You would not really compute this for the weak interaction on its own. The weak interactions with Ws and Zs are residuals from the breaking of the electroweak gauge symmetry U(1)xSU(2). Above the scale of electroweak symmetry breaking, the running of the hypercharge U(1) and the SU(2)_L are separate and you can compute the beta functions of these based on the number of fermions and scalars transforming under different representations of the gauge symmetries. This should be described in any intermediate text on quantum field theory such as Peskin-Schröder.nigelscott said:How can I derive the running coupling for the weak interaction. I have found derivations for QED and QCD that involve the β function http://www.physics.umd.edu/courses/Phys741/xji/chapter1.pdf but I can't find anything specific for the WI. Thanks.
The running coupling for weak interaction is a mathematical concept that describes how the strength of the weak nuclear force changes at different energy scales. It is represented by the Greek letter alpha (α) and is a dimensionless quantity.
The running coupling for weak interaction is calculated through a process known as renormalization, which involves integrating out high-energy particles in order to obtain a more accurate description of the force at lower energies. This calculation is done using quantum field theory and is a highly complex and mathematically rigorous process.
The running coupling for weak interaction plays a crucial role in understanding the behavior of particles and their interactions at different energy scales. It helps to explain phenomena such as particle decay, radioactive decay, and the behavior of subatomic particles in high-energy collisions. Without a proper understanding of the running coupling, these phenomena cannot be accurately predicted or explained.
The running coupling for weak interaction is unique in that it changes significantly at different energy scales, unlike the running coupling for other fundamental forces such as electromagnetism and the strong nuclear force, which remain relatively constant. This is due to the fact that the weak nuclear force is carried by massive particles, whereas the other forces are carried by massless particles.
The running coupling for weak interaction can be experimentally measured through a variety of methods, including high-energy particle collisions and precision measurements of the properties of subatomic particles. These experiments can help to confirm the predictions of quantum field theory and provide insights into the behavior of the weak nuclear force at different energy scales.