Ecole normale supérieure (ENS) de Lyon. Institut de Physique Nucléaire d Orsay. Groupe de Physique Théorique

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1 Ecole normale supérieure (ENS) de Lyon Institut de Physique Nucléaire d Orsay Groupe de Physique Théorique Master Stage Beyond-mean-field theories and zero-range effective interactions. A way to handle the ultraviolet divergence. Author: Kassem Moghrabi Supervisor: Marcella Grasso September 7, 1

2 Abstract Phenomenological effective interactions are commonly used to derive a Hamiltonian density (density functional) in the mean-field framework. When correlations are introduced to describe a many-body system, corrections beyond the standard mean-field theory have to be explicitly evaluated. If phenomenological contact interactions (local density functionals) are employed in beyond-mean-field models, two important issues have to be addressed: 1. Defining at which level (mean field or beyond mean field) the parameters of the interaction have to be fitted and which constraints have to be included in the procedure;. Dealing with an artificial ultraviolet divergence that is generated by the zero-range of the interaction and appears in many beyond-mean-field theories. In this work, we apply to a simple case a general strategy to account for the points 1) and ). We evaluate with a simple zero-range interaction the second-order correction to the equation of state of symmetric nuclear matter. The fitting criterium is to have a reasonable equation of state with the corrected beyond-mean-field expression. In this procedure, the energy cutoff Λ is included among the parameters and the new sets of parameters can be used for beyond-mean-field calculations. The ultraviolet divergence is absorbed in the fitting criteria (reasonable equation of state in the present case). We have found that the fit quality is always reasonably good for physically meaningful values of Λ and does not deteriorate with increasing values of the cutoff (χ. in all cases). 1

3 Contents 1 Introduction 3 Second-Order Energy Correction 6.1 Skyrme effective interaction; t-t3 model and the equation of state Comments The Corrected Equation of State Comments Mathematical Formalism Kinetic energy Hamiltonian density for central and density-dependent terms First- and second- order energy Hartree-Fock Second-order energy correction Conclusions and Perspectives. 3

4 Chapter 1 Introduction In general it is not possible to solve a many-body problem exactly; one has to employ some simple approximate methods and many of them are based on the Independent Particle approximation. This approximation assumes that a particle moves inside a certain potential well (which keeps it bound to the system) independently from the other particles. This amounts to replacing a N-body problem by N one-body problems. This essential simplification of the problem is the cornerstone of mean-field theories, which are very currently employed in the nuclear many-body problem. This problem presents an additional difficulty with respect to other many-body problems. The nucleon-nucleon interaction V NN is not definitely known, although in recent years there has been much progress in its determination. In fact, there are many ways to introduce V NN. Phenomenological effective interactions are widely used within the nuclear meanfield framework. A way to introduce a phenomenological interaction is to parameterize the nuclear potential by an appropriate mathematical function. Historically, this procedure was applied with the greatest success by Sven Gösta Nilsson, who used as a potential a (deformed) harmonic oscillator potential [5]. The self-consistent Hartree Fock approach aims to deduce mathematically the nuclear potential from the nucleon-nucleon interaction, for instance, from a phenomenological effective interaction. This technique implies a resolution of the Schrodinger equation in an iterative fashion, since the potential depends there 3

5 upon the wave-functions to be determined. The parameters of the effective nucleon-nucleon interaction are obtained by fitting the Hartree-Fock mean-field results to the experimental data (binding energies and radii). In this work, we will concentrate on the non-relativistic Hartree-Fock approach to study nuclear matter. The self-consistent approaches of the Hartree-Fock type. Our starting point in the Hartree Fock approach of the N-body problem, is the many-body Hamiltonian containing the kinetic energy term, and the interaction term (-body interaction). Then, we assume that the wave-function of the system can be written as a Slater determinant, that can be determined by the energy minimization condition. This will lead to N-equations (the Hartree Fock equations) that gives the wave-functions and the individual energy levels of nucleons. To be more precise, it should be mentioned that the energy density (Hamiltonian density) is a functional of the density, defined as the sum of the individual squared wave-functions. In contrast with other domains of the many-body physics where the interaction is known, the nucleonnucleon interaction within the nucleus is not known analytically because it is very difficult to derive it from "QCD". This strong interaction is well understood in the framework of the Standard Model at high energies in vacuum, but it is much more complicated for low-energy nucleons interacting in a nuclear medium. Thus, there is no fundamental theory allowing one to deduce the nucleon-nucleon interaction from the quark-quark interaction. It has thus been very helpful to invent the concept of effective phenomenological interaction. The latter is basically a mathematical function with several arbitrary parameters, which are adjusted to agree with some experimental data. Examples of such interactions are: Gogny (finite-range with Gaussians) and Skyrme (zero-range) effective interactions. This report is dealing with: 1. The infinite-symmetric nuclear matter which is an ideal system composed of nucleons with the following properties: - Equal number of protons and neutrons (N = Z). - No Coulomb interaction between protons. 4

6 - No correlations between nucleons. - Neglecting the surface forces. - Constant density ρ at equilibrium.. Evaluating beyond-mean-field corrections of the equation of state of symmetric nuclear matter with Skyrme-type interaction. 3. Refitting the parameters of the effective interaction at a beyond-mean-field level. 4. Treating the ultraviolet divergence that exists at beyond-mean-field level due to the zero-range Skyrme interaction. 5. Least Square Root Fit to absorb the ultraviolet divergence in the fitting procedure. 5

7 Chapter Second-Order Energy Correction.1 Skyrme effective interaction; t-t3 model and the equation of state In 1956 Skyrme proposed an approximate representation of the effective nuclear force with a zero-range two-body and three-body term [1]: V = V 1 (i, j) + V 13 (i, j, k) i<j i<j<k Later the three-body part has been replaced by a density-dependent term [4]: 1 6 t 3(1 + x 3 P σ )δ( r 1 r )ρ α ( ) r1 + r ( α=1 corresponds to a 3-body force ). So our effective-phenomenological interaction becomes: V = t (1 + x P σ )δ( r 1 r ) + 1 ( ) 6 t 3(1 + x 3 P σ )δ( r 1 r )ρ α r1 + r + 1 [ ] t 1 (1 + x 1 P σ ) δ( r 1 r ) k + k δ( r 1 r ) + t (1 + x P σ ) k δ( r 1 r ) k +iw σ k δ( r 1 r ) k with the usual notations: r = r 1 r, k = 1 i ( 1 ), P σ = 1 (1 + σ 1 σ ) and σ = σ 1 + σ The ten constants t, t 1, t, t 3, x, x 1, x, x 3, α and W are adjusted to the experimental binding energies 6

8 and radii of nuclei and to have a reasonable equation of state of matter. The t term describes a pure central δ force with a spin exchange; t 1 and t terms simulate an effective range ; the W term decries a two-body spin orbit interaction and the t 3 term is the density-dependent term (necessary to obtain the saturation point). We will restrict ourselves in this report to the t t 3 terms only and we use a two-body interaction written as: V 1 = t δ( r 1 r ) + 1 ( ) 6 t 3δ( r 1 r )ρ α r1 + r ( = t + 1 ) 6 t 3ρ α ( R) δ( r) = gδ( r), where R = r 1 + r, r = r 1 r. }{{} g Let us suppose that the ground state is represented by a Slater determinant ϕ of single-particle states ϕ i : ϕ 1 (x 1 ) ϕ 1 (x )... ϕ(x 1, x, x A ) = 1 det ϕ i (x j ) = 1 A! A! ϕ (x 1 ) ϕ (x ) where x denotes the set r, σ, q of space, spin and isospin coordinates (q = + 1 for a proton, 1 for a neutron). The expectation value of the total energy is: E = i < i p m i > +1 < ij V 1 (r 1, r ) ij >= ij d 3 r H( r). For the t t 3 Skyrme model, the energy density H( r), is an algebraic function of the nucleon densities ρ p and ρ n and the kinetic energy density. The nucleon density depends on the single-particle states ϕ i : ρ q ( r) = i,σ ϕ i ( r, σ, q). In symmetric nuclear matter with Z = N (ρ p = ρ n = ρ ) and no Coulomb field, the equation of state becomes 1 : E (ρ) A ( ) = 3 h 3π 3 1m ρ t ρ t 3ρ α+1. (.1) 1 See chapter 3, sections 3.1, 3. and 3.3 for more details 7

9 One way to go beyond-mean-field is to calculate the second order correction to the energy: Figure.1: Feynman diagrams for the first-and second-order energy corrections The waved lines describe the interaction (transferred momenta); whereas the solid lines describe the Green s functions. The second order energy corrections E/A in symmetric nuclear matter is given by : E A (ρ) = π3 kf 7 ( = 3k7 F m g 4π 6 h ρ ( 8 m 1 (π) 9 h ρ ) I code (u max ) ) g I code (u max ) = χ(ρ) I code (u max ), where }{{} (.) U.V divergence χ(ρ) = 3k7 F 4π 6 m h g ρ.. Comments 1. I code (u max ) is an integral 3 which is being calculated up to u max = Λ/(k F ); u max is density-dependent because it depends on k F = cutoff Λ. ( 3π ) 1 ρ 3. The integral is done inside a density loop. As input we fix the. It should be noted that the second-order energy correction is cutoff dependent and ultraviolet divergent. 3. The factor 3/4 takes into account the sum direct + exchange. 4. To analyze how the correction behaves as a function of the cutoff Λ and the density ρ, the analytical solution of the integrals in eq. (.) has to be considered: See chapter 3, section See Section

10 Figure.: (a): Density-dependent χ coefficient. Other panels: I code (λ, ρ) for Λ =.5 (b), 1 (c) and 1.5 (d) fm 1 ; ρ is in units of fm 3. For small values of Λ (.5 fm 1 for example): The value of I code and thus the second-order energy correction is almost zero. For higher values of Λ, I code (Λ, ρ) and consequently E A = χ(ρ) I code(λ, ρ) and increases linearly with Λ. This can be viewed from the fact that the second-order energy correction is ultraviolet divergent..3 The Corrected Equation of State The beyond-mean-field equation of state of symmetric nuclear matter, can be written as follows: E A (ρ) = E E (ρ) + A A (ρ) (.3) ( ) = 3 h 3π 3 1m ρ t ρ t 3ρ α+1 3k7 F m ( t }{{}}{{} 4π 6 h ρ + t 3 36 ρα ) 3 t t 3 ρ α 1 I code (u max ) }{{} Kinetic energy 1 st order nd order correction 9

11 Taking the parameters t, t 3 and α of the Skyrme interaction "SkP"[6], t = MeV fm 3, t 3 = MeV fm 3+3α, α = 1/6, (.4) We are taking the ground state SkP as our reference because: 1. Different models provide different equations of state with different properties (saturation point, compressibility, ).. We need a reasonable equation of state. 3. We choose as a reference the SkP equation of state: a- It is a reasonable equation of state; b- SkP is a complete parameterizations (also with the t 1 t terms), but it turns out that the parameters are such that the t 1 t terms do not contribute to the equation of state of symmetric matter. Only the t t 3 terms contribute. the equation of states (.1) and (.3) are plotted in the Figure below for different values of the cutoff Λ (for the second-order correction). Upon regularizing the second-order integral loop, we introduced a cutoff Λ that can take any value. Indeed, this cutoff Λ must take finite values in such a manner that the integral can be calculated. For instance, for low-energy nuclear physics problems, Λ takes a maximum value: Λ max fm 1 because Λ must be smaller than the momentum associated with the nucleon size. Thus, all the values below fm 1 will be compatible with low-energy nuclear physics. In this report, we consider and display results also for larger values of Λ, but only for purely mathematical illustration. 1

12 Figure.3: E/A as a function of the density ρ (in units of fm 3 ) for different values of the cutoff Λ (in fm 1 ) with SkP parameters. Clearly from Fig.(.3), the energy correction increases when Λ increases; this is due to the fact that the corrected energy diverges linearly. For example, the binding energy per nucleon with SkP parameters at ρ = ρ =.16 fm 3 is Mev, whereas that for Λ = 4 fm 1, E/A = 75 Mev. So there is about 59 Mev energy difference. To solve this problem i.e treating this U.V divergence, we will refit our parameters t, t 3 and α by using a Least Square Root Fit program (Python Language). This program needs an initial guess of the parameters (we will consider the "EoS" with SkP parameters as a reference for our equation of state) and a set of points to be fitted( ρ /4, ρ /, 3ρ /4, ρ, 3ρ /). On Fig.(.4), We show the corrected equations of state with the refitted parameters for different values of Λ : 11

13 Figure.4: Refitted second-order-corrected equations of state compared with the reference "SkP"-mean-field equation of state. For unphysical huge values of Λ, starting from Λ 35 fm 1, the corrected equation of state with the refitted parameters follows very well the "SkP" equation of state except for very low densities: An example is illustrated in Fig..5. Table.1 lists the values of the refitted parameters t, t 3 and α for different values of Λ: 1

14 Table.1: Values of t, t 3 and α for # values of Λ Λ(fm 1 ) t (Mev fm 3 ) t 3 (Mev fm 3+3α ) α Figure.5: Refitted second-order-corrected equations of state compared with the reference "SkP"-mean-field equation of state for extreme cases of Λ = 35 and 4. 13

15 .4 Comments 1. The quality of the fits can be judged by the corresponding values of χ = 5 i=1 (O i E i ) E i, where: E i are the expected energies of the fitted points and O i are the observed energies given by eq. (.3). As shown by Table., the values of χ << 1, so the quality of the fit is always good for all values of Λ: Table.: Values of the fit quality χ at different values of Λ Λ =.5 Λ = 1 fm 1 Λ = fm 1 Λ = 4 fm 1 Λ = 35 fm 1 χ The behavior of I code (u max ) for large values of the cutoff Lambda, can be approximated as: I code (u max ) 1 Λ ( 11 + log ) + k ( ) F 15 9k F 45Λ 8k3 F k 5 55Λ 3 + O F Λ 5. Clearly from the above expression, the correction is linear in the cutoff, i.e, we have a linear ultraviolet divergence. 3. The second-order energy correction integral diverges because of the zero-range of the Skyrme force. The ultraviolet (UV) divergences have often unphysical effects that can be removed by regularization and renormalization. 4. Table.3 shows the values of I code (Λ, ρ) evaluated at ρ =.16 fm 3 for different values of Λ: Table.3: Λ (fm 1 ) I code (Λ, ρ) For low densities (k F ), the corrected energy can be approximated: E 5Λ (ρ) + χ(ρ) A 3k F Cte ρ g Λ 14

16 For Λ < 35 fm 1, E A (k F ), while for Λ 35 fm 1, E/A. Table.4 represents the equation of states E/A at ρ = ρ 1 for different values of Λ : Table.4: Λ (fm 1 ) E A (Mev) Thus for large values of Lambda and at low densities, there is some kind of divergence; but we will not deal with this divergence because it manifests itself at huge and unphysical values of the cutoff Λ. 15

17 Chapter 3 Mathematical Formalism In this chapter, we are going to derive the kinetic energy, Hartree-Fock energy (first-order energy) and the second-order energy of the t t 3 model. 3.1 Kinetic energy Consider a three-dimensional cubical box of volume V. The single-particle energies whose states are labeled by three quantum numbers n x, n y, and n z, are given by: E k = h π mv 3 ( n x + n y + n ) z. The number of states with energy less than E f is equal to the number of states that lie within a sphere of radius n f in the region of n-space where n x, n y, n z are positive. In the ground state this number equals the number of fermions in the system: N = πn3 f = π 3 n3 f. The factor and 1/8 are due to Pauli-exclusion principle and to the fact that n x, n y and n z are positive. Thus, the relationship between the Fermi energy and the number of particles N is given by: E F ermi = h π m 16 ( ) 3N 3. πv

18 The total energy of a Fermi sphere of N fermions per volume is given by: Consequently, E N V = E(N )dn ( ) ( ) = 3 h N 3π 3 N = 3 h 3π 3 1m V V 1m ρ ρ W = E ( ) 3 h 3π 3 (ρ) = A 1m ρ. 3. Hamiltonian density for central and density-dependent terms In all the calculations below, we will assume that the subspace of occupied states is invariant under time reversal, i.e, we assume that if a single-particle state i >is occupied, the time-reversed state is also occupied. The wave function of the time-reversed state is just: i > = iσ y i > ϕ ī ( r, σ, q) = i σ < σ σ y σ > ϕ ( r, σ, q) = σ z ϕ i ( r, σ, q). where ϕ i ( r) is a four-component spinor in the spin and isospin space. The density of occupied states, based on our assumption is equal to: ϕ ( r, σ 1, q)ϕ i ( r, σ, q) = 1 ϕ i ( r, σ 1, q)ϕ i ( r, σ, q) + 1 ϕ ī ( r, σ 1, q)ϕ ī ( r, σ, q) i i i [ ] = 1 ϕ i ( r, σ 1, q)ϕ i ( r, σ, q) + 4σ 1 zσzϕ i ( r, σ 1, q)ϕ i ( r, σ, q) i if σ 1 = σ 1 = = 1 i,σ ϕ i( r, σ, q) δ σ 1,σ ρ q ( r). if σ 1 = +σ Due to the fact that the spin Pauli matrices are traceless, we get, ϕ i ( r) σϕ i ( r) = The Hartree-Fock energy is given by: i i,σ 1,σ ϕ i ( r, σ 1, q) < σ 1 σ σ > ϕ i ( r, σ, q) = (3.1) < V (r 1, r ) > = 1 < ij gδ( r)(1 + x P σ )(1 P σ P τ ) ij > ij = 1 d 3 rδ( r) ϕ i ( r)ϕ j ( r)g ( 1 P σ P τ + x P σ x P ) σp τ ϕi ( r)ϕ j ( r) ij 17

19 Using equation (3.1) and the fact that P σ = 1 (1 + σ 1 σ ) and P τ = δ qi,q j, we get: < V (r 1, r ) > = 1 = g d 3 rδ( r) ( ϕ i ( r)ϕ j ( r)g 1 1 δ q i,q j + x ij [ ( 1 + x ) ( ) 1 (ρ ρ + x p + ρ ) ] n x δ qi,q j In nuclear symmetric matter with Z = N i.e (ρ p = ρ n = ρ/) and no Coulomb field, we get: ) ϕ i ( r)ϕ j ( r) < V (r 1, r ) >= 3 8 gρ = 3 8 t ρ t 3ρ α First- and second- order energy The first- and second- order energy correction (with the exchange diagrams) can be calculated from Feynman diagrams: Diagram dictionary for ground state energy of interacting fermion system: 18

20 3.3.1 Hartree-Fock The simplest approximation to the ground state energy is the Hartree-Fock energy which corresponds to the first-order of the perturbation theory. In terms of diagrams it may be written as: The first order energy W 1 (direct) and W x 1 (exchange) can be calculated as: W 1 = 16 1 Ω (π) 6 d 3 k1 d 3 k v( k 1, k, k 1, k ) k 1,k <k F W1 x = 4 1 Ω (π) 6 d 3 k1 d 3 k v( k 1, k, k, k 1 ) k 1,k <k F where there is a sum over spin and isospin. V = gδ(r 1 r ) v( k 1, k, k 3, k 4 ) = g Ω δ k1+ k, k 3+ k 4. The Hartree-Fock energy turns out to be: W 1 + W1 x = 1 1 Ω (π) 6 d 3 k1 d 3 g k k 1,k <k F Ω = 6 ( ) 4 16 gω (π 3 d 3 k ) k<k F = 6 ( ) A 16 gω = 3 Ω 8 gωρ Therefore the zero order plus the first order energy per nucleon is: E A (ρ) = W + W 1 + W1 x ρ = 3 h 1m (3π ρ) t ρ t 3ρ α Second-order energy correction In terms of Feynman diagrams, the second order energy correction is given by: 19

21 Using the previous table, the second order energy correction E (direct) and E x (exchange) are: E = 16 1 Ω 3 (π) 9 d 3 k1 d 3 k d 3 v (q) q C I ϵ k1 + ϵ k ϵ k1 +q ϵ k q E x = 4 1 Ω 3 (π) 9 d 3 k1 d 3 k d 3 v( q)v( k q 1 k + q) C I ϵ k1 + ϵ k ϵ k1 +q ϵ k q where C I = { k 1, k < k F, k 1 + q > k F, k } q > k F. It should be noted that the factors 16 = 4 and 4 = in the above equations stand for the sum over the spin and isospin states in the direct and exchange terms, respectively. As a result, the total energy contribution E = E + E x, is: Calculation of I(q): E = 1 Ω 3 m (π) 9 h d 3 k1 d 3 k d 3 q 16v (q) 4v(q)v(k 1 k + q) C I k 1 + k ( k1 + q) ( k q) = 6 Ω3 m ( g ) (π) 9 h d 3 k1 d 3 k d 3 1 q Ω C I q + q ( k 1 k ) = 3 ( g 4 C E d Ω) 3 q I(q), where: C I C E = 8Ω3 m (π) 9 h and I(q) = d 3 k1 d 3 1 k C I q + q ( k 1 k ). The denominator in the integrand is replaced by: 1 A = dα e αa. Then: I(q) = = C I d 3 k1 d 3 k dα e αq dα e α[ q + q ( k 1 k )] d 3 k1 e α q k1 k 1 <k F k 1 +q >k F Performing the change of variables: q q k F, k 1 k1 k F, k k k F I(q) = k 4 F dα e αq k 1 <1 k 1+q >1 For the region < q <, see the article of Euler [3]: d 3 k1 e α q k 1 d 3 k e α q k k <k F k q >k F and α kf α, I(q) becomes: k <1 k q >1 d 3 k e α q k k <1 k±q >1 d 3 k e α q k = π q = 1 (π) (α q ) 3 1 dβ dx x e α q x e αβq q β [ αq e αq + (1 + α q ) e α q (1 + α q ) e α q ( q 1) ]

22 As a result, I 1 (q) = k 4 F = k4 F q dα e αq (π) (α q ) 6 [ αq e αq + (1 + α q ) e α q (1 + α q ) e α q ( q 1) ] [ y q (π) dy e y 6 y q e q y + (1 + y) e y (1 + y) e y( q 1)] Introducing the change of variable: u = q/, we get: I 1 (u) = k 4 F (π) 3u where y = α q. [ ( u 5u3 + 3 u5 ) log(1 + u) + ( u3 3 ] u5 ) log(1 u) + 9u 3u 4 4u log For the region where q >, one can easily check using the triangular identity that: 1 < p ± q < p + q p > 1 q p >. Therefore, ( p < 1) ( p + q > 1) = ( p < 1). I (q) = k 4 F = k 4 F = k4 F q Consequently, after taking u = q/, we get: I (u) = k 4 F ( ) dα e αq d 3 k e α q k k <1 [ dα e αq (π) (α q ) 6 (α q 1) e α q + (α q + 1) e α q ] q y (π) [ dy e (y 1) e y y 6 + (y + 1) e y] where y = α q. (π) 3u [(4 u u 3 + 4u 5 ) log(u + 1) + ( 4 + u u 3 + 4u 5 ) log(u 1) +u + 4u 3 + (4u 3 8u 5 ) log(u)] Clearly u I (u) is analytic and well defined at u = 1 because lim x x log x =. Now, let us check the behavior of u I (u) near u : u I (u ) = (π) 3 k 4 F [ u + 3 ( )] 1 35u 4 + O u 6 It is obvious from the above expression that the second order energy correction diverges linearly near u. To treat this ultraviolet divergence, we introduce an energy cutoff Λ which will be one of the input parameters. The energy is cutoff dependent. We define: u max = Λ k F. 1

23 Finally, the second-order energy correction per nucleon is: E A (ρ) = 3 4 3π k3 F I code (u max ) = 1 15 = π3 k 7 F +1 C E A [ 1 du u I 1 (u) + ( 8 (π) 9 m h 1 ρ + 1 ] du u I (u) ) g I code (u max ), where : du u[( u 5u3 + 3 u5 ) log(1 + u) + ( u3 3 u5 ) log(1 u) +9u 3u 4 4u log ] + 1 Λ k F du u[(4 u u 3 + 4u 5 ) log(u + 1) + ( 4 + u u 3 + 4u 5 ) log(u 1) u + 4u 3 + (4u 3 8u 5 ) log(u)]. After long manipulations and calculations (one can check this result using mathematica), we get an exact form of the I code (u max ) without any approximations: I code (u max ) = 1 (43 46 log[]) Λ + 11Λ3 35k F 1kF 3 [ Λ 3kF [ 1 35 Λ 3kF + Λ4 48k 4 F + Λ4 48k 4 F + Λ5 84k 5 F Λ5 1kF 5 + Λ5 1k 5 F + 16 log 35 + Λ7 336k 7 F Λ7 336kF 7 ( Λ 5 + 6kF 5 ] log Λ7 168kF 7 ) ( + ΛkF ] ) log ( + ΛkF ) ( ) Λ log k F

24 Chapter 4 Conclusions and Perspectives. In this report, our effective interaction for the nucleon-nucleon interaction is the Skyrme zero-range force with t and t 3 parameters only. Our system is symmetric nuclear matter. We employ the mean-field approximation to calculate the Hartree-Fock energy (first-order). After that, we go beyond the mean-field level and calculate the second-order corrected equation of state for symmetric nuclear matter. This corrected energy diverges linearly (ultraviolet divergence) with high momenta. In order to treat this divergence, we introduce an energy cutoff Λ and do the regularization procedure for the integral loop. Consequently the corrected equation of state is cutoff dependent. We refit our parameters using a Least Square Root fit program. This program fits the corrected equation of state to the ground state EoS with SkP parameters (reference equation of state). We succeed in the fitting process uptill Λ 35 fm 1 (very high cutoff), whereas for Λ 35 fm 1 and at low densities there is a divergence that cannot be removed by the refit of the parameters. However, 35 fm 1 is an unphysical huge value of Λ. Our idea was to fit a set of parameters at a beyond-mean-field level (second-order correction) to have a reasonable equation of state. The cutoff Λ is treated as one of the parameters to get free from the ultraviolet divergence. The fit quality is always good for reasonable values of Λ (χ.). 3

25 We plan to continue this work by adding the t 1 t velocity-dependent terms and apply this general strategy to handle the ultraviolet divergence (fit at beyond-mean-field level for each value of Λ) for beyond-meanfield calculations for finite nuclei, for instance calculations where the coupling between individual degrees of freedom and the collective coordinates (particle-phonon coupling) is explicitly considered. This work is summarized in the article: K. Moghrabi, M. Grasso, G. Colò, and N. Van Giai, in preparation. 4

26 Acknowledgment I would like to thank my research advisor, Marcella Grasso, for her guidance, enthusiasm and endless encouragement over the past 4 months. I would like to express my thankfulness to the Institut de Physique Nucléaire d Orsay and especially the Groupe de Physique Théorique where I will do my thesis. 5

27 Bibliography [1] P. Ring and P. Schuck, The Nuclear Many-Body Problem, Springer-Verlag Berlin Heidelberg New York, 198. [] R.D Mattuck, A Guide to Feynman diagrams in the many-body problem, Dover, New York, 199. [3] H. Euler, Z. Physik 15, 553 (1937). [4] D. Vautherin and D. M. Brink, Phys. Rev. C 5, 66 (197). [5] arxiv:79.355v1, N. Schwierz, I. Wiedenhover, A. Volya. [6] J. Dobaczewski, H. Flocard, J. Treiner, Nucl. Phys. A 4, 13 (1984). 6