20
12. Global Non-Abelian Symmetries
which also has the ‘generalized phase transformation’ character of (12.35), now
with eight ‘phase angles’. Thus W is parametrized as W = exp(iα · λ/2).
As in the case of SU(2) f , exact symmetry under SU(3) f would imply that
the three states u, d and s were degenerate in mass. Actually, of course, this
is not the case: in particular, while the u and d quark masses are of order 1-5
MeV, the s quark mass is greater, of order 100 MeV. Nevertheless it is still
possible to regard this as relatively small on a typical hadronic mass scale, so
we may proceed to explore the physical consequences of this (approximate)
SU(3) f flavour symmetry.
Such a symmetry implies that the eigenvalues of the λ’s are constants
of the motion, but because of the commutation relations (12.73) not all of
these operators have simultaneous eigenstates. This happened for SU(2) too,
but there the very close analogy with SO(3) told us how the states were
to be correctly classified, by the eigenvalues of the relevant complete set of
mutually commuting operators. Here it is more involved – for a start, there
are 8 matrices λ a . A glance at appendix M, section M.4.5, shows that two of
the λ’s are diagonal (in the chosen representation), namely λ 3 and λ 8 . This
means physically that for SU(3) there are two additively conserved quantum
numbers, which in this case are of course the third component of hadronic
isospin (since λ 3 is simply τ 3 bordered by zeros), and a quantity related to
strangeness. Defining the hadronic hypercharge Y by Y = B + S, where B is
the baryon number (
1 for each quark) and the strangeness values are S(u) =
3
S(d) = 0, S(s) = −1, we find that the physically required eigenvalues imply
3
that the matrix representing the hypercharge operator is Y
( ) = √ 1 λ 8 , in this
3
fundamental (three-dimensional) representation, denoted by the symbol 3.
(3)
Identifying T 3 =
1 λ 3 then gives the Gell-Mann–Nishijima relation Q =
2
T 3 + Y/2 for the quark charges in units of | e |.
So λ 3 and λ 8 are analogous to τ 3 ; what about the analogue of τ
2 , which
is diagonalizable simultaneously with τ 3 in the case of SU(2)? Indeed, (cf
(12.41)) τ
2 is a multiple of the 2 × 2 unit matrix. In just the same way one
finds that λ
2 is also proportional to the unit matrix:
∑ 8
4
(λ/2)
2 =
(λ a /2)
2 = 1 3 ,
(12.75)
3
a=1
as can be verified from the explicit forms of the λ-matrices given in appendix
M, section M.4.5. Thus we may characterize the ‘fundamental triplet’ (12.70)
by the eigenvalues of (λ/2)
2 , λ 3 and λ 8 . The conventional way of representing
this pictorially is to plot the states in a Y − T 3 diagram, as shown in figure
12.3.
We may now consider other representations of SU(3) f . The first important one is that to which the antiquarks belong. If we denote the fundamental
three-dimensional representation accommodating the quarks by 3, then the
antiquarks have quantum numbers appropriate to the ‘complex conjugate’ of
12. Global Non-Abelian Symmetries
which also has the ‘generalized phase transformation’ character of (12.35), now
with eight ‘phase angles’. Thus W is parametrized as W = exp(iα · λ/2).
As in the case of SU(2) f , exact symmetry under SU(3) f would imply that
the three states u, d and s were degenerate in mass. Actually, of course, this
is not the case: in particular, while the u and d quark masses are of order 1-5
MeV, the s quark mass is greater, of order 100 MeV. Nevertheless it is still
possible to regard this as relatively small on a typical hadronic mass scale, so
we may proceed to explore the physical consequences of this (approximate)
SU(3) f flavour symmetry.
Such a symmetry implies that the eigenvalues of the λ’s are constants
of the motion, but because of the commutation relations (12.73) not all of
these operators have simultaneous eigenstates. This happened for SU(2) too,
but there the very close analogy with SO(3) told us how the states were
to be correctly classified, by the eigenvalues of the relevant complete set of
mutually commuting operators. Here it is more involved – for a start, there
are 8 matrices λ a . A glance at appendix M, section M.4.5, shows that two of
the λ’s are diagonal (in the chosen representation), namely λ 3 and λ 8 . This
means physically that for SU(3) there are two additively conserved quantum
numbers, which in this case are of course the third component of hadronic
isospin (since λ 3 is simply τ 3 bordered by zeros), and a quantity related to
strangeness. Defining the hadronic hypercharge Y by Y = B + S, where B is
the baryon number (
1 for each quark) and the strangeness values are S(u) =
3
S(d) = 0, S(s) = −1, we find that the physically required eigenvalues imply
3
that the matrix representing the hypercharge operator is Y
( ) = √ 1 λ 8 , in this
3
fundamental (three-dimensional) representation, denoted by the symbol 3.
(3)
Identifying T 3 =
1 λ 3 then gives the Gell-Mann–Nishijima relation Q =
2
T 3 + Y/2 for the quark charges in units of | e |.
So λ 3 and λ 8 are analogous to τ 3 ; what about the analogue of τ
2 , which
is diagonalizable simultaneously with τ 3 in the case of SU(2)? Indeed, (cf
(12.41)) τ
2 is a multiple of the 2 × 2 unit matrix. In just the same way one
finds that λ
2 is also proportional to the unit matrix:
∑ 8
4
(λ/2)
2 =
(λ a /2)
2 = 1 3 ,
(12.75)
3
a=1
as can be verified from the explicit forms of the λ-matrices given in appendix
M, section M.4.5. Thus we may characterize the ‘fundamental triplet’ (12.70)
by the eigenvalues of (λ/2)
2 , λ 3 and λ 8 . The conventional way of representing
this pictorially is to plot the states in a Y − T 3 diagram, as shown in figure
12.3.
We may now consider other representations of SU(3) f . The first important one is that to which the antiquarks belong. If we denote the fundamental
three-dimensional representation accommodating the quarks by 3, then the
antiquarks have quantum numbers appropriate to the ‘complex conjugate’ of