# Isospin Breaking in Lattice QCD Computations of
Decay Amplitudes††thanks: Contribution to the 60th Jubilee Kraków School of
Theoretical Physics

C T Sachrajda  Department of Physics and Astronomy
University of Southampton, Southampton SO17 1BJ, UK

###### Abstract

The remarkable recent progress in the precision of Lattice QCD computations
for a number of physical quantities relevant for flavour physics has motivated
the introduction of isospin-breaking effects, including in particular
electromagnetic corrections, to the computations. The isospin breaking
corrections are necessary to fully exploit this improved precision for the
determination of the fundamental parameters of the Standard Model, including
the CKM matrix elements, and to look for deviations from experimental
measurements which might signal the presence of new physics. Together with
colleagues from Rome, we have developed and implemented a framework for
including isospin-breaking corrections in leptonic decays
$P\to\ell\bar{\nu}_{\ell}(\gamma)$, where $P$ is a pseudoscalar meson and
$\ell$ a charged lepton, and the theoretical framework and numerical results
are reviewed below. The status of our studies to extend this framework to
semileptonic decays $P_{1}\to P_{2}\ell\bar{\nu}_{\ell}(\gamma)$, where
$P_{1,2}$ are pseudoscalar mesons, is also presented.

##  1 Introduction

Lattice QCD is proving to be a powerful and precise tool for quantitative
studies in a wide range of non-perturbative hadronic processes in general and
in flavour physics and the extraction of CKM matrix elements from experimental
measurements in particular. Until relatively recently almost all lattice
simulations were performed in isosymmetric QCD, i.e. neglecting
electromagnetic interactions and with equal up and down quark masses
($m_{u}=m_{d}\equiv m_{ud}$). Over the past decade however, the precision of
lattice computations of hadronic quantities relevant for flavour physics
phenomenology has reached such an impressive level of precision that both
electromagnetic and strong isospin-breaking effects can no longer be
neglected. For a review of recent results and references to the original
literature see the latest report from the _Flavour Physics Lattice Averaging
Group, FLAG_[1]. The aim of this report is to discuss the theoretical issues
which arise when isospin breaking effects, and electromagnetic corrections in
particular, are included and to review the development and implementation of
the framework which, together with colleagues from Rome, we have been
developing to handle these issues.

Isospin breaking effects are given in terms of two small parameters
$O(\alpha_{\mathrm{em}})$ and $(m_{d}-m_{u})/\Lambda_{\mathrm{QCD}}$, each of
which as a first approximation we take to be of $O(1\%)$ (unless there are
particular reasons to expect an enhancement or suppression of these effects).
In this review, I will follow the RM123 approach of Refs.[2, 3], in which
physical observables are evaluated at first order in these two small
parameters. Alternatively, one might add QED directly to the action and
perform QCD+QED simulations at a number of values of the electric charge (see
for example Refs[4, 5]). An advantage of the RM123 method is that the two
small expansion parameters are factorised out, so that one can get relatively
large numerical signals for the corrections, computed directly in isosymmetric
QCD.

Formulating QED in a finite spatial box ($V=L^{3}$) raises some significant
issues. For example, with the frequently used periodic boundary conditions
Gauss’ Law is not satisfied for a charged particle in the box. The electric
flux across the boundary is zero in this situation. There have been a number
of proposed approaches to circumvent or mitigate this. A pragmatic approach,
and one which we will follow in the following discussion, is to implement the
QEDL formulation, defined by omitting the three-momentum zero mode from the
sum over the photon’s momentum $\vec{k}$[6, 4]:

$\int\frac{d^{\hskip
1.5pt3}{k}}{(2\pi)^{3}}f(\vec{k})\to\frac{1}{L^{3}}\sum_{\vec{k}\neq\vec{0}}f(\vec{k})\,,$
(1)

where $f(\vec{k})$ is some function of $\vec{k}$. On the left-hand side of Eq.
(1) we have the infinite-volume integral over the photon’s momentum, which
corresponds to the physical result we are attempting to derive. On the right-
hand side, with periodic boundary conditions the sum is over the discrete
momenta $\vec{k}=(2\pi/L)\vec{n}$, where $\vec{n}$ is a vector of integers,
with the contribution from $\vec{n}=\vec{0}$ omitted. The key question is what
is the difference between the lattice results obtained using the QEDL
regulator of the zero mode (right-hand side of (1)) and the physical result
(left-hand side of (1)). We will address this question in the following, but
in general the difference decreases only as inverse powers of the volume, and
not exponentially.

We note that other interesting approaches to formulating QED in a finite
volume include the use of $C^{\ast}$ boundary conditions which allow for a
non-zero electric flux across the boundary of the volume[7, 8, 9, 10] and the
infinite-volume reconstruction method in which correlation functions at large
time separations are obtained from computations at moderate separations with
only exponentially small finite-volume corrections[11, 12].

Accurate lattice results including electromagnetic and strong isosping
breaking effects have been obtained for the hadron spectrum, for example for
the mass splitting between charged and neutral pseudoscalar mesons and baryons
(see, e.g., Refs.[2, 4]). The calculation of electromagnetic effects in the
hadron spectrum does not suffer from the presence of infrared divergences. The
same is not true however, in the case of hadronic amplitudes, where
electromagnetic infrared divergences are present and cancel for well defined,
measurable physical quantities only after including diagrams containing both
real and virtual photons [13]. This is the case, for example, for the leptonic
decays $\pi_{\ell 2}$ (i.e. $\pi\to\ell\bar{\nu}_{\ell}$, where $\ell$ is a
charged lepton, $e$ or $\mu$) and $K_{\ell 2}$ as well as the semileptonic
$K_{\ell 3}$ decays (i.e. $K\to\pi\ell\bar{\nu}_{\ell}$). These decays play a
central role in the accurate determination of the Cabibbo-Kobayashi-Maskawa
(CKM) entries $|V_{us}/V_{ud}|$ and $|V_{us}|$[14].

In a recent series of papers we have developed a framework for the evaluation
of first-order isospin-breaking corrections to leptonic decays of pseudoscalar
mesons[15], calculated the corresponding finite-volume corrections up to and
including $O(1/(m_{P}L))$ (where $m_{P}$ is the mass of the meson and the
spatial volume of the lattice is $V=L^{3}$) using the QEDL regulator of the
zero mode in the photon propagator[16] and successfully implemented the
framework in the study of the leptonic decays
$K,\pi\to\mu\nu_{\mu}(\gamma)$[17, 18]. The theoretical framework will be
summarised in the sections below together with a sketch of the numerical
results.

The plan for the remainder of this paper is as follows. In the following
section we look at the question of what is meant by isospin breaking
corrections and how one might calculate them in principle. This may seem to be
a surprising question but at the level of the $O(1\%)$ effects we are
considering it is necessary to define what we mean by QCD and in particular
what the quark and hadron masses are in QCD without QED. In Sec. 3 we discuss
infrared divergences which are present in the leptonic and semileptonic
processes discussed later and this is followed in Sec. 4 by a discussion of
the finite-volume corrections in the QEDL formulation. Sections 5 and 6
contain the applications of the framework to leptonic and semileptonic decays
respectively. Sec. 7 contains a brief summary and conclusions.

The material presented below is intended to be an introduction for a general
theory audience to the problem of including isospin breaking effects, and
electromagnetic corrections in particular, in a finite Euclidean volume.
Although the motivation for these studies is to include isospin breaking
effects in lattice computations, which are necessarily performed in a finite
volume, the focus will be on the long-distance aspects rather than on ultra-
violet issues associated with the finite lattice spacing. Moreover, although
we do discuss and include strong isospin breaking effects in the computations,
the principal theoretical difficulties concern the inclusion of the propagator
of a zero-mass photon and so naturally most of the presentation is devoted to
this.

##  2 What is meant by isospin breaking corrections?

When performing lattice QCD computations, with or without QED corrections, we
need to choose a discretisation of the field theory and the numerical values
of the parameters of the Standard Model, the masses and coupling constants.
“Physical” values of the bare quark masses are determined by requiring that
the results for a chosen set of physical quantities correctly reproduce their
experimentally measured values.

It is important to note that, once isospin breaking effects, including
electromagnetism, are introduced into QCD computations, it is only the full
QCD+QED theory which is unambiguous. Strong isospin breaking implies that
there is a difference in the masses of the up and down quarks, $m_{d}\neq
m_{u}$. However since the electric charges of the $u$ and $d$ quarks are
different, electromagnetic corrections themselves induce a difference between
$m_{u}$ and $m_{d}$, so that asking the question of how much of the isospin
breaking, not only for the quark masses but in general, is attributed to
different input masses in QCD and how much to electromagnetism cannot be
answered without introducing a prescription. Physical results, of course, must
be independent of the prescription.

In this section I describe how the quark masses and the lattice spacing, $a$,
are determined using lattice QCD, both in isosymmetric QCD (Sec. 2.1) and in
the _full theory_ in which both electromagnetism and strong isospin breaking
is included (Sec. 2.2). In Sec. 2.3 I explain how QCD and hence isospin
breaking corrections might be defined in the full theory. The discussion in
this section follows Sec.II of Ref.[18] where more details can be found.

###  2.1 Calibrating the lattice in isosymmetric QCD

Imagine that we wish to compute some physical quantities in a lattice QCD
computation with $N_{f}=2+1+1$ flavours of quarks, in the isosymmetric limit,
i.e. with $m_{u}=m_{d}\equiv m_{ud}$ and without including electromagnetic
effects. To perform the computations it is necessary to choose a value for the
(dimensionless) strong coupling constant $g_{s}(a)$ and the corresponding
parameter is then the (dimensionful) lattice spacing $a$. The four parameters
to be determined are the bare quark masses $m_{ud},m_{s},m_{c}$ and the
lattice spacing $a$. This requires us to sacrifice the possibility of making
predictions for four physical quantities and instead imagine tuning the bare
quark masses in the lattice QCD action to ensure that the lattice results for
these quantities reproduce their physical values. To illustrate the procedure
imagine that we have found values of $m_{ud},\,m_{s}$ and $m_{c}$ such that
the dimensionless ratios

$\frac{am_{\pi^{0}}}{am_{\Omega}},\quad\frac{am_{K^{0}}}{am_{\Omega}},\quad\frac{am_{D^{0}}}{am_{\Omega}}$
(2)

reproduce the values in the Particle Data Group[19]. At this value of
$\alpha_{s}(a)$ we will use these quark masses to determine all other physical
quantities in which we are interested. In the numerators and denominators of
Eq. (2), the hadron masses are written in the form $am_{H}$ to underline the
point that they are obtained in lattice units from the computations. In order
to determine the lattice spacing, we need to compare the lattice result for a
dimensionful quantity, for example the mass of the $\Omega$ baryon in lattice
units ($am_{\Omega}$) with its physical value in conventional units such as
GeV:

$a=\frac{am_{\Omega}}{m_{\Omega}^{\mathrm{phys}}}=\frac{am_{\Omega}}{1.672\,\mathrm{GeV}}\,.$
(3)

(In Sec. 2.3 the lattice spacing obtained using this procedure will be denoted
by $a_{0}^{\mathrm{ISO}}$ where the subscript $0$ denotes that it has been
obtained in QCD without QED and the superscript ISO indicates that
$m_{u}=m_{d}$.) Having determined the bare-quark masses and the lattice
spacing in this way, other physical quantities can be computed. They will be
subject to systematic uncertainties, including discretisation errors (”lattice
artefacts”), which are proportional to $a^{2}$ in most currently used lattice
formulations of QCD.

The choice of the ratios in Eq. (2) to determine the “physical” quark masses
and the use of $m_{\Omega}$ to set the scale, is convenient and introduced for
illustration. It is certainly not unique, four other physical quantities can
be used for this calibration instead.

The presentation in this subsection describes an idealised situation in which
we can afford to perform a scan of the results with different input quark
masses to determine the ones which reproduce the ratios in Eq. (2) correctly.
In practice this is not possible and some level of interpolation and
extrapolation is necessary.

###  2.2 Calibration of the full theory

The main difference in the steps required to calibrate the full theory (i.e.
QCD+QED) compared to the procedure in isosymmetric QCD is the presence of the
photon as well as the fact that $m_{u}\neq m_{d}$. The presence of the
massless photon implies that the finite-volume (FV) corrections appear as
inverse powers of $L$. By contrast, in QCD for leptonic and semileptonic
decays the FV corrections are exponentially small in the volume.

A possible strategy for the determination of the quark masses and lattice
spacing in principle is the following:
1\. Using a four flavour theory for illustration, choose a value of the strong
coupling constant $g_{s}$, the bare quark masses
$\mathbf{m}=\\{m_{u},m_{d},m_{s},m_{c}\\}$ and the number of lattice points
$N$, e.g. $T=2aN$ and $L=aN$. (The specific choice $T=2L$ is convenient for
illustration but not necessary for the following argument.)
2\. In order eventually to determine the four physical bare quark masses and
the lattice spacing, we compute five quantities, e.g. the four dimensionless
ratios

$\displaystyle
R_{1}(aN;\textbf{m})=\frac{am_{\pi^{+}}}{am_{\Omega}}(aN;\textbf{m})\,,\quad
R_{2}(aN;\textbf{m})=\frac{am_{K^{0}}}{am_{\Omega}}(aN;\textbf{m})$ (4)
$\displaystyle
R_{3}(aN;\textbf{m})=\frac{am_{D_{s}}}{am_{\Omega}}(aN;\textbf{m})\,,\quad
R_{4}(aN;\textbf{m})=\frac{am_{K^{+}}-aM_{K^{0}}}{am_{\Omega}}(aN;\textbf{m})\,,$

as well as a dimensionful quantity, e.g. the mass of the $\Omega$ baryon,
computed in lattice units, from which the lattice spacing will be determined
after extrapolation to the infinite volume limit (see below),

$R_{0}(aN;\textbf{m})=\frac{am_{\Omega}(aN;\textbf{m})}{m_{\Omega}^{\textrm{phys}}}~{},$
(5)

where $m_{\Omega}^{\textrm{phys}}=1.672$ GeV is the physical value of the mass
of the $\Omega$ baryon. For illustration we are considering the masses of
QCD$+$QED stable pseudoscalar mesons in the numerators of the dimensionless
ratios (4) and using $m_{\Omega}^{\mathrm{phys}}$ to determine the lattice
spacing, but of course other quantities can be used instead. In Eqs. (4) - (5)
we have used $aN$ instead of $L$ to highlight that the infinite-volume limit
should be taken at fixed lattice spacing (see Eq. (6) below).
3\. Up to this point the procedure is the natural generalisation of that used
in isosymmetric QCD simulations, as described in Sec. 2.1. The difference here
is the presence of FV effects which behave as inverse powers of $L$. We
therefore envisage extrapolating the ratios $R_{i}$ to the infinite-volume
limit:

$R_{i}(\textbf{m})\equiv\lim_{N\to\infty}R_{i}(aN;\textbf{m})~{},\qquad
i=0,1,2,3,4\,.$ (6)

4\. For a given discretisation and choice of the strong coupling constant
$g_{s}$, the physical bare quark masses, $\textbf{m}^{\textrm{phys}}$, are
defined by requiring that the four ratios $R_{1,2,3,4}$ take their physical
values

$R_{i}(\textbf{m}^{\textrm{phys}}(g_{s}))=R_{i}^{\textrm{phys}}\,,\qquad
i=1,2,3,4\,.$ (7)

In practice, of course, this will require some extrapolations of results
obtained at different values of the bare quark masses.
5\. The lattice spacing $a$ at this value of the coupling $g_{s}$ is now
defined to be

$a=R_{0}(\textbf{m}^{\textrm{phys}}(g_{s}))\,.$ (8)

Note that with such a procedure the bare parameters and the lattice spacing
$a$ do not depend on the lattice volume.
6\. At first order in isospin breaking, i.e.
${\cal{O}}(\alpha_{\mathrm{em}},m_{d}-m_{u})$, the renormalisation of the
lepton masses is performed perturbatively, by requiring that the on-shell
masses correspond to the physical ones.

###  2.3 Defining observables in QCD

As mentioned above, once strong isospin breaking effects and electromagnetism
are included, then it only the full QCD+QED theory which is unambiguous. If in
this context we wish to define separately what we mean by QCD and what we mean
by electromagnetic corrections then we have to introduce a prescription. One
possibility, an example of what we call hadronic schemes, is to determine the
quark masses in QCD by following the same procedure as for the full theory
described in Sec. 2.2, i.e. using the ratios $R_{0}$ \- $R_{4}$ in Eqs. (4)
and (5). This is one possible definition of QCD using a hadronic scheme. (By
hadronic schemes we mean ones which are defined in terms of experimentally
measurable hadronic quantities. This is in contrast to possible schemes such
as the GRS scheme which is defined in terms of quark and gluon Green functions
renormalised at a chosen scale in the $\overline{\footnotesize\textrm{MS}}$
renormalisation scheme[20].) We denote the lattice spacing obtained in this
way by $a_{0}$ to distinguish it from $a=a_{0}+\delta a$, the spacing in the
full theory:

$a_{0}=\frac{\langle
a_{0}M_{\Omega}\rangle^{\textrm{QCD}}}{M_{\Omega}^{\textrm{phys}}}\quad\textrm{and}\quad
a=\frac{\langle
aM_{\Omega}\rangle^{\textrm{full}}}{M_{\Omega}^{\textrm{phys}}}\equiv
a_{0}(1+\delta a)~{}.$ (9)

When we add electromagnetism to QCD as defined above, the hadron masses used
in the calibration, i.e. those in Eqs. (4) and (5), will change away from
their physical values (indeed the shift will be logarithmically ultra-violet
divergent). To cancel this shift we introduce mass counterterms for the quark
masses, which then have to be included in all correlation functions.

To illustrate the procedure imagine that we wish to calculate an observable
$O$ of mass dimension 1, for example the mass of a hadron which has not been
used in the calibration. The generalisation to other cases is straightforward
and presented in Ref.[18]. At a fixed value of the strong coupling, which we
choose to be the same in QCD and in QCD+QED, we denote the best estimate of
the observable $O$, which is the one obtained in the full theory, by
$O^{\textrm{phys}}$, and that obtained in QCD as defined above by
$O^{\textrm{QCD}}$:

$O^{\textrm{phys}}\equiv\frac{\langle
aO\rangle^{\textrm{full}}}{a}\quad\textrm{and}\quad
O^{\textrm{QCD}}\equiv\frac{\langle a_{0}O\rangle^{\textrm{QCD}}}{a_{0}}~{}.$
(10)

We _define_ the difference of the two as being due to QED effects, $\delta
O^{\textrm{QED}}\equiv O^{\textrm{phys}}-O^{\textrm{QCD}}$. There are 3
contributions to $\delta O^{\textrm{QED}}$:

  1. 1.

The first contribution comes from the diagrams which contain the explicit
exchange of virtual photons.

  2. 2.

The second contribution comes from the fact that the bare quark masses
appearing in QCD and the full theory are different. The corresponding quark-
mass counterterms must therefore be inserted into the correlation functions
used to determine $O^{\textrm{phys}}$. We stress that the need to include
quark-mass counterterms is generic and arises from the requirement that the
conditions being used to determine the quark masses must be satisfied both in
the full theory and in QCD (for the hadronic scheme being used for
illustration we impose that the conditions in Eq. (7) are satisfied in both
theories).

  3. 3.

Finally we must account for the difference in the lattice spacings $\delta
a=a-a_{0}$ in the full theory and QCD.

Combining these contributions we arrive at

$O^{\textrm{phys}}=O^{\textrm{QCD}}+\frac{\langle a_{0}\,\delta
O\rangle^{\textrm{QCD}}}{a_{0}}-\frac{\delta a}{a_{0}^{2}}\langle
a_{0}\,\\!O\rangle^{\textrm{QCD}}~{},$ (11)

where we have combined the contributions to the correlation functions from the
exchange of virtual photons and from the insertion of the mass counterterms
into $\langle a_{0}\delta O\rangle^{\textrm{QCD}}$. The first term on the
right-hand side is one that can be calculated within QCD alone. It has a well
defined continuum limit as does the sum of all the terms in Eq. (11). This
term allows us to define what is the difference between QCD (defined as above)
and the full theory in the hadronic scheme: $\delta
O^{\mathrm{QED}}=O^{\textrm{phys}}-O^{\textrm{QCD}}$. An important feature of
the RM123 approach, is that the ${\cal{O}}(\alpha_{\mathrm{em}})$ terms are
computed explicitly and so we do not have to take the difference between
numerical calculations performed in the full theory and in QCD. Each of the
terms on the right-hand side of Eq. (11) is calculated directly.

We have devoted a considerable discussion to the definition of the isospin-
breaking effects due to electromagnetism, $\delta O^{\textrm{QED}}$. Having
done this, the subsequent definition of the strong isospin breaking effects is
straightforward. To do this however, we need to define the isosymmetric theory
by imposing appropriate conditions to determine the bare quark masses and the
lattice spacing. A convenient possibility is to use the procedure sketched in
Sec. 2.1. The strong isospin breaking correction $\delta O^{\textrm{SIB}}$ to
the observable $O$ can now be defined by

$\delta O^{\textrm{SIB}}=O^{\textrm{QCD}}-O^{\textrm{ISO}}~{},$ (12)

where $O^{\textrm{ISO}}=\frac{\langle
a_{0}^{\textrm{ISO}}O\rangle^{\textrm{ISO}}}{a_{0}^{\textrm{ISO}}}$ is the
value of the observable obtained in isosymmetric QCD. With these definitions
we have the natural relation $O^{\textrm{phys}}=O^{\textrm{ISO}}+\delta
O^{\textrm{QED}}+\delta O^{\textrm{SIB}}$. We underline however that $\delta
O^{\textrm{SIB}}$ depends on the quantities used for calibration, both in
4-flavour QCD and in isosymmetric QCD.

##  3 Infrared Divergences

Figure 1: Examples of diagrams contributing to electromagnetic corrections to
the mass ((a) and (b)) and to a decay amplitude ((c)).

In Sec. 1 I simply stated that infrared divergences are absent in the
evaluation of electromagnetic corrections to the spectrum, while they are
present in the decay amplitudes. To illustrate the absence of infrared
divergences in the spectrum consider the diagrams in Figs. 1(a) and (b). In
the diagrams the solid line represents an elementary charged scalar meson of
mass $m$. These diagrams contribute to the electromagnetic mass shift. The
contribution from diagram (a) is proportional to the integral

$I_{a}=i\int\frac{d^{\hskip
1.5pt4}{k}}{(2\pi)^{4}}\,\frac{(2p-k)^{2}}{[k^{2}+i\epsilon][(p-k)^{2}-m^{2}+i\epsilon]}\,,$
(13)

evaluated at $p^{2}=m^{2}$. At small $k$, the integrand behaves as $1/k^{3}$
and so the four-dimensional integral is infrared convergent. The integrand of
diagram (b) is $1/k^{2}$ and is therefore also infrared convergent.

The diagram of Fig. 1(c) is an example of a particle decaying into two
elementary charged scalar particles of masses $m_{1}$ and $m_{2}$. Now the
corresponding integral is

$I_{c}=i\int\frac{d^{\hskip
1.5pt4}{k}}{(2\pi)^{4}}\,\frac{(2p_{1}-k)\cdot(2p_{2}+k)}{[k^{2}+i\epsilon][(p_{1}-k)^{2}-m_{1}^{2}+i\epsilon][(p_{2}+k)^{2}-m_{2}^{2}+i\epsilon]}\,,$
(14)

so that at small $k$ the integrand behaves as $1/k^{4}$ and the four
dimensional integral is infrared divergent.

The treatment of infrared divergences in evaluating decay widths or scattering
cross sections was first understood by Bloch and Nordsieck in 1937[13].
Diagrams with virtual photons must be combined with those corresponding to the
emission of real photons; in this way the infrared divergences cancel. In
intermediate stages of perturbative calculations, an infrared regulator, such
as a small photon mass $m_{\gamma}$, is introduced and the divergences
manifest themselves as factors of $\log(m^{2}/m_{\gamma}^{2})$, where $m$ is a
finite mass scale. In lattice computations the volume is finite, $V=L^{3}$,
and the volume itself acts as a regulator with factors of $\log(mL)$. In
section 5 I present our framework for the evaluation of the widths for
leptonic decays of pseudoscalar mesons $P$,
$P\to\ell\bar{\nu}_{\ell}(\gamma)$, fully consistent of course with the Bloch-
Nordsieck mechanism for the cancelation of infrared divergences. Before that
however, we discuss the central issue of finite-volume corrections.

##  4 Finite-Volume Corrections

Lattice computations are necessarily performed in finite-volumes, $V=L^{3}$
say, which implies that the momenta of the photon and other particles are
discrete. Integrals, such as those in Eqs. (13) and (14) are replaced by
momentum sums. With periodic boundary conditions for the photon, we repeat Eq.
(1) writing

$\int\\!\frac{d^{\hskip
1.5pt3}{k}}{(2\pi)^{3}}\,f(\vec{k})\to\sum_{\vec{k}\neq\vec{0}}f(\vec{k})\,,$
(15)

where the sum is over $\vec{k}=(2\pi/L)\,\vec{n}$ and $\vec{n}$ is a vector of
integers. The powerful tool for evaluating the relationship between finite-
volume sums and infinite-volume integrals is the Poisson summation formula
which can be written in the form

$\sum_{\vec{k}=\frac{2\pi}{L}\vec{n}}\\!\\!f(\vec{k})=\int\\!\frac{d^{\hskip
1.5pt3}{k}}{(2\pi)^{3}}\,f(\vec{k})\,+\,\sum_{\vec{m}\neq\vec{0}}\int\\!\frac{d^{\hskip
1.5pt3}{k}}{(2\pi)^{3}}\,f(\vec{k})e^{i\vec{k}\cdot\vec{m}L}\,.$ (16)

If the function $f$ has no singularities, then the oscillating exponential in
the second term on the right-hand side of Eq. (16) suppresses the integrals
for large $L$ and the finite-volume sum is equal to the infinite-volume
integral up to terms which are exponentially suppressed in the volume. On the
other-hand, if $f$ contains singularities, which must be regulated, then the
oscillating behaviour of the exponential factors is overcome by the abrupt
behaviour at the singularity and the finite-volume effects may decrease only
as inverse powers of $L$.

For integrals such as those in Eqs. (13) and (14) the integrands are singular
at $k=0$. In the absence of other singularities, a practical rule summarising
the relation between the power of the finite-volume corrections and the
leading singularity of the integrand at $k=0$ is the scaling law derived in
[16]:

$\xi^{\prime}=\int\frac{dk_{0}}{2\pi}\,\left(\frac{1}{L^{3}}\sum_{\vec{k}\neq
0}\,-\int\frac{d^{3}k}{(2\pi)^{3}}\right)\frac{1}{(k^{2})^{n/2}}=O\left(\frac{1}{L^{4-n}}\right)\,,$
(17)

where the $1/(k^{2})^{n/2}$ simply represents the leading behaviour as $k\to
0$. Thus for example, the integrand in $I_{a}$ in Eq. (13) is proportional to
$1/k^{3}$ at small $k$, i.e. $n=3$, so that the leading finite-volume
correction is of $O(1/(mL))$. The integrand in $I_{c}$ in Eq. (14) on the
other hand behaves as $1/k^{4}$ at small $k$ corresponding to infrared
divergent terms containing factors of $\log(mL)$.

Although the discussion above was presented for structureless elementary
particles, it contains a number of important points which are valid also for
composite particles. At small momenta the photon couples to the charge of the
particle, independently of its internal structure. Thus we would expect that
the leading finite-volume corrections are _universal_ and this is indeed the
case. Studies of the possible higher-order couplings of the photon, such as
those to the electric dipole moment of the mesons, reveal that the next-to-
leading order finite-volume effects are also universal for the spectrum[6, 4]
(see Sec. 4.1) and leptonic decay amplitudes[16] (see Sec 5).

The Minkowski-space integral $I_{c}$ in Eq. (14) contains an imaginary part,
corresponding to a cut through the two internal propagators. This leads to
additional singularities from those at $k=0$ which must be treated separately.
Such cuts are absent in the calculation of the spectrum and leptonic decay
rates, where there is a single particle in the final state to which the photon
can couple. They are present however, in the study of semileptonic decays and
we comment on this in Sec. 6.

We postpone further discussion of the finite-volume corrections to leptonic
decay rates until Sec. 5, but now discuss the corrections to the spectrum.

###  4.1 Leading finite-volume corrections to hadron masses

When calculating the electromagnetic corrections to the mass of hadron H, the
finite-volume corrections decrease only as powers of $1/L$, starting at
$O(1/(m_{H}L))$, and not exponentially as is the case for many physical
quantities in QCD. As mentioned above, with QEDL the situation is made
somewhat easier in that the leading two terms, i.e. those of $O(1/(m_{H}L))$
and $O(1/(m_{H}L)^{2})$ are independent of the structure of the hadron. Thus
if the FV corrections of order ${\cal{O}}(e^{2}/(m_{H}L)^{3})$ can be
neglected then the extrapolation to the infinite-volume limit can be avoided
by making use of the formula[6, 4] (similar formulae also exist for other
finite-volume formulations of the theory[9])

$\frac{am_{H}(L)}{am_{H}}=1-\kappa\,\alpha_{\mathrm{em}}\,e_{H}^{2}\left\\{\frac{1}{2L\,m_{H}}+\frac{1}{L^{2}\,m^{2}_{H}}\right\\}~{},$
(18)

where $e_{H}$ is the charge of the hadron $H$, $m_{H}(L)$ and $m_{H}$ are the
masses of the hadron in the finite and infinite volume respectively and
$\kappa=2.837297\,(1)$. Equation (18) can be used to determine the infinite-
volume mass of the hadron $H$ from the value measured on the finite-volume
$L^{3}$, up to corrections of order of ${\cal{O}}(e^{2}/(m_{H}L)^{3})$. Even
if one wishes to study the behaviour with $L$ by performing simulations at
different volumes, the subtraction of the universal $O(e^{2}/(m_{H}L))$ and
$O(e^{2}/(m_{H}L)^{2})$ terms using Eq. (18) is a useful starting point; the
residual leading behaviour of hadronic masses is then of
$O(e^{2}/(m_{H}L)^{3})$. For reviews of isospin-breaking contributions to the
spectrum and discussions of the different approaches used to perform QCD+QED
computations of the spectrum see, for example, Refs. [21, 22, 23].

##  5 Leptonic Decays

In this section we briefly review the framework which we have developed and
implemented in the series of papers [15, 16, 17, 18]. In the absence of
electromagnetic corrections, the width for the decay of a pseudoscar meson $P$
into a charged lepton $\ell$ and its neutrino, $P\to\ell\bar{\nu}_{\ell}$, is
given by

$\Gamma(P\to\ell\bar{\nu}_{\ell})=\frac{G_{F}^{2}|V_{\mathrm{CKM}}|^{2}f_{P}^{2}}{8\pi}\,m_{P}m_{\ell}^{2}\left(1-\frac{m_{\ell}^{2}}{m_{P}^{2}}\right)^{2}\,,$
(19)

where $G_{F}$ is the Fermi constant, $V_{\mathrm{CKM}}$ is the CKM matrix
element corresponding to the flavours of the valence quarks of $P$, and
$f_{P}$ is the decay constant given by the matrix element of the corresponding
axial current. For example for the decay of a kaon, $V_{\mathrm{CKM}}=V_{us}$
and $f_{K}$ is given by

$\langle
0|\bar{u}\gamma^{\mu}\gamma^{5}s|K^{-}(p_{K})\rangle=if_{K}p_{K}^{\mu}\,,$
(20)

so that all hadronic effects are contained in the single number $f_{K}$, or
more generally $f_{P}$. There are now a very large number of lattice
computations of the $f_{P}$ at the percent or sub-percent level[1].

If one wishes to include the electromagnetic corrections to the width $\Gamma$
and hence to access the CKM matrix element with greater precision, one needs
to include contributions from the amplitude with a real photon in the final
state:

$\Gamma(\Delta E_{\gamma})=\Gamma_{0}+\Gamma_{1}(\Delta E_{\gamma})\,,$ (21)

where the subscripts $0$ and $1$ indicate the number of photons in the final
state and $\Delta E_{\gamma}$ is the maximum detected energy of the emitted
real photon (in the meson rest-frame). The calculations are performed up to
$O(\alpha_{\mathrm{em}})$. Both $\Gamma_{0}$ and $\Gamma_{1}$ are individually
infrared divergent, but the divergences cancel in the sum. In Sec. 5.2 we
describe how one might handle the infrared divergences and their cancelation
in lattice computations. Before this we introduce the effective Hamiltonian
for leptonic and semileptonic decays.

###  5.1 The effective Hamiltonian

For illustration, consider the Fermi Hamiltonian for the leptonic decay
$K^{-}\to\mu^{-}\bar{\nu}_{\mu}$; this is given by
$H_{F}=\frac{G_{F}}{\sqrt{2}}\,V_{us}\,\big{[}\bar{u}\gamma^{\rho}(1-\gamma^{5})s\big{]}\,\big{[}\bar{\mu}\gamma_{\rho}(1-\gamma^{5})\nu_{\mu}\big{]}$
, where $G_{F}$ is the Fermi constant and is generally obtained from muon
$\beta$-decay. Since we aim to calculate the $O(\alpha_{\mathrm{em}})$
corrections to leptonic decay rates, we need to ensure that the definition and
determination of $G_{F}$ is consistent with our procedure at this order. We
use the formula for the muon lifetime $\tau_{\mu}$[24, 25]

$\frac{1}{\tau_{\mu}}=\frac{G_{F}^{2}m_{\mu}^{5}}{192\pi^{3}}\left(1-\frac{8m_{e}^{2}}{m_{\mu}^{2}}\right)\,\left[1+\frac{\alpha_{\mathrm{em}}}{2\pi}\left(\frac{25}{4}-\pi^{2}\right)+O(\alpha_{\mathrm{em}}^{2})\right]$
(22)

from which, together with the measured value of $\tau_{\mu}$, one deduces the
value $G_{F}=1.16632(2)\times 10^{-5}$. Many electroweak corrections are
absorbed into the definition of $G_{F}$; the explicit
$O(\alpha_{\mathrm{em}})$ corrections on the right-hand side of Eq.(22) come
from the diagrams in Fig.2. The diagrams are evaluated with the
$W$-regularisation in which the photon propagator is modified by[26]:

$\frac{1}{k^{2}}\to\frac{1}{k^{2}}-\frac{1}{k^{2}-M_{W}^{2}}=\frac{M_{W}^{2}}{M_{W}^{2}-k^{2}}\,\frac{1}{k^{2}}\,.$
(23)

Figure 2: Diagrams contributing at $O(\alpha_{\mathrm{em}})$ to the right-hand
side of Eq. (22).

Many of the electroweak corrections which are absorbed in $G_{F}$ are common
to leptonic and semileptonic decays which leads to a factor in the amplitude
of $(1+(\alpha_{\mathrm{em}}/\pi)\log(M_{Z}/M_{W}))$[27, 28] and the effective
Hamiltonian for the leptonic or semi-leptonic decay of a $K^{-}$ meson is

$H_{\mathrm{eff}}=\frac{G_{F}}{\sqrt{2}}\,V_{us}\left(1+\frac{\alpha}{\pi}\log\frac{M_{Z}}{M_{W}}\right)O_{1}^{W}$
(24)

where
$O_{1}^{W}=(\bar{u}\gamma^{\mu}(1-\gamma^{5})s)(\bar{\ell}\,\gamma_{\mu}\,(1-\gamma^{5})\nu_{\ell})$
renormalised in the W-regularisation scheme. Its matrix elements are finite,
but depend on $M_{W}$.

In lattice computations we evaluate the matrix elements of operators in the
bare theory defined by a chosen lattice discretisation of QCD with the lattice
spacing $a$ as the ultraviolet cut-off. In order to obtain matrix elements of
$O_{1}^{W}$ we therefore have to perform the renormalisation into the
W-regularisation scheme. If the lattice theory breaks chiral symmetry, then
$O_{1}^{W}$ is a linear combination of the lattice operator $O_{1}^{L}$ and
four other lattice four-fermion operators which transform under different
chiral representations:

$O_{1}^{W}=\sum_{i=1}^{5}Z_{1i}O^{L}_{i}\,,$

where at one-loop order only $Z_{11}$ is divergent (proportional to
$\log[aM_{W}]$). Since in current simulations $a^{-1}\ll M_{W}$ it is not
feasible to perform the renormalisation fully non-perturbatively (even with
step-scaling) and we employ a combination of non-perturbative renormalisation
and perturbative running and matching. For a recent report on the current
status of the renormalisation of lattice operators into the W-regularisation
scheme see Ref.[29].

###  5.2 Infrared divergences in lattice computations of radiative corrections
to leptonic decays

In practice, it is convenient to rewrite Eq. (21) in the form

$\displaystyle\Gamma(\Delta E_{\gamma})$ $\displaystyle=$
$\displaystyle\displaystyle\lim_{L\to\infty}\left[\Gamma_{0}(L)-\Gamma_{0}^{{\rm
pt}}(L)\right]+\displaystyle\lim_{m_{\gamma}\to 0}\left[\Gamma_{0}^{{\rm
pt}}(m_{\gamma})+\Gamma_{1}^{{\rm pt}}(m_{\gamma},\Delta
E_{\gamma})\right]~{}$ (25) $\displaystyle\hskip
36.135pt+\,\Gamma_{1}^{\mathrm{SD}}(\Delta
E_{\gamma})+\Gamma_{1}^{\mathrm{INT}}(\Delta E_{\gamma})\,.$

The superscript pt indicates that $\Gamma_{0,1}^{\mathrm{pt}}$ are calculated
perturbatively in the point-like approximation. We have written
$\Gamma_{1}=\Gamma_{1}^{\mathrm{pt}}+\Gamma_{1}^{\mathrm{SD}}+\Gamma_{1}^{\mathrm{INT}}$,
where the superscripts SD and INT refer respectively to the _Structure
Dependent_ contribution and to that from the _Interference_ between the
structure dependent and point-like contributions to the amplitude. These terms
are given in terms of the vector and axial-vector form factors in the
decomposition of the non-local matrix element:

$\displaystyle H^{\alpha r}_{W}(k,\vec{p})$ $\displaystyle=$
$\displaystyle\epsilon_{\mu}^{r}(k)\,\int d^{\hskip 1.5pt4}{y}\,e^{ik\cdot
y}\,{\mathrm{T}}\,\langle{0}|\,j_{W}^{\alpha}(0)j^{\mu}_{\mathrm{em}}(y)|{P(\vec{p})}\rangle$
(26)
$\displaystyle=\epsilon_{\mu}^{r}(k)\Bigg{\\{}H_{1}\,\left[k^{2}g^{\mu\alpha}-k^{\mu}k^{\alpha}\right]+H_{2}\,\left[(p\cdot
k-k^{2})k^{\mu}-k^{2}(p-k)^{\mu}\right](p-k)^{\alpha}]]$
$\displaystyle-i\frac{F_{V}}{m_{P}}\varepsilon^{\mu\alpha\gamma\beta}k_{\gamma}p_{\beta}+\frac{F_{A}}{m_{P}}\left[(p\cdot
k-k^{2})g^{\mu\alpha}-(p-k)^{\mu}k^{\alpha}\right]$ $\displaystyle\hskip
14.45377pt+f_{P}\left[g^{\mu\alpha}+\frac{(2p-k)^{\mu}(p-k)^{\alpha}}{2p\cdot
k-k^{2}}\right]\Bigg{\\}}\;.$

In Eq. (26), $\epsilon^{r}_{\mu}$ is the polarisation vector of the photon
with polarisation state $r$, $j_{\mathrm{em}}^{\mu}$ is the electromagnetic
current to which the photon couples and $j_{W}^{\alpha}$ is the hadronic
component of the weak operator. For decays into a real photon, for which
$k^{2}=0$ and $\varepsilon\cdot k=0$, only the decay constant $f_{P}$ and the
structure-dependent vector and axial form factors $F_{V}(x_{\gamma})$ and
$F_{A}(x_{\gamma})$ are needed to specify the amplitude, where
$x_{\gamma}=2p\cdot k/m_{P}$. The final term on the right-hand side of Eq.
(26) is the point-like (or inner bremsstrahlung) contribution.

We now discuss each of the terms on the right hand side of Eq. (25).

1\. $\Gamma_{0}(L)$ is the contribution to the width which includes all the
finite-volume modes of the photon’s momentum except for $\vec{k}=0$ and
therefore depends on the structure of the meson and must be computed
nonperturbatively. At small photon momenta, for which the photon couples to
the charge of the meson, $\Gamma_{0}(L)\to\Gamma_{0}^{\mathrm{pt}}(L)$, and
the infrared divergences cancel in the difference
$\Gamma_{0}(L)-\Gamma_{0}^{{\rm pt}}(L)$. While in our calculations we use the
volume as the infrared regulator, $\Gamma_{0}(L)-\Gamma_{0}^{{\rm pt}}(L)$ is
independent of the regulator.

2\. The second term on the right-hand side of Eq. (25)

$\displaystyle\lim_{m_{\gamma}\to 0}\left[\Gamma_{0}^{{\rm
pt}}(m_{\gamma})+\Gamma_{1}^{{\rm pt}}(m_{\gamma},\Delta E_{\gamma})\right]$

is purely perturbative and can be calculated directly in infinite volume. Each
of the two terms are infrared divergent, so a regulator, such as a photon mass
$m_{\gamma}$, has to be introduced. The divergences cancel in the sum of the
two terms and the result is independent of the regulator.

3\. The infrared divergence in $\Gamma_{1}$ comes from the point-like coupling
of the photon and so the term on the second line of Eq. (25),
$\Gamma_{1}^{\mathrm{SD}}(\Delta E_{\gamma})+\Gamma_{1}^{\mathrm{INT}}(\Delta
E_{\gamma})$, is infrared convergent. It can be computed directly in infinite
volume requiring knowledge of the structure-dependent form factors,
$F_{A}(x_{\gamma})$ and $F_{V}(x_{\gamma})$, and of the meson decay constant
$f_{P}$[30, 31].

Originally we had proposed to perform the calculations with a cut-off $\Delta
E_{\gamma}$ which was sufficiently small for structure-dependent effects to be
negligible, but with $\Delta E_{\gamma}$ large enough to allow for
experimental measurements of $\Gamma(\Delta E_{\gamma})$ to be possible (20
MeV or so). While this is practicable for the decays of pions and kaons,
particularly into muons for which the rate for large $E_{\gamma}$ is
suppressed[15], this is not the case for the decays of heavy mesons. More
recently we have demonstrated that the structure-dependent contributions to
$\Gamma_{1}$ can be calculated[30, 32], thus extending the framework to the
decays of heavy mesons.

###  5.3 Finite-volume corrections to leptonic decay rates

We reported in Sec. 4.1 that the leading and next-to-leading finite-volume
effects in the calculation of electromagnetic corrections to the spectrum are
of $O(1/(m_{H}L))$ and $O(1/(m_{H}L)^{2})$ with coefficients which are
universal, i.e. independent of the structure of the hadron $H$, see Eq. (18).
For leptonic decays of a pseudoscalar meson $P$, $P\to\ell\nu_{\ell}$, we
organise the calculation as in Eq. (25) and have found that
$\Gamma^{\mathrm{pt}}_{0}(L)$ takes the form:

$\Gamma_{0}^{\mathrm{pt}}(L)=C_{0}(r_{\ell})+\tilde{C}_{0}(r_{\ell})\log\left(m_{P}L\right)+\frac{C_{1}(r_{\ell})}{m_{P}L}+\dots\,,$
(27)

where $r_{\ell}=m_{\ell}/m_{P}$ and $m_{\ell}$ is the mass of the final-state
charged lepton[16]. The exhibited $L$-dependent terms are _universal_ , i.e.
independent of the structure of the meson, and in Ref.[16] we have calculated
the coefficients $C_{0},\,\tilde{C}_{0}$ and $C_{1}$. The leading structure-
dependent FV effects in $\Gamma_{0}(L)-\Gamma_{0}^{\textrm{pt}}(L)$ are
therefore of $O(1/(m_{P}L)^{2})$. If necessary, these can be determined by
extrapolating results obtained on different volumes (see for example Fig. 3).

###  5.4 Numerical Results

In order to demonstrate that the framework presented above is practicable we
briefly present some numerical results for the $K_{\mu 2}$ and $\pi_{\mu 2}$
decays[17, 18] obtained using gauge ensembles generated by the European
Twisted Mass Collaboration (ETMC) with $N_{f}=2+1+1$ dynamical quarks[33, 34]
in the quenched QED approximation in which the charges of the sea quarks are
set to 0. In Ref.[17] we started by calculating the electromagnetic and
strong isospin breaking corrections to the ratio of $K_{\mu 2}$ and $\pi_{\mu
2}$ decay rates. This ratio is less sensitive to various sources of
uncertainty than the isospin breaking corrections to $\pi_{\mu 2}$ and $K_{\mu
2}$ decay rates separately. In Ref.[18] we provided a more complete
description of the calculation and did evaluate the electromagnetic and strong
isospin breaking corrections to the decay processes $\pi_{\mu 2}$ and $K_{\mu
2}$ separately. Since the corresponding experimental rates are fully inclusive
in the energy of the final state photon, structure-dependent contributions to
the real photon emission should be included, however the Chiral Perturbation
Theory (ChPT) predictions of Ref.[35] indicate that these structure-dependent
contributions are negligible for both kaon and pion decays into muons, an
expectation explicitly verified in a recent lattice computation[32], where the
structure dependent contributions to $\Gamma_{1}$ were shown to be negligible.
The same is not true to the same extent for decays into final-state electrons
(see Ref.[15]) and so we focus here on decays into muons.

For a detailed presentation of our study of isospin breaking contributions to
$K_{\mu 2}$ and $\pi_{\mu 2}$ decays, including many important technical
issues, please see Ref.[18]. Here we focus on two general points: i) a check
that the leading finite-volume corrections are indeed of $O(1/(m_{P}L)^{2})$
($P=K$ or $\pi$) and ii) the phenomenological implications of our calculations
and in particular the determination of the CKM matrix element $V_{us}$.

Figure 3: Volume dependence of $\delta R_{\pi}$ and $\delta R_{K}$ for a pion
of mass 320 MeV and a kaon of mass 580 MeV. The data come from computations on
4 different volumes at the same value of the lattice spacing $a$ and are
consistent with the expectation that the leading behaviour should be linear in
$1/L^{2}$ [18].

For the leptonic decay $P\to\ell\bar{\nu}_{\ell}(\gamma)$ we choose to define
the isospin-breaking correction to the rate, $\delta R_{P}$, by

$\Gamma(P\to\ell\bar{\nu}_{\ell}(\gamma))=\frac{G_{F}^{2}}{8\pi}|V_{q_{1}q_{2}}|^{2}m_{\ell}^{2}m_{P}\left(1-\frac{m_{\ell}^{2}}{m_{P}^{2}}\right)(f_{P}^{(0)})^{2}\,[1+\delta
R_{P}]\,,$ (28)

where $q_{1,2}$ are the valence quarks of the meson $P$, $m_{P}$ is its mass
and $f_{P}^{(0)}$ is its decay constant obtained within isosymmetric QCD using

$\langle 0|\bar{q}_{2}\gamma_{0}\gamma_{5}q_{1}|P(\vec{0})\rangle\equiv
f_{P}^{(0)}m_{P}^{(0)}\,,$ (29)

where the initial-state meson $P$ is at rest. Here $m_{P}^{(0)}$ is the mass
of $P$ in QCD. As discussed above, $f_{P}^{(0)}$ is prescription dependent. In
order to be able to exploit existing ETMC correlation functions (which for
example do not include correlation functions for the $\Omega^{-}$ baryon), in
the isosymmetric theory we have adopted a ”FLAG Scheme”[1] taking
$m_{\pi}^{(0)}=134.98$ MeV, $m_{K}^{(0)}=494.2(3)$ MeV and
$f_{\pi}^{(0)}=130.41$ MeV (as well as $m_{D_{s}^{+}}=1969.0(1.4)$ MeV).
Having used $f_{\pi}^{(0)}$ as part of the calibration means that we sacrifice
the possibility of determining $V_{ud}$. Such a scheme turns out to be
numerically equivalent within the uncertainties (although theoretically
different) to the GRS scheme[20] used in the chiral perturbation theory
study[36]. The GRS scheme is defined by imposing values for the renormalised
strong coupling and masses in the $\overline{\footnotesize\textrm{MS}}$ scheme
at a scale of $2$ GeV. The numerical near equivalence of the two schemes is
convenient for comparison of the lattice and ChPT results.

In Fig. 3 we show the results for $\delta R_{K}$ and $\delta R_{\pi}$ obtained
at four different values of the volume at the same value of the lattice
spacing. The data correspond to meson masses $m_{\pi}\simeq 320$ MeV and
$m_{K}\simeq 580$ MeV. The expectation is that after the subtraction of the
universal terms, the results should be largely linear in $1/L^{2}$ and the
data is nicely consistent with this.

Extrapolating our lattice results to physical quark masses and to the
continuum and infinite-volume limits we found:

$\delta R_{\pi}^{\mathrm{phys}}=+0.0159(20)\quad\mathrm{and}\quad\delta
R_{K}^{\mathrm{phys}}=+0.0032(11)\,.$ (30)

Our results in Eq. (30) can be compared with the ChPT predictions $\delta
R_{\pi}^{\textrm{phys}}=0.0176(21)$ and $\delta
R_{K}^{\textrm{phys}}=0.0064(24)$ obtained in Ref.[36] and adopted by the
PDG[37, 38]. The difference is within one standard deviation for $\delta
R_{\pi}^{\textrm{phys}}$ and a little larger for $\delta
R_{K}^{\textrm{phys}}$.

Since, as mentioned above, we have used $f_{\pi}^{(0)}$ in the determination
of the lattice spacing, we cannot use our calculation to obtain $V_{ud}$. For
the kaon on the other hand, adopting the best lattice determination of the QCD
kaon decay constant, $f_{K}^{(0)}=156.11(21)$ MeV [1, 39, 40, 41] (after
subtracting the strong isospin breaking effects) and combining it with the
experimental result
$\Gamma(K^{-}\to\mu^{-}\bar{\nu}_{\mu}[\gamma])=5.134(11)\cdot 10^{7}$ s-1
from the PDG[37], we obtain the very precise result:

$|V_{us}|=0.22561(26)_{\mathrm{exp}}\,(33)_{\mathrm{th}}=0.22561(42)\,.$ (31)

Following Ref.[17], we can also determine $|V_{us}|$ from the ratio of the
pion and kaon experimental decay rates which yields

$\frac{|V_{us}|}{|V_{ud}|}\frac{f_{K}^{(0)}}{f_{\pi}^{(0)}}=0.27677\,(29)_{\mathrm{exp}}\,(20)_{\mathrm{th}}=0.27677\,(35)\,.$
(32)

Using the best $N_{f}=2+1+1$ lattice determination of the ratio of the QCD
kaon and pion decay constants, $f_{K}^{(0)}/f_{\pi}^{(0)}=1.1966~{}(13)$[1,
39, 40, 41], we find

$\frac{|V_{us}|}{|V_{ud}|}=0.23130\,(24)_{\mathrm{exp}}\,(30)_{\mathrm{th}}=0.23130\,(38)\,.$
(33)

Taking the updated value $|V_{ud}|=0.97420\,(21)$ from super-allowed nuclear
beta decays[42], Eq. (33) yields the following value for the CKM element
$|V_{us}|$:

$|V_{us}|=0.22533\,(24)_{\mathrm{exp}}\,(30)_{\mathrm{th}}=0.22533(38)\,,$
(34)

which agrees with our result (31) within the errors. Note that our result (34)
agrees with the latest estimate $|V_{us}|=0.2252(5)$, recently updated by the
PDG[19].

Taking the values $|V_{ub}|=0.00413(49)$[37] and $|V_{ud}|=0.97420(21)$[42]
our result in Eq. (34) implies that the unitarity of the first-row of the CKM
matrix is confirmed to better than the per-mille level

$|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}=0.99986\,(44)\,.$ (35)

##  6 Semileptonic Decays

Figure 4: (a) Diagram at $O(\alpha_{\mathrm{em}})$ contributing to the
semileptonic decay $K\to\pi\ell\nu_{\ell}$; (b) diagram contributing to the
leptonic decay of a kaon.

In this section we discuss our ongoing work to develop a framework for the
evaluation in a finite Euclidean volume of electromagnetic contributions to
amplitudes for semileptonic decays $P_{1}\to
P_{2}\ell\bar{\nu}_{\ell}(\gamma)$, where $P_{1}$ and $P_{2}$ are pseudoscalar
mesons and $\ell$ is a charged lepton. A discussion of the issues has
previously been presented in Ref.[43]. Throughout this section, we illustrate
the issues by considering $K_{\ell 3}$ decays, e.g.
$K^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$ decays where $\ell=\mu$ or $e$, but the
discussion is general to all semileptonic decays. In QCD without
electromagnetic corrections, the amplitudes are given by two invariant form
factors, which for $K_{\ell 3}$ decays for example can be defined by

$\langle\,\pi^{-}(p_{\pi})\,|\bar{s}\gamma_{\mu}u\,|\,K^{0}(p_{K})\,\rangle=f_{+}(q^{2})\,(p_{K}+p_{\pi})_{\mu}+f_{-}(q^{2})\,(p_{K}-p_{\pi})_{\mu}\,,$
(36)

where the momentum transfer $q=p_{K}-p_{\pi}$.

When computing electromagnetic corrections, for which contributions to the
rate with a photon in the final state must be included, an appropriate
measurable quantity to consider is

$\frac{d^{2}\Gamma}{dq^{2}ds_{\pi\ell}},$

where $q^{2}=(p_{K}-p_{\pi})^{2}$ and $s_{\pi\ell}=(p_{\pi}+p_{\ell})^{2}$.
Much of the discussion in Sec.5 applies also to semileptonic decays, however
there is an additional significant complication which arises due to the
presence of two particles in the final state to which the photon can couple.
This leads to additional non-exponential finite-volume effects, analogous to
those due to QCD re-scattering effects in nonleptonic $K\to\pi\pi$ decays
which are corrected by the Lellouch-Lüscher factor[44, 45]. Consider, for
example, the contribution to the $K_{\ell 3}$ decay amplitude illustrated in
the diagram of Fig. 4(a). In Minkowski space, this diagram contains an
imaginary part corresponding to the cut over the internal pion and lepton
propagators. In order to relate the physical amplitude to the results from a
computation on a finite Euclidean lattice, we imagine first performing the
$k_{0}$ integration. The imaginary part arises because the internal energy
with on-shell particles can be smaller than the external energy, i.e. $\Delta
E>0$ where

$\Delta
E\equiv\omega_{\pi}+\omega_{\ell}-(\omega_{\pi}^{\prime}+\omega_{\ell}^{\prime})\,,$
(37)

$\omega_{\pi}=\sqrt{\vec{p}_{\pi}^{\hskip
2.5pt2}+m_{\pi}^{2}},\,\omega_{\ell}=\sqrt{\vec{p}_{\ell}^{\hskip
2.5pt2}+m_{\ell}^{2}},\,\omega_{\pi}^{\prime}=\sqrt{\vec{p}_{\pi}^{\hskip
2.5pt\prime\,2}+m_{\pi}^{2}}$ and
$\omega_{\ell}^{\prime}=\sqrt{\vec{p}_{\ell}^{\hskip
2.5pt\prime\,2}+m_{\ell}^{2}}$ . The presence of the imaginary part manifests
itself by a term with a factor of $\frac{1}{\Delta E+i\epsilon}$ in the
integrand of the integration over $\vec{k}$. The singularity at $\Delta E=0$
is present in the region of integration and the corresponding
$\delta$-function leads to an imaginary contribution.

The presence of points with $\Delta E\geq 0$ in the integration region in
Minkowski space, presents a number of significant difficulties in the
evaluation of finite-volume Euclidean correlation functions.

1\. In lattice computations of the diagram in Fig. 4(a), the weak Hamiltonian
and interpolating operators which create the kaon and annihilate the pion and
lepton are inserted at fixed times. The correlation functions contain terms
which are proportional to
$e^{-(\omega_{\pi}^{\prime}+\omega_{\ell}^{\prime})\,t}$ where $t$ is the time
interval between the insertions of the weak Hamiltonian and the interpolating
operators which annihilate the pion and lepton. Energy is therefore not
conserved and the correlation functions are, as usual, dominated by the
intermediate states of lowest energy. If $\Delta E>0$ the dominant component
will provide matrix elements different from those contributing to the physical
decay amplitude which we wish to evaluate. These exponentially dominant, but
unphysical, contributions have therefore to be subtracted in order to obtain
the physical result. This is the issue raised in 1990 by Maiani and Testa in
the context of QCD final-state interactions[46].

2\. Assuming that after the subtraction the matrix element with the correct
energy can be extracted, the most significant theoretical issue is to
determine the non-exponential finite-volume corrections. The finite-volume
matrix element contains terms which take the schematic form

$\frac{1}{L^{3}}\sum_{\vec{k}}\mkern 0.75mu\raise
6.45831pt\hbox{\footnotesize$\prime$}~{}\frac{f(\vec{k})}{\Delta E}\,,$ (38)

where the prime on the summation indicates that in QEDL the term with
$\vec{k}=0$ is omitted and that possible other terms corresponding to $\Delta
E=0$ are also not included. The theoretical challenge is to relate the sum in
Eq.(38) to the real part of the corresponding infinite-volume integral:

$\mathrm{Re}\int\frac{d^{\hskip
1.5pt3}{k}}{(2\pi)^{3}}\,\frac{f(\vec{k})}{\Delta E+i\epsilon}\,.$ (39)

with controlled finite-volume corrections. We are currently working towards
this goal; here we simply note that the necessary subtractions require
knowledge of the pion’s electromagnetic form factor and the $K\to\pi$
transition form factors in QCD, both for a range of momentum transfers.

3\. We note that the $1/\Delta E$ singularity and related difficulties are
also present in the semileptonic decays of charged mesons, e.g.
$K^{+}\to\pi^{0}\ell^{+}\nu_{\ell}$ decays in which the final-state pion is
neutral. The photon still couples to the neutral pion, e.g. to its dipole
moment, so that diagrams such as that in Fig. 4(a) are also present in this
case and it remains to be seen whether the numerical effects are less severe.

4\. Finite-volume effects which decrease only as inverse powers of $L$, do not
only arise because of the presence of the $1/\Delta E$ factor discussed above.
Indeed we have seen in Sec. 5 that such effects are also present in the
computation of electromagnetic corrections to the spectrum[47, 4] and to
leptonic decay amplitudes (see for example the diagram in Fig. 4(b)), where
they arise due to terms in the summand which diverge sufficiently as
$|\vec{k}|\to 0$ (see the scaling law in Eq. (17)). However, the denominator
of each such term in the summand only vanishes at the single point
$|\vec{k}|=0$ and we have developed the techniques necessary to calculate the
corresponding power-law finite-volume corrections[16].

5\. From the above discussion it follows that the computation of semileptonic
decay rates is considerably simpler at the edge of phase space,
$(p_{\pi}+p_{\ell})^{2}=(m_{\pi}+m_{\ell})^{2}$, where the cuts leading to the
imaginary part of the amplitude are absent. In this case the finite-volume
effects which decrease only as inverse powers of $L$ still occur because the
denominators of terms in the summand vanish, but now only at the single point
$k\equiv|\vec{k}|=0$. This is a similar situation to the computation of
electromagnetic corrections to the spectrum[47, 4] and to leptonic decay
amplitudes (see for example the diagram in Fig. 4(b)) and we have the
techniques to compute the universal finite-volume corrections. In Sec. 5 we
explained that the two leading FV corrections to the spectrum and to leptonic
decay amplitudes are universal, independent of the structure of the mesons. In
the case of leptonic decays they only require the knowledge of the decay
constant of the pseudoscalar meson, $f_{P}$, computed in QCD. For semileptonic
decays there is also a universality in the leading two terms, but the
coefficient of the $1/(m_{P}L)$ corrections requires knowledge of the
derivative of the form factors, $\partial f_{\pm}(q^{2})/\partial q^{2}$. The
reason for this can be understood as follows. The most singular summand in the
sum over $\vec{k}$ is proportional to $1/k^{3}$ and leads to the infrared
divergent terms proportional to $\log[m_{P}L]$. The terms proportional to
$1/k^{2}$, which lead to corrections of $O(1/(m_{P}L))$, therefore require the
leading $\vec{k}$-dependent term in the form-factors which is proportional to
the derivative.

In summary, the techniques developed to include electromagnetic corrections to
leptonic decays of pseudoscalar mesons can also be applied to semileptonic
decays. At the edge of phase-space, for example for $K_{\ell 3}$ decays for
$s_{\pi\ell}\equiv(p_{\pi}+p_{\ell})^{2}=(m_{\pi}+m_{\ell})^{2}$, so that the
only singular term in the summand is at $|\vec{k}|=0$, these techniques can be
applied directly. The cancelation of infrared divergences occurs as for
leptonic decays, the $O(1/(m_{K}L))$ finite-volume corrections are also
”universal” but the coefficients depend on the derivatives of the form-
factors, $\partial f_{\pm}/\partial q^{2}$, which are physical quantities,
computed in QCD. For $s_{\pi\ell}>(m_{\pi}+m_{\ell})^{2}$, the physical
(Minkowski) amplitude has an imaginary part which corresponds to a $1/\Delta
E$ singularity and poles away from $|\vec{k}|=0$, requiring knowledge of the
electromagnetic form-factor of the pion and $f_{\pm}$ for a range of values of
momentum transfer. We are currently investigating the optimal way to implement
the necessary subtractions. For illustration in Fig. 5 we exhibit the physical
phase space for $K^{0}\to\pi^{-}\mu^{+}\bar{\nu}_{\mu}$ decays and the point
with $s_{\pi\ell}=(m_{\pi^{-}}+m_{\ell^{+}})^{2}$.

Figure 5: The physical region for $K_{\mu 3}$ decays. The quantities
$s_{\pi\mu}$ and $q^{2}$ are given in GeV2. The black circle represents the
point at the minimum value of $s_{\pi\mu}$ .

##  7 Summary and Conclusions

The remarkable recent improvement in the precision of lattice QCD results for
many quantities relevant for flavour physics has necessitated the inclusion of
isospin breaking effects, and electromagnetic corrections in particular, into
the computations. The presence of a zero-mass photon leads to significant
long-distance issues, including infrared divergences and finite-volume effects
which decrease only as inverse powers of $L$ and not exponentially. In this
paper I have reviewed the issues and the status of the framework which,
together with colleagues from Rome, we have been developing and implementing
in leptonic and semileptonic decays of pseudoscalar mesons.

As explained in Sec. 5, for leptonic decays
$P\to\ell\bar{\nu}_{\ell}(\gamma)$, the framework is complete and has been
successfully implemented for $\pi_{\mu 2}$ and $K_{\mu 2}$ decays. We are able
to handle the cancelation of infrared divergences and the subtraction of the
universal finite-volume corrections which are of $O(1/(m_{P}L))$. We have
demonstrated that after this subtraction, the expectation that the leading
residual (structure-dependent) finite-volume corrections, are of
$O(1/(m_{P}L)^{2})$ is satisfied numerically. We have been able to determine
$V_{us}$ with excellent precision (see Eqs. (31) and (34)) and to verify the
unitarity of the first row of the CKM matrix to better than per-mille accuracy
(see Eq. (35)). The most recent development has been the calculation of the
radiative decays $P\to\ell\bar{\nu}_{\ell}\gamma$ for light and charmed
mesons, and a phenomenological comparison of our results with those from
experimental measurements[30, 32].

For semileptonic decays $P_{1}\to P_{2}\ell\bar{\nu}_{\ell}(\gamma)$, where
$P_{1}$ and $P_{2}$ are pseudoscalar mesons, there are additional non-
exponential finite-volume effects associated with diagrams such as that in
Fig. 4(a) which in Minkowski space contain an imaginary part. As explained in
Sec. 6 the subtraction of these additional finite-volume effects requires
knowledge of the electromagnetic form factor of the $P_{2}$ meson and the weak
$P_{1}\to P_{2}$ transition form factors, both for a range of momentum
transfer and studies of how best to perform this subtraction are currently in
progress. This difficulty is generic and relevant for most decay processes;
leptonic decays are a rare exception. The techniques developed for leptonic
decays can however, be directly applied to semileptonic decays at the edge of
phase space where the invariant mass of the $P_{2}$ \- $\ell$ pair is
$m_{P_{2}}+m_{\ell}$.

### Personal Note

It has been an honour and pleasure to have been invited to make this
contribution to the volume celebrating 60 years of the Kraków Schools in
Theoretical Physics. I have very fond recollections, both scientific and
personal, of the four previous times I have lectured at the school: 1977
(_Asymptotic Freedom and Deep Inelastic Electroproduction_), 1991 (_Heavy
Quark Physics from Lattice QCD_), 2006 (_Lattice Flavour Dynamics_) and 2014
(_Flavour Physics_).

I warmly congratulate all the organisers, from Professor Andrzej Białas who
organised the first School through to Professor Michał Praszałowicz who has
organised this 60th one, for creating and maintaining such an important and
high-quality forum for the presentation and discussion of the latest
developments in theoretical physics. On this anniversary, I wish the School
the traditional Polish _Sto Lat_ (a hundred years).

### Acknowledgements

I warmly thank my collaborators from the Universities of Rome _La Sapienza_ ,
_Tor Vergata_ and _Roma Tre_ with whom the ideas discussed in this paper were
developed and implemented. I was partially supported by an Emeritus Fellowship
from the Leverhulme Trust and by STFC (UK) grants ST/P000711/1 and
ST/T000775/1.

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