Particle Physics: Introduction to the Standard Model

Transcription

1 Particle Physics: Introduction to the Standard Model Overview of the Standard Model Frédéric Machefert Laboratoire de l accélérateur linéaire (CNRS) Cours de l École Normale Supérieure 24, rue Lhomond, Paris January 12th, / 23

2 Part I Overview of the Standard Model 2 / 23

3 1 Relativity recapitulation Fermions... and Bosons Properties: Electric charge Properties: Color charge Comparison of the interaction intensities 2 3 / 23

4 Relativity recapitulation Fermions... and Bosons Properties: Electric charge Properties: Color charge Comparison of the interaction intensities Metric A four-vector x is attributed to a particular space-time point. x 0 ( ) x = (x µ ) = x 1 t x 2 = x x 3 Greek letters are for four-vectors Roman letters for spatial coordinates (vectors) The scalar product is defined thanks to the metric tensor g µν g = (g µν) = by x.y = g µνx µ y ν = x µ y µ = x µy µ 4 / 23

5 Relativity recapitulation Fermions... and Bosons Properties: Electric charge Properties: Color charge Comparison of the interaction intensities Lorentz transformation A transformation(λ, a) defines the transition from an inertia frame to another (Λ, a) : x µ x µ = Λ µ νx ν + a µ The energy and 3-momentum p of a particle of mass m form a four-vector whose square p.p = m 2 In the course, we will apply the Einstein summation rule on greek indices The velocity of the particle is β = v/c = p/e and the Lorentz factor is γ = 1 1 β 2 The energy and momentum (E, p ) viewed from a frame moving with velocity β f are given by ( )( ) E γf γ = f β f E γ f β f γ f p p 5 / 23

6 Relativity recapitulation Fermions... and Bosons Properties: Electric charge Properties: Color charge Comparison of the interaction intensities Special relativity - Space-time coordinates t γ γβ 0 0 x y = γβ γ z with β = v/c and γ = 1 1 β 2 t x y z 6 / 23

7 Relativity recapitulation Fermions... and Bosons Properties: Electric charge Properties: Color charge Comparison of the interaction intensities Matter = fermions (Spin- 1 2 particles): Electrons with two spin orientations: L and R Neutrinos (L) Quarks L and R (proton=uud, neutron=udd) Three families = heavier copies of the first family ( ul d L ( νel e L cl s L νµl µ L tl b L ντl τ L u R c R t R d R s R b R e R µ R τ R ) ) 7 / 23

8 Relativity recapitulation Fermions... and Bosons Properties: Electric charge Properties: Color charge Comparison of the interaction intensities Interactions = bosons (Spin 0 or 1 particles): Electromagnetism: Spin 1 massless Strong interaction (p=uud): Spin 1 massless Weak interaction: Spin 1 massive Masses: Spin 0 massive ( ul d L ( νel e L cl s L νµl µ L tl b L ντl τ L u R c R t R d R s R b R e R µ R τ R γ g W ±, Z H ) ) 8 / 23

9 Relativity recapitulation Fermions... and Bosons Properties: Electric charge Properties: Color charge Comparison of the interaction intensities Fractional charges not observed in nature Strong interaction: uud, udd ( ul d L ( νel e L cl s L νµl µ L tl b L ντl τ L ) ) 2 3 u R c R t R 1 3 d R s R b R 1 e R µ R τ R 0 γ 0 g ±1, 0 W ±, Z 0 H 9 / 23

10 Relativity recapitulation Fermions... and Bosons Properties: Electric charge Properties: Color charge Comparison of the interaction intensities Sum of colors (RGB) white R+G+B= (qqq =baryon) Color+anti-color= White (qq =meson) Gluon carries color+anti-color 8 different gluons (not 9) C C ( ul d L ( νel e L cl s L νµl µ L tl b L ντl τ L C u R c R t R C d R s R b R e R µ R τ R γ C + C g W ±, Z H ) ) 10 / 23

11 Relativity recapitulation Fermions... and Bosons Properties: Electric charge Properties: Color charge Comparison of the interaction intensities Rule of thumb for interactions Interaction Carrier Relative strength Gravitation Graviton (G) Weak Weak Bosons (W ±,Z ) 10 7 Electromagnetic Photon (γ) 10 2 Strong Gluon (g) 1 Forget about Gravitation in particle physics problems The course will lead us to understand how the model describes the interactions and their strength. 11 / 23

12 a = (E a, p a) = (p 0, p 1, p 2, p 3 ) E a E a p a p a = m 2 a g µν p µp ν = m 2 a Conservation of E and p g µµ = (1, 1, 1, 1) for µ ν : g µν = 0 Mandelstam Variables therefore a+b = c+d a c = d b a+b c + d s = (a+b) 2 t = (a c) 2 u = (a d) 2 12 / 23

13 Theorem s + t + u = m 2 a + m 2 b + m 2 c + m 2 d = 0 High energy approx (E m 0, E = p ) CM-frame ( p a = p b ) E a = E b = E c = E d = s/2 Proof. s = a 2 + b a b = ma 2 + mb 2 + 2(Ea E b p a p b ) = 2(E a E b p a p b ) = 2(E a E a + p a p a) = 2(Ea 2 + Ea) 2 = 4Ea 2 t = 2(E a E c p a p c) u = 2(E a E d p a p d ) = 2(E a E c + p a p c) 13 / 23

14 Proof. t + u = 2(2 E a E c) = 2(2 E a E a) s + t + u = 4 E a E a 4 E a E a = 0 2 particle reaction 2 independent variables! 14 / 23

15 Useful relationships t = 2(E a E c p a p c) u t = 2( s 2 s 2 s 2 s 2 cosθ) = s 2 (1 cosθ) = s (1+cosθ) 2 = 2( s s E s c mc 2 cosθ) = 2( s 2 s 2 s 2 s/4 m 2 c cosθ) = 2( s 2 s 2 s 2 s 2 1 4m 2 c/s cosθ) = s (1 β cosθ) 2 massless massless initial state and massive final state of identical particles 15 / 23

16 a+b c + d a+c b + d s = (a+b) 2 s = (a+c) 2 = (a c) 2 = t t = (a c) 2 t = (a b) 2 = (a+b) 2 = s u = (a d) 2 u = (a d) 2 = (a d) 2 = u Calculate a process as function of s,t,u Derive crossed process by s t, t s, u u We can express one process in the kinematic variables of another process (Xcheck) Global factor 1 for each fermion line crossed (will see an example) in Tutorial 16 / 23

17 s channel: annihiliation e + + e γ µ + µ t channel: scattering e + A e + A e µ e e + t µ + A t q γ = p e + p e + s = q 2 γ (CM) = (E e + E e +) 2 > 0 the photon is massive (virtual) time-like p e i t = q γ + p e o = q 2 γ = me 2 + me 2 2 p e p i eo 2(E i E o p i p o cosθ) 2E i E o(1 cosθ) 0 the photon is massive space-like 17 / 23

18 Cross Section The cross section σ is the ratio of the transition rate and the flux of incoming particles. Its unit is cm 2 1b = cm 2 (puts barn in perspective, doesn t it?) Two ingredients: the interaction transforming initial state i to a final state f of m particles with four-vectors p i kinematics (including Lorentz-Invariant phase space element) dσ = 1 2S 12 m d 3 p i (2π) 4 δ(p (2π) 3 2E i p m p 1 p 2 ) M 2 i=1 with (originating from flux) S 12 = (s (m 1 + m 2 ) 2 )(s (m 1 m 2 ) 2 ) 18 / 23

19 Total Width or Decay Rate Total width is the inverse of the lifetime of the particle unit: energy, e.g., GeV. Closely related, but not identical to the cross section dγ = 1 2E m d 3 p i (2π) 4 δ(p (2π) 3 2E i p m p 1 ) M 2 i=1 For the decay of an unpolarized particle of mass M into two particles (in the CM frame p 1 = p 2): dγ = 1 p 1 32π 2 M 2 M 2 dω where Ω is the solid angle with dω = dφd cosθ 19 / 23

20 for a final state with 2 particles Cross section 2 2 reaction with four massless particles: dσ = 1 M 2 64π 2 s dω Width of a massive particle ( s = M) decaying to two massless particles in the final state p 1 = s/2: dγ = 1 64π 2 M 2 s dω Study of the phase space in Problem Solving with applications to 2-body. 20 / 23

21 Example pp H γγ (EW 2013) 21 / 23 Particles: plane waves ψ( x, t) exp im 0 t m 0 m 0 iγ/2 N(t) = N 0 exp t/τ Γ = 1/τ Fourrier transform to momentum space: A A 2 1 (m m 0 )+iγ/2 1 (m m 0 ) 2 +Γ 2 /4 Γ: full width half maximum Similarity to classical mechanics: resonance Example e e + Z q q σ had [nb] ALEPH DELPHI L3 OPAL measurements (error bars increased by factor 10) σ from fit QED corrected E cm [GeV] lifetime too short to be measured directly: measure mass via decay products q q cross section measurement Γ Z M Z σ 0

22 Suppose that we have two (and exactly two) possible decays for the particle a: a b + c a d + e then: Γ = Γ bc +Γ de If a particle of a given mass can decay to more final states than another one with the same mass, it will have a shorter lifetime Branching ratio B(a b+c) = Γ bc /Γ The branching ratio: Of N decays of particle a, a fraction B will be the final state with the particles b and c. Γ bc is a partial width of particle a. Remember: for the calculationγall final states (partial widths) have to be considered. 22 / 23

23 What do we know? Names of particles Kinematic description of interactions Definition of cross section and decay width What is next? Electromagnetic interactions (QED) Strong interaction (QCD) Electroweak interactions 23 / 23