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Phenomenology of neutron oscillations
Volume 94B, number 2
PHYSICS LETTERS
28 July 1980
PHENOMENOLOGY OF NEUTRON OSCILLATIONS ~ R.N. MOHAPATRA Department of Physics, City College of the City University of N ew York, New York, N Y 10031, USA and R.E. MARSHAK Department o f Physics, VirginiaPolytechnic Institute and State University, Blaeksburg, VA 24061, USA Received 14 April 1980
We have recently argued that gauge models with spontaneously broken local B - L symmetry can lead to large AB = 2 nucleon transition amplitudes for processes such as n-E oscillations and n + p ~ pions. We discuss the phenomenology of neutron oscillations both in vacuum and in the presence of external fields. These considerations indicate that neutron oscillation experiments can be much more sensitive to the AB --- 2 amplitude than tests of nucleon instability.
1. Introduction. The issue o f baryon number nonconservation has been the subject o f discussion for a long time [1 ]. It has attracted a great deal o f attention in recent years due to the fact that attempts to unify strong, weak and electromagnetic interactions within a gauge theory framework almost invariably lead to violation o f baryon number and hence to the interesting possibility o f proton decay. Once one accepts the possibility o f baryon number being only an approximately conserved quantum number, two basic questions arise: (i) the magnitudes o f various B-violating amplitudes; and (ii) the kind o f selection rules governing these processes. The present experimental limit [1,2] on the nucleon decay lifetime of r N ~> 1030 years, implies that the strengths o f Bviolating processes in nuclei are very weak indeed. The current wisdom is that the electroweak group is SU(2)L × U(1) (where SU(2)L is the left-handed weak isospin group and U(1) is the weak hypercharge group) and that this group, together with the color group SU(3)c representing the strong interaction, is embedded in the grand unification group SU(5). If this gauge
'~ Work supported by National Science Foundation Grant No. Phy-78-24888 and CUNY PSC-BHE research award No. RF 13096.
theoretical picture is correct, it can be shown that baryon number non-conserving processes must nevertheless conserve [3] B - L. This means that AB = --1, 2xL = - 1 processes like p -+ e+lr0, n ~ e+n - can take place and that their decay probabilities will be extremely small and directly correlated with the lifetime for nuclear stability. Taking their cue from the B - L conservation law, experiments are underway to detect proton decay [4]. Recently, we have shown [5] that the l e f t - r i g h t symmetric electroweak group SU(2)L × SU(2)R × U(1)B_L, while giving equivalent predictions to those of the standard electroweak group at energies thus far studied, has consequences for baryon number nonconserving processes that differ from those of the standard electroweak group. In particular, if the aforementioned l e f t - r i g h t symmetric group is embedded in the "partial unification" group SU(2)L × SU(2)R × S U ( 4 ' ) [where S U ( 4 ' ) contains U(1)B_ L × SU(3)c as a sub-group and B - L is the fourth color], the dominant baryon number non-conserving process does not conserve B - L. More specifically, the dominant process is a AB = 2, AL = 0 nucleon transition rather than the AB 1, AL = -- 1 nucleon--antilepton transition. The possibility of differentiating between the standard gauge theory and the l e f t - r i g h t symmetric gauge =
- -
183
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theory by means o f baryon number non-conserving processes is most interesting [6] *~ but equally important is the possibility o f distinguishing experimentally between a "free" AB -- 2 nucleon transition and a " b o u n d " 2xB = 2 nucleon transition. The "free" A B = 2 nucleon transition is characterized b y the process n ~ ~ (neutron oscillation) while the " b o u n d " AB = 2 nucleon transition is represented by the process N 1 + N 2 ~ pions (with N 1 and N 2 two nucleons inside a nucleus), The "free" A B = 2 nucleon transition can be represented by a six-quark diagram whereas the " b o u n d " AB = 2 nucleon transition requires the addition of a spectator quark line (see ref. [5]). The experimental detection o f b o t h types o f AB = 2 nucleon transitions may be possible but, as it will turn out, the first method will be more sensitive under suitable conditions. This note is devoted to the basic phenomenology o f neutron oscillations which, in our view, should be searched for quite independently o f the particular gauge theory. In section 2 we consider free neutron oscillations, in section 3 the effect o f an external field on neutron oscillations is treated, in particular the effect o f an external magnetic field on the opposite magnetic moments o f n and ~. Section 4 speaks to the possibility o f detecting neutron oscillations in the presence o f a reduced earth's magnetic field.
28 July 1980
Thus consider the question o f coherent regeneration o f antineutrons starting with a beam of neutrons, in free space. The starting point is the n - f i mass matrix exactly as in the case of the K 0 - K 0 system [7]: 2~mass = ~ 3 4 ~ , where
and
In eq. (2), 6 m is the AB = 2 transition mass between n and ~ states. The equality o f the diagonal elements follows from C P T invariance and that of the off-diagonal elements from CP invariance. The eigenstates of M can be written as nl, 2 = (n -+fi)/,v/2 ,
of neutral kaon oscillations is well known and this A S = 2 weak transition has played an important role in the theory o f weak interactions [7]. The 2xB = 2 n - ~ transition can be treated in a similar fashion in the absence o f an external field. As soon as an external field is present, whether it is a magnetic field, say o f the earth, or the nuclear interaction (which results from the proximity of other nucleons in an atomic nucleus), the phenomenology o f neutron oscillations is changed dramatically. To see how this comes about and affects the possibilities for experimental detection, we recapitulate briefly the theory o f free neutron oscillations. *! Table 1 in t e l [6] shows that whereas the grand unification group SU(5) predicts proton decay but no neutron oscillation (even with an extra Higgs boson in the {15} representation - contrary to a statement in ref. [8] - the reverse is true for the "partial unification" group SU(2) L x SU(2) R X SU (4'). The deteetion of neutron o sciUations thereby becomes an important test of unification models. 184
(3)
with masses m l , 2 = A +-6rn .
(4)
Next, the amplitude for finding an fi at time t starting with a beam o f neutrons at t = 0 is: In(t)) --- exp ( - 7 t / 2 )
2. Free neutron oscillations. The phenomenology
(1)
×
(5)
[ I n l ( O ) ) e x p ( - i r n t t ) + In2(O)) exp ( - i m 2 t )
t
where we have assumed that the decay widths of n 1and n 2 are equal and denoted by 7- The probability for finding an ~ at time t, i.e. P~(t), follows from eq. (1) P~(t) = ½ e -~/t (1 - cos 2 6 m t ) .
(6)
If t ~ 1~fro, then we obtain P-~(t) ~- e - ~ t ( 6 m t ) 2 .
(7)
The upper limit on 6 m can be estimated from the upper limit on the AB = 2 " b o u n d " nucleon transition rate, F/,B= 2 by means of the relation [8] *~: 5m ~ (F ~B=zM)I/2 ,
(8)
where M is a typical hadronic mass, say 10 GeV. Using [2] 10 20 yr as the lower limit on the lifetime for 4:2 Essentially the same relation was deduced by Kuz'min [9].
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PHYSICS LETTERS
nuclear stability [6] and allowing a factor 10 - 2 for the overlap effect o f the wave functions o f the two nucleons in the nucleus [ I 0 ] , we find 6rn ~ 10 -20 eV, which corresponds to a mixing time rn_ ~ ~> 105 s. It follows that if a beam o f N free neutrons is allowed to travel for a time T before hitting a target, the number of antineutrons,/V, at the target after T should be (assuming ")'T ~ I)
~ N(Tlrn_~)2 .
(9)
With reasonable values of N and T, eq. (9) would be rather promising for a "free" neutron oscillation experiment. However, there is a complicating factor resulting from the presence of the earth's magnetic field which shifts the energy levels of n and fi by opposite amounts -+/.tB ~ 10 -11 eV, which is much larger than 6m. We deal with this situation in the next section.
3. Oscillation o f neutrons in an external field. If the neutron oscillation takes place in an external field (such as the earth's magnetic field or any other field) so that the CPT theorem need not be respected, the picture outlined in the previous section changes and we have M=
(A t 8rn) 8rn A 2 "
in 2 ) ~ - - 0 [ n ) + [ ~ ) ,
where
0 ~- 8 m l A M ,
(11)
where A M -= M 1 - M 2 ~ A 1 - A 2 i f A 1 - A 2 >> fro. The analog of eq. (6) for this case is:
P~(t) -~ ½0 2 [1 - cos A g t ] .
(12)
Eq. (12) is interesting in two limiting cases: (a) AMt >> 1 and (b) A M t ~ 1. In case (a), the second term oscillates rapidly and the average probability for finding an h, (P~)av, becomes (P~)av -~ 1(SIn/AM)2 .
(13)
In case (b), we get:
P~(t)-~1 (Smt)2 ,
4. Expetqmental detection o f neutron oscillations. Eqs. (13) and (14) can now be used to suggest optimal conditions for a proposed reactor experiment to detect neutron oscillations [11]. In such an experiment with thermal neutrons, t ~ 10 - 2 s and if the magnetic field of the earth is not shielded, &Mt >> 1 and eq. (13) applies. On the other hand, if the earth's magnetic field is "degaussed" by a factor o f 10 3 or more, eq. (14) applies and the experiment becomes quite favorable. The sensitivity o f the experiment depends, o f course, on the detailed design, but N = 5 X 1013 neutrons/s seems possible [11 ] and with the numbers of eq. (14) corresponding to a "degaussing" factor o f 10 3, values of N as large as 10 6 yr -1 are not incompatible with the present lower limit on the lifetime for nuclear stability. Whether AB = 2 nucleon transitions take place or not, the above considerations do indicate that the search for "free" neutron oscillations (n ~ 5) in a shielded earth's magnetic field can, in principle, yield a sensitivity many orders o f magnitude greater than the search for " b o u n d " AB = 2 nucleon transitions, e.g. through the process n + p ~ rr's in a nucleus [8].
(10)
The neutron mass eigenstates in this case are In 1 ) ~ l n ) + 0 [ ~ ) ,
28 July 1980
(14)
which recaptures the field-free result (7) (as long as ~ t ~ 1).
We thank L.W. Mo, Riazuddin and G. Senjanovic for very useful discussions. We are especially indebted to Professor Richard Wilson for asking the questions that led to this paper. As this paper was going to press, a preprint by M. Baldo-Ceolin arrived with similar resuits and a proposed experiment with "cold" neutrons from the Grenoble reactor. The measurement of neutron oscillations as a test of unification models is not considered.
References [1] For a review and earlier references, see: M. Goldhaber, in: Unification of elementary forces and gauge theories, eds. D. Cline and F. Mills (Academic Press, New York, 1977) p. 531; H.S. Gurr, W.R. Kropp, F. Reines and B.S. Meyer, Phys. Rev. 158 (1967) 1321; F. Reines and M,F. Crouch, Phys. Rev. Lett. 32 (1974) 493. [2] J. Learned, F. Reines and A. Soni, Phys. Rev. Lett. 43 (1979) 907; F. Reines, private communication. [3] S. Weinberg, Phys. Rev. Lett. 43 (1979) 1566; F. Wilczek and A. Zee, Phys. Rev. Lett. 43 (1979) 157l. [4] For a review of the proposed experiments, see: L. Sulak, Proc. Weak interaction Workshop (Virginia Polytechnic Institute, 1979). 185
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[5] R.N. Mohapatra and R.E. Maxshak, VPI-HEP-80/1, to be published in Phys. Rev. Lett. [6] R.E. Marshak and R.N. Mohapatra, VPI-HEP-80/2, to be published in Proc. Orbis Scientiae (Coral Gables, 1980). [7] Cf. R.E. Marshak, Riazuddin and C.P. Ryan, Theory of weak interactions in particle physics (Wiley-lnterscience, New York, 1969); also P.K. Kabir, CP puzzle (Academic Press, New York, 1967).
186
28 July 1980
[81 S.L. Glashow, Harvard preprint HUTP-79/A059. [9] V.A. Kuz'min, Sov. Phys. JETP 12 (1970) 228. [10] G. Feinberg, M. Goldhaber and B. Steigman, Phys. Rev. 18D (1979) 1602. [ 11 ] R. Wilson, private communication.
PHYSICS LETTERS
28 July 1980
PHENOMENOLOGY OF NEUTRON OSCILLATIONS ~ R.N. MOHAPATRA Department of Physics, City College of the City University of N ew York, New York, N Y 10031, USA and R.E. MARSHAK Department o f Physics, VirginiaPolytechnic Institute and State University, Blaeksburg, VA 24061, USA Received 14 April 1980
We have recently argued that gauge models with spontaneously broken local B - L symmetry can lead to large AB = 2 nucleon transition amplitudes for processes such as n-E oscillations and n + p ~ pions. We discuss the phenomenology of neutron oscillations both in vacuum and in the presence of external fields. These considerations indicate that neutron oscillation experiments can be much more sensitive to the AB --- 2 amplitude than tests of nucleon instability.
1. Introduction. The issue o f baryon number nonconservation has been the subject o f discussion for a long time [1 ]. It has attracted a great deal o f attention in recent years due to the fact that attempts to unify strong, weak and electromagnetic interactions within a gauge theory framework almost invariably lead to violation o f baryon number and hence to the interesting possibility o f proton decay. Once one accepts the possibility o f baryon number being only an approximately conserved quantum number, two basic questions arise: (i) the magnitudes o f various B-violating amplitudes; and (ii) the kind o f selection rules governing these processes. The present experimental limit [1,2] on the nucleon decay lifetime of r N ~> 1030 years, implies that the strengths o f Bviolating processes in nuclei are very weak indeed. The current wisdom is that the electroweak group is SU(2)L × U(1) (where SU(2)L is the left-handed weak isospin group and U(1) is the weak hypercharge group) and that this group, together with the color group SU(3)c representing the strong interaction, is embedded in the grand unification group SU(5). If this gauge
'~ Work supported by National Science Foundation Grant No. Phy-78-24888 and CUNY PSC-BHE research award No. RF 13096.
theoretical picture is correct, it can be shown that baryon number non-conserving processes must nevertheless conserve [3] B - L. This means that AB = --1, 2xL = - 1 processes like p -+ e+lr0, n ~ e+n - can take place and that their decay probabilities will be extremely small and directly correlated with the lifetime for nuclear stability. Taking their cue from the B - L conservation law, experiments are underway to detect proton decay [4]. Recently, we have shown [5] that the l e f t - r i g h t symmetric electroweak group SU(2)L × SU(2)R × U(1)B_L, while giving equivalent predictions to those of the standard electroweak group at energies thus far studied, has consequences for baryon number nonconserving processes that differ from those of the standard electroweak group. In particular, if the aforementioned l e f t - r i g h t symmetric group is embedded in the "partial unification" group SU(2)L × SU(2)R × S U ( 4 ' ) [where S U ( 4 ' ) contains U(1)B_ L × SU(3)c as a sub-group and B - L is the fourth color], the dominant baryon number non-conserving process does not conserve B - L. More specifically, the dominant process is a AB = 2, AL = 0 nucleon transition rather than the AB 1, AL = -- 1 nucleon--antilepton transition. The possibility of differentiating between the standard gauge theory and the l e f t - r i g h t symmetric gauge =
- -
183
Volume 94B, number 2
PHYSICS LETTERS
theory by means o f baryon number non-conserving processes is most interesting [6] *~ but equally important is the possibility o f distinguishing experimentally between a "free" AB -- 2 nucleon transition and a " b o u n d " 2xB = 2 nucleon transition. The "free" A B = 2 nucleon transition is characterized b y the process n ~ ~ (neutron oscillation) while the " b o u n d " AB = 2 nucleon transition is represented by the process N 1 + N 2 ~ pions (with N 1 and N 2 two nucleons inside a nucleus), The "free" A B = 2 nucleon transition can be represented by a six-quark diagram whereas the " b o u n d " AB = 2 nucleon transition requires the addition of a spectator quark line (see ref. [5]). The experimental detection o f b o t h types o f AB = 2 nucleon transitions may be possible but, as it will turn out, the first method will be more sensitive under suitable conditions. This note is devoted to the basic phenomenology o f neutron oscillations which, in our view, should be searched for quite independently o f the particular gauge theory. In section 2 we consider free neutron oscillations, in section 3 the effect o f an external field on neutron oscillations is treated, in particular the effect o f an external magnetic field on the opposite magnetic moments o f n and ~. Section 4 speaks to the possibility o f detecting neutron oscillations in the presence o f a reduced earth's magnetic field.
28 July 1980
Thus consider the question o f coherent regeneration o f antineutrons starting with a beam of neutrons, in free space. The starting point is the n - f i mass matrix exactly as in the case of the K 0 - K 0 system [7]: 2~mass = ~ 3 4 ~ , where
and
In eq. (2), 6 m is the AB = 2 transition mass between n and ~ states. The equality o f the diagonal elements follows from C P T invariance and that of the off-diagonal elements from CP invariance. The eigenstates of M can be written as nl, 2 = (n -+fi)/,v/2 ,
of neutral kaon oscillations is well known and this A S = 2 weak transition has played an important role in the theory o f weak interactions [7]. The 2xB = 2 n - ~ transition can be treated in a similar fashion in the absence o f an external field. As soon as an external field is present, whether it is a magnetic field, say o f the earth, or the nuclear interaction (which results from the proximity of other nucleons in an atomic nucleus), the phenomenology o f neutron oscillations is changed dramatically. To see how this comes about and affects the possibilities for experimental detection, we recapitulate briefly the theory o f free neutron oscillations. *! Table 1 in t e l [6] shows that whereas the grand unification group SU(5) predicts proton decay but no neutron oscillation (even with an extra Higgs boson in the {15} representation - contrary to a statement in ref. [8] - the reverse is true for the "partial unification" group SU(2) L x SU(2) R X SU (4'). The deteetion of neutron o sciUations thereby becomes an important test of unification models. 184
(3)
with masses m l , 2 = A +-6rn .
(4)
Next, the amplitude for finding an fi at time t starting with a beam o f neutrons at t = 0 is: In(t)) --- exp ( - 7 t / 2 )
2. Free neutron oscillations. The phenomenology
(1)
×
(5)
[ I n l ( O ) ) e x p ( - i r n t t ) + In2(O)) exp ( - i m 2 t )
t
where we have assumed that the decay widths of n 1and n 2 are equal and denoted by 7- The probability for finding an ~ at time t, i.e. P~(t), follows from eq. (1) P~(t) = ½ e -~/t (1 - cos 2 6 m t ) .
(6)
If t ~ 1~fro, then we obtain P-~(t) ~- e - ~ t ( 6 m t ) 2 .
(7)
The upper limit on 6 m can be estimated from the upper limit on the AB = 2 " b o u n d " nucleon transition rate, F/,B= 2 by means of the relation [8] *~: 5m ~ (F ~B=zM)I/2 ,
(8)
where M is a typical hadronic mass, say 10 GeV. Using [2] 10 20 yr as the lower limit on the lifetime for 4:2 Essentially the same relation was deduced by Kuz'min [9].
Volume 94B, number 2
PHYSICS LETTERS
nuclear stability [6] and allowing a factor 10 - 2 for the overlap effect o f the wave functions o f the two nucleons in the nucleus [ I 0 ] , we find 6rn ~ 10 -20 eV, which corresponds to a mixing time rn_ ~ ~> 105 s. It follows that if a beam o f N free neutrons is allowed to travel for a time T before hitting a target, the number of antineutrons,/V, at the target after T should be (assuming ")'T ~ I)
~ N(Tlrn_~)2 .
(9)
With reasonable values of N and T, eq. (9) would be rather promising for a "free" neutron oscillation experiment. However, there is a complicating factor resulting from the presence of the earth's magnetic field which shifts the energy levels of n and fi by opposite amounts -+/.tB ~ 10 -11 eV, which is much larger than 6m. We deal with this situation in the next section.
3. Oscillation o f neutrons in an external field. If the neutron oscillation takes place in an external field (such as the earth's magnetic field or any other field) so that the CPT theorem need not be respected, the picture outlined in the previous section changes and we have M=
(A t 8rn) 8rn A 2 "
in 2 ) ~ - - 0 [ n ) + [ ~ ) ,
where
0 ~- 8 m l A M ,
(11)
where A M -= M 1 - M 2 ~ A 1 - A 2 i f A 1 - A 2 >> fro. The analog of eq. (6) for this case is:
P~(t) -~ ½0 2 [1 - cos A g t ] .
(12)
Eq. (12) is interesting in two limiting cases: (a) AMt >> 1 and (b) A M t ~ 1. In case (a), the second term oscillates rapidly and the average probability for finding an h, (P~)av, becomes (P~)av -~ 1(SIn/AM)2 .
(13)
In case (b), we get:
P~(t)-~1 (Smt)2 ,
4. Expetqmental detection o f neutron oscillations. Eqs. (13) and (14) can now be used to suggest optimal conditions for a proposed reactor experiment to detect neutron oscillations [11]. In such an experiment with thermal neutrons, t ~ 10 - 2 s and if the magnetic field of the earth is not shielded, &Mt >> 1 and eq. (13) applies. On the other hand, if the earth's magnetic field is "degaussed" by a factor o f 10 3 or more, eq. (14) applies and the experiment becomes quite favorable. The sensitivity o f the experiment depends, o f course, on the detailed design, but N = 5 X 1013 neutrons/s seems possible [11 ] and with the numbers of eq. (14) corresponding to a "degaussing" factor o f 10 3, values of N as large as 10 6 yr -1 are not incompatible with the present lower limit on the lifetime for nuclear stability. Whether AB = 2 nucleon transitions take place or not, the above considerations do indicate that the search for "free" neutron oscillations (n ~ 5) in a shielded earth's magnetic field can, in principle, yield a sensitivity many orders o f magnitude greater than the search for " b o u n d " AB = 2 nucleon transitions, e.g. through the process n + p ~ rr's in a nucleus [8].
(10)
The neutron mass eigenstates in this case are In 1 ) ~ l n ) + 0 [ ~ ) ,
28 July 1980
(14)
which recaptures the field-free result (7) (as long as ~ t ~ 1).
We thank L.W. Mo, Riazuddin and G. Senjanovic for very useful discussions. We are especially indebted to Professor Richard Wilson for asking the questions that led to this paper. As this paper was going to press, a preprint by M. Baldo-Ceolin arrived with similar resuits and a proposed experiment with "cold" neutrons from the Grenoble reactor. The measurement of neutron oscillations as a test of unification models is not considered.
References [1] For a review and earlier references, see: M. Goldhaber, in: Unification of elementary forces and gauge theories, eds. D. Cline and F. Mills (Academic Press, New York, 1977) p. 531; H.S. Gurr, W.R. Kropp, F. Reines and B.S. Meyer, Phys. Rev. 158 (1967) 1321; F. Reines and M,F. Crouch, Phys. Rev. Lett. 32 (1974) 493. [2] J. Learned, F. Reines and A. Soni, Phys. Rev. Lett. 43 (1979) 907; F. Reines, private communication. [3] S. Weinberg, Phys. Rev. Lett. 43 (1979) 1566; F. Wilczek and A. Zee, Phys. Rev. Lett. 43 (1979) 157l. [4] For a review of the proposed experiments, see: L. Sulak, Proc. Weak interaction Workshop (Virginia Polytechnic Institute, 1979). 185
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[5] R.N. Mohapatra and R.E. Maxshak, VPI-HEP-80/1, to be published in Phys. Rev. Lett. [6] R.E. Marshak and R.N. Mohapatra, VPI-HEP-80/2, to be published in Proc. Orbis Scientiae (Coral Gables, 1980). [7] Cf. R.E. Marshak, Riazuddin and C.P. Ryan, Theory of weak interactions in particle physics (Wiley-lnterscience, New York, 1969); also P.K. Kabir, CP puzzle (Academic Press, New York, 1967).
186
28 July 1980
[81 S.L. Glashow, Harvard preprint HUTP-79/A059. [9] V.A. Kuz'min, Sov. Phys. JETP 12 (1970) 228. [10] G. Feinberg, M. Goldhaber and B. Steigman, Phys. Rev. 18D (1979) 1602. [ 11 ] R. Wilson, private communication.