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Calculating the scalar part of the D-term in the MSSM

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Hi. I wish to calculate the scalar part of the Lagrangian for the MSSM. However, using the expression,

$$V_{D} = 1/2\sum_a {g_a}^2 (\sum_\alpha \phi^\dagger_\alpha T^a \phi_\alpha)^2
$$

I am able to calculate the U(1) term but I am not able to calculate SU(2)/SU(3) terms. Does $T^a$ stand for all the generators in the gauge group or refers only to the diagonal generators? Using only the two Higgs, the down squark and the selectron the result that is obtained for the SU(2) part should be

$$ g_2^2/8 (|h_d|^2 - |h_u|^2 - |d|^2 - |e|^2)
$$
and for the SU(3) part is

$$ g_3^2/6 (|d|^2 - |d^C|^2)
 $$

Please explain why this is so. Thanks in advance..

asked Nov 12, 2015 in Theoretical Physics by Joyshaitan (20 points) [ no revision ]
reshown Nov 15, 2015 by dimension10

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