68. CP Violation in KL0 Decays

1

68. CP Violation in KL0 Decays
Revised August 2023 by C.-J. Lin (LBNL).
The symmetries C (particle-antiparticle interchange) and P (space inversion) hold for strong
and electromagnetic interactions. After the discovery of large C and P violation in the weakinteractions, it appeared that the product CP was a good symmetry. In 1964 CP violation wasobserved in K 0 decays at a level given by the parameter  ≈ 2.3 × 10−3 .
A unified treatment of CP violation in K, D, B, and Bs mesons is given in “CP Violationin Meson Decays” by D. Kirkby and Y. Nir in this Review. A more detailed review including athorough discussion of the experimental techniques used to determine CP violation parameters isgiven in a book by K. Kleinknecht [1]. Here we give a concise summary of the formalism needed
to define the parameters of CP violation in KL decays, and a description of our fits for the bestvalues of these parameters.

68.1 Formalism for CP violation in Kaon decay
CP violation has been observed in the semi-leptonic decays KL0 → π ∓ `± ν, and in the nonleptonic decay KL0 → 2π. The experimental numbers that have been measured are
AL =Γ (KL0 → π − `+ ν) − Γ (KL0 → π + `− ν)
Γ (KL0 → π − `+ ν) + Γ (KL0 → π + `− ν)(68.1a)η+− = A(KL0 → π + π − )/A(KS0 → π + π − )= |η+− | eiφ+−(68.1b)η00 = A(KL0 → π 0 π 0 )/A(KS0 → π 0 π 0 )= |η00 | eiφ00 .

(68.1c)0

CP violation can occur either in the K 0 – K mixing or in the decay amplitudes. Assuming CP T
invariance, the mass eigenstates of the K 0 –K 0 system can be written|KS i = p|K 0 i + q|K 0 i ,|KL i = p|K 0 i − q|K 0 i .

(68.2)

If CP invariance held, we would have q = p so that KS would be CP -even and KL CP -odd. (Wedefine |K 0 i as CP |K 0 i). CP violation in K 0 –K 0 mixing is then given by the parameter e wherep(1 + e)=.
q(1 − e)(68.3)

CP violation can also occur in the decay amplitudes
A K 0 → ππ(I) = AI eiδI ,

A K 0 → ππ(I) = A∗I eiδI ,(68.4)

where I is the isospin of ππ, δI is the final-state phase shift, and AI would be real if CP invarianceheld. The CP -violating observables are usually expressed in terms of  and 0 defined by
η+− =  + 0 ,

η00 =  − 20 .

(68.5a)One can then show [2]√ = e + i Im A0 /Re A0 ,02 = iei(δ2 −δ0 )(68.5b)(Re A2 /Re A0 ) Im A2 /Re A2 − Im A0 /Re A0 ,(68.5c)2(68.5d)AL = 2Re /(1 + || ) ≈ 2Re  .
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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68. CP Violation in KL0 Decays

In Eq. (68.5a), small corrections [3] of order 0 × Re (A2 /A0 ) are neglected, and Eq. (68.5d) assumesthe ∆S = ∆Q rule.
The quantities Im A0 , Im A2 , and Im e depend on the choice of phase convention, since one can
0change the phases of K 0 and K by a transformation of the strange quark state |si → |si eiα ; ofcourse, observables are unchanged. It is possible by a choice of phase convention to set Im A0 or
Im A2 or Im e to zero, but none of these is zero with the usual phase conventions in the Standard
Model. The choice Im A0 = 0 is called the Wu-Yang phase convention [4], in which case  = e. The
value of 0 is independent of phase convention, and a nonzero value demonstrates CP violation in
the decay amplitudes, referred to as direct CP violation. The possibility that direct CP violation
is essentially zero, and that CP violation occurs only in the mixing matrix, was referred to as thesuperweak theory [5].
By applying CP T invariance and unitarity the phase of  is given approximately byφ ≈ tan−12(mKL − mKS )≈ 43.52 ± 0.05◦ ,ΓKS − ΓKL

(68.6a)while Eq. (68.5c) gives the phase of 0 to beφ0 =δ2 − δ0 +π
≈ 42.3 ± 1.5◦ ,2(68.6b)where the numerical value is based on an analysis of π–π scattering using chiral perturbation
theory [6]. The approximation in Eq. (68.6a) depends on the assumption that direct CP violationis very small in all K 0 decays. This is expected to be good to a few tenths of a degree, as
indicated by the small value of 0 and of η+−0 and η000 , the CP -violation parameters in the decays
KS → π + π − π 0 [7], and KS → π 0 π 0 π 0 [8]. The relation in Eq. (68.6a) is exact in the superweak
theory, so this is sometimes called the superweak-phase φSW . An important point for the analysis
is that cos(φ0 –φ ) ' 1. The consequence is that only two real quantities need be measured, themagnitude of  and the value of (0 /), including its sign. The measured quantity |η00 /η+− |2 is very
close to unity so that we can write|η00 /η+− |2 ≈ 1 − 6Re (0 /) ≈ 1 − 60 / ,1
Re(0 /) ≈ (1 − |η00 /η+− |) .
3

(68.7a)(68.7b)From the experimental measurements in this edition of the Review, and the fits discussed in thenext section, one finds|| = (2.228 ± 0.011) × 10−3 ,(68.8a)◦φ = (43.5 ± 0.5) ,00(68.8b)−3

Re( /) ≈  / = (1.66 ± 0.23) × 10◦,

(68.8c)φ+− = (43.4 ± 0.5) ,(68.8d)◦(68.8e)φ00 –φ+− = (0.34 ± 0.32) ,AL = (3.32 ± 0.06) × 10−3 .

(68.8f)Direct CP violation, as indicated by 0 /, is expected in the Standard Model. However, the numerical value cannot be reliably predicted because of theoretical uncertainties [9]. The value of AL
agrees with Eq. (68.5d). The values of φ+− and φ00 − φ+− are used to set limits on CP T violation[see “Tests of Conservation Laws"].
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368. CP Violation in KL0 Decays

68.2 Fits for KL0 CP -violation parameters

In recent years, KL0 CP -violation experiments have improved our knowledge of CP -violation
parameters,
and their consistency with the expectations of CP T invariance and unitarity. To determine thebest values of the CP -violation parameters in KL0 → π + π − and π 0 π 0 decay, we make two typesof fits, one for the phases φ+− and φ00 jointly with ∆m and τS , and the other for the amplitudes|η+− | and |η00 | jointly with the KL0 → ππ branching fractions.
68.2.1 Fits to φ+− , φ00 , ∆φ, ∆m, and τS data
These are joint fits to the data on φ+− , φ00 , the phase difference ∆φ = φ00 – φ+− , the KL0 – KS0
mass difference ∆m, and the KS0 mean life τS , including the effects of correlations.
Measurements of φ+− and φ00 are highly correlated with ∆m and τS . Some measurements of
τS are correlated with ∆m. The correlations are given in the footnotes of the φ+− and φ00 sectionsof the KL0 Listings, and the τS section of the KS0 Listings.
In most cases, the correlations are quoted as 100%, with the value and error of φ+− or φ00 given
at a fixed value of ∆m and τS , with additional terms specifying the dependence of the value on∆m and τS . These cases lead to diagonal bands in Figs. 68.1 and 68.2. The KTeV experiment [10]quotes its results as values of ∆m, τS , φ , Re(0 /), and Im(0 /) with correlations, leading to theellipses labeled “b.” The correlations for the KTeV measurements are given in the Im(0 /) sectionof the KL0 Listings. For small |0 /|, φ+− ≈ φ + Im(0 /).
Table 68.1: References, Document ID’s, and sources corresponding to
the letter labels in the figures. The data are given in the φ+− and ∆msections of the KL Listings, and the τS section of the KS Listings.
Label SourceaThis Reviewb
FNAL KTeVc
CERN
CPLEARd
FNAL E773
e
FNAL E731
f
CERN
g
CERN NA31h
CERN NA48i
CERN NA31j
This Review

PDG Document ID
OUR FITABOUZAID 11
APOSTOLAKIS 99C
SCHWINGENHEUER 95
GIBBONS 93,93C
GEWENIGER 74B,74C
CAROSI 90
LAI 02CBERTANZA 97
SUPERWEAK 16

Ref.
[10]
[11]
[12]
[13], [14][15], [16]
[17]
[18]
[19]

The data on τS , ∆m, and φ+− shown in Figs. 68.1 and 68.2 are combined with data on φ00
and φ00 – φ+− in two fits, one without assuming CP T , and the other with this assumption. The
results without assuming CP T are shown as ellipses labeled “a.” These ellipses are seen to be ingood agreement with the superweak phase
φSW = tan−1



2∆m
∆Γ



= tan−12∆mτS τL~(τL – τS )

.

(68.9)In Figs. 68.1 and 68.2, φSW is shown as narrow bands labeled “j.”Table 68.2 column 2, “Fit w/o CP T ,” gives the resulting fitted parameters, while Table 68.3
gives the correlation matrix for this fit. The white ellipses labeled “a” in Figure 68.1 and 68.2 arethe χ2 = 1 contours for this fit.
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68. CP Violation in KL0 Decays

4
48

c

g
e

f

d

φ+ (degrees)

46

c
d

44

b

j

a
e

42

f

40

38
5150

5200

5250

5300
6

5350

5400

-1

mKL - mKs (10 hs )Figure 68.1: φ+− vs ∆m for experiments which do not assume CP T invariance. ∆m measurements appear as vertical bands spanning ∆m ± 1σ, cut near the top and bottom to aid theeye. Most φ+− measurements appear as diagonal bands spanning φ+− ± σφ . Data are labeled byletters: “b”–FNAL KTeV, “c”–CERN CPLEAR, “d”–FNAL E773, “e”–FNAL E731, “f”–CERN,“g”–CERN NA31, and are cited in Table 68.1. The narrow band “j” shows φSW . The ellipse “a”
shows the χ2 = 1 contour of the fit result.
For experiments which have dependencies on unseen fit parameters, that is, parameters otherthan those shown on the x or y axis of the figure, their band positions are evaluated using the fit
results and their band widths include the fitted uncertainty in the unseen parameters. This is also
true for the φSW bands.
If CP T invariance and unitarity are assumed, then by Eq. (68.6a), the phase of  is constrained
to be approximately equal toφSW = (43.50258 ± 0.00021)◦ + 54.1(∆m − 0.5289)◦ + 32.0(τS − 0.89564)(68.10)where we have linearized the ∆m and τS dependence of Eq. (68.9). The error ±0.00021 is due tothe uncertainty in τL . Here ∆m has units 1010 ~ s−1 and τS has units 10−10 s.
If in addition we use the observation that Re(0 /)  1 and cos(φ0 − φ ) ' 1, as well as thenumerical value of φ0 given in Eq. (68.6b), then Eqs. 68.5a, which are sketched in Fig 68.3, lead
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68. CP Violation in KL0 Decays

5
48

i
h
g

φ+ (degrees)

46

b
a

44

f
j
c

42

d
40

e
38
0.888

0.892

0.896
-10

0.9

0.904

τKs (10 s)
Figure 68.2: φ+− vs τS . τS measurements appear as vertical bands spanning τS ± 1σ, some ofwhich are cut near the top and bottom to aid the eye. Most φ+− measurements appear as diagonal or horizontal bands spanning φ+− ± σφ . Data are labeled by letters: “b”–FNAL KTeV,“c”–CERN CPLEAR, “d”–FNAL E773, “e”–FNAL E731, “f”–CERN, “g”–CERN NA31, “h”–
CERN NA48, “i”–CERN NA31, and are cited in Table 68.1. The narrow band “j” shows φSW .
The ellipse “a” shows the fit result’s χ2 = 1 contour.
to the constraint
 0



 0

≈ −3 Retan(φ0 – φ )
≈ 0.006◦ ± 0.008◦ ,φ00 – φ+− ≈ −3 Im(68.11)so that φ+− ≈ φ00 ≈ φ ≈ φSW .
In the fit assuming CP T , we constrain φ = φSW using the linear expression in Eq. (68.10),
and constrain φ00 − φ+− using Eq. (68.11). These constraints are inserted into the Listings with
the Document ID of SUPERWEAK 16. Some additional data for which the authors assumed CP T
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668. CP Violation in KL0 Decays
Table 68.2: Fit results for φ+− , ∆m, τS , φ00 , ∆φ = φ00 − φ+− , and φ
without and with the CP T assumption
Quantity(units)φ+− (◦ )∆m(1010 ~ s−1 )τS (10−10 s)
φ00 (◦ )∆φ(◦ )φ (◦ )
χ2
# Deg. Free.

Fit w/o CP T
43.4 ± 0.5 (S=1.2)
0.5289 ± 0.0010
0.89564 ± 0.00033
43.7 ± 0.6 (S=1.2)
0.34 ± 0.32
43.5 ± 0.5 (S=1.3)
16.4
14

Fit w/ CP T
43.51 ± 0.05 (S=1.2)
0.5293 ± 0.0009 (S=1.3)
0.8954 ± 0.0004 (S=1.1)
43.52 ± 0.05 (S=1.3)
0.006 ± 0.014 (S=1.7)
43.52 ± 0.05 (S=1.2)
20.0
16

are added to this fit or substitute for other less precise data for which the authors did not makethis assumption. See the Listings for details.

Figure 68.3: Sketch of Eqs. 68.5a. Not to scale
The results of this fit are shown in Table 68.2, column 3, “Fit w/CP T ,” and the correlation
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68. CP Violation in KL0 Decays

7matrix is shown in Table 68.4. The ∆m precision is improved by the CP T assumption.
Table 68.3: Correlation matrix for the results of the fit without the CP T
assumption

φ+−
∆m
τS
φ00
∆φ
φ

φ+−
1.000
0.596
-0.4880.827-0.040
0.976

∆m
0.596
1.000
-0.5720.487-0.0350.580

τS-0.488-0.5721.000-0.423-0.014-0.484

φ00
0.827
0.487
-0.423
1.000
0.529
0.929

∆φ
-0.040-0.035-0.014
0.529
1.000
0.178

φ
0.976
0.580
-0.484
0.929
0.178
1.000

Table 68.4: Correlation matrix for the results of the fit with the CP T
assumption

φ+−
∆m
τS
φ00
∆φ
φ

φ+−
1.000
0.972
-0.311
0.957-0.105
0.995

∆m
0.972
1.000
-0.5090.958-0.0070.977

τS-0.311-0.5091.000-0.3060.004-0.312

φ00
0.957
0.958
-0.306
1.000
0.189
0.981

∆φ
-0.105-0.007
0.004
0.189
1.000
-0.006

φ
0.995
0.977
-0.3120.981-0.0061.000

68.2.2 Fits for 0 /, |η+− |, |η00 |, and B(KL → ππ)We list measurements of |η+− |, |η00 |, |η00 /η+− |, and 0 /. Independent information on |η+− |and |η00 | can be obtained from measurements of the KL0 and KS0 lifetimes (τL , τS ), and branchingratios (B) to ππ, using the relations"

B(KL0 → π + π − )τS|η+− | =τL
B(KS0 → π + π − )"B(KL0 → π 0 π 0 )τS|η00 | =0
τL
B(KS → π 0 π 0 )

#1/2

,

(68.12a)#1/2.

(68.12b)

For historical reasons, the branching ratio fits and the CP -violation fits are done separately, but wewant to include the influence of |η+− |, |η00 |, |η00 /η+− |, and 0 / measurements on B(KL0 → π + π − )and B(KL0 → π 0 π 0 ) and vice versa. We approximate a global fit to all of these measurements by first
performing two independent fits: 1) BRFIT, a fit to the KL0 branching ratios, rates, and mean life,and 2) ETAFIT, a fit to the |η+− |, |η00 |, |η+− /η00 |, and 0 / measurements. The results from fit 1,along with the KS0 values from this edition, are used to compute values of |η+− | and |η00 |, which
are included as measurements in the |η00 | and |η+− | sections with a document ID of BRFIT 16.
Thus, the fit values of |η+− | and |η00 | given in this edition include both the direct measurements
and the results from the branching ratio fit.
The process is reversed in order to include the direct | η | measurements in the branching ratio
fit. The results from fit 2 above (before including BRFIT 16 values) are used along with the KL0
and KS0 mean lives and the KS0 → ππ branching fractions to compute the KL0 branching ratio
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868. CP Violation in KL0 Decays

Γ (KL0 → π 0 π 0 )/Γ (KL0 → π + π − ). This branching ratio value is included as a measurement in the
branching ratio section with a document ID of ETAFIT 16. Thus, the KL0 branching ratio fit values
in this edition include the results of the direct measurement of |η00 /η+− | and 0 /. Most individual
measurements of |η+− | and |η00 | enter our fits directly via the corresponding measurements ofΓ (KL0 → π + π − )/Γ (total) and Γ (KL0 → π 0 π 0 )/Γ (total), and those that do not have too large errors
to have any influence on the fitted values of these branching ratios. A more detailed discussion ofthese fits is given in the 1990 edition of this Review [20].
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