# The Trivial Higgs at LHC

###### Abstract

We further elaborate on our proposal for the Trivial Higgs that within the Standard Model is the unique possibility to implement the spontaneous symmetry breaking of the local gauge symmetry by elementary local scalar fields. The Trivial Higgs boson turns out to be rather heavy with mass GeV. We discuss the experimental signatures of our Trivial Higgs and compare with the recent data from ATLAS and CMS collaborations based on a total integrated luminosity between 1 fb and 2.3 fb. We suggest that the available experimental data could be consistent with our scenario.

## I Introduction

In a previous paper Cea:2009 () we have enlightened the scenario where the Higgs boson without self-interaction (Trivial Higgs) could coexist with spontaneous symmetry breaking. Due to the peculiar rescaling of the Higgs condensate, the relation between and the physical is not the same as in perturbation theory. According to this picture one expects that the ratio would be a cutoff-independent constant. In fact, our numerical results Cea:2009 () showed that the extrapolation to the continuum limit leads to the quite simple result:

(1) |

pointing to a rather massive Trivial Higgs boson GeV.

A cornerstone of the Standard Model is the mechanism of spontaneous symmetry breaking that is mediated by the Higgs boson.
In fact, the discovery of the Higgs boson is the highest priority of the Large Hadron Collider (LHC).
Recently, both the ATLAS and CMS collaborations Nisati:2011 (); Sharma:2011 () reported the experimental results for the search of the Higgs boson at the Large Hadron Collider
running at TeV, based on a total integrated luminosity between 1 fb and 2.3 fb.

The aim of the present paper is to further elaborate on the production mechanisms of our Trivial Higgs and to compare the theoretical expectations with selected available data
from LHC.

## Ii The Trivial Higgs

For Higgs mass in the range the main production mechanism at LHC is the gluon fusion . The theoretical estimate of the production cross section at LHC for centre of mass energy is Dittmaier:2011ti () :

(2) |

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 non-zero value for large loop-quark masses. On the other hand, we already argued Cea:2009 () that the non trivial rescaling of the Higgs condensate means that, if the fermions acquires a finite mass through the Yukawa couplings, then the coupling of the physical Higgs field to the fermions could be very different from the perturbative Standard Model Higgs boson. However, the coupling of the Higgs field to the gauge vector bosons is fixed by the gauge symmetries, therefore the coupling of the Trivial 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 the gluon fusion Higgs production Dittmaier:2011ti ():

(3) |

The main difficulty in the experimental identification of a very heavy Higgs () resides in the large width which makes impossible to observe a mass peak. In fact, the expected mass spectrum of our trivial Higgs should be proportional to the Lorentzian distribution:

(4) |

where we assume the central value of the Higgs mass according to Eq. (1) and is the Higgs total width.
Note that Eq. (4) is the simplest distribution consistent with the Heisenberg uncertainty principle and the finite lifetime
.

According to 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:

(5) |

Since the coupling of the Trivial Higgs to the gauge vector bosons is fixed by the local gauge symmetry, the decay width into gauge vector bosons is the same as in the case of the perturbative Higgs boson, consequently Djouadi:2005gi ()

(6) |

Equation (6) shows that in the high mass region GeV the Higgs total width depends strongly on . To take care of the energy dependence of the width in Eq. (4) we need as a function of E. To this end we may follow the calculations of Higgs total width within the Standard Model performed in Ref. Dittmaier:2011ti ().

In Fig. 1 (left) we show the energy dependence of the Higgs total width according to Tables 28 and 29 of Ref. Dittmaier:2011ti (). We have fitted the tabulated values of with the phenomenological relation (full blue line in Fig. 1, left):

(7) |

We obtain, therefore, the following Lorentzin distribution:

(8) |

where is given by Eq. (7), and the normalization is such that:

(9) |

In the limit , reduces to . In Fig. 1 (right) we display the Lorentzian distribution as a function of the energy E.

To evaluate the Higgs event production at LHC we need the Higgs production total cross section. As already discussed,
for large Higgs masses the main production processes are by vector-boson fusion and gluon-gluon fusion. The Trivial Higgs production cross section by vector-boson fusion is almost the same as in the perturbative Standard Model calculations.

In Fig. 2 (left) we show the energy dependence of the perturbative Higgs boson vector-boson fusion cross section
within the Standard Model at = 7 TeV reported in Table 11 of Ref. Dittmaier:2011ti (). We parametrize the
energy dependence of the cross section as (full blue line in Fig. 2, left):

(10) |

As concern the gluon fusion cross section, in Fig. 2 (right) we display the Standard Model Higgs production cross section at = 7 TeV reported in Table 5 of Ref. Dittmaier:2011ti (). Again we parametrize the energy dependence of the cross section as (full blue line in Fig. 2, right):

(11) |

where is expressed in GeV.

The gluon coupling to the perturbative Higgs boson in the Standard Model is mediated by triangular loops of top and bottom quarks. On the other hand, as already discussed, the coupling of the Trivial Higgs to the fermions is not fixed by the local gauge symmetry so that there is no a priori reasons to expect that these couplings would be proportional to the fermion masses. However, in the relevant high mass region GeV the energy dependence of is given essentially by the gluon distribution function. Thus, we may safely assume that the Trivial Higgs production cross section by gluon-gluon fusion is proportional to as given by Eq. (11).

To summarize, our main approximation for the total production cross section of the Trivial Higgs is:

(12) |

where the parameter takes care of our ignorance on the Yukawa couplings of the Trivial Higgs to the fermions.

To compare the invariant mass spectrum of our Trivial Higgs with the experimental data, we observe that:

(13) |

where is the number of Higgs events in the energy interval , corresponding to an integrated luminosity , in the given channel with branching ratio . The parameter accounts for the efficiency of trigger, acceptance of the detectors, the kinematic selections, and so on. Thus, in general depends on the energy, the selected channel and the detector. In our preliminary study we shall adopt the rather crude approximation:

(14) |

As concern the branching ratios BR(E), in the relevant Higgs mass region GeV the Higgs decays mainly into pairs of massive gauge bosons. The most important decay channels are , , and . Since the relevant branching ratios are almost independent on the Higgs mass and, assuming the Standard Model values, we are led to use the following values:

(15) |

The above approximations should be sufficient for the preliminary study of interest here.

## Iii

(Right) Comparison of the background subtracted experimental data corresponding to an integrated luminosity of 1.04 fb (data taken from Ref. Murray:2011 ()) with the Higgs event distribution according to Eq. (13) binned in energy intervals of 20 GeV assuming (red line) and (blue line).

For high Higgs mass the decay process has the largest branching ratio. Moreover,
the presence of charged lepton allows to obtain a good rejection of the QCD processes. The main background is given by the production of W + jet which, however, should be suppressed for large invariant mass .

Preliminary results from the ATLAS collaboration ATLAS:2011a () reported the experimental search of the Higgs boson at the Large Hadron Collider running at TeV, based on a total integrated luminosity of about 40 pb. In particular, in Fig. 3 (left) we display the distribution of the invariant mass for the
Higgs boson candidates corresponding to the process . According to
Ref. ATLAS:2011a (), the events were selected requiring exactly one lepton with GeV. The missing transverse energy in the event were required to be GeV. The invariant mass continuum background is parametrized as a falling exponential function. It is amusing to see that there are some events in the high invariant mass region which, however, are
not statistically significant due to the low integrated luminosity.
Recently, an update of search for the Higgs boson in this channel from the ATLAS collaboration corresponding to an integrated luminosity of 1.04 fb have been presented in Ref. Murray:2011 (). In Fig. 3 (right) we display the background-subtracted data. The large statistical uncertainties, mainly due to the background subtraction, do not show evidence of any structure in the invariant mass distribution.

To compare our theoretical expectation we use Eq. (13) with the branching ratios in Eq. (15).
In Fig. 3 (right) we compare the the Higgs event distribution binned in energy intervals of 20 GeV assuming
(red line), i.e. the Standard Model perturbative Higgs production cross section, and integrated luminosity of
1.04 fb. We see that our theoretical distribution is compatible with experimental data.
Interestingly enough, we find that the experimental data allow an enhanced gluon-gluon fusion cross section, as can be inspected
in Fig. 3 (right) where we also display the theoretical Higgs event distribution assuming the total production cross section of the Trivial Higgs is given by Eq. (12) with (blue line).

In the following Sections we consider the most important Higgs decay channels in order
to determine if the enhanced Higgs production by the gluon-gluon fusion process is compatible with
available experimental observations.

## Iv

(Right) Comparison of the distribution of the invariant mass for the process corresponding to an integrated luminosity of 1.6 fb with the Higgs event distribution according to Eq. (13) binned in energy intervals of 20 GeV assuming (blue line). The data from the CMS Collaboration have been extracted from Ref Sharma:2011 ().

The decay channel has the highest rate among all the processes where the Higgs boson decays into two Z bosons. The search strategy is to find some structures in the invariant mass . The major background is due to processes with production of Z + jet. In Fig. 4 we report the experimental data from ATLAS (left) and CMS (right) collaborations. Our theoretical expectations obtained from Eq. (13) are compared with the data (compare also with Fig. 10, panel c) of Ref. CMS:2011a ()). We see that our Higgs event distribution is not in contradiction with the experimental data, even though we cannot exclude that the data are compatible with the background-only hypothesis.

## V &

(Right) Comparison of the distribution of the invariant mass for the process corresponding to an integrated luminosity of 1.96 - 2.28 fb with the Higgs event distribution according to Eq. (13) binned in energy intervals of 20 GeV assuming and fb (blue line). The data have been extracted from Fig. 1, panel b) of Ref. ATLAS:2011b ().

The channels and have the
lowest branching ratios, see Eq. (15). Nevertheless, the presence of leptons allows to efficiently reduce the huge
background due mainly to diboson production.

In Fig. 5 (left) we display the distribution of the missing transverse energy for the channel
corresponding to an integrated luminosity of 1.04 fb. The data
from the ATLAS collaboration have been taken from Ref. ATLAS:2011b (). In the high mass region
GeV, where the background is strongly suppressed, there are a few events which compare well
with our theoretical prediction assuming an enhanced gluon fusion cross section, = 10.

In Fig. 5 (right) we report the invariant mass distribution for the golden channel
with integrated luminosity 1.96 - 2.28 fb from the ATLAS
collaboration ATLAS:2011b (). Even in this channel there are a few events in the high invariant mass region
which are in fair agreement with our Higgs event distribution with = 10 and fb.

## Vi Conclusion

To conclude, we argued that strictly local scalar fields are compatible with spontaneous symmetry breaking.
We stress that within the Standard Model our proposal is the unique possibility to implement the
spontaneous symmetry breaking of the local gauge symmetry by elementary local scalar fields.
Our Trivial Higgs boson turns out to be rather heavy. We compared our proposal with the recent results from ATLAS and
CMS collaborations and gave some evidence that experimental data are not in contradiction with our scenario. Of course, it should be emphasized
that the data could be compatible with the background-only hypothesis. We suggested that the Yukawa couplings of the
Trivial Higgs could be very different from the perturbative Standard Model Higgs boson. In particular, there are
no compelling reasons to expect that the couplings of the Trivial Higgs to fermions are proportional to the
fermion mass. As a consequence, we do not have enough arguments to set constraints on the Higgs production
by the gluon-gluon fusion processes. Nevertheless, it is remarkable that the available
data allow us to set an upper limit to the gluon fusion Higgs production cross section.

We are confident that in the near future forthcoming data from LHC will confirm our Trivial Higgs proposal.
In the most favourable case of an enhanced gluon fusion cross section an integrated
luminosity of order 10 fb will be enough to confirm our scenario. On the other hand, if the gluon
fusion cross section turns out to be suppressed with respect to the Standard Model perturbative Higgs boson,
, then we must wait for an integrated luminosity of several hundred of fb.

## References

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- (3) V. Sharma, Higgs Searches at CMS, The XXV International Symposium on Lepton Photon Interactions at High Energies, Mumbai, India, 22 - 27 August 2011.
- (4) S. Dittmaier et al., Handbook of LHC Higgs Cross Sections: 1. Inclusive Observables arXiv:1101.0593.
- (5) A. Djouadi, Phys. Rept. 457 (2008) 1.
- (6) The ATLAS Collaboration, Limits on the production of the Standard Model Higgs Boson in collisions at TeV with the ATLAS detector, arXiv:1106.2748.
- (7) B. Murray, LHC Higgs, International Europhysics Conference on High Energy Physics, Grenoble, France, 21 - 27 July 2011.
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