High Energy Physics (3HEP) aka “particle physics”

1. High Energy Physics (3HEP)aka “particle physics” Dr. Paul D. Stevenson 2009

2. Outline of Course • Relativistic kinematics. Particle detectors, beam characteristics. • Classification of elementary particles. Leptons, hadrons and bosons. Symmetry and anitsymmetry. Interaction between particles. Conservation laws. The four fundamental forces – strong, weak, electromagnetic and gravitational; • Composite particles: quark and gluon structure of mesons and baryons, colour charge, Pauli exclusion principle. Quantum numbers: spin, isospin, parity, strangeness, charm, bottomness, topness. Production and decay of particles. Particles and antiparticles. The evidence for all these quantities. • Scattering experiments; evidence for parton structure; deep inelastic scattering off nuclei, nucleons; • Limits of the Standard Model: evidence for number of particle families, CKM matrix and unitarity. The Higgs boson.

3. 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

4. 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

5. 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

6. 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

7. 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

8. Collisions and Decays Most experimental particle physics is concerned with collisions and decays. Collisions: a+b->c+d Given the initial particles we can have different final states: these are called channels. We can observe many things in a given collision: the directions of c and d, their polarisation etc. We are interested also in the cross-section of the process. For each different channel we can define a partial cross-section.

9. Decays Decay processes are a -> b + c + d + … The main quantity of interest is the decay rate in the measured final state. This is also known as the partial width Γbcd to final state b + c + d The total width is the sum of all partial widths and is denoted Γ. It is the reciprocal of the lifetime Γ=1/τ The branching ratio of a into b + c + d is the ratio Rabc = Γbcd/Γ. In unnatural units, we have Γ=ħ/τ.

10. Beam Characteristics Consider a beam colliding with a target The beam consists of particles of a definite type, moving approximately in the same direction. The beam intensity Ib is the number of incident particles per unit time. The beam flux Φb is the intensity per unit normal section. The target is a piece of matter with scattering centres, which might be nuclei, nucleons, quarks or electrons depending on what we are trying to look at. Let nt be the number of scattering centres per unit volume, Nt be their total number The interaction rate Ri is the number of interactions per unit time. By definition the cross-section σiis Ri = σNtΦb = WNt. (W=rate per particle in target)

14. Antiparticles 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?

15. 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).

16. 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

17. 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

18. 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

19. Anderson, Physical Review 43, 491 (1933)

20. 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:

21. 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.

22. 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( )

23. 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

24. Disallowed processes Various processes are allowed by energy conservation, but not by some of the empirical conservations laws, e.g.

25. 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

26. 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

28. 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.

29. 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

30. 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

31. 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 (123) but linear combinations of them. For simplicity, let’s consider the mixing of two states in the following way:

32. 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

33. 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.

34. Quarks The six quark types, or flavours, know to exist are given below:

35. 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)

36. 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

37. 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

38. 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

39. 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:

40. Isospin

41. 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

42. 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)

43. Space-Time Symmetries As well as the quantum numbers associated directly with intrinsic quark properties, particles (including the leptons) have quantum numbers associated with spatial properties. These are Spin, J, Parity, P, and Charge Conjugation, C. SPIN We use the word spin to characterise the intrinsic angular momentum of a particle whether or not it is composite. If the particle is elementary, the total spin J is equal to the intrinsic spin S, e.g. 1/2 for any of the fundamental fermions (quarks & leptons)

44. Meson Angular Momentum Mesons are made a quark and an antiquark which have intrinsic spin of their own, and can have orbital angular momentum about their centre of mass If we first consider the case that there is no orbital angular momentum (this is the case for the lowest masss mesons) then we just have J=S and S is made by coupling together two single spins of 1/21. When coupling an angular momentum I1 to another I2 the allowed values of the result I areI = |I1-I2|, |I1-I2|+1,…,I1+I2-1, I1+I2 Coupling 1/2 with 1/2 gives only 0 or 1 as possibilities for S and hence for J when L=0. 1 we are not including units of c or hbar. Angular momentum has units of hbar

45. Meson Angular Momentum If the quark and antiquark in the meson have a relative angular momentum (which is quantized in integer units of hbar) L=1, 2, 3, … then this L couples with the coupled intrinsic spin S according to the normal rules of AM coupling, I.e. the total spin J can lie anywhere (in integer steps) between |L-S| and L+S. We label the mesons in spectroscopic notation: 2S+1LJ where L is written as the usual identifying letter: S=0, P=1, D=2, F=3, G=4 etc.. and S and J are given as numbers. So a pion, which has L=S=0 has an angular momentum configuration 1S0 . For a given quark-antiquark pair, there are many allowed angular momentum configurations, which are identified as different particles. For example there is a state made of u (like a +) but with J=1, which is labelled as +. It can be thought of as an excited state of +, but since it has different quantum numbers, it is also thought of as a particle in its own right.

46. Baryon Spin In the case of baryons, the three spins of 1/2 can couple to either 3/2 or 1/2 which then couples to the orbital angular momentum, which can be any integer value. q3 q1 L3 L=L12+L3 L12 q2 what are the allowed baryon states for L=0, L=1?

47. Parity The parity transformation is xixi’=-xi, i.e. the position vector of every particle is reflected in the origin. If we define a parity operator P as P(x,t) = Pa(-x,t) where Pa is a phase factor. Since acting with the parity operator twice must give us back the original state, we have P2(x,t) = (x,t) =Pa2(x,t) and so Pa=1

48. Intrinsic Parity Consider an eigenfunction of momentum then so when a particle is at rest (p=0), it is an exact eigenstate of the parity operator with eigenvalue Pa. For this reason, Pa is called the intrinsic parity of particle a, or more commonly just the parity of particle a. If a Hamiltonian is invariant under the parity transformation then [H,P]=0 and in a reaction governed by the Hamiltonian, the initial and final parities must be the same, and parity is conserved. This is true in the strong and electromagnetic interactions, but not the weak interaction.

49. Angular Momentum & Parity orbital angular momentum wavefunction are written in spatial coordinates as spherical harmonics in the angular part. These feature in the usual separable solutions to central-field potential in quantum mechanics: (r,,) = R(r)Ylm(,) The first few spherical harmonics are Y00 = 1/sqrt(4) Y10 = sqrt(3/4) cos Y11 = -sqrt(3/8)sin exp(i)

50. Spherical Harmonics Using x=rsincos, y=rsinsin, z=rcos the parity transformation in polar coordinates become r  r’ = r •  ’ = - •  ’ = + from which it can be shown that Ylm(,)  Ylm(-, +) =(-1)l Ylm(,) and (r,,) is an eigenstate of parity with eigenvalue Pa(-1)l (where Pa is the intrinsic parity ignoring angular momentum)