One-loop renormalization of vector boson scattering with the electroweak chiral Lagrangian in covariant gauges

This work presents a first full one-loop computation of vector boson scattering (VBS) within the non-linear effective field theory given by the bosonic sector of the usually called electroweak chiral Lagrangian (EChL). The computation is performed in the most general case of covariant $R_{\xi }$ gauges and is compared through all this work with the Standard Model case, whose computation in these covariant gauges is also novel and is presented also here. The calculation of the one-loop VBS amplitude is performed using the diagrammatic method by means of the one-particle-irreducible (1PI) Green functions that are involved in these scattering processes. The central part of this work is then devoted to the renormalization of all the $n$-legs one-loop 1PI Green functions involved. This renormalization is performed in the most general off-shell case with arbitrary external legs momenta. We then describe in full detail the renormalization program, which within this context of the EChL, implies to derive all the counterterms for both the electroweak parameters, like boson masses and gauge couplings, and those for the EChL coefficients. These later are crucial for the renormalization of the new divergences typically appearing when computing loops with the lowest chiral dimension Lagrangian. We present here the full list of involved divergences and counterterms in the $R_{\xi }$ gauges and derive the complete set of renormalization group equations for the EChL coefficients. In the last part of this work, we present the EChL numerical results for the one-loop cross section in the $WZ$ channel and compare them with the SM results.

Figure 1

Full 1PI functions (black balls) contributing to the full one-loop amplitude $\mathcal{A}(WZ\to WZ)$.

Figure 2

Diagrammatic Lorentz conventions corresponding to the self-energies of two gauge bosons (left), two scalars (middle) and gauge boson with a scalar (right). Incoming momenta are understood.

Figure 3

Diagrams at LO contributing to $W^{+}Z\to W^{+}Z$ in the covariant $R_{\xi }$ gauges.

Figure 4

Diagrammatic Lorentz conventions corresponding to the 3-legs ($i{\widehat{\mathrm{\Gamma }}}_{SV{V}^{\prime }}^{\mu \nu }$, $i{\widehat{\mathrm{\Gamma }}}_{WWV}^{\mu \nu \rho }$) and 4-legs ($i{\widehat{\mathrm{\Gamma }}}_{V{V}^{\prime }WW}^{\mu \nu \rho \sigma }$) Green functions, relevant for our observables of interest. The generic scalar here, $S$, can be a Goldstone boson or a Higgs boson. The vector bosons $V$ here, refer generically to the corresponding EW gauge boson, $W$, $Z$, $\gamma$. All momenta are incoming.

Figure 5

Cross-section predictions for $W_L^{+}{Z}_L\to W_L^{+}{Z}_L$ within the EChL corresponding to $a=1$, $b=a^2$, ${\kappa }_3={\kappa }_4=1$ and varying $a_4$ (left) and $a_5$ (right). The rest of $a_i$’s are set to zero. The $\mu$ scale is set to $\mu =1\mathrm{TeV}$. Dashed lines are the tree level predictions. Solid lines are the full one-loop predictions. The solid red (pink) lines correspond to the full (tree) SM prediction. The relative size of the one-loop prediction respect to the tree level one defined by means of ${\delta }_{1-\mathrm{loop}}$ is also shown (lower plots in the upper panels). The predictions for the lowest partial wave $a_J$ with $J=0$ are also displayed (lower panels). The unitarity violating regions in these plots are indicated by the shadowed areas.

Figure 6

Cross-section predictions for $W_L^{+}{Z}_L\to W_L^{+}{Z}_L$ within the EChL corresponding to $a=0.9$, $b=a^2$, ${\kappa }_3={\kappa }_4=1$ and varying $a_4$ (left) and $a_5$ (right). The rest of $a_i$’s are set to zero. The $\mu$ scale is set to $\mu =1\mathrm{TeV}$. Dashed lines are the tree level predictions. Solid lines are the full one-loop predictions. The solid red (pink) lines correspond to the full (tree) SM prediction. The relative size of the one-loop prediction respect to the tree level one defined by means of ${\delta }_{1-\mathrm{loop}}$ is also shown (lower plots in the upper pannels). The predictions for the lowest partial wave $a_J$ with $J=0$ are also displayed (lower pannels). The unitarity violating regions in these plots are indicated by the shadowed areas.

Figure 7

Cross-section predictions for $W_L^{+}{Z}_L\to W_L^{+}{Z}_L$ within the EChL corresponding to $a=1$ (upper-left), $a=0.9$ (upper-right), $b=a^2$, ${\kappa }_3={\kappa }_4=1$ and varying $a_4$ and $a_5$ simultaneously. The rest of $a_i$’s are set to zero. The $\mu$ scale is set to $\mu =1\mathrm{TeV}$. Dashed lines are the tree level predictions. Solid lines are the full one-loop predictions. The solid red (pink) lines correspond to the full (tree) SM prediction. The relative size of the one-loop prediction respect to the tree level one defined by means of ${\delta }_{1-\mathrm{loop}}$ is also shown (lower plots in the upper panels). The predictions for the lowest partial wave $a_J$ with $J=0$ are also displayed (lower panels). The unitarity violating regions in these plots are indicated by the shadowed areas.

Figure 8

Check of perturbative unitarity in the one-loop predictions for the lowest partial wave $a_0$ in both the EChL with $a=0.9$ and the SM. The 10 blue (red) points are the EChL (SM) predictions for the ten chosen energies, $\sqrt{s}[\mathrm{GeV}]=300$, 600, 900, 1200, 1500, 1800, 2100, 2400, 2700, 3000, from left to right along the diagonal line in this plot.

Figure 9

Generic loop diagrams for the self-energies in the EChL. Dashed lines represent the Higgs boson and the GB bosons, wavy lines represent the EW gauge bosons, and dotted lines are the ghost fields. The same notation is used in the loop diagrams of all the figures.

Figure 10

Generic loop diagrams for the $WWZ$, $ZZA$ and $ZZZ$ Green functions in the EChL.

Figure 11

Generic loop diagrams for the $\pi WZ$ Green functions in the EChL.

Figure 12

Generic loop diagrams for the $HWW$ and $HZZ$ Green functions in the EChL.

Figure 13

Generic loop diagrams for the $WZWZ$ Green functions in the EChL.