CP Violation and Arrows of Time:
Evolution of a Neutral K or B Meson from an
Incoherent to a Coherent State
Ch. Berger
I. Physikalisches Institut der RWTH, Aachen, Germany
L. M. Sehgal∗
Institut für Theoretische Physik (E) der RWTH, Aachen, Germany
October 24, 2018
Abstract
We study the evolution of a neutral K meson prepared as an in-
coherent equal mixture of K0 and K̄0. Denoting the density matrix
by ρ(t) = 1
1+ ~ζ(t) · ~σ
, the norm of the state N(t) is found
to decrease monotonically from one to zero, while the magnitude of
the Stokes vector |~ζ(t)| increases monotonically from zero to one. This
property qualifies these observables as arrows of time. Requiring mono-
tonic behaviour of N(t) for arbitrary values of γL, γS and ∆m yields a
bound on the CP-violating overlap δ = 〈KL |KS 〉, which is similar to,
but weaker than, the known unitarity bound. A similar requirement
on |~ζ(t)| yields a new bound, δ2 < 1
which is
particularly effective in limiting the CP-violating overlap in the B0-B̄0
system. We obtain the Stokes parameter ζ3(t) which shows how the
average strangeness of the beam evolves from zero to δ. The evolu-
tion of the Stokes vector from |~ζ| = 0 to |~ζ| = 1 has a resemblance
to an order parameter of a system undergoing spontaneous symmetry
breaking.
1 Introduction
We examine in this paper the time evolution of a neutral K meson prepared
as an equal incoherent mixture of K0 and K̄0. Such a state is easily obtained
in a reaction such as e+e− → φ(1020) → K0K̄0, when only one of the kaons
in the final state is observed. (Our considerations apply equally to B mesons
e-mail: sehgal@physik.rwth-aachen.de
http://arxiv.org/abs/0704.1232v2
produced in e+e− → Υ(4s) → B0B̄0.) An incoherent beam of this type is
characterized by a density matrix, which we write in the K0-K̄0 basis as
ρ(t) =
1+ ~ζ(t) · ~σ
The evolution is described by a normalization function N(t), which is the
intensity of the beam at time t, and a Stokes vector ~ζ which characterizes
the polarization state of the system with respect to strangeness. The beam,
which has |~ζ(0)| = 0 at the time of production evolves ultimately into a pure
state corresponding to the long-lived K meson KL, with a Stokes vector of
unit length: |~ζ(∞)| = 1.
In this sense, the system can be regarded as possessing two dynamical func-
tions: N(t) which varies from one to zero, and |~ζ(t)| which goes from zero
to one. This evolution touches on interesting issues such as the role of
CP-violation, and the extent to which the functions N(t) and |~ζ(t)| define
arrows of time. The requirement that these functions are monotonic yields
constraints on the CP-violating parameter δ = 〈KL |KS 〉. The fact that
the incoherent initial state is completely neutral with respect to strangeness
and CP quantum numbers is of significance in this regard. In addition the
component ζ3(t) of the Stokes vector describes the manner in which the
strangeness of the state evolves from zero to final value δ = 3.27× 10−3 and
serves as a model for flavour-genesis induced by CP violation in a decaying
system. Finally, the evolution of the system from an initial “amorphous”
state with ~ζ(0) = 0 to a final “crystalline” state described by a three-
dimensional Stokes vector ~ζ(t) with unit length, is suggestive of a phase
transition, with ~ζ(t) playing the role of an order parameter of a system
undergoing spontaneous symmetry breaking.
2 Density Matrix
An arbitrary state of the K meson can be desccribed by a 2 x 2 density
matrix which we write, in the K0-K̄0 basis, as [1, 2, 3]
ρ(t) =
1+ ~ζ(t) · ~σ
Here N(t) is the intensity or norm of the state at time t, calculated from
the trace of ρ,
N(t) = tr ρ(t) (2)
and ~ζ(t) is the Stokes vector, whose components can be expressed as
ζi(t) = tr [ρ(t)σi] /trρ(t) (3)
An initial state which is a 1:1 incoherent mixture of K0 and K̄0 has the
density matrix
ρ(t) =
|K0 〉〈K0 |+
| K̄0 〉〈 K̄0 |
which corresponds to an initial Stokes vector ~ζ(0) = 0. To determine the
time evolution, we note that [4]
|K0 〉 →
|ψ(t) 〉 =
|KS 〉e
−λSt + |KL 〉e
| K̄0 〉 →
| ψ̄(t) 〉 =
|KS 〉e
−λSt − |KL 〉e
where we have introduced the eigenstates
|KL 〉 = p|K
0 〉 − q| K̄0 〉
|KS 〉 = p|K
0 〉+ q| K̄0 〉
(|p|2 + |q|2 = 1)
with eigenvalues
λL,S =
γL,S + imL,S (7)
The overlap of the states |KL 〉 and |KS 〉 is given by the CP-violating
parameter
δ = 〈KL |KS 〉 = (|p|
2 − |q|2)/(|p|2 + |q|2) = 3.27 × 10−3 (8)
The resulting density matrix at time t is
ρ(t) =
ρ11(t) ρ12(t)
ρ21(t) ρ22(t)
ρ11(t) =
4(1− δ)
e−γSt + e−γLt − 2δe−
(γS+γL)t cos∆mt
ρ22(t) =
4(1 + δ)
e−γSt + e−γLt + 2δe−
(γS+γL)t cos∆mt
ρ12(t) =
e−γSt − e−γLt + i2δe−
(γS+γL)t sin∆mt
ρ21(t) = ρ
12(t) (10)
Using Eqs. (2) and (3), we derive
N(t) =
2(1 − δ2)
e−γSt + e−γLt − 2δ2e−
(γS+γL)t cos∆mt
ζ1(t) =
Re (pq∗)
e−γSt − e−γLt
− Im (pq∗) 2δe−
(γS+γL)t · sin∆mt
e−γSt + e−γLt − 2δ2e−
(γS+γL)t cos∆mt
ζ2(t) =
Im (pq∗)
e−γSt − e−γLt
+Re (pq∗) 2δe−
(γS+γL)t · sin∆mt
e−γSt + e−γLt − 2δ2e−
(γS+γL)t cos∆mt
ζ3(t) = δ
e−γSt + e−γLt − 2e−
(γS+γL)t cos∆mt
e−γS t + e−γLt − 2δ2e−
(γS+γL)t cos∆mt
. (12)
Note that the components ζ1,2(t) involve Re (pq
∗) and Im (pq∗) where p
and q are the coefficients in the definition of KL,S in eq.(6). These are
convention-dependent, since the relative phase of p and q can be changed by
a phase transformation |K0 〉 → eiα|K0 〉, | K̄0 〉 → e−iα| K̄0 〉. A quantity
independent of phase convention is
ζ21 + ζ
2 = (1− δ
e−γSt − e−γLt
+ 4δ2e−(γS+γL)t sin2 ∆mt
e−γSt + e−γLt − 2δ2e−
(γS+γL)t cos∆mt
Thus the length of the Stokes vector is
|~ζ(t)| =
ζ21 (t) + ζ
2 (t) + ζ
3 (t)
2N(t)(1 − δ2)
e−γSt + e−γLt − 2e−
(γS+γL)t cos∆mt
+ (1− δ2)
e−γSt − e−γLt
+ 4δ2e−
(γS+γL)t · sin2∆mt
N(t)2
e−(γS+γL)t
This equation provides a simple relation between the magnitude of the
Stokes vector |~ζ(t)| and the normalization function N(t).
In the CP-invariant limit, δ → 0, the density matrix reduces to
ρ(t) −→
e−γSt + e−γLt e−γSt − e−γLt
e−γSt − e−γLt e−γSt + e−γLt
and the limiting form of N(t) and ~ζ(t) is
2 4 6 8 10 12 14 16
0 2 4 6 8 10 12 14 16
dN/dt
Figure 1: Normalization function N(t) (a) and its time derivative (b) as function
of time in units of τS for a state prepared as incoherent mixture of K
0 and K̄0.
N(t) −→
e−γSt + e−γLt
ζ12 ≡
ζ21 (t) + ζ
2 (t)
e−γLt − e−γSt
e−γSt + e−γLt
ζ3(t) −→ 0 (16)
The behaviour of N(t) and dN/dt for the K0-K̄0 system is shown in fig.1.
The behaviour of the functions ζ212(t), ζ
3 (t) and ζ
2(t) = ζ212(t) + ζ
3 (t) and
their time derivatives is shown in fig.2. The function ζ23 is clearly nonmono-
tonic, and its derivative has a number of zeros (e.g. t/τS = 4.95, 11.6, 18.2, ...).
By comparison the derivative of ζ212 has a distant zero at t/τS = 25.7.
As seen in fig.2b these two nonmonotonic functions combine to produce a
Stokes vector ζ2(t) which is strictly monotonic, the asymptotic values being
ζ212 → (1− δ
2), ζ23 → δ
2, ζ2 → 1.
3 Arrows of time
3.1 The Normalization Arrow N(t)
The normalization of the kaon state is given in eq.(11). As seen in fig.1, this
function is indeed monotonic for the parameters of the K meson system.
This monotonic (unidirectional) property implies that N(t) behaves as an
arrow of time. In the absence of CP-violation (δ = 0), the function N(t) is
simply the sum of two exponentials (e−γSt + e−γLt)/2, and the monotonic
decrease is ensured by the requirement γS, γL > 0 (positivity of the decay
matrix). The third term in eq.(11), appearing when δ 6= 0, indicates a KL-
KS interference effect. It implies that an incoherent K
0-K̄0 mixture does
2 4 6 8 10 12 14 16
|ζ |2
· 1/2
· 20000
–5e–11
5e–11
1e–10
1.5e–10
2e–10
24 26 28 30 32 34 36
|ζ̇|2
Figure 2: Behaviour of |ζ|2, ζ2
and ζ2
(a) and their time derivatives (b) as function
of time. Note the different scale for ζ2
and ζ2
in (a). In (b) only the tail of the
time dependence is shown.
not evolve like an incoherent KL-KS mixture. Notice however, that the
coefficient of the interference term is quadratic in δ, so that the function
N(t) is CP-even, remaining unchanged under δ → −δ. Nevertheless the
presence of the δ2 term is decisive in determining whether or not N(t) is
monotonic, and hence an arrow of time. If we require the function N(t) to
be monotonic (dN/dt < 0) then we have from (11) (see also [5]) that
2(1 − δ2)
−γSt + γLe
−γLt − 2δ2e−
(γS+γL)t
γS + γL
cos∆mt+∆m sin∆mt
< 0 (17)
from which it follows, as sufficient condition, that
(γS + γL)
/4 + ∆m2
or δ2 ≤
(1 + r)
/4 + µ2
where we have introduced the notation r = γL/γS , µ = ∆m/γS . This con-
straint is analogous to, but weaker than, the unitarity constraint derived
in [6, 7], which reads
δ2unit ≤
(1 + r)
/4 + µ2
. (19)
2 4 6 8 10 12 14 16
0 2 4 6 8 10 12 14 16
dN/dt
Figure 3: Behaviour of N(t) (a) and dN/dt (b) for values of parameters µ =
µK , r = 0.01, δ = 0.6 violating the monotonicity bound (18).
It is clear that the unitarity bound is interesting for a system like K0-K̄0,
where r = γL/γS is small, but does not provide a useful constraint for B
0-B̄0,
where r is close to 1.
To see what happens if the parameters δ, r and µ are allowed to vary, we
show in fig.3 the behaviour of N(t) and dN/dt for δ = 0.6, r = 0.01, keep-
ing µ at its standard K meson value, µK = 0.47. The function N(t) shows
fluctuations, and the derivative dN/dt changes sign. Such a behaviour re-
sults from the violation of the bound (18). The fluctuations in N(t) may
be regarded as fluctuations in the direction of the time arrow (we call this
phenomenon “Zeitzitter”), and can occur when the CP-violating parameter
δ2 exceeds the limit (18). This is the manner in which CP violation impacts
on the time arrow, even though the function N(t) is CP-even.
From the point of view of an observer monitoring the intensity of the kaon
beam (for example by measuring the rate of leptonic decays π∓ℓ±ν) the fluc-
tuation in N would appear as an inexplicable enhancement or suppression
of the beam intensity in certain intervals of time. The effect can be regarded
equivalently as violation of unitarity or a flutter in the arrow of time.
3.2 The Stokes Arrow
The magnitude of the Stokes vector |~ζ(t)|2 calculated in eq.(14), is a measure
of the coherence of the state, and is plotted in fig.2 for the physical K-meson
parameters. One sees that the function |~ζ|2 evolves monotonically from 0 to
1, and its derivative remains positive at all times. Thus the Stokes parameter
|~ζ(t)| qualifies as an arrow of time. To see how this arrow is affected if the
parameters δ, r and µ are allowed to vary, we look at the derivative of the
function ζ(t). Writing
|~ζ(t)| =
e−(γS+γL)t
N(t)2
we find that the monotonicity condition d|~ζ(t)|/dt > 0 is equivalent to the
condition
(γS + γL)N
≤ 0 (21)
which implies
e∆γt/2 − e−∆γt/2
+ 4δ2
sin∆mt ≥ 0 (22)
where ∆γ = γS − γL. From this we derive a new upper bound on δ
or δ2 <
(1− r)
This bound is obtained from the requirement that |~ζ(t)| be monotonic (an
arrow of time) just as the bound in eq.(18) was derived from the mono-
tonicity of N(t). The bound (23) is particularly effective in constraining
the value of the overlap parameter in the B0-B̄0 system, in which the decay
widths of the two eigenstates are close together, r → 1. In this respect the
bound in eq.(23) is complementary to the unitarity bound in eq.(19) which
is effective when r → 0. The contrast between the two bounds is highlighted
in fig.4. Taking the parameters of the B0-B̄0 system to be r = 0.99, µ = 0.7,
we obtain from (23)
δB = 〈B
L 〉 . 0.0155 . (24)
We wish to stress that for a B0-like system the bound in (23) is not just a
sufficient condition for monotonic behaviour of |ζ(t)|2, but almost a critical
value separating the monotonic and nonmonotonic domains. As an illustra-
tion we show in fig.5 the transition in the behaviour of |ζ(t)|2 for a system
with parameters µ = 0.7, r = 0.9, as δ is varied from a value 0.1, below the
critical value of δcrit = 0.156, to a value 0.2 above δcrit.
The fluctuations in |ζ(t)|2, shown in fig.5 are the analog of the fluctuations
in N(t), shown in fig.3, which arise when the parameters of the system vi-
olate the bound in eq.(18). Whereas the fluctuation in N(t) would reveal
itself as an inexplicable Zitter in the beam intensity, the fluctuation in |~ζ(t)|
would show up as an unaccountable Zitter in the coherence of the beam.
0.2 0.4 0.6 0.8 1
constraints on δ
Figure 4: Constraints on δ in the δ-r plane resulting from unitarity and monotoni-
city of |ζ(t)| for a B0-like system with µ = 0.7. The thick line represents the
unitarity bound (19) and the thin line our new bound evaluated from (23). The
numerical evaluation of (22) (open circles) yields values very close to the approxi-
mation given in eq.(23).
In both cases, the effect results from a breakdown in the monotonicity of a
function, associated with a loss of directionality in an arrow of time.
4 Evolution of Strangeness
The component ζ3(t) of the Stokes vector has a special significance: it is
the expectation value of σ3, which can be identified with the strangeness
operator with eigenvalues +1 for K0 and −1 for K̄0. Thus a measurement
of ζ3(t) is simply a measurement of the decay asymmetry into the channels
π−ℓ+ν and π+ℓ−ν̄ :
ζ3(t) =
Γ (π−ℓ+ν; t)− Γ (π+ℓ−ν̄; t)
Γ (π−ℓ+ν; t) + Γ (π+ℓ−ν̄; t)
2 4 6 8 10 12 14 16
|ζ(t)|2
Figure 5: Evolution of the Stokes vector |ζ(t)|2 for a B0-like system with para-
meters µB = 0.7, r = 0.9. The middle curve corresponds to δcrit = 0.156 obtained
from the bound (23). The nonmonotonic upper curve is obtained for δ = 0.2 and
the monotonic lower curve for δ = 0.1.
Referring to eq.(12), we observe that ζ3(t) is a pure CP-violating observable,
since it changes sign under δ → −δ (By contrast, the functions N(t) and
|~ζ(t)|, are invariant under δ → −δ). Writing ζ3(t) explicitly as
ζ3(t) = δ
e−γSt + e−γLt − 2e−
(γL+γS)t cos∆mt
2 (1− δ2)N(t)
we note that it is a quotient of a function that contains an oscillating term
and a monotonic function N(t). The average strangeness ζ3(t) is thus clearly
not a monotonic function of time. This is visible in fig.6, where we also
show the derivative dζ3(t)/dt. We have here an explicit example of a CP-
odd observable emerging from an initial state that has no preferred CP
direction. Such observables are not monotonic, and cannot be associated
with an arrow of time.
0.001
0.002
0.003
0.004
2 4 6 8 10 12 14 16
–0.0002
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0 2 4 6 8 10 12 14 16
dζ3/dt
Figure 6: Evolution of ζ3(t) (a) and its derivative (b) in the K
0-K̄0 system.
5 Summary
(1) We have shown that the evolution of an incoherent K0-K̄0 mixture
is characterized by two time-dependent functions, the norm N(t) and
the magnitude of the Stokes vector |~ζ(t)| both of which evolve mono-
tonically and may therefore be associated with microscopic arrows of
time. It should be stressed that we are discussing here conditional ar-
rows of time, whose existence depends on the degree of CP violation,
and not simply on the positivity of the decay widths γL,S.
(2) If the parameters γL, γS ,∆m and δ = 〈KL |KS 〉 are allowed to vary,
the requirement of monotonic behaviour of N(t) leads to the bound
in eq.(18), which is similar to, but weaker than, the unitarity bound
(19), derived in [6, 7]. The requirement of monotonicity for |~ζ(t)|
leads to a new bound on δ2 given in eq.(23), which is complementary
to the unitarity bound (19), and far more restrictive for systems such
as B0-B̄0 with r = γL/γS close to unity.
(3) A violation of the bounds in eq.(18) and (23) leads to fluctuations
in N(t) and ζ(t) associated with fluctuations in the arrow of time
(“Zeitzitter”) and a violation of unitarity.
(4) It is worth noting that the product N2(1− |~ζ|2) is equal to e−(γL+γS)t
and therefore monotonic for all values of δ, r and µ. This product is
just four times the determinant of the density matrix ρ(t),
(5) The time-dependence of ζ3(t) describes the evolution of strangeness
in a beam that is initially an equal mixture of K0 and K̄0. It is
an example of flavour-genesis induced by CP violation in a decaying
system.
(6) The emergence of a non-zero three-dimensional Stokes vector ~ζ(t) from
a state that is initially “amorphous” (~ζ(0) = 0), is suggestive of a
phase transition. The evolution of the Stokes vector from zero to unit
length is reminiscent of an order parameter for a system undergoing
spontaneous symmetry breaking.
(7) All our considerations have been in the framework of ordinary quan-
tum mechanics and CPT invariance. Discussions that involve violation
of quantum mechanics and/or CPT symmetry may be found, for ex-
ample, in [8]. An early discussion of the arrow of time in connection
with K meson decays is given in [9]. Broader issues connected with the
arrow of time are discussed, for instance, in [10]. Finally, experimental
investigations of discrete symmetries in the decays of K mesons and
B mesons produced in e+e− or pp̄ collisions are described in [11].
Acknowledgement: One of us (LMS) wishes to thank Dagmar Bruss
(University of Düsseldorf) for a useful correspondence.
References
[1] U. Fano, Rev. Mod. Phys. 29, 74 (1957)
[2] R. G. Sachs, Physics of Time Reversal, University of Chicago Press,
[3] L. M. Sehgal Density Matrix Description of Neutral K Meson Decay,
Tata Institute report, TIFR - TH- 70-35 (1970)
[4] T. D. Lee, R. Oehme and C. N. Yang, Phys. Rev. 105, 1671 (1957)
[5] L. M. Sehgal, Decays of Neutral K Mesons Produced in pp̄ Annihilation.
A Comment, Aachen preprint, 1973 (unpublished)
[6] T. D. Lee and L. Wolfenstein, Phys. Rev. 138, B1490 (1965)
[7] J. S. Bell and J. Steinberger, in Proc. Oxford International Conference
on Elementary Particles, 1965, pp. 195-222
[8] P. Huet, M. E. Peskin, Nucl. Phys. B 434, 3 (1995)
J. Ellis, N. E. Mavromatos and D. V. Nanopoulos, Phys. Lett. B 293,
142 (1992)
[9] A Aharony, Ann. Phys. 67, 1 (1971); ibid 68, 163 (1971);
A. Aharony and Y. Ne’eman, Int. J. Theor. Phys. 3, 437 (1970)
[10] Physical Origins of Time Asymmetry, Eds. Jose Angel Sanchez Asiain,
et al, Cambridge University Press (1996)
[11] KLOE Collaboration, F. Ambrosino et al, Phys.Lett. B642, 315 (2006)
Babar Collaboration (B.Aubert et al), Phys.Rev.Lett. 96, 251802
(2006)
Belle Collaboration (E.Nakano et al), Phys.Rev. D73, 112002 (2006)
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(1997);
K. Kleinknecht Uncovering CP Violation, Springer-Verlag, Berlin, Hei-
delberg (2003).
	Introduction
	Density Matrix
	Arrows of time
	The Normalization Arrow N(t)
	The Stokes Arrow
	Evolution of Strangeness
	Summary

  We study the evolution of a neutral $K$ meson prepared as an incoherent equal
mixture of $K^0$ and $\bar{K^0}$. Denoting the density matrix by $\rho(t) =
{1/2} N(t) [\1 + \vec{\zeta}(t) \cdot \vec{\sigma} ] $, the norm of the state
$N(t)$ is found to decrease monotonically from one to zero, while the magnitude
of the Stokes vector $|\vec{\zeta}(t)|$ increases monotonically from zero to
one. This property qualifies these observables as arrows of time. Requiring
monotonic behaviour of $N(t)$ for arbitrary values of $\gamma_L, \gamma_S$ and
$\Delta m$ yields a bound on the CP-violating overlap $\delta =
\braket{K_L}{K_S}$, which is similar to, but weaker than, the known unitarity
bound. A similar requirement on $|\vec{\zeta}(t)|$ yields a new bound,
$\delta^2 < {1/2} (\frac{\Delta \gamma}{\Delta m}) \sinh (\frac{3\pi}{4}
\frac{\Delta \gamma}{\Delta m})$ which is particularly effective in limiting
the CP-violating overlap in the $B^0$-$\bar{B^0}$ system. We obtain the Stokes
parameter $\zeta_3(t)$ which shows how the average strangeness of the beam
evolves from zero to $\delta$. The evolution of the Stokes vector from
$|\vec{\zeta}| = 0$ to $|\vec{\zeta}| = 1$ has a resemblance to an order
parameter of a system undergoing spontaneous symmetry breaking.

Introduction
We examine in this paper the time evolution of a neutral K meson prepared
as an equal incoherent mixture of K0 and K̄0. Such a state is easily obtained
in a reaction such as e+e− → φ(1020) → K0K̄0, when only one of the kaons
in the final state is observed. (Our considerations apply equally to B mesons
e-mail: sehgal@physik.rwth-aachen.de
http://arxiv.org/abs/0704.1232v2
produced in e+e− → Υ(4s) → B0B̄0.) An incoherent beam of this type is
characterized by a density matrix, which we write in the K0-K̄0 basis as
ρ(t) =
1+ ~ζ(t) · ~σ
The evolution is described by a normalization function N(t), which is the
intensity of the beam at time t, and a Stokes vector ~ζ which characterizes
the polarization state of the system with respect to strangeness. The beam,
which has |~ζ(0)| = 0 at the time of production evolves ultimately into a pure
state corresponding to the long-lived K meson KL, with a Stokes vector of
unit length: |~ζ(∞)| = 1.
In this sense, the system can be regarded as possessing two dynamical func-
tions: N(t) which varies from one to zero, and |~ζ(t)| which goes from zero
to one. This evolution touches on interesting issues such as the role of
CP-violation, and the extent to which the functions N(t) and |~ζ(t)| define
arrows of time. The requirement that these functions are monotonic yields
constraints on the CP-violating parameter δ = 〈KL |KS 〉. The fact that
the incoherent initial state is completely neutral with respect to strangeness
and CP quantum numbers is of significance in this regard. In addition the
component ζ3(t) of the Stokes vector describes the manner in which the
strangeness of the state evolves from zero to final value δ = 3.27× 10−3 and
serves as a model for flavour-genesis induced by CP violation in a decaying
system. Finally, the evolution of the system from an initial “amorphous”
state with ~ζ(0) = 0 to a final “crystalline” state described by a three-
dimensional Stokes vector ~ζ(t) with unit length, is suggestive of a phase
transition, with ~ζ(t) playing the role of an order parameter of a system
undergoing spontaneous symmetry breaking.
2 Density Matrix
An arbitrary state of the K meson can be desccribed by a 2 x 2 density
matrix which we write, in the K0-K̄0 basis, as [1, 2, 3]
ρ(t) =
1+ ~ζ(t) · ~σ
Here N(t) is the intensity or norm of the state at time t, calculated from
the trace of ρ,
N(t) = tr ρ(t) (2)
and ~ζ(t) is the Stokes vector, whose components can be expressed as
ζi(t) = tr [ρ(t)σi] /trρ(t) (3)
An initial state which is a 1:1 incoherent mixture of K0 and K̄0 has the
density matrix
ρ(t) =
|K0 〉〈K0 |+
| K̄0 〉〈 K̄0 |
which corresponds to an initial Stokes vector ~ζ(0) = 0. To determine the
time evolution, we note that [4]
|K0 〉 →
|ψ(t) 〉 =
|KS 〉e
−λSt + |KL 〉e
| K̄0 〉 →
| ψ̄(t) 〉 =
|KS 〉e
−λSt − |KL 〉e
where we have introduced the eigenstates
|KL 〉 = p|K
0 〉 − q| K̄0 〉
|KS 〉 = p|K
0 〉+ q| K̄0 〉
(|p|2 + |q|2 = 1)
with eigenvalues
λL,S =
γL,S + imL,S (7)
The overlap of the states |KL 〉 and |KS 〉 is given by the CP-violating
parameter
δ = 〈KL |KS 〉 = (|p|
2 − |q|2)/(|p|2 + |q|2) = 3.27 × 10−3 (8)
The resulting density matrix at time t is
ρ(t) =
ρ11(t) ρ12(t)
ρ21(t) ρ22(t)
ρ11(t) =
4(1− δ)
e−γSt + e−γLt − 2δe−
(γS+γL)t cos∆mt
ρ22(t) =
4(1 + δ)
e−γSt + e−γLt + 2δe−
(γS+γL)t cos∆mt
ρ12(t) =
e−γSt − e−γLt + i2δe−
(γS+γL)t sin∆mt
ρ21(t) = ρ
12(t) (10)
Using Eqs. (2) and (3), we derive
N(t) =
2(1 − δ2)
e−γSt + e−γLt − 2δ2e−
(γS+γL)t cos∆mt
ζ1(t) =
Re (pq∗)
e−γSt − e−γLt
− Im (pq∗) 2δe−
(γS+γL)t · sin∆mt
e−γSt + e−γLt − 2δ2e−
(γS+γL)t cos∆mt
ζ2(t) =
Im (pq∗)
e−γSt − e−γLt
+Re (pq∗) 2δe−
(γS+γL)t · sin∆mt
e−γSt + e−γLt − 2δ2e−
(γS+γL)t cos∆mt
ζ3(t) = δ
e−γSt + e−γLt − 2e−
(γS+γL)t cos∆mt
e−γS t + e−γLt − 2δ2e−
(γS+γL)t cos∆mt
. (12)
Note that the components ζ1,2(t) involve Re (pq
∗) and Im (pq∗) where p
and q are the coefficients in the definition of KL,S in eq.(6). These are
convention-dependent, since the relative phase of p and q can be changed by
a phase transformation |K0 〉 → eiα|K0 〉, | K̄0 〉 → e−iα| K̄0 〉. A quantity
independent of phase convention is
ζ21 + ζ
2 = (1− δ
e−γSt − e−γLt
+ 4δ2e−(γS+γL)t sin2 ∆mt
e−γSt + e−γLt − 2δ2e−
(γS+γL)t cos∆mt
Thus the length of the Stokes vector is
|~ζ(t)| =
ζ21 (t) + ζ
2 (t) + ζ
3 (t)
2N(t)(1 − δ2)
e−γSt + e−γLt − 2e−
(γS+γL)t cos∆mt
+ (1− δ2)
e−γSt − e−γLt
+ 4δ2e−
(γS+γL)t · sin2∆mt
N(t)2
e−(γS+γL)t
This equation provides a simple relation between the magnitude of the
Stokes vector |~ζ(t)| and the normalization function N(t).
In the CP-invariant limit, δ → 0, the density matrix reduces to
ρ(t) −→
e−γSt + e−γLt e−γSt − e−γLt
e−γSt − e−γLt e−γSt + e−γLt
and the limiting form of N(t) and ~ζ(t) is
2 4 6 8 10 12 14 16
0 2 4 6 8 10 12 14 16
dN/dt
Figure 1: Normalization function N(t) (a) and its time derivative (b) as function
of time in units of τS for a state prepared as incoherent mixture of K
0 and K̄0.
N(t) −→
e−γSt + e−γLt
ζ12 ≡
ζ21 (t) + ζ
2 (t)
e−γLt − e−γSt
e−γSt + e−γLt
ζ3(t) −→ 0 (16)
The behaviour of N(t) and dN/dt for the K0-K̄0 system is shown in fig.1.
The behaviour of the functions ζ212(t), ζ
3 (t) and ζ
2(t) = ζ212(t) + ζ
3 (t) and
their time derivatives is shown in fig.2. The function ζ23 is clearly nonmono-
tonic, and its derivative has a number of zeros (e.g. t/τS = 4.95, 11.6, 18.2, ...).
By comparison the derivative of ζ212 has a distant zero at t/τS = 25.7.
As seen in fig.2b these two nonmonotonic functions combine to produce a
Stokes vector ζ2(t) which is strictly monotonic, the asymptotic values being
ζ212 → (1− δ
2), ζ23 → δ
2, ζ2 → 1.
3 Arrows of time
3.1 The Normalization Arrow N(t)
The normalization of the kaon state is given in eq.(11). As seen in fig.1, this
function is indeed monotonic for the parameters of the K meson system.
This monotonic (unidirectional) property implies that N(t) behaves as an
arrow of time. In the absence of CP-violation (δ = 0), the function N(t) is
simply the sum of two exponentials (e−γSt + e−γLt)/2, and the monotonic
decrease is ensured by the requirement γS, γL > 0 (positivity of the decay
matrix). The third term in eq.(11), appearing when δ 6= 0, indicates a KL-
KS interference effect. It implies that an incoherent K
0-K̄0 mixture does
2 4 6 8 10 12 14 16
|ζ |2
· 1/2
· 20000
–5e–11
5e–11
1e–10
1.5e–10
2e–10
24 26 28 30 32 34 36
|ζ̇|2
Figure 2: Behaviour of |ζ|2, ζ2
and ζ2
(a) and their time derivatives (b) as function
of time. Note the different scale for ζ2
and ζ2
in (a). In (b) only the tail of the
time dependence is shown.
not evolve like an incoherent KL-KS mixture. Notice however, that the
coefficient of the interference term is quadratic in δ, so that the function
N(t) is CP-even, remaining unchanged under δ → −δ. Nevertheless the
presence of the δ2 term is decisive in determining whether or not N(t) is
monotonic, and hence an arrow of time. If we require the function N(t) to
be monotonic (dN/dt < 0) then we have from (11) (see also [5]) that
2(1 − δ2)
−γSt + γLe
−γLt − 2δ2e−
(γS+γL)t
γS + γL
cos∆mt+∆m sin∆mt
< 0 (17)
from which it follows, as sufficient condition, that
(γS + γL)
/4 + ∆m2
or δ2 ≤
(1 + r)
/4 + µ2
where we have introduced the notation r = γL/γS , µ = ∆m/γS . This con-
straint is analogous to, but weaker than, the unitarity constraint derived
in [6, 7], which reads
δ2unit ≤
(1 + r)
/4 + µ2
. (19)
2 4 6 8 10 12 14 16
0 2 4 6 8 10 12 14 16
dN/dt
Figure 3: Behaviour of N(t) (a) and dN/dt (b) for values of parameters µ =
µK , r = 0.01, δ = 0.6 violating the monotonicity bound (18).
It is clear that the unitarity bound is interesting for a system like K0-K̄0,
where r = γL/γS is small, but does not provide a useful constraint for B
0-B̄0,
where r is close to 1.
To see what happens if the parameters δ, r and µ are allowed to vary, we
show in fig.3 the behaviour of N(t) and dN/dt for δ = 0.6, r = 0.01, keep-
ing µ at its standard K meson value, µK = 0.47. The function N(t) shows
fluctuations, and the derivative dN/dt changes sign. Such a behaviour re-
sults from the violation of the bound (18). The fluctuations in N(t) may
be regarded as fluctuations in the direction of the time arrow (we call this
phenomenon “Zeitzitter”), and can occur when the CP-violating parameter
δ2 exceeds the limit (18). This is the manner in which CP violation impacts
on the time arrow, even though the function N(t) is CP-even.
From the point of view of an observer monitoring the intensity of the kaon
beam (for example by measuring the rate of leptonic decays π∓ℓ±ν) the fluc-
tuation in N would appear as an inexplicable enhancement or suppression
of the beam intensity in certain intervals of time. The effect can be regarded
equivalently as violation of unitarity or a flutter in the arrow of time.
3.2 The Stokes Arrow
The magnitude of the Stokes vector |~ζ(t)|2 calculated in eq.(14), is a measure
of the coherence of the state, and is plotted in fig.2 for the physical K-meson
parameters. One sees that the function |~ζ|2 evolves monotonically from 0 to
1, and its derivative remains positive at all times. Thus the Stokes parameter
|~ζ(t)| qualifies as an arrow of time. To see how this arrow is affected if the
parameters δ, r and µ are allowed to vary, we look at the derivative of the
function ζ(t). Writing
|~ζ(t)| =
e−(γS+γL)t
N(t)2
we find that the monotonicity condition d|~ζ(t)|/dt > 0 is equivalent to the
condition
(γS + γL)N
≤ 0 (21)
which implies
e∆γt/2 − e−∆γt/2
+ 4δ2
sin∆mt ≥ 0 (22)
where ∆γ = γS − γL. From this we derive a new upper bound on δ
or δ2 <
(1− r)
This bound is obtained from the requirement that |~ζ(t)| be monotonic (an
arrow of time) just as the bound in eq.(18) was derived from the mono-
tonicity of N(t). The bound (23) is particularly effective in constraining
the value of the overlap parameter in the B0-B̄0 system, in which the decay
widths of the two eigenstates are close together, r → 1. In this respect the
bound in eq.(23) is complementary to the unitarity bound in eq.(19) which
is effective when r → 0. The contrast between the two bounds is highlighted
in fig.4. Taking the parameters of the B0-B̄0 system to be r = 0.99, µ = 0.7,
we obtain from (23)
δB = 〈B
L 〉 . 0.0155 . (24)
We wish to stress that for a B0-like system the bound in (23) is not just a
sufficient condition for monotonic behaviour of |ζ(t)|2, but almost a critical
value separating the monotonic and nonmonotonic domains. As an illustra-
tion we show in fig.5 the transition in the behaviour of |ζ(t)|2 for a system
with parameters µ = 0.7, r = 0.9, as δ is varied from a value 0.1, below the
critical value of δcrit = 0.156, to a value 0.2 above δcrit.
The fluctuations in |ζ(t)|2, shown in fig.5 are the analog of the fluctuations
in N(t), shown in fig.3, which arise when the parameters of the system vi-
olate the bound in eq.(18). Whereas the fluctuation in N(t) would reveal
itself as an inexplicable Zitter in the beam intensity, the fluctuation in |~ζ(t)|
would show up as an unaccountable Zitter in the coherence of the beam.
0.2 0.4 0.6 0.8 1
constraints on δ
Figure 4: Constraints on δ in the δ-r plane resulting from unitarity and monotoni-
city of |ζ(t)| for a B0-like system with µ = 0.7. The thick line represents the
unitarity bound (19) and the thin line our new bound evaluated from (23). The
numerical evaluation of (22) (open circles) yields values very close to the approxi-
mation given in eq.(23).
In both cases, the effect results from a breakdown in the monotonicity of a
function, associated with a loss of directionality in an arrow of time.
4 Evolution of Strangeness
The component ζ3(t) of the Stokes vector has a special significance: it is
the expectation value of σ3, which can be identified with the strangeness
operator with eigenvalues +1 for K0 and −1 for K̄0. Thus a measurement
of ζ3(t) is simply a measurement of the decay asymmetry into the channels
π−ℓ+ν and π+ℓ−ν̄ :
ζ3(t) =
Γ (π−ℓ+ν; t)− Γ (π+ℓ−ν̄; t)
Γ (π−ℓ+ν; t) + Γ (π+ℓ−ν̄; t)
2 4 6 8 10 12 14 16
|ζ(t)|2
Figure 5: Evolution of the Stokes vector |ζ(t)|2 for a B0-like system with para-
meters µB = 0.7, r = 0.9. The middle curve corresponds to δcrit = 0.156 obtained
from the bound (23). The nonmonotonic upper curve is obtained for δ = 0.2 and
the monotonic lower curve for δ = 0.1.
Referring to eq.(12), we observe that ζ3(t) is a pure CP-violating observable,
since it changes sign under δ → −δ (By contrast, the functions N(t) and
|~ζ(t)|, are invariant under δ → −δ). Writing ζ3(t) explicitly as
ζ3(t) = δ
e−γSt + e−γLt − 2e−
(γL+γS)t cos∆mt
2 (1− δ2)N(t)
we note that it is a quotient of a function that contains an oscillating term
and a monotonic function N(t). The average strangeness ζ3(t) is thus clearly
not a monotonic function of time. This is visible in fig.6, where we also
show the derivative dζ3(t)/dt. We have here an explicit example of a CP-
odd observable emerging from an initial state that has no preferred CP
direction. Such observables are not monotonic, and cannot be associated
with an arrow of time.
0.001
0.002
0.003
0.004
2 4 6 8 10 12 14 16
–0.0002
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0 2 4 6 8 10 12 14 16
dζ3/dt
Figure 6: Evolution of ζ3(t) (a) and its derivative (b) in the K
0-K̄0 system.
5 Summary
(1) We have shown that the evolution of an incoherent K0-K̄0 mixture
is characterized by two time-dependent functions, the norm N(t) and
the magnitude of the Stokes vector |~ζ(t)| both of which evolve mono-
tonically and may therefore be associated with microscopic arrows of
time. It should be stressed that we are discussing here conditional ar-
rows of time, whose existence depends on the degree of CP violation,
and not simply on the positivity of the decay widths γL,S.
(2) If the parameters γL, γS ,∆m and δ = 〈KL |KS 〉 are allowed to vary,
the requirement of monotonic behaviour of N(t) leads to the bound
in eq.(18), which is similar to, but weaker than, the unitarity bound
(19), derived in [6, 7]. The requirement of monotonicity for |~ζ(t)|
leads to a new bound on δ2 given in eq.(23), which is complementary
to the unitarity bound (19), and far more restrictive for systems such
as B0-B̄0 with r = γL/γS close to unity.
(3) A violation of the bounds in eq.(18) and (23) leads to fluctuations
in N(t) and ζ(t) associated with fluctuations in the arrow of time
(“Zeitzitter”) and a violation of unitarity.
(4) It is worth noting that the product N2(1− |~ζ|2) is equal to e−(γL+γS)t
and therefore monotonic for all values of δ, r and µ. This product is
just four times the determinant of the density matrix ρ(t),
(5) The time-dependence of ζ3(t) describes the evolution of strangeness
in a beam that is initially an equal mixture of K0 and K̄0. It is
an example of flavour-genesis induced by CP violation in a decaying
system.
(6) The emergence of a non-zero three-dimensional Stokes vector ~ζ(t) from
a state that is initially “amorphous” (~ζ(0) = 0), is suggestive of a
phase transition. The evolution of the Stokes vector from zero to unit
length is reminiscent of an order parameter for a system undergoing
spontaneous symmetry breaking.
(7) All our considerations have been in the framework of ordinary quan-
tum mechanics and CPT invariance. Discussions that involve violation
of quantum mechanics and/or CPT symmetry may be found, for ex-
ample, in [8]. An early discussion of the arrow of time in connection
with K meson decays is given in [9]. Broader issues connected with the
arrow of time are discussed, for instance, in [10]. Finally, experimental
investigations of discrete symmetries in the decays of K mesons and
B mesons produced in e+e− or pp̄ collisions are described in [11].
Acknowledgement: One of us (LMS) wishes to thank Dagmar Bruss
(University of Düsseldorf) for a useful correspondence.
References
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	Introduction
	Density Matrix
	Arrows of time
	The Normalization Arrow N(t)
	The Stokes Arrow
	Evolution of Strangeness
	Summary
