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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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