Supercritically charged objects and electron-positron pair creation
SSupercritically charged objects and electron-positron pair creation
Cheng-Jun Xia , ∗ She-Sheng Xue , † Ren-Xin Xu , , ‡ and Shan-Gui Zhou , , , § School of Information Science and Engineering,Zhejiang University Ningbo Institute of Technology, Ningbo 315100, China ICRANet and Department of Physics, Sapienza University of Rome, Rome 00185, Italy School of Physics, Peking University, Beijing 100871, China Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China Synergetic Innovation Center for Quantum Effects and Application,Hunan Normal University, Changsha 410081, China (Dated: May 25, 2020)We investigate the stability and e + e − pair creation of supercritically charged superheavy nuclei, ud QM nuggets, strangelets, and strangeon nuggets based on the Thomas-Fermi approximation. Themodel parameters are fixed by reproducing masses and charge properties of these supercriticallycharged objects reported in earlier publications. It is found that ud QM nuggets, strangelets, andstrangeon nuggets may be more stable than Fe at the baryon number
A > ∼ × , and 1 . × ,respectively. For those stable against neutron emission, the most massive superheavy element hasa baryon number ∼ ud QM nuggets, strangelets, and strangeon nuggets need to havebaryon numbers larger than 39, 433, and 2 . × . The e + e − pair creation will inevitably start forsuperheavy nuclei with charge numbers Z ≥ ud QM nuggets with Z ≥ Z ≥ Z ≥ Q/R e = ( m e − ¯ µ e ) /α is obtained at a given electron chemical potential ¯ µ e , where Q is the total charge and R e the radiusof electron cloud. The maximum number of Q without causing e + e − pair creation is then fixed bytaking ¯ µ e = − m e . For supercritically charged objects with ¯ µ e < − m e , the decay rate for e + e − pairproduction is estimated based on the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) approximation.It is found that most positrons are emitted at t < ∼ − s, while a long lasting positron emission canbe observed for large objects with R > ∼ γ -rayburst during the merger of binary compact stars, the 511 keV continuum emission, as well as thenarrow faint emission lines in X-ray spectra from galaxies and galaxy clusters. PACS numbers: 21.60.-n, 12.39.-x, 97.60.Jd, 98.70.Rz
I. INTRODUCTION
The possible existence of objects heavier than thecurrently known nuclei has been a long-standing andintriguing question. As early as in 1960s, it was suggestedthat there may exist unusually stable or long-livedsuperheavy nuclei due to quantum shell effects, i.e., theisland of stability of superheavy nuclei [1–3]. Based oncold and hot fusion reactions, superheavy elements withcharge number Z up to 118 have been synthesized [4–7]. The quest to obtain heavier elements is still ongoing,which is focused both on their properties [8–14] andsynthesis mechanism [15–25]. Meanwhile, there existother possibilities. For example, it was argued thatstrange quark matter (SQM) comprised of approximatelyequal numbers of u , d , and s quarks may be more ∗ Electronic address: [email protected] † Electronic address: [email protected]; [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] stable than nuclear matter (NM) [26–28]. This indi-cates the possible existence of stable SQM objects suchas strangelets [29–32], nuclearites [33, 34], meteorlikecompact ultradense objects [35], and strange stars [36–38]. Nevertheless, if we consider the dynamical chiralsymmetry breaking [39, 40], the stability window of SQMvanishes. An interesting proposition was raised recentlysuggesting that quark matter comprised of only u and d quarks ( ud QM) may be more stable [41]. It wasshown that the energy per baryon of ud QM nuggetsbecome smaller than 930 MeV at
A > ∼
300 [41], while theproperties of nonstrange quark stars are still consistentwith current pulsar observations [42, 43]. Inspiredby various astrophysical observations [44], instead ofdeconfined quark matter, it was proposed that a solidstate comprised of strangeons (quark-clusters with three-light-flavor symmetry) can be the true ground state [45,46], then small strangeon nuggets could also be stableand persist in the universe [47].To synthesize these heavy objects with terrestrialexperiments is very difficult. The fusion evaporation-residue cross sections in producing superheavy elementswith
Z >
118 are extremely small and synthesizing them a r X i v : . [ nu c l - t h ] M a y requires great efforts [22–25]. The possible productionof strangelets via heavy-ion collisions was proposed inthe 1980s [48, 49], while up till now no evidence of theirexistence is obtained [50, 51]. Meanwhile, the ud QMnuggets and strangeon nuggets have not been observedin any of the heavy-ion collision experiments either.The situation may be very different in astrophysicalenvironments. Being one type of the most dense celestialobjects in the universe, pulsars provide natural labo-ratories for strongly interacting matter (termed simplystrong matter) at the highest densities. As discussedin numerous investigations, pulsars are often recognizedas neutron stars comprised of nuclear matter. Due toa first-order liquid-gas phase transition at subsaturationdensities, nuclear matter could form pasta phase in theinner crust region of a neutron star [52–54], where giantnuclei with Z up to 10 are expected [55, 56]. Meanwhile,if any of the arguments on SQM, ud QM, or strangeonmatter (SM) is true, pulsars may in fact be strangestars [36–38, 57–62], nonstrange quark stars [42, 43], orstrangeon stars [44–46].The matter inside compact stars can be released duringthe merger of a binary system by both tidal disruptionand squeezing as the stars come into contact [63, 64].With a simple estimation on the balance between thetidal force and surface tension σ , the mass of the heaviestobjects ejected into space is M max ≈ R σ/GM c , where R c is the distance to the centre and M c the total massof the binary system. Nevertheless, in such a violentenvironment, the ejecta is heated and further collisionsbetween those objects are expected, then most of theheavy objects are expected to decay. For example, inthe binary neutron star merger event GW170817 [65],the ejecta quickly evolves into a standard neutron-richenvironment for r-process nucleosynthesis and producesthe transient counterpart AT2017gfo [64, 66], which isrecently confirmed by the identification of the neutron-capture element strontium [67]. For the merger of strangestars, strangelets are ejected but quickly evaporate intonucleons due to neutrino heating [68]. Strangeon nuggetsare formed during the merger of binary strangeon stars,and their decay provides an important energy sourcefor the bolometric light curve of the following strangeonkilonova [69].In such cases, even if heavy objects are ejected fromcompact stars, they may not survive since most of themdecay into neutrons. However, if the charge numberof those objects is large enough, a supercritical electricfield can be built around them and lead to e + e − pairproduction via the Schwinger mechanism [70]. Duringthe merger of a binary system, large amount of matter( ∼ − –10 − M (cid:12) ) are ejected into space within a fewseconds [63, 64]. Objects with various sizes are thenformed in the ejecta, which will collide with each otherand are usually heated. In such a catastrophic event, theelectrons of those objects may be stripped away, whichinvolve various possible mechanisms. For example: 1.The thermal ionization process should take effect at a high temperature [71]; 2. When those objects cross areaswith strong magnetic fields , electrons are trapped alongthe magnetic field lines while the massive core passesthrough, i.e., the Lorentz ionization [72]; 3. The collisionwith other objects, charged particles, and photons couldexcite the bound electrons into the continuum of freeelectron states [73, 74]; 4. The Goldreich-Julian effectof electric charge separation should also play a role if thecentral merger remnant does not collapse promptly into ablack hole [75]. In such cases, the charge number of thoseobjects may increase significantly and exceed the criticalvalues for e + e − pair creation. Depending on the timeof their creation, the emitted positrons may produce adistinct photon signature via positronium decay [76, 77],or form an electron-positron plasma. Meanwhile, dueto back-reaction the e + e − pairs may create alternat-ing electric fields in time, which emit electromagneticradiations with the peak frequency located around 4keV [78]. The corresponding signals for the existenceof heavy objects may be identified based on variousastrophysical observations. For the gravitational-waveevent GW170817, a short γ -ray burst GRB 170817A thatlasted about 2 s was observed shortly after (1 . ± . e + e − pair creations for thoseheavy objects. The paper is organized as follows. InSec. II, we present our theoretical framework to modelthe properties of NM, SQM, ud QM, and SM aroundtheir energy minimum. The properties of finite nuclei, ud QM nuggets, strangelets, and strangeon nuggets arethen obtained in Sec. III based on the method adoptedin our previous publications [82–85], and the e + e − pair creations for supercritically charged objects areinvestigated in Sec. IV. Our conclusion is given in Sec. V. II. PROPERTIES OF STRONG MATTER
The properties of various types of strong matterforming the supercritically charged objects can be wellapproximated by expanding the energy per baryon to thesecond order, i.e., E DM n b = ε + K (cid:18) n b n − (cid:19) + 4 ε s ( f Z − f Z ) . (1)Here E DM is the energy density, n b the baryon numberdensity, and f Z the charge fraction with the chargedensity f Z n b . The parameter ε is the minimum energyper baryon at saturation density n and charge fraction f Z , while K is the incompressibility parameter and The minimum magnetic field strength to create supercriticallycharged objects in this scenario is roughly 3 . × G, which isobtained by equating the the Coulomb and Lorentz forces withthe objects moving in a typical speed of 0.1 c [64]. ε s the symmetry energy. The exact values for thoseparameters are fixed according to the properties of strongmatter obtained based on various studies. Note thatEq. (1) does not involve any information on the particlesthat the strong matter is made of, where the evolutionof their masses and coupling constants are not explicitlyshown. To obtain those properties, one should refer tothe models that determine the parameters of Eq. (1). Inthis work, we adopt four representative parameter setsfor NM, ud QM, SQM, and SM, which are summarized inTable I.The baryon chemical potential µ b = ∂E DM ∂n b and chargechemical potential µ Q = n b ∂E DM ∂f Z of strong matter areobtained with µ b = ε + K (cid:18) n n − n b n + 1 (cid:19) + 4 ε s (cid:0) f Z − f Z (cid:1) , (2) µ Q = 8 ε s ( f Z − f Z ) . (3)Then the pressure is fixed according to the basic thermo-dynamic relations, i.e., P DM = µ b n b + µ Q f Z n b − E DM = K n n ( n b − n ) . (4)In nuclear matter, the minimum energy per baryonis obtained at f Z = f Z = 0 . n ≈ . .
16 fm − , where ε = m N − B withthe binding energy B ≈
16 MeV, the incompressibility K = 240 ±
20 MeV [86], and the symmetry energy ε s =31 . ± . n = 0 .
16 fm − , ε = 922 MeV, K = 240 MeV, and ε s = 31 . ud QM obtained with linear sigmamodel in Ref. [41] can be well reproduced if we take n = 0 .
22 fm − , ε = 887 MeV, K = 2500 MeV,and ε s = 17 .
35 MeV with f Z = 0 .
5. Note that thesymmetry energy ε s adopted here is small and containsonly the kinetic term. In fact, extensive investigationson the values of ε s were carried out in the past few years,e.g., those in Refs. [89–93], where one may find a differentvalue for ε s .To fix the properties of SQM, we adopt the pQCDthermodynamic potential density with non-perturbativecorrections [85], i.e., Ω = Ω pt + B, (5)where Ω pt is the pQCD thermodynamic potential densityup to the order of α s in the MS scheme [94]. The scaledependence of the strong coupling constant and quarkmasses is given by α s (¯Λ) = 1 β L (cid:18) − β ln Lβ L (cid:19) , (6) m i (¯Λ) = ˆ m i α γ β s (cid:20) (cid:18) γ β − β γ β (cid:19) α s (cid:21) , (7) TABLE I: The adopted parameter sets in Eq. (1) for nuclearmatter (NM) [87, 88], ud quark matter ( ud QM) [41], strangequark matter (SQM) [85], and strangeon matter (SM) [95]. n f Z ε K ε s σ fm − MeV MeV MeV MeV/fm NM 0.16 0.5 922 240 31.7 1.34 ud QM 0.22 0.5 887 2500 17.35 19.35SQM 0.296 0.1 924.9 2266 18.2 15SM 0.27 0.0063 927.6 4268 250 100 where β = 9 / π and β = 4 /π for the β -function, γ = 1 /π and γ = 91 / π for the γ -function, and L = 2 ln (cid:16) ¯ΛΛ MS (cid:17) with Λ MS being the MS renormalizationpoint. The renormalization scale ¯Λ is expanded withrespect to the average value of quark chemical potentials.Its value to the first order is¯Λ = C + C µ u + µ d + µ s ) . (8)In this work we take C = 1 GeV, C = 4, and B / = 138 MeV, so that the most massive strangestar can reach a mass of 2 M (cid:12) [85]. The parameters inEq. (1) are then obtained by varying µ u and µ d ( µ d = µ s )around the minimum energy per baryon, which is fixedat zero external pressure P = − Ω = 0 and chemicalequilibrium µ u = µ d = µ s for infinite strange quarkmatter. The Coulomb interaction is neglected here,which will be considered for finite sized objects. Thisgives n = 0 .
296 fm − , ε = 924 . K = 2266MeV, and ε s = 18 . f Z = 0 . E SM = 2 U (cid:0) . r n − . r n (cid:1) + M q n, (9)where n = n b /A q is the number density of strangeons.In this work we take the potential depth U = 50MeV, the range of interaction r = 2 .
63 fm, the baryonnumber of a strangeon A q = 6, and the mass of astrangeon M q = 975 A q MeV. The obtained propertiesof strangeon stars well reproduce the current constraintson pulsar-like compact objects [96]. The energy densityobtained with Eq. (9) around the saturation density canbe approximated with Eq. (1) if we take n = 0 .
27 fm − , ε = 927 . K = 4268 MeV. Meanwhile, sincestable strangeon matter is slightly positively chargeddue to the larger current mass of s -quarks, we take f Z = 0 . ε s = 250 MeV.Since the strong matter considered here is positivelycharged with f Z >
0, the contribution of electronsshould be considered due to the attractive Coulombinteraction. The electron energy density is obtained with E e = (cid:90) ν e p π (cid:112) p + m e d p = m e π (cid:104) x e (2 x e + 1) (cid:112) x e + 1 − arcsh( x e ) (cid:105) . (10)Here x e ≡ ν e /m e with ν e being the Fermi momentumof electrons and m e = 0 .
511 MeV the electron mass.The number density, chemical potential, and pressure ofelectron gas are given by n e = ν e / π , (11) µ e = (cid:112) ν e + m e , (12) P e = µ e n e − E e . (13)To reach the energy minimum, electrons interact withstrong matter and the β -stability condition should befulfilled, i.e., µ e = − µ Q . (14) III. FINITE-SIZED OBJECTS
To investigate the properties of finite-sized objects, weassume they are spherically symmetric and each of themconsists of a core of strong matter surrounded by anelectron cloud. We thus adopt a unified description thatwas previously intended for SQM objects, i.e., the UDSmodel [82–85]. The mass M , total baryon number A , netcharge number Z , total charge number Q , and electronnumber N e of the object are determined by M = (cid:90) ∞ (cid:34) πr E ( r ) + r α (cid:18) d ϕ d r (cid:19) (cid:35) d r + 4 πR σ, (15) A = (cid:90) R πr n b ( r )d r, (16) Z = (cid:90) R πr f Z ( r ) n b ( r )d r, (17) Q = (cid:90) ∞ πr n ch ( r )d r, (18) N e = (cid:90) ∞ πr n e ( r )d r = Z − Q. (19)Note that the local energy density is obtained with E = E DM + E e and charge density n ch = f Z n b − n e at r ≤ R ,while the region at r > R is occupied by electrons with E = E e and n ch = − n e . The energy densities for strongmatter E DM and electrons E e are obtained with Eqs. (1)and (10), while the electron density is determined byEq. (11). The finite-size effects are treated with a surfacetension σ , which accounts for the energy contributionfrom density gradient terms of strong interaction. Byminimizing the mass in Eq. (15) based on the Thomas-Fermi approximation, we obtain the density distributions n b ( r ), f Z ( r ) n b ( r ), and n e ( r ) ( µ e = − µ Q ), which follows µ b ( r ) = constant , (20)¯ µ e = µ e ( r ) − ϕ ( r ) = constant , (21) with the electric potential ϕ ( r ) determined by r d ϕ d r + 2 r d ϕ d r + 4 παr n ch = 0 . (22)Here µ b and ¯ µ e correspond to the respective chemicalpotentials of finite-sized objects. The charge density isobtained with n ch ( r ) = f Z ( r ) n b ( r ) − n e ( r ). With thelocal chemical potentials determined by Eq. (21), thelocal density profiles are then obtained based on theproperties predicted in Sec. II. At a given surface tensionvalue σ , the radius of the core R is fixed according to thedynamic stability of the hadron/quark-vacuum interface,i.e., P DM ( R ) = 2 σR . (23)In our calculation, electrons are trapped within theCoulomb potential of the core and ¯ µ e represents the topof the Fermi sea for electrons. By increasing ¯ µ e , thetotal number of electrons N e increases, which reduces thetotal charge number with Q = Z − N e . The boundaryof the electron cloud R e is fixed at vanishing n e , i.e., µ e ( R e ) = m e . In fact, since there is no electron persistsat r > R e , the Coulomb potential is simply ϕ ( r ) = αQ/r .According to Eq. (21), at given ¯ µ e one obtains thefollowing relation QR e = m e − ¯ µ e α . (24)If the core radius exceeds the Bohr radius (e.g., R > ∼ fm), we have R ≈ R e and a direct correlation between Q and ¯ µ e can be obtained with Q = ( m e − ¯ µ e ) Rα . (25)Based on the parameter sets indicated in Table I,we can study finite-sized objects comprised of NM, ud QM, SQM, and SM, i.e., finite nuclei, ud QM nuggets,strangelets, and strangeon nuggets. For finite nuclei, toreproduce the masses of known atomic nuclei [97–99],we take σ = 1 .
34 MeV / fm . The surface tension valuefor ud QM nuggets is indicated in Ref. [41] with σ =19 .
35 MeV / fm . For strangelets, it was shown that thecurvature term is important for small strangelets [100].However, small strangelets are unstable according to ourprevious calculation [85], we thus neglect the curvatureterm and take σ = 15 MeV / fm , which well reproducesthe strangelets’ masses at A > ∼ σ for strangeon nuggets is not determinedand should be fixed based on the interaction betweenstrangeons [101]. In this work, however, we take areasonable surface tension value σ = 100 MeV/fm sincestrangeon matter is in a solid-state. The adopted surfacetension values are summarized in Table I.At given µ b and ¯ µ e , Eq. (22) is solved numerically andthe density profiles are obtained according to Eq. (21). (cid:1) - s t a b l e n u c l e i M / A (MeV) A u d Q M n u g g e t s s t r a n g e le t s strangeon nuggets f i n i t e n u c l e i FIG. 1: Energy per baryon for four types of finite-sized objects as functions of the baryon number A . Theexperimental data for β -stable nuclei are indicated with solidsquares, which are obtained from the 2016 Atomic MassEvaluation [97–99]. The properties of a finite-sized object is then fixed basedon Eqs. (18-23). It is found that varying ¯ µ e has littleimpact on the obtained masses of finite-sized objects.To investigate the properties of supercritically chargedobjects, we thus adopt ¯ µ e = − m e in our calculation.In Fig. 1 we present the energy per baryon of finite-sized objects fulfilling the β -stability condition. Theexperimental values for finite nuclei obtained from the2016 Atomic Mass Evaluation [97–99] are well reproducedin our framework. A minimum value corresponding to Fe is identified with
M/A = 930 MeV, which is mainlydue to the small surface tension of nuclear matter. Forother exotic objects such as ud QM nuggets, strangelets,and strangeon nuggets, the obtained energy per baryonis decreasing with A due to the dominant surface energycorrection. As indicated in Table II, a critical baryonnumber A crit can then be fixed for those objects, whereat A > A crit they become more stable than Fe, i.e.,
M/A <
930 MeV. Note that the critical baryon numbermay vary with surface tension. In fact, if a small enough σ is adopted, it was shown there also exists a local energyminimum for strangelets, where strangelets of a certainsize are more stable than others [58, 102, 103]. Similarsituations may occur for other exotic objects. A crossingbetween the curves of finite nuclei and ud QM nuggetsis found at A ≈ Og synthesized by far [104], producing ud QMnuggets may be imminent via heavy ion collisions or thedecay of superheavy elements if ud QM is the true groundstate for strong matter.The stability of those objects against particle emissioncan be observed through their chemical potentials. InFig. 2 we present the baryon chemical potential µ b TABLE II: The ranges of baryon ( A ) and/or charge ( Z )numbers for objects that are stable against decaying into Fewith
M/A <
930 MeV, neutron emission with S n >
0, and e + e − pair creation with Z − Q < MA < S n > Z − Q < A A Z A finite nuclei < < < ∼ ud QM nuggets > > < < ∼ > × > < < ∼ > . × > . × < < ∼ (cid:1) b (MeV) A f i n i t e n u c l e i u d Q M n u g g e t s s t r a n g e l e t s s t r a n g e o n n u g g e t s m n FIG. 2: Baryon chemical potential of finite-sized objects asfunctions of the baryon number A . The mass of a free neutron m n is indicated with the horizontal line. as functions of the baryon number A . The neutronseparation energy is then obtained with S n = m n − µ b ,which becomes negative once µ b > m n and spontaneousneutron emission is thus inevitable for those objectsonce ejected into space. The corresponding baryonnumber ranges for objects that are stable against neutronemission ( S n >
0) are listed in Table II. For the emissionof charged particles such as protons and α particles, theexistence of a Coulomb barrier effectively reduces therate of emission, which is less significant compared withneutron emissions at S n <
0. For superheavy elementswith
A < µ e , the structuresof the core and electron cloud are obtained by solvingEq. (22). The net charge number Z of the core isdetermined by subtracting the contributions of electrons,while the total charge number Q includes contributionsof all charged particles. As indicated in Eq. (24), taking¯ µ e = m e neutralizes the core entirely and corresponds to Q = A Z Q (cid:1) - s t a b l e n u c l e i
Charge number A f i n i t e n u c l e i u d Q M n u g g e t s s t r a n g e l e t s s t r a n g e o n n u g g e t s FIG. 3: The net ( Z ) and maximum ( Q = Z − N e ) chargenumbers of finite-sized objects as functions of the baryonnumber A , obtained by taking ¯ µ e = − m e .TABLE III: The charge properties of maximum chargedobjects obtained at ¯ µ e = − m e , i.e., the net charge-to-massratios Z/A , the surface charge density of the core Q ( R ) /R ( R in fm), and the ratio of maximum charge number to baryonnumber Q/A / . Z/A Q ( R ) /R Q/A / A < ∼ A > ∼ A > ∼ A > ∼ finite nuclei 0.5 0.047 1.4 0.81 ud QM nuggets 0.5 0.0064 0.56 0.73strangelets 0.1 4 . × − . × − the global charge neutrality condition with Q = 0, whilehere we have adopted ¯ µ e = − m e , i.e., the upper edgeof the electron Dirac sea. In such cases, Q representsthe maximum charge number without causing e + e − pair creation. The obtained net and maximum chargenumbers are presented in Fig. 3. The predicted protonnumbers for nuclei coincide with the experimental β -stability line as indicated with solid squares. For ud QMnuggets, the obtained charge numbers are slightly smallerthan finite nuclei, which is mainly due to the smallsymmetry energy adopted here. By taking f Z = 0 . f Z at A < ∼
100 to smallvalues at
A > ∼ , which are presented in Table III.Meanwhile, as was discussed in our previous works [82–85], a constant surface charge density Q ( R ) /R (asindicated in Table III) is obtained at A > ∼ if we alsoconsider the contribution of electrons in the core.Since the single particle levels for electrons are degen-erate in spin, a critical charge number Z crit is obtained
11 01 0 01 0 0 01 0 0 0 0 R e R Radius (fm) A f i n i t e n u c l e i u d Q M n u g g e t s s t r a n g e l e t s s t r a n g e o n n u g g e t s FIG. 4: Radii of the core R and electron cloud R e asfunctions of the baryon number A , obtained by taking ¯ µ e = − m e . at Z − Q = 2 according to Fig. 3. The correspondingupper limits of baryon and charge numbers for objectsthat are stable against e + e − pair creation with Z − Q ≤ Z , with the critical electric field built around thecore, electrons will inevitably appear due to e + e − paircreation, which effectively reduces the charge numberfrom Z to Q ( Q < Z ). The corresponding decay ratescan be estimated by Eq. (27). Note that the criticalcharge number for superheavy elements was a long-standing problem and many efforts were made in thepast decades. For example, the critical charge number Z crit = 137 is obtained for a pointlike nucleus [108, 109].For more realistic cases, adopting different radii for finite-sized nuclei predicts various critical charge numbers with Z crit = 171–178 [110–112], while our prediction in Fig. 3with Z crit = 177 lies within this range. Finally, themaximum charge numbers Q for different types of objectsare converging at A > ∼ or R > ∼ R and electroncloud R e obtained at ¯ µ e = − m e . At A < ∼ r = R/A / is increasingwith A , which arises from the Coulomb repulsion and adecrease of pressure from surface energy as in Eq. (23).In fact, such a decrease of baryon density was pointedout in previous studies, e.g., the bubble-like structuresfound in very heavy nuclei embedded in an electronbackground [113]. Based on Eq. (24), the maximumcharge an object can carry without causing e + e − pairproduction can then be obtained by taking ¯ µ e = − m e ,which gives Q = 0 . R e ( R e in fm) [114]. The radiiof electron cloud R e are thus linked with the maximumcharge number Q , which is indeed the case according toour numerical calculation. The relation also predicts the (cid:2) e F e r m i s e a D i r a c s e a - m e + V ( r ) m e + V ( r ) e - (cid:1) Q = r + r - e + Energy (MeV) r ( p m ) FIG. 5: Positive and negative energy spectra for electrons inthe Coulomb potential of a charged object with Q = 1000. trend on the maximum charge numbers with Q = 0 . R (or Q = 0 . r A / with r ≈ / πn ) as we increase A , which should be valid at R > ∼ fm or A > ∼ with R and R e being nearly the same. For finite nuclei,as indicated in Fig. 3, adopting n = 0 .
16 fm − gives Q = 0 . A / . For other exotic objects, as indicated inTable III, Q is smaller due to larger values for n . IV. e + e − PAIR PRODUCTION
For e + e − pair production in the electric field of apositively charged object, an example of the tunnelingprocess is illustrated in Fig. 5. Electrons located in theDirac sea propagate into the Fermi sea (from r − to r + ),leaving behind a hole at r − , i.e., positrons. The electronchemical potential of the system is ¯ µ e ( ≤ − m e ), withthe total charge number Q . A potential for electronsis then obtained with V ( r ) = − ϕ ( r ) = − αQ/r for r ≥ R e . Note that the screening effects of electronsare included in the total charge number, where thecharge number without electrons Z is larger than Q .The tunneling process is only possible for electrons withenergy ¯ µ e ≤ ε ≤ − m e , where the levels at ε ≤ ¯ µ e are already occupied. According to the Thomas-Fermiapproximation, a boundary for electrons is obtained at r = R e with ¯ µ e = V ( R e ) + m e , beyond which electronsdo not exist. The relation between Q , R e , and ¯ µ e isindicated in Eq. (24), while the maximum charge anobject can carry without causing e + e − pair productionwas obtained by taking ¯ µ e = − m e [114].The decay rate of the vacuum for e + e − pair productionin an arbitrary constant electric field E is given by [70]Γ V = α E π ∞ (cid:88) n =1 n exp (cid:18) − nπ E c E (cid:19) , (26)where the critical electric field is E c = m e /e = m e / √ πα . For a supercritically charged object, the decay ratecan then be estimated based on the JWKB approxima-tion [110, 115], i.e.,Γ = 1 π (cid:90) − m e V (0)+ m e l max (cid:88) l =0 (2 l + 1) f ( ε ) P JWKB ( ε, l )d ε, (27)with the electron transmission probability at given en-ergy ε and angular momentum l being P JWKB = exp (cid:90) r + r − (cid:115) l ( l + 1) r + m e − (cid:18) ε + αQr (cid:19) d r . (28)Here f ( ε ) predicts the empty states of electrons. If the e + e − pair creation rate is much smaller than the rate ofelectron thermalization, we can adopt the Fermi-Diracdistribution of electrons and have f ( ε ) = 1 − (cid:20) (cid:18) ε − ¯ µ e T (cid:19)(cid:21) − , (29)where a lower limit ¯ µ e in the integral of Eq. (27)is obtained for zero temperature cases ( T = 0) dueto the requirement of Pauli exclusion principle, andthe maximum angular momentum is given by l max =Int (cid:16)(cid:112) α Q + 1 / − / (cid:17) . The two real turning points r ± are obtained by solving ε + αQr ± = ± (cid:115) l ( l + 1) r ± + m e , (30)which gives r ± = − αQε ± (cid:112) α Q m e + l ( l + 1) ( ε − m e ) ε − m e . (31)Note that the turning points may become smaller thanthe electron-vacuum boundary ( r ± < R e ) at l >
0. Thetunneling process for ε > ¯ µ e is still possible withoutviolating the Pauli exclusion principle. However, theCoulomb potential V ( r ) = − αQ/r is not valid at r < R e ,since the charge number enclosed within the sphere ofradius r becomes larger than Q [116]. In such cases, r + may become slightly larger and the transmissionprobability P JWKB ( ε, l ) at l > r < R e . The integral in Eq. (28) can thenbe obtained with P JWKB = exp (cid:34) π (cid:112) α Q − l ( l + 1) + 2 παQε (cid:112) ε − m e (cid:35) . (32)By taking l as continuum values, the summation inEq. (27) can be obtained via integration and givesΓ = 12 π [1 + (2 παQ −
1) exp (2 παQ )] × (cid:90) − m e ¯ µ e exp (cid:32) παQε (cid:112) ε − m e (cid:33) d ε. (33) - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 Z = 3 0 0 f i n i t e n u c l e i u d Q M n u g g e t s s t r a n g e l e t s s t r a n g e o n n u g g e t s G (MeV) Q - 1 5 - 1 2 - 9 - 6 - 3 Z = 1 0 0 0 f i n i t e n u c l e i u d Q M n u g g e t s s t r a n g e l e t s s t r a n g e o n n u g g e t s G (MeV) Q - 2 5 - 2 0 - 1 5 - 1 0 - 5 Z = 1 0 f i n i t e n u c l e i u d Q M n u g g e t s s t r a n g e l e t s s t r a n g e o n n u g g e t s G (MeV) Q ( 1 0 ) - 3 0 - 2 5 - 2 0 - 1 5 - 1 0 - 5 Z = 1 0 f i n i t e n u c l e i u d Q M n u g g e t s s t r a n g e l e t s s t r a n g e o n n u g g e t s G (MeV) Q ( 1 0 ) FIG. 6: The decay rates of e + e − pair creation for objects with Z = 300, 1000, 10 , and 10 , where the total charge Q is fixedat a given ¯ µ e . Assuming a constant Coulomb potential inside a coreof radius R and net charge number Z , the electrondistributions at given ¯ µ e can be obtained based onEqs. (21) and (22). Note that for the ultra-relativisticcases with ϕ ( r ) (cid:29) m e , an analytical solution is obtainedfor ϕ ( r ) [117]. The values of R and Z for various typesof objects are fixed according to the results indicated inFigs. 3 and 4, where ¯ µ e = − m e was adopted.The e + e − pair production rate is predicted by Eq. (33),where the total charge number Q is fixed at a given ¯ µ e with ¯ µ e ≤ − m e . In Fig. 6 we present our results for su-percritically charged nuclei, ud QM nuggets, strangelets,and strangeon nuggets with Z = 300, 1000, 10 , and 10 .For a supercritically charged object carrying a net charge Z , as indicated in Eq. (24), the total charge number Q decreases from Z as electrons are created and fill inthe Fermi sea, while the corresponding positrons leavethe system due to Coulomb repulsion. The variation of Q for supercritically charged objects becomes small at Γ < ∼ − MeV. This suggests that the e + e − pair creationis most effective at t < ∼ − s under the assumption thatpositions are emitted sequentially and Γ does not deviatemuch from those indicated in Fig. 6. During the mergerof binary compact stars, the positron emission due tothe release of supercritically charged objects may thusbe partially responsible for the short γ -ray burst [79, 80].For a fixed net charge number Z , more e + e − pairs areproduced by objects with smaller R , where R increases inthe order of finite nuclei, ud QM nuggets, strangelets, andstrangeon nuggets. For the superheavy nucleus e + e − pair takes at least a few 10 − swith the decay rate on the order of MeV, while longerduration is expected for smaller Z [118]. Note that atsmall charge numbers such as Z = 300, the pair creationquickly stops at Γ < ∼ − MeV since the Coulomb fieldis easily screened by electrons with Q − < Q ¯ µ e = − m e as indicated in Fig. 3. This is not the case for largerobjects, where the positron emission tends to last much - 5 0 - 4 0 - 3 0 - 2 0 - 1 0 R = 1 0 f m f i n i t e n u c l e i u d Q M n u g g e t s s t r a n g e l e t s s t r a n g e o n n u g g e t s G (MeV) Q ( 1 0 ) FIG. 7: Same as Fig. 6 but for supercritically charged objectswith R = 1000 fm. longer since they possess larger charge numbers.For larger objects, as an example, we consider the caseswith R = 1000 fm, which correspond to the net chargenumbers Z = 3 . × , 6 . × , 91698, and 60487 forfinite nuclei, ud QM nuggets, strangelets, and strangeonnuggets, respectively. The decay rates as functions ofthe charge number Q are presented in Fig. 7, which areincreasing with Q . At Q ≈ Q . As Q decreases, the decay rates for e + e − pair creation becomes much smaller, i.e., a continuedsource of positron emission. Comparing with the chargenumbers Q (= Z − N e ) indicated in Fig. 6, the valuesobtained here for objects with same radii are close toeach other and possess similar decay widths, which iswhat we have observed in Fig. 3 for objects with A > ∼ or R > ∼ Q increases with Z . As was discussed inFig. 4, a universal relation Q/R = ( m e − ¯ µ e ) /α can beobtained based on Eq. (24) for very large objects with R > ∼ fm or A > ∼ . By substituting this relationinto Eq. (33), the decay rate for objects with R > ∼ fmcan be determined.With most e + e − pairs created at t < ∼ − s, themaximum number of positrons emitted by supercriticallycharged objects at T = 0 can be obtained with N e + ≈ Z − Q ¯ µ e = − m e based on the charge numbers indicatedin Fig. 3. In Fig. 8 a rough estimation on the energyrelease ( E ≈ m e N e + M ej /M A ) of positron annihilationduring the merger of binary compact stars is presented,where we have assumed M ej = 0 . M (cid:12) for the totalmass of ejected objects with baryon number A andmass M A as determined by Eq. (15). It should bementioned that the ejected mass ( ∼ − -10 − M (cid:12) )and its composition depend on binary parameters andthe equation of state of dense stellar matter [119, 120],which are likely to deviate from our current assumption. M e j = 1 0 - 3 M (cid:1) f i n i t e n u c l e i u d Q M n u g g e t s s t r a n g e l e t s s t r a n g e o n n u g g e t s Isotropic energy release (erg) A FIG. 8: Isotropic energy release in γ -rays via the process ofpositrons annihilating with electrons. The obtained isotropic energy release for the ejectedsuperheavy nuclei and ud QM nuggets are comparablewith the estimated value (3 . ± . × erg of GRB170817A [79, 80], while smaller values for strangelets andstrangeon nuggets are obtained. With such a substantialamount of e + e − pairs produced within a compact regionand a short period of time, drastic collisions amongelectrons, positrons, photons, various types of particles,and supercritically charged objects take place, whichbecome optically thick to γ -rays and would reach thermalequilibrium if the thermalization time is shorter than theescape time. A reduction of photon peak energy frompositron annihilation (511 keV) is thus expected, e.g., ablackbody spectrum with a high-energy tail [121]. Notethat there is a 1.7 second delay between the trigger timesof the gravitational-wave signal GW170817 and the γ -rayburst GRB 170817A [79, 80], which may be attributedto two main reasons [80]: 1. the intrinsic delay betweenthe moment of binary coalescence and the productionof an emitting region, e.g., the time it takes for theionization process to take effects and/or the launchingof a relativistic jet; 2. the time elapsed for the emittingregion to become transparent to γ -rays, e.g., the requiredtime for the fireball to expand and become optically thinto γ -rays and/or the propagation of the jet to break outof the dense gaseous environment.It is worth mentioning that the temperature can beas high as T ≈
50 MeV during the merger of a binarysystem, e.g., those indicated in Ref. [122]. The thermalelectron-positron pairs will thus be produced abundantly,where the number density of positrons can be fixed by n e + ≈ . × − T with n e + in fm − and T ( > ∼ m e )in MeV. If we suppose there is a heated spherical region( T = 50 MeV) with a radius ∼ e + e − pairs is E ≈ . × erg. This value may0become larger if we consider the other regions of ejecta,though most of the energy may be converted into kineticenergy as ejecta expands [80]. Meanwhile, we shouldmention there may be other important energy sources,e.g., the thermonuclear reactions, the thermal radiationsuch as the outflowing ν ¯ ν [123] and/or e + e − [71] fluxes,the decay of strangelets [68] and strangeon nuggets [69],etc. In such cases, the energy release in γ -rays duringthe merger of binary strange stars or strangeon starscan be attributed to those processes instead of positronemissions from strangelets or strangeon nuggets.A substantial amount of positrons and supercriticallycharged objects may finally escape the binary system,which later create the 511 keV continuum emissionobserved in the Galaxy via positronium decay [76, 77]. Infact, it was shown that the observed positron annihilationmainly comes from the bulge with a large bulge-to-disk ratio around 1.4 [77], which seems to correlatewith the distribution of binary systems in the MilkyWay. Such kinds of correlations have recently beenadopted as tracers of binary neutron star mergers [124].Meanwhile, before the emission of positrons, the e + e − pairs produced around the surfaces of supercriticallycharged objects would oscillate with alternating electricfield for a short time, and emit electromagnetic radiationswith a characteristic frequency around 4 keV [78]. Wesuspect these radiations are actually responsible for thenarrow faint emission lines around 3.5, 8.7, 9.4 and 10.1keV observed in the Milky Way center, nearby galaxiesand galaxy clusters [125, 126]. V. CONCLUSION
We study the properties of finite-sized objects thatare heavier than the currently known nuclei, i.e., super-heavy nuclei, ud QM nuggets, strangelets, and strangeonnuggets. The structures of those objects are obtainedbased on the UDS model [82–85], where the Thomas-Fermi approximation is adopted. The local properties ofnuclear matter, ud quark matter, strange quark matter,and strangeon matter are determined by expanding theenergy per baryon to the second order, while a surfacetension is introduced for the hadron/quark-vacuum in-terface. The parameters are fixed by reproducing themasses and charge properties of β -stable nuclei [97–99], ud QM nuggets [41], large strangelets [85], and strangeonmatter [95].Comparing with the most stable nucleus Fe, ud QMnuggets, strangelets, and strangeon nuggets are morestable at
A > A crit with A crit ≈ × , and1 . × , respectively. The masses of finite nucleiand ud QM nuggets become similar at A ≈ ud QM nuggetsvia heavy ion collisions. The stability of those objectsis investigated by examining their chemical potentials,where we have obtained a maximum baryon number forsuperheavy elements with A max ≈ A min ≈
39, 433, and 2 . × for ud QMnuggets, strangelets, and strangeon nuggets that arestable against neutron emission. The charge propertiesof those objects are obtained, where the net chargefraction (
Z/A ) vary smoothly from 0.5, 0.5, 0.1, and0.0063 (
A < ∼ . × − , and3 . × − ( A > ∼ ) for finite nuclei, ud QM nuggets,strangelets, and strangeon nuggets, respectively. Forobjects with large enough net charge numbers Z ≥ Z crit , e + e − pair creation inevitably starts, where Z crit = 163,177, 192, and 212 for ud QM nuggets ( e + e − pair creation areinvestigated, which increase with Z and are convergingat R > ∼ A > ∼ for different types of objects.A universal relation Q/R e = ( m e − ¯ µ e ) /α is obtainedat given ¯ µ e , where Q the charge and R e the radius ofelectron cloud. The maximum charge can be obtained bytaking ¯ µ e = − m e . At R > ∼ fm or A > ∼ , R ≈ R e and the universal charge radius relation is obtained with Q = 0 . R , which is consistent with those predicted inRef. [114].For supercritically charged objects, the decay rate for e + e − pair production is estimated based on the JWKBapproximation [110, 115]. It is found that most positronsare emitted at t < ∼ − s, which should be partiallyresponsible for the short γ -ray burst due to the releaseof supercritically charged objects during the merger ofbinary compact stars [79, 80]. For the superheavy nucleus e + e − pair requires at least few10 − s, while longer duration is expected for smaller Z . The e + e − pair creation for small objects ( Z = 300)quickly stops due to the screening effects of electrons.For larger objects, positron emission last much longer,which may be responsible for the 511 keV emission frompositron annihilation in the Galaxy [76, 77] as well asthe narrow faint emission lines in X-ray spectra observedin the Milky Way center, nearby galaxies and galaxyclusters [125, 126].Finally, it is worth mentioning that the temperature ofnewly created supercritically charged objects may reachup to ∼
50 MeV during the merger of a binary system [63].In such cases, the rate of e + e − pair creation becomesmuch larger since the electronic states with ε < ¯ µ e maynot be completely occupied as predicted in Eq. (29).The thermal ionization should also be considered, wherebound electrons are excited to the continuum of freeelectron states so that the charge Q of those objectsis increased. In fact, the emission of positrons due to e + e − pair creation combined with the evaporation ofthermalized electrons was shown to create an outflowingplasma of ∼ ergs/s on strange stars’ surfaces with T ≈ K [71]. Meanwhile, the environment ofthese objects created during the merger of a binarysystem may be filled with e + e − plasma, which couldreduce Q by capturing the surrounding electrons. Insuch cases, to determine the final state of those charged1objects, more detailed studies on the evolution of Q with e + e − pair creation, thermal ionization, and electroncapturing combined with the time evolution of theirsurrounding environment are necessary, which is intendedin our future works. Due to the requirement of chargeconservation, same amount of electrons N e = Q areejected from the charged object. Some of the electronswill recombine with the positively charged objects, or ex-perience a positronium decay with the positrons emittedby supercritically charged objects, while the rest of themforms a e + e − plasma or trapped along magnetic fieldlines and emit synchrotron radiation. All of which areexpected to contribute to the electromagnetic signal ofthe short γ -ray bursts. Nevertheless, we do not know forsure how many of those supercritically charged objectsare created or the exact charge number Q they carry,in which case a detailed dynamical simulation on thoseprocesses needs to be carried out. ACKNOWLEDGMENTS
C.J.X. would like to thank Prof. Bao-An Lifor fruitful discussions. This work was supported by National Natural Science Foundation of China(Grants No. 11705163, No. 11875052, No. 11673002,No. 11525524, No. 11621131001, No. 11947302, andNo. 11961141004), Ningbo Natural Science Foundation(Grant No. 2019A610066), the National Key R&DProgram of China (Grant No. 2018YFA0404402), theKey Research Program of Frontier Sciences of ChineseAcademy of Sciences (No. QYZDB-SSWSYS013), andthe Strategic Priority Research Program of ChineseAcademy of Sciences (Grant No. XDB34010000). Thecomputation for this work was supported by the HPCCluster of ITP-CAS and the Supercomputing Center,Computer Network Information Center of ChineseAcademy of Sciences. [1] W. D. Myers and W. J. Swiatecki, Nucl. Phys. , 1(1966).[2] A. Sobiczewski, F. Gareev, and B. Kalinkin, Phys. Lett. , 500 (1966).[3] H. 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