The theory and discovery of the Higgs boson

Inhaltsverzeichnis

1 Standard Model of particle physics
1.1 History of the Standard Model
1.2 Theoretical description
1.3 Physical interactions

2 Models of the Higgs mechanism
2.1 Abelian Higgs Model
2.2 Weinberg-Salam Model

3 The Higgs boson
3.1 Mass of the Higgs boson
3.2 Production and decay of the Higgs boson .

4 Experimental Search
4.1 History of events
4.2 The Large Hadron Collider LHC
4.3 Experimental Data from ATLAS and CMS

5 Conclusion

6 Bibliography

1 Standard Model of particle physics

1.1 History of the Standard Model

The Standard Model is a gauge quantum field theory and owes its modern form to many contributors in its history of development. Quantum electrodynamics QED can be seen as the first Quantum field the- ory, it was popularised by Wolfgang Pauli in the 1940s. QED describes the electromagnetic field and explains its effect on charged quantum mechanical particles. In the light of QED Chen Ning Yang and Robert Mills proposed a non-Abelian gauge field theory in 1954 to describe the weak interaction¹. In 1961 Sheldon Glashow published ”Partial-symmetries of weak interac- tions”², in which he described a way to combine electromagnetic with weak interactions. Three years later François Englert and Robert Brout³ and Pe- ter Higgs⁴ independently proposed a mechanism, that would allow symmetry breaking. It is commonly known as the Higgs mechanism today and was incorpo- rated in 1967/1968 by Weinberg and Salam into Glashow’s electroweak theory, forming the basis of the modern Standard Model^{5 6}. The last major addition to the Standard Model was the inclusion of the strong interactions in the 1970s.

The Standard Model⁷ consists of 16 elementary particles, which are divided into 3 groups: 6 Quarks, 6 Leptons and 4 Bosons/force carriers The quark and leptons are also devided in 3 generations:

- first generation: Up quark, Down quark, electron, electron neutrino
- second generation: Charm quark, Strange quark, muon, muon neutrino
- third generation: Top quark, Bottom quark, tau, tau neutrino

The difference between corresponding particles in different generations is their mass energy. The higher the generation the heavier the particle is. The quarks and leptons are the matter particles, that the universe is build of. They are usually also referred to as fermions. The bosons are the so-called force carriers, due to the fact that they mediate forces between fermions. In addition to the 6 quarks, 6 leptons and 4 bosons, there exist their cor-responding anti-matter particles. The antiparticle has the same mass as its matter counterpart, but opposite electromagnetic charge and other properties. In collisions of matter and antimatter, they will annihilate each other and create energy, in accordance to the energy-matter equivalence.

The Higgs boson is a unique particle of the Standard Model. Its importance is to give the other particles their mass.

1.2 Theoretical description

In the Standard Model⁷ the elementary particles are described as quantised excitations of physical fields, it is thus a Quantum field theory. The particles can interact with each other via their fields and therefore form a dynamical

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Figure 1: The 16 elementary particles and the Higgs boson⁸

system. A dynamical system is usually described by its kinetic energy T and potential energy V, the quantity L = T − V is called the Lagrangian. The Lagrangian of the Standard Model is⁹

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The hc. mean the corresponding Hermitian conjugates.

This is a very simplified version, with L f = ψ D ψ + hc. being the Lagrangian

of the fermions, L Y coupling and − ¹

= ψ i y ij ψ j φ + hc. being the Lagrangian of the Yukawa

4 F μ v F μ v beingthegaugetermofthebosons.

The bosons form a mathematical Lie-group[[10]] called SU(³ ) × SU(2) × U(1). A unitary group U(n) is a group of unitary matrices that have the size n × n. The group operation is the matrix multiplication. The inverse of a unitary matrix is its Hermitian conjugate.

The special unitary groups SU(n) are the subgroups of U(n), in which all element matrices have detM = 1.

The SU(3) × SU(2) × U(1) group has 8 × 3 × 1 generators respectively.

Mathematically, it is the direct product of three groups, which forms another group.

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Physically, it can be interpreted as 8 gluons, 3 bosons (W + , W − , Z ⁰ ) and 1 photon γ.

The gauge bosons have properties that have to be conserved in interactions. A Quantum field theory, in which the Lagrangian of a system is invariant under certain symmetry gauge transformations, is called a Gauge theory. Applying a gauge transformation on the Lagrangian and checking for its invariance will tell us if a physical quantity is conserved in the system. This is a result of Noether’s theorems^{11 12}.

The gauge transforms that correlate to the conservation of electric charge can be written as

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One can see that a simple mass term m 22 A μ A μ fortheLangrangiandoesnot hold the gauge invariance and thus electric charge will not be conserved.

m ²

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Consequently, the theory dictates that the particles have to be massless. How- ever, everyday experience tells us that there is mass. To solve this problem a mechanism has to be added into the Lagrangian, that will give mass to the particles. It is called the Higgs mechanism and will be introduced in section 2.

1.3 Physical interactions

We know 4 fundamental forces in nature: Gravity, Weak force, Strong force and the electromagnetic force

Excluding gravity, the Standard Model⁷ describes the other 3 forces as force fields in space whose excitation would correspond to bosons. Those bosons can be seen as particles whch interacts between fermions.

The electromagnetic force is carried by the photon, the weak force is carried by the W and Z bosons and the stong force is carried by the gluons.

The fundamental interactions between bosons and fermions can be described by Feynman diagrams in figure 2:

- quark and anti-quark mediated by a gluon
- fermion and anti-fermion mediated by a Z-boson
- electron, muon, tau (electron generations e ′) and their corresponding antineutrino ν ′ mediated by a W-boson
- Up, Charm, Top quark (Up quark generations u ′) and Down, Strange, Bottom quark (Down quark generations d ′) mediated by a W-boson
- charged fermions f ′ and their anti-particles f ′ mediated by a photon γ

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Figure 2: The possible interactions being bosons and fermions

2 Models of the Higgs mechanism

2.1 Abelian Higgs Model

In the Abelian Higgs model^{13 14 15} the Lagrangian takes the form

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The first term describes the field strength of a simple gauge field of U(1), the second and third term add the kinetic and potential term of a complex scalar field.

D μ is called the covariant derivative, μ and λ are constants and φ is the wave function associated to the particle.

In the U(1) symmetry the local gauge invariances that conserve electromagnetic charge are

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Inserting (10) into (8) and (11) into (9) will give

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Going back to (9), two cases can be distinguished: μ ² is positive or negative. If it is positive, V will look like a parabola with a minimum at zero. The minimum is called the Vacuum expectation value(VEV) as it is the state of lowest energy for the system. A parabola-like shaped potential with the VEV being at zero does not break any symmetry.

If it is negative, V will look like what is called a mexican hat potential. The minimum of the potential can be calculated

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As the potential shown below in figure 3, the φ = 0 solution will lead to a local maximum. Thus leaving

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Figure 3: A 2-D comparison of V(φ) for μ ² > 0 and μ ² < 0¹⁶

Since there are two different φ with the lowest energy possible for the system, after some time has passed the system has to decide for one or the other, at this point the symmetry has been broken.

For convenience we will define φ as

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with χ and h being real field with no VEV.

Substituing (23) back into the Lagrangian (6) we will get the following

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The Lagrangian (²⁴ ) now describes a photon with mass M A = ev, a scalar field h with mass −μ ² > 0, a massless field χ and a mixed term (χ− A μ)-contribution, that can be removed via a gauge transformation

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