Contribution to beta-function from wave-function renormalisation

[najimuddin]

Hi,
I have tried to get the RGEs of R-parity violating couplings in the presence of extra field TH. We have defined
SuperFields[[8]] = {TH, 3, conj[THR], 1/3, 1, -3} and written the Lagrangian as,

SuperPotential = Yu u.q.Hu - Yd d.q.Hd - Ye e.l.Hd + \[Mu] Hu.Hd + \[Epsilon] l.Hu + L1/2 l.l.e + L2 l.q.d + L3/2 u.d.d + LPPH/2 u.d.TH + LPH l.q.TH;

I am getting the extra contribution to the beta-function of L3[i1, i2, i3] as

"sum[j3, 1, 3, LPPH[i1, i2, j3]*sum[j2, 1, 3, sum[j1, 1, 3, conj[LPH[j1, j2, j3]]*L2[j1, j2, i3]]]]".

I think it is coming from the attached diagram (rough).
As there is no mixing between TH and d (right-handed down quarks) at the tree-level, this contribution should be zero. How this contribution is there in the beta-function of L3? Am I missing something? Please clarify this at your convenience.
Best regards,
Najim

[FStaub]

The diagram you have shown seems to have the correct form. I don't understand why this should be zero just because you assume that there is no tree-level mixing between TH and d. The shown diagram is not proportional to any tree-level mixing.

[najimuddin]

Hi,
As per my understanding, the beta function is the combination of the correction of the vertex and the wave-function renormalization of the external fields. In this diagram, there is loop correction between TH and d. It is neither the wave-function renormalization of the field TH or d. Then, how it is coming?. I am unable to understand this results using my poor knowledge. Please help me to understand this.
Best regards
Najim

[FStaub]

It's obviously an off-diagonal wave-function renormalisation.
I'm sorry, but I can only offer support related to my tools. (Even that takes sometimes too much time which I could spent on own projects). For anything beyond that, please ask your supervisor or collaborators.

[najimuddin]

Thank you for the comments "off-diagonal wave-function renormalisation".