Elementary Particles” Lecture 3

“Elementary Particles” Lecture 3

Niels Tuning Harry van der Graaf

Niels Tuning (1) Plan

Theory Detection and sensor techn.

Quantum Quantum Forces Mechanics Field Theory

Interactions Light Scintillators with Matter PM

Tipsy Accelerators Bethe Bloch Medical Imag. Cyclotron Photo effect X-ray Compton, pair p. Proton therapy Bremstrahlung Cherenkov Experiments Fundamental Astrophysics Charged Particles Physics Cosmics Particles ATLAS Grav Waves Km3Net Si Neutrinos Virgo Gaseous Lisa Pixel …

Special General Gravity Optics Relativity Relativity Laser

Niels Tuning (2) Plan

Theory Today Detection and sensor techn.

Niels 2) Niels 2) Niels 7) + 10) Quantum Quantum Forces Mechanics Field Theory 5) + 8) 4) Harry Particles 3) Harry

Light RelativisticIn

teractions 6) + 9) with Matter Ernst-Jan 1) Harry 11) +12) Martin Fundamental 6) Ernst-Jan Accelerators Martin Physics 13) + 14) Astrophysics Charged Excursions Particles Experiments

1) Niels 9) Ernst-Jan 9) Ernst-Jan 9) Ernst-Jan Special General Gravity Optics Relativity Relativity

Niels Tuning (3) Schedule

1) 11 Feb: Accelerators (Harry vd Graaf) + Special relativity (Niels Tuning) 2) 18 Feb: Quantum Mechanics (Niels Tuning) 3) 25 Feb: Interactions with Matter (Harry vd Graaf) 4) 3 Mar: Light detection (Harry vd Graaf) 5) 10 Mar: Particles and cosmics (Niels Tuning) 6) 17 Mar: Astrophysics and Dark Matter (Ernst-Jan Buis) 7) 24 Mar: Forces (Niels Tuning) break 8) 21 Apr: e+e- and ep scattering (Niels Tuning)

9) 28 Apr: Gravitational Waves (Ernst-Jan Buis) 10) 12 May: Higgs and big picture (Niels Tuning) 11) 19 May: Charged particle detection (Martin Franse) 12) 26 May: Applications: experiments and medical (Martin Franse)

13) 2 Jun: Nikhef excursie 14) 8 Jun: CERN excursie

Niels Tuning (4) Plan

1) Intro: Standard Model & Relativity 11 Feb 1900-1940 2) Basis 18 Feb 1) Atom model, strong and weak force 2) Scattering theory 1945-1965 3) Hadrons 10 Mar 1) Isospin, strangeness 2) Quark model, GIM 1965-1975 4) Standard Model 24 Mar 1) QED 2) Parity, neutrinos, weak inteaction 3) QCD

1975-2000 5) e+e- and DIS 21 Apr

2000-2015 6) Higgs and CKM 12 May

Niels Tuning (5) Thanks

• Ik ben schatplichtig aan: – Dr. Ivo van Vulpen (UvA) – Prof. dr. ir. Bob van Eijk (UT) – Prof. dr. M. Merk (VU)

Niels Tuning (6) Exercises Lecture 2: QM and Scattering

Niels Tuning (7) Exercises Lecture 2: QM and Scattering

Niels Tuning (8) Answers Homework Theory 2

1 Celebrating Bohr

a) v = (↵qeqp)/(mr) L = mvr = p↵qeqpmr ) 2 b) L =pp↵qeqpmr = ~ r = ~ /(↵qeqpm). ) 9 2 2 19 2 (I tried SI units: ↵qq = keqq =(9 10 Nm /C ) (1.6 10 C) ) 2 34 2 ⇥9 ⇥19 2 ⇥ 30 10 r = ~ /(↵qeqpm)=(10 ) /(9 10 (1.6 10 ) 10 )=0.4 10 m ⇥ ⇥ ⇥ ⇥ ⇥ ↵qeqp ↵qeqp ↵qeqp Etot = Ekin + Epot = 2r r = 2r 2 m(↵qeqp) Etot = 2 . 2~ I tried natural units here: E = ↵2m/2) = (1/137)2 0.5 MeV / 2 = -13.3 eV Exercisestot Lecture 2: ⇥QM and Scattering 3 Spinors 2 Yukawa’s massive force carrier We saw that the requirement of a relativistically correct, but linear equation led to the Dirac equation, (iµ@ m) = 0, with being a four component spinor. 15 µ 15 8 24 a) r 10 m t =r/c =10 /(3 10 )=3 10 s. ⇠ ) ⇥ ⇥ 2 2 2 a) H =(~↵ p~ + m) 22 gives E = p + m24 if the matrices anticommute, ↵i, ↵j = b) E ~/t =6·.6 10 MeVs/(3 10 s) = 200 MeV. { } ⇠↵i↵j + ↵j↵i =⇥ 0. Usually we use⇥ the matrices, =(, ~↵). Show that indeed 12 = 21, using the Pauli-Dirac representation,

- 0 0 ~ = ; ~↵ = . 3 Spinors 0 ~ 0 ✓ ◆ ✓ ◆

ipx µ a)With12 == u(p2)e1 : and pµ i@µ we get ( pµ m)u(p) = 0. Looking for the eigenvectors, it is easier to go back to the! original form; Hu=(~↵ p~ + m)u = Eu, which leads to 10 0 0 0001· 0001 m ~ p~ u u Hu01= 0· 0 A0010= E A 0010 1 = ↵1 = 0 ~ p~ m 1 0uB uB1 = 0 1 00✓ · 10 ◆✓ 0100◆ ✓ ◆ 0 100 with ~ the Pauli matrices,B 00 0 1 C B 1000C B 1000C B C B C B C @ A @ A @ A 01 0 i 10 10 1 = ; 102 = 0 0 ; 3000= i; = 000. i 10 i 0 0 1 01 ✓ ◆ 01✓ 0 0◆ 00✓ i ◆0 ✓ 00◆ i 0 2 = ↵2 = 0 1 0 1 = 0 1 And thus: 00 10 0 i 00 0 i 00 B 00 0 1 C B i 000C B i 00 0C B (~ p~)uB C=(BE m)uA C B (1) C @ · A @ A @ A (~ p~)u =(E + m)u , (2) i 000· A B i 000 where uA and uB are two-component0 i 00 objects. Let’s inspect this two-fold0 i degeneracy,00 and 12 = 0 1 21 = 0 1 ﬁnd the observable that distinguishes00 thei 0 two components. 00i 0 B 000i C B 000 i C B C B C Niels Tuning (9) b) Consider an electron@ with the momentumA in the z-direction,@ p~ =(0, 0,p). WhatA do you ﬁnd for ~ p~ ? · 1 c) What is the eigenvalue of 1~ pˆ for the eigenfunction 2 · 0 = 1 ✓ ◆ withp ˆ = p/~ p~ the vector in the direction of p~ with unit length. What does this value correspond| | to, you think?

1 d) Supposep ˆ can point in any direction, what is then the meaning of 2~ pˆ? What are the possible eigenvalues? ·

2 3 Spinors

We saw that the requirement of a relativistically correct, but linear equation led to the Dirac equation, (iµ@ m) = 0, with being a four component spinor. µ

a) H =(~↵ p~ + m) gives E2 = p2 + m2 if the matrices anticommute, ↵ , ↵ = · { i j} ↵i↵j + ↵j↵i = 0. Usually we use the matrices, =(, ~↵). Show that indeed 12 = 21, using the Pauli-Dirac representation,

0 0 ~ = ; ~↵ = . 0 ~ 0 ✓ ◆ ✓ ◆

ipx µ With = u(p)e and pµ i@µ we get ( pµ m)u(p) = 0. Looking for the eigenvectors, it is easier to go back to the! original form; Hu=(~↵ p~ + m)u = Eu, which leads to · m ~ p~ u u Hu = A = E A ~ p~ m· u u ✓ · ◆✓ B ◆ ✓ B ◆ with ~ the Pauli matrices,

01 0 i 10 10 = ; = ; = ; = . 1 10 2 i 0 3 0 1 01 ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ And thus:

(~ p~)u =(E m)u (1) · B A (~ p~)u =(E + m)u , (2) Exercises Lecture 2: QM· andA ScatteringB where uA and uB are two-component objects. Let’s inspect this two-fold degeneracy, and ﬁnd the observable that distinguishes the two components.

b) Consider an electron with the momentum in the z-direction, p~ =(0, 0,p). What do you ﬁnd for ~ p~ ? · c) What is the eigenvalue of 1~ pˆ for the eigenfunction 2 · 0 = 1 ✓ ◆ withp ˆ = p/~ p~ the vector in the direction of p~ with unit length. What does this value correspond| | to, you think?

1 d) Supposep ˆ can point in any direction, what is then the meaning of 2~ pˆ? What are the possible eigenvalues? ·

b) 2 p 0 ~ p~ = p = · 3 0 p ✓ ◆ c) 1 1 0 1 0 ( ~ pˆ) = = 2 · 2 3 1 2 1 ✓ ◆ ✓ ◆ The eigenvalue -1/2 is the z-component of the spin.

d) 1~ pˆ is the spin component in the direction of motion. 2 · Possible eigenvalue: 1/2. Niels Tuning (10) ± e) Positive and negative helicity:

1000 1 (1) ~ (1) 3 0 uA 0 10 0 0 (1) (⌃ pˆ)u = (1) = 0 1 0 1 =+u · 0 3 u 0010 pz/(E + m) ✓ ◆ B ! B 000 1 C B 0 C B C B C @ A @ A (⌃~ pˆ)u(2) = u(2) ·

4 Rutherford scattering and Cross section

e) d Z Z ↵ 2 1 1 2 = 1 2 =(2mZ Z ↵)2 2 4 ✓ 1 2 2 2 2 ✓ d⌦ mv0 4 sin 4m v0 sin ⇣ ⌘ 2 ⇣ 2 ⌘ 0 2 2 0 2 2 2 2 2 2 2 ✓ q = 0 1 =(mv0) (sin ✓+(1 cos ✓) )=2(mv0) (1 cos ✓)=4m v0 sin mv0 sin ✓ 2 B mv (1 cos ✓) C B 0 C @ A

2 Exercises Lecture 2: QM and Scattering e) Let’s consider the operator

~ pˆ 0 ⌃~ pˆ · , · ⌘ 0 ~ pˆ ✓ · ◆ b) What are its eigenvalues for p 0 ~ (1)p~ = 3p = (2) (1) u· A (2) 0 uAp u = (1) ,u✓= (2) ◆ uB ! uB ! c) where: 1 1 0 1 0 ( ~ pˆ) = = 3 1 1 1 2 · 12 20 0 u(1) = ,u(1) = ~ p/~ (E+m) ✓ u◆(2) = ✓ ,u◆ (2) = ~ p/~ (E+m) TheA eigenvalue0 -1/2B is· the z-component0 of theA spin. 1 B · 1 ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ d) (Hint:1~ pˆ is rotate the spin your component frame such thatin thep~ points direction along of motion.the z-axis, such that you only 2 · needPossible to worry eigenvalue: about p3.)1/2. ± e) Positive and negative helicity: 4 Rutherford scattering 1000 1 (1) ~ (1) 3 0 uA 0 10 0 0 (1) We calculate(⌃ pˆ)u the= distribution of scattering(1) = angles0 for charged particles1 0 on a charged1 tar-=+u · 0 3 u 0010 pz/(E + m) get, like alpha particles✓ scattering◆ o↵B gold! nuclei as done by Ernest Rutherford in 1913. B 000 1 C B 0 C B C B C @ A @ A (⌃~ pˆ)u(2) = u(2) · Niels Tuning (11)

4 Rutherford scattering and Cross section

e) d Z Z ↵ 2 1 1 2 = 1 2 =(2mZ Z ↵)2 2 4 ✓ 1 2 2 2 2 ✓ d⌦ mv0 4 sin 4m v0 sin ⇣ ⌘ 2 ⇣ 2 ⌘ 0 2 2 0 2 2 2 2 2 2 2 ✓ q = 0 1 =(mv0) (sin ✓+(1 cos ✓) )=2(mv0) (1 cos ✓)=4m v0 sin mv0 sin ✓ 2 a) The incoming particle arrives with an impact parameter b, and initial velocity v0. B mv0(1 cos ✓) C The angularB momentum of theC initial state is L = mbv0, whereas the angular momentum somewhere@ after the scatterA can be given by L = mrv = mr d/dr ? Express r in terms of b.

b) The force perpendicular to the direction of the incoming particle is given by Fy = 2 mdvy/dt, and Fy = F sin =(Z1Z2↵/r ) sin . Give the expression for dvy/dt, as a function of b (using the result from a).

2 e) Let’s consider the operator

~ pˆ 0 ⌃~ pˆ · , · ⌘ 0 ~ pˆ ✓ · ◆ What are its eigenvalues for

4 Rutherford scattering

We calculate the distribution of scattering angles for charged particles on a charged tar- get, like alpha particles scattering o↵ gold nuclei as done by Ernest Rutherford in 1913.

b) p 0 ~ p~ = 3p = e) Let’s consider the operator · 0 p ✓ ◆ ~ pˆ 0 ⌃~ pˆ · , · ⌘ 0 ~ pˆ c) ✓ · ◆ What are its eigenvalues for

(1) (2) 1 1 0 1 0 (1) uA (2) uA u = (1) ,u = (2) ( ~ pˆ) = 3 = u u B ! B ! 2 · 2 1 2 1 where: ✓ ◆ ✓ ◆ 1 1 0 0 u(1) = The,u(1) eigenvalue= ~ p/~ (E+m) -1/2u(2) = is the,u(2) z-component= ~ p/~ (E+m) of the spin. A 0 B · 0 A 1 B · 1 ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ (Hint: rotate1 your frame such that p~ points along the z-axis, such that you only Exercisesneedd) to worry2~ aboutpˆpis3.) the Lecture spin component 2: inQM the directionand Scattering of motion. Possible· eigenvalue: 1/2. 4 Rutherford scattering ± a) The incoming particle arrives with an impact parameter b, and initial velocity v . We calculatee) the Positive distribution of scattering and negative angles for charged particleshelicity: on a charged tar- 0 get, like alpha particles scattering o↵ gold nuclei as done by Ernest Rutherford in 1913. The angular momentum of the initial state is L = mbv0, whereas the angular mo- 1000mentum somewhere after1 the scatter can be given by L = mrv = mr d/dr (1) ? ~ (1) 3 0 uA 0Express10r 0in terms of 0b. (1) (⌃ pˆ)u = (1) = 0 1 0 1 =+u · 0 3 u 0010 pz/(E + m) ✓ ◆ B ! Bb)000 The force perpendicular1 C B 0 to theC direction of the incoming particle is given by F = B BeforeC B C y @ mdv /dt, andA @F = F sin =(A Z Z ↵/r2) sin . ~ (2) y (2) y 1 2 (⌃ pˆ)u Give= theu expression for dv /dt, as a function of b (using the result from a). a) The incoming particle arrives with an impact parameter b, and initial velocity v0. · y The angular momentum of the initial state is L = mbv0, whereas the angular mo- After: mentum somewhere after the scatter can be given by L = mrv = mr d/dr ? Express r in terms of b. dϕ/dt r b) The force perpendicular to the direction of the incoming particle is given by Fy = 3 2 4mdvy/dt Rutherford, and Fy = F sin =(Z1Z2↵/r ) sin . scattering Give the expression for dvy/dt, as a function of b (using the result from a).

3 a) d L = mv b = mr r 0 dt ! 1 r2 = bv 0 d/dt

b) dvy Z1Z2↵ Z1Z2↵ d Fy = m = F sin = 2 sin = sin Niels Tuning (12) dt r bv0 dt c)

v0 sin ✓ Z Z ↵ cos ✓ dv = 1 2 d cos ✓ (1) y mbv Z0 0 Zcos ⇡ Z1Z2↵ v0 sin ✓ = (cos ✓ +1) (2) mbv0 sin ✓ Z Z ↵ = 1 2 (3) cos ✓ 1 mbv2 0 d)

2 e) Let’s consider the operator

~ pˆ 0 ⌃~ pˆ · , · ⌘ 0 ~ pˆ ✓ · ◆ What are its eigenvalues for

(1) (2) (1) uA (2) uA u = (1) ,u = (2) uB ! uB ! where: 1 1 0 0 u(1) = ,u(1) = ~ p/~ (E+m) u(2) = ,u(2) = ~ p/~ (E+m) A 0 B · 0 A 1 B · 1 ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ (Hint: rotate your frame such that p~ points along the z-axis, such that you only need to worry about p3.) b) p 0 4 Rutherford scattering ~ p~ = p = · 3 0 p ✓ ◆ We calculate the distribution of scattering angles for charged particles on a charged tar- get, likec) alpha particles scattering o↵ gold nuclei as done by Ernest Rutherford in 1913. 1 1 0 1 0 ( ~ pˆ) = = 2 · 2 3 1 2 1 ✓ ◆ ✓ ◆ The eigenvalue -1/2 is the z-component of the spin.

1 d) 2~ pˆ is the spin component in the direction of motion. Possible· eigenvalue: 1/2. ± e) Positive and negative helicity:

1000 1 (1) ~ (1) 3 0 uA 0 10 0 0 (1) (⌃ pˆ)u = (1) = 0 1 0 1 =+u a) The incoming· particle arrives0 with3 an impactu parameter 0010b, and initial velocity v0. pz/(E + m) ✓ ◆ B ! The angular momentum of the initial state is L = mbvB0,000 whereas the angular1 C mo-B 0 C mentum somewhere after the scatter can be given by BL = mrv = mr d/dr C B C Exercises Lecture 2: QM? and Scattering Express r in terms of b. @ A @ A (⌃~ pˆ)u(2) = u(2) b) The force perpendicular to the direction of the incoming· particle is given by Fy = 2 mdvy/dt, and Fy = F sin =(Z1Z2↵/r ) sin . Give the expression for dvy/dt, as a function of b (using the result from a).

4 Rutherford scattering3 c) We now multiply both sides with dt, and perform the integral from the start until a)the end, so the velocity on the left-hand side ranges from vy =0tovy = v0 sin ✓, and the angle on the right-hand side ranges from =0to = ✓. d Show that L = mv0b = mr r sin(✓/2) Z1Z2↵ 1 dt ! = 2 cos(✓/2) mv0 b 1 r2 = bv d) For a given surface (ring) of possible incoming particles, d 0=dbdbd/dt, the particle is scattered in a certain solid angle d⌦ = sin ✓d✓d. Show that the expression for b)the di↵erential cross section is given by, 2 d b dvdby Z1Z2↵ 1 Z1Z2↵ Z1Z2↵ d Fy == m == F sin2 = 4 ✓ sin = sin d⌦ sin ✓ d✓ mv0 4 sin 2 dt ⇣ ⌘ 2 r bv0 dt

e) Use the 4-vectors pi =(E,0, 0, mv0) and po =(E,0, mv0 sin ✓,mv0 cos ✓) for the in- comingc) and outgoing particle, respectively, and express the di↵erential cross section in terms of the 4-momentum transfer q = p p , instead of ✓. o i v0 sin ✓ Z Z ↵ cos ✓ dv = 1 2 d cos ✓ (1) y mbv 5 Cross section Z0 0 Zcos ⇡ Z1Z2↵ v0 sin ✓ = (cos ✓ +1) (2) Let’s juggle a bit with cross sections and luminosities. mbv0 sin ✓ Z Z ↵ = 1 2 (3) a) The total cross section for proton-proton scattering at the LHC2 is about tot = cos ✓ 1 mbv0 28 2 60 mb. To what surface does this cross section correspond? (1 barn = 10 m .) Niels Tuning (13) What is the size of an object with similar surface? d) b) The cross section for Higgs production at the LHC is approximately pp H+X = ! 30 pb. The “luminosity” is the number of particles produced for a given cross- section, and is an important characteristic of the performance of an accelerator. How 1 many Higgs particles are then produced for a total luminosity of =10fb ? 2 Ltot 34 1 2 c) The “instantaneous” luminosity at the LHC is about inst =10 s cm . How many Higgs particles are thus produced per hour? L

d) Compare the total proton-proton cross section with the cross section for Higgs production. In what fraction of the proton-proton collisions is a Higgs particle produced?

4 c) WeExercises now multiply both Lecture sides with dt ,2: and performQM theand integral Scattering from the start until the end, so the velocity on the left-hand side ranges from vy =0tovy = v0 sin ✓, and the angle on the right-hand side ranges from =0to = ✓. Show that sin(✓/2) Z1Z2↵ 1 = 2 cos(✓/2) mv0 b d) For a given surface (ring) of possible incoming particles, d = bdbd, the particle is scattered in a certain solid angle d⌦ = sin ✓d✓d. Show that the expression for the di↵erential cross section is given by,

2 d b db Z1Z2↵ 1 = = 2 4 ✓ d⌦ sin ✓ d✓ mv0 4 sin ⇣ ⌘ 2

e) Use the 4-vectors pi =(E,0, 0, mv0) and po =(E,0, mv0 sin ✓,mv0 cos ✓) for the incoming and outgoing particle, respectively, and express the di↵erential cross section in terms of the 4-momentum transfer q = p p , instead of ✓. o i

5 Cross section

Let’s juggle a bit with cross sections and luminosities.

a) The total cross section for proton-proton scattering at the LHC is about tot = 28 2 60 mb. To what surface does this cross section correspond? (1 barn = 10 m .) What is the size of an object with similar surface?

b) The cross section for Higgs production at the LHC is approximately pp H+X = ! 30 pb. The “luminosity” is the number of particles produced for a given cross- section, and is an important characteristic of the performance of an accelerator. How Niels Tuning (14) 1 many Higgs particles are then produced for a total luminosity of =10fb ? Ltot 34 1 2 c) The “instantaneous” luminosity at the LHC is about inst =10 s cm . How many Higgs particles are thus produced per hour? L

d) Compare the total proton-proton cross section with the cross section for Higgs production. In what fraction of the proton-proton collisions is a Higgs particle produced?

4 Rutherford

Ø 3d: incoming particle “sees” surface dσ, and scatters off solid angle dΩ Before § Conservation of angular momentum: a) After: § Force: using L conservation Replace r by b, b) c) d)

=(cosθ + 1)

Niels Tuning (15) c) We now multiply both sides with dt, and perform the integral from the start until the end, so the velocity on the left-hand side ranges from vy =0tovy = v0 sin ✓, and the angle on the right-hand side ranges from =0to = ✓. Show that sin(✓/2) Z1Z2↵ 1 = 2 cos(✓/2) mv0 b d)Exercises For a given surface Lecture (ring) of 2: possible QM incoming and particles,Scatteringd = bdbd , the particle is scattered in a certain solid angle d⌦ = sin ✓d✓d. Show that the expression for the di↵erential cross section is given by,

2 d b db Z1Z2↵ 1 = = 2 4 ✓ d⌦ sin ✓ d✓ mv0 4 sin ⇣ ⌘ 2

5 Cross section e) Let’s juggle a bit with cross sections and luminosities.

b) The cross section for Higgs production at the LHC is approximately pp H+X = ! 30 pb. The “luminosity” is the number of particles produced for a given cross- section, and is an important characteristic of the performance of an accelerator. How 1 many Higgs particles are then produced for a total luminosity of tot =10fb ? L Niels Tuning (16) 34 1 2 c) The “instantaneous” luminosity at the LHC is about inst =10 s cm . How many Higgs particles are thus produced per hour? L

d) Compare the total proton-proton cross section with the cross section for Higgs production. In what fraction of the proton-proton collisions is a Higgs particle produced?

4 c) We now multiply both sides with dt, and perform the integral from the start until the end, so the velocity on the left-hand side ranges from vy =0tovy = v0 sin ✓, and the angle on the right-hand side ranges from =0to = ✓. Show that sin(✓/2) Z1Z2↵ 1 = 2 cos(✓/2) mv0 b d) For a given surface (ring) of possible incoming particles, d = bdbd, the particle is scattered in a certain solid angle d⌦ = sin ✓d✓d. Show that the expression for the di↵erential cross section is given by,

2 d b db Z1Z2↵ 1 = = 2 4 ✓ d⌦ sin ✓ d✓ mv0 4 sin ⇣ ⌘ 2

Let’s juggle a bit with cross sections and luminosities.

b) The cross section for Higgs production at the LHC is approximately pp H+X = ! 30 pb. The “luminosity” is the number of particles produced for a given cross- section, and is an important characteristic of the performance of an accelerator. How 1 many Higgs particles are then produced for a total luminosity of =10fb ? Ltot 34 1 2 c) The “instantaneous” luminosity at the LHC is about inst =10 s cm . How many Higgs particles are thus produced per hour? L

d) Compare the total proton-proton cross section with the cross section for Higgs production. In what fraction of the proton-proton collisions is a Higgs particle produced?

Niels Tuning (17) Lecture 1: Standard Model & Relativity

• Standard Model Lagrangian

• Standard Model Particles

Niels Tuning (18) Lecture 1: Accelerators & Relativity

• Theory of relativity – Lorentz transformations (“boost”) – Calculate energy in collissions

• 4-vector calculus

• High energies needed to make (new) particles

Niels Tuning (19) Lecture 2: Quantum Mechanics & Scattering

• Schrödinger equation – Time-dependence of wave function

• Klein-Gordon equation – Relativistic equation of motion of scalar particles

Ø Dirac equation – Relativistically correct, and linear – Equation of motion for spin-1/2 particles – Prediction of anti-matter

Niels Tuning (20) Lecture 2: Quantum Mechanics & Scattering

• Scattering Theory – (Relative) probability for certain process to happen – Cross section

Classic Scattering amplitude in Quantum Field Theory • Fermi’s Golden Rule

– Decay: “decay width” Γ a → b + c – Scattering: “cross section” σ a + b → c + d

Niels Tuning (21) Resonances Quantum mechanical description of decay

State with energy E0 ( ) and lifetime τ To allow for decay, we need to change the time-dependence:

What is the wavefunction in terms of energy (instead of time) ? Ø Infinite sum of flat waves, each with own energy

Ø Fourier transformation:

1 = Ψ0 ⎛ i ⎞ i⎜(E0 − E)− Γ⎟ ⎝ 2 ⎠ Resonance

P Probability to find particle with max Breit-Wigner energy E:

Pmax/2

E0-Γ/2 E0 E0-Γ/2

Resonance-structure contains information on: § Mass § Lifetime § Decay possibilities Rutherford

Ø 3d: incoming particle “sees” surface dσ, and scatters off solid angle dΩ Ø Calculate:

Niels Tuning (25) Scattering Theory

Let’s try some potentials

• Yukawa: (Pion exchange)

• Coulomb: (Elastic scattering)

• Centrifugal Barier: (Resonances)

Niels Tuning (26) Well-known resonances e+e- cross-section e+e-→R→ e+e-

Z-boson

J/ψ Outline for today

• Resonances • Quarkmodel – Strangeness – Color • Symmetries – Isospin – Adding spin – Clebsch Gordan coefficients

Niels Tuning (28) Lecture 1: Standard Model & Relativity

• Standard Model Lagrangian

• Standard Model Particles

Niels Tuning (29) Particles

• Quarks and leptons…:

Niels Tuning (30) Particles…

Niels Tuning (31) The number of ‘elementary’ particles

1932: 1936: 1947: electron electron electron proton proton proton neutron neutron neutron muon muon pion

1947

§ 1932: the positron had been observed to confirm Dirac’s theory, § 1947: and the pion had been identified as Yukawa’s strong force carrier,

Ø So, things seemed under control!?

§ Ok, the muon was a bit of a mystery…

§ Rabi: “Who ordered that?” Quark model Discovery strange particles

Discovery strange particles Discovery strange particles