Φdoublet were transferred to theW±and Z0 bosons, which by gaining mass, obtain an additional degree of freedom in the form of a possible longitudinal polarisation. Expanding the kinetic term in Eq.1.20using Eq.1.21, leads to new terms that include interactions between theW±,Z0andHbosons as well as the mass terms, where the masses of the bosons are:
mW = gv
2, mZ= gv
2 cosθW, mH = rvλ
2 . (1.22)
In addition, the mass termsmΨΨ¯ of fundamental fermions also break theSU(2)Lsymmetry. This problem is cured in the SM by introducingSU(2)L-covariantYukawainteraction terms∼gfΨ¯LΦΨR terms, where thegf are theYukawacouplings that express the relative strength of the interaction of the fermions with the Higgs field. After the SSB, the Lagrangian includes terms∼ gfvΨΨ¯ , which are mass terms where the mass of the particle is given by the Yukawa coupling and the VEV. Both the Yukawa couplings and the masses of theW±andZ0bosons are free parameters, that are determined experimentally.
The Higgs boson(15)was discovered at the LHC by the ATLAS [14] and CMS [15] experiments in 2012, with a mass of approximately 125 GeV, concluding almost a 50-year period since the proposal of the SSB mechanism in the SM.
1.5 Strong interaction
The strong interaction within the SM is described by QCD, a theory based on theSU(3)Ccolour gauge symmetry. The theory describes strong interactions between quarks, and the interaction mediators;
gluons. The formulation of the theory can be tracked back to the work of Fritzsch, Gell-Mann and Leutwyler [16] and Gross, Wilczek and Politzer [17,18], employing the previous work of Yang and Mills [4] on the non-Abelian gauge symmetries in QFT. In QCD, quarks carry a colour charge(16). The strong interaction is mediated by eight massless gluons, determined by the dimension of representation of theSU(3)Cgroup. Similarly toSU(2)non-Abelian group, gluons also interact with each other and carry colour charge.
The kinetic and gluon field terms in the Lagrangian are:
L=X
q
iΨ¯qjD/jkΨkq−1
4GaµνGµνa , (1.23)
with the sum overqdenoting the sum over all quark flavours and indices j,k =1,2,3 sum over colour states. The gauge-covariant derivative is defined as:
Dµjk= δjk∂µ−igs(Ta)jkGµa, (1.24) wheregs is the strong coupling constant andTaare Gell-Mann’s matrices, generators of theSU(3)C representation, and theGµaare the gluon fields. The field-strength tensorGµνa for the gluon fields is defined analogously as in Eq.1.7.
(15)The discovered particle appears to have properties consistent with the SM-predicted Higgs boson.
(16)The namecolouris motivated by the fact that there are three unit colours in contrast to, for example, one unit electric charge.
1. The Standard Model of elementary particles
Let us give a very brief overview of history of the strong interactions to highlight a particularly important aspect of the QCD theory. By the 1960s a large number of particles, today known as hadrons, were discovered. These particles had a large production cross-section, indicating that these particles were interacting via a strong interaction. It was hypothesised that this was the same interaction responsible for processes in atomic nuclei. It was noticed that the masses of the hadrons seemed to follow a hierarchy of some underlying symmetry. Gell-Mann [19] and Neeman [20] realised the particles formed mass multiplets of irreducible representations of theSU(3) group, today referred to as theSU(3)flavourgroup. This symmetry is today considered “accidental” because of the very similar masses of the up, down and strange quark. These quarks formed the bound states of all the known mesons and baryons discovered by the 1960s. Gell-Mann and Zweig [21] indeed argued, that theSU(3) flavour symmetry had a physical meaning, that the hadrons were bound states of particles, that Gell-Mann referred to as quarks. The problem with this hypothesis at that time was that quarks were never observed. Studies of structure of the proton via deep inelastic scattering (DIS) experiments during 1960s and 1970s also suggested that proton is a composite particle containing more fundamental point-like constituents.
It was discovered by Gross, Wilczek [17] and Politzer [18] in 1973, that the QCD theory exhibits so-calledasymptotic freedom – the strength of the interaction decreases with momentum transfer.
The relative strength of an interaction in QFT is determined by its coupling constant. Despite the confusing name, the coupling constant is a function of an energy scale at which it is probed. This is a consequence of higher-order corrections in QFT which also impact the coupling constant itself. The so-calledbarequantities that enter the Lagrangian do not correspond to the physical quantities at a particular energy scale of an experiment. The procedure of renormalisation is used to redefine the perturbative expansions in terms of measurable quantities. The dependence of the strong coupling αS(17)in QCD on the energy scaleQ2in the next-to-leading order of perturbative series is:
αS(Q2) = αs(µ2R)
1+ β0αs(µ2R)log(Q2/µ2R), β0= 11nC−2nf
12π , (1.25)
whereαS(µ2R) is the coupling constant at a particular renormalisation scale µ2R, nC is the number of colours (3 in SM) andnf is the number of quark flavours (6 in SM). The equation relates the value of the coupling constant at some probed energy scaleQ2with a value of the constant at some renormalisation scale(18) µ2R.
It can be shown thatαSdiverges if the denominator goes to zero, which happens for a characteristic scale referred to as theΛQCD ≈200 MeV. Eq.1.25can be rewritten in terms ofΛ2QCD
αS(Q2)= 1
β0log(Q2/Λ2QCD). (1.26)
Eq.1.26points at two features of the strong interactions. As mentioned, QCD exhibits asymptotic freedom, where in theQ2→ ∞limitαSconverges to zero. Secondly, for low energies, neither quarks
(17)TheαSand thegSin the Lagrangian are often both referred to as the coupling constant. The relation between the two in natural units (~=c=1) isgs=p4παS.
(18)This scale dependence is a result of the limited-order perturbative expansion. The cross-section of a process does not depend on the choice of some arbitrary scale, hence the scale dependence of the coupling constant and the scale dependence of the amplitude compensate each other.
1.5. Strong interaction
nor gluons can be observed as free particles, because they areconfined to bound states, which are colourless. Confinement is indeed supported by the experimental evidence, though exact theoretical proof does not yet exist. The divergence ofαS forQ2 →Λ2QCDonly demonstrates that the perturbative expansion for αS is not valid in this kinematic region. At low energies, other non-perturbative approaches are generally used to describe quark and gluon interactions, such as the lattice QCD [22]
which has also provided clues for the quark confinement.
The behaviour of the strong coupling implies further phenomena particularly relevant to high-energy collider experiments. High-energy quarks and gluons initiatejets, collimated showers of hadrons. Due to the strong coupling, hard quarks lose energy by radiating further gluons, and gluons split intoqq¯ pairs. This process can be described via perturbative QCD. At a sufficiently small energy, the showering stops. The produced quarks bind into mesons and baryons, a process referred to as thehadronisation. The hadronisation is a non-perturbative process, typically described by phenomenological model, such as the string model [23] or the cluster model [24].
1.5.1 Parton model and the structure of proton
The parton model was proposed by Feynman [24] to describe the structure of hadrons. It is based on the evidence from electron-proton DIS experiments which showed that at sufficiently high momentum transfer, electrons scattering on the proton appeared to scatter on point-like constituents of the proton.
In the parton model, the DIS on a proton is described as an elastic scattering on a parton which carries a fraction xof the momentumpof the proton. The probability that the parton entering the scattering carries momentumxpis given by aparton density function f(x). The observation that f(x) does not depend on the momentum transferQ2at sufficiently high energies, is calledBjorken scaling, discovered by Bjorken in 1968 [25].
After the discovery of the asymptotic freedom of QCD, it was thought that the partons in the proton areuanddquarks. With increasing energy of the DIS experiments, however, it was realised that the energy scale independence of the parton density functions (PDFs) was violated. This is because the proton contains not only theuudquarks, but also a “sea” of other (anti-)quarks and gluons. From this point, when referring to partons in a proton, all of the quark/anti-quark types and gluons are meant.
To provide predictions about the structure of the proton requires non-perturbative approaches to QCD calculations. Because this is very challenging, the PDFs f(x,Q2)are measured experimentally.
It is, however, possible to calculate scale dependence of the PDFs in perturbative QCD via Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [26–28]. These functions allow to extrapolate PDF predictions from measurements performed at some energy scale to a different energy scale. An illustration of the dependence of the PDFs on the energy scale is shown in Fig.1.2.
In this thesis we focus on the studies ofhard-scatteringprocesses, in which partons from thepp collision interact and produce different particlesX, for example a top quark–anti-quark pair. Therefore, the proton PDFs are a crucial ingredient because the cross-section of the pp → X production is a convolution ofx1x2→X processes, wherex1andx2are the incoming partons. The convolution is described by thefactorization theorem:
σp p→X =X
i,j
Z dxi
Z
dxjfi(xi, µF)fj(xj, µF)σi j→X(i,j, µR, µF). (1.27)
1. The Standard Model of elementary particles
The partonic cross-sections σi j→X is summed over all types of partons in the proton and over all possible momentum fractionsxi and xj of the incoming partons, weighted by their corresponding PDFs fi, fj. The factorization theorem assumes that the partons are behaving as free particles, which is a valid assumption for processes involving high-momentum transfer based on the asymptotic freedom of QCD. The partonic cross-sectionσi j→X(i,j, µR, µF) depends on the momenta(19)of the incoming partonsi, jand also on the choice of the renormalisation scale µR and the factorisation scaleµF. The factorisation scale defines the scale which separates perturbative and non-perturbative parts in the evolution of the PDFs and in the partonic cross-section.
x
Fig. 1.2: Proton parton density functions from the NNPDF collaboration [29] at two different energy scales. At higher energy scale (right) the relative contribution of gluons (red line) and sea quarks at low momentum fractionxincreases.