Volume 205, number 2,3 PHYSICS LETTERS B 28 April 1988

GENERATIONS OF ABNORMAL QED STATES

Chun Wa WONG Department of Physics, University of California, Los Angeles, CA 90024, USA

Received 19 December 1987

Generations of abnormal QED states containing heavy leptons might be expected if a postulated abnormal QED vacuum con- fines these leptons.

It has been suggested recently [ 1 ] that the five nar- row states of masses 1-2 MeV/c 2 observed in low- energy heavy-ion collisions [2-7] might be the 'So and the four P-states of an e+e - pair trapped in bags of abnormal QED vacuum [ 8-11 ]. I f this abnormal QED vacuum exists and if electrons and photons can be confined in it, a rich spectroscopy of abnormal QED states containing electrons and photons might be expected in a way reminiscent of hadron spectros- copy. The question naturally arises as to whether this postulated abnormal QED vacuum might also con- fine the muon and the tau lepton as well.

It must be pointed out that the existence of these abnormal QED states has neither been experimen- tally confirmed nor theoretically proven at the pres- ent time. An unambiguous experimental confirmation of the proposed interpretation and a theoretical un- derstanding of what the abnormal vacuum really is remain the basic questions which must be adressed. Given sufficient interest, answers to these important questions should be forthcoming in the near future.

In the meantime, it is useful to speculate on how the higher generations of abnormal QED spectros- copy might look like should the abnormal vacuum confine the heavier leptons as well. Such a study might give us some idea what to look for in the eventuality that these states could be produced, for example, in relativistic heavy-ion collisions (see ref. [12] and references quoted therein). In this short note, I would like to give a simple bag-model description [ 1,13 ] of some of these possible abnormal leptonic states in the

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bag, as well as estimates of the 27 decay widths of the n J So states.

I shall assume that the actual masses rni of the heavy leptons will not be changed significantly inside the bag. The actual changes Am~, no matter how small, will be important in determining the absolute masses of the states to be calculated; this difficult question will be left to the very end of this note. Given the large masses involved, the leptonic motion inside the bag will be non-relativistic (NR). This has the ad- vantage that the resulting calculations become totally elementary and transparent.

The total energy of leptons and bag is

E ( R ) = ~ ( m i + A m , ) + A E ( R ) , (1)

where

A E ( R ) = 4 nBR 3_ Z o / R +pE/2/t,

and/l = m l m 2 / ( m t + m 2 ) is the reduced mass. I f it is assumed for simplicity that the confinement is achieved by using an infinite square-well potential, then p = X / R l 2 , where x is a zero of a spherical Bessel function and R12 is the bag radius in the relative co- ordinate . R t2 is related to the bag radius R around the center of mass (CM) of the system by the expression

Rl2 = ( l + ml /m2 ) R = f l R , (2)

if we assume that the lighter particle 1 with ml ~ m2 controls the size of the bag by virtue of its being fur- ther away from the CM.

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Volume 205, number 2,3 PHYSICS LETTERS B 28 April 1988

The system described by eq. ( 1 ) saturates at the mass

M = ~ (miWAmi)WAM,

with

AM= ~ (4 ~BR3),

R = B - '/4 (x2B l/4/47tltf12 ) 1/5,

if the Zo term is neglected. This approximate analytic solution permits the calculation to be done by hand. The 27 decay width of the IS 0 state of heavy lepton- ium, calculated with the help of the usual positron- ium formula, is

F( Is 0-027) = ] (or~m)2 xE/R 3.

Using the bag-model parameter of B1/4= 0.28 MeV of ref. [1], and m, ( m 0 = 1 0 5 . 7 (1784) MeV, we obtain the results shown in table 1. Here excitation energies are in keV, while widths are in eV. The bag radius of the 1S state turns out to be 180 ( 100, 200) fm for the ],L 2 (X 2, I.~X) system. The ~tx bags are the biggest and their excitation energies are the largest because the lighter lepton is furthest away from the CM.

There are other leptonic systems as well. The most interesting of these are probably elx, ex, 711 and "/x, where one of the particles is massless. The parameter fl in eq. (2) is then 1. (The expression forfl should actually be corrected for relativistic effects, but the result should not be very different from 1. ) The de- scription of these systems then simplifies because the massive lepton is roughly at rest in the CM, and is simply surrounded by the massless particle and the

bag. The energy of the light particle is now just ( x - Z o ) / R , where R=RI2 is measured from the massive particle at the CM. This is the same expres- sion as that for two identical massless particles with or without the massive lepton at the CM. The reason is that the factor 2 in 2x/R12 is then cancelled by fl= 2 in R12. (A related cancellation appears also in NR systems, but there the cancellation is not complete because the NR kinetic energy is quadratic in p. The residual reduced-mass effect then distinguishes the positronium spectrum from that of the hydrogen atom. ) As a result, the excitation energies of elx, ex, e2kt and e2x are identical, and the same as those for the abnormal trapped positronium (TPs) states dis- cussed in ref. [ 1 ]. In a similar way, the excitation energies of 7~t, 7z, 721x and 72"r can be read off from those of the lightball states 72. These theoretical ener- gies, obtained from ref. [1 ], are also shown in the table.

The actual masses of these states depend of course on the mass changes Ami of the leptons and photons in the bag. It would be interesting if all leptonic masses decrease by a universal amount, say one electron mass. Then the abnormal e~t, ex, e2~ and eEz states will be above the breakup threshold by the same amounts as the corresponding TPs states are above theirs. For the ~t 2 and x 2 systems on the other hand, the abnormal states will be mostly below the normal states. This possibility might have observable conse- quences on the properties of the normal states themselves.

Another possibility is that the masses of the heavy leptons do not change very much inside the bag. In that case, the abnormal states will be above the nor-

Table 1 Excitation energies and 27 widths of heavy leptonium states trapped in bags of abnormal QED vacuum.

nl AMc 2 (keY) F('So-,27) (eV)

~2 x2 gx e(~,x) 7(~,x) ~2 x2 e2(~,x) 72(~,x)

Zo 0 0 0 0,31 o

IS 48 9 7o lO6O 17oo 1.6× lO -2 3.0× lO -4 1P 73 13 110 1640 2200 1D 100 18 140 2110 2600 2S 110 20 160 2380 2800 2.8 5.3 3S 180 33 260 3420 3800 3.8 7.3

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Volume 205, number 2,3 PHYSICS LETTERS B 28 April 1988

mal ones. An interest ing range of possibi l i t ies for Ami appears to be between 0 and - me. This seems to im- ply that there are relat ively well-defined ranges o f masses where the lightest states o f each type might be found. F o r states with const i tuent photons , there is an addi t ional uncertainty due to the unknown masses

o f the t r apped photons. Rela t iv is t ic heavy- ion coll isions should be copious

sources o f these states, i f they exist. However , 2y spectroscopy might not be as useful here because o f the small 2y decay widths. The invar iant mass spec- t ra o f charged states might be more informat ive , i f they could be measured. The muonic charged states could also be p roduced by the inelast ic scattering o f muons f rom heavy nuclear targets. It would be inter- esting to s tudy the feasibi l i ty of this react ion in me-

son factories. In summary, generat ions o f abnormal heavy lep-

tonic states might be expected i f the pos tu la ted ab- normal QED vacuum confines these leptons too. There is no compel l ing exper imenta l or theoret ical reason at the present t ime to suppor t thei r existence. But it is useful to know how to recognize them when

they are seen.

This work is suppor ted in part by N S F grant PHY86-12604.

References

[ 1 ] C.W. Wong, Electrons and photons trapped in bags of ab- normal QED vacuum, UCLA preprint ( i 987).

[ 2 ] W. Greiner, ed., Physics of strong fields (Plenum, New York, 1987).

[3] T.E. Cowan et al., in: Physics of strong fields, ed. W. Grei- ner (Plenum, New York, 1987) p. I 11; H. Bokemeyer, in: Physics of strong fields, ed. W. Greiner (Plenum, New York, 1987 ) p. 195.

[4] T. Cowan et al., Phys. Rev. Lett. 56 (1986) 444. [5] T. Cowan et al., Phys. Rev. Lett. 54 (1985) 1761. [6] H. Tsertos et al., Z. Phys. A 326 (1987) 235. [7] K. Danzmann el al., Phys. Rev. Lett. 59 (1987) 1885. [ 8 ] LS. Celenza, V.K. Mishra, C.M. Shakin and K.F. Liu, Phys.

Rev. Lett. 57 (1986) 55. [9] LS. Celenza, C.-R. Ji and C.M. Shakin, Phys. Rev. D 36

(1987) 2144. [ 10 ] D.G. Caldi and A. Chodos, Phys. Rev. D 36 ( 1987 ) 2876. [ 11 ] Y.J. Ng and Y. Kikuchi, Phys. Rev. D 36 (1987) 2880. [ 12 ] C. Bottcher and M.R. Strayer, in: Physics of strong fields,

ed. W. Greiner (Plenum, New York, 1987 ) p. 629. [ 13] T. DeGrand et al., Phys. Rev. D 12 (1975) 2060.

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