65. Rare Kaon Decays

1

65. Rare Kaon Decays
Revised August 2023 by L. Littenberg (BNL) and G. Valencia (Monash U.).

65.1 Introduction
There are several useful reviews on rare kaon decays and related topics [1–13]. Activity in rarekaon decays can be divided roughly into four categories:
1. Searches for explicit violations of the Standard Model (SM)
2. The golden modes: K → πν ν̄
3. Other constraints on SM parameters
4. Studies of strong interactions at low energy.
The paradigm of Category 1 is the lepton flavor violating decay KL → µe. Category 2 includes
the two modes that can be calculated with negligible theoretical uncertainty, K + → π + νν andKL → π 0 νν. These modes can lead to precision determinations of CKM parameters or, in combination with other measurements of these parameters, they can constrain new interactions. They
constitute the main focus of the current experimental kaon program. The search for new light particles through the reaction K → πX 0 is a byproduct of these measurements. Category 3 is focusedon decays with charged leptons, such as KL → π 0 `+ `− or KL → `+ `− where ` ≡ e, µ. These modes
are sensitive to CKM parameters, but they suffer from multiple hadronic uncertainties that canbe addressed, at least in part, through a systematic study of the peripheral modes indicated inFig. 65.1.

Figure 65.1: Role of rare kaon decays in determining the unitarity triangle. The solid arrows
point to auxiliary modes needed to interpret the main results, or potential backgrounds to them.

The interplay between Categories 3-4 and their complementarity to Category 2 is illustrated
in the figure. Category 4 includes reactions like K + → π + `+ `− where long distance contributions
are dominant and which constitute a testing ground for the ideas of chiral perturbation theory.
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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65. Rare Kaon Decays

2

Other decays in this category are KL → π 0 γγ and KL → `+ `− γ. The former is important inunderstanding a CP -conserving contribution to KL → π 0 `+ `− , whereas the latter could shed lighton long distance contributions to KL → µ+ µ− .

65.2 Explicit violations of the Standard Model
Much activity has focussed on searches for lepton flavor violation (LFV). This is motivated by
the fact that many extensions of the minimal Standard Model violate lepton flavor and by thepotential to access very high energy scales. For example, the tree-level exchange of a LFV vectorboson of mass MX that couples to left-handed fermions with electroweak strength and without
mixing angles yields B(KL → µe) = 4.7 × 10−12 (148 TeV/MX )4 [2]. This simple dimensional
analysis may be used to read from Table 65.1 that the reaction KL → µe is already probing scalesof over 100 TeV. Table 65.1 summarizes the present experimental situation vis-à-vis LFV. The
decays KL → µ± e∓ and K + → π + e∓ µ± (or KL → π 0 e∓ µ± ) provide complementary information on
potential family number violating interactions, since the former is sensitive to parity-odd couplings
and the latter is sensitive to parity-even couplings.
Limits on certain lepton-number violating (LNV) kaon decays also have been obtained, withspecial interest arising from their role in constraining possible extensions of the neutrino sector[14, 15], and we list those in the table as well. Related searches in µ and τ processes are discussed
in our section “Tests of Conservation Laws.”
Table 65.1: Searches for lepton flavor and lepton number violation in K
decay
LFV mode
K + → π + e− µ+
K + → π + e+ µ−
KL → µe
KL → π 0 eµ
KL → π 0 π 0 eµ
K + → π − e+ e+
K + → π − π 0 e+ e+
K + → π − µ+ µ+
KL → e± e± µ∓ µ∓
K + → π − µ+ e+
K + → µ− νe+ e+

90% CLupper limit
1.3×10−11
6.6×10−11
4.7×10−12
7.6×10−11
1.7×10−10
5.3×10−11
8.5×10−10
4.2×10−11
4.12×10−11
4.2×10−11
8.1×10−11

Experiment

Yr./Ref.

Type

BNL-865
NA62
BNL-871
KTeV
KTeV
NA62
NA62
NA62
KTeV
NA62
NA62

2005/ [16]2021/ [17]1998/ [18]2008/ [19]2008/ [19]2022/ [20]2022/ [20]2019/ [21]2003/ [22]2021/ [17]2023/ [23]

LFV
LFV
LFV
LFV
LFV
LNV
LNV
LNV
LNV
LNFV
LNV/LFV

65.3 The golden modes: K → πν ν̄
In the SM, the decay K + → π + νν is dominated by one-loop diagrams with top-quark intermediate states while long-distance contributions are known to be quite small [24–26]. This permitsa precise calculation of this rate in terms of SM parameters. Studies of this process are thus motivated by the possibility of detecting non-SM physics when comparing with the results of global
fits [27, 28].
The branching ratio can be written in a compact form that exhibits the different ingredients
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65. Rare Kaon Decays

3that go into the calculation [29],


+

+

"



B K → π νν(γ) = κ+ (1 + ∆EM )Im(Vts? Vtd )
Xt
λ5

2



+

Re(Vcs? Vcd )Re(Vts? Vtd )
Pc +
Xt
λ
λ5

2 #

.
(65.1)The parameters in Eq. 65.1 incorporate the a priori unknown hadronic matrix element in terms ofthe very well-measured Ke3 rate [24] in κ+ ; long distance QED corrections in ∆EM [30]; the InamiLim function for the short distance top-quark contribution [31] including NLO QCD corrections [32,33] and the two-loop electroweak correction [29], all in Xt ; and the charm-quark contributions due
to short distance effects including NNLO QCD corrections [34,35] and NLO electroweak corrections
via Pc [36], which also includes certain long distance terms dubbed δPc,u [26, 37]. An interesting
approximate way to cast this result in terms of the CKM parameters λ, Vcb , ρ and η (see our Section
on “The Cabibbo-Kobayashi-Maskawa mixing matrix”) [9] is:B(K + → π + νν) ≈ 1.6 × 10−5 |Vcb |4 [(ση)2 + (ρc − ρ)2 ],(65.2)

where ρc ≈ 1.45 and σ ≡ (1− 11λ2 )2 . Thus, B(K + → π + νν) determines an ellipse in the ρ, η plane2r1

with center (ρc , 0) and semiaxes ≈ |Vcb |2

B(K + →π+ νν) and1.6×10−5r
1
σ|Vcb |2

B(K + →π+ νν) .

1.6×10−5
BNL-787 observed two candidate events [38, 39] in the clean high π + momentum and one

event [40] in the low-momentum region. The successor experiment BNL-949 observed one morein the high-momentum region [41] and three more in the low-momentum region [42], yielding a−10 [43].
branching ratio of (1.73+1.15−1.05 ) × 10
NA62 was commissioned in 2015 and took data in 2016, 2017 and 2018 and published resultsfrom those runs [44]. Twenty candidate events were observed, including an estimated background+4.0
of 7.03, yielding B(K+ → π + νν) = (10.6−3.4|stat ± 0.9sys ) × 10−11 . They resumed data taking in
2023 after some upgrades to the detector with the goal of achieving a 20% measurement of the
branching ratio by the next long shutdown of the accelerator.
Using the latest CKMfitter input (Jan. 2022) [27], we estimate B(K + → π + νν)SM = (8.1 ±0.4) × 10−11 , near the lower end of the measurements of BNL-787/949 and NA62. Current parametric uncertainty in the CKM angles can result in numbers with central values differing from thisone by up to 10% [45].
The second golden mode is the neutral counterpart to our preceeding discussion: KL → π 0 νν. It
is dominantly CP -violating and free of hadronic uncertainties [24, 46, 47]. In the Standard Model,
this mode is dominated by an intermediate top-quark state and does not suffer from the small
uncertainty associated with the charm-quark intermediate state that affects K + → π + νν. The
branching ratio is given by [9]:B(KL → π 0 νν) = κL

Im(Vts? Vtd )
Xt
λ5

2

≈ 7.6 × 10−5 |Vcb |4 η 2 .

(65.3)

As with the charged mode, the hadronic matrix element can be related to that measured in K`3
decay and is parameterized in κL .
Our estimate for the branching ratio within the SM, using the latest CKMfitter input (Jan.
2022) [27], is (2.8 ± 0.2) × 10−11 . Similarly to the charged kaon case, parametric uncertainty in the
CKM angles can result in a central value that differs from this one by up to almost 20% [45].
Grossman and Nir (GN) [48] pointed out that, in a nearly model-independent manner, the twogolden modes satisfy the relation B(KL → π 0 νν) . 4.3 B(K + → π + νν). Using the BNL 787/949
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65. Rare Kaon Decays

4

90% CL bound on K + → π + νν, GN then predict B(KL → π 0 νν) < 1.46 × 10−9 . Using instead
the latest NA62 result, the GN upper bound becomes B(KL → π 0 νν) < 8.14 × 10−10 [13].
The KOTO experiment at J-PARC, whose goal is to make an initial observation of this decay,
has been in operation since 2013. It has made a series of data collection runs and has reported
on three data sets thus far. In 2018 it published a 90% CL upper limit of 3.0 × 10−9 [49], basedon 2015 data. They have run almost every year since, making incremental improvements to theexperimental configuration between runs. In 2021 they published a result on the data taken between 2016 and 2018 in which three events were observed in the signal region [50], whereas the
expected background was 1.22 events, mainly from charged kaons produced by KL interactions in
the downstream collimator. Conservatively assuming the three events were due to background, they
extracted a 90% CL upper limit of 4.9 × 10−9 . Subsequent hardware and analysis improvements
reduced the expected background to 0.26 in a sample of similar sensitivity taken in 2021. No eventswere observed in the signal region and a 90% CL upper limit of 2.0 × 10−9 was established [51]. An
increase of 3-5 in sensitivity is anticipated from runs planned in the next three years.
The current theoretical and experimental situation for the golden modes is summarized inFig. 65.2. The red area corresponds to the 1σ SM prediction we obtain with the latest input
available from CKMfitter (January 2022) [27]. The yellow region shows the result establishedby the combined BNL-787 and BNL-949 results, whereas the green region marks the new NA62
result for K + → π + ν ν̄. The black shaded region marks the GN exclusion, which lies significantly
above the SM expectation leaving a large window for discovery of new physics contributions by
experiments seeking to measure B(KL → π 0 νν). The 90% CL upper bound on this mode from the
KOTO result published in 2018 [49] is shown as a dashed orange line, and is seen to still lie in the
GN excluded zone.

B(KL  π0 νν)1011

500

KOTO 90% CL (2023)
GN exclusion

100
50
10
5

BNL
787/949

NA62
SM

1
0

5

10

15

20

25

30

35

B(K +  π+ νν)1011
Figure 65.2: Summary of current situation for the golden modes K → πν ν̄. The red ellipse
shows the 1σ SM prediction with input from CKMfitter; the green (yellow) region corresponds to
the NA62 (BNL787/949) 1σ measurement; and the dashed orange line marks the 90% CL KOTO
upper bound. The black shaded region shows the GN exclusion.

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65. Rare Kaon Decays

590% cl B(K  π X 0 )Much theoretical work has explored beyond the SM scenarios that can populate this window
as well as their correlations with other rare processes outside kaon physics. Although it would be
relatively straightforward to establish the existence of new physics by observing deviations fromtheir SM values in the K → πν ν̄ modes, it would take much more extensive global fits to pinpoint
the origin of any such deviation. Partial summaries with references focusing on quark flavor physics
can be found in Refs. [9,52–56]. Different possibilities emphasizing an interpretation of these modes
as K → π + Emiss can be found in Refs. [57–59].
There is a subtlety in converting the GN bound extracted from measurements of K + → π + ν ν̄
into an upper bound for KL → π 0 ν ν̄ as applied to KOTO [60]. This is due to a lack of sensitivity of
the charged kaon experiments to a kinematic window near the pion mass. New physics appearing
through a two body mode K + → π + X 0 , for an invisible X 0 with mass near the pion mass, has amuch weaker constraint. The situation is illustrated in Fig. 65.3 which shows the BNL787/949 [43]and NA62 [61] 90% c.l. limits on K + → π + X 0 as a function of mX 0 assuming X 0 is stable. The
implied GN bound for KL → π 0 X 0 , B(KL → π 0 X 0 ) . 4.3B(K + → π + X 0 ) is shown as a blue line
and can be compared with the actual KOTO [49] constraint on KL → π 0 X 0 in red. Notice that theupper bound derived from NA62 is lower than that derived from BNL787/949 except for a small
mass window. For 120 . mX 0 . 150 MeV, KOTO already has better sensitivity than that impliedby the GN bound in this case. The possibility of new physics in this scenario has generated muchtheoretical speculation [62–72].

10-8
10-9

KOTO
GN

10

-10

10-11

BNL
NA62
0

50

100
150
mX 0 (MeV)

200

250

Figure 65.3: Summary of current situation for K → πX 0 . The solid (dashed) black line showsthe exclusion limit for K + → π + X 0 from NA62 (BNL787/949) assuming X 0 is stable. The blueline marks the constraint implied for KL → π 0 X 0 by the Grossman-Nir relation and the red line
marks the current KOTO limit on K + → π + X 0 .

The search for a new light particle X 0 in K + → π + X 0 has a long tradition covering both longlived particles (e.g., hyperphoton, axion, familon, etc.), and short-lived ones that decay to muon,
electron, photon or neutrino pairs. The longstanding 90% CL upper limit on K + → π + X 0 from
BNL787/949 7.3 × 10−11 [41] for the case of massless X 0 has been slightly improved by NA62 to
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65. Rare Kaon Decays

6

5 × 10−11 as can be read from Fig. 65.3. These limits can also be reinterpreted in connection witha dark photon [73] or dark Z [74], and in this context NA48/2 also constrained the mode B(K ± →π ± A0 → π ± e+ e− ) [75]. Complementary searches for new light particles in kaon experiments use
modes with two pions and include the KTeV bound B(KL → π 0 π 0 X 0 → π 0 π 0 µ+ µ− ) < 1×10−10 [76]and the E391a bound B(KL → π 0 π 0 X 0 → π 0 π 0 γγ) < 2.4 × 10−7 [77].
Neutrino pair modes with one extra pion, K → ππν ν̄, are similarly dominated by short distancecontributions in the SM [78–80]. Even though they are theoretically clean, they occur with very
low rates with branching ratios of order 10−13 . The current best bound comes from KEK-391a,
B(KL → π 0 π 0 ν ν̄) < 8.1 × 10−7 at 90% CL [81]. There is also a bound for the charged kaon mode
B(K + → π + π 0 ν ν̄) < 4.3 × 10−5 at 90% CL [82] from BNL-787. New physics contributions to thesemodes are discussed in [83].

65.4 Other constraints on Standard Model parameters
The decay KL → µ+ µ− has a short distance contribution sensitive to the CKM parameter ρ,
given by [5]:
BSD (KL → µ+ µ− ) ≈ 2.7 × 10−4 |Vcb |4 (ρ0c − ρ)2(65.4)

where ρ0c depends on the charm quark mass and is approximately 1.2. This decay, however, isdominated by a long-distance contribution from a two-photon intermediate state. The absorptive(imaginary) part of the long-distance component is determined by the measured rate for KL → γγ
to be Babs (KL → µ+ µ− ) = (6.64 ± 0.07) × 10−9 ; and it almost completely saturates the observed
rate B(KL → µ+ µ− ) = (6.84 ± 0.11) × 10−9 [84]. The difference between the observed rate andthe absorptive component can be attributed to the (coherent) sum of the short-distance amplitude
and the real part of the long-distance amplitude. The latter cannot be derived directly fromexperiment [85], but can be estimated with certain assumptions [86, 87].
By contrast, the decay KL → e+ e− is completely dominated by long distance physics and iseasier to estimate. The result, B(KL → e+ e− ) ∼ 9 × 10−12 [85, 88], is in good agreement with the−12 [89].
BNL-871 measurement, (8.7 +5.7
−4.1 ) × 10+−
The mode KS → µ µ has a short distance contribution proportional to the square of the
CKM parameter η̄ entering at the 10−13 level [10]. It also has long distance contributions arisingfrom the two photon intermediate state which result in a rate B(KS → µ+ µ− )LD = 5.1×10−12 [10].
There is a 90% CL limit B(KS → µ+ µ− ) < 2.1 × 10−10 from LHCb [90, 91]. The interplay between
KL → µ+ µ− and KS → µ+ µ− has been the subject of [92, 93], and it has been pointed out that ameasurement of time-dependent interference effects could be used to extract information on CKM
angles from K → µ+ µ− measurements [94, 95].
The decay KL → π 0 e+ e− is sensitive to the CKM parameter η through its CP -violating component. There are both direct and indirect CP -violating amplitudes that can interfere. The direct
CP -violating amplitude is short distance dominated and has been calculated in detail within theSM [5]. The indirect CP -violating amplitude can be inferred from a measurement of KS → π 0 e+ e− .
The complete CP -violating contribution to the rate can be written as [96–98]:
|Vcb |2 η
BCPV ≈ 15.7|aS |2 ± 1.4
10−4!

|Vcb |2 η|aS | + 0.12
10−4

!2 
 × 10−12(65.5)

where the three terms correspond to the indirect CP violation, the interference, and the direct
CP violation, respectively. The parameter aS has been extracted by NA48/1 from a measurement+0.26
of KS → π 0 e+ e− with the result |aS | = 1.06−0.21± 0.07 [99], as well as from a measurement of+0.40
KS → π 0 µ+ µ− with the result |aS | = 1.54−0.32± 0.06 [100]. With current constraints on the
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7

65. Rare Kaon Decays

CKM parameters, and assuming a positive sign for the interference term [98, 101], this implies that
BCPV (KL → π 0 e+ e− ) ≈ (3.1 ± 0.9)×10−11 , where the three contributions to the central value fromindirect, interference and direct CP violation are (1.76, 0.9, 0.45) × 10−11 respectively. It should be
noted that more recent studies suggest a much larger uncertainty in the value of aS [102].
KL → π 0 e+ e− also has a CP -conserving component dominated by a two-photon intermediate
state. This component can be decomposed into an absorptive and a dispersive part. The absorptive
part can be extracted from the measurement of the low mγγ region of the KL → π 0 γγ spectrum.
The rate and the shape of the distribution dΓ/dmγγ in KL → π 0 γγ are well described in chiralperturbation theory in terms of three (a priori) unknown parameters [103–106].
Both KTeV and NA48 have studied the mode KL → π 0 γγ, reporting similar results. KTeV finds
B(KL → π 0 γγ) = (1.29±0.03stat ±0.05sys )×10−6 [107], while NA48 finds B(KL → π 0 γγ) = (1.36±
0.03stat ± 0.03sys ± 0.03norm ) × 10−6 [108]. Both experiments are consistent with a negligible ratein the low mγγ region, suggesting a very small CP -conserving component BCP (KL → π 0 e+ e− ) ∼O(10−13 ) [98, 106, 108]. There remains some model dependence in the estimate of the dispersive
part of the CP -conserving KL → π 0 e+ e− [98].
The related process, KL → π 0 γe+ e− , is potentially an additional background to KL → π 0 e+ e−
in some region of phase space [109]. This process has been observed with a branching ratio of(1.62 ± 0.14stat ± 0.09sys ) × 10−8 [110].
The decay KL → γγe+ e− constitutes the dominant background to KL → π 0 e+ e− . It was firstobserved by BNL-845 [111], and subsequently confirmed with a much larger sample by KTeV [112].
It has been estimated that this background will enter at about the 10−10 level [113,114], comparable
to or larger than the signal level. Because of this, the observation of KL → π 0 e+ e− at the SMlevel will depend on background subtraction with good statistics. Possible alternative strategies
are discussed in [98] and references cited therein.
The 90% CL upper bound for the process KL → π 0 e+ e− is 2.8 × 10−10 [114]. For the closely
related muonic process, the published upper bound is B(KL → π 0 µ+ µ− ) ≤ 3.8 × 10−10 [115],
compared with the SM prediction of (1.9 ± 0.5) × 10−11 . The latter assumes positive interference
between the direct- and indirect-CP violating components and includes a CP-conserving component
which contributes about 30% of the total [10, 116].
A study of KL → π 0 µ+ µ− has indicated that it might be possible to extract the direct CP violating contribution by a joint study of the Dalitz plot variables and the components of the µ+polarization [117]. The latter tends to be quite substantial so that large statistics may not benecessary.
Combined information from KL → π 0 `+ `− as well as KL → µ+ µ− complements the K → πννmeasurements in constraining physics beyond the SM [118].

65.5 Other long distance dominated modes
The rates and spectra for the decays K + → π + `+ `− (` = e or µ) have been measured [119–124]
and are successfully described with a parameterization inspired by chiral perturbation theory butwith large corrections beyond leading order [96, 97] (see [102, 125] for a review). The most recent
NA62 result shows good agreement between the electron and muon mode form factors [126]. NA62
has also obtained a first measurement of the forward-backward asymmetry in the muon mode [126].
The mode K + → π + π 0 e+ e− , analyzed by NA48/2 [127], is also dominated by long distancephysics but it has been argued that measuring asymmetries can provide information on the shortdistance components [128]. The current status of these modes is discussed in [129]. Contributions
to K + → π + e+ e− e+ e− beyond the double Dalitz decay in K + → π + π 0 are considered in [130], andhave been measured by NA62 [131].
The decay K + → π + γγ can be predicted in terms of one unknown parameter to leading order in
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865. Rare Kaon Decays

χPT, dubbed ĉ, resulting in a correlation between the rate and the diphoton mass spectrum [132].
Certain important corrections at the next order are also known [133]. The rate and spectrum
were first measured by E787 [134], followed by NA48/2 [135] and NA62 [136]. The most recent
measurement from NA62 [137], based on a sample of over 4000 events, finds B(K + → π + γγ) =(9.73 ± 0.17 ± 0.08) × 10−7 and a precise value for ĉ.
Much information has been recorded by KTeV and NA48 on the rates and spectrum for the
Dalitz pair conversion modes KL → `+ `− γ [138, 139], and KL → `+ `− `0+ `0− for `, `0 = e or µ[22, 140]. LHCb has recently placed limits on KL,S → µ+ µ− µ+ µ− [141]. All these results are used
to test hadronic models and should eventually help unravel the underlying physics in KL → µ+ µ−[87, 92, 142].
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