Radioactive beams and inverse kinematics: probing the quantal texture of the nuclear vacuum

TL;DR

This paper explores the quantum vacuum in nuclear physics, proposing that certain nuclear reactions may serve as analogues to Hawking radiation, revealing the quantal texture of the nuclear vacuum through experimental signals.

Contribution

It introduces a novel analogy between nuclear reactions and Hawking radiation, suggesting experimental signatures of the nuclear vacuum's virtual processes.

Findings

Identification of nuclear reactions as potential analogues of Hawking radiation

Detection of signals indicating virtual particle processes in the nuclear vacuum

Proposal of experimental methods to probe the quantal texture of the nuclear vacuum

Abstract

The properties of the quantum electrodynamic (QED) vacuum in general, and of the nuclear vacuum (ground) state in particular are determined by virtual processes implying the excitation of a photon and of an electron--positron pair in the first case and of, for example, the excitation of a collective quadrupole surface vibration and a particle--hole pair in the nuclear case. Signals of these processes can be detected in the laboratory in terms of what can be considered a nuclear analogue of Hawking radiation. An analogy which extends to other physical processes involving QED vacuum fluctuations like the Lamb shift, pair creation by $\gamma-$rays, van der Waals forces and the Casimir effect, to the extent that one concentrates on the eventual outcome resulting by forcing a virtual process to become real, and not on the role of the black hole role in defining the event horizon. In the nuclear case, the role of this event is taken over at a microscopic, fully quantum mechanical level, by nuclear probes (reactions) acting on a virtual particle of the zero point fluctuation (ZPF) of the nuclear vacuum in a similar irreversible, no--return, fashion as the event horizon does, letting the other particle, entangled with the first one, escape to infinity, and eventually be detected. With this proviso in mind one can posit that the reactions $^1$H($^{11}$Be,$^{10}$Be$(2^+$;3.37 ${\rm MeV}$))$^2$H and $^{1}$H($^{11}$Li,$^9$Li($1/2^-$; 2.69 ${\rm MeV}$))$^3$H together with the associated $\gamma-$decay processes indicate a possible nuclear analogy of Hawking radiation.

Tables

Table 1. Table 1: Binding energy B ​ E 𝐵 𝐸 BE and binding energy per nucleon B ​ E / A 𝐵 𝐸 𝐴 BE/A (both in keV) (see Bachelet:08 and Suppl. material of this reference) associated with the nuclei 11 Be and 11 Li. The ZPF contribution to the binding energy of these two nuclei (see ref. Baroni:04 for details of the general framework) arises from dipole, quadrupole and octupole ( p , h ) 𝑝 ℎ (p,h) vibrations and from monopole pair vibrational modes (see also Potel:14 ).

	$BE/A$(keV)	$B{E}_{ZPF}$ (keV)	$BE$(keV)	$B{E}_{ZPF}/BE$ ($\times {10}^{-2}$)
11Be	5952.54$\pm 22$	4110	65477.94	6.3
11Li	4155.38$\pm 6$	2150	45709.18	4.7

Equations

$$(m_e{c}^2+T_{e^{-}}+(U_G{)}_{e^{-}-bh})$$

$$+(m_e{c}^2+T_{e^{+}}+(U_G{)}_{e^{+}-bh})=0,$$

$$\left(h{\nu }_0+(U_G{)}_{ph-bh}\right)+(m_e{c}^2+T_{e^{-}}+(U_G{)}_{e^{-}-bh})$$

$$+(m_e{c}^2+T_{e^{+}}+(U_G{)}_{e^{+}-bh})=0,$$

$$h{\nu }_{\infty }=h{\nu }_0{\left(1-\frac{r_s}{r_0}\right)}^{1/2}=h{\nu }_0+(U_G(r_0){)}_{ph-bh}+...$$

$$M{(}_Z{X}_f)c^2=M{(}_Z{X}_i)c^2-h{\nu }_{2^{+}}+((\Delta m)c^2+\Delta T),$$

$$\Delta m=\{\begin{array}{l}(m_p-m_d)[({(}_Z{X}_i,{}_Z{X}_f)\equiv {(}_4^{11}\text{Be},{}_4^{10}\text{Be}))],\\ (m_p-m_t)[({(}_Z{X}_i,{}_Z{X}_f)\equiv {(}_3^{11}\text{Li},{}_3^9\text{Li}))],\end{array}$$

$$\Delta T=\{\begin{array}{l}(T{(}^{11}\text{Be})-T{(}^{10}\text{Be})-T_d,[(Z=4)],\\ T{(}^{11}\text{Li})-T{(}^9\text{Li})-T_t,[(Z=3)],\end{array}$$

$$P_2={\left|\frac{e^{i{\varphi }_1}\sqrt{P_1}+e^{-i{\varphi }_2}\sqrt{P_1}}{2}\right|}^2=\frac{P_1}{2}(1+\cos ({\varphi }_1+{\varphi }_2)),$$

$$\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-\bar{\Psi }({\gamma }^{\mu }[{\partial }_{\mu }+ie{A}_{\mu }]+m)\Psi$$

$$F_{\mu \nu }(x)={\partial }_{\mu }{A}_{\nu }(x)-{\partial }_{\nu }{A}_{\mu }(x),$$

$$J^{\mu }=\frac{\partial \mathcal{L}}{\partial A_{\mu }}=-ie\bar{\Psi }{\gamma }^{\mu }\Psi .$$

$$V(t)=ie\int d^{3}x(\bar{\Psi }(\vec{x},t){\gamma }^{\mu }\Psi (\vec{x},t))a_{\mu }(\vec{x},t)+V_{coul},$$

$$\mathbf{A}(\mathbf{x})=\sum _{\lambda =\pm }\int \widetilde{dk}\left[{ϵ}_{\lambda }^{\ast }(\mathbf{k})a_{\lambda }(\mathbf{k})e^{ikx}+{ϵ}_{\lambda }(\mathbf{k})a_{\lambda }^{†}(\mathbf{k})e^{-ikx}\right],$$

$${\Delta }_{\mu \nu }^{eff}(x-y)=(2\pi {)}^{-4}\int d^{4}q\frac{{\eta }_{\mu \nu }}{q^2-iϵ}{e}^{iq(x-y)}$$

$$\frac{-i}{(2\pi {)}^4}\frac{{\eta }_{\mu \nu }}{q^2-iϵ},$$

Full text

1 Introduction

At the basis of quantum mechanics one finds Heisenberg’s indeterminacy relations, Born–Jordan commutation rules, Pauli principle, Born probability interpretation of Schrödinger wave function and Dirac transformation theory. All these elements find natural imagery in Feynman diagrams, and call for the existence of a vacuum state, whose structure is determined by virtual processes. These processes, which do not conserve energy, modulate the quantum vacuum through the transient presence of fermionic particles and antiparticles and of bosonic quanta. In the case of the electromagnetic vacuum permeating space, these are virtual off–shell electron–positron pairs and photons (Fig. 1(I)(a)). If some of these elements are modified through the action of an external field which provides in the process energy, angular and linear momentum, etc., the remaining particles can become on-shell and thus the vacuum radiates. Reaching the detector, this radiation provides information on the virtual states, and thus on the texture of the vacuum, let alone on the event triggering the radiation.

Let us build the case one step at a time, and start considering pair creation in the laboratory by a photon (left wavy line and vertex, Fig. 2(a)). Because the created pair has invariant finite mass, while the photon has zero mass, a second interaction is necessary. In Fig. 2(a), it is provided by a second photon (lower right wavy line) and associated vertex, reflecting the action of a massive charge $Z$ (cross labeled Z) needed for momentum conservation. It is of notice that in all vertices one finds three particles. This is in keeping with the fact that in QED the interaction acting at each vertex is bilinear in fermions (electrons, positrons) and linear in bosons (photons) (see App. A). Within this context, the process associated with the vertex to the left in Fig. 2(a) plays the same role as the lower vertex of Fig. 1(a)(I). Returning to Fig. 2 (a), if one allows the electron to annihilate with the positron (closing the loop) and absorb the photon (left wavy line), one obtains the Feynman diagram of Fig. 1(a)(I), as the presence of the massive charge $Z$ and associated photon is not needed.

By collapsing the two vertices and the massive charge of the QED Feynman diagram of Fig. 2 (a) into a single vertex assumed to result from the action of the curved gravitational space associated with a black hole (cylinder), one can adapt 2 (a) to Hawking’s heuristic diagram (Fig. 2 (b); see Hawking:75 ; Hawking:77 ). We return to this point below, but before let us consider the possibility to subject the QED vacuum to a supercritical atomic nucleus of effective charge $Z>Z{crit}\approx 180$, resulting from a quasimolecular state transiently formed in a heavy ion collision. Under such conditions the QED vacuum state is expected to become charged, positrons being emitted at the same time. The vacuum rearranges in such a way so a to minimise the effect of the applied ”external” field. That is, the vacuum acts as a screening medium. A schematic representation of such a process is given in Fig. 2(c). The heavy grey lines provide a schematic representation of the two ions at the distance of closest approach, of the order of 16 fm. The transient, quasimolecular state of charge $Z=Z_{1}+Z_{2}>Z_{crit}$ leads to a multi photon process of pair creation Muller:72 ; Rafelski:74 ; Soff:77 .

As in the black hole case, the $Z>Z_{crit}$ situation shows a preferred distance (radius) for the occurrence of the phenomenon leading to particle emission. In the simplest black hole description it is the Schwarzchild radius, while in the heavy ion collision leading to $Z_{crit}$ it is that of the radius of the 1s orbital with $\epsilon_{s_{1/2}}<-2m_{e}c^{2}$ of the quasimolecular system.

In a similar way in which Hawking’s approach Hawking:75 is based on a classical (relativistic) gravitational picture, the calculation of the heavy ion collision eventually leading to the $Z_{1}+Z_{2}=Z_{cr}$ is described in a time-dependent semiclassical approximation up to the distance of closest approach. But from that point on, the pair production and associated radiation process is carried out quantum mechanically, as testified by the QED Feynman diagram (c) of Fig. 2 . Within this context it is in principle thinkable to follow a similar approach in dealing with Hawking radiation and instead of (b) Fig. 2, use a diagram similar to (c). It is of notice that while the $Z_{crit}$ heavy ion reaction positron production phenomenon did not lead, in spite of much experimental effort, to a conclusive answer Rafelski:16 , a similar ”shake off” phenomenon of the QED vacuum was observed in the process $\omega+n\omega_{0}\to e^{+}e^{-}$ for $n\geq 4$ laser photons of wavelength 527 nm colliding with a photon energy of 29 GeV Burke:97 . Within this context see also Dumlu:10 ; Yu:19 and references therein.

In the above scenario nothing precludes the possibility of the presence of both photons and gravitons in the corresponding Feynman diagrams. In fact this seems to be the most likely situation, as shown in Parentani:99 in connection with photon emission from a charged particle falling into a black hole, described within the framework of QED and found to be of Hawking type, although some nontrivial differences with the “classical” result were found.

QED vacuum fluctuations play also a central role in the Lamb shift Lamb:47 ; Bethe:47 ; Welton:48 ; Kroll:49 ; French:49 , as seen from Figs. 1 (II)(b)-(d) and Fig. 3 (C) (see also Weinberg:96 ). The corresponding self-energy processes depend on the atomic orbital occupied by the electron. In the case of hydrogen it leads to a splitting between the ${}^{2}P_{1/2}$ and ${}^{2}S_{1/2}$ orbital of 1058 MHz. Within this context, we mention that Hawking refers to the Lamb shift as a phenomenon which provides confirmation of virtual fluctuations of QED Hawking:77 . As mentioned above, this phenomenon implies processes involving photons, aside from electrons and positrons (Figs. 1 (II) (b)-(d)) and Fig. 3). The fact that Hawking states that quantum mechanics implies that the whole of space (and not only that close to the event horizon of black holes) is filled with pairs of ”virtual” particles and antiparticles that are constantly materialising in pairs must imply that he views the process displayed in Fig. 1 (a) (right, i.e. with a photon) equivalent to that shown on the left, where the photon is represented by the associated electromagnetic field.

Let us now briefly mention the Casimir effect Casimir:48 ; Casimir:48b ; Casimir:48c ; Jaffe:05 , originally intended to provide a quantum mechanical description of the van der Waals force London:30 ; London:37 ; Lifschitz:55 ; Bjorken:98 ; Israelachvili:85 ; Pauling:63 acting between two non-polar molecules, taking into account retardation effects. The results obtained correspond to the long–wavelength limit of the Feynman diagram shown in Fig. 4 (d), and connected with vacuum ZPF. In other words, while it is true that the Casimir energy can be expressed in terms of Feynman diagrams with external legs Jaffe:05 , this does not mean that they are not a direct consequence of QED vacuum zero point fluctuations (within this context see Figs. 1 (II)(b) and (d) and 4 (a)-4 (d)). The Casimir effect is referred to, if not by name, quite explicitly on p.202 of Hawking:75 , in connection with the statement that the black hole being an excited state of the gravitational field can decay quantum mechanically and that because of quantal fluctuations, energy should be able to tunnel out of the corresponding potential well, a particle creation analogous to that caused by a deep potential well in flat space (confinement of two infinite walls) Bjorken:98 . It is of notice that a detailed determination of the Casimir effect requires surface plasmons to be considered Intravaia:07 . Within the nuclear connection it is closely connected with induced nuclear interaction, in particular induced pairing interaction Barranco:99 ; Terasaki:02a ; Brink:05 .

2 Hawking radiation

After the above has been stated, we can use Fig. 1 I (b) in what follows. We start with the left hand side representation of electromagnetic quantum fluctuations (Fig. 1 I (a)), for then use the QED description (right hand side Feynman diagram).

At the zero point fluctuation (ZPF) domain centered around $r_{0}$, all particles can, in principle, become real (on shell). Adopting a simple Newtonian description, this can happen in the case of the electron-positron pair, provided

[TABLE]

where $(U_{G})_{i-bh}$ is the gravitational interaction energy between the black hole ($bh$, of mass $m_{bh}$) and the electron and positron ($e^{-},e^{+},m_{e})$. The quantities $T_{e^{-}}$ and $T_{e^{+}}$ are the kinetic energies associated with these fermions. The functions $U_{G}(r)$ vanish at $r=\infty$, the remaining quantities being all positive. Thus, both particles cannot be emitted together as $HR$. But if one of them falls behind the event horizon characterised by the Schwarzchild radius ($r_{s}=2Gm_{bh}/c^{2})$, and the associated gravitational energy $(U_{G})_{i-bh}$ is sufficiently negative, the on-shell condition can eventually be fulfilled and HR emitted.

Assuming the electron escapes to infinity ($(U_{G})_{e^{-}-bh}$=0) with kinetic energy $T_{e^{-}}$, the trapped (infalling) positron-black hole gravitational interaction provides the negative energy necessary to fulfill global energy conservation. Because the subsystem ($bh+e^{+}$) has less energy than the original $bh$, one can posit that the $bh$ has lost mass which has been emitted as an electron (HR).

Let us now consider the case in which the pair materialises through an elementary QED vacuum fluctuation (Fig. 1 (I)(b) right hand side diagram). The above equation should then include the photon energy and associated gravitational interaction with the $bh$ field,

[TABLE]

where $h\nu_{0}$ is the photon energy while $m_{ph}=h\nu/c^{2}$ is the photon mass entering $(U_{G})_{ph-bh}$. Being three the particles present in the vacuum ZPF (Fig. 1 I (a) right), a variety of escape combinations are possible. It is sensible to think that a $bh$ radiates as a black body. Thus, all possible particles combinations as well as final-state interactions are expected to be present, similarly to what happens in QED pair creation by a supercritical Coulomb field (Fig. 2 (c)). Because not only gravitons can induce pair production, but also photons in presence of the $bh$, the situation resembles that of the ”classical” result for the Casimir force per unit area between two parallel plates separated by a distance $d$ ($F_{C}=-\hbar c\pi^{2}/(240\times d^{4}))$ Lifschitz:55 . In fact, this expression is only valid in the limit $\alpha\to 0$ of the fine structure constant, and assuming perfect conductivity. As mentioned above, the QED expression of the Van der Waals force between two metallic plates depends on the corresponding surface plasma, let alone on $\alpha$ Jaffe:05 ; Intravaia:07 .

In the case in which the photon escapes as $HR$ (Fig. 1 (I)(b)), it will eventually be observed that the original emission frequency undergoes a strong gravitational red shift. Being emitted near the event horizon, the asymptotic ($r=\infty$) frequency is

[TABLE]

Regarding the process in which the photon associated with the vacuum ZPF falls behind the horizon, $E_{HR}=2m_{e}c^{2}+T_{e^{-}}+T_{e^{+}}$ (Fig. 1 (I)(c)). Assuming electron-positron pair annihilation takes place (in presence of the massive object which ensures linear momentum conservation) , may lead to photon production of frequency $\nu^{\prime}\geq 2m_{e}c^{2}/h$.

Predicted more than forty years ago Hawking:75 , Hawking radiation through which black holes lose energy and mass, eventually evaporating (primordial black holes), still awaits experimental confirmation. In fact, it is difficult if not impossible to observe Hawking radiation from a real black hole, (see however An:18 ) and analogue black–hole experiments are being studied in search for alternative examples of it (cf. Steinhauer:16 ; Castelvecchi:16 and refs. therein).

3 Nuclear Field Theory: structure and reactions of exotic nuclei

Quantum electrodynamics (QED) in Feynman formulation provides a detailed description of the electromagnetic vacuum, paradigm of the quantum vacuum Schweber:94 . Nuclear field theory (NFT), tailored after Feynman’s graphical version of QED, supplemented by renormalization, allows for a quantal description of nuclear structure in general and of the nuclear vacuum in particular Bes:74 ; Bortignon:77 ; Bes:77c ; Broglia:16 .

In this description particles ($p$) and holes ($h$), namely nucleons moving above and missing from the Fermi sea respectively, play the role of electrons and positrons. They are to be calculated as solutions of the Hartree-Fock mean field. Collective vibrations, play the role of photons. The strength of the particle-vibration coupling vertices play the role of the fine structure constant. Such vertices are to be summed up to infinite order to calculate the vibrations which, at variance to the photon, are composite modes. A further difference is that the “nuclear photons” come in a number of species, namely of $ph$-type (e.g. surface vibrations) of, $pp$-type (pairing vibrations), as well as a variety of spin, isospin, etc. quantum numbers.

Worked out in the seventies in connection with nuclear structure NFT has been further developed to systematically deal with spontaneously broken symmetries and associated phase transitions and Goldstone modes Bes:90 , and generalized to deal, on equal footing, with structure and reactions Broglia:04a ; Broglia:75 . Making use of renormalization techniques, convergence in non–perturbative situations can be ensured Broglia:16 . NFT has been applied to deal with a wide variety of phenomena throughout the mass table, providing an overall account of the experimental findings at the 10% level Barranco:17 ; Barranco:19 , and predictions which tested, were found in accordance with observations at a similar level of accuracy Barranco:01 ; Tanihata:08 ; Potel:10 .

Because of its graphical rules, NFT allows to make parallels and find unexpected connections with many–body theories of condensed matter and cluster physics Mahaux:85 ; Broglia:04b , let alone QED. In particular, in connection with analogues to the Lamb shift in the systematic probing of the nuclear quantum vacuum (ground state) (see Fig. 3 and e.g. Barranco:17 ). This is the reason why it appears natural to elaborate on a possible parallel between nuclear phenomena (structure and reactions), and Hawking radiation. Connection extended to the Casimir effect triggered by the remark found on p. 202 of Hawking:75 , namely: “This particle creation is directly analogous to that caused by a deep potential well in flat space–time”. In other words, pair production of QED vacuum under stress (constrain).

In this connection we also note that the realistic description of the Casimir effect involves the consideration of the fluctuations of the QED vacuum (exchange of virtual photons). Generalizing these phenomena to the dynamical Casimir effect (conducting plates in relative acceleration) the connection with HR through Einstein’s equivalence principle emerges in a natural fashion Nation:12 ; Nugayev:87 . Given the parallel existing between NFT and QED, replacing the moving plates by the colliding nuclei in a nuclear reaction, the nuclear analogue of HR seems permissible.

Reactions using exotic radioactive nuclei in inverse kinematics and active cell targets setups have brought the study of the nuclear structure and reactions to unexpected heights and technical refinements. This is mainly a consequence of the efforts made to achieve a complete description of the nuclei under study, reflected in the use of a wide variety of probes leading to Coulomb excitation and inelastic scattering and associated $\gamma-$decay, as well as inducing one- and two-nucleon transfer reactions. This is particularly so in the probing of nuclei lying at the edge of matter stability as is the case of neutron drip line systems. Paradigmatic examples of such developments are studies carried out at TRIUMPH Tanihata:13 , Saclay and GANIL Keeley:04 and RIKEN Motobayashi:12 , which have provided, among other things, detailed information on the vacuum state of exotic nuclei. The reason for concentrating our attention on these nuclei is because, being weakly bound and close to the neutron drip line, they display very large fluctuations.

The zero point fluctuations associated with 11Be and 11Li cores (see Fig. 1 (II)(a) as well as boxed inset), self energy contribution of the parity inverted (Fig. 3 ) ground state $1/2^{+}$ of 11Be (Fig. 1 (II) (d)), see also Fig. 5 (I) (a)) and of the induced pairing correlation of the halo neutrons of 11Li (Figs. 6 (b) where also a 1- vibration is to be considered in this last case), contribute approximately 6.3% and 4.7% of the corresponding binding energies, respectively (Table 1). A major fraction of the associated ZPF mass defect in these nuclei is contributed by processes which involve the quadrupole modes: 86% in the case of 11Be and 74% in that of 11Li (for details of the general framework see e.g. Baroni:04 and refs. therein).

Direct experimental insight into the mechanism at the basis of the above results in particular, and of the structure properties of the two halo nuclei in question can be obtained through one- and two-nucleon processes, namely Winfield:01 1H(11Be,10Be(($2^{+}$;3.368 MeV))2H and Tanihata:08

1H(11Li,9Li($1/2^{-}$;2.69 MeV))3H.

The mass relations, which parallel (2) are in these cases (see Fig. 5 (I)(b),(d)–(f) in relation to the first reaction and Fig. 6 (c) in connection with the second one),

[TABLE]

with

[TABLE]

and

[TABLE]

where $p,d,$ and $t$ label proton, deuteron and triton respectively. The term inside parentheses in the left hand side of (3) takes into account the kinetic energy of the projectile inducing the nuclear reaction and of the resulting outgoing particles, the other term being associated with the reaction $Q-$value. Although the outcome of the $\gamma-$coincidence experiment (related to the $h\nu_{2^{+}}$ term in (4)) can be taken for granted, its actual measurement in processes based on inverse kinematics like the ones under consideration, is technically quite trying and has not yet been measured. Be as it may, the fact that the calculated absolute transfer differential cross sections provide an overall account of the experimental findings Barranco:17 ; Potel:10 gives direct insight into the soundness, the (renormalised) NFT picture of the nuclear vacuum state, has (see Fig. 5 (I)(c) and lower inset of Fig. 6).

Let us now return to Fig. 1 (II). The bare properties of an odd nucleon moving around the core (Fig. 1 (II)(b)) get modified though Pauli principle corrections (Fig. 1 (II)(c)) and through the associated dressing process resulting from its time ordering (Fig. 1 (II)(d)). Within the scenario of quantum electrodynamics (QED) where Feynman diagrams were developed, and in keeping with the symmetry existing between positron and electron phase spaces, N-like and self-energy-like Schweber:94 processes (Figs. 1 (II)(c) and (d)) are operative on equal footing. Observation of any of the associated virtual processes dressing the electron by interrupting it through the action of an external field (e.g. Fig. 1 (III)(b)), carries similar information concerning both contributions II(c) and (d). Because of spatial quantisation, finite nuclei display an asymmetry between occupied and empty states (particles and holes). As a consequence process (c) of Fig. 1 (II) may be allowed and not process (d), or viceversa. This is particularly true for light nuclei, for example 11Be Barranco:17 .

In the core of 11Be, namely ${}^{10}_{4}$Be6, six neutrons occupy the $1s_{1/2}$ and 1$p_{3/2}$ levels (Fig. 3). The dominant ZPF is of quadrupole type, the main neutron component being associated with the $((p_{1/2},p^{-1}_{3/2})\otimes 2^{+})_{0^{+}}$ ZPF (Fig. 5(II)(a)). Because $\epsilon_{p1/2}-\epsilon_{p3/2}\approx 3.38$ MeV and $\hbar\omega_{2^{+}}=$3.368 MeV, the largest amplitude of the wavefunction of the quadrupole mode is associated with the neutron particle-hole excitation $(p_{1/2},p^{-1}_{3/2})_{2^{+}}$. The repulsion due to Pauli principle correction (Fig. 3 inset (A))is $\approx 2.86$ MeV. The clothing of the $2s_{1/2}$ bare level by the quadrupole mode (Fig. 3 inset (B)) makes it heavier, lowering its energy by about 0.5 MeV (570 keV). The result of the two processes mentioned above is parity inversion, and the appearance of the $N=6$ new magic number together with the melting away of the $N=8$ standard one. In a similar way in which the Lamb shift (Fig. 3, inset C) provides a measure of the fluctuations of the QED vacuum (see Pais:86 , p. 451), parity inversion measures ZPF of the nuclear vacuum (ground) state. In this last case further information can be obtained as compared with the atomic case, through particle transfer reactions.

Let us elaborate on this point. Interpreting the arrowed lines of Fig. 1 (III)(a) as an electron and a positron, the wavy curve as a photon and the external field (cross + dashed line) as the event horizon of a black hole (see Fig. 1 (I)(b)) one has a Feynman representation of Hawking radiation. A nuclear analogue of such radiation, to the extent that one considers only the wavy line and the detector click, is provided by graph (b) of Fig. 1 (III), if one interprets the arrowed line as a nucleon, the wavy line as a nuclear vibration and the external field (open square+ dashed line) as a irreversible and nucleon pickup reaction intervening the self–energy process shown in Fig. 1 (II)(d) at a time $t$ fulfilling $t_{0}<t<t_{1}$. A concrete example of the above parlance is provided by the no return event corresponding to the one neutron pickup reaction of the single-halo valence nucleon of 11Be, leading to the population of the low-lying quadrupole, first excited (vibrational) state of the core 10Be, as shown in Figs. 5 (I)(b),(e) and (c) (see also Fig. 5 (II) in relation with the spontaneous 111Spontaneous $\gamma$-decay is a direct consequence of the ZPF of the nuclear vacuum (through its proton component) due to the presence of the ZPF of the electromagnetic field.

$\gamma-$decay of the $2^{+}$ state, in coincidence with the reaction process).

Light nuclei at the drip line provide another paradigmatic example of parity inversion and of a nuclear analogue, again in the sense of a virtual process becoming real through the irreversible action of an external field. The nucleus is 11Li, the no return event in question the process 1H(11Li,9Li($1/2^{-}$;2.69 MeV))3H (Fig. 6 (c)). The $|{{}^{11}\text{Li(gs)}}\rangle$ can be viewed as a two-neutron halo pair addition mode (double arrowed line) and a proton moving in the $p_{3/2}$ orbital which acts as a spectator. In Figs. 6 (a) and (b), virtual processes associated with self-energy and induced pairing interaction (vertex corrections) are shown (for details see ref. Barranco:01 ). Acting with an external two-nucleon pickup field at a time $t$ such that $t_{0}<t<t_{1}$, leads to the population of the $|{{}^{11}\text{Li}(1/2^{-};2.69\text{ MeV})}\rangle$ first excited state with an absolute differential cross section (Fig. 6, lower part) accounting for the experimental findings (see ref. Tanihata:08 ).

4 Entanglement and correlations

The characteristic trait of quantum mechanics is the fact that when two systems, of which one knows the states through their respective wavefunctions enter into temporary physical interaction, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, as they have become entangled. After restablishing one wavefunction by observation, the other one can be inferred simultaneously Schrodinger:35 . With the proviso that detector sensitivity is adequate to cope with background noise, let alone set up to pick up the specific signal of the phenomenon under study. A macroscopic manifestation of quantum entanglement is provided by superconductivity in bulk metals at low temperature, and by Josephson current through an unbiased junction. The Josephson effect provides a macroscopic manifestation of quantum entanglement. But to detect the supercurrent circulating through an unbiased junction between two weakly coupled superconductors it is necessary to go from standard 100$\Omega$ junctions easy to operate with, to 1$\Omega$ ones, let alone eliminate the earth magnetic field, as well as to carry out quantitative investigations to distinguish the effect from tiny superconducting shorts Anderson:64b ; McMillan:69 .

Within this context, arguably, is it possible to set in the proper perspective the failure to detect the QED vacuum instability through collisions between very heavy ions. This is in keeping with the complexity of calculating absolute cross sections in such cases Broglia:04a , let alone analyze experiments associated with highly excited, massive nuclei which eventually can convert their many–body energy into pairs $(e^{-},e^{+})$ Rafelski:16 . At variance, in the case of direct reactions, in particular those under discussion $((p,d),\,(p,t))$, carried out at moderate bombarding energies (3 MeV/$A$), only few channels and elementary modes of excitation are open and active respectively. Furthermore one, in these cases, knows how to calculate absolute cross sections which reproduce the experimental findings within a 10% error.

Within this context it is of notice that the probabilities of populating the final state $1/2^{-}$ in the reaction 11Li$(p,t)^{9}$Li($1/2^{-}$; 2.691 MeV) through channels alternative to the direct, one–step ones, are considerably less important. They lead to cross sections which are three orders of magnitude smaller than experimentally observed (see Fig. 6 (d), and Table I of ref. Potel:10 ). A similar situation is expected in the case of the population of the first $2^{+}$ excited state of 11Be in the reaction 11Be$(p,d)^{10}$Be($2^{+}$; 3.368 MeV) Winfield:01 . Concerning entanglement of the escaping Hawking particle (detected $\gamma$–ray) with its partner(s) swallowed in the black hole (picked up in the no–return reaction process), the nuclear examples under discussion are amenable to a technically trying, but straightforward control, known as coincidence experiments. Namely, to accept events in which the photon ($\hbar\nu_{2^{+}}$=3.368 MeV) and the deuteron (Fig 5 (e)) or the photon ($\hbar\nu_{2^{+}}$=2.691 MeV) and the triton (Fig. 6 (c)) are recorded gating the corresponding detectors at the energy $\hbar\nu_{2^{+}}$ and at that resulting from (4) respectively. Entanglement which extends over the physical dimensions of modern RIB laboratory detector setups.

An alternative, simpler experiment, which carries equal bona fide quantum mechanical entanglement information but is arguably less technically demanding is the following. Identify only the nature of the outgoing particle, or set up a $\gamma$–detector array to record a single line of frequency $\nu_{2^{+}_{1}}$ and intensity $I_{\gamma}\pm a$. The quantity $I_{\gamma}$ is related to the absolute transfer cross section and $a$ to the associated experimental error. In this way one eliminates any possible contributions from other channels but the direct one (see e.g. Fig. 6 (d)).

From a quantum mechanical point of view, once the click in the $\gamma$–detector has disentangled the outgoing ($2^{+}_{1}$ vibration$\to$ ($\gamma$–decay), see Fig. 6 (d)) particle wavefunction $\Psi_{\gamma}$ from that of the two halo neutrons $\Psi_{2n}$, one knows also this one, and the no return event (although most likely the $2n$ system long before has ended up as heat in the accelerator shielding). Namely, the falling of $2n$ into the no–return triton potential leading to $\Psi_{t}=f(\Psi_{p},\Psi_{2n})$ ($\Psi_{p}$ describing the proton beam), and thus to its ($2n$) ultimate fate. Viceversa, observing $\Psi_{t}$ but not measuring neither the energy nor the momentum of the triton, provides complete information on $\Psi_{\gamma}$, and of the presence of a $\gamma$–ray of frequency $\nu_{2^{+}_{1}}$ and intensity $I_{\gamma}$. Whether it reaches the detector or ends up contributing to the (local) background radiation or detector shielding heat, is a question of detector budget.

But the possibilities within the scenario of entanglement and correlation in nuclear structure and reactions are richer than anticipated above. In fact, by changing the bombarding energy of the proton one expects a resonant behaviour when the de Broglie wavelength matches a value related to the wavelength of $h\nu_{2^{+}}$ in each of the reactions considered (self energy processes Figs. 5 (a) and (d), and Fig. 6 (a)) but also tht of the dipole mode in the second one (vertex correction, induced pairing, Fig. 6 (b)) which essentially provides all of the small but finite ($S_{2n}=380$ keV) energy, binding the two halo neutrons to the core 9Li. By making the proton beam oscillate between the differential cross sections resonant behaviour bombarding energies associated with $h\nu_{2^{+}}$ and $h\nu_{1^{-}}$, one would mimic a kind of self–amplifying Hawking radiation. Technical difficulties likely restricts this to remain an only gedanken eksperiment.

Nonetheless, in the nuclear case, there are further degrees of entanglements. At the level of nuclear structure, in keeping with the fact that the bosonic elementary modes of excitation are not elementary but composite two–

quasiparticle–like collective excitations. At the level of nuclear reactions in which case two–nucleon transfer is completely dominated by successive transfer, due to the fact that the correlation energy of Cooper pairs is much smaller than the Fermi energy ($\approx 10^{-2}$ in the case of 11Li), and that the correlation length between members of the pair is larger than nuclear dimensions.

Let us elaborate on these points, using as examples $|{{}^{11}\text{Li (gs)}}\rangle$, $|{{}^{11}\text{Li}(1^{-};750\text{ keV})}\rangle$ and $|{{}^{9}\text{Li}(2^{+};3.368\text{ MeV})}\rangle$ and the reaction 11Li$(p,t)^{9}$Li $(1^{-}$; 2.691 MeV). In all these states it is assumed that the $p_{3/2}(\pi)$ odd proton acts as a spectator and thus we do not write it for simplicity. We start with $|{{}^{11}\text{Li (gs)}}\rangle$, namely the two neutron halo pairing correlated system. Making use of the microscopic random phase approximation (RPA) and of the quasiparticle RPA (QRPA) description of $|{{}^{9}\text{Li}(2^{+};3.368\text{ MeV})}\rangle$ and of $|{{}^{11}\text{Li}(1^{-};750\text{ keV})}\rangle$ respectively, one can calculate the self–energy contributions and the induced pairing interaction (vertex corrections) shown in Figs. 6 (a) and (b), using also the $v_{14}$ Argonne potential as the bare $NN$–${}^{1}S_{0}$ bare pairing interaction. Propagating these processes to infinite order by solving a Dyson–like equation, one obtains an accurate description of the experimental findings (for details see Broglia:16 and Potel:10 ).

In Fig. 7 (a) and (b) we display the resulting spatial correlations of the two neutrons in $|{{}^{11}\text{Li (gs)}}\rangle$ and compare it with the pure configuration $1p^{2}_{1/2}(0)$, an important component of the ground state of 11Li. Similar results are shown in connection with the $2^{+}$ of 9Li and the $1^{-}$ of 11Li (Figs. 7 (c) and (d) and (e), (f)). The importance of the correlation is apparent.

Let us concentrate now on entanglement regarding the two–nucleon transfer process. As seen in Fig. 6 (d), the transfer of one nucleon at a time, that is successive transfer, constitutes the main contribution to the transfer process. From the particle–particle correlation displayed in Fig. 7 (a) and (b), and the fact that the two neutrons in the triton are close by ($\sim 2$ fm) one would have expected simultaneous transfer to be the main component. Now, the probability of neutron tunneling decreases exponentially with the square root of the mass. Because pairing correlations have a coherence length larger than nuclear dimensions, it is thus profitable that one nucleon tunnels at a time. Said it differently, to calculate the probability $P_{2}$ of a two–particle transfer process of a pair of correlated nucleons, one has to add the phased single–particle tunneling probability amplitudes, before taking the absolute square value, that is

[TABLE]

and thus $P_{2}\approx P_{1}$ $(\phi_{1}+\phi_{2}\approx 0)$, a result which parallels that found by Anderson Anderson:64b in connection with the Josephson effect. Typical examples of $P_{2}\approx P_{1}$ in the nuclear case are provided by Tanihata:08

$d\sigma(^{9}\text{Li}(d,p)^{10}\text{Li}(1/2^{-}))/d\Omega|_{\theta_{max}}\approx 0.8$ mb/sr, as compared to Cavallaro:17 $d\sigma(^{11}\text{Li}(p,t)^{9}\text{Li}(1/2^{-}))/d\Omega|_{\theta_{max}}\approx 1$ mb/sr, Fortune:94 10Be$(t,p)^{12}$Be(gs) $(\sigma=1.9\pm 0.5$ mb, $4.4^{\circ}\leq\theta_{cm}\leq 54.4^{\circ})$ as compared to Schmitt:13 10Be$(d,p)^{11}$Be($1/2^{+}$) $(\sigma=2.4\pm 0.013$ mb, $5^{\circ}\leq\theta_{cm}\leq 39^{\circ})$ in the case of light nuclei around closed $(N=6)$ shell, and Bassani:65 120Sn$(p,t)^{118}$Sn(gs) $(\sigma=3.024\pm 0.907$ mb, $5^{\circ}\leq\theta_{cm}\leq 40^{\circ})$ as compared to Bechara:75 120Sn$(d,p)^{121}$Sn($7/2^{+}$) $(\sigma=5.2\pm 0.6$ mb, $2^{\circ}\leq\theta_{cm}\leq 58^{\circ})$.

Let us now discuss the correlation between particles and holes ($ph$) associated with the two quasiparticle ($pp,ph,hh$) states $2^{+}_{1}$ and $1^{-}$, and at the basis of the phenomena of core polarization responsible for the dressing of particles (self energy) and the renormalization of the $ph$ and ($pp,hh$) interactions (vertex and pairing renormalization).

As seen from Figs. 7 (c), (d) and (e), (f), the ($ph$) become closer together when correlated by the quadrupole and the dipole residual interaction, respectively. Emitted and reabsorbed by single nucleons (Fig. 3, Fig. 5 I (a), (d), Fig. 6 (a)) they give rise to the quasiparticle degrees of freedom carrying effective masses (energies) and spectroscopic amplitudes (single–particle content), as experimentally observed (see e.g. Barranco:17 ; Barranco:19 and refs. therein). Exchanged between nucleons they renormalise the bare nucleon–nucleon interaction, in particular the ${}^{1}S_{0}$ pairing interaction (see e.g. Barranco:99 ; Terasaki:02a and refs. therein), effects which can be treated in nuclear field theory also to infinite order of perturbation if needed, in particular in the case of superfluid nuclei, but also of halo nuclei like 11Li. Ground state correlations of $ph$ collective modes and associated renormalization effects provide non negligible contributions to the binding energies (see Figs. 5 (II) (a)–(c), Table 1 and e.g. Baroni:04 and refs. therein). They are also essential in reproducing the experimental value of the electromagnetic transition probabilities. In fact, dressing the collective vibrations, e.g. the collective quadrupole mode of 120Sn, leads to conspicuous increase in the $B(E2)$–value associated with the decay into the ground state Barranco:04 , in overall agreement with the experimental findings.

Clearly, this can hardly be connected with whether particles and holes are close in space, as the wavelength of $\gamma$–rays of 1–2 MeV are orders of magnitude larger than nuclear dimensions. In fact, it is related to the fact that the components of the wavefunctions of collective states are phase–correlated, as is the case in $pp$ correlated states (pairing vibrations like $|{{}^{11}\text{Li (gs)}}\rangle$) and associated two–particle, mainly successive, transfer.

A summary of correlation and entanglement simultaneously operative at the level of structure and reaction discussed above, is shown in Fig. 8, for the case of the process ${}^{A+2}X+p\to^{A}X(J^{\pi};E_{x})+t$ (e.g 11Li+$p\to^{9}$Li($1/2-;2.69$ MeV)+$t$). The small (grey) ellipses focus on the particle–particle (neutron–neutron) correlations. That is, a structure property which is calculated for the systems ($A+2$) and $t$ ($\equiv^{3}$H) in isolation. The corresponding wavefunctions describe the effect of both ($pp$)–correlation (weak), as well as that of the external single–particle field (strong). The large ellipse focus on the $(pp)$ entanglement taking place in the transfer process, dominated by the mechanism of successive transfer. The outgoing triton and $\gamma$–ray (resulting from the $E2$–decay of the quadrupole mode of the core $A$ (9Li) are entangled and bring the specific information regarding the correlation existing between the fermionic partners of the Cooper pair, closely connected with the transfer formfactor. The $(p,t)$ is an irreversible, no–return process providing the energy, momentum both linear and angular for the $\gamma$ ray to become on shell. The variety of processes are treated fully quantum mechanically and on equal footing, within the full single–particle space, described by both bound and continuum states.

5 Conclusions

The vacuum state of a quantal system contains, through zero point fluctuations, virtual information concerning the particles (elementary modes of excitation in the case of a many-body system) building the system, and their interactions (interweaving). To bring this information to the detector, one needs to intervene the virtual states, in a no–return fashion, with external fields which share the properties one wants to observe. In the nuclear case, one-particle transfer to learn about single-particle motion, Cooper pair transfer to get information concerning the mechanisms by which gauge invariance can be violated (Cooper pair binding). Doing so in the case of light halo nuclei we have learned that, in a similar way in which the Lamb shift provided in the H-atom a definitive answer to Rabi’s question of whether the polarisation of a QED vacuum could be measured, parity inversion in nuclei provides a definitive answer of the central role collective vibrations play both in the dressing processes of valence nucleons, as well as in the induced pairing interaction acting among them, as testified by the Hawking-like radiation observed in the 1H(11Be,10Be($2^{+}$))2H and 1H(11Li,9Li($1/2^{-}$))3H reaction processes, respectively. To be able to recover information contained in the vacuum associated with the field theoretical description of the nuclear structure $(s)$, NFT had to be extended to be able to describe also reaction processes $(r)$ to the same level of accuracy, and making use of the same language Broglia:16 . In particular, treating on equal footing non-orthogonality and non-locality of the elementary modes of excitation as well as simultaneous, successive and non-orthogonality (non-local-)contributions to Cooper pair tunnelling. Within this context, one can refer to the simultaneous renormalisation of single-particle energies and transfer form factors as a further consequence of the above $(s+r)$ unification requirement.

Say it differently, we have critically assessed experimental information shedding light on one- and two-neutron halo nuclei 11Be Winfield:01 and 11Li Tanihata:08 respectively, discuss the texture of the associated nuclear vacuum and point to possible nuclear parallels to Hawking radiation in the sense of the abstract. It is of notice that in spite of much effort, a theory of quantum gravity which unifies general relativity and quantum mechanics does not yet exist. The prediction of Hawking radiation results from a combination of these two theories. In the nuclear case, a unified quantal description of structure and reactions (which play the role of the irreversible, no-return event), taking into account nonlocality and retardation both concerning correlations and transfer mechanisms and involving also electromagnetic decay, is available within the framework of renormalised NFT Broglia:16 .

It is our hope that the subjects discussed in the present paper, presenting a new view of nuclear dynamics in connection with black hole Hawking radiation can act as intellectual stimulus concerning a full quantum treatment of the phenomena involved. From our part we consider this the first step of a major challenge which we plan to follow up in future publications.

6 Acknowledgment

R.A.B. is grateful to L. Mandelli for discussions as well as to C. Pethick for suggestions. F.B. and E.V. acknowledge funding from the European Union Horizon 2020 research and innovation program, under Grant Agreement No. 654002. F.B. acknowledges funding from the Spanish Ministerio de Economìa under Grant Agreement FIS2014-53448-C2-1-P.

Appendix A Feynman rules for calculating the S-matrix in QED

One considers electrons, positrons and photons. A possible gauge- and Lorentz-invariant Lagrangian for QED is

[TABLE]

where

[TABLE]

$m$ being the electron mass, $\gamma^{\mu}$ Dirac matrices and one sums over indices like $\mu$ and $\nu$ which appear twice, one upstairs and the other downstairs. The electric current four-vector is

[TABLE]

The interaction is

[TABLE]

where electrons are created and annihilated by fields ${\bar{\Psi}}$ and $\Psi$, while the photon is created and annihilated by fields $a_{\mu}$, the instantaneous Coulomb field $V_{coul}(t)$ just serving to cancel the part of the photon propagator that is non-covariant and local in time, and

[TABLE]

where $k^{0}=\omega=|\mathbf{k}|$, $\widetilde{dk}=d\mathbf{k}/2(\pi)^{3}2\omega$ and $\epsilon_{+}$ and $\epsilon_{-}$ are polarization vectors. Let us now remind some of the Feynman rules for calculating the connected part of the $S-$matrix of quantum electrodynamics. In particular the first one: (i) draw all Feynman diagrams with up to some number of vertices. The diagrams consist of electron lines carrying arrows, with the lines joined at the vertices, a each of which there is one incoming and one outgoing electron line and one photon line. Examples of diagrams containing two vertices are provided by lowest-order Compton scattering and electron-positron scattering (Fig. 9 (a) and (b), respectively).

Another two-vertex diagram is obtained by joining in graph (a) the photon lines and the electron lines, in this last case going backwards in time, and the 1’ - 2’ and $1-2$ electron line in (b). The resulting diagram (e) describes the lowest order vacuum fluctuation.

In drawing these diagrams one should exclude disconnected diagrams, that is, diagrams in which any $V(t)$ operator or any initial or final particle is not connected to every other one by a sequence of particle creations and annihilations. Examples are provided by diagrams (c),(d) and (f) of Fig. 9.

If one views the disconnected electron line in (f) as describing the electron of an hydrogen atom, virtual fluctuations can affect the energy levels, in particular that of the $2s_{1/2},2p_{1/2}$ levels predicted by Dirac equation to have the same energy. This is because the electron in the zero point fluctuation (ZPF) of the vacuum may partially occupy the same state occupied by the electron of the hydrogen atom.

The exchange of the virtual and the disconnected electron lines correct for Pauli principle violation, leading to the connected two-vertex diagram (g) (identical to (h)) and, by time ordering, to diagram (i), The energy denominators, associated with these diagrams, difference of the initial and of the intermediate state energy are $Den_{(h)}=-(E_{N}+E_{M}+\omega)$ and $Den_{(i)}=[E_{N}-(E_{M}+\omega$)] respectively. In these expressions $E_{N}$ is the energy of the electron in the initial state, the energy of the intermediate state being either the sum of the energy $E_{M}$ of an electron and a photon of energy $\omega$ (Fig. 9 (i)), or else a positron of energy $E_{M}$, a photon of energy $\omega$, plus $2E_{N}$, sum of the energy of both initial and final electrons (Fig. 9 (h)).

The resulting value of the difference between the summed contributions $(h)+(i)$ associated with the $2s_{1/2}$ and $2p_{1/2}$ states of the hydrogen atom from relativistic calculations leads to $[\delta E]_{2s}-[\delta E]_{2p_{1/2}}=1052.19$ MHz Weinberg:96 (see also Welton:48 ; Kroll:49 ; French:49 , and Bethe:47 ) as compared to the experimental value 1057.845(9) MHz Lundeen:81 , the value reported by the first experiment and (non-relativistic) theoretical calculations being 1000 MHz Lamb:47 and 1040 MHz respectively Bethe:47 . As exemplified by the Feynman diagrams shown in Fig. 9, by construction and as a direct consequence of the interaction (A.4) which is bilinear in electron fields and linear in photon fields, associated with each vertex there are two electron (positron) lines an one photon line. In particular in the case of the process shown in Fig. 9 (e), namely a two vertex Feynman diagram describing the zero point fluctuations of the QED vacuum.

As recounted by Pais Pais:86 ; Pais:00 , Lamb provided a quantitative answer, both experimentally and theoretically Kroll:49 ; Lamb:47 to the question of Rabi of whether the polarisation of the vacuum could be measured. According to quantum mechanics, intervening a virtual process as a result of a conservation law or a physical principle (external field), in the present case the exclusion principle, one can probe the structure of the associated off-the-energy-shell process. Within this context and of the Lamb shift phenomenon, the need of a three lines, two electrons (positrons) and one photon, virtual process in connection with QED vacuum fluctuations is apparent.

It could be argued that an electron-positron vacuum virtual excitation can be a valid QED Feynman diagram interpreting the vertices as the result of the action of a Coulomb field. As stated in connection with Eq. (A.5), the violation of Lorentz invariance by the instantaneous Coulomb interaction, is cancelled by another apparent violation of Lorentz invariance connected with the fact that the photon fields $a_{\mu}$ are not four vectors, and therefore have a non-covariant propagator. The important point is that the photon propagator is taken effectively as a covariant quantity

[TABLE]

with the Coulomb interaction dropped. From a practical point of view, the main issue s that in the momentum space Feynman rules, the contribution of an internal photon line is given by

[TABLE]

and the Coulomb interaction dropped, as reflected by (A.5) (for details see Weinberg:96 ).