Post on 18-Dec-2015
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)
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
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