Kaon Experiments

By 1973, when the paper by Kobayashi and Maskawa was published, a charge asymmetry in $K_{L}\to\pi^{\pm}e^{\mp}\nu$ had also been observedke3delta , proving that in $K_{L}$, the amplitude of $K^{0}$ is slightly larger than the amplitude of $\overline{K}^{0}$.²²2The initial state can be tagged by the lepton charge, as $K^{0}\to\pi^{-}e^{+}\nu$ and $\overline{K}^{0}\to\pi^{+}e^{-}\overline{\nu}$. The asymmetry also shows that $K_{L}$ is not a pure $CP$-odd state, $|K_{\mathrm{odd}}\rangle=(|K^{0}\rangle-|\overline{K}^{0}\rangle)/\sqrt{2}$, but it also has a small admixture ($\epsilon$) of $CP$-even state, $|K_{\mathrm{even}}\rangle=(|K^{0}\rangle+|\overline{K}^{0}\rangle)/\sqrt{2}$, as

	$\displaystyle|K_{L}\rangle$	$\displaystyle\simeq$	$\displaystyle\frac{1}{\sqrt{2}}\left((1+\epsilon)|K^{0}\rangle-(1-\epsilon)|\overline{K}^{0}\rangle\right)$		(1)
		$\displaystyle=$	$\displaystyle|K_{\mathrm{odd}}\rangle+\epsilon|K_{\mathrm{even}}\rangle.$		(2)

The $CP$-violation found in the $K_{L}\to\pi\pi$ decays was explained by the $K_{\mathrm{even}}$ component decaying to the $CP$-even $\pi\pi$ state. This type of $CP$ violation is called “indirect $CP$ violation”, but the origin of $\epsilon$ was unknown.

The fact that $K_{L}$ stays in this unbalanced equilibrium state means that the overall rates of transitions between $K^{0}$ and $\overline{K}^{0}$ are the same, as

$\displaystyle|(1+\epsilon)A(K^{0}\to\overline{K}^{0})|^{2}$

$\displaystyle=$

$\displaystyle|(1-\epsilon)A(\overline{K}^{0}\to K^{0})|^{2},$

(3)

where $A(x\to y)$ represents the amplitude for transition $x\to y$. Because $\epsilon\neq 0$, $|A(K^{0}\to\overline{K}^{0})|\neq|A(\overline{K}^{0}\to K^{0})|$. The transition amplitudes are determined by the Schrödinger equation for the two-component wave function of a neutral kaon system,

	$\displaystyle i\frac{\partial}{\partial t}\begin{pmatrix}K^{0}(t)\\ \overline{K}^{0}(t)\end{pmatrix}$	$\displaystyle=$	$\displaystyle H\begin{pmatrix}K^{0}(t)\\ \overline{K}^{0}(t)\end{pmatrix}$		(4)
		$\displaystyle=$	$\displaystyle\begin{bmatrix}M_{0}-i\Gamma_{0}/2&M_{12}-i\Gamma_{12}/2\\ M^{*}_{12}-i\Gamma^{*}_{12}/2&M_{0}-i\Gamma_{0}/2\end{bmatrix}\begin{pmatrix}K^{0}(t)\\ \overline{K}^{0}(t)\end{pmatrix},$		(5)

where $K^{0}(t)$ ($\overline{K}^{0}(t)$) is the amplitude of $K^{0}$ ($\overline{K}^{0}$) state at time $t$. The transition amplitude for $K^{0}$ returning to the same state, $A(K^{0}\to K^{0})$, is proportional to $M_{0}-i\Gamma_{0}/2$. The $\Gamma_{0}$ is determined by the sum of amplitudes of $K^{0}\to f\to K^{0}$ where $f$ represents the decay final states, and $M_{0}$ is determined by the sum of amplitudes of $K^{0}\to i\to K^{0}$ where $i$ represents the intermediate virtual states. The transition amplitude $A(\overline{K}^{0}\to K^{0})$ is proportional to $M_{12}-i\Gamma_{12}/2$, where $\Gamma_{12}$ is determined by the sum of amplitudes of $\overline{K}^{0}\to f\to K^{0}$, and $M_{12}$ is determined by the sum of amplitudes of $\overline{K}^{0}\to i\to K^{0}$, where $f$ and $i$ are the final and intermediate states, respectively, which are common to both $K^{0}$ and $\overline{K}^{0}$. If, for example, $\Gamma_{12}$ is real but $M_{12}$ has an imaginary component³³3To be more exact, if $\Gamma_{12}$ and $M_{12}$ are not parallel in the complex plane., then $|M_{12}-i\Gamma_{12}/2|$ which determines $|A(\overline{K}^{0}\to K^{0})|$ would be different from $|M_{12}^{*}-i\Gamma_{12}^{*}/2|$ which determines $|A(K^{0}\to\overline{K}^{0})|$. Thus, the indirect $CP$ violation in neutral kaon system can occur if there is an imaginary part in $M_{12}$.

What was not known back in 1973 was the source of the imaginary part in $M_{12}$. The Superweak model explained that there is a very weak unknown interaction that changes the strangeness by 2 ($\overline{K}^{0}\to K^{0}$), and that interaction introduces an imaginary part in $M_{12}$. Kobayashi and Maskawa explained that such an imaginary part can be naturally introduced by mixings in three generations of quarks. Figure 2 shows such an example. The question then was, which model is correct? One way to answer it was to check whether the $CP$ is violated in the decay process itself. The Superweak model cannot violate $CP$ in the decay process because it cannot contribute to a $\Delta S=1$ process. However, the Kobayashi-Maskawa model can naturally introduce an imaginary part in $\Delta S=1$ decay transition if three generation of quarks are involved in the decay, such as in a penguin diagram shown in Fig.2. Such $CP$ violation in decay processes is called “direct $CP$ violation” in kaon physics. If one could show that the direct $CP$ violation exists, then one could reject the Superweak model and support the Kobayashi-Maskwa model.

In 1980’s, there were two experiments which measured $Re(\epsilon^{\prime}/\epsilon)$; NA31 at CERN, and E731 at Fermilab.

The NA31 experiment collected $\pi^{+}\pi^{-}$ and $\pi^{0}\pi^{0}$ events simultaneously, and $K_{L}$ and $K_{S}$ runs separately. The four photons from the $\pi^{0}\pi^{0}$ decays were detected by a liquid Argon calorimeter. The energies and positions of the charged pions from the $\pi^{+}\pi^{-}$ decays were measured with a hadronic calorimeter. To make the final state topology similar to that of $\pi^{0}\pi^{0}$ events, a magnetic spectrometer was not used. To mimic the long decay position distribution of $K_{L}$, for the $K_{S}$ run, a target to produce $K_{S}$ was inserted and moved along the beam line inside the decay region, as shown in Fig. 3. The events from the four decay modes were grouped by kaon energy and decay position bins. The double ratio $R$ was calculated in each bin and then averaged.

The E731 experiment collected the $K_{L}$ and $K_{S}$ decays simultaneously. By placing a material in one of the two $K_{L}$ beams as shown in Fig. 4 to regenerate $K_{S}$⁵⁵5Because materials are made of quarks, $\overline{K}^{0}$ containing $s$ quark has a higher cross section than $K^{0}$ containing $\bar{s}$ quark, to conserve the baryon number. Thus if $|K_{L}\rangle\sim|K_{\mathrm{odd}}\rangle\propto|K^{0}\rangle-|\overline{K}^{0}\rangle$ passes through a material, and the wave function becomes $|\psi\rangle=(a|K^{0}\rangle-b|\overline{K}^{0}\rangle)/\sqrt{2}=(a+b)/\sqrt{2}\ |K_{L}\rangle+(a-b)/\sqrt{2}\ |K_{S}\rangle$, the $K_{S}$ component appears., E731 effectively created $K_{L}$ and $K_{S}$ beams with a fixed $N_{K_{L}}/N_{K_{S}}$ ratio. The $K_{L}$ and $K_{S}$ events were identified by reconstructing the event and finding in which beam the kaon decayed. Note that because charged pions and photons from the decays spread out, the detectors themselves were insensitive to in which beam the parent $K$ decayed. Initially, $\pi^{+}\pi^{-}$ and $\pi^{0}\pi^{0}$ events were collected separately, but later, all four decay modes were collected simultaneously. To confirm that the acceptances were understood, the decay position distributions of high-statistics samples of $K_{L}\to\pi^{\pm}e^{\mp}\nu$ and $K_{L}\to\pi^{0}\pi^{0}\pi^{0}$ decays were compared between data and MC.

The NA31 measured $Re(\epsilon^{\prime}/\epsilon)=(23\pm 6.5)\times 10^{-4}$, $3.5\sigma$ away from zero barr1993 , whereas the E731 measured $Re(\epsilon^{\prime}/\epsilon)=(7.4\pm 6.0)\times 10^{-4}$, consistent with zero with $1.2\sigma$ gibbons . Many critical questions on various systematic effects were raised between the groups, but none of them could resolve the difference in the results. At the end, the both groups decided to build new experiments with an order of magnitude higher precision.

Fermilab KTeV E832 experiment pursued the double $K_{L}$ and $K_{S}$ beam technique and collected the four decay modes simultaneously. A new high intensity beam line with a better collimation scheme was built to increase the kaon rates and to reduce accidental hits in the detectors. The electromagnetic calorimeter was newly built with 3100 CsI crystals with improved energy and position resolutions to reduce systematic uncertainties. The waveforms of the calorimeter signals were digitized at 53 MHz and recorded. A new fast data acquisition system allowed to collect high statistics sample of $K_{L}\to\pi e\nu$ and $K_{L}\to 3\pi^{0}$ events to study systematic effects, in addition to the $K\to\pi\pi$ events.

CERN NA48 experiment introduced a new technique to collect $K_{L}$ and $K_{S}$ events. A small fraction of protons passing through the $K_{L}$ production target were guided to another production target far downstream to produce $K_{S}$. The $K_{L}$ and $K_{S}$ beams were designed to overlap in the detector region to reduce systematic uncertainties. The events whose timings were correlated with the timings of the protons hitting the $K_{S}$ production target were identified as $K_{S}$. The momentum of charged pions were measured with a magnetic spectrometer this time. A new Liquid Krypton calorimeter was built to improve the photon energy resolution.

In 1999, with partial dataset, the Fermilab KTeV E832 experiment published $Re(\epsilon^{\prime}/\epsilon)=(28.0\pm 4.1)\times 10^{-4}$ktev_1999 , 7$\sigma$ away from zero, and CERN N48 also published $Re(\epsilon^{\prime}/\epsilon)=(18.5\pm 7.3)\times 10^{-4}$na48_1999 , both before $CP$ violation in B-meson system was first observed in 2001 babar_cp ; belle_cp . At the end, the combined result based on full datasets of both experiments was $Re(\epsilon^{\prime}/\epsilon)=(16.8\pm 1.4)\times 10^{-4}$ ktev_2011 , and the PDG’s fit result is $Re(\epsilon^{\prime}/\epsilon)=(16.6\pm 2.3)\times 10^{-4}$ pdg_epoe . The number of $K_{L}\to\pi^{0}\pi^{0}$ were $6\times 10^{6}$ for KTeV and $1.5\times 10^{6}$ for NA48. This put an end to the long awaited question. The kaon experiments rejected the Superweak model as a sole source of $CP$ violation, and supported the Kobayashi-Maskawa model. Figure 5 shows how the measured $Re(\epsilon^{\prime}/\epsilon)$ has improved over the years. The experimental results from $K$ and $B$ mesons gave solid foundations for the Kobayashi-Maskawa model to be a part of the Standard Model.

Major backgrounds for the $K^{+}\to\pi^{+}\nu\overline{\nu}$ decay are the $K^{+}\to\mu^{+}\nu$ decay in which $\mu^{+}$ is misidentified as $\pi^{+}$, and the $K^{+}\to\pi^{+}\pi^{0}$ decay in which the two photons from the $\pi^{0}$ escaped detection.

One of the early experiments that searched for the $K^{+}\to\pi^{+}\nu\overline{\nu}$ decay was KEK E10. The charged kaons were stopped in a target, and the decayed $\pi^{+}$ was identified by its path length in detectors, and observing the $\pi^{+}\to\mu^{+}\to e^{+}$ decay chain. The $K^{+}\to\pi^{+}\pi^{0}$ background was suppressed by detecting photons with lead glass blocks located on the other side of charged pion tracking detectors. The experiment set an upper limit on the branching fraction as $BR<1.4\times 10^{-7}$ (90% CL) asano .

To increase the acceptance of the $K^{+}\to\pi^{+}\nu\overline{\nu}$ decay, BNL E787/E949 experiments surrounded the $K^{+}$ stopping target with a cylindrical spectrometer and range counters to measure the momentum and energy of $\pi^{+}$, respectively. The waveforms in the range counters were recorded to track the $\pi^{+}\to\mu^{+}\to e^{+}$ decay chain to identify $\pi^{+}$. The target was also surrounded by photon veto counters to suppress $K^{+}\to\pi^{+}\pi^{0}$ background. At the end, BNL E787/E949 measured $BR=(1.73^{+1.15}_{-1.05})\times 10^{-10}$e949 based on 7 observed events where 0.93 background events were expected.

To increase the number of events by an order of magnitude, experiments proposed later chose to use high-momentum $K^{+}$’s decaying in flight. By not having a stopping target in a high intensity beam, one can avoid producing secondary particles in the target. Also, vetoing the $K^{+}\to\mu^{+}\nu$ background is easier because high-momentum muons can easily penetrate through materials whereas charged pions cannot: this removed the need to track the $\pi^{+}\to\mu^{+}\to e^{+}$ decay chain for microseconds. The $K^{+}\to\pi^{+}\pi^{0}$ background was suppressed by vetoing high energy photons decaying toward downstream. Another background is caused by $\pi^{+}$ scattered from the beam. Earlier stopping $K^{+}$ experiments suppressed $\pi^{+}$’s with a mass separator, but the same technique cannot be used for high momentum $K^{+}$’s and $\pi^{+}$’s.

Fermilab CKM experiment planned to apply high frequency electric fields at two locations on a momentum-selected beam to kick out $\pi^{+}$’s which fly with a speed different from $K^{+}$’s. The experiment was however, canceled during preparation.

CERN NA62 experiment decided not to remove $\pi^{+}$’s from the beam, but to identify $K^{+}$ with a fast differential Cherenkov counter. Also, the momentum, direction, and timing of each $K^{+}$ is measured with pixel detectors and dipole magnets. The decayed $\pi^{+}$ is identified by a ring-imaging Cherenkov counter, and its momentum is measured with a magnetic spectrometer in vacuum. The background from $K^{+}\to\mu^{+}\nu$ decays is suppressed by identifying muons with NA48’s Liquid Krypton calorimeter and scintillators between and behind iron plates. The background from $K^{+}\to\pi^{+}\pi^{0}$ is suppressed by detecting photons from the decay with ring-shaped lead glass modules at multiple locations in the decay region, a liquid Krypton calorimeter placed downstream, and lead/scintillator calorimeter in the beam. In the data sample collected in 2016-2018, NA62 observed 20 events where 7 background events were expected, and measured $BR=(1.06^{+0.40}_{-0.34}\pm 0.09)\times 10^{-10}$ na62_jhep06 . NA62 will improve the sensitivity with more data taken with improved beam line and detectors.

For the $K_{L}\to\pi^{0}\nu\overline{\nu}$ decay, the major issues is how to identify the decay because only visible particles are two photons from the $\pi^{0}$. The major background is the $K_{L}\to\pi^{0}\pi^{0}$ decay where two of the four photons in the final state escaped detection, but there are also many unexpected backgrounds because observables are limited.

The very first upper limit on the branching fraction was set as $BR<7.6\times 10^{-3}$ (90% CL)littenberg by using the photon energy spectrum measured in an old experiment on $K_{L}\to\pi^{0}\pi^{0}$. Fermilab KTeV E799-II experiment had a one-day special run to collect two-photon events, and set $BR<1.6\times 10^{-6}$ (90% CL) ktev2g . Later, KTeV E799-II searched for the decay by using the $\pi^{0}\to e^{+}e^{-}\gamma$ decay to identify the $\pi^{0}$, reconstruct the decay vertex and the transverse momentum of $\pi^{0}$, and set $BR<5.9\times 10^{-7}$ (90% CL) kteveeg . Although the technique using the $\pi^{0}\to e^{+}e^{-}\gamma$ decay is cleaner, it is less sensitive than the technique using the $\pi^{0}\to\gamma\gamma$ decay because of the small branching fraction of the $\pi^{0}\to e^{+}e^{-}\gamma$ decay (1.2%), and a small acceptance for detecting $e^{+}e^{-}$ pairs with a small opening angle. Later experiments thus chose to use $\pi^{0}\to\gamma\gamma$ to search for the $K_{L}\to\pi^{0}\nu\overline{\nu}$ decay.

KEK E391a using 12 GeV protons was the first experiment dedicated for the $K_{L}\to\pi^{0}\nu\overline{\nu}$ decay. To suppress the background caused by the $K_{L}\to\pi^{0}\pi^{0}$ decay with two missing photons, and to collect events with only two photons, the decay region was surrounded by a hermetic photon veto detector system and a CsI calorimeter covering downstream. To avoid a background caused by a photon lost in a beam pipe, the beam pipe was removed and the entire photon veto system and the calorimeter were placed inside a vacuum tank, instead. The decay vertex ($Z$) was reconstructed by assuming that the two photons were originated from a $\pi^{0}$. The transverse momentum of $\pi^{0}$ ($P_{T}$) was then reconstructed based on the decay vertex and the energies and hit positions of the two photons assuming that the $K_{L}$ decayed at the center of the beam line. A narrow $K_{L}$ beam was made to have a small $P_{T}$ resolution while keeping the necessary $K_{L}$ yield which is proportional to the beam size. Because some momentum was carried away by neutrino pairs, $P_{T}$ was required to have a finite value. E391a set $BR<2.6\times 10^{-8}$ (90% CL) based on no observed events in the signal region defined in a $P_{T}-Z$ planee391a .

As the construction of the J-PARC accelerator facility began, KOTO experiment was built to utilize its intense slow-extracted 30-GeV proton beam. A new narrow $K_{L}$ beam line with a clean collimation scheme was built. The E391a’s vacuum tank and cylindrical photon veto counters were moved to J-PARC. The electromagnetic calorimeter was replaced with 2716 CsI crystals originally used in FNAL KTeV to improve the detection efficiency and shower shape measurements. A new data acquisition system records waveforms from all the detectors channels at 125 MHz (some at 500 MHz) to identify overlapping pulses.

Many unforeseen backgrounds were encountered as the sensitivity increased, and they were solved one by one. One new background was caused by a neutron hitting the calorimeter and producing another neutron, making two clusters. This background was suppressed by installing thin photosensors on the upstream surface of the calorimeter to select photons which interact near the upstream end. Another background was found to be caused by a small contamination ($O(10^{-5})$) of $K^{\pm}$ in the beam decaying into $\pi^{0}e^{\pm}\nu$ with undetected $e^{\pm}$ koto2016_2018 . This was suppressed by installing a thin scintillation detector in the beam, and later, by installing another sweeping magnet in the beam. The current best limit set by KOTO is $BR<3.0\times 10^{-9}$ (90% CL) koto2015 . The experiment continues to collect more data with improved detectors and higher beam power to search for an order of magnitude enhancement on the branching fraction by new physics.

A new experiment called KOTO II is being planned to search for a smaller enhancement by new physics by increasing the sensitivity by more than two orders of magnitude. To increase the $K_{L}$ yield, $K_{L}$ will be extracted at a smaller angle ($5^{\circ}$, compared 16^∘ at KOTO) from the proton beam. To increase the acceptance of the signal events, the decay region will be extended from 2 m to 12 m, and the calorimeter will be enlarged from 2 m to 3 m in diameter. In 36 months of running, the experiment expects to observe 35 Standard Model signal events on top of 56 background events with the 4.7$\sigma$ significancekoto2 . If the measured branching fraction deviates from the Standard Model prediction by 44% due to new physics, it can be claimed with the 90% confidence level.

Since 1989, the upper limit on the branching fraction of $K_{L}\to\pi^{0}\nu\overline{\nu}$ has been reduced by 5 orders of magnitudes, as shown in Fig. 8, and KOTO II is being prepared to observe the events.