Bottom-up approach within the electroweak effective theory: Constraining heavy resonances

The LHC has confirmed the existence of a mass gap between the known particles and possible new states. Effective field theory is then the appropriate tool to search for low-energy signals of physics beyond the Standard Model. We adopt the general formalism of the electroweak effective theory, with a nonlinear realization of the electroweak symmetry breaking, where the Higgs is a singlet with independent couplings. At higher energies we consider a generic resonance Lagrangian which follows the above-mentioned nonlinear realization and couples the light particles to bosonic heavy resonances with $J^P=0^{\pm }$ and $J^P=1^{\pm }$. Integrating out the resonances and assuming a proper short-distance behavior, it is possible to determine or to constrain most of the bosonic low-energy constants in terms of resonance masses. Therefore, the current experimental bounds on these bosonic low-energy constants allow us to constrain the resonance masses above the TeV scale, by following a typical bottom-up approach, i.e., the fit of the low-energy constants to precise experimental data enables us to learn about the high-energy scales, the underlying theory behind the Standard Model.

Figure 1

Predicted values for the LECs ${\mathcal{F}}_1$, ${\mathcal{F}}_3$, ${\mathcal{F}}_4$ and ${\mathcal{F}}_4+{\mathcal{F}}_5$, from Tables 3 and 4, as a function of the corresponding resonance mass ($M_V$, $M_A$ or $M_{S_1^1}v/c_d$). The green area covers the experimentally allowed region, at 95% CL, and it is further extended by a dashed green band that accounts for our estimated one-loop running uncertainties in Eq. (16). If there is a dependence on $M_V$ and $M_A$, the gray and/or brown regions cover all possible values for $M_A>M_V$. If the 2nd WSR has been considered, it is explicitly indicated in the plot, with the corresponding lines for $M_A=M_V$ (orange), $M_A=1.1M_V$ (blue), $M_A=1.2M_V$ (red) and $M_A\to \infty$ (dark gray). In the case without the 2nd WSR, the theoretically allowed region for ${\mathcal{F}}_1$ is given by the gray and brown regions. In case of using only the even-parity operators, we indicate it in the plot.

Figure 2

Predicted values for the LECs ${\mathcal{F}}_6$, ${\mathcal{F}}_6+{\mathcal{F}}_7$ and ${\mathcal{F}}_9$, from Tables 3 and 4, as a function of the corresponding resonance mass ($M_V$, $M_A$ or $M_{P}v/d_P$). If there is a dependence on $M_V$ and $M_A$, the gray regions cover all possible values for $M_A>M_V$. The ${\mathcal{F}}_6$ plot assumes the 2nd WSR and shows the corresponding lines for $M_A=M_V$ (orange), $M_A=1.1M_V$ (blue), $M_A=1.2M_V$ (red) and $M_A\to \infty$ (dark gray). The lines are thick due to the experimental uncertainty on ${\kappa }_W$. In case of using only the even-parity operators, we indicate it in the plot.