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The Electromagnetic Structure of Hadrons

The Electromagnetic Structure of Hadrons. Elastic scattering of spinless electrons by (pointlike) nuclei (Rutherford scattering). A. Z a.  a. 1/q 2. A. Mott Scattering. Suppression at backward angles for relativistic particles due to helicity conservation. Target recoil.

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The Electromagnetic Structure of Hadrons

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  1. The Electromagnetic Structure of Hadrons Elastic scattering of spinless electrons by (pointlike) nuclei (Rutherford scattering) A Za a 1/q2 A

  2. Mott Scattering Suppression at backward angles for relativistic particles due to helicity conservation Target recoil

  3. Form Factors Scattering on an extended charge distribution ~a/q2 for r(r)=d(r) FF is the Fourier transform of the charge distribution

  4. Special case: Pointlike charge distribu- tion has a constant FF

  5. Form Factors (an Afterword) Gauss´s theorm: V is a vector field Green´s theorm: if u and v are scalar functions we have the identies: Subtracting these and using Gauss´s theorm we have If u and v drop off fast enough, then The Fourier Transform interpretation is only valid for long wavelengths

  6. Elastic e- Scattering on the Nucleon There is a magnetic interaction with the nucleon due to its magnetic moment For spin ½ particles with no inner structure (Dirac particles) g=2 from Dirac Equation The relative strength of the magnetic interaction is largest at large Q2 and backward angles: Mott suppresses backward angles and the spinflip suppresses forward angles. (Dipole B~1/r3 E~1/r2)

  7. Rosenbluth-Formula Due to their inner structure, nucleons have an anomales magnetic moment (g2). mp=+2.79mN mn=-1.91mN (1:0 expected) Two form factors are now needed. At Q2=0 the form factors must equal the static electric and magnetic moments: GpE(0)=1, GpM(0)=2.79, GnE(0)=0, GnM(0)=-1.91

  8. Spacelike Proton Form Factors The form factors are determined the differential cross section versus tanQ/2 at different values of Q2. The form factors have dipole behavior (i.e. exponential charge distribution) with the same mean charge radius. (0.81 fm) (N.B. small deviations from dipole)

  9. Neutron Electric Form Factor Even though the neutron is electrically neutral, it has a finite form factor at Q2>0 [GE(Q2=0)=0 is the charge] and thus has a rms electric radius <r2>=-0.11fm2 Density distribution Similarly, GES(Q2=0)=0 and GMS(Q2=0)=ms

  10. Mean Charge Radius (I) FF is FT of charge distribution Inverse Fourier Transform Long wavelength approximation Taylor expansion

  11. Mean Charge Radius (II) Mean quadratic charge radius Proton FF measurements are difficult on the neutron (no n target!). Either do e- scattering on deuteron (but pn interaction!) or low energy neutrons from a reactor on atomic e-.

  12. Virtual Photons Virtual particles do not fulfill the relationship: E2 = m2c4 + p2c2 (DEDt ~) Feynman diagram for the elastic scattering of two electrons Xa Xb (4-Vectors) X = Xb – Xa ct x

  13. Light Cone Lorentz Invariant X2 = (ct)2 – x2 = Const Timelike (ct)2 – x2 > 0 Lightlike (ct)2 – x2 = 0 Spacelike (ct)2 – x2 < 0 ct x ( P2 = (E/c)2 – p2 = Const = q2 )

  14. Examples Spacelike: For elastic scattering momentum is transferred but energy is not (in CM) Timelike: For particle annihilation energy is transferred but momentum is not (in CM) (E/c)2 – p2 < 0 (E/c)2 – p2 > 0

  15. Vector Dominance Model (VDM) A photon can appear for a short time as a q qbar pair of the same quantum numbers. This state (vector meson) has a large probability to interact with another hadron. The intermediate state can be either space-like or time-like, where there is a large kinematically forbidden region

  16. Pion Form Factor Mean charge radius from the spacelike kinematic region. There is a kinematically forbidden region between 0 < q2 < 4mp2 Timelike kinmatic region r-w mixing L.M. Barkov et al., Nucl. Phys. B256, 365 (1985).

  17. Kaon Form Factor Contributions from r, w, and f are needed to explain the data mean charge radius = 0.58 fm (0.81 for proton)

  18. Timelike Nucleon Form Factor Large kinematically for- bidden region from 0<q2<4Mp2, exactly where the vector meson poles are. The interference from many vector mesons can produce a dipole FF, even though the BreitWigner is not a dipole. Similarly, 2 close el. charges of opposite sign have a 1/r2 potential (dipole) although it is 1/r for a single charge.

  19. Transition Form Factors Since the photon has negative C-parity it can not couple to pairs of neutral mesons (e.g gww). But transitions are allowed where the products have opposite C parity. Dalitz decay The decay of the off-shell photon is called internal conversion The ee spectrum can be separated into 2 parts: the 1st describes the coupling of the virtual photon to a point charge and the second describes the spatial distribution of the hadron.

  20. VDM and Transition FF VDM seems to work for some channels: p, (N), h, and h´ N h h´ mm used: although a smaller branching ratio, more at high invariant masses Max. background correction

  21. Transition Form Factors e+e-p0w wp0m+m- R.I.Dzhelyadin et al., Phys. Lett. B102, 296 (1981). V.P.Druzhinin et al., Preprint, INP84-93 Novosibirsk. L.G.Lansberg, Phys.Rep. 128, 301 (1985). F.Klingel,N.Kaiser,W.Weise,Z.Phys.A356,193 (1996).

  22. Problem: Large Forbidden Region Near r-Pole • Mgmax = Mv –Mp = 0.65 GeV for w-Dalitz and • = 0.89 GeV for f-Dalitz • Meson has more decay phase space! But low cross sections and small branching ratios

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