Comments on Testing Charge Conservation and the Pauli Exclusion Principle
A short critic:i l revi ew i · given or experiments aimed nt testing electric charge con ~c rvat i on nnd the exdusion principle. Lt is stressed tbat there is no ~c lt-co u s i s l c 11t th eo ry which could describe , eve n phcnomcnologicall , violations of charge con crvation und/or of the c clusi 11 prindplc. In the case or charge-nom.:unscrva tion the d cc i ~ iv rol •of Ion itutlinnl photons is unde rlined. New :;uggcslion · to l c..~ t th e Pauli principle arc dlscusscd.
Key Words: bosons, fermions, Pauli principle, charge conservation
There exist about 30 papers dealing wilh the possibility of violati n of charge conservation and/or the excl usion principle, pu lished during la. t 30 years. A short review of the e papers i given bel w. The two subjects of the review are interconnected b cause the two phenomena are often searched for in the same experiments. Both principles-electric charge conservation and the exclusion principle-are among the most fundamental one in modern phys ics. As will be clear from the revi w they are singled out by the orists' inability to construct a self-consistent phenomenol gica l framework for a quantitative description of the degree of accuracy with which these principl s have been tested. The review cons.is ts of four parts: 1. Experiments Already Done; 2. Theoretical Papers on Charge Non-conservation; 3. Theoretical
Originally appeared as Institute of Theoretical Physics Preprint ITEP 89-22 (1989) .
Co 111111e111.t Nucl. Part. f'liy•-. © 1989 Gordon and Breach, 1989, Vol. l9. No. J . pp. 99- 116 Science Publishers, Inc. Rep tints avnllablc dirccrl fm111 the publisher Printed in Great Britain Photocopying permmed b liccn c only
99 Papers on the Pauli Principle Violation; 4. Proposals for Future Experiments.
1. EXPERIMENTS ALREADY DONE
l.A. Exclusive Experiments with Electrons Approximately 30 years ago G. Feinberg and M. Goldhaber1 car ried out an experiment with the NaI detector aimed at testing electron stability. They looked for characteristic x-radiation ex pected to be emitted in the process of filling the vacancy in the atomic shell of iodine (see Fig. 1) and deduced a lower limit for the electron lifetime around 1018 years. In 1965 M. Moe and F. Reines2 raised this limit to 1020 years; and by searching for monochromatic -y-rays with energy m)2 they deduced the lower 22 limit for the lifetime for the process e ___.,. v-y: T(e ___.,. v-y) > 4 x 10 yr. In 1974 F. Reines and H. Sobel3 used the result2 of their search for characteristic iodine x-rays to put a limit on possible violation of the Pauli principle. This time they considered a transition not to a vacancy but to a filled atomic shell (see Fig. 2). A similar search for x-radiation carried out in 1975 by R. Stein berg et al. 4 with a germanium detector gave Te > 5 x 1021 yr. In 1979 E. Kovalchuk, A. Pomansky and A. Smolnikov5 raised the limit to 2 x 1022 yr (again with NaI); and in 1983 E. Bellotti et al. 6 obtained the same result with Ge. ZP----
a) b) c) FIGURE 1 (a) Filled ls- and 2p-shells of iodine. (b) Electron mysteriously dis appears from ls-shell, violating charge conservation. (c) Electron from p-shell jumps to ls-shell emitting a characteristic x-ray.
100 2.p
is---~
a) b) FIGURE 2 (a) Filled ls- and 2p-shells of iodine. (b) Electron from 2p-shell jumps to ls-shell, violating the Pauli principle.
In 1986 F. Avignone III et al. 7 repeated the 1965 search by M. Moe and F. Reines for e ~ vy decay, this time with a ger manium detector, and concluded that T(e ~ v')') > 1.5 x 1025 yr. All experiments described above have tested electrons: they searched for x-rays or ')'-rays caused by the decay of an electron or x-lines caused by violation of the exclusion principle for elec trons.
l.B. An Exclusive Experiment with Nucleons The above considerations were applied not only to electrons, but to nucleons as well. In 1979 B. Logan and A. Ljubicic8 tested the Pauli principle by searching for ')'-rays with energy of the order of 20 MeV. Such ')'-rays were expected to signal the transition of a nucleon in the 12C-nucleus from the 2p-shell to a filled ls-shell. They obtained a lower limit for the characteristic time of such a transition and hence for the creation of a "non-Paulian" nucleus of carbon 12C with five nucleons on the lowest s-shell: T(12C ~ 12t'Y) > 2 x 1020 yr.
l.C. Inclusive Experiments with Nucleons Inclusive experiments differ from exclusive ones by not choosing a given mechanism through which the phenomenon under inves tigation is realized. In the case of electric charge it looks like a certain kind of a "black box." Charge Q1 "enters" the box and charge Q2 leaves it (see Fig. 3). The first experiment of inclusive type was done in 1979 by
101 18
FIGURE 3 The "black box," symbolizing the unknown mechanism of charge non conservation.
9 87 B. Norman and A. Seamster, who established that T( Rb ~ 87Sr) > 1.9 x 1018 yr. In 1980 I. Barabanov et al. 10 established for 1 another charge-non-conserving transformation T(7 Ga ~ 71 Ge) > 2.3 x 1023 yr. This result was obtained as a byproduct of developing radiochemical technique for the gallium-germanium detector of low energy solar neutrinos which is under construction at the Bak san Neutrino Observatory. l.D. Global Limit for Charge Non-conservation The idea of a global approach to the possible charge non-conser vation was advanced in 1976 by A. Pomansky11 who considered the balance of electric currents in the atmosphere of the Earth and concluded that the imbalance current due to decay of electrons or more generally due to charge non-conservation in the atoms of the Earth cannot be larger than 200 A. Taking into account that 51 the Earth contains 2 x 10 electrons, be obtained T .. > 5 x 1022 yr. A review of experimental tests of the Pauli exclusion principle and of charge non-conservation was given in 1980 by F. Reines and H. Sobel. 12
2. THEORETICAL PAPERS ON CHARGE NON-CONSERVATION
In 1978 in the papers by Ya. B. Zeldovich, M. B. Voloshin and 13 14 myself • a number of problems was considered which arise when one tries to construct a self-consistent phenomenological descrip tion of non-conservation of electric charge. The main conclusions of our papers are summarized below in Sections 2.A-2.C.
102 2.A. Impossibility of Spontaneous Breaking of Charge Conservation It was shown that unlike spontaneous breaking of electroweak symmetry, electric charge non-conservation cannot be realized spontaneously because the photon, unlike the Z-boson, is ex tremely light or (which is even worse) massless. As is well known the Higgs mechanism of spontaneous breaking of a U(l) gauge symmetry calls for the existence of a charged scalar field. After the breaking the imaginary part of this field becomes the third (longitudinal) component of the now massive vector boson, while the real part becomes a Higgs boson. The characteristic mass parameter of the charged scalar field determines the mass of the Higgs field and of the gauge boson. In the case of electroweak theory this mass parameter is very large (of the order of W, Z-boson masses). But in the case of charge non-conservation it has to be extremely small, of the order of the photon mass. As photons are practically massless, the charged scalar boson must also be practically massless. Emission and absorption of such charged almost massless bosons would drastically change the whole electrodynamics. So their existence in our world is definitely ruled out. On the other hand, the non-spontaneous, explicit breaking of charge-conservation would lead to catastrophic bremsstrahlung of longitudinal photons.
2.B. Catastrophic Bremsstrahlung in the Case of Explicit Charge-Non-conservation When charge (and current) is conserved the amplitude of emission of the longitudinal photon is negligibly small, being proportional to em./w, where e is electric charge, m'Y is the mass of the photon and w the photon's energy (we use h, c = 1 units). When the charge is not conserved the situation is opposite: the amplitude of emission of a longitudinal photon is proportional to ewlm'Y and therefore is extremely large. As a result the probability of emission of two longitudinal photons is larger than that of one, of three is larger than of two, and so on.
103 )' 1 " e • /J··· / /.-(.. e--<: v
a) b)
FIGURE 4 (a) Hypothetical decay e -> vvv which violates charge conservation. (b) Catastroph ic bremsstrahlung, accompanying decay e -> vvv.
If we assume that an electron can decay into three neutrinos with an extremely small coupling constant g (see Fig. 4(a)), then such decay occurs with the emission of an immense number of longitudinal photons (see Fig. 4(b)). The energy released in the decay which is equal to the electron mass is carried away by these photons, not by neutrinos, and the energy of each of these photons is extremely small. The same refers to the decay e ~ v-y, which is transformed into decay e ~ v + N'Y-y (see Fig. 5) , and one can show that
2) 1/3 N = 3 ~. m, = 1014_1021. 'Y ( 41T m ~
Here the smaller number (10 14) corresponds to the upper limit of m-y derived from the measurement of the magnetic field of Jupiter. The larger number (1021 ) corresponds to a less certain limit
. I• LI!·/(.~
a) b) FIGURE 5 (a) Hypothetical decay e-> V'Y, which violates charge conservation. (b) Catastrophic bremsstrahlung, accompanying decay e-> vy .
104 llm-y ~ 1022 cm derived from the observed dimensions of the gal actic magnetic fields. The probability of the electron decay is given (up to unknown preexponent factor) by the expression:
We see that all energy which is rel ased in the decay is carried away by infra-infra . . . infrared photons, that is by a practically static field. T ber f re there shouJd be no )'-line with energy m)2, and there shouJd be no characteristic x-lines when electron disappears in an atom (the dimension of an atom are negligible compared with the characteri tic wavelength of the longitudinal photons, o the atom may 1 e consid r "d t be point-like) , and the alm st tatic field of longitudinal phot ns is practicall y unobserv able. Thus, one has to conclude that all exclusive experiments dis cussed above were unable to discover electron decay or charge non-conserving nuclear transformations of the type discussed above even if one believes that such phenomena do occur in Nature.* Only the limits obtained by inclusive non-spectroscopic searches and the global limit remain valid.
2.C. "Self-healing" by Radiative Corrections The previous section may create an impression that explicit break ing of charge conservation is a reasonable basis for a self-consistent theory of this phenomenon. But such an impression is a deceiving one. The point is that large probability of emitting real longitudinal photons is accompanied by large probability of emission and ab sorption of virtual longitudinal photons by the same particle. That means that radiative corrections are expected to be so huge that the term "corrections" can be used only by tradition. For real longitudinal photons it can be shown that their emission takes place at that point of the Feynman diagram at which charge conservation is violated (see Fig. 6). For virtual photons both the
•Atomic x-rays could signal decay of an electron into so-called minicharged particles (see below).
105 rr -rr oJ !L-1t· <::
a) b) FIGURE 6 Diagrams of type (a) are equivalent to diagrams of type (b), where all photons are emitted from a single vertex. emission and absorption take place at that point (Fig. 7). As a result the bare coupling constant g of the decay e ~ vvv is renor malized and becomes
g = ge - «/4'1T(A 2 /m~) where A is the parameter of ultraviolet cut-off. It is obvious that A >> me and therefore the enhancement factor produced by real photons is overcompensated by the suppression factor produced by virtual photons, so that an infinite bare coupling g has to be taken as a starting point of the theory, which does not seem to be plausible. We see that the fact that the photon is (practically) massless "heals" the theory. The lightness of light is on guard for charge conservation.
2.D. Minicharged Particles and Spontaneous Breaking of Charge Conservation As was explained in Section 2.A the possibility of spontaneous
~~... FIGURE 7 Virtual longitudinal photons renormalize the vertex e--> vvv.
106 breaking of charge conservation by a scalar field with unit charge (Q
Q, - QjNq,, Q. - 2QjN, Q. - 3QjNq,, ...
Q. - (N<1> - l)QjNq, Q,
with a "ladder" of vertices:
According to Ref. 14, N ;;;:: 100. According to an estimate by M. B. Voloshin (see Ref. 21) N ;;;:: 108 ; otherwise ordinary ca pacitors would discharge too swiftly by emitting -pairs.
2.E. Theoretical Papers of the Last Two Years The last two years have witnessed a definite revival of interest in the problem of charge non-conservation. The possibility of con structing a renormalizable theory with explicitly non-conserved electromagnetic current was discussed in 1986 by H. Nakazato et al. 16 Three papers appeared in 1987. An attempt to sponta neously break charge conservation in the framework of broken SU(S)-symmetry was undertaken by J. Huang.17 S. Nussinov has considered the influence of an external potential on electron decay. 18 R. Mohapatra19 has proposed a theoretical model according to which charge non-conservation is caused by electron-positron vac uum oscillations and has conjectured that such a theory is only logarithmically divergent. In 1988 a preprint by M. Suzuki20 ap peared which discussed minicharged particles. Recently (1988) all these papers (except Ref. 20) were reviewed and critically analyzed by M. Tsypin21 whose main conclusion is that the verdict of Refs. 13 and 14 is not refuted.
107 3. THEORETICAL PAPERS ON THE PAULI PRINCIPLE VIOLATION
3.A. Years 1930-1980 A non-conformist approach to the Pauli principle could be traced to remarks by P.A. M. Dirac, W. Pauli and E. Fermi. By carefully reading the famous books by Dirac22 and Pauli,23 one can conclude that in the framework of Quantum Mechanics with a Hamiltonian which is permutationally invariant, transitions to a filled shell are forbidden independent of the validity of the Pauli principle, be cause such transitions would change the permutational symmetry of a wave function of a given set of particles. In 1934 E. Fermi discussed in one of his popular science articles24 the possibility that electrons are a "little bit" non-identical. He pointed out that a tiny non-identity would drastically change the properties of atoms during the billions of years of their existence. In 1971 V. L. Luboshitz and M. I. Podgoretskii25 discussed a model in which electron presents a superposition of a large number of almost degenerate mass eigenstates. In this model the properties of the electron have slightly changed with time. In 1980 R. Amado and H. Primakoff26 applied arguments, anal og us to the above argument of irac and Pauli, to the interpre tati on of experiments by F. Reine and H. Sobel3 and by B. Logan and A. Ljubicic.11 The conclusion f Ref. 26 was that in the frame work o~ Quantum Mechanics the Pauli-forbidden transitions searched for in Refs 3 and 8 are forbidden even if the Pauli principle is violated.
3.B. Papers of the Last Two Years A new idea was introduced in the field by A. Yu. Ignatiev and V. A. Kuzmin in 1987. To explain the essence of the idea let us be reminded that standard operators of creation and annihilation of fermions a+ and a resemble 2 x 2 Pauli matrices with one nonvanishing element in each matrix:
a~+ (0O O'1) a ~ (01 o0) ·
108 Ignatiev and Kuzmin27 introduced 3 x 3 matrices with two non vanishing matrix elements:
a+ - 0 013 10) , a - (0 0 00 ) ( 13 0 0 0 0 1 0
where 13 is a small parameter: 13 <<< 1. With such matrices they described one level, which can be either empty or filled by one electron, or (with a small amplitude 13) filled by two electrons. (When 13 equals zero we regain the standard fermionic behavior.) The generalization of this one level model to the field theory (to infinite number of levels) was attempted by 0. W. Greenberg and R. N. Mohapatra28 ·29 and by myself. 30 - 32 In Refs. 30 and 31 the operators at and a; for each level were constructed as explicit tensor products of 3 x 3 matrices, the anticommutators of the operators, {a+, a +}+ and {a, a}+ being proportional to the small parameter 13. But it became immediately obvious that such a theory violates locality, the superposition prin ciple and, last but not least, in such a theory there is no smooth transition from two infinitesimally close states to one state. This last property was conspicuous for the so-called ferbons (fermionic bosons). 32 Creation operators of ferbons anticommute when two states do not coincide, but the number of particles in a given state can be arbitrarily large. In short, my conclusions concerning the possibility of construction of a reasonable theory which violates the Pauli principle were pessimistic. On the contrary the paper by 0. W. Greenberg and R. N. Mohapatra28 was optimistic. They started with a trilinear com mutation relation:
where indices i, j, k label different levels (states) and parameters c1 and c2 are functions of 13:
C1 (2132 - 1)/(134 - 132 + 1),
C2 (132 - 2)/(134 - 132 + 1) .
109 The operator of the number of particles in a given state i has the form
The properties of the vacuum are determined by relations
By using these relations and the trilinear commutator one can construct an arbitrary many-particle state. The theory is local and preserves the superposition principle. However, it has its own fatal disease, pointed out by A. B. Govorkov33 in 1988 on the basis of his earlier more general work. 33 As was pointed out by Govorkov some of the many-particle states in the Greenberg-Mohapatra model have negative norm (negative probability). The simplest of these states has four electrons, three of them being on one level and the fourth on some other level. A straightforward calculation reveals that
2 a.+(a+a.+ - 0 l t k I a.+a+)a+IO)lI k I < 0
In their latest paper34 Greenberg and Mohapatra discuss the arguments by Govorkov and arrive at the conclusion: "Thus it is impossible to construct a free field theory for small violation of Fermi or Bose statistics. We don't expect interactions to change this situation." The failure of attempts to violate (on paper) the Pauli principle is a consequence of rather general theorems a ed n fundamental properties of field theory. Relevant (and compl mentary) Ii t. of references can be found in Ref . 31 and 34. By some accident the. lists do not contain a very important paper by G. Liiders and B. Zumino. 35 One can find an excellent explanation of how the Pauli principle makes QED self-consistent in the last lecture pub lished by R. P. Feynman.36
4. PROPOSALS FOR FUTURE EXPERIMENTS
A number of new experimental searches for Pauli principle vio 27 37 38 lation have been suggested during the last two years. - 31 • • Among
110 the suggested objects are stable non-Paulian molecules, atoms, atomic nuclei and hadrons. Let us consider some of them. 3 The ground state S1 of orthohelium can be searched for either by electron-spin resonance29 - 31 or by Zeeman splitting of an atomic beam. 37 An atom of Na with three electrons on the K-shell will lack its active valence electron and will chemically resemble Ne, but the optical spectrum of this false neon will differ from the spectrum of the genuine neon. So after separation and enrichment the false neon could be searched for with tunable lasers by well developed techniques, especially by the methods of resonance excitation and 30 31 photoionization · •38 or by using the neutron-activation analysis. 38 There is also a proposal to search for x-rays or Auger electrons from a piece of matter when bringing to this place "new" electrons via strong electric current. 28 ·29 If the exclusion principle is violated at the quark level28 unusual baryons belonging to ans-wave 70-plet of SU(6) should exist (among them an octet with JP = ~ + and a decuplet with JP = ~ +) and some of them should be stable. There is a Russian saying, "New is a well forgotten old." Some of the experiments suggested during the last year are similar to the experiments done many years ago, when physicists (at least some of them) were not absolutely sure that beta-particles are identical to the ordinary electrons, or that there is no other not yet discovered particles of the same mass and charge as ordinary electrons. For instance, in 1948 M. Goldhaber and G. Scharff Goldhaber39 stopped [3-rays from 14C in lead and looked for K shell x-rays from lead. They were able to set a 3% upper limit on the existence of such x-rays and therefore concluded that beta particles are identical to electrons. (Earlier studies on the identity of beta-particles and electrons were described in a review by 40 H. Crane. ) In 1968 E. Fishbach, T. Kirsten and 0. Shaeffer41 searched for a "false 9He" which they called 9Be1-a Be-atom with two ordinary electrons and two false electrons e', all of them on the K-shell. They established that abundance in the atmosphere of such "false 9He" is less than 10-6 of that of normal 4He. At present we have no doubts that there is only one particle with the mass and charge of an electron-that is, the electron itself. The second electron would be abundantly produced by col-
111 liders; it would destroy the excellent agreement of QED with a lot of experiments. So these old searches may be considered searches for the violation of the exclusion principle. Turning now from the Pauli principle to charge conservation, let us stress the great potential of gallium - germanium detectors at Baksan (60 tons of Ga) and Gran Sasso (30 tons of Ga). These detectors will be able to raise the lower limit for Ga-Ge sponta neous transformation time from 1023 yr to 1026 -1027 yr. In spite of the fact that at present we have no theoretical self consistent framework for a description of violation of charge con servation and/or the exclusion principle, I do not think that ex perimentalists should stop testing these fundamental concepts of modern physics. In fundamental physics if something can be tested it should be tested.
5. POSTSCRIPT
The above text was written almost a year ago and its Russian version has recently been published in the June 1989 issue of the Journal, "Uspekhi Fizicheskikh Nauk. " 42 After completion of this text, I learned about five other papers dealing with possible vio 43 44 lation of the Pauli Exclusion Principle. Three of them • •47 discuss or describe experimental tests and two others48.49 propose theo retical schemes for the violation of the Pauli Principle. V. Novikov and A. Pomansky43 suggest a search for those non Paulian isotopes, with atomic charge Z + 1, whose chemical an alogs with atomic charge Z have very low abundance; for instance, to look for non-Paulian carbon, which chemically appears like boron. As the abundance of normal boron is known to be 6 orders of magnitude smaller than the abundance of carbon, this would give an enhancement factor of the order of 106 in the search for "false boron." Especially promising are mass-spectroscopic searches for false " 12B" (the non-Paulian 12C), as ordinary 12B does not occur in nature. Other promising pairs of elements are fluorine neon and chlorine-argon. The search for false "F" and "Cl" using accelerator mass spectrometry is discussed by V. Novikov, A. Po mansky and E. Nolte. 44 The technique of accelerator mass-spec trometry is rather advanced (see for instance, Refs. 45 and 46
112 which present results of searches for some rare istopes at the level 14 16 of sensitivity 10 - -10- ). With this technique, lower limits for the lifetimes of Pauli-forbidden transitions in the ballpark of 1031 years could be achieved. An attempt to introduce a large number of "fresh" electrons into a copper sample and to observe X-rays was undertaken re cently by E. Ramberg and G. Snow. 47 In principle this experiment is similar to that of M. Goldhaber and G. Scharff-Goldhaber.38 However, this time, the "fresh" electrons were supplied not by a radioactive 13-source, but by a strong electric current. The integral of the current over time was about 15 x 106 coulombs = 1026 electrons. No X-rays were observed and the authors arrived at an upper limit
2 26 13 ::; 1.6 x 10- at 95% c.l. where 13 is the Pauli principle violating parameter, defined in Sec tion 3.B. 8 11 Let u now turn t the th oretical papers.'' -* · • Bi d nbarn P. ruini and H. van Dam 4~ consider the algebra of fgnati v and Kuzmin 27 using the formalism f J rd n pair . A a physical ex ampl e of such a pair they tak an ele tr n and a mu n. According to them, their model leads to the sam • apparent violation , yet preserves the Pauli principle [and) a tiny interacti n mixing muons into ele tron lates w uld lead lo a tripl t spin component in the helium atom gr und late, ignalling physically aJ1 apparent (but not actual violation of the Pauli principle .. . Any apparent vio lation is physically significant and implie new degree of free dom." Unfortunately, the electron-muon mixing cannot lead to the existence of orthohelium or to other apparent violations of the Pauli principle, because the mass of the electron and of the muon is different. On the other hand, the existence of another electron degenerate in ma s with the ordinary one is excluded by experi ment; this is mentioned in Section 4 above. o when referring to new degrees of freedom one has to consider "o cilia ting electrons' a la Fermi24 and Luboshitz and P dgoretskii. 25 A was explained in R f . 31 the mass eigen tates in t11is case have no definite electric charge.
113 The paper by V. Rahal and A. Campa49 considers possible vi olation of the Pauli principle in the framework of non-relativistic Quantum Mechanics, wiithout touching on the extremely difficult problems connected with Quantum Field Theory. As is well known in ordinary Quantum Mechanics all ~ 1080 electrons in the universe are in an antisymmetric state described by a Young tableau, which represents a column of N = 1080 boxes. Rahal and Campa assume that the actual world electron wave function consists of two col umns with the second column being much, much shorter than the first one. The smallness of the ratio of the number of boxes in the two columns determines the smallness of the violation of the Pauli Principle. Unfortunately such a wave function is non-invariant with respect to some permutations which means that in this scheme some elec trons are "more equal than others."
Acknowledgments
It is a pleasure to thank for its kind hospitality the Aspen Center for Physics where this postscript was written, Glenn Starkman for a helpful discussion and Heather Tharp for typing the postscript. L.B. OKUN Institute of Theoretical and Experimental Physics, Moscow, USSR
References
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