QCD A BRIDGE BETWEEN PARTONS HADRONS Adnan Bashir

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

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

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

The SU(3) Group • The group SU(3) has 32 -1=8 generators. • The number of generators for SU(N) is N 2 -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.

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.

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 [F 3 , F 8]=0, we can label the states with the eigenvalues of F 3 and F 8. • The group elements of the of SU(3) are of the form:

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

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 T 3 and Y are:

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:

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

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

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.

Anti-Quarks • Thus in the anti-triplet representation: • And the isospin ladder operators are: • We thus have the actions of T 3 and T : • We need anti-quarks to construct mesons

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.

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 T 3=0 state: • T=1 states are transformed into each other through isospin transformations. T=0 isospin state is a singlet.

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+ U+ V+ raises T 3 by 1 & Y is unchanged lowers T 3 by ½ & raises Y by 1 raises T 3 by ½ & raises Y by 1.

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

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

SU(3) and Baryons • The irreducible representations for Baryons:

SU(3) and Baryons • The Eight Fold Way:

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

Masses and Symmetry Breaking • The SU(2) symmetry: • Mn-Mp= 1. 293 Me. V. Excellent isospin symmetry is exhibited because up and down masses are nearly equal.

Masses and Symmetry Breaking • The mass differences along Y-axis ~ 200 Me. V. • 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.

Masses and Symmetry Breaking • The mass differences of baryons in J=3/2 decouplet. • Discovery of the Ω. 1969 Nobel prize to Gell-Mann.

Gell-Mann Okubo Mass Formulae • We estimate the hadron masses in the quark model by assuming that a part HI of the total Hamiltonian H 0+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:

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 Ge. V 2, LHS ~ 0. 98 Ge. V 2. Thus the relation is good to a few percent. • We can repeat the exercise for ½+ baryons:

Gell-Mann Okubo Mass Formulae • Experimentally, LHS is 2. 23 Ge. V and RHS is 2. 25 Ge. V. • 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.

The Eightfold Way

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.

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?

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 Ge. V were scattered off the protons in a deep inelastic scattering process. • The plot of cross-section versus invariant momentum transfer to the proton Q 2=2 EE’(1 -cosθ) showed that it dropped much more slowly than the same quantity for elastic scattering.

The Quark Model – Tested Cross-section for inelastic electron-proton scattering measured at 6 o 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.

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 the spins aligned to give a total spin isospin part is: content is uuu with 3/2. So the spin- • In the lowest state, space part must be symmetric too. This is in contradiction with spin-statistics as quarks have spin ½.

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.

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, 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.

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 1974. It had a mass 3 Ge. V, 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 10 -23 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.

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.

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

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

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?