- #1

Trifis

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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:

[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?