Help with Feynman diagrams involving Majorana Fermions

[Trifis]

Hello everyone,

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

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:
$$\def\pds{\kern+0.1em /\kern-0.55em p} \overline {{d_L}} \frac{{\left( {\pds + M} \right)C}}{{{p^2} - {M^2}}}{\left( {\overline {{d_L}} } \right)^T}{\left( {{s_L}} \right)^T}\frac{{{C^{ - 1}}\left( {\pds + M} \right)}}{{{p^2} - {M^2}}}{s_L}{\left( {\frac{1}{{{p^2} - {m^2}}}} \right)^2}$$

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:

$${Q_1} = \left( {\overline {{d_L}} {\gamma ^\mu }{s_L}} \right)\left( {\overline {{d_L}} {\gamma _\mu }{s_L}} \right)$$

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:

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

 

[AndyW]

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