How to observe the vacuum decay in low-energy heavy-ion collisions

TL;DR

This paper proposes a method to observe vacuum decay in low-energy heavy-ion collisions by measuring pair-production probabilities, providing a feasible way to detect this fundamental quantum phenomenon.

Contribution

It introduces impact-sensitive measurements of pair-production probabilities as a novel approach to detect vacuum decay in heavy-ion collisions.

Findings

Numerical calculations demonstrate the feasibility of observing vacuum decay.

Impact-sensitive measurements can distinguish vacuum decay from background signals.

The method provides a new experimental pathway to confirm spontaneous vacuum decay phenomena.

Abstract

In slow collisions of two bare nuclei with the total charge larger than the critical value $Z_{\rm cr} \approx 173$, the initially neutral vacuum can spontaneously decay into the charged vacuum and two positrons. Detection of the spontaneous emission of positrons would be the direct evidence of this fundamental phenomenon. However, the spontaneously produced particles are indistinguishable from the dynamical background in the positron spectra. We show that the vacuum decay can nevertheless be observed via impact-sensitive measurements of pair-production probabilities. Possibility of such observation is demonstrated using numerical calculations of pair production in low-energy collisions of heavy nuclei.

Equations

$${\dot{R}}_{\alpha }(t)=\alpha \dot{R}(t).$$

$$\begin{array}{rl}& R=a\left(e\cosh \xi +1\right),\\ & t=\sqrt{\frac{M_r{a}^3}{Z_1{Z}_2}}\left(e\sinh \xi +\xi \right),\end{array}$$

$$a=\frac{Z_1{Z}_2}{2E},e={\left(1+\frac{b^2}{a^2}\right)}^{1/2},\xi \in (-\infty ,\infty ),$$

$$R_{\min }=a(e+1).$$

$$b^2=R_{\min }^2-R_{\min }\frac{Z_1{Z}_2}{E}$$

$$E_0=\frac{Z_1{Z}_2}{R_{\min }},$$

$$V_{\rm mon}(r,t)=\frac{1}{4\pi}\int d\Omega\,V_{\rm TC}({\mbox{\boldmath$r$}},t).$$

$$\psi_{\kappa m}({\mbox{\boldmath$r$}},t)=\left(\begin{array}[]{l}\displaystyle\,\,\frac{~{}G_{\kappa}(r,t)}{r}\,\chi_{\kappa m}(\Omega)\\[11.38109pt] \displaystyle i\,\frac{F_{\kappa}(r,t)}{r}\,\chi_{-\kappa m}(\Omega)\end{array}\right)\,,$$

Full text

In relativistic quantum mechanics, the energy levels of hydrogenlike ions are described by the Dirac equation. For the pointlike nucleus this equation has a solution for the 1$s$ state only if the nuclear charge $Z$ is not greater than $Z_{0}=137$. Therefore the energy of this state is bounded from below by $E(Z_{0})=0$. However, for an extended nucleus $E(Z)$ decreases further as $Z$ increases and eventually crosses the value $-mc^{2}$ at the critical charge $Z_{\rm cr}\approx 173$ Pomeranchuk_45 ; Gershtein_70 ; Greiner_69 ; Zeldovich_71 ; Greiner_85 ; Mueller_94 ; Godunov_17 . After the crossing the level “dives” into the negative-energy Dirac continuum becoming a resonance. If this supercritical resonance state was initially vacant then it can be occupied by two electrons from the negative-energy continuum with emission of two positrons Gershtein_70 ; Greiner_69 ; Zeldovich_71 ; Greiner_85 ; Mueller_94 . This process can be interpreted as a spontaneous decay of the old neutral vacuum with formation of a new “charged” vacuum.

Obviously, the required critical charge $Z_{\rm cr}\approx 173$ is much larger than the charge of the heaviest nuclei produced so far. However, two heavy colliding ions can form a quasimolecular system with the total charge $Z_{\rm tot}=Z_{1}+Z_{2}$ large enough for the ground state to reach the negative-energy continuum. Observation of the electron-positron pairs spontaneously produced during the collision would be the direct evidence of the vacuum decay. But in heavy-ion collisions the pair production is also induced by the ion dynamics. In order to detect the vacuum decay, one has to distinguish the spontaneous pair production from the dynamical one.

The experiments on low-energy heavy-ion collisions were intensively performed many years ago at GSI (Darmstadt, Germany). However, no sign of the spontaneous pair production or the diving phenomenon had been found Mueller_94 ; Ahmad_99 . There are several proposals for investigation of supercritical collisions at the upcoming accelerator facilities Gumbaridze_09 ; Ter-Akopian_15 ; Ma_17 , which will allow to perform the experiments on an entirely new level. In particular, experiments on low-energy collisions of heavy bare nuclei are anticipated at these facilities. But so far it was not clear whether or not there exists a theoretical possibility of the diving phenomenon detection.

To date, the pair production in low-energy ion collisions was investigated using various theoretical approaches Gershtein_73 ; Popov_73 ; Peitz_73 ; Lee_16 ; Khriplovich_16_17 ; Reinhardt_81 ; Muller_88 ; Ackad_07_08 ; Maltsev_15 ; Bondarev_15 ; Maltsev_17 ; RV_Popov_18 ; Maltsev_18 . As was found by the Frankfurt group, the pair-production probability as a function of the total nuclear charge and the impact parameter has no threshold effects at the border of the supercritical region, where the spontaneous mechanism should start to work Reinhardt_81 . It was also shown that the energy-differential spectra of the emitted positrons do not exhibit any feature which can be associated with the spontaneous pair production. The calculations were performed using so-called monopole approximation, in which only the spherical part of the two-center ion potential is taken into account. Recently the obtained results were confirmed with the monopole approximation Maltsev_15 as well as beyond it Maltsev_17 ; RV_Popov_18 ; Maltsev_18 .

The absence of any signature of the spontaneous mechanism in the calculated pair-production probabilities and in the positron spectra led the Frankfurt group to the conclusion that the vacuum decay could only be observed in collisions with nuclear sticking, in which the nuclei are bound to each other for some period of time by nuclear forces Soff_mem . In such collisions, there should be a visible effect of the vacuum decay due to increase of the diving time. In numerical calculations, the nuclear sticking can be taken into account via introducing the time delay at the point of the closest nuclear approach. It was demonstrated that the time delay leads to the enhancement of the pair-production probability in the supercritical case that can be explained only with the spontaneous mechanism (see, e.g., Ref. Muller_88 ). However, to date there is no robust evidence of existence of the sufficiently long nuclear sticking.

In this Letter, we show that the vacuum decay can be detected experimentally even without any nuclear sticking. The idea of the detection is based on the different behavior of the pair-production probability as a function of nuclear velocities in the supercritical and subcritical cases.

Let us consider first hypothetical collisions with the modified velocity Maltsev_15 :

[TABLE]

Here $R(t)$ is the internuclear distance which depends on time in accordance with the classical Rutherford scattering:

[TABLE]

where

[TABLE]

$E$ is the collision energy in the center-of-mass frame, $M_{r}$ is the reduced mass of the nuclei, and $b$ is the impact parameter. Varying the parameter $\alpha$, we can change the nuclear velocity in numerical calculations. Figure 1 presents the pair-production probability $P$ as a function of $\alpha$ obtained in Ref. Maltsev_15 . The calculations were performed for subcritical Fr–Fr and supercritical U–U head-on collisions of bare nuclei at energy about the Coulomb barrier. As one can see from the figure, the behavior of the curves at small values of $\alpha$ is remarkably different. As $\alpha$ decreases, $P(\alpha)$ decreases in the subcritical case and drastically increases in the supercritical one, which indicates the existence of the spontaneous pair production mechanism. It should be emphasized that the subcritical curve always rises with increase of $\alpha$ and the supercritical curve has quite a simple shape with one minimum.

Of course, it is impossible to modify the collisions according to Eq. (1) in real experiments. However, there exists a way to investigate the dependence of the pair-production probability on the ion velocity using the pure Rutherford kinematics defined by Eq. (2). Let us fix the nuclear charges $Z_{1}$, $Z_{2}$ and the distance of the closest nuclear approach

[TABLE]

One can vary the collision energy $E$ with changing the impact parameter $b$ according to the equation

[TABLE]

with fixed $R_{\rm min}$. The collision energy is bounded from below by the value

[TABLE]

which corresponds to the head-on collision ($b$ = 0). Using Eqs. (5) and (2), for the range of available energies, $E\geq E_{0}$, one can define the set of functions $R_{E}(t)$ which have the same minimum but different durations of the supercritical regime. For the case of U–U collision, these functions are displayed in Fig. 2. It can be seen that the supercritical time period decreases monotonically with increase of $E$. Employing the defined set of $R_{E}(t)$ it is possible to investigate the pair-production probability as a function of nuclear velocity keeping the range of internuclear distances fixed ($R_{\rm min}\leq R(t)<\infty$). The major limitation of this approach is that the nuclei cannot be slowed down more than it is allowed by the condition $E\geq E_{0}$.

In order to find the desired difference in pair production between subcritical and supercritical systems, we performed calculations using the method described in Ref. Maltsev_15 . The method is based on the numerical solving of the time-dependent Dirac equation in the monopole approximation, according to which the two-center nuclear potential V_{\rm TC}({\mbox{\boldmathr}},t) is approximated by its spherically symmetric part

[TABLE]

This approximation allows us to consider the radial Dirac equation instead of the two-center one. The corresponding electron wave function can be represented as

[TABLE]

where $\chi_{\pm\kappa m}(\Omega)$ are the spherical spinors, $F_{\kappa}(r,t)$ and $G_{\kappa}(r,t)$ are the small and large radial components, respectively, $m$ is the projection of the total angular momentum, and $\kappa$ is the relativistic angular quantum number. We take into account the electronic states with $\kappa=\pm 1$, which are expected to give the major contribution to the pair production. Since there is no coupling between these two sets of states, the corresponding contributions can be calculated independently.

For simplicity, we consider the collision of two identical bare nuclei with $Z_{1}=Z_{2}=Z_{\rm nucl}$. The closest nuclear approach is fixed to $R_{\rm min}=16.5$ fm. At such a distance the nuclei are about 1–2 fm away from touching each other. The calculations are performed for subcritical and supercritical collisions at different energies $E$ for different values of $Z_{\rm nucl}$. In Figure 3, we present the obtained results for the pair-production probability $P$ as a function of the $\eta=E/E_{0}$ ratio for Fr–Fr ($Z_{\rm nucl}=$ 87), U–U ($Z_{\rm nucl}=$ 92), and Cm–Cm ($Z_{\rm nucl}=$ 96) collisions. The Fr–Fr system is subcritical (it is the heaviest subcritical system), the U–U and Cm–Cm systems are supercritical. As can be seen from the figure, the Fr–Fr curve goes monotonically down with decrease of $E$ as in the case of the modified collisions (see Fig. 1). Such a behavior takes place for all collisions with $Z_{\rm nucl}\leq 87$. In contrast, the pair-production probability in the supercritical Cm–Cm collision starts to increase as $\eta$ approaches unity. In the U–U collision, which is also supercritical, the function $P(\eta)$ has only a slight increase as $\eta\rightarrow 1$ but exhibits clearly different behavior compared to the subcritical case.

To clarify this point, let us consider the U–U collision in more details. It should be noted that, in our calculations, the total pair-production probability is the sum of two independent contributions: $P_{\kappa=-1}$ and $P_{\kappa=1}$, which correspond to creation of particles in the states with $\kappa=-1$ and $\kappa=1$, respectively. Only the channel with $\kappa=-1$ is supercritical, because it includes the diving 1$s$ state. In Figure 4, we depict the calculated values of $P_{\kappa=-1}$, $P_{\kappa=1}$, and the total probability for the U–U collision. The curve corresponding to the supercritical ($\kappa=-1$) results has a rather pronounced minimum while the subcritical ($\kappa=1$) one is monotonic. But in the sum $P_{\kappa=-1}+P_{\kappa=1}$ for $\eta\rightarrow 1$, an increase in $P_{\kappa=-1}$ and a decrease in $P_{\kappa=1}$ almost cancel each other out, which leads to a much less pronounced minimum.

It is clear that all the calculations can be easily extended to asymmetric collisions of bare nuclei. In Figure 5, we present the corresponding results for the pair-production probability as a function of $\eta$ for the U–Cm collision ($Z_{1}+Z_{2}=188$). As one can see, there is a clear signal of the spontaneous pair production for this system as well.

So far we considered only collisions of bare nuclei. On the one hand, the calculations for collisions of bare nuclei are the simplest to demonstrate the principle possibility of vacuum decay detection in the proposed scenario and, on the other hand, the experiments with such systems would be most favorable for such detection. Proposals for experimental investigation of collisions of bare nuclei up to Cm–Cm system were considered, e.g., in Ref. Ter-Akopian_15 . However, we would like to note that the same scenario can be potentially used for collisions of bare nuclei with atoms having the filled K-shell. An estimation of the bound quasimolecular level occupation probability in collisions of bare uranium nuclei on uranium and curium atoms with a filled K-shell, based on the methods developed in Refs. Tup_10 ; Tup_12 ; Kozhedub_14 , revealed that the filled K-shell can only lead to decrease of the pair-production probability roughly by a factor within the range 0.25 - 0.6. We also do not expect that a possible energy dependence of the corresponding suppression factor will qualitatively change the main conclusions concerning the observation of the effect of interest. We note that even if such a dependence is noticeable, it can be analyzed and isolated by means of more accurate calculations of the effects of the filled K-shell. The refined treatment requires more elaborate many-electron two-center calculations of the pair-creation probabilities and is currently under way.

In Figure 6, for symmetric collisions, we show the derivative $dP/d\eta$ taken at $\eta=1$ as a function of $Z_{\rm nucl}$. As one can see from the figure, the function changes its behavior after transition to the supercritical domain. It starts to decrease and finally crosses the zero line that corresponds to the appearance of the minimum on the graph of $P(\eta)$. The derivative becomes negative at $Z_{\rm nucl}\approx 92$.

From comparing the subcritical and supercritical scenarios, we conclude that there is the qualitative difference in behavior of the pair-production probability in the subcritical and the supercritical cases. If the distance of the closest approach is fixed, the increase of this probability with decrease of the collision energy can be observed only in the supercritical collisions. Moreover, even a pronounced decrease of $dP/d\eta$ at $\eta\approx 1$ as a function of $Z_{\rm nucl}$, which takes place already at $Z_{\rm nucl}=92$ (see Fig. 6), must be considered as a clear evidence of the vacuum decay at supercritical field.

Although the calculations in the present paper are mainly restricted to the monopole approximation, our recent study Maltsev_17 ; RV_Popov_18 ; Maltsev_18 clearly showed that effects beyond the monopole approximation only slightly change the pair-creation probabilities in the region of small impact parameters, where the derivative presented in Fig. 6 is calculated. We can state that effects beyond the monopole approximation will not change the main results obtained in this Letter. We believe, however, that further studies of the pair-production probabilities and the corresponding positron spectra beyond the monopole approximation can be very useful for finding the most promising experimental scenarios that allow for the determination of the angular distribution of the emitted positrons Maltsev_18 .

We hope that the results obtained in this Letter will promote new efforts for the experimental detection of the vacuum decay in a supercritical Coulomb field. In particular, such experiments seem feasible with the CRYRING facility at GSI/FAIR Lestinsky_16 ; Hagmann_unub , where storing of bare uranium nuclei at low energies is anticipated in the near future.

We thank I. B. Khriplovich and Yu. Ts. Oganessian for stimulating discussions. This work was supported by RFBR (Grants No. 18-32-00723 and No. 18-32-20063), RFBR-NSFC (Grants No. 17-52-53136 and No. 11611530684), and by SPbSU-DFG (Grants No. 11.65.41.2017 and No. STO 346/5-1). I.A.M. and R.V.P. acknowledge support from TU Dresden via the DAAD Programm Ostpartnerschaften. The work of V.M.S. and R.V.P. was supported by the Foundation for the advancement of theoretical physics and mathematics “BASIS”. V.M.S. also acknowledges the support of the CAS President International Fellowship Initiative (PIFI) and of SPbSU (COLLAB 2019: No 37722582). The research was carried out using computational resources provided by Resource Center “Computer Center of SPbSU”.