New phase of QED?

We discuss the speculation that the sharp positron lines and correlated e+e and yy signals seen in heavy-ion collisions may be evidence for a new phase of QED. We examine several characteristics of the data which argue for this interpretation and point out further experimental observations which would favor this hypothesis. However, we detail also theoretical difficulties and experimental contradictions which considerably weaken the basis for this speculation. In particular, we argue that for the formation of a new phase or a solitonlike structure in QED it is necessary that nonlinear effects in electrodynamics become important. Even though Za is large, these effects always entail a suppression factor of a, which is difficult to overcome.

In the collision of very heavy ions, at energies close to the Coulomb barrier, one produces sufficiently strong electromagnetic fields that positron emission can be induced. In addition, for sufficiently high ionic charges Z & +Z2 )173, since the lowest bound state of the combined system dips below -2m, one expects that it should be possible for spontaneous positron creation to occur.
Experimentally, quasiatomic positron production was observed soon after the beginning of UNILAC operation at GSI (Ref. 3). However, some evidence for an anomalous line structure in the positron spectrum was reported early on, which was subsequently confirmed by detailed investigations by the EPOS (Ref. 5) and ORANGE (Ref. 6) Collaborations.
The presence of this not-understood line structure has spurred additional measurements, which have revealed even more puzzling phenomena. The EPOS Collaboration, using a double solenoid spectrometer, recently presented evidence for a peak structure in the spectrum of correlated positrons and electrons with the same energy, emitted opposite each other. In fact, more detailed investigations appear to show three correlated e+e structures, at sum energies of approximately 1630, 1780, and 1830. Multiple peak structures in the single positron spectrum have also been reported recently by the ORANGE Collaboration, at positron kinetic energies around 250, 340, and 410 keV. Finally, researchers in a very recent experiment in the super HILAC at LBL, looking at U+ Th collisions at the Coulomb barrier, have reported a correlated back-to-back yy signal, at a sum energy of 1060 keV (Ref. 10).
The energy distribution of the quasiatomic-induced positrons is rather broad, with a width of the order of 500 keV. In contrast, the peak structure observed and, particularly, the correlated e+e and yy signals are very narrow. Typically, for the positron lines I +-50 -80 keV (Ref. 9), while the sum energy e+e peaks have widths of the order of I + -25-40 keV (Ref. 8). The yy correlated peak is extremely sharp, with I z&-2. 5 keV (Ref. 10). Furthermore, both the location of the positron lines, as well as their width, seem to be largely independent of the total charge Z=Z, +Z2 of the colliding ions, although the strength of the lines has some dependence on the precise parameters of the scattering process. ' Since the peak phenomena is seen for both Z & 173 as well as Z & 173 (Ref. 9), there appears to be no direct correlation between these observations and the possibility of spontaneous positron creation.
A great many theoretical explanations have been put forward concerning the origin of the positron lines and correlated e+e signals. " It is fair to say, however, that no wholly satisfactory solution is yet in sight. A particularly intriguing early suggestion associated the positron signal with the production and subsequent decay of a real elementary particlean axion. ' However, this explanation was rendered moot by the observation of the multiple-peak structures and was eliminated altogether by experiments which showed that, in electron beam dumps, no such elementary excitation was produced. ' A much more conventional possibility is that the positron peaks are the result of some interference among different amplitudes contributing to the positron production. ' ' This possibility is difficult to negate out of hand, since the actual production mechanism is very complex. Thus to calculate the emitted positron spectrum one necessarily must resort to truncations and extensive numerical evaluations of the time-evolution operator. ' This said, however, it is difFicult to see where another time scale, besides the Rutherford scattering time, enters into the problem. Furthermore, although interference effects could modulate the positron spectrum, it is unclear how they could produce the correlated e+e signals. Obviously this explanation would have no bearing in the yy correlations, which would have to be accidental. Perhaps the most intriguing, and bold, explanation for the puzzling heavy-ion data so far, has been suggested recently by three different groups. ' ' These authors argued that the strong electromagnetic fields in the collision cause the formation of a new phase of QED. The correlated e+e and yy signals are then associated with the decay of a set of discrete bound states produced in this new QED phase. Although this idea is very interesting, only some phenomenological arguments, but little theoretical evidence supporting it, is presented by its proponents. The purpose of this paper is to examine this What one needs to imagine is that, as a function of an order parameter related to the strong external electromagnetic fields provided by the heavy ions, the phase transition point in QED, if it exists, moves down from a, -1 to a =, 37 To our knowledge, however, no theoretical evidence exists suggesting that, in the presence of an external electromagnetic field, the possible phase-transition point of QED moves to weak coupling.
There are several aspects of the heavy-ion data which fit very well with the idea that some phase-transition phenomena have occurred. ' (i) The peaks seem to be peculiar to heavy-ion collisions and so are naturally connected to the presence of the strong electromagnetic fields.
(ii) If in the new phase extended objects are formed, ' these will naturally have various excitation modes. Thus multiple-peak structures are expected. ' (iii) The energy sharpness of the signals can be understood if the strong fields trigger the formation of a new phase, which then persists as a false vacuum. ' This would explain why the peaks appear to originate from the decay of a neutral object, produced essentially at rest in the c.m. system. The lifetime and mass of the various modes excited is then independent of the precise formation characteristics.
It is useful to elaborate somewhat on the last point above, which is particularly important. In Rutherford collisions of heavy ions at the Coulomb barrier, the heavy ions experience substantial acceleration only for a limited time, as shown in Fig

3.2
Although the strong electromagnetic fields during the Rutherford time tz may trigger the formation of a new phase of QED in a volume of order (1/m )s, these fields diminish rapidly as the heavy ions move away and are totally negligible by tr . This behavior is illustrated in Fig.  2, where the dimensionless electric field P=eE/m at r= 1/m and r =1/2m, at a given angle, is plotted as a function of time.
The above considerations make it obvious that to associate the y y signals with the decay of a new phase of QED requires that this phase be self-sustained for a considerable time, after the strong fields of the ions have ceased to be important. The only sensible picture is that an extended bubble of the new QED phase is formed and that this solitonlike structure survives long after the triggering fields are gone. The decay time argument is diferent for the positron peaks and correlated e + e signals since at t + -20m ' = 10 sec the electromagnetic fields in the central region are still substantial. If electrons and positrons would be produced at t + from Z, Z2a(M, +M2) 3 t~-10 = -=2X10 ' sec, (1) p(pf iM2 m where Pc=0.11 is the typical relative velocity of the heavy ions in the experiments A signal of intrinsic width less than 2.5 keV, as the yy correlation peak observed in Ref. 10, by the uncertainty principle is associated to very much larger time scales, t~~& 2.5 X 10 ' sec. At t~~t he ions themselves are separated by almost 10 fm and the residual fields in the interaction region are very small. the decay of a neutral "particle" at rest, they would feel the strong potential of the nearby ions (typically Vm at t + ) and therefore experience difFerent accelerations. It e+e is difficult to imagine that sharp peaks in the correlated energy spectrum are produced unless the ions are already much further away at the time of decay. The sharpness of the peaks suggests that at the decay time the electric potential from the ions, in the central region, is (I + The true intrinsic width is therefore expected in the order of at most a few keV.
The different masses of the correlated signals are associated with excitations within this new phase. The spatial extent of the new QED phase cannot be substantially greater than a Compton length m ', since the electromagnetic fields decrease rapidly outside a central volume of this size. The Compton length is also the characteristic size of possible bound states in a new QED phase. Thus one should visualize the various distinct peaks as solitons with different quantum numbers, ' rather than as local excitations propagating in an extended new QED phase.
Two significant conclusions can be drawn from these considerations, which have experimental importance.
(a) The formation of a bubble of the new QED phase is triggered by some order parameter, connected with the strong electromagnetic fields. Thus one can expect that the production cross section for the observed signals be sensitive to the detailed characteristics of the heavy-ion reactions, including the total charge Z=Z&+Zz, the ion's scattering angle 8, , and the ion's velocity po.
However, variations in the signal induced by changing some parameter, such as Z, are correlated with similar variations produced by changing another parameter, say c.m. ' (b) Since the soliton bubbles must survive to times I when the external fields are insignificant, 2 «1, the masses and widths of these excitations should be independent of detailed characteristics of the heavy-ion reactions: Z, 8, , and po. Once the soliton einerges from soine specific initial configuration, its intrinsic properties are governed by the asymptotic (quasistationary) behavior, which should be independent of how it was formed.
As we shall see, broadly speaking, the heavy-ion data appear to have these properties. However, in detail there are numerous contradictions, which considerably weaken the phenomenological support for the hypothesis of the formation of bubbles of a new QED phase.
It is conceivable that the soliton production sets in only if the electromagnetic fields have passed some critical value. Then the peaks should appear only for Z &Z, and we would expect a strong Z dependence of the production cross section for total charges in the "threshold region" just above Z, . For Z»Z"however, the production should only depend weakly on Z. However, the characteristics of the electromagnetic field during the collision depend not only on Z, but also on the distance of closest approach of the ions, which is connected with 8, and po. If there is threshold behavior in Z, one expects a similar threshold in 8, and po. On the other hand, if the dependence on e, is weak, one also expects a correlated weak dependence on Z. Unfortunately, the experimental situation is somewhat obscured, since the scattered ions are not identified and a threshold for small e, may be difficult to detect.
We do not know what a suitable order parameter for the triggering of the new phase should be. As an example, which we feel should be sensible beyond the threshold region, we focus on the interaction energy of the heavy ions, in a sphere of radius r =1/m, around the in- Here 2R (t) is the distance between the two ions. As is shown in Fig. 3 The experimental data in the range measured, from 8, -30' to 8, -90', is approximately independent of 8, . The prediction of Eq. (4) nicely reproduces this behavior, as is seen in Fig. 4. Note that P(8, ) decreases for smaller ion scattering angles. However, this behavior is compensated for by the increase in the Rutherfordscattering cross section. A formula quite similar to Eq.
(4) was proposed independently earlier by Bang, Hansteen, and Kocbach, who also were trying to relate the formation of the positron lines with the time-varying Coulomb fields of the heavy-ion reaction. Since it is not possible to distinguish between ejectiles and recoils in the experiment, what is measured is the symmetrized convolution of P with the Rutherford cross section If W;", is the correct order parameter to consider, then the Z and PD dependence of the cross section for producing the lines is fixed by Eq. (4}. It is not difficult to convince oneself that the quantity do /dQ, , with P given by Eq. (4), is also not terribly strongly dependent on Z or P0. Indeed, the weak Z dependence can be directly seen in Fig. 3, where we plot W;", for both U+U and Pb+ Pb collisions, at two different angles 8, , for the heavy-ion scattering. Unfortunately, these expectations do not seem to be in agreement with data obtained rather recently by the EPOS and ORANGE Collaborations. However, to add to the confusion, these experiments do not seem to agree with each other in these very important details.
To be more precise, the ORANGE Collaboration sees an increase of about an order of magnitude in the strength of the positron lines between the measurements done in Pb+ Pb and in U+ U (do'/dQ, =0.46+0.1 pb/sr vs do/dQ, =3.5+1 pb/sr, for the 340 keV line}. Our simple formula, for the angular range studied, predicts only a small change. On the other hand, the EPOS Collaboration2 seems to see very little Z dependence in their data and gives a value of do/dQ, -10 pb/sr for all the systems studied. The situation is reversed with regard to the P0 dependence. The ORANGE Collaboration, studying U+ U collisions, have purposely varied the bombarding energy from 5.6 MeV/nucleon to 5.9 MeV/nucleon (Ref. 28). Their results show little change in the single-positron peak intensity. This is in agreement with our expectations, but in contradiction to what has been reported by the EPOS Collaboration. The strength of the correlated e+e peaks observed in this experiment apparently is very sensitively dependent on the initial bombarding energy, with variations of a few hundredths of an MeV/nucleon being important.
Clearly it is very important to resolve the above experimental discrepancies before reaching premature conclusions regarding the possible existence of a new phase of QED. The rapid Z variation of the ORANGE Collaboration data and the rapid P0 variation of the EPOS Collaboration data, if confirmed, could indicate a threshold behavior which should also appear in the angular dependence. However, an irregular dependence on the collision parameters 8, , Z, and P0 is much more likely to be obtained through detailed atomic processes, so that a lack of correlations in these parameters would favor some interference origin for the positron lines. ' ' ' The most distinct experimental characteristic of the soliton interpretation is the predicted independence of mass and width from the production parameters Z, 8, and P0. In contrast, an interference type or other atomic or nuclear explanation would lead to dependence of the invariant mass of the peaks on Z, 8, , or PD, even if this dependence is only weak. Experimentally, there is a disquieting drift in time of the location of the positron peaks. Although these peaks are essentially Z independent within a given set of experiments, the data of different runs do not seem to always reproduce the same structures. Also there does not seem to be a strict oneto-one correspondence between the ORANGE spectrometer positron peaks and the EPOS correlated peaks.
Naively, one could think that strong nonlinearities arise for eElm -1/a, which occurs in a volume with radius -40 fm during the heavy-ion collision. These nonlinearities are, however, an artifact of an invalid expansion. If we neglect for a moment the higher derivative corrections, we can use the exact one-loop effective action for arbitrary strong fields, calculated by Schwinger. i' Schwinger's formula is valid to all orders in the external fields, but only to lowest order in a: A glance at the summary table of all the observations to date (Table IV in Ref. 26) could very well lead one to conclude that the peaks are randomly scattered in an energy interval between 230 and 400 keV. We do not want to be too critical, but we do feel that the data need urgent clarification.
Purely theoretically, of course, it is even harder to answer the question if a new phase of QED could be obtained in the heavy-ion reactions. It is often stated that since the electric fields are strong (Za & 1) this obviously signals a breakdown of the perturbative regime of QED.
However, nonlinear effects in QED arise only indirectly. The external fields can couple to the photon 6eld strengths only via electron loops, and these necessarily involve a. In fact, as we shall demonstrate below, just having strong homogeneous fields P =eE/m »1 in no way causes profound modifications to QED.
For any "new phase" or "soliton" state in QED, it is crucial that Coulomb's law for the interaction between charged particles gets qualitatively modified. E~ae). Although the expansion in 2 and 8 breaks down, this does not lead to a breakdown of the expansion in a, which would be needed for strong nonlinear effects.
We suspect that this feature remains true for higher loops in strong fields which are suppressed by further powers of a. We conclude that an extended new QED phase with weakly varying strong electromagnetic fields seems very unlikely.
The situation here is qualitatively different from the case of strong electromagnetic coupling for which nonlinearities indeed become important.
During the collision of the heavy ions, the expansion in V E/Em breaks down at a distance =1/m from the center of mass. The same will be true for a possible "soliton configuration" with size m '. Thus the real problein of the peaks in heavy-ion collisions cannot be treated with the constant-field approximation. In particular, the formula (8) gives only information on the photon propagator for q~0 whereas we need the behavior for q~& m . We therefore cannot exclude a soliton interpretation of the GSI peaks on theoretical grounds so far. However, we can point to a serious difBculty already encountered above: the breakdown in the derivative approxiination must be so strong that it overrides the factors of a necessarily appearing in all modifications of Maxwell's equations. Naively, n, »1 appears to be a manifestation of a breakdown of QED. If one imagines that the e+e pairs are produced nearly at rest, then it is not possible to pack many such pairs in a Compton volume, per Compton time, so that n, »1 would be contradictory. However, in the case at hand, as 2 grows so does the energy of the pairs and many more pairs "fit" in a Compton volume.
So n, can be much greater than unity, without signifying anything amiss in perturbative QED.
The approximation of constant P does not apply in our case. One is interested in knowing n, for the rapidly varying Coulomb field with charge Z, +Z2 and also the kinetic energy distribution of the produced pairs. A large number of pairs produced with small kinetic energy could indicate an instability or breakdown of standard QED. For example, the system could respond to this abnormality by forming a chiral condensate. Unfortunately, we do not know how to compute n, for this realistic situation. A small example shows that n, not only depends on the value of P (averaged in some region of space} but also in a crucial way on the spatial distribution of X'. The pair production in static fields can be understood qualitatively as a quantum-mechanical tunneling phenomenon. (This is quite different from pair production by time-varying electromagnetic fields. ) A positron bound in a deep well, Vo ---2m, when an external constant field E is applied, can tunnel through the potential barrier. The tunneling probability, computed with the usual WKB approximation, provides the damping factor e ' in Eq. (10}. By applying the same sort of reasoning to the Coulomb potential, one may obtain a rough estimate for the exponential factor in a more realistic situation. The problem is analogous to a disintegration in nuclear physics and the probability of tunneling through a Coulomb barrier, between ao and a &, is e,wher e In our problem the height of the barrier, Za/ao -Za/a"should be 2m. Defining Z, =2mao/tz one finds a nonvanishing probability only for Z & Z"a typical threshold behavior with strong Z dependence near Z"and e near one for Z beyond the threshold region: a(ZZ, )'" W= [arccosX -X(1 -X )'~] X (12) We would like to thank E. Dagotto, M. Luscher, and T. T. Wu for useful conversations. One of us (R.D.P.) is also grateful to H. Bokemeyer for helpful information on the experimental data.