Invariance Principles • Invariance of states under operations implies conservation laws – Invariance of energy under space translation → momentum conservation – Invariance of energy under space rotation → angular momentum conservation • Continuous Space-Time Transformations – Translations – Rotations – Extension of Poincare group to include fermionic anticommuting spinors (SUSY) • Discrete Transformations – Space Time Inversion (Parity=P) – Particle-Antiparticle Interchange (Charge Conjugation=C) – Time Reversal (T) – Combinations of these: CP, CPT • Continuous Transformations of Internal Symmetries – Isospin – SU(3)flavor – SU(3)color – Weak Isospin – These are all gauge symmetries Symmetries and Conservation Laws SU(2) SU(3)
8 generators; 2 can be diagonalized at the same time: rd I3 … 3 component of isospin Y … hypercharge SU(3) Raising and lowering operators Y = +1/2√3
Y = +1/√3 SU(3) SU(3) Adding 2 quarks SU(3) Adding 3 quarks Quantum Numbers
• Electric Charge • Baryon Number • Lepton Number • Strangeness • Spin • Isospin • Parity • Charge Conjugation Electric Charge Quantum Numbers are quantised properties of particles that are subject to constraints. They are often related to symmetries
Electric Charge Q is conserved in all interactions
Strong ✔ Interaction
Weak ✔ Interaction Baryon Number Baryon number is the net number of baryons or the net number of quarks ÷ 3 Baryons have B = +1 Quarks have B = +⅓ Antibaryons have B = -1 or Antiquarks have B = -⅓ Everything else has B = 0 Everything else has B = 0
Baryons = qqq = ⅓ + ⅓ + ⅓ = 1 Mesons = qq = ⅓ + (-⅓) = 0
Baryon Number B is conserved in Strong, EM and Weak interactions
Total (quarks – antiquarks) is constant Baryon Number
Strong ✔ Interaction
Weak ✔ Interaction
Since the proton is the lightest baryon it cannot decay if B is conserved e.g:
Leptons have L = +1 Antileptons have L = -1 Everything else has L = 0
Lepton Number L is conserved in Strong, EM and Weak interactions but is also separately conserved within lepton families:
– + e and ѵe have Le = 1 e and ѵe have Le= -1 μ– and ѵ have L = 1 + μ μ μ and ѵμ have Lμ = -1 �– and ѵ have L = 1 + � � � and ѵ� have L� = -1
Le, L μ and L � are separately conserved Lepton Number
Pair ✔ Production
Pion Decay ✔
Muon Decay ✔ Lepton Number
Radiative Forbidden ✗ Decay ✗ OK ✔
L is conserved but neither Le or Lμ separately
The decay has not been observed and has a Branching Ratio < 10-9 Spin
Spin is an intrinsic property of all particles: 0ħ, 1ħ, 2ħ, 3ħ, . . . Bosons ½ħ, 3/2ħ, 5/2ħ, . . . Fermions
Spin is like angular momentum but a Quantum Mechanical effect. For spin S there are 2S+1 states of different Sz (like 2J+1 in Angular Momentum)
For Spin S = ½, Sz can be +½ or -½ (2 states)
For Spin S = 1, Sz can be +1, 0, -1 (3 states)
For a process a + b → c + d the cross section is
This can be used to determine the spin of unknown particles Helicity
• All particles have spin and three momentum p • For fermions S=½, two possible arrangements
p! p! Left Right Handed Handed f f Spin vector Spin vector opposite same as momentum momentum vector vector Photon Helicity
• The photon has spin 1. Therefore JZ =-1, +1 • Massless photon cannot have 0 helicity
p! p! p!
JZ= −1 JZ=0 JZ= 1 γ γ γ Isospin
Used mostly in Nuclear Physics from charge independence of nuclear force p ↔ p = n ↔ n = p ↔ n sometimes called Isobaric Spin/Isotopic Spin T (or t!)
Isospin is represented by a 'spin' vector I with component I3 along some axis
I = ½ : p has I3 = +½ (↑), n has I3 = -½ (↓)
– 3 0 3 + 3 I = 1 : π has I = -1, π has I = 0, π has I = +1
u has I3 = +½ u has I3 = -½
d has I3 = -½ d has I3 = +½
I3 really only counts the number of u and d quarks
- p = uud = ½ + ½ + (-½) = ½ π = ud = -½ + (-½) = -1 Isospin Symmetry Proton and neutron are different states of the nucleon, with the quantum number of Isospin (I = 1/2) Heisenberg Complete analogy with the quantum number of spin. Strong interactions conserve isospin (I) and do not
depend on I3: do not distinguish p from n
• Observed equivalence of n-p, n-n, p-p forces once EM effects are subtracted (the proton is heavier than the neutron thanks to the energy needed to “bring” electric charge on the proton) • The isospin symmetry is transferred from the p-n equality to the quark level u-d → their nearly mass equality is the source of the isospin symmetry • Therefore, the symmetry extend to all baryons and mesons that are associated by u-d quark exchange
π + = ud
1 + - π 0 = (dd − uu) Mass of π and π (equal for C-symmetry): 140 MeV 2 Mass of π0: 135 MeV € π − = ud
€ € Isospin
I3 can be related to charge Q and baryon number B:
For a proton B = 1, I3 = +½ and hence Q = 1
Since the Strong Interaction doesn't distinguish p from n or u from d, I and I3 are conserved in Strong Interactions
This is equivalent to saying that the number of (u – u) – (d – d) = constant
In Weak Interactions where u ⇌ d, I and I3 are NOT conserved
In EM Interactions u and d are not changed but because of the different charges, u and d can be distinguished. Hence I3 is conserved but I is NOT conserved Isospin
EM and weak interactions do not conserve I Strangeness
Associated production of 'strange' particles π– + p → K0 + Λ0
K0 and Λ0 'Strange' – decay weakly not strongly Strangeness
Assume pair production of new quark s and antiquark s by Strong Interaction but once produced s and s can only decay weakly
Strong Production Weak Decay
Strange particles are produced in pairs (ΔS = 0) via strong interaction but they decay weakly (ΔS = ±1) Strangeness
0 0 0 Λ is uds Λ0 is uds The K has an antiparticle the K 0 0 although it is neutral, unlike the K is ds K is ds π0 which is its own antiparticle K+ is us K– is us Strangeness can be combined with Isospin if Gell Mann – Nishijima relation The s quark has strangeness S = -1 Strangeness is conserved in Strong and EM Interactions but NOT in Weak Interactions Likewise charm, bottom, top quantum numbers
Strong and EM Interactions do not change quark flavours. Number of (u – u), (d – d), (s – s), (c – c), (b – b) , (t – t) constant Weak Interaction changes one quark type to another Isospin, Strangeness and Hypercharge There is a compact way to express the relation between electric charge, third component of the isospin and baryon number: Q B = I + e 3 2
N(q) − N(q) (its conservation implies the stability of Baryon number is B = 3 matter: no proton decay) € If we also include the other quantum number associated to the s quark (strangeness S):
€
Q B + S Y = I + = I + With Y = B + S defined as hypercharge | e | 3 2 3 2
€ Strangeness
Strangeness is conserved in strong and EM interactions, not in weak processes
strong
EM
weak
weak
Associated production of a ss quark pair, weak decay of the s (s) quark Parity
Parity is a Quantum Mechanical concept
For a wavefunction ѱ(r) and Parity If an operator Ô acts on a operator P, the Parity Operator wavefunction ѱ such that ѱ reverses the coordinates r to –r is unchanged Pψ(r) =ψ(−r)
ѱ is an Eigenfunction of Ô and λ is the Eigenvalue € Hence the eigenvalues of Parity are +1 (even) and -1 (odd)
Global and discrete operations: e.g. translation r → r+δr is global, reflection through the origin of coordinates: x, y, z → -x, -y, -z is discrete. Parity of Particles • Intrinsic parity – Fermions – Consider electrons and positrons represented by a wave function Ψ. Pˆ x!,t P x!,t Ψ( ) = e± Ψ(− ) – Dirac equation is satisfied by a wave function representing both electrons and positrons ⇒ related and it can be shown P P 1 e+ e− = − – Strong and EM reactions always produce e+e- pairs. – Arbitrarily have to set one =1 and the other = -1.
• P2 = 1 (P is unitary operator) and its eigenvalues (if any) are ±1 (the parity of the system) • Parity is a multiplicative quantum number: if ψ = φ + η then Pψ = Pφ x Pη Parity
The Parity Operator reverses the coordinates r to –r
Equivalent to a reflection in the x-y plane followed by a rotation about the z axis
Reflection in x-y plane
Rotation about z axis ψ = cos x ⇒ Pψ = cos(−x) = cos x = +ψ ⇒ ψ is even (P = +1) ψ = sin x ⇒ Pψ = sin(−x) = −sin(x) = −ψ ⇒ ψ is odd (P = −1) ψ = cos x + sin x ⇒ Pψ = cos x − sin x ≠ ±ψ ⇒ ψ has no defined P Parity
The hydrogen atom with a potential V(r) = V(-r) must have a well defined parity Its wave functions are the product of radial and angular functions (spherical harmonics): m ψ(r,θ,φ) = χ(r) Yl (θ,φ) r → −r ⇒ θ → π −θ, φ → π + θ m l and P Yl = (−1) Electric dipole transitions with photon emission have Δl= ± 1. In order to conserve the parity of the global system (atom + photon) the latter must have NEGATIVE PARITY. € Parity IS FOUND to be conserved in EM and strong interactions, but not in weak € interactions
• The “intrinsic” parity of the proton and of the neutron are assumed by convention +1 (baryons are conserved) • The “intrinsic” parity of the charged pion is -1: from an experiment on π- + d → n + n • While pions can be created singly, particles carrying “strange quarks” are created in pairs, whose parity is -1
Link between the total angular momentum of a particle: J = L + S and the parity:
J P =0+ scalar particle J P =0- pseudoscalar particle J P =1- vector particle J P =1+ axial-vector particle Parity Parity is a multiplicative quantum number. The parity of a composite system is equal to the product of the parities of the parts:
One can show that a state with angular momentum ℓ has parity
For a system of particles:
For Fermions P (antiparticle) = (-1) × P (particle) For Bosons P (antiparticle) = P (particle)
Arbitrarily assign p, n → P = +1 p, n → P = -1 Others determined from experiment (angular distributions) Parity of π+, π–, π0 → P = -1
Parity We label mesons by JP – SpinParity corresponding to how their wavefunctions behave: JP = 0– Pseudoscalar (Pressure,...) 0+ Scalar (Mass, time, wavelength,...) 1– Vector (Momentum, position,...) 1+ Axial Vector (Spin, angular momentum,...) 2+ Tensor (Stress in a material,...)
Vector r → -r ∴ P = -1 Axial Vector r → r ∴ P = +1
Parity is conserved Strong and EM Interactions but NOT Weak Vectors and Axial Vectors Examples: scalar
pseudoscalar
vector
axial-vector
Parity of a meson with no angular momentum
Meson with angular momentum
Baryon with angular momentum
Anti-baryon with angular momentum Parity of Fermions • Assign positive parity state to particles, negative to antiparticles: P = P = P =1 e− µ − τ − P = P = P = −1 e+ µ + τ + • Make same assumption about quarks to be consistent:
Pd = Pu = Ps = Pc = Pb = Pt =1
Pd = Pu = Ps = Pc = Pb = Pt = −1 Parity of Mesons
• Mesons are quark antiquark pairs: L L+1 PM = PqPq′(−1) = (−1)
• For L=0 then P = -1. – Look at charged pion interactions with nucleons – Complex relationship and not straightforward. Parity of Baryons • Baryons contain three quarks:
L L PB = PqPqPq (−1) = (−1) L L+1 PB = Pq Pq Pq (−1) = (−1)
• Dealing with ground state Baryons in most cases, hence L=0 Parity of Photons
• Photons: PΨ(x,t) =-Ψ(-x,t) – If applied to electric field reverses field lines direction. – Hence the photon wave function must be odd under the parity transformation. – Photon also has spin 1 parity conservation not straight forward. – Neutral pion decays. Problematic just under P transformations Some Comments on Parity
• Parity effects different particles/fields depending on their type: – Spin-1 particles: • (Polar) Vectors V(x) PV(x) = V(-x) = -V(x) PV = -1 (e.g. linear momentum p) • Axial Vectors are of type V(x) x V’(x) PA = (-1) x (-1) = +1 (e.g. angular momentum L = r x p) – Spin-0 particles:
• Scalars no dimension and transform to themselves PS = +1 • Pseudoscalars are of type V(x).V’(x)xV’’(x) PPS = (-1)³ = -1 – Spin-2 particles:
• Tensors transform with ∴ PT = +1 Vector Reflections
Cross Product
Mirror: Parity
Vector Forces: Particle Exchange • The Strong interaction, mediated by the exchange of spin-1 gluons, a vector interaction • The EM interaction, mediated by the exchange of spin- 1 photons, is a vector interaction • The weak interaction, mediated by the exchange of spin-1 W and Z bosons, is a mixed vector and axial- vector interaction!
Force Carriers γ Z W g q=0 q=0 q=1 q=0 S=1 S=1 S=1 S=1 What about π0 → γγ?
• We need to keep in mind that π0 has • Spin 0 & parity of -1
• Photons • Spin 1 & parity of -1
So, why we can have π0→γγ Charge Conjugation
• Changes: – Charge – Baryon number – Lepton number – Etc… • Does not change: – Mass, Energy, momentum – Spin Charge Conjugation The Charge Conjugation operator reverses the sign of electric charge and magnetic moment (μ) This implies particle ⇌ antiparticle Swaps Particles with Antiparticles proton ⇌ antiproton Q = +e C Q = -e B |χ⟩ is (Dirac) bra/ket = +1 B = -1 notation for ѱX i.e. μ -μ + |π ⟩ ≡ ѱπ Cπ + → π − ≠ π + Hence π ± not eigenstates of C C only has definite eigenvalues for neutral systems such as the π0
If we apply two subsequent Charge Conjugations we do get the original particle back. ∴ λ = ± 1 Cπ 0 → ± π 0 Charge Conjugation
EM fields come from moving charges which change sign under
Charge Conjugation ∴ C� = -1 C • If swap charge that produces a photon, then we γ → ± γ get a reversed electric field. n photons have C = (-1)n • Implies that photon is anti-symmetric under charge changes. 0 Since π → ɣɣ this implies Cπ0 = +1 (assuming C invariance in EM decays)
0 π → 2γ Note π0 →ɣɣɣ is then forbidden 2 C 0 = C C = (−1) π γ γ The η (eta) meson (mass 550 MeV/c2) η → ɣɣ The π0 decay in 3 photons occurs at the level of 10-8 w.r.t. the decay in 2 η ↛ ɣɣɣ photons, because EM interactions are C-invariant i.e. Cη = +1
C is conserved in Strong and EM Interactions but NOT in Weak CP Weak interactions are not C-invariant: (Right-handed neutrinos do not exist) But, if we apply both P and C operations (CP transformation):
Existing Left-handed neutrino
However, as we will see later on, CP is violated at the level of 10-4
If we also apply the time reversal transformation (T), the CPT theorem tells us that all interactions are invariant under CPT transformation.
The theorem is based on very general assumptions. It can be verified by comparing the properties of particles and antiparticles, e.g.: Particles and Helicity
• (Most) Fermions come in left and right-handed verities: fL and fR. • How are they effected by the C and P operations
P fL fR CP C
fL f R Neutrinos – Weird Particles P ν L ν R CP C
ν L ν R • The neutrinos only come in one helicity – Particle Left Handed – Antiparticle Right Handed Evidence for Parity Violation CP Transformations: Nickel W-Boson
• In fact this is a property of the W boson – Only sees left handed particles – Only sees right handed anti-particles – Hence if right handed neutrinos exist they would not interact with W’s and hence not weakly
• Massive particles: – Not pure helicity states – Right handed = +1, Left Handed –1 – Left handed massive particle has helicity: -v/c Charged Pion Decays
+ + π → µ ν µ BR ≈100% + −4 → e ν e BR ≈1.27 ×10 • Pion has spin 0. – Spins of decay particles must sum to zero.
+ + νµ π µ CP Conservation • In general the Weak interaction does not conserve Parity or Charge Conjugation. P • It does however almost ν L ν R satisfy CP Conservation CP – This is a C and P operation C one after the other – Take product of C and P to calculate ν L ν R Conserved Quantum Numbers
Quantity Strong EM Weak Charge Q ✔ ✔ ✔ Baryon Number B ✔ ✔ ✔ Lepton Number L ✔ ✔ ✔ Strangeness S ✔ ✔ ✗ Isospin I ✔ ✗ ✗
I3 ✔ ✔ ✗ Parity P ✔ ✔ ✗ Charge Conjugation C ✔ ✔ ✗