Name: Aliaksandr Dubrouski Date: 05/05/15-04:45:54 PM Z

2 Lingxiao Xu:

I do not see any color factors and gluon color summation in your
calculation. You must take into account that gluons carry color in your
calculation and color is not directly observable. The rest of calculation
seems to be correct.

2015-05-05 10:49 GMT+03:00 Lingxiao Xu <[noreply_at_HIDDEN-E-MAIL]>:

> 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).
>
> «HighEnergyPhysics`FeynCalc`
>
> 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[
> GAD[mu].(GSD[k] + mq).GAD[
> 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[
> GAD[nu].(GSD[k] + mq).GAD[
> 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 SubscriptC, 0+\!\(
> \*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 SubscriptB, 0-4 SubscriptB, 0+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]) SubscriptC, 0+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])
> SubscriptC, 0+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]) SubscriptC, 0+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
>
>

-- 
Regards,
            Aliaksandr Dubrouski