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Summary Contents UNIVERSITY OF ZURICH NUCLEAR AND PARTICLE PHYSICS Summary Contents 1 Recap1 1.1 Four vectors...............................1 1.1.1 Space-vector..........................1 1.1.2 Velocity-vector.........................2 1.1.3 Momentum-vector.......................2 1.1.4 Mandelstam variables.....................3 1.1.5 Rapidity.............................3 1.2 Quantum Mechanics..........................4 1.2.1 Clebsch-Gordon coefficients..................5 2 Introduction8 3 Fermi’s golden rule8 3.1 Lifetime.................................8 3.2 Cross section..............................9 3.3 Luminosity...............................9 3.4 Mean free path............................. 10 3.5 Decay rate................................ 10 3.5.1 Lorentz invariance....................... 13 4 Liquid Drop Model 15 4.1 Harmonic oscillator........................... 16 5 Nuclear decay 17 5.1 α-decay................................. 17 5.1.1 Gamov model.......................... 17 5.2 β-decay................................. 18 5.3 γ-decay................................. 19 6 Nuclear Fission and Fusion 22 6.1 Fission.................................. 22 6.2 Fusion.................................. 23 7 The Standard Model 24 7.1 Quarks.................................. 25 7.2 Leptons................................. 25 7.3 Gauge bosons.............................. 25 7.4 Scalar bosons.............................. 25 8 Quantum Field Theory 26 8.1 The Klein-Gordon Equation...................... 26 8.2 The Dirac Equation........................... 27 8.3 Probability density........................... 29 8.4 Covariant form............................. 29 8.5 Solution of the Dirac Equation..................... 30 8.6 Antiparticles............................... 32 8.7 Antiparticle spinors........................... 34 8.8 Charge conjugation........................... 34 9 Spin 35 9.1 Helicity................................. 35 9.2 Parity.................................. 37 10 Particle exchange 38 10.1 Time-ordered perturbation theory.................... 38 10.2 Scattering in a potential......................... 40 11 QED 41 11.1 Feynman rules for QED......................... 43 11.2 t-channel Diagram........................... 44 11.3 s-channel Diagram........................... 45 11.4 u-channel Diagram........................... 46 11.5 Example: Compton scattering..................... 47 11.6 Example: Bhabha scattering...................... 48 11.7 Electron-Positron annihilation..................... 49 11.8 Spin sums................................ 49 12 QCD 51 12.1 Global gauge symmetry......................... 51 12.2 The local gauge principle........................ 51 12.3 Strong interaction............................ 52 12.4 Hadronization and jet.......................... 53 12.5 e+e− collisions............................. 53 12.6 Charged current: Leptons........................ 55 12.7 Charged current: Quarks........................ 55 12.8 Neutral current: Quarks......................... 56 12.9 Neutral current: Quarks......................... 57 12.10Quark mixing.............................. 57 12.11Baryon number conservation...................... 58 12.12Lepton number conservation...................... 58 12.13Lepton flavor conservation....................... 58 12.14Example: Beta decay.......................... 59 CONTENTS CONTENTS 13 Isospin 61 3 1 RECAP 1 Recap 1.1 Four vectors Four vectors are used in special relativity, as an example we use space and time coor- dinates in a four vector as well as energy and momentum coordinates. We write: aµ = (a0, a1, a2, a3) for the contravariant 0 1 2 3 aµ = (a0, a1, a2, a3) = (a , −a , −a , −a ) for the covariant 1.1.1 Space-vector The space-vector in fourvector-form consists of the time-coordinate t as well as the space -coordiante ~r = (x, y, z). For the dimensions to match, time-coordinate gets multiplied by c: xµ = (ct, x, y, z) = (ct, ~r) xµ is a contravariant fourvector, because it is a coordinate-vector to an orthonormal base of the minkowski space and thus with a base-change it changes contravariant through a lorentz-transformation. In this metric of flat spacetime, t has a different sign to the space-coordiantes: ds2 = c2dt2 − dx2 − dy2 − dz2 The contravariant fourvector a under the Lorentztransform Λ goes to a0 = Λa The covariant fourvector b goes to: b0 = Λ−1T b where γ −γβ 0 0 −γβ γ 0 0 Λ = 0 0 1 0 0 0 0 1 Further we have Λ−1T = gΛg−1 with 1 0 0 0 0 −1 0 0 µν gµν = diag(1, −1, −1, −1) = = g 0 0 −1 0 0 0 0 −1 1 1.1 Four vectors 1 RECAP Thus ν 0 1 2 3 aµ = (a0, a1, a2, a3) = gµνa = ga = (a , −a , −a , −a ) The product of two fourvectors in minkowski-space is given by: ν aµb = aµgµνbν = a0b0 − a1b1 − a2b2 − a3b3 If a is a fourvector we have: − a2 > 0 =⇒ aµ is called timelike − a2 < 0 =⇒ aµ is called spacelike − a2 = 0 =⇒ aµ is called lightlike If the distance between two events (∆xµ) is spacelike, they can not be in causal rela- tion. 1.1.2 Velocity-vector The velocity-vector uµ is given by differentiation xµ by proper time τ: dxµ uµ = dτ Where proper time τ is the smaller amount of time between two events and given by: 1 dτ = dt γ with 1 1 |~v | γ = = √ β = r 2 | ~v | 2 1 − β c 1 − c Thus d uµ = γ (ct, x, y, z) = γ(c, x,˙ y,˙ z˙) = γ( c,~v ) dt with its norm being q µ q µ ν q µ 2 2 2 |u | = u u gµν = uµu = γ (c − |~v | ) = c 1.1.3 Momentum-vector The momentum-vector in fourvector-form is defined as p µ = muµ = (γmc, γm~v ) where m is the rest mass of the body. Using E = γmc2 we get p µ = (E/c, ~p ) ~p = γm~v From where we can derive the energy-momentum relationship: E2 − | ~p |2c2 = m2c4 2 1.1 Four vectors 1 RECAP 1.1.4 Mandelstam variables In Particle physics there are three really useful Lorentz invariant quantities: s, t and u. We consider the scattering process: 1 + 2 → 3 + 4 we can then define 2 2 2 s = (p1 + p2) , t = (p1 − p3) , u = (p1 − p4) we note 2 2 2 2 s + t + u = m1 + m2 + m3 + m4 and we note that s is a scalar product of two four vectors: 2 2 2 s = (p1 + p2) = (E1 + E2) − (~p1 − ~p2) And since these quantities are Lorentz invariant we can evaluate them in any frame, thus we choose the most convenient one (center of mass frame): ∗ ∗ ∗ ∗ ∗ ∗ p1 = (E1 , ~p ) p2 = (E2 , −~p ) ∗ ∗ 2 =⇒ s = (E1 + E2 ) 1.1.5 Rapidity ??? 3 1.2 Quantum Mechanics 1 RECAP 1.2 Quantum Mechanics In quantum mechanics the state of a system is defined by a wave function |φi which is a complex number. This function is normalized as hφ|φi = 1. Measurements are represented by operators O, and the possible outcomes of a mea- surement are its eigenvalues λi. And all its eigenvectors |ψii are orthonormal. P 2 If a state φ = i αi|ψii is given, the quantity |αi| is the probability to measure the value λi when measuring O. Also αi = hψi|φi. Since we use bra-ket notation (dirac), we have X ∗ hφ| = αi hψi| i P ∗ and therefore: αiαi = 1. The expectation value of an operator O is defined as hφ|O|φi We could also use the function notation instead of the dirac notation, thus we have: Z hψ|φi = ψ∗(~x)φ(~x)d3x The expectation value: Z ψ∗(~x)Oφ(~x)d3x Some important operators: ∂ pˆ = −i ∇ Eˆ = ih xˆ = x ~ ∂t If two operators commute, then [A, B] = AB − BA = 0 Then they can be measured in whatever order and they have a complete set of eigen- states in common. Hermitian operators O† = O have real eigenvalues, orthogonal eigenvectors and are associated with measurable quantities. 4 1.2 Quantum Mechanics 1 RECAP 1.2.1 Clebsch-Gordon coefficients In classical mechanics we know all the components of the angular momentum, but in quantum mechanics the various components of the angular momentum do not com- mute [Li,Lj] = i~ijkLk At best we can measure L2 and one of the components, moreover only certain values are allowed: 2 2 hψ|L |ψi = `(` + 1)~ ` = 1, 2, 3, ··· It might be that we are not interested in L and S separately but only in the total angular momentum J = L + S. Or you might want to add the spin of two particles to get the spin of a composite particle (in case the relative orbital angular momentum is zero). We cannot simply add component by component as they do not commute. But there is a handy little trick called Clebsch-Gordon coefficients, they are given in fancy tables, we use one small part of it: L S state X 5/2 2 × 1/2 5/2 5/2 3/2 2 1/2 1 3/2 3/2 2 −1/2 1/5 4/5 5/2 3/2 X 1 1/2 4/5 −1/5 1/2 1/2 Xz 1 −1/2 2/5 3/5 0 1/2 3/5 −2/5 P Lz Sz Figure 1: Part of the Clebsch-Gordon coefficients table. In the shaded rectangle above we have the possibilities (without the square root) for the states (given on top of the rectangle) and the associated z-component of the angular momentum and spin (given on the left of the rectangle). We will explain this using an example: 5 1.2 Quantum Mechanics 1 RECAP Example 1: Angular momentum combination We have a particle with spin S = 1/2 and angular momentum L = 2. In a mea- surement we discover that the particle is in the state of total angular momentum 3 1 X = , 2 2 We want to find out the probability to find the particle with spin up (+1/2) and to find it in the angular momentum state of Lz = 0. How do we do this? Since we are dealing with a particle with spin 1/2 and angular momentum 2, we look for the Clebsch-Gordon table 2 × 1/2 (given above). And sin we are looking at the state X we search this column in the table (red arrow). Now we are able to write the combination for the state X: |X, Xzi = P1|L1,L1,zi|S1,S1z i + P2|L2,L2,zi|S2,S2,zi when we put in the numbers we get s s 3 1 3 1 1 2 1 1 , = |2, 1i , − − |2, 0i , 2 2 5 2 2 5 2 2 | {z } It is important to note that these coefficients Pi in the table (only the ones cir- cled by thin lines, in our case inside the shaded rectangle) are actually under a square-root, except for the minus sign, if there is a minus sign present it is always in front of the root.
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