Nuclear and Particle Physics Some basic concepts Natural units Cross sections Particle lifetime and width Types of forces Range of forces Types of particles Kinematics – 4-vectors Feynman diagrams Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Natural units Particle physics : Energy in electron-volt: eV, keV, MeV, GeV, TeV In natural units: all quantities – dimension of a power of energy: Mass(m), momentum (mc) , energy (mc2) have units of GeV length (ħ/mc), time (ħ/mc2) have units of GeV-1. Usefull #s: ħc = 6.58*10-22 MeV s * 3*1010 cm s-1 = 197*10-13 MeV cm = 197 MeV fm Example: Compton wavelength of particle with energy mc2=197 MeV: R in natural units: Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Cross section (s) Definition (Wordnet Dictionary): the probability that a particular interaction will take place between particles. Is a measure of the probability for a reaction to occur; may be calculated if form of basic interaction between particles is known. a + b → c + d na density of particles of type a impinge on target, thickness dx, containing density nb of particles of type b Flux F of particles a through the target ≡ # particles/unit area/unit time: F = na vi Reaction rate per target particle: W = F s s - effective cross section of b ; probability that any particle a hit target particle b ~ fraction of target area obscured by the b particles: s nb dx → number of reactions per unit time: F s nb dx s = W / F Cross section (per target particle) = reaction rate/ flux Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Cross Section (2) Total Cross Section of an object determines how “big” it is and how likely it is to collide with projectile thrown towards it randomly. Small cross section Large cross section Nuclear and Particle Physics Feb. 2011

Scattering Cross Section Differential Cross Section dW - solid angle Flux q – scattering angle Target Unit Area Total Cross Section Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Cross section (3) Reaction rate W – given by (non-relativistic) perturbation theory, known also as Fermi’s Golden Rule : Mif – matrix element between initial and final states. Effectively it is the overlap integral over volume, ∫ψ*fUψidV , between initial-state and final-state wavefunctions, brought about by the interaction potential U. Mif – include coupling constant, propagator, angular dependence of the reaction rate. rf – energy density dN/dE of final states. Nuclear and Particle Physics Feb. 2011

Particle life-time or width Most particles studied in particle physics or high energy nuclear physics are unstable and decay within a finite life-time. Some exceptions include the electron, and the proton. Particles decay randomly (stochastically) in time. The time of their decay cannot be predicted. Only the the probability of the decay can be determined. The probability of decay (in a certain time interval) depends on the life-time of the particle. In traditional nuclear physics, the concept of half-life is commonly used. In particle physics and high energy nuclear physics, the concept of mean life time or simple life time is usually used. The two are connected by a simple multiplicative constant. Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Half-Life Nuclear and Particle Physics Feb. 2011

Half-Life and Mean-Life The number of particle (nuclei) left after a certain time t can be expressed as follows: where t is the mean life time of the particle t can be related to the half-life t1/2 via the simple relation: Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Particle Widths By virtue of the fact that a particle decays, its mass or energy (E=mc2), cannot be determined with infinite precision, it has a certain width noted G. The width of an unstable particle is related to its life time by the simple relation h is the Planck constant. Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Examples - particles Mass (MeV/c2) t or G c t Type Proton (p) 938.2723 >1.6x1025 y Very long… Baryon Neutron (n) 939.5656 887.0 s 2.659x108 km N(1440) 1440 350 MeV Very short Baryon resonance D(1232) 1232 120 MeV L 1115.68 2.632x10-10 s 7.89 cm Strange Baryon resonance Pion (p+-) 139.56995 2.603x10-8 s 7.804 m Meson Rho - r(770) 769.9 151.2 MeV Kaon (K+-) 493.677 1.2371 x 10-8 s 3.709 m Strange meson D+- 1869.4 1.057x10-12 s 317 mm Charmed meson Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Examples - Nuclei Radioactive Decay Reactions Used to date rocks Parent Nucleus Daughter Nucleus Half-Life (billion year) Samarium (147Sa) Neodymium (143Nd) 106 Rubidium (87Ru) Strontium (87Sr) 48.8 Thorium (232Th) Lead (208Pb) 14.0 Uranium (238U) Lead (206Pb) 4.47 Potassium (40K) Argon (40Ar) 1.31 Nuclear and Particle Physics Feb. 2011

Decay Widths and Branching Fractions In general, particles can decay in many ways (modes). Each of the decay modes have a certain relative probability, called branching fraction or branching ratio. Example (K0s) Neutral Kaon (Short) Mean life time = (0.8926±0.0012)x10-10 s ct = 2.676 cm Decay modes and fractions mode Gi/ G p+ p- (68.61 ± 0.28) % p0 p0 (31.39 ± 0.28) % p+ p- g (1.78 ± 0.05) x10-3 Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Types of forces 4 types of forces: gravitational, weak nuclear, electromagnetic and strong Force between two particles described by exchange diagram. Strength of force is determined by a constant, characteristic of a given interaction, and by the properties of the exchanged particle. Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Types of forces(2) Gravitational force Attraction between 2 particles ~ gravitational ‘charge’ (=mass); long range; controls motion of planets + galaxies, etc. Gravitational potential between 2 protons: V=GNm2p/r GN = 4.17*10-5 GeVcm/gm2 = 0.67*10-38 GeV-2 (Newton constant) using mp ≈ 1 GeV and r = 10-13 cm ≈ 5 GeV-1 → V ≈ 10-39 GeV The force is very weak for protons but can be important if: mass is very large, m ≈ 1/√GN ≡ MP = 1019 GeV (MP – Planck mass) distance is very small, r ≈ 10-33 cm. Force carrier – graviton (??) Nuclear and Particle Physics Feb. 2011

Types of forces (3) Weak nuclear forces Responsible for radioactivity: n → p + e- + e O14 → N14 + e- + e (half-life ~ 71.4 sec) The strength is determined by the weak coupling, which is proportional to the Fermi constant ( GF = 0.896*10-7 GeV fm3 = 1.166*10-5 GeV-2 ): for mw ≈ 80 GeV g2/ħc ≈ 1/240 Force carriers - W±, Z0 Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Types of forces(4) Electromagnetic forces Proportional to electric charge. Responsible for binding atoms. The electromagnetic (em) potential energy: Strength of force determined by ‘fine-structure constant’  = e2M/(ħc) = 1/137 (In Heaviside-Lorentz units  = e2/(4pħc) = 1/137) |E1| = ½ 4 (mc2) = 14 eV Bohr: binding energy of e- in hydrogen atom Force binding atoms into stable molecules – residual em force Force carrier -  Nuclear and Particle Physics Feb. 2011

The em coupling constant  Expression of  depends on the units of the electric charge – but numerically its value is always 1/137.036….. Standard units (SI) (charge in Coulombs): Gaussian units (G) (charge in electrostatic units): Heaviside-Lorentz (0=0=1): Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Types of forces (5) Strong forces Holds quarks together. Works on the color charge. Force gets stronger as quarks get further apart. Quarks cannot exist individually – they are confined. Nucleons bound in the nucleus by a residual strong force. Binding energy in the deuteron ≈ 2 MeV. Force carrier - gluon Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Forces - summary Gravitational : Weak : Electromagnetic : Strong ≈ 10-40 : 10-7 : 10-2 : 1 Electroweak Grand unified theory (GUT) : at very high energies – electroweak and strong unite Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Ranges of forces Heisenberg: DE Dt ≈ ħ → R ≡ c Dt ≈ ħc / DE DE ≥ Mexc2 → R ≈ ħ / Mexc R ~ ∞ RW ~ 2*10-3 fm Rp ~ 1.4 fm (Since the individual gluons and quarks are contained within the proton or neutron, the masses attributed to them cannot be used in the range relationship to predict the range of the force.) Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Types of particles electron e and neutrino  do not experience strong interactions - leptons particles experiencing strong interactions – hadrons hadrons subdivided into mesons and baryons. mesons: bosons with integral spin; lightest π (140 MeV) baryons: fermions with half integer spin; lightest p (938 MeV) hadrons – not fundamental (SLAC, DIS) Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Types of particles (2) Symmetry between number of leptons and number of quarks 3 families (generations) of doublets Why 3 generations?? mesons – q q baryons – q q q Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Coloured quarks The problem of - : contains 3 s quarks, each is a fermion of spin 1/2 . Spin of - is 3/2 → all s quarks have sz = +1/2 → violation of Pauli principle Answer: colour Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics gluons quarks carry colour ‘charge’ → theory of strong interactions called Quantum Chromo Dynamics (QCD); force → colour force carrier of colour force - gluon quarks and gluons – partons. Partons turn into hadrons, which remember the direction of the parton → in experiment see jets : Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Four-vectors Extend idea of 3-component vector x to 4-component vector (ct, x), by including the time component in addition to the space one. Get a four-vector, the square of which is defined as: c2t2 - x2 - y2 - z2 which is Lorentz invariant. Same is true for momentum and energy p = (E, pc) p2 = E2 – px2c2 – py2c2 – pz2c2 = m2c4 The sum of two four-vectors A = (A0, A) and B = (B0, B) is A + B = (A0+B0, A+B) Scalar product of two four-vectors A · B = A0 B0 – A · B Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Feynman diagrams Feynman developed pictorial representation of processes used originally in QED. There is a precise and quantitative correspondence between a given diagram and a specific mathematical expression for a quantum amplitude. Some basic em processes: Each process has an associated probability ~ to the strength of the em coupling constant  (≈1/137) Note:  may depend on the scale at which the process takes place. Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Feynman diagrams (2) Any diagram which can be built using the basic processes is a possible process provided conservation of energy momentum is required at every vertex lines entering or leaving the diagram represent real particles and must have E2 = p2 + m2 lines in intermediate stages in the diagram represent “virtual particles” which do not need to have the right relationship between E, p, and m, but which can never be observed if they do not. Nuclear and Particle Physics Feb. 2011

Møller scattering e-+e-→e-+e- Exchanged  ( 4-vector q = k - k’ ) is virtual: it has a ‘mass’ which is different from zero. Assume E » me, denote  - scatt. angle  Cross section  ~ |amplitude|2 . Amplitude A gets factor of √ at each vertex. → A ~  . →  ~ |A|2 ~ 2 . Nuclear and Particle Physics Feb. 2011

Møller scattering e-+e-→e-+e- (2) Process can proceed also via two-photon exchange Amplitude |A| ~ 2 →  ~ 4 . But  « 1 → two-photon exchange diagram contributes much less than one-photon exchange. Depending on the required accuracy, it can be neglected. Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Annihilation process Compare two annihilation processes: e+ e- →   , e+ e- →    Confirmed experimentally. Nuclear and Particle Physics Feb. 2011

Nuclear and Particle Physics Open questions Where is the Higgs particle? Why are there 3 generations? Are quarks and leptons really elementary? Are there other fundamental forces? Can one unify electroweak and strong forces (GUT)? Can one unify all forces (including gravity)? Need to increase energy to (a) improve resolving power, (b) get closer to the unification scale (‘reproduce the “big-bang” conditions) Nuclear and Particle Physics Feb. 2011