HPQCD collaboration

# Standard Model predictions for $B\to K\ell^{+}\ell^{-}$, $B\to
K\ell_{1}^{-}\ell_{2}^{+}$ and $B\to K\nu\bar{\nu}$ using form factors from
$N_{f}=2+1+1$ lattice QCD

W. G. Parrott<EMAIL_ADDRESS>SUPA, School of Physics and
Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK  C. Bouchard
<EMAIL_ADDRESS>SUPA, School of Physics and Astronomy, University
of Glasgow, Glasgow, G12 8QQ, UK  C. T. H. Davies
<EMAIL_ADDRESS>SUPA, School of Physics and Astronomy,
University of Glasgow, Glasgow, G12 8QQ, UK http://www.physics.gla.ac.uk/HPQCD

###### Abstract

We use HPQCD’s recent lattice QCD determination of $B\to K$ scalar, vector and
tensor form factors to determine Standard Model differential branching
fractions for $B\to K\ell^{+}\ell^{-}$, $B\to K\ell_{1}^{+}\ell_{2}^{-}$ and
$B\to K\nu\overline{\nu}$. These form factors are calculated across the full
$q^{2}$ range of the decay and have smaller uncertainties than previous work,
particularly at low $q^{2}$. For $B\to K\ell^{+}\ell^{-}$ we find the Standard
Model branching fraction in the $q^{2}$ region below the squared $J/\psi$ mass
to exceed the LHCb results, with tensions as high as $4.2\sigma$ for $B^{+}\to
K^{+}\mu^{+}\mu^{-}$. For the high $q^{2}$ region we see $2.7\sigma$ tensions.
The tensions are much reduced by applying shifts to Wilson coefficients
$C_{9}$ and $C_{10}$ in the effective weak Hamiltonian, moving them away from
their Standard Model values consistent with those indicated by other $B$
phenomenology. We also update results for lepton-flavour ratios $R^{\mu}_{e}$
and $R^{\tau}_{\mu}$ and the ‘flat term’, $F_{H}^{\ell}$ in the differential
branching fraction for $\ell\in\\{e,\mu,\tau\\}$. Our results for the form-
factor-dependent contributions needed for searches for lepton-flavour-
violating decays $B\to K\ell^{-}_{1}\ell^{+}_{2}$ achieve uncertainties of 7%.
We also compute the branching fraction $\mathcal{B}(B\to K\nu\bar{\nu})$ with
an uncertainty below 10%, for comparison with future experimental results.

##  I Introduction

The weak $B\to K$ transition involves a $b\to s$ flavour changing neutral
current (FCNC) which proceeds at loop level in the Standard Model (SM), where
contributions are suppressed by loop factors and associated elements of the
Cabbibo-Kobayashi-Maskawa (CKM) matrix [1, 2]. This makes FCNC processes
promising places to look for effects from new physics.

Tests for new physics require the accurate calculation in the SM of quantities
that can be compared to experimental results. Because the weak transition
occurs between hadronic bound states of the $b$ and $s$ quarks, key inputs to
the SM calculation are the hadronic matrix elements of the weak currents,
parameterised by form factors. The form factors are functions of the
4-momentum transfer, $q$, between $B$ and $K$ and depend on the strong
interaction effects that bind the quarks inside the mesons. To calculate the
form factors from first principles in quantum chromodynamics (QCD) requires
the use of lattice QCD. Here we will update the phenomenology for
$B\rightarrow K$ decays in the SM using improved form factors calculated using
lattice QCD by the HPQCD collaboration [3] and compare to existing
experimental results. We also provide tables of results in a variety of
$q^{2}$ bins that will be useful for comparison to future experimental
analyses.

The lattice QCD form factors that we use [3] improve on previous calculations
in a number of ways. They are the first to be calculated on gluon field
configurations (generated by the MILC collaboration [4]) that include $u$,
$d$, $s$ and $c$ quarks in the sea ($N_{f}=2+1+1$). They use a fully
relativistic discretisation of the QCD Dirac equation on the lattice [5] which
allows accurate normalisation of the lattice weak currents to their continuum
counterparts [6, 7], rather than the $\mathcal{O}(\alpha_{s})$ perturbative
normalisation used in earlier lattice QCD calculations [8, 9]. The calculation
must be done at a range of masses for the heavy quark, from that of the $c$
quark upwards, and at a range of values of the lattice spacing. Very fine
lattices must be used to get close to the $b$ quark mass for the heavy quark.
Fits to the results then enable form factors for $B\to K$ to be determined and
smoothly connected (since only the heavy quark mass changes) to those for
$D\to K$. Our form factors for $D\to K$ agree well as a function of $q^{2}$
with those inferred from the experimental results for the tree-level weak
decay $D\to K\ell\overline{\nu}$ [10], also giving a better-than-1%
determination of $V_{cs}$. We do not expect to see new physics in this decay,
so this is a strong test of the agreement of (lattice) QCD with experiment for
a SM weak process. Our $B\to K$ form factors build on this. They are also the
first to cover the full physical $q^{2}$ range of the $B\to K$ decay; this is
possible because of the use of very fine lattices. This means in particular
that our form factors have smaller uncertainties in the low $q^{2}$ region,
allowing more accurate SM phenomenology here than has been possible before.

The $B\to K$ process studied most to date is $B\to K\ell^{+}\ell^{-}$, for
lepton $\ell=e$ or $\mu$. $B\to K\ell^{+}\ell^{-}$ experimental data has been
collected by BaBar [11, 12, 13], Belle [14, 15], CDF [16] and LHCb [17, 18,
19, 20, 21, 22, 23]. We hope to see more experimental data for $\tau$ leptons
in the final state in future.

The most direct comparison between the SM and experiment is for the
differential branching fraction as a function of $q^{2}$. A complication is
that of contamination by long-distance contributions from charmonium
resonances in the intermediate $q^{2}$ region [24]. These are not easily
included in the SM analysis (and will not be included here) and so the key
comparisons to be made are at low $q^{2}$, below the resonance region, or at
high $q^{2}$, above the resonance region. Tensions between the SM and
experimental results have been seen in earlier work [25, 26, 27, 28, 8, 9].
Here we will provide an improved analysis with smaller theory uncertainties
from the form factors, particularly in the low $q^{2}$ region. We will also
update SM results for the lepton-universality violating ratios, $R^{\mu}_{e}$
and $R^{\tau}_{\mu}$ (although form factor effects largely cancel in these
ratios) and give results for the ‘flat term’, $F^{\ell}_{H}$.

The lepton-number-violating decay $B\to K\ell_{1}^{-}\ell_{2}^{+}$ is of
interest for new physics constraints and upper limits on this decay have been
obtained by experiment [29, 30, 15, 31]. Here we provide the hadronic input
needed for future analyses of this decay.

The $B\to K\nu\bar{\nu}$ decay promises to be a useful mode for study in
future, because it is free of charmonium resonance contamination. There is a
small amount of experimental data on this from Belle [32], Belle-II [33] and
BaBar [34], with the promise of more to come [35]. Here we provide the SM
predictions for the differential and total branching fractions for this decay
mode with an improved level of accuracy compared to previous calculations.

As well as comparing to experimental results, where available, we will also
compare to earlier theoretical calculations [25, 26, 27, 36, 28, 37, 8, 9].
These use a variety of form factors from light-cone sum-rules [38, 39],
lattice QCD (with $N_{f}=2+1$ sea quarks) [40, 41] and a combination of the
two, also including dispersive bounds [42].

The rest of the paper is organised as follows. Most of the paper is dedicated
to the well-studied $B\to K\ell^{-}\ell^{+}$ decays. We begin with these
decays in Section II, which outlines the calculation of SM observables in
Section II.1 (including discussions of how we deal with resonances, and the
effects from isospin, QED and lepton flavour), comparisons with experiment in
Section II.2 and comparisons with previous theoretical results in Section
II.3. Section III looks at the lepton flavour number violating $B\to
K\ell^{-}_{1}\ell^{+}_{2}$ decay, calculating several parameters relevant to
this channel. In Section IV we calculate the branching fraction for the $B\to
K\nu\bar{\nu}$ decay, comparing with experimental bounds and other theoretical
work. We conclude in Section V.

This version of our paper contains a correction of the scale used for
$\alpha_{\rm EW}$, compared to the published version. Instead of
$1/\alpha_{\rm EW}(M_{Z})=127.952(9)$ [43], we take $1/\alpha_{\rm
EW}(4.2~{}\mathrm{GeV})=132.32(5)$, in line with the findings of [44]. This
change affects our branching fractions for $B\to K\ell^{+}\ell^{-}$ and our
results for $B\to K\ell_{1}^{+}\ell_{2}^{-}$, by an overall multiplicative
factor of $\alpha^{2}_{\rm EW}(4.2~{}\mathrm{GeV})/\alpha^{2}_{\rm
EW}(M_{Z})\approx 0.94$. For most of our results, this leads to a change of
approximately $1\sigma$, which does not significantly impact our conclusions.
All affected figures and tables have been updated accordingly, and will appear
as an erratum.

There is no effect on ratios of branching fractions such as $R^{\mu}_{e}$ and
$F_{H}$, nor on $B\to K\nu\bar{\nu}$, as this uses $\alpha_{\rm EW}(M_{Z})$
[45].

Additionally, we have removed a factor of $\eta_{\mathrm{EW}}=1.007(2)$, which
was incorrectly applied to $G_{\mathrm{F}}$ in the original paper. This
results in a modest reduction of 1.4% to all branching fractions.

##  II $B\to K\ell^{+}\ell^{-}$

The majority of theoretical and experimental work on $B\to K$ weak decays has
focused on $B\to K\ell^{+}\ell^{-}$. An example of a Feynman diagram for this
process, sometimes called a penguin diagram, is shown in Figure 1. The
analysis of such decays in the SM starts from the effective weak Hamiltonian
constructed from 4-fermion operators multiplied by Wilson coefficients
($C_{1}$–$C_{10}$), with the matrix elements of the 4-fermion operators
expressed in terms of form factors [46].

Figure 1: An example of a Feynman diagram for the process $B^{+}\to
K^{+}\ell^{+}\ell^{-}$.

This section discusses the calculation of SM observables related to these
decays in Sec. II.1, reviewing the inputs used in the evaluation of
observables in Sec. II.1.1 and focusing on the Wilson coefficients in Sec.
II.1.2. Section II.1.3 then discusses how we handle resonance contributions in
a way that allows comparison to experiment. Sections II.1.4 and II.1.5 then
discuss the uncertainties to account for the fact that our form factors are
determined in a pure QCD calculation on the lattice in the strong-isospin
limit ($m_{u}=m_{d}$). The discussion of uncertainties there is also relevant
to our input form factors for Sections III and IV. Section II.2 gives our main
results for SM observables and their comparison to experiment. Section II.3
closes the section with a comparison to earlier theoretical results.

###  II.1 Calculating SM observables

The SM differential decay rate for $B\to K\ell^{+}\ell^{-}$ for lepton $\ell$
is constructed as follows, where we use the notation in [40, 47].

$\frac{d\Gamma_{\ell}}{dq^{2}}=2a_{\ell}+\frac{2}{3}c_{\ell},$ (1)

where $a_{\ell}$ and $c_{\ell}$ are given by

$\begin{split}a_{\ell}&=\mathcal{C}\Big{[}q^{2}|F_{P}|^{2}+\frac{\lambda(q,M_{B},M_{K})}{4}(|F_{A}|^{2}+|F_{V}|^{2})\\\
&+4m_{\ell}^{2}M_{B}^{2}|F_{A}|^{2}+2m_{\ell}(M_{B}^{2}-M_{K}^{2}+q^{2}){\rm
Re}(F_{P}F_{A}^{*})\Big{]},\\\
c_{\ell}&=-\frac{\mathcal{C}\lambda(q,M_{B},M_{K})\beta_{\ell}^{2}}{4}(|F_{A}|^{2}+|F_{V}|^{2}),\end{split}$
(2)

with

$\displaystyle\mathcal{C}$
$\displaystyle=\frac{G_{F}^{2}\alpha_{EW}^{2}|V_{tb}V_{ts}^{*}|^{2}}{2^{9}\pi^{5}M_{B}^{3}}\beta_{\ell}\sqrt{\lambda(q,M_{B},M_{K})},$
$\displaystyle\beta_{\ell}$ $\displaystyle=\sqrt{1-4m_{\ell}^{2}/q^{2}},$ (3)
$\displaystyle\lambda(a,b,c)$
$\displaystyle=a^{4}+b^{4}+c^{4}-2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2}).$

$F_{P,V,A}$ are constructed from the scalar, vector and tensor form factors
$f_{0}$, $f_{+}$ and $f_{T}$ respectively, by

$\begin{split}F_{P}&=-m_{\ell}C_{10}\Big{[}f_{+}-\frac{M_{B}^{2}-M_{K}^{2}}{q^{2}}(f_{0}-f_{+})\Big{]},\\\
F_{V}&=C_{9}^{\rm
eff,1}f_{+}+\frac{2m_{b}^{\overline{\mathrm{MS}}}(\mu_{b})}{M_{B}+M_{K}}C_{7}^{\rm
eff,1}f_{T}(\mu_{b}),\\\ F_{A}&=C_{10}f_{+}\,.\end{split}$ (4)

The Wilson coefficient $C_{9}^{\rm eff,1}=C_{9}^{\rm eff,0}+\Delta C_{9}^{\rm
eff}+\delta C_{9}^{\rm eff}$ includes nonfactorisable corrections in $\Delta
C_{9}^{\rm eff}$, as well as $\mathcal{O}(\alpha_{s})$ and more heavily
suppressed corrections in $\delta C_{9}^{\rm eff}$. Similarly, $C_{7}^{\rm
eff,1}=C_{7}^{\rm eff,0}+\delta C_{7}^{\rm eff}$ contains
$\mathcal{O}(\alpha_{s})$ corrections in $\delta C_{7}^{\rm eff}$. These
corrections are discussed in detail in Appendix B.

$C_{9}^{\rm eff,0}$ ($\equiv C_{9}(\mu_{b})+Y(q^{2})$) is a function of
$q^{2}$ through

$\begin{split}Y(q^{2})&=\frac{4}{3}C_{3}+\frac{64}{9}C_{5}+\frac{64}{27}C_{6}\\\
&-\frac{1}{2}h(q^{2},0)\left(C_{3}+\frac{4}{3}C_{4}+16C_{5}+\frac{64}{3}C_{6}\right)\\\
&+h(q^{2},m_{c})\left(\frac{4}{3}C_{1}+C_{2}+6C_{3}+60C_{5}\right)\\\
&-\frac{1}{2}h(q^{2},m_{b})\left(7C_{3}+\frac{4}{3}C_{4}+76C_{5}+\frac{64}{3}C_{6}\right),\end{split}$
(5)

where

$\begin{split}h(q^{2},m)&=-\frac{4}{9}\left(\ln\frac{m^{2}}{\mu^{2}}-\frac{2}{3}-x\right)-\frac{4}{9}(2+x)\\\
&\times\begin{cases}\sqrt{x-1}\arctan\frac{1}{\sqrt{x-1}}&\\!\\!,x>1\vspace{0.02in}\\\
\sqrt{1-x}\left(\ln\frac{1+\sqrt{1-x}}{\sqrt{x}}-\frac{i\pi}{2}\right)&\\!\\!,x\leq
1,\end{cases}\end{split}$ (6)

with $x=4m^{2}/q^{2}$.

####  II.1.1 Inputs

Parameter | Value | Reference
---|---|---
$G_{F}$ |  $1.1663787(6)\times 10^{-5}$ $\text{GeV}^{-2}$ | [43]
$m_{c}^{\overline{\mathrm{MS}}}(m_{c}^{\overline{\mathrm{MS}}})$ | 1.2719(78) GeV | See caption
$m_{b}^{\overline{\mathrm{MS}}}(\mu_{b})$ | 4.209(21) GeV | [48]
$m_{c}$ | 1.68(20) GeV | -
$m_{b}$ | 4.87(20) GeV | -
$f_{K^{+}}$ | 0.1557(3) GeV | [49, 50, 51, 52]
$f_{B^{+}}$ | 0.1894(14) GeV | [53]
$\tau_{B^{0}}$ | 1.519(4) ps | [54]
$\tau_{B^{\pm}}$ | 1.638(4) ps | [54]
$1/\alpha_{\rm EW}(\mu_{b})$ | 132.32(5) | -
$|V_{tb}V_{ts}^{*}|$ | 0.04185(93) | [55]
$C_{1}(\mu_{b})$ | -0.294(9) | [56]
$C_{2}(\mu_{b})$ | 1.017(1) | [56]
$C_{3}(\mu_{b})$ | -0.0059(2) | [56]
$C_{4}(\mu_{b})$ | -0.087(1) | [56]
$C_{5}(\mu_{b})$ | 0.0004 | [56]
$C_{6}(\mu_{b})$ | 0.0011(1) | [56]
$C_{7}^{\rm eff,0}(\mu_{b})$ | -0.2957(5) | [56]
$C_{8}^{\mathrm{eff}}(\mu_{b})$ | -0.1630(6) | [56]
$C_{9}(\mu_{b})$ | $4.114(14)$ | [56]
$C_{9}^{\rm eff,0}(\mu_{b})$ | $C_{9}(\mu_{b})+Y(q^{2})$ | -
$C_{10}(\mu_{b})$ | -4.193(33) | [56]
Table 1: Input parameters used to calculate SM observables. Details of the
error on the Wilson coefficients are given in [56], and these are quoted at
$\mu_{b}=4.2\,\mathrm{GeV}$. For more details on Wilson coefficients and their
uncertainties, see Section II.1.2.
$m_{c}^{\overline{\mathrm{MS}}}(m_{c}^{\overline{\mathrm{MS}}})$ is obtained
using $m_{c}^{\overline{\mathrm{MS}}}(3~{}\mathrm{GeV})=0.9841(51)$ [57] and
running the scale to its own mass. $m_{c}$ and $m_{b}$ are the $c$ and $b$
quark pole masses obtained from the masses in the $\overline{\mathrm{MS}}$
scheme at three loops (see Appendix A for details).

Key inputs to our determination of the differential rate are our recently
calculated scalar, vector and tensor form factors [3]. These are plotted as a
function of $q^{2}$ in Fig. 2. The plot shows the $\pm 1\sigma$ error bands,
smaller than those for previous results, especially in the low $q^{2}$ region.
See [3] for details on how these form factors were calculated. Note that we
run $f_{T}$ (calculated in [3] at scale 4.8 GeV) to a scale $\mu_{b}=4.2$ GeV,
to match the scale used in the Wilson coefficients of Table 1. This is
achieved by following [7] and multiplying our form factor result
$f_{T}(\mu_{b}=4.8\,\mathrm{GeV})$ by a factor of $1.00933(16)$. The value for
$\alpha_{\rm EW}(\mu_{b}=4.2\,\mathrm{GeV})$ is run from $1/\alpha_{\rm
EW}(4.8\,\mathrm{GeV})=132.14(4)$ [58].

Figure 2: Form factors used in this analysis, from [3].

Further external input parameters needed for Eq. (1) are provided in Table 1.
Except in the case of Equation (4), all expressions above (and in Appendix B)
use the pole masses $m_{c}$ and $m_{b}$ given in Table 1. These pole masses
are derived from the masses given in the same table in the
$\overline{\mathrm{MS}}$ scheme, determined to high accuracy using lattice QCD
calculations [57, 48]. We convert to the pole mass using the standard
perturbative matching factor at three-loop order [59, 60]. For more details,
see Appendix A. The perturbation series in this expression suffers from the
presence of a renormalon in the pole mass. To account for renormalon-induced
uncertainties [61] in the pole mass, we allow an error of 200 MeV which dwarfs
any other uncertainty in the mass. We assess the impact of this uncertainty by
shifting the central values of $m_{c}$ and $m_{b}$ down by 200 MeV. In the
case of $m_{b}$, we find almost no effect on the branching fractions in the
important $1.1\leq q^{2}/\mathrm{GeV}^{2}\leq 6$ and $15\leq
q^{2}/\mathrm{GeV}^{2}\leq 22$ regions. In the case of $m_{c}$, a maximal
effect of $0.3\,\sigma$ is seen.

To obtain the branching fraction $\mathcal{B}$, we note that
$\mathcal{B}_{\ell}=\Gamma_{\ell}\tau_{B}$, where $\tau_{B}$ is the lifetime
of the $B$ meson. The values for $\tau$ used are given in Table 1.

The value of $|V_{tb}V^{*}_{ts}|$ that we use comes from the recent HPQCD
lattice QCD calculation [55] of the mixing matrix elements for the $B_{s}$
meson, the first to be done with $N_{f}=2+1+1$ quarks in the sea.
$|V_{tb}V_{ts}^{*}|$ is obtained by combining the lattice QCD result with the
experimentally-determined $B_{s}$ oscillation rate [43]. The oscillation rate
can be determined to high precision, so the uncertainty in
$|V_{tb}V_{ts}^{*}|$ is dominated by that from the lattice QCD result, where
the uncertainty is smaller than that for previous lattice QCD calculations.
The value of $|V_{ts}|$ quoted in [55] used $|V_{tb}|=0.999093$ from [62]. The
value of $|V_{ts}|$ that we use is about 1$\sigma$ higher than values based on
CKM unitarity that do not use the information from the $B_{s}$ oscillation
rate, for example that in [63].

####  II.1.2 Wilson coefficients

The Wilson coefficients we use are from Table I of [56]. They are quoted at
$\mu_{b}=4.2~{}\mathrm{GeV}$ with uncertainties given for higher order
corrections, as well as for a variation (doubling and halving) of the matching
scale at the top quark mass, $m_{t}$. These uncertainties do not allow for
possible scale dependence in $\mu_{b}$, however. Whilst our differential decay
rates should be scale independent, we test this over a reasonable variation of
$\mu_{b}$ and include an uncertainty to account for any variation. We look at
the effect of changing $\mu_{b}$ to $4.8~{}\mathrm{GeV}$ on branching
fractions (for both meson charges and $\ell\in\\{e,\mu\\}$) in the key ranges
$1.1~{}\mathrm{GeV}^{2}\leq q^{2}\leq 6~{}\mathrm{GeV}^{2}$ and
$15~{}\mathrm{GeV}^{2}\leq q^{2}\leq 22~{}\mathrm{GeV}^{2}$. This involves the
following changes:

1) Removing the running of $f_{T}(\mu_{b})$ described above, so we have
$f_{T}(\mu_{b}=4.8~{}\mathrm{GeV})$.

2) Running the $m_{b}^{\overline{\mathrm{MS}}}(\mu_{b})$ given in Table 1 to
$m_{b}^{\overline{\mathrm{MS}}}(\mu_{b}=4.8~{}\mathrm{GeV})=4.108(24)~{}\mathrm{GeV}$.

3) Shifting the central values of the Wilson coefficients to the
$4.8~{}\mathrm{GeV}$ values given Table I of [56].

4) Setting $1/\alpha_{\rm EW}(4.8\,\mathrm{GeV})=132.14(4)$ [58].

After making these changes, we observe a change of the same absolute size and
sign in the branching fractions for both $q^{2}$ regions. This corresponds to
a change of between $1.4\%$ and $2.6\%$ on the branching fraction, a maximum
of $0.3\sigma$. Though a modest effect, we account for this by including a
$2.5\%$ uncertainty on $a_{\ell}$ and $c_{\ell}$ across the full $q^{2}$ range
to allow for reasonable variations in $\mu_{b}$.

####  II.1.3 Resonances and vetoed regions

From Fig. 1 it is easy to see how a resonance that couples to the photon can
affect the differential rate for $B\to K\ell^{+}\ell^{-}$, causing a spike at
$q^{2}=M^{2}$ for a resonance of mass $M$. These ‘long-distance’ resonance
effects are not included in the ‘short-distance’ SM calculation of Eqs.
(1)–(6). The resonances of concern are $u\overline{u}$ and $d\overline{d}$
resonances (the $\rho$ and $\omega$), which appear at very low $q^{2}$, below
$1~{}\mathrm{GeV}^{2}$, the $s\overline{s}$ $\phi$ close to
$1~{}\mathrm{GeV}^{2}$ and $c\overline{c}$ resonances, which affect the
intermediate $q^{2}$ region. The $c\bar{c}$ resonances that dominate are the
$J/\psi$ and $\psi(2S)$ and experimentalists typically discard their data in
the vicinity of these resonances for a $B\to K\ell^{+}\ell^{-}$ analysis.
Whilst the regions blocked out vary somewhat from experiment to experiment, we
will use the choices $8.68\leq q^{2}/\text{GeV}^{2}\leq 10.11$ and $12.86\leq
q^{2}/\text{GeV}^{2}\leq 14.18$, which are adopted by several experiments, and
mark these as ‘vetoed regions’ (greyed out in our plots). In these two regions
and between them, the resonance contributions are large [22] and there are
strong violations of quark hadron duality [24]. The decay rate is dominated by
$B\to K\Psi\to K\ell^{+}\ell^{-}$ [64] and is approximately two orders of
magnitude larger than the short-distance rate of Eqs. (1)–(6).

To avoid these resonances in the comparison between the SM short-distance
calculations and experiment it is common practice to restrict the $q^{2}$
range to two ‘well-behaved’ regions, giving separate results for the integrals
over $1.1\leq q^{2}/\text{GeV}^{2}\leq 6$ and $15\leq q^{2}/\text{GeV}^{2}\leq
22$ (or similar). The lower $q^{2}$ region here is above the light-quark
resonances but below the $c\overline{c}$ resonances. The higher $q^{2}$ region
is above the dominant $c\overline{c}$ resonances, but small contributions are
seen in the experimental data from higher resonances in this region [22].
Following [9], we allow for an uncertainty of 2% on the differential rate in
this high $q^{2}$ region to allow for these additional resonance
contributions. We do this by applying a factor of $1.00(2)$ to $a_{\ell}$ and
$c_{\ell}$ (Eq. (1)) above $q^{2}=14.18~{}\mathrm{GeV}^{2}$. We will adopt
these two $q^{2}$ regions here and our key results will be a comparison with
experiment for these two regions. This allows us to test for the presence of
new physics in as clean a way as possible.

It is also informative, however, to quote the total branching fraction for the
decay, integrating over the full $q^{2}$ range. This means that some approach
must be adopted for the $q^{2}$ regions where the resonances sit so that a
comparison can be made between the short-distance SM branching fraction and
experiment. In the very low $q^{2}$ region this is not an issue; the light-
quark resonances contribute only at the 1% level [22]. The narrow $\phi$
resonance can be cut out of the experimental data, where the data is
sufficiently accurate to warrant it [19]. The $c\overline{c}$ resonances in
the intermediate $q^{2}$ region have a bigger impact over a larger $q^{2}$
range. The integration over this region must then be handled carefully in a
way that is consistent with what is done in experimental analyses. To cover
the $c\overline{c}$ resonance region, we linearly interpolate $a_{\ell}$ and
$c_{\ell}$ across the whole of the two vetoed regions and the gap between
them, i.e. for $8.68\leq q^{2}/\text{GeV}^{2}\leq 14.18$. This is only
necessary for the case $\ell\in\\{e,\mu\\}$, as $\ell=\tau$ sits mostly above
this region. This approach is used in [9] and similar interpolations are
performed in experimental papers (see [19], the interpolation of which will be
compared with ours below). This means that the total branching fraction is
then also comparable between theory and experiment. Of course, comparisons
with other theoretical calculations that are based on Eqs. (1)–(6) can be done
at any $q^{2}$ value.

Finally, it should be noted that the $m_{c}$ value we use (Table 1) is such
that $4m_{c}^{2}=11.34\,\mathrm{GeV}^{2}$. This sits between the two vetoed
regions and therefore inside the region in which we perform the linear
interpolation described. This means that the cusp arising at this point from
the function $h$ of Eq. (6) is not visible in our plots.

####  II.1.4 Isospin and lepton flavour

The $B\to K$ decays we wish to study are for $B^{0}\to K^{0}\ell^{+}\ell^{-}$
and $B^{+}\to K^{+}\ell^{+}\ell^{-}$. The charged and neutral $B$ meson cases
differ only in the flavour of the light spectator quark, which can either be
an up or a down. The masses of the $B^{0/+}$ and $K^{0/+}$ differ slightly as
a result of this (along with QED effects). We take these masses from the
Particle Data Tables [43] and use these in the equations of Sec. II.1 for each
case; it has negligible effect on the differential rates. The difference in
branching fractions between the charged and neutral cases is much larger,
however, because of the difference in $B^{+}$ and $B^{0}$ lifetimes, which we
take from [54] and give in Table 1.

We must also include an uncertainty for the fact that our form factors [3]
were calculated in pure lattice QCD, with no QED effects included and with
degenerate $u$ and $d$ quark masses (as were previous lattice QCD calculations
of these form factors [40, 41]). Instead of having separate values for $m_{u}$
and $m_{d}$ we used $m_{u}=m_{d}=m_{l}$, where $m_{l}$ is denoted ‘the light
quark mass’. For this reason, our form factors represent the case with
$M_{B}=(M_{B^{0}}+M_{B^{+}})/2$ and $M_{K}=(M_{K^{0}}+M_{K^{+}})/2$ with the
light quark mass tuned to the physical point $m_{s}/m_{l}=27.18(10)$
corresponding to $m_{l}=(m_{u}+m_{d})/2$. In order to work out how much our
form factors would change if the $u$ and $d$ quark masses were different we
follow the procedure adopted in HPQCD’s $D\to K$ analysis [10].

We can repeat our analysis to fit the $B\to K$ form factors from [3] but
changing the physical value of $m_{l}$ so that it corresponds to either
$m_{u}$ or $m_{d}$ (this means taking $m_{s}/m_{l}=27.18\times 3/2$ or
$27.18\times 3/4$ respectively). Note that this will give an overestimate of
the effect because this change refers to all the light quarks in the
simulation, whereas in fact we just want to change the valence quark in the
$B$ and $K$. At the same time as making this change to $m_{l}$ we also change
the meson masses that appear in the fit function to their appropriate values
corresponding to a $u$ or $d$ spectator quark. For details on the fit function
for the form factors see [3].

Looking at the effects of these tests on the $f_{0}$, $f_{+}$ and $f_{T}$ form
factors across the $q^{2}$ range, we find a maximal effect of $\approx 0.5\%$
[3]. To allow for the full effect as an uncertainty on our form factors for
the analysis here, we apply an uncorrelated multiplicative factor of
$1.000(5)$ to each of our form factors. We include this uncertainty in our
form factors for all of the calculations discussed here, both in this Section
on $B\to K\ell^{+}\ell^{-}$ and Sections III and IV.

The lepton masses, $m_{e}$, $m_{\mu}$ and $m_{\tau}$ as appropriate, also
appear in the equations to determine the differential rate (see Eqs. (2) and
(II.1)). One of the notable changes between leptons is in the minimum $q^{2}$
value, since $q_{\text{min}}^{2}=4m_{\ell}^{2}$. In practice, in the case of
the differential decay rate, our final result is almost identical for $\ell=e$
and $\ell=\mu$ because their masses are both so small. We sometimes treat
these cases together, simply using the $\ell=\mu$ case where a lepton is not
explicitly stated. Larger differences are seen for the $\ell=\tau$ case and we
show results for that case in Sec. II.3 where we compare to other theory
calculations.

####  II.1.5 QED effects

The largest QED effects in the decay rate for $B\to K\ell^{+}\ell^{-}$ are
expected to be from final-state interactions of the charged leptons. The
photon radiation produced can cause several issues within the experimental
analysis, particularly in the hadronic environment of the LHC. These are
handled using the PHOTOS software [65] to correct the differential rate to a
fully photon-inclusive rate that can be compared with theory. It is also used
in the LHC context to obtain a value for $q^{2}$. Theoretical tests [66, 67]
indicate that this is a robust procedure.

We should allow an uncertainty for QED effects in our comparison with
experiment, however, and we will base it on the size of QED final-state
radiative corrections [66, 67]. This will then comfortably include the
uncertainty from other QED effects missing from our calculation, such as that
from the fact that the quarks inside the $B$ and $K$ mesons have electric
charge and that there are additional structure-dependent radiative effects
from the $K$ mesons. We will quote the uncertainty for QED effects as a
separate uncertainty in Tables where we quote values so that its impact is
explicit. We will not include it in plots of the differential rate, nor in
comparison with other theory results that do not include a QED uncertainty.

From [66, 67], we take an overall multiplicative 5% (2%) uncertainty to allow
for missing QED effects in the case of final state electrons (muons). This is
a conservative move, but we shall see that this uncertainty has minimal effect
on our results. We take the QED uncertainty for final state $\tau$ to be
negligible because the $\tau$ is so heavy.

For the case of the lepton-flavour-universality-violating ratio $R^{\mu}_{e}$,
there has been a lot of analysis of QED effects [68, 69, 67]. There is
agreement that QED effects are well accounted for using PHOTOS [65]. An
additional 1% uncertainty for QED effects in the comparison between theory and
experiment is suggested in [68] and we adopt that here when discussing
$R^{\mu}_{e}$ in Section II.2.3.

###  II.2 Comparison to experiment

Label | Reference
---|---
Experiment |
BaBar ’08 | [11]
BaBar ’12 | [12]
BaBar ’16 | [13]
Belle ’09 | [14]
Belle ’19 | [15]
CDF ’11 | [16]
LHCb ’12A | [17]
LHCb ’12B | [18]
LHCb ’14A | [19]
LHCb ’14B | [20]
LHCb ’14C | [21]
LHCb ’16 | [22]
LHCb ’21 | [23]
Theory |
AS ’12 | [36]
BHP ’07 | [25]
BHDW ’11 | [26]
BHD ’12 | [27]
FNAL/MILC ’15 | [9]
GRDV ’22 | [70]
HPQCD ’13 | [8]
HPQCD ’22 | This work
KMW ’12 | [28]
WX ’12 | [37]
Table 2: Labels used in figures and the corresponding references.

In this section we will give our results for the SM short-distance
differential branching fractions (and integrals thereof) for $B\to
K\ell^{+}\ell^{-}$ using Eqs. (1)–(6) and the improved $B\to K$ form factors
of [3]. We will compare to experimental results and show where tensions exist,
and how significant they are.

There are numerous experimental results in the literature from BaBar [11, 12,
13], Belle [14, 15], CDF [16] and LHCb [17, 18, 19, 20, 21, 22, 23], for
decays $B^{0}\to K^{0}\ell^{+}\ell^{-}$ and $B^{+}\to K^{+}\ell^{+}\ell^{-}$
where $\ell\in\\{e,\mu\\}$. There are few for $\ell=\tau$, which we will treat
separately. Experiments are labelled in figures and Table 2 provides a
correspondence between references and labels. Naturally, different analyses
from one particular experiment are likely to be correlated, and should be
viewed as such. One could take the most recent data from each of the
experiments. However, owing in part to the large number of different choices
made with regard to $q^{2}$ limits, meson charge and lepton flavour, we find
it most informative to display all available results in general, even in the
case where older measurements have been superseded by more recent analyses.
This allows the reader to observe the progress of experimental results, in
comparison with our own numbers. In the discussion of branching fraction
$\mathcal{B}$ (for $\ell\in\\{e,\mu\\}$) that follows, we will group
experimental data by $B/K$ charge, as our results are sensitive to this. We
will not group results by $e$ or $\mu$ lepton flavour because our results (in
the SM) are not sensitive to this (see Appendix C for tables including our
results for different $\ell$). In the case of lepton flavour, experiments
sometimes give $e$ and $\mu$ values separately and sometimes both together. In
this latter scenario we will call the lepton $\ell$ and take an average of $e$
and $\mu$. All these will be treated together for a given $B/K$ charge.
Similarly, results are given for a fixed meson charge, as well as for the
average of the charged and uncharged case. We will use the following labelling
conventions, which are adopted in many experimental papers. Where a charge is
specified, it will be labelled, with the unlabelled case indicating the
average of the charged and neutral cases. Similarly, $e$ and $\mu$ leptons
will be labelled explicitly, with $\ell$ indicating an average of the two.
Where possible in plots comparing values with experiment, one set of charges
or lepton flavours will be given in red, the other in blue, and the average in
purple.

Figure 3: Differential branching fraction for $B^{+}\to
K^{+}\ell^{+}\ell^{-}$, with our result in blue, compared with experimental
results [16, 18, 19, 21, 15, 23]. Note that Belle ’19, and LHCb ’14C and ’21
have $\ell=e$, whilst otherwise $\ell=\mu$. Horizontal error bars indicate bin
widths. Figure 4: Differential branching fraction for $B^{0}\to
K^{0}\ell^{+}\ell^{-}$, with our result in blue, compared with experimental
results [16, 17, 19]. All experimental results take $\ell=\mu$. Horizontal
error bars indicate bin widths. Figure 5: Differential branching fraction for
$B\to K\ell^{+}\ell^{-}$, with our result in blue, compared with experimental
results [14, 12, 16]. CDF ’11 takes $\ell=\mu$, whilst Belle ’09 and Babar ’12
do not differentiate $e$ from $\mu$. Horizontal error bars indicate bin
widths.

####  II.2.1 Differential branching fraction and integrals over low and high
$q^{2}$ regions avoiding $c\overline{c}$ resonances

Figures 3, 4 and 5 show our calculation (as a blue band) of the differential
branching fractions for the cases of positively charged mesons, neutral
mesons, and the average of the two, respectively, across the full physical
$q^{2}$ range. Note that the different charges give shapes which vary not just
because of the overall factor of the lifetime $\tau_{B^{0/+}}$, but also
because of the charge dependence of the corrections to
$C_{9}^{\mathrm{eff},0}$ detailed in Appendix B. The black vertical bands
indicate regions vetoed because of the $J/\psi$ and $\psi(2S)$ resonances, not
included in our calculation. As discussed in Section II.1.3, our result is
interpolated across these regions and the gap between them.

Experimental results for decays to electrons, muons or both (averaged) are
displayed in each case as coloured points, with the results shown for each
experimental $q^{2}$ bin. The horizontal error bars on the experimental
results reflect the width of the bin. Some of the experimental results are for
$\ell=e$ and some for $\ell=\mu$; our results are insensitive to the
difference. The experiments ignore data taken in the black vetoed regions, but
there are results in between these regions. However, we cannot make a reliable
comparison between our short-distance SM results and the experimental results
between the vetoed regions.

Figures 3, 4 and 5 show that our results are somewhat higher than experiment
in most cases, particularly in the region $4\leq q^{2}/\mathrm{GeV^{2}}\leq
8.68$. This is most clearly visible in Figure 3, where the tension between our
result and the most precise data from LHCb is obvious.

Channel | Result |  $q^{2}/\mathrm{GeV}^{2}$ range | $\mathcal{B}\times 10^{7}$ | Tension with HPQCD ’22
---|---|---|---|---
$B^{+}\to K^{+}e^{+}e^{-}$ | LHCb ’21 | $(1.1,6)$ | $1.401^{+0.074}_{-0.069}\pm 0.064$ | $-2.7\sigma~{}(-2.4\sigma)$
$B^{+}\to K^{+}e^{+}e^{-}$ | HPQCD ’22 | $(1.1,6)$ | $1.91\pm 0.16(\pm 0.095)_{\mathrm{QED}}$ | -
$B^{+}\to K^{+}e^{+}e^{-}$ | Belle ’19 | $(1,6)$ | $1.66^{+0.32}_{-0.29}\pm 0.04$ | $-0.8\sigma~{}(-0.8\sigma)$
$B^{+}\to K^{+}e^{+}e^{-}$ | HPQCD ’22 | $(1,6)$ | $1.94\pm 0.16(\pm 0.097)_{\mathrm{QED}}$ | -
$B^{0}\to K^{0}\mu^{+}\mu^{-}$ | LHCb ’14A | $(1.1,6)$ | $0.92^{+0.17}_{-0.15}\pm 0.044$ | $-3.1\sigma~{}(-3.1\sigma)$
$B^{0}\to K^{0}\mu^{+}\mu^{-}$ | HPQCD ’22 | $(1.1,6)$ | $1.60\pm 0.14(\pm 0.032)_{\mathrm{QED}}$ | -
$B^{0}\to K^{0}\mu^{+}\mu^{-}$ | LHCb ’14A | $(15,22)$ | $0.67^{+0.11}_{-0.11}\pm 0.035$ | $-2.7\sigma~{}(-2.7\sigma)$
$B^{0}\to K^{0}\mu^{+}\mu^{-}$ | HPQCD ’22 | $(15,22)$ | $1.070\pm 0.095(\pm 0.021)_{\mathrm{QED}}$ | -
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ | Belle ’19 | $(1,6)$ | $2.30^{+0.41}_{-0.38}\pm 0.05$ | $+0.8\sigma~{}(+0.8\sigma)$
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ | HPQCD ’22 | $(1,6)$ | $1.95\pm 0.16(\pm 0.039)_{\mathrm{QED}}$ | -
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ | LHCb ’14A | $(1.1,6)$ | $1.186\pm 0.034\pm 0.059$ | $-4.2\sigma~{}(-4.1\sigma)$
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ | HPQCD ’22 | $(1.1,6)$ | $1.91\pm 0.16(\pm 0.038)_{\mathrm{QED}}$ | -
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ | LHCb ’14A | $(15,22)$ | $0.847\pm 0.028\pm 0.042$ | $-2.8\sigma~{}(-2.7\sigma)$
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ | HPQCD ’22 | $(15,22)$ | $1.16\pm 0.10(\pm 0.023)_{\mathrm{QED}}$ | -
Table 3: Comparison of branching fractions with recent experimental results
[19, 15, 23] in well behaved regions of $q^{2}$. Note that the $B^{+}\to
K^{+}e^{+}e^{-}$ result quoted here from LHCb ’21 is obtained using the
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ result from LHCb ’14A, combined with the ratio
determined in LHCb ’21. In the fifth column, the tension is given by
mean(Experiment - HPQCD)/$\sigma$(Experiment - HPQCD). An additional
uncertainty for QED of 5% for $e$ and 2% for $\mu$ is included in brackets on
each branching fraction (see Section II.1.5). Additionally, the final column
gives the tensions with and without QED uncertainty - those including this
uncertainty are in brackets. Figure 6: Comparison of branching fractions with
recent experimental results [19, 15, 23] in low and high regions of $q^{2}$
away from the charmonium resonance region. Here we show the ratio of the
experimental branching fraction to our results, compared to the black vertical
line at the value 1. The error bars are $5\sigma$ long, with markers at 1, 3
and 5$\sigma$. Note that the $\sigma$ here are for the ratio, so not the same
as those calculated for the difference in Table 3. On the right, labels
indicate the colours of the $q^{2}$ bins in units of $\mathrm{GeV}^{2}$. No
uncertainty from QED is included in this plot.

To examine this tension in more detail, we integrate over two well-behaved
$q^{2}$ regions, one above and one below the $c\overline{c}$ resonances, as
discussed in Section II.1.3. For these regions we can make a reliable
comparison with experiment. We show the results in Table 3; these constitute
our main numerical results. In Table 3, we compare our branching fractions
with the most recent experimental results available for $B\to Ke^{+}e^{-}$ and
$B\to K\mu^{+}\mu^{-}$. Note that our relative uncertainties are comparable to
those from the experiments for most of the values. We have larger
uncertainties than those for LHCb ’14A for $B^{+}\to K^{+}\mu^{+}\mu^{-}$ but
smaller uncertainties than those from Belle ’19. Our uncertainties are
dominated by those from the form factors, followed by those from the CKM
elements $|V_{tb}V_{ts}^{*}|$.

We find our partial branching fractions to be significantly higher than the
experimental values, typically with tensions exceeding $2.7\sigma$ and with
LHCb’14A [19] in the low $q^{2}$ region with a tension reaching $4.2\sigma$
for $B^{+}\to K^{+}\mu^{+}\mu^{-}$. We do not see any tension with the
Belle’19 results [15], which are themselves in tension with LHCb ’14A for
$B^{+}\to K^{+}\mu^{+}\mu^{-}$. Belle ’19 have considerably larger
uncertainties than those from LHCb and those from our results. The
uncertainties we allow for QED make little difference to the tensions that we
see.

The same results, plotted in the form of a ratio of the experimental branching
fraction to our result, are shown in Figure 6, with error bars indicating 1, 3
and 5$\sigma$ differences from a value for the ratio of 1. This makes the
tensions graphically clearer and also shows the difference in the size of
uncertainties in the comparison with different experiments. Note that the
tensions in the comparison of the ratio to the value 1 are not exactly the
same as in the differences between our result and experiment shown in Table 3.

It is worth noting that the corrections to $C_{7}^{\mathrm{eff},0}$ and
$C_{9}^{\mathrm{eff},0}$ (discussed in Appendix B) shift our results by
amounts of between 0.3 and 1$\sigma$, depending on the $q^{2}$ region. In the
high $q^{2}$ region, this is mostly driven by corrections to
$C_{7}^{\mathrm{eff},0}$. In the low $q^{2}$ region the size and direction of
the shift depends on the meson charge. The impact of the corrections on the
tension between our SM theory and experiment given in Table 3 then amounts at
most to 0.6$\sigma$.

Our numerical results for integration of our differential branching fraction
for a variety of $q^{2}$ ranges are given in Table 7 in Appendix C.

In Section V we discuss the effect on our results, and the tensions with
experimental values, of changing the values of the Wilson coefficients $C_{9}$
and $C_{10}$ that we use in Eq. (4). This tests the impact of possible new
physics at a high scale that would change the effective weak Hamiltonian.

####  II.2.2 Total branching fractions

In this section we discuss the total branching fraction, i.e.
$d\mathcal{B}/dq^{2}$ integrated across the physical $q^{2}$ range from
$q^{2}_{\text{min}}=4m_{\ell}^{2}$ to $q^{2}_{\text{max}}=(M_{B}-M_{K})^{2}$.
By convention, experimentally measured branching fractions are extrapolated to
the full physical $q^{2}$ range, ignoring resonances. This is achieved using
correction factors based on an assumed $q^{2}$ distribution (see e.g. [71]) to
the branching fraction integrated over the experimentally measured bins (see
discussion in [19]). This achieves the same result as our interpolation across
the $c\overline{c}$ resonance region (discussed in Section II.1.3) and this
should make our short-distance theory results comparable to experiment here.

Figures 7, 8 and 9 show this comparison for the $\ell\in\\{e,\mu\\}$ case.
Again, results are split up by meson charge and our result (HPQCD ’22) is
given at the top, denoted by a black star. The grey band carries our result
down the plot. The experimental results are split up according to lepton
flavour, but as discussed earlier, our results are insensitive to this
distinction. As we do not have information about correlations between the
various results for each experimental collaboration we do not give an
experimental average here for comparison.

Figure 7: The total branching fraction for $B^{+}\to K^{+}\ell^{+}\ell^{-}$.
Our result (HPQCD ’22) is given by the black star and grey band, as compared
with experimental results [11, 14, 19, 18, 16, 22, 15]. Dashed lines indicate
the effect of adding QED uncertainty (see Section II.1.5) to our result.
Figure 8: The total branching fraction for $B^{0}\to K^{0}\ell^{+}\ell^{-}$.
Our result (HPQCD ’22) is given by the black star and grey band, as compared
with experimental results [11, 14, 19, 17, 16, 15]. Dashed lines indicate the
effect of adding QED uncertainty (see Section II.1.5) to our result. Figure 9:
The total branching fraction for $B\to K\ell^{+}\ell^{-}$. Our result (HPQCD
’22) is given by the black star and grey band, as compared with experimental
results [12, 11, 14, 16]. Dashed lines indicate the effect of adding QED
uncertainty (see Section II.1.5) to our result.

Our result is in general larger than the experimental values, reflecting the
pattern in Figures 3, 4 and 5, which do not contain exactly the same
experiments. This is clearest in the case of the neutral meson decay, but both
the charged and neutral LHCb ’14A results with $\mu$ in the final state in
Figures 7 and 8 show clear tension: $3.9(4.0)\sigma$ for $B^{+}$ and
$4.2(4.2)\sigma$ for $B^{0}$, with(without) QED uncertainty.

To test the effect of the LHCb ’14A $q^{2}$ interpolation across the vetoed
regions, we compare our total branching fraction (without QED uncertainty)
with that when we exclude the regions vetoed by LHCb ’14A ($8\leq
q^{2}/\text{GeV}^{2}\leq 11$ and $12.5\leq q^{2}/\text{GeV}^{2}\leq 15$), we
obtain ratios $1.3814(84)$ and $1.3951(84)$ for the $B^{+}$ and $B^{0}$ cases,
respectively. This compares very well with the factor of $1.39$ used by LHCb,
and indicates that the interpolation used by them is consistent with our
linear interpolation approach. This adds confidence that the comparison
between theory and experiment here is a reliable one.

We conclude that, in agreement with what was seen in the low and high $q^{2}$
regions in Sec. II.2.1, significant tension (exceeding 4$\sigma$) is also seen
between the total short-distance SM branching fraction and the most accurate
(LHCb) results for $B\to K\ell^{+}\ell^{-}$.

For the decay with $\tau$ in the final state, experimental results are much
less mature. The BaBar collaboration reported the first measurement of the
total $B^{+}\to K^{+}\tau^{+}\tau^{-}$ branching fraction in [13]. Their
result of $(1.31^{+0.66+0.35}_{-0.61-0.25})\times 10^{-3}$, (where errors are
statistical and systematic respectively) is consistent with no signal. It is
also consistent with our 7%-accurate SM result of
$\mathcal{B}^{(+)}_{\tau}=1.68(12)\times 10^{-7}$, where we do not include an
uncertainty for QED effects since the $\tau$ is so heavy. Note that the
$q^{2}$ range available for the case with the $\tau$ is much smaller than for
the lighter leptons. We will discuss the $\tau$ case further in Sec. II.3.

Our numerical results for the total branching fraction for
$\ell\in\\{e,\mu\\}$ are given in Table 7 in Appendix C. Results for $\tau$ in
the final state are given in Table 10 in the same Appendix.

Figure 10: The ratio of branching fractions to muons and electrons, as defined
in Equation (7), with various $q^{2}$ limits. Our result (HPQCD ’22) has
negligible uncertainty compared to experiment [14, 12, 15, 23], and does not
differ visibly when integrated over different $q^{2}$ ranges. Note that for
the ratios of total branching fractions (top two panes), we have used a lower
limit on the $q^{2}$ integral, $q^{2}_{\text{min}}$, that differs between $e$
and $\mu$ cases in the ratio. This was done to match what was done in the
experiment, but has no visible effect. We include a 1% uncertainty to allow
for possible QED corrections [68] as discussed in the text. This completely
dominates our theoretical error here. The vertical dotted line is at 1. Figure
11: The flat term $F_{H}^{\ell}$ for the $B^{+}\to K^{+}\mu^{+}\mu^{-}$ decay.
Our result (HPQCD ’22) is given for bins between integer values of $q^{2}$,
with the uncertainty included but too small to be visible. Experimental
results [18, 20] are included for comparison.

####  II.2.3 $R^{\mu}_{e}$

We define the ratio of branching fractions for decays to different leptons,

$R^{\ell_{1}}_{\ell_{2}}(q^{2}_{\text{low}},q^{2}_{\text{upp}})=\frac{\int_{q^{2}_{\text{low}}}^{q^{2}_{\text{upp}}}\frac{d\mathcal{B}_{\ell_{1}}}{dq^{2}}dq^{2}}{\int_{q^{2}_{\text{low}}}^{q^{2}_{\text{upp}}}\frac{d\mathcal{B}_{\ell_{2}}}{dq^{2}}dq^{2}}.$
(7)

We calculate this separately for the charged and neutral meson cases,
$R^{\mu(0/+)}_{e}$. We also give the charge-averaged result, $R^{\mu}_{e}$,
which is obtained from performing a charge average in each of the numerator
and denominator of Eq. (7).

As discussed above, the effect of changing lepton flavour between electrons
and muons on our branching fraction is very small. The factors of $\beta$ in
Eqs. (2) and (II.1) mean that $R^{\mu}_{e}<1$ at very small values of $q^{2}$
and the factors of $m_{\ell}$ mean that $R^{\mu}_{e}>1$ at large values of
$q^{2}$. However, for almost the entire $q^{2}$ range $R^{\mu}_{e}$ is very
close to 1. This also means that effects from the form factors and from the
resonances cancel between numerator and denominator and the theory
uncertainties in $R^{\mu}_{e}$ are small. Our results are tabulated in Table 8
in Appendix C for a variety of $q^{2}$ ranges; we find $R^{\mu}_{e}$ to differ
from 1 by a few parts in 1000.

Radiative corrections from QED (see Sec. II.1.5) affect the $\mu$ and $e$
cases differently but are well understood [68, 66]. Tests show that the
corrections applied on the experimental side using PHOTOS [65] should be
reliable up to $\pm$1% [68]. We will take a 1% uncertainty on our results to
allow for missing QED effects in the comparison of our results with
experiment. However, where this uncertainty is already included in the
experimental results this will amount to a double-counting of this source of
error. This QED uncertainty dominates all other sources of uncertainty in our
determination of $R^{\mu}_{e}$.

Figure 10 shows a comparison of our result for $R^{\mu}_{e}$ (including the 1%
uncertainty) with those from experiment. The most notable experimental results
are the recent values from LHCb ’21 [23] with relatively small uncertainties
for the charged meson decay, integrated over the low $q^{2}$ region. The LHCb
’21 result differs from 1 (taken as the SM result when QED corrections are
removed [68]) by 3.1$\sigma$. The relative uncertainty from theory is
negligible so having improved form factors, as we do here, makes no difference
to this tension. When we come to compare with other theoretical work in
Section II.3, potential deviations of $R^{\mu}_{e}$ from 1 will be studied in
more detail.

####  II.2.4 $F^{\ell}_{H}$

Two additional experimentally measured quantities with which we can compare
are based on studying the angular distribution of decay products. The
differential distribution in angle is given by [25]

$\frac{1}{\Gamma_{\ell}}\frac{d\Gamma_{\ell}}{d\cos\theta}=\frac{3}{4}(1-F_{H}^{\ell})(1-\cos^{2}\theta)+\frac{1}{2}F_{H}^{\ell}+A^{\ell}_{\text{FB}}\cos\theta\,,$
(8)

where $\theta$ is the angle between $B$ and $\ell$ as measured in the dilepton
rest frame. Both the flat term, $F_{H}^{\ell}/2$, and the forward-backward
asymmetry, $A^{\ell}_{\text{FB}}$, are small, and sensitive to new physics.
$A^{\ell}_{\text{FB}}=0$ in the SM, up to QED corrections, so we will not
consider it here. We can construct $F^{\ell}_{H}$ from [25]

$F^{\ell}_{H}(q^{2}_{\text{low}},q^{2}_{\text{upp}})=\frac{\int_{q^{2}_{\text{low}}}^{q^{2}_{\text{upp}}}(a_{\ell}+c_{\ell})dq^{2}}{\int_{q^{2}_{\text{low}}}^{q^{2}_{\text{upp}}}(a_{\ell}+c_{\ell}/3)dq^{2}}.$
(9)

Figure 11 shows $F^{\ell}_{H}$ for the $B^{+}\to K^{+}\mu^{+}\mu^{-}$ decay.
The black stars show our result, calculated in bins between integer values of
$q^{2}/\text{GeV}^{2}$, with the exception of the first bin, which starts at
$q^{2}_{\text{min}}$. Experimental results from [18] are included for
comparison, as are those from [20]. In the latter case, central values are not
given for the data, and so in order to add the systematic error to the
statistical confidence interval, this interval is assumed Gaussian, and the
central value taken as the middle of the interval. The effect of this
approximation is negligible, owing to the small systematic errors, and does
not affect the comparison with our results, as the overall confidence interval
is broad.

As with $R^{\mu}_{e}$, the cancellation of correlated uncertainties in Eq. (9)
means that our result here is very precise compared with experiment (our error
bars are included in Figure 11 but not visible). We can see that a large
reduction in experimental uncertainty will be required in order to reveal any
meaningful inconsistency (if one exists) between SM theory and experiment.

Our results for $F_{H}^{\ell}$ for $\ell\in\\{e,\mu\\}$ are tabulated in Table
9 in Appendix C for a variety of $q^{2}$ ranges; we will discuss the flat term
for the $\tau$ case in Section II.3.

###  II.3 Comparison to theory

In this section we compare to previous theoretical results, both from previous
lattice QCD calculations [8, 9] and others [25, 26, 27, 28, 36, 37, 70].
Again, Table 2 provides a correspondence between references and labels in
figures. Different theory calculations differ both in the form factors they
use and in their input parameters used to determine the differential rates. In
[3] we compare the improved lattice QCD form factors that we use here with
earlier lattice QCD form factors. Here we will show comparisons of
differential and total branching fractions, flat terms etc. as for the
comparison with experiment in Section II.2. There are many more theory results
for the case of $\tau$ leptons in the final state, so we can make more
detailed comparison of that case.

####  II.3.1 Differential and total branching fractions

We compare differential branching fractions for $\ell\in\\{e,\mu\\}$ in the
same manner that we did for experiment, allowing us to factor in the different
$q^{2}$ bin choices made by different authors. As before, lepton flavour makes
an insignificant difference for $\ell\in\\{e,\mu\\}$, so we treat all cases
together. However, we split the cases up by meson charge. The three charge
cases (charged $B$, neutral $B$ and the average of the two) are shown in
Figures 12, 13 and 14. Again, note the effect of the charge dependence of the
corrections to $C_{9}^{\mathrm{eff},0}$ detailed in Appendix B.

We see good agreement with previous work, although our uncertainties are
generally a lot smaller. We do not agree with the results of AS’12 [36] at low
$q^{2}$ shown in Figure 14. Our central values are slightly higher than most
previous results. In the comparison with FNAL/MILC ’15 [9], this $\approx
0.5\sigma$ effect can be traced to the larger value of $|V_{ts}|$ that we use,
based on more recent $B$-mixing results (see Section II.1.1); we agree well on
the form factor central values (see [3]). GRDV ’22 [70] fix their form factors
from a combination of earlier lattice QCD results (at high $q^{2}$) and values
derived from light-cone sum rules (at low $q^{2}$). Our form factors have
lower central values than theirs but this is partly offset by the difference
of values of $|V_{ts}|$ used. Note that GRDV ’22 also include non-local form
factors in their calculation that account for charmonium resonance
contributions. The uncertainty in their differential branching fraction is
dominated by that from their local form factors.

Figure 12: Differential branching fraction for $B^{\pm}\to
K^{\pm}\ell^{+}\ell^{-}$, with our result in blue, compared with earlier work
[25, 26, 27, 9]. Error bars in $q^{2}$ indicate bin widths. Figure 13:
Differential branching fraction for $B^{0}\to K^{0}\ell^{+}\ell^{-}$, with our
result in blue, compared with earlier work [25, 26, 27, 9, 28]. Error bars in
$q^{2}$ indicate bin widths. Figure 14: Differential branching fraction for
$B\to K\ell^{+}\ell^{-}$ (the average of that for charged and neutral $B$),
with our result in blue, compared with earlier work [8, 36]. Error bars in
$q^{2}$ indicate bin widths. Figure 15: Differential branching fraction for
$B\to K\tau^{+}\tau^{-}$, with our result as a blue band ($B^{0}\to K^{0}$)
and a red band ($B^{+}\to K^{+}$), compared to earlier results from FNAL/MILC
’15 [9] (blue and red filled triangles). Results from HPQCD ’13 [8] are for
the average of the charged and neutral $B$ meson cases. Error bars in $q^{2}$
indicate bin widths.

We compare our results for the differential branching fraction for the case
$\ell=\tau$ to those of FNAL/MILC ’15 [9] and HPQCD ’13 [8] in Figure 15. The
$q^{2}$ range is limited by the much larger value of $q^{2}_{\text{min}}$ in
this case (set by $m_{\tau}$), and so no interpolation across the vetoed
$c\overline{c}$ resonance region is required. We include both the charged and
neutral meson cases on the same plot, in red and blue respectively. We see
again that our results are somewhat, but not significantly, higher than the
earlier results largely because of the change in $|V_{ts}|$.

Figure 16: The total branching fraction for $B\to K\ell^{+}\ell^{-}$
($\ell\in\\{e,\mu\\}$), integrated from $q^{2}_{\text{min}}$ to
$q^{2}_{\text{max}}$ compared with [26, 8, 9, 37]. Different meson charges are
treated separately. The lowest panel shows a comparison to BHDW ’11 in which
the integration over the differential branching fraction starts at
$q^{2}_{\text{min}}=14.18\,\text{GeV}^{2}$. Figure 17: The total branching
fraction for $B\to K\tau^{+}\tau^{-}$, integrated from $q^{2}_{\text{min}}$ to
$q^{2}_{\text{max}}$ compared with [26, 8, 9, 37]. Different meson charges are
treated separately. The lowest two panels show a comparison to BHDW ’11 and
HPQCD ’13 in which the integration over the differential branching fraction
starts at $q^{2}_{\text{min}}=14.18\,\text{GeV}^{2}$.

Figures 16 and 17 show the total branching fraction for the
$\ell\in\\{e,\mu\\}$ and $\ell=\tau$ cases respectively. The figures are
broken down according to meson charge, with the integral extending from
$q^{2}_{\text{min}}$ to $q^{2}_{\text{max}}$, except as indicated otherwise in
the plot. In HPQCD ’13 [8] the branching fraction to electrons is also given,
but (as is the case for us) is insignificantly different from the muon case
shown, so we omit it. Again, our central values are somewhat higher than
previous work, but not by a significant margin. Our results provide an
improvement in uncertainty over earlier values.

Figure 18: The deviation of the ratio $R^{\mu}_{e}$ (Eq. (7)) from 1. Our
results (HPQCD ’22) are shown as open black stars for the charged $B$ case,
evaluated in $q^{2}$ bins defined by $0.1,2,4,6,\dots,22\,\text{GeV}^{2}$.
Error bars in $q^{2}$ indicate bin widths. Earlier theory results are from
[25, 8, 9]. We do not include any uncertainty from possible QED corrections
that would affect experimental results.

####  II.3.2 $R^{\mu}_{e}$ and $R^{\tau}_{\mu}$

We now look in more detail at the ratio of branching fractions to different
flavours of leptons, defined in Eq. (7). Figure 18 shows our results (as black
open stars) for the deviation of the ratio $R^{\mu}_{e}$ from 1.0, as a
function of $q^{2}$. We use $q^{2}$ bins of width 2 $\text{GeV}^{2}$,
following the pattern 2–4 $\text{GeV}^{2}$, 4–6 $\text{GeV}^{2}$ etc, except
for the first bin which covers 0.1–2 $\text{GeV}^{2}$. We plot our results for
the charged $B$ case; values for the neutral $B$ case are indistinguishable,
except in the lowest $q^{2}$ bin (numerical values for each bin are given in
Table 8 in Appendix C). As discussed earlier, the ratio is very well
determined theoretically in a pure QCD calculation; uncertainties from
possible QED corrections that would affect experiment are not included in Fig.
18. We see that $R^{\mu}_{e}$ differs from 1 at the level of $10^{-3}$, with a
larger deviation at large $q^{2}$. There is also a sizeable effect in the
smallest $q^{2}$ bin where the kinematic cut-off from the $\mu$ mass starts to
have an effect.

Figure 18 also shows a comparison to earlier theory results. Good agreement is
seen between the different calculations.

Figure 19: The ratio $R^{\tau}_{\mu}$ (Equation (7)). Our results (HPQCD ’22,
open black stars) are evaluated in $q^{2}$ bins defined by
$14.18,16,18,20,22\,\text{GeV}^{2}$, for the charged $B$ meson case. We
compare with HPQCD ’13 [8] and FNAL/MILC ’15 [9]. No uncertainty is included
for QED corrections that would affect experiment. Figure 20: Our result for
$R^{\mu}_{e}$ (filled blue star) determined from Eq. (7) using the full
$q^{2}$ range and shown as the deviation from 1 in units of $10^{-2}$. The
filled red star gives our result for $R^{\tau}_{\mu}$ using the $q^{2}$ range
from 14.18 $\text{GeV}^{2}$ to $q^{2}_{\text{max}}$. We compare to results
from [8] (filled hexagons). No uncertainty for QED corrections is included
here.

Figure 19 shows our results for the ratio $R^{\tau}_{\mu}$ (Eq. (7)) as open
black stars, calculated in $q^{2}$ bins of width 2 ${\text{GeV}}^{2}$. The
lowest bin is from the top of the $c\overline{c}$ resonance region at 14.18
${\text{GeV}}^{2}$ (slightly above the kinematic end-point of $4m_{\tau}^{2}$)
to 16 ${\text{GeV}}^{2}$, the next 14–16 ${\text{GeV}}^{2}$ and so on. Results
are shown for the charged meson case, but the neutral meson results are
indistinguishable, see the numerical values given in Table 11 in Appendix C.
$R^{\tau}_{\mu}$ deviates from 1.0 much more strongly than $R^{\mu}_{e}$;
given the lepton masses involved this is not surprising. The results show a
similar pattern to $R^{\mu}_{e}$ with lepton mass effects pushing $R$ values
upwards as $q^{2}$ increases. $R^{\tau}_{e}$ results would not differ visibly
and so are not shown. Again, the comparison to earlier results shows good
agreement.

In Fig. 20 we give our result for $R^{\mu}_{e}$ using the full $q^{2}$ range
from $4m_{\mu}^{2}$ to $q^{2}_{\text{max}}$. This is shown as the deviation
from 1 in units of $10^{-2}$. Note that in forming this ratio we integrate
both the numerator and denominator of Eq. (7) through the vetoed region as
discussed in Sec. II.1.3. We also give our result for $R^{\tau}_{\mu}$ using
the range from 14.18 $\text{GeV}^{2}$ (the upper edge of the vetoed region) to
$q^{2}_{\text{max}}$. Both of our results are consistent with zero deviation
from 1 at the 2$\sigma$ level. We compare to HPQCD ’13 [8] and see reasonable
agreement; our result for $R^{\mu}_{e}$ has half the theory uncertainty of
their earlier value.

Figure 21: Open black stars show our results for $F^{e}_{H}$ for the charged
$B$ case in bins of $q^{2}$ defined by $0.1,2,4,6,\dots,22\,\text{GeV}^{2}$.
We compare to HPQCD ’13 [8] and FNAL/MILC ’15 [9].  Figure 22: Open black
stars show our results for $F^{\mu}_{H}$ for the charged $B$ case in bins of
$q^{2}$ defined by $0.1,2,4,6,\dots,22\,\text{GeV}^{2}$. We compare to HPQCD
’13 [8], FNAL/MILC ’15 [9] and BHD ’12 [27].  Figure 23: Open black stars show
our results for $F^{\tau}_{H}$ for the charged $B$ case in bins of $q^{2}$
defined by $14.18,16,18,20,22\,\text{GeV}^{2}$. We compare to HPQCD ’13 [8]
and FNAL/MILC ’15 [9].

####  II.3.3 $F_{H}^{e}$, $F_{H}^{\mu}$ and $F_{H}^{\tau}$

Here we examine in more detail the quantity $F_{H}^{\ell}$, twice the flat
term of Eq. (8), for different lepton flavours. $F_{H}^{\ell}$ is determined
from the ratio of Eq. (9) in bins of $q^{2}$. Figures 21, 22 and 23 show our
results for $e$, $\mu$ and $\tau$ cases respectively as open black stars. We
use $q^{2}$ bins of width 2 $\text{GeV}^{2}$, as for $R$ in the previous
section. Since meson charge does not affect the results, except in the
smallest $q^{2}$ bin, we show our results for the charged $B$ cases. Our
results are given in detail in Tables 9 and 12 in Appendix C. No additional
uncertainties from QED effects that would affect experiment are included in
the plots.

For the light, $e$ and $\mu$, lepton cases $F_{H}^{\ell}$ is small across most
of the $q^{2}$ range but rises rapidly towards the low $q^{2}$ end. For small
values of $m_{\ell}$ and $q^{2}$ the integrand in the numerator of
$F_{H}^{\ell}$ (Eq. (9)) is proportional to $\beta(1-\beta^{2})$ (see Eq.
(2)). For $q^{2}=\mathcal{O}(4m_{\ell}^{2})$, this has no suppression by
$m_{\ell}^{2}$ and gives a large contribution to the integral. This means that
great care must be taken in integrating the numerator in the region $q^{2}\to
4m_{\ell}^{2}$ (see Appendix C.1).

In Figures 21, 22 and 23 we also compare to earlier (mainly lattice QCD)
results, plotting the values for the charged meson case where both are given.
There is good agreement between the different calculations in general; our
results have an improvement in uncertainty. We see some tension (2.8$\sigma$)
for $F_{H}^{e}$ and $F_{H}^{\mu}$ with results from FNAL/MILC ’15 in the
smallest $q^{2}$ bin for the charged $B$ case, but agree well for the neutral
$B$ case. This can be seen by comparing our Table 9 with Tables IX and X of
[9]. For $F_{H}^{\tau}$ we see some tension with HPQCD ’13 [8].

Figure 24: Filled stars in the top and third-from-top panels show our results
(‘HPQCD ’22’) for $F_{H}^{e}$, integrated over the full $q^{2}$ range (from
$4m_{e}^{2}$ to $q^{2}_{\mathrm{max}}$) for the charged and neutral meson
cases respectively. Our results for this case are compared with [9] (filled
triangles). We also give values for our results integrated from $4m_{\mu}^{2}$
to $q^{2}_{\mathrm{max}}$ (open stars, second-from-top and lowest panels) to
illustrate the large contribution of the integral from very small $q^{2}$
values. Figure 25: $F_{H}^{\mu}$, integrated over the full $q^{2}$ range, for
the charged and neutral meson cases, and over a reduced range from
$14.18\,\text{GeV}^{2}$ in the case of the average of the charged and neutral
cases (lowest panel). Our results are shown as the filled stars and denoted
‘HPQCD ’22’. This work is compared with [26, 9], shown as filled triangles and
circles respectively.  Figure 26: $F_{H}^{\tau}$, integrated over the full
$q^{2}$ range, for the charged and neutral meson cases, and over a reduced
range from $14.18\,\text{GeV}^{2}$ in the case of the average of the charged
and neutral cases (lowest panel). Our results are shown as the filled stars
and denoted ‘HPQCD ’22’. This work is compared with [26, 8, 9], shown as
filled hexagons, triangles and circles.

Figures 24, 25 and 26 show $F^{\ell}_{H}$ as defined above, integrated over
the full $q^{2}$ range $q^{2}_{\text{min}}\leq q^{2}\leq q^{2}_{\text{max}}$,
except in cases where the range is stated explicitly. These plots are split up
according to meson charge, although in most cases the difference is
insignificant. A significant difference is seen for $F^{e}_{H}$ between the
charged and neutral cases when integrating from $4m_{e}^{2}$ up to
$q^{2}_{\text{max}}$. This difference comes from the nonfactorisable
corrections to $C_{9}^{\text{eff}}$ discussed in Appendix B. These depend on
meson charge and have a sizeable impact at very small value of $q^{2}$,
relevant to $F^{e}_{H}$ (see below) but not for other lepton flavours or to
other quantities.

We see good agreement with previous work for $F_{H}^{\mu}$ and $F_{H}^{\tau}$,
with our results showing a considerable improvement in uncertainty in most
cases. For $F^{e}_{H}$ our results are in tension with those of FNAL [9] when
we integrate from $4m_{e}^{2}$ up to $q^{2}_{\text{max}}$. Given the good
agreement seen for most of the $q^{2}$ range in Fig. 21 it seems likely that
the tension is a result of behaviour in the integrand of the numerator towards
the lower limit of the $q^{2}$ integral, discussed above. We illustrate this
in Fig. 24 by showing results also for the case where the lower cut-off on the
$q^{2}$ integral is $4m_{\mu}^{2}$ rather than $4m_{e}^{2}$. Doing this gives
considerably lower values for $F_{H}^{e}$ because the integral in the
numerator of $F_{H}^{e}$ changes; the denominator is half the total branching
fraction and very insensitive to small changes in $q^{2}_{\text{min}}$. We
conclude that determining $F^{e}_{H}$ at small $q^{2}$ values or integrated
over the full kinematic range requires care over integration and is sensitive
to small-$q^{2}$ corrections. However, away from small $q^{2}$ values there
are no issues with determining $F^{e}_{H}$ (see Figure 24) and this is the
experimentally more accessible region.

##  III $B\to K\ell_{1}^{-}\ell_{2}^{+}$

Figure 27: $\phi_{9,10}(q^{2})\times 10^{9}$ for the $B^{0}\to K^{0}\mu\tau$
decay. Figure 28: $\phi_{S,P}(q^{2})\times 10^{9}$ for the $B^{0}\to
K^{0}\mu\tau$ decay.

We also investigate the rare lepton flavour number violating (LFV) $B\to
K\ell^{-}_{1}\ell^{+}_{2}$ decay, with different final state leptons
$\ell_{1}\neq\ell_{2}$ (with masses $m_{1}$ and $m_{2}$). The differential
branching fraction for this decay is given in Eq. (9) of [72] in terms of a
number of Wilson coefficients and functions $\varphi_{i}(q^{2})$ (Eq. (10) of
[72]). The knowledge of these functions in the SM can be combined with
experimental measurements to set limits on the Wilson Coefficients [73].

In the case that the LFV in $B\to K\ell^{-}_{1}\ell^{+}_{2}$ arises from
purely vector or purely scalar operators, we can write simple forms for the
branching fraction, in terms of Wilson coefficients $C^{(^{\prime})}_{9}$,
$C^{(^{\prime})}_{10}$, $C^{(^{\prime})}_{S}$ and $C^{(^{\prime})}_{P}$ and
lepton flavour dependent coefficients $a_{K}^{12}$, $b_{K}^{12}$, $e_{K}^{12}$
and $f_{K}^{12}$.

$10^{9}\mathcal{B}(B\to
K\ell_{1}\ell_{2})=a_{K}^{12}|C_{9}+C_{9}^{\prime}|^{2}+b_{K}^{12}|C_{10}+C_{10}^{\prime}|^{2}$
(10)

for the vector case and

$10^{9}\mathcal{B}(B\to
K\ell_{1}\ell_{2})=e_{K}^{12}|C_{S}+C_{S}^{\prime}|^{2}+f_{K}^{12}|C_{P}+C_{P}^{\prime}|^{2}$
(11)

for the scalar. The factor of $10^{9}$ makes the coefficients
$\mathcal{O}(10)$.

Here, we will construct the functions $\varphi_{9}$, $\varphi_{10}$,
$\varphi_{S}$, $\varphi_{P}$, which, once multiplied by $|N_{K}(q^{2})|^{2}$,
can be integrated to give the coefficients above [72] .

$\begin{split}&\varphi_{9,10}(q^{2})=\\\ &\frac{1}{2}|f_{0}|^{2}(m_{1}\mp
m_{2})^{2}\frac{(M_{B}^{2}-M_{K}^{2})^{2}}{q^{2}}\Big{(}1-\frac{(m_{1}\pm
m_{2})^{2}}{q^{2}}\Big{)}+\\\
&\frac{1}{2}|f_{+}|^{2}\lambda(q,M_{B},M_{K})\Big{(}1-\frac{(m_{1}\mp
m_{2})^{2}}{q^{2}}-\frac{\lambda(q,m_{1},m_{2})}{3q^{4}}\Big{)},\end{split}$
(12)
$\begin{split}\varphi_{S,P}(q^{2})&=\frac{q^{2}|f_{0}|^{2}}{2(m^{\overline{\mathrm{MS}}}_{b}(\mu_{b})-m^{\overline{\mathrm{MS}}}_{s}(\mu_{b}))^{2}}\times\\\
&(M_{B}^{2}-M_{K}^{2})^{2}\Big{(}1-\frac{(m_{1}\pm
m_{2})^{2}}{q^{2}}\Big{)},\end{split}$ (13)
$\begin{split}&|N_{K}(q^{2})|^{2}=\\\
&\tau_{B}\frac{\alpha^{2}_{\mathrm{EW}}G_{F}^{2}|V_{tb}V^{*}_{ts}|^{2}}{512\pi^{5}M_{B}^{3}}\times\\\
&\frac{\sqrt{\lambda(q,m_{1},m_{2})}}{q^{2}}\sqrt{\lambda(q,M_{B},M_{K})},\end{split}$
(14)

where the $\pm$s are matched with $9,10$ and $S,P$. We use our improved
$f_{+}$ and $f_{0}$ form factors [3] to determine these functions, including
uncertainties for strong-isospin breaking discussed in Section II.1.4. For the
$\overline{\mathrm{MS}}$ $s$ mass in Eq. (13) we take
$m^{\overline{\mathrm{MS}}}_{s}(\mu_{b}\equiv
4.2\,\mathrm{GeV})=0.07966(80)\,\mathrm{GeV}$, run from
$m^{\overline{\mathrm{MS}}}_{s}(\mu=3\,\mathrm{GeV})=0.08536(85)\,\mathrm{GeV}$
[74]. $m^{\overline{\mathrm{MS}}}_{b}(\mu_{b})$ is given in Table 1, along
with the other parameters used. None of the $\varphi$ functions are sensitive
to interchanging $1\leftrightarrow 2$, so neither are the resulting
coefficients ($a_{K}^{12}=a_{K}^{21}$, etc.).

We plot the functions
$\phi_{i}(q^{2})\equiv|N_{K}(q^{2})|^{2}\varphi_{i}(q^{2})$ in Figures 27 and
28, for the choice $\ell_{1}=\mu$, $\ell_{2}=\tau$, and find good agreement
with similar plots in [72]. Note that there is no issue with $c\overline{c}$
resonances in this case. The functions are insensitive to $B$ meson charge, so
the neutral case is shown here - the charged case differs mainly by the
overall multiplicative factor of $\tau_{B^{0/+}}$ used in Eq. (14). Plots for
lepton combination $e\tau$ are nearly identical to those for $\mu\tau$, and
are not shown.

The shape of $\phi_{S,P}(q^{2})$ varies little with lepton flavour choices, so
we show only the one example in Fig. 28. The shape of $\phi_{9,10}$ is more
sensitive, however and we plot the case of $e\mu$ in Fig. 29. It shows the
functions falling much more abruptly towards the minimal
$q^{2}=(m_{1}+m_{2})^{2}$ value than when one lepton is a $\tau$.

Figure 29: $\phi_{9,10}(q^{2})\times 10^{9}$ for the $B^{0}\to K^{0}e\mu$
decay.

Integrating $\phi_{9,10,S,P}$ across the full physical $q^{2}$ range for each
$\ell_{1},\ell_{2}$ combination yields $a_{K}^{12}$, $b_{K}^{12}$,
$e_{K}^{12}$ and $f_{K}^{12}$ respectively, and we report their values in
Table 4. These are updates to the values in [72] (which took LCSR form factors
from [38]). We agree well with the results found in [72], with a modest
improvement in $a_{K}^{12}$ and $b_{K}^{12}$ uncertainties. Table 4 gives
results for both charged and neutral mesons, which are slightly different
because of the change in lifetime $\tau_{B^{0/+}}$.

The results of Table 4 can be inserted into Eqs. (10) or (11) to obtain a
branching fraction for given Wilson coefficient values. Current experimental
limits for $\mathcal{B}(B\to Ke\mu)$ are $7.0\times 10^{-9}$ for $B^{+}$ [31]
and $3.8\times 10^{-8}$ for $B^{0}$ [15]. These limits start to provide
constraints on beyond the Standard Model (BSM) physics, for example models
with leptoquarks [75]. Current experimental limits for $B^{+}\to
h^{+}\tau\ell$ are a few times $10^{-5}$ [30] and so need to be reduced by
several orders of magnitude to provide a test of this decay mode.

$\ell_{1}\ell_{2}$ | $a^{12}_{K}$ | $b^{12}_{K}$ | $e^{12}_{K}$ | $f^{12}_{K}$
---|---|---|---|---
$B^{0}\to K^{0}e\mu$ | 17.7(1.2) | 17.7(1.2) | 24.5(1.4) | 24.5(1.4)
$B^{0}\to K^{0}e\tau$ | 11.08(69) | 11.08(69) | 13.97(79) | 13.97(79)
$B^{0}\to K^{0}\mu\tau$ | 10.85(69) | 11.26(69) | 13.44(76) | 14.41(82)
$B^{+}\to K^{+}e\mu$ | 19.2(1.3) | 19.2(1.3) | 26.6(1.5) | 26.6(1.5)
$B^{+}\to K^{+}e\tau$ | 11.99(75) | 11.99(75) | 15.14(86) | 15.15(86)
$B^{+}\to K^{+}\mu\tau$ | 11.73(75) | 12.18(75) | 14.57(83) | 15.62(89)
Table 4: Results for the coefficients of Eqs. (10) and (11) for $B^{0}\to
K^{0}\ell_{1}\ell_{2}$ and $B^{+}\to K^{+}\ell_{1}\ell_{2}$ decay. No
additional uncertainty is included for QED effects.

##  IV $B\to K\nu\bar{\nu}$

Lastly, we study the rare decays $B\to K\nu\bar{\nu}$, which are of
phenomenological interest [76]. Precise measurements in this channel could
shed light on new physics models to explain lepton flavour universality
violation [77, 78].

There is limited experimental data at the present time, and currently no
experimental results have conclusively demonstrated a non-zero branching
fraction. However, more experimental results are anticipated in the near
future, and the upper limit on the branching fraction may soon be driven down
close to theoretical determinations [35].

We can calculate the short distance (SD) contribution to the differential
branching fraction (summed over neutrino flavours) [9, 79, 45], which depends
only on the vector form factor.

$\begin{split}\frac{d\mathcal{B}(B\to
K\nu\bar{\nu})_{\text{SD}}}{dq^{2}}=&\frac{G_{F}^{2}\alpha_{\mathrm{EW}}^{2}(M_{Z})X_{t}^{2}}{32\pi^{5}\sin^{4}\theta_{W}}\\\
&\times\tau_{B}|V_{tb}V^{*}_{ts}|^{2}|\vec{p}_{K}|^{3}f^{2}_{+}(q^{2}),\end{split}$
(15)

with $X_{t}=1.469(17)$ [80], $\sin^{2}\theta_{W}=0.23121(4)$ [43] and
$1/\alpha_{\rm EW}(M_{Z})=127.952(9)$ [43]. Other parameters are given in
Table 1. We evaluate this expression using our improved vector form factor
determined in [3], including uncertainties for strong-isospin breaking
discussed in Section II.1.4. If the reader wishes to reproduce this,
$f_{+}(q^{2})$ can be easily obtained using the code attached to [3]. Note
that no QED uncertainty is required in this case.

This short distance expression is sufficient to describe the case of $B^{0}\to
K^{0}\nu\bar{\nu}$ decays and dominates the branching fraction in $B^{+}\to
K^{+}\nu_{\ell}\bar{\nu_{\ell}}$ decays. In the case of $B^{+}\to
K^{+}\nu_{\tau}\bar{\nu_{\tau}}$ however, long distance (LD) effects from a
double charged-current interaction have a noticeable contribution. These are
given by [81]

$\begin{split}\mathcal{B}(B^{+}\to
K^{+}\nu_{\tau}\bar{\nu}_{\tau})_{\mathrm{LD}}=&\frac{|G_{F}^{2}V_{ub}V^{*}_{us}f_{K^{+}}f_{B^{+}}|^{2}}{128\pi^{2}M^{3}_{B^{+}}}\\\
&\times\frac{m_{\tau}(M^{2}_{B^{+}}-m_{\tau}^{2})^{2}(M^{2}_{K^{+}}-m_{\tau}^{2})^{2}}{\Gamma_{\tau}\Gamma_{B^{+}}}.\end{split}$
(16)

We take the following parameter values to determine this branching fraction.
$|V_{ub}|=0.00370(10)(12)$ is given from exclusive $B\to\pi\ell\nu$ decays in
[43, 54], based on experimental data and lattice QCD form factors from [82,
83]. Other values, $|V_{us}|f_{K^{+}}=0.03509(4)(4)\,\mathrm{GeV}$ [43, 84,
85], $f_{B^{+}}=0.1894(8)(11)(3)(1)\,\mathrm{GeV}$ [53] and
$\tau_{\tau}=0.2903(5)\,\mathrm{ps}$ [43] are used to obtain

$\mathcal{B}(B^{+}\to
K^{+}\nu_{\tau}\bar{\nu_{\tau}})_{\text{LD}}=6.09(53)\times 10^{-7}\,.$ (17)

This is a minor update on the $6.22(60)\times 10^{-7}$ calculated in [9], and
represents an adjustment of about +10% (or roughly $1.5\sigma$) to our
$\mathcal{B}(B^{+}\to K^{+}\nu_{\tau}\bar{\nu_{\tau}})_{\text{SD}}$ result.

Our results for the differential branching fractions for the short-distance
contribution are shown in Figure 30 for both the charged and neutral $B$ meson
cases. Note that there is no issue with $c\overline{c}$ resonances or
nonfactorisable terms in this case so the SM calculation is very
straightforward and simply requires an accurate vector form factor. From the
differential branching fraction we calculate the total branching fractions,
adding in the long-distance contribution to the charged case. These values are
given in Table 5, and show a total uncertainty of around 7%. Of this
uncertainty 5% comes from the vector form factor, 4.4% from the CKM elements
$|V_{tb}V_{ts}^{*}|$ and 2.3% from $X_{t}^{2}$.

Table 5 also compares our values to experimental limits and earlier
theoretical results. Our results sit slightly higher than previous theoretical
determinations and have smaller uncertainty. The comparison of our $B^{+}\to
K^{+}\nu\bar{\nu}$ branching fraction result with other theoretical values is
shown in Figure 31. Results in various $q^{2}$ bins are presented in Table 6.

Decay | $\mathcal{B}\times 10^{6}$ | Reference
---|---|---
$B^{0}\to K_{S}^{0}\nu\bar{\nu}$ |  $<13$ (90% CL) Exp. | [32]
$B^{0}\to K^{0}\nu\bar{\nu}$ |  $<49$ (90% CL) Exp. | [34]
4.01(49) | [9]
$4.1^{+1.3}_{-1.0}$ | [37]
4.60(34) | HPQCD ’22
$B^{+}\to K^{+}\nu\bar{\nu}$ |  $<16$ (90% CL) Exp. | [34]
$<19$ (90% CL) Exp. | [32]
$<41$ (90% CL) Exp. | [33]
5.10(80) | [79, 81]
$4.4^{+1.4}_{-1.1}$ | [37]
3.98(47) | [45]
4.94(52) | [9]
| 4.53(64) | [86]
| 4.65(62) | [87]
| 5.58(37) | HPQCD ’22
Table 5: Summary of experimental and theoretical results for the branching
fraction $\mathcal{B}(B\to K\nu\bar{\nu})$, summed over neutrino flavours. Our
results are denoted by ‘HPQCD ’22’ and include the long-distance contribution
of Eq. (17) in the charged $B$ case. Experimental results are given as 90%
confidence limit and marked ‘Exp.’. In the case of [86], we use
$|V_{tb}V_{ts}^{*}|$ (Table 1) to obtain the result. Figure 30: The short
distance contribution to $d\mathcal{B}(B\to K\nu\bar{\nu})/dq^{2}$, for both
the charged and neutral $B$ meson cases. Figure 31: Our result for
$\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu})$, including long distance effects,
compared with other theory calculations from Table 5 [79, 81, 37, 45, 9, 86,
87]. Figure 32: Our result (black filled star) for $\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})$, summing short- and long-distance effects, compared with
current 90% confidence limits from BaBar [34] and Belle II [33] (blue lines).
Red filled squares with error bars show the expected precision to be achieved
by Belle II with 5 $\text{ab}^{-1}$ and 50 $\text{ab}^{-1}$ [35], assuming our
result is obtained for the central value.

Current experimental bounds on the $B\to K\nu\overline{\nu}$ branching
fraction are roughly an order of magnitude larger than the SM expectation
given by the theory results of Table 5. The uncertainty that we have achieved
here, however, complements the $\approx 10\%$ error expected from Belle II
with 50 $\text{ab}^{-1}$ [35], allowing for a more stringent test of this
quantity in future. This is demonstrated in Figure 32, which compares our
result with the 90% confidence limits set by BaBar [34] and Belle II [33] in
Table 5, as well as the precision forecast by Belle II with 5 $\text{ab}^{-1}$
and 50 $\text{ab}^{-1}$ [35], assuming a result centred on our value.

##  V Conclusions

We have used our improved scalar, vector and tensor form factors calculated in
$N_{f}=2+1+1$ lattice QCD [3] to determine SM observables for $B\to
K\ell^{+}\ell^{-}$, $B\to K\ell^{-}_{1}\ell^{+}_{2}$ and $B\to K\nu\bar{\nu}$
decays. The form factors were calculated in a fully relativistic approach
which enables the full $q^{2}$ range of the decay to be covered. This approach
also gives better control of the current normalisation than in previous
calculations. Our form factors then have smaller uncertainties, particularly
at low $q^{2}$, than earlier work so that we can improve QCD uncertainties on
the SM observables. We have tabulated all of our results across a variety of
$q^{2}$ bins in Appendix C, for future use.

For $B\to K\ell^{+}\ell^{-}$ we determine the differential rate for both
charged and neutral meson cases, including an analysis of nonfactorisable
corrections at low $q^{2}$ (see Appendix B). We then compare to the wealth of
experimental information (see Figures 3, 4 and 5) focussing on the low and
high $q^{2}$ ranges below and above the region where $c\overline{c}$
resonances make a large contribution to experimental results, but are not
included in our calculation. We allow an uncertainty for QED effects missing
from our calculation in this comparison.

Previous lattice QCD calculations by the Fermilab/MILC collaboration [9] saw a
2$\sigma$ tension between the SM branching fraction and LHCb results [19] in
the low and high $q^{2}$ regions, with the SM results being higher than
experiment in both regions. Our improved form factors significantly sharpen
this tension, which becomes particularly strong in the region below the
charmonium resonances. For $1.1<q^{2}<6\text{GeV}^{2}$ we find a tension of
$4.2\sigma$ for $B^{+}\to K^{+}\mu^{+}\mu^{-}$ and $3.1\sigma$ for $B^{0}\to
K^{0}\mu^{+}\mu^{-}$ with LHCb ’14A [19] and $2.7\sigma$ for $B^{+}\to
K^{+}e^{+}e^{-}$ with LHCb ’21 [23] (see Table 3 and Fig. 6). We see no
tension with results from Belle ’19 [15] in this low $q^{2}$ region, but they
have much larger uncertainties. The tension between our results and LHCb in
the high $q^{2}$ region ($15<q^{2}<22{\text{GeV}}^{2}$) amounts to
$2.7\sigma$, again with our SM results being higher than experiment.

The tension between the SM and LHCb results in the ‘safe’ $q^{2}$ regions away
from narrow $c\overline{c}$ resonances points to the possibility of BSM
physics that would have the effect of changing the Wilson coefficients of the
effective weak Hamiltonian. Recent analyses (see, for example, [88, 70]) focus
on BSM changes to $C_{9}$ and $C_{10}$ and include a number of different $B$
decay processes in providing a best fit result for these BSM effects.
Gubernari et al [70] give values $\Delta C_{9}^{\text{BSM}}=-1.0$ and $\Delta
C_{10}^{\text{BSM}}=+0.4$ at $\mu=4.2\,\text{GeV}$ from comparison of their SM
results to experiment in three separate analyses, one of which is a combined
one using $B\to K\mu^{+}\mu^{-}$ and $B\to\mu^{+}\mu^{-}$ and the other two
are for $B\to K^{*}\mu^{+}\mu^{-}$ and $B_{s}\to\phi\mu^{+}\mu^{-}$. If we
repeat our analysis changing $C_{9}$ and $C_{10}$ by these BSM shifts we see
significantly lowered values for the differential rate for $B\to
K\ell^{+}\ell^{-}$ because $C_{9}$ and $C_{10}$ now have smaller magnitudes.
We find the tension with LHCb ’14A in the low $q^{2}$ region (1.1–6
$\text{GeV}^{2}$) to be reduced from 4.2$\sigma$ to 1.1$\sigma$. All of the
other tensions seen in Table 3 where the SM result exceeds the experimental
value are now reduced below 1$\sigma$. At the same time a 2.2$\sigma$ tension
with the Belle ’19 result for $B^{+}\to K^{+}\mu^{+}\mu^{-}$ develops in the
opposite direction. We conclude that BSM shifts to $C_{9}$ and $C_{10}$ of
-1.0 and +0.4 respectively give improved agreement between our results and the
experimental values for the $B\to K\ell^{+}\ell^{-}$ branching fractions.

We update results for ratios of $B\to K\ell^{+}\ell^{-}$ branching fractions
to leptons of different flavour, $\ell$, giving results for $q^{2}$ bins
across the kinematic range in Section II.3.2. The ‘flat term’ in the lepton
angular distribution is also examined in detail in Section II.3.3 for
different $\ell$.

We have also provided improved results for the form-factor-dependent pieces
that, multiplied by appropriate Wilson coefficients, give the branching
fraction for the lepton flavour number violating decay $B\to
K\ell_{1}^{+}\ell_{2}^{-}$. Table 4 gives our results for all 3 lepton flavour
pairs and for charged and neutral $B$ mesons, with uncertainties of 6–7%.
These can be used to constrain BSM models in future searches for this decay.

Finally, Table 5 gives our improved SM values for $\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})$ and $\mathcal{B}(B^{0}\to K^{0}\nu\bar{\nu})$. We have
uncertainties below 10%, commensurate with that expected from Belle II with
$50~{}\mathrm{ab}^{-1}$ worth of data. This decay mode, expected to reach
3$\sigma$ significance at Belle II with $5~{}\mathrm{ab}^{-1}$ of data, is an
exciting possibility for future tests for BSM physics, since it is
theoretically ‘cleaner’ than $B\to K\ell^{+}\ell^{-}$ across the full $q^{2}$
range with no need for vetoed regions.

We have shown here the improvements to $B\to K$ semileptonic decay
phenomenology that result from our improved lattice QCD form factors covering
the full kinematic $q^{2}$ range. For $B\to K\ell^{+}\ell^{-}$ our relative
uncertainties are comparable with those from experiment. Our uncertainties are
dominated by those from the form factors so further improvements in the
lattice QCD calculation of the form factors will reduce them further. This is
possible in future with increased computational power enabling smaller
statistical errors and a push towards smaller values of the lattice spacing
reducing residual discretisation effects. A more detailed comparison will then
be possible with future experimental results with smaller uncertainties for
$B\to K\ell^{+}\ell^{-}$ and future differential rates for $B\to
K\nu\overline{\nu}$. In combination with analysis of other decay processes for
hadrons containing $b$ quarks, this is a promising way forward to uncover
physics beyond the Standard Model.

##  VI Acknowledgements

We thank D. van Dyk, P. Owen and E. Lunghi for clarification regarding data in
their papers and Rafael Coutinho for help understanding the use of PHOTOS in
LHCb results. We also thank L. Grillo and R. Zwicky for useful discussions, as
well as Quim Matias and Sébastien Descotes-Genon. Computing was done on the
Cambridge Service for Data Driven Discovery (CSD3) supercomputer, part of
which is operated by the University of Cambridge Research Computing Service on
behalf of the UK Science and Technology Facilities Council (STFC) DiRAC HPC
Facility. The DiRAC component of CSD3 was funded by BEIS via STFC capital
grants and is operated by STFC operations grants. We are grateful to the CSD3
support staff for assistance. Funding for this work came from STFC.

Note added in proof \- Whilst this paper was being prepared for publication, a
set of new results for $R_{K}$ and $R^{*}_{K}$ appeared from LHCb [89]. These
results agree with Standard Model expectations (see Sec. II.2.3) and supersede
their earlier values.

##  Appendix A Calculating pole masses

The three loop relation between quark masses in the $\overline{\mathrm{MS}}$
scheme and the pole mass scheme is [60]

$\begin{split}\frac{\overline{m}(m)}{m}&=1+A\Big{(}\frac{\alpha_{s}}{\pi}\Big{)}+B\Big{(}\frac{\alpha_{s}}{\pi}\Big{)}^{2}+C\Big{(}\frac{\alpha_{s}}{\pi}\Big{)}^{3},\\\
A&=-\frac{4}{3},\\\ B&=1.0414N_{L}-14.3323,\\\
C&=-0.65269N_{L}^{2}+26.9239N_{L}-198.7068,\end{split}$ (18)

where $\overline{m}$ is the $\overline{\mathrm{MS}}$ mass, $m$ is the pole
mass and $N_{L}$ is the number of active light quarks. We evaluate
$\alpha_{s}$ at scale $m$ and use $N_{L}=3$ for $c$ and $N_{L}=4$ for $b$.

Inverting (18) gives

$\begin{split}m=&\overline{m}(m)\Big{[}1-A\Big{(}\frac{\alpha_{s}}{\pi}\Big{)}+(A^{2}-B)\Big{(}\frac{\alpha_{s}}{\pi}\Big{)}^{2}\\\
&+(-A^{3}+2AB-C)\Big{(}\frac{\alpha_{s}}{\pi}\Big{)}^{3}\Big{]}.\end{split}$
(19)

We solve Equation (19) iteratively, making an initial guess for $m$ then
evaluating $\alpha_{s}(m)$ by running from
$\alpha_{s}(5.0\,\mathrm{GeV})=0.2128(25)$ [90] and $\overline{m}(m)$ by
running from the
$m_{c}^{\overline{\mathrm{MS}}}(m_{c}^{\overline{\mathrm{MS}}})$ or
$m_{b}^{\overline{\mathrm{MS}}}(m_{b}^{\overline{\mathrm{MS}}})$ value in
Table 1. Plugging these results into Equation (19) results in an updated value
for $m$. The initial guess for $m$ is adjusted to reduce the difference
between it and the value obtained from Equation (19). This process is repeated
until the values of $m$ converge.

Using this method, we obtain the pole masses $m_{c}=1.684(22)\,\mathrm{GeV}$
and $m_{b}=4.874(32)\,\mathrm{GeV}$. The perturbation series in this
expression suffers from the presence of a renormalon in the pole mass [61], so
we take a $200\,\mathrm{MeV}$ uncertainty on both numbers. The effect of this
uncertainty is described in Section II.1.1. We note that
$4m_{c}^{2}=11.34\,\mathrm{GeV}^{2}$ falls within the vetoed charmonium
resonance region.

##  Appendix B Corrections to $C_{7}^{\mathrm{eff},0}$ and
$C_{9}^{\mathrm{eff},0}$

As mentioned in Section II.1, corrections need to be applied to
$C_{7}^{\mathrm{eff},0}$ and $C_{9}^{\mathrm{eff},0}$ to obtain the values of
$C_{7}^{\mathrm{eff},1}$ and $C_{9}^{\mathrm{eff},1}$ that enter Equation (4).
The corrections are defined by

$\displaystyle C_{7}^{\mathrm{eff},1}$
$\displaystyle=C_{7}^{\mathrm{eff},0}+\delta C_{7}^{\mathrm{eff}},$ (20)
$\displaystyle C_{9}^{\mathrm{eff},1}$
$\displaystyle=C_{9}^{\mathrm{eff},0}+\Delta C_{9}^{\mathrm{eff}}+\delta
C_{9}^{\mathrm{eff}},$ (21)

where $C_{7}^{\mathrm{eff},0}$ and $C_{9}^{\mathrm{eff},0}$ are given in Table
1. The corrections are discussed in Appendix B of [9], which compiles results
from [26, 27, 91, 92, 93, 94, 64, 95, 96, 97, 98, 99, 100, 101]. We do not
repeat formulae from that reference here, but direct the reader there for more
detail; our aim here is to assess the significance of these corrections to the
decay rates. Below we outline the form of the corrections, plot them versus
$q^{2}$ and discuss their sizes relative to $C_{7}^{\mathrm{eff},0}$ and
$C_{9}^{\mathrm{eff},0}$. All numerical inputs not explicitly stated below can
be found in Table 1.

The leading contribution to the correction $\delta C_{7}^{\mathrm{eff}}$ is
from $\mathcal{O}(\alpha_{s})$ effects,

$\delta
C_{7}^{\mathrm{eff}}=-\frac{\alpha_{s}}{4\pi}\Big{(}(C_{1}-6C_{2})F_{1,c}^{(7)}+C_{8}F_{8}^{(7)}\Big{)},$
(22)

where the expression for $F_{1,c}^{(7)}$ is lengthy and provided in the C`++`
header files of [92]. $F_{8}^{(7)}$ is given in Appendix B of [9]. We use
$\alpha_{s}(4.2\,\mathrm{GeV})=0.2253(28)$, which is run from
$\alpha_{s}(5.0\,\mathrm{GeV})=0.2128(25)$ [90]. The next higher order
contribution, which we neglect, is suppressed by a factor of
$\lambda_{u}^{(s)}=\frac{V_{us}^{*}V_{ub}}{V^{*}_{ts}V_{tb}}=0.01980(62)$ [9]
and is $\mathcal{O}(\alpha_{s}\lambda_{u}^{(s)})$. The leading order
contributions to the correction $\delta C_{9}^{\mathrm{eff}}$ are given by

$\begin{split}\delta
C_{9}^{\mathrm{eff}}=-&\frac{\alpha_{s}}{4\pi}\left(C_{1}F_{1,c}^{(9)}+C_{2}F_{2,c}^{(9)}+C_{8}F_{8}^{(9)}\right)\\\
+&\lambda_{u}^{(s)}\left(h(q^{2},m_{c})-h(q^{2},0)\right)\Big{(}\frac{4}{3}C_{1}+C_{2}\Big{)}.\end{split}$
(23)

We neglect the $\mathcal{O}(\alpha_{s}\lambda_{u}^{(s)})$ term, which is even
smaller (see Equation (B11) of [9] for more details). The function
$h(q^{2},m)$ is defined in Equation (6) and $F_{8}^{(9)}$ is given in Appendix
B of [9]. Expressions for $F_{1,c}^{(9)}$ and $F_{2,c}^{(9)}$ are also
provided in the C`++` header files of [92]. The corrections $\delta
C_{7}^{\mathrm{eff}}$ and $\delta C_{9}^{\mathrm{eff}}$ are applicable across
the full $q^{2}$ range.

Figure 33: The real and imaginary parts of $\delta C_{7}^{\mathrm{eff}}$ and
of the $\mathcal{O}(\alpha_{s})$ and $\mathcal{O}(\lambda_{u}^{(s)})$
contributions to $\delta C_{9}^{\mathrm{eff}}$, as defined in Equations (23)
and (22).

We plot the real and imaginary parts of $\delta C_{7}^{\mathrm{eff}}$ and
$\delta C_{9}^{\mathrm{eff}}$ in Figure 33, showing separately the
$\mathcal{O}(\alpha_{s})$ and $\mathcal{O}(\lambda_{u}^{(s)})$ contributions
to $\delta C_{9}^{\mathrm{eff}}$. The correction Re$[\delta
C_{7}^{\mathrm{eff}}]$ is approximately 20% of
Re$[C_{7}^{\mathrm{eff},0}]\approx-0.3$ (Table 1). The magnitude of Im$[\delta
C_{7}^{\mathrm{eff}}]$ is small in comparison to $|C_{7}^{\mathrm{eff},1}|$.
The $\mathcal{O}(\lambda_{u}^{(s)})$ contributions to $\delta
C_{9}^{\mathrm{eff}}$ are negligible in comparison to
$C_{9}^{\mathrm{eff},0}$, which is approximately 4 across the full $q^{2}$
range (Table 1). The $\mathcal{O}(\alpha_{s})$ contributions to $\delta
C_{9}^{\mathrm{eff}}$ peak at $q^{2}\approx 10\,\mathrm{GeV}^{2}$, due to the
behaviour of the functions $F_{1,c}^{(7)}$, $F_{2,c}^{(7)}$, $F_{1,c}^{(9)}$
and $F_{2,c}^{(9)}$. This peak occurs within the experimentally vetoed
$J/\Psi$ resonance region and is largely contained within the region of
$q^{2}$ between the $J/\Psi$ and $\Psi(2S)$ resonances, outside of which the
contributions are modest. As a result, it has minimal impact on results in the
well behaved regions of $q^{2}$ below the $J/\Psi$ and above the $\Psi(2S)$.
The uncertainty in Re$[\delta C_{7}^{\mathrm{eff}}]$ and Re$[\delta
C_{9}^{\mathrm{eff}}]$ grows rapidly towards $q^{2}_{\mathrm{max}}$ because of
the behaviour of $F_{8}^{(7/9)}$ [9]. This effect is suppressed in observables
by the fact that the differential decay rate vanishes at $q^{2}_{\rm max}$ and
so it is not noticeable in plots of observables versus $q^{2}$ or in
uncertainties of observables in the largest $q^{2}$ bins.

Non-factorisable corrections are accounted for via $\Delta
C_{9}^{\mathrm{eff}}$, which is dependent on the meson charge (see below).
Following the notation of [9],

$\begin{split}&\Delta
C_{9}^{\mathrm{eff}}=\frac{2\pi^{2}m_{b}f_{B}f_{K}}{3M^{2}_{B}f_{+}}\times\\\
&\sum_{\pm}\int_{0}^{\infty}\frac{d\omega}{\omega}\Phi_{B,\pm}(\omega)\int_{0}^{1}du\Phi_{K}(u)\big{[}T_{K,\pm}^{(0)}+\frac{\alpha_{s}}{4\pi}C_{F}T_{K,\pm}^{(\mathrm{nf})}\big{]},\end{split}$
(24)

where $C_{F}=4/3$, and $T_{K,+}^{(0)}=0$. The functions
$\Phi_{B,\pm}(\omega)$, $\Phi_{K}$, and $T_{K,\pm}^{(0/\mathrm{nf})}$ are
given in [91] and Equation (24) is discussed in detail in Appendix B of [9].
We evaluate the expressions for $\Phi_{B,\pm}(\omega)$, $\Phi_{K}$, and
$T_{K,\pm}^{(0/\mathrm{nf})}$ using the following inputs:
$\omega_{0}^{-1}=3(1)\,{\rm GeV}^{-1}$ [91], $a_{1}^{K}=0.0453(30)$, and
$a_{2}^{K}=0.175(50)$ [102] (we take $\Phi_{K}$ to second order). The non-
factorisable corrections are valid for small $q^{2}$, and we turn them off at
$q^{2}=8.68\,\mathrm{GeV}^{2}$, the start of the vetoed $J/\Psi$ resonance
region. We do not calculate through this region. Instead, we linearly
interpolate, beginning from the point where the corrections are turned off
through the $\Psi(2S)$ resonance, so the differential branching fraction is a
smooth function of $q^{2}$ (see, e.g., Figure 13).

Figure 34: The real parts of the contributions to $f_{+}\Delta
C_{9}^{\mathrm{eff}}$ from each of the three non-zero terms, $T_{K,-}^{(0)}$
and $T_{K,\pm}^{(\mathrm{nf})}$ (Equation (24)). A decay channel-specific
factor of $e_{q}\in\\{2/3,-1/3\\}$ is removed from $T_{K,-}^{(0)}$ and
$T_{K,-}^{(\mathrm{nf})}$. Figure 35: The imaginary parts of the contributions
to $f_{+}\Delta C_{9}^{\mathrm{eff}}$ from each of the three non-zero terms,
$T_{K,-}^{(0)}$ and $T_{K,\pm}^{(\mathrm{nf})}$ (Equation (24)). A decay
channel-specific factor of $e_{q}\in\\{2/3,-1/3\\}$ is removed from
$T_{K,-}^{(0)}$ and $T_{K,-}^{(\mathrm{nf})}$. Figure 36: The real and
imaginary parts of $f_{+}\Delta C_{9}^{\mathrm{eff}}$ for both the $B^{0}$ and
$B^{+}$ cases.

The contribution from $\Delta C_{9}^{\mathrm{eff}}$ to observables is via the
term $f_{+}\Delta C_{9}^{\mathrm{eff},1}$ in Equation (4). This has the effect
of cancelling the dependence of $f_{+}\Delta C_{9}^{\mathrm{eff}}$ on the form
factor $f_{+}$. The real and imaginary parts of the three nonzero
contributions to $f_{+}\Delta C_{9}^{\mathrm{eff}}$ in Equation (24),
corresponding to $T_{K,-}^{(0)}$ and $T_{K,\pm}^{(0/\mathrm{nf})}$, are
plotted in Figures 34 and 35. In these plots we remove a decay channel-
specific factor of the light quark charge, $e_{q}$, which is $2/3$ for
$B^{+}\to K^{+}$ and $-1/3$ for $B^{0}\to K^{0}$. Among these terms, the
$T_{K,-}^{(0)}$ contribution is dominant, especially for $q^{2}\lesssim 1$
GeV2.

The combined effect of these terms to $f_{+}\Delta C_{9}^{\mathrm{eff}}$ is
shown in Figure 36, where the real and imaginary parts are plotted separately
for both the $B^{0}$ and $B^{+}$ cases. Both the real and imaginary parts are
smooth functions of $q^{2}$ in the region below the $J/\Psi$ resonance where
they are considered ($4m_{\ell}^{2}\leq q^{2}\leq 8.68$ GeV2), and are small
for $q^{2}>1\,\mathrm{GeV}^{2}$. For $q^{2}<1\,\mathrm{GeV}^{2}$, where the
nonfactorisable corrections are largest, they also have little impact. This is
because the differential decay rate falls rapidly as $q^{2}$ approaches
$m_{l}^{2}$ for kinematic reasons (see the factor of
$\beta=\sqrt{1-4m_{\ell}^{2}/q^{2}}$ in Eq. (II.1)). As a result, the
corrections do not make a significant contribution to the differential
branching fraction.

Figure 37: The real and imaginary parts of $C_{7}^{\mathrm{eff},0}$ and
$C_{7}^{\mathrm{eff},1}$ (see Equation (20)), showing the combined effect of
the nonfactorisable and $\mathcal{O}(\alpha_{s})$ corrections. Note that this
plot is independent of the meson charge. Corrected values are shown with solid
lines and dark fill colour, while uncorrected values are shown with dotted
lines and light fill colour.

Figure 38: The real and imaginary parts of $C_{9}^{\mathrm{eff},0}$ and
$C_{9}^{\mathrm{eff},1}$ (see Equation (21)) for $B^{0}\to K^{0}$ (top) and
$B^{+}\to K^{+}$ (bottom), showing the combined effect of the nonfactorisable
and $\mathcal{O}(\alpha_{s})$ corrections. Corrected values are shown with
solid lines and dark fill colour, while uncorrected values are shown with
dotted lines and light fill colour. $C_{9}=C_{9}^{\mathrm{eff},0}-Y(q^{2})$
(Table 1) is included for comparison.

The overall modest contributions of the above corrections are shown for
$C_{7}^{\mathrm{eff}}$ in Figure 37 and for $C_{9}^{\mathrm{eff}}$ in Figure
38. In each plot, both corrected and uncorrected values are plotted. In the
case of $C_{9}^{\mathrm{eff}}$, the difference between the $B^{0}\to K^{0}$
and $B^{+}\to K^{+}$ corrections, arising from the charge dependence of
$\Delta C_{9}^{\mathrm{eff}}$, is responsible for the slightly different
shapes of the differential branching fractions (see e.g. Figure 3 and 4).
Cusps in $C_{7}^{\mathrm{eff}}$ and $C_{9}^{\mathrm{eff}}$ either occur within
the vetoed $J/\Psi$ resonance region near $q^{2}\approx 10$ GeV2 or at
$q^{2}=4m_{c}^{2}\approx 11.3$ GeV2, between the $J/\Psi$ and $\Psi(2S)$
resonances. In this region between the resonances, we linearly interpolate the
differential decay rate and are therefore unaffected by the cusp at
$q^{2}=4m_{c}^{2}$. The most significant effect of the corrections is an
approximately 20% shift to Re$[C_{7}^{\mathrm{eff},1}]$ arising from
Re$[\delta C_{7}^{\mathrm{eff}}]$. In fact the total effect of all corrections
on our results in the well-behaved regions is small (Section II.2), so we do
not include additional uncertainty on the corrections. The growth of
corrections and in their uncertainties at high and low $q^{2}$ is suppressed
by kinematic factors in the decay rate, resulting in modest impact in the
well-behaved regions of $q^{2}$ where we give our main results (Table 3), as
discussed.

##  Appendix C Numerical Results

The Tables in this appendix list our numerical results in a number of $q^{2}$
bins, aimed at matching those appearing in previous work and providing good
coverage for future work to compare with. The charged and neutral meson case,
as well as the average (no charge label) is given. Similarly the $e$, $\mu$,
average of $e$ and $\mu$ (labelled as $\ell$) and $\tau$ cases are all given.
In all cases, the average is taken (including correlations) after all other
calculation and integration has taken place. In most cases the meson charge
and/or light lepton flavour makes no significant difference. We include all
cases for completeness, and ease of comparison with future results.

###  C.1 Integration

When performing integrals to generate the data in the following tables, as
well as to generate the figures above, we use the trapezoid rule. That is to
say,

$\begin{split}\int_{q^{2}_{\mathrm{low}}}^{q^{2}_{\mathrm{upp}}}&f(q^{2})dq^{2}\approx\\\
&\frac{\Delta
q^{2}}{2}\Big{(}f(q^{2}_{\mathrm{low}})+2f(q^{2}_{1})+...+2f(q^{2}_{N-1})+f(q^{2}_{\mathrm{upp}})\Big{)}\end{split}$
(25)

where $f$ is evaluated at $N+1$ evenly spaced $q^{2}$ values in total. In
order to ensure accurate results, we start with $N=16$, and double $N$
repeatedly until the result for $2N$ differs from that for $N$ by less than
$0.02\,\sigma$ (as uncertainties are given to two significant figures, this is
the point at which on average the last digit of the mean has only changed by
one). Using this method allows us to confirm that our integrals have
converged, and most do so for $N\leq 1024$. An exception is $F^{e}_{H}$. In
this case, the integrand in the numerator (Eq. (9)) changes very rapidly as we
approach the lower limit $4m_{e}^{2}\approx 10^{-6}\,\mathrm{GeV}^{2}$ and a
naive approach will not allow the integrals to converge without very large
values of $N$. This is not the case for $F^{\mu}_{H}$, as the integrals do not
span so many orders of magnitude for $4m_{\mu}^{2}\approx
0.05\,\mathrm{GeV}^{2}$. To solve this problem, we split the integral limits
up into different orders of magnitude, integrating from
$q^{2}_{\mathrm{low}}=4m_{e}^{2}$ to $10q^{2}_{\mathrm{low}}$, then from
$10q^{2}_{\mathrm{low}}$ to $100q^{2}_{\mathrm{low}}$ and so on, up to
$q^{2}_{\mathrm{upp}}$. In each case we increase $N$ until the result is
stable, as above, and then we sum the (correlated) sub-integrals to get the
final value for both the numerator and denominator. This method concentrates
the number of evaluations of $f$ in regions where they are most needed, and so
reduces the total number of evaluations required for the whole integral,
making it tractable.

To assess the suitability of this method, and check that we are not
introducing bias with our algorithm, we checked the integration against the
vegas python package [103], and confirmed that the results agree.

$q^{2}$ bin | $(0.001,4)$ | $(4,8)$ | $(8,12)$ | $(12,16)$ | $(16,20)$ | $(20,q^{2}_{\text{max}})$
---|---|---|---|---|---|---
$10^{6}\mathcal{B}(B^{+}\to K^{+}\nu\bar{\nu}_{~{}\text{SD}})$ | 1.173(95) | 1.139(89) | 1.056(83) | 0.892(71) | 0.589(47) | 0.125(11)
$10^{6}\mathcal{B}(B^{0}\to K^{0}\nu\bar{\nu}_{~{}\text{SD}})$ | 1.087(88) | 1.056(82) | 0.979(77) | 0.826(65) | 0.543(43) | 0.1133(96)
Table 6: Short distance contributions to branching fractions
$\mathcal{B}^{(0/+)}=\mathcal{B}(B^{0/+}\to K^{0/+}\nu\bar{\nu})_{\text{SD}}$
(Equation (15)) for the rare decay $B\to K\nu\bar{\nu}$ integrated over
various $q^{2}$ bins. Numerical values for $q^{2}$ bins are in units of GeV2.

$q^{2}$ bin | $(4m_{\ell}^{2},q^{2}_{\text{max}})$ | $(0.05,2)$ | $(1,6)$ | $(2,4.3)$ | $(4.3,8.68)$ | $(14.18,16)$ | $(16,18)$ | $(18,22)$
---|---|---|---|---|---|---|---|---
$10^{7}\mathcal{B}(B^{+}\to K^{+}e^{+}e^{-})$ | 6.50(50) [5.59(43)] | 0.752(65) | 1.94(16) | 0.902(76) | 1.63(15) | 0.491(44) | 0.441(39) | 0.458(41)
$10^{7}\mathcal{B}(B^{0}\to K^{0}e^{+}e^{-})$ | 5.79(45) [4.97(39)] | 0.655(57) | 1.64(14) | 0.751(64) | 1.44(13) | 0.454(41) | 0.407(36) | 0.419(38)
$10^{7}\mathcal{B}(B\to Ke^{+}e^{-})$ | 6.14(48) [5.28(41)] | 0.703(61) | 1.79(15) | 0.826(69) | 1.54(13) | 0.473(42) | 0.424(38) | 0.438(40)
$10^{7}\mathcal{B}(B^{+}\to K^{+}\mu^{+}\mu^{-})$ | 6.48(50) [5.59(43)] | 0.746(64) | 1.95(16) | 0.902(75) | 1.63(15) | 0.492(44) | 0.442(39) | 0.459(41)
$10^{7}\mathcal{B}(B^{0}\to K^{0}\mu^{+}\mu^{-})$ | 5.77(45) [4.95(38)] | 0.648(57) | 1.64(14) | 0.751(64) | 1.44(13) | 0.455(41) | 0.408(36) | 0.421(38)
$10^{7}\mathcal{B}(B\to K\mu^{+}\mu^{-})$ | 6.13(47) [5.27(40)] | 0.697(60) | 1.79(15) | 0.827(69) | 1.54(13) | 0.473(42) | 0.425(38) | 0.440(40)
$10^{7}\mathcal{B}(B^{+}\to K^{+}\ell^{+}\ell^{-})$ | 6.49(50) [5.59(43)] | 0.749(65) | 1.94(16) | 0.902(76) | 1.63(15) | 0.492(44) | 0.442(39) | 0.458(41)
$10^{7}\mathcal{B}(B^{0}\to K^{0}\ell^{+}\ell^{-})$ | 5.78(45) [4.96(39)] | 0.652(57) | 1.64(14) | 0.751(64) | 1.44(13) | 0.455(41) | 0.408(36) | 0.420(38)
$10^{7}\mathcal{B}(B\to K\ell^{+}\ell^{-})$ | 6.14(47) [5.27(40)] | 0.700(60) | 1.79(15) | 0.827(69) | 1.54(13) | 0.473(42) | 0.425(38) | 0.439(40)

$q^{2}$ bin | $(0.1,2)$ | $(2,4)$ | $(4,6)$ | $(6,8)$ | $(15,17)$ | $(17,19)$ | $(19,22)$ | $(1.1,6)$ | $(15,22)$
---|---|---|---|---|---|---|---|---|---
$10^{7}\mathcal{B}(B^{+}\to K^{+}e^{+}e^{-})$ | 0.736(64) | 0.786(66) | 0.761(66) | 0.740(66) | 0.495(44) | 0.380(34) | 0.284(26) | 1.91(16) | 1.16(10)
$10^{7}\mathcal{B}(B^{0}\to K^{0}e^{+}e^{-})$ | 0.636(55) | 0.653(56) | 0.658(57) | 0.658(58) | 0.458(41) | 0.350(31) | 0.259(24) | 1.60(14) | 1.067(95)
$10^{7}\mathcal{B}(B\to Ke^{+}e^{-})$ | 0.686(59) | 0.719(60) | 0.709(59) | 0.699(61) | 0.477(42) | 0.365(33) | 0.271(25) | 1.75(15) | 1.113(99)
$10^{7}\mathcal{B}(B^{+}\to K^{+}\mu^{+}\mu^{-})$ | 0.733(63) | 0.786(66) | 0.762(66) | 0.741(66) | 0.496(44) | 0.381(34) | 0.285(26) | 1.91(16) | 1.16(10)
$10^{7}\mathcal{B}(B^{0}\to K^{0}\mu^{+}\mu^{-})$ | 0.633(55) | 0.653(56) | 0.658(57) | 0.658(58) | 0.459(41) | 0.351(31) | 0.260(24) | 1.60(14) | 1.070(95)
$10^{7}\mathcal{B}(B\to K\mu^{+}\mu^{-})$ | 0.683(59) | 0.720(60) | 0.710(59) | 0.700(61) | 0.477(42) | 0.366(33) | 0.273(25) | 1.76(15) | 1.116(99)
$10^{7}\mathcal{B}(B^{+}\to K^{+}\ell^{+}\ell^{-})$ | 0.734(63) | 0.786(66) | 0.761(66) | 0.741(66) | 0.496(44) | 0.380(34) | 0.285(26) | 1.91(16) | 1.16(10)
$10^{7}\mathcal{B}(B^{0}\to K^{0}\ell^{+}\ell^{-})$ | 0.635(55) | 0.653(56) | 0.658(57) | 0.658(58) | 0.458(41) | 0.351(31) | 0.260(24) | 1.60(14) | 1.068(95)
$10^{7}\mathcal{B}(B\to K\ell^{+}\ell^{-})$ | 0.685(59) | 0.719(60) | 0.710(59) | 0.699(61) | 0.477(42) | 0.365(33) | 0.272(25) | 1.76(15) | 1.114(99)

$q^{2}$ bin | $(0.1,4)$ | $(4,8.12)$ | $(10.2,12.8)$ | $(14.18,q^{2}_{\text{max}})$ | $(10.09,12.86)$ | $(16,q^{2}_{\text{max}})$
---|---|---|---|---|---|---
$10^{7}\mathcal{B}(B^{+}\to K^{+}e^{+}e^{-})$ | 1.52(13) | 1.55(14) | 0.847(70) | 1.40(12) | 0.903(75) | 0.910(81)
$10^{7}\mathcal{B}(B^{0}\to K^{0}e^{+}e^{-})$ | 1.29(11) | 1.35(12) | 0.772(64) | 1.29(11) | 0.823(68) | 0.836(75)
$10^{7}\mathcal{B}(B\to Ke^{+}e^{-})$ | 1.41(12) | 1.45(12) | 0.809(67) | 1.35(12) | 0.863(71) | 0.873(78)
$10^{7}\mathcal{B}(B^{+}\to K^{+}\mu^{+}\mu^{-})$ | 1.52(13) | 1.55(14) | 0.847(70) | 1.40(12) | 0.904(75) | 0.913(81)
$10^{7}\mathcal{B}(B^{0}\to K^{0}\mu^{+}\mu^{-})$ | 1.29(11) | 1.36(12) | 0.773(64) | 1.29(11) | 0.824(68) | 0.838(75)
$10^{7}\mathcal{B}(B\to K\mu^{+}\mu^{-})$ | 1.40(12) | 1.45(12) | 0.810(67) | 1.35(12) | 0.864(71) | 0.876(78)
$10^{7}\mathcal{B}(B^{+}\to K^{+}\ell^{+}\ell^{-})$ | 1.52(13) | 1.55(14) | 0.847(70) | 1.40(12) | 0.903(75) | 0.911(81)
$10^{7}\mathcal{B}(B^{0}\to K^{0}\ell^{+}\ell^{-})$ | 1.29(11) | 1.36(12) | 0.773(64) | 1.29(11) | 0.824(68) | 0.837(75)
$10^{7}\mathcal{B}(B\to K\ell^{+}\ell^{-})$ | 1.40(12) | 1.45(12) | 0.810(67) | 1.35(12) | 0.864(71) | 0.874(78)

Table 7: Branching fractions integrated over some commonly used $q^{2}$
binning schemes for the electron, muon and $\ell$, which is the average of the
two. In the first bin of the top panel, giving the branching fraction
integrated over the full $q^{2}$ range, the first number is the whole
integral, whilst the one which follows in square brackets is the result when
the ranges 8.68-10.11 $\text{GeV}^{2}$ and 12.86-14.18 $\text{GeV}^{2}$ are
excluded. Numerical values for $q^{2}$ bins are in units of GeV2. QED
corrections discussed in Section II.1.5 are not included here.

$q^{2}$ bin | $(4m_{\mu}^{2},q^{2}_{\text{max}})$ | $(0.05,2)$ | $(1,6)$ | $(2,4.3)$ | $(4.3,8.68)$ | $(14.18,16)$ | $(16,18)$ | $(18,22)$
---|---|---|---|---|---|---|---|---
$10^{3}(R^{\mu(+)}_{e}-1)$ | -0.08(28) | -7.84(95) | 0.39(30) | 0.41(30) | 0.62(25) | 1.43(23) | 1.91(25) | 3.81(37)
$10^{3}(R^{\mu(0)}_{e}-1)$ | -0.81(29) | -10.30(96) | 0.44(34) | 0.47(33) | 0.65(27) | 1.43(23) | 1.92(25) | 3.85(37)
$10^{3}(R^{\mu}_{e}-1)$ | -0.44(26) | -9.03(90) | 0.41(32) | 0.44(31) | 0.63(26) | 1.43(23) | 1.92(25) | 3.83(37)

$q^{2}$ bin | $(0.1,2)$ | $(2,4)$ | $(4,6)$ | $(6,8)$ | $(15,17)$ | $(17,19)$ | $(19,22)$ | $(1.1,6)$ | $(15,22)$
---|---|---|---|---|---|---|---|---|---
$10^{3}(R^{\mu(+)}_{e}-1)$ | -3.29(76) | 0.40(30) | 0.54(27) | 0.65(25) | 1.63(24) | 2.31(27) | 4.57(42) | 0.40(30) | 2.57(29)
$10^{3}(R^{\mu(0)}_{e}-1)$ | -4.25(82) | 0.46(34) | 0.58(29) | 0.68(26) | 1.64(24) | 2.32(27) | 4.63(43) | 0.45(33) | 2.59(29)
$10^{3}(R^{\mu}_{e}-1)$ | -3.75(79) | 0.43(32) | 0.56(28) | 0.66(25) | 1.63(24) | 2.32(27) | 4.60(42) | 0.43(32) | 2.58(29)

$q^{2}$ bin | $(0.1,4)$ | $(4,8.12)$ | $(10.2,12.8)$ | $(14.18,q^{2}_{\text{max}})$ | $(10.09,12.86)$ | $(16,q^{2}_{\text{max}})$
---|---|---|---|---|---|---
$10^{3}(R^{\mu(+)}_{e}-1)$ | -1.35(50) | 0.60(26) | 0.98(23) | 2.50(29) | 0.97(23) | 3.08(32)
$10^{3}(R^{\mu(0)}_{e}-1)$ | -2.30(54) | 0.63(27) | 0.99(23) | 2.51(29) | 0.99(23) | 3.10(32)
$10^{3}(R^{\mu}_{e}-1)$ | -1.80(52) | 0.61(26) | 0.98(23) | 2.50(29) | 0.98(23) | 3.09(32)

Table 8: The ratio $R^{\mu}_{e}$ (Equation 7) integrated over some commonly
used $q^{2}$ binning schemes. We give results for the charged meson case, the
neutral meson case and the charge-averaged case (defined as the ratio of the
two charge-averaged integrals). Numerical values for $q^{2}$ bins are in units
of GeV2. The 1% uncertainty from QED effects [68] (see Sec. II.2.3) is not
included in these numbers.

$q^{2}$ bin | $(4m_{\ell}^{2},q^{2}_{\text{max}})$ | $(0.05,2)$ | $(1,6)$ | $(2,4.3)$ | $(4.3,8.68)$ | $(14.18,16)$ | $(16,18)$ | $(18,22)$
---|---|---|---|---|---|---|---|---
$10^{8}F_{H}^{e(+)}$ | 119(18) | 279.9(3.3) | 57.8(1.0) | 53.44(75) | 26.71(63) | 13.76(55) | 13.74(59) | 16.97(86)
$10^{8}F_{H}^{e(0)}$ | 226(32) | 312.4(5.0) | 57.4(1.1) | 53.42(83) | 26.71(66) | 13.78(55) | 13.77(59) | 17.06(87)
$10^{3}F_{H}^{\mu(+)}$ | 21.61(89) | 105.0(1.1) | 24.46(43) | 22.66(31) | 11.37(26) | 5.87(23) | 5.85(25) | 7.22(36)
$10^{3}F_{H}^{\mu(0)}$ | 22.35(92) | 115.1(1.5) | 24.29(44) | 22.65(34) | 11.37(28) | 5.87(23) | 5.87(25) | 7.26(37)

$q^{2}$ bin | $(0.1,2)$ | $(2,4)$ | $(4,6)$ | $(6,8)$ | $(15,17)$ | $(17,19)$ | $(19,22)$ | $(1.1,6)$ | $(15,22)$
---|---|---|---|---|---|---|---|---|---
$10^{8}F_{H}^{e(+)}$ | 238.4(2.5) | 55.56(75) | 33.14(64) | 24.10(59) | 13.65(56) | 14.15(64) | 18.48(99) | 55.91(99) | 15.00(68)
$10^{8}F_{H}^{e(0)}$ | 256.3(3.1) | 55.57(83) | 33.20(68) | 24.16(62) | 13.67(56) | 14.19(64) | 18.6(1.0) | 55.5(1.0) | 15.04(68)
$10^{3}F_{H}^{\mu(+)}$ | 94.88(90) | 23.55(31) | 14.10(27) | 10.26(25) | 5.82(24) | 6.03(27) | 7.86(42) | 23.66(41) | 6.39(29)
$10^{3}F_{H}^{\mu(0)}$ | 101.4(1.1) | 23.56(34) | 14.12(29) | 10.29(26) | 5.82(24) | 6.04(27) | 7.92(42) | 23.49(42) | 6.41(29)

$q^{2}$ bin | $(0.1,4)$ | $(4,8.12)$ | $(10.2,12.8)$ | $(14.18,q^{2}_{\text{max}})$ | $(10.09,12.86)$ | $(16,q^{2}_{\text{max}})$
---|---|---|---|---|---|---
$10^{8}F_{H}^{e(+)}$ | 143.9(2.4) | 28.47(64) | 17.15(56) | 15.13(67) | 17.18(56) | 15.87(75)
$10^{8}F_{H}^{e(0)}$ | 154.7(2.3) | 28.46(67) | 17.15(57) | 15.16(67) | 17.17(57) | 15.91(76)
$10^{3}F_{H}^{\mu(+)}$ | 57.97(94) | 12.12(27) | 7.31(24) | 6.44(28) | 7.32(24) | 6.75(32)
$10^{3}F_{H}^{\mu(0)}$ | 61.89(89) | 12.11(28) | 7.31(24) | 6.45(29) | 7.32(24) | 6.77(32)

Table 9:  $F_{H}$ (Eq. (9)), which is double the ‘flat term’, integrated over some commonly used $q^{2}$ binning schemes for the $e$ and $\mu$ final-state cases for charged and neutral meson decays. Numerical values for $q^{2}$ bins are in units of GeV2. No uncertainty to allow for QED is included in these numbers. $q^{2}$ bin | $(4m_{\tau}^{2},q^{2}_{\text{max}})$ | $(14.18,q^{2}_{\text{max}})$ | $(14.18,16)$ | $(16,18)$ | $(18,22)$ | $(15,17)$ | $(17,19)$ | $(19,22)$ | $(15,22)$
---|---|---|---|---|---|---|---|---|---
$10^{7}\mathcal{B}(B^{+}\to K^{+}\tau^{+}\tau^{-})$ | 1.68(12) | 1.49(10) | 0.358(26) | 0.426(30) | 0.656(46) | 0.419(30) | 0.414(29) | 0.454(32) | 1.288(90)
$10^{7}\mathcal{B}(B^{0}\to K^{0}\tau^{+}\tau^{-})$ | 1.55(11) | 1.375(95) | 0.332(24) | 0.394(28) | 0.604(42) | 0.388(28) | 0.383(27) | 0.417(29) | 1.188(83)
$10^{7}\mathcal{B}(B\to K\tau^{+}\tau^{-})$ | 1.62(11) | 1.43(10) | 0.345(25) | 0.410(29) | 0.630(44) | 0.403(29) | 0.399(28) | 0.436(30) | 1.238(86)
Table 10: Branching fractions for decays to $\tau$ leptons, integrated over some commonly used $q^{2}$ bins. Numerical values for $q^{2}$ bins are in units of GeV2. No uncertainty to allow for QED is included in these numbers. $q^{2}$ bin | $(4m_{\tau}^{2},q^{2}_{\text{max}})$ | $(14.18,q^{2}_{\text{max}})$ | $(14.18,16)$ | $(16,18)$ | $(18,22)$ | $(15,17)$ | $(17,19)$ | $(19,22)$ | $(15,22)$
---|---|---|---|---|---|---|---|---|---
$R^{\tau(+)}_{e}$ | 0.902(34) | 1.065(42) | 0.729(26) | 0.966(36) | 1.434(62) | 0.846(31) | 1.091(42) | 1.600(74) | 1.111(44)
$R^{\tau(0)}_{e}$ | 0.902(35) | 1.066(42) | 0.730(26) | 0.967(36) | 1.441(63) | 0.847(31) | 1.094(42) | 1.610(74) | 1.113(44)
$R^{\tau}_{e}$ | 0.902(35) | 1.065(42) | 0.729(26) | 0.967(36) | 1.438(63) | 0.847(31) | 1.092(42) | 1.605(74) | 1.112(44)
$R^{\tau(+)}_{\mu}$ | 0.900(34) | 1.063(42) | 0.728(26) | 0.964(36) | 1.429(62) | 0.845(31) | 1.089(42) | 1.592(73) | 1.108(44)
$R^{\tau(0)}_{\mu}$ | 0.900(34) | 1.063(42) | 0.729(26) | 0.966(36) | 1.436(62) | 0.846(31) | 1.091(42) | 1.603(73) | 1.110(44)
$R^{\tau}_{\mu}$ | 0.900(34) | 1.063(42) | 0.728(26) | 0.965(36) | 1.432(62) | 0.845(31) | 1.090(42) | 1.597(73) | 1.109(44)
$R^{\tau(+)}_{\ell}$ | 0.901(34) | 1.064(42) | 0.729(26) | 0.965(36) | 1.432(62) | 0.845(31) | 1.090(42) | 1.596(73) | 1.110(44)
$R^{\tau(0)}_{\ell}$ | 0.901(34) | 1.064(42) | 0.729(26) | 0.967(36) | 1.438(63) | 0.846(31) | 1.092(42) | 1.606(74) | 1.112(44)
$R^{\tau}_{\ell}$ | 0.901(34) | 1.064(42) | 0.729(26) | 0.966(36) | 1.435(62) | 0.846(31) | 1.091(42) | 1.601(74) | 1.111(44)
Table 11: The ratio $R$ (Eq. (7)) for the $\tau$ case to the electron, muon and $\ell$, (which is the average of the two) cases, integrated over some commonly used $q^{2}$ bins. Numerical values for $q^{2}$ bins are in units of GeV2. No uncertainty to allow for QED effects is included in these numbers. $q^{2}$ bin | $(4m_{\tau}^{2},q^{2}_{\text{max}})$ | $(14.18,q^{2}_{\text{max}})$ | $(14.18,16)$ | $(16,18)$ | $(18,22)$ | $(15,17)$ | $(17,19)$ | $(19,22)$ | $(15,22)$
---|---|---|---|---|---|---|---|---|---
$F_{H}^{\tau(+)}$ | 0.8843(46) | 0.8743(50) | 0.9105(32) | 0.8667(50) | 0.8543(64) | 0.8871(41) | 0.8525(57) | 0.8576(66) | 0.8656(54)
$F_{H}^{\tau(0)}$ | 0.8848(46) | 0.8747(50) | 0.9106(32) | 0.8669(50) | 0.8551(64) | 0.8873(41) | 0.8529(57) | 0.8586(66) | 0.8661(54)
Table 12: $F^{\tau}_{H}$ (Eq. (9)) integrated over some commonly used $q^{2}$
bins for charged and neutral $B$ meson decays. Numerical values for $q^{2}$
bins are in units of GeV2. No uncertainty to allow for QED effects is included
in these numbers.

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