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

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

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

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

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

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 )

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

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)

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

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

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

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

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

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

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

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

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

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.

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.

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  ´´

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    

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