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Dear Dr. Mertig:

Could you please help to resolve a puzzling result that the OneLoop FeynCalc calculation leads to incorrect results (in comparison with my hand-calculations and those in the literature (e.g.,Politzer's) for the 3rd. order Feynman diagram involving 3 ghost propagators and 3 external gluon lines [with one external gluon momentum set to be zero for simplicity] in the calculation of the asymptotic freedom)? Thank you very much for your help.

I used Mathematica 11 for calculations(with SU(2) in mind). My short program is attached for your reference.

Comments (do not run the .nb before reading these comments):

1. The vertices g1*g2*g3 contains 2 terms. For clarity, I first calculate the first term by using the vertex g1*g2*g3t. The OneLoop[...] gives the result A1, which is only half of the correct result A10 (see In[2182]).

2. I calculate the second term by using g1*g2*g3tt, which lead to the oneloop result A2. Only when it is multiplied by(3/2), one gets the correct result. See In[2198].

3. In[2198]: By observation, A1*(4/2)+A2*(3/2) give the correct result.

4. For the crossed diagram (of external gluons with non-zero momenta), a similar corrections for the one-loop results (T1 and T) are done for the 2 terms: B1=2*T1, B2=(3/2)*T.

5. The corrected result, {A1*2 + A2*(3/2)}+{2*T1 + (3/2)*T}, leads to the correct structure and magnitude of the diagram involving 3 ghost internal lines, as shown in the coefficient of B0 in Out[2231] and in Out[2232].

HSU Jongping,

Chancellor Professor

Department of Physics

Univ. of Massachusetts Dartmouth,

North Dartmouth, MA 02747. FAX (508)999-9115

http://www.umassd.edu/engineering/phy/people/facultyandstaff/jong-pinghsu/

recent monograph: Space-Time Symmetry and Quantum Yangâ€“Mills Gravity (https://sites.google.com/site/yangmillsgravity123/)

- application/mathematica attachment: SU2Ghostloop.nb

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