KCL-PH-TH/2023-51
MITP-24-043
TUM-HEP-1508/24

$CP$ conservation in the strong interactions

Wen-Yuan Ai,a Björn Garbrechtb and Carlos Tamaritc

aTheoretical Particle Physics and Cosmology, King’s College London,
Strand, London WC2R 2LS, UK
bPhysik-Department T70, Technische Universität München,
James-Franck-Straße, 85748 Garching, Germany
cPRISMA+ Cluster of Excellence and MITP, Johannes Gutenberg University,
D-55099 Mainz, Germany

###### Abstract

We discuss matters related to the point that topological quantization in the
strong interaction is a consequence of an infinite spacetime volume. Because
of the ensuing order of limits, i.e. infinite volume prior to summing over
topological sectors, $CP$ is conserved. Here, we show that this reasoning is
consistent with the construction of the path integral from steepest-descent
contours. We reply to some objections that aim to support the case for $CP$
violation in the strong interactions that are based on the role of the
$CP$-odd theta-parameter in three-form effective theories, the correct
sampling of all configurations in the dilute instanton gas approximation and
the volume dependence of the partition function. We also show that the chiral
effective field theory derived from taking the volume to infinity first is in
no contradiction with analyses based on partially conserved axial currents.

###### Contents

  1. 1 Introduction
  2. 2 Summary and outline
  3. 3 Path integral and topological quantization
  4. 4 Finite Euclidean spacetimes
  5. 5 Dilute instanton gas approximation
  6. 6 Thermodynamic limit and cluster decomposition
  7. 7 Effective theories and effective operators
  8. 8 Some objections and answers to these
  9. 9 Conclusions

##  1 Introduction

The strong interactions conserve charge–parity ($CP$). This has been
established through many observations, to the greatest precision in searches
for the permanent electric dipole moment (EDM) of the neutron [1, 2], and
calls for an explanation of certain theoretical aspects.

Since only a few years after the discovery of quantum chromodynamics (QCD)
being the theory of the strong interactions [3] it has been suggested that due
to the generic presence of a topological term in the action, the charge–parity
symmetry $CP$ should be violated [4, 5, 6]. To the present day, there has been
no such observation, so arguably the coefficient $\theta$ of the topological
term (more precisely $\bar{\theta}$ as we introduce below) would need to be
zero, corresponding to unnatural tuning and therefore a problem.

The present paper adds to the discussion following Ref. [7]. There, it has
been brought up that the effective decomposition (quantization) of gauge field
configurations of finite action into topological sectors of integer winding
number without imposing ad hoc boundary conditions can only be derived when
taking the volume of Euclidean spacetime to infinity. In contrast, there is no
physical motivation for fixed boundary conditions on a finite surface in
Euclidean space. As a consequence, the limit of infinite spacetime volume must
be taken before summing over topological sectors, and it turns out that $CP$
remains conserved this way.

One of the main points of the present paper is to extend this line of
reasoning by demonstrating that this order of limits corresponds to a well-
defined integration contour in the path integral constructed from the
steepest-descent flows. In contrast, the opposite conventional order of limits
results in an integration that is inequivalent in the sense of the Cauchy
theorem. Another purpose of this article is to address some criticisms
regarding Ref. [7]. This article also partly serves as a review of the related
topics, though without including a complete list of references. For some other
reviews, see e.g. Refs. [8, 9, 10].

Without going into extensive technical detail, the basic reasons for $CP$
conservation in the strong interaction are given in the summary of Section 2.
There, we refer to the sections in the remainder of the paper where the
statements of Section 2 are supported at a more technical level.

##  2 Summary and outline

Strong interactions are described by a Yang–Mills theory which generally
involves $CP$-odd parameters through the masses of the quarks as well as
through the topological term. The Lagrangian in the Euclidean spacetime is

$\displaystyle{\cal L}=\frac{1}{2g^{2}}{\rm
tr}F_{\mu\nu}F_{\mu\nu}+\bar{\psi}\left(\hat{\gamma}_{\mu}D_{\mu}+m{\rm
e}^{{\rm i}\alpha\gamma_{5}}\right)\psi-\frac{{\rm i}}{16\pi^{2}}\theta\,{\rm
tr}F_{\mu\nu}\widetilde{F}_{\mu\nu}\,,$ (1)

where we use the convention ${\rm Tr}(T^{a}T^{b})=\delta^{ab}/2$,
$[T^{a},T^{b}]={\rm i}f^{abc}T^{c}$ for the Lie algebra generators $T^{a}$ and
the structure constants $f^{abc}$. Above, $F_{\mu\nu}=F_{\mu\nu}^{a}T^{a}$
with
$F_{\mu\nu}^{a}\equiv\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A_{\mu}^{a}+f^{abc}A^{b}_{\mu}A^{c}_{\nu}$
being the field strength tensors.
$\widetilde{F}^{a}_{\mu\nu}\equiv\frac{1}{2}\varepsilon_{\mu\nu\alpha\beta}F^{a}_{\alpha\beta}$
($\varepsilon_{1234}=1$) is the Hodge dual of $F^{a}_{\mu\nu}$. The covariant
derivative takes the form

$\displaystyle D_{\mu}\psi_{i}=\left(\partial_{\mu}-{\rm
i}A_{\mu}^{a}T^{a}\right)\psi_{i}$ (2)

when $\psi_{i}$ lives in the fundamental representation of the gauge group and

$\displaystyle D_{\mu}\psi_{i}=\partial_{\mu}\psi_{i}-{\rm
i}A_{\mu}^{a}[T^{a},\psi_{i}]$ (3)

when $\psi_{i}$ lives in the adjoint representation.

The Euclidean gamma matrices $\hat{\gamma}_{\mu}$ are obtained from the
Minkowskian counterparts

$\displaystyle\gamma^{\mu}=\begin{pmatrix}0&\sigma^{\mu}\\\
\bar{\sigma}^{\mu}&0\end{pmatrix}\,,$ (4)

(where $\sigma^{\mu}=\left(\mathsf{1}_{2},\vec{\sigma}\right)$ and
$\bar{\sigma}^{\mu}=(\mathsf{1}_{2},-\vec{\sigma})$ with $\vec{\sigma}^{i}$
the Pauli matrices) via

$\displaystyle\hat{\gamma}_{4}=\gamma_{0}=\gamma^{0}\,,\quad\hat{\gamma}_{i}={\rm
i}\gamma_{i}=-{\rm i}\gamma^{i}\,.$ (5)

These matrices satisfy the Clifford algebras

$\displaystyle\\{\gamma^{\mu},\gamma^{\nu}\\}=2g^{\mu\nu}\mathsf{1}_{4}\,,\qquad\\{\hat{\gamma}_{\mu},\hat{\gamma}_{\nu}\\}=2\delta_{\mu\nu}\mathsf{1}_{4}\,.$
(6)

Following Ref. [7], we use the same $\gamma^{5}$ in Euclidean spacetime as for
Minkowski spacetime111In Ref. [7], it is mistakenly stated that
$\gamma^{5}=\hat{\gamma}_{1}\hat{\gamma}_{2}\hat{\gamma}_{3}\hat{\gamma}_{4}$.

$\displaystyle\gamma^{5}\equiv{\rm
i}\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=-\hat{\gamma}_{1}\hat{\gamma}_{2}\hat{\gamma}_{3}\hat{\gamma}_{4}\,.$
(7)

Since in this paper we mostly work in Euclidean space, from now on we remove
the hat on the Euclidean gamma matrices. Note that the $\gamma^{5}$ in the
Euclidean spacetime used here may differ from that used in some other papers,
e.g. Ref. [11], by a minus sign. The chirality in Euclidean spacetime in these
two notations is thus defined oppositely. The only effect of this appears in
applying the Atiyah–Singer index theorem [12] when e.g. deriving the anomalous
axial current by counting the zero modes of the massless Dirac operator.

We note that the terms ${\rm tr}F_{\mu\nu}\widetilde{F}_{\mu\nu}$ and
$\bar{\psi}{\rm i}\gamma^{5}\psi$ are both parity-odd and charge-conjugation
even [13]. The $CP$-odd parameters are therefore $\alpha$, the phase
pertaining to the mass $m$ of the fermion $\psi$, and $\theta$, the
coefficient of the topological term. The fermion $\psi$ in the fundamental
representation is referred to as quark. To focus on the principal aspects, for
the most part of the discussion we take here just one quark flavour and the
gauge group to be ${\rm SU}(2)$. This is the minimal setup that allows to
study the interplay of the $CP$-odd parameters $\alpha$ and $\theta$. In
particular, we are interested in how $\alpha$ and $\theta$ appear in the
effective interaction (known as ‘t Hooft operator) that captures
nonperturbative effects associated with the chiral anomaly. In the single-
flavour model, this operator has the same form as a quark mass term, and
therefore both would be hard to discern. Nonetheless, the main question of
whether and how $\alpha$ and $\theta$ appear in the effective ‘t Hooft
operator can be answered within this setup. The generalization to the
phenomenologically relevant case of several flavours is presented in Ref. [7].
In the strong interactions, the group ${\rm SU}(2)$ can be viewed as embedded
within the ${\rm SU}(3)$ colour group. The technical details of this
construction are reviewed in Ref. [14].

It is well known that the presence of a $CP$-odd Lagrangian term does not
readily imply $CP$-violating physical effects. A necessary condition is the
existence of a $CP$-odd combination of the Lagrangian terms that is invariant
under field redefinitions. In the present case, this condition is met as the
parameter

$\displaystyle\bar{\theta}=\theta+\alpha$ (8)

is invariant under redefinitions of the quark fields, in particular through
anomalous chiral transformations. The chiral anomaly [15, 16] also implies
that $\theta$ is an angular variable, i.e. all observables must be
$2\pi$-periodic in $\theta$. Therefore, the integral $1/(16\pi^{2})\int{\rm
d}^{4}x\,{\rm tr}F\widetilde{F}$ that multiplies $\theta$ must be an integer
in order to contribute to the action and thereby to the partition function.

Here, we ask the question of whether a nonvanishing $\bar{\theta}$ is also a
sufficient condition for $CP$ violation in strong interactions. Does
$\bar{\theta}$ have physical effects, in particular, is there a neutron EDM
depending on its value? Evidently, $\bar{\theta}$ has no impact on the
classical equations of motion since the topological term is a total
derivative. Nonetheless, under certain assumptions, based on the fact that the
third homotopy group of the gauge group or one of its ${\rm SU}(2)$ subgroups
is $\pi_{3}({\rm SU}(2))=\mathbbm{Z}$, the energy functional is periodic under
so-called large gauge transformations. The situation is therefore reminiscent
of a periodic quantum mechanical potential in a crystal, and $\bar{\theta}$
would then correspond to the crystal momentum [4, 5, 6, 11].

One way to see how far the analogy goes is to study canonical quantization of
the gauge theory [4]. Since non-Abelian gauge theories are typically handled
through functional quantization this possibility has not yet been investigated
in all aspects pertinent to the present questions. We briefly comment on
canonical quantization in Section 4 and in more detail in a separate paper
[17].

For the time being, we focus on functional quantization since it has been the
principal method to carry out calculations on $CP$ violation in the strong
interactions ever since the matter was brought up [5, 6, 11, 18]. In the
functional approach, we take the partition function

$\displaystyle Z[\eta,\bar{\eta}]=\int{\cal D}\bar{\psi}\,{\cal D}\psi\,{\cal
D}A\,{\rm e}^{-\lim\limits_{\Omega\to\infty}\int_{\Omega}{\rm
d}^{4}x\,\left({\cal L}-\bar{\eta}\psi-\bar{\psi}\eta\right)}$ (9)

as the defining point of the quantum field theory, where $A$ is the gauge
potential. We write this as a functional of external fermionic sources
$\eta(x)$, $\bar{\eta}(x)$ as a provision in order to derive quark correlation
functions that may or may not exhibit $CP$ invariance. Furthermore, we make
explicit that the integral over Euclidean spacetime is understood as a
limiting procedure, taking the spacetime volume $\Omega$ to infinity. It is
this limit that allows us to state Eq. (9) without specifying boundary
conditions on the path integral and that, moreover, this way we obtain the
vacuum correlation functions of the theory [19]. We shall review this point in
Section 3.

Now, as argued in Ref. [11], the partition function (9) in infinite spacetime
volume $\Omega\to\infty$ receives its nonvanishing contributions from saddle
points of finite action and fluctuations about these. For these field
configurations, the winding number $\Delta n$ is an integer that labels the
topological sector:

$\displaystyle\Delta n=\lim\limits_{\Omega\to\infty}\int_{\Omega}{\rm
d}^{4}x\,\frac{1}{16\pi^{2}}{\rm tr}F\widetilde{F}\in\mathbbm{Z}\quad\text{for
nonvanishing contributions to}\;Z\,.$ (10)

This is the desired outcome because it is consistent with $\theta$ being an
angular variable, as required by the chiral anomaly. As topological
quantization, i.e. integer $\Delta n$, is a consequence of $\Omega\to\infty$,
we must carry out this limit before summing over topological sectors.

Suppose now that it is valid to organize the calculation of the path integral
by adding contributions from the individual topological sectors. Then, Eq. (9)
implies that the partition function should be evaluated as

$\displaystyle Z[\eta,\bar{\eta}]=\sum\limits_{\Delta n}\int{\cal
D}\bar{\psi}\,{\cal D}\psi\,{\cal D}A_{\Delta n}{\rm
e}^{-\lim\limits_{\Omega\to\infty}\int_{\Omega}{\rm d}^{4}x\,\left({\cal
L}-\bar{\eta}\psi-\bar{\psi}\eta\right)}\,.$ (11)

The subscript $\Delta n$ on ${\cal D}A_{\Delta n}$ indicates that the path
integral is supposed to cover the configurations with given $\Delta n$, i.e.
to sweep over the given topological sector. The contributions of the
topological sectors are evaluated with the limit $\Omega\rightarrow\infty$
before they are added together. A rearrangement of the limits will in general
lead to different results and is therefore not justified.

In Section 3, we expand on this argument and put it on a more formal footing.
The path integrals within individual topological sectors correspond to
steepest-descent contours for the exponent of the Euclidean path integral. For
different $\Delta n$, these contours can only be connected by configurations
of infinite action. It thus follows that the arrangement of limits in Eq. (11)
indeed corresponds to a good contour for the path integral in Eq. (9).
Further, this formally establishes that the decomposition of the path integral
into topological sectors is valid in the first place.

This brings us to the salient point: In Ref. [7], it has been shown that the
limit $\Omega\to\infty$ does not commute with the sum over $\Delta n$ in Eq.
(11), with the consequence that the quark correlations do not exhibit $CP$
violation. While we review this technical argument in Section 5, the basic
reason is that $\lim_{\Delta n\to\pm\infty}\lim_{\Omega\to\infty}\Delta
n/\Omega=0$, see also Section 3. As explained in Section 7, one can thus
conclude that there is no EDM for the neutron, no matter what the value of
$\bar{\theta}$ is. Since in the sum over $\Delta n$ all integer values are
taken, this is where the analogy with the quantum-mechanical crystal breaks
down as for the latter the number of potential minima may be large but remains
finite. Therefore, the order of the path integral and the limit of an infinite
spacetime volume is not an issue in that case.

After all, the strong interactions are complete without the necessity of
tuning the parameter $\bar{\theta}$ to be small or extending the theory by
additional scalar fields and nonrenormalizable operators. While this is a
gratifying conclusion, a scrutiny of the argument is warranted, not least
because the prevalent line of reasoning arrives at the contrary verdict: In
order to deduce $CP$-violation in the strong interactions, one would have to
impose that the limit $\Omega\to\infty$ is taken last, i.e.

$\displaystyle
Z[\eta,\bar{\eta}^{\dagger}]\stackrel{{\scriptstyle?}}{{=}}\lim\limits_{\Omega\to\infty}\int{\cal
D}\bar{\psi}\,{\cal D}\psi\,{\cal D}A\,{\rm e}^{-\int_{\Omega}{\rm
d}^{4}x\,\left({\cal L}-\bar{\eta}\psi-\bar{\psi}\eta\right)}\,,$ (12)

and at the same time specify boundary conditions on the finite surfaces
$\partial\Omega$ such as

$\displaystyle A_{\mu}={\rm i}\omega\partial_{\mu}\omega^{-1}\;\text{on some
finite}\;\partial\Omega\,,$ (13)

where $\omega(x)\in{\rm}{\rm SU}(2)$, which corresponds to a pure gauge and
implies topological quantization, i.e. $\Delta n\in\mathbbm{Z}$. Equations
(12) and (13) together are either directly or indirectly implied in the bulk
of the existing literature, including the initial papers on the topic [5, 6,
11, 18]. Although Eq. (13) may be motivated by considering fields in the
classical ground state, i.e. of vanishing classical energy, on the initial and
final spatial hypersurfaces, the quantum ground state also receives
contributions from other field configurations. Therefore, the boundary
condition (13) does not follow from the partition function (12) (note that Eq.
(13) moreover assumes that three-dimensional space is finite), unlike the
topological quantization (10) that is implied by the partition function (9),
see Section 3 for more detail.

To our knowledge, the published papers do not provide a conclusive reason for
how the procedure given by Eqs. (12) and (13) might be deduced from the
functional (9) that defines the theory. Given that the limits do not commute,
i.e., that Eq. (11) is not equivalent with Eqs. (12) and (13), as shown
explicitly in Section 5, there also cannot be such a derivation. (Note that
while in Ref. [11] it is shown that the correlators in a large but finite box
depend only on the boundary conditions through the Chern–Simons flux, the
issue that the limits do not commute is not addressed in that work.) Neither
are we aware of an argument why Eq. (9), which is the standard textbook
expression (up to the fact that we write the infinite-spacetime limit
explicitly, which is a purely notational matter), might be incorrect to start
with. Unless taking $\Omega\to\infty$ there is also no apparent reason why the
boundary condition (13) should be physical.

In the simplest terms, the reason for $CP$-conservation in the strong
interactions can thus be stated as follows: • For $\theta$ to be physical, we
must have $\Delta n\in\mathbbm{Z}$ since $\theta$ is an angular variable.
Since fixed boundary conditions on a finite surface are not physical, this
topological quantization can only follow from $\Omega\to\infty$. • Given the
order of limits that is thus implied, there is no $CP$-violation for
$\Omega\to\infty$ since $\lim_{\Delta
n\to\pm\infty}\lim_{\Omega\to\infty}\Delta n/\Omega=0$.

While the main conclusions and technicalities on the absence of $CP$ violation
in the strong interactions have been presented in Ref. [7], one objective of
the present paper is to add a more formal interpretation of the difference
between Eqs. (9) and (11) versus Eqs. (12) and (13). In Section 3, we recall
to that end the reason for taking Euclidean time to infinity in the first
place. Then, we show that Eq. (11) corresponds to a contour integration that
can be derived and assembled from steepest-descent flows, while the
prescription of Eqs. (12) and (13) does not correspond to a connected
integration contour. Since the reasoning in the present work is based on
infinite Euclidean spacetime as the analytic continuation of Minkowski
spacetime, we briefly comment in Section 4 on how calculations in finite
Euclidean spacetimes with and without boundaries can be made meaningful, but
we leave a detailed discussion to a separate paper. Next, as it allows for an
explicit demonstration of $CP$ conservation as a consequence of Eq. (11) and
for an intuitive interpretation of the matter, we review in Section 5 the
dilute instanton gas calculation of Ref. [7]. This lays the ground to address
objections concerning the volume dependence of the partition function in
Section 6. Since there is no physical interpretation of fixed boundary
conditions on finite Euclidean surfaces, we show that the partition function
in fact shows the expected behaviour when evaluated in finite volumes with
open boundary conditions. Further objections are based on effective field
theory (EFT) descriptions, which is why we review the role of the $\theta$
parameter in the effective ‘t Hooft vertex as well as in chiral perturbation
theory in Section 7. With this preparation, we can reply in Section 8 to an
objection using the topological term in hadronic matrix elements and the role
of $\theta$ in the effective description of the dynamics of the topological
current. After some additional comments on the sampling of topological
configurations in the different orders of limits and the recent literature, we
wrap up and conclude in Section 9. Except for Section 7 where we work in
Minkowski spacetime, we work in Euclidean spacetime throughout all other
sections.

##  3 Path integral and topological quantization

### Euclidean partition function for infinite volume

As a defining point of the quantum field theory, one may take the partition
function (9). Such a partition function based on an infinite Euclidean volume
is a common starting point for a wide range of calculations. Here, we go for
variations with respect to the sources $\eta(x)$ and $\bar{\eta}(x)$ that
yield Euclidean correlation functions for the quarks. Eventually, these can be
analytically continued to Minkowski spacetime and interpreted as vacuum
correlations.

Nonetheless, as some of these matters are contested in the context of $CP$
conservation in the strong interactions, we shall briefly revisit here the
reasons for taking an infinite Euclidean spacetime volume, or, more precisely,
why we take imaginary time to infinity. For more discussion, see Ref. [19]. In
short, it allows one to obtain vacuum correlation functions without specifying
the vacuum in terms of a wave functional (which appears to be practically
impossible in quantum field theory at the nonlinear level).

Imaginary time arises from the analytic continuation of real time. The
corresponding Wick rotation is straightforward for any spacetime that is
stationary in real time. Note that taking the spacetime volume $\Omega$ to
infinity comes as a consequence of the limit of infinite Euclidean time. The
spatial geometry is not the decisive reason for this even though we take here
unbounded Cartesian space $\mathbbm{R}^{3}$, for definiteness. The reason for
taking time to infinity is as follows: By Wick rotation, the correlations
derived from $Z$ for infinite imaginary time correspond to the analytically
continued expectation values for the state of the lowest energy that is
accessible given the conservation laws. This remains true also for clockwise
rotations of the real time axis by an angle $0<\vartheta\leq\pi/2$ in the
complex plane provided the infinite time limit is applied prior to taking
$\vartheta$ to zero. (Do not confuse $\vartheta$ here with the angle $\theta$
as the coefficient of the topological term.) That is, for given initial and
final states $|i,f\rangle$ (which here may be taken as the usual linear
combinations of field eigenstates of integer Chern–Simons number to comply
with gauge invariance under large gauge transformations, i.e. the so-called
$\theta$-vacua) the path integral corresponds to

$\displaystyle Z_{\vartheta}=\lim\limits_{t\to\infty}\langle f|{\rm e}^{-{\rm
i}Ht\exp(-{\rm i}\vartheta)}|i\rangle\,,$ (14)

where $H$ is the Hamiltonian. It therefore projects on the lowest energy
eigenstate that is accessible. For $\vartheta=\pi/2$, one gets the Euclidean
path integral. In this sense, the Minkowskian vacuum correlation functions are
analytic continuations of the Euclidean ones with a discontinuity on the real
time axis.

If instead we were keeping time purely real or still complex in general but
finite, then we would have to weigh each path contributing to the partition
function by the ground state wave functional $\Psi$ evaluated at the endpoints
of these paths, as we discuss in Section 4. The main reason for taking complex
time to infinity is therefore to avoid this complication because then we can
evaluate the path integral without explicit use of the vacuum state. In
particular, since the ground state wave functional in Yang–Mills theory is not
known to an approximation that addresses the present purpose, taking the
imaginary part of time to infinity is the main method of making the analytic
approximation of vacuum correlators feasible.

### Evaluation of the partition function and topological quantization

Having specified the problem through the partition function (9), we now turn
to its evaluation. In the following, we show that Eq. (9) implies an
integration contour that joins together the steepest-descent paths for each
topological sector $\Delta n$. The field configurations on these steepest-
descent contours are not bound by any finite spacetime volume, i.e. there is
no value $R$ so that $F_{\mu\nu}(x)F_{\mu\nu}(x)=0$ (at least to some
approximation) for $|x|>R$ for all field configurations on a given steepest
descent. Therefore, we must evaluate the path integrals for the different
topological sectors $\Delta n$ in the limit $\Omega\to\infty$ before
interfering these [7]. This also implies that organizing the evaluation of the
path integral in terms of a sum over contributions from the individual
topological sectors is valid in the first place. Further, we argue that the
order of limits in Eqs. (12) and (13) is opposite to what is implied by the
form of the correct integration contour.

To start, we state why it is necessary to specify an appropriate integration
contour. Since the integrand in Eq. (9) for $\bar{\theta}\not=0\mod 2\pi$ is
not positive definite, explicitly because of the topological term and
implicitly because of fermion determinants, we must determine the integration
contours to leave the integral well defined. These specify the order in which
the integral $\int{\cal D}A$ must be carried out so that we can derive how to
sum over the topological sectors. In particular, it will allow us to discern
whether it is Eq. (11) or Eqs. (12) and (13) that correspond to the correct
procedure.

To determine the contours, we first note that in general, the real part of the
Euclidean Yang–Mills action

$\displaystyle S\supset S_{\rm YM}=\frac{1}{2g^{2}}\int_{\Omega}{\rm
d}^{4}x\,{\rm tr}F_{\mu\nu}F_{\mu\nu}$ (15)

should be bounded from below on its domain and thus have global or local
minima that may or may not exhibit degeneracies. If $\Omega$ is infinite, this
implies that the vector potential at such minima reduces to pure gauge
configurations at infinity, i.e.

$\displaystyle A_{\mu}(x)\rightarrow{\rm
i}\omega(x)\partial_{\mu}\omega^{-1}(x)\;\text{for}\;|x|\to\infty\quad\text{for
local minima of}\;{\rm Re}[S]\,,$ (16)

where $\omega(x)\in{\rm SU}(2)$. Because the surface at infinity is
homeomorphic to $S^{3}$, and the third homotopy group of the gauge symmetry is
$\pi_{3}({\rm SU}(2))=\mathbbm{Z}$, this immediately implies topological
quantization as in Eq. (10) for these minima, i.e. integer winding number
$\Delta n$. In addition, for each of these minima, there are degeneracies,
i.e. flat directions of the action, parameterized by moduli [20]. Note that
this reasoning applies without imposing boundary conditions ad hoc (as in Eq.
(13)) because the limit $\Omega\to\infty$ appears inside the path integral
(9). This is in line with standard introductions of path integrals in infinite
spacetimes that do not impose particular boundary conditions, see e.g. Ref.
[21] where this point is mentioned explicitly.

Figure 1:  Upper panel: The integration contour given by the steepest-descent
trajectories for the partition function (9) may be followed by successive full
integrations (i.e. over all field configurations in the infinite spacetime),
along the particular thimbles. This corresponds to Eq. (11). The thimbles can
be thought of as being connected via configurations of infinite action,
leading to a connected integrated contour represented by the blue line. Lower
panel: Equations (12) and (13) amount to a contour (represented again by the
blue lines) that is not connected because there is no continuous deformation
from one topological sector into another in finite $\Omega$. The upper and
lower contours are therefore not equivalent in the sense of the Cauchy
theorem.

Given the minima of the real part of the classical Euclidean action with
integer winding number, the contributions to the path integral for the
individual topological sectors $\Delta n$ can then be evaluated on steepest-
descent contours (of the negative action $-S$) passing through these minima.
The contours are Lefschetz thimbles and are determined through flow equations
[22, 23, 24, 25]. By this reasoning of steepest-descent contours, upon dealing
with the usual ultraviolet divergences and the vacuum contributions in the
infinite spacetime volume, the path integrals in the individual topological
sectors are convergent. The steepest-descent contours for different $\Delta n$
do not intersect for finite $S$ because, in the infinite spacetime volume,
solutions of different winding number are separated by infinite action
barriers. The integration contours over the different sectors $\Delta n$ can
therefore only be connected via configurations of infinite action $S$ that
give no contribution to the partition function (9). Note that these infinite
action configurations connecting the sectors are allowed precisely because we
do not impose boundary conditions when evaluating the partition function. For
comparison, in finite volumes with fixed boundary conditions (13) the field
configurations in different topological sectors are not continuously
connected, not even via a path through configurations of infinite action since
paths with a noninteger winding number are forbidden by the boundary
conditions (13), no matter whether $S$ is finite or infinite. In Figure 1, we
schematically illustrate these integration paths and their crucial
differences.

Therefore, we first carry out the integration over the entire infinite
spacetime in a given topological sector $\Delta n$ individually. Then, we can
connect this integration with the steepest-descent contour for a different
$\Delta n$ via configurations of infinite action that do not contribute to the
path integral. This determines the integration contour that should be used in
order to evaluate Eq. (9) and that is given by the order of limits specified
in Eq. (11). Any alternative contour must be a continuous deformation in
compliance with the prerequisites of the Cauchy theorem, a criterion that is
not met with Eqs. (12) and (13), which consequently lead to a different
result. In practice, this means that, without specifying ad hoc boundary
conditions, we must not interfere the different sectors before taking
$\Omega\to\infty$. Otherwise, we would partition and rearrange the full
integration contour in a non-continuous way that in general leads to an
inequivalent result because the integrand, or more specifically, the sum over
the topological sectors, is not positive definite and not absolutely
convergent.

We can therefore write the path integral as in Eq. (11) and as indicated in
Figure 1. We emphasize that the decomposition into topological sectors follows
from $\Omega\to\infty$ and is not a consequence of the saddle point
approximation (which only comes into play in the dilute instanton gas
approximation in Section 5). In turn, finite surfaces $\partial\Omega$ imply
that topological charge is not quantized, i.e. it can flow in and out of the
volume $\Omega$, unless imposing unphysical constraints.

Since the winding number density $\Delta n/\Omega$ apparently is a measure of
$CP$ violation, it is already clear that for each term of the series (11),
there is no $CP$-violating contribution, as $\lim_{\Omega\to\infty}\Delta
n/\Omega=0$. For the quark system, the corresponding calculation is explicitly
presented in Section 5.

### Commuting the order of limits

The decisive point in the present discussion regarding the $CP$ symmetry of
the strong interactions is whether Eqs. (12) and (13) are consistent with Eq.
(9) and the integration contour that it implies. As we shall review in Section
5, the conclusion that the $CP$ symmetry is violated when
$\alpha+\theta\not=0\mod\pi$ relies on a partition function as in Eq. (12) in
conjunction with the ad hoc boundary conditions (13). In Eq. (12) the order of
limits is therefore opposite to Eq. (11) that we have derived from the
starting point given by Eq. (9).

Now to further (beyond the apparent contradiction with Eq. (9) regarding $CP$)
assess the validity of Eqs. (12) and (13), we take $\Omega$ to be finite and
first assume that the boundary conditions from Eq. (13) are not imposed. In
particular, one can then move topological charge (i.e. instantons in the weak
coupling limit) across the boundary $\partial\Omega$ so that there is no
topological quantization into sectors with integer winding number $\Delta n$
and also no conservation of topological charge within $\Omega$. Without these,
nontrivial minima of the action do not exist also. To bring topology back into
the picture, one might therefore impose the boundary conditions given in Eq.
(13). These boundary conditions can be derived as a consequence of the
infinite spacetime volume $\Omega\to\infty$ in the partition function (9)
[11]. However, imposing these boundary conditions and computing the partition
function according to Eqs. (12) and (13) requires commuting the limit of
infinite spacetime volume with the sum over infinitely many topological
sectors which is however not justified.

Equations (12) and (13) therefore do not follow from Eq. (9) so that one may
keep looking for alternative arguments. However, while topological
quantization also emerges for fixed boundary conditions on a given compact
$\partial\Omega$ as in Eq. (13), there does not appear to be a valid reason
for imposing such configurations. In fact, the vacuum wave functional has
nonvanishing support on configurations that do not observe Eq. (13) and thus
have a nonvanishing classical energy. Since the field operators do not commute
with the Hamiltonian, the configurations obeying Eq. (13) are as good or as
bad as any other field configuration subject to a different boundary condition
on $\partial\Omega$. (For the boundary conditions given in Eq. (13), $\Delta
n$ are integers whereas, for more general boundary conditions specified up to
gauge transformations, $\Delta n$ are given by a fixed real number plus any
integer.) So there is no preference for choosing pure gauges as a boundary
condition on a sequence of finite surfaces, even as these surfaces are taken
to infinity. For example, if $\partial\Omega$ is spherical, instantons could
be placed at certain angles and close to the radius of $\partial\Omega$, what
defines boundary conditions with noninteger $\Delta n$. Except that there is
not a valid reason to impose Eqs. (12) and (13), whose consequence for the
outcome of the calculation is material, there also is no a priori
justification why $\Omega$ should be taken to infinity at the same rate for
all topological sectors in Eq. (12).

Finally, note that for examples that do not involve noncommuting limits,
correlations from fixed boundary conditions on finite $\partial\Omega$
converge in imaginary time to the vacuum correlators as $\Omega\to\infty$.
However, this does not imply by analogy that in the present case, where we
must sum over infinitely many topological sectors, Eqs. (12) and (13) yield
the correct vacuum correlation functions.

##  4 Finite Euclidean spacetimes

Above, we have reviewed the reasoning for computing the path integral without
specifying boundary conditions in favour of taking complex time to infinity.
While yielding the physical correlation functions, taking time to infinity
clearly is a mathematical trick. However, as we have discussed in Section 3,
simply using Eqs. (12) and (13) is not a valid procedure. Nonetheless, it
should still be possible, at least in principle, to carry out the calculation
in a finite spacetime volume.

To this end, we see three ways of doing this. All of these turn out to require
the replacement of the fixed boundary conditions (13) with different
configurations that again lead to the same conclusion of $CP$ conservation.
These particular possibilities are:

  * •

We can take a finite time interval at the price of having to project on a
vacuum wave functional (see the present section).

  * •

In order to avoid the projection on the wave functional, we can stay within
functional quantization and consider a finite subvolume of infinite Euclidean
space. Then, we have to integrate over all possible boundary configurations on
the subvolume (see Section 6).

  * •

We can take compact spacetimes without boundaries. For definiteness, consider
here a four-torus with a finite Euclidean time interval of length $\beta=1/T$,
where $T$ is the temperature. The relation with Minkowski-spacetime is given
by its correspondence with the canonical thermodynamic partition function.
This once again requires canonical quantization that restricts the form of the
wave functionals (see the present section).

### Projecting on the wave functional

Regarding the first option, unless taking complex time to infinity or assuming
finite temperature, we need to specify the ground state in order to get the
correct boundary conditions for the path integral. That is, when restricting
to a real time interval from $t^{\prime}$ to $t$, we must weigh each path
contributing to the partition function

$\displaystyle Z_{\rm M}(t_{2},t_{1})=$ $\displaystyle\int{\cal
D}\bar{\psi}\,{\cal D}\psi\,{\cal
D}A\,\Psi^{*}[A_{\mu}(t_{2},\mathbf{x}),\psi(t_{2},\mathbf{x})]\,\Psi[A_{\mu}(t_{1},\mathbf{x}),\bar{\psi}(t_{1},\mathbf{x})]$
$\displaystyle\times$ $\displaystyle{\rm e}^{{\rm i}\int_{\Omega_{12}}{\rm
d}^{4}x\,\left({\cal L}_{\rm M}-\bar{\eta}\psi-\bar{\psi}\eta\right)}\,,$ (17)

by the unknown ground state wave functional

$\displaystyle|\Psi,t\rangle=\Psi[A_{\mu}(\mathbf{x},t),\bar{\psi}(\mathbf{x},t))]\,|A_{\mu}(\mathbf{x},t)\rangle\otimes|\psi(\mathbf{x},t)\rangle\,$
(18)

evaluated at the endpoints of these paths. Here, ${\cal L}_{\rm M}$ is the
Minkowskian Lagrangian, $|A_{\mu}(\mathbf{x},t)\rangle$, and
$|\psi(\mathbf{x},t)\rangle$ are field eigenstates and $\Omega_{12}$ is the
four-volume bounded by the three-volumes at the times $t_{1}$ and $t_{2}$.

In turn, for physical boundary conditions that do account for fluctuations
about the classical minimal-energy states, topological quantization cannot be
assumed. Note while this means that the action generally receives
contributions from field configurations of noninteger $\Delta n$, this does
not preclude the wave functional from transforming with a phase factor under
large gauge transformations. However one may wonder whether one can still
assume topological quantization in an approximate sense, so that one might
still obtain a good result from the sum over path integrals with boundary
conditions corresponding to pure gauges imposed on some finite volume. To
settle this question, we would have to find the ground state wave functional
in canonical quantization of Yang–Mills theory [26], which does not appear to
be practically possible. Finite temperature field theory also relies on
canonical quantization, but tangible conclusions may be drawn in that context,
as we discuss next.

### Compact spacetimes without boundary

As for introducing finite spacetime volume through temperature, a notable
example is de Sitter space, where the Euclidean counterpart is a sphere, and
hence $\Omega$ is finite for this spacetime. The latter can be interpreted as
the representation of a canonical ensemble of states over a static patch of de
Sitter space with a Lorentzian signature. Another important situation where
$\Omega$ is finite is when space is a three torus and we consider a canonical
ensemble as well. Then, the Euclidean representation is a four-torus and
corresponds to the continuum limit of computations in lattice QCD. Both of
these very relevant examples of finite Euclidean volume therefore require
first the canonical quantization in the respective background geometries in
order to connect these with physical observables in Lorentzian spacetimes.
This is mandatory in order to evaluate the trace of the canonical density
matrix from which the path integral representation can then be derived.

In canonical quantization of gauge theory, large gauge transformations on the
spatial section can play a special role. These are gauge transformations that
are not continuously connected with the identity but give rise to equivalence
classes that are a representation of the homotopy group. This also happens for
infinite volumes (provided that the gauge field configurations approach a
unique value at spatial infinity) and leads to the well-known $\theta$-vacua
[4, 5, 6]. However, the infinite volume limit taken inside the path integral
readily implies $CP$ conservation so that it is interesting to further focus
on finite spatial volumes.

On finite spatial volumes, large gauge transformations are only singled out
when fixing the gauge corresponding to periodic (i.e. single-valued) gauge
potentials on the torus or single-valued gauge potentials on the sphere.
Imposing single-valuedness, we find that the canonical quantization on these
finite spatial volumes only admits states that are invariant under large gauge
transformations, i.e. without any phase incurred, and that there hence can be
no $CP$ violation. This also resolves the matter of renormalizability of the
states raised in Refs. [27, 28]. On the other hand, without imposing single-
valuedness, all gauge transformations on the spatial sections are continuously
connected with the identity transformation, and again no $CP$-violating
effects can be deduced. The details of this argument shall be published
elsewhere [17].

##  5 Dilute instanton gas approximation

The most sensitive probe of possible $CP$ violation associated with the strong
interactions is the EDM of the neutron. At the relevant energy scale, QCD is
deeply in the nonperturbative regime. This is a well-known and obvious
drawback for any analytical approximation. Yet, one can observe from
semiclassical calculations that instantons play a central role in the
spontaneous breaking of chiral symmetry as well as in mediating the effects
from the anomalous axial ${\rm U}(1)_{\rm A}$ symmetry that notably explains
the large mass of the $\eta^{\prime}$ meson [29, 30]. It is therefore also
strongly indicated that the role of the topological term can be understood
from a semiclassical evaluation of the effective fermion interaction mediated
by instantons, i.e. the ‘t Hooft operator [29, 30]. This corresponds to the
expectation that the presence or absence of $CP$ violation should prevail when
crossing between the strongly and weakly coupled regimes at low and high
energies, respectively. Moreover, the generic arguments in Section 3 as well
as in Refs. [7, 19], where cluster decomposition and the index theorem are
used, do not refer to the semiclassical approximation.

The semiclassical approximation therefore remains of substantial interest,
being the only analytic procedure to make quantitative statements about $CP$
violation in the strong interactions. Also, it offers a very useful
perspective on the central issues with this topic.

In the present context, the semiclassical approach is given by the dilute
instanton gas approximation. Stationary and quasi-stationary points of the
action are described in terms of instantons and their individual collective
coordinates [29, 30, 11, 10]. Stationary points are the classical solutions.
These are the minima of the action for each topological sector characterized
by winding number $\Delta n$. For $\Delta n=\pm 1$, they are given by
Belavin–Polyakov–Schwarz–Tyupkin (BPST) (anti-)instanton solutions [31] whose
classical Yang–Mills action is

$\displaystyle S_{\rm BPST}=\frac{8\pi^{2}}{g^{2}}\,.$ (19)

Explicitly, the BPST instanton reads in the regular gauge

$\displaystyle
A^{a}_{\mu}=2\eta_{a\mu\nu}\frac{(x-x_{0})_{\nu}}{(x-x_{0})^{2}+\rho^{2}},$
(20)

where $x_{0}$ and $\rho$ are free parameters corresponding to the center
location of the instanton and its size, respectively. Here $\eta_{a\mu\nu}$
are the ‘t Hooft symbols [30]

$\displaystyle\eta_{a\mu\nu}=\begin{cases}\varepsilon_{a\mu\nu},\quad&\mu,\nu=1,2,3\\\
-\delta_{a\nu},&\mu=4\\\ \delta_{a\mu},&\nu=4\\\ 0,&\mu=\nu=4\end{cases}\,.$
(21)

Similarly one can define $\bar{\eta}_{a\mu\nu}$ by a change in the sign of
$\delta$ in the above equation. For the anti-instanton, we should replace
$\eta_{a\mu\nu}$ by $\bar{\eta}_{a\mu\nu}$ in Eq. (20).

To visualize the BPST instanton (in analogy to Ref. [32] for one-dimensional
instantons), we consider some explicit expressions with $x_{0}=0$, i.e., with
the center set at the origin. For example,

$\displaystyle A_{1}^{3}(x)=\frac{2x_{2}}{x^{2}+\rho^{2}}\,,$ (22)
$\displaystyle A_{2}^{3}(x)=-\frac{2x_{1}}{x^{2}+\rho^{2}}\,.$ (23)

$A_{1}^{3}(x)$ is symmetric in the hyperplane $\\{x_{1},x_{3},x_{4}\\}$ and
$A_{2}^{3}$ symmetric in the hyperplane $\\{x_{2},x_{3},x_{4}\\}$. Without
loss of generality, in Figure 2, we show these as a function of $x_{1},x_{2}$
by taking $x_{3}=x_{4}=0$ in arbitrary units. The field strength components
read

$\displaystyle
F_{\mu\nu}^{a}=-4\eta_{a\mu\nu}\frac{\rho^{2}}{x^{2}+\rho^{2}}\,.$ (24)

As an example, we plot $F_{21}^{3}$ as a function of $x_{1}$ and $x_{2}$ in
Figure 3. These quantities are gauge dependent. The gauge-independent
quantities are

$\displaystyle{\rm tr}F_{\mu\nu}F^{\mu\nu}={\rm
tr}F_{\mu\nu}\widetilde{F}^{\mu\nu}=-\frac{96\rho^{4}}{[x^{2}+\rho^{2}]^{4}}\,.$
(25)

For the anti-instanton, one would have ${\rm tr}F_{\mu\nu}F^{\mu\nu}=-{\rm
tr}F_{\mu\nu}\widetilde{F}^{\mu\nu}$. We plot these in Figure 4. From the
graph, one can see that the instanton indeed has a radius characterized by the
value of $\rho$ ($\rho=1$ in the plot).

Figure 2: Visualization of the gauge potential components $A_{1}^{3}$ and
$A_{2}^{3}$ in the BPST instanton solution on the $\\{x_{1},x_{2}\\}$ plane
($\rho=1$) . Figure 3: The field strength component $F_{21}^{3}$ for the BPST
instanton solution as a function of $x_{1}$ and $x_{2}$ with $x_{3}=x_{4}=0$,
$\rho=1$. Figure 4: ${\rm tr}F_{\mu\nu}F^{\mu\nu}={\rm
tr}F_{\mu\nu}\widetilde{F}^{\mu\nu}$ as a function of $|x|$ with $\rho=1$ for
the BPST instanton.

For $|\Delta n|>1$, they should be obtained from the
Atiyah–Drinfeld–Hitchin–Manin (ADHM) construction [33]. In the dilute gas
picture, they correspond to $\Delta n$ instantons, no anti-instantons for
$\Delta n>0$ and $-\Delta n$ anti-instantons, no instantons for $\Delta n<0$.
The collective coordinates for the individual instantons describe their size,
their gauge orientation as well as their position in Euclidean spacetime. As
we follow the steepest-descent contours, the action $S$ evolves towards larger
values, and we can encounter quasi-stationary points. These can be described
in terms of the number $n$ of instantons and $\bar{n}$ of anti-instantons,
where both of these objects can coexist within such configurations. In the
sector (i.e. on the thimble) characterized by $\Delta n$, it must hold that
$n-\bar{n}=\Delta n$. Each of these individual instantons and anti-instantons
is again parametrized in terms of the aforementioned collective coordinates.
In Figure 5, we illustrate the typical structure of the thimbles in the
semiclassical approximation.

Figure 5:  Some projection of steepest-descent contours (thimbles) in the
semiclassical approximation. Within each topological sector, there is a set of
points with minimal action. This is continuously connected via steps of height
$2S_{\rm BPST}$, corresponding to the addition of pairs of instantons and
anti-instantons, to additional plateaus. Double lines indicate that these
plateaus extend over infinite sets in field space, in the direction of the
(approximate) collective coordinates.

Now, we aim to integrate out the gluon fields in order to see explicitly what
quark correlations breaking the anomalous chiral symmetry ${\rm U}(1)_{A}$
they leave behind. We follow Ref. [7], but here we work with Euclidean time
for simplicity.

The relevant quark correlation function is given by

$\displaystyle\langle\psi(x)\bar{\psi}(x^{\prime})\rangle=-\left.\frac{\delta^{2}\log
Z[\eta,\bar{\eta}]}{\delta\bar{\eta}(x)\delta\eta(x^{\prime})}\right|_{\eta=\bar{\eta}=0}=\frac{\int{\cal
D}\bar{\psi}\,{\cal D}\psi\,{\cal D}A\,\psi(x)\bar{\psi}(x^{\prime}){\rm
e}^{-\lim\limits_{\Omega\to\infty}\int_{\Omega}{\rm d}^{4}x\,{\cal
L}}}{\int{\cal D}\bar{\psi}\,{\cal D}\psi\,{\cal D}A\,{\rm
e}^{-\lim\limits_{\Omega\to\infty}\int_{\Omega}{\rm d}^{4}x\,{\cal L}}}\,.$
(26)

While this is a standard expression one should note that the numerator and
denominator in this equation are not well defined in the thermodynamic limit
$\Omega\to\infty$, even when ultraviolet divergences have been renormalized.
However, this does not force us to keep $\Omega$ finite. Rather, divergent
extensive contributions in the numerator and denominator from spacetime
regions far away from $x$ and $x^{\prime}$ cancel. In standard perturbation
theory, these contributions are represented by vacuum diagrams where the
divergence results from their overall invariance under spacetime translations.
We go into more detail regarding this point in Section 6. In the present
semiclassical evaluation, we shall see how to deal with these extensive
contributions a bit further down the line of argument.

To proceed with the evaluation of Eq. (26), we approximate the Green’s
function of the quarks in the background of one anti-instanton ($\Delta n=-1$
as)

$\displaystyle
S(x,x^{\prime})=\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}\frac{\hat{\psi}_{\bar{\lambda}}(x)\hat{\psi}^{\dagger}_{\bar{\lambda}}(x^{\prime})}{\bar{\lambda}}\approx\frac{\hat{\psi}_{0\rm
L}(x)\hat{\psi}_{0{\rm L}}^{\dagger}(x^{\prime})}{m\,{\rm e}^{-{\rm
i}\alpha}}+\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}\limits_{\lambda\not=0}\frac{\hat{\psi}_{\lambda}(x)\hat{\psi}^{\dagger}_{\lambda}(x^{\prime})}{\lambda}\,.$
(43)

The middle expression is the exact spectral sum representation in terms of the
eigenvalues $\bar{\lambda}$ of the Dirac operator of massive quarks in the
anti-instanton background, and $\hat{\psi}_{\bar{\lambda}}$ are the
corresponding eigenfunctions. As for the approximation on the right, by
$\hat{\psi}_{0{\rm L,R}}$ we denote the ‘t Hooft zero modes of the massless
Dirac operator in the corresponding one anti-instanton or instanton background
[29, 30], which are purely chiral and where their handedness is indicated by
${\rm L,R}$. The nonzero eigenvalues of the massless Dirac operator are given
by $\lambda$ and the pertaining eigenmodes by $\psi_{\lambda}$. Note that the
contribution breaking chiral symmetry, i.e. the first term in the approximate
expression, aligns with the $CP$ phase $\alpha$ pertaining to the quark mass
and not with the angle $\theta$. For real masses, this approximation has been
used in Refs. [34, 35].

In the semiclassical approximation, we carry out the path integral by taking
the quasistationary configurations of the action, i.e. with $n$ instantons and
$\bar{n}$ anti-instantons and evaluate the leading fluctuations, i.e. the
functional determinants corresponding to one-loop order. For such a
quasistationary background, the Green’s function for the quarks should be well
approximated by [35]

$\displaystyle S_{n,\bar{n}}(x,x^{\prime})\approx S_{0}(x,x^{\prime}){+}$
$\displaystyle\sum_{\bar{\nu}=1}^{\bar{n}}\frac{\hat{\psi}_{0{\rm
L}}(x-x_{0,\bar{\nu}}){\hat{\psi}^{\dagger}_{0{\rm
L}}}(x^{\prime}-x_{0,\bar{\nu}})}{m\,{\rm e}^{-{\rm i}\alpha}}$
$\displaystyle{+}$ $\displaystyle\sum_{\nu=1}^{n}\frac{\hat{\psi}_{0{\rm
R}}(x-x_{0,\nu}){\hat{\psi}^{\dagger}_{0{\rm
R}}}(x^{\prime}-x_{0,\nu})}{m\,{\rm e}^{{\rm i}\alpha}}\,.$ (44)

Here $x_{0,\nu}$ and $x_{0,\bar{\nu}}$ are the locations of instantons and
anti-instantons, respectively, $S_{0}$ is the Green’s function of a Dirac
fermion with mass $m\exp({\rm i}\alpha\gamma^{5})$ in a translation-invariant
(i.e., void of instantons) background. This approximation neglects
contributions from overlapping instantons which are more suppressed as the
instanton gas becomes more dilute. While the Green’s function close to the
individual instantons and anti-instantons is dominated and therefore
approximated by the ‘t Hooft zero modes, sufficiently far away from the points
$x_{0,\nu}$ and $x_{0,\bar{\nu}}$ the Green’s function is given by the form in
the background without instantons, i.e.

$\displaystyle S_{0}(x,x^{\prime})=(-\gamma_{\mu}\partial_{\mu}+m\,{\rm
e}^{-{\rm i}\alpha\gamma^{5}})\int\frac{{\rm d}^{4}p}{(2\pi)^{4}}\,{\rm
e}^{-{\rm i}p(x-x^{\prime})}\frac{1}{p^{2}+m^{2}}\,.$ (45)

In Eq. (44), we note the alignment of the instanton-induced breaking of chiral
symmetry with the quark masses so that there is no indication of $CP$
violation at this level but also note that $\theta$ has not yet entered into
the calculation.

Given the Green’s functions (44), we can proceed with evaluating the fermion
correlation on the thimble (or equivalently in a fixed topological sector)
characterized by the winding number $\Delta n$:

$\displaystyle\langle\psi(x)\bar{\psi}(x^{\prime})\rangle_{\Delta n}$
$\displaystyle=$ $\displaystyle\\!\\!\\!\sum\limits_{\bar{n},n\geq 0\atop
n-\bar{n}=\Delta n}\\!\\!\int{\cal D}A_{\bar{n},n}{\cal D}\bar{\psi}{\cal
D}\psi\,\psi(x)\bar{\psi}(x^{\prime}){\rm e}^{-S[A,\bar{\psi},\psi]}$
$\displaystyle=$ $\displaystyle\sum\limits_{\bar{n},n\geq 0\atop
n-\bar{n}=\Delta
n}\frac{1}{\bar{n}!n!}\left(\prod\limits_{\bar{\nu}=1}^{\bar{n}}\;\int_{\Omega}{\rm
d}^{4}x_{0,\bar{\nu}}{\rm
d}\Sigma_{\bar{\nu}}J_{\bar{\nu}}\right)\left(\prod\limits_{\nu=1}^{n}\;\int_{\Omega}{\rm
d}^{4}x_{0,\nu}{\rm d}\Sigma_{\nu}J_{\nu}\right)S_{\bar{n},n}(x,x^{\prime})$
$\displaystyle\hskip 142.26378pt\times\,{\rm e}^{-S_{\rm
BPST}\,(\bar{n}+n)}{\rm e}^{-{\rm
i}(\bar{n}-n)(\alpha+\theta)}(-\Theta\varpi)^{(\bar{n}+n)}\,.$ (46)

The symbol ${\cal D}A_{\bar{n},n}$ implies that the path integral is evaluated
in terms of fluctuations and moduli about the classical background, i.e.
(quasi-)stationary point, made up from $n$ instantons and $\bar{n}$ anti-
instantons. The integration over collective coordinates other than the
locations of the instantons and anti-instantons are denoted by ${\rm
d}\Sigma_{\nu,\bar{\nu}}$, and the Jacobians from the transformation of the
zero modes in the path integral in favour of the collective coordinates are
denoted by $J_{\nu,\bar{\nu}}$. The one-loop determinant of the gauge field
about a single instanton or anti-instanton (denoted by $\bar{A}$ below) with
the zero modes omitted and divided by the gauge field determinant in the
background $A=0$ is given by

$\displaystyle\varpi\equiv\frac{1}{\sqrt{{\det}^{\prime}_{\bar{A}}/{\det}_{A=0}}}\,,$
(47)

where the prime on the determinant indicates the omission of zero eigenvalues.
In an analogous manner, $\Theta$ represents the modulus of the ratio of the
fermionic determinants in the one-(anti)instanton and $A=0$ backgrounds,

$\displaystyle\Theta\equiv\left|\frac{\det(-\not{D}-m{\rm e}^{{\rm
i}\alpha\gamma^{5}})}{\det(-\not{\partial}-m{\rm e}^{{\rm
i}\alpha\gamma^{5}})}\right|\,.$ (48)

As usual, the partition function diverges in the thermodynamic limit so that
we keep the spacetime volume $\Omega$ finite for now. Nonetheless, we need to
take $\Omega\to\infty$ before eventually summing over the topological sectors
$\Delta n$ as the latter are only a consequence of infinite spacetime volume
and to remain true to the integration contour implied by Eq. (9), cf. the
discussion in Section 3.

In order to normalize, i.e., to divide out vacuum contributions, we also need
the partition function in a fixed topological sector. Proceeding as for the
fermion correlation, we obtain

$\displaystyle\begin{aligned} Z_{\Delta n}=&\sum\limits_{\bar{n},n\geq 0\atop
n-\bar{n}=\Delta n}\int{\cal D}A_{\bar{n},n}{\cal D}\bar{\psi}{\cal
D}\psi\,{\rm e}^{-S[A,\bar{\psi},\psi]}\\\ =&\sum\limits_{\bar{n},n\geq 0\atop
n-\bar{n}=\Delta n}\frac{1}{\bar{n}!n!}\left(-\Omega J\,\Theta\,\varpi\,{\rm
e}^{-S_{\rm BPST}}{\textstyle\int}\\!{\rm d}\Sigma\right)^{(\bar{n}+n)}{\rm
e}^{-{\rm i}(\bar{n}-n)(\alpha+\theta)}\,.\end{aligned}$ (49)

Next, we turn to the collective coordinates and integrate out the location of
a single anti-instanton as

$\displaystyle\begin{aligned} &\int_{\Omega}{\rm d}^{4}x_{0,\bar{\nu}}\
S(x,x^{\prime})\\\\[-5.69054pt] \approx&\,\int_{\Omega}{\rm
d}^{4}x_{0,\bar{\nu}}\left[S_{0}(x,x^{\prime}){+}\frac{\hat{\psi}_{0{\rm
L}}(x-x_{0,\bar{\nu}})\hat{\psi}_{0{\rm
L}}^{\dagger}(x^{\prime}-x_{0,\bar{\nu}})}{m{\rm e}^{-{\rm
i}\alpha}}+\cdots\right]\\\
=&\Omega\,(S_{0}(x,x^{\prime})+\cdots){+}m^{-1}{\rm e}^{{\rm
i}\alpha}h(x,x^{\prime})P_{\rm L}\,.\end{aligned}$ (50)

The dots above represent contributions from the zero modes of the
(anti)-instantons whose centers were not integrated over. This expression
defines the overlap function $h(x,x^{\prime})$—a rank-two tensor in spinor
space:

$\displaystyle h(x,x^{\prime})P_{\rm L}=$ $\displaystyle\,\int_{\Omega}{\rm
d}^{4}x_{0,\bar{\nu}}\,\hat{\psi}_{0{\rm
L}}(x-x_{0,\bar{\nu}})\hat{\psi}_{0{\rm
L}}^{\dagger}(x^{\prime}-x_{0,\bar{\nu}})\,,$ (51) $\displaystyle
h(x,x^{\prime})P_{\rm R}=$ $\displaystyle\,\int_{\Omega}{\rm
d}^{4}x_{0,\nu}\,\hat{\psi}_{0{\rm R}}(x-x_{0,\nu})\hat{\psi}_{0{\rm
R}}^{\dagger}(x^{\prime}-x_{0,\nu})\,.$ (52)

Further, we integrate over the remaining collective coordinates as

$\displaystyle\bar{h}(x,x^{\prime})\equiv$ $\displaystyle\frac{\int{\rm
d}\Sigma\,h(x,x^{\prime})}{\int{\rm d}\Sigma}\,.$ (53)

Notice that we ignore here the fact that for the classical instanton, the
integral over the dilatational mode is divergent. The running coupling will
however render the correlations finite in a more complete calculation.

Strictly speaking, the dilute instanton gas approximation is only applicable
when the integral over the dilatational mode converges, i.e. when
contributions from both, large and small instantons are cut off. This is
naturally the case in the ultraviolet, where small instantons are suppressed
for asymptotically free theories as $g\to 0$. In the infrared, one may attempt
to keep $g$ perturbatively small by a bespoke particle content that controls
the running coupling. As a matter of principle, an infrared cutoff can also be
enforced when the gauge symmetry is spontaneously broken so that the size of
the instantons is limited by the inverse gauge boson mass [29, 30]. While none
of this applies to the strong interactions, such considerations show that the
dilute instanton gas is a meaningful concept. Note that an infrared cutoff for
the instanton size has no implications for the integration over the locations
$x_{0,\nu},x_{0,\bar{\nu}}$ of the instanton centers. The preservation of
Poincaré symmetry, and in particular Lorentz invariance, demands that the
values of $x_{0,\nu},x_{0,\bar{\nu}}$ should remain unconstrained. Hence, the
dependence of the results on the spacetime volume is unchanged in the presence
of a cutoff for the instanton size. As a consequence, the former has no
consequence for the order of limits of infinite spacetime volume and infinite
maximal absolute value of the topological charge. Note that even with the
aforementioned size cutoff it is clear that either expression (11) or (12) can
be technically evaluated. Further, the presence or absence of divergences from
infrared instantons does not decide which order of limits must be taken
because the presence of sectors of integer $\Delta n$ is a topological
argument that does not depend on the validity of the semiclassical expansion.
We eventually note here that the validity of the dilute instanton gas and its
generalization toward the inclusion of interactions between instantons has
been addressed in Refs. [9, 10].

The present point of view is that the saddle point approximation in the dilute
instanton gas approach, while not quantitatively applicable to the strong
interactions, yields information about the symmetries that are respected by
the theory. This does not only apply to the present work that argues in favour
of the evaluation of the partition function according to Eq. (11) but also to
Refs. [5, 18] that assume Eqs. (12) and (13). To our knowledge, Ref. [18] is
the only paper that explicitly evaluates the ’t Hooft operator for nonzero
$\bar{\theta}$. While the saddle point approximation is an important cross-
check, we note that the conclusion about the absence of $CP$ violation does
not rely on it, cf. the boxed argument in Section 2 and in the part of Section
3 on the evaluation of the partition function and topological quantization as
well as Refs. [7, 19]. Furthermore, as will be reviewed at the end of Section
6, the results obtained with the dilute instanton gas can be recovered from
general arguments based on cluster decomposition and the index theorem,
without making use of the dilute gas approximation.

Integrating now over all locations of instantons and anti-instantons, we
obtain the correlation function for fixed $\Delta n$:

$\displaystyle\langle\psi(x)\bar{\psi}(x^{\prime})\rangle_{\\!\Delta n}$
$\displaystyle=$ $\displaystyle\sum\limits_{\bar{n},n\geq 0\atop
n-\bar{n}=\Delta
n}\frac{1}{\bar{n}!n!}\Big{[}{\,\bar{h}(x,x^{\prime})}\left(\frac{\bar{n}}{m{\rm
e}^{-{\rm i}\alpha}}P_{\rm L}+\frac{n}{m{\rm e}^{{\rm i}\alpha}}P_{\rm
R}\right)\Omega^{\bar{n}+n-1}+S_{0}(x,x^{\prime})\Omega^{\bar{n}+n}\Big{]}$
$\displaystyle\hskip 227.62204pt\times\kappa^{\bar{n}+n}(-1)^{n+\bar{n}}{\rm
e}^{{\rm i}\Delta n(\alpha+\theta)}$ $\displaystyle=$
$\displaystyle\left[\\!\left({\rm e}^{{\rm i}\alpha}\\!I_{\Delta
n+1}(2\kappa\Omega)P_{\rm L}+{\rm e}^{\\!\\!-{\rm i}\alpha}\\!I_{\Delta
n-1}(2\kappa\Omega)P_{\rm
R}\right)\frac{\kappa}{m}\,\bar{h}(x,x^{\prime})+I_{\Delta
n}(2\kappa\Omega)S_{0}(x,x^{\prime})\\!\right]$ $\displaystyle\hskip
227.62204pt\times{(-1)^{\Delta n}{\rm e}^{{\rm i}\Delta n(\alpha+\theta)}}\,,$
(54)

where $\kappa={\textstyle\int}\\!{\rm d}\Sigma\,J\,\,\Theta\,\varpi\,{\rm
e}^{-S_{\rm BPST}}$ is the instanton density per spacetime volume222Note that
in Minkowskian spacetime, we define ${\rm i}\kappa=\int{\rm
d}\Sigma\,J\Theta\varpi{\rm e}^{-S_{\rm BPST}}$ in Ref. [7]. The $\kappa$ in
both cases is the same and is real due to the fact that the Jacobian $J$ in
Minkowski spacetime contains an additional factor of ${\rm i}$ compared to its
Euclidean counterpart. and $I_{\nu}(x)$ is the modified Bessel function.

The terms involving the overlap function $\bar{h}$ are due to the instanton
effects on the quarks and break chiral symmetry. While we should expect that
these scale in the same way with the spacetime volume $\Omega$ as the term
with $S_{0}$, i.e. the contribution from regions between instantons, the
explicit dependence on $\Omega$ in Eq. (54) is different. However, we see that
the scaling after all is the same. Relax for the moment the constraint of
fixed $\Delta n$ and use that $\kappa$ may be interpreted as the likelihood
for finding an instanton in a unit four-volume. Then, for large $\Omega$ the
sum is dominated by particular value of $n\approx\bar{n}$:

$\displaystyle\langle
n\rangle=\frac{\sum_{n=0}^{\infty}n\frac{(\kappa\Omega)^{n}}{n!}}{\sum_{n=0}^{\infty}\frac{(\kappa\Omega)^{n}}{n!}}=\kappa\Omega\,.$
(55)

Moreover, the relative fluctuation vanishes in the infinite-volume limit [7]:

$\displaystyle\frac{\sqrt{\langle(n-\langle n\rangle)^{2}\rangle}}{\langle
n\rangle}=\frac{1}{\sqrt{\kappa\Omega}}\,.$ (56)

This means that in the coefficients in front of the chiral projection
operators within the middle expression in Eq. (54), we can replace
$n,\bar{n}\to\kappa\Omega$. This basic behaviour, i.e. that the central value
for the number of instantons is given by $\kappa\Omega$, is also reflected by
the fact that for large arguments, the modified Bessel functions become
independent of their index, i.e. $\lim_{x\to\infty}I_{\Delta n}(x)/I_{\Delta
n^{\prime}}(x)=1$. Since all the modified Bessel functions in Eq. (54) tend to
the same value, we see directly from this expression that there is no relative
$CP$ phase between the terms from the quark masses and instanton-induced
breaking of chiral symmetry in the infinite-volume limit. Correspondingly, the
partition function for fixed $\Delta n$ turns out as [36]

$\displaystyle Z_{\Delta n}=I_{\Delta n}(2\kappa\Omega)\,(-1)^{\Delta n}{\rm
e}^{{\rm i}\Delta n(\alpha+\theta)}\,.$ (57)

Now, when calculating the correlation function as the sum over the topological
sectors, we have to take the limit $\Omega\to\infty$ first for the reasons
explained in Section 3. Because of the divergence in the thermodynamic limit,
the numerator and denominator have to be treated together, and we obtain

$\displaystyle\langle\psi(x)\bar{\psi}(x^{\prime})\rangle=$
$\displaystyle\lim_{N\to\infty\atop
N\in\mathbbm{N}}\lim_{\Omega\to\infty}\frac{\sum_{\Delta
n=-N}^{N}\langle\psi(x)\bar{\psi}(x^{\prime})\rangle_{\Delta n}}{\sum_{\Delta
n=-N}^{N}Z_{\Delta n}}$ $\displaystyle=$ $\displaystyle
S_{0\text{inst}}(x,x^{\prime})+\kappa\bar{h}(x,x^{\prime})m^{-1}{\rm e}^{-{\rm
i}\alpha\gamma^{5}}\,.$ (58)

In Section 6, we show that this procedure amounts to dropping the divergent
extensive contributions that correspond to the vacuum diagrams in standard
perturbation theory. In this final result, the phase from the quark mass in
$S_{0\rm inst}$, cf. Eq. (45), is aligned with the phase from the instanton-
induced effects in the term with the overlap function $\bar{h}$, so that there
are no $CP$-violating effects.

One may wonder about Eq. (58) why we take the limit $N\to\infty$ in front of
the fraction whereas by Eq. (9), it appears that it should hold for numerator
and denominator separately in the first place. As we have noted though,
without normalization by vacuum contributions, the partition function is not
well defined in the thermodynamic limit. The present procedure is necessary to
divide out the extensive contributions causing the divergence. It is unique in
the sense that we carry out the integrals over each steepest-descent contour
before interfering them. Doing otherwise would correspond to a partitioning
and reordering of the full integration contour that consists of the steepest-
descent contours connected via configurations of infinite action, see Figure 1
for illustration. This amounts to an incorrect manipulation of a path integral
that is not absolutely convergent.

Now we consider what happens when the limits are ordered the other way around,
i.e. sum over the topological sectors before taking $\Omega\to\infty$,
according to Eqs. (12) and (13). We reiterate though that this procedure is
not valid because topological quantization can only be deduced in infinite
spacetime volume. As for the fermion correlation, one obtains

$\displaystyle\sum\limits_{\bar{n},n\geq
0}\frac{1}{\bar{n}!n!}\Big{[}{\,\bar{h}(x,x^{\prime})}(\bar{n}\,m^{-1}{\rm
e}^{{\rm i}\alpha}P_{\rm L}+n\,m^{-1}{\rm e}^{-{\rm i}\alpha}P_{\rm
R})\left(\Omega\right)^{\bar{n}+n-1}\\!\\!+S_{0}(x,x^{\prime})\left(\Omega\right)^{\bar{n}+n}\\!\Big{]}$
$\displaystyle\hskip 227.62204pt\times(-\kappa)^{\bar{n}+n}{\rm e}^{{\rm
i}\Delta n(\alpha+\theta)}$ $\displaystyle=$ $\displaystyle\left[-\left({\rm
e}^{-i\theta}P_{\rm L}+{\rm e}^{{\rm i}\theta}P_{\rm
R}\right)\frac{\kappa}{m}\bar{h}(x,x^{\prime})+S_{0}(x,x^{\prime})\right]{\rm
e}^{-2\kappa\Omega\cos(\alpha+\theta)}\,,$ (59)

and for the partition function

$\displaystyle\sum_{n,\bar{n}}\frac{1}{n!\bar{n}!}(-\kappa\Omega)^{\bar{n}+n}{\rm
e}^{-{\rm i}(\bar{n}-n)(\alpha+\theta)}={\rm
e}^{-2\kappa\Omega\cos(\alpha+\theta)}\,.$ (60)

Taking the ratio, the overall exponential factors cancel but now there is a
misalignment between the phases in $S_{0\rm inst}$ and in the instanton-
induced term. This means that as $\Omega\to\infty$, there is an infinite
amount of destructive interference that suppresses the statistically more
likely contributions with approximately equal numbers of $n$ and $\bar{n}$
(see Eq. (55)) in favour of outliers for which $\Delta n/\Omega$ does not go
to zero. Equations (5) and (60), if they were correct, would signal
$CP$-violating effects. Note that in either result, terms that break the ${\rm
U(1)}_{A}$ symmetry from both, instanton-mediated effects and the quark mass
$m$ are present. When we turn to the phenomenology of the strong interactions
and generalization to several flavours in Section 7, we shall recall that
both, the breaking of chiral symmetry through the quark masses and from
instantons are necessary in order to explain the spectrum of mesons, in
particular why the $\eta^{\prime}$-meson is much heavier than the pions [29,
30, 18]. In either order of limits, this phenomenology is explained. So the
meson spectrum alone cannot be used in order to conclude the the correct order
of limits.

##  6 Thermodynamic limit and cluster decomposition

We establish here that with the limiting procedure in Eq. (58), contributions
that are divergent due to the infinite spacetime volume cancel between
numerator and denominator. This corresponds to the usual cancellation of
vacuum diagrams when evaluating connected correlation functions in standard
perturbation theory (i.e. without expanding around nontrivial classical
solutions).

The present argument is also interesting for what concerns Eq. (9). The
partition function is defined in the limit $\Omega\to\infty$ in the first
place. This appears as an obstacle to using $\log Z$ as an extensive, volume-
dependent quantity in line with what is familiar from thermodynamics. We shall
see here that an expression with such a property can nonetheless be defined
when restricting $Z$ to some subvolume of $\Omega$. Note that as such a
restriction is arbitrary, no boundary conditions on the subvolume can be
placed.

While we are working here at zero temperature, we note that one can use the
Polyakov line at finite temperature in order to control and study the
deconfinement phase transition, including contributions from the gradient
expansion of the quark determinant [37].

We use a well known line of reasoning [38] and consider the expectation value
of an operator $\cal O$ in an infinite spacetime volume $\Omega$, and
interfere different topological sectors $\Delta n$ as

$\displaystyle\langle{\cal O}\rangle=\lim_{N\to\infty\atop
N\in\mathbbm{N}}\lim_{\Omega\to\infty}\frac{\sum\limits_{\Delta
n=-N}^{N}f(\Delta n)\int\limits_{\Delta n}{\cal D}\phi\,{\cal O}\,{\rm
e}^{-S_{\Omega}[\phi]}}{\sum\limits_{\Delta n=-N}^{N}f(\Delta
n)\int\limits_{\Delta n}{\cal D}\phi\,{\rm e}^{-S_{\Omega}[\phi]}}\,,$ (61)

where ${\cal D}\phi$ is the path integral measure over all fields involved.
Now, let ${\cal O}$ be an operator corresponding to a correlation function
evaluated for some spacetime points. For example, in Eq. (58), ${\cal
O}=\psi(x)\bar{\psi}(x^{\prime})$. For the action, we write $S_{\Omega}$ to
indicate that it is obtained from integrating the Lagrangian over the
spacetime volume $\Omega$. As for the Lagrangian, we take it not to include
the topological term $-{\rm i}\theta{\rm tr}F\widetilde{F}/(16\pi^{2})$.
Rather, we have the function $f(\Delta n)$ taking care of the dependence on
the topological sector.

Now, consider partitioning the spacetime volume as
$\Omega=\Omega_{1}\cup\Omega_{2}$ so that $\Delta n(\Omega)=\Delta
n_{1}(\Omega_{1})+\Delta n_{2}(\Omega_{2})$. We further assume that the
spacetime arguments of the operator fall within $\Omega_{1}$, and we write
${\cal O}_{1}$ in favour of ${\cal O}$ to indicate this. We can thus write

$\displaystyle\langle{\cal O}_{1}\rangle=\lim_{N_{1},N_{2}\to\infty\atop
N_{1},N_{2}\in\mathbbm{N}}\lim_{\Omega\to\infty}\frac{\sum\limits_{\Delta
n_{1}=-N_{1}}^{N_{1}}\sum\limits_{\Delta
n_{2}=-N_{2}}^{N_{2}}\\!\\!\\!\\!f(\Delta n_{1}+\Delta
n_{2})\int\limits_{\Delta n_{1}}{\cal D}\phi\,{\cal O}_{1}\,{\rm
e}^{-S_{\Omega_{1}}[\phi]}\int\limits_{\Delta n_{2}}{\cal D}\phi\,{\rm
e}^{-S_{\Omega_{2}}[\phi]}}{\sum\limits_{\Delta
n_{1}=-N_{1}}^{N_{1}}\sum\limits_{\Delta
n_{2}=-N_{2}}^{N_{2}}\\!\\!\\!\\!f(\Delta n_{1}+\Delta
n_{2})\int\limits_{\Delta n_{1}}{\cal D}\phi\,{\rm
e}^{-S_{\Omega_{1}}[\phi]}\int\limits_{\Delta n_{2}}{\cal D}\phi\,{\rm
e}^{-S_{\Omega_{2}}[\phi]}}\,.$ (62)

Since there may be instantons sitting right at the boundaries of the two
subvolumes, $\Delta n_{1,2}$ will not be strictly integer. However, if the
instanton gas is sufficiently dilute, integer winding numbers may still
correspond to an adequate approximation.

Now, as required by the cluster decomposition principle, provided $\Omega_{1}$
is chosen large enough, $\langle{\cal O}_{1}\rangle$ must not depend on
contributions from $\Omega_{2}$ to the path integral. This is generally the
case when the numerator and the denominator decompose into factors that only
depend on $\Delta n_{1}$, $\Omega_{1}$ or $\Delta n_{2}$, $\Omega_{2}$,
respectively. Then the contributions from the volume $\Omega_{2}$ can be
reduced from the fractions. This generally happens when

$\displaystyle f(\Delta n_{1}+\Delta n_{2})=f(\Delta n_{1})f(\Delta
n_{2})\Rightarrow f(\Delta n)={\rm e}^{{\rm i}\Delta n\theta}\,.$ (63)

So the contributions from the topological term, that we have left aside thus
far, can indeed be accounted for through $f(\Delta n)$. Note that the argument
holds for either order of limits, i.e. the one from Eqs. (9), that implies Eq.
(11) which is imposed here, as well as for the commuted version from Eq. (12).

To carry out the limits, we write Eq. (62) as

$\displaystyle\langle{\cal O}_{1}\rangle=$
$\displaystyle\lim_{N_{1},N\to\infty\atop
N_{1},N\in\mathbbm{N}}\lim_{\Omega\to\infty}\ \frac{\sum\limits_{\Delta
n=-N}^{N}\sum\limits_{\Delta n_{1}=-N_{1}}^{N_{1}}\\!\\!\\!f(\Delta
n)\int\limits_{\Delta n_{1}}{\cal D}\phi\,{\cal O}_{1}\,{\rm
e}^{-S_{\Omega_{1}}[\phi]}\int\limits_{\Delta n_{2}=\Delta n-\Delta
n_{1}}{\cal D}\phi\,{\rm e}^{-S_{\Omega_{2}}[\phi]}}{\sum\limits_{\Delta
n=-N}^{N}\sum\limits_{\Delta n_{1}=-N_{1}}^{N_{1}}\\!\\!\\!f(\Delta
n)\int\limits_{\Delta n_{1}}{\cal D}\phi\,{\rm
e}^{-S_{\Omega_{1}}[\phi]}\int\limits_{\Delta n_{2}=\Delta n-\Delta
n_{1}}{\cal D}\phi\,{\rm e}^{-S_{\Omega_{2}}[\phi]}}\,.$ (64)

Corresponding to Eq. (57), the integrations over the volume $\Omega_{2}$ lead
to

$\displaystyle\langle{\cal O}_{1}\rangle=$
$\displaystyle\lim_{N_{1},N\to\infty\atop
N_{1},N\in\mathbbm{N}}\lim_{\Omega_{2}\to\infty}$
$\displaystyle\hskip-11.38092pt\frac{\sum\limits_{\Delta
n=-N}^{N}\sum\limits_{\Delta n_{1}=-N_{1}}^{N_{1}}\\!\\!\\!f(\Delta
n)\,I_{\Delta n-\Delta n_{1}}(2\kappa\Omega_{2}){(-1)^{N_{f}(\Delta n-\Delta
n_{1})}{\rm e}^{{\rm i}\,\alpha(\Delta n-\Delta n_{1})}}\int\limits_{\Delta
n_{1}}{\cal D}\phi\,{\cal O}_{1}\,{\rm
e}^{-S_{\Omega_{1}}[\phi]}}{\sum\limits_{\Delta n=-N}^{N}\sum\limits_{\Delta
n_{1}=-N_{1}}^{N_{1}}\\!\\!\\!f(\Delta n)\,I_{\Delta n-\Delta
n_{1}}(2\kappa\Omega_{2}){(-1)^{N_{f}(\Delta n-\Delta n_{1})}{\rm e}^{{\rm
i}\,\alpha(\Delta n-\Delta n_{1})}}\int\limits_{\Delta n_{1}}{\cal
D}\phi\,{\rm e}^{-S_{\Omega_{1}}[\phi]}}\,.$ (65)

The explicit exponential factors here are phases from the fermion
determinants. Note that these have not been absorbed in $\kappa$, which we
have defined to be real.

Now we are aiming for an expression for $\langle{\cal O}_{1}\rangle$ with
finite $\Omega_{1}$, without making reference to $\Omega_{2}$. This proves to
be possible because the contributions from $\Omega_{2}$ can be interpreted as
vacuum factors that reduce out from the normalized expectation value.

Since we must take $\Omega\to\infty$ to have well-defined integer $\Delta n$,
we also have to take here $\Omega_{2}\to\infty$. The Bessel functions with a
factor $\Omega_{2}$ in their argument then go to a common limit so that we can
factorize out the sum over $\Delta n$. We are left with

$\displaystyle\langle{\cal O}_{1}\rangle=$
$\displaystyle\frac{\sum\limits_{\Delta
n_{1}=-\infty}^{\infty}\int\limits_{\Delta n_{1}}{\cal
D}\phi\,{{(-1)^{-N_{f}\Delta n_{1}}{\rm e}^{-{\rm i}\,\alpha\Delta
n_{1}}}}{\cal O}_{1}\,{\rm e}^{-S_{\Omega_{1}}[\phi]}}{\sum\limits_{\Delta
n_{1}=-\infty}^{\infty}\int\limits_{\Delta n_{1}}{\cal
D}\phi\,{{(-1)^{-N_{f}\Delta n_{1}}{\rm e}^{-{\rm i}\,\alpha\Delta
n_{1}}}}{\rm e}^{-S_{\Omega_{1}}[\phi]}}\,.$ (66)

We therefore see that taking the limits as in Eq. (58) leads to the correct
cancellation of “disconnected” terms, in particular those that originate from
regions that are far separated from the spacetime arguments of the observable
${\cal O}_{1}$.

Moreover, in Eq. (66) the $\theta$-angle from the function $f(\Delta n)$ does
not occur anymore. We can see this as a consequence of phases incurred in
$\Omega_{1}$ being canceled against complementary phases from
$\Omega\setminus\Omega_{1}$. The remaining explicit dependence on the
unphysical phase $\alpha$ cancels when the fermionic part of the path integral
${\cal D}\phi$ is carried out. Since the path integral here is restricted to
$\Omega_{1}$, which is finite, we can compute the expectation values in finite
volumes after all from a partition function in the form of Eq. (12) but with
the parameter $\bar{\theta}$ set to zero. This way, the logarithm of the
partition function can be taken as an extensive quantity.

In the previous derivation, when going from Eq. (64) to (65) we made use of
the result for the partition function in the dilute instanton gas
approximation, Eq. (57). However, it is worth pointing out that the latter
result can be derived from the cluster decomposition principle alone, without
making use of the dilute instanton gas approximation. One can start by noting
that the factorization of the path integrals in the denominator in Eq. (62)
can be written in terms of the following relations between the partition
functions $Z_{\Delta n}(\Omega)$ in the full volume and their counterparts
$Z_{\Delta n_{1}}(\Omega_{1})$, $Z_{\Delta n_{2}}(\Omega_{2})$ for the
subvolumes $\Omega_{1},\Omega_{2}$,

$\displaystyle Z_{\Delta n}(\Omega)=\sum_{\Delta n_{1}}Z_{\Delta
n_{1}}(\Omega_{1})Z_{\Delta n-\Delta n_{1}}(\Omega_{2})\,.$ (67)

Equation (67) is an infinite set of identities that can be used to solve for
$Z_{\Delta n}(\Omega)$ from a set of minimal assumptions. First, we note that
$Z_{\Delta n}(\Omega)$ are complex. For starters, they receive a phase ${\rm
e}^{\rm i\theta\Delta n}$ due to the-$\theta$ term. Further complex phases in
$Z_{\Delta n}(\Omega)$ can only come from the phases $\alpha_{i}$ of the
fermion masses. At least at the leading order, the fermionic path integration
yields determinants of the massive Dirac operator in a background of
topological charge $\Delta n$, which can be fully general and is not assumed
to be precisely captured by the dilute instanton gas approximation. The phase
of the total fermionic determinant is then fixed by the Atiyah-Singer index
theorem [12], and for a single fermion is given by ${\rm e}^{{\rm
i}\alpha\Delta n}$. As a consequence, one can write

$\displaystyle Z_{\Delta n}(\Omega)={\rm e}^{{\rm i}(\theta+\alpha)\Delta
n}\tilde{g}_{\Delta n}(\Omega)\,,\quad\tilde{g}_{\Delta
n}(\Omega)\in\mathbb{R}\,.$ (68)

Parity considerations and appropriate limits of the cluster-decomposition
relation (67) can be used to motivate the simple ansatz [7]

$\displaystyle\tilde{g}_{\Delta n}(\Omega)=\Omega^{|\Delta n|}f_{\Delta
n}(\Omega^{2}),\quad f_{\Delta n}(0)\neq 0\,.$ (69)

Notably, the previous ansatz together with the assumption of analyticity in
$\Omega$ give rise to a unique solution for the infinite tower of identities
in Eq. (67), which can be written as

$\displaystyle Z_{\Delta n}(\Omega)={\rm e}^{{\rm i}(\theta+\alpha)}I_{\Delta
n}(2\beta\Omega)\,,\quad\beta\equiv f_{\Delta n=1}(0)\in\mathbb{R}\,.$ (70)

As advertised, this recovers the result of Eq. (57) without making use of the
dilute instanton gas approximation.

Equation (70) can be taken even further, as it allows one to rederive the
phases of fermionic correlators and confirm the conclusions of this section
without using the dilute instanton gas approximation. Defining a complex mass
parameter $\mathfrak{m}$ as

$\displaystyle\mathfrak{m}\equiv m{\rm e}^{{\rm i}\alpha}\,,$ (71)

the mass terms in the Lagrangian of Eq. (1) can be written as

$\displaystyle{\cal
L}\supset\mathfrak{m}\bar{\psi}P_{R}\psi+\mathfrak{m}^{*}\bar{\psi}P_{L}\psi\,.$
(72)

Then, one can view the complex mass parameters as sources for integrated
correlators,

$\displaystyle\begin{aligned} \frac{\partial}{\partial\mathfrak{m}}Z_{\Delta
n}=\,-\int{\rm d}^{4}x\,\langle\bar{\psi}P_{\rm R}\psi\rangle_{\Delta
n}\,,\quad\frac{\partial}{\partial\mathfrak{m}^{*}}Z_{\Delta n}=\,-\int{\rm
d}^{4}x\,\langle\bar{\psi}P_{\rm L}\psi\rangle_{\Delta n}\,.\end{aligned}$
(73)

As the partition functions of Eq. (70) have been derived on general grounds,
the previous correlators are meant to include nonperturbative effects. Noting
that the reality condition in the $\beta$ parameter of Eq. (70) implies
$\beta=\beta(\mathfrak{m}\mathfrak{m}^{*})$ and writing $\alpha=-({\rm
i}/2)\log(\mathfrak{m}/\mathfrak{m}^{*})$ yields the following spacetime
averaged correlators [7]

$\displaystyle\begin{aligned} \frac{1}{\Omega}\int{\rm
d}^{4}x\,\langle\bar{\psi}P_{\rm
R}\psi\rangle=&\,\lim_{N\rightarrow\infty}\lim_{\Omega\rightarrow\infty}\frac{\sum_{|\Delta
n|<N}\int{\rm d}^{4}x\,\langle\bar{\psi}P_{\rm R}\psi\rangle_{\Delta
n}}{\Omega\sum_{|\Delta m|<N}Z_{\Delta
m}}={-}2\mathfrak{m}^{*}\,\partial_{\mathfrak{m}\mathfrak{m}^{*}}\beta(\mathfrak{m}\mathfrak{m}^{*})\,,\\\
\frac{1}{\Omega}\int{\rm
d}^{4}x\,\langle\bar{\psi}P_{L}\psi\rangle=&\,\lim_{N\rightarrow\infty}\lim_{\Omega\rightarrow\infty}\frac{\sum_{|\Delta
n|<N}\int{\rm d}^{4}x\,\langle\bar{\psi}P_{\rm L}\psi\rangle_{\Delta
n}}{\Omega\sum_{|\Delta m|<N}Z_{\Delta
m}}={-}2\mathfrak{m}\,\partial_{\mathfrak{m}\mathfrak{m}^{*}}\beta(\mathfrak{m}\mathfrak{m}^{*})\,.\end{aligned}$
(74)

It is readily seen that the total phase of the fermionic correlators,
including nonperturbative effects, is aligned with the phases of the tree-
level masses in the Lagrangian. This generalizes the result of Eq. (58) and
leads again to the conclusion of no $CP$ violation. Again, the order of limits
plays a crucial role in Eq. (74).

##  7 Effective theories and effective operators

We shall now draw the connection from the results of the semiclassical
approximation corresponding to integrating out gluons from Section 5 with
observables probing $CP$ conservation or violation in the strong interactions.
The main object of interest in that context is the ‘t Hooft vertex, which can
be inferred from the correlation function (58) as the Lagrangian term

$\displaystyle-\bar{\psi}(x)\Gamma{\rm e}^{{\rm i}\alpha\gamma^{5}}\psi(x)\,.$
(75)

This vertex generates the same correlation functions as in Eq. (58) for the
EFT where gluons have been integrated out. Figure 6 illustrates how such a
model fits into the picture of the different EFTs discussed in the present
context. In addition, there will also be in general nonlocal operators from
the long-range interactions of the gluons, because with quark degrees of
freedom still in the theory, there is no cutoff parameter that allows for a
local expansion. The new operators appear in favour of the gluon kinetic term
$F_{\mu\nu}F_{\mu\nu}$ as well as the topological term
$F_{\mu\nu}\widetilde{F}_{\mu\nu}$ which disappear together with the gluons.

Figure 6:  Schematic representation of the different effective field theories
(EFTs) discussed in the present article and their relation with QCD. Note that
the nonlocal operators due to gluon dynamics are all but under calculational
control.

In the dilute instanton gas approximation in Section 5, one has integrated out
gluons in the semiclassical approximation. For this to be valid, the theory
should be perturbative throughout, which can be achieved in principle by
adding a bespoke matter content that controls the renormalization group
evolution in this peculiar way. Certainly, however, this is neither the case
for the theory specified in Eq. (1) with the gauge group ${\rm SU}(2)$ and one
quark flavour nor for QCD with ${\rm SU}(3)$ and three flavours of light
quarks. As a consequence, one should expect substantial deviations from the
correlation function given in Eq. (58), in particular for large distances
between $x$ and $x^{\prime}$. Nonetheless, there should still be a small
distance, high energy contribution of this form. When drawing conclusions
about $CP$ conservation or violation, one therefore must make the assumption
that the $CP$-odd coefficient of the ‘t Hooft vertex appears in the same way
within the extra operators that have to be added in principle to account for
the low-energy behaviour. Note however that this shortcoming applies to the
conclusions based on either order of limits when the calculation is carried
out semiclassically.

To some extent, the above matter is addressed by the concluding argument from
Section 5, where the leading fermion correlations are constrained without the
dilute instanton gas approximation but using instead cluster decomposition and
the index theorem. There, no assumption about the fermion correlation is made
but for its ${\rm U}(1)_{A}$-violating form. While the resulting fermion
correlation then can only be stated in the coincident limit, the conclusions
about $CP$ conservation based on the order of the infinite volume limit and
the sum over topological sectors should therefore extend to the
nonperturbative low-energy regime as well.

The underlying theory that we are concerned with after all is QCD, which is
specified (now with the gauge group ${\rm SU}(3)$, $N_{f}$ flavours of light
quarks and in Minkowski spacetime) as (we choose $\varepsilon_{1230}=+1$ in
Minkowski spacetime)

$\displaystyle{\cal L}_{\rm M}\supset-\frac{1}{2g^{2}}{\rm
tr}F_{\mu\nu}F^{\mu\nu}+\sum\limits_{i,j=1}^{N_{f}}\bar{\psi}_{i}\left({\rm
i}\gamma^{\mu}D_{\mu}\delta_{ij}-M_{ij}P_{\rm R}-M^{\dagger}_{ij}P_{\rm
L}\right)\psi_{j}+\frac{1}{16\pi^{2}}\theta\,{\rm
tr}F_{\mu\nu}\widetilde{F}^{\mu\nu}\,,$ (76)

where in the mass-diagonal basis

$\displaystyle M_{ij}=\delta_{ij}m_{j}\quad\text{(no sum over $j$)}\,.$ (77)

For the model as in Eq. (76), the vertex corresponding to Eq. (75) is

$\displaystyle-\Gamma_{N_{f}}{\rm e}^{-{\rm
i}\bar{\alpha}}\prod_{j=1}^{N_{f}}(\bar{\psi}_{j}P_{\rm
L}\psi_{j})-\Gamma_{N_{f}}{\rm e}^{{\rm
i}\bar{\alpha}}\prod_{j=1}^{N_{f}}(\bar{\psi}_{j}P_{\rm R}\psi_{j})\,,$ (78)

where

$\displaystyle\bar{\alpha}=\sum\limits_{i}^{N_{f}}\alpha_{i}=\arg\det M\,.$
(79)

We need to sort in what way (cf. Figure 6) this is connected to the EFT of
hadrons that is valid at low energies and should describe those possible
$CP$-violating effects that are accessible by current precision experiments. A
principal obstacle to systematically deriving quantitative predictions lies of
course within the circumstance that perturbation theory is not valid anymore
at low energies.

Yet, the symmetries, even when realized approximately only, offer a standard
method of constraining the EFT. In the fundamental theory as well as on the
EFT side one can introduce operators of the physical fields coupled to
external sources (sometimes called spurions) so that these operators are
invariant under local symmetry transformations. On the side of the EFT, the
coefficient of these operators has to be obtained through computational or
experimental matching. Variation with respect to these sources then allows one
to express matrix elements of the fundamental theory in terms of parameters of
the EFT.

In the present case, we can apply this method by perceiving the quark masses
that break the chiral flavour symmetries ${\rm SU}(N_{f})_{A}$ as well as the
operators breaking ${\rm U}(1)_{A}$ as external sources that transform
according to these explicitly broken symmetries. In the following discussion,
we occasionally let $N_{f}=2$, meaning that only up and down quarks are
considered, for simplicity. But for expressions explicitly depending on
$N_{f}$, we keep $N_{f}$ general. First, we parametrize a chiral
transformation as

$\displaystyle\psi\to LP_{\rm L}\psi+RP_{\rm R}\psi\,,$ (80)

where $P_{\rm L,R}=(1\mp\gamma^{5})/2$ and $L,R$ are independent unitary
matrices. For an axial transformation, $R=L^{-1}$ so that the ${\rm
SU}(2)_{A}$ transformations are given by

$\displaystyle\psi_{i}\to[{\rm e}^{{\rm
i}\vec{\gamma}\cdot\vec{\sigma}\gamma^{5}}]_{ij}\psi_{j}\,.$ (81)

The Lagrangian (76) would remain invariant if the mass matrix transformed as

$\displaystyle M\to LMR^{\dagger}={\rm e}^{-{\rm
i}\vec{\gamma}\cdot\vec{\sigma}}M{\rm e}^{-{\rm
i}\vec{\gamma}\cdot\vec{\sigma}}\,.$ (82)

In this transformation, $M$ corresponds to a spurion field.

The corresponding EFT Lagrangian (cf. Figure 6) with the lowest-order terms is
(see, e.g., Refs. [39, 40] and Ref. [8] where the effective theory is derived
from integrating out quark fields)

$\displaystyle{\cal L}^{\rm EFT}_{\rm M}=\frac{f_{\pi}^{2}}{4}{\rm
Tr}\,\partial_{\mu}U\partial^{\mu}U^{\dagger}+\frac{f_{\pi}^{2}B_{0}}{2}\,{\rm
Tr}(MU+U^{\dagger}M^{\dagger})+|\lambda|{\rm e}^{-{\rm i}\xi}f_{\pi}^{4}\,{\rm
det}\,U+|\lambda|{\rm e}^{{\rm i}\xi}f_{\pi}^{4}\,{\rm det}\,U^{\dagger}\,,$
(83)

where

$\displaystyle U=U_{0}{\rm e}^{\frac{{\rm
i}}{f_{\pi}}\Phi}=U_{0}\tilde{U}\,,\quad U_{0}=\langle
U\rangle=\left(\begin{array}[]{cc}{\rm e}^{{\rm i}\varphi_{u}}&0\\\ 0&{\rm
e}^{{\rm
i}\varphi_{d}}\end{array}\right)\,,\quad\Phi=\left[\begin{array}[]{cc}\pi^{0}+\eta^{\prime}&\sqrt{2}\,\pi^{+}\\\
\sqrt{2}\,\pi^{-}&-\pi^{0}+\eta^{\prime}\end{array}\right]\,.$ (88)

In the equations above, $f_{\pi}$ is the pion decay constant and $\lambda$,
$B_{0}$ are EFT coefficients to be determined experimentally or
computationally and $B_{0}$ is directly related to the magnitude of the chiral
quark condensate. The phases of the latter correspond to
$\varphi_{u},\varphi_{d}$, and we have assumed a diagonal mass matrix $M$. The
squared pion and $\eta^{\prime}$ masses are then given by

$\displaystyle\begin{aligned}
m^{2}_{\pi}=&\,B_{0}(m_{u}\cos(\alpha_{u}+\varphi_{u})+m_{d}\cos(\alpha_{d}+\varphi_{d}))\,,\\\
m^{2}_{\eta^{\prime}}\approx&\,16|\lambda|f^{2}_{\pi}\cos(\xi-\varphi_{u}-\varphi_{d})\,,\end{aligned}$
(89)

where we have made the phenomenologically valid approximation that
$m_{\pi}^{2}\ll m^{2}_{\eta^{\prime}}$. We leave the terms with the parameter
$\lambda$ aside just yet as $\det U$ is invariant under ${\rm SU}(2)_{A}$ but
transforms with ${\rm U}(1)_{A}$ in a way that we shall get to shortly. In
correspondence with Eq. (81), the meson fields behave under axial
transformations as

$\displaystyle U\to RUL^{\dagger}={\rm e}^{{\rm
i}\vec{\gamma}\cdot\vec{\sigma}}U{\rm e}^{{\rm
i}\vec{\gamma}\cdot\vec{\sigma}}$ (90)

The term with the parameter $B_{0}$ should be matched so that the correct
correlation functions are produced. Corresponding to the invariance of the
underlying theory (76), the Lagrangian (83) is invariant under the
simultaneous ${\rm SU}(2)_{A}$ transformations (90) and (82).

Now, after all, the (up and down) quark masses do not transform under ${\rm
SU}(2)_{A}$, they rather break this symmetry explicitly. We can still perceive
these as local sources though, that perturb the correlators of the theory
about the case with full ${\rm SU}(2)_{A}$ symmetry. For the local source, we
can then take the fixed physical values of $M$ so that Eq. (83) accounts for
the perturbation through the quark masses to linear order. In the EFT, one can
continue this to higher orders pending on the precision that is aimed for.

Now, consider ${\rm U}(1)_{A}$ transformations

$\displaystyle\psi_{i}\to{\rm e}^{{\rm i}\beta\gamma^{5}}\psi_{i}$ (91)

and recall the expression for $\bar{\alpha}$ from Eq. (79). The fundamental
theory (76) would remain invariant if the quark mass transformed as

$\displaystyle M\to{\rm e}^{-2{\rm i}\beta}M\,,\quad\text{so
that}\quad\bar{\alpha}\to\bar{\alpha}-2\beta N_{f}\,.$ (92)

The chiral anomaly requires that the coefficient $\theta$ of the topological
term goes as

$\theta\to\theta+2N_{f}\beta$ (93)

in order to keep the Lagrangian invariant. Note that this implies that the
combination

$\displaystyle\bar{\theta}=\theta+\bar{\alpha}$ (94)

is invariant under chiral rephasings and in general is nonzero. The presence
of such an invariant does however not yet guarantee that it leads to physical
effects.

We thus see that there are two local sources that transform under the symmetry
${\rm U}(1)_{A}$: $\theta$ and $-\alpha$. Noting that under this symmetry

$\det U\to{\rm e}^{2{\rm i}N_{f}\beta}\det U\,,$ (95)

the EFT Lagrangian (83) remains invariant if either

$\displaystyle\xi=\begin{cases}-\bar{\alpha}\\\ \ \;\theta\end{cases}\,.$ (96)

In principle, one may also allow linear combinations of the parameters
$-\bar{\alpha}$ and $\theta$. As this does not follow from either order of
limits for the sum over topological sectors and spacetime volume that we
discuss here, we do not consider this combination option further.

We also note that the operator with the coefficient $B_{0}$ breaks ${\rm
U}(1)_{A}$. So instead of the quark mass phase in $M$, one could also use
$\theta$ to write this as an invariant operator with the help of chiral-
variant source fields. However, the symmetric theory should respond to ${\rm
U}(1)_{A}$-breaking perturbations through a quark mass term in the same way as
it does for ${\rm SU}(2)_{A}$-breaking. In this sense, the term with $B_{0}$
is unique to linear order in $M$. The explicit breaking of ${\rm U}(1)_{A}$
through instantons is independent of the quark masses, cf. Eq. (58) together
with the fact that $\kappa={\cal O}(|M|)$, and therefore $M$ does not appear
in the terms with $\det U$.

Now recall that Eq. (58) leads to the effective vertex (78) in the theory
where gluons have been integrated out. At this level, $\theta$ has disappeared
so that the only option for the EFT Lagrangian (83) is

$\displaystyle\xi=-\bar{\alpha}\,.$ (97)

The $CP$-odd coefficients can then be removed by an overall field
redefinition. On the other hand, if it were $\xi=\theta$, there would be a
residual $CP$-odd term.

Further, note that Eq. (89) shows that the mass of the $\eta^{\prime}$ in
general does not vanish in the limit of $m_{u,d}\to 0$, no matter which of the
values $\xi$ takes in Eq. (96). In turn, the fact that the $\eta^{\prime}$ is
heavy compared to the pions as such does not lead to a conclusion about which
is the correct order of limits.

Finally, the parameter $\xi$ in the coefficient of the ‘t Hooft operator
enters the calculation of the nucleon EDM as follows: Given the EFT Lagrangian
(83) and choosing a basis in which $M$ is diagonal, the minimum of the field
$U$ is given by $U_{0}$ as in Eq. (88) where in the limit of $|\lambda|\gg
B_{0}m_{d}/f_{\pi}^{4}$, and for $\xi+\alpha_{u}+\alpha_{d}$ in the first
quadrant, one has [41, 42]

$\displaystyle
m_{u}\sin(\varphi_{u}+\alpha_{u})=m_{d}\sin(\varphi_{d}+\alpha_{d})=\frac{\sin(\xi+\alpha_{u}+\alpha_{d})}{\sqrt{\frac{1}{m_{u}^{2}}+\frac{1}{m_{d}^{2}}+\frac{2\cos(\xi+\alpha_{u}+\alpha_{d})}{m_{u}m_{d}}}}\,.$
(98)

Going beyond the assumption $|\lambda|\gg B_{0}m_{d}/f_{\pi}^{4}$ leads to a
mixing of the flavour eigenstates $\pi^{0}$ and $\eta^{\prime}$ within the
mass eigenstates.

In order to expand in terms of the meson fields, following Eqs. (88) and (98)
leads to

$\displaystyle{\rm Tr}[MU+U^{\dagger}M^{\dagger}]=$
$\displaystyle\frac{1}{2}\left[m_{u}\cos(\varphi_{u}+\alpha_{u})+m_{d}\cos(\varphi_{d}+\alpha_{d})\right]{\rm
Tr}[\tilde{U}+\tilde{U}^{\dagger}]$ $\displaystyle+$ $\displaystyle\frac{\rm
i}{2}\left[m_{u}\sin(\varphi_{u}+\alpha_{u})+m_{d}\sin(\varphi_{d}+\alpha_{d})\right]{\rm
Tr}[\tilde{U}-\tilde{U}^{\dagger}]$ $\displaystyle+$ $\displaystyle\text{terms
mixing $\pi^{0}$ with $\eta^{\prime}$}\,.$ (99)

The operators in the second line are $CP$ odd, and we note that

$\displaystyle{\rm Tr}\left[\tilde{U}-\tilde{U}^{\dagger}\right]=\frac{2\rm
i}{f_{\pi}}\,{\rm Tr}[\Phi]-\frac{\rm i}{3f_{\pi}^{3}}{\rm
Tr}\left[\Phi^{3}\right]=\frac{4\rm i}{f_{\pi}}\,\eta^{\prime}-\frac{\rm
i}{f_{\pi}^{3}}\left(\frac{2}{3}{\eta^{\prime}}^{3}+2\eta^{\prime}\left[(\pi^{0})^{2}+2\pi^{+}\pi^{-}\right]\right)\,.$
(100)

Substituting this into the term with $B_{0}$ in Eq. (83) and expanding in the
meson fields, one generally would obtain $CP$-violating effects if
$\xi\not=-\bar{\alpha}$, the most immediate consequence of which would be
$\eta^{\prime}\to 2\pi$ (recall that the meson fields are $CP$-odd) through
the interaction term

$\displaystyle{\cal L}^{\rm EFT}_{\rm M}\supset$
$\displaystyle\frac{B_{0}\sin(\xi+\alpha_{u}+\alpha_{d})}{f_{\pi}\sqrt{\frac{1}{m_{u}^{2}}+\frac{1}{m_{d}^{2}}+\frac{2\cos(\xi+\alpha_{u}+\alpha_{d})}{m_{u}m_{d}}}}\left[(\pi^{0})^{2}+2\pi^{+}\pi^{-}\right]\eta^{\prime}$
$\displaystyle\approx$
$\displaystyle\frac{m_{u}m_{d}(\xi+\alpha_{u}+\alpha_{d})}{(m_{u}+m_{d})^{2}}\frac{m_{\pi}^{2}}{6f_{\pi}}{\rm
Tr}[\Phi^{3}]\,,$ (101)

The latter expression (which follows when assuming
$\xi+\alpha_{u}+\alpha_{d}\ll 1$) is shown here for comparison with Eq. (8) of
Ref. [43]. In the latter, the quark masses are taken as real, (see Eq. (5) in
[43]) which means that the parameter $\theta$ of that reference should
correspond to $\pm\bar{\theta}$ in Eq. (94). Matching the resulting signs for
the phases of the quark masses in Eq. (8) of Ref. [43] leads to an
identification with $-\bar{\theta}$. It then follows that up to the central
issue that which value in Eq. (96) is taken by $\xi$, the results from the
present EFT description and the partially conserved axial currents in Ref.
[43] are therefore in agreement, as they should be. We further compare the two
approaches given different values of $\xi$ in Section 8. Finally, let us note
also that the coefficient in Eq. (101) is different in the approximation of
three light flavours where an extra factor of $\sqrt{2/3}$ occurs.

To see what the above $CP$-odd interactions of the pions and $\eta^{\prime}$
would imply for the nucleons, one can add their interactions to the EFT
Lagrangian as

$\displaystyle{\cal L}^{\pi,{\rm p},{\rm n}}_{\rm M}\supset{\rm
i}\bar{N}\not{\partial}N-\left(m_{N}\bar{N}UP_{\rm L}N+{\rm
i}c\bar{N}U^{\dagger}\not{\partial}UP_{\rm L}N+d\bar{N}M^{\dagger}P_{\rm
L}N+e\bar{N}UMUP_{L}N+{\rm h.c.}\right)\,,$ (102)

where the nucleon doublet transforms as

$\displaystyle N\to L\,P_{\rm L}N+R\,P_{\rm R}N\,.$ (103)

Again, promoting $M$ to a source that transforms under the axial symmetries
rather than breaking these, this Lagrangian is invariant. Substituting the
expectation value of the chiral condensate (88), (98) for small
$\xi+\bar{\alpha}$, expanding in the meson field and applying field
redefinitions $N\to\cal N$ so as to obtain the canonically normalized flavour
eigenstates of the nucleons, one finds the interaction terms [41]

$\displaystyle{\cal L}^{\text{neutron}}_{\rm M}\supset
c_{1}\partial_{\mu}\pi^{a}\bar{\cal
N}\frac{\tau^{a}}{2}\gamma^{\mu}\gamma^{5}{\cal
N}+c_{2}(\xi+\bar{\alpha})\bar{\cal N}\pi^{a}\frac{\tau^{a}}{2}{\cal N}\,.$
(104)

The first of these is $CP$ even, as it couples two axial currents, $\pi^{a}$
being a pseudoscalar field. The second term is $CP$ odd, as it couples a
scalar density with a pseudoscalar field. At one loop level, if it were
$\xi\not=-\bar{\alpha}$, this would induce an EDM through the famous diagrams
shown in Figure 7. Note that the weak interactions make an additional
contribution to the neutron EDM [44, 45], which however is too small and
usually neglected in the discussion of $CP$ in the strong interactions.

$n$$p$$n$$\pi^{+}$$\pi^{+}$$\gamma$

\+ $n$$p$$n$$\pi^{+}$$\pi^{+}$$\gamma$

Figure 7: Leading order contribution to the neutron EDM from the strong
interactions for $\xi\neq-\bar{\alpha}$. The $CP$-violating vertex is
indicated by a cross. A similar diagram involves $\pi^{-}$.

##  8 Some objections and answers to these

We review here and reply to some objections that we have been made aware of,
mostly to the extent that these can be related to (partly earlier) articles or
to conference talks that have been published online.

### Instanton configurations in the different limits

As argued in Section 3 the different orders of limits correspond to a
partitioning and rearrangement of the integration contour that leads to
inequivalent results for the path integration. Still, upon taking the limits,
the path integral covers the same field configurations. Yet, arguments have
been put forward that partitioning the contour should be harmless or that the
order of limits as in Eqs. (9) and (11) does not include all relevant
contributions as opposed to Eqs. (12) and (13) [46].

Regarding the partitioning of the contour, one may state that for a given
configuration of finite action, one can find a radius $R$ so that [11]

$\displaystyle A_{\mu}(x)={\rm i}\omega(x)\partial_{\mu}\omega^{-1}(x)+{\cal
O}(1/R^{2})$ (105)

and, consequently,

$\displaystyle\Delta n=\frac{1}{16\pi^{2}}\int\limits_{|x|<R}{\rm
d}^{4}x\,{\rm tr}F\widetilde{F}=\text{integer}+{\cal O}(1/R)\,.$ (106)

So in this sense, even for finite volumes one can at least approximately
categorize certain configurations by an integer $\Delta n$. However, $R$ is
not universal and not even a function of $\Delta n$ since the individual units
of winding number can be separated arbitrarily far for given $\Delta n$. So we
cannot take this as an argument for the calculation in fixed volumes $\Omega$
to be equivalent to the full result.

Concerning Eq. (11) perhaps not accounting for all relevant configurations,
one may attempt to argue as follows: Consider the dilute instanton gas picture
and let $\rho$ be the radius of an instanton. Then, demanding that the
instantons and anti-instantons do not overlap, the maximum number of
instantons and anti-instantons $n+\bar{n}$ satisfies $|\Delta n|\leq
N\leq(n+\bar{n})_{\rm max}\approx\ \Omega/\rho^{4}$. Therefore, there is room
to take $N\to\infty$ as $\Omega\to\infty$ and it seems not to be appropriate
to cap $\Delta n$ by some finite value in an infinite volume as Eq. (11)
suggests.

However, such a cap is not forced by Eq. (11) in the following sense:
Decompose the full spacetime into subvolumes, e.g.,
$\Omega=\Omega_{1}\cap\Omega_{2}$. For each topological sector $\Delta n$,
there is a constraint $\Delta n_{1}+\Delta n_{2}=\Delta n$ where $\Delta
n_{1},\Delta n_{2}$ are the winding numbers (which may not be integers
precisely) in the subvolumes, but no constraint on $\Delta n_{1}$ and $\Delta
n_{2}$ separately. Therefore, $\Delta n_{1}$ can be arbitrarily large. For an
observer in a subvolume, say $\Omega_{1}$, the path integral (11) therefore
includes configurations of arbitrarily large winding number density $\Delta
n_{1}/\Omega_{1}$ within $\Omega_{1}$. Hence there is no cap on $\Delta n_{1}$
in the finite subvolume $\Omega_{1}$. Indeed, this is already clear from Eq.
(66) which is derived from the cluster decomposition principle.

### Chiral limit

In a theory with at least one massless quark, the parameter $\bar{\alpha}$ can
be chosen arbitrarily with no consequence to the Lagrangian. Without further
ado, this implies that $\bar{\theta}$ in Eq. (8) cannot be physical and that
there is no $CP$ violation in such a model. In the dilute instanton gas
picture, this behaviour results from the suppression of single instantons
through the zero-mode from the fermion determinant that makes the factor
$\varpi$ in Section 5 and consequently $\kappa$ vanishes proportionally to the
absolute value of the quark mass determinant.

In this sense, as $\kappa\sim m$, we can take in Eq. (58) $m\to 0$ and obtain
a well-defined limit. Since the gluons and therefore the topological term have
been integrated out, the parameter $\alpha$ in this expression is still
arbitrary but unphysical, as it can be removed by a chiral rotation of the
fermion field. The same reasoning applies to the effective operator (78).

### Effective theory for the topological current

We respond here to comments concerning Ref. [7] that have been made in Ref.
[47], see also Ref. [48]. It is argued there that finite spacetime volumes
$\Omega$ lead to an unphysical breaking of the conservation of the winding
number $\Delta n$ (or the density $\Delta n/\Omega$) so that the absence of
$CP$ violation would be an artifact of such regularization. But apparently, in
Eq. (9) and consequently Eq. (11) that lead us here as well as in Ref. [7] to
the conclusion of no $CP$ violation, $\Omega\to\infty$ is the first limit that
is taken. This is in contrast to Eqs. (12) and (13), which lead to
$CP$-violating observables but where the topological sectors are fixed prior
to taking $\Omega\to\infty$. So the criticism of producing finite volume
artifacts would rather be an issue for the latter prescription, and in fact it
is, as we have discussed in the previous sections.

While this comes as a rather immediate conclusion, it is of interest to see
how the EFT for the topological current, which is introduced in Refs. [49, 50,
47], fits into the present considerations of imposing boundary conditions and
orders of limits. (See Figure 6 for where it stands in relation to the other
EFTs that are discussed here.) The topological current can be defined as

$\displaystyle K_{\mu}=\frac{1}{4\pi^{2}}\epsilon_{\mu\nu\alpha\beta}{\rm
tr}\left[\frac{1}{2}A_{\nu}\partial_{\alpha}A_{\beta}-\frac{\rm
i}{3}A_{\nu}A_{\alpha}A_{\beta}\right]\,,$ (107)

where we recall ${\rm tr}(T^{a}T^{b}T^{c})={\rm i}f^{abc}/4$ for ${\rm
SU}(2)$. The topological charge density and hence the topological term can be
explicitly written as a total divergence

$\displaystyle q=\frac{1}{16\pi^{2}}{\rm
tr}F_{\mu\nu}\widetilde{F}_{\mu\nu}=\partial_{\mu}K_{\mu}\,.$ (108)

One of the interesting points concerning the current $K_{\mu}$ is the form of
its two-point correlation. Some information can be extracted from the chiral
susceptibility

$\displaystyle\chi_{\Omega}=$
$\displaystyle\frac{1}{\Omega}\,\left.\langle\Delta
n^{2}\rangle\right|_{\bar{\theta}=1/2(1-(-1)^{N_{f}})\pi}$ $\displaystyle=$
$\displaystyle\frac{1}{\Omega}\,\left.\left\langle\left(\int_{\Omega}{\rm
d}^{4}x\,q(x)\right)^{2}\right\rangle\right|_{\bar{\theta}=1/2(1-(-1)^{N_{f}})\pi}=\int_{\Omega}{\rm
d}^{4}x\left.\langle
q(x)q(0)\rangle\right|_{\bar{\theta}=1/2(1-(-1)^{N_{f}})\pi}\,.$ (109)

We evaluate this for the $CP$-even values of
$\bar{\theta}=1/2(1-(-1)^{N_{f}})\pi$. When summing over topological sectors
in a finite spacetime volume with fixed boundary conditions according to Eqs.
(12) and (13), the vacuum energy is minimized at the chosen value of
$\bar{\theta}$ and $\chi$ remains positive. Choosing instead the starting
point Eq. (9) and consequently Eq. (11), the value of $\bar{\theta}$ is
irrelevant by the arguments of Sections 5 and 6. We attach a subscript on
$\chi$ in order to indicate that it is important which volume is referred to.
Significant differences can arise when $\Omega$ corresponds to a subvolume of
the spacetime as opposed to the full spacetime, in particular when boundary
conditions control the overall topological fluctuations. For expressions that
apply to all volumes we omit the subscript.

One should also pay attention to the fact that $\chi$ will in general have
connected and disconnected parts. If there is $CP$ violation in a certain
setup, e.g. imposed by unphysical boundary conditions, then $\langle\Delta
n\rangle/\Omega\not=0$. When one aims to characterize the volume-scaled
variance of the topological charge, one should then subtract the disconnected
contributions, i.e. consider

$\displaystyle\chi_{\Omega}-\frac{\langle\Delta n\rangle^{2}}{\Omega}\,.$
(110)

Of course, one may also define $\chi$ without the disconnected pieces to start
with, which we do not do here for the sake of simpler expressions.

Applying the result (66) to Eq. (109) yields

$\displaystyle\chi_{\Omega_{1}}\equiv$
$\displaystyle\frac{1}{\Omega_{1}}\left\langle\left(\int_{\Omega_{1}}{\rm
d}^{4}x\,q(x)\right)^{2}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\Omega_{1}}\frac{\sum\limits_{{\Delta
n_{1}=-\infty}}^{\infty}\int{\cal D}A_{\Delta n_{1}}{{\cal D}\bar{\psi}{\cal
D}\psi}\int_{\Omega_{1}}{\rm d}^{4}z\,q(z)\int_{\Omega_{1}}{\rm
d}^{4}z^{\prime}\,q(z^{\prime}){\rm
e}^{-S_{\Omega_{1}}[A_{\mu}]}{{(-1)^{-N_{f}\Delta n_{1}}{\rm e}^{-{\rm
i}\,\bar{\alpha}\Delta n_{1}}}}}{\sum\limits_{{\Delta
n_{1}=-\infty}}^{\infty}\int{\cal D}A_{\Delta n_{1}}{{\cal D}\bar{\psi}{\cal
D}\psi}{\rm e}^{-S_{\Omega_{1}}[A_{\mu}]}{{(-1)^{-N_{f}\Delta n_{1}}{\rm
e}^{-{\rm i}\,\bar{\alpha}\Delta n_{1}}}}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Omega_{1}}\frac{\sum\limits_{n_{1}=0}^{\infty}\sum\limits_{\bar{n}_{1}=0}^{\infty}{\frac{1}{n_{1}!\bar{n}_{1}!}}(n_{1}-\bar{n}_{1})^{2}(\kappa\Omega_{1})^{n_{1}+\bar{n}_{1}}}{\sum\limits_{n_{1}=0}^{\infty}\sum\limits_{\bar{n}_{1}=0}^{\infty}{\frac{1}{n_{1}!\bar{n}_{1}!}}(\kappa\Omega_{1})^{n_{1}+\bar{n}_{1}}}=2\kappa\,.$
(111)

That is, when calculating the susceptibility from the partition function (9)
but evaluating Eq. (109) in a finite subvolume $\Omega_{1}$ of infinite
Euclidean spacetime, $\chi_{\Omega_{1}}$ is nonzero. On the other hand, as
$\lim_{\Omega\to\infty}\Delta n/\Omega=0$, in infinite volume we have

$\displaystyle\chi_{\Omega\to\infty}=0\,.$ (112)

One next introduces the Fourier transforms of the correlation functions, which
we indicate here by a tilde:

$\displaystyle\widetilde{\langle qq\rangle}(p)=\int{\rm d}^{4}(x-y)\,{\rm
e}^{{\rm i}p(x-y)}\langle q(x)q(y)\rangle=p_{\mu}p_{\nu}\widetilde{\langle
K_{\mu}K_{\nu}\rangle}(p)\,.$ (113)

From Eq. (109), one can conclude the following infrared behaviour [51]:

$\displaystyle\lim\limits_{p\to 0}p_{\mu}p_{\nu}\widetilde{\langle
K_{\mu}K_{\nu}\rangle}(p)=\int{\rm
d}^{4}x\,\langle\partial_{\mu}K_{\mu}(x)\partial_{\nu}K_{\nu}(y)\rangle=\chi\,.$
(114)

The former equation implies e.g. in transverse gauge [51], where
$\partial_{\mu}\epsilon_{\mu\nu\rho\sigma}A_{\sigma}=0$,

$\displaystyle\langle\widetilde{K_{\mu}K_{\nu}}\rangle(p)=\frac{\chi\,p_{\mu}p_{\nu}}{p^{4}}+{\cal
O}(p^{2})$ (115)

so that one would expect for $\chi\not=0$ a simple massless pole in
$\widetilde{\langle K_{\mu}K_{\nu}\rangle}(p)$ for $p_{\mu}\to 0$. Here we
have further dropped the spacetime volume subscript on $\chi$.

Now per Eq. (107), $K_{\mu}$ can be understood as the Hodge dual of a three-
form field. So it is of interest to compare it with the action for a massless
three-form $L_{\mu}$,

$\displaystyle S=\int_{\Omega}{\rm d}^{4}x\,(\partial_{\mu}L_{\mu})^{2}\,.$
(116)

The expression “massless” refers here to the absence of a mass term in the
action and is not supposed to indicate that there is a massless propagating
degree of freedom, which in fact there is not. The variation of this action is
given by

$\displaystyle\delta S=-2\int_{\Omega}{\rm
d}^{4}x\,\left(\partial_{\nu}\partial_{\mu}L_{\mu}\right)\delta
L_{\nu}+2\int_{\partial\Omega}{\rm
d}a_{\nu}\,\left(\partial_{\mu}L_{\mu}\right)\delta L_{\nu}\,,$ (117)

where $a_{\nu}$ is a normal surface element on $\Omega$. Discarding the
boundary term for the moment, one would conclude that
$\partial_{\mu}L_{\mu}=\text{const.}$ But then again, dropping the boundary
term is in general not justified because it in general contributes to the
equations of motion. The situation here is different from the model specified
by the Lagrangian (1) in an infinite spacetime volume $\Omega$: There,
solutions of finite action exist while the topological term does not need to
vanish and can be written per Eqs. (10) and (108) as a boundary term. However,
within a given topological sector the boundary term gives a fixed contribution
$-{\rm i}\theta\Delta n$ to the action. Therefore it has no impact on the
equations of motion.

Back to the action (116), we must therefore make further assumptions about the
boundary term. While there are many different options, we follow Ref. [50]
that works with solutions

$\displaystyle\partial_{\mu}L_{\mu}=\theta_{L}\,,$ (118)

where $\theta_{L}$ is an integration constant. While not stated explicitly in
Ref. [50], this appears to require boundary conditions $\delta L_{\mu}=0$ on
$\partial\Omega$ and

$\displaystyle\theta_{L}=\frac{1}{\Omega}\int_{\partial\Omega}{\rm
d}{a_{\nu}}\,L_{\nu}\,.$ (119)

Interpreting for the moment $L_{\mu}$ as a four-dimensional electric field in
the background of a constant charge density $\theta_{L}$, it should be clear
that Eq. (118) has solutions that satisfy the condition (119). Note that for a
given $\theta_{L}$, $L_{\mu}\sim\theta_{L}\Omega^{1/4}$. To keep the boundary
conditions well defined in terms of finite $L_{\nu}$, we therefore have to
impose these on finite $\Omega$ or we have to take $\theta_{L}=0$.

Given the boundary condition $\delta L_{\mu}=0$, we can also derive the
equation of motion

$\displaystyle\partial_{\mu}^{x}\partial_{\nu}^{x}\langle
L_{\nu}(x)L_{\rho}(y)\rangle=-\delta_{\mu\rho}\delta^{4}(x-y)$ (120)

for the correlation function. It further implies

$\displaystyle\langle(\partial_{\mu}L_{\mu}(x))(\partial_{\nu}L_{\nu}(y))\rangle=4\delta^{4}(x-y)+\text{const.}$
(121)

To arrive at the above equation, one uses the translation invariance to deduce
that the correlator is a function of $(x-y)$. Then one can take
$\partial^{x}_{\mu}=-\partial^{y}_{\mu}$ and contract $\mu$ and $\rho$ in Eq.
(120).

Having stated all this, it is interesting to follow Ref. [50] and consider the
action (116) as an EFT of the full model (1), of which all degrees of freedom
but $K_{\mu}$ have been integrated out. The effective action should generally
feature higher order terms in $\partial_{\mu}L_{\mu}$ beyond Eq. (116) as well
as nonlocal expressions that account for the rich phenomenology of QCD
compared to the present simplistic model [50]. Yet, it appears that the
following characteristic features of QCD are recovered.

Equation (121) indicates that the topological fluctuations correspond to white
noise. In that sense, the model (116) without higher order and nonlocal terms
captures the infrared behaviour on scales where the fluctuations are indeed
expected to be uncorrelated. We may use the $\delta$-function term in relation
(121) to identify

$\displaystyle L_{\mu}=\frac{2K_{\mu}}{\sqrt{\chi}}\,.$ (122)

Indeed, substituting the above into Eq. (121) (ignoring the constant term for
the moment) and taking a Fourier transform one recovers Eq. (114).

On the basis of these correspondences, the argument of Ref. [50] then goes
that the integration constant $\theta_{L}$ selects a vacuum state with a
nonzero value of the $CP$-odd observable $\partial_{\mu}L_{\mu}$ that it is
therefore proportional, at least at the linearized level, to the parameter
$\theta$ in the microscopic theory, which would thus turn out to be physical.
However, as we shall see next, we disagree with this interpretation. Because
of the different boundary conditions, the EFT for $L_{\mu}$ cannot correspond
to either of the QCD partition functions (12) or (9), due to reasons that will
also make clear why $\theta_{L}$ is not proportional to an angle.

Translating the boundary condition (119) back to the fundamental theory
implies that one fixes

$\displaystyle\Delta n=\int_{\partial\Omega}{\rm
d}a_{\nu}\,K_{\nu}=\frac{1}{16\pi^{2}}\int_{\Omega}{\rm
d}^{4}x\,F\widetilde{F}\,.$ (123)

Moreover, these boundary conditions do not imply that the physical fields
vanish on $\partial\Omega$, and in fact per Eq. (118) they do not for
$\theta_{L}\not=0$, because
$\theta_{L}\propto\partial_{\mu}L_{\mu}\propto\partial_{\mu}K_{\mu}$, which is
gauge invariant. The present setup is therefore not an EFT for the partition
function specified through Eqs. (12) and (13). For the latter, one samples
over all integer values of $\Delta n$ and the physical fields vanish on
$\partial\Omega$.

A consequence of this discrepancy is the fact that from Eqs. (12) and (13),
one concludes that $\langle\Delta n\rangle/\Omega=2{\rm i}(-1)^{\rm
N_{f}}\kappa\sin\bar{\theta}$ is purely imaginary [7] whereas in the EFT and
for the boundary conditions that are assumed here, $\Delta n$ is given by the
fixed real value (123). The two setups therefore are not only different, but
they also predict distinctly inequivalent results for the observables. There
should also be no relation between $\theta$ from the model of Eqs. (12) and
(13) and $\theta_{L}$, which explains why the EFT in terms of the latter does
not show any periodic behaviour as would be required for an angular variable.

One should notice that while $\Delta n$ is fixed over the full volume
$\Omega$, it will nonetheless fluctuate in any subvolume, cf. Eq. (121).
Still, the boundary conditions on $\partial\Omega$ are required, and these are
very different from those in Eq. (13). Further, if we interpret $L_{\mu}$ as
an effective infrared field, nonvanishing $\partial_{\mu}L_{\mu}$ on
$\partial\Omega$ does not conflict with vanishing microscopic fields on that
surface. But following the above remark concerning real versus imaginary
$\langle\Delta n\rangle$, the expectation values for $\Delta n$ in $\Omega$
still do not agree between the EFT and the setup specified through Eqs. (12)
and (13).

It is also clear now that the present EFT (116) with the boundary condition
(119) is also not equivalent to the partition function (9) that we take as the
object to theoretically define the strong interactions. The EFT corresponds to
imposing a fixed flux (119)—not necessarily an integer—on a finite surface.
This in contrast to the case of Eq. (9), for which we see from Eq. (11) that
all integer values of the topological flux at infinity are accounted for in
the action. Imposing a fixed flux on a finite surface cannot yield the correct
vacuum correlations, not least since a general finite Euclidean
$\partial\Omega$ does not have a geometrically meaningful continuation to some
domain in Minkowski space.

Though as just has been argued the EFT is not compatible with the usual QCD
partition functions, one may nevertheless point out that the topological
susceptibility in the EFT can have a similar infrared behaviour as in the
theory (9). To see this, consider the case $\theta_{L}=0$ for simplicity. The
generalization can be carried out by accounting for disconnected contributions
to the correlations in Eq. (109). Since the topological flux $\Delta n$
vanishes, this means by Eq. (109) that $\chi_{\Omega}=0$. In fact, this
requirement fixes the constant term in Eq. (121). This clearly resembles the
behaviour exhibited in Eq. (112). For either setup, there is no problem with
$\chi_{\Omega}=0$ on the full volume $\Omega$ because there is no inhibition
of topological fluctuations in subvolumes. Note that in the EFT for the
topological current, for $\theta_{L}\not=0$ it follows that
$\chi_{\Omega}=\langle\Delta n\rangle^{2}/\Omega$ is purely disconnected
(because the winding number in $\Omega$ is fixed through the flux on
$\partial\Omega$) and still does not correspond to the value that is locally
observed in a subvolume and includes connected contributions.

Now in Ref. [47], it is suggested that the calculations in Ref. [7] (i.e.
using the partition function (11)) are equivalent to choosing boundary
conditions on a finite surface that lift the mass of the three-form,
supposedly preventing to capture the true massless dynamics. However, as
discussed above, the vanishing of the chiral susceptibility from the partition
function (11) for the full volume in Eq. (112) also occurs for the EFT for the
topological current, up to disconnected contributions. In either case, this
has the implication that the massless simple pole in $\widetilde{\langle
K_{\mu}K_{\nu}\rangle}(p)$ disappears when evaluating Eq. (114) for the full
spacetime. In this sense, both models behave very similarly and no conclusion
can be drawn that the EFT captures dynamics that Eq. (11) does not. In
particular, one cannot support the statement from Ref. [47] that with Eq.
(11), one loses crucial information about $CP$-violating vacuum states based
on some ill-behaved infrared behaviour of e.g. $\chi$ compared to the EFT.

Though also for the three-form EFT, Eq. (114) does not yield a massless simple
pole, it is of interest to give a more direct argument for why the presence of
such a pole is not necessary for consistency with observed topological
fluctuations to start with. First observe that Eq. (114) is only true when the
spacetime integral is taken over the full spacetime; otherwise, the Fourier
transform is not simply projected into its zero momentum limit. To see this,
consider a topological susceptibility defined by integrating over a finite
subvolume $\Omega_{1}\subset\Omega$

$\displaystyle\chi_{\Omega_{1}}=\int_{\Omega_{1}}{\rm
d}^{4}x\,\langle\partial_{\mu}K_{\mu}(x)\partial_{\nu}K_{\nu}(y)\rangle=\int_{\Omega_{1}}{\rm
d}^{4}x\,\int\frac{{\rm d}^{4}p}{(2\pi)^{4}}\,{\rm e}^{-{\rm
i}p(x-y)}p_{\mu}p_{\nu}\langle\widetilde{K_{\mu}K_{\nu}}\rangle(p)\,.$ (124)

In an infinite spacetime $\Omega$, when $\Omega_{1}=\Omega$ is also infinite,
the spacetime integral yields a delta-function and one recovers the result of
Eq. (114). For a finite $\Omega_{1}\subset\Omega$, however the topological
susceptibility will receive contributions from the correlator
$\langle\widetilde{K_{\mu}K_{\nu}}\rangle(p)$ evaluated at nonzero momentum.

Lattice calculations in frozen topologies (that have $\chi_{\Omega}=0$ up to
disconnected contributions) still observe nonzero topological susceptibility
when sampling over a subvolume of the full lattice [52]. Hence, there is the
logical possibility of a topological susceptibility that vanishes for the full
spacetime, yet is nonzero when defined over subvolumes. This is exactly what
happens with QCD defined as in Eq. (9), as can be seen in Eqs. (8), (112). Now
with $\chi$ vanishing in the full spacetime volume, Eq. (114) implies that the
correlator $\widetilde{\langle K_{\mu}K_{\nu}\rangle}(p)$ cannot have a
massless pole. In Ref. [47] it is stated without proof that this implies a
massive pole instead. Let us adopt this assumption or more loosely introduce
$m_{K}$ as an infrared regulator that may or may not correspond exactly to the
effect from the finite-volume cutoff on the correlation function. We may also
have $m_{K}$ stipulate the absence of the simple pole in Eq. (115). Either
way,

$\displaystyle\langle\widetilde{K_{\mu}K_{\nu}}\rangle(p)=\frac{\chi_{0}\,g_{\mu\nu}}{p^{2}+m_{K}^{2}}\,.$
(125)

Provided such a massive pole is consistent with nonzero $\chi_{\Omega_{1}}$,
there is then no motivation for a massless pole for the three-form EFT of Ref.
[47]. Hence any conclusion derived from insisting on a massless pole in the
EFT does not apply.

One should therefore demonstrate how one can reconcile the topological
susceptibility being zero at infinite volume while remaining nonzero at finite
subvolumes. It turns out that such behaviour can actually be derived from the
massive correlator of Eq. (125). For estimating the susceptibility defined
over a finite region $\Omega_{R}$ of radius $R$, one can consider an
integration with a cutoff function ${\rm e}^{-R^{2}/r^{2}}$ which suppresses
the fluctuations for $r>R$,

$\displaystyle\chi_{\Omega_{\Lambda}}\equiv\int\frac{{\rm
d}^{4}p}{(2\pi)^{4}}\int_{\mathbb{R}^{4}}{\rm d}^{4}x\,{\rm
e}^{-R^{2}/r^{2}}{\rm e}^{-{\rm
i}p(x-y)}p_{\mu}p_{\nu}\langle\widetilde{K_{\mu}K_{\nu}}\rangle(p)\,.$ (126)

Substituting Eq. (125) and considering $y=0$, the result is

$\displaystyle\chi_{\Omega_{R}}=\chi_{0}\left(1-\frac{R^{4}m^{4}}{16}\,{\rm
e}^{\frac{R^{2}m_{K}^{2}}{4}}\text{Ei}\left(-\frac{1}{4}m_{K}^{2}R^{2}\right)-\frac{1}{4}\,R^{2}m_{K}^{2}\right)\sim\left\\{\begin{array}[]{c}\chi_{0},\quad
R\ll\frac{1}{m_{K}},\\\ 0,\quad R\gg\frac{1}{m_{K}}.\end{array}\right.$ (129)

Above, ${\rm Ei}(z)=-\int_{-z}^{\infty}{\rm e}^{-t}/t\,{\rm d}t$ is the
exponential integral function.

As advertised, one gets a zero topological susceptibility in the full volume,
but also constant susceptibility for subvolumes with associated length scales
$L\ll 1/m_{K}$. This behaviour is compatible with lattice results [52] and
matches the result of Eqs. (8), (112), which are based on the dilute instanton
gas approximation.

### Taking $\theta$ as a perturbation and PCAC

Sometimes it is argued that $\theta$ can be seen to be physical without making
topological considerations by referring to calculations of possible
$CP$-violating effects using identities for partially conserved axial currents
(PCAC), e.g. in the discussion of Section 3 of Ref. [53] or in Ref. [43].
(Note though that these papers do not state explicitly that their calculations
would allow them to come to this result without making topological arguments.)
However, we show here that such conclusions ultimately rely on the assumption
$\xi=\theta$ for Eq. (96). The answer to the question of whether
$\xi=\bar{\alpha}$ or $\xi=\theta$ relies however on topology so that PCAC
methods or related arguments cannot be used in order to bypass this step.

To review such points, we express the anomalous divergence of the axial ${\rm
U}(1)$ current as

$\displaystyle\partial_{\mu}\sum\limits_{j=1}^{N_{f}}\langle\psi_{j}^{\dagger}\gamma^{5}\gamma_{\mu}\psi_{j}\rangle=\frac{2N_{f}}{16\pi^{2}}{\rm
tr}F\widetilde{F}+2\sum\limits_{j=1}^{N_{f}}\langle\psi_{j}^{\dagger}\gamma^{5}m_{j}{\rm
e}^{{\rm i}\alpha_{i}\gamma^{5}}\psi_{j}\rangle\,.$ (130)

Using this relation, we can therefore see that

$\displaystyle\langle A|\frac{\theta}{16\pi^{2}}{\rm
tr}F\widetilde{F}|B\rangle=-\frac{\theta}{N_{f}}\langle
A|\sum\limits_{j=1}^{N_{f}}\psi_{j}^{\dagger}\gamma^{5}m_{j}{\rm e}^{{\rm
i}\alpha_{i}\gamma^{5}}\psi_{j}|B\rangle\,,$ (131)

where we have dropped the total divergence of the axial current. We have
inserted a factor $\theta$ on each side in view of the subsequent discussion
which is however arbitrary at this point.

Now, we first follow Ref. [53] and carry out chiral field redefinitions so
that in Eq. (76) the mass matrices are purely real (and diagonal) and in
particular $\bar{\alpha}=0$ and $\theta=\bar{\theta}$. Moreover, one can take
the freedom of non-anomalous ${\rm SU}(2)_{A}$ chiral field redefinitions to
set $\alpha_{j}=0$ individually. Assuming $\theta$ to be small one may view
the left-hand side of Eq. (131) as a perturbative insertion of the $CP$-odd
topological term into a hadronic transition matrix element. When $|A\rangle$
and $|B\rangle$ are eigenstates of $CP$ with opposite eigenvalues, a nonzero
result would then signal $CP$ violation.

Before proceeding, we note that we could just as well have set $\theta=0$,
$\bar{\alpha}=+\bar{\theta}$ and treated the axial ${\rm U}(1)_{A}$ phase of
the quark masses as the perturbation. This is an equivalent point of view that
is taken e.g. in Ref. [43] and perhaps more true to the circumstance that the
gluons have been integrated out in the chiral EFT so that the operator $\theta
F\widetilde{F}$ has been removed as well. Either way, the right-hand side of
Eq. (131) turns out as

$\displaystyle-\frac{\bar{\theta}}{N_{f}}\langle
A|\sum\limits_{j=1}^{N_{f}}\psi_{j}^{\dagger}\gamma^{5}m_{j}\psi_{j}|B\rangle\,.$
(132)