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The existence of a mass gap between the Standard Model (SM) and possible new states encourages us to use effective field theories. Here we follow the non-linear realization of the electroweak symmetry breaking: the electroweak effective theory (EWET), also known as Higgs effective field theory (HEFT) or electroweak chiral Lagrangian (EWChL). At short distances an effective resonance Lagrangian which couples the SM states to bosonic and fermionic resonances is considered. After integrating out the resonances and assuming a well-behaved high-energy behavior, we estimate or bound purely bosonic low-energy constants in terms of only resonance masses. Current experimental information on these low-energy constants allows us to constrain the high-energy resonance masses.

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 JP ¼ 0 and JP ¼ 1 . 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.

The couplings of the electroweak e ective theory contain information on the heavy-mass scales which are no-longer present in the low-energy Lagrangian. We build a general e ective Lagrangian, implementing the electroweak chiral symmetry breaking SU(2)L SU(2)R ! SU(2)L+R, which couples the known particle elds to heavier states with bosonic quantum numbers JP = 0 and 1 . We consider colour-singlet heavy elds that are in singlet or triplet representations of the electroweak group. Integrating out these heavy scales, we analyze the pattern of low-energy couplings among the light elds which are generated by the massive states. We adopt a generic non-linear realization of the electroweak symmetry breaking with a singlet Higgs, without making any assumption about its possible doublet structure. Special attention is given to the di erent possible descriptions of massive spin-1 elds and the di erences arising from naive implementations of these formalisms, showing their full equivalence once a proper short-distance behaviour is required.

Accepting that there is a mass gap above the electroweak scale, the Electroweak E ective Theory (EWET) is an appropriate tool to describe this situation. Since the EWET couplings contain information on the unknown high-energy dynamics, we consider a generic strongly-coupled scenario of electroweak symmetry breaking, where the known particle fields are coupled to heavier states. Then, and by integrating out these heavy fields, we study the tracks of the lightest resonances into the couplings. The determination of the low-energy couplings (LECs) in terms of resonance parameters can be made more precise by considering a proper short-distance behaviour on the Lagrangian with heavy states, since the number of resonance couplings is then reduced. Notice that we adopt a generic non-linear realization of the electroweak symmetry breaking with a singlet Higgs.

Due to the mass gap between the Standard Model and possible New Physics states, electroweak effective approaches are appropriate. Although a linear realization of the electroweak symmetry breaking with the Higgs forming a doublet together with the Goldstone bosons of the EWSB is a first possibility (SMEFT), we adopt the more general non-linear realization, where the Higgs is a singlet with independent couplings (EWET, HEFT or EWChL). We present the effective Lagrangian at low energies (the EWET, with only the SM fields) and at high energies (the resonance theory, with also a set of resonances). Taking into account the high scale of these resonances, their experimental searches seem to be more accessible by considering their imprints at low-energies, i.e., their imprints in the Low Energy Constants (LECs) of the EWET at energies lower than the resonance masses. We give some examples of these phenomenological connections.

Contrary to what is sometimes stated, the current electroweak precision data easily allow for massive composite resonance states at the natural EW scale, i.e., well over the TeV. The oblique parameters S and T are analyzed by means of an effective Lagrangian that implements the SU(2)L⊗SU(2)R→SU(2)L+R pattern of electroweak symmetry breaking. They are computed at the one-loop level and incorporating the newly discovered Higgs-like boson and possible spin–1 composite resonances. Imposing a proper ultraviolet behaviour is crucial and allows us to de- termine S and T at next-to-leading order in terms of a few resonance parameters. Electroweak precision data force the vector and axial-vector states to have masses above the TeV scale and suggest that the W+W− and ZZ couplings to the Higgs-like scalar should be close to the Stan- dard Model value. Our findings are generic: they only rely on symmetry principles and soft requirements on the short-distance properties of the underlying strongly-coupled theory, which are widely satisfied in more specific scenarios.

Taking into account the negative results of the searches for New Physics at the LHC, electroweak effective theories are appropriate to deal with current energies. Tracks of new, higher scales can be studied through next-to-leading order corrections of the electroweak effective theory. We assume a generic non-linear realization of the electroweak symmetry breaking with a singlet Higgs and a strongly-coupled UV-completion. We further consider a high-energy Lagrangian that incorporates explicitly a general set of new heavy fields. After integrating out these heavy resonances, we study the pattern of low-energy constants among the light fields, which are generated by the massive states.

In the light of the mass gap between Standard Model (SM) states and possible new particles, effective field theories are a suitable approach. We take on the non-linear realization of the electroweak symmetry breaking: the electroweak effective theory (EWET), also known as Higgs effective field theory (HEFT) or electroweak chiral Lagrangian (EWChL). At higher scales we consider a resonance electroweak Lagrangian, coupling SM fields to resonances. Integrating out these resonances and assuming a well-behaved high-energy behavior, some of the bosonic lowenergy constants are determined or constrained in terms of resonance masses. Present experimental bounds on these low-energy constants allow us to push the resonance mass scale to the TeV range, MR>2 TeV, in good agreement with previous estimations.

Taking into account the negative results of direct searches for beyond the Standard Model fields and the consequent mass gap between Standard Model and possible unknown states, the use of electroweak effective theories is justified. Whereas at low energies we consider a non-linear realization of the electroweak symmetry breaking with a singlet Higgs and a strongly-coupled ultraviolet completion, at higher energies the known particles are assumed to be coupled to heavy states: bosonic fields with JP = 0 and JP = 1 (in electroweak triplets or singlets and in QCD octets or singlets) and fermionic states with J = 1 2 (in electroweak doublets and in QCD triplets or singlets). By integrating out these heavy resonances, the pattern of next-to-leading order lowenergy constants among the light fields can be studied. A phenomenological study trying to estimate the scale of these resonances is also shown.