6-point NMHV Gluon Amplitude Using BCFW Recursion Diagrams

Introduction

The BCFW (Britto-Cachazo-Feng-Witten) recursion is a powerful tool for calculating scattering amplitudes in gauge theories, particularly in the context of Yang-Mills theory. It provides a systematic way to decompose the amplitude into simpler building blocks, known as recursion diagrams, which can be evaluated using spinor-helicity formalism. In this article, we will focus on the 6-point NMHV (next-to-maximally helicity-violating) gluon amplitude using BCFW recursion diagrams.

BCFW Recursion and Spinor-Helicity Formalism

The BCFW recursion is based on the idea of introducing a shift in the momenta of two particles in the amplitude, which allows us to rewrite the amplitude in terms of a sum of simpler amplitudes. This shift is typically denoted as

$$\lambda_{\mu} \rightarrow \lambda_{\mu} + z \lambda_{\nu}$$

where $\lambda_{\mu}$ and $\lambda_{\nu}$ are the spinors associated with the two particles being shifted, and $z$ is a complex parameter. The spinor-helicity formalism is a powerful tool for evaluating the amplitudes resulting from this shift, as it allows us to express the amplitudes in terms of spinor products and helicity labels.

6-point NMHV Gluon Amplitude

The 6-point NMHV gluon amplitude is a particularly interesting case, as it involves a non-trivial combination of helicity labels and spinor products. The amplitude can be written as

$$\mathcal{A}_6 = \sum_{i=1}^6 \frac{\mathcal{A}_{i-1,i+1}}{\langle i-1,i+1 \rangle}$$

where $\mathcal{A}_{i-1,i+1}$ is the amplitude for the $(i-1)$-th and $(i+1)$-th particles, and $\langle i-1,i+1 \rangle$ is the spinor product associated with the $(i-1)$-th and $(i+1)$-th particles.

Valid Recursion Diagrams

To determine which recursion diagrams are valid, we need to consider the following conditions:

• Kinematic constraints: The momenta of the particles must satisfy the kinematic constraints of the amplitude.
• Spinor product constraints: The spinor products associated with the particles must satisfy certain constraints, such as the vanishing of the spinor product for certain pairs of particles.
• Helicity constraints: The helicity labels of the particles must satisfy certain constraints, such as the conservation of helicity.

Invalid Recursion Diagrams

There are several types of recursion diagrams that are invalid, including:

• Degenerate diagrams: These are diagrams where the spinor product associated with a pair of particles vanishes.
• Non-physical diagrams: These are diagrams where the helicity labels of the particles do not satisfy the kinematic constraints.
• Overlapping diagrams: These are diagrams where the spinor products associated with two pairs of particles overlap.

Example of a Valid Recursion Diagram

Consider the following diagram for the 6-point NMHV gluon amplitude:

$$\mathcal{A}_6 = \frac{\mathcal{A}_{1,3}}{\langle 1,3 \rangle} + \frac{\mathcal{A}_{2,4}}{\langle 2,4 \rangle} + \frac{\mathcal{A}_{3,5}}{\langle 3,5 \rangle} + \frac{\mathcal{A}_{4,6}}{\langle 4,6 \rangle}$$

This diagram is valid because it satisfies the kinematic constraints, spinor product constraints, and helicity constraints.

Conclusion

In conclusion, the 6-point NMHV gluon amplitude using BCFW recursion diagrams is a complex and non-trivial case that requires careful consideration of the kinematic constraints, spinor product constraints, and helicity constraints. By understanding the valid and invalid recursion diagrams, we can systematically evaluate the amplitude and gain insights into the underlying physics of the theory.

References

• Britto, R., Cachazo, F., Feng, B., & Witten, E. (2005). Direct proof of the tree-level scattering amplitude recursion relation in Yang-Mills theory. Physical Review Letters, 95(22), 221601.
• Cachazo, F., & Svrcek, P. (2004). MHV vertices and tree amplitudes in gauge theory. Journal of High Energy Physics, 2004(9), 006.
• Elvang, H., & Huang, Y. T. (2013). Scattering amplitudes. Journal of Physics A: Mathematical and Theoretical, 46(19), 193001.

Appendix

A.1 Spinor Products

The spinor products are defined as

$$\langle i,j \rangle = \lambda_i \cdot \tilde{\lambda}_j$$

where $\lambda_i$ and $\tilde{\lambda}_j$ are the spinors associated with the $i$-th and $j$-th particles, respectively.

A.2 Helicity Labels

The helicity labels are defined as

$$\epsilon_i = \frac{\lambda_i \cdot p_i}{\langle i,i+1 \rangle}$$

where $p_i$ is the momentum of the $i$-th particle.

A.3 Kinematic Constraints

The kinematic constraints are defined as

$$\sum_{i=1}^6 p_i = 0$$

where $p_i$ is the momentum of the $i$-th particle.

A.4 Spinor Product Constraints

The spinor product constraints are defined as

$$\langle i,j \rangle = 0 \quad \text{for} \quad i \neq j$$

where $\langle i,j \rangle$ is the spinor product associated with the $i$-th and $j$-th particles.

A.5 Helicity Constraints

The helicity constraints are defined as

$$\sum_{i=1}^6 \epsilon_i = 0$$

$$$$

Q: What is the BCFW recursion and how does it relate to the 6-point NMHV gluon amplitude?

A: The BCFW recursion is a powerful tool for calculating scattering amplitudes in gauge theories, particularly in the context of Yang-Mills theory. It provides a systematic way to decompose the amplitude into simpler building blocks, known as recursion diagrams, which can be evaluated using spinor-helicity formalism. The 6-point NMHV gluon amplitude is a particularly interesting case, as it involves a non-trivial combination of helicity labels and spinor products.

Q: What are the kinematic constraints for the 6-point NMHV gluon amplitude?

A: The kinematic constraints for the 6-point NMHV gluon amplitude are defined as

$$\sum_{i=1}^6 p_i = 0$$

where $p_i$ is the momentum of the $i$-th particle.

Q: What are the spinor product constraints for the 6-point NMHV gluon amplitude?

A: The spinor product constraints for the 6-point NMHV gluon amplitude are defined as

$$\langle i,j \rangle = 0 \quad \text{for} \quad i \neq j$$

where $\langle i,j \rangle$ is the spinor product associated with the $i$-th and $j$-th particles.

Q: What are the helicity constraints for the 6-point NMHV gluon amplitude?

A: The helicity constraints for the 6-point NMHV gluon amplitude are defined as

$$\sum_{i=1}^6 \epsilon_i = 0$$

where $\epsilon_i$ is the helicity label of the $i$-th particle.

Q: How do I determine which recursion diagrams are valid for the 6-point NMHV gluon amplitude?

A: To determine which recursion diagrams are valid, you need to consider the following conditions:

• Kinematic constraints: The momenta of the particles must satisfy the kinematic constraints of the amplitude.
• Spinor product constraints: The spinor products associated with the particles must satisfy certain constraints, such as the vanishing of the spinor product for certain pairs of particles.
• Helicity constraints: The helicity labels of the particles must satisfy certain constraints, such as the conservation of helicity.

Q: What are the different types of recursion diagrams that are invalid for the 6-point NMHV gluon amplitude?

A: There are several types of recursion diagrams that are invalid, including:

• Degenerate diagrams: These are diagrams where the spinor product associated with a pair of particles vanishes.
• Non-physical diagrams: These are diagrams where the helicity labels of the particles do not satisfy the kinematic constraints.
• Overlapping diagrams: These are diagrams where the spinor products associated with two pairs of particles overlap.

Q: Can you provide an example of a valid recursion diagram for the 6-point NMHV gluon amplitude?

A Consider the following diagram for the 6-point NMHV gluon amplitude:

$$\mathcal{A}_6 = \frac{\mathcal{A}_{1,3}}{\langle 1,3 \rangle} + \frac{\mathcal{A}_{2,4}}{\langle 2,4 \rangle} + \frac{\mathcal{A}_{3,5}}{\langle 3,5 \rangle} + \frac{\mathcal{A}_{4,6}}{\langle 4,6 \rangle}$$

This diagram is valid because it satisfies the kinematic constraints, spinor product constraints, and helicity constraints.

Q: How can I use the BCFW recursion to evaluate the 6-point NMHV gluon amplitude?

A: To evaluate the 6-point NMHV gluon amplitude using the BCFW recursion, you need to:

1. Choose a pair of particles: Choose a pair of particles to shift, such as particles 1 and 2.
2. Introduce a shift: Introduce a shift in the momenta of the chosen pair of particles, such as

$$\lambda_{1} \rightarrow \lambda_{1} + z \lambda_{2}$$

3. Evaluate the amplitude: Evaluate the amplitude for the shifted particles using spinor-helicity formalism.
4. Repeat the process: Repeat the process for different pairs of particles to obtain the final amplitude.

Q: What are the benefits of using the BCFW recursion to evaluate the 6-point NMHV gluon amplitude?

A: The benefits of using the BCFW recursion to evaluate the 6-point NMHV gluon amplitude include:

• Systematic evaluation: The BCFW recursion provides a systematic way to evaluate the amplitude, which can be challenging to do manually.
• Improved accuracy: The BCFW recursion can improve the accuracy of the amplitude by reducing the number of terms and simplifying the expressions.
• Increased efficiency: The BCFW recursion can increase the efficiency of the evaluation process by reducing the number of calculations required.

Q: What are the limitations of using the BCFW recursion to evaluate the 6-point NMHV gluon amplitude?

A: The limitations of using the BCFW recursion to evaluate the 6-point NMHV gluon amplitude include:

• Complexity: The BCFW recursion can be complex to implement, particularly for higher-point amplitudes.
• Numerical instability: The BCFW recursion can be numerically unstable, particularly for amplitudes with large momentum transfers.
• Limited applicability: The BCFW recursion is limited to amplitudes with a specific structure, such as the 6-point NMHV gluon amplitude.