Lecture 3: Invariance Principles, Conservation Laws

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Lecture 3Invariance Principles,Conservation
Laws
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Symmetries

• Symmetries of nature provide strong constraints
  to what nature can do.
• They lead to conserved quantities
• Parity, C-parity, CP, Charge, Isospin
• Even simple ideas lead to non-trivial qualitative
  and quantitative predictions

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Simple Symmetries

• Time translation invariance
• ? Conservation of energy
• Space translation invariance
• Conservation of linear momentum

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Parity

• One would expect physical laws to look the same
  in a mirror
• That is, if we changed our coordinate system to
  X?X, Y?Y, Z?-Z
• Parity is the most extreme case
• X?-X, Y?-Y, Z?-Z
• In quantum mechanics, it corresponds to
• Discrete transformation ? not simply a rotation
• PP1 so Unitary with eigenvalues of 1 or 1
• Not all wave functions have well-defined parity

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Hydrogen Wavefunctions

• The wave functions you see in an intro QM course
  are still useful here
• Represent the angular distribution of two bodies
  in motion around each other
• Parity on these wave functions corresponds to

p-q
q
fp
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Parity on angular functions

• So working through the components of the
  spherical harmonics
• In other words, the parity is only determined by
  the total angular momentum matters, not the
  particular state

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Rules for Parity

• Parity of a system is the product of the parity
  of its parts
• Parity is conserved in strong and electromagnetic
  interactions only

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Intrinsic Parity

• We find that parity is conserved in strong and
  electromagnetic interactions
• We assign an intrinsic parity to particles to
  make sure it balances in reactions
• We will discuss the pion, but baryons are
  conserved (well also discuss this!) so their
  parity is a matter of convention
• Parity of proton and neutron P1
• Some reactions, like pp?pLK which involve
  strange particles always find particles produced
  in pairs. Can only measure pair parity

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Example Parity of p

• I must admit, these examples are tricky, but its
  worth working through slowly, just to appreciate
  the power of conservation laws
• We will first establish the spin, and then the
  parity of the pion, by studying simple reactions
• It is also worth noting that during the dawn of
  particle physics, nothing was assumed.
• Today, we know pions have a quark and anti-quark,
  which have opposite parities, and so P-1
• Back then, everything had to be learned from
  scratch!

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Spin of p

• Measured in study of reversible reaction
• Rates should be the same if carried out with the
  same CMS energy by symmetry of matrix element
• But sd1 and momenta are equal so

So we measurespin of pion!
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Parity of p

• Parity determined by capture in deuterium
• Perkins tells us that capture proceeds through
  S-state of p relative to d
• Since sd1 and sp0, then initial state has J1
• Final state has two neutrons, JLS
• NR QM lets us factorize space and spin of nn
  state

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Parity of p , cont

• Spin wave functions have well defined symmetry
  under interchange of spins
• Spin function gets (-1)S1 under interchange of
  spin
• Space function gets (-1)L under spatial
  interchange
• So full interchange gives us (-1)LS1
• But nn wave function must be fully antisymmetric
  under interchange ? LS is even!

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Almost there!

• So we have two conditions
• J1 in initial state
• LS even in final state
• Satisfied by three possibilities
• L0, S1 ? LS odd
• L1, S0 or 1 ? LS even for L,S1
• L2, S1 ? LS odd
• So final neutron state have L1
• Thus, parity of final state P1
• P(n) and P(d)1, so P(p)-1 QED!

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Parity of Anti-particles

• Diracs theory predicts that particles and anti
  particles have opposite parities
• Proven by experiment for e and e-
• Good rule of thumb for parities of hadrons
• Pions ? quark and antiquark in S state, so
  Pp(-1)L1 -1
• Rho goes to two pions, but J1, so L1 in final
  state, so Pr-1 just from conservation of orbital
  angular momentum

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Is Parity always conserved?

• No
• Weak interactions break the symmetry as much as
  they can, e.g. RH neutrinos do not exist!
• All observation of parity violation in studies
  (nominally) of strong and EM can be attributed to
  small components of weak interactions in the
  hamiltonian
• Not too surprising cant completely isolate
  yourself!

RH
LH
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Charge Conjugation Invariance

• C operator reverses charge (and thus magnetic
  moment) of a particle
• For fermions in Dirac, theory, C exchanges
  particle and anti-particle
• So the C operator results in many illegal
  transitions ? not a good symmetry for baryons and
  leptons
• However, some hadronic and photonic states do
  have good C-parity

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C-parity states

• Particle-antiparticle annihilation
• Strong interactions conserve charge
• Neutral bosons
• Sign determined by Cg
• Photons determined by charges, which couple to
  fields as eA. C symmetry requires A to change
  sign Cg-1
• So p0?gg ? Cp1
• This implies that p0?ggg is forbidden, even if it
  does not violate charge conservation

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Trick for Charge Conjugation

• To think about C symmetry for multiparticle
  systems, just remember that a C-transformation
  corresponds to an exchange of particles
• I.e. it looks a lot like a parity flip

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CP Symmetry

• Weak interactions also violate C for the same
  reason no LH anti-neutrinos!
• However, CP corresponds to
• Flip the handedness of the neutrino LH?RH
• Flip the charge of the neutrino?anti-neutrino
• Thus, you have a RH anti-neutrino, which does
  exist
• CP looks like a good symmetry of the weak
  interaction.
• Its not, but well get to this later!

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Charge Conservation

• Charge conservation is something we learn quite
  early
• I1I2I3 in circuit theory
• It is obeyed to an obscene level
• Any net violation would give measureable net
  charge of the earth, which is not seen
• How does it come about?
• Typically, things which dont happen cant
  happen!
• How does this conservation law come about?

I1
I3
I2
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Gauge Invariance

• We know that Maxwells equations are invariant
  under transformations
• These gauge transformations leave the fields
  the same, so any observable quantities are
  unchanged
• We never observe the absolute value of the scalar
  and vector potentials

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Gauge Invariance in QM

• Double slit experiment measures interference
  between two waves
• Depends on relative phases
• If we changed both phases by a constant over t
  x, no change ? no problem!
• If the phase could change with x, then the
  physics at C would change

B
C
A
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Gauge Symmetry

• How do we deal with this?
• Let phase be a smooth function
• However, lets allow the particle to couple to a
  field (e.g. EM), which affects the phase
• Then our problem looks like this
• But EM is gauge invariant, so
  gives the same physics as A itself

And physics is unchanged!
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Implications

• The fact that physics should be invariant under
  local changes in phase implies two critical
  features of nature
• Charges must be coupled to a long-range field
  which can change the phases of the electrons
• Charge is conserved
• Gauge invariance implies a very robust theory
• Many of the problems encountered with field
  theories in general disappear when gauge symmetry
  is applied

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Isospin

• Heisenberg (1932) suggested that the similarity
  of the neutron and proton (M938 MeV, spin ½)
  indicated that they were two states of the same
  particle the nucleon
• The formalism was the same as for spin-1/2, thus
  the name isospin (self spin)
• Isospin was found to be conserved in strong
  interactions, not in weak EM
• Only the magnitude I matters, not particular Iz
• Evidence
• Equivalence of nn, np, pp interactions
• Equivalence of mirror nuclei

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Isospin in the Quark Model

• Isospin can be understood in the quark model of
  hadrons
• The only difference betweenn and p is
  interchange d?u
• Is d u had the same mass, this would be a true
  symmetry
• No feature except charge would distinguish p/n
• But its not perfect
• We know that Mn-Mp 1.3 MeV
• Coulomb effects should be stronger for p than n
• So we conclude that DMqMd-Mu2-3 MeV
• And DMq/MN.2 so a small effect!

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Isospin in other hadrons

• We now know the fundamental doublet for isospin
  is
• The comparable doublet for anti-particles is
• We can combine two spin-1/2 states just like spin

Bigger chargehas larger I3
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Isospin in the N-N system

• Consider a system of 2 nucleons
• Nucleon, remember, is p or n
• We again combine the isospins into triplet and
  singlet combinations

Symmetric
Anti-symmetric
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Symmetry of NN wavefunction

• Consider full wavefunction
• For a deuteron
• Spin function is symmetric (J1)
• Space function is L0,2 so symmetric
• Overall antisymmetry?I0!
• Now consider and
• Each final state has I1, so initial state must
  have that as well
• Thus, only ½ initial states available

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Pion-Nucleon scattering

• If strong reactions only care about total
  isospin, then six reactionscan be described by
  two amplitudes!
• We can classify them into groups
• Processes are pure I3/2
  and so have the same cross section
• The other processes will have I31/2 and thus
  I1/2 or I3/2
• Now we need Clebsch-Gordon coefficients

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Clebsch-Gordon Coefficients

• Not going to make a formal introduction, but we
  can get a long way with 4 simple rules for
  angular momentum in QM
• If we act on the I3/2, I33/2 state, we get

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CGC Table
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Comparing Reaction Rates

• Consider three similar processes of pion-nucleon
  scattering
• What can we say about their reaction rates just
  from isospin considerations?

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Matrix elements

• Cross section goes as the matrix element squared
• H is a Hamiltonian which has pieces connecting
  different I states

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Elastic Scattering (Pure)

• The reactionis pure I3/2, I33/2 so
• There is a leading factor (phase space, spins,
  etc.) and then a matrix element squared,
  incorporating the force
• We dont have a fundamental model for this, but
  you will see how even presenting it as an unknown
  variable is useful sometimes!

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Elastic Scattering (Mixed)

• However, is a
  combination of I3/2 and I1/2 states

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Charge Exchange

• Now we have to consider two different wave
  functions, one for initial and one for final state

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Cross-Section Ratios

• We can now look at ratios of the cross sections
• Consider limiting cases, when one or other
  isospin exchange process dominates

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Hypercharge

• There is a compact description of the charges of
  pions and nucleons
• Adding strangeness to the mix is easy

Hypercharge
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G-parity

• There is another operator used to characterize
  hadron states
• For nucleon-antinucleon system
• The neutral pion has G-parity 1
• All of the pions are chosen to have the same
  G-parity as the neutral one

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Selection Rule for G

• So ultimately we have a nice rule for the
  multi-pion states
• Constrains the number of pions a particular
  isospin state of nucleons can decay into!

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Conservation laws

• All forces we know of conserve
• Energy-momentum
• Charge
• Baryon Number
• Lepton Number
• CPT
• Weak force violates P,C,CP(or T)
• Strong and EM Conserve them
• Only strong force obeys isospin
• Weak and EM violate them