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The Quark Model

The Quark Model. 1 . Costituents of Matter 2 . Fundamental Forces 3. Particle Detectors 4. Symmetries and Conservation Laws 5. Relativistic Kinematics 6. The Quark Model 7. The Weak Interaction 8. Introduction to the Standard Model.

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The Quark Model

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  1. The Quark Model 1. CostituentsofMatter 2. FundamentalForces 3. ParticleDetectors 4. Symmetries and ConservationLaws 5. RelativisticKinematics 6. The Quark Model 7. The WeakInteraction 8. Introduction to the Standard Model The Piedra del Sol, an aztec monolith, located a the Museo Nacional de Antropologia of Mexico City, is also called «TheTenochtitlan Stone". It has a circular shape with 3.6 m diameter and a weight of 25 tons. It was discovered in 1790 below the south side of the Main Square of Mexico City (the "Zocalo"). The Piedra del Sol has a strongly symbolic meaning, centered around the Sun figure, center of the Stone and of the Universe, Mediator between Mankind and the Heavens.

  2. Starting point: the discovery of many particles, both baryons and mesons. Regularities are interpreted in terms of combination of Quarks. The Quark hypotesis was put forward in 1964 by Gell-Mann and Zweig Classification based on regularities and an underlying (approximate) SU(3) symmetry Three quarks are used to classify all hadrons • From the dynamic viewpoint: • Parton Model • Deep Inelastic Scattering test • --> Partons = Quarks

  3. The Elementary Particles Zoo The rate of discovery of new particles increased dramatically after 1945 The proliferation concerned the strongly interacting particles (hadrons).

  4. Evidence of internal structure also from the magnetic moment of particles There is plenty of evidence for hadrons that are not fundamental

  5. How to classify strongly interacting particles ? • General idea of an underlying symmetry. • Subdivide strongly interacting particles (Hadrons) in Mesons (quark-antiquark states) and Baryons (3-quarks combinations). • States are grouped in isotopic spin multiples. • Different multiplets have different strangeness. • Isotopic spin multiples contain states that are equivalent with respect to the Strong Interaction. • Inside an isospin multiplet, different I(3) values correspond to different states (rotational symmetry in Isospin space). • Degeneracy in a multiplet is removed by the Electromagnetic Interaction. Proposal by Gell-Mann and Ne’emann (1961)  SU(3) as the symmetry group SU(3) flavor : three light quarks to explain all the observed hadronic states Mesons: 1 quark and 1 antiquark Baryons: 3 quarks (The «Eightfold Way»)

  6. The Baryon Decuplet The ten lowest lying baryon states having JP = 3/2+ 4-plet of Isospin 3-plet of Isospin S = -1 Doublet of Isospin S = -2 Singlet of Isospin S = -3

  7. The mass difference between states in I-spin multiplet members is of the order of a few MeV  this is typical of electromagnetic mass differences. I = 3/2 152 MeV I = 1 149 MeV I = 1/2 139 MeV I = 0 An additional s quark typically entails an increase of mass of about 145 MeV

  8. About the mass difference between an s quark and an u,d quark Reasonable values for the naked quark masses For a proton : For a Lambda: Assuming

  9. This Quark Model allowed to predict the existence of the Ω baryon, whose discovery took place in 1964 Production and decay of the Ω particle A cascade of weak decays Strangeness changes (from a multiplet to another multiplet) take place by means of the Weak Interaction

  10. Quark Spin and Color Baryon decuplet states are spin 3/2 Baryons of lowest mass, with no orbital angular momentum. Their spins are parallel, adding up to 3/2. These are states with a wave function which is parallel with respect to space (l=0), spin (parallel). Flavor can be symmetrized to account for particle indistinguishability (e.g. in case of uud): But, since they are fermions, every ψ must be antisymmetric For instance in the case: A new quantum number (the color) is necessary:

  11. More evidence for color: • The Pi-zero decay rate • The value of the cross section In the Pi-zero decay rate, color alters the axial quark current Let us now introduce the ratio R, by considering decays into hadrons or leptons starting from an e+e- initial state Behaviour of R as a function of the energyavailable in the center of mass : • Sensitivity to the totalnumber of colors • Sensitivity to the new kinematicallypossible quark production (c and b)

  12. Z peak C threshold B threshold The ratio displays resonances due to Vector Meson production (and the Z)

  13. Crossing the Charm Quark production threshold R value is sensitive to the number of quark states active at that energy

  14. Crossing the Beauty Quark production threshold

  15. The Baryon Octet The lowest lying eight baryonic states with JP = 1/2+ Isospin doublet S=0 Isospin tripletS = -1 Isospin singlet S = -1 Isospin doublet S = -2

  16. The actual particles: Proton Neutron I = 1/2 177 MeV I = 1 I = 0 202 MeV I = 1/2

  17. Mesons: general ideas Baryons are formed by 3 quarks and do have anti-multiplets (antibaryons) Mesons: a multiplet already contains quarks and antiquarks Meson families have di 32=9 stati Triplet states: J=1, paralleli spin, vector mesons Singlet states: J=0, antiparallel spins, pseudoscalar mesons With only u and d we can form:

  18. Isospin formalism in quark systems (in analogy with the angular momentum) raising & lowering Acting on quark states : Acting on two quark states: Therefore, for what concern particles :

  19. And similarly: These are the lowest mass combinations – the pseudoscalar mesons They are pseudoscalars, since parities of fermions and antifermions are opposite The last combination is a singlet : To be identified with the η (550) meson

  20. The presence of the s quark generates 32=9 states 8 1 Octet-singlet mixing:

  21. Pseudoscalar mesons Lowest lying mesons with JP=0- I = 1/2 I = 1 I = 0 I = 1/2

  22. The Vector Mesons They are mesons with l=0 and parallel spins (triplets): JP= 1- They also feature an octet-singlet mixing Singlet Octet Physical states are obtained with a rotation:

  23. The Vector Mesons The lowest lying mesons with mass JP=1- I = 1/2 I = 1 I = 0 I = 1/2

  24. Vector Mesons: they have the same quantum number of the photon Decays of vector mesons : Two possibilities: Zweig suppression

  25. Leptonic Decays of Vector Mesons They constitute a test of the quark composition of mesons Dilepton decays (Van Royen - Weisskopf) Since vector meson masses are similar, at high energies the following factors will be comparable :

  26. Drell-Yan process: a case study This is another process where the cross section depends on the charge of the quarks. Using the C-12 nucleus ,one has 18u+18d as the quark mixture Negative pion beam: u / anti-u annihilation Positive pion beam : d / anti-d annihilation Experimental result :

  27. Total pion-nucleon cross section at high energy Predictions of the Quark Model on the cross sections Under the assumption that one can incoherently sum the amplitudes of the scattering on constituent quarks Nucleon: made of three quarks Meson: composed by quark and antiquark So, the model predicts : And experiments say, at an energy of 60 GeV for the incoming beam :

  28. Hyperfine Interaction and masses • Mass differences in the Static Quark Model are due to: • Differences between bare masses of constituent quarks costituenti (an s substituting an u or a d) • Changes in the color binding energy • Hyperfine color interaction between quarks (e.g. decuplet-octet difference for baryons) • Hyperfine electromagnetic interaction between quarks Hyperfine interaction in the case of two fermions (electromagnetic) :

  29. This interaction is of the order of MeV ( it cannot explain the baryon octet-decuplet difference, for instance). However, the hyperfine color splitting is instead : The interaction depends on the spin state and it is different between the octet and the decuplet. In the case of two quarks:

  30. In the case of baryons, three quarks : Different sign for octet and decuplet For instance in the case N (spin ½) and ∆ (spin 3/2): Octet-decuplet mass difference in Baryons can be explained at the few % level by the hyperfine color splitting !

  31. In the case of mesons the correctionis more importantbecause : • In the hyperfine color splitting formula, a quark-antiquark termisbiggerthan a quark-quark term : • The factoristypicallybigger for mesons (radius of ~0.6 fm) than for baryons (radius of ~0.8 fm) Using these factors, also vector-pseudoscalar (spin1-spin 0) mass differences can be estimated reasonably well. These effects are generally a factor of two more important for mesons. Experimentally : 636 MeV

  32. More on masses • The mass of a hadron is composed by: • Color binding energy (Strong Interaction) • Of the order of a GeV in the case of Baryons • Mass of its constituents (the bare quark mass) • Introducing differences of order 150 MeV • Strong Interaction hyperfine term (How are the spins oriented? • Baryons: Decuplet.vs.Octet. Mesons: Vector.vs.Pseudoscalar ) • Difference of order 300 MeV for Baryons and 500 MeV for Mesons • Electromagnetic correction inside the same multiplet Physical origin and typical values ?

  33. For instance in the case of the Baryon Octet: So, electromagnetic mass differences are small. They are originated by two effects : 1. Coulomb Energy due to differenc charges of the quarks. Estimate of what happens when you have a different charge over a Fermi : 2. Electromagnetic hyperfine energy:

  34. Beyond the Octet Way. The fourth Quark. Electroweak Interactions and the GIM Mechanism. A little preview of the Cabibbo Mixing Theory

  35. Beyond the Octet Way. The fourth Quark. Electroweak Interactions and the GIM Mechanism. In 1970 Glashow, Iliopolous and Maiani (GIM) predicted the existence of a fourth quark: Charm. The prediction was based on the absence of strangeness-changing neutral currents. Here’s a puzzle ! The 3 quarks Neutral Current has the form : And the third quark enters in the (Cabibbo-rotated) combination

  36. And taking into account Cabibbo-angle mixing matrix style with just 2 flavors The introduction of the fourth quark removes the term driving strangeness-changing Neutral Currents (This is the GIM Mechanism)

  37. Heavy Quarks: Charm 1974: the «november revolution». The discovery of the J/ψ particle by two experiments: Brookhaven experiment: 28 GeV protons hitting a fixed target SLAC experiment: electron-positron collisions Final State Invariant Mass Distribution

  38. The observed width was dominated by the experimental resolution Intrinsic width obtained from the knowledge of the cross section and the branching ratio Resonance width: Γ= 0.093 MeV Lifetime of 10-20 s

  39. Final states: e+e-  hadrons, e+e-  e+e-, e+e  μ+μ-

  40. The J/ψ as an experimental «problem»: Hadronic resonances are normally WIDE since they decay by Strong Interaction and have very short lifetimes: As a comparison : How can a resonance be 100/1000 times smaller than usual and still be a strongly interacting particle ? To answerthisquestion, weshould first knowsomethingaboutotherparticlescontaining the charm quark.

  41. The J/Psi production energyrangeactuallyturns out to be rich in severalotherstructures

  42. The J/ψ contains a new quark, charm. Hidden charm Particles with «open» charm (C not zero) were discovered at SLAC in the following years : And now, a J/ψ decay like : Cannot take place because:

  43. 1 gluon? J/psi is colorless while a gluon is colored • 2 gluons? Angular momentum must be conserved For J/ψ-like states such that : 3-gluons decay Zweig suppression For J/ψ-like states such that: Excited states have enough mass to decay in charmed (having open charm) particles

  44. Quarkonium The J/ψ as quarkonium : a non relativistic state with a potential of the form Systems made by heavy quark-antiquark pairs have masses much higher than the Strong Interaction scale parameter (Λ ≈ 200 MeV). One can use the nonrelativistic Schoedinger Equation to study bound states :

  45. Onia systems !

  46. Charmonium system studies Study of the transitions between charmonium states require the ability to detect gammas (example: the Crystal Ball detector at Stanford). The Crystal Ball was then used at DESY for b-physics

  47. Charm Particles Lightest charmed mesons Main decay modes: c  s type, by means of Weak Interactions. For instance: Mesons with charm and strangeness: Typical decay, with cs: Charmed Baryons: Typical decay, with cs:

  48. Charm Physics: as an example, a classification of non-strange charmed baryons ! Different amount of charm are possible (3,2,1,0) and Weak Interactions makes it possible to change the Charm quantum number. Antibaryons with anticharm are of course there! The ordinary spin-3/2 baryon decuplet is just the first floor The ordinary spin-1/2 baryon octed is just the first floor

  49. Heavy Quark Physics : welcome to the complexity ! • Lifetimes of the order of 10-13 s (Weak Decays) • Many possible decay modes (each with a low branching ratio) • Complex Topology of the event • Not trivial to reconstruct Tracks in detectors downstream Tracks in detectors downstream

  50. Charm introduces an additional (flavor) degree of freedom in the particle classification scheme Before charm, just two families were known. It was also known that they entered in Weak Interaction through a rotation (Cabibbo angle) : Charm was predicted in 1970. Charm was discovered in 1974-1977. The Beauty quark was discovered in 1977, at about the same time as the Tau lepton. Third family First hints of a third family: the beauty quark Actually, the mixing involves all flavors (CKM Matrix, introduced in 1973)

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