An upper limit on CP violation in the $B^0_s-\bar{B}^0_s$ system

0.1 MB·English

by  Ch. Berger

An upper limit on CP violation in the B0 − B¯0
s s
system
Ch. Berger ∗
I. Physikalisches Institut, RWTH Aachen University, Germany
1
1 L. M. Sehgal†
0
Institute for Theoretical Particle Physics and Cosmology,
2
RWTH Aachen University, Germany
n
a
J

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2 Abstract

In a previous publication we noted that the time dependence of
]
h an incoherent B0−B¯0 mixture undergoes a qualitative change when
p
the magnitude of CP violation δ exceeds a critical value. Requiring,
-
p on physical grounds,that the system evolve from an initial incoherent
e statetoafinalpurestateinamonotonicway,yieldsanewupperlimit
h
forδ. Therecentmeasurementofthewrongchargesemileptonicasym-
[ metry of Bs0 mesons presented by the D0 collaboration is outside this
3 bound by one standarddeviation. If this result is confirmed it implies
v the existence of a new quantum mechanical oscillation phenomenon.
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9 Inapreviouspaper[1]westudiedthetimeevolutionofanincoherentB0−B¯0
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mixture as a function of the strength of the CP-violating parameter
2
.
7 |p|2−|q|2
0 δ = hB |B i = , (1)
0 L S |p|2+|q|2
1
: where BL and BS denote the long-lived and short-lived eigenstates of the
v system which can be expressed as superpositions of B0 and B¯0 with the
i
X
coefficients p,q:
r
a 1
|B i = (p|B0i−q|B¯0i)
L
|p|2+|q|2
1
|B i = p (p|B0i+q|B¯0i) (2)
S
|p|2+|q|2
The density matrix, in the Bp0−B¯0 basis, was written as
1
ρ(t)= N(t) +ζ~(t)·~σ , (3)
2 1
h i
∗e-mail: [email protected]
†e-mail: [email protected]
1
wherethe normalisation function N(t) and the Stokes vector ζ~(t) hadinitial
values N(0) = 1, ζ~(0) = 0 respectively. By explicit calculation one finds
N(t)= 1 e−γSt+e−γLt−2δ2e−12(γS+γL)tcos∆mt . (4)
2(1−δ2)
h i
Furthermore, the magnitude of the Stokes vector is found to have a remark-
able relation to N(t):
1
1 2
|ζ~(t)| = 1− e−(γS+γL)t . (5)
N(t)2
(cid:20) (cid:21)
ItshouldbestressedthatEqs.(4,5)arevalidforanysystemsuchasK0−K¯0,
B0−B¯0 or B0−B¯0 satisfying the Lee-Oehme-Yang equation of motion [2].
d d s s
In particular the relation (5) between |ζ~(t)| and N(t) does not involve the
parameters δ, ∆γ or ∆m where ∆γ and ∆m denote the differences in the
decay widths and masses of the eigenstates. Our results are independent of
the sign of ∆γ/∆m.
An upper bound on δ2 arises from the requirement that the normalisation
function N(t) should be a monotonically decreasing function of time, i.e.
dN/dt < 0 for all t. This leads to the constraint
1
γ γ 2
δ2 ≤ S L . (6)
(γ +γ )2/4+∆m2
(cid:18) S L (cid:19)
In fact, a stronger constraint is obtained by demanding that the time-
dependent norm of an arbitrary pure state α|B0i+β|B¯0i decrease mono-
tonically with time. This bound was obtained by Lee and Wolfenstein [3]
and by Bell and Steinberger [4], and is referred to as the unitarity bound
γ γ
δ2 ≤ S L . (7)
unit (γ +γ )2/4+∆m2
(cid:18) S L (cid:19)
In [1] we derived a complementary bound from the requirement that the
function |ζ~(t)| evolve from its initial value |ζ~(0)| = 0 to its final value
|ζ~(∞)| = 1 (representing the long-lived pure state B ) in a monotonic way.
L
This amounts to the assumption that the transformation of the B0 − B¯0
mixture from its incoherent (high entropy) beginning to its final pure (low
entropy) end-state is unidirectional in time. The common requirement of
monotonicity for N(t) and |ζ~(t)| is equivalent to the statement that both of
these observables should behave as arrows of time [1].
The monotonicity condition d|ζ~(t)|/dt ≥ 0 can be expressed as a condition
onN(t)makinguseoftherelation (5). Asshownin[1]thisimpliesanupper
bound
1 ∆γ 3π ∆γ
δ2 ≤ sinh . (8)
2 ∆m 4 ∆m
(cid:18) (cid:19) (cid:18) (cid:19)
2
which reduces for small ∆γ/∆m, as found in the B0−B¯0 system, to
3π ∆γ
|δ| ≤ . (9)
8 ∆m
r (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
A useful experimental observable is the “wrong charge” semileptonic asym-
(cid:12) (cid:12)
metry defined (with q = d,s) as
Γ(B¯0 → µ+X)−Γ(B0 → µ−X)
q q q
a = (10)
sl Γ(B¯0 → µ+X)+Γ(B0 → µ−X)
q q
which is directly related to the CP violating parameter (for recent reviews
see [5])
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1− q
q p 2δ
a = = (11)
sl 1+(cid:12)(cid:12)q(cid:12)(cid:12)4 1+δ2
(cid:12)p(cid:12)
q (cid:12) (cid:12)
reducing to a = 2δ for small δ. (cid:12) (cid:12)
sl (cid:12) (cid:12)
The “coherence bound” for |as | calculated from (8) is plotted in Fig. 1
sl
versus |∆γ/∆m| together with the unitarity bound obtained from (7). It is
immediately seen that for |∆γ/∆m| → 0 the coherence bound is the more
effective of the two, providing stringent upper limits.
In a recent analysis of like-sign dimuon pairs in the D0 experiment [6] an
asymmetry was measured between µ+µ+ and µ−µ− final states,
N++−N−−
Ab = b b = −(0.957±0.251±0.146)% . (12)
sl N+++N−−
b b
This asymmetry was determined to be a linear combination of the semilep-
tonic wrong charge asymmetry associated with B and B mesons
d s
Ab = (0.506±0.043)ad +(0.494±0.043)as . (13)
sl sl sl
Data from B -factories [7] provide an estimate of the asymmetry parameter
d
ad
sl
ad = −(0.47±0.46)% . (14)
sl
Combining this information with Eq. (13) the D0 experiment has extracted
a value for the parameter as which is negative in sign and has the modulus
sl
|as | = (1.46±0.75)% . (15)
sl
This result is included in Fig. 1. The horizontal error bar reflects the uncer-
tainty in the measured value [8] of |∆γ/∆m| = 0.0035±0.002. The figure
shows that the experimental value of |as | is clearly outside the allowed re-
sl
gion violating the coherence bound by about one standard deviation.
3
0.1
|as |
sl
0.08
0.06

0.04
0.02
(cid:12) (cid:12)
(cid:12)∆γ(cid:12)
(cid:12)∆m(cid:12)
0
0.01 0.02 0.03 0.04

Figure1: Constraintson|as |inthe|as |−|∆γ/∆m|planeresultingfromunitarity
sl sl
and monotonicity of |ζ(t)| for the B0 −B¯0 system. The thin line represents the
s s
unitaritybound(7)with∆m/γS =26.2andthethicklineournewboundevaluated
from (8). The cross represents the experimental result of the D0 experiment for
|as | with the horizontal error bar indicating the uncertainty of |∆γ/∆m|
sl
If the Hamiltonian governing the time-dependence of the B0−B¯0 system is
written as a 2×2 matrix M −ıΓ /2, where M and Γ are hermitian, the
q q q q
origin of CP violation resides in the phase
M12
q
Φ = arg − . (16)
q Γ12
q !
In terms of this phase the wrong charge asymmetry parameter is given by
∆γ
q
a = tanΦ (17)
sl ∆m q
neglecting higher orders in ∆γ/∆m. The linear dependence of (17) on
∆γ/∆mallows animmediatecombination with(9)leadingtotheconstraint
3π
|tanΦ | ≤ = 2.17 . (18)
s
2
r
Note that the standard model value for Φ is very small [9] ΦSM = 0.24◦.
s s
The bound (18) may serve as a new constraint on theories that go beyond
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the standard model (see for example, the papers in [10]). It should be
stressed that variations of Φ must in any case respect the unitarity limit
s
shown in Fig. 1. The upper limit as < 7.6% implies the unitarity bound
sl
|tanΦ | < 22.
s
Our analysis based on elementary quantum mechanical reasoning (requiring
no assumption about the Hamiltonian other than CPT invariance) shows
interesting new bounds on the CP violating parameters of the B0 − B¯0
system. It is, however, not granted that nature respects the monotonicity
bound following from (8). In this case the D0 result – if true – implies the
existence of a new type of quantum mechanical oscillation.
0.05
~
|ζ(t)|
0.04
0.03

0.02
0.01
t/τs
S
0
0.2 0.4 0.6 0.8 1

Figure 2: Plot of |ζ~(t)| versus t in units of the lifetime τs. The thick line is
S
calculated for the nominal D0 value of δ =0.0073. The strictly monotonic dashed
line is obtained for the standard model value of δ. The thin line represents the
behaviourof|ζ~(t)| atthe criticalvalue δcrit =0.0038113i.e. the transitionbetween
monotonic and nonmonotonic regimes.
To exhibit this new phenomenon we examine in Fig. 2 the characteristics of
|ζ~(t)| as a function of time in units of τs, the lifetime of the short lived B0
S s
meson. The thick line calculated for the nominal D0 value of δ = 0.0073
exhibits two clear oscillations within the first half of the B0 lifetime. By
s
contrast the dashed line evaluated for the standard model with δ ≈ 10−5 is
strictlymonotonicandinthechosenrangeoft/τs even linearlyincreasingin
S
time. At the critical value δ =0.0038113 (found by numerical evaluation
crit
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of the condition d|ζ(t)|/dt ≥ 0) the evolution of |ζ~(t)| in Fig. 2 follows the
middle curve which marks the line of transition between the monotonic and
nonmonotonic regimes.
If the D0 result is confirmed it should be possible, in principle, to insert
precise data on N(t) into (5) in order to reveal the coherence oscillations
shown in the upper curve of Fig. 2. Since the whole effect is induced by
the term proportional to δ2 in (4), the precision required would be at least
O(10−4), which may be beyond reach. It is interesting, nevertheless, that
precise knowledge of N(t) would allow one to probe the coherence function
ζ(t), and determine whether or not the evolution of coherence in a B0−B¯0
s s
system follows an arrow of time. The D0 asymmetry, taken at face value,
suggests that this arrow is broken.
Note added in proof: Since submission of this paper the Heavy Flavor Av-
eraging Group has reanalyzed [11] the available published [6, 12, 13] and
unpublished [14] experimental data related to as and extracted an average
sl
valueofas = (−0.85±0.58)%. Thecentralvalueismuchcloser toourlimit.
sl
In view of the large error a precise determination of as is eagerly awaited.
sl
References
[1] Ch. Berger and L.M. Sehgal, Phys. Rev. D 76, 036003 (2007);
arXiv:0704.1232 [hep-ph].
[2] T.D. Lee, R. Oehme and C.N. Yang, Phys. Rev. 105, 1671 (1967).
[3] T.D. Lee and L. Wolfenstein, Phys. Rev. 138, B1490 (1965).
[4] J. S. Bell and J. Steinberger, in Proc. Oxford International Conference
on Elementary Particles, 1965, pp. 195-222.
[5] Y.Nir,arXiv:hep-ph/0510413 (2005); Y.Grossman,arXiv:1006.3534v1
[hep-ph].
[6] V.M.Abazov etal.(D0 Collaboration), Phys.Rev.D82, 03201 (2010),
arXiv:1005.2757 [hep-ex];V.M.Abazovetal.(D0Collaboration), Phys.
Rev. Lett. 105, 081801 (2010), arXiv:1007.0395 [hep-ex].
[7] E. Barberio et al. (Heavy Flavour Averaging Group), arXiv:0808.1297.
[8] C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1 (2008),
and 2009 partial update for the 2010 edition.
[9] M. Beneke, G. Buchalla, A. Lenz and U. Nierste, Phys. Lett. B 576,
173 (2003); A. Lenz and U. Nierste, J. High Energy Physics, 0706, 072
(2007).
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[10] A. Dighe, A. Kundu and S. Nandi, arXiv:1005.4051v2 [hep-ph]; B. A.
Dobrescu, P. J. Fox and A.Martin, arXiv:1005.4238v2 [hep-ph]; A. J.
Buras, M. V. Carlucci, S. Gori and G. Isidori, arXiv:1005.5310v2 [hep-
ph]; Z. Ligeti, M. Papucci, G. Perez and J. Zupan, arXiv:1006.0432v1
[hep-ph]; N. Deshpande, X.-G. He and G. Valencia, Phys. Rev D
82, 017301 (2010) arXiv:1006.1682v2 [hep-ph]; D. Choudhury, D. K.
Ghosh, arXiv:1006.2171v1 [hep-ph]; J.K. Parry, arXiv:1006.5331v1
[hep-ph]; Y. Bai, A. E. Nelson, arXiv:1007.0596v1 [hep-ph]; K. Blum,
Y. Hochberg and Y. Nir, arXiv:1007.1872v1 [hep-ph]; B. Ren, X. He
and P. Xie, arXiv 1009.3398v3 [hep-ph].
[11] D. Asner et al. (Heavy Flavor Averaging Group), arXiv:1010.1589v2
[hep-ex].
[12] V.M. Abazov et al. (D0 Collaboration), Phys. Rev. D 82, 012003
(2010), arXiv:0904.3907 [hep-ex].
[13] F. Abe et al. (CDF Collaboration), Phys. Rev. D 55, 2546 (2007).
[14] CDF Collaboration, CDF note 9015 (2007),
http://www.cdf.fnal.gov/physics/new/bottom/070816.blessed-acp-bsemil/.
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