Searches for new physics with high-intensity kaon beams

Contributed paper to Snowmass 2021

The KOTO111Contact persons: Hajime Nanjo (nanjo@champ.hep.sci.osaka-u.ac.jp),
Tadashi Nomura (tadashi.nomura@kek.jp), Y.W. Wah(ywah@uchicago.edu),
LHCb222Contact persons: Francesco Dettori (francesco.dettori@cern.ch), Diego
Martinez Santos (diego.martinez.santos@cern.ch), and NA62/KLEVER333Contact
persons: Cristina Lazzeroni (cristina.lazzeroni@cern.ch), Matthew Moulson
<EMAIL_ADDRESS>Collaborations
and the US Kaon Interest Group444W. Altmannshofer, L. Bellantoni, E. Blucher,
G.J. Bock, N.H. Christ, D. Denisov, Y. Grossman, S.H. Kettell, P. Laycock,
J.D. Lewis, H. Nguyen, R. Ray, J.L. Ritchie, P. Rubin, R. Tschirhart, Y. Wah,
E. Worcester, E.D. Zimmerman; Contact person: Elizabeth Worcester
<EMAIL_ADDRESS>
28 April 2022

###### Abstract

The availability of high-intensity kaon beams at the J-PARC Hadron
Experimental Facility and the CERN SPS North Area, together with the abundant
forward production of kaons at the LHC, gives rise to unique possibilities for
sensitive tests of the Standard Model in the quark flavor sector. Precise
measurements of the branching ratios for the flavor-changing neutral current
decays $K\to\pi\nu\bar{\nu}$ can provide unique constraints on CKM unitarity
and, potentially, evidence for new physics. Building on the success of the
current generation of fixed-target experiments, initiatives are taking shape
in both Europe and Japan to measure the branching ratio for
$K^{+}\to\pi^{+}\nu\bar{\nu}$ to $\sim$5% and for
$K_{L}\to\pi^{0}\nu\bar{\nu}$ to $\sim$20% precision. These planned
experiments would also carry out lepton flavor universality tests, lepton
number and flavor conservation tests, and perform other precision measurements
in the kaon sector, as well as searches for exotic particles in kaon decays.
Meanwhile, the LHCb experiment is ready to restart data taking with a trigger
upgrade that will vastly increase its sensitivity for rare $K_{S}$ decays and
complementary hyperon decays. We overview the initiatives for next-generation
experiments in kaon physics in Europe and Japan, identifying potential
contributions from the US high-energy physics community.

###### Contents

  1. 1 Introduction
    1. 1.1 A very brief history
    2. 1.2 The golden kaon decay modes
      1. 1.2.1 $K\to\pi\nu\nu$ in the Standard Model
      2. 1.2.2 $K\to\pi\nu\bar{\nu}$ and the search for new physics
      3. 1.2.3 Experimental status for $K^{+}\to\pi^{+}\nu\bar{\nu}$
      4. 1.2.4 Experimental status for $K_{L}\to\pi^{0}\nu\bar{\nu}$
      5. 1.2.5 $K_{L(S)}\to\mu^{+}\mu^{-}$
      6. 1.2.6 $K_{L(S)}\to\pi^{0}\ell^{+}\ell^{-}$
    3. 1.3 Broader physics programme
      1. 1.3.1 Interplay of kaon physics with lepton flavour anomalies
      2. 1.3.2 Rare kaon decay measurements
      3. 1.3.3 Neutral pion decays
      4. 1.3.4 Searches for dark sectors in kaon decays
      5. 1.3.5 Hyperon decays
  2. 2 Experiments at the J-PARC Hadron Experimental Facility
    1. 2.1 The KOTO experiment until 2021
    2. 2.2 The KOTO experiment from 2022
      1. 2.2.1 Upgrade of trigger and DAQ system
      2. 2.2.2 Upgrade of upstream charged particle veto counter
      3. 2.2.3 Installation of second sweeping magnet
    3. 2.3 KOTO step-2 experiment
      1. 2.3.1 Concept of KOTO step-2
      2. 2.3.2 KL2 beamline
      3. 2.3.3 Activities in the experimental area behind the beam dump
      4. 2.3.4 KOTO step-2 detector
      5. 2.3.5 Beam Condition and Event Selection
      6. 2.3.6 Signal sensitivity estimation
      7. 2.3.7 Background estimation
      8. 2.3.8 Outlook at J-PARC
  3. 3 Experiments at the CERN SPS
    1. 3.1 NA62 until LS3
    2. 3.2 Kaon physics at the SPS after LS3
    3. 3.3 High-intensity beams
      1. 3.3.1 Primary proton beam
      2. 3.3.2 Particle production at the target
      3. 3.3.3 Charged kaon beam
      4. 3.3.4 Neutral kaon beam
      5. 3.3.5 Neutral kaon beam with extended beamline
    4. 3.4 A high intensity $K^{+}$ experiment
      1. 3.4.1 Beam detectors
      2. 3.4.2 Final state detectors
    5. 3.5 $K_{L}\to\pi^{0}\nu\bar{\nu}$: KLEVER
      1. 3.5.1 Layout
      2. 3.5.2 Rates and timing performance
      3. 3.5.3 Main electromagnetic calorimeter (MEC)
      4. 3.5.4 Upstream veto (UV) and active final collimator (AFC)
      5. 3.5.5 Large-angle photon vetoes
      6. 3.5.6 Small-angle calorimeters
      7. 3.5.7 Charged-particle rejection
      8. 3.5.8 Preshower detector
      9. 3.5.9 Expected performance for $K_{L}\to\pi^{0}\nu\bar{\nu}$
    6. 3.6 An intermediate experimental phase targeting $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays
    7. 3.7 Trigger and data acquisition
  4. 4 Experiments at the CERN LHC
    1. 4.1 LHCb in Run-I and II
    2. 4.2 Future results of LHCb and its Upgraded detectors
  5. 5 Outlook and Opportunities for US Collaborators

##  1 Introduction

###  1.1 A very brief history

In 1947, with a cloud chamber observing cosmic rays, George Rochester and
Clifford Butler discovered the neutral kaon [1], dubbed the ‘strange’ particle
at the time. In 1949, Rosemary Brown and Cecil Powell observed the three-pion
decay of the charged kaon [2], dubbed the ‘tau’ particle. In the early
fifties, Murray Gell-Mann recalled [3] that “…Although the presence of Enrico
(Fermi) in the back row was terrifying, nothing went seriously wrong. One day,
in the course of explaining the strangeness idea, I discussed the neutral
meson $K^{0}$ and $\overline{K^{0}}$, which are antiparticle of each other.
Enrico objected that nothing new was involved, since one could take, instead
of the $K^{0}$ and $\overline{K^{0}}$ fields, their sum and difference, which
would then describe two neutral spin zero particle, each its own antiparticle
… I was able to answer Enrico by saying that in certain decays of this neutral
strange particle, the sum and difference would indeed be the relevant fields,
but that in the production process a single $K^{0}$ could accompany a lambda
hyperon, which a $\overline{K^{0}}$ could not. Later (with A. Pais, 1955 [4]),
when I wrote up the neutral K particle situation, I thanked Fermi for his
question…”. $K_{1}$ and $K_{2}$ are the admixtures of $K^{0}$ and
$\overline{K^{0}}$ with even and odd respectively, under the CP
transformation. With the assumption of CP invariance, the $K_{2}$ was
forbidden to decay to two pions and it lives a lot longer than the $K_{1}$.

However, the person who really introduced this particle to the public was Leon
M. Lederman and his collaborators. Their work on the long-lived neutral $K$
mesons was published in 1956 [5]. The paper contained a description of the
long-lived V-particle found in the Brookhaven Cosmotron. They established its
lifetime, and observed some of its decays including semi-leptonic (so called
$K_{e3}$ and $K_{\mu 3}$) and three-pion final states. In late 1956, parity
violation was experimentally observed by C.S. Wu [6], and it shook the physics
world that parity is violated in the weak interaction.

In 1964, CP violation was discovered by James Christenson, James Cronin, Val
Fitch and René Turlay with about 50 samples of $K_{2}$ to two pion decays [7].
The experiment was designed to study the regeneration of the neutral kaons.
When asked, Cronin said the two pions final state study was the very last item
to be studied on a list of more than twenty physics analyses. The neutral kaon
system, via its quantum mechanical interference (mass mixing), provides the
precision measurements of the mass difference and decay width. The question of
the origin of CP violation beyond mixing (the so-called $\epsilon^{\prime}$
direct CP violating effect) was settled some 35 years later. The NA31
experiment at CERN found the first evidence of direct CP violation in 1988
with three standard deviations from zero. However shortly after, the E731
experiment at Fermilab reported a measurement consistent with zero. A better
precision was needed and a new generation of detectors was built. Direct CP
violation was finally established by the NA48 experiment at CERN and the KTeV
experiment at Fermilab in the late 1990s. The history of kaon physics
parallels particle physics, and there may be further surprises as discussed.

For over 70 years, experimental studies of kaons have played a singular role
in propelling the development of the Standard Model (SM). As in other branches
of flavor physics, the continuing experimental interest in the kaon sector
derives from the possibility of conducting precision measurements,
particularly of suppressed or rare processes, that may reveal the effects of
new physics with mass-scale sensitivity exceeding that which can be explored
directly, e.g., at the LHC, or even at a next-generation hadron collider.
Because of the relatively small number of kaon decay modes and the relatively
simple final states, combined with the relative ease of producing intense kaon
beams, kaon decay experiments are in many ways the quintessential intensity-
frontier experiments.

###  1.2 The golden kaon decay modes

Rare kaon decays provide information on the unitary triangle, as illustrated
in Figure 1. These are flavor-changing neutral current processes (FCNC) that
probe the $s\to d\nu\bar{\nu}$ or $s\to d\ell^{+}\ell^{-}$ transitions. They
are highly GIM-suppressed and their SM rates are very small. Complications
from long-distance (LD) physics affect the modes unevenly. As seen from Table
1, the $K\to\pi\nu\bar{\nu}$ decays are the least affected. The branching
ratios (BRs) for the $K\to\pi\nu\bar{\nu}$ decays are among the observable
quantities in the quark-flavor sector most sensitive to new physics.

Figure 1: Determination of the unitary triangle with rare kaon decays. Decay | $\Gamma_{\rm SD}$ | Theory error | SM BR $\times 10^{11}$ | Experimental BR $\times 10^{11}$
---|---|---|---|---
$K_{L}\to\mu^{+}\mu^{-}$ | 10% | 30% |  $79\pm 12$ (SD) | $684\pm 11$
$K_{L}\to\pi^{0}e^{+}e^{-}$ | 40% | 10% | $3.2\pm 1.0$ | $<28$
$K_{L}\to\pi^{0}\mu^{+}\mu^{-}$ | 30% | 15% | $1.5\pm 0.3$ | $<38$
$K^{+}\to\pi^{+}\nu\bar{\nu}$ | 90% | 4% | $8.4\pm 1.0$ | $10.6^{+4.1}_{-3.5}$
$K_{L}\to\pi^{0}\nu\bar{\nu}$ | >99% | 2% | $3.4\pm 0.6$ | $<300$
Table 1: FCNC kaon decays with high sensitivity to short-distance (SD)
physics. Theory errors refer to approximate uncertainty on LD-subtracted rate,
excluding parametric contributions. Experimental upper limits on decay BRs are
quoted at 90% CL.

####  1.2.1 $K\to\pi\nu\nu$ in the Standard Model

The $K\to\pi\nu\bar{\nu}$ decays probe the $s\to d\nu\bar{\nu}$ transition via
the $Z$-penguin and box diagrams shown in Figure 2. For several reasons, the
SM calculation for their BRs is particularly clean [8].

  * •

The loop amplitudes are dominated by the top-quark contributions. The $K_{L}$
decay violates $CP$; its amplitude involves the top-quark contribution only.
In fact, ${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$ gives a direct measurement of
the height (and therefore the area, $J$) of the unitary triangle. Small
corrections to the amplitudes from the lighter quarks come into play for the
charged channel.

  * •

The hadronic matrix elements for these decays can be obtained from the precise
experimental measurements of the $K\to\pi e\nu$ ($K_{e3}$) decay rates.

  * •

There are no long-distance (LD) contributions from processes with intermediate
photons. This point in particular distinguishes the $K\to\pi\nu\bar{\nu}$
decays from the other modes appearing in Figure 1.

Figure 2: Diagrams contributing to the process $K\to\pi\nu\bar{\nu}$.

In the SM, the uncertainties on the $K\to\pi\nu\bar{\nu}$ rates are entirely
dominated by the uncertainties on the CKM matrix elements $|V_{ub}|$ and
$|V_{cb}|$ and the angle $\gamma$. Using values for these parameters from the
analysis of tree-level observables, Buras et al. obtain [9]

$\begin{split}{\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})&=(3.4\pm 0.6)\times
10^{-11},\\\ {\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})&=(8.4\pm 1.0)\times
10^{-11},\end{split}$ (1)

where the uncertainties are dominated by the external contributions from
$V_{cb}$ and $V_{ub}$. These uncertainties can be reduced by adding
information on the CKM parameters from the analysis of loop-level observables
[9]. The CKM analysis has been revisited several times, with values of the
relevant CKM parameters taken from the PDG’s global CKM fit [10], or from
approaches optimized to optimize the uncertainty on the $K\to\pi\nu\bar{\nu}$
BRs [11]. The results of these analyses for the BRs vary within the bracket
expected from the CKM uncertainties. Assuming no new-physics effects in
$\epsilon_{K}$ and $S_{\psi K}$, the BRs can be determined independently of
$|V_{cb}|$ as ${\rm BR}(K^{0}\to\pi^{0}\nu\bar{\nu})=(2.94\pm 0.15)\times
10^{-11}$ and ${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})=(8.60\pm 0.42)\times
10^{-11}$ [11]. In order to avoid the potential for bias to creep in from new
physics sources contributing to loop-level observables, we continue to use the
evaluation of Eqs. (1) as our reference value for the SM BRs. As a
demonstration of the size of the intrinsic theoretical uncertainties, if the
CKM parameters are taken to be exact, the SM BRs of Eqs. (1) become $(3.36\pm
0.05)\times 10^{-11}$ and $(8.39\pm 0.30)\times 10^{-11}$, respectively.
Because of the corrections from lighter-quark contributions, these
uncertainties are larger for the $K^{+}$ case.

####  1.2.2 $K\to\pi\nu\bar{\nu}$ and the search for new physics

Because the $K\to\pi\nu\bar{\nu}$ decays are strongly suppressed and
calculated very precisely in the SM, their BRs are potentially sensitive to
mass scales of hundreds of TeV, surpassing the sensitivity of $B$ decays in
most SM extensions [12].

Because the amplitude for $K^{+}\to\pi^{+}\nu\bar{\nu}$ has both real and
imaginary parts, while the amplitude for $K_{L}\to\pi^{0}\nu\bar{\nu}$ is
purely imaginary, the decays have different sensitivity to new sources of CP
violation. Measurements of both BRs would therefore be extremely useful not
only to uncover evidence of new physics in the quark-flavor sector, but also
to distinguish between new physics models.

Figure 3: Scheme for BSM modifications of $K\to\pi\nu\bar{\nu}$ BRs.

Figure 3, reproduced from Ref. [13], illustrates a general scheme for the
expected correlation between the $K^{+}$ and $K_{L}$ decays under various
scenarios. If the new physics has a CKM-like structure of flavor interactions,
with only couplings to left-handed quark currents, the values for ${\rm
BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ and ${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$
will lie along the green band in the figure. Specifically, for new physics
models with minimal flavor violation, constraints from other flavor
observables limit the expected size of the deviations of the BRs to within
about 10% of their SM values [14]. If the new physics contains only left-
handed or right-handed couplings to the quark currents, it can affect
$K\bar{K}$ mixing ($\Delta F=2$) as well as $K\to\pi\nu\bar{\nu}$ decays
($\Delta F=1$). Then, to respect constraints from the observed value of the
$K\bar{K}$ mixing parameter $\epsilon_{K}$, the BRs for the $K^{+}$ and
$K_{L}$ decays must lie on one of the two branches of the blue cross-shaped
region [15]. This is characteristic of littlest-Higgs models with $T$ parity
[16], as well as models with flavor-changing $Z$ or $Z^{\prime}$ bosons and
pure right-handed or left-handed couplings [12, 17]. In the most general case,
if the new physics has an arbitrary flavor structure and both left-handed and
right-handed couplings, the constraint from $\epsilon_{K}$ is evaded and there
is little correlation, as illustrated by the red region. This is the case for
the general MSSM framework (without minimal flavor violation) [14] and in
Randall-Sundrum models [18].

As discussed in Section 1.2.3, ${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ has
recently been measured to 40% precision by NA62, following upon the collection
of a few candidate events in the late 1990s and early 2000s by the Brookhaven
E787 and E949 experiments. The BR for the neutral decay
$K_{L}\to\pi^{0}\nu\bar{\nu}$ has not yet been measured. Up to small
corrections, considerations of isospin symmetry lead to the model-independent
bound
$\Gamma(K_{L}\to\pi^{0}\nu\bar{\nu})/\Gamma(K^{+}\to\pi^{+}\nu\bar{\nu})<1$
[19], known as the Grossman-Nir bound. Thus the limit on ${\rm
BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ from NA62 discussed in Section 1.2.3 implies
${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})<0.81\times 10^{-9}$.

The observation of three candidate events in the KOTO 2016–2018 data set,
although ultimately found to be statistically compatible with the expected
background, initially gave rise to some theoretical speculation about
mechanisms for evading the Grossman-Nir bound, mainly involving the apparent
enhancement of ${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$ by the contribution
from $K_{L}$ decays into final states containing a $\pi^{0}$ (or two photons)
and one or more new, light particles with very weak SM couplings, the most
studied case being $K_{L}\to\pi^{0}S$, with $S$ a light scalar. Since the
Grossman-Nir relation is based on general isospin symmetry arguments that
relate the decay amplitudes of charged and neutral kaons, it would be expected
to hold also for $K_{L}\to\pi^{0}S$ and $K^{+}\to\pi^{+}S$. Indeed, in most of
these studies, the Grossman-Nir bound is evaded for experimental reasons: NA62
might not detect $K^{+}\to\pi^{+}S$ events with $m_{S}\approx m_{\pi^{0}}$
because of cuts to eliminate $K^{+}\to\pi^{+}\pi^{0}$ background, for example
[20, 21, 22], or because of issues related to the different acceptances of
KOTO and NA62 for different values of the lifetime of $S$ [21]. New physics
scenarios have also been proposed in which the Grossman-Nir bound is violated
at the level of the decay amplitudes [23]. Although, as discussed in Section
1.2.4, there is no evidence for the enhancement of ${\rm
BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$ at this time, with the increased sensitivity
expected from improvements to KOTO, and especially, from the next generation
of $K_{L}\to\pi^{0}\nu\bar{\nu}$ searches, some of these scenarios may regain
relevance.

Like ${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$, the parameter
$\epsilon^{\prime}_{K}$ also gives a direct measurement of the height of the
unitary triangle. Experimentally, ${\rm
Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}$ has been measured by NA48 [24] and
KTeV [25], leading to the average ${\rm
Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}=(16.6\pm 2.3)\times 10^{-4}$ [26]. In
principle, the experimental value of ${\rm
Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}$ significantly constrains the value of
${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$ to be expected in any given new-
physics scenario. However, because of the delicate balance between the
amplitudes for the hadronic matrix elements of different operators, it is
difficult to perform a reliable calculation of ${\rm
Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}$ in the SM. In recent years, there has
been significant progress in evaluating the hadronic matrix elements on the
lattice. In 2015, the RBC-UKQCD Collaboration obtained the result ${\rm
Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}=(1.38\pm 5.15\pm 4.59)\times 10^{-4}$
from a lattice calculation [27], 2.1$\sigma$ less than the experimental value.
However, the detailed systematic analysis of the components of this result, in
particular, the evaluation of the $\Delta I=1/2$ $K\to\pi\pi$ amplitude
$A_{0}$, led to significant improvements in the calculation. The updated
result, ${\rm Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}=(21.7\pm 2.6\pm 6.2\pm
5.0)\times 10^{-4}$, where the third uncertainty is from neglected isospin-
breaking corrections, is much more consistent with the SM expectation [28]. An
estimate based on the same calculation of the hadronic matrix elements, with
evaluation of the isospin-breaking corrections from chiral-perturbation theory
as well as NNLO QCD contributions to electroweak and QCD penguins, gives ${\rm
Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}=(13.9\pm 5.2)\times 10^{-4}$ [29], in
agreement with a recent estimate from chiral-perturbation theory including
isospin-breaking corrections ${\rm
Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}=(14\pm 5)\times 10^{-4}$ [30].
Although at the moment the experimental result is in agreement with SM
expectations, the uncertainties on the expected value remain larger than those
of the measurement by about a factor of two, leaving significant room for
contributions from new physics to the value of ${\rm
Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}$. As an example, the effects on the
BRs for the $K\to\pi\nu\bar{\nu}$ decays from an O(10-TeV) $Z^{\prime}$ boson
with flavor-violating quark couplings or from modified $Z$ couplings from
renormalization-group mixing with the $Z^{\prime}$ have been explored with an
effective field theory framework, with present constraints from
$\epsilon_{K}$, ${\rm Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}$, and $\Delta
m_{K}$ [31, 32]. These studies show that enhancements of 100% or more are
possible for ${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$, with smaller
enhancements ($<50\%$) for ${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$, depending
on the details of the scenario assumed.

As discussed in Section 1.3.1, observations of lepton-flavor-universality-
violating phenomena are mounting in the $B$ sector. Many explanations for such
phenomena predict strong third-generation couplings and thus significant
changes to the $K\to\pi\nu\bar{\nu}$ BRs through couplings to final states
with $\tau$ neutrinos [33]. Measurements of the $K\to\pi\nu\bar{\nu}$ BRs are
therefore critical to interpreting the data from rare $B$ decays, and may
demonstrate that these effects are a manifestation of new degrees of freedom
such as leptoquarks [34, 35, 36]. See Section 1.3.1 for more general
considerations about the interplay of strange physics and lepton flavor
anomalies.

####  1.2.3 Experimental status for $K^{+}\to\pi^{+}\nu\bar{\nu}$

During the period from the mid 1990s to the mid 2000s, the American physics
community demonstrated a particularly keen interest in measuring the BRs for
the $K\to\pi\nu\bar{\nu}$ decays. Although many of the planned experiments
were never realized, a considerable amount of R&D was conducted, resulting in
important inputs for the conceptual design of any new $K\to\pi\nu\bar{\nu}$
experiment.

From 1995 to 2002, the Brookhaven experiments E787 and E949 made the first
measurement of ${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ using low-energy kaons
from the AGS stopped in an active target surrounded by a hermetic array of
detectors for energy and range measurements to identify the $\pi^{+}$ for
candidate signal events. An innovative readout system was used for full
waveform digitization of the signals from the detectors, providing additional
constraints. The first-phase experiment, E787, took data from 1995 to 1998 and
observed two signal candidates [37]; in 2002, the upgraded experiment, E949,
was terminated 12 weeks into a scheduled 60-week AGS run after having
collected an additional five candidate events [38]. In late 2011, ORKA, a
stopped-kaon experiment based on the substantially upgraded E949 apparatus
relocated to the Fermilab Main Injector, was proposed [39]. ORKA aimed to
collect as many as 1000 $K^{+}\to\pi^{+}\nu\bar{\nu}$ events in a five-year
run, but development was terminated by the US DOE’s P5 Panel in May 2014.

The CKM experiment, a decay-in-flight experiment to measure ${\rm
BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ at Fermilab with a 22-GeV, RF-separated
beam, was proposed in 2001 [40]. CKM, with its 100-m length and 25-m-long
fiducial volume, Cherenkov particle identification for both beam and secondary
particles, straw-chamber tracking in vacuum, and hermetic photon vetoes, was
in many ways a precursor to the NA62 experiment at CERN. Unfortunately, the P5
Panel could not identify funding for the experiment in 2003, and an attempt to
reformulate the proposal to reduce costs, largely by abandoning the idea of
the RF-separated beam [41], was not successful.

Figure 4: Schematic diagram (side view) of the NA62 setup.

Following on the successes of NA31 and NA48, the NA62 experiment at the CERN
SPS was proposed [42] in 2005 to measure ${\rm
BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ with a precision of about 10%. Like its
predecessors, NA62 [43, 44] is installed in the ECN3 experimental hall in the
SPS North Area. The experimental setup is schematically illustrated in Figure
4. $K^{+}\to\pi^{+}\nu\bar{\nu}$ decays are observed in flight. The
experimental signature is an upstream $K^{+}$ coming into the experiment and
decaying to a $\pi^{+}$, with no other particles present. Precise
reconstruction of the primary and secondary tracks allows the abundant two-
body decays to be rejected on the basis of the missing mass. The remainder of
the experiment’s rejection power (in particular, for decays without closed
kinematics) comes from redundant particle identification systems and highly-
efficient, hermetic photon veto detectors.

A 400-GeV primary proton beam from the SPS at a nominal intensity of
$3.3\times 10^{12}$ protons per pulse (ppp) is collided on a beryllium target
to produce the 75-GeV unseparated positive secondary beam, which has a total
rate of 750 MHz and consists of about 6% $K^{+}$. The beamline opens into the
vacuum tank about 100 m downstream of the target. The vacuum tank is about 110
m long; the fiducial volume (FV) occupies the first 60 m. About 10% of the
$K^{+}$s entering the experiment decay in the FV, for a rate of 4.5 MHz.

Both the beam particle and the decay secondary are identified and tracked. A
differential Cherenkov counter (KTAG) identifies the kaons in the beam; the
beam spectrometer (Gigatracker, or GTK), consisting of a series of hybrid
silicon pixel tracking detectors installed in an achromat in the beam line,
measures the track parameters. The magnetic spectrometer for the secondary
particles consists of four straw chambers operated inside the vacuum tank.
$\pi/\mu$ separation for secondary particles is provided by the three
downstream hadronic calorimeters/muon vetoes (MUV) and an 18-m-long ring-
imaging Cherenkov counter (RICH) at the end of the vacuum tank.

Rejection of photons from $\pi^{0}\to\gamma\gamma$ decays is important for the
elimination of many background channels, in particular
$K^{+}\to\pi^{+}\pi^{0}$. For these decays, requiring the secondary $\pi^{+}$
to have $p<35$ GeV guarantees that the two photons from the $\pi^{0}$ have a
total energy of at least 40 GeV. The missing-mass cut provides a rejection
power of $10^{4}$, so the probability for the photon vetoes to miss both
photons must be less than $10^{-8}$. Ring-shaped large-angle photon vetoes
(LAVs) are placed at 12 stations along the vacuum volume. Downstream of the
vacuum volume, the NA48 liquid-krypton calorimeter (LKr) vetoes forward, high-
energy photons. Smaller calorimeters around and at the downstream end of the
beam pipe complete the coverage for photons at very small angles.

After commissioning periods in 2014 and 2015, NA62 took physics data in 2016
[45], 2017 [46], and 2018 [47], collecting $2.2\times 10^{18}$ protons on
target (POT) corresponding to $6\times 10^{12}$ $K^{+}$ decays in the fiducial
decay volume, as summarized in Table 2.

| 2016 | 2017 | 2018
---|---|---|---
Days of running | 45 | 160 | 217
Typical intensity, % of nominal | 40 | 55 | 65
Kaon decays in FV | $1.2\times 10^{11}$ | $1.5\times 10^{12}$ | $2.7\times 10^{12}$
Sensitivity ($10^{-10}$) | $3.15\pm 0.24$ | $0.389\pm 0.024$ | $0.111\pm 0.007$
Expected signal (SM) | $0.267\pm 0.037$ | $2.16\pm 0.28$ | $7.6\pm 1.0$
Expected background | $0.15^{+0.09}_{-0.04}$ | $1.46\pm 0.30$ | $5.4^{+1.0}_{-0.8}$
Candidates observed | 1 | 2 | 17
Table 2: Summary of NA62 data taking in 2016 [45], 2017 [46], and 2018 [47].

The trigger for the signal-event (PNN) sample was designed to recognize the
1-track + missing energy topology: a hardware trigger at level 0 used the
signal from the RICH as a time reference with additional conditions on
multiplicities in the veto hodoscopes, the number of clusters and energy in
the LKr and the number of hits in MUV3, while a software trigger at level 1
performed fast online event reconstruction to select events with a single,
positive track, a $K^{+}$ signal from the KTAG, and conditions on the LAVs.
The normalization and control sample was collected with a simple trigger with
loose multiplicity cuts in the trigger hodoscopes. The principal steps for
offline event selection were the reconstruction of the vertex between beam and
secondary tracks, kinematic selection, pion identification and muon rejection,
and vetoes on any additional activity in the LAVs, LKr, small-angle vetoes,
trigger hodoscopes, and tracking detectors.

The single-track topology was reconstructed starting from the beam track in
the GTK in time with a signal from the KTAG. Good events were required to have
a single positive track in the straw chambers and spectrometer. Upstream-
downstream track matching was based on timing criteria ($\sigma_{t}\sim 100$
ps) and geometrical compatibility (distance of closest approach between
upstream and downstream tracks less than 4 mm, with the decay vertex in the
fiducial volume, $110<z<165$ m). The missing mass in the $\pi^{+}$ track-
identification hypothesis was calculated with a resolution $\sigma(m^{2}_{\rm
miss})$ of about 0.001 GeV2, allowing the definition of blinded signal and
control regions as illustrated in Figure 5:

$\displaystyle{\rm Region~{}1:}$ $\displaystyle 0<m^{2}_{\rm miss}<0.10~{}{\rm
GeV}^{2},\>\>15<p_{\pi^{+}}<35~{}{\rm GeV};$ $\displaystyle{\rm Region~{}2:}$
$\displaystyle 0.26<m^{2}_{\rm miss}<0.68~{}{\rm
GeV}^{2},\>\>15<p_{\pi^{+}}<35~{}{\rm GeV}\>\>(45~{}{\rm GeV~{}for~{}2018}).$

Figure 5: Signal, background, and control regions in the $p_{\pi^{+}}$ vs.
$m^{2}_{\rm miss}$ plane used for the NA62 analysis of
$K^{+}\to\pi^{+}\nu\bar{\nu}$ with 2016–2017 data (for 2018, region 2 was
extended to $p_{\pi^{+}}=45$ GeV).

The RICH was used to identify pions with $>75\%$ efficiency and to suppress
muons at the level of 99.5% over the fiducial momentum interval. Information
on the energy sharing and cluster shape in the LKr and the hadron calorimeters
MUV1–2, in combination with the veto on activity in MUV3, provided several
orders of magnitude of additional muon rejection ($6\times 10^{-6}$ at
$p_{\pi^{+}}=15~{}{\rm GeV}$ and improving at higher momentum), with $\sim$80%
efficiency for pion tracks. The photon veto detectors (LAVs, LKR, IRC, and
SAC), together with conditions on extra activity in the straws and trigger
hodoscopes, were used to reject $K^{+}\to\pi^{+}\pi^{0}(\gamma)$ events with
$p_{\pi^{+}}$ in the fiducial range down to the level of $2.7\times 10^{-8}$,
as verified by tag-and-probe measurements of the single-photon detection
efficiencies folded with $K^{+}\to\pi^{+}\pi^{0}(\gamma)$ kinematics by Monte
Carlo simulation [48]. The veto efficiency for events with extra activity
allowed a separate limit to be placed on the invisible branching ratio of the
$\pi^{0}$: $<4.4\times 10^{-9}$ at 90% CL.

The remaining backgrounds from $K^{+}\to\mu^{+}\nu$, $K^{+}\to\pi^{+}\pi^{0}$,
and $K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decays were estimated from the fractions
of the missing-mass distributions in the tails extending into the signal
regions measured with control samples. For example, the control sample of
$K^{+}\to\mu^{+}\nu$ decays was obtained by applying the $\pi\nu\bar{\nu}$
selection criteria to the data collected with the control trigger, but with
the $\pi^{+}/\mu^{+}$ separation criteria inverted. The control sample of
$K^{+}\to\pi^{+}\pi^{0}$ was obtained by applying the single-track and
$\pi^{+}$ identification criteria to the control sample, but without applying
the photon veto criteria, instead reconstructing the $\pi^{0}$ in the LKr and
using the missing mass calculated from the $\pi^{0}$ momentum and the nominal
$K^{+}$ momentum for event selection. The $K^{+}\to\pi^{+}\pi^{0}$ control
sample does not include $K^{+}\to\pi^{+}\pi^{0}\gamma$ events with the energy
of the radiative photon high enough to shift the value of the missing mass
into signal region 2. Such events are much more likely to be rejected than
non-radiative events due to the presence of the additional photon, and the
very small corresponding background contribution is estimated with Monte Carlo
studies folding the single-photon detection efficiency with the
$K^{+}\to\pi^{+}\pi^{0}\gamma$ kinematics.

Process | Expected events | Expected events
---|---|---
| 2017 data | 2018 data
$K^{+}\to\pi^{+}\nu\bar{\nu}$ | $2.16\pm 0.13\pm 0.26_{\rm ext}$ | $7.58\pm 0.40\pm 0.90_{\rm ext}$
$K^{+}\to\pi^{+}\pi^{0}(\gamma)$ | $0.29\pm 0.04$ | $0.75\pm 0.05$
$K^{+}\to\mu^{+}\nu(\gamma)$ | $0.15\pm 0.04$ | $0.64\pm 0.08$
$K^{+}\to\pi^{+}\pi^{-}e^{+}\nu$ | $0.12\pm 0.08$ | $0.51\pm 0.10$
$K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ | $0.008\pm 0.008$ | $0.22\pm 0.08$
$K^{+}\to\pi^{+}\gamma\gamma$ | $0.005\pm 0.005$ | $<0.01$
$K^{+}\to\pi^{0}\ell^{+}\nu$ | $<0.001$ | $<0.001$
Upstream background | $0.89\pm 0.31$ | $3.30^{+0.98}_{-0.73}$
Total background | $1.46\pm 0.33$ | $5.42^{+0.99}_{-0.75}$
Table 3: Expected numbers of signal and background events in NA62 data in the
2017 [46] and 2018 [47] datasets. Figure 6: Distribution in the
$(p_{\pi^{+}},m^{2}_{\rm miss})$ plane of events in 2018 data passing all
selection cuts, before unblinding of the signal regions. The observed and
expected numbers of background events are indicated for each control region
[47]. Figure 7: Distribution in the $(p_{\pi^{+}},m^{2}_{\rm miss})$ plane of
events in 2018 data passing all selection cuts, showing 17 events in the
signal regions after unblinding [47].

Accidental matching of a beam track and a $\pi^{+}$ track in the spectrometer
can occur when a $K^{+}$ decays or interacts upstream of the last GTK station.
For this reason, events are discarded in which the $\pi^{+}$ track
extrapolates upstream to the exit from the final collimator (the “box cut”).
However, some of these “upstream background” events may not be eliminated.
Consider the following scenario:

  * •

A $K^{+}$ identified as a kaon in the KTAG decays to $\pi^{+}\pi^{0}$ inside
the GTK, and the photons from the $\pi^{0}$ are not observed. The $\pi^{+}$
from the decay makes it through or around the final collimator and is seen in
the spectrometer, but a beam track is not reconstructed or is reconstructed
poorly in the GTK.

  * •

A $\pi^{+}$ in accidental coincidence with the $K^{+}$ leaves a track in the
GTK but does not decay and does not emerge from the beam pipe, and so is not
observed in the spectrometer.

  * •

The $\pi^{+}$ from the $K^{+}$ decay undergoes multiple scattering in the
first tracking station, so that its track is reconstructed after the
scattering. Offline, it is extrapolated upstream and matched to the track from
the accidental $\pi^{+}$, forming a vertex in the fiducial volume. Because of
the scattering, it does not extrapolate to the exit of the final collimator
and is not rejected.

This background was quantified by developing a Monte Carlo model for the
upstream background and validating it against control data samples on the
basis of distributions of such quantities as the distance of closest approach
between beam and spectrometer tracks. Then, the final estimates of the
upstream background were validated by comparison with subsamples defined by
different inversions of background suppression cuts, such as the box cut, the
missing-mass cuts, a veto on extra GTK activity, and a veto on the activity in
the CHANTI (a charged-particle veto around the beamline located just
downstream of the GTK). Through the start of the 2018 run, a large region
around the exit of the final collimator was used for upstream background
rejection, keeping the background level acceptable, but incurring a
significant loss in signal acceptance. A new final collimator was installed
just after the start of 2018 run, allowing the use of a much smaller rejection
region. Table 3 compares the number of expected events for the signal and most
important background channels in 2017 and 2018, demonstrating the effect of
the new collimator and other improvements to the analysis, such as the
extension of the upper limit of the fiducial momentum region from 35 GeV to 45
GeV for signal selection region 2, as well as of moderate increases in running
time and intensity. The distribution in the $(p_{\pi^{+}},m^{2}_{\rm miss})$
plane of the events in 2018 data passing all selection cuts, before unblinding
of the signal regions, is presented in Figure 6. The observed and expected
numbers of background events in each control region are indicated and
demonstrate satisfactory agreement. The same distribution with the signal
regions unblinded, showing the 17 $K^{+}\to\pi^{+}\nu\bar{\nu}$ candidates in
2018 data, is presented in Figure 7.

In all, from the 20 candidate events observed in the 2016–2018 data (see Table
2), NA62 obtains

${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})=\left(10.6\,^{+4.0}_{-3.4}\left|{}_{\rm
stat}\pm 0.9_{\rm syst}\right.\right)\times 10^{-11}.$ (2)

This result, the first measurement of ${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$,
is in agreement with the SM prediction, and provides evidence for the
existence of the decay with $3.4\sigma$ significance. In data-taking during
LHC Run III, which started in July 2021 and will continue until the start of
Long Shutdown III, NA62 will reduce the uncertainty on this result, aiming to
collect enough events to reach order 10% precision [49, 50], as discussed in
Section 3.1.

####  1.2.4 Experimental status for $K_{L}\to\pi^{0}\nu\bar{\nu}$

The history of searches for the $K_{L}\to\pi^{0}\nu\overline{\nu}$ decay is
shown in Fig. 8 with the upper limits and the single event sensitivities. The
decay was firstly studied by interpreting $K_{L}\to\pi^{0}\pi^{0}$ data [51].
The E731, the E799, the KTeV experiments at Fermilab and the KEK E391a
experiment updated the upper limit of the branching ratio [52, 53, 54, 55, 56,
57, 58]. Currently, the KOTO experiment, a successor of the KEK E391a
experiment, is the only running experiment. KOTO has been improving the
sensitivity [59, 60, 61].

Figure 8: History of $K_{L}\to\pi^{0}\nu\bar{\nu}$ searches [51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61].

As was the case for the $K^{+}$ decay, there were complementary proposals for
measurements of the neutral decay at Brookhaven and at Fermilab during the
late 1990s and early 2000s. As in the case for the charged decay, neither
experiment was ultimately realized, but substantial R&D work was carried out,
of considerable interest for the design of new experiments.

The KOPIO experiment at Brookhaven [62] was designed to exploit the capability
of the AGS to provide microbunched slow-extracted proton beam to create a low-
momentum neutral beam with a time structure allowing the use of time-of-flight
constraints for signal identification. During the extraction flat top, the
primary beam was to be delivered in 200 ps pulses every 40 ns, with an
extinction of $10^{-3}$ between pulses. The neutral secondary beam, of peak
momentum 0.7 GeV, was tightly collimated in the vertical plane only, to
maximize the beam solid angle (and therefore flux) while preserving useful
geometrical constraints. A calorimeter at the downstream end of the decay
volume was used to precisely measure the arrival times and energies of photons
from candidate signal decays; a preradiator in front of the calorimeter was
used to obtain the angles of incidence of these photons as well. This
information, together with the a priori knowlegde of the time structure of the
beam, allowed for event-by-event reconstruction of the $K_{L}$ momentum by
time-of-flight, providing redundant constraints for signal identification and
background rejection. The highly innovative KOPIO proposal was supported by a
decade of planning and R&D; ultimately, however, funding for rare decay
physics at Brookhaven was terminated in 2005.

The proposed high-energy experiment at Fermilab, KAMI [63], took a more
conventional approach. The 120-GeV primary proton beam from the Main Injector
was to be used to obtain a tightly collimated neutral secondary beam at an
angle of 12–20 mrad, with a mean momentum of about 20 GeV. The total length of
the beamline and detector was about 200 m, including an evacuated volume
hermetically enclosed by photon vetoes of about 90 m. The fiducial volume
spanned approximately the first 65 m of the vacuum tank. A unique feature of
KAMI was the presence of a charged-particle spectrometer consisting of five
stations of scintillating fiber planes straddling a large-gap dipole magnet,
making possible a complete physics program in rare $K_{L}$ decays, as well as
providing additional constraints in the reconstruction of control samples of
abundant $K_{L}$ decays to keep measurement systematics under control. The
KAMI proposal was rejected by the Fermilab PAC in 2001 because of concerns
that the target sensitivity (100 events) might not be reached due to
insufficient redundancy to guarantee rejection of background from
unanticipated sources.

The E391a experiment at KEK 12-GeV Proton Synchrotron was the first dedicated
experiment to search for the $K_{L}\to\pi^{0}\nu\overline{\nu}$ decay. With
the data taken in 2005, the single event sensitivity of $1.1\times 10^{-8}$
was achieved, and the upper limit of the branching ratio was set to be
$2.6\times 10^{-8}$ at 90% CL [58]. A narrow $K_{L}$ beam (“pencil beam”) was
prepared with 10-m-long beam line. At the downstream of the beam line, the
E391a detector with a decay volume was prepared. Most of the sub-detectors and
the decay volume were in the 4-m-diameter 9-m-long evacuated vacuum tank. The
observed particles are required to be only two photons from a $\pi^{0}$ to
identify the signal. Two photons from a $\pi^{0}$ were detected with an endcap
calorimeter downstream of the decay volume. A hermetic veto system surrounding
the decay volume ensured no other particles detected. The $\pi^{0}$ decay
vertex was reconstructed assuming the two photons from a $\pi^{0}$ decay. The
decay vertex could be assumed on the beam axis owing to the "pencil beam". The
four vectors of the two photons and the $\pi^{0}$ were fully reconstructed.
The transverse momentum ($p_{T}$) of the $\pi^{0}$ can be larger due to
missing two neutrinos. Therefore, the $p_{T}$ can be used to identify the
signal. This “pencil beam” method was demonstrated in the E391a experiment,
which gave a basis of the next high sensitivity experiments.

Figure 9: Schematic diagram of the KOTO detector.

The KOTO experiment at J-PARC was designed with the “pencil beam” method. The
salient features of the experiment are the use of a highly collimated, neutral
beam and very high performance hermetic calorimetry and photon vetoing. The
30-GeV primary proton beam is interacted on a gold target, and secondary
particles are generated. Neutral beam is prepared from the secondaries with
20-m-long beam line. Two collimators in the beam line shape the beam in a
solid angle of 7.8 $\mu$sr at an extraction angle of $16^{\circ}$, which
provides $K_{L}$ beam with a peak momentum of 1.4 $\mathrm{GeV}/c$ in a
$8\times 8~{}\mathrm{cm}^{2}$ cross-section at the beam line exit.
Contamination of short-lived and charged particles in the beam is suppressed
with the long beam line and a sweeping magnet, while the photon flux is
reduced using a 7-cm long lead absorber. At the downstream of the beam line,
the KOTO detector with the decay volume is located (Fig. 9). Most of the sub-
detectors and the decay volume are in the 4-m diameter 9-m long evaculated
chamber. Two photons from a $\pi^{0}$ are detected with a calotimeter (CsI)
composed of undoped CsI crystals. The $\pi^{0}$ vertex and the $p_{T}$ are
reconstructed with the “pencile beam” method. The reconstructed vertex
position and $p_{T}$ of the $\pi^{0}$ are used to define the signal region.
The $K_{L}\to 2\gamma$ decay is suppressed by requiring higher $p_{T}$ of the
reconstructed $\pi^{0}$. Other backgrounds from $K_{L}$ decays involving
charged particles or additional photons are reduced using the hermetic veto
system surrounding the decay volume.

The J-PARC has achieved 64 kW beam power in the 30-GeV-proton slow extraction,
which corresponds to $7\times 10^{13}$ protons per 5.2-s cycle. Shorter cycle
of 4.2 s and more protons in a spill are expected after an on-going
accelerator upgrade in 2021-2022, and 100-kW beam power is expected to be
achieved.

The KOTO experiment currently set an upper limit of the branching ratio at
$3\times 10^{-9}$ (90% CL) with no candidate events observed from the analysis
of data collected in 2015 [60]. From the analysis of data collected in
2016-2018, KOTO observed three candidate events at the single event
sensitivity of $7.2\times 10^{-10}$, which is consistent to the expected
number of background 1.2 [61].

Figure 10: Distribution of events in reconstructed $p_{T}$ and $z$ vertex
plain after imposing the event selection [61]. The region surrounded with
dotted lines is the signal region. Black dots corresponds to the observed
events. The shaded contour shows the distribution of
$K_{L}\to\pi^{0}\nu\overline{\nu}$ MC. The number of observed (expected)
events in each region is shown with black (red) number. Table 4: Summary of
the numbers of background events with a central value estimate [61]

[h]  source Number of events $K_{L}$ $K_{L}\to\pi^{0}\pi^{0}\pi^{0}$ 0.01
$\pm$ 0.01 $K_{L}\to 2\gamma$ (beam halo) 0.26 $\pm$ 0.07 a Other $K_{L}$
decays 0.005 $\pm$ 0.005 $K^{\pm}$ 0.87 $\pm$ 0.25 a Neutron Hadron cluster
0.017 $\pm$ 0.002 CV $\eta$ 0.03 $\pm$ 0.01 Upstream $\pi^{0}$ 0.03 $\pm$ 0.03
total 1.22 $\pm$ 0.26

The contribution of each background source is shown in Table 4. The $K^{\pm}$
background contributes dominantly. In this background, KOTO found charged kaon
contaminates the neutral beam slightly, and the decay
$K^{\pm}\to\pi^{0}e^{\pm}\nu$ mainly fakes the signal with $e^{\pm}$ missed
due to detector inefficiency. This charged kaon background is reduced with a
low-mass in-beam charged particle counter at the entrance of the detector. A
(prototype) counter composed of plastic scintillating fibers was already
installed and used in (2020) 2021. Currently it reduces the background by one
order or more. In 2022, a new counter with 0.2-mm-thick scintilltor film will
be installed with possible reduction factor of 100. The second largest one is
$K_{L}\to 2\gamma$ decay in the beam halo region, which gives fake $p_{T}$.
Multivariate analysis using cluster energies, positions, and shapes can be
used to reduce this background, because actual vertex position of $\pi^{0}$ is
far from the beam axis. Currently the background can be reduced by a factor of
30 or more. KOTO will reach the single event sensitivity of $O(10^{-11})$ in
mid 2020s with possible 10 times larger data owing to the accelerator upgrade.
To go beyond this sensitivity, a next-generation experiment is needed in order
to actually measure the branching ratio.

####  1.2.5 $K_{L(S)}\to\mu^{+}\mu^{-}$

The $K_{L(S)}\to\mu^{+}\mu^{-}$ decays are suppressed in the SM, which
predicts [64, 65, 66, 67]:

${\rm BR}(K_{L}\to\mu^{+}\mu^{-})=\begin{cases}&(6.85\pm 0.80_{\textrm{LD}}\pm
0.06_{\textrm{SD}})\times 10^{-9}\\\ &(8.11\pm 1.49_{\textrm{LD}}\pm
0.1_{\textrm{SD}})\times 10^{-9}\end{cases}$ (3)

and

${\rm BR}(K_{S}\to\mu^{+}\mu^{-})=(5.18\pm 1.50_{\textrm{LD}}\pm
0.02_{\textrm{SD}})\times 10^{-12},$ (4)

where SD stands for Short-Distance, LD for Long-Distance, and the two ${\rm
BR}(K_{L}\to\mu^{+}\mu^{-})$ possibilities correspond to the two values of the
unknown sign of the LD amplitude $A_{L}^{\gamma\gamma}$. These decays are very
sensitive to dynamics beyond the standard model, such as scalar or vector
Leptoquarks [68, 69, 70] or the Minimal Supersymmetric Standard Model (MSSM)
[71]. The current experimental measurement [72, 73, 74, 26],

$\textrm{BR}_{\rm exp}(K_{L}\to\mu^{+}\mu^{-})=\left(6.84\pm 0.11\right)\times
10^{-9},$ (5)

agrees with one of the two predictions of the SM within the theoretical
uncertainties, leaving no room for a BSM discovery in this decay alone unless
the theory significantly improves. On the other hand, the current experimental
upper bound set by LHCb [75],

${\rm BR}(K_{S}\to\mu^{+}\mu^{-})<2.1\times 10^{-10},$ (6)

leaves still an almost two orders-of-magnitude uncharted territory. In
addition, a positive signal could allow a measurement of the $K_{S}/K_{L}$
interference term by means of a tagged analysis [66]. This can be done in LHCb
by tagging the $K^{0}$ flavor using accompanying particles in reactions like
$pp\rightarrow K^{0}K^{-}X$ or $pp\rightarrow K^{0}\Lambda^{0}X$, where the
experiment would measure an effective yield:

$\displaystyle\textrm{BR}(K_{S}\to\mu^{+}\mu^{-})_{\rm
eff}=\tau_{S}\left(\int^{t_{{\rm max}}}_{t_{{\rm
min}}}dte^{-\Gamma_{S}t}\varepsilon(t)\right)^{-1}\Biggl{[}\int^{t_{{\rm
max}}}_{t_{{\rm
min}}}dt\Biggl{\\{}\Gamma(K_{S}\to\mu^{+}\mu^{-})e^{-\Gamma_{S}t}$
$\displaystyle~{}~{}+\frac{Df_{K}^{2}M_{K}^{3}\beta_{\mu}}{{8}\pi}\textrm{Re}\left[i\left(A_{S}A_{L}-\beta_{\mu}^{2}{B_{S}^{\ast}}B_{L}\right)e^{-i\Delta
M_{K}t}\right]e^{-\frac{\Gamma_{S}+\Gamma_{L}}{2}t}\Biggr{\\}}\varepsilon(t)\Biggr{]},$
(7)

as a function of the dilution factor:

$\displaystyle D=\frac{K^{0}-\bar{K^{0}}}{K^{0}+\bar{K^{0}}}.$ (8)

At the LHC the $K^{0}$ and $\bar{K^{0}}$ are produced approximately in equal
amounts due to the high energy, but the dilution factor can be controlled
experimentally if flavor tagging is provided by channels like the ones
mentioned above. A measurement of the interference also allows determining the
scalar amplitude of the decay, which is theoretically extremely clean;
moreover it is suggested that the interference can also be measured using
$K_{S}$ regeneration in high intensity $K_{L}$ beams [76].

####  1.2.6 $K_{L(S)}\to\pi^{0}\ell^{+}\ell^{-}$

The FCNC transitions $K\to\pi\ell^{+}\ell^{-}$ ($\ell=e,\mu$) are dominated by
single virtual-photon exchange $K\to\pi\gamma^{*}\to\pi\ell^{+}\ell^{-}$ if
allowed by CP invariance as in $K_{S}$ and $K^{\pm}$ decays. However, this
contribution is CP violating for $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ and
consequently this process has become a point of reference for studying the CP-
violating sector of the SM [8].

Like $K_{L}\to\pi^{0}\nu\bar{\nu}$, the short-distance contribution to the
$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays depends on $|V_{ts}^{*}V_{td}|$ and
thus measures the height of the unitarity triangle shown in Fig. 1. To the
extent that there is little constraint on the CP-violating phase of the $s\to
d\ell^{+}\ell^{-}$ transition, measurement of the
$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ BRs may reveal the effects of new physics
[77]. However, unlike in the case of $K_{L}\to\pi^{0}\nu\bar{\nu}$, there are
long-distance contributions to the decay amplitude that are of the same order
of magnitude as the short-distance component. First, there is an indirect CP-
violating contribution of the type $K_{L}\to K_{S}\to\pi^{0}\ell^{+}\ell^{-}$
the magnitude of which can be obtained from measurements of the $K_{S}$
decays. Second, for $K_{L}\to\pi^{0}\mu^{+}\mu^{-}$, there is a CP-conserving
long-distance contribution mediated by $K_{L}\to\pi^{0}\gamma\gamma$ (this is
helicity suppressed for the $e^{+}e^{-}$ mode). In the SM, ${\rm
BR}(K_{L}\to\pi^{0}e^{+}e^{-})=3.2\left({}^{+8}_{-9}\right)\times 10^{-11}$
and ${\rm BR}(K_{L}\to\pi^{0}\mu^{+}\mu^{-})=1.29(24)\times 10^{-11}$ [78].
The best experimental limits are from KTeV: ${\rm
BR}(K_{L}\to\pi^{0}e^{+}e^{-})<28\times 10^{-11}$ [79] and ${\rm
BR}(K_{L}\to\pi^{0}\mu^{+}\mu^{-})<38\times 10^{-11}$ [80] at 90% CL.

Branching ratios of the $K_{S}\to\pi^{0}\ell^{+}\ell^{-}$ decays were measured
by NA48/1 [81, 82]:

${\rm BR}(K_{S}\to\pi^{0}e^{+}e^{-})=\left(3.0^{+1.5}_{-1.2}\right)\times
10^{-9},\quad{\rm
BR}(K_{S}\to\pi^{0}\mu^{+}\mu^{-})=\left(2.9^{+1.5}_{-1.2}\right)\times
10^{-9}.$ (9)

In the $e^{+}e^{-}$ case, there are complications from photon-conversion
background from $K_{S}\to\pi^{0}\pi^{0}$, but the level of precision achieved
is similar to that for $\mu^{+}\mu^{-}$. Significant progress on the
measurements of these decays is required to be able to provide a more precise
determination of the indirect CP-violating contribution to the
$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decay rates.

###  1.3 Broader physics programme

The large datasets of ${\cal O}(10^{14})$ kaon decays to be collected by the
future experiments aiming at $K\to\pi\nu\bar{\nu}$ and
$K_{L}\to(\pi^{0})\ell^{+}\ell^{-}$ measurements, as well as the state-of-the-
art detectors involving precision tracking, redundant particle identification,
high performance calorimetry and hermetic photon vetoes, lead to a broad
physics program in addition to the “golden modes” discussed in Section 1.2. A
review of rare kaon decays is provided in Ref. [8], while the relevant hidden-
sector scenarios are reviewed in Ref. [83]. Moreover, future $K^{+}$ decay
experiments may serve as a proof of concept for a tagged neutrino facility
[84].

####  1.3.1 Interplay of kaon physics with lepton flavour anomalies

Significant hints for violation of lepton universality have been seen at
various experiments, usually referred to as ‘anomalies’. These include the
$(g-2)_{\mu}$ anomaly [85], the $B^{+}$ anomalies in $b\to c\ell\nu$ reported
by LHCb, Belle, and BaBar, and the $B^{0}$ anomalies observed by $b\to
s\ell\ell$ at LHCb. The latter gets further strengthened by small (but
consistent) departures of $b\to s\mu\mu$ exclusive branching fractions with
respect to the SM prediction, including differential branching fractions in
$B^{0}\to K^{*}\mu\mu$ decays. The combined global significance for BSM
dynamics in $b\to s\ell\ell$ transitions is at present at the 4.3–5.5$\sigma$
level [86, 87]. A broad set of models are being proposed by the theory
community to address those, most notably leptoquark models, but also other
possibilities like $Z^{\prime}$ models or RPV SUSY models. The strangeness
decays are needed in this scenario in order to determine the dynamics behind
those anomalies: the $s\to d\ell\ell$ transition is the equivalent to $b\to
s\ell\ell$ for the second generation of quarks, and, similarly, $s\to
u\ell\nu$ is the equivalent of $b\to c\ell\nu$. Thus, strange decays allow to
understand how the BSM dynamics couples to each quark generation or, in other
words, its flavor structure. Some recent examples for the interplay between
these anomalies and the strangeness decays can be found in [69, 36, 88].
Lepton flavor universality violation is related to lepton flavor violation,
and from that perspective, bounds of lepton flavor violation in strange decays
are also required to constrain possible models for the $B$ anomalies [89].

####  1.3.2 Rare kaon decay measurements

##### Lepton flavour universality (LFU) tests

Leptonic kaon decays represent a very sensitive LFU probe. In the SM, these
decays are mediated by a $W$ boson with universal coupling to leptons. They
are helicity suppressed, due to the $V-A$ structure of the charged current
coupling, and the ratio of their amplitudes

$R_{K}=\frac{\Gamma(K^{+}\to
e^{+}\nu)}{\Gamma(K^{+}\to\mu^{+}\nu)}=\frac{m_{e}^{2}}{m_{\mu}^{2}}\cdot{\left(\frac{m_{K}^{2}-m_{e}^{2}}{m_{K}^{2}-m_{\mu}^{2}}\right)}^{2}\cdot\left(1+\delta^{K}_{\rm
EM}\right)$ (10)

is very clean theoretically, as the strong interaction dynamics cancels out at
the first order, and the hadronic structure dependence appears only through
the electroweak corrections $\delta^{K}_{\rm EM}$. In particular, the SM
prediction for $R_{K}$ is currently known at the sub-permille level [90]:

$R_{K}=(2.477\pm 0.001)\times 10^{-5}.$ (11)

Lepton universality can be violated in many extensions of the SM with non-
trivial flavor structure. Models with mass-dependent couplings, for instance,
such as models with an extended Higgs sector [91], or leptoquarks [92, 93] can
be constrained by precise tests of LFU. Moreover, $R_{K}$ is sensitive to the
neutrino mixing parameters within SM extensions involving a fourth generation
of quarks and leptons [94] or sterile neutrinos [95].

The first measurements of $R_{K}$ were performed in the 1970s [96, 97, 98].
The current PDG world average [26],

$R_{K}^{\rm exp}=(2.488\pm 0.009)\times 10^{-5},$ (12)

is dominated by a result from an early phase of the NA62 experiment [99] and
is consistent with the earlier measurements and with the SM expectation. The
experimental precision on $R_{K}$ is still an order of magnitude larger than
the uncertainty in the SM prediction, which motivates further measurements
with improved accuracy.

Compared to the leptonic modes, the semi-leptonic decays are polluted by the
hadronic structure dependence, typically described by a form factor with free
parameters. However, the flavour-changing neutral current mediated
$K\to\pi\ell\ell$ processes allow probing lepton universality at the one-loop
level. If LFU holds, the form factor parameters have to be equal in both the
electron and the muon channels [100].

The physics of $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ ($\ell=e,\mu$) decays [101,
102, 103] is similar to that of $K_{S}\to\pi^{0}\ell^{+}\ell^{-}$ discussed in
Section 1.2.6. In the $K^{+}$ decay case, interference with the long-distance
amplitude from $K^{+}\to\pi^{+}\gamma^{*}$ leads to a CP violating charge
asymmetry for the $e^{+}e^{-}$ channel, however the rate asymmetry in the SM
is too small to be observed [102]. The $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$
amplitudes include the common weak form factor
$W(z;a_{+},b_{+})=G_{F}m_{K}^{2}(a_{+}+b_{+}z)+W^{\pi\pi}(z)$, where
$z=m_{\mu\mu}^{2}/m_{K}^{2}$, $a_{+}$ and $b_{+}$ are free real parameters,
and $W^{\pi\pi}(z)$ is a complex function describing the contribution from the
three-pion intermediate state with a $\pi^{+}\pi^{-}\to\gamma^{*}$ transition.
The most precise form factor and BR measurements to date has been performed by
the E865 and NA48/2 experiments [104, 105, 106]:

$\displaystyle\text{E865: }{\rm BR}(K^{+}\to\pi^{+}e^{+}e^{-})$
$\displaystyle=2.94(15)\times 10^{-7},\,a_{+}=-0.587(10),\,b_{+}=-0.655(44),$
(13) $\displaystyle\text{NA48/2: }{\rm BR}(K^{+}\to\pi^{+}e^{+}e^{-})$
$\displaystyle=3.11(13)\times 10^{-7},\,a_{+}=-0.578(16),\,b_{+}=-0.779(66),$
(14) $\displaystyle\text{NA48/2: }{\rm BR}(K^{+}\to\pi^{+}\mu^{+}\mu^{-})$
$\displaystyle=9.62(25)\times 10^{-7},\,a_{+}=-0.575(39),\,b_{+}=-0.813(145).$
(15)

These results are compatible with LFU within uncertainties. Further
measurements are in progress with the NA62 Run 1 dataset.

##### Lepton flavor and lepton number conservation tests

Discovery of lepton number (LN) or lepton flavor number (LF) violation would
be a clear indication of new physics. The minimal Type-I seesaw model [107]
provides a source of LN violation through the exchange of Majorana neutrinos,
while processes violating LF conservation can occur via the exchange of
leptoquarks or $Z^{\prime}$ boson, as well as in SM extensions with light
pseudoscalar bosons. Using the dataset collected in 2016–18 using prescaled
di-lepton trigger chains, the NA62 experiment has published ${\cal
O}(10^{-11})$ upper limits on the branching fractions of LF and LN violating
decays $K^{+}\to\pi^{-}(\pi^{0})\ell^{+}\ell^{+}$,
$K^{+}\to\pi^{\pm}\mu^{\mp}e^{+}$ and $\pi^{0}\to\mu^{-}e^{+}$ [108, 109,
110]. Most of these searches are not limited by backgrounds, and the
corresponding sensitivities are expected to improve linearly with the
integrated kaon flux. This, along with the possible improvements to the
trigger scheme, would allow probing these and other LN/LF violating processes
at the ${\cal O}(10^{-12})$ or lower level at future $K^{+}$ experiments.

LF violating observables in the $K_{L}$ sector include the branching fractions
of the $K_{L}\to(\pi^{0})(\pi^{0})\mu^{\pm}e^{\mp}$ and $K_{L}\to
e^{\pm}e^{\pm}\mu^{\mp}\mu^{\mp}$ decays. These could be accessed at the
${\cal O}(10^{-12})$ sensitivity level at a high-intensity $K_{L}$ experiment
with a magnetic spectrometer (Section 3.6), improving on the earlier limits
reported by the BNL-E871 and KTeV experiments.

##### Chiral perturbation theory tests

The non-perturbative nature of the strong interaction in kaon physics presents
a major theoretical challenge for describing most of the kaon decays. Chiral
perturbation theory (ChPT) provides a universal framework for treating
leptonic, semileptonic and nonleptonic decays, including rare and radiative
modes. Hadronic uncertainties are parametrized in ChPT by a number of
parameters, the so-called low-energy constants [8]. Precision measurements of
kaon decay rates and differential distributions represent important tests of
the ChPT predictions, and provide crucial inputs to the theoretical
development of low-energy hadron physics.

The rare and radiative decays that could be studied at future kaon experiments
include, among others: $K^{+}\to e^{+}\nu\gamma$,
$K^{+}\to\pi^{0}e^{+}\nu\gamma$, $K_{L}\to\pi^{-}e^{+}\nu\gamma$ [111],
$K\to\pi\gamma\gamma$ [112], $K\to\pi\gamma\ell^{+}\ell^{-}$ [113, 114],
$K\to\pi\pi\ell\nu$ [115], $K^{+}\to\ell_{1}^{+}\nu\ell_{2}^{+}\ell_{2}^{-}$
[111], and $K\to\pi\pi\pi\gamma$ [116].

####  1.3.3 Neutral pion decays

High intensity rare kaon decay experiments also represent $\pi^{0}$ factories,
since neutral pions are produced in about 30% of both $K^{+}$ and $K_{L}$
decays. Moreover, the $\pi^{0}$ production can be tagged by other (charged)
particles originating from the kaon decay vertex. The $\pi^{0}$ decays proceed
via the $\pi^{0}\to\gamma^{*}\gamma^{*}$ vertex, described by the transition
form factor, which also enters the computation of the hadronic light-by-light
scattering contributing to the muon anomalous magnetic moment [117]. Future
kaon experiments with dedicated di-electron trigger chains would be able to
improve the precision of the measurements of the rare decays $\pi^{0}\to
e^{+}e^{-}e^{+}e^{-}$ [118] and $\pi^{0}\to e^{+}e^{-}$ [119], and thus
provide an important input to the theoretical modeling of the form factor. In
addition, a new precise measurement of the branching ratio for the latter
decay would resolve an existing $2\sigma$ tension with the SM prediction
[120].

####  1.3.4 Searches for dark sectors in kaon decays

Measurements of rare kaon decays, including dedicated studies of differential
distributions, represent crucial probes of light dark sectors. The sensitivity
provided by the kaon sector owes to both the suppression of the total kaon
decay width, and the availability of the very large dataset. Dark sector
searches in kaon decays have attracted much interest recently. The status of
current searches over a broad range of hidden sector scenarios, the impact of
current experiments on the parameter space of these scenarios, and the future
projections are discussed in a recent comprehensive review [83].

The NA62 collaboration is pursuing a dedicated program of dark-sector searches
using the Run 1 dataset collected in 2016–18. The results published so far
include the world’s most stringent upper limits on the dark scalar ($\varphi$)
mixing parameter obtained from searches for the $K^{+}\to\pi^{+}\varphi$ decay
[47, 48, 121], heavy neutral lepton ($N$) mixing parameters $|U_{\ell 4}|^{2}$
($\ell=e;\mu$) obtained from searches for the $K^{+}\to\ell^{+}N$ decays [122,
123, 124], and the dark photon ($A^{\prime}$) mixing parameter obtained from a
search for the $\pi^{0}\to\gamma A^{\prime}$ decay [125]. The sensitivity of
these searches is expected to improve in future at least as the square root of
the integrated $K^{+}$ flux. Possible improvements to the data collection and
analysis (e.g. software trigger development) would bring further improvements
in sensitivity.

Similarly, the KOTO experiment has published a search for the
$K_{L}\to\pi^{0}\varphi$ decay, and is searching for other hidden-sector
scenarios with photons in the final state ($K_{L}\to\gamma\gamma X_{\rm inv}$,
$K_{L}\to\pi^{0}\pi^{0}X_{\rm inv}$, $K_{L}\to\gamma A^{\prime}$). Next-
generation $K_{L}\to\pi^{0}\nu\bar{\nu}$ experiments would improve
significantly the sensitivity of these searches, while a $K_{L}$ experiment
with a magnetic spectrometer (Section 3.6) would widen the physics reach by
addressing additional dark-sector scenarios.

####  1.3.5 Hyperon decays

Hyperons physics has been always complementary to kaon physics, thanks to
baryon number and spin properties which change the sensitivity to SM and new
interactions. While CP violation in kaon decays was discovered early, that in
hyperons is yet to be established. Important efforts in this direction were
done by the HyperCP dedicated experiment [126] at Fermilab. In fact, the
recent sudden change in the average of the $\alpha$ decay parameter value of
the $\Lambda\to p\pi^{-}$ decays, thanks to the BESIII and CLAS experiment
measurements [127, 128], has renewed the interest in possible measurements of
CP violation in hyperon decays.

The $\Sigma^{+}\to p\mu^{+}\mu^{-}$ decay represents an $s\to d\mu^{+}\mu^{-}$
transition and provides complementary information to the equivalents from kaon
decays [129] and attracted theoreticians attention since 1962 [130] as probe
of neutral currents. An evidence of this decay was first obtained at HyperCP
[131]. While the branching fraction measured by HyperCP is consistent with the
SM prediction, the dimuon mass spectrum showed hints of a resonance near the
dimuon mass threshold. More recently, LHCb also found an evidence for this
decay, obtaining the result [132]:

${\rm BR}(\Sigma^{+}\to p\mu^{+}\mu^{-})=\left(2.2_{-1.3}^{+1.8}\right)\times
10^{-8},$ (16)

although with no sign of any new resonance, excluding the central value of the
HyperCP branching fraction with intermediate particle hypothesis.

##  2 Experiments at the J-PARC Hadron Experimental Facility

###  2.1 The KOTO experiment until 2021

The KOTO collaboration is searching for the $K_{L}\to\pi^{0}\nu\overline{\nu}$
decay at the J-PARC KOTO experiment. For the branching ratio of the decay, we
set an upper limit of $3.0\times 10^{-9}$ at 90% CL with the data collected in
2015, and reached a single event sensitivity of $7.2\times 10^{-10}$ with the
data collected in 2016–18. We took 2 times more data in 2019–21 than in
2016–18 with detector upgrades, and we are analyzing the data.

###  2.2 The KOTO experiment from 2022

The J-PARC accelerator achieved the beam power 65 kW ($7\times 10^{13}$
protons per 5.2 s cycle) in 2021 in the slow extraction. With an upgrade of
J-PARC accelerator in 2021–22, shorter repetition cycle (4.2 s) and smoother
beam-spill will be expected. The beam power of 80 kW is expected in 2022, and
is expected to be increased to 100 kW in the following 2–3 years. For the
effect of the instantaneous beam intensity on the accidental loss of
$K_{L}\to\pi^{0}\nu\overline{\nu}$ acceptance, we expect that the 100-kW beam
will give better situation compared to the 65-kW beam in 2021. We expect to
reach the single event sensitivity at $O(10^{-11})$ around 2025 with the
following DAQ and detector upgrades.

####  2.2.1 Upgrade of trigger and DAQ system

KOTO will upgrade the trigger and DAQ system in 2022 in order to handle higher
data rate form higher beam power with the accelerator upgrade.

Waveforms for all the detector are recorded with 125-MHz or 500-MHz ADCs. In
the current trigger, KOTO used a level-1 trigger from the energy and timing
information of each detector component, and used a cluster-counting trigger
for the number of calorimeter clusters. With the trigger, the waveform data
were sent to a level-2 module (currently without any triggering function), and
then to a level-3 PC cluster at J-PARC. Event building was performed at the PC
cluster, and the data were sent to a computing center at KEK.

For the trigger and DAQ upgrade, the level-2 module with a bottle-neck was
replaced with a FPGA-based event builder. The data after event building will
be send to a new PC cluster with GPUs. The GPU will perform waveform-based
data compression, and the CPU power can be used for complicated event
selections. With those efforts, KOTO plan to keep the same data rate from
J-PARC to the KEK computing center even with twice higher input data rate.

####  2.2.2 Upgrade of upstream charged particle veto counter

In the analysis of data taken in 2016–18, KOTO found a background source from
charged kaon in the beam. Charged kaons are produced from the interaction of
$K_{L}$ or pions in the beam line materials. Charged kaons produced in the
beamline downstream of a sweeping magnet can enter the KOTO detector. The flux
of the charged kaon was $O(10^{-5})$ compared to the $K_{L}$, but
$K^{\pm}\to\pi^{\pm}e^{\mp}\nu$ becomes a background when $e^{\pm}$ is not
detected. This was a main background in the analysis of 2016-18 data. In 2020,
we installed a charged particle counter called UCV in the beam to detect
charged kaons in the beam, and veto the event. In 2021, we installed a new
version of UCV with a sheet of 0.5-mm square plastic scintillating fibers. The
efficiency for the charged particle is 90% or more. Beam particles scatters in
the UCV and increase the hit rate of detectors near the UCV, which increases
the accidental loss of the signal acceptance. $K_{L}$s or neutrons scatters in
the UCV into the beam halo region, which increases halo-$K_{L}$ or halo-
neutron background. In order to reduce these effects, KOTO plans to install
another UCV with 160-mm-square 0.2-mm-thick plastic scintillator film. The
scintillation photons escaping out from the scintillator surface is collected
with aluminized-mylar film, and PMTs are used to detect the photons. The
charged particle efficiency of 99% was achieved with a prototype counter.

####  2.2.3 Installation of second sweeping magnet

KOTO plans to install a new sweeping magnet at the downstream-end of the beam
line in order to reduce the charged kaons. This will reduce the charged-kaon
flux by a factor of ten or more, and the charged-kaon background can be
negligible with both the new UCV counter and the second magnet.

###  2.3 KOTO step-2 experiment

####  2.3.1 Concept of KOTO step-2

Although KOTO will reach the sensitivity at $O(10^{-11})$ around 2025, it
would take longer time toward the sensitivity predicted by the Standard Model,
$3\times 10^{-11}$, considering the operation plan of the Main Ring
accelerator and the expected running time in future. We thus should have a new
experiment that can discover and observe a large number of
$K_{L}\to\pi^{0}\nu\overline{\nu}$ events and measure its branching ratio.

An idea of such a next-step experiment, called KOTO step-2, was already
mentioned in the KOTO proposal in 2006 [133]. Now we have gained experiences
in the KOTO experiment and are ready to consider and design the next-
generation experiment more realistically. We eagerly consider the realization
of KOTO step-2 in the early phase of the extension of the Hadron Experimental
Facility.

To achieve a higher experimental sensitivity for the
$K_{L}\to\pi^{0}\nu\overline{\nu}$ measurement, we must consider to maximize
the $K_{L}$ flux, the detection acceptance of the signal, and the signal-to-
background ratio. The $K_{L}$ flux is determined by the production angle and
the solid angle of the secondary neutral beam. Other parameters are the
achievable intensity of the primary proton beam and the target properties
(material, thickness, etc.). The production angle is defined as the angle
between the primary proton beam and secondary neutral beam directions. Figure
11 shows the $K_{L}$ and neutron yields and the neutron-to-$K_{L}$ flux ratio
as functions of the production angle when a 102-mm-long gold target is used.

Figure 11: Simulated $K_{L}$ and neutron yields (left) and their ratio (right)
as functions of the production angle [134]. The yields were evaluated at 1 m
downstream of the target, normalized by the solid angle ($\mu$str). Black,
red, and blue points indicate the results when selecting neutrons with their
energies of more than 1, 100, and 300 MeV, respectively.

KOTO step-2 chooses the production angle of 5 degrees as an optimum point for
a higher $K_{L}$ flux with a smaller neutron fraction in the beam. In case of
the KOTO experiment, the production angle is 16-degree, as shown in Fig. 12,
which was chosen to utilize the T1 target with the experimental area away from
the primary beam.

Figure 12: Schematic drawing of the KL beam line for the KOTO experiment in
the current Hadron Experimental Facility.

In order to realize the 5-degree production while keeping the solid angle of
the neutral beam as large as possible, $\textit{i}.\textit{e}.$ with the
shortest beam line, a new experimental area behind the primary beam dump and a
new target station close to the dump are necessary. Figure 13 shows a possible
configuration in the Extended Hadron Experimental Facility, utilizing the
second target (T2).

Figure 13: Schematic drawing of the KOTO step-2 beam line in the Extended
Hadron Experimental Facility. The experimental area is located behind the beam
dump. Distance between the T2 target and the experimental area is assumed to
be 43 m in the present studies. A modeled cylindrical detector with 20 m in
length and 3 m in diameter is described in the experimental area as a
reference.

Here we introduce the KOTO step-2 detector base design. The acceptance of the
signal is primarily determined by the detector geometry. The basic detector
configuration is same as the current KOTO experiment, a cylindrical detector
system with an electromagnetic calorimeter to detect two photons from the
$\pi^{0}$ decay at the end cap. A longer $K_{L}$ decay region and a larger
diameter of the calorimeter are being considered to obtain a larger
acceptance. The 5-degree production also provides a benefit in view of the
signal acceptance; a harder $K_{L}$ spectrum than KOTO is expected, and two
photons are boosted more in the forward direction, and thus the acceptance
gain by a longer decay region can be utilized.

The ability of the signal discovery and the precision of the branching ratio
measurement depend on the signal-to-background ratio, as well as the expected
number of observed events. The background level is affected by many factors
such as the beam size at the detector, the flux of beam particles (neutrons,
$K_{L}$) leaking outside the beam (beam-halo), charged kaons in the neutral
beam, and detector performances.

####  2.3.2 KL2 beamline

Table 5 summarizes the beam parameters for KOTO step-2 and those in the
current KOTO experiment.

Table 5: Beam parameters for KOTO step-2 and the current KOTO experiment.

| KOTO step-2* | KOTO
---|---|---
Beam power | 100 kW | 64 kW (100 kW in future)
Target | 102-mm-long gold | 60-mm-long gold
Production angle | 5∘ | 16∘
Beam line length | 43 m | 20 m
Solid angle | 4.8 $\mu$sr | 7.8 $\mu$sr

  * *

Note the parameters for step-2 are tentative for this study.

A T2 target used in the KOTO step-2 study was a gold rod with its diameter of
10 mm and length of 102 mm, which corresponds to 1$\lambda_{I}$ (interaction
length). Spectra and relative yields of $K_{L}$ at 1 m from the target were
obtained from simulations as shown in Fig. 14 (left), where GEANT3, GEANT4
(10.5.1 with a physics list of QGSP_BERT or FTFP_BERT), and FLUKA (2020.0.3)
results are shown for comparison. The resultant $K_{L}$ fluxes were found to
agree with each other within 30%. GEANT3 provided the smallest $K_{L}$ yield
and thus is considered to be a conservative choice in the discussion of the
sensitivity.

Figure 14:  $K_{L}$ spectra at 1 m from the T2 target by the target simulation
(left) and at the exit of the KL2 beam line at 43 m from the T2 target by the
beam line simulation (right). In the left plot, the results by using various
simulation packages are also shown, as well as the result by the GEANT3-based
simulation (labeled “G3 GFLUKA”) which is our default in this study.

The beam line of KOTO step-2 (KL2 beam line) (Fig. 15) consists of two stages
of 5-m-long collimators, a photon absorber in the beam, and two sweeping
magnets to sweep out charged particles from the target or produced in the beam
line. The photon absorber, made of 7-cm-long lead, is located at 7 m
downstream of the target. The first collimator, starting from 20 m from the
target, defines the solid angle and shape of the neutral beam. The solid angle
is set to be 4.8 $\mu$str. The second collimator, starting from 38 m from the
target, cut the particles coming from the interactions at the photon absorber
and the beam-defining edge of the first collimator. The bore shape of the
second collimator is designed not to be seen from the target so that particles
coming directly from the target do not hit the inner surface and thus do not
generate particles leaking outside the beam. The first sweeping magnet is
located upstream of the first collimator, and the second sweeping magnet is
located downstream of the second collimator (The effect of the second sweeping
magnet was evaluated independently from the beam line without the second
sweeping magnet).

Figure 15: Design of the KL2 beam line in x-z (left) and y-z (right) views.
The short vertical lines at the upstream show the photon absorber, the solid
lines in the middle show the two collimators, the shaded regions show the two
sweeping magnets, the dahsed lines show the 4.8-$\mu$str collimation lines
defined with the first collimator, and the black filled regions at the
downstream show inner region of the calorimeter of the KOTO step-2 detector,
where the beam size is 15-cm square.

Figures 14 (right) and 16 show the simulated spectra of $K_{L}$, neutrons, and
photons at the exit of the KL2 beam line, respectively.

Figure 16: Simulated neutron (left) and photon (right) spectra at the exit of
the beam line.

The $K_{L}$ yield was evaluated to be $1.1\times 10^{7}$ per $2\times 10^{13}$
protons on the target (POT). Note that the beam power of 100 kW corresponds to
$2\times 10^{13}$ POT per second with 30 GeV protons. The resultant $K_{L}$
flux per POT is 2.6 times higher than that of the current KOTO experiment. The
$K_{L}$ spectrum peaks at 3 GeV/c, while it is 1.4 GeV/c in the current KOTO
experiment. The simulated particle fluxes are summarized in Table 6.

Table 6: Expected particle yields estimated by the simulations.

Particle | Energy range | Yield | On-spill rate
---|---|---|---
(per $2\times 10^{13}$ POT) | (MHz)
$K_{L}$ |  | $1.1\times 10^{7}$ | 24
Photon |  $>$10 MeV | $5.3\times 10^{7}$ | 110
$>$100 MeV | $1.2\times 10^{7}$ | 24
Neutron |  $>$0.1 GeV | $3.1\times 10^{8}$ | 660
$>$1 GeV | $2.1\times 10^{8}$ | 450

  * The beam power of 100 kW corresponds to $2\times 10^{13}$ POT/s with 30 GeV protons. The on-spill rate means the instantaneous rate during the beam spill, assuming 2-second beam spill every 4.2 seconds.

##### Halo neutrons

Figure 17 shows the neutron profile at the assumed calorimeter location, 64 m
from the T2 target. As shown in the figure, the neutral beam is shaped so as
to be a square at the calorimeter location.

Figure 17: Beam shape at the end-cap plane, represented by the neutron
profile. The left figure shows the distribution in $\pm 15$ cm of the beam
center, and the right top (bottom) histogram indicates the horizontal
(vertical) distribution in $\pm 50$ cm of the beam center.

Evaluation of neutrons spreading to the beam halo region, called “halo
neutron”, is important since they are potential sources of backgrounds due to
their interaction with the detector materials. Here we define the core and
halo neutrons as those inside and outside the $\pm$10 cm region at the
calorimeter, respectively. The ratio of the halo neutron yield to the core
yield was found to be $1.8\times 10^{-4}$.

##### Charged kaons in the neutral beam

The contamination of the charged kaons in the neutral beam line is harmful,
since $K^{\pm}$ decays such as $K^{\pm}\to\pi^{0}e^{\pm}\nu$ in the detector
region can mimic the signal, which were pointed out in the analysis of the
KOTO experiment [61]. The major production point of charged kaons is the
second collimator. Neutral particles ($K_{L}$ and neutrons) hit the inner
surface of the collimator and the produced charged kaon in the interactions
can reach the end of the beam line. Charged pions from $K_{L}$ decays hitting
the collimator also can produce charged kaons. According to the beam line
simulation, the flux ratio of the charged kaon and $K_{L}$ entering the decay
region is $R(K^{\pm}/K_{L})=4.1\times 10^{-6}$ without the second sweeping
magnet. To evaluate the reduction by the second sweeping magnet, we conducted
another beam line simulation with a sweeping magnet that provides a magnetic
field of 2 Tesla in 1.5 m long at the end of the beam line. We confirmed that
it can reduce the ratio to $R<1.1\times 10^{-6}$, which is limited by the
simulation statistics.

####  2.3.3 Activities in the experimental area behind the beam dump

To realize the production angle of 5 degree, the experimental area must be
located behind the primary beam dump. There is a concern that many particles
from interactions in the beam dump penetrate the shield and reach the
experimental area, which cause a high rate of accidental hits in the detector.
To evaluate the flux in the experimental area, a GEANT3-based simulation of
the beam dump was conducted. Figure 18 illustrates the model of the current
beam dump.

Figure 18: Model of the current primary beam dump used in the simulation study
in horizontal (left) and vertical cutaway views.

The main body of the dump is made of copper, on which a tapered hole exists to
spread the proton hit position along the beam direction and thus distribute
the heat dissipation. Steel and concrete shields follow behind the copper
dump. In this simulation, the proton beam with the size ($\sigma$) of 3 cm is
injected into the dump, which roughly indicate the parameter in the current
operation.

Almost all the particles entering the area were muons. As shown in Fig. 20,
the KOTO step-2 detector is not in the most intense region but still in the
region where the flux is high.

Figure 19: Simulated development of the muon tracks behind the current beam
dump, distributed in the horizontal (x) and the beam direction (z). The red
box indicates the assumed location of the KOTO step-2 detector, and the red
dashed line indicates the center of the KL2 beam line.

Figure 20: Energy distribution of muons whose extrapolated trajectory crosses
the KOTO step-2 detector region in case of the current dump (blue) and the
modified dump (red).

The on-spill counting rate due to the muons which cross the detector region
was estimated to be 15 MHz when the 100 kW beam is fully injected into the
dump. The contribution of other particles such as neutrons were found to be
small, about 25 kHz even integrated over the whole region behind the dump.

To reduce the muon flux, a part of the concrete shield must be replaced with
steel shield. Replacing the material of the 4 m (horizontal) $\times$ 3 m
(vertical) $\times$ 7 m (beam direction) volume from concrete to steel reduces
the flux by an order of magnitude to 1.3 MHz. Figure 20 shows the energy
distribution of the muons crossing the detector region before and after the
replacement.

####  2.3.4 KOTO step-2 detector

A conceptual detector used in the base design is shown in Fig. 21.

Figure 21: Conceptual KOTO step-2 detector. The upstream edge of the Front
Barrel Counter is 44 m from T2 target.

The $z$ axis is defined on the beam axis pointing downstream with the origin
at the upstream surface of the Front Barrel Counter, which is 44 m from the T2
target (43-m long beam line and 1-m long space). The following conceptual
detector is designed; the diameter of the calorimeter is 3 m to gain the
signal acceptance. The beam hole in the calorimeter is 20 cm $\times$ 20 cm to
accept the 15 cm $\times$ 15 cm beam size. The $z$ position of the calorimeter
is 20 m with a larger decay volume to enhance the $K_{L}$ decay. The larger
decay volume is effectvie in the KOTO step-2, because the higher $K_{L}$
momentum keeps the signal acceptance.

The Charged Veto Counter is a charged-particle veto counter at 30 cm upstream
of the calorimeter to veto $K_{L}\to\pi^{\pm}e^{\mp}\nu$,
$K_{L}\to\pi^{\pm}\mu^{\mp}\nu$, or $K_{L}\to\pi^{+}\pi^{-}\pi^{0}$. The beam
hole at the Charged Veto Counter is 24 cm $\times$ 24 cm to avoid $\pi^{0}$ or
$\eta$ produced by the interaction with neutrons. The Front Barrel Counter is
1.75-m long and the Upstream Collar Counter is 0.5-m long to veto $K_{L}\to
3\pi^{0}$ decay at upstream of and inside the Front Barrel Counter. The
Central Barrel Counter is 20-m long mainly to veto $K_{L}\to 2\pi^{0}$ by
detecting extra photons from the decay. The Downstream Collar Counter is 4-m
long to veto particles passing through the beam hole in the calorimeter but
going outside the beam region. The Beam Hole Counter covers in-beam region
starting from 24.5 m to veto particles escaping through the calorimeter beam
hole. The Front Barrel Counter, the Upstream Collar Counter, the Central
Barrel Counter, and the Downstream Collar Counter, act as either photon and
charged-particle veto for the conceptual design. For the Beam Hole Counter, we
introduce two separate counters, a beam-hole charged-veto counter at the
upstream and a beam-hole photon-veto counter. The signal region is defined in
a 12-m region from 3 m to 15 m in $z$.

The modeling of the calorimeter was based on the 50-cm long CsI-crystal
calorimeter in the current KOTO detector. Those of the front and central
barrel counters, and the upstream and downstream collar counters were based on
lead/plastic-scintillator sandwich counters. Those of the beam hole charged /
photon veto counters were based on the current KOTO in-beam detectors.

####  2.3.5 Beam Condition and Event Selection

##### Beam conditions

The beam condition and running time as shown in Table 7 were assumed in the
study.

Table 7: Assumed beam and running time. Beam power | 100 kW | (at 1-interaction-length T2 target)
---|---|---
|  | ($1.1\times 10^{7}K_{L}/2\times 10^{13}~{}\mathrm{POT}$)
Repetition cycle | 4.2 s |
Spill length | 2 s |
Running time |  $3\times 10^{7}$ s |

##### Signal Reconstruction

We evaluated yields of the signal and backgrounds with Monte Carlo
simulations. The calorimeter response was simulated either with model
responses or with shower simulations in the calorimeter. We assumed 50-cm-long
CsI crystals for the calorimeter material in the shower simulations.

A shower is generated by a particle incident on the calorimeter. A cluster is
formed based on energy deposits in the calorimeter segmented in x-y
directions. Assuming the incident particle to be a photon, the energy and
position are reconstructed. We treat it as a photon in the later analysis
regardless of the original particle species.

A $\pi^{0}$ is reconstructed from the two photons on the calorimeter; The
opening angle of two photon-momentum directions ($\theta$) can be evaluated
with the energies of the two photons ($E_{0},E_{1}$) from 4-momentum
conservation:

$\displaystyle{p}_{\pi^{0}}=$ $\displaystyle{p}_{0}+{p}_{1},$ $\displaystyle
m_{\pi^{0}}^{2}=$ $\displaystyle 2E_{0}E_{1}(1-\cos\theta),$

where ${p}_{\pi^{0}}$ is four-momentum of the $\pi^{0}$, ${p}_{0}$ and
${p}_{1}$ are four-momenta of two photons, and $m_{\pi^{0}}$ is the nominal
$\pi^{0}$ mass. The vertex position of the $\pi^{0}$ is assumed to be on the
$z$ axis owing to the narrow beam, and the $z$ vertex position
($z_{\mathrm{vtx}}$) is calculated from the geometrical relation among
$\theta$ and hit positions ${\bf r}_{0}=(x_{0},y_{0})$, ${\bf
r_{1}}=(x_{1},y_{1})$ on the calorimeter as shown in Fig. 22. The
$z_{\mathrm{vtx}}$ gives the momenta of two photons, and the sum of the
momenta gives momentum of $\pi^{0}$. Accordingly, the $\pi^{0}$ transverse
momentum ($p_{\mathrm{T}}$) is obtained.

Figure 22: Geometrical relation in the vertex reconstruction.

##### Event selection

We use the following event selections for the events with two clusters in the
calorimeter.

  1. 1.

Sum of two photon energies : $E_{0}+E_{1}>500~{}\mathrm{MeV}$.

  2. 2.

Calorimeter fiducial area : $\sqrt{x_{0}^{2}+y_{0}^{2}}<1350~{}\mathrm{mm}$,
$\sqrt{x_{1}^{2}+y_{1}^{2}}<1350~{}\mathrm{mm}$.

  3. 3.

Calorimeter fiducial area : $\mathrm{max}(|x_{0}|,|y_{0}|)>175~{}\mathrm{mm}$,
$\mathrm{max}(|x_{1}|,|y_{1}|)>175~{}\mathrm{mm}$.

  4. 4.

Photon energy : $E_{0}>100~{}\mathrm{MeV}$, $E_{1}>100~{}\mathrm{MeV}$.

  5. 5.

Distance between two photons : $|{\bf r}_{1}-{\bf r}_{0}|>300~{}\mathrm{mm}$.

  6. 6.

Projection angle ($\theta_{\mathrm{proj}}$ as shown in Fig. 22) :
$\theta_{\mathrm{proj}}\equiv\mathrm{acos}\left(\frac{{\bf r}_{0}\cdot{\bf
r}_{1}}{|{\bf r}_{0}||{\bf r}_{1}|}\right)<150^{\circ}$.

  7. 7.

$\pi^{0}$ decay vertex : $3~{}\mathrm{m}<z_{\mathrm{vtx}}<15~{}\mathrm{m}$.

  8. 8.

$\pi^{0}$ transverse momentum :
$130~{}\mathrm{MeV}/c<p_{\mathrm{T}}<250~{}\mathrm{MeV}/c$.

  9. 9.

Tighter $\pi^{0}$ $p_{\mathrm{T}}$ criteria in the downstream (Fig. 23):
$\frac{p_{\mathrm{T}}}{(\mathrm{MeV}/c)}>\frac{z_{\mathrm{vtx}}}{(\mathrm{mm})}\times
0.008+50$.

  10. 10.

Selection to reduce hadron cluster background
In order to reduce neutron clusters, cluster shape, pulse shape, and depth
information of the hits in the calorimeter are used as in the analysis of the
KOTO step-1. The signal selection efficiency of $0.9^{3}=0.73$ is assumed.

  11. 11.

Selection to reduce halo $K_{L}\to 2\gamma$ background
The photon incident-angle information is used to reduce the halo $K_{L}\to
2\gamma$ background as in the KOTO step-1. The signal selection efficiency of
$0.9$ is assumed.

The first 5 selections ensure the quality of the photon cluster; The sum of
the photon energies is useful to reduce a trigger bias, because we plan to use
the sum of the calorimeter energy for the trigger. The edge region of the
calorimeter is avoided to reduce the energy leak outside the calorimeter.
Higher energy photons give good resolution. Large distance between the two
photons reduces the overlap of two clusters.

The next four are kinematic selections. The projection angle selection
requires no back-to-back configuration of the two photons to reduce $K_{L}\to
2\gamma$. Larger $\pi^{0}$ $p_{\mathrm{T}}$ is required to match the
kinematics of the signal. The tighter $p_{\mathrm{T}}$ selection is required
in the downstream region, because the reconstructed $p_{\mathrm{T}}$ tends to
be larger due to worse $p_{\mathrm{T}}$ resolution for the decay near the
calorimeter.

The last two are the identification criteria with the calorimeter to
discriminate photon and neutron clusters, or to discriminate correct and
incorrect photon-incident angles.

Figure 23: The shaded area shows the $p_{\mathrm{T}}$ criteria in the
$z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane. The blue dotted line shows the
tighter $p_{\mathrm{T}}$ criteria in the downstream region.

####  2.3.6 Signal sensitivity estimation

##### Decay probability and geometrical acceptance of two photons at the
calorimeter

The decay probability is found to be

$\displaystyle P^{\mathrm{truth}}_{\mathrm{decay}}=$
$\displaystyle\frac{\text{Number of $K_{L}$'s that decayed in
$3~{}\mathrm{m}<z<15~{}\mathrm{m}$}}{\text{Total number of $K_{L}$'s at
$z=-1$~{}m}}=9.9\%.$

The geometrical acceptance is evaluated to be

$\displaystyle A^{\mathrm{truth}}_{\mathrm{geom}}=$
$\displaystyle\frac{\text{Number of $K_{L}$'s with 2$\gamma$'s in the
calorimeter that decayed in $3<z<15$~{}m}}{\text{Number of $K_{L}$'s that
decayed in $3~{}\mathrm{m}<z<15~{}\mathrm{m}$}}=24\%.$

##### Cut acceptance

The overall acceptance for the conditions 1–6 and 8–9555The seventh condition,
$z_{\mathrm{vtx}}$ selection, is already treated in the previous section.
listed in Section 2.3.5 is 40%. The assumed acceptance for all the additional
cuts to reduce the hadron-cluster background and halo $K_{L}\to 2\gamma$ is
$0.9^{4}=66\%$. Including all the above, the cut acceptance is 26%.

##### Accidental loss

In order to veto background events, we set a timing window (veto window) to
detect extra particles with respect to the two-photon hit timing at the
calorimeter. The width of the veto window is set to 40 ns for the Central
Barrel Counter, 30 ns for the beam-hole charged-veto counter, 6 ns for the
beam-hole photon-veto counter, and 20 ns for the other counters.

When the $K_{L}\to\pi^{0}\nu\overline{\nu}$ signal is detected with the
calorimeter, another $K_{L}$ might decay accidentally and its daughter-
particle may hit a counter at the same time. Similarly, a photon or neutron in
the beam might hit the Beam Hole Counter at the same time. These accidental
hits will veto the signal if the hit timing is within the veto window. We call
this type of signal loss as “accidental loss”.

The hit rates of counters other than Beam Hole Counters were evaluated with
$K_{L}$-decay simulations, where any hits in the counters were included.

For the Beam-Hole Charged Veto Counter, three layers of a MWPC-type wire
chamber [135] are assumed: the thickness of the gas volume for each layer is
2.8 mm, and the cathode plane is a 50-$\mu$m-thick graphite-coated polyimide
film. This design reduces the hit rate from neutral particles, such as
photons, neutrons, and $K_{L}$. The layer-hit is defined as the energy deposit
larger than 1/4 of the minimum-ionizing-particle peak. The counter-hit is
defined as two coincident layer-hits out of three layers, which maintains the
charged-particle efficiency to be better than 99.5% with less contribution
from neutral particles.

For the Beam-Hole Photon Veto Counter, 25 modules of lead-aerogel Cherenkov
counters [136] are assumed. A high-energy photon generates an $e^{+}e^{-}$
pair in the lead plate, and these generate Cherenkov light in the aerogel
radiator. The Cherenkov light is guided by mirrors to a PMT. The individual
module-hit is defined with a 5.5-photoelectron (p.e.) threshold. The counter-
hit is defined with the consecutive three-module coincidence.

The hit rate ($r_{i}$), the veto width $(w_{i})$, and the individual loss
$(1-\exp\left(-w_{i}r_{i}\right))$ for each detector are summarized in Table
8. The total accidental loss is evaluated:

$\displaystyle\text{Accidental loss}=$ $\displaystyle
1-\exp\left(-\sum_{i}w_{i}r_{i}\right)$ $\displaystyle=$ $\displaystyle 39\%.$

Table 8: Summary of rate, veto width, and individual accidental loss. Detector | Rate(MHz) | Veto width (ns) | Individual loss (%)
---|---|---|---
Front Barrel Counter | 0.18 | 20 | 0.4
Upstream Collar Counter | 0.80 | 20 | 1.6
Central Barrel Counter | 2.21 | 40 | 8.5
Calorimeter | 3.45 | 20 | 6.7
Downstream Collar Counter | 0.97 | 20 | 1.9
Beam-hole charged-veto | 2.9 | 30 | 8.3
Beam-hole photon-veto | 35.2 | 6 | 19

##### Shower-leakage loss

When two photons from the $K_{L}\to\pi^{0}\nu\overline{\nu}$ decay are
detected in the calorimeter, the shower can leak both the downstream and
upstream of the calorimeter, and make hits on the other counters such as the
Central Barrel Counter. Such hits will veto the signal if the hit timing is
within the veto window. We call this signal loss “shower-leakage loss”. In
particular, we call the loss caused by shower leakage toward the upstream
“backsplash loss”.

The Charged Veto Counter covers the upstream side of the calorimeter, and
would suffer from the backsplash. This backsplash loss was neglected assuming
that the back-splash contribution can be distinguished with the timing
difference between the Charged Veot Counter and the calorimeter.

The large coverage of the Central Barrel Counter gives significant effect on
the backsplash loss. First, we introduce a timing definition of the Central
Barrel Counter ($t_{\mathrm{BarrelVeto}}$):

$\displaystyle t_{\mathrm{BarrelVeto}}=$ $\displaystyle
t_{\mathrm{BarrelHit}}-\left[t_{\mathrm{CalorimeterHit}}-\frac{z_{\mathrm{Calorimeter}}-z_{\mathrm{BarrelHit}}}{c}\right].$

Figure 24 shows the incident timing of the backsplash particles on the Central
Barrel Counter as a function of the incident $z$ position. The timing-smearing
is applied with the timing resolution depending on the incident energy. Events
with larger $t_{\mathrm{BarrelVeto}}$ at $z\sim 20000~{}\mathrm{mm}$ are
generated with neutrons from the electromagnetic shower. Events with smaller
$z$ tend to have larger $t_{\mathrm{BarrelVeto}}$ due to longer flight
distance. This clear correlation can be used to exclude the backsplash
particles from the veto to reduce the backsplash loss. We loosen the veto
criteria at the downstream region: for $z$ from 12.5 m to 17 m, the largest
timing of the veto window is changed from 35 ns to 4 ns linearly, and for
$z>17~{}\mathrm{m}$, it is 4 ns. The region within the two lines in the figure
shows the new veto region, which gives the survival probability of $91\%$,
equivalently the backsplash loss of 9%.

In addition, if we added $z$ smearing using a Gaussian with the $\sigma$ of
500 mm, the same survival probability is obtained.

Figure 24: Incident timing of the backsplash particles on the Central Barrel
Counter as a function of the incident $z$ position. The smearing with the
timing resolution is applied depending on the incident energy.

##### Signal yield

We evaluate the number of the signal ($S$) in $3\times 10^{7}$ s running time
with $\mathrm{BR}_{\mathrm{K_{L}\to\pi^{0}\nu\overline{\nu}}}=3\times
10^{-11}$:

$\displaystyle S=$ $\displaystyle\frac{(\text{beam power})\times(\text{running
time})}{(\text{beam energy})}\times(\text{number of }K_{L}\text{/POT})$
$\displaystyle\times P_{\mathrm{decay}}\times A_{\mathrm{geom}}\times
A_{\mathrm{cut}}\times(\text{1-accidental loss})\times(\text{1-backsplash
loss})\times\mathcal{B}_{K_{L}\to\pi^{0}\nu\overline{\nu}}$ $\displaystyle=$
$\displaystyle\frac{(100~{}\mathrm{kW})\times(3\times
10^{7}~{}\mathrm{s})}{(30~{}\mathrm{GeV})}\times\frac{(1.1\times
10^{7}K_{L})}{(2\times 10^{13}~{}\mathrm{POT})}$ $\displaystyle\times
9.9\%\times 24\%\times 26\%\times(1-39\%)\times 91\%\times(3\times 10^{-11})$
$\displaystyle=$ $\displaystyle 35.$

Here, $P_{\mathrm{decay}}$ is the decay probability, $A_{\mathrm{geom}}$ is
the geometrical acceptance for the two photons entering the calorimeter,
$A_{\mathrm{cut}}$ is the cut acceptance, and
$\mathcal{B}_{K_{L}\to\pi^{0}\nu\overline{\nu}}$ is the branching fraction of
$K_{L}\to\pi^{0}\nu\overline{\nu}$. Distribution in the
$z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane is shown in Fig. 25.

Figure 25: Distribution in the $z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane for
$K_{L}\to\pi^{0}\nu\overline{\nu}$ for the running time of $3\times 10^{7}$ s.
All the cuts other than $p_{\mathrm{T}}$ and $z_{\mathrm{vtx}}$ cuts are
applied.

####  2.3.7 Background estimation

##### $K_{L}\to\pi^{0}\pi^{0}$

$K_{L}\to\pi^{0}\pi^{0}$ becomes a background when two clusters are formed at
the calorimeter and the other photons are missed in the following cases.

  1. 1.

Fusion background: Three photons enter the calorimeter, and two of them are
fused into one cluster. The other one photon is missed due to the detector
inefficiency.

  2. 2.

Even-pairing background: Two photons from a $\pi^{0}$-decay form two clusters
in the calorimeter. Two photons from the other $\pi^{0}$ are missed due to the
detector inefficiency.

  3. 3.

Odd-pairing background : One photon from a $\pi^{0}$ and one photon from the
other $\pi^{0}$ form two clusters in the calorimeter. The other two photons
are missed due to the detector inefficiency.

The number of this background is evaluated to be 33.2, in which 2 event come
from the fusion, 27 events from the even-pairing, 4 events from the odd-
pairing background. Fig. 26 shows the distribution in the
$z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane with all the cuts other than the
$z_{\mathrm{vtx}}$ and $p_{\mathrm{T}}$ selections.

Among the 33.2 background events, both two photons are missed in the Central
Barrel Counter for 29 events, which gives the dominant contribution.

Figure 26: Distribution in the $z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane for
$K_{L}\to\pi^{0}\pi^{0}$ background events for the running time of $3\times
10^{7}$ s. All the cuts other than $p_{\mathrm{T}}$ and $z_{\mathrm{vtx}}$
cuts are applied.

##### $K_{L}\to\pi^{+}\pi^{-}\pi^{0}$

$K_{L}\to\pi^{+}\pi^{-}\pi^{0}$ becomes a background when $\pi^{+}$ and
$\pi^{-}$ are not detected. The number of this background events is evaluated
to be 2.5 as shown in Fig. 27, where one charged pion is lost in the Charged
Veto Counter, and the other is lost in the Beam Hole Counter. The maximum
$p_{\mathrm{T}}$ of the reconstructed $\pi^{0}$ is limited with the kinematics
and the $p_{\mathrm{T}}$ resolution. The tighter $p_{\mathrm{T}}$ selection in
the downstream makes the pentagonal cut in the
$z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane as shown in the figure. This cut
reduces the background because the $p_{\mathrm{T}}$ resolution is worse in the
downstream.

Figure 27: Distribution in the $z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane for
$K_{L}\to\pi^{+}\pi^{-}\pi^{0}$ background for the running time of $3\times
10^{7}$ s. All the cuts other than $p_{\mathrm{T}}$ and $z_{\mathrm{vtx}}$
cuts are applied.

#####  $K_{L}\to\pi^{\pm}e^{\mp}\nu$ (Ke3)

The Ke3 background happens when the electron and the charged pion are not
identified with the Charged Veto Counter. The number of background events is
evaluated to be 0.08 with $10^{-12}$ reduction with the Charged Veto Counter
as shown in Fig. 28.

Figure 28: Distribution in the $z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane for
Ke3 background for the running time of $3\times 10^{7}$ s. All the cuts other
than $p_{\mathrm{T}}$ and $z_{\mathrm{vtx}}$ cuts are applied.

#####  $K_{L}\to 2\gamma$ for halo $K_{L}$

$K_{L}$ in the beam scatters at the beam line components, and exists in the
beam halo region. When such a halo $K_{L}$ decays into two photons, larger
$p_{\mathrm{T}}$ is possible due to the assumption of the vertex on the
$z$-axis. The decay vertex is wrongly reconstructed with the nominal pion mass
assumption. This fake vertex gives a wrong photon-incident angle. Therefore,
this halo $K_{L}\to 2\gamma$ background can be reduced with incident-angle
information at the calorimeter. We can reconstruct another vertex with the
nominal $K_{L}$ mass assumption, which gives a correct photon-incident angle.
By comparing the observed cluster shape to those from the incorrect and
correct photon-incident angles, this background is reduced to be 10% in the
KOTO step-1, while keeping 90% signal efficiency. In this report, we assume
that the background is reduced to be 1%, because the higher energy photon in
the step-2 will give a better resolution in the photon-incident angle. We will
study it more in the future. The number of this background is evaluated to be
4.8 as shown in Fig. 29. For the halo $K_{L}$ generation in this simulation,
the core $K_{L}$ momentum spectrum and the halo neutron directions were used.
Systematic uncertainties on the flux and spectrum are also one of the future
studies.

Figure 29: Distribution in the $z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane for
halo $K_{L}\to 2\gamma$ background for the running time of $3\times 10^{7}$ s.
All the cuts other than $p_{\mathrm{T}}$ and $z_{\mathrm{vtx}}$ cuts are
applied.

##### $K^{\pm}\to\pi^{0}e^{\pm}\nu$

$K^{\pm}$ is generated in the interaction of $K_{L}$, neutron, or $\pi^{\pm}$
at the collimator in the beam line. Here we assume that the second sweeping
magnet near the entrance of the detector will reduce the contribution to be
10%.

Higher momentum $K^{\pm}$ can survive in the downstream of the second magnet,
and $K^{\pm}\to\pi^{0}e^{\pm}\nu$ decay occurs in the detector. This becomes a
background if $e^{\pm}$ is undetected. The kinematics of $\pi^{0}$ is similar
to $K_{L}\to\pi^{0}\nu\overline{\nu}$, therefore this is one of the serious
backgrounds. Detection of $e^{\pm}$ is one of the keys to reduce the
background.

We evaluated the number of the background events to be 4.0 as shown in Fig.
30. In the current beam line simulation, statistics is not large enough. We
use the $K_{L}$ momentum spectrum and directions for the $K^{\pm}$ generation.
We will evaluate these with more statistics with the beam line simulation in
the future.

Figure 30: Distribution in the $z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane for
$K^{\pm}\to\pi^{0}e^{\pm}\nu$ for the running time of $3\times 10^{7}$ s. All
the cuts other than $p_{\mathrm{T}}$ and $z_{\mathrm{vtx}}$ cuts are applied.

##### Hadron cluster

A halo neutron hits the calorimeter to produce a first hadronic shower, and
another neutron in the shower travels inside the calorimeter, and produces a
second hadronic shower apart from the first one. These two hadronic clusters
mimic the signal.

We evaluated this background using halo neutrons prepared with the beam line
simulation, and with a calorimeter composed of 50-cm-long CsI crystals. A
full-shower simulation was performed with those neutrons.

We evaluated the number of background events to be 3.0 as shown in Fig. 31.
Here we assume $10^{-7}$ reduction with the cluster-shape information, pulse-
shape information, and shower depth information in the calorimeter. In the
KOTO step-1, we achieved to reduce the background to be $\times((2.5\pm
0.01)\times 10^{-6})$ with 72% signal efficiency by using cluster and pulse
shapes. By using the shower depth information, we also achieved to reduce it
to be $\times(2.1\times 10^{-2})$ with 90% signal efficiency with small
correlation to the cluster and pulse shape cuts. In total, $10^{7}$ reduction
is feasible. This reduction power is one of the requirements on the
calorimeter design.

Figure 31: Distribution in the $z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane of
the hadron cluster background for the running time of $3\times 10^{7}$ s. All
the cuts other than $p_{\mathrm{T}}$ and $z_{\mathrm{vtx}}$ cuts are applied.

#####  $\pi^{0}$ production at the Upstream Collar Counter

If a halo neutron hits the Upstream Collar Counter, and produces a $\pi^{0}$,
which decays into two photons, it mimics the signal.

Halo neutrons obtained from the beam line simulation are used to simulate the
$\pi^{0}$ production in the Upstream Collar Counter. We assume fully-active
CsI crystals for the detector. Other particles produced in the $\pi^{0}$
production can deposit energy in the detector, and can veto the event. In the
simulation, such events were discarded at first. In the next step, only the
$\pi^{0}$-decay was generated in the Upstream Collar Counter. Two photons from
the $\pi^{0}$ can also interact the Upstream Collar Counter, and deposit
energy in the counter. Such events were also discarded. The $\pi^{0}$
production near the downstream surface of the Upstream Collar Counter mainly
survives. Finally when the two photons hit the calorimeter, a full shower
simulation was performed. In this process, photon energy can be mis-measured
due to photo-nuclear interaction. Accordingly the distribution of the events
in the $z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane was obtained as shown in Fig.
32. We evaluated the number of background events to be 0.19.

Figure 32: Distribution in the $z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane of
the background from $\pi^{0}$ production at the Upstream Collar Counter for
the running time of $3\times 10^{7}$ s. All the cuts other than
$p_{\mathrm{T}}$ and $z_{\mathrm{vtx}}$ cuts are applied.

#####  $\eta$ production at the Charged Veto Counter

A halo neutron hits the Charged Veto Counter, and produces a $\eta$. The
$\eta$ decays into two photons with the branching fraction of 39.4%, which can
mimic the signal. The decay vertex will be reconstructed at the upstream of
the Charged Veto Counter because the $\eta$ mass is four times larger than the
$\pi^{0}$ mass.

Halo neutrons obtained from the beam line simulation are used to simulate the
$\eta$ production in the Charged Veto Counter. We assume a 3-mm-thick plastic
scintillator at 30-cm upstream of the calorimeter. Other particles produced in
the $\eta$ production can deposit energy in the Charged Veto Counter, and can
veto the event. In the simulation, such events were discarded at first. In the
next step, only the $\eta$-decay was generated. When two photons from the
$\eta$ hit the calorimeter, a full-shower simulation was performed. Two
clusters were formed, and the distribution of the events in the
$z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane was obtained as shown in Fig. 33. We
evaluated the number of background events to be 8.2.

Figure 33: Distribution in the $z_{\mathrm{vtx}}$-$p_{\mathrm{T}}$ plane of
the background from $\eta$ production at the Charged Veto Counter for the
running time of $3\times 10^{7}$ s. All the cuts other than $p_{\mathrm{T}}$
and $z_{\mathrm{vtx}}$ cuts are applied.

##### Summary of the background estimation

A summary of the background estimations is shown in Table 9. The total number
of background events is $56.0\pm 2.8$, where $33.2\pm 1.3$ comes from the
$K_{L}\to\pi^{0}\pi^{0}$ decay.

Table 9: Summary of background estimations. Background | Number |
---|---|---
$K_{L}\to\pi^{0}\pi^{0}$ | 33.2 |  $\pm$1.3
$K_{L}\to\pi^{+}\pi^{-}\pi^{0}$ | 2.5 |  $\pm$0.4
$K_{L}\to\pi^{\pm}e^{\mp}\nu$ | 0.08 |  $\pm$0.0006
halo $K_{L}\to 2\gamma$ | 4.8 |  $\pm$0.2
$K^{\pm}\to\pi^{0}e^{\pm}\nu$ | 4.0 |  $\pm$0.4
hadron cluster | 3.0 |  $\pm$0.5
$\pi^{0}$ at upstream | 0.2 |  $\pm$0.1
$\eta$ at downstream | 8.2 |  $\pm$2.3
Total | 56.0 |  $\pm$2.8

####  2.3.8 Outlook at J-PARC

We assume $3\times 10^{7}$ s running time with 100 kW beam on a 1-interaction-
length T2 target, where the $K_{L}$ flux is $1.1\times 10^{7}$ per $2\times
10^{13}$ protons on target.

The sensitivity and the impacts are calculated and summarized in Table 10.
Here we assume that the statistical uncertainties in the numbers of events are
dominant.

The single event sensitivity is evaluated to be $8.5\times 10^{-13}$. The
expected number of background events is 56. With the SM branching fraction of
$3\times 10^{-11}$, 35 signal events are expected with a signal-to-background
ratio (S/B) of 0.63.

  * •

$4.7$-$\sigma$ observation is expected for the signal branching fraction $\sim
3\times 10^{-11}$.

  * •

It indicates new physics at the 90% confidence level (C.L.) if the new physics
gives $44\%$ deviation on the BR from the SM prediction.

  * •

It corresponds to $14\%$ measurement of the CP-violating CKM parameter $\eta$
in the SM (The branching fraction is proportional to $\eta^{2}$).

Table 10: Summary of the sensitivity and the impact.

| Formula | Value
---|---|---
Signal (branching fraction : $3\times 10^{-11}$) | $S$ | $35.3\pm 0.4$
Background | $B$ | $56.0\pm 2.8$
Single event sensitivity | $(3\times 10^{-11})/S$ | $8.5\times 10^{-13}$
Signal-to-background ratio | $S/B$ | 0.63
Significance of the observation | $S/\sqrt{B}$ | $4.7\sigma$
90%-C.L. excess / deficit | $1.64\times\sqrt{S+B}$ | 16 events
| $1.64\times\sqrt{S+B}/S$ | 44% of SM
Precision on branching fraction | $\sqrt{S+B}/S$ | 27%
Precision on CKM parameter $\eta$ | $0.5\times\sqrt{S+B}/S$ | 14%

  * *

Running time of $3\times 10^{7}$ s is assumed in the calculation.

##  3 Experiments at the CERN SPS

###  3.1 NA62 until LS3

As discussed in Section 1.2.3, NA62 has performed the first measurement of the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ branching ratio with $3.4\sigma$ significance,
having collected 20 candidate events in 2016–2018 with an expected $S/B$ of
about 1.4. As a result, ${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ is now
experimentally known to about 40% precision. The original NA62 goal, however,
was to measure the BR to within just over 10%. To fully realize the potential
of the experiment to achieve its original goal, in October 2019, NA62
submitted an addendum to its original proposal [49] to the CERN SPSC to extend
the data taking throughout LHC Run III, until Long Shutdown 3 (LS3), currently
foreseen at the end of 2025 [137]. Additional data taking was initially
approved for one year and began in July 2021. Finally, in 2021 NA62 was
approved until LS3.

In advance of the 2021 run, numerous improvements to the experiment were made
in order to allow data to be taken efficiently at the nominal beam intensity,
including improvements to the efficiency and stability of the data-acquisition
system, the introduction of additional detectors to provide increased
rejection power against specific backgrounds, the reduction of losses from
accidental coincidence vetoes, and the further optimization of the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ analysis.

Figure 34: Layout of the new GTK achromat and beamline elements for NA62 in
2021. The main changes include optimization of the achromat and the
installation of additional detectors: a fourth GTK station (GTK0) placed next
to GTK1, a new veto counter (Veto 1–3) around the beam pipe before and after
the final collimator, and a new hodoscope (Anti 0) to veto muon halo
background.

As noted in Section 1.2.3, the largest source of background in NA62 is from
upstream $K^{+}$ decays and interactions. To reduce this background, the
beamline elements were rearranged around the GTK achromat, and a fourth
station was added to the GTK beam tracker to reduce the influence of
combinatorics in GTK track reconstruction (Figure 34). An upstream charged-
particle veto counter, consisting of three stations of 2-cm wide scintillator
tiles mounted immediately above and below the beam pipe, was installed just
upstream and downstream of the final collimator to detect photons and charged
pions from upstream $K^{+}$ decays. This veto counter is read out with a new
digital front-end board data-acquisition chain consisting of a new, custom,
on-detector FPGA TDC board that sends data over high-speed links to an off-
detector front-end server equipped with FELIX data receiving PCIe cards. In
addition, a new hodoscope (ANTI0) was installed at the entrance of the
fiducial volume, which mainly serves to veto the muon halo background for new
physics searches in beam dump mode. In standard running with the $K^{+}$ beam,
the ANTI0 is expected to assist with the reduction of background trigger
rates.

The above improvements allow the present selection criteria against the
upstream background to be loosened, regaining signal acceptance. Steps to
reduce losses from accidental coincidences in the veto detectors include
improvements to the use of time-clustering information for LKr reconstruction
and the adoption of multivariate analysis techniques to refine the veto from
the LAV detectors and to make better used of the information from the small-
angle photon vetoes and from the detectors used to reject events with charged
products from photon interactions. These optimizations are currently being
refined with the 2016–2018 dataset, for which the analysis has been completed.

###  3.2 Kaon physics at the SPS after LS3

Following upon the successful application at NA62 of the in-flight technique
to measure ${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$, we envision a
comprehensive program for the study of the rare decay modes of both $K^{+}$
and $K_{L}$ mesons, to be carried out with high-intensity kaon beams from the
CERN SPS in multiple phases, including an experiment to measure ${\rm
BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ at the 5% level and an experiment to measure
${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$ at the 20% level. The detectors could
also be reconfigured to allow measurements of $K_{L}$ decays with charged
particles, such as $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$.

The success of this program depends on the delivery of a high-intensity, slow-
extracted 400-GeV/$c$ proton beam from the SPS to the ECN3 experimental hall
(where NA62 is located), which is ideally suited for next-generation kaon
experiments. Up to $10^{19}$ protons on target per year will be required.
Studies indicate that, with duty-cycle optimization, sufficient proton fluxes
can be delivered to allow a high-intensity kaon experiment to run as part of a
robust fixed-target program in the CERN North Area [138]. The feasibility of
upgrades to the primary beamline tunnel, target gallery, and experimental
cavern to handle an intensity of up to six times that in NA62 has been studied
as part of the Physics Beyond Colliders initiative at CERN, and preliminary
indications are positive [139, 140].

To measure ${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ to $5\%$ accuracy matching
the irreducible theory precision, the experiment must be able to handle a beam
intensity of at least four times that in NA62. The entire complement of NA62
detectors would require substantial upgrades in order to provide clean event
reconstruction at the expected rates. Ideas for the new detectors are inspired
by R&D efforts for HL-LHC.

The baseline design for the $K_{L}$ experiment is the KLEVER project [141],
presented as part of the Physics Beyond Colliders initiative [142]. The KLEVER
goal is to measure ${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$ to within 20%. The
boost from the high-energy neutral beam facilitates the rejection of
background channels such as $K_{L}\to\pi^{0}\pi^{0}$ by detection of the
additional photons in the final state. Background from $\Lambda\to n\pi^{0}$
decays in the fiducial volume must be kept under control, and recent work has
focused on the possibility of a beamline extension and other adaptations of
the experiment to ensure sufficient rejection of this channel. The layout
poses challenges for the design of the small-angle vetoes, which must reject
photons from $K_{L}$ decays escaping through the beam exit amidst an intense
background from soft photons and neutrons in the beam.

The $K^{+}$ and $K_{L}$ experiments would run consecutively, with the order to
be determined, and use interchangeable detectors. The forward calorimetry and
photon veto systems would be common to both experiments. Detector and beamline
configuration changes between phases would be scheduled during LHC shutdown
periods. As in NA62, the broad physics program of the high-intensity $K^{+}$
experiment includes studies of rare kaon decays, searches for lepton-flavor or
lepton-number violating decays, and searches for hidden sectors in kaon and
pion decays (Section 1.3). In the $K_{L}\to\pi^{0}\nu\bar{\nu}$ configuration,
to optimize the measurement, the $K_{L}$ experiment would have no secondary
tracking and limited PID capability. We foresee however the possibility of
taking data with the downstream detectors configured for charged-particle
measurements (secondary tracking and PID) but with a neutral beam. This will
allow searches for radiative and lepton-flavor violating $K_{L}$ decays,
$K_{L}$ decays to exotic particles, and particularly, measurements of the
$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays, for which only limits exist at
present. These measurements would overconstrain the kaon unitarity triangle;
to the extent that there is little constraint on the CP-violating phase of the
$s\to d\ell^{+}\ell^{-}$ transition, the $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$
measurements may reveal the effects of new physics [77].

###  3.3 High-intensity beams

Future experiments at the SPS for precision measurements of the BRs of the
$K\to\pi\nu\bar{\nu}$ decays will require a very intense slow-extracted proton
beam.

At a nominal intensity of $1.1\times 10^{12}$ protons on target per effective
second of spill, the current NA62 beam, with an angular acceptance of 12
$\mu$sr and a momentum acceptance of $\delta p/p\sim 1\%$, has a total rate of
about 750 MHz and a kaon content of about 6%. When going from charged to
neutral beam, the momentum acceptance is defined by the fiducial volume and
analysis cuts and is a considerable fraction of the total momentum
distribution. However, the increase in flux is offset by the very tight
collimation required for neutral beams (at most 0.5 $\mu$sr, in order to
conserve $p_{\perp}$ constraints for background rejection, in the case of
KLEVER). Then, because of the optimization of the production angle for the
$K_{L}$ experiment, the smaller fiducial-volume acceptance for $K_{L}$ decays
due to the longer lifetime, and the smaller standard-model BR for the
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay, an increase of intensity by an order of
magnitude is required to obtain the same rate of decays in the fiducial
volume. Therefore, the statistical needs of the neutral-kaon experiment drive
the intensity request.

The assumed proton beam intensity for KLEVER of $6.7\times 10^{12}$ protons on
target per effective second of spill ($2\times 10^{13}$ ppp assuming a 4.8-s
flat-top and a 3-s effective spill) is as high as considered reasonably
achievable in an existing North Area installation, as described below. For the
KLEVER beam parameters, the total beam rate (excluding low-energy photons) is
just under 700 MHz, and the total rate at which events must be vetoed while
maintaining high signal efficiency is around 150 MHz.

####  3.3.1 Primary proton beam

The proton beam for the North Area experiments is extracted from the SPS with
electromagnetic (ZS) and magnetic (MST) septa in the LSS2 area and the beam is
brought to the North Area targets via the TT20 transfer tunnel. Before
arriving at the TCC2 target cave, the beam is split twice into three beams,
which are brought to interact on targets T2, T4, and T6 inside TCC2. Targets
T2 and T4 are “wobbling” targets, at which multiple secondary beams can be
produced. The T4 target produces (in addition to the H6 and H8 beams for the
EHN1 beamlines) the P42 proton beam, which is the primary beam for kaon
experiments. The P42 beam is transported over 800 m to the TCC8 target cave,
which houses the T10 target used to produce the secondary beam for the ECN3
hall (where the NA62 experiment is currently installed).

We assume that the proposed $K\to\pi\nu\bar{\nu}$ experiments will be
installed in ECN3. This will require upgrades to the P42 transport line, and
TCC8 target gallery. These upgrades have been studied by the Conventional
Beams working group in the context of the Physics Beyond Colliders initiative
[139, 140]. Preliminary indications from these studies are described below.

##### Slow extraction to T4

In 2018, the maximum intensity accelerated in the SPS under normal operation
was $3.5\times 10^{13}$ ppp, obtained using the multi-turn extraction scheme
for the PS-to-SPS transfer [138]. This intensity is more than sufficient to
allow both a high-intensity kaon experiment and a robust North Area fixed-
target program to run concurrently. In reality, however, the success of the
experiment depends on the total number of protons delivered to target during
the running period, as well as the quality of the spill. In 2018, about
$1.25\times 10^{19}$ protons were delivered to the TCC2 targets. To observe 60
SM $K_{L}\to\pi^{0}\nu\bar{\nu}$ events, KLEVER will require a total
integrated intensity of $1\times 10^{19}$ pot/year for five years.

The SPS Losses and Activation Working Group and Physics Beyond Colliders Beam
Dump Facility Working Group have made general progress on issues related to
the slow extraction of high-intensity proton beams to the North Area. The
emphasis of these studies has been on how to extract $4\times 10^{19}$ pot
onto the BDF target while delivering roughly the same number of protons
($1\times 10^{19}$) per year to the TCC2 targets as in past years.

The principal limitation on the sustained intensity on target (i.e., in
pot/year) is from beam losses at the extraction septa, leading to machine
activation, reduced component lifetime, and limitations on personnel access
and maintenance. Several percent of the beam is lost in the extraction
process. Primary proton scattering on the septum wires is the dominant source
of the beam loss. To keep the residual activation levels at around 5 mSv/h
after 30 hours of cool-down while extracting around $5\times 10^{19}$ pot per
year will require a reduction in the activation per proton of approximately a
factor four. The BDF study developed and tested methods to reduce the losses
per proton at extraction by a factor of four. A significant effort was made on
several fronts to conceive, design, deploy and test where possible methods to
reduce the extraction beam losses or to mitigate their effects through
different material choice or handling. It is expected that, through a
combination of concepts presented in this study, a factor of four reduction of
the prompt extraction losses in LSS2 is within reach for operational
scenarios.

The same extraction system in LSS2 and transfer line to TT20 are used for the
BDF and the TCC2 targets, with the first TCC2 splitter (splitting the beam
between the T2-T4 and the T6 targets) replaced to allow switching between the
destinations BDF and North Area on a cycle-by-cycle basis. Therefore, the
factor of four in loss reduction should apply to both BDF and North Area
extraction. This substantially relaxes the limitation on the total number of
protons extracted per year (to whatever target), to $5\times 10^{19}$ pot.

For the TCC2 targets, the next limiting factor then becomes losses on the
Lamberson splitter septa used to split the beam first between T2/T4 and T6,
and then between T2 and T4. The fraction of beam lost on the splitters is at
least 3%, as estimated by intensity monitors in the NA. Activation from
splitter losses would then represent the limitation on the total number of
protons delivered per year. It may be possible to reduce the losses per proton
during the splitting process to the North Area; this question certainly merits
further study.

Duty cycle optimization for the BDF and North Area has been studied under the
assumptions that extraction losses can be reduced by a factor of four but that
splitter losses limit the total number of protons per year delivered to the
North Area targets to current levels [138]. The studies were performed taking
into account realistic supercycle compositions and respecting the SPS limits
on power dissipation in the magnets. Two values of the flat-top length for the
North Area targets were considered, 4.9 and 9.7 s, with the more favorable
results being obtained for a 4.9-s flat top. The results are summarized in
Figure 35.

Figure 35: Future proton sharing scenarios with (green) and without (blue) ion
operation, with a 4.9-s flat top (FT) for the TCC2 experiments. Reproduced
from Ref. [138].

As noted above, activation from splitter losses limit the North Area targets
to 1–1.5$\times 10^{19}$ pot/year. Limits from radiation resistance of cables
in the extraction region are also illustrated. The blue and green lines
indicate proton sharing between the TCC2 targets and BDF, with and without ion
running, assuming 80% SPS availability. These results indicate that with small
compromises, BDF, K experiments, and the North Area test beam program might be
able to run concurrently. It does not currently seem possible to run the BDF,
K experiments, and a robust fixed-target program requiring significant
intensity all at the same time without significant compromises.

It is important to note that this study was performed with the primary
objective of establishing the feasibility of delivering $4\times 10^{19}$