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Limit on the constant of T-nonconserving nucleon-nucleon interaction
Volume 162B, number 4,5,6
PHYSICS LETTERS
14 November 1985
LIMIT ON THE CONSTANT OF T-NONCONSERVING NUCLEON-NUCLEON INTERACTION V.V. F L A M B A U M , I.B. K H R I P L O V I C H and O.P. S U S H K O V Institute of Nuclear Physics, 630090, Novosibirsk, USSR Received 1 July 1985
T-odd moments of the xenon, mercury and thallium nuclei, due to T-nonconserving nuclear forces, are calculated. From the measurement of the electric dipole moment (EDM) of the 129Xe atom we extract a limit on the T-noninvariant nucleon-nucleon interaction. This limit approaches in its implications the limit on the neutron EDM.
In ref. [1] we have shown that under reasonable assumptions there is a regular enhancement factor for nuclear electric dipole moment (EDM) caused by Todd nuclear forces in comparison with the neutron EDM. This factor equals in order of magnitude 3rr(m21ulr30 ) - 1 ~ 60.
(1)
Here m n = 140 MeV is the n-meson mass, lul ~ 45 MeV is the characteristic depth o f the nuclear potential, r 0 ~- 1.15 fm. In ref. [1] we have calculated EDM, magnetic quadrupole moment (MQM) and the so called Schiffmoment (SM) for some nuclei. The SM Q is defined by the relations Q = ~ e _ ((r2rp) _ R2(rp)), p 10 p
= 4nOv~(r),
(2)
where ~ is the scalar potential created by the SM, e is the proton charge,R 0 = r0A 1/3 is the nuclear radius, the summation is carried out over all protons, the brackets (...) denote the expectation value in the nuclear state. The SM and MQM induce atomic and molecular EDM. The nuclear EDM by itself is useless in this respect: in a stationary atomic or molecular state the average electric field on the nucleus vanishes, so that the interaction - d ~ is absent (for a detailed discussion and relevant references see e.g. ref. [2] ). The expression for a T-odd nuclear multipole can be presented as 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
T= ~ !01TIk>(klTI0> k E0 - Ek '
(3)
where 1" = Za'f'a, 7"a is the operator o f the nucleon EDM, MQM or SM, W is T-odd interaction. In ref. [ 1] T-odd nuclear characteristics were calculated in the approximation where only the excitations Ik) of external, non-paired nucleons were taken into account. In other words, only an operator corresponding to the external nucleon was taken into account in the sum T = Za Ta" In the present paper the contribution to these nuclear multipoles from the nucleons o f the core is considered. This contribution is shown to be in general no smaller than that o f the external nucleon. This conclusion seems to be interesting by itself * 1. It is important also for the following, applied, reason. The most stringent limit on the T-violation in atomic physics was obtained from the experiment on the measurement o f the EDM o f the 129Me atom in the ground state [4] : d(129Xe) = ( - 0 . 3 -+ 1.1) X 10 -26 e cm.
(4)
In the same ref. [4] the feasibility of further increase o f accuracy in the experiments with xenon and mercury is discussed. In the ground state o f these atoms the electron shells are closed so that the nuclear MQM *1 After this work was over we found ref. [3] which contains the conclusion that the contribution from the paired nucleons excitations is important for usualP-odd effects in nuclear "r-transitions. 213
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PHYSICS LETTERS
does not induce an EDM of the atom [1]. Therefore, the nuclear SM is here the only nuclear multipole that leads to an atomic EDM. However, in the case of even - o d d nuclei the contribution of the external nucleon, the neutron, to the nuclear SM vanishes. This conclusion is valid also when recoil effects are taken into account, at least within the shell model of nucleus [1]. The calculation of the xenon SM taking account of the paired nucleon excitations in the present work, allows one to obtain from (4) a limit on the constant of the T-odd nucleon-nucleon interaction. The result (4) is shown to approach as a source of information on the nature of CP-violation, the best limit on the neutron EDM [51:
Idn/el < 4 × 10 -25 cm.
(5)
We start from consideration of the T-odd nucleonnucleon interaction. To first order in velocities p/m this interaction can be presented as [Cab =
(G/x/2) (1/2 m) ((r/abff a
- r/b aff b) Va~ (r a - r b )
+ r/'abff a X II b {(Pa - P b ) ' 5(ra - rb)})"
(6)
Here { , } is an anticommutator, m is the nucleon mass, lI, r and p are the spins, coordinates and momenta of the nucleons a and b. In this approximate local description of the T-odd nucleon-nucleon interaction the exchange terms can be reduced by means of a Fierz transformation to the form (6). Therefore, their role is reduced to the redefit nition of the dimensionless constants r/ab, r/ab characterizing the magnitude of the T-odd interaction in units of the Fermi constant G. The calculation of the SM for the nuclei 129,131Xe, 199,201Hg and 203,205 T1, which are of present experimental interest, was performed by us using wave functions and Green functions in the Woods-Saxon potential with spin-orbit correction. The results are pre. sented in table 1. Note that all the found values are of
Table 1 Values of Schiff moments (Q/efm s) x l0 s . Constants r/axe the parameters of T-odd interaction (6).
129Xe 131Xe
199Hg
201Hg 203,205,i, 1
1.75 rtnp -2.6 rlnp
-1.4 nnp
2.4 r~np 1.2 r~pp - 1.4 npn
214
14 November 1985
the natural order of magnitude for heavy nuclei [1] : l 20 ~ er/X 10 -9 fm3A 2/3 ~ e 0 X 10 -8 fm 3. Q ~ i--6eXR
(7) Here X = (G/x/~)(1/2m) p/lul = 2 × 10 -8 fm is the dimensional parameter characterizing T-odd nuclear multipoles. The nuclear EDM and MQM are expressed through it as follows [ 1] :
O "" kerl,
M "" la(e/m)(2I - 1) k~,
where/a e/2m is the nucleon magnetic moment, I is the nuclear spin. In the first four nuclei the non-paired nucleon is a neutron, therefore the SM is caused by the internal proton excitations and is expressed through the constant ~/nD" In thallium the internal nucleons contribution to the SM (1.23 r/pp in the units 10 -8 e fm 3) is comparable in order o f magnitude with the external proton contribution (-1.4~/pn - 0.04r/pp in the same units). The last number is somewhat different from that presented in ref. [1] since there the empirical density parametrization was used, and here the density was expressed through the wave functions in the Woods-Saxon potential. The conclusion that the contribution of internal nucleons to T-odd nuclear multipoles is as important as the contribution of external ones is confirmed also by the analysis within the oscillator model performed by us in ref. [6]. We shall get now the limit on the parameter r/n p of the T-odd potential (6) that follows from the experimental result (4). The Hartree-Fock calculation carded out in ref. [7] has shown that the xenon atom EDM is connected with the nuclear SM as d(Xe) = 2.7 X 10 -18 (Q/e fm 3) e cm.
(8)
From (4), (8) and the value of Q(129Xe) presented in table 1 we get the following limit on the constant r/np: r/np = - 0 . 0 6 + 0.23.
(9)
From the experiments with the TIF molecule [8,9] and the calculations [10] the following limit on Q(203,205TI) arises: Q(2°3'205T1) = (0.8 + 1.2) X 10 -8 e fm 3.
(10)
Unfortunately, the calculation of Q(T1) contains large uncertainties connected with the estimate of the giant resonance contribution (see ref. [6]). Therefore, the
Volume 162B, number 4,5,6
PHYSICS LETTERS
limit following from (10) and the value of Q(T1) from table 1 1.2r/pp - 1.4¢/pn = 0.8 + 1.2,
(11)
is an order of magnitude estimate only. The natural question arises: what are the physical implications of these limits? We shall try to answer this question grounding for definiteness on the Kobayashi -Maskawa model [ 11 ]. Now it seems to be the most natural gauge scheme of CP-invariance violation. The theoretical prediction for the neutron EDM in this model [12,13] d n "" 10 -31 e cm is about seven orders of magnitude smaller than the experimental limit (5). As to the T-odd nucleon-nucleon interaction, the corresponding dimensionless constants in the KobayashiMaskawa model constitute [1] r/ab "~ 10 -8 . The gap between this theoretical prediction and the limit (9) is only slightly larger than in the case of the neutron EDM. Therefore, the result (4) in its physical implications indeed approaches the limit (5) on the neutron EDM. As to the tremendous gap between the experimental limits and theoretical predictions for both quantities compared, one should note that the Kobayashi-Maskawa model is only the most simple scheme of CP-violation and that the true magnitude of T-odd effects can be much larger. One can try to compare the results of the measurements of the neutron and xenon EDM purely phenomenologically, grounding on the one-boson exchange model popular in the calculations of the usual weak interactions. The lowest intermediate state contributing to the constant r/nD is the *r0-meson. Its contribution is given by the relation ~IOG/X/~ = - g ~ 0 m ~ 2 where g and~ are the constants of the strong and T-odd ,rmeson-nucleon interaction:
14 November 1985
Here M " mp is the scale at which 7r.meson loop converges. From the limit on the neutron EDM (5) it follows Ig~_t < 4.5 × 10 -1°.
(15)
The limits (13) and (15) refer in general to different quantities. But, e.g., in the model of T-invariance violation with the 0-term *: Ig~0 t= Lgg'_t = 0.36 101 [14] Unlike the Kobayashi-Maskawa model, in this case the limit on the parameter 0 following from the neutron EDM measurements is considerably stronger than those from the xenon EDM. However, even the most optimistic of the modem projects of the neutron EDM measurements do not expect an accuracy in the determination ofdn/e better than 10 -26 cm, whereas the authors of ref. [4] plan to increase the accuracy of atomic experiments by four orders of magnitude. Note also that by changing from xenon to mercury one could gain, due to the larger Z, an order of magnitude in the value of the atomic EDM. Using the atomic calculation [16] one can show that the EDM of mercury atom is d(Hg) = - 4 × 10 -17
(Q/e fm 3) e cm.
(16)
Therefore, the prognoses in atomic physics look more optimistic than in the neutron one, especially since the transition to other heavy atoms, as well as molecules, with larger EDM is possible (see, e.g. ref. [1]). We are grateful to V.F. Dmitriev, PaN. Isaev, V~B. Telytsyn and V.G. Zelevinsky for numerous discussions and the programs for the computation of wave functions. ,2 The nuclear T-invarianeenonconservation was considered within this model in ref. [15].
(giant/5 ~/n + ~0 ~p ~ p)~O~rO
References + x/r2 ( g i ~ q'5 ~n + g - ~ p ~ n )~°+- "
(12)
From (9) the limit follows: gg,0 = (1.0 + 3.7) × 10 -8.
(13)
The virtual ,r--meson creation leads also to the neutron EDM [14]
d n = (eg~_/4,r2m) In(Mimer) = 0.9 X 10 -15 e cmgg_~
(14)
[ 1] O.P. Sushkov, V.V. Flambaum and I.B. Khriplovich, Zh. Eksp. Teor. Fiz. (JETP) 87 (1984) 1521. [2] I.B. Khriplovich, Parity noneonservation in atomic phenomena (Nanka, Moscow, 1981). [3] B. Desplanques, Phys. Lett. 47B (1973) 212. [4] T.G. Void, E.J. Raab, B. Heckel and E.N. Fortson, Phys. Rev. Lett. 52 (1984) 2229. [5] V.M. Lobashov and A.P. Serebrov, J. de Phys. 45 (1984) C3-11. [6] V.V. Flambaum, I.B. Khriplovich and O.P. Sushkov, Nucl. Phys. A, to be published. 215
Volume 162B, number 4,5,6
PHYSICS LETTERS
[7] V.A. Dzuba, V.V. Flambaum and P.G. SilvestTov, Phys. Lett. 154B (1985) 93. [8] E.A. Hinds and P.G.H. Sandars, Phys. Rev. A21 (1980) 480. [9] D.A. Wilkening, N.F. Ramsey and DJ. Larson, Phys. Rev. A29 (1984) 425. [10] P.V. Coveney and P.G.H. Sandats, J. Phys. B16 (1983) 3727. [ 11 ] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652.
216
14 November 1985
[12] M.B. Gavela, A. Le Yaouanc, L. Oliver, O. P~ne, J.-C. Raynal and T.N. Pham, Phys. Lett. 109B (1982) 215. [ 13] I.B. Khriplovieh and A.R. Zhitnltsky, Phys. Lett. 109B (1982) 490. [14] R.J. Crewther, P. Di Veeehia, G. Veneziano and E. Witten, Phys. Lett. 88B (1979) 123; 91B (1980) 487. [ 15] W.C. Haxton and E.M. Henley, Phys. Rev. Lett. 51 (1983) 1937. [16] A.-M. Martensson-Pendrill,Phys. Rev. Lett. 54 (1985) 1153.
PHYSICS LETTERS
14 November 1985
LIMIT ON THE CONSTANT OF T-NONCONSERVING NUCLEON-NUCLEON INTERACTION V.V. F L A M B A U M , I.B. K H R I P L O V I C H and O.P. S U S H K O V Institute of Nuclear Physics, 630090, Novosibirsk, USSR Received 1 July 1985
T-odd moments of the xenon, mercury and thallium nuclei, due to T-nonconserving nuclear forces, are calculated. From the measurement of the electric dipole moment (EDM) of the 129Xe atom we extract a limit on the T-noninvariant nucleon-nucleon interaction. This limit approaches in its implications the limit on the neutron EDM.
In ref. [1] we have shown that under reasonable assumptions there is a regular enhancement factor for nuclear electric dipole moment (EDM) caused by Todd nuclear forces in comparison with the neutron EDM. This factor equals in order of magnitude 3rr(m21ulr30 ) - 1 ~ 60.
(1)
Here m n = 140 MeV is the n-meson mass, lul ~ 45 MeV is the characteristic depth o f the nuclear potential, r 0 ~- 1.15 fm. In ref. [1] we have calculated EDM, magnetic quadrupole moment (MQM) and the so called Schiffmoment (SM) for some nuclei. The SM Q is defined by the relations Q = ~ e _ ((r2rp) _ R2(rp)), p 10 p
= 4nOv~(r),
(2)
where ~ is the scalar potential created by the SM, e is the proton charge,R 0 = r0A 1/3 is the nuclear radius, the summation is carried out over all protons, the brackets (...) denote the expectation value in the nuclear state. The SM and MQM induce atomic and molecular EDM. The nuclear EDM by itself is useless in this respect: in a stationary atomic or molecular state the average electric field on the nucleus vanishes, so that the interaction - d ~ is absent (for a detailed discussion and relevant references see e.g. ref. [2] ). The expression for a T-odd nuclear multipole can be presented as 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
T= ~ !01TIk>
(3)
where 1" = Za'f'a, 7"a is the operator o f the nucleon EDM, MQM or SM, W is T-odd interaction. In ref. [ 1] T-odd nuclear characteristics were calculated in the approximation where only the excitations Ik) of external, non-paired nucleons were taken into account. In other words, only an operator corresponding to the external nucleon was taken into account in the sum T = Za Ta" In the present paper the contribution to these nuclear multipoles from the nucleons o f the core is considered. This contribution is shown to be in general no smaller than that o f the external nucleon. This conclusion seems to be interesting by itself * 1. It is important also for the following, applied, reason. The most stringent limit on the T-violation in atomic physics was obtained from the experiment on the measurement o f the EDM o f the 129Me atom in the ground state [4] : d(129Xe) = ( - 0 . 3 -+ 1.1) X 10 -26 e cm.
(4)
In the same ref. [4] the feasibility of further increase o f accuracy in the experiments with xenon and mercury is discussed. In the ground state o f these atoms the electron shells are closed so that the nuclear MQM *1 After this work was over we found ref. [3] which contains the conclusion that the contribution from the paired nucleons excitations is important for usualP-odd effects in nuclear "r-transitions. 213
Volume 162B, number 4,5,6
PHYSICS LETTERS
does not induce an EDM of the atom [1]. Therefore, the nuclear SM is here the only nuclear multipole that leads to an atomic EDM. However, in the case of even - o d d nuclei the contribution of the external nucleon, the neutron, to the nuclear SM vanishes. This conclusion is valid also when recoil effects are taken into account, at least within the shell model of nucleus [1]. The calculation of the xenon SM taking account of the paired nucleon excitations in the present work, allows one to obtain from (4) a limit on the constant of the T-odd nucleon-nucleon interaction. The result (4) is shown to approach as a source of information on the nature of CP-violation, the best limit on the neutron EDM [51:
Idn/el < 4 × 10 -25 cm.
(5)
We start from consideration of the T-odd nucleonnucleon interaction. To first order in velocities p/m this interaction can be presented as [Cab =
(G/x/2) (1/2 m) ((r/abff a
- r/b aff b) Va~ (r a - r b )
+ r/'abff a X II b {(Pa - P b ) ' 5(ra - rb)})"
(6)
Here { , } is an anticommutator, m is the nucleon mass, lI, r and p are the spins, coordinates and momenta of the nucleons a and b. In this approximate local description of the T-odd nucleon-nucleon interaction the exchange terms can be reduced by means of a Fierz transformation to the form (6). Therefore, their role is reduced to the redefit nition of the dimensionless constants r/ab, r/ab characterizing the magnitude of the T-odd interaction in units of the Fermi constant G. The calculation of the SM for the nuclei 129,131Xe, 199,201Hg and 203,205 T1, which are of present experimental interest, was performed by us using wave functions and Green functions in the Woods-Saxon potential with spin-orbit correction. The results are pre. sented in table 1. Note that all the found values are of
Table 1 Values of Schiff moments (Q/efm s) x l0 s . Constants r/axe the parameters of T-odd interaction (6).
129Xe 131Xe
199Hg
201Hg 203,205,i, 1
1.75 rtnp -2.6 rlnp
-1.4 nnp
2.4 r~np 1.2 r~pp - 1.4 npn
214
14 November 1985
the natural order of magnitude for heavy nuclei [1] : l 20 ~ er/X 10 -9 fm3A 2/3 ~ e 0 X 10 -8 fm 3. Q ~ i--6eXR
(7) Here X = (G/x/~)(1/2m) p/lul = 2 × 10 -8 fm is the dimensional parameter characterizing T-odd nuclear multipoles. The nuclear EDM and MQM are expressed through it as follows [ 1] :
O "" kerl,
M "" la(e/m)(2I - 1) k~,
where/a e/2m is the nucleon magnetic moment, I is the nuclear spin. In the first four nuclei the non-paired nucleon is a neutron, therefore the SM is caused by the internal proton excitations and is expressed through the constant ~/nD" In thallium the internal nucleons contribution to the SM (1.23 r/pp in the units 10 -8 e fm 3) is comparable in order o f magnitude with the external proton contribution (-1.4~/pn - 0.04r/pp in the same units). The last number is somewhat different from that presented in ref. [1] since there the empirical density parametrization was used, and here the density was expressed through the wave functions in the Woods-Saxon potential. The conclusion that the contribution of internal nucleons to T-odd nuclear multipoles is as important as the contribution of external ones is confirmed also by the analysis within the oscillator model performed by us in ref. [6]. We shall get now the limit on the parameter r/n p of the T-odd potential (6) that follows from the experimental result (4). The Hartree-Fock calculation carded out in ref. [7] has shown that the xenon atom EDM is connected with the nuclear SM as d(Xe) = 2.7 X 10 -18 (Q/e fm 3) e cm.
(8)
From (4), (8) and the value of Q(129Xe) presented in table 1 we get the following limit on the constant r/np: r/np = - 0 . 0 6 + 0.23.
(9)
From the experiments with the TIF molecule [8,9] and the calculations [10] the following limit on Q(203,205TI) arises: Q(2°3'205T1) = (0.8 + 1.2) X 10 -8 e fm 3.
(10)
Unfortunately, the calculation of Q(T1) contains large uncertainties connected with the estimate of the giant resonance contribution (see ref. [6]). Therefore, the
Volume 162B, number 4,5,6
PHYSICS LETTERS
limit following from (10) and the value of Q(T1) from table 1 1.2r/pp - 1.4¢/pn = 0.8 + 1.2,
(11)
is an order of magnitude estimate only. The natural question arises: what are the physical implications of these limits? We shall try to answer this question grounding for definiteness on the Kobayashi -Maskawa model [ 11 ]. Now it seems to be the most natural gauge scheme of CP-invariance violation. The theoretical prediction for the neutron EDM in this model [12,13] d n "" 10 -31 e cm is about seven orders of magnitude smaller than the experimental limit (5). As to the T-odd nucleon-nucleon interaction, the corresponding dimensionless constants in the KobayashiMaskawa model constitute [1] r/ab "~ 10 -8 . The gap between this theoretical prediction and the limit (9) is only slightly larger than in the case of the neutron EDM. Therefore, the result (4) in its physical implications indeed approaches the limit (5) on the neutron EDM. As to the tremendous gap between the experimental limits and theoretical predictions for both quantities compared, one should note that the Kobayashi-Maskawa model is only the most simple scheme of CP-violation and that the true magnitude of T-odd effects can be much larger. One can try to compare the results of the measurements of the neutron and xenon EDM purely phenomenologically, grounding on the one-boson exchange model popular in the calculations of the usual weak interactions. The lowest intermediate state contributing to the constant r/nD is the *r0-meson. Its contribution is given by the relation ~IOG/X/~ = - g ~ 0 m ~ 2 where g and~ are the constants of the strong and T-odd ,rmeson-nucleon interaction:
14 November 1985
Here M " mp is the scale at which 7r.meson loop converges. From the limit on the neutron EDM (5) it follows Ig~_t < 4.5 × 10 -1°.
(15)
The limits (13) and (15) refer in general to different quantities. But, e.g., in the model of T-invariance violation with the 0-term *: Ig~0 t= Lgg'_t = 0.36 101 [14] Unlike the Kobayashi-Maskawa model, in this case the limit on the parameter 0 following from the neutron EDM measurements is considerably stronger than those from the xenon EDM. However, even the most optimistic of the modem projects of the neutron EDM measurements do not expect an accuracy in the determination ofdn/e better than 10 -26 cm, whereas the authors of ref. [4] plan to increase the accuracy of atomic experiments by four orders of magnitude. Note also that by changing from xenon to mercury one could gain, due to the larger Z, an order of magnitude in the value of the atomic EDM. Using the atomic calculation [16] one can show that the EDM of mercury atom is d(Hg) = - 4 × 10 -17
(Q/e fm 3) e cm.
(16)
Therefore, the prognoses in atomic physics look more optimistic than in the neutron one, especially since the transition to other heavy atoms, as well as molecules, with larger EDM is possible (see, e.g. ref. [1]). We are grateful to V.F. Dmitriev, PaN. Isaev, V~B. Telytsyn and V.G. Zelevinsky for numerous discussions and the programs for the computation of wave functions. ,2 The nuclear T-invarianeenonconservation was considered within this model in ref. [15].
(giant/5 ~/n + ~0 ~p ~ p)~O~rO
References + x/r2 ( g i ~ q'5 ~n + g - ~ p ~ n )~°+- "
(12)
From (9) the limit follows: gg,0 = (1.0 + 3.7) × 10 -8.
(13)
The virtual ,r--meson creation leads also to the neutron EDM [14]
d n = (eg~_/4,r2m) In(Mimer) = 0.9 X 10 -15 e cmgg_~
(14)
[ 1] O.P. Sushkov, V.V. Flambaum and I.B. Khriplovich, Zh. Eksp. Teor. Fiz. (JETP) 87 (1984) 1521. [2] I.B. Khriplovich, Parity noneonservation in atomic phenomena (Nanka, Moscow, 1981). [3] B. Desplanques, Phys. Lett. 47B (1973) 212. [4] T.G. Void, E.J. Raab, B. Heckel and E.N. Fortson, Phys. Rev. Lett. 52 (1984) 2229. [5] V.M. Lobashov and A.P. Serebrov, J. de Phys. 45 (1984) C3-11. [6] V.V. Flambaum, I.B. Khriplovich and O.P. Sushkov, Nucl. Phys. A, to be published. 215
Volume 162B, number 4,5,6
PHYSICS LETTERS
[7] V.A. Dzuba, V.V. Flambaum and P.G. SilvestTov, Phys. Lett. 154B (1985) 93. [8] E.A. Hinds and P.G.H. Sandars, Phys. Rev. A21 (1980) 480. [9] D.A. Wilkening, N.F. Ramsey and DJ. Larson, Phys. Rev. A29 (1984) 425. [10] P.V. Coveney and P.G.H. Sandats, J. Phys. B16 (1983) 3727. [ 11 ] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652.
216
14 November 1985
[12] M.B. Gavela, A. Le Yaouanc, L. Oliver, O. P~ne, J.-C. Raynal and T.N. Pham, Phys. Lett. 109B (1982) 215. [ 13] I.B. Khriplovieh and A.R. Zhitnltsky, Phys. Lett. 109B (1982) 490. [14] R.J. Crewther, P. Di Veeehia, G. Veneziano and E. Witten, Phys. Lett. 88B (1979) 123; 91B (1980) 487. [ 15] W.C. Haxton and E.M. Henley, Phys. Rev. Lett. 51 (1983) 1937. [16] A.-M. Martensson-Pendrill,Phys. Rev. Lett. 54 (1985) 1153.