The Large Hadron Collider (LHC) [1] constructed at 27 km long underground tunnel at CERN is the most powerful accelerator and collider of protons and heavy ions worldwide.
The LHC orbit consists of eight circle arcs, four interaction sections hosting the LHC experiments and of four straight sections. One straight section hosts the accelerating cavities, two straight sections are used to steer the beams and one is dedicated for beams dump.
Ex 3. Protons at LHC are kept on LHC orbit by the Lorentz force. The force is applied in 1232 magnetic dipoles 15 m long each. Calculate the value of the magnetic field necessary to keep 4 TeV protons on the circular part of the LHC orbit.
First we derive the relation between the proton momentum , magnetic field and radius of the circular orbit R. Let us assume that proton velocity is perpendicular to the magnetic field, the Lorentz force induces the acceleration of protons:
(M is the rest mass of the particle and γ is its Lorentz factor). The relation can be further edited:
Where we introduced the momentum in energy units (Pc) and expressed it in electron-volts.
Because the value of we obtain very simple equation:
or in TeV and km units:
The length of the circular part (eight arcs) of the LHC orbit is determined by 1232 dipole magnets 15 m long each, i.e. 1232 · 15 m = 18.48 km that imply the radius R = 2.94 km.
Using the formula:
we easily find the value of B ~ 4.5 T.
Maximal designed magnetic field of ~8 T of the LHC dipoles will allow colliding 7 TeV 7 TeV protons.
Two general purpose experiments ATLAS [2] and CMS [3] were built to measure secondary particles created in proton-proton collisions. One of the main goals of the LHC experiments is to search for the Higgs boson. Because the creation of the Higgs boson is very rare event, the LHC parameters must allow for hundreds of millions of pp interactions per second in order to accumulate enough statistics of the Higgs boson decays. In 2012 the protons has been accelerated to 4 TeV - the highest energy ever reached at accelerators.
Accelerated protons are concentrated in bunches separated by 15 m. Each bunch contains
~100 billion of protons and at the collision region in the centers of the ATLAS and CMS detectors bunches are squeezed to perpendicular sizes of the order of 0.03 mm.
Probability of interactions and the cross section
Probability p that the particle interacts with the target is proportional to the density n of target particles and to the length L of the target:
with the coefficient of proportionality σ called the cross section. The cross section has the dimension of cm2, units of the cross section are barns (1 barn = 10-24 cm2).
In the experiment we can tune the interaction probability by changing the target parameters; however the value of the cross section however is the intrinsic characteristic of a particular particle interaction.
Ex 4. The value of the (total) cross section σ of LHC 4 TeV protons with 4 TeV protons is 0.1 barn. Calculate how many protons interact in one bunch crossing.
Let us assume N=100 billion of protons in each bunch of cylindrical shape with the radius r of 0.03 mm and L=10 cm. Probability that one proton from the first bunch interacts with N protons of the second bunch is:
and the number of interaction per one bunch crossing is simply given by:
Because the total number of protons in the bunches decreases due to pp interactions, the total number of interactions per bunch decreases during the run. The mean value of ~20 interactions per bunch crossing has been measured in the ATLAS experiment.
The Higgs boson
The weak and electromagnetic interactions have very similar strengths at high energies. At low energies weak interactions are suppressed because, contrary to massless photon, intermediate bosons W and Z have large masses. Main theoretical motivation to introduce the Higgs boson is to explain non-zero masses of intermediate bosons W and Z. In its simplest form, the theory predicts one electrically neutral spin-less Higgs boson. The Higgs boson is predicted to decay almost immediately into lighter particles. These could eventually decay subsequently to stable or long-living particles.
Theory does not predict the value of the Higgs boson mass, it has to be measured experimentally. The comparison of various experimental data with the theory indicates that if the Higgs boson in its simplest form exists, it shall have the mass around 100 GeV.
It is very important that for each possible value of the Higgs boson mass, theory can predict how often the Higgs boson will be created in pp collisions and with what probabilities it will decay to its final states. Predictions can be illustrated for 125 GeV Higgs boson. The 125 GeV Higgs boson will be created in 1 out of ~5 bilions (109) 4 TeV x 4 TeV pp interactions and it will decay to various pairs of particles with following probabilities .
58% to pairs of quarks (the heaviest quarks to which 125 GeV Higgs could decay);
21% to pairs of intermediate bosons ; 8.6% to pairs of gluons;
6.3% to pairs of the heaviest leptons ; 2.9% to pairs of quarks;
2.6% to pairs of intermediate bosons ; 0.23% to pairs of γ γ.
Ex 5. Theory predicts the probability of the Higgs boson decays to leptons pairs (in general to all fermion pairs) being proportional to squares of leptons (fermions) masses. Estimate the probabilities of 125 GeV Higgs boson decays to pair of muons and pair of electrons.
The probability of the 125 GeV Higgs boson decay to a pair of tau leptons is 6.3%. Masses of leptons are 1778 MeV, 106 MeV and 0.5 MeV for tauons, muons and electrons respectively.
Predicted probability for the Higgs boson decay to a pair of muons is therefore:
6.3% (106 MeV / 1778 MeV)2 ~ 0.022 %. Probability of the Higgs boson decay to electrons is negligibly small.
How to discover particles with very short life time
Unstable particles like the Higgs boson that decays almost instantaneously can be discovered using the energy and momentum conservations. Let us assume that the unstable particle with the mass M, the total energy E and the momentum decays to n secondary particles with masses mi, the total energies ei and momenta . Using the energy and momentum conservation:
the relation between the particle rest energy (mass) M, the total energy E and the momentum can be used to calculate the mass M by measured energies and momenta of secondary particles:
This variable is frequently called the invariant mass.
In 2012 the experiments ATLAS and CMS have announced the discovery of a new boson with the mass around 125 GeV [4].
The discovery of a Higgs boson in decays to pair of gammas is using exactly the formula derived in Ex.6 below.
Ex 6. Derive the formula for the invariant mass of pair of gamma particles with momenta forming the angle θ12
For gamma particles the energies equal to the sizes of momenta ( ):
Energies and directions of gammas are measured using ATLAS or CMS detectors. After careful selection of events with two gammas, the signature of an unstable particle is observed as an excess of events in the distribution of invariant masses in the region of the
mass of an unstable particle. On Figure 5 one can see such signals observed with the ATLAS detector [5].
Figure 5. The distribution of invariant masses of two gammas for selected events shows statistically significant excess of events in the region of 125 GeV. The figure is taken from
the publication [5].
Similar formula for the invariant mass of four leptons ( , and ) is used to evaluate and plot four-lepton invariant mass on Figure 6.
Figure 6. The distribution of the invariant masses of four leptons. Clear signal of the decay of a Higgs particle in red is observed above blue and violet distribution expected without
a Higgs boson. The figure is taken from the publication [5].
Summary of Part II
In 2012 the experiments ATLAS and CMS have announced the discovery of a new boson with the mass around 125 GeV.
The mass of the discovered particle (~126 GeV) is precisely determined by the position of peaks in the invariant mass distributions.
The new particle is electrically neutral; its charge is equal to the sum of charges of decays products.
Because the particle decays to two gammas (bosons) or four leptons (fermions), it must have an integer spin, i.e. it is the boson. Moreover the decay to two gammas excludes the value 1 of the spin a new boson.
More detailed study of decays agrees with the hypothesis of spin-less particle with positive parity as predicted for a Higgs boson.