Download Physics 557 – Lecture 8 Quantum numbers of the Standard Model

Physics 557 – Lecture 8
Quantum numbers of the Standard Model – (See Chapter 2 in Griffiths and Part B
and Chapter 15 in Rolnick.) We want to organize the experimental results of the last
70 years produced by the accelerators and experiments introduced in Lecture 7 in
terms of the “particle” content – the building blocks. To this end we will make
extensive use of the idea of quantum numbers and symmetries. Recall that in
quantum mechanics (and quantum field theory) we can characterize a complete set of
basis states in terms of their eigenvalues with respect to a maximal set of commuting
operators. Generally this set of commuting operators includes the full Hamiltonian,
i.e., the states have definite (real) energy so that the states do not decay and the
quantum numbers in question are conserved. For our purposes it is useful to expand
these ideas a bit (to allow decay). In particular we imagine the Hamiltonian as having
3 distinct parts
H ~ H strong  H EM  H weak .
(8.1)
For now we will ignore the role of both gravity and the Higgs sector. The former can
be ignored on the grounds that it is a very weak interaction and, in any case, we do
not have an adequate quantum description. The Higgs sector has yet to be established
experimentally (although there have been hints) and we will return to it later. The
separation into 3 sectors makes sense since the three interactions of the Standard
model have distinctly different “strengths”, i.e., gstrong ~ 1, gEM ~ 10-1, gweak ~ 10-7, and
preserve different quantum numbers (commute with different charges). Of course,
the second two sectors are intimately related through electro-weak unification and
symmetry-breaking but first we want to understand the organization of the observed
particles. From this standpoint the masses of the quarks and massive vector bosons
are determined by externally provided masses terms. The masses of the observed
hadrons (strongly interaction particles) are “explained” by Hstrong and in that sense we
think of Hstrong as the dominant member of this trio. The other two terms enter our
discussion essentially as the explanation for particle decays. (Of course, HEM is also
the source of atomic physics and thus of chemistry and life!). We will be interested
in those generators of symmetry operations that commute with Hstrong, but not
necessarily with HEM or Hweak.
The idea of resonances will play an important in our understanding of the structure of
the standard model. As discussed earlier, resonances correspond to bona fide states
of the underlying theory, which, however, are produced and decay via the strong
interaction. Thus their lifetimes (~ 10-23 s) are so short that these states are not
Lecture 8
1
Physics 557 Autumn 2012
detected as observable tracks in detectors. Rather their existence is detected via
enhanced interaction rates in specific channels, i.e., channels with specific quantum
numbers. They play an essential role in our counting of the possible degrees of
freedom and the confirmation of the Standard Model.
In contrast, “particles” are characterized by living long enough to be “seen” as tracks
(or at least displaced vertices) in detectors. However, it is still true that most of these
particles eventually decay (often inside of today’s large detectors) and it is precisely
by the analysis of these decays that we come to understand the identity of these
particles and the structure of the Standard Model. Note that only the electron, the
proton and the neutrinos are apparently free from decay and that this observation is
presumably of limited validity, i.e., limited by the quality of our experiments. If the
idea of Grand Unification is valid, there must be (extremely weak!) interactions (i.e.,
symmetry generators) that connect even the light quarks to the leptons and allow
protons to decay. Likewise, since at least some of the neutrinos are massive,
oscillations between different kinds of neutrinos are possible, and have been
observed.
The observed hierarchy of decay times is understood in terms of the conservation of
certain quantum numbers by some of the interactions but not by the others. In
particular, the strong interactions respect (commute with the charge operators of)
essentially all readily observable quantum numbers. The underlying operators of the
strong interaction operate on (i.e., do not commute with) the color charge, but for
distances > 1 fm we observe only color singlets. Thus the strong interactions cannot
contribute to processes that change any of the interesting quantum numbers (quark
flavors, electric charge, parity, etc.). On the other hand, the weak interaction
respects very few quantum numbers.
Once we have discussed the general properties of quantum numbers, we will proceed
to discuss the various particles observed, essentially in historical order, and describe
how the various relevant quantum numbers were introduced. We can think of the
quantum numbers as being the eigenvalues of some operator acting on the appropriate
“single” (or multi-) particle state (with the “” ’s to remind us that we will include
resonances in the counting of 1-particle states). Generally, for the case of an
eigenstate of an operator, we can write
Op X   XOp X .
Lecture 8
2
(8.2)
Physics 557 Autumn 2012
The quantum numbers of interest will generally fall into two classes – multiplicative
and additive. This distinction arises from the way in which the quantum numbers are
combined when we consider 2 (or more) particle states. Thus for multiplicative
quantum numbers we simply multiply the quantum numbers of the two particles
together to find the corresponding quantum number for the 2-particle state (ignoring
for now the impact of any interactions and the possible influence of the composite
state), 1  2  3: Op3 3  Op1  Op2 1 2 . Such quantum numbers are typically
associated with discrete symmetry groups (see Chapter 4 in Griffiths and Chapter 12
of Rolnick) and the typical values of  are 1. Examples are parity P, charge
conjugation C, the combination CP and time reversal T. The symmetry group for all
of these operations is isomorphic to the two-element group we discussed earlier. P,1
G : P-1 = P, P2 = 1. Additive quantum numbers, on the other hand, are, as the name
implies, additive, 1  2  3: Op3 3   Op1  Op2  1 2 . Examples are electric charge,
baryon number, lepton number, etc. In these cases the quantum numbers of
composite states are the scalar sums of the quantum numbers of the parts. More
generally, the addition of the quantum numbers may be of a vector nature or be even
more complex, as with SU(3) quantum numbers. Examples include momentum,
angular momentum, isospin, etc. The symmetry group associated with additive
quantum numbers is generally a continuous group, which may or may not be a local
symmetry.
Multiplicative Quantum Numbers (and discrete symmetries)
Parity, P: the operation of parity inverts all vectors
through the origin. We can think of the corresponding
passive operation as reflecting each of the unit vectors
through the origin.
Thus in the transformed frame the directions of all
(real) vectors are opposite, i.e., the signs of all components have changed by a factor
of -1:



r 
 r   r ,
P
  x, y, z     x,  y,  z  .
 x, y, z  
P
(8.3)
Thus a usual 3-vector is odd under P, i.e., has eigenvalue = –1. The usual scalar, e.g.,
the scalar product of two vectors
Lecture 8
3
Physics 557 Autumn 2012
 
s  r1  r2 

P
 


 
s  r1 r2   r1    r2   r1  r2  s,
(8.4)
is unchanged by P and thus has eigenvalue = +1. As we know there are quantities,
with which we are already familiar, that behave in a different manner – pseudovectors
(or axial vectors) and pseudoscalars. A good example of the former is angular
momentum, which involves two vectors:
  
L  r  p 

P
  


  
L   r   p    r   p  r  p   L.
(8.5)
Unlike a vector, a pseudovector has P eigenvalue +1. With 3 independent (ordinary)
vectors available we can construct a pseudoscalar using the familiar scalar triple
product, which transforms as
  
ps  r1   r2  r3  

P
  



ps  r1  r2  r3    r1    r2  r3    ps.
(8.6)
Unlike a true scalar, a pseudoscalar has P eigenvalue –1. (Note that this pseudoscalar
product is constructed with our old friend jkl.)
When we combine two (or more) particles, the parity of the resulting composite state
will depend on the product of the intrinsic parities of the two particles, times any
collective parity of the two particle state. This latter factor is easy to determine for
states of definite angular momentum, l. Such states are described by the spherical
harmonic functions Yl,m(,). In the transformed frame    = -,    = +
and it is easy to verify that
Yl ,m  ,   

P
Yl ,m  ,     Yl , m    ,       1 Yl ,m  ,   .
l
(8.7)
Thus the parity eigenvalue of the composite state of two particles with relative
angular momentum l and intrinsic parities P1 and P2 is the product
P
P P
composite
12   1 1  2 .
l
Lecture 8
4
(8.8)
Physics 557 Autumn 2012
An extremely useful application of this form is the parity eigenvalue for a state
composed of a spin ½ fermion and its anti-particle. It is straightforward to verify that
the structure of the Dirac equation requires that a fermion and its antiparticle have
opposite intrinsic parity (see page 141 in Griffiths and Eq. 12.8 and problem 12.1 in

Rolnick). In particular, as we will see in more detail later, if   t , r  is a solution of

0
the Dirac equation in the original frame, then    t , r  is a solution in the frame
reached after a parity transformation. Due to the opposite sign of 0 for upper
(particle) and lower (antiparticle) components, the intrinsic parity properties of the
two components are opposite. States formed of particle – antiparticle fermion pairs
with definite angular momentum, such as positronium (e+e- pairs) or mesons (quark –
antiquark pairs) must have parity
P f f pair,l   1
l 1
f f pair,l .
(8.9)
Thus we expect that a quark – antiquark pair in the ground state with l = 0 will exhibit
quantum numbers
 Pseudoscalar, P = -1, JP = 0-, 1S0 (spin = 0, l = 0, J = 0), the ’s, K’s and ;
 Vector, P = -1, JP = 1-, 3S1 (spin = 1, l = 0, J = 1), the ’s, K*’s and .
For bosons the particles and antiparticles have the same intrinsic parity and the
corresponding formula for a state composed of a scalar particle – antiparticle pair
does not have the extra (-1) factor of Eq. (8.9),
P b0 b0 pair,l   1 b0 b0 pair,l .
l
(8.10)
Interactions that respect parity, i.e., are invariant under a parity transformation, must
treat left-handed and right-handed particles in the same way. This characterization
applies to the strong and electromagnetic interactions but not the weak. As we have
already noted, the weak interactions involve left-handed fermions and right-handed
antifermions but not the inverse. (It is also interesting to note that biology seems to
give special roles to molecules of definite handedness.)
ASIDE: It is useful to note that the simple relationship between orbital angular
momentum and parity in Eqs. (8.9) and (8.10) obtains only for 2 particle states. We
will eventually be interested in 3 (and more) particle states where the relationship
Lecture 8
5
Physics 557 Autumn 2012
between the total angular momentum (i.e., how the state rotates) and parity (i.e.,
how that state looks after reflection through the origin) is more complex.
Charge Conjugation, C: The charge conjugation operation is a bit more subtle. It
does not operate in configuration space but rather changes a particle into its
antiparticle, i.e., it changes the sign of all of the additive quantum numbers describing
the particle: electric charge, weak charge, color charge, baryon number, lepton
number, etc. (Charge conjugation can also introduce an arbitrary, unobservable
 ei q , which is chosen by convention, but there exist different choices
phase, q 
C
for this same convention.) Thus not many (single) particles can be eigenstates of C,
i.e., be their own antiparticle. Clearly only particles with vanishing additive quantum
satisfy this requirement. The photon is the easiest example to identify. Since the
photon couples to a current of electric charge and that charge changes sign under C,
we find that a photon is odd under C
C     ; C n   1 n .
n
(8.11)
Likewise the fermion – antifermion states discussed above can also be eigenstates C,
the operator simply switches the fermion and antifermion labels in the spatial and

spin wave functions. Since the spin triplet state, S = 1, is symmetric ,


  
,  
2

under interchange of the two fermions while the singlet state, S = 0, is antisymmetric
    
S+1

 , the spin component of the wave function contributes a factor (-1) to the
2 

C eigenvalue of the composite state. The spatial symmetry of the state of definite
angular momentum is given by (-1)l. Finally there is an extra factor of (-1) from
Fermi statistics. The operator C exchanges the fermion and antifermion creation
operators in the definition of the state and we must switch their order back to return to
the original state,
a†f a†f 0 
 a†f a†f 0  a†f a†f 0 .
C
(8.12)
Since fermion operators anticommute, this introduces another factor of (-1). Pulling
these facts together we have
C f f pair,l   1
Lecture 8
6
l S
f f pair,l .
(8.13)
Physics 557 Autumn 2012
For example, the 0 composed of a quark-antiquark pair with l = 0, S = 0, obeys
C 0   0 .
(8.14)
So what about the s-wave 2-photon state into which we know the 0 decays? From
above the 2-photon state has C = +1 suggesting the (true!) result that the
electromagnetic interactions conserve C. This explains why 0   is not observed
(at least not at the level of 3 parts in 108). More generally for a particle-antiparticle
pair of bosons it is the states of even spin that are symmetric under interchange of the
two bosons (i.e., opposite to the behavior of the half-integer spin fermions – we will
check this below). At the same time, the creation operators commute rather than
anticommute. Thus there are two extra factors of (-1) compared to the fermion case
and the result for a particle-antiparticle boson pair is unchanged from the fermion
case (b here stands for a boson and not the bottom quark),
C bb pair,l   1
l S
bb pair,l .
(8.15)
You might also ask, what happened with the issue of parity for the 2-photon decay of
the 0? The product of the intrinsic parities is +1 and the
2-photon state must be symmetrical by Bose-Einstein
statistics, so where is the oddness under parity? It comes
from the polarizations of the photons. The picture of what
is happening at short distances is indicated in the figure,
where we are looking at the u-quark component of the


wave function. If the photons have polarizations vectors 1 and  2 and momenta


   
k1  k2 (with 1  k1   2  k2  0 ), the matrix element must be linear in each of the
polarizations (i.e., the figure shows one vertex for each photon), be a scalar under
rotations and be symmetric under the interchange of the two photons. The possible
forms (“What else can it be?”) for such a matrix element include both a true scalar
and a pseudoscalar
 
M s   1   2  ,
  
(8.16)
M ps  k1   1   2  .
The first form corresponds to the two photons having their polarizations aligned
while in the second the polarizations are orthogonal. In fact, it was the experimental
observation that the polarizations of the two photons are always orthogonal (as
required by the cross product) that confirmed the negative parity of the 0.
Lecture 8
7
Physics 557 Autumn 2012
Summarizing, both the strong interactions and the electromagnetic interactions
preserve C and P. Thus states of definite C and P will remain states of definite C and
P even if they experience strong or EM interactions. The same cannot be said for the
weak interactions. As we have already mentioned, the weak interactions are observed
to involve couplings to only left-handed neutrinos and right-handed antineutrinos.
This indicates that the weak interactions are symmetric under neither P (switching
left and right handedness) nor C (switching particle and antiparticle). In some sense
the violation of these two symmetries is maximal. We will discuss this in more detail
when we discuss the field theoretic structure of the weak interactions. On the other
hand, the joint operation CP would seem to be respected by the weak interactions
based on what we have said so far, i.e., a left-handed neutrino is equivalent to a righthanded antineutrino. This is true for the neutrinos (based on current measurements,
but watch for future measurements!) but, as we will see when we discuss the neutral
K-mesons (and the corresponding B-mesons), CP is, in fact, observed to be violated
at a very low level!
Time reversal, T: Above we considered what happens when we look at particles “in a
mirror” or switch all particles to antiparticles. Now consider the result of running the
movie of physics backwards. While thermodynamics clearly shows an “arrow of
time”, interactions of particles on the smallest scale exhibit considerable symmetry
with respect to time inversion, e.g., second-order equations of motion are invariant
under t  -t. Note that time inversion has an extra feature compared to spatial
inversion. While the later switches left and right, the former changes incoming states
into outgoing states and vice versa. If we define
T  
T
,
(8.17)
then time reversal invariance (the same physics forward and backward in time)
requires
T
 
T
     
.
*
(8.18)
The last step follows from the definition of such products of states, i.e., that the state
in the “bra” is complex conjugated. An operator with this property is called
antiunitary and it implies that
  a1 1  a2  2  T   a1*T 1  a2*T 2 ,
Lecture 8
8
Physics 557 Autumn 2012
(8.19)
i.e., the T operation takes the complex conjugate of any coefficients. The
representations of such antiunitary operators take the form of a product of operators,
UK, where K is the operator that takes the complex conjugate and U is a unitary
operator. The clear implication is that interactions that introduce complex numbers,
i.e., phases, can lead to T invariance violation. This is precisely what seems to be at
work in the weak interactions. It is interesting to note that the presence of a nontrivial phase in the quark mixing (MKS) matrix (allowing time reversal invariance
violation) is possible only because there are three (or more) generations. Maybe this
is the underlying reason for this number of generations – God wants T violation.
Actually we do too since, as far as we know, the product CPT is a good symmetry of
all interactions (it is true in all standard field theories) and we need CP (and thus T)
violation in order to understand the abundance of baryons (us) over antibaryons in the
universe (although the current understanding of the magnitude of CP violation is not
sufficient to explain the universe we live in).
Returning to less philosophical issues, we see that the operation of time inversion will


 r  but changes the sign of quantities that
leave spatial locations unchanged  r 
T

 
 



 -v, p 
 - p, L 
 r   p  L  . This last
involve single time derivatives  v 
T
T
T

 

  S , J 
 J  .
result applies also to all angular momenta, including spin,  S 
T
T
This suggests a typical experimental test of T invariance. Consider an electrically
neutral particle, e.g., the neutron or an atom. If the positive charges in the system are
slightly displaced from the negative charges, the system will still be electrically

neutral but exhibit a nonzero electric dipole p , proportional to the separation of the
charge. This dipole with interact with an electric field as
 
H ED  p  E.
(8.20)

Now we ask, in what direction can p point? Consider the neutron. There is only one

direction defined, the direction of the neutron’s spin, S . So by our usual, “what else


can it be?” argument, we know that pn must behave like S n and change sign under T.
Thus, since the electric field does not change under T, the electric dipole interaction
must change sign under T, and thus violate time invariance. (Actually, since the
electric field changes sign under P and the dipole does not, this interaction can also
signal P violation.) The current limit on the neutron electric dipole is < 0.29 x 10-25 e
cm. UW groups are involved in both the neutron measurement and measurements of
edm’s in atomic systems. On theoretical grounds we expect a neutron edm of order
Lecture 8
9
Physics 557 Autumn 2012
10-32 e cm in the context of the Standard model, while SUSY extensions can yield
much larger numbers (by factors of 104 to 108).
As noted earlier, the symmetries of CP and T are tightly coupled on theoretical
grounds. A quantum field theory that a) is Lorentz invariant, b) has a well defined
lowest energy state (the vacuum) and c) obeys microcausality (i.e., all field theories
of immediate interest) is CPT invariant. (This may not apply to all string theories.)
This result has several implications. Stated simply (a la Feynman) a forward moving
in time positive energy particle is the same as a backward moving in time, in the
opposite spatial direction, negative energy (i.e., anti-) particle. This implies that
particles and their antiparticles must have the same mass, the same decay width and
the same magnitude (but opposite direction) of magnetic moment. The measurement
of these properties is also a local specialty and literally “carved in stone” on the
outside of the physics building. The magnitude of the magnetic moment (“g-2”) of
the electron and positron are observed to be equal at the level of 2 parts in 1012 (and
to be in comparable agreement with the theory of QED).
Additive Quantum Numbers (and continuous symmetries)
Now let’s turn to the more familiar additive quantum numbers. The most familiar is
probably electric charge, associated with the U(1) symmetry of EM. That symmetry
ensures that the EM current is conserved
  J   0.
(8.21)
The electric charges of systems can change only when individual charges move
between systems. The charges of the elementary particles are unchanging. Further,
the total electric charge of a system is the algebraic sum of its charged components.
Electric charge is conserved by all interactions. A subtle but important issue
concerning electric charge is the question of why the charge of the proton is so well
matched to that of the electron : |Qp/Qe| = 1.0 to 1 part in 1021. This is a good
argument for the Grand Unification idea. In GUTS the lepton and quarks are in the
same representations of the larger unified symmetry. Since there are symmetry
generators that mix these states, all members of the representations must have the
same basic quantum of electrical charge.
Similar additive quantum numbers are baryon number B (and the related quark
number) and lepton number L. These numbers simply count the number of baryons
(quarks) and leptons minus the number of antibaryons (antiquarks) and antileptons.
Lecture 8
10
Physics 557 Autumn 2012
The three known interactions of the Standard Model conserve these quantum
numbers. Only the mesons, with zero baryon and quark number can decay entirely
into leptons and photons. At present we do not associate this conservation with any
underlying symmetry. If the idea of Grand Unification is true, only B-L is conserved
when we include the super-weak interactions of the larger symmetry. These
interactions allow baryons (quarks) to decay into leptons and protons do eventually
decay. Observationally the mode independent lifetime for the proton is bounded
below by about 2 x 1029 years. Mode dependent results go as high as 1033 years.
The conserved quark number is also usefully broken down into separate quark flavor
numbers, which are separately conserved by the strong (and electromagnetic)
interactions but not the weak interactions. Thus we characterize hadronic states by
the number of strange quarks, the number of charm quarks, the number of bottom
quarks and the number of top quarks. Of course, before quarks became “real” we
treated strangeness as a quantum number of the hadrons. (Note that the up and down
quarks are be handled separately by the concept of isospin.) There is a similar
generation dependent separate conserved lepton quantum number for the electron and
electron neutrino, muon and muon neutrino and tau and tau neutrino, Le, L and L,
respectively. The separate conservation of these numbers is violated by the neutrino
oscillation results.
We are also familiar with conserved vector quantum numbers like 3-momentum and
4-monentum, associated with translational invariance in space and time. Again
composite systems have momenta that are simple vector sums of the momenta of the
components. We learn in quantum mechanics, however, that the addition of angular
momentum is somewhat more complex. Instead of the infinite dimensional,
continuous representations that characterize the translation group, spin is quantized
and appears in discrete, finite dimensional representations of SU(2) (or SO(3) for
orbital angular momentum). This means that the group structure plays a role in the
addition process, i.e., the issue is that of adding these finite representations.
Let us a take a moment to review the subject of ladder operators and Clebsch-Gordan
coefficients as they help us to add spin. Consider the result of adding two states with


spins J1 and J 2 and third components of spin m1 and m2, i.e., the eigenvalues of Jz. We
know from the general rules of the addition of angular momentum that the resulting
 


total angular momentum will lie in the range || J1 |-| J 2 || to || J1 |+| J 2 ||. We work in the
orthonormal basis of the eigenstates of the total angular momentum operator and the
third component operator
Lecture 8
11
Physics 557 Autumn 2012

J 2 j, m  j  j  1 j, m , J z j, m  m j, m .
(8.22)
The states are normalized as
j, m j, m   mm .
(8.23)
Recall that the components of J satisfy the algebra
 J x , J y   iJ z , J   J x  iJ y ,  J z , J     J  .
(8.24)
It follows that the raising and lowering operators (or ladder operators) do just what
their names imply
J z J  j, m   m  1 J  j, m .
(8.25)
So we define the corresponding coefficients
J  j, m  C j, m  1 , J  j, m  1  C j, m
(8.26)
and solve
C   j , m  1 J  j , m , C  j , m J  j , m  1
 j, m  1 J  j, m
†
(8.27)
 C*  C  C ,
where the last step involves setting the unphysical phase of C+ to zero. We can
obtain an explicit expression for C by considering the operator

J  J   J x2  J y2  i  J x J y  J y J x   J 2  J z2  J z ,
(8.28)
J  J  j , m  C 2 j , m   j  j  1  m2  m  j , m .
(8.29)
which tells us that
Thus the Clebsch-Gordan coefficients are given by
Lecture 8
12
Physics 557 Autumn 2012
J  j, m 
j  j  1  m  m  1 j , m  1  J  j , j  0,
J  j, m 
j  j  1  m  m  1 j , m  1  J  j ,  j  0.
(8.30)
The two special results verify that the raising and lowering processes truncate at the
boundary of the representation and so the representation is of finite size.
Consider the simple example of combining spin 1 (3 states) with spin ½ (2 states).
We expect that the resulting 6 states will correspond to spin 3/2 (4 states) and spin ½
(2 states),
3  2  4  2.
(8.31)
The highest J and Jz states are unique and we can always start with one of them,
1,1 1 2 ,1 2  3 2 , 3 2 ,
(8.32)
1, 1 1 2 ,  1 2  3 2 ,  3 2 .
To find the remaining members of the spin 3/2 multiplet we can just apply J+ or J- to
the appropriate starting state (recall that J is additive, i.e., J TOT   J1  J 2 )
J 3   J 1
2
J  3 2,3 2 
 1  J 2  
3 5 3 1
   3 2,1 2  3 3 2,1 2 ,
2 2 2 2
 1  2  0   1 1, 0 1 2,1 2 
1 3 1  1
      1,1 1 2, 1 2
2 2 2  2
(8.33)
 2 1, 0 1 2,1 2  1,1 1 2, 1 2 
3 2,1 2 
2
1
1, 0 1 2,1 2 
1,1 1 2, 1 2 .
3
3
Similarly we find
3 2, 1 2 
1
2
1, 1 1 2,1 2 
1, 0 1 2, 1 2 .
3
3
Now what about the corresponding spin ½ state? It has the general form
Lecture 8
13
Physics 557 Autumn 2012
(8.34)
1 2,1 2   1,1 1 2, 1 2   1, 0 1 2,1 2 ,
(8.35)
where 2 + 2 = 1. We can proceed by requiring that this state be orthogonal to the
previous spin 3/2 state. Alternatively we can require that
J  1 2,1 2   1,1 1 2,1 2  2  1,1 1 2,1 2  0
    2    
(8.36)
2
1
,   .
3
3
Both procedures yield this same result with an overall sign ambiguity. As usual with
such issues, we fix the sign by convention. Here we use the Condon-Shortley
convention to agree with the usage in the PDG tables of Clebsch-Gordan coefficients.
Using the same order as in Eq.(8.33) to make clear the orthogonality we have
1 2,1 2  
1
2
1, 0 1 2,1 2 
1,1 1 2, 1 2 ,
3
3
2
1
1 2,  1 2  
1, 1 1 2 ,1 2 
1, 0 1 2, 1 2 .
3
3
(8.37)
The interested student can verify from the PDG table that, as noted above, when
adding identical integer spins (bosons) the resulting states with even spin are even
under interchange of the two initial spins, while the odd spins are odd.
It should be noted that similar techniques hold for larger symmetry groups, e.g.,
SU(3), and we will discuss them when they are needed. They will exhibit the added
complication that the representations are “higher dimension” than those for SU(2)
(SO(3)). For example, SU(3) has “2-D representations” (instead of 1-D) in the sense
that we will need two independent ladder operators to explore an entire representation
and a 2-D figure to display the representation (the issue is how many generators in
the algebra commute – the size, or the rank, of the Abelian subalgebra).
While the role of representations is familiar for transformations in configuration
space, it was a major step to recognize the usefulness of symmetry groups in more
abstract spaces. The first major application was to the concept of isospin (see
Chapter 4.3 in Griffiths and Chapter 6.1 in Rolnick). One of the obvious features of
nuclear physics is that the mass of nuclei depends primarily on the total number of
nucleons, A, and not separately on the numbers of protons, Z, and neutrons. Likewise
Lecture 8
14
Physics 557 Autumn 2012
the particles themselves are nearly degenerate, mp = 938.272 MeV while mn =
939.565 MeV (mn – mp = 1.293 MeV). The difference is small both compared to the
masses themselves and to the characteristic scale of nuclear physics interactions ~
100 MeV. These facts suggested that the nuclear (strong) interactions are invariant
under a transformation that interchanges protons and neutron. Such a 2-component
representation clearly suggests a similarity to ordinary spin and thus to SU(2)
symmetry. In the analysis above we simply replace spin S (or J) by isospin I. The
structure of states and matrix elements will be the same even though the spaces in
which the transformations act are different (i.e., we are rotating the axes in isospin
space rather than configuration space). (Remember the words of Richard Feynman,
“The same equations have the same solutions.”) Thus we have in isospin space
3
1
p , Iz p  p ;
4
2
3
1
n  1 2, 1 2 , I 2 n  n , I z p   n .
4
2
p  1 2,1 2 , I 2 p 
(8.38)
When we combine a neutron and proton we expect to find both an I = 1 state and an I
= 0 state. However, only the antisymmetric (in isospace) I = 0 np state of the
deuteron is bound, i.e., a stable nucleus. The symmetric pp, np, nn I = 1 states are not
bound. Similarly the u and d quarks form a doublet under isospin. In fact, the masses
of the u and d quarks are nearly degenerate (mu ~ 1.7 to 3.1 MeV, md ~ 4.1 to 5.7
MeV). The observed isospin symmetry of the non-strange hadrons arises from the
fact that mu and md are much smaller than the scale of the strong interactions (QCD ~
200 MeV) and not from their near equality. The quark mass difference, md > mu,
however, does help to explain the mass splitting mn > mp, which is very important to
our existence, i.e., so that the neutron, and not the proton, decays weakly. Combining
a quark and antiquark yields both isovectors (’s and ’s, spin 0 and 1, respectively)
and isoscalars (’s and ’s, again spin 0 and 1). (Note that the antiquarks are
members of doublets just like the quarks, i.e., an “anti-doublet” is just like a doublet
except for the issue of phases. This similarity of representations for “things” with
representations for “anti-things” will not carry over to larger symmetries.) For the 3
quarks in a baryon we combine 3 isospin ½ objects to find isospin 3/2, the ’s, and
isospin ½, the nucleons. The global SU(2) of isospin described here is the so-called
strong isospin (and there are no corresponding gauge bosons). The isospin
corresponding to the local SU(2) symmetry of the weak interactions is slightly
different (e.g., it has gauge bosons) and will be described later in a more detailed
discussion of the weak interactions.
Lecture 8
15
Physics 557 Autumn 2012
Note another difference from the application of SU(2) to (real) spin. In the latter case
it is easy to appreciate what we mean by a rotated reference frame or observer. In
isospace such an observer sees a mixed state of a neutron and a proton. This clearly
does not conserve electric charge (electromagnetic interactions violate isospin
invariance) but does make sense quantum mechanically. However, we will seldom
use such concepts. The real value of the idea of isospin invariance is the prediction
that hadrons must appear in complete (and approximately mass degenerate) multiplets
of SU(2) and that the strong interactions themselves should be isoscalars. This latter
constraint will provide information on relative couplings of different channels.
Consider the following processes
1) p  p  D    ,
(8.39)
2) p  n  D   0 ,
where D is the deuteron. As noted above the deuteron is I = 0 while the pion is I = 1.
Thus the final state is I = 1 for both reactions. On the other hand the initial state for
the first process is pure I = 1 while the second process has a 50/50 mixture of I = 1
and I = 0,
p n  n p
p n n p
1, 0 
; 0, 0 
.
(8.40)
2
2
The isospin invariance of the strong interactions says that the initial isospin and the
final isospin must be identical and thus only the I = 1 component of the initial state
can contribute. We have
D  H s pp
1

2
D o H s pn

1
2
2

I  1, I z  1 I  1, I z  1
I  1, I z  0
 I  1, I
z
2
 0  I  0, I z  0

2
2
(8.41)
2
2
 2,
1
2
as is observed experimentally.
Before proceeding to the discussion of the particles, we should introduce two more
quantities. First note that we can expect that there is some relationship between
isospin and electric charge. (It is already clear that the electric charge operator cannot
Lecture 8
16
Physics 557 Autumn 2012
commute with the isospin ladder operators.) Conventionally this relationship for the
electric charges of hadrons is expressed as (where I here is the strong isospin)
Q  Iz 
Y
.
2
(8.42)
We have introduced a new additive quantum, Y, the hypercharge, which will
eventually be associated with a local U(1) symmetry. Initially Y was observed to be
equal to the baryon number, Y = B, i.e., 1 for the proton and neutron and 0 for the
mesons. Thus the charge of the proton is ½ + ½ = 1, while the neutron is – ½ + ½ =
0. Likewise for the u and d quarks with baryon number 1/3 and hypercharge 1/6, we
have charges ½ + 1/6 = 2/3 and – ½ + 1/6 = - 1/3, respectively. For the mesons, the
electric charge is just the Iz value, e.g., +1, 0 and –1 for the three components of the
pion isovector. (When we include strangeness, we will find that Y = B + S, in the
(historical) context of strong isospin where strangeness is an isospin singlet. In the
modern context of quarks and the local SU(2) symmetry of the weak interactions,
where all quarks are in doublets, Y = B. The above expression about for Q stays the
same throughout.)
Finally the concept of G-parity, which combines isospin and charge conjugation, was
found to be useful in analyzing the strong interactions (see Chapter 4.4.2 in Griffiths
and Appendix G of Rolnick). In our discussion of C we noted that it was conserved
by the strong (and electromagnetic) interactions but that electrically charged states,
or, in fact, states with any nonzero value for one of the additive quantum numbers
cannot be eigenstates. G-parity was introduced to study states that have zero internal
scalar additive quantum numbers, except possibly Iz, i.e., states with “meson-like”
quantum numbers. Such states have Y = 0 and thus Q = Iz. Since C changes the sign
of Q, we have the operator statement
CI z   I z C.
(8.43)
As expected, C and I do not commute. Consider instead the G-parity operator
defined as
i I
G  Ce y ,
(8.44)
which first rotates a state by  (you will see both choices
+ and - in the literature – it does not matter) about the
Iy axis and then performs charge conjugation. Since the
two factors separately are conserved by the strong
Lecture 8
17
Physics 557 Autumn 2012
interactions, this is an invariance of the strong interactions. Due to the isospin
transformation, G-parity it is not an invariance of the electromagnetic (or weak)
interactions. We can understand the result of the isospin rotation by thinking about
what happens to spins under a rotation about the y-axis (remember the words of
Feynman),
 ,       ,     ,
(8.45)
i.e., the directions of both the x-axis and z-axis have flipped. Consider first neutral
states with Iz = 0. We know from our earlier discussion of parity (and the Ylm’s) that
the transformation above has the following impact on a state with Iz = 0 (Yl0 is
independent of ),
I
G I , 0   C  1 I , 0 ,
(8.46)
where C is the C quantum number of the state. So for our friend the 0, with C = +1
and I = 1, we find G = -1. Now, if Iz  0 for the state of interest, the isospin rotation
will flip the sign of Iz. Then C will just flip it back (recall Iz is the only nonzero
internal scalar additive quantum), adding at most an overall phase. This phase that
arises when C is applied to a charged state is undefined and can be chosen so that
entire isospin multiplet has the same G-parity quantum number,
G I , I z   C  1 I , I z ,
I
(8.47)
with C the same as for the Iz = 0 component (i.e., the eigenstate of C). Thus for the
pion multiplet, +, -, 0 all members have G = -1. In the restricted space of such
“nearly” neutral states we have

G, I   0
(8.48)
and G-parity is useful for analyzing strong decays between the mesons, as we shall
see. Note that, like the photons with the operator C, we have
G   1  , G n   1  .
n
(8.49)
You will sometimes see this information for the quantum number information of the
pion multiplet summarized as JPC IG = 0-+ 1-.
Lecture 8
18
Physics 557 Autumn 2012
Clearly, since we were really discussing quark-antiquark pairs above, we can also
apply G-parity to the nucleon-antinucleon channel. From our study of the C
properties of such states we have
G NN  l , S , I    1
lS I
NN  l , S , I  .
(8.50)
Combining our various discussions, we find a useful relationship for any pair of
identical particles, or for any two particles from the same isospin multiplet,
 1
l  S  I Y
2
 1.
(8.51)
Such states containing pairs of identical particles must satisfy the spin statistics
connection: the interchange of two identical fermions must produce an overall –1 (the
state is antisymmetric and no two identical fermions can be in the same state); the
interchange of two bosons must produce a +1 (the state is symmetric and bosons
“want” to all be in the same state). This result is encoded in the above formula. For
example, if we look at the two pion state with S = Y = 0, we must have l + I = even.
This means that odd angular momentum (the antisymmetric spatial wave function)
goes with odd isospin and even with even. This constraint guarantees that the isospin
wave function that is odd under interchange of the two pions goes with the spatial
wave function that is also odd and even isospin wave functions go with even spatial
wave functions. (For integer isospin the symmetry structure of the isospin wave
function is just like that for integer angular momentum – the same equations have the
same solutions.) The result is an overall wave function that is even under interchange
of the pions as required.
Lecture 8
19
Physics 557 Autumn 2012
We summarize the various symmetries of the three Standard Model interactions in the
following table.
Conservation Summary:
Conserved quantity
Additive
Energy-momentum
Electric charge
Baryon number
Lepton number
Angular Momentum
Isospin - I
Multiplicative
Parity - P
Charge Conjugation - C
Time Reversal – T or CP
CPT
G - parity
Lecture 8
Strong
Interaction
Electromagnetic
Weak
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
Yes
Yes
Yes
Yes
Yes
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
No
No
~ 10-3 viol
Yes
No
20
Physics 557 Autumn 2012