Name: Lingxiao Xu (email_not_shown)
Date: 05/05/15-09:49:08 AM Z

Dear developers and users of FeynCalc:

I've calculated the process "Higgs decay into two gluons" at the leading order by FeynCalc. In the final result, I just got a C_0(0,0,m_h^2,m_q^2,m_q^2,m_q^2) which should be -1/(2m_q^2) at the limit of zero Higgs mass (see my code below).

<< HighEnergyPhysicsFeynCalc

onshell = {ScalarProduct[p1, p1] -> 0, ScalarProduct[p2, p2] -> 0,
ScalarProduct[p1, p2] -> Subscript[m, h]^2/2};
SetOptions[OneLoop, Dimension -> D];

num1 = -I mq/v (I Subscript[g, s])^2 (I) DiracTrace[
nu].(GSD[k] + GSD[p2] + mq).(GSD[k] - GSD[p1] + mq)] /.
DiracTrace -> TR //
Simplify; num2 = -I mq/v (I Subscript[g, s])^2 (I) DiracTrace[
mu].(GSD[k] + GSD[p1] + mq).(GSD[k] - GSD[p2] + mq)] /.
DiracTrace -> TR // Simplify;
amp1 = num1 FAD[{k, mq}, {k + p2, mq}, {k - p1, mq}]/(2 Pi)^D // FCI;
amp2 = num2 FAD[{k, mq}, {k + p1, mq}, {k - p2, mq}]/(2 Pi)^D // FCI;
amp = (OneLoop[k, amp1 + amp2] // PaVeReduce) /. onshell // Simplify

1/(4 \[Pi]^2 v Subsuperscript[m, h, 2]) I mq^2 Subsuperscript[g, s, 2] (2 p1^mu p2^nu (4 mq^2 Subscript[C, 0](0,0,Subsuperscript[m, h, 2],mq^2,mq^2,mq^2)+\!$$\*SubsuperscriptBox[\(m$$, $$h$$, $$2$$]\ $$\(TraditionalForm\ \*SubscriptBox[\("C"$$, $$"0"$$]\)(TraditionalForm\0, TraditionalForm\0, TraditionalForm\
\*SubsuperscriptBox[$$m$$, $$h$$, $$2$$], TraditionalForm\
\*SuperscriptBox[$$mq$$, $$2$$], TraditionalForm\
\*SuperscriptBox[$$mq$$, $$2$$], TraditionalForm\
\*SuperscriptBox[$$mq$$, $$2$$])\)\)+4 Subscript[B, 0](Subsuperscript[m, h, 2],mq^2,mq^2)-4 Subscript[B, 0](0,mq^2,mq^2)+2)+\!$$\*SubsuperscriptBox[\(m$$, $$h$$, $$2$$]\
\*SuperscriptBox[$$g$$, $$mu nu$$]\ $$(\(( \*SubsuperscriptBox[\(m$$, $$h$$, $$2$$] - 4\
\*SuperscriptBox[$$mq$$, $$2$$])\)\ $$\(TraditionalForm\ \*SubscriptBox[\("C"$$, $$"0"$$]\)(TraditionalForm\0, TraditionalForm\0, TraditionalForm\
\*SubsuperscriptBox[$$m$$, $$h$$, $$2$$], TraditionalForm\
\*SuperscriptBox[$$mq$$, $$2$$], TraditionalForm\
\*SuperscriptBox[$$mq$$, $$2$$], TraditionalForm\
\*SuperscriptBox[$$mq$$, $$2$$])\) - 2)\)\)+2 p2^mu p1^nu ((4 mq^2-Subsuperscript[m, h, 2]) Subscript[C, 0](0,0,Subsuperscript[m, h, 2],mq^2,mq^2,mq^2)+2))

msq = 2 (amp (ComplexConjugate[amp] /. {mu -> rho,
nu -> sigma}) PolarizationSum[mu, rho, p1,
p2] PolarizationSum[nu, sigma, p2, p1] // Contract) /.
onshell /. Subscript[g, s] -> Sqrt[4 Pi Subscript[\[Alpha], s]] //
Simplify

(4 mq^4 Subsuperscript[\[Alpha], s, 2] ((4 mq^2-Subsuperscript[m, h, 2]) Subscript[C, 0](0,0,Subsuperscript[m, h, 2],mq^2,mq^2,mq^2)+2)^2)/(\[Pi]^2 v^2)

\[CapitalGamma]HGG =
1/(2 8 Pi) 1/(2 Subscript[m, h]) msq /.
v -> Sqrt[Subscript[m, W]^2 SW^2/(Pi \[Alpha])]

(\[Alpha] mq^4 Subsuperscript[\[Alpha], s, 2] ((4 mq^2-Subsuperscript[m, h, 2]) Subscript[C, 0](0,0,Subsuperscript[m, h, 2],mq^2,mq^2,mq^2)+2)^2)/(8 \[Pi]^2 SW^2 Subscript[m, h] Subsuperscript[m, W, 2])

On the other hand, we can check the result with the analytical side(for example, Peskin and Schroeder's Final Project 3) and a problem comes. In Peskin, a factor I_f(\tau_q) is defined. By the definition of I_f(\tau_q), it contains an extra factor"3" to make itself become 1 at the limit m_h->0. So that the amplitude squared should contain a factor 1/9.

My problem is that I can't find such a factor 1/9 in the result got by FeynCalc.

Best Regards, Thanks for the help!
Lingxiao Xu

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