#### 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
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