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 Scattering amplitude in Classic Quantum Field Theory • Fermi’s Golden Rule a → b + c – Decay: “decay width” Γ – Scattering: “cross section” σ a + b → c + d Niels Tuning (21)
Resonances
Quantum mechanical description of decay State with energy E 0 ( ) 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 = Ψ 0 i ⎛ ⎞ ( ) i E E − − Γ ⎜ ⎟ 0 2 ⎝ ⎠
Resonance P max Probability to find particle with Breit-Wigner energy E: P max /2 E 0 - Γ /2 E 0 E 0 - Γ /2 Resonance-structure contains information on: § Mass § Lifetime § Decay possibilities
Rutherford Ø 3d: incoming particle “sees” surface d σ , and scatters off solid angle d Ω 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 1936: 1947: 1932: 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 • Why were these particles called strange ? Ø Large production cross section (10 -27 cm 2 ) Ø Long lifetime (corresponding to process with cross section 10 -40 cm 2 ) Niels Tuning (36)
Discovery strange particles • Why were these particles called strange ? Ø Large production cross section (10 -27 cm 2 ) Ø Long lifetime (corresponding to process with cross section 10 -40 cm 2 ) • Associated production! Niels Tuning (37)
Discovery strange particles • Why were these particles called strange ? Ø Large production cross section (10 -27 cm 2 ) Ø Long lifetime (corresponding to process with cross section 10 -40 cm 2 ) • Associated production! Niels Tuning (38)
Discovery strange particles • Why were these particles called strange ? Ø Large production cross section (10 -27 cm 2 ) Ø Long lifetime (corresponding to process with cross section 10 -40 cm 2 ) • Associated production! π New quantum number: Ø Strangeness, S π Ø Conserved in the strong K π interaction, Δ S=0 p Particles with S=+1 and § Λ S=-1 simultaneously π produced Ø Not conserved in individual decay, Δ S=1 Niels Tuning (39)
Discovery strange particles • Why were these particles called strange ? Ø Large production cross section (10 -27 cm 2 ) Ø Long lifetime (corresponding to process with cross section 10 -40 cm 2 ) Production: • Associated production! π - p → K 0 Λ 0 π Decay: New quantum number: K 0 → π - π + Ø Strangeness, S Λ 0 → π - p π Ø Conserved in the strong K π interaction, Δ S=0 p Particles with S=+1 and § Λ S=-1 simultaneously π produced Ø Not conserved in individual decay, Δ S=1 Niels Tuning (40)
Intermezzo: conserved quantities • What is conserved in interactions? – Decays & Scattering Ø Energy, momentum Ø Electric charge Ø Total angular momentum (not just spin) • Strangeness? • Baryon number • Lepton flavour • Colour? • Parity? • CP ? • … Niels Tuning (41)
Kinematics π - π + K 0 S m 1 m 2 Specific (m 1 =m 2 =m): M before after What is the energy of final-state particles?
Kinematics p Λ 0 π - m 1 m 2 M Specific: (m 1 =m 2 =m) What if masses of final-state particles differ, m 1 ≠ m 2 ? General: ( ) 2 2 M m ± Δ p 1,2 =? E = 1 , 2 2 M
Strange particles Strangeness Mesons Baryons Particle Mass S Particle Mass S K 0 497.7 +1 Σ + 1189.4 -1 K + 493.6 +1 What is different…? 1192.6 -1 Σ 0 K - 493.6 -1 1197.4 -1 Σ - K 0 497.7 -1 Λ 0 1115.6 -1 1314.9 -2 Ξ 0 1321.3 -2 Ξ - Corresponding anti-baryons have positive Strangeness
50’s – 60’s • Many particles discovered à ‘particle zoo’ • Will Lamb: “ The finder of a new particle used to be awarded the Nobel Prize, but such a discovery now ought to be punished with a $10,000 fine. ” • Enrico Fermi: “If I could remember the names of all these particles, I'd be a botanist.” • Wolfgang Pauli: “Had I foreseen that, I would have gone into botany." Niels Tuning (45)
The number of ‘elementary’ particles “Particle Zoo”
Strange particles The 8 lightest strange baryons: baryon octet Particle Mass S n 938.3 0 p 939.6 0 Σ + 1189.4 -1 Σ 0 1192.6 -1 1197.4 -1 Σ - Λ 0 1115.6 -1 1314.9 -2 Ξ 0 Ξ - 1321.3 -2 Breakthrough in 1961 (Murray Gell-Mann): “ The eight-fold way ” (Nobel prize 1969) Also works for: Eight lightest mesons - meson octet Other baryons - baryon decuplet
Strange particles The Noble Eightfold Path is one of the principal The 8 lightest strange baryons: baryon octet teachings of the Buddha, who described it as the way leading to the cessation of suffering and the achievement of self-awakening. Particle Mass S n 938.3 0 p 939.6 0 Σ + 1189.4 -1 Σ 0 1192.6 -1 1197.4 -1 Σ - Λ 0 1115.6 -1 1314.9 -2 Ξ 0 Ξ - 1321.3 -2 Breakthrough in 1961 (Murray Gell-Mann): “ The eight-fold way ” (Nobel prize 1969) Also works for: Eight lightest mesons - meson octet Other baryons - baryon decuplet
Strange particles The 8 lightest strange baryons: baryon octet strangeness: Particle Mass S n 938.3 0 p 939.6 0 Σ + 1189.4 -1 Σ 0 1192.6 -1 1197.4 -1 Σ - Λ 0 1115.6 -1 1314.9 -2 Ξ 0 Ξ - 1321.3 -2 Breakthrough in 1961 (Murray Gell-Mann): “ The eight-fold way ” (Nobel prize 1969) Also works for: Eight lightest mesons - meson octet Other baryons - baryon decuplet
Discovery of Ω - Not all multiplets complete… 1232 MeV 1385 MeV 1533 MeV Gell-Mann and Zweig predicted the Ω - … and its properties
Discovery of Ω - Not all multiplets complete… 1232 MeV 1385 MeV 1533 MeV 1680 MeV Gell-Mann and Zweig predicted the Ω - … and its properties
Discovery of Ω - Discovered in 1964: K – + p à Ω – + K + + K 0 Not all multiplets complete… Ξ + π 1232 MeV 1385 MeV 1533 MeV 1680 MeV Gell-Mann and Zweig predicted the Ω - … and its properties
Discovery of Ω - Discovered in 1964: K – + p à Ω – + K + + K 0 Not all multiplets complete… Ξ + π 1232 MeV 1385 MeV 1533 MeV 1680 MeV Gell-Mann and Zweig predicted the Ω - … and its properties
Quark model Gell-Mann en Zweig (1964): “ All multiplet patterns can be explained if you assume hadrons are composite particles built from more elementary constituents: quarks ” § First quark model: 3 types: up, down en strange (and anti-quarks) § Baryons: 3 quarks § 26 � 3+3 Mesons: 2 quarks § mesonen up down strange baryonen p = uud n = udd Σ + = uus Λ 0 = uds Ξ 0 = uss
Quark model • Mesons: – Octet • Baryons: – Octet – Decuplet Niels Tuning (55)
New last year: Ω c 0 (css) • Just discovered 5 excited (ccs) states • Still active research! + with K – : + = csu state 2. Combine Ξ c 1. Reconstruct Ξ c + K – + à p K – π + Strong decay: Ω c 0 à Ξ c Ξ c 0 states ! 5 narrow Ω c 0 = css state Ω c Spectrum Ξ c+ sideband Niels Tuning (56)
The number of ‘elementary’ particles
“Problems” 1) Are quarks ‘real’ or a mathematical tric? 2) How can a baryon exist, like Δ ++ with (u ↑ u ↑ u ↑ ), given the Pauli exclusion principle? Niels Tuning (58)
“ Problem ” of quark model s s Intrinsic spin: = symmetric Ω - s quarks: = symmetric u u Intrinsic spin: = symmetric Δ ++ u quarks: = symmetric J=3/2, ie. fermion, ie. obey Fermi-Dirac statistics: anti-symmetric wavefunction New quantum number: color! s s s s - 3 values: red, green, blue � - Only quarks, not the leptons
The Particle Zoo mass Force carier: γ <1x 10 -18 eV Leptons: e - ,µ - , τ - , υ e , υ µ , υ τ ~0 – 1.8 GeV Mesons: π + , π 0 , π - ,K + ,K - ,K 0 , ρ + , ρ 0 , ρ - 0.1-1 GeV Baryons: p,n, Λ , Σ + , Σ - , Σ 0 , Δ ++ , Δ + , Δ 0 , Δ - , Ω ,… 1-few GeV http://pdg.lbl.gov/
Protons and neutrons Proton and neutron identical under strong interaction proton neutron m p = 938.272 MeV m n = 939.565 MeV ? Nucleon + internal degree of freedom to distinguish the two
Multiplets Pattern (mass degeneracy) suggest internal degree of freedom m = 1232 MeV m = 1385 MeV m = 1530 MeV m = 1672 MeV Baryon decuplet
Eightfold way • Introduction of quarks • Introduction of quantum numbers – Strangeness – Isospin Niels Tuning (63)
Tetra- and pentaquarks ?? • Tetraquark discovered in 2003 – X(3872) – Also charged cc and bb states… • Pentaquark discovered in 2016 – P c + (4450) ce, called hereafter the Λ ⇤ decay chain matrix element. Ne c ! ψ p , ψ ! µ + µ � decay sequence, e Λ 0 b ! P + c K � , P + Niels Tuning (64)
Timeline • Active research…: Niels Tuning (65)
In the news last year In the News Niels Tuning (66) Patrick Koppenburg Pentaquarks at hadron colliders 18/01/2017 — Physics at Veldhoven [2 / 33]
[LHCb, Phys. Rev. Lett. 115 (2015) 072001, arXiv:1507.03414] What is a Pentaquark? > 300 papers citing the result, with many possible interpretations. Niels Tuning (67) Patrick Koppenburg Pentaquarks at hadron colliders 18/01/2017 — Physics at Veldhoven [26 / 33]
Symmetries
Conserved quantities Time dependence of observable U: Hamilton formalism: If U commutes with H, [U,H]=0 (and if U does not depend on time, dU/dt=0) Then U is conserved: d/dt<U> = 0 U conserved à U generates a symmetry of the system Niels Tuning (69)
Other symmetries: Transformation Conserved quantity Translation (space) Momentum Translation (time) Energy Rotation (space) Orbital momentum Rotation (iso-spin) Iso-spin
Quantum mechanics: orbital momentum L x and L y cannot be known simultaneously Sequence matters! L 2 and L i (i=x,y,z) can be known simultaneously Can both be used to label states Provided V = V(r), ie not θ dependent [ L 2 , H ] = [ L z , H ] = 0 L 2 and L z label eigenstates
Quantum mechanics: orbital momentum f l m =Y l m spherical harmonics L z m = -l, -l+1, …, 0, … , l-1, l 2 2 1 Different notation: 0 L y -1 L x -2
Quantum mechanics: (intrinsic) spin Spin is characterized by: - total spin S - spin projection S Z Rotations: SO(3) group similar Internal symmetry: SU(2) group Spin is quantized, Eigenfunctions |s,m s >: just as orbital momentum
spin- ½ particles spin- spin- up down
spin- ½ particles Complex numbers general | α | 2 prob for S z = + | β | 2 prob for S z = - - Pauli matrices: any complex 2x2 matrix can be written as: A = a σ 1 +b σ 2 +c σ 3
Isospin
Protons and neutrons Proton and neutron identical under strong interaction proton neutron m p = 938.272 MeV m n = 939.565 MeV ? Nucleon + internal degree of freedom to distinguish the two
Protons and neutrons: Isospin Proton and neutron identical under strong interaction proton neutron m p = 938.272 MeV m n = 939.565 MeV Introduce new quantum number: isospin Proton and neutron (‘nucleons’): I en I 3 Nucleon + internal degree of freedom Isospin ’up’ Isospin ‘down’
Possible states for given value of the Isospin
Possible states for given value of the Isospin I z = +3/2 I z = +1 I z = +1/2 I z = +1/2 I z = 0 I z = -1/2 I z = -1/2 I z = -1 I z = -3/2
Possible states for given value of the Isospin I z = +3/2 I z = +1 I z = +1/2 I z = +1/2 I z = 0 I z = -1/2 I z = -1/2 I z = -1 I z = -3/2 Δ ++ π + proton Δ + π 0 Δ 0 neutron π - Δ - m p ~ 939 MeV m π ~ 140 MeV m Δ ~ 1232 MeV
I z =-1 I z =0 I z =+1 I = 3/2 π + π 0 π - I = 1 p n I = 1/2 I = 0 Baryon decuplet
Adding spin
Quantum mechanica: adding spin |s 1 ,m 1 > + |s 2 ,m 2 > à |s,m> 1) Conditions: - S z add up m= m 1 + m 2 S = |s 1 -s 2 |, |s 1 -s 2 |+1, .. , s 1 +s 2 -1 , s 1 +s 2 - S can vary between difference and sum 2) Notation: C: Clebsch-Gordan coefficient
Adding spin of two spin- ½ particles S z S (1) +1 1 (2) 0 ? (3) -1 1
Adding spin of two spin- ½ particles S z S (1) +1 1 ( ) + 1 0 (2a) ( ) - (2b) 0 0 -1 1 (3)
Adding spin of two spin- ½ particles 2 2 3 1 ⊗ = ⊕ ( ) Triplet S=1 + (symmetric) ( ) - Singlet S=0 (anti-symmetric)
Adding spin of two spin- ½ particles 2 2 3 1 ⊗ = ⊕ Triplet (symmetric) Singlet (anti-symmetric)
Quantum mechanics: adding spin Clebsch-Gordan coefficient Specific: adding spin of two spin-1/2 particles: Triplet ( ) + (symmetric) ( ) - Singlet (anti-symmetric)
Why is and not ? S z 1 ? 0 S y S x - 1 Griffiths Par 4.4.3
Clebsch-Gordan coefficients Coefficients can be used “both ways”: 1) add |s 1 ,m 1 > + |s 2 ,m 2 > à |s,m> 2) decay |s,m> à |s 1 ,m 1 > + |s 2 ,m 2 >
Clebsch-Gordan coefficients A) Find out yourself (doable, but bit messy…)
decay Every coefficient has sqrt
Every coefficient has sqrt scattering
Every coefficient has sqrt
Every coefficient has sqrt |s,m> à |s 1 ,m 1 > + |s 2 ,m 2 > Decay:
Every coefficient has sqrt |s 1 ,m 1 > + |s 2 ,m 2 > à |s,m> 2-particle process:
Example: π p scattering 1) π + p → π + p § I z = 3/2 § è Pure I = 3/2 ! 2) π - p → π - p § I z = 3/2 1 1 1 3 1 2 1 1 | 1 , 1 | , | , | , − 〉 〉 = − 〉 − − 〉 2 2 3 2 2 3 2 2 § è Mixed I ! What is relative cross section to make the I=3/2 resonance? Niels Tuning (98)
Example: π p scattering ( ) p p ~ 200 mb + + + + σ π → Δ → π Compare Δ resonance in elastic scattering: 1) π + p → π + p 2) π - p → π - p ( ) 0 p p ~ 25 mb − − σ π → Δ → π Niels Tuning (99)
Group theory 3 3 8 1 ⊗ = ⊕ • Mesons: – 2 quarks, with 3 possible flavours: u, d, s – 3 2 =9 possibilities = 8 + 1 q=1 Niels Tuning (100)
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