solutions of the dehnen-shahin relativistic equations for positronium

Nuclear Physics A562 (1993) 598-616 North-Holland

NUCLEAR PHYSICS A

Solutions of the Dehnen-Shahin relativistic equations for positronium

Chun Wa Wong

Cheuk-Yin Wong

Oak Rtige National Laboratory, Oak Ridge, TN 37831-6373, USA

Received 11 March 1993

Abstract A pole singularity is found in the Kemmer-Fermi-Yang relativistic equation for positronium-

like systems under favorable conditions. We show that its presence in the ‘Se state for the ~si~onium interaction used by Dehnen and Shahin does not give rise to any resonance. Resonances of zero width appear only if the electromagnetic interaction strength is increased 160..fold or more. However, the same singularity, even in a weak potential, shows an unacceptable infrared pathology in which a number of spurious bound states appear near zero energy. Examination of the phase shifts confirms the absence of resonances in the Dehnen-Shahin equation in both ‘Se and 3Pa states.

1. Introduction

It has been realized for some time that strong electromagnetic fields might give rise to novel non-perturbative phenomena [l]. An example is the possible decay of the normal neutral QED vacuum to a stable charged vacuum by spontaneous positron emission [2] under the following circumstance: The energy of an electron around a finite nucleus of charge Z can fall below -m (with c = l), where m is the electronic mass, when Z is sufficiently large. If such an electronic state is empty, it can be populated by two electrons from the negative-energy sea. When this happens, we see two positrons in the continuum, together with a doubly charged atom.

Positrons have in fact been detected in heavy-ion collisions where ultra-large Coulomb fields are generated [3]. However, the detected positrons do not seem to have the properties expected of the particles emitted by the vacuum decay described above, They have instead characteristic energies which seem to be

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C. U! Wang, C.-Y. Wong / Dehnen -Shahin relativistic equations 599

independent of 2. Furthermore they appear in coincidence with electrons of similar energies [4]. If both positron and electron come from the decay of a neutral particle, the total energies of these experimental structures would be in the range of 1.5-1.8 MeV. They might represent structures in the positronium (Ps) continuum. If so, their lifetimes should be shorter than about lo-r0 s; otherwise they

would escape undetected from the detector. The interest in these experimental structures has led in recent years to a

number of attempts to look for these resonances directly in elastic e+e- (Bhabha) scattering [5,6]. The experimental lower limit for the lifetime of a Ps resonance around 1.8 MeV has now been improved to 5 X lo-l1 s (if J = 0) and 1 X 10-i’ s (if J = 1) [6]. This seems to have ruled out the explanation of the observed structures as Ps resonances. Recent reviews of low-energy Bhabha scattering and heavy-ion positrons are given in refs. [5,7].

We should add that QED has been checked in both atomic and high-energy regimes: The theoretical calculation [8] of the anomalous magnetic moment of the electron, including all terms of order a! 4, is in agreement with the recent high-pre- cision experimental measurement [9], while the measured differential cross section for high-energy Bhabha scattering agrees with the result of the standard model for a point electron with a compositeness mass scale of not less than 1 TeV [lo]. In addition, we do not know of any serious discrepancy in normal nuclear properties involving the electromagnetic interactions.

On the theoretical side, the possibility of magnetic resonances appearing in the Ps continuum has been studied in the past fll-131. These could arise if the relativistic interaction between the anomalous magnetic moments [12] or magnetic moments [13] of the interacting particles at small distances becomes sufficiently attractive for some spin and angular momentum combinations such as the “PO state 1131. Unfortunately, a more careful examination of the relativistic interaction at short distances using Dirac’s constraint dynamics has shown that although the attractive interaction at short distances in the 3Po state is strong enough to overwhelm the centrifugal barrier, it is not strong enough to hold a resonance [14].

Such strong interactions should also give rise to non-linear self-energies and other effects which cannot be treated with any degree of confidence. Furthermore, the ~nihilation diagram has not been included in ref. [13]. consequently, no definite conclusions have been drawn from such studies. However, the belief persists that self-energy effects at small distances could lead to a phase transition in QED and the possible appearance of resonances 1151.

It has been suggested recently that Ps resonances might already be present even in the absence of unusual interaction terms: First, Arbuzov et al. [16] have solved a two-body relativistic wave equation in scalar QED (the Wick-Cutkosky model) in momentum space and found S-wave resonances not only in the Ps continuum, but also in the electron-electron and proton-proton continua. However, a similar equation has been solved by Walet, Klein and Dreizler [17] who find no reso-

600 C. W Wang, C.-Y Wong / Dehnen - Shahin relativistic equations

nances. They believe that the results of ref. [16] are spurious effects of the momentum basis used in the calculation. Similarly Horbatsch [18] has solved equations of similar structure without finding any resonance.

Secondly, Dehnen and Shahin have solved the Breit equation for Ps (DSl [19]) and a similar equation (DS2 [20]) proposed by Barut et al. [21], both in coordinate space. They find a number of resonances of finite width in the Ps continuum in at least the 3P,, and ‘S, states beginning at a total energy of 1.3 MeV, i.e. the same energy region as the experimental structures.

Finally, 3P0 resonances in the Ps continuum, rather similar to those of DS, have been found by Spence and Vary (SVl [221 and SV2 [23]) in three momentum-space relativistic wave equations derived from QED: a Tamm-Dancoff equation, a no-pair form of the Breit equation (but containing positive-energy projection operators), and a Blankenbecler-Sugar (i.e. instantaneous) form of the no-pair, projected Breit equation. There are significant differences between these and the DS resonances. The SV resonances disappear if the tensor interaction is turned off. They have zero width, unlike the finite widths of the DS resonances, but it is found in SV2 that when the Coulomb term is dropped, the SV resonances develop large widths of = 10 keV, comparable to those of DS resonances. (The resonance lifetimes of the order of 10e2’ s given by DSl and DS2 should read lo-l9 s if they are to be consistent with the widths of about 10 keV shown in their figures.)

The SV resonances have been criticized by Horbatsch [181 who finds no resonances in similar momentum-space equations. Horbatsch has further stated that scattering solutions which do not include the on-shell momentum as a basis point will introduce spurious narrow structures at the arbitrarily chosen momentum basis and that the treatment of the infinite-range Coulomb potential in momentum space is not simple. Scepticism has also been expressed by Crater et al. [14] because their own relativistic equations, which give results in agreement with QED to order cx4, do not show resonances.

We should note that all three theoretical claims of Ps resonances are based on numerical solutions about which some questions have been raised, as summarized above. All suffer from the serious omission that the physical resonance mechanism has not been identified [16,19,20,22,23].

Dehnen and Shahin call their structures magnetic resonances, but they have not shown that these are caused by magnetic interactions. They do point out the presence of a pole singularity on a spherical shell of finite radius in the effective potential for the ‘So state of parity (- l)J+’ (misidentified as the 3Po state of parity (- l)J in both DSl and DS2) when the e+-e- separation is near the classical electron radius. However, they have not demonstrated that their resonances come specifically from this singularity. Furthermore, this singularity is absent in the 3Po state (misidentified as the ‘So state in both DSl and DS2), and yet similar resonances are found in this state in DSl.

The presence of the pole singularities later studied by DS has already been

C. WO Wang? C.-Y. Wong / Dehnen - Shahin relativistic equations 601

noted by a number of authors including Koide f251, Childers 1261, Krohkowski f27], Brayshaw [28], Barut et al. 1211 and Geiger et al. t241. Koide @5] chooses the interactions carefully in order to avoid the singularities. Childers [26] has similarly suggested that they should be avoided if possible, and treated only as a perturbation if they cannot be avoided. This is consistent with the textbook warning 1291 that the Breit equation shouId not be used beyond first-order perturbation theory. ~olikowski [27] repeats the “widely accepted’* proposal of adding projection operators to change the Breit equation ta the Salpeter equation in order to avoid inconsistency with the results of the Dirac hole theory. He states further that “this inconsistency disease . . . coincides with the non-regularity disease pointed out by Childers . . . “, but admits that a proof is lacking. (He also calls the pole si~~ari~ a ~~non-no~aIizable~J singularity.) Brayshaw [2S] claims that changing the Breit equation into a Tamm-Dancoff equation (shown as Eq. (38.13) in ref. [29]) by restricting the Hilbert space to the subspace of positive-energy states will take care of the problem, but offers no proof. Finally, Geiger et al. [24] dismiss these pole singularities as ‘%trictly a mathematical artifact” originating from “the reduction of the coupled first-order equations _ _ _ to the second-order Schr&%nger type” of equations.

In view of the experimental non-observation of Ps resonances, it seems necessary to clarify the theoretical situation. In this paper, we concentrate on the DS and similar equations because they involve only local potentials in coordinate space. They are therefore much easier to solve than the momen~m-space equations mentioned previously. We would like to clarify the nature of the solution of the DS equation if and when the pole singularities mentioned previousIy cannot be avoided. We want to show in particular if they give rise to resonances and if they give rise to unacceptable pathologies.

We shall begin in sect. 2 by finding the circumstan~s under which DS poles would appear in the Kemmer-Fermi-Yang (KFY) equation [30] for Fs-like systems with a very general fermion-antifermion interaction. The DS equations whose solutions we want to study are then presented in sect. 3 using the conve- nient energy-dependent dimensionless coordinate of DS. In sect. 4, we show how resonance solutions are related to the DS pole and why they could appear only for interactions much stronger than those found in the Ps. Phase shifts for the DS equations are calculated in sect. 5 to confirm the non-appearance of resonances in the Ps continuum. The behavior of phase shifts for interactions strong enough to give resonances is also elucidated. A serious difficulty for Ps equations containing DS poles is pointed out in sect. 6. These equations contain spurious bound states near zero total energy; hence they cannot give a reliable description of Ps, Our conclusions are very briefly summarized in sect. 7.

A summary of some of our results has already been reported elsewhere [313. The present paper gives a more detailed report.

602 C. W. Wang, C.-Y. Wang / Dehnen - Shahin relativistic equations

2. Origin of the DS pole

The DS pole singularities arises only under favorable circumstances. Consider, for example, the KFY equation [30] for fermion-antifermion systems such as Ps,

[(“1-~2).~+Plml+P2m,+V-E]~=0, (1)

with an interaction

which is more general than that considered by Koide [25]. Like many other relativistic wave functions for Ps, the wave function (w.f.) $ has

16 spinor components. We have derived the resulting 16 coupled first-order differential equations. (The details are given in the appendix.) As expected, they separate into two disjoint groups of eight, with parity (- #+’ and (- ljJ respec- tively, for each value of the total angular momentum J. Four of the equations in each group can be written as algebraic equations for properly chosen w.f.‘s. In the notation of Koide [25] and our appendix, those for parity P = (- ljJ states are

(E + &-)a, - 2Aa, = 0,

(E + U,,-) b, - 2Ab, = 0,

(E - iI&+)& + 2&T, + 2 ma _.

r 3- 9 (3)

where a,, a3,. . . , g, are the eight w.f. components of this group. The U potentials are linear combinations of the I/ and A potentials of Eq. (2). (Explicit definitions of the w.f. components and the U potentials are given in the appendix, but they are not needed in the discussion which follows.) The remaining quantities in Eq. (3) are

2M=m,+m,, 24 =m,-m,. (4)

C. W. Wong, C.-Y. Wong / Dehnen - Shahin relativistic equations 603

The four P=(-1) ‘+ ’ algebraic equations are

(E - UzO+)al + 2Ma, = 0,

(E- U,,+)b,+ 2Mb, =o,

ml) =o

(E+~,f-)fo-24f3-2 r I 5

(E+U,,-)g,-2Ag,+ ma _.

r z- .

The remaining equations of each group are coupled first-order differential equations, given in the appendix. They are not directly involved in the discussion in this

section. Each of the algebraic equations can be used to eliminate the w.f. component

appearing in its first term from the set of eight coupled equations. When this is done, the coefficient of the eliminated w.f. component in the original algebraic equation appears in a denominator in one or more terms of the remaining equations. Hence, a pole singularity appears whenever this denominator vanishes. For example, the use of the first algebraic equation in Eq. (3) to eliminate a, gives a pole at a zero of the coefficient E + IV..,-. This leads to the eight singularity conditions for positive energy E

E+ U,,-=O, E-k U,,-= 0, . . . . E + &-=O. (6)

We should add that the pole does not come from the superficial change of expressing coupled first-order differential equations as coupled or uncoupled second-order differential equations. Consequently, it is not a mathematical artifact, but must be taken into account once the original wave equations are chosen for study. (It is a totally different issue whether the equation chosen gives a correct description of the physical problem.)

The presence of a pole requires that an appropriate w.f. vanish there. We shall show in sect. 4 that this has important consequences.

For the simpler e+-e- interaction

the singularity conditions simplify to

3a+b>l, for all P = ( - l)‘+’ states,

a+b>l, for all states,

b-a>l, for all states except ‘S,. (8)

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Eq. (8) shows that a pole singularity appears in the ‘S, state of both DSl where a = $, b = 3 (corresponding to the Breit interaction), and in DS2 where a = 1, b = 0 (corresponding to the Barut interaction). However, no such pole singularity appears in the 3P0 DS equation. The Breit interaction is also used in SV in the 3Pa state. Hence the pole is also absent there.

It is worth mentioning that the Breit equation is known to give erroneous results when taken beyond first-order perturbation theory [29]. 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 Barut equations also lack sufficient accuracy for spin-singlet states [24]. Here the negative-energy states must be treated in a special way [15]. The question of finding the best relativistic wave equations for Ps is a complicated issue which remains unresolved. We are con- cerned here only with the possibility of resonances in the DS equations.

3. The dimensionless equations of DS

Only three of the eight w.f. components in the ‘S, state are non-zero. (Two of these components are absent because the vector spherical harmonic %; = 0, L = 0, s = 1 vanishes identically. The remaining three components do not appear for equal-mass particles.) In the DS2 notation (also defined in the appendix), the particle number density takes on the form

~+~=t(lU212+IU212+Ifi I’)* (9)

These w.f.‘s are actually related to one another. For the DS2 (i.e. the Barut) interaction, they satisfy the equations

u2= -[2-/(G$)]U2,

f3 = - ;a,,,, 2

i 1 .+; v2+2mu2-2 ar+; f3=0, i 1

where

(10)

(11)

(12)

E = 2\lm2 -t p2 , (13)

C. W. Wang, C.-Y. Wong / Dehnen - Shahin relativistic equations 605

(with h = c = 1) is the total Ps energy. It is therefore necessary to study only one of these w.f.3, say uz, in our discussion.

The wave equation satisfied by uZ is

i

a~+~~r+p’-;{-E+[2m2/(E-3]))u2=o. (14)

The effective potential contains not only the familiar Coulomb attraction but also a singular term with a pole at

rPle = 2e2/E, (15)

which comes from the use of Eq. (10) to eliminate u2. The pole term is attractive inside rpole, and repulsive outside, being Coulomb-like

on each side of rpole. It therefore gives rise to an infinite bowl-shaped potential in which resonances could appear. However, rpole itself shrinks to zero as the total energy E increases to infinity.

The effect of the shrinking radius can be displayed more clearly by using the dimensionless (but energy-dependent) distance of DS

n= (E/a)r, (16)

where (Y = e2 is the fine-structure constant. The wave equation for the radial wave function F(x) =xu2 is then

F”+(k2-V)F=O, (17)

where

c=E/m,

V(x) = -A/x-B/(2-x), (19)

B = 2cu2/& A=a2. (20)

Thus the change of the variable simplifies the equation by fixing the pole radius at x = 2 at all energies. The effect of the originally shrinking radius is now transferred to the dimensionless wave number k: It is always small and reaches a maximum value of only k,, = +a = 4 X 10V3 at infinite energy! We are now in a position to understand if resonances could appear in the continuum of the DS equation.

606 C. W. Wang, C.-Y. Wong / Dehnen - Shahin relativistic equations

4. Resonant solutions of the DS equation

The boundary conditions (b.c.5) satisfied by Eq. (17) are

F(x) =o, at x=Oand x=2. (21)

The condition at x = 2 is needed to ensure that uz is finite there. (This is in fact the textbook solution of the “non-normalizable” problem noted in ref. [27].) With this b.c., the problem becomes identical to that of an infinite square-well potential with an inside potential V(x). The resulting energy spectrum is always discrete. It contains states which are bound states embedded in the continuum, or equivalently resonances of zero width.

We have shown previously [31] that since the interior potential is very weak ((u* being of the order 10-4), the perturbative energies

k,2=($m)2-(A+B)C,, (22)

where

C, = /o’ksin’($zrrx) dx (23)

are very accurate. Thus, k, is very close to $zr. The first non-trivial (i.e. non-zero) solution in the inside region (x < 2) therefore appears at k, = in-. According to Eq. (18), the value of the variable k actually does not exceed its small maximum value k,, in the Ps continuum. Hence Eq. (17) cannot have any resonant interior solution in the Ps continuum. The interior solution has to be non-resonant, and therefore it can only be the trivial (i.e. zero) solution.

In other words, while the b.c.‘s (21) have a tendency to induce resonances in the interior region, the potential there 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 for a weak interior potential. All resonances are thus successfully evaded.

However, it is obvious that if we allow the potential to become arbitrarily strong, it will eventually trap resonances in the Ps continuum even under the kinematical constraint (18). An interesting way to study this possibility is to consider a model in which one of the particles has a charge 2 (the “effective charge”) so that the parameter (Y in the problem has to be replaced by a2 = Za everywhere. The perturbative solution is no longer reliable for these strong potentials. We look instead for exact solutions by expanding the w.f. in power series about both x = 0 and 2. The two power series are matched at an intermedi- ate point (say x = 1) to find the discrete eigenvalue spectrum in the interior region. These energies are found, as expected, in pairs with opposite signs.

C. W Wang, C.-X Wang / Dehnen - Shahin relativistic equations 601

Table 1 Resonances in the subtracted Dehnen-Sbahin potential [Eq. (2411 for large effective charges 2. The energy of the highest bound state (HBS) is also shown

Z E - 2m CkeV)

HBS Resonances

237.4 - 327 317 418.7 - 136 317 7408 594.3 -35 317 12.51 768.2 - 154 28 317 884 3629 941.1 -94 71 317 732 1705

To be more definite, we look for the effective charges for which the DS equation has a resonance at the kinetic energy K = E - 2m = 317 keV where DS2 have their first resonance. The potential used by DS2 in their numerical calculations is not the potential shown in Eqs. (19) and (201, but is instead the subtracted potential

made up of the pole term minus its long-range Coulomb ~mponent. This can also be obtained from Eq. (19) by taking A = B instead.

There are an infinite number of Z values at which K= 317 keV is a resonance. The first five of these solutions are shown in Table 1, which also gives all other resonances (if any> in the Ps continuum, as well as the energy of the highest bound state (HBS). A number of features should be noted:

6) If K = 317 keV is the first resonance, there can be at most one other resonance in the positronium continuum.

(ii> All resonances have zero width. (iii) All values of Z shown give potentials which are so strong that they give rise

to additional bound states. Table 1 gives only the highest of these bound states. (iv) Because of the unusual b.c. at x = 2, these additional bound states rise into

the positronium continuum as the potential strength (or 2) increases. The first resonance rises through zero energy into the positronium continuum when Z increases through the value of 193.4.

Feature (iv> is different from the familiar result for ordinary bound states in ener~-independent potentials, in which bound states first appear at zero energy and then fall deeper into the attractive potentials as its strength increases. Our unusual features come about in the following way: As the interior attraction increases, the w.f. bends over faster, while rpole at the same energy increases with the potential strength. Hence the w.f. now reaches zero before x = 2. By increasing

608 C. W. Wang, C.-Y. Wong / Dehnen - Shahin relativistic equations

the energy, we decrease rpole and make it chase the w.f. zero in order to satisfy the b.c. at x = 2.

Resonances also appear in the effect-charge version of the original potential (191, the smallest 2 at which a resonance appears at 317 (0) keV being 2 = 163.2 (157.4). The detailed behavior of these resonances differs from that for the potential (24), however. For example, there are now also resonances which appear first at infinite energy as 2 increases, their energies decreasing with increasing 2. This is, of course, a more familiar type of bound-state behavior.

Since there is no experimental evidence that bound states of this type exist in nature, the DS models with strong potentials based on either Eq. (19) or (24) are not valid models of Ps states.

5. Phase shifts for the DS equations

The DS claim of resonances is based on their study of the phase shifts of the regularized potential, with the singular factor l/(2 - X) replaced by the finite form (2 - x)/K2 - xj2 + s21. w e now calculate these phase shifts and examine them for resonances.

Since the regularized potential is no longer singular, the phase shifts are easily calculated by simply integrating the differential equation across x = 2 with the help of standard stepwise integrators. (This works best if S is not too small and if the potential is not too strong.) We use the Runge-Kutta-Gill program ODEINT from Numerical recipes [32], and note that the wave function F bends down to zero at x = 2 only for k very near to $nrr, in complete agreement with the results described after Eq. (23).

However, w.f. zeros do not signify the presence of resonances in finite potentials. Resonances appear only when the scattering phase shift rises rapidly through $r (modulo r>. These phase shifts are easily calculated for the subtracted and regularized DS potential. A further simplification is made by truncating the weak potential at x,, = 20. The resulting phase shifts are all very small in the Ps continuum: - 1.0 x lop6 (- 4.1 x lo-*) rad at a kinetic energy E - 2m of 1 (10) MeV and 6 = 10A6. They are also stable against changes in 6 in the range we have studied, 10e6 < 6 Q 10-l. They do not show any resonant behavior. (The sign convention for these phase shifts is the usual one of negative phase shifts for repulsive potentials. This is opposite in sign to the convention followed by DSl and DS2.1

We have also calculated the phase shifts for arbitrary k values above the Ps continuum, i.e. with k > k,,. For definiteness, we use the energy parameter E = 2 in Eq. (20), eliminate the kinematical constraint (18) on k, and truncate the potential at x,, = 20. The resulting phase shifts are all very small, typically of the

C. W Wang, C.-Y. Wong / Dehnen -Shahin relativistic equations 609

order of 10e5 rad. These results are consistent with the fact that the potential in Eq. (17) has a strength of only a2 in the unusual dimensionless units used.

The phase shifts for the original, i.e. unregularized, potential can also be calculated easily. Since the boundary condition F(x = 2) = 0 must be imposed, the phase shift can be written as - 2k + A, where - 2k is the “hard-core” phase shift (X = 2 being the effective hard-core radius) and A the additional contribution from the external potential. Numerical integration from x = 2 to x,, = 20 gives A = - 5.7 x 10e7 (- 2.2 x lops) rad for E - 2m = 1 (10) MeV. The additional phase shift A remains small in the k-continuum (again with E = 2), being -2.0 X 10m5 (- 1.3 x 10-5, -9.6 x 10m6) at k = $T CT, 5 r). Since the resonance widths are zero, one cannot tell by examining these phase shifts that there are resonances. The same qualitative behavior is found when larger values of x,, are used.

These results indicate that the phase shift might behave in a rather complicated way as 6 approaches zero. We have not tried to establish how the hard-core behavior is recovered as 6 approaches zero.

We have noted that zero-width resonances appear in the Ps continuum only when the effective charge Z is made sufficiently large. It is interesting to ask what happens to them when these strong potentials are regularized.

With regularization, the interior and exterior regions “communicate”. The resonance width can be expected to become finite. The actual behavior turns out to be more complex and interesting. Since both the external repulsion and the interior attraction now come into play, the phase shifts vary with the choice of x,, where the potential is truncated and the phase shift is determined. We use

20 GX,, G 40000 and 10e5 B 6 Q 10-i, with most calculations using x,, = 1000 and 6 = lo-‘, but the qualitative nature of the solutions is found to be independent of the choice of parameters within the ranges studied. For the regularized subtracted DS potential with effective charge Z, we find that for Z < N 120, the phase shift decreases monotonically from 0 at E = 0 to a minimum value above $r before increasing monotonically to zero again as E + to. Between Z = 120 and the first solution Z, = 237 of Table 1, it decreases right through - $r, reaching down to -rr as E + ~0. (A more useful convention would be to insist that 6,s + 0 as E + 03. This requires that S,,(E = 0) = rzz-, with n = 1 in the last example.) A phase shift failing through - $7 (modulo rr) has been called an echo [33]. The cross section reaches the unitarity limit at that energy as in a resonance, but unlike a resonance, the time delay of a scattered wave packet is not positive but negative. This means that the main body of the wave packet is scattered out before it gets into the potential region. There is also an oscillation of 6,s at high energies as the potential gets stronger.

We next examine the phase shifts for the five strong potentials shown in Table 1. The results can be summarized by listing the number of echoes and resonances as the energy increases: At Z = 237.4, one echo; at Z = 418.7 and 594.3, two echoes; at Z = 768.2 and 941.4, three echoes, followed by a resonance and finally

610 C. W Wang, C.-Y. Wong / Dehnen - Shahin relativistic equations

an echo. The resonance appears at the kinetic energy of 420 (1780) keV for Z = 768.2 (941.4). The resonance is very broad, with a width of about 700 (3000) keV when estimated from the formula $r/(dSps/dE). This phase-shift resonance seems to be an average over the four resonances of the exact solutions shown in Table 1. The exact resonances are not reproduced even for 6 as small as lo-‘.

Finally we have calculated the phase shifts for the 3P,, Breit equations solved in sect. 4.2 of DSl, where the Breit interaction is used. We first confirm that the equation in question agrees with our own derivation based on Eq. (11, provided that the parity misidentification made by DSl is corrected. The w.f. of interest also satisfies Eq. (17), but the effective potential is now

where

ff2 2 I+)=, 2-1 +;.

i 1 (26)

The potential V,(x) contains all the long-range Coulombic potentials as well as the centrifugal potential for the P-wave. It is obvious that the phase shift 6, from the weakly attractive Coulomb potential, plus the normal repulsive centrifugal potential in Vi, is non-resonant. The additional phase AS = 6,, - 6, caused by the remaining potential terms turns out to be very small: It is only 3.0 x lop4 rad at the kinetic energy K = E - 2m of 1 keV. It increases monotonically with K, reaching a value of only 5.5 x 10e3 rad at 1 GeV. [It does not vanish at infinite energy because of the small attractive 1/x2 term in V(x) - ~,(n).l Hence, we find that the 3P0 equation in DSl also does not give rise to resonances in the Ps continuum, again in total disagreement with DSl.

6. Spurious bound states near zero energy

We have seen that the DS2 Eq. (17) does not have resonant solutions in the Ps continuum for the weak potentials appearing in the Ps system, in total disagreement with the results reported by DS2. The only solutions consistent with the stated b.c.‘s at all physical energies are the non-resonant solutions in which F is completely excluded from the interior region. (That is, F(x) = 0 for 0 <X f 2.) According to Eqs. (10) and (111, u2 and f3 are excluded from the interior region as well. The fractional excluded volume in the normal 23P0 Ps state can be estimated

C. W. Wang, C.-Y. Wong / Dehnen - Shahin relativistic equations 611

readily. At its non-relativistic energy, rpole = e2/m 2: 3 fm; the fractional excluded

volume turns out to be only about 4 X 10e5a6 2: 10-i’. This is very small indeed, but is even this small effect real?

To answer this question, we return to Eq. (17). The pole term in the potential is

proportional to l m2, where E = E/m; it therefore becomes infinitely strong as E (or E) approaches zero, while k goes to negative infinity. It does look as if there will be bound states near and perhaps at zero total energy E even when the mass m is finite.

The question of additional bound states can readily be answered by looking for them numerically. For the original potential (20), the first 13 of these bound states are found to be ( E 1 = 3.22569, 1.72487, 1.17865, . . . , 0.30650, and 0.28323 keV in order of decreasing 1 E I. (The numerical accuracy of our program becomes questionable at lower values of I E I. It appears to be limited by the finite number of terms in our power-series solution allowed by our simple Turbo Pascal program.)

Note that the spacing between successive eigenvalues decreases as ( E I decreases. It is probable that there is an accumulation point as E --f 0. However, E = 0 itself is not a solution. This can be seen by writing Eq. (14) as an equation for the w.f. f= rv2:

At E = 0, only the first term survives. The resulting equation has solutions of the form a + br, which cannot satisfy the b.c.‘s f = 0 at both r = 0 and rpole = co.

In any case, the Ps ground state would have been found at these lower energies if these interior bound states had existed, and characteristic y-rays coming from the transitions between normal and abnormal states would also be present. Unless the experimental situation changes, we must conclude that these interior bound states are spurious and unphysical. Thus the exclusion effect described previously cannot be real.

7. Concluding remarks

To summarize, we find that the pole singularity at a finite distance from the origin appearing in the DS2 ‘S, wave equations for the Barut interaction does not give rise to a resonance in the Ps continuum. Resonances can appear only when the e+-e- interaction is made much stronger. These resonances have zero width and are similar to resonances in an infinite square-well potential. However, the pole singularity is found to give rise to additional bound states near zero energy which are clearly non-physical. Hence the equations containing such pole singularities cannot give an acceptable description of Ps.

612 C. W. Wang, C.-Y. Wong / Dehnen-Shahin relativistic equations

The pole singularity also appears in the ‘S, DSl equations for the Breit interaction. Therefore these equations can be expected to have the same qualitative behavior as those of DS2.

We have also examined the phase shifts of the is,, DS2 equations. They are all consistent with the very small effects normally expected of the week e+-e- interaction, and they do not show any resonant behavior. Only when the potential is made much stronger will the phase shifts begin to show resonances.

There is no pole singularity in the 3P0 equations of both DSl and DS2. We show explicitly that the 3P0 phase shifts for DSl are small and non-resonant, in total disagreement with the results of DSl. Since the DS2 3P0 equations are qualita- tively similar to those of DSl, it can be expected that their phase shifts are similarly small and non-resonant. This expectation has explicitly been confirmed recently [341.

A final comment on the pole singularity in the KFY equation might be appropriate. Since it gives rise to additional bound states near zero energy which do not appear to be physical, it must be avoided by a careful choice of the interaction parameters appearing in the KFY equation. In the case of the electromagnetic interaction, such a choice might not be possible. This can only mean that the simple KFY dynamics of the system is unrealistic and must be modified. It would be interesting to determine if the singularities and their associated spurious bound states could really be avoided by using a Tamm-Dancoff equation or the Salpeter equation. These are other formulations of the problem, such as that based on the relativistic constraint dynamics of Crater et al. [14], which give Schrodinger- like equations with non-singular potentials. In particular, the Crater-Van-Alstine equation [14,35] for charged systems can be written in the KFY form with non-singular effective potentials [36]. The understanding of how the unpleasant singularities are avoided in each of these formalisms should be of great interest.

This research is supported in part by the US DOE under Contract Number DE-AC05-840R21400 managed by Martin Marietta Energy Systems, Inc.

Appendix. Derivation of the Kemmer-Fermi-Yang equations

In this appendix, we reduce the Kemmer-Fermi-Yang (KFY) equation (1) to two sets of eight coupled first-order equations. We use the formalism and notation of Koide [21] for a simpler interaction.

The wave function is first written as

(A.1)

C. W. Wong, C.-Y. Wong / Dehnen - Shahin relativistic equations 613

where (G;L,, contains two-dimensional spinors of particles 1 and 2. For the P = ( - ljJ states, they have the angular-momentum structure

+I,+= i(f* -frE%-, + i(g* -gI@?r(+)7

4--= i(fi +f&Q-, + i(g* +gr)‘%(+),

*-+= (a3 - %>%Jo+ (4 - bl)%Jl~

9+-= (a3 + %)%JO + (b, + bi)%Jl7

where

1’2

(A.3

(A.3)

and yJLS is a vector spherical harmonic. After some algebra, the resulting eight coupled first-order equations are found to consist of the four algebraic equations (3) and the following four coupled differential equations:

(E + &-)a3 - 2Aa, + 2 “‘“:“g2+2(f$+23co,

(E + U,,-)b, - 2Ab, + 2 ~fl+2(~+k+o,

(E-u3f+)f2+2Mfl-2J$=0,

(E - U,,+)g, + 2m, -2(2+9=0.

For the P = ( - lk’+ ’ states, the w.f.‘s are

*++= i(a2 -%)&%o + i(b2 - h)%.Tl,

$--= it a2 + %>%.m+ i(b2 + &)%;,I7

ICI-+= (f3 -fob%-, + (g3 -&d%(+,Y

(A.4)

*+-= (f3 +f&%,-, + (g3 +&L%,+,. (A.51

614 C. W. Wang, C.-Y Wong / Dehnen - Shahin relativistic equations

The resulting eight coupled equations are the four algebraic equations given in Eq. (5) and the four differential equations

(E - U,,+)b, + 2Mb, - 2

(E + U&s - 2Afa + 22 = 0,

(E+U&a-2AgS+2($+;)=0. (A.6)

The U potentials appearing in these equations are linear combinations of the I/ and A interactions of Eq. (2):

VI,,= Vs + VP - 6Vr & (VI - 31/, +A, -3A,),

o;,*= V, - VP * (VI + 35 -A, - 3A,),

U 26*= v, - VP f (VI - v* - 2V, -A, +A*),

U zf+= v, - VP * (VI - v2 + 4V, -A, +/I,),

U 2,*=vs-v,+(v,-v,-2~~--A,+A,),

u,,*=v,+v,+2V,_+(V,+V,+2t/,+A,+A,)=U,,*,

$*= v, + VP + 2Vr f (V, + V, - 41/, +A, +F12). (A-7)

Table 2 The correspondence between the wave functions of Koide, DS2, DSl and Geiger et al.

States Authors Ref.

P=(-1)J P=(-l)JC’ Koide DS2 DSl Geiger Koide DS2 DSl Geiger [211 WI [151 PO1 Pll 1161 [151 DOI

a0 g2 f,‘F’ k al k? fi’+’ -g a3 f2 fJ_’ d a2 u2 f4” f bo g1 ig$-’ c h u1 g$+) b

b, f* -igf-) h u1 g(l+j -a

fl u3 fd” it

-fJ” aybl f3 g3 -ifi-) cl

f2 u3 f3 f3(_) 4 g1 u4 -ig$” -b, go g4-) c2

g2 u4 ig$+’ a2 g3 -ig$-) h,

C. W. Wang, C.-X Wang / D&men - ~hah~~ refatiuistic equations 615

These coupled equations can be written in the notation of DS2, DSl, or Geiger

et al. [20] by using the w.f. correspondence shown in Table 2. In this way we have found that our equations agree with those of DSl and DS2 for their choices of interactions, except for Eq. (2.10) of DS2. (Our results for the second and third equations of this set differ from theirs in the signs of the g, and u4 terms.) Our results should agree with those of Geiger et al. when their anomalous moments are set to zero, but they are found to differ for both P = (- l)J and P = (- #+’ states. We would therefore recommend caution in using those results.

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616 C. K Wang, C.-Y. Wang / Dehnen - Shuhi~ relativktic equutions

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solutions of the dehnen-shahin relativistic equations for positronium

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