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Nikhef-2011-026

Effective Lifetimes of BsDecays and their

Constraints on the B0

s–¯ B0

sMixing Parameters

Robert Fleischer and Robert Knegjens

Nikhef, Science Park 105, NL-1098 XG Amsterdam, The Netherlands

Abstract

Measurements of the effective lifetimes of Bs-meson decays, which only require

untagged rate analyses, allow us to probe the width difference ∆Γsand the CP-

violating phase φs of B0

smixing. We point out that the dependence of the

effective lifetime on non-linear terms in ∆Γsallows for a determination of φsand

∆Γsgiven a pair of Bsdecays into CP-even and CP-odd final states. Using recent

lifetime measurements of B0

this method and show how it complements the constraints in the φs–∆Γsplane

from other observables.

s–¯B0

s→ K+K−and B0

s→ J/ψf0(980) decays, we illustrate

September 2011

arXiv:1109.5115v1 [hep-ph] 23 Sep 2011

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1 Introduction

A promising avenue for New Physics (NP) to enter the observables of Bs-meson decays

is given by B0

originates from box topologies and is strongly suppressed. In the presence of NP, new

particles could give rise to additional box topologies or even contribute at the tree level.

Should these NP contributions also involve new CP-violating phases, the B0

phase φscould differ sizably from the tiny SM value of −2.1◦(see, for instance, Refs. [1,2]

and references therein).

A key channel for addressing this exciting possibility is B0

feature of this channel is that its final state contains two vector mesons and thereby

requires a time-dependent angular analysis of the J/ψ → µ+µ−and φ → K+K−decay

products [2,3]. Over the last couple of years, measurements at the Tevatron of CP-

violating asymmetries in “tagged” analyses (distinguishing between initially present B0

or¯B0

[4–6]. These results are complemented by the measurement of the anomalous like-sign

dimuon charge asymmetry at DØ, which was found to differ by 3.9σ from the SM

prediction [7]. This summer, the LHCb collaboration has also joined the arena, reporting,

however, results that disfavour large NP effects [8]. The above measurements, which we

will discuss in more detail below, are typically shown in the φs–∆Γsplane, where ∆Γs

is the width difference between the mass eigenstates of the Bs-meson system.

In this paper, we point out a new method for determining further constraints in the

φs–∆Γsplane using measurements of the effective lifetimes of Bsdecays. In particular, we

show that the information provided by the lifetimes of a pair of decays into CP-even and

CP-odd final states is sufficient to determine φsand ∆Γs. The advantage of this strategy

is that it only requires an “untagged” analysis, i.e. it is not necessary to distinguish

between initially present B0

Specifically, we will consider the B0

decays, which have final states with the CP eigenvalues +1 and −1, respectively. From

here on we shall abbreviate the latter decay as B0

the effective lifetimes of these channels are already available from the CDF and LHCb

collaborations [13–15]. For the theoretical interpretation of these results we also need

to address hadronic uncertainties. A closer look will reveal that these decays are well

suited in this respect. We will illustrate our method with the most recent data and shall

compare the resulting constraints in the φs–∆Γsplane with those from the alternative

measurements listed above.

The outline is as follows: in Section 2, we discuss the general formalism to calculate

effective lifetimes and show in Section 3 how the corresponding measurements can be

converted into contours in the φs–∆Γs plane. In Section 4, we turn to the hadronic

uncertainties affecting this analysis and their control through experimental data. The

constraints on the B0

tive lifetimes of the B0

where we also illustrate the impact of future lifetime measurements with errors at the

1% level. In Section 6, we give a collection of additional Bsdecays that can be added to

this analysis in the future. Finally, we summarize our conclusions in Section 7.

s–¯B0

smixing. In the Standard Model (SM), the phenomenon of mixing

s–¯B0

smixing

s→ J/ψφ. A characteristic

s

smesons) of the B0

s→ J/ψφ channel indicate possible NP effects in B0

s–¯B0

smixing

sor¯B0

smesons, which is experimentally advantageous.

s→ K+K−[9,10] and B0

s→ J/ψf0(980) [11,12]

s→ J/ψf0. First measurements of

s–¯B0

s→ K+K−and B0

smixing parameters arising from the current data for the effec-

s→ J/ψf0channels are explored in Section 5,

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2General Formalism

We will consider a Bs→ f transition with a final state f into which both a B0

meson can decay. The corresponding untagged rate can then be written as follows [2]:

sand a¯B0

s

?Γ(Bs(t) → f)? ≡ Γ(B0

s(t) → f) + Γ(¯B0

He−Γ(s)

s(t) → f)

= Rf

Ht+ Rf

Le−Γ(s)

Lt, (1)

where L and H denote the light and heavy Bsmass eigenstates, respectively. Using

Γs≡Γ(s)

L+ Γ(s)

2

H

= τ−1

Bs,∆Γs≡ Γ(s)

L− Γ(s)

H, (2)

we can straightforwardly write (1) as

?Γ(Bs(t) → f)? ∝ e−Γst

?

cosh

?∆Γst

2

?

+ Af

∆Γsinh

?∆Γst

2

??

(3)

with

Af

∆Γ≡Rf

H− Rf

Rf

L

H+ Rf

L

.(4)

We define the effective lifetime of the decay B0

the untagged rate [10],

?∞

s→ f as the time expectation value of

τf≡

0t ?Γ(Bs(t) → f)? dt

?∞

0?Γ(Bs(t) → f)? dt

=Rf

L/Γ(s)2

Rf

L

+ Rf

H/Γ(s)2

H/Γ(s)

H

L/Γ(s)

L+ Rf

H

,(5)

which is equivalent to the lifetime that results from fitting the two exponentials in (1)

to a single exponential [16]. By making the usual definition

ys≡∆Γs

2Γs, (6)

we can express the effective lifetime as

τf

τBs

=

1

1 − y2

s

?

1 + 2Af

1 + Af

?

∆Γys+ y2

∆Γys

s

?

= 1 + Af

∆Γys+2 − (Af

∆Γ)2?

y2

s+ O(y3

s), (7)

where we have also given the expansion in powers of ysup to cubic corrections.

We proceed to consider the case where f is a CP eigenstate with eigenvalue ηf. In the

SM, the decay amplitude can be written, without loss of generality (using the unitarity

of the Cabibbo–Kobayashi–Maskawa (CKM) matrix), as

A(B0

s→ f) = Af

1eiδf

1eiϕf

1+ Af

2eiδf

2eiϕf

2, (8)

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where the Af

weak phases, respectively. Using the standard B0

1,2are real and the δf

1,2and ϕf

1,2are CP-conserving strong and CP-violating

s–¯B0

smixing formalism [17], we have

Af

∆Γ=

2Reξ(s)

1 +??ξ(s)

e−iϕf

eiϕf

f

f

??2,(9)

where

ξ(s)

f

= −ηfe−iφs

?

1+ hfeiδfe−iϕf

1 + hfeiδfeiϕf

2

2

?

. (10)

Here we have introduced the abbreviation

hfeiδf≡Af

2

Af

1

ei(δf

2−δf

1), (11)

and φsdenotes the B0

s–¯B0

smixing phase, which is given by

φs≡ φSM

s

+ φNP

s, (12)

where φSM

following discussion to introduce the direct CP asymmetry of the Bs→ f decay [17]:

Cf≡1 − |ξf|2

s

and φNP

s

are the SM and NP pieces, respectively. It is convenient for the

1 + |ξf|2=2hfsinδfsin(ϕf

1− ϕf

2)

Nf

, (13)

where

Nf≡ 1 + 2hfcosδfcos(ϕf

1− ϕf

2) + h2

f.(14)

Subsequently, we may write

2ξ(s)

f

1 +??ξ(s)

f

??2= −ηf

?

1 − C2

fe−i(φs+∆φf).(15)

Here ∆φfis a hadronic phase shift, which is given by

sin∆φf=sin2ϕf

1+ 2hfcosδfsin(ϕf

1+ ϕf

2) + h2

fsin2ϕf

2

Nf

?

1 − C2

f

(16)

cos∆φf=cos2ϕf

1+ 2hfcosδfcos(ϕf

1+ ϕf

2) + h2

fcos2ϕf

2

Nf

?

1 − C2

f

,(17)

yielding

tan∆φf=

sin2ϕf

cos2ϕf

1+ 2hfcosδfsin(ϕf

1+ 2hfcosδfcos(ϕf

1+ ϕf

1+ ϕf

2) + h2

fsin2ϕf

fcos2ϕf

2

2) + h2

2

.(18)

The twofold ambiguity for ∆φfarising from the latter expression can be resolved using

sign information from sin∆φf or cos∆φf. These expressions generalize those given in

Refs. [12,18].

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Using (9) and (15), we thus obtain

Af

∆Γ= −ηf

?

1 − C2

fcos(φs+ ∆φf). (19)

As we will see in Section 4, there are fortunate Bsdecays into CP eigenstates where the

hadronic parameter hfeiδfand the resulting phase shift ∆φf can be controlled through

experimental data. For these decays, we can hence use the corresponding lifetime mea-

surements to constrain ys(or ∆Γs) with respect to φs.

3Lifetime Contours in the φs–∆ΓsPlane

Let us now have a closer look at (7), which we can write as the following cubic equation

for the real parameter ys:

y3

s+ a2y2

s+ a1ys+ a0= 0,(20)

where

a0≡

τBs− τf

τfAf

∆Γ

,a1≡

2τBs− τf

τf

,a2≡

τBs+ τf

τfAf

∆Γ

. (21)

In order to solve this cubic equation, it is useful to rewrite it in the “reduced” form

?

with

P ≡ a1−a2

Applying Cardano’s formula then yields the solutions

?

with ω ∈ {0,2π/3,−2π/3}, where

R ≡ −Q

54

?P

For Af

thermore, the above expressions may prove cumbersome to use in practice. A convenient

approximate solution is obtained by solving the expansion in (7) up to quadratic order

in ys:

?

2 − (Af

ys+a2

3

?3

+ P

?

ys+a2

3

?

+ Q = 0(22)

2

3,Q ≡2a3

2

27−a2a1

3

+ a0. (23)

ys= −a2

3+ eiω

3

R +

√D + e−iω

3

?

R −

√D (24)

2=

1

?9a1a2− 27a0− 2a3

0− 18a0a1a2+ 4a0a3

2

?

(25)

D ≡

3

?3

+

?Q

2

?2

=

1

108

?27a2

2+ 4a3

1− a2

1a2

2

?. (26)

∆Γ= 0, this solution is not valid as (7) is then a quadratic equation in ys. Fur-

ys≈ −1

2

Af

∆Γ

∆Γ)2

?

±1

2

?

?

?

?

?

Af

∆Γ

2 − (Af

∆Γ)2

?2

+

4

τBs

?

τf− τBs

2 − (Af

∆Γ)2

?

. (27)

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−180

−135

−90

−450 4590 135 180

φs[deg]

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

∆Γs[ps−1]

τf+= 1.38 ps

τf−= 1.58 ps

∆ΓSM

s/Γs= 0.133 ± 0.032

Figure 1: Illustration of the lifetimes that are compatible with the SM value of ∆Γs/Γs

given in (29) for CP-even and CP-odd final states f+and f−, respectively. The decay

amplitudes are assumed to have no CP-violating phases. We also show the constraint

from the theoretical value of ∆ΓSM

s /Γsgiven in (29), as discussed in the text.

This quadratic solution is in excellent agreement with the corresponding branches of the

exact solution (24) for the numerical analyses discussed below.

For illustration we consider two Bsdecays to CP eigenstates, Bs→ f+and Bs→ f−,

with positive and negative CP eigenvalues, respectively. Further, we assume

hf±= 0,ϕf±

1 = 0(28)

for these decays, yielding Cf±= 0 and ∆φf±= 0. In Fig. 1, we show the lifetime

constraints that are compatible with the theoretical SM calculation of ∆Γs[19],

∆ΓSM

Γs

s

= 2ySM

s

= 0.133 ± 0.032, (29)

and the SM value of the B0

s–¯B0

smixing phase, which is given as follows [20]:

φSM

s

≡ −2βs= −(2.08 ± 0.09)◦. (30)

Throughout this paper, we shall use [21]

τBs=?1.477+0.021

−0.022

?ps(31)

for the Bslifetime introduced in (2), resulting in the SM effective lifetimes τf+= 1.38ps

and τf−= 1.58ps. The difference in behaviour for CP-odd and CP-even eigenstates is

due to the non-linear dependence on ysin (7). Said differently, if (7) is expanded and

only terms up to linear order in ysare kept the two curves in Fig. 1 would overlap.

In Fig. 1, we have included another constraint, which is related to the theoretical

value in (29) as follows: if we assume that NP can only affect ∆Γs through B0

mixing, which is a very plausible assumption, we have [22]

s–¯B0

s

ys=∆ΓSM

s

cos˜φs

2Γs

= ySM

s

cos˜φs, (32)

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where

˜φs≡˜φSM

s

+ φNP

s. (33)

Here φNP

Af

s

is the NP B0

s–¯B0

smixing phase, which also enters φsdefined in (12) on which

∆Γdepends, whereas the SM piece takes the following value [19]:

˜φSM

s

= (0.22 ± 0.06)◦. (34)

The formalism developed above is also valid for non-CP eigenstates provided the

final state is accessible to both B0

states are Bs → D±

replaced by (−1)L, where L denotes the relative orbital angular momentum of the decay

products [23].

sand¯B0

sso that mixing is possible. Examples of such

sK(∗)∓. For these decays the CP eigenvalue ηf in (19) should be

4 Hadronic Corrections and Their Control

Examples of effective lifetimes that have been measured for Bsdecays to CP-even and

CP-odd final states are B0

pothetical examples from the previous section, however, the decay amplitudes of these

decays are not devoid of weak phases, and can be written in the SM as follows [9,10,12]:

s→ K+K−and B0

s→ J/ψf0, respectively. Unlike our hy-

A(B0

s→ K+K−) = λC

?

?

eiγ+1

?deiθ

?

(35)

A(B0

s→ J/ψf0) =

?

1 −λ2

2

A?1 + ?beiϑeiγ?. (36)

Here λ ≡ |Vus| = 0.2252 ± 0.0009 is the Wolfenstein parameter of the CKM matrix [24],

λ2

1 − λ2= 0.0534 ± 0.0005

and γ is the usual angle of the unitarity triangle whereas C, deiθand A, beiθare hadronic,

CP-conserving parameters.

Consequently, the parameters introduced in (9) and (11) take the forms

? ≡

(37)

hK+K− = d/?,δK+K− = θ,ϕK+K−

1

= γ,ϕK+K−

2

= 0(38)

and

hJ/ψf0= ?b,δJ/ψf0= ϑ,ϕJ/ψf0

1

= 0,ϕJ/ψf0

2

= γ,(39)

so that the hadronic phase shifts can be obtained from

tan∆φK+K− = 2?

?

?

dcosθ + ?cosγ

d2+ 2?dcosθcosγ + ?2cos2γ

?

?

sinγ, (40)

tan∆φJ/ψf0= 2?b

cosϑ + ?bcosγ

1 + 2?bcosϑcosγ + ?2b2cos2γ

sinγ.(41)

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We observe that these phases are proportional to the tiny ? parameter, i.e. are doubly

Cabibbo-suppressed. Consequently, the Af

respect to hadronic uncertainties for the decays at hand.

In the case of the B0

interactions to relate it to the B0

γ as well as d and θ [9]. The current status of an analysis along these lines, using also

the direct CP violation in Bd→ π∓K±, is given by [10]:

γ = (68 ± 7)◦,

where the errors include the uncertainties of the relevant input quantities and estimates of

U-spin-breaking corrections. The γ result is in excellent agreement with the current fits

of the unitarity triangle [25,26], thereby excluding large CP-violating NP contributions

to the B0

?

where we have added the errors in quadrature. Similarly, we also find CK+K− = 0.09+0.05

Unfortunately, as discussed in Ref. [12], it is much more involved to control the

hadronic effects in the B0

control channel B0

B0

Ref. [12], we use the conservative range 0 ≤ b ≤ 0.5 and leave ϑ unconstrained. Using

moreover the value for γ in (42), we find

∆Γobservable given in (19) is robust with

s→ K+K−channel, we can use the U-spin symmetry of strong

d→ π+π−decay, which thereby allows us to determine

d = 0.50+0.12

−0.11,θ = (154+11

−14)◦, (42)

s→ K+K−decay amplitude. Using these numbers in (40), we find

∆φK+K− = −

10.5+0.3

−0.5

??

γ

+2.9

−2.1

??

d

+0.9

−1.7

??

θ

?◦

= −?10.5+3.1

−2.8

?◦, (43)

−0.04.

s→ J/ψf0decay through experimental data, and the potential

d→ J/ψf0has not yet been observed. On the other hand, contrary to

s→ K+K−, the denominator of (41) is equal to one at leading order in ?. Following

∆φJ/ψf0∈ [−2.9◦,2.8◦](44)

and

??CJ/ψf0

??? 0.05, which has, just like CK+K−, a negligible impact on (19).

be so robust with respect to the hadronic effects and the weak phase γ, suffering from

uncertainties of only ∼ 3◦. Future data should allow us to determine them with even

higher precision.

It is remarkable that the hadronic phase shifts ∆φK+K− and ∆φJ/ψf0turn out to

5Constraints from Current and Future Data

In the previous section we presented all the ingredients necessary to compute AK+K−

AJ/ψf0

allows us to draw contours for the lifetime measurements on the φs–∆Γs plane. The

first measurement of the effective B0

collaboration in 2006 [13]. In the spring of 2011, the LHCb collaboration reported their

first measurement of this observable [14]:

∆Γ

and

∆Γ

as functions of φs. Inserting them into the solution for ysdiscussed in Section 3

s→ K+K−lifetime was performed by the CDF

τK+K− = [1.44 ± 0.096(stat) ± 0.010(syst)]ps,

which is currently the most precise. The first measurement of the B0

has recently been made by the CDF collaboration [15]:

τJ/ψf0=?1.70+0.12

(45)

s→ J/ψf0lifetime

−0.11(stat) ± 0.03(syst)?ps.

7

(46)

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−180

−135

−90

−4504590135180

φs[deg]

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

∆Γs[ps−1]

τJ/ψf0=?1.70+0.12

τK+K− = [1.44 ± 0.096 ± 0.010] ps

39% CL of χ2fit

−0.11± 0.03?

ps

−180

−135

−90

−450 4590135180

φs[deg]

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

∆Γs[ps−1]

† Errors are for illustration

τJ/ψf0= 1.70 ps ± 1%†

τK+K− = 1.44 ps ± 1%†

∆ΓSM

s/Γs= 0.133 ± 0.032

Figure 2: The measurements of the effective B0

projected onto the φs–∆Γsplane. Left panel: analysis of the current data, where the

shaded bands give the 1σ uncertainties of the lifetimes; the 39% confidence regions

originating from a χ2fit are also shown. Right panel: illustration of how the situation

improves for unchanged central values if the uncertainties were improved to 1% accuracy,

including also the constraint from the theoretical value of ∆ΓSM

s→ K+K−and B0

s→ J/ψf0lifetimes

s /Γs.

−180

−135

−90

−4504590135180

φs[deg]

0.0

0.1

0.2

0.3

0.4

∆Γs[ps−1]

τK+K− = 1.44 ps

τJ/ψf0= 1.70 ps

∆ΦJ/ψf0= [−2.9◦,2.8◦]

∆ΦK+K− = −?10.5+3.1

−2.8

ps

?◦

τBs=?1.477+0.022

−0.021

?

Figure 3: Illustration of the errors of the hadronic phase shifts ∆φK+K− and ∆φJ/ψf0on

the contours in the φs–∆Γsplane for the central values of the lifetime measurements.

We also shown the impact of the present error of the Bslifetime.

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-3-210123

φs[rad]

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

∆Γs[ps−1]

τK+K−

τJ/ψf0

D0: 68% & 95% CL (prelim.)

CDF: 68% & 95% CL (prelim.)

LHCb: 68% & 95% CL (prelim.)

Lifetimes: 39% CL

Figure 4: The fitted lifetime regions in the φs–∆Γsplane from the left panel of Fig. 2

added to a compilation of measurements as obtained in Ref. [29]. The DØ, CDF and

LHCb allowed regions refer to tagged analyses of B0

region includes also the result for the like-sign dimuon asymmetry while LHCb has also

included a first analysis of CP violation in B0

s→ J/ψφ. In addition, the DØ

s→ J/ψf0.

In the left panel of Fig. 2, we show the current measurements of the effective lifetimes

of the B0

also show the 39% confidence region resulting from a χ2fit of these two results. The

individual fitted values for the φsand ∆Γsparameters are given as follows:

φs= −?52+19

φs=?71+14

where the errors are 68% confidence levels corresponding to a χ2fit of the lifetimes. Each

solution has a two-fold ambiguity given by the transformation

s→ K+K−and B0

s→ J/ψf0decays as constraints on the φs–∆Γsplane. We

−43

?◦,

?◦,

∆Γs=?0.23+0.08

∆Γs=?0.28+0.08

−0.12

?ps−1

?ps−1,

(47)

−27

−0.14

(48)

φs→ φs+ 180◦,∆Γs→ −∆Γs. (49)

Both lifetime measurements currently have an error of about 7%. However, it seems

feasible to reduce the uncertainty of the τK+K− measurement at LHCb to the few-percent

level [27]. In the right panel of Fig. 2, we show – for illustration – the impact of measure-

ments of the B0

change in the central values. Clearly, at this level of accuracy, the lifetime measurements

could strongly constrain φsand ∆Γs.

Using (32), we also include the band corresponding to the theoretical value of ∆ΓSM

given in (29). We observe, as also noted in Ref. [12], that the central value of the τJ/ψf0

measurement is too large in comparison with this constraint. To spoil the relation in

(32) either large NP effects are required, a very contrived scenario in our opinion, or the

width difference ∆Γsmust be affected by hadronic long-distance effects, which are not

included in the SM calculation of (29). The B0

the SM calculation is τJ/ψf0= (1.582 ± 0.036)ps [12].

s→ K+K−and B0

s→ J/ψf0lifetimes with 1% uncertainty, assuming no

s /Γs

s→ J/ψf0effective lifetime predicted by

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The uncertainties of the hadronic phase shifts given in (43) and (44) as well as

the error of the Bslifetime in (31) were not included in Fig. 2 or in the fit results in

(47) and (48). In Fig. 3, we illustrate the impact of these uncertainties on the lifetime

contours in the φs–Γsplane. Comparing with the error bands in Fig. 2, we observe that

the effects of these uncertainties are marginal with respect to the current errors of the

effective lifetime measurement. More sophisticated fits should take these uncertainties

into account as well.

It is interesting to compare our fitted results to recent measurements of CP violation

in the B0

φs∈ [−177.6◦,−123.8◦] ∨ [−59.6◦,−2.3◦] (68% C.L.), while DØ has recently reported

φs= −?31.5+20.6

thermore, LHCb has presented a first tagged analysis of the CP-violating asymmetry of

the B0

A compilation of the preliminary results from the DØ, CDF and LHCb collaborations

as constraints in the φs–∆Γsplane has recently been performed in Ref. [29]. In Fig. 4,

we have overlaid on the corresponding plot the lifetime contours and fit results of the

analysis described above. It is intriguing to see how well the lifetime allowed region

overlaps with those from DØ and CDF.

The current errors leave space for interesting future developments. Should the central

values of the CP-even and CP-odd lifetimes approach the theoretical SM point, the power

to pinpoint φsis lost, as illustrated by Fig. 1. However, because the curves are flat at

this point, ∆Γscould still be determined accurately in this case.

s→ J/ψφ channel, where the current picture of φslooks as follows: CDF finds

?◦[6]. The LHCb collaboration has also entered the arena, reporting

−21.8

φs= +(7.4 ± 10.3 ± 4.0)◦and ∆Γs= [0.123 ± 0.029(stat) ± 0.008(syst)]ps−1[8]. Fur-

s→ J/ψf0channel, yielding φs= −(25 ± 25 ± 1)◦[28].

6 Further Promising BsDecays

So far we have discussed the effective lifetimes for the B0

channels, which have both been measured in first analyses by the CDF and LHCb col-

laborations. These channels have final states with opposite CP eigenvalues and happen

to be well paired for obtaining constraints in the φs–∆Γsplane using the strategy pro-

posed in this paper. The hadronic corrections in B0

better in the future through precise measurements of the CP-violating observables of the

B0

on the lifetime analysis. A potential control channel is B0

situation is much more involved than in B0

structure of the scalar f0(980) state [12].

Another interesting decay that can soon be added to this picture is B0

[30,31], which has been observed by the CDF and LHCb collaborations [32,33]. This

channel has a final state with CP eigenvalue −1 and is caused by¯b → ¯ cc¯d quark-level

processes, i.e. it has a CKM structure that is different from the decays considered above.

In particular, the relevant hadronic parameter does not enter in a doubly Cabibbo-

suppressed way. However, the uncertainties can be controlled through B0

are found to have a moderate impact on the effective B0

has not yet been measured.

s→ K+K−and B0

s→ J/ψf0

s→ K+K−can be controlled even

d→ π+π−channel. Regarding B0

s→ J/ψf0, hadronic corrections have a minor impact

d→ J/ψf0, although here the

s→ K+K−due to the unsettled hadronic

s→ J/ψKS

d→ J/ψπ0and

s→ J/ψKSlifetime [34], which

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Another Bsdecay with a CP-even final state is B0

corrections are again doubly Cabibbo-suppressed and can be controlled with the help of

the U-spin-related B0

lifetime of B0

states where a similar analysis can be performed are the B0

Decays of Bsmesons into CP-selfconjugate final states with two vector mesons or

higher resonances offer another laboratory for lifetime analyses. In this case the B0

and B0

already been observed experimentally [6,8,37], are penguin modes. Their final states

are mixtures of CP-even and CP-odd eigenstates and can be disentangled by means of

angular analyses. It would be interesting to perform measurements of the lifetimes for

the CP-even and CP-odd final-state configurations and to add them as contours to the

φs–Γsplane along the lines of the strategy proposed above.

A similar comment applies to the Bs→ J/ψφ channel, where it would also be de-

sirable to determine the individual lifetimes for the CP-even and CP-odd final-state

configurations separately instead of making a fit to the whole time-dependent angular

distribution. This can be done by means of the moment analysis proposed in Ref. [3].

The hadronic uncertainties of the B0

such as B0

In the future, also decays with final states that are not CP-selfconjugate can be added

to the agenda to further constrain φsand ∆Γs, provided both a B0

decay into the same final state (see Section 2). Prime examples are the Bs→ D±

channels. Their effective lifetimes can be used to constrain φs+ γ with respect to ∆Γs

(see Ref. [23] for an overview of the observables of these decays).

s→ D+

sD−

s. Here the hadronic

d→ D+D−decay [30]. A first theoretical analysis of the effective

sD−

s→ D+

swas performed in Ref. [35]. Further decays into CP-even final

s(d)→ J/ψη(?)channels [36].

s→ φφ

s→ K∗0¯K∗0channels look particularly interesting. These decays, which have

s→ J/ψφ channel can be controlled by channels

d→ J/ψρ0[18].

s→ J/ψ¯K∗0and B0

sand a¯B0

smeson can

sK(∗)∓

7 Conclusions

Thanks to the sizable width difference ∆Γsof the Bs-meson system, effective lifetimes of

Bsdecays offer interesting probes of B0

require only untagged data samples and are advantageous from an experimental point

of view. Thanks to non-linear terms in ∆Γs, a pair of Bs decays into CP-even and

CP-odd final states is sufficient to determine the B0

difference. Prime examples for implementing this strategy in practice are the decays

B0

respect to hadronic uncertainties, which can be controlled or constrained with the help

of further experimental data.

We have calculated the constraints in the φs–∆Γsplane following from the current

measurements of the B0

∼ 7% uncertainties. The resulting picture is consistent with other constraints following

from Tevatron measurements, which have not been supported by recent LHCb data.

The uncertainties still preclude us from drawing definite conclusions but leave space for

interesting future developments. Lifetime measurements with 1% precision would allow

us to obtain much stronger constraints in the φs–∆Γsplane as we have illustrated in our

study. Should the central values of the lifetimes for the CP-even and CP-odd final states

s–¯B0

smixing. The corresponding measurements

s–¯B0

smixing phase and the width

s→ K+K−and B0

s→ J/ψf0. Their effective lifetimes turn out to be robust with

s→ K+K−and B0

s→ J/ψf0lifetimes, which both suffer from

11

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approach the SM values, the lifetime contours would loose their power to determine φs.

However, the width difference could still be determined in a precise way in this case.

An interesting trend of the current data is that it favours a value of ∆Γs that is

larger than the one calculated in QCD. With the plausible assumption that NP affects

this observable only through B0

hence the discrepancy will be even larger for sizable mixing phases. This feature raises

the question of whether the SM calculation of ∆Γs fully includes all hadronic long-

distance contributions. It will be interesting to see if this trend will be supported by

future data or if it will eventually disappear.

In the future, also other effective lifetime measurements of Bs-meson decays can be

added to the φs–∆Γsplane, allowing us to “overconstrain” the mixing parameters in the

same spirit as the determination of the apex of the unitarity triangle. This information

will be complementary to the tagged analyses of CP violation in the Bssystem. It will

be intriguing to see at which point of the φs–∆Γsplane all measurements will eventually

converge.

s–¯B0

smixing, its absolute value can only decrease and

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