Nuclear interactions with modern three-body forces lead to the instability of neutron matter and neutron stars
Dmitry K. Gridnev, Stefan Schramm, Walter Greiner, Konstantin Gridnev
aa r X i v : . [ nu c l - t h ] J u l EPJ manuscript No. (will be inserted by the editor)
Nuclear interactions with modern three-body forces lead to theinstability of neutron matter and neutron stars
Dmitry K. Gridnev , Stefan Schramm , Konstantin A. Gridnev , and Walter Greiner FIAS, Ruth-Moufang-Straße 1, Frankfurt am Main, Germany Saint Petersburg State University, Uljanovskaja 1, Saint Petersburg, RussiaReceived: date / Revised version: date
Abstract.
It is shown that the neutron matter interacting through Argonne V18 pair-potential plus modernvariants of Urbana or Illinois three-body forces is unstable. For the energy of N neutrons E ( N ), whichinteract through these forces, we prove mathematically that E ( N ) = − cN + O ( N / ), where c > N neutrons exist for N large enough. The neutron matter collapse is possibledue to the form of the repulsive core in three-body forces, which vanishes when three nucleons occupy thesame site in space. The old variant of the forces Urbana VI, where the phenomenological repulsive coredoes not vanish at the origin, resolves this problem. We prove that to prevent the collapse one should adda repulsive term to the Urbana IX potential, which should be larger than 50 MeV when 3 nucleons occupythe same spatial position. PACS.
It is a common place that two liters of water contain twiceas much energy as one liter. Thermodynamically speaking,this is the result of the energy being an extensive quan-tity [1]. From the quantum mechanical point of view thisextensivity of the energy can be stated as the followingresult: let E ( N ) be the energy of N atoms or molecules,where all nuclei and electrons are treated as point parti-cles interacting solely through the Coulomb forces. Thenthe limit lim N →∞ E ( N ) /N is supposed to exist, that isthe energy per atom (molecule) approaches a limit in themany-body problem. A formidable task is to prove thatthe energy can be linearly bounded from up and frombelow cN ≤ E ≤ CN . This type of inequality provesthe stability of matter (the most difficult part here is toprove the lower bound). In their seminal paper [2] Dysonand Lennard proved the stability of non-relativistic mat-ter made of pointwise nuclei and electrons, see also [3] onthe history of this subject. Lieb and Thirring [4] simplifiedthe argument and improved the value of the constant c byorders of magnitude. The corresponding mathematical is-sues are enlightened in detail in [3]. The proof of Dysonand Lennard also demonstrated the vital role of the Pauliprinciple for the stability matter: if electrons were bosonsthen the energy would not grow linearly in N , but ratheras E ( N ) ∼ N / , see [3] for the proof.Nuclear substance formed by protons and neutronsalso forms stable matter. For finite nuclei this is best man-ifested in the Bethe-Weizs¨acker formula [5]. For symmetric nuclear matter (number of protons is equal to the numberof neutrons) the energy per particle is approximately equal16 MeV and the nuclear density is ρ ≃ .
16 fm − [5]. Nu-clei that are composed solely from neutrons are believed tohave positive energy, however, the question of existence ofbound state of N neutrons, where N is large, is still notultimately resolved [6,7]. Adding to the strong interac-tion gravitational forces enables the creation of neutronstars, which contain about 10 neutrons. The astrophys-ical data regarding masses and radii of these stars makesus conclude that the neutrons inside them form neutronmatter. Let us remark that for stability of nuclear matterthe Pauli principle is absolutely essential.The basic model of a nuclear system assumes that theHamiltonian H = T + P i Below we shall construct the upper bound on the energy ofinteracting neutrons. In the framework of non-relativisticquantum mechanics related bounds were obtained in [20,21,22,23,24], where the authors investigated the questionof existence of bound states of N identical particles, whichlie below dissociation thresholds. In [22] Zhislin has provedthe following result. Let E ( N ) denote the ground stateenergy of N fermions (or bosons) that interact through thescalar pair potential v ( r ) satisfying the following condition Z r , r ∈ K v ( r − r ) d r d r < K is a fixed arbitrary finite cube in the three-dimensional space. Then E ( N ) < − cN for N > N ,where c, N > N fermions (or bosons)this implies that: (i) for N large enough there always ex-ists a negative energy bound state of N particles irre-spectively of a given particle mass; (ii) the particles donot form stable matter, that is the energy per particle di-verges if N → ∞ . The condition Eq. (1) can be improvedif instead of one cubic box in [22,23] one takes two disjointcubes of equal size K ∩ K = ∅ and requires that Z r , r ∈ K v ( r − r ) d r d r + Z r ∈ K , r ∈ K v ( r − r ) d r d r < . (2)Instead of cubes one could use rectangular boxes. We donot prove that the bound E ( N ) < − cN follows fromEq. (2) explicitly, but the proof practically mimics theconstruction below that we use to prove the collapse ofneutron matter with modern nucleon interactions and issimilar to the proof in [23]. Note that the condition Eq. (2)is fulfilled by the following potential. Suppose that v ( r ) iscontinuous, falling off at infinity faster than | r | − − δ , where δ > v (0) = 0 and v ( r ) < 0, where r is afixed three-dimensional vector. To see that Eq. (2) withsuch v ( r ) can be satisfied one can take two cubes K , each with the side length L , where K and K have theircenters at the origin and at r respectively. Taking L smallenough one ensures that Eq. (2) holds. Let us remark thatcondition Eq. (2) is satisfied by simplified neutron-neutronpair interactions like Minnesota [25] or Volkov [26]. ForMinnesota interaction we found a bound multineutron,which contains 2364 neutrons and has a nuclear density ρ ≃ ρ [27].Now let us consider 2 N neutrons that are described bythe following Hamiltonian H = − ~ m N X i =1 ∆ r i + V b + V b = T + V b + V b . (3)The kinetic energy operator T includes the center of massmotion; m is the neutron mass and r i for i = 1 , . . . , N are neutrons’ position vectors. The term V b = P i The neutrons are placed into two disjoint cubes K u , K d each with the side length L (subscripts u, d stand for “up” and“down” respectively). The upper cube is shifted by a distance D along the Z -axis with respect to the lower cube. described by Eq. (3). Let us take two cubes K u , K d eachwith the side length L and place the first cube at the ori-gin as it is shown in Fig. 1 and the second over the firstone so that the center of K u is shifted by the distance D along the Z axis compared to the center of K d . In or-der to prevent overlapping of the cubes we require that D > L . The trial function would depend on three param-eters L, D, ω > 0, where ω is an integer. Following [22,23] let us first construct N one particle orthogonal wavefunctions. For any p = 1 , , . . . and x ∈ R we set ϕ p ( x ) = (cid:26) ( L/ − / sin (cid:0) πpL − ωx (cid:1) if x ∈ [0 , L] , / ∈ [0 , L] (4)Let us fix the an integer n in a way that makes the in-equality n ≤ N < ( n + 1) hold. For each t = 1 , . . . , N let us choose a triple of positive integers { t , t , t } so that1 ≤ t , t , t ≤ n + 1 and | t − t ′ | + | t − t ′ | + | t − t ′ | 6 = 0 for t = t ′ . (5)That is all N triples should be different (for example,triples { } and { } are different). Using these tripleswe define the one particle states for t = 1 , . . . , N as follows f t ( r ) := ϕ t ( r x ) ϕ t ( r y ) ϕ t ( r z ) , (6)where r x , r y , r z are the Cartesian components of the vec-tor r .Each cube confines N neutrons, which form an excitedstate of the Fermi gas. Let us set Ψ Π ( r , . . . , r N ) := [ f ( r ) f ( r ) · · · f N ( r N )] × [ f ( r N +1 − D ) f ( r N +2 − D ) · · · f N ( r N − D )] , (7) where D := (0 , , D ) is a three-dimensional vector.Let S N denote the permutation group for 2 N parti-cles, whose elements g ∈ S N permute only the spatialcoordinates. We define the antisymmetrizer A as A = 1(2 N )! X g ∈ S N ( − π ( g ) g, (8)where π ( g ) denotes the parity of the permutation g . Nowwe construct the trial function for 2 N neutrons as˜ Ψ A = Ψ A ( r , . . . , r N ) | n ↑i| n ↑i · · · | n ↑i , (9)where the spatial part of the wave function is Ψ A := p (2 N )! A Ψ Π .In Eq. (9) | n ↑i denotes the isospin-spin state of eachnucleon (isospin down for neutron and spin up). In thefully polarized trial function we set all neutrons into thestate of a neutron with spin up because this simplifiesthe antisymmetrization. It is easy to see that ˜ Ψ A is fullyantisymmetric and normalized because the cubes K u , K d are disjoint. By the variational principle E (2 N ) ≤ h ˜ Ψ A | H | ˜ Ψ A i = h ˜ Ψ A | T | ˜ Ψ A i + h ˜ Ψ A | V b | ˜ Ψ A i + h ˜ Ψ A | V b | ˜ Ψ A i (10)where E (2 N ) is the ground state energy of 2 N neutronsdescribed by Eq. (3). Let us first consider the contributionfrom the three-body term. Substituting the interactionsfrom Eqs. (2.1), (2.7) in [9] and using that both cubes arefilled with the same states we get h ˜ Ψ A | V b | ˜ Ψ A i = X ≤ i 0) = 0, whereas the integrand on the right-handside of Eq. (16) would be close to W (0 , , D ). The graphof the function W (0 , , D ), which depends on D is shownin Fig. 2. One can set D = 1 fm in order to ensure thatthe integrand in Eq. (16) would be less than − 12 MeV.Therefore Q > L is fixed sufficiently small thoughdifferent from zero.It remains to fix the last parameter ω in the trial func-tion. For convenience of notation let us introduce the nine-dimensional vector s = ( r , r , r ) so that s ∈ R hasthe components s = ( s , . . . , s ) = ( r x , r y , . . . , r y , r z ). Let D ⊂ R denote the subset of all non-zero vectors with inte-ger coordinates (that is for any d ∈ D all d i for i = 1 , . . . , P i =1 d i = 0). And let us define Υ ( ω ) = sup d ∈D (cid:12)(cid:12)(cid:12) Re Z L0 ds . . . Z L0 ds W( s ) × exp (cid:0) i πL − ω ( d · s ) (cid:1)(cid:12)(cid:12)(cid:12) . (18)Note that W ( r , r , r ) is square integrable in the cube,that is Z L ds . . . Z L ds (cid:12)(cid:12) W ( s ) (cid:12)(cid:12) < ∞ . (19)It is a trivial consequence of the Bessel’s inequality that Υ ( ω ) → ω → ∞ , since the integral in Eq. (18)is proportional to the Fourier coefficient of the function W ( s ). Similarly, we define Υ ( ω ) = sup d ∈D (cid:12)(cid:12)(cid:12) Re Z L0 ds . . . Z L0 ds ˜W( s ) × exp (cid:0) i πL − ω ( d · s ) (cid:1)(cid:12)(cid:12)(cid:12) , (20) where by definition ˜ W ( s ) = W ( r , r + D , r + D ), and Υ ( ω ) → ω → ∞ as well. Therefore, we can fix ω requiring that(3 + 2 ) (cid:2) Υ ( ω ) + Υ ( ω ) (cid:3) ≤ Q/ , (21)where Q is defined in Eq. (17).Now we turn back to Eq. (11). On the right-hand side(rhs) of Eq. (11) the antisymmetrization operator Eq. (8)enters two times. It is easy to check that in the first termon the rhs of Eq. (11) only 6 permutations g in Eq. (8)produce a non-vanishing contribution. These are 6 permu-tations, which permute the indices { i, j, k } and leave allother 2 N − j, k and do not permute the other 2 N − h ˜ Ψ A | V b | ˜ Ψ A i = 2 X ≤ i 1) = O ( N ). To derive anupper bound on h ˜ Ψ a | V b | ˜ Ψ A i it suffices to consider the con-tribution from the interaction V P i The inequalities in Eq. (33) are understood as operatorinequalities. For self-adjoint operators A, B the inequality A ≤ B means that h f | A | f i ≤ h f | B | f i for all admissible f . In Eq. (33) we have used the operator inequality ( A − B ) ≤ A + 2 B , which easily follows from the obvious( A + B ) ≥ L ij D. K. Gridnev, S. Schramm, K. A. Gridnev and W. Greiner: Instability of neutron matter with 3-body forces we finally obtain V h ˜ Ψ A | X i The plot of the function W (0 , , D ) (where D ≡ (0 , , D )) versus parameter D . leads to even larger estimate of | E (2 N ) | ! Another techni-cal trick is to set neutrons into a highly dense state in thetrial function. From the proof, which uses the variationalprinciple, one may get a false impression that we applythe modern nuclear Hamiltonian to a media with extremedensities, which do not occur in Nature. This is, however,not true. Instead we prove that ultra high densities in-evitably result when N neutrons are in the ground stateand N is large! We do not claim that such densities ap-pear in Nature. On the contrary, we claim that the forcehas to be corrected in order to avoid the appearance ofunphysical densities. That is we demonstrate mathemat-ically that the repulsive core in the Urbana and Illinois3-body interactions is wrong, since it leads to the growthof the binding energy according to the law of N .It is important to show that the corrections of the 3-body interaction that are required for stability of neutronmatter are substantial. For that let us consider a hypo-thetical change in the Urbana IX interaction, which canbe written as ˜ V b = V b + V NR , (37)where V b is defined as above and V NR has the form ofthe repulsive term that is used in Urbana VI interaction,see Eqs. (2.8)-(2.10) in [9]. That is V NR ( r , r , r ) = U C W ( r ) W ( r ) W ( r ) (38)where U C is a constant and W ( r ) = (cid:2) (cid:0) ( r − R ) c − (cid:1)(cid:3) − (39)with R = 0 . c = 0 . . K. Gridnev, S. Schramm, K. A. Gridnev and W. Greiner: Instability of neutron matter with 3-body forces 7 reads13 B + B + 13 Z r , r , r ∈ K d V NR ( r , r , r ) d r d r d r + Z r , r , r ∈ K d V NR ( r , r + D , r + D ) d r d r d r ≥ , (40)where integrals B , are given in Eqs. (15) and (16) re-spectively. Inequality (40) should hold for all values of theconstants L, D > D − L > D = 0 . W (0 , , D ) ≃ − . L → U C in (38), which is U C > 51 MeV. Vary-ing the sizes of the cubes one may obtain a better valueand the minimal value of U C indeed becomes larger if oneconstructs an non-polarized trial function.Stability, radius and masses of neutron stars are largelygoverned by the equation of state (EOS) of nuclear matter,see f. e. [29,30]. The present result shows a dramatic effectof the repulsive core on the EOS at high densities. Let usnote that recently there were calculations of neutron mat-ter [31] with potentials derived from chiral perturbationtheory [32,33]. We were not able to reach the conclusions,whether such instability occurs for such interactions; it isimportant to generalize the stability condition derived inthis paper to interactions in momentum space like in [32]. It is a standard practice to study the neutron matter byconsidering the ground state of N neutrons, which are setin an external trap [34]. We prove that the neutrons wouldcollapse with large N if one uses modern 3-body forces.The mathematical proof is absolutely rigorous, the col-lapse is derived from the Schr¨odinger equation. The rea-son for the collapse is the presence of form-factors in theinteractions, which make 3-body force vanish when 3 nu-cleons occupy the same position is space. The neutrondensity is most probably zero for N < 100 (100 neutronsseem to have no bound states). As N increases the firstbound state of N neutrons emerges at some point and thedensity starts growing with N . This happens without anyexternal compression and it is a mathematical fact. By us-ing the mathematical approach we come to the conclusionthat in order for the modern nuclear Hamiltonian to workone should constrain the number of particles. Otherwisefor N ' n = N/ n = N/ V very small. A free gas wave function in each blobwould give you the kinetic energy n times the E F (Fermienergy), or proportional to n / . Since T ( r ) and V ( r ) inUrbana and Illinois become small for small r , for smallblobs, the only three body interaction is when 1 particle ofa triplet is in one blob, and the other two are in the secondblob. Pick a distance between the blobs where this three-body interaction is attractive. You then get n triplets,multiplied by this attractive interaction. 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