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PARTICLE PRODUCTION IN SUPERCRITICAL ELECTRIC FIELDS CREATED IN HEAVY ION
COLLISIONS
K. Geiger, M. Grabiak, W. Greiner, B. Müller, J. Reinhardt, S. Schramm, G.
Soff
To cite this version:
K. Geiger, M. Grabiak, W. Greiner, B. Müller, J. Reinhardt, et al.. PARTICLE PRODUCTION IN SUPERCRITICAL ELECTRIC FIELDS CREATED IN HEAVY ION COLLISIONS. Journal de Physique Colloques, 1987, 48 (C9), pp.C9-147-C9-155. �10.1051/jphyscol:1987924�. �jpa-00227342�
PARTICLE PRODUCTION IN SUPERCRITICAL ELECTRIC FIELDS CREATED IN HEAVY ION COLLISIONS
K. GEIGER, M. GRABIAK, W. GREINER, B. M ~ ~ L L E R , J. REINHARDT, S. SCHRAMM and G. SOFF'
Institut fiir Theoretische Physik, Joh. Wolfg. Goethe-
Universitdt, Postfach 111932, 0-6000 Frankfurt-am-Main, F.R.G.
"~esellschaft fiir Schwerionenforschung, Planckstrasse 1.
0-6100 Darmstadt, F.R.G.
Abstract:
In the past decade narrow line structures have been discovered in the spectra of posi- trons and electrons emitted in heavy ion collisions by several experimental groups.
These peaks have not yet found a satisfactory explanation. Various theoretical models have been proposed to explain the effect. In this contribution we will report on some of the latest theoretical work and discuss models of a new extended, neutral particle decaying into electron-positron pairs.
I. Introduction
Positron emission in heavy ion collisions has been thoroughly investigated during the past decade because it is unique as a tool for the study of QED processes in very strong electric fields, most notably spontaneous pair production in supercritical fields C 11. Recently, the observation of sharp, peaklike structures in the energy spectrum of positrons emitted in such collisions has attracted wide interest C21.
Today, inspite of many attempts to understand their origin, these positron peaks still remain a puzzle.
Since the first discovery of the narrow lines numerous experiments have been carried out by several groups that yielded detailed informations on position and width of the peaks as well as their dependence on various parameters of the colliding ions. Here we will not review the actual experimental situation but refer to ref. [11[21.
The main features of the present experimental findings are the following:
i) Several peaks are observed simultaneously
ii) there is no obvious Z -dependence of the peak positions, where Zu=Z,+Z2 is the combined nucleyr charge of the collision system.
i i i ) for each positron peak there is a correlated line in the electron spectrum of equal energy and intensity. Positron and electron seem to be emitted back-to-back.
To illustrate these statements we quote two recent experimental results. Fig. 1 shows a positron spectrum obtained by the ORANGE group C3,41. Very similar spectra were found in U+Au and Pb+Pb collisions at bombarding energies of 5.9 MeVIu and 5.7 MeVIu.
In the Figure both spectra were added to enhance the statistical significance. In addition to a smooth spectrum which consists of nuclear pair conversion background and dynamically induced positrons C71 further structures are visible. The smooth spectrum has been subtracted in part b of the Figure, leaving two distinct peaks centered at Ee+ = 240 keV and 315 keV with width AEe+ a 30 key.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987924
C9-148 JOURNAL DE PHYSIQUE
The EPOS group using a two-armed solenoidal spectrometer has discovered coincident peaks in the spectra of both positrons and electrons C51. In Fig. 2 the doubly dif- ferentjal spectrum d2PldE -dE
+
for U+U collisions is shown along two cuts through the E -E-
energy plane,ekee/?ing constant the energy difference (left part) and the sum efiergf (right part) C61. Peaks are observed at sum energies xE = 620keV and 810 keV and at equal electron and positrons, AE = 0. The upper and lower spectra are gated with respect to different flight times of the positrons through the spectro- meter, which is an indirect measure for the angular distribution.In addition to the positron data very recently indications for correlated 27 emission in U+Th collisions at E = 1062 keVhave.been found C81.
Most models aimed to explain the observed effect involving conventional physics are not compatible with the experimental data or suffer from other inconsistencies (e-g.
nuclear pair conversion, spontaneous pair production C51, atomic interference ef- fects C91C101).
In view of the experimental facts quoted above, the most promising explanation sfems to be the hypothesis of the production of neutral particles that decay into an e e- pair. However, from previous theoretical studies it has become clear that this sup- posed new "particle" X cannot be elementary, because existing bounds on such particles from precision experiments put upper limits to the production cross section that fall short of the observed cross sections by several orders of magnitude [Ill-C141.
Fig. 1 Positron spectrum by the Orange group:
a) Sum spectrum of U-AU and Pb-Pb results b) Spectrum after subtraction of the
background (smooth curve in a)).
In order to explain the experimental facts in terms of the formation of a new "par- ticle" XO a theoretical model has to satisfy several requirements:
a) The conjectured new object X must have a complex structure with internal degrees of freedom allowing for a spectrum of excited states (alterna- tively, a "family" of new particles could be considered as well, although this assumption appears less natural).
b ) The X-particles must have a mass in the MeV region. Their decay has to pro- vide a source of monoenergetic positrons with energies independent of the details of the formation process corresponding to the observed constancy of the peak positions for various collision systems.
d) The model should not lead to contradictions with other established experi- mental facts, in particular high precision atomic physics data.
In the following, we will review recent theoretical work in this direction:
Section I1 deals with the interpretation of these "particle states" in terms of mag- netic resonances formed by an electron-positron.pair. We present strong arguments
500 1000 1500 -500 0 500
E,. + E,-
[keVI
E,. - E,-Fig. 2 Pysitron spectrum by the EPOS group:
e e-coincidence measurements in U-Th coll isions.
a) ,c) sum energy spectrum, b) ,d) energy difference spectrum.
that such a model is unsuitable to explain the positron peaks. In section I11 we describe a bag model for an extended particle composed of two or more electrically charged constituents that are bound by some novel confining force.
11. Micropositronium Resonances
It has been proposed that the hypothetical "particle states" may be interpreted as magnetic resonances between an electron and a positron C151C161. In this model the particles are assumed to be in a highly localized quasibound state due to their strong mutual magnetic interaction at short distances. It was claimed the magnetic interaction, when included in the Hamiltonian non-perturbatively, leads to $ very narrow (a few fermis) and deep (-GeV) "potential pocket" in the effective e e-inter- action which can support one or more quasibound resonant states.
Unfortunately, there exists no practical method to describe a system of two spin 1 particles interacting via the electromagnetic field in a rigorous, non-perturbatze way. To be sure, the Bethe-Salpeter equation is the adequate two-body equation, derived from the principles of QED. Although formally exact, its practical applica- tion to problems of two fermions interacting through vector coupling is restricted to perturbative calculations. The ladder approximation permits to sum an infinite set of diagrams, but neglects important non-perturbative effects, e.g. concerning the radiative form factor of the particles.
C9-150 JOURNAL DE PHYSIQUE
A possible non-perturbative approach has originally been suggested by Barut et al.
1171, who formulated the two-fermion problem within a semiclassical framework. The main features of this approach are: (i) the particles are represented by first quantized Dirac-wave functions, (ii) the electromagnetic field of the interaction is assumed to be created by the expectation value of the particles currents and is treated classically, (i i i) radiative corrections are partly considered by taking into account the anomalous magnetic moments of the particles. From a classical action principle the authors derive a relativistic covariant 16 component two body equation that reads in the CM frame
with the interaction potential
e ~ e z
V(7) = -(I-GIG2)
T
e.
where e - and a. = v denote the charge and the anomalous magnetic moment of the particl$ j , (jJl,27
fl)
contains spin-spin, spin-orbit, spin-anomalous magnetic moment interactions as well as the Coulomb term.Depending on the angular momentum quantum numbers eq. (2.2) can be decomposed into sets of coupled radial differential equations. An analysis of these equations C141 has revealed the following problem: The spectrum of ordinary (lightly bound) posi- tronium is not described correctly already in the lowest non-trivial order O ( a 2 ) .
This can be traced to an incorrect coefficient in the spin-spin interaction term as a consequence of the semiclassical treatment.
Close to the origin the radial equations do not have normalizable solutions, similar to the "collapse" encountered in the Coulomb problem for point nuclei with charge 2>137. This problem is due to the strongly singular behaviour of the magnetic inter- action and could be alleviated by introducing a form factor suppressing the anomalous magnetic moment in the limit r-o. Such an effect is indeed expected from perturbative
QED, since the anomalous magnetic moment is driven by radiative corrections rather than being an intrinsic property of the electron. The evaluation of the effective potential U , however, has not revealed a potential pocket, independent of the chosen cut-&FF parameter of the formfactor. This was true for all studied singlet
('So; 'PI) and triplet OS,; 3Po.,
.,
) states. This means that the outlined semiclas-, - , -
sical approach in terms of eq. (2. I), (2.2) does not support magnetic resonances of the conjectured type C181.
However, we emphasize that this negative result does not necessarily carry over to the
exact
Bethe-Salpeter equation. It is valid within the semiclassical framework, which exhibits several inconsistencies for both, the ordinary atomic positronium as well as the hypothetical "micropositronium", and lacks a physically satisfactoryargument [ I 9 1 quite independent of any specific model assumptions stating that within the framework of QED, no narrow resonances can be formed by a e+e-pair above thres- hold (i.e. above 1.022 MeV).
Our argument, which we will sketch in the following, is based on the virial theorem.
Since every eigenstate of the Hamiltonian yields an extremum of the energy, the total energy must be stationary against any small variations. In particular it follows that the Hamiltonian of QED (using the Coulomb gauge)
is invariant under scale transformations x
-
Ax except for the mass term which sets the scale. That means, replacing m-
Am and scaling everything else, the energy E will also be changed to AE. The virial theorem then states thatfor bound (i-e. spatially localized) states. As we shall discuss, this also holds for sharp resonances up to a small-deviation which is of the same order of magnitude as the resonance width. For an e e state, the rhs of (2.4) is not larger than twice the electron mass so that an e'e-resonance with an energy around 1.8 MeV would con- tradict eq. (2.4).
It is important to mention that the validity of (2.4) is essentially based on the fact that the scale in QED is set by the physical mass of the electron and not by renormalisation effects. As a consequence of the small QED coupling constant the latter are negligible. In QCD, where a priori there is no dimensionful scale para- meter, this argument does not hold, since renormalisation corrections become essen- tial.
According to (2.4), the energy of a state dominated by a single e+e-pair (and an arbitrary configuration of the photon field) is
where b(c,s) and d(c,s) denote annihilation operators for an electron or a positron, respectively, and wD = The expectation value in (2.4) without the factor
8 8 ,
; would just give the number of particles (i.e.2). Since $he weight factor is smal- l@r than one, the virial theorem states that a localized e e state has an energy below 2m. If the wavefunction has a spatial extension smaller than the Compton wave- length llm, i.e. it contains high momentum components, the energy will be much reduced due to the factor m/wp.
This result strictly holds for truly localized boundstates. However, in order to extend the above argument to resonances, we consider a quasibound state in a poten- tial pocket that can decay by tunneling through the confining potential barrier.
C9-152 JOURNAL DE PHYSIQUE
The energy eigenfunction therefore is not localized but exhibits an oscillating tail outside the potential pocket. The idea now is to split the Hamiltonian into a part supporting a true bound state plus a part responsible for the decay of the resonance, i.e. H = H
+
(H-HR) : HR+HD. Thus under a scale transormation x-
Ax the virial theorem no8 readsThe first term is due to the fact that under a change of the scale we must rescale all dimensioned parameters. For QED this term is just in <Jd3xq+> (omitting renormal- isation corrections). The second contribution describes the scale dependence of the decaying part H (expressed in terms of the confining barrier) giving rise to an energy shift. ~Bwever, for a narrow resonance this term will be small since it is of the order of the resonance width r. For a resonance which is to be identified with the observed positron lines at E=1.8 MeV the second term in (2.6) must be larger than 0.8 MeV. However, from Bhabha scattering one knows that the resonance width does not exceed a value of r
+
-- lo-'
MeV! On the other hand, if configurations with more than two particlesew&re admixed significantly to the state one had to explain astrong binding for which no origin is apparent. Thus the hypothesis of magnetic re- sonances seems to untenable.
We close this section Qy-quoting one further point. As mentioned in section I, the experimptal data of e e correlation measurements point to a back-to-back emission of an e e-pair,+leading to the conclusion that the two-body decay of an essentially free object X-e +e is observed. Thus, if the object X exists in free spaces as an e+e-resonance, it should be possible ty invert the decay mechanism and to produce the X-particle in Bhabha scattering, e +e -X. In ref. C201 the corresponding scatter- ing cross-section has been studied by calculating the first order Feynman graphs for Bhabha scattering through the exchange of a photon or a hypothetical X-meson, re- spectively. When compared with the ordinary QED Bhabba cross section it is concluded that if an experimental energy resolution of a few KeV can be attained, Bhabha scat- tering should be sensitive to the formation of resonances with masses around 1.8 MeV as required by the positron peak data. Recent experimental investigation of Bhabha scattering at energies around 2 MeV C211 has not yet provided fvidence for definite structures in the cross section that could be interpreted as e e-resonances, but the required resolution will only be reached in the next round of experiments.
111. Confined elementary particles as a model for an extended new object X
In this Section we report on recent theoretical developments trying to explain the supposed new particle X in terms of a soliton type object.
We first present a model that assumes the X-particle as a neutral meson-like bound sqate of two previously unkown electrically charged elementary constituents y and y , most likely fermions being bound by some novel confining fyrce similar to the color -SU(3) force of the strong interaction C221. Since the y and y particles are though to couple electromagnetically to the nuclear charge, this model could also explain why the X-particle should preferentially be produced in heavy ion collisions.
Assume the existence of charge conjugate fermions y- and yf with mass m contained in a bag that is formed by the action of a vacuum pressure B
.
The total energy of the bag is supposed t o correspond to the experimentally o6served peak energies.The energy of a spherical bag with radius R, is given by
with the linear boundary condition
The stability condition for the bag requires the radius R to be chosen such tQat Eba attains a minimum value for e.g. , E = 1.8 MeV, c6rresponding to the e e - llng energies at about 400 keV. This condifaon yields a functional relation between the vacuum pressure B and the constituent mass m
.
By selecting a suitable mass m the level spacing canXbe adju~ted to values that 8re not in contradiction to the experimental results o f the e e d a t a C21.The extension of the bag is quite large on hadronic length scales, corresponding to
-
30 keV, so that the motion of the yk-particles in the the small bag constant Bxbag is essentially now-relativistic and a neglect of the mutual interaction inside the bag should be a fairly good approximation.
One may also co~sider the possibility that the supposed X-meson is composed of four constituents (y-spin up and down). The calculation is analogous to the two particle case. The result is, that the single pair meson has a mass around 1 MeV, which is slightly higher than half the mass of the four partille state. One might therefore be tempted to identify the two pair meson with the e e-peaks C21 while the coresponding single-pair meson may be attached to the y-Y correlation events reported in ref. C31.
The central idea for the discussed model is the explanation of how the mesonic yty- state could be easily produced in heavy ion collisions. Consider two high-Z nuclei surrounded by an X-bag centered in the nuclear c.m.-frame. The negatively charged Y is strongly attracted to the nuclei by the Coulomb force whereas the y+ is re- pelled towards the boundary o f the bag. The gain in binding energy, however, is found to dominate the energy balance. In order to calculate the energy of this configura- tion eq. (3.11, (3.2) are solved again, but now including the nuclear Coulomb field using the monopole approximation in the Dirac equation (3.2). The energy of the bag is found to be drastically reduced by the presence of the Coulomb field, approalhlng zero energy, i.e. the threshold for energyless, spontaneous production of the y y bound state from the vacuum. This mechanism may be responsible for many of the charac- teristic features observed in the experiments, which are otherwise hard to under- stand C231.
The four-particle bag is even supercritically bound (it has negative total energy) so that it can spontaneously be produced from the vacuum. Thus one can expect that the production probability of the bag is strongly enhanced in the field of the ions. Since the whole bag is bound by the internuclear Coulomb field the state should preferen- tially be produced at rest with respect to the cm-frame of the nuclei as required by the experiments. Furthermore, for a convenient choice of the constituent mass m the Xrbag is not bound by a single nucleus so that the bag may decay into a correlaxed e e pair of equal energy, after the nuclei separate. Treating the decay in analogy to
JOURNAL DE PHYSIQUE
Fig 3: The y'y--bag structure centered in the CM-frame of the two nuclei with mass A and charge Z: The negatively charged y is strongly attracted ty the nuclei by the Coulomb force whereas the y is repelled towards the boundary of the bag.
that of charmonium, the typical lifetime of the bag is found to be - 10-14-10-13s.
Thus the model can reproduce some of the main features displayed by the exqeriments and therefore may point into the direction of a valid explanation of the e e data.
However, one major yet unsolved problem remains: Since the y-constituents are elec- trically charged their virtual excitation can contribute to radiative corrections like vacuum polarisation and anomalous magnetic moment, thus spoiling the agreement between theory and experiment. It is not obvious whether the presence of the con- fining interaction will suppress such effects.
We must admit that the ad hoc introduction of previously unknown particles confined by a novel force is a quite radical assumption, but since any conventional explana- tiop of the e e coincidences has failed so far, and the exposed model fits the experimental data quite we1 1, such a Step could point the right direction for future studies. It is amusing to 4o;e that y- pair production would not show up in jet phenomena at hiqh-energy e e colliders, in contrast to t h e case of qual-k p a i r s . TIrt:
reason lies in the smallness of the associated bag constant B , compared with the constituent mass m To wit, the rate of string breaking in tie framework of the
Y'
flux-tube model is proportional to exp (-nmZ/o )
-
e-1200 where o =[
- cr Y$'iZ B is the1y4-'
string tension and we took m = 880 keV, Bx - 25 keV and ay = I . The y+ pair can therefore only decay by annihilation back into a virtual photon- Y
phase transition in the strong coupling limit. It is assumed that this new phase is induced by the intense electromagnetic fields of the large-Z ions during the col- lision. This new vacuuqj state would be a condensate of electrons and positrons as well as photons. The e e peaks are then identified with quasiparticle excitations in the new vacuum.
However, it is not clear how an expected QED phase transition at a critical coupling e from a weak coupling require e < e to a strong coupling region e > e is related t6 QED in strong external fields wherg the strong coupling strength Ze2 fs governed by the large external nuclear charge and not by eZ. Although these models have the advantage that they dispense with the need to postulate any new particles or fields, the connection between strongly coupled QED and the strong electromagnetic field in heavy ion collisions remains rather mysterious.
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