Tensor optimized antisymmetrized molecular dynamics for relativistic nuclear matter

Chinese Journal of Physics 55 (2017) 28–46

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Tensor optimized antisymmetrized molecular dynamics for relativistic nuclear matter Hiroshi Toki a,∗, Jinniu Hu b a b

Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka 567-0047, Japan School of Physics, Nankai University, Tianjin 300071, China

a r t i c l e

i n f o

Article history: Received 11 June 2016 Revised 9 October 2016 Accepted 14 October 2016 Available online 9 December 2016 Keywords: TOAMD Relativistic nuclear matter EOS

a b s t r a c t We apply a newly developed many-body theory, tensor optimized antisymmetrized molecular dynamics (TOAMD), to nuclear matter using a relativistic bare nucleon-nucleon interaction in the relativistic framework. It becomes evident that the tensor interaction plays an important role in nuclear many-body system due to the role of the pion in a strongly interacting system. We take the relativistic nuclear matter (RNM) wave function as a basic state and add tensor and short-range correlation operators in the form of pion and omega-meson correlation functions acting on the RNM wave function using the concept of TOAMD. We use the Monte Carlo (Metropolis) method based on the Gaussian integration and the second quantization method for antisymmetrization to calculate all the matrix elements of the many-body Hamiltonian. We write the whole formula of the TOAMD method for numerical calculations of the nuclear binding and saturation properties of nuclear matter using one-boson exchange potential. © 2017 Published by Elsevier B.V. on behalf of The Physical Society of the Republic of China (Taiwan).

1. Introduction It is very important to describe nuclear many-body system using the bare nucleon-nucleon (NN) interaction. The NN interaction has a strong tensor force due to the pion exchange and a strong short-range repulsion due to the quark structure of the nucleon. These two features are important to determine the structure of the deuteron made of a proton and a neutron. We are able to solve the two-body system using the Schroedinger equation. The resulting S-wave component has a strong depletion in the central region due to the repulsion and the D-wave component has large momentum components [1]. The central force is not enough to bind the deuteron and the tensor force provides the dominant attraction through the coupling of the S- and D-wave components. It is definitely necessary to develop a theoretical framework to include these features in the many-body wave function for complex nuclei. They were treated in a few-body framework for S-shell nuclei as 3 He and 4 He by explicitly minimizing the total energy of few-body variational wave functions. It was demonstrated that these two features are absolutely important for the structures of S-shell nuclei [2]. However, it is difficult to go beyond the S-shell nuclei in the few-body framework due to the difficulty to treat antisymmetrization. The Argonne group developed the Green’s Function Monte Carlo (GFMC) method to treat heavier nuclei including P-shell nuclei [3]. They were able to solve many-body systems using the NN interaction and additionally the three-body interaction

∗

Corresponding author. E-mail addresses: [email protected] (H. Toki), [email protected] (J. Hu).

http://dx.doi.org/10.1016/j.cjph.2016.12.001 0577-9073/© 2017 Published by Elsevier B.V. on behalf of The Physical Society of the Republic of China (Taiwan).

H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

29

through  excitation. They found the importance of the role of pion exchange for the binding energy of nuclei. Hence, it is very important to treat the pion exchange interaction, where the difficulty of the many-body calculation lies on the treatment of the tensor force. At the present, they are able to extend their calculations to heavy nuclei as the ground state and the Hoyle state of 12 C [4]. The difficulty to go beyond this nucleus in the GFMC method comes from the limitation of the computer resources. As an effort to reduce the demand on the computational resources, we introduced the tensor optimized shell model (TOSM) to treat the tensor correlation [5]. In addition to the shell model state, we include two-particle two-hole (2p2h) states in order to allow the tensor force to have large matrix elements. TOSM is successful to describe shell model states in light nuclei, but it does not provide enough binding energy for cluster states [6]. Following the success of TOSM, a powerful theoretical framework was constructed by combining TOSM and the antisymmetrized molecular dynamics (AMD) so as to treat the cluster structures [7]. This combined framework was named as tensor optimized antisymmetrized molecular dynamics (TOAMD), where the antisymmetrization was treated in the momentum space and all the matrix elements were calculated by the analytical gaussian integration method. As for nuclear matter, the Brueckner–Hartree–Fock (BHF) calculation of nuclear matter was performed by Brockmann and Machleidt in the relativistic framework [8]. They used the meson exchange interaction (Bonn potential) for the NN interaction, and showed that the relativistic effect had a strongly repulsive effect at higher densities and the saturation point was shifted to a reasonable density. Hence, it is important to use the relativistic kinematics for the study of nuclear matter. In the non-relativistic framework, Akmal, Pandharipande and Ravenhall studied nuclear matter using the variational chain summation (VCS) method [9]. They had to include the three-body repulsive interaction and the boost effect caused by the relativistic kinematics. Including further the three-body interaction through the  excitation, they could get reasonable equations of state (EOS) for symmetric and pure neutron matter. Recently Hu, Toki and Ogawa applied the TOSM method for the calculation of nuclear matter in the relativistic framework [10]. Using Bonn-B potential they obtained the EOS of nuclear matter at a reasonable density. However, the binding energy was about 5 MeV short from the empirical value. This amount should be attributed to the three-body interaction through the  excitation. This is the motivation of the present work to apply the TOAMD framework to nuclear matter, since the inclusion of the three-body interaction is straightforward in TOAMD. In this paper, we apply the TOAMD framework to the formulation of the relativistic nuclear matter. The basic wave function of nuclear matter is simple because the momentum is a good quantum number and the spin space is saturated. The momentum states are occupied by nucleons up to the Fermi surface. The application of the tensor and short-range correlations makes the wave function to have the realistic features. The correlation functions contain variational parameters to be fixed by the energy minimization. Section 2 provides the basic ingredient of the new formulation TOAMD for relativistic nuclear matter. Section 3 provides the calculation method of the matrix elements with short-range and tensor correlations. In Section 4, we write various useful tools to calculate the matrix elements of multi-body terms in the RNM wave function. Section 5 is devoted to the summary of this paper. We add few appendices to elucidate the details of formulations. 2. Tensor optimized antisymmetrized molecular dynamics for relativistic nuclear matter We write here the basic idea of the tensor optimized relativistic nuclear matter (TORNM) theory, which is a straightforward application of TOAMD developed for finite nuclei to nuclear matter [7]. We write all the ingredients in this section as TORNM wave function, Hamiltonian and the matrix elements of the Hamiltonian based on the RNM wave function. 2.1. Wave function of TORNM In this subsection, we introduce the TORNM wave function for nuclear matter. In the framework of the tensor optimized shell model (TOSM), it is important to prepare a basic wave function to represent a correct ground state profile with low momentum components, and add high momentum components excited by the strong tensor interaction [5]. In TOSM, the high momentum components are introduced in terms of 2p2h excitations from the low momentum components [5]. Instead of the 2p2h excitations of the TOSM, we introduce the short-range and tensor correlation operators applied to the RNM wave function in the TOAMD framework [7]:

|  = (1 + FS )(1 + FD )|RNM = (1 + FD + FS + FS FD )|RNM.

(1)

Here, |RNM describes nucleons in the Fermi sea with low momentum components and FS |RNM represents high momentum states made by the short-range correlation operator FS . FD |RNM represents high momentum states made by the tensor correlation operator FD on the basic RNM wave function. The state FS FD |RNM should represent interference terms of the short-range and tensor correlations. In principle, we can add more correlation operators to the above wave function to get energetically most favorable ground state. The correlation operators FS and FD contain variational parameters to be fixed by the energy minimization. |RNM is a RNM wave function for infinite matter:

|RNM = A Ai=1 ψ pi (ri )ξ pi (ti ),

(2)

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where pi denotes various quantum numbers as the momentum, spin and isospin. Here, A is the number of nucleons, related with the Fermi momentum pF , which defines filled states in the RNM wave function. The antisymmetrizer A makes sure that the exchange of particle coordinates among all particles have an opposite sign from the original wave function (slater determinant). The spatial spinor wave function is written in terms of a relativistic spinor wave function specified by a momentum (plane wave):



ψ pi (xi ) =



E p∗i + m∗pi 2E p∗i

 1 χsi ξti √ ei p i xi .

1

σ ·pi E p∗ +m∗p i

V

i

(3)

 is the Pauli spin operator. We take plane waves in The particle coordinate is written as xi for all nucleons i = 1, ..A. Here, σ a box V = L3 and all the spatial wave functions are discretized as pxi = 2Lπ ix with ix = 0, ±1, ±2.... The number of nucleons A in a box V is fixed by the density ρ = A/V . The effective mass m∗pi is obtained from the scalar self-energy by the method of 2 1/2 . This wave function satisfies Brockmann and Machleidt [8] and the single particle energy is expressed as E pi = (m∗2 pi + pi ) the following Dirac equation:

 + β m∗ )ψ p (r ) = E ∗ ψ p (r ).  ·∇ (−iα p

(4)

We assume here that the effective mass does not depend on the momentum [8]. γ μ is four-dimensional gamma matrix and αi = γ0 γi and β = γ0 are the Dirac matrices [11]. The vector density normalization is applied as ψ p |γ0 |ψq  = δ p,q , where † the bra-state is ψ¯ p = ψ γ0 , which is the adjoint conjugate of ψ p . Here, p is three-components momentum. p

The spin wave function is written as:

  1 0

χsi (s ) =

 

0 . 1

or

(5)

The spin wave function is simply a spin-up or spin-down spinor. As for the isospin we take pure proton and neutron states:

ξti (t ) = | proton or |neutron.

(6)

Here, |proton and |neutron are pure proton and neutron states. This means that the spin and isospin saturated wave function is the ground state of symmetric nuclear matter. As for pure neutron matter, we simply consider the proton Fermi momentum is zero. In the non-relativistic framework, we can introduce the tensor correlation operator in TOAMD, which is important for a large attractive contribution to the total energy [1]. In the relativistic framework, we should better express the correlation operator in the form of meson exchange interaction so that we are able to use the Feynman-rule to be discussed in Section 4. Instead of the tensor correlation operator we use the pion correlation operator in the form of the isovector-pseudovector interaction as:

FD (ri j ) =

1  1  fD (k )eik(ri −r j ) γ γ x kx γ5 j γ jy ky τi · τ j . 2 ( 2 m ) 2 x 5i i i= j

(7)

k

The pion exchange interaction is the source of the strong tensor interaction and we use the suffix D for the pion correlation operator. The variational function of the pion correlation operator is written as the sum of Gaussian functions in the momentum space:

f D (k ) =



Cμ e−k

μ

2

/k2μ

.

(8)

Here, Cμ and kμ are the variational parameters. In the continuum limit, the spatial correlation function is expressed by the Fourier transform:

f D (r ) = =

 μ

 μ



Cμ Cμ

d3 k −k2 /k2μ −ikr e e ( 2π )3 1

( 2π )3

(π k2μ )3/2 e−r

kμ /4

2 2

.

(9)

In this sense, we express the variational wave function of nuclear matter in the Gaussian basis, which is advantageous of having analytical integration formula. The relation of the box normalization in the many-body theory and the continuum limit is explained in Appendix A. Since we use the variational principle, we can introduce other correlation functions as the ρ meson exchange operator if necessary in the same framework. We include further the short-range correlation operator. For this we take the temporal part of the ω meson exchange:

FS (ri j ) =

1  fS (k )eik(ri −r j ) γi0 γ j0 . 2 i= j

k

(10)

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31

We may also consider the scalar correlation function. The variational function for the short-range correlation operator is written by the sum of Gaussian functions:

f S (k ) =

 μ

Cμ e−k

2

/k2μ

.

(11)

All the correlation functions in the wave function (1) contain variational parameters. They are fixed by the energy minimization of the many-body Hamiltonian:

E=

RNM|(1 + FD + FS + FS FD )† H (1 + FD + FS + FS FD )|RNM . RNM|(1 + FD + FS + FS FD )† (1 + FD + FS + FS FD )|RNM

(12)

Here, H denotes the many-body relativistic Hamiltonian, which includes the kinetic, two- and three-body interactions. The denominator is the overlap matrix element for normalization. We have to calculate all the necessary matrix elements for † † the kinetic, two- and three-body interactions using the TORNM wave function. We write FD = FD and FS = FS hereafter.

2.2. Second quantization formalism of TORNM We introduce the ground state wave function and operators in the second quantization formalism to obtain systematically the matrix elements using the Wick theorem. The many-body wave function for nuclear matter can be written as:

|RNM =  ppFi a†pi |0.

(13)

Here |0 denotes the vacuum state, and the nuclear matter wave function contains particles up to the Fermi momentum pF . We take the many-body Hamiltonian as a summation of the kinetic (Tˆ ), two-body (Vˆ ) and three-body (Uˆ ) interactions:

H = Tˆ + Vˆ + Uˆ .

(14)

Here Tˆ is the one-body kinetic energy operator:

Tˆ =

 αβ

α|γ · p + m|βa†α aβ .

(15)

Vˆ is the two-body interaction operator:

Vˆ =

1  αβ|V |γ δa†α a†β aδ aγ . 2

(16)

αβγ δ

Uˆ is the three-body interaction operator:

Uˆ =

 αβγ δη

αβγ |U |δηa†α a†β a†γ aη a aδ .

(17)

Here we use the natural unit: h ¯ = c = 1. The two-body interaction in the meson exchange model is written as [8]:

αβ|V |γ δ =



 d 3 x1

 d 3 x2

d3 ku¯ α (x1 )eikx1  uγ (x1 )V (k )u¯ β (x2 )e−ikx2  uδ (x2 ) ,

(18)

where V(k) contains the coupling constant g and the form factor of each meson exchange interaction:

 V (k ) = ±g

2

2 − M2 2 + k2

2

1 . k2 + M 2

(19)

Here, the gamma matrix operator  includes the transferred momentum and isospin operators.  represents the vertex of interaction or correlation and specifies the type of interaction or correlation. The sign ± in front of the coupling constant g2 depends on the exchanged bosons [8]. k2 denotes the three momentum square: k2 = |k|2 .

2.3. One-, two- and three-body matrix elements We write here how to calculate one-, two-, and three-body matrix elements. We want to show all the necessary ingredients for the matrix elements of the RNM wave function.

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2.3.1. Kinetic energy The matrix element of the individual kinetic energy for the RNM wave function is written as:

RNM|Tˆ |RNM =



 p|γ · p + m| p.

(20)

p

We can explicitly work out the single particle matrix element of the kinetic energy as:

RNM|Tˆ |RNM =

A  p2 + mm∗p

E p∗

p

.

(21)

This is the kinetic energy of nuclear matter, where m is the mass of the nucleon in the Hamiltonian and m∗p is the effective mass to be fixed by the scalar self-energy following the method of Brockmann and Machleidt [8]. If we were to measure the energy from the nucleon mass, we have to subtract the bare mass m from the above expression. 2.3.2. Two-body interaction We take the matrix elements of two-body interaction in the RNM wave function:

RNM|Vˆ |RNM =

1  αβ|Vˆ |γ δ0| p a p a†α a†β aδ aγ  p a†p |0 2 αβγ δ

1 = 2



C ( p1 p2 : q1 q2 ) p1 p2 |V |q1 q2 .

(22)

p1 p2 q1 q2

The antisymmetrization coefficient C ( p1 p2 : q1 q2 ) = δ p1 ,q1 δ p2 ,q2 − δ p1 ,q2 δ p2 ,q1 comes from the two body creation and annihilation operator matrix element:

0| pi a pi a†α a†β aδ aγ  p j a†p j |0 = δγ ,p1 δδ,p2 [δα,p1 δβ ,p2 − δα,p2 δβ ,p1 ].

(23)

We can write the matrix element as [7]:

RNM|Vˆ |RNM =

A 1  u¯ ( p1 )eikx  u( p1 )V (k )u¯ ( p2 )e−ikx  u( p2 ) 2p p 1

k

2

A 1  − u¯ ( p1 )eikx  u( p2 )V (k )u¯ ( p2 )e−ikx  u( p1 ) 2p p 1

2

k

A A 1 1 = u¯ ( p1 ) u( p1 )V (0 )u¯ ( p2 ) u( p2 ) − u¯ ( p1 ) u( p2 )V ( p1 − p2 )u¯ ( p2 ) u( p1 ) 2p p 2p p 1

2



 



1

2





kF kF   1 1 /p + m /p + m /p + m /p2 + m = Tr 1  Tr 2  V (0 ) − Tr 1   V ( p1 − p2 ). 2 2 E 2 E 2 2 E p1 2 E p2 p p 1 2 p p p p 1

2

1

(24)

2

We do not write here the isospin matrix elements, unless it becomes necessary. As for the gamma matrices for various mesons, we can summarize them as follows: •

Scalar meson

=1 , •

for σ providing isoscalar interaction and for δ isovector interaction. Pseudoscalar meson

= •

(25)

1 γ5 γμ kμ , 2m

(26)

for π providing isovector interaction and for η isoscalar interaction. Vector meson

 = γμ +

f σμν kν , 2m

(27)

for ω providing isoscalar interaction and for ρ isovector interaction. In the trace theorem, we can use the following properties of the gamma matrices:

γ0 γ μ γ0 = γ μ† , γ5 γ μ γ5 = −γ μ .

(28)

Once we fix the type of the interaction, we are able to obtain the trace explicitly. The matrix elements of this section after the trace is calculated are written in Appendix B.

H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

33

2.3.3. Three-body interaction We write the explicit form of the three-body interaction due to the  excitation [12]. The details are written in Appendix C and the final expression is written here:



RNM|Uˆ |RNM =

C ( p1 p2 p3 : q1 q2 q3 )

p1 p2 p3 :q1 q2 q3

μ u¯ ( p2 )k1μ G ( p )k2



 f2 f2 1 π N π NN ¯ u( p1 )γ5 k/1 τ a u(q1 ) 2 m4π k1 + m2π

k1 k2 p



i 2 1 δ − ε τ c u ( q2 ) 2 u¯ ( p3 )γ5 k/2 τ b u(q3 ) 3 ab 3 abc k2 + m2π

δ p1 ,k1 +q1 δ p2 +k1 ,p δ p +k2 ,q2 δ p3 +k2 ,q3 .

(29)

Here, the permutation term is:

C ( p1 p2 p3 : q1 q2 q3 ) =

δ p1 ,q1 δ p2 ,q2 δ p3 ,q3 − δ p1 ,q1 δ p2 ,q3 δ p3 ,q2 + δ p1 ,q2 δ p2 ,q3 δ p3 ,q1 − δ p1 ,q2 δ p2 ,q1 δ p3 ,q3 + δ p1 ,q3 δ p2 ,q1 δ p3 ,q2 − δ p1 ,q3 δ p2 ,q2 δ p3 ,q1 .

(30)

The antisymmetrization coefficient C(p1 p2 p3 : q1 q2 q3 ) comes from the three-body matrix element:

0| pi a pi a†α a†β a†γ aη a aδ  p j a†p j |0

(31)

The π N and π NN coupling constants are denoted as fπ N and fπ NN , respectively. We write the matrix elements of the three body interaction explicitly by taking explicitly the initial and final momentum states. There are six matrix elements:

RNM|Uˆ |RNM p1 p2 p3 = 0.

(32)

RNM|Uˆ |RNM p1 p3 p2 = 0.

(33)

RNM|Uˆ |RNM p2 p1 p3 = 0.

(34)

RNM|Uˆ |RNM p2 p3 p1 =

 f2 f2 1 1 π N π NN 4 2 + m2 ( p − p )2 + m2 m ( p − p ) 1 2 1 3 π π π p p p 1

 2

3

/p + m /p + m /p + m Tr 1 γ5 ( /p1 − /p2 ) 2 ( p1μ − p2μ )G ( p1 )( pμ1 − pμ3 ) 3 γ5 ( /p1 − /p3 ) 2 E p1 2 E p2 2 E p3

    i a 2 c Tr τ δab − εabc τ τ b . 3

RNM|Uˆ |RNM p3 p1 p2 =



(35)

3

 f2 f2 1 1 π N π NN 4 2 + m2 ( p − p )2 + m2 m ( p − p ) 1 3 2 3 π π π p p p 1

 2

Tr



/p + m /p1 + m /p + m γ5 ( /p1 − /p3 ) 3 γ5 ( /p2 − /p3 ) 2 ( p1μ − p3μ )G ( p1 + p2 − p3 )( pμ2 − pμ3 ) 2 E p1 2 E p3 2 E p2



τ aτ b

Tr

RNM|Uˆ |RNM p3 p2 p1 = −

3

i 2 δ − ε τc 3 ab 3 abc





.

(36)

 f2 f2 1 1 π N π NN 4 2 + m2 ( p − p )2 + m2 m ( p − p ) 1 3 1 3 π π π p p p

 1

2



3



/p + m /p + m Tr 1 γ5 ( /p1 − /p3 ) 3 γ5 ( /p1 − /p3 ) Tr τ a τ b 2 E p1 2 E p3 

/p + m Tr 2 ( p1μ − p3μ )G ( p1 + p2 − p3 )( pμ1 − pμ3 ) 2 E p2







i 2 Tr δ − ε τc . 3 ab 3 abc

(37)

We write here a comparison with other three-body interactions in the literature. In this paper, we take only the Fujita– Miyazawa (FM) three-body interaction through  excitation [12]. The main term of the Tucson-Melbourne [13] and Urbana

34

H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

Fig. 1. One interaction and one correlation operator provide the two-body term (left) and the three-body term (right). The outgoing momenta are denoted as p1 , p2 , p3 and the incoming momenta as q1 , q2 , q3 . The intermediate momenta are denoted as k1 , k2 and the internal nucleon momenta as l1 , l2 . These momenta are related each other by the momentum conservation.

[3] three-body interaction is the FM three-body interaction. They introduce further three-body ring diagrams, where the central three-body interaction is provided by the FM interaction and two nucleons among the three nucleons are connected by the pion exchange interaction. In our framework, we do not take the three-body ring diagrams, but instead we use the D-wave correlation function for the role of the additional pion exchange interaction. Additionally, they introduce the threebody repulsive interaction due to the relativistic effect, which is treated here explicitly using the relativistic framework. In the literature, there is a phenomenological approach based on the chiral effective theory, where the three-body interaction appears in the chiral expansion [14]. In the chiral effective theory, the role of the tensor force due to the pion exchange is largely suppressed by taking a small cut-off parameter, and multi-body interactions appear in the effective Hamiltonian. We do not take this chiral effective interaction, since we believe the tensor force in the two-body interaction is important for nuclear structure and introduce high momentum components in the wave function. 3. Two-body interaction with many correlation operators The many-body wave functions are written as a linear combination of the basic wave function and correlated wave functions. The Hamiltonian with the correlation operators provide complicated multi-body terms to be sandwiched by the basic wave function. We discuss here the form of matrix elements in the TOAMD formalism. We start with the matrix element of the two-body interaction and one correlation operator in the basic wave function. The forms of correlation operators are the same as the interaction:

Fˆi =

1  αβ|Fi |γ δa†α a†β aδ aγ , 2

(38)

αβγ δ

where Fi is FD or FS . The Feynman graph of this case is shown in Fig. 1. Since the two-body interaction Vˆ and the shortrange and tensor correlation operators Fˆi have the same operator structure, we write each of them as Fi and a combined operator as F1 F2 :

RNM|Fˆ1 Fˆ2 |RNM =

1 4



αβ|F1 |γ δα  β  |F2 |γ  δ  

(39)

αβγ δα  β  γ  δ 

0| pi a pi a†α a†β aδ aγ a†α a†β  aδ aγ   p j a†p j |0. The creation and annihilation terms can be rewritten as:

a†α a†β aδ aγ a†α  a†β  aδ  aγ  = a†α a†β aδ (δγ α  − a†α  aγ )a†β  aδ  aγ  = a†α a†β [δγ α  (δδβ  − a†β  aδ ) − aδ a†α  aγ a†β  ]aδ  aγ  = a†α a†β [δγ α  δδβ  − δγ α  a†β  aδ − (δδα  − a†α  aδ )(δγ β  − a†β  aγ )]aδ  aγ  = a†α a†β [δγ α  δδβ  − δδα  δγ β  − δγ α  a†β  aδ + δδα  a†β  aγ + δγ β  a†α  aδ − δδβ  a†α  aγ + a†α  a†β  aδ aγ ]aδ  aγ  .

(40)

Hence, we get:

(T wo-body term ) = (δγ α δδβ  − δδα δγ β  )a†α a†β aδ aγ  , (T hree-body term ) = a†α a†β [−δγ α a†β  aδ + δδα a†β  aγ + δγ β  a†α aδ − δδβ  a†α aγ ]aδ aγ  , (F our-body term ) = a†α a†β a†α a†β  aδ aγ aδ aγ  .

(41)

H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

35

3.1. Two-body term We can write the matrix element of a two-body term as:

RNM|Fˆ1 Fˆ2 |RNMT wo =

1 4

 αβγ δα  β  γ  δ 

αβ|F1 |γ δα  β  |F2 |γ  δ  0| pi a pi a†α a†β [δγ α δδβ  − δδα δγ β  ]aδ aγ   p j a†p j |0

1  = [ p1 p2 |F1 |l1 l2 l1 l2 |F2 | p1 p2  −  p1 p2 |F1 |l1 l2 l1 l2 |F2 | p2 p1 ]. 2

(42)

p1 p2 l1 l2

We write this result in a unified way as:



RNM|Fˆ1 Fˆ2 |RNMT wo = S

C ( p1 p2 : q1 q2 ) p1 p2 |F1 |l1 l2 l1 l2 |F2 |q1 q2 .

(43)

p1 p2 l1 l2 q1 q2

Here, S is called a symmetry factor and C(p1 p2 : q1 q2 ) is called an anti-symmetrization coefficient. For the two-body term as shown here, S = 12 and C ( p1 p2 : q1 q2 ) = δ p1 ,q1 δ p2 ,q2 − δ p1 ,q2 δ p2 ,q1 . We tabulate the configurations and the corresponding symmetry factors in Appendix D. As for the configuration, we write (12)2 to say that two F operate on the particle coordinates 1 and 2. We insert all the operators in the above matrix element:

RNM|Fˆ1 Fˆ2 |RNM =

1  f1 (k1 ) f2 (k2 )δ p1 ,l1 +k1 δl1 ,p1 +k2 δ p2 +k1 ,l2 δk2 +l2 ,p2 2p p 1



2 l1 l2 k1 k2

 



/p1 + m /l 1 + m /p + m /l 2 + m 1 2 Tr 2 1 2 2 E p1 2El1 2 E p2 2El2 1  − f (k1 ) f (k2 )δ p1 ,l1 +k1 δl1 ,p2 +k2 δ p2 +k1 ,l2 δk2 +l2 ,p1 2p p Tr



Tr

1

2 l1 l2 k1 k2



/p1 + m /l 1 + m /p + m /l 2 + m 1 2 2 1 2 . 2 E p1 2El1 2 E p2 2El2

(44)

 2 2 Here, f(k1 ) is V(k1 ) for the two-body interaction and μ Cμ e−k1 /kμ for correlation operators.  represents the vertex of the operators and correlations and contains the Dirac matrix, isospin and transfer momentum. 3.2. Three-body term We shall work out three-body terms:

RNM|Fˆ1 (12 )Fˆ2 (13 )|RNMT hree =

1 4

 αβγ δα  β  γ  δ 

αβ|F1 |γ δα  β  |F2 |γ  δ  0| pi a pi a†α a†β [−δγ α a†β  aδ

+ δδα  a†β  aγ + δγ β  a†α  aδ − δδβ  a†α  aγ ]aδ  aγ   p j a†p j |0  = C ( p1 p2 p3 : q1 q2 q3 ) p1 p2 |F1 |l1 q2 l1 p3 |F2 |q1 q3 .

(45)

p1 p2 p3 l1 q1 q2 q3

Here, the symmetry factor is S = 1 and the antisymmetrization coefficient C(p1 p2 p3 : q1 q2 q3 ) provide six permutation terms. This configuration is tabulated in Appendix D as (12: 13), where F1 operates particle 1 and 2 and F2 operates particle 1 and 3. We write all the matrix elements obtained by the six permutations:

RNM|Fˆ1 (12 )Fˆ2 (13 )|RNM p1 p2 p3 =

 

f1 (k1 ) f2 (k2 )δ p1 ,l1 +k1 δl1 ,p1 +k2 δ p2 +k1 ,p2 δ p3 +k2 ,p3

p1 p2 p3 l1 k1 k2



 

 



/p + m /p + m /l 1 + m /p + m Tr 1 1 2 Tr 2 1 Tr 3 2 . 2 E p1 2El1 2 E p2 2 E p3

RNM|Fˆ1 (12 )Fˆ2 (13 )|RNM p1 p3 p2 = −

  p1 p2 p3 l1 k1 k2



(46)

f1 (k1 ) f2 (k2 )δ p1 ,l1 +k1 δl1 ,p1 +k2 δ p2 +k1 ,p3 δ p3 +k2 ,p2

 



/p + m /p + m /l 1 + m /p + m Tr 1 1 2 Tr 2 1 3 2 . 2 E p1 2El1 2 E p2 2 E p3

(47)

36

H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

Fig. 2. One interaction and two and three correlation operators provide the two-body terms with three internal horizontal lines (left) and with four internal horizontal lines (right). The outgoing momenta are denoted as p1 , p2 and the incoming momenta as q1 , q2 . The intermediate momenta are denoted as k1 , k2 , k3 , k4 and the internal nucleon momenta as l1 , l2 , l3 , l4 , l5 , l6 . These momenta are related each other by the momentum conservation.

RNM|Fˆ1 (12 )Fˆ2 (13 )|RNM p2 p3 p1 =

 

f1 (k1 ) f2 (k2 )δ p1 ,l1 +k1 δl1 ,p2 +k2 δ p2 +k1 ,p3 δ p3 +k2 ,p1

p1 p2 p3 l1 k1 k2



Tr

RNM|Fˆ1 (12 )Fˆ2 (13 )|RNM p2 p1 p3 = −



/p + m /p1 + m /l 1 + m /p + m 1 2 2 1 3 2 . 2 E p1 2El1 2 E p2 2 E p3

  p1 p2 p3 l1 k1 k2



(48)

f1 (k1 ) f2 (k2 )δ p1 ,l1 +k1 δl1 ,p2 +k2 δ p2 +k1 ,p1 δ p3 +k2 ,p3

 



/p + m /p + m /l 1 + m /p + m Tr 1 1 2 2 1 Tr 3 2 . 2 E p1 2El1 2 E p2 2 E p3

RNM|Fˆ1 (12 )Fˆ1 (13 )|RNM p3 p1 p2 =

 

f1 (k1 ) f2 (k2 )δ p1 ,l1 +k1 δl1 ,p3 +k2 δ p2 +k1 ,p1 δ p3 +k2 ,p2

p1 p2 p3 l1 k1 k2





/p + m /p + m /l 1 + m /p + m Tr 1 1 2 3 2 2 1 . 2 E p1 2El1 2 E p3 2 E p2

RNM|Fˆ1 (12 )Fˆ1 (13 )|RNM p3 p2 p1 = −

  p1 p2 p3 l1 k1 k2



(49)

(50)

f1 (k1 ) f2 (k2 )δ p1 ,l1 +k1 δl1 ,p3 +k2 δ p2 +k1 ,p2 δ p3 +k2 ,p1

 



/p + m /p + m /l 1 + m /p + m Tr 1 1 2 3 2 Tr 2 1 . 2 E p1 2El1 2 E p3 2 E p2

(51)

3.3. Four-body term We calculate the four-body term:

RNM|Fˆ1 Fˆ2 |RNMF our = =

1 4 1 4

 αβγ δα  β  γ  δ 



αβ|F1 |γ δα  β  |F2 |γ  δ  0| pi a pi a†α a†β a†α a†β  aδ aγ aδ aγ   p j a†p j |0

[ p1 p2 |F1 | p1 p2  −  p1 p2 |F1 | p2 p1 ][ p3 p4 |F2 | p3 p4  p3 p4 |F2 | p4 p3 ].

(52)

p1 p2 p3 p4

In the second equality we have used the fact that the quantum numbers αβ have to be equal to γ δ and α  β  have to be equal to γ  δ  and separate the matrix element to two matrices. This is a term where the interaction matrix element and the correlation matrix element are disconnected. Due to the normalization to be discussed in the Feynman rule, we need not calculate the disconnected four-body term. 3.4. Two-body terms for multiple correlation operators We increase the number of correlation operators. The next one corresponds to two-body interaction with two correlation operators. We write here explicitly only a two-body term. All the other matrix elements are obtained by using the Feynman rule written in the next section. We write them as F1 F2 F3 operators and the configuration is denoted as (12)3 . Here, one of the Fi is V. The Feynman graph is shown in Fig. 2.

H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

RNM|Fˆ1 Fˆ2 Fˆ3 |RNMT wo = =



1 2

p1 p2 q1 q2

1 2p p 1





C ( p1 p2 : q1 q2 )

37

 p1 p2 |F1 |l1 l3 l1 l3 |F2 |l2 l4 l2 l4 |F3 |q1 q2 

l1 l2 l3 l4 k1 k2 k3



f1 (k1 ) f2 (k2 ) f3 (k3 ) δ p1 ,l1 +k1 δl1 ,l2 +k2 δl2 ,p1 +k3 δ p2 +k1 ,l3 δl3 +k2 ,l4 δl4 +k3 ,p2

2 l1 l2 l3 l4 k1 k2 k3

 

/ + m /l 2 + m /p1 + m 1 /p + m /l 3 + m /l 4 + m 1 1 2 3 Tr 2 1 2 3 2 E p1 2El1 2El2 2 E p2 2El3 2El4 1  − f 1 ( k1 ) f 2 ( k2 ) f 3 ( k3 ) 2p p



Tr

1

2

k1 k2 k3

δ p1 ,l1 +k1 δl1 ,l2 +k2 δl2 ,p2 +k3 δ p2 +k1 ,l3 δl3 +k2 ,l4 δl4 +k3 ,p1   /p + m /l 1 + m /l 2 + m /p + m /l 3 + m /l 4 + m Tr 1 1 2 3 2 1 2 3 . 2 E p1

2El1

2El2

2 E p2

2El3

(53)

2El4

We write explicitly the two-body term for two-body interaction V with three correlation operators, whose configurations are denoted as (12)4 , shown in Fig. 2. In this case one of the Fi is V:

RNM|Fˆ1 Fˆ2 Fˆ3 Fˆ4 |RNMT wo =

1 2





C ( p1 p2 : q1 q2 )

p1 p2 q1 q2

 p1 p2 |F1 |l1 l4 l1 l4 |F2 |l2 l5 

l1 l2 l3 l4 l5 l6 k1 k2 k3 k4

l2 l5 |F3 |l3 l6 l3 l6 |F4 |q1 q2  =

1 2p p 1



f 1 ( k1 ) f 2 ( k2 ) f 3 ( k3 ) f 4 ( k4 )

2 l1 l2 l3 l4 l5 l6 k1 k2 k3 k4

δ p1 ,l1 +k1 δl1 ,l2 +k2 δl2 ,l3 +k3 δl3 ,p1 +k4 δ p2 +k1 ,l4 δl4 +k2 ,l5 δl5 +k3 ,l6 δl6 +k4 ,p2   /p + m /l 1 + m /l 2 + m /l 3 + m Tr 1 1 2 3 4 2 E p1 2El1 2El2 2El3   /p2 + m /l 4 + m /l 5 + m /l 6 + m Tr 1 2 3 4 2 E p2  1 − 2p p 1

2El4 

2El5

2El6

f 1 ( k1 ) f 2 ( k2 ) f 3 ( k3 ) f 4 ( k4 )

2 l1 l2 l3 l4 l5 l6 k1 k2 k3 k4

δ p1 ,l1 +k1 δl1 ,l2 +k2 δl2 ,l3 +k3 δl3 ,p2 +k4 δ p2 +k1 ,l4 δl4 +k2 ,l5 δl5 +k3 ,l6 δl6 +k4 ,p1   /p + m /l 1 + m /l 2 + m /l 3 + m /p + m /l 4 + m /l 5 + m /l 6 + m Tr 1 1 2 3 4 2 1 2 3 4 . 2 E p1

2El1

2El2

2El3

2 E p2

2El4

2El5

2El6

(54) We get matrix elements in a systematic manner with the trace theorem. The trace of gamma matrices is straightforward and we write explicitly the calculated results. We are left with multi-fold momentum integration. We should continue getting many matrix elements. We do not write all of them explicitly. Rather in the next section, we write a systematic method to calculate all the necessary matrix elements using the graphical method. 4. Feynman rules for many-body terms We ought to calculate many matrix elements in the present TOAMD formulation. The calculation is straightforward, but we have to repeat many times similar calculations. Since we have written all the ingredients in a similar way to free space, we use Feynman rules for the matrix elements of many-body terms. We write here how to get all the necessary matrix elements using graphical method. 1. 2. 3. 4. 5. 6.

We write all the possible Feynman diagrams considering the following rules Omission of disconnected diagrams Avoid double counting of diagrams Antisymmetrization Feynman rule construction of matrix elements Monte Carlo multiple momentum integrations

4.1. Writing Feynman diagrams We should take care of all the necessary orders of the interaction and correlations. The order of V, FS and FD is important, although all the calculations look similar. We assign all the momenta to the lines as shown in Fig. 3.

38

H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

Fig. 3. One interaction F1 and three correlation operators F2 , F3 , F4 provide a five-body term, as an example. This diagram is categorized as (12: 13: 24: 45) in the appendix. The outgoing momenta are denoted as p1 , p2 , p3 , p4 , p5 and the incoming momenta as q1 , q2 , q3 , q4 , q5 . The intermediate momenta are denoted as k1 , k2 , k3 , k4 and the internal nucleon momenta as l1 , l2 , l3 . These momenta are related each other by the momentum conservation. The incoming momenta are fixed by all the permutations of p1 , p2 , p3 , p4 , p5 with proper signs.

4.2. Omission of disconnected diagrams We should not calculate disconnected diagrams. In addition, we calculate all the terms where the components of the Hamiltonian are connected with the correlation terms. This rule comes from the definition of the many-body energy:

E=

RMF |(1 + F )H (1 + F )|RMF  . RMF |(1 + F )(1 + F )|RMF 

(55)

In the denominator, any diagrams where the correlation operators are connected are calculated. Hence, we have to calculate only the terms where the components of the Hamiltonian are connected with the correlation terms. This omission of disconnected diagrams becomes strictly true when the total energy converges to the real vacuum. 4.3. Avoid double counting of diagrams This part arises from the Wick theorem of the creation and annihilation operators discussed for some simple cases. We have to perform exchange of the creation and annihilation operator to bring the operator in the normal order. There appear many multi-body terms. It is quite cumbersome to get all the terms without mistake. Hence, we write all the possible combinations of the interaction and several correlation operators in the Appendix D. 4.4. Antisymmetrization We have to identify the single particle states in the basic wave function p1 , p2 .. for the out going lines from the left most line. We should choose the incoming momenta q1 , q2 .. following the antisymmetrization coefficient C(p1 p2 ..: q1 q2 ..). If the number of lines is n, there are n! = n(n − 1 )..1 ways to identify the final states. 4.5. Feynman rule Once all the momenta are identified as the outer lines by p1 , p2 .., pn and the internal interaction and the correlation operators by k1 , k2 .., km , and the internal nucleon lines by l1 , l2 , .., li , we can essentially follow the standard Feynman rules to calculate matrix elements [11]. The momenta of the internal nucleon lines are fixed by the momenta pi and kj using the momentum conservation. 4.6. Momentum integration using Monte Carlo method After getting all the expressions of matrix elements, we have to perform integrations of the momenta in the matrix elements. The momentum integration is multi-dimensional and quite time consuming. The integrations of matrix elements are written in general as:

I = npi m kj



pF pi

d 3 pi ( 2π )3



∞

−∞

d3 k j −k2 /k2 e j μ j f ( { pi }, {k j } ). ( 2π )3

(56)

where f({pi }, {kj }) contains momentum conservation δ functions, and the number of which is written as l. The number of the momentum integrations is n + m − l. Here, we have taken the continuum limit and written all the summation over momenta by the integral over momenta. For this we write Appendix A to express summations in terms of integrals in the continuum limit with proper factors.

H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

39

The basic integration after consideration of the momentum δ functions is: −l d = npi m k j

=

4π 3

p3F



pF

d 3 p1 ( 2π )3

pi

n

1

( 2π )

3



∞

−∞

mj −l

d3 k j −k2 /k2 e j μj ( 2π )3

(k2μ j π )3/2 . ( 2π )3

(57)

The integral with a complicated function of momenta can be calculated by the Monte Carlo method with random numbers generated with the weight:

I = d × lj e−k j /kμ j f ({ pi }, {k j } ). 2

2

(58)

4.7. Some example using Feynman rule We prepared all the methods to calculate matrix elements necessary for the TOAMD calculation. The table given in Appendix D is important to systematically obtain the matrix elements. As an example, we choose one configuration categorized as (12: 13: 24: 45) with a two-body interaction and three correlation operators arranged as F1 , F2 , F3 , F4 from above, where the vertices of each interaction-correlation are  1 ,  2 ,  3 ,  4 , as shown in Fig. 3. Here, Fi is either V, or FS , or FD . We have to take several orderings of V, FS and FD and the symmetry factor is S = 1. We first fix all the momenta as pi for the outgoing momenta, qi for the incoming momenta, ki for the interaction-correlation momenta, and li for the intermediate nucleon momenta. We can use the Feynman rule to write down the matrix element of this diagram:



RNM|F1 F2 F3 F4 |RNM = S



C ( p1 p2 p3 p4 p5 : q1 q2 q3 q4 q5 )

p1 p2 p3 p4 p5 :q1 q2 q3 q4 q5

f ( k1 ) f ( k2 ) f ( k3 ) f ( k4 )

l1 l2 l3 :k1 k2 k3 k4

δ p1 ,l1 +k1 δl1 ,q1 +k2 δ p2 +k1 ,l2 δl2 ,q2 +k3 δ p3 +k2 ,q3 δ p4 +k3 ,l3 δl3 ,q4 +k4 δ p5 +k4 ,q4 u¯ ( p1 )1 u(l1 )u¯ (l1 )2 u(q1 )u¯ ( p2 )1 u(l2 )u¯ (l2 )3 u(q2 )u¯ ( p3 )2 u(q3 ) u¯ ( p4 )3 u(l3 )u¯ (l3 )4 u(q4 )u¯ ( p5 )4 u(q5 ).

(59)

In order to perform the spin summation leading to the trace expression, we have to specify the relation of qi to pi . We choose one of the permutation as (q1 , q2 , q3 , q4 , q5 ) = ( p2 , p1 , p3 , p5 , p4 ) as an example. This is an even permutation from the standard ordering; (p1 , p2 , p3 , p4 , p5 ). The matrix element is now expressed using the trace expression:

RNM|F1 F2 F3 F4 |RNM p2 p1 p3 p5 p4 =





f ( k1 ) f ( k2 ) f ( k3 ) f ( k4 )

p1 p2 p3 p4 p5 l1 l2 l3 :k1 k2 k3 k4

δ p1 ,l1 +k1 δl1 ,q1 +k2 δ p2 +k1 ,l2 δl2 ,q2 +k3 δ p3 +k2 ,q3 δ p4 +k3 ,l3 δl3 ,q4 +k4 δ p5 +k4 ,q4     /l 1 + m /l 2 + m /p + m /p1 + m /p2 + m Tr 1 2 1 3 Tr 3 2 2E ( p1 ) 2E ( l1 ) 2E ( p2 ) 2E ( l2 ) 2E ( p3 )   /l + m /p + m /p4 + m Tr 3 3 4 5 4 . 2E ( p4 ) 2E ( l3 ) 2E ( p5 )

(60)

We can work out this matrix element using the trace theorem with Mathematica and the resulting expressions are calculated using the Monte Carlo integration. 5. Conclusion It is important to calculate finite nuclei and nuclear matter using the bare nucleon-nucleon interaction, and three-body interaction with  excitation. Such an effort has been done by Argonne–Illinois group using the GFMC method for light nuclei and the VCS method for nuclear matter. The variational methods are extremely powerful to provide the minimum energy of nuclear system by fixing many variational parameters. These variational methods, however, used a large amount of computer resources. In order to calculate heavier nuclear systems, we ought to develop an economical method to calculate the total energy. We have been looking for a more economical formulation to perform many-body calculations and found that the tensor optimized antisymmetrized molecular dynamics (TOAMD) is suited for this purpose. The idea of the TOAMD comes from the fact that the largest attraction in nuclear system is caused by the tensor interaction, which connects a simple shell model or cluster state with low momentum components to tensor correlated state with high momentum components. The calculation of this transition matrix element can be performed in short time. This TOAMD method was formulated for finite nuclei and being applied for light nuclei [7]. In this paper, we proposed to use the TOAMD method for nuclear matter. It is well known that the relativistic framework is able to provide a three-body repulsive contribution for the stabilization of the EOS at a reasonable density. Hence, it is promising to get a good result for nuclear matter by calculating the total energy with the bare nucleon-nucleon interaction

40

H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

and further by adding the three-body interaction through  excitation. We provide all the details of calculation of nuclear matter in the TOAMD method using the relativistic wave function and relativistic nucleon-nucleon interaction. We have to calculate many matrix elements in this formulation. The calculation of each matrix element is not time consuming and further to formulate the matrix elements are also not difficult, since we can use the Feynman rule to obtain all the necessary matrix elements. We start to perform numerical calculations for nuclear matter and pure neutron matter. It should not take too long time before getting the final results, which will be published in near future. Acknowledgment We are grateful to Takayuki Myo, Hisashi Horiuchi, Kiyomi Ikeda and Tadahiro Suhara for the TOAMD collaboration and fruitful discussions. We appreciate the hospitality of the RCNP theory group of Osaka University and the Nuclear Theory group of Nankai University. We are supported by the National Natural Science Foundation of China (Grants No. 11405090) and the Fundamental Research Funds for the Central Universities. Appendix A. Discretization in infinite matter If we want to use the field theory, we can use the Feynman rule for calculations of any observables. In our work, we use the TOAMD and many-body wave function, where it is better to use discrete states normalized in a box to write the manybody wave functions. On the other hand, the box is an object introduced by hand, and the results should not depend on the box size. Hence, after working out all the matrix elements in the box, we shall make the box size infinity so that we get all the numbers per unit volume. In this paper, we write the relation of the discrete states and the states with continuum momenta in the infinite volume limit. We write only the spatial part by taking the momentum conservation and neglect spin and isospin parts. The wave function in a box with length L and volume V = L3 can be written as



ψn (x ) =

1 i 2π n x e L . V

(A.1)

 and x are vectors, where the components nx , ny , nz are integer. This state is ortho-normalized: Here, n



 = n |m

L/2 −L/2

d3 xψn∗ (x )ψm (x ) = δn ,m .

(A.2)

Now let us calculate the matrix elements of an interaction V (x1 − x2 ):

 1m  2 . n 1 n 2 |V (x1 − x2 )|m

(A.3)

We calculate this matrix element using the Fourier series expansion of this interaction:

V (x1 − x2 ) =

∞ 

ck eihk(x1 −x2 ) , 

(A.4)

k=−∞

where we have written h = 2Lπ and k is a vector of integer numbers. Then we can calculate ck as:

ck =

1 V



L/2

−L/2



d3 xV (x )e−ihkx .

(A.5)

Even V (x ) is a Gauss-function, it is a complicated expression for ck due to the finite size. For a while we just write ck without giving an explicit form. The dimension of ck is the same as the interaction V (x ). Now we calculate the above matrix element in the k space:

 1m  2 = n 1 n 2 |V (x1 − x2 )|m



1n  2 |eihk(x1 −x2 ) |m  1m  2 ck n 

k

=



 1 |eihkx |m  1 n  2 |e−ihkx |m  2 ck n 



k

=



ck δn 1 ,k+m 1 δn 2 ,−k+m 2 = cn 1 −m 1 δn 1 +n 2 ,m 1 +m 2 .

(A.6)

k

The Cronecker delta in the last expression denotes the momentum conservation in the infinite limit. The dimension of ck is energy and this matrix element has the dimension of energy. We should work out all the matrix elements in this manner and the matrix element for nuclear matter (NM) is obtained using the matter wave function:

|NM = AAm=1 ψm (xm ).

(A.7)

H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

41

We then take the matrix element of the interaction:

N M |

1 1 V (x1 − x2 )|N M = 2 2 ij

1 = 2



1n 2 : m  1m  2) C (n

1n  2 :m  1m 2 n





1n  2 |eihk(x1 −x2 ) |m  1m  2 ck n

k

1n 2 : m  1m  2 )cn 1 −m 1 δn 1 +n 2 ,m 1 +m 2 . C (n

(A.8)

1n  2 :m  1m 2 n

1n 2 : m  1m  2 ) = δn 1 ,m Since C (n  1 δn  2 ,m  2 − δn  1 ,m  2 δn  2 ,m  1 , we get the Hartree–Fock matrix element:

1 1 V (x1 − x2 )|N M = [c0 − cn 1 −n2 ]. 2 2 

N M |

(A.9)

n1 n2

ij

Now we should know how to calculate the full energy of infinite matter. We should obtain energy per volume: A 

→

n

 A 1 → V n

pF

d3 p . ( 2π )3

(A.10)

In the last step we have used the relation between the momentum px and the node quantum number nx as px = 2Lπ nx . Since L3 /V = 1, we can take the infinite limit. We have to go to the infinite volume limit for ck also. For this we need a detailed explanation. We have used the Fourier series expansion:

V (x ) =

∞ 



ck eihkx ,

(A.11)

k=−∞

where we have written h = 2Lπ . Then we can calculate ck as

ck =

1 V



L/2



d3 xV (x )e−ihkx .

−L/2

(A.12)

We want to bring h → 0 so that L → ∞. For this purpose, we insert the second Eq. (A.12) to the first one (A.11):

V (x ) =



∞  k=−∞

Writing

 π /h

−π /h

V (x ) =

h3 ( 2π )3

 π /h −π /h



d xV (x )e 3

−ihkx



eihkx .

(A.13)



d3 xV (x )e−ihkx = F (hk ), we can obtain:



∞  k=−∞



h3 1  F (hk ) eihkx = ( 2π )3 ( 2π )3

 p

 )ei px , d3 pF ( p

(A.14)

 = hk. Here, we can take the limit h → 0 and get: where p

V (x ) = ) = F (p



1

( 2π )3



∞

−∞

 )ei px , d3 pF ( p

d3 xV (x )e−i px .

(A.15)

This is the familiar expression of the Fourier transformation. Now let us see the dimension. In this limiting procedure, we introduced some dimensions: the dimension of V (x ) = [E], and therefore F ( p) = [L3 ][E]. Since p = [E] = [L−1 ], we get back the dimension of V (x ) = [E]. We should rewrite the nuclear matter energy:

 1 1 1 N M | V ( x1 − x2 )|N M  → V 2 2 ij

kF

d 3 p1 ( 2π )3



kF

d 3 p2 [ F ( 0 ) − F ( p 1 − p 2 )] . ( 2π )3

(A.16)

We get an additional 1/V in the replacement of ck by V1 F (k ). In this way we get the dimension [L−3 ][E] for the matrix element. Interesting is the case of one interaction and one correlation. We introduce a two-body correlation C and take a matrix element.

C (x ) =





Ck eihkx .

k

We assume here that C is dimensionless. Now we calculate

(A.17)

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H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

NM|

1 1 Vi j (xi − x j )C (xi − x j )|NM = 2 2 ij

1 = 2

 2 n 2 |e−ihk1 x2 e−ihk2 x2 |m 





1n 2 : m  1m  2) C (n

1n  2 :m  1m 2 n





 1 |eihk1 x1 eihk2 x1 |m  1 ck1 Ck2 n 



k1k2

1n 2 : m  1m  2) C (n

1n  2 :m  1m 2 n



ck1 Cn 1 −m 1 −k1 .

(A.18)

k1

The limiting expression is

NM|

1 Vi j (xi − x j )C (xi − x j )|NM 2 ij

1 2

→



pF



d 3 p1 ( 2π )2

pF

d 3 p2 ( 2π )2



∞

d 3 k1  2 − k1 )]. 1 − p [F (k1 )G(−k1 ) − F (k1 )G( p ( 2π )2

Another interesting case is the three-body term:



N M |



Vi j (xi − x j )C (x j − xk )|N M =

C ( n1 n2 n3 : m1 m2 m3 )

n1 n2 n3 :m1 m2 m3

i jk

n 2 |e =

e

 1 |eihk1 x1 |m  1 ck1 Ck2 n 

k1 k2

 3 |m C (n1 n2 n3 : m1 m2 m3 )cn 1 −m 1 C−n3 +m 3 .

−ihk1 x2 ihk2 x2





(A.19)

 2 n  3 |e |m

−ihk2 x2

(A.20)

n1 n2 n3 :m1 m2 m3

The limiting expression is

NM|

1 Vi j (xi − x j )C jk (x j − xk )|NM 2 i jk



kF

→

d 3 p1 ( 2π )3



kF

d 3 p2 ( 2π )3



kF

d 3 p3 C ( p1 p2 p3 : q1 q2 q3 )F ( p1 − q1 )G(−p3 + q3 ). ( 2π )3

(A.21)

Appendix B. Matrix elements after trace calculations In Section 2, we have introduced general expressions for two-body matrix elements without working out the trace of gamma matrices. To write explicitly we have to specify the type of interaction. Scalar meson exchange

1 2

RNM|Vσ |RNM = − Vσ (0 )ρS2 +

1 2



kF 0

1 · p  2 + m2 d 3 p1 d 3 p2 E p1 E p2 − p Vσ ( p1 − p2 ) , E p1 E p2 ( 2π )6

(B.1)

Here, the scalar density ρ S is:

ρS =



kF 0

d 3 p1 m . ( 2 π )3 E p1

(B.2)

Omega meson exchange

1 2

RNM|Vω |RNM = Vω (0 )ρ 2 −

1 2



kF 0

 2 + 4m2 1 · p d3 p1 d3 p2 −2E p1 E p2 + 2 p Vω ( p1 − p2 ) , E p1 E p2 ( 2π )6

(B.3)

Here, the vector density ρ is:

ρ=



kF 0

d 3 p1

( 2π )3

.

(B.4)

B0.3. Rho meson exchange with vector coupling

1 2

RNM|Vρv |RNM = Vρ (0 )ρ32 −

1 2

 0

kF

 2 + 4m2 1 · p d3 p1 d3 p2 −2E p1 E p2 + 2 p Vρ ( p1 − p2 ). E p1 E p2 ( 2π )6

(B.5)

Here, the third component of vector density ρ 3 is:

ρ3 = ρn − ρ p .

(B.6)

where ρ n and ρ p are the neutron and proton density, respectively. This is the case where we consider the isospin part.

H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

43

Fig. C.1. Three-body diagram through  excitation.

Rho meson exchange with tensor coupling

RNM|VρT |RNM = −



1 2

kF 0

p  + ( E p1 E p2 − p 1 · p  2 + 3m2 )q 1 · q 2 · q 2 T d 3 p1 d 3 p2 4 p Vρ ( p1 − p2 ). E p1 E p2 ( 2π )6

(B.7)

Rho meson exchange with vector-tensor coupling

RNM|VρV T |RNM = −

1 2



kF

0

−p 2 · q ) V T 1 · q d 3 p1 d 3 p2 6m ( p Vρ ( p1 − p2 ). 6 E p1 E p2 ( 2π )

(B.8)

Pion exchange with pseudovector coupling

1 2

RNM|Vπ |RNM =



kF

0

p  + ( E p1 E p2 − p 1 · p  2 + m2 )q 1 · q 2 · q 2 d 3 p1 d 3 p2 2 p Vπ ( p1 − p2 ). 6 E p1 E p2 ( 2π )

(B.9)

=p 1 − p 2. where q As for isovector mesons, we also should consider the isospin matrix element summation for the Fock term:

Tr {τ1 · τ2 } = 2 − δτ1 ,τ2 .

(B.10)

Appendix C. Three-body interaction through  excitation We have to include the three-body interaction through  excitation like Fig. C.1. The  state has a spin 3/2 and isospin 3/2. It strongly couples with the nucleon and the pion :

f π N ¯ μ f ψ μν T ψ · ∂ ν φ π + π N ψ¯ μν T † ψ μ · ∂ ν φ π , mπ mπ

Lπ N  =

(C.1)

where the  wave function is written as:

ψ μ ( p, s ) =

 1 λs

3 s1λ| s eμ ( p, λ )ψ ( p, s ). 2 2

(C.2)

where eμ (p, λ) is the basis vector. μν is written as:

μν = gμν + xγμ γν ,

(C.3)

with x the off-shell parameter. The Rarita-Schwinger  isobar propagator is:



μν

G ( p ) = −



1 1 2 pμ pν 1 γ μ pν − γ ν pμ gμν − γ μ γ ν − − . 2 3 3 M 3 M /p − M

(C.4)

Together with the π NN vertex:

Lπ NN =

fπ NN ¯ ψ γ5 γμ τ ψ · ∂ μ φ π . mπ

(C.5)

and the pion propagator:

Gπ ( p ) =

1 . p2 − m2π

(C.6)

we are able to calculate the three-body interaction through  excitation. We calculate the three-body interaction using the π N coupling and the  propagator as:

44

H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

f2 f2 1 U ( p1 p2 p3 : q1 q2 q3 ) = π N4 u¯ ( p1 )γ5 k/1 τ a u(q1 ) 2 mπ k1 − m2π 

μμ u¯ ( p2 )μν kν1 T a G ( p )μ ν  kν2 T b† u(q2 ) 

1 u¯ ( p3 )γ5 k/2 τ b u(q3 ) k22 − m2π

(2π )16 δ ( p1 − k1 − q1 )δ ( p2 + k1 − p )δ ( p − k2 − q2 )δ ( p3 + k2 − q3 ).

(C.7)

The isospin part is written as:



T a |t t |T b† =

t

i 2 δ − ε τ c. 3 ab 3 abc

(C.8)

We take no μν dependence in the  factor, and obtain:

f2 f2 1 U ( p1 p2 p3 : q1 q2 q3 ) = π N 4 π NN u¯ ( p1 )γ5 k/1 τ a u(q1 ) 2 mπ k1 − m2π μ u¯ ( p2 )k1μ G ( p )k2 (

i 2 1 δ − ε τ c )u ( q2 ) 2 u¯ ( p3 )γ5 k/2 τ b u(q3 ) 3 ab 3 abc k2 − m2π

(2π )16 δ ( p1 − k1 − q1 )δ ( p2 + k1 − p )δ ( p − k2 − q2 )δ ( p3 + k2 − q3 ).

(C.9)

Appendix D. All the possible combinations of matrix elements We have to take all the possible matrix elements of many-body terms caused by the correlation operators and interaction. It is very cumbersome to work out possible matrix elements using the Wick theorem. Hence, we write the final results for all the configurations of operators. Here we write Fi for the correlation operators FS or FD and interactions V. As for the symmetry factor S, it is S = 12 for two-body terms and S = 1 for higher-body terms. We have to take all the permutation terms following the antisymmetrization factor C({pi }: {qi }) with proper signs ± for each configuration (ab: cd: ef..), where a, b etc. are particle numbers. For example, when we write a configuration ((12)2 : 13), it means there are three correlations F1 , F2 , F3 , one of which might be interaction, and F1 and F2 act on particles 1 and 2, and F3 acts on particles 1 and 3. Since it is a three-body term, the symmetry factor S is 1. In this tabulation, we write also an intermediate step to come to final configurations so that we should not skip any configurations or not make any double counting. For example, F1 F2 (12 )2 + F3 means F1 and F2 act on particles 1 and 2, and a correlation F3 is added to derive the final configurations of (12)3 and ((12)2 : 13). Other cases are similar to this procedure.

AMD|Fˆ1 |AMD F1 (12): (12)

AMD|Fˆ1 Fˆ2 |AMD F1 (12 ) + F2 : (12)2 ; (12: 13) AMD|Fˆ1 Fˆ2 Fˆ3 |AMD F1 F2 (12 )2 + F3 : (12)3 ; ((12)2 : 13) F1 F2 (12 : 13 ) + F3 : (12: 13: 12); (12: (13)2 ); (12: 13: 23); (12: 13: 14); (12: 13: 24); (12: 13: 34)

AMD|Fˆ1 Fˆ2 Fˆ3 Fˆ4 |AMD F1 F2 F3 (12 )3 + F4 : ((12)4 ); ((12)3 : 13) F1 F2 F3 (12 )2 : 13 + F4 : ((12)2 : 13: 12); ((12)2 : (13)2 ); ((12)2 : 13: 23); ((12)2 : 13: 14); ((12)2 : 13: 24); ((12)2 : 13: 34) F1 F2 F3 (12 : 13 : 12 ) + F4: (12: 13: 12: 12); (12: 13: 12: 13); (12: 13: 12: 23); (12: 13: 12: 14); (12: 13: 12: 24); (12: 13: 12: 34) F1 F2 F3 (12 : (13 )2 ) + F4 : (12: (13)2 : 12); (12: (13)2 : 13); (12: (13)2 : 23); (12: (13)2 : 14); (12: (13)2 : 24); (12: (13)2 : 34) F1 F2 F3 (12 : 13 : 23 ) + F4: (12: 13: 23: 12); (12: 13: 23: 13); (12: 13: 23: 23); (12: 13: 23: 14); (12: 13: 23: 24); (12: 13: 23: 34) F1 F2 F3 (12 : 13 : 14 ) + F4: (12: 13: 14: 12); (12: 13: 14: 13); (12: 13: 14: 14); (12: 13: 14: 23); (12: 13: 14: 24); (12: 13: 14: 34); (12: 13: 14: 15); (12: 13: 14: 25); (12: 13: 14: 35); (12: 13: 14: 45) F1 F2 F3 (12 : 13 : 24 ) + F4: (12: 13: 24: 12); (12: 13: 24: 13); (12: 13: 24: 14); (12: 13: 24: 23); (12: 13: 24: 24); (12: 13: 24: 34); (12: 13: 24: 15); (12: 13: 24: 25); (12: 13: 24: 35); (12: 13: 24: 45)

H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

45

F1 F2 F3 (12 : 13 : 34 ) + F4: (12: 13: 34: 12); (12: 13: 34: 13); (12: 13: 34: 14); (12: 13: 34: 23); (12: 13: 34: 24); (12: 13: 34: 34); (12: 13: 34: 15); (12: 13: 34: 25); (12: 13: 34: 35); (12: 13: 34: 45)

AMD|Fˆ1 Fˆ2 Fˆ3 Fˆ4 Fˆ5 |AMD F1 F2 F3 F4 (12 )4 + F5 : ((12)5 ); ((12)4 : 13) F1 F2 F3 F4 ((12 )3 : 13 ) + F5 : ((12)3 : 13: 12); ((12)3 : 13: 13); ((12)3 : 13: 23); ((12)3 : 13: 14); ((12)3 : 13: 24); ((12)3 : 13: 34) F1 F2 F3 F4 ((12 )2 : 13 : 12 ) + F5: ((12)2 : 13: (12)2 ); ((12)2 : 13: 12: 13); ((12)2 : 13: 12: 23); ((12)2 : 13: 12: 14); ((12)2 : 13: 12: 24); ((12)2 : 13: 12: 34) F1 F2 F3 F4 ((12 )2 : (13 )2 ) + F5: ((12)2 : (13)2 : 12); ((12)2 : (13)2 : 13); ((12)2 : (13)2 : 23); ((12)2 : (13)2 : 14); ((12)2 : (13)2 : 24); ((12)2 : (13)2 : 34) F1 F2 F3 F4 ((12 )2 : 13 : 23 ) + F5: ((12)2 : 13: 23: 12); ((12)2 : 13: 23: 13); ((12)2 : 13: (23)2 ); ((12)2 : 13: 23: 14); ((12)2 : 13: 23: 24); ((12)2 : 13: 23: 34) F1 F2 F3 F4 ((12 )2 : 13 : 14 ) + F5: ((12)2 : 13: 14: 12); ((12)2 : 13: 14: 13); ((12)2 : 13: (14)2 ); ((12)2 : 13: 14: 23); ((12)2 : 13: 14: 24); ((12)2 : 13: 14: 34); ((12)2 : 13: 14: 15); ((12)2 : 13: 14: 25); ((12)2 : 13: 14: 35); ((12)2 : 13: 14: 45) F1 F2 F3 F4 ((12 )2 : 13 : 24 ) + F5: ((12)2 : 13: 24: 12); ((12)2 : 13: 24: 13); ((12)2 : 13: 24: 14); ((12)2 : 13: 24: 23); ((12)2 : 13: 24: 24); ((12)2 : 13: 24: 34); ((12)2 : 13: 24: 15); ((12)2 : 13: 24: 25); ((12)2 : 13: 24: 35); ((12)2 : 13: 24: 45) F1 F2 F3 F4 ((12 )2 : 13 : 34 ) + F5: ((12)2 : 13: 34: 12); ((12)2 : 13: 34: 13); ((12)2 : 13: 34: 14); ((12)2 : 13: 34: 23); ((12)2 : 13: 34: 24); ((12)2 : 13: 34: 34); ((12)2 : 13: 34: 15); ((12)2 : 13: 34: 25); ((12)2 : 13: 34: 35); ((12)2 : 13: 34: 45) F1 F2 F3 F4 (12 : 13 : (12 )2 ) + F5: (12: 13: (12)3 ); (12: 13: (12)2 : 13); (12: 13: (12)2 : 23); (12: 13: (12)2 : 14); (12: 13: (12)2 : 24); (12: 13: (12)2 : 34) F1 F2 F3 F4 (12 : 13 : 12 : 13 ) + F5: (12: 13: 12: 13: 12); (12: 13: 12: 13: 13); (12: 13: 12: 13: 23); (12: 13: 12: 13: 14); (12: 13: 12: 13: 24); (12: 13: 12: 13: 34) F1 F2 F3 F4 (12 : 13 : 12 : 23 ) + F5: (12: 13: 12: 23: 12); (12: 13: 12: 23: 13); (12: 13: 12: 23: 23); (12: 13: 12: 23: 14); (12: 13: 12: 23: 24); (12: 13: 12: 23: 34) F1 F2 F3 F4 (12 : 13 : 12 : 14 ) + F5: (12: 13: 12: 14: 12); (12: 13: 12: 14: 13); (12: 13: 12: 14: 14); (12: 13: 12: 14: 23); (12: 13: 12: 14: 24); (12: 13: 12: 14: 34); (12: 13: 12: 14: 15); (12: 13: 12: 14: 25); (12: 13: 12: 14: 35); (12: 13: 12: 14: 45) F1 F2 F3 F4 (12 : 13 : 12 : 24 ) + F5: (12: 13: 12: 24: 12); (12: 13: 12: 24: 13); (12: 13: 12: 24: 14); (12: 13: 12: 24: 23); (12: 13: 12: 24: 24); (12: 13: 12: 24: 34); (12: 13: 12: 24: 15); (12: 13: 12: 24: 25); (12: 13: 12: 24: 35); (12: 13: 12: 24: 45) F1 F2 F3 F4 (12 : 13 : 12 : 34 ) + F5: (12: 13: 12: 34: 12); (12: 13: 12: 34: 13); (12: 13: 12: 34: 14); (12: 13: 12: 34: 23); (12: 13: 12: 34: 24); (12: 13: 12: 34: 34); (12: 13: 12: 34: 15); (12: 13: 12: 34: 25); (12: 13: 12: 34: 35); (12: 13: 12: 34: 45) F1 F2 F3 F4 (12 : (13 )2 : 12 ) + F5: (12: (13)2 : 12: 12); (12: (13)2 : 12: 13); (12: (13)2 : 12: 14); (12: (13)2 : 12: 23); (12: (13)2 : 12: 24); (12: (13)2 : 12: 34); (12: (13)2 : 12: 15); (12: (13)2 : 12: 25); (12: (13)2 : 12: 35); (12: (13)2 : 12: 45) F1 F2 F3 F4 (12 : (13 )2 : 13 ) + F5: (12: (13)2 : 13: 12); (12: (13)2 : 13: 13); (12: (13)2 : 13: 23); (12: (13)2 : 13: 14); (12: (13)2 : 13: 24); (12: (13)2 : 13: 34) F1 F2 F3 F4 (12 : (13 )2 : 23 ) + F5: (12: (13)2 : 23: 12); (12: (13)2 : 23: 13); (12: (13)2 : 23: 23); (12: (13)2 : 23: 14); (12: (13)2 : 23: 24); (12: (13)2 : 23: 34) F1 F2 F3 F4 (12 : (13 )2 : 14 ) + F5: (12: (13)2 : 14: 12); (12: (13)2 : 14: 13); (12: (13)2 : 14: 14); (12: (13)2 : 14: 23); (12: (13)2 : 14: 24); (12: (13)2 : 14: 34); (12: (13)2 : 14: 15); (12: (13)2 : 14: 25); (12: (13)2 : 14: 35); (12: (13)2 : 14: 45) F1 F2 F3 F4 (12 : (13 )2 : 24 ) + F5: (12: (13)2 : 24: 12); (12: (13)2 : 24: 13); (12: (13)2 : 24: 14); (12: (13)2 : 24: 23); (12: (13)2 : 24: 24); (12: (13)2 : 24: 34); (12: (13)2 : 24: 15); (12: (13)2 : 24: 25); (12: (13)2 : 24: 35); (12: (13)2 : 24: 45) F1 F2 F3 F4 (12 : (13 )2 : 34 ) + F5: (12: (13)2 : 34: 12); (12: (13)2 : 34: 13); (12: (13)2 : 34: 14); (12: (13)2 : 34: 23); (12: (13)2 : 34: 24); (12: (13)2 : 34: 34); (12: (13)2 : 34: 15); (12: (13)2 : 34: 25); (12: (13)2 : 34: 35); (12: (13)2 : 34: 45) F1 F2 F3 F4 (12 : 13 : 23 : 12 ) + F5: (12: 13: 23: 12: 12); (12: 13: 23: 12: 13); (12: 13: 23: 12: 23); (12: 13: 23: 12: 14); (12: 13: 23: 12: 24); (12: 13: 23: 12: 34) F1 F2 F3 F4 (12 : 13 : 23 : 13 ) + F5: (12: 13: 23: 13: 12); (12: 13: 23: 13: 13); (12: 13: 23: 13: 23); (12: 13: 23: 13: 14); (12: 13: 23: 13: 24); (12: 13: 23: 13: 34)

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H. Toki, J. Hu / Chinese Journal of Physics 55 (2017) 28–46

F1 F2 F3 F4 (12 : 13 : 23 : 23 ) + F5: (12: 13: 23: 23: 12); (12: 13: 23: 23: 13); (12: 13: 23: 23: 23); (12: 13: 23: 23: 14); (12: 13: 23: 23: 24); (12: 13: 23: 23: 34) F1 F2 F3 F4 (12 : 13 : 23 : 14 ) + F5: (12: 13: 23: 14: 12); (12: 13: 23: 14: 13); (12: 13: 23: 14: 14); (12: 13: 23: 14: 23); (12: 13: 23: 14: 24); (12: 13: 23: 14: 34); (12: 13: 23: 14: 15); (12: 13: 23: 14: 25); (12: 13: 23: 14: 35); (12: 13: 23: 14: 45) F1 F2 F3 F4 (12 : 13 : 23 : 24 ) + F5: (12: 13: 23: 24: 12); (12: 13: 23: 24: 13); (12: 13: 23: 24: 14); (12: 13: 23: 24: 23); (12: 13: 23: 24: 24); (12: 13: 23: 24: 34); (12: 13: 23: 24: 15); (12: 13: 23: 24: 25); (12: 13: 23: 24: 35); (12: 13: 23: 24: 45) F1 F2 F3 F4 (12 : 13 : 23 : 34 ) + F5: (12: 13: 23: 34: 12); (12: 13: 23: 34: 13); (12: 13: 23: 34: 14); (12: 13: 23: 34: 23); (12: 13: 23: 34: 24); (12: 13: 23: 34: 34); (12: 13: 23: 34: 15); (12: 13: 23: 34: 25); (12: 13: 23: 34: 35); (12: 13: 23: 34: 45) F1 F2 F3 F4 (12 : 13 : 14 : 12 ) + F5: (12: 13: 14: 12: 12); (12: 13: 14: 12: 13); (12: 13: 14: 12: 14); (12: 13: 14: 12: 23); (12: 13: 14: 12: 24); (12: 13: 14: 12: 34); (12: 13: 14: 12: 15); (12: 13: 14: 12: 25); (12: 13: 14: 12: 35); (12: 13: 14: 12: 45) F1 F2 F3 F4 (12 : 13 : 14 : 13 ) + F5: (12: 13: 14: 13: 12); (12: 13: 14: 13: 13); (12: 13: 14: 13: 14); (12: 13: 14: 13: 23); (12: 13: 14: 13: 24); (12: 13: 14: 13: 34); (12: 13: 14: 13: 15); (12: 13: 14: 13: 25); (12: 13: 14: 13: 35); (12: 13: 14: 13: 45) F1 F2 F3 F4 (12 : 13 : 14 : 14 ) + F5: (12: 13: 14: 14: 12); (12: 13: 14: 14: 13); (12: 13: 14: 14: 14); (12: 13: 14: 14: 23); (12: 13: 14: 14: 24); (12: 13: 14: 14: 34); (12: 13: 14: 14: 15); (12: 13: 14: 14: 25); (12: 13: 14: 14: 35); (12: 13: 14: 14: 45) F1 F2 F3 F4 (12 : 13 : 14 : 23 ) + F5: (12: 13: 14: 23: 12); (12: 13: 14: 23: 13); (12: 13: 14: 23: 14); (12: 13: 14: 23: 23); (12: 13: 14: 23: 24); (12: 13: 14: 23: 34); (12: 13: 14: 23: 15); (12: 13: 14: 23: 25); (12: 13: 14: 23: 35); (12: 13: 14: 23: 45) F1 F2 F3 F4 (12 : 13 : 14 : 24 ) + F5: (12: 13: 14: 24: 12); (12: 13: 14: 24: 13); (12: 13: 14: 24: 14); (12: 13: 14: 24: 23); (12: 13: 14: 24: 24); (12: 13: 14: 24: 34); (12: 13: 14: 24: 15); (12: 13: 14: 24: 25); (12: 13: 14: 24: 35); (12: 13: 14: 24: 45) F1 F2 F3 F4 (12 : 13 : 14 : 34 ) + F5: (12: 13: 14: 34: 12); (12: 13: 14: 34: 13); (12: 13: 14: 34: 14); (12: 13: 14: 34: 23); (12: 13: 14: 34: 24); (12: 13: 14: 34: 34); (12: 13: 14: 34: 15); (12: 13: 14: 34: 25); (12: 13: 14: 34: 35); (12: 13: 14: 34: 45) F1 F2 F3 F4 (12 : 13 : 14 : 15 ) + F5: (12: 13: 14: 15: 12); (12: 13: 14: 15: 13); (12: 13: 14: 15: 14); (12: 13: 14: 15: 15); (12: 13: 14: 15: 23); (12: 13: 14: 15: 24); (12: 13: 14: 15: 25); (12: 13: 14: 15: 34); (12: 13: 14: 15: 35); (12: 13: 14: 15: 45); (12: 13: 14: 15: 16); (12: 13: 14: 15: 26); (12: 13: 14: 15: 36); (12: 13: 14: 15: 46); (12: 13: 14: 15: 56) F1 F2 F3 F4 (12 : 13 : 14 : 25 ) + F5: (12: 13: 14: 25: 12); (12: 13: 14: 25: 13); (12: 13: 14: 25: 14); (12: 13: 14: 25: 15); (12: 13: 14: 25: 23); (12: 13: 14: 25: 24); (12: 13: 14: 25: 25); (12: 13: 14: 25: 34); (12: 13: 14: 25: 35); (12: 13: 14: 25: 45); (12: 13: 14: 25: 16); (12: 13: 14: 25: 26); (12: 13: 14: 25: 36); (12: 13: 14: 25: 46); (12: 13: 14: 25: 56) F1 F2 F3 F4 (12 : 13 : 14 : 35 ) + F5: (12: 13: 14: 35: 12); (12: 13: 14: 35: 13); (12: 13: 14: 35: 14); (12: 13: 14: 35: 15); (12: 13: 14: 35: 23); (12: 13: 14: 35: 24); (12: 13: 14: 35: 25); (12: 13: 14: 35: 34); (12: 13: 14: 35: 35); (12: 13: 14: 35: 45); (12: 13: 14: 35: 16); (12: 13: 14: 35: 26); (12: 13: 14: 35: 36); (12: 13: 14: 35: 46); (12: 13: 14: 35: 56) F1 F2 F3 F4 (12 : 13 : 14 : 45 ) + F5: (12: 13: 14: 45: 12); (12: 13: 14: 45: 13); (12: 13: 14: 45: 14); (12: 13: 14: 45: 15); (12: 13: 14: 45: 23); (12: 13: 14: 45: 24); (12: 13: 14: 45: 25); (12: 13: 14: 45: 34); (12: 13: 14: 45: 35); (12: 13: 14: 45: 45); (12: 13: 14: 45: 16); (12: 13: 14: 45: 26); (12: 13: 14: 45: 36); (12: 13: 14: 45: 46); (12: 13: 14: 45: 56) We can write the following configurations in the similar manner, but in order to save the space we skip writing them explicitly. F1 F2 F3 F4 (ab : cd : e f : gh ) + F5 (12: 13: 24: 12); (12: 13: 24: 13); (12: 13: 24: 14); (12: 13: 24: 23); (12: 13: 24: 24); (12: 13: 24: 34); (12: 13: 24: 15); (12: 13: 24: 25); (12: 13: 24: 35); (12: 13: 24: 45); (12: 13: 34: 12); (12: 13: 34: 13); (12: 13: 34: 14); (12: 13: 34: 23); (12: 13: 34: 24); (12: 13: 34: 34); (12: 13: 34: 15); (12: 13: 34: 25); (12: 13: 34: 35); (12: 13: 34: 45) Here, all the configurations (ab: cd: ef: gh) are listed here and the rule to write the final expressions can be learned from the above listings. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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