The Quark & Bag Models

1. The Quark & Bag Models Simona Stoica KVI, September 17, 2008

2. Outline • The Quark Model • Original Quark Model • Additions to the Original Quark Model • How to form mesons and baryons • Color • Quantum Chromodynamics (QCD) • Color Charge • Quark confinement • M.I.T. Bag Model • Assumptions • Predictions • Failures of the MIT Bag model • Heavy quark spectra

3. The Quark Model • By the early 60’s there was a large zoo of particle found in bubble chamber experiments

4. Sorting them out • We could classify them by various quantum numbers • Mass • Spin • Parity • C parity • Isospin • Strangeness

5. First steps It was realized that even these new particles fit certain patterns: pions: p+(140 MeV) p-(140 MeV) po(135 MeV) kaons: k+(496 MeV) k-(496 MeV) ko(498 MeV) Þ If mass difference between proton neutrons, pions, and kaons is due to electromagnetism then how come: Mn > Mp and Mko > Mk+but Mp+ > Mpo Lots of models concocted to try to explain why these particles exist: Þ Model of Fermi and Yang (late 1940’s-early 50’s): pion is composed of nucleons and anti-nucleons (used SU(2) symmetry) note this model was proposed before discovery of anti-proton !

6. First steps With the discovery of new unstable particles (L, k) a new quantum number was invented: Þ strangeness Gell-Mann, Nakano, Nishijima realized that electric charge (Q) of all particles could be related to isospin (3rd component), Baryon number (B) and Strangeness (S): Q = I3 +(S + B)/2= I3 +Y/2 hypercharge (Y) = (S+B) Interesting patterns started to emerge when I3 was plotted vs. Y Y I3

7. Original Quark Model 1964 The model was proposed independently by Gell-Mann and Zweig Three fundamental building blocks 1960’s (p,n,l) Þ 1970’s (u,d,s) mesons are bound states of a of quark and anti-quark: Can make up "wave functions" by combing quarks: baryons are bound state of 3 quarks: proton = (uud), neutron = (udd), L= (uds) anti-baryons are bound states of 3 anti-quarks: Λ= (uds)

8. Quarks • These quark objects are: • point like • spin 1/2 fermions • parity = +1 (-1 for anti-quarks) • two quarks are in isospin doublet (u and d), s is an • iso-singlet (=0) • Obey Q = I3 +1/2(S+B) = I3 +Y/2 • Group Structure is SU(3) • For every quark there is an anti-quark • The anti-quark has opposite charge, baryon number and strangeness • Quarks feel all interactions (have mass, electric charge, etc)

9. Early 1960’s Quarks • Successes of 1960’s Quark Model: • Classify all known (in the early 1960’s) particles in terms of 3 building blocks • predict new particles (e.g. W-) • explain why certain particles don’t exist (e.g. baryons with spin 1) • explain mass splitting between meson and baryons • explain/predict magnetic moments of mesons and baryons • explain/predict scattering cross sections (e.g. spp/spp = 2/3) • Failures of the 1960's model: • No evidence for free quarks (fixed up by QCD) • Pauli principle violated (D++= (uuu) wave function is totally symmetric) (fixed up by color) • What holds quarks together in a proton ? (gluons! ) • How many different types of quarks exist ? (6?)

10. Additions to the Original Quark Model – Charm • Another quark was needed to account for some discrepancies between predictions of the model and experimental results • Charm would be conserved in strong and electromagnetic interactions, but not in weak interactions • In 1974, a new meson, the J/Ψ was discovered that was shown to be a charm quark and charm antiquark pair

11. More Additions – Top and Bottom • Discovery led to the need for a more elaborate quark model • This need led to the proposal of two new quarks • t – top (or truth) • b – bottom (or beauty) • Added quantum numbers of topness and bottomness • Verification • b quark was found in a  meson in 1977 • t quark was found in 1995 at Fermilab

12. Numbers of Particles • At the present, physicists believe the “building blocks” of matter are complete • Six quarks with their antiparticles • Six leptons with their antiparticles

13. Number of particles The additive quark quantum numbers are given below: Quantum # u d s c b t electric charge 2/3 -1/3 -1/3 2/3 -1/3 2/3 I3 1/2 -1/2 0 0 0 0 Strangeness 0 0 -1 0 0 0 Charm 0 0 0 1 0 0 bottom 0 0 0 0 -1 0 top 0 0 0 0 0 1 Baryon number 1/3 1/3 1/3 1/3 1/3 1/3 Lepton number 0 0 0 0 0 0

14. How to form mesons?

15. Baryons?

16. Color • Baryon decuplet (10) states consist of lowest mass J=3/2 states,  assume that the quarks are in the spatially symmetric ground state (=0) • To make J=3/2, the quark spins must be ‘parallel’ (ex) ++ = u u u • The ++ wave function is symmetric

17. Color • Pauli exclusion principle? • two or more identical fermions may not exist in the same quantum state • what about the u quarks in D++ ? It must be antisymmetric under Pauli principle! • More questions on the quark model

18. Color • Another internal degree of freedom was needed “COLOR” • Postulates • quarks exist in three colors: • hadrons built from quarks have net zero color (otherwise, color would be a measurable property) • We overcome the spin-statistics problem by dropping the concept of identical quarks; now distinguished by color ++ = uRuGuB

19. Color & strong interactions • We have assigned a “hidden” color quantum # to quarks. • “hidden” because detectable particles are all “colorless” • It solves the embarrassment of fermion statistics problem for otherwise successful quark model. • Most importantly, color is the charge of strong interactions

20. Quantum Chromodynamics (QCD) • QCD gave a new theory of how quarks interact with each other by means of color charge • The strong force between quarks is often called the color force • The strong force between quarks is carried by gluons • Gluons are massless particles • There are 8 gluons, all with color charge • When a quark emits or absorbs a gluon, its color changes

21. More About Color Charge • Like colors repel and unlike colors attract • Different colors attract, but not as strongly as a color and its opposite colors of quark and antiquark • The color force between color-neutral hadrons (like a proton and a neutron) is negligible at large separations • The strong color force between the constituent quarks does not exactly cancel at small separations • This residual strong force is the nuclear force that binds the protons and neutrons to form nuclei

22. Quantum Chromodynamics (QCD) • Asymptotic freedom • Quarks move quasi-free inside the nucleon • Perturbation theoretical tools can be applied in this regime • Quark confinement • No single free quark has been observed in experiments • Color force increases with increasing distance • Chiral symmetry

23. Quark confinement • Spatial confinement • Quarks cannot leave a certain region in space • String confinement • The attractive( color singlet) quark-antiquark • Color confinement • The quark propagator has no poles

24. M.I.T. Bag Model • Developed in 1974 at Massachusetts Institute of Technology • It models spatial confinement only • Quarks are forced by a fixed external pressure to move only inside a given spatial region • Quarks occupy single particle orbitals • The shape of the bag is spherical, if all the quarks are in ground state

25. M.I.T Bag Model • Inside the bag, quarks are allowed to move quasi-free. • An appropriate boundary condition at the bag surface guarantees that no quark can leave the bag • This implies that there are no quarks outside the bag

26. M.I.T. Bag Model • The boundary condition generates discrete energy eigenvalues. R - radius of the Bag x1=2.04 Nq = # of quarks inside the bag B – bag constant that reflects the bag pressure

27. M.I.T. Bag Model • Minimizing E(R), one gets the equilibrium radius of the system Fixing the only parameter of the model B, by fitting the mass of the nucleon to 938MeV we have first order predictions

28. One gluon exchange • Model so far excluded all interactions between the quarks • There should be some effective interaction that is not contained in B( how do we know that?) αs – the strong coupling constant Mq depends on the quantum no. of the coupled quarks

29. The Casimir Term • The zero point energy of the vacuum • The Casimir term improves the predictions of the MIT bag model. • However, theory suggests the term to be negative • Best fits provide a slightly positive value

30. Predictions The masses of N, Δ, Ω, ω were used to fit the parameters.

31. Quark confinement

32. Color confinement • The non-perturbative vacuum can be described by a color dielectric function k(r) that vanishes for r→∞. • The total energy Wc of the color electric field Ec of a color charge Qc is • Integral diverges, unless Qc=0

33. Failures of the Bag Model • Chiral symmetry is explicitly broken on the bag surface( static boundary condition) • Chiral extensions of the MIT-Bag model have been suggested: Cloudy bag model • Introduces a pion field that couples to the quarks at the surface.

34. Heavy quarks. Positronium Results • Positronium is an e+e- state that forms an “atom” • Two important decay modes • Two photon (singlet) • J=0 by Bose Symmetry • C=1 since C(photon)=-1 • Three photon • J=1 • C=-1

35. Postrionium Energy Levels • Can be done with non-relativistic Schrodinger equation & Coulomb Potential • Principal quantum number n=1,2,3… • Reduced mass • So result for positronium is

36. Relativistic Corrections • Spin-orbit couplings • Fine structure • Spin-spin couplings • Hyperfine structure • These interactions split levels into • Triplet (3S1) (orthopositronium) • Singlet (1S0) (parapositronium)

37. Positronium Levels S=1 L=1 n=2 S=0 L=0 S=1 S=0 S=1 n=1 L=0 S=0

38. Comparison with Charmonium

39. Why should these be similar? • Coulomb Potential has been shown before: mediated by massless photons • QCD has been found numerically to have a similar form

40. Conclusions • The quark model • classifies all known particles in terms of 6 building blocks • Explains mass splitting between meson and baryons • Explain/predict magnetic moments of mesons and baryons • Explain/predict scattering cross sections • The MIT Bag Model • predicts fairly accurate masses of the particles • Explains color confinement • Helps predict heavy quark spectrum Simple models can give us a very good picture!

41. Bibliography • Y. IWAMURA and Y. NOGAMI, IL NUOVO CIMENTO VOL. 89 A, N. 3(1985) • Peter HASENFRATZ and Julius KUTI, PHYSICS REPORTS (Section C of Physics Letters) 40, No. 2 (1978) 75-179. • T. Barnes, arXiv:hep-ph/0406327v1 • Carleton E. DeTar, John 12. Donoghue, Ann. Rev. Nucl. Part. Sci. (1983) • E. Eichten et al. , Phys. Rev. D, 203 (1980) • E. Eichten et al. , Phys. Rev. Lett, 369 (1975) • Stephan Hartmann, Models and Stories in Hadron Physics