Prepared By

A.K.M. Moinul Haque Meaze

Student ID: 2003419008

Center for High Energy Physics

Kyungpook National University

DaeguDaegu 702-701

Republic of Korea

Course Title: Phenomenology

Honorable Professor Kihyeon Cho

DISCRETE SYMMETRIES

November 11, 2003

Symmetry is usually associated with an action or transformation of a system or object such that after carrying the operation the system or object in a state indistinguishable from that which it had prior to carrying out the action or transformation. The existence of symmetries implies that it is impossible to devise an experiment to distinguish before and after the situation. So the symmetry or invariance, of physical laws describing a system undergoing some operation is one of the most important concepts in physics.

• Discrete symmetries, such as reflections, inversions, time reversal, charge conjugation, parity, finite rotations, permutations etc. associated with multiplicative or phase-like quantum numbers

• Continuous symmetries, such as translations and rotations are associated with additive quantum numbers (e.g., angular momentum J or linear momentum p)

SYMMETRIES

Types of Symmetries

* Global, Local, Dynamical, Internal-are also classes of symmetries.

November 11, 2003

Charge conjugation C transforms each particle into its antiparticle.

Here C2=1, So C=±1, i.e., it has eigen value ±1 Any particle that is an eigen state of C must be its own anti particle, since C|particle>=|anti particle>=±|particle>

C

ppC C

Charge Conjugation

November 11, 2003

Parity is the act of reflecting a system in a mirror.

),,,(),,,( zyxtzyxtP VVP )(

WVWVWVPWVP .)).(()).(().(

Parity invariance means the probability of a particle process occurring is exactly the same as the probability of the same process occurring with the position vector and directions of travel of all particle reversed.

Parity

1000

0100

0010

0001

)(),,,(),,,( PzyxtzyxtP

P=±1

November 11, 2003

This doodle pad was used by T.D. Lee during talks with C.N. Yang, while both were visiting scientists at Brookhaven in the summer of 1956. These discussions led to questioning the conservation of parity in weak interactions and resulted in their being awarded the Nobel prize in1957

November 11, 2003

)()()( ttatT

),(),(),( trtrKtrT

Time reversal invariance is a theory predicting that if a process is governed by a physical theory, then the same physical theory applied in the reverse direction of

time.

 

Time Reversal Invariance

1000

0100

0010

0001

),,,(),,,( TzyxtzyxtT

Time reversal invariance, T reverses the time coordinate.

But T does not satisfy the simple eigen value equation

Instead

here K is the complex conjugation operator.

* For example direction of motion of particles

T violation means that the rate for a particle interaction is different for the time reversal process.November 11,

2003

CP Violation Must Exist!

Our planet is made out of matter and a block of antimatter can not exist. In fact we have not found any evidence of anti matter in the whole universe! If this universe originated from the big bang, which creates equal amount of matter and anti matter, the forces responsible for the expansion and cooling of our universe must violate particle-antiparticle symmetry. CP violation must exit!!!

If Big Bang is to produce unequal amount of matter and anti matter CP must be violated.

November 11, 2003

CP In Weak Interaction

Fig. Transformations on pion decay

Under a combined operation CP

LH Neutrino RH Anti neutrino

So CP is conserved in weak interaction even though C and P separately are not conserved

November 11, 2003

Then

CP|K1>=|k1>

CP|K2>=-|K2>

CP Violation in K Meson

November 11, 2003

CP Violation in K Meson

The short lived kaon decay into two pions while long lived pions would decay into three pions, but in some cases two pions, thus violating the CP symmetry.

November 11, 2003

CPT Theorem

* This is one of the most important and generally valid theorems in quantum field theory.

* This appears to be an exact invariance of all process. This means that any process has a related process which an identical rate, the process to which it is converted by making three replacements of C,P & T.

* All interactions are invariant under combined C, P & T.

* Invariance under Lorentz transformation implies CPT invariance showed by George Luders, Wolfgang Pauli and Julian Schwinger.

* Implies particle and antiparticle have equal masses and lifetimes.

* Implies all the internal quantum numbers of antiparticles are opposite to those of particles.

November 11, 2003

Quantity Symbol T P

Position r r -r

Momentum p -p -p

Spin σ - σ σ

Electric Field E E -E

Magnetic field B -B B

Magnetic dipole moment

σ.B σ.B σ.B

Electric dipole moment

σ.E -σ.E -σ.E

Longitudinal polarization

σ.p σ.p -σ.p

Transverse polarization

σ.(p1×p2) -σ.(p1×p2) σ.(p1×p2)

Common Quantities under T & P

November 11, 2003

We can write the Hamiltonians HM, HE that describe the interaction of μ

with B, and of D with E, in the non relativistic limit 

Under space inversion (P) the axial vector σ and B remain unchanged, but the polar vector E changes sign. Hence under P, HM is invariant, while HE is not.

Under time reversal (T), σ and B changes sign, while E remains unchanged. Hence under T, HM is once again invariant, but HE changes sign.

BHM .EDHE .

Therefore, non zero EDM violates both T and P symmetries

General Remark on EDM

November 11, 2003

Fig. A bit history of EDM [Ref. E. A. Hinds, Electric dipole moments: Theory and Experimenthttp://blois.in2p3.fr/2002/plenary/friday21/ cpt/Blois2002Hinds.pdf ]The limits of neutron electric dipole moments are:

dn <1.2x10^ -25ecm[Ref. P.F. Smith et al, Phy Lett B234(1990)19]

dn <6.3x10^ -25ecm[Ref. P. G. Harris et al, Phy Rev Lett 82(1999)904]

November 11, 2003

November 11, 2003

Symmetry

Obeyed By

Strong Force

Electromagnetic Force

Weak Force

Gravitational Force

Charge Conjugation, C

Yes Yes No Yes

Parity, P Yes Yes No Yes

Time Reversal, T

Yes Yes Almost Yes

CPYes Yes Almost Yes

CPT Yes Yes Yes Yes

Sym

metrie

s on

Force

s

SYM

METR

IES-A

T A

GLA

NC

E

Symmetry Action

Charge Conjugation, C Particle→ Anti-particle

Parity, P x→-x, y→-y, z→-z

Time reversal, T t→-t

CP Particle→ Anti-particlex→-x, y→-y, z→-z

CPT Particle→ Anti-particlex→-x, y→-y, z→-z, t→-t

SymmetryAction On

Position TimeVelocity, Momentum

Spin

Charge Conjugation, C Unchange

d Unchanged

Unchanged Unchanged

Parity, PReversed

Unchanged

Reversed Unchanged

Time reversal, TUnchanged

Reversed Reversed Reversed

CPReversed

Unchanged

Reversed Unchanged

CPT Reversed Reversed Unchanged ReversedNovember 11, 2003