Equation of State for Neutron Matter in the Quark ... - Springer Link

ISSN 1547-4771, Physics of Particles and Nuclei Letters, 2017, Vol. 14, No. 6, pp. 849–856. © Pleiades Publishing, Ltd., 2017. Original Russian Text © M.I. Krivoruchenko, 2017, published in Pis’ma v Zhurnal Fizika Elementarnykh Chastits i Atomnogo Yadra, 2017.

PHYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY

Equation of State for Neutron Matter in the Quark Compound Bag Model M. I. Krivoruchenkoa, b, c aMoscow

Institute of Physics and Technology, Dolgoprudnyi, Moscow oblast, 141700 Russia Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia cInstitute for Theoretical and Experimental Physics, Moscow, 117218 Russia e-mail: [email protected]

b

Received July 2, 2016

Abstract⎯The equation of state for neutron matter is derived in the framework of the Quark Compound Bag model, in which the nucleon–nucleon interaction is generated by the s-channel exchange of six-quark Jaffe– Low primitives. DOI: 10.1134/S1547477117060188

1. INTRODUCTION The nucleon–nucleon interaction is known to be repulsive at small distances and attractive at large ones. In the one-boson-exchange (OBE) models, the repulsive and attractive forces are largely generated by the ω and σ-meson exchanges, respectively. Various OBE models rely on different sets of exchanged mesons that help describe the large amount of accumulated data (see, e.g., [1–4]). The nucleon–nucleon interaction occurs, however, at distances comparable with the nucleon and meson sizes. Geometrically, the dominance of the meson-exchange mechanism is questionable as soon as the overlap between the particles is on the order of unity. At the same time, simple estimates demonstrate that configurations of overlapping nucleons play an important role in nuclei [4].1 When describing the system of overlapping nucleons, one should take into account the quark degrees of freedom. Two nucleons driven close together form a six-quark state. Therefore, the nucleon–nucleon interaction can be graphically visualized as follows: the initial nucleons propagate to form a six-quark state, which then propagates as a whole and finally breaks up to two free nucleons. 1 Using

the Holtsmark distribution, for nuclear matter at saturation density the mean distance between a nucleon and its closest neighbor nucleon is estimated as r = 1.02 fm, with a statistical uncertainty of ±0.37 fm [14]. This distance is surprisingly small when compared to the proton charge radius of rπ2 1/2 = 0.875 ± 0.007 fm and the pion charge radius of rπ2 1/2 = 0.659 ± 0.025 fm [5]. The strongly overlapping nucleons leave practically no space for the mesons to mediate the t-exchange processes.

Such an interaction mechanism was originally studied by T.D. Lee [6] for purposes not related to the problem of the nucleon–nucleon interaction. The modified Lee model involving resonance exchanges in the s-channel was then considered by Dyson [7] towards revealing the physical nature of the CDD poles [8]. The Dyson–Lee models involve bound states and resonances, and therefore refer to the systems dominated by attractive forces. At the same time, short-range nucleon–nucleon interactions are dominated by repulsive forces. In Ref. [9], Simonov extended the scope of Dyson–Lee models by including the so-called Jaffe–Low primitives [10] in the dynamics. This rendered the models of the Dyson– Lee type capable of describing the systems with repulsive forces, as mediated by the primitives. Primitives may be viewed as the resonances whose widths tend to zero on the mass shell. Still, these participate in particle interactions, since they have nonzero widths out of the mass shell. The primitives correspond to the zeros of the scattering phase with a negative slope. The P-matrix shows poles at nucleon energies equal to the primitive masses, while the S-matrix shows no singularities. Among the models of the Simonov–Dyson type, the most detailed studies are made within the framework of the Quark Compound Bad (QCB) model. The QCB model allows one to adequately reproduce the phases of the nucleon–nucleon, meson–nucleon, and meson–meson scattering and the properties of fewnucleon systems [9, 11–14]. In this paper, we derive the neutron-matter equation of state for the version [13] of the QCB formalism.

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The nuclear-matter equation of state is important for the astrophysics of compact objects. The discovery of neutron stars with masses about 2M [15, 16] allowed exclusion of a broad set of the soft equation of state (EoS) of nuclear matter for which neutron stars lose gravitational stability at the lower values of masses. At the same time, the results of laboratory experiments suggest that the EoS of nuclear matter should be rather soft. This apparent contradiction has been actively discussed over the last several years. The nuclear-matter asymmetry coefficient, whereby the matter stiffness increases with the neutron fraction, plays an important role in the EoS. The nuclear-matter EoS is additionally softened by the production of hyperons at the center of a massive neutron star arising because of the chemical equilibrium with respect to weak interactions (see, e.g., [17, 18]). In [19–21], it is argued that the nuclear matter EoS can be made stiffer by introducing light weakly interacting bosons beyond the Standard-Model framework. In Ref. [20], it is also noted that the repulsion between hyperons can receive a contribution from the φ-meson exchange, which is suppressed by the OZI rule for nonstrange baryons. This additional repulsion leads to the suppression of hyperon formation.2 The existing laboratory data still leave some room for a stiff equation of state of the equilibrium ultradense nuclear matter, which can be based either on the existing hadron-physics concepts or on those beyond the Standard Model. In the mean-field approximation (MF), the exotic degrees of freedom soften the EoS of nuclear matter. The 2M upper limit on the neutron-star mass implies strong restrictions on the critical density of the phase transition to quark matter and, if the latter exists in the neutron-star cores, on its EoS. Beyond the MF approximation, exotic states can induce opposite effects. In quantum theory, the exotic degrees of freedom are virtually present even if the critical density of the phase transition is large. Therefore, they should be taken into account already at the saturation density through a renormalization of phenomenological parameters. With increasing density, the sign of the effect is generally unknown. This can be illustrated by the analysis [24] of the dibaryon Bose-condensation in nuclear matter in the relativistic Hartree approximation.3 In connection with this, one can expect the recently detected dibaryon d*(2380) [25–30] to strongly affect the EoS of nuclear matter, despite its relatively large mass. This is because the spin of this resonance is as high as J = 3, whereas the Casimir effect, which consists in modifying the zero-oscillation energy of the d*(2380) boson field in nuclear matter, is proportional to 2J + 1. The Bose condensation 2 This

mechanism was studied numerically in [23, 24], where its viability was confirmed. 3 The data of Fig. 2 demonstrate that, below the Bose condensation point, the exotic dibaryon phase is stiffer than the dibaryonfree phase.

of dibaryons in nuclear matter was studied in [14, 25, 31–42]. The implications of Simonov–Dyson models for the properties of nuclear matter have not yet been investigated. In this paper, we formulate the equations for deriving the EoS of neutron matter for version [13] of the QCB model taking into account the neutron superfluidity. The dependences of the neutron-matter pairing gap, pressure, and energy density on the matter density are obtained. The discussed version of the QCB model has been proposed towards demonstrating the existence of CDD poles associated with the primitives. The parameter values were selected so as to best reproduce the moderately low momentum component of the NN-scattering phases in the 1S0 and 3S1 channels (up to 350 MeV for the 1S0 channel). Using this approach, one can quantitatively describe the nuclear matter only at densities below that of the saturation. One may hope that these predictions can be consistently extrapolated to higher nuclear-matter densities where, in qualitative contrast with the OBE predictions, the Bose condensation of six-quark states may occur. The basic concepts of the model are formulated in the next section. Section 3 gives the system of equations whereby the EoS of nuclear matter can be derived taking into account the effect of neutron pairing. In Section 4, these equations are solved numerically, the results of the calculus are presented, and our predictions for the pairing gap are compared with those of the OBE models. In the concluding section, we summarize the main results of this analysis and discuss the prospects for refining the derivation of the nuclear-matter EoS in the QCB framework. 2. MODEL In the QCB model [9], nucleon scattering proceeds through the formation of an intermediate compound state with mass M α > s 0 = 2m. The D-function of the process can be written as [13]

D(s) = Λ(s) − Π(s),

(1)

where −1

Λ (s) =

∑

α +∞

Π(s) = − 1 π Here, Φ 2(s) = πp*

∫

s0

g α2 + G, s − M α2

(2)

F 2(s ') ds '. Φ 2(s ') s' − s

s is the relativistic phase space,

p* = s − 4m 2 is the momentum of either nucleon in the center-of-mass frame, g α is the coupling constant between the primitive dα and the nucleons, and 2

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EQUATION OF STATE FOR NEUTRON MATTER

F(s) is the formfactor of the dαNN vertex. The S-matrix has the form S=e

2i δ( s )

=

D(s − i 0) . D(s + i 0)

=

+

(3) Fig. 1. Dyson’s equation for the compound-state propagator. The double line corresponds to the bare propagator Λ –1(s). The double line with a shaded block depicts the full compound-state propagator. The loop is formed by nucleon propagators.

The poles of Λ(s) are referred to as CDD poles [10]. These are positioned in between the zeros of the Λ(s) function, which in turn correspond to the masses of compound states: Λ(s = M α2 ) = 0. Upon including the coupling with the continuum, the compound states acquire the forms of bound states, resonances, or primitives.

=

The D-function (1) is a generalized R-function [13] which has no complex zeros on the first sheet of the Riemann surface. Under the condition D(s0) < 0 for

(4)

located under the unitary cut on the unphysical sheet of the Riemann surface, are identified as the resonances. The roots located on the real half-axis (s0,+∞) are identiﬁed as primitives where the real and imaginary parts of D(s) have zeros of the first and second order, respectively.

–

+

Fig. 2. Graphic representation of the two-nucleon Green’s function. The full compound-state propagator, depicted by a double line with a shaded block, defines the T-matrix of NN scattering.

s0 < M α2, it also has no zeros corresponding to bound states on the real half-axis (–∞, s0). Simple roots of the equation

D(s) = 0,

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In the Born approximation, we have

s p* U (q) = −Φ 2(s)F(s)Λ −1(s)F(s), (6) 8π where q = p' – p is the momentum transfer and U(q) is the Fourier image of the potential. Therefore, the kinematic factors are reconstructed using the correspondence A(s) = −

(7) U (q) → 8π F(s)Λ −1(s)F(s). s In separable models, the potential is represented in the form U (q) = f ν ( p ') f ν ( p) . 2

Quantity 1/D can be interpreted as the propagator of the compound state. The system of equations for deriving the compound-state full propagator is schematically shown in Fig. 1. The bare propagator, depicted by a double solid line, corresponds to Λ–1(s). The loop depicts the dispersive part of the D-function, i.e., the quantity Π(s). The more general scheme also includes the contact four-fermion interaction described by the coupling costant G in Eq. (2).

i + iG2 , and 2 s − M α gα 1/D refer to the dαNN vertex, the bare propagator, and the full propagator, respectively. By definition, the bare propagator includes the contact-interaction vertex. The formfactor F(s) depends on the difference between the nucleon three-momenta in the center-ofmass frame, whereas the propagator is a function of the total four-momentum squared, s = (p1 + p2)2. For the on-mass-shell nucleons, F(s) is a function of the nucleon momentum: F(p 2 ) = F(s = 4(m 2 + p 2 )). Thuswise, quantities −ig αF(s) ,

The process of nucleon–nucleon scattering is graphically represented in Fig. 2. The scattering amplitude has the form

A(s) = e

i δ( s )

sin δ(s) = −

Φ 2(s)F 2(s) . D(s)


∑

ν

3. NEUTRON PAIRING AND THE EQUATION OF STATE The superfluidity equations for separable potentials were discussed in Ref. [43] and, more recently, in Ref. [44]. Taking into account the correspondence rule (7), we have

1= −

∫

d p 2π 2 2 −1 2 F(p )Λ (s)F(p ) (2π)3 E 2(p)

(8)

1 . × 2 (E (p) − μ) 2 + Δ 2(s, p) s =4μ 2 Here, s is the squared energy of the Cooper pair in the medium rest frame. The quantity s is assigned the value of 4μ2, where μ is the chemical potential of neutrons, and E (p) = p 2 + m 2 according to the relativistic dispersion law. The pairing gap is obtained as

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2π F(p 2 )Λ −1(4μ 2 ) Ξ . E (p)

(9)

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nucleon mass. Therefore, the dispersion law can be written in the form Fig. 3. Mass operator Σ for nuclear medium. The loop is formed by the Fermi sphere hole (the single line with a shaded block) and the primitive (the double line with a shaded block).

Quantity Ξ* is defined as the integral

i Ξ* =

∫

d 4 p 2π F(p 2 )F †( p), 4 (2π) E (p)

(10)

where F †( p) is the anomalous Green’s function in the momentum representation [45],

ε αβF †( p) = d 4 xe ip( x − y)ε αβF †( x − y)

∫

= d xe 4

∫

ip( x − y )

†

where ε12 = −ε 21 = 1, ε11 = ε 22 = 0. Solving the superfluidity equation, for Green’s function we obtain

G ( p) =

v 2p u 2p , + ω − ε(p) + i 0 ω + ε(p) − i 0 ⎛ u 2p ⎞ 1 ⎛ ηp ⎞ ⎜⎜ 2 ⎟⎟ = ⎜1 ± ⎟, ⎝v p ⎠ 2 ⎝ ε(p) ⎠

(11)

where η p = E (p) − μ and ε(p) = η2p + Δ 2(4μ 2, p). For the anomalous Green’s function, we obtain

F †( p) = − 2π F(p 2 )Λ −1(4μ 2 ) E (p) (12) Ξ* . × (ω − ε(p) + i 0)(ω + ε(p) − i 0) Substituting (12) in Eq. (10), we finally arrive at condition (8). The particle-number density is expressed through Green’s function (11) as d ω d p −iω t N = − 2i e G (ω, p) lim 4 V t →− 0 (2π) (13) dp 2 ∂Ω . =2 = − v 3 p ∂μ (2π) Upon integrating in the chemical potential, one derives the thermodynamic potential Ω and obtains the system energy as (14) E = Ω + μN .

∫

∫

As is shown in Fig. 3, the exchange of the 1S0 primitive contributes to the nucleon self-energy in matter. The contribution is proportional to (1 + γ0)/2, where γ0 is the Dirac gamma matrix [14]. The mass operator Σ redefines the nucleon energy represented by the chemical potential and yields a contribution to the

(15)

The mass operator has vector and scalar components with respect to the Lorentz group. In order to discriminate between these components, one has to go beyond the nonrelativistic approximation (see the discussion in [14]). In full generality, the mass operator depends on the nucleon momentum and is defined out of the mass shell, which renders the form of the dispersion law more complex than that in Eq. (15). Here, we neglect the effects of off-shell nucleons and momentum dependence. The mass operator is derived assuming a zero nucleon momentum. The diagram of Fig. 3 corresponds to the expression

Σ=

(−i ) T Ψ α ( x)Ψ β( y) , †

E (p) = Σ 2 + p 2 + (m + Σ 2)2.

(− 2π)2 2d p F 2(s) 1 2 v p, m2 (2π)3 D(s) 4

∫

(16)

where function v p2 is the probability for the nucleon to have the corresponding momentum. This is a generalization of the known equation of the optical-potential method. In Eq. (16), p is the nucleon momentum in the medium rest frame and s = (m + E(p))2 – p2 is the squared energy of the nucleons in their center-of-mass frame. Since the T-matrix modification in the medium is not considered, the imaginary part of the self-energy is neglected. In free-nucleon theory, the Fermi momentum is expressed through the chemical potential as

pF[0] = μ 2 − m 2 . The momentum corresponding to minimum energy of the quasiparticle, or E(p) = μ, is obtained from Eq. (15) as pF[1] = (μ − Σ 2) 2 − (m + Σ 2) 2. As soon as the particle-number density is known, the Fermi momentum can be obtained from the equation 2 4π p 3 . (17) F (2π)3 3 These three values of the momentum are no longer equal in the theory with interacting nucleons. The minimum value of ε(p) determines the pairing gap in the spectrum of one-particle excitations:

n=

Δ F ≡ Δ(4μ 2, pF[1]). 4. NUMERICAL RESULTS The strategy of the calculations consists in varying the chemical potential towards solving Eq. (8) for Ξ*. As a zero approximation for Σ, integral (16) is computed assuming v p2 = 1 and v p2 = 0 for momenta within and beyond the Fermi sphere, respectively. From the


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853

5

ΔF, MeV

3.0 2.5 2.0 1.5

QCB model

Paris Argonnre V14 Argonnre V18 CD-Bonn Nĳmegen I Nĳmegen II

0 ΔF, MeV

3.5

1.0

–5 –10 –15

0.5 –20 0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 pF, fm–1

0.5

1.0

1.5 p, fm–1

2.0

2.5

Fig. 5. Fermi-momentum dependence of ΔF for neutron matter in the QCB model shown with a reduced scale. A new branch of solutions for anomalous Green’s functions emerges at pF > 2.0 fm–1. The limit value pF = 2.5 fm–1 corresponds to the primitive’s mass of 2007 MeV.

Fig. 4. The pairing gap ΔF as a function of Fermi momentum pF in neutron matter. The prediction of the Quark Compound Bag (QCB) model is compared with those of various OBE schemes with free meson spectral functions.

given Σ, we derive the value of Ξ * . Then, we estimate v p2 , compute the self-energy Σ and again calculate Ξ*. This iterative procedure is continued until a convincing convergence is reached. At the next stage, the particle-number density is obtained using Eq. (13). Then, the thermodynamic potential (equal to the pressure with the opposite sign) is obtained by integrating the density in the chemical potential. Finally, the energy of the system is derived using Eq. (14). Substituting the parameter values as determined in [13] and solving Eq. (8), we obtain the pairing gap Δ F as a function of Fermi momentum (see Fig. 4). This prediction of the QCB model is compared with those of the most sophisticated OBE schemes [46]. In the discussed model, the pairing gap vanishes at Fermi momenta pF > 1.6 fm–1 because of a zero of the

F(s) formfactor at p* = 353 MeV. As momentum pF[1] approaches p*, in the integrand in Eq. (8) the smallness of the denominator is compensated by that of the numerator. As a result, both the integral and the solutions tend to zero. With further increasing the chemical potential, pF[1] shifts away from the formfactor zero at p* = 353 MeV and an additional branch of solutions emerges at high densities. This is shown in Fig. 5 with a different Fermi momentum scale. The pairing gap is negative because the formfactor changes its sign (note that the pairing gap as a function of momentum is proportional to the formfactor, see Eq. (9)). In this region, twice of the chemical potential still falls short of the boundary PHYSICS OF PARTICLES AND NUCLEI LETTERS

0

value of 2007 MeV; therefore, the solutions are physically meaningful. Since the effective constant of the interaction is proportional to Λ −1(4μ 2 ), one can expect the pairing gap to increase with μ. This effect is clearly seen in Fig. 5 in region pF > 2.0 fm–1, and also reveals itself in the very emergence of a new branch of solutions. Such behavior can be explained, in particular, by repulsion in a two-level system. The first level is formed by the Cooper pair, and the higher second level is the primitive (the compound state) with the same quantum numbers. The interaction between these levels results in the repulsive force irrespectively of the potential sign. The mass of the primitive remains unchanged as soon as its dynamics are not considered. However, the binding energy of the Cooper pair increases, leading to a reduction in the system energy and a corresponding increase in the pairing gap.

At Fermi momenta below 1.6 fm–1, our predictions for the pairing-gap size agree with those of the OBE models of NN interactions. This is understandable, since the long-range interactions corresponding to small densities are thoroughly investigated and adequately parametrized in all model approaches. For the nucleon-pairing energy in nuclei, the Bethe–Weizsacker formula yields a value of 15 MeV/А1/2. For the heavy nuclei with A = 100, this results in a pairing gap of 1.5 MeV. Theoretically reproducing the pairing gaps on the order of 1 MeV meets with some problems discussed in the literature, but these have probably been overcome in the most sophisticated OBE schemes. In the first approximation and without taking into account the boundary

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15

–10

12 P, MeV fm–3

0

Σ, MeV

–20 –30 –40 –50

9 6 3

0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 pF, fm–1

0.1

0

Fig. 6. Neutron self-energy as a function of Fermi momentum.

0.2

0.3 0.4 –3 n, fm

0.5

0.6

Fig. 7. Pressure as a function of the neutron-number density.

975

600

970 500 E/A, MeV

ε, MeV fm–3

965 400 300 200

960 955 950 945

100 0

940 0.1

0.2

0.3 0.4 n, fm–3

0.5

0.6

935

0

0.1

0.2

0.3 0.4 n, fm–3

0.5

0.6

Fig. 8. Energy density of neutron matter as a function of its particle-number density.

Fig. 9. Energy per nucleon in neutron matter as a function of particle-number density.

effects and the proton component, the discussed model yields ∆ = 0.2 MeV for the Fermi momentum of pF = 1.4 fm–1 . In any case, our predictions for the pairing gap behavior at small densities quantitatively agree with those of the OBE schemes for the uniform nuclear medium. In Figs. 4–6, the abscissa shows the Fermi momentum estimated from the particle-number density (17). The applicability range of the model is restricted in momentum to p < pF = 2.5 fm–1. At Fermi momenta above some 2.5 fm–1, dibaryon Bose condensation probably occurs. The neutron self-energy is shown in Fig. 6 as a function of Fermi momentum. The in-medium modification of the neutron mass is seen to be less than that predicted by the MF schemes. The pressure and the energy density are shown as functions of particle-number density in Figs. 7 and 8,

respectively. In Fig. 8, the almost linear dependence reflects the dominant contribution of the neutron rest mass. Figure 9 shows the energy per neutron, E/A, as a function of particle-number density. The density of 0.55 fm–3 corresponds to the threshold value of Fermi momentum for the dibaryon Bose condensation. Note that this value is model-dependent. 5. CONCLUSIONS The nuclear-matter equation of state is derived for the first time for the Quark Compound Bag. The analysis is based on the parametrization [13] of the available data on the nucleon–nucleon scattering phases. The short-range repulsion between the nucleons is underestimated, since in [13] the phases were fitted only in the moderately low momentum region. This


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results in a rather soft equation of state. Our quantitative description of neutron matter is valid at densities below the saturation density. The pairing gap of neutrons below the saturation density was found to be fairly consistent with previous estimates in the OBE models. Our results for the high densities find a transparent qualitative interpretation and include the effects induced by the second branch of the superfluid state. In the discussed scheme, Bose condensation of the primitives occurs at a critical density of 0.55 fm–3, which is three times higher than the saturation density. We foresee several avenues for refining and further developing the discussed formalism: (i) In Refs. [9, 11, 12, 14], the high-momentum components of scattering phases were reproduced by adding a second primitive. In this case, the single equation (8) transforms to a system of two equations. This is unlikely to affect the basic features of the solutions, but will render the EoS stiffer. In our analysis, the treatment of neutron scattering is restricted to the 1S channel. A more realistic treatment should also 0 include other investigated channels. (ii) Given that properties of the symmetric nuclear matter at the saturation density are experimentally well known, their description becomes a mandatory test of any model of dense nuclear matter. (iii) In this analysis, the primitive is not treated dynamically and, correspondingly, the in-medium modification of the T-matrix is not taken into account. For this reason, the calculations are restricted to neutron-matter densities of n < 0.55 fm–3 and our results for the pairing gap are compared with the OBE predictions for vacuum spectral functions. The T-matrix modification affects the pairing gap and has other important implications. In the absence of additional constraints, the primitive affected by the medium is generally expected to move away from the unitary cut transforming to a resonance. This results in the Bose condensation of six-quark resonances. On the other hand, in the OBE-type schemes the primitive is strictly bound to the unitary cut. In QCB models, the same behavior can be reproduced by either fine-tuning the parameters or introducing additional constraints. In this case, the OBE and QCB predictions for dense nuclear matter are expected to be similar. In principle, the stability of primitives with respect to perturbations can be tested experimentally [47]. (iv) Thus, in the QCB framework one still needs to investigate the in-medium modification of primitives and the properties of nuclear matter at densities above that for the possible formation of a Bose condensate. (v) In this paper, the Dyson–Gor’kov equations are solved approximately taking into account the pairing of neutrons, and the neutron self-energy is derived using the T-matrix. A more realistic approach should rely on the G-matrix based on the solutions of EliashPHYSICS OF PARTICLES AND NUCLEI LETTERS

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berg’s equations [45, 48]. In QCB-type schemes, Eliashberg’s equations can apparently be solved exactly owing to the relatively simple structure of the s-channel-exchange diagrams. This situation is in stark contrast with that in the t-channel schemes. Indeed, even in the most sophisticated OBE schemes the summation is restricted to the contributions of ladder diagrams describing the nucleon–nucleon interactions. At the same time, the QCB schemes are exactly solvable in the sense that all contributing diagrams can be summed up. Note that the QCB scheme is a generalization of the Lee model, which is exactly solvable. (vi) In order to derive the EoS for dense nuclear matter, we will have to go beyond the nonrelativistic approximation when formulating the NN-interaction model and the equations for Green’s functions in nuclear medium. In summary, we have demonstrated that nuclear matter can be realistically described, assuming that the NN interaction is mediated by the Jaffe–Low primitives. The proposed model adequately reproduces the properties of neutron matter at densities below the saturation density, including the effect of neutron pairing. In order to pass over to the region of higher nuclear densities, one has to rely on more sophisticated versions of the QCB model describing the NN interactions at high momentum transfers and determine the in-medium Green’s functions in a self-consistent manner. ACKNOWLEDGMENTS The author is grateful to Yu.A. Simonov, F. Simkovic, and A.V. Yudin for useful discussions. This work was supported in part by the Russian Foundation for Basic Research under grant 16-02-01104 and by the Heisenberg–Landau Program under grant HLP2015-18. REFERENCES 1. V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, and J. J. de Swart, Phys. Rev. C 49, 2950 (1994); R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995). 2. R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C. 51, 38 (1995) 3. F. Gross and A. Stadler, Phys. Rev. C 78, 014005 (2008). 4. R. Machleidt and I. Slaus, J. Phys. G 27, R69 (2001). 5. K. A. Olive et al. (Particle Data Group), Chin. Phys. C 38, 09000 (2014). 6. T. D. Lee, Phys. Rev. 95, 1329 (1954). 7. F. Dyson, Phys. Rev. 106, 157 (1957). 8. L. Castillejo, R. Dalitz, and F. Dyson, Phys. Rev. 101, 543 (1956). 9. Yu. A. Simonov, Phys. Lett. B 107, 1 (1981). Nucl. Phys. A 416, 109c (1984); Nucl. Phys. A 463, 231c

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Translated by A. Asratyan

Vol. 14

No. 6

2017