High-energy behavior of non-Abelian Quantum
Field Gauge Theories
A well-known result regarding the scattering of monochromatic electromagnetic waves (in a classical setting) is
that the total scattering cross section approaches a constant as the frequency becomes higher and higher.
This
result follows from asymptotics on the scalar and vector Helmholtz PDEs, which form
reduced (time-independent) versions
of corresponding
wave equations in free space, with appropriate boundary conditions. This
behavior of the total scattering cross section characterizes, under reasonably general conditions,
other contexts of physical scattering in wave mechanics as well:
for example, it persists in
non-relativistic and relativistic quantum
mechanics,
where the starting points of a mathematical description are the Schrödinger and Dirac equations for particle wave functions.
A fundamental, yet vastly unexplored, question is: How does the total scattering cross section behave to leading order if the frequency
of the incident beam of light or particles, which we call a ``probe'', becomes so high (thus, their wavelength becomes so small)
that the probe starts seeing the subatomic structure
of the obstacle? At such high frequencies, standard wave mechanics
(in terms of
PDEs for wave motion) does not hold.
In fact, the appropriate description is furnished by Quantum Field Theories, where (roughly)
the wavefunctions of ordinary quantum mechanics must be replaced by operators in Hilbert spaces with arbitrary number of particles.
In this framework, particles can be created and destroyed, in compliance with
principles of relativity and quantization.
In the late 1960s and early 1970s, Cheng and Wu (C&W) answered the above question in the context of the Abelian Quantum Field Theory for
photons and electrons. Their surprising (at that time) result was that the total scattering cross section grows as the square
of the logarithm of
the
energy of the probe
to
leading order. Their study relied on a rather elaborate application of perturbation theory
at high frequencies. There are two small parameters: the coupling constant (proportional to the electron charge) of the Lagrangian and the wavelength (1/frequency) of
the probe. The perturbative calculations were finally
expressed in terms of an infinite number of
Feynman diagrams of a specific structure,
whose contributions were summed up.
To this date, experimental data of cross sections for
particle collisions are consistent with
the C&W
asymptotic result, predicting rising total cross sections in high-energy scattering.
See, for example the book
H. Cheng and T. T. Wu, ``Expanding Protons: Scattering at High Energies'', The MIT Press, 1987.
Part of my research, in collaboration with
T. T. Wu, has focused on the study of the high-energy behavior
of the SU(2)
Yang-Mills Quantum Field Theory. In this setting, the Lagrangian is invariant under gauge transformation of a non-Abelian (SU(2)) group.
Because of the ensuing, more intricate structure of the
Yang-Mills Lagrangian, the application of perturbation theory in the form of diagrammatic computations leads to great difficulties.
In fact, it turns
out that certain simplifications, which previously occured in the Abelian case by C&W, do not work in the
case of the
Yang-Mills theory;
a richer variety of Feynman
diagrams contribute to the leading order in the energy. A goal is to identify Feynman diagrams that contribute to leading order in the energy,
asymptotically evaluate each of
these diagrams
to arbitrary order in the coupling constant (Yang-Mills "charge"),
and sum them up.