High Energy Physics (3HEP)aka “particle physics” Dr. Paul D. Stevenson 2007
Outline of Course • Basic building blocks, particles and antiparticles, hole theory, feynman diagrams • forces and particle exchange, Yukawa potential, units • leptons: their interactions and decays, lepton universality, • neutrino mass, neutrino oscillations • quarks, mesons and hadrons, baryon spectroscopy • baryon quantum numbers and selection rules • space-time symetries, color • QCD, jets • partons & deep inelastic scattering • lepton-quark symmetry, quark mixing
What we know about the universe This course is about the following basic building blocks: Electron, electron neutrino, up & down quarks Muon, mu neutrino, charm & strange quarks Tauon, tau neutrino, bottom & top quarks And… Photon, W & Z bosons, gluons Higgs boson (maybe) But not… The other 96% of the universe
Results from WMAP http://map.gsfc.nasa.gov/m_mm.html We don’t know much about Dark Energy, but will touch on some Dark Matter candidates later on in the course
Antiparticles (see sec 1.2) Assume that a particle in free space is described by a de Broglie wavefunction: Then using E=p2/2m, it is seen that the wavefunction obeys the Schrödinger equation: But we now know (from relativity) that E2=p2c2+m2c4 So what is a suitable form for a relativistic wave equation?
Klein-Gordon Equation But for any plane wave solution, there is an equivalent one with the opposite sign of energy. The negative energy solutions are a consequence of the quadratic mass-energy relations and cannot be avoided Dirac extended the above equation to apply to spin-1/2 objects (e.g. electrons), and the results agreed impressively with experiment (e.g by predicting the correct magnetic moment).
Dirac Hole Theory The negative energy states remained, and Dirac picture the vacuum as a “sea” of negative energy electron states combining to give a total energy, momentum and spin of zero. Vacuumstatemeans negative energy states filled with electrons
Positrons When a “hole” is created in the negative energy sea, Dirac postulated that this corresponds to a particle with positive charge, but with mass equal to the electron, called the positron or anti-electron. γ
Note that removing an electron with energy E=-Ep<0, momentum -p, and charge -e from the vacuum (which has E=0, p=0, Q=0) leaves the state with a positive energy and momentum and with a positive charge. • This state cannot be distinguished in any measurement from the situation in which an equivalent positive energy particle is added to the system. • Dirac postulated the existence of the positron based on this hole theory in 1928 • Not everyone took this idea seriously: • “Dirac has tried to identify holes with antielectrons … We do not believe that this explanation can be seriously considered” (Handbuch der Physik, 24, 246 (1933)) • The positron was discovered experimentally in 1933
Antiparticles in General • Any particle with spin-1/2 obeys the Dirac equation, and the hole theory applies. This means every spin-1/2 particle has an antiparticle. • Spin-0 particles do not obey the Dirac equation and do not feel the Pauli Exclusion Princliple on which hole-theory depends, and do not (necessarily) have antiparticles. Need to extend the list of fundamental particles:
Feynman Diagrams Feynman introduced a way of drawing processes in elementary particle physics which can be interpreted in an physical way, as well as being used in calculations. e.g, consider e-→ e-+ γ We draw it like this, with time going left to right:
Positrons • In Feynman diagrams, an antiparticle is drawn with an arrow pointing to the left, i.e. back in time, e.g. e- + e+→ γ is drawn as Thereare8basic process where electrons and positrons interact with photons. The arrow direction convention reflects the conservation of charge. What about energy and momentum?
Energy and Momentum Conservation • Consider our first process (e-→ e-+ γ) in the rest frame of the electron. We write their four-momenta to describe the conservation of both energy and momentum: • e-(E0,0) → e-(Ek,-k)+ γ(ck,k) Where we have already made sure momentum is conserved by construction. In free space (e.g. not in a an electric field from which can do work on the electron): E0=mc2, Ek=(k2c2+m2c4)1/2 so, for finite k, ΔE≡Ek+kc-E0 satisfies kc < ΔE < 2kc And energy is not conserved.
Real and virtual processes These8basicverticesallfailtoconserveenergy,andarecalledvirtualprocesses. Theycannothappeninisolation,butmustbecombinedwithotherprocessessothattheviolatedenergyisrecoveredwithinatime τΔE∼ħ andtheinitialandfinalstatesinparticularmusthaveequalenergyandmomentum. Youcanthinkoftheinitialandfinalstatesasbeingabletobestudiedinthedistantpastandfuturerespectively
Example: elastic electron scattering In principle, both these “time-orderings” should be included, but in practice, usually one example time-ordering is shown and the others are implied.
Higher-order Diagrams Each vertex in the 8 diagrams contributes a probablility of Hence the more vertices, the smaller the contribution to the scattering amplitude
Particle Exchange Interactions take place by exchange of other particles. In this case, electrons are interacting by exchanging photons. Photons are massless, but in general, the exchanged particle can have mass.
Particle Exchange suppose two particles of mass MA and MB interact by exchanging a particle X of mass MX. The reaction at the upper vertex, including the four-momentum in the rest frame of particle A is A(MAc2,0) → A(EA,p) + X(EX,-p) and the energy difference between initial and final state is ΔE = EX+EA-MAc2 →2pc as p → infinity → MXc2 as p → 0 This is always at leastMXc2
Range of Force The energy violation can persist only for a time τ∼ħ/ΔE during which time the force-carrying particle can travel a maximum distance R ∼ c τ = ħ/MXc This maximum distance over which the particle can propagate gives the range of the interaction. A photon is massless, so electromagnetic interactions are infinite-ranged. The weak force is mediated by W and Z bosons which have MW = 80.3 GeV/c2 and MZ = 91.2 GeV/c2 so RW = ħ/MWc ~ 2 x 10-3 fm
Point interaction approximation In the case of the weak force, the range is so short - often smaller than the de Broglie wavelength of the particles involved in an interaction (e.g. a nucleon undergoing beta decay) so that the interaction can be approximated to a zero-range point interaction corresponding to the MX→infinitylimit
Yukawa Potential In the limit that MA becomes large, we can regard B as being scattered by a static potential of which A is the source. In general this potential will be spin-dependent, however we can consider the simplest case of neglecting spin (so applying to the exchange of spinless bosons) obeying the Klein-Gordon equation Which for static (in time) solutions becomes
Yukawa Potential In this equation, φ plays the role of a static potential. Note that if MX=0 the equation is Laplace’s equation for the electrostatic potential, in which case the potential is In the case of finite MX, the solution turns out to be where R = ħ/MXc as expected and a coupling strength g, analogous to the electric charge has been included. This is the Yukawa potential, introduced by Hideki Yukawa in 1935
Scattering Amplitude To obtain quantitative information from the Yukawa picture, we need to calculate scattering amplitudes. These are the probabilities of scattering a particle from an initial momentum qi to a final momentum qf by a potential V(x). In first-order perturbation theory, this is given by where q=qf-qi is the momentum transfer This gives… This is the Born approximation and corresponds to the Feynman diagrams with one particle being exchanged
Zero-range Approximation If the range R = ħ/MXc is indeed small compared with the de Broglie wavelength of any particle involved and we make the zero-range approximation, then we have |q|2<<MX2c2 and the scattering amplitude reduces to a constant M(q)=-G where in units of inverse energy squared. This approximation is very useful in the theory of beta-decay, in which the appropriate coupling constant which has been measured is the Fermi coupling constant
Units In particle physics, a somewhat curious system of units is used which are called “natural” units and set the units of velocity to be the speed of light, c, and the standard unit of action to be ħ. In other words ħ=c=1. The units of energy used are electron volts (eV) and its multiples (keV, GeV etc). By choosing these “natural units,” all occurrences of ħ and c are omitted from equations so that E2=p2c2+m2c4 becomes E2=p2+m2 and the Fermi coupling constant is now written GF=1.166 x 10-5 GeV-2
Units All quantities now have the dimension of some power of energy since they can be expressed as some combination of ħ, c and energy. For example, mass, length and time can be expressed as M=E/c2, L=ħc/E, T=ħ/E Because of this, note that if the SI dimension of a quantity are MpLqTr then in natural units they are En=Ep-q-r. To convert expressions back to “normal” units, factors of ħ and c are inserted by dimensional arguments and then ħ = 6.528 x 10-22 MeV s
Example The cross section for Thomson Scattering (photon scattering from free electrons when the photon energy is much less than me) is To turn this into practical units, we write and demand that the cross section has units of length-squared. We find a=2, b=-2 so
Leptons The fundamental fermions (particles with spin-1/2) which do not feel the strong force are called leptons. They are § neutrino masses non-zero but small. See later. # neutrinos do not decay, but oscillate. see later, too.
Interactions The charged leptons interact via the electromagnetic interaction (like all charged particles) and the weak interaction (like all particles) The uncharged leptons (i.e. the neutrinos) interact only via the weak interaction, mediated by W and Z bosons In all interactions, it is observed that the following numbers are conserved: Le = N(e-) - N(e+) + N(ve) - N( ) Lµ = N(µ-) - N(µ+) + N(vµ)-N( ) L = N(-) - N(+) + N(v) - N( )
Lepton processes Since neutrinos do not feel the electromagnetic interaction, in electromagnetic processes, the Le conservation rule reduces to the conservation of N(e-)-N(e+). This implies that in electromagnetic processes, electrons and positrons can only be created or annihilated in pairs. In weak interactions, other possibilities exist. For example, the following beta-decay process is allowed: n → p + e- + because the total Le number on the left (0) is equal to the number on the right (1-1=0) Charge is also conserved, which is required in all physical processes
Disallowed processes Various processes are allowed by energy conservation, but not by some of the empirical conservations laws, e.g.
Neutrino interactions Neutrinos are very hard to detect, because they interact only via the weak interaction. This means that the scattering amplitude (determining the probability of reactions to occur) is small because of the large mass of the W and Z particles. Its existence was postulated in 1930 by Fermi to appear in ß-decay. If ß-decay had a two-body final state (Z,N) → (Z+1,N-1) + e-then the energy of the electron would be uniquely determined by However, experimentally, electrons are observed to have a range of energies. If a neutrino is also present the energy of the electron can lie in the range
Neutrino detection Though postulated in 1930, neutrinos were not detected until 1956. They can be detected in the following processes: These have a cross section of around 10-47m2 which means that a neutrino would typically have to travel through many light years of matter before interaction. If the neutrino flux is large enough, however, sufficient events can be seen in much smaller (!) detectors. see article from G. L. Trigg, Landmark Experiments in 20th Century Physics
Heavier Leptons electrons are the lightest charged particles, and as such can not decay (since charge must be conserved). Muons behave very much like electrons, except that their mass is much larger (105.7 MeV/c2 compared to 0.511 MeV/c2). In particular, they satisfy the Dirac equation for point-like spin-1/2 particles, and their magnetic moment is µ=(e/mµ)S indicating that there is no substructure, and the muon is an elementary particle.
Lepton Decay Muons decay with a lifetime of 2.2x10-6s via Taus decay to many different final states. This is because there mass is sufficiently high that many different allowed combinations of particles have a lower rest-mass energy than the tau. The different decays are characterised by their branching ratio, which gives the fraction of decays to a given final state compared to all decays. For taus we have
Lepton Universality electrons are light and stable, and are stopped by a modest thickness of lead. muons are about 200 times heavier and are very penetrating. taus are much heavier still and has a lifetime many orders of magnitude below the muon. Nevertheless, all experimental data is consistent with the assumption that the interactions of the electron with its neutrino, the muon with its neutrino and the tau with its neutrino are identical, provided the mass difference is taken into account. This is know as Lepton Universality. e.g. the decay rate for a weak process is predicted to be proportional to the Q-value multiplied by the same GF2 independent of kind of lepton: for example, In excellent agreement with experiment
Neutrino Mass The masses of particles is not predicted by the standard model. Neutrinos certainly have a very small mass (e.g. as measured by the electron energy distribution in beta decay, and had usually been considered to be zero. If the neutrino masses are not zero, neutrino mixing can occur. This happens because the weak neutrino states (e) need not be the same as the mass neutrino states (123) but linear combinations of them. For simplicity, let’s consider the mixing of two states in the following way:
Oscillations Consider an electron neutrino created at time t=0 with momentum p as After a time t it will be in a state corresponding to a probability of having changed weak eigenstate of
Detection When cosmic ray protons collide with atoms in the atmosphere, they create many pions which decay to neutrinos via One would expect then to have twice as many muon neutrinos as electron neutrinos, but the observed ratio was about 1.3:1 Neutrinos do oscillate.
Quarks (section 2.2) The six quark types, or flavours, know to exist are given below:
Quarks Unlike the leptons, quarks are never seen as free particles but only in the following combinations: baryons: qqq antibaryons: mesons: Collectively, these resulting particles are known as hadrons. No other combination are seen, though it is thought that other possibilities could exist (e.g. pentaquark states)
Quarks and Interactions Quarks (and hence hadrons) feel the weak interaction (because all known particles do), the electromagnetic interaction (because they are charged) and also the strong interaction, mediated by gluons. Despite the fact that they are never seen in isolation, the evidence for their existence is compelling because of: • hadron spectroscopy • deep-inelastic scattering • Jets
Spectroscopy Combining quarks in the allowed forms of baryons or mesons correctly reproduces all known hadrons, e.g. p = uud n = udd • = uds + = K+ = B- = etc. Additive properties like charge are seen to agree straightforwardly with the quark model. A large part of the mass is associated with the binding energy, and is not easy to calculate. Young man, if I could remember the names of all these particles, I would have been a botanist -- Enrico Fermi
Hadron quantum numbers Various quantum numbers are associated with hadrons. Some are related to symmetry operations, and some are intrinsic to the quarks. The quark quantum numbers are the electric charge Q, and baryon number B, which are conserved in all known interactions, and the strangeness S, charm C, beauty B’ and truth T, which are conserved in strong and electromagnetic, but not in weak, interactions
Isospin For partly historical reasons, the internal quantum number associated with the up and down quarks works a little differently. Heisenberg thought that the similarity in mass between the neutron and the proton meant that there was some underlying symmetry which was only broken by the charge on the proton vs the lack of charge on the neutron. He called this isospin symmetry, because it behaves like an angular momentum, and thought of the nucleon as an isospin-1/2 system, with the proton being the +1/2 projection and the neutron the -1/2 projection on the third axis. Similarly for other families of particles with similar masses and other properties:
quark isospin u & d quarks are an isodoublet, both with I=1/2, but d has I3=Iz=-1/2, u has I3=Iz=+1/2 the equivalent antiquarks are also an isodoublet with I=1/2, but with opposite third-components. Another quantum number is often used, which is derived from the known ones, called hypercharge: Y=B+S+C+B’+T and then for a hadron, I3=Q-Y/2
Example: How new particles can be identified by the quantum numbers and conservation laws. When a K- is fired at a proton target, they can interact (with a cross section appropriate to the strong interaction to give an - and a new particle +, which decays with weak interaction lifetimes to + + + n, or + 0 + p. Applying conservation of B,S,C,B’,T to the K-+p - + + reaction, and using the known values for K- (B=0, S=-1, C=B’=T=0), p ( B=1, S=0, C=B’=T=0) and - (B=0, S=C=B’=T=0) tells us that for the +, B=1, S=-1, C=B’=T=0. From the relationship between hypercharge and isospin, we also infer I3=1. It must therefore be part of an isotriplet, and partners of similar mass but different isospin should exist, as indeed they do. We can infer the quark structure from the quantum numbers as uus (n.b components of vectors are always additive)