# Peskin Schroeder Problem 5.6 (b) Weyl spinors

• philipke
In summary, the solution to the problem is to use the Fierz identity to expand the first term of the amplitude and then apply it again to the second term to get the final expression.
philipke
Hey! I have a problem with problem 5.6 (b) from Peskin + Schroeder. Maybe I just don't see how it works, but I hope somebody can help me!

## Homework Statement

We are asked to calculate the amplitude for the annihilation of a positron electron pair into two photons in the high-energy limit. The high-energy limit is assumed by taking massless Weyl spinor for the electron and positron.

## Homework Equations

I get in my amplitude a factor that looks like this.

$$\bar{u_R}(p_2)\frac{-\gamma^{\nu}k_2\!\!\!/\gamma^{\mu} + 2\gamma^{\nu}p_1^{\mu}}{-2p_1k_2}u_R(p_1) = C$$

I call it C to use it below. I am supposed to apply the Fierz identiy (from problem 5.3)

$$\bar{u}_L(p_1)\gamma^{\mu}u_L(p_2)[\gamma_{\mu}]_{ab} = 2[u_L(p_2)\bar{u}_L(p_1) + u_R(p_1)\bar{u}_R(p_2)]_{ab}$$

I have not really an idea how to apply this identity on the first term in the factor of my amplitude. The complete amplitude looks like

$$A(\gamma_{\nu})*B(\gamma_{\mu})*(C + C(\mu <-> \nu$$ and $$k_1 <-> k_2))$$

where A and B are sums of bilinears of u_R and u_L (The polarization vectors from the problems's description)

The solution is to use the Fierz identity to expand the first term of your amplitude. The Fierz identity states that \bar{u}_L(p_1)\gamma^{\mu}u_L(p_2)[\gamma_{\mu}]_{ab} = 2[u_L(p_2)\bar{u}_L(p_1) + u_R(p_1)\bar{u}_R(p_2)]_{ab}So this means that the first term of your amplitude can be written as\bar{u_R}(p_2)\frac{-\gamma^{\nu}k_2\!\!\!/\gamma^{\mu} + 2\gamma^{\nu}p_1^{\mu}}{-2p_1k_2}u_R(p_1) = \frac{-2u_R(p_2)\bar{u_R}(p_1)\gamma^{\nu}k_2\!\!\!/ + 4u_R(p_2)\bar{u_L}(p_1)\gamma^{\nu}p_1^{\mu}}{-2p_1k_2} You can then apply the Fierz identity again to the second term in order to get your final expression.

## 1. What is the significance of the Weyl spinor in Peskin Schroeder Problem 5.6 (b)?

The Weyl spinor is a fundamental object in quantum field theory, representing a spin-1/2 particle. In Problem 5.6 (b), we are tasked with finding the explicit form of the Weyl spinor in terms of the Pauli matrices.

## 2. How does the Weyl spinor relate to the Dirac spinor?

The Weyl spinor is a half of the Dirac spinor, which describes a spin-1/2 particle with both left- and right-handed components. The Weyl spinor only contains one of these components, representing a particle with definite chirality.

## 3. What is the general form of the Weyl spinor in Problem 5.6 (b)?

In Problem 5.6 (b), we are considering a Weyl spinor with a fixed momentum in the z-direction. The general form of this spinor can be written as a linear combination of the two basis spinors, with coefficients determined by the particle's momentum and mass.

## 4. How does the Weyl spinor transform under Lorentz transformations?

The Weyl spinor transforms under Lorentz transformations in the same way as a spin-1/2 particle, with a rotation and a boost acting on its two components. However, unlike the Dirac spinor, the Weyl spinor is not invariant under a parity transformation.

## 5. What is the physical interpretation of the Weyl spinor?

The Weyl spinor describes a fundamental particle with half-integer spin, which is a crucial building block of our understanding of the universe. In particular, it is a key component in the Standard Model of particle physics, describing the behavior of fermions such as quarks and leptons.

• High Energy, Nuclear, Particle Physics
Replies
8
Views
3K
Replies
15
Views
2K
• High Energy, Nuclear, Particle Physics
Replies
3
Views
2K
Replies
1
Views
3K
Replies
1
Views
3K
Replies
1
Views
3K
• Quantum Physics
Replies
6
Views
1K
Replies
1
Views
1K
• High Energy, Nuclear, Particle Physics
Replies
7
Views
2K