Particle Physics

1
Particle Physics
4th Handout

• Symmetries,Invariances and Conservation laws
• (Or how to decide whether to shake hands with an
  alien!)
• Conserved quantities in QM
• Parity
• Scalars,Vectors and pseudo-S,axial-V
• CP,T

http//ppewww.ph.gla.ac.uk/parkes/teaching/PP/PP.
html
Chris Parkes
2
Symmetries and Conservation Laws
In classical physics there are a number of
quantities which are conserved momentum,
energy, angular momentum Conservation theorems
also occur in QM In classical physics the
conservation laws tend to be the starting
points (there are also more sophisticated way of
deriving them) In QM however the conservation
laws are deeply related to the principle of
superposition of amplitudes and the symmetry of
the system. we will deal with both continuous
(e.g. displacement in space/time) and discrete
symmetries (e.g. mirror like)
Quantum numbers are associated with the conserved
observables Some are universal laws of nature
(p,E,L,CPT), others are valid only in
approximations (e.g. parity - valid for strong/EM
force but not weak)
Emmy Noether
Noethers theorem Symmetries (invariances)
naturally lead to conserved quantities
3
P
Charge Inversion Particle-antiparticle mirror
C
Parity Inversion Spatial mirror
CP
4
Conserved quantities in QM - Revision
Any operator, Â, which is time independent (e.g.
p) and commutes with the Hamiltonian is
associated with a conserved quantity.
Hamiltonian
Expectation value Of operator
Conservation requires (e.g. momentum)
Minus sign from complex conjugate
Thus if A is indep.of time the expectation value
is constant, as long as A,H commute
5
Translational Invariance? linear momentum
conservation
MS 4.1
i.e. wish to show p operator independent of time
0
Invariance All positions in space are physically
indistinguishable Consider moving a particle a
small distance
Depends on derivatives not on position (natural
units)
define operator D that performs this translation
Higher order terms
Since in natural units
Hence, linear momentum is conserved and is a
good quantum number
Consider wavefunction
1) 2)
1, i?x just numbers so
Comparing 1)2),
6
Rotational Invariance?Angular Momentum
Conservation
MS 4.2

• All directions in space are physically
  indistinguishable
• Rotating a system of particles around its CM to a
  new orientation
• leaves its physical properties unchanged
• ? Proof is very similar to translation, see
  lectures

This proof considers only L, in general must also
consider spin Get

• Laws of physics independent
• of time

Translations in time? Energy Conservation
Define an operator for time evolution, A
N.B. time translation invariance is different
from time reversal operator T discussed later
But wait, we already have this operator, it is
the hamiltonian
TISE
And H commutes with itself
So time translation is the symmetry, H is
operator, E is the conserved quantity
7
Other conserved quantities
Electric Charge, Colour Charge, Baryon number,
lepton number, strangeness..
First three always conserved (strong,EM,weak) Last
one not conserved in weak e.g.
Other Discrete Symmetry operators
Parity (P) spatial Inversion Charge
conjugation (C) particles ?anti-particles re
verses charge magnetic moments baryon
number strangeness Only particles that are
their own anti-particles are eigenstates of
C Time (T) - Time reversal
CPT combined is a fundamental symmetry of QFTs,
arising from very basic assumptions like Lorentz
invariance
Q) Is there a difference in behaviour between
matter and anti-matter ?
Discuss P,C,CP/T, particularly for the weak force
8
Parity - Spatial Inversion
Discrete symmetry
Parity conserved when Hamiltonian invariant under
Parity transformation (strong,EM)
P operator acts on a state y(r, t)gt as
Hence for eigenstates P1
e.g. hydrogen atom wavefn y(r,?,?
)gt?(r)Ylm(?,?) Ylm(?,?) Ylm(?-?,??) (-1)l
Ylm(?,?) So atomic s,d ve, p,f ve P
y(r, t)gt cos x has P1, even
y(r, t)gt sin x has P-1, odd
y(r, t)gt cos x sin x, no eigenvalue
Hence, Electric dipole transition ?l1?P?- 1
9
Parity Examples
Conventions quarks and leptons have ve
parity Anti-quarks and anti-leptons have ve
parity

• Parity multiplicative
• For a meson made from q qbar pair with orbital
  angular momentum l
• ?gt ?a ?b, PPaPb(-1)l
• For ground state (l0) Pmeson-1, expect ve
  parity for light mesons

?-,?o,K-,Ko all P-1
q1
q3
l3
l12
For baryons
So, expectve parity for low lying states
q2
For anti-baryons
expect-ve parity for low lying states
10
Scalars,Vectors,Pseudo-Scalars,Axial Vectors
Scalar unaffected by parity (ve parity) Vector
reverses (-ve parity)
Can also form quantities from . and X
products of vectors. How do the resultant
scalars/vectors behave ?
p
Axial vector consider cross-product of two
vectors
-r
Both reverse under parity, so L unaltered
r
-p
In a parity conserving theory you cant add an
axial vector to a vector
Pseudo-scalar consider dot product of two
vectors
Acts like a scalar
Now, consider dot product of vector, axial vector
Changes sign , a pseudo-scalar
This leads to parity violation in weak
interactions
11
Weak Force Parity Violation Discovery ?-?
problem
Revision
Actually K Postulated Yang Lee, 1956

• Same mass, same lifetime, BUT
• ??????, (21) P ? 1
• ??????-, (6) P ? -1

Experimental discovery
C.S. Wu et. al., Phys. Rev. 105, 1413 (1957)
?
e- (E,p)
B field
?
Co60Nuclei spin aligned Beta decay to Ni60
Spin axial vector -gt maximal violation V-A
theory, neutrino handedness
Parity
e- (E,-p)
12
Helicity and the neutrino

• In angular momentum we choose the axis of
  quantisation to be the z axis.
• Lets choose this axis along the particle momentum
  direction.
• Helicity is the component of the spin along the
  momentum direction.
• A spin ½ particle can thus have helicity 1 (ms
  ½) or 1 (ms- ½ )

p
p
1
-1
s
Right-handed
s
Left-handed
Not so interesting for a massive particle, as not
Lorentz invaraint, but consider the neutrino.

• Only left-handed neutrinos exist and right-handed
  anti-?
• Helicity is a pseudo-scalar

Operating with P on this reverses p, not spin,
produces a right-handed neutrino. Do not observe
Operating with C on this produces a left-handed
anti-neutrino. Do not observe
Operating with C and P on this produces a
right-handed anti-neutrino. Do observe!
Weak force violates Parity, but CP OK?
13
Measuring Helicity of the Neutrino
Goldhaber et. al. 1958
Consider the following decay
Electron capture K shell, l0
photon emission

• Momenta, p

Eu at rest
Neutrino, Sm In opposite dirns
Select photons in Sm dirn

• spin

e-
?
?
S ½
S 1
right-handed
right-handed
OR
S- ½
S- 1
Left-handed
Left-handed

• Helicities of forward photon and neutrino same
• Measure photon helicity, find neutrino helicity

14
Neutrino Helicity Experiment

• Tricky bit identify forward ?
• Use resonant scattering!
• Measure ? polarisation with different B-field
  orientations

Vary magnetic field to vary photon
absorbtion. Photons absorbed by e- in iron only
if spins of photon and electron opposite.
Fe
Forward photons, (opposite p to neutrino), Have
slightly higher p than backward and cause
resonant scattering
Only left-handed neutrinos exist
Similar experiment with Hg carried out for
anti-neutrinos
15
CP Violation

• Parity is violated by weak force
• But neutrino analysis shows CP looks OK.

History repeats itself, just as we expected
parity to be conserved, we then expected CP to
be conserved. Actually violated by a tiny amount
currently a hot research topic CPT is
conserved so CP violation is equivalent to T
violation
QM relativity Gave us matter/anti-matter
symmetry So why is our world full of
protons,neutrons,electrons and not anti-protons,
anti-neutrons, positrons? Historical accident
that our corner of universe has more matter than
anti-matter ? No, astronomical evidence tells us
that observable universe is all made of
matter. CP/T violation is the key.
16
P
Charge Inversion Particle-antiparticle mirror
C
Parity Inversion Spatial mirror
CP
17
Time reversal T
Leaving all position vectors x unchanged but p,J
reverse
Particle are eigenstates of P, neutral particles
can be of C, but cannot be identical to itself
going backwards in time
Detailed balance
Compare reaction
Where m are spin Z-components
With time reversed counter-part
Conserved for Strong,EM We wish to test this for
the weak force Inverse experiments are difficult
to do with the weak force, need to avoid
strong/EM contamination e.g.
reversed would be Would be dominated by
strong interaction of proton,pion. Neutrino expt.
would be possible, but difficult and looking for
a small effect
18
Let us have a quick look at nature....
Neutral kaon system
KS
mass eigenstates
are a mixture of flavour eigenstates
KL
Short lifetime
Mainly Decays to two pions (99.9)
Long lifetime
Mainly Decays to three pions (34) 3 x m(pion)
m(Kaon)
Time dilation - ? factor needed for actual
flight distance in lab
19
CPLEAR- some parameters
Kaon Oscillation

• Particle can turn into anti-particle. So say at
  t0, pure Ko, later a superposition of states

d
s
W-
u, c, t
u, c, t
_
_
W
-
d
s
K0
K0
u, c, t
_
d
s
W-
W
_
_
_
_
u, c, t
d
s
Rate difference Ko ?Ko ? Ko ?Ko is T violation

• Beam 106 anti-protons /s into Hydrogen target
• Fast online trigger selection of events 103/s
• Ability to separate charged pions / kaons using
  Cherenkov, dE/dx, Time of flight
• discriminate in momentum range 350-700 MeV/c
• Can detect and reconstruct Ks vertex to 60
  lifetimes c?2.6 cm
• Observe events over 4?
• Magnetic field (0.4T) and tracking leads to
  particle momentum determination
• (5 accuracy)

20
CPLEAR T invariance test
Initial state at t 0
S 0
1) Identify Ko / Ko at production produced in
association with K/K- 2) Identify Ko / Ko at
decay observe leptonic decay
Get positron
Or electron
21
Experiment at LEAR ring at CERN 1990-1996
Pions from kaon decay
22
Discovery of T violation
CPLEAR,1998

• Currently the only direct observation of T
  violation
• Measure asymmetry in rates

Number of lifetimes

• T, or equivalently CP, violated by this tiny
  amount

23
CP ViolationWhy is it interesting ?

• Fundamental The Alien test
• C violation does not distinguish between
  matter/anti-matter
• Left-handed / right-handed are simply
  conventions
• We cannot define what we mean by Co60 e-
  emission asymmetry unless we can define
  difference from anti-Co60 (or charge)
• CP violaton says preferred decay KL?eve?-
• Never shake hands with an alien whose electrons
  are the preferred decay state of the long lived
  kaon!
• Hence, it allows us to unambiguously distinguish
  between matter and anti-matter.
• Least Understood CP Violation is add-on in SM
• Parity violation naturally imbedded from coupling
  structure
• Left-handed and right-handed couplings
• There is a matrix (CKM matrix) that tells us how
  likely transitions are from one quark generation
  to another e.g. b quark to decay to a c quark or
  a u quark.
• CP violation can be accommodated in this matrix
  by adding a complex phase. It is an add-on
  justified only by the observation of CP
  violation.

24
CP Why ?
CP Violation key dates

• 1964 CP Violation discovery in Kaons
• 1973 KM predict 3 or more families
• ..
• ..ermnotmuch
• .
• 1999 Direct CP Violation NA48/KTeV
• 2001 BaBar/Belle CP Violation in B mesons
• 200? LHCb physics beyond the SM?

• Powerful delicately broken symmetry
• Very sensitive to New Physics models
• Historical Predicted 3rd generation !
• Baryogenesis there is more matter !
• N(antibaryon) ltlt N(baryon) ltlt N(photons)
• Fortunately! 1 109
• Sakharov (1968) Conditions for matter dominated
  universe
• Baryon number violation
• CP violation
• Not in thermal equilibrium

Are we just the left over matter after CP
violating matter/ anti-matter annihilation
processes?
Assuming not initial conditions, but
dynamic. Cannot allow all inverse reactions to
have happened