# Baryon Fields with $U_{L}(3)\times U_{R}(3)$ Chiral Symmetry: Axial Currents
of Nucleons and Hyperons

Hua-Xing Chen1<EMAIL_ADDRESS>V. Dmitrašinović2<EMAIL_ADDRESS>Atsushi Hosaka3<EMAIL_ADDRESS>1Department of Physics and State Key
Laboratory of Nuclear Physics and Technology Peking University, Beijing
100871, China
2 Vinča Institute of Nuclear Sciences, lab 010, P.O.Box 522, 11001 Beograd,
Serbia
3 Research Center for Nuclear Physics, Osaka University, Ibaraki 567–0047,
Japan

###### Abstract

We use the conventional $F$ and $D$ octet and decimet generator matrices to
reformulate chiral properties of local (non-derivative) and one-derivative
non-local fields of baryons consisting of three quarks with flavor $SU(3)$
symmetry that were expressed in $SU(3)$ tensor form in Ref. Chen:2008qv . We
show explicitly the chiral transformations of the $[(6,3)\oplus(3,6)]$ chiral
multiplet in the “$SU(3)$ particle basis”, for the first time to our
knowledge, as well as those of the $(3,\overline{3})\oplus(\overline{3},3)$,
$(8,1)\oplus(1,8)$ multiplets, which have been recorded before in Refs.
Lee:1968 ; Bardeen:1969ra . We derive the vector and axial-vector Noether
currents, and show explicitly that their zeroth (charge-like) components close
the $SU_{L}(3)\times SU_{R}(3)$ chiral algebra. We use these results to study
the effects of mixing of (three-quark) chiral multiplets on the axial current
matrix elements of hyperons and nucleons. We show, in particular, that there
is a strong correlation, indeed a definite relation between the flavor-singlet
(i.e. the zeroth), the isovector (the third) and the eighth flavor component
of the axial current, which is in decent agreement with the measured ones.

baryon, chiral symmetry, axial current, $F$/$D$ values

###### pacs:

14.20.-c, 11.30.Rd, 11.40.Dw

##  I Introduction

Axial current “coupling constants” of the baryon flavor octet Okun:1982ap  are
well known by now, see Ref. Yamanishi:2007zza 111for history and other
references, see Chapter 6.7 of Okun’s book Okun:1982ap  and PDG tables
Amsler:2008zzb . The zeroth (time-like) components of these axial currents are
generators of the $SU_{L}(3)\times SU_{R}(3)$ chiral symmetry that is one of
the fundamental symmetries of QCD. The general flavor $SU_{F}(3)$ symmetric
form of the nucleon axial current contains two free parameters, the so called
$F$ and $D$ couplings, which are empirically determined as $F$=$0.459\pm
0.008$ and $D$=$0.798\pm 0.008$, see Ref. Yamanishi:2007zza . The conventional
models of (linearly realized) chiral $SU_{L}(3)\times SU_{R}(3)$ symmetry,
Refs. Lee:1968 ; Bardeen:1969ra , on the other hand appear to fix these
parameters at either ($F$=0,$D$=1), which case goes by the name of
$[(3,\overline{3})\oplus(\overline{3},3)]$, or at ($F$=1,$D$=0), which case
goes by the name of $[(8,1)\oplus(1,8)]$ representation. Both of these chiral
representations suffer from the shortcoming that $F$+$D$=1$\neq
g_{A}^{(3)}=$1.267 without derivative couplings. But, even with derivative
interactions, one cannot change the value of the vanishing coupling, e.g. of
$F$=0, in $[(3,\overline{3})\oplus(\overline{3},3)]$, or of $D$=0, in
$[(8,1)\oplus(1,8)]$. Rather, one can only renormalize the non-vanishing
coupling to 1.267.

Attempts at a reconciliation of the measured values of axial couplings with
the (broken) $SU_{L}(3)\times SU_{R}(3)$ chiral symmetry go back at least 40
years Hara:1965 ; Lee:1968 ; Bardeen:1969ra ; Harari:1966yq ; Harari:1966jz ;
Gerstein:1966zz ; Weinberg:1969hw , but, none have been successful to our
knowledge thus far. As noted above, perhaps the most troublesome problem are
the $SU(3)$ axial current’s $F$,$D$ values, which problem has repercussions
for the meson-baryon interaction $F$,$D$ values, with far-reaching
consequences for hyper-nuclear physics and even astrophysics. Another, perhaps
equally important and difficult problem is that of the flavor-singlet axial
coupling of the nucleon Bass:2007zzb . This is widely thought of as being
disconnected from the $F$,$D$ problem, but we shall show that the three-quark
interpolating fields cast some perhaps unexpected light on this problem. We
shall attack both of these problems from Weinberg’s Weinberg:1969hw  point of
view, viz. chiral representation mixing, extended to the $SU_{L}(3)\times
SU_{R}(3)$ and $U_{L}(1)\times U_{R}(1)$ chiral symmetries, with added input
from three-quark baryon interpolating fields Chen:2008qv  that are ordinarily
used in QCD calculations.

The basic idea is simple: a mixture of two baryon fields belonging to
different chiral representations/multiplets has axial couplings that lie
between the extreme values determined by the two chiral multiplets that are
being mixed, and depend on the mixing angle, of course. Weinberg used this
idea to fit the iso-vector axial coupling of the nucleon using the
$[(1/2,0)\oplus(0,1/2)]$ and $[(1,1/2)\oplus(1/2,1)]$ multiplets of the
$SU_{L}(2)\times SU_{R}(2)$ chiral symmetry, but the same idea may be used on
any baryon belonging to the same octet, e.g. for the $\Lambda,\Sigma$ and
$\Xi$ hyperons. In other words, the $F$ and $D$ values of the mixture can be
determined from the $F$ and $D$ values of the $SU_{L}(3)\times SU_{R}(3)$
representations corresponding to the $[(1/2,0)\oplus(0,1/2)]$ and
$[(1,1/2)\oplus(1/2,1)]$ multiplets, viz.
$[(3,\overline{3})\oplus(\overline{3},3)]$ or $[(8,1)\oplus(1,8)]$, and
$[(6,3)\oplus(3,6)]$, respectively. The same principle holds for the
$U_{L}(1)\times U_{R}(1)$ symmetry “multiplets” and the value(s) of the flavor
singlet axial charge.

The $SU_{L}(3)\times SU_{R}(3)$ and $U_{L}(1)\times U_{R}(1)$ chiral
transformation properties of three-quark baryon interpolating fields, that are
commonly used in various QCD (lattice, sum rules) calculations, and that have
recently been determined in Ref. Chen:2008qv  will be used here as input into
the chiral mixing formalism, so as to deduce as much phenomenological
information about the axial currents of hyperons and nucleons as possible. As
a result we find three “optimal” scenarios all with identical $F$,$D$ values
(see Sect. IV).

First we recast our previous results Chen:2008qv  into the language that is
conventional for axial currents, i.e. in terms of octet $F$ and $D$ couplings.
A large part of the present paper is devoted to this notational conversion
(change of basis) and the subsequent check whether and how the resulting
chiral charges actually satisfy the $SU(3)\times SU(3)$ chiral algebra. That
is a non-trivial task for the $[(3,\overline{3})\oplus(\overline{3},3)]$ and
$(6,3)\oplus(3,6)$ representations, because they involve off-diagonal terms,
and in the latter case one of the diagonal terms in the axial current is
multiplied by a fractional coefficient, that appears to spoil the closure of
the $SU(3)\times SU(3)$ chiral algebra; the off-diagonal terms in the axial
current make crucial contributions that restore the closure. Thus, the afore-
mentioned fractional coefficient is uniquely determined.

We use these results to study the effects of mixing of (three-quark) chiral
multiplets on the axial current matrix elements of hyperons and nucleons. We
show, in particular, that there is a strong correlation between the flavor-
singlet (i.e. the zeroth), the isovector (the third) and the eighth flavor
component of the axial current. There are, in principle, three independent
observables here: the flavor-singlet (i.e. the zeroth), the isovector (the
third) and the eighth flavor component of the axial current of the nucleon. By
fitting just one mixing angle to one of these values, e.g. the (best known)
isovector coupling, we predict the other two. These predictions may differ
widely depending on the field that one assumes to be mixed with the
$(6,3)\oplus(3,6)$ field (which must be present if the isovector axial
coupling has any chance of being fit). If one assumes mixing of three fields
(again, always keeping the $(6,3)\oplus(3,6)$ as one of the three) and fits
the flavor-singlet and the isovector axial couplings, then one finds a unique
prediction for the $F$, $D$ values, which is in decent agreement with the
measured ones, modulo $SU(3)$ symmetry breaking corrections, which may be
important (for a recent fit, see Ref. Yamanishi:2007zza ). The uniqueness of
this result is a consequence of a remarkable relation, $g_{A}^{(0)}=3F-D$ that
holds for all three (five) chiral multiplets involved here, and which leads to
the relation: $g_{A}^{(0)}=\sqrt{3}g_{A}^{(8)}$, see Sect. IV.

Most of the ideas used in this paper, such as that of chiral multiplet mixing,
have been presented in mid- to late 1960’s, Refs. Harari:1966yq ;
Weinberg:1969hw ; Harari:1966jz ; Gerstein:1966zz , with the (obvious)
exception of the use of QCD interpolating fields, which arrived only a decade
afterwards/later, and the (perhaps less obvious) question of baryons’ flavor-
singlet axial current (a.k.a. the $U_{A}(1)$), which was (seriously) raised
yet another decade later.

The present paper consists of five parts: after the present Introduction, in
Sect. II we define the $SU(3)\times SU(3)$ chiral transformations of three-
quark baryon fields, with special emphasis on the $SU(3)$ phase conventions
that ensure standard $SU(2)$ isospin conventions for the isospin sub-
multiplets, and we define the ($SU(3)$ symmetric) vector and axial-vector
Noether currents of three-quark baryon fields. In Sect. III we prove the
closure of the chiral $SU_{L}(3)\times SU_{R}(3)$ algebra. In Sect. IV we
apply chiral mixing formalism to the hyperons’ axial currents and discuss the
results. Finally, in Sect. V we offer a summary and an outlook on future
developments.

##  II $SU(3)\times SU(3)$ Chiral Transformations of Three-quark Baryon Fields
and their Noether Currents

We must make sure that our conventions ensure that identical isospin
multiplets in different $SU(3)$ multiplets, such as the octet and the
decuplet, have identical isospin algebras/generators. That is a relatively
simple matter of definition, but was not the case with the octet conventions
used in Ref. Chen:2008qv . Our new definitions of the octet and decuplet
fields avoid these problems.

###  II.1 Octet and Decuplet State Definition

The new $\Xi^{-}$ wave function comes with a minus sign: that is precisely the
convention used in Eqs. (18) and (19) in Sect. 18 of Gasiorowicz’s textbook
gas . But then we must also adjust the $8\times 10$ $SU(3)$-spurion matrices
for this modification.

$\displaystyle\Sigma^{\mp}\sim{\pm 1\over\sqrt{2}}(N^{1}\pm
iN^{2})\,,\;\;\;N^{3}\sim\Sigma^{0}\,,\;\;\;N^{8}\sim\Lambda_{8}\,,$ (1)
$\displaystyle\left(\begin{array}[]{c}~{}\Xi^{-}\\\
p\end{array}\right)\sim{\mp 1\over\sqrt{2}}(N^{4}\pm
iN^{5}),\;\;\left(\begin{array}[]{c}~{}\Xi^{0}\\\
n\end{array}\right)\;\sim\,{1\over\sqrt{2}}(N^{6}\pm iN^{7}).$ (6)
$\displaystyle\left(\begin{array}[]{c}p\\\ n\\\ \Sigma^{+}\\\ \Sigma^{0}\\\
\Sigma^{-}\\\ \Xi^{0}\\\ \Xi^{-}\\\ \Lambda_{8}\end{array}\right)$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&\frac{1}{\sqrt{2}}&\frac{-i}{\sqrt{2}}&0&0&0\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&\frac{-i}{\sqrt{2}}&0\\\
\frac{-1}{\sqrt{2}}&\frac{i}{\sqrt{2}}&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\
\frac{1}{\sqrt{2}}&\frac{i}{\sqrt{2}}&0&0&0&0&0&0\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&\frac{i}{\sqrt{2}}&0\\\
0&0&0&\frac{-1}{\sqrt{2}}&\frac{-i}{\sqrt{2}}&0&0&0\\\
0&0&0&0&0&0&0&1\end{array}\right)\left(\begin{array}[]{c}N^{1}\\\ N^{2}\\\
N^{3}\\\ N^{4}\\\ N^{5}\\\ N^{6}\\\ N^{7}\\\ N^{8}\end{array}\right)\,,$ (31)

or put them into the $3\times 3$ baryon matrix as follows

$\displaystyle{\mathfrak{N}}=\left(\begin{array}[]{c c
c}{\Sigma^{0}\over\sqrt{2}}+{\Lambda^{8}\over\sqrt{6}}&-\Sigma^{+}&p\\\
\Sigma^{-}&-{\Sigma^{0}\over\sqrt{2}}+{\Lambda^{8}\over\sqrt{6}}&n\\\
-\Xi^{-}&\Xi^{0}&-{2\over\sqrt{6}}\Lambda^{8}\end{array}\right)\,.$ (35)

Note the minus signs in front of $\Xi^{-}$ and $\Sigma^{+}$. We also use a new
normalization of the decuplet fields:

$\displaystyle{\Delta^{1}}\sim-{1\over\sqrt{3}}\Delta^{++}\,,{\Delta^{7}}\sim-{1\over\sqrt{3}}\Delta^{-}\,,{\Delta^{10}}\sim-{1\over\sqrt{3}}\Omega^{-}\,,$
(36)
$\displaystyle\Delta^{2}\sim-\Delta^{+}\,,\Delta^{4}\sim-\Delta^{0}\,,\Delta^{3}\sim-\Sigma^{*+}\,,\Delta^{8}\sim-\Sigma^{*-}\,,\Delta^{6}\sim-\Xi^{*0}\,,\Delta^{9}\sim-\Xi^{*-}\,,$
$\displaystyle\Delta^{5}\sim-\sqrt{2}\Sigma^{*0}$

For the singlet $\Lambda$, we use the normalization:

$\Lambda_{1}=\Lambda_{phy}={2\sqrt{2}\over\sqrt{3}}\Lambda\,.$ (37)

For simplicity, we will just use $\Lambda_{1}$ instead of $\Lambda_{phy}$ in
the following sections.

We define the flavor octet and decuplet matrices/column vectors as

$\displaystyle N$ $\displaystyle=$
$\displaystyle(p,n,\Sigma^{+},\Sigma^{0},\Sigma^{-},\Xi^{0},\Xi^{-},\Lambda_{8})^{T}\,,$
(38) $\displaystyle\Delta$ $\displaystyle=$
$\displaystyle(\Delta^{++},\Delta^{+},\Delta^{0},\Delta^{-},\Sigma^{*+},\Sigma^{*0},\Sigma^{*-},\Xi^{*0},\Xi^{*-},\Omega)^{T}$
(39)

In our previous paper, Ref. Chen:2008qv , we found that the baryon
interpolating fields $N_{+}^{a}=N^{a}_{1}+N^{a}_{2}$ belong to the chiral
representation $(\mathbf{8},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})$;
$\Lambda$ and $N_{-}^{a}=N^{a}_{1}-N^{a}_{2}$ belong to the chiral
representation
$(\mathbf{3},\mathbf{\overline{3}})\oplus(\mathbf{\overline{3}},\mathbf{3})$;
$N^{a}_{\mu}$ and $\Delta^{P}_{\mu}$ belong to the chiral representation
$(\mathbf{6},\mathbf{3})\oplus(\mathbf{3},\mathbf{6})$; and
$\Delta^{P}_{\mu\nu}$ belong to the chiral representation
$(\mathbf{10},\mathbf{1})\oplus(\mathbf{1},\mathbf{10})$. Here $N^{a}_{1}$ and
$N^{a}_{2}$ are the two independent kinds of nucleon fields. $N^{a}_{1}$
contains the “scalar diquark” and $N^{a}_{2}$ contains the “pseudoscalar
diquark”. Moreover, we calculated their chiral transformations in Ref.
Chen:2008qv . That form, however, is not conventionally used for the axial
currents. So in the following subsections, we use different conventions,
listed above, and display the chiral transformations in these bases.

###  II.2 Chiral Transformations of Three-Quark Interpolating Fields

####  II.2.1 $(8,1)\oplus(1,8)$ Chiral Transformations

This chiral representation contains the flavor octet representation
$\mathbf{8}$. For the octet baryon field $N^{a}$ ($a=1,\cdots,8$), chiral
transformations are given by:

$\displaystyle\delta_{5}^{\vec{b}}N_{+}$ $\displaystyle=$ $\displaystyle
i\gamma_{5}b^{a}{\bf F}_{(8)}^{a}N_{+}\,,\ $ (40)

The $SU(3)$-spurion matrices ${\bf F}_{(8)}^{a}$ are listed in the Appendix
A.2. This corresponds to the chiral transformations of Ref. Chen:2008qv :

$\displaystyle\delta_{5}^{\vec{b}}(N^{a}_{1}+N^{a}_{2})$ $\displaystyle=$
$\displaystyle\gamma_{5}b^{b}f^{bac}(N^{c}_{1}+N^{c}_{2})\,.$

The coefficients $f^{abc}$ are the standard antisymmetric “structure
constants” of $SU(3)$. For completeness’ sake, we show the following equation
which defines the $f$ and $d$ coefficients

$\displaystyle\lambda^{a}_{AB}\lambda^{b}_{BC}$ $\displaystyle=$
$\displaystyle(\lambda^{a}\lambda^{b})_{AC}={1\over
2}\\{\lambda^{a},\lambda^{b}\\}_{AC}+{1\over 2}[\lambda^{a},\lambda^{b}]_{AC}$
(41) $\displaystyle=$ $\displaystyle{2\over
3}\delta^{ab}\delta_{AC}+(d^{abc}+if^{abc})\lambda^{c}_{AC}\,.$

####  II.2.2 $(3,\overline{3})\oplus(\overline{3},3)$ Chiral Transformations

This chiral representation contains the flavor octet and singlet
representations $\mathbf{\bar{3}}\otimes\mathbf{3}=\mathbf{8}\oplus\mathbf{1}$
$\sim(N^{a},\Lambda)$. These two flavor representations are mixed under chiral
transformations as

$\displaystyle\delta_{5}^{\vec{b}}\Lambda_{1}$ $\displaystyle=$ $\displaystyle
i\gamma_{5}b^{a}\sqrt{2\over 3}{\rm\bf T}^{a}_{1/8}N_{-}$
$\displaystyle\delta_{5}^{\vec{b}}N_{-}$ $\displaystyle=$ $\displaystyle
i\gamma_{5}b^{a}\left({\rm{\bf D}}^{a}N_{-}+\sqrt{2\over 3}{\rm\bf
T}^{a\dagger}_{1/8}\Lambda_{1}\right)\,.\ $ (42)

where ${\rm{\bf D}}^{a}$ are defined in the Appendix A.1. The $SU(3)$-spurion
matrices ${\rm\bf T}^{a}_{1/8}$ have the following properties

$\displaystyle{\rm\bf T}^{a}_{1/8}{\rm\bf T}^{a\dagger}_{1/8}$
$\displaystyle=$ $\displaystyle 8$ $\displaystyle{\rm\bf
T}^{a\dagger}_{1/8}{\rm\bf T}^{a}_{1/8}$ $\displaystyle=$
$\displaystyle{\mathbf{1}}_{8\times 8}\,,$ (43)

and are listed in the Appendix A.4. Here ${\mathbf{1}}_{8\times 8}$ is a unit
matrix of $8\times 8$ dimensions.

####  II.2.3 $(6,3)\oplus(3,6)$ Chiral Transformations

This chiral representation contains flavor octet and decuplet representations
$\mathbf{6}\otimes\mathbf{3}=\mathbf{8}\oplus\mathbf{10}$
$\sim(N^{a},\Delta^{b})$. For their chiral transformations we use the results
from Ref. Chen:2008qv , where they were expressed in terms of coefficients
$g$, $g^{\prime}$, $g^{\prime\prime}$ and $g^{\prime\prime\prime}$ that were
tabulated in Table II. For off-diagonal terms (between octet and decuplet),
there is a (new) factor $1\over 6$, which comes from the different
normalization of octet and decuplet. Here we show the final result:

$\displaystyle\delta_{5}^{\vec{b}}N$ $\displaystyle=$ $\displaystyle
i\gamma_{5}b^{a}\left({\rm({\bf D}^{a}+{2\over 3}{\bf
F}_{(8)}^{a})}N+\frac{2}{\sqrt{3}}{\rm{\bf T}}^{a}\Delta\right)\,,\ $
$\displaystyle\delta_{5}^{\vec{b}}\Delta$ $\displaystyle=$ $\displaystyle
i\gamma_{5}b^{a}\left(\frac{2}{\sqrt{3}}{\rm{\bf
T}}^{a\dagger}N+\frac{1}{3}{\rm{\bf F}}_{(10)}^{a}\Delta\right)\,.\ $ (44)

These $SU(3)$-spurion matrices ${\bf T}^{a}$ (sometimes we use ${\bf
T}^{a}_{10/8}$) and ${\bf F}_{(10)}^{a}$ have the following properties

$\displaystyle{\rm{\bf F}}_{(10)}^{a}$ $\displaystyle=$
$\displaystyle-\,i\,f^{abc}{\bf T}^{b\dagger}{\bf T}^{c}\,$ $\displaystyle{\bf
T}^{a}{\bf T}^{a\dagger}$ $\displaystyle=$ $\displaystyle\,\frac{5}{2}\
{\mathbf{1}}_{8\times 8}$ $\displaystyle{\bf T}^{a\dagger}{\bf T}^{a}$
$\displaystyle=$ $\displaystyle\,2\ {\mathbf{1}}_{10\times 10}\,,$ (45)

These transition matrices ${\bf T}^{c}$ and the decuplet generators ${\rm{\bf
F}}_{(10)}^{a}$ are listed in Appendices A.2 and A.3, respectively.

###  II.3 Noether Currents of the Chiral $SU_{L}(3)\times SU_{R}(3)$ Symmetry

The chiral $SU_{L}(3)\times SU_{R}(3)$ transformations of the baryon fields
$B_{i}$ define eight components of the baryon isovector axial current ${\bf
J}_{\mu 5}^{a}$, by way of Noether’s theorem:

$\displaystyle-{\bm{b}}\cdot{\bm{J}_{\mu 5}}$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{\partial{\cal
L}}{\partial\partial^{\mu}B_{i}}\delta_{5}^{\vec{b}}B_{i}.\ $ (46)

Similarly, the flavor $SU(3)$ transformations $\delta^{\vec{a}}B_{i}$ define
the Lorentz-vector Noether (flavor) current

$\displaystyle-{\bm{a}}\cdot{\bm{J}_{\mu}}$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{\partial{\cal
L}}{\partial\partial^{\mu}B_{i}}\delta^{\vec{a}}B_{i}.\ $ (47)

####  II.3.1 The Axial Current in the $(8,1)\oplus(1,8)$ Multiplet

Eqs. (40), the chiral $SU_{L}(3)\times SU_{R}(3)$ transformation rules of the
$B_{i}=N_{+}^{i}$ baryons in the $(8,1)\oplus(1,8)$ chiral multiplet, define
the eight components of the (hyperon) flavor octet axial current ${\bf J}_{\mu
5}^{a}$, by way of Noether’s theorem, Eq. (46), where $B_{i}$ are the octet
$N^{i}$ baryon fields. The axial current ${\bf J}_{\mu 5}$ is

$\displaystyle{\bf J}_{\mu 5}^{a}$ $\displaystyle=$
$\displaystyle\,\overline{N}\gamma_{\mu}\gamma_{5}{\bf F}_{(8)}^{a}N~{}.\ $
(48)

Here ${\bf F}_{(8)}^{i}$ are the $SU(3)$ octet matrices/generators. The
Lorentz vector Noether (flavor-octet) current in this multiplet reads

$\displaystyle{\bf J}_{\mu}^{a}$ $\displaystyle=$
$\displaystyle\overline{N}\gamma_{\mu}\,{\bf F}_{(8)}^{a}\,N~{},\ $ (49)

which are valid if the interactions do not contain derivatives.

####  II.3.2 The Axial Current in the $(\overline{3},3)\oplus(3,\overline{3})$
Multiplet

Eqs. (42), the chiral $SU_{L}(3)\times SU_{R}(3)$ transformation rules of the
$B_{i}=(N_{-}^{i},\,\Lambda)$ baryons in the
$(\overline{3},3)\oplus(3,\overline{3})$ chiral multiplet, define the eight
components of the (hyperon) flavor octet axial current ${\bf J}_{\mu 5}^{a}$,
by way of Noether’s theorem, Eq. (46), where $B_{i}$ are the flavor octet
$N_{-}^{i}$ and the flavor singlet $\Lambda_{1}$ baryon fields. The axial
current ${\bf J}_{\mu 5}$ is

$\displaystyle{\bf J}_{\mu 5}^{a}$ $\displaystyle=$
$\displaystyle\,\overline{N}\gamma_{\mu}\gamma_{5}\left({\rm{\bf
D}}^{a}N+\sqrt{2\over 3}{\rm\bf T}^{a\dagger}_{1/8}\Lambda_{1}\right)$ (50)
$\displaystyle+$
$\displaystyle\,\overline{\Lambda}_{1}\gamma_{\mu}\gamma_{5}\sqrt{2\over
3}{\rm\bf T}^{a}_{1/8}N~{}.\ $

Here ${\bf D}^{i}$ are the $SU(3)$ octet matrices/generators. The Lorentz
vector Noether (flavor-octet) current in this multiplet reads

$\displaystyle{\bf J}_{\mu}^{a}$ $\displaystyle=$
$\displaystyle\overline{N}\gamma_{\mu}\,{\bf F}_{(8)}^{a}\,N~{}.\ $ (51)

####  II.3.3 Axial Current in the $(3,6)\oplus(6,3)$ Multiplet

The chiral $SU_{L}(3)\times SU_{R}(3)$ transformation rules of the
$B_{i}=(N^{i},\,\Delta^{j})$ baryons, Eqs. (II.2.3), in the $(3,6)\oplus(6,3)$
chiral multiplet, define the eight components of the (hyperon) flavor octet
axial current ${\bf J}_{\mu 5}^{a}$, by way of Noether’s theorem (Eq. (46)),
where $B_{i}$ are the octet $N^{i}$ and the decuplet $\Delta^{j}$ baryon
fields. The axial current ${\bf J}_{\mu 5}$ is

$\displaystyle{\bf J}_{\mu 5}^{a}$ $\displaystyle=$
$\displaystyle\,\overline{N}\gamma_{\mu}\gamma_{5}\left({\rm({\bf
D}^{a}+{2\over 3}{\bf F}_{(8)}^{a})}N+\frac{2}{\sqrt{3}}{\rm{\bf
T}}^{a}\Delta\right)$ (52) $\displaystyle+$
$\displaystyle\,\overline{\Delta}\gamma_{\mu}\gamma_{5}\left(\frac{2}{\sqrt{3}}{\rm{\bf
T}}^{a\dagger}N+\frac{1}{3}{\rm{\bf F}}_{(10)}^{a}\Delta\right)~{}.\ $

Here ${\bf D}^{i}$ and ${\bf F}_{(8)}^{i}$ are the $SU(3)$ octet
matrices/generators ${\bf D}^{a}$ and ${\bf F}_{(8)}^{a}$, respectively, ${\bf
F}_{(10)}^{i}$ are the $SU(3)$ decuplet generators, and ${\bf T}^{i}$ are the
so-called $SU(3)$-spurion matrices. The Lorentz vector Noether (flavor-octet)
current in this multiplet reads

$\displaystyle{\bf J}_{\mu}^{a}$ $\displaystyle=$
$\displaystyle\left(\overline{N}\gamma_{\mu}\,{\bf
F}_{(8)}^{a}\,N\right)+\left(\overline{\Delta}\gamma_{\mu}\,{\bf
F}_{(10)}^{a}\,\Delta\right)~{}.\ $ (53)

##  III Closure of the chiral $SU_{L}(3)\times SU_{R}(3)$ algebra

The $SU(3)$ vector charges $Q^{a}=\int d{\bf x}J_{0}^{a}(t,{\bf x})$ defined
by Eq. (47), together with the axial charges $Q_{5}^{a}=\int d{\bf
x}J_{05}^{a}(t,{\bf x})$ defined by Eq. (46) ought to close the chiral algebra

$\displaystyle\left[Q^{a},Q^{b}\right]$ $\displaystyle=$ $\displaystyle
if^{abc}Q^{c}$ (54) $\displaystyle\left[Q_{5}^{a},Q^{b}\right]$
$\displaystyle=$ $\displaystyle if^{abc}Q_{5}^{c}$ (55)
$\displaystyle\left[Q_{5}^{a},Q_{5}^{b}\right]$ $\displaystyle=$
$\displaystyle if^{abc}Q^{c}~{}.\ $ (56)

where $f^{abc}$ are the SU(3) structure constants. Eqs. (54) and (55) usually
hold automatically, as a consequence of the canonical (anti)commutation
relations between Dirac baryon fields $B_{i}$, whereas Eq. (56) is not trivial
for the chiral multiplets that are different from the $[(8,1)\oplus(1,8)]$,
because of the (nominally) fractional axial charges and the presence of the
off-diagonal components. When taking a matrix element of Eq. (56) by baryon
states in a certain chiral representation, the axial charge mixes different
flavor states within the same chiral representation. This is an algebraic
version of the Adler-Weisburger sum rule Weinberg:1969hw . In the following we
shall check and confirm the validity of Eq. (56) in the three multiplets of
SU(3)${}_{L}\times$SU(3)R.

###  III.1 Closure of the Chiral $SU_{L}(3)\times SU_{R}(3)$ Algebra in the
$(8,1)\oplus(1,8)$ Multiplet

Due to the absence of fractional coefficients in the $(8,1)\oplus(1,8)$
multiplet’s axial charge $Q_{5}^{a}=\int d{\bf x}J_{05}^{a}(t,{\bf x})$
defined by the current given in Eq. (48), the vector charge $Q^{a}=\int d{\bf
x}J_{0}^{a}(t,{\bf x})$ defined by the current given in Eq. (49) and the axial
charge close the chiral algebra defined by Eqs. (54), (55) and (56). The same
comments holds for the $(10,1)\oplus(1,10)$ chiral multiplet for the same
reasons as in the example shown above.

###  III.2 Closure of the Chiral $SU_{L}(3)\times SU_{R}(3)$ Algebra in the
$(3,\overline{3})\oplus(\overline{3},3)$ Multiplet

The vector charge $Q^{a}=\int d{\bf x}J_{0}^{a}(t,{\bf x})$ defined by the
current given in Eq. (51), together with the axial charge $Q_{5}^{a}=\int
d{\bf x}J_{05}^{a}(t,{\bf x})$ defined by the current given in Eq. (50) ought
to close the chiral algebra defined by Eqs. (54), (55) and (56). Eqs. (54) and
(55) hold here, whereas Eq. (56) is the non-trivial one: the diagonal $D$
charge of $N$ ($Q_{5D}^{a}(N)$) axial charge,

$\displaystyle Q_{5D}^{a}(N)$ $\displaystyle=$ $\displaystyle~{}~{}~{}\int
d{\bf x}\,\left(\overline{N}\gamma_{0}\gamma_{5}\,{\bf D}^{a}\,N\right)\,,$
(57) $\displaystyle Q_{D}^{a}(N)$ $\displaystyle=$ $\displaystyle~{}~{}~{}\int
d{\bf x}\,\left(\overline{N}\gamma_{0}\,{\bf D}^{a}\,N\right)\,,\ $ (58)

lead to

$\displaystyle\left[Q_{5D}^{a}(N),Q_{5D}^{b}(N)\right]$ $\displaystyle=$
$\displaystyle\int d{\bf x}\left(\overline{N}\gamma_{0}\,\left({\bf D}^{a}{\bf
D}^{b}-{\bf D}^{b}{\bf D}^{a}\right)N\right)\,.\ $ (59)

It turns out that the off-diagonal terms in the axial charge

$\displaystyle Q_{5}^{a}(N,\Lambda)$ $\displaystyle=$ $\displaystyle\int d{\bf
x}\,\Bigg{(}\sqrt{\frac{2}{3}}\left(\overline{N}\gamma_{0}\gamma_{5}\,{\bf
T}^{a\dagger}_{1/8}\,\Lambda+\overline{\Lambda}\gamma_{0}\gamma_{5}\,{\bf
T}^{a}_{1/8}\,N\right)\Bigg{)}~{},\ $ (60)

play a crucial role in the closure of the chiral commutator Eq. (56). The
additional terms in the commutator add up to

$\displaystyle\left[Q_{5}^{a}(N,\Delta),Q_{5}^{b}(N,\Delta)\right]$
$\displaystyle=$ $\displaystyle{\frac{2}{3}}\int d{\bf
x}\overline{N}\gamma_{0}\,\left({\bf T}^{a\dagger}_{1/8}{\bf T}^{b}_{1/8}-{\bf
T}^{b\dagger}_{1/8}{\bf T}^{a}_{1/8}\right)\,N\,,\ $ (61)

which provide the “missing” factors due to the following properties of the
off-diagonal isospin operators ${\bf T}^{i}_{1/8}$ and ${\bf D}^{i}$ matrices

$\displaystyle i\,f^{ijk}({\bf F}_{(8)}^{k})$ $\displaystyle=$
$\displaystyle({\bf D}^{i}{\bf D}^{j}-{\bf D}^{j}{\bf D}^{i})+{2\over 3}({\bf
T}^{i\dagger}_{1/8}{\bf T}^{j}_{1/8}-{\bf T}^{j\dagger}_{1/8}{\bf
T}^{i}_{1/8})\,.$ (62)

Therefore, the chiral algebra Eqs. (54), (55) and (56) close.

###  III.3 Closure of the Chiral $SU_{L}(3)\times SU_{R}(3)$ Algebra in the
$(3,6)\oplus(6,3)$ Multiplet

The vector charge $Q^{a}=\int d{\bf x}J_{0}^{a}(t,{\bf x})$ defined by the
current in Eq. (53), together with the axial charge $Q_{5}^{a}=\int d{\bf
x}J_{05}^{a}(t,{\bf x})$ defined by the current in Eq. (52) ought to close the
chiral algebra defined by Eqs. (54), (55) and (56). Eqs. (54) and (55) hold
here, whereas Eq. (56) is once again the non-trivial one: the fractions
$\frac{2}{3}$ and $\frac{1}{3}$ in the diagonal $F$ charge of $N$
($Q_{5}^{a}(N)$) and $\Delta$ axial charges, respectively, and the diagonal
$D$ charge of $N$ ($Q_{5}^{a}(N)$):

$\displaystyle Q_{5F}^{a}(N)$ $\displaystyle=$ $\displaystyle\frac{2}{3}\int
d{\bf x}\,\left(\overline{N}\gamma_{0}\gamma_{5}\,{\bf
F}_{(8)}^{a}\,N\right)~{}\,,$ (63) $\displaystyle Q_{5F}^{a}(\Delta)$
$\displaystyle=$ $\displaystyle\frac{1}{3}\int d{\bf
x}\,\left(\overline{\Delta}\gamma_{0}\gamma_{5}\,{\bf
F}_{(10)}^{a}\,\Delta\right)~{}\,,$ (64) $\displaystyle Q_{5D}^{a}(N)$
$\displaystyle=$ $\displaystyle~{}~{}~{}\int d{\bf
x}\,\left(\overline{N}\gamma_{0}\gamma_{5}\,{\bf D}^{a}\,N\right)~{}\,,$ (65)

lead to

$\displaystyle\left[Q_{5D+F}^{a}(N),Q_{5D+F}^{b}(N)\right]$ $\displaystyle=$
$\displaystyle\int d{\bf x}\Bigg{(}\overline{N}\gamma_{0}\,\Big{(}\big{(}{\bf
D}^{a}+\frac{2}{3}{\bf F}_{(8)}^{a}\big{)}\big{(}{\bf D}^{b}+\frac{2}{3}{\bf
F}_{(8)}^{b}\big{)}$ $\displaystyle-\big{(}{\bf D}^{b}+\frac{2}{3}{\bf
F}_{(8)}^{b}\big{)}\big{(}{\bf D}^{a}+\frac{2}{3}{\bf
F}_{(8)}^{a}\big{)}\Big{)}N\Bigg{)}\,,$
$\displaystyle\left[Q_{5F}^{a}(\Delta),Q_{5F}^{b}(\Delta)\right]$
$\displaystyle=$ $\displaystyle if^{abc}\frac{1}{9}Q^{c}(\Delta)\,,\ $ (67)

lead to ”only” one part of the N and Delta vector charges respectively, on the
right-hand side of Eqs. (III.3) and (67).

Once again, it turns out that the off-diagonal terms in the axial charge

$\displaystyle Q_{5}^{a}(N,\Delta)$ $\displaystyle=$ $\displaystyle\int d{\bf
x}\,\Bigg{(}\frac{2}{\sqrt{3}}\left(\overline{N}\gamma_{0}\gamma_{5}\,{\bf
T}^{a}\,\Delta+\overline{\Delta}\gamma_{0}\gamma_{5}\,{\bf
T}^{a\dagger}\,N\right)\Bigg{)}\,,$ (68)

play a crucial role in the closure of the chiral algebra Eq. (56). The
additional terms in the commutator add up to

$\displaystyle\left[Q_{5}^{a}(N,\Delta),Q_{5}^{b}(N,\Delta)\right]$
$\displaystyle=$ $\displaystyle\frac{4}{3}\int d{\bf
x}\left(\overline{N}\gamma_{0}\,\left({\bf T}^{a}{\bf T}^{b\dagger}-{\bf
T}^{b}{\bf T}^{a\dagger}\right)\,N+\overline{\Delta}\gamma_{0}\,\left({\bf
T}^{a\dagger}{\bf T}^{b}-{\bf T}^{b\dagger}{\bf T}^{a}\right)\Delta\right)\,,$
(69)

which provide the “missing” factors due to the following properties of the
off-diagonal flavor operators ${\bf T}^{i}$ and ${\bf D}^{i}$ matrices

$\displaystyle i\,f^{ijk}({\bf F}_{(8)}^{k})$ $\displaystyle=$
$\displaystyle\Big{(}\big{(}{\bf D}^{i}+\frac{2}{3}{\bf
F}_{(8)}^{i}\big{)}\big{(}{\bf D}^{j}+\frac{2}{3}{\bf
F}_{(8)}^{j}\big{)}-\big{(}{\bf D}^{j}+\frac{2}{3}{\bf
F}_{(8)}^{j}\big{)}\big{(}{\bf D}^{i}+\frac{2}{3}{\bf
F}_{(8)}^{i}\big{)}\Big{)}+{4\over 3}({\bf T}^{i}_{10/8}{\bf
T}^{j\dagger}_{10/8}-{\bf T}^{j}_{10/8}{\bf T}^{i\dagger}_{10/8})\,,$
$\displaystyle i\frac{2}{3}\,f^{ijk}{\bf F}_{(10)}^{k}$ $\displaystyle=$
$\displaystyle{\bf T}_{10/8}^{i\dagger}{\bf T}_{10/8}^{j}-{\bf
T}_{10/8}^{j\dagger}{\bf T}_{10/8}^{i}\,.$ (70)

Therefore, the chiral algebra Eqs. (54), (55) and (56) closes in spite, or
perhaps because of the apparent fractional axial charges ($\frac{2}{3}$ and
$\frac{1}{3}$).

##  IV Chiral mixing and the axial current

A unique feature of the use of the linear chiral representation is that the
axial coupling is determined by the chiral representations, as given by the
coefficients of the axial transformations. For the nucleon (proton and
neutron), chiral representations of $SU_{L}(2)\times SU_{R}(2)$,
$(\frac{1}{2},0)(\sim(8,1),(3,\bar{3}))$ and $(1,\frac{1}{2})(\sim(6,3))$
provide the nucleon isovector axial coupling $g_{A}^{(3)}=1$ and $5/3$
respectively. Therefore, the mixing of chiral $(\frac{1}{2},0)$ and
$(1,\frac{1}{2})$ nucleons leads to the axial coupling

$\displaystyle 1.267$ $\displaystyle=$ $\displaystyle
g_{A~{}(\frac{1}{2},0)}^{(1)}~{}\cos^{2}\theta+g_{A~{}(1,\frac{1}{2})}^{(1)}~{}\sin^{2}\theta$
(71) $\displaystyle=$ $\displaystyle
g_{A~{}(\frac{1}{2},0)}^{(1)}~{}\cos^{2}\theta+\frac{5}{3}~{}\sin^{2}\theta\,,$

Table 1: The Abelian and the non-Abelian axial charges (+ sign indicates “naive”, - sign “mirror” transformation properties) and the non-Abelian chiral multiplets of $J^{P}=\frac{1}{2}$, Lorentz representation $(\frac{1}{2},0)$ nucleon and $\Delta$ fields, see Refs. Nagata:2007di ; Nagata:2008zzc ; Dmitrasinovic:2009vp ; Dmitrasinovic:2009vy . case | field | $g_{A}^{(0)}$ | $g_{A}^{(1)}$ | $F$ | $D$ | $SU_{L}(3)\times SU_{R}(3)$
---|---|---|---|---|---|---
I | $N_{1}-N_{2}$ | $-1$ | $+1$ | $~{}~{}0$ | $+1$ | $(3,\overline{3})\oplus(\overline{3},3)$
II | $N_{1}+N_{2}$ | $+3$ | $+1$ | $+1$ | $~{}~{}0$ | $(8,1)\oplus(1,8)$
III | $N_{1}^{{}^{\prime}}-N_{2}^{{}^{\prime}}$ | $+1$ | $-1$ | $~{}~{}0$ | $-1$ | $(\overline{3},3)\oplus(3,\overline{3})$
IV | $N_{1}^{{}^{\prime}}+N_{2}^{{}^{\prime}}$ | $-3$ | $-1$ | $-1$ | $~{}~{}0$ | $(1,8)\oplus(8,1)$
0 | $\partial_{\mu}(N_{3}^{\mu}+\frac{1}{3}N_{4}^{\mu})$ | $+1$ | $+\frac{5}{3}$ | $+\frac{2}{3}$ | $+1$ | $(6,3)\oplus(3,6)$

Three-quark nucleon interpolating fields in QCD have also well-defined, if
perhaps unexpected $U_{A}(1)$ chiral transformation properties, see Table 1,
that can be used to predict the isoscalar axial coupling $g_{A~{}\rm
mix.}^{(0)}$

$\displaystyle g_{A~{}\rm mix.}^{(0)}$ $\displaystyle=$ $\displaystyle
g_{A~{}(\frac{1}{2},0)}^{(0)}~{}\cos^{2}\theta+g_{A~{}(1,\frac{1}{2})}^{(0)}~{}\sin^{2}\theta$
(72) $\displaystyle=$ $\displaystyle
g_{A~{}(\frac{1}{2},0)}^{(0)}~{}\cos^{2}\theta+\sin^{2}\theta,$

together with the mixing angle $\theta$ extracted from Eq. (71). Note,
however, that due to the different (bare) non-Abelian $g_{A}^{(1)}$ and
Abelian $g_{A}^{(0)}$ axial couplings, see Table 1, the mixing formulae Eq.
(72) give substantially different predictions from one case to another, see
Table 2.

Table 2: The values of the baryon isoscalar axial coupling constant predicted from the naive mixing and $g_{A~{}\rm expt.}^{(1)}=1.267$; compare with $g_{A~{}\rm expt.}^{(0)}=0.33\pm 0.03\pm 0.05$, $F$=$0.459\pm 0.008$ and $D$=$0.798\pm 0.008$, leading to $F/D=0.571\pm 0.005$, Ref. Yamanishi:2007zza . case | ($g_{A}^{(1)}$,$g_{A}^{(0)}$) | $g_{A~{}\rm mix.}^{(1)}$ | $\theta_{i}$ | $g_{A~{}\rm mix.}^{(0)}$ | $g_{A~{}\rm mix.}^{(0)}$ | $F$ |  $F$/$D$
---|---|---|---|---|---|---|---
I | $(+1,-1)$ | $\frac{1}{3}(4-\cos 2\theta)$ | $39.3^{o}$ | $-\cos 2\theta$ | -0.20 | 0.267 | 0.267
II | $(+1,+3)$ | $\frac{1}{3}(4-\cos 2\theta)$ | $39.3^{o}$ | $(2\cos 2\theta+1)$ | 2.20 | 0.866 | 2.16
III | $(-1,+1)$ | $\frac{1}{3}(1-4\cos 2\theta)$ | $67.2^{o}$ | $1$ | 1.00 | 0.567 | 0.81
IV | $(-1,-3)$ | $\frac{1}{3}(1-4\cos 2\theta)$ | $67.2^{o}$ | $-(2\cos 2\theta+1)$ | 0.40 | 0.417 | 0.491

We can see in Table 2 that the two best candidates are cases I and IV, with
$g_{A}^{(0)}=-0.2$ and $g_{A}^{(0)}=0.4$, respectively, the latter being
within the error bars of the measured value $g_{A~{}\rm expt.}^{(0)}=0.33\pm
0.08$, Bass:2007zzb ; Ageev:2007du . Moreover, this scheme predicts the $F$
and $D$ values, as well:

$\displaystyle F$ $\displaystyle=$ $\displaystyle
F_{(\frac{1}{2},0)}~{}\cos^{2}\theta+F_{(1,\frac{1}{2})}^{(1)}~{}\sin^{2}\theta,$
(73) $\displaystyle=$ $\displaystyle
F_{(\frac{1}{2},0)}~{}\cos^{2}\theta+\frac{2}{3}~{}\sin^{2}\theta$
$\displaystyle D$ $\displaystyle=$ $\displaystyle
D_{(\frac{1}{2},0)}~{}\cos^{2}\theta+D_{(1,\frac{1}{2})}~{}\sin^{2}\theta$
(74) $\displaystyle=$ $\displaystyle
D_{(\frac{1}{2},0)}~{}\cos^{2}\theta+\sin^{2}\theta,$

where we have used the $F$ and $D$ values for different chiral multiplets as
listed in Table 1.

Cases I and IV, with $F$/$D$ = 0.267 and 0.491, respectively, ought to be
compared with $F$/$D$ = $0.571\pm 0.005$ 222Note that the Ref.
Yamanishi:2007zza  values add up to F+D = $1.312\pm 0.002$, more than
2-$\sigma$ away from the experimental constraint $\neq 1.269\pm 0.002$.. Case
I is, of course, the well-known “Ioffe current”, which reproduces the
nucleon’s properties in QCD lattice and sum rules calculations. The latter is
a “mirror” opposite of the orthogonal complement to the Ioffe current, an
interpolating field that, to our knowledge, has not been used in QCD thus far.

Manifestly, a linear superposition of any three fields (except for the
mixtures of cases II and III, IV above, which yield complex mixing angles)
should give a perfect fit to the central values of the experimental axial
couplings and predict the $F$ and $D$ values. Such a three-field admixture
introduces new free parameters (besides the two already introduced mixing
angles, e.g. $\theta_{1}$ and $\theta_{4}$, we have the relative/mutual mixing
angle $\theta_{14}$, as the two nucleon fields I and IV may also mix). One may
subsume the sum and the difference of the two angles $\theta_{1}$ and
$\theta_{4}$ into the new angle $\theta$, and define
$\varphi\doteq\theta_{14}$ (this relationship depends on the precise
definition of the mixing angles $\theta_{1}$, $\theta_{4}$ and $\theta_{14}$);
thus we find two equations with two unknowns of the general form:

$\displaystyle\frac{5}{3}\,{\sin}^{2}\theta+{\cos}^{2}\theta\,\left(g_{A}^{(1)}{\cos}^{2}\varphi+g_{A}^{(1)\prime}{\sin}^{2}\varphi\right)$
$\displaystyle=1.267$ (75)
$\displaystyle{\sin}^{2}\theta+{\cos}^{2}\theta\,\left(g_{A}^{(0)}{\cos}^{2}\varphi+g_{A}^{(0)\prime}\,{\sin}^{2}\varphi\right)$
$\displaystyle=0.33\pm 0.08$ (76)

The solutions to these equations (the values of the mixing angles
$\theta,\varphi$) provide, at the same time, input for the prediction of $F$
and $D$:

$\displaystyle\cos^{2}\theta\,\left(F\,{\cos}^{2}\varphi+F^{\prime}\,{\sin}^{2}\varphi\right)+\frac{2}{3}~{}\sin^{2}\theta$
$\displaystyle=F$ (77)
$\displaystyle\cos^{2}\theta\,\left(D\,{\cos}^{2}\varphi+D^{\prime}\,{\sin}^{2}\varphi\right)+\sin^{2}\theta$
$\displaystyle=D.$ (78)

The values of the mixing angles ($\theta,\varphi$) obtained from this
straightforward fit to the baryon axial coupling constants are shown in Table
3.

Table 3: The values of the mixing angles obtained from the simple fit to the baryon axial coupling constants and the predicted values of axial $F$ and $D$ couplings. The experimental values are $F$=$0.459\pm 0.008$ and $D$=$0.798\pm 0.008$, leading to $F/D=0.571\pm 0.005$, Ref. Yamanishi:2007zza . case | $g_{A~{}\rm expt.}^{(3)}$ | $g_{A~{}\rm expt.}^{(0)}$ | $\theta$ | $\varphi$ | $F$ | $D$ |  $F$/$D$
---|---|---|---|---|---|---|---
I-II | 1.267 | $0.33$ | $39.3^{o}$ | $28.0^{o}\pm 2.3^{o}$ | $0.399\pm 0.02$ | 0.868$\mp 0.02$ | $0.460\pm 0.04$
I-III | 1.267 | $0.33$ | $50.7^{o}\pm 1.8^{o}$ | $23.9^{o}\pm 2.9^{o}$ | $0.399\pm 0.02$ | 0.868$\mp 0.02$ | $0.460\pm 0.04$
I-IV | 1.267 | $0.33$ | $63.2^{o}\pm 4.0^{o}$ | $54^{o}\pm 23^{o}$ | $0.399\pm 0.02$ | 0.868$\mp 0.02$ | $0.460\pm 0.04$

Note that all three admissible scenarios (i.e. choices of pairs of fields
admixed to the (6,3) one that lead to real mixing angles) yield the same
values of $F$ and $D$. This is due to the fact that all three-quark baryon
fields satisfy the following relation $g_{A}^{(0)}=3F-D=\sqrt{3}g_{A}^{(8)}$
Jido09 . The first relation $g_{A}^{(0)}=3F-D$ was not expected, as the
flavor-singlet properties, such as $g_{A}^{(0)}$ are generally expected to be
independent of the flavor-octet ones, such as $F,D$. Yet, it is not unnatural,
either, as it indicates the absence of polarized $s\bar{s}$ pairs in these
SU(3) symmetric, three-quark nucleon interpolators. In order to show that, we
define $g_{A}^{(0)}=\Delta u+\Delta d+\Delta s$ and
$g_{A}^{(8)}={1\over\sqrt{3}}(\Delta u+\Delta d-2\Delta s)$, where $\Delta q$
are the (corresponding flavor) quark contributions to the matrix element of
the nucleon’s axial vector current $\Delta q=\langle
N|\bar{q}\gamma_{\mu}\gamma_{5}q|N\rangle$. We see that $g_{A}^{(0)}\sim
g_{A}^{(8)}$ only if $\Delta s=0$.

Thus, the relation $g_{A}^{(0)}=3F-D$ appears to depend on the choice of
three-quark interpolating fields as a source of admixed mirror fields and may
well change when one considers other interpolating fields, such as the five-
quark (“pentaquark”) ones for example333Note, however, that five- and more
quark, and derivative interpolating fields are not the only ones that can
produce mirror fields, however: so can the one-gluon-three-quark “hybrid
baryon” interpolators, which necessarily have the same chiral properties as
the corresponding three-quark fields.. In that sense a deviation of the
measured values of $g_{A}^{(0)}$ and $g_{A}^{(8)}=\frac{1}{\sqrt{3}}(3F-D)$
from this relation may well be seen as a measure of the contribution of
higher-order configurations’ to the baryon ground state. It seems very
difficult, however, to evaluate $F$ and $D$ for specific higher-order
configurations without going through the procedure outlined in Ref.
Chen:2008qv  for the “pentaquark” interpolator chiral multiplets 444If one
were to assign one particular source of mirror fields, for example some
“pentaquark” interpolators, then one could try to determine the contribution
of $s\bar{s}$ pairs to the flavor singlet axial coupling..

Some of the ideas used above have also been used in some of the following
early papers: two-chiral-multiplet mixing was considered long ago by Harari
Harari:1966yq , and by Weinberg Weinberg:1969hw , for example. Moreover,
special cases of three-field/configuration chiral mixing have been considered
by Harari Harari:1966jz , and by Gerstein and Lee Gerstein:1966zz  in the
context of the (“collinear”) $U(3)\times U(3)$ current algebra at infinite
momentum. One (obvious) distinction from these early precedents is our use of
QCD interpolating fields, which appeared only in the early 1980’s, and the
(perhaps less obvious) issue of baryons’ flavor-singlet axial current (a.k.a.
the $U_{A}(1)$), that was (seriously) raised yet another decade later. We
emphasize here that our results are based on the $U_{L}(3)\times U_{R}(3)$
chiral algebra of space-integrated time components of currents, without any
assumptions about saturation of this algebra by one-particle states, or its
dependence on any one particular frame of reference. Indeed, our nucleon
interpolating fields transform as the $(\frac{1}{2},0)+(0,\frac{1}{2})$
representation of the Lorentz group, thus making the Noether currents (fully)
Lorentz covariant, so that our results hold in any frame.

##  V Summary and Outlook

We have re-organized the results of our previous paper Chen:2008qv  into the
(perhaps more) conventional form for the baryon octet using $F$ and $D$
coupling ($SU(3)$ structure constants). This means that, inter alia, we have
explicitly written down (perhaps for the first time) the chiral
transformations of the $(6,3)\oplus(3,6)$ octet and decimet fields in the
$SU(3)$ particle (octet and decimet) basis.

In the process we have independently constructed $SU(3)$ generators of the
decimet and derived a set of $SU(3)$ Clebsch-Gordan coefficients in the
“natural” convention, which means that all isospin $SU(2)$ sub-multiplets of
the octet and the decimet have standard isospin $SU(2)$ generators.

Then we used the above mentioned $SU(3)$ Clebsch-Gordan coefficients to
explicitly check the closure of the $SU_{L}(3)\times SU_{R}(3)$ chiral algebra
in the $SU(3)$ particle basis, which forms an independent check/confirmation
of the calculation.

Next, we investigated the phenomenological consequence for the baryon axial
currents, of the chiral $[(6,3)\oplus(3,6)]$ multiplet mixing with other
three-quark baryon field multiplets, such as the
$[(3,\overline{3})\oplus(\overline{3},3)]$ and $[(8,1)\oplus(1,8)]$. The
results of the three-field (“two-angle”) mixing are interesting: all
permissible combinations fields lead to the same $F$/$D$ prediction, that is
in reasonable agreement with experiment. This identity of results is a
consequence of the relation $g_{A}^{(0)}=3F-D$ between the flavor singlet
axial coupling $g_{A}^{(0)}$ and the (previously unrelated) flavor octet $F$
and $D$ values. That relation is a unique feature of the three-quark
interpolating fields and any potential departures from it may be attributed to
fields with a number of quarks higher than three.

The next step, left for the future, is to investigate $SU_{L}(3)\times
SU_{R}(3)$ chiral invariant interactions and the $SU(3)\times SU(3)\to
SU(2)\times SU(2)$ symmetry breaking/reduction and to the study of the chiral
$SU(2)\times SU(2)$ properties of hyperons. Then one may consider explicit
chiral symmetry breaking corrections to the axial and the vector currents,
which are related to the $SU(3)\times SU(3)$ symmetry breaking meson-nucleon
derivative interactions, not just the explicit $SU(3)$ symmetry breaking ones
that have been considered thus far (see Ref. Yamanishi:2007zza  and the
previous subsection, above).

## Acknowledgments

We wish to thank Profs. Daisuke Jido, Akira Ohnishi and Makoto Oka for
valuable conversations regarding the present work. One of us (V.D.) wishes to
thank the RCNP, Osaka University, under whose auspices this work was begun,
and the Yukawa Institute for Theoretical Physics, Kyoto, (molecular workshop
“Algebraic aspects of chiral symmetry for the study of excited baryons”) where
it was finished, for kind hospitality and financial support.

##  Appendix A SU(3) Octet, Decimet Generators and 8x10 Transition Matrices

###  A.1 Octet “Generator” 8x8 Matrices ${\rm{\bf D}^{a}},~{}{\rm{\bf
F}_{(8)}^{a}}$ in the Particle Basis

$\displaystyle{\rm({\bf D}^{1}+{2\over 3}{\bf F}_{(8)}^{1})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll|l}0&\frac{5}{6}&0&0&0&0&0&0&p\\\
\frac{5}{6}&0&0&0&0&0&0&0&n\\\
0&0&0&\frac{\sqrt{2}}{3}&0&0&0&-\frac{1}{\sqrt{6}}&\Sigma^{+}\\\
0&0&\frac{\sqrt{2}}{3}&0&\frac{\sqrt{2}}{3}&0&0&0&\Sigma^{0}\\\
0&0&0&\frac{\sqrt{2}}{3}&0&0&0&\frac{1}{\sqrt{6}}&\Sigma^{-}\\\
0&0&0&0&0&0&-\frac{1}{6}&0&\Xi^{0}\\\ 0&0&0&0&0&-\frac{1}{6}&0&0&\Xi^{-}\\\
0&0&-\frac{1}{\sqrt{6}}&0&\frac{1}{\sqrt{6}}&0&0&0&\Lambda_{8}\\\ \hline\cr\\\
p&n&\Sigma^{+}&\Sigma^{0}&\Sigma^{-}&\Xi^{0}&\Xi^{-}&\Lambda_{8}\end{array}\right)$
(89) $\displaystyle{\rm{\bf D}^{1}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&\frac{1}{2}&0&0&0&0&0&0\\\
\frac{1}{2}&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&-\frac{1}{\sqrt{6}}\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&\frac{1}{\sqrt{6}}\\\
0&0&0&0&0&0&-\frac{1}{2}&0\\\ 0&0&0&0&0&-\frac{1}{2}&0&0\\\
0&0&-\frac{1}{\sqrt{6}}&0&\frac{1}{\sqrt{6}}&0&0&0\end{array}\right)$ (98)
$\displaystyle{\bf F}_{(8)}^{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&\frac{1}{2}&0&0&0&0&0&0\\\
\frac{1}{2}&0&0&0&0&0&0&0\\\ 0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\
0&0&\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}&0&0&0\\\
0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\ 0&0&0&0&0&0&\frac{1}{2}&0\\\
0&0&0&0&0&\frac{1}{2}&0&0\\\ 0&0&0&0&0&0&0&0\end{array}\right)$ (107)

$\displaystyle{\rm({\bf D}^{2}+{2\over 3}{\bf F}_{(8)}^{2})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&-\frac{5i}{6}&0&0&0&0&0&0\\\
\frac{5i}{6}&0&0&0&0&0&0&0\\\
0&0&0&-\frac{i\sqrt{2}}{3}&0&0&0&\frac{i}{\sqrt{6}}\\\
0&0&\frac{i\sqrt{2}}{3}&0&-\frac{i\sqrt{2}}{3}&0&0&0\\\
0&0&0&\frac{i\sqrt{2}}{3}&0&0&0&\frac{i}{\sqrt{6}}\\\
0&0&0&0&0&0&\frac{i}{6}&0\\\ 0&0&0&0&0&-\frac{i}{6}&0&0\\\
0&0&-\frac{i}{\sqrt{6}}&0&-\frac{i}{\sqrt{6}}&0&0&0\end{array}\right)$ (116)
$\displaystyle{\rm{\bf D}^{2}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&-\frac{i}{2}&0&0&0&0&0&0\\\
\frac{i}{2}&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&\frac{i}{\sqrt{6}}\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&\frac{i}{\sqrt{6}}\\\
0&0&0&0&0&0&\frac{i}{2}&0\\\ 0&0&0&0&0&-\frac{i}{2}&0&0\\\
0&0&-\frac{i}{\sqrt{6}}&0&-\frac{i}{\sqrt{6}}&0&0&0\end{array}\right)$ (125)
$\displaystyle{\bf F}_{(8)}^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&-\frac{i}{2}&0&0&0&0&0&0\\\
\frac{i}{2}&0&0&0&0&0&0&0\\\ 0&0&0&-\frac{i}{\sqrt{2}}&0&0&0&0\\\
0&0&\frac{i}{\sqrt{2}}&0&-\frac{i}{\sqrt{2}}&0&0&0\\\
0&0&0&\frac{i}{\sqrt{2}}&0&0&0&0\\\ 0&0&0&0&0&0&-\frac{i}{2}&0\\\
0&0&0&0&0&\frac{i}{2}&0&0\\\ 0&0&0&0&0&0&0&0\end{array}\right)$ (134)

$\displaystyle{\rm({\bf D}^{3}+{2\over 3}{\bf F}_{(8)}^{3})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}\frac{5}{6}&0&0&0&0&0&0&0\\\
0&-\frac{5}{6}&0&0&0&0&0&0\\\ 0&0&\frac{2}{3}&0&0&0&0&0\\\
0&0&0&0&0&0&0&\frac{1}{\sqrt{3}}\\\ 0&0&0&0&-\frac{2}{3}&0&0&0\\\
0&0&0&0&0&-\frac{1}{6}&0&0\\\ 0&0&0&0&0&0&\frac{1}{6}&0\\\
0&0&0&\frac{1}{\sqrt{3}}&0&0&0&0\end{array}\right)$ (143)
$\displaystyle{\rm{\bf D}^{3}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}\frac{1}{2}&0&0&0&0&0&0&0\\\
0&-\frac{1}{2}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&\frac{1}{\sqrt{3}}\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&-\frac{1}{2}&0&0\\\ 0&0&0&0&0&0&\frac{1}{2}&0\\\
0&0&0&\frac{1}{\sqrt{3}}&0&0&0&0\end{array}\right)$ (152) $\displaystyle{\bf
F}_{(8)}^{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}\frac{1}{2}&0&0&0&0&0&0&0\\\
0&-\frac{1}{2}&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&-1&0&0&0\\\ 0&0&0&0&0&\frac{1}{2}&0&0\\\ 0&0&0&0&0&0&-\frac{1}{2}&0\\\
0&0&0&0&0&0&0&0\end{array}\right)$ (161)

$\displaystyle{\rm({\bf D}^{4}+{2\over 3}{\bf F}_{(8)}^{4})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&\frac{1}{6\sqrt{2}}&0&0&0&-\frac{\sqrt{\frac{3}{2}}}{2}\\\
0&0&0&0&\frac{1}{6}&0&0&0\\\ 0&0&0&0&0&-\frac{5}{6}&0&0\\\
\frac{1}{6\sqrt{2}}&0&0&0&0&0&-\frac{5}{6\sqrt{2}}&0\\\
0&\frac{1}{6}&0&0&0&0&0&0\\\ 0&0&-\frac{5}{6}&0&0&0&0&0\\\
0&0&0&-\frac{5}{6\sqrt{2}}&0&0&0&-\frac{1}{2\sqrt{6}}\\\
-\frac{\sqrt{\frac{3}{2}}}{2}&0&0&0&0&0&-\frac{1}{2\sqrt{6}}&0\end{array}\right)$
(170) $\displaystyle{\rm{\bf D}^{4}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&\frac{1}{2\sqrt{2}}&0&0&0&-\frac{1}{2\sqrt{6}}\\\
0&0&0&0&\frac{1}{2}&0&0&0\\\ 0&0&0&0&0&-\frac{1}{2}&0&0\\\
\frac{1}{2\sqrt{2}}&0&0&0&0&0&-\frac{1}{2\sqrt{2}}&0\\\
0&\frac{1}{2}&0&0&0&0&0&0\\\ 0&0&-\frac{1}{2}&0&0&0&0&0\\\
0&0&0&-\frac{1}{2\sqrt{2}}&0&0&0&\frac{1}{2\sqrt{6}}\\\
-\frac{1}{2\sqrt{6}}&0&0&0&0&0&\frac{1}{2\sqrt{6}}&0\end{array}\right)$ (179)
$\displaystyle{\bf F}_{(8)}^{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&-\frac{1}{2\sqrt{2}}&0&0&0&-\frac{\sqrt{\frac{3}{2}}}{2}\\\
0&0&0&0&-\frac{1}{2}&0&0&0\\\ 0&0&0&0&0&-\frac{1}{2}&0&0\\\
-\frac{1}{2\sqrt{2}}&0&0&0&0&0&-\frac{1}{2\sqrt{2}}&0\\\
0&-\frac{1}{2}&0&0&0&0&0&0\\\ 0&0&-\frac{1}{2}&0&0&0&0&0\\\
0&0&0&-\frac{1}{2\sqrt{2}}&0&0&0&-\frac{\sqrt{\frac{3}{2}}}{2}\\\
-\frac{\sqrt{\frac{3}{2}}}{2}&0&0&0&0&0&-\frac{\sqrt{\frac{3}{2}}}{2}&0\end{array}\right)$
(188)

$\displaystyle{\rm({\bf D}^{5}+{2\over 3}{\bf F}_{(8)}^{5})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&-\frac{i}{6\sqrt{2}}&0&0&0&\frac{1}{2}i\sqrt{\frac{3}{2}}\\\
0&0&0&0&-\frac{i}{6}&0&0&0\\\ 0&0&0&0&0&\frac{5i}{6}&0&0\\\
\frac{i}{6\sqrt{2}}&0&0&0&0&0&\frac{5i}{6\sqrt{2}}&0\\\
0&\frac{i}{6}&0&0&0&0&0&0\\\ 0&0&-\frac{5i}{6}&0&0&0&0&0\\\
0&0&0&-\frac{5i}{6\sqrt{2}}&0&0&0&-\frac{i}{2\sqrt{6}}\\\
-\frac{1}{2}i\sqrt{\frac{3}{2}}&0&0&0&0&0&\frac{i}{2\sqrt{6}}&0\end{array}\right)$
(197) $\displaystyle{\rm{\bf D}^{5}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&-\frac{i}{2\sqrt{2}}&0&0&0&\frac{i}{2\sqrt{6}}\\\
0&0&0&0&-\frac{i}{2}&0&0&0\\\ 0&0&0&0&0&\frac{i}{2}&0&0\\\
\frac{i}{2\sqrt{2}}&0&0&0&0&0&\frac{i}{2\sqrt{2}}&0\\\
0&\frac{i}{2}&0&0&0&0&0&0\\\ 0&0&-\frac{i}{2}&0&0&0&0&0\\\
0&0&0&-\frac{i}{2\sqrt{2}}&0&0&0&\frac{i}{2\sqrt{6}}\\\
-\frac{i}{2\sqrt{6}}&0&0&0&0&0&-\frac{i}{2\sqrt{6}}&0\end{array}\right)$ (206)
$\displaystyle{\bf F}_{(8)}^{5}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&0&\frac{i}{2\sqrt{2}}&0&0&0&\frac{1}{2}i\sqrt{\frac{3}{2}}\\\
0&0&0&0&\frac{i}{2}&0&0&0\\\ 0&0&0&0&0&\frac{i}{2}&0&0\\\
-\frac{i}{2\sqrt{2}}&0&0&0&0&0&\frac{i}{2\sqrt{2}}&0\\\
0&-\frac{i}{2}&0&0&0&0&0&0\\\ 0&0&-\frac{i}{2}&0&0&0&0&0\\\
0&0&0&-\frac{i}{2\sqrt{2}}&0&0&0&-\frac{1}{2}i\sqrt{\frac{3}{2}}\\\
-\frac{1}{2}i\sqrt{\frac{3}{2}}&0&0&0&0&0&\frac{1}{2}i\sqrt{\frac{3}{2}}&0\end{array}\right)$
(215)

$\displaystyle{\rm({\bf D}^{6}+{2\over 3}{\bf F}_{(8)}^{6})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&-\frac{1}{6}&0&0&0&0&0\\\
0&0&0&-\frac{1}{6\sqrt{2}}&0&0&0&-\frac{\sqrt{\frac{3}{2}}}{2}\\\
-\frac{1}{6}&0&0&0&0&0&0&0\\\
0&-\frac{1}{6\sqrt{2}}&0&0&0&-\frac{5}{6\sqrt{2}}&0&0\\\
0&0&0&0&0&0&-\frac{5}{6}&0\\\
0&0&0&-\frac{5}{6\sqrt{2}}&0&0&0&\frac{1}{2\sqrt{6}}\\\
0&0&0&0&-\frac{5}{6}&0&0&0\\\
0&-\frac{\sqrt{\frac{3}{2}}}{2}&0&0&0&\frac{1}{2\sqrt{6}}&0&0\end{array}\right)$
(224) $\displaystyle{\rm{\bf D}^{6}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&-\frac{1}{2}&0&0&0&0&0\\\
0&0&0&-\frac{1}{2\sqrt{2}}&0&0&0&-\frac{1}{2\sqrt{6}}\\\
-\frac{1}{2}&0&0&0&0&0&0&0\\\
0&-\frac{1}{2\sqrt{2}}&0&0&0&-\frac{1}{2\sqrt{2}}&0&0\\\
0&0&0&0&0&0&-\frac{1}{2}&0\\\
0&0&0&-\frac{1}{2\sqrt{2}}&0&0&0&-\frac{1}{2\sqrt{6}}\\\
0&0&0&0&-\frac{1}{2}&0&0&0\\\
0&-\frac{1}{2\sqrt{6}}&0&0&0&-\frac{1}{2\sqrt{6}}&0&0\end{array}\right)$ (233)
$\displaystyle{\bf F}_{(8)}^{6}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&\frac{1}{2}&0&0&0&0&0\\\
0&0&0&\frac{1}{2\sqrt{2}}&0&0&0&-\frac{\sqrt{\frac{3}{2}}}{2}\\\
\frac{1}{2}&0&0&0&0&0&0&0\\\
0&\frac{1}{2\sqrt{2}}&0&0&0&-\frac{1}{2\sqrt{2}}&0&0\\\
0&0&0&0&0&0&-\frac{1}{2}&0\\\
0&0&0&-\frac{1}{2\sqrt{2}}&0&0&0&\frac{\sqrt{\frac{3}{2}}}{2}\\\
0&0&0&0&-\frac{1}{2}&0&0&0\\\
0&-\frac{\sqrt{\frac{3}{2}}}{2}&0&0&0&\frac{\sqrt{\frac{3}{2}}}{2}&0&0\end{array}\right)$
(242)

$\displaystyle{\rm({\bf D}^{7}+{2\over 3}{\bf F}_{(8)}^{7})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&\frac{i}{6}&0&0&0&0&0\\\
0&0&0&\frac{i}{6\sqrt{2}}&0&0&0&\frac{1}{2}i\sqrt{\frac{3}{2}}\\\
-\frac{i}{6}&0&0&0&0&0&0&0\\\
0&-\frac{i}{6\sqrt{2}}&0&0&0&\frac{5i}{6\sqrt{2}}&0&0\\\
0&0&0&0&0&0&\frac{5i}{6}&0\\\
0&0&0&-\frac{5i}{6\sqrt{2}}&0&0&0&\frac{i}{2\sqrt{6}}\\\
0&0&0&0&-\frac{5i}{6}&0&0&0\\\
0&-\frac{1}{2}i\sqrt{\frac{3}{2}}&0&0&0&-\frac{i}{2\sqrt{6}}&0&0\end{array}\right)$
(251) $\displaystyle{\rm{\bf D}^{7}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&\frac{i}{2}&0&0&0&0&0\\\
0&0&0&\frac{i}{2\sqrt{2}}&0&0&0&\frac{i}{2\sqrt{6}}\\\
-\frac{i}{2}&0&0&0&0&0&0&0\\\
0&-\frac{i}{2\sqrt{2}}&0&0&0&\frac{i}{2\sqrt{2}}&0&0\\\
0&0&0&0&0&0&\frac{i}{2}&0\\\
0&0&0&-\frac{i}{2\sqrt{2}}&0&0&0&-\frac{i}{2\sqrt{6}}\\\
0&0&0&0&-\frac{i}{2}&0&0&0\\\
0&-\frac{i}{2\sqrt{6}}&0&0&0&\frac{i}{2\sqrt{6}}&0&0\end{array}\right)$ (260)
$\displaystyle{\bf F}_{(8)}^{7}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}0&0&-\frac{i}{2}&0&0&0&0&0\\\
0&0&0&-\frac{i}{2\sqrt{2}}&0&0&0&\frac{1}{2}i\sqrt{\frac{3}{2}}\\\
\frac{i}{2}&0&0&0&0&0&0&0\\\
0&\frac{i}{2\sqrt{2}}&0&0&0&\frac{i}{2\sqrt{2}}&0&0\\\
0&0&0&0&0&0&\frac{i}{2}&0\\\
0&0&0&-\frac{i}{2\sqrt{2}}&0&0&0&\frac{1}{2}i\sqrt{\frac{3}{2}}\\\
0&0&0&0&-\frac{i}{2}&0&0&0\\\
0&-\frac{1}{2}i\sqrt{\frac{3}{2}}&0&0&0&-\frac{1}{2}i\sqrt{\frac{3}{2}}&0&0\end{array}\right)$
(269)

$\displaystyle{\rm({\bf D}^{8}+{2\over 3}{\bf F}_{(8)}^{2})}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}\frac{1}{2\sqrt{3}}&0&0&0&0&0&0&0\\\
0&\frac{1}{2\sqrt{3}}&0&0&0&0&0&0\\\ 0&0&\frac{1}{\sqrt{3}}&0&0&0&0&0\\\
0&0&0&\frac{1}{\sqrt{3}}&0&0&0&0\\\ 0&0&0&0&\frac{1}{\sqrt{3}}&0&0&0\\\
0&0&0&0&0&-\frac{\sqrt{3}}{2}&0&0\\\ 0&0&0&0&0&0&-\frac{\sqrt{3}}{2}&0\\\
0&0&0&0&0&0&0&-\frac{1}{\sqrt{3}}\end{array}\right)$ (278)
$\displaystyle{\rm{\bf D}^{8}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}-\frac{1}{2\sqrt{3}}&0&0&0&0&0&0&0\\\
0&-\frac{1}{2\sqrt{3}}&0&0&0&0&0&0\\\ 0&0&\frac{1}{\sqrt{3}}&0&0&0&0&0\\\
0&0&0&\frac{1}{\sqrt{3}}&0&0&0&0\\\ 0&0&0&0&\frac{1}{\sqrt{3}}&0&0&0\\\
0&0&0&0&0&-\frac{1}{2\sqrt{3}}&0&0\\\ 0&0&0&0&0&0&-\frac{1}{2\sqrt{3}}&0\\\
0&0&0&0&0&0&0&-\frac{1}{\sqrt{3}}\end{array}\right)$ (287) $\displaystyle{\bf
F}_{(8)}^{8}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllll}\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0\\\
0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&-\frac{\sqrt{3}}{2}&0&0\\\
0&0&0&0&0&0&-\frac{\sqrt{3}}{2}&0\\\ 0&0&0&0&0&0&0&0\end{array}\right)$ (296)

###  A.2 Octet-Decimet 8x10 Transition Matrices $T^{a}$

$\displaystyle
T_{1}=\left(\begin{array}[]{llllllllll|l}-\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{6}}&0&0&0&0&0&0&0&p\\\
0&-\frac{1}{\sqrt{6}}&0&\frac{1}{\sqrt{2}}&0&0&0&0&0&0&n\\\
0&0&0&0&0&\frac{1}{2\sqrt{3}}&0&0&0&0&\Sigma^{+}\\\
0&0&0&0&\frac{1}{2\sqrt{3}}&0&\frac{1}{2\sqrt{3}}&0&0&0&\Sigma^{0}\\\
0&0&0&0&0&\frac{1}{2\sqrt{3}}&0&0&0&0&\Sigma^{-}\\\
0&0&0&0&0&0&0&0&-\frac{1}{\sqrt{6}}&0&\Xi^{0}\\\
0&0&0&0&0&0&0&-\frac{1}{\sqrt{6}}&0&0&\Xi^{-}\\\
0&0&0&0&\frac{1}{2}&0&-\frac{1}{2}&0&0&0&\Lambda_{8}\\\ \hline\cr\\\
\Delta^{++}&\Delta^{+}&\Delta^{0}&\Delta^{-}&\Sigma^{*+}&\Sigma^{*0}&\Sigma^{*-}&\Xi^{*0}&\Xi^{*-}&\Omega\end{array}\right)$
(307)

$\displaystyle
T_{2}=\left(\begin{array}[]{llllllllll}-\frac{i}{\sqrt{2}}&0&-\frac{i}{\sqrt{6}}&0&0&0&0&0&0&0\\\
0&-\frac{i}{\sqrt{6}}&0&-\frac{i}{\sqrt{2}}&0&0&0&0&0&0\\\
0&0&0&0&0&-\frac{i}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&\frac{i}{2\sqrt{3}}&0&-\frac{i}{2\sqrt{3}}&0&0&0\\\
0&0&0&0&0&\frac{i}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&0&0&\frac{i}{\sqrt{6}}&0\\\
0&0&0&0&0&0&0&-\frac{i}{\sqrt{6}}&0&0\\\
0&0&0&0&\frac{i}{2}&0&\frac{i}{2}&0&0&0\end{array}\right)$ (316)

$\displaystyle
T_{3}=\left(\begin{array}[]{llllllllll}0&\sqrt{\frac{2}{3}}&0&0&0&0&0&0&0&0\\\
0&0&\sqrt{\frac{2}{3}}&0&0&0&0&0&0&0\\\
0&0&0&0&\frac{1}{\sqrt{6}}&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&-\frac{1}{\sqrt{6}}&0&0&0\\\
0&0&0&0&0&0&0&-\frac{1}{\sqrt{6}}&0&0\\\
0&0&0&0&0&0&0&0&\frac{1}{\sqrt{6}}&0\\\
0&0&0&0&0&-\frac{1}{\sqrt{2}}&0&0&0&0\end{array}\right)$ (325)

$\displaystyle
T_{4}=\left(\begin{array}[]{llllllllll}0&0&0&0&0&\frac{1}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&\frac{1}{\sqrt{6}}&0&0&0\\\
-\frac{1}{\sqrt{2}}&0&0&0&0&0&0&\frac{1}{\sqrt{6}}&0&0\\\
0&-\frac{1}{\sqrt{3}}&0&0&0&0&0&0&\frac{1}{2\sqrt{3}}&0\\\
0&0&-\frac{1}{\sqrt{6}}&0&0&0&0&0&0&0\\\
0&0&0&0&\frac{1}{\sqrt{6}}&0&0&0&0&-\frac{1}{\sqrt{2}}\\\
0&0&0&0&0&\frac{1}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&0&0&-\frac{1}{2}&0\end{array}\right)$ (334)

$\displaystyle
T_{5}=\left(\begin{array}[]{llllllllll}0&0&0&0&0&-\frac{i}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&-\frac{i}{\sqrt{6}}&0&0&0\\\
-\frac{i}{\sqrt{2}}&0&0&0&0&0&0&-\frac{i}{\sqrt{6}}&0&0\\\
0&-\frac{i}{\sqrt{3}}&0&0&0&0&0&0&-\frac{i}{2\sqrt{3}}&0\\\
0&0&-\frac{i}{\sqrt{6}}&0&0&0&0&0&0&0\\\
0&0&0&0&\frac{i}{\sqrt{6}}&0&0&0&0&\frac{i}{\sqrt{2}}\\\
0&0&0&0&0&\frac{i}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&0&0&\frac{i}{2}&0\end{array}\right)$ (343)

$\displaystyle
T_{6}=\left(\begin{array}[]{llllllllll}0&0&0&0&-\frac{1}{\sqrt{6}}&0&0&0&0&0\\\
0&0&0&0&0&-\frac{1}{2\sqrt{3}}&0&0&0&0\\\
0&-\frac{1}{\sqrt{6}}&0&0&0&0&0&0&0&0\\\
0&0&-\frac{1}{\sqrt{3}}&0&0&0&0&\frac{1}{2\sqrt{3}}&0&0\\\
0&0&0&-\frac{1}{\sqrt{2}}&0&0&0&0&\frac{1}{\sqrt{6}}&0\\\
0&0&0&0&0&\frac{1}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&\frac{1}{\sqrt{6}}&0&0&-\frac{1}{\sqrt{2}}\\\
0&0&0&0&0&0&0&\frac{1}{2}&0&0\end{array}\right)$ (352)

$\displaystyle
T_{7}=\left(\begin{array}[]{llllllllll}0&0&0&0&\frac{i}{\sqrt{6}}&0&0&0&0&0\\\
0&0&0&0&0&\frac{i}{2\sqrt{3}}&0&0&0&0\\\
0&-\frac{i}{\sqrt{6}}&0&0&0&0&0&0&0&0\\\
0&0&-\frac{i}{\sqrt{3}}&0&0&0&0&-\frac{i}{2\sqrt{3}}&0&0\\\
0&0&0&-\frac{i}{\sqrt{2}}&0&0&0&0&-\frac{i}{\sqrt{6}}&0\\\
0&0&0&0&0&\frac{i}{2\sqrt{3}}&0&0&0&0\\\
0&0&0&0&0&0&\frac{i}{\sqrt{6}}&0&0&\frac{i}{\sqrt{2}}\\\
0&0&0&0&0&0&0&-\frac{i}{2}&0&0\end{array}\right)$ (361)

$\displaystyle T_{8}=\left(\begin{array}[]{llllllllll}0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0&0\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0\\\
0&0&0&0&0&0&0&-\frac{1}{\sqrt{2}}&0&0\\\
0&0&0&0&0&0&0&0&-\frac{1}{\sqrt{2}}&0\\\
0&0&0&0&0&0&0&0&0&0\end{array}\right)$ (370)

###  A.3 Decimet Generator 10x10 Matrices ${\bf F}_{(10)}^{a}$

$\displaystyle{\bf F}_{(10)}^{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll|l}0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0&0&\Delta^{++}\\\
\frac{\sqrt{3}}{2}&0&1&0&0&0&0&0&0&0&\Delta^{+}\\\
0&1&0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0&\Delta^{0}\\\
0&0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0&\Delta^{-}\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0&\Sigma^{*+}\\\
0&0&0&0&\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}&0&0&0&\Sigma^{*0}\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0&\Sigma^{*-}\\\
0&0&0&0&0&0&0&0&\frac{1}{2}&0&\Xi^{*0}\\\
0&0&0&0&0&0&0&\frac{1}{2}&0&0&\Xi^{*-}\\\ 0&0&0&0&0&0&0&0&0&0&\Omega\\\
\hline\cr\\\
\Delta^{++}&\Delta^{+}&\Delta^{0}&\Delta^{-}&\Sigma^{*+}&\Sigma^{*0}&\Sigma^{*-}&\Xi^{*0}&\Xi^{*-}&\Omega\end{array}\right)$
(383)

$\displaystyle{\bf F}_{(10)}^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}0&-\frac{i\sqrt{3}}{2}&0&0&0&0&0&0&0&0\\\
\frac{i\sqrt{3}}{2}&0&-i&0&0&0&0&0&0&0\\\
0&i&0&-\frac{i\sqrt{3}}{2}&0&0&0&0&0&0\\\
0&0&\frac{i\sqrt{3}}{2}&0&0&0&0&0&0&0\\\
0&0&0&0&0&-\frac{i}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&\frac{i}{\sqrt{2}}&0&-\frac{i}{\sqrt{2}}&0&0&0\\\
0&0&0&0&0&\frac{i}{\sqrt{2}}&0&0&0&0\\\ 0&0&0&0&0&0&0&0&-\frac{i}{2}&0\\\
0&0&0&0&0&0&0&\frac{i}{2}&0&0\\\ 0&0&0&0&0&0&0&0&0&0\end{array}\right)$ (394)

$\displaystyle{\bf F}_{(10)}^{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}\frac{3}{2}&0&0&0&0&0&0&0&0&0\\\
0&\frac{1}{2}&0&0&0&0&0&0&0&0\\\ 0&0&-\frac{1}{2}&0&0&0&0&0&0&0\\\
0&0&0&-\frac{3}{2}&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&-1&0&0&0\\\
0&0&0&0&0&0&0&\frac{1}{2}&0&0\\\ 0&0&0&0&0&0&0&0&-\frac{1}{2}&0\\\
0&0&0&0&0&0&0&0&0&0\end{array}\right)$ (405)

$\displaystyle{\bf F}_{(10)}^{4}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}0&0&0&0&\frac{\sqrt{3}}{2}&0&0&0&0&0\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\ 0&0&0&0&0&0&\frac{1}{2}&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ \frac{\sqrt{3}}{2}&0&0&0&0&0&0&1&0&0\\\
0&\frac{1}{\sqrt{2}}&0&0&0&0&0&0&\frac{1}{\sqrt{2}}&0\\\
0&0&\frac{1}{2}&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&\frac{\sqrt{3}}{2}\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&0&\frac{\sqrt{3}}{2}&0&0\end{array}\right)$ (416)

$\displaystyle{\bf F}_{(10)}^{5}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}0&0&0&0&-\frac{i\sqrt{3}}{2}&0&0&0&0&0\\\
0&0&0&0&0&-\frac{i}{\sqrt{2}}&0&0&0&0\\\ 0&0&0&0&0&0&-\frac{i}{2}&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ \frac{i\sqrt{3}}{2}&0&0&0&0&0&0&-i&0&0\\\
0&\frac{i}{\sqrt{2}}&0&0&0&0&0&0&-\frac{i}{\sqrt{2}}&0\\\
0&0&\frac{i}{2}&0&0&0&0&0&0&0\\\ 0&0&0&0&i&0&0&0&0&-\frac{i\sqrt{3}}{2}\\\
0&0&0&0&0&\frac{i}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&0&\frac{i\sqrt{3}}{2}&0&0\end{array}\right)$ (427)

$\displaystyle{\bf F}_{(10)}^{6}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&\frac{1}{2}&0&0&0&0&0\\\ 0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&\frac{\sqrt{3}}{2}&0&0&0\\\ 0&\frac{1}{2}&0&0&0&0&0&0&0&0\\\
0&0&\frac{1}{\sqrt{2}}&0&0&0&0&\frac{1}{\sqrt{2}}&0&0\\\
0&0&0&\frac{\sqrt{3}}{2}&0&0&0&0&1&0\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&1&0&0&\frac{\sqrt{3}}{2}\\\
0&0&0&0&0&0&0&0&\frac{\sqrt{3}}{2}&0\end{array}\right)$ (438)

$\displaystyle{\bf F}_{(10)}^{7}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&-\frac{i}{2}&0&0&0&0&0\\\ 0&0&0&0&0&-\frac{i}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&-\frac{i\sqrt{3}}{2}&0&0&0\\\ 0&\frac{i}{2}&0&0&0&0&0&0&0&0\\\
0&0&\frac{i}{\sqrt{2}}&0&0&0&0&-\frac{i}{\sqrt{2}}&0&0\\\
0&0&0&\frac{i\sqrt{3}}{2}&0&0&0&0&-i&0\\\
0&0&0&0&0&\frac{i}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&0&0&i&0&0&-\frac{i\sqrt{3}}{2}\\\
0&0&0&0&0&0&0&0&\frac{i\sqrt{3}}{2}&0\end{array}\right)$ (449)

$\displaystyle{\bf F}_{(10)}^{8}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{llllllllll}\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0&0&0\\\
0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0&0\\\
0&0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0&0\\\
0&0&0&\frac{\sqrt{3}}{2}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&-\frac{\sqrt{3}}{2}&0&0\\\
0&0&0&0&0&0&0&0&-\frac{\sqrt{3}}{2}&0\\\
0&0&0&0&0&0&0&0&0&-\sqrt{3}\end{array}\right)$ (460)

###  A.4 Singlet-Octet 1x8 Transition Matrices $T^{a}_{1/8}$

$\displaystyle{\rm\bf
T}^{1}_{1/8}=\left(\begin{array}[]{llllllll|l}0&0&-{1\over\sqrt{2}}&0&{1\over\sqrt{2}}&0&0&0&\Lambda_{1}\\\
\hline\cr\\\
p&n&\Sigma^{+}&\Sigma^{0}&\Sigma^{-}&\Xi^{0}&\Xi^{-}&\Lambda_{8}\end{array}\right)$
(464)

$\displaystyle{\rm\bf
T}^{2}_{1/8}=\left(\begin{array}[]{llllllll}0&0&-{i\over\sqrt{2}}&0&-{i\over\sqrt{2}}&0&0&0\end{array}\right)$
(466)

$\displaystyle{\rm\bf
T}^{3}_{1/8}=\left(\begin{array}[]{llllllll}0&0&0&1&0&0&0&0\end{array}\right)$
(468)

$\displaystyle{\rm\bf
T}^{4}_{1/8}=\left(\begin{array}[]{llllllll}{1\over\sqrt{2}}&0&0&0&0&0&-{1\over\sqrt{2}}&0\end{array}\right)$
(470)

$\displaystyle{\rm\bf
T}^{5}_{1/8}=\left(\begin{array}[]{llllllll}{i\over\sqrt{2}}&0&0&0&0&0&{i\over\sqrt{2}}&0\end{array}\right)$
(472)

$\displaystyle{\rm\bf
T}^{6}_{1/8}=\left(\begin{array}[]{llllllll}0&{1\over\sqrt{2}}&0&0&0&{1\over\sqrt{2}}&0&0\end{array}\right)$
(474)

$\displaystyle{\rm\bf
T}^{7}_{1/8}=\left(\begin{array}[]{llllllll}0&{i\over\sqrt{2}}&0&0&0&-{i\over\sqrt{2}}&0&0\end{array}\right)$
(476)

$\displaystyle{\rm\bf
T}^{8}_{1/8}=\left(\begin{array}[]{llllllll}0&0&0&0&0&0&0&1\end{array}\right)$
(478)

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