Hugonweb | Effective Higgs Field Theory

This allows for $H \gamma \gamma$ and $Hgg$ to be effective tree level calculations instead of loop diagrams. This can simplify multi-loop calculations, and allows these processes to be put into tree level only software.

Simple Legrangian terms are¹

$$\mathcal{L}_{H \gamma \gamma} = \frac{e^2_q \alpha}{2 \pi} \left( \sqrt{2} G_F \right)^{1/2}\left[ 1 - \frac{\alpha_s}{\pi} \right] \times F_{\mu \nu} F_{\mu \nu} H$$

$$\mathcal{L}_{H g g} = \frac{\alpha_s}{12 \pi}(\sqrt{2} G_F)^{1/2}\left[ 1 + \frac{11 \alpha_s}{4 \pi} \right] \times G_{\mu \nu}^a G_{\mu \nu}^a H$$

Using LanHEP, we can find Feynman vertex factors for the terms. First, the factor for the $\gamma \gamma H$ vertex:

$$-2 \frac{e_q \alpha}{\pi^2} (\sqrt{2} G_F)^{1/2} ( \pi p_1^\rho p_2^\rho g^{\mu \nu} - \pi p_1^\nu p_2^\mu - \alpha_s p_1^\rho p_2^\rho g^{\mu \nu} + \alpha_s p_1^\nu p_2^\mu )$$

and then the factor for the $ggH$ vertex:

$$-\frac{\alpha_s}{12 \pi^2} (\sqrt{2} G_F)^{1/2} \times \big( 4 \pi p_1^\rho p_2^\rho g^{\mu \nu} - 4 \pi p_1^\nu p_2^\mu -11 \alpha_s p_1^\rho p_2^\rho g^{\mu \nu} + 11 \alpha_s p_1^\nu p_2^\mu \big)$$

These vertices need to be added in CalcHEP:

G    |G    |h    |     |-HFA*alfSMZ/(12*pi^2)         |4*pi*p1.p2*m1.m2-4*pi*m2.p1*m1.p2-11*alfSMZ*(p1.p2*m1.m2-m2.p1*m1.p2)

A    |A    |h    |     |-2*EQ*HFA*alfEMZ/pi^2         |pi*p1.p2*m1.m2-pi*m2.p1*m1.p2-alfSMZ*p1.p2*m1.m2+alfSMZ*m2.p1*m1.p2

These parameters:

HFA           |0.0040935657 |sqrt(sqrt(2)*GF)

and these constraints:

pi         |acos(-1)                      %Pi
EQ         |(2/3)^2               %Charge of Heaviest Quark (For HEFT)