The Higgs Boson: From the Lattice to LHC

Abstract

We discuss the triviality and spontaneous symmetry breaking scenario where the Higgs boson without self-interaction coexists with spontaneous symmetry breaking. We argue that nonperturbative 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±20 (stat) ±20 (syst) GeV and the Higgs total width Γ(𝐻)≃340 GeV.

1. 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^2𝐻, is proportional to 𝜆_𝑅𝑣^2𝑅, 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 [4–7]. As a consequence, within the perturbative approach, these theories represent just an effective description, valid only up to some cut-off 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 [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∼lnΛ in such a way to compensate the 1/lnΛ from 𝜆_𝑅. As a consequence, the ratio 𝑚_𝐻/𝑣_𝑅 would be a cutoff-independent constant. In other words, one should have𝑚_𝐻=𝜉𝑣_𝑅,(1.1) where 𝜉 is a cutoff-independent constant.

It is noteworthy to point out that (1.1) can be checked by nonperturbative 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 (1.1). Moreover, our numerical results showed that the extrapolation to the continuum limit leads to the quite simple result:𝑚_𝐻≃𝜋𝑣_𝑅,(1.2) pointing to a rather massive Higgs boson without self-interactions (triviality).

The plan of the paper is as follows. In Section 2, we illustrate that triviality could coexist with spontaneous symmetry breaking within the simplest self-interacting scalar field theory in four dimensions. In Section 3, we briefly review the lattice indications for the nonperturbative interpretation of triviality in self-interacting four-dimensional scalar field theories and furnish our best numerical determination of the constant 𝜉 in (1.1). Section 4 is devoted to discuss some experimental signatures of the Higgs boson at LHC. Finally, our conclusions are drawn in Section 5.

2. 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 𝜆Φ⁴ in four dimensions:ℒ=12𝜕^𝜇Φ₀²−14𝜆⁰Φ⁴⁰,(2.1) where 𝜆₀ and Φ₀ 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:𝑉_1−loop𝜙₀=14𝜆⁰𝜙⁴⁰−𝑖2𝑑⁴𝑘(2𝜋)⁴ln−𝑘²⁰+𝑘²+3𝜆₀𝜙²⁰−𝑖𝜖.(2.2) Integrating over 𝑘₀ and discarding a (infinite) constant give𝑉_1−loop𝜙₀=14𝜆⁰𝜙⁴⁰+12𝑑³𝑘(2𝜋)³𝑘²+3𝜆₀𝜙²⁰.(2.3) This last equation can be interpreted as the vacuum energy of the shifted field:Φ₀=𝜙₀+𝜂(2.4) in the quadratic approximation. Indeed, in this approximation, the hamiltonian of the fluctuation 𝜂 over the background 𝜙₀ isℋ₀=12Π^𝜂²+12∇⃗𝜂²+123𝜆⁰𝜙²⁰𝜂²+14𝜆⁰𝜙⁴⁰.(2.5) Introducing an ultraviolet cutoff Λ, we obtain, from(2.3)𝑉_1−loop𝜙₀=14𝜆⁰𝜙⁴⁰+𝜔⁴64𝜋²ln𝜔²Λ²,𝜔²=3𝜆₀𝜙²⁰.(2.6) It is easy to see that the one-loop effective potential displays a minimum at3𝜆₀𝑣²⁰=Λ²√𝑒exp−196𝜆𝜋²⁰.(2.7) Moreover,𝑉_1−loop𝑣₀=−𝜔⁴128𝜋²,(2.8) so that𝑉_1−loop𝜙₀=𝜔⁴64𝜋²ln𝜙𝑣²⁰²⁰−21.(2.9) According to the renormalization group invariance, we impose that, for Λ→∞,Λ𝜕𝜕Λ+𝛽𝜕𝜕𝜆₀+𝛾𝜙₀𝜕𝜕𝜙₀𝑉^1−loop𝜙₀=0.(2.10) Within perturbation theory, one finds𝛾=0,𝛽=98𝜋²𝜆²⁰.(2.11) 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 1, 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 [8], we argue below that spontaneous symmetry breaking could be compatible with triviality. The arguments go as follows. WriteΦ₀=𝜙₀+𝜂,(2.12) where 𝜙₀ is the bare uniform scalar condensate; thus, triviality implies that the fluctuation field 𝜂 is a free field with mass 𝜔(𝜙₀). This means that the exact effective potential is𝑉_eff𝜙₀=14𝜆⁰𝜙⁴⁰+12𝑑³𝑘(2𝜋)³𝑘²+𝜔²𝜙₀=14𝜆⁰𝜙⁴⁰+𝜔⁴𝜙₀64𝜋²ln𝜔²𝜙₀Λ².(2.13) Moreover, the mechanism of spontaneous symmetry breaking implies that the mass of the fluctuation is related to the scalar condensate as𝜔²𝜙₀=3𝜆̃𝜙²⁰,𝜆̃=𝑎₁𝜆₀,(2.14) where 𝑎₁ is some numerical constant.

Now, the problem is to see if it exists the continuum limit Λ→∞. Obviously, we must haveΛ𝜕𝜕Λ+𝛽𝜆⁰𝜕𝜕𝜆₀+𝛾𝜆₀𝜙₀𝜕𝜕𝜙₀𝑉^eff𝜙₀=0.(2.15) Note that now we cannot use perturbation theory to determine 𝛽(𝜆₀) and 𝛾(𝜆₀). As in the previous case, the effective potential displays a minimum at3𝜆̃𝑣²⁰=Λ²𝑉_eff√𝑒exp−196𝜆𝜋̃²,𝑣₀=−𝑚^4𝐻128𝜋²,𝑚^2𝐻=𝜔²𝑣₀.(2.16) Using (2.15) at the minimum 𝑣₀, we getΛ𝜕𝜕Λ+𝛽𝜆⁰𝜕𝜕𝜆₀𝑚^2𝐻=0,(2.17) which in turn gives𝛽𝜆₀=−𝑎₁98𝜋²𝜆̃².(2.18) This last equation implies that the theory is free asymptotically for Λ→∞ in agreement with triviality:𝜆̃∼16𝜋²9𝑎₁1lnΛ²/𝑚^2𝐻.(2.19) Inserting now (2.18) into (2.15), we obtain𝛾𝜆₀=𝑎²¹916𝜋²𝜆̃.(2.20) This last equation assures that 𝜆̃𝜙²⁰ is a renormalization group invariant. Rewriting the effective potential as𝑉_eff𝜙₀=3𝜆̃𝜙²⁰²64𝜋²ln3𝑚𝜆̃^2𝐻𝜙²⁰−12,(2.21) we see that 𝑉_eff 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, namely𝑍_𝜂=1.(2.22) On the other hand, for the scalar condensate, according to (2.20) we have𝜙_𝑅=𝑍^𝜙−1/2𝜙₀,𝑍_𝜙∼𝜆⁰⁻¹∼lnΛ𝑚_𝐻.(2.23) As a consequence, we get that the physical mass 𝑚_𝐻 is finitely related to the renormalized vacuum expectation scalar field value 𝑣_𝑅:𝑚_𝐻=𝜉𝑣_𝑅.(2.24) 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 nonperturbative lattice approach.

3. 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 nonperturbative methods. Starting from the classical Lagrangian (2.1), one obtains the lattice theory defined by the Euclidean action:𝑆=_𝑥⎡⎢⎢⎣12𝜇Φ𝑥+𝜇−Φ(𝑥)²+𝑟⁰2Φ²(𝑥)+𝜆⁰4Φ⁴(𝑥)⎤⎥⎥⎦,(3.1) 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 𝜆₀→∞. In this limit, the one-component scalar field theory becomes governed by the lattice action𝑆_Ising=−𝜅_𝑥_𝜇𝜙𝑥+𝑒_𝜇̂𝜙(𝑥)+𝜙𝑥−𝑒_𝜇̂𝜙(𝑥)(3.2) with Φ(𝑥)=√2𝜅𝜙(𝑥) and where 𝜙(𝑥) takes only the values +1 or −1.

It is known that there is a critical coupling [13]:𝜅_𝑐=0.074834(15),(3.3) 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 1, the triviality of the scalar theory means that the renormalized self-coupling vanishes as 1/ln(Λ²/𝑚^2𝐻) 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 have𝜆∼1lnΛ/𝑚_𝐻∼ln𝜋1/𝑎𝑚_𝐻=ln𝜋1/𝑚_latt.(3.4) 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 2, 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 interpretations of triviality lead to different logarithmic correction to the Gaussian scaling laws which can be checked with numerical simulations on the lattice.

In [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 expectation value):𝑣_latt=||𝜙||,𝜙≡1𝐿⁴_𝑥𝜙(𝑥),(3.5) and the bare zero-momentum susceptibility:𝜒_latt=𝐿⁴||𝜙||²−||𝜙||²,(3.6) have been computed. According to the perturbative scheme of [4, 5], one expects𝑣²_latt𝜒_latt∼||ln𝜅−𝜅_𝑐||,𝜅⟶𝜅^+𝑐.(3.7) On the other hand, since, in the triviality and spontaneous symmetry breaking scenario, one expects that 𝑍_𝜑∼ln(Λ/𝑚_𝐻)∼|ln(𝜅−𝜅_𝑐)|, we have𝑣²_latt𝜒_latt∼||ln𝜅−𝜅_𝑐||²,𝜅⟶𝜅^+𝑐.(3.8) The predictions in (3.8) can be directly compared with the lattice data reported in [10] and displayed in Figure 1. We fitted the data to the 2-parameter form:𝑣²_latt𝜒_latt=𝛼||ln𝜅−𝜅_𝑐||².(3.9) We obtain a rather good fit of the lattice data (full line in Figure 1) with𝛼=0.07560(49),𝜅_𝑐=0.074821(12),𝜒²_dof≃1.5.(3.10) Note that our precise determinations of the critical coupling 𝜅_𝑐 in (3.10) are in good agreement with the value obtained in [13] (see (3.3)).

Figure 1 

We show the lattice data for 𝑣2latt𝜒latt together with the fit (3.9) (solid line) and the two-loop fit (3.11) (dashed line) where the fit parameters 𝑎1 and 𝑎2 are allowed to vary inside their theoretical uncertainties (3.12).

On the other hand, the prediction based on 2-loop renormalized perturbation theory is [5, 16] (𝑙=|ln(𝜅−𝜅_𝑐)|):𝑣²_latt𝜒_latt_2-loop=𝑎₁𝑙−2275ln𝑙+𝑎²(3.11) together with the theoretical relations:𝑎₁=1.20(3),𝑎₂=−1.6(5).(3.12) We fitted the lattice data to (3.11) by allowing the fit parameters 𝑎₁ and 𝑎₂ to vary inside their theoretical uncertainties (3.12). The fit resulted in (dashed line in Figure 1)𝑎₁=1.17,𝑎₂=−2.10,𝜅_𝑐=0.074800(1),𝜒²_dof≃132.(3.13) It is evident from Figure 1 that the quality of the 2-loop fit is poor. However, these results have been criticized by the authors of [16] and have given rise to an intense debate in the recent literature [17–21].

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 that𝑍_𝜙≡2𝜅𝑚²_latt𝜒_latt.(3.14) In Figure 2 we display the lattice data obtained in [10] for 𝑍_𝜙, as defined in (3.14) versus 𝑚_latt reported in [5] at the various values of 𝜅. For comparison, we also report the perturbative prediction of 𝑍_𝜂 taken from [5]. We try to fit the lattice data with𝑍_𝜙=𝐴ln𝜋𝑚_latt.(3.15) Indeed, we obtain a satisfying fit to the lattice data (solid line in Figure 2):𝐴=0.498(5),𝜒²_dof≃4.1.(3.16)

Figure 2 

The lattice data for 𝑍𝜙, as defined in (3.14), and the perturbative prediction 𝑍𝜂 versus 𝑚latt. The solid line is the fit to (3.15).

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]:𝑚_𝐻𝑣_𝑅≡𝜆_𝑅3,(3.17) becomes𝑚_𝐻𝑣_𝑅=𝜆_𝑅3𝑍𝑍^𝜑_𝜂≡𝜉(3.18) obtained by replacing 𝑍_𝜂 with 𝑍_𝜑 in [5] and correcting for the perturbative 𝑍_𝜂. Using the values of 𝜆_𝑅 reported in [5] and our values of 𝑍_𝜑, we display, in Figure 3, the values of 𝑚_𝐻 as defined through (3.18) versus 𝑚_latt for 𝑣_𝑅=246GeV. The error band corresponds to a one standard deviation error in the determination of 𝑚_𝐻 through a fit with a constant function. As one can see, the 𝑍_𝜑∼lnΛ trend observed in Figure 2 compensates the 1/lnΛ from 𝜆_𝑅 so that 𝜉 turns out to be a cutoff-independent constant:𝜉=3.065(80),𝜒²_dof≃3.0,(3.19) which corresponds to𝑚_𝐻=754±20±20GeV,(3.20) where the last error is our estimate of systematic effects.

Figure 3 

The values of 𝑚𝐻 as defined through (3.18) versus 𝑚latt assuming 𝑣𝑅=246GeV. The error band corresponds to one standard deviation error in the determination of 𝑚𝐻 through a fit with a constant function.

One could object that our lattice estimate of the Higgs mass (3.20) 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.

4. The Higgs Physics at LHC

Recently, both the ATLAS and CMS collaborations [22, 23] reported the experimental results for the search of the Higgs boson at the Large Hadron Collider running at √𝑠=7TeV, based on a total integrated luminosity between 1 fb⁻¹ and 2.3 fb⁻¹.

It is worthwhile to briefly discuss the main physical properties of our proposal for the trivial Higgs boson. For Higgs mass in the range 700−800GeV, 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=7TeV is [24]𝜎(𝑔𝑔⟶𝐻)≃0.06−0.14pb,700GeV<𝑚_𝐻<800GeV.(4.1) 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, balancing the decrease of the triangle amplitude, the effective gluon coupling approaches a nonzero 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 acquire a finite mass through the Yukawa couplings, then we are led to conclude that the coupling of the physical Higgs field to the fermions could be very different from the Standard Model Higgs boson. On the other hand, the coupling of the Higgs field to the gauge vector bosons is fixed by the gauge symmetries. So the coupling of our Higgs boson to the gauge vector bosons is the same as for the Standard Model Higgs boson. For large Higgs masses, the vector-boson fusion mechanism becomes competitive to gluon fusion Higgs production [24]:𝜎𝑊⁺𝑊⁻⟶𝐻≃0.02−0.03pb,700GeV<𝑚_𝐻<800GeV.(4.2)

The main difficulty in the experimental identification of a very heavy Standard Model Higgs (𝑚_𝐻>650GeV) 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 widthΓ(𝐻)≃Γ𝐻⟶𝑊⁺𝑊⁻+Γ𝐻⟶𝑍⁰𝑍⁰,(4.3) where [1, 2]Γ𝐻⟶𝑊⁺𝑊⁻≃𝐺^𝐹𝑚^3𝐻8√2𝜋√1−4𝑥_𝑊1−4𝑥_𝑊+12𝑥^2𝑊,𝑥_𝑊=𝑚^2𝑊𝑚^2𝐻Γ𝐻⟶𝑍₀,𝑍⁰≃𝐺^𝐹𝑚^3𝐻16√2𝜋√1−4𝑥_𝑍1−4𝑥_𝑍+12𝑥^2𝑍,𝑥_𝑍=𝑚^2𝑍𝑚^2𝐻.(4.4) Assuming 𝑚_𝐻≃750GeV, 𝑚_𝑊≃80GeV, and 𝑚_𝑍≃91GeV, we obtainΓ(𝐻)≃340GeV.(4.5)

A thorough discussion of the experimental signatures of our trivial Higgs is presented in [25] where we compare our proposal with the recent data from ATLAS and CMS collaborations based on a total integrated luminosity between 1 fb⁻¹ and 2.3 fb⁻¹. In fact, we argue that the available experimental data seem to be consistent with our scenario.

5. Conclusions

The Standard Model requires the existence of a scalar Higgs boson to break electroweak symmetry and provide mass terms to gauge bosons and fermion fields. 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 is proportional to the self-coupling. However, there exist several results which point to vanishing scalar self-coupling. Therefore, within the perturbative approach, scalar field theories represent just an effective description valid only up to some cutoff scale, for without a cutoff, there would be no scalar self-interactions and without them no symmetry breaking. In other words, spontaneous symmetry breaking is incompatible with strictly local scalar fields in the perturbative approach.

In this paper, we have shown that local scalar fields are compatible with spontaneous symmetry breaking. In this case, the continuum dynamics is governed by an ultraviolet Gaussian fixed point (triviality) and a nontrivial rescaling of the scalar condensate. We argued that nonperturbative lattice simulations are consistent with this scenario. Moreover, we find that the Higgs boson is rather heavy. Finally, the nontrivial rescaling of the Higgs condensate suggests that the whole issue of generation of fermion masses through the Yukawa couplings must be reconsidered.