The Higgs boson: from the lattice to LHC

We discuss the triviality and spontaneous symmetry breaking scenario where the Higgs boson without self-interaction coexists with spontaneous symmetry breaking. We argue that non perturbative lattice investigations support this scenario. Moreover, from lattice simulations we predict that the Higgs boson is rather heavy. We estimate the Higgs boson mass $m_H = 754 \pm 20 {\text{(stat)}} \pm 20 {\text{(syst)}} {\text{GeV}}$ and the Higgs total width $\Gamma(H) \simeq 340 {\text{GeV}}$.


I. INTRODUCTION
A cornerstone of the Standard Model is the mechanism of spontaneous symmetry breaking that, as is well known, is mediated by the Higgs boson.Then, the discovery of the Higgs boson is the highest priority of the Large Hadron Collider (LHC) [1,2].
Usually the spontaneous symmetry breaking in the Standard Model is implemented within the perturbation theory which leads to predict that the Higgs boson mass squared, m 2 H , is proportional to λ R v 2 R , where v R is the known weak scale (246 GeV) and λ R is the renormalized scalar self-coupling.However, it has been conjectured since long time [3] that self-interacting four dimensional scalar field theories are trivial, namely λ R → 0 when Λ → ∞ (Λ ultraviolet cutoff).Even though no rigorous proof of triviality exists, there exist several results which leave little doubt on the triviality conjecture [4,5,6,7].As a consequence, within the perturbative approach these theories represent just an effective description, valid only up to some cutoff scale Λ, for without a cutoff there would be no scalar self-interactions and without them no symmetry breaking.However, within the variational gaussian approximation it has been suggested in Ref. [8] that this conclusion could not be true.The point is that the Higgs condensate and its quantum fluctuations could undergo different rescalings when changing the ultraviolet cutoff.Therefore, the relation between m H and the physical v R is not the same as in perturbation theory.Indeed, according to this picture one expects that the condensate rescales as Z ϕ ∼ ln Λ in such a way to compensate the 1/ ln Λ from λ R .As a consequence the ratio m H /v R would be a cutoff-independent constant.In other words, one should have: where ξ is a cutoff-independent constant.
It is noteworthy to point out that Eq. ( 1) can be checked by non-perturbative numerical simulations of self-interacting four dimensional scalar field theories on the lattice.Indeed, in previous studies [9,10] we found numerical evidences in support of Eq. (1).Moreover, our numerical results showed that the extrapolation to the continuum limit leads to the quite simple result: pointing to a rather massive Higgs boson without self-interactions (triviality).
The plan of the paper is as follows.In Sect.II we illustrate as triviality could coexist with spontaneous symmetry breaking within the simplest self-interacting scalar field theory in four dimensions.In Sect.III we briefly review the lattice indications for the non perturbative interpretation of triviality in self-interacting four dimensional scalar field theories and furnish our best numerical determination of the constant ξ in Eq. ( 1).Section IV is devoted to discuss some experimental signatures of the Higgs boson at LHC.Finally, our conclusions are drawn in Sect.V.

II. TRIVIALITY AND SPONTANEOUS SYMMETRY BREAKING
In this section we discuss the triviality and spontaneous symmetry breaking scenario within the simplest scalar field theory, namely a massless real scalar field Φ with quartic self-interaction λΦ 4 in four dimensions: where λ 0 and Φ 0 are the bare coupling and field respectively.As it is well known [11,12], the one-loop effective potential is given by summing the vacuum diagrams: Integrating over k 0 and discarding a (infinite) constant gives: This last equation can be interpreted as the vacuum energy of the shifted field: in the quadratic approximation.Indeed, in this approximation the hamiltonian of the fluctuation η over the background φ 0 is: Introducing an ultraviolet cutoff Λ we obtain from Eq. ( 5): It is easy to see that the one-loop effective potential displays a minimum at: Moreover so that According to the renormalization group invariance we impose that for Λ → ∞ Within perturbation theory one finds: Thus, the one-loop corrections have generated spontaneous symmetry breaking.However, the minimum of the effective potential lies outside the expected range of validity of the one-loop approximation and it must be rejected as an artefact of the approximation.On the other hand, as discussed in Section I, there is no doubt on the triviality of the theory.As a consequence, within perturbation theory there is no room for symmetry breaking.However, following the suggestion of Ref. [8] we argue below that spontaneous symmetry breaking could be compatible with triviality.The arguments go as follows.Write: where φ 0 is the bare uniform scalar condensate, thus triviality implies that the fluctuation field η is a free field with mass ω(φ 0 ).This means that the exact effective potential is: Moreover, the mechanism of spontaneous symmetry breaking implies that the mass of the fluctuation is related to the scalar condensate as: where a 1 is some numerical constant.Now the problem is to see if it exists the continuum limit Λ → ∞.Obviously, we must have: Note that now we cannot use perturbation theory to determine β(λ 0 ) and γ(λ 0 ).As in the previuos case the effective potential displays a minimum at: and Using Eq. ( 17) at the minumum v 0 we get: which in turns gives: This last equation implies that the theory is free asymptotically for Λ → ∞ in agreement with triviality: Inserting now Eq. ( 21) into Eq.( 17) we obtain: This last equation assures that λ φ 2 0 is a renormalization group invariant.Rewriting the effective potential as: we see that V ef f is manifestly renormalization group invariant.
Let us introduce the renormalized field η R and condensate φ R .Since the fluctuation η is a free field we have η R = η, namely: On the other hand, for the scalar condensate according to Eq. ( 23) we have: As a consequence we get that the physical mass m H is finitely related to the renormalized vacumm expectation scalar field value v R : It should be clear that the physical mass m H is an arbitrary parameter of the theory (dimensional transmutation).On the other hand the parameter ξ being a pure number can be determined in the non perturbative lattice approach.

III. THE HIGGS BOSON MASS
The lattice approach to quantum field theories offers us the unique opportunity to study a quantum field theory by means of non perturbative methods.Starting from the classical Lagrangian Eq.( 3) one obtains the lattice theory defined by the Euclidean action: where x denotes a generic lattice site and, unless otherwise stated, lattice units are understood.It is customary to perform numerical simulations in the so-called Ising limit.The Ising limit corresponds to λ 0 → ∞.In this limit, the one-component scalar field theory becomes governed by the lattice action with Φ(x) = √ 2κ φ(x) and where φ(x) takes only the values +1 or −1.
It is known that there is a critical coupling [13]: such that for κ > κ c the theory is in the broken phase, while for κ < κ c it is in the symmetric phase.The continuum limit corresponds to κ → κ c where m latt ≡ am H → 0, a being the lattice spacing.
As discussed in Section I, the triviality of the scalar theory means that the renormalized self coupling vanishes as when Λ → ∞.As a consequence in the continuum limit the theory admits a gaussian fixed point.
On the lattice the ultraviolet cutoff is Λ = π a so that we have: The perturbative interpretation of triviality [4,5] assumes that in the continuum limit there is an infrared gaussian fixed point where the limit m latt → 0 corresponds to m H → 0. On the other hand, according to Section II, in the triviality and spontaneous symmetry breaking scenario the continuum dynamics is governed by an ultraviolet gaussian fixed point where m latt → 0 corresponds to a → 0. As we discuss below, these two different interpretation of triviality lead to different logarithmic correction to the gaussian scaling laws which can be checked with numerical simulations on the lattice.
In Ref. [10] extensive numerical lattice simulations of the one-component scalar field theory in the Ising limit have been performed.In particular, using the Swendsen-Wang [14] and Wolff [15] cluster algorithms the bare magnetization (vacuum expection value): and the bare zero-momentum susceptibility: have been computed.According to the perturbative scheme of Refs.[4,5] one expects On the other hand, since in the triviality and spontaneous symmetry breaking scenario one The predictions in Eq. ( 35) can be directly compared with the lattice data reported in Ref. [10] and displayed in Fig. 1.We fitted the data to the 2-parameter form: We obtain a rather good fit of the lattice data (full line in Fig. 1) with α = 0.07560(49) , κ c = 0.074821(12) , χ 2 dof ≃ 1.5 . (37) Note that our precise determinations of the critical coupling κ c in Eq. ( 37) is in good agreement with the value obtained in Ref. [13] (see Eq. ( 30) ).
On the other hand, the prediction based on 2-loop renormalized perturbation theory is [5,16] (l = | ln(κ − κ c )|): together with the theoretical relations: We fitted the lattice data to Eq. ( 38) by allowing the fit parameters a 1 and a 2 to vary inside their theoretical uncertainties Eq. (39).The fit resulted in (dashed line in Fig. 1): It is evident from Fig. 1 that the quality of the 2-loop fit is poor.However, these results have been criticized by the authors of Ref. [16] and have given rise to an intense debate in the recent literature [17,18,19,20,21].
Additional numerical evidences would come from the direct detection of the condensate rescaling Z φ ∼ | ln(κ − κ c )| on the lattice.To this end, we note that: In Fig. 2 we display the lattice data obtained in Ref. [10] for Z φ , as defined in Eq. ( 41) versus m latt reported in Ref. [5] at the various values of κ.For comparison we also report the perturbative prediction of Z η taken from Ref. [5].We try to fit the lattice data with: Indeed, we obtain a satisfying fit to the lattice data (solid line in Fig. 2): By adopting this alternative interpretation of triviality there are important phenomenological implications.In fact, assuming to know the value of v R , the ratio ξ = m H /v R is now a cutoff-independent quantity.Indeed, the physical v R has to be computed from the bare v B through Z = Z ϕ rather than through the perturbative Z = Z η .In this case the perturbative relation [5]: obtained by replacing Z η with Z ϕ in Ref. [5] and correcting for the perturbative Z η .Using the values of λ R reported in Ref. [5] and our values of Z ϕ , we display in Fig. 3  which corresponds to: where the last error is our estimate of systematic effects.
One could object that our lattice estimate of the Higgs mass Eq. ( 47) is not relevant for the physical Higgs boson.Indeed, the scalar theory relevant for the Standard Model is the O(4)-symmetric self-interacting theory.However, the Higgs mechanism eliminates three scalar fields leaving as physical Higgs field the radial excitation whose dynamics is described by the one-component self-interacting scalar field theory.Therefore, we are confident that our determination of the Higgs mass applies also to the Standard Model Higgs boson.

IV. THE HIGGS PHYSICS AT LHC
For Higgs mass in the range 700 − 800 GeV the main production mechanism at LHC is the gluon fusion gg → H.The theoretical estimate of the production cross section at LHC for centre of mass energy √ s = 14 TeV and top quark mass m t = 178 GeV [2] is: The gluon coupling to the Higgs boson in the Standard Model is mediated by triangular loops of top and bottom quarks.Since the Yukawa coupling of the Higgs particle to heavy quarks grows with quark mass, thus bilancing the decrease of the triangle amplitude, the effective gluon coupling approaches a non-zero value for large loop-quark masses.On the other hand, we argued that the Higgs condensate rescales with Z φ .This means that, if the fermions acquires a finite mass through the Yukawa couplings, then we are led to conclude that the coupling of the physical Higgs field to the fermions must vanishes or be strongly suppressed.Fortunately, for large Higgs masses the vector-boson fusion mechanism becomes competitive to gluon fusion Higgs production [2]: σ(W + W − → H) ≃ 0.2 − 0.3 pb , 700 GeV < m H < 800 GeV .
In any case, for the luminosities expected at the LHC we expect several Standard Model Higgs events.
The main difficulty in the experimental identification of a very heavy Standard Model Higgs (m H > 650 GeV) resides in the large width which makes impossible to observe a mass peak.However, in the triviality and spontaneous symmetry breaking scenario the Higgs self-coupling vanishes so that the decay width is mainly given by the decays into pairs of massive gauge bosons.Since the Higgs is trivial there are no loop corrections due to the Higgs self-coupling and we obtain for the Higgs total width: where [1,2] Γ(H

FIG. 1 :
FIG.1: We show the lattice data for v 2 latt χ latt together with the fit Eq. (36) (solid line) and the two-loop fit Eq. (38) (dashed line) where the fit parameters a 1 and a 2 are allowed to vary inside their theoretical uncertainties Eq. (39).

FIG. 2 :
FIG.2:The lattice data for Z φ , as defined in Eq. (41), and the perturbative prediction Z η versus m latt .The solid line is the fit to Eq. (42).

FIG. 3 :
FIG. 3: The values of m H as defined through Eq. (45) versus m latt assuming v R = 246 GeV.The error band corresponds to one standard deviation error in the determination of m H through a fit with a constant function.