- Email: [email protected]
Searching for granularity of the vacuum
22 July 1999
Physics Letters B 459 Ž1999. 86–90
Searching for granularity of the vacuum D. Eichler Ben-Gurion UniÕersity, Beer-SheÕa 84105, Israel Received 23 February 1999; received in revised form 5 May 1999 Editor: P.V. Landshoff
Abstract The hypothesis is considered that the vacuum is a Lorentz non-invariant foam in which translational symmetry is spontaneously broken at the Planck or compactification scale. This could possibly be observed via Rayleigh scattering of ultrahigh energy quanta, and it appears to be ruled out for spin 1r2 leptons by SN 1987a, and independently testable for photons. A weaker version of the hypothesis predicts Žfor otherwise massless neutrinos. irreversible neutrino mixing over a length scale of l Planck Ž EnrEPlanck .y2 in comoving cosmic coordinates for a purely four dimensional universe, and, for neutrinos that have spin 3r2 in a higher dimensional manifold above the energy scale h E Planck , the mixing is predicted to take place over a scale of hy4 times this length. Such neutrino mixing might also be observable with atmospheric and solar neutrino experiments, and account for some of the observed deficits. Long distance terrestrial experiments would test the latter hypothesis unequivocably. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction Translational invariance of the vacuum is a sacred tenet of physics. One might suspect that, near the Planck scale, space may be grainy due, say, to condensates, quantum black holes, a spatially varying u parameter, or phase decorrelation of virtual modes. This raises the question of whether on such small scales translational symmetry is preserved or spontaneously broken. A grainy vacuum would suggest that physics at macroscopic scales is to some Žpossibly small. degree unpredictable. Hawking w1,2x has suggested that the formation and evaporation of quantum black holes leads to such unpredictability. The question has also been raised as to whether wavefunction collapse in quantum mechanics, an apparent breakdown of determinism, has anything to
do with strong non-linear effects in quantum gravity, though no self-consistent picture has been constructed along these lines. More generally, any Lagrangian that has non-renormalizable terms that contain field derivatives of higher order than the usual kinetic term Em fE mf may imply spontaneous breaking of translational symmetry. Probably, any breaking of translational invariance also involves breaking of Lorentz invariance. This is because otherwise there would be energy non-conservation, which, if due to Planck scale effects, would be unacceptably conspicuous. On the other hand, a Lorentz invariant vacuum that contains quantum black holes could contain an infinite density of them because of the infinitude of Lorentz boosts, so, if they were to have any physical effect on objects of arbitrarily high energy, then a high energy regular-
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 6 9 3 - 0
D. Eichlerr Physics Letters B 459 (1999) 86–90
ization procedure would surely be a prerequisite for describing such an effect, and such a procedure might discard Lorentz invariance at sufficiently high energy. It is pointed out here that observational limits on Rayleigh scattering of spin 1r2 particles rule out a Lorentz non-invariant grainy vacuum. Although the discussion is hampered by the lack of a specific theory of quantum gravity, we make a simple model in which the conclusions are quantitatively robust. The results of the Einstein–Padolsky–Rosen experiment tell us that nature can coordinate phases even over space-like separations when it is necessary to do so to protect the most basic kinematical conservation laws. But this result would not be obvious without the experimental results. Given the logical possibility that quantum gravity may be more complicated than already existing theories, which could make it ‘‘harder’’ to coordinate phases, there is a priori motivation for doing other EPR-type experiments. Looking for Rayleigh scattering of extremely high energy quanta that traverse large distances is proposed in this spirit. Similarly, it is conceivable that phases are coordinated over space-like separations in such a way as to conserve momentum exactly, but not to conserve neutrino flavor. There is an extensive literature on non-Lorentz invariant effects andror information-losing effects in the vacuum w3–9x Že.g. Refs. w3–9x and references within., sometimes called ‘‘quantum friction’’. Formalisms to describe such effects typically involve an additional operator that introduces an uncertainty in the phase of any wavefunction that would otherwise be described by the usual Schroedinger equation. For example, Ellis et al. w3x ŽEHNS. invoke an additional term to the usual quantum mechanical Hamiltonian that contributes to the evolution of the density matrix. As off diagonal terms in the density matrix are affected, the phase correlation between different pure states is affected in some unpredictable way. Although the formalism is most developed for two independent states, it is quite general, and one might expect that many states could be involved. If the pure states are eigenstates of the position operator, then phase decorrelation suggests corrugation of the wavefront, i.e. scattering. Gambini and Pullin discuss possible systematic, parity violating effects of operators associated with a noisy vacuum, but note that
87
the term they consider would vanish if the regions of the correlation scale D have ‘‘random orientations’’ in their parity violation. On the other hand, a granulation or weave pattern that was entirely random would still produce opalescence due to fluctuations. The purpose of this paper is to address the issue of opalescence. The only previous discussion on this point that we are aware of is the claim by Hawking w12x that the vacuum should be opaque to scalar particles because of quantum black hole effects, and that the Higgs particle, if fundamental, will not be found. Fundamental particles with spin are more robust against opalescence, essentially because they are gyroscopically stabilized. They are considered here.
2. Spontaneous breaking of translational invariance It is assumed here that the interaction between a quantum and ‘‘grain’’ of vacuum preserves unitarity and merely changes the phase of wavefunctions in some 4 q n dimensional space that includes the four external space time dimensions and internal dimensions as well. ŽPossibly, one could attribute the observed local gauge symmetries in nature to the grains themselves, for any low energy physics that survives their effects must be robust to local changes of phase.. The effects of the grains are assumed to be local to a very good approximation so that, although the wavefront is corrugated by the grains, the angular momentum relative to any particular point is conserved by the interaction with a grain at that same point. One might suppose that the particular mechanism of Planck scale quantum black holes would be hard to reconcile with Lorentz invariance at all scales. If Lorentz invariance is an exact symmetry of the vacuum, then there are an infinite number of quantum black holes at any point, corresponding to the infinity of velocity space at each spatial coordinate. The ability of quantum black holes to swallow an object would not decrease with relative velocity, so their opacities would add up. Moreover, the noise generated by the quantum black holes gives rise to a sort of Olber’s paradox: Any point at any time is subjected to the output of black hole evaporation at
D. Eichlerr Physics Letters B 459 (1999) 86–90
88
all other points within its backward light cone. Hence any regularization procedure that deals with this difficulty might have to include some sort of cosmological assumption, and define a select frame of reference. There is some literature on the possibility that Lorentz invariance is merely a low energy symmetry w13x. There may, however, be some differences between those proposals and what is considered here. In any case, the dimensional argument given below does not make sense if Lorentz invariance is exact at arbitrarily high energies. In this paper we merely assume that energies and hence wavelengths l are defined in comoving cosmological coordinates, and that the sum of the effects of all scattering grains on any point in space time can be roughly represented by a spherical scatterer with a scattering length of order the Planck length. Consider, then, the scattering of a massless particle by a more or less closely packed ‘‘foam’’ of individual grains Že.g. quantum black holes. and let us model the grains as something like packed spheres of radius l P ; 10y3 3 cm, with a potential of comparable range and height of order gly2 P , where g is dimensionless and Planck’s constant and c are set to unity. The scattering of a massless particle of spin s requires a finite amplitude for a helicity flip of 2 s, but conservation of angular momentum relative to the cell that scatters thus requires a change in orbital angular momentum of 2 s, identifying the dominant partial wave contribution to the total cross section in the low energy limit as l s 2 s. The scattering cross section is thus of order
s Ž k. ;
4p k2
Ž 2 l q 1 . sin2d l .
Ž 1.
As the low energy phase shift of the lth partial wave is
d l ; Ž kl P .
2 lq1
g,
Ž 2.
the total cross section is of order 4s
s ; g 2 l P2 Ž l Prl . .
Ž 3.
As quantum black holes would have a strong if any effect on whatever comes near them, g is probably close to unity. But large numerical factors Žw2 l q 1x!!y4 in the case of the delta shell model. that enter
into the value of the cross section for exact models should render this approximation useful only where estimates yield conclusions by a comfortable margin. Consider photons: s s 1. In traversing cosmological distances, a photon traverses of order 10 60 l P , so the universe might plausibly become opaque to quanta above ; 10y1 5 m P ; 10 13 eV. Curiously, although photons just below this energy have been detected from point sources, there is no firm evidence that any high energy quanta above this energy propagate freely as they are all charged particles that are in any case deflected by cosmic magnetic fields Žbut see Ref. w14x.. Below 10 13 eV, although the optical depth to large angle Rayleigh scattering is in any case less than unity, there may still be small angle refractive scattering. Consider first the limit on deviations from Newton’s first law that comes from ultrahigh energy astronomy, which at present can detect TeV gamma rays from distances of order 100 Mpc ŽMarkarian 501. with a variability timescale of order 10 3 seconds w15x. This implies that the angular deflection is less than 10y6 .5. If one assumes that the index of refraction of the vacuum is significantly affected by the presence of Planck scale foam, that is, the presence of the foam has a significant effect on the propagation of light relative to the ‘‘bare’’ vacuum, then fluctuations in the index of refraction over a scale of order wavelength l are of order Ž lrl P .y3 r2 . Over one wavelength l, a photon is ‘‘wing’’ Rayleigh scattered by an angle of order Ž lrl P .y3 r2 . In traversing a cosmological distance D, the photon is scattered of order Ž Drl. times, yielding a net scattering angle of order l P3r2 D 1r2rl2 . For TeV photons Ž l ; 10y1 6 cm., this is about 10y4 , somewhat in excess of the observed limits. Now consider spin 1r2 particles. Here the scattering cross section scales as l P2 Ž l Prl. 2 . Identifying neutrinos as massless spin 1r2 particles implies a cosmic mean free path for 10 MeV neutrinos of only 10 9 cm. This is difficult to reconcile with solar neutrino data and especially with the supernova 1987A neutrinos. If the familiar spin 1r2 fermions are in fact spin 3r2 gravitinos in the 4 q n dimensional space w16x one must consider that the scattering grains are small compared to the compactified dimensions. In this
D. Eichlerr Physics Letters B 459 (1999) 86–90
case, the scattering cross section scales as Ž l Prl. 6 on scales smaller than the compactification scale. Thus, for a compactification scale that is hy1 l P , the mean free path of the ’’familiar spin 1r2 ’’ particle is increased by hy4 , up to 10 17 cm at En s 10 MeV for h as low as 1r100. This is still ruled out by SN 1987A. Note that a Lorentz invariant version of translational symmetry breaking would necessarily imply energy non-conservation and could be tested on essentially every electron, taking the energy of the electron to be its rest mass. The existence of stable atoms over a Hubble time appears to immediately rule out this possibility. In recent theories of a noisy vacuum, the putative scatterers are polymer like, rather than grainy. In general, this makes the opalescence problem worse than for grains, unless there is some mechanism for establishing long range order. It is not clear, however, that long range order is compatible with information loss to quantum black holes.
3. Neutrino mixing We now consider the possibility that the phases of the grains are coordinated such that linear momentum and energy are conserved as well as charge and angular momentum. In this case, interactions with the grains Že.g. absorption and reemission by quantum black holes. can alter only the non-conserved quantum numbers of a particle such as the generation type of a massless neutrino. This is more or less equivalent to representing the grains as internal lines, but possibly with some loss of phase information in their propagators. We can imagine as a rough ansatz a vacuum polarization type loop diagram with the internal loop consisting of a scalar grain and a spin 1r2 grain Že.g. a spinless quantum black hole and one that, having swallowed a neutrino, has spin 1r2.. The external lines are neutrinos of different types. Although the diagram is divergent, the introduction of a suitably shaped Lorentz non-invariant cutoff at about the Planck scale could conceivably give – in addition to large terms that are presumably cancelled by some counterterms – a Lorentz non-in-
89
variant term to the amplitude of order EnrMpl , as proposed by many previous authors. For example, it gives an energy dependent speed for an otherwise massless neutrino. The freedom in choosing such a cutoff without detailed knowledge of the Planck scale physics is large enough that there is little predictive power here. The point is merely to note that there exists a mathematical possibility for a theory that mixes neutrino types while conserving momentum and energy exactly. The cross section associated with such a scattering would then be proportional to En2rM pl2 . It is here that the loss of information posited by Hawking becomes important. The matrix element mixing neutrino flavors would be proportional to E. If there were no loss of coherence, it would lead to an oscillation rate proportional to E, as in many other previous works. However, loss of coherence would require that a proper description consider the cross section, which is proportional to E 2 , for each ‘‘grain’’, and then sum the cross sections. Alternatively, one could imagine a cutoff shaped so that the leading Lorentz non-invariant term was only of order Ž EnrMpl . 2 , but with no loss of coherence, so that the neutrino oscillations were reversible but with an oscillation length scaling as the square of the particle energy. In this case the formalism would probably be more like that of Reznik w17x than that of EHNS. This would give similar but not identical neutrino phenomenology to irreversible mixing with the same energy scaling Žbasically more low-energy neutrinos, see below.. If neutrinos are massless Žexcept for the interaction with the grains. they would in the above picture simply be scattered among different generations so that the neutrino flux of any type emitted by a source would eventually approach the average over the 3 generations. This would be similar to oscillations except that the change in neutrino type would be irreversible, and the mass matrix would have a random component with imaginary expected eigenvalues. Applying the simple dimensional arguments discussed above and assuming neutrinos to be spin 1r2 down to the Planck scale, the mean free path for a 30 GeV neutrino would be about 10 2 cm. This is apparently ruled out by post-beam dump experiments at large accelerators, which set a minimum oscillation length of order a km for 30 GeV neutrinos.
90
D. Eichlerr Physics Letters B 459 (1999) 86–90
If the neutrinos have spin 3r2 in a higher dimensional manifold that has dimensions hy1 times the Planck length, then the same dimensional arguments raise the neutrino mixing length by hy4 . If h f 10y3 r4 , then the mixing length for 30 GeV neutrinos is of order 10 5 cm, probably allowed by present experiments but testable with next generation experiments. The range of possible values for h still has enough flexibility that, together with the rough nature of the arguments, accurate quantitative predictions are not attempted here. The predicted level of atmospheric neutrino mixing could be detectable for reasonable values of h , as the accelerator experiments allow an Ey2 mixing length to be less than an Earth diameter for E G 0.5 GeV Žand implying in the present context a compactification length scale within an order of magnitude or so of the Planck scale., but an a priori quantitative prediction is not attempted here. We note, however, that if atmospheric neutrinos are produced in the ratio of two muon types, no tau types to one electron type, the post-mixing distribution is equal numbers of each, as if the muon neutrinos had oscillated with tau neutrinos. Differences between the two hypotheses would arise, however, in the ratio of neutrinos to anti-neutrinos, and in the energy dependence of the mixing length. This idea has been considered independently by Chang et al. w10x and even earlier by some of the same authors for solar neutrinos. The hypothesis of neutrinos mixing over a length that is proportional to Ey2 implies that if atmospheric neutrinos are mixed over a length scale of order 0.1 Earth radius, then solar Boron neutrinos Ž E ; 10 MeV. are mixed over a scale of less than 1 astronomical unit, whereas the pp neutrinos Ž E ; 0.26 MeV. have a range that exceeds an A.U. Thus, the Boron neutrino flux at Earth would be reduced by a factor of 3, whereas the pp neutrino flux at Earth would be nearly unchanged. A quantitative fit to the GALLEX, Homestake, and SuperKamiokande data is made in a companion paper w18x.
Another prediction is that ultrahigh energy neutrinos from extragalactic sources would certainly be mixed among the three known neutrino types w19x. Finally, 30 GeV neutrinos would be mixed thoroughly over the CERN - Gran Sasso distance or the Fermilab - SOUDAN distance, as also noted by Chang et al. w10x. Acknowledgements I am grateful to Y. Avishai, R. Brustein, M. Berry, J. Bekenstein, Y. Band, A. Dar,x E. Gedalin, E. Guendelman, E. Nissimov, S. Nussinov, D. Owen, Z. Seidov, G. Veneziano and V. Usov for vital discussions. I thank the referee for pointing out work by Chang, Liu and coworkers. I acknowledge support from the Israel-US BSF and from the Israeli Science Foundation. References w1x S.W. Hawking, Nature 248 Ž1974. 30. w2x S.W. Hawking, Commun. Math. Phys. 43 Ž1975. 199. w3x J. Ellis, J.S. Hagelin, D.V. Nanopoulos, M. Sredicki, Nucl. Phys. B 241 Ž1984. 381. w4x G. Amelino-Camelia, Nature 393 Ž1998. 763. w5x S. Biller et al., gr-qcr9810044. w6x R. Gambini, J. Pullin, gr-qcr9809038. w7x A. Ashtekar, gr-qcr9901023. w8x G. Amelino-Camelia, gr-qcr9808029. w9x D. Colladay, V.A. Kostelecky, Phys. Rev. D 58 Ž1998. 116002. w10x C. Chang et al., hep-phr9809371. w11x Y. Liu, J. Chen, M. Ge, J. Phys. G 24 Ž1998. 2289. w12x S.W. Hawking, in: Huggat ŽEd.., Proc. the Geometric Universe, Oxford University Press, 1996. w13x C.D. Froggat, H.B. Nielsen ŽEds.., Origin of Symmetries, World Scientific, 1991. w14x M. Milgrom, V. Usov, ApJ. Lett. 449 Ž1995. L3. w15x E. Lorenz, talk presented at MAGIC Workshop, Barcelona, May 1997. w16x P.G.O. Freund, Introduction to Supersymmetry, Cambridge, Monographs on Mathematical Physics, 1986. w17x B. Reznik, Phys. Rev. Lett. 76 Ž1996. 1192. w18x D. Eichler, Z. Seidov, Z. Physics Letters, submitted. w19x Waxman, E., J. Bahcall, astro-ph 970231.
Physics Letters B 459 Ž1999. 86–90
Searching for granularity of the vacuum D. Eichler Ben-Gurion UniÕersity, Beer-SheÕa 84105, Israel Received 23 February 1999; received in revised form 5 May 1999 Editor: P.V. Landshoff
Abstract The hypothesis is considered that the vacuum is a Lorentz non-invariant foam in which translational symmetry is spontaneously broken at the Planck or compactification scale. This could possibly be observed via Rayleigh scattering of ultrahigh energy quanta, and it appears to be ruled out for spin 1r2 leptons by SN 1987a, and independently testable for photons. A weaker version of the hypothesis predicts Žfor otherwise massless neutrinos. irreversible neutrino mixing over a length scale of l Planck Ž EnrEPlanck .y2 in comoving cosmic coordinates for a purely four dimensional universe, and, for neutrinos that have spin 3r2 in a higher dimensional manifold above the energy scale h E Planck , the mixing is predicted to take place over a scale of hy4 times this length. Such neutrino mixing might also be observable with atmospheric and solar neutrino experiments, and account for some of the observed deficits. Long distance terrestrial experiments would test the latter hypothesis unequivocably. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction Translational invariance of the vacuum is a sacred tenet of physics. One might suspect that, near the Planck scale, space may be grainy due, say, to condensates, quantum black holes, a spatially varying u parameter, or phase decorrelation of virtual modes. This raises the question of whether on such small scales translational symmetry is preserved or spontaneously broken. A grainy vacuum would suggest that physics at macroscopic scales is to some Žpossibly small. degree unpredictable. Hawking w1,2x has suggested that the formation and evaporation of quantum black holes leads to such unpredictability. The question has also been raised as to whether wavefunction collapse in quantum mechanics, an apparent breakdown of determinism, has anything to
do with strong non-linear effects in quantum gravity, though no self-consistent picture has been constructed along these lines. More generally, any Lagrangian that has non-renormalizable terms that contain field derivatives of higher order than the usual kinetic term Em fE mf may imply spontaneous breaking of translational symmetry. Probably, any breaking of translational invariance also involves breaking of Lorentz invariance. This is because otherwise there would be energy non-conservation, which, if due to Planck scale effects, would be unacceptably conspicuous. On the other hand, a Lorentz invariant vacuum that contains quantum black holes could contain an infinite density of them because of the infinitude of Lorentz boosts, so, if they were to have any physical effect on objects of arbitrarily high energy, then a high energy regular-
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 6 9 3 - 0
D. Eichlerr Physics Letters B 459 (1999) 86–90
ization procedure would surely be a prerequisite for describing such an effect, and such a procedure might discard Lorentz invariance at sufficiently high energy. It is pointed out here that observational limits on Rayleigh scattering of spin 1r2 particles rule out a Lorentz non-invariant grainy vacuum. Although the discussion is hampered by the lack of a specific theory of quantum gravity, we make a simple model in which the conclusions are quantitatively robust. The results of the Einstein–Padolsky–Rosen experiment tell us that nature can coordinate phases even over space-like separations when it is necessary to do so to protect the most basic kinematical conservation laws. But this result would not be obvious without the experimental results. Given the logical possibility that quantum gravity may be more complicated than already existing theories, which could make it ‘‘harder’’ to coordinate phases, there is a priori motivation for doing other EPR-type experiments. Looking for Rayleigh scattering of extremely high energy quanta that traverse large distances is proposed in this spirit. Similarly, it is conceivable that phases are coordinated over space-like separations in such a way as to conserve momentum exactly, but not to conserve neutrino flavor. There is an extensive literature on non-Lorentz invariant effects andror information-losing effects in the vacuum w3–9x Že.g. Refs. w3–9x and references within., sometimes called ‘‘quantum friction’’. Formalisms to describe such effects typically involve an additional operator that introduces an uncertainty in the phase of any wavefunction that would otherwise be described by the usual Schroedinger equation. For example, Ellis et al. w3x ŽEHNS. invoke an additional term to the usual quantum mechanical Hamiltonian that contributes to the evolution of the density matrix. As off diagonal terms in the density matrix are affected, the phase correlation between different pure states is affected in some unpredictable way. Although the formalism is most developed for two independent states, it is quite general, and one might expect that many states could be involved. If the pure states are eigenstates of the position operator, then phase decorrelation suggests corrugation of the wavefront, i.e. scattering. Gambini and Pullin discuss possible systematic, parity violating effects of operators associated with a noisy vacuum, but note that
87
the term they consider would vanish if the regions of the correlation scale D have ‘‘random orientations’’ in their parity violation. On the other hand, a granulation or weave pattern that was entirely random would still produce opalescence due to fluctuations. The purpose of this paper is to address the issue of opalescence. The only previous discussion on this point that we are aware of is the claim by Hawking w12x that the vacuum should be opaque to scalar particles because of quantum black hole effects, and that the Higgs particle, if fundamental, will not be found. Fundamental particles with spin are more robust against opalescence, essentially because they are gyroscopically stabilized. They are considered here.
2. Spontaneous breaking of translational invariance It is assumed here that the interaction between a quantum and ‘‘grain’’ of vacuum preserves unitarity and merely changes the phase of wavefunctions in some 4 q n dimensional space that includes the four external space time dimensions and internal dimensions as well. ŽPossibly, one could attribute the observed local gauge symmetries in nature to the grains themselves, for any low energy physics that survives their effects must be robust to local changes of phase.. The effects of the grains are assumed to be local to a very good approximation so that, although the wavefront is corrugated by the grains, the angular momentum relative to any particular point is conserved by the interaction with a grain at that same point. One might suppose that the particular mechanism of Planck scale quantum black holes would be hard to reconcile with Lorentz invariance at all scales. If Lorentz invariance is an exact symmetry of the vacuum, then there are an infinite number of quantum black holes at any point, corresponding to the infinity of velocity space at each spatial coordinate. The ability of quantum black holes to swallow an object would not decrease with relative velocity, so their opacities would add up. Moreover, the noise generated by the quantum black holes gives rise to a sort of Olber’s paradox: Any point at any time is subjected to the output of black hole evaporation at
D. Eichlerr Physics Letters B 459 (1999) 86–90
88
all other points within its backward light cone. Hence any regularization procedure that deals with this difficulty might have to include some sort of cosmological assumption, and define a select frame of reference. There is some literature on the possibility that Lorentz invariance is merely a low energy symmetry w13x. There may, however, be some differences between those proposals and what is considered here. In any case, the dimensional argument given below does not make sense if Lorentz invariance is exact at arbitrarily high energies. In this paper we merely assume that energies and hence wavelengths l are defined in comoving cosmological coordinates, and that the sum of the effects of all scattering grains on any point in space time can be roughly represented by a spherical scatterer with a scattering length of order the Planck length. Consider, then, the scattering of a massless particle by a more or less closely packed ‘‘foam’’ of individual grains Že.g. quantum black holes. and let us model the grains as something like packed spheres of radius l P ; 10y3 3 cm, with a potential of comparable range and height of order gly2 P , where g is dimensionless and Planck’s constant and c are set to unity. The scattering of a massless particle of spin s requires a finite amplitude for a helicity flip of 2 s, but conservation of angular momentum relative to the cell that scatters thus requires a change in orbital angular momentum of 2 s, identifying the dominant partial wave contribution to the total cross section in the low energy limit as l s 2 s. The scattering cross section is thus of order
s Ž k. ;
4p k2
Ž 2 l q 1 . sin2d l .
Ž 1.
As the low energy phase shift of the lth partial wave is
d l ; Ž kl P .
2 lq1
g,
Ž 2.
the total cross section is of order 4s
s ; g 2 l P2 Ž l Prl . .
Ž 3.
As quantum black holes would have a strong if any effect on whatever comes near them, g is probably close to unity. But large numerical factors Žw2 l q 1x!!y4 in the case of the delta shell model. that enter
into the value of the cross section for exact models should render this approximation useful only where estimates yield conclusions by a comfortable margin. Consider photons: s s 1. In traversing cosmological distances, a photon traverses of order 10 60 l P , so the universe might plausibly become opaque to quanta above ; 10y1 5 m P ; 10 13 eV. Curiously, although photons just below this energy have been detected from point sources, there is no firm evidence that any high energy quanta above this energy propagate freely as they are all charged particles that are in any case deflected by cosmic magnetic fields Žbut see Ref. w14x.. Below 10 13 eV, although the optical depth to large angle Rayleigh scattering is in any case less than unity, there may still be small angle refractive scattering. Consider first the limit on deviations from Newton’s first law that comes from ultrahigh energy astronomy, which at present can detect TeV gamma rays from distances of order 100 Mpc ŽMarkarian 501. with a variability timescale of order 10 3 seconds w15x. This implies that the angular deflection is less than 10y6 .5. If one assumes that the index of refraction of the vacuum is significantly affected by the presence of Planck scale foam, that is, the presence of the foam has a significant effect on the propagation of light relative to the ‘‘bare’’ vacuum, then fluctuations in the index of refraction over a scale of order wavelength l are of order Ž lrl P .y3 r2 . Over one wavelength l, a photon is ‘‘wing’’ Rayleigh scattered by an angle of order Ž lrl P .y3 r2 . In traversing a cosmological distance D, the photon is scattered of order Ž Drl. times, yielding a net scattering angle of order l P3r2 D 1r2rl2 . For TeV photons Ž l ; 10y1 6 cm., this is about 10y4 , somewhat in excess of the observed limits. Now consider spin 1r2 particles. Here the scattering cross section scales as l P2 Ž l Prl. 2 . Identifying neutrinos as massless spin 1r2 particles implies a cosmic mean free path for 10 MeV neutrinos of only 10 9 cm. This is difficult to reconcile with solar neutrino data and especially with the supernova 1987A neutrinos. If the familiar spin 1r2 fermions are in fact spin 3r2 gravitinos in the 4 q n dimensional space w16x one must consider that the scattering grains are small compared to the compactified dimensions. In this
D. Eichlerr Physics Letters B 459 (1999) 86–90
case, the scattering cross section scales as Ž l Prl. 6 on scales smaller than the compactification scale. Thus, for a compactification scale that is hy1 l P , the mean free path of the ’’familiar spin 1r2 ’’ particle is increased by hy4 , up to 10 17 cm at En s 10 MeV for h as low as 1r100. This is still ruled out by SN 1987A. Note that a Lorentz invariant version of translational symmetry breaking would necessarily imply energy non-conservation and could be tested on essentially every electron, taking the energy of the electron to be its rest mass. The existence of stable atoms over a Hubble time appears to immediately rule out this possibility. In recent theories of a noisy vacuum, the putative scatterers are polymer like, rather than grainy. In general, this makes the opalescence problem worse than for grains, unless there is some mechanism for establishing long range order. It is not clear, however, that long range order is compatible with information loss to quantum black holes.
3. Neutrino mixing We now consider the possibility that the phases of the grains are coordinated such that linear momentum and energy are conserved as well as charge and angular momentum. In this case, interactions with the grains Že.g. absorption and reemission by quantum black holes. can alter only the non-conserved quantum numbers of a particle such as the generation type of a massless neutrino. This is more or less equivalent to representing the grains as internal lines, but possibly with some loss of phase information in their propagators. We can imagine as a rough ansatz a vacuum polarization type loop diagram with the internal loop consisting of a scalar grain and a spin 1r2 grain Že.g. a spinless quantum black hole and one that, having swallowed a neutrino, has spin 1r2.. The external lines are neutrinos of different types. Although the diagram is divergent, the introduction of a suitably shaped Lorentz non-invariant cutoff at about the Planck scale could conceivably give – in addition to large terms that are presumably cancelled by some counterterms – a Lorentz non-in-
89
variant term to the amplitude of order EnrMpl , as proposed by many previous authors. For example, it gives an energy dependent speed for an otherwise massless neutrino. The freedom in choosing such a cutoff without detailed knowledge of the Planck scale physics is large enough that there is little predictive power here. The point is merely to note that there exists a mathematical possibility for a theory that mixes neutrino types while conserving momentum and energy exactly. The cross section associated with such a scattering would then be proportional to En2rM pl2 . It is here that the loss of information posited by Hawking becomes important. The matrix element mixing neutrino flavors would be proportional to E. If there were no loss of coherence, it would lead to an oscillation rate proportional to E, as in many other previous works. However, loss of coherence would require that a proper description consider the cross section, which is proportional to E 2 , for each ‘‘grain’’, and then sum the cross sections. Alternatively, one could imagine a cutoff shaped so that the leading Lorentz non-invariant term was only of order Ž EnrMpl . 2 , but with no loss of coherence, so that the neutrino oscillations were reversible but with an oscillation length scaling as the square of the particle energy. In this case the formalism would probably be more like that of Reznik w17x than that of EHNS. This would give similar but not identical neutrino phenomenology to irreversible mixing with the same energy scaling Žbasically more low-energy neutrinos, see below.. If neutrinos are massless Žexcept for the interaction with the grains. they would in the above picture simply be scattered among different generations so that the neutrino flux of any type emitted by a source would eventually approach the average over the 3 generations. This would be similar to oscillations except that the change in neutrino type would be irreversible, and the mass matrix would have a random component with imaginary expected eigenvalues. Applying the simple dimensional arguments discussed above and assuming neutrinos to be spin 1r2 down to the Planck scale, the mean free path for a 30 GeV neutrino would be about 10 2 cm. This is apparently ruled out by post-beam dump experiments at large accelerators, which set a minimum oscillation length of order a km for 30 GeV neutrinos.
90
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If the neutrinos have spin 3r2 in a higher dimensional manifold that has dimensions hy1 times the Planck length, then the same dimensional arguments raise the neutrino mixing length by hy4 . If h f 10y3 r4 , then the mixing length for 30 GeV neutrinos is of order 10 5 cm, probably allowed by present experiments but testable with next generation experiments. The range of possible values for h still has enough flexibility that, together with the rough nature of the arguments, accurate quantitative predictions are not attempted here. The predicted level of atmospheric neutrino mixing could be detectable for reasonable values of h , as the accelerator experiments allow an Ey2 mixing length to be less than an Earth diameter for E G 0.5 GeV Žand implying in the present context a compactification length scale within an order of magnitude or so of the Planck scale., but an a priori quantitative prediction is not attempted here. We note, however, that if atmospheric neutrinos are produced in the ratio of two muon types, no tau types to one electron type, the post-mixing distribution is equal numbers of each, as if the muon neutrinos had oscillated with tau neutrinos. Differences between the two hypotheses would arise, however, in the ratio of neutrinos to anti-neutrinos, and in the energy dependence of the mixing length. This idea has been considered independently by Chang et al. w10x and even earlier by some of the same authors for solar neutrinos. The hypothesis of neutrinos mixing over a length that is proportional to Ey2 implies that if atmospheric neutrinos are mixed over a length scale of order 0.1 Earth radius, then solar Boron neutrinos Ž E ; 10 MeV. are mixed over a scale of less than 1 astronomical unit, whereas the pp neutrinos Ž E ; 0.26 MeV. have a range that exceeds an A.U. Thus, the Boron neutrino flux at Earth would be reduced by a factor of 3, whereas the pp neutrino flux at Earth would be nearly unchanged. A quantitative fit to the GALLEX, Homestake, and SuperKamiokande data is made in a companion paper w18x.
Another prediction is that ultrahigh energy neutrinos from extragalactic sources would certainly be mixed among the three known neutrino types w19x. Finally, 30 GeV neutrinos would be mixed thoroughly over the CERN - Gran Sasso distance or the Fermilab - SOUDAN distance, as also noted by Chang et al. w10x. Acknowledgements I am grateful to Y. Avishai, R. Brustein, M. Berry, J. Bekenstein, Y. Band, A. Dar,x E. Gedalin, E. Guendelman, E. Nissimov, S. Nussinov, D. Owen, Z. Seidov, G. Veneziano and V. Usov for vital discussions. I thank the referee for pointing out work by Chang, Liu and coworkers. I acknowledge support from the Israel-US BSF and from the Israeli Science Foundation. References w1x S.W. Hawking, Nature 248 Ž1974. 30. w2x S.W. Hawking, Commun. Math. Phys. 43 Ž1975. 199. w3x J. Ellis, J.S. Hagelin, D.V. Nanopoulos, M. Sredicki, Nucl. Phys. B 241 Ž1984. 381. w4x G. Amelino-Camelia, Nature 393 Ž1998. 763. w5x S. Biller et al., gr-qcr9810044. w6x R. Gambini, J. Pullin, gr-qcr9809038. w7x A. Ashtekar, gr-qcr9901023. w8x G. Amelino-Camelia, gr-qcr9808029. w9x D. Colladay, V.A. Kostelecky, Phys. Rev. D 58 Ž1998. 116002. w10x C. Chang et al., hep-phr9809371. w11x Y. Liu, J. Chen, M. Ge, J. Phys. G 24 Ž1998. 2289. w12x S.W. Hawking, in: Huggat ŽEd.., Proc. the Geometric Universe, Oxford University Press, 1996. w13x C.D. Froggat, H.B. Nielsen ŽEds.., Origin of Symmetries, World Scientific, 1991. w14x M. Milgrom, V. Usov, ApJ. Lett. 449 Ž1995. L3. w15x E. Lorenz, talk presented at MAGIC Workshop, Barcelona, May 1997. w16x P.G.O. Freund, Introduction to Supersymmetry, Cambridge, Monographs on Mathematical Physics, 1986. w17x B. Reznik, Phys. Rev. Lett. 76 Ž1996. 1192. w18x D. Eichler, Z. Seidov, Z. Physics Letters, submitted. w19x Waxman, E., J. Bahcall, astro-ph 970231.