Kaon physics - or an introduction to flavor oscillations

Kaon mixing and eigenstatesNeutral kaons and CP violation

Neutral kaons and flavor oscillations

Kaon physics. . . or an introduction to flavor oscillations and CP violation

Harri Waltari

University of Helsinki & Helsinki Institute of PhysicsUniversity of Southampton & Rutherford Appleton Laboratory

Autumn 2018

H. Waltari Kaon physics



Contents

Kaons are the lightest strange mesons and relatively long-lived. They arefrom the experimental point of view the easiest platform to study manyof the interesting features of flavor physics with high precision. In thislecture we shall

study the flavor and CP eigenstates of kaons

study how CP violation emerges in kaon physics

study the flavor oscillations between K 0 and K0

This lecture corresponds to chapters 14.4, 14.5.1, 14.5.2 and 14.5.4 ofThomson’s book.




Weak interactions mix the neutral kaon states K 0 and K0

Neutral kaons are produced in two different quark compositions K 0

(ds) and K0

(ds) — notice that for historical reasons, thestrangeness quantum number is so defined that S(K 0) = +1, i.e.the antiquark has positive strangeness

As neutral kaons propagate, they can transform between each other

through box diagrams, so neither K 0 nor K0

is an eigenstate of thefull Hamiltonian




The physical eigenstates have differing lifetimes

Experimentally we observe that there are two different neutral kaonstates with (almost) the same mass (close to 500 MeV) but differentlifetimes, one having τ ' 0.1 ns and the other with τ ' 50 ns, theseare called KS and KL (for short and long)

We know that in the leptonic sector CP is (at least within currentexperimental precision) conserved so we may expect the CPeigenstates to be (at least close to) the stationary states

Kaons are JP = 0− mesons so P|K 0〉 = −|K 0〉 and P|K 0〉 = −|K 0〉The flavor eigenstates have opposite flavor contents so

C |K 0〉 = e iζ |K 0〉 and C |K 0〉 = e−iζ |K 0〉, where ζ is anunobservable phase

If we choose ζ = π (which is a common convention)

C |K 0〉 = −|K 0〉 and C |K 0〉 = −|K 0〉

Hence CP|K 0〉 = |K 0〉 and CP|K 0〉 = |K 0〉




The CP eigenstates are combinations of K 0 and K0

From the CP properties of K 0 and K0

it is easy to construct the CPeigenstates:

CP eigenstates of neutral kaons

|K1〉 = 1√2

(|K 0〉+ |K 0〉) with CP = +1 and

|K2〉 = 1√2

(|K 0〉 − |K 0〉) with CP = −1

In weak interactions CP is nearly conserved so these almost coincidewith the physical eigenstates KS and KL

On the other hand flavor eigenstates are linear combinations of |K1〉and |K2〉 but this combination will have a non-trivial time evolution

As we shall see, some of the decays are spesific to flavor eigenstatesand some to CP eigenstates, which will give us interesting probes onthe kaon system




Decays to pions determine the CP properties of KS and KL

Experimentally we see that KS decays mainly to π0π0 or π+π−,while KL decays to π0π0π0 or π+π0π−

These decay modes determine the lifetimes — for the three piondecay there is less phase space available so the decay is slower

As both kaons and pions are JP = 0− mesons, the pions must be inan ` = 0 state in the decay K 0 → π0π0, π+π− to conserve angularmomentum

Hence the parity of the final state isP(π0π0) = (−1)`P(π0)P(π0) = 1 · (−1) · (−1) = 1, similarly forπ+π−

The neutral pion is a state |π0〉 = 1√2

(uu − dd) so its C-parity is +1

as C transforms the pion to itself

For π+π− C is equivalent to the exchange of particles and forbosons this does not change the sign

Hence CP(π0π0) = +1 and CP(π+π−) = +1 implying that KS

could be identified with K1





The angular momentum argument is trickier for the three pion finalstate

We first take the angular momentum of two pions labeling it ~L1 andthen the angular momentum of the third with respect to thecenter-of-mass of the pair (~L2)

The total angular momentum is ~L = ~L1 + ~L2, which must be zerodue to angular momentum conservation (the spins are all zero),giving |L1| = |L2|Hence the parity is (−1)L1 · (−1)L2 · (P(π0))3 = −1





The C-parities of the three-pion final states are +1 by the samearguments as for the two pion case

Hence the CP for the three pion state is negative, which impliesCP(KL) = −1

Thus |KL〉 can be identified with |K2〉In 1964 Christenson, Cronin, Fitch and Turlay observed the decayKL → π+π−, which was the first (and together with other similarkaon decays, the only for more than 30 years) indication of CPviolation in weak interactions (Fitch and Cronin got the 1980 NobelPrize for this discovery)




Particle decays can be described by complex energyeigenvalues

We shall first look at kaon state vectors in the limit of CPconservation

Kaons are produced as flavor eigenstates, e.g. K 0 = 1√2

(|KS〉+ |KL〉)The time evolution can be properly described if we attach animaginary part to the eigenvalue of the Hamiltonian, e.g.|KS(t)〉 = |KS(0)〉e−imS t−ΓS t/2

In such a case 〈KS(t)|KS(t)〉 = e−ΓS t so the wave function dies outin the timescale τS = 1/ΓS

As there is a factor of 500 between the lifetimes of KS and KL, aftera reasonably long propagation time a beam initially in a flavoreigenstate has become a pure |KL〉 state




CP is nearly conserved — only a tiny fraction of decays areCP violating

KS decays BR KL decays BRKS → π+π− 69.2% KL → π+π− 0.20%KS → π0π0 30.7% KL → π0π0 0.09%KS → π0π0π0 < 2.6× 10−8 KL → π0π0π0 19.5%KS → π+π−π0 3.5× 10−7 KL → π+π−π0 12.5%KS → π−e+νe 0.03% KL → π−e+νe 20.3%KS → π+e−νe 0.03% KL → π+e−νe 20.3%

Here the leptonic modes have equal decay rates but due to thelarger total rate of KS , the branching ratios are smaller

There is a clear difference in the order of magnitude of the CPviolating decays, 2 orders of magnitude when kinematics arefavorable and 6 orders of magnitude when they are not




CP is violated both in kaon mixing and decays

There are two possible sources for CP violation: The Hamiltoniancan introduce CP violation in the kaon mixing so that the physicaleigenstates do not coincide with the CP ones, it could also bepossible for the kaon decays to be directly affected by the CPviolating phase

In the first case the eigenstates are parametrized as

|KS〉 =1√

1 + |ε|2(|K1〉+ ε|K2〉), |KL〉 =

1√1 + |ε|2

(|K2〉+ ε|K1〉),

where ε is a small complex parameter

The second case is parametrized by ε′ = Γ(K2 → ππ)/Γ(K2 → πππ)

Both are nonzero and small and <(ε′/ε) ' 1.65× 10−3




The Hamiltonian gets contributions from interactionpotentials and decays

We shall next go through the kaon mixing and see how most of the CPviolation is generated. The mixing is a second order effect in weakinteractions, i.e. proportional to G 2

F .

The Hamiltonian equation of motion for, say, |K 0〉 in its rest frameis i ∂∂t |K

0(t)〉 = (m − i2 Γ)|K 0(t)〉

The mass term is the sum of the quark masses and the potentialenergy of the interactions between quarks:

m = md+ms+〈K 0|HQCD+HQED+HW |K 0〉+∑j

〈K 0|HW |j〉〈j |HW |K 0〉Ej −mK

The decay rate is determined by the Fermi golden rule:Γ = 2π

∑f |〈f |HW |K 0〉|2ρf , where the sum goes over all possible

final states |f 〉 and ρf is the corresponding density of states




Weak interactions introduce mixing terms to theHamiltonian

To introduce K 0–K0

mixing, we have to generalize the Hamiltonianto a matrix

We parametrize the state as |K (t)〉 = a(t)|K 0〉+ b(t)|K 0〉The equation of motion is generalized to

i∂

∂t

(a(t)|K 0〉b(t)|K 0〉

)=

(M11 − i

2 Γ11 M12 − i2 Γ12

M21 − i2 Γ21 M22 − i

2 Γ22

)(a(t)|K 0〉b(t)|K 0〉

)

The mass terms M11 and M22 are similar to the ones before, butM12 and M21 arise from the box diagrams:

M12 = M∗21 =∑j

〈K 0|HW |j〉〈j |HW |K 0〉Ej −mK




Weak interactions introduce mixing terms to theHamiltonian

The decay term can be written as∑f

|〈f |HW |K (t)〉|2 =∑f

|〈f |HW |(a(t)|K 0〉+ b(t)|K 0〉)|2 =∑f

|a(t)|2〈K 0|HW |f 〉〈f |HW |K 0〉+ |b(t)|2〈K 0|HW |f 〉〈f |HW |K0〉+

a∗(t)b(t)〈K 0|HW |f 〉〈f |HW |K 0〉+ a(t)b∗(t)〈K 0|HW |f 〉〈f |HW |K0〉

The first two terms can be identified as Γ11 and Γ22, whereas the twolatter ones are Γ12 and Γ21 = Γ∗12.




Diagonalizing the Hamiltonian leads to the physicaleigenstates

From the definition we see that M11, M22, Γ11 and Γ22 are real,whereas M12 and Γ12 are complex (and are the source of CPviolation)

CPT invariance requires M11 = M22 = M and Γ11 = Γ22 = Γ so theequation of motion simplifies to

i∂

∂t

(a(t)|K 0〉b(t)|K 0〉

)=

(M − i

2 Γ M12 − i2 Γ12

M∗12 − i2 Γ∗12 M − i

2 Γ

)(a(t)|K 0〉b(t)|K 0〉

)We may solve the eigenvalues of the Hamiltonian giving

E± = M − i

2Γ±

√(M∗12 −

i

2Γ∗12

)(M12 −

i

2Γ12

),

the real parts giving the masses and the imaginary parts giving thedecay widths




The eigenstates differ slightly from |K1,2〉The eigenvectors of the Hamiltonian (i.e. the stationary states) can

be solved and they are 1√1+|ξ|2

(|K 0〉+ ξ|K 0〉 and

1√1+|ξ|2

(|K 0〉 − ξ|K 0〉, where

ξ =

(M∗12 − i

2 Γ∗12

M12 − i2 Γ12

)1/2

If M12 and Γ12 were real, ξ = 1 and the eigenstates would coincidewith |K1,2〉, however they are complex, which leads to CP violation

Rewriting ξ = 1−ε1+ε leads to eigenstates

|KS〉 =1√

1 + |ε|2(|K1〉+ ε|K2〉)

|KL〉 =1√

1 + |ε|2(|K2〉+ ε|K1〉)




The kaon eigenstates have a tiny mass difference

From the eigenvalues we get

E+ − E− = 2√(

M∗12 − i2 Γ∗12

) (M12 − i

2 Γ12

)We may write E± = M ±∆M/2− i

2 (Γ±∆Γ/2), where∆M = |<(E+ − E−)| and ∆Γ = ±|2=(E+ − E−)|, where the sign of∆Γ depends on the experimental data on the lifetimes

Experimentally the heavier state has the longer lifetime, so it isassociated with |KL〉The states KS and KL have a tiny mass difference∆mK = 3.5× 10−15 GeV (we will see soon, how can this bemeasured — the masses are not known to this precision)




The kaons oscillate between K 0 and K0

and relax to astate with equal weights

Kaons are produced in flavor eigenstates, but since they are not theeigenstates of the Hamiltonian, they have a nontrivial timeevolution, which leads to flavor oscillationsThe oscillations are due to the box diagrams with two W -bosonsSince the KS component decays faster, the oscillations will

eventually end as the KL has fixed proportions of K 0 and K0

Measuring flavor oscillations is possible because the semileptonicdecays show the flavor of the kaon: Only K 0 → π−e+νe and

K0 → π+e−νe are possible, so the charge of the electron (or pion)

tells us, which flavor the kaon had




The time evolution is dictated by the CP eigenstates

We’ll first neglect CP violation as it is a small effect. Look at a stateproduced as K 0:

The initial state is |K 0〉 = 1√2

(|KS〉+ |KL〉)This evolves in time as|K (t)〉 = 1√

2(θS |KS〉+ θL|KL〉) = 1

2 (θS + θL)|K 0〉+ 12 (θS − θL)|K 0〉

Here θS,L = e−imS,Lt−ΓS,Lt/2 are the phase factors for decayingeigenstates of the Hamiltonian

The probability of observing a kaon as K 0 isP(K 0

t=0 → K 0t ) = |〈K 0|K (t)〉|2 = 1

4 |θS + θL|2

Similarly the probability of observing a K0

is

P(K 0t=0 → K

0

t ) = |〈K 0|K (t)〉|2 = 14 |θS − θL|

2




The oscillation probability is sensitive to the massdifference

Expanding these gives |θS ± θL|2 = |θS |2︸︷︷︸=e−ΓS t

+ |θL|2︸︷︷︸=e−ΓLt

±2<(θSθ∗L)

The last term is 2<(e−(ΓS+ΓL)t/2−i∆mK t) = 2e−(ΓS+ΓL)t/2 cos(∆mK t)Hence the oscillation period is sensitive to the kaon mass difference(like neutrino oscillations), although the amplitude is decayingrapidlyThe mass difference is so small that the oscillation period is longerthan the KS lifetime so only the first oscillation is somewhat visible




The CPLEAR experiment measured kaon flavor oscillations

Kaons were produced by pp → π+K−K 0, π−K+K0, the kaon charge

gives the neutral kaon flavor

Figure: The CPLEAR collaboration, Phys. Rept. 403 (2004) 303




Cerenkov counters were used to identify kaons

The kinematics were such that pions, electrons and muons gave aCerenkov signal, while kaons did not — helps in triggering andparticle identificationThe low energy meant that the neutral kaons decayed within thedetector, their flight distance indicating the lifetimeThe events of interest were the semileptonic decays of kaons, wherethe charge of the lepton gave the flavor of the kaon




Many uncertainties cancel when the relative difference inrates is considered

The probability of K 0 decaying as a K 0 is measured from eventswith a K− and e+ in the final state, this will be proportional toe−ΓS t + e−ΓLt + 2e−(ΓS+ΓL)t/2 cos(∆mK t)

The probability of K 0 oscillating to a K0

before decaying ismeasured from events with a K− and e− in the final state, which isproportional to e−ΓS t + e−ΓLt − 2e−(ΓS+ΓL)t/2 cos(∆mK t)

When considering the ratio

A(t) =P(K 0

t=0 → K 0t )− P(K 0

t=0 → K0

t )

P(K 0t=0 → K 0

t ) + P(K 0t=0 → K

0

t )

uncertainties related to the production rate cancel, it is possible to

include also the similar rates for K0




The asymmetry in the decays allows us to measure ∆mK

Plugging in the expressions for the various rates leads to theexpression

A(t) =2e−(ΓS+ΓL)t/2 cos(∆mK t)

e−ΓS t + e−ΓLt

for the asymmetry in the kaon decaysThe measured asymmetry is consistent with∆mK = 3.5× 10−15 GeV, whereas the masses themselves have beenmeasured with a precision of 10−5 GeV




CP violation can be seen in semileptonic decays of KL

We may observe CP violation in the semileptonic decays if we lookat events far from the interaction point, where the state is a pure|KL〉 state

Since |KL〉 = 1√2(1+|ε|2)

[(1 + ε)|K 0〉 − (1− ε)|K 0〉], there will be a

slight difference in the rates of the semileptonic decays

We have Γ(KL → π±e∓ν) ∝ |1∓ ε|2 ' 1∓ 2<(ε)

This allows the measurement of CP violation in terms of theasymmetry parameter (writing ε = |ε|e iφ)

δ =Γ(KL → π−e+νe)− Γ(KL → π+e−νe)

Γ(KL → π−e+νe) + Γ(KL → π+e−νe)' 2<(ε) = 2|ε| cosφ

Experimentally we have δ = 0.327± 0.012% showing again that CPis violated in weak interactions (|ε| & 1.6× 10−3)

Next time: How to measure the full ε and not just its real part




Summary

Kaons are usually produced as flavor eigenstates, they propagatenearly as CP eigenstates (modulo small CP violation) and they maydecay as either of them

Most of the CP violation in the neutral kaon system comes from the

K 0–K0

mixing through the box diagrams

The flavor oscillations can be seen in semileptonic decays and theyare sensitive to the tiny mass difference between KS and KL

Also CP violation can be seen in the semileptonic decays of kaons


Documents

Kaon physics - or an introduction to flavor oscillations