Higgs Effective couplings to gluons (and photons)

The Higgs effective field theory (heft) model is an 'extension' of the Standard Model, where the Higgs boson couples directly to gluons (and photons). In the SM these couplings are present through a heavy (top) quark loop. For a not too heavy Higgs (m_h<2 m_t), it is a good approximation to take the mass of the heavy quark in the loop to infinity (For this approximation to hold, not only should the Higgs mass be smaller than twice the top mass, also all other kinematic variables, such as the transverse momentum of the Higgs boson, should be smaller than 2m_t.) This results in effective couplings between gluons and Higgs bosons.

The effective vertices can be derived from the effective dimension five Lagrangian:

$\mathcal{L}_{h}=-\frac{1}{4}g_hG_{\mu\nu}^aG_{\mu\nu}^a \Phi,$
\texttt{{\small where}} $G^a_{\mu\nu}=\partial_{\mu}A^a_{\nu}-\partial_{\nu}A^a_{\mu}+gf^{abc}A^b_{\mu}A^c_{\nu}$.

The coupling constant g_h is given by:

$g_h=\frac{\alpha_s}{3\pi v}\Big(1+ \frac{7}{30}\tau + \frac{2}{21}\tau^2+ \frac{26}{525}\tau^3\Big),$ \texttt{{\small with}} $\tau=m_h^2/(4m_t^2)$

and higher orders in τ have been neglected. Due to the non-abelian nature of the SU(3)_C color group the effective vertices do not only include two, but also three and four gluons coupling to the Higgs boson. Since MadGraph can work only with three- and four-point vertices, the four-gluon interactions in the heft model are obtained by rewriting the QCD four-gluon interaction in terms of three-point vertices with an extra non-propagating internal tensor particle, T. This trick can be easily understood by noting that the usual (text-book) form of the four-gluon interaction is the sum of three terms, whose color and Lorentz structure correspond to 2 → 2 diagrams where a color octet tensor is exchanged in the s, t, u channels. With the introduction of this extra particle, the four-gluon-Higgs vertices can be reduced to diagrams with at most four-point vertices. To get the standard diagrammatic visualization of four-gluon and four-gluon-Higgs vertices it is sufficient to contract the T particle lines to a single point.

The gluon couplings to a pseudo-scalar Higgs are also implemented. The name of the pseudo-scalar Higgs in MadGraph is h3 ( i.e. , the same as in the 2HDM and MSSM models). The effective dimension five Lagrangian for the pseudo-scalar Higgs coupling to the gluons is

$\mathcal{L}_{A}=\frac{1}{2}g_AG_{\mu\nu}^a\tilde{G}_{\mu\nu}^a \Phi_A,$ \texttt{{\small where}} $\tilde{G}_{\mu\nu}^a$ \texttt{{\small is the dual of}} $G_{\mu\nu}^a$, $\tilde{G}_{\mu\nu}^a=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}G_{\rho\sigma}^a$. 

The effective coupling constant g_A is given by

$g_A=\frac{\alpha_s}{2\pi v}\Big(1+ \frac{1}{3}\tau+ \frac{8}{45}\tau^2 + \frac{4}{35}\tau^3\Big)$,

where the higher orders in τ have been neglected. The pseudo-scalar Higgs has only effective couplings to two or three gluons. The four-gluon-pseudo-scalar Higgs vertex is absent due to the anti-symmetry of the epsilon tensor ε^μνρσ. If a mixed Higgs with no definite CP parity is needed, it sufficient to change the couplings of the Higgs to the gluons. First generate the process with the SM Higgs, then, after downloading the code, change the coupling in the ./Source/Model/couplings.f file. The coupling constant is defined as a two-dimensional object, where the first and second elements are the CP-even and CP-odd couplings of the Higgs to the gluons, respectively. The HELAS subroutines automatically use the correct kinematics for odd-, even- or mixed CP Higgs's coupling to the gluons. At present, the implementation allows production of only one Higgs-boson. The effective couplings of two Higgs bosons to gluons are available in HELAS, but not yet included in the HEFT model.

The non-propagating auxiliary particle T

To describe the four-gluon-Higgs coupling a vertex with 5 external lines is needed. This cannot be done with MadGraph. However, there is a way to circumvent this problem by introducing non-propagating auxiliary particles. It is possible to rewrite the four-gluon interactions as two three-point interactions connected by the new auxiliary tensor particle (called tn internally, and shows as T in the MG diagrams, the PDG code is 99). The troublesome five-point interaction (between the four gluons and the Higgs) reduces then to three three-point interactions.

Notice that the color part of the four-gluon vertex is exactly like the sum of the s-, t-, and u-channel exchange diagram, where the exchanged particle is in the adjoint representation of color, i.e. an octet.

Second, notice that the 'Feynman part' (i.e. the part with the metric tensors) is almost (except for a factor of two) an projection operator

$(g^{\mu\rho}g^{\nu\sigma}-g^{\mu\sigma}g^{\nu\rho})(g^{\rho\tau}g^{\sigma o}-g^{\rho o}g^{\sigma\tau})=2(g^{\mu\tau}g^{\nu o}-g^{\mu o}g^{\nu\tau}).$

This means that we have the following new Feynman rules. The gluon-gluon-tensor vertex is given by

$V_2^T=gf^{abc}(g^{\mu\rho}g^{\nu\sigma}-g^{\mu\sigma}g^{\nu\rho})/\sqrt{2}$ 

and the tensor propagator is

$\Delta^{\mu\nu,\rho\sigma}_{ab}=-i g^{\mu\rho}g^{\nu\sigma} \delta_{ab}.$

Notice that the new particle is a rank-2 tensor (hence the name 'tensor'). It does not propagate, because there is no momentum dependence in the propagator. And it only exist as an internal particle, it can never be seen as an external one.

With these new rules we can make six tree-level diagrams with four external gluon lines: three with two gluon-gluon-gluon vertices and three with two gluon-gluon-tensor vertices. The sum of these diagrams is exactly the same as the four diagrams in the original QCD: the three diagrams with two gluon-gluon-gluon vertices have not changed, and the four-gluon interaction is equal to the sum of the three diagrams with the two gluon-gluon-tensor vertices.

The only thing needed to implement the four-gloun-Higgs coupling is a coupling between the tensor particle and the Higgs boson. A short derivations leads to the following tensor-tensor-Higgs vertex

$V^{TTH}= ig_H g^{\mu\rho}g^{\nu\sigma} \delta^{ab}.$

Higgs coupling to photons

The coupling between the scalar Higgs to photons is mediated by a top quark loop and by a W boson loop. In the limit of small Higgs masses (lower than approx 150 GeV), the loop induced interaction can approximately described by the Lagrangian

$$L_{\textrm{heft}}=-\frac{1}{4}gF_{\mu\nu} F_{\mu\nu} H.$$

Due to the abelian nature of QED there is only one effective vertex between photons and Higgs bosons. The value for the coupling constant in the heft model is given by

$$g=-\frac{\alpha}{\pi v} \frac{47}{18}\Big( 1+ \frac{66}{235}\tau_w +\frac{228 }{1645}\tau_w^2+\frac{696}{8225}\tau_w^3+\frac{5248}{90475} \tau_w^4+\frac{1280}{29939}\tau_w^5+\frac{54528}{1646645}\tau_w^6-\frac{56}{705} \tau_t-\frac{32}{987}\tau_t^2\Big),$$

where

$\tau_t=m_h^2/(4m_t^2)$ \texttt{{\small and}} $\tau_w=m_h^2/(4m_W^2)$. 

Higher order in τ_t and τ_w have been neglected.

-- Main.RikkertFrederix - 25 Oct 2007