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1 64. Rare Kaon Decays

64. Rare Kaon Decays

Revised August 2019 by L. Littenberg (BNL) and G. Valencia (Monash U.). 64.1 Introduction There are several useful reviews on rare kaon decays and related topics [1–12]. Activity in rare kaon decays can be divided roughly into four categories: 1. Searches for explicit violations of the (SM) 2. The golden modes: K → πνν¯ 3. Other constraints on SM parameters 4. Studies of strong interactions at low energy.

The paradigm of Category 1 is the flavor violating decay KL → µe. Category 2 in- cludes the two modes that can be calculated with negligible theoretical uncertainty, K+ → π+νν 0 and KL → π 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. Category 3 is focused 0 + − + − on decays with charged , such as KL → π ` ` or KL → ` ` where ` ≡ e, µ. These modes are sensitive to CKM parameters but they suffer from multiple hadronic uncertainties that can be addressed, at least in part, through a systematic study of the peripheral modes indicated in Fig. 64.1. 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 con- tributions are dominant and which constitute a testing ground for the ideas of chiral perturbation 0 + − theory. Other decays in this category are KL → π γγ and KL → ` ` γ. The former is important 0 + − in understanding a CP -conserving contribution to KL → π ` ` , whereas the latter could shed + − light on long distance contributions to KL → µ µ . 64.2 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 the potential to access very high energy scales. For example, the tree-level exchange of a LFV vector of mass MX that couples to left-handed with electroweak strength and without −12 4 mixing angles yields B(KL → µe) = 4.7 × 10 (148 TeV/MX ) [2]. This simple dimensional analysis may be used to read from Table 64.1 that the reaction KL → µe is already probing scales of over 100 TeV. Table 64.1 summarizes the present experimental situation vis-à-vis LFV. The ± ∓ + + ∓ ± 0 ∓ ± decays KL → µ e and K → π e µ (or KL → π e µ ) provide complementary information on potential family number violating interactions, since the former is sensitive to -odd couplings and the latter is sensitive to parity-even couplings. Limits on certain lepton-number violating (LNV) kaon decays also have been obtained, with recent interest arising from their role in constraining possible extensions of the sector [13, 14], 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.” Physics beyond the SM is also pursued through the search for K+ → π+X0, where X0 is a new light . The searches cover both long-lived (e.g., hyperphoton, , familon, etc.), and short-lived ones that decay to , or pairs. The 90% CL upper limit on K+ → π+X0 is 7.3 × 10−11 [22] for the case of massless X0; additional results as a function of the X0 mass can be found in [23]. Recently these limits have been reinterpreted in connection with a [24] or dark Z [25]. Such vectors have also been sought in their e+e− decay mode by

P.A. Zyla et al. (), Prog. Theor. Exp. Phys. 2020, 083C01 (2020) 1st June, 2020 8:31am 2 64. Rare Kaon Decays

Figure 64.1: 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.

Table 64.1: Searches for lepton flavor and lepton number violation in K decay

LFV mode 90% CL upper limit Experiment Yr./Ref. Type K+ →π+e−µ+ 1.3×10−11 BNL-865 2005/Ref. [15] LFV K+ →π+e+µ− 5.2×10−10 BNL-865 2000/Ref. [16] LFV −12 KL →µe 4.7×10 BNL-871 1998/Ref. [17] LFV 0 −11 KL →π eµ 7.6×10 KTeV 2008/Ref. [18] LFV 0 0 −10 KL →π π eµ 1.7×10 KTeV 2008/Ref. [18] LFV K+ →π−e+e+ 2.2×10−10 NA-62 2019/Ref. [19] LNV K+ →π−µ+µ+ 4.2×10−11 NA-62 2019/Ref. [19] LNV K± →π∓µ±µ± 8.6×10−11 NA48/2 2017/Ref. [20] LNV ± ± ∓ ∓ −11 KL →e e µ µ 4.12×10 KTeV 2003/Ref. [21] LNV K+ →π−µ+e+ 5.0×10−10 BNL-865 2000/Ref. [16] LNFV

NA48/2 [26]. Additional bounds for a short lived pseudoscalar X0 decaying to or 0 0 + − −10 0 0 −7 are B(KL → π π µ µ ) < 1 × 10 [27] and B(KL → π π γγ) < 2.4 × 10 [28]. 64.3 The golden modes: K → πνν¯ In the SM, the decay K+ → π+νν is dominated by one-loop diagrams with top- interme- diate states while long-distance contributions are known to be quite small [29–31]. This permits a precise calculation of this rate in terms of SM parameters. Studies of this process are thus mo- tivated by the possibility of detecting non-SM physics when comparing with the results of global fits [32, 33]. The branching ratio can be written in a compact form that exhibits the different ingredients

1st June, 2020 8:31am 3 64. Rare Kaon Decays that go into the calculation [34], "   Im(V ?V ) 2 B K+ → π+νν(γ) = κ (1 + ∆ ) ts td X + EM λ5 t # Re(V ? V ) Re(V ?V ) 2 + cs cd (P + δP ) + ts td X . (64.1) λ c c,u λ5 t The parameters in Eq. 64.1 incorporate the a priori unknown hadronic matrix element in terms of the very well-measured Ke3 rate [29] in κ+; long distance QED corrections in ∆EM [35]; the Inami- Lim function for the short distance top-quark contribution [36] including NLO QCD corrections [37, 38] and the two-loop electroweak correction [34], all in Xt; and the -quark contributions due to short distance effects including NNLO QCD corrections [39,40] and NLO electroweak corrections via Pc [41], as well as certain long distance effects via δPc,u [31, 42]. 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”) [43] is: + + −5 4 2 2 B(K → π νν) ≈ 1.6 × 10 |Vcb| [ση + (ρc − ρ) ], (64.2) where ρ ≈ 1.45 and σ ≡ 1 . Thus, B(K+ → π+νν) determines an ellipse in the ρ, η plane c (1− 1 λ2)2 2 r r 1 B(K+→π+νν) 1 B(K+→π+νν) with center (ρc, 0) and semiaxes ≈ 2 −5 and 2 −5 . |Vcb| 1.6×10 σ|Vcb| 1.6×10 BNL-787 observed two candidate events [44, 45] in the clean high π+ momentum and one event [46] in the low-momentum region. The successor experiment BNL-949 observed one more in the high-momentum region [22] and three more in the low-momentum region [47], yielding a +1.15 −10 branching ratio of (1.73−1.05) × 10 [23]. The NA62 experiment, performed with in-flight decays at CERN, aims to reach a sensitivity of ∼ 10−12/event. NA62 was commissioned in 2015 and has taken data in 2016, 2017 and 2018. The short 2016 run resulted in the observation of one event over an expected background of 0.152 events, which allowed a 95% C.L. limit of 14 × 10−10 to be set [48]. Very recently, preliminary results from the 2017 run have been announced. Two additional events were observed, over an expected background of 1.5 events. Combining this data with that of 2016, they obtain a a 90% C.L. limit of 1.85 × 10−10 [49]. Using the latest CKMfitter input [32], we estimate B(K+ → π+νν) = (8.5 ± 0.5) × 10−11, near the lower end of the measurement of BNL-787 and 949. However, current parametric uncertainty in the CKM angles can result in numbers with central values differing from this one by up to 10% [50]. 0 The second golden mode is the neutral counterpart to our preceeding discussion: KL → π νν. It is dominantly CP -violating and free of hadronic uncertainties [29, 51, 52]. 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 Ref. [43]: Im(V ?V ) 2 B(K → π0νν) = κ ts td X L L λ5 t −5 4 2 ≈ 7.6 × 10 |Vcb| η . (64.3)

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, using the latest CKMfitter input [32], is (3.0±0.2)×10−11. But 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% [50].

1st June, 2020 8:31am 4 64. Rare Kaon Decays

Grossman and Nir (GN) [53] pointed out that, in a nearly model-independent manner, the two 0 + + golden modes satisfy the relation B(KL → π νν) . 4.4 B(K → π νν). Using the BNL 787/949 + + 0 −9 90% CL bound on K → π νν, GN then predict B(KL → π νν) < 1.46 × 10 . Using instead the 0 −10 latest NA62 result, the GN upper bound becomes B(KL → π νν) < 8.14 × 10 as can be seen in Figure 64.2. The KOTO experiment at J-PARC, whose initial goal is to observe this decay, has been running since 2013 and in 2018 published a 90% CL upper limit of 3.0×10−9 [54], based on their 2015 data. They have run every year since, making incremental upgrades to the experimental configuration between runs. It was pointed out in a recent paper that the GN bound obtained from the BNL result applies to 0 0 0 the three body decay KL → π νν¯ and not necessarily to two body modes such as KL → π X . In this case KOTO can provide interesting constraints on new physics even at the current sensitivity level [55]. Using the 2015 data, they have established a 90% CL upper limit of 2.4 × 10−9 on 0 0 KL → π X for mX0 ≈ mπ0 [54]. ��� KOTO 90%CL

�� GN exclusion ���

) ⨯ �� �� νν � ��

π → BNL � � 787/949 � ( SM �� �� % ���� � � � �� �� �� �� �� �� �(�+→ π+ νν)⨯����

Figure 64.2: Summary of current situation for the golden modes K → πνν¯. The red ellipse shows the 1σ SM prediction with input from CKMfitter; the yellow region corresponds to the BNL 1σ measurement; the vertical green line marks the 90% CL upper bound from NA62, with the corresponding exclusion region shaded green; and the dashed orange line marks the 90% CL KOTO upper bound. The black shaded region shows the GN exclusion.

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 straight forward to establish the existence of new physics by observing deviations from their SM values in the K → πνν¯ modes, it would take much more extensive global fits to pinpoint the origin of any such deviation. Partial summaries with references can be found in Refs. [9,56–58]. There has also been a recent classification of the different possibilities in terms of the neutrino couplings [59], and several studies on the effect of with different lepton flavor. [60–62].

1st June, 2020 8:31am 5 64. Rare Kaon Decays

The current theoretical and experimental situation for the golden modes is summarized in Fig. 64.2. The red area corresponds to the 1σ SM prediction we obtain with the latest input available from CKMfitter (summer 2018) [32]. The yellow region shows the result established by the combined BNL-787 and BNL-949 results, whereas the green region marks the new 90% CL upper bound from NA62 for K+ → π+νν¯. The black shaded region marks the GN exclusion, which lies significantly above the SM expectation leaving a large window for discovery of new physics 0 contributions by experiments seeking to measure B(KL → π νν). The 90% CL upper bound on this mode from KOTO is shown as a dashed orange line, and is seen to still lie in the GN excluded zone. Related modes with one extra , K → ππνν¯, are similarly dominated by short distance contributions [63–65]. However, they occur at much lower rates with branching ratios of order −13 0 0 −7 10 . The current best bound comes from KEK-391a, it is B(KL → π π νν¯) < 8.1 × 10 at 90% CL [66]. There is also a bound B(K+ → π+π0νν¯) < 4.3 × 10−5 at 90% CL [67] from BNL-787. 64.4 Other constraints on Standard Model parameters + − The decay KL → µ µ has a short distance contribution sensitive to the CKM parameter ρ, given by [43]:

+ − −4 4 0 2 BSD(KL → µ µ ) ≈ 2.7 × 10 |Vcb| (ρc − ρ) (64.4)

0 where ρc depends on the mass and is approximately 1.2. This decay, however, is dominated 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 + − −9 to be Babs(KL → µ µ ) = (6.64 ± 0.07) × 10 ; and it almost completely saturates the observed + − −9 rate B(KL → µ µ ) = (6.84 ± 0.11) × 10 [68]. The difference between the observed rate and the 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 from experiment [69], but can be estimated with certain assumptions [70, 71]. + − By contrast, the decay KL → e e is completely dominated by long distance physics and is + − −12 easier to estimate. The result, B(KL → e e ) ∼ 9 × 10 [69, 72], is in good agreement with the +5.7 −12 BNL-871 measurement, (8.7 −4.1 ) × 10 [73]. + − The mode KS → µ µ similarly has a short distance contribution proportional to the square of the CKM parameter η¯ entering at the 10−13 level [10]. It also has long distance contributions arising + − −12 from the two photon intermediate state which result in a rate B(KS → µ µ )LD = 5.1×10 [10]. + − −10 A 90% CL limit B(KS → µ µ ) < 2.1 × 10 was recently obtained by LHCb [74, 75]. The + − + − interplay between KL → µ µ and KS → µ µ has been the subject of [76, 77]. 0 + − The decay KL → π e e is sensitive to the CKM parameter η through its CP -violating com- ponent. 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 the 0 + − SM [5]. The indirect CP -violating amplitude can be inferred from a measurement of KS → π e e . The complete CP -violating contribution to the rate can be written as [78–80]:   2 ! 2 !2 −12 2 |Vcb| η |Vcb| η BCPV ≈ 10 15.7|aS| ± 1.4 |aS| + 0.12  (64.5) 10−4 10−4 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 + − +0.26 of KS → π e e with the result |aS| = 1.06−0.21 ± 0.07 [81], as well as from a measurement 0 + − +0.40 of KS → π µ µ with the result |aS| = 1.54−0.32 ± 0.06 [82]. With current constraints on the

1st June, 2020 8:31am 6 64. Rare Kaon Decays

CKM parameters, and assuming a positive sign for the interference term [80, 83], this implies that 0 + − −11 BCPV(KL → π e e ) ≈ (3.1 ± 0.9) × 10 , where the three contributions to the central value from indirect, 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 [84]. 0 + − The complete CP violating amplitude for the related mode KL → π µ µ is predicted to be 0 + − −11 BCPV(KL → π µ µ ) ≈ (1.4 ± 0.5) × 10 [10, 85]. 0 + − KL → π 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 0 part can be extracted from the measurement of the low mγγ region of the KL → π γγ spectrum. 0 The rate and the shape of the distribution dΓ/dmγγ in KL → π γγ are well described in chiral perturbation theory in terms of three (a priori) unknown parameters [86–89]. 0 Both KTeV and NA48 have studied the mode KL → π γγ, reporting similar results. KTeV 0 −6 0 finds B(KL → π γγ) = (1.29 ± 0.03stat ± 0.05sys) × 10 [90], while NA48 finds B(KL → π γγ) = −6 (1.36±0.03stat±0.03sys±0.03norm)×10 [91]. Both experiments are consistent with a negligible rate 0 + − in the low mγγ region, suggesting a very small CP -conserving component BCP(KL → π e e ) ∼ O(10−13) [80,89,91]. There remains some model dependence in the estimate of the dispersive part 0 + − of the CP -conserving KL → π e e [80]. 0 + − 0 + − The related process, KL → π γe e , is potentially an additional background to KL → π e e in some region of phase space [92]. This process has been observed with a branching ratio of −8 (1.62 ± 0.14stat ± 0.09sys) × 10 [93]. + − 0 + − The decay KL → γγe e constitutes the dominant background to KL → π e e . It was first observed by BNL-845 [94], and subsequently confirmed with a much larger sample by KTeV [95]. It has been estimated that this background will enter at about the 10−10 level [96,97], comparable 0 + − to or larger than the signal level. Because of this, the observation of KL → π e e at the SM level will depend on background subtraction with good statistics. Possible alternative strategies are discussed in Ref. [80] and references cited therein. 0 + − −10 The 90% CL upper bound for the process KL → π e e is 2.8 × 10 [97]. For the closely 0 + − −10 related muonic process, the published upper bound is B(KL → π µ µ ) ≤ 3.8 × 10 [98], compared with the SM prediction of (1.5±0.3)×10−11 [85] (assuming positive interference between the direct- and indirect-CP violating components). 0 + − A study of KL → π 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 [99]. The latter tends to be quite substantial so that large statistics may not be necessary. 0 + − + − Combined information from KL → π ` ` as well as KL → µ µ complements the K → πνν measurements in constraining physics beyond the SM [100]. 64.5 Other long distance dominated modes The decays K+ → π+`+`− (` = e or µ) have received considerable attention. The rate and spectrum have been measured for both the electron and muon modes [101–106]. The theoretical status of these modes has been summarized recently [84, 107]. The measurements have been used to exclude new physics such as a dark photon [24]. Ref. [78, 79] has proposed a parameterization inspired by chiral perturbation theory, which provides a suc- cessful description of data but indicates the presence of large corrections beyond leading order. More work is needed to fully understand the origin of these large corrections. The mode K+ → π+π0e+e−, recently analyzed by NA48/2 [108], is also dominated by long distance physics but it has been ar- gued that measuring can provide information on the short distance components [109]. + − + − The related mode KS → π π e e , which was measured by NA48/1 [110], has received new in-

1st June, 2020 8:31am 7 64. Rare Kaon Decays terest by LHCb [111] as an important background to other rare decays. An update of the theory for these modes can be found in [112]. The decay K+ → π+γγ can be predicted in terms of one unknown parameter to leading order in χPT resulting in a correlation between the rate and the diphoton mass spectrum [113]. Certain important corrections at the next order are also known [114]. The rate was first measured by E787 [115], and more recently NA48/2 [116] has obtained a more precise result with a 6% error, as well as the corresponding spectrum fits. The most recent, and precise, result is from NA62 based on a sample of 232 events [117] but is still insufficient to distinguish between the leading order and next order χPT parameterizations. Much information has been recorded by KTeV and NA48 on the rates and spectrum for the + − + − 0+ 0− 0 Dalitz pair conversion modes KL → ` ` γ [118, 119], and KL → ` ` ` ` for `, ` = e or µ + − 0+ 0− [21, 120]. More recently, LHCb has performed preliminary studies of KS → ` ` ` ` [111]. All these results are used to test hadronic models and should eventually help unravel the underlying + − physics in KL → µ µ [71, 76, 121]. References [1] L. Littenberg and G. Valencia, Ann. Rev. Nucl. Part. Sci. 43, 729 (1993), [hep-ph/9303225]. [2] J. L. Ritchie and S. G. Wojcicki, Rev. Mod. Phys. 65, 1149 (1993). [3] B. Winstein and L. Wolfenstein, Rev. Mod. Phys. 65, 1113 (1993). [4] A. Pich, Rept. Prog. Phys. 58, 563 (1995), [hep-ph/9502366]. [5] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996), [hep- ph/9512380]. [6] G. D’Ambrosio and G. Isidori, Int. J. Mod. Phys. A13, 1 (1998), [hep-ph/9611284]. [7] P. Buchholz and B. Renk, Prog. Part. Nucl. Phys. 39, 253 (1997). [8] A. R. Barker and S. H. Kettell, Ann. Rev. Nucl. Part. Sci. 50, 249 (2000), [hep-ex/0009024]. [9] A. J. Buras, F. Schwab and S. Uhlig, Rev. Mod. Phys. 80, 965 (2008), [hep-ph/0405132]. [10] V. Cirigliano et al., Rev. Mod. Phys. 84, 399 (2012), [arXiv:1107.6001]. [11] D. Bryman et al., Ann. Rev. Nucl. Part. Sci. 61, 331 (2011). [12] T. K. Komatsubara, Prog. Part. Nucl. Phys. 67, 995 (2012), [arXiv:1203.6437]. [13] A. Atre et al., JHEP 05, 030 (2009), [arXiv:0901.3589]. [14] L. S. Littenberg and R. Shrock, Phys. Lett. B491, 285 (2000), [hep-ph/0005285]. [15] A. Sher et al., Phys. Rev. D72, 012005 (2005), [hep-ex/0502020]. [16] R. Appel et al., Phys. Rev. Lett. 85, 2877 (2000), [hep-ex/0006003]. [17] D. Ambrose et al. (BNL), Phys. Rev. Lett. 81, 5734 (1998), [hep-ex/9811038]. [18] E. Abouzaid et al. (KTeV), Phys. Rev. Lett. 100, 131803 (2008), [arXiv:0711.3472]. [19] E. Cortina Gil et al. (NA62), Phys. Lett. B797, 134794 (2019), [arXiv:1905.07770]. [20] J. R. Batley et al. (NA48/2), Phys. Lett. B769, 67 (2017), [arXiv:1612.04723]. [21] A. Alavi-Harati et al. (KTeV), Phys. Rev. Lett. 90, 141801 (2003), [hep-ex/0212002]. [22] V. V. Anisimovsky et al. (E949), Phys. Rev. Lett. 93, 031801 (2004), [hep-ex/0403036]. [23] A. V. Artamonov et al. (BNL-E949), Phys. Rev. D79, 092004 (2009), [arXiv:0903.0030]. [24] M. Pospelov, Phys. Rev. D80, 095002 (2009), [arXiv:0811.1030]. [25] H. Davoudiasl, H.-S. Lee and W. J. Marciano, Phys. Rev. D89, 9, 095006 (2014), [arXiv:1402.3620].

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