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Example from Classical Mechanics: Momentum Conservation

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Example from Classical Mechanics: Momentum Conservation IntroIntro toto NuclearNuclear andand ParticleParticle PhysicsPhysics (5110)(5110) Apr 15, 2009 Selection Rules Symmetries and Conservation Laws 4/15/2009 1 WhatWhat dodo thethe patternspatterns reveal?reveal? “Flavor symmetries” and the existence of quarks! Isospin symmetry reflects the equality of the strong interactions of quarks and the near equality of the masses of the u and d quarks Phenomenology provided the roadmap, waiting for inspiration to put it all together Example: Gell-Mann-Nishijima Formula Y Y is the strong hypercharge QI=+3 2 … are the additional flavor quantum numbers Y = B + S + C + B'+T discovered after strangeness B=baryon #, S=Strangeness, C=charm, B’=bottomness/beauty, T=topness/truth Isospin and flavor quantum numbers, and therefore strong hypercharge, are conserved by the strong interaction. Weak decays, with long lifetimes and lepton production, are the hallmarks of a new flavor 4/15/2009 2 Reaction/DecayReaction/Decay ExamplesExamples -- GenericGeneric ab+→c+d • These reactions are connected by “crossing symmetry” ac+→b+d • If any one of these is allowed, the others are allowed too ad+→c+b • Interactions are all the same ⇒ the matrix elements are the same. Kinematics (phase space) can be ab→+c+d different cd+→a+b “Principle of Detailed Balance” follows from microscopic reversibility (“Time Equality of first and last: Reversal” invariance), subject to the requirement of energy conservation Multiple ways to get at the same physics 4/15/2009 3 Reaction/DecayReaction/Decay ExamplesExamples -- SpecificSpecific • It happens! It’s a weak semileptonic decay − –Conserves charge and baryon #. Must conserve E, p, ang. mom. np→+e+ν e • By happening it tells us… – Neutrino exists (E, p, lepton #) – Neutrino is spin-1/2 (ang. mom.) – Related processes must occur + + pn→+e+ν e β decay (bound protons only) νe charged current reaction – − tiny cross section (~10-38 cm2 or ν e +→ne+p 10-14 barn, compared to tens of mbarn for hadrons) 4/15/2009 4 OtherOther possiblepossible reactionreaction examplesexamples • Is it observed? Should it be observed? –Conserves q, baryon #, E, p, ang. mom. + – Neutrino tells us it must be weak, but even ν µ +→pnµ + weak won’t do because of lepton # (Lµ is +1 + for νµ and -1 for µ ) ⇒ Reaction Forbidden! • Is it observed? Should it be observed? –Conserves q, E, p. Λ→e −++π +ν – Neutrino tells us it must be weak, as does e disappearance of strangeness, but it still can’t happen because it violates baryon # (B is +1 for Λ, 0 for others) ⇒ Reaction Forbidden! • Is it observed? Should it be observed? –Conserves q, E, p. No baryons – Neutrino tells us it must be weak, as does −−0 disappearance of strangeness. Lepton # is - K →+µ πν+µ OK (Lµ is +1 for µ and -1 for anti-νµ) ⇒ Reaction Allowed! 4/15/2009 5 StillStill moremore possiblepossible reactionsreactions Kp+++→Λ+π +π +Kp− + →+π 0 Λ K 0 →+π + π − A B C • All three of these are purely hadronic processes that don’t violate any of the sacred principles. Since it would be dominant, consider the strong interaction first A. Not possible by strong – violates strangeness (+1→-1) B. OK by strong (strangeness -1 on both sides) C. Not possible by strong – violates strangeness (+1→0) • Next consider EM – briefly! – EM conserves S, so it can’t help with A and C. No competitive way for EM to contribute to B • Next consider weak A. This could happen, but the weak-interaction cross section is too small to see. (Swamped by SI processes in meson-baryon collisions) B. Since strong does it, who cares about tiny weak contribution? C. Hadronic weak decay – can happen and does happen a lot 4/15/2009 6 SelectionSelection RulesRules –– ProgressProgress ReportReport • We can use the selection rules we have so far to declare some potential processes as forbidden and to classify the processes we see based on the responsible interaction •Lotsof production/decay processes violate isospin (by EM, weak) and strangeness (by weak). The latter can be hadronic, semileptonic and leptonic. Their details reveal the rules of the Standard Model and quark mixing – A quark of one flavor decays to another flavor because the weak interaction sees it as a mixture of quarks. The piece of another quark provides a route through which a decay can occur – Before we can describe this satisfactorily, we need to collect some additional properties and conservation principles that will help us organize the observed particles, leading to quarks 4/15/2009 7 SymmetriesSymmetries andand ConservationConservation LawsLaws • Symmetries are rooted in the dynamics of Every symmetry is systems → Laws of Physics associated with a – Typically we note the symmetries first transformation that (conservation rules) and they are the guide to leaves the system’s the interactions Lagrangian or • Noether’s theorem: Hamiltonian “To every symmetry there is a corresponding invariant conservation law, and vice versa.” • Two kinds of symmetry transformations/conservation laws: – Continuous symmetry ↔ additive conservation law x → x + dx θ→θ+ dθ Momentum, energy, angular momentum, etc. – Discontinuous symmetry ↔ multiplicative conservation law Parity: (x,y,z) → (−x,−y,−z) Time reversal: tt→− Charge conjugation: A → A 4/15/2009 8 ExampleExample fromfrom classicalclassical mechanics:mechanics: MomentumMomentum ConservationConservation (Physics(Physics 44104410 Version)Version) pi Momentum coordinates HH= ()pii,,qt qi Space coordinates Effect of an arbitrary transformation: 33∂∂HH∂H dH =+∑∑dqiidp +dt ii==11∂∂qpii∂t Make it a spatial translation: dpi = dt = 0 Hamilton's Equations : 3 ∂H 3 ∂∂HH dH = ∑ dqi dH =− p&iidq qp&&ii== − ∑ i=1 ∂qi i=1 ∂∂pqii If H is invariant under dqi, then Each momentum 3 d 3 dH =− p =0 p = 0 component is ∑ &i ∑ i constant in time ⇒ i=1 dt i=1 4/15/2009 p is conserved 9 ExampleExample fromfrom classicalclassical mechanics:mechanics: MomentumMomentum ConservationConservation (Physics(Physics 22102210 Version)Version) dp Newton’s Second Law of Motion: F = dt Differential form of Work-Energy − dU F = Theorem: dx Translational Invariance: U (x) = U (x − ∆x) for any position x and displacement ∆x ⇒ F = 0 dp Conservation of Momentum = 0 dt 4/15/2009 10 ConservationConservation LawsLaws inin QuantumQuantum MechanicsMechanics Physical quantities whose operators commute with the Hamiltonian H are conserved Ψ=Ψ(rtr, ) Expectation value UU=Ψ∗ Ψd3r ∫ r r UU= ()r,,r& t How does it change in time? dd ∂Ψ∗ ∂U ∂Ψ UU=Ψ∗ Ψd3r=Ψ∫∫Ud33r+Ψ∗∗Ψdr+∫ΨU d3r dt dt ∫ ∂∂tt∂t ∗ ∂Ψ ∂Ψ ∗ † Schrödinger Eqn: iHhh=Ψ -i=ΨH ∂∂tt dU11∂ U =− ∫∫Ψ∗∗H†3UΨdr+ Ψ H†Ψdr3+ ∫Ψ∗UHΨdr3 dt ihh∂t i Since HH is Hermitian = H† , rewrite ∂U ( ) If = 0, and []UH = 0, dU∗ ∂ 1 3 ∂t UU=Ψ∫ + [], HΨdr dt ∂t ih then U is conserved 4/15/2009 11 ConservationConservation ofof ElectricElectric Charge:Charge: GaugeGauge InvarianceInvariance (Physics(Physics 44204420 Version)Version) ∑QQif=∑ e− → γν Electrons bound in (iodine) atoms – no X-ray e 25 photons signaling decays →τe > 10 yr There is a general connection Maxwell's equations are invariant under between charge conservation, rrr AA→=′ A+∇Λ Vector potential gauge invariance and quantum 1 ∂Λ field theory with its roots in Φ→Φ=Φ− Scalar potential Maxwell ct∂ A Lagrangian that is invariant under transformation U = eiα is said to be gauge invariant. There are two types of gauge transformation: Global: α = const → Charge conservation Local: αα=→()rr EM force, massless photon 4/15/2009 12 ConservationConservation ofof ElectricElectric Charge:Charge: GaugeGauge InvarianceInvariance (Physics(Physics 22202220 Version)Version) Sorry: There is no Physics 2220 Version for Gauge Invariance 4/15/2009 13.
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