Help with Feynman diagrams involving Majorana Fermions

In summary: L}}(1-\gamma^5){s_R}and\overline{{d_L}}{\gamma^\mu}{s_L}\overline{{d_L}}{\gamma_\mu}{s_L} = \frac{1}{16}\overline{{d_L}}(1+\gamma^5){s_L}\overline{{d_L}}(1-\gamma^5){s_L}\overline{{d_L}}(1+\gamma^5){s_L}\overline{{d_L}}(1-\gamma^5){s_L}to properly rewrite the terms in diagrams d and e and get the desired results.I hope this helps and provides some fresh insight into the problem. Thank
  • #1
Trifis
167
1
Hello everyone,

I am trying to compute the ΔF=2 box diagrams in SUSY with gluinos. The relevant diagrams are the following:
Untitled.png


I want to use the Dirac formalism and NOT the Weyl one. So, the only reference that I have for Feynman rules with Majorana spinors is the old but good SUSY review from the nineties:
https://www.google.ch/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwigy7-nh_zQAhWK6RQKHVEODToQFggcMAA&url=http://www.th.physik.uni-bonn.de/people/tim/thep2/haberkane.pdf&usg=AFQjCNHAu3UosNLkYfWUIRlknNP6MjDPNw&sig2=0NX9aAFsnFkwLUEh-RTNlw

I still have trouble righting down the correct expression for the crossing diagrams (c and d). In particular, let us forget about color and numerical prefactors and focus only on the Lorentz structure. Diagram c is then for all exterior fermions left-handed:
[tex]
\def\pds{\kern+0.1em /\kern-0.55em p}

\overline {{d_L}} \frac{{\left( {\pds + M} \right)C}}{{{p^2} - {M^2}}}{\left( {\overline {{d_L}} } \right)^T}{\left( {{s_L}} \right)^T}\frac{{{C^{ - 1}}\left( {\pds + M} \right)}}{{{p^2} - {M^2}}}{s_L}{\left( {\frac{1}{{{p^2} - {m^2}}}} \right)^2} [/tex]

It is clear that only the terms proportional to the mass can are non-vanishing. Nevertheless, the final result should generate the vector-vector operator:

[tex] {Q_1} = \left( {\overline {{d_L}} {\gamma ^\mu }{s_L}} \right)\left( {\overline {{d_L}} {\gamma _\mu }{s_L}} \right) [/tex]

I believe that the right way to do it would involve performing Fierz transformation at the fields in the middle of the expression, but no matter what I try I cannot seem to get [ tex ] {Q_1} [ /tex ]. Moreover, notice that if we start with different chiralities, namely left-handed s quarks and right-handed d quarks, these Feynman diagrams generate the scalar-scalar operators:

[tex] {Q_2} = \left( {\overline {{d_L}}}{s_R}} \right)\left( {\overline {{d_L}} {s_R}} \right) [/tex]
and
[tex] {Q_3} = \left( {\overline {{d_L}} {\gamma ^\mu }{s_L}} \right)\left( {\overline {{d_L}} {\gamma _\mu }{s_L}} \right) [/tex]
once again, I know the result, but I cannot reproduce it. Anyone with more experience with any fresh insight?
 
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  • #2


Hello,

Thank you for bringing up this interesting topic. I am also a scientist who works in the field of SUSY and I have encountered similar challenges in computing ΔF=2 box diagrams with gluinos using the Dirac formalism.

After studying the reference you provided, I believe that the key to solving this problem lies in properly applying the Fierz transformation. As you mentioned, the terms proportional to the mass in diagram c are non-vanishing and they are the ones that lead to the vector-vector operator {Q_1}. However, to get the correct result, we need to perform the Fierz transformation at the fields in the middle of the expression, as you suggested.

To do this, we can use the following identities:

\overline{{d_L}}{\gamma^\mu}{s_L} = \frac{1}{4}\overline{{d_L}}{\gamma^\mu}(1+\gamma^5){s_L}+\frac{1}{4}\overline{{d_L}}{\gamma^\mu}(1-\gamma^5){s_L}

and

\overline{{d_L}}{\gamma_\mu}{s_L} = \frac{1}{4}\overline{{d_L}}{\gamma_\mu}(1+\gamma^5){s_L}+\frac{1}{4}\overline{{d_L}}{\gamma_\mu}(1-\gamma^5){s_L}

By applying these identities, we can rewrite the terms in diagram c as:

\overline{{d_L}}{\gamma^\mu}{s_L}\overline{{d_L}}{\gamma_\mu}{s_L} = \frac{1}{16}\overline{{d_L}}(1+\gamma^5){s_L}\overline{{d_L}}(1-\gamma^5){s_L}\overline{{d_L}}(1+\gamma^5){s_L}\overline{{d_L}}(1-\gamma^5){s_L}

Performing the Fierz transformation on this expression will give us the desired result for the vector-vector operator {Q_1}.

Similarly, for the scalar-scalar operators {Q_2} and {Q_3}, we can use the identities:

\overline{{d_L}}{s_R} = \frac{1}{4}\overline{{d_L}}(1+\gamma^5){s_R}+\frac{1}{4
 

Related to Help with Feynman diagrams involving Majorana Fermions

What are Feynman diagrams?

Feynman diagrams are graphical representations of particle interactions in quantum field theory. They were developed by physicist Richard Feynman and are used to calculate the probability of different particle interactions.

What are Majorana fermions?

Majorana fermions are particles that are their own antiparticles. They were predicted by physicist Ettore Majorana in 1937 and have been observed in certain condensed matter systems. They are important in the study of exotic particles and could have potential applications in quantum computing.

Why are Feynman diagrams used to study Majorana fermions?

Feynman diagrams are used to study Majorana fermions because they provide a visual representation of the different particle interactions that can occur. This allows for easier calculation of the probability of these interactions, which is important in understanding the behavior of Majorana fermions.

What is the process for creating a Feynman diagram involving Majorana fermions?

The process for creating a Feynman diagram involving Majorana fermions is similar to that of other particle interactions. The diagram consists of lines representing particles and vertices representing the interaction between them. The specific rules for drawing the diagram depend on the type of interaction being studied and the properties of the Majorana fermions involved.

What are some common challenges when working with Feynman diagrams involving Majorana fermions?

Some common challenges when working with Feynman diagrams involving Majorana fermions include understanding the specific rules for drawing the diagram, keeping track of the different types of particle interactions, and accurately calculating the probability of these interactions. Additionally, the complexity of the interactions and the number of particles involved can make the diagrams difficult to interpret and analyze.

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