Symmetry and conservation Laws Submitted: Pro. Cho By Adil Khan

conservation laws conservation laws, in physics, basic laws that together determine which processes can or cannot occur in nature; each law maintains that the total value of the quantity governed by that law, e.g., mass or energy, remains unchanged during physical processes. Conservation laws have the broadest possible application of all laws in physics and are thus considered by many scientists to be the most fundamental laws in nature.

In developing the standard model for particles, certain types of interactions and decays are observed to be common and others seem to be forbidden. The study of interactions has led to a number of conservation laws which govern them. These conservation laws are in addition to the classical conservation laws such as conservation of energy, charge, etc., Strong overall conservation laws are the conservation of baryon number and the conservation of lepton number. Specific quantum numbers have been assigned to the different fundamental particles, and other conservation laws are associated with those quantum numbers.

Conservation of Baryon Number Nature has specific rules for particle interactions and decays, and these rules have been summarized in terms of conservation laws. One of the most important of these is the conservation of baryon number. Each of the baryons is assigned a baryon number B=1. This can be considered to be equivalent to assigning each quark a baryon number of 1/3. This implies that the mesons, with one quark and one antiquark, have a baryon number B=0. No known decay process or interaction in nature changes the net baryon number The neutron and all heavier baryons decay directly to protons, the proton being the least massive baryon. This implies that the proton has nowhere to go without violating the conservation of baryon number, so if the conservation of baryon number holds exactly, the proton is completely stable against decay. One prediction of grand unification of forces is that the proton would have the possibility of decay, so that possibility is being investigated experimentally.

Conservation of Baryon Number Conservation of baryon number prohibits a decay of the type p + n → p +  + +  - B = ≠ but with sufficient energy permits pair production in the reaction p + n → p + n + p + p - B = =

Conservation of Lepton Number One of the most important of these is the conservation of lepton number. This rule is a little more complicated than the conservation of baryon number because there is a separate requirement for each of the three sets of leptons, the electron, muon and tau and their associated neutrinos. The first significant example was found in the decay of the neutron. When the decay of the neutron into a proton and an electron was observed, it did not fit the pattern of two-particle decay. That is, the electron emitted does not have a definite energy as is required by conservation of energy and momentum for a two-body decay. This implied the emission of a third particle, which we now identify as the electron antineutrino.

Conservation of Lepton Number n → p + + e - n → p + + e - + The assignment of a lepton number of 1 to the electron and -1 to the electron antineutrino keeps the lepton number equal to zero on both sides of the second reacton above, while the first reaction does not conserve lepton number. The observation of the following two decay processes leads to the conclusion that there is a separate lepton number for muons which must also be conserved.

Table of lepton properties particles Symbo l Anti- particle Restmass Mev / c^2 L(e)L(muon)L(tau)Lifetime (second) electron e-e- e+e stable Neutrino Electron νeνe νeνe 0(<7 * stable muon -- + * Neutrino muon νν νν 0(<0.27)0+10stable tau t-t- t+t Neutrino tau νtνt νtνt 0(<31)00+1stable

Some Reaction Examples a) Consider the reaction v u +p →  + + n What force is involved here? Sinc e neutrinos are involved, it must be WEAK interaction. Is this reaction allowed or forbidden? Consider quantities conserved by weak interaction: lepton #, baryon #, q, E, p, L, etc. muon lepton number of v u =1,  + =-1 (particle Vs. anti-particle) Reaction not allowed! b) Consider the reaction v e + p → e - + π + +p Must be weak interaction since neutrino is involved. conserves all weak interaction quantities Reaction is allowed c) Consider the reaction Λ →e - +p + + (anti-v e ) Must be weak interaction since neutrino is involved. conserves electron lepton #, but not baryon # (1 → 0) Reaction is not allowed d) Consider the reaction K + →  - + π 0 + (anti-v  ) Must be weak interaction since neutrino is involved. conserves all weak interaction (e.g. muon lepton #) quantities Reaction is allowed

Symmetries and Conservation Laws General Assumptions in Physics 1) Continuous symmetries: –Space is homogeneous (uniform)  conservation of momentum and energy –Space is isotropic (no preferred direction)  conservation of angular momentum –Properties of space do not depend on the observer  special relativity 2) Discrete symmetries: Space reflection (mirror symmetry)  parity conservation, P Particle-antiparticle symmetry  charge conjugation, C Time reversal  T True for strong, electromagnetic, violated in weak interactions. Conservation of C, P and T  matter-antimatter symmetry

Three Important Discrete Symmetries Parity, P –Parity reflects a system through the origin. Converts right-handed coordinate systems to left-handed ones. –Vectors change sign but axial vectors remain unchanged x  -xL  L Charge Conjugation, C –Charge conjugation turns a particle into its anti-particle e+  e-K-  K+g  g Time Reversal, T –Changes the direction of motion of particles in time t  -t CPT theorem –One of the most important and generally valid theorems in quantum field theory. –All interactions are invariant under combined C, P and T transformations. –Implies particle and anti-particle have equal masses and lifetimes  

Parity Let us examine the parity operator (P) and its eigenvalues. The parity operator acting on a wavefunction is defined by: PY(x, y, z) = Y(-x, -y, -z) P 2 Y(x, y, z) = PY(-x, -y, -z) = Y(x, y, z) Therefore P 2 = I and the parity operator is unitary. If the interaction Hamiltonian (H) conserves parity then [H,P]=0, and: PY(x, y, z) = Y(-x, -y, -z) = nY(x, y, z) with n = eigenvalue of P P 2 Y(x, y, z) = PPY(x, y, z) = nPY(x, y, z) = n 2 Y(x, y, z) Y(x, y, z) = n2Y(x, y, z) => n 2 = ±1 so, n=1 or n=-1. The quantum number n is called the intrinsic parity of a particle. If n= 1 the particle has even parity. If n= -1 the particle has odd parity. In addition, if the overall wavefunction of a particle (or system of particles) contains spherical harmonics (Y L m ) then we must take this into account to get the total parity of the particle (or system of particles). The parity of Y L m is: PY L m = (-1) L Y L m. For a wavefunction Y(r, θ,φ)=R(r)Y L m (θ, φ) the eigenvalues of the parity operator are: PY(r, θ,φ)= PR(r)Y L m (θ, φ) = (-1) L R(r)Y L m (θ, φ) The parity of the particle would then be: n(-1) L Note: Parity is a multiplicative quantum number

Parity The parity of a state consisting of particles a and b is: (-1) L n a n b where L is their relative orbital momentum and n a and n b are the intrinsic parity of each of the two particle. Note: strictly speaking parity is only defined in the system where the total momentum (p) =0 How do we know the parity of a particle ? By convention we assign positive intrinsic parity (+) to spin 1/2 fermions: +parity: proton, neutron, electron, muon (m-) Anti-fermions have opposite intrinsic parity -parity: anti-proton, anti-neutron, positron, anti-muon (m+) Bosons and their anti-particles have the same intrinsic parity. What about the photon? Strictly speaking, we can not assign a parity to the photon since it is never at rest. By convention the parity of the photon is given by the radiation field involved: electric dipole transitions have + parity magnetic dipole transitions have - parity We determine the parity of other particles (π, K..) using the above conventions and assuming parity is conserved in the strong and electromagnetic interaction.

Parity Example: determination of the parity of the π using π -d → nn. For this reaction we know many things: a) s π =0, s n =1/2, s d =1, orbital angular momentum L d =0, J d =1 b) We know (from experiment) that the π is captured by the d in an s-wave state. Thus the total angular momentum of the initial state is just that of the d (J=1). c) The isospin of the nn system is 1 since d is an isosinglet and the π - has I=|1,-1> Let’s use these facts to pin down the intrinsic parity of the π. L=1 consistent with angular momentum conservation: nn has s=1, L=1, J=1 → 3 P 1 The parity of the final state is: n n n n (-1) L = (+)(+)(-1) 1 = - The parity of the initial state is: n π n d (-1) L = n π (+)(-1)0 = n π Parity conservation gives: n n n n (-1) L = n π n d (-1) L => n π = -

Parity Some use “spin-parity” buzz words: buzzwordspinparityparticle pseudoscalar 0 - π, k scalar 0 +higgs (none observed) vector 1 -g,φ, Ψ, ρ pseudovector 1 +A1 How well is parity conserved? Very well in strong and electromagnetic interactions not at all in the weak interaction! The θ-t puzzle and the downfall of parity in the weak interaction In the mid-1950’s it was noticed that there were 2 charged particles that had (experimentally) consistent masses, lifetimes and spin = 0, but very different weak decay modes: θ + → π + + π 0 t + → π + π - π + The parity of θ + = + while the parity of t+ = - Some physicists said the θ + and t+ were different particles, and parity was conserved. Lee and Yang said they were the same particle but parity was not conserved in weak interaction! Lee and Yang win Nobel Prize when parity violation was discovered. Note: θ + / t+ is now known as the K+.

Charge Conjugation Charge Conjugation (C) turns particles into anti-particles and visa versa. C(proton) → anti-proton C(anti-proton) → proton C(electron) → positron C(positron) → electron The operation of Charge Conjugation changes the sign of all intrinsic additive quantum numbers: electric charge, baryon #, lepton #, strangeness, etc.. Variables such as spin and momentum do not change sign under C. The eigenvalues of C are ± 1 and, like parity, C is a multiplicative quantum number. It the interaction conserves C then C commutes with the Hamiltonian, [H,C]A=0. Most particles are NOT eigenstates of C. Consider a proton with electric charge = q. Let Q= charge operator, then: Q|q>=q|q> and C|q>=|-q> CQ|q>=qC|q> = q|-q> and QC|q>=Q|-q> = -q|-q> [C,Q]|q>=2q|q> Thus C and Q do not commute unless q=0. We get the same result for all additive quantum numbers! ”.

Charge Conjugation How do we assign c to particles that are eigenstates of C? a) photon. Consider the interaction of the photon with the electric field. the interaction Lagrangian of a photon is: L EM =J u A u with J u the electromagnetic current density and A u the vector potential. By definition, C changes the sign of the EM field and thus J transforms as: CJC -1 =-J Since C is conserved by the EM interaction we have: CL EM C -1 = L EM CJ u A u C -1 = J u A u CJ u C -1 C A u C -1 = -J u C A u C -1 = J u A u C A u C-1 = -A u Thus the photon (as described by A) has c = -1. A state that is a collection of n photons has c = (-1) n. b) The π 0 : Experimentally we find that the π 0 decays to 2 gs and not 3 gs. This is an electromagnetic decay so C is conserved. Therefore we have: C π 0 = (-1) 2 = +1

Charge Conjugation and Parity Experimentally we find that all neutrinos are left handed and anti-neutrinos are right handed (assuming massless neutrinos). By left or right handed we mean: left handed: spin and z component of momentum are anti-parallel right handed: spin and z component of momentum are parallel In the strong and EM interaction C and P are conserved separately. In the weak interaction we know that C and P are not conserved separately. BUT the combination of CP should be conserved! Consider how a neutrino (and anti-neutrino) transforms under C, P, and CP. Experimentally we find that all neutrinos are left handed and anti-neutrinos are right handed. C C CP P P αPαP αPαP αPαP αPαP So, CP should be a good symmetry right handed ν left handed ν

The strong and electromagnetic interactions are invariant under C, P and T transformations. This is not true of the weak interaction, as can be seen by considering neutrinos (which are only involved in weak interactions). Neutrinos are always left-handed, i.e. their spin is antiparallel to their direction of motion. The P operator reverses momentum but not spin, so when applied to a neutrino would produce a right-handed neutrino, which is not observed, Similarly C applied to a neutrino produces an unobserved left-handed antineutrino. Weak interactions therefore violate C and P. The combination CP, however, applied to a left- handed neutrino produces a right-handed antineutrino, which is observed. Therefore (to a good approximation) weak interactions are invariant under the combined transformation CP. The weak interaction, and all other interactions, are exactly invariant under the combination CPT.

summery InvarianceConserved Quantity Gravitation, weak, electromagnetic and strong interactions are independent of: translation in spacelinear momentum rotations in spaceangular momentum translations in timeenergy EM gauge transformationelectric charge CPT(product of parities below) Gravitation, electromagnetic and strong interactions are independent of: spatial inversion Pspatial parity charge conjugation C"charge parity" time reversal T"time parity""time parity"

PARITY VIOLATION A Parity (P) operation on a system of interacting particles means to replace that system with its mirror image. It is a spatial inversion operation that has the effect of changing left-handed particles to right-handed ones and vice versa. P violation occurs when the rate for a particle interaction is different for the mirror image of that interaction. Electromagnetic and strong nuclear forces have the same strength for left-handed and right-handed particles. So parity is a good symmetry for these interactions and is said to be conserved by them. But the weak nuclear force is asymmetric for right-handed and left-handed particles and thus violates parity. This was first observed in charged current (exchange of W+ or W- particles) interactions in 1956, by Madame Wu and collaborators studying the radioactive decay of 60Co (isotope 60 of Cobalt). Parity violation in neutral current (exchange of Z0 particles) interactions was first observed at SLAC in 1978, by Charles Prescott and collaborators studying the scattering of electrons from protons in a liquid hydrogen target.

here is Tother experimental evidence that the parity of the p is -: the reaction π -d → nn π 0 is not observed. (axial vector) There is other experimental evidence that the parity of the π is -: particles with the same quantum numbers as the photon (g,φ, Ψ, ρ) have c = -1. particles with the same quantum numbers as the π 0 (h, h¢) have c = +1.