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Fractionally charged particles and one Dirac charge magnetic monopoles: Are they compatible?
Volume 120B, number 1,2,3
PHYSICS LETTERS
6 January 1983
FRACTIONALLY CHARGED PARTICLES AND ONE DIRAC CHARGE MAGNETIC MONOPOLES: ARE THEY COMPATIBLE ? V.A.
RUBAKOV
Institute for Nuclear Research o f the Academy o f Sciences o f the USSR, 60th October prospect, 7a, Moscow 117312, USSR Received 7 September 1982
The simultaneous existence o f fractional electric charges and one Dirac charge magnetic monopoles implies the existence o f a long-ranged force different from electromagnetism. This may be either unconfined colour or/and some new gauge interaction. In the latter case, ordinary matter could (and, if colour is unbroken, should) carry new charge. This charge, however small the coupling constant be, could be experimentally observed in interactions o f monopoles with matter. An experiment for checking this possibility is suggested.
Recently, much interest has been raised by two experiments performed in SLAC. Namely, Fairbank et al. [ 1 ] claimed that there exist particles with electric charges ½ e, and Cabrera [2] reported on an event which can be interpreted as a monopole of one Dirac magnetic charge gD = 27r/e , passing through the superconducting coil. These results taken together imply the violation of the well known Dirac quantization rule [3] which states that magnetic charge g of any monopole and electric charge q of any particle should obey the following relation g - q = 2rrn ,
(1)
where n is some integer number. Though both SLAC results have not yet been confirmed by other groups, it might be interesting to discuss possible consequences of the violation of the Dirac rule (1). This is the main purpose of the present paper. The Dirac rule is based on very solid theoretical grounds (see, e.g. ref. [4] and references therein). In order to save these grounds one has to go beyond the pure electromagnetism, so some exotics is unavoidable. We fred that, provided there exist one Dirac i charge monopoles and free particles of charge 5 e, the following possibilities can take place: (1) Ifcolour is unbroken and quarks are fractionally 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
charged, then either (i) monopoles with g = g D possess ehromomagnetic charge and the particles of charge 51 e are coloured *~ (so that colour is not confmed), or/and (ii) there exists new massless vector boson(s), (e.g., the "baryonic photon" of Lee and Yang [8]), associated, say, with an unbroken U(1)' group; monopoles with g = gD carry "primed magnetic charge" and ordinary matter (as well as the particles with q -- l e5) carries non-vanishing "primed charge". (2) If colour is broken and quarks are integrally charged ,2 [9] then either (i) there exists an unbroken subgroup of SU(3) c (say, U(1) or SU(2)), the monopoles withg = gD carry "magnetic charge" corresponding to this subgroup and the particles with charges ~ e are not singlets under this subgroup or/and (ii) there exists a new long-ranged interactions (as in the case (1 .ii); ordinary matter need not, however, carry "primed charge". ,1 This possibility could be realized in the Georgi-Glashow SU(5) model [5 ]. In this model the fundamental monopole [6] o f 't Hooft-Polyakov type [7] carries one Dirac unit o f magnetic charge and non-vanishing chromomagnetic charge. 1 The particles of charge ge could be viewed as "dressed" dantiquarks. ,2 Note that theories with integrally charged quarks are not experimentally excluded [ 10].
191
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All these possibilities are rather non-standard. Among them, the most exotic one is the possibility of non-vanishing "primed charge" of ordinary matter. Surprisingly enough, this possibility can be easily checked experimentally, provided that the monopole flux is sufficiently large (say, compatible with the Cabrera event [2]). We discuss this point in the end of this paper. To drive at the above conclusions, we begin with the observation that in theories with unbroken gauge groups larger than U(1)EM, the quantization rule is no longer valid. Though this fact has been recognized for some time (see, e.g. ref. [11]), we recall the derivation in order to stress the relevance of long ranged interactions different from the electromagnetism. First, let us remind the basic arguments leading to (1). The vector potential of a magnetic monopole has a singularity on a line (Dirac string) going from the center to inffmity, (along, say, a positive z semiaxis). The singular part of A is A s = (g/27r)V ~0,
(2)
where ~0is the polar angle. As follows from the Stokes theorem, g is just the magnetic charge of the monopole. From (2) it is clear that the wave function ff of a particle with an electric charge q has a singularity of the form ~s ~x exp
[i(qg/2rr)~o]
.
For the Diract string to be unobservable, this singularity should be of a pure gauge type, i.e. the phase factor exp [i(qg/2zr)~o] should be a single-valued function. This is possible only if (1) is satisfied. Note that essentially the same arguments lead to (1) in the W u Yang [12] "stringless" monopole theory. Note also that these arguments apply to 't H o o f t - P o l y a k o v monopoles as well, since in the unitary gauge [13] their structure at long distances coincides with that of Dirac ones (for further discussion see ref. [11]). Let an unbroken group be U(1)E M × U(1)', where U(1)' is associated with a long-range force different from the electromagnetism. Then any monopole and any particle are characterized by a pair of "magnetic charges" (g, g ' ) and "electric charges" (q, q ' ) respectively. The straightforward generalization of the above arguments leads to the following generalization of(l) g - q +g'- q' = 21rn . 192
(3)
6 January 1983
Therefore, in this case therecan exist monopoles with the Dirac magnetic charge g = gD a n d particles with the electric charge q -1 5 e. However, these monopoles a n d these particles should have non-vanishing "primed magnetic charges" g' and "primed electric charges" q' respectively. The above possibilities are clearly seen just from eq. (3). The cases (1 .i) and (2.i) correspond to the group U(1)' embedded into colour SU(3), while in the cases (1 .ii) and (2.ii) the group U(1)' has nothing to do with colour. In models with integrally charged quarks, the quark charges can obey eq. (3) and t
I
qu=qd=0, simultaneously, so in the case (2.ii) ordinary matter need not be "primed charged". What remains to be discussed in the "primed charge" of ordinary matter in the case (1 .ii). In this case, the notions of "photon" and "primed photon" require some clarification. Indeed, any linear combination of two massless vector fields is also a massless vector field. Without loss of generality we may assume that electron does not carry "primed charge", qet = 0 (cf. ref. [14]). With this prescription, the searches for magnetic monopoles through ionization [15] or induction of electric current in superconductors [ 16,2] are sensible just to the magnetic charge g. In particular, Cabrera candidate event corresponds to g = gD" The quantization rule (3) applied to the monopole withg = gD and to u and d quarks reads g qI ut = 27r(nu
2 ~),
, , = 2rr(n d + ½)
gqd
Introducing "unit primed charge" e' = 2 t r i g ' we find that the proton and neutron "primed charges" are t
qp' = 2qu' + qd' = e ( 2 n u + n d - l ) t
qn' = qu' + 2qd' = e ( n u + 2 n d ) Note that ifn u = n d + 1, the proton and neutron "primed charges" are equal, so the "primed photon" is just the "baryonic photon" of Lee and Yang [8]. For any integer n u and no the average "primed charge" of a nucleon, . . .+qn) . qN, = tgLqp
t
= ~l e ( 3 n
u+3n d
-
1)
,
(4)
does not vanish, which is the desired result. As discussed by Lee and Yang [8] (see also, refs. [14,17]), the value of a' -= e'2/4rr is severely limited
Volume 120B, number 1,2,3
PHYSICS LETTERS
by EStvSs-Dicke Braginsky experiments [18]. For n u, n d = O(1) one has [8,14,17] o~'~ 10 4 7 . Nevertheless, the m o n o p o l e - b a r y o n coupling, g " q ~ is 0 ( 1 ) , so it can be experimentally observed, provided that the monopole flux is sufficiently large * 3. The idea of the experunent may be as follows. Consider the monopole passing through the electrically neutral ring of radius r, which can rotate essentially without friction. Assume for simplicity that the monopole moves along the ring axis. Then the induced "primed electric field" causes the rotation of the ring. The resulting angular m o m e n t u m of the ring is easily calculated to be !
t
L : ( g " qN/27r)N, where N is the number of nucleons inside the ring (so f that qN "N is the total "primed charge" of the ring), N ~ 2V/ring/Mproton. Therefore, the angular velocity of the ring, co = L/(Mring" r 2) , is essentially independent of the ring mass, co ~ g " q'N /( 2 nMproton, r 2 ) . For g ' - q ~ = zr [which is the minimal value according to (4)] and r = 10 cm (so that the size of the ring exceeds that of Cabrera coil), the angular velocity would be roughly one angular second per second, and the velocity of the ring would be rougtfly v = car ~ 5 × 10-5 c m / s . This is definitely a measurable quantity. In conclusion, we wish to emphasize that the mere existence of one Dirac charge monopoles and free fractionally charged particles implies the existence of a long-ranged interaction different from electromagnetism. It might be either unconfined colour or some new force. Both these possibilities could in principle be
,3 we do not discuss here a question whether the flux of these monopoles can be large enough. The effective monopolemonopole coupling is enormous,g'2/4u >>.1047. Therefore, if the monopole density were large (say, O(10 -16 cm -3) [2] ), the monopoles would interact very strongly. At present we are unable to analyse this possibility in a more or less rigorous way.
6 January 1983
checked experimentally, and magnetic monopoles could help to do this .4 . I thank N.V. Krasnikov, V.A. Kuzmin, V.A. Matveev, L.B. Okun, M.E. Shaposhnikov, A.N. Tavkhelidze, I.I. Tkachev, V.F. Tokarev and A.V. Veryaskin for helpful discussions. ,4 If one were able to construct a material with the sufficiently large density of particles with q = -~e (and with the negligible density of their antiparticles), the experiment described above could provide an evidence for the new long-ranged force. References [1 ] W.M. Fairbank, G.S. La Rue and A.F. Hebard, Phys. Rev. Lett. 38 (1977) 1011; W.M. Fairbank, G.S. La Rue and J.D. Phillips, Phys. Rev. Lett. 42 (1979) 142. [2] B. Cabrera, Phys. Rev. Lett. 48 (1982) 82. [3] P.A.M. Dirac, Phys. Rev. 74 (1948) 817. [4] V.I. Strazhev and L.M. Tomil'chik, Fiz. Elem. Chastits. At. Yadra 4 (1973) 187. [5] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [6] C.P. Dokos and T.N. Tomaras, Phys. Rev. D21 (1980) 2940; M. Daniel, G. Lazarides and Q. Shaft, Nucl. Phys. B170 [FS1] (1980) 156. [7] GI 't Hooft, Nucl. Phys. B79 (1974) 276; A.M. Polyakov, Pis'ma Zh. Eksp. Teor. Fiz. 20 (1974) 430. [8] T.D. Lee and C.N. Yang, Phys. Rev. 98 (1955) 1501. [9] N.N. Bogoliubov, B.V. Struminsky and A.N. Tavkhelidze, JINR preprint D-1968 (1965); M.Y. Han and Y. Nambu, Phys. Rev. 139B (1965) 1038; Y. Miyamoto, Prog. Theor. Phys. Suppl., extra No. (1965) 187. [10] A.Yu. Ignatiev et al., Teor. Mat. Fiz. 47 (1981) 147; K.G. Chetyrkin, A.Yu. Ignatiev, N.V. Krasnikov, V.A. Matveev and M.E. Shaposhnikov, Inst. Nucl. Res. preprint P-0212 (1981); K.G. Chetyrkin, A.Yu. Ignatiev, V.A. Matveev, A.N. Tavkhelidze and M.E. Shaposhnikov, Report Intern. Seminar Quarks 82 (Sukhumi, May 1982), and references therein. [11] P. Goddard, J. Nuits and D. Olive, Nucl. Phys. B125 (1977) 1; P. Goddard and D. Olive, Rep. Prog. Phys. 41 (1978) 1357. [12] T.T. Wu and C.N. Yang, Phys. Rev. D12 (1975) 3845. [13] J.F. Englert and P. Windey, Phys. Rev. D14 (1976) 2728; A.S. Schwarz, Nucl. Phys. B112 (1976) 358. [141 L.B. Okun, Yad. Fiz. 10 (1969) 358.
193
Volume 120B, number 1,2,3
PHYSICS LETTERS
[15] J.D. Ullman, Phys. Rev. Lett. 47 (1981) 289; E.N. Alexeev et al., Report Intern. Conf. on Baryon nonconservation (ICOBAN) (Bombay, January 1982); Report Intern. Seminar Quarks 82 (Sukhumi, May 1982); R. Bonarelli et al., Phys. Lett. l12B (1982) 100; for earlier references see: Ya. Ruzliichka and V.P. Zrelov, JINR preprint D2-81675 (1981). [16] P.H. Eberhard, R.R. Ross, L.W. Alwarez and R.D. Watt, Phys. Rev. D4 (1974) 3260; D.B. Cline, Talk Intern. Seminar Quarks 82 (Sukhumi, May 1982).
194
6 January 1983
[17] L.B. Okun, Leptons and quarks (Moscow, Nauka, 1981) p. 154. [18] R.V. E~Stvgs, D. Pekar and E. Fekete,Ann. Physik 68 (1922) 11; R.H. Dicka, P.G. Poll and G. Krotkov, Ann. Phys. (NY) 26 (1964) 422; V.B. Braginsky and V.1. Panov, Zh. Eksp. Teor. Fiz. 61 (1971) 873.
PHYSICS LETTERS
6 January 1983
FRACTIONALLY CHARGED PARTICLES AND ONE DIRAC CHARGE MAGNETIC MONOPOLES: ARE THEY COMPATIBLE ? V.A.
RUBAKOV
Institute for Nuclear Research o f the Academy o f Sciences o f the USSR, 60th October prospect, 7a, Moscow 117312, USSR Received 7 September 1982
The simultaneous existence o f fractional electric charges and one Dirac charge magnetic monopoles implies the existence o f a long-ranged force different from electromagnetism. This may be either unconfined colour or/and some new gauge interaction. In the latter case, ordinary matter could (and, if colour is unbroken, should) carry new charge. This charge, however small the coupling constant be, could be experimentally observed in interactions o f monopoles with matter. An experiment for checking this possibility is suggested.
Recently, much interest has been raised by two experiments performed in SLAC. Namely, Fairbank et al. [ 1 ] claimed that there exist particles with electric charges ½ e, and Cabrera [2] reported on an event which can be interpreted as a monopole of one Dirac magnetic charge gD = 27r/e , passing through the superconducting coil. These results taken together imply the violation of the well known Dirac quantization rule [3] which states that magnetic charge g of any monopole and electric charge q of any particle should obey the following relation g - q = 2rrn ,
(1)
where n is some integer number. Though both SLAC results have not yet been confirmed by other groups, it might be interesting to discuss possible consequences of the violation of the Dirac rule (1). This is the main purpose of the present paper. The Dirac rule is based on very solid theoretical grounds (see, e.g. ref. [4] and references therein). In order to save these grounds one has to go beyond the pure electromagnetism, so some exotics is unavoidable. We fred that, provided there exist one Dirac i charge monopoles and free particles of charge 5 e, the following possibilities can take place: (1) Ifcolour is unbroken and quarks are fractionally 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
charged, then either (i) monopoles with g = g D possess ehromomagnetic charge and the particles of charge 51 e are coloured *~ (so that colour is not confmed), or/and (ii) there exists new massless vector boson(s), (e.g., the "baryonic photon" of Lee and Yang [8]), associated, say, with an unbroken U(1)' group; monopoles with g = gD carry "primed magnetic charge" and ordinary matter (as well as the particles with q -- l e5) carries non-vanishing "primed charge". (2) If colour is broken and quarks are integrally charged ,2 [9] then either (i) there exists an unbroken subgroup of SU(3) c (say, U(1) or SU(2)), the monopoles withg = gD carry "magnetic charge" corresponding to this subgroup and the particles with charges ~ e are not singlets under this subgroup or/and (ii) there exists a new long-ranged interactions (as in the case (1 .ii); ordinary matter need not, however, carry "primed charge". ,1 This possibility could be realized in the Georgi-Glashow SU(5) model [5 ]. In this model the fundamental monopole [6] o f 't Hooft-Polyakov type [7] carries one Dirac unit o f magnetic charge and non-vanishing chromomagnetic charge. 1 The particles of charge ge could be viewed as "dressed" dantiquarks. ,2 Note that theories with integrally charged quarks are not experimentally excluded [ 10].
191
Volume 120B, number 1,2,3
PHYSICS LETTERS
All these possibilities are rather non-standard. Among them, the most exotic one is the possibility of non-vanishing "primed charge" of ordinary matter. Surprisingly enough, this possibility can be easily checked experimentally, provided that the monopole flux is sufficiently large (say, compatible with the Cabrera event [2]). We discuss this point in the end of this paper. To drive at the above conclusions, we begin with the observation that in theories with unbroken gauge groups larger than U(1)EM, the quantization rule is no longer valid. Though this fact has been recognized for some time (see, e.g. ref. [11]), we recall the derivation in order to stress the relevance of long ranged interactions different from the electromagnetism. First, let us remind the basic arguments leading to (1). The vector potential of a magnetic monopole has a singularity on a line (Dirac string) going from the center to inffmity, (along, say, a positive z semiaxis). The singular part of A is A s = (g/27r)V ~0,
(2)
where ~0is the polar angle. As follows from the Stokes theorem, g is just the magnetic charge of the monopole. From (2) it is clear that the wave function ff of a particle with an electric charge q has a singularity of the form ~s ~x exp
[i(qg/2rr)~o]
.
For the Diract string to be unobservable, this singularity should be of a pure gauge type, i.e. the phase factor exp [i(qg/2zr)~o] should be a single-valued function. This is possible only if (1) is satisfied. Note that essentially the same arguments lead to (1) in the W u Yang [12] "stringless" monopole theory. Note also that these arguments apply to 't H o o f t - P o l y a k o v monopoles as well, since in the unitary gauge [13] their structure at long distances coincides with that of Dirac ones (for further discussion see ref. [11]). Let an unbroken group be U(1)E M × U(1)', where U(1)' is associated with a long-range force different from the electromagnetism. Then any monopole and any particle are characterized by a pair of "magnetic charges" (g, g ' ) and "electric charges" (q, q ' ) respectively. The straightforward generalization of the above arguments leads to the following generalization of(l) g - q +g'- q' = 21rn . 192
(3)
6 January 1983
Therefore, in this case therecan exist monopoles with the Dirac magnetic charge g = gD a n d particles with the electric charge q -1 5 e. However, these monopoles a n d these particles should have non-vanishing "primed magnetic charges" g' and "primed electric charges" q' respectively. The above possibilities are clearly seen just from eq. (3). The cases (1 .i) and (2.i) correspond to the group U(1)' embedded into colour SU(3), while in the cases (1 .ii) and (2.ii) the group U(1)' has nothing to do with colour. In models with integrally charged quarks, the quark charges can obey eq. (3) and t
I
qu=qd=0, simultaneously, so in the case (2.ii) ordinary matter need not be "primed charged". What remains to be discussed in the "primed charge" of ordinary matter in the case (1 .ii). In this case, the notions of "photon" and "primed photon" require some clarification. Indeed, any linear combination of two massless vector fields is also a massless vector field. Without loss of generality we may assume that electron does not carry "primed charge", qet = 0 (cf. ref. [14]). With this prescription, the searches for magnetic monopoles through ionization [15] or induction of electric current in superconductors [ 16,2] are sensible just to the magnetic charge g. In particular, Cabrera candidate event corresponds to g = gD" The quantization rule (3) applied to the monopole withg = gD and to u and d quarks reads g qI ut = 27r(nu
2 ~),
, , = 2rr(n d + ½)
gqd
Introducing "unit primed charge" e' = 2 t r i g ' we find that the proton and neutron "primed charges" are t
qp' = 2qu' + qd' = e ( 2 n u + n d - l ) t
qn' = qu' + 2qd' = e ( n u + 2 n d ) Note that ifn u = n d + 1, the proton and neutron "primed charges" are equal, so the "primed photon" is just the "baryonic photon" of Lee and Yang [8]. For any integer n u and no the average "primed charge" of a nucleon, . . .+qn) . qN, = tgLqp
t
= ~l e ( 3 n
u+3n d
-
1)
,
(4)
does not vanish, which is the desired result. As discussed by Lee and Yang [8] (see also, refs. [14,17]), the value of a' -= e'2/4rr is severely limited
Volume 120B, number 1,2,3
PHYSICS LETTERS
by EStvSs-Dicke Braginsky experiments [18]. For n u, n d = O(1) one has [8,14,17] o~'~ 10 4 7 . Nevertheless, the m o n o p o l e - b a r y o n coupling, g " q ~ is 0 ( 1 ) , so it can be experimentally observed, provided that the monopole flux is sufficiently large * 3. The idea of the experunent may be as follows. Consider the monopole passing through the electrically neutral ring of radius r, which can rotate essentially without friction. Assume for simplicity that the monopole moves along the ring axis. Then the induced "primed electric field" causes the rotation of the ring. The resulting angular m o m e n t u m of the ring is easily calculated to be !
t
L : ( g " qN/27r)N, where N is the number of nucleons inside the ring (so f that qN "N is the total "primed charge" of the ring), N ~ 2V/ring/Mproton. Therefore, the angular velocity of the ring, co = L/(Mring" r 2) , is essentially independent of the ring mass, co ~ g " q'N /( 2 nMproton, r 2 ) . For g ' - q ~ = zr [which is the minimal value according to (4)] and r = 10 cm (so that the size of the ring exceeds that of Cabrera coil), the angular velocity would be roughly one angular second per second, and the velocity of the ring would be rougtfly v = car ~ 5 × 10-5 c m / s . This is definitely a measurable quantity. In conclusion, we wish to emphasize that the mere existence of one Dirac charge monopoles and free fractionally charged particles implies the existence of a long-ranged interaction different from electromagnetism. It might be either unconfined colour or some new force. Both these possibilities could in principle be
,3 we do not discuss here a question whether the flux of these monopoles can be large enough. The effective monopolemonopole coupling is enormous,g'2/4u >>.1047. Therefore, if the monopole density were large (say, O(10 -16 cm -3) [2] ), the monopoles would interact very strongly. At present we are unable to analyse this possibility in a more or less rigorous way.
6 January 1983
checked experimentally, and magnetic monopoles could help to do this .4 . I thank N.V. Krasnikov, V.A. Kuzmin, V.A. Matveev, L.B. Okun, M.E. Shaposhnikov, A.N. Tavkhelidze, I.I. Tkachev, V.F. Tokarev and A.V. Veryaskin for helpful discussions. ,4 If one were able to construct a material with the sufficiently large density of particles with q = -~e (and with the negligible density of their antiparticles), the experiment described above could provide an evidence for the new long-ranged force. References [1 ] W.M. Fairbank, G.S. La Rue and A.F. Hebard, Phys. Rev. Lett. 38 (1977) 1011; W.M. Fairbank, G.S. La Rue and J.D. Phillips, Phys. Rev. Lett. 42 (1979) 142. [2] B. Cabrera, Phys. Rev. Lett. 48 (1982) 82. [3] P.A.M. Dirac, Phys. Rev. 74 (1948) 817. [4] V.I. Strazhev and L.M. Tomil'chik, Fiz. Elem. Chastits. At. Yadra 4 (1973) 187. [5] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [6] C.P. Dokos and T.N. Tomaras, Phys. Rev. D21 (1980) 2940; M. Daniel, G. Lazarides and Q. Shaft, Nucl. Phys. B170 [FS1] (1980) 156. [7] GI 't Hooft, Nucl. Phys. B79 (1974) 276; A.M. Polyakov, Pis'ma Zh. Eksp. Teor. Fiz. 20 (1974) 430. [8] T.D. Lee and C.N. Yang, Phys. Rev. 98 (1955) 1501. [9] N.N. Bogoliubov, B.V. Struminsky and A.N. Tavkhelidze, JINR preprint D-1968 (1965); M.Y. Han and Y. Nambu, Phys. Rev. 139B (1965) 1038; Y. Miyamoto, Prog. Theor. Phys. Suppl., extra No. (1965) 187. [10] A.Yu. Ignatiev et al., Teor. Mat. Fiz. 47 (1981) 147; K.G. Chetyrkin, A.Yu. Ignatiev, N.V. Krasnikov, V.A. Matveev and M.E. Shaposhnikov, Inst. Nucl. Res. preprint P-0212 (1981); K.G. Chetyrkin, A.Yu. Ignatiev, V.A. Matveev, A.N. Tavkhelidze and M.E. Shaposhnikov, Report Intern. Seminar Quarks 82 (Sukhumi, May 1982), and references therein. [11] P. Goddard, J. Nuits and D. Olive, Nucl. Phys. B125 (1977) 1; P. Goddard and D. Olive, Rep. Prog. Phys. 41 (1978) 1357. [12] T.T. Wu and C.N. Yang, Phys. Rev. D12 (1975) 3845. [13] J.F. Englert and P. Windey, Phys. Rev. D14 (1976) 2728; A.S. Schwarz, Nucl. Phys. B112 (1976) 358. [141 L.B. Okun, Yad. Fiz. 10 (1969) 358.
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[15] J.D. Ullman, Phys. Rev. Lett. 47 (1981) 289; E.N. Alexeev et al., Report Intern. Conf. on Baryon nonconservation (ICOBAN) (Bombay, January 1982); Report Intern. Seminar Quarks 82 (Sukhumi, May 1982); R. Bonarelli et al., Phys. Lett. l12B (1982) 100; for earlier references see: Ya. Ruzliichka and V.P. Zrelov, JINR preprint D2-81675 (1981). [16] P.H. Eberhard, R.R. Ross, L.W. Alwarez and R.D. Watt, Phys. Rev. D4 (1974) 3260; D.B. Cline, Talk Intern. Seminar Quarks 82 (Sukhumi, May 1982).
194
6 January 1983
[17] L.B. Okun, Leptons and quarks (Moscow, Nauka, 1981) p. 154. [18] R.V. E~Stvgs, D. Pekar and E. Fekete,Ann. Physik 68 (1922) 11; R.H. Dicka, P.G. Poll and G. Krotkov, Ann. Phys. (NY) 26 (1964) 422; V.B. Braginsky and V.1. Panov, Zh. Eksp. Teor. Fiz. 61 (1971) 873.