Omm$#2\text{\,}#\mathrm{3}$

HIKE, High Intensity Kaon Experiments
at the CERN SPS

Letter of Intent

The HIKE Collaboration***Address all correspondence to<EMAIL_ADDRESS>
###### Abstract

A timely and long-term programme of kaon decay measurements at a new level of
precision is presented, leveraging the capabilities of the CERN Super Proton
Synchrotron (SPS). The proposed programme is firmly anchored on the experience
built up studying kaon decays at the SPS over the past four decades, and
includes rare processes, CP violation, dark sectors, symmetry tests and other
tests of the Standard Model. The experimental programme is based on a staged
approach involving experiments with charged and neutral kaon beams, as well as
operation in beam-dump mode. The various phases will rely on a common
infrastructure and set of detectors.

The HIKE collaboration

E. Cortina Gil1, J. Jerhot1, N. Lurkin1, T. Numao2, B. Velghe2, V. W. S.
Wong2, D. Bryman3, L. Bician4, Z. Hives4, T. Husek4, K. Kampf4, M. Koval4, A.
T. Akmete5, R. Aliberti5, V. Büscher5, L. Di Lella5, N. Doble5, L. Peruzzo5,
M. Schott5, H. Wahl5, R. Wanke5, B. Döbrich6, L. Montalto7, D. Rinaldi7, F.
Dettori8,9, A. Cardini9, A. Lai9, L. Bomben10, S. Carsi10, M. Prest10, A.
Selmi10, G. Lezzani11, P. Monti-Guarnieri11, L. Perna11, P. Dalpiaz12,13, V.
Guidi,12,13, A. Mazzolari12,13, I. Neri12,13, F. Petrucci12,13, M.
Soldani12,13, L. Bandiera13, A. Cotta Ramusino13, A. Gianoli13, M.
Romagnoni13, A. Sytov13, M. Lenti14,15, I. Panichi14,15, G. Ruggiero14,15, A.
Bizzeti15, F. Bucci15, A. Antonelli16, E. Di Meco16,30, G. Lanfranchi16, S.
Martellotti16, M. Martini16,31, M. Moulson16, D. Paesani16,30, I. Sarra16, T.
Spadaro16, G. Tinti16, E. Vallazza17, F. Ambrosino18,20, R. Giordano18,20, P.
Massarotti18,20, M. Napolitano18,20, G. Saracino18,20, C. Di Donato19,20, G.
D’Ambrosio20, M. D’Errico20, M. Mirra20, S. Neshatpour20, R. Fiorenza21, I.
Rosa21, D. De Salvador22, F. Sgarbossa22, G. Anzivino23,24, S. Germani23,24,
R. Volpe23,24, P. Cenci24, S. Cutini24, V. Duk24, P. Lubrano24, M. Pepe24, M.
Piccini24, F. Costantini25,26, S. Donati25,26, M. Giorgi25,26, S.
Giudici25,26, G. Lamanna25,26, E. Pedreschi25,26, J. Pinzino25,26, M.
Sozzi25,26, R. Fantechi26, V. Giusti26, F. Spinella26, I. Mannelli27, M.
Raggi28,29, A. Biagioni29, P. Cretaro29, O. Frezza29, F. Lo Cicero29, A.
Lonardo29, M. Turisini29, P. Vicini29, R. Ammendola30, V. Bonaiuto30, A.
Fucci30, A. Salamon30, F. Sargeni30, R. Arcidiacono32,33, B. Bloch-Devaux32,
E. Menichetti32,33, E. Migliore32,33, C. Biino33, F. Marchetto33, D.
Baigarashev34, Y. Kambar34, D. Kereibay34, Y. Mukhamejanov34, S. Sakhiyev34,
A. Briano Olvera35, J. Engelfried35, N. Estrada-Tristan35, R. Piandani35, M.
A. Reyes Santos35, K. A. Rodriguez Rivera35, P. C. Boboc36, A. M.
Bragadireanu36, S. A. Ghinescu36, O. E. Hutanu36, T. Blazek37, V. Cerny37, A.
Kleimenova37, Z. Kucerova37, D. Martinez Santos38, C. Prouve39, M. Boretto40,
F. Brizioli40, A. Ceccucci40, M. Corvino40, H. Danielsson40, F. Duval40, E.
Gamberini40, R. Guida40, E. B. Holzer40, B. Jenninger40, G. Lehmann Miotto40,
P. Lichard40, K. Massri40, E. Minucci40, M. Perrin-Terrin40, V. Ryjov40, J.
Swallow40, M. Van Dijk40, M. Zamkovsky40, R. Marchevski41, A. Gerbershagen42,
J. R. Fry43, F. Gonnella43, E. Goudzovski43, J. Henshaw43, C. Kenworthy43, C.
Lazzeroni43, C. Parkinson43, A. Romano43, J. Sanders43, A. Shaikhiev43, A.
Tomczak43, H. Heath44, D. Britton45, A. Norton45, D. Protopopescu45, J. B.
Dainton46, R. W. L. Jones46, A. De Santo47, F. Salvatore47, P. Cooper48, D.
Coward48, P. Rubin48

1 Université Catholique de Louvain, B-1348 Louvain-La-Neuve, Belgium
2 TRIUMF, Vancouver, British Columbia, V6T 2A3, Canada
3 University of British Columbia, Vancouver, British Columbia, V6T 1Z4, Canada
4 Charles University, 116 36 Prague 1, Czech Republic
5 Johannes Gutenberg Universität Mainz, D-55099 Mainz, Germany
6 Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), D-80805
München, Germany
7 Dipartimento di Scienze e Ingegneria della Materia, dell’Ambiente ed
Urbanistica, Università Politecnica delle Marche, I-60131 Ancona, Italy
8 Dipartimento di Fisica dell’Università, I-09042 Cagliari, Italy
9 INFN, Sezione di Cagliari, I-09042 Cagliari, Italy
10 Università degli Studi dell’Insubria, 22100, Como and INFN Sezione Milano
Bicocca, I-20126 Milano, Italy
11 Dipartimento di Scienza e Alta Tecnologia, Università degli Studi
dell’Insubria, I-22100, Como, Italy
12 Dipartimento di Fisica e Scienze della Terra dell’Università, I-44122
Ferrara, Italy
13 INFN, Sezione di Ferrara, I-44122 Ferrara, Italy
14 Dipartimento di Fisica e Astronomia dell’Università, I-50019 Sesto
Fiorentino, Italy
15 INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Italy
16 INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
17 INFN Sezione di Milano Bicocca, I-20126 Milano, Italy
18 Dipartimento di Fisica “Ettore Pancini” dell’Università degli Studi di
Napoli Federico II, I-80126 Napoli, Italy
19 Dipartimento di Ingegneria, Università degli Studi di Napoli Parthenope,
I-80143 Napoli
20 INFN, Sezione di Napoli, I-80126 Napoli, Italy
21 Scuola Superiore Meridionale e INFN, Sezione di Napoli, I-80138 Napoli,
Italy
22 Dipartimento di Fisica e Astronomia dell’ Università degli Studi di Padova,
I-35131 Padova e INFN Laboratori Nazionali di Legnaro, I-35020 Legnaro, Italy
23 Dipartimento di Fisica e Geologia dell’Università, I-06100 Perugia, Italy
24 INFN, Sezione di Perugia, I-06100 Perugia, Italy
25 Dipartimento di Fisica dell’Università, I-56100 Pisa, Italy
26 INFN, Sezione di Pisa, I-56100 Pisa, Italy
27 Scuola Normale Superiore e INFN, Sezione di Pisa, I-56100 Pisa, Italy
28 Dipartimento di Fisica, Sapienza Università di Roma, I-00185 Roma, Italy
29 INFN, Sezione di Roma I, I-00185 Roma, Italy
30 Dipartimento di Fisica, Università degli Studi di Roma Tor Vergata, I-00133
Roma, Italy
31 Dipartimento di Scienze Ingegneristiche, Università degli Studi Guglielmo
Marconi, I-00193 Roma, Italy
32 Dipartimento di Fisica dell’Università, I-10125 Torino, Italy
33 INFN, Sezione di Torino, I-10125 Torino, Italy
34 Institute of Nuclear Physics, 050032 Almaty, Kazakhstan
35 Instituto de Física, Universidad Autónoma de San Luis Potosí, 78240 San
Luis Potosí, Mexico
36 Horia Hulubei National Institute for R&D in Physics and Nuclear
Engineering, 077125 Bucharest-Magurele, Romania
37 Faculty of Mathematics, Physics and Informatics, Comenius University, 842
48, Bratislava, Slovakia
38 Axencia Galega de Innovacion, Conselleria de Economia e Industria, Xunta de
Galicia, 15704 Santiago de Compostela, Spain
39 Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, 15782 Santiago de Compostela, Spain
40 CERN, European Organization for Nuclear Research, CH-1211 Geneva 23,
Switzerland
41 Ecole polytechnique fédérale de Lausanne (EPFL), CH-1015 Lausanne,
Switzerland
42 PARTREC, UMCG, University of Groningen, 9747 AA Groningen, The Netherlands
43 School of Physics and Astronomy, University of Birmingham, Edgbaston,
Birmingham, B15 2TT, UK
44 School of Physics, University of Bristol, Bristol, BS8 1TH, UK
45 School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ,
UK
46 Faculty of Science and Technology, University of Lancaster, Lancaster, LA1
4YW, UK
47 Department of Physics and Astronomy, University of Sussex, Brighton, BN1
4GE, UK
48 Physics and Astronomy Department, George Mason University, Fairfax, VA
22030, USA

###### Contents

  1. 1 Overview of HIKE
    1. 1.1 The scientific goals of HIKE
  2. 2 Scientific context
    1. 2.1 Experimental context for high-scale BSM physics
    2. 2.2 Experimental context for feebly-interacting particles
    3. 2.3 The role of HIKE
  3. 3 Physics with high-intensity kaon beams
    1. 3.1 Flavour-changing neutral currents
    2. 3.2 Lepton flavour universality tests
    3. 3.3 Lepton flavour and lepton number violation
    4. 3.4 Tests of low-energy QCD
    5. 3.5 Test of first-row CKM unitarity
    6. 3.6 Production of feebly-interacting particles in kaon decays
  4. 4 Physics in the beam-dump mode
  5. 5 High-intensity beams
    1. 5.1 Beam delivery to the kaon production target
    2. 5.2 Charged kaon beamline
    3. 5.3 Neutral kaon beams
  6. 6 Phase 1: a multi-purpose $K^{+}$ decay experiment
    1. 6.1 Experimental layout
    2. 6.2 Physics sensitivity
  7. 7 Phase 2: a multi-purpose $K_{L}$ decay experiment
    1. 7.1 Experimental layout
    2. 7.2 Physics sensitivity
  8. 8 Phase 3 (KLEVER): measurement of the $K_{L}\to\pi^{0}\nu\bar{\nu}$ decay
    1. 8.1 Experimental layout
    2. 8.2 Rates and timing performance
    3. 8.3 Expected performance for $K_{L}\to\pi^{0}\nu\bar{\nu}$
  9. 9 Operation in beam-dump mode
    1. 9.1 Experimental layout
    2. 9.2 Physics sensitivity
  10. 10 The HIKE detectors
    1. 10.1 Detectors upstream of the fiducial decay volume
    2. 10.2 Fiducial decay volume and its detectors
    3. 10.3 Detectors downstream of the fiducial decay volume
    4. 10.4 Detectors specific to KLEVER
  11. 11 Data acquisition and high-level trigger
    1. 11.1 Readout boards
    2. 11.2 Streaming readout
    3. 11.3 Triggered readout
    4. 11.4 Event-building farm and event filter
  12. 12 Online and offline computing
    1. 12.1 Online monitoring
    2. 12.2 Calibration and data quality
    3. 12.3 Data processing model
    4. 12.4 Distributed computing model
    5. 12.5 Data reduction model
  13. 13 Infrastructure and safety
  14. 14 Summary and conclusions

##  1 Overview of HIKE

Over the past 75 years, experimental studies of kaon decays have played a
singular role in propelling the development of the Standard Model. As in other
branches of flavour physics, the continuing experimental interest in the kaon
sector derives from the possibility of conducting precision measurements,
particularly of suppressed 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 next-generation colliders. 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.

Over the past four decades, the CERN North Area has been the host to a
successful series of precision kaon decay experiments. Among the many results
obtained by these experiments is the discovery of direct CP violation, widely
quoted to be among the major discoveries made at CERN.

Continuation of high-intensity kaon experiments at CERN, including future
upgrades (providing a unique probe into BSM physics complementary to the
$B$-sector) has been identified as an essential scientific activity in the
2020 Update of the European Strategy for particle physics, and is strongly
supported in the national roadmaps across Europe.

The High Intensity Kaon Experiments (HIKE) project represents a broad, long-
term programme at CERN after LS3, based in the North Area ECN3 experimental
hall, covering all the main aspects of rare kaon decays and searches
accessible via kaon physics, from ultra-rare kaon decays to precision
measurements and searches for new phenomena. HIKE is intended to continue the
very successful tradition of kaon experiments at CERN in ECN3, the latest of
which is the currently operating NA62 experiment. HIKE will profit from a beam
intensity increase by a factor between four and six with respect to NA62, and
cutting-edge detector technologies. This will allow HIKE to play a pivotal
role in the quest for New Physics (NP) at the sensitivity required by the
present experimental limits and theoretical models, in a wide range of
possible masses and interaction couplings. The programme will consist of
multiple phases, first with charged and then neutral kaon beams, as well as
periods in beam dump mode. The long decay volume and detector characteristics
needed for kaon physics make HIKE suitable to search for new feebly-
interacting, long-lived particles, providing unique sensitivity to forward
processes. The detector design is challenging but at least one technological
solution exists for each subsystem, thanks also to the synergy with HL-LHC.
The various phases allow for insertion, repositioning or removal of specific
detector elements depending on the physics requirements, while the overall
setup remains broadly the same.

###  1.1 The scientific goals of HIKE

The HIKE programme consists of several phases using shared detectors and
infrastructure: a charged kaon phase between LS3 and LS4, and neutral kaon
experiments commencing after LS4 including a phase with tracking, and a phase
optimised specifically for the $K_{L}\to\pi^{0}\nu\bar{\nu}$ measurement
(KLEVER).

Phase 1: a multi-purpose $K^{+}$ decay experiment

The physics of $K^{+}$ decays will be scrutinised with the highest precision,
expanding and improving the physics programme of the NA62 experiment presently
running in ECN3. New detectors will replace those of NA62 with the goal of
improving the performance and sustaining higher rates.

The principal goals of this phase are:

  * •

Measurement of the $K^{+}\to\pi^{+}\nu\bar{\nu}$ branching ratio to 5%
relative precision, matching the SM theoretical uncertainty.

  * •

Precision measurements of $K^{+}\to\pi^{+}e^{+}e^{-}$ and
$K^{+}\to\pi^{+}\mu^{+}\mu^{-}$ decays, and a precision lepton universality
test.

  * •

Searches for lepton flavour/number violating decays $K^{+}\to\pi(\pi^{0})\mu
e$, $K^{+}\to\pi^{-}(\pi^{0})\ell^{+}\ell^{+}$,
$K^{+}\to\ell_{1}^{-}\nu\ell_{2}^{+}\ell_{2}^{+}$ and
$\pi^{0}\to\mu^{\pm}e^{\mp}$ at the ${\cal O}(10^{-12})$ sensitivity.

  * •

Measurement of the quantity $R_{K}=\Gamma(K^{+}\to
e^{+}\nu)/\Gamma(K^{+}\to\mu^{+}\nu)$ to 0.1% precision.

  * •

Measurement of the ratios of the branching ratios of the main decay modes
$K^{+}\to\pi^{0}\ell^{+}\nu$, $K^{+}\to\pi^{+}\pi^{0}$,
$K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ to a relative precision of a few per mille.

  * •

Improvements of existing measurements of rare decays, including
$K^{+}\to\pi^{+}\gamma\gamma$, $K^{+}\to\pi^{+}\gamma\ell^{+}\ell^{-}$,
$K^{+}\to\pi^{+}\pi^{0}\gamma$, $K^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$.

  * •

Searches for production of feebly-interacting particles in $K^{+}$ decays.

  * •

Collection of a dataset in the beam-dump mode (with appropriate time sharing
with the kaon mode), aiming for a factor 10 improvement in sensitivity to
decays of feebly-interacting particles with respect to what NA62 can achieve
by LS3.

Phase 2: a multi-purpose $K_{L}$ decay experiment

The physics of the $K_{L}$ decays will be addressed, including $K_{L}$ decays
into charged particles in the final state. This mode of operation will require
some modifications of the detector layout with respect to the charged mode.
The main objectives of this phase are:

  * •

Observation of the ultra-rare decays $K_{L}\to\pi^{0}e^{+}e^{-}$ and
$K_{L}\to\pi^{0}\mu^{+}\mu^{-}$, or establishment of stringent upper limits on
the branching ratios of these decays at the ${\cal O}(10^{-11})$ level.

  * •

Measurement of the $K_{L}\to\mu^{+}\mu^{-}$ decay branching ratio to 1%
relative precision.

  * •

Searches for lepton flavour violating decays including
$K_{L}\to(\pi^{0})(\pi^{0})\mu^{\pm}e^{\mp}$ and $K_{L}\to
e^{\pm}e^{\pm}\mu^{\mp}\mu^{\mp}$ with ${\cal O}(10^{-12})$ sensitivity.

  * •

Measurement of the ratios of the branching ratios of the main decay modes
$K_{L}\to\pi^{\pm}\ell^{\mp}\nu$, $K_{L}\to\pi^{+}\pi^{-}(\pi^{0})$,
$K_{L}\to\pi^{0}\pi^{0}(\pi^{0})$ to a relative precision of a few per mille.

  * •

Collection of a further dataset in the beam-dump mode (with appropriate time
sharing with the kaon mode), and characterisation of the neutral beam
necessary to proceed to the third phase of HIKE.

Phase 3 (KLEVER): measurement of the $K_{L}\to\pi^{0}\nu\bar{\nu}$ decay

The experimental apparatus will be modified to specifically address the
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay, and to measure its branching ratio to a
20% relative precision. To optimise the measurement, the tracking systems will
be removed from the fiducial volume and additional photon veto detectors
installed. Although this phase will focus on $K_{L}\to\pi^{0}\nu\bar{\nu}$,
the setup will allow study of additional topics:

  * •

Searches for additional flavour-changing neutral current decays, such as
$K_{L}\to\pi^{0}\pi^{0}\nu\bar{\nu}$.

  * •

Limits on the production of a long-lived, feebly interacting particle $X$ in
$K_{L}\to\pi^{0}X$ decays, as a by-product of the measurement of
$K_{L}\to\pi^{0}\nu\bar{\nu}$.

  * •

Searches for $K_{L}$ decays to feebly-interacting particles $X$ such as
$K_{L}\to XX$, with $X\to\gamma\gamma$.

  * •

Searches for forbidden decays such as $K_{L}\to\pi^{0}\gamma$.

##  2 Scientific context

The Standard Model (SM) of particle physics describes the experimental results
obtained so far by experiments with exceptional precision. The culmination of
this success was the discovery of the Higgs boson by the ATLAS and CMS
collaborations at the LHC in 2012 [1, 2]. Thus the SM provides a solid
foundation for the present understanding of elementary particles and their
interactions. Despite this success, cosmological observations, experimental
tensions, and theoretical motivations strongly suggest that the SM is an
approximation of a more fundamental theory. One paradigm assumes that this
theory lies above the electroweak (EW) mass scale and manifests itself in
terms of new particles with masses well above the Higgs boson mass and having
sizeable interactions with the SM fields. Another paradigm accounts for
extensions of the SM that predict particles with masses below the EW scale,
feebly interacting with the SM fields. Such a low-energy NP paradigm is
usually referred to as a Feebly Interacting Particles (FIPs) scenario. Both
high-scale BSM and FIPs models provide explanations for the main open
questions in modern physics: the nature of dark matter, the baryon asymmetry
of the universe, cosmological inflation, the origin of neutrino masses and
oscillations, the strong CP problem, the hierarchical structure of the Yukawa
couplings, the hierarchy problem, and the cosmological constant.

HIKE intends to study NP in both high-scale BSM and FIPs directions. In this
respect, HIKE is perfectly aligned with the recommendation of the 2020 Update
of the European Strategy for Particle Physics [3], where a whole chapter is
devoted to flavour physics research with the statement: “Experimental hints
for deviations from SM predictions in flavour processes are one of our best
hopes to direct research towards the right energy scale where NP can be
found”.

###  2.1 Experimental context for high-scale BSM physics

The experimental techniques to investigate the presence of BSM physics at high
mass scales are direct searches for massive particles or processes forbidden
by the SM, and precision measurements of observables precisely known within
the SM. The first method is usually referred to as direct searches; the second
as indirect searches.

Direct searches allow unambiguous identification of NP already in single
processes. After the observation and the study of the Higgs boson, direct
searches for BSM particles were a primary goal of the LHC experiments ATLAS
and CMS. Still, the LHC centre-of-mass energy limits the sensitivity of this
technique to the TeV scale, or slightly above. Besides, a direct observation
alone gives little information on the BSM structure. The absence of any
significant direct observations of NP so far sets strong bounds on several
types of BSM models up to the TeV scale. The bounds are still statistically
limited in the TeV region, and the LHC high-luminosity programme will allow
them to be extended to the kinematic limits, if no observations emerge in the
meantime. Another type of direct searches focuses on the study of processes
forbidden by the SM due to accidental SM symmetries like such as charged
lepton flavour (CLF) or lepton number (LN) conservation. These searches
typically involve the study of hadron or lepton decays forbidden by the SM,
for example, $\tau\to\mu\mu\mu$ and $K_{L}\to\mu^{\pm}e^{\mp}$. Presently,
these searches are statistically limited, and experiments at the high-
intensity frontier are planned with increased sensitivity.

Indirect searches exploit the possibility that NP particles affect low-energy
observables virtually, via loop-level corrections to the SM prediction. The
golden modes of indirect searches are theoretically clean observables of rare
processes in the SM, i.e. occurring at loop-level at the lowest order, because
NP here may favorably compete with SM. Indirect searches allow sensitivities
to NP up to mass scales well above the TeV, but may require measurements of
several observables to provide conclusive evidence of NP and to determine its
nature. Given the limits to NP at TeV-scale already set by LHC through direct
searches, indirect searches represent a promising tool to further boost BSM
searches. Experimental flavour physics, measurements of Higgs couplings, and
measurements in the gauge sectors of the SM ($W$, $Z$, top masses and gauge
couplings) are typical examples of indirect searches. The existing
measurements are broadly consistent with the SM and already suggest that NP
may occur well beyond the TeV scale. Nevertheless, tensions between
experiments and the SM exist mainly in flavour physics observables, and
statistical limitations or the lack of measurements of rare processes
demonstrate the need for additional efforts and motivate future directions in
experimental particle physics. Presently, the most significant experimental
tensions with the SM are:

  * •

the violation of lepton flavour universality in $b\to s$ and $b\to c$
transitions reported by LHCb and $B$-factories [4, 5, 6, 7, 8, 9, 10], with a
global significance of about 3–4$\sigma$ [11, 12, 13, 14];

  * •

the violation of CKM unitarity in the first row, known as the “Cabibbo angle
anomaly”, with a significance of about 2–3$\sigma$ [15, 16, 17];

  * •

the inconsistency between the exclusive and inclusive measurements of the CKM
matrix element $|V_{cb}|$ at the level above $3\sigma$ [18];

  * •

the inconsistency between the experiment and the SM prediction for the
anomalous magnetic dipole moment of the muon at the level of 4.2$\sigma$ [19].

The sensitivity of the indirect searches to BSM physics scales with the size
of the datasets, which is particularly important in case of rare processes.
Moving to higher intensities is the way to increase statistics, however this
requires improvements of the detector performance to cope with harsher
experimental environments. Upgraded detectors also allow the improvement of
the accuracy of already precise measurements, overcoming systematic
uncertainties whenever they are the main limiting factor. The indirect search
approach is at the heart of the LHC intensity frontier programme, the
motivation for the upgrade of the LHCb experiment and for the continuation of
the $B$-factory programme with the Belle II experiment.

###  2.2 Experimental context for feebly-interacting particles

Direct searches for new low-mass particles can provide experimental insight
into the models involving dark sectors and predicting the existence of FIPs
[20, 21]. FIPs can be produced in meson decays or hadronic interactions of
protons, and can decay to SM particles. They are produced in rare processes
and are expected to have long lifetimes with respect to typical experimental
scales as a result of their feeble couplings to the SM fields.

Experiments may search for FIP production or decays. The first method focuses
on the associated production of FIPs with SM particles in meson decays; see
Ref. [22] for a review of possible production processes in kaon decays. The
meson mass ($\pi$, $K$, $D$, $B$) determines the kinematic region that can be
explored. Some examples are production of heavy neutral leptons in
$K^{+}\to\ell^{+}N$, dark photons in $\pi^{0}\to\gamma X$, and dark scalars in
$B\to KS$ or $K\to\pi S$. These kinds of searches can be pursued by flavour
experiments, like LHCb and NA62, but also at ATLAS and CMS. The second method
is based on FIP production, mainly in proton-nucleon interactions, and the
detection of the subsequent FIP decays. Forward configurations and long
experimental setups are best suited for this type of search, as those of
fixed-target experiments, but also LHCb. The solid angle covered by the
detector defines the kinematic range of the FIPs that can be studied. Examples
of FIP decays include dark scalars or dark photon decays
$S/A^{\prime}\to\mu^{+}\mu^{-}$ and heavy neutral lepton decays
$N\to\pi^{\pm}\mu^{\mp}$. Searches for FIPs performed so far have not reported
any hints of NP, and only limits in the plane of the coupling versus FIP mass
have been set within the framework of various NP models. Nevertheless, it is
worth continuing to explore the parameter space for FIP models. High beam
intensities and experiments with appropriate geometries are necessary for the
pursuit of these studies.

###  2.3 The role of HIKE

The HIKE experiment is primarily a flavour physics experiment, with a fixed
target configuration appropriate for the study of decays of kaons at high
energies, on the order of 100 GeV. The study of the kaon sector makes HIKE
complementary to LHCb and Belle II: kaons provide different, and in some cases
higher, sensitivity to NP than $B$ and $D$ mesons. Measurements of kaon
observables and correlations among them can help to find NP through the
comparison with corresponding SM expectations and investigation of the present
experimental inconsistencies with SM. More generally, the comparison between
the flavour picture emerging from kaons with that from $B$ mesons is a
powerful tool to investigate the indirect effects of NP: to shed light on
existing experimental anomalies, push the sensitivity to mass scales beyond
those attainable by $B$ and $D$ only, and provide insights to the flavour
structure of possible NP models.

Several models of BSM physics predict effects on kaon observables, ranging
from supersymmetric to non-supersymmetric models, NP involving heavy
$Z^{\prime}$ bosons, vector-like couplings, leptoquarks, and extra dimensions,
in both minimally flavour violating (MFV) scenarios or with new sources of
flavour violation. A common feature is that bounds on the parameter space of
these models coming from direct searches at LHC and $B$ and $D$ physics have
only a marginal impact on the possible effects on kaon observables, even more
so if the models are non-MFV and non-supersymmetric. In contrast, the main
constraints on the effects of NP on kaon observables come from the precisely
measured parameters $\varepsilon_{K}$ and $\Delta M_{K}$. Table 1 gives a
selected list of the most relevant NP models that may affect significantly
kaon observables [23].

Table 1: Compendium of new-physics models relevant for kaon processes [23]. NP scenarios | Process | References
---|---|---
Z-FCNC |  $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$, $\varepsilon^{\prime}/\varepsilon$ | [24, 25, 26, 27, 28]
Z′ |  $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$, $\varepsilon^{\prime}/\varepsilon$, $\Delta M_{K}$ | [29, 24, 25, 30, 31]
Simplified models |  $K_{L}\to\pi^{0}\nu\bar{\nu}$, $\varepsilon^{\prime}/\varepsilon$ | [32]
LHT | All $K$ decays | [33, 34, 35]
331 models | Small effects in $K\to\pi\nu\bar{\nu}$ | [30]
Vector-like quarks |  $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$, $\Delta M_{K}$ | [36]
Supersymmetry |  $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$ | [37, 38, 39, 40, 41, 42, 43, 44, 45]
2HDM |  $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$ | [46, 47]
Universal extra dimensions |  $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$ | [48, 49]
Randall-Sundrum models | All rare $K$ decays | [50, 51, 52, 53, 54]
Leptoquarks | All rare $K$ decays | [55, 56, 57]
SMEFT | Several processes in $K$ system | [25, 58]
SU(8) |  $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$ | [59]
Diquarks |  $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$, $\varepsilon_{K}$ | [60, 61]
Vector-like compositeness |  $K^{+}\to\pi^{+}\nu\bar{\nu}$, $K_{L}\to\pi^{0}\nu\bar{\nu}$, $\varepsilon_{K}$ | [62]

Presently, the main limitation to the investigation of the above models, and
more generally to the quest for NP with kaons, comes from the experimental
precision of the measurements of the kaon observables. To a lesser extent, the
limitation is due to theoretical uncertainties and the precision of the
measurements of the SM parameters, which will eventually be improved in the
upcoming years. The primary goal of HIKE is to improve the accuracy of the
measurements, wherever they exist, in order to match and possibly challenge
the theory precision, to study and measure for the first time channels not yet
observed, and to search with unprecedented sensitivity for kaon decays
forbidden by the SM.

A natural by-product of HIKE is the possibility to search for kaon decays to
FIPs, a line of inquiry that is already being pursued with success by NA62. In
addition, the fixed-target configuration and the high beam intensity
requirement make HIKE suitable to search for decays of FIPs, exploring regions
mainly below the $D$ mass, but with unprecedented sensitivity.

##  3 Physics with high-intensity kaon beams

Kaons offer a unique opportunity to investigate BSM physics in a way
complementary to $B$ or $D$ mesons, lepton physics, and EDM searches. The
strategy is to perform multiple experimental studies ranging from measurements
of branching ratios and form factors of kaon decays to searches for processes
forbidden or not existing within the SM. A global fit to the experimental
inputs provides the best sensitivity to NP, although single measurements also
put significant constraints. Evidence for NP can also show up as
inconsistencies between the observed flavour pattern and that expected from
the SM, violation of accidental SM symmetries like lepton flavour universality
(LFU), lepton flavour or number (LFV or LNV), and direct production of new
low-mass particles in kaon decays.

Because the BSM paradigm assumes NP at high masses, NP is expected to affect
kaon processes at short-distance (SD) scales. The link between kaon-physics
observables and BSM physics relies on the SM theoretical knowledge of these
observables. The development of the theoretical understanding of the kaon
sector is focused exclusively on calculations of observables relevant to pin
down SD contributions from experimental results. Examples are: the SM
calculation of the $K^{+}\to\pi^{+}\nu\bar{\nu}$ and
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay branching ratios at NNLO leading to a few
percent precision on these quantities; the recent calculation of the CP
violation parameter $\varepsilon^{\prime}/\varepsilon$; computations of the
branching ratios for the rare decays $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$,
$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$, $K_{L,S}\to\mu^{+}\mu^{-}$.

It is convenient to group kaon decays into the following classes.

  * •

Flavour-changing neutral currents (Section 3.1). The $K\to\pi\nu\bar{\nu}$,
$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$, $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ and
$K_{L}\to\mu^{+}\mu^{-}$ decays are suppressed in the SM, occurring at loop-
level at the lowest order. The corresponding SD contributions are related to
CKM matrix elements and are generally calculated with high precision within
the SM. Together with $\varepsilon_{K}$, rare decays allow the unitary
triangle to be over-constrained, independently from $B$ processes. With the
exception of $K\to\pi\nu\bar{\nu}$ decays, the above decays receive important
contributions from long distance (LD) physics. The theoretical framework used
to study these processes involves low-energy effective theories, like Chiral
Perturbation Theory (ChPT), which require experimental inputs (chiral
parameters) to provide and validate the predictions, as well as lattice QCD.

  * •

Lepton flavour universality tests (Section 3.2). The ratios ${\cal
B}(K^{+}\to\pi^{+}e^{+}e^{-})/{\cal B}(K^{+}\to\pi^{+}\mu^{+}\mu^{-})$ and
${\cal B}(K^{+}\to e^{+}\nu)/{\cal B}(K^{+}\to\mu^{+}\nu)$ are extremely
sensitive probes of lepton flavour universality. Another sensitive test is
based on the ratio ${\cal B}(K\to\pi e\nu)/{\cal B}(K\to\pi\mu\nu)$, where
both neutral and charged kaon decays ($K_{L}\to\pi^{\pm}\ell^{\mp}\nu$ and
$K^{+}\to\pi^{0}\ell^{+}\nu$) can be used.

  * •

Lepton flavour and number violating decays (Section 3.3). Decay modes
forbidden by the SM by either lepton flavour or number conservation include
$K^{+}\to\pi^{-}(\pi^{0})\ell_{1}^{+}\ell_{2}^{+}$,
$K^{+}\to\pi^{+}\mu^{\pm}e^{\mp}$,
$K^{+}\to\ell_{1}^{-}\nu\ell_{2}^{+}\ell_{2}^{+}$,
$K_{L}\to(\pi^{0})(\pi^{0})\mu^{\pm}e^{\mp}$ and $\pi^{0}\to\mu^{\pm}e^{\mp}$.
Any experimental signal of these processes would constitute direct evidence
for NP. It should be noted that kaon beams represent sources of tagged,
Lorentz-boosted $\pi^{0}$ mesons.

  * •

Tests of low-energy QCD (Section 3.4). The abundant kaon decays $K\to\pi\pi$
and $K\to\pi\pi\pi$ occur at the tree level and are dominated by LD physics.
The decays are described by ChPT which is the low-energy effective field
theory of QCD. Global fits to the measured branching ratios and kaon lifetimes
of the $K\to\pi\pi$ and $K\to\pi\pi\pi$ decays play a crucial role in the
extraction of the ChPT low-energy constants. Rare and radiative decays
including $K\to\pi\gamma\gamma$, $K\to\ell\nu\gamma$, $K\to\pi\ell\nu\gamma$,
$K\to\pi\pi\gamma$, $K\to\pi\pi\pi\gamma$, $K\to\pi\pi e^{+}e^{-}$, and
$K\to\pi\pi\ell\nu$ are also dominated by LD physics and similarly provide
access to the ChPT parameters.

  * •

CKM first-row unitarity tests (Section 3.5). Branching ratios and form factors
of the tree-level $K^{+}\to(\pi^{0})\ell^{+}\nu$ and
$K_{L}\to\pi^{\pm}\ell^{\mp}\nu$ decays represent the principal experimental
inputs for the determination of CKM matrix element $V_{us}$, with input from
lattice QCD.

  * •

Production of feebly-interacting particles in kaon decays (Section 3.6) is
predicted by FIP models involving new particles below the kaon mass scale.
Examples include production of a heavy neutral lepton ($K^{+}\to\ell^{+}N$),
or a dark scalar ($K^{+}\to\pi^{+}S$).

HIKE intends to address experimentally all these processes at a new level of
sensitivity, improving the existing experimental picture to fully exploit the
complementarity of kaon with respect to heavy-quark physics in the quest for
NP.

###  3.1 Flavour-changing neutral currents

Kaon decays proceeding at loop level in the SM are denoted “rare”, and
typically have branching ratios below $10^{-7}$. They provide a tool
independent from $B$ and $D$ physics to test the flavour structure of the SM
by constraining the unitary triangle via loop-level observables.

The link between rare kaon decays and flavour physics in the $\rho-\eta$ plane
is schematically shown in Fig. 1. CP-violating short distance (SD) physics
contributes to the $K_{L}\to\pi^{0}\nu\bar{\nu}$, $K_{L}\to\pi^{0}e^{+}e^{-}$,
$K_{L}\to\pi^{0}\mu^{+}\mu^{-}$, and $K_{S}\to\mu^{+}\mu^{-}$ decays. The
amplitudes of these decays thus depend on the height, $\eta$, of the unitary
triangle. The SD physics contributing to the decay $K_{L}\to\mu^{+}\mu^{-}$ is
CP-conserving, and its amplitude depends on the base, $\rho$, of the unitary
triangle. The amplitude of the rare decay $K^{+}\to\pi^{+}\nu\bar{\nu}$
contains terms depending on both $\eta$ and $\rho$.

Figure 1: Relation between kaon rare decay modes and the parameters $\rho$ and
$\eta$ of the unitary triangle (UT). The direct link between decay modes and
the UT indicates short distance terms dependent on $\rho$ or $\eta$
contributing to the corresponding decay amplitudes. Decays not directly
connected to the UT are relevant to interpret the experimental results of the
decay modes to which they are related.

The $K_{L}\to\pi^{0}e^{+}e^{-}$, $K_{L}\to\pi^{0}\mu^{+}\mu^{-}$ and
$K_{L}\to\mu^{+}\mu^{-}$ decays are LD dominated. The extraction of their SD
components from experimental data proceeds through the study of ancillary
decay modes, listed in Fig. 1 in association with the rare decays mentioned
above. In contrast, LD contributions are sub-dominant to the amplitudes of the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ decay, and negligible for the
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay. As a matter of fact, these two decay
modes belong to the theoretically cleanest probes of the flavour structure of
the SM among all the kaon and $B$ meson decays.

####  3.1.1 $K\to\pi\nu\bar{\nu}$ decays

The $K\to\pi\nu\bar{\nu}$ decays involve $s\to d$ quark transitions. They
proceed through box and penguin diagrams, which are SD dominated because of
the leading contribution of the virtual $t$ quark exchange. The quadratic GIM-
mechanism and the $t\to d$ Cabibbo suppression make place these processes
among the rarest meson decays in the SM. Their decay amplitudes can be
parameterised in terms of the precisely measured $K^{+}\to\pi^{0}e^{+}\nu$
decay, allowing theoretical computation free from hadronic uncertainties. The
SM predictions of the $K^{+}\to\pi^{+}\nu\bar{\nu}$ and
$K_{L}\to\pi^{0}\nu\bar{\nu}$ branching ratios can be written in terms of CKM
parameters as [63]

$\displaystyle{\cal B}(K^{+}\to\pi^{+}\nu\bar{\nu})$ $\displaystyle=(8.39\pm
0.30)\times
10^{-11}\left(\frac{|V_{cb}|}{0.0407}\right)^{2.8}\left(\frac{\gamma}{73.2^{\circ}}\right)^{0.74},$
(1) $\displaystyle{\cal B}(K_{L}\to\pi^{0}\nu\bar{\nu})$
$\displaystyle=(3.36\pm 0.05)\times 10^{-11}\left(\frac{|V_{ub}|}{3.88\times
10^{-3}}\right)^{2}\left(\frac{|V_{cb}|}{0.0407}\right)^{2}\left(\frac{\sin{\gamma}}{\sin{73.2^{\circ}}}\right)^{2}.$
(2)

Here $V_{cb}$ and $V_{ub}$ are elements of the CKM matrix, and $\gamma$ is the
angle of the unitary triangle defined as
$\arg[(-V_{ud}V_{ub}^{*})/(V_{cd}V_{cb}^{*})]$. The theoretical uncertainty
depends on the NNLO approximation of the computation of the Feynmann diagrams,
on the radiative corrections, and on the estimates of the long distance
contributions due to the exchange of the $u,d,c$ quarks. The prediction for
the neutral mode prediction is more precise than that for the charged mode
because the neutral amplitude is purely imaginary, allowing cancellation of
the long distance corrections. In either case, the main contribution to the
uncertainties for the branching ratio calculation comes from the precision of
the measurements of the CKM matrix elements, known as the parametric
uncertainty, and could be as large as 9% if the current measurements of
$V_{cb}$, $V_{ub}$ and $\gamma$ from the CKM fit are considered. A suitable
combination of external parameters nearly independent of new physics
contributions allows the reduction of the parametric uncertainty, reducing the
overall relative uncertainties in the branching ratios to 5% [64]. Other
recent calculations reach a similar quantitative conclusion [65]. A global fit
to the CKM unitary triangle, instead, allows the parametric uncertainty to be
factored out, and offers a powerful tool to compare the SM flavour structure
arising from kaon and $B$ physics. Equations 1 and 2 also indicate that
precise measurements of $K\to\pi\nu\bar{\nu}$ decays would shed light on CKM
parameters historically measured with $B$ mesons and suffering from long-
standing experimental tensions like $|V_{cb}|$.

The extreme SM suppression makes these decays particularly sensitive to NP.
From a model-independent point of view, the $K\to\pi\nu\bar{\nu}$ decays probe
NP at the highest mass scales [63], of the order of hundreds of TeV. Existing
experimental constraints on NP affect the $K\to\pi\nu\bar{\nu}$ branching
ratio weakly. Model-dependent scenarios predict sizeable deviations of the
branching ratios from the SM, as well as correlations between the branching
ratios for the charged and neutral modes, depending on the model [46, 55, 36,
28, 45, 44, 35, 66] (see also Table 1).

The NA62 experiment at CERN has observed $20$ candidate events for the decay
$K^{+}\to\pi^{+}\nu\bar{\nu}$ with 7 expected background events and 10
expected SM signal events [67], leading to the measurement
$\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})=(10.6^{+4.0}_{-3.4}|_{\text{stat}}\pm
0.9_{\text{syst}})\times 10^{-11}$ at $68\%$ confidence level. This represents
the most precise measurement to date of this process, providing first evidence
for its existence and falsifying the background-only hypothesis with
$3.4\sigma$ significance. The experiment is currently taking data in Run 2
(2021–LS3) with the aim of reaching a ${\cal O}(10\%)$ uncertainty
measurement, and has demonstrated its ability to sustain nominal beam
intensity. The NA62 experiment has therefore shown that the decay-in-flight
technique works well and is scalable to larger data samples.

Beyond the $\mathcal{B}(K^{+}\to\pi^{+}\nu\bar{\nu})$ measurement, it is
important to establish if the decay has a purely vector nature as expected
within the SM. In BSM scenarios, an additional scalar contribution to the
decay is predicted [68, 69], leading to possibly LNV/LFV
$K^{+}\to\pi^{+}\nu\nu$ contributions, and since the neutrinos are not
detected, a $\nu\bar{\nu}$ pair cannot be directly distinguished from a
$\nu\nu$ pair, and the measurement of the branching ratio alone is
insufficient to understand the nature of the process. Because of the
suppression from the small neutrino mass, there is a negligible interference
between vector and scalar contributions, and the experimentally measured
branching ratio is given by [68, 23]:

$\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\overset{\textbf{(---)}}{\nu})=\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})_{\rm
SM}+\sum_{i\leq j}^{3}\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu_{i}\nu_{j})_{\rm
LNV}.$ (3)

To fully test the SM prediction of a purely vector, lepton-number-conserving
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay, the momentum-transfer spectrum
must be studied to rule out the additional scalar BSM
$K^{+}\rightarrow\pi^{+}\nu\nu$ contribution to the overall experimentally
measured rate. Such a measurement requires a much larger dataset, as offered
within the HIKE programme.

The current upper limit on the branching ratio of the
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay is $3\times 10^{-9}$ at 90% CL, set by the
KOTO experiment at J-PARC [70]. The KOTO experiment is currently taking data
with the goal of reaching ${\cal O}(10^{-11})$ sensitivity in the next 5
years. In the longer term, an ambitious upgrade, KOTO Step-2, is planned to
begin construction after 2025 in a proposed extension of the Hadron
Experimental Facility, with the goal of measuring the branching ratio to
within $\sim$20% in about three years of data taking [71].

####  3.1.2 $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays

The ultra-rare $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays represent another set
of theoretically clean “golden modes” in kaon physics, second only to
$K\to\pi\nu\bar{\nu}$ in terms of their importance, and allowing for the
direct exploration of new physics contributions in $s\to d\ell\ell$
transitions (to be compared to $b\to s\ell\ell$ transitions). In the SM, they
are dominated by the indirect (mixing-induced) CP violation and its
interference with the direct CP-violating contribution. The decay rates can be
enhanced significantly in the presence of large new CP-violating phases, in a
correlated way with the effects in $K_{L}\to\pi^{0}\nu\bar{\nu}$ and
$\varepsilon^{\prime}/\varepsilon$ [23]. The SM expectation for the decay
branching ratios is [72, 73, 74]

$\displaystyle{\cal B}(K_{L}\to\pi^{0}e^{+}e^{-})$ $\displaystyle=$
$\displaystyle 3.54^{+0.98}_{-0.85}~{}\left(1.56^{+0.62}_{-0.49}\right)\times
10^{-11},$ $\displaystyle{\cal B}(K_{L}\to\pi^{0}\mu^{+}\mu^{-})$
$\displaystyle=$ $\displaystyle
1.41^{+0.28}_{-0.26}~{}\left(0.95^{+0.22}_{-0.21}\right)\times 10^{-11},$

with the two sets of values corresponding to constructive (destructive)
interference between direct and indirect CP-violating contributions.

Experimentally, the most stringent upper limits (at 90% CL) of the decay
branching ratios have been obtained by the KTeV experiment [75, 76]:

${\cal B}(K_{L}\to\pi^{0}e^{+}e^{-})<28\times 10^{-11},\quad{\cal
B}(K_{L}\to\pi^{0}\mu^{+}\mu^{-})<38\times 10^{-11}.$

The irreducible $K^{+}\to\gamma\gamma\ell^{+}\ell^{-}$ background, whose rate
vastly exceeds that of the signal decays $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$,
represents a significant problem in achieving experimental sensitivity at the
SM level, especially for $K_{L}\to\pi^{0}e^{+}e^{-}$ [77].

####  3.1.3 $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ decays

The amplitudes of the $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ decays ($\ell=e,\mu$)
are LD-dominated and their branching ratios are $\mathcal{O}(10^{-7})$. The
differential decay amplitude of these decays with respect to the di-lepton
invariant mass depends on two form factor parameters, denoted $a_{+}$ and
$b_{+}$, that lepton universality predicts to be independent from the flavour
of the leptons. Differences between $a_{+}$ and $b_{+}$ between the electron
and muon channels can be correlated to LUV effects that can explain the
possible observed lepton anomalies in $B$ physics [78, 79].

The NA62 experiment has recently reported a new measurement of the form
factors of the $K^{+}\to\pi^{+}\mu^{+}\mu^{-}$ decay using data collected in
2017–2018 [80]. The excellent kinematic resolution allowed a nearly
background-free selection of $3\times 10^{4}$ decays. The decay
$K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ was used for normalisation, and the parameter
values $a_{+}=-0.575\pm 0.013$ and $b_{+}=-0.722\pm 0.043$ were extracted from
a fit to the di-lepton mass spectrum. This improves significantly on the
previous measurements in the muon channel while also consistent within
uncertainty with measurements of the same parameters performed both in the
muon and electron channels [81, 82, 83, 84]. The NA62 experiment is expected
to improve the precision of the LFU test in the next years, addressing also
the measurement of the $K^{+}\to\pi^{+}e^{+}e^{-}$ form factors. However a
significant increase in precision is required to perform a LFU test with
comparable sensitivity to NP models as in current results from $B$ physics
[78, 85]. HIKE will offer a significantly larger data sample, further reducing
statistical uncertainties, which remain a large fraction of the total
uncertainty in the most recent NA62 measurement [80].

####  3.1.4 $K_{L}\to\mu^{+}\mu^{-}$ decays

The $K_{L}\to\mu^{+}\mu^{-}$ is a rare and helicity suppressed decay. Its
branching ratio is driven by LD contributions, with a sub-dominant CP-
conserving SD component [86]. The SM prediction exhibits large uncertainties
due to the sign ambiguity of the interference between the LD and SD terms
contributing to the decay amplitude. The sign ambiguity leads to two SM
predictions:

$\text{[LD$+$]:}~{}~{}\left(6.82^{+0.77}_{-0.24}\pm 0.04\right)\times
10^{-9},~{}~{}~{}\text{[LD$-$]:}~{}~{}\left(8.04^{+1.66}_{-0.97}\pm
0.04\right)\times 10^{-9}.$ (4)

The first uncertainty comes from the calculation of the LD terms, dominates
the overall precision, and makes the uncertainty quite asymmetrical. This
asymmetry is reflected in the computation of the $K_{L}\to\mu^{+}\mu^{-}$ rate
with the inclusion of NP. This decay has been studied experimentally and the
measured branching ratio is $(6.84\pm 0.11)\times 10^{-9}$ [87]. The sign
ambiguity and the LD uncertainties prevent a clean theoretical interpretation
of this result. Theoretical efforts are foreseen in the next few years to
resolve the sign ambiguity through an appropriate matching between LD and SD
contributions and to reduce the LD uncertainty by at least a factor of two.
This could allow a sizeable reduction of the uncertainty of the SM prediction,
opening the possibility to exploit the high sensitivity of this decay to BSM
physics models.

The $K_{S}\to\mu^{+}\mu^{-}$ decay provides a complementary, theoretically
clean observable [88, 89]. While the total $K\to\mu^{+}\mu^{-}$ rate is
dominated by long-distance (LD) physics, the measurement (in a dedicated high-
intensity $K_{S}$ experiment) of CP violation in the interference of mixing
and decay via the time-dependent rate enables the extraction of the purely CP
odd short-distance amplitude, which is predicted within the SM with an ${\cal
O}(1\%)$ uncertainty.

The study of the $K_{S}\to\mu^{+}\mu^{-}$ decay is the main goal of the kaon
physics programme of the LHCb experiment. The high production rate of kaons in
proton-proton collisions at 13 TeV partly compensates the small LHCb
acceptance for kaon decays due to the long lifetime. LHCb has recently
reported an upper limit on the $K_{S}\to\mu^{+}\mu^{-}$ branching fraction
using the combined Run 1+2 dataset, ${\cal
B}(K_{S}\to\mu^{+}\mu^{-})<2.1\times 10^{-10}$ 90% CL [90], to be compared to
the SM prediction, ${\cal B}_{\rm SM}(K_{S}\to\mu^{+}\mu^{-})=(5.18\pm
1.50)\times 10^{-12}$ [91, 88]. The ultimate expected LHCb sensitivity for
this decay is close to the SM branching ratio and is expected to be limited by
the statistical uncertainty on the background subtraction and the signal yield
[92].

####  3.1.5 Global sensitivity to New Physics

The rare decays discussed above offer the possibility to look for NP using a
global fit technique. As an example, Fig. 2 shows the result of this technique
using a low energy effective theory approach [85]. In this approximation, the
physics is described in terms of an effective Hamiltonian, which is an
expansion series of fermion current operators. The coefficient of this
expansion is the Wilson coefficient $C_{k}$, where $k$ is an index running
over the expansion terms. The assumption is that NP modifies $C_{k}$ with
respect to the SM values. The Wilson coefficients are then fitted using
experimental values of observables from rare kaon decays as input variables.
The results shown in Fig. 2 assume NP scenarios compatible with the LFUV
anomalies observed in $B$ decays.

Figure 2: Global fit to current data (purple, labelled as “current fit” in the
legend) and the results of two projections discussed in Ref. [85]: A, light-
red, where central values of observables currently only constrained with upper
bounds are set to the SM predictions; B, dark-red, where central values for
all of the observables are best-fit projections from existing data. The SM and
best-fit point with the current data are indicated with black and purple
crosses, respectively. The two figures represent the two possible signs of the
long-distance contribution to the $K_{L}\to\mu^{+}\mu^{-}$ process; the sign
ambiguity may be resolved with future data.

###  3.2 Lepton flavour universality tests

Lepton flavour universality (LFU) is a cornerstone of the SM postulating that
the lepton coupling to gauge bosons is independent of lepton type (“flavour”),
in contrast with the flavour-dependence of quark interactions. The origin of
the observed LFU, which is not associated with any known symmetry and is thus
not a fundamental conservation law, is a major question in modern physics.
Recently observed tensions between the SM predictions and experimental results
such as the measurement of the anomalous magnetic moment of the muon [93], the
Cabibbo angle anomaly [94], and the $B$-meson anomalies [8, 95, 6], suggest
possible LFU violation and motivate further the quest for improved
experimental LFU measurements.

The kaon sector offers opportunities for precision lepton flavour universality
tests by measuring the ratio ${\cal B}(K^{+}\to\pi^{+}e^{+}e^{-})/{\cal
B}(K^{+}\to\pi^{+}\mu^{+}\mu^{-})$ as discussed in the framework of a global
fit in a low-energy effective theory approach in Sections 3.1.3 and 3.1.5, as
well as ${\cal B}(K^{+}\to e^{+}\nu)/{\cal B}(K^{+}\to\mu^{+}\nu)$ and ${\cal
B}(K\to\pi e\nu)/{\cal B}(K\to\pi\mu\nu)$, exploiting cancellation of hadronic
effects [17, 78].

The ratio of purely leptonic decay rates of the charged kaon
$R_{K}=\Gamma(K^{+}\to e^{+}\nu)/\Gamma(K^{+}\to\mu^{+}\nu)$ is strongly
suppressed in the SM by conservation of angular momentum and an extremely
sensitive probe of LFU. Its SM expectation $R^{\rm SM}_{K}=(2.477\pm
0.001)\times 10^{-5}$ [96] is known to excellent (0.4‰) precision and the
currently most precise (4‰) experimental result is $R_{K}=(2.488\pm 0.007_{\rm
stat}\pm 0.007_{\rm syst})\times 10^{-5}$ [97]. The ratio $R_{K}$ is highly
sensitive to the possible violation of LFU naturally arising in new physics
scenarios involving sterile neutrinos [98, 99], leptoquarks [100], massive
gauge bosons [101], or an extended Higgs sector [102, 103]. Variations of
$R_{K}$ up to a few per mille from its SM expectation are predicted by these
models, without contradicting any present experimental constraints.

An analogous LFU test is based on the ratio of semileptonic kaon decay rates
$R_{K}^{(\pi)}=\Gamma(K\to\pi e\nu)/\Gamma(K\to\pi\mu\nu)$, where both neutral
and charged kaon decays ($K_{L}\to\pi^{\pm}\ell^{\mp}\nu$ and
$K^{+}\to\pi^{0}\ell^{+}\nu$) can be used. For a given neutral or charged
initial state kaon, the Fermi constant, $V_{us}$, short-distance radiative
corrections, and the hadronic form factor at zero momentum transfer cancel out
when taking the ratio $R_{K}^{(\pi)}$ [17]. Therefore, in the SM this ratio is
entirely determined by phase space factors and long-distance radiative
corrections [104, 105, 106, 107, 16].

###  3.3 Lepton flavour and lepton number violation

In the SM, lepton number and lepton flavour are conserved due to accidental
global symmetries. Additionally, the SM requires neutrinos to be strictly
massless due to the absence of right-handed chiral states. However the
discovery of neutrino oscillations has demonstrated non-zero neutrino mass and
non-conservation of lepton flavour in the neutrino sector. On the other hand,
no evidence has yet been found for lepton flavour violation (LFV) in the
charged lepton sector, or lepton number violating (LNV) processes. Any such
observation would be a clear indication of NP.

Mixing of active neutrinos can, in principle, mediate LFV decays in the SM.
However the branching ratios are vanishingly small, and therefore any
observation of LFV kaon decays would be clear evidence of new physics. Such
LFV kaon decays are predicted in BSM scenarios with light pseudoscalar bosons
(ALPs) [108], a $Z^{\prime}$ boson [109, 110] or leptoquarks [111, 112]. As
discussed in Section 2.1, several hints of violations of lepton flavour
universality (LFUV) have been reported. Models which predict LFUV and explain
these potential anomalies in general naturally predict LFV processes [113,
114, 115]. Therefore search for LFV processes are of strong interest.

In BSM scenarios, LNV kaon decays are predicted for example in models with
Majorana neutrinos. In the minimal Type-I seesaw model [116, 117, 118], the
neutrino is a Majorana fermion with a mass term that violates lepton number by
two units. Non-zero neutrino masses make it possible, in principle, to
distinguish experimentally between the possible Dirac and Majorana natures of
the neutrino. Strong evidence for the Majorana nature of the neutrino would be
provided by the observation of lepton number violating (LNV) processes,
including kaon decays.

The recent results of LFV/LNV searches with the NA62 Run 1 (2016–2018) dataset
[119, 120, 121] include upper limits on the
$K^{+}\to\pi^{-}\ell^{+}_{1}\ell^{+}_{2}$ decay branching ratios at the ${\cal
O}(10^{-11})$ level, already leading to stringent constraints on active-
sterile mixing angles between Majorana neutrinos. Below the kaon mass, these
constraints are competitive with those obtained from neutrinoless double beta
decay [122, 123, 124, 125]. The NA62 searches are not limited by backgrounds,
and indicate that HIKE Phase 1 would make significant improvements in
sensitivity to LFV and LNV $K^{+}$ decays, extending the limits to the ${\cal
O}(10^{-12})$ level. On the other hand, the existing limit on the LFV
$K_{L}\to\mu^{\pm}e^{\mp}$ decay [126], which represents a stringent
constraint on several BSM scenarios, would be improved significantly by the
HIKE Phase 2; further details are provided in Section 7.2.

###  3.4 Tests of low-energy QCD

Most kaon decays are governed by physics at long distances. These decays can
be described by chiral perturbation theory (ChPT), which is the low-energy
effective field theory of QCD. The ChPT framework determines the kaon decay
amplitudes in terms of the so-called low-energy constants, which are
determined using experimental data. Therefore, measurements of kaon decay
rates and form factor parameters of various decay channels represent essential
tests of the ChPT predictions and crucial inputs to the theory at the same
time. A comprehensive overview of kaon decays and their relation to the ChPT
can be found in Ref. [127].

The HIKE dataset will provide a unique opportunity to perform precision
measurements of rare and radiative decays including $K\to\pi\gamma\gamma$,
$K\to\ell\nu\gamma$, $K\to\pi\ell\nu\gamma$, $K\to\pi\pi\gamma$,
$K\to\pi\pi\pi\gamma$, $K\to\pi\pi e^{+}e^{-}$, and $K\to\pi\pi\ell\nu$ of
both $K^{+}$ and $K_{L}$ mesons. Possible studies of the abundant decay modes
$K\to\pi\pi$ and $K\to\pi\pi\pi$ will also provide important inputs to future
ChPT parameter fits.

###  3.5 Test of first-row CKM unitarity

Measurements of semileptonic kaon decays $K\to\pi\ell\nu$ provide the
principal input for the extraction of the CKM parameter $V_{us}$, while the
ratios of (semi)leptonic $K^{+}$ and $\pi^{+}$ decay rates are used to extract
the ratio $V_{us}/V_{ud}$, with inputs provided from lattice QCD [128].
Determination of $V_{us}$ from kaon, pion, and $\tau$ decays, combined with
$V_{ud}$ measurement from super-allowed beta decays [129] and neutron decays
[130, 131], gives rise to a $3\sigma$ deficit in first-row CKM unitarity
relation known as the Cabibbo angle anomaly; a tension of similar significance
is observed between $K\to\ell\nu$ and $K\to\pi\ell\nu$ rates [15, 17]. The
uncertainty in $V_{us}$ comes in equal parts from the experimental errors and
theoretical uncertainties in the ratio of decay constants, $f_{K}/f_{\pi}$,
and the $K\to\pi\ell\nu$ form factor, $f_{+}(0)$. Substantial improvements in
the lattice QCD calculations of these hadronic factors are expected in the
next five years, thanks to decreased lattice spacing and accurate evaluation
of electromagnetic effects [132, 133, 134, 135]. Significant progress on the
calculation of radiative corrections has been achieved recently, reducing the
uncertainties on the long-distance electromagnetic corrections to a negligible
level [136, 107, 137, 16]. In order to shed light on the Cabibbo angle
anomaly, improved measurements of the principal $K^{+}$ and $K_{L,S}$
branching ratios are essential (and it should be noted that no $K_{L}$ decay
measurements have been made in the past decade). In particular a precision
measurement of the ratio of $K^{+}\to\pi^{0}\mu^{+}\nu$ and
$K^{+}\to\mu^{+}\nu$ rates is well-motivated [15]. In case the Cabibbo angle
anomaly persists, HIKE would be able to make precision measurements of
(semi)leptonic $K^{+}$ and $K_{L}$ decays based on special datasets collected
with minimum-bias triggers at low beam intensity.

It should be noted that the Belle II experiment is planning to extract
$V_{us}$ at an improved precision from a suite of inclusive and exclusive
measurements of $\tau$ decays, including ${\cal B}(\tau^{-}\to K^{-}\nu)/{\cal
B}(\tau^{-}\to\pi^{-}\nu)$, combined with theory improvements [138]. This
would provide a possibility to cross-check the kaon results.

###  3.6 Production of feebly-interacting particles in kaon decays

An alternative paradigm to the searches for NP at high energy is the search
for feebly-interacting particles (FIPs), which would be the manifestations of
weakly-interacting NP potentially at much lower energy scales. Interactions of
FIPs with SM fields are possible through BSM extensions involving low-
dimension operators, the so-called portals [20, 21], the simplest of which are
summarised in Table 2. For further discussion of the benchmark scenarios, see
also Section 9.2.

Table 2: Summary of generic portal models with FIPs [20, 21]. Portal | FIPs
---|---
Vector | Dark photon, $A^{\prime}$
Scalar | Dark Higgs/scalar, $S$
Fermion | Heavy neutral lepton (HNL), $N$
Pseudoscalar | Axion/axion-like particle (ALP), $a$

Due to the availability of large datasets and the suppression of the total
kaon decay width, kaon decays represent uniquely sensitive probes of light
hidden sectors via FIP production searches. The possible search strategies
have been reviewed recently [22], and the following have been identified as
the most promising. Searches for the $K\to\pi X_{\rm inv}$ decay by extension
of the $K\to\pi\nu\bar{\nu}$ measurements, where $X_{\rm inv}$ represents an
invisible particle, represent a unique probe into the dark-scalar and ALP
parameter space. Searches for heavy neutral lepton ($N$) production in
$K^{+}\to\ell^{+}N$ decays are approaching the seesaw neutrino mass models
with ${\cal O}(100~{}{\rm MeV})$ sterile neutrinos [139]. Searches for
resonances in the $K\to\pi\ell^{+}\ell^{-}$ and $K\to\pi\gamma\gamma$ decay
spectra are complementary to searches at beam-dump experiments for a
significant ALP mass range. Finally, searches for a leptonic force mediator
($X$) in $K^{+}\to\mu^{+}\nu X$ decays can probe a region of parameter space
providing an explanation for the muon $g-2$ anomaly [140].

Experimentally, the NA62 Run 1 dataset has been used to establish the analysis
methods for searches for dark-scalar and HNL production in $K^{+}$ decays,
producing world-leading limits on the dark-scalar coupling [67, 141, 142] and
HNL mixing parameters [143, 144] below the kaon mass. Further details on the
experimental aspects and HIKE sensitivity projections are provided in Sections
6.2.3 and 6.2.4.

##  4 Physics in the beam-dump mode

Thanks to the high-intensity beam and high-performance detectors (redundant
particle identification capability, highly efficient veto systems and high
resolution measurements of momentum, time, and energy), HIKE can achieve the
sensitivities required for competitive searches for production and decay of
long-lived light mediators in a variety of new-physics scenarios.

A number of possible Standard Model (SM) extensions aimed at explaining the
abundance of dark matter in our universe predict a new “hidden sector” with
mediator fields or matter fields in the MeV–GeV mass range. The realisations
of such scenarios are usually classified in terms of the Lagrangian operators
(the so-called “portals” listed in Table 2) connecting SM particles to the new
mediators. The related coupling constants are small, thus evading the existing
exclusion bounds and justifying the label of “feebly interacting” given to
these new-physics models [20, 21]. The mass of the mediator and hidden-sector
matter fields and the couplings are free parameters of the models. The
relevant features of the phenomenology of such models are:

  * •

The mediators can be produced in proton-nucleus interactions through a number
of mechanisms, the most important being direct Primakoff production and meson-
mediated tertiary production. The specific production mechanism for each
mediator type differs in terms of production cross section and momentum-angle
spectra of the mediator emitted.

  * •

At the 400-GeV energy of SPS protons, mediators with momenta above 10 GeV have
decay lengths ranging from tens of meters to tens of kilometres in a suitable,
interesting interval of the feeble coupling.

  * •

Due to the feeble interaction with the SM particles, the emitted mediators can
reach the decay volume after punching through tens of meters of traversed
material before decaying.

As demonstrated by the NA62 Run 2 operation [145], the intense P42 400 GeV
proton beam extracted from the CERN SPS and the HIKE setup can be exploited to
search for production and decay of the emitted mediators. For this purpose,
HIKE can be operated in the so-called beam-dump mode. Considering that HIKE
will have the same type of forward geometry as NA62, the HIKE beam-dump
operation will be particularly sensitive to specific physics cases in which
the hidden sector mediator is produced by Primakoff scattering or light-meson
mediated decays. Prominent examples are the vector and some axion-like
particle scenarios. We address a number of new physics scenarios in Section 9,
tentatively assuming a total integrated statistics of $10^{19}$ POT for HIKE
Phase 1, and $5\times 10^{19}$ POT for the complete HIKE programme.

##  5 High-intensity beams

As the acronym HIKE suggests, the beam line must provide significantly higher
beam intensities than those that are now delivered to the NA62 experiment, on
the order of 4 to 6 times of that currently provided, which would correspond
to between 1.2 and $2\times 10^{13}$ protons on the T10 target per SPS
extraction. Here, we assume that a typical running year has 200 days with 3000
spills each, including an SPS uptime of about 80%, and a flat top length of
4.8 seconds.

The proposed HIKE program includes both a charged beam ($K^{+}$) phase and a
neutral beam ($K_{L}$) phase. For both parts of the program, protons at much
higher intensity have to be transported via the P42 beam line from the T4
target to the kaon production target T10. Conceptually, the $K^{+}$ beam is
proposed to be the same as the present beam line, but consolidation and
upgrades are needed to handle the higher beam intensities. Improved beam
instrumentation will keep the beam losses better under control. A more precise
calibration of the primary beam intensity measurements by the secondary
emission monitors at T4 and T10 is important for beam optimisation, and for
normalisation purposes in dump mode. Efforts in this respect have already been
started by the BI and EA groups at CERN. The neutral beamline is an entirely
new design and requires minor local modifications to the last section of the
P42 line.

###  5.1 Beam delivery to the kaon production target

The P42 beam line started operation in 1980 and hardly has been modified since
then [146, Chapter 5]. Most of the equipment dates from then and is showing
strong effects of ageing. Many issues and suggestions for mitigation have been
described in the report from the PBC Conventional Beams working group [147],
which has recently been updated. The first phase of a vast consolidation
program for the CERN North Area, titled NACONS, has recently been approved and
funded, and the second phase has been prepared [148, 149, 150]. This will
allow restoration of the equipment and beam infrastructure to a reliable
state. Some upgrades are required to cope with the up to 6 times higher beam
intensity, including operational requirements from the rest of the North Area
complex and with stricter safety and radiation protection requirements
compared to the original installation. Many studies have been performed by the
SPS Losses and Activation working group (SLAWG) [151, 152, 153], the Physics
Beyond Collider (PBC) Proton Delivery working group [154], the PBC
Conventional Beams working group [147], and more recently, by the PBC ECN3
beam delivery task force [155], focusing on the possibility to extract and
deliver such high intensities towards TCC8 and ECN3.

The present optics of the P42 beam is shown in Fig. 4. In order to reduce the
attenuation of the primary beam in the T4 target, several options are
considered. One is to increase the vertical beam size so that most of the beam
bypasses the target without interacting. Another option is to run dedicated
cycles for ECN3 in which the beam is bumped to pass above or below the target
plate. This option could allow in principle even higher intensities than
$2\times 10^{13}$ ppp on the T10 target. At present, the beam is provided in
spills of 4.8 seconds, typically twice per super-cycle of about 40 to 45
seconds during day time, and two equally-spaced spills during 33.6 seconds
during nights. One spill every 14.4 seconds is also technically possible and
will be the standard cycle adopted during the night from 2023 (Fig. 3).

Figure 3: The spill configuration used on 22 October 2022, as seen from SPS
Page 1.

In recent years, on average 3000 spills per day are delivered, including the
typical availability of SPS beams of about 80%. The figure of merit is the
time on flat top per year, which is proportional to the duty cycle and the
running time. It is also important that the beam intensity is delivered as
uniformly as possible. At the time of the NA62 proposal one assumed an
effective spill of 3.3 seconds, but thanks to recent improvements we
anticipate that it can be significantly better, ideally better than 4 seconds
for a 4.8 second spill. The spill length can be modified to some extent, but
it is important to maintain the duty cycle as well as the instantaneous
intensity, respecting both the RMS power limitation of the SPS and transfer
lines as well as the maximum intensity that the SPS is able to accelerate. The
latter implies that the intensity per spill shall be kept proportional to the
spill length.

Studies by the SY-STI group in the PBC context suggest that the TAXs in P42
and K12 are already running close to the maximum intensity allowed [147]. An
upgrade is being studied at the technical level in the framework of the ECN3
task force. This implies a substantial upgrade of the T10 target-TAX complex
to withstand the higher beam power and to be compatible with RP and
maintenance requirements. A new target design could be inspired by the
concepts of the CNGS target. The target material might be changed from
beryllium to carbon for better handling. However, also more dense target
materials can be considered as an option outside of the present baseline, as
they could be advantageous in particular for the neutral beamline. The new K12
TAX will be based on the concept that is developed for the P42 TAX. TAX holes
will be defined to ensure compatibility with the charged beam as well as the
neutral beam. It will probably require a different choice of material in the
TAX blocks and cooling closer to the beam impact point. The present
interlocks, P0-Survey and Dump Control, protect both the beam line equipment
and the NA62 experiment against incorrect magnet currents, T10 target and K12
TAX cooling problems and closed vacuum valves in the K12 beam line. For the
higher intensities in the future, this system must be sped up. This is planned
within the NACONS program baseline, independently of the proposed high
intensity upgrade for ECN3, profiting from a new generation of power
converters. On the shorter time scale, some improvements are already under
consideration, compatible with the presented HIKE timeline. Many studies are
on-going concerning radiation dose levels to equipment and on surface, which
must remain under control also at the higher intensities. All these issues
will be addressed in the forthcoming report from the ECN3 beam delivery task
force.

For a $K_{L}\to\pi^{0}\nu\bar{\nu}$ experiment, the production angle of the
kaon beam has to be increased to 8 mrad instead of the zero production angle
used for the charged mode. This can be achieved by a minor modification of the
P42 beam line, i.e. realigning the three last dipoles as documented in the
Conventional Beams Working Group Report [147], shown in Fig. 5. The high-
intensity proton beam will ultimately impinge on the kaon production target,
followed by the TAX beam dumps.

###  5.2 Charged kaon beamline

The HIKE charged kaon beamline will remain conceptually as it is now, with
essentially the same beam properties apart from the beam intensity (Fig. 6).
Consolidation of some equipment is required in the framework of NACONS, such
as the K12 machine protection system in synergy with the P42 interlock
upgrade. The main upgrades concern the T10 production target and the K12 TAX,
including the needed shielding. A pre-study was presented in the Conventional
Beams Working Group report [147] and is being followed up further in the
working group.

The K12 beam is a mixed beam containing 6% of kaons at 75 GeV/$c$. A study has
been performed to evaluate if the beam can be enriched by RF separation. RF
separation requires a much longer beamline, implying that more kaons decay
before the fiducial volume. This decreases not only the kaon flux but also the
initial kaon fraction in the beam before enrichment. An RF-separated beam
cannot fulfil the HIKE requirements, even with state-of-the-art RF systems
[156].

Figure 4: Beam optics for the P42 primary beam line, delivering protons from
the T4 production target to the T10 kaon production target. The red line
depicts the local values of the R12 (horizontal) and R34 (vertical) beam
transport matrix terms. The dotted blue line shows the local dispersion terms
R16 (horizontal) and R36 (vertical). Figure 5: Modifications of the T10 target
steering dipoles for increasing the production angle to 8 mrad, as required
for the KLEVER phase of the HIKE programme.

Figure 6: Beam optics for the K12 beamline. The red line depicts the local
values of the R12 (horizontal) and R34 (vertical) beam transport matrix terms.
The dotted black line shows the local dispersion terms R16 (horizontal) and
R36 (vertical).

###  5.3 Neutral kaon beams

The original design and simulations for the KLEVER beamline were performed by
the Conventional Beams Working Group [147]. The design is based on the
experience with the $K_{L}$ beam lines for NA31 and NA48, following guidelines
laid out in Ref. [157], and is shown schematically in Fig. 7. It forms the
basis for the beamline to be used also for HIKE Phase 2, and will be extended
in length for the precise measurement of $K_{L}\to\pi^{0}\nu\bar{\nu}$
(KLEVER).

Figure 7: Baseline design of neutral beamline for HIKE Phases 2 and 3.

The primary proton beam impinges on the T10 target with a downward vertical
angle of 8 mrad for KLEVER, and possibly 2.4 mrad for the $K_{L}$ experiment
with tracking. These choices balance several factors for the measurements of
$K_{L}\to\pi^{0}\nu\bar{\nu}$ and $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$,
respectively. As seen from Figs. 8 and 9, a smaller production angle increases
$K_{L}$ flux and hardens the $K_{L}$ spectrum, facilitating the task of
vetoing background channels with extra photons, such as
$K_{L}\to\pi^{0}\pi^{0}$. On the other hand, with harder input momentum
spectra, a smaller $K_{L}$ fraction and a larger $\Lambda$ faction decay
inside the fiducial volume. For the $K_{L}\to\pi^{0}\nu\bar{\nu}$ measurement,
$\Lambda\to n\pi^{0}$ decays constitute a potentially dangerous background.
Neutron interactions on residual gas in the vacuum tank and from halo neutrons
striking the detector constitute another dangerous background, which is
reduced by going to larger angle. The production angle of 8 mrad softens the
$\Lambda$ spectrum, so that almost all $\Lambda$ hyperons decay before the
start of the fiducial volume. For the $K_{L}\to\pi^{0}\nu\bar{\nu}$
measurement, the reduced $K_{L}$ flux is compensated to some extent by an
increase in the fraction of $K_{L}$ mesons that decay in the fiducial volume.
However, this is not true for $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$, and moreover,
background from $\Lambda$ decays and neutron interactions is not a concern in
this case. A production angle as small as possible is therefore preferred for
HIKE Phase 2, subject to limitations from the neutron flux in the beam at very
small angles. A production angle of 2.4 mrad (the same as for the $K_{L}$ beam
in NA48) is a likely choice. The $K_{L}$ and $\Lambda$ momentum spectra for
production angles of 2.4 and 8 mrad are shown in Fig. 10, and the fluxes and
other parameters are tabulated in Table 3.

Figure 8: Mean (left) and peak (right) momentum for $K_{L}$ (red) and
$\Lambda$ (blue) components vs beam production angle, from FLUKA (squares) and
Geant4 (circles), with solid (dashed) curves from fits to the form in Ref.
[158].

Figure 9: Left: neutral beam fluxes per POT per $\mu$sr of acceptance for
$K_{L}$ (blue) and $\Lambda$ (red) vs beam production angle. Right: ratios of
$\gamma$ (green), $n$ (red), and $\Lambda$ (blue) to $K_{L}$ fluxes vs beam
production angle. FLUKA (Geant4) results are shown with squares (circles),
with solid (dashed) curves from fits.

Figure 10: $K_{L}$ (blue) and $\Lambda$ (red) beam momentum spectrum (number of particles in beam per POT per $\mu$sr of neutral beam acceptance) for production angles of 2.4 mrad (left) and 8 mrad (right). The solid curves show a parameterisation [158] of FLUKA results at production; dashed curves show the momentum spectra for particles decaying in an indicative fiducial volume of the experiment extending from 135 m to 195 m downstream of the target. Table 3: Parameters of $K_{L}$, $n$, $\Lambda$ production at angles of 2.4 and 8.0 mrad, obtained from the FLUKA simulation of the beamline. An indicative fiducial volume (FV) extending from 135 m to 195 m from the production target is considered. | 2.4 mrad | 8.0 mrad
---|---|---
Mean $K_{L}$ momentum (GeV/$c$) |  |
at production | 79 | 39
in FV | 46 | 26.4
Mean $\Lambda$ momentum (GeV/$c$) |  |
at production | 151 | 66
in FV | 285 | 251
$K_{L}$ rate ($10^{-6}$/pot/$\mu$sr) |  |
at production | 187 | 73
in FV | 6.8 | 4.1
$\Lambda$ rate ($10^{-6}$/pot/$\mu$sr) |  |
at production | 310 | 37
in FV | 0.053 | 0.000132
Table 4: Particle fluxes in the neutral beam, obtained from the FLUKA simulation. The rates in MHz/GHz assume a primary beam intensity of $6.7\times 10^{12}$ pot/s. Beam component | Before converter | After converter | After final collimator
---|---|---|---
|  pot-1 | GHz |  pot-1 | MHz |  pot-1 | MHz
$\gamma$ |  |  |  |  |  |
2em $E>1$ GeV | $4.2\times 10^{-3}$ | 27.9 | $9.1\times 10^{-4}$ | 6100 | $2.97\times 10^{-5}$ | 198
2em $E>5$ GeV | $2.19\times 10^{-3}$ | 14.6 | $7.1\times 10^{-5}$ | 470 | $7.9\times 10^{-6}$ | 53
2em $E>10$ GeV | $1.55\times 10^{-3}$ | 10.3 | $1.81\times 10^{-5}$ | 121 | $3.15\times 10^{-6}$ | 21
2em $E>30$ GeV | $6.3\times 10^{-4}$ | 4.2 | $2.06\times 10^{-6}$ | 13.7 | $6.2\times 10^{-7}$ | 4.1
$n$ |  |  |  |  |  |
2em $E>1$ GeV | $4.3\times 10^{-4}$ | 2.88 | $4.2\times 10^{-4}$ | 2820 | $6.7\times 10^{-5}$ | 440
$K_{L}$ |  |  |  |  |  |
2em $E>1$ GeV | $1.37\times 10^{-4}$ | 0.91 | $1.29\times 10^{-4}$ | 870 | $2.11\times 10^{-5}$ | 140

The target is immediately followed by a first collimator that stops hadrons
outside the beam acceptance before they decay into muons. The non-interacting
protons are swept further downward by a strong dipole magnet. Protons and
charged secondaries are dumped in the TAX, which has apertures that allow the
beam to pass. A photon converter consisting of 9$X_{0}$ of high-$Z$ material
is positioned at the centre of the TAX between the two modules and reduces the
flux of high-energy photons ($E>5$ GeV) in the neutral beam by two orders of
magnitude. A thinner oriented crystal converter is an optional possibility
under study. Shower products emerging from the TAX are swept horizontally by a
dipole magnet. Three collimators define the beam acceptance in a clean way.
The defining collimator is located at $1/3$ of the distance to the final
collimator and defines the beam angular acceptance to not more than $\pm$0.4
mrad, which is necessary to measure the transverse momentum of the $\pi^{0}$
to sufficient precision in order to reject background from
$K_{L}\to\pi^{0}\pi^{0}$ decays. A cleaning collimator stops debris from
scatterings in the jaws of the defining collimator, and a final collimator
stops scattering products from the cleaning collimator. The defining and
cleaning collimator are each followed by strong sweeping magnets. The final
collimator will be active, and will define the start of the fiducial volume.
Sufficient space is available to add more collimation stages, depending on
future design iterations within the Conventional Beams Working Group.

####  5.3.1 Simulation of the $K_{L}$ beamline

A detailed FLUKA simulation of the entire beamline has been developed by the
Conventional Beams Working Group. Fig. 11 shows the momentum distributions for
$K_{L}$, photons, and neutrons in the beam at various stages of the beamline
simulation: at generation (at the exit from the target), after the converter
and each of the collimators, and at the end of the beamline, at the front face
of the small-angle calorimeter (SAC). The normalisation of these distributions
provides the particle fluxes in the neutral beam per incident proton. Of
particular interest are the photon and neutron fluxes after the final
collimator. The fluxes of photons, neutrons and $K_{L}$ in the beam are
summarised in Table 4. For the purposes of the sensitivity estimates for the
KLEVER phase, $2.1\times 10^{-5}$ $K_{L}$ mesons enter the detector per proton
incident on the target. The beam rates in the table assume a primary beam
intensity of $2\times 10^{13}$ ppp with a (pessimistic) effective spill length
of 3 seconds. In addition to the 140 MHz of $K_{L}$ mesons entering the
detector, there are 440 MHz of neutrons: the $n/K_{L}$ ratio of about 3
observed for particle production at 8 mrad is not significantly changed during
transport of the neutral beam. In addition, there are about 50 MHz of photons
with $E>5$ GeV entering the detector, most of which are incident on the SAC at
the downstream end of the detector (the condition $E<5$ GeV corresponds to the
SAC threshold for KLEVER). The FLUKA simulation also contains an idealised
representation of the KLEVER experimental setup, for the purposes of
evaluating rates on the detectors from beam halo. These rates, with specific
reference to the KLEVER experimental geometry, are discussed in Section 8.2.

As seen from Fig. 11 and Table 4, the photon converter in the TAX dramatically
reduces the photon flux in the beam, especially for high energy photons. This
is critical to avoid blinding the SAC at the downstream end of the beamline.
In the baseline design, the converter is a tungsten prism of 32.9 mm
thickness, corresponding to 9.4$X_{0}$, or 7.3 photon conversion lengths. This
thickness is chosen to keep the rate of photons with $E>5$ GeV below 40 MHz at
the entrance to the SAC. On the other hand, this thickness corresponds to 58%
of a nuclear collision length and 33% of an interaction length, so that about
35% of $K_{L}$ mesons and 40% of neutrons interact in or are scattered out of
the beam by the converter.

Figure 11: Momentum distributions for $K_{L}$ mesons (top), photons (bottom
left), and neutrons (bottom right) in the neutral beam at various points along
the neutral beamline. Figure 12: Photon fluxes (for $E>5$ GeV) at the SAC as
functions of photon converter thickness for beryllium, copper, and lead
targets. The dotted line indicates the flux corresponding to a rate of 40 MHz,
considered to be the maximum tolerable rate for high-energy photons on the
SAC.

The use of a high-$Z$ material for the target is an interesting alternative to
the current baseline. Most of the prompt photons come from $\pi^{0}$ decays
within the target, and a high-$Z$ material causes many of these photons to be
converted immediately, while the overall thickness of the target in nuclear
interaction lengths is unchanged. No large differences are observed in terms
of hadronic production for targets of identical thickness in nuclear
interaction lengths [159]. Photon rates at the SAC for beryllium, copper and
tungsten targets (each with a nuclear interaction length equivalent to 400 mm
Be), determined with the FLUKA simulation, are shown in Fig. 12. The required
converter thicknesses for beryllium, copper, and tungsten targets of
equivalent thickness are 7.3, 5.2, and 3.8 photon conversion lengths,
respectively, resulting an effective increase in the $K_{L}$ flux into the
detector of 15% (28%) for a copper (tungsten) target relative to a beryllium
target. Changing the target material strongly impacts the target design, as
the energy deposit per unit of volume is much higher in a high-$Z$ target than
in beryllium, adding to cost and complexity of the new target design.

Another promising technique to reduce the photon content in the beam is to use
a crystal metal photon converter. If the crystal axis is aligned with the
incoming photon direction, the coherent effect of the crystal lattice promotes
pair production, leading to an effective decrease in the photon conversion
length [160, 161, 162]. The effects of coherent interactions increase with
photon energy and for decreasing angle of incidence. A series of exploratory
tests was performed with a set of tungsten crystals at the CERN SPS in summer
2018, together with the AXIAL collaboration [163]. In particular, a commercial
quality tungsten crystal of 10-mm thickness was targeted with a tagged photon
beam. When the <111> crystal axis was aligned with the beam to within 2.5
mrad, the multiplicity of charged particles was found to be enhanced by a
factor 1.6–2.3 for photon energies over the range of 30–100 GeV. Ref. [163]
also reports simulations validated by this result that suggest that a crystal
of this type could be used to reduce the thickness of the photon converter by
15–20% at no cost in effectiveness. Such a solution appears to be relatively
easy to implement.

####  5.3.2 Beam for Phase 2 (a multi-purpose $K_{L}$ experiment)

The proposed HIKE Phase 2 is based on the $K_{L}$ beam, but a detector very
similar to the Phase 1 experiment, including a tracking system. This phase of
the programme will make use of the neutral beamline described above with a
production angle chosen to optimise the statistics for the measurement of
$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays, likely 2.4 mrad. Alternative
beamline options can be studied. The full FLUKA simulation will be performed
when the choice of production angle is finalised. For the sensitivity
estimates in Section 7, the entries in Table 4 are scaled to account for
production angle of 2.4 mrad (Table 3).

####  5.3.3 Beam for Phase 3 (KLEVER: measurement of
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay)

After the initial neutral beam studies for the KLEVER phase of HIKE were
completed, it was discovered that the background from $\Lambda\to\pi^{0}n$
decays is more difficult to suppress than previously thought. Therefore, the
amount of $\Lambda$ decays in the fiducial region has to be reduced further by
several orders of magnitude, whilst preserving as much as possible the kaon
flux. Various options were considered, such as lengthening the distance
between the target and the start of the fiducial volume, reducing the primary
proton beam energy (for example, from 400 to 300 GeV), and increasing the
production angle (for example, from 8 mrad to 20 mrad). Lengthening the beam
line by 150 m seems to be the best option, and several possibilities are under
consideration. Moving the T10 target further upstream is one of them, but this
may have complex radiation protection implications and might require the whole
experiment to be installed on a slope, depending on the exact target location.
The more straightforward option might be to prolong the ECN3 cavern by 150 m,
implying civil engineering works. Simulations show that this brings the
background from $\Lambda$ decays to an acceptable level (Section 8.3). The
technical details and costs of the different options are under discussion with
the Civil Engineering group of CERN.

An extension of the ECN3 cavern implies that the detector will be installed
150 m downstream of the present NA62 location. At the moment, we do not
contemplate changing the transverse dimensions of the setup. The positions of
the beamline elements would change as noted in Table 5. The beam solid angle
and hence the $K_{L}$ flux decreases by a factor of $(0.256/0.400)^{2}\approx
0.41$. Since the $p_{\perp}$ reconstruction at the downstream end of the FV
depends on the beam spot size, it is not trivial to recover the sensitivity of
the experiment by enlarging its transverse dimensions, and we favour the other
approaches discussed in Section 8.3. The full FLUKA simulation of the extended
configuration will be performed as soon as the design is finalised. For the
sensitivity estimates, the rates in Table 4 can be scaled to account for the
reduction in solid angle from the extension.

In case resources allow to advance the construction of the ECN3 extension,
HIKE Phase 2 can be performed with the extended beamline and the detector in
the new location. This would eliminate the need for infrastructural work to
change beamline configurations between Phase 2 and KLEVER and allow precise
information on the beam composition and halo to be obtained during Phase 2,
thereby improving the understanding of beam-related backgrounds in KLEVER.

Table 5: Collimator positions and other parameters for standard and extended KLEVER beamlines. Configuration | Standard | Extended
---|---|---
Target position [m] | 0 (T10) | 0 (T10)
Defining collimator position [m] | 40 | 90
Cleaning collimator position [m] | 80 | 180
Final collimator position [m] | 120 | 270
Beam opening angle [mrad] | 0.400 | 0.256

##  6 Phase 1: a multi-purpose $K^{+}$ decay experiment

###  6.1 Experimental layout

While the success of the NA62 experiment has proven that its layout is
suitable for a precision ${\cal B}(K^{+}\to\pi^{+}\nu\bar{\nu})$ measurement,
new or upgraded detectors will replace those of NA62 with the goal of
sustaining secondary-beam rates four times higher than those of NA62, to
substantially boost the statistical sensitivity. To this end, new technologies
will be used, in synergy with the upgrades of the LHC experiments. While some
detectors will be renovated for the start of the HIKE $K^{+}$ phase (Phase 1),
others are already intrinsically fast in terms of detector technologies but
will need readout upgrades. Therefore a staged approach will be used, where
renovated or new detectors are inserted as soon as they are needed and ready,
while maintaining the general principle that changes must serve the remaining
phases of the programme once they are applied.

In the baseline configuration, the $K^{+}$ beam is produced by $1.2\times
10^{13}$ $400~{}{\rm GeV}/c$ protons/spill at zero angle on target. The mean
$K^{+}$ momentum at the entrance to the decay volume is 75 GeV/$c$. The layout
(Fig. 13) includes a beam tracker, a kaon-identification detector, and a veto
counter upstream of the decay volume; a main tracking system with the MNP33
magnet in the decay volume; calorimeters, vetoes and particle identification
detectors downstream of the decay volume; and a large-angle veto system
surrounding the decay volume and part of the region downstream.

HIKE Phase 1 ($K^{+}$)

Figure 13: HIKE Phase 1 layout, with an aspect ratio of 1:10.

###  6.2 Physics sensitivity

####  6.2.1 $K^{+}\to\pi^{+}\nu\bar{\nu}$ measurement

The first phase of HIKE will be dedicated to a detailed measurement of the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ decay, with a target of reaching a branching
ratio measurement with $\mathcal{O}(5\%)$ precision, competitive with the
theoretical uncertainty. The approach towards this goal will continue, and
build upon, the successful strategy of the NA62 experiment. The keystones of
the measurement, required to suppress backgrounds at the
$\mathcal{O}(10^{11})$ level, are the following:

  * •

High efficiency and high-precision tracking of both the $K^{+}$ upstream and
$\pi^{+}$ downstream. This, coupled with a careful choice of signal regions,
will allow kinematic suppression of backgrounds by a factor of
$\mathcal{O}(10^{3})$.

  * •

High precision time measurements, allowing time-matching between upstream and
downstream detectors with $\mathcal{O}(20~{}\text{ps})$ precision. Simulations
show that with this time-matching performance, the effect of the higher
intensity can be totally compensated in the matching between upstream and
downstream tracks, without losing efficiency for the signal selection and
without increasing the relative contamination of background coming from
upstream decays and interactions with respect to NA62.

  * •

Comprehensive and hermetic veto systems:

    * –

Photon veto detectors with hermetic coverage of the 0–50 mrad range for
photons from $\pi^{0}\to\gamma\gamma$ decays, allowing photon suppression by a
factor of $\mathcal{O}(10^{8})$.

    * –

Veto detectors to cover downstream regions not covered by the principal
detectors.

    * –

Rejection of upstream decays and interactions, for suppressing possible
upstream backgrounds by an additional factor of $\mathcal{O}(10)$, with
changes to the beam line setup and new systems dedicated to the detection of
the upstream background events.

  * •

High-performance particle identification system (for $\pi$/$\mu$
discrimination), suppressing backgrounds with muons by a factor
$\mathcal{O}(10^{7})$.

Crucial to the study of the $K^{+}\to\pi^{+}\nu\bar{\nu}$ decay at HIKE will
be the exploitation of the high-intensity environment, taking advantage of the
high $K^{+}$ flux while mitigating detector pileup effects. A critical
performance indicator is the “random veto efficiency”, i.e., the fraction of
events passing signal selection criteria that are sensitive to accidental
activity in the detector systems. The random veto efficiency measured within
the NA62 selection for the $K^{+}\to\pi^{+}\nu\bar{\nu}$ decay at the current
intensity working-point is $\varepsilon_{\rm RV}\approx 65\%$. This is
dependent on selection criteria that are limited by the timing precision of
the detectors. The random veto efficiency is approximately linear as a
function of the instantaneous beam intensity. At HIKE, this random veto
efficiency must be maintained or improved, requiring an improvement in time
resolution by the same factor as the intensity increase. With this
performance, a projection for the random veto efficiency as a function of
intensity is shown in Fig. 14. Individual contributions from photon veto
subsystems (SAV, LAV, ECAL) are indicated which in total give the ‘photon
rejection’ curve. The curve labelled ‘multiplicity rejection’ relates to the
selection criteria required to reject additional activity in an event (outside
the photon veto detectors). Finally, the total random veto efficiency,
combining photon and multiplicity rejection, is shown. At a secondary beam
intensity 4 times higher than NA62, with the detector updates and
corresponding selection changes shrinking veto windows by a factor of 4, the
random veto efficiency is maintained at the same level as currently achievable
at NA62.

Figure 14: Expected HIKE Phase 1 random veto efficiency $\varepsilon_{\rm RV}$
for the $K^{+}\to\pi^{+}\nu\bar{\nu}$ analysis as a function of instantaneous
beam intensity: the total effect (grey band) and the contributions from the
individual photon/multiplicity veto conditions. The NA62 graph is also shown
for comparison: the lower $\varepsilon_{\rm RV}$ values with respect to HIKE
Phase 1 are due to the worse time resolution.

The HIKE sensitivity for the ${\cal B}(K^{+}\to\pi^{+}\nu\bar{\nu})$
measurement is estimated as follows.

  * •

The effective number of collected $K^{+}$ decays per year is $N_{K}/{\rm
year}\simeq 2\times 10^{13}$. This can be extrapolated from the data already
collected by the NA62 experiment, scaling for an intensity four times larger
than the NA62 nominal intensity and 200 days of data-taking per year. This
automatically includes effects already present in the current setup, such as
duty cycle, SPS beam availability, secondary beamline downtime, and detector
and DAQ efficiencies.

  * •

Acceptance of the signal selection: $\varepsilon_{A}\simeq 0.1$. The
improvements on the time-matching performance and on the resolution of the
charged-particle identification system will lead to an increase of about 50%
relative to that at the current NA62 working point.

  * •

Random veto efficiency ($\varepsilon_{\rm RV}$) and trigger efficiency
($\varepsilon_{\rm trig}$), which according to the projections already
described should remain the same as for the current NA62 working point even at
higher intensity: $\varepsilon_{\rm RV}\simeq 0.7$, $\varepsilon_{\rm
trig}\simeq 0.9$.

Under these conditions, the single event sensitivity attainable in one year of
data-taking is found to be ${\cal B}_{\rm SES}(1\;{\rm year})=(N_{K}/{\rm
year}\cdot\varepsilon_{A}\cdot\varepsilon_{\rm RV}\cdot\varepsilon_{\rm
trig})^{-1}\simeq 8\times 10^{-13}$, leading to the number of expected SM
signal events per year:

$N_{\pi\nu\bar{\nu}}/{\rm year}\simeq 100.$

With an $O(10\%)$ relative background contamination in the signal sample and
systematic uncertainty sources under control to $O(1\%)$ or better, a
measurement of the branching ratio $\mathcal{B}(K^{+}\to\pi^{+}\nu\bar{\nu})$
to $\mathcal{O}(5\%)$ precision can be reached by HIKE in 4 years of data-
taking. Assuming that the number of observed events is equal to the SM
expectation, values of ${\cal B}(K^{+}\to\pi^{+}\nu\bar{\nu})$ higher than 10%
with respect to the SM prediction will be excluded at 95% CL.

####  6.2.2 Test for scalar amplitudes in the $K^{+}\to\pi^{+}\nu\bar{\nu}$
decay

As introduced in Section 3.1.1, in BSM scenarios the measurable branching
ratio of $K^{+}\to\pi^{+}\nu\bar{\nu}$ is formed from two components: the SM
process $K^{+}\to\pi^{+}\nu\bar{\nu}$ with a purely vector nature and a
possible BSM process $K^{+}\to\pi^{+}\nu\nu$ with a scalar nature [68, 69].
Therefore the experimentally measured branching ratio is given by [23, 68]:

$\mathcal{B}(K^{+}\to\pi^{+}\nu\bar{\nu})=\mathcal{B}_{\rm
SM}(K^{+}\to\pi^{+}\nu\bar{\nu})+\sum_{i\leq j}^{3}\mathcal{B}_{\rm
LNV}(K^{+}\to\pi^{+}\nu_{i}\nu_{j}).$ (5)

The SM and BSM contributions are predicted to have different kinematic
distributions (Fig. 15). To identify the nature of the decay, an investigation
of the shape of the distribution of selected signal candidates as a function
of kinematic variables will be performed. In absolute terms, within the signal
regions (as defined in the current NA62 analysis), the acceptance for the BSM
mode is approximately 75% of the acceptance for the SM mode. With a sample of
several hundred $K^{+}\to\pi^{+}\nu\bar{\nu}$ candidates, HIKE will be able to
study the shape of the kinematic distributions as well as compare sets of sub-
categories of phase-space to distinguish between, or constrain the relative
contributions of, the possible vector and scalar natures of the process.

Figure 15: Simulated distributions of the squared missing mass $m_{\rm
miss}^{2}=(P_{K^{+}}-P_{\pi^{+}})^{2}$ vs pion momentum in the laboratory
frame for the SM $K^{+}\to\pi^{+}\nu\bar{\nu}$ decay with a vector nature
(left) and the LNV $K^{+}\to\pi^{+}\nu\nu$ decay with a scalar nature (right).
Black boxes indicate the signal regions for which experimental sensitivity is
highest.

####  6.2.3 Feebly-interacting particle production in $K^{+}\to\pi^{+}X_{\rm
inv}$ decays

As described in Section 3.6, searches for the $K^{+}\to\pi^{+}X_{\rm inv}$
decay, where $X_{\rm inv}$ is a FIP (either a dark scalar or ALP), can be
performed within the analysis framework developed for the study of the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ decay. This is possible because the signature of
these two decays is identical except that the $K^{+}\to\pi^{+}X_{\rm inv}$
decay produces a peak in the $m_{\rm miss}^{2}=(P_{K^{+}}-P_{\pi^{+}})^{2}$
variable at $m_{X}^{2}$, where $m_{X}$ is the mass of the FIP. Searches are
therefore performed by looking for evidence of a peaking signal on top of the
background dominated by the SM $K^{+}\to\pi^{+}\nu\bar{\nu}$ decay. The search
strategy is described in Ref. [141], which furthermore presents results based
on the analysis of 2017 data, which has been updated for the full Run 1
(2016–2018) dataset in Ref. [67]. Based on this experience, a sensitivity
projection for HIKE Phase 1 has been performed assuming a 40-fold increase in
the size of the data sample with respect to NA62 Run 1.

The principal background comes from the SM $K^{+}\to\pi^{+}\nu\bar{\nu}$
decay, and other backgrounds are negligible in comparison. The selection
acceptance and efficiency are assumed to be consistent with the results
obtained for the NA62 2018 data [67]. In this way realistic projections are
performed, accounting for all first-order effects. The results are shown in
Fig. 16 for the excluded regions in the interpretations of $X_{\rm inv}$ as
either a dark scalar mixing with the Higgs boson or an ALP with fermionic
couplings. Additional specialisation of the selection used for the
$K^{+}\to\pi^{+}X_{\rm inv}$ decays might be possible, further enhancing
sensitivity. HIKE Phase 3 sensitivity projections for the search for the
$K_{L}\to\pi^{0}X_{\rm inv}$ decay, refined with respect to Ref. [20], are
also shown in Fig. 16. HIKE Phases 1 and 3 provide complementary
sensitivities, allowing for exclusion of, or discovery within, new phase-space
regions which are otherwise difficult to cover.

Figure 16: Left: excluded regions at 90% CL of the $(m_{S},\sin^{2}\theta)$
parameter space for a dark scalar, $S$, of the BC4 model [20]. Right: excluded
regions of the parameter space $(m_{a},g_{Y})$ for an ALP, $a$, of the BC10
model [20]. The exclusion bound from analysis of NA62 Run 1 data [141, 142,
67] is shown in purple; the projected exclusion for HIKE Phase 1 and Phase 3
(KLEVER, assuming invisible $S$ or $a$ and updated from Ref. [20]) is shown in
red and light blue, respectively. Other exclusion bounds are shown in grey,
including results from E949 [164], CHARM [165, 166], NA48/2 [167], LHCb [168,
169], Belle [170], $K_{\mu 2}$ experiment [171], CLEO [172], KTEV [76],
MicroBooNE [173].

####  6.2.4 Heavy neutral lepton production in $K^{+}\to\ell^{+}N$ decays

Searches for heavy neutral lepton (HNL) production in $K^{+}\to\ell^{+}N$
decays have been well established by the NA62 experiment using the main
$K^{+}\to\pi^{+}\nu\bar{\nu}$ trigger chain for the $K^{+}\to e^{+}N$ case,
and a downscaled control trigger chain for the $K^{+}\to\mu^{+}N$ case. NA62
has published world-leading exclusion on the HNL mixing parameters $|U_{\ell
4}|^{2}$ over much of the accessible mass range of 144–462 MeV/$c^{2}$ with
the Run 1 dataset [143, 144]. Both searches are limited by background. In
particular, the $K^{+}\to\mu^{+}\nu$ decay followed by $\mu^{+}\to
e^{+}\nu\bar{\nu}$ decay in flight, and the $\pi^{+}\to e^{+}\nu$ decay of the
pions in the unseparated beam, represent irreducible backgrounds to the
$K^{+}\to e^{+}N$ process. The peaking nature of the $K^{+}\to\ell^{+}N$
signal in terms of the reconstructed missing mass allows for data-driven
background evaluation, reducing the systematic uncertainties in the background
estimates.

HIKE Phase 1 sensitivity projections for HNL production searches in
$K^{+}\to\ell^{+}N$ decays, obtained from a detailed analysis assuming
NA62-like trigger chains and conditions, are shown in Fig. 17. HIKE Phase 1
offers world-leading sensitivity to $|U_{e4}|^{2}$ in the region
$m_{N}>140~{}{\rm MeV}/c^{2}$, approaching the seesaw bound and complementing
future long-baseline neutrino experiments [139], and is expected to improve on
the state-of-the-art for lower HNL masses via searches for the
$K^{+}\to\pi^{0}e^{+}N$ [174] and $\pi^{+}\to e^{+}N$ decays. HIKE Phase 1
also provides competitive sensitivity to $|U_{\mu 4}|^{2}$. The projection for
$|U_{\mu 4}|^{2}$ assumes data collection with a highly-downscaled control
trigger, and may improve by an order of magnitude in case a software trigger
based on the streaming readout (Section 11.2) is employed.

Figure 17: Summary of the upper limits at 90% CL of the HNL mixing parameters
$|U_{e4}|^{2}$ and $|U_{\mu 4}|^{2}$ obtained from production searches, and
HIKE Phase 1 sensitivity with $K^{+}\to\ell^{+}N$ decays. HIKE projection for
$|U_{\mu 4}|^{2}$ assumes data collection with a highly-downscaled control
trigger, and may improve by an order of magnitude in case a software trigger
is employed.

####  6.2.5 Other measurements

HIKE Phase 1 is expected to increase the world samples of many rare $K^{+}$
decays by an order of magnitude, developing upon the trigger strategies
developed for the NA62 experiment. This would bring many rare decay
measurements to a new level of precision. In particular, we expect to collect
background-free samples of several times $10^{5}$ events of both
$K^{+}\to\pi^{+}e^{+}e^{-}$ and $K^{+}\to\pi^{+}\mu^{+}\mu^{-}$ decays,
leading to a powerful test of lepton universality (LFU) by comparing the form-
factor parameters ($a_{+}$, $b_{+}$) of the two decay modes. This is of
particular interest in the framework of the global NP fit [85] to the kaon
decay data (Section 3.1.5, Fig. 2). Considering that the precision of the
recent NA62 measurement in the muon channel [80] is limited by the size of the
dataset, HIKE Phase 1 is expected to measure the difference of the form-factor
parameter $a_{+}$ between the two decay modes, $\Delta a_{+}^{e\mu}$, to a
precision of $\pm 0.007$, and the corresponding difference $\Delta
b_{+}^{e\mu}$ to a precision of $\pm 0.015$. Exploitation of the correlations
between the measured $a_{+}$ and $b_{+}$ parameters in each mode will provide
additional power in terms of LFU tests.

In terms of searches for $K^{+}$ and $\pi^{0}$ decays violating lepton number
or flavour conservation, including
$K^{+}\to\pi^{-}(\pi^{0})\ell_{1}^{+}\ell_{2}^{+}$,
$K^{+}\to\pi^{+}\mu^{\pm}e^{\mp}$,
$K^{+}\to\ell_{1}^{-}\nu\ell_{2}^{+}\ell_{2}^{+}$ and
$\pi^{0}\to\mu^{\pm}e^{\mp}$, the experimental technique based on dedicated
di-lepton trigger chains has been firmly established by the NA62 experiment,
leading to world-leading upper limits of ${\cal O}(10^{-11})$ on the branching
ratios of a number of processes with the NA62 Run 1 dataset [119, 120, 121].
These searches are not limited by background, and HIKE Phase 1 sensitivity is
expected to improve in the future almost linearly with the size of the dataset
to the ${\cal O}(10^{-12})$ level.

##  7 Phase 2: a multi-purpose $K_{L}$ decay experiment

###  7.1 Experimental layout

The baseline design of the multi-purpose $K_{L}$ decay experiment includes a
120 m long neutral beamline with the secondary beam opening angle of 0.4 mrad,
proposed and studied extensively for the KLEVER phase of the HIKE programme.
The beamline involves four stages of collimation, and provides adequate
suppression of the short-lived $K_{S}$ and $\Lambda$ components. A detailed
description of the beamline, including the expected $K_{L}$ yields and
momentum spectra, is provided in Section 5.3 Unlike the KLEVER phase, HIKE
Phase 2 detector will be equipped with a tracker, which offers the opportunity
to perform characterisation of the KLEVER beam. For the forthcoming proposal,
alternative beamline configurations can be studied.

The proposed $K_{L}$ production angle is 2.4 mrad (to be compared to 8 mrad
for the KLEVER phase), which improves the $K_{L}$ yield and increases the mean
$K_{L}$ momentum (Figs. 8, 10) therefore improving the acceptances for
$K_{L}\to(\pi^{0})\ell^{+}\ell^{-}$ decays. Thanks to the relatively compact
detector, HIKE Phase 2 allows for a 90 m long fiducial decay volume to be
accommodated in the present ECN3 experimental hall, and no major civil
engineering work is required in preparation for this phase of the programme.

It is proposed to use the HIKE Phase 1 experimental setup with minimal
modifications (Fig. 18). The GTK, KTAG, RICH and SAC detectors will be
removed, the STRAW spectrometer will be shortened to a total length of 25 m,
and central holes of the STRAW chambers will be realigned on the neutral beam
axis (Section 10.2.3). Reduction of the magnetic field of the spectrometer
dipole magnet by about 20%, leading to a momentum kick of 210 MeV/$c$, is
possible without significant degradation of the mass resolution.

HIKE Phase 2 ($K_{L}$)

Figure 18: HIKE Phase 2 layout, with an aspect ratio of 1:10.

###  7.2 Physics sensitivity

The expected $K_{L}$ yield in the beam is $5.4\times 10^{-5}$ per proton on
target, and the assumed integrated proton flux of $1.2\times 10^{19}$ pot/year
leads to the number of $K_{L}$ decays in the decay volume of $3.8\times
10^{13}$/year. The mean momentum of $K_{L}$ mesons entering the decay volume
is 79 GeV/$c$, while the mean momentum of decaying $K_{L}$ mesons is 46
GeV/$c$ (Table 3, Fig. 8). The expected signal yields for several principal
rare and forbidden $K_{L}$ decays have been evaluated using a full
Geant4-based simulation, reconstruction and analysis chain within the flexible
HIKE offline software platform being developed on the basis of the NA62
software (Section 12). The results are summarised in Table 6, and acceptances
in bins of the longitudinal coordinate of the decay vertex and $K_{L}$
momentum are shown in Fig. 19.

Table 6: HIKE Phase 2 sensitivity estimate for the principal rare and forbidden $K_{L}$ decays: assumed branching ratios (see Section 3.1.2 for details), acceptances and expected signal yields in five years of data taking. Mode | Assumed branching ratio | Acceptance | Signal yield in five years
---|---|---|---
$K_{L}\to\pi^{0}e^{+}e^{-}$ | $3.5\times 10^{-11}$ | 2.1% | 140
$K_{L}\to\pi^{0}\mu^{+}\mu^{-}$ | $1.4\times 10^{-11}$ | 6.0% | 160
$K_{L}\to\mu^{+}\mu^{-}$ | $7\times 10^{-9}$ | 17% | $2.3\times 10^{5}$
$K_{L}\to\mu^{\pm}e^{\mp}$ | – | 16% | –

Figure 19: Acceptances for rare $K_{L}$ decays for HIKE Phase 2, in bins of
the decay vertex longitudinal coordinate (left) and $K_{L}$ momentum (right).
Momentum distribution of the $K_{L}$ mesons decaying in the decay volume is
shown by a continuous curve in the right panel.

For the $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays, the expected SM Dalitz plot
distributions [72] are used in the simulation, and a selection condition
$m_{ee}>140~{}{\rm MeV}/c^{2}$ is applied in the $K_{L}\to\pi^{0}e^{+}e^{-}$
case as required to suppress backgrounds from major $K_{L}$ decays into
neutral pions followed by $\pi^{0}\to\gamma e^{+}e^{-}$ decays. The resulting
single-event sensitivities for $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays (Table
6) improve by more than two orders of magnitude on the previous searches at
the kTeV experiment [75, 76]. Suppression of the
$K_{L}\to\gamma\gamma\ell^{+}\ell^{-}$ background, which represents a
significant limitation especially in the $K_{L}\to\pi^{0}e^{+}e^{-}$ case
[77], relies on the good photon energy resolution provided by the EM
calorimeter. The current NA62 LKr calorimeter [175] would provide a 2.2 MeV
di-photon mass resolution for the proposed HIKE Phase 2 setup. Further
potential backgrounds to $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays come from
major $K_{L}$ decays such as $K_{L}\to\pi^{+}\pi^{-}\pi^{0}$ with $\pi^{\pm}$
misidentification or decay in flight, and from pileup effects.

Background estimation for the $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays is
currently in progress, however the extensive experience of the NA62 lepton
flavour/number violation programme in terms of background reduction [119, 120,
121] suggests that the proposed experimental technique allows for background
suppression to the level of ${\cal O}(10^{-11})$ for rare and forbidden decays
into multiple tracks and photons, for a well-collimated beam. Considering the
expected signal yields in excess of 100 events (Table 6), we conclude that the
first observation of the $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decays is likely,
and a sensitivity at the SM branching ratio level of ${\cal O}(10^{-11})$ can
certainly be reached. Detailed HIKE Phase 2 sensitivity studies are in
progress for the forthcoming proposal.

The expected $K_{L}\to\mu^{+}\mu^{-}$ signal yield suggests a 0.2% statistical
precision on the measurement of the decay branching ratio. We expect that
overall a precision better than 1% is attainable when normalising to the
$K_{L}\to\pi^{+}\pi^{-}$ decay, which improves on the previous measurement by
the BNL-E871 experiment [87].

HIKE Phase 2 offers sensitivities of ${\cal O}(10^{-12})$ for branching ratios
of a broad range of rare and forbidden $K_{L}$ decay modes. As an example, the
acceptance for the lepton flavour violating $K_{L}\to\mu^{\pm}e^{\mp}$ decays
(Table 6) leads to expected upper limits of both decay branching ratios of
$8\times 10^{-14}$ at 90% CL in five years of operation (the null hypothesis
and assuming low background), improving the limit by a factor of 60 on the
previous search by the BNL-E871 experiment [126].

##  8 Phase 3 (KLEVER): measurement of the $K_{L}\to\pi^{0}\nu\bar{\nu}$ decay

Exploratory work on an experiment to measure ${\cal
B}(K_{L}\to\pi^{0}\nu\bar{\nu})$ in a future stage of the ECN3 high-intensity
kaon physics programme began even before the start of NA62 data taking. The
basic design work for the KLEVER experiment ($K_{L}$ Experiment for VEry Rare
events) was carried out as a part of the first Physics Beyond Colliders
initiative at CERN, in preparation for the 2020 update of the European
Strategy for Particle Physics [176].

As in the other phases of the programme, KLEVER makes use of the 400-GeV
primary proton beam from the SPS, slow extracted and transported to the T10
target (Section 5.1) at an intensity of $2\times 10^{13}$ protons per pulse
(ppp), corresponding to about six times the nominal NA62 intensity. The
neutral secondary beam for KLEVER is derived at an angle of 8 mrad; the
beamline design is described in Section 5.3. In practice, the choice of
production angle has been optimised together with the limits of the fiducial
volume (FV) to maximise the signal sensitivity and minimise
$K_{L}\to\pi^{0}\pi^{0}$ and $\Lambda\to n\pi^{0}$ backgrounds. For a
production angle of 8 mrad, the neutral beam has a mean $K_{L}$ momentum of 40
GeV, so that 5% of $K_{L}$ decay inside an FV extending from 130 m to 190 m
downstream of the target. The $K_{L}$ mesons decaying inside the FV have a
mean momentum of 27 GeV/$c$. The boost from the high-energy 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. On the other hand, the
layout poses particular 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.
Background from $\Lambda\to n\pi^{0}$ decays in the beam must also be kept
under control.

The KLEVER goal is to achieve a sensitivity of about 60 events for the
$K_{L}\to\pi^{0}\nu\bar{\nu}$ decay at the SM BR with an $S/B=1$. At the SM
BR, this would correspond to a relative uncertainty of about 20%. We would
expect to be able to observe a discrepancy with SM predictions with $5\sigma$
significance if the true BR is a bit more than twice or less than one-quarter
of the SM BR, or with $3\sigma$ significance if the true BR is less than half
of the SM rate. As noted in Section 5.3, with an opening angle of 0.4 mrad,
the $K_{L}$ yield in the beam is $2.1\times 10^{-5}$ per proton on target
(pot). With a fiducial-volume acceptance of 5% and a selection efficiency of
3%, collection of 60 SM events would require a total primary flux of $6\times
10^{19}$ pot, corresponding to five years of running at an intensity of
$2\times 10^{13}$ ppp under NA62-like slow-extraction conditions with 3000
spills per day for 200 days per year.

The scheduling of the KLEVER phase is influenced by two considerations.
Firstly, to provide maximum protection from $\Lambda\to n\pi^{0}$ background,
the beamline would need to be lengthened by 150 m (Section 5.3.3). Secondly,
despite the efforts to benchmark the KLEVER beam simulation by comparison with
existing data on inclusive particle production by protons on lightweight
targets at SPS energies (Section 5.3.1), confidence in the simulations and
background estimates would be greatly increased by the possibility to acquire
particle production data and measure inclusive rates in the KLEVER beamline
setup with a tracking experiment in place. This would be possible in HIKE
Phase 2 (Section 7). Therefore, although in principle KLEVER could aim to
start data taking after LS4 (for which injector physics is currently foreseen
to begin in early 2034), the most natural timescale might envisage KLEVER as
running after Phase 2. We are studying scheduling options to allow KLEVER data
taking to begin before LS5 (foreseen to start in 2038).

###  8.1 Experimental layout

The KLEVER setup largely consists of a collection of high-efficiency photon
detectors arranged around a 160-m-long vacuum volume to guarantee hermetic
coverage for photons from $K_{L}$ decays emitted at polar angles out to 100
mrad and to provide a nearly free path through vacuum up to the main
electromagnetic calorimeter (MEC) for photons emitted into a cone of at least
7.5 mrad. The fiducial volume (FV) spans about 60 m just downstream of the
active final collimator (AFC), but the photon veto coverage extends along the
entire length up to the MEC. The layout of the detector elements is
schematically illustrated in Fig. 20.

HIKE Phase 3 (KLEVER)

Figure 20: HIKE Phase 3 (KLEVER) layout, with an aspect ratio of 1:10. The
baseline configuration, without the extension of the beamline by 150 m is
shown.

The largest elements are about 3 m in diameter. The beginning of the vacuum
volume is immediately downstream of the cleaning collimator at $z=80$ m, i.e.,
80 m downstream of the T10 target. A 40-m vacuum decay region allows the
upstream veto calorimeter (UV) surrounding the AFC to have an unobstructed
view for the rejection of $K_{L}$ decays occurring upstream of the detector
volume. The UV and AFC at $z=120$ m define the start of the detector volume,
which is lined with 25 large-angle photon and charged-particle veto stations
(LAV) in five different sizes, placed at intervals of 4 to 6 m to guarantee
hermeticity for decay particles with polar angles out to 100 mrad. The MEC, at
the downstream end of the vacuum volume, replaces the NA48 LKr calorimeter
used in NA62: it reconstructs the $\pi^{0}$ vertex for signal events and helps
to reject events with extra photons. A charged-particle veto detector (CPV) in
front of the MEC rejects $K_{L}\to\pi^{\pm}e^{\mp}\nu$ and
$K_{L}\to\pi^{+}\pi^{-}\pi^{0}$ backgrounds, and a preshower detector (PSD)
allows reconstruction of the angles of incidence for photons, providing
additional constraints on signal candidates even if only one photon converts.
The small-angle vetoes on the downstream side of the MEC intercept photons
from $K_{L}$ decays that pass through the beam pipe. The small-angle
calorimeter (SAC) intercepts the neutral beam; its angular coverage as seen
from the downstream end of the FV extends to $\pm 2$ mrad. Because of the high
neutron and photon rate in the beam, the SAC design is one of the most
challenging aspects of the experiment. The intermediate-ring calorimeter (IRC)
is a ring-shaped detector between the SAC and MEC and intercepts photons from
downstream decays that make it through the calorimeter bore at slightly larger
angles. In addition to the photon vetoes, the experiment makes use of hadronic
calorimeters downstream of the MEC, to help reject background from hadron
interactions and the copious $K_{L}$ decays into charged particles such as
muons and pions.

Because of the experimental challenges involved in the
$K_{L}\to\pi^{0}\nu\bar{\nu}$ measurement, and in particular, the very high
efficiency required for the photon veto systems, the performance requirements
for KLEVER drive the specifications for many of the new detectors for HIKE. In
this case, the LAVs, MEC, IRC, SAC, and hadronic calorimeters built for Phases
1 and 2 would all be able to be used for KLEVER in Phase 3. The UV, CPV, and
PSD would have to be built new for KLEVER, possibly with recycled components
from the analogous detectors for the earlier phases.

In the baseline design with the original 120 m beamline, the FV covers the
region $130~{}{\rm m}<z<190$ m. The positioning of the FV significantly
upstream of the calorimeter, together with the relatively high $K_{L}$
momentum, is key to obtaining sufficient $K_{L}\to\pi^{0}\pi^{0}$ background
rejection with the ring-shaped LAV geometry covering polar angles out to 100
mrad. This comes at a cost in acceptance for signal decays, which increases
significantly as the FV is moved closer to the calorimeter. Extending the
downstream limit of the FV would increase the sensitivity of the experiment
but will require improvements in background rejection; further optimisation
along these lines is under study.

Fig. 20 shows the experimental configuration without a lengthened beamline to
suppress background from $\Lambda$ decays. Lengthening the beamline would
change the distance from the cleaning collimator to the AFC, so that the
cleaning collimator would be upstream of the start of the vacuum tank instead
of right at its entrance. The detector configuration would be otherwise
unchanged.

###  8.2 Rates and timing performance

Table 7: Rates for events with hits on KLEVER detectors, by detector system (left) and for certain event classes (right), evaluated for the 120 m beamline configuration. Detector | Event rate (MHz) | Event class | Rate (MHz)
---|---|---|---
AFC | 2.3 | Exactly 2 hits on MEC | 4.8
UV | 7.1 | Exactly 2 photons on MEC | 1.0
LAV | 14 | 2 hits on MEC with UV, LAV veto | 3.1
MEC | 18 | 2 hits on MEC, no other hits | 0.007
IRC | 22 |  |
SAC | 95 |  |

The FLUKA simulation of the beamline performed by the Conventional Beams
Working Group (Section 5.3) contains an idealised representation of the
experimental setup for the purposes of evaluating rates on the detectors, both
from the decays of $K_{L}$ mesons and from the beam halo. The estimated rates
by detector system are listed in Table 7, assuming a primary intensity of
$2\times 10^{13}$ ppp and a (pessimistic) 3 s effective spill. The simulation
was performed for the standard, 120 m beamline (without extension). The rates
are obtained with a geometrical representation of the detector volumes and
assumed probabilities for hadrons to register signals. For the AFC, UV, LAV,
and MEC, the probability for registering signals is assumed to be 25% for
neutrons and $K_{L}$ mesons, and 100% for all other particles. For the IRC and
SAC, the assumed probability is 10% for neutral hadrons, and 100% for all
other particles. The particle tracking threshold is 1 GeV. Since a particle
detected in any of the detectors would effectively constitute a veto (the
rates of signal candidates being negligible for these purposes), this provides
an estimate of the total veto rate: 134 MHz, of which 104 MHz from events with
hits on a single detector and 30 MHz from events with hits on multiple
detectors. The total rate is dominated by the rate of interactions of beam
particles in the SAC, consisting of 13 MHz of hits from $K_{L}$ mesons, 44 MHz
of hits from neutrons, and 40 MHz of hits from beamline photons with $E>5$
GeV. Hadronic interactions can be efficiently recognised offline and we assume
only 10% of the corresponding rate contributes to the inefficiency from
accidental coincidence. The remaining veto rate from accidentals is then at
most 100 MHz, which we take as a figure of merit. The event time is obtained
from the $\pi^{0}$ candidate reconstructed in the MEC, and the accidental
coincidence rate is dominated by events on the SAC. Assuming a $\pm
5\sigma_{t}$ coincidence window, to limit the inefficiency from accidental
coincidence to below 25%, the time resolution $\sigma_{t}$ on the coincidence
must be better than 250 ps, and the detector design must minimise any
contributions from non-gaussian fluctuations to the timing distribution. These
considerations lead to the specification that the nominal time resolution for
the most critical detectors (the MEC and the SAC) must be better than 100 ps.

As noted above, the full FLUKA simulation was performed with the original, 120
m beamline configuration. Many of the rates in Table 7 would be expected to
decrease by the ratio of beam solid angle, i.e., by a factor of 0.41, when the
beamline is extended. This scaling may not be exact for the upstream detectors
(AFC, UV, LAV), as the rates on those detectors are determined in part by the
details of the collimation and sweeping scheme, but is is certainly true for
the downstream detectors, and in particular, for the SAC, which drives the
random veto rate for the experiment. Thus, the downstream extension not only
provides a significant general decrease in the rates on the detectors, but
would lower the estimated random veto rate from 25% to 10%.

The rates for certain classes of events to be acquired are also listed in
Table 7: for events with exactly two hits and with exactly two photons on the
calorimeter, independently of the other detectors; for events with two hits on
the calorimeter without hits on the UV or LAV (presumed to be in online veto);
and for events with two hits on the calorimeter and no hits on any of the
other detectors. These are estimates of the rates for the physical
events—there is no simulation of the detector response apart from the
efficiency assumptions outlined above—but these results do suggest that the
total rates for event classes that would approximately correspond to level-0
trigger conditions in NA62 are in fact of about the same order of magnitude as
in NA62.

Figure 21: Radial distribution of particle fluxes at the MEC for KLEVER, from
FLUKA beamline simulation.

Fig. 21 shows the radial distribution of the particle fluxes at the front face
of the MEC, as estimated with the FLUKA simulation of the neutral beam. The
beam halo from photons and neutral hadrons drops off rapidly for $10<r<20$ cm,
with $r$ the radial distance from the beam axis. For neutrons, the halo-to-
core ratio decreases from $2.5\times 10^{-2}$ to $8.8\times 10^{-5}$ over this
interval, corresponding to a decrease in the total neutron rate in the halo
from 12 MHz to 43 kHz. For $r>20$ cm, the neutral hadron fluxes are on the
order of a few Hz/cm2. At $r=20$ cm, there are 1 kHz/cm2 of photons and
charged pions and a few hundred Hz/cm2 of muons, dropping off more or less
linearly with radius.

Once the extended beamline layout is finalised, it will be necessary to re-run
the FLUKA-based simulation to obtain definitive estimates of the beam flux,
halo, and veto rates in the detectors.

###  8.3 Expected performance for $K_{L}\to\pi^{0}\nu\bar{\nu}$

Simulations of the experiment in the standard configuration carried out with
fast-simulation techniques (idealised geometry, parameterised detector
response, etc.) suggest that the target sensitivity (60 SM events with
$S/B=1$) is achievable. However, due to the the reduction in the beam solid
angle, the $K_{L}$ flux is decreased by 60% when the beamline is extended by
150 m. We are exploring options to increase the acceptance for signal decays.
The acceptance for $K_{L}\to\pi^{0}\nu\bar{\nu}$ decays in the fiducial volume