Download Internal Symmetries of Strong Interactions {intsymm

H. Kleinert, PARTICLES AND QUANTUM FIELDS
November 19, 2016 ( /home/kleinert/kleinert/books/qft/nachspa1.tex)
In science one tries to tell people something
that no one ever knew before.
But in poetry, it’s the exact opposite.
Paul Dirac (1902–1984)
24
Internal Symmetries of Strong Interactions
If we open up a standard table of the presently known elementary particles, we see
a confusing variety of them with many different properties. Before trying to develop
a more detailed theory of their interactions it is useful to find certain organizing
principles which correlate their quantum numbers and mass spectra [1].
24.1
{
intsym
{@spontsb}
Classification of Elementary Particles
The most fundamental distinction concerns their statistical properties. They can
be of the Fermi-Dirac or Bose-Einstein type, in which case the particles are called
fermions or bosons, respectively. Historically, the fermions were subdivided into
light and heavy fermions. The former are called leptons (named after the Greek
λεπτóς=“fine, small, thin”), which were historically electrons, muons, and neutrinos.
Recently, also heavier particles were found which should be considered as leptons.
This is due to the similarity of their interactions with those of the traditional, truely
light leptons. In fact, the characteristic property of leptons is now that they are
fermions with no strong interactions. The distinction according to mass is no longer
significant.
The most important of the traditional heavy fermions are those particles which
provide most of the mass of to atomic nuclei. These are the so-called nucleons, which
can be protons or neutrons. These particles have strong interactions. They can be
excited in nuclear collisions, thereby producing short-lived resonances. Also these
are classified as elementary heavy fermions. The set of all heavy fermions with strong
interactions are called baryons (particles named after the Greek βαρύς = “heavy”).
All baryons can be produced in collision or decay processes, sometimes in the form
of antiparticles which, according to the spin-statistics theorem of Section 7.10, have
exactly the same mass and spin as the corresponding particles, except for some
quantum numbers such as the charge. They carry an opposite sign.
The particles with Bose-Einstein statistics are separated into those that don’t
and those that do possess strong interactions. The former play an important role in
the theoretical description of gravitational, electromagnetic, and weak interactions.
1330
{CLASSEP}
1331
24.1 Classification of Elementary Particles
They are called gravitons, photons, and intermediate vector bosons W and Z. The
bosons with strong interactions are called mesons (middle-heavy particles after the
Greek µέσóς). The set of all particles with strong interactions, baryons and mesons,
are called hadrons (after the Greek ὰδρóς = “stout”).
Only few of all these particles are completely stable. The most prominent ones
make up the stable matter of the universe. These are the protons, electrons, photons,
neutrinos, and gravitons. The proton is the only stable baryon. The other important
nuclear constituents, the neutrons, live on the average only 898 ± 16 seconds. After
this, a neutron decays into a proton, an electron, and an antineutrino.
Among the leptons, only electrons and neutrinos are stable particles. The muons
are particles very similar to the electrons, but much heavier. They decay in about
10−6 seconds into an electron, a neutrino, and an antineutrino.
The proton is not only the only stable heavy fermion, it is the only stable strongly
interacting particle. All mesons are unstable. The decays proceeding over a “long”
time, as in the case of the neutron, are called weak. Even lifetimes of 10−13 seconds
are still considered as “long” in elementary particle physics, and the decay is called
weak. A particle with such a lifetime down to 10−11 sec leaves an observable trace
in a bubble chamber.
Most of the hadrons which decay via strong interaction processes live only for a
much shorter time than that, so short that they do travel any visible distance in a
bubble chamber. They are observable only as resonances in scattering cross sections.
The most prominent example is the first resonance in pion nucleon scattering shown
in Figs. 24.1 and 24.2. The resonance is called ∆(1232). It is found at a pion beam
momentum
pπ = 0.34 GeV/c.
(24.1).6.1 {nolabel}
In the center of mass coordinate frame, the energy is equal to the mass of the
resonance M,
ECM = M =
q
(Eπ + mp )2 − p2π =
q
m2π + m2p + 2mp Eπ
(24.2).6.2 {nolabel}
q
where Eπ = p2π + m2π is beam energy and mπ , mp are the masses of pion and
proton. From pπ = 0.34 GeV/c one finds M = 1232 MeV/c2 which is the number
given in the parenthesis of ∆(1232). The large width in the total cross section shows
the rather short lifetime. The peak is of the Lorentz shape and can be parametrized
as
Γ2
σtot ∝ 2πλ̄2
,
(24.3).6.3 {nolabel}
(ECM − M)2 + (Γ/2)2
where λ̄ ≡ h̄/pπ is the Compton wavelength of the incoming pion beam. At the
energies ECM = M ± Γ/2, the cross section has fallen to half its peak value. The
parameter Γ is therefore identified with the width of the resonance. From the experimental curve we extract the width
Γ ≈ 115 MeV.
(24.4).6.4 {nolabel}
1332
24 Internal Symmetries of Strong Interactions
In the quantum mechanical theory of resonance scattering it is known that such a
cross section indicates an exponentially decaying wave function of the form
Ψ ∝ e−i(M −iΓ/2)t/h̄ .
(24.5).6.5 {nolabel}
Indeed, such a wave function leads to a scattering amplitude which contains a pole
in the energy plane below the real axis, proportional to
1
.
ECM − M + iΓ/2
(24.6).6.6 {nolabel}
Its absolute square gives the above Lorentzian cross section. The lifetime of such a
wave function is
τ = 2h̄/Γ.
(24.7).6.7a {nolabel}
This is why the inverse width is sometimes called half-life. In physical units, the
right-hand side has to be multiplied with h̄ if Γ is measured in MeV and τ in seconds.
The ∆(1231)-resonance has therefore a lifetime of
τ = 2h̄/Γ = 0.6583 × 10−21
≈ 5.72 × 10−24 sec.
MeV
sec
Γ
(24.8).6.8 {nolabel}
Figure 24.1 Total and elastic π + -proton cross section showing clearly the first nucleon
resonance called ∆(1232).
H. Kleinert, PARTICLES AND QUANTUM FIELDS
{Fig. 6.1A}
24.1 Classification of Elementary Particles
1333
Figure 24.2 Total and elastic π − -proton cross section of the proton in a beam of negatively charged pions π − . It shows once more the first nucleon resonance called ∆(1232)
as in the previous Figure 24.1, but with only a third of the height. This agrees with the
isospin-3/2 assignment and the amplitude relations (24.44) if one invokes the optical theorem (9.36), according to which the total cross section is proportional to the imaginary
part of the forward scattering amplitude.
It decays mostly into a pion and a nucleon. Such a short-lived decay is called strong
decay.
A small fraction, however, about 0.6% of the decays, goes into a nucleon and a
photon. This is called the radiative decay channel.
Since the ∆(1232)-resonance can decay into a nucleon and a photon, we should
expect that it can also be produced in a collision taking place from the final states,
i.e., when a photon hits a nucleon. Indeed, the cross sections in Fig. 24.3 clearly
show this resonance.
In the past twenty years, various simple organizing principles have been found
according to which the variety of particles and resonances can be classified and remembered and their interactions be related. These principles are based on symmetry
groups associated with certain characteristic invariances of the different interactions.
{Fig. 6.1B}
1334
24 Internal Symmetries of Strong Interactions
{Fig. 6.2ABC
Figure 24.3 Photon-proton and photon-deuteron total cross sections showing clearly the
first nucleon resonance ∆(1232) as well as a second resonance called N (1520).
{photonproto
They have helped in understanding the quantum numbers, the energy spectra, the
various decay channels, and the different scattering cross sections.
24.2
Isospin in Nuclear Physics
If one compares binding energies of so-called mirror nuclei (which are obtained
from each other by the interchange of protons and neutrons), one finds that the
difference may be attributed entirely to the different electromagnetic interactions of
protons and neutrons. For instance, 3 H= nnp and 3 He= ppn have binding energies
BH3 = 8.452 MeV and B3 He = 7.728 MeV, respectively, with the difference being
caused by the proton-neutron mass difference mn − mp = 1.293323 ± .000016 MeV
and by the different Coulomb field energy around the nuclei. Indeed, if the Coulomb
energy is approximated by that of a uniformly charged sphere of charge Z and radius
R, the field energy is
ECoul = (1/2)(Z − 1)
6 αh̄c
6 e2
= (1/2)Z(Z − 1)
m,
5 4πR
5 R
(24.9).6.9 {nolabel}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
24.2 Isospin in Nuclear Physics
1335
Figure 24.4 Mirror nuclei 5 B11 and 6 C11 with their excited states (the numbers are
the excitation energies in MeV). The ground state of 6 C11 decays into that of 5 B11 by a
positive β-decay (i.e., emission of a positron and a neutrino) as indicated by the arrow. An
estimate of the Coulomb energy via formula (24.9) explains roughly the energy difference
of 1.98MeV between the two level schemes.
{f11.3}
with α = e2 /4πh̄c ≈ 1/137 being the fine-structure constant. Using the estimate
R ≈ 1.45 A1/3 × 10−13 cm for the nuclear radius, where A is the atomic number, we
can write
6
ECoul ≈ Z(Z − 1) · A−1/3 MeV.
5
(24.10).6.10 {nolabel}
For A = 3, Z = 2 we estimate ECoul ≈ 0.57 MeV, and find the difference of the
binding energies to be
BH3 − BHe3 ≈ 1.29 MeV − 0.57 MeV ≈ 0.72 MeV,
(24.11).6.10b {nolabel}
in rough agreement with the observed 0.76 MeV.
Not only the binding energies, but also the excitation spectra of mirror nuclei
differ merely by a common electromagnetic shift, as seen in the example in Fig. 24.4.
This property of mirror nuclei has led nuclear theorists to postulate that, if it were
possible to switch off the electromagnetic interactions, the potentials Vpp and Vnn
would turn out to be exactly equal. This symmetry is called charge symmetry.
The charge symmetry possesses a natural extension. When analyzing the data
of nucleon-nucleon scattering at low energy, it was found that also Vpp ≈ Vpn . This
led Heisenberg to postulate that within purely nuclear interactions, protons and
neutrons should be two indistinguishable states of the same particle, the nucleon N.
He described the two states in analogy with the two spin states of a spin 1/2 object
by a two-component object
!
p
,
(24.12).6.11 {nolabel}
N=
n
1336
24 Internal Symmetries of Strong Interactions
and he called this object an isospinor state. Heisenberg considered the states as a
representation of a rotation group in a fictitious space, the space of isotopic spin.
Its generators are denoted by Ii . They obey the commutation rules of the rotation
group
[Ii , Ij ] = iǫabc Ik .
(24.13).6.12 {nolabel}
The two-component isospinors are rotated by the 2 × 2 representation of Ii , given
by Ii = σ i /2. Proton and neutron are eigenstates of the third component I3 = σ 3 /2,
with eigenvalue 1/2 and −1/2.
The exchange of protons and neutrons corresponds to a rotation around the 2 axes
of isospin by 1800:
iπI2
e
p
n
!
iπσ2 /2
=e
p
n
!
=
0 1
−1 0
!
p
n
!
=
n
−p
!
.(24.14).6.13a {nolabel}
The postulate of indistinguishability of protons and neutrons can then be formulated mathematically as the commutativity of the generators of isospin Ii with the
Hamiltonian of strong interactions
[Ii , Hstrong ] = 0.
(24.15).6.14 {nolabel}
This is called isotopic spin symmetry or isospin invariance of strong interactions.
If |Ai is a nuclear n body wave function, then the isospin acts additively on each
nucleon, just as ordinary angular momentum (compare Appendix 3B)
Ii |Ai = (Ii × 1 × . . . × 1 + . . . + 1 × 1 × 1 × . . . × Ii )|Ai.
(24.16).6.15 {nolabel}
Nuclei, which possess a large difference in their number of protons np and neutrons
nn , have a third component of isospin
1
I3 |Ai = (np − nn )|Ai.
2
(24.17).6.16 {nolabel}
Their total isospin I must be at least equal to I3 . The ground state has I = I3 , the
excited states have a higher I. If the Coulomb energies are taken into account, they
can be seen to be members of isospin multiplets formed together with neighboring
nuclei of equal total number of nucleons and different np − nn (see Fig. 24.5). With
the help of isospin rotations, the charge symmetry operation may be represented as
a rotation around the second axis by an angle π, namely as exp(iπI2 ).
The postulate of isospin symmetry leads immediately to interesting selection
rules for decay processes by nuclear interactions, such as α decay or emission of
deuterons. Consider a nucleus |Ai that has an equal number of protons and neutrons,
i.e. a nucleus that is charge symmetric to itself, a so-called self-conjugate nucleus.
Since it consists of an even number of isospin 1/2 nucleons, its total isospin is
necessarily integer, and the third component of its isospin is zero. The operation
exp(iπI2 ) must therefore produce the same state, except for a phase factor η:
exp(iπI2 )|Ai = η|Ai.
(24.18).6.7 {nolabel}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1337
24.2 Isospin in Nuclear Physics
Figure 24.5 Singlets and triplets of isospin in the nuclei 6 C14 , 7 N14 , 8 O14 (with the excitation energies in MeV). The dotted lines connect levels that are roughly degenerate after
subtracting the Coulomb energy according to the estimate (24.9).
{Fig. 6.3}
Applying this operation twice must lead back to the original state. Hence the phase
factor η can only be ±1. It is called the intrinsic charge parity of the nucleus. Every
charge-symmetric nucleus has a definite charge parity.
The charge parity η is calculable uniquely as a function of the total isospin.
We use the property of the eigenfunctions of angular momentum Ylm (Θ, ϕ) with
m = 0 to pick up a phase (−)l , when rotated around the y-axis by an angle π [see
the matrix elements of the rotation matrix (4.883)]. This is also obvious from the
representation [(I +I3 )!(I −I3 )!]1/2 ap†(I+I3 ) an†(I−I3 ) |0i of an isospin I, I3 -state in terms
of creation operators a†p , a†n of isospin up and down, as constructed in Eq. (4.871).
Under an isospin rotation around the 2-axis by an angle π, the operators a†p and a†n
go over into a†n and −a†p , respectively. A charge symmetric state with I3 = 0 has
therefore the charge parity
η = (−)I .
(24.19) {@}
The charge parity gives rise to interesting section rules for decay processes via nuclear
interactions. If the Hamiltonian commutes with all generators of isospin, the initial
and final states must have equal charge parities. Typical forbidden processes are
∗10 ր
5B
ց
3 Li
6
8
4 Be
+ α,
+ d,
(24.20).6.17 {nolabel}
where 5 B∗10 can be the 4.5 MeV or the 6.5 MeV resonance of 5 B10 . Now, the excited
nucleus 5 B∗10 of excitation energy 4.5 MeV is a member of an isospin triplet formed
together with ground states of the isotopes 4 Be10 and 6 C10 . It therefore has a charge
parity η = −1. On the other hand, the nuclei 3 Li6 and 4 Be8 have isospin 0 and
therefore a positive charge parity η = +1. The two processes (24.20) are thus
forbidden processes, which have indeed never been observed in the laboratory.
1338
24 Internal Symmetries of Strong Interactions
Let us now study the consequences of the larger isotopic spin symmetry in nuclear
reactions. A deuteron in its ground state has the orbit in a symmetric s-wave, and
the spin states are given by the three symmetric triplet states
| ↑↓i,
1
√ | ↑↓ + ↓↑i,
2
| ↓↓i.
(24.21).6.17 {nolabel}
The antisymmetry of the wave function requires the isotopic wave function to be
antisymmetric, i.e., it must be
1
√ |pn − npi.
2
(24.22).6.17 {nolabel}
This is an I = 0 -state of isospin.
If a deuteron collides with 4 Be9 , which has I = 12 , I3 = − 12 , the isospin of the
two-particle system is necessarily I = 21 , I3 = − 12 . We can now compare two possible
final states, namely
∗10
(24.23).6.17 {nolabel}
5 B1.7MeV + n
and
10
4 Be
+ p.
(24.24).6.17 {nolabel}
If we denote the isospins of the two particles by I (1) , I (2) , the isospin states are in
the first case
(1)
(2)
|I (1) , I3 ; I (2) , I3 i = |1, 0; 21 , − 12 i,
(24.25).6.18 {nolabel}
and in the second
(1)
(2)
|I (1) , I3 ; I (2) , I3 i = |1, −1; 1, 12 i.
(24.26).6.19 {nolabel}
The strengths, with which these two final states are coupled to the initial state
| 12 , − 12 i, are proportional to the Clebsch-Gordan coefficients
1
h 21 , − 12 |1, 0; 12 , − 21 i = √ ,
3
s
2
,
h 12 , − 12 |1, −1; 1, 21 i = −
3
(24.27).6.20 {nolabel}
(24.28).6.20 {nolabel}
respectively (recall Table 4.2). Hence the first reaction cross section should be half
as big as the second, a fact borne out by experiment.
24.3
Isospin in Pion Physics
{ISOSPIN
In 1935, Hideki Yukawa [2] postulated the existence of a Bose particle which should
mediate the strong interactions between nucleons in a similar way as the photons
do between charges. Since the size of nuclei are of the order of 1013 cm, Yukawa
concluded that instead of a Coulomb potential
v(x) =
1
,
4πr
(24.29).6.21a {nolabel}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1339
24.3 Isospin in Pion Physics
which solves the field equation
−∂x 2 v(x) = δ (3) (x),
(24.30).6.21b {nolabel}
the potential between nucleons should have a finite range and read
v Y (x) =
e−µr
.
4πr
(24.31).6.22 {nolabel}
This solves the differential equation
(−∂x 2 + µ2 )v Y (x) = δ (3) (x).
(24.32).6.22b {nolabel}
He identified the inverse range µ with the mass of the particle. This particle was
discovered experimentally around 1946, and is now called the pion. It exists in three
charge states π + , π 0 , π − . The neutral pion has a mass
mπ0 = 134.9642 ± .0038 MeV,
(24.33).6.13 {nolabel}
from which the masses of the charged pions differ by
mπ± − mπ0 = 4.6043 ± .00037 MeV.
(24.34) {@}
The π-mesons exist only for a small time. The neutral π 0 meson decays mainly into
two photons, with a lifetime
τπ0 ≈ (0.87 ± .04) × 10−16 sec.
(24.35).6.24a {nolabel}
This relatively fast decay is caused purely by electromagnetic interactions. The
charged π ± -states decay mainly into µ, νµ with a lifetime of
τπ± ≈ (2.6030 ± 0.0023) × 10−8 sec.
(24.36).6.23 {nolabel}
This decay is so slow that it proceeds via weak interactions.
Since nuclear forces are supposed to arise mainly from pion interactions, the
isosymmetry of nuclear forces implies that the pion-nucleon coupling should be symmetric under isospin rotations. The pion itself is obviously an isospin 1 object. If it
interacts with a nucleon of isospin 1/2, the total isospin can be either 32 or 21 . The
Clebsch-Gordan coefficients for the different channels of total isospin |I, I3 i are the
following (see Table 4.2)
|π + pi = |1, 1; 12 , 21 i
|π + ni = |1, 1; 21 , − 12 i
|π 0 pi
|π 0 ni
= |1, 0; 21 , 21 i
= |1, 0; 21 , − 12 i
= | 32 , 23 i,
q
= √13 | 32 , 21 i + 23 | 21 , 21 i,
=
=
|π − pi = |1, −1; 21 , 21 i
=
1
1
−
|π ni = |1, −1; 2 , − 2 i =
q
2 3 1
| 2 , 2 i − √13 | 12 , 21 i,
2 3
| , − 12 i + √13 | 12 , − 21 i,
3 2
q
2 1
√1 | 3 , − 1 i −
| , − 12 i,
2
3 2
3 2
| 23 , − 32 i.
q3
(24.37)Q {nolabel}
1340
24 Internal Symmetries of Strong Interactions
From these we find the amplitudes for scattering processes to be all given in terms
of two amplitudes, one for total isospin I = 23 and one for total isospin I = 21 . The
amplitudes are given by the so-called scattering matrix, whose matrix elements are
obtained by evaluating the scattering operator S between incoming and outgoing
states. The detailed theory of the scattering matrix has been given in Chapter 9.
From that theory it follows that if all generators Ii of isospin rotations commute with
the Hamiltonian operator of strong interactions Hstrong , they also commute with the
scattering operator S. Then the amplitudes do not depend on the I3 components of
the total isospin:
[Ii , S] = 0.
(24.38).6.25 {nolabel}
We therefore can use I+ inside the matrix elements hI, I3 |S|I, I3i to change I3 to
any allowed value |I3 | ≤ I. From
hI, I3 |I+ S|I, I3 − 1i = hI, I3|SI+ |I, I3 − 1i
(24.39) {@}
it follows that
q
hI, I3 − 1|S|I, I3 − 1i (I + I3 )(I − I3 + 1)
q
= hI, I3 |S|I, I3i (I + I3 )(I − I3 + 1)
(24.40).6.26 {nolabel}
[see Eqs. (4.370), (4.374)]. Thus the diagonal matrix elements hI, I3 |S|I, I3i are
independent of I3 . Similarly one sees that the non-diagonal elements vanish. We
therefore define the reduced scattering amplitude S I of a given isospin I by the
equation
hI, I3′ |S|I, I3i ≡ δI3′ I3 S I .
(24.41) {wigeck0}
This statement is a special case of the Wigner-Eckart theorem (4.899) for the
matrix elements of an arbitrary spherical tensor operator TI,I3 of isospin I. Such a
tensor operator is defined by the transformation law [compare (4.895)]
′
[Ii , TI,I3 ] = TI,I3′ D I (Ii )I3 I3 ,
(24.42) {translaw}
′
where D I (Ii )I3 I3 are the representation matrices of the generators Ii of the isospin
rotation group. According to the Wigner-Eckart theorem, the matrix elements of
a tensor operator between the representation states |I, I3 i are related by ClebschGordan coefficients:
hI ′′ , I3′′ |TI ′ ,I3′ |I, I3i = hI ′′ , I3′′ |I ′ , I3′ ; I, I3 ihI ′′ ||T ||Ii.
(24.43) {wigeck}
These account for the dependence on the quantum numbers I3 , I3′ , I3′′. The matrix
elements depend only on a few reduced matrix elements hI ′′ ||T ||Ii which are nonzero
if the isospins I and I ′ can be coupled to I ′′ . Within this definition, the scattering
operator is a trivial tensor operator of isospin zero, in which (24.43) reduces to
(24.41).
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1341
24.4 SU(3)-Symmetry
Note that (24.41) is also a manifestation of Schur’s Lemma, according to which
a matrix commuting with an irreducible set of matrices must be proportional to the
unit matrix.
As an application of (24.41), the scattering amplitudes for various pion-nucleon
scattering processes are given by the following combinations of only two amplitudes,
S 3/2 and S 1/2 :
Sπ+ p→π+p = S 3/2 ,
Sπ+ n→π+ n = (1/3)S 3/2 + (2/3)S 1/2 ,
Sπ0 p→π0p = (2/3)S 3/2 + (1/3)S 1/2 ,
√
√
Sπ+ n↔π0p = ( 2/3)S 3/2 − ( 2/3)S 1/2 ,
Sπ0 n→π0 n = (2/3)S 3/2 + (1/3)S 1/2 ,
Sπ− p→π− p = (1/3)S 3/2 + (2/3)S 1/2 ,
√
√
Sπ0 n↔π− p = ( 2/3)S 3/2 − ( 2/3)S 1/2 ,
Sπ− n→π− n = S 3/2 .
(24.44).6.27 {nolabel}
The ↔ symbols in the charge-exchange reactions exhibit time-reversal invariance.
By taking the absolute squares of the amplitudes, we see that the cross sections
must satisfy
2σπ0 p→π0p = σπ+ p→π+ p + σπ− p→π− p .
(24.45).6.28 {nolabel}
This is borne out by experiment.
24.4
SU(3)-Symmetry
In 1944, the set of fundamental particles was enriched by a particle which was a
heavy neutral meson of mass ≈ 500 MeV [3], with a lifetime of τ ≈ 10−10 sec. It
became known as a strange particle [4] under the name K 0 . It is observed to decay
mainly into π + , π − and has a mass and a lifetime
mK 0 ≈ 497.72 ± 0.07 MeV ,
τK 0 ≈ (0.8323 ± 0.0022) × 10−10 sec.
{@SU3S}
(24.46).6.29 {nolabel}
In 1951, a charged partner of this particle was discovered which decays mostly into
a µ+ and a neutrino of the meson type, namely the K + meson. It has a mass and
a lifetime
mK + ≈ 493.667 ± 0.014 MeV ,
τK + ≈ (1.2371 ± 0.0026) × 10−8 sec.
(24.47).6.30 {nolabel}
Later also a K − meson was found, that decays into a negatively charged meson and
a neutrino with the same lifetime. In 1955 one discovered, moreover, that there
1342
24 Internal Symmetries of Strong Interactions
are actually two K 0 -mesons, now called K 0 and K̄ 0 . These mesons could soon be
produced abundantly in particle accelerators and it became possible to study their
interactions with nucleons. In the course of this it was found that not only the
mesons but also the nucleons possess strange partners. Moreover, if one assigned to
the strange mesons a quantum number S = 1, called strangeness, and to the strange
partner of the baryons in the associate production a quantum number S = −1, then
strangeness is a conserved quantum number in a nuclear scattering process. If an
ordinary non-strange meson such as a pion hits a nucleon, the result is either a pion
and a nucleon, both of strangeness zero, or a strange meson with S = 1 together
with a strange partner of the nucleon with S = −1. For example
π−p → π− p
(24.48).6.31a {nolabel}
K 0 Λ0 ,
K + Σ− .
(24.49).6.31b {nolabel}
or
π−p
ր
ց
There are also associate production processes where two K’s are produced in addition to a nucleon:
π − p → K − K 0 p,
(24.50).6.31c {nolabel}
π + p → K + K̄ 0 p.
(24.51) {@}
K − p → K + Ξ− ,
(24.52).6.31d {nolabel}
These allow to deduce that the strangeness of K + , K − is S = 1, −1, respectively.
By scattering further a strange meson K − on a proton,
one is able to produce a new particle Ξ− of strangeness S = −2. This particle is
called cascade since it decays in a cascade-like process with the first step
Ξ− → Λ 0 π −
(24.53) {@}
and the second steps
Λ0 → pπ − ,
π − → µ− ν̄µ .
(24.54) {@}
(24.55) {nolabel}
Note that in contrast to the production process, the decay of the strange particles
violates strangeness. For instance:
Σ− → π − n
S : −1
0
0.
(24.56).6.32 {nolabel}
However, this violation can be blamed on another interaction, the weak interaction,
since the decay proceeds quite slowly, with a lifetime of roughly 10−10 sec.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1343
24.4 SU(3)-Symmetry
Figure 24.6 Pseudoscalar meson octet states associated with the triplet of pions. The
same picture holds for the vector meson octet states with the replacement (24.62).
{F6.4}
While non-strange particles have a charge
Q = N/2 + I3 ,
(24.57).6.33 {nolabel}
where N is the total nucleon-number, the charge of strange particles is given by
Q = Y /2 + I3 ,
(24.58).6.34 {nolabel}
with Y being defined as a convenient combination of N, and S, called hypercharge
Y ≡ N + S.
(24.59).6.35 {nolabel}
Murray Gell-Mann plotted the known ordinary and strange pseudoscalar meson
states in the I3 , Y plane and found the multiplet of eight particles shown in Fig. 24.6
(there and in subsequent similar plots the ket symbols | . . .i of the states are omitted). The center contains, beside the π 0 -meson, another pseudoscalar neutral meson
called the η meson. Its mass is
mη = 548.8 ± 0.6 MeV
(24.60).6.37 {nolabel}
Γ ≈ 1.05 ± 0.15keV,
(24.61).6.38 {nolabel}
and it decays with a width of
principally into γγ and 3π 0 . A ninth much heavier pseudoscalar meson η ′ (958) of
width 8.5 MeV, known at that time, did not fit into the scheme by having a larger
mass, and no partners.
Gell-Mann applied the same extension to the spin 1 meson resonances of negative
parity to nine pseudoscalar particles, replacing
K
π
η
η′
→
→
→
→
K ∗ (892),
ρ(770),
φ(1020),
ω(783).
(24.62).6.vm {nolabel}
1344
24 Internal Symmetries of Strong Interactions
He continued to organize the nucleons and their strange partners in this way and
found again an octet, shown in Fig. 24.7, with the masses and lifetimes given in
Table 24.1.
Figure 24.7 Baryon octet states associated with isodoublet of nucleons.
{F6.5}
Table 24.1 Masses and lifetimes of the octet states associated with the isodoublet of
nucleons.
p
n
Σ+
Σ−
Σ0
Λ0
Ξ0
Ξ−
m( MeV)
τ (10−10 sec)
938.2796± 0.0027
stable
939.5731± 0.0027 (898 ± 16) × 1010
1189.37 ± 0.06
0.8 ± 0.004
1197.34 ± 0.05
1.482± 0.011
1192± 0.08 (5.8± 1.3) × 10−10
1115.60 ± 0.05
2.632 ± 0.020
1314.9 ± 0.6
1.642 ± 0.015
1321.32 ± 0.13
1.642 ± 0.015
Gell-Mann further noticed that there exists no similar regularity for the most prominent excited states of the nucleon, the resonance ∆(1232) of mass 1232 MeV and
width 115±5 MeV. They exist with charges −, 0, +, ++ and possess strange partners
Σ(1385) of width ≈ 37 MeV with charges −, 0, +, and Ξ(1530) of width ≈ 10 MeV
with charges −, 0. When he looked at the masses of these excited states, he observed
an about equal spacing on the Y -axis. When plotted in an I3 −Y plane, the particles
fill an equilateral triangle except for a missing lower corner. See Fig. 24.8. Starting
from these observations he put forward the hypothesis that just as nuclear forces
are independent under isospin rotations, which form the group of unitary matrices in two dimensions, they should also be approximately invariant under the most
direct extension of this group, which includes the additional quantum number of
strangeness. So he postulated an approximate invariance of the strong interactions
under the group of unitary matrices in three dimensions, SU(3). Since the mass
splittings are much larger than in the former SU(2)-multiplets, this symmetry is
much more broken in nature than the isotopic symmetry.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
{T6.1}
1345
24.4 SU(3)-Symmetry
Figure 24.8 Baryon decuplet states associated with the first resonance of nucleons.
{F6.6}
In field theory, the group theoretic implications of SU(3)-symmetry are studied
by giving the relativistic fields an extra SU(3)-subscript. One introduces a fundamental Dirac field with subscripts α = 1, 2, 3:


ψ1 (x)


ψ(x) → ψα (x) =  ψ2 (x)  ,
ψ3 (x)
(24.63).6.40 {nolabel}
3 : qα → Uα β qβ ,
(24.64).6.41 {nolabel}
which changes under SU(3)-transformations by group elements from the fundamental three-dimensional representation of SU(3), called 3. Such a Dirac field is called
a quark field, and will be denoted by qα (x). Omitting the spacetime arguments, the
transformation law is:
U ∈ SU(3).
The Hermitian adjoint of a quark field, (qα )† ≡ (q † )α . It transforms according to
the complex conjugate representation, called 3∗ . To emphasize the different transformation behavior we write [Uα β ]† as (U ∗ )α β so that
3∗ : q † α → U ∗α β q † β ,
U ∈ SU(3).
(24.65).6.41b {nolabel}
The unitary scalar product q † q ≡ (q † )α qα is an invariant, likewise the contraction of
qα with any other particle field, which transforms like (q † )α . For Dirac fields it is
more convenient to work with the fields q̄(x) = q † (x)γ 0 rather than q † (x) because
of their more favorable Lorentz transformation properties, in particular the Lorentz
invariance of the scalar product q̄(x)q(x). Thus we shall work with conjugate quark
fields
q̄ α = (q̄ 1 , q̄ 2 , q̄ 3 ).
(24.66) {@}
1346
24 Internal Symmetries of Strong Interactions
This field is said to transform according to the representation 3̄ of SU(3). The
contraction q̄ α qα is obviously invariant under Lorentz- and SU(3)-transformations:
q̄ α qα = invariant.
(24.67).6.42 {nolabel}
We have seen in Eq. (7.308) that the charge-conjugate Dirac field is directly related
to q̄ by a similarity transformation of the Dirac indices, which for quark fields reads
qC (x) = C q̄ T (x).
(24.68) {@}
The fields q̄ annihilate antiparticles, and will therefore be referred to as antiquark
fields, ignoring at this point the similarity transformation.
The traceless tensor field obtained from the direct product of qα and q̄ β ,
Mα β ≡ qα q̄ β − 31 δα β qγ q̄ γ ,
(24.69).6.43 {nolabel}
is in general a bilocal field containing qα (x, t) and q̄ β (x′ , t′ ) at different spacetime
points which are irrelevant to the present discussion and therefore omitted. This
field forms an invariant 8-dimensional representation space of SU(3), which GellMann identified with the 8 meson states described above. This octet representation
is irreducible. This means that every state of it can be reached by performing group
operations on an arbitrary single fixed state. Gell-Mann called the fields up, down,
and strange quark fields and denoted them by u, d, s :

Similarly for the antiquarks:

u


qα =  d  .
s
(24.70) {@}
¯ s̄).
q̄ α = (ū, d,
(24.71) {@}
He associated the octet fields (24.69) with the pseudoscalar mesons as shown in
Fig. 24.9. The field M α β annihilates the corresponding particle, for example π + ≡
¯ annihilates the particle π + . In contrast to isospin, the SU(3)-symmetry
M 1 2 = du
is strongly broken. The pion and the K mesons have quite different masses, namely
135 MeV and 490 MeV.
The operator which measures the third component of isospin, I3 , is the number
operator
1
Iˆ3 = [(N̂u − N̂d ) − (N̂ū − N̂d¯)].
(24.72).6.45 {nolabel}
2
Here and in the sequel we denote by N̂u,d,s the total number operator of the u, d, squarks and their antiparticles, respectively. Thus we have the commutation rules
−[N̂u,d , qα ] = nu,d qα ,
α(u, d, s)
(24.73) {@}
with the eigenvalues
nu = (1, 0, 0),
nd = (0, 1, 0),
(24.74) {nolabel}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1347
24.4 SU(3)-Symmetry
Figure 24.9 Quark content of the pseudoscalar meson octet. The particle- and quarksymbols denote the annihilation parts of the corresponding fields.
{F6.7}
¯ s̄). {@}
and similar relations between the number operators and the antiquarks q̄ α = (ū, d,
The hypercharge of the mesons is given by
1
Ŷ = [(N̂u + N̂d − 2N̂s ) − (N̂ū + N̂d¯ − 2N̂s̄ )],
3
(24.75).6.46 {nolabel}
where Ns counts the number of strange quarks, i.e., it is defined to have, for (u, d, s)
and (ū, d̄, s̄), the eigenvalues ns = (0, 0, 1) and (0, 0, −1), respectively, with corresponding eigenvalues of Ns̄ for the antiquarks.
The group SU(3) is a 3 · 3 − 1 = 8-parameter group. Its Lie algebra possesses
8 traceless Hermitian generators which are conventionally denoted by λa /2 (a =
1, · · · , 8) with the 3 × 3 traceless Hermitian matrices

λ1 = 


λ4 = 


0
1
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0



,
λ2 = 

,
λ5 = 






0
i
0
0
0
i
0
0
0
−i 0
0 0
0 0
0 −i
0 0
0 0
0 0
0 −i
i 0



1
0 0



3
 , λ =  0 −1 0  ,
0
0 0


,



1 0
0





1
8
7
6
0 
λ =
 , λ = √3  0 1
, λ = 
.
0 0 −2
(24.76).6.47 {nolabel}
Note that they have the same trace orthonormality relations as the Pauli matrices:
tr(λa λb ) = 2δab .
a
(24.77) {@}
It is easy to see that the 3 × 3 -matrices U = eiαa λ /2 span the space to unitary 3 × 3
-matrices with unit determinants. The first three λa ’s are the direct extension of
the σ i -matrices of isospin into three dimensions.
1348
24 Internal Symmetries of Strong Interactions
Table 24.2 Structure constants of SU(3). All elements not given in the table follow from
the total antisymmetry.
abc
fabc
123
1
147
1/2
156 −1/2
246
1/2
257
1/2
345
1/2
367 √
−1/2
458 √3/2
678
3/2
{t6.2X}
The group structure is specified by the commutation rules of the generators
[λa , λb ] = 2ifabc λc ,
(24.78).6.48 {nolabel}
where the structure constants fabc are totally antisymmetric in abc, due to the Jacobi
identity satisfied by the commutators. Their values are given in Table 24.2. Let Ga
be the operators in Hilbert space associated with these Lie algebra elements. These
act on the three quark field operators of the fundamental representation as follows:
h
i
− Ĝa , qα = (λa /2)α β qβ .
(24.79).6.49 {nolabel}
An explicit operator expression for the generators can be given by assigning canonical
commutation or anticommutation relations to the q-fields:
[qα , q †β ] = δα β
(24.80).6.50 {nolabel}
Then we can express the generators Ĝi simply as follows [recall Section 2.5]
Ga = q †α (λa /2)α β qβ .
(24.81).6.51 {nolabel}
It is easy to see that these operators Ĝa have indeed the same commutation relation
as the matrices λa /2:
[Ĝa , Ĝb ] = ifabc Ĝc .
(24.82).6.52 {nolabel}
Besides these commutators, some calculations will involve the anticommutators of
the λa -matrices. They are given by
n
o
λa , λb = 2dabc λc + (4/3)δab ,
(24.83).6.53 {nolabel}
where dabc are completely symmetric in abc and have the nonzero matrix elements
shown in Table 24.3.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1349
24.4 SU(3)-Symmetry
Table 24.3 The symmetric couplings dabc . All values not given explicitly are obtained
from the total symmetry in abc.
abc
118
146
157
228
247
256
338
344
dabc
√
1/ 3
1/2
1/2√
1/ 3
-1/2
1/2√
1/ 3
1/2
abc
dabc
355
1/2
366
-1/2
377
-1/2√
448 -1/2 √ 3
558 -1/2 √3
668 -1/2 √3
778 -1/2√ 3
888 -1/ 3
{Table 6.3}
Consider now the antiquark fields, which transform according to the representation 3∗ of SU(3). Under commutation with Ĝa , the antiquark fields behave as
follows
−[Ĝa , q̄ α ] = −q̄ β (λa /2)β α = −(λa∗ /2)α β q̄ β .
(24.84).6.54 {nolabel}
The antiquark fields have the same commutation rules as the Hermitian conjugate
of the quark fields, i.e., they commute like (q̄ † )α ≡ (q̄ α )† :
[q̄ β , q̄α† ] = δ β α .
(24.85).6.55 {nolabel}
Thus, the generators of SU(3) have, for quark and antiquark fields, the operator
representation:
Ĝa = q † (λa /2)q − q̄ † (λa∗ /2)q̄.
(24.86).6.56 {nolabel}
The generator of the third component of isospin is
Iˆ3 = Ĝ3 = q † (λ3 /2)q − q̄ † (λ3 /2)q̄
1
1
¯
= (u† u − d† d) − (ū† ū − d¯† d).
2
2
(24.87).6.57 {nolabel}
Similarly, the hypercharge is given by
i
2
2 h
Ŷ = √ Ĝ8 = √ q † (λ8 /2)q − q̄ † (λ8 /2)q̄
3
3
h
i
1
(u† u + d† d − 2s† s) − (ū† ū + d¯† d¯ − 2s̄† s̄) ,
=
3
(24.88).6.58 {nolabel}
and the charge by
Q̂ = Y /2 + I3
i
1h †
(2u u − d† d − s† s) − (2ū† ū − d¯† d¯ − s̄† s̄) .
=
3
(24.89).6.58b {nolabel}
1350
24 Internal Symmetries of Strong Interactions
The raising and lowering of the third component is done with the operators
Iˆ± =
h
q † λ1±i2 q − q̄ † (λ1±i2 )∗ q̄
(
)
u† d − d¯† ū
=
,
d† u − ū† d¯
1
2
i
(24.90).6.59 {nolabel}
where λ1±i2 is a short notation for λ1 ± iλ2 . A pair of operators which raise and
lower the hypercharge by a unit (thereby also raising and lowering I3 by half a unit)
is1
V̂± =
=
1
2
(
h
q † λ4±i5 q − q̄ † (λ4±i5 )∗ q̄
u† s − ū† s̄
s† u − s̄† ū
)
i
.
(24.91).6.60 {nolabel}
The effect of these operators on the quark and antiquark states |qα i (created by the
operators (q † )α when acting on the vacuum |0i ) is illustrated in Fig. 24.10 [this
being the analog to the illustration in Fig. 4.3]. Note that there exists a related pair
Figure 24.10 Effect of raising and lowering operators on quark and antiquark states.
Here and in the subsequent weight diagrams the ket symbols of the states are omitted.
{Fig. 6.8}
of operators which raise and lower hypercharge by a unit while acting oppositely to
V± in the isospin:
h
q † λ6±i7 q − q̄ † (λ6±i7 )∗ q̄
(
)
d† s − d¯† s̄
=
.
s† d − s̄† d¯
Û± =
1
2
i
(24.92).6.60b {nolabel}
Note that there exists a third component to the two operators V± , namely V3 = 12 (u† u −
s s) − 12 (ū† ū − s̄† s̄), which commutes with V± in the same way as the isospin operators. The three
operators form the so-called V -spin subgroup of SU(3). A similar set of commutation rules holds
for the U± operators of (24.92) and U3 = 21 (d† d − s† s) − 21 (d¯† d¯ − s̄† s̄).
1
†
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1351
24.4 SU(3)-Symmetry
These, however, will play no role in the future discussion.
Consider now the composite states associated with the product representation
3 × 3̄ of a quark and an antiquark assigned to the pseudoscalar mesons in (24.69).
The quantum numbers I3 and Y are additive so that the quantum numbers of the
meson octet can be found by vector addition, i.e. by placing the 3̄ diagram with
the (I3 , Y ) origin once on each state of the 3 diagram and by adding up the vectors
as shown in Fig. 24.11. In the theory of group representations the vectors |I3 , Y i
of quarks and antiquarks are referred to as the fundamental weights. Using the
s̄d
s̄u
¯
dd
ūd
ūu
¯
du
s̄s
¯
ds
ūs
Figure 24.11 Addition of the fundamental weights in product representation space of 3
and 3̄ vectors.
{F6.9}
raising and lowering operators I± , V± , it is easy to find all states of an irreducible
representation starting out from any fixed state. For instance, the operator Iˆ+ acts
upon the particle state |π − i = |dūi as follows:
√
¯ ≡ 2|π 0 i,
Iˆ+ |π − i = |ūu − ddi
(24.93).6.61 {nolabel}
whereas
√
√
¯ ≡ − 2|π † i.
Iˆ+ |π 0 i = − 2|dui
(24.94).6.61b {nolabel}
Similarly, the application of V+ to |π − i = |ūdi yields
V̂+ |π − i = −|s̄di = −|K 0 i,
(24.95).6.62 {nolabel}
whereas applying V̂+ to |K̄0 i we find
V̂+ |K̄ 0 i = |ūu − s̄si.
(24.96).6.63 {nolabel}
This state has the same (I3 , Y )-quantum numbers as |π 0 i, but it is not an eigenstate
of total isospin I 2 . Such a state is given by
|ηi =
q
1
6
¯ − 2s̄si,
|ūu + dd
(24.97).6.64 {nolabel}
1352
24 Internal Symmetries of Strong Interactions
so that V̂+ |K̄ 0 i can be decomposed into a π 0 - and an η-state as follows:
V̂+ |K̄ 0 i =
=
1
2
¯ + 12 |ūu + dd
¯ − 2s̄si
|ūu − ddi
q
1
2
|π 0 i +
q
3
2
|ηi.
(24.98).6.65 {nolabel}
It has become customary (de Swart-convention2 ) to change the phases to the q̄states so that the matrix elements of I± and V± are all positive. This is achieved by
changing the phases of the states of the representation 3∗ as shown in Fig. 24.12. For
−|s̄i
2
3
V±
Y
1
1
− 31 |ūi
¯
−|di
I±
− 21
+ 12
0
T3
Figure 24.12 States of the 3̄-representation with phases in the de Swart-convention.
{F6.10}
the SU(3)-representation matrices U ∗ , these phase-changes on the states correspond
to a similarity transformation. The new matrices will be denoted by Ū :
U ∗ → Ū.
(24.99) {@}
Correspondingly, we shall say that the antiquark fields q̄, after the appropriate phase
changes, transform according to the 3̄-representation.
The phases of all higher representations are adjusted accordingly. The meson
octet states associated with the product of 3 and 3̄-states are shown in Fig. 24.13.
As a check we calculate a couple of phases:
V̂+ |π 0 i = V̂+
q
q
= −
1
2
1
2
¯
|ūu − ddi
|s̄ui =
q
1
2
|K + i
(24.100).6.67a {nolabel}
and
¯ = −|dui
¯ = |π + i,
V̂+ |K̄ 0 i = V̂+ | − dsi
2
(24.101).6.67 {nolabel}
J.J. de Swart, Rev. Mod. Phys. 35 , 916 (1963).
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1353
24.4 SU(3)-Symmetry
K 0 = −s̄d
+1
π0 =
S 0
−
π = ūd
η=
−1
K − = ūs
0
−1
T3
K + = −s̄u
¯
− dd)
¯
π + = −du
¯ − 2s̄s)
(ūu + dd
√1 (ūu
2
√1
6
¯
K̄ 0 = −ds
+1
Figure 24.13 Quark-antiquark content of the meson octet states with phases in the
de Swart-convention.
{F6.11}
which are indeed positive.
Note that these quark-antiquark states can also be obtained by forming certain
combinations of λa -matrices and sandwiching them between the creation operator
parts of the quark and antiquark fields
q † α = (u† , d†, s† ), (q̄ † )α = (ū† , d¯† , s̄† ),
(24.102).6.pof0 {nolabel}
and applying them to the vacuum. The creation operators of the meson octet states
are
√
M † a = q † (λa / 2)q̄ † ,
(24.103).6.pof {nolabel}
with the combinations indicated in Fig. 24.14. These combinations of λa indices are
often used to specify octet states instead of the quark content. Any Hermitian
octet
√
†
operator Ôa (for instance Ĝa itself and M a ) with the indices −(1 + i2)/ 2 adds to
a given state the quantum numbers of a π + . Let us verify the de Swart-phases of
the assignments in Fig. 24.14 by applying Iˆ± , V̂± to the meson states |Ma i = M † a |0i
with the above index combinations. Using the commutation rules
[Ĝa , M † b ] = ifabc M † c
(24.104).6.70 {nolabel}
√
√
[I+ , M † π− ] =
2f123 M † 3 = 2M † π0 ,
√
[I+ , M † π0 ] = if123 (−M † 2 + iM † 1 ) = − 2M † π+ ,
[I+ , M † K 0 ] = (f174 M † 4 − f165 M † 5 ) = M † K + .
(24.105).6.71 {nolabel}
we have, for example,
There is another simple way of constructing an octet representation of SU(3)
from quark states without the use of antiquarks. By applying the product of three
fields qᆠq↠qㆠat three different places3 to the vacuum one obtains 27-states. If these
3
This assumption of different places is necessary to distinguish all 27-states, for otherwise some
states will coincide due to particle identity.
1354
24 Internal Symmetries of Strong Interactions
√
K 0 = −(6 + i7)/ 2
+1
S
0
√
π − = (1 − i2)/ 2
√
K + = −(4 + i5)/ 2
π0 = 3
√
π + = −(1 + i2)/ 2
η=8
√
K − = (4 − i5)/ 2
−1
−1
0
T3
√
K̄ 0 = −(6 − i7)/ 2
+1
Figure 24.14 Combination
of indices a in the√pseudoscalar octet field M † a of (24.103).
√
Here −(1 ± i2)/ 2-stands for −(M † 1 ± iM † 2 )/ 2.
{F6.12}
are decomposed into the irreducible contents of SU(3), they lead to the following
multiplets
3 × 3 × 3 = 10 + 8 + 8 + 1.
(24.106) {@}
The octet states can be identified with the nucleon octet states. In the decuplet there
are doubly charged non-strange states which can be identified with the I3 = 3/2
component of the first nucleon resonance ∆(1236). It is therefore suggestive to
identify the other 10-states with the strange partners of this resonance. At the time
when Gell-Mann proposed this identification, he could only match 9 of the decuplet
states with known particles and resonances. There was one more state of strangeness
3 (at the bottom of the weight diagram to be constructed in Fig. 24.17) which closes
it to a triangle. Since the masses within the two rows are almost degenerate and
since they grow from row to row by about the same amount
m∆ − mΣ∗ ≈ −150,
mΣ∗ − mΞ∗ ≈ −145,
with the masses given in units MeV, Gell-Mann extrapolated the mass of the tenth
particle to be
M ≈ M∆ + 3(MΣ∗ − M∆ )
≈ 1232 + 3 · 1567 ≈ 1682.
(24.107).6.72 {nolabel}
He therefore postulated the existence of a particle with this mass, a negative charge,
and a hypercharge −2 as it resulted from the three quark content of the states. He
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1355
24.4 SU(3)-Symmetry
called this particle Ω− . It was indeed found in 1964 at a mass of 1675 ± 8 Mev.
This was celebrated as a triumph of the approximate SU(3)-symmetry hypothesis
and the 10-assignment of the excited nucleons and their strange partners.
Let us explicitly construct the irreducible representations contained in the product 3 × 3 × 3 and assign them to physical states. We perform the multiplication
successively
3 × 3 × 3 = 3 × (3 × 3).
(24.108).6.73 {nolabel}
The product of two states 3 × 3 is easily reduced to
3 × 3 = 6 + 3̄.
(24.109) {@}
To see this we write down the symmetric and antisymmetric combinations as shown
in Fig. 24.15. The first is a 6-representation, the second transforms in the same way
dd
u
d
d
×
=
uu
u
=
us, su
ds, sd
s
3
√
dd (ud + du)/ 2 uu
s
3
ud, du
×
ss
√
−(ud − du)/ 2
√
(us + su)/ 2 +
√
(ds − sd)/ 2
√
(ds + sd)/ 2
ss
6
+
√
(us − su)/ 2
3̄
Figure 24.15 Quark content in the reduction of the product 3 × 3 = 6 + 3̄.
{F6.13}
as an antiquark representation 3̄. As before we have assigned the phases to satisfy the
de Swart-convention. After forming these products we continue the multiplication:
3 × 3 × 3 = 3 × (6 + 3̄) = 3 × 6 + 3 × 3̄.
(24.110).6.76 {nolabel}
Let us now form 3 × 6 and 3 × 3̄. The product 3 × 3̄ has been calculated before in
terms of quark-antiquark states:
3 × 3̄ = 8 + 1.
(24.111) {87.6}
Now the octet states contain three quarks, as shown in Fig. 24.16. The octet states
can be identified with the nucleons and their strange partners.
1356
24 Internal Symmetries of Strong Interactions
√
−(ud − du)/ 2
u
d
×
s
√
(us − su)/ 2
√
(ds − sd)/ 2
×
3
3̄
√
−d(ud − du)/ 2
√
− u(ud − du)/ 2
√
= d(ds − sd)/ 2
√
u(us − su)/ 2
√
s(ds − sd)/ 2
+
√
s(us − su)/ 2




√1 
6


+uds
−usd
+dsu
−dus
+sud
−sdu
+
8








1
Figure 24.16 Octet and singlet states obtained from 3 × 3̄ in the product space of threequarks.
{F6.14}
The completely antisymmetric combination of uds-states is a singlet state under
SU(3)-transformations:
1
|Λi = √ (|udsi + |dsui + |sudi + |dusi + |sdui + |usdi).
3!
(24.112) {@}
The proton state is
1
|pi = √ |u(ud − du)i.
(24.113).6.pro {nolabel}
2
The state |Σ0 i in the octet is found by applying Iˆ− to the right most state |Σ† i =
−|u(us − su)i, thereby obtaining
1
|Σ0 i = − |d(us − su) + u(ds − sd)i.
2
(24.114) {@}
√
The isosinglet state |Λ0 i is obtained by applying V̂ − to |pi = |u(ud − du)i/ 2 and
by separating the result into a linear combination of |Σ0 i and an orthogonal state,
which must be |Λ0 i. First we have
V̂ − |pi =
√ 1
2 |sud + usd − sdu − udsi.
2
(24.115).6.78 {nolabel}
The scalar product with |Σ0 i gives
1
hΣ0 |V̂ − |pi = √ .
2
(24.116).6.79 {nolabel}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1357
24.4 SU(3)-Symmetry
√
(ud + du)/ 2 uu
dd
u
d
√
(us + su)/ 2
√
× (ds + sd)/ 2
s
ss
3
×
6
uuu
ddd
√
suu, u(us + su)/ 2
=
√
uss, s(us + su)/ 2
sss
10 + 8
Figure 24.17 Irreducible three-quark states 10 and 8 in the product 3 × 6 (the symbol
(. . .)s denotes complete symmetrization).
Hence we rewrite
V − |pi =
and identify
√ 0
2 12 |Σ i +
√
3
2
|Λ0i
q
|Λ0 i ≡ − 1/12|u(ds − sd) − d(us − su) − 2s(ud − du)i.
{F6.15}
(24.117).6.80 {nolabel}
(24.118).6.81 {nolabel}
Consider now the product 3 × 6. It gives the states shown in Fig. 24.17. The state
|uuui is identified with the 1232 MeV resonance ∆++ (1232) of isospin and parity
3/2† . By applying the operators I± and V± to |uuui we find the ten decuplet states
|∆++ i
|∆+ i
|∆0 i
|∆− i
|Σ+ i
|Σ0 i
|Σ− i
|Ξ0 i
|Ξ− i
|Ω− i
=
=
=
=
=
=
=
=
=
=
|uuui,
√1
|uud + udu + duui,
√ 31
|udd + dud + uddi,
3
|dddi,
√1
|uus + usu + suui,
√ 31
|uds + usd + dsu + dus + sud + sdui,
√ 61
|dds + dsd + sddi,
3
√1
|uss + sus + ssui,
√ 31
|dss + sds + ssdi,
6
|sssi
(24.119).6.83 {nolabel}
1358
24 Internal Symmetries of Strong Interactions
In addition to this decuplet, there is a further octet of particle states which can
alternatively be assigned to the nucleon octet. Its quark states are found by constructing one state, say the proton, and applying the operators I± ,√
V± to it. The
proton state is found by linearly composing |uudi and |u(ud + du)i/ 2 so that the
result is orthogonal to |∆+ i in (24.119) and to |pi of (24.113):
|p′ i =
q
1
6
|u(ud + du) − 2duui.
(24.120).6.84 {nolabel}
The prime in |p′i indicates that the quark wave function differs from (24.113). All
other octet states associated with |p′ i are then obtained by applying I± , V± to |p′ i,
for instance
q
|n′ i = Iˆ− |p′ i = −
1
6
|d(ud + du) − 2uddi.
(24.121).6.86 {nolabel}
As a first test of the octet assignment we calculate the mass differences within
the nucleon octet assuming that the mass operator transforms like an SU(3) singlet,
plus a term which behaves like a neutral isosinglet member of an octet, i.e., like η.
Fortunately, the Clebsch-Gordan coefficients for the matrix elements
h8a |8b |8c i
(24.122).6.87 {nolabel}
are already in our position. They are of two types, related to the traces
tr([λa , λb ]λc ) = fabc and tr({λa λb }λc ) = dabc . The first is antisymmetric, the second
symmetric in abc. We therefore expand
h8a |8b|8c i = F fabc + Ddabc ,
(24.123).6.88 {nolabel}
where F and D are two unknown irreducible matrix elements. To relate these to
particles, we only have to multiply them with the SU(3)-polarization vectors ǫa (m)
of the particles whose assignments are clear from Fig. (24.14). For instance
1
ǫa (π + ) = − √ (1, i, 0, 0, 0, 0, 0, 0).
2
(24.124) {@}
Then we have between octet states:
h8m2 |8m|8m1 i = ǫ∗a (m2 )ǫ∗b (m)(F fabc + Ddabc )ǫc (m1 ).
(24.125).6.88b {nolabel}
We may also write
h8m2 |8m|8m1 i = F fm2 ∗m∗ m1 + Ddm∗2 m∗ m1 ,
(24.126).6.88bc {nolabel}
with obvious notation. The matrix associated with the b = 8 =
ˆ η component is
assumed to explain three mass differences. Two of them may be used to determine
the two irreducible matrix elements F and D. The symmetry breaking will then
allows us to derive one relation between the masses. Using the values of fabc and
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1359
24.4 SU(3)-Symmetry
dabc with b = 8 from Table 24.3, we have for the differences of the masses from some
common SU(3) singlet mass msg :
√
√
∆mN = hp|8η |pi
= F f485 + Dd484 = F 3/2 − D/2√3,
∆mΣ = hΣ+ |8η |Σ+ i =
Dd181 =
D/√3,
(24.127).6.89a {nolabel}
0
0
∆mΛ = hΛ |8η |Λ i =
Dd888 =
−D/√3,
√
∆mΞ = hΞ0 |8η |Ξ0 i = F f687 + Dd686 = −F 3/2 − D/2 3.
Using these equations we readily find that the common singlet mass is
msg = 12 (mΣ + mΛ ) = 21 (mN + mΞ ) + 41 (mΣ − mΛ ),
(24.128).6.89 {nolabel}
so that the masses have to satisfy the famous Gell-Mann–Okubo relation
1
2
(mN + mΞ ) = 14 (3mΛ + mΣ ).
(24.129) {@}
It holds reasonably well, the left-hand side being approximately equal to 1.128 GeV,
the right-hand side equal to 1.134 GeV.
For the decuplet, the same ansatz reproduces the equal mass splittings between
the rows that had led Gell-Mann to his prediction of the Ω-meson.
For the pseudoscalar meson octet the relative mass splitting is very large and
the mass relation cannot be expected to be as good as for the baryon octet. If we
use, however, the square masses, it is nevertheless in surprisingly good agreement
with experiment:
m2K = 14 (3m2η + m2π ).
(24.130).6.psm {nolabel}
The left-hand side is equal to 0.25 GeV2 , the right-hand side to 0.23 GeV2 . An
argument for using square-masses is that, in contrast to the fermion Lagrangian,
the boson Lagrangian has a mass term proportional to m2 .
The vector meson octet has a smaller relative mass splitting and the mass relation
should be good. If we insert, however, the experimental masses
m2K ∗ = 14 (3m2ω + m2ρ ),
(24.131).6.GO {nolabel}
there is a bad surprise: The left-hand side is equal to 0.80 GeV2 , considerably larger
than the value 0.61 GeV2 on the right hand. This disagreement can be resolved
by postulating that the symmetry breaking part of the Lagrangian mixes the isosinglet octet state with the SU(3) singlet state [5]. This is always possible with the
symmetry breaking transforming like an η-meson. Specifically, if ω0 , φ0 denote the
SU(3)-states, the physical particles can be
ω = cos θω0 − sin θφ0 ,
φ = sin θω0 + cos θφ0 ,
(24.132).6.phs {nolabel}
with some unknown mixing angle θ. The symmetry breaking Lagrangian can contain
the mass terms
Lsymm
br
= −m28 ω02 − m21 φ20 − m218 φ0 ω0 + . . . ,
(24.133) {@}
1360
24 Internal Symmetries of Strong Interactions
with some SU(3) octet, singlet, and mixing mass parameters and the associated
SU(3) fields. The octet mass parameter is determined from the Gell-Mann–Okubo
relation (24.131) as
m8 = 930 MeV.
(24.134) {@}
Between the physical states (24.132) this becomes diagonal, if the mixing angle θ is
given by
1
m2
θ = arctan 2 18 2 .
(24.135) {@}
2
m9 − m1
The diagonal masses are
m2ω = 12 (m28 + m21 ) − 21 [(m28 − m21 )2 − m418 ],
m2φ = 12 (m28 + m21 ) + 21 [(m28 − m21 )2 − m418 ].
(24.136) {@}
(24.137) {nolabel}
From the experimental numbers we determine the parameters
m18 ≈ 0.65 MeV,
cos θ ≈ 0.77.
(24.138) {@}
The mixing angle can be tested by experiment, since it determines the decay rate
of the processes
φ → K +K −,
(24.139) {@}
which occurs with a width 2.2 MeV. Since the vector meson has spin 1, the pseudoscalar kaons must come out in a p-wave. Their orbital wave function is therefore
antisymmetric. But for bosons the total wave function must be symmetric, which
implies that the SU(3)-quantum numbers have to be coupled antisymmetrically.
This, however, excludes the SU(3)-singlet part of φ from the decay (since its coupling would be proportional to δab in the SU(3)-indices of the pseudoscalar mesons).
The amplitude for the decay is therefore proportional to the SU(3)-Clebsch-Gordan
√
√
coefficient cos θ(8ω0 |8K + |8K − )∝cos θh0, 0| 21 , 21 , − 12 if845 = cos θ × (1/ 2) × ( 3/2).
−
This should be compared with the decay ρ0 → π +
√π , whose amplitude is proportional to (8ρ0 |8π− |8π+ )∝h1, 0|1, 1; 1, −1if123 = (1/ 2) × 1. Taking the squares, we
find the ratio
S(φ → K + K − ) 2
3
= cos2 θ ≈ 0.44.
(24.140) {9.as}
S(ρ0 → π + π − ) 4
The decay rates contain a phase space factor
Γ∝
q3
,
M2
(24.141) {9.ps}
where q is the momentum of the emerging pseudoscalar mesons, and M the mass
of the decaying vector meson. In the processes at hand we have qK ≈ 127, Mφ ≈
1020, qπ ≈ 359, Mρ ≈ 770 MeV, so that the phase space factors are 0.0020 MeV and
0.078 MeV, respectively. Together with (24.140) this gives the ratio of decay rates
γ(φ → K + K − ) 2
γ(ρ0 → π + π − ) ≈ 0.011.
(24.142) {9.dr}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1361
24.5 Newer Quarks
The experimental ratio is
γ(φ → K + K − ) 2
γ(ρ0 → π + π − ) exp
≈ 0.014,
(24.143) {9.dre}
in reasonable agreement with SU(3)-symmetry.
For the pseudoscalar mesons the same type of mixing can occur between η- and
η ′ - mesons. However, since their mass difference is relatively large compared to that
between ω and φ, their mixing angle is much smaller, which explains why no mixing
was needed to satisfy the Gell-Mann–Okubo relation (24.130).
As a further test of the octet assignment we may calculate branching ratios for
the decay of the decuplet resonances in the nucleon and the meson octets. Also here
the agreement with experimental data is reasonably good.
24.5
Newer Quarks
Particles discovered in the 1970s and the
teractions and decay properties suggested
named charmed quark, with the symbol c,
quark, with the symbol b. In Table 24.4
numbers of the quarks discovered so far.
theoretical attempts to explain their inthe existence of more quarks. They were
top quark, with the symbol t, and bottom
we have listed the masses and quantum
Table 24.4 List of Quarks and their properties taken from the Particle Data Group [11].
Quark
u
d
s
c
t
b
(I, I3 )
1
1
2, 2
1
1
2, -2
0,0
0,0
0,0
0,0
Charge
2
3
1
3
1
3
2
3
2
3
1
3
−
−
−
Strangeness
0
0
−1
0
0
0
Mass (MeV)
2.3+0.7
−0.5
4.8+0.5
−0.3
95 ± 5
1275 ± 25
173070 ± 0.890
4180 ± 30
It was tempting to incorporate these quarks into the above broken symmetry
considerations and extend the basic quark antiquark SU(3)-multiplet by these additional quarks and antiquarks to higher multiplets, with an even larger broken
symmetry group. This has led indeed to a reasonable particle classification if we
add the lightest of the additional quarks, the charmed quark c, to the other three
and forming a quartet u, d, s, c. This quartet can then be treated very approximately
as a fundamental representation 4 of an internal symmetry group SU(4). Clearly,
the corresponding antiquarks must be transformed according to the representation
4̄. The weight diagram of these representations is three-dimensional and contains
an additional “charm” axis labeled by C in Fig. 24.18.
{TPDGRP2}
1362
24 Internal Symmetries of Strong Interactions
Figure 24.18 The four lowest quarks u, d, s, c and their position in the three-dimensional
weight space with the quantum number “charm”. By combining a quark and an antiquark
one finds the 16 states on the right-hand side, of which 15 form an irreducible representation of SU(4). The new states D 0 , D + , F + and their antiparticles have been found in the
laboratory.
24.6
{f11.ch}
Tensor Representations and Young Tableaux
A systematic construction of the tensor representation in direct product spaces such
as 3 × 3 × 3 is possible with the help of Young tableaux. The irreducible representations are obtained by forming tensors which transform irreducibly under permutations of the indices. They are of a definite symmetry type specified by the Young
tableau introduced in Appendix 2A. The tableau 1 2 3 indicates a tensor of com1
plete symmetry transforming like a 10-representation, while 2 corresponds to a
3
complete antisymmetric tensor transforming like an SU(3) singlet. For the mixed
1 2
tableau
we adopt the convention to first symmetrize the state |qα qβ qγ i in
3
the first two indices, then antisymmetrize in the first and the third index. Equi1 2
2 1
, or
. Each gives an octet
valently we can use another tableau, say
3
3
representation. For instance, the proton wave function (24.113) arises from the
2 1
symmetry operations of the Young tableau
applied to the state |uudi. The
3
symmetrization in the first two indices gives
|uudi → 2|uudi
(symmetry in 21),
(24.144).6.90a {nolabel}
the antisymmetrization in the second and third index (of the original tensor |qα qβ qγ i)
|uudi → 2|u(ud − du)i
(antisymmetry in 23).
(24.145).6.91 {nolabel}
√
2 1
|uudi = 2|u(ud − du)i = 2 2|pi.
3
(24.146).6.92 {nolabel}
Hence:
H. Kleinert, PARTICLES AND QUANTUM FIELDS
24.6 Tensor Representations and Young Tableaux
1363
Equivalent proton states could have been obtained from the tableau
1 2
|uudi = 2|uud − duui,
3
1 2
|udui = 2|udu − duui,
(24.147).6.93 {nolabel}
3
or the sum of the two which gives
|u(ud + du) − 2duui.
(24.148).6.94 {nolabel}
This is once more the proton state |p′ i of Eq. (24.120) found in the product 6 × 3.
It cannot be decided by SU(3)-symmetry alone, which mixture of the two independent three-quark wave functions corresponds to the proton in nature. This
has has to emerge from a dynamical model [6,7]. We shall see later that the larger
symmetry group SU(6) makes definite predictions on this.
In general, the dimensionality of the SU(n)-representation of a tableau is obtained from the following formula
dSU(n) =
D(l1 , l2 , . . . , ln )
,
D(n − 1, n − 2, . . . , 0)
(24.149).6.95 {nolabel}
where
Y
(xi − xj ),
(24.150) {@}
i = 1, 2, 3, . . . , n,
(24.151) {@}
D(x1 , . . . , xn ) ≡
i<j
and
li ≡ n + mi − i,
with mi being the number of boxes in the ith row of the tableau. Note that also
rows with no boxes have to be counted, up to i = n.
A useful alternative formula for the dimension is
dSU(n) =
l1 !l2 ! · · · ln !
1
,
Q
0!1! · · · (n − 1)! i,j hij
(24.152).6.95’ {nolabel}
where hij is the sum of the number of boxes to the right of the element ij in the
tableau plus the number of boxes below ij plus 1.
Take some examples. The trivial Young tableau
has m1 = 1, m2 = 0, m3 =
0, . . . , mn = 0 and thus l1 = n, l2 = n − 2, . . . , ln = 0. Hence
d
= n.
(24.153).6.90 {nolabel}
The first non-trivial one
has m1 = 2, m2 = 0, . . . , mn = 0 and thus l1 =
n + 1, l2 = n − 2, l3 = n − 3, . . . , ln = 0. Hence
d
= n(n + 1)/2,
(24.154).6.97 {nolabel}
= n(n − 1)/2,
(24.155).6.98 {nolabel}
which gives 6 for SU(3). Further
d
1364
24 Internal Symmetries of Strong Interactions
which gives 3 for SU(3), the dimension of the representation 3̄.
For 3-quark states, the dimensions are
d
d
d
= n(n + 1)(n + 2)/6,
(24.156).6.99 {nolabel}
= (n − 1)n(n + 1)/3,
(24.157).6.100 {nolabel}
= n(n − 1)(n − 2)/6,
(24.158).6.101 {nolabel}
which take the values 10, 8, 1, respectively, for SU(3). Somewhat more generally, the
dimensions of some simple tableaux are
d
1 2 3
d
···
k
1 2 3
n+k−1
k
=
···
k
!
,
(24.159).6.102 {nolabel}
n+k−1
k+1
=
!
!
n
.
=
d
1
k
2
..
.
k,
(24.160) {@}
(24.161).6.104 {nolabel}
k
A column with n boxes is completely antisymmetric in the corresponding indices
and thus proportional to the invariant antisymmetric tensor ǫi1 i2 ,...,in . This is why
its dimensionality is 1. In constructing all Young tableaux any such column can be
omitted. For instance, in SU(3),
=
.
(24.162).6.105a {nolabel}
One often indicates this cancellation by crossing out any complete column with a
vertical line:
→
.
(24.163).6.105 {nolabel}
The representation associated with each Young tableaux occurs as often as dimensionality of the associated representation of the permutation group. This follows
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1365
24.6 Tensor Representations and Young Tableaux
from the fact that the representation matrices of the permutation group commute
with those of SU(3). They can therefore be brought simultaneously to a block form

 D1












D1
D2
..
.






.





(24.164).6.106 {nolabel}
Along the diagonal there are ni identical irreducible representation blocks Di of dimensionality di , etc. Now, according to Schur’s lemma, any matrix commuting with
such a block matrix must be itself a block matrix with the dual form, that consists
of di identical blocks of dimensionality ni . The formulas for the dimensionality of
the irreducible representations of the permutation group were given in App. 2A. If
the Young tableau has r boxes, the dimensionality is given by
r!
,
ij hij
(24.165).6.107 {nolabel}
d Sr = Q
where the numbers hij are the same as in (24.149).
Using this formula we find, for instance, that the product of three-quark states
×
×
decompose into one SU(3) decuplet
and one singlet
, two octets
,
:
3 × 3 × 3 = 10 + 8 + 8 + 1.
(24.166).6.108 {nolabel}
The Young tableau can also be used to find the irreducible contents in a product
of two or more irreducible representations. For instance, suppose we want to know
the irreducible contents of 8 × 8. We take the two Young tableaux
×
and distinguish the rows of boxes in the second factor by a, b, c, . . ., in the example
a a
. Then we add the lettered boxes to the first tableau in the following way:
b
a) add all a’s in such a way that one obtains all proper tableaux
(i.e., with mi+1 ≥ mi ) which have no more than one a in each column;
b) add the b’s, following the same rule;
c) add the c’s, following the same rule.
1366
24 Internal Symmetries of Strong Interactions
In our example we thus obtain the expansion
×
a a
b
a a
a a
=
+
b
8×8
b
27
10
a
a
a b
+
a
+
b
10
(24.167).6.108b {nolabel}
8
a
+
a
a b
1
+
b
a
8
In accordance with the rule given earlier we have dropped all complete columns.
Note that due to the antisymmetry of any complete column, an incomplete column
is equivalent to the complex conjugate representation, i.e., a tableau that would be
obtained from the missing boxes necessary to complete it (indicated by a box with
a circle), fo instance
=
◦
◦
◦
.
(24.168).6.108b {nolabel}
3̄
This is how we can see immediately that the representation associated with the
tableau
is a 10
=
◦ ◦ ◦
◦ ◦ ◦
◦ ◦ ◦
.
(24.169).6.108b {nolabel}
10
The octet representation, being the adjoint representation of SU(3) is real [its generators are (Ga )bc = −ifabc with real antisymmetric matrices (fa )bc ; recall (4.83)].
From the Young tableau of the octet
this is obvious since the missing boxes
◦ ◦
◦ form once more the same tableau
, so the representation is equal
◦
◦ ◦
to the complex conjugate of itself and thus real.
in
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1367
24.7 Effective Interactions among Hadrons
24.7
Effective Interactions among Hadrons
There exists a great variety of strongly interacting particles with many possible
interactions between them. Let us study a few of them.
24.7.1
The Pion-Nucleon Interaction
The most important strong interaction is the one between pions and nucleons since
it gives rise to the dominant part of the forces which keep the nuclei together. If we
neglect electromagnetic interactions, this interaction exhibits isospin symmetry of
pions and nucleons, so that a system of fields πa and N has the following effective
action
A = Aπ + AN + AπN N ,
(24.170) {pinu}
with
Aπ =
AN =
Z
Z
d4 x
o
1n
[∂πa (x)]2 − m2π πa2 (x) ,
2
d4 x N̄(x)(i/
∂ − MN )N(x),
AπN N = gπN N
Z
d4 x N̄ (x)iγ5 τ a N(x)πa (x).
(24.171) {nolabel}
(24.172) {nolabel}
(24.173) {pinu2}
Here πa (x) is an isovector field and N(x) a Dirac spinor field with an additional
isospinor index, which is not explicitly written down. The matrices τ a are 2×2 Pauli
matrices acting on the isospinor indices of N(x). Studies of pion nucleon scattering
amplitudes determine the size of the pseudoscalar coupling constant gπN N to be
2
gπN
N
≈ 14.4 ± .4 .
4π
(24.174) {x104}
In some analyses, the pion nucleon interaction (24.173) is parametrized in the axial
vector form
AπN N =
fπN N Z 4
d xN̄ (x)γ µ γ5 τ a N(x)∂µ πa .
Mπ
(24.175) {@}
For physical nucleons on the mass shell, the two interactions are the same with a
relation between the two coupling constants
fπN N = −
Mπ
g.
2MN
(24.176) {@}
Thus
2
fπN
N = 0.081 ± 0.002.
(24.177) {@}
In either form, the coupling strength is too large to perform perturbative calculations
with the interaction AπN N .
1368
24 Internal Symmetries of Strong Interactions
It is useful to find the SU(3)-symmetric extension of the pion nucleon interaction.
In SU(3), there are two ways of coupling three octets with each other, an antisymmetric one fijk and a symmetric one, dijk . The easiest way to do the calculation is
by writing the nucleon octet as a 3 × 3 -matrix.
+ √16 Λ0
Σ+
Σ−
− √12 Σ0 +
−Ξ−
Ξ0
√1 Σ0
2


B=

p
n
√1 Λ0
6
− √26 Λ0
and the pseudoscalar octet as

√1 π 0
2
+ √16 η 0
π+
π−
− √12 π 0 +
K−
K̄ 0

P =

√1 η 0
6


,

(24.178) {@}
K+

K0 
.
− √26 η 0

(24.179) {@}
√
In this context the phases of π ± are chosen as π ± ≡ (π1 ± iπ2 )/ 2 so that P is a
Hermitian matrix. Then the interaction Lagrangian has a simple explicit form
LPBB = g
F
D
tr([B̄, B]P ) + g
tr({B̄, B}P ),
F +D
F +D
(24.180) {@}
where g ≡ gπN N is the pion nucleon coupling constant and F/D the so-called “F over-D -ratio”, which gives the relative strength of antisymmetric and symmetric
coupling. Working out the different components, this yields the expansion in terms
of isospinors
+ (1 − 2α)(Ξ̄Ξ) · + 3 −√34α N̄Nη
3 − 2α
− √ Ξ̄Ξη − (1 − 2α)(N̄ K) · − (Ξ̄K ) · 3
2α
¯ · (K N) − ¯ · (K Ξ) − 3 −
√ N̄ΛK
−(1 − 2α)
3
0
Lint = g (N̄ N) ·
c
0
†
c†
3 − 4α
3 − 4α
3 − 2α
− √ K † Λ̄N + √ Ξ̄ΛK c + √ K c† Λ̄Ξ
3
3
3
2α
2α
+i(1 − α)( ¯ × ) · + √ ( ¯ Λ) · + √ (Λ̄ ) ·
3
3
2α ¯
2α
+ √ ( · )η 0 − √ Λ̄Λη 0 .
3
3
(24.181) {su3l}
Here we have used the isospinors
N≡
p
n
!
,
Ξ≡
Ξ0
Ξ−
!
,
(24.182) {@}
and
K≡
K+
K0
!
,
c
K ≡
K̄ 0
− K−
!
.
(24.183) {@}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
24.7 Effective Interactions among Hadrons
1369
Note that K c is the charge conjugate spinor C K̄ ∗ . The parameter α is defined by
D
F +D
(24.184) {@}
F/D = (1 − α)/α.
(24.185) {@}
α≡
so that
The two numbers g = gπN N and F/D parametrize all couplings.
The pion nucleon coupling was written down in Eq. (24.174). The F/D-ratio is
determined experimentally from an analysis of the production process
π − p → Σ0 K +
(24.186) {@}
which contains the coupling
√
¯ − 2g(1 − 2α)
gnΣ− K + =
√ F −D
.
= − 2g
F +D
(24.187) {nsigk}
From the experimental data one finds
F
2
≈ .
D
3
(24.188) {@24.182}
For the coupling between pions and Σ-particles, the interaction Lagrangian
(24.181) yields
F
gΣΣπ = g(1 − α) = g
.
(24.189) {sisipi}
F +D
The SU(3)-relations between coupling constants in the Lagrangian (24.181) are,
of course, in agreement with standard SU(3)-Clebsch-Gordan coefficients. The coefficients of the SU(2)-subgroup can be taken from Table 4.2 in Chapter 4.1, The
isoscalar factors from Table 24.5 in Appendix 24A on p. 1378.
Take, for √
instance, the matrix element hp|π 0|pi. The SU(2) Clebsch-Gordon
√ √ coefficient is√1/ √3. The antisymmetric
and symmetric isoscalar factors are 3/ 12 =
√
1/2 and 9/ 20 = 3/2 5, respectively. Hence, with the normalization factors of
(24A.13), we have
F +D
.
(24.190) {@}
hp|π 0|pi =
2
√
+ 0
+
2 and the
The matrix element
hΣ
|π
|Σ
i
has
an
SU(2)-Clebsch-Gordan
1/
q
isoscalar factors 2/3 and 0, respectively, so that
hΣ+ |π 0 |Σ+ i = F,
just as before in (24.189).
(24.191) {sisipip}
1370
24 Internal Symmetries of Strong Interactions
Let usqalso calculate the coupling gnΣ− K + for which the Clebsch-Gordan coeffi√
cient is − 2/3 and the isoscalar factors are 1/2 and −3/2 5, respectively. Thus
1
hn|K + |Σ− i = √ (F − D),
2
(24.192) {@}
and
gnΣ− K + = −g
√ F −D
,
2
F +D
(24.193) {@}
as in (24.187).
In Appendix 24A we have collected a few useful formulas for SU(3)-calculations.
24.7.2
The Decay ∆(1232) → Nπ
The interactions between the various members of the nucleon octet with pseudoscalar
mesons and the decuplet resonances of spin-parity J P = 3/2+ are all determined by
only one SU(3)-invariant matrix element. This, in turn, may be chosen to coincide
with the coupling between the proton, the π + , and the ∆++ (1232)-resonance. The
Lagrangian for this specific interaction is conventionally written as
L∆++ pn+ =
g ∗ ++ +
ψ̄µ ψ ∂µ π + + c.c. ,
MN
(24.194) {del}
where ψµ is a Rarita-Schwinger spinor for the spin 3/2 particle ∆++ (1232). Some
people write 4h/Mπ instead of g ∗/MN .
The isospin structure can be incorporated by multiplying the Lagrangian by the
Clebsch-Gordan factor
h 23 c| 12 a1bi
,
h 23 23 | 12 21 11i
(24.195) {@}
where c, a, b are the isospin orientations of ∆, N and π. Alternatively, we can use
Rarita-Schwinger isospinors ∆a for the ∆-particle (each of the three components
∆1 , ∆2 , ∆3 is a two-component isospinor), and we can write in isospace (suppressing
the Lorentz indices):
L=
g∗ ¯
N − N̄
MN
· .
(24.196) {@}
Indeed, the isospinor for the ∆++ -resonance has only a + -component with isospin
¯ ++ pπ + , as in
up, so that ¯ N contains the particles ∆++ , p, and π + in the form ∆
(24.194).
For calculations it is useful to record the completeness relation of RaritaSchwinger isospinors
X
s3
¯ b = δab 1 − 1 τa τb = 2 δab 1 − 1 [τa , τb ],
∆a ∆
3
3
6
(24.197) {@}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1371
24.7 Effective Interactions among Hadrons
where 1 stands for the 2 × 2 unit matrix in isospace.
The coupling g ∗ can be determined by an analysis of the strength of the ∆++
resonance in the π + p scattering amplitude. This gives
g ∗ ≈ 12.43 g ≈ 0.92 g
(24.198) {x9.21}
h ≈ 0.457 g.
(24.199) {x9.22}
corresponding to
If we assume the ∆ resonance to be very narrow, we can immediately calculate the
decay rate from the interaction (24.194). The decay amplitude is found as follows:
We consider a ∆++ -resonance in its rest frame and use helicity amplitudes along the
direction of the momentum of the emerging proton, which we assume to run in the
3-direction. The ∆ helicity can be ±3/2 or ±1/2. Only the latter two orientations
allow to conserve angular momentum with the helicities of the proton being ±1/2.
The two orientations decay with the same rate, so we only need to study one case, say
the helicity +1/2. The Rarita-Schwinger spinor is a Clebsch-Gordan combination
of spin J, M = 1, 0 with 1/2, 1/2 and of J, M = 1, 1 with 1/2 − 1/2:
uµ (0; 32 , 21 ) =
s

2


3
0
0
0
1



 u(0,

1
2

1 

)− √ 
6
0
1
i
0



 u(0, − 12 ).

(24.200) {@}
Between the state
a†π+ a†p |0i
(24.201) {@}
and a ∆-state at rest h0|a∆ with unit normalization in a finite volume V , the wave
functions for the pion and the nucleon are
√
1
1
e−iqπ x √ u(pN , s′3 ),
2Eπ V
V
(24.202) {@}
and for the ∆-resonance
1
√ uµ (p∆ , s3 ).
V
(24.203) {@}
We therefore find the decay amplitude [matrix element of the t-matrix (9.290)]:
t =
1
1
g∗
1
q
ūµ u q µ √ 3 √
MN
2Eπ EN /MN
V
g∗
=
MN
s
ζ
1
1
1
2
,
cosh pN √
√ 3q
3
2
2Eπ V
EN /MN
(24.204) {nolabel}
(24.205) {@}
1372
24 Internal Symmetries of Strong Interactions
where
u(pN , 12 )u(0, 21 ) = cosh
ζ
1
=
(EN + MN ),
2
2MN
(24.206) {@}
with EN , Eπ being the energies of proton and pion in the rest frame of the decaying
∆-resonance, and pN the momentum of the nucleon emerging in the z-direction. The
decay rate is found from formula (9.338), yielding for the helicity h = 1/2 -state:
EN Eπ
1
2π
dΩ|t|2
3
(2π)
M∆
4π
pN g ∗2 2 2 1
=
2π
p (EN + MN )
(2π)3 M∆ MN2 3 N 4
g ∗ 2 2 p3N
(EN + MN ).
=
4π 3 MN2 M∆
Z
Γ∆h=1/2 →N π =
(24.207) {half}
Since the resonance ∆ is, with probability 1/2, in the helicity-3/2 state, where it
cannot decay, the final decay rate is only one half of the value (24.207), i.e.,
Γ=
g ∗2 1 p3N
(EN + MN ).
4π 3 MN2 M∆
(24.208) {@}
In terms of the three particle masses involved, the energy EN is given by
EN =
M∆ 2 + MN2 − Mπ2
,
2M∆
(24.209) {@}
and the momentum pN of the outcoming proton (equal to that of the pion) is
pN =
q
[M∆2 − (MN + Mπ )2 ][M∆2 − (MN − Mπ )2 ]/2M∆ .
(24.210) {@}
Using the experimental width
Γ∆ ≈ 115MeV,
(24.211) {@}
we obtain
g ∗ ≈ 14.43 ≈ 1.07g ,
(24.212) {@24.205}
corresponding to
h ≈ 0.53,
(24.213) {@}
in reasonable agreement with the independent determination (24.199) from the pionnucleon scattering.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1373
24.7 Effective Interactions among Hadrons
24.7.3
Vector Meson Decay ρ(770) →ππ
Among mesons, the most directly observable coupling is the ρππ-coupling. It governs
the strength of the most prominent resonance in the p-wave pion-pion scattering
amplitude. Since ρ is a vector meson of isospin 1, it is coupled to the two pions in
a p-wave. The Lagrangian interaction is given by
L=
gρππ
2
· × ∂ µ
↔
µ
.
(24.214) {@}
For the decay ρ+ → π 0 π − , the invariant t-matrix element [defined in (9.290)] is given
by
1
1
q
2pµπ ǫµ (pρ , s3 ),
hπ 0 π + |t|ρ+ i = −gρππ q
2
2V Epi 2V Mρ
(24.215) {@}
where ǫµ (pρ , s3 ) is the polarization vector of the ρ meson and the decay rate is,
according to formula (9.339),
2
Γρ+ →π0 π+ = gρππ
4π pCM 1
4|pµ ǫµ |2 .
(2π)2 Mρ2 23 π
(24.216) {@GammRho
The average over the initial polarization gives
pµ pν
1 µ νX µ
1
ǫ (pρ , s3 )ǫν (pρ , s3 ) = − pµπ pνπ gµν − ρ 2 ρ
pπ pπ
3
3
pρ
s3
!
1
= p2CM π ,
3
(24.217) {@}
where pCM π is the momentum of the pions in the rest frame of the decaying ρ-meson:
pCM π =
1/2
1 2
Mρ − 4m2π
.
2
(24.218) {@}
This brings (24.216) to the form
Γρ+ →π0 π+
2
2
2
2
gρππ
gρππ
2 p3CM π
2 Mρ − 4mπ
=
=
4π 3 Mρ2
4π 3
8Mρ2
3/2
.
(24.219) {@}
The experimental decay width is
Γρππ ≈ 153MeV.
(24.220) {@RhOWi}
From this one finds the coupling constant
2
gρππ
≈ 2.85.
4π
(24.221) {@}
1374
24 Internal Symmetries of Strong Interactions
24.7.4
Vector Meson Decays ω(783) →ρπ and ω(783) → πππ
Another important coupling is that between ω, ρ, and π:
Lωρπ = gωρπ ǫµνλκ ∂ µ ω ν ∂ λ
· .
κ
(24.222) {@}
It is responsible for the decay
ω → πππ,
(24.223) {@}
Γω→πππ = 9.8 ± 0.3 MeV.
(24.224) {@}
which proceeds at a rate
The ππ -channels are dominated by the ρ-resonance. One can therefore assume a
sequential decay, finding
Γω→3π = (Mω − 3Mπ )4 Mρ2 − 4Mπ2
≈ 10.3 MeV
2
2
gρππ
gωρπ
,
4π 4π
−2
2
2
gωρπ
1 1 gρππ
× 3.56
Mω Mπ2 √
27 4 4π 4π
(24.225) {@}
where the number 3.56 is due to the numeric integration of a phase space integral
[8]. The experimental rate gives
2
gρππ
0.95
,
≈
4π
Mπ2
(24.226) {@}
and hence
2
gωρπ
1
0.117
.
≈ 6.14
≈
2
2
gρππ
Mπ
GeV2
24.7.5
(24.227) {@}
Vector Meson Decays K∗ (892) →Kπ
In πK -scattering one observes the SU(3)-partner of the ρ-meson, the strange vector
meson K ∗ (892). Its coupling is written as
(24.228) {@}
2 p3CM K∗
,
3 MK2
(24.229) {@}
↔
1
LK ∗ Kπ = gK ∗ Kπ (K †µ K)i ∂ µ .
2
Its decay width is
ΓK ∗ Kπ = 3
1
g ∗
2 K Kπ
4π
2
where
pCM K ∗
1
=
2MK ∗
rh
MK2 ∗ − (MK ∗ + Mπ )2
ih
MK2 ∗ − (MK ∗ − Mπ )2
i
(24.230) {@}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1375
24.7 Effective Interactions among Hadrons
is the momentum of the decay products in the rest frame of K ∗ . This implies a ratio
with respect to the ρππ -width:
2
2
3
2
ΓK ∗ Kπ
3 gK
3 gK
∗ Kπ Mρ pCM K ∗
∗ Kπ
× 0.385.
=
≈
2
3
2
2
Γρππ
4 gρππ MK ∗ pCM ρ
4 gρππ
(24.231) {@RATIOF}
The experimental width of K ∗ is
ΓK ∗ Kπ = (51.8 ± 0.8) MeV,
(24.232) {@}
so that its ratio with respect to the ρ-width (24.220) is 0.3406. Inserting this into
(24.231), we extract a ratio of coupling constants
gK ∗Kπ
≈ 1.086.
gρππ
(24.233) {@}
By SU(3)-symmetry, this ratio is predicted to be unity, in good agreement with
experiments.
24.7.6
Axial Vector Meson Decay a1 (1270)→ ρπ
A more involved coupling governs the decay of the axial vector meson a1 (1270). It
is seen as a resonance of mass 1270 MeV and width
Γa,ρπ ≈ 316 ± 45MeV
(24.234) {@}
in the ρπ-scattering amplitude. Angular momentum and parity allow s- and dwave interactions, so that there exist two independent coupling constants with a
Lagrangian density
L = ga1 ρπ (aµ ×
)·+h
µ
a1 ρπ
(∂µ aν × ∂ν
) · .
µ
(24.235) {@}
Let ǫ(p), ǫ(q) denote the polarization vectors of the a1 and ρ meson, respectively,
and k the momentum of the outgoing pion. Then the invariant t-matrix element for
A+ → ρ− π + reads
tba = √
1
2V
√
3
h
i
1
ga1 ρπ ǫ∗µ (p)ǫµ (q) − ha1 ρπ qν ǫ∗ν (p)pµ ǫµ (q) .
p0 q0 k0
(24.236) {@}
For the calculation of decay rates it is better to use a more complicated-looking
decomposition into a longitudinal and a transverse helicity amplitude
tL
tT
1
!
!
pν pκ
qµ qλ
1
= √ 3√
gνκ − 2 ǫ∗ ∗ κ(p)ǫλ (q),
gL Pµ q ν gµλ − 2
q
p
p0 q0 k0
2V
gT µα′ β ′ λαβγ
(24.237) {@}
qβ pα qβ ′ pα′ ǫ∗λ (p)ǫµ (q).
= − 2ǫ
γǫ
ma
1376
24 Internal Symmetries of Strong Interactions
The relation between them is
−ha1 ρπ
m2a1 + m2ρ − m2π
= gL + gT
,
2m2a1
−ga1 ρπ
gT  m2a1 + m2ρ − m2π
=
m2a1
2

!2

− m2a1 m2ρ  .
(24.238) {@}
The advantage of these amplitudes is, that they are decoupled in the decay process.
A somewhat tedious calculation (see Appendix 24B) gives
Γa1 ρπ = ΓLa1 ρπ + ΓTa1 ρπ ,
(24.239) {@}
gL2 1 p5ρ
,
nπ 3 m2ρ
(24.240) {@}
where
ΓLa1 ρπ =
ΓTa1 ρπ =
gT2 2 p5ρ
.
nπ 3 m2a1
Since the experimental masses have the ratio
m2a1
≈ 2.72,
m2ρ
(24.241) {@}
we get
Γa1 ρπ ≈ 5.5 × 10−5 gL2 + 2.735 gT2 GeV,
(24.242) {@}
and find for the coupling constants the relation
gL2 +
2m2ρ 2
g ≈ gL2 + 0.68gT2 ≈ 75.1.
m2a1 T
(24.243) {@}
The ratio between the coupling constants gL and gT has to be determined by some
other method.
24.7.7
Coupling of ρ(770) to Nucleons
The coupling is defined by the non-minimal Lagrangian density
L = gρN N
ψ̄γ µ
µ
κρN N
(∂µ
ψ+
2M
ν
− ∂ν
)ψ̄σ ψ
µ
µν
.
(24.244) {@}
An analysis of the phase shifts in ππ → N N̄ yields the coupling strengths [9]
2
gρN
N
= 0.6 ± 0.1,
4π
κρN N = 6.6 ± 1.0.
(24.245) {@}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1377
Appendix 24A
Useful SU(3)-Formulas
Appendix 24A
Useful SU(3)-Formulas
The matrix elements of an octet operator between two octets labeled by the real indices i =
1, 2, 3, . . . , 8 of the adjoint representation 8 can be written as follows:
h8i |8k 8j i =
F (−ifkij ) + Ddkij = F (F̂k )ij + D(D̂k )ij ,
{USEFULL}
(24A.1) {@}
where fijk are the structure constants of SU(3), and dijk those of the anticommutators
4
{λi , λj } = dijk λk + δij .
3
The SU(3)-vector ei associated with a specific particle is found from the assignment of the nucleon
octet to the 3 × 3 -matrix elements

 1 0
√ Σ + √1 Λ
Σ+
p
2
6


Σ−
− √12 Σ0 + √16 Λ
n
(24A.2) {@}
N = 
,
1
−
0
Ξ
Ξ
− √6 2Λ
by contraction with the λ-matrices:
1
ei = √ tr λi N .
2
(24A.3) {@}
The proton, for example, corresponds to the octet vector
 

0 0 1
1
ei (p) = √ tr λi  0 0 0  .
2
0 0 0
(24A.4) {@}
It has the non-zero components
1
e4 (p) = √ ,
2
i
e5 (p) = √ .
2
(24A.5) {@}
For the π 0 -couplings we calculate the matrix elements F̂0ij and D̂0ij between two proton states as
follows:
hp|F̂0 |pi =
1
,
2
hp|D̂0 |pi =
1
.
2
(24A.6) {@}
Hence
F +D
.
hp|π0 |pi =
2
+
Similarly, the SU(3)-vector of the Σ -particle is
 

0 1 0
1
ei (Σ+ ) = √ tr λi  0 0 0  .
2
0 0 0
(24A.7) {@}
(24A.8) {@}
It has the non-zero components
1
e 1 Σ+ = √ ,
2
i
e 2 Σ+ = √ .
2
(24A.9) {@}
For the coupling of π 0 to Σ-particles we need the matrix elements (F̂0 )ij and (D̂0 )ij between
Σ+ -states:
hΣ+ |F̂0 |Σ+ i = 1,
hΣ+ |D̂0 |Σ+ i = 0.
(24A.10) {@}
1378
24 Internal Symmetries of Strong Interactions
Hence
hΣ+ |π 0 |Σ+ i = F.
(24A.11) {@}
The matrix elements of π + between Σ+ Σ+ and pp have therefore the ratio
2F
Σ+ π + Σ+
=
,
pπp
F +D
(24A.12) {@}
just as in (24.189), (24.191), and (25.100).
The SU(3)-matrix elements may also be calculated by using tables of the SU(3)-Clebsch-Gordan
coefficients. The matrices F̂i D̂i correspond to the coefficients
r
√
5
F̂i = 3h8a |88i,
h8s |8 8i.
(24A.13) {normf}
D̂i =
3
The Clebsch-Gordan coefficients can be decomposed into a product of an isospin SU(2)-ClebschGordan coefficient and a so-called isoscalar factor
h8a i|8j; 8ki = hh8a Yi |8Yj ; 8Yk ii(T i , T3i |T j , T3j ; T k , T3k ),
(24A.14) {@}
where T i , T3i is the isospin content of the SU(3)-index i. The isoscalar factors are listed in Table
24.5.
Table 24.5 Isoscalar factors of SU(3). We have omitted the square roots of the number
√
in the arrays on the right-hand sides, i.e., −3 stands for − 3. This is indicated by the
superscript 1/2 on each array. The table is from the Particle Data Properties Booklet [10].
1 →8⊗8
1
(Λ) → (N K̄ Σπ Λη ΞK) = √
8
81 → 8 ⊗ 8
2
3 −1 −2
1/2

N π N η ΣK ΛK
N K̄ Σπ Λπ Ση ΞK
N K̄ Σπ Λη ΞK
ΣK̄ ΛK̄ Ξπ Ξη


9 −1
1
−6 0
= √ 
2 − 12
20
9 −1

N π N η ΣK ΛK
N K̄ Σπ Λπ Ση ΞK
N K̄ Σπ Λη ΞK
ΣK̄ ΛK̄ Ξπ Ξη

1
= √
12

∆
Nπ Nη
 Σ  →  N K̄ Σπ Λπ Ση ΞK
Ξ
ΣK̄ ΛK̄ Ξπ Ξη
Ω
ΞK̄
8 → 10 ⊗ 8

1
√
12



N
Σ
Λ
Σ

→
83 → 8 ⊗ 8


N
Σ
Λ
Ξ

→
10 → 8 ⊗ 8



N
Σ
Λ
Ξ

∆
Σ
Ξ
Ω
3
2
6
3
−9 −1
4 4 −6
−4 −2
−9 −1
3
−6 6
−2 2 −3 3 3
3 −3 3 3
12



∆π ΣK
∆K̄ Σπ Ση ΞK
Σπ ΞK
ΣK̄ Ξπ Ξη ΩK


−12 3
8 −2 −3 2
−9 6
3 −3 −3 6

∆π ∆π ΣK
∆K̄ Σπ Ση ΞK
ΣK̄ Ξπ Ξη ΩK
Ξ K̄Ωη


15
8 8
12 3
12
→

→
=
 = √1
15
=
1
√
24



3
1/2

−3
8
0
0 −2
0
0
6
3
3 −3

10 → 10 ⊗ 8



3 −6
0 −8
−3 −6
− 12
1/2

1/2

1/2

1/2

H. Kleinert, PARTICLES AND QUANTUM FIELDS
{isosc}
Appendix 24B
Decay Rate for a1 → ρπ
1379
Appendix 24B
Decay Rate for a1 →ρπ
{appa1}
Denoting the polarization vectors ǫ(p) and ǫ(q) of a1 and ρ mesons by ǫ and ǫ′ , and using the
transversality properties√pµ ǫµ = 0, qµ ǫ′µ = 0, respectively, the amplitudes are (dropping the
normalization factors 1/ 2N q0 associated with the three particles)
′∗
ǫ ǫ ǫ′∗ q ǫ′∗ p g
T
q2
qp (24B.1) {nolabel}
t = gL (pǫ′∗ )(qǫ) + 2 ǫq
M a1 ǫp
qp
p2 gT = gL (pǫ′ )(qǫ) + 2 Ma21 Mρ2 (ǫ′∗ ǫ) + (pǫ′∗ )qp − (ǫ′∗ ǫ)(qp)2 .
M a1
This expression shows directly the relation of the couplings gL and gT with the couplings ga1 ,ρπ
and ha1 ρπ . We now put a1 in its rest frame, so that pµ = (Ma1 , 0, 0, 0), and we find the scalar
products
ǫ′∗ (q, 1)ǫ∗ (p, 1) = −1,
ǫ′∗ (q, 2)ǫ∗ (p, 2) = −1,
ǫ′∗ (q, 3)ǫ(p, 3) = −
q0
,
Mρ
and further
(qǫ(p, 1)) = 0,
(pǫ′∗ (q, 1)) = 0,
(qǫ(p, 2)) = 0,
(pǫ′∗ (q, 2)) = 0,
pCM
M a1 ,
(pǫ′∗ (q, 3)) =
Mρ
(qǫ(p, 3)) = −pCM ,
(24B.2) {@}
where
pCM =
1 q 2
[Ma1 − (Mρ − Mπ )2 ][Ma21 − (Mρ − Mπ )2 ]
2Ma1
(24B.3) {@}
is the center-of-mass momentum of the ρ-meson. We therefore find the helicity amplitudes (writing
the helicities as superscripts in parentheses)
hρ(0) π1 |t|a1 (0)i =
1
1
hρ 2 π|t|a12 i =
where
−gL p2CM
M a1
,
Mρ
(24B.4) {nolabel}
gT
2
Ma21 = gT p2CM ,
−Ma21 Mρ2 + ECM
Ma21
ECM =
Ma21 + Mρ2 − Mπ2
2Ma1
(24B.5) {@}
is the energy of the ρ meson in the center-of-mass frame. From this we obtain directly the longitudinal width
ΓL
a1 →ρπ =
2
2 p5CM
gL
,
4π 6 Mρ2
(24B.6) {@}
having inserted a factor 2 due to isospin, and a transversal width
ΓTa1 →ρπ =
gT2 2 p5CM
·2
.
4π 6 Ma21
Notes and References
For discussion of internal symmetries see the book by
S. Gasiorowicz, Elementary Particle Physics, John Wiley & Sons, N.Y. 1966.
The individual citations refer to:
(24B.7) {FEYANMARP
1380
24 Internal Symmetries of Strong Interactions
[1] For an introduction see
H. Fritzsch, Elementary Particles, Building Blocks of Matter, World Scientific, Singapore,
2005.
[2] H. Yukawa, Proc. Phys. Math. Soc., Japan 17, 48 (1935).
[3] For a brief early history of the discovery of the strange mesons see
H.S. Bridge, in Progress in Cosmic Ray Physics, North-Holland, Amsterdam, 1956.
[4] M. Gell-Mann, Phys. Rev. 92, 833 (1952);
T. Nakano and K. Nishijima, Progr. Theor. Phys. 10, 581 (1953).
[5] J.J. Sakurai, Phys. Rev. Letters 9, 472 (1962).
{FEYANMARP
[6] An example for such a model is
R.P. Feynman, M. Kislinger, and F. Ravndal, Phys. Rev. D 3 , 2706 (1971).
[7] Other quark models are discussed in
M. Danilov, R. Mizuk, (arXiv:0704.3531);
K. Carter, Symmetry Magazine, September 2006 (http://www.symmetrymagazine.org/
cms/?pid=1000377);
F. Buccella, P. Sorba, Mod. Phys. Lett. A 19, 1547 (2004);
F. Buccella, H. Hogaasen, J.M. Richard, and P. Sorba, Eur. Phys. J. C 49, 743 (2007).
[8] For details see M. Gell-Mann, D. Sharp, W.G. Wagner, Phys. Rev. Lett. 8, 261 (1962).
[9] G.E. Brown, R. Machleidt, Phys. Rev. 50, 1731 (1994).
[10] This booklet can be downloaded from the internet address http://pdg.lbl.
gov/2013/reviews/rpp2013-rev-su3-isoscalar-factors.pdf.
[11] J. Beringer at al. Phys. Rev. D 86, 010001 (20014) (http://pdg.lbl.gov/2013/tables/
rpp2013-sum-quarks.pdf).
H. Kleinert, PARTICLES AND QUANTUM FIELDS
{FEY}
{FEYANMARP