Symmetric-group decomposition of SU(N) group-theory constraints on four-, five-, and six-point color-ordered amplitudes

0.39 MB·English

by  Alexander C. Edison

BOW-PH-155
Symmetric-group decomposition of
SU(N) group-theory constraints on
four-, five-, and six-point color-ordered amplitudes
2
1
at all loop orders
0
2

l
u
J

3 Alexander C. Edison and Stephen G. Naculich1
2

] Department of Physics
h
t Bowdoin College
-
p Brunswick, ME 04011, USA
e
h
[email protected]
[

[email protected]
1
v
1
1
5
5
.
7 Abstract
0
2 Color-ordered amplitudes for the scattering of n particles in the adjoint representation
1
of SU(N) gauge theory satisfy constraints that arise from group theory alone. These con-
:
v
straints break into subsets associated with irreducible representations of the symmetric
i
X group S , which allows them to be presented in a compact and natural way. Using an
n
r iterative approach, we derive the constraints for six-point amplitudes at all loop orders,
a
extending earlier results for n = 4 and n = 5. We then decompose the four-, five-, and
six-point group-theory constraints into their irreducible S subspaces. We comment briefly
n
on higher-point two-loop amplitudes.
1 Research supported in part by the National Science Foundation under Grant No. PHY10-67961.
1
1 Introduction
Scattering amplitudes in gauge theory may be given a gauge-invariant decomposition in terms
of color-ordered (or partial) amplitudes [1,2]. Color-ordered amplitudes are not independent
but satisfy a number of constraints, some of which follow from the recently-discovered color-
kinematic duality [3–5] and were proven in refs. [6–9]. Other constraints among the color-ordered
amplitudes, however, followdirectlyfromgrouptheory, andhavebeenknownforovertwodecades
at tree-level [10] and at one loop [11]. Four-point color-ordered amplitudes were also known to
obey group-theory relations at two loops [12] and these were recently generalized to all loop
orders [13]. All-loop-order group-theory constraints were also recently derived for five-point
amplitudes [14]. Other recent work on loop-order relations among color-ordered amplitudes
includes refs. [15–20].
In this paper, we derive the SU(N) group-theory constraints for six-point color-ordered am-
plitudes at all loop orders, following the iterative approach of refs. [13,14] and generalizing the
known relations at tree level [10] and at one loop [11]. The other major goal of this paper is
to describe how group-theory constraints on n-point color-ordered amplitudes naturally fall into
sets associated with irreducible representations of the symmetric group S . We decompose the
n
constraints for n = 4, 5, and 6 into their irreducible S subspaces, thus presenting them in a
n
compact way.
The derivation of group-theory constraints can be cast as a straightforward problem in linear
algebra2 that emerges from two alternative ways of expressing the color structure of a gauge
theory amplitude. One way is to decompose the amplitude into a color basis [22,23]
= a c (1.1)
i i
A
i
X
where a carries the momentum- and polarization-dependence of the amplitude, and the color
i
factors c are obtained by sewing together the gauge theory factors from all the vertices of the
i
contributingFeynmandiagrams. Inatheorythatcontainsonlyfieldsintheadjointrepresentation
of SU(N), such as pure or supersymmetric Yang-Mills theory, each cubic vertex contributes a
factor of the SU(N) structure constants fabc, and each quartic vertex contributes factors of
fabefcde, facefbde, and fadefbce, which are equivalent from a purely color point-of-view to a pair
of cubic vertices sewn along one leg. Hence a complete set of color factors c can be constructed
i
{ }
from diagrams with cubic vertices only. An independent basis of color factors3 for tree-level and
one-loop n-point amplitudes was described in refs. [22,23].
The alternative trace decomposition [1,2]
= A t (1.2)
λ λ
A
λ
X
expresses the amplitude in terms of gauge-invariant color-ordered amplitudes A with respect
λ
to a basis t of single and (at loop level) multiple traces of gauge group generators Ta in the
λ
{ }
2 A similar approachwas taken to the BCJ relations [3] in ref. [21].
3 Thecolorfactorsconstructedfromthesetofallcubicdiagramsaregenerallynotindependentbutarerelated
byJacobirelations. Suchanovercompletesetis usuallyrequiredto makecolor-kinematicdualitymanifest[3,24].
2
defining representation of SU(N). We will characterize this basis more explicitly in sec. 2. The
dimension of the trace basis, however, is larger than that of the independent color basis4, so there
is redundancy among the color-ordered amplitudes.
The color (1.1) and trace (1.2) decompositions can be related by using5
f˜abc = i√2fabc = Tr([Ta,Tb]Tc), (1.3)
together with the SU(N) relations given later in eq. (6.20), to write each color factor c as a
i
linear combination of trace factors
c = M t . (1.4)
i iλ λ
λ
X
Since the dimension of the trace space is larger than the number of independent color factors,
the linear combinations given by eq. (1.4) span a proper subspace (which we will refer to as the
color space) of the trace space. Consequently, the transformation matrix M possesses a set of
iλ
independent null eigenvectors
M r = 0, m = 1, ,n (1.5)
iλ λm null
···
λ
X
whose number n is the difference between the dimensions of the trace space and the color
null
space. The null vectors r = r t are orthogonal to the color factors c with respect to the
m λ λm λ i
inner product
P
(tλ,tλ′) = δλλ′ (1.6)
and hence span the orthogonal complement of the color space, which we will refer to as the null
space. One combines eq. (1.4) with eqs. (1.1) and (1.2) to express the color-ordered amplitudes
as
A = a M . (1.7)
λ i iλ
i
X
The existence of the null eigenvectors (1.5) implies a set of constraints
A r = 0, m = 1, ,n (1.8)
λ λm null
···
λ
X
which we refer to as group-theory relations. Hence, specifying the null space is equivalent to
specifying the complete set of group-theory relations satisfied by the color-ordered amplitudes.
For tree-level n-point amplitudes, the relations (1.8), which number 1(n 3)(n 2)!, are the
2 − −
Kleiss-Kuijf [10,23] relations. For one-loop n-point amplitudes, the group-theory relations were
given in refs. [11,23]. At two loops and above, the full set of constraints on n-point color-ordered
amplitudes has yet to be identified6, although all-loopgroup-theory relations for n = 4 and n = 5
have been derived using an iterative approach in refs. [13,14].
4 For example, the number of independent color factors for tree-level n-point amplitudes is (n 2)! while the
−
dimension of the trace basis is 1(n 1)!.
2 −
5 The generators are normalized using Tr(TaTb)=δab.
6 See refs. [12,16] for partial results at two loops.
3
For four-point amplitudes, there are 4 group-theory relations at each loop order L 2 (with
≥
1 at tree level, and 3 at one loop). The form of the constraints begins to repeat after three loops,
so that the even-loop constraints for L 4 are essentially equivalent to the two-loop constraints,
≥
and the odd-loop constraints for L 5 are equivalent to the three-loop constraints [13].
≥
For five-point amplitudes, there are 10 group-theory relations for odd L, and 12 for even L 2
≥
(with 6 at tree level). Again the form of the constraints repeats after two loops, so that odd-loop
constraints for L 3 are equivalent to the one-loop constraints, and even-loop constraints for
≥
L 4 are equivalent to the two-loop constraints [14].
≥
In this paper we will derive the group-theory relations for six-point amplitudes at all loop
orders using the same iterative approach. We will find that there are 76 group-theory relations
for L 3, with 36 at tree level, 65 at one loop, and 80 at two loops. Once again the form
≥
of the constraints presents a repeating pattern, which only begins, however, after five loops, so
that even-loop constraints for L 6 are equivalent to the four-loop constraints, and odd-loop
≥
constraints for L 7 are equivalent to the five-loop constraints.
≥
A subset of the group-theory relations obeyed by color-ordered amplitudes can be derived
as U(1) decoupling relations [2,25–27]. To obtain these, one enlarges the gauge group from
SU(N) to U(N). The trace decomposition is expanded to include terms with factors of Tr(Ta)
(which automatically vanish in the SU(N) trace decomposition). Next, one observes that an
n-point amplitude containing any number of external U(1) gauge bosons vanishes because the
associated structure constants are zero. By setting the corresponding U(N) generators in the
full U(N) trace decomposition equal to the unit matrix, one obtains a series of equations that
relatetheremainingSU(N)amplitudes. ThesearetheU(1)decouplingrelations. Throughoutthe
paper, we will indicate which of the group-theory relations can be obtained from U(1) decoupling
relations and which cannot.
The group-theory relations at four and five points were presented in refs. [13,14] by specifying
(L) (L)
a complete set of L-loop null vectors r in terms of their components r with respect to
{ m } λm
(L)
an explicit L-loop trace basis t . While this approach has the advantage of explicitness, it
{ λ }
quickly becomes unwieldy as the dimensions of both the null space and the trace space grow
with n. For example, listing the 200 coefficients of the 80 null vectors for two-loop six-point
amplitudes would hardly be enlightening. Clearly, a more streamlined approach is called for.
Such a compact approach for presenting the null vectors of n-point amplitudes is provided by
the representation theory of the symmetric group S . Since a permutation of the arguments of
n
an element of the trace basis t for n-point amplitudes yields another element of the trace basis,
λ
the trace space constitutes a (reducible) representation of the symmetric group S . Similarly, a
n
permutation of the external legs of a diagram corresponding to a color factor c yields a diagram
i
corresponding to another color factor, hence the color space constitutes an invariant subspace
(with respect to permutations) of the trace space. The orthogonal complement of the color space,
namely the null space, is therefore also an invariant subspace, since the inner product (1.6) is
invariantunder permutations. Thus thenull spaceformsa(reducible) representation ofS , which
n
can be decomposed into a direct sum of irreducible representations. Indeed, the null vectors for
five-point amplitudes found in ref. [14] naturally broke into sets of 6 and 4, and correspond to
irreducible representations of S . Thus, a compact and efficient way to characterize the null space
5
4
is to describe its constituent irreducible representations.
Inthispaper, wewill present thenullspaces offour-, five-, andsix-point amplitudesatallloop
orders in terms of their irreducible subspaces with respect to S . We will find, for example, that
n
the 80 vectors spanning the null space for two-loop six-point amplitudes can be classified into 13
irreducible representations whose dimensions vary from 1 to 16. In addition to compactness, this
method has the advantage that the results are presented in a form independent of the specific
choice of trace basis.
We also briefly examine whether the tree-level Kleiss-Kuijf relations also apply to subleading-
color single-trace n-point amplitudes at loop level. We confirm the results of ref. [16] that at two
loops, the KK relations hold for n 7, but fail for eight-point amplitudes. We check that they
≤
fail for nine-point amplitudes as well.
We begin in section 2 with an explicit look at the trace basis for n-point amplitudes through
two loops. In section 3, we review some representation theory of the symmetric group, showing
how Young tableaux can be used to construct projection operators onto irreducible representa-
tions for regular and induced representations. In sections 4, 5, and 6, we describe the decomposi-
tion of the null spaces of four-, five-, and six-point amplitudes into irreducible representations of
S . Section 7 discusses Kleiss-Kuijf relations at looplevel, and section 8 contains our conclusions.
n
Various additional details can be found in two appendices.
2 Trace bases for gauge-theory amplitudes
In this section, we spell out the explicit form of the trace basis for an n-point amplitude through
two loops as a prelude to examining its decomposition into irreducible representations of S later
n
in the paper.
(0)
First we consider tree-level n-point amplitudes . One can use eq. (1.3) together with
n
f˜abcTa = [Tb,Tc] to decompose the color factor c buAilt from any tree-level color diagram into a
i
linear combination of single-trace terms, so that the amplitude may be expressed as [1]
(0) = A(σ(1), ,σ(n))Tr(σ(1) σ(n)) (2.1)
An ··· ···
σ∈XSn/ZZn
whereZZ denotesthesubgroupofcyclicpermutations. Foreasier readabilitywewriteTr(12 n)
n
···
for Tr(Ta1Ta2 Tan). In addition to being invariant under cyclic permutations of their argu-
···
ments (due to the cyclicity of the trace), color-ordered tree amplitudes are invariant (up to sign)
under reversal of the arguments
A(σ(n), ,σ(2),σ(1)) = ( 1)nA(σ(1),σ(2), ,σ(n)) (2.2)
··· − ···
as can be seen by using the antisymmetry of f˜abc. Hence, the tree-level n-point amplitude can
be written
ns
(0) = A(0)T (2.3)
An λ λ
λ=1
X
5
in terms of a basis T of dimension n = 1(n 1)!, consisting of sums/differences of the form
{ λ} s 2 −
Tr(12 n)+( 1)nTr(n 21) (2.4)
··· − ···
and permutations thereof.
At one loop and above, the amplitude also contains double-trace terms
A(σ(1), ,σ(p); σ(p+1), σ(n)) Tr(σ(1) σ(p)) Tr(σ(p+1) σ(n)), σ S (2.5)
n
··· ··· ··· ··· ∈
with 2 p n/2 . The color-ordered amplitudes again satisfy a reflection property
≤ ≤ ⌊ ⌋
A(σ(p), ,σ(1); σ(n), ,σ(p+1)) = ( 1)nA(σ(1), ,σ(p); σ(p+1), ,σ(n)) (2.6)
··· ··· − ··· ···
allowing the amplitude to be written in terms of a basis containing terms of the form
Tr(1 p)Tr((p+1) n)+( 1)nTr(p 1)Tr(n (p+1)) (2.7)
··· ··· − ··· ···
and permutations thereof, which we label as T with λ = n +1, ,n +n . The number of
λ s s d
{ } ···
independent terms of the form (2.7) is n = 1 n−2n!/p(n p), except for n = 4, in which case
d 4 p=2 −
there are 3 such terms. The full one-loop n-point amplitude can then be expressed as [2]
P
ns+nd NT , λ = 1, ,n
(1) = A(1)t(1), t(1) = λ ··· s (2.8)
An λ λ λ (Tλ, λ = ns +1, ,ns +nd.
λ=1 ···
X
Beginning at two loops (and n 6), the trace basis also requires triple-trace terms of the
≥
form
Tr(1 p)Tr((p+1) q)Tr((q +1) n)+( 1)nTr(p 1)Tr(q (p+1))Tr(n (q +1))
··· ··· ··· − ··· ··· ···
(2.9)
and permutations thereof, which we label as T with λ = n +n +1, ,n +n +n . The
λ s d s d t
{ } ···
full two-loop n-point amplitude can be written
N2T , λ = 1, ,n
λ s
···
2ns+nd+nt NT , λ = n +1, ,n +n
(2) = A(2)t(2), t(2) =  λ s ··· s d
An Xλ=1 λ λ λ Tλ, λ = ns +nd +1,··· ,ns +nd +nt
T , λ = n +n +n +1, ,2n +n +n
λ−ns−nd−nt s d t ··· s d t
 (2.10)


which includes subleading-color single-trace amplitudes, as well as leading-color single-trace am-
plitudes and double- and triple-trace amplitudes.
One can continue this procedure at higher loops, where the L-loop trace basis will include (for
general n) up to (L+1)-trace terms, as well as additional subleading-color terms. In secs. 4-6, we
will characterize the general L-loop basis for n = 4, 5, and 6, which only require up to triple-trace
terms.
We now turn to the decomposition of the single- and multiple-trace bases into irreducible
representation of S .
n
6
3 Projection operators for representations of S
n
In this section, we review some standard results from the representation theory of the symmetric
group S , which can be found in many group theory textbooks. We found ref. [28] particularly
n
useful. First we will describe a method to project the reducible regular representation onto its
irreducible subspaces. Then we will turn our attention to the induced representations spanned
by the trace bases described in sec. 2.
3.1 Regular representation
We begin by recalling that irreducible representations of S are labeled by τ Y , where Y
n n n
∈
denotes the set of Young tableaux with n boxes. The dimension of the representation labeled by
τ is given by d = n!/H, where H is the product of all the hook lengths associated with τ.
τ
Then!-dimensional regularrepresentationofS isreducible, andcontainsalloftheirreducible
n
representations τ of S , each with a multiplicity equal to d , which implies that d2 = n!.
n τ τ∈Yn τ
For example, the regular representation of S reduces into
4
P
Rreg = 3 2 3
S4 ⊕ ⊕ ⊕ ⊕ . (3.1)
dim = 24 = 1 + 3 3 + 2 2 + 3 3 + 1
· · ·
A basis f for the regular representation of S can be constructed from an arbitrary function
σ n
{ }
of n variables f(x ,x , ,x ) by permuting the arguments
1 2 n
···
f f(x ,x , ,x ), σ S . (3.2)
σ σ(1) σ(2) σ(n) n
≡ ··· ∈
Then a permutation ρ S acts on f by7
n σ
∈
ρ·fσ ≡ fρ·σ = fσ′Dσre′gσ(ρ) (3.3)
σX′∈Sn
where Dreg(ρ) are the matrices of the regular representation. We can put Dreg(ρ) into block
σ′σ σ′σ
diagonal form by choosing a different basis fτ defined by
{ ij}
fτ = eτ f(x ,x , ,x ), τ Y , i,j = 1, ,d (3.4)
ij ij · 1 2 ··· n ∈ n ··· τ
where eτ areasetofn!elementsofthegroupalgebra(i.e., linearcombinationsofpermutations)
{ ij}
that satisfy (for each τ) the simple matrix algebra
eτ eτ′ = δττ′δ eτ . (3.5)
kl ij li kj
7 Our conventionis that ρ σ denotes the permutationobtained by acting firstwith σ and then with ρ. Hence
·
if σ = (12) and ρ = (23) then ρ σ = (132), where we use the cycle notation for permutations. Note that the
·
permutations act on the labels rather than the positions of the arguments, so that (123) f(x ,x ,x ,x ) =
4 3 2 1
·
f(x ,x ,x ,x ) and not f(x ,x ,x ,x ).
4 1 3 2 2 4 3 1
7
A permutation ρ acts on this basis as8
dτ
ρ fτ = fτ Dτ (ρ) (3.6)
· ij kj ki
k=1
X
showing that each subset of functions
fτ i = 1, ,d (3.7)
{ ij | ··· τ}
spans anirreducible subspace of theregular representation. Hence, eτ act as projectionoperators
ij
from the regular representation onto irreducible representations of S . The index j = 1, ,d
n τ
···
labels distinct copies of the representation τ, each of which transforms according to the same set
of matrices Dτ (ρ).
ki
Explicit expressions for eτ may be constructed from Young tableaux as follows [28,29]. For
ij
any representation τ, define the set of standard tableaux τ , i = 1, ,d , using the ordering
i τ
···
given on p. 139 of ref. [28]. Then for each standard tableau τ , define the row symmetrizer P
i i
as the sum of permutations that exchange only symbols within the same row, and the column
antisymmetrizer Q as the (signed) sum of permutations that exchange only symbols within the
i
same column, where the sign is given by the parity of the permutation. Finally, define s as the
ij
permutation of symbols that converts tableau τ into τ . For example, the standard tableaux for
j i
the representation of S are given by
4
1 2 1 3
τ = 3 4 τ = 2 4 (3.8)
1 2
so that
P = 1+(12)+(34)+(12)(34), Q = 1 (13) (24)+(13)(24), s = (23),
1 1 12
− −
P = 1+(13)+(24)+(13)(24), Q = 1 (12) (34)+(12)(34), s = (23). (3.9)
2 2 21
− −
With these definitions, one can construct projection operators
j−1
d
eτ = τ Q s P M , M = 1 eτ (3.10)
ij n! i ij j j j − kk
(cid:18) (cid:19) k=1
X
that obey the simple matrix algebra (3.5). The diagonal elements eτ are mutually annulling
ii
idempotents
eτ eτ′ = δττ′δ eτ (3.11)
kk ii ki ii
known as Young operators.
In the next subsection, we show that eτ can also be used to project single- and multiple-trace
ij
spaces onto irreducible representations of S .
n
8 Equation (3.6) follows from eq. (3.5) and the fact that any ρ is a linear combination of the eτ .
kl
8
3.2 Induced representations
As discussed in sec. 2, gauge theory amplitudes can be expressed in terms of a basis of single and
multiple traces of SU(N) generators. Since these sets of traces are closed under permutations of
their arguments, they form representations of the symmetric group. These single- and multiple-
trace representations are induced by one-dimensional representations of various subgroups of
the symmetric group. They are in general reducible, and in this section we describe how to
determine their decomposition into irreducible representations, and how to use the projection
operators introduced in sec. 3.1 to find their irreducible subspaces.
First consider an element of the single-trace basis introduced in sec. 2
Tr(12 n)+( 1)nTr(n 21). (3.12)
··· − ···
Permutations act on eq. (3.12) in the same way as on f(x , ,x ), namely
1 n
···
ρ Tr(12 n)+( 1)nTr(n 21) = Tr(ρ(1)ρ(2) ρ(n))+( 1)nTr(ρ(n) ρ(2)ρ(1))
· ··· − ··· ··· − ···
h i (3.13)
thus generating a (reducible) representation of S , which we will refer to as the single-trace
n
representation R(ind). Equation (3.12) is invariant9 under the dihedral subgroup D S of
n n n
⊂
(ind)
order 2n (generated by cyclic permutations and reversal of indices), so the dimension of R
n
is 1(n 1)!. The superscript (ind) denotes that R(ind) is the representation induced by a one-
2 − n
(ind)
dimensional representation of the dihedral group. When n is even, R is induced by the trivial
n
representation of the dihedral subgroup; when n is odd, it is induced by the representation of
the dihedral subgroup that assigns +1 to cyclic permutations and 1 to reversals.
(ind) −
The single-trace representation R is a subset of the regular representation, and contains
n
m d copies of the irreducible representation τ of S , where the multiplicities m may be
τ τ n τ
≤
determined by character analysis (see appendix A). As in sec. 3.1, we can explicitly construct
(ind)
the irreducible subspaces of R using the projection operators (3.10). For each τ Y , we
n n
∈
construct a basis of the associated irreducible representation(s) via
v ( n τ ,j ) eτ [Tr(1 n)+( 1)nTr(n 1)], i = 1, ,d . (3.14)
i | ≡ ij ··· − ··· ··· τ
For each τ, there are several sets of basis elements, labeled by j = 1, ,d , but if τ appears m
τ τ
(ind) ···
times in R , then only m of these sets will be linearly independent, and we will use only the
n τ
first m (independent) values of j. (If m = 0, then all of the associated v ( n τ ,j ) vanish.)
τ τ i
|
If m = 1, then we will use the lowest value of j for which v (n τ,j) does not vanish, generally
τ i
|
j = 1, and will omit j from the argument, i.e. v (n τ).
i
|
Let us illustrate this with an example. Consider the four-point single-trace representation
(ind)
R spanned by
4
Tr(1234)+Tr(1432), Tr(1243)+Tr(1342), Tr(1324)+Tr(1423) . (3.15)
{ }
9up to a sign when n is odd
9
Using character analysis, as described in appendix A, we find that this reduces into
(ind)
R = .
4 ⊕ (3.16)
3 = 1 + 2
Since e = 1 σ, the one-dimensional subspace corresponding to consists of the com-
24 σ∈S4
pletely symmetric sum of single-trace terms
P
v( 4 ) = e Tr(1234)+Tr(4321) (3.17)
|
= 1 Tr(1234)+Tr(1432)+Tr(1243)+Tr(1342)+Tr(1324)+Tr(1432)
3 (cid:2) (cid:3)
while the two-dimensio(cid:2)nal subspace corresponding to is spanned by (cid:3)
1 [Tr(1243)+Tr(1342) Tr(1324) Tr(1423)], i = 1,
3 − −
v ( 4 ) = e Tr(1234)+Tr(4321) =
i | i1 
1 [Tr(1243)+Tr(1342) Tr(1234) Tr(1432)], i = 2.
(cid:2) (cid:3) 3 − −
(3.18)

 (ind)
Asexplainedabove, since appearsonlyonceinR ,v ( 4 ,2 )isproportionaltov ( 4 )
4 i | i | ≡
v ( 4 ,1 ).
i
|
Next we consider double-trace representations. An element of the form
Tr(1 p)Tr((p+1) n)+( 1)nTr(p 1)Tr(n (p+1)) (3.19)
··· ··· − ··· ···
is invariant (up to sign when n is odd) under the subgroup of S generated by cyclic permuta-
n
tions of the arguments of each trace and by simultaneous reversal of the arguments within each
(ind)
trace (and, if p = n/2, by an exchange of the two traces). We will denote by R the S
n;p−n n
representation induced by the representation of this subgroup that leaves eq. (3.19) invariant.
(ind)
The multiplicities of the irreducible representations τ contained in R may again be ob-
n;p−n
tained by character analysis, and the irreducible subspace(s) corresponding to τ are spanned
by
v ( p;n p τ ,j ) eτ Tr(1 p)Tr((p+1) n)+( 1)nTr(p 1)Tr(n (p+1)) .
i − | ≡ ij ··· ··· − ··· ···
(3.20)
(cid:2) (cid:3)
(ind)
For example, the four-point double-trace representation R is spanned by
2;2
2 Tr(12)Tr(34), 2 Tr(13)Tr(24), 2 Tr(14)Tr(23) (3.21)
{ }
and reduces into
(ind)
R = .
2;2 ⊕ (3.22)
3 = 1 + 2
The one-dimensional subspace corresponding to consists of thesymmetric sum of double-trace
terms
v( 2;2 ) = e 2Tr(12)Tr(34) = 1 [2Tr(12)Tr(34)+2Tr(13)Tr(24)+2Tr(14)Tr(24)]
| 3
(3.23)
(cid:2) (cid:3)
10