aa r X i v : . [ nu c l - t h ] O c t Light nuclei quasiparticle energy shifts in hot and dense nuclearmatter
G. R¨opke
Universit¨at Rostock, Institut f¨ur Physik, 18051 Rostock, Germany (Dated: November 17, 2018)
Abstract
Nuclei in dense matter are influenced by the medium. In the cluster mean field approximation,an effective Schr¨odinger equation for the A -particle cluster is obtained accounting for the effects ofthe correlated medium such as self-energy, Pauli blocking and Bose enhancement. Similar to thesingle-baryon states (free neutrons and protons), the light elements (2 ≤ A ≤
4, internal quantumstate ν ) are treated as quasiparticles with energies E A,ν ( ~P ; T, n n , n p ). These energies depend on thecenter of mass momentum ~P , as well as temperature T and the total densities n n , n p of neutronsand protons, respectively. No β equilibrium is considered so that n n , n p (or the correspondingchemical potentials µ n , µ p ) are fixed independently.For the single nucleon quasiparticle energy shift, different approximate expressions such asSkyrme or relativistic mean field approaches are well known. Treating the A -particle problemin appropriate approximations, results for the cluster quasiparticle shifts are given. Properties ofdense nuclear matter at moderate temperatures in the subsaturation density region considered hereare influenced by the composition. This in turn is determined by the cluster quasiparticle energies,in particular the formation of clusters at low densities when the temperature decreases, and theirdissolution due to Pauli blocking as the density increases. Our finite-temperature Green functionapproach covers different limiting cases: The low-density region where the model of nuclear statis-tical equilibrium and virial expansions can be applied, and the saturation density region where amean field approach is possible. PACS numbers: 21.65.-f,21.45.-v . INTRODUCTION A well established tool in treating many-particle systems is the quasiparticle concept.In contrast to free particles, the properties of quasiparticles such as the dispersion relationare modified due to the interaction with other particles. This relation between energy andmomentum is often characterized by an energy shift and an effective mass which dependson temperature and density of the medium. A significant part of the interaction can betaken into account by introducing this approach. The quasiparticle concept is not restrictedto elementary particles like nucleons only, but it can also be applied to composed particles,i.e. nuclei. (Note that the nucleons itselves are also composed particles.) In this work weevaluate these cluster quasiparticle energies within a microscopic approach.Nuclear matter is a strongly interacting quantum fluid. In order to treat warm anddilute matter (i.e. at subsaturation baryon densities n B = n n + n p . .
16 fm − and tem-peratures T .
20 MeV) within a systematic quantum statistical approach, we start from anonrelativistic Hamiltonian H = X E (1) a † a + 12 X , ′ ′ V (12 , ′ ′ ) a † a † a ′ a ′ , (1)where { } denotes momentum ~ ~P , spin σ , and isospin τ characterizing the neutron ( n )or proton ( p ) state. The kinetic energy in fermion second quantization a † , a contains E (1) = ~ P / m , whereas the potential energy contains the matrix element V (12 , ′ ′ )of the nucleon-nucleon interaction.Since there is no fundamental expression for the nucleon-nucleon interaction (like, e.g.,the Coulomb interaction in charged particle systems [1]), it is taken to reproduce empiricaldata such as the nucleon scattering phase shifts. Different parametrization are in use. Simplepotentials as proposed by Yukawa, Yamaguchi, Mongan, Gogny and others are based on two-nucleon phase shifts and can be used for exploratory calculations. For detailed calculationsone can use more sophisticated potentials such as PARIS and BONN or their separablerepresentations [2]. To obtain the empirical parameter values of nuclear matter at saturationdensity, three-body forces have been introduced in the Hamiltonian (1). In particular, theArgonne AV18/UIX potential [3] has been used to calculate light nuclei [4].Near saturation density, nucleons can be treated as quasiparticles. Semiempirical ap-proaches such as the Skyrme contact pseudopotential [5] and relativistic mean field (RMF)2pproaches [6] parametrize the quasiparticle shift as a function of densities and tempera-ture. Microscopic approaches such as Dirac-Brueckner Hartree Fock (DBHF) [7] give anappropriate description of the thermodynamic properties of warm and dense matter. For arecent review of the nuclear matter equation of state (EoS) see Ref. [8].In the low-density limit, nuclear matter at finite temperature is a mixture of free nucle-ons and nuclei in chemical equilibrium as described by a mass action law. This chemicalpicture, where bound states are treated as new species, is also denoted as nuclear statisticalequilibrium (NSE) and should be recovered as the low-density limit of a quantum statisticalapproach to nuclear matter at finite temperatures.At increasing densities, the simple NSE approach becomes invalid because of interactionsbetween the nucleons and nuclei. A systematic coherent description should include scatteringstates, avoid double counting of many-particle effects, and avoid semiempirical concepts suchas excluded volume. This can be achieved within a physical picture where bound statesare produced by the interaction, investigating the few-particle propagator in the medium.The chemical picture can be used as a guideline to select important contributions within aperturbative approach and performing partial summations of the corresponding Feynmandiagrams. In the case of charged particle systems, where the interaction is given by theCoulomb potential, the use of the chemical picture to introduce bound states in the partiallyionized plasma has been extensively worked out, see Ref. [1].The in-medium wave equation for a system of A nucleons is derived in Sec. II. Boundstates describe nuclei with energy eigenvalues E A,ν ( P ; T, n p , n n ) where A denotes the massnumber. The index ν specifies the internal state of the A -nucleon system such as spin, isospinand excitation. In a homogeneous system, the center-of-mass momentum ~P of the clusteris conserved and can be used as quantum number. Assuming (local) thermal equilibrium,due to the influence of the surrounding matter these energy eigenvalues will depend onthree parameter: The temperature T and the total number densities n p , n n of protons andneutrons, respectively [or the baryon density n B = n n + n p and the asymmetry parameter α = ( n n − n p ) /n B ]. We do not consider β -equilibrium due to weak interaction processes.Explicit results for the corresponding quasiparticle shifts are given for H (deuteron d ), H(triton t ), He (helion h ), and He ( α -particle) in Sec. IV. As an application, the generalizedBeth-Uhlenbeck formula is outlined in Sec. V, and the nuclear matter EoS is consideredwhich contains the quasiparticle energies of the single-nucleon states as well as of the light3uclei. II. MANY PARTICLE APPROACHA. Single-particle spectral function and quasiparticles
The basic equations of a Green function approach to the many-nucleon system can befound in different textbooks and papers, see [9, 10]. Here we give only some final results.The propagation of a single-nucleon excitation h a † ( t ) a ′ i can be expressed in terms of thesingle-particle spectral function S (1 , ω ) as h a † ( t ) a ′ i = δ , ′ Z dω π e iωt f ,Z ( ω ) S (1 , ω ) , (2)where f A,Z ( ω ) = [exp( β ( ω − Zµ p − ( A − Z ) µ n )) − ( − A ] − (3)is the Fermi or Bose distribution function which depends on the inverse temperature β =1 / ( k B T ) and the chemical potentials µ p , µ n (instead of the isospin quantum number τ weuse the charge number Z ). A special case is the single-particle density matrix h a † a ′ i whichcan be used to evaluate the equation of state (EoS) for the nucleon density n τ ( β, µ p , µ n ) = 1Ω X h a † a i δ τ,τ = 2 Z d P (2 π ) Z ∞−∞ dω π f ,Z ( ω ) S (1 , ω ) , (4)where Ω is the system volume, and summation over spin direction is collected in the factor 2.Both the Fermi distribution function and the spectral function depend on the temperatureand the chemical potentials µ p , µ n not given explicitly. We work with a grand canonicalensemble and have to use the EoS (4) to replace the chemical potentials by the densities n p , n n . For this equation of state, expressions such as the Beth-Uhlenbeck formula arediscussed below in Sec. V.The spectral function is related to the self-energy according to S (1 , ω ) = 2ImΣ(1 , ω − i ω − E (1) − ReΣ(1 , ω )] + [ImΣ(1 , ω − i , (5)where the imaginary part has to be taken for a small negative imaginary part in the fre-quency. The solution of the relation E qu1 (1) = E (1) + ReΣ[1 , E qu1 (1)] (6)4 IG. 1: Cluster decomposition of the single-nucleon self-energy. The index ’ex’ denotes full anti-symmetrization including all exchange diagrams. For bound states with A ≤ σ, τ . The Fock term results from T , ex (first order in V , exchange term,see also App. B). defines the single-nucleon quasiparticle energies E qu1 (1) = E (1) + ∆ E SE (1). Expanding forsmall ImΣ(1 , z ), the spectral function yields a δ -like contribution, and the densities arecalculated from Fermi distributions with the quasiparticle energies so that n qu τ ( β, µ p , µ n ) = 2Ω X P f ,Z [ E qu1 (1)] (7)follows for the EoS in mean field approximation. This result does not contain the contri-bution of bound states and therefore fails to be correct in the low-temperature, low-densitylimit where the NSE describes the nuclear matter EoS.As shown in Refs. [10, 11], the bound state contributions are obtained from the polesof ImΣ(1 , z ) which cannot be neglected expanding the spectral function. A cluster decom-position of the self-energy has been proposed, see Fig. 1. The diagrams are calculatedas Σ(1 , z ν ) = X A> X Ω λ , ...A G (0) A − (2 , ..., A, Ω λ − z ν )T A (1 ...A, ′ ...A ′ , Ω λ ) . (8)The free ( A −
1) quasiparticle propagator G (0) A − and the T A matrix are given in App. A. TheT A matrices are related to the A -particle Green functions which read in bilinear expansion G A (1 ...A, ′ . . . A ′ , z A ) = X νP ψ AνP (1 . . . A ) 1 z A − E qu A,ν ( P ) ψ ∗ AνP (1 ′ . . . A ′ ) . (9)The A -particle wave function ψ AνP (1 . . . A ) and the corresponding eigenvalues E qu A,ν ( P ) resultfrom solving the in-medium Schr¨odinger equation, see the following Sec. II B. Besides thebound states, the summation over the internal quantum states ν includes also the scatteringstates. 5he evaluation of the equation of state in the low-density limit is straightforward, seeApp. A. Considering only the bound-state contributions, we obtain the result n p ( T, µ p , µ n ) = 1Ω X A,ν,P Zf A,Z [ E qu A,ν ( P ; T, µ p , µ n )] ,n n ( T, µ p , µ n ) = 1Ω X A,ν,P ( A − Z ) f A,Z [ E qu A,ν ( P ; T, µ p , µ n )] (10)for the EoS describing a mixture of components (cluster quasiparticles) obeying Fermi orBose statistics. The NSE is obtained in the low-density limit if the in-medium energies E qu A,ν ( P ; T, µ p , µ n ) can be replaced by the binding energies of the isolated nuclei E (0) A,ν ( P ).Note that at low temperatures Bose-Einstein condensation may occur.We discuss the cluster decomposition of the single-nucleon self-energy, Fig. 1, in com-parison with other approximations. Restricting the cluster decomposition only to the con-tribution of two-particle correlations, we obtain the so-called T G approximation. In thisapproximation, the Beth-Uhlenbeck formula is obtained for the EoS, as shown in [10, 11].The relation between this T matrix approach and the Brueckner G matrix approach wasdiscussed in detail in Ref. [12]. Extended work has been performed using sophisticatedinteraction potentials to evaluate the quasiparticle energies in the DBHF approximation, forrecent reviews see Refs. [7, 13, 14].Replacing the T matrix in Born approximation with the interaction potential V , weobtain the Hartree-Fock approximation∆ HF (1) = X [ V (12 , − V (12 , f ,τ [ E (2) + ∆ HF (2)] . (11)In this approximation, all correlations in the medium are neglected. The self-energy doesnot depend on frequency, i.e. it is instantaneous in time, with vanishing imaginary part. B. Effective wave equation for the A -nucleon cluster We consider the propagation of an A -nucleon cluster in warm, dense matter which isdescribed by the A -particle Green function G A . The solution will not only enter the clusterdecomposition of the single-nucleon self-energy as considered in Sec. II A to calculate thecontribution of bound states to the nuclear matter EoS in a systematic way. It determinesalso the cluster decomposition of other quantities such as the polarization function, the6ynamical structure factor etc. We can proceed as above in the single-nucleon case andinvestigate higher order correlation functions. The A -nucleon spectral function S A can beintroduced which is related to G A . Cluster quasiparticle excitations are determined by δ -likepeaks in the spectral function S A .For the A -particle Green function, the perturbation expansion can be represented byFeynman diagrams. New elements such as the A -particle self-energy can be introduced, seeRef. [1, 11, 15]. In the low-density limit, the bound and scattering states of the A -particlecluster are obtained performing the partial summation of ladder diagrams.The interaction of the A -nucleon cluster with the surrounding nucleons can be consideredin the cluster-mean field approximation given in App. B. Since nuclear matter can formclusters, the interaction of the A -nucleon cluster under consideration is taken with an arbi-trary cluster B of the surrounding matter in first order of the nucleon-nucleon interaction,but full antisymmetrization of all nucleons in the cluster A and B . As a result, not onlythe Hartree shift due to the interaction with the surrounding nucleons in free as well as incluster states is obtained. Also the the Pauli blocking terms due to the occupation of thephase space by free nucleons as well as nucleons which are bound in clusters is taken intoaccount. In higher orders, the Bose-enhancement or Pauli blocking of nucleonic clusters isobtained.At present, the full self-consistent solution of the cluster-mean field approximation is outof reach. Only special cases can be solved such as α cluster nuclei [16] where an α cluster isconsidered in a surroundings consisting of a few α clusters, with full antisymmetrization onthe nucleonic level.We calculate the modification of the A -nucleon cluster due to the surroundings consid-ering only self-energy and Pauli-blocking terms in the cluster-mean field approximation.Correlations in the medium are neglected. In momentum space, the distribution of freenucleons, forming a Fermi sphere, and the bound states which are characterized by a wavefunction, are different functions, but the total amount of the occupied phase space is deter-mined by the total density of the protons or neutrons. Considering the uncorrelated medium,the phase space occupation is described by a Fermi distribution function normalized to thetotal density of nucleons, see App. B,˜ f (1) = 1exp[ E qu1 (1) /T − ˜ µ τ /T ] + 1 ≈ n τ (cid:18) π ~ mT (cid:19) / e − E qu1 (1) /T (12)7n the low-density, non-degenerate limit (˜ µ τ < µ τ isdetermined by the normalization condition 2Ω − P p ˜ f ( p ) = n τ , where τ denotes isospin(neutron or proton), and has to be expressed in terms of these densities.In ladder approximation, the A -particle Green function obeys a Bethe-Salpeter equation(BSE) G A (1 ...A, ′ . . . A ′ , z A ) = G (0) A (1 ...A, z A ) δ ′ . . . δ AA ′ + X ′′ ...A ′′ G (0) A (1 ...A, z A ) V A (1 ...A, ′′ . . . A ′′ ) G A (1 ′′ ...A ′′ , ′ . . . A ′ , z A ) , (13)where V A (1 ...A, ′ . . . A ′ ) = P i The quasiparticle energies in warm and dense nuclear matter, E qu Aν ( P ; T, n p , n n ), are welldefined functions of temperature and total proton and neutron densities, given by a peak inthe A -particle spectral function. To evaluate these quantities for infinite matter from a first-principle quantum statistical approach, we have to perform some approximations. Notethat the quasiparticle shifts can also be introduced phenomenologically fitting empiricaldata. This is well-known for the single-nucleon quasiparticle energy which can be adaptedto reproduce the structure of nuclei [6].The solution of the few-body in-medium Schr¨odinger equation (15) for separable interac-tion is simple in the case A =2. For A =3, 4, a Faddeev approach can be used, see [18, 19].To obtain explicit expressions for the quasiparticle energy shifts, we will apply perturbationtheory, which can be justified in the low-density region. Denoting the unperturbed wavefunction of the A -nucleon cluster with ϕ AνP (1 . . . A ), we have E qu Aν ( P ) ≈ X ...A, ′ ...A ′ ϕ AνP (1 . . . A ) H eff (1 . . . A, ′ . . . A ′ ) ϕ AνP (1 ′ . . . A ′ ) , (16)where the form of H eff (1 . . . A, ′ . . . A ′ ) is given by Eq. (15) after symmetrization.Before calculating the in-medium quasiparticle energy eigenvalues (16), A ≤ 4, wefirst consider the isolated A-nucleon problem to determine the unperturbed wave function ϕ AνP (1 . . . A ). Extended analysis has been carried out by Wiringa et al. using Green’s func-tion Monte Carlo calculations [4], where the AV18/UIX potential was used. Some propertiesare given in Tab. I. There is excellent agreement between theory and experimental data[20].In contrast to the second virial coefficient of the EoS given below in Sec. V, which isdetermined by on-shell properties (binding energy and scattering phase shifts of the two-nucleon problem) determined from experiments, the nucleon interaction potential and the9 ABLE I: Light cluster properties at zero densitybinding mass spin rms-radius rms-radiusenergy (charge) (point)[MeV] [MeV/ c ] [fm] [fm] n p ( H) 0 938.783 1/2 0.87 0 d ( H) -2.225 1876.12 1 2.14 1.96 t ( H) -8.482 2809.43 1/2 1.77 1.59 h ( He) -7.718 2809.41 1/2 1.97 1.76 α ( He) -28.30 3728.40 0 1.68 1.45 bound state wave function must be known to evaluate the quasiparticle energies . Thispotential is derived fitting empirical data which have to be reproduced. In previous work[10], the PARIS potential [2] has been used. Alternatively, the AV18/UIX potential [4]can be taken. Here, we will use simple separable interaction potentials, which are fittedto the binding energy and the rms radius value of the respective cluster, in order to obtainanalytical expressions for the quasiparticle shifts as function of P, T, n p , n n . Details are givenin App. C. 1. Gaussian wave function approach For the separable pair interaction in the A -nucleon cluster a Gaussian form is taken, V ν (12 , ′ ′ ) = λ ν e − ( ~p − ~p γ ν e − ( ~p ′ − ~p ′ γ ν δ p + p ,p ′ + p ′ δ σ ,σ ′ δ σ ,σ ′ δ τ ,τ ′ δ τ ,τ ′ , (17)where ν = { A, Z } := { d, t, h, α } denotes the cluster under consideration.To solve the A -particle Schr¨odinger equation, a variational approach is used. Two dif-ferent classes of functions are considered. First, a Gaussian wave function is taken whichallows for analytical expressions in evaluating the shifts. In the following Sec. III A 2 abetter, but more complex Jastrow ansatz is considered which, however, allows in generalonly for numerical evaluation of the shifts.For Gaussian wave functions, the center of mass motion can be easily separated. For10 ABLE II: Light cluster wave function parameter at zero density from the Gaussian approach ν ˜ λ ν ˜ γ ν B ν [MeV fm ] [fm − ] [fm − ] d ( H) -3677.2 0.753 0.625 t ( H) -1670.0 1.083 0.889 h ( He) -1957.5 0.960 0.804 α ( He) -1449.6 1.152 1.034 vanishing center-of-mass motion, P = 0, we have ϕ Gauss ν ( p ...p A ) = 1norm ν e − ( p + ... + p A ) /B ν δ ~p + ... + ~p A , , (18)with the normalization P p ...p A | ϕ Gauss ν ( p ...p A ) | = 1. The parameter B ν is fixed by thenucleonic point rms radius p h r i ν , given in Tab. I, B ν = 3( A − A h r i ν , (19)see App. C. Values for B ν are presented in Tab. II.Next we are interested in the parameter values ˜ λ ν , ˜ γ ν of the potential (17) which yield thebinding energy E (0) ν of the nucleus as well as the nucleonic point rms radius, given in Tab.I, using the variational ansatz (18). Calculating the kinetic energy as well as the potentialenergy for a Gaussian wave function with range parameter B ν , we have E (0) ν = 3( A − ~ m B ν + A ( A − λ ν γ ν B ν π / ( B ν + 2 γ ν ) . (20)Varying B ν , the potential parameters which have the minimum energy E (0) ν at B ν , consistentwith the rms radius value, are shown in Tab. II. 2. Jastrow wave function approach Of course, the variational solution of the separable Gauss potential (17) by using Gaus-sians is not optimal, which is clearly seen for the two-nucleon system where the exact solutionis known. A Jastrow ansatz which reproduces this exact solution for A = 2 is given by ϕ Jastrow ν ( ~p . . . ~p A ) = 1 N ν Y i The prefactor N ν is determined by the normalization condition. The results for the pa-rameter values λ ν , γ ν of the interaction potential, which reproduce the binding energies andrms values of the cluster (see Tab. I), as well as the parameter values a ν , b ν characteriz-ing the wave function (21) within the variational approach, are given in Tab. III. For thecalculations see App. C. B. Expressions for the quasiparticle shifts 1. Nucleon quasiparticles The in-medium single-nucleon dispersion relation E qu1 ( P ) can be expanded for small mo-menta P as E qu1 ( P ) = ~ m P + ∆ E SE1 ( P ) = ∆ E SE1 (0) + ~ m ∗ P + O ( P ) , (22)where the quasiparticle energies are shifted by ∆ E SE1 (0), and m ∗ denotes the effective massof neutrons ( τ = n ) or protons ( τ = p ). Both quantities, ∆ E SE1 (0) and m ∗ , are functionsof T, n p , n n characterizing the surrounding matter.Different expressions are used to parametrize the nucleon quasiparticle shift at subsatura-tion density. The Skyrme parametrization [5] of ∆ E SE τ (0), see App. C, is used in a standardapproach to the nuclear matter EoS by Lattimer and Swesty [21]. Different improvementshave been performed to optimise the calculation for various nuclei. Alternatively, relativisticmean-field (RMF) approaches have been developed starting from a model Lagrangian whichcouples the nucleons to mesons. The relativistic quasiparticle energy is given by E qu τ ( P ) = q [ m τ c − S ( n n , n p , T )] + ~ c P + V τ ( n n , n p , T ) . (23)12xpressions for S and V τ can be given [22] determining the nucleon quasiparticle shift andthe effective mass, which aim to obtain the nuclear matter EoS for supernovae collapses.Recent work on RMF parametrization [6, 7] intend to reproduce properties of nuclei, butare also in agreement with microscopic DBHF calculations.The EoS of asymmetric nuclear matter has been investigated in the low-density regionbelow the nuclear saturation density [14], and expansions of the quasiparticle shift in powersof the density have been given. The Dirac mass and the Landau mass are considered [13].A more detailed discussion of the nucleon quasiparticle approach which is very successful indescribing nuclear matter near saturation density cannot be given here. 2. Deuteron quasiparticles In the low-density limit, the shift of the deuteron binding energy can be calculated fromthe in-medium Schr¨odinger equation (15) taking the medium modifications as correction.Within perturbation theory, the shifts of the solution of this medium modified wave equationare given by the self-energy term and the Pauli blocking term, E qu d ( P ) = E (0) d + ~ m d P + ∆ E SE d ( P ) + ∆ E Pauli d ( P ) , (24) m d ≈ m is the deuteron rest mass. After separation of the center-of mass motion, thefree deuteron wave function ϕ d ( ~q ) is taken as a function of the relative momentum ~q =( ~p − ~p ) / E SE1 ( P ) = E qu1 ( P ) − E (1)in the medium part of the effective Hamiltonian (15)∆ E SE d ( P ) = 1 N d X ~q ϕ ∗ d ( ~q ) " ∆ E SE n ~P ~q ! + ∆ E SE p ~P − ~q ! ϕ d ( ~q ) . (25) N d = P ~q | ϕ d ( ~q ) | is the normalization. For a separable interaction, the Pauli blockingterm reads∆ E Pauli d ( P ) = − N d X ~q ,~q ′ ϕ ∗ d ( ~q ) " ˜ f n ~P ~q ! + ˜ f p ~P − ~q ! V ( ~q , ~q ′ ) ϕ d ( ~q ′ ) , (26)where the distribution function ˜ f ( ~p ) is given by Eq. (12). After applying the unperturbedSchr¨odinger equation, the interaction in Eq. (26) can be eliminated. With the reduced mass13 / E Pauli d ( P ) = 1 N d X ~q | ϕ d ( ~q ) | " ˜ f n ~P ~q ! + ˜ f p ~P − ~q ! ~ m q − E (0) d (cid:19) . (27)We give some more explicit expressions. Similar to the effective mass representation forthe single-nucleon quasiparticle states, we can introduce the deuteron shift and effectivemass according to E qu d ( P ) = E (0) d + ∆ E d (0) + ~ m ∗ d P + O ( P ) . (28)The self-energy contribution to the deuteron shift contains the contribution of the single-nucleon shift ∆ E rigid shift d (0) = ∆ E SE n (0) + ∆ E SE p (0) . (29)The rigid shift of the nucleons is also present in the continuum states and will not changethe binding energy of the bound states. It can be incorporated in the chemical potentials µ n , µ p , similar to the rest mass of the nucleons.If we consider the effective mass of the nucleons m ∗ τ = m τ , we get the contribution∆ E eff . mass d ( P ) = 1 N d X ~q | ϕ d ( ~q ) | ~ m (cid:18) m p m ∗ p + m n m ∗ n − (cid:19) (cid:18) q + P (cid:19) . (30)The first term contributes to the deuteron shift ∆ E SE d (0), the last term ( ∝ P ) to thedeuteron effective mass. With ∆ m d = m ∗ d − m d we find for the self-energy contribution∆ m SE d m d = − (cid:18) m p m ∗ p + m n m ∗ n − (cid:19) . (31)For the further evaluation, we need the free deuteron wave function. For the solution (21)of the Gaussian interaction (17), used in the Jastrow ansatz with parameter values givenin Tab. III, we obtain for the Pauli blocking shift (27) the expression ∆ E Pauli , Jastrow d ( P ) =( n p + n n ) δE Pauli , J d ( P ) with δE Pauli , J d ( P ) = 12 (cid:18) π ~ mT (cid:19) / Z dqq e − q /a d ( q /b d + 1) e − ~ mT “ P + q ” (32) × mT ~ P q (cid:18) e ~ Pq mT − e − ~ Pq mT (cid:19) (cid:18) ~ m q − E (0) d (cid:19) √ a d √ πb d − e b da d √ π √ bdad (1 + 4 b d a d )erfc[ √ b d a d ] ! , P = 0 δE Pauli , J d (0) = 12 (cid:18) π ~ mT (cid:19) / ~ a d m × h ~ a d mT i − / − e b da d „ ~ a d mT « √ πb d a d erfc (cid:20) √ b d a d q ~ a d mT (cid:21) − e b da d √ πa d √ b d (cid:16) b d a d (cid:17) erfc h √ b d a d i . (33)Values for the deuteron quasiparticle shift are given below in Tab. IVTo obtain the Pauli blocking contribution to the deuteron effective mass, we can expandthe Pauli blocking shift (32) for small values of P . We will not give the correspondingexpressions here. The effective mass approximation for the deuterons in warm dense matteris limited to small values of P . For arbitrary P , after averaging over the direction between ~q and ~P , we get the approximation∆ E Pauli , Jastrow d ( P ) ≈ ∆ E Pauli , Jastrow d (0) e − ~ mT P . (34)Together with Eqs. (33), (30), (29), we obtain the momentum-dependent quasiparticleenergy (24) describing the deuteron in matter.In contrast to the self-energy shift, the Pauli blocking shift depends strongly on tempera-ture by the following reasons. The binding energy per nucleon characterizes the extension ofthe wave function in momentum space. The form of the Fermi distribution function is deter-mined by the temperature T which we consider here to be of the same order as the bindingenergy per nucleon. Further, the Pauli blocking term is strongly depending on the centerof mass momentum P because of the overlap of the deuteron wave function in momentumspace with the Fermi sphere. Therefore, the bound states with high momentum P are lessmodified by the Pauli blocking effects. The evaluation of the deuteron quasiparticle energyis further improved using a better wave function, an appropriate interaction potential andavoiding perturbation expansions. Comparison with other potentials such as the YamaguchiLorentzian formfactor has been performed which give only small changes (below 10 %) tothe deuteron quasiparticle shift. 15 . Tritium and Helium in matter We now consider the clusters with A =3 ( t, h ) and A =4 ( α ). Some of the relations givenfor the deuteron case in Sec. III B 2 can be generalized to higher values of A . In the low-density limit, the shift of the cluster binding energy can be calculated from the effectiveSchr¨odinger equation (15) taking the medium modifications as correction, E qu ν ( P ) = E (0) ν + ~ m ν P + ∆ E ν ( P ) (35)with ν = { A, Z } = { t, h, α } , and m ν ≈ Am denoting the rest mass of the cluster. Withinperturbation theory, the shift ∆ E ν ( P ) = ∆ E SE ν ( P ) + ∆ E Pauli ν ( P ) consists of the self-energyand Pauli-blocking term which can be calculated from the effective wave equation (15) withthe unperturbed wave function ϕ ν ( ~p , . . . , ~p A ). After separation of the center-of mass motion, ϕ ν ( ~q , . . . , ~q A − ) is a function of the remaining Jacobian momenta ~q i , see App. C. We findfor the energy shifts due to the single-nucleon self-energy shift in the wave equation (15)∆ E SE ν ( P ) = 1 N ν X ~q i | ϕ ν ( ~q i ) | (cid:2) ∆ E SE n ( ~p ) + · · · + ∆ E SE n ( ~p A ) (cid:3) . (36)and for the Pauli blocking term∆ E Pauli ν ( P ) = − N ν X ~q i ,~q i ′ ϕ ∗ ν ( ~q i ) X i 4, andwe have to evaluate expression (37). Analytical expressions for the Pauli blocking shift willbe given in the following Sec. IV, based on the more simple Gaussian ansatz for the unper-turbed wave function of the cluster. For the Jastrow ansatz (21), results for δE Pauli , J ν (0) aregiven below in Tab. IV.As already discussed for the deuteron case, after angular averaging the Pauli blockingshift can be approximated as∆ E Pauli ν ( P ) ≈ ∆ E Pauli ν (0) e − ~ P A mT . (40)It leads to the dissolution of the bound states below saturation density, starting at theMott baryon density n ν, Mott B ( T, α ) depending on temperature T and asymmetry parameter α , where the A -nucleon bound state with P = 0 merges in the continuum of scattering states[11]. IV. RESULTS FOR THE CLUSTER QUASIPARTICLE SHIFTS AT LOW DEN-SITIES We collect some results for the quasiparticle energy E qu ν ( P ; T, n n , n p ) = E (0) ν + ~ P Am + ∆ E SE ν ( P ) + ∆ E Pauli ν ( P ) + ∆ E Coul ν ( P ) (41)for the light elements ν = { d, t, h, α } = { H , H , He , He } in warm, dense nuclear mat-ter. In addition to the single-particle self-energy shift ∆ E SE ν ( P ) and the Pauli blockingterm ∆ E Pauli ν ( P ), the quasiparticle energy shift contains also the Coulomb shift ∆ E Coul ν ( P )which will not be elaborated here. The remaining two contributions are calculated withinperturbation theory.The quasiparticle self-energy shift ∆ E SE ν ( P ) is caused by the self-energy shift ∆ E SE1 ( P )of the single-nucleon energies. For the A -nucleon system, it contributes to the bound state17nergies (nuclei) as well as to the energy of scattering states, in particular the edge of thecontinuum of scattering states, see Ref. [1]. In addition to the rigid shift, the quasiparticleself-energy shift at zero momentum ∆ E SE ν (0) = ∆ E rigid shift ν + 3( A − ~ B ν ( m − m ∗ ) / (8 m )contains the contribution of the effective nucleon masses, calculated for Gaussian wave func-tions where B ν is given by Eq. (19). Similarly, the self-energy contribution to the clustereffective mass can be calculated.The Pauli blocking contribution ∆ E Pauli ν ( P ) may be taken in the approximation (40)so that we discuss only the cluster quasiparticle shift ∆ E Pauli ν (0) here. The Hamiltoniandescribing the unperturbed cluster contains parameters for the interaction potential whichare determined such that the correct binding energy and point rms radius of the nuclei,given in Tab. I, are reproduced. Two different approximations are considered.(i) The variational function to solve the isolated cluster Schr¨odinger equation is taken asa Gaussian wave function with the parameter B ν . A Gaussian potential (17) is constructedwhich reproduces the correct values for the binding energies and the cluster point rms radii.The corresponding values for ˜ λ ν , ˜ γ ν are given in Tab. II. Now, having the interaction andthe wave function to our disposal, we can calculate the shift of the binding energies in thelow-density region using perturbation theory. Expanding with respect to the densities n n , n p ,we have ∆ E Pauli ν (0) = [(1 + α ν ) n n + (1 − α ν ) n p ] δE Pauli ν ( T ) + O ( n B ) (42)with α d = 0 , α t = 1 / , α h = − / , α α = 0. In the non-degenerate case ( µ τ < δE Pauli ν ( T ) are obtained in analytic form as δE Pauli , G ν ( T ) = − A ( A − π / (cid:18) π ~ mT (cid:19) / ˜ λ ν B ν ˜ γ ν ( B ν + 2˜ γ ν ) / ( B ν + 2˜ γ ν + ~ B ν c ν mT ( B ν + d ν ˜ γ ν )) / , (43)where c d = 2 , c t = c h = 24 , c α = 16; d d = 0 , d t = d h = 14 , d α = 10.(ii) The same procedure as in (i), only the Jastrow function (21) is taken as variationalansatz for the wave function. The Gaussian potential (17) with parameter values λ ν , γ ν , seeTab. III, reproduces the correct values for the binding energies and the cluster point rmsradii. This variational ansatz gives the exact solution (33) of the two-nucleon problem, sothat it is expected to yield better results also for the higher clusters. However, no analyticexpressions are obtained for A > 2. 18 ABLE IV: Temperature dependence of the first order Pauli blocking shift for A = 2, 3, 4. Com-parison between the Gauss (G) and the Jastrow (J) approach. T in [MeV], δE Pauli ν in [MeV fm ]. T δE Pauli , G d δE Pauli , J d δE Pauli , G t δE Pauli , J t δE Pauli , G h δE Pauli , J h δE Pauli , G α δE Pauli , J α 20 157.9 172.9 482.7 470.77 438.9 432.52 967.7 950.1615 228.1 235.4 651.7 628.95 604.0 588.70 1263.8 1237.7210 371.4 352.5 967.7 905.37 906.7 871.81 1749.1 1711.839 418.8 389.2 1037.9 986.54 998.4 957.23 1884.2 1844.908 477.1 433.3 1140.0 1081.19 1106.6 1058.08 2037.5 1996.847 550.3 487.7 1260.0 1192.82 1236.0 1178.72 2212.9 2171.866 644.4 556.2 1402.7 1326.29 1392.6 1325.31 2415.0 2375.565 768.8 645.2 1574.4 1488.50 1585.4 1506.83 2649.8 2615.754 939.0 766.2 1784.3 1689.66 1827.2 1736.97 2925.4 2903.943 1182.8 941.0 2045.4 1945.59 2137.3 2037.73 3252.1 3259.112 1553.9 1212.0 2377.0 2282.48 2546.2 2447.21 3644.7 3720.671 2169.8 1756.9 2808.9 2747.92 3104.5 3038.96 4123.0 4434.30 In Tab. IV the comparison between the Gaussian ( δE Pauli , G ν ) and the Jastrow ansatz( δE Pauli , J ν ) is shown for different temperatures. The differences are small (below 5 %) for A = 3 , A = 2 because the deuteron wave function is not well approximated by a Gaussian. However,for A = 2 we can take the Jastrow result which is given analytically in Eq. (33).The wave functions of the isolated clusters determine the quasiparticle shifts within per-turbation theory, in particular the Pauli blocking shift. The correct reproduction of therms radii of the nuclei is important in order to estimate the region in phase space, which isneeded to form the bound state. This region, however, may already be occupied by nucleonsof the medium, leading to the Pauli blocking contribution in the quasiparticle shift. Im-provements in calculating the wave functions of the isolated nuclei are possible using moresophisticated potentials such as BONN, PARIS [2] or AV18/UIX [3] and applying advancedmethods for the solution of the few-body problem such as the Faddeev-Yakubovski approach[18, 19] or the Green’s function Monte Carlo method [4]. For tritium, a comparison withcalculations of the wave function by Wiringa has been performed, and reasonable agreement19ith the Jastrow ansatz used here has been found. Further improvements of our results forthe quasiparticle shifts using more advanced approaches to the few-nucleon problem may bethe subject of future considerations. V. APPLICATION: GENERALIZED BETH-UHLENBECK EQUATION ANDCLUSTER VIRIAL EXPANSION The nuclear matter equation of state (EoS) is obtained from Eq. (4) after specifying theself-energy Σ(1 , z ). Taking into account only two-particle correlations in ladder approxima-tion (8), A = 2, we obtain the generalized Beth-Uhlenbeck formula [10, 11] n B ( T, µ p , µ n ) = n ( T, µ p , µ n ) + n ( T, µ p , µ n ) . (44)The single-quasiparticle contribution is n = n free n + n free p , where n free τ ( T, µ n , µ p ) =2(2 π ) − R d P f τ ( E qu τ ( P )) describes the free quasi-particle contributions of neutrons andprotons, see also Eq. (7). The two-particle contributions n = n bound2 + n scat2 contains thecontribution of deuteron-like quasiparticles (spin factor 3) n bound2 ( T, µ p , µ n ) = 3 Z P >P Mott d d P (2 π ) f d ( E qu d ( P ; T, µ p , µ n )) , (45)with f d ( E ) = [ e ( E − µ p − µ n ) /T − − , and scattering states of the isospin singlet and tripletchannel τ (degeneration factor γ τ ) n scat2 ( T, µ p , µ n ) = X τ γ τ Z d P (2 π ) Z ∞ dE π f τ (∆ E SE d ( P ) + E ) sin δ qu τ ( E, P ) ddE δ qu τ ( E, P ) . (46)∆ E SE d ( P ) = ∆ E SE n ( P/ 2) + ∆ E SE p ( P/ 2) is the shift of the continuum edge (self-energiesat momentum P/ δ qu τ ( E, P ) denotes the in-medium two-nucleon scattering phaseshift in the channel τ with relative energy E and center of mass momentum P . Thequasideuteron binding energy E qu d ( P ; T, µ p , µ n ) depends on temperature and nucleon den-sities of the medium, expressions are given in Sec. III B 2. The shift of the quasiparticlebinding energy increases with nuclear matter density so that the bound state may mergewith the continuum of scattering states (Mott effect). P Mott d ( T, µ p , µ n ) denotes the momen-tum P where, at given temperature and nucleon density, the binding energy of the deuteronbound state vanishes. Above the Mott density defined by P Mott d = 0, the integral over P in20he bound state contribution n bound2 (45) is restricted to the region where bound states canexist.The generalized Beth-Uhlenbeck formula (44) can be considered as a virial expansion ofthe EoS and was first applied to nuclear matter in Ref. [11]. Results using the PEST4interaction potential are shown in Ref. [10]. Due to the inclusion of medium effects suchas Pauli blocking and self-energy, a smooth transition has been obtained from the lowdensity limit, describing nuclear statistical equilibrium between nucleons and deuterons,to high densities, where the nucleon quasiparticle picture can be used. Neglecting themedium effects, the ordinary Beth-Uhlenbeck formula is recovered, where the single nucleoncontribution contains the energy E (1) = ~ P / m instead of the quasiparticle energy of freenucleons. In the correlated density n , the free deuteron binding energy and scattering phaseshifts enter the ordinary Beth-Uhlenbeck formula. Here, the term sin δ ( E ) preventing doublecounting if quasiparticle energies are taken in n , is absent, see [10]. Then, the evaluation ofthe second virial coefficient can be traced back to on-shell properties such as binding energyand scattering phase shifts. This has been performed recently by Horowitz and Schwenk [23]using directly the empirical data, instead an interaction potential which is constructed toreproduce these data. However, the ordinary Beth-Uhlenbeck formula is restricted to onlylow densities.As also shown in Ref. [11], the contribution of higher clusters to the EoS can be includedafter a cluster decomposition of the self-energy Σ(1 , z ), Eq. (8). Neglecting the contributionof the scattering states, a generalized form of the ordinary NSE is obtained, as given by Eqs.(10). The quasiparticle shift of the nuclei can be calculated in cluster-mean field approxi-mation, see Eq. (15). Results for light clusters, A ≤ 4, have been given in the previous Sec.IV. Due to the medium dependence of the quasiparticle energies, the contribution of A > A > 4. Es-timates of the quasiparticle shift due to the interaction of the heavier cluster with the sur-rounding nuclear matter have been given in Ref. [24]. For large proton number Z , Coulombeffects become important. Calculating the composition of warm dense nuclear matter, themass fraction of nucleons bound in heavy clusters is increasing with decreasing temperatureand increasing density. This limits the region in the phase space where the EoS is determinedonly by clusters with A ≤ A -nucleon cluster (15). A term which isof second order with respect to the densities has been considered in Ref. [25]. Another effectis the formation of quantum condensates due to the Bose distribution function occurringin the bound state contribution (45), but also in the scattering contribution (46) to thedensity. Degeneracy effects are included in the quantum statistical approach used in thiswork. However, the evaluation of the EoS including quantum condensates such as pairingor quartetting needs further consideration.We shortly comment the question of thermal stability in connection with the EoS (10).It is well-known that symmetric matter at zero temperature becomes instable against phaseseparation below saturation density. The conditions of thermodynamic stability are directlyrelated to the behavior of the chemical potential as a function of the densities and will not bedetailed here. Consequently, the EoS (10) describes nuclear matter in thermal equilibriumonly outside the instability region. Parameter values belonging to metastable states orunstable states may occur in nonequilibrium situations or in inhomogeneous systems whena local density approach is considered. They have also to be considered when the Maxwellconstruction is performed to determine the region of instability in the phase diagram.The liquid-gas like phase transition in nuclear matter has been considered in the EoS ofLattimer and Swesty [21] and in the EoS of Shen et al . [22]. Including Coulomb interaction,an optimum spherical density profile in a Wigner-Seitz cell approach was determined whichcan be interpreted as representative for heavy nuclei, but also for droplet formation. The α particle as representative of light clusters has been included. To mimic the density effects forthe α particle, both approaches used the concept of the excluded volume which cannot be22igorously derived in a quantum statistical approach. In contrast, Pauli blocking consideredhere defines a microscopic process for the medium effects of light clusters. VI. CONCLUSIONS In the low density limit, the nuclear statistical equilibrium (NSE) with binding energiesof the isolated nuclei is obtained from the quantum statistical approach to nuclear matter.We consider only light clusters, A ≤ 4, but heavier cluster may become more important astemperature goes down and density increases. Thus, focusing on only light elements in theEoS restricts the temperature and density parameter to the region where the mass fractionof heavy elements is small, but our approach may also be extended to heavier nuclei, seeRef. [24].Deviation from the NSE are due to medium effects, which become relevant once thebaryon density exceeds 10 − fm − . A -nucleon correlation effects are described by the A -nucleon spectral function which defines the A -cluster quasiparticles. The dependence ofthe cluster quasiparticle energy on temperature and nucleon densities is approximated byanalytical expressions, Eq. (33) for A = 2 and Eq. (43) for A = 3 , 4, combined with themomentum dependence according to Eq. (40). Compared with more accurate numericalcalculations, deviations are of the order of 5 %. Analytical expressions for the clusterquasiparticle shifts are convenient for calculating the thermodynamic properties of nuclearmatter in a large parameter range. The evaluation of the cluster-quasiparticle shifts isfurther improved considering more sophisticated potentials and wave functions as obtained,e.g., in Green’s function Monte Carlo approaches. The values given in the present work areapproximate estimations, similar as the Skyrme or RMF approaches for the single-nucleonquasiparticle case.With the shift of the quasiparticle energies, properties such as the EoS (10) can be deter-mined in the subsaturation region. It is possible to interpolate between the low-density limitwhere the NSE is valid, and the saturation density where the single nucleon quasiparticlepicture can be applied.To go beyond the quasiparticle picture, the full A -nucleon spectral function S A shouldbe explored. Instead of δ -like quasiparticle structures, S A accounts for weakly bound statesas well as scattering phase shifts including resonances consistently. The full solution of23he cluster-mean field approximation would be an important step in this direction. Theconstruction of a nuclear matter EoS remains a challenging topic not only in the high-densityregion, see [8], but also in the low-density region where the concept of nuclear quasiparticlesgiven here may be a valuable ingredient. Acknowledgments The author thanks R. B. Wiringa (Argonne) for discussions on few-nucleon propertiesand for providing data of the bound state wave function, the Compstar collaboration, in par-ticular D. Blaschke, for intensive discussions on the EoS, and the Yukawa Institute (Kyoto,YIPQS International Molecule Workshop), particularly H. Horiuchi, for hospitality whencompleting this work. APPENDIX A: T-MATRIX APPROACH TO THE SELF-ENERGY A detailed derivation of the expressions for the self-energy and the EoS can be found inthe literature, see, e.g., Refs. [1, 9, 10, 11]. We give here only some relations, using theshort notations E = E qu1 (1), E AνP = E qu A,ν ( P ).In Eq. (8), the free ( A − 1) quasiparticle propagator is G (0) A − (2 , ..., A, z ) = 1 z − E − ... − E A f ,Z (2) ...f ,Z A ( A ) f A − ,Z A − ( E + ... + E A ) . (A1)The T A matrices are related to the A -particle Green functionsT A (1 . . . A, ′ . . . A ′ , z ) = V A (1 . . . A, ′ . . . A ′ )+ V A (1 . . . A, ′′ . . . A ′′ ) G A (1 ′′ . . . A ′′ , ′′′ . . . A ′′′ , z ) V A (1 ′′′ . . . A ′′′ , ′ . . . A ′ ) (A2)with the potential V A (1 . . . A, ′ . . . A ′ ) = P i 24e can perform the Ω λ summation in Eq. (8). We obtain the result X Ω λ λ − z ν − E − ... − E A (Ω λ − E − ... − E A )( E AνP − E − ... − E A )Ω λ − E AνP = (A4) f A − ( E + ... + E A ) z ν − E z ν + E + ... + E A − E AνP − f A ( E AνP ) E + ... + E A − E AνP z ν + E + ... + E A − E AνP . Taking Im Σ(1 , z ) and integrating the δ function arising from the pole in the denominator,the leading term in density is f ( E AνP − E − ... − E A ) f A − ( E + ... + E A ) = f A ( E AνP ).Neglecting the contribution of the scattering states, we obtain the generalized form (10) ofthe NSE. APPENDIX B: THE CLUSTER-MEAN FIELD APPROXIMATION The cluster-mean field approximation [15] is inspired by the chemical picture where boundstates are considered as new species, to be treated on the same level as free particles. Weconsider the propagation of an A -particle cluster ( { A, ν, P } ) in a correlated medium. Thecorresponding A -particle cluster self-energy is treated to first order in the interaction withthe single particles as well as with the B -particle cluster states ( { B, ¯ ν, ¯ P } ) in the medium.The B -clusters in the surrounding medium are distributed according to Eq. (3). Full anti-symmetrization between both clusters A and B has to be performed, in analogy to the Fockterm in the single-nucleon case.For the A -particle problem, the effective wave equation reads[ E (1) + . . . E ( A ) − E AνP ] ψ AνP (1 . . . A ) + X ′ ...A ′ A X i The nucleon quasiparticle energy shift (22) contains the nucleon quasiparticle shift ∆ E SE τ and the effective nucleon mass m ∗ τ . We expand with respect to the baryon density,∆ E τ ( P ; T, n B , α ) = δE τ ( P ; T, α ) n B + O ( n B ) ,m ∗ τ m τ ( T, n B , α ) = 1 + δm (0) τ ( T, α ) n B + O ( n B ) . (C1)To illustrate the quasiparticle approach in the single-nucleon case, we give the SkyrmeI parametrization by Vautherin and Brink [5, 21] which represents an analytical expressionfor the quasiparticle shift of the single nucleon states,∆ E SE n (0) = t − x ) n n + t x ) n p + t (cid:0) n n n p + n p (cid:1) + (cid:18) t t (cid:19) τ n + (cid:18) t t (cid:19) τ p , (C2)26nd the effective mass m ∗ n m n = (cid:26) (cid:20)(cid:18) t t (cid:19) n n + (cid:18) t t (cid:19) n p (cid:21) m n ~ (cid:27) − , (C3)with the parameter values t = − . , t = 235 . , t = − 100 MeVfm , t = 14463 . , x = 0 . τ n denotes the neutron kinetic energy per particle.Expressions for the protons are found by interchanging n and p . For symmetric matter wefind the shift δE τ ( T, 0) = − . The temperature dependence is only weak.Recently, the EoS of asymmetric nuclear matter has been investigated in the low-densityregion below the nuclear saturation density [14]. Microscopic calculations based on theDirac-Brueckner Hartree Fock approach with realistic nucleon- nucleon potentials are used toadjust a low-density energy functional. This functional is constructed on a density expansionof the relativistic mean-field theory. An improved version of the relativistic mean fieldapproach to reproduce properties of nuclei in wide range of A and Z was worked out by Typel[6]. The relativistic quasiparticle energy (23) reads in the non-relativistic limit ∆ E SE τ (0) = − S ( n n , n p , T ) + V τ ( n n , n p , T ) and m ∗ τ /m τ = 1 − S ( n n , n p , T ) / ( m τ c ). For the low-densitylimit, expansions of the scalar and vector potentials with respect to the neutron and protondensities can be given. 2. Quasideuteron In the deuteron case, we introduce Jacobi coordinates such that q = ( p − p ) / p rel , q = p + p = P or p = − q + q / , p = q + q / 2. The kinetic energy is KE = ~ m (2 q + q ), the interaction is parametrized by the separable Gaussian (17) V ( q , q , q ′ , q ′ ) = λe − q γ e − q ′ γ δ q ,q ′ . (C4)The deuteron wave function (21) results as ϕ ( q ) ∝ e − q /a / ( q /b + 1). The value of therms radius follows asrms = 12 h [( r − R ) + ( r − R ) ] i = 14 Z d q (cid:20) ∂∂q ϕ d ( q , q ) (cid:21) . (C5)As in the case of single nucleons, the quasiparticle shifts can be expanded as power seriesof the densities, ∆ E d ( P ; T, n B , α ) = δE d ( P ; T, α ) n B + O ( n B ) (C6)27he first order term δE d consists of the self-energy contribution and the Pauli-blocking con-tribution, see Sec. III B 2, in particular δE rigid shift d = δE p (0; T, α )+ δE n (0; T, α ). Furthermorewe have m ∗ d m d ( T, n B , α ) = 1 + δm d ( T, α ) n B + O ( n B ) . (C7)Values for δE Pauli d (0; T ) are given in Tab. IV. For T =10 MeV we have δm d (10 , , whereas for T =4 MeV the value δm d (4 , results. Due to the Pauli blockingboth quantities are strongly temperature dependent. At zero temperature and low densities,we find for the Gaussian interaction δE Pauli d ( P ; 0) = − λ d ψ d ( P/ e − P γ d R d q e − q /γ d ψ d ( q ) R d q | ψ d ( q ) | . (C8) 3. Quasitriton/helion Next we consider A = 3 ( t, h ). Jacobi coordinates are q = ( p − p ) , q = ( − p − p + p ) , q = p + p + p or p = − q − q + q , p = q − q + q , p = q + q . Thekinetic energy is KE = ~ m (2 q + q + q ).We start from a Gaussian pair interaction (17) which gives in Jacobian coordinates thethree-nucleon interaction V pair3 ( q , q , q , q ′ , q ′ , q ′ ) = λδ q ,q ′ (cid:26) δ q ,q ′ e − q γ e − q ′ γ + δ q ′ ,q + q ′ − q e − ( q 1+ 32 q γ e − ( q − q q ′ γ + δ q ′ ,q − q ′ − q e − ( q − q γ e − ( q 1+ 12 q − q ′ γ (cid:27) . (C9)The Gaussian variational ansatz (18) reads after introducing Jacobians (indices in B, a, b are omitted) ϕ Gauss3 ( q , q , q ) ∝ e − q B e − q B δ q ,P . (C10)The Jastrow variational ansatz (21) motivated by the solution of the two-particle problem,reads after introduction of the reduced Jacobian coordinates ~x i = ~q i /b and choosing thecoordinates as ~x = x { (1 − z ) / , , z } , ~x = x { , , } ϕ Jastrow3 ( x , x , z ) ∝ e − b a x − b a x ( x + 1)( x + x + x x z + 1)( x + x − x x z + 1) (C11)The kinetic energy follows as KE = ~ m b N Z dx x Z dx x (cid:18) x + 34 x (cid:19) Z − dzϕ ( x , x , z ) (C12)28ith the norm N = Z dx x Z dx x Z − dzφ ( x , x , z ) . (C13)For the potential energy we obtain P E = 3 λ b π N Z dx x (cid:20)Z dx x e − x b γ Z − dzϕ ( x , x , z ) (cid:21) . (C14)The nucleonic point rms radius follows asrms = 1 b N Z dx x Z dx x Z − dz " (cid:18) ∂ϕ ∂~x (cid:19) + 29 (cid:18) ∂ϕ ∂~x (cid:19) , (C15)in particular rms = 2 /B for the Gaussian ansatz (18).At finite temperature, the Pauli blocking contribution to the quasiparticle shift is givenby δE Pauli , J3 ( P ) = 3 λ b π N (cid:18) π ~ mT (cid:19) / Z dx x (C16) × Z dx x Z dz ϕ ( x , x , z ) e − x b γ e − ~ b mT ( ~x + ~x − ~x ) Z dx x Z dz ϕ ( x , x , z ) e − x b γ . At zero temperature where p ≈ ~x = ~x / − ~x / ~x = 0), the shift is given by δE Pauli , J3 (0) = 3 λ N Z dx x e − x ( b a + b γ ) ( x + 1)( x + 1) Z dx x Z dz ϕ ( x , x , z ) e − x b γ . (C17) 4. The α -quasiparticle To solve the four-nucleon Schr¨odinger equation in the zero-density limit, we separatethe center-of-mass motion from the internal motion introducing Jacobian coordinates, q = − p + p , q = − p − p + p , q = − p − p − p + p , q = p + p + p + p .The inverse transformation is p = q − q − q − q , p = q − q − q + q , p = q − q + q , p = q + q .The Schr¨odinger equation separates with ϕ α,P (1 , , , 4) = ϕ ( ~q , ~q , ~q ) δ ~P ,~q . The kinetic29nergy is KE = ~ m (2 q + q + q + P ). The potential energy follows as V pair4 ( q , q , q , q , q ′ , q ′ , q ′ , q ′ ) = λ α δ q ,q ′ ( δ q ,q ′ δ q ,q ′ e − q γ α e − q ′ γ α + δ q ,q ′ δ q ′ ,q + ( q ′ − q ) e − ( q 1+ 32 q γ α e − ( q − q q ′ γ α + δ q ,q ′ δ q ′ ,q − ( q ′ − q ) e − ( q − q γ α e − ( q 1+ 12 q − q ′ γ α + δ q ,q ′ δ q ′ ,q + ( q − q ′ ) e − ( − q 2+ 43 q γ α e − ( − q − q q ′ γ α + δ q ′ ,q − ( q − q ′ ) δ q ′ ,q − ( q − q ′ ) e − ( q 1+ 12 q 2+ 43 q γ α e − (6 q q − q q ′ γ α + δ q ′ ,q + ( q − q ′ ) δ q ′ ,q − ( q − q ′ ) e − ( − q 1+ 12 q 2+ 43 q γ α e − ( − q q − q q ′ γ α ) . (C18)To solve the internal motion we use a variational ansatz for the wave function. TheGaussian variational ansatz (18) reads after introducing Jacobians ϕ Gauss α ( q , q , q , q ) ∝ e − q B e − q B e − q B δ q ,P . (C19)The Jastrow variational ansatz (21) ϕ Jastrow α,P ( ~p , ~p , ~p , ~p ) ∝ e − a ( p − p ) ( p − p ) b + 1 . . . e − a ( p − p ) ( p − p ) b + 1 δ ~p + ~p + ~p + ~p , ~P . (C20)reads after introduction of the reduced Jacobian momenta ~x i = ~q i /b , with ~x = x { p − z cos( φ ) , p − z sin( φ ) , z } ; ~x = x { , , } ; ~x = x { p − z , , z } , ϕ ( x , z , φ , x , x , z ) = e − b a (2 x + x + x ) ( x + 1)[( x + x + 1) − x x z ]( x + x − x x z + 1)1[( x + x + x + x x z + 1) − ( x x z + x x { z z + p − z p − z cos φ } ) ] . (C21)We evaluate the norm as N = Z ∞ dx x Z − dz Z π dφ π Z ∞ dx x Z ∞ dx x Z − dz ϕ . (C22)The kinetic energy of the internal motion is calculated fromKE = ~ b m N Z ∞ dx x Z − dz Z π dφ π Z ∞ dx x Z ∞ dx x Z − dz × (cid:18) x + 34 x + 23 x (cid:19) ϕ (C23)30he potential energy is (all six terms give the same contribution; we take the first one asrepresentative)PE = 6 λb π N Z ∞ dx x Z ∞ dx x Z − dz (cid:18)Z ∞ dx x Z − dz Z π dφ π e − b γ x ϕ (cid:19) . (C24)The integrals over φ can be performed analytically.The nucleonic point rms radius follows asrms = 1 b N Z ∞ dx x Z − dz Z π dφ π Z ∞ dx x Z ∞ dx x Z − dz " (cid:18) ∂ϕ ∂~x (cid:19) + 16 (cid:18) ∂ϕ ∂~x (cid:19) + 316 (cid:18) ∂ϕ ∂~x (cid:19) , (C25)in particular rms = 9 / B for the Gaussian ansatz (18).Calculating the rms radius for the Jastrow ansatz, the three terms give the same contri-bution so that we take the last one as representativerms = 916 b N Z ∞ dx x Z − dz Z π dφ π Z ∞ dx x Z ∞ dx x Z − dz ϕ f ( f − f ) × ( x (cid:20) b a f ( f − f ) + 329 ( f − f ) + 649 f f (cid:21) + x (cid:20) − 83 ( f − f ) + 83 f f (cid:21) + x (cid:20) − f f (cid:21) +2 x x z (cid:20) b a f ( f − f ) + 329 ( f − f ) + 649 f f (cid:21) (cid:20) − 83 ( f − f ) + 83 f f (cid:21) +2 x x ( z z + q − z q − z cos φ ) (cid:20) β f ( f − f ) + 329 ( f − f ) + 649 f f (cid:21) (cid:20) − f f (cid:21) +2 x x z (cid:20) − 83 ( f − f ) + 83 f f (cid:21) (cid:20) − f f (cid:21)(cid:27) , (C26)where we used the abbreviations f = ( x + x − x x z + 1) , f = ( x + x + x + x x z + 1) , f = ( x x z + x x { z z + p − z p − z cos φ } ).At finite densities, we take into account self-energies and Pauli blocking due to themedium. We obtain 12 terms which differ only by the isospin ( p, n ) dependence so that∆ E Pauli α ( P ) = − X , ′ ′ ϕ ∗ α,P (1 , , , f p (1) + f n (1)] V (1 , 2; 1 ′ , ′ ) ϕ α,P (1 ′ , ′ , , . (C27)First we consider P = 0. At T = 0, the Fermi distribution function is replaced by the step31unction. In the low-density limit where ~p = 0 or ~x = − ~x − ~x , we have δE Pauli , J4 (0; 0) = − λ N Z − dz Z ∞ dx x Z ∞ dx x × e − b a (2 x + x + x x z ) e − b γ ( x + x + x x z ) ( x + x + x x z + 1)[( x + x + x x z + 1) − x + x x z ) ] × x + x − x x z + 1)[( x + x + x x z + 1) − ( x + x + x x z ) ] × (cid:26)Z ∞ dx x Z − dz Z π dφ π e − b γ x ϕ ( x , x , x , z , z , φ ) (cid:27) . 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