Particle Particle PhysicsPhysics

Chris Parkes

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

4th Handout

http://ppewww.ph.gla.ac.uk/~parkes/teaching/PP/PP.html

Symmetries and Symmetries and Conservation LawsConservation Laws

Quantum numbers are associated with the conserved observablesSome 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)

In classical physics there are a number of quantities which are conserved –momentum, energy, angular momentumConservation 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)

Emmy Noethe

rNoether’s theorem – Symmetries (invariances) naturally lead to

conserved quantities

C P

CPParity InversionSpatialmirror

Charge InversionParticle-antiparticlemirror

Conserved quantities in QMConserved quantities in QM- Revision- Revision

Any operator, Â, which is time independent (e.g. p) and commutes with the Hamiltonian is associated with a conserved quantity.

HAidt

Ad

AHHAit

A

dt

Ad

AHHAit

A

dt

Ad

Att

At

A

dt

Ad

dt

Ad

dAA

ˆ,ˆ10

ˆ

ˆˆˆˆ1ˆˆ

ˆˆˆˆ1ˆˆ

ˆˆˆ

0ˆ

)(ˆ)(ˆ

***

***

*

xxxExpectation valueOf operator

Conservation requires(e.g. momentum)

Thus if A is indep.of time the expectation value is constant, as long as A,H commute

)(ˆ1)(

)(ˆ1)(

**

tHit

t

tHit

t

Hamiltonian

Minus sign from complex conjugate

Translational InvarianceTranslational Invariance linear momentum conservation linear momentum conservation

M&S 4.1

i.e. wish to show p operator independent of time 0ˆ,ˆ1ˆˆ

Hitdt

dp

pp

Invariance: All positions in space are physically indistinguishableConsider moving a particle a small distance xxxx '

)()'(

)'()()(2

21

xx

xxx

HH

H

HHxH

m

Depends on derivatives not on position (natural units)

define operator D that performs this translation

)()ˆ1()(ˆ

....)()()()(ˆ

xpxx

xxxxxx

iD

D Higher order terms

ip̂Since in natural units

)(ˆ)(ˆ)()(ˆ)(')('ˆ

)()(ˆˆ)('ˆ

xxxxxxxxx

xxx

DHHD

HDD

)()(ˆ)(' xxx HConsider wavefunction

1)2)

Comparing 1)&2),

0]ˆ),ˆ1[(

0]ˆ,ˆ[

)(ˆ)(ˆ)()(ˆˆ

Hi

HD

DHHD

px

xxxx 1, ix just numbers so

=0

0ˆ,ˆ Hp

Hence, linear momentum is conserved and is a good quantum number

Rotational InvarianceRotational InvarianceAngular Momentum ConservationAngular Momentum Conservation

M&S 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 spinGet

SLJ 0ˆ,ˆˆ,ˆ

0ˆ,ˆ

HH

H

SL

J

Translations in timeTranslations in timeEnergy ConservationEnergy Conservation

)()()(ˆ

)()()()(ˆ

tEtt

itH

tdt

tttttA

Define an operator for time evolution, A

But wait, we already have this operator, it is the hamiltonianTISE

And H commutes with itself 0ˆ,ˆ HHSo time translation is the symmetry, H is operator, E is the conserved quantity

N.B. time translation invariance is different from time reversal operator T– discussed later

•Laws of physics independent of time

Other conserved quantitiesOther conserved quantitiesElectric Charge, Colour Charge, Baryon number, lepton number, strangeness..

First three always conserved (strong,EM,weak)Last one not conserved in weak e.g. p0

Other Discrete Symmetry operatorsOther Discrete Symmetry operatorsParity (P) – spatial Inversion

Charge conjugation (C) – particles anti-particlesreverses: charge

magnetic momentsbaryon numberstrangeness

Only particles that are their own anti-particles are eigenstates of C

Time (T) - Time reversal

Discuss P,C,CP/T, particularly for the weak force

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 ?

Parity - Spatial InversionParity - Spatial Inversion

P operator acts on a state |(r, t)> as

),(),(

),(),(

2 ttP

ttP P

rr

rr

Hence for eigenstates P=±1

(r, t)>= cos x has P=+1, even

(r, t)>= sin x has P=-1, odd

(r, t)>= cos x + sin x, no eigenvalue

e.g. hydrogen atom wavefn

(r,, )>=(r)Ylm(,)

Ylm(,)= Yl

m(-,+)

=(-1)l Ylm(,)

So atomic s,d +ve, p,f –ve P

Hence, Electric dipole transition l=1P=- 1

Discrete symmetry 0ˆ, HP

Parity conserved when Hamiltonian invariant under Parity transformation (strong,EM)

Parity ExamplesParity Examples

Conventions – quarks and leptons have +ve parityAnti-quarks and anti-leptons have –ve parity

•Parity multiplicative

•For a meson made from q qbar pair with orbital angular momentum l

•|> = a b, P=PaPb(-1)l

For ground state (l=0) Pmeson=-1, expect –ve parity for light mesons

-,o,K-,Ko all P=-1

For baryons:

0)1()1(

)1()1)()()((

312

312

ll

llcbaBaryon PPPP

So, expect+ve parity for low lying states

For anti-baryons:

1)1)(1(

)1)()()((

312

312

ll

ll

cbaBaryonPPPP

expect-ve parity for low lying states

l12l3

q1 q3

q2

Scalars,Vectors,Pseudo-Scalars,Axial VectorsScalars,Vectors,Pseudo-Scalars,Axial VectorsScalar – unaffected by parity (+ve parity)Vector – reverses (-ve parity) pprr ,

Can also form quantities from ‘.’ and ‘X’ products of vectors.How do the resultant scalars/vectors behave ?

Axial vector: consider cross-product of two vectors

LprL

prL

)()()(PBoth reverse under parity, so L unaltered

p

r-r

-p

Pseudo-scalar: consider dot product of two vectors

In a parity conserving theory you can’t add an axial vector to a vector

bababa )()()(P Acts like a scalar

Now, consider dot product of vector, axial vector

pLpLpL )()()(P Changes sign , a pseudo-scalar

This leads to parity violation in weak interactions

Weak Force Parity Violation Weak Force Parity Violation Discovery “Discovery “--” problem” problem

Same mass, same lifetime, Same mass, same lifetime, BUTBUT ++, , (21%)(21%) P P =+1 =+1 ++++-, -, (6%) (6%) P P =-1 =-1

Actually K+

Postulated Yang& Lee, 1956

C.S. Wu et. al., Phys. Rev. 105,

1413 (1957) B field

e- (E,p)

Co60Nuclei

spin aligned

Beta decay to Ni*60

e- (E,-p)

Parity

Spin axial vector

-> maximal violation

V-A theory, neutrino handedness

Experimental discovery

Revision

Operating with P on this reverses p, not spin, produces a right-handed neutrino.Do not observe:

Helicity and the neutrinoHelicity and the neutrinoIn 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 ) E

pσ ˆˆ

Not so interesting for a massive particle, as not Lorentz invaraint, but consider the neutrino.

p

s+1 -1

p

sRight-handed Left-handed

1) Only left-handed neutrinos exist and right-handed anti-2) Helicity is a pseudo-scalar

Operating with C on this produces a left-handed anti-neutrino.Do not observe: LLC ˆ

RLP ˆ

RRC ˆ

LRP ˆ

Operating with C and P on this produces a right-handed anti-neutrino.Do observe! RRL CPC )(ˆ)ˆ(ˆ

Weak force violates Parity, but CP OK?

Measuring Helicity of the NeutrinoMeasuring Helicity of the Neutrino

152 152 * 152Eu Sm Sm (960 KeV)

J=0 1 1/2 0 1

ee

152 * 152Sm Sm

J= 1 0 1

Goldhaber et. al. 1958

Electron captureK shell, l=0

photon emission

Consider the following decay:Consider the following decay:

Eu at restSelect photons in Sm* dirn

Neutrino, SmIn opposite dirns

e-

•Momenta, p

•spin

OR

S=+ ½

S=- ½Left-handed

S=+ 1

S=- 1

right-handed

Left-handed

right-handed

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

Neutrino Helicity ExperimentNeutrino Helicity Experiment

Tricky bit: identify forward Tricky bit: identify forward γ γ Use resonant scattering!Use resonant scattering!

Measure γ polarisation with different B-field orientationsMeasure γ polarisation with different B-field orientations

152 152 * 152Sm Sm Sm

magnetic field

Pb

NaI

PMT

152Sm152Sm

152Eu

γγ

Fe

Similar experiment with Hg carried out for anti-neutrinos

Vary magnetic field to vary photon absorbtion.Photons absorbed by e- in iron only if spins of photon and electronopposite.

)2

1()

2

1()1(

)2

1()

2

1()1(

'

ee SSS

Forward photons,(opposite p to neutrino),Have slightly higher p than backwardand cause resonant scattering

Only left-handed neutrinos exist

CP ViolationCP 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,electronsand 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….

C P

CPParity InversionSpatialmirror

Charge InversionParticle-antiparticlemirror

Time reversal – TTime reversal – Tttt ' Leaving all position vectors x unchanged

but p,J reverse

Detailed balanceDetailed balance

),(),(),(),( ddccbbaa mdmcmbma pppp

Compare reaction

With time reversed counter-part

),(),(),(),( bbaaddcc mbmamdmc pppp

Where m are spinZ-components

Conserved for Strong,EMWe wish to test this for the weak force

Inverse experiments are difficult to do with the weak force, need to avoid strong/EM contaminatione.g. reversed would beWould be dominated by strong interaction of proton,pion.Neutrino expt. would be possible, but difficult and looking for a small effect

Particle are eigenstates of P, neutral particles can be of C, but cannot be identical to itself going backwards in time

0 p p0

Let us have a quick look at nature....Let us have a quick look at nature....

Neutral kaon system sdK0 sdK0 flavour eigenstates CP conjugated

m

cm

6.15c

s1004.017.51

7.2c

s100008.08934.01

L

8

LL

S

10

SS

mass eigenstatesKS

KL

Short lifetime

Long lifetime

are a mixture of flavour eigenstates

Time dilation - factor needed for actual flight distance in lab

Mainly Decays to three pions (34%)3 x m(pion) ~ m(Kaon)

Mainly Decays to two pions (99.9%)

CPLEAR- some parametersCPLEAR- some parameters•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)

Kaon OscillationKaon Oscillation

Rate difference Ko Ko Ko Ko is T violation

Particle can turn into anti-particle. So say at t=0, pure Ko, later a superposition of states

d

su, c, t W

W_

s

d_u, c, t

ds

u, c, tW W

_ sd_

u, c, t___

K0K0

1) Identify Ko / Ko at production:produced in association with K+/K-

2) Identify Ko / Ko at decay:observe leptonic decay

CPLEAR T invariance testCPLEAR T invariance test

Initial state at t = 0

KK

KKpp

0

0

S = 0 S = 0)su(K

)su(K

Get positron: Or electron:

Experiment at LEAR ring

at CERN 1990-1996

Pions from kaon decay

Discovery of T violationDiscovery of T violation Currently the only direct observation of T violationCurrently the only direct observation of T violation

Measure asymmetry in rates

3106.16.6 TA

CPLEAR,1998

)()(

)()(0000

0000

KKRKKR

KKRKKRAT

Number of lifetimes

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

CP Violation:CP Violation:Why is it interesting ?Why is it interesting ?

Fundamental:Fundamental: The Alien test 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 KLe+ve-

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 UnderstoodLeast Understood: CP Violation is ‘add-on’ in SM: 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.

CP: Why ? CP: Why ? Powerful: Powerful: delicately broken symmetrydelicately broken symmetry

Very sensitive to New Physics models Historical: Predicted 3rd generation !

Baryogenesis: Baryogenesis: there is more matter !there is more matter ! N(antibaryon) << N(baryon) << N(photons) N(antibaryon) << N(baryon) << N(photons)

Fortunately! 1 : 109

Sakharov (1968) Conditions for matter dominated universeSakharov (1968) Conditions for matter dominated universe Baryon number violation CP violation Not in thermal equilibrium

Assuming not initial conditions, but dynamic.Cannot allow all inverse

reactions to have happened

CP Violation key datesCP Violation key dates

19641964 CP Violation CP Violation discovery in Kaonsdiscovery in Kaons

19731973 KM predict 3 or more KM predict 3 or more familiesfamilies

… ….... … …..erm…not…much…..erm…not…much… …….. 19991999 Direct CP Violation Direct CP Violation

NA48/KTeVNA48/KTeV 20012001 BaBar/Belle CP BaBar/Belle CP

Violation in B mesonsViolation in B mesons

200?200? LHCb physics LHCb physics beyond the SM?beyond the SM?

Are we just the left over matter after CP violating matter/ anti-matter annihilation processes?