nonappearance of dehnen-shahin resonances in the positronium continuum

Physics Letters B 301 (1993) 1-5 PHYSICS LETTERS B North-Holland

Nonappearance of Dehnen-Shahin resonances in the positronium continuum

Chun Wa Wong Department of Physics, University of California, Los Angeles, CA 90024-1547, USA

and

Cheuk-Yin Wong Oak Ridge National Laboratory, Oak Ridge, TN 37831-6373, USA

Received 27 October 1992; revised manuscript received 16 December 1992

The Dehnen-Shahin relativistic equations for the positronium are found to have no resonant solutions in both the ~So and the 3P o positronium continua. A certain pole singularity appearing in the ~So potential gives rise to resonances, but only if the electro- magnetic interaction strength is increased 160-fold or more. These resonances are found to have zero width. However, the same singnlarity, even in a weak potential, gives unacceptable nonphysical bound states near zero energy. The strange similarity of Spence-Vary 3P 0 resonances to resonances in an infinite square-well potential is noted.

The possible appearance of resonant states in the pos i t ron ium (Ps) con t inuum has been a quest ion o f considerable interest in the past. Thei r appearance would indicate a need to modi fy or f ine-tune our understanding of QED. Interest in this question has been enhanced in recent years by persistent reports o f resonance-l ike structures or composi te states in heavy- ion collisions where ultra-large Coulomb fields are generated [ 1,2 ]. Groups of electrons and posi t rons o f identical energies have been seen in coincidence. The total energies of these experimental structures are in the range of 1.5-1.8 MeV. They might represent structures in the Ps cont inuum.

One o f the ideas s tudied in the past in this connec- t ion is the appearance o f magnet ic resonances in the 3Po state [ 3-5 ]. But a more recent reexaminat ion [6 ] has shown that the potent ia l is not strong enough to hold a resonance.

Three related, but technically different, ideas have recently appeared. First , Arbuzov et al. [7 ] have solved a two-body relat ivist ic wave equat ion in sca- lar QED (the Wick-Cutkosky mode l ) in m o m e n t u m space and found resonances not only in the Ps con- t inuum, but also in the e - - e - and p r o t o n - p r o t o n

continua. However, a s imilar equat ion has been solved by Walet, Klein and Dreizler [ 8 ] who can find no resonance. They believe that the results ofref . [ 7 ] are the spurious effects coming from the chosen momen tum basis used in the calculation. Similarly, Horbatsch [9] cannot f ind resonances in s imilar equations.

Secondly, Dehnen and Shahin have solved the Breit equat ion for the Ps (DS1 [ 10] ) and a s imilar equa- t ion (DS2 [ 11 ] ) proposed by Barut et al. [ 12 ], both in coordinate space. They f ind a number o f resonances o f finite widths in the Ps con t inuum in at least the 3po and ~So states beginning at the total energy of 1.3 MeV, i.e. the same energy region as the experimental structures.

Finally, zero-width resonances rather s imilar to those o f DS have been found by Spence and Vary (SV 1 [ 13 ] and SV2 [ 14 ] ) in three momentum-space relativistic wave equations derived from QED. These resonances have been cri t icized by Horbatsch [9 ] who finds no resonances in s imilar momentum-space equations. Horbatsch further states that scattering solutions which do not include the on-shell momentum as a basis point will in t roduce spurious narrow

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Volume 301, number I PHYSICS LETTERS B 25 February 1993

structures at the arbitrarily chosen momentum basis and that the treatment of the infinite-range Coulomb potential in momentum space might not be simple.

We should add that the physical origin of these claimed resonances have not been studied or identi- fied in refs. [ 7,10,1 l, 13,14 ]. DS do mention a pole singularity on a spherical shell of finite radius in the effective potential for the ~So state of parity ( - 1 )J+ (misidentified as the 3P o state of parity ( - 1 )Sin both ref. [ 10] and ref. [ 11 ] ) when the e + - e - separation is near the classical electron radius. However, they have not shown that their resonances come specifi- cally from this singularity. Furthermore, this singularity is absent in the 3Po state (misidentified as the IS o state in both ref. [10] and ref. [ l l ] ) , and yet similar resonances are found in this state in ref. [ 10 ].

In view of the continued interest concerning the possible appearance of resonances in the Ps continuum in both experiment and theory, it would be of interest to clarify the theoretical situation with re- spect to the DS resonances. Furthermore, the DS wave equations involve only local potentials in coordinate space; they are therefore much easier to solve than the momentum-spaced equations mentioned previously. Finally, it is an interesting question whether the DS pole singularity could give rise to resonances at all.

We would like to prove here that the DS pole singularity does not give rise to a resonance in the Ps continuum. Resonances appear only when the interaction is made much stronger. They have zero width and are similar to resonances in an infinite square- well potential. We find, furthermore, that the pole singularity gives rise to unphysical bound states near zero energy. Hence equations containing such poles cannot give an acceptable description of the Ps.

In the 3P o state where the pole singularity is absent, our phase-shift solutions for the DS equations show no resonant behavior, only the very small effects normally expected of the weak e + - e - interaction.

In common with many other relativistic wave functions (w.f.s) for the Ps, the DS w.f.s have 16 spi- nor components. The resulting 16 coupled first-order differential equations separate into two disjoint groups of eight, with parity ( - 1 )J+~ and ( - 1 )J, re- spectively. Four of the eight equations in each group can be written as algebraic equations for properly chosen w.f.s. They can be used to eliminate four of

the w.f.s, leaving four coupled, first-order, differential equations with effective potentials arising from the algebraic elimination. We shall show explicitly that these effective potentials could have unusual singularities where certain w.f.s acquire unusual boundary conditions (b. c. s ) because of the algebraic elimination.

For the ~So states, only three of the eight DS w.f.s (denoted us, Vs andfa in DS2) are nonzero, so that the particle number density takes on the form

C~u=l ( lu212+ Ivs 12+ If31 z) . (1)

These w.f.s are actually related to one another:

2m u2= E_2e2/rvs, (2)

10~V2, (3) A=-~

(E+-~)v2+2mu2-2(Or+2)f3=O, (4)

where m is the electronic mass and E = 2 ~ - - / + p 2 is the total Ps energy (with h = c = l ) . It is therefore necessary to study only one of these w.f.s, say vz.

The wave equation satisfied by v2 is

0~+20r+V2-r -e+E_--ffe~/r)Jvs=o. (5) The effective potential contains an unusual term with a pole at rpole= 2eS/E coming from the elimination of u2. This is the pole singularity mentioned earlier in the paper.

The pole term is attractive inside rpo~e, and repul- sive outside, being Coulomb-like on each side of rw~,. It therefore gives rise to an infinite bowl-shape potential in which resonances could appear. However, rpol~ itself shrinks to zero as the total energy E in- creases to infinity. We shall now show that the situation is not unlike resonances in an infinite square- well potential, but that because of the shrinking radius, no resonance is trapped inside a weak potential.

The mathematical analysis for a shrinking radius is facilitated by using the dimensionless (but energy- dependent) distance x = (E/ot)r, where ot = e s is the fine structure constant. The wave equation for the ra- dial w.f. F(x) =xvs is then

F " + (k s - V ) F = 0 , (6)

Volume 301, number 1 PHYSICS LETTERS B 25 February 1993

where

k2 = ~ a 2 ( 1_ 4~ p 2 ~ ,

A B V ( x ) =

x 2 - x '

E

m (7)

(8)

2o¢ 2 B = g''T-, A = O t 2 " (9)

Eq. (6) is just eq. (3.11 a) of ref. [ 1 ]. It represents the starting point of our investigation.

In terms of x, the pole appears at x- -2 at all energies. It is the dimensionless wave number k which now has an unusual energy dependence: It is always small and reaches a maximum value of only lot at infinite energy! Thus the shrinking radius has the same effect as the limitation o fk in eq. (6) to small values.

Eq. (6) is to be solved subject to the b.c.s:

F ( x ) = 0 , a t x = 0 , x = 2 . (10)

The condition at x = 2, needed to ensure that u2 is finite there, reduces the problem to that of an infinite square-well potential, plus a very weak inside potential V(x). We therefore perturb from the discrete states of the infinite square-well potential to obtain the following first-order discrete spectrum:

k 2 = (½ni t ) 2 - (A+B)Cn, (11 )

where 2

,12, 0

Since A and B are both very small ( ~ 10-4), kn is very close to ½ nrc. These are resonances of zero width embedded in the k-continuum 0 < k < ~ .

The first nontrivial (nonzero) solution in the inside region ( x < 2 ) therefore appears at k~ ~ ½~. Ac- cording to eq. (7), k cannot exceed ½ oL ~ 4 × 10- 3 in the Ps continuum. Hence there is no nonzero or resonant solution in the inside region in the physical Ps continuum.

In other words, while the b.c. (10) has a tendency to induce resonances in the inside region, the inside potential is so weak and the shrinkage of the pole position with increasing energy is so rapid that the usual resonance (or standing-wave) condition cannot be satisfied. All resonances are thus successfully evaded.

To confirm these perturbative results and to treat stronger potentials, we obtain exact solutions to eq. (6) for arbitrarily strong potentials by expanding the w.f. in power series about both x = 0 and x=2 . The two power series are matched at an intermediate point (say x = 1 ) to find the discrete eigenvalue spectrum in the inside region. In this way, we verify that the Coulomb perturbative result ofeq. ( 1 1 ) is very good. For example, if we take e=2 (i.e. A----O/2, B= ½a 2)

and a=1/137.0359895, and let k take any value unconstrained by the kinematical relation (7), the exact result for the dimensionless eigenmomentum shift Akn=kn-½nn for n = l is found to be -3 .0989794×10 -5. The perturbative result is - 3.0989463 X l0 -5, about 3X l0 -~° too high. Both the sign and the order of magnitude [ ~ ( A k l ) 2 ] of the error in this and other perturbative results are consistent with the known behavior of perturbation theory.

It is obvious that if we allow the potential to be- come arbitrarily strong, sooner or later it will trap resonances in the Ps continuum even under the kinematical constraint (7). This possibility can be studied by using a model where one of the particles has a charge (the "effective charge") Z so that a becomes Otz= Z a everywhere. We look for the smallest effective charge which will reproduce the first DS resonance at the reported kinetic energy K = E - 2 m = 317 keV. The answer is Z = 237 for the subtracted potential B [ ( 2 - x) - i - - X -- 1 ] actually used by DS2, and Z-- 163 for the original potential shown in eq. (9).

Similar conclusions are obtained by examining the phase shifts of both the ~So equation in ref. [ 10] and the 3P o equation in ref. [ 11 ]. We have done calculations both with the original potentials and with the regularized potentials actually used by DS (with 1/ (2 - x) regularized to (2 - x) / [ (2 - x) 2 .]_ ~2 ] ). Typ- ical phase shifts for different k values are of the order l0 -8 to l0 -4 rad when the outside potential is trun- cated at Xm~ss = 20 to 20 000. These results are consistent with the fact that all these potentials are very weak, with strengths of O (or 2) in the chosen dimensionless unit. Resonances in the phase shifts begin to appear only when the potentials are made abnor- mally strong, with effective charges of the order of 200 or more. Details of these calculations will be given elsewhere [ 15 ].

For these weak potentials, the only solutions to eq.


(6) consistent with the stated b.c.s at all physical energies are the nonresonant solutions in which all w.f.s (F, u2 and f3) are completely excluded from the inside region. The fractional excluded volume in the normal 23po Ps state can be shown to be of the order 4 × 10-5ot6__ 10-,7. But is even this very small effect real?

To answer this question, we reexamine eq. (6). The pole term in the potential is proportional to e -2, where e=E/m; it therefore becomes infinitely strong as e (or E) approaches zero. We find by numerical calculation [ 15 ] that this strong potential gives rise to additional bound states near E = 0 . The first of these abnormal bound states appears at E = 3.23 keV for the original potential (9). We find ten other bound states down to 0,33 keV. Below this energy, our numerical solution begins to be unstable.

The origin of these nonphysical bound states can be seen in a more revealing way in eq. (5). As the total energy E (which includes the finite rest energy rn) approaches 0, the pole position rpo,e becomes larger and larger so that the inside region could trap more standing waves under favorable conditions. Since these additional bound states do not appear in nature, the corresponding DS equations cannot be used to describe the physical Ps. Hence the exclusion effect described previously is not real.

The unphysical DS pole potential appears only for certain e + - e - interactions and in certain states. If the interaction has the form

e 2 V ( r ) = - - - ( 1 - a ~ , " ~ 2 -b~ l "~ ~2"~) , (13)

r

we can show that the DS pole appears if [ 15 ]

3a + b> 1, for all P = ( - 1 )J+ ~ states,

a+b> 1, for all states,

b - a > 1, for all states except ~ So. ( 14 )

Thus a DS pole appears in the 1S o state of both ref. [ 10] where a = ½, b= ½ (corresponding to the Breit interaction), and in ref. [ 11 ], where a = 1, b = 0 (corresponding to the Barut interaction). This means that the ~So DS equations based on either interaction are not acceptable. However, no DS pole singularity appears in the 3po DS equations.

We should add that the Breit equation is known to give erroneous results when taken beyond first-order

perturbation [ 16 ]. To get correct results known from QED, it must be changed into a Salpeter equation by projecting out the negative-energy plane-wave states.

The requisite positive-energy projection operator does appear in all three 3P o w a v e equations studied in ref. [ 13 ]. They are a Tamm-Dancof f equation, a no-pair form of the Breit equation, and a Blanken- becler-Sugar (i.e. instantaneous) form of the no-pair Breit equation. The interaction used in model 1 dif- fers somewhat from the Breit interaction, but it is just the Breit interaction in the other two models. Since the DS pole does not appear in any of these 3P o equations, we must conclude that the zero-width SV resonances can only come from the positive-energy projection operator itself.

The presence of projection operators reduces the Hilbert space and reduces the overall attraction. This would normally reduce the likelihood of resonance formation. Furthermore, zero-width resonances im- ply a complete trapping of the w.f.s in restricted re- gions of space. Thus a persuasive case for SV resonances may require an understanding of how this complete trapping is achieved by the positive-energy projection operator.

Much can also be learned from the resonance energies themselves. The SV resonances appear at the same total energies for all three models, starting at E = 1351 keV [13]. The corresponding eigenmomenta p , start at 442 keV/c and are equally spaced thereafter. They are approximately given by Pn"" (n+4)Ap, where Ap= 106 keV. All these features (an equally spaced momentum spectrum, zero resonance width, and insensitivity to differences in the weak e +- e - interaction) could be reproduced readily by an infinite square-well potential model with the radius of E=n/Ap= 5850 fm, provided that the first four eigenmomenta are projected out. This result suggests that the reported SV resonances, in order to satisfy the uncertainty principle, should have a spatial ex- tension of about 6000 fm, not the 30 fm given in ref. [ 13 ]. In view of these puzzling features, a more de- tailed study of the SV equations might be of interest.

This research is supported in part by the USDOE under Contract Number DE-AC05-84OR21400 managed by Martin Marietta Energy Systems, Inc.


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nonappearance of dehnen-shahin resonances in the positronium continuum

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