General operator form of the non-local three-nucleon force

The European Physical Journal A, Sep 2017

This paper describes a procedure to obtain the general form of the three-nucleon force. The result is an operator form where the momentum space matrix element of the three-nucleon potential is written as a linear combination of 320 isospin-spin-momentum operators and scalar functions of momenta. Any spatial and isospin rotation invariant three-nucleon force can be written in this way and in order for the potential to be Hermitian, symmetric under parity inversion, time reversal and particle exchange, the scalar functions must have definite transformation properties under these discrete operations. A complete list of the isospin-spin-momentum operators and scalar function transformation properties is given.

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General operator form of the non-local three-nucleon force

Eur. Phys. J. A General operator form of the non-local three-nucleon force K. Topolnicki 0 0 M. Smoluchowski Institute of Physics, Jagiellonian University , PL-30348, Krak ́ow , Poland This paper describes a procedure to obtain the general form of the three-nucleon force. The result is an operator form where the momentum space matrix element of the three-nucleon potential is written as a linear combination of 320 isospin-spin-momentum operators and scalar functions of momenta. Any spatial and isospin rotation invariant three-nucleon force can be written in this way and in order for the potential to be Hermitian, symmetric under parity inversion, time reversal and particle exchange, the scalar functions must have definite transformation properties under these discrete operations. A complete list of the isospin-spin-momentum operators and scalar function transformation properties is given. 1 Introduction Three-nucleon (3N) forces are becoming an increasingly important ingredient in few-nucleon calculations. It is becoming clear that 3N potentials must be utilized in order to precisely describe experimental data. For instance, if only two-nucleon forces are used the binding energy of 3N systems is underestimated by 0.5–1.0 MeV and large discrepancies arise for the vector analyzing power in elastic neutron-deuteron scattering at low energies, for more details see, e.g., [1] and references therein. The rich operator structure of 3N potentials [2–7] motivates considerations of the general structure of these forces. Furthermore, discrepancies between theory and experiment still exist and it is possible that this is caused by not utilizing the full structure of the 3N force. In this paper, the general form of the three-nucleon potential, constructed to be invariant under spatial rotations, isospin rotations and discrete symmetry operations (parity inversion, time reversal, particle exchange, and Hermitian conjugation) is developed. This general form is compatible with any model of the 3N force that satisfies the appropriate symmetries making it useful for a verity of practical applications. I will follow the approach from [8, 9], where a local 3N force was considered, and use the algorithm from [10], where the general form of the total momentum dependent two-nucleon potential was developed, to generate the spatial rotation invariant operator form of the 3N potential. The general form is meant to become an important ingredient in the so called “three-dimensional” (3D) calculations. In this approach, the three-dimensional, vector, degrees of freedom of the nucleons are treated directly without resorting to angular momentum decomposition. The biggest advantage of the 3D formalism is the possibility to avoid the complicated numerics of partial wave representations at higher energies. Additionally, calculations performed within this formalism are flexible and allow different models of few-nucleon forces to be used. This is especially important since new models of few-nucleon potentials are constantly being derived from chiral effective field theory [2–7] in a form directly suitable for 3D calculations. A good overview of the 3D approach can be found in [1]. An introduction to these calculations can be found in earlier works, e.g. [11, 12]. More detailed information about the 3D formalism, with emphasis on few-nucleon bound and scattering states, can be found in works by the Krako´w, Bohum, Tehran, Ohio, and University of Iowa groups [13–28]. More traditional approaches that employ partial wave decomposition can also benefit from the possibility to represent different models of 3N forces using a common template. This useful property has important practical implications. It might result in numerical codes that are more general and can be applied to test a large verity of fewnucleon force models. Especially in the new effective methods of partial wave decomposition [29, 30] used to obtain matrix elements in the 3N partial wave basis. It should be emphasized that the discussion presented in this paper is applicable also to operators that depend on the total momentum of the 3N system. This opens the door for applications in calculations that include relativistic corrections. The paper is organized as follows. Section 2 discusses the symmetry of the 3N force with respect to spatial rotations in spin space. Next, sect. 3 extends the potential to 3N isospin space and adds symmetry with respect to isospin rotations. Section 4 contains considerations related to discrete symmetries. Section 5 explicitly gives the final general operator form of the 3N potential. Finally, sect. 6 contains a summary and appendixes A, B, C, D.1, and D.2 contain additional materials necessary to construct the general form including a list of the 320 operators. expressed as a linear combination of the remaining 63 operators and scalar functions of momenta. Furthermore, a product of any two (or more) operators from this set can be expressed as a linear combination of the 64 operators and scalar functions making the set complete. An additional observation can be made about the [Oˇi] operators. If the matrix element of Vˇ is allowed to depend also on the total momentum K, 2 Invariance under spatial rotations A modified version of the method from sect. 2 of ref. [10] is used to generate the general spatial rotation invariant form of the momentum space matrix element of the 3N potential operator Vˇ ˇ p q | V | pq , where p , q and p, q are Jacobi momenta in the final and initial state respectively. In the spin space of the 3N system (for a given isospin in the initial and final state) the operator form of Vˇ is 64 i=1 with square brackets being used (here and in the following) to denote a matrix representation, [Oˇi(p , q , p)] being 8 × 8 matrices representing given spin operators (appendix A contains a complete list) and fi(p , q , p, q) being scalar functions of momenta. Note that the [Oˇi(p , q , p)] operators depend on only three of the four Jacobi momenta. Since the momentum vectors have three spatial dimensions the potential dependence of [Oˇ] on some forth momentum vector x can be written entirely in terms of the angles p · x, q · x, p · x and x2. This results in the additional momentum dependence being separated out from [Oˇ] and pushed into the scalar functions f . In general, it is possible to construct spatial rotation invariant operator forms with sets of 64 operators that depend on any combination of three of the four Jacobi momenta. The momentum dependence of the spin operators in the new sets will be the same as the momentum dependence in [Oˇi(p , q , p)] except with p , q , p directly replaced by a different combination of three vectors. The choice of p , q , p used in this paper is arbitrary. The algorithm given in [10] uses the observation that any scalar expression can be written as a product of two types of elements —a scalar product of two vectors (a · b) and a scalar product of a vector and a vector product (a · b × c). This observation can be verified using simple vector identities. In the present case a, b, c are the momentum vectors p , q , p, q or vectors of spin operators σ( 1 ), σ( 2 ), σ( 3 ) acting in the spaces of particles 1, 2, 3. As it turns out combining the two types of elements (a · b, a · b × c) results in only a finite number of independent operators —the 64 operators from appendix A. Independence means that none of the 64 operators can be ˇ p q K | V | pqK , then in the spatial rotation invariant, operator form in 3N spin space (for a given isospin in the initial and final state) ˇ p q K | V | pqK = 64 i=1 fi(p , q , p, q, K) Oˇi(p , q , p) 8×8 the set of [Oˇi=1,...,64(p , q , p)] remains the same. The additional momentum dependence appears in the new scalar functions fi(p , q , p, q, K). This is again a reflection of the three dimensional nature of space and the [Oˇi(p , q , p)] operators being composed of only three of the momentum vectors. This property can be used to make the following discussion, where the dependence on K is omitted, more general. 3 Invariance under isospin rotations In order to preserve symmetry with respect to isospin rotations one of the following five operators: I1 = 1ˇ, Iˇ2 = τˇ( 1 ) · τˇ( 2 ), Iˇ3 = τˇ( 1 ) · τˇ( 3 ), ˇ ˇ I4 = τˇ( 2 ) · τˇ( 3 ), Iˇ5 = τˇ( 1 ) · (τˇ( 2 ) × τˇ( 3 )), is appended to each [Oˇi=1,...,64(p , q , p)] where τ (i) is a vector isospin operator of particle i = 1, 2, 3. This results in the following operator form: ˇ p q | V | pq where gk=5(j−1)+i ≡ gi j and [Qˇk=5(j−1)+i] ≡ [Iˇi ⊗ Oˇj ] are operators in the isospin-spin-momentum space of the 3N system with a 64×64 matrix representation. A list of these 320 operators is provided in appendix B. In the following sections I will show that, after taking into account discrete symmetries, the general form of the 3N force will also consist of 320 operators. This number is much greater than the 80 operators in the local version of the 3N force [8, 9]. The local potential depends only on two momentum transfer vectors which leads to a reduced number of operators in the rotation invariant form. This in turn translates into a reduced number of operators in the final form of the local 3N force. ( 1 ) ( 2 ) ( 3 ) 4 Discrete symmetries – time reflection (Rˇt) – parity inversion (Rˇs) – Hermitian conjugate (Rˇh) – particle exchange (Pˇ ∈ 3 S ) The potential is additionally required to be symmetric with respect to: All of these operations commute and the first three operations form simple cyclic groups. This results in the combined group being a direct product of three cyclic groups Z2 and S3 In order to enforce the discrete symmetries the method from [8, 9], where a local 3N force was considered, is extended from S3 to G. First the general operator form ( 3 ) , ( 4 ) where the isospin-spin-momentum operators Girj ([Qˇk]) are constructed from [Qˇk] in such a way as that they transform according to specific representations “r” of the group G and the indexes i, j take on the value 1 for one-dimensional representations “r” or 1, 2 for two-dimensional representations “r”. Next, knowing the transformation properties of Girj ([Qˇk]) under operations Rˇ ∈ G, the scalar functions r hk;ij are required to compensate for this behaviour and make the whole operator symmetric. There are two representations for each of the three Z2 groups and three representations for S3 (given, e.g., in [8, 9]). The notation r = (rt, rs, rh, rp) will be used with the value rt, rs, rh = 1, 2 denoting the representations of the three Z2 groups (for time reversal, parity inversion, and the Hermitian conjugate) and rp = 1, 2, 3 denoting the representations for S3 (particle exchange). This gives a total of 24 representations of G. Finally, the function G is defined as r Gij where Dirj (Rˇ) is the matrix representation (or just a single number for one-dimensional representations) of the group element Rˇ ∈ G for a given representation “r” and Rˇ([Qˇk]) is the action of the discrete operation Rˇ on [Qˇk]. The new operators, constructed according to ( 5 ), will transform under symmetry operations Rˇ ∈ G as (see, e.g., [8, 9]): It is easy to work out that if the scalar functions hrk;ij satisfy r hk;ij = Dirl(Rˇ)Rˇ(hrk;lj ) ( 7 ) for all Rˇ ∈ G, then they compensate for the transformations of the Glrj ([Qˇk]) operators and make ( 4 ) invariant under the discrete symmetry operations. The two following subsections discuss the matrix representations and the implementation of discrete symmetries in more detail. 4.1 Matrix representations of G There are two irreducible linear representations for the cyclic group Z2. Both are 1 × 1 dimensional matrices and the notation DZ1,2 will be used to denote these matrices for 2 the two representations. The first representation is trivial, DZ12 (Rˇt) = DZ12 (Rˇs) = DZ12 (Rˇh) = (−1) and the second one changes the sign, DZ22 (Rˇt) = DZ22 (Rˇs) = DZ22 (Rˇh) = (−1). Next, there are three representations for the S3 group of particle permutations in the 3N system [8, 9]. The cycle representation for permutations will be used with (ij) being a permutation exchanging particles i, j = 1, 2, 3: i → j, j → i and (ijk) being a permutation changing particles i, j, k = 1, 2, 3: i → j, j → k, k → i. Two representations are one-dimensional; DS1,2 is used to denote 1 × 1 3 matrices belonging to these two representations. The first representation is trivial, and the second representation changes the sign, The third representation is two-dimensional. The 2 × 2 matrices DS33 for this representation are DS33 (( 1 )) = These four types of representations for the four discrete symmetries can be combined using the Kronecker product ⊗: Dr=(rt,rs,rh,rp) = DZrt2 ⊗ DZrs2 ⊗ DZrh2 ⊗ DSr3p ( 8 ) and in practice, since the first three Drt , Drs , Drh are 1 × 1 matrices, the Kronecker product can be replaced by a regular multiplication. The linear representation of G, D(rt,rs,rh,rp), is a 1 × 1 matrix for all cases except when rp = 3 that is when it is a 2 × 2 matrix. 4.2 Implementation of discrete transformations In the proposed approach, the discrete symmetries are implemented as operations on the momentum space matrix element of an operator Xˇ . This element p q | X | pq ˇ is a function of four Jacobi momenta and an operator in the isospin-spin-momentum space of the 3N system. In practice discrete symmetry operators are realized as operations on the 64 × 64 (23 isospin states and 23 spin states) matrix representation of the isospin-spin-momentum operator p q | Xˇ | pq ≡ [Xˇ (p , q , p, q)]. Time reversal is implemented using ˇt R ˇ1 ⊗ ˇ1 ⊗ ˇ1 ⊗ [iσy] ⊗ [iσy] ⊗ [iσy] † T , ( 9 ) where the identity operators [1ˇ]⊗[1ˇ]⊗[1ˇ] act in the isospin space of the 3N system and [iσy] ⊗ [iσy] ⊗ [iσy] act in the spin space. If dependence on the total momentum of the 3N system K is considered then −K will appear in the momentum space matrix element after the application of time reversal. I would like to take this opportunity to correct a misprint, found in our paper [10]. The implementation of time reversal in equation ( 10 ) of [10] should, of course, be supplemented by a transposition. Parity inversion is implemented as ˇs R Xˇ (p , q , p, q) = Xˇ (−p , −q , −p, −q) . ( 10 ) Similarly as before, if dependence on the total momentum of the 3N system K is considered then −K will appear in the momentum space matrix element after the application of the spatial reflection. Hermitian conjugation has a straightforward implementation Rˇh Xˇ (p , q , p, q) = Xˇ (p, q, p , q ) † ( 11 ) and, if dependence on the total momentum of the 3N system K is considered, then the same vector K will appear in the momentum space matrix element after the application of the symmetry operation. Particle exchange is more complicated since there are six operations to implement. In general for Pˇ ∈ S3 ˇ P Xˇ (p , q , p, q) = [P ]T Xˇ (JAPˇ (p , q ), JBPˇ (p , q ), JAPˇ (p, q), JBPˇ (p, q)) [P ] ( 12 ) where [P ] is a 64 × 64 matrix performing a particle permutation in the isospin-spin-momentum space of the 3N system. J Pˇ and J Pˇ are functions that transform the Ja A B cobi momenta to implement the appropriate particle perˇ mutation. The construction of [P ] and the functions J P A and JBPˇ are given in appendix C. Again if dependence on the total momentum of the 3N system K is considered then the same vector K will appear in the momentum space matrix element after the application of the particle permutation. 5 Removing redundant operators The above considerations show that there are potentially 320 × 2 × 2 × 2 × 2 × 1 = 5120 of Gir=1 j=1([Qˇk=1,...,320]) operators that transform according to one-dimensional representations r = (rt = 1, 2, rs = 1, 2, rh = 1, 2, rp = 1, 2) of G and 320 × 2 × 2 × 2 × 1 × 4 = 10240 of Gir=1,2 j=1,2 ([Qˇk=1,...,320]) operators that transform according to twodimensional representations r = (rt = 1, 2, rs = 1, 2, rh = 1, 2, rp = 3) of G. It was numerically verified that out of the 15360 possible Girj ([Qˇk]) operators only 3507 (about 23%) are nonzero. This still leaves a number of redundant operators that should be removed from the final operator form ( 4 ) since only 320 operators are independent. If any operator Xˇ from the set of all nonzero {Girj ([Qˇk]) = 0} can be expressed as a linear combination of operators from {Girj ([Qˇk]) = 0} \ Xˇ and scalar functions of momenta then it is not independent and can be eliminated. It is not immediately obvious that this is true and to demonstrate this a situation where the operators Girj ([Qˇk]) can be written as ˇ Qk = xii jjk kr r Gi j r will be considered with xii jjk kr r being scalar functions As a consequence operators Girj ( Qˇk ) should be added hk;ij xii jjk kr r ⎠ r of momenta and xii jj kk rr = 0 to ensure that the operators in groups with a given j: {Gr11([Qˇk]), Gr21([Qˇk])}, Gir j ([Qˇk ]) are chosen from {Gir j ([Qˇk ]) = 0}\Girj ([Qˇk]). {Gr12([Qˇk]), Gr22([Qˇk])}. This guarantees thatr the transThe general operator form ( 4 ) now reads forrmation properties of the scalar functions {hk;11, hrk;21}, {hk;12, hrk;22} are easy to work out. p q | Vˇ | pq = k r ij hrk;ij aacncdoMrDdy.i2nc.hgAoitpcoepoeonnfdteh-idxeim3D2.e01noslipiosetnrsaaaltolrlresoppirseelrsiaestntoetdrastiintohnaasptpotefrnaGdnis.xfoADrpm.1xii jjk kr r Gir j Qˇk . ( 14 ) pendix D.2 lists all operators that transform according to k r i j two-dimensional representations of G, and this set is split into two additional categories. In the first one, there are Rearranging the terms in ( 14 ) all the operators from the first column Girj=1([Qˇk]) and in p q | Vˇ | pq = tGhirej=s2e(c[oQˇnkd])o. nTeogaellththere aolpltehraetsoerospferroamtortshecasnecboencdomcobl uinmedn k r i j to the general form of the 3N force that is invariant with ⎛ ⎞ respect to spatial rotations, isospin rotations, and discrete Qˇk symmetries r i j ( 15 ) ( 16 ) h kr ;i j Gi j r This equation defines new scalar functions h kr ;i j = hk;ij xii jjk kr r r k r ij and the invariance of the potential with respect to discrete symmetry operations implies that also the new scalar functions h kr ;i j satisfy ( 7 ). As a consequence of this, if ( 13 ) can be solved for a particular operator Girj ([Qˇk]) such that xii jj kk rr = 0 (Gir j ([Qˇk ]) are chosen from {Gir j ([Qˇk ]) = 0} \ Girj ([Qˇk])) then this operator is not independent and can safely be removed from the operator form ( 4 ) since it does not bring any new structures. In practice equation ( 13 ) is solved numerically by substituting random numbers for the momentum vector components. There is another possibility to construct the set of 320 independent operators. Instead of eliminating nonindependent operators from the set of 3507 non-zero Girj ([Qˇk]) it is possible to start with an empty set and add, to this set, operators from {Girj ([Qˇk]) = 0} one by one or in small groups, checking each time if all newly added operators are independent (i.e. no solution to ( 13 ) with xii jj kk rr = 0 exists). This process does not lead to a unique general form and the additional freedom allows the consideration of some practical issues related to the final set of 320 independent operators in ( 4 ). In particular, it is important to be able to easily work out the transformation properties of all the scalar functions. This is not a problem for one-dimensional representations of G. For two-dimensional operators, however, the operators need to be added in groups of 2. This is a result of ( 7 ) and scalar r functions hk;ij from a single column (with a given j) being transformed into scalar functions from the same column hk;i1 = Dir1(Rˇ)Rˇ(hrk;11) + Dir2(Rˇ)Rˇ(hrk;21), r hk;i2 = Dir1(Rˇ)Rˇ(hrk;12) + Dir2(Rˇ)Rˇ(hrk;22). r where the transformation properties of the scalar functions hk and operators [Sˇk] can be read off from appendix D.1 and D.2. 6 Summary The construction began with the general spatial and isospin rotation symmetric form of the three nucleon potential ˇ p q | V | pq with [Qˇk] being three-nucleon isospin-spin-momentum operators having 64×64 matrix representations and gk being scalar functions of Jacobi momenta in the initial p, q and final p , q states. Next, in order to take into account discrete symmetries, this operator form was transformed into where the operators Girj ([Qˇk]) are constructed from ˇ Qk r Gij using the matrix representation Dirj (Rˇ) of the symmetry group transformations Rˇ for a given representation “r” and the indices i, j take on a single value 1 for onedimensional representations and 1, 2 for two-dimensional representations. The Girj ([Qˇk]) operators have simple, known transformation properties with respect to time reversal, parity, Hermitian conjugation and particle exchange, that are determined by one of the 24 representations “r” of the symmetry group These constraints compensate for the behavior of Girj ([Qˇk]) under symmetry transformations and make the whole operator invariant. Finally, knowing that there are only 320 operators Girj ([Qˇk]) that are independent —they cannot be expressed as linear combinations of each other and scalar functions— a subset of 320 operators from {Girj ([Qˇk]) = 0} is chosen. The choice is dictated by practical considerations, namely, it is important that the transformation properties of the scalar functions hk in the final operator form are easy to work out. These transformation properties, together with all the [Sˇk] operators are listed in the appendixes. As mentioned in the beginning of this paper, the general form of the three-nucleon force can easily be extended to operators that depend on the total momentum of the system by adding new arguments to the scalar functions. This opens the door for applications in calculations that include relativistic corrections. The general form has potential to become an important ingredient in the, so called, “three-dimensional” formalism, where instead of relying on angular momentum decomposition, the threedimensional degrees of freedom of the nucleons are used directly. Additionally, being able to represent different models of three-nucleon forces using the same template is a very useful property which might also be utilized in more traditional, partial wave based, calculations. The author would like to thank Prof Jacek Golak, Dr Roman Skibin´ski and Prof Henryk Witala for fruitful discussions and help in preparing the manuscript. This work was supported by the National Science Center, Poland, under Grants No. 2016/22/M/ST2/00173 and No. 2016/21/D/ST2/01120. Appendix A. Operators in the general form invariant under spatial rotations Below is a list of the 64 operators that make the spatial rotation invariant form of the 3N potential (they also appear in the general form of the 3N scattering amplitude [31] but with the names of vectors changed). In the 3N spin space (for a given isospin in the initial and final state) the momentum space matrix element of the 3N potential between an initial state with Jacobi momenta p, q and a final state with Jacobi momenta p , q it has an 8 × 8 matrix representation and can be written as where fk(p , q, p, q) are scalar functions and the [Oˇk(p , q , p)] operators only depend on three of the four momenta with the additional momentum dependence transferred to the scalar functions. In the list below σ(i) are spin operators acting in the spaces of particles i = 1, 2, 3. An electronic version of these operators is available upon request from . Oˇ27 = (p · σ( 1 ))(q · σ( 3 )) Oˇ63 = (q · σ( 1 ))(q · σ( 2 ))(q · σ( 3 )) Oˇ64 = (q · σ( 1 ))(q · σ( 2 ))(p · σ( 3 )) 320 k=1 Appendix B. Operators in general form invariant under spatial and isospin rotations Below is a list of the 320 operators that make up the spatial and isospin rotation invariant form of the 3N potential where p , q , p, q are Jacobi momenta in the initial and final state, fk(p , q , p, q) are scalar functions and square brackets are used to mark a matrix representation in the isospin-spin-momentum space of the 3N system. The [Qˇk(p , q , p)] operators only depend on three of the four momenta with the additional momentum dependence transfered to the scalar functions. In the list below τ (i), σ(i) are isospin and spin operators acting in the spaces of particles i = 1, 2, 3. Qˇ67 = (τ ( 1 ) · τ ( 2 ))(p × σ( 1 ) · σ( 2 )) Qˇ68 = (τ ( 1 ) · τ ( 3 ))(p × σ( 1 ) · σ( 2 )) Qˇ69 = (τ ( 2 ) · τ ( 3 ))(p × σ( 1 ) · σ( 2 )) Qˇ70 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p × σ( 1 ) · σ( 2 )) Qˇ71 = p × σ( 1 ) · σ( 3 ) Qˇ72 = (τ ( 1 ) · τ ( 2 ))(p × σ( 1 ) · σ( 3 )) Qˇ73 = (τ ( 1 ) · τ ( 3 ))(p × σ( 1 ) · σ( 3 )) Qˇ74 = (τ ( 2 ) · τ ( 3 ))(p × σ( 1 ) · σ( 3 )) Qˇ75 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p × σ( 1 ) · σ( 3 )) Qˇ76 = p × σ( 2 ) · σ( 3 ) Qˇ77 = (τ ( 1 ) · τ ( 2 ))(p × σ( 2 ) · σ( 3 )) Qˇ78 = (τ ( 1 ) · τ ( 3 ))(p × σ( 2 ) · σ( 3 )) Qˇ79 = (τ ( 2 ) · τ ( 3 ))(p × σ( 2 ) · σ( 3 )) Qˇ80 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p × σ( 2 ) · σ( 3 )) Qˇ81 = q × σ( 1 ) · σ( 2 ) Qˇ82 = (τ ( 1 ) · τ ( 2 ))(q × σ( 1 ) · σ( 2 )) Qˇ83 = (τ ( 1 ) · τ ( 3 ))(q × σ( 1 ) · σ( 2 )) Qˇ84 = (τ ( 2 ) · τ ( 3 ))(q × σ( 1 ) · σ( 2 )) Qˇ85 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(q × σ( 1 ) · σ( 2 )) Qˇ86 = q × σ( 1 ) · σ( 3 ) Qˇ87 = (τ ( 1 ) · τ ( 2 ))(q × σ( 1 ) · σ( 3 )) Qˇ88 = (τ ( 1 ) · τ ( 3 ))(q × σ( 1 ) · σ( 3 )) Qˇ89 = (τ ( 2 ) · τ ( 3 ))(q × σ( 1 ) · σ( 3 )) Qˇ90 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(q × σ( 1 ) · σ( 3 )) Qˇ91 = q × σ( 2 ) · σ( 3 ) Qˇ92 = (τ ( 1 ) · τ ( 2 ))(q × σ( 2 ) · σ( 3 )) Qˇ93 = (τ ( 1 ) · τ ( 3 ))(q × σ( 2 ) · σ( 3 )) Qˇ94 = (τ ( 2 ) · τ ( 3 ))(q × σ( 2 ) · σ( 3 )) Qˇ95 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(q × σ( 2 ) · σ( 3 )) Qˇ96 = p × σ( 1 ) · σ( 2 ) Qˇ97 = (τ ( 1 ) · τ ( 2 ))(p × σ( 1 ) · σ( 2 )) Qˇ98 = (τ ( 1 ) · τ ( 3 ))(p × σ( 1 ) · σ( 2 )) Qˇ99 = (τ ( 2 ) · τ ( 3 ))(p × σ( 1 ) · σ( 2 )) Qˇ100 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p × σ( 1 ) · σ( 2 )) Qˇ101 = p × σ( 1 ) · σ( 3 ) Qˇ102 = (τ ( 1 ) · τ ( 2 ))(p × σ( 1 ) · σ( 3 )) Qˇ103 = (τ ( 1 ) · τ ( 3 ))(p × σ( 1 ) · σ( 3 )) Qˇ104 = (τ ( 2 ) · τ ( 3 ))(p × σ( 1 ) · σ( 3 )) Qˇ105 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p × σ( 1 ) · σ( 3 )) Qˇ106 = p × σ( 2 ) · σ( 3 ) Qˇ107 = (τ ( 1 ) · τ ( 2 ))(p × σ( 2 ) · σ( 3 )) Qˇ108 = (τ ( 1 ) · τ ( 3 ))(p × σ( 2 ) · σ( 3 )) Qˇ109 = (τ ( 2 ) · τ ( 3 ))(p × σ( 2 ) · σ( 3 )) Qˇ110 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p × σ( 2 ) · σ( 3 )) Qˇ111 = σ( 1 ) × σ( 2 ) · σ( 3 ) Qˇ112 = (τ ( 1 ) · τ ( 2 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ113 = (τ ( 1 ) · τ ( 3 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ114 = (τ ( 2 ) · τ ( 3 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ115 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ116 = (p · σ( 1 ))(p · σ( 2 )) Qˇ117 = (τ ( 1 ) · τ ( 2 ))(p · σ( 1 ))(p · σ( 2 )) Qˇ118 = (τ ( 1 ) · τ ( 3 ))(p · σ( 1 ))(p · σ( 2 )) Qˇ119 = (τ ( 2 ) · τ ( 3 ))(p · σ( 1 ))(p · σ( 2 )) Qˇ120 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p · σ( 1 ))(p · σ( 2 )) Qˇ121 = (p · σ( 1 ))(p · σ( 3 )) Qˇ122 = (τ ( 1 ) · τ ( 2 ))(p · σ( 1 ))(p · σ( 3 )) Qˇ123 = (τ ( 1 ) · τ ( 3 ))(p · σ( 1 ))(p · σ( 3 )) Qˇ124 = (τ ( 2 ) · τ ( 3 ))(p · σ( 1 ))(p · σ( 3 )) Qˇ125 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p · σ( 1 ))(p · σ( 3 )) Qˇ126 = (p · σ( 1 ))(q · σ( 2 )) Qˇ127 = (τ ( 1 ) · τ ( 2 ))(p · σ( 1 ))(q · σ( 2 )) Qˇ128 = (τ ( 1 ) · τ ( 3 ))(p · σ( 1 ))(q · σ( 2 )) Qˇ129 = (τ ( 2 ) · τ ( 3 ))(p · σ( 1 ))(q · σ( 2 )) Qˇ130 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p · σ( 1 ))(q · σ( 2 )) Qˇ131 = (p · σ( 1 ))(q · σ( 3 )) Qˇ132 = (τ ( 1 ) · τ ( 2 ))(p · σ( 1 ))(q · σ( 3 )) Qˇ133 = (τ ( 1 ) · τ ( 3 ))(p · σ( 1 ))(q · σ( 3 )) Qˇ134 = (τ ( 2 ) · τ ( 3 ))(p · σ( 1 ))(q · σ( 3 )) Qˇ135 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p · σ( 1 ))(q · σ( 3 )) Qˇ136 = (p · σ( 1 ))(p · σ( 2 )) Qˇ137 = (τ ( 1 ) · τ ( 2 ))(p · σ( 1 ))(p · σ( 2 )) Qˇ138 = (τ ( 1 ) · τ ( 3 ))(p · σ( 1 ))(p · σ( 2 )) Qˇ139 = (τ ( 2 ) · τ ( 3 ))(p · σ( 1 ))(p · σ( 2 )) Qˇ140 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p · σ( 1 ))(p · σ( 2 )) Qˇ141 = (p · σ( 1 ))(p · σ( 3 )) Qˇ142 = (τ ( 1 ) · τ ( 2 ))(p · σ( 1 ))(p · σ( 3 )) Qˇ143 = (τ ( 1 ) · τ ( 3 ))(p · σ( 1 ))(p · σ( 3 )) Qˇ144 = (τ ( 2 ) · τ ( 3 ))(p · σ( 1 ))(p · σ( 3 )) Qˇ145 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p · σ( 1 ))(p · σ( 3 )) Qˇ146 = (p · σ( 1 ))(σ( 2 ) · σ( 3 )) Qˇ147 = (τ ( 1 ) · τ ( 2 ))(p · σ( 1 ))(σ( 2 ) · σ( 3 )) Qˇ148 = (τ ( 1 ) · τ ( 3 ))(p · σ( 1 ))(σ( 2 ) · σ( 3 )) Qˇ149 = (τ ( 2 ) · τ ( 3 ))(p · σ( 1 ))(σ( 2 ) · σ( 3 )) Qˇ150 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p · σ( 1 ))(σ( 2 ) · σ( 3 )) Qˇ151 = (p · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ152 = (τ ( 1 ) · τ ( 2 ))(p · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ153 = (τ ( 1 ) · τ ( 3 ))(p · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ154 = (τ ( 2 ) · τ ( 3 ))(p · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ155 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 )) (p · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ156 = (p · σ( 1 ))(q × σ( 2 ) · σ( 3 )) Qˇ157 = (τ ( 1 ) · τ ( 2 ))(p · σ( 1 ))(q × σ( 2 ) · σ( 3 )) Qˇ158 = (τ ( 1 ) · τ ( 3 ))(p · σ( 1 ))(q × σ( 2 ) · σ( 3 )) Qˇ159 = (τ ( 2 ) · τ ( 3 ))(p · σ( 1 ))(q × σ( 2 ) · σ( 3 )) Qˇ160 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 )) (p · σ( 1 ))(q × σ( 2 ) · σ( 3 )) Qˇ161 = (p · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ162 = (τ ( 1 ) · τ ( 2 ))(p · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ163 = (τ ( 1 ) · τ ( 3 ))(p · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ164 = (τ ( 2 ) · τ ( 3 ))(p · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ165 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 )) (p · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ166 = (p · σ( 1 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ167 = (τ ( 1 ) · τ ( 2 ))(p · σ( 1 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ168 = (τ ( 1 ) · τ ( 3 ))(p · σ( 1 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ169 = (τ ( 2 ) · τ ( 3 ))(p · σ( 1 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ170 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 )) (p · σ( 1 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ171 = (p · σ( 2 ))(p · σ( 3 )) Qˇ172 = (τ ( 1 ) · τ ( 2 ))(p · σ( 2 ))(p · σ( 3 )) Qˇ173 = (τ ( 1 ) · τ ( 3 ))(p · σ( 2 ))(p · σ( 3 )) Qˇ174 = (τ ( 2 ) · τ ( 3 ))(p · σ( 2 ))(p · σ( 3 )) Qˇ175 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p · σ( 2 ))(p · σ( 3 )) Qˇ176 = (p · σ( 2 ))(q · σ( 3 )) Qˇ177 = (τ ( 1 ) · τ ( 2 ))(p · σ( 2 ))(q · σ( 3 )) Qˇ178 = (τ ( 1 ) · τ ( 3 ))(p · σ( 2 ))(q · σ( 3 )) Qˇ179 = (τ ( 2 ) · τ ( 3 ))(p · σ( 2 ))(q · σ( 3 )) Qˇ180 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(p · σ( 2 ))(q · σ( 3 )) Qˇ181 = (p · σ( 2 ))(p · σ( 3 )) Qˇ182 = (τ ( 1 ) · τ ( 2 ))(p · σ( 2 ))(p · σ( 3 )) Qˇ183 = (τ ( 1 ) · τ ( 3 ))(p · σ( 2 ))(p · σ( 3 )) Qˇ184 = (τ ( 2 ) · τ ( 3 ))(p · σ( 2 ))(p · σ( 3 )) Qˇ224 = (τ ( 2 ) · τ ( 3 ))(q · σ( 1 ))(p · σ( 3 )) Qˇ225 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(q · σ( 1 ))(p · σ( 3 )) Qˇ226 = (q · σ( 1 ))(σ( 2 ) · σ( 3 )) Qˇ227 = (τ ( 1 ) · τ ( 2 ))(q · σ( 1 ))(σ( 2 ) · σ( 3 )) Qˇ228 = (τ ( 1 ) · τ ( 3 ))(q · σ( 1 ))(σ( 2 ) · σ( 3 )) Qˇ229 = (τ ( 2 ) · τ ( 3 ))(q · σ( 1 ))(σ( 2 ) · σ( 3 )) Qˇ230 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(q · σ( 1 ))(σ( 2 ) · σ( 3 )) Qˇ231 = (q · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ232 = (τ ( 1 ) · τ ( 2 ))(q · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ233 = (τ ( 1 ) · τ ( 3 ))(q · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ234 = (τ ( 2 ) · τ ( 3 ))(q · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ235 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 )) (q · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ236 = (q · σ( 1 ))(q × σ( 2 ) · σ( 3 )) Qˇ237 = (τ ( 1 ) · τ ( 2 ))(q · σ( 1 ))(q × σ( 2 ) · σ( 3 )) Qˇ238 = (τ ( 1 ) · τ ( 3 ))(q · σ( 1 ))(q × σ( 2 ) · σ( 3 )) Qˇ239 = (τ ( 2 ) · τ ( 3 ))(q · σ( 1 ))(q × σ( 2 ) · σ( 3 )) Qˇ240 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 )) (q · σ( 1 ))(q × σ( 2 ) · σ( 3 )) Qˇ241 = (q · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ242 = (τ ( 1 ) · τ ( 2 ))(q · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ243 = (τ ( 1 ) · τ ( 3 ))(q · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ244 = (τ ( 2 ) · τ ( 3 ))(q · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ245 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 )) (q · σ( 1 ))(p × σ( 2 ) · σ( 3 )) Qˇ246 = (q · σ( 1 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ247 = (τ ( 1 ) · τ ( 2 ))(q · σ( 1 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ248 = (τ ( 1 ) · τ ( 3 ))(q · σ( 1 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ249 = (τ ( 2 ) · τ ( 3 ))(q · σ( 1 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ250 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 )) (q · σ( 1 ))(σ( 1 ) × σ( 2 ) · σ( 3 )) Qˇ251 = (q · σ( 2 ))(q · σ( 3 )) Qˇ252 = (τ ( 1 ) · τ ( 2 ))(q · σ( 2 ))(q · σ( 3 )) Qˇ253 = (τ ( 1 ) · τ ( 3 ))(q · σ( 2 ))(q · σ( 3 )) Qˇ254 = (τ ( 2 ) · τ ( 3 ))(q · σ( 2 ))(q · σ( 3 )) Qˇ255 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(q · σ( 2 ))(q · σ( 3 )) Qˇ256 = (q · σ( 2 ))(p · σ( 3 )) Qˇ257 = (τ ( 1 ) · τ ( 2 ))(q · σ( 2 ))(p · σ( 3 )) Qˇ258 = (τ ( 1 ) · τ ( 3 ))(q · σ( 2 ))(p · σ( 3 )) Qˇ259 = (τ ( 2 ) · τ ( 3 ))(q · σ( 2 ))(p · σ( 3 )) Qˇ260 = (τ ( 1 ) · τ ( 2 ) × τ ( 3 ))(q · σ( 2 ))(p · σ( 3 )) Qˇ261 = (q · σ( 2 ))(σ( 1 ) · σ( 3 )) [( 12 )]is = ⎜⎜⎜ 0 0 1 0 0 0 0 0⎟⎟⎟ and work in a basis of 8 spin (isospin) states | i = 1, . . . ,8 such that | i = 4(12 − m1) + 2(12 − m2) + (21 − m3) + 1 = | 12 m1 ⊗ | 21 m2 ⊗ | 21 m3 where m1,2,3 = ±21 are single particle spin (isospin) 12 projections. Finally, the functions that are used to permute the momentum vectors have the form where Girj([Qˇk]) depends on the representation “r” and takes one of the operators from appendix B creating operators that transform according to one-dimensional (i, j = 1) or two-dimensional (i, j = 1, 2) representations of G. Invariance under symmetry transformations is preserved if for every Rˇ ∈ G the scalar functions hir;jk(p , q , p, q) hrk;ij = Dirl(Rˇ)Rˇ(hrk;lj). l Below is a list with a choice for the final 320 operators [Sˇk]. An electronic version of this list is available upon request from . Page 14 of 15 Operators from the second column (j = 2): r ( 1, 1, 1, 3 ) ( 1, 1, 2, 3 ) ( 1, 2, 1, 3 ) ( 1, 2, 2, 3 ) ( 2, 1, 1, 3 ) ( 2, 1, 2, 3 ) ( 2, 2, 1, 3 ) ( 2, 2, 2, 3 ) ˆSˇ127˜˜ == GGr22 22 `ˆQˇ61˜˜´´ r `ˆQˇ3˜´ ˆSˇ131˜ = Gr2 2 `ˆQˇ117˜´ ˆSˇ135˜ = Gr2 2 `ˆQˇ129˜´ ˆSˇ139˜ = Gr2 2 `ˆQˇ133˜´ ˆSˇ143˜ = Gr2 2 `ˆQˇ138˜´ ˆSˇ147˜ = Gr2 2 `ˆQˇ177˜´ ˆSˇ151˜ = Gr2 2 `ˆQˇ182˜´ ˆSˇ155˜ = Gr2 2 `ˆQˇ205˜´ ˆSˇ159˜ = Gr2 2 `ˆQˇ207˜´ ˆSˇ163˜ = Gr2 2 `ˆQˇ218˜´ ˆSˇ167˜ = Gr2 2 `ˆQˇ245˜´ ˆSˇ171˜ = Gr2 2 `ˆQˇ256˜´ ˆˆSSˇˇ117795˜ = Grr2 2 ``ˆˆQQˇˇ128705˜´ ˆˆSSˇˇ118851˜˜˜ === GGGrr222 222 ``ˆˆQQˇˇ483˜˜˜´´´ ˆSˇ189˜ = Gr2 2 `ˆQˇ75 ˜´ ˆSˇ193˜ = Gr2 2 `ˆQˇ284˜´ ˆSˇ197˜ = Gr2 2 `ˆQˇ306˜´ ˆSˇ201˜ = Gr2 2 `ˆQˇ311˜´ ˆˆˆˆˆSSSSSˇˇˇˇˇ222221110059159˜˜˜˜˜ ===== GGGGGrrrrr22222 22222 `````ˆˆˆˆˆQQQQQˇˇˇˇˇ5213254815959˜˜˜˜˜´´´´´ ˆˆˆSSSˇˇˇ222322173˜˜˜ === GGGrrr222 222 ```ˆˆˆQQQˇˇˇ121422409˜˜˜´´´ ˆSˇ235˜ = Gr2 2 `ˆQˇ221˜´ ˆSˇ239˜ = Gr2 2 `ˆQˇ253˜´ ˆˆˆSSSˇˇˇ222444973˜˜˜ === GGGrrr222 222 ```ˆˆˆQQQˇˇˇ12204775˜˜˜´´´ ˆSˇ253 r `ˆQˇ305 ˆˆSSˇˇ226517˜˜˜ === GGGrr222 222 ``ˆˆQQˇˇ187˜˜˜´´´ ˆSˇ265˜ = Gr2 2 `ˆQˇ42 ˜´ ˆSˇ269˜ = Gr2 2 `ˆQˇ170˜´ ˆSˇ273˜ = Gr2 2 `ˆQˇ226˜´ ˆSˇ277˜ = Gr2 2 `ˆQˇ281˜´ ˆSˇ281˜ = Gr2 2 `ˆQˇ291˜´ ˆSˇ285˜ = Gr2 2 `ˆQˇ293˜´ ˆSˇ289˜ = Gr2 2 `ˆQˇ297˜´ ˆSˇ293˜ = Gr2 2 `ˆQˇ299˜´ ˆSˇ297˜ = Gr2 2 `ˆQˇ306˜´ ˆSˇ301˜ = Gr2 2 `ˆQˇ308˜´ ˆSˇ305˜ = Gr2 2 `ˆQˇ311˜´ ˆSˇ309˜ = Gr2 2 `ˆQˇ316˜´ ˆSˇ313 r `ˆQˇ318 ˆSˇ317˜ = G2 2 ˆSˇ128˜˜ == GGr11 22 `ˆQˇ61˜˜´´ r `ˆQˇ3˜´ ˆSˇ132˜ = Gr1 2 `ˆQˇ117˜´ ˆSˇ136˜ = Gr1 2 `ˆQˇ129˜´ ˆSˇ140˜ = Gr1 2 `ˆQˇ133˜´ ˆSˇ144˜ = Gr1 2 `ˆQˇ138˜´ ˆSˇ148˜ = Gr1 2 `ˆQˇ177˜´ ˆSˇ152˜ = Gr1 2 `ˆQˇ182˜´ ˆSˇ156˜ = Gr1 2 `ˆQˇ205˜´ ˆSˇ160˜ = Gr1 2 `ˆQˇ207˜´ ˆSˇ164˜ = Gr1 2 `ˆQˇ218˜´ ˆSˇ168˜ = Gr1 2 `ˆQˇ245˜´ ˆSˇ172˜ = Gr1 2 `ˆQˇ256˜´ ˆˆSSˇˇ117860˜ = Grr1 2 ``ˆˆQQˇˇ217850˜´ ˆˆSSˇˇ118826˜˜˜ === GGGrr111 222 ``ˆˆQQˇˇ843˜˜˜´´´ ˆSˇ190˜ = Gr1 2 `ˆQˇ75 ˜´ ˆSˇ194˜ = Gr1 2 `ˆQˇ284˜´ ˆSˇ198˜ = Gr1 2 `ˆQˇ306˜´ ˆSˇ202˜ = Gr1 2 `ˆQˇ311˜´ ˆˆˆˆˆSSSSSˇˇˇˇˇ222221211060206˜˜˜˜˜ ===== GGGGGrrrrr11111 22222 `````ˆˆˆˆˆQQQQQˇˇˇˇˇ5122358451599˜˜˜˜˜´´´´´ ˆˆˆSSSˇˇˇ222322284˜˜˜ === GGGrrr111 222 ```ˆˆˆQQQˇˇˇ112422490˜˜˜´´´ ˆSˇ236˜ = Gr1 2 `ˆQˇ221˜´ ˆSˇ240˜ = Gr1 2 `ˆQˇ253˜´ ˆˆˆSSSˇˇˇ222544084˜˜˜ === GGGrrr111 222 ```ˆˆˆQQQˇˇˇ21240775˜˜˜´´´ ˆSˇ254 r `ˆQˇ305 ˆˆSSˇˇ226528˜˜˜ === GGGrr111 222 ``ˆˆQQˇˇ187˜˜˜´´´ ˆSˇ266˜ = Gr1 2 `ˆQˇ42 ˜´ ˆSˇ270˜ = Gr1 2 `ˆQˇ170˜´ ˆSˇ274˜ = Gr1 2 `ˆQˇ226˜´ ˆSˇ278˜ = Gr1 2 `ˆQˇ281˜´ ˆSˇ282˜ = Gr1 2 `ˆQˇ291˜´ ˆSˇ286˜ = Gr1 2 `ˆQˇ293˜´ ˆSˇ290˜ = Gr1 2 `ˆQˇ297˜´ ˆSˇ294˜ = Gr1 2 `ˆQˇ299˜´ ˆSˇ298˜ = Gr1 2 `ˆQˇ306˜´ ˆSˇ302˜ = Gr1 2 `ˆQˇ308˜´ ˆSˇ306˜ = Gr1 2 `ˆQˇ311˜´ ˆSˇ310˜ = Gr1 2 `ˆQˇ316˜´ ˆSˇ314 r `ˆQˇ318 ˆSˇ318˜ = G1 2 Open Access This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. 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K. Topolnicki. General operator form of the non-local three-nucleon force, The European Physical Journal A, 2017, 181, DOI: 10.1140/epja/i2017-12376-4