2.2. The Standard Model of Particle Physics
2.2.1. Electroweak unification
One of the main goals of particle physics is to provide a unified picture of fundamental particles and their interactions. In the 19th century, Maxwell presented electricity and magnetism as different aspects of a unified theory of electromagnetism. In the 1960s, Glashow, Salam and Weinberg [7, 8, 9] (GSW) developed a unified picture of electromagnetism and weak interaction, known as electroweak theory. The electroweak interaction is described by the SU(2)L⊗U(1)Y
group, as can be seen in Table 2.4. To understand the implications of this unification, it is necessary to understand the characteristics of each symmetry group separately.
The electromagnetic theory (QED) is based on the unitary group U(1)Q, where Q denotes the electric charge. The charged-current weak interaction is invariant under SU(2) local phase transformations:
ϕ(x)→ϕ0(x) = exp[igα(x)·T]ϕ(x), (2.2) whereTare the three generators of theSU(2) group that can be written in terms of the Pauli spin matrices, σ, as T= 12σ, andα(x) are the three functions which specify the local phase at each point in space-time. The three gauge fieldsW1,W2, andW3, are introduced to satisfy the required local gauge invariance. Since the generators of theSU(2) gauge transformation are the 2×2 Pauli spin-matrices, the wavefunction ϕ(x) in Equation 2.2 must be written in terms of two components, and thereforeϕ(x) is denoted as the weak isospin doublet.
Fermion fields are described by spinors (u(p)) containing four components. The left-handed component is obtained by the projection of the operator 1−γ2 5 and the right-handed component is obtained by the projection of the operator 1+γ2 5, where γ5 represents the product of the four Dirac matrices. While QED and QCD are vector interactions with a current of the form jµ= ¯u(p0)γµu(p), the weak-charged current is a vector minus axial vector (V - A) interaction, and the four-vector current is given by
jµ= g
√2u(p¯ 0)1
2γµ(1−γ5)u(p). (2.3)
The weak-charged current already contains the left-handed chiral projection operator. Given the properties of theγ5matrix, the weak-charged current interaction only couples to left-handed
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2.2. The Standard Model of Particle Physics
(LH) chiral particle states and right-handed (RH) chiral antiparticle states. Therefore, the weak isospin doublets are only composed of LH chiral particle states and RH chiral antiparticle states and, for this reason, the symmetry group of the weak interaction is referred asSU(2)L.
The physical W-bosons can be identified as the linear combinations ofW1 and W2: Wµ±= 1
√2(Wµ1∓iWµ2). (2.4)
From the observed decay rates of muons and tau leptons, it can be shown that the strength of the weak-charged interaction is the same for all lepton flavours (lepton universality). According to the Cabibbo hypothesis, the weak interactions of quarks have the same strength as the leptons, but their weak eigenstates differ from the mass eigenstates. The unitary Cabibbo-Kobayashi-Maskawa [10, 11](CKM) matrix relates the weak and the mass eigenstates of quarks by:
d0 s0 b0
=
Vud Vus Vub
Vcd Vcs Vcb Vtd Vts Vtb
d s b
· (2.5)
The CKM matrix elements can be most precisely determined by a global fit that uses all available measurements and imposes the SM constraints (i.e. three generation unitarity). The fit results for the magnitudes of all nine quark mixing parameters are [4]:
d0 s0 b0
=
0.97427±0.00015 0.22534±0.00065 0.00351+0.00015−0.00014 0.22520±0.00065 0.97344±0.00016 0.0412+0.0011−0.0005
0.00867+0.00029−0.00031 0.0404+0.0011−0.0005 0.999146+0.000021−0.000046
d s b
· (2.6) TheSU(2)Lsymmetry of the weak interaction implies the existence of a weak-neutral current, the one corresponding to theW3. Nevertheless, that neutral current cannot be identified simply as the one due to the exchange of the Z-boson, since it was shown experimentally that the physical Z-boson couples to both left- and right-handed chiral states.
Since both the photon and Z-boson, with the corresponding fieldsAµ and Zµ, are neutral, it is reasonable that they can be expressed in terms of quantum states formed from two neutral bosons (Wµ3 associated with the SU(2)L local gauge symmetry, and Bµ associated with the U(1)Y local gauge symmetry) as follows:
Aµ= +BµcosθW +Wµ3sinθW, (2.7)
Zµ=−BµsinθW +Wµ3cosθW, (2.8) where θW is the weak mixing angle.
By imposing invariance under SU(2)Land U(1)Y local gauge transformations and respecting the electric charge and the third component of the weak isospin of the LH and RH chiral particle states, the following relationship between couplings is derived:
e=gsinθW =g0cosθW, (2.9)
and the weak hypercharge is given by:
Y = 2(Q−I3). (2.10)
2. Physics
Defining the coupling to the physical Z-boson as:
gZ= g
cosθW ≡ e
sinθWcosθW, (2.11)
the Z-boson interaction vertex factor can be expressed as:
−i1
2gZγµ[cV −cAγ5], (2.12)
where the vector and axial-vector couplings of the Z-boson are:
cV =I3−2Qsin2θW and cA=I3. (2.13) 2.2.2. The Higgs mechanism
The local gauge principle provides an elegant description of the interactions in the SM. How-ever, the required local gauge invariance is broken by terms in the Lagrangian corresponding to the particle masses. This is not a problem for QED and QCD, where the gauge bosons are massless, but it is not supported by the observation of the large masses of the W- and Z-bosons. Nevertheless, as shown by ’t Hooft [12, 13, 14], only theories with local gauge invariance are renormalisable, such that the cancellation of all infinities takes place among only a finite number of interactions. The Higgs mechanism generates the masses of the electroweak gauge bosons in a way that it preserves the local gauge invariance of the SM. It also gives mass to the fundamental fermions. As a consequence, a new field, the Higgs field, is added to the SM Lagrangian [15, 16, 17, 18].
In the Salam-Weinberg model, the Higgs mechanism is embedded in theU(1)Y⊗SU(2)Llocal gauge symmetry of the electroweak sector of the SM. The simplest Higgs model consists of two complex scalar fields, placed in a weak isospin doublet:
φ= φ+
φ0
= 1
√2
φ1+iφ2 φ3+iφ4
, (2.14)
whereφ0 is a neutral scalar field andφ+is a charged scalar field, such thatφ+and (φ+)∗ =φ− give the longitudinal degrees of freedom of theW+and W−. The Lagrangian of this doublet of the complex scalar is:
L= (∂µφ)†(∂µφ)−V(φ), (2.15)
with the Higgs potential expressed as:
V(φ) =µ2φ†φ+λ(φ†φ)2. (2.16) The shape of the potential depends on the sign ofµ2(it is required thatλ >0 for the potential to have a finite minimum), as can be seen in a simplified example in Figure 2.2. Forµ2 <0, the potential has an infinite set of degenerate minima satisfying:
φ†φ=−µ2 2λ = v2
2, (2.17)
where v is the non-zero vacuum expectation value of the Higgs field. The choice of the vacuum state breaks the symmetry of the Lagrangian, a process known asspontaneous symmetry breaking.
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2.2. The Standard Model of Particle Physics
Figure 2.2.: Graphic representation of the potentialV(φ) = 12µ2φ2+14λφ4 of a scalar real field φfor (left): µ2>0 and (right): µ2<0 [19].
In order to keep the neutral photon massless, the minimum of the potential must correspond to a non-zero expectation value only for the neutral scalar field φ0. The fields can be expanded about this minimum in the form:
φ(x) = 1
√2
φ1+iφ2 v+η(x) +iφ4
. (2.18)
From this spontaneous symmetry breaking, the creation of a massive scalar and three massless Goldstone bosons arises, which give the longitudinal degrees of freedom of theW±andZ bosons.
The Higgs doublet can be written in a unitary gauge3 as:
φ(x) = 1
√2 0
v+h(x)
, (2.19)
where h(x) is the Higgs field.
The mass terms can be identified by writing the Lagrangian of Equation 2.15 such that it respects theSU(2)L⊗U(1)Y local gauge symmetry of the electroweak model and are expressed in terms of the vacuum expectation value as:
mW = 1
2gv, (2.20)
mZ= 1 2vp
g2+g02, (2.21)
and
mγ = 0. (2.22)
Using the relationship between couplings from Equation 2.9, the masses of the Z- and the W-boson are related to one another as:
mW mZ
= cosθW. (2.23)
3The gauge in which the Goldstone fields are eliminated from the Lagrangian.
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The experimental verification of this relation provides a strong argument in favour of the validity of the Higgs mechanism.
In summary, the GSW model is described by four parameters: the gauge couplings g and g0 and the two free parameters of the Higgs potential µ and λ, which are related to the vacuum expectation value of the Higgs field,v, and the mass of the Higgs boson,mH, by:
v2 = −µ2
λ and m2H = 2λv2. (2.24)
The vacuum expectation value of the Higgs field can be calculated using Equation 2.20 and the measured values for mW and gW, resulting in v = 246 GeV. The remaining parameter, λ, can be obtained from the measured Higgs mass at the LHC.
The spontaneous symmetry breaking of the SU(2)L⊗U(1)Y gauge group of the SM can also be used to generate the masses of the fermions. The Yukawa couplings of the fermions to the Higgs field are given by:
gf =√ 2mf
v . (2.25)
It is interesting to see that for the top quark, with measured mass given in Table 2.1 and v= 246 GeV, the Yukawa coupling is close to unity. This indicates that the top quark is closely connected to the electroweak symmetry breaking.
2.2.3. Quantum Chromodynamics
Quantum Chromodynamics (QCD) is the Quantum Field Theory of the strong interaction.
The associated underlying symmetry is invariant under SU(3) local phase transformations, ψ(x)→ψ0(x) = exp [igsα(x)·T]ψ(x),ˆ (2.26) where ˆT=Taare the eight generators of the SU(3) symmetry group, which are related to the Gell-Mann matrices by Ta = 12λa, and αa(x) are eight functions of the space-time coordinate x. The required local gauge invariance can be fulfilled by introducing eight new fields Gaµ(x), where the index a = 1, . . . ,8, each corresponding to one of the eight generators of the SU(3) symmetry group. These eight new fields are the massless gluons of QCD.
The Dirac equation, including the interactions with the new gauge fields, is invariant under local SU(3) phase transformations, provided the new fields transform as:
Gkµ→Gkµ0 =Gkµ−∂µαk−gSfijkαiGjµ. (2.27) The last term in Equation 2.27 arises because the generators of the SU(3) group do not commute 4, thus allowing gluon self-interactions. fijk are the structure constants of the SU(3) group, defined by the commutation relations [λi, λj] = 2ifijkλk, andgS is the coupling constant of the strong interaction.
The quantum number of the strong interaction is called colour, and comes in three types:
red, green, and blue. Only particles that have non-zero colour charge couple to gluons. The quarks, unlike the leptons, carry colour charge and exist in three orthogonal colour states.
Also the gluons, unlike the photon, which are electrically neutral, carry colour charge5 and can therefore interact among themselves. The SU(3) colour symmetry is exact and QCD is invariant
4Therefore, QCD is known as a non-Abelian gauge theory.
5To be more precise, gluons carry simultaneously both colour charge and anticolour charge.
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2.2. The Standard Model of Particle Physics
under unitary transformations in colour space. Therefore, the strength of QCD interactions is independent of the colour charge of the quark.
Free quarks have never been observed directly. This is explained by the hypothesis of colour confinement, which states that coloured objects are always confined to colour singlet states and that no objects with non-zero colour charge can propagate as free particles. Colour confinement is believed to originate from the gluon-gluon self-interactions that arise because the gluons carry colour charge. Because the energy stored in the colour field between two quarks increases linearly with distance, it would require an infinite amount of energy to separate two quarks.
As a result, it becomes energetically preferable to break the colour string and create another pair of quarks with opposite-colour charge from the vacuum. Consequently, coloured objects arrange themselves into colourless bound hadronic states with no confining colour field between them. To date, all confirmed observed hadronic states correspond to colour singlets either in the form of mesons (qq),¯ baryons (qqq) or antibaryons (¯qq¯q). Also as a consequence of colour¯ confinement, the high-energy quarks produced in processes such ase+e−→qq¯do not propagate freely but are observed asjets of colourless particles. The process by which high-energy quarks (and gluons) produce jets is known as hadronisation.
The coupling constantgS is related toαS as:
αS= gS2
4π. (2.28)
It is important to note that the coupling constant (orαS) is not constant and its value depends on the energy scale of each interaction. The evolution of αS(q2) is given to lowest order by:
αS(q2) = αS(µ2) 1 +BαS(µ2) ln
q2 µ2
, (2.29)
with
B = 11NC−2Nf
12π , (2.30)
and µbeing a chosen scale at which the coupling constant is known6.
For NC = 3 colours and Nf ≤ 6 quarks, B is greater than zero and αS decreases with increasing q2. This behaviour of the coupling constant has important implications. At low energies (|q| ∼ 1 GeV), αS is of O(1) and perturbation theory cannot be used 7. At high energies (|q|>100 GeV),αS ∼0.1, which is sufficiently small so that perturbation theory can be used. This property of QCD is known as asymptotic freedom [20, 21, 22], meaning that at high energies quarks can be treated as quasi-free particles. Nevertheless, αS ∼0.1 is not small.
As a result, higher-order corrections cannot be neglected. For this reason, QCD calculations for processes at the LHC are usually calculated beyond lowest order.
2.2.4. The SM and beyond
Despite the profound and elegant theoretical ideas that sustain the pillars of the SM, and the numerous experimental results confirming its unquestionable success in explaining a wide-range of phenomenas at energies up to the electroweak scale, the SM is still an ad hoc compilation of theories, put together in such a way that it reproduces the experimental data.
6A common scale is ΛQCD, which effectively controls the hadron masses (ΛQCD≈200 MeV).
7This non-perturbative regime applies to the latter stages in the hadronisation process.
2. Physics
If neutrinos are assumed to be Dirac fermions, the SM has 25 free parameters8 that are not predicted, and must be measured in experiments. These parameters can be classified as [1]:
• Associated with the Higgs field
– 12 masses of fermions (or 12 Yukawa couplings to the Higgs field) – 2 parameters describing the Higgs potential: v and mH
• 3 coupling constants describing the strengths of the gauge interactions: g0, g and gS
• 8 mixing angles of the PMNS 9 and CKM matrices.
Besides the large number of free parameters of the SM, there are a handful of outstanding issues that the SM is not able to explain. Some of the open questions include:
• Observational facts unexplained by the SM:
– What is dark matter?
– What explains the matter-antimatter asymmetry of the Universe?
• Fine-tuning problems:
– The hierarchy problem associated with the Higgs: Why is the EW scale (mW) so small, in units of the (assumed) cutoff (MP lanck)? What prevents quantities at the EW scale, such as the Higgs boson mass, from getting quantum corrections on the order of the Planck scale?
– The flavour problem: Why are the fermion masses, mixing angles and CP (charge-parity) violating phases undetermined?
– The strong CP problem: Why is there so little strong CP violation?
• Why are there three generations of fermions?
• Why is there such a fermion mass hierarchy?
• Are neutrinos Majorana10 or Dirac particles?
• Is there gauge coupling unification at higher energies?
Because of these various open issues, the SM could be a low-energy approximation of the ultimate theory of particle physics. Other alternative theories beyond the SM, such as a su-persymmetry [23] or large-scale extra dimensions [24, 25], provide solutions to some of these questions, as well as reproduce the complete range of current phenomena that the SM has successfully explained.
8It would be 26 parameters if the Lagrangian of QCD would contain a phase, denoted by θCP, different from zero that would lead to CP (charge-parity) violation in the strong interaction.
9Unitary matrix that relates the three neutrino weak eigenstates (νe, νµandντ) to the neutrino mass eigenstates (ν1, ν2 andν3)
10Contrary to the Dirac neutrinos, Majorana neutrinos would be their own antiparticles.
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2.3. Top Quark Physics
2.3. Top Quark Physics
The top quark is the chargeQ= +23 and weak isospinT3= +12 partner of the b-quark in the third generation weak isospin doublet. Once theb-quark was discovered in 1977, there were many reasons to expect the existence of the top quark. Indirect evidence of its existence was obtained from limits on FCNC decays of theb-quark [26, 27] as well as from the absence of tree-level (lowest order) mixing in theBd0−B¯d0 system [28, 29, 30, 31], discarding the possibility of an isosinglet b-quark. Before it was discovered by the CDF and D0 Collaborations in 1995 [32, 33], the top quark mass was predicted by the electroweak precision data with very large uncertainty [34].
The top quark has two main properties that make it a particularly interesting and unique quark in the elementary-particle zoo:
• The Heaviest: the top quark is by far the heaviest known quark. The recently released first combination of the top quark mass measurements from the two experiments, CDF and D0, at the Tevatron, and from the two experiments, ATLAS and CMS, at the LHC, measures mtop = 173.34 ±0.27(stat.)±0.71(syst.) GeV, with a total uncertainty of 0.76 GeV, corresponding to a precision of 0.4% [5].
This large top quark mass translates into a large Yukawa coupling close to unity. Therefore, it is expected that the top quark may play a special role in electroweak symmetry breaking (EWSB) and might open a window to new physics. Due to its large mass, the top quark also gives a significant contribution to the Higgs self-energy. Before the Higgs boson discovery [35, 36], the top quark mass, together with the W-boson mass, gave indirect constraints on the possible Higgs boson mass.
• The Quickest to Decay : as a consequence of its large mass, the top quark has a very short lifetime (≈0.5×10−24 s [4]), and on average it decays before it can hadronise and before its spin is depolarised by the strong interaction. This “bare” quark transfers its properties to its decay products. This allows the study of its properties, such as the top quark polarisation or spin via the angular distribution of its decay products. Top quark physics is therefore described by perturbative QCD.
In the remainder of this section, the production mechanisms and decay modes of the top quark are described. Reviews of top quark physics at hadron colliders can be found in References [4, 37, 38, 39].
2.3.1. Top Quark Production
At hadron colliders, top quarks are dominantly produced in pairs via the strong interaction, or, with smaller cross section, as single top quarks via the electroweak interaction.
Top Quark Pair Production
Top quark pair production can be described by perturbative QCD. According to this approach, a hard scattering between two incoming hadrons (proton or anti-proton) is effectively the inter-action between its constituents (quarks or gluons), denoted as partons, each carrying a certain fractionx of the initial momenta of the incoming hadrons. Using the factorisation theorem, the
2. Physics
inclusive production cross section of the process pp → tt¯at the LHC can be expressed as the convolution of parton distribution functions (PDF) and a partonic cross section ˆσ [38, 40]:
σpp→t¯t(s, mt, µf, µr) = X
i,j=q,¯q,g
Z
dxidxjfi(xi, µ2f)fj(xj, µ2f)×σˆij→t¯t(ˆs, mt, µf, µr, αs(µ2r)), (2.31) where:
• parton distribution functions (PDF) fi(xi, µ2f): probability density to observe a parton flavour i with longitudinal momentum fraction xi in the incoming hadron, when probed at a factorisation scaleµf [37]. Figure 2.3 shows the momentum densities in the proton of the quarks, antiquarks and gluons for the CT10 PDF set [41, 42] atQ2 =µ2f = 100 GeV2.
• factorisation scale µf: scale that separates the hard scattering regime from the PDF associated with the incoming hadrons.
• renormalisation scale µr: artificial scale introduced by a renormalisation procedure, that redefines fields and parameters in order to eliminate ultra-violet divergences from the QCD Lagrangian. It is common to use the same scale for bothµr and µf.
• s: effective centre-of-mass energy squared of the parton-parton process, related to theˆ centre-of-mass energy squared of thepp process, s, by ˆs=xixjs.
In order to produce t¯t pairs, it is required that ˆs≥4m2t. Given the available high centre-of-mass energy, the large gluon density at smallx, as can be seen in Figure 2.3, and the presence of antiquarks in protons only as sea quarks, the tt¯production at the LHC is dominated by gluon-gluon fusion (≈80% at√
s= 7 TeV) (Figure 2.4(a)), with a minor remaining contribution from quark-antiquark annihilation (Figure 2.4(b)). For opposite reasons, the main t¯t production at the Tevatron (pp¯ collisions) is quark-antiquark annihilation (≈ 85% at √
s = 1.96 TeV).
Furthermore, given the high centre-of-mass energy at the LHC, top pairs are typically produced above the mass threshold, whereas at Tevatron energies (1.8 / 1.96 TeV at Run I / Run II), top pairs are typically produced at rest.
The theoretical calculation of thet¯tproduction cross section inppcollision at√
s= 8 TeV used in the main analysis of this thesis is performed at full next-to-next-to-leading-order (NNLO) in
s= 8 TeV used in the main analysis of this thesis is performed at full next-to-next-to-leading-order (NNLO) in