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Limit on “isappearance” of orthopositronium in vacuum
5 May 1994 PHYSICS LETTERS B
Physics Letters B 326 (1994) 317-319
ELSEVIER
Limit on "disappearance" of orthopositronium in vacuum S.N. G n i n e n k o Institutefor Nuclear Research of the Russian Academy of Sciences, 117312 Moscow, Russia
Received 28 June 1993;revised manuscript received 9 March 1994 Editor: L. Montanet
Abstract
If mixing between orthopositronium (o-Ps) and its mirror analogue o-Ps' exists it could be the origin of the present discrepancy between theory and experiment in the o-Ps lifetime in vacuum. It is pointed out that a recent experiment in search of the invisible decay of o-Ps provides considerably less stringent limits both on branching ratio for decay of o-Ps as mirror matter in vacuum, BR(o-Ps~o-Ps') < 10 -2, and on photon-mirror photon mixing strength ~< 10 - 6 than was claimed. The new experimental limits BR(o-Ps ~ o-Ps') < 10-3 and e < 2.8 X 10-7 were obtained from measurements of the o-Ps lifetime in vacuum and low pressure gases.
Recently it was found that values of the orthopositronium (o-Ps) decay rate in vacuum, obtained from the experiment in low-pressure gases, Ag = 7 . 0 5 1 4 _ 0.0014 MHz [1], and vacuum experiment, A~ = 7 . 0 4 8 2 + 0 . 0 0 1 6 MHz [2], are in agreement and exceed the theoretical value AT = 7.03830-t-0.00007 MHz calculated within QED to the order (c~/~) 2 [ 3,4 ] by (1900-t-190) ppm and ( 1 4 0 0 + 2 2 7 ) ppm respectively. This discrepancy led to the speculation on possible exotic decay modes of o-Ps, that are not taken into account in the calculation of AT and give a relative contribution to the decay rate of o-Ps at the level = 10 -3. No indications on the existence of exotic decays were found; for a review see [ 5 ], and references therein. One of the most intriguing ways to resolve the puzzle may be found in the paper of Glashow [6], who using the idea o f a "mirror universe" [7] has considered the contribution to the decay rate of o-Ps due to the mixing between ordinary o-Ps and its mirror analogue o-Ps' based on the mechanism suggested by Holdom [8]. According to Holdom, a third (mixed) form of matter Elsevier Science B.V. SSD10370-2693 ( 94 ) 00349-C
couples both ordinary and mirror forces. An example of an interaction ,,~ with this effect is a loop of such particles which generates an effective mixing between the photon and mirror photon:
e.gr,, ~ J ~,,.,
(1)
and yields a matrix element for the o-Ps ~ o-Ps' transition equal to (o-Ps I,,~¢'lo-Ps ') = 2~ref= 8 / 2 ,
(2)
where e is the mixing strength, and f = 8.7 × 10 4 MHz is the contribution to the ortho-para splitting from the one-photon annihilation diagram involving o-Ps [6]. If the interaction (1) exists, the vacuum energy eigenstates will not be o-Ps and o-Ps', but (o-Ps + o-Ps') and ( o - P s - o - P s ' ) , which are split in energy by A E = 8, giving rise to o-Ps-o-Ps' oscillation of characteristic frequency e Xf. A state which at t = 0 is pure o-Ps will contain an admixture of mirror o-Ps with probability
[6] P ( t ) = [ 1 - cos2(St/2) ] exp( - ATt) •
(3)
318
S.N. Gninenko /Physics Letters B 326 (1994) 317-319
An indication of the existence of the mirror universe could be obtained from the observation of a "disappearance" of o-Ps in our universe, because the fraction (3) of orthopositronium would decay invisibly as mirror matter. Let us note that the effect of the o-Ps ~ oPs' oscillation is analogous to another process which may occur in a pure leptonic hydrogen like system muonium (M) ~ antimuonium (K/I) conversion. Further, we will base our discussion on a detailed analysis of the M ~ 1VIprocess in various environments made in [9]. According to this paper the relative probability of an invisible decay of o-Ps in vacuum may be defined
the grain surfaces. The time of thermalization is of order I/AT and the energy loss per collision is about 10 -5 eV >> AE [ 13]. The mirror o-Ps practically does not interact with the grains because of the small electric charge of the mirror electron (positron) e ' ~ ~e [6]. The significant difference in electromagnetic properties of o-Ps and o-Ps' leads to the effect that after each collision the o-Ps system is an incoherent mixture of oPs and o-Ps'. Using Eq. (3) it is easy to show that if one starts with pure o-Ps, the relative probability of an invisible decay of o-Ps will be [9]
as
Br(o-Ps ~ o-Ps') = 32/2toeAT,
B R ( o - P s ~ o - P s ' ) = ½ t~2/(t~2 q- A2 + A2) .
(4)
where A is any additional splitting of o-Ps and o-Ps' by the external fields. As was shown by Carlson and Glashow [ 10] the upper limit of ~< 3 × 10 -8 on the photon-mirror photon mixing strength can be obtained from the theory of big band nucleosynthesis. This result seems to exclude mirror matter as an explanation of the o-Ps lifetime [10]. Indeed, for A = 0 and ~ < 3 × 1 0 -8, BR(oP s ~ o - P s ' ) =82/2A~ < 1 0 -2, which is too small to explain the discrepancy. Nevertheless, two experiments were performed on the "disappearance" of oPs [ 11,12]. The authors of a recent experiment [ 12] performed a more sensitive search for the invisible decay of o-Ps giving an upper limit of B R ( o - P s ~ o Ps') < 2.8 × 10 -6 on the branching ratio for decay of o-Ps as mirror matter in vacuum. They also claim a new limit of e < 1.5 × 10 -8 on the photon-mirror photon mixing strength. Using the upper limit of E < 3 × 10 -8 obtained in [ 10], we can see that the splitting energy A E < 10 -12 eV is so small that the effects due to the environment of the o-Ps system become important for the rate of oPs ~ o-Ps'. A measurement in which the o - P s - o-Ps' system remains in matter will give a much lower probability for o-Ps ~ o-Ps' transitions. In the experiment [ 12] o-Ps were formed in a target of silica aerogel (density 0.1 g / c m 3 and typical grain size ---50/~); not in vacuum. The behavior of o-Ps in silica aerogel is fairly well understood. Positronium is found inside the ultrafine grains and diffuses to the grain surface where it is ejected into the free space between the grains [ 13]. After ejection, the o-Ps is thermalized loosing its energy through collisions with
(5)
where toc is the o-Ps-grains collision rate. Thus, the transition rate will be reduced roughly by a factor AT/ toc = 1 / N compared to the vacuum rate, where N is the number of o-Ps-grains collisions in an o-Ps lifetime in silica aerogel, which is about 1/hr. The collision frequency of the o-Ps with grains is to~ = n~rR2v ,
(6)
where n is the number of grains per unit volume, R is the radius of the grains, and v is the mean o-Ps velocity. We assume that the grains are spherical and uniform in radius. For values n - 1 0 1 6 c m -3, R~--50/~, v ~- 107 cm/s (o-Ps kinetic energy =0.1 eV, [13]) we get tOe ~ 1011 S-1, or N ~ 104 collisions in a o-Ps lifetime of 135 ns in silica aerogel [ 12]. Thus, taking the suppression factor 1 / N ~ . 10 - 4 into account we actually have B R ( o - P s ~ o - P s ' ) < 10 -2 and ~< 10 - 6 instead of the values given above. A more stringent experimental limit for decay of oPs as mirror matter in vacuum can be obtained from the comparison of the results of the experiments [ 1 ] and [2]. In the gas experiment [1] the vacuum decay rate A~=7.0514+0.0014 MHz was obtained by extrapolation to zero gas density. The o-Ps-gas collisions make the amplitudes for developing mirror o-Ps in the period between collisions add incoherently, thus the possible contribution to )t~- due to the transitions oPs ---)o-Ps' is suppressed by the high positronium-gas collision rate. The factor 1/N is about 10- 3 even at the lowest gas pressure ( = 100 Torr) used in the experiment. Using the dependence of the measured o-Ps lifetime on the gas pressure [ 1 ] one can show that within the 1tr error limit on A~ the corresponding uncertainty in the number of collisions in the o-Ps lifetime is of the
S.N. Gninenko /Physics Letters B 326 (1994) 317-319
order of 30. In this case we expect that there is no significant contribution to the decay rate A~ from the effect of a mirror universe. In the vacuum experiment [2] the average number of o-Ps collisions with the walls of a vacuum cavity was in the range 3-10 in the o-Ps lifetime depending on the cavity volume. The o-Ps vacuum decay rate was obtained by extrapolation to an infinite volume of the cavity giving A~ = 7.0482 + 0.0016 MHz. Again, taking into account the dependence of the measured o-Ps lifetime value on the cavity volume [2] we see that the asymptotic value A~- corresponds to the number of collisions N = 3. External fields do not seem to contribute significantly to B R ( o - P s ~ o - P s ' ) in (4). The contribution due to the quadratic Stark effect in o-Ps is A---~ X $" (in atomic units) for the ground state. This gives A << AT ( = 4 × 10 - 9 e V ) for the electric field ~---100 V / c m used in the experiment. For M---,/f4 conversion the Stark effect does not contribute because both M and 1VIinteract with the electric field producing the same shift [9]. To avoid magnetic quenching the magnetic field H must be below about 0.1G in order to keep the splitting A = eH/2me
(7)
319
For the limit on the photon-mirror photon mixing strength it gives E< 2.8 × 10 -7. It should be noted that these limits are experimental limits, which cannot exclude mirror matter as an explanation of the o-Ps lifetime. The account of higher order correction in oL may, possibly, resolve the problem, but whether one accepts this assumption or not, it is interesting to try to exclude the effect of a mirror universe on the o-Ps lifetime by direct experiment on the "disappearance" of o-Ps in vacuum. The author thanks M. Baldo-Ceolin for helpful discussions, and D.W. Gidley and A.A. Poblaguev for useful remarks. The help of CERN/PPE on the supporting part of this work is gratefully acknowledged.
References [ 1] C.I. Westbrook, D.W. Gidley, R.S. Conti and A. Rich, Phys. Rev. Lett. 58 (1987) 1328; Phys. Rev. A40 (1989) 5489. [2] J.C. Nico, D.W. Gidley, A. Rich and P.W. Zitzewitz, Phys. Rev. Lett. 65 (1990) 1344. [3] W.E. Caswell and G.P. Lepage, Phys. Rev. A 20 (1979) 36. [4] G.S. Adkins, Ann. Phys. (NY) 146 (1983) 78. [5] M.I. Dobroliubov, S.N. Gninenko, A.Yu. Ignatiev and V.A. Matveev, Intern. J. Mod. Phys. A 8 (1993) 2859. [6] S.L. Glashov, Phys. Lett. B 167 (1986) 35. [7] I.Y. Kobzarev, L.B. Okun and I.Y. Pomeranchuk, Yad. Fiz. 3 (1966) 1154 [Soy. J. Nucl. Phys. 3 (1966) 837]. [8] B. Holdom, Phys. Lett. B 166 (1986) 196; B 178 (1986) 65. [9] G. Feinberg and S. Weinberg, Phys. Rev. Lett. 6 (1961) 381; Phys. Rev. 123 (1961) 1439. [ 10] E.D. Carlson and S.L. Glashow, Phys. Lett. B 193 (1987) 168. [ 11 ] G.S. Atoyan, S.N. Gninenko, V.I. Razin and Yu.V. Ryahov, Phys. Lett. B 220 (1989) 317. [ 12] T. Mitsui et al., Phys. Rev. Lett. 70 (1993) 2265. [ 13] T. Chang, M. Xu and X. Zeng, Phys. Lett. A 126 (1987) 189. [14] J.C. Nico, Ph.D. thesis, University of Michigan (Ann Arbor, 1991 ), unpublished. [15] Particle Data Group, K. Hikasa et al., Review of Particle Properties, Phys. Rev. D 45 (1992) Part II, Ill.39.
Physics Letters B 326 (1994) 317-319
ELSEVIER
Limit on "disappearance" of orthopositronium in vacuum S.N. G n i n e n k o Institutefor Nuclear Research of the Russian Academy of Sciences, 117312 Moscow, Russia
Received 28 June 1993;revised manuscript received 9 March 1994 Editor: L. Montanet
Abstract
If mixing between orthopositronium (o-Ps) and its mirror analogue o-Ps' exists it could be the origin of the present discrepancy between theory and experiment in the o-Ps lifetime in vacuum. It is pointed out that a recent experiment in search of the invisible decay of o-Ps provides considerably less stringent limits both on branching ratio for decay of o-Ps as mirror matter in vacuum, BR(o-Ps~o-Ps') < 10 -2, and on photon-mirror photon mixing strength ~< 10 - 6 than was claimed. The new experimental limits BR(o-Ps ~ o-Ps') < 10-3 and e < 2.8 X 10-7 were obtained from measurements of the o-Ps lifetime in vacuum and low pressure gases.
Recently it was found that values of the orthopositronium (o-Ps) decay rate in vacuum, obtained from the experiment in low-pressure gases, Ag = 7 . 0 5 1 4 _ 0.0014 MHz [1], and vacuum experiment, A~ = 7 . 0 4 8 2 + 0 . 0 0 1 6 MHz [2], are in agreement and exceed the theoretical value AT = 7.03830-t-0.00007 MHz calculated within QED to the order (c~/~) 2 [ 3,4 ] by (1900-t-190) ppm and ( 1 4 0 0 + 2 2 7 ) ppm respectively. This discrepancy led to the speculation on possible exotic decay modes of o-Ps, that are not taken into account in the calculation of AT and give a relative contribution to the decay rate of o-Ps at the level = 10 -3. No indications on the existence of exotic decays were found; for a review see [ 5 ], and references therein. One of the most intriguing ways to resolve the puzzle may be found in the paper of Glashow [6], who using the idea o f a "mirror universe" [7] has considered the contribution to the decay rate of o-Ps due to the mixing between ordinary o-Ps and its mirror analogue o-Ps' based on the mechanism suggested by Holdom [8]. According to Holdom, a third (mixed) form of matter Elsevier Science B.V. SSD10370-2693 ( 94 ) 00349-C
couples both ordinary and mirror forces. An example of an interaction ,,~ with this effect is a loop of such particles which generates an effective mixing between the photon and mirror photon:
e.gr,, ~ J ~,,.,
(1)
and yields a matrix element for the o-Ps ~ o-Ps' transition equal to (o-Ps I,,~¢'lo-Ps ') = 2~ref= 8 / 2 ,
(2)
where e is the mixing strength, and f = 8.7 × 10 4 MHz is the contribution to the ortho-para splitting from the one-photon annihilation diagram involving o-Ps [6]. If the interaction (1) exists, the vacuum energy eigenstates will not be o-Ps and o-Ps', but (o-Ps + o-Ps') and ( o - P s - o - P s ' ) , which are split in energy by A E = 8, giving rise to o-Ps-o-Ps' oscillation of characteristic frequency e Xf. A state which at t = 0 is pure o-Ps will contain an admixture of mirror o-Ps with probability
[6] P ( t ) = [ 1 - cos2(St/2) ] exp( - ATt) •
(3)
318
S.N. Gninenko /Physics Letters B 326 (1994) 317-319
An indication of the existence of the mirror universe could be obtained from the observation of a "disappearance" of o-Ps in our universe, because the fraction (3) of orthopositronium would decay invisibly as mirror matter. Let us note that the effect of the o-Ps ~ oPs' oscillation is analogous to another process which may occur in a pure leptonic hydrogen like system muonium (M) ~ antimuonium (K/I) conversion. Further, we will base our discussion on a detailed analysis of the M ~ 1VIprocess in various environments made in [9]. According to this paper the relative probability of an invisible decay of o-Ps in vacuum may be defined
the grain surfaces. The time of thermalization is of order I/AT and the energy loss per collision is about 10 -5 eV >> AE [ 13]. The mirror o-Ps practically does not interact with the grains because of the small electric charge of the mirror electron (positron) e ' ~ ~e [6]. The significant difference in electromagnetic properties of o-Ps and o-Ps' leads to the effect that after each collision the o-Ps system is an incoherent mixture of oPs and o-Ps'. Using Eq. (3) it is easy to show that if one starts with pure o-Ps, the relative probability of an invisible decay of o-Ps will be [9]
as
Br(o-Ps ~ o-Ps') = 32/2toeAT,
B R ( o - P s ~ o - P s ' ) = ½ t~2/(t~2 q- A2 + A2) .
(4)
where A is any additional splitting of o-Ps and o-Ps' by the external fields. As was shown by Carlson and Glashow [ 10] the upper limit of ~< 3 × 10 -8 on the photon-mirror photon mixing strength can be obtained from the theory of big band nucleosynthesis. This result seems to exclude mirror matter as an explanation of the o-Ps lifetime [10]. Indeed, for A = 0 and ~ < 3 × 1 0 -8, BR(oP s ~ o - P s ' ) =82/2A~ < 1 0 -2, which is too small to explain the discrepancy. Nevertheless, two experiments were performed on the "disappearance" of oPs [ 11,12]. The authors of a recent experiment [ 12] performed a more sensitive search for the invisible decay of o-Ps giving an upper limit of B R ( o - P s ~ o Ps') < 2.8 × 10 -6 on the branching ratio for decay of o-Ps as mirror matter in vacuum. They also claim a new limit of e < 1.5 × 10 -8 on the photon-mirror photon mixing strength. Using the upper limit of E < 3 × 10 -8 obtained in [ 10], we can see that the splitting energy A E < 10 -12 eV is so small that the effects due to the environment of the o-Ps system become important for the rate of oPs ~ o-Ps'. A measurement in which the o - P s - o-Ps' system remains in matter will give a much lower probability for o-Ps ~ o-Ps' transitions. In the experiment [ 12] o-Ps were formed in a target of silica aerogel (density 0.1 g / c m 3 and typical grain size ---50/~); not in vacuum. The behavior of o-Ps in silica aerogel is fairly well understood. Positronium is found inside the ultrafine grains and diffuses to the grain surface where it is ejected into the free space between the grains [ 13]. After ejection, the o-Ps is thermalized loosing its energy through collisions with
(5)
where toc is the o-Ps-grains collision rate. Thus, the transition rate will be reduced roughly by a factor AT/ toc = 1 / N compared to the vacuum rate, where N is the number of o-Ps-grains collisions in an o-Ps lifetime in silica aerogel, which is about 1/hr. The collision frequency of the o-Ps with grains is to~ = n~rR2v ,
(6)
where n is the number of grains per unit volume, R is the radius of the grains, and v is the mean o-Ps velocity. We assume that the grains are spherical and uniform in radius. For values n - 1 0 1 6 c m -3, R~--50/~, v ~- 107 cm/s (o-Ps kinetic energy =0.1 eV, [13]) we get tOe ~ 1011 S-1, or N ~ 104 collisions in a o-Ps lifetime of 135 ns in silica aerogel [ 12]. Thus, taking the suppression factor 1 / N ~ . 10 - 4 into account we actually have B R ( o - P s ~ o - P s ' ) < 10 -2 and ~< 10 - 6 instead of the values given above. A more stringent experimental limit for decay of oPs as mirror matter in vacuum can be obtained from the comparison of the results of the experiments [ 1 ] and [2]. In the gas experiment [1] the vacuum decay rate A~=7.0514+0.0014 MHz was obtained by extrapolation to zero gas density. The o-Ps-gas collisions make the amplitudes for developing mirror o-Ps in the period between collisions add incoherently, thus the possible contribution to )t~- due to the transitions oPs ---)o-Ps' is suppressed by the high positronium-gas collision rate. The factor 1/N is about 10- 3 even at the lowest gas pressure ( = 100 Torr) used in the experiment. Using the dependence of the measured o-Ps lifetime on the gas pressure [ 1 ] one can show that within the 1tr error limit on A~ the corresponding uncertainty in the number of collisions in the o-Ps lifetime is of the
S.N. Gninenko /Physics Letters B 326 (1994) 317-319
order of 30. In this case we expect that there is no significant contribution to the decay rate A~ from the effect of a mirror universe. In the vacuum experiment [2] the average number of o-Ps collisions with the walls of a vacuum cavity was in the range 3-10 in the o-Ps lifetime depending on the cavity volume. The o-Ps vacuum decay rate was obtained by extrapolation to an infinite volume of the cavity giving A~ = 7.0482 + 0.0016 MHz. Again, taking into account the dependence of the measured o-Ps lifetime value on the cavity volume [2] we see that the asymptotic value A~- corresponds to the number of collisions N = 3. External fields do not seem to contribute significantly to B R ( o - P s ~ o - P s ' ) in (4). The contribution due to the quadratic Stark effect in o-Ps is A---~ X $" (in atomic units) for the ground state. This gives A << AT ( = 4 × 10 - 9 e V ) for the electric field ~---100 V / c m used in the experiment. For M---,/f4 conversion the Stark effect does not contribute because both M and 1VIinteract with the electric field producing the same shift [9]. To avoid magnetic quenching the magnetic field H must be below about 0.1G in order to keep the splitting A = eH/2me
(7)
319
For the limit on the photon-mirror photon mixing strength it gives E< 2.8 × 10 -7. It should be noted that these limits are experimental limits, which cannot exclude mirror matter as an explanation of the o-Ps lifetime. The account of higher order correction in oL may, possibly, resolve the problem, but whether one accepts this assumption or not, it is interesting to try to exclude the effect of a mirror universe on the o-Ps lifetime by direct experiment on the "disappearance" of o-Ps in vacuum. The author thanks M. Baldo-Ceolin for helpful discussions, and D.W. Gidley and A.A. Poblaguev for useful remarks. The help of CERN/PPE on the supporting part of this work is gratefully acknowledged.
References [ 1] C.I. Westbrook, D.W. Gidley, R.S. Conti and A. Rich, Phys. Rev. Lett. 58 (1987) 1328; Phys. Rev. A40 (1989) 5489. [2] J.C. Nico, D.W. Gidley, A. Rich and P.W. Zitzewitz, Phys. Rev. Lett. 65 (1990) 1344. [3] W.E. Caswell and G.P. Lepage, Phys. Rev. A 20 (1979) 36. [4] G.S. Adkins, Ann. Phys. (NY) 146 (1983) 78. [5] M.I. Dobroliubov, S.N. Gninenko, A.Yu. Ignatiev and V.A. Matveev, Intern. J. Mod. Phys. A 8 (1993) 2859. [6] S.L. Glashov, Phys. Lett. B 167 (1986) 35. [7] I.Y. Kobzarev, L.B. Okun and I.Y. Pomeranchuk, Yad. Fiz. 3 (1966) 1154 [Soy. J. Nucl. Phys. 3 (1966) 837]. [8] B. Holdom, Phys. Lett. B 166 (1986) 196; B 178 (1986) 65. [9] G. Feinberg and S. Weinberg, Phys. Rev. Lett. 6 (1961) 381; Phys. Rev. 123 (1961) 1439. [ 10] E.D. Carlson and S.L. Glashow, Phys. Lett. B 193 (1987) 168. [ 11 ] G.S. Atoyan, S.N. Gninenko, V.I. Razin and Yu.V. Ryahov, Phys. Lett. B 220 (1989) 317. [ 12] T. Mitsui et al., Phys. Rev. Lett. 70 (1993) 2265. [ 13] T. Chang, M. Xu and X. Zeng, Phys. Lett. A 126 (1987) 189. [14] J.C. Nico, Ph.D. thesis, University of Michigan (Ann Arbor, 1991 ), unpublished. [15] Particle Data Group, K. Hikasa et al., Review of Particle Properties, Phys. Rev. D 45 (1992) Part II, Ill.39.