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Understanding Quantum Chromo-Dynamics and Hadronic Physics in Strong Interactions

Explore the renormalizable non-abelian gauge theory of Quantum Chromo-Dynamics (QCD) and its implications, including the interplay of screening, antiscreening, and confinement. Study hadronic systems, such as the Quark Parton Model, through effective approaches induced by QCD. Learn about the historical origins and key concepts in hadronic physics, including Deep Inelastic Scattering and the Quark Parton Model.

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Understanding Quantum Chromo-Dynamics and Hadronic Physics in Strong Interactions

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  1. “Strong Interactions” Underlying field theory is a renormalizable non-abelian gauge theory named QuantumChromo-Dynamics (QCD) QCDLagrangian: with ψ Dirac field and Aμ vector field and a,b,c color indices, others understood covariant derivative (makes Dirac-vector field interaction locally gauge-invariant) non-abelian theory t = generators of gauge transformations f = fine structure constants

  2. QCD Lagrangian ⇒ trilinear and quadrilinear couplings Consequence? “Maxwell” eqs. for vector fields for  = 0 “Gauss” law for color charge adistributed with density a and generating color electric field Eai = Fa0i

  3. density from pointlike color charge a=1 vacuumfluctuation “sink” of field E 3 dipole charge a=1 pointing to source getting away from source the color charge a=1 looks stronger!antiscreening

  4. QED : screening QCD : screening + antiscreening (≫) confinement ? asymptotic freedom (only non-abelian 4-dim. gauge field theories display it) ~ ΛQCD

  5. Regimes ΛQCD ≪ : perturbative regime, calculable with techniques mutuated from QED ΛQCD ≲ : non-perturbative regime, not directly calculable Structure of hadrons realized at scale ~ ΛQCD hadrons cannot be deduced directly from the Lagrangian describing the forces that make them ! Alternative: compute QCD on lattice statistical approach, no direct access to dynamics Hadronic Physics : study hadronic systems using effective approaches induced by QCD ✦ spectroscopy ✦ dynamical structure ☜

  6. Historical origin End of ‘60’s: famous SLAC experiment of Deep Inelastic Scattering (DIS) on proton targets at 7 ≲ Q2 ≲ 10 (GeV/c)2 and 6o < e < 10o ☛ scaling = the target response does no longer depend on momentum transferred ☛ isolated events of diffusion at very large angles ☛ the proton behaves like an ensemble of pointlike scattering centers, each one moving independently from the others ☛ birth of the Quark Parton Model (QPM) Bloom et al., P.R.L. 23 (69) 930 Feynman, P.R.L. 23 (69) 1415 Friedman, Kendall,Taylor 1990 NOBEL laureates

  7. Scattering lepton -- hadron (electron, neutrino, muon) (nucleon, nucleus, photon) • Quantum ElectroDynamics (QED) known at any order • leptonic probe explores the whole target volume • em ~ fine structure constant is small → perturbative expansion • Born approximation (exchange of one photon only) works well • virtual photon (* ): (q,) independent, two different * polarizations • (longitudinal and transverse) → two different target responses 3 independent 4-vectors k, k’, P + spin S e scattering angle prototype reaction e+p -> e’+X

  8. definitions and kinematics e- ultrarelativistic me≪ |k|, |k’| Target Rest Frame (TRF) kinematical invariants

  9. (cont’ed) elastic limit final invariant mass anelastic limit

  10. Q is our “lense” N.B. 1 fm = (200 MeV)-1

  11. Frois, Nucl. Phys. A434 (’85) 57c nucleus MA nucleon M unaccessible domain

  12. Cross Section no events per unit time, scattering center, solid angle no incident particles per unit time, area J  flux phase space scattering amplitude

  13. Leptonic and Hadronic Tensors 2 J  = leptonic tensor hadronic tensor

  14. Inclusive Scattering hadronic tensor  X cross section for inclusive scattering (general formula) large angles are suppressed !

  15. Inclusive Elastic Scattering W ’=(P+q)2=M2 hadronic tensor ↔Q : scaling various cases

  16. target = free scalar particle 2 independent 4-vectors: R=P+P ’, q=P-P ’ ⇒ J » F1 R + F2 q  F1,2(q2,P 2,P ’2) = F1,2 (q2) current conservation qJ  = 0 ⇒ define : N.B. for on-shell particles q•R = 0 ; but in general for off-shell

  17. Inclusive elastic scattering on free scalar target elastic Coulomb scattering on pointlike target target recoil target structure

  18. Breit frame → target form factor P = - q/2 R = (2E, 0) q = ( 0, q) = 0 J = (J0, 0) ≈ (2E F1(Q2), 0) P’ = + q/2 F1(Q2) → F1(|q|2) = ∫ dr(r) e iq∙r form factor for charge matter ….. distribution of charge matter ….. works only in the non-relativistic limit in fact, ρ is a static density, while Breit frame changes with Q2=q2 : boost makes | P’=+q/2 > ≠ | P=−q/2 > ⇒ density interpretation is lost

  19. target = pointlike free Dirac particle Example: e- + -→ e-’+ - spin magnetic interaction with*

  20. target = free Dirac particle with structure 3 independent 4-vectors P, P ’,  (+ parity and time-reversal invariance) current conservation qJ = 0 Dirac eq.

  21. Gordon Decomposition (on-shell) • proof flow-chart • from right handside, insert def. of  • use Dirac eq. • use {,} = 2 g • use Dirac eq. → left handside namely R ⇔ 2M – iq

  22. target = free Dirac particle with structure …… cross section internal structure (not easy to extract)

  23. Rosenbluth formula Define Sachs form factors (Yennie, 1957) N.B. reason: in Breit frame + non-rel. reduction → charge / magnetic target distribution easier to handle

  24. Rosenbluth separation • at large e (large Q2) → extract GM • small e (small Q2) → extract GE by difference • Rosenbluth plot linear transverse polarization of * measurements at different (E, e) → plot in  at fixed Q2 crossing at =0 → GM slope in → GE

  25. Rosenbluth plot pQCD scaling JLAB data (obtained with more precise e- N → e- N ) Q2~ 10 (GeV/c)2not yet perturbative regime?? → →

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