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The Electromagnetic Structure of Hadrons Elastic scattering of spinless electrons by (pointlike) nuclei (Rutherford scattering) A A ZZ 1/q 2

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

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Form Factors Scattering on an extended charge distribution FF is the Fourier transform of the charge distribution ~ /q 2 for (r)= (r)

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

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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

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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 Q 2 and backward angles: Mott suppresses backward angles and the spinflip suppresses forward angles. (Dipole B~1/r 3 E~1/r 2 )

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Rosenbluth-Formula Due to their inner structure, nucleons have an anomales magnetic moment (g 2). p =+2.79 N n =-1.91 N (1:0 expected) Two form factors are now needed. At Q 2 =0 the form factors must equal the static electric and magnetic moments: G p E (0)=1, G p M (0)=2.79, G n E (0)=0, G n M (0)=-1.91

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Spacelike Proton Form Factors The form factors are determined the differential cross section versus tan /2 at different values of Q 2. 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)

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Neutron Electric Form Factor Even though the neutron is electrically neutral, it has a finite form factor at Q 2 >0 [G E (Q 2 =0)=0 is the charge] and thus has a rms electric radius =-0.11fm 2 Density distribution Similarly, G E S (Q 2 =0)=0 and G M S (Q 2 =0)= s

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

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Mean Charge Radius (II) Mean quadratic charge radius 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 -. Proton

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Virtual Photons Virtual particles do not fulfill the relationship: E 2 = m 2 c 4 + p 2 c 2 ( E t ~ ) ct x Feynman diagram for the elastic scattering of two electrons XaXa XbXb (4-Vectors) X = X b – X a

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Lorentz Invariant X 2 = (ct) 2 – x 2 = Const Timelike (ct) 2 – x 2 > 0 Lightlike (ct) 2 – x 2 = 0 Spacelike (ct) 2 – x 2 < 0 x ct ( P 2 = (E/c) 2 – p 2 = Const = q 2 ) Light Cone

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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 – p 2 < 0(E/c) 2 – p 2 > 0 Examples

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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

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Pion Form Factor Mean charge radius from the spacelike kinematic region. There is a kinematically forbidden region between 0 < q 2 < 4m 2 L.M. Barkov et al., Nucl. Phys. B256, 365 (1985). Timelike kinmatic region mixing

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

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Timelike Nucleon Form Factor Large kinematically for- bidden region from 0<q 2 <4M p 2, 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/r 2 potential (dipole) although it is 1/r for a single charge.

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Transition Form Factors Since the photon has negative C- parity it can not couple to pairs of neutral mesons (e.g ). But transitions are allowed where the products have opposite C parity. The decay of the off-shell photon is called internal conversion Dalitz decay 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.

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

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Transition Form Factors 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). e e

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Problem: Large Forbidden Region Near -Pole M max = M v –M = 0.65 GeV for -Dalitz and = 0.89 GeV for -Dalitz Meson has more decay phase space! But low cross sections and small branching ratios

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