Gauge Boson Propagators in Spontaneously Broken Gauge Theories

In summary, the propagator for gauge bosons in a spontaneously broken (non-abelian) gauge theory in the R_\xi gauge is given by \tilde{D}^{\mu\nu}_F(k)^{ab}=\frac{-i}{k^2-M^{ab}}\left[g^{\mu\nu}-(1-\xi)\frac{k^\mu k^\nu}{k^2-\xi M^{ab}}\right], where M^{ab} is the gauge boson mass matrix and \xi is the gauge fixing parameter. To simplify perturbative calculations, the mass matrix should be diagonalized and the propagator can be written in terms of the eigenvalues. It is possible to rational
  • #1
TriTertButoxy
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The propagator for gauge bosons in a spontaneously broken (non-abelian) gauge theory in the [itex]R_\xi[/itex] gauge is (see Peskin and Schroeder eqn. 21.53)

[tex]
\tilde{D}^{\mu\nu}_F(k)^{ab}=\frac{-i}{k^2-M^{ab}}\left[g^{\mu\nu}-(1-\xi)\frac{k^\mu k^\nu}{k^2-\xi M^{ab}}\right]\,,
[/tex]​

where [itex]M^{ab}[/itex] is the gauge boson mass matrix, and [itex]\xi[/itex] is the gauge fixing parameter. The matrices in the denominator should be interpreted as matrix inverses. To make perturbative calculations, I am supposed to diagonalize the mass matrix [itex]M^{ab}[/itex], and write the propagator in terms of the eigenvalues.

I would like to make my calculations as general as possible, and avoid having to go to a particular model to diagonalize the mass matrix. Is there a way to rationalize the propagator above so that the matrices are in the numerator?
 
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  • #2
I would like to write the propagator as\tilde{D}^{\mu\nu}_F(k)^{ab}=\frac{-i}{k^2}\left[g^{\mu\nu}+(1-\xi)\frac{M^{ab}}{k^2-\xi M^{ab}}k^\mu k^\nu\right]\,.If this is not possible, what is the best way to proceed? Is it just necessary to choose a particular model and diagonalize the mass matrix?
 
  • #3


Thank you for sharing this interesting topic. The propagator for gauge bosons in spontaneously broken gauge theories is a crucial concept in understanding the dynamics of these theories. In the R_\xi gauge, the propagator is given by the equation you mentioned, where M^{ab} is the gauge boson mass matrix and \xi is the gauge fixing parameter. This equation can be simplified by diagonalizing the mass matrix, which is a common approach in perturbative calculations.

However, as you mentioned, this approach requires us to go to a specific model to diagonalize the mass matrix. In order to make the calculations more general, it would be beneficial to have a rationalization of the propagator where the matrices are in the numerator instead of the denominator. Fortunately, there is a way to achieve this by using the concept of inverse matrices.

By using the property of matrix inverses, we can rewrite the denominator of the propagator as a numerator. This is achieved by multiplying both the numerator and denominator by the inverse of the mass matrix, denoted as (M^{ab})^{-1}. This results in a new propagator given by:

\tilde{D}^{\mu\nu}_F(k)^{ab}=\frac{-i}{k^2-M^{ab}}\left[g^{\mu\nu}-(1-\xi)\frac{(M^{ab})^{-1}k^\mu k^\nu}{1-\xi (M^{ab})^{-1}k^2}\right]\,.

This rationalized propagator still contains the mass matrix, but now it is in the numerator instead of the denominator. This allows for a more general approach to perturbative calculations, as we are not constrained by a specific model for diagonalizing the mass matrix. Instead, we can use any method to obtain the inverse of the mass matrix and use it in the propagator.

In summary, by using the concept of inverse matrices, we can rationalize the propagator for gauge bosons in spontaneously broken gauge theories and have the mass matrix in the numerator instead of the denominator. This allows for a more general approach to perturbative calculations and avoids the need to go to a specific model to diagonalize the mass matrix. I hope this explanation helps in your understanding of gauge boson propagators in spontaneously broken gauge theories.
 

What is a gauge boson propagator?

A gauge boson propagator is a mathematical description of the behavior of a gauge boson, which is a type of elementary particle that carries a fundamental force. In particle physics, gauge boson propagators are used to calculate the probability of interactions between particles mediated by gauge bosons.

How do gauge boson propagators work in spontaneously broken gauge theories?

In spontaneously broken gauge theories, the gauge boson propagator describes the behavior of gauge bosons in a system where the symmetry of the theory has been spontaneously broken. This means that the particles in the theory no longer exhibit the same symmetries as the equations that describe them, leading to different interactions and behaviors.

What are the main properties of gauge boson propagators?

The main properties of gauge boson propagators include their mass, spin, and polarization. The mass of a gauge boson propagator determines the strength of its interaction with other particles, while its spin determines its angular momentum. Polarization refers to the orientation of the gauge boson's spin with respect to its direction of motion.

Can gauge boson propagators be experimentally observed?

No, gauge boson propagators cannot be directly observed in experiments. They are purely mathematical constructs used to describe the behavior of gauge bosons, and cannot be directly measured. However, their effects can be observed through the interactions between particles that they mediate.

What are the practical applications of studying gauge boson propagators in spontaneously broken gauge theories?

Studying gauge boson propagators in spontaneously broken gauge theories is important for understanding the fundamental forces that govern the behavior of particles in the universe. This knowledge can have practical applications in fields such as particle physics, cosmology, and materials science. Additionally, it can help scientists develop more accurate theories and models to explain the behavior of particles and their interactions.

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