CP Violation

Standard Model contains only left-handed neutrinos and right-handed anti-neutrinos

C ν LL= ν charge conjugation not a symmetry of the weak interaction

P ν LR= ν parity also not conserved in weak interactions

what about the combined CP transformation…..looks OK ? CP ν LR= ν Kaon Oscillations (K0-K0 Mixing)

K 0 K 0 s d s d

W−

u

u u W− W− u

W− d s d s K 0 K 0

Neutral mesons can “oscillate” back and forth between particle and antiparticle states e.g. equivalent diagrams mix D0 with D0 , B0 with B0

This (second-order) process mixes states of strangeness = 1, strangeness = -1 Neutral Kaons are produced in states of definite strangeness CP Operations on Neutral Kaons

Neutral kaons are ground state spin-0 sd mesons, so are pseudoscalars [ P =− (1 ) A + 1 ]

0000⎧ 00 PK =−K PK =−K ⎪ CPK =−K ⎨ CK00==K CK0K0 CPK0=−K0 ⎩⎪

Normalized eigenstates of CP are therefore:

1 00 K11=−(KK) CPK=+K1 2

1 00 K22=+()KK CPK=−K2 2 Decays of Neutral Kaons

If CP is conserved by the weak interaction:

K1 decays only to CP = +1 final states (CP even)

K2 decays only to CP = -1 final states (CP odd)

Neutral kaons decay are the lightest neutral mesons containing a strange quark.

0 - 2 Cannot decay via strong interaction: M(K ) – M(K ) ~ 4 MeV/c << Mπ Cannot decay electromagnetically. Dominant decays are weak decays ( s Æ u transition) into two or three pions System of n pions (without orbital angular momentum) has P = (-1)n

MM−≈2 220 MeV/c2 K1 → ππ K 0 π

MM−≈385MeV/c2 K2 → πππ K 0 π Decays of Neutral Kaons

Decay of K1 takes place 500 times faster than K2 (phase space considerations)

Imagine a beam of neutral kaons (produced via the strong interaction)

1 KK0 =+()K 2 12

Mean decay length (in lab frame) is gβcτ(g the same for K1 and K2) so mean decay length of K2 is ~500 times longer that for K1

K1 component of beam will decay very quickly after production.

Expect to see many K0Æ ππ decays near the production point

Far downstream expect an almost pure beam of K2 CP Violation in Neutral Kaon Decays

By going very far downstream from K0 beam production point, expect an arbitrarily pure beam of K2, if CP is indeed conserved by the weak interaction

Cronin and Fitch, 1964 (Nobel Prize 1980) performed this experiment

Found small admixture (1/500) of 2π decays very far from the production point

Long-lived neutral kaon state (K0 Long) is NOT a CP eigenstate

0 1 KKL =+()21ε K 1+ ε 2

CP is therefore NOT a good symmetry of the weak interaction, although the level at which it is broken is rather small ( ε = 2.3 x 10-3 ). Other Manifestations of CP Violation in Kaon System

0 If K L were a CP eigenstate (eg ε = 0) then the two decays

0 −+ 0 +− KeL → π ν e KeL → π ν e

Would occur with equal probability: instead

00−+ +− BR()KLe→−πνe BR()KLe→πνe 00−+ +− = .0033 BR()KLe→+πνe BR()KLe→πνe Quark Mixing and the Cabibbo Angle

Weak decays of s quarks requires su coupling (i.e. generational transition) unlike in pion decays (for example) Cabibbo introduced quark mixing to explain relative branching ratios weak decays of mesons into leptons:

µ νµ µ νµ

WW

u d u s π+ K+

Relative strengths imply smaller coupling at suW vertex than at udW Quark Mixing and the Cabibbo Angle

Cabibbos hypothesis (only u,d, and s quarks were “known” at the time).

dd′ =+cosθccssinθ ⎛⎞dd′ ⎛⎞cosθθccsin ⎛⎞ ⎜⎟= ⎜⎟⎜⎟ sd′ =− sinθcc+scosθ ⎝⎠ss′ ⎝⎠−sinθθcccos ⎝⎠

θc is the Cabibbo angle ( ~ 13º )

u d u s

sinθc cosθc

w+ w+

Cabibbo-favoured Cabibbo-suppressed ++ ++ π → µνµ K → µ ν µ The GIM Mechanism (1970)

Glashow, Iliopoulos and Maini postulating the existence of a fourth quark 0-+− 9 (charm) on the basis of the measured braching fraction: BR()KL →=µµ 7x10

Should occur at a much higher rate via the process:

u d u s µµ

cosθc sinθc ν w+ w+ W− W− u c s c d

cosθc −sinθc d s K 0 L w+ w+ The GIM Mechanism (1970)

Contribution from process with internal u quark exchange partially cancelled by same diagram but with internal c quark exchange

µµµµ

ν ν

W− W− W− W− u c

d s d s 0 0 KL KL

If the masses of the u and c quarks were identical, these contributions would exactly cancel. Since they are not, this merely leads to the (observed) suppression of such processes. The Kobayashi Maskawa Prediction (1973)

Later we will see, in the weak interaction Lagrangian:

⎛⎞d µ 5 ⎜⎟ (,uc,t)γγ(1− )V s quark mixing matrix complex N x N matrix has ⎜⎟ 2 ⎜⎟ 2N real free parameters ⎝⎠b

Unitarity eliminates N2 2N (number of quarks) can be absorbed into the phases in the quark wavefunctions. Not sensitive to an overall phase, so this eliminates 2N-1 This means N2-2N+1 = (N-1)2 are required to parametrize the mixing matrix N(N-1)/2 real mixing angles + (N-1)(N-2)/2 complex phases

A complex phase in the quark mixing matrix allows for CP violation in the SM.

Based on this, Kobayashi and Maskawa predicted a third generation in 1973 Cabibbo Kobayashi Maskawa Matrix

⎛⎞dV′ ⎛ud Vus Vub ⎞⎛d⎞ ⎜⎟⎜ ⎟⎜⎟ s′ = VVV sCKM mixing matrix for three generation of quarks ⎜⎟⎜cd cs cb ⎟⎜⎟ ⎜⎟bV′ ⎜ VV⎟⎜b⎟ ⎝⎠⎝td ts tb ⎠⎝⎠ 1,2,3 label mixing angles, δ is the phase

s1 = sin(angle 1), c1 = cos(angle 1) etc

⎛⎞cs11c1 s1c3 ⎛⎞10 0⎛cs110⎞⎛1001⎞⎛00⎞ ⎜⎟iiδδ⎜⎟⎜⎟⎜⎟⎜⎟ −−sc ccc s se ccs +sce 00cs−sc 0100cs ⎜⎟12 123 23 123 23 = ⎜⎟22⎜11⎟⎜⎟⎜33⎟ ⎜⎟iiδδ⎜⎟00−−s ce⎜01⎟⎜00iδ ⎟⎜0sc⎟ ⎝⎠s12s −−cs12c3 c2s3e −cs12s3+c2c3e ⎝⎠22⎝⎠⎝⎠⎝33⎠

Three rotations and a phase

2 ⎛⎞1(−−λ λλAi3 ρη) ⎜⎟2 ⎜⎟2 Wolfenstein parametrization emphasizes the λ 1(−+λ AOλλ34) ⎜⎟2 magnitudes of the off-diagonal elements ⎜⎟33 ⎜⎟Aiλρ(1 −−η) −Aλ 1 ⎝⎠ λ = sinθc ~ 0.225 Experimental Determination of CKM Matrix Elements

⎛⎞VVud us Vub ⎛⎞0.9739 −−0.9751 0.221 0.227 0.0029 −0.0045 ⎜⎟⎜⎟ VVV 0.221−−0.227 0.9730 0.9744 0.039 −0.044 ⎜⎟cd cs cb = ⎜⎟ ⎜⎟⎜⎟ ⎝⎠VVtd ts Vtb ⎝⎠0.0048 −−0.014 0.037 0.043 0.9990 −0.9992

2 ⎛⎞1(−−λ λ Aλρ3 iη) ⎜⎟2 ⎜⎟2 λ = sinθ λλ1− λ A 2 c ⎜⎟2 ~ 0.225 ⎜⎟32 ⎜⎟Aiλρ(1 −−η) −Aλ 1 ⎝⎠ CP Violation in the K System

K 0 s d ⎛⎞Vud Vus Vub ⎜⎟ Vcd Vcs Vcb Vus − Vud ⎜⎟ W ⎜⎟V V V u u ⎝⎠td ts tb

W− There are also equivalent diagrams with c Vud Vus and t quarks replacing the u quarks in the loop. d s K 0 CP Violation in the K System

K 0 s d ⎛⎞Vud Vus Vub ⎜⎟ Vcd Vcs Vcb Vcs − Vcd ⎜⎟ W ⎜⎟V V V c c ⎝⎠td ts tb

W− Vcd Vcs

d s K 0 CP Violation in the K System

K 0 s d ⎛⎞Vud Vus Vub ⎜⎟ Vcd Vcs Vcb Vts − Vtd ⎜⎟ W ⎜⎟V V V t t ⎝⎠td ts tb

W− Vtd Vts

d s K 0 Unitarity Constraints

*** 2 ⎛⎞ ⎛⎞1(−−λ λ Aλρ3 iη) VVud cd Vtd ⎛⎞Vud Vus Vub ⎛100⎞⎜⎟2 ⎜⎟***⎜⎟⎜⎟⎜⎟2 VVV VVV= 010 λλ1− λ A 2 ⎜⎟us cs ts ⎜⎟cd cs cb ⎜⎟⎜⎟2 ***⎜⎟⎜⎟⎜⎟32 ⎜⎟VVV VVV 001 ⎜⎟Aiλρ(1 −−η) −Aλ 1 ⎝⎠ub cb tb ⎝⎠td ts tb ⎝⎠⎝⎠

η *** VVus ud ++VcsVcd VtsVtd =0 ρ * “Unitarity” triangle in ρη plane *5 VVus ud ~ λ VVts td ~ λ

* VVcs cd ~ λ

This angle is a measure of the degree to which CP is 54 violated in the kaon system ~ λλ/ = λ~ .0025 CP Violation in the B System

B0 B0 b d b d

W−

u,c,t u,c,t u,c,t W− W− u,c,t

W− d b d b B0 B0

Mixing of neutral B meson occurs in the same way as in the kaon system CP Violation in the B System B0

b d ⎛⎞Vud Vus Vub ⎜⎟ Vcd Vcs Vcb Vub − Vud ⎜⎟ W ⎜⎟V V V u u ⎝⎠td ts tb

W− There are also equivalent diagrams with c Vud Vub and t quarks replacing the u quarks in the loop. d b B 0 CP Violation in the B System B0

b d ⎛⎞Vud Vus Vub ⎜⎟ Vcd Vcs Vcb VVcb − cd ⎜⎟ W ⎜⎟V V V c c ⎝⎠td ts tb

W− VVcd cb

d b B 0 CP Violation in the B System B0

b d ⎛⎞Vud Vus Vub ⎜⎟ Vcd Vcs Vcb Vtb − Vtd ⎜⎟ W ⎜⎟V V V t t ⎝⎠td ts tb

W− Vtd Vtb

d b B 0 CP Violation in the B Meson System

*** 2 ⎛⎞ ⎛⎞1(−−λ λ Aλρ3 iη) VVud cd Vtd ⎛⎞Vud Vus Vub ⎛100⎞⎜⎟2 ⎜⎟***⎜⎟⎜⎟⎜⎟2 VVV VVV= 010 λλ1− λ A 2 ⎜⎟us cs ts ⎜⎟cd cs cb ⎜⎟⎜⎟2 ***⎜⎟⎜⎟⎜⎟32 ⎜⎟VVV VVV 001 ⎜⎟Aiλρ(1 −−η) −Aλ 1 ⎝⎠ub cb tb ⎝⎠td ts tb ⎝⎠⎝⎠

η

*** ρ VVub ud ++VcbVcd VtbVtd =0 *3 *3 VVcs cb ~ λ VVub ud ~ λ “Unitarity” triangle in ρη plane

*3 VVcb cd ~ λ

CP Violation expected to be large in the B system. ~/λλ33=1 This has been an active field of research at the so-called B factories since the mid-late nineties (and remains so) CP Violation and Baryon Number Asymmetry

Our universe appears to be made of matter only. Astronomical observations rule out “domains” of anti-matter dominance In the Big Bang, matter and antimatter would have been created in equal amounts. Need some mechanism for creating the observed asymmetry Need CP violation (which is an asymmetry between matter and anti-matter) Mechanism of CP violation in Standard Model not enough to explain this. Also need baryon number non-conserving processes (GUTs) Search for Proton Decay (violates baryon number conservation)

SuperKamiokande Detector, Japan Alpha Magnetic Spectrometer (on Space Station)

Amongst other studies, AMS will look for anti- matter potentially left over from the Big Bang Time Reversal Invariance

Quantum Field Theory (QFT): CPT theorem says that for any theory consistent with Lorentz invariance, quantum mechanics and the idea that interactions are carried by fields, the combined operations of C, P and T (in any order) form an exact symmetry of any interaction. Since CP is known to be violated by weak interaction, T must be as well Experimental tests of this are hard: can’t easily run a weak interaction backwards (principle of detailed balance) Instead, look at quantities that are identically zero if T is conserved Popular example is electric dipole moment of the neutron (see text) Current limit from 2002: e · (0.069 ± 0.074) x 10-26 cm