The Electromagnetic Structure of HadronsElastic scattering of spinless electrons by (pointlike) nuclei (Rutherford scattering)

cos11

2sin2

1

8

2

2

22

McE

EE

pppq

Zq

M

p

p

s

Mgc

d

d

fi

if

i

fiff

A

A

Z1/q2

4

224

q

EZ

d

d

Rutherford

Mott ScatteringSuppression at backward angles for relativistic particles due to helicity conservation

22 sin1

1

E

E

d

d

d

d

ps

psh

RuthefordMott

Target recoil

Form FactorsScattering on an extended charge distribution

10

1

H

0uvvu

H

3/2

0

3

3/2int

/2

2/23

3/int

FrderconstqF

RdRrr

rderqV

e

eq

erd

rdreV

eM

rqi

rqifi

rqirqi

rqifiif

FF is the Fourier transform of the charge distribution

22

exp

qFd

d

d

d

Mott

~/q2 for (r)=(r)

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

Pointlikeexp

d

d

d

d

Form Factors (an Afterword)Gauss´s theorm:

VS

VddV V is a vector field

Green´s theorm: if u and v are scalar functions we have the identies:

uvuvuv

vuvuvu

Subtracting these and using Gauss´s theorm we have

SV

duvvuduvvu If u and v drop off fast enough, then

0S duvvu The Fourier Transform interpretation is only valid for long wavelengths

Elastic e- Scattering on the NucleonThere is a magnetic interaction with the nucleon due to its magnetic moment

For spin ½ particles with no inner structure (Dirac particles)

N

M

eg

22

g=2 from Dirac Equation

2

22

42tan21M

Qd

d

d

d

MottDirac

The relative strength of the magnetic interaction is largest at large Q2 and backward angles: Mott suppresses backward anglesand the spinflip suppresses forward angles.

(Dipole B~1/r3 E~1/r2)

Rosenbluth-FormulaDue to their inner structure, nucleons have an anomales magneticmoment (g2). p=+2.79N n=-1.91N (1:0 expected)Two form factors are now needed.

2tan2

1222

2222

QGQGQG

d

d

d

dM

ME

Mott

At Q2=0 the form factors must equal the static electric and magnetic moments:

GpE(0)=1, Gp

M(0)=2.79, GnE(0)=0, Gn

M(0)=-1.91

Spacelike Proton Form FactorsThe form factors are determined the differential cross section versus tan/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)

22

6

11 rqqF

Neutron Electric Form FactorEven 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 GM

S(Q2=0)=s

Mean Charge Radius (I)

drrrfq

drrrf

drrddRq

rf

rdRqi

nrfqF

Rq

eqFrdrf

rdrfeqF

n

n

rqi

rqi

0

42

2

0

2

0

21

1

2

0

2

3

0 0

2

/233

3/2

46

14

coscos

2

11

cos

!

1

1

2

1

FF is FT of charge distribution

Inverse Fourier Transform

Long wavelength approximation

Taylor expansion

Mean Charge Radius (II)

Mean quadratic charge radius

2

0

2

222

2

22

2

0

222

66.06

6

11

4

2

fmdq

qdFr

rqqF

drrrfrr

q

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 PhotonsVirtual particles do not fulfill the relationship:

E2 = m2c4 + p2c2 (Et ~ )

ct

x

Feynman diagram for the elastic scattering of two electrons Xa Xb

(4-Vectors) X = Xb – Xa

Lorentz Invariant

X2 = (ct)2 – x2 = Const

Timelike

(ct)2 – x2 > 0

Lightlike

(ct)2 – x2 = 0

Spacelike

(ct)2 – x2 < 0 x

ct

( P2 = (E/c)2 – p2 = Const = q2 )

Light Cone

Spacelike:For elastic scatteringmomentum is transferredbut energy is not (in CM)

Timelike:For particle annihilationenergy is transferred butmomentum is not (in CM)

(E/c)2 – p2 < 0 (E/c)2 – p2 > 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 kinematicallyforbidden region between0 < q2 < 4m

2

GeVq2

L.M. Barkov et al., Nucl. Phys. B256, 365 (1985).

Timelike kinmatic region

222

22

qimqm

mqF

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 from0<q2<4Mp

2, exactly where the vector meson poles are.

The interference from many vector mesons can producea 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.

Transition Form Factors

2223

222

222

22

2

22

221

2

2

2

0

41

121

41

3)(

)(

AB

AB

BA

A

BA

ee

f

qf

mm

qm

mm

q

qq

m

q

m

BAdq

eBeAd

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).

ee

22

2

2

2

3

40

MQs

sfpsee

23

22

44

s

mmsmmssp

Problem: Large Forbidden Region Near -Pole

Mmax = Mv –M= 0.65 GeV for -Dalitz and

= 0.89 GeV for -Dalitz Meson has more decay phase space!

But low cross sections andsmall branching ratios