# Structural fixed-point theorems

Brian Rabern and Landon Rabern

[Editorial Comment, added 2021] This note was originally written sometime in
2007 with various updates in 2008 and 2009, and then updated into its current
state in 2014. The authors had planned on expanding it and submitting for
publication, but other interests got in the way and it sat dormant.
Unfortunately, Landon Rabern died in October 2020, so the authors will not be
expanding it. Since the core ideas are interesting the first author has
decided to make the paper available online. See [1], appendix A (The Global
Function) for an alternative, but much condensed presentation. For more recent
frameworks applying graph-theoretic tools to semantic paradoxes see Beringer,
T., & Schindler, T. (2017). A graph-theoretic analysis of the semantic
paradoxes. The Bulletin of Symbolic Logic, 23(4), 442-492; Hsiung, M. (2020).
What paradoxes depend on. Synthese, 197(2), 887-913.

##  1\. Introduction

The semantic paradoxes are often associated with self-reference or referential
circularity. However, Yablo has shown in [2] that there are infinitary
versions of the paradoxes that do not involve this form of circularity. It
remains an open question what relations of reference between collections of
sentences afford the structure necessary for paradoxicality. In [1] we laid
the groundwork for a general investigation into the nature of reference
structures that support the semantic paradoxes. The remaining task is to
classify the so-called _dangerous_ directed graphs. In appendix A of [1], we
sketched a reformulation of the problem in terms of fixed points of certain
functions. Here we expand on this reformulation, removing all syntactic
considerations to get a purely mathematical problem. It is definitely possible
that the problem’s solution depends on the axioms of set theory we choose—this
would be an interesting outcome.

For sets $A$ and $B$, we write $A^{B}$ for all functions from $B$ to $A$. We
use standard von Neumann ordinals, so
$2\mathrel{\mathop{:}}=\left\\{0,1\right\\}$, etc. When it is convenient to
visualize $g\in A^{B}$ as a sequence of elements from $A$ indexed by $B$, we
use the notation $g_{v}$ to mean $g(v)$ for $v\in B$.

##  2\. Dangerous graphs

A _directed graph_ is a pair $G\mathrel{\mathop{:}}=(V,E)$ where $V$ is any
set and $E\subseteq V^{2}$. For all $\alpha\in 2^{V}$ and $I\subseteq V$, put

$\alpha^{I}\mathrel{\mathop{:}}=\left\\{\beta\in 2^{V}\mid\forall_{x\in
V-I}\beta(x)=\alpha(x)\right\\}.$

A function $g\in 2^{2^{V}}$ is _independent of_ $I\subseteq V$ if $g$ is
constant on $\alpha^{I}$ for each $\alpha\in 2^{V}$. A function
$f\in\left(2^{V}\right)^{2^{V}}$ _respects_ $G$ if for each $v\in V$, the
function $f^{v}$ given by $f^{v}(x)\mathrel{\mathop{:}}=f(x)_{v}$ is
independent of $V-N^{+}(v)$. We let $\mathfrak{R}(G)$ be all the
$f\in\left(2^{V}\right)^{2^{V}}$ respecting $G$.

######  Definition 1.

A graph $G$ is _dangerous_ if $\mathfrak{R}(G)$ contains a fixed-point-free
function.

It is not difficult to see that on this definition, the same graphs come out
as dangerous as those in [1]. Define an inverse operation for
$f\in\left(2^{V}\right)^{2^{V}}$ by
$\mathfrak{R}^{-1}(f)\mathrel{\mathop{:}}=\left\\{G\mid
f\in\mathfrak{R}(G)\right\\}$. Let
$\mathfrak{F}\subseteq\left(2^{V}\right)^{2^{V}}$ be the fixed-point-free
functions. Then $\mathfrak{R}^{-1}(\mathfrak{F})$ is the set of all dangerous
graphs (on $V$). It is easy to see that a graph is dangerous if any subgraph
of it is and thus the complement of $\mathfrak{R}^{-1}(\mathfrak{F})$ can be
defined by excluding subgraphs (excluding all of
$\mathfrak{R}^{-1}(\mathfrak{F})$ is one such description). The problem is to
find such a description using a “small” set of subgraphs. When $V$ is finite,
the complement of $\mathfrak{R}^{-1}(\mathfrak{F})$ is given precisely by
excluding the set of all directed cycles (on $V$). The next case to figure out
is when $V=\omega$.

##  3\. An example: locally finite graphs

In [1], the dangerous locally finite graphs we’re classified using Gödel
compactness. In fact, a larger class of dangerous graphs was classified by a
direct application of Zorn’s lemma. This same work can be done in the setting
of fixed points of functions using Tychonoff’s theorem. This section gives an
example of this method.

The graphs in this section all have $V=2^{\omega}$. Adorn $2^{\omega}$ with
the product topology where each copy of $2$ has the discrete topology. By
Tychonoff’s theorem, $2^{\omega}$ is compact. Also, this topology is generated
by the $p$-adic metric on $2^{\omega}$; that is, for sequences
$(x_{i})_{i\in\omega}$ and $(y_{i})_{i\in\omega}$ define $d((x_{i}),(y_{i}))$
as $\frac{1}{k}$ where $k\in\omega$ is minimal such that $x_{k}\neq y_{k}$.
Using this metric, the following is easy to see.

We call a graph $G$ _locally finite_ if $\left|N^{+}(v)\right|$ is finite for
all $v\in V(G)$.

######  Lemma 1.

All functions in $\mathfrak{R}(G)$ are continuous iff $G$ is locally finite.

We’ll need a simple fixed point lemma.

######  Lemma 2.

Let $(M,d)$ be a compact metric space. If $f\in M^{M}$ is continuous and for
every $\epsilon>0$ there exists $x\in M$ such that $d(f(x),x)<\epsilon$, then
$f$ has a fixed point.

###### Proof.

Let $f\in M^{M}$ be such a function. For $k\in\mathbb{N}$, pick $x_{k}\in M$
such that $d(f(x_{k}),x_{k})<\frac{1}{k}$. Since $M$ is compact, the sequence
$x_{1},x_{2},\ldots$ has a convergent subsequence $x_{n_{1}},x_{n_{2}},\ldots$
converging to $x$, say. Since $f$ is continuous, the sequence
$f(x_{n_{1}}),f(x_{n_{2}}),\ldots$ converges to $f(x)$. We claim that
$f(x)=x$. Suppose not, then we have $\epsilon>0$ such that
$d(f(x),x)>\epsilon$. Since these two sequences converge, we can choose
$k\in\mathbb{N}$ so that
$\max\left\\{\frac{1}{n_{k}},d(x_{k},x),d(f(x),f(x_{k}))\right\\}<\frac{\epsilon}{3}$.
But then we have $d(f(x),x)\leq d(f(x),f(x_{k}))+d(f(x_{k}),x)\leq
d(f(x),f(x_{k}))+d(f(x_{k}),x_{k})+d(x_{k},x)<\epsilon$, a contradiction. ∎

Now the characterization of dangerous graphs follows easily.

######  Theorem 3.

A locally finite graph is dangerous iff it contains a directed cycle.

###### Proof.

Plainly, if a graph contains a directed cycle it is dangerous. For the other
direction, let $G$ be an acyclic locally finite graph and
$f\in\mathfrak{R}(G)$. By Lemma 1, $f$ is continuous. Since acyclic finite
graphs are not dangerous, for each $k\in\omega$ and any choice of
$x_{k+1},x_{k+2},x_{k+3},\ldots$ we have $x\in 2^{\omega}$ such that
$f(x)_{i}=x_{i}$ for $i\in\left[k\right]$. Therefore, in the $p$-adic metric,
$d(f(x),x)<\frac{1}{k}$. Now applying Lemma 2 shows that $f$ has a fixed
point. Hence $G$ is not dangerous. ∎

We note that by using ultrafilter convergence, the use of metric spaces can be
done away with and Theorem 3 can be proved in a similar way for all $V$. This
more general result is proved using Gödel compactness in [1].

## References

  * [1] Landon Rabern, Brian Rabern, and Matthew Macauley, _Dangerous reference graphs and semantic paradoxes_ , Journal of Philosophical Logic 42 (2013), no. 5, 727–765.
  * [2] Steven Yablo, _Paradox without self-reference_ , Analysis 53 (1993), no. 4, 251–252.


[a,b]Matteo Di Carlo

# Isospin-breaking corrections to weak decays:
the current status and a new infrared improvement

###### Abstract

We provide an overview of the current state of lattice calculations concerning
isospin-breaking corrections in weak decays, focusing in particular on
leptonic decays of light pseudoscalar mesons. We examine the two currently
existing calculations, placing a particular emphasis on the role of
electromagnetic finite-volume corrections. Given the large systematic
uncertainty associated with such corrections, we propose a novel method to
improve the finite-volume scaling of leptonic decay rates and other hadronic
observables, like hadron masses and the hadronic vacuum polarization
contribution to the muon $g{-}2$. We introduce a new prescription for the QED
action, referred to as QEDr, which effectively removes finite-volume
corrections at $\mathcal{O}(1/L^{3})$ and consequently shifts the systematic
uncertainty to a higher order. The specifics of this action, the practical
implementation of the infrared improvement, and the ongoing numerical tests
are also discussed.

##  1 Introduction

Lattice QCD is an extremely solid and successful theoretical framework and has
recently become essential for precision physics calculations, as it is able to
provide a number of hadronic matrix elements and Standard Model parameters
with uncertainties in some cases even smaller than one percent [1]. At this
level of precision, small effects such as the electromagnetic (e.m.)
interactions of quarks and leptons, as well as the difference in mass between
up and down quarks, which are both expected to amount to $\mathcal{O}(1\%)$
corrections, play a crucial role. In this work we discuss the calculation of
such isospin-breaking (IB) effects in the context of weak processes, in
particular of leptonic decays of pseudoscalar mesons. These processes are
mediated by charged-weak currents and hence the combination of a precise
experimental determination of their decay rates and the theoretical
calculation of the relevant hadronic amplitudes, including IB effects, can
provide precise estimates of the corresponding Cabibbo-Kobayashi-Maskawa (CKM)
matrix elements. The unitarity contraints on the CKM matrix imposed by the
Standard Model offer a unique opportunity for indirect searches of new
physics. Significant deviations from unitarity could in fact hint at
contributions of new particles or interactions which are not accounted in the
theoretical calculations.

The evaluation of IB corrections to hadronic observables requires the
inclusion of QED in numerical lattice QCD calculations, posing the theoretical
challenge of how to define a photon in a finite volume with periodic boundary
conditions, which are commonly employed in lattice calculations. In fact,
Gauss’ law forbids states with non-zero electric charge in a box with periodic
boundary conditions. In order to circumvent this problem, many regularizations
of the lattice QED action have been proposed, that either put contraints on
the Fourier modes of the photon field, like QED${}_{\text{L}}$ [2] and its
infrared improvements [3], QED${}_{\text{TL}}$ [4] and QED${}_{\text{SF}}$
[5], or provide a small mass to the photon like QED${}_{\text{m}}$ [6], or
adopt instead different boundary conditions like QED${}_{\text{C}}$ [7]. For a
detailed review discussing all the above prescriptions, see ref. [8]. A
different alternative approach has also been recently developed, in which
radiative corrections are determined as a convolution of hadronic correlators
with infinite-volume QED kernels [9, 10]. In this case, effects due to the
finite extent of the lattice are expected to come only from QCD hadronic
matrix elements, and hence decay exponentially as the lattice extent $L$ is
increased. In contrast, when QED is formulated in a finite volume with a
massless photon, the long-range nature of e.m. interactions leads to power-
like finite-volume effects. In this work we will focus on lattice calculations
in which QED has been defined in a finite volume and regularized using the
QED${}_{\text{L}}$ prescription, which consists in removing the contributions
of the spatial zero modes of the photon, and we will put particular emphasis
on the crucial role played by such power-like finite-volume corrections.

There exist two approaches currently adopted to include QED effects in lattice
QCD calculations. In the all-order or “stochastic” approach QED is added
directly to the lattice action, and dedicated QCD+QED simulations are
performed, with the result of including the IB corrections to all orders (see,
e.g., refs. [11, 12, 13, 14]). In the perturbative approach, presented in
refs. [15, 16] (and often referred to as the “RM123 approach”), the lattice
path-integral is expanded at first order in the two small parameters
$\alpha_{\mathrm{em}}$ and
$(m_{\mathrm{d}}-m_{\mathrm{u}})/\Lambda_{\text{QCD}}$. Being leading order
corrections sufficient for many phenomenological applications, the two small
parameters are factorized out and their coefficients can be determined
directly from simulations of isosymmetric QCD, with no need to perform new
dedicated simulations. So far, due to the computational challenge of
evaluating quark disconnected diagrams, all calculations using the
perturbative method have been performed in the electro-quenched approximation,
in which sea quarks are treated as electrically neutral. Although deviations
from this approximation are expected to be small, this consitutes a major
source of systematic uncertainty and therefore work is in progress to overcome
it [17].

Lattice QCD+QED is a new frontier for precise numerical calculations and many
collaborations have successfully produced results for IB corrections to
different hadronic observables: the hadron spectrum [11, 18, 19, 20, 21], the
anomalous muon $g{-}2$ [22, 23, 24, 25, 26, 27] (see also refs. [28, 29]) and
weak leptonic decays [30, 31, 32]. In the latter case, additional challenges
arise. In fact, when separating the contributions to the decay rate from the
exchange of virtual photons and those from the emission of real ones,
logarithmic infrared divergences appear which have to be properly treated. The
strategy for evaluating leading e.m. and strong IB effects to leptonic decay
rates on the lattice has been developed by the RM123-Southampton (RM123S)
collaboration in ref. [33] and then applied by the same group to the leptonic
decay of pions and kaons into muons and neutrinos (referred to as $\pi_{\mu
2}$ and $K_{\mu 2}$, respectively) in refs. [30, 31]. More recently, the
RBC/UKQCD collaboration also computed the IB correction to the ratio of kaon
and pion decay rates, $\Gamma(K_{\mu 2})/\Gamma(\pi_{\mu 2})$, obtaining
results in agreement with RM123S, but with a larger systematic uncertainty
related to finite-volume effects. The finite-volume dependence of this
observable has been thoroughly studied in the last few years. The structure-
independent logarithmic infrared divergence and the e.m. finite-volume effects
to decay amplitudes up to $\mathcal{O}(1/L^{2})$ have been first computed in
refs. [34, 35]. More recently, the structure-dependent $\mathcal{O}(1/L^{2})$
corrections, as well as the point-like contribution at $\mathcal{O}(1/L^{3})$,
have been evaluated in ref. [36], while the functional form of the structure-
dependent effects at $\mathcal{O}(1/L^{3})$ has been presented at this
conference in ref. [37].

In this work, after briefly describing how to compute IB corrections to
leptonic decays on the lattice and evaluate the corresponding leading finite-
volume effects, we will discuss the two existing calculations of the ratio
$\Gamma(K_{\ell 2})/\Gamma(\pi_{\ell 2})$ performed by the RM123S and the
RBC/UKQCD groups using the QED${}_{\text{L}}$ regularization. We will focus on
the role of finite-volume effects in the calculations and how further studies
of the correction at $\mathcal{O}(1/L^{3})$ are necessary to improve the
precision on such observable. With this aim, we dedicate the rest of the work
to present a new finite-volume regularization of the QED action, which differs
from QED${}_{\text{L}}$ in the treatment of the photon zero mode and it is
expected to reduce finite-volume effects on a number of hadronic observables.

##  2 QED and isospin-breaking corrections to weak decays

Let us consider the decay of a charged pseudoscalar meson, $P^{+}$, into a
muon-neutrino pair. When electromagnetism is included, photons can be
exchanged between quarks and leptons or emitted by them. As mentioned above,
this introduces new infrared divergences in the decay rate, which only cancel
when summing virtual ($\Gamma_{0}$) and real ($\Gamma_{1}$) photon
contributions [38]. A practical strategy for lattice calculations of the decay
rate, including leading QED corrections, has been proposed by RM123S in ref.
[33] and consists in defining the decay rate as the sum of infrared-finite
contributions, namely

$\displaystyle\Gamma(P^{+}\to\mu^{+}\nu_{\mu}[\gamma])$
$\displaystyle=\lim_{L\to\infty}\big{[}\Gamma_{0}(L)-\Gamma_{0}^{\mathrm{uni}}(L)\big{]}+\lim_{m_{\gamma}\to\infty}\big{[}\Gamma_{0}^{\mathrm{uni}}(m_{\gamma})+\Gamma_{1}^{\mathrm{uni}}(m_{\gamma})\big{]}$
(1)
$\displaystyle+\lim_{L\to\infty}\big{[}\Gamma_{1}(L)-\Gamma_{1}^{\mathrm{uni}}(L)\big{]}\,.$

In the first bracketed term the virtual decay rate is computed on the lattice
using the finite volume with the QED${}_{\text{L}}$ prescription as an IR
regulator, with the universal (structure-independent) logarithmic infrared
divergence removed, as well as finite volume effects up to $\mathcal{O}(1/L)$
[34, 35]. The finite-volume scaling of this term can be further improved by
replacing $\Gamma_{0}^{\mathrm{uni}}(L)$ with $\Gamma_{0}^{(2)}(L)$, which
includes the QED${}_{\text{L}}$ finite-volume corrections up to
$\mathcal{O}(1/L^{2})$, computed in ref. [36]. The second term is the decay
rate in the point-like approximation, which has been evaluated in perturbation
theory using a photon mass as a regulator in ref. [33]. Finally, the third
term corresponds to the structure-dependent part of the real decay rate. This
contribution has been studied in refs. [39, 40, 41, 42, 43, 44, 45] and it is
relevant for decays of pions and kaons into electrons or decays of heavy
mesons, while it can be neglected in the calculations of $\pi_{\mu 2}$ and
$K_{\mu 2}$. This method has been applied to two different numerical
calculations, first by the RM123S group to the rates of $\pi_{\mu 2}$ and
$K_{\mu 2}$ [30, 31], and more recently by the RBC/UKQCD collaboration to the
ratio $\Gamma(K_{\mu 2})/\Gamma(\pi_{\mu 2})$ [32]. In the following we focus
on the latter quantity and compare it between the two calculations.

For a given choice of scheme for the iso-symmetric QCD theory
($\alpha_{\mathrm{em}}=0$, $m_{\mathrm{u}}=m_{\mathrm{d}}$), in which the
decay constants $f_{\pi}$ and $f_{K}$ are defined, one can write the ratio of
$K_{\mu 2}$ and $\pi_{\mu 2}$ decay rates as

$\frac{\Gamma(K^{+}\to\mu^{+}\nu_{\mu}[\gamma])}{\Gamma(\pi^{+}\to\mu^{+}\nu_{\mu}[\gamma])}=\frac{|V_{\mathrm{us}}|^{2}}{|V_{\mathrm{ud}}|^{2}}\frac{m_{\pi}}{m_{K}}\frac{(m_{K}^{2}-m_{\mu}^{2})}{(m_{\pi}^{2}-m_{\mu}^{2})}\frac{f_{K}^{2}}{f_{\pi}^{2}}\big{(}1+\delta
R_{K\pi}\big{)}+\mathcal{O}(\epsilon^{2})\,,$ (2)

where $\delta R_{K\pi}=\delta R_{K}-\delta R_{\pi}$ denotes the (scheme-
dependent) leading IB correction and $\mathcal{O}(\epsilon^{2})$ is understood
as a second-order correction in
$(\alpha_{\mathrm{em}},(m_{\mathrm{d}}^{\mathrm{R}}-m_{\mathrm{u}}^{\mathrm{R}})/\Lambda_{\text{QCD}})$.111We
stress that the definition of the isospin symmetric QCD theory is scheme
dependent, as is the separation of strong and e.m. IB effects. Although
differences between commonly adopted schemes are expected to be small,
discussions are ongoing for the definition of a common reference scheme, which
would facilitate the comparison of lattice results between different
collaborations [46]. As discussed in ref. [32], the schemes adopted in the two
calculations discussed in this work can be considered to be equivalent. As
discussed in refs. [30, 31, 32], the IB correction $\delta R_{K\pi}$ can be
obtained by computing corrections to the bare matrix elements and to the meson
masses, which can be extracted from the long-distance behaviour of suitable
Euclidean lattice correlators in the time-momentum representation. Note that
$\delta R_{K\pi}$ depends on the the velocities of the final state leptons in
the pion and kaon decays, whose absolute value is fixed by energy-momentum
conservation and injected in the lattice correlators via twisted boundary
conditions. In both calculations the RM123 perturbative method has been
adopted to evaluate QED and strong IB effects in the electro-quenched
approximation. This consists, in practice, in computing a number of connected
correlation functions, with photons exchanged in all possible ways and mass
corrections inserted along all quark lines.

The lattices used in the two calculations are rather different. On the one
hand, RM123S employed ensembles with twisted mass fermions at three different
lattice spacings and multiple volumes. However, unphysical quark masses have
been simulated, corresponding to pion masses above 230 MeV, and a chiral
extrapolation has been performed to reach the physical point. One complication
introduced by the use of an action that breaks chiral symmetry is a non-
trivial mixing of operators under renormalization when including QED
corrections. This has been studied in ref. [31], but the issue does not apply
to the calculation of $\delta R_{K\pi}$, as the contributions from the
renormalization of the matrix elements cancel in the ratio. On the other hand,
the RBC/UKQCD calculation used the domain wall fermion action with physical
quark masses. However, due to the high computational cost of such simulation,
this has been performed for the moment at a single value of lattice spacing
and volume.

The results obtained by the two collaborations are the following [31, 32]

$\delta R_{K\pi}^{\mathrm{RM123S}}=-0.0126\,(21)\,,\qquad\delta
R_{K\pi}^{\mathrm{RBC/UKQCD}}=-0.0086\,(13)(39)_{\mathrm{vol.}}\,,$ (3)

where the error in the first bracket is a combination of statistical and
systematic uncertainties, while for the RBC/UKQCD result we factor out the
systematic error due to finite-volume effects. The two results are in
agreement with each other, as well as with a previous calculation obtained in
chiral perturbation theory, $\delta R_{K\pi}^{\mathrm{\chi PT}}=-0.0112\,(21)$
[47]. This is a noteworthy result, emphasizing the capability of the lattice
in reliably computing such observables with a precision systematically
improvable to the percent level.

The origin of the large finite-volume systematic uncertainty in the RBC/UKQCD
result is explained in ref. [32] and additional details are also given in ref.
[37]. This is due to the partial knowledge of finite-volume corrections to the
decay rate at $\mathcal{O}(1/L^{3})$. While the scaling in inverse powers of
$L$ is known for this quantity up to $\mathcal{O}(1/L^{2})$, including
structure-dependent contributions, the $\mathcal{O}(1/L^{3})$ term is only
known in the point-like approximation. It can be shown that the structure-
dependent part receives contributions from branch-cuts in hadronic amplitudes,
which are difficult to evaluate numerically. The appearance of these
contributions is a combined effect of the breaking of spatial locality, due to
the use of the QED${}_{\text{L}}$ action, and of rotational symmetry breaking
in a finite volume, as it will be discussed in the next section. In addition,
the known point-like contribution at $\mathcal{O}(1/L^{3})$ is found to be
sizeable (and with opposite sign) compared to the lower order corrections,
thus giving rise to a final large value of the systematic uncertainty. It is
worth mentioning that the systematic error assigned by RBC/UKQCD is likely
over-conservative, while the RM123S one might be slightly underestimated. With
the improved understanding of finite-volume corrections to $\delta R_{K\pi}$
since the publication of the RM123S result, an updated determination is
certainly within reach for the RM123S group. Reducing the finite-volume
systematic error requires a better knowledge of the finite-volume scaling of
$\delta R_{K\pi}$. This can be studied with a combination of numerical and
analytical calculations, for example by repeating the calculation on a number
of different volumes and extrapolating the data to the infinite-volume limit
(undoubtedly the cleanest but most expensive approach), while trying to
estimate the missing terms in the $1/L$ expansion. In this work we propose an
additional approach to improve the finite-volume scaling of leptonic decay
rates and other hadronic observables, which consists in a modification of the
finite-volume QED action. This new prescriptions, which we call
QED${}_{\text{r}}$, leads to vanishing contributions at
$\mathcal{O}(1/L^{3})$, promising a substantial improvement in the precision
of $\delta R_{K\pi}$.

##  3 QED${}_{\text{r}}$: an infrared improved QED regularization

The removal of the spatial zero modes from the photon propagator in the
QED${}_{\text{L}}$ regularization corresponds to placing a uniform charge
density in the volume, such that the finite-volume Gauss’ law is respected
even with a non-zero electric charge in the volume. As a result, hadronic
observables are affected by finite-volume effects at order $1/L^{3}$ which, as
mentioned in the previous section, are related to branch-cuts in hadronic
amplitudes and therefore difficult to estimate. We propose here a variation of
the QED${}_{\text{L}}$ regularization that restores the zero-mode contribution
in the infinite-volume limit and does not generate $\mathcal{O}(1/L^{3})$
corrections. This proposal can be seen as a particular case of the infrared
improvement of the QED${}_{\text{L}}$ action introduced in ref. [3], in which
a finite number of photon momentum modes are reweighted to obtain a better
finite-volume scaling of a given observable, without altering its infinite-
volume limit. While the coefficients introduced in ref. [3] were improving
finite-volume effects only in a process-dependent fashion, the new choice we
present here allows for a universal removal of momentum-independent $1/L^{3}$
effects. Our proposal, which we refer to as QED${}_{\text{r}}$, consists in
redistributing the zero mode, ${\mathbf{k}}=\boldsymbol{0}$, over the set of
neighbouring modes, ${\mathbf{k}}\in\mathcal{S}_{R}$, lying on a sphere of
radius $\tfrac{2\pi}{L}R$, namely

$\displaystyle\mathcal{S}_{R}$
$\displaystyle=\big{\\{}\,{\mathbf{k}}\in\tfrac{2\pi}{L}\mathbb{Z}^{3}\ \
\big{|}\ \ |{\mathbf{k}}|=\tfrac{2\pi}{L}R\big{\\}}\,.$ (4)

In fig. 1 we show a schematic visualisation of such redistribution of the zero
mode onto the nearest neighour modes on the shell with radius $R=1$.

Figure 1: Visualisation of the redistribution of the spatial zero mode
${\mathbf{k}}=\boldsymbol{0}$ over the nearest neighbouring modes on the shell
with radius $R=1$ with equal weights.

The infrared-improved Euclidean QED${}_{\text{r}}$ propagator is implemented,
in practice, by adding to the QED${}_{\text{L}}$ propagator the contributions
of the modes ${\mathbf{k}}\in\mathcal{S}_{R}$. Denoting
${\mathbf{k}}=\tfrac{2\pi}{L}{\mathbf{n}}$, the QED${}_{\text{r}}$ propagator
in Feynman gauge reads

$D^{\mu\nu}(k_{0},{\mathbf{k}})=\delta^{\mu\nu}\
\frac{1-\delta_{{\mathbf{k}},\boldsymbol{0}}}{k_{0}^{2}+{\mathbf{k}}^{2}}\
\left[1+h({\mathbf{k}},R)\right]$ (5)

where the first term in the sum corresponds to the QED${}_{\text{L}}$
propagator and

$h({\mathbf{k}},R)=w({\mathbf{k}},R)\,\delta_{{\mathbf{n}}^{2},R^{2}}\,.$ (6)

The function $w({\mathbf{k}},R)$ denotes the weight assigned to the mode
${\mathbf{k}}$ on the shell $\mathcal{S}_{R}$ and is defined such that the sum
of all weights equals one,

$\sum_{{\mathbf{k}}\in\mathcal{S}_{R}}w({\mathbf{k}},R)=1\,,$ (7)

which implies that

$\sum_{\mathbf{k}}h({\mathbf{k}},R)=1\,.$ (8)

Later in this section, we will demonstrate that the condition in eq. 8 is
crucial for the cancellation of the finite-volume corrections at
$\mathcal{O}(1/L^{3})$. In the case of an isotropic system with zero net
velocity the natural choice is to assign equal weights to the modes
${\mathbf{k}}\in\mathcal{S}_{R}$ with values
$w({\mathbf{k}},R)=1/r_{3}(R^{2})$, where the function
$r_{3}(R^{2})=\sum_{{\mathbf{k}}}\delta_{{\mathbf{k}}^{2},R^{2}}$ counts the
number of representation of $R^{2}$ as the sum of 3 squares. Following the
definitions above, it is clear that in the QED${}_{\text{r}}$ regularization
the electric charge density in the volume is not uniform as in
QED${}_{\text{L}}$, but it is a function of the spatial coordinates with a
periodicity that depends on the radius $R$, as shown in fig. 2.

Figure 2: Contour plot of the charge density for different choices of the
shell radius $R$.

In principle, the definition of the propagator in eq. 5 can be extended to the
case of the zero mode redistributed over multiple shells with radii
$\boldsymbol{R}=\\{R_{1},R_{2},\dots\\}$ as follows

$D^{\mu\nu}(k_{0},{\mathbf{k}})=\delta^{\mu\nu}\
\frac{1-\delta_{{\mathbf{k}},\boldsymbol{0}}}{k_{0}^{2}+{\mathbf{k}}^{2}}\
\left[1+h({\mathbf{k}},\boldsymbol{R})\right]\,,$ (9)

where this time

$h({\mathbf{k}},\boldsymbol{R})=\sum_{\alpha}\omega(\alpha)\,h({\mathbf{k}},\alpha)\,,$
(10)

with $\omega(\alpha)$ denoting the weight of the shell $|{\mathbf{n}}|=\alpha$
and defined such that $\sum_{\alpha}\omega(\alpha)=1$.

In the following, we will focus on the simplest implementation of
QED${}_{\text{r}}$, corresponding to a redistribution of the zero mode over a
single shell $\mathcal{S}_{R}$ with radius $R$ with all the modes equally
weighted, namely with
$h({\mathbf{k}},R)=\delta_{{\mathbf{n}}^{2},R^{2}}/r_{3}(R^{2})$. In this
case, the propagator reads

$D^{\mu\nu}(k_{0},{\mathbf{k}})=\delta^{\mu\nu}\
\frac{1}{k_{0}^{2}+{\mathbf{k}}^{2}}\
\left[(1-\delta_{{\mathbf{k}},\boldsymbol{0}})+h({\mathbf{k}},R)\right]\equiv
D_{\mathrm{L}}^{\mu\nu}(k_{0},{\mathbf{k}})+D_{\mathrm{r}}^{\mu\nu}(k_{0},{\mathbf{k}})\,,$
(11)

where in the second equation we have identified the QED${}_{\text{L}}$
propagator and the QED${}_{\text{r}}$ correction.

We now turn to discuss the impact of the QED${}_{\text{r}}$ infrared
improvement on the finite-volume scaling of hadronic observables. We first
study observables with no dependence on an external spatial momentum. This is
the case, for instance, of the e.m. corrections to hadron masses or to the HVP
contribution to the anomalous muon $g{-}2$. Then, we will discuss the more
complicated scenario of a system that depends on some external non-zero
spatial momentum, like the case of IB corrections to leptonic decays, where
the decay amplitude depends on the velocity of the final-state lepton.

###  3.1 Momentum-independent observables

Let us consider the case of e.m. finite-volume corrections to the mass of a
charged hadron. These have been studied in details in the QED${}_{\text{L}}$
regularization in refs. [11, 48, 36], as well as in QED${}_{\text{C}}$ [7]. We
find it convenient to adopt here the notation of ref. [7], to which we refer
for further discussions on the analytical properties of the functions
appearing below.

In QED${}_{\text{r}}$, finite-volume corrections to a charged hadron mass can
be written as

$\Delta m^{2}(L)=\Delta m_{\mathrm{L}}^{2}(L)+\Delta m_{\mathrm{r}}^{2}(L)\,,$
(12)

where, similarly to eq. 11, the first term on the right hand side corresponds
to the QED${}_{\text{L}}$ correction, while the second term is the additional
contribution from the redistributed zero mode.

The QED${}_{\text{L}}$ e.m. finite-volume effects can then be obtained as
follows

$\Delta
m^{2}_{\mathrm{L}}(L)=e^{2}\,\Delta_{\mathbf{k}}^{\prime}\int\frac{\mathrm{d}k_{0}}{2\pi}\frac{{{M_{\mu}}^{\mu}}(\mathrm{i}k_{0},{\mathbf{k}})}{k_{0}^{2}+{\mathbf{k}}^{2}}=\frac{e^{2}}{2}\,\Delta_{\mathbf{k}}^{\prime}\frac{{{M_{\mu}}^{\mu}}(-|{\mathbf{k}}|,{\mathbf{k}})}{|{\mathbf{k}}|}\,,$
(13)

where
$\Delta_{\mathbf{k}}^{\prime}=\Big{[}\tfrac{1}{L^{3}}\sum_{{\mathbf{k}}\neq\boldsymbol{0}}-\int\tfrac{\mathrm{d}^{3}{\mathbf{k}}}{(2\pi)^{3}}\Big{]}$
and

${M_{\mu}}^{\mu}(\mathrm{i}k_{0},{\mathbf{k}})=\frac{Z_{\mathrm{1P}}({\mathbf{k}}^{2})}{\sqrt{m^{2}+{\mathbf{k}}^{2}}-m-\mathrm{i}k_{0}}+Z_{\mathrm{MP}}(\mathrm{i}k_{0},{\mathbf{k}}^{2})$
(14)

is the retarded Compton tensor, split into single-particle (1P) and multi-
particle (MP) contributions. Once evaluated at
$(-|{\mathbf{k}}|,{\mathbf{k}})$, it can be written as

${M_{\mu}}^{\mu}(-|{\mathbf{k}}|,{\mathbf{k}})=\frac{Z_{\mathrm{1P}}(0)}{|{\mathbf{k}}|}+\mathcal{M}(|{\mathbf{k}}|)\,,$
(15)

with $\mathcal{M}(|{\mathbf{k}}|)$ being a regular function of
$|{\mathbf{k}}|$. Substituting $|{\mathbf{k}}|=\tfrac{2\pi}{L}|{\mathbf{n}}|$
in eq. 13 and expanding for large values of $L$, one obtains

$\Delta
m^{2}_{\mathrm{L}}(L)=\frac{e^{2}}{2}\,\bigg{[}c_{1}\frac{Z_{\mathrm{1P}}(0)}{4\pi
L}+c_{2}\,\frac{\mathcal{M}(0)}{2\pi^{2}L^{2}}+c_{0}\,\frac{\mathcal{M}^{\prime}(0)}{L^{3}}-\frac{1}{2\pi^{2}}\sum_{\ell=0}^{\infty}\frac{(-1)^{\ell}\,c_{4+2\ell}}{L^{4+2\ell}}\,\mathcal{M}^{(2+2\ell)}(0)\bigg{]}\
,$ (16)

where the finite-volume coefficients

$c_{j}=\Big{[}\sum_{{\mathbf{n}}\neq\boldsymbol{0}}-\int\mathrm{d}^{3}{\mathbf{n}}\Big{]}\frac{1}{|{\mathbf{n}}|^{j}}$
(17)

have been introduced in ref. [48] and discussed in refs. [3, 36]. We stress
that the finite-volume correction at $\mathcal{O}(1/L^{3})$ does not vanish in
QED${}_{\text{L}}$ since $c_{0}=-1$. While the terms $Z_{\mathrm{1P}}(0)$ and
$\mathcal{M}(0)$ are universal and only depend on charge and mass of the
hadron, the contribution at $\mathcal{O}(1/L^{3})$ and beyond depend also on
the internal structure of the particle. As discussed in ref. [36], evaluating
the contribution denoted here as $\mathcal{M}^{\prime}(0)$ requires computing
an integral over the branch cut of the forward Compton amplitude, which makes
the estimation of the $\mathcal{O}(1/L^{3})$ correction rather complicated.
However, we can prove now that such effect is cancelled in QED${}_{\text{r}}$.
In fact, the additional contribution to the e.m. finite-volume effects to the
hadron mass is obtained by following similar steps and evaluating

$\displaystyle\Delta
m^{2}_{\mathrm{r}}(L)=e^{2}\,\sum_{{\mathbf{k}}\neq\boldsymbol{0}}\,h({\mathbf{k}},R)\int\frac{\mathrm{d}k_{0}}{2\pi}\frac{{{M_{\mu}}^{\mu}}(\mathrm{i}k_{0},{\mathbf{k}})}{k_{0}^{2}+{\mathbf{k}}^{2}}=\left.\frac{e^{2}}{2}\,\frac{{{M_{\mu}}^{\mu}}(-|{\mathbf{p}}|,{\mathbf{p}})}{|{\mathbf{p}}|}\right|_{|{\mathbf{p}}|=2\pi
R/L}\,.$ (18)

Substituting now $|{\mathbf{p}}|=\tfrac{2\pi}{L}R$ and expanding again for
large values of $L$ yields

$\displaystyle\Delta m^{2}_{\mathrm{r}}(L)$
$\displaystyle=\frac{e^{2}}{2}\,\bigg{[}\frac{R^{-2}}{\pi}\frac{Z_{\mathrm{1P}}(0)}{4\pi
L}+(\pi
R^{-1})\,\frac{\mathcal{M}(0)}{2\pi^{2}L^{2}}+\frac{\mathcal{M}^{\prime}(0)}{L^{3}}\,+$
(19) $\displaystyle\qquad\qquad\quad+\sum_{\ell=0}^{\infty}\frac{(2\pi
R)^{2\ell+1}}{(2\ell+2)!\,L^{4+2\ell}}\mathcal{M}^{(2\ell+2)}(0)+\sum_{\ell=0}^{\infty}\frac{(2\pi
R)^{2\ell+2}}{(2\ell+3)!\,L^{5+2\ell}}\mathcal{M}^{(2\ell+3)}(0)\bigg{]}\,.$

Combining this result with the QED${}_{\text{L}}$ correction in eq. 16 we
obtain the full e.m. finite-volume correction to the mass

$\displaystyle\Delta m^{2}(L)$
$\displaystyle=\frac{e^{2}}{2}\,\bigg{[}\bar{c}_{1}(R)\frac{Z_{\mathrm{1P}}(0)}{4\pi
L}+\bar{c}_{2}(R)\,\frac{\mathcal{M}(0)}{2\pi^{2}L^{2}}+\bar{c}_{0}\,\frac{\mathcal{M}^{\prime}(0)}{L^{3}}+$
(20)
$\displaystyle\qquad\qquad\quad-\frac{1}{2\pi^{2}}\sum_{\ell=0}^{\infty}\frac{(-1)^{\ell}\,\bar{c}_{4+2\ell}(R)}{L^{4+2\ell}}\mathcal{M}^{(2\ell+2)}(0)+\sum_{\ell=0}^{\infty}\frac{\bar{c}_{5+2\ell}(R)}{L^{5+2\ell}}\,\mathcal{M}^{(3+2\ell)}(0)\bigg{]}\,,$

where

$\displaystyle\boxed{\bar{c}_{0}=c_{0}+1=0}\,,\quad\bar{c}_{1}(R)=c_{1}+\frac{R^{-2}}{\pi}\,,\quad\bar{c}_{2}(R)=c_{2}+\pi
R^{-1}\,,$
$\displaystyle\bar{c}_{4+2\ell}(R)={c}_{4+2\ell}+\frac{(-1)^{1-\ell}\pi(2\pi)^{2\ell+2}}{(2\ell+2)!}R^{1+2\ell}\,,$
(21)
$\displaystyle\bar{c}_{5+2\ell}(R)=\frac{(2\pi)^{2\ell+2}}{(2\ell+3)!}R^{2\ell+2}\,.$
(22)

Some comments on these results are in order. First, we note that the
coefficient associated with the zero mode vanishes, namely $\bar{c}_{0}=0$. As
anticipated below eq. 8, this is a direct consequence of the constraint on the
weights $\sum_{\mathbf{k}}h({\mathbf{k}},R)=1$, which characterizes
QED${}_{\text{r}}$ among all possible infrared improvements of the QED action
[3]. We can interpret this result as follows: as the volume increases, the
Fourier space becomes denser and, in the infinite-volume limit, the
redistributed modes “reproduce” the zero-mode contribution, which in
QED${}_{\text{L}}$ would be simply removed. Then, we note that the other
QED${}_{\text{L}}$ finite-volume coefficients are shifted by an amount that
depends on the radius $R$. For these coefficients, choosing the shell $R=1$
($|{\mathbf{p}}|={2\pi}/{L}$ and $w({\mathbf{k}},1)=1/6$) seems to be the
optimal choice. This implementation of QED${}_{\text{r}}$ is the one we
consider _standard_ and that we adopt in the rest of the work. Finally, we
observe that new (but more suppressed) finite-volume effects arise at
$\mathcal{O}(1/L^{5})$ and higher odd inverse powers of $L$. This phenomenon
is associated with the redistribution of the zero mode and to the fact that
spatial locality is still broken in QED${}_{\text{r}}$, as it is in
QED${}_{\text{L}}$. In fact, in a local finite-volume formulation of QED, like
QED${}_{\text{C}}$, such contributions would not arise [7]. Nonetheless, the
emergence of these new, unknown, higher-order finite-volume effects does not
raise any practical concerns or limitations. This is because, even at
$\mathcal{O}(1/L^{4})$, our knowledge of structure-dependent finite-volume
effects is incomplete. The notable advantage of QED${}_{\text{r}}$ over
QED${}_{\text{L}}$ is that systematic uncertainties to finite-volume effects
on hadron masses are pushed to a higher order, specifically to
$\mathcal{O}(1/L^{4})$. And for typical lattice extents of $m_{\pi}L\gtrsim
4$, the size of such residual power-like finite-volume effects might be
comparable with that of exponentially suppressed finite-volume effects, which
are commonly neglected.

In this section, we have focused on hadron masses to illustrate the impact of
QED${}_{\text{r}}$ on observables that do not depend on an external momentum.
However, similar conclusions can be drawn for other quantities, such as the
e.m. corrections to the HVP contribution to the anomalous muon $g{-}2$. The
e.m. finite-volume corrections to the two-pion contribution to the HVP,
computed in QED${}_{\text{L}}$ within the point-like approximation and in the
photon rest frame, are detailed in ref. [49] and amount to

$\Delta\hat{\Pi}(q_{0}^{2})=\frac{c_{0}}{(m_{\pi}L)^{3}}\,\Omega\big{(}{q_{0}^{2}}/{m_{\pi}^{2}}\big{)}+\mathcal{O}\Big{(}\frac{1}{L^{4}}\Big{)}\,,$
(23)

where $\Omega(z)$ is a dimensionless function. From this result we deduce that
QED${}_{\text{r}}$ finite-volume corrections to the $\pi\pi$ contribution to
the HVP only start at $\mathcal{O}(1/L^{4})$. As the inclusion of IB
corrections becomes essential for achieving high precision in lattice
calculations of the muon $g{-}2$ (see ref. [28] for a recent review), the use
of QED${}_{\text{r}}$ can be beneficial in mitigating systematic uncertainties
related to e.m. finite-volume effects.

###  3.2 Momentum-dependent observables

Let us study now the case of an observable that depends on an external spatial
momentum. We consider here the leptonic decay rate of a meson, which depends
on the velocity ${\mathbf{v}}_{\ell}={\mathbf{p}}_{\ell}/\omega_{\ell}$ of the
final charged lepton. As discussed in section 2, the calculation of the
leading IB effects to this quantity suffers from a large systematic
uncertainty due to the only partial knowledge of its finite-volume corrections
of $\mathcal{O}(1/L^{3})$. In this case, the realization of the
QED${}_{\text{r}}$ improvement is much less straightforward and not automatic.
This is due to the appearance in the finite-volume expansion of the decay rate
of finite-volume coefficients like

$c_{j}({\mathbf{v}})=\bigg{[}\sum_{{\mathbf{n}}\neq\boldsymbol{0}}-\int\mathrm{d}^{3}{\mathbf{n}}\bigg{]}\,\frac{1}{|{\mathbf{n}}|^{j}(1-{\mathbf{v}}\cdot\hat{{\mathbf{n}}})}\,.$
(24)

As shown in ref. [37], the QED${}_{\text{L}}$ finite-volume correction of
$\mathcal{O}(1/L^{3})$ to leptonic decay rates contains both terms
proportional to $c_{0}$ and terms proportional to $c_{0}({\mathbf{v}})$. While
the former are cancelled in QED${}_{\text{r}}$ as an effect of the condition
$\sum_{\mathbf{k}}h({\mathbf{k}},R)=1$, the latter are not, since the
coefficient

$\bar{c}_{0}({\mathbf{v}})={c}_{0}({\mathbf{v}})+\sum_{\mathbf{k}}\frac{h({\mathbf{k}},R)}{1-{\mathbf{v}}\cdot\hat{{\mathbf{n}}}}$
(25)

is not necessarily zero. As studied in ref. [3], the dependence of the
QED${}_{\text{L}}$ coefficients $c_{j}({\mathbf{v}})$ on the direction
$\hat{{\mathbf{v}}}$ is a direct consequence of rotational symmetry breaking
on a lattice. In fact, these coefficients can be rewritten as

$c_{j}({\mathbf{v}})=\frac{\mathrm{arctanh}(|{\mathbf{v}}|)}{|{\mathbf{v}}|}\,c_{j}\,+\,f_{j}({\mathbf{v}})\,,$
(26)

with the functions $f_{j}({\mathbf{v}})$ encoding direction-dependent
corrections, which get stronger as $|{\mathbf{v}}|\to 1$ and vanish once
averaged over the solid angle of ${\mathbf{v}}$, namely
$\tfrac{1}{4\pi}\int\mathrm{d}\Omega_{{\mathbf{v}}}\,f_{j}({\mathbf{v}})=0$ .
This implies that $c_{0}({\mathbf{v}})$ is proportional to $c_{0}$, up to
rotational breaking effects. Since the additional QED${}_{\text{r}}$
correction in eq. 25, once averaged over the directions of ${\mathbf{v}}$,
yields

$\frac{1}{4\pi}\sum_{\mathbf{k}}\int\mathrm{d}\Omega_{{\mathbf{v}}}\,\frac{h({\mathbf{k}},R)}{1-{\mathbf{v}}\cdot\hat{{\mathbf{n}}}}=\frac{\mathrm{arctanh}(|{\mathbf{v}}|)}{|{\mathbf{v}}|}\sum_{\mathbf{k}}h({\mathbf{k}},R)=\frac{\mathrm{arctanh}(|{\mathbf{v}}|)}{|{\mathbf{v}}|}\,,$
(27)

then, considering that $c_{0}=-1$, the QED${}_{\text{r}}$ coefficient
$\bar{c}_{0}({\mathbf{v}})$ is zero up to rotational breaking effects.

Figure 3: Angular dependence of the velocity-dependent coefficient
$\bar{c}_{0}({\mathbf{v}})$ for three different values of $|{\mathbf{v}}|$.
Positive and negative values of the coefficient are coloured in blue and red,
respectively, while white regions correspond to the directions for which
$\bar{c}_{0}({\mathbf{v}})=0$. Figures are retrieved from ref. [50].

The angular dependence of $\bar{c}_{0}({\mathbf{v}})$ is shown in fig. 3 for
different values of $|{\mathbf{v}}|$ and for the standard implementation of
QED${}_{\text{r}}$ with $R=1$.222The pictures reported in fig. 3 have been
taken from A. Portelli’s talk at this conference [50] and have been generated
using a recently developed software that allows a fast evaluation of velocity-
dependent finite-volume coefficients [51]. As we see, directions always exist
for which $\bar{c}_{0}({\mathbf{v}})=0$, corresponding to the white regions in
the figure. Moreover, we note that as the velocity $|{\mathbf{v}}|$ increases,
a non-trivial fractal pattern arises, which is likely related to number-
theoretical properties of the components of the vector $\hat{{\mathbf{v}}}$.
While $|\min\bar{c}_{0}({\mathbf{v}})|$ remains small,
$\max\bar{c}_{0}({\mathbf{v}})$ diverges as $|{\mathbf{v}}|\to 1$, but at the
same time the corresponding positive (blue) regions in fig. 3 get more
localized and compensated by larger (red) regions where
$\bar{c}_{0}({\mathbf{v}})$ is negative. This visually verifies that the
angular average of $\bar{c}_{0}({\mathbf{v}})$ vanishes. We can then summarize
two fundamental properties of QED${}_{\text{r}}$, which hold for any value of
$|{\mathbf{v}}|$:

  1. 1.

the average of $\bar{c}_{0}({\mathbf{v}})$ over the solid angle of the
velocity is zero;

  2. 2.

there always exists a direction $\hat{{\mathbf{v}}}^{\star}$ such that
$\bar{c}_{0}({\mathbf{v}}^{\star})=0$.

We propose here two possible ways, each related to one of the properties
above, to implement the QED${}_{\text{r}}$ improvement and set the coefficient
$\bar{c}_{0}({\mathbf{v}})$ to zero in a numerical lattice calculation of
momentum-dependent observables.

One simple approach, which is easily implementable, consists in selecting the
velocity with a direction $\hat{{\mathbf{v}}}^{\star}$ such that
$\bar{c}_{0}({\mathbf{v}}^{\star})=0$. This process is facilitated by
utilizing the C++ code QedFvCoef [51], which enables a fast evaluation of
velocity-dependent finite-volume coefficients, using an auto-tuned algorithm
based on that proposed in ref. [3]. Additionally, it offers a Python binding
with a suite of useful tools, including a notebook that conducts an angle scan
and identifies directions for which $\bar{c}_{0}({\mathbf{v}}^{\star})=0$.

Figure 4:  Stochastic direction average of the QED${}_{\text{r}}$ finite-
volume coefficient $\bar{c}_{0}({\mathbf{v}})$ at $|{\mathbf{v}}|=0.994$. The
figure is retrieved from ref. [50].

Another possibility uses the fact that averaging over the solid angle of the
velocity sets $\bar{c}_{0}({\mathbf{v}}^{\star})$ to zero. In a lattice
calculation, we can achieve this result by implementing a stochastic average
of the velocity directions, which are drawn randomly for each measurement. In
this way, in the limit of large statistics the stochastic average

$\langle\bar{c}_{0}({\mathbf{v}})\rangle_{\hat{{\mathbf{v}}}}=\frac{1}{N_{\mathrm{meas}}}\sum_{n=1}^{N_{\mathrm{meas}}}\bar{c}_{0}({\mathbf{v}}_{n})$
(28)

converges to zero. This behaviour is shown in fig. 4, where the stochastic
average is computed for $|{\mathbf{v}}|=0.994$ and for an increasing number of
random directions. We see that few thousands angles are needed to reproduce
the desired result, the number of samples also depending on the size of the
absolute value $|{\mathbf{v}}|$. Further details on the numerical
implementation of this procedure, as well as proposals to improve the
direction sampling will be discussed in a separate publication. While this
procedure comes with a higher computational cost, its notable advantage lies
in eliminating rotational-breaking corrections not only to the coefficient
$\bar{c}_{0}({\mathbf{v}})$ but also to all other coefficients
$\bar{c}_{j}({\mathbf{v}})$.

There is, in principle, also a third way to achieve the QED${}_{\text{r}}$
improvement, that we briefly mention here before closing the section. While
the previous two strategies use properties of QED${}_{\text{r}}$ defined in
its standard implementation, with weights assigned in an isotropic way to the
modes on the first shell of radius $R=1$, one could define the
QED${}_{\text{r}}$ action with anisotropic weights that depend on the velocity
${\mathbf{v}}$ of the system studied. If the velocity is aligned along a
lattice vector, namely ${\mathbf{v}}=\alpha{\mathbf{x}}$, with
${\mathbf{x}}=(x_{1},x_{2},x_{3})\in\mathbb{Z}^{3}$ and
$\mathrm{GCD}(x_{1},x_{2},x_{3})=1$, we identify the modes on the shell of
radius $R=|{\mathbf{x}}|$ and assign equal weight $w_{1}$ to the two modes
along the direction of ${\mathbf{v}}$ and a different weight $w_{2}$ to all
the others. We can then tune $w_{1}$ and $w_{2}$ in such a way that the
following two equations are satisfied

$\begin{cases}\ \
\bar{c}_{0}({\mathbf{v}})=c_{0}({\mathbf{v}})+w_{1}\,\Big{(}\frac{1}{1-|{\mathbf{v}}|}+\frac{1}{1+|{\mathbf{v}}|}\Big{)}+w_{2}\,\sum_{\mathbf{k}}^{\prime}\frac{1}{1-{\mathbf{v}}\cdot\hat{{\mathbf{n}}}}\equiv
0\\\ \ \
\sum_{\mathbf{k}}h({\mathbf{k}},R)=2w_{1}+(r_{3}(R^{2})-2)\,w_{2}\equiv
1\,.\end{cases}$ (29)

The sum $\sum_{\mathbf{k}}^{\prime}$ in the first equation is over the modes
not parallel to ${\mathbf{v}}$ and the second equation guarantees that
$c_{0}=0$. Although potentially effective, the disadvantage of this
implementation is that one needs to tune the lattice QED action for any
specific velocity ${\mathbf{v}}$. Since we are interested in computing in the
same simulation leptonic decay rates of different hadrons and hence with
different final lepton velocities, we do not employ this implementation in our
numerical studies.

###  3.3 Ongoing numerical investigations

In order to study the properties of QED${}_{\text{r}}$ regularization outlined
in the previous sections, numerical investigations are ongoing. On the one
hand, a study at unphysical pion masses is being performed to compare the
scaling of hadron masses and leptonic decay amplitudes with the volume in
QED${}_{\text{L}}$ and QED${}_{\text{r}}$. Dedicated ensembles with Möbius
domain wall fermions have been generated at pion masses of around
$340~{}\mathrm{MeV}$ and a total of four ensembles are available with lattice
sizes $L/a=\\{16,20,24,32\\}$. Although the size of finite-volume effects
depends on the pion mass, a comparison of QED${}_{\text{L}}$ and
QED${}_{\text{r}}$ effects is possible also away from the physical point.
Different velocities have also been simulated to test numerically the
improvement in choosing a special direction such that
$\bar{c}_{0}({\mathbf{v}}^{\star})=0$, to be compared with a “naive” one. On
the other hand, parallel tests of the stochastic direction average are under
study on a physical point domain wall fermions ensemble at a lattice spacing
different from the one used in ref. [32], in the context of a large scale
calculation of IB corrections to decay rates of $\pi$, $K$, $D$ and $D_{s}$
mesons into muons and neutrinos. This calculation will also allow, in the
future, to reduce systematic uncertainties on $\delta R_{K\pi}$ associated
with discretization effects. Further details on these investigations and on
the ensembles used will be given in separate publications.

##  4 Conclusions

In this work we have discussed the current status of lattice calculations of
IB corrections to weak decays, focusing on the two existing calculations of
leptonic decay rates of pions and kaons into muons. Progress is also being
made on other weak processes, and a summary of recent works in the context of
kaon decays can be found in ref. [52]. Both calculations discussed in this
work have been performed using the RM123S strategy, where the infrared
divergences in the virtual and real decay rates are regularized by the
introduction of an infrared cutoff. The virtual corrections to the decay rate
require a non perturbative evaluation on the lattice and therefore the finite
volume with the QED${}_{\text{L}}$ prescription for the photon action is used
as an infrared regulator. The removal of the spatial zero modes of the photon
in the QED${}_{\text{L}}$ action generates finite-volume corrections at
$\mathcal{O}(1/L^{3})$, which would be otherwise absent in a local theory like
QED${}_{\text{C}}$. While corrections to lower orders are known, the
$\mathcal{O}(1/L^{3})$ contribution is only understood in the point-like
approximation, ignoring the internal structure of the decaying meson. Such
partial knowledge of finite-volume corrections can generate large systematic
uncertainties, as observed in ref. [32]. In this work, we have introduced a
novel approach to improve the finite-volume scaling of IB corrections in
leptonic decay rates and other hadronic observables. This approach consists in
a different treatment of the photon modes in the finite-volume QED action,
that we call QED${}_{\text{r}}$. Instead of removing the spatial zero modes of
the photon, ${\mathbf{k}}=\boldsymbol{0}$, these are redistributed over
neighbouring modes on a shell of given radius
$|{\mathbf{k}}|=\tfrac{2\pi}{L}R$. This strategy eliminates the
$\mathcal{O}(1/L^{3})$ correction, shifting the systematic uncertainty to a
higher order. While this improvement occurs automatically for hadronic
observables independent of external spatial momenta, such as hadron masses or
the HVP contribution to the muon $g{-}2$, it doesn’t apply to momentum-
dependent observables like leptonic decay rates. In this case, the breaking of
rotational symmetry in a finite volume makes the cancellation of the terms at
$\mathcal{O}(1/L^{3})$ more intricate. Two methods are proposed to implement
the infrared improvement in a lattice calculation. Both methods are currently
under study to test numerically the differences between QED${}_{\text{L}}$ and
QED${}_{\text{r}}$ prescriptions and the results will be discussed in
forthcoming publications. Given that implementing the QED${}_{\text{r}}$
action is a straightforward modification of the QED${}_{\text{L}}$
prescription, we anticipate this new method could significantly contribute to
reducing systematic uncertainties in various calculations of IB corrections.

## Acknowledgments

I am grateful to all my collaborators, in particular M. T. Hansen, N.
Hermansson-Truedsson and A. Portelli, whose contributions significantly shaped
the content of this paper, as well as F. Erben and F. Joswig for the precious
collaboration on setting up the numerical investigation of QED${}_{\text{r}}$.
I also wish to thank A. Patella and N. Tantalo for the useful discussions
ahead of the conference. I have been supported in part by the UK STFC grant
ST/P000630/1. This project has also received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme under grant agreements No 757646 and No 813942, and from
the European Union’s Horizon Europe research and innovation programme under
the Marie Sklodowska-Curie grant agreement No 101108006.

## References

  * [1] Y. Aoki, T. Blum, G. Colangelo, S. Collins, M.D. Morte, P. Dimopoulos et al., _Flag review 2021_ , _The European Physical Journal C_ 82 (2022) .
  * [2] M. Hayakawa and S. Uno, _QED in finite volume and finite size scaling effect on electromagnetic properties of hadrons_ , _Prog. Theor. Phys._ 120 (2008) 413 [0804.2044].
  * [3] Z. Davoudi, J. Harrison, A. Jüttner, A. Portelli and M.J. Savage, _Theoretical aspects of quantum electrodynamics in a finite volume with periodic boundary conditions_ , _Phys. Rev. D_ 99 (2019) 034510 [1810.05923].
  * [4] A. Duncan, E. Eichten and H. Thacker, _Electromagnetic splittings and light quark masses in lattice QCD_ , _Phys. Rev. Lett._ 76 (1996) 3894 [hep-lat/9602005].
  * [5] M. Göckeler, R. Horsley, E. Laermann, P. Rakow, G. Schierholz, R. Sommer et al., _Qed — a lattice investigation of the chiral phase transition and the nature of the continuum limit_ , _Nuclear Physics B_ 334 (1990) 527.
  * [6] M.G. Endres, A. Shindler, B.C. Tiburzi and A. Walker-Loud, _Massive photons: an infrared regularization scheme for lattice QCD+QED_ , _Phys. Rev. Lett._ 117 (2016) 072002 [1507.08916].
  * [7] B. Lucini, A. Patella, A. Ramos and N. Tantalo, _Charged hadrons in local finite-volume QED+QCD with C ∗ boundary conditions_, _JHEP_ 02 (2016) 076 [1509.01636].
  * [8] A. Patella, _QED Corrections to Hadronic Observables_ , _PoS_ LATTICE2016 (2017) 020 [1702.03857].
  * [9] X. Feng and L. Jin, _QED self energies from lattice QCD without power-law finite-volume errors_ , _Phys. Rev. D_ 100 (2019) 094509 [1812.09817].
  * [10] N.H. Christ, X. Feng, L.-C. Jin, C.T. Sachrajda and T. Wang, _Radiative corrections to leptonic decays using infinite-volume reconstruction_ , _Phys. Rev. D_ 108 (2023) 014501 [2304.08026].
  * [11] S. Borsanyi, S. Durr, Z. Fodor, C. Hoelbling, S.D. Katz, S. Krieg et al., _Ab initio calculation of the neutron-proton mass difference_ , _Science_ 347 (2015) 1452–1455.
  * [12] P. Boyle, V. Gülpers, J. Harrison, A. Jüttner, C. Lehner, A. Portelli et al., _Isospin breaking corrections to meson masses and the hadronic vacuum polarization: a comparative study_ , _JHEP_ 09 (2017) 153 [1706.05293].
  * [13] M. Hansen, B. Lucini, A. Patella and N. Tantalo, _Gauge invariant determination of charged hadron masses_ , _JHEP_ 05 (2018) 146 [1802.05474].
  * [14] RCstar collaboration, _First results on QCD+QED with C ∗ boundary conditions_, _JHEP_ 03 (2023) 012 [2209.13183].
  * [15] G.M. de Divitiis et al., _Isospin breaking effects due to the up-down mass difference in Lattice QCD_ , _JHEP_ 04 (2012) 124 [1110.6294].
  * [16] RM123 collaboration, _Leading isospin breaking effects on the lattice_ , _Phys. Rev. D_ 87 (2013) 114505 [1303.4896].
  * [17] T. Harris, V. Gülpers, A. Portelli and J. Richings, _Efficiently unquenching QCD+QED at $\mathrm{O}(\alpha)$_, _PoS_ LATTICE2022 (2023) 013 [2301.03995].
  * [18] Z. Fodor, C. Hoelbling, S. Krieg, L. Lellouch, T. Lippert, A. Portelli et al., _Up and down quark masses and corrections to dashen’s theorem from lattice qcd and quenched qed_ , _Physical Review Letters_ 117 (2016) .
  * [19] D. Giusti, V. Lubicz, C. Tarantino, G. Martinelli, F. Sanfilippo, S. Simula et al., _Leading isospin-breaking corrections to pion, kaon and charmed-meson masses with Twisted-Mass fermions_ , _Phys. Rev. D_ 95 (2017) 114504 [1704.06561].
  * [20] X. Feng, L. Jin and M.J. Riberdy, _Lattice QCD Calculation of the Pion Mass Splitting_ , _Phys. Rev. Lett._ 128 (2022) 052003 [2108.05311].
  * [21] R. Frezzotti, G. Gagliardi, V. Lubicz, G. Martinelli, F. Sanfilippo and S. Simula, _Lattice calculation of the pion mass difference M $\pi$+-M$\pi$0 at order O($\alpha$em)_, _Phys. Rev. D_ 106 (2022) 014502 [2202.11970].
  * [22] RBC, UKQCD collaboration, _Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment_ , _Phys. Rev. Lett._ 121 (2018) 022003 [1801.07224].
  * [23] D. Giusti, V. Lubicz, G. Martinelli, F. Sanfilippo and S. Simula, _Electromagnetic and strong isospin-breaking corrections to the muon $g-2$ from Lattice QCD+QED_, _Phys. Rev. D_ 99 (2019) 114502 [1901.10462].
  * [24] S. Borsanyi et al., _Leading hadronic contribution to the muon magnetic moment from lattice QCD_ , _Nature_ 593 (2021) 51 [2002.12347].
  * [25] M. Cè et al., _Window observable for the hadronic vacuum polarization contribution to the muon g-2 from lattice QCD_ , _Phys. Rev. D_ 106 (2022) 114502 [2206.06582].
  * [26] V. Biloshytskyi, E.-H. Chao, A. Gérardin, J.R. Green, F. Hagelstein, H.B. Meyer et al., _Forward light-by-light scattering and electromagnetic correction to hadronic vacuum polarization_ , _JHEP_ 03 (2023) 194 [2209.02149].
  * [27] E.-H. Chao, H.B. Meyer and J. Parrino, _Coordinate-space calculation of QED corrections to the hadronic vacuum polarization contribution to $(g-2)_{\mu}$_, in _40th International Symposium on Lattice Field Theory_ , 10, 2023 [2310.20556].
  * [28] S. Kuberski, _Muon $g-2$: Lattice calculations of the hadronic vacuum polarization_, _PoS_ LATTICE2023 (2024) 125 [2312.13753].
  * [29] T. Aoyama et al., _The anomalous magnetic moment of the muon in the Standard Model_ , _Phys. Rept._ 887 (2020) 1 [2006.04822].
  * [30] D. Giusti, V. Lubicz, G. Martinelli, C.T. Sachrajda, F. Sanfilippo, S. Simula et al., _First lattice calculation of the QED corrections to leptonic decay rates_ , _Phys. Rev. Lett._ 120 (2018) 072001 [1711.06537].
  * [31] M. Di Carlo, D. Giusti, V. Lubicz, G. Martinelli, C.T. Sachrajda, F. Sanfilippo et al., _Light-meson leptonic decay rates in lattice QCD+QED_ , _Phys. Rev. D_ 100 (2019) 034514 [1904.08731].
  * [32] P. Boyle, M. Di Carlo, F. Erben, V. Gülpers, M.T. Hansen, T. Harris et al., _Isospin-breaking corrections to light-meson leptonic decays from lattice simulations at physical quark masses_ , _Journal of High Energy Physics_ 2023 (2023) .
  * [33] N. Carrasco, V. Lubicz, G. Martinelli, C.T. Sachrajda, N. Tantalo, C. Tarantino et al., _QED Corrections to Hadronic Processes in Lattice QCD_ , _Phys. Rev. D_ 91 (2015) 074506 [1502.00257].
  * [34] V. Lubicz, G. Martinelli, C.T. Sachrajda, F. Sanfilippo, S. Simula and N. Tantalo, _Finite-Volume QED Corrections to Decay Amplitudes in Lattice QCD_ , _Phys. Rev. D_ 95 (2017) 034504 [1611.08497].
  * [35] N. Tantalo, V. Lubicz, G. Martinelli, C.T. Sachrajda, F. Sanfilippo and S. Simula, _Electromagnetic corrections to leptonic decay rates of charged pseudoscalar mesons: finite-volume effects_ , 1612.00199v2.
  * [36] M. Di Carlo, M.T. Hansen, A. Portelli and N. Hermansson-Truedsson, _Relativistic, model-independent determination of electromagnetic finite-size effects beyond the pointlike approximation_ , _Phys. Rev. D_ 105 (2022) 074509 [2109.05002].
  * [37] M. Di Carlo, M.T. Hansen, N. Hermansson-Truedsson and A. Portelli, _Structure-dependent electromagnetic finite-volume effects through order $1/L^{3}$_, in _40th International Symposium on Lattice Field Theory_ , 10, 2023 [2310.13358].
  * [38] F. Bloch and A. Nordsieck, _Note on the Radiation Field of the electron_ , _Phys. Rev._ 52 (1937) 54.
  * [39] G.M. de Divitiis et al., _Real photon emissions in leptonic decays_ , 1908.10160.
  * [40] C. Kane, C. Lehner, S. Meinel and A. Soni, _Radiative leptonic decays on the lattice_ , _PoS_ LATTICE2019 (2019) 134 [1907.00279].
  * [41] C. Kane, D. Giusti, C. Lehner, S. Meinel and A. Soni, _Controlling unwanted exponentials in lattice calculations of radiative leptonic decays_ , _PoS_ LATTICE2021 (2022) 162 [2110.13196].
  * [42] R. Frezzotti, M. Garofalo, V. Lubicz, G. Martinelli, C.T. Sachrajda, F. Sanfilippo et al., _Comparison of lattice QCD+QED predictions for radiative leptonic decays of light mesons with experimental data_ , _Phys. Rev. D_ 103 (2021) 053005 [2012.02120].
  * [43] D. Giusti, C.F. Kane, C. Lehner, S. Meinel and A. Soni, _Methods for high-precision determinations of radiative-leptonic decay form factors using lattice QCD_ , _Phys. Rev. D_ 107 (2023) 074507 [2302.01298].
  * [44] A. Desiderio et al., _First lattice calculation of radiative leptonic decay rates of pseudoscalar mesons_ , _Phys. Rev. D_ 103 (2021) 014502 [2006.05358].
  * [45] R. Frezzotti, N. Tantalo, G. Gagliardi, F. Sanfilippo, S. Simula, V. Lubicz et al., _Lattice calculation of the Ds meson radiative form factors over the full kinematical range_ , _Phys. Rev. D_ 108 (2023) 074505 [2306.05904].
  * [46] N. Tantalo, _Matching lattice QC+ED to Nature_ , _PoS_ LATTICE2022 (2023) 249 [2301.02097].
  * [47] V. Cirigliano and H. Neufeld, _A note on isospin violation in Pl2(gamma) decays_ , _Phys. Lett. B_ 700 (2011) 7 [1102.0563].
  * [48] Z. Davoudi and M.J. Savage, _Finite-Volume Electromagnetic Corrections to the Masses of Mesons, Baryons and Nuclei_ , _Phys. Rev. D_ 90 (2014) 054503 [1402.6741].
  * [49] J. Bijnens, J. Harrison, N. Hermansson-Truedsson, T. Janowski, A. Jüttner and A. Portelli, _Electromagnetic finite-size effects to the hadronic vacuum polarization_ , _Phys. Rev. D_ 100 (2019) 014508 [1903.10591].
  * [50] A. Portelli, “Finite-volume collinear divergences in radiative corrections to meson leptonic decays.” Talk at 40th International Symposium on Lattice Field Theory, 2023.
  * [51] M. Di Carlo and A. Portelli. https://github.com/aportelli/QedFvCoef, 2023\.
  * [52] G. Anzivino et al., _Workshop summary – Kaons@CERN 2023_ , in _Kaons@CERN 2023_ , 11, 2023 [2311.02923].