \qquad (5.17) $$

where $x_i^t = x_1 + t(x_i - x_1)$. The first term gives a local quartic interaction with

$$ \lambda = \int_{x_1=0} d^d\mathbf{x}F(\mathbf{x}), \qquad (5.18) $$

which defines $T_{4L}^{4,0}$ in (5.12).<fnref n="26" /> The second term in (5.17) gives a sum of interactions where one of $\psi_b, \psi_c, \psi_e$ is differentiated: this defines $T_{4R}^{4,0}(H_{\text{eff}})_{4,0}$.

The $l = 2$ maps $T_{2L}^{2,0}$, $T_{2R}^{2,1}$, $T_{2R}^{2,0}$ are defined analogously. For $T_{2R}^{2,0}$ one needs to apply interpolation twice, to get from a term with no derivatives to a term where both fields carry derivatives. See appendix C for the full construction of these maps, and for the analysis of how they behave with respect to the norms measuring the size of interaction kernels.

<footnote n="26">²⁶This equation can be equivalently written in momentum space as $\lambda = \hat{F}(0, 0, 0, 0)$, i.e. evaluating the kernel with all external momenta set to zero.</footnote>
**Remark 5.2** Note that our trimming map $T$ is just one of infinitely many possible trimming maps, corresponding to different choices of interpolation identities. E.g. instead of (5.17) we could have used

$$ \psi_a(x_1)\psi_b(x_2)\psi_c(x_3)\psi_e(x_4) = \psi_a(x_1) \left[ \psi_b(x_1) + \int_0^1 dt \frac{d\psi_b}{dt}(x_2^t) \right] \left[ \psi_c(x_1) + \int_0^1 ds \frac{d\psi_c}{ds}(x_3^s) \right] \left[ \psi_e(x_1) + \int_0^1 du \frac{d\psi_e}{du}(x_4^u) \right]. \quad (5.19) $$

Such alternative trimming maps $\tilde{T}$ differ from $T$ by a map which is null (gives a null sequence of couplings when applied to any interactions). In our construction we will use $T$, but any other trimming map satisfying the same norm bounds (see appendix C) would work equally well.

## 5.3 Dilatation

After having integrated out the fluctuation field and rearranged the result so that it is equivalently rewritten in trimmed form, we rescale the fields, see (2.15)–(2.16). We call this rescaling step *dilatation*, and denote it by $D$. Note that $D$ preserves the trimmed representation. The action on the kernels is:

$$ D: H_{\ell,p}(\mathbf{x}) \mapsto \gamma^{-D_l-p}\gamma^{d(l-1)}H_{\ell,p}(\gamma\mathbf{x}), \quad (5.20) $$

where we recall that $l = |\ell|$, $p$ denotes the number of derivatives in the interaction term, and we denoted

$$ D_l = l[\psi] - d = l(d/4 - \varepsilon/2) - d. \quad (5.21) $$

For the special cases $\ell \in \{2\mathrm{L}, 4\mathrm{L}, 6\mathrm{SL}\}$ eq. (5.20) becomes:

$$ \nu \mapsto \gamma^{-D_2}\nu = \gamma^{\frac{d}{2}+\varepsilon}\nu, \quad \lambda \mapsto \gamma^{-D_4}\lambda = \gamma^{2\varepsilon}\lambda, \quad \mathcal{X}(x) \mapsto \gamma^{-D_6}\gamma^d\mathcal{X}(\gamma x). \quad (5.22) $$

In terms of the norms of section 4.2.1, the irrelevance condition for $H_{\ell,p}$ will be $D_l + p > 0$, see eq. (5.37) below. As stated in eq. (2.8) we are assuming $d \in \{1, 2, 3\}$ and $0 < \varepsilon < d/6$. Under these conditions it's easy to check that $D_2, D_4 < 0$, so that $\nu, \lambda$ are relevant, while

$$ D_2 + 2, D_4 + 1 > 0, \quad D_l \ge D_6 > 0 \quad (l \ge 6). \quad (5.23) $$

so that $H_{2\mathrm{R}}, H_{4\mathrm{R}}, H_{6\mathrm{SL}}, H_{6\mathrm{R}}$ and $H_\ell$ ($\ell \ge 8$) are irrelevant. These interactions comprise $H_{\mathrm{IRR}}$ in (2.9).

## 5.4 Renormalization map in the trimmed representation

The renormalization map $R$ is obtained composing the three operations: integrating-out, then trimming, then dilatation. It is the map defined in section 2.1 but now written in a specific set of coordinates (the trimmed representation). Summarizing sections 5.1, 5.2.2, 5.3, we represent $R = R(\varepsilon, \gamma)$ follows: if $H \in B_{\mathrm{trim}}$ (the vector space of sequences of trimmed couplings), then $R: H \mapsto H' \in B_{\mathrm{trim}}$, with

$$ H'_{\ell} = \sum_{(\ell_i)_1^n} \mathcal{R}^{\ell_1, \dots, \ell_n}(H), \quad (5.24) $$
where $\mathcal{R}_{\ell}^{\ell_1, \dots, \ell_n}$ is a homogeneous map of degree $n$ obtained by identifying the arguments in a multilinear map $R_{\ell}^{\ell_1, \dots, \ell_n}$:

$$ \mathcal{R}_{\ell}^{\ell_1, \dots, \ell_n}(H) = R_{\ell}^{\ell_1, \dots, \ell_n}(h_1, \dots, h_n), \quad h_i = H_{\ell_i}. \qquad (5.25) $$

This multilinear map can be written explicitly as follows. For $(n; (\ell_1, \dots, \ell_n)) = (1; \ell)$ we have

$$ R_{\ell}^{\ell} = D, \qquad (5.26) $$

since in this case $S_{\ell}^{\ell} = 1$ and trimming is not needed. In all the other cases $(n; (\ell_1, \dots, \ell_n)) \neq (1; \ell)$, recalling that $l = |\ell|$, we have

$$ \begin{aligned} R_{\ell}^{\ell_1, \dots, \ell_n} &= D \begin{cases} S_{\ell}^{\ell_1, \dots, \ell_n} & \ell \ge 8 \\ T_{\ell}^{\ell} S_{\ell}^{\ell_1, \dots, \ell_n} & \ell \in \{2\text{L}, 2\text{R}, 4\text{L}, 4\text{R}\} \end{cases} \\ R_{6\text{SL}}^{\ell_1, \dots, \ell_n} &= D \begin{cases} S_{6}^{4\text{L}, 4\text{L}} & (\ell_i)_1^n = (4\text{L}, 4\text{L}) \\ 0 & \text{otherwise} \end{cases} \qquad (5.27) \\ R_{6\text{R}}^{\ell_1, \dots, \ell_n} &= D \begin{cases} S_{6}^{\ell_1, \dots, \ell_n} & (\ell_i)_1^n \neq (6\text{SL}), (4\text{L}, 4\text{L}) \\ 0 & \text{otherwise} \end{cases} \end{aligned} $$

where $T_{\ell}^{l} : B_{l} \to B_{\ell}$ is the trimming map whose various components are defined by equations (5.12), (5.13), (5.14), and see eq. (5.11) for $\ell \in \{6\text{SL}, 6\text{R}\}$.

Just as $S_{\ell}^{\ell_1, \dots, \ell_n}$, the map $R_{\ell}^{\ell_1, \dots, \ell_n}$ is symmetric (invariant under the interchanges of indices $\ell_i$ accompanied by the simultaneous interchange of arguments), and it vanishes unless $\sum_i |\ell_i| \ge l + 2(n - 1)$.

## 5.5 Fixed point equation

The fixed point equation (FPE) that we will study is

$$ (H'_{\ell}) = (H_{\ell}). \qquad (5.28) $$

with $(H'_{\ell})$ given by (5.24). If we distinguish the components $\ell = 2\text{L}, 4\text{L}, 6\text{SL}$ from the other couplings, denoted $u = (H_{\ell})_{\ell \neq 2\text{L}, 4\text{L}, 6\text{SL}} \equiv (u_{\ell})_{\ell \in \{2\text{R}, 4\text{R}, 6\text{R}, 8, 10, \dots\}}$, it reads:

$$ \begin{aligned} \nu &= \gamma^{\frac{d}{2}+\epsilon}\nu + R_{2\text{L}}^{4\text{L}}(\lambda) + \sum_{(\ell_i)_1^n \neq (2\text{L}), (4\text{L})} R_{2\text{L}}^{\ell_1, \dots, \ell_n}(H_{\ell_1}, \dots, H_{\ell_n}), \\ \lambda &= \gamma^{2\epsilon}\lambda + R_{4\text{L}}^{4\text{L}, 4\text{L}}(\lambda, \lambda) + R_{4\text{L}}^{6\text{SL}}(\mathcal{X}) + \sum_{(\ell_i)_1^n \neq (4\text{L}), (4\text{L}, 4\text{L}), (6\text{SL})} R_{4\text{L}}^{\ell_1, \dots, \ell_n}(H_{\ell_1}, \dots, H_{\ell_n}), \\ \mathcal{X}(x) &= R_{6\text{SL}}^{6\text{SL}}(\mathcal{X}) + R_{6\text{SL}}^{4\text{L}, 4\text{L}}(\lambda, \lambda) = \gamma^{2d-6[\psi]}[\mathcal{X}(x\gamma) - 8\lambda^2 g(x\gamma)], \\ u_{\ell} &= \sum_{(\ell_i)_1^n} R_{\ell}^{\ell_1, \dots, \ell_n}(H_{\ell_1}, \dots, H_{\ell_n}), \quad \text{if } \ell \neq 2\text{L}, 4\text{L}, 6\text{SL}. \end{aligned} \qquad (5.29) $$

We already observed that, given $\lambda$, the FPE for $\mathcal{X}$ is solved exactly by $\mathcal{X} = \mathcal{X}_*$, with $\mathcal{X}_*$ as in (3.12). Substituting $\mathcal{X}(x) = \mathcal{X}_*(x)$ in the remaining equations, the variable $\mathcal{X}$ is
eliminated.<fnref n="27" /> Denoting $y = (\nu, \lambda, u)$, we are left with the fixed point equation $y = R(y)$, or in components:

$$ 
\begin{align} 
\nu &= \gamma^{\frac{d}{2}+\epsilon}(\nu + I_1\lambda) + e_{\nu}^{(0)}(y), \\ 
\lambda &= \gamma^{2\epsilon}(\lambda + I_2\lambda^2) + e_{\lambda}^{(0)}(y), \\ 
u &= e_u(y), 
\end{align} 
\tag{5.30} $$

with $e_{\nu}^{(0)}$, $e_{\lambda}^{(0)}$, $e_u$ defined via the infinite sums in the right sides of (5.29), and $I_1$, $I_2$ defined by

$$ R_{2L}^{4L}(\lambda) = \gamma^{\frac{d}{2}+\epsilon} I_1 \lambda, \quad R_{4L}^{4L,4L}(\lambda, \lambda) + R_{4L}^{6SL}(\mathcal{X}_*) = \gamma^{2\epsilon} I_2 \lambda^2. \tag{5.31} $$

These coefficients $I_1$, $I_2$ are the same as in section 3. They are given by one-loop Feynman integrals evaluated in appendix G.

Moving the l.h.s. into the r.h.s. and rescaling, we rewrite the system (5.30) as

$$ f(y) = 0, \quad f(y) := \begin{pmatrix} \nu + a\lambda + e_{\nu}(y) \\ \varepsilon\lambda + b\lambda^2 + e_{\lambda}(y) \\ u - e_{u}(y) \end{pmatrix}, \tag{5.32} $$

where

$$ 
\begin{align} 
(a, e_{\nu}) &= \frac{1}{1 - \gamma^{-d/2-\epsilon}} (I_1, \gamma^{-\frac{d}{2}-\epsilon} e_{\nu}^{(0)}), \\ 
(b, e_{\lambda}) &= \frac{\epsilon}{1 - \gamma^{-2\epsilon}} (I_2, \gamma^{-2\epsilon} e_{\lambda}^{(0)}). 
\end{align} 
\tag{5.33} $$

Eq. (5.30) or its equivalent eq. (5.32) are the main equations that we will be solving. Of course, part of the problem is to show that these equations make sense: that is, we need to prove that the infinite sums entering the definitions of $e_{\nu}^{(0)}$, $e_{\lambda}^{(0)}$, $e_u$ are convergent. We will actually show that, if $y = (\nu, \lambda, u)$ has bounded norm (4.19), say $\|y\|_Y \le 1$, with $\delta$ sufficiently small, then the sums defining $e_{\nu}^{(0)}$, $e_{\lambda}^{(0)}$, $e_u$ are absolutely convergent and $e_u$ is contractive. This will be proved in sections 6 and 7 below. In preparation to this, in the next subsection we state the norm bounds satisfied by the multilinear operators $R_{\ell}^{\ell_1, \dots, \ell_n}$, which will be central for our proof of convergence.

**Remark 5.3** From a more general viewpoint, a fixed point is a sequence of couplings ($H_{\ell}$) such that $(H'_{\ell}) = R(\epsilon, \gamma)[(H_{\ell})]$ given by (5.24) describes the same interaction as ($H_{\ell}$). This will be the case if $(H'_{\ell}) = (H_{\ell})$, as stated in (5.28), or, more generally, if the two sequences differ by a null sequence of couplings (see section 5.2.1). In this sense, the FPE (5.28) discussed above is not the most general we could (and should) consider: the general FPE to be considered reads $(H'_{\ell}) = (H_{\ell}) + (N_{\ell})$, with $(N_{\ell})$ a null sequence. In this paper, for simplicity, we focus only on the restricted FPE (5.28), and we will show that it has a non-trivial, non-null, solution,<fnref n="28" /> which is unique in some neighborhood. The same methods of

<footnote n="27">²⁷Note that although in this paper we take advantage of this possibility, in principle we could have treated $\mathcal{X}$ on par with all the other irrelevant couplings. The RG map would end up contractive also in the $\mathcal{X}$ direction, and RG iterations would converge to the same solution $\mathcal{X} = \mathcal{X}_*$.</footnote>

<footnote n="28">²⁸The fixed points we will construct will have nonzero $\lambda$ and $\nu$, and will therefore be nontrivial. **Lemma.** Any trimmed coupling sequence with nonzero $\nu$ and/or $\lambda$ is not null. Proof is left as an exercise.</footnote>
proof would allow us to show that, for each sufficiently small ($N_\ell$), the general FPE has a unique solution, which differs from the one with $N_\ell \equiv 0$ by a null sequence ($N'_\ell$). In this sense, we expect that there is a unique interaction (equivalent class of couplings) solving the general FPE. This remains to be shown in full detail, but we prefer not to present this additional proof here, in order not to overwhelm the presentation.

**Remark 5.4** Recall that we are considering renormalization maps $R = R(\epsilon, \gamma)$ with rescaling factor $\gamma \ge 2$, in particular $\gamma$ is separated from 1. Such RG transformations are called “finite” or “discrete”. Eq. (5.29) thus sets to zero the “beta-functions” expressing the change of the interaction under a finite RG transformation. In theoretical physics, it is more common to take the limit $\gamma \to 1$ and define an “infinitesimal” or “continuous” RG transformation formally given by the derivative $(d/d\gamma)R_\gamma$ at $\gamma = 1$. At a formal level the continuous RG equation (Polchinski’s equation [59]) has fewer terms and looks much simpler than the discrete RG. However, so far it has not been possible to take advantage of this formal simplicity in rigorous constructions of RG fixed points. The problem is to show that solutions to Polchinski’s equation have sufficiently good boundedness properties in a Banach space of interactions, and it is not known how to do this without dealing with the finite RG at the intermediate steps of the argument, which brings back the complexity. This problem is open even in fermionic theories.<fnref n="29" />

## 5.6 Norm bounds

As anticipated in the previous subsection, we now state the norm bounds satisfied by the multilinear operators $R_\ell^{\ell_1, \dots, \ell_n}$ entering the definitions of $e_\nu^{(0)}$, $e_\lambda^{(0)}$, $e_u$. In the case $(n; (\ell_1, \dots, \ell_n)) = (1; \ell)$, in which $R_\ell^\ell$ is defined as in (5.26), we have

$$ \|R_\ell^\ell(H_\ell)\|_w \le \begin{cases} \gamma^{-D_2-2} \|H_{2R}\|_w & \text{if } \ell = 2R, \\ \gamma^{-D_4-1} \|H_{4R}\|_w & \text{if } \ell = 4R, \\ \gamma^{-D_l} \|H_\ell\|_w & \text{if } l = |\ell| \ge 6, \end{cases} \quad (5.34) $$

while $|R_{2L}^{2L}(\nu)| = \gamma^{-D_2}|\nu|$ and $|R_{4L}^{4L}(\lambda)| = \gamma^{-D_4}|\lambda|$. In all the other cases $(n; (\ell_1, \dots, \ell_n)) \ne (1; \ell)$, in which $R_\ell^{\ell_1, \dots, \ell_n}$ is defined as in (5.27), we have

$$ \|R_\ell^{\ell_1, \dots, \ell_n}(h_1, \dots, h_n)\|_w \le \gamma^{-D_l} \rho_l(h_1, \dots, h_n), \quad h_i \in B_{\ell_i}, \quad (5.35) $$

$$ \rho_l(h_1, \dots, h_n) := \begin{cases} C_\gamma^{n-1} \prod_{i=1}^n C_0^{|\ell_i|} \|h_i\|_w & \text{if } \sum_i |\ell_i| \ge l + 2(n-1), \\ 0 & \text{otherwise} \end{cases}, \quad (5.36) $$

where, as usual, $l = |\ell|$, and, in (5.36), $C_\gamma$, $C_0$ are constants independent of $l, n, \ell_i$. In addition, $C_0$ does not depend on $\gamma$, while $C_\gamma$ does.

<footnote n="29">²⁹See [72] for some global solvability results for Polchinski’s equation in bosonic theories with bounded interactions. Ref. [73] attempted to prove local solvability for fermionic theories but their argument has a gap, see [74]. See also an interesting discussion in the conclusions of [75]. Ref. [76] considered continuous RG in a fermionic theory, although that construction was not fully based on continuous RG: they define the effective action via a convergent tree expansion (morally equivalent to using a finite RG), then verify that the continuous RG equations hold when applied to this effective action.</footnote>
The proof of (5.34) readily follows from the definition of $R_\ell^\ell$, see (5.26), and from the fact that, using the definition of $D$, see (5.20), and of weighted norm, see section 4.2.1, we have:

$$ \|DH_{\ell,p}\|_w = \gamma^{-D_l-p} \|H_{\ell,p}\|_{w(\cdot/\gamma)} \le \gamma^{-D_l-p} \|H_{\ell,p}\|_w. \quad (5.37) $$

Besides proving (5.34), this justifies the rule stated in section 5.3 that the terms with $D_l + p > 0$ are irrelevant. Since $D_l + p = l[\psi] + p$, this rule turns out the same as for the local interactions (see footnote 9).

The proof of (5.35) is more subtle, see the next two subsections, 5.6.1 and 5.6.2.

### 5.6.1 Bounds for $S_l^{\ell_1, \dots, \ell_n}$

Recall that, if $(n; (\ell_1, \dots, \ell_n)) \ne (1; \ell)$, then $R_\ell^{\ell_1, \dots, \ell_n}$ is defined in terms of $S_l^{\ell_1, \dots, \ell_n}$ via (5.27). Therefore, in order to prove (5.35), we first need a bound on $S_l^{\ell_1, \dots, \ell_n}$. This is similar to (5.27), with the important difference that there is no scaling factor $\gamma^{-D_l}$ in the right side:

$$ \|S_l^{\ell_1, \dots, \ell_n}(h_1, \dots, h_n)\|_w \le \rho_l(h_1, \dots, h_n), \quad h_i \in B_{\ell_i}, \quad (5.38) $$

with the same $\rho_l$ as in (5.36) (with, possibly, a different constant $C_0$). For the full proof of (5.38) see appendix E. Here are the main ideas: from its definition, the map $S_l^{\ell_1, \dots, \ell_n}$ is an integral operator whose kernel is the connected expectation $C(\mathbf{x}_{\bar{\mathbf{B}}})$ (more precisely, it is a sum of $O(\mathrm{const}\sum l_i)$ integral operators corresponding to different choices of $\mathbf{B}_i$ and $\mathbf{A}_i$). The fermionic connected expectation $C(\mathbf{x}_{\bar{\mathbf{B}}})$ satisfies a crucial bound due to Gawedzki-Kupiainen-Lesniewski (appendix D.4):

$$ |C(\mathbf{x}_{\bar{\mathbf{B}}})| = \left| \langle \Phi(\bar{\mathbf{B}}_1, \mathbf{x}_{\bar{\mathbf{B}}_1}); \dots; \Phi(\bar{\mathbf{B}}_n, \mathbf{x}_{\bar{\mathbf{B}}_n}) \rangle_c \right| \le C^s \sum_{\mathcal{T}} \prod_{(xx') \in \mathcal{T}} M(x - x'), \quad (5.39) $$

where the sum is over all “anchored trees $\mathcal{T}$ on $n$ groups of points $\mathbf{x}_{\bar{\mathbf{B}}_i}$”. These are graphs which become connected trees when each group of points $\mathbf{x}_{\bar{\mathbf{B}}_i}$ is collapsed to a point. There is at least one anchored tree within each connected Wick contraction, and bounding each propagator along the anchored tree by (4.15) we get the product in (5.39). The contribution of remaining $s = \frac{1}{2}(\sum |\bar{\mathbf{B}}_i| - 2(n-1))$ propagators is bounded by $C^s$. This explains the general structure of (5.39), but the full proof is rather more subtle. The sum of connected graphs defining the connected expectation has to be rewritten as a sum over anchored trees without double counting. For each anchored tree, we then have to sum over the remaining propagator choices, and this whole sum with factorially many terms has to be bounded by $C^s$. This turns out possible due to fermionic cancelations.

The number of anchored trees is $\le n! 4 \sum_{i} |\bar{\mathbf{B}}_i|$ (appendix D.5), which by the way is much smaller than the total number of connected graphs. This $n!$ cancels with $1/n!$ in (5.3), leaving only exponential factors. When evaluating the weighted norm, the product of $M$'s in (5.39) gives the factor $C_\gamma^{n-1}$ with $C_\gamma = \|M\|_w = O(\gamma^d)$ by (4.17). This finishes our brief exposition of (5.38); see appendix E for the details.

By (5.38), the multilinear map $S_l^{\ell_1, \dots, \ell_n}$ is continuous. The homogeneous map $S_l^{\ell_1, \dots, \ell_n}$ related to $S_l^{\ell_1, \dots, \ell_n}$ by identifying some arguments, eq. (5.7), is also continuous.
We will also need Frechet derivatives of these maps.<fnref n="30" /> Since $S_l^{\ell_1, \dots, \ell_n}$ is a multilinear map, it is Frechet-differentiable and its Frechet derivative in each argument coincides with the map itself. The homogeneous map $S_l^{\ell_1, \dots, \ell_n}$ is also Frechet-differentiable. The derivative $\nabla S_l^{\ell_1, \dots, \ell_n}(H)$ is, for a fixed $H$, a linear operator from $B_{\text{trim}}$ to $B_l$. Using eq. (5.7), the value of this operator on $\delta H \in B_{\text{trim}}$ is:

$$[\nabla_H S_l^{\ell_1, \dots, \ell_n}(H)]\delta H = \sum_{i=1}^n S_l^{\ell_1, \dots, \ell_n}(H_{\ell_1}, \dots, \delta H_{\ell_i}, \dots, H_{\ell_n}). \quad (5.40)$$

Estimating each term in the r.h.s. via (5.38) we get a bound:

$$\|[\nabla_H S_l^{\ell_1, \dots, \ell_n}(H)]\delta H\|_w \le \sum_{i=1}^n \rho_l(H_{\ell_1}, \dots, \delta H_{\ell_i}, \dots, H_{\ell_n}). \quad (5.41)$$

### 5.6.2 Bounds for $R_l^{\ell_1, \dots, \ell_n}$

From the definition (5.27), we have that $R_l^{\ell_1, \dots, \ell_n}$ is related to $S_l^{\ell_1, \dots, \ell_n}$ via the dilatation operator, which we already bounded in (5.37), and via the trimming operator $T$ introduced in section 5.2.2, whose components we still need to bound. The easiest components to bound are the localization maps $T_{2L}^{2,0}$ and $T_{4L}^{4,0}$, which map kernels to local kernels and do not increase the norm (see (C.15)):

$$\|(H_{\text{eff}})_{2L}\|_w \le \|(H_{\text{eff}})_{2,0}\|_w, \quad \|(H_{\text{eff}})_{4L}\|_w \le \|(H_{\text{eff}})_{4,0}\|_w. \quad (5.42)$$

In turn, the interpolation maps satisfy the bounds (see appendix C):

$$\begin{align}
\|T_{4R}^{4,0}(H_{\text{eff}})_{4,0}\|_{w(\cdot/\gamma)} &\le C_R \gamma \|(H_{\text{eff}})_{4,0}\|_w, \nonumber \\
\|T_{2R}^{2,1}(H_{\text{eff}})_{2,1}\|_{w(\cdot/\gamma)} &\le C_R \gamma \|(H_{\text{eff}})_{2,1}\|_w, \tag{5.43} \\
\|T_{2R}^{2,0}(H_{\text{eff}})_{2,0}\|_{w(\cdot/\gamma)} &\le C_R \gamma^2 \|(H_{\text{eff}})_{2,0}\|_w, \nonumber
\end{align}$$

where $C_R$ depends on $C_w$ and $\sigma$ in (4.16) but not on $\gamma$, and we use the fact that $\gamma \ge 2$.

Putting together (5.37), (5.38), (5.42) and (5.43), we readily obtain (5.35). In fact, for $\ell \ge 8$ and $\ell \in \{6R, 6SL\}$, (5.35) is a consequence of (5.38), and (5.37) with the worst possible $p=0$. For $\ell \in \{2L, 4L\}$ we also need to use (5.42). Finally, for $\ell \in \{2R, 4R\}$ we additionally have to use (5.43) and rely on the first equality in (5.37). A power of $\gamma$ that we lose in the r.h.s. of (5.43) is compensated during dilatation, due to the presence of derivatives in the couplings $H_{2R}, H_{4R}$. Because of the sums in the r.h.s. of (5.13) and (5.14), we get eq. (5.35) with an extra factor of $\gamma^{-2} + \gamma^{-1}C_R + C_R \le 1 + 2C_R$ for $\ell = 2R$ and

$$\lim_{\|\delta x\|_Z \to 0} \frac{\|f(x + \delta x) - f(x) - \nabla f(x) \delta x\|_{Z'}}{\|\delta x\|_Z} = 0.$$

In some of our cases of interest, one of the two spaces $Z$ or $Z'$ may be $\mathbb{R}$. When $Z = \mathbb{R}$ we have $\nabla f(x) \in Z'$, and when $Z' = \mathbb{R}$ we have $\nabla f(x) \in \mathcal{L}(Z, \mathbb{R})$, i.e. a linear functional on $Z$.

<footnote n="30">³⁰Recall that Frechet derivative is a generalization of ordinary derivative to Banach spaces. In general, for a map $f(x)$ from a Banach space $Z$ to another space $Z'$, its Frechet derivative at a point $x$ is defined as a linear operator $\nabla f(x) \in \mathcal{L}(Z, Z')$ having the property that</footnote>
$\gamma^{-1} + C_R \le 1 + C_R$ for $\ell = 4R$. We absorb this factor by increasing the constant $C_0$ in the definition of the function $\rho_l$ in the right side of (5.35).$^{31}$

We will also need Frechet derivatives of the homogeneous map $\mathcal{R}_{\ell}^{\ell_1, \dots, \ell_n}$. Since $\mathcal{R}_{\ell}^{\ell} = R_{\ell}^{\ell}$ is a linear map, its Frechet derivative coincides with it and satisfies the same bound (5.34). In all the other cases $(n; (\ell_1, \dots, \ell_n)) \ne (1; \ell)$ we have the bound

$$ \|[\nabla_H \mathcal{R}_{\ell}^{\ell_1, \dots, \ell_n}(H)] \delta H\|_w \le \gamma^{-D_l} \sum_{i=1}^{n} \rho_l(H_{\ell_i}, \dots, \delta H_{\ell_i}, \dots, H_{\ell_n}), \quad (5.44) $$

which follows from (5.41) just as (5.35) followed from (5.38).

# 6 Construction of the fixed point

In this and the following section, we finally construct a solution of the FPE $f(y) = 0$, see (5.32), and discuss its uniqueness and regularity properties. The presentation is organized as follows: in section 6.1, we state the main bound on the components of $f(y)$, whose proof (which is one of the main technical contributions of this paper, and uses in a crucial way the bounds stated in section 5.6) is postponed to section 7. Given the bounds of section 6.1, existence and uniqueness at fixed $\gamma$ of the fixed point follow by a rather general and straightforward argument, discussed in sections 6.2 and 6.3. The independence of the fixed point from $\gamma$ and its analyticity in $\varepsilon$ are simple but remarkable corollaries of our construction, discussed in sections 6.4 and 6.5, respectively.

## 6.1 Key lemma

In this subsection we formulate, as promised, the estimates for the functions $e_{\nu}^{(0)}$, $e_{\lambda}^{(0)}$, $e_u$ entering the definition of $f(y)$, see (5.32)–(5.33). We will assume that $\gamma$ is large enough and that the norm of $y$, see (4.19), is bounded, say smaller than 1; the constants $A_0, A_0^R, A_1^R, A_2^R, A$ in (4.19) will be fixed in a suitable, $\gamma$-dependent, way, and the parameter $\delta$ will be chosen sufficiently small (in a $\gamma$-dependent way). The smallness of $\delta$ is conceptually independent from any stringent requirement on the physical parameter $\varepsilon$: the only needed condition on $\varepsilon$ will be that all $u_\ell$ directions are irrelevant, as guaranteed by eqs. (2.8), (5.23). The conditions that $\lambda$ is weakly relevant ($\varepsilon$ small), and that its one-loop beta-function does not vanish ($N \ne 8$) won't be used here. To emphasize that for the moment the smallness of $\varepsilon$ is not used, here we assume that $\delta$ is independent of $\varepsilon$. Eventually, the smallness of $\varepsilon$ will come back into play in the full contraction argument involving all couplings $\nu, \lambda, u_\ell$ (sections 6.2 and 6.3): there, $\delta$ will be identified with $\varepsilon$ up to a constant factor, but here it is logically convenient to keep them separate.

Given $\gamma$-dependent constants $A_0 = A_0(\gamma)$, $A_0^R = A_0^R(\gamma)$, $A_1^R = A_1^R(\gamma)$, $A_2^R = A_2^R(\gamma)$, $A = A(\gamma)$, we denote by $\|u\|_{B(\gamma, \delta)}$ the following norm of a vector $u = (H_\ell)_{\ell \ne 2L, 4L, 6SL} \equiv (u_\ell)_{\ell \in \{2R, 4R, 6R, 8, 10, \dots\}}$ of irrelevant components (6SL excluded):

$$ \|u\|_{B(\gamma, \delta)} = \max \left\{ \frac{\|u_{2R}\|_w}{A_0^R(\gamma)\delta^2}, \frac{\|u_{4R}\|_w}{A_1^R(\gamma)\delta^2}, \frac{\|u_{6R}\|_w}{A_2^R(\gamma)\delta^3}, \sup_{\ell \ge 8} \frac{\|u_\ell\|_w}{A(\gamma)\delta^{k(\ell)}} \right\}, \quad (6.1) $$

<footnote>$^{31}$Since $\sum |\ell_i| \ge 4$ in any of these cases, it's enough to increase $C_0 \to C_0(1 + 2C_R)^{1/4}$.</footnote>
where $k(\ell) = |\ell|/2 - 1$, in terms of which the norm (4.19) of $y = (\nu, \lambda, u)$ can be rewritten

$$ \|y\|_{Y(\gamma, \delta)} = \max \left\{ \frac{|\nu|}{A_0(\gamma)\delta}, \frac{|\lambda|}{A_0(\gamma)\delta}, \|u\|_{B(\gamma, \delta)} \right\}. \quad (6.2) $$

Note that, compared with (4.19), the symbol $Y$ in (6.2) has an explicit dependence upon $\gamma$ and $\delta$; the dependence on $\delta$ is obvious, the one on $\gamma$ is meant to emphasize the $\gamma$-dependence of the constants $A, A_0$, etc. and of the weight $w$ (see (4.16)). We are now ready to state the main result of this section.

**Lemma 6.1 (Key lemma)** Choose $d \in \{1, 2, 3\}$, cutoff $\chi$, $N \ge 4$, and an $\varepsilon$ satisfying (2.8). There exists $\gamma_{\text{key}} \ge 2$ and

$$ \delta_0(\gamma), A_0(\gamma), \{A_k^R(\gamma)\}_{k=0,1,2}, A(\gamma), E_0(\gamma), E_1(\gamma), \quad (6.3) $$

positive continuous functions on $\gamma \ge \gamma_{\text{key}}$ [whose dependence on $\gamma$ is omitted in eqs. (6.5)–(6.6) below], with the following property. Take any $\gamma \ge \gamma_{\text{key}}$, any $0 < \delta \le \delta_0(\gamma)$, and any sequence $y = (\nu, \lambda, u)$ satisfying

$$ \|y\|_{Y(\gamma, \delta)} \le 1. \quad (6.4) $$

Then the infinite sums defining the functions $e_\nu^{(0)}$, $e_\lambda^{(0)}$, $e_u$ in the right side of (5.30), see (5.29) and following lines, are absolutely convergent, and their sums satisfy:

$$ |e_\nu^{(0)}(y)| \le E_0\delta^2, \quad |e_\lambda^{(0)}(y)| \le E_1\delta^3, \quad \|e_u(y)\|_{B(\gamma, \delta)} \le \gamma^{-\bar{D}}, \quad (6.5) $$

where $\bar{D} = \frac{1}{2} \min\{D_2 + 2, D_4 + 1, D_6\}$. In addition,

$$ \begin{aligned} & |\partial_i e_\nu^{(0)}(y)| \le E_0 \delta^2 / (A_0 \delta), && |\partial_i e_\lambda^{(0)}(y)| \le E_1 \delta^3 / (A_0 \delta), \\ & \|\partial_i e_u(y)\|_B \le \gamma^{-\bar{D}} / (A_0 \delta) \quad (i = \nu, \lambda), \\ & \|\partial_u e_\nu^{(0)}(y)\|_{\mathcal{L}(B, \mathbb{R})} \le E_0 \delta^2, && \|\partial_u e_\lambda^{(0)}(y)\|_{\mathcal{L}(B, \mathbb{R})} \le E_1 \delta^3, \\ & \|\partial_u e_u(y)\|_{\mathcal{L}(B, B)} \le \gamma^{-\bar{D}}, && \end{aligned} \quad (6.6) $$

where $B = B(\gamma, \delta)$, $\mathcal{L}(B, \mathbb{R})$ is the space of linear operators from $B$ to $\mathbb{R}$, and similarly for $\mathcal{L}(B, B)$.

We wrote (6.6) in the form which makes apparent that the $u$-derivatives satisfy the same bounds as the functions themselves, while the bounds for $\nu, \lambda$-derivatives are worse by $1/(A_0\delta)$ factor. This pattern is natural in view of the assumptions $|\nu| \le A_0\delta$, $|\lambda| \le A_0\delta$ and $\|u\|_B \le 1$ (which are the same as $\|y\|_Y \le 1$). Before presenting the proof of the Key lemma, which is postponed to section 7, we will show that its bounds straightforwardly imply that the FPE $f(y) = 0$ has a unique solution in a suitable neighborhood of the Banach space $Y$; see the next two subsections, 6.2 and 6.3.

**Remark 6.1** The third inequality in (6.5) means that the RG map restricted to the irrelevant directions $\ell = 2\mathbb{R}, 4\mathbb{R}, 6\mathbb{R}, 8, 10, \dots$ is contractive, as it is natural to expect. Contractivity along the directions with $|\ell| \ge 6$ is “easy” to establish: it follows straightforwardly
from the bounds (5.34) and (5.35); note, in fact, that $\gamma^{-D_l} \le \gamma^{-\bar{D}} < 1$ for $l = |\ell| \ge 6$. See sections 7.1 and 7.2 for the full proof. On the other hand, contractivity along the directions 2R and 4R is more subtle to prove, due to the factor $\gamma^{-D_l}$, which is larger than 1 for $l = 2, 4$, in the right side of (5.35). In these cases, we take advantage of the fact that the linearization of the RG map, (5.34), has the good factor $\gamma^{-D_2-2}$ and $\gamma^{-D_4-1}$ in the directions $\ell = 2R$ and 4R, respectively. On the other hand, the nonlinear contributions bounded in (5.35) are small because they are of higher order: loosely speaking, the higher order can be used to compensate the additional bad factor $\gamma^2$ or $\gamma$, which ultimately originates from the bounds (5.43). More technically, here is where we use the freedom in the choice of the constants $A, A_0, A_0^R$ etc., entering the definition the norm (4.19) defining the Banach space: by carefully playing with these $\gamma$-dependent constants, we can reabsorb the bad factors $\gamma^2$ or $\gamma$ into their definitions, see sections 7.3, 7.4 and 7.7 for the technical details.

**Remark 6.2** In connection with the end of previous remark, we note that the use of a norm involving several constants $A, A_0, A_0^R$ etc, rather than a single one, is one original aspect of our proof, and it is the key ingredient allowing us to choose an optimal powers of $\delta$ in (4.19) (recall that eventually $\delta$ will be chosen proportional to $\varepsilon$, and that the $\delta$-exponents $2, 2, 3, k(\ell) = |\ell|/2-1$ in the right side of (4.19) are dictated by the lowest order contributions to $u_\ell$ in perturbation theory and cannot be improved; see the discussion at the beginning of section 4.2.2). If we tried to repeat the proof of Key lemma with a simplified norm with $A = A_0 = A_0^R = \dots$ we would not succeed in proving the analogues of (6.5) and (6.6). One can however use a simplified norm, and a simplified proof of Key lemma, if one changes the optimal powers of $\delta$ to sub-optimal ones, strictly smaller than $2, 2, 3, k(\ell)$. This was the strategy followed in [27] (see the non-optimal powers in their eq. (2.17)). Naively, this strategy leads to an estimate on the fixed-point couplings (analogue of corollary 6.1 below) with sub-optimal powers. However, armed with our analyticity argument from section 6.5, this limitation can be overcome. Namely, once the fixed point existence is proven by working in the sub-optimal Banach space, the argument from section 6.5 still works and shows that it is analytic in a disk around $\varepsilon = 0$. From analyticity, we could then recover the optimal estimates on the fixed-point couplings. Although such a mixed real/complex strategy is possible, here we prefer to keep these two lines of development independent. So, we work with the optimal powers from the start and obtain the optimal estimates with purely real methods (even though it leads to some mild complications in the proof of Key Lemma).

## 6.2 Abstract analysis

Recall that we are solving $f(y) = 0$ with $y = (\nu, \lambda, u)$ and $f$ given in (5.32). In this subsection we consider $u$ as a vector living in an abstract Banach space $B$ endowed with some norm $\|u\|_B$. This norm will be used to state conditions on the maps $e_j$ guaranteeing the existence and uniqueness of a solution in some neighborhood of $y_0$, see eq. (6.10). In the next subsection these conditions will be verified with the help of Key Lemma, identifying the norm $\|u\|_B$ with (6.1).
Concerning the rescalings (5.33), note that (the O symbols here and in (6.8) have $\gamma$- and $\varepsilon$-independent constants):

$$ \frac{\varepsilon}{1 - \gamma^{-2\varepsilon}} = (2 \log \gamma)^{-1} (1 + O(\varepsilon \log \gamma)). \qquad (6.7) $$

By the small $\varepsilon$ asymptotics of $I_1, I_2$ from Lemma G.1 in appendix G, we have

$$ \begin{aligned} a &= 2(N-2) \left[ \int \frac{d^d k}{(2\pi)^d} \frac{\chi(k)}{|k|^{d/2}} + O(\varepsilon \log \gamma) \right], \\ b &= -2(N-8) \left[ \frac{S_d}{(2\pi)^d} + O(\varepsilon \log \gamma) \right]. \end{aligned} \qquad (6.8) $$

As mentioned in the introduction, we are assuming $N \neq 8$ so that $b \neq 0$. We will also assume $\varepsilon \le c/\log \gamma$ where c is a small $\gamma$-independent constant. Under these conditions $a, b, b^{-1} = O(1)$. In particular $b \neq 0$.

Setting $e_j$ ($j = \nu, \lambda, u$) to zero in (5.32), we get an “approximate equation”

$$ f_0(y) = 0, \quad f_0(y) = \begin{pmatrix} \nu + a\lambda \\ \varepsilon\lambda + b\lambda^2 \\ u \end{pmatrix}, \qquad (6.9) $$

which has a nontrivial solution

$$ y_0 = (\nu_0, \lambda_0, u_0) = \left( \frac{a}{b}\varepsilon, -\frac{1}{b}\varepsilon, 0 \right). \qquad (6.10) $$

Our goal will be to show that the full equation $f(y) = 0$ has a solution of the form $y_0+O(\varepsilon^2)$. Aiming to apply a contraction argument, we rewrite equation $f(y) = 0$ one last time as a fixed point equation for a map $F(y)$. We choose the following rewrite:

$$ f(y) = 0 \iff y = F(y), \quad F(y) = y - G^{-1}f(y), \qquad (6.11) $$

with G an arbitrary invertible linear operator. We would like to choose G so that $F(y)$ is a contraction in a small neighborhood of $y_0$. Recall that Newton's method for solving nonlinear equations would correspond to $G = \nabla f(y)$. We do not want to deal with the full gradient of the complicated map $f(y)$, and we will instead choose $G = \nabla f_0(y_0)$, cf. (6.9). This “approximated gradient” will be sufficient to make $F(y)$ a contraction. We have

$$ G = \begin{pmatrix} 1 & a & 0 \\ 0 & -\varepsilon & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad G^{-1} = \begin{pmatrix} 1 & a\varepsilon^{-1} & 0 \\ 0 & -\varepsilon^{-1} & 0 \\ 0 & 0 & 1 \end{pmatrix}. \qquad (6.12) $$

With this choice, the map $F(y)$ takes the form:

$$ F(y) = \begin{pmatrix} F^{\nu}(y) \\ F^{\lambda}(y) \\ F^{u}(y) \end{pmatrix} = \begin{pmatrix} -2a\lambda - ab\frac{\lambda^2}{\varepsilon} - e_{\nu} - a\frac{e_{\lambda}}{\varepsilon} \\ 2\lambda + b\frac{\lambda^2}{\varepsilon} + \frac{e_{\lambda}}{\varepsilon} \\ e_{u} \end{pmatrix}. \qquad (6.13) $$
**Remark 6.3** The reader may be puzzled: why introduce the new map $F$ rather than use in its place the renormalization map itself, given that eq. (5.30) already has the fixed point form $y = R(y)$? The reason is that contraction argument cannot be applied directly to $R$, since it is not fully contracting: it is contracting near the fixed point along all directions except $\nu$. Note as well that $R$ is only “barely contracting” in direction $\lambda$: its linearization around $y_0$ has the corresponding eigenvalue equal to $2 - \gamma^{2\epsilon} = 1 - 2\epsilon \log \gamma + \cdots$, smaller than 1 but only by $O(\epsilon)$. This “barely contracting” direction is the reason why we apply the contraction argument in a neighborhood of size $\epsilon^2$ of $y_0$ (outside of which even $F$ would not be contracting).

We will aim to apply a contraction argument to $F(y)$ in a neighborhood $Y_0$ of $y_0$ defined as

$$Y_0 = \{y : |\nu - \nu_0| \le M_0\epsilon^2, |\lambda - \lambda_0| \le M_0\epsilon^2, \|u\|_B \le 1\}, \quad (6.14)$$

whose size depends on $\epsilon$ and on an additional parameter $M_0$. First of all let us arrange that $F$ maps $Y_0$ to itself. Writing $\lambda = \lambda_0 + \delta\lambda$, we express $F(y)$ as

$$F(y) = \begin{pmatrix} \nu_0 - \frac{ab}{\epsilon}(\delta\lambda)^2 - e_\nu - a \frac{e_\lambda}{\epsilon} \\ \lambda_0 + \frac{b}{\epsilon}(\delta\lambda)^2 + \frac{e_\lambda}{\epsilon} \\ e_u \end{pmatrix}. \quad (6.15)$$

We see that $F(Y_0) \subset Y_0$ provided that for any $y \in Y_0$

$$K_1 \max(M_0^2 \epsilon^3, |e_\nu(y)|, \epsilon^{-1}|e_\lambda(y)|) \le M_0 \epsilon^2, \quad \|e_u(y)\|_B \le 1, \quad (6.16)$$

where

$$K_1 = K_1(a, b) = \max(1 + |a| + |ab|, 1 + |b|). \quad (6.17)$$

Then (6.16) are satisfied as long as $K_1 M_0 \epsilon \le 1$ and provided that

$$|e_\nu(y)| \le \frac{M_0}{K_1} \epsilon^2, \quad |e_\lambda(y)| \le \frac{M_0}{K_1} \epsilon^3, \quad \|e_u(y)\|_B \le 1 \quad (y \in Y_0). \quad (6.18)$$

We next proceed to arrange that $F$ is a contraction in $Y_0$. For this we need to specify a Banach space norm on $y = (\nu, \lambda, u)$. We will use the norm

$$\|y\|_Y = \max\{\tilde{\epsilon}^{-1}|\nu|, \tilde{\epsilon}^{-1}|\lambda|, \|u\|_B\}, \quad (6.19)$$

depending on a parameter $\tilde{\epsilon}$. Since in our application $\nu$ and $\lambda$ are $O(\epsilon)$, while $\|u\|_B$ will be $O(1)$ when identified with (6.1), the natural value for $\tilde{\epsilon}$ is order $\epsilon$ so that all terms in (6.19) have the same order. Eventually in section 6.3 we will fix $\tilde{\epsilon} = A_0\delta$ so that this norm will coincide with (6.2). However in this section let us keep the ratio $\tilde{\epsilon}/\epsilon$ as a free parameter.

We will next study the gradient $\nabla F$ and arrange that its operator norm is less than 1. Here the gradient $\nabla F$ is the Frechet derivative which was already discussed in section 6.1.

From (6.13), we compute the gradient $\nabla F$ in components:

$$\frac{\partial(F^\nu, F^\lambda, F^u)}{\partial(\nu, \lambda, u)} = \begin{pmatrix} -\partial_\nu e_\nu - a \frac{\partial_\nu e_\lambda}{\epsilon} - \frac{2ab}{\epsilon}(\lambda - \lambda_0) - \partial_\lambda e_\nu - a \frac{\partial_\lambda e_\lambda}{\epsilon} - \partial_\nu e_\nu - a \frac{\partial_\nu e_\lambda}{\epsilon} \\ \frac{\partial_\nu e_\lambda}{\epsilon} \\ \partial_\nu e_u \end{pmatrix} + \frac{2b}{\epsilon}(\lambda - \lambda_0) + \frac{\partial_\lambda e_\lambda}{\epsilon} \begin{pmatrix} \partial_\lambda e_\nu \\ \partial_\lambda e_u \\ \partial_\lambda e_u \end{pmatrix}. \quad (6.20)$$
Various partial derivatives of maps $e_j(\nu, \lambda, u)$ are understood as Frechet derivatives, sometimes with $Z$ or $Z'$ being equal to $\mathbb{R}$. E.g. $\partial_\nu e_u$ is, just like $e_u$, a $B$-valued function on $Y$. On the other hand $\partial_u e_\lambda \in \mathcal{L}(B, \mathbb{R})$, a linear functional on $B$.

We next proceed to study the norm of $\nabla F(y) \in \mathcal{L}(Y, Y)$ where $y \in Y_0$. Let $\delta y \in Y$, $\|\delta y\|_Y \le 1$ which means

$$|\delta\nu| \le \tilde{\epsilon}, \quad |\delta\lambda| \le \tilde{\epsilon}, \quad \|\delta u\|_B \le 1. \qquad (6.21)$$

We have

$$\nabla F(y)\delta y = \begin{pmatrix} \partial_\nu F^\nu \delta\nu + \partial_\lambda F^\nu \delta\lambda + \partial_u F^\nu \delta u \\ \partial_\nu F^\lambda \delta\nu + \partial_\lambda F^\lambda \delta\lambda + \partial_u F^\lambda \delta u \\ \partial_\nu F^u \delta\nu + \partial_\lambda F^u \delta\lambda + \partial_u F^u \delta u \end{pmatrix}, \qquad (6.22)$$

where all partial derivatives in the r.h.s. are evaluated at $y$. This implies

$$\begin{align*}
\|\nabla F(y)\|_{\mathcal{L}(Y,Y)} &= \sup_{\|\delta y\|_Y \le 1} \|\nabla F(y)(\delta y)\|_Y \\
&= \sup_{\|\delta y\|_Y \le 1} \max \left( \begin{array}{l} \tilde{\epsilon}^{-1} |\partial_\nu F^\nu \delta\nu + \partial_\lambda F^\nu \delta\lambda + \partial_u F^\nu \delta u| \\ \tilde{\epsilon}^{-1} |\partial_\nu F^\lambda \delta\nu + \partial_\lambda F^\lambda \delta\lambda + \partial_u F^\lambda \delta u| \\ \|\partial_\nu F^u \delta\nu + \partial_\lambda F^u \delta\lambda + \partial_u F^u \delta u\|_B \end{array} \right) \\
&\le \max \left( \begin{array}{l} |\partial_\nu F^\nu| + |\partial_\lambda F^\nu| + \tilde{\epsilon}^{-1} \|\partial_u F^\nu\|_{\mathcal{L}(B, \mathbb{R})} \\ |\partial_\nu F^\lambda| + |\partial_\lambda F^\lambda| + \tilde{\epsilon}^{-1} \|\partial_u F^\lambda\|_{\mathcal{L}(B, \mathbb{R})} \\ \tilde{\epsilon} \|\partial_\nu F^u\|_B + \tilde{\epsilon} \|\partial_\lambda F^u\|_B + \|\partial_u F^u\|_{\mathcal{L}(B,B)} \end{array} \right) \tag{6.23}
\end{align*}$$

Finally using the explicit form of $\nabla F$ components we get that for $y \in Y_0$

$$\|\nabla F(y)\|_{\mathcal{L}(Y,Y)} \le K_2 \max\left\{ |\partial_\nu e_\nu|, \frac{|\partial_\nu e_\lambda|}{\epsilon}, M_0\epsilon, |\partial_\lambda e_\nu|, \frac{|\partial_\lambda e_\lambda|}{\epsilon}, \frac{\|\partial_u e_\nu\|_{\mathcal{L}(B,\mathbb{R})}}{\tilde{\epsilon}}, \frac{\|\partial_u e_\lambda\|_{\mathcal{L}(B,\mathbb{R})}}{\epsilon\tilde{\epsilon}}, \right. \\
\left. \tilde{\epsilon}\|\partial_\nu e_u\|_B, \tilde{\epsilon}\|\partial_\lambda e_u\|_B, \|\partial_u e_u\|_{\mathcal{L}(B,B)} \right\}, \qquad (6.24)$$

where we used that $|\lambda - \lambda_0| \le M_0\epsilon^2$ in $Y_0$ and defined a constant

$$K_2(a, b) = \max(3 + 3|a| + 2|ab|, 3 + 2|b|). \qquad (6.25)$$

We will demand that the following conditions hold uniformly for $y \in Y_0$:

$$\begin{align}
& |\partial_i e_\nu| \le M_0 \epsilon, \qquad |\partial_i e_\lambda| \le M_0 \epsilon^2, \qquad \|\partial_i e_u\|_B \le \alpha \tilde{\epsilon}^{-1} \quad (i = \nu, \lambda), \nonumber \\
& \|\partial_u e_\nu\|_{\mathcal{L}(B, \mathbb{R})} \le M_0 \epsilon \tilde{\epsilon}, \qquad \|\partial_u e_\lambda\|_{\mathcal{L}(B, \mathbb{R})} \le M_0 \epsilon^2 \tilde{\epsilon}, \qquad \|\partial_u e_u\|_{\mathcal{L}(B, B)} \le \alpha, \tag{6.26}
\end{align}$$

where $\alpha$ is yet another parameter. Under these conditions eq. (6.24) implies:

$$\|\nabla F(y)\|_{\mathcal{L}(Y,Y)} \le \max(K_2 M_0 \epsilon, K_2 \alpha) \quad (y \in Y_0). \qquad (6.27)$$

We restate the conclusions of the above discussion as

**Lemma 6.2 (Abstract Lemma)** Suppose that, for a given $\epsilon$, the constants $M_0, \tilde{\epsilon}, \alpha$ are such that maps $e_j$ satisfy bounds (6.18) and (6.26) everywhere in $Y_0$ defined by (6.14). Suppose in addition that (see (6.17), (6.25) for the definition of $K_1$ and $K_2$)

$$K_1 M_0 \epsilon \le 1, \quad K_2 M_0 \epsilon \le 1/2, \quad K_2 \alpha \le 1/2. \qquad (6.28)$$

Then $F(Y_0) \subset Y_0$ and $\|\nabla F(y)\|_{\mathcal{L}(Y,Y)} \le 1/2$ in $Y_0$, so that $F$ is a contraction in $Y_0$ and has a unique fixed point there.
### 6.2.1 Complex version of the Abstract Lemma

By a few minor modifications of the proof of the Abstract Lemma we can get a complex-$\epsilon$ version thereof. This is needed in the proof of fixed point analyticity (section 6.5) and is not used anywhere else. We let $\epsilon \in \mathbb{C}$, $y$ be an element of the complex Banach space $\mathbb{Y}$ with the norm (6.19), and $\mathbb{Y}_0$ (the complex analogue of $Y_0$, see (6.14)) be defined as:

$$ \mathbb{Y}_0 = \{y : |\nu - \nu_0| \le M_0|\epsilon|^2, |\lambda - \lambda_0| \le M_0|\epsilon|^2, \|u\|_B \le 1\}. \quad (6.29) $$

Then the following generalization of Lemma 6.2 holds.

**Lemma 6.3 (Complex Abstract Lemma)** Suppose that, for a given $\epsilon \in \mathbb{C}$, the constants $M_0, \tilde{\epsilon}, \alpha$ are such that maps $e_j$ satisfy bounds the complex analogues of (6.18) and (6.26), i.e.,

$$ |e_{\nu}(y)| \le \frac{M_0}{K_1}|\epsilon|^2, \quad |e_{\lambda}(y)| \le \frac{M_0}{K_1}|\epsilon|^3, \quad \|e_u(y)\|_B \le 1 \quad (y \in \mathbb{Y}_0), \quad (6.30) $$

and

$$ \begin{aligned} & |\partial_i e_\nu| \le M_0 |\epsilon|, & |\partial_i e_\lambda| \le M_0 |\epsilon|^2, & \|\partial_i e_u\|_B \le \alpha \tilde{\epsilon}^{-1} \quad (i = \nu, \lambda), \\ & \|\partial_u e_\nu\|_{\mathcal{L}(B, \mathbb{R})} \le M_0 |\epsilon| \tilde{\epsilon}, & \|\partial_u e_\lambda\|_{\mathcal{L}(B, \mathbb{R})} \le M_0 |\epsilon|^2 \tilde{\epsilon}, & \|\partial_u e_u\|_{\mathcal{L}(B, B)} \le \alpha, \end{aligned} \quad (6.31) $$

everywhere in $\mathbb{Y}_0$ defined by (6.29). Suppose in addition that (see (6.17), (6.25) for the definition of $K_1$ and $K_2$)

$$ K_1 M_0 |\epsilon| \le 1, \quad K_2 M_0 |\epsilon| \le 1/2, \quad K_2 \alpha \le 1/2. \quad (6.32) $$

Then $F(\mathbb{Y}_0) \subset \mathbb{Y}_0$ and $\|\nabla F(y)\|_{\mathcal{L}(Y, Y)} \le 1/2$ in $\mathbb{Y}_0$, so that $F$ is a contraction in $\mathbb{Y}_0$ and has a unique fixed point there.

*Proof.* The proof of this lemma is a straightforward repetition of the one of Lemma 6.2, modulo the replacement of $\epsilon$ by $|\epsilon|$ in a few inequalities. More precisely, a simple critical rereading of the proof shows that, if we leave the definitions of $F$, see (6.15), and of $\nabla F$, see (6.20) and (6.22), as they are, and we replace $Y_0$ by $\mathbb{Y}_0$ and $\epsilon$ by $|\epsilon|$ everywhere in the rest of the proof (in particular in the following places: 1 line after (6.17); in eq. (6.18); in eq. (6.24); in eq. (6.26); and in eq. (6.27)), then we readily obtain the desired claim.

## 6.3 Fixed point theorem

In this section we will put Key Lemma and Abstract Lemma together and will finally show that the FPE (5.32) has a solution. Namely, we will prove the following result:

**Theorem 6.1** There exists a $\gamma_0 \ge 2$ and a positive continuous function $\epsilon_0(\gamma)$ defined for $\gamma \ge \gamma_0$ such that for each $\gamma \ge \gamma_0$ and $0 < \epsilon \le \epsilon_0(\gamma)$ the fixed point equation (5.32) has a nontrivial solution.
Proof. We will show, with the help of Key Lemma 6.1, that for $\gamma \ge \gamma_0$ and for $0 < \varepsilon \le \varepsilon_0(\gamma)$ conditions of Abstract Lemma 6.2 can be satisfied.

We thus identify the abstract Banach space $B$ in section 6.2 with the space $B(\gamma, \delta)$ in (6.1). We also put

$$ \tilde{\varepsilon} = A_0\delta, \qquad (6.33) $$

and identify the space $Y$ from (6.19) with $Y(\gamma, \delta)$ in (6.2). The parameter $\delta$ in the Key Lemma will be chosen proportional to $\varepsilon$:

$$ \delta = h\varepsilon, \qquad (6.34) $$

with $h$ to be fixed momentarily.

Abstract Lemma requires us to examine the neighborhood $Y_0$ defined in (6.14). By $K_1 M_0 \varepsilon \le 1$, the first of conditions (6.28) (we will make sure to satisfy all of these conditions below), the points of $Y_0$ will satisfy

$$ |\nu|, |\lambda| \le K_3\varepsilon, \qquad K_3 = K_3(a, b) = \max\left(\left|\frac{a}{b}\right| + \frac{1}{K_1}, \frac{1}{|b|} + \frac{1}{K_1}\right). \qquad (6.35) $$

Let us choose

$$ h = K_3/A_0. \qquad (6.36) $$

By (6.35), we have

$$ Y_0 \subset \{y : \|y\|_Y \le 1\}. \qquad (6.37) $$

Thus, the basic assumption (6.4) holds in $Y_0$, and we can use Key Lemma to estimate $e_j$ and their derivatives in $Y_0$.

We will also fix (see Key Lemma for the definition of $\bar{D}$)

$$ \alpha = \gamma^{-\bar{D}}. \qquad (6.38) $$

With this identification and (6.33), the bounds on the derivatives of $\partial_i e_u$, $\partial_u e_u$ requested in (6.26) coincide with the bounds for the same derivatives in (6.6) of the Key Lemma. The request $\|e_u\|_B \le 1$ in $Y_0$ (eq. (6.18)) is also satisfied by the bound on $\|e_u\|_B$ in (6.5).

Furthermore, we choose $\gamma_0$ as

$$ \gamma_0 = \max(\gamma_{\text{key}}, (2K_2)^{1/\bar{D}}). \qquad (6.39) $$

Then for $\gamma \ge \gamma_0$ we have $\gamma \ge \gamma_{\text{key}}$ so that we can use Key Lemma, and in addition we satisfy the third condition in (6.28).

Let us now arrange for the conditions in (6.18) and (6.26) concerning $e_\nu, e_\lambda$, and their derivatives. By eq. (5.33), $e_\nu, e_\lambda$ equal $e_\nu^{(0)}, e_\lambda^{(0)}$ times factors bounded by a $\gamma$-dependent constant $f_\gamma$. Key Lemma gives estimates for $e_\nu^{(0)}, e_\lambda^{(0)}$ and their derivatives with constants $E_0, E_1$ in the r.h.s., and $e_\nu, e_\lambda$ will satisfy the same estimates with $E_i \to E'_i = f_\gamma E_i$. Using the proportionality (6.34) between $\delta$ and $\varepsilon$, these estimates take the form

$$ \begin{aligned} & |e_\nu| \le E'_0 h^2 \varepsilon^2, \\ & |\partial_i e_\nu| \le (E'_0/A_0)h\varepsilon, \\ & \|\partial_u e_\nu\|_{\mathcal{L}(B, \mathbb{R})} \le E'_0 h^2 \varepsilon^2, \end{aligned} \qquad \begin{aligned} & |e_\lambda| \le E'_1 h^3 \varepsilon^3, \\ & |\partial_i e_\lambda| \le (E'_1/A_0)h^2 \varepsilon^2 \quad (i = \nu, \lambda), \\ & \|\partial_u e_\lambda(\nu, \lambda, u)\|_{\mathcal{L}(B, \mathbb{R})} \le E'_1 h^3 \varepsilon^3. \end{aligned} \qquad (6.40) $$
These have the same scaling in $\epsilon$ as the corresponding estimates in (6.18), (6.26) (recall that $\tilde{\epsilon}/\epsilon = A_0h$). So, to satisfy (6.18), (6.26), we simply choose $M_0$ sufficiently large, namely:

$$M_0 = \max(K_1 E'_0 h^2, K_1 E'_1 h^3, (E'_0/A_0)h, (E'_1/A_0)h^2). \quad (6.41)$$

We still have to satisfy the first two conditions in (6.28), as well as to make sure that $\delta = h\epsilon \le \delta_0$. We achieve this by choosing

$$\epsilon_0(\gamma) = \min \left( \frac{\delta_0}{h}, \frac{1}{K_1 M_0}, \frac{1}{2K_2 M_0} \right). \quad (6.42)$$

For any $0 < \epsilon \le \epsilon_0(\gamma)$, conditions of Abstract Lemma are satisfied, and hence a fixed point exist.

**Corollary 6.1** *The fixed point whose existence we proved belongs to the neighborhood*

$$\begin{gathered} |\nu - \nu_0|, |\lambda - \lambda_0| \le M_0\epsilon^2, \\ \|H_{2R}\|_w \le A_0^R h^2 \epsilon^2, \|H_{4R}\|_w \le A_1^R h^2 \epsilon^2, \|H_{6R}\|_w \le A_2^R h^3 \epsilon^3, \|H_l\|_w \le Ah^{l/2-1}\epsilon^{l/2-1}, \end{gathered} \quad (6.43)$$

where $C_0$, $A$, $A_k^R$, $h$ are some $\gamma$-dependent quantities. Moreover in this neighborhood this is a unique solution of the fixed point equation.

This follows from writing in full the condition $\|u\|_B \le 1$.

## 6.4 Semigroup property and $\gamma$-independence

Theorem 6.1 shows that the renormalization map $R(\epsilon, \gamma)$ has a fixed point provided that $\gamma \ge \gamma_0$ is sufficiently large and $\epsilon \le \epsilon_0(\gamma)$ is sufficiently small. We would now like to study how this fixed point depends on various parameters. In this section we will discuss $\gamma$-independence, while in the next one we will show that it depends on $\epsilon$ analytically.

The $\gamma$-independence at $O(\epsilon)$ is visible in eq. (6.10), since both $a$ and $b$ become $\gamma$-independent as $\epsilon \to 0$. That it should hold in general can be suspected from the semigroup property (2.17). Indeed, if a certain interaction $H_*$ is a fixed point of $R(\epsilon, \gamma)$, then by the semigroup property it is also a fixed point of $R(\epsilon, \gamma^n)$ for any $n \ge 2$, as long as $R(\epsilon, \gamma^n)$ is defined on $H_*$ as a continuous map acting on a neighborhood of a Banach space to which $H_*$ belongs. Combining this simple argument with continuity in $\gamma$, one should be able to prove that, in fact, the fixed point is unique and completely independent of $\gamma$, at least on a suitable interval of values of $\gamma$, such as the one denoted by $J$ below. A full proof of this fact requires a critical re-reading of the proofs of the Key Lemma, of the Abstract Lemma and of the Fixed point theorem, as well as a generalization thereof, providing existence and uniqueness of the general FPE in the space of equivalency classes of couplings modulo null sequences, see remark 5.3 and remark 6.4 below. We won't belabor all the required details here, but we will provide all the elements sufficient for a willing reader to sit down and check the various claims, most of which are just straightforward corollaries of the previous discussion.
Fix an interval $I = [\gamma_{\text{key}}, \bar{\gamma}]$, with $\gamma_{\text{key}}$ the constant of the Key Lemma. From the proof the Key Lemma, see in particular section 7.7, we see that, with no loss of generality, all the functions in (6.3) can be chosen to be decreasing in $\gamma$, so that their smallest values in $I$ are those at $\gamma = \bar{\gamma}$, which we denote by $\bar{\delta}_0, \bar{A}_0, \{\bar{A}_k^R\}_{k=0,1,2}, \bar{A}, \bar{E}_0, \bar{E}_1$. It is easy to check that the Key Lemma 6.1 admits the following “uniform” version on $I$: take $\gamma \in I$, $0 \le \delta \le \bar{\delta}_0$, and any sequence $(\nu, \lambda, u)$ satisfying the inequalities (6.4) with $A_0$ replaced by $\bar{A}_0$ and $\|u\|_B$ replaced by $\|u\|_{\bar{B}} = \|u\|_{B(\bar{\gamma}, \delta)}$; then the conclusions of the lemma, (6.5) and (6.6) hold, with $\delta_0, A_0, \{A_k^R\}_{k=0,1,2}, A, E_0, E_1$ replaced by $\bar{\delta}_0, \bar{A}_0, \{\bar{A}_k^R\}_{k=0,1,2}, \bar{A}, \bar{E}_0, \bar{E}_1$, and $B$ replaced by $\bar{B}$.

Similarly, we can easily obtain a uniform version of the Abstract Lemma and Fixed point theorem for $0 < \varepsilon \le \bar{\varepsilon}_0$, with $\bar{\varepsilon}_0 = \min_{\gamma \in I} \varepsilon_0(\gamma)$, and $\gamma \in I_0 \equiv [\bar{\gamma}_0, \bar{\gamma}]$, with $\bar{\gamma}_0$ defined by the analogue of (6.39) with $K_2$ replaced by $\bar{K}_2 = \min_{0 < \varepsilon \le \bar{\varepsilon}_0, \gamma \in I} K_2$ (note that $I_0$ is non empty for $\bar{\gamma}$ large enough). We denote by $F_\gamma$ the function $F$ of section 6.2, in order to emphasize its dependence upon $\gamma$. We define $\bar{\nu}_0 = \frac{a_0}{b_0} \varepsilon$ and $\bar{\lambda}_0 = -\frac{1}{b_0} \varepsilon$, with $a_0 = a|_{\varepsilon=0}$ and $b_0 = b|_{\varepsilon=0}$, see (6.8), and let

$$ \bar{Y}_0 = \{y : |\nu - \bar{\nu}_0| \le \bar{M}_0 \varepsilon^2, |\lambda - \bar{\lambda}_0| \le \bar{M}_0 \varepsilon^2, \|u\|_{\bar{B}} \le 1\}. \quad (6.44) $$

A critical re-reading of the Abstract Lemma and of the Fixed point theorem shows that there exist constants $\bar{h}$ and $\bar{M}_0$ such that, fixing $\delta = \bar{h}\varepsilon$ and using the uniform version of the Key Lemma, then, for any $\gamma \in I_0$, $F_\gamma(\bar{Y}_0) \subset \bar{Y}_0$ and $F_\gamma$ is continuous for $\gamma \in I_0$. Moreover, letting $\bar{Y}$ be the Banach space with norm

$$ \|y\|_{\bar{Y}} = \max\{(\bar{A}_0\delta)^{-1}|\nu|, (\bar{A}_0\delta)^{-1}|\lambda|, \|u\|_{\bar{B}}\}, \quad (6.45) $$

we have $\|\nabla F_\gamma(y)\|_{L(\bar{Y}, \bar{Y})} \le 1/2$ in $\bar{Y}_0$, so that, for any $\gamma \in I_0$, $F_\gamma$ is a contraction in $\bar{Y}_0$, uniformly in $\gamma$, and has a unique fixed point there, denoted $y_*(\gamma)$. Of course, $y_*(\gamma) = \lim_{n \to \infty} F_\gamma^n(\bar{y}_0)$, with $\bar{y}_0 = (\bar{\nu}_0, \bar{\lambda}_0, 0)$. Recalling that $F_\gamma$ is continuous in $\gamma$ for $\gamma \in I_0$ and is uniformly contractive there, we find that $y_*(\gamma)$ is continuous in $\gamma$ for $\gamma \in I_0$, being the uniform limit of a sequence of uniformly continuous functions.

**Remark 6.4** The previous discussion, as well as the one of the previous sections, shows that $y_*(\gamma)$ is the unique solution of the restricted FPE (5.28), in the sense of remark 5.3. As discussed there, we expect that a generalization of the methods of this paper will allow us to prove the uniqueness of the solution of the general FPE $(H'_\ell) = (H_\ell) + (N_\ell)$ modulo null couplings, provided the null sequence $(N_\ell)$ is sufficiently small in norm. We will denote by the symbol $h_*(\gamma)$ such a (presumed) unique solution in the space of equivalency classes of couplings. Of course, continuity of $y_*(\gamma)$ implies the continuity of $h_*(\gamma)$ in the appropriate topology.

By construction, $y_*(\gamma), \gamma \in I_0$, is the unique solution in $\bar{Y}_0$ to the fixed point equation $y = R_\gamma(y)$, where $R_\gamma$ is the original form of the RG map (before the manipulations (6.11)), given by the right side of (5.30). By its very definition, $R_\gamma$ satisfies the semigroup property $R_\gamma \circ R_{\gamma'} = R_{\gamma \cdot \gamma'} + \text{null}$, so that, if $h_*(\gamma) = h_*(\gamma')$, then $h_*(\gamma) = h_*(\gamma') = h_*(\gamma \cdot \gamma')$. From this, it follows that $h_*(\gamma) = h_*(\bar{\gamma})$ for all the values $\gamma$ in the subset $X$ of $I_0$ characterized
by the following properties: (i) $\bar{\gamma} \in X$; (ii) if $\gamma \in X$, then $\gamma^{1/n} \in X$, for all natural $n$ such that $\gamma^{1/n} \in I_0$; (iii) if $\gamma, \gamma' \in X$, then $\gamma \cdot \gamma' \in X$, as long as $\gamma \cdot \gamma' \in I_0$. Of course, by the continuity of $h_*(\gamma)$, the fixed point is constant and equal to $h_*(\bar{\gamma})$ on the closure of $X$, as well. For $\bar{\gamma} \ge (\bar{\gamma}_0)^3$, the closure of $X$ contains the sub-interval $J = [\bar{\gamma}^{2/3}, \bar{\gamma}]$.<fnref n="32" /> This proves the independence of the fixed point from $\gamma$, for any $\gamma \in J$.

**Remark 6.5** Another parameter which entered into the renormalization map is the cut-off function $\chi$. The fixed point coupling $\nu_*$ depends on $\chi$ already at $O(\varepsilon)$, as seen from eq. (6.10), because $a$ depends on $\chi$. That $\lambda_*$ is $\chi$-independent at $O(\varepsilon)$ is in agreement with the usual lore that the beta-functions for near-marginal couplings and the corresponding fixed-point coupling should not depend on the UV regularization scheme at the first non-trivial order. In higher orders in $\varepsilon$ we expect that all couplings will acquire $\chi$ dependence. So, in contrast with the $\gamma$-independence, the fixed point does depend on $\chi$. In spite of this, we expect on physical grounds that the critical exponents (i.e. eigenvalues of the renormalization map linearized near the fixed point) should be $\chi$-independent. Showing this rigorously is one of the open problems for the future (see section 8).

## 6.5 Analyticity

In view of the Complex Abstract Lemma 6.3, and of the complex version of the Key Lemma, stated and proved in section 7, see Lemma 7.1, it is easy to show that the fixed point of theorem 6.1 can be extended to an analytic function of $\varepsilon$ in a small neighborhood of the origin. More precisely, we get the following:

**Theorem 6.2 (Analytic Fixed Point Theorem)** There exists a $\gamma_0 \ge 2$ and a positive continuous function $\varepsilon_0(\gamma)$ defined for $\gamma \ge \gamma_0$ such that for each $\gamma \ge \gamma_0$ and $\varepsilon \in \{z \in \mathbb{C} : |z| \le \varepsilon_0(\gamma)\} \equiv \mathbb{E}_0$ the fixed point equation (5.30) has a solution, analytic in $\varepsilon$, extending the one of theorem 6.1. For any $\varepsilon \in \mathbb{E}_0$, such a solution is the unique solution of the fixed point equation in the complex neighborhood defined by the analogue of (6.43) with $|\varepsilon|$ replacing $\varepsilon$.

*Proof.* We let $\delta = h|\varepsilon|$, with $h$ the same as in (6.36). By proceeding as in the proof of theorem 6.1, with $\bar{D}$ defined as in (7.3), we find that $F$ is a contraction on $\mathbb{Y}_0$ for each $\varepsilon \in \mathbb{E}_0 \setminus \{0\}$. Moreover, by Lemma 7.1, $F$ is analytic in $\varepsilon$ on the punctured disk $\bar{\mathbb{Y}}_0(\gamma) = \cup_{0<|\varepsilon|\le\varepsilon_0(\gamma)} \mathbb{Y}_0$, and so is $y_0$. Therefore, $y_n \equiv F^n(y_0)$ is analytic in $\varepsilon$ on $\bar{\mathbb{Y}}_0(\gamma)$. Since $F$ is a contraction, $y_n$ converges to a fixed point, call it $y_*(\varepsilon)$, as $n \to \infty$, for any $\varepsilon \in \mathbb{E}_0 \setminus \{0\}$; for any such $\varepsilon$, $y_*(\varepsilon)$ is the unique solution of the fixed point equation in $\mathbb{Y}_0$. By Vitali's theorem on the convergence of sequences of analytic functions, $y_*(\varepsilon)$ is holomorphic in $\varepsilon$ on $\bar{\mathbb{Y}}_0(\gamma)$ (monodromy follows from the uniqueness of the solution to the fixed point equation in $\mathbb{Y}_0$). Note that $\lim_{\varepsilon \to 0} \varepsilon y_*(\varepsilon) = 0$; therefore, by Riemann's theorem on removable singularities, $y_*(\varepsilon)$ can be extended to an analytic function of $\varepsilon$ on the complex disk of radius $\varepsilon_0(\gamma)$ by letting $y_*(0) = 0$.

<footnote n="32">³²For $\bar{\gamma} \ge (\bar{\gamma}_0)^3$, we have $\bar{\gamma}^{1/3} \in X$ by (ii), and then $\bar{\gamma}^{2/3} \in X$ by (iii). So both endpoints of $J$ are in $X$. Also, if $\gamma_1, \gamma_2 \in J$, then both $\gamma_1^{1/2}, \gamma_2^{1/2} \in X$ by (ii) and hence the geometric mean $(\gamma_1\gamma_2)^{1/2} \in X$ by (iii). Applying this last statement recursively starting from the endpoints of $J$, we obtain that $X$ is dense in $J$.</footnote>
This result has various consequences. One clear consequence is that since the fixed point is analytic around $\epsilon = 0$, it has a convergent power series expansion around this point. This is just the perturbative $\epsilon$-expansion discussed at the level of formal power series in appendix I which is therefore convergent. Another consequence is that the fixed points with real $\epsilon > 0$ analytically continue to the fixed points with $\epsilon < 0$. For negative real $\epsilon$, the quartic interaction is an irrelevant perturbation of the gaussian fixed point (at the linearized level). Thus, the $\epsilon < 0$ fixed points should be interpreted as UV fixed points: one can RG-flow from them to the gaussian theory, not the other way around. We expect analyticity to be valid also in the long-range Gross-Neveu model of [27] (see the introduction), and in other similar models. See also appendix J for an alternative proof of fixed point analyticity via the tree expansion.

## 7 Proof of Key lemma

Here we finally prove the Key lemma that, as seen above, is the crucial ingredient for showing the existence and uniqueness of the nontrivial RG fixed point. Rather than proving the Key lemma in the formulation of section 6.1, here we state and prove a generalization of the lemma with complex $\epsilon$, which is the version used in section 6.5 in the discussion on the analyticity of the fixed point. This does not create any additional complications in the proof.

Let us start by observing that both the fluctuation propagator (2.10) and the rescaling factor $\gamma^{-[\psi]}$ in (2.16) depend analytically on $\epsilon$. So each individual term $R_{\ell}^{\ell_1, \dots, \ell_n}$ is analytic in $\epsilon$, and the sum (5.24) will be analytic when convergent. Let $T$ be a compact subset of the half-plane (see eq. (2.8))

$$ T \subset \{\epsilon \in \mathbb{C} : \mathrm{Re} \, \epsilon < d/6\}. \qquad (7.1) $$

By Lemma D.3, the constant $C_{GH}$ is uniformly bounded for $\epsilon \in T$. As a result the multi-linear maps $S_l^{\ell_1, \dots, \ell_n}$ will satisfy estimates (5.38) with uniform ($T$-dependent) constants for $\epsilon \in T$. The action of dilatation for complex $\epsilon$ is still given by (5.20), where $D_l = D_l(\epsilon)$ are complex. We have to replace $D_l \to \mathrm{Re} \, D_l$ in the norm bounds (5.37) for dilatation, which become

$$ \|DH_{\ell,p}\|_w = \gamma^{-\mathrm{Re} D_l - p} \|H_{\ell,p}\|_{w(\cdot/\gamma)} \le \gamma^{-\mathrm{Re} D_l - p} \|H_{\ell,p}\|_w \quad (\epsilon \in \mathbb{C}) \qquad (7.2) $$

The criterion for irrelevance becomes $\mathrm{Re} \, D_l - p > 0$. The same replacement has to be done in the right-hand-sides of the estimates for multilinear maps $R_{\ell}^{\ell_1, \dots, \ell_n}$ in section 5.6, see eqs. (5.34) and (5.35).

The parameter $\bar{D}$ from section 6.1 is redefined for complex $\epsilon \in T$ as

$$ \bar{D} = \bar{D}(T) = \frac{1}{2} \min_{\epsilon \in T} \{\mathrm{Re} \, D_2(\epsilon) + 2, \mathrm{Re} \, D_4(\epsilon) + 1, \mathrm{Re} \, D_6(\epsilon)\}. \qquad (7.3) $$

Note that $\bar{D} > 0$ by assumptions on $T$. We can now state the generalization of Lemma 6.1 to $\epsilon \in \mathbb{C}$.
**Lemma 7.1 (Complex Key Lemma)** Choose $d \in \{1, 2, 3\}$, cutoff $\chi$, $N \ge 4$, and a compact set $T \subset \mathbb{C}$ satisfying (7.1). There exists $\gamma_{\text{key}} \ge 2$ and positive continuous functions (6.3) on $\gamma \ge \gamma_{\text{key}}$, with the following property. Take any $\gamma \ge \gamma_{\text{key}}$, any $0 < \delta \le \delta_0(\gamma)$ and any sequence $y = (\nu, \lambda, u)$ satisfying $\|y\|_{Y(\gamma, \delta)} \le 1$, and apply to it the renormalization map $R(\varepsilon, \gamma)$ with any $\varepsilon \in T$. Then the functions $e_{\nu}^{(0)}$, $e_{\lambda}^{(0)}$, $e_u$ in eq. (5.30) and their derivatives satisfy the bounds (6.5) and (6.6) uniformly in $\varepsilon \in T$. These functions are analytic in $\varepsilon$, being given by convergent series consisting of analytic terms.

The proof of the Complex Key Lemma is presented in the next subsections, distinguishing various subcases. For instance, in order to prove that $\|e_u\|_{B(\gamma, \delta)} \le \gamma^{-\bar{D}}$, recalling the definition (6.1) of the norm, we will separately prove that $\|(e_u)_{\ell}\|_w \le A(\gamma)\delta^{k(\ell)}$ for all $\ell \ge 8$, that $\|(e_u)_{6R}\|_w \le A_2^R(\gamma)\delta^3$, $\|(e_u)_{4R}\|_w \le A_1^R(\gamma)\delta^2$, and $\|(e_u)_{2R}\|_w \le A_0^R(\gamma)\delta^2$. For ease of notation, we will drop the dependence on $\gamma$ from the constants $A(\gamma)$, $A_0(\gamma)$, etc, and simply denote them by $A$, $A_0$, etc. Similarly for $Y(\gamma, \delta)$ and $B(\gamma, \delta)$, to be denoted by $Y$ and $B$, respectively.

## 7.1 Case $\ell \ge 8$

We start from the bound on $\|(e_u(y))_\ell\|_w$ with $\ell \ge 8$. From the definitions, see (5.29) and (5.30), we have

$$ (e_u(y))_\ell = \sum_{(\ell_i)_1^n} R_\ell^{\ell_1, \dots, \ell_n} (H_{\ell_1}, \dots, H_{\ell_n}). \quad (7.4) $$

Using bounds (5.34) and (5.35) on $R_\ell^{\ell_1, \dots, \ell_n}$ collected we find that

$$ \|(e_u(y))_\ell\|_w \le \gamma^{-\text{Re} D_\ell} \|u_\ell\|_w + \gamma^{-\text{Re} D_\ell} \sum_{(\ell_i)_1^n \ne (\ell)} \rho_l[(\ell_i)_1^n], \quad (7.5) $$

where we denoted

$$ \rho_l[(\ell_i)_1^n] = \rho_l(H_{\ell_1}, \dots, H_{\ell_n}), \quad (7.6) $$

and $\rho_l(H_{\ell_1}, \dots, H_{\ell_n})$ is given in eq. (5.36). Here $H_{\ell_i}$ should be interpreted as equal to: $\nu$, if $\ell_i = 2L$; $\lambda$, if $\ell_i = 4L$; $\mathcal{X}_*$, if $\ell_i = 6SL$; $u_{\ell_i}$, otherwise. Recall that $\rho_l = 0$ unless $\sum_i |\ell_i| \ge l + 2(n-1)$.

By using the assumption $\|y\|_{Y(\gamma, \delta)} \le 1$ of Key lemma, writing in full the meaning of this condition (recall the definition of $\|y\|_{Y(\gamma, \delta)}$, eq. (4.19)), we find:

$$ \begin{aligned} & \|H_{2L}\|_w + \|H_{2R}\|_w \le A_0\delta + A_0^R\delta^2 =: b_0, \\ & \|H_{4L}\|_w + \|H_{4R}\|_w \le A_0\delta + A_1^R\delta^2 =: b_1, \\ & \|H_{6SL}\|_w + \|H_{6R}\|_w \le C_{\gamma 3} A_0^2 \delta^2 + A_2^R \delta^3 =: b_2, \\ & \|H_\ell\|_w \le A\delta^{k(\ell)} =: b_{k(\ell)}, \quad \text{if} \quad \ell \ge 8. \end{aligned} \quad (7.7) $$

It will be convenient to arrange so that

$$ b_k \le A\delta^{\max\{k,1\}}, \quad k \ge 0. \quad (7.8) $$
For $k \ge 3$ this is true as an equality by the definition of $b_k$. To have this for $k = 0, 1, 2$ as well, we will assume (we will see later how to satisfy simultaneously all ♠-constraints):

$$ (\spadesuit) \quad 2 \max(A_0, A_0^R \delta_0, A_1^R \delta_0, C_{\gamma 3} A_0^2 + A_2^R \delta_0) \le A. \qquad (7.9) $$

Using these bounds in (7.5) we find that:

$$ \|(e_u(y))_\ell\|_w \le \gamma^{-\mathrm{Re} D_\ell} [A\delta^{k(\ell)} + \Delta_{k(\ell)}^{(1)} + \Delta_{k(\ell)}^{(2)}], \qquad (7.10) $$

where we defined [here $C = C_2^2$]:

$$ \begin{aligned} \Delta_k^{(1)} &= \sum_{k'=k+1}^{\infty} C^{k'+1} b_{k'}, \\ \Delta_k^{(2)} &= \sum_{(k_i)_i^n, n \ge 2} F_k[(k_i)_1^n], \end{aligned} \qquad (7.11) $$

$$ F_k[(k_i)_1^n] = \begin{cases} C_\gamma^{n-1} \prod_{i=1}^n C^{k_i+1} b_{k_i} & \text{if } \sum_i k_i \ge k, \\ 0 & \text{otherwise.} \end{cases} \qquad (7.12) $$

We will estimate these sums with the help of the following lemma, imposing assumptions (7.13) which we will arrange in the end by choosing $\delta_0$ and $A$ appropriately. For the proof see appendix F.

**Lemma 7.2** Suppose the nonnegative constants $C_\gamma, C, \delta, A$ satisfy

$$ (\spadesuit) \quad C\delta \le 1/4, \quad C_\gamma C A\delta \le 1/2, \quad C_\gamma C A \le 1/2, \qquad (7.13) $$

and that $0 \le b_k \le A\delta^{\max\{k,1\}}$ for all $k \ge 0$. Then $\Delta_k^{(1)}$ and $\Delta_k^{(2)}$ defined in terms of $C, C_\gamma, b_k$ by (7.11), (7.11) satisfy

$$ \Delta_k^{(1)} \le A\delta^{k+1}(2C^{k+2}), \qquad (7.14) $$

$$ \Delta_k^{(2)} \le A\delta^{\max\{k,2\}} \cdot \begin{cases} C_0 = 4C + 8C^2 + 16C^3 & \text{if } k = 0,1 \\ 2(2C)^{k+1} & \text{if } k \ge 2, \end{cases} \qquad (7.15) $$

Using (7.14), (7.15) in (7.10), and recalling that we are assuming $\ell \ge 8$ (so that $k(\ell) \ge 3$), we find

$$ \|(e_u(y))_\ell\|_w \le \gamma^{-\mathrm{Re} D_\ell} A \delta^{k(\ell)} [1 + 2C^{k(\ell)+2}\delta + 2(2C)^{k(\ell)+1}], \qquad (7.16) $$

It follows that

$$ \|(e_u(y))_\ell\|_w \le \gamma^{-\bar{D}} A \delta^{k(\ell)} \quad (\ell \ge 8), \qquad (7.17) $$

as long as we impose

$$ 1 + C^{k(\ell)+1} + 2(2C)^{k(\ell)+1} \le \gamma^{\mathrm{Re} D_\ell - \bar{D}}. \qquad (7.18) $$

Given the form of this inequality, it is sufficient to check that it holds for $\ell = 8$, and that the l.h.s. grows slower than r.h.s. as $\ell \to \ell + 2$, which amounts to two requirements:

$$ (\spadesuit) \quad 1 + C^4 + 2(2C)^4 \le \gamma^{\mathrm{Re} D_8 - \bar{D}}, \quad \max(1, C, 2C) \le \gamma^{d/2 - \mathrm{Re} \varepsilon}. \qquad (7.19) $$
Next let us estimate derivatives. Consider a vector $\delta y = (\delta\nu, \delta\lambda, \delta u)$ satisfying $\|\delta y\|_Y \le 1$. Consider also a trimmed coupling sequence $\delta H_\ell$ which contains the couplings in $\delta y$ and, in addition, the coupling $\delta H_{6SL}$ corresponding to the variation of $\mathcal{X}_*$. We have

$$ \nabla_y(e_u(y))_{\ell} \delta y = \sum_{(\ell_i)_1^n} \sum_{i=1}^{n} R_{\ell}^{\ell_1, \dots, \ell_n}(H_{\ell_1}, \dots, \delta H_{\ell_i}, \dots, H_{\ell_n}), \quad (7.20) $$

and thus

$$ \|\nabla_y(e_u(y))_{\ell} \delta y\|_w \le \gamma^{-\mathrm{Re} D_{\ell}} \|\delta H_{\ell}\|_w + \gamma^{-\mathrm{Re} D_{\ell}} \sum_{(\ell_i)_1^n \ne (\ell)} \sum_{i=1}^{n} \rho_l(H_{\ell_1}, \dots, \delta H_{\ell_i}, \dots, H_{\ell_n}) \quad (7.21) $$

Note that $\|\delta H_{6SL}\|_w \le 2C_{3\gamma}A_0^2\delta^2$. We will increase $C_{3\gamma}$ by factor 2. Then all couplings $\delta H_\ell$ satisfies the same bounds as the bounds on couplings $H_\ell$ used to estimate $\|(e_u(y))_\ell\|_w$. It follows that the functions $\rho_l$ in the r.h.s. of (7.21) can be estimated in exactly the same way. This gives an estimate of the same form as (7.10), namely

$$ \|\nabla_y(e_u(y))_{\ell} \delta y\|_w \le \gamma^{-\mathrm{Re} D_{\ell}} [A \delta^{k(\ell)} + \Delta_{k(\ell)}^{(1)} + \tilde{\Delta}_{k(\ell)}^{(2)}], \quad (7.22) $$

where $\tilde{\Delta}_k^{(2)}$ differs from $\Delta_k^{(2)}$ in that $F_k[(k_i)]$ is replaced by

$$ \tilde{F}_k[(k_i)_1^n] = n F_k[(k_i)_1^n], \quad (7.23) $$

where the factor $n$ accounts for the sum $\sum_{i=1}^n$ in (7.21). We will increase $C_\gamma$ in (7.12) by 2 to absorb this factor (note $n \le 2^{n-1}$), so that both $\tilde{F}_k$ and $F_k$ can be considered to satisfy the same bound (7.12).

Then, under the same assumptions that (7.17) was obtained, we will have

$$ \|\nabla_y(e_u(y))_{\ell} \delta y\|_w \le \gamma^{-\bar{D}} A \delta^{k(\ell)} \quad (\ell \ge 8). \quad (7.24) $$

Taking into account the assumed bounds on couplings $\delta y$, this inequality is precisely what is asserted in the last line of (6.6) concerning the part of $e_u$ with $\ell \ge 8$.

Incidentally, convergence of the series (7.21) also proves that the functions $e_u(y)$ are in fact Frechet differentiable.

The shown method of bounding derivatives is general and will apply to all the other functions that we still have to consider, i.e. $(e_u)_{2R}, (e_u)_{4R}, (e_u)_{6R}, e_\nu^{(0)}, e_\lambda^{(0)}$. They are all given by sums of multilinear operators applied to the sequence $H_\ell$, and will be estimated using the basic bound (5.36). Whenever we manage to bound such a function by an $X$, the shown method will naturally bound its $u$-derivative by the same $X$, while its $\nu, \lambda$ derivatives by $X/(A_0\delta)$. Note that all bounds (6.6) are of precisely such a form. So we no longer need to discuss derivative bounds, but can focus on estimating the functions themselves.

## 7.2 Case $\ell = 6R$

From the definitions (see (5.29) and (5.30) and the third of (5.27)) and the bounds on $R_{6R}^{\ell_1, \dots, \ell_n}$ we find that

$$ \|(e_u(y))_{6R}\|_w \le \gamma^{-\mathrm{Re} D_6} \|u_{6R}\|_w + \gamma^{-\mathrm{Re} D_6} \sum_{(\ell_i)_1^n \ne (6\mathrm{SL}), (6\mathrm{R}), (4\mathrm{L}, 4\mathrm{L})} \rho_6[(\ell_i)_1^n]. \quad (7.25) $$
By repeating a discussion analogous to that of section 7.1, we get the analogue of (7.10), namely

$$ \|(e_u(y))_{6R}\|_w \le \gamma^{-\text{Re} D_6} [A_2^R \delta^3 + \Delta_2^{(1)} + \Delta_{2;6R}^{(2)}], \quad (7.26) $$

where $\Delta_{2;6R}^{(2)}$ is defined analogously to $\Delta_2^{(2)}$, modulo the fact that the contribution from the sequence $(k_i)_{i=1}^n = (1, 1)$ is now proportional to $b_1 b_1^R$, with $b_1^R = A_1^R \delta^2$, rather than to $b_1^2$ (this comes from the constraint $(\ell_i)_{i=1}^n \ne (4\text{L}, 4\text{L})$ in (7.25)):

$$ \Delta_{2;6R}^{(2)} = 2C_{\gamma}C^4 b_1 b_1^R + \sum_{(k_i)_1^n \ne (1,1)}^{n \ge 2} F_2[(k_i)_1^n]. \quad (7.27) $$

It is convenient to define, for any sequence $\varkappa = (k_i)_1^n$,

$$ F_{\text{ext}}[\varkappa] = \sum_{\varkappa' : \text{extends } \varkappa \text{ by } \ge 0 \text{ zeros}} F[\varkappa']. \quad (7.28) $$

Using this definition, we split the second term in the r.h.s. of (7.27) into (a) the contributions of sequences $(1, 1, 0)$, $(2, 0)$, their permutations and extensions by zero and (b) sequences with $\sum k_i \ge 3$ which form $\Delta_3^{(2)}$. We get

$$ \Delta_{2;6R}^{(2)} = 2C_{\gamma}C^4 b_1 b_1^R + 2F_{\text{ext}}[(2,0)] + 3F_{\text{ext}}[(1,1,0)] + \Delta_3^{(2)}. \quad (7.29) $$

It is shown in appendix F, see eq. (F.11), that, in the assumptions of Lemma 7.2,

$$ F_{\text{ext}}[(k_i)_1^n] \le 4C^{k+1}A\delta^{k+m}, \quad (7.30) $$

where $k = \sum k_i$ and $m$ is the number of zeros in the sequence $(k_i)_1^n$.

Using (7.15) for $\Delta_3^{(2)}$, the basic estimates $b_1^R \le A_1^R \delta^2$, $C_\gamma C A \le 1/2$, and (7.30) we get

$$ \Delta_{2;6R}^{(2)} \le (C^3 A_1^R + [8C^3 + 12C^3 + 2(2C)^4]A)\delta^3, \quad (7.31) $$

so that

$$ \|(e_u(y))_{6R}\|_w \le \gamma^{-\text{Re} D_6} \delta^3 \{A_2^R + 2C^4 A + C^3 A_1^R + [20C^3 + 2(2C)^4]A\}, \quad (7.32) $$

which is smaller than $\gamma^{-\bar{D}} A_2^R \delta^3$, provided that

$$ (\spadesuit) \quad A_2^R + 2C^4 A + C^3 A_1^R + [20C^3 + 2(2C)^4]A \le \gamma^{\text{Re} D_6 - \bar{D}} A_2^R. \quad (7.33) $$

### 7.3 Case $\ell = 4R$

From the definitions and the bounds on $R_{4R}^{\ell_1, \dots, \ell_n}$ we find that

$$ \|(e_u(y))_{4R}\|_w \le \gamma^{-\text{Re} D_4 - 1} \|u_{4R}\|_w + \gamma^{-\text{Re} D_4} \sum_{(\ell_i)_1^n \ne (4\text{L}), (4\text{R})} \rho_4[(\ell_i)_1^n], \quad (7.34) $$

(the condition $(\ell_i)_1^n \ne (4\text{L})$ comes from the fact that $R_{4R}^{4L}$ is identically zero, see the first of (5.27) and the definition of $T_{4R}^4$ in section 5.2.2; note in particular that, by construction, $T_{4R}^{4,0}$ annihilates the local quartic kernel associated with $H_{4L}$) so that

$$ \|(e_u(y))_{4R}\|_w \le \gamma^{-\text{Re} D_4 - 1} [A_1^R \delta^2 + \gamma \Delta_1^{(1)} + \gamma \Delta_1^{(2)}], \quad (7.35) $$
which gives

$$ \|(e_u(y))_{4R}\|_w \le \gamma^{-\mathrm{Re} D_4-1} \delta^2 [A_1^R + \gamma A(2C^3) + \gamma C_0 A]. \quad (7.36) $$

This is smaller than $\gamma^{-\bar{D}}A_1^R\delta^2$, provided that

$$ (\spadesuit) \quad A_1^R + \gamma A(2C^3 + C_0) \le \gamma^{\mathrm{Re} D_4+1-\bar{D}} A_1^R. \quad (7.37) $$

## 7.4 Case $\ell = 2R$

From the definitions and the bounds on $R_{2R}^{\ell_1, \dots, \ell_n}$ we find that

$$ \|(e_u(y))_{2R}\|_w \le \gamma^{-\mathrm{Re} D_2-2} \|u_{2R}\|_w + \gamma^{-\mathrm{Re} D_2} \sum_{(\ell_i)^n \ne (2L), (2R), (4L)} \rho_2[(\ell_i)^n], \quad (7.38) $$

(the conditions $(\ell_i)^n \ne (2L), (4L)$ come from the fact that $R_{2R}^{2L}$ and $R_{2R}^{4L}$ are identically zero, see the first of (5.27) and the definition of $T_{2R}^2$ in section 5.2.2; note in particular that, by construction, $T_{2R}^{2,0}$ annihilates the local quadratic kernels associated with $H_{2L}$ and with $S_2^{4L}(H_{4L})$ so that

$$ \|(e_u(y))_{2R}\|_w \le \gamma^{-\mathrm{Re} D_2-2} [A_0^R \delta^2 + \gamma^2 \Delta_{0;2R}^{(1)} + \gamma^2 \Delta_0^{(2)}], \quad (7.39) $$

where

$$ \Delta_{0;2R}^{(1)} = C^2 b_1^R + \Delta_1^{(1)} \le C^2 A_1^R \delta^2 + A \delta^2 (2C_3^3). \quad (7.40) $$

Therefore,

$$ \|(e_u(y))_{2R}\|_w \le \gamma^{-\mathrm{Re} D_2-2} \delta^2 [A_0^R + \gamma^2(C^2 A_1^R + 2C^3 A + C_0 A)]. \quad (7.41) $$

This is smaller than $\gamma^{-\bar{D}}A_0^R\delta^2$, provided that

$$ (\spadesuit) \quad A_0^R + \gamma^2(C^2 A_1^R + 2C^3 A + C_0 A) \le \gamma^{\mathrm{Re} D_2+2-\bar{D}} A_0^R. \quad (7.42) $$

## 7.5 $e_\nu^{(0)}$

From the definitions and the bounds on $R_{2L}^{\ell_1, \dots, \ell_n}$ we find that

$$ |e_\nu^{(0)}(y)| \le \gamma^{-\mathrm{Re} D_2} \sum_{(\ell_i)^n \ne (2L), (2R), (4L)} \rho_2[(\ell_i)^n], \quad (7.43) $$

(the conditions $(\ell_i)^n \ne (2L), (4L)$ come directly from the definition of $e_\nu^{(0)}$, see (5.29) and (5.30), while $(\ell_i)^n \ne (2R)$ comes from the fact that $R_{2L}^{2R}$ is identically zero, see the first of (5.27) and the definition of $T_{2L}^2$ in section 5.2.2; note in particular that, by construction, $T_{2L}^{2,0}$ annihilates the nonlocal quadratic kernel associated with $H_{2R}$ so that

$$ |e_\nu^{(0)}(y)| \le \gamma^{-\mathrm{Re} D_2} [\Delta_{0;2R}^{(1)} + \Delta_0^{(2)}], \quad (7.44) $$

which gives

$$ |e_\nu^{(0)}(y)| \le \gamma^{-\mathrm{Re} D_2} \delta^2 [C^2 A_1^R + 2C^3 A + C_0 A]. \quad (7.45) $$

We thus get the first of (6.5), with

$$ E_0 = \gamma^{-\mathrm{Re} D_2} [C^2 A_1^R + 2C^3 A + C_0 A]. \quad (7.46) $$
## 7.6 $e_{\lambda}^{(0)}$

From the definitions and the bounds on $R_{4L}^{\ell_1, \dots, \ell_n}$ we find that

$$|e_{\lambda}^{(0)}(y)| \le \gamma^{-\mathrm{Re} D_4} \sum_{(\ell_i)_1^n \ne (4\mathrm{L}), (4\mathrm{R}), (6\mathrm{SL}), (4\mathrm{L}, 4\mathrm{L}), (4\mathrm{L}, 2\mathrm{L}), (4\mathrm{R}, 2\mathrm{L})} \rho_4[(\ell_i)_1^n], \quad (7.47)$$

(the conditions $(\ell_i)_1^n \ne (4\mathrm{L}), (6\mathrm{SL}), (4\mathrm{L}, 4\mathrm{L})$ come directly from the definition of $e_{\lambda}^{(0)}$, see (5.29) and (5.30), while $(\ell_i)_1^n \ne (4\mathrm{R}), (4\mathrm{L}, 2\mathrm{L}), (4\mathrm{R}, 2\mathrm{L})$ come from the fact that $R_{4\mathrm{L}}^{4\mathrm{R}}, R_{4\mathrm{L}}^{4\mathrm{L},2\mathrm{L}}$ and $R_{4\mathrm{L}}^{4\mathrm{R},2\mathrm{L}}$ are identically zero, see the first of (5.27) and the definition of $T_{4\mathrm{L}}^4$ in section 5.2.2; note in particular that, by construction, $T_{4\mathrm{L}}^{4,0}$ annihilates the nonlocal quartic kernels associated with $H_{4\mathrm{R}}, S_4^{4\mathrm{L},2\mathrm{L}}(H_{4\mathrm{L}}, H_{2\mathrm{L}})$ and $S_4^{4\mathrm{R},2\mathrm{L}}(H_{4\mathrm{R}}, H_{2\mathrm{L}})$ so that

$$|e_{\lambda}^{(0)}(y)| \le \gamma^{-\mathrm{Re} D_4} [\Delta_{1;\lambda}^{(1)} + \Delta_{1;\lambda}^{(2)}], \quad (7.48)$$

where

$$\Delta_{1;\lambda}^{(1)} = C^3 b_2^R + \Delta_2^{(1)} \le C^3 A_2^R \delta^3 + A \delta^3 (2C^4), \quad (7.49)$$

$$\Delta_{1;\lambda}^{(2)} = 2C_{\gamma}C^4 b_1 b_1^R + \sum_{(k_i)_1^n \ne (1,1)}^{n \ge 2} F_2[(k_i)_1^n]. \quad (7.50)$$

The sequences with $\sum k_i = 1$ such as $(1,0), (1,0,0)$, etc are excluded from the second term because $b_0$ insertions then happen on the external legs of a quartic interaction, and they give rise to a vertex with a vanishing local part. We see that $\Delta_{1;\lambda}^{(2)}$ is identical to (7.27) and therefore satisfies the same bound (7.31)

$$\Delta_{1;\lambda}^{(2)} \le (C^3 A_1^R + [8C^3 + 12C^3 + 2(2C)^4]A)\delta^3, \quad (7.51)$$

Therefore we get

$$|e_{\lambda}^{(0)}(y)| \le \gamma^{-\mathrm{Re} D_4} \delta^3 [C^3 A_2^R + 2C^4 A + C^3 A_1^R + [20C^3 + 2(2C)^4]A]. \quad (7.52)$$

We thus get the second equation of (6.5), with

$$E_1 = \gamma^{-\mathrm{Re} D_4} [C^3 A_2^R + 2C^4 A + C^3 A_1^R + [20C^3 + 2(2C)^4]A]. \quad (7.53)$$

## 7.7 Possibility of all choices

Finally, we need to show that all the ♠-constraints above can be satisfied consistently: eqs. (7.9), (7.13), (7.19), (7.33), (7.37), (7.42). To write them in a more manageable form, let us replace all $\gamma$-independent constants in the l.h.s. of the ♠-constraints by their maximum $\bar{C}$ (Recall that $C_0$ was fixed in terms of $C$ in (7.15)). Also let $\bar{C}_{\gamma} = \max(C_{\gamma}, C_{\gamma 3})$. Finally let $Z$ be the minimal of the exponents of $\gamma$ in the r.h.s. of (7.19), (7.33), (7.37), (7.42) over $\varepsilon \in T$:

$$Z = \min_{\varepsilon \in T} \{\mathrm{Re} D_8 - \bar{D}, d/2 - \mathrm{Re} \varepsilon, \mathrm{Re} D_6 - \bar{D}, \mathrm{Re} D_4 + 1 - \bar{D}, \mathrm{Re} D_2 + 2 - \bar{D}\}. \quad (7.54)$$
Crucially $Z > 0$ by the assumption on $T$ and the definition of $\bar{D}$. We then get the following list of constraints which, if satisfied, imply the ♠-constraints for any $\varepsilon \in T$:

$$ \bar{C} \le \gamma^Z, \qquad (7.55) $$

$$ \bar{C}\delta_0 \le 1, \quad \bar{C}\bar{C}_\gamma A \le 1, \qquad (7.56) $$

$$ \max(A_0, A_0^R \delta_0, A_1^R \delta_0, \bar{C}_\gamma A_0^2 + A_2^R \delta_0) \le A/2, \qquad (7.57) $$

$$ A_2^R + \bar{C}(A_1^R + A) \le \gamma^Z A_2^R, \quad A_1^R + \bar{C}\gamma A \le \gamma^Z A_1^R, \quad A_0^R + \bar{C}\gamma^2(A_1^R + A) \le \gamma^Z A_0^R. \qquad (7.58) $$

The only remaining varying parameter is $\gamma$. We should now choose $\gamma_{\text{key}}$ and $\delta_0, A_0, \{A_k^R\}_{k=0,1,2}, A, E_0, E_1$, which are $\gamma$-dependent and positive, so that all these constraints hold for $\gamma \ge \gamma_{\text{key}}$.

We can satisfy the first two lines taking $\gamma$ large, then $A$ and $\delta_0$ small (in this order, because $\bar{C}_\gamma$ depends on $\gamma$). The remaining constraints are a bit more subtle because $A$ and $A_k^R$ occur both in the l.h.s. and in the r.h.s. To satisfy (7.58) we will require:

$$ A_1^R, A \le A_2^R, \quad \gamma A \le A_1^R, \quad \gamma^2 A, \gamma^2 A_1^R \le A_0^R, \qquad (7.59) $$

$$ 1 + 2\bar{C} \le \gamma^Z, \quad 1 + 2\bar{C} \le \gamma^Z, \quad 1 + \bar{C} \le \gamma^Z. \qquad (7.60) $$

The last three constraints on $\gamma$ are of the same type as (7.55). Joining inequalities in (7.59) to (7.57), the resulting set of constraints reduces to:

$$ A_0 \le 0.5A, \quad \bar{C}_\gamma A_0^2 \le 0.25A, \qquad (7.61) $$

$$ A_2^R \in [A, 0.25\delta_0^{-1}A], \quad A_1^R \in [\gamma A, 0.5\delta_0^{-1}A], \quad A_0^R \in [\gamma^2 A, 0.5\delta_0^{-1}A], \qquad (7.62) $$

$$ A_1^R \le A_2^R, \quad \gamma^2 A_1^R \le A_0^R. \qquad (7.63) $$

Here's then the final order in which all choices have to be made: $\gamma_{\text{key}}$ is chosen as the minimal $\gamma \ge 2$ satisfying (7.55) and (7.60). We then pick any $\gamma \ge \gamma_{\text{key}}$ and compute the constant $\bar{C}_\gamma$. We then satisfy (7.56) by choosing:

$$ A = (\bar{C}\bar{C}_\gamma)^{-1}. \qquad (7.64) $$

We then choose $A_0$ sufficiently small to satisfy (7.61). Finally, we choose

$$ \delta_0 = \min(\bar{C}^{-1}, 1/(2\gamma^3)), \qquad (7.65) $$

which satisfies (7.56) and at the same time, thanks to $\delta_0 \le 1/(2\gamma^3)$, allows us to choose $A_k^R$ as follows:

$$ A_2^R = 0.25\delta_0^{-1}A, \quad A_1^R = \gamma A, \quad A_0^R = 0.5\delta_0^{-1}A. \qquad (7.66) $$

Then (7.62) is satisfied, and (7.63) holds as well. Key lemma is proved.

## 8 Discussion and open problems

In this paper, we discussed what is perhaps the simplest theoretical model to study field-theoretic non-Gaussian fixed points, which is amenable to rigorous analysis: **symplectic**
**fermions with a long-range kinetic term and local quartic interaction.** Our model is translation and rotation invariant, and the structure of the RG equations is quite similar to models with local kinetic term. This makes our model more realistic than, for example, models with hierarchical interactions (see [77] for an introduction).

Our model depends on 3 physical parameters: the number of dimensions $d$, the number of fermion species $N$ (assuming $\text{Sp}(N)$ invariance), and a parameter $\varepsilon$ in the long-range fermion propagator, which controls the relevance of the quartic fermion interaction. For $0 < \varepsilon \ll 1$ this interaction is weakly relevant, and the beta-function equation for the quartic coupling $\lambda$ takes the forms $\beta_\lambda = -\varepsilon\lambda + \text{const} \cdot (N-8)\lambda^2 + \dots$. One thus has the right to expect that, for $N \neq 8$, there exists an RG fixed point with $\lambda = O(\varepsilon)$. Our main result (theorem 6.1) establishes the existence of this fixed point rigorously and non-perturbatively.

Although the path towards this rigorous result was somewhat long, most of the ingredients are rather natural. We introduce an infinite-dimensional Banach space of interactions, whose kernels are essentially local (have to decay very fast at point separation). We work with a smooth momentum space cutoff, so that the UV and IR-cutoff fermion propagator decays very fast in position space, and the almost-locality of the interaction is preserved by an RG step.

An essential feature of our model is that a single RG step leads to a convergent effective action (for weak coupling). Intuitively, this property of fermionic models is due to the Pauli principle, or, equivalently, to fermionic signs leading to cancellations between Feynman diagrams. The formal derivation is somewhat delicate, and we review it pedagogically in appendix D. This is standard in the constructive field theory community, but may appear unexpected to the others. A related detail is that exhibiting these fermionic cancellations requires considering a finite RG step with a rescaling parameter $\gamma > 1$. That's what we do in this paper, as opposed to performing continuous RG à la Polchinski's equation (see remark 5.4).

With these ingredients, we show that the Wilsonian RG map is a well-defined nonlinear operator in the Banach space of interactions, and is a contraction (has derivative whose operator norm is less than 1) along the irrelevant directions. The behavior along the mass direction $\nu$ and the quartic $\lambda$ has to be analyzed separately. These directions are both relevant at the linearized level, with $\lambda$ becoming irrelevant near the approximate one-loop fixed point. Rigorous bounds on error terms show that these statements remain true at the nonlinear level, at weak coupling. The proofs of these results rely just on some elementary combinatorics, geometric series convergence, and chasing $\gamma^{-D_l}$ factors suppressing the irrelevant interactions. Given one relevant and infinitely many irrelevant directions, the fixed point equation can then be rigorously solved (for $\varepsilon$ sufficiently small) via a variant of Newton's method, appealing to the Banach fixed point theorem.

Although our main interest is in $\varepsilon$ real and positive, in which case the fixed point can be thought of as the IR fixed point of an RG flow originating at the gaussian theory, our methods apply as long as $\varepsilon$ is small and nonzero. E.g. we can also consider $\varepsilon < 0$. In this case the quartic interaction is irrelevant around the gaussian theory, but relevant around the fixed point whose existence we can prove (which in this case is classified as a UV fixed point). We can also consider complex nonzero $\varepsilon$. Although perhaps lacking clear physical
meaning, we can use this as a formal device to show that the fixed point is analytic in $\epsilon$ in a punctured disk around the origin, and thus in the whole disk by Riemann's removable singularity theorem (section 6.5). This is a dramatic conclusion, which implies that our fixed points can be obtained via the convergent perturbative $\epsilon$-expansion around $\epsilon = 0$.

## 8.1 Open questions

We will now list many open questions raised by our work. Some of them are theoretical, while others have potential practical applications to numerical calculations of critical exponents.

### 8.1.1 Extensions to other nonlocal models

It should be relatively easy to extend our results to many other similar models:

* Models with a symmetry group $G \subset \text{Sp}(N)$, which have several mass terms and quartic couplings consistent with this symmetry. One should be able to find a non-perturbative fixed point in a neighborhood of any isolated solution to the one-loop beta-function equations (as long as all quartic directions have eigenvalues $O(\epsilon)$, the condition which generalizes non-vanishing one-loop beta-function used in this work).

* Models where different species of fermions have different propagator scaling (different $\epsilon$). This may include models where some fermions $\psi$ have local kinetic terms (and thus a fixed scaling dimension for a given $d$), while others $\psi'$ are long-range with tunable dimensions, so that the interaction $\psi^2(\psi')^2$ can be made near-marginal.

* Models with a vanishing one-loop beta function, like our model with $N = 8$. As discussed in appendix G, the two-loop beta-function term $\lambda^3$ has a nonzero coefficient [78], giving a perturbative fixed point with $\lambda = O(\sqrt{\epsilon})$. The non-perturbative existence of such a fixed point and its analyticity properties in $\epsilon$ can be understood almost immediately using the tree expansion method described in appendix J, and a contraction argument should also be possible.

* Our model in $d = 4$. Compared to $d \in \{1, 2, 3\}$ treated here, the local term $(\partial\psi)^2$ would be (weakly) relevant for $\epsilon > 0$. One thus has to treat it on equal footing with the local $\psi^2$ and $(\psi^2)^2$ terms. One should be able to construct a non-perturbative fixed point for small $\epsilon$, helped by the fact that the new coupling is quadratic in $\psi$. This would be the fermionic analogue of the bosonic problem considered in [22].

* Models where the sextic or higher power $(\psi^2)^p$ is near-marginal, i.e. $[\psi] \approx d/(2p)$, $p \ge 3$.

More ambitiously, time may be ripe for a “general theory of fermionic fixed points with scale-invariant kinetic terms (local or long-range) and near-marginal local polynomial interactions”. One should be able to prove that any such fixed point showing up in perturbative analysis exists non-perturbatively, rather than writing a new paper for each particular model. The main challenge is to choose an efficient notation, and to cleanly separate the
algebraic and analytic aspects of the problem.<fnref n="33" /> This future general theory should cover all the above examples, as well as fermionic fields transforming in other rotation representations (e.g. spinors [27]), and even non-rotationally invariant (Lifshitz-type) fixed points having anisotropic scaling.

### 8.1.2 Further properties of the RG fixed points

Here we proved that the RG fixed points exist, and established a few of there basic properties such as $\gamma$-independence (modulo some loose ends), and analyticity in $\varepsilon$. Future work should investigate several other interesting properties, such as:

* Uniqueness of the fixed point as an equivalence class of interactions (i.e. uniqueness of solutions of the general fixed point equation; see remark 5.3);

* Dependence of the fixed point on the UV cutoff function $\chi$ (see remark 6.5). In spite of this dependence, the critical exponents are expected to be $\chi$-independent. It is instructive to compare the family of long-range models discussed here with one-parameter families of short-range fixed points, such as the Ashkin-Teller model, 6- and 8-vertex models, and interacting dimer models (see e.g. [35, 39]). In the latter case, the deformation parameter is an exactly marginal coupling, which can renormalize along the RG flow, and so the critical exponents depend on the microscopic details, although if one critical exponent is known, others can be expressed via it (the so called weak universality). In our case, $\varepsilon$ is not a coupling but a parameter controlling the nonlocal part of the action, so it does not renormalize. Therefore, the situation is similar to the usual universality, and all critical exponents should be universal functions of $\varepsilon$ independent of microscopic details such as the UV cutoff $\chi$.<fnref n="34" /> It would be interesting to establish this rigorously. See [81] for a classic intuitive discussion of these issues, in the context of local models.

* Critical exponents. These can be defined, most generally, as eigenvalues of the RG transformation linearized around the fixed point (removing the eigenvalues corresponding to the “redundant operators” [81]).<fnref n="35" /> From the densities of the corresponding eigenvectors, one should be able to define the “scaling operators”, whose correlation functions with respect to the fixed point interaction have exact scale invariance. One can also study correlation functions of simple operators such as $\psi^2$. While not exactly scale invariant, they should become so at asymptotically long distances.

<footnote n="33">³³One may be inspired by how somewhat similar difficulties have been solved for nonlinear stochastic partial differential equations, another problem which involves renormalization [79, 80].</footnote>

<footnote n="34">³⁴Note in this respect that the IR scaling dimension of $\psi$ is exactly known and equal to its UV dimension $d/4 - \varepsilon/2$. Therefore the exponent $\eta$ is trivially known as a function of $\varepsilon$. Even for weak universality, all exponents can be found if one exponent is known, making the conclusion that in our situation all exponents are universal functions of $\varepsilon$ less surprising.</footnote>

<footnote n="35">³⁵Sometimes this is equivalently expressed by introducing perturbing “source terms” and studying their beta-functions.</footnote>
* Full RG trajectory. By this we mean the theory which interpolates between the gaussian fixed point at short distances and the non-gaussian fixed point at long distances (for $\epsilon > 0$, while for $\epsilon < 0$ it is the other way around).

### 8.1.3 Conformal invariance

The RG fixed points constructed here are expected to be conformally invariant, based on the same intuitive arguments as for the long-range bosonic models [82]. Conformal invariance means the invariance of correlation functions of scaling operators (see above) under the finite-dimensional conformal group $SO(d+1, 1)$. For $d=1$ these are Möbius transformations, and for $d=2$ the product of holomorphic and antiholomorphic Möbius transformations.<fnref n="36" /> This invariance also implies correspondence between correlation functions in infinite volume as studied here, and correlation functions on a sphere of finite radius (which for $d=1$ is just a circle with periodic boundary conditions), putting the two manifolds in correspondence via the stereographic projection. Such properties are expected to be generally true based on intuitive physics arguments, and it would be very interesting to see how they emerge rigorously in an explicit model such as ours. In particular, this would provide the first rigorous non-gaussian conformal theory in $d=3$.<fnref n="37" />

Conserved stress tensor operator plays key role in intuitive discussions of conformal invariance of local theories. Our model being nonlocal (long-range), it does not possess a local stress tensor in $d$ dimensions. One way around this difficulty is to represent the nonlocal kinetic term as arising from a local quadratic action in the $(d+1)$-dimensional Anti-de-Sitter (AdS) space, of which the $d$-dimensional space is the boundary, where the quadratic, quartic, and all the irrelevant interactions are localized. This construction is useful for intuitive understanding of conformal invariance (as discussed for bosonic models in [82]), and perhaps also for proving it rigorously.<fnref n="38" />

A key property of local conformally invariant theories is the convergent Operator Product Expansion (OPE).<fnref n="39" /> Though nonlocal (long-range), our model also should have this property due to the local AdS representation.<fnref n="40" /> It would be very interesting to establish this rigorously. This appears somewhat nontrivial due to the fact that the scaling operators, introduced as densities of linearized RG eigenvectors (see above), will not be exactly local but “mildly nonlocal”, with kernels of stretched-exponential decay. It is puzzling why this mild nonlocality does not invalidate the usual intuitive arguments for the OPE, which

<footnote n="36">³⁶Because of the nonlocal kinetic term, there will be no invariance under more general holomorphic transformations, unlike in the case of fixed points of fully local models.</footnote>

<footnote n="37">³⁷In the last 20 years, starting with Smirnov [83], there was significant progress in showing rigorously conformal invariance of various critical observables of specific 2d lattice models (see [84] for review). Many of these models are exactly solvable in infinite volume, and the main challenge is to show conformal invariance of correlators defined in an arbitrary planar region (see e.g. [85] for the 2d Ising model). A key method used in these works is discrete holomorphicity, which is limited to 2d and to specific models, while RG does not play much of a role. The proof of conformal invariance of our RG fixed points will require very different methods, which should work for any $d$.</footnote>

<footnote n="38">³⁸On the contrary it is probably hard to make rigorous sense of the Caffarelli-Silvestre construction from [82], where the higher-dimensional ambient space is flat, but it has non-integer dimension.</footnote>

<footnote n="39">³⁹See [86] for an introduction for physicists, and [87] for a more mathematical one.</footnote>

<footnote n="40">⁴⁰Bosonic cousins of our model have been studied via the numerical conformal bootstrap in [88].</footnote>
treat scaling operators as living at a point. Somehow, the mild nonlocalities of the scaling
operators and of the fixed-point interaction should conspire to produce a fully local OPE.
Note that this issue is not specific to our model with a long-range kinetic term, as mild
nonlocality of scaling operators would be present also for fully local models such as the 3D
Ising model.

8.1.4 Relations with analytic regularization

Analytic dependence of our fixed point on ε implies that the critical exponents should also be analytic in ε. Analyticity of the correlation-length and correction-to-scaling exponents ν and ω follows easily from the tree expansion construction (appendix J), as they can be computed by linearizing the analytic right sides of (J.12) near the fixed point.<fnref n="41" /> The exponent η is trivial in our model due to the absence of wavefunction renormalization. Higher exponents may have a subtle analytic structure because of degeneracies of linearized RG eigenvalues at ε = 0. Since our model is non-unitary, some higher critical exponents may become complex even for real ε, forming complex-conjugate pairs.<fnref n="42" />

As already mentioned, it would be interesting to show that the critical exponents are
χ-independent. Another problem is to prove rigorously that our critical exponents agree
with perturbative techniques by which these exponents are computed in theoretical physics.
This is especially interesting given that, as we have shown, perturbation theory converges
in the problem at hand.

In theoretical physics, higher-order perturbative computations of critical exponents are usually done by working with a bare action containing only the relevant and nearly-marginal couplings. This uses the fact that, due to the renormalizability of the theory at short distances, one can always find an RG trajectory leading to the fixed point from such a UV theory where all irrelevant couplings are set to zero. Furthermore, theoretical physics calculations are greatly simplified by choosing a “mass-independent regularization scheme”, which allows to simply set the mass terms to zero. Examples of such schemes are dimensional regularization and analytic regularization with minimal subtraction, which amount to analytically continue Feynman diagrams in ε, dropping the poles. It is universally believed that any scheme, and in particular a mass independent one, should give the same power series in ε for the critical exponents, but to our knowledge this has never been discussed in full rigor.

8.1.5 Increasing the range of ε

Existence proofs of fixed points in this paper work for |ε| ≤ ε₀(γ). We have not attempted to evaluate ε₀(γ) explicitly, although it would be straightforward to do this, following step-by-step our arguments. This may be a good exercise for someone wishing to understand our methods in depth. Both of our methods (contraction and the tree expansion) can be

<footnote n="41">⁴¹These series have been computed, via another regulator, for the bosonic $O(N_b)$ long-range models in [78] up to three loops. We thank Dario Benedetti for sharing a Mathematica notebook. Fermionic series should be obtainable by setting $N_b = -N$. These series are not sufficiently long to test our claim that they are convergent in the fermionic case.</footnote>

<footnote n="42">⁴²This is similar to how some higher Wilson-Fisher critical exponents become complex in 4 − ε dimensions [89].</footnote>
optimized to enlarge the range of $\epsilon$ where the fixed point is under control. One simple strategy is to increase the number of terms which are computed explicitly, or estimated more carefully than what is currently done. For $\epsilon$ of order 1, one might have to resort to computer-assisted methods.<fnref n="43" />

An interesting feature which might be revealed by such exploration is the cross-over to the short-range universality class. Namely consider the local symplectic fermion model with the bare action (cf (1.1))

$$ \int d^d x (\Omega_{ab} \partial_\mu \psi^a \partial^\mu \psi^b + \nu \psi^2 + \lambda \psi^4). \qquad (8.1) $$

Some literature concerning such models was cited in the Introduction. This model is expected to flow to a non-gaussian fixed point for $d = 3$ (although, by the usual arguments, not for $d \le 2$). This fixed point is strongly coupled, and we cannot access it using the techniques of this paper.<fnref n="44" /> Physicists study such fixed points by the usual Wilson-Fisher $\epsilon$-expansion working formally in $d = 4 - \epsilon$ and then resumming the series at $\epsilon = 1$. [We will use $\epsilon$ to denote $4 - d$ as opposed to the long-range parameter $\epsilon$.] As mentioned in appendix G, $\epsilon$-expansion for these models is perturbatively equivalent to the $\epsilon$-expansion of bosonic $O(N_b)$ models with $N_b = -N$.

So for $d = 3$ we have a family of long-range fixed points studied here whose critical exponents depend on $\epsilon$, and the fixed point of (8.1) which we will call “short-range”. The scaling dimension of $\psi$ is $[\psi]_{LR}(\epsilon) = d/4 - \epsilon/2$ in our models, while it is $[\psi]_{SR} = d/2 - 1 + \eta_{SR}/2$ at the fixed point of (8.1). The short-range $\eta_{SR}$ is given by $\eta_{SR} = \epsilon^2 \frac{2+N_b}{2(8+N_b)^2} + O(\epsilon^3)$, $N_b = -N$, with the series which needs to be Borel-resummed at $\epsilon = 1$.<fnref n="45" />

The subsequent discussion applies for any $N$ for which $[\psi]_{SR} < d/4$, as appears to be the case at least for $N = 4$ (see footnote 45). For such $N$, we will have $[\psi]_{LR}(\epsilon) = [\psi]_{SR}$ for $\epsilon = \epsilon_* = 2(d/4 - [\psi]_{SR}) > 0$. It can then be conjectured that, for $\epsilon = \epsilon_*$, the long-range to short-range crossover will take place. Namely, the long-range fixed point at $\epsilon = \epsilon_*$ should become identical to the short-range fixed-point plus a non-interacting gaussian theory of an $Sp(N)$ symplectic fermion $\zeta$ of scaling dimension $d - [\psi]_{LR}(\epsilon_*)$. This would be analogous to the bosonic long-range models, for which such a crossover has been studied since a long time theoretically (starting in [93, 94], reviewed in [95], section 4.3) and is supported by Monte Carlo simulations [96, 97]. The extra gaussian field $\zeta$ is expected by the same arguments as in [58, 98] for the bosonic case.<fnref n="46" /> Note that the operator $\psi\zeta$ is marginal for $\epsilon = \epsilon_*$ (it

<footnote n="43">⁴³Inspired by Lanford’s construction of the Feigenbaum fixed point [90].</footnote>

<footnote n="44">⁴⁴We could still prove a result like Key Lemma 6.1, but we would not be able to derive the Fixed Point Theorem 6.1, for lack of a small parameter analogous to $\epsilon$. Perhaps a computer-assisted method could help.</footnote>

<footnote n="45">⁴⁵This series is known up to 7 loops [91, 92], see [6] for the earlier 6 loop results. It is tempting to speculate, by analogy with the long-range case, that the $\epsilon$-expansion series have a finite radius of convergence for negative $N_b$, while they are known to be only Borel-summable for positive $N_b$. Numerically, the 6-loop series for $\nu_{SR}, \eta_{SR}, \omega_{SR}$ for $N_b = -4$ seem to be remarkably well behaved. E.g. $\eta_{SR}(N_b = -4) = -0.25\epsilon^3 - 0.25\epsilon^2 - 0.535957\epsilon^4 - 1.25122\epsilon^5 - 3.14893\epsilon^6$. We are grateful to Kay Wiese for communicating this to us.</footnote>

<footnote n="46">⁴⁶We will give three reasons: (1) Since the LR (long-range) theory is nonlocal, the theory to which it crosses over cannot be fully local. (2) (Counting of degrees of freedom) The leading spin 2 operator is not conserved in the LR theory. At the crossover its dimension goes to $d$, which is the stress-tensor dimension of</footnote>