JHEP01(2019)007
mass term for the complicated fields mixes different flavors by means of the matrices
˜
S
(1)
i
and
˜
S
(2)
i
and we will have to diagonalize the color and flavor part simultaneously. Note that
compared to the solution where only three scalar fields get non-trivial SO(3)-symmetric vevs
studied in [
13
], all scalars
˜
φ
i
are now complicated bosons and only the three components of
the gauge fields
A
0
, A
1
, A
2
and the ghost field remain easy. We will denote the eigenvalues
of the matrices inside the trace in (2.13) by m
2
.
2.2.1 Decomposition of the color matrices and easy fields
In order to proceed with the diagonalization, we decompose the color part of a generic field
Φ in blocks:
Φ = [Φ]
n,n
0
E
n
n
0
+ [Φ]
n,a
E
n
a
+ [Φ]
a,n
E
a
n
+ [Φ]
a,a
0
E
a
a
0
, (2.15)
with
n, n
0
= 1
, . . . , k
1
k
2
and
a, a
0
=
k
1
k
2
+ 1
, . . . , N
. Here
E
n
n
0
are
N ×N
matrices with a
single non-vanishing entry, namely a 1 at position (
n, n
0
). The fields [Φ]
n,a
and [Φ]
a,n
will
often be referred to as fields in the off-diagonal block.
The fields [Φ]
a,a
0
in the (N − k
1
k
2
) × (N −k
1
k
2
) block are massless since
L
(1)
i
E
a
a
0
=
h
t
k
1
i
⊗ 1
k
2
⊕ 0
N−k
1
k
2
, E
a
a
0
i
= 0, (2.16)
and similarly for
L
(2)
i
. One can think of this result as the statement that the indices
a
and
a
0
are singlets under su(2) × su (2).
The matrices
E
n
a
and
E
a
n
transform in the (
k
1
×k
2
)-dimensional irreducible represen-
tation of su(2) × su(2),
L
(1)
i
E
n
a
= E
n
0
a
[t
k
1
i
⊗ 1
k
2
]
n
0
,n
, L
(1)
i
E
a
n
= −[t
k
1
i
⊗ 1
k
2
]
n,n
0
E
a
n
0
,
L
(2)
i
E
n
a
= E
n
0
a
[1
k
1
⊗ t
k
2
i
]
n
0
,n
, L
(2)
i
E
a
n
= −[1
k
1
⊗ t
k
2
i
]
n,n
0
E
a
n
0
.
(2.17)
Equivalently, each index
n
transforms in the same representation as
t
i
, namely the one
with spins
`
1
=
k
1
−1
2
and
`
2
=
k
2
−1
2
. It follows that the matrices
E
n
a
and
E
a
n
already
diagonalize the quadratic Casimir operators,
(L
(1)
)
2
E
n
a
=
k
2
1
− 1
4
E
n
a
, (L
(1)
)
2
E
a
n
=
k
2
1
− 1
4
E
a
n
, (2.18)
and analogously for (
L
(2)
)
2
. The matrices
E
n
a
and
E
a
n
transform into each other under
Hermitian conjugation, and this behavior carries over to the fields [Φ]
n,a
and [Φ]
a,n
in the
off-diagonal block:
(E
n
a
)
†
= E
a
n
, [Φ]
†
n,a
≡
[Φ]
n,a
†
= [Φ]
a,n
. (2.19)
Moreover, they are orthogonal and normalized in the sense that
tr
h
(E
n
a
)
†
E
n
0
a
0
i
= δ
nn
0
δ
aa
0
, tr
h
(E
a
n
)
†
E
n
0
a
0
i
= 0,
tr
h
(E
a
n
)
†
E
a
0
n
0
i
= δ
aa
0
δ
nn
0
, tr
h
(E
n
a
)
†
E
a
0
n
0
i
= 0.
(2.20)
– 8 –
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