More than 50 years ago experiments proved that the interaction of a hard
photon with a proton is about the same as its interaction with a neutron.
Furthermore, the intensity of a photon-nucleon interaction is much stronger
than that of a photon interacting with the nucleon charge constituents. A
theoretical concept called Vector Meson Dominance (VMD) has been used for
providing an explanation of the data.
The expression hadronic structure of the photon is a
generalization of VMD.
(For reading a review article, see [1].)
Herein, VMD is used as a shorthand also for the
concepts called Vector Dominance Models (VDM) and the idea called
hadronic structure of the photon.
VMD argues that a physical photon
is a linear combination of a pure electromagnetic photon and a hadron
(see [1] or [2], pp. 296-304)
|φ> = a0|φ0> +
ah|φh>,
where the 0,h subscripts denote the pure electromagnetic and the hadronic
components of the photon, respectively, and the numerical coefficients
satisfy
a02 + ah2 = 1.
VMD argues that the hadronic component of a low energy photon practically
vanishes whereas a photon whose energy corresponds to the mass of a vector
meson has a significant hadronic component (see the foregoing
review article [1] or [2], pp. 296-304).
The following argument proves that VMD contradicts Special Relativity.
Light rays intersection
Consider two intersecting rays of light (see the figure) which are
emitted from S1 and S2,
respectively. The energy of the optical photons
of these rays is E0 = 2 eV.
The position of the rays' source is at x= ±1; y=0, respectively,
and the figure is embedded in the (x,y) plane.
The energy of the optical photons is extremely smaller than that
of the lightest meson.
Therefore, according to VMD
ah ≅ 0.
For this reason photons of the two rays do not
interact with each other. Let
Σ0 denote the inertial frame where the
light sources are at rest.
Hence, in Σ0
all photons continue moving in their original direction.
Now consider another inertial
frame Σ that moves relativistically in the negative direction of
the y-axis. Let γ denote the relativistic factor of Σ,
β2 = γ2-1,
and s = sin α, where α is the angle between
the light rays and the x-axis in frame Σ0.
In frame Σ the photon energy
is
E = E0 (γ + βs).
Evidently, one can fix the velocity of the inertial frame Σ so that
E = 770 MeV, which corresponds to the
ρ meson energy. It follows that if VMD is correct then
in the inertial frame Σ the hadronic component of the photons
cannot be ignored. Hence, some of the photons are expected to interact
strongly at the intersection point O
and to elastically
scatter away from their original direction. This outcome contradicts
the result which has been found in frame Σ0.
This analysis proves that VMD is inconsistent with Special Relativity. As
a matter of fact, the discussion presented herein is just an illustration
of Wigner's general analysis of the irreducible representations of the
inhomogeneous Lorentz group [3,4]. According to his results, a massless
particle (like the photon) and a massive particle (like a hadron)
belong to two different categories. It means that no particle
can be represented as a linear combination of a massless photon
and a massive particle. Hence, VMD violates Special Relativity.
Conclusion:
The Standard Model has no consistent explanation for the interaction of a hard
photon with nucleons.
For reading a full article which discusses this issue,
click here.
References
[1] T. H. Bauer, R. D. Spital, D. R. Yennie and F. M. Pipkin,
Rev. Mod. Phys.50, 261 (1978).
[2] H. Frauenfelder and E. M. Henley, Subatomic Physics (Prentice Hall,
Upper Saddle River, New Jersey, 1991).
[3]
E. P. Wigner, Annals of Math., 40, 149 (1939).
[4] S. S. Schweber, An Introduction to Relativistic
Quantum Field Theory, (Harper & Row, New York, 1964). Pp. 44-53.
