1. Adnan Bashir, UMSNH, Mexico
QCD: A BRIDGE BETWEEN
PARTONS & HADRONS
Adnan Bashir, UMSNH, Mexico
November 2017
CINVESTAV

2. Hadron Physics & QCD Part 1: From Hadrons to Quarks:
Mesons & Baryons - A historical preview
Particles and quantum numbers
Symmetries and cross-sections
The Quark Model – The eightfold way
SU(3) and hadrons, SU(4), SU(5)
Chiral symmetry and its breaking
Pions as Goldstone bosons
Linear sigma model
Explicit symmetry breaking
Non-linear sigma model

3. Contents The SU(3) Group Quarks and Anti-quarks SU(3) and Mesons
SU(3) and Baryons
Masses and Symmetry Breaking
Gell-Mann Okubo Mass Formulae
Quark-Model Tested
The Color Quantum Number
The Charm and SU(4)
Beyond SU(4)
Pions and Chiral Symmetry Breaking

4. The SU(3) Group The group SU(3) has 32-1=8 generators.
The number of generators for SU(N) is N2-1.
The generators can be taken to be traceless and
Hermition which implies that the elements of the group
are unitary and have unit determinant.
Out of 8 generators, at most two are diagonal.
The number of commuting generators is equal to the rank
of the group. The rank of SU(N) is N-1. For SU(3), it
is 2.
There are 2 Casimir operators. For Lie groups, number of
Casmir operators is equal to the rank of the group.

5. The SU(3) Group Standard choice of the generators of the fundamental
representations are Fi= ½ i, i are Gell-Mann matrices.
3 and 8 are diagonal matrices. Note that 1, 2 and 3
generate the SU(2) group.

6. The SU(3) Group The properties of the SU(3) group are defined by the
commutator:
The structure constants fijk are completely anti-symmetric
in the indices.
Since [F3 ,F8]=0, we can label the states with the eigen-
values of F3 and F8.
The group elements of the of SU(3) are of the form:

7. The SU(3) Group As a basis for the fundamental representation in SU(3)
we choose the eigenstates of λ3 and λ8.
The eigenvalues of 3 and 8 are:

8. The SU(3) Group What can SU(3) possibly have to do with mesons and
baryons discovered so far?
Let us define the matrices for isospin and hypercharge as:
Then eigenvalues of T3 and Y are:

9. Quarks Due to the relation: charge Q of the three basis states is:
Let us also define the ladder operators:
For the fundamental representation of SU(2), T does:

10. Quarks We define: We thus have for all basis states, B:
Thus SU(3) triplet in isospin-hypercharge plane is:

11. Anti-Quarks
SU(3) anti-triplet in the isospin-hypercharge plane:

12. Anti-Quarks We choose the basis states in the anti-triplet space as:
Note that they should be distinguished from the
basis states in the triplet space.
The generators in the conjugate representation:
As the isospin and hypercharge of quarks and anti-quarks
are reversed while states are represented by same vectors
SU(3) algebra fixes the remaining matrices.

13. Anti-Quarks Thus in the anti-triplet representation:
And the isospin ladder operators are:
We thus have the actions of T3 and T:
We need anti-quarks to construct mesons

14. Mesons In the quark model, mesons are constructed from quark &
anti-quark states.
The corresponding multiplets are obtained by combining a
3 and a 3.
The baryon number of a meson is 1/3 – 1/3 =0.
The total spin is J=L+S. Since S=0,1, the mesons have
integer spin and hence are bosons.
Let us start with SU(2) where we just have u and d-quarks.
We can construct a meson with charge 1 by combining a
u and a d.

15. Mesons The isospin projection is given by:
Thus |u d> belongs to T=1
isotriplet (pions):
T=0 state is orthogonal to T=1 and T3=0 state:
T=1 states are transformed into each other through
isospin transformations. T=0 isospin state is a singlet.

16. SU(3) and Mesons
For SU(3) group, we have U and V spins in addition to the
isospin T:
More mesons are formed from quark anti-quark states:
T+ raises T3 by 1 & Y is unchanged
U+ lowers T3 by ½ & raises Y by 1
V+ raises T3 by ½ & raises Y by 1.

17. SU(3) and Mesons
Ground state
Mesons in SU(3)

18. SU(3) and Mesons
JP=1--Mesons
In SU(3)

19. SU(3) and Baryons
The irreducible representations for Baryons:

20. SU(3) and Baryons
The Eight Fold Way:

21. SU(3) and Baryons
J=3/2
baryon
decouplet:

22. Masses and Symmetry Breaking
The SU(2) symmetry:
Mn-Mp= MeV. Excellent isospin symmetry is
exhibited because up and down masses are nearly equal.

23. Masses and Symmetry Breaking
The mass differences along Y-axis ~ 200 MeV.
It is still much smaller than the masses themselves. So
we still have an SU(3) flavor symmetry though not as good
as SU(2) isospin symmetry.

24. Masses and Symmetry Breaking
The mass differences of baryons in J=3/2 decouplet.
Discovery of the Ω Nobel prize to Gell-Mann.

25. Gell-Mann Okubo Mass Formulae
We estimate the hadron masses in the quark model by
assuming that a part HI of the total Hamiltonian H0+HI
breaks SU(3) symmetry.
Let us assume that the isospin symmetry remains intact.
Thus mu=md. Assume that HI ~ Y.
Let us also assume that the binding energy of hadrons is
independent of quark flavor and the mass difference is
entirely due to quark mass differences.
Consider 0- meson multiplet:

26. Gell-Mann Okubo Mass Formulae
Note that we have used the quadratic mass for mesons.
It fits much better than linear mass and it is related to
chiral symmetry breaking in QCD. Thus
Experimentally: RHS ~ 0.92 GeV2, LHS ~ 0.98 GeV2.
Thus the relation is good to a few percent.
We can repeat the exercise for ½+ baryons:

27. Gell-Mann Okubo Mass Formulae
Experimentally, LHS is 2.23 GeV and RHS is 2.25 GeV.
We also have the relation:
For 3/2 baryon decouplet, one can also derive the equal
spacing relations:
This relation was used by Gell-Mann to predict the
existence, nature and mass of the  particle.

28. The Eightfold Way

29. The Eightfold Way Nobel Prize (1969):
Gell-Mann has also found that "The Eightfold Way" can be described very simply by assuming that all particles which interact strongly with each other are composed of only three kinds of particles which he called quarks and of the corresponding antiparticles. The quarks are peculiar in particular because their charges are fractions of the proton charge which according to all experience up to now is the indivisible elementary charge. It has not yet been possible to find individual quarks although they have been eagerly looked for. Gell-Mann's idea is none the less of great heuristic value.

30. The Quark Model – Tested
If the quark model were not correct, one could try to
look for exotic mesons (say of strangeness -3, like the
baryon Ω- or of charge +2 like the Δ++) or exotic
baryons (S=0 and Q=-2). None of these exotics were
found.
The failure to produce isolated quarks in experiments
made people very skeptical about the quark model.
Those who wanted to stick to the quark model invented
confinement of quarks.
Could the experiments like the Rutherford scattering
indicate the existence of quarks?

31. The Quark Model – Tested
The inelastic electron-nucleus scattering experiments
conducted between 1967 and 1973 at the Stanford
Linear Accelerator Center provided a key evidence for
the existence of quarks.
The leaders of these experiments were J. Friedman and
H. Kendall of MIT & R. Taylor of SLAC. They were
awarded the 1990 Nobel Prize in Physics.
The electrons of 20 GeV were scattered off the protons
in a deep inelastic scattering process.
The plot of cross-section versus invariant momentum
transfer to the proton Q2=2EE’(1-cosθ) showed that it
dropped much more slowly than the same quantity for
elastic scattering.

32. The Quark Model – Tested
Cross-section for inelastic
electron-proton scattering
measured at 6o in the first
MIT-SLAC experiment,
normalized by those expected
for Mott scattering. The data
points are given for two values
of W, the invariant mass of
the unobserved final state of
hadrons.

33. The Color Quantum Number
There was another objection to the Quark Model. It
seemed to violate Pauli exclusion principle.
The hadron wave-function can be decomposed into 3 parts,
the space, spin and isospin parts:
Let us consider the Δ++ whose quark content is uuu with
the spins aligned to give a total spin 3/2. So the spin-
isospin part is:
In the lowest state, space part must be symmetric too.
This is in contradiction with spin-statistics as quarks have
spin ½.

34. The Color Quantum Number
This problem was solved by the works of Greenberg (1964)
and Han & Nambu (1965).
A new quantum number color associated with a symmetry
group SU(3)c was proposed.
The quarks lie in the fundamental representation of this
group. They can have any of the three colors, say green,
red and blue.
As we can differentiate between the quarks through their
color, Pauli exclusion principle does not apply to them.

35. The Color Quantum Number
With the hypothesis of the color, the number of quarks
has increased from 3 to 9. However, the number of
observed hadrons remains the same!
It is because it is further assumed that all hadrons must
be color singlet. Thus Δ++ is a bound state of three up
quarks with different colors (ur, ug, ub). We never find
a Δ++ with (ur, ur, ub) for example.
The only colorless combibations of quarks that can be
made are q q, q q q or q q q.
Of course we can also have bound state of six quarks but
it can be interpreted as a bound state of two baryons.

36. The Charm and SU(4)
Bjorken and Glashow observed a mismatch between leptons
and quarks:
e, νe, μ, νμ
u, d, s
Demanding lepton quark symmetry, they expected the
existence of a new charm quark. (GIM)
A new meson J/ψ was discovered in It had a mass
3 GeV, 3 times as heavy as that of a proton.
It was electrically neutral and had a life-time of about
10-20 seconds while all other hadrons of that mass range
had a life time of about seconds. ?
It came out to be the bound state of quarks charm and
anti-charm, a charmonium. Richter and Ting were awarded
Nobel prize of 1976 for its discovery.

37. The Charm and SU(4)
The multiplets of S(4) mesons and baryons can readily be
constructed. The nonets of light mesons occupy the
central plane.

38. The Charm and SU(4)
Similarly, one can construct the SU(4) baryon multiplets

39. Beyond SU(4)
We now know there are six quark flavors: u,d,s,c,b,t

40. What Next? How do quarks interact with each other?
Do they exchange any particles like electrons interact
through the exchange of photons?
Can we develop a gauge theory of interactions between
quarks just as quantum electrodynamics?