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9.2. Bjorken scaling and the parton model
FIGURE 9.6
Photon–parton interaction in the Breit frame.
and x = Q
2 /2M ν finite, as Q
2 and ν grow large. We find
4π
2 α
σ T →
F 1 (x)
(9.54)
M K
and
σ S → (4π
2 α/M K)(1/2x)(F 2 − 2xF 1 )
(9.55)
where we have neglected a term of order M F 2 /ν in the last expression. Thus
the Callan–Gross relation corresponds to the result
σ S /σ T → 0
(9.56)
in terms of photon cross sections.
A parton calculation using point-like spin-0 partons shows the opposite
result, namely
σ T /σ S → 0.
(9.57)
Both these results may be understood by considering the helicities of partons
and photons in the so-called parton Breit or ‘brick-wall’ frame. The particular frame is the one in which the photon and parton are collinear and the
3-momentum of the parton is exactly reversed by the collision (see figure 9.6).
In this frame, the photon transfers no energy, only 3-momentum. The vanishing of transverse photon cross sections for scalar partons is now obvious.
The transverse photons bring in ±1 units of the z-component of angular momentum: spin-0 partons cannot absorb this. Thus only the scalar λ = 0 cross
section is non-zero. For spin1 partons the argument is slightly more compli2
cated in that it depends on the helicity properties of the γ μ coupling of the
parton to the photon. As is shown in problem 9.4, for massless spin1 particles
2
the γ μ coupling conserves helicity – i.e. the projection of spin along the direction of motion of the particle. Thus in the Breit frame, and neglecting parton
masses, conservation of helicity necessitates a change in the z-component of
the parton’s angular momentum by ±1 unit, thereby requiring the absorption of a transverse photon (figure 9.7). The Lorentz transformation from the
parton Breit frame to the ‘laboratory’ frame does not affect the ratio of transverse to longitudinal photons, if we neglect the parton transverse momenta.
9.2. Bjorken scaling and the parton model
FIGURE 9.6
Photon–parton interaction in the Breit frame.
and x = Q
2 /2M ν finite, as Q
2 and ν grow large. We find
4π
2 α
σ T →
F 1 (x)
(9.54)
M K
and
σ S → (4π
2 α/M K)(1/2x)(F 2 − 2xF 1 )
(9.55)
where we have neglected a term of order M F 2 /ν in the last expression. Thus
the Callan–Gross relation corresponds to the result
σ S /σ T → 0
(9.56)
in terms of photon cross sections.
A parton calculation using point-like spin-0 partons shows the opposite
result, namely
σ T /σ S → 0.
(9.57)
Both these results may be understood by considering the helicities of partons
and photons in the so-called parton Breit or ‘brick-wall’ frame. The particular frame is the one in which the photon and parton are collinear and the
3-momentum of the parton is exactly reversed by the collision (see figure 9.6).
In this frame, the photon transfers no energy, only 3-momentum. The vanishing of transverse photon cross sections for scalar partons is now obvious.
The transverse photons bring in ±1 units of the z-component of angular momentum: spin-0 partons cannot absorb this. Thus only the scalar λ = 0 cross
section is non-zero. For spin1 partons the argument is slightly more compli2
cated in that it depends on the helicity properties of the γ μ coupling of the
parton to the photon. As is shown in problem 9.4, for massless spin1 particles
2
the γ μ coupling conserves helicity – i.e. the projection of spin along the direction of motion of the particle. Thus in the Breit frame, and neglecting parton
masses, conservation of helicity necessitates a change in the z-component of
the parton’s angular momentum by ±1 unit, thereby requiring the absorption of a transverse photon (figure 9.7). The Lorentz transformation from the
parton Breit frame to the ‘laboratory’ frame does not affect the ratio of transverse to longitudinal photons, if we neglect the parton transverse momenta.