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A guide to CP violation

Chris Parkes

12/01/2006

:Section 1:

Symmetries

Symmetries• Role of symmetries in physics

– e.g. translational -> momentum conservation– rotational -> angular momentum conservation– Time -> energy conservation

• Fundamental Symmetries we will study– Parity (P) – spatial inversion– Charge Conjugation (C) – particle/ anti-

particle– CP– CPT

Emmy Noether

Parity - Spatial InversionP 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

Parity cont.• Conserved in strong & emag. Interactions• Parity multiplicative |> = a b, P=PaPb

• Proton– Convention Pp=+1

• QFT– Parity fermion -> opposite parity anti-fermion– Parity boson -> same parity anti-particle

• Angular momentum– Use intrisnic parity with GROUND STATES– Also multiply spatial config. Term (-1) l

scalar, pseudo-scalar, Vector, axial(pseudo)-vector,

Jp = 0+ , 0-, 1-, 1+ -,o,K-,Ko all 0- , photon 1-

Parity Violation Discovery“-” problem

• Same mass, same lifetime, BUT+, (21%) P =+1

++-, (6%) P =-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

Charge Conjugation

C operator acts on a state |(x, t)> as),(),(

),(),(

2 ttC

ttC C

rr

rr

Particle to anti-particle

Only a particle that is its own anti-particle can be eigenstate of C,

e.g. C |o> = ±1 |o>

o + A = J, hence C=-1

Thus, C|o> =(-1)2 |o> = +1 |o>

G , isospin rotation I3 ->-I3, e.g. + -> -

Neutrino helicity

left-handed

right-handed

Parity ->

left-handed

right-handed

Charge & Parity ->

•Massless approximation

•Goldhaber et al. Phys Rev 109 1015 (1958)

Time

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

Neutral kaon system sdK0 sdK0 flavour eigenstates CP conjugated

110SL

8

LL

10

SS

s100014.05301.0=

s1004.017.51

s100008.08934.01

mmm

mass eigenstatesKS

KL

Three pion decay, very little phase

space

CPLEAR T invariance testInitial state at t = 0

KK

KKpp

0

0

S = 0 S = 0)su(K

)su(K

Rate difference Ko Ko Ko Ko is T violation

Experiment at LEAR ring

at CERN 1990-1996

Discovery of T violation• direct observation of T violation

– Detailed balance expts difficult due to strong/em. effects

3106.16.6 TA

Electric Dipole Moments• Energy shift due to say, neutron, being in

weak electric field– e.d.m. (measured in e cm)

– TdT-1= d, but only available direction is J so– d=const.J– TJT-1= -J, hence d=0

• Also for electron, and (less obviously) atomic nuclei – (linear term in E not present)

ii

ie rd

Spin precession fequency of ultracold neutrons in a weak magnetic field.

d(n) 6.3x10-26 ecm, also d(e) 1.6x10-27 ecm (sussex)

CPT Invariance• Particle->anti-particle, reverse

time, invert space.• CPT |(r,t)> = |(-r,-t)> • Lagrangian invariant under CPT

– Lorentz invariant– Unique ground state– Spin-statistics (Fermi/Bose)….

• No appealing theory of CPT violation exists

CPT Consequences(1)• Particle/anti-particle mass equality

m

PHHP

PCPTHHCPTP

PCPTHHCPTP

PCPTCPTHHCPTCPTP

PHHPm

ems

ems

ems

ems

ems

))(()(

))(()(

)())()(()(

†

†

††

CPT Consequences (2)• Particle/anti-particle width equality

:Section 2:

Introducing CP in SM

CP Violation Introduction:Why is it interesting ?

• Fundamental: The Martian test– C violation does not distinguish between

matter/anti-matter. LH /RH are conventions– CP says preferred decay KLe+ve-

• Least Understood: CP Violation is ‘add-on’ in SM– Parity violation naturally imbedded from V-A

coupling structure– CP requires a complex phase in 3 generation

CKM matrix, allowed but not natural

CP: Why ? cont.• Powerful: delicately broken symmetry

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

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

– Fortunately! 1 : 109

• Sakharov (1968) Conditions– 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 dates

• 1964 CP Violation discovery in Kaons• 1973 KM predict 3 or more families• …..• …..erm…not…much…• ….• 1999 Direct CP Violation NA48/KTeV• 2001 BaBar/Belle CP Violation in B’s

• 200? LHCb physics beyond the SM?

CP Violation in SM: CKM matrix• SM weak charged current

– V-A form LH states

L Vij Ui () Dj W

• Vij is the quark mixing matrix, the CKM matrix• for 3 famillies this is a 3x3 matrix

• U,D are up/down type quark vectors

U =

uct

D =

dsb

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

e.g.

W-

c d

Coupling Vcd

CKM continued

• Cabibbo (1963) and Kobayashi & Maskawa (1973)

• Realised mass and flavour eigenstates – need not be the same

• Weak interaction generations

• Related to physical quark states by CKM matrix d’

s’b’

dsb

= VCKM

ud’

c s’

t b’

Values of elements a

purely experimental

matter

Number of Parameters in CKM

• n x n complex matrix, – 2n2 parameters

• Unitarity n2 constraints– n2 parameters

• Phases of quark fields can be rotated freely – (n-1)2 parameters

• Real parameters, rotation (Euler) angles – n(n-1)/2 real

• Phases– (n-1)(n-2)/2 phases

ikjkj

ijVV *

n=2, 1 real, 0 phasen=3, 3 real, 1 phase

K&M Predict 3 famillies (Prog. Theor. Phys. 49, 652(1973) )

• Only 3 quarks discovered– Charm predicted by GIM mechanism – CP violation discovered

• Phase ei(wt+) Tei(-wt+)

– i.e. Violates T/CP

• Hence predict three (or more) famillies!

• Now parameterize 3x3 CKM in 4 parameters

PDG, 3 angles + phase

C12 S12 0

-S12 C12 0

0 0 1

1 0 00 C23 S23

0 -S23 C23

3 angles 12, 23, 13 phase

Cij= cos ij

Sij=sin ij

VCKM = R23 x R13 x R12

R12 = R23 =

R13 =

C13 0 S13 e-i

0 1 0

-S13 e-i 0 C13

Wolfenstein’s parameters

1ˆˆ12

1

21

423

22

52

32

iAAiA

AiA

iA

A ~ 1, ~ 0.22, ≠ 0 but ≠ 0 ???

21ˆ ,

21ˆ

22

VCKM

= S12, A=S23/S212, =S13cos/ S13S23, =S13sin/ S12S23

VCKM(3) terms in up to 3

CKM terms in 4,5

Unitarity conditions

ikjkj

ijVV *

hence 6 triangles in complex plane

123

1

i

ijV j=1,3

No phase info.

3

1

* 0i

ikijVV j,k =1,3 jk

0

0

0

0

0

0

***

***

***

***

***

***

cbubcsuscdud

tbcbtscstdcd

tbubtsustdud

tstdcscdusud

tbtdcbcsubus

tbtdcbcdubud

VVVVVV

VVVVVV

VVVVVV

VVVVVV

VVVVVV

VVVVVVdb:

sb:

ds:

ut:

ct:

uc:

More triangles•Area of all the triangles is the same (6A2)

•Two triangles (db) and (ut) have sides of similar size

•Easier to measure, (db) is often called the unitarity triangle

*ubudVV

*tbtdVV

*cbcdVV

Bottom side A3 normalised to 1

(,)

*ubtbVV

*udtdVV

*ustsVV

’=, =-’, =-’ = -arg(Vts)

= arg(Vts)

CP in SM summary

• Study of CP violation is the analysis of the CKM matrix to verify if it is consistent with the standard model.

• If not New Physics!

• Will CP lead to SM ?