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CP-invariance and the phase of K0L−K0S regeneration
Volume 63B, number 4
PHYSICS LETTERS
16 August 1976
C P - I N V A R I A N C E A N D T H E P H A S E O F K 0L-K 0s R E G E N E R A T I O N J. VOTRUBA Institute of Physics, CSA V, Prague, Czechoslovakia Received 11 June 1976 In the limit of the CP-invarianceof the neutral kaon complex in material medium a simple formula for the phase of the difference of forward scattering amplitudes of K°M and K.°M is proposed. The coherent transmission regeneration KL-K S 0 0 in matter has been studied experimentally [1,2] for different momenta p of primary K 0 mesons and for the hydrogen, deuterium, carbon, copper and lead regenerators. The results of these experiments are the modulus IAI and the phase ~p= arg A of the difference A = fKM(P) --f-~M (P) of the forward scattering amplitudes of K 0 and 1~0 on the corresponding scatterer (denoted here by subscript M). These experiments show that the value of ~0in the limits of experimental errors is independent of the kind of the scatterer and it has the constant value ~o~ -(37r/4). Both mentioned facts are explainable with the aid of the single pole Regge model which uses the 60, p-exchange [3]: ~o= -(1r/2)(1 +¢~(0)). Another method [4] based on the assumption of asymptotical behavior A(s) ~ sa(ln s)~ gives in the limit of s ~ ~ and for a < 1 a constant phase ~0(s) -+ cotan(ncq2). There are also optical model calculations and dispersion relation calculations. The survey of experimental and theoretical results can be found in the report by Kleinknecht [ 1]. In this paper the problem of the regeneration phase ~0is considered from the point of view of discrete symmetries. Since the existence of the symmetry operators such as CP, T or CPT is deduced from the alleged equivalence of certain reference systems, the presence of phenomena violating that equivalence implies, that the operators themselves cannot be exactly defined [5]. It will be therefore necessary to assume, in the main part of our considerations, that the CP, T and CPT are exact operations of symmetry in all interactions. The propagation of the neutral kaon beam in a material medium is (in the basis [K) = (1), IK) = (0)) described by the well known equation [6, 7] :
i~-dy
= ~
,
O)
where -
m (O
f_e_M ( P ) / "
(21
Equation (1) is clearly non-invariant under the simple CP-operation I K ) ~ IK), i.e. (~) -* Ol(~). This, however, is not surprising. Our basic assumption about the CPsymmetry of all fundamental interactions means namely, that in the case of our system (including the material medium which is not CP.invariant) the true CP-transformation must also include the conversion of the medium M into anti-medium M [8] :
(CP)M : K ~ ~:, M -~ ~ .
(3)
This true CP-operation on our system consist in the linear transformation o I followed by the substitution fKM ~ f K ~ ' fRM ~ f g ~ " Using the CPT-relations
fY(~(P) = fKM(P),
f~,M(P) = f K ~ ( p ) ,
(4)
we see that the operation (3) leaves the mass matrix (2) invafiant. Although the matrix (2) is non-hermitian it has two independent eigenstates IK1M) = (2(1 + [pl2)) -1/2
IKM)=(2(1+10i2))-1/2
(I
_ +p
(5) '
where rtN(fKM(P) - fKM(P)) P = m ((m 1 -- mz) -- (i/Z)(3'1 -- 3'2))"
(6)
Here the 71, ')'2 and m 1 - m 2 denote the total decay rates and the difference of the mean rest masses of the short- and long-lived kaons in the CP-invariance limit, 439
Volume 63B, number 4
PHYSICS LETTERS
respectively. Applying the operation (3) on the states (5) we find
(CP)MIK > =
(7)
We see that these states have the same symmetry properties as the states IK1) and IK2). In contradistinction to them the states IK~~) and IK M) propagate independently in matter analogously to the states [K S) and IK L) which propagate independently in vacuum. It is well known that the states IK s) and IK L) are not orthogonal if the CP-invariance of the mass matrix in vacuum is violated but orthogonal in the CP-invariance limit. About the orthogonality of the states IK1M) and [K M) decides the quantity Im p, since (KMIK M) = 2i Im p/(1 + 1012). The mass matrix (2) is actually CPinvariant, however. This circumstance leads us to the idea to introduce, analogously, the orthogonality of the states (5) i.e. Imp = 0,
(8)
as an additional assumption. We shall see that this assumption is justified by experimental data. Using the expressions (6) and (8) we obtain the following simple formula for the regeneration phase ~: Im(fKM(P) --fl~M(P)) _ 7'1 -- 3/2 t a n ~ = Re(fKM(P ) _ f R u ( p ) ) 2(m2_ml ) •
(9)
Before the comparison of this formula with experiment we must first discuss the realistic case with the CPviolating (super)weak interaction. One can expect that the CP-violation has practically no influence on the forward scattering amplitudes fKM and fl~Mwhich are essentially "strong interaction quantities". Moreover, the CP-violation can be considered as a small perturbation in the mass matrix and the quantities 7 1 , 7 2 and m 2 - m 1 remain unchanged in the first approximation [6]. Therefore, to great degree of accuracy, we can substitute (m 2 - m l ) ~ ( m L - m s ) , )'1 ~3'S, 7'2 ~ 7L. Inserting the experimental values [9] into the right hand side of eq. (9) we find ¢ = 46.197+0.007 °
440
16 August 1976
or -(133.803-+0.007°). The latter solution is in agreement with the available experimental results [ 1,2]. The most precise measurements of the phase ~ have been performed with the liquid hydrogen used as the scattering medium. Brandenburg et al. [1] give the value ~ = -133.9-+4.0 °, the latest result of Birulev et al. [2] is ¢ = - 1 3 2 + 5 ° and the world mean value for hydrogen is ¢ --- - 1 3 3 +3.2 °. Agreement of our formula (9) with the experimental data indicates, conversely, that the states IK f ) and IK M) are indeed orthogonal justifying our orthogonality assumption. Formula (9) represents the connection between the strong and the weak interaction quantities. Besides that, there is also the connection between the phase and the phase ~e = arctan 2(m 2 - ml)/('yt -'Y2) of the superweak interaction mixing coefficient e: tan % . tan ~ = 1.
References
[i] K. Kleinknecht, Fortsch. d. Phys. 21 (1973) 57; V.K. Birulev et al., Phys. Lett. 38B (1972) 452; [2] V.K. Birulev et al., Dubna preprint P1-9434 (1976), submitted to Yad. Fiz.; G.W. Brandenburg et al., Phys. Rev. D9 (1974) 1939; D. Freytag et al., Phys. Rev. Lett. 35 (1975) 412; K.F. Albrecht et al., Phys. Lett. 48 (1974) 257; Nucl. Phys. B93 (1975). [3] F.J. Gilman, Phys. Rev. 171 (1968) 1453; V.I. Lisin et al., Nucl. Phys. B40 (1972) 298. [4] Nguyen Van Hieu, Theor. Math. Phys. 8 (1971) 354 (Soy. Phys.). [5] T.D. Lee, G.C. Wick, Phys. Rev. 148 (1966) 1385. [6] C.P. Enz, R.R. Lewis, Helv. Phys. Acta 38 (1965) 860. [7] See also J. Steinberger, CERN 70-1, 9 January 1970; G. Charpak, M. Gourdin, CERN 67-18, 11 July 1967; T.D. Lee, C.S. Wu, Ann. Rev. Nuel. Sci. 16 (1966) 511. [8] N. Cabibbo in "Symmetries in Elementary Particle Physics" 1964, International School of Physics Ettore Majorana, p. 272. Editor: A. Zichichi, Acad. Press, New York and London 1965. [9] Particle Data Group, Phys. Lett. 50B (1974) 2.
PHYSICS LETTERS
16 August 1976
C P - I N V A R I A N C E A N D T H E P H A S E O F K 0L-K 0s R E G E N E R A T I O N J. VOTRUBA Institute of Physics, CSA V, Prague, Czechoslovakia Received 11 June 1976 In the limit of the CP-invarianceof the neutral kaon complex in material medium a simple formula for the phase of the difference of forward scattering amplitudes of K°M and K.°M is proposed. The coherent transmission regeneration KL-K S 0 0 in matter has been studied experimentally [1,2] for different momenta p of primary K 0 mesons and for the hydrogen, deuterium, carbon, copper and lead regenerators. The results of these experiments are the modulus IAI and the phase ~p= arg A of the difference A = fKM(P) --f-~M (P) of the forward scattering amplitudes of K 0 and 1~0 on the corresponding scatterer (denoted here by subscript M). These experiments show that the value of ~0in the limits of experimental errors is independent of the kind of the scatterer and it has the constant value ~o~ -(37r/4). Both mentioned facts are explainable with the aid of the single pole Regge model which uses the 60, p-exchange [3]: ~o= -(1r/2)(1 +¢~(0)). Another method [4] based on the assumption of asymptotical behavior A(s) ~ sa(ln s)~ gives in the limit of s ~ ~ and for a < 1 a constant phase ~0(s) -+ cotan(ncq2). There are also optical model calculations and dispersion relation calculations. The survey of experimental and theoretical results can be found in the report by Kleinknecht [ 1]. In this paper the problem of the regeneration phase ~0is considered from the point of view of discrete symmetries. Since the existence of the symmetry operators such as CP, T or CPT is deduced from the alleged equivalence of certain reference systems, the presence of phenomena violating that equivalence implies, that the operators themselves cannot be exactly defined [5]. It will be therefore necessary to assume, in the main part of our considerations, that the CP, T and CPT are exact operations of symmetry in all interactions. The propagation of the neutral kaon beam in a material medium is (in the basis [K) = (1), IK) = (0)) described by the well known equation [6, 7] :
i~-dy
= ~
,
O)
where -
m (O
f_e_M ( P ) / "
(21
Equation (1) is clearly non-invariant under the simple CP-operation I K ) ~ IK), i.e. (~) -* Ol(~). This, however, is not surprising. Our basic assumption about the CPsymmetry of all fundamental interactions means namely, that in the case of our system (including the material medium which is not CP.invariant) the true CP-transformation must also include the conversion of the medium M into anti-medium M [8] :
(CP)M : K ~ ~:, M -~ ~ .
(3)
This true CP-operation on our system consist in the linear transformation o I followed by the substitution fKM ~ f K ~ ' fRM ~ f g ~ " Using the CPT-relations
fY(~(P) = fKM(P),
f~,M(P) = f K ~ ( p ) ,
(4)
we see that the operation (3) leaves the mass matrix (2) invafiant. Although the matrix (2) is non-hermitian it has two independent eigenstates IK1M) = (2(1 + [pl2)) -1/2
IKM)=(2(1+10i2))-1/2
(I
_ +p
(5) '
where rtN(fKM(P) - fKM(P)) P = m ((m 1 -- mz) -- (i/Z)(3'1 -- 3'2))"
(6)
Here the 71, ')'2 and m 1 - m 2 denote the total decay rates and the difference of the mean rest masses of the short- and long-lived kaons in the CP-invariance limit, 439
Volume 63B, number 4
PHYSICS LETTERS
respectively. Applying the operation (3) on the states (5) we find
(CP)MIK > =
(7)
We see that these states have the same symmetry properties as the states IK1) and IK2). In contradistinction to them the states IK~~) and IK M) propagate independently in matter analogously to the states [K S) and IK L) which propagate independently in vacuum. It is well known that the states IK s) and IK L) are not orthogonal if the CP-invariance of the mass matrix in vacuum is violated but orthogonal in the CP-invariance limit. About the orthogonality of the states IK1M) and [K M) decides the quantity Im p, since (KMIK M) = 2i Im p/(1 + 1012). The mass matrix (2) is actually CPinvariant, however. This circumstance leads us to the idea to introduce, analogously, the orthogonality of the states (5) i.e. Imp = 0,
(8)
as an additional assumption. We shall see that this assumption is justified by experimental data. Using the expressions (6) and (8) we obtain the following simple formula for the regeneration phase ~: Im(fKM(P) --fl~M(P)) _ 7'1 -- 3/2 t a n ~ = Re(fKM(P ) _ f R u ( p ) ) 2(m2_ml ) •
(9)
Before the comparison of this formula with experiment we must first discuss the realistic case with the CPviolating (super)weak interaction. One can expect that the CP-violation has practically no influence on the forward scattering amplitudes fKM and fl~Mwhich are essentially "strong interaction quantities". Moreover, the CP-violation can be considered as a small perturbation in the mass matrix and the quantities 7 1 , 7 2 and m 2 - m 1 remain unchanged in the first approximation [6]. Therefore, to great degree of accuracy, we can substitute (m 2 - m l ) ~ ( m L - m s ) , )'1 ~3'S, 7'2 ~ 7L. Inserting the experimental values [9] into the right hand side of eq. (9) we find ¢ = 46.197+0.007 °
440
16 August 1976
or -(133.803-+0.007°). The latter solution is in agreement with the available experimental results [ 1,2]. The most precise measurements of the phase ~ have been performed with the liquid hydrogen used as the scattering medium. Brandenburg et al. [1] give the value ~ = -133.9-+4.0 °, the latest result of Birulev et al. [2] is ¢ = - 1 3 2 + 5 ° and the world mean value for hydrogen is ¢ --- - 1 3 3 +3.2 °. Agreement of our formula (9) with the experimental data indicates, conversely, that the states IK f ) and IK M) are indeed orthogonal justifying our orthogonality assumption. Formula (9) represents the connection between the strong and the weak interaction quantities. Besides that, there is also the connection between the phase and the phase ~e = arctan 2(m 2 - ml)/('yt -'Y2) of the superweak interaction mixing coefficient e: tan % . tan ~ = 1.
References
[i] K. Kleinknecht, Fortsch. d. Phys. 21 (1973) 57; V.K. Birulev et al., Phys. Lett. 38B (1972) 452; [2] V.K. Birulev et al., Dubna preprint P1-9434 (1976), submitted to Yad. Fiz.; G.W. Brandenburg et al., Phys. Rev. D9 (1974) 1939; D. Freytag et al., Phys. Rev. Lett. 35 (1975) 412; K.F. Albrecht et al., Phys. Lett. 48 (1974) 257; Nucl. Phys. B93 (1975). [3] F.J. Gilman, Phys. Rev. 171 (1968) 1453; V.I. Lisin et al., Nucl. Phys. B40 (1972) 298. [4] Nguyen Van Hieu, Theor. Math. Phys. 8 (1971) 354 (Soy. Phys.). [5] T.D. Lee, G.C. Wick, Phys. Rev. 148 (1966) 1385. [6] C.P. Enz, R.R. Lewis, Helv. Phys. Acta 38 (1965) 860. [7] See also J. Steinberger, CERN 70-1, 9 January 1970; G. Charpak, M. Gourdin, CERN 67-18, 11 July 1967; T.D. Lee, C.S. Wu, Ann. Rev. Nuel. Sci. 16 (1966) 511. [8] N. Cabibbo in "Symmetries in Elementary Particle Physics" 1964, International School of Physics Ettore Majorana, p. 272. Editor: A. Zichichi, Acad. Press, New York and London 1965. [9] Particle Data Group, Phys. Lett. 50B (1974) 2.