# PhD Thesis Chapter I

March 2, 2011 Leave a comment

**Chapter 1**

**Physics at HERA**

**1.1 The Standard Model**

Physics contains four fundamental forces: gravity, electromagnetism, strong and weak forces. The current understanding of particle physics is embodied in the ‘standard model’, which combines three of these forces in the framework of ‘gauge theories’. ^{[1]} Quantum Electrodynamics (QED) describes forces between charged particles in terms of photon exchange between them. This idea is extended to include the weak force (‘electroweak unification’), mediated by heavy W^{+}, W^{–}, Z^{0} bosons. Finally, in quantum chromodynamics (QCD), gluons mediate the attraction which binds quarks in hadrons.

The model is in excellent agreement with experimental results, but contains more than twenty arbitrary parameters which must be adjusted to fit the data. It is hoped that as progress is made, this number will be reduced. Additionally, further steps towards a ‘grand unified theory’ containing all known forces are hoped for.

At the moment, the standard model envisages three families of quarks and leptons. The quarks are arranged in doublets. Bound states of two or three quarks form mesons or baryons respectively. For example, two up quarks and a down form a proton. Leptons are currently thought to be elementary and are also arranged in doublets as shown in table 1.2. Here, heavy fermions are each accompanied by a neutrino.

**Table 1.1: Quark doublets.**

**Table 1.2: Lepton doublets.**

**1.1 The Standard Model**

**1.1.1 QED**

The Klein-Gordon (equation 1.1) and Dirac (equation 1.2) equations were devised as relativistic substitutes for the Schrödinger wave equation for fermions and bosons respectively.

where θ and φ are the wave-functions of their particles, m their mass, the s are matrices constructed from Pauli spin matrices, and γ^{mu}.γ_{mu} = γ^{0} δ/δt + γ . □

Quantum mechanics postulates that wave-functions may have arbitrary phase since phases do not influence any observable quantities. The requirements that the behavior of particles under the equations is invariant under phase transformations constitutes the powerful ‘gauge principle’. In particular, there may be ‘local’ transformations: phase changes dependent on space-time coordinates may be introduced. The gauge principle is equivalent to the demand that there should be invariance under the local transformation in equation 1.3.

**1.1 The Standard Model**

**Figure 1.1: Feynman diagrams for electron-positron scattering in QED.**

However, the differential operators in both of the relativistic field equations equation 1.1 and equation 1.2 will now operate on the phase factor α so the invariance is lost. It transpires that it is necessary to introduce vector potentials in which the particles described move in order to offset the changes and restore the invariance.

In QED, these potentials may then be expanded in a perturbation series whose expansion parameter at each order is α, the fine structure constant. This is small: α = e^{2}/4π = 1/137. The expansion allows the computation of amplitudes at a given order of α for scattering processes via consideration of four-vectors for incoming and outgoing particles and matrix elements representing the transition probabilities between initial and final states.

The amplitudes were represented in diagrammatic form by Feynman. A scattering process such as e^{+}e^{–} → e^{+}e^{–} was thus viewed at lowest order as involving exchange of a virtual photon (figure 1.1(a)).

Each part of a diagram for a given process is related to a corresponding term in the amplitude. There is a propagator term referring to the internal photon. Incoming and outgoing spinors are introduced for external particle lines, and there are polarization vectors for any photons in the initial or final states.

Higher order correction terms in the expansion take the form of additional lines in the diagrams. For example, the diagram in figure 1.1(b) allows for the possibility of virtual pair creation in the propagator. These corrections proved problematical: the relevant series diverge leading to unphysical infinite cross-sections.

The solution to this difficulty was found in the idea of renormalization. The integrals corresponding to loop corrections diverge at high momentum. Renormalization involves choosing an energy scale Λ_{QED} above which no contribution to the amplitude will be considered. This is equivalent to truncating the perturbation series after a fixed number of terms. It emerges that the infinities now cancel at each order: their effects are subsumed into the properties of the particle in question. It is not possible ever to measure ‘bare’ charge and mass because of these vacuum polarization effects.

Renormalized QED has shown remarkable predictive power. For example, the magnetic moment μ of the electron is given by equation 1.4

where g is the ‘gyromagnetic ratio’. The lowest order Dirac treatment predicts that g = 2 exactly, but in the broader picture an electron must be regarded as an entity surrounded by virtual pairs. As discussed above, these alter its apparent properties and mean that summing corrections to higher orders produces a different prediction for g. This prediction and the measurement agree to ten significant figures.

**1.1.2 Weak Interactions**

The weak interaction is responsible for β-decay. Fermi postulated a point-like process involving a proton becoming a neutron together with the emission of a positron to conserve charge and a neutrino to explain the observed energy spectrum. This picture has been modified (it is now based rather on quark transitions) but retains its validity.

In the framework of gauge theories, forces require a quantum to transmit their effects; intermediate vector bosons were postulated to fill this role. These are now known to be charged W^{+} and W^{–} together with the neutral Z^{0}. These play an analogous part in the weak interaction to that of the photon in QED.

By the Heisenberg uncertainty principle, the range of a force is related to the inverse mass of its quanta. The masses (greater/equal to 80 GeV/c^{2}) of the Ws and Zs show that the range of the weak force is relatively small. Further, the propagator term in the cross-section formula depends on M^{-2}_{W}; _{Z}, so the strength of the weak interaction is also comparatively small for low energy processes.

Weak interactions violate parity conservation: no process has so far been observed which involves a right-handed (i:e: positive helicity) neutrino. In the formalism, operators are formed from γ_{5} which is a product of Dirac γ matrices. The ‘V-A’ term (1-γ_{5}) projects out negative helicity states. Changing the sign of γ_{5} is equivalent to introducing a ‘V+A’ component and would result in a projection of positive helicity states. This would then allow processes producing the unobserved right-handed neutrinos. Since these are not observed the framework describing charged current weak interactions is known as ‘V-A’ theory.

**1.1.3 Electro-weak unification**

The unification of the electromagnetic and weak sectors is embodied in the theory of Glashow, Salam and Weinberg.^{[2], [3], [4]} This required i) the devising of a mechanism to generate mass for the weak bosons and ii) the identification of an appropriate gauge group.

Mass generation involves substitutions of derivatives analogous to equation 1.5

where □ is given by (1.6) so the interacting Maxwell equation for a mass-less photon (1.7) becomes (1.8).

which is the equation for a free massive vector field. Considering only spatial components, this means it is necessary for the ‘screening current condition’ to hold: i:e: that the current has a component proportional to the vector field.

This can only occur if an additional field is introduced. The Higgs field^{[5]} screens out the infinite range weak field which would result from having mass-less weak bosons.

It transpires that the correct gauge group here is SU(2) U(1). SU(2) is the ‘weak isospin’ space in which there is a symmetry of the weak sector and U(1) represents the standard phase invariance of electromagnetism. The conserved quantity in the whole of this space is hypercharge y given by

where Q is the electromagnetic charge and t_{3} is the third component of the weak isospin quantum number.

There are two gauge fields in the resulting wave equation each of which have their own coupling strength. The linear combination of fields

corresponds to the SU(2) and the U(1) parts of the overall gauge group. This represents a mass-less photon and a large mass W boson. The W mass is given by

where f is the vacuum expectation value of the Higgs field. The Z mass is related to this in terms of an angle which is the main free parameter of the theory

This angle also fixes the relative strengths of the unified parts via

The theory was vindicated with the discovery at CERN of the W and Z bosons with the correct masses. It has successfully predicted a large number of relevant cross-sections (e:g: for ee, ep scattering) and the decay width for the Z.

**1.1.4 QCD**

Considering baryons to be made up of three quarks had been shown to be productive prior to the advent of QCD. However, the Pauli exclusion principle was to force an extension to the simple quark model. The principle requires the wave-function of a fermion to be antisymmetric; that is, under exchange of a pair of fermions, the wave-function must change sign. Particles made up of an odd number of fermions are themselves fermions and thus the combined wave-function of a baryon must change sign under exchange of one of its quark components. However, the Δ^{++} resonance consists of three u quarks in identical spin states. It was therefore necessary to introduce an additional ‘internal’ degree of freedom to distinguish the quarks. This was termed ‘color’. It is important to remember that free color has never been observed, so all particles must be formed from color neutral superpositions.

QCD is the theoretical framework which describes the strong interactions between quarks in terms of this color charge. This is mediated by gluons which are themselves colored and so can feel the influence of other gluons. This results in the phenomenon of ‘color anti-screening’ in QCD. As the distance scale probed decreases, apparent electric charge increases in QED. However, gluons reduce the effective coupling at smaller distances. This is why the ideas of perturbation theory, developed for weak forces, are applicable to QCD which describes the strong force. The theory is described as ‘asymptotically free’, meaning that as the distance scale probed grows smaller, so does the coupling. Conversely, this has the important consequence that quarks cannot exist in the free state: as two quarks are separated the bond strength between them increases to the point where pair production of new quarks takes place.

**1.2 Types of events at HERA**

**1.2.1 Introduction**

HERA will collide 820 GeV protons with 30 GeV electrons. Physics of interest will lie in the extension of measurements to a much larger kinematic range than has been previously available.^{[7]} This section will outline the processes relevant to the work described later in this thesis. These fall into three main sections. Deep inelastic scattering events (DIS) are the mainstay of HERA physics. Secondly, many processes of interest take place via the mechanism of boson-gluon fusion (BGF). Finally more exotic processes are outlined.

The amount of data taken by an experiment is often quoted in units of inverse picobarns. A barn is 10^{-28} m^{2}. ZEUS is expected to accumulate 100 pb^{-1} for each year of operation. This figure of ‘integrated luminosity’ may be multiplied by the cross-section for a given process to deduce the number of such events to be expected in a sample.

For example, the cross-section for top-quark pair production via photon-gluon fusion (see section 1.2.3.1) is dependent on the mass of the top quark. This now means that is unlikely to be much larger than 0.01 pb and ZEUS is now thought to be unlikely to observe these events.

**1.2.2 Deep Inelastic Scattering Events**

**1.2.2.1 Introduction**

All elements of the standard model are necessary to understand DIS events. In the quark-parton model, these events are viewed as being due to the exchange of a boson between the incoming lepton and a quark. As mentioned previously, it is impossible to observe free quarks: at some separation the binding energy becomes sufficient to enable pair creation. By processes not at present fully understood, the scattered quark ‘hadronizes’ forming a ‘current jet’ of many energetic, strongly interacting particles.

DIS events are classed as ‘charged current’ (CC) if the intermediate boson is a charged W, neutral current (NC) if it is a Z or γ. If the event is of NC type, the scattered electron may be observed in the detector; CC events contain a neutrino which will escape from the detector without interacting.

The topology of these events is generally described in terms of a particular formalism, described in the next section.

**1.2.2.2 General Kinematics**

Several variables are used to describe event kinematics. Q^{2} is the squared four-momentum transfer between the quark and the outgoing lepton.

In these equations p_{e}, p_{l} and P are the four-momenta of the incoming and scattered lepton, and the incoming proton respectively, and ν = q.p/m_{p} is the energy transferred by the current in the proton rest frame.

In the limit of small lepton masses the variables Q^{2}, x and y can be determined from the outgoing lepton energy E_{l} and the lepton scattering angle θ_{l} (measured with respect to the electron direction).

In CC events the neutrino energy and scattering angle cannot be measured by the detector but Q^{2}, x and y can be calculated from the energy E_{j} and production angle θ_{j} of the current jet (measured with respect to the proton direction).

Physically, x is the fraction of the proton momentum carried by the struck quark and can thus take values between 0 and 1. It is related to Q^{2} by a well known relation which is shown in equation 1.24.

The interpretation of y at HERA is less straightforward. In fixed target experiments, it is the fractional energy transfer in the laboratory frame and is given by dividing the energy of the exchanged boson by the incoming lepton energy. In a lepton-quark frame in which ŝ = xs is the squared sub-process total energy, y = Q^{2}/ŝ.

The value of s at the HERA nominal beam energies is given by equation 1.25.

Clearly for Q^{2} to take this maximum value is it required that x = y = 1 meaning that all of the proton momentum is carried by the struck quark; y = 1 corresponds to maximum Q^{2} for the particular struck quark.

**1.2.2.3 Jacquet-Blondel Kinematics**

Jacquet-Blondel kinematics consists of a parametrization of the above variables. The standard equations^{[8]} express Q^{2} in terms of jet angle θ_{jet} and energy E_{jet} as follows:

This formalism is applicable to all CC and NC processes. At low Q^{2} the scattering is dominated by the pure electromagnetic term. As Q^{2} increases, the γ/Z^{0} interference term becomes important and finally above around Q^{2} = 104 (GeV/c)^{2} the pure weak term dominates.

DIS events are defined to be those which have high Q^{2} and high ν in distinction from elastic processes. [If the scattering is elastic, then the four-momentum of the proton is unchanged by the collision: p = p’. By conservation at the vertex, p + q = p’ so (p’)^{2} = p^{2}+2p.q+q^{2} and 2p.q = -q^{2} = Q^{2} = 2m_{p}.ν holds for elastic scattering.] The relevant Feynman diagram is shown in figure 1.2.

**Figure 1.2: Feynman diagram for DIS.**

**1.2.2.4 Structure Functions and Scaling**

The kinematical dependency of the DIS cross-section factorizes into leptonic and hadronic parts, each represented by a tensor.

where q, p are four-vectors for the intermediate boson and the incoming hadron respectively. This leads to the following form for the NC cross-section

in which three new parameters have been introduced. F1, F2 and F3 are the structure functions of the proton.

Bjorken scaling postulates that in the DIS regime, the Q^{2} dependence of the structure functions should disappear, leaving only the x variation. Intuitively, this is pictured as being related to the idea that at high Q^{2} the virtual boson interacts at short distances inside the proton, essentially with only one parton. This may then be regarded as free on the short timescales involved and the scattering is elastic.

Then y = x and the structure functions depend only on one variable. QCD predicts a violation of this scaling behavior due to the addition of gluon loops.

**1.2.3 Boson-gluon Fusion**

**1.2.3.1 Heavy-Flavor Pair Production**

The phrase heavy-flavor conventionally refers to events involving the three heaviest quarks, that is the charm, bottom and the so far unobserved top quark. Events are mediated by the exchange of photons or bosons which fuse with a gluon radiated by the proton. This gives the BGF mechanism its name. The lowest order QCD diagrams are shown in figure 1.3.

The CC process is important for top production. However the total cross-section for all processes has a strong dependence on the quark mass.^{[8]}

**Figure 1.3: The two lowest order QCD diagrams for BGF. A quark/antiquark pair is formed.**

At present, the top quark mass is thought to be 122+41-32 GeV/c^{2} so it is no longer expected that HERA experiments will be able to observe any top quark pair events.

NC processes (in fact mostly gamma-gluon exchanges at low Q^{2}) dominate the BGF cross-section.

**1.2.3.2 J/ψ Production**

J/ψ particles may also be produced by the BGF mechanisms shown in figure 1.4.

Perturbative QCD has been used extensively to make calculations concerning ccbar pair production as a whole. This is a useful approach because the strong coupling constant S becomes relatively small at the charmed quark mass scale. This can be seen from the leading logarithm approximation for S as a running coupling constant (this form is valid only for six quark flavors).

in which Λ is a QCD renormalization parameter.

**Figure 1.4: Lowest order diagram for inelastic J/ψ production.**

The running of the coupling constant arises, analogously with the QED case, from choosing a mass scale at which to cut off higher-order diagrams with many loops. The running of the coupling constant is a consequence of the color anti-screening mentioned previously. Clearly Q^{2} can be regarded as a measure of the penetration of the probe so it is expected that the coupling will decrease with higher Q^{2}.

Λ_{QCD} has been found to be between 0.1 and 0.2 GeV. The cross-section is dominated by almost real photons: Q^{2}_photon = 0 so the gluon must have Q^{2}_gluon ~ M_{ψ}^{2}.

From equation 1.31 it can be seen that the coupling is relatively weak at the relevant mass scale.

Data so far available (e.g. EMC) supports the ‘color singlet’ model of Berger and Jones^{[9]} as the mechanism for J/ψ production. The model successfully reproduces the transverse momentum and Q^{2} dependence of the EMC data.^{[10]}

In order to extrapolate cross-sections down to the very low Q^{2} domain of J/ψ production, the Weizsäcker-Williams approximation^{[11]} is important. It states that the cross-section for reactions with initial state photon *bremsstrahlung* may be factorized into two terms. One is the cross-section at the total energy after photon emission. The other is dependent on the emitted photon and initial particle energies.

The J/ψ cross-section is sharply peaked in x just above x ~ (m_{ψ}^{2})/s.^{[12]}

Because HERA is a high energy machine with large and variable s, the peaks occur at much lower values of x than at previous experiments so these events will be useful to probe the gluon distribution^{[13]} in a new domain. They will have extremely low Q^{2} (10^{-4} GeV^{2} < Q^{2} < 10^{-2} GeV^{2} for example) and hence a very small scattered electron angle.

**1.2.4 Exotica**

Three main avenues for investigation of exotic physics^{[14]} exist at HERA. These are searched for excited electrons, leptoquarks/leptogluons and supersymmetry. The crucial parameter in this context at HERA is √s = 314 GeV, which is the amount of energy available in the center of mass frame for the creation of new states.

**1.2.4.1 Excited Electrons**

The existence of excitations would indicate that the presence of a previously undetected substructure. This would require another internal degree of freedom like color: all observed states would be ‘hypercolor’ neutral. Hypergluon exchange would confine preons on some compositeness scale. The excited states decay (e.g. l* → e + γ or q* → q + W^{+} or W_{–}) so construction of invariant mass plots from the decay products should prove a useful method of investigation.

**1.2.4.2 Leptoquarks and Leptogluons**

These are resonant states between leptons and partons.^{[15]} Leptoquark production occurs at fixed x (x = m_{leptoquark} = s). The events are similar to DIS, which forms the major background. The DIS cross-sections show a Q^{-4} dependency whereas leptoquark decays are isotropic: they will show no Q^{2} dependence. Thus selecting events with Q^{2} greater than or equal to 1,000 will produce a clean signature with good event rates if leptoquarks exist.

HERA will be able to observe leptoquarks up to √s, the kinematic limit.

**1.2.4.3 Supersymmetry**

Supersymmetry, or SUSY, envisages a more broad symmetry than the usual multiplet schemes. Here, fermions and bosons may be members of the same gauge group multiplet. As usual, however, the supersymmetry must be broken in a manner consistent with low-energy phenomenology.

The minimal model gives each particle a SUSY partner so that there are now eight gluinos as well as the familiar gluons and also there are squarks and sleptons. Processes analogous to standard DIS are envisaged in which the gauge boson is replaced by a gaugino (e.g. a Zino or a Wino). This leads to a squark and a selectron in the final state.

The cross-sections for production at HERA should be sufficient to observe SUSY-NC processes providing that m_{ebar} + m_{qbar} is less than or equal to 200 GeV/c^{2} (1.32).