075. B 0 –B Mixing

1075. B 0 –B Mixing
Revised March 2024 by O. Schneider (EPFL).
0

0

0

0

There are two neutral B 0 –B meson systems, Bd0 –B d and Bs0 –B s (generically denoted Bq0 –B q ,q = s, d), which exhibit particle-antiparticle mixing [1]. This mixing phenomenon is describedin Ref. [2]. In the following, we adopt the notation introduced in Ref. [2], and assume CP T
conservation throughout. In each system, the light (L) and heavy (H) mass eigenstates,|BL,H i = p|Bq0 i ± q|B 0q i,(75.1)

have a mass difference ∆mq = mH − mL > 0, a total decay width difference ∆Γq = ΓL − ΓH and an
average decay width Γq = (ΓL + ΓH )/2. In the absence of CP violation in the mixing, |q/p| = 1, thedifferences are given by ∆mq = 2|M12 | and |∆Γq | = 2|Γ12 |, where M12 and Γ12 are the off-diagonal
elements of the mass and decay matrices [2]. The evolution of a pure |Bq0 i or |B 0q i state at t = 0 is
given byq
|Bq0 (t)i =g+ (t) |Bq0 i + g− (t) |B 0q i,(75.2)
pp|B 0q (t)i =g+ (t) |B 0q i + g− (t) |B 0q i,(75.3)qwhich means that the flavor states remain unchanged (+) or oscillate into each other (−) with
time-dependent probabilities proportional to|g± (t)|2 =∆Γqe−Γq tcosh
t ± cos(∆mq t) .
2
2(75.4)

R|g− (t)|2 dt/( |g− (t)|2 dt +









In the absence of CP violation, the time-integrated mixing probabilityR|g+ (t)|2 dt) is given by
χq =x2q + yq2,2(x2q + 1)

where xq =

∆mq
,
Γq

yq =

∆Γq
.
2Γq

R

(75.5)

75.1 Standard Model predictions and phenomenology
In the Standard Model, the transitions Bq0 → B 0q and B 0q → Bq0 are due to the weak interaction.
They are described, at the lowest order, by box diagrams involving two W bosons and two up-type
quarks (see Fig. 75.1), as is the case for K 0 − K 0 mixing. However, the long range interactions
arising from intermediate virtual states are negligible for the neutral B meson systems, because
the large B mass is off the region of hadronic resonances. The calculation of the dispersive andabsorptive parts of the box diagrams yields the following predictions for the off-diagonal elementof the mass and decay matrices [3],M12 = −
Γ12 =G2F m2W ηB mBq BBq fB2 q

12π 2
2
2
0
GF mb ηB mBq BBq fB2 q

S0 (m2t /m2W ) (Vtq∗ Vtb )2 ,(75.6)8π"

×(Vtq∗ Vtb )2 + Vtq∗ Vtb Vcq∗ Vcb O+ (Vcq∗ Vcb )2 O

m4c
m4b

m2c
m2b

!

!#

,

S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:12am(75.7)

075. B 0 –B Mixing

2

_
b
q

W
t

+

t

W−

_
q

_
b

b

q

_
t
W

_
q
W

t

b

0

Figure 75.1: Dominant box diagrams for the Bq0 → Bq transitions (q = d or s). Similar diagrams
exist where one or both t quarks are replaced with c or u quarks.
where GF is the Fermi constant, mW the W boson mass, and mi the mass of quark i; mBq , fBq
and BBq are the Bq0 mass, weak decay constant and bag parameter, respectively. The known
function S0 (xt ) can be approximated very well by 0.784 xt0.76 [4], and Vij are the elements of the
0 are of order unity. The only non-negligible
CKM matrix [5]. The QCD corrections ηB and ηB
contributions to M12 are from box diagrams involving two top quarks. The phases of M12 and Γ12
satisfy!
m2c
φM − φΓ = π + O,(75.8)
m2b
implying that the mass eigenstates have mass and width differences of opposite signs. This means
that, like in the K 0 –K 0 system, the heavy state is expected to have a smaller decay width thanthat of the light state: ΓH < ΓL . Hence, ∆Γq = ΓL − ΓH is expected to be positive in the Standard
Model.
Furthermore, the quantity
Γ121
3π m2b'∼O
22
M122 mW S0 (mt /m2W )

m2b
m2t

!

(75.9)

is small, and a power expansion of |q/p|2 yields
q 2
Γ12
=1+
sin(φM − φΓ ) + O
p
M12

Γ12 2
M12

!

.

(75.10)

Therefore, considering both Eqs. (75.8) and (75.9), the CP -violating parameter
Γ12
q 2
' Im
p
M12


1−



(75.11)00

is expected to be very small: ∼ O(10−3 ) for the Bd0 –B d system and . O(10−4 ) for the Bs0 –B ssystem [6].
In the approximation of negligible CP violation in mixing, the ratio ∆Γq /∆mq is equal to
the small quantity |Γ12 /M12 | of Eq. (75.9); it is hence independent of CKM matrix elements, i.e.,00
the same for the Bd0 –B d and Bs0 –B s systems. Calculations [7] yield ∼ 5 × 10−3 with a ∼ 15%
uncertainty. Given the published experimental knowledge [8] on the mixing parameter xq
0(xd = 0.7697 ± 0.0035 (Bd0 –B d system),0xs = 26.99 ± 0.09
(Bs0 –B s system)
31st May, 2024

(75.12)

075. B 0 –B Mixing

3

the Standard Model thus predicts that ∆Γd /Γd is very small (below 1%), but ∆Γs /Γs considerablylarger (∼ 10%). These width differences are caused by the existence of final states to which boththe Bq0 and B 0q mesons can decay. Such decays involve b → ccq quark-level transitions, which are
Cabibbo-suppressed if q = d and Cabibbo-allowed if q = s.
0
A complete set of Standard Model predictions for all mixing parameters in both the Bd0 –B d and0
Bs0 –B s systems can be found in Ref. [7].

75.2 Experimental issues and methods for oscillation analyses
0Time-integrated measurements of B 0 –B mixing were published for the first time in 1987 by
UA1 [9] and ARGUS [10], and since then by many other experiments. These measurements are
typically based on counting same-sign and opposite-sign lepton pairs from the semileptonic decayof the produced bb pairs. Such analyses cannot easily separate the contributions from the different
b-hadron species, therefore, the clean environment of Υ (4S) machines (where only Bd0 and charged
Bu mesons are produced) is in principle best suited to measure χd .
However, better sensitivity is obtained from time-dependent analyses aiming at the direct measurement of the oscillation frequencies ∆md and ∆ms , from the proper time distributions of Bd0 or
Bs0 candidates identified through their decay in (mostly) flavor-specific modes, and suitably tagged0
as mixed or unmixed. This is particularly true for the Bs0 –B s system, where the large value ofxs implies maximal mixing, i.e., χs ' 1/2. In such analyses, the Bd0 or Bs0 mesons are either
fully reconstructed, partially reconstructed from a charm meson, selected from a lepton with thecharacteristics of a b → `− decay, or selected from a reconstructed displaced vertex. At high-energy
colliders (LEP, SLC, Tevatron, LHC), the proper time t = mpB L is measured from the distance L
between the production vertex and the B decay vertex, and from an estimate of the B momentum
p. At asymmetric B factories (SuperKEKB, KEKB, PEP-II), producing e+ e− → Υ (4S) → Bd0 Bd 0
events with a boost βγ (= 0.28, 0.425, 0.55), the proper time difference between the two B can∆zdidates is estimated as ∆t ' βγc, where ∆z is the spatial separation between the two B decay
vertices along the boost direction. In all cases, the good resolution needed on the vertex positions
is obtained with silicon detectors.
The average statistical significance S of a Bq0 oscillation signal can be approximated as [11]
S≈

q

2

N/2 fsig (1 − 2η) e−(∆mq σt ) /2 ,(75.13)where N is the number of selected and tagged candidates, fsig is the fraction of signal in thatsample, η is the total mistag probability, and σt is the resolution on proper time (or proper timedifference). The quantity S decreases very quickly as ∆mq increases; this dependence is controlled
by σt , which is therefore a critical parameter for ∆ms analyses. At high-energy colliders, the proper
σpBtime resolution σt ∼ m
hpi σL ⊕t p includes a constant contribution due to the decay length resolution
σL (typically 0.04–0.3 ps), and a term due to the relative momentum resolution σp /p (typically 10–
20% for partially reconstructed decays), which increases with proper time. At B factories, theboost of the B mesons is estimated from the known beam energies, and the term due to the spatial
resolution dominates (typically 0.7–1.5 ps because of the much smaller B boost).
In order to tag a Bq0 candidate as mixed or unmixed, it is necessary to determine its flavor bothin the initial state and in the final state. The initial and final state mistag probabilities, ηi and ηf ,degrade S by a total factor (1 − 2η) = (1 − 2ηi )(1 − 2ηf ). In lepton-based analyses, the final stateis tagged by the charge of the lepton from b → `− decays; the largest contribution to ηf is then due
to b → c → `− decays. Alternatively, the charge of a reconstructed charm meson (D∗− from Bd0 or
Ds− from Bs0 ), or that of a kaon hypothesized to come from a b → c → s decay [12], can be used.
For fully-inclusive analyses based on topological vertexing, final-state tagging techniques include
31st May, 2024

075. B 0 –B Mixing

4jet-charge [13] and charge-dipole [14,15] methods. At high-energy colliders, the methods to tag theinitial state (i.e., the state at production), can be divided into two groups: the ones that tag theinitial charge of the b quark contained in the Bq0 candidate itself (same-side tag), and the ones thattag the initial charge of the other b quark produced in the event (opposite-side tag). On the sameside, the sign of a charged pion, kaon or proton from the primary vertex is correlated with the
production state of the Bq0 meson if that particle is a decay product of a B ∗∗ state or the first in
the fragmentation chain [16, 17]. Jet- and vertex-charge techniques work on both sides and on the
opposite side, respectively. Finally, the charge of a lepton from b → `− , of a kaon from b → c → s
or of a charm hadron from b → c [18] can be used as an opposite-side tag, keeping in mind thatits performance is degraded due to integrated mixing. At SLC, the beam polarization produced asizeable forward-backward asymmetry in the Z → bb decays, and provided another very interestingand effective initial state tag based on the polar angle of the Bq0 candidate [14]. Initial state tags
have also been combined to reach ηi ∼ 26% at LEP [17, 19] or 22% at SLD [14] with full efficiency.
In the case ηf = 0, this corresponds to an effective tagging efficiency Q = D2 = (1 − 2η)2 , where is the tagging efficiency, in the range 23 − 31%. The equivalent figure achieved by CDF during
Tevatron Run I was ∼ 3.5% (see tagging summary on page 160 of Ref. [20]), reflecting the fact thattagging is more difficult at hadron colliders. The CDF and DØ analyses of Tevatron Run II datareached D2 = (1.8 ± 0.1)% [21] and (2.5 ± 0.2)% [22] for opposite-side tagging, while same-sidekaon tagging (for Bs0 analyses) contributed an additional 3.7 − 4.8% at CDF [21], and pushed the
combined performance to (4.7 ± 0.5)% at DØ [23]. LHCb, operating in the forward region at the
LHC where the environment is different in terms of track multiplicity and b-hadron production
kinematics, has reported D2 = (2.10 ± 0.25)% [24] for opposite-side tagging, (1.80 ± 0.26)% [25]for same-side kaon tagging, and (2.11 ± 0.11)% [26] for same-side pion and proton tagging: thecombined figure ranges typically between (3.73 ± 0.15)% [27] and (6.3 ± 0.5)% [28] depending on
the mode in which the tagged Bs0 meson is reconstructed, and reaches up to (8.1 ± 0.6)% [29] forhadronic Bd0 modes. ATLAS [30] and CMS [31] have reported D2 ∼ 1.75% and D2 ∼ 10% using
opposite-side tagging of Bs0 → J/ψφ decays.
At B factories, the flavor of a Bd0 meson at production cannot be determined, since the twoneutral B mesons produced in a Υ (4S) decay evolve in a coherent P -wave state where they keep
opposite flavors at any time. However, as soon as one of them decays, the other follows a timeevolution given by Eqs. (75.2) or (75.3), where t is replaced with ∆t (which will take negative values
half of the time). Hence, the “initial state” tag of a B can be taken as the final-state tag of theother B. Effective tagging efficiencies of 30% are achieved by BaBar and Belle [32], using different
techniques including b → `− and b → c → s tags. It is worth noting that, in this case, mixing of
the other B (i.e., the coherent mixing occurring before the first B decay) does not contribute to
the mistag probability.
Before the experimental observation of a decay-width difference, oscillation analyses typicallyneglected ∆Γq in Eq. (75.4), and described the time dependence with the functions Γq e−Γq t (1 ±cos(∆mq t))/2 (high-energy colliders) or Γd e−Γd |∆t| (1 ± cos(∆md ∆t))/4 (asymmetric Υ (4S) machines). As can be seen from Eq. (75.4), a non-zero value of ∆Γq would effectively reduce theoscillation amplitude with a small time-dependent factor that would be very difficult to distinguishfrom time resolution effects. Measurements of ∆mq are usually extracted from the data using amaximum likelihood fit.

75.3 ∆md and ∆Γd measurements
0

Many Bd0 –B d oscillations analyses have been published [33] by the ALEPH [34], DELPHI [15,35], L3 [36], OPAL [37, 38], CDF [16], DØ [22], BaBar [39], Belle [40], Belle II [41], and LHCb
[42–45] collaborations. Although a variety of different techniques have been used, the individual
31st May, 2024

5075. B 0 –B Mixing∆md results obtained at LEP and Tevatron have remarkably similar precision. Their average
is compatible with the more precise measurements at the asymmetric B factories and the LHC.
The systematic uncertainties are not negligible; they are often dominated by sample composition,
mistag probability, or b-hadron lifetime contributions. Before being combined, the measurements
are adjusted on the basis of a common set of input values, including the b-hadron lifetimes andfractions published in this Review. Some measurements are statistically correlated. Systematic
correlations arise both from common physics sources (fragmentation fractions, lifetimes, branchingratios of b hadrons), and from purely experimental or algorithmic effects (efficiency, resolution,
tagging, background description). Combining all measurements [15, 16, 22, 34–45] and accounting
for all identified correlations yields ∆md = 0.5069 ± 0.0016(stat) ± 0.0011(syst) ps−1 [8], a result
dominated by the latest LHCb measurement with B 0 → D(∗)− µ+ νµ X decays [45].
On the other hand, ARGUS and CLEO have published time-integrated measurements [46–48],
which average to χd = 0.182 ± 0.015. Following Ref. [48], the width difference ∆Γd could in
principle be extracted from the measured value of 1/Γd and the above averages for ∆md and χd(see Eq. (75.5)), provided that ∆Γd has a negligible impact on the ∆md and 1/Γd analyses thathave assumed ∆Γd = 0. However, ∆Γd /Γd is too small and the knowledge of χd too imprecise toprovide useful sensitivity on ∆Γd /Γd . Direct time-dependent studies published by DELPHI [15],BaBar [49], Belle [50], LHCb [51], ATLAS [52] and CMS [53] provide stronger constraints, which
can be combined to yield [8]∆Γd /Γd = +0.001 ± 0.010 .
(75.14)This determination is compatible both with zero and with the Standard Model prediction of (4.0 ±0.9) × 10−3 [54].
Assuming ∆Γd = 0 and no CP violation in mixing, and using the Bd0 lifetime average of
1.517 ± 0.004 ps [8], the ∆md and χd results are combined to yield the world average∆md = 0.5069 ± 0.0019 ps−1

(75.15)χd = 0.1860 ± 0.0011 .

(75.16)or, equivalently,
This ∆md value provides an estimate of 2|M12 |, and can be used with Eq. (75.6) to extract |Vtd |within the Standard Model [55]. The main experimental uncertainties on the result come from mtand ∆md , but are still
completely negligible with respect to the uncertainty due to the hadronicpmatrix element fBd BBd = 225 ± 9 MeV [56] obtained from three-flavor lattice QCD calculations.

75.4 ∆ms and ∆Γs measurements0

After many years of intense search at LEP and SLC, Bs0 –B s oscillations were first observed0
in 2006 by CDF using 1 fb−1 of Tevatron Run II data [21]. LHCb then observed Bs0 –B s oscillations independently with Bs0 → Ds− π + [42, 57], Bs0 → Ds− µ+ νX [44] and Bs0 → J/ψK + K − [27]decays using up to 3 fb−1 of LHC Run 1 data. More recently measurements based on additional
LHC Run 2 data have been published by CMS with Bs0 → J/ψφ decays [31], and by LHCb with
Bs0 → Ds− π + π − π + [58], Bs0 → Ds− π + [59] and Bs0 → J/ψK + K − [60] decays. Taking systematic correlations into account, the average [8] of all published measurements of ∆ms [21,27,31,42,44,57–60]
is∆ms = 17.765 ± 0.004(stat) ± 0.004(syst) ps−1 ,(75.17)
with an impressive precision dominated by the most recent Bs0 → Ds− π + result [59] (see Fig. 75.2).

31st May, 2024

075. B 0 –B Mixing

6.2+vb f UyXy9 TbVӷ֎Ј ݂ ӹ֎਷ ᅺ

lMi;;2/ӷЈ֎ ݂ ӷ֎Ј ݂ ӹ֎਷ ᅺ






G>*#



ϩ 7#਷ȯ










ԣ <Tb>

Figure 75.2: Proper decay-time distributions of background-subtracted Bs0 → Ds− π + decays
tagged as unmixed (blue), tagged as mixed (red) or untagged (grey) in the Run 2 data of the LHCb
0
experiment, displaying Bs0 –B s oscillations [59].
The information on |Vts | obtained in the framework of the Standard Model is hampered by thehadronic uncertainty, as in the Bd0 case. However, several uncertainties cancel in the frequency
ratio
∆ms
mBs 2 Vts 2=,(75.18)
ξ
∆md
mBd
Vtd
p

p

where the SU(3) flavor-symmetry breaking factor ξ = (fBs BBs )/(fBd BBd ) is obtained as 1.206±
0.017 from a combination of three-flavor lattice QCD calculations [56] dominated by the results ofRef. [61], or as 1.2014+0.0065−0.0072 from QCD sum rules [62]. Using the measurements of Eqs. (75.15)
and (75.17), one can extract
Vtd
=
Vts

(

0.2054 ± 0.0004 ± 0.0029 (lattice QCD),
0.2045 ± 0.0004+0.0011−0.0012 (QCD sum rules)(75.19)in good agreement with (but much more precise than) the value obtained from the ratio of theb → dγ and b → sγ transition rates observed at the B factories [55].
The CKM matrix can be constrained using experimental results on observables such as ∆md ,∆ms , |Vub /Vcb |, K , and sin(2β) together with theoretical inputs and unitarity conditions [55,63,64].
The constraint from our knowledge on the ratio ∆ms /∆md is more effective in limiting the positionof the apex of the CKM unitarity triangle than the one obtained from the ∆md measurements
alone, due to the reduced hadronic uncertainty in Eq. (75.18). We also note that the experimental
31st May, 2024

7

075. B 0 –B Mixing

value of ∆ms from Eq. (75.17) is consistent with the Standard Model predictions obtained from
CKM fits where no experimental information on ∆ms is used, e.g., 17.89 ± 0.65 ps−1 [63] or−1 [64]. It is also consistent with the prediction 18.23 ± 0.62 ps−1 from Ref. [7].
17.26 +0.63−0.41 ps
Information on ∆Γs can be obtained from the study of the proper time distribution of untagged
Bs0 samples [65]. In the case of an inclusive Bs0 selection [66], or a flavor-specific (semileptonic or
hadronic) Bs0 decay selection [19, 67–69], both the short- and long-lived components are present,
and the proper decay-time distribution is a superposition of two exponentials with decay constants
ΓL,H = Γs ± ∆Γs /2. In principle, this provides sensitivity to both Γs and (∆Γs /Γs )2 . Ignoring ∆Γs
and fitting for a single exponential leads to an estimate of 1/Γs (called effective lifetime) with arelative bias proportional to (∆Γs /Γs )2 . An alternative approach, sensitive to first order in ∆Γs /Γs ,is to determine the effective lifetime of untagged Bs0 decays to pure CP eigenstates; measurements
exist for Bs0 → Ds+ Ds− [68], Bs0 → K + K − [69, 70], Bs0 → J/ψη [71, 72], Bs0 → J/ψf0 (980) [73],Bs0 → J/ψπ + π − [53, 74, 75], Bs0 → J/ψKS0 [76], and Bs0 → µ+ µ− [77]. The extraction of 1/Γs and∆Γs from such measurements, discussed in detail in Ref. [78], requires additional information in
the form of theoretical assumptions or external inputs on weak phases and hadronic parameters. In
what follows, we only use the effective lifetimes of decays to CP -even (Ds+ Ds− , J/ψη) and CP -odd
(J/ψf0 (980), J/ψπ + π − ) final states where CP conservation can be assumed. In addition ∆Γs canbe extracted from the decay-time distributions of Bs0 decays to CP -even and CP -odd final states,
as has been done by LHCb with Bs0 → J/ψη 0 and Bs0 → J/ψπ + π − , respectively [79].
The best sensitivity to 1/Γs and ∆Γs is achieved by the time-dependent measurements of theBs0 → J/ψK + K − (including Bs0 → J/ψφ) and Bs0 → ψ(2S)φ decay rates performed at CDF [80],DØ [81], ATLAS [30, 82], CMS [31, 83] and LHCb [27, 60, 84–86], where the CP -even and CP -odd
amplitudes are separated statistically through a full angular analysis (see Fig. 75.3). The LHCb
collaboration analyzes the Bs0 → J/ψK + K − decay considering that the K + K − system can be ina P-wave or S-wave state, and measures the dependence of the strong phase difference between the
P-wave and S-wave amplitudes as a function of the K + K − invariant mass [27, 60, 87]; this allows
the unambiguous determination of the sign of ∆Γs , which is found to be positive. All these studies
use both untagged and tagged Bs0 candidates and are optimized for the measurement of the phase
0
0
φcc̄s
s that describes CP violation in the interference between Bs –B s mixing and decay in b → cc̄stransitions. The published Bs0 → J/ψK + K − , J/ψφ and ψ(2S)φ analyses [27, 30, 31, 60, 80–86]
are combined in a multi-dimensional fit including all measured parameters and their correlations.
To account for a tension in the time and angular parameters, scale factors are applied on the
combined uncertainty of each parameter where a discrepancy arise. For example, the scale factors
on the uncertainties of ∆Γs , Γs and φcc̄s
are 1.84, 2.45 and 1.00, respectively. The averages aresthen further refined by applying constraints from the published lifetime measurements with flavorspecific [19, 67–69] and pure CP [53, 68, 71–75] final states, to yield [8, 88]∆Γs = +0.083 ± 0.005 ps−1 and 1/Γs = 1.520 ± 0.005 ps ,(75.20)

1/ΓL = 1.429 ± 0.006 ps(75.21)or, equivalently,
and1/ΓH = 1.622 ± 0.008 ps ,in good agreement with the Standard Model predictions ∆ΓsSM = +0.091 ± 0.015 ps−1 [7] and∆ΓsSM = +0.076 ± 0.017 ps−1 [89]. Estimates of ∆Γs /Γs obtained from measurements of the(∗)+ (∗)−
Bs0 → Ds Dsbranching fractions are not included in the average, since they are based on thequestionable [90] assumption that these decays account for all CP -even final states.
From Eqs. (75.5), (75.17) and (75.20), one getsχs = 0.499318 ± 0.000005 .
31st May, 2024

(75.22)

0

Candidates/(0.01 ps)

105

Candidates/0.02

75. B 0 –B Mixing

8

LHCb Run 2, 6 fb-1
Data
Total fit
CP-evenCP-oddS-wave

104
103

Data
Total fit
CP-evenCP-oddS-wave

10000

102

LHCb Run 2, 6 fb-1

5000

10
1

5

0

10

−0.5

0

0.5

Candidates/0.02

10000cosθ K
Candidates/(0.06 rad)Decay time [ps]LHCb Run 2, 6 fb-1

8000
6000
Data
Total fit
CP-evenCP-oddS-wave

4000
2000
0

−0.5

0

LHCb Run 2, 6 fb-1

8000
6000

Data
Total fit
CP-evenCP-oddS-wave

4000
2000
0

0.5

cosθ µ−202

φ h [rad]Figure75.3:
Distributions
the properdistributions
decay-time for
andbackground-subtractedof the three decay angles
of J/
backgroundFigure
2: Decay-timeand ofdecay-angle
Bs0 !
K +K
+
−
0
K decaysprojections
reconstructedby PDF
LHCb
themaximum-likelihood
vicinity of the φ resonance[60].
subtracted
Bs →decays with
theJ/ψK
one-dimensionalof theatinthepoint. The
Thedatakaonandpair
ispredominantlyinaP-wave
state
(from
the
φ
decay)
whichcanbeCP-evenorCPfit projections for the diﬀerent samples considered [data-taking year, trigger and tagging+
odd,categories,
while a small
stateareis CP-odd.
m(KS-waveK ) bins]
combined.The curves show the projections of a four-dimensional
fit allowing the various components to be disentangled. The different lifetimes of the CP-even and
CP-odd components is evident from the decay-time distribution.

Around 1.4% of the selected events have multiple candidates; the eﬀect of such cases is
consideredin systematic
uncertainties by choosing one candidate randomly and repeating
75.5
CP -violation
studiesthe fit. Systematic uncertainties
are0 assigned due to the limited size of the PID calibration
Evidence for CP violation in Bq0 –B q mixing has been searched for, both with flavor-specific andsamples. Diﬀerent models of the S-wave line shape based on the results in Ref. [34] areinclusive Bq0 decays, in samples where the initial flavor state is tagged, usually with a lepton fromused to evaluate the CSP factors
and assign systematic uncertainties.
the other b-hadron in the event.
In the case of semileptonic (or other flavor-specific) decays, where
A systematic
uncertainty
translationof the [2]decay-time resolution calibration
the final-state
tag is alsoavailable,forthethefollowing
asymmetryfrom the control sample to signal is derived using simulation. A minor systematic0− X)`+ ν` X)in−the
N (B
N (B 0q (t) → eﬀects
uncertainty due qto non-Gaussian
decay-time
resolution
is assigned. Systematic`q (t) → ` νASL =' 1 − |q/p|2q(75.23)00+−uncertainties accountingthe` limited
sample for the decayN (B qfor(t) →ν` X) + sizes
N (Bq of(t) the→ ` calibrationν ` X)
time resolution and the simulated samples for the angular eﬃciencies are estimated byhas been measured either in time-integrated analyses at CLEO [48, 91], BaBar [92], CDF [93],varying the calibration parameters and eﬃciencies according to the statistical covarianceDØ [94–96] and LHCb [97], or in time-dependent analyses at LEP [38, 98], BaBar [49, 99] andmatrices. The eﬀect of ignoring the angular resolutions in the fit is estimated by performing
Belle [100]. In the inclusive case, also investigated at LEP [98, 101], no final-state tag is used, andseparate fits to the generated and reconstructed angular variables from simulation, and thethe asymmetry [102]diﬀerences are taken as systematic uncertainties. The eﬀect of the specific configuration
0 (t) → all)of the gradient-boosted tree
appliedto q0correct
the kinematics of simulation in the
N (Bmethod− N (B(t) → all)q0
0time and angular eﬃciency
isestimatedbyapplying
100
N (B q (t) → all) + N (Bq (t) → all) alternative configurations.
 has a relative uncertaintyThe longitudinal scale of the vertex ∆mdetectorof 0.022% [50,51]
xq
qt
q
2
' ASL issinassigned by−scalingsin(∆mq t)
and a systematic uncertainty
the
track
parameters
with(75.24)
this22
uncertainty. The systematic uncertainty associated with the track momentum scale
31st May, 2024

6

9

0

75. B 0 –B Mixing

must be measured as a function of the proper time to extract information on CP violation. In
addition LHCb has studied the time dependence of the charge asymmetry of B 0 → D(∗)− µ+ νµ X
decays without tagging the initial state [103], which would be equal to
N (D(∗)− µ+ νµ X) − N (D(∗)+ µ− ν̄µ X)
1 − cos(∆md t)= AdSL
(∗)−
+
(∗)+
−
2
N (D
µ νµ X) + N (Dµ ν̄µ X)

(75.25)in absence of detection and production asymmetries.
The DØ collaboration measured a like-sign dimuon charge asymmetry in semileptonic b decaysthat deviates by 2.8 σ from the tiny Standard Model prediction and concluded, from a more refined
analysis in bins of muon impact parameters, that the overall discrepancy is at the level of 3.6 σ [94].
In all other cases, asymmetries compatible with zero (and the Standard Model [7]) have been found,with a precision limited by the available statistics. Several of the analyses at high energy don’t
disentangle the Bd0 and Bs0 contributions, and either quote a mean asymmetry or a measurement of
AdSL assuming AsSL = 0: we no longer include these in the average. An exception is the dimuon DØ
analysis [94], which separates the two contributions by exploiting their dependence on the muon
impact parameter cut. The resulting measurements of AdSL and AsSL are then both compatible with
the Standard Model. They are also correlated. We therefore perform a two-dimensional average ofthe measurements of Refs. [48, 49, 91, 92, 94–97, 99, 100, 103] and obtain [8]AdSL = −0.0021 ± 0.0017 ⇔ |q/p|d = 1.0010 ± 0.0008 ,(75.26)AsSL = −0.0006 ± 0.0028 ⇔ |q/p|s = 1.0003 ± 0.0014 ,(75.27)

with a correlation coefficient of −0.054 between AdSL and AsSL . These results show no evidenceof CP violation and are compatible with the very small Standard Model predictions, Ad,SM
=
SL
s,SM
−4
−5
−(5.1 ± 0.5) × 10 and ASL = +(2.2 ± 0.2) × 10 [7], but have insufficient precision yet toconstrain the Standard Model.
0
CP violation induced by Bs0 –B s mixing in b → cc̄s decays is controlled by the small weak phase
cc̄s
0
φcc̄s
s . Measuring φs requires tagging the initial flavour of the decaying Bs meson. In addition to
the previously mentioned Bs0 → J/ψK + K − (including Bs0 → J/ψφ) and Bs0 → ψ(2S)φ studies,
the decay modes Bs0 → J/ψπ + π − (including Bs0 → J/ψf0 (980)) [75, 105] and Bs0 → Ds+ Ds− [104]have also been analyzed by LHCb to measure φcc̄s
s , without the need for an angular analysis. The
J/ψπ + π − final state has been shown indeed to be (very close to) a pure CP -odd state [106]. In
the Bs0 → J/ψφ and Bs0 → J/ψK + K − analyses, φcc̄s
is obtained together with several others
observables, including ∆Γs , Γs , the longitudinal and perpendicular φ polarisation amplitudes, theS-wave amplitude, and strong phases. In order to account for all correlations, the full sets ofmeasurements provided by the different analyses are combined in a multi-dimensional fit [8] of which
0
φcc̄ss is just one of the free parameters. As already mentioned the Bs → J/ψφ analyses of ATLAS,
CMS and LHCb show a poor overall compatibility, mostly in the lifetime and angular parameters,
corresponding approximately to 3 standard deviations. Therefore, scale factors are applied on the
combined uncertainty of each parameter where a discrepancy arises. For the parameters alreadyin agreement, such as φcc̄s
s , no scale factor is applied. The combined result based on all publishedanalyses [27, 30, 31, 60, 75, 80–86, 104, 105] is
φcc̄ss = −0.040 ± 0.016 .

(75.28)

A two-dimensional projection of the overall situation in the (φcc̄ss , ∆Γs ) plane is shown in Fig. 75.4.
The experimental determination of φscc̄s is still statistically limited. It is consistent with the Standard Model prediction, which is equal to −2βs = −2 arg(−(Vts Vtb∗ )/(Vcs Vcb∗ )) = −0.0367±0.0010 [63]or −0.0376 +0.0006−0.0005 [64] assuming negligible Penguin pollution.
31st May, 2024

075. B 0 –B Mixing

10

Figure 75.4: 68% CL contours in the (φcc̄ss , ∆Γs ) plane, showing all measurements from CDF [80],DØ [81], ATLAS [30,82], CMS [31,83] and LHCb [27,60,75,84–86,104,105] using Bs0 decays governedby the b → cc̄s transition. Their average [8] is represented as the black ellipse, where the combined
uncertainty on ∆Γs has been multiplied by 1.88. The very thin white rectangle represents theStandard Model predictions of −2βs [63, 64] and ∆Γs [7].
0CP violation induced by Bs0 –B s mixing in b → ss̄s decays is controlled by the weak phase φsss̄s ,which is an observable different from φscc̄s . In the Standard Model, such pure gluonic Penguin decays
have an amplitude with a CKM phase that cancels that of the mixing amplitude, and hence φsss̄sis expected to be essentially zero. LHCb has performed a flavour-tagged time-dependent angularanalysis of Bs0 → φφ decays [28, 107], similar to that of Bs0 → J/ψφ decays, and measured
φsss̄s = −0.074 ± 0.069 ,(75.29)in agreement with the Standard Model prediction.

75.6 Summary00

B 0 –B mixing has been and still is a field of intense study. The mass differences in the Bd0 –B d0
and Bs0 –B s systems are known to relative precisions of 0.38% and 0.03%, respectively. The non0
zero decay width difference in the Bs0 –B s system is well established, with a relative difference0of ∆Γs /Γs = (12.7 ± 0.7)%, meaning that the heavy state of the Bs0 –B s system lives ∼ 14%0longer than the light state. In contrast, the relative decay width difference in the Bd0 –B d system,00∆Γd /Γd = (0.1 ± 1.0)%, is still consistent with zero. CP violation in Bd0 –B d or Bs0 –B s mixing has
not been observed yet, with precisions on the semileptonic asymmetries below 0.3%. CP violation0
induced by Bs0 –B s mixing in Bs0 decays has not yet been observed either, with uncertainties on the
ss̄s
φcc̄s
s and φs phases of 16 mrad and 69 mrad, respectively. All observations so far remain consistent
31st May, 2024

11

075. B 0 –B Mixing

with the Standard Model expectations.
However, the measurements where New Physics might show up are still statistically limited.
More results are awaited from the LHC experiments and Belle II, with promising prospects for theinvestigation of the CP -violating phase arg(−M12 /Γ12 ) and improved determination of the φcc̄ss
and φss̄sphases.
s
Mixing studies have clearly reached the stage of precision measurements, where much effort isneeded, both on the experimental and theoretical sides, in particular to further reduce the hadronic
uncertainties of lattice QCD calculations. In the long term, a stringent check of the consistency of
the Bd0 and Bs0 mixing amplitudes (magnitudes and phases) with all other measured flavor-physics
observables will be possible within the Standard Model, leading to very tight limits on (or otherwise
a long-awaited surprise about) New Physics.
References[1] T. D. Lee and C. S. Wu, Ann. Rev. Nucl. Part. Sci. 16, 511 (1966); I. I. Bigi and A. I. Sanda,
Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 9, 1 (2009), second edition, first publishedin 2000; G. C. Branco, L. Lavoura and J. P. Silva, Int. Ser. Monogr. Phys. 103, 1 (1999).
[2] See the review on CP violation in the quark sector by T. Gershon and Y. Nir in this publication.
[3] A. J. Buras, W. Slominski and H. Steger, Nucl. Phys. B245, 369 (1984).
[4] T. Inami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981), erratum ibid. 65, 1772 (1981); forthe power-like approximation, see A. J. Buras and R. Fleischer, page 91 in “Heavy Flavours
II,” eds. A. J. Buras and M. Lindner, Singapore World Scientific (1998).
[5] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
[6] I. I. Bigi et al., in “CP violation,” ed. C. Jarlskog, Singapore World Scientific (1989).
[7] J. Albrecht, F. Bernlochner, A. Lenz and A. Rusov (2024), [arXiv:2402.04224].
[8] Y. S. Amhis et al. (HFLAV), Phys. Rev. D 107, 052008 (2023), [arXiv:2206.07501]; thecombined results on b-hadron lifetimes and mixing parameters published in this Review havebeen obtained by the corresponding sub-group of the Heavy Flavor Averaging (HFLAV) groupin March 2024 as updates of the averages published in Chapter 5 of the above paper; for more
information, see https://hflav.web.cern.ch/.
[9] C. Albajar et al. (UA1), Phys. Lett. B186, 247 (1987), erratum ibid. B197, 565 (1987).
[10] H. Albrecht et al. (ARGUS), Phys. Lett. B192, 245 (1987).
[11] H. G. Moser and A. Roussarie, Nucl. Instrum. Meth. A384, 491 (1997).
[12] SLD collab., SLAC-PUB-7228, SLAC-PUB-7229, and SLAC-PUB-7230, 28th Int. Conf. on
High Energy Physics, Warsaw (1996); J. L. Wittlin, A measurement of the time dependence0of Bd0 –B d mixing with kaon tagging, Ph.D. thesis, Massachusetts U., Amherst (2001), URL
http://wwwlib.umi.com/dissertations/fullcit?p3027272.
[13] ALEPH collab., contrib. 596 to Int. Europhysics Conf. on High Energy Physics, Jerusalem(1997).
[14] K. Abe et al. (SLD), Phys. Rev. D67, 012006 (2003), [hep-ex/0209002].
[15] J. Abdallah et al. (DELPHI), Eur. Phys. J. C28, 155 (2003), [hep-ex/0303032].
[16] F. Abe et al. (CDF), Phys. Rev. Lett. 80, 2057 (1998), [hep-ex/9712004]; F. Abe et al.
(CDF), Phys. Rev. D59, 032001 (1999), [hep-ex/9806026]; F. Abe et al. (CDF), Phys. Rev.
D60, 051101 (1999); F. Abe et al. (CDF), Phys. Rev. D60, 072003 (1999), [hep-ex/9903011];
T. Affolder et al. (CDF), Phys. Rev. D60, 112004 (1999), [hep-ex/9907053].
31st May, 2024

12075. B 0 –B Mixing[17] R. Barate et al. (ALEPH), Eur. Phys. J. C4, 367 (1998); R. Barate et al. (ALEPH), Eur.
Phys. J. C7, 553 (1999), [hep-ex/9811018].
[18] R. Aaij et al. (LHCb), JINST 10, P10005 (2015), [arXiv:1507.07892].
[19] P. Abreu et al. (DELPHI), Eur. Phys. J. C16, 555 (2000), [hep-ex/0107077].
[20] K. Anikeev et al. (2002), [hep-ph/0201071].
[21] A. Abulencia et al. (CDF), Phys. Rev. Lett. 97, 242003 (2006), [hep-ex/0609040].
[22] V. M. Abazov et al. (DØ), Phys. Rev. D74, 112002 (2006), [hep-ex/0609034].
[23] V. M. Abazov et al. (DØ), Phys. Rev. Lett. 101, 241801 (2008), [arXiv:0802.2255].
[24] R. Aaij et al. (LHCb), Eur. Phys. J. C72, 2022 (2012), [arXiv:1202.4979].
[25] R. Aaij et al. (LHCb), JINST 11, P05010 (2016), [arXiv:1602.07252].
[26] R. Aaij et al. (LHCb), Eur. Phys. J. C77, 238 (2017), [arXiv:1610.06019].
[27] R. Aaij et al. (LHCb), Phys. Rev. Lett. 114, 041801 (2015), [arXiv:1411.3104].
[28] R. Aaij et al. (LHCb), Phys. Rev. Lett. 131, 17, 171802 (2023), [arXiv:2304.06198].
[29] R. Aaij et al. (LHCb), Phys. Rev. Lett. 117, 261801 (2016), [arXiv:1608.06620].
[30] G. Aad et al. (ATLAS), Eur. Phys. J. C81, 342 (2021), [arXiv:2001.07115].
[31] A. M. Sirunyan et al. (CMS), Phys. Lett. B816, 136188 (2021), [arXiv:2007.02434].
[32] B. Aubert et al. (BaBar), Phys. Rev. Lett. 94, 161803 (2005), [hep-ex/0408127]; K. F. Chenet al. (Belle), Phys. Rev. D72, 012004 (2005), [hep-ex/0504023].
[33] Throughout this document we omit references of results that have been replaced by new
published measurements.
[34] D. Buskulic et al. (ALEPH), Z. Phys. C75, 397 (1997).
[35] P. Abreu et al. (DELPHI), Z. Phys. C76, 579 (1997).
[36] M. Acciarri et al. (L3), Eur. Phys. J. C5, 195 (1998).
[37] G. Alexander et al. (OPAL), Z. Phys. C72, 377 (1996); K. Ackerstaff et al. (OPAL), Z.
Phys. C76, 417 (1997), [hep-ex/9707010]; G. Abbiendi et al. (OPAL), Phys. Lett. B493, 266(2000), [hep-ex/0010013].
[38] K. Ackerstaff et al. (OPAL), Z. Phys. C76, 401 (1997), [hep-ex/9707009].
[39] B. Aubert et al. (BaBar), Phys. Rev. Lett. 88, 221802 (2002), [hep-ex/0112044]; B. Aubert
et al. (BaBar), Phys. Rev. D66, 032003 (2002), [hep-ex/0201020]; B. Aubert et al. (BaBar),
Phys. Rev. Lett. 88, 221803 (2002), [hep-ex/0112045]; B. Aubert et al. (BaBar), Phys. Rev.
D67, 072002 (2003), [hep-ex/0212017]; B. Aubert et al. (BaBar), Phys. Rev. D73, 012004(2006), [hep-ex/0507054].
[40] N. C. Hastings et al. (Belle), Phys. Rev. D67, 052004 (2003), [hep-ex/0212033]; Y. Zheng
et al. (Belle), Phys. Rev. D67, 092004 (2003), [hep-ex/0211065]; K. Abe et al. (Belle), Phys.
Rev. D71, 072003 (2005), erratum ibid. D71, 079903 (2005), [hep-ex/0408111].
[41] F. Abudinén et al. (Belle II), Phys. Rev. D107, L091102 (2023), [arXiv:2302.12791].
[42] R. Aaij et al. (LHCb), Phys. Lett. B709, 177 (2012), [arXiv:1112.4311].
[43] R. Aaij et al. (LHCb), Phys. Lett. B719, 318 (2013), [arXiv:1210.6750].
[44] R. Aaij et al. (LHCb), Eur. Phys. J. C73, 2655 (2013), [arXiv:1308.1302].
[45] R. Aaij et al. (LHCb), Eur. Phys. J. C76, 412 (2016), [arXiv:1604.03475].
[46] H. Albrecht et al. (ARGUS), Z. Phys. C55, 357 (1992); H. Albrecht et al. (ARGUS), Phys.
Lett. B324, 249 (1994).
31st May, 2024

13

075. B 0 –B Mixing[47] J. E. Bartelt et al. (CLEO), Phys. Rev. Lett. 71, 1680 (1993).
[48] B. H. Behrens et al. (CLEO), Phys. Lett. B490, 36 (2000), [hep-ex/0005013].
[49] B. Aubert et al. (BaBar), Phys. Rev. Lett. 92, 181801 (2004), [hep-ex/0311037]; B. Aubert
et al. (BaBar), Phys. Rev. D70, 012007 (2004), [hep-ex/0403002].
[50] T. Higuchi et al., Phys. Rev. D85, 071105 (2012), [arXiv:1203.0930].
[51] R. Aaij et al. (LHCb), JHEP 04, 114 (2014), [arXiv:1402.2554].
[52] M. Aaboud et al. (ATLAS), JHEP 06, 081 (2016), [arXiv:1605.07485].
[53] A. M. Sirunyan et al. (CMS), Eur. Phys. J. C78, 457 (2018), erratum ibid. C78, 561 (2018),[arXiv:1710.08949].
[54] M. Artuso, G. Borissov and A. Lenz, Rev. Mod. Phys. 88, 045002 (2016), [arXiv:1511.09466].
[55] See the review on the CKM quark-mixing matrix by A. Ceccucci, Z. Ligeti, and Y. Sakai inthis publication.
[56] S. Aoki et al. (Flavour Lattice Averaging Group) (2019), [arXiv:1902.08191].
[57] R. Aaij et al. (LHCb), New J. Phys. 15, 053021 (2013), [arXiv:1304.4741].
[58] R. Aaij et al. (LHCb), JHEP 03, 137 (2021), [arXiv:2011.12041].
[59] R. Aaij et al. (LHCb), Nature Phys. 18, 1 (2022), [arXiv:2104.04421].
[60] R. Aaij et al. (LHCb), Phys. Rev. Lett. 132, 051802 (2024), [arXiv:2308.01468].
[61] A. Bazavov et al. (Fermilab Lattice and MILC), Phys. Rev. D93, 113016 (2016),[arXiv:1602.03560].
[62] D. King, A. Lenz and T. Rauh, JHEP 05, 034 (2019), [arXiv:1904.00940].
[63] M. Bona et al. (UTfit), Rend. Lincei Sci. Fis. Nat. 34, 37 (2023), [arXiv:2212.03894]; updated
results at http://www.utfit.org/.
[64] J. Charles et al. (CKMfitter), Phys. Rev. D91, 073007 (2015), [arXiv:1501.05013]; updated
results at http://ckmfitter.in2p3.fr/.
[65] K. Hartkorn and H. G. Moser, Eur. Phys. J. C8, 381 (1999).
[66] M. Acciarri et al. (L3), Phys. Lett. B438, 417 (1998).
[67] D. Buskulic et al. (ALEPH), Phys. Lett. B377, 205 (1996); K. Ackerstaff et al. (OPAL),
Phys. Lett. B426, 161 (1998), [hep-ex/9802002]; F. Abe et al. (CDF), Phys. Rev.
D59, 032004 (1999), [hep-ex/9808003]; V. M. Abazov et al. (DØ), Phys. Rev. Lett.
114, 062001 (2015), [arXiv:1410.1568]; T. Aaltonen et al. (CDF), Phys. Rev. Lett. 107,
272001 (2011), [arXiv:1103.1864]; R. Aaij et al. (LHCb), Phys. Rev. Lett. 113, 172001(2014), [arXiv:1407.5873]; R. Aaij et al. (LHCb), Phys. Rev. Lett. 119, 101801 (2017),[arXiv:1705.03475].
[68] R. Aaij et al. (LHCb), Phys. Rev. Lett. 112, 111802 (2014), [arXiv:1312.1217].
[69] R. Aaij et al. (LHCb), Phys. Lett. B736, 446 (2014), [arXiv:1406.7204].
[70] R. Aaij et al. (LHCb), Phys. Lett. B707, 349 (2012), [arXiv:1111.0521].
[71] R. Aaij et al. (LHCb), Phys. Lett. B762, 484 (2016), [arXiv:1607.06314].
[72] R. Aaij et al. (LHCb), Eur. Phys. J. C83, 7, 629 (2023), [arXiv:2206.03088].
[73] T. Aaltonen et al. (CDF), Phys. Rev. D84, 052012 (2011), [arXiv:1106.3682]; V. M. Abazov
et al. (DØ), Phys. Rev. D94, 012001 (2016), [arXiv:1603.01302].
[74] R. Aaij et al. (LHCb), Phys. Rev. D87, 112010 (2013), [arXiv:1304.2600].
[75] R. Aaij et al. (LHCb), Phys. Lett. B797, 134789 (2019), [arXiv:1903.05530].
31st May, 2024

14

075. B 0 –B Mixing[76] R. Aaij et al. (LHCb), Nucl. Phys. B873, 275 (2013), [arXiv:1304.4500].
[77] R. Aaij et al. (LHCb), Phys. Rev. Lett. 128, 041801 (2022), [arXiv:2108.09284]; R. Aaij et al.
(LHCb), Phys. Rev. D105, 012010 (2022), [arXiv:2108.09283]; A. Tumasyan et al. (CMS),
Phys. Lett. B842, 137955 (2023), [arXiv:2212.10311]; G. Aad et al. (ATLAS), JHEP 09, 199(2023), [arXiv:2308.01171].
[78] R. Fleischer and R. Knegjens, Eur. Phys. J. C71, 1789 (2011), [arXiv:1109.5115].
[79] R. Aaij et al. (LHCb), submitted to JHEP, [arXiv:2310.12649].
[80] T. Aaltonen et al. (CDF), Phys. Rev. Lett. 109, 171802 (2012), [arXiv:1208.2967].
[81] V. M. Abazov et al. (DØ), Phys. Rev. D85, 032006 (2012), [arXiv:1109.3166].
[82] G. Aad et al. (ATLAS), Phys. Rev. D90, 052007 (2014), [arXiv:1407.1796]; G. Aad et al.
(ATLAS), JHEP 08, 147 (2016), [arXiv:1601.03297].
[83] V. Khachatryan et al. (CMS), Phys. Lett. B757, 97 (2016), [arXiv:1507.07527].
[84] R. Aaij et al. (LHCb), Eur. Phys. J. C81, 1026 (2021), [arXiv:2105.14738].
[85] R. Aaij et al. (LHCb), JHEP 08, 037 (2017), [arXiv:1704.08217].
[86] R. Aaij et al. (LHCb), Phys. Lett. B762, 253 (2016), [arXiv:1608.04855].
[87] R. Aaij et al. (LHCb), Phys. Rev. Lett. 108, 241801 (2012), [arXiv:1202.4717].
[88] Including the result of Ref. [79], which has not been accepted by a journal at the time ofwriting this review, changes ∆Γs to +0.084 ± 0.004 ps−1 and 1/ΓH to 1.623 ± 0.008 ps.
[89] M. Gerlach, U. Nierste, P. Reeck, V. Shtabovenko and M. Steinhauser (2024),[arXiv:2403.08316].
[90] A. Lenz and U. Nierste, JHEP 06, 072 (2007), [hep-ph/0612167].
[91] D. E. Jaffe et al. (CLEO), Phys. Rev. Lett. 86, 5000 (2001), [hep-ex/0101006].
[92] J. P. Lees et al. (BaBar), Phys. Rev. Lett. 114, 081801 (2015), [arXiv:1411.1842].
[93] F. Abe et al. (CDF), Phys. Rev. D55, 2546 (1997).
[94] V. M. Abazov et al. (DØ), Phys. Rev. D89, 012002 (2014), [arXiv:1310.0447].
[95] V. M. Abazov et al. (DØ), Phys. Rev. D86, 072009 (2012), [arXiv:1208.5813].
[96] V. M. Abazov et al. (DØ), Phys. Rev. Lett. 110, 011801 (2013), [arXiv:1207.1769].
[97] R. Aaij et al. (LHCb), Phys. Rev. Lett. 117, 061803 (2016), erratum ibid. 118, 129903 (2017),[arXiv:1605.09768].
[98] R. Barate et al. (ALEPH), Eur. Phys. J. C20, 431 (2001).
[99] J. P. Lees et al. (BaBar), Phys. Rev. Lett. 111, 101802 (2013), erratum ibid. 111, 159901(2013), [arXiv:1305.1575].
[100] E. Nakano et al. (Belle), Phys. Rev. D73, 112002 (2006), [hep-ex/0505017].
[101] G. Abbiendi et al. (OPAL), Eur. Phys. J. C12, 609 (2000), [hep-ex/9901017].
[102] M. Beneke, G. Buchalla and I. Dunietz, Phys. Lett. B393, 132 (1997), [hep-ph/9609357];
I. Dunietz, Eur. Phys. J. C7, 197 (1999), [hep-ph/9806521].
[103] R. Aaij et al. (LHCb), Phys. Rev. Lett. 114, 041601 (2015), [arXiv:1409.8586].
[104] R. Aaij et al. (LHCb), Phys. Rev. Lett. 113, 211801 (2014), [arXiv:1409.4619].
[105] R. Aaij et al. (LHCb), Phys. Lett. B736, 186 (2014), [arXiv:1405.4140].
[106] R. Aaij et al. (LHCb), Phys. Rev. D86, 052006 (2012), [arXiv:1204.5643].
[107] R. Aaij et al. (LHCb), Phys. Rev. D90, 052011 (2014), [arXiv:1407.2222].

31st May, 2024