Fortschritte der Physik / Progress of Physics: Band 14, Heft 11 1966 [Reprint 2021 ed.] 9783112500408, 9783112500392

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R PHYSIKALISCHEN GESELLSCHAFT IN D E R DEUTSCHEN DEMOKRATISCHEN R E P U B L I K VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

B.YND 14 • I I E F T 11 • 1966

A K A D E M I E

•

VERLAG

•

B E R L I N

1 -N 11 A 1. I

C. MAHN: K" on.l < l>

ij'J")

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Fortschritte der Physik 14, 6 9 5 - 7 3 9 (1966)

K° and CP1) G . MABX

Institute

for Theoretical

Physics

of the Roland Eotvos

University,

Budapest

Abstract. The phenomenological theory of the neutral K-mesons (kaons) is summarized. The consequences of the C, CP, CPT symmetries are discussed in details. Finally, the different suggestions are reviewed, concerning the explanation of the CP violating KJ, -> 7t+7t_ decay.

Content § 1. Dicsovery of the C P violation § 2. The eigenstates of the neutral kaon § 3. The CP symmetry § 4. The decay K->2TT § 5. The decay K § 6. The decay K TT"i+v § 7. The decay K tc+i'v § 8. The T symmetry § 9. The CPT symmetry § 10. The behaviour of kaons in media §11. Superweak C P violation? § 12. Model for a superweak violation of CP and CPT § 13. Model for a superweak violation of CP and T . § 14. Unitary symmetry properties of the C P violation § 15. Weak CP violation? § 16. Electromagnetic C violation? § 17. Strong C violation? Appendix: The Lee-Oehme-Yang theorem

695 701 708 709 712 712 713 715 717 720 725 726 728 729 730 731 733 737

§ 1. Discovery of the C P Violation U n t i l 1956 it w a s generally accepted t h a t t h e total H a m i l t o n i a n H ^ of N a t u r e is invariant under b o t h space reflection (the operator P exchanging left b y right) a n d charge conjugation (the operator C exchanging particles b y antiparticles). Expressed i n a different w a y : I t w a s assumed that t h e righthanded a n d l e f t h a n d e d configurations, as well as t h e particles a n d antiparticles are completely equivalent i n Nature. Lecture at the Hungarian Winter School on Weak Interactions, Balatonkenese, January 1966, organized by the Hungarian Physical Society 51

Zeitschrift „Fortschritte der Physik", Heft 11

696

G. MARX

The first indication contradicting the P symmetry was the decay of the K+ meson into 7t+7r° and 7t+ rc+ n~ mesons. The spin of the K-meson is 0, the spin of the nmesons is also 0, their parity equals — 1. Because of the conservation of the angular momentum in the 7t+ state the orbital angular momentum of the two mesons is I = 0, thus their parity equals P 2lt = ( — l) 2+i = + 1 . I n the n + n~ state, on the other hand, the relative orbital angular momentum of the two 7r+ mesons (I) and the orbital angular momentum of the TI~ meson, refered to the center of mass of the 7r+7t+ system (1') are equal, and give a vanishing resultant (Fig. 1). (The uniformly

Fig. 1. Angular orbital momenta in the 3 —finalstate

populated Dalitz plot indicates that I = I' = 0 with an overwhelming probability.) Consequently the parity of the final state 7t+7t+7r~ is equal to P 3 „ = (—1)3+/+;.' = = — 1. Decays of the K + into states of parity + 1 and — 1 are equally observed; on that ground T. D. L E E and C. N . Y A N G recognized that weak interactions do violate the conservation of the parity P [J]. The more detailed investigation of the leptonic weak interaction proved that weak interactions also violate the C symmetry. I n 1 9 5 7 , however, A B D U S SALAM, L E V L A N D A U , L E E and Y A N G showed that the product C P is conserved even in the weak interactions. After the discovery of the V-A coupling this became generally accepted: (CFy^Hm{CP)

=

Hw.

(1)

(H„ is the Hamiltonian of the weak interactions.) Thus the following information had been coolected about the symmetry properties of weak interactions : C, P are not conserved;

(2a)

CP, T and CPT are strictly conserved;

(2b)

j r = 0,±l;

(2 c)

M J | = l, 1/2;

(2d)

± 1 , ±1/2.

(2e)

(Here T stand for the operator of time reversal, / f o r the vector of the isospin, I3 for its third component, Y for the hypercharge.) I t was evident to interpret the C P conservation by saying that the particles and antiparticles should be considered completely equivalent, but in order to arrive at a physical state, after the particle-antiparticle exchange one has to perform simultaneously a left-right reflection, too. Talking about the equivalence of particles and antiparticles it used to be customary in elementary courses to mention the equality of respective masses, life times and

K° and CP

697

branching ratios. I t has been proved, hovewer, by T. D. L E E , T. O E H M E and C . N . Y A N G [2] t h a t the equality of these data is already a consequence of the CPT symmetry, a separate C symmetry is not needed. (See the Appendix.) According to the CPT theorem a local expression (or its volume integral, e.g. a Hamiltonian) is invariant under the combined CPT transformation, if it is constructed from causal operators (which commute or anticommute at points separeted by space-like intervals) in a Lorentz-invariant way [3]. Thus CPT symmetry is a much weaker condition than C or CP symmetry. The experimental fact t h a t the numerical properties of particles and antiparticles do agree up to a few decimals proves nothing more than t h a t an eventual CPT-violating interaction cannot be very strong, e.g. it cannot be comparable with the intensity of the electromagnetic interaction. A more detailed check of the C and CP symmetry is made difficult by t h a t particles and antiparticles differ in the sign of one of the strictly conserved charges, so t h a t as a consequence, a superselection separates them from each other [4], A particle and its antiparticle cannot build u p a coherently superposed state, their unforseen little differences cannot be recognized by making use of the very sensitive interference experiments. Table 1 Quantum numbers of kaons

Y

Q K+ K° K° K-

+ 1 0 0 -1

} - J Ì }

I 1/2 1/2

h + 1/2 -1/2 + 1/2 -1/2

The only fortunate exception is offered by the neutral kaon. I t is consequence of the isospin-scheme of G E L L - M A N N , t h a t the K-mesons form an isodoublet, the K-mesons form another isodoublet, the two doublets differing in the sign of the hypercharge Y (Table 1). This means that the neutral kaon at rest has two degrees of freedom: F | K ° > = + |K°>,

(3a)

Y | K°) = — | K°>. (3b) I t is of decisive importance, t h a t the hypercharge does not obey a strict conservation rule: it is conserved in strong and electromagnetic interactions but not in the weak ones (see (2 c)): [Hw, Y] ^ 0. (4) If we include weak interactions, the superselection between K° and K° dissolves, the two states become coherent. This exceptional situation makes the neutral kaons to uniquely useful test bodies in exploring the C, CP, CPT symmetries of Nature. Since {CP, 7 } = 0, we can fix the relativ phase of the states K° and K° by the convention |K°) = C P | K ° ) . 51*

(5) (6)

698

G. Maux

The kaons are created in the collision of hadrons, which are F-eigenstates, because of the F-conserving strong interactions. Consequently F-eigenstates, |K°) or |K°) are created. The lowest thereshold energy belongs to the reaction TV-

+ p+ - > A 0 + K °

(7)

(7 = 0 + 1 = 0 + 1), but making use of the F-conservation also a pure K°-beam can be produced, e.g. by K-+p+^n°+K° (8) ( 7 = - 1 + 1 = + 1 -1). The main decay channel of the neutral kaons is K - > 7ZK Table 2 Experimental properties of the kaons Mass K± K,?

M

m± s m m

Life time

= 493.78 ± 0 . 1 7 MeV 0 = 497.70 ± 0.30 MeV = 497.70 ± 0.30 MeV

8 ± = (1.229 ± 0.008) • IO- lso

T

S

T

L

—

=

(0.881 ± 0.010) • 10~ s (5.77 ± 0 . 5 9 ) - 10-"s

Branching ratios

Ki^v

TT 7T~ 71 7^71° 71° 7t°[Jl±V

7r°e±V T^Tt* e ± v

TC* (X* V 71*71° Y TT^T^Y

63.5 ± 0.7 % 21.6 ± 0.6 % 5.59 ± 0.011% 1.68 ± 0.06 % 3.17 ± 0.35 % 4.49 ± 0.25 % 0.0036 % 0.00076 % 0.022± 0.007% 0.01 ± 0.04 %

K°. -> 71+71" 7t°7t°

69.1 30.9

K% 7t°7t°7t» 7T+7T-7T0

23.2 ± 2 . 0 % 11.8 ± 0 . 5 % 26.6 ± 1 . 3 % 38.4 ± 1 . 4 % 0.150 ± 0 . 0 0 9 % 0.438 ± 0.084% 0.074 ± 0 . 0 1 6 %

7t+7t_ 7t°7r° YÏ

±2.2 ± 2.2

% %

with a probability larger than 10 s (Table 2). In view of the conservation of angular momentum, the relative angular momentum of the two 7r-mesons is still I = 0 so that in both of the final states tc+ 7T and TZ°-K° we have 10 -1

P2ji = ( - 1)W = + 1 , CiK = +1, consequently (CP)2„ = + 1 . The n-rz final state is, according to (9), a CP-eigenstate. When Eq. (1) is fulfilled by the weak interaction, responsible for the K decay, only that component, which is the CP-eigenstate with eigenvalue C P = + 1 , is allowed to decay into mz [5]. Because of Eq. (5) the operators F and C P do not possess common eigenfunctions, but the eigenfunctions of Y, introduced in (3) can be expanded according to the eigenfunctions of the Hermitian C P operator, |K°) = - L [|K°) + |K°)], /2

CP |KJ> = + |K°),

(10a)

|K») = T i - [ | K 0 ) - | K » > ] ,

CP |Kg) = — |K®),

(10b)

i y2

(9)

K° and CP

699

as follows (Fig. 5): |K°)==-1[|K°)+;|K°>],

F | K » ) = + IK»),

(11a)

|K°) = - L [|K°) - i IK")], V2

Y |K°) = —|K°).

(lib)

p

Either K°-s are produced by the reaction (7), or K°-s by (8), from both beam the K® component fades out quickly decaying into the final states TI+TZ~, 7I°TC°. According to the selection rule (2d) this final state is characterised by the isospin 7 = 0, i.e. \I = 0) =

+ | TT-tc*) + |

V6

,

(12)

so that we expect a branching ratio F(K®

7i+ 7c-):r(K® —> 7r°7r°) = 2

(13)

in good agreement with experience (Table 2). The |Kj) component left with 5 0 % probability after the decays KJ —> TZ TZ can decay only into 37t or leptonic final states which are much less convenient because of the phase space conditions. The 37t final state can have an isospin 7 = 0 or 7 = 1 , by (2d). According to the experiments T(K+

71+71+71") = 7"(K° -> 7r07t°TC°),

so it lies at hand to describe both the K+ 3 TC and the K® -> 3 7t decays with 7 = 1 in the final state. The uniform population of the Dalitz-plot indicates I = V = 0 (Fig. 1), consequently P,» = ( - 1 ) 3+!+r = - 1 .

C3n = ( - If = + 1 ,

so finally (CP)^ = - 1 .

The K!J -»• 3 n transition is absolutely C P allowed, the KJ solutely C P forbidden.

n -p+-~A°K°

K~p*-~-

(14)

3 it0 decay is ab-

nK°

Fig. 2. The kaon producing nuclear reactions act like "optical polarisators": they produce kaon beams in Y eigenstates cays are dominant, Fig. 4.) Historically, the long-lived component was discovered just in accordance with the theoretical predictions [5]. After this success of the theory it is completely understa,ndable that the observed K® mesons had been identified with the predicted C P eigenstates K®, K® and that the decay forms

eooo tff—Zn:

Kj—3n

(CP= +1)

(CP= -1)

v

(y=

K

~~l'y

cy= -1)

+v

Fig. 3. The kaon decays as "optical analysators" select the CP and Y eigenstates (KJ, Kg, K°, beam and exhaust them selectively

•

from the kaon

of the neutral kaons were regarded as the most direct proof of the C P symmetry (1). In order to make this proof more accurate, V. L. FITCH and his co-workers in Princeton investigated if any 2n decays of the long-lived K£ component could

Fig. 4. Sketch on the occurence of K

K^

K°L K.2

\ / \ / jf

/\

decays

\

CP{\

Fig. 5. The degrees of freedom in the Hilbert space (the kaon subspace). The dashed lines represent the K J, eigensolutions. The dotted line shows the motion in time of the state vector of a K° meson

/

\

\jK? K

/ \\ \\ / •

\\

1

be observed at all. To the greatest surprise of the scientific world, in the spring of 1964 they discovered the K° —7T+ 7T~

(15a)

decay, admittedly with a very low branching ratio: r(K° L ->• 7T+7r-):r(K®) = (1.58 ± 0.12) • 10"3.

(15b)

The experience the same particle, the K® meson is able to decay into a C P eigenstate with eigenvalue C P = + 1 and also into an other one with eigenvalue C P = — 1, indicates in a straightforward manner that either C P changes during the transmission, or theK® energy eigenstate is not an eigenstate of OP. Both possibilities force us to recognise that the Hamiltonian, describing the kaon dynamics, cannot show a CP symmetry. One understands the commotion about this Result. I t

K° and CP

701

seems that the positive-negative charge symmetry, which played an important role since the dawn of electrodynamics, must finally be given up. The CP violation means also that i) either a T asymmetry is present, i.e. also the elementary microprocesses are irreversible (the principle of the detailed balance is no longer valid), ii) or CPT is violated, which means the violation of the principle of relativity or that of the principle of local causality, because of the CPT theorem. The most surprising in the whole story is that these asymmetries are so well hidden in Nature : they did not show themselves until 1964. The Fitch-effect, the result (15) concentrated the attention on the kaons a second time. In order to find the way to the clarification of the nature of C P violation, a new analysis of the creation, scattering and decay phenomena has become necessary, with more general assumptions than used up to now. I n this paper we attempt to review the investigations, performed in this field during the years 1964 and 1965. (The most important experimental results of the year 1966 have been taken into account in the proof.) § 2. The Eigenstates of the Neutral Kaon Let us split the complete Hamiltonian H t a t into two parts: Htot = H 0 -)- H

(16)

and let up suppose about H0 that [H0, CP] = 0 ,

(17)

[H 0 , Y] = 0.

(18)

Concerning the symmetry properties of H no assumption is made, but the matrix elements of H shall be considered as small compared to the eigenvalues of H 0 , thus the consequences of H will be taken into account by perturbation theory. The Hamiltonian H 0 may contain the kinetic and rest energies, the strong and electromagnetic interaction. The Hamiltonian H may contain the weak interaction and a so-far unknown term, which produces the C P violation. The time dependence of an arbitrarily normalized state vector is described in Schrodinger picture by the equation d - | i > + i # t o t | i > = 0-

(19)

Let 11) be expanded with respect to the eigenvectors of H0 : \t) = I where

n

c„(t)e->^

H0\n)=En\n).

\ n),

(20) (21)

Substituting the expression (20) into Eq. (19), taking Eqs. (16) and (21) into account, proceeding in the way known from perturbation theory the following system

702

G . MARX

of equations is obtained: * Jt CM( .

(24c)

Let us now assume that the state vector (23) consists of only a neutral kaon at the time t = 0, i.e. 6 r (0) = 0. Then, from Eq. (24 c) we get by integration t br(t) = ~if

o

-i~Z(r

L(r \H\K°)a+{t')

a_(i')]el «_(1, ]. (38 b)

|K£> = I Y2

The differential equations (24) are linear, so an arbitrary linear combination of the two eigensolutions gives also a solution. Consequently the time dependence of the state vector (23), describing the kaon, can be written as | = cs| K|)

+

C£

|

k

®

+

2" br(t) \ r) erW.

•R

(39)

706

G. MARX

Let us project this normalized state vector into the kaon subspace: |K(i)> = Pk \t) = cs I K ^ e - ^ e - " 2 ^ + cL | K D e r ^ e - V * ^

(40)

(Here P K means the projection operator of the kaon subspace.) This shows that any neutral kaon state appears as a superposition of two eigenstates; one of them, K° s , has a mass (41 a)

ms = m0 + A ms

and a mean life time rs; the other one,

has a mass (41 b)

mj, = m0 + dmL

and a mean life time TL. When the labeling of the eigenstates is such that RS


= |K°) = ^ 2 [JV31 (cos 0 + sin 0) \ K°) + iN?(c0S V then according to Eq. (40) this varies in time as J_

0 - sin 0 ) |K°>],

|K°( [ . ^ ( c o s 0 + sin 0) |K°> er^er*!*^ yf

+

+

cos 0 - sin 0) |Kl) e ~ im Lt e -tliirL].

(42)

Having been restricted to the kaon subspace, the truncated state vector (42) shrinks in time: (K°(i) | K° ( 71+71- decay, has failed. 3

707

K° and OP

t h e n according to Eq. (40) the state vector varies in time as | K° (i)) = - L2 e-'® [ I V (cos 0 - sin 0)

-

y

— iNl1

(cos 0 + s i n 0) | Kl)e-imL'e-'l2^].

(45)

The norm of this truncated state vector is decreasing in time: