166. CPT Invariance Tests in Neutral Kaon Decay

66. CPT Invariance Tests in Neutral Kaon Decay
Revised August 2023 by M. Antonelli (INFN, Frascati), G. D’Ambrosio (INFN, Napoli) and M.S.
Sozzi (Pisa U.).
CP T theorem is based on three assumptions: quantum field theory, locality, and Lorentz invariance, and thus it is a fundamental probe of our basic understanding of particle physics. Strangeness0oscillation in K 0 − K system, described by the equation"#"d K0K0i0 = [M − iΓ/2]
0dt KK#,

where M and Γ are hermitian matrices (see PDG review [1], references [2, 3], and KLOE paper [4]for notations and previous literature), allows a very accurate test of CP T symmetry; indeed since
CP T requires M11 = M22 and Γ11 = Γ22 , the mass and width eigenstates, KS,L , have a CP T violating piece, δ, in addition to the usual CP T -conserving parameter :hi
10KS,L = q(1 + S,L ) K 0 ± (1 − S,L ) K
2 (1 + |S,L |2 )hS,L =−i= (M12 ) − 12 = (Γ12 ) ∓ 12 M11 − M22 − 2i (Γ11 − Γ22 )imL − mS + i(ΓS − ΓL )/2≡  ± δ.

(66.1)

L −mS )
Using the phase convention =(Γ12 ) = 0, we determine the phase of  to be ϕSW ≡ arctan 2(mΓS −ΓL .

0Imposing unitarity to an arbitrary combination of K 0 and K wave functions, we obtain the BellSteinberger relation [5] connecting CP and CP T violation in the mass matrix to CP and CP T
violation in the decay; in fact, neglecting O() corrections to the coefficient of the CP T -violating
parameter, δ, we can write [4] ΓS + ΓL

ΓS − ΓL+ i tan φSW ][<()− i=(δ)] =1 + ||2X
1AL (f )A∗S (f ),ΓS − ΓL f(66.2)

where AL,S (f ) ≡ A(KL,S → f ). We stress that this relation is phase-convention-independent. The
advantage of the neutral kaon system is that only a few decay modes give significant contributions
to the r.h.s. in Eq. (66.2); in fact, defining for the hadronic modes
1hAL (i)A∗S (i)i = ηi B(KS → i),
ΓSi = π 0 π 0 , π + π − (γ), 3π 0 , π 0 π + π − (γ),αi ≡(66.3)

the recent data from CPLEAR, KLOE, KTeV, and NA48 have led to the following determinations(the analysis described in Ref. [4] has been updated by using the recent measurements of KL
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:13am

266. CPT Invariance Tests in Neutral Kaon Decay

branching ratios from KTeV [6, 7], NA48 [8, 9], the results described in the CP violation in KL
decays minireview, and the KLOE result [10])απ+ π− = ((1.121 ± 0.010) + i(1.061 ± 0.010)) × 10−3 ,απ0 π0 = ((0.493 ± 0.005) + i(0.471 ± 0.005)) × 10−3 ,απ+ π− π0 = ((0 ± 2) + i(0 ± 2)) × 10−6 ,|απ0 π0 π0 | < 1.5 × 10−6 at 95% CL .

(66.4)The semileptonic contribution to the right-handed side of Eq. (66.2) requires the determination ofseveral observables: we define [2, 3]
A(K 0 → π − l+ ν) = A0 (1 − y) ,A(K 0 → π + l− ν) = A∗0 (1 + y ∗ )(x+ − x− )∗ ,0A(K → π + l− ν) = A∗0 (1 + y ∗ ) ,0A(K → π − l+ ν) = A0 (1 − y)(x+ + x− ) ,(66.5)

where x+ (x− ) describes the violation of the ∆S = ∆Q rule in CP T -conserving (violating) decay
amplitudes, and y parameterizes CP T violation for ∆S = ∆Q transitions. Taking advantage of0
their tagged K 0 (K ) beams, CPLEAR has measured =(x+ ), <(x− ), =(δ), and <(δ) [11]. These determinations have been improved in Ref. [4] by including the information AS −AL = 4[<(δ)+<(x− )](valid at first order in the small parameters), where AL,S are the KL and KS semileptonic charge
asymmetries, respectively, from the PDG [12] and the new KLOE semileptonic measurement [13].
Here we are also including the T -violating asymmetry measurement from CPLEAR [14] with afiner binning than appearing in the published article.
Table 66.1:
Values, errors, and correlation coefficients for<(δ), =(δ), <(x− ), =(x+ ), and AS + AL obtained from a combined fit,including KLOE [4, 13] and CPLEAR [14].
value
<(δ)
(4.3 ± 2.7) × 10−4=(δ)(−0.9 ± 0.6) × 10−2<(x− )(−0.22 ± 0.10) × 10−2=(x+ )(0.06 ± 0.19) × 10−2
AS + AL (−0.23 ± 0.38) × 10−2

Correlations coefficients
1−0.40 1−0.14 −0.30 1−0.12 −0.02 0.34 1−0.12 −0.29 0.94 0.18 1

The value AS + AL in Table 66.1 can be directely included in the semileptonic contributions tothe Bell Steinberger relations in Eq. (66.2)
hAL (π`ν)A∗S (π`ν)iXπ`ν= 2Γ (KL → π`ν)(<() − <(y) − i(=(x+ ) + =(δ)))= 2Γ (KL → π`ν)((AS + AL )/4 − i(=(x+ ) + =(δ))) .

(66.6)

Definingαπ`ν ≡1 XτK
hAL (π`ν)A∗S (π`ν)i + 2i S B(KL → π`ν)=(δ) ,ΓS π`ντKL
31st May, 2024

(66.7)

3

66. CPT Invariance Tests in Neutral Kaon Decay

we find:απ`ν = ((−0.1 ± 0.2) + i(−0.1 ± 0.5)) × 10−5 .

(66.8)

Figure 66.1: Top: allowed region at 68% and 95% C.L. in the <(), =(δ) plane. Bottom: allowed
region at 68% and 95% C.L. in the ∆M, ∆Γ plane.
The analysis of semileptonic decay asymmetries implicitly assumes Lepton Flavour Universality(LFU) of any effect violating CPT or ∆S = ∆Q, through the use of a single set of x, y parameters,
31st May, 2024

466. CPT Invariance Tests in Neutral Kaon Decay
Table 66.2: Summary of results: values, errors, and correlation coefficients for <(), =(δ), <(δ), and <(x− ).

<()=(δ)<(δ)<(x− )

value(161.2 ± 0.5) × 10−5(−0.3 ± 1.4) × 10−5(2.6 ± 2.5) × 10−4(−2.7 ± 1.0) × 10−3

Correlations coefficients+1+0.08 1+0.00 −0.05 1+0.05 0.13 −0.30 1

consistently with the availability of experimental information on AS for the electron mode only. Theexplicit LFU assumption in the input BR measurements has been lifted exploiting the measurementof BR(KS → πµν) Ref. [15], with no effect on the numerical results.
Inserting the values of the α parameters into Eq. (66.2), we find<() = (161.2 ± 0.5) × 10−5 ,=(δ) = (−0.3 ± 1.4) × 10−5 .

(66.9)

The complete information on Eq. (66.9) is given in Table 66.2.
Now the agreement with CP T conservation, =(δ) = <(δ) = <(x− ) = 0, is at 18% C.L.
The allowed region in the <() − =(δ) plane at 68% CL and 95% C.L. is shown in the top panelof Fig. 66.1.
The process giving the largest contribution to the size of the allowed region is KL → π + π − ,through the uncertainty on φ+− .
0The limits on =(δ) and <(δ) can be used to constrain the K 0 − K mass and width differenceδ=i(mK 0 − mK 0 ) + 21 (ΓK 0 − ΓK 0 )ΓS − ΓLcos φSW eiφSW [1 + O()] .

The allowed region in the ∆M = (mK 0 −mK 0 ), ∆Γ = (ΓK 0 −ΓK 0 ) plane is shown in the bottom
panel of Fig. 66.1. As a result, we improve on the previous limits (see for instance, P. Bloch inRef. [12]) and in the limit ΓK 0 − ΓK 0 = 0 we obtain−4.0 × 10−19 GeV < mK 0 − mK 0 < 4.0 × 10−19 GeV

at 95 % C.L .

References[1] See the “CP Violation in Meson Decays,” in this Review.
[2] L. Maiani, “CP And CP T Violation in Neutral Kaon Decays,” L. Maiani, G. Pancheri, and
N. Paver, The Second DAΦNE Physics Handbook, Vol. 1,2.
[3] G. D’Ambrosio, G. Isidori and A. Pugliese, in “2nd DAΦNE Physics Handbook:63-96,”
63–96 (1994), [hep-ph/9411389], URL http://preprints.cern.ch/cgi-bin/setlink?base=preprint&categ=cern&id=th-7504-94.
[4] G. D’Ambrosio and G. Isidori (KLOE), JHEP 12, 011 (2006), [hep-ex/0610034].
[5] J. S. Bell and J. Steinberger, in “Wolfenstein, L. (eq.): CP violation, 42-57. (In Oxford International Symposium Conference on Elementary Particles) ,” 195–208, 221–222 (1966), (SeeBook Index).
[6] T. Alexopoulos et al. (KTeV), Phys. Rev. D70, 092006 (2004), [hep-ex/0406002].
[7] E. Abouzaid et al. (KTeV), Phys. Rev. D83, 092001 (2011), [arXiv:1011.0127].
31st May, 2024

566. CPT Invariance Tests in Neutral Kaon Decay[8] A. Lai et al. (NA48), Phys. Lett. B645, 26 (2007), [hep-ex/0611052]; A. Lai et al. (NA48),
Phys. Lett. B602, 41 (2004), [hep-ex/0410059].
[9] We thanks G. Isidori and M. Palutan for their contribution to the original analysis [4] performed
with KLOE data.

[10] D. Babusci et al. (KLOE), Phys. Lett. B723, 54 (2013), [arXiv:1301.7623].
[11] A. Angelopoulos et al. (CPLEAR), Phys. Lett. B444, 52 (1998).
[12] W. M. Yao et al. (Particle Data Group), J. Phys. G33, 1 (2006).
[13] A. Anastasi et al. (KLOE-2), JHEP 09, 021 (2018), [arXiv:1806.08654].
[14] P. Bloch, M. Fidecaro, private communication of the data in a finer binning format; A. Angelopoulos et al. (CPLEAR), Phys. Lett. B444, 43 (1998).
[15] D. Babusci et al. (KLOE-2), Phys. Lett. B 804, 135378 (2020), [arXiv:1912.05990].

31st May, 2024