W Boson Propagator in Unitary Gauge

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SUMMARY

The W boson propagator in the unitary gauge is defined by the formula: \frac{-i}{k^2-M^2} [g_{\mu\nu}- k_\mu k_\nu/ M^2 ] when \zeta approaches infinity. This transformation is derived from the original propagator expression \frac{-i}{k^2-M^2} [g_{\mu\nu}+ (\zeta-1) k_\mu k_\nu/(k^2 - \zeta M^2)] by dividing both the numerator and denominator by \zeta and taking the limit as \zeta tends to infinity. The discussion clarifies the mathematical steps involved in this limit process.

PREREQUISITES
  • Understanding of quantum field theory
  • Familiarity with the W boson and its properties
  • Knowledge of gauge theories, specifically unitary gauge
  • Proficiency in mathematical limits and calculus
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  • Study the derivation of the W boson propagator in different gauges
  • Explore the implications of gauge invariance in quantum field theories
  • Learn about the role of \zeta in propagator calculations
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Physicists, particularly those specializing in particle physics and quantum field theory, as well as students seeking to deepen their understanding of gauge theories and propagators.

Safinaz
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Hi there,

As in Ta-Pei Cheng and Li's book for example the W boson propagator is given by :

## \frac{-i}{k^2-M^2} [g_{\mu\nu}+ (\zeta-1) k_\mu k_\nu/(k^2 - \zeta M^2)] ##

At the unitary gauge ## \zeta = \infty ##, where the W propagator becomes :

## \frac{-i}{k^2-M^2} [g_{\mu\nu}- k_\mu k_\nu/ M^2 ] ##

It is not clear for me how when we set ## \zeta ## to ## \infty ##, we got the last formula in the unitary gauge ?

Bests.
 
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You can divide numerator and denominator by ζ and then simply take the limit of both.
 

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