#  Numerical fluid dynamics for FRG flow equations:
Zero-dimensional QFTs as numerical test cases.
I. The $O(N)$ model

Adrian Koenigstein<EMAIL_ADDRESS>Institut für
Theoretische Physik, Goethe University,
Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany  Martin J. Steil
<EMAIL_ADDRESS>Technische Universität Darmstadt,
Department of Physics, Institut für Kernphysik, Theoriezentrum,
Schlossgartenstraße 2, D-64289 Darmstadt, Germany  Nicolas Wink
<EMAIL_ADDRESS>Institut für Theoretische Physik, University
Heidelberg,
Philosophenweg 16, D-69120 Heidelberg, Germany  Eduardo Grossi
<EMAIL_ADDRESS>Center for Nuclear Theory, Department of Physics
and Astronomy,
Stony Brook University, Stony Brook, NY 11794, U.S.A.  Jens Braun
<EMAIL_ADDRESS>Technische Universität Darmstadt, Department
of Physics, Institut für Kernphysik, Theoriezentrum,
Schlossgartenstraße 2, D-64289 Darmstadt, Germany  Helmholtz Research Academy
Hesse for FAIR, Campus Darmstadt,
D-64289 Darmstadt, Germany  ExtreMe Matter Institute EMMI, GSI,
Planckstraße 1, D-64291 Darmstadt, Germany  Michael Buballa
<EMAIL_ADDRESS>Technische Universität Darmstadt,
Department of Physics, Institut für Kernphysik, Theoriezentrum,
Schlossgartenstraße 2, D-64289 Darmstadt, Germany  Helmholtz Research Academy
Hesse for FAIR, Campus Darmstadt,
D-64289 Darmstadt, Germany  Dirk H. Rischke<EMAIL_ADDRESS>frankfurt.de Institut für Theoretische Physik, Goethe University,
Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany  Helmholtz Research
Academy Hesse for FAIR, Campus Riedberg,
Max-von-Laue-Straße 12, D-60438 Frankfurt am Main, Germany

###### Abstract

The functional renormalization group (FRG) approach is a powerful tool for
studies of a large variety of systems, ranging from statistical physics over
the theory of the strong interaction to gravity. The practical application of
this approach relies on the derivation of so-called flow equations, which
describe the change of the quantum effective action under the variation of a
coarse-graining parameter. In the present work, we discuss in detail a novel
approach to solve such flow equations. This approach relies on the fact that
RG equations can be rewritten such that they exhibit similarities with the
conservation laws of fluid dynamics. This observation can be exploited in
different ways. First of all, we show that this allows to employ powerful
numerical techniques developed in the context of fluid dynamics to solve RG
equations. In particular, it allows to reliably treat the emergence of non-
analytic behavior in the RG flow of the effective action as it is expected to
occur in studies of, e.g., spontaneous symmetry breaking. Second, the analogy
between RG equations and fluid dynamics offers the opportunity to gain novel
insights into RG flows and their interpretation in general, including the
irreversibility of RG flows. We work out this connection in practice by
applying it to zero-dimensional quantum-field theoretical models. The
generalization to higher-dimensional models is also discussed. Our findings
are expected to help improving future FRG studies of quantum field theories in
higher dimensions both on a qualitative and quantitative level.

Functional Renormalization Group, conservation laws, numerical fluid dynamics,
$O(N)$ model, zero-dimensional QFT

###### Contents

  1. I Introduction
  2. II The Functional Renormalization Group – an introduction in zero dimensions
    1. II.1 The partition function in zero dimensions
    2. II.2 Solving integrals with flow equations
      1. II.2.1 The scale-dependent partition function
      2. II.2.2 A flow equation for the scale-dependent partition function
    3. II.3 The Functional Renormalization Group equation
      1. II.3.1 The scale-dependent Schwinger functional
      2. II.3.2 The scale-dependent effective action
      3. II.3.3 The Exact Renormalization Group equation
    4. II.4 Contextualization with FRG in higher-dimensional space-time
    5. II.5 $n$-point correlation functions
  3. III The $O(N)$ model in zero dimensions and its treatment within the FRG
    1. III.1 The zero-dimensional $O(N)$ model
    2. III.2 Symmetry restoration during the RG flow
    3. III.3 FRG formulation and flow equations
      1. III.3.1 The exact RG flow equation of the zero-dimensional $O(N)$ model
      2. III.3.2 FRG Taylor (vertex) expansion of the $O(N)$ model
  4. IV FRG flow equations and (numerical) fluid dynamics
    1. IV.1 Conservative form of FRG flow equations – advection-diffusion equations
      1. IV.1.1 The conservative form
      2. IV.1.2 Advection-diffusion equation, irreversibility of RG flows, and entropy production
    2. IV.2 Finite-volume method
    3. IV.3 Kurganov-Tadmor (KT) central scheme
    4. IV.4 Boundary conditions and computational domain in FRG flow equations
      1. IV.4.1 Boundary condition at $\sigma=0$
      2. IV.4.2 Boundary condition at $\sigma\rightarrow\infty$
  5. V Zero-dimensional field theory as testing ground for FRG
    1. V.1 Test case I: Non-analytic initial condition
      1. V.1.1 General discussion of the FRG flow – advection and diffusion
      2. V.1.2 Tests of the spatial resolution $\Delta x$
      3. V.1.3 Tests of the size of the computational domain
      4. V.1.4 Tests of the UV and IR scales
    2. V.2 Test case II: $\phi^{4}$ theory
      1. V.2.1 Results obtained using the KT scheme
      2. V.2.2 Results obtained using the FRG Taylor (vertex) expansion
    3. V.3 Test case III: $\phi^{6}$ potential
    4. V.4 Test case IV: the $\sigma=0$ boundary
  6. VI Conclusions and outlook
  7. A Numerical derivatives
  8. B Coleman-Mermin-Wagner-Hohenberg theorem in zero dimensions: Absence of spontaneous symmetry breaking and of phase transitions
    1. B.0.1 Ehrenfest classification of phase transitions
    2. B.0.2 Landau’s theory of phase transitions
    3. B.0.3 The Coleman-Mermin-Wagner-Hohenberg theorem
    4. B.0.4 Phase transitions during the RG flow

##  I Introduction

In statistical mechanics and quantum field theory (QFT) the central objective
is to compute the expectation values of physical observables from a partition
of probabilities among the various microscopic states of a given model or
theory. On a technical level, the calculation of expectation values oftentimes
corresponds to the evaluation of nested sums (for discrete systems) or
complicated high-dimensional integrals (for continuous systems) in the
framework of partition functions or functional integrals. In most cases such
computations cannot be done analytically. Various methods were developed to
overcome this difficulty. Focusing on high-energy physics, stochastic methods
have been developed to study Quantum Chromodynamics from first principles (see
Refs. [1, 2, 3, 4] for reviews), but also systematic approximation schemes
such as (chiral) perturbation theory (see Refs. [5, 6] for reviews) or the
large-$N$ expansion [7, 8, 9] have been employed, where (at least) parts of
the calculations can still be performed analytically. Within the last decades
non-perturbative holographic and functional methods, such as the AdS/CFT
correspondence [10, 11], Dyson-Schwinger equations (see Ref. [12] for a
review), and the (Functional) Renormalization Group ((F)RG) (see Ref. [13] for
a recent review) have significantly gained importance and nowadays provide a
viable complement to Monte-Carlo simulations and semi-analytic methods.
However, despite great success within various areas of physics, holographic
and functional methods are sometimes still criticized for the lack of
providing reliable systematic and numerical error estimates. In this work, we
will provide important steps to amend this shortcoming for the FRG approach.

Although the mathematical formulation of the FRG approach is in principle
exact, a first source of systematic errors is introduced by the fact that one
has to make certain approximations (truncations) in order to actually perform
calculations. However, since the method is non-perturbative, the
identification of, e.g., a small expansion parameter is challenging, if at all
possible. A lot of work has already been invested into this question, e.g.,
approximation errors can be evaluated by comparing different truncation
schemes and truncation orders against each other [14, 15, 16]. Furthermore,
the comparison with other non-perturbative methods [17], effective field
theories [18, 19, 20, 14, 21], or with Monte-Carlo studies [22, 23, 24, 25]
can provide estimates on the reliability of the results.

A second source of systematic errors arises from the way the RG flow equations
are solved in practice. In recent work by two of us and collaborators [26,
27], it was pointed out that the possible appearance of non-analytic behavior
in field space as well as the influence of the boundary conditions require
great care in the numerical solution of RG flow equations. In particular, it
was shown that these equations can be cast into a conservative form, such that
analogies to fluid-dynamical flow equations become manifest and allow to
access to the highly developed toolbox of numerical fluid dynamics, e.g., in
the case of Refs. [26, 27] including the discontinuous Galerkin method. In
consequence, this suggests that a systematic analysis of the quality of the
different numerical methods to solve RG flow equations as well as an analysis
of the structure of the RG flow equations themselves is in order. The question
of numerical errors in FRG calculations was systematically addressed in Ref.
[26] by a comparison of numerical results with analytically known solutions
for the $O(N)$ model in the large-$N$ limit [28, 29, 30]. Furthermore,
phenomena like shock waves in the derivative of the effective potential along
the field space direction during the RG flow, which are directly related to
phase transitions [26, 27, 31, 32], were resolved and interpreted in a fluid-
dynamical framework.

The goal of the present work is threefold. On the one hand, we will continue
to elaborate on the analogies between RG flow equations and (numeric) fluid
dynamics, including precision and stability tests for numerical schemes. On
the other hand, we will contribute to the ongoing discussion on truncation
schemes of the FRG framework. In addition, this article is supposed to provide
a low-level introduction to the FRG method within the fluid-dynamic mindset
also for non-experts and (under)graduate students.

In order to provide reliable estimates of the precision of numerical methods
and the quality of truncation schemes, the standard approach is to compare
numerical results and/or results from truncations against analytically known
results. However, analytically known results for non-trivial QFTs or
statistical mechanics are scarce. Fortunately, there is a class of non-trivial
QFTs, where either analytic results are known or numerical results can be
easily obtained with arbitrary precision: zero-dimensional QFTs. In this work,
we choose the zero-dimensional $O(N)$ model as a testing ground to
systematically analyze the precision of the numerical methods which are used
to solve the RG flow equations. Furthermore, we will use zero-dimensional QFT
to demonstrate the similarities between RG flow equations and conservation
laws from fluid dynamics (which also generalize to an arbitrary number of
space-time dimensions and different field content). We will elucidate the
different roles played by advective and diffusive contributions in the RG flow
equations as partial differential equations (PDEs). Furthermore, we start a
discussion of the relation between the RG time, entropy production in the RG
flow, the dissipative character of the FRG equation, and the irreversibility
of RG transformations during the RG flow. This discussion is deepened in part
II and III of this series of publications [33, 31].

In order to numerically solve the RG flow equations, in this work we apply the
Kurganov-Tadmor scheme, a finite-volume method which is well-established in
numerical fluid dynamics. We test the accuracy of the FRG results against
direct evaluations of expectation values from the partition function, which
can be calculated to in principle arbitrary precision in zero space-time
dimensions. We note that the RG flow equations arising in the FRG framework
for certain zero-dimensional models, and in particular the $O(N)$ model, are
exact PDEs. Therefore, they do not involve any systematic error of the first
kind mentioned above, namely truncation errors. Possible errors are therefore
solely of the second kind, introduced by the numerical scheme used to solve
the flow equations.

As a next step, we will analyze the FRG Taylor expansion as a truncation to
the FRG approach and contrast our findings with the general properties of the
FRG equation as a non-linear PDE in zero space-time dimensions. In a follow-up
publication, we will also introduce more elaborate zero-dimensional models
including Grassmann numbers (mimicking fermionic degrees of freedom in $d=0$)
[34]. In this context, we will apply the methods developed in the present work
to investigate several truncation schemes by comparing against exact results
for a constructed fermion-boson-model. Generalizing our findings from zero
dimensions to higher-dimensional QFTs is not necessarily trivial.
Nevertheless, we will comment on this issue at various places throughout this
work. We thus hope that this paper will contribute to ongoing debates on
subtleties of the RG flow equations. Furthermore, we hope to establish
reliable minimal requirements for numerical methods to solve RG flow
equations, which can be used as benchmark tests for future numerical
toolboxes.

The length of this paper is explained by the fact that we have tried to make
the presentation self-contained as much as possible. This should enable the
reader not familiar with the FRG approach to understand all arguments and
intermediate steps without resorting to the literature. The more experienced
reader can certainly skip or skim over some parts, as indicated below.

The remainder of this paper is organized as follows. In Sec. II we give an
introduction to the FRG approach for zero-dimensional QFTs. In Sec. III we
focus on the zero-dimensional $O(N)$ model and its respective RG flow
equation. Readers familiar with the FRG approach and the $O(N)$ model can omit
these two sections. The relationship between RG flow equations and fluid
dynamics is discussed in Sec. IV. Readers familiar with fluid dynamics may be
interested in the analogy between the FRG and fluid dynamics discussed in
Sub.Sec. IV.1, but can skip over the remainder of this section that focuses on
details of the numeric implementation. Section V presents our numerical
results. Readers familiar with both the FRG approach and fluid dynamics should
focus on this section and the Sub.Sec. IV.1. We conclude this work with a
discussion and an outlook for future studies in Sec. VI. In the Appendices, we
list useful formulas for the calculation of numerical derivatives and present
a discussion of the absence of spontaneous symmetry breaking in zero space-
time dimensions.

##  II The Functional Renormalization Group – an introduction in zero
dimensions

This section provides an introduction to the Functional Renormalization Group
and a detailed derivation of the FRG equation [35, 36, 37] for a zero-
dimensional QFT. Our discussion is geared towards non-experts. Readers who are
familiar with the FRG method might still find this discussion instructive,
because we will introduce the FRG without any direct reference to
regularization and renormalization, only based on properties of (functional)
integrals. This sheds light on the details and structure of the flow equations
and the technical subtleties in their solution. In addition we use this
introduction to establish some notation and special features of zero-
dimensional field theory.

As already mentioned in the introduction, the efficient and sufficiently
precise calculation of correlation functions is key to understanding the
properties of a particular model or theory. Usually this is done by
introducing a partition function or functional integral that provides a
probability distribution for the microstates of the model and serves as a
generating functional for the $n$-point-correlation functions [38, 39, 40,
41]. The partition function is based on an energy function that can be a
discrete or continuous Hamilton function or an action, which determines the
microscopic properties of the model. Another way of calculating the $n$-point
correlation functions is to calculate the effective infrared action of the
model, for example via the FRG equation. Both methods are discussed in this
section.

###  II.1 The partition function in zero dimensions

Consider a zero-dimensional QFT with a single real bosonic scalar field or
degree of freedom $\phi$. While all definitions generalize to arbitrary QFTs
in zero or higher dimensions and arbitrary space-time backgrounds, in zero
dimensions the field $\phi$ does not depend on the space-time position. The
same applies to derivatives of the field or space-time integrals, which simply
do not exist. This implies that the action $\mathcal{S}[\phi]$ of the model is
identical to the Lagrangian $\mathcal{L}[\phi]$. The action, the Lagrangian,
and also the Hamiltonian $\mathcal{H}[\phi]$ are simply functions of $\phi$
instead of functionals111Nevertheless, we will stick to the notation of
functionals using square brackets, in order to facilitate the generalization
to a nonzero number of space-time dimensions, as long as we do not focus on
particular zero-dimensional examples. . Furthermore, because of the absence of
a space-time derivative and thus of kinetic terms,
$\mathcal{S}[\phi]=\mathcal{L}[\phi]=\mathcal{H}[\phi]=U(\phi)$, where
$U(\phi)$ is the potential. Therefore, the only requirement for these
functions is that they must be bounded from below, in order to exclude
“negative-energy states”222 We put “negative-energy states” in quotation
marks, because all quantities in zero-dimensional field theory are
dimensionless, hence bare numbers without physical dimensions. For
convenience, we will still use the well-established notions from higher-
dimensional QFT in our discussion.  and to obtain positive normalizable
probability distributions. Apart from this requirement, for the moment we do
not demand any additional properties, like symmetries (e.g., $\mathbb{Z}_{2}$,
$\phi\rightarrow-\phi$) or analyticity.

If we choose a specific model with action $\mathcal{S}[\phi]$ all expectation
values of arbitrary functions $f(\phi)$ that do not grow exponentially in
$\phi$ are defined and can be calculated via the following expression

$\displaystyle\langle
f(\phi)\rangle\equiv\frac{\int_{-\infty}^{\infty}\mathrm{d}\phi\,f(\phi)\,\mathrm{e}^{-\mathcal{S}[\phi]}}{\int_{-\infty}^{\infty}\mathrm{d}\phi\,\mathrm{e}^{-\mathcal{S}[\phi]}}\,,$
(1)

where $\mathrm{e}^{-\mathcal{S}[\phi]}$ provides the partition of
probabilities among the microstates. Note that due to the zero-dimensional
nature all expectation values for such a model reduce to proper one-
dimensional integrals over $\phi$. Such integrals can be computed to extremely
high precision using standard techniques of numerical integration [42, 43]. It
is worth emphasizing that the current discussion holds also for non-analytic
$\mathcal{S}[\phi]$ and/or $f(\phi)$. Some specific choices of
$\mathcal{S}[\phi]$ and $f(\phi)$ even allow for an analytic evaluation of Eq.
(1), see, e.g., Ref. [44]. The possibility to compute expectation values to
high precision makes zero-dimensional field theory of great interest as a
testing ground for approximations and/or numerical methods.

Some explicit examples of zero-dimensional field theories used as a testing
ground for methods in statistical mechanics and QFT can be found in Refs. [45,
46, 47, 48, 49, 50, 44, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63,
64, 65, 66]. In Ref. [55], for example, the asymptotic convergence and the
vanishing convergence radius of perturbation theory of $\phi^{4}$-theory is
discussed. Approximation schemes such as the large-$N$, the FRG vertex
expansion, or the FRG Taylor expansion were analyzed in Ref. [44]. Zero-
dimensional field theory was also used to study density-functional theory [56,
57, 59] and applied to fermionic fields [54]. Recently, it was used to study
and visualize 2PI effective actions [60] – also in the FRG framework [61, 63,
64].

Usually the calculation of expectation values is facilitated by a suitably
defined generating functional

$\displaystyle\mathcal{Z}[J]\equiv\mathcal{N}\int_{-\infty}^{\infty}\mathrm{d}\phi\,\mathrm{e}^{-\mathcal{S}[\phi]+J\,\phi}\,,$
(2)

from which one can derive all correlation functions by taking the
corresponding number of derivatives with respect to the external source $J$,

$\displaystyle\langle f(\phi)\rangle=\frac{f(\tfrac{\delta}{\delta
J})\,\mathcal{Z}[J]}{\mathcal{Z}[J]}\bigg{|}_{J=0}\,.$ (3)

One should note that if $f(\phi)$ is non-analytic, then Eq. (3) is to be
understood symbolically. Otherwise, it is defined through a Taylor series in
$\frac{\delta}{\delta J}$. Irrespective of that, Eqs. (1) and (2) are always
well defined and Eq. (2) can be always calculated for arbitrary $J$. One can
even show in zero dimensions that $\mathcal{Z}[J]\in C^{\infty}$, hence,
$\mathcal{Z}[J]$ is a smooth function, see Ref. [52] and App. B. We shall come
back to this crucial point later on in our discussion of the Coleman-Mermin-
Wagner-Hohenberg theorem [67, 68, 69].

The normalization $\mathcal{N}$ is not an observable quantity. For our
purposes, it is convenient to choose

$\displaystyle\mathcal{Z}[0]\overset{!}{=}1\,,$
$\displaystyle\longleftrightarrow$
$\displaystyle\mathcal{N}^{-1}=\int_{-\infty}^{\infty}\mathrm{d}\phi\,\mathrm{e}^{-\mathcal{S}[\phi]}\,.$
(4)

As already mentioned above, calculating expectation values in a zero-
dimensional QFT via Eq. (1) is (numerically) rather straightforward. In
contrast, for higher-dimensional models or theories with non-trivial field-
content etc. calculating functional integrals similar to Eq. (1) with
sufficient precision is usually extremely challenging or might even be
impossible with limited computational resources. Therefore, alternative
methods or approximation schemes apart from “direct numerical integration”,
like in lattice simulations, are of great interest. One of these alternatives,
which is at the heart of this work, is the FRG.

In the following, we will therefore focus on the FRG as a specific method for
calculating $n$-point correlation functions in QFT and statistical mechanics.
In contrast to the usual motivation of the FRG, arising in the discussion of
renormalization and the integration of momentum shells from ultraviolet to
infrared energy scales, we will take a different path to arrive at the FRG
equation, which does not require any knowledge of renormalization. To this
end, we will follow and extend the discussion in Refs. [44, 51, 52, 53, 54]
and discuss its technical properties as an alternative way of solving the
integrals in Eqs. (1) and (2).

###  II.2 Solving integrals with flow equations

The starting point is the observation that there is one well-known non-trivial
class of actions $\mathcal{S}[\phi]$ for which the calculation of integrals
like Eq. (1) is straightforward, even in higher dimensions and even for more
complicated field content. These actions are QFTs for “(massive) free
particles” and correspond to Gaussian-type integrals. In the present case the
Gaussian-type action takes the following simple form,

$\displaystyle\mathcal{S}[\phi]=\tfrac{m^{2}}{2}\,\phi^{2}\,.$ (5)

where $m$ is called a “mass” for convenience, although it is actually a
dimensionless quantity in zero space-time dimensions.

For non-trivial actions $\mathcal{S}[\phi]$, Eq. (1) can still be approximated
by a Gaussian integral, as long as $\mathcal{S}[\phi]$ contains a mass term
(5) with a coefficient $m^{2}$ that is much larger than all other scales
contained in $\mathcal{S}[\phi]$. If this is the case, the Gaussian part of
the integrand $\mathrm{e}^{-\mathcal{S}[\phi]}$ completely dominates the
integrals in Eqs. (1) and (2). The reason is that the mass term $\sim\phi^{2}$
is dominant for small and moderate $\phi$, and most of the area under the
curve $\mathrm{e}^{-\mathcal{S}[\phi]}$ lies in the region of small $\phi$,
similar to a pure Gaussian integral. For very large values of $\phi$ other
terms in the action $\mathcal{S}[\phi]$ may become more important.
Nevertheless, if $m^{2}$ is large enough, the corresponding area under the
curve $\mathrm{e}^{-\mathcal{S}[\phi]}$ is completely negligible in regions
where $\phi$ is large, because $\mathcal{S}[\phi]$ is bounded from below such
that $\mathrm{e}^{-\mathcal{S}[\phi]}$ tends to zero exponentially fast for
$\phi\to\infty$. In summary, the Gaussian part with the huge mass term
dominates the integral and even non-trivial $\mathcal{S}[\phi]$ can be
approximated by Gaussian integrals.

This observation generalizes to higher dimensions and arbitrary field content,
but is more apparent in a zero-dimensional field theory with one degree of
freedom. This is illustrated in Figs. 1 and 2, which are discussed in the
following subsubsection.

####  II.2.1 The scale-dependent partition function

Based on the above observation, let us now introduce the following quantity:

$\displaystyle\mathcal{Z}_{t}[J]\equiv\mathcal{N}\,\int_{-\infty}^{\infty}\mathrm{d}\phi\,\mathrm{e}^{-\mathcal{S}[\phi]-\Delta\mathcal{S}_{t}[\phi]+J\,\phi}\,,$
(6)

which is called the scale-dependent generating functional or scale-dependent
partition function. It differs from the usual partition function (2) only by a
scale-dependent mass term

$\displaystyle\Delta\mathcal{S}_{t}[\phi]\equiv\tfrac{1}{2}\,r(t)\,\phi^{2}\,.$
(7)

We directly adopt the common notation from the FRG community and call $r(t)$
the regulator (shape) function, which depends on the RG scale (“time”)
$t\in[\,0,\infty)$, see, e.g., Refs. [70, 71]. We will discuss this
interpretation of $r(t)$ and $t$ in Sub.Sec. II.4. For now, we only demand
that the function $r(t)$ has such properties that $\mathcal{Z}_{t}[J]$
interpolates between an almost Gaussian-type partition function333This is also
why the UV fixed point of RG flows is denoted as the trivial or Gaussian fixed
point.  with extremely massive free fields at $t=0$ and the actual partition
function $\mathcal{Z}[J]$ that we are interested in at $t\rightarrow\infty$.
In order to achieve this behavior, $r(t)$ has to have the following
properties:

  1. 1.

In the limit of $t\rightarrow 0$, $r(t)$ ($\mathcal{S}[\phi]$) should behave
like a mass (term), similar to what we discussed at the beginning of this
section, and be much larger than all other scales in $\mathcal{S}[\phi]$.
Oftentimes in the literature $r(t)$ is set to infinity at $t=0$. We will see,
cf. Sub.Sec. II.3, that this is not suitable.

  2. 2.

For $t\rightarrow\infty$, $r(t)$ is supposed to vanish, such that
$\lim_{t\rightarrow\infty}\mathcal{Z}_{t}[J]=\mathcal{Z}[J]$. The same applies
to expectation values calculated from $\mathcal{Z}_{t}[J]$, which become
expectation values of $\mathcal{Z}[J]$. For practical calculations it is
sufficient to assume that, for $t\rightarrow\infty$, $r(t)$ becomes much
smaller than all scales in $\mathcal{S}[\phi]$, because then the contribution
$\Delta\mathcal{S}_{t}[\phi]$ to the whole integrand
$\mathrm{e}^{-\mathcal{S}[\phi]-\Delta\mathcal{S}_{t}[\phi]}$ is negligible
and the integrand is almost identical to $\mathrm{e}^{-\mathcal{S}[\phi]}$.
The value $\lim_{t\rightarrow\infty}r(t)=r_{\mathrm{IR}}\gtrsim 0$ is usually
referred to as (numerical) infrared (IR) cutoff.

  3. 3.

The interpretation of $r(t)$ ($\mathcal{S}[\phi]$) as a mass (term) is
guaranteed by further demanding monotonicity, ${\partial_{t}r(t)\leq
0\quad\forall t}$. We will provide additional arguments for monotonicity in
Sub.Sub.Sec. II.2.2.

  4. 4.

In order to be able to smoothly deform the integral in Eq. (2) and for the
following derivation of evolution equations, we further require $r(t)\in
C^{1}$.

Apart from these four properties there are no further requirements on $r(t)$
in zero dimensions.444For the subtleties associated with the choice of
regulators in higher-dimensional theories, we refer the interested reader to
Refs. [70, 71, 72, 73, 74, 75]. Note that for higher-dimensional field
theories the fourth requirement turns into $\Delta\mathcal{S}_{t}[\phi]\in
C^{1}$. A specific choice which is used in large parts of our work is the so-
called exponential regulator (shape) function

$\displaystyle r(t)=\Lambda\,\mathrm{e}^{-t}\,,$ (8)

with an ultraviolet (UV) cutoff $\Lambda$, which must be chosen much larger
than all scales in $\mathcal{S}[\phi]$.

Figure 1: The integrand (upper panel) and exponent (lower panel) from Eq. (6)
(at $J=0$) as a function of the field variable $\phi$ for various RG times
$t=0,1,2,\ldots,15$ and for the action (9). We choose the exponential
regulator (8) with UV cutoff $\Lambda=10^{3}$, which is notably larger than
the absolute value of the mass term and the quartic coupling. The IR cutoff
scale $r_{\mathrm{IR}}$ was chosen at $t=15$, which corresponds to
$r_{\mathrm{IR}}\simeq 3.06\cdot 10^{-4}$. This value is significantly smaller
than all scales in $\mathcal{S}[\phi]$.

In order to get a better intuition of the effect of $r(t)$ on the integral
(6), in Fig. 1 we show the integrand at $J=0$,
$\mathrm{e}^{-\mathcal{S}[\phi]-\Delta\mathcal{S}_{t}[\phi]}$, and the
respective exponent for different values of $t$ for the analytic action

$\displaystyle\mathcal{S}(\phi)=-\tfrac{1}{2}\,\phi^{2}+\tfrac{1}{4!}\,\phi^{4}$
(9)

and in Fig. 2 the same quantities for the non-analytic action

$\displaystyle\mathcal{S}(\phi)=\begin{cases}-\phi^{2}\,,&\text{if}\quad|\phi|\leq\tfrac{5}{4}\,,\vphantom{\bigg{(}\bigg{)}}\\\
-\big{(}\tfrac{5}{4}\big{)}^{2}\,,&\text{if}\quad\tfrac{5}{4}<|\phi|\leq
2\,,\vphantom{\bigg{(}\bigg{)}}\\\
\tfrac{1}{48}\,\big{(}\phi^{4}-91\big{)}&\text{if}\quad
2<|\phi|\,.\vphantom{\bigg{(}\bigg{)}}\\\ \end{cases}$ (10)

The figures show how the integrands are deformed from Gaussian-shaped
integrands to the integrands $\mathrm{e}^{-\mathcal{S}[\phi]}$. One observes
that, as long as $r(t)$ is much larger than all other parameters in
$\mathcal{S}[\phi]$, the Gaussian-like mass term dominates, while for
increasing $t$ the regulator $r(t)$ becomes negligible. The most interesting
part, where the integrands change their shapes significantly, is where $r(t)$
is of the same order as the scales in $\mathcal{S}(\phi)$.

Figure 2: The same as in Fig. 1, but for the action (10).

####  II.2.2 A flow equation for the scale-dependent partition function

The change of the integrals with $t$ between the two limiting cases at $t=0$
and $t\rightarrow\infty$ is called RG flow. If this RG flow is known, we can
obtain the function
${\mathcal{Z}(J)\equiv\lim_{t\rightarrow\infty}\mathcal{Z}_{t}[J]=\mathcal{Z}[J]}$
right from the Gaussian-like partition function $\mathcal{Z}_{t=0}[J]$ without
the need to calculate the $\phi$-integral in the partition function (2)
directly. For zero dimensions this does not seem to be an advantage, because
the integrals in field space are (at least numerically) simple to compute. For
higher dimensions, however, circumventing the challenging functional
integration is a tremendous benefit.

The RG flow of $\mathcal{Z}_{t}[J]$ is characterized by taking the derivative
with respect to the RG time $t$,

$\displaystyle\partial_{t}\mathcal{Z}_{t}[J]=\vphantom{\bigg{(}\bigg{)}}$ (11)
$\displaystyle=\,$
$\displaystyle-\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\mathcal{N}\int_{-\infty}^{\infty}\mathrm{d}\phi\,\phi^{2}\,\mathrm{e}^{-\mathcal{S}[\phi]-\Delta\mathcal{S}[\phi]+J\,\phi}=\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle=\,$
$\displaystyle-\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\frac{\delta^{2}\mathcal{Z}_{t}[J]}{\delta
J\,\delta J}\equiv\vphantom{\bigg{(}\bigg{)}}$ $\displaystyle\equiv\,$
$\displaystyle-\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\mathcal{Z}^{(2)}_{t,JJ}[J]\,,\vphantom{\bigg{(}\bigg{)}}$

which is a PDE for a function $\mathcal{Z}(t,J)$ in the $(t,J)$-plane,

$\displaystyle\partial_{t}\mathcal{Z}(t,J)=-\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\partial_{J}^{2}\mathcal{Z}(t,J)\,.$
(12)

With slight modifications, this also applies to higher-dimensional QFTs.
Solving this equation with appropriate initial and boundary conditions results
in a function $\mathcal{Z}(J)$ from which one can calculate expectation values
by taking ordinary (numerical) derivatives with respect to $J$ at $J=0$, cf.
Eq. (3).

The structure of this equation is that of a linear one-dimensional diffusion
equation (heat equation) [74, 54, 76, 77], where $t$ corresponds to the
temporal direction, while $J$ corresponds to the spatial direction. The term
$-\tfrac{1}{2}\,\partial_{t}r(t)$ corresponds to a time-dependent (positive
definite) diffusion coefficient555Note that in zero dimensions one can get rid
of $\partial_{t}r(t)$ by an appropriate reparametrization of the time
coordinate $t$, which nevertheless keeps the structure of the equation
unchanged. In higher dimensions this elimination of $r(t)$ is in general not
possible. The positivity of the diffusion coefficient is directly related to
the stability of solutions of the heat equation [78, 79] and positivity – here
guaranteed by the regulator properties – is necessary for a stable solution
[74, 75]. . This also motivates the name RG “time” for the parameter $t$. We
will come back to the concept of RG “time” in the true sense of the word and
the diffusive, irreversible character of RG flows in Sub.Sec. IV.1.

In zero dimensions, the Eq. (12) is a PDE in two variables. For the remainder
of this subsection we will discuss properties and practical issues considering
this exact PDE. We will neither discuss any kind of expansions in $J$ nor its
application in higher dimensions. However, some of the issues and questions
raised in the following are also relevant for higher-dimensional theories.

Finding the correct initial and boundary conditions for numerical solutions of
Eq. (12) as an exact PDE is challenging. By construction
$\mathcal{Z}_{t=0}[J]$ approaches a Gaussian integral,

$\displaystyle\mathcal{Z}_{t=0}[J]=\vphantom{\bigg{(}\bigg{)}}$ (13)
$\displaystyle=\,$
$\displaystyle\mathcal{N}\,\int_{-\infty}^{\infty}\mathrm{d}\phi\,\mathrm{e}^{-\frac{1}{2}r(0)\,\phi^{2}+J\,\phi}\,\mathrm{e}^{-\mathcal{S}(\phi)}=\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle=\,$
$\displaystyle\mathcal{N}\,\int_{-\infty}^{\infty}\tfrac{\mathrm{d}\tilde{\phi}}{\sqrt{r(0)}}\,\mathrm{e}^{-\frac{1}{2}\,\tilde{\phi}^{2}+J\,\frac{\tilde{\phi}}{\sqrt{r(0)}}}\,\Big{[}1-\mathcal{O}\big{(}\mathcal{S}\big{(}r(0)^{-\frac{1}{2}}\big{)}\big{)}\Big{]}$
$\displaystyle=\,$
$\displaystyle\mathcal{N}\,\sqrt{\tfrac{2\pi}{r(0)}}\,\mathrm{e}^{\frac{J^{2}}{2r(0)}}\,\Big{[}1-\mathcal{O}\big{(}\mathcal{S}\big{(}r(0)^{-\frac{1}{2}}\big{)}\big{)}\Big{]}\,,\vphantom{\bigg{(}\bigg{)}}$

with $\tilde{\phi}\equiv\sqrt{r(0)}\,\phi$ and independent of the explicit
shape of $\mathcal{S}[\phi]$. Considering different actions
$\mathcal{S}[\phi]$ with couplings of the same order of magnitude we can
choose the same regulator with an $r(0)$ larger than all internal scales
involved in the different actions. The initial condition $\mathcal{Z}(0,J)$ is
then independent of the explicit action under consideration.

According to the integral formulation (6), $\mathcal{Z}(t,J)$ changes for
different actions when $t>0$. In the differential formulation of the Eq. (12)
those changes are generated by the diffusion term on the right-hand side.
However, we argued that it is permissible to use identical initial conditions
$\mathcal{Z}(0,J)$ for different actions involving similar scales (as long as
these are much smaller than $r(0)$). This then results in an identical
diffusion on the right-hand side of Eq. (12) when the latter is computed by
means of a second derivative of $\mathcal{Z}(0,J)$. If one uses identical
large-$J$ boundary conditions for the solution of the PDE (12) for different
actions, this would imply that, despite different $\mathcal{S}[\phi]$, the RG
time evolution leads to identical $\mathcal{Z}(J)$ for $t\rightarrow\infty$,
which in general cannot be correct.

In order to resolve this problem, particular action-dependent spatial boundary
conditions seem to be necessary for a direct numerical solution starting at
$t=0$ with a Gaussian for $\mathcal{Z}(0,J)$. It is not obvious how to derive
or formulate such boundary conditions from the asymptotics of Eq. (12) alone.
In light of this, a numerical solution of Eq. (12) in the $(t,J)$ plane by
means of a spatial discretization in $J$ direction and an integration in $t$
direction appears to be conceptually questionable.

However, this invalidates by no means the flow equation for $\mathcal{Z}(t,J)$
in general. Augmenting it (at $t=0$) with information from the integral
formulation (6) or, equivalently, other additional information, could enable
practical computations using the PDE (12). But it is at this point (at least
to us) not obvious how one would implement a numerical solution strategy for
the PDE (12) avoiding integrals of the action.

There is another well-known drawback in using the partition function
$\mathcal{Z}[J]$ for calculating $n$-point correlation functions (or
expectation values) $\langle\phi^{n}\rangle$. The latter are rather
inefficient in storing information, because they contain redundant information
in the form of disconnected and reducible terms, see Refs. [80, 40, 38, 39] or
the mathematical theory of moment- and cumulant-generating functionals in
statistics for details [81]. This is further discussed in Sub.Sec. II.5.
However, the redundant information in $\langle\phi^{n}\rangle$ is not
necessarily a strong argument against the use of the flow equation (12) in
practical computations, since the irreducible information can be extracted
from the correlation functions $\langle\phi^{n}\rangle$.

In order to resolve both the problem of initial and boundary conditions for
$\mathcal{Z}(t,J)$ as well as the issue of redundant information in
$\langle\phi^{n}\rangle$, we now consider two different generating
functionals, which are better suited for practical calculations of $n$-point
correlation functions or expectation values, respectively. To this end, we
employ the Schwinger functional,

$\displaystyle\mathcal{W}[J]\equiv\ln\mathcal{Z}[J]\,,$
$\displaystyle\mathcal{W}[0]=0\,,$ (14)

and its Legendre transform, the effective action,

$\displaystyle\Gamma[\varphi]\equiv\mathop{\mathrm{sup}}_{J}\big{\\{}J\,\varphi-\mathcal{W}[J]\big{\\}}\,.$
(15)

Here, “$\mathrm{sup}$” denotes the supremum with respect to the source $J$.
The Schwinger functional generates all connected $n$-point correlation
functions while the effective action generates all one-particle irreducible
(1PI) $n$-point vertex (correlation) functions, see Sub.Sec. II.5 or Refs.
[80, 40, 38, 39] for details.

In general $\mathcal{W}[J]$ is convex with a positive definite Hessian
$\mathcal{W}^{(2)}_{JJ}[J]$, which implies convexity for $\Gamma[\varphi]$,
since the Legendre transform of a convex function is convex by definition, see
e.g., Refs. [82, 83] for details. In the present case the convexity of
$\mathcal{W}[J]=\mathcal{W}(J)$ becomes apparent considering its second
derivative,

$\displaystyle\partial_{J}^{2}\mathcal{W}(J)=\langle\phi^{2}\rangle_{J}-\langle\phi\rangle_{J}\langle\phi\rangle_{J}=\langle(\phi-\langle\phi\rangle_{J})^{2}\rangle_{J}\,,$
(16)

which, as the expectation value of a positive quantity, is always
positive.666Note that also $\mathcal{Z}[J]$ is convex, which can be seen by
investigating its second derivative.  In zero dimensions, also smoothness,
$\mathcal{Z}[J]\in C^{\infty}$, directly translates to $\mathcal{W}[J]\in
C^{\infty}$ and $\Gamma[\varphi]\in C^{\infty}$, because all derivatives
$\mathcal{W}[J]$ and $\Gamma[\varphi]$ can be entirely expressed in terms of
derivatives of $\mathcal{Z}[J]$, see Sub.Sec. II.5. We will need both
properties several times during our discussion, see also the discussion in
App. B.

Having these definitions at hand, we shall start the next section by defining
scale-dependent generating functionals $\mathcal{W}_{t}[J]$ and
$\Gamma_{t}[\varphi]$. From these, we will also derive and discuss two flow
equations which are similar to Eq. (11). The final result of the next
subsection is the FRG equation (known as Wetterich equation), which is the
exact analogue to Eq. (11) on the level of $\Gamma_{t}[\varphi]$. It provides
the opportunity to circumvent the direct calculation of integrals of type (1).

###  II.3 The Functional Renormalization Group equation

In this subsection we derive and discuss the Functional Renormalization Group
(FRG) equation [35, 36, 37] (also known as Exact Renormalization Group
equation) for our zero-dimensional toy-model QFT. All formulas presented in
this section can be generalized to higher dimensions and arbitrary field
content, see e.g., Refs. [84, 85, 86, 87, 73, 88].

####  II.3.1 The scale-dependent Schwinger functional

We begin the derivation by introducing the scale-dependent Schwinger
functional starting from definition (6),

$\displaystyle\mathcal{W}_{t}[J]\equiv\ln\mathcal{Z}_{t}[J]\,.$ (17)

It follows from our previous discussion that for $t\rightarrow\infty$ the
Schwinger functional (14) is recovered,

$\displaystyle\lim\limits_{t\rightarrow\infty}\mathcal{W}_{t}[J]=\mathcal{W}[J]\,,$
(18)

while $\mathcal{W}_{t=0}[J]$ is given by the logarithm of Eq. (13).

The insertion of the regulator (7) into $\mathcal{Z}_{t}[J]$ does not spoil
the convexity and smoothness (in zero dimensions) of the Schwinger functional:
$\mathcal{W}_{t}[J]$ and $\mathcal{Z}_{t}[J]$ are convex and smooth for all
$t$.

Completely analogous to Eq. (11) one can derive a PDE for
$\mathcal{W}_{t}[J]=\mathcal{W}(t,J)$ in the $(t,J)$ plane,

$\displaystyle\partial_{t}\mathcal{W}(t,J)=\vphantom{\bigg{(}\bigg{)}}$ (19)
$\displaystyle=\,$
$\displaystyle-\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\Big{(}\partial_{J}^{2}\mathcal{W}(t,J)+\big{[}\partial_{J}\mathcal{W}(t,J)\big{]}^{2}\Big{)}\,.\vphantom{\bigg{(}\bigg{)}}$

which describes the flow of $\mathcal{W}(t,J)$ from $t=0$ to
$t\rightarrow\infty$777In terms of its structure Eq. (19) is also known as the
Polchinski equation in the context of the RG for higher-dimensional QFTs.
However, in the original work [89] an effective action $L(\Lambda,\phi)$ takes
the role of $\mathcal{W}$ and it is formulated in terms of the fields $\phi$
instead of the sources $J$. For relations between the original Polchinski
equation and the flow equations studied in this work and selected applications
of the Polchinski equation, see, e.g., Refs. [90, 91, 92, 93]..

We could now repeat the discussion about the issues of initial and boundary
conditions for the solution of this PDE. However, the problems are almost
identical to those of Eq. (12), because on the level of the PDE, we only
substituted the function $\mathcal{Z}(t,J)$ by $\mathcal{W}(t,J)$ via the
logarithm, which does not change the structure of the problem fundamentally.
Formulating appropriate initial and boundary conditions in the spatial $J$
direction therefore remains as complicated as before. Note that the PDE (19)
became more complicated when compared to Eq. (12) due to the non-linear term
on the right-hand side. In summary, the scale-dependent Schwinger functional
is, from a practical point of view, as badly suited as $\mathcal{Z}(t,J)$ to
perform the (numeric) calculation of the functional integral via a flow
equation starting from a Gaussian-type integral.

In the following we will focus on the scale-dependent effective (average)
action and its respective flow equation, which does not suffer from the issues
of particular initial and boundary conditions. As an added benefit, the
effective action is also the most efficient functional in terms of storing
information of a theory at hand. Formulating proper initial and boundary
conditions for the flow equations for $\mathcal{Z}(t,J)$ and
$\mathcal{W}(t,J)$ and if possible implementing adequate numerical schemes in
the context of zero-dimensional field theories would certainly be interesting
from an academic point of view. Translating the initial and boundary
conditions for the scale-dependent effective (average) action to
$\mathcal{Z}(t,J)$ and $\mathcal{W}(t,J)$ could be a possible and potentially
feasible strategy. A comparison of the flows of $\mathcal{Z}(t,J)$,
$\mathcal{W}(t,J)$, and $\Gamma(t,\varphi)$, both conceptually and for
explicitly specified actions, is a worthwhile subject of future work.

####  II.3.2 The scale-dependent effective action

We now define the scale-dependent effective action $\Gamma_{t}[\varphi]$ via
the Legendre transform of Eq. (17) with respect to the sources $J$ at a RG
timescale $t$,

$\displaystyle\Gamma_{t}[\varphi]\equiv\,$
$\displaystyle\mathop{\mathrm{sup}}_{J}\big{\\{}J\,\varphi-\mathcal{W}_{t}[J]\big{\\}}\equiv\vphantom{\bigg{(}\bigg{)}}$
(20) $\displaystyle\equiv\,$ $\displaystyle
J_{t}(\varphi)\,\varphi-\mathcal{W}_{t}[J_{t}(\varphi)]\,,\vphantom{\bigg{(}\bigg{)}}$
(21)

where we introduced the source $J_{t}(\varphi)$ which realizes the supremum.

Note that, analogous to $\mathcal{Z}_{t}[J]$ and $\mathcal{W}_{t}[J]$, the
convexity and smoothness (in zero dimensions) of $\Gamma_{t}[\varphi]$ is not
spoiled by the $t$ dependence, because the properties of the Legendre
transformation still ensure both.

To obtain an explicit relation for the scale-dependent source
$J_{t}(\varphi)$, which realizes the supremum in Eq. (20), we consider the
functional derivative of Eq. (20) at the supremum to find the important
relation

$\displaystyle\mathcal{W}_{t,J}^{(1)}[J_{t}(\varphi)]\equiv\frac{\delta\mathcal{W}_{t}[J]}{\delta
J}\bigg{|}_{J=J_{t}(\varphi)}=\varphi\,,$ (22)

which will be used frequently in the following. Taking the functional
derivative of Eq. (21) with respect to $\varphi$ and using Eq. (22) we
ultimately find

$\displaystyle\Gamma^{(1)}_{t,\varphi}[\varphi]\equiv\frac{\delta\Gamma_{t}[\varphi]}{\delta\varphi}=J_{t}(\varphi)\,,$
(23)

which is referred to as quantum equation of motion. Due to the strict
convexity of $\Gamma_{t}[\varphi]$ the function $J_{t}(\varphi)$ is bijective
and as such can be inverted, which can be achieved by considering Eq. (22) at
fixed value $J$ for $J_{t}$:

$\displaystyle\varphi_{t}(J)\equiv\frac{\delta\mathcal{W}_{t}[J]}{\delta
J}\,,$ (24)

where $\varphi_{t}(J)$ is the so-called scale-dependent classical field
(sometimes also referred to as scale-dependent mean field).

The subtle relations between, and scale dependences of, $\varphi_{t}(J)$ and
$J_{t}(\varphi)$ are rarely discussed in literature and usually suppressed in
the notation. The relation between $\varphi_{t}$ and $J_{t}$ will be of
particular importance in the discussion of $n$-point correlation functions in
Sub.Sec. II.5. The scale dependence of $\varphi_{t}(J)$ from Eq. (24) is not
related to a rescaling (RG transformation) using, e.g., a wave-function
renormalization for $\varphi$.

Before we derive the FRG equation, which is the flow equation for
$\Gamma_{t}[\varphi]$ and a PDE for the function $\Gamma(t,\varphi)$ in the
$(t,\varphi)$ plane, we check whether we will run into the same issues
(related to initial and boundary conditions) as before. Hence, first of all,
we must derive the initial condition for the PDE for $\Gamma(t,\varphi)$. To
this end, we study the limit $t\rightarrow 0$ of $\Gamma_{t}[\varphi]$. We use
the definitions (6), (17), (20), and (21) to obtain

$\displaystyle\mathrm{e}^{-\Gamma_{t}[\varphi]}=\,$
$\displaystyle\mathrm{e}^{-\mathop{\mathrm{sup}}_{J}\\{J\,\varphi-\mathcal{W}_{t}[J]\\}}=\vphantom{\bigg{(}\bigg{)}}$
(25) $\displaystyle=\,$
$\displaystyle\mathrm{e}^{\ln\mathcal{Z}_{t}[J_{t}(\varphi)]-J_{t}(\varphi)\,\varphi}=\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle=\,$
$\displaystyle\mathcal{N}\int_{-\infty}^{\infty}\mathrm{d}\phi\,\mathrm{e}^{-\mathcal{S}[\phi]-\Delta\mathcal{S}_{t}[\phi]+J_{t}(\varphi)\,(\phi-\varphi)}\,.\vphantom{\bigg{(}\bigg{)}}$

We now shift the integration variable888It is the same shift that is used in
the background field formalism [94, 95], where the full fluctuating quantum
field $\phi$ is split into a background field configuration $\varphi$ and
additional fluctuations $\phi^{\prime}$ about the background field. This is
why $\varphi$ is called the classical or mean field.
$\phi\mapsto\phi^{\prime}=\phi-\varphi$. Using Eq. (7), we find

$\displaystyle\mathrm{e}^{-\Gamma_{t}[\varphi]+\Delta\mathcal{S}_{t}[\varphi]}=\vphantom{\bigg{(}\bigg{)}}$
(26) $\displaystyle=\,$
$\displaystyle\mathcal{N}\int_{-\infty}^{\infty}\mathrm{d}\phi^{\prime}\,\mathrm{e}^{-\mathcal{S}[\phi^{\prime}+\varphi]-\Delta\mathcal{S}_{t}[\phi^{\prime}]-r(t)\,\phi^{\prime}\,\varphi+\Gamma^{(1)}_{t,\varphi}[\varphi]\,\phi^{\prime}}\,.\vphantom{\bigg{(}\bigg{)}}$

In the next step, we introduce the scale-dependent effective average action,

$\displaystyle\bar{\Gamma}_{t}[\varphi]\equiv\Gamma_{t}[\varphi]-\Delta\mathcal{S}_{t}[\varphi]\,,$
(27)

which also tends to the effective action $\Gamma[\varphi]$ for
$t\rightarrow\infty$, because the second term vanishes in this limit, cf. Eq.
(8).

At any finite value of $t$ (including $t=0$), $\bar{\Gamma}_{t}[\varphi]$
differs from $\Gamma_{t}[\varphi]$ and is no longer guaranteed to be convex,
which can be seen directly from the second term in Eq. (27). Convexity is only
recovered for $t\rightarrow\infty$. However, the second term in Eq. (27) does
not violate the smoothness of $\bar{\Gamma}_{t}[\varphi]$ in zero dimensions
for all $t$, because
$\Delta\mathcal{S}_{t}[\varphi]\equiv\mathcal{S}_{t}(\varphi)\in C^{\infty}$
in $\varphi$.

We express Eq. (26) in terms of the scale-dependent effective average action
(27) and, for the sake of convenience, revert the notation
$\phi^{\prime}\rightarrow\phi$,

$\displaystyle\mathrm{e}^{-\bar{\Gamma}_{t}[\varphi]}=\mathcal{N}\,\int_{-\infty}^{\infty}\mathrm{d}\phi\,\mathrm{e}^{-\mathcal{S}[\phi+\varphi]-\Delta\mathcal{S}_{t}[\phi]+\bar{\Gamma}^{(1)}_{t,\varphi}[\varphi]\,\phi}\,.$
(28)

In the next step one formally introduces the normalization of a Gaussian
integral with mass $r(t)$ and takes the logarithm, which results in

$\displaystyle\bar{\Gamma}_{t}[\varphi]=\vphantom{\bigg{(}\bigg{)}}$ (29)
$\displaystyle=\,$
$\displaystyle-\ln\int_{-\infty}^{\infty}\mathrm{d}\phi\,\sqrt{\tfrac{r(t)}{2\pi}}\,\mathrm{e}^{-\mathcal{S}[\phi+\varphi]-\frac{1}{2}r(t)\,\phi^{2}+\bar{\Gamma}^{(1)}_{t,\varphi}[\varphi]\,\phi}-\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle-\ln\Big{[}\mathcal{N}\sqrt{\tfrac{2\pi}{r(t)}}\Big{]}\,.\vphantom{\bigg{(}\bigg{)}}$

We are now ready to study the limit $t\rightarrow 0$, which corresponds to the
initial condition for a possible flow equation for $\Gamma_{t}[\varphi]$ or
$\bar{\Gamma}_{t}[\varphi]$, respectively. Focusing on the $\phi$ integral in
the first term on the right-hand side of Eq. (29), we employ the fact that the
regulator terms act like a Gaussian representation of the Dirac delta
distribution,

$\displaystyle\lim\limits_{t\rightarrow
0}\sqrt{\tfrac{r(t)}{2\pi}}\,\mathrm{e}^{-\frac{1}{2}r(t)\,\phi^{2}}\approx\delta(\phi)\,,$
(30)

as long as $r(t)$ is much larger than all scales in $\mathcal{S}[\phi]$. Thus,
denoting

$\displaystyle
c(t)\equiv-\ln\Big{[}\mathcal{N}\,\sqrt{\tfrac{2\pi}{r(t)}}\,\Big{]}\,,$ (31)

we find as $t\rightarrow 0$

$\displaystyle\bar{\Gamma}_{t}[\varphi]\rightarrow\,$
$\displaystyle-\ln\int_{-\infty}^{\infty}\mathrm{d}\phi\,\delta(\phi)\,\mathrm{e}^{-\mathcal{S}[\phi+\varphi]+\bar{\Gamma}^{(1)}_{t,\varphi}[\varphi]\,\phi}+c(t)=\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle=\,$
$\displaystyle\mathcal{S}[\varphi]+c(t)\,,\qquad(t\rightarrow\infty)\vphantom{\bigg{(}\bigg{)}}$
(32)

This means that the initial condition for a flow of
$\bar{\Gamma}_{t}[\varphi]$ is given by the classical action $\mathcal{S}$
evaluated for the classical field $\varphi$ and some additional $t$ dependent,
but $\varphi$ independent term $c(t)$. This choice for an initial condition of
a PDE for $\bar{\Gamma}_{t}[\varphi]$ has subtle consequences:

Although $c(t)$ does not depend on $\varphi$, it is large,
$c(t)\sim\frac{1}{2}\ln r(t)$. Consequently, as far as the initial condition
for the PDE for $\Gamma_{t}[\varphi]$ or $\bar{\Gamma}_{t}[\varphi]$ is
concerned, it seems as if we run into the same problem as before: The initial
condition is dominated by the artificial mass of the regulator $r(t)$,
independent of the specific action $\mathcal{S}[\phi]$, and differences in the
specific choice for $\mathcal{S}[\phi]$ enter the initial condition only as
small deviations from the large term $c(t)$. Furthermore, $c(t)$ contains the
normalization constant $\mathcal{N}$, which was fixed according to Eq. (4).

However, precisely because $c(t)$ appears like the normalization
$\mathcal{N}$, it should be irrelevant for all physical observables. Indeed
this is the case, because all $\varphi$ independent terms in
$\Gamma_{t}[\varphi]$ do not enter the $n$-point correlation functions, since
the latter are calculated as derivatives of $\Gamma[\varphi]$ with respect to
$\varphi$ at $t\rightarrow\infty$, see Sub.Sec. II.5. This implies that an
additive, $\varphi$ independent term in the three effective actions
$\Gamma[\varphi]$, $\Gamma_{t}[\varphi]$, and $\bar{\Gamma}_{t}[\varphi]$ is
irrelevant and only relative differences in the effective actions are
observable. Therefore, we can simply omit $c(t)$ and take as initial condition
for the PDE for $\bar{\Gamma}_{t}[\varphi]$ the value $\mathcal{S}[\varphi]$,
which perfectly incorporates the difference between different models with
distinct actions $\mathcal{S}[\phi]$.

One problem in disregarding $c(t)$ remains: one has to ensure that a PDE for
$\bar{\Gamma}_{t}[\varphi]$ must not contain any terms without field
derivatives of $\bar{\Gamma}_{t}[\varphi]$. Otherwise $c(t)$ would influence
the flow in a time-dependent manner. Fortunately, this does not happen, as we
will see later, and the FRG equation (37) does not contain terms without field
derivatives of $\bar{\Gamma}_{t}[\varphi]$ on the right-hand side.

This, however, brings up another question: After Eq. (27) we argued that
$\bar{\Gamma}_{t}[\varphi]$ does not need to be convex, but must still be
smooth for all $t$. Let us for example consider the non-analytic action (10)
as an initial condition, $\bar{\Gamma}_{t=0}[\varphi]=\mathcal{S}[\varphi]$.
This action does not cause any problems for the convexity and the smoothness
of $\mathcal{Z}_{t}[J]$ and $\mathcal{W}_{t}[J]$ at arbitrary $t$, see for
example App. B and Fig. 36. The non-convexity of $\mathcal{S}[\varphi]$ is
also not a problem for $\bar{\Gamma}_{t}[\varphi]$, which does not necessarily
need to be convex at finite $t$. Nevertheless, the smoothness of
$\bar{\Gamma}_{t}[\varphi]$ is violated by this choice of
$\mathcal{S}[\varphi]$ at $t=0$. This issue originates from relation (30),
which is exactly fulfilled only in the limit $\Lambda\rightarrow\infty$ for
the UV cutoff. This, however, leads to a trivial theory of infinitely massive
particles at $t=0$, cf. Eq. (6). If one chooses a reasonably large but finite
$\Lambda$ and does not use Eq. (30), one would ensure that
$\bar{\Gamma}_{t}[\varphi]$ is also smooth at $t=0$. However, then the initial
condition is not exactly $\mathcal{S}[\varphi]$, but rather an extremely
complicated expression. In consequence, if we use the approximation (30) even
for finite $\Lambda$, one has to pay the price of introducing errors into the
initial condition as well as violating the smoothness of
$\bar{\Gamma}_{t}[\varphi]$ at $t=0$. In return one has a well-defined initial
condition $\mathcal{S}[\varphi]$ for the PDE for $\bar{\Gamma}_{t}[\varphi]$.
However, if $\Lambda$ is chosen to be much larger than all scales in
$\mathcal{S}[\phi]$, the errors from the initial condition are minor and
expected to be of magnitude

$\displaystyle\mathrm{error}\approx\frac{\text{largest scale
in\,\,}\mathcal{S}}{\Lambda}\,,$ (33)

We will come back to this issue in Sec. V in the context of RG consistency
[96, 97, 98, 99, 100].

Additionally, we will find that also the smoothness of
$\bar{\Gamma}_{t}[\varphi]$ is recovered automatically for all $t>0$ by the
structure of the PDE for $\bar{\Gamma}_{t}[\varphi]$, because it always
contains diffusive contributions which immediately smear out kinks in the
initial condition right in the first time step. We will also come back to this
issue later on, after we have derived the FRG equation (37) and discussed its
diffusive, irreversible character.

####  II.3.3 The Exact Renormalization Group equation

In analogy to the previous flow equations, the FRG equation, which is the flow
equation for $\bar{\Gamma}_{t}[\varphi]$, is obtained by taking the derivative
of $\bar{\Gamma}_{t}[\varphi]$ with respect to $t$ and using the definitions
(20) and (27) to express the derivative of $\bar{\Gamma}_{t}[\varphi]$ by the
scale-dependent Schwinger functional,

$\displaystyle\partial_{t}\bar{\Gamma}_{t}[\varphi]=\,$
$\displaystyle\partial_{t}\,\big{(}\Gamma_{t}[\varphi]-\Delta\mathcal{S}_{t}[\varphi]\big{)}=\vphantom{\bigg{(}\bigg{)}}$
(34) $\displaystyle=\,$
$\displaystyle\partial_{t}\,\big{(}J_{t}(\varphi)\,\varphi-\mathcal{W}_{t}[J_{t}(\varphi)]-\Delta\mathcal{S}_{t}[\varphi]\big{)}=\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle=\,$
$\displaystyle[\partial_{t}J_{t}(\varphi)]\,\varphi-\partial_{t}\mathcal{W}_{t}[J_{t}(\varphi)]-\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle-[\partial_{t}J_{t}(\varphi)]\,\mathcal{W}^{(1)}_{t,J_{t}}[J_{t}]-\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\varphi^{2}=\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle=\,$
$\displaystyle-\partial_{t}\mathcal{W}_{t}[J_{t}(\varphi)]-\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\varphi^{2}\,,\vphantom{\bigg{(}\bigg{)}}$

where we used the chainrule and Eq. (22).

We now use the flow equation for the Schwinger functional (19) to substitute
the first term on the right-hand side. Again employing the identity (22), the
last term in the last line of Eq. (34) cancels with the non-linear term in Eq.
(19), such that

$\displaystyle\partial_{t}\bar{\Gamma}_{t}[\varphi]=\,$
$\displaystyle\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\mathcal{W}_{t,JJ}^{(2)}[J_{t}(\varphi)]\,.$
(35)

It remains to replace the second derivative of the scale-dependent Schwinger
functional by a corresponding derivative of $\bar{\Gamma}_{t}[\varphi]$. This
is done via the identity

$\displaystyle 1=\frac{\delta
J_{t}(\varphi)}{\delta\varphi}\,\frac{\delta\varphi}{\delta
J_{t}(\varphi)}=\Gamma^{(2)}_{t,\varphi\varphi}[\varphi]\,\mathcal{W}^{(2)}_{t,JJ}[J_{t}(\varphi)]\,,$
(36)

which follows from Eqs. (22) and (23). Plugging this into Eq. (35) and using
Eq. (27) with Eq. (7) we obtain the FRG equation, Exact Renormalization Group
equation or Wetterich equation [35, 36, 37]

$\displaystyle\partial_{t}\bar{\Gamma}_{t}[\varphi]=\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\big{[}\bar{\Gamma}^{(2)}_{t,\varphi\varphi}[\varphi]+r(t)\big{]}^{-1}\,,$
(37)

which is a flow equation – a PDE – for the scale-dependent effective average
action $\bar{\Gamma}(t,\varphi)$ in the $(t,\varphi)$ plane,

$\displaystyle\partial_{t}\bar{\Gamma}(t,\varphi)=\frac{\frac{1}{2}\,\partial_{t}r(t)}{\partial_{\varphi}^{2}\bar{\Gamma}(t,\varphi)+r(t)}\,,$
(38)

with the initial condition $\bar{\Gamma}(t=0,\varphi)=\mathcal{S}[\varphi]$.
Some remarks are in order:

  1. 1.

In contrast to the PDEs for $\mathcal{Z}(t,J)$ and $\mathcal{W}(t,J)$ the FRG
equation can be initialized with a suitable initial condition at $t=0$ that
produces distinct flows for different actions $\mathcal{S}[\phi]$, as was
discussed in the previous subsubsection.

  2. 2.

The spatial boundary conditions, i.e., for $\varphi\rightarrow\pm\infty$ are
provided by the asymptotics of the FRG equation (38) itself and by the
requirement that $\mathcal{S}[\varphi]$ must be bounded from below: The action
$\mathcal{S}[\varphi]$ of an (interacting) field theory must at least grow
like $\varphi^{2}$ for large $|\varphi|$ and the dominant contribution for
large $|\varphi|$ must be even in $\varphi$. For actions
$\mathcal{S}[\varphi]$ that grow asymptotically faster than $\varphi^{2}$ the
denominator on the right-hand side of the PDE (38) already diverges at
$t\approx 0$, such that

$\displaystyle\lim\limits_{|\varphi|\rightarrow\infty}\partial_{t}\bar{\Gamma}(t,\varphi)\approx
0\,.$ (39)

It follows that for $|\varphi|\rightarrow\infty$ the function
$\bar{\Gamma}(t,\varphi)$ does not change at all, but keeps its initial value
$\mathcal{S}[\varphi]$. These are perfectly valid boundary conditions for a
PDE. The scenario for initial conditions with
$\lim\limits_{|\varphi|\rightarrow\infty}\mathcal{S}[\varphi]\sim\varphi^{2}$
is more delicate. We will return to this issue and a detailed discussion of
boundary conditions, when we discuss the numerical implementation and solution
of Eq. (38) in Sub.Sec. IV.4 in the context of numerical fluid dynamics.

  3. 3.

The structure of the PDE (38) is again a diffusion equation. In contrast to
the PDEs (12) and (19) it is non-linear in the second-order spatial
derivatives of $\Gamma(t,\varphi)$ that appear in the denominator. By applying
the same formalism to models with different field content, the FRG equation
can also acquire convective/advective terms and source terms. We will thus
find that the FRG equation shares many properties with other notable
advection-diffusion equations, e.g., the Navier-Stokes equation [101]. This is
discussed in Sec. IV, where our numerical approach to the FRG equation is
presented in more detail. However, it should be already mentioned at this
point that analyzing and solving non-linear advection-diffusion-source/sink
equations like Eq. (38) is a state-of-the-art problem in numerical
mathematics. Thus, some care is required in the search for well-established
numerical solution schemes for PDEs of this type.

  4. 4.

In zero dimensions, similar to the flow equations for $\mathcal{Z}(t,J)$ and
$\mathcal{W}(t,J)$, one can reparameterize the flow time $t$ in terms of $r$
in Eq. (38) and get rid of the prefactor $\partial_{t}r(t)$. Additionally, one
could eliminate $r(t)$ in the denominator in Eq. (38) by shifting
$\bar{\Gamma}(t,\varphi)\rightarrow\bar{\Gamma}(r,\varphi)-\tfrac{1}{2}\,r\,\varphi^{2}$
and switching from $t$ to $r$ as flow parameter, which corresponds to the
zero-dimensional analogue of the rescaled “dimensionless” flow equation in
fixed-point form, but is not suited for the practical calculations in this
work.

This reparameterization effectively corresponds to different choices of
regulator (shape) functions in zero dimensions. However, for higher-
dimensional problems, different choices of regulators do not need to be
related to each other via simple reparametrization of the RG time. In any
case, the effective dynamics in the PDE during the RG flow strongly depends on
the parametrization of the RG scale as well as the explicit choice of
regulator, which has two direct consequences: First, although the dynamics and
$t$ evolution of observables (the $n$-point correlation functions) during the
RG flow might be highly interesting and must also be studied to ensure that
the UV and IR cutoff scales are chosen appropriately, one must clearly state
that only the IR value of $\Gamma[\varphi]$ is mathematically and physically
meaningful and suitable for extracting information on the $n$-point
correlation functions. This is demonstrated and discussed again in the context
of numerical precision tests of the $O(N)$ model in Sec. V. Second, from a
numerical point of view, some parametrizations or choices of regulators might
be more challenging for the numerical integrators than others and must be
adopted to the specific problems at hand. On the level of the PDE this
corresponds to the time-dependent strength of the diffusion, see below.

  5. 5.

Unrelated to the present discussion, a formulation of the FRG equation using
mean fields carrying an explicit scale dependence (in higher dimensions often
related to a running wave-function renormalization) is also possible with a
careful consideration and distinction between total and partial derivatives
with respect to $t$. Generalizations including composite mean fields are also
possible, see, e.g., Ref. [73].

Using a zero-dimensional field theory with one degree of freedom, we have
therefore demonstrated that it is possible to transform the problem of solving
functional integrals like Eqs. (1) and (2) for a model with action
$\mathcal{S}[\phi]$ into solving the PDE (38) in $t$ and $\varphi$ with
initial condition $\mathcal{S}[\varphi]$. The FRG equation thus directly
implements the idea of transforming Gaussian-type functional integrals into
arbitrary functional integrals, but on the level of the effective action
$\Gamma[\varphi]$ rather than the partition function $\mathcal{Z}[J]$. Both
formulations of the problem of calculating $n$-point correlation functions –
the functional-integral formulation and the FRG formulation – are
mathematically equivalent. This, however, is, as we have seen, a highly non-
trivial statement and demands numerical precision tests, which are part of
this work.

In Ref. [73] it is shown that the FRG framework can be generalized to models
or theories with arbitrary field content in arbitrary dimensions and space-
time background (even a formulation for space-time itself, i.e., quantum
gravity is possible [85, 86], see Ref. [13] for a recent review).

Before we introduce the zero-dimensional $O(N)$ model and explain the relation
of the FRG to fluid dynamics, followed by our main discussion of zero-
dimensional QFTs as a testing ground for numerical methods and truncation
schemes, we discuss two further issues. The first contextualizes our previous
discussion with an interpretation of the FRG from the RG perspective (also for
higher-dimensional field theories). Furthermore, it briefly discusses the
generalization of the FRG equation to different field content. This can also
be found in Refs. [73, 74, 13, 87, 88, 102, 103, 104]. The second issue
discusses the relation between the $n$-point correlation functions of the
different generating functionals $\mathcal{Z}_{(t)}[J]$,
$\mathcal{W}_{(t)}[J]$, and $\Gamma_{(t)}[\varphi]$. This is needed for a
comparison of the exact results from the partition function $\mathcal{Z}[J]$
with our results from the FRG based on $\Gamma[\varphi]$. Readers familiar
with these issues may skip the following two sections.

###  II.4 Contextualization with FRG in higher-dimensional space-time

The structure of the FRG equation (38) is already very general and extends
with only minor modifications to arbitrary fields and dimensions. Derivations
can be found in, e.g., Refs. [73, 102, 88]. The FRG equation reads

$\displaystyle\partial_{t}\bar{\Gamma}_{t}[\Phi]=\mathrm{STr}\Big{[}\big{(}\tfrac{1}{2}\,\partial_{t}R_{t}\big{)}\,\big{(}\bar{\Gamma}^{(2)}_{t}[\Phi]+R_{t}\big{)}^{-1}\Big{]}\,.$
(40)

The supertrace in Eq. (40) entails sums over internal indices and different
fields and integrals over momenta, taking minus signs for fermionic fields
properly into account. The fundamental difference between ERG Eq. (40) and its
counterpart in zero dimensions (37), is that the ERG equation in $d>0$ is a
functional differential equation for the classical fields $\Phi$. It does not
naturally present as a PDE which necessitates truncations in practical
computation to project the ERG equation onto a finite set of coupled ODEs
and/or PDEs. The regulator $R_{t}$ for computations in $d>0$ is no longer a
simple scalar function but an operator with a particular, non-trivial
structure in position/momentum space. While different regulator choices are
still possible in higher dimensions, corresponding RG flows are no-longer
related by simple rescaling and a suitable regulator choice for the problem at
hand becomes particularity important when considering explicit truncated FRG
flow equations see, e.g., Refs. [70, 71]. More details can be found in, e.g.,
Refs. [73, 74, 87, 102, 103, 104, 88, 13, 105]. The equation is based upon
momentum locality, i.e., the integrand of the momentum integral on the right-
hand side is peaked around the RG scale $k\approx q$ (for conventional
regulators), see, e.g., Fig. 1 in Ref. [102] or Fig. 3.1 of Ref. [106], where
$q$ is the loop momentum and

$\displaystyle t=-\ln\big{(}\tfrac{k}{\Lambda}\big{)}\,.$ (41)

The FRG equation can be interpreted as a direct implementation of Wilson’s
approach to the RG [107, 108, 109].

In general, the space-time dimensionality has to be taken into account when
considering the convergence properties of different expansion schemes. For
example, the vertex expansion is believed to work very well for QCD in $d=4$
dimensions (see, e.g., Ref. [110] for a recent overview), however, as we will
discuss below, the convergence of the expansion is in general not guaranteed.
The vertex expansion is an expansion in terms of moments of the quantum
effective action, explained in detail in Sub.Sub.Sec. III.3.2. Here, the
moments are the irreducible parts of scattering kernels.

The convergence of this expansion is given by two main ingredients,

  1. 1.

phase-space suppression,

  2. 2.

finite couplings.

The first point means that higher-order vertices, which originate from quantum
effects and are typically not present in the classical action, come with
increasing suppression factors, e.g., due to the angular integrations. The
second point simply relates to the fact that all couplings have to stay
finite. Otherwise the argument related to phase-space suppression simply does
not work. There are several scenarios where this can be the case. The main one
being the presence of resonances, where couplings can be divergent. Also large
densities might circumvent the effect of phase-space suppression, but are not
our main concern in this work. The last, and for this work most important
effect, is that of the dimension.

In particular, for zero-dimensional space-time the angular integrations are
not present, and hence the entire argument of phase-space suppression does not
work. Zero-dimensional QFT is ultra-local – defined only in a single point –
and thus extremely coupled in field space. This, of course, has to be kept in
mind when considering convergence properties of vertex expansions.

Still, also a parallel work in $1+1$ spacetime dimensions by some of us and
collaborators [32] generically supports these statements and the increasing
importance of local interactions in low spacetime dimensions.

###  II.5 $n$-point correlation functions

In this section we discuss the scale-dependent correlation functions, which
can be extracted from the (scale-)dependent generating functionals
$\mathcal{Z}_{(t)}$, $\mathcal{W}_{(t)}$, and $\Gamma_{(t)}$. We restrict the
discussion to a zero-dimensional quantum theory with a single real scalar. The
concepts and expressions can be generalized to theories including arbitrary
fields and generalize to higher dimensions. For a broader discussion in the
context of QTFs we refer the interested reader to the textbooks [39, 38, 94,
111, 40]. For a comprehensive discussion of correlation functions and their
relations in the FRG see, e.g., Refs. [83, 103].

Correlation functions can be extracted by taking successive functional
derivatives of the generating functional, cf. Eq. (1):

$\displaystyle\langle\phi^{n}\rangle_{t,J}\equiv\frac{\mathcal{Z}^{(n)}_{t,J\cdots
J}[J]}{\mathcal{Z}_{t}[J]}\,.$ (42)

(The non-observable normalization, which we fixed by means of Eq. (4)
cancels.)

According to Eq. (24), the one-point correlation function

$\displaystyle\langle\phi\rangle_{t,J}=\frac{\mathcal{Z}^{(1)}_{t,J}[J]}{\mathcal{Z}_{t}[J]}=\mathcal{W}^{(1)}_{t,J}[J]$
(43)

equals the scale-dependent classical field $\varphi_{t}(J)$.

The two-point correlation function

$\displaystyle\langle\phi^{2}\rangle_{t,J}=\frac{\mathcal{Z}^{(2)}_{t,JJ}[J]}{\mathcal{Z}_{t}[J]}=\mathcal{W}^{(2)}_{t,JJ}[J]+\langle\phi\rangle_{t,J}^{2}$
(44)

is of particular interest in QFT since it is related to the transition
amplitude between two states. In $d>0$ such an amplitude between $\phi(x_{1})$
and $\phi(x_{2})$ encodes the particle motion between the space-time points
$x_{1}$ and $x_{2}$. $\langle\phi^{2}\rangle_{t,J}$ includes the
disconnected999“Connected” and “disconnected” in this context refers to the
connectivity of the Feynman-diagram representation of the correlation
functions. In a connected Feynman diagram all external lines are connected in
the diagram through at least one internal line.  contribution
$\langle\phi\rangle_{t,J}^{2}$. This information is already stored in the
$1$-point correlation function. Higher-order $n$-point correlation functions
include disconnected parts consisting of products of lower $m$-point functions
with $m<n$ [81]. The disconnected contributions correspond to scattering
processes where only a subset of the fields interact with each other and are
as such irrelevant for observables. Loosely speaking, $\mathcal{Z}_{t}[J]$
contains redundant information in the form of these disconnected diagrams.

The Schwinger functional $\mathcal{W}_{t}[J]$ does not contain this redundant
information. Functional derivatives of $\mathcal{W}_{t}[J]$ generate connected
$n$-point functions:

$\displaystyle\langle\phi^{n}\rangle^{c}_{t,J}\equiv\mathcal{W}^{(n)}_{t,J\ldots
J}[J]\,.$ (45)

The first two connected $n$-point functions are

$\displaystyle\langle\phi\rangle^{c}_{t,J}=\,$
$\displaystyle\langle\phi\rangle_{t,J}=\mathcal{W}^{(1)}_{t,J}[J]\,,\vphantom{\bigg{(}\bigg{)}}$
(46) $\displaystyle\langle\phi^{2}\rangle^{c}_{t,J}=\,$
$\displaystyle\langle\phi^{2}\rangle_{t,J}-\langle\phi\rangle_{t,J}^{2}=\mathcal{W}^{(2)}_{t,JJ}[J]\,.\vphantom{\bigg{(}\bigg{)}}$
(47)

Higher-order $n$-point functions are interpreted as interaction vertices. For
example, the connected three-point correlation function is given by

$\displaystyle\langle\phi^{3}\rangle^{c}_{t,J}=\,$
$\displaystyle\langle\phi^{3}\rangle_{t,J}-3\,\langle\phi^{2}\rangle_{t,J}\,\langle\phi\rangle_{t,J}+2\,\langle\phi\rangle_{t,J}^{3}\,.$
(48)

The Schwinger functional, as the generating functional of connected
correlation functions, still contains redundant information since connected
correlation functions can be decomposed into 1PI101010One-particle irreducible
(1PI) in this context refers to Feynman diagrams, which cannot be split into
two disconnected diagrams by cutting a single internal line.  vertex
functions. 1PI vertex functions encode all information about a QFT.

The effective action $\Gamma_{t}[\varphi]$ is the generating functional of 1PI
vertex functions [112, 94, 80, 111, 38, 39, 40, 113]. We now introduce a
central object in functional approaches to QFT: the full scale-dependent
propagator

$\displaystyle
G_{t}^{\varphi\varphi}[\varphi_{t}]\equiv\mathcal{W}^{(2)}_{t,J_{t}J_{t}}[J_{t}]=\big{(}\Gamma^{(2)}_{t,\varphi_{t}\varphi_{t}}[\varphi_{t}]\big{)}^{-1}\,,$
(49)

where the last equality follows from Eq. (36). Recalling Eqs. (23) and (24) we
then obtain

$\displaystyle\frac{\delta}{\delta J_{t}}=\frac{\delta\varphi_{t}}{\delta
J_{t}}\,\frac{\delta}{\delta\varphi_{t}}=\,$
$\displaystyle\mathcal{W}^{(2)}_{t,J_{t}J_{t}}[J_{t}]\,\frac{\delta}{\delta\varphi_{t}}\equiv
G_{t}^{\varphi\varphi}[\varphi_{t}]\,\frac{\delta}{\delta\varphi_{t}}\,,$ (50)

Here we dropped the explicit $\varphi$ ($J$) dependence of the source
realizing the supremum $J_{t}$ (the scale-dependent mean field $\varphi_{t}$)
for readability only and will do so for the remainder of this section.
Equation (50) is basically a chain rule, which allows to convert functional
$J_{t}$ derivatives into $\varphi_{t}$ derivatives. The correlation function
$\langle\phi^{n}\rangle_{t,J_{t}}$ for $n\geq 1$ can be rewritten by
successively pulling out functional $J_{t}$ derivatives,

$\displaystyle\langle\phi^{n}\rangle_{t,J_{t}}=\,$
$\displaystyle\frac{\mathcal{Z}^{(n)}_{t,J_{t}\cdots
J_{t}}[J_{t}]}{\mathcal{Z}_{t}[J_{t}]}=\bigg{(}\frac{\delta}{\delta
J_{t}}+\varphi_{t}\bigg{)}\,\frac{\mathcal{Z}^{(n-1)}_{t,J_{t}\ldots
J_{t}}[J_{t}]}{\mathcal{Z}_{t}[J_{t}]}=\vphantom{\bigg{(}\Bigg{)}}$
$\displaystyle=\,$
$\displaystyle\Bigg{(}\prod_{i=1}^{n-1}\bigg{(}\frac{\delta}{\delta
J_{t}}+\varphi_{t}\bigg{)}\Bigg{)}\,\varphi_{t}\,,\vphantom{\bigg{(}\bigg{)}}$
(51)

where the $\varphi_{t}$ terms account for the derivatives of the normalization
$1/\mathcal{Z}_{t}[J_{t}]$. Using the chain rule (50) in Eq. (51) we arrive at

$\displaystyle\langle\phi^{n}\rangle_{t,J_{t}}=\Bigg{(}\prod_{i=1}^{n-1}\bigg{(}G_{t}^{\varphi\varphi}[\varphi_{t}]\,\frac{\delta}{\delta\varphi_{t}}+\varphi_{t}\bigg{)}\Bigg{)}\,\varphi_{t}\,,$
(52)

which expresses the correlation function $\langle\phi^{n}\rangle_{t,J_{t}}$
completely in terms of $\varphi_{t}$, $G_{t}^{\varphi\varphi}[\varphi_{t}]$,
and 1PI vertices for $n\geq 3$. The higher ($n\geq 3$) 1PI vertices
$\Gamma^{(n)}_{t,\varphi_{t}\cdots\varphi_{t}}[\varphi_{t}]$ emerge in Eq.
(52) from the functional derivatives of the propagator. Taking the
$\varphi_{t}$ derivative of Eq. (36) (for $\varphi\equiv\varphi_{t}$, $J\equiv
J_{t}$), we derive

$\displaystyle\frac{\delta}{\delta\varphi_{t}}\,G_{t}^{\varphi\varphi}[\varphi_{t}]=\,$
$\displaystyle\frac{\delta}{\delta\varphi_{t}}\,\big{(}\Gamma^{(2)}_{t,\varphi_{t}\varphi_{t}}[\varphi_{t}]\big{)}^{-1}\vphantom{\bigg{(}\bigg{)}}$
(53) $\displaystyle=\,$ $\displaystyle-
G_{t}^{\varphi\varphi}[\varphi_{t}]\,\Gamma^{(3)}_{t,\varphi_{t}\varphi_{t}\varphi_{t}}[\varphi_{t}]\,G_{t}^{\varphi\varphi}[\varphi_{t}]\,,\vphantom{\bigg{(}\bigg{)}}$

where we have used Eq. (49) and where

$\displaystyle\frac{\delta}{\delta\varphi_{t}}\,\Gamma^{(n)}_{t,\varphi_{t}\cdots\varphi_{t}}[\varphi_{t}]=\Gamma^{(n+1)}_{t,\varphi_{t}\cdots\varphi_{t}\varphi_{t}}[\varphi_{t}]\,.$
(54)

From the definition (45) and Eq. (50) it is even simpler to derive

$\displaystyle\langle\phi^{n}\rangle_{t,J_{t}}^{c}=\,$
$\displaystyle\Bigg{(}\prod_{i=1}^{n-1}\bigg{(}G_{t}^{\varphi\varphi}[\varphi_{t}]\,\frac{\delta}{\delta\varphi_{t}}\bigg{)}\Bigg{)}\,\varphi_{t}\,,$
(55)

which establishes a decomposition of connected correlation functions in terms
of $\varphi_{t}$, $G_{t}^{\varphi\varphi}[\varphi_{t}]$, and 1PI vertices for
$n\geq 3$. Equation (55) is simpler than Eq. (52) because disconnected
contributions arising from the term $\sim\varphi_{t}$ in the parenthesis in
Eq. (52) are absent.

In terms of $\Gamma_{t}[\varphi_{t}]$ the first three (connected) correlation
functions are given by

$\displaystyle\langle\phi^{1}\rangle_{t,J_{t}}=\,$
$\displaystyle\langle\phi^{1}\rangle_{t,J_{t}}^{c}=\varphi_{t}\,,\vphantom{\bigg{(}\bigg{)}}$
(56) $\displaystyle\langle\phi^{2}\rangle_{t,J_{t}}^{c}=\,$ $\displaystyle
G_{t}^{\varphi\varphi}[\varphi_{t}]\,,\vphantom{\bigg{(}\bigg{)}}$ (57)
$\displaystyle\langle\phi^{2}\rangle_{t,J_{t}}=\,$ $\displaystyle
G_{t}^{\varphi\varphi}[\varphi_{t}]+\varphi_{t}^{2}\,,\vphantom{\bigg{(}\bigg{)}}$
(58) $\displaystyle\langle\phi^{3}\rangle_{t,J_{t}}^{c}=\,$
$\displaystyle-\big{(}G_{t}^{\varphi\varphi}[\varphi_{t}]\big{)}^{3}\,\Gamma^{(3)}_{t,\varphi_{t}\varphi_{t}\varphi_{t}}[\varphi_{t}]\,,\vphantom{\bigg{(}\bigg{)}}$
(59) $\displaystyle\langle\phi^{3}\rangle_{t,J_{t}}=\,$
$\displaystyle-\big{(}G_{t}^{\varphi\varphi}[\varphi_{t}]\big{)}^{3}\,\Gamma^{(3)}_{t,\varphi_{t}\varphi_{t}\varphi_{t}}[\varphi_{t}]+\vphantom{\bigg{(}\bigg{)}}$
(60)
$\displaystyle+3\,G_{t}^{\varphi\varphi}[\varphi_{t}]\varphi_{t}+\varphi_{t}^{3}\,.\vphantom{\bigg{(}\bigg{)}}$

We will need these relations among the different $n$-point correlation
functions to compare our numerical results from solving the RG flow equation
with fluid-dynamical methods to the direct computation of the correlation
functions from the partition function $\mathcal{Z}[J]$.

##  III The $O(N)$ model in zero dimensions and its treatment within the FRG

Zero-dimensional $O(N)$ models are predominantly studied for pedagogical and
conceptual purposes [45, 46, 47, 48, 49, 50, 62, 53, 44, 54, 52, 51, 55, 56,
58, 60, 63, 64]. In Ref. [44] the model was used to compare the quality of
perturbation theory, the large-$N$ expansion, and the FRG vertex/Taylor
expansion with the exact result. The primary focus of the present work is to
push this analysis even further and to study the limits of untruncated RG flow
equations as well as the FRG Taylor expansion.

$O(N)$ models in higher dimensions play an important role in understanding
spin systems, like the Ising model [114, 115, 104], and magnetization
phenomena. Furthermore, they are often used as toy models and are of utmost
importance for understanding the Anderson-Brout-Englert-Guralnik-Hagen-Higgs-
Kibble mechanism and the formation of a chiral condensate in strong-
interaction matter. In the context of numerical methods for the FRG, two of us
used the $O(N)$ model in $d=3$ to study numerical solutions of RG flow
equations in the large-$N$ limit [26].

This section is structured as follows: In Sub.Sec. III.1 we introduce the
$O(N)$ model on the level of the classical action and the functional integral.
We further comment on the calculation of expectation values and 1PI vertex
functions from the functional integral, which are our observables of interest.
Thereafter, in Sub.Sec. III.2, we comment on symmetry restoration during the
RG flow, for scenarios in which the classical action
$\mathcal{S}[\vec{\varphi}\,]=U(t_{\mathrm{UV}},\vec{\varphi}\,)$ possesses a
non-trivial minimum. In Sub.Sec. III.3, we introduce the exact FRG formulation
of the model, which includes the derivation of the RG flow equation as an
exact PDE and generalization of Eq. (38). We close this section by deriving
the FRG Taylor expansion for the $O(N)$ model, which is a commonly used
expansion scheme in FRG studies.

###  III.1 The zero-dimensional $O(N)$ model

Consider a zero-dimensional theory of $N$ bosonic scalars $\phi_{a}$, which
transform according to

$\displaystyle\phi_{a}\mapsto\phi^{\prime}_{a}=O_{ab}\,\phi_{b}\,,$ (61)

where $O\in O(N)$ and $a,b\in\\{1,\,\ldots,\,N\\}$. In vector notation, this
reads

$\displaystyle\vec{\phi}\mapsto\vec{\phi}^{\,\prime}=O\,\vec{\phi}\,,$ (62)

where $\vec{\phi}=(\phi_{1},\,\phi_{2},\,\ldots,\,\phi_{N})$. If the action
$\mathcal{S}[\vec{\phi}\,]$ of the model possesses an $O(N)$ symmetry, it can
contain all possible terms that are functions of the $O(N)$ invariant

$\displaystyle\rho\equiv\tfrac{1}{2}\,\phi_{a}\,\phi_{a}\equiv\tfrac{1}{2}\,\vec{\phi}^{\,2}$
(63)

This implies that the most general action obeying this symmetry is given by

$\displaystyle\mathcal{S}[\vec{\phi}\,]=U(\vec{\phi}\,)=U(\rho)\,,$ (64)

where $U(\rho)$ is the effective potential, in analogy to models from higher-
dimensional space-times. This effective potential might for example include a
bosonic “mass term” $m^{2}\rho$ as well as other interaction terms containing
arbitrary powers of $\rho$. Although one may now be tempted to assume that the
effective potential $U(\rho)$ must be a power series or an analytic function
of $\rho$, as long as it fulfills all symmetries it can be any continuous
function of $\rho$ which is bounded from below, cf. the discussion in Sec. II
for the special case of the $O(1)$ model.

In the remainder of this section we will summarize relevant relations for the
$O(N)$ model. For a more detailed discussion, we refer the interested reader
to Ref. [44] and references therein.

All generating functionals of the theory retain the $O(N)$ symmetry of the
action, which makes them functionals of the invariants
$\tfrac{1}{2}\,\vec{J}^{\,2}$ for $\mathcal{Z}$ and $\mathcal{W}$ and
$\varrho\equiv\frac{1}{2}\,\vec{\varphi}^{\,2}$ for $\Gamma$. This entails
that all $n$-point correlation functions for odd $n$ vanish by symmetry and
all $n$-point correlation functions of a given order of even $n$ are
proportional to each other, e.g., for the four-point function we find

$\displaystyle\langle\phi_{i}\,\phi_{i}\,\phi_{j}\,\phi_{j}\rangle=\tfrac{1}{3}\,\langle\phi_{i}\,\phi_{i}\,\phi_{i}\,\phi_{i}\rangle\,,$
(65)

for $i\neq j$ and $i,j\in\\{1,\ldots,N\\}$ (no summation over repeated indices
implied here). For the proof, use that $\tfrac{\delta}{\delta
J_{i}}\mathcal{Z}\big{(}\tfrac{1}{2}\vec{J}^{\,2}\big{)}=J_{i}\,\mathcal{Z}^{\prime}\big{(}\tfrac{1}{2}\,\vec{J}^{\,2}\big{)}$
and set the source $\vec{J}=0$ at the end of the calculation. Using the $O(N)$
symmetry on the right-hand side of

$\displaystyle\langle\phi_{i_{1}}\,\cdots\,\phi_{i_{n}}\rangle=\vphantom{\bigg{(}\bigg{)}}$
(66) $\displaystyle=\,$
$\displaystyle\frac{1}{\mathcal{Z}[0]}\,\int_{-\infty}^{\infty}\mathrm{d}^{N}\phi\,\phi_{i_{1}}\,\cdots\,\phi_{i_{n}}\,\mathrm{e}^{-U(\vec{\phi}^{\,2}/2)}\,,\vphantom{\bigg{(}\bigg{)}}$

one can relate correlation functions of even order $2n$ to the expectation
value $\langle(\vec{\phi}^{\,2})^{n}\rangle$. For the two-, four-, and six-
point functions, which are studied in this work, we find

$\displaystyle\langle\phi_{i}\,\phi_{j}\,\rangle=\,$
$\displaystyle\frac{1}{N}\,\delta_{ij}\,\langle\vec{\phi}^{\,2}\rangle\,,\vphantom{\bigg{(}\bigg{)}}$
(67) $\displaystyle\langle\phi_{i}\,\phi_{j}\,\phi_{k}\,\phi_{l}\rangle=\,$
$\displaystyle\frac{1}{N(N+2)}\,(\delta_{ij}\,\delta_{kl}+\delta_{ik}\,\delta_{jl}+\delta_{il}\,\delta_{jk})\,\langle(\vec{\phi}^{\,2})^{2}\rangle\,,\vphantom{\bigg{(}\bigg{)}}$
(68)
$\displaystyle\langle\phi_{i}\,\phi_{j}\,\phi_{k}\,\phi_{l}\,\phi_{m}\,\phi_{n}\rangle=\,$
$\displaystyle\frac{1}{N(N+2)(N+4)}\,(\delta_{ij}\,\delta_{kl}\,\delta_{mn}+\mathrm{all\
permutations})\,\langle(\vec{\phi}^{\,2})^{3}\rangle\,.\vphantom{\bigg{(}\bigg{)}}$
(69)

Connected correlation functions and 1PI vertex functions are related to
correlation functions as outlined in Sub.Sec. II.5. Using the fact that, for
odd $n$, all $n$-point correlation functions and all $n$-point 1PI vertex
functions vanish by symmetry, the following relations hold for the two-,
four-, and six-point functions (no summation over repeated indices):

$\displaystyle\langle\phi_{i}\,\phi_{i}\rangle^{c}=\,$
$\displaystyle\langle\phi_{i}\,\phi_{i}\rangle=\big{(}\Gamma^{(2)}_{\varphi_{i}\varphi_{i}}\big{)}^{-1}\,,\vphantom{\bigg{(}\bigg{)}}$
(70)
$\displaystyle\langle\phi_{i}\,\phi_{i}\,\phi_{i}\,\phi_{i}\rangle^{c}=\,$
$\displaystyle\langle\phi_{i}\,\phi_{i}\,\phi_{i}\,\phi_{i}\rangle-3\,\langle\phi_{i}\,\phi_{i}\rangle^{2}=-\langle\phi_{i}\phi_{i}\rangle^{4}\,\Gamma^{(4)}_{\varphi_{i}\varphi_{i}\varphi_{i}\varphi_{i}}\,,\vphantom{\bigg{(}\bigg{)}}$
(71)
$\displaystyle\langle\phi_{i}\,\phi_{i}\,\phi_{i}\,\phi_{i}\,\phi_{i}\,\phi_{i}\rangle^{c}=\,$
$\displaystyle\langle\phi_{i}\,\phi_{i}\,\phi_{i}\,\phi_{i}\,\phi_{i}\,\phi_{i}\rangle-15\,\langle\phi_{i}\,\phi_{i}\,\phi_{i}\,\phi_{i}\rangle\,\langle\phi_{i}\,\phi_{i}\rangle+30\,\langle\phi_{i}\,\phi_{i}\rangle^{3}=\vphantom{\bigg{(}\bigg{)}}$
(72) $\displaystyle=\,$
$\displaystyle-\langle\phi_{i}\,\phi_{i}\rangle^{6}\,\Gamma^{(6)}_{\varphi_{i}\varphi_{i}\varphi_{i}\varphi_{i}\varphi_{i}\varphi_{i}}+10\,\langle\phi_{i}\,\phi_{i}\rangle^{-1}\,(\langle\phi_{i}\,\phi_{i}\,\phi_{i}\,\phi_{i}\rangle^{c})^{2}\,.\vphantom{\bigg{(}\bigg{)}}$

Inserting Eqs. (67) – (69) into Eqs. (70) – (72) and solving for the 1PI
vertex functions yields

$\displaystyle\Gamma^{(2)}\equiv\,$
$\displaystyle\Gamma^{(2)}_{\varphi_{i}\varphi_{i}}=N\,\frac{1}{\langle\vec{\phi}^{\,2}\rangle}\,,\vphantom{\bigg{(}\bigg{)}}$
(73) $\displaystyle\Gamma^{(4)}\equiv\,$
$\displaystyle\Gamma^{(4)}_{\varphi_{i}\varphi_{i}\varphi_{i}\varphi_{i}}=3\,N^{2}\,\frac{1}{\langle\vec{\phi}^{\,2}\rangle^{2}}\,\bigg{[}1-\frac{N}{N+2}\,\frac{\langle(\vec{\phi}^{\,2})^{2}\rangle}{\langle\vec{\phi}^{\,2}\rangle^{2}}\bigg{]}\,,\vphantom{\bigg{(}\bigg{)}}$
(74) $\displaystyle\Gamma^{(6)}\equiv\,$
$\displaystyle\Gamma^{(6)}_{\varphi_{i}\ldots\varphi_{i}}=60\,N^{3}\,\frac{1}{\langle\vec{\phi}^{\,2}\rangle^{3}}\bigg{[}1-\frac{9\,N}{4\,(N+2)}\,\frac{\langle(\vec{\phi}^{\,2})^{2}\rangle}{\langle\vec{\phi}^{\,2}\rangle^{2}}+\frac{3\,N^{2}}{2\,(N+2)^{2}}\,\frac{\langle(\vec{\phi}^{\,2})^{2}\rangle^{2}}{\langle\vec{\phi}^{\,2}\rangle^{4}}-\frac{N^{2}}{4\,(N+2)\,(N+4)}\,\frac{\langle(\vec{\phi}^{\,2})^{3}\rangle}{\langle\vec{\phi}^{\,2}\rangle^{3}}\bigg{]}\,.\vphantom{\bigg{(}\bigg{)}}$
(75)

In summary, computing arbitrary correlation functions (or 1PI vertex
functions) of the zero-dimensional $O(N)$ model boils down to computing
expectation values $\langle(\vec{\phi}^{\,2})^{n}\rangle$. The latter can be
computed using Eq. (66). Because of the $O(N)$ symmetry of the integrand, this
is most easily done in spherical coordinates. Performing the integration over
spherical coordinates, we have

$\displaystyle\int_{-\infty}^{\infty}\mathrm{d}\phi_{1}\cdots\int_{-\infty}^{\infty}\mathrm{d}\phi_{N}=\frac{2\,\pi^{\frac{N}{2}}}{\Gamma\big{(}\frac{N}{2}\big{)}}\int_{0}^{\infty}\mathrm{d}\rho\,(2\rho)^{\frac{N}{2}-1}\,,$
(76)

Then the expectation value is a simple one-dimensional integral,

$\displaystyle\langle(\vec{\phi}^{\,2})^{n}\rangle=\frac{2^{n}\int_{0}^{\infty}\mathrm{d}\rho\,\rho^{\frac{N}{2}-1}\,\rho^{n}\,\mathrm{e}^{-U(\rho)}}{\int_{0}^{\infty}\mathrm{d}\rho\,\rho^{\frac{N}{2}-1}\,\mathrm{e}^{-U(\rho)}}\,.$
(77)

For certain potentials $U(\rho)$, the integral (77) can even be computed
symbolically in terms of known functions [44, 56, 31], whereas for general
$U(\rho)$ a numerical evaluation to high precision is straightforward using
standard methods [42, 43]. Thus, the zero-dimensional $O(N)$ model is an ideal
testing ground for alternative methods to calculate correlation functions,
such as, e.g., the FRG.

###  III.2 Symmetry restoration during the RG flow

Besides being invariant under $O(N)$ transformations the classical action
(potential) $\mathcal{S}[\vec{\phi}\,]=U(\vec{\phi}\,)$ is also invariant
under the discrete $\mathbb{Z}_{2}$ transformation

$\displaystyle\phi_{a}\rightarrow-\phi_{a}\,,$ (78)

which, as already mentioned above, implies that all $n$-point functions with
odd $n$ vanish, e.g., the one-point function
$\varphi_{a}=\langle\phi_{a}\rangle=0$.

However, it is possible to consider actions (potentials)
$\mathcal{S}[\rho]=U(\rho)$ which possess non-trivial minima $\rho_{0}\neq 0$.
This means that the RG flow of $\bar{\Gamma}_{t}[\vec{\varphi}\,]$ of such
models is initialized in a symmetry-broken regime in the UV, where the $O(N)$
symmetry is broken to its $O(N-1)$ subgroup. (For the $O(1)$ model, this
reduces to a breaking of the $\mathbb{Z}_{2}$ symmetry.) Following the
discussion in App. B, this property of the classical action neither translates
to the full quantum effective action $\Gamma[\vec{\varphi}\,]$ in the IR nor
to the $n$-point functions, due to a limiting case of the Coleman-Mermin-
Wagner-Hohenberg theorem [67, 68, 69]. The theorem states that there is no
long-range order in $d\leq 2$ dimensions if the interactions between the
constituents are sufficiently short of range. Therefore, there is no breaking
of a (continuous) symmetry in such systems in the IR, i.e., after integrating
out all quantum fluctuations, even when starting with a classical action in
the UV that has non-trivial minima. This is the equivalent of the statement
that $\varphi_{a}=\langle\phi_{a}\rangle=0$. The “Nambu-Goldstone modes” [116,
117, 112]111111We put the term “Nambu-Goldstone modes” in quotation marks,
because in zero dimensions the concept of “massless modes” can only refer to
the curvature masses in the corresponding bosonic field direction, which are
obtained from the effective potential $U(\rho)$. But the actual particle
masses in a higher-dimensional QFT are derived from the poles of the real-time
propagators, which simply do not exist in zero dimensions. , which we will
also call pions121212We adopt the high-energy terminology. Condensed-matter
physicists associate the pions with quasiparticles – the Anderson-Bogoliubov
modes. $\vec{\pi}$ in the zero-dimensional $O(N)$ model, and the radial
$\sigma$ mode “vaporize” any condensate and smear out all cusps in
$\bar{\Gamma}_{t}[\vec{\varphi}\,]$ during the RG flow. In the IR all modes
are then “massive” again.

There are two reasons, why this feature of symmetry restoration on the level
of $\bar{\Gamma}_{t}[\vec{\varphi}\,]$ is desirable for our numerical tests:

  1. 1.

Symmetry breaking/restoration associated with condensation/“vaporization” is
an essential property of all kinds of QFTs [38, 39, 40] and we have to show
that it is correctly captured by our numerical tools. This is especially
important, because it was shown by two of us and collaborators [26, 27] that
non-analytic behavior in the effective potential $U(t,\vec{\varphi}\,)$, cf.
Refs. [22, 118], which is directly associated with dynamical symmetry
breaking/restoration, is realized as shock and rarefaction waves in field
space during the RG flow.

  2. 2.

The possibility of dynamical symmetry restoration on the level of
$\bar{\Gamma}_{t}[\vec{\varphi}\,]$ is also a desired feature in order to
demonstrate that it is of utmost importance to choose the UV cutoff $\Lambda$
and the IR cutoff $r_{\mathrm{IR}}$ as well as initial and boundary conditions
in numerical FRG-flow calculations carefully. For our example it is expected
that if the IR cutoff time $t_{\mathrm{IR}}$ is chosen too small, such that
the regulator $r(t)$ is still too large, the system might still be in the
symmetry-broken phase (indicated by a non-trivial minimum). This means that
the scale-dependent effective average action
$\bar{\Gamma}_{t_{\mathrm{IR}}}[\vec{\varphi}\,]$ at this RG scale cannot be
interpreted as the full quantum effective action $\Gamma[\vec{\varphi}\,]$,
because the Coleman-Mermin-Wagner-Hohenberg theorem is still violated. The
same applies to a problematic implementation boundary conditions, especially
at $\varrho=0$, which can lead to a violation of the Coleman-Mermin-Wagner-
Hohenberg theorem, such that the system is not in the restored phase in the
IR.

For a direct physical consequences of these subtleties, we refer to the
parallel works [32, 31] by two of us and collaborators.

In a follow-up publication [34], we will generalize the zero-dimensional
$O(N)$ model to a model involving fermions (Grassmann numbers) and bosons. The
more complicated interactions may also allow for dynamical symmetry breaking
via attractive fermion interactions during the RG flow. Of course, the system
must return to the restored phase in the limit $t\to\infty$.

###  III.3 FRG formulation and flow equations

This subsection is dedicated to the FRG formulation of the $O(N)$ model of the
previous Sub.Sec. III.1. To this end, we demonstrate how to arrive at the
exact untruncated RG flow equation of the $O(N)$ model. Furthermore, we
introduce a commonly used truncation scheme for RG flow equations – the FRG
Taylor expansion, see, e.g., Refs. [87, 102, 104, 88, 44, 74, 13]. We start
our discussion with general remarks on the derivation of RG flow equations and
truncation schemes.

From Sec. II and especially Sub.Secs. II.3 and II.4 we have learned that the
FRG equation (40) constitutes an exact PDE for the RG time evolution of the
full field-dependent effective average action $\bar{\Gamma}_{t}[\Phi]$ with
initial condition $\bar{\Gamma}_{t=0}[\Phi]=\mathcal{S}[\Phi]$. Here, $\Phi$
stands for the field space vector of all fields of the specific model under
consideration. However, if there is more than one field space degree of
freedom, the direct (numerical) solution of the FRG equation (40) as a PDE is
exceedingly difficult, because of the high dimensionality of the field space.
In higher space-time dimensions, space-time or momentum dependences of the
fields complicate this issue and promote Eq. (40) to a functional integro-
partial-differential equation with a functional $\mathcal{S}[\Phi(x)]$ or
$\mathcal{S}[\tilde{\Phi}(p)]$ as initial condition.

Instead of solving Eq. (40) directly (independent of the dimensionality and
the field content), one usually specifies some ansatz function for the
effective average action $\bar{\Gamma}_{t}[\Phi]$, which involves only a
finite number of $t$ dependent couplings (vertices). The ansatz function for
$\bar{\Gamma}_{t}[\Phi]$ must respect all symmetries of the model and the
functional integral. Afterwards, one works out a projection prescription,
which extracts these couplings from $\bar{\Gamma}_{t}[\Phi]$. Usually this is
done by

  1. 1.

Taking a suitable number of (functional) derivatives in field (and/or
momentum) space,

  2. 2.

Evaluating the resulting expression on a specific (usually constant) field
configuration (and/or at specific external momenta, energies etc.),

  3. 3.

Applying contractions of open field space and space-time indices with suitable
tensors.

Thus, inserting the ansatz for $\bar{\Gamma}_{t}[\Phi]$ into the FRG equation
(40) and applying these projection rules to both sides of the equation yields
a coupled set of PDEs and/or ODEs for the couplings. This system of
differential equations must be initialized at $t=0$ with the values of the
couplings taken from the specific choice of the classical action
$\mathcal{S}[\Phi]$. The system for the $t$ dependent couplings is then
evolved to $t\rightarrow\infty$. If needed, the values of the couplings at
$t\rightarrow\infty$ can afterwards be reinserted in $\bar{\Gamma}_{t}[\Phi]$
to obtain the effective action $\Gamma[\Phi]$ in the IR. We will present this
procedure explicitly for the zero-dimensional $O(N)$ model in the next
paragraphs.

However, by considering an ansatz function for $\bar{\Gamma}_{t}[\Phi]$, which
consists of a finite number (of usually an infinite set) of all the possible
interaction terms that respect the symmetries of the system, one effectively
introduces an approximation. In the context of the FRG this is called a
truncation. The concept of a truncation of the system can directly be seen
from Eq. (40): Taking an appropriate number of field space derivatives of this
equation to project on a specific coupling, the right-hand side of this
equation depends on higher-order interaction vertices. These are up to two
orders higher than the ones on the left-hand side, because of
$\bar{\Gamma}_{t}^{(2)}[\Phi]$ already involves two field space derivatives.
The highest-order couplings in the system of PDEs for the couplings are,
however, set to zero by definition via the ansatz for
$\bar{\Gamma}_{t}[\Phi]$, because only a finite number of couplings is evolved
with $t$. As a result Eq. (40), which originally corresponds to a coupled
system of infinitely many ODEs and PDEs for couplings of all orders in field
and momentum or position space, is reduced to a finite set of PDEs for the
couplings involved in the ansatz for $\bar{\Gamma}_{t}[\Phi]$, see Refs. [87,
102, 104, 88, 74, 13] for general discussions or, e.g., Refs. [119, 120, 121,
18, 14] for specific applications. After all, the quality of the ansatz
completely determines the quality of the approximation to the actual IR
effective action $\Gamma[\Phi]$ after the RG flow of the truncated system is
solved.

In general, finding reliable truncations for a given problem is a challenging
problem. In particular, the identification of a small parameter to justify the
truncations is a difficult task. In fact, such a parameter may not even exist.
It may also turn out that a given truncation yields reliable results for one
observable but not for another. The latter observation may even be considered
a feature as it allows to identify mechanisms underlying specific phenomena.
In any case, there are construction schemes for systematic ansätze for the
effective action. Commonly used truncation schemes are for example the
derivative expansion [87, 122, 115, 123], which relies on the expansion of
$\bar{\Gamma}_{t}[\Phi]$ in powers of derivatives (momenta) but includes all
orders of field-dependent vertices at the same momentum order. Another
expansion scheme is the vertex expansion, which expands
$\bar{\Gamma}_{t}[\Phi]$ in terms of (momentum-dependent) $n$-point functions.
Oftentimes different expansion schemes are combined, in order to keep the
system of PDEs tractable [18, 19, 20, 14]. Moreover, truncations can always be
benchmarked against perturbative studies, see, e.g., Refs. [124, 102] for
instructive examples.

One measure for the quality of these expansion schemes is comparing terms of
different order. It is expected and can also be observed for certain systems
and situations, see e.g., Refs. [125, 15, 126, 127, 128, 16, 14], that the
expansions seem to converge and deviations in the observables are decreasing
by increasing the expansion order. In the FRG community, this is often
referred to as apparent convergence. Another indication for the quality of the
truncation is the comparison of FRG results with results from other methods
[23, 129, 22, 130, 131, 17], e.g., Monte-Carlo simulations, or the comparison
of critical exponents derived from the FRG and other methods.

In this context, zero-dimensional QFTs play a very special role: Due to the
absence of space-time and momentum dependences of the fields, the effective
average action $\bar{\Gamma}_{t}[\Phi]=\bar{\Gamma}_{t}(\Phi)$ is merely a
function (not a functional) of the fields $\Phi$ and of the $t$ dependent
couplings accompanying all possible terms which respect the symmetry of the
model. This structure can, however, be summarized in terms of effective $\Phi$
and $t$ dependent terms. It is therefore possible to express the effective
average action in terms of a finite amount of terms, which nevertheless
incorporate all possible interactions to all orders in the fields and do not
even need to be analytic functions of the fields. In consequence, truncating
the system is superfluous and the PDEs, which are derived via projections from
the FRG equation, constitute an exact and complete system. Solving this system
must therefore lead to the exact effective action $\Gamma[\Phi]$ in the IR and
is therefore completely equivalent to solving the functional integral. In
other words, calculating $n$-point correlation functions via the (functional)
integral or via the FRG equation (if done properly) must yield identical
results without truncation errors.

This feature makes zero-dimensional QFT particularly interesting for several
reasons:

  1. 1.

It can be used to test the quality of numerical schemes which are used to
solve the flow equations.

  2. 2.

It can be used to estimate the errors resulting from the choices of various
parameters entering the RG flow equations like UV and IR cutoff scales, etc..

  3. 3.

It can be used to test commonly used truncation schemes by artificially
truncating the system to a non-complete set of ordinary first-order
differential equations.

All these tests can be performed on a quantitative level, by studying the
relative errors of the FRG results for $n$-point correlation functions
compared to the exact results from the functional integral. We provide results
for various precision tests in Sec. V.

For the remainder of this section, we will proceed as follows: First, we will
derive the untruncated exact RG flow equation for the zero-dimensional $O(N)$
model. Afterwards, we introduce a commonly used truncation scheme – the FRG
Taylor (vertex) expansion.

####  III.3.1 The exact RG flow equation of the zero-dimensional $O(N)$ model

For the special case of the zero-dimensional $O(N)$ model, the most general
ansatz for the effective average action is given by a scale-dependent
effective potential

$\displaystyle\bar{\Gamma}_{t}[\vec{\varphi}\,]=U(t,\vec{\varphi}\,)=U(t,\varrho)\,.$
(79)

This ansatz can describe arbitrary $O(N)$ invariant effective actions and can
include terms at all orders of $\varrho=\tfrac{1}{2}\,\vec{\varphi}^{\,2}$.
However, it is in principle not restricted to analytic (Taylor-expandable)
functions. Truncations of $\bar{\Gamma}_{t}[\vec{\varphi}\,]$ are not
required.

In order to arrive at the exact flow equation for $U(t,\vec{\varphi}\,)$ one
has to perform the following steps:

  1. 1.

Insert the function (79) into the FRG equation (40).

  2. 2.

Invert the full field-dependent two-point function

$\displaystyle\big{(}\bar{\Gamma}^{(2)}_{t,\varphi\varphi}[\vec{\varphi}\,]+R_{t}\big{)}_{ij}\,.$
(80)

  3. 3.

Take the trace in field space.

  4. 4.

Remove the redundant $N-1$ field space directions in $\vec{\varphi}$.

For the last step, the RG flow equation can be evaluated on a constant
background field configuration131313Here we adopt terminology from higher-
dimensional FRG: The word “constant” is therefore somewhat misleading in a QFT
which cannot vary in space-time, but it is used anyhow.
$\varphi_{1}=\ldots=\varphi_{N-1}=0$ and $\varphi_{N}=\sigma$. Without loss of
generality, the $\varphi_{N}$ direction was singled out as the direction of
the radial $\sigma$ mode and the constant background field.

The inversion of the full field-dependent two-point function (80) can be
performed analytically [28, 132, 103, 104] by introducing the complete,
orthogonal, and idempotent field space projection operators

$\displaystyle\mathcal{P}^{\perp}_{ij}(\vec{\varphi}\,)\equiv\delta_{ij}-\frac{\varphi_{i}\,\varphi_{j}}{\vec{\varphi}^{\,2}}\,,$
$\displaystyle\mathcal{P}^{\parallel}_{ij}(\vec{\varphi}\,)\equiv\frac{\varphi_{i}\,\varphi_{j}}{\vec{\varphi}^{\,2}}\,.$
(81)

The projection operators are used to decompose the full field-dependent two-
point function (80) into components perpendicular ($\perp$) and parallel
($\parallel$) to $\vec{\varphi}$, which can be inverted separately. The
regulator $R_{t}$ is matrix-valued and diagonal in field space,

$\displaystyle(R_{t})_{ij}=\delta_{ij}\,r(t)\,,$ (82)

where $r(t)$ again is denoted as regulator shape function, cf. Eqs. (7) and
(8). One finds that

$\displaystyle\big{(}\bar{\Gamma}^{(2)}_{t,\varphi\varphi}[\vec{\varphi}\,]+R_{t}\big{)}^{-1}_{ij}=\vphantom{\bigg{(}\bigg{)}}$
(83) $\displaystyle=\,$
$\displaystyle\mathcal{P}^{\parallel}_{ij}(\vec{\varphi}\,)\,\frac{1}{r(t)+\partial_{\varrho}U(t,\varrho)+2\varrho\,\partial_{\varrho}^{2}U(t,\varrho)}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\mathcal{P}^{\perp}_{ij}(\vec{\varphi}\,)\,\frac{1}{r(t)+\partial_{\varrho}U(t,\varrho)}\,,\vphantom{\bigg{(}\bigg{)}}$

which can be inserted directly into the FRG equation (40).

After taking the field space trace and evaluating the resulting equation on
the constant background field configuration, we arrive at the RG flow equation
for the effective potential

$\displaystyle\partial_{t}U(t,\sigma)=\,$
$\displaystyle\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\frac{N-1}{r(t)+\frac{1}{\sigma}\,\partial_{\sigma}U(t,\sigma)}+\vphantom{\Bigg{(}\Bigg{)}}$
(84)
$\displaystyle+\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\frac{1}{r(t)+\partial_{\sigma}^{2}U(t,\sigma)}=\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle=\,$
$\displaystyle\begin{gathered}\includegraphics{diagrams/potential_pi.pdf}\end{gathered}+\begin{gathered}\includegraphics{diagrams/potential_sigma.pdf}\end{gathered}\,.$
(87)

This RG flow equation is an exact non-linear PDE for the effective potential
$U(t,\sigma)$, which is of first order in RG time $t$ and of first and second
order in the field space direction $\sigma$. It also includes an explicit
$\sigma$ dependence. A detailed analysis of the structure of this PDE,
including its relation to conservation equations and fluid dynamics is
provided in Sub.Sec. IV.1.

For now, we conclude this section with a few comments on the widely used
diagrammatic notation of the PDE and its relation to the RG flow equation (38)
from Sec. II: Similar to Feynman diagrams which are commonly used in
perturbation theory, the propagators141414The term “propagator” is of course
misleading for a QFT in a single point, where “propagation” in the true sense
of the word is not possible. Nevertheless, we again adopt the notation from
higher-dimensional QFT and statistical mechanics.  are depicted as lines;
blue-jagged lines for the $\sigma$ propagator,

$\displaystyle\frac{1}{r(t)+\partial_{\sigma}^{2}U(t,\sigma)}\,,$ (88)

and red-dashed lines for the $\pi$ propagators

$\displaystyle\frac{1}{r(t)+\frac{1}{\sigma}\,\partial_{\sigma}U(t,\sigma)}\,.$
(89)

The crossed circle ($\otimes$) stands for the regulator insertion
$\frac{1}{2}\,\partial_{t}r(t)$. (The factor $\frac{1}{2}$ is often not
included in the regulator insertion, but written in front of the diagrams.
See, e.g., Refs. [24, 16, 133, 134, 102, 104, 44] for different notations.)
The factor $N-1$ is the multiplicity of the pion-loop contribution (indicated
by the vector over the pion field in the diagram, cf. Eq. (84)) and
corresponds to the number of pions in the system.

For the special case $N=1$, the $O(N)$ model reduces to the $O(1)$ model. Such
a theory of a single scalar field in zero dimensions, was used in the
introductory section II on FRG. In this limit, the pion contributions to the
flow equation vanish. As already stated in Sec. II, we find that for non-zero
pion contributions ($N>1$) the flow equation for $U(t,\sigma)$ acquires a term
that is of first order in the spatial derivative,
$\partial_{\sigma}U(t,\sigma)$, which no longer has diffusive character, but
corresponds to advection in field space. This is further discussed in Sec. IV.

####  III.3.2 FRG Taylor (vertex) expansion of the $O(N)$ model

The FRG Taylor (vertex) expansion is based on the assumption that the
effective (average) action $\bar{\Gamma}_{t}[\vec{\varphi}\,]$ can be expanded
in a series in field space with RG-time dependent expansion coefficients [87].
In zero dimensions, this effectively reduces to an expansion of the effective
potential $U(t,\varrho)$, cf. Eq. (79). Consequently, it is also equivalent to
a Taylor expansion of the effective potential, which is well-known from
higher-dimensional truncation schemes [135, 87, 133, 16, 15, 136, 14, 18, 20,
19, 122, 104]. Throughout this work, we will therefore use the term “FRG
Taylor expansion” to refer to this approach. The RG-scale dependent expansion
coefficients $\bar{\Gamma}^{(2n)}(t)$ correspond directly to the scale-
dependent vertex functions
$\bar{\Gamma}^{(2n)}_{t,\varphi_{i}\ldots\varphi_{j}}$ of the QFT. For $d>0$,
these expansion coefficients are usually position or momentum dependent
whereas in $d=0$ the coefficients depend only on the RG time $t$.

The assumption of expandability and thus differentiability significantly
restricts the form of the effective action
$\bar{\Gamma}_{t}[\vec{\varphi}\,]=U(t,\vec{\varphi}\,)$, cf. Refs. [130,
131]. In fact, it neither allows for the formation of any non-analytic
behavior throughout the RG flow nor for any non-analytic initial conditions.
However, non-analytic initial conditions are not forbidden, as we will see in
Sec. V. Furthermore, it is well known that non-analyticities can (and in some
models have to) form in the effective potential during the RG flow [118, 26,
27, 137]. Considering these caveats, an expansion in vertices of a given
theory has always to be considered with care. Still, this expansion scheme is
used in certain applications.

In our work, we restrict our analysis of the precision of this truncation
scheme to RG flows with rather specific properties: We study initial
conditions that are analytic. Furthermore, we know, cf. App. B, that the IR
effective action is smooth for the special case of zero dimensions, which is a
necessary condition for the convergence of a (Taylor) series. It should,
however, be noted that smoothness is only a necessary but not a sufficient
condition for the convergence of a Taylor series151515A textbook example for a
smooth function which has a non-converging Taylor series around $x=0$ is
$f(x)=\begin{cases}\mathrm{e}^{-1/x}&\text{if }x>0,\\\
0&\text{else.}\end{cases}$ . Only analyticity would formally imply the
convergence of a Taylor series at all $\vec{\varphi}$. Additionally, we argue
that for sufficiently small $N$, the diffusive contributions to the RG flow
are important, which smear out any possible cusps. In summary, we expect that
for these extremely special scenarios it is unlikely that non-analyticities
will form and disappear again during the RG flow. Nevertheless, we do not know
if a small finite number of expansion coefficients is always enough to reach a
reliable approximation of $\bar{\Gamma}_{t}[\vec{\varphi}\,]$ during the RG
flow or if it is always necessary to flow the effective potential as a PDE
without additional assumptions. This (rather limited) applicability of the FRG
Taylor expansion to analytic initial conditions will be tested by calculating
the relative errors of 1PI $n$-point vertex functions in the FRG Taylor
expansion in comparison with the exact results and the results from the flows
of a full field-dependent $U(t,\sigma)$ in Sec. V.

The FRG Taylor expansion of the zero-dimensional $O(N)$ model is given by the
following ansatz [44, 52, 51, 56],

$\displaystyle\bar{\Gamma}_{t}[\vec{\varphi}\,]=\,$
$\displaystyle\sum_{n=0}^{m}\frac{\bar{\Gamma}^{(2n)}(t)}{(2n-1)!!}\,\frac{1}{n!}\,\bigg{(}\frac{\vec{\varphi}^{\,2}}{2}\bigg{)}^{n}=\vphantom{\bigg{(}\bigg{)}^{2}}$
(90) $\displaystyle=\,$
$\displaystyle\bar{\Gamma}^{(0)}(t)+\bar{\Gamma}^{(2)}(t)\,\frac{\vec{\varphi}^{\,2}}{2}+\frac{\bar{\Gamma}^{(4)}(t)}{3}\,\frac{1}{2}\,\bigg{(}\frac{\vec{\varphi}^{\,2}}{2}\bigg{)}^{2}+\ldots\,,\vphantom{\bigg{(}\bigg{)}^{2}}$

where $\bar{\Gamma}^{(2n)}(t)$ are $t$ dependent expansion coefficients and
$m$ is the truncation order. The factors of $(2n-1)!!$ and $n!$ were
introduced in order to have
$\bar{\Gamma}^{(2n)}(t_{\mathrm{IR}})=\Gamma^{(2n)}_{\varphi_{i}\ldots\varphi_{i}}$
in the IR, where $\Gamma^{(2n)}_{\varphi_{i}\ldots\varphi_{i}}$ are the 1PI
$2n$-point vertex functions in the IR, with all indices being identical (no
summation over $i$ here), see also Eqs. (73) – (75). In order to arrive at the
corresponding flow equations, we proceed in a similar manner as before in
Sub.Sub.Sec. III.3.1: We insert our ansatz (90) into the full field-dependent
two-point function (80) and use the field space projection operators (81) to
invert the latter. We obtain

$\displaystyle\big{(}\bar{\Gamma}^{(2)}_{t,\varphi\varphi}[\vec{\varphi}\,]+R_{t}\big{)}^{-1}_{ij}=\vphantom{\bigg{(}\bigg{)}}$
(91) $\displaystyle=\,$
$\displaystyle\mathcal{P}^{\perp}_{ij}(\vec{\varphi}\,)\,G^{\pi\pi}_{t}(\vec{\varphi}\,)+\mathcal{P}^{\parallel}_{ij}(\vec{\varphi}\,)\,G^{\sigma\sigma}_{t}(\vec{\varphi}\,)\,,\vphantom{\bigg{(}\bigg{)}}$

where

$\displaystyle G^{\pi\pi}_{t}(\vec{\varphi}\,)\equiv\,$
$\displaystyle\Bigg{[}r(t)+\sum_{n=1}^{m+1}\frac{\bar{\Gamma}^{(2n)}(t)}{(2n-1)!!}\,\frac{1}{(n-1)!}\,\bigg{(}\frac{\vec{\varphi}^{\,2}}{2}\bigg{)}^{n-1}\Bigg{]}^{-1}\,,$
$\displaystyle G^{\sigma\sigma}_{t}(\vec{\varphi}\,)\equiv\,$
$\displaystyle\Bigg{[}r(t)+\sum_{n=1}^{m+1}\frac{\bar{\Gamma}^{(2n)}(t)}{(2n-3)!!}\,\frac{1}{(n-1)!}\,\bigg{(}\frac{\vec{\varphi}^{\,2}}{2}\bigg{)}^{n-1}\Bigg{]}^{-1}\,,$

are the field-dependent propagators of the pion and sigma field in the Taylor
expansion.

This result can be inserted into the FRG equation (40), where the trace in
field space is evaluated to

$\displaystyle\partial_{t}\,\bar{\Gamma}_{t}[\vec{\varphi}\,]=\vphantom{\bigg{(}\bigg{)}}$
(92) $\displaystyle=\,$
$\displaystyle\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\big{[}(N-1)\,G^{\pi\pi}_{t}(\vec{\varphi}\,)+G^{\sigma\sigma}_{t}(\vec{\varphi}\,)\big{]}\,.\vphantom{\bigg{(}\bigg{)}}$

Finally, we insert the ansatz (90) for the effective average action into the
left-hand side of this equation and expand the propagators
$G^{\circ\circ}_{t}(\vec{\varphi}\,)$ up to order $n=m$ in the expansion
coefficients $\bar{\Gamma}^{(2n)}(t)$. This can also be achieved by
successively taking derivatives with respect to the fields and setting
$\vec{\varphi}=0$ afterwards. By comparing the expansion coefficients on the
left- and right-hand sides of the equation, one arrives at a coupled set of
ordinary differential equations for the $\bar{\Gamma}^{(2n)}(t)$ with $0\leq
n\leq m$. The flow equation for $\bar{\Gamma}^{(2m)}(t)$ contains
$\bar{\Gamma}^{(2m+2)}(t)$ on the right-hand side. We truncate the system by
neglecting the flow of $\bar{\Gamma}^{(2m+2)}(t)$, taking
$\partial_{t}\bar{\Gamma}^{(2m+2)}(t)=0$.

For an automatization of the derivation of the flow equations (the system of
ODEs) via computer algebra routines such as Mathematica [138], it is advisable
to formulate the FRG Taylor expansion in the invariant
${\varrho=\tfrac{1}{2}\,\vec{\varphi}^{\,2}}$,

$\displaystyle\bar{\Gamma}_{t}[\varrho]=\,$
$\displaystyle\sum_{n=0}^{m}\frac{\bar{\Gamma}^{(2n)}(t)}{(2n-1)!!}\,\frac{\varrho^{n}}{n!}\,.\vphantom{\bigg{(}\bigg{)}}$
(93)

Equation (92) becomes

$\displaystyle\partial_{t}\,\bar{\Gamma}_{t}[\varrho]=\,$
$\displaystyle\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\big{[}(N-1)\,G^{\pi\pi}_{t}(\varrho)+G^{\sigma\sigma}_{t}(\varrho)\big{]}\,,$
(94)

while

$\displaystyle G^{\pi\pi}_{t}(\varrho)\equiv\,$
$\displaystyle\bigg{[}r(t)+\sum_{n=1}^{m+1}\frac{\bar{\Gamma}^{(2n)}(t)}{(2n-1)!!}\,\frac{\varrho^{n-1}}{(n-1)!}\bigg{]}^{-1}\,,$
(95) $\displaystyle G^{\sigma\sigma}_{t}(\varrho)\equiv\,$
$\displaystyle\bigg{[}r(t)+\sum_{n=1}^{m+1}\frac{\bar{\Gamma}^{(2n)}(t)}{(2n-3)!!}\,\frac{\varrho^{n-1}}{(n-1)!}\bigg{]}^{-1}\,.$
(96)

The coupled set of ODEs for the expansion coefficients
$\bar{\Gamma}^{(2n)}(t)$ is given by [44, 52]161616We do not indicate $t$
dependences of the $\bar{\Gamma}^{(2n)}(t)$ for reasons of readability. ,

$\displaystyle\partial_{t}\bar{\Gamma}^{(0)}=\,$
$\displaystyle\frac{N}{2}\,\,\frac{\partial_{t}r(t)}{r(t)+\bar{\Gamma}^{(2)}}\,,\vphantom{\Bigg{(}\Bigg{)}}$
(97) $\displaystyle\partial_{t}\bar{\Gamma}^{(2)}=\,$
$\displaystyle-\frac{N+2}{6}\,\frac{\partial_{t}r(t)}{\big{[}r(t)+\bar{\Gamma}^{(2)}\big{]}^{2}}\,\bar{\Gamma}^{(4)}\,,\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle\partial_{t}\bar{\Gamma}^{(4)}=\,$
$\displaystyle\frac{N+8}{3}\,\frac{\partial_{t}r(t)}{\big{[}r(t)+\bar{\Gamma}^{(2)}\big{]}^{3}}\,\big{[}\bar{\Gamma}^{(4)}\big{]}^{2}-\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle-\frac{N+4}{10}\,\frac{\partial_{t}r(t)}{\big{[}r(t)+\bar{\Gamma}^{(2)}\big{]}^{2}}\,\bar{\Gamma}^{(6)}\,,\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle\vdots\,\,$

Recall that

$\displaystyle\partial_{t}\bar{\Gamma}^{(n)}=0\,$ (98)

for $n\geq 2m+2$ in this approximation.

##  IV FRG flow equations and (numerical) fluid dynamics

In this section, we discuss the formulation of the RG flow equation as an
advection-diffusion equation, as well as its interpretation in the context of
fluid dynamics, including its numerical implementation.

The fluid-dynamical formulation of the exact RG flow equation for the
effective potential $U(t,\varrho)$ of models of $O(N)$ type (in the large-$N$
limit [28]) is also presented in a recent and a parallel publication by some
of us and collaborators [26, 27]. It was shown that the RG flow equation can
be recast in the form of a pure advection equation (a hyperbolic conservation
law) for the derivative of the effective potential
$u(t,\varrho)=\partial_{\varrho}U(t,\varrho)$, where $u(t,\varrho)$ serves as
the conserved quantity (the fluid), the RG time $t$ as a temporal coordinate
and $\varrho$ as a spatial coordinate. In this section, we generalize this
result and discuss various consequences for the numerical implementation and
interpretation of FRG flow equations.171717Generalizations of the fluid-
dynamical picture of FRG flow equations from the large-$N$ results of Ref.
[26] to systems with finite $N$ as well as the inclusion of fermions were
already presented by us in various talks (see, e.g., Refs. [139, 140]) and
discussed in a master thesis [141] co-supervised by some of us, as well as a
PhD thesis by one of us [142], see also Ref. [27]. Furthermore, also in Ref.
[118] a formulation of the flow equation as a conservation law and a
discussion of shock waves based on the characteristics is presented, however,
without really elaborating on a fluid-dynamical interpretation and its
consequences.

###  IV.1 Conservative form of FRG flow equations – advection-diffusion
equations

The formulation of FRG flow equations in terms of a fluid-dynamical language
has two major advantages:

  1. 1.

It provides an intuitive explanation for all kinds of phenomena observed in
FRG flow equations, e.g., the flattening of the effective potential for small
$\sigma$ in the IR, which occurs in conjunction with a non-differentiable
point of the effective potential at the ground state. Such non-analytic
behavior cannot be handled and systematically analyzed by commonly used
numerical schemes such as the Taylor expansion or related discretization
schemes for the effective potential, since the latter strongly rely on
differentiability. However, these phenomena have a direct impact on the
physics, for instance on the occurrence of phase transitions [118, 22, 130,
131, 26, 83, 143, 27], and therefore must be resolved and analyzed accurately
also on a numerical level.

  2. 2.

The formulation of the FRG flow equations in terms of fluid-dynamical concepts
provides access to the highly developed and extremely powerful toolbox of
numerical fluid dynamics, which finds applications in a wide area of fields,
ranging from the natural sciences and engineering all the way to economics.
How to adopt these methods to flow equations arising in the FRG framework is
discussed in detail in Sub.Secs. IV.2 and IV.3.

Interestingly, the idea of interpreting RG flow equations as “flow” equations
in the true sense of the word is not new and explains the term “RG flow
equations”: A discussion of analogies between “RG flow” and hydro-dynamical
flow can be found in widely used textbooks [39, 144] and is discussed via the
example of field-independent coupling constants in the context of perturbative
renormalization. Furthermore, the RG flow was already associated with gradient
flow and dissipative processes in Refs. [145, 146, 147, 148, 149, 148, 74],
even though a stringent fluid-dynamical interpretation and formulation was not
presented.

It is therefore also not accidental that the (F)RG community has chosen the
term “RG time” for the logarithm of the RG scale $k$ over the UV cutoff
$\Lambda$, $\tilde{t}=\ln\big{(}\tfrac{k}{\Lambda}\big{)}$. In contrast, we
find that $t=-\tilde{t}\in[0,\,\infty)$ can be naturally identified as a
temporal coordinate in the fluid-dynamical picture of (F)RG flow equations,
see below.

It was also discussed, see, e.g., Refs. [74, 76, 54], that – on the level of
the scale-dependent generation functionals $\mathcal{Z}_{t}[J]$ or
$\mathcal{W}_{t}[J]$ – the corresponding PDEs can be considered as a (non-
linear) functional diffusion equations for the source fields $J$ (cf. Eqs.
(11) and (19) for the respective zero-dimensional versions). Sometimes Eq.
(11) is even explicitly denoted as a (non-linear) heat equation, which is also
a specific fluid-dynamical problem [77, 78, 79, 150].

Considering the obvious analogies between RG flow equations arising in the FRG
framework and fluid-dynamical equations, it is remarkable that the FRG
equation (40) was so far not more systematically investigated and compared to
equations well-known from fluid dynamics. For the related RG flow equations
the situation is slightly different and the mathematical analysis on the level
of PDEs was more systematic, see, e.g., Refs. [151, 152, 149, 148, 74].
Furthermore, certain phenomena well-known in fluid dynamics, such as
discontinuities (shock waves), rarefaction waves, or cusps, occur in the
solution of such PDEs. These require a careful numerical treatment to resolve
them, but their occurrence was very often ignored by numerical approaches to
solve the FRG equations by erroneously assuming that the solution
$U(t,\sigma)$ is continuous and differentiable. Still, there are some
publications which use numerical schemes to systematically capture non-
analytic behavior or discuss the limitations of numerical methods in the
presence of these effects, see, e.g., Refs. [137, 118].

In order to make the fluid-dynamical analogy more apparent, we present a
formulation of the RG flow equation (84) for the effective potential
$U(t,\sigma)$ in terms of a conservation law. Furthermore, we discuss its
fluid-dynamical interpretation on a qualitative level and classify the various
contributions to the PDE (the RG flow) in the fluid-dynamical picture. This
sets the stage for an adequate qualitative interpretation of the RG flow
equation and possible numerical approaches, which are presented in the next
two Sub.Secs. IV.2 and IV.3.

####  IV.1.1 The conservative form

Starting from the RG flow equation (84) of the effective potential
$U(t,\sigma)$, we have several options to recast the flow equation in a
conservative form, two of which are:

  1. 1.

Following Refs. [118, 26, 27, 139, 142, 141], we can take an overall
derivative of Eq. (84) with respect to the $O(N)$ invariant
$\varrho=\tfrac{1}{2}\,\sigma^{2}$ and express the resulting equation in terms
of $\varrho$ and $u(t,\varrho)\equiv\partial_{\varrho}U(t,\varrho)$,

$\displaystyle\partial_{t}u(t,\varrho)=\frac{\mathrm{d}}{\mathrm{d}\varrho}\,\bigg{(}\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\frac{N-1}{r(t)+u(t,\varrho)}+\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\frac{1}{r(t)+u(t,\varrho)+2\varrho\,\partial_{\varrho}u(t,\varrho)}\bigg{)}\,.$
(99)

  2. 2.

Another option is to formulate the problem on the level of the background
field $\sigma$ itself [140] and by alternatively defining
$u(t,\sigma)\equiv\partial_{\sigma}U(t,\sigma)$. Taking an overall derivative
of Eq. (84) with respect to $\sigma$ yields,

$\displaystyle\partial_{t}u(t,\sigma)=\frac{\mathrm{d}}{\mathrm{d}\sigma}\,\bigg{(}\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\frac{N-1}{r(t)+\frac{1}{\sigma}\,u(t,\sigma)}+\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\frac{1}{r(t)+\partial_{\sigma}u(t,\sigma)}\bigg{)}\,.$
(100)

In both cases one ends up with a one-dimensional conservation law, where $u$
plays the role of the conserved quantity (the fluid), $t$ can be identified
with the time variable and $\varrho$ or $\sigma$ are identified as the spatial
variable.

The conservative form of the RG flow equation (84) for the effective potential
$U$ on the level of its derivative $u$ is not restricted to zero space-time
dimensions or models with purely bosonic field content, see also Refs. [118,
26, 27, 139, 140, 142, 141]. As a matter of fact, this formulation generalizes
to arbitrary dimensions and also to models which include fermionic degrees of
freedom on the level of the local potential approximation (LPA). In
particular, the flow equation for the effective potential for models of
strong-interaction matter, such as the quark-meson, the Nambu-Jona-Lasinio,
and the Gross-Neveu(-Yukawa) model can be formulated in this
fashion181818Meanwhile, we and our collaborators [141, 27, 153, 32] were also
working on the conservative formulation of (F)RG flow equations in higher
dimensions in more advanced truncations as well as on conservative
formulations of (F)RG flow equations for zero-dimensional systems involving
fermions (Grassmann numbers) [34]. .

In this context, it is also worthwhile to note that Eq. (100) can be derived
not only by taking a derivative of the FRG flow equation for the effective