Anatomy of Higgs boson decays into $\gamma \gamma$ and $\gamma Z$ within the electroweak chiral Lagrangian in the $R_{\xi }$ gauges

In this work, we study the Higgs boson decays into two photons and into one photon and one $Z$ gauge boson within the context of the nonlinear effective field theory called the electroweak chiral Lagrangian (EChL). We present a detailed computation of the corresponding amplitudes to one-loop level in the covariant $R_{\xi }$ gauges. We assume that the fermionic loop contributions are as in the Standard Model and focus here just in the computation of the bosonic loop contributions. Our renormalization program and the anatomy of the various contributions participating in the $R_{\xi }$ gauges are fully explored. With this present computation, we demonstrate the gauge invariance of the EChL result, not only for the case of on-shell Higgs boson, but also for the most general and interesting case of off-shell Higgs boson. We finally analyze and conclude on the special relevance of the Goldstone boson loops, in good agreement with the expected chiral loops behavior in chiral Lagrangians. We perform a systematic comparison with the corresponding computation of the Standard Model in the $R_{\xi }$ gauges and with the previous EChL results in the unitary gauge. This work represents the first computation within the EChL of these observables to one loop, for both on- and off-shell Higgs boson, in the most general $R_{\xi }$ gauges and with a full renormalization program description, not yet fully explored in the previous literature and which is different to the most frequently used in the linear effective field theory.

Figure 1

Contributions of the two- and three-point renormalized Green functions (denoted by gray circles). The upper line corresponds to $H\to \gamma \gamma$ and the lower line to $H\to \gamma Z$.

Figure 2

Bosonic loop diagrams contributing to ${\mathrm{\Sigma }}_{ZA}^{\mathrm{L}}(q^2)$ in the $R_{\xi }$ gauges. We denote the internal $W$ bosons, charged GBs, and charged ghosts by wavy, dashed, and dotted lines, respectively.

Figure 3

One-loop diagrams contributing to ${\mathcal{M}}_{\gamma V}^{\text{gauge}}$ in both the EChL and the SM cases. Negative charge flows in the same direction of the momentum $p$.

Figure 4

One-loop diagrams contributing to ${\mathcal{M}}_{\gamma V}^{\mathrm{mix}}$ in both the EChL and the SM cases. Negative charge flows in the same direction of the momentum $p$.

Figure 5

One-loop diagrams contributing to ${\mathcal{M}}_{\gamma V}^{\mathrm{GB}}$ in the EChL case. Only diagrams (h), (h’), and (i) are present in the SM. Negative charge flows in the same direction of the momentum $p$.

Figure 6

One-loop diagrams contributing to ${\mathcal{M}}_{\gamma V}^{\text{ghost}}$ in the SM case. There are not such diagrams in the EChL. Negative charge flows in the same direction of the momentum $p$.

Figure 7

EChL partial width (fermionic loop included) corresponding to the $H\to \gamma \gamma$ (left) and $H\to \gamma Z$ (right) decays as function of $\mathrm{\Delta }a=a-1$ for different values of $c_{H\gamma V}$. The SM value is shown for reference (green star).

Figure 8

Contour lines for the ratio of the EChL partial width respect to the SM one, corresponding to the $H\to \gamma \gamma$ (left) and $H\to \gamma Z$ (right) decays. The various lines correspond to specific settings for the deviations respect to the SM value, $({\mathrm{\Gamma }}^{\mathrm{EChL}}-{\mathrm{\Gamma }}^{\mathrm{SM}})/{\mathrm{\Gamma }}^{\mathrm{SM}}$, of 0% (solid blue line), $\pm 10\%$ (dotted), $\pm 20\%$ (dashed), $\pm 30\%$ (dot-dashed), and $\pm 50\%$ (solid) purple (green), respectively.

Figure 9

Anatomy of the various contributions to the EChL partial widths as functions of $\mathrm{\Delta }a=a-1$ in the Landau and Feynman gauges, for the $H\to \gamma \gamma$ (left) and $H\to \gamma Z$ (right) decays. The predictions for the complete partial width (solid), the GB loop contribution (dashed), and the contributions from the rest (dotted) are shown separately.

Figure 10

Anatomy of the EChL vertex functions, $V_{H\gamma \gamma }$ (left) and $V_{H\gamma Z}$ (right), split into the various boson loop contributions corresponding to the total bosonic (solid lines) and GB loop contributions (dashed lines), as function of the Higgs boson momentum $\sqrt{q^2}$. We set $c_{H\gamma \gamma }=c_{H\gamma Z}=0$ and normalize the resulting vertex functions respect to the $a$ parameter. The real and imaginary parts (solid blue and light blue lines, respectively) of the total vertex functions are displayed separately. We include predictions for both gauge choices, $\xi =0$ (dashed green) and $\xi =1$ (dashed pink and light pink for its real and imaginary parts). The predictions from the approximate vertex functions at large $q^2$ are also shown (dotted lines).

Figure 11

Relevant EChL Feynman rules from ${\mathcal{L}}_4$ (not present in the SM). The interaction vertex for $H\gamma V$ is denoted in this case by a shadowed little box. The first and second arguments in $\{,\}$ correspond to the photon and $Z$ boson, respectively.

Figure 12

Relevant EChL Feynman rules from all the counterterms involved in the computation. A crossed circle is drawn to denote these counterterms. We use the short notation $\delta {\mathcal{Z}}_{ZA}=s_{\mathrm{w}}{c}_{\mathrm{w}}(\delta {\mathcal{Z}}_W-\delta {\mathcal{Z}}_B)$ for the contribution from the counterterm of the wave function renormalization of the mixing.