Name: Vladyslav Shtabovenko Date: 12/14/16-04:53:26 PM Z
Hi,
sorry, somehow I completely overlooked this message.
I’ve just pushed an example that reproduces this calculation from
Peskin
to the development version:
<https://github.com/FeynCalc/feyncalc/blob/master/FeynCalc/Examples/EW/EWHiggsToTwoGluonsOneLoop.m>
Furthermore, there is a similar example present in FeynHelpers,
where the final result is compared with the literature (Spira et al.)
<https://github.com/FeynCalc/feynhelpers/blob/master/Examples/EW/EWHiggsToTwoGluonsOneLoop.m>
It is also described in the accompanying preprint:
https://arxiv.org/abs/1611.06793
Sec 5.3
Cheers,
Vladyslav
> 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