Light Higgs bosons from a strongly interacting Higgs sector

The mass and the decay width of a Higgs boson in the minimal standard model are evaluated by a variational method in the limit of strong self-coupling interaction. The non-perturbative technique provides an interpolation scheme between strong-coupling regime and weak-coupling limit where the standard perturbative results are recovered. In the strong-coupling limit the physical mass and the decay width of the Higgs boson are found to be very small as a consequence of mass renormalization. Thus it is argued that the eventual detection of a light Higgs boson would not rule out the existence of a strongly interacting Higgs sector. Typeset using REVTEX 1 The impressive success of the Standard Model (SM) has enforced the common believing that the Higgs boson will be soon detected by the new generation of accelerators [1]. In fact there are two unknown parameters in the SM that wait for their experimental determination: the mass m of the Higgs boson and the strength of its self-coupling interaction λ. This last one determines the bare Higgs mass m 0 = λv/3 where v is the vacuum expectation value for the scalar field which is fixed by the known strength of weak interactions. Thus at tree level perturbation theory predicts a light Higgs mass m ≈ m0 if the coupling λ is small enough. Conversely, in the strong coupling limit, perturbation theory breaks down and there is no simple relation between m and λ. A light weakly interacting Higgs boson has been strongly desired, mainly because perturbation theory would be reliable, and the Higgs boson would be detectable at a reasonable energy threshold. However, if nature had chosen for a strongly interacting boson, the physics would be richer and more interesting. Actually, the physics of such a strongly interacting Higgs boson has been explored in the last twenty years, and interesting proposals have been discussed ranging from the existence of bound states [2–8] to unconventional descriptions of the symmetry breaking mechanism [9]. During the last years the possibility of a strongly interacting Higgs boson has been rejected for two main reasons: i) A large λ is believed to imply a large mass, in contrast with the recent phenomenological evidence [1] for a light m ≈ 100 − 200 GeV; ii) For a strongly interacting Higgs boson the decay width Γ has been predicted to be very large [10,11] compared to the mass, and such very large resonance could hardly be regarded as a true particle. In this letter we point out that both the statements i) and ii) have a perturbative nature and cannot be trusted in the strong coupling limit. At tree level m and Γ are small if the coupling λ is small, which is consistent in the framework of perturbation theory. However if λ is very large any perturbative argument breaks down and fails to predict what m and Γ are. In fact, by use of a variational method we show that both m and Γ are small in the strong coupling limit. The existence of a saturation of m at strong coupling has been shown by several nonperturbative techniques as 1/N expansions [11], variational methods [12] and Bethe-Salpeter 2 equation [13]. We have shown that a further increase of the coupling strength yields a decrease of the mass [12], and this has also been confirmed by recent Bethe-Salpeter calculations [13]. The physical reason is very simple: at tree level m is proportional to λ; however the interaction renormalizes the mass, since the attractive self-coupling reduces the energy of a free boson. At some stage this reduction overcomes the tree level increase, and the renormalized mass decreases for some very large self coupling. As a result a light Higgs boson could be a very strongly interacting particle whose ground state could even be a Higgs-Higgs bound state. A light self-interacting Higgs boson would not make any sense as a free particle if its decay width Γ would be so large and increasing with λ as found by 1/N expansion calculations [11]. However in the real world the goldstone bosons of the O(N) model do not play any physical role, while the Higgs sector is coupled with the gauge bosons through a quite weak interaction which does not increase with λ. As could be expected, we show that for very large couplings and a reasonable choice of the cut-off, a light Higgs boson would be characterized by a very small decay width: thus the experimantal knowledge of m and Γ would not say the last word on the strength of the self-interaction. The eventual detection of a light Higgs with a narrow decay width would be consistent with both a perturbative weakly interacting and a non-perturbative strongly interacting theory. In order to deal with the non-perturbative limit we use a variational method in the Hamiltonian formalism [14–16,6,8]. The method has the advantage of yielding the known perturbative results in the weak-coupling limit [16,8] (e.g. masses, decay widths and binding energies), while it can be safely extended to the non-perturbative strong coupling regime. The results achieved by such method have not been appreciated in the past since the variational equations have been usually approximated by perturbative methods [17] thus spoiling their most important advantages. In the framework of a study on bound states we have recently shown [12] that the variational equations can be decoupled exactly, giving important consequences on mass renormalization. In this letter we show that the same method can be used for decoupling the variational equations arising from a more complete trial state, 3 describing a Higgs field h which interacts with a neutral gauge vector field Z: |Ψ〉 = |h〉+ |hh〉+ |hhh〉+ |ZZ〉 (1)