Lecture 16: Mixing in the K0 and B Systems

October 26, 2016

Role of Kaons in Development of SM

• Kaons played essential role in understanding particle physics

• Have already seen two examples:

I K+ → 2π, 3π: Parity violation

I K0 → µ+µ+: Very low BR due to GIM mechanism

• Today and Thursday, explore special role of neutral K system

I Mixing

I CP Violation

• Will also include some discussion of neutral B mesons, which exhibitsimilar phenomena

Reminder: Strange Particle Phenomenology

• Stange pseudoscale mesons: 2 isodoublets(K+

K0

) (K

0

K−

)where K+ = su, K− = us, K0 = sd, K

0= ds

• Because strangeness conserved in SI, K0 and K0

are distinct particles

• Strange particles are pair produced via SI

π−p → ΛK0

π+p → pK+K0

• First reaction has much lower threshold than second

I Can produce a pure K0 beam

• Today, neutal K beams produced using high intensity proton beams

hitting targets

I Background a big issue for K0 experiments, most notably from neutrons

Neutral Kaon Decays

• m(K0) = 497 MeV. Not many decay modes open

I Fully leptonic decays highly suppressed (GIM)I K0 → π−`+ν` (and charge conjugate) occurs

I Since parity not conserved in WI, both 2π and 3π decays possible

• Since both K0 and K0

can decay to same states, they can mix throughvirtual decays

K0 ←→⟨

πππππ

⟩←→ K

0

• These are 2nd order WI with ∆S = 2

• If we start with a pure K0 state at t = 0, it will at some later time have a

combination of K0 and K0

|K(t)〉 = α(t)∣∣K0⟩+ β(t)

∣∣∣K0⟩

with√α2 + β2 = 1

Neutral Kaon Mixing (I)

• Can describe this 2nd order WI in quark terms:

• If there were no WI, K0 and K0

would be degenerate in mass

• WI break the degeneracy

• Because physical observables (eg mass, lifetime) are eigenstates ofcomplete Hamiltonian (SI+WI), must select correct linear combination of

K0 and K0

define the states that propagate and decay

The mass and lifetime eigenstates are not the flavor eigenstates!

Neutral Kaon Mixing (I)

• We can almost guess the correct basis to use

I We know WI don’t conserve P since ν are LH and ν are RHI Parity would turn a LH ν into a RH νI But Charge Conjugation turns a ν into a ν

I Hence, CP turns a a LH ν into a RH ν

• WI Lagrangian appears to be CP invariant

• In fact, CP is violated in CKM matrix (∼ 10−3 effect)

• But the CP basis is close to the correct one and that’s what we’ll use

today

I We’ll add the small CP violating piece on Thurs

Neutral Kaon Mixing (II)

• Neutral Kaons transform under CP (not unique definition)

CP∣∣K0⟩ =

∣∣∣K0⟩

CP∣∣∣K0

⟩=

∣∣K0⟩• Therefore, we can write

|K1〉 =1√2

(∣∣K0⟩+∣∣∣K0

⟩)CP |K1〉 = |K1〉

|K2〉 =1√2

(∣∣K0⟩− ∣∣∣K0⟩)

CP |K2〉 = − |K2〉

• |K1〉 and |K2〉 are CP eigenstates and almost the physical basis

CP of Possible Hadronic Decays

π0π0:

I Bose symmetry: WF symmetric

P (π0π0) → π0π0

Cπ0 → π0

CP (π0π0) → +1(π0π0)

π+π−:

I Bose symmetry: WF symmetric

P (π+(~p)π−(−~p)) → π+(−~p)π−(~p)

Cπ± → π∓

CP (π+π−) → +1(π+π−)

π0π0π0:

I Any two π0 combo must haveeven ` (Bose stats)

I J = 0 so ` of 3rd π0 also evenwrt other two

I But π has intrinsic parity P = −1

P (π0π0π0) → (−1)3π0π0π0

Cπ0 → π0

CP (π0π0π0) → −1(π0π0π0)

π+π−π0:

I Small Q suggests ` = 0. If so,same argument as above

I Both CP states allowed but

CP (π+π−π0) = −(π+π−π0)

state highly dominant

2π states have CP = +1 and 3π states have CP = −1

Hadronic Decays of the |K1〉 and |K2〉

• Associating the CP states with the decays:

|K1〉 → 2π

|K2〉 → 3π

• However, very little phase space for 3π decay: Lifetime of |K2〉 muchlonger than of |K1〉

• Physical states called “K-long” and “K-short”:

τ(KS) = 0.9× 10−10 sec

τ(KL) = 0.5× 10−7 sec

• We’ll use distinction that |K1〉, |K2〉 are the CP eigenstates and |KS〉,|KL〉 are true mass eigenstates (including CP violation)

A More Formal Treatment of Mixing

• Write our state ψ as linear combination of K0 and K0:

ψ = α∣∣K0⟩+ β

∣∣∣K0⟩

=

(αβ

)• Schrodinger eq tells us

idψ

dt= Hψ

where H is Hermitian matrix: “generalized mass matrix”

• In matrix form:

H =

(M − i

2Γ M12 − i

2Γ12

M∗12 − i2Γ∗12 M − i

2Γ

)• Diagonal elements equal from CPT

• If CP is a good symmetry, M12 and Γ12 are real

• Find eigenstates by diagonalizing the matrix

M = (m1 +m2)/2 ∆m ≡M12 = (m1 −m2)/2Γ ≡ Γ12 = (Γ1 + Γ2)/2 ∆Γ = (Γ1 − Γ2)/2

Time Dependence (I)

• Write wave functions

|K1(t)〉 = e−im1t−Γ1t/2 |K1〉

|K2(t)〉 = e−im2t−Γ2t/2 |K2〉

• Writing this in terms of strong eigenstates∣∣K0⟩

=1√

2

[e−im1t−Γ1t/2 |K1〉+ e−im2t−Γ2t/2 |K2〉

]∣∣∣K0

⟩=

1√

2

[e−im1t−Γ1t/2 |K1〉 − e−im2t−Γ2t/2 |K2〉

]• If a state ψ that is purely

∣∣K0⟩

is produced at t = 0, at a later time it

will be a combination of∣∣K0

⟩and

∣∣∣K0⟩

:

⟨K0∣∣ ∣∣K0

⟩=

1

2

[e−im1t−Γ1t/2 + e−im2t−Γ2t/2

]⟨K

0∣∣∣ ∣∣∣K0

⟩=

1

2

[e−im1t−Γ1t/2 − e−im2t−Γ2t/2

]

Time Dependence (II)

• Square to get probability:∣∣⟨K0∣∣ ∣∣K0

⟩∣∣2 =1

4

[e−Γ1t + e−Γ2t + 2e−(Γ1+Γ2)t/2 cos(∆mt)

]∣∣∣⟨K0

∣∣∣ ∣∣K0⟩∣∣∣2 =

1

4

[e−Γ1t + e−Γ2t − 2e−(Γ1+Γ2)t/2 cos(∆mt)

]

• The∣∣K0

⟩and

∣∣∣K0⟩

oscillate with frequency ∆m and at the same time

they decay

Observing the Oscillation

• Oscillation provides a way to measure ∆M

• Also demonstrates that this QM phenomenon is happening

• How do we see it?

1. Start with pure K0 beam (low energy)

Look at time dependence of hyperon yield in interactions (K0p→ Λπ)

∆mτ1 = 0.477± 0.2

2. Look for decays that tag the flavor: semileptonic

Observe time dependence in `+ vs `− rate

• Note: Phenomenology of K mixing depends on two things

I Large lifetime difference: time to mix before decaying

I Small mass difference: short oscillation frequency

• In B system things look somewhat different (we’ll discuss later)

What we expect to see as a function of ∆M

∣∣⟨K0∣∣ ∣∣K0

⟩∣∣2 =1

4

[e−Γ1t + e−Γ2t + 2e−(Γ1+Γ2)t/2 cos(∆mt)

]∣∣∣⟨K0

∣∣∣ ∣∣K0⟩∣∣∣2 =

1

4

[e−Γ1t + e−Γ2t − 2e−(Γ1+Γ2)t/2 cos(∆mt)

]

For the measured ∆M

• Start with a K0 beam

• After many KS lifetimes, have a pure KL beamI In absence of CP violation, equal parts K0 and K0

Observation of K0 −K0 Oscillation using semileptonic decays

• Use lepton flavor to distinguish K0 and K0

• Plot shows asymmetry N(`+)−N(`−)N(`+)+N(`−)

• Removes (trivial) lifetime dependence• We’ll come back to the non-zero value at large times

Thursday (CP violation)

Regeneration

• Suppose we put our K beam into matter

• K0 and K0

(SI) interaction rates very different (can make Λ from protons butnot Λ)

• Different absorption cross sections and different forward scattering amplitudes

• Define a complex index of refraction (see Jackson’s E&M book)

n = 1 +2πN

k2f(0)

(similar for n)

• Optical theorm

σtot =4π

kIm (f(0))

• Can include this complex index of refraction into Schrodinger Eq as a phase

∆(phase) =2πNvf

k(1− v2)12

• This changes the eigenstates to∣∣K′1,2⟩ = |K1,2〉 ± r |K2,1〉

• We’ll see an important application of this next time

How About the B system?

• Again, second order in weak interactions

• Different CKM matrix elements for B0 and BsI Larger ∆M for Bs than Bd

• Many possible final states for the decayI Difference in lifetime of the BL and BS states small

Example of B Mixing (B0 and Bs)

• ∆M for B0 = 0.510± 0.003± 0.002 ps−1

• ∆M for Bs = 17.761± 0.021± 0.007 ps−1