Lectures on Polarized DIS by mifei


									Elastic Scattering in Electromagnetism
In this lecture we will practice some more with calculating cross sections, playing special attention to the differential cross section in the laboratory frame. In Lab frame

pb  (M ,0,0,0)

We assume that me<< M (a good assumption for the cases of physical interest) so that we can ignore me in what follows If

E  M

the target is able to absorb momentum without effectively changing its energy and one can neglect the effects of target recoil.
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In the LAB frame with (essentially) massless electrons we have

We will often use E, E’ and  L as our variables. However, note that they are not independent.

The last expression makes explicit the point that, when E  M , E  E '
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Next we need to derive the form of the differential cross section in the LAB frame. In the last lecture we had

Noting that the variable t has both an explicit cos L dependence and an implicit dependence through E’, we find that



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For comparison the CM result is The remaining issue is to evaluate the matrix element M . The simplest case, and the one studied first historically, is that of a scalar scattering on scalar, i.e., two spin 0 bosons scattering via the exchange of a photon.

The amplitude squared (no spin summing or averaging needed here)


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Thus the differential cross section looks like

The first factor in this last expression is called the “Rutherford cross section”, while the last factor is clearly a “recoil” correction, which becomes unity in the limit E  M
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The next step is to scatter “real” spin ½ electrons but still from a scalar target

Thus the spin averaged and summed squared amplitude becomes

Using the appropriate trace identity we find


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Thus the differential cross section can be expressed as


Note, that

d ( L   )  0 d L
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We recognize the first factor as our previous result for scalar-scalar scattering (the Rutherford result) and the last factor as a (slightly different) recoil correction. Historically the above result (modulo the recoil correction) was referred to as the Mott cross section. We can easily work out the form of this cross section in CM variables,

where we have used the relation

Note that the cross sections in the 2 reference frames have very similar structure.
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Recall that the standard fermion-QED vertex is helicity conserving.
1 2 1  R  (1   5 ) ;  R   2 1 (1   5 ) 2 1 (1   5 ) 2


 L  (1   5 ) ;  L  

Thus, if the incident electron is right handed, the scattered electron will be right handed also For backward scattering the exchanged photon must have J3 = +1, helicity +1, in order to conserve angular momentum at the electron vertex. Such a photon cannot be absorbed by a spin zero particle and conserve total angular momentum at the scalar vertex. The vanishing of the factor cos2  L 2 in this limit accounts for this fact.
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The next level of complexity is illustrated by the case of fermion-fermion scattering. Ignoring the mass of the electron

This is identical to the electron-scalar result except for the final term. In terms of LAB frame variables we have

Note, that in this case

d ( L   )  0 d L
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Compared to our previous result for an electron scattering from a scalar this result for scattering from another spin ½ particle exhibits a new possible type of process. In this case the spin of the “stationary” target particle can flip, a process not present in the scalar case.

This configuration allows the absorption of the helicity ±1 photons described above. It is this process that leads to the sin 4  l 2 term in the numerator, which dominates at  L   Note that this term vanishes in the no recoil limit where

E M 


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The final step of complexity is the elastic scattering of an electron from a proton, which is not a point particle. In general, the matrix  element M  J  ( t arg et )  J ( beam) , where
J   ( p  p' )  for pointlike scalars J   u (p' )  u(p) for poinlike fermions

Now, for objects with structure J   ( p  p' )  F(q 2 ) for scalars
q 2  q  q  , q  p  p '

four-momentum transfer

Object with extended charge distribution

d d pointlike 2 2   F q  d d
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In nonrelativistic (three-dimensional) case

F q 2    d 3 r e iqr  r  F ( 0)  1 d 3 r  r   1 
where (r) – charge distribution
(q  r ) 2 1 2 2 2 3 F (q )   d r (1  iq  r     )  (r )  1  q  r      2 6

Mean square charge radius:

 dF (q 2 )   r 2   6  2  dq q 2 0
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A conventional normalization for the electromagnetic current of the proton in relativistic case is

where k is the normalized anomalous magnetic moment of the proton, k  1.79 N and the form factors are normalized to have limits F1(0) = F2(0) = 1. The usual limits for a point Dirac particle are F1 = 1, F2 = 0 (all q 2 ). Other definitions often seen in the literature include “electric” and “magnetic” form factors

which describe the full complexity of the (separate) elastic electric and magnetic couplings of the proton.
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Using the above expression for the photon-proton coupling we find a LAB frame differential cross section of the form

In terms of the electric and magnetic form factors this is often written (the Rosenbluth form) in terms of the variable

The last term, spin flip arises from the magnetic interaction
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Measurement of the elastic electron-proton scattering
McAllister and Hofstadter 1956 - 188 MeV and 236 MeV electron beam from linear accelerator at Stanford Hydrogen and helium (gas) targets Magnetic spectrometer Measurements of scattering angle and energy
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Elastic profiles

Typical elastic profiles obtained in the scattering experiment with hydrogen gas

The elastic profiles were used to obtain cross-section of scattering the cross-section at a particular scattering angle is proportional to the numb of observed events at this angle (area under the curve)
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Energy estimates
Energy of scattered electrons as a function of scattering angle in the laboratory frame Energy was measured as peak value of the elastic profile (from spectrometer data) Theoretical curve was obtained using relativistic kinematics


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Cross-section of elastic scattering
a - Mott curve for spinless point-like proton b - Rosenbluth curve for a point-like proton with the Dirac magnetic moment (without anomalous magnetic moment) F1 (q2) = 1, F2 (q2) = 0 c - Rosenbluth curve with contribution from anomalous magnetic moment for point-like proton F1 (q2) = 1, F2 (q2) =  The deviation of experimental data from curve (c) were interpreted as an effect from proton form-factors - finite size proton
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The size of the proton
The experiment was not sensitive enough to the q2 dependence of the form factors. All that could be determined was a mean square radius of the nucleon. The best fit was achieved for <r2>1/2 = (0.70 ±0.24)10-13 cm. With 236 MeV electrons <r2>1/2 = (0.78±0.20)10-13cm. Combined result - <r2>1/2 = (0.74±0.24)10-13 cm.
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Sometime in literature another normalization is used:
Q2  2 4mN


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No free neutron: extract from e-D elastic scattering


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